LMFDB - Embedded newform 273.12.a.c.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,12,Mod(1,273)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(273, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 12, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("273.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 273 = 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	273.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$209.757688293$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} - 26640 x^{14} - 38138 x^{13} + 279954048 x^{12} + 940095168 x^{11} + \cdots + 45\!\cdots\!68 $ x^16 - x^15 - 26640x^14 - 38138x^13 + 279954048x^12 + 940095168x^11 - 1473489774048x^10 - 6951601887328x^9 + 4072481535854976x^8 + 21355761647276288x^7 - 5703452025696599040x^6 - 25041700255043457024x^5 + 3737994671267475603456x^4 + 6711995200063092490240x^3 - 963518272786595267739648x^2 + 1287697466669064032092160x + 45555354651594891689197568
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{12}\cdot 5\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-23.9940$ of defining polynomial
Character	$\chi$	$=$	273.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.9940 q^{2} -243.000 q^{3} -1264.33 q^{4} -7765.11 q^{5} +6802.55 q^{6} +16807.0 q^{7} +92725.6 q^{8} +59049.0 q^{9} +O(q^{10})$	\(q-27.9940 q^{2} -243.000 q^{3} -1264.33 q^{4} -7765.11 q^{5} +6802.55 q^{6} +16807.0 q^{7} +92725.6 q^{8} +59049.0 q^{9} +217377. q^{10} +67931.3 q^{11} +307233. q^{12} -371293. q^{13} -470496. q^{14} +1.88692e6 q^{15} -6408.51 q^{16} -3.74683e6 q^{17} -1.65302e6 q^{18} -8.63030e6 q^{19} +9.81769e6 q^{20} -4.08410e6 q^{21} -1.90167e6 q^{22} -4.83626e6 q^{23} -2.25323e7 q^{24} +1.14688e7 q^{25} +1.03940e7 q^{26} -1.43489e7 q^{27} -2.12497e7 q^{28} -9.95736e7 q^{29} -5.28225e7 q^{30} -2.31224e7 q^{31} -1.89723e8 q^{32} -1.65073e7 q^{33} +1.04889e8 q^{34} -1.30508e8 q^{35} -7.46576e7 q^{36} +1.11079e8 q^{37} +2.41597e8 q^{38} +9.02242e7 q^{39} -7.20024e8 q^{40} -5.91074e8 q^{41} +1.14330e8 q^{42} -1.14229e8 q^{43} -8.58878e7 q^{44} -4.58522e8 q^{45} +1.35386e8 q^{46} +3.04238e9 q^{47} +1.55727e6 q^{48} +2.82475e8 q^{49} -3.21057e8 q^{50} +9.10480e8 q^{51} +4.69438e8 q^{52} +2.10800e9 q^{53} +4.01684e8 q^{54} -5.27494e8 q^{55} +1.55844e9 q^{56} +2.09716e9 q^{57} +2.78747e9 q^{58} +1.98209e8 q^{59} -2.38570e9 q^{60} -2.68281e9 q^{61} +6.47289e8 q^{62} +9.92437e8 q^{63} +5.32423e9 q^{64} +2.88313e9 q^{65} +4.62106e8 q^{66} +9.22410e9 q^{67} +4.73725e9 q^{68} +1.17521e9 q^{69} +3.65345e9 q^{70} +1.63258e10 q^{71} +5.47535e9 q^{72} +2.98009e10 q^{73} -3.10956e9 q^{74} -2.78691e9 q^{75} +1.09116e10 q^{76} +1.14172e9 q^{77} -2.52574e9 q^{78} -1.17864e10 q^{79} +4.97628e7 q^{80} +3.48678e9 q^{81} +1.65466e10 q^{82} +1.06107e10 q^{83} +5.16367e9 q^{84} +2.90945e10 q^{85} +3.19774e9 q^{86} +2.41964e10 q^{87} +6.29897e9 q^{88} -8.40634e10 q^{89} +1.28359e10 q^{90} -6.24032e9 q^{91} +6.11464e9 q^{92} +5.61874e9 q^{93} -8.51686e10 q^{94} +6.70152e10 q^{95} +4.61026e10 q^{96} -2.50839e10 q^{97} -7.90762e9 q^{98} +4.01127e9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 63 q^{2} - 3888 q^{3} + 20761 q^{4} - 3168 q^{5} + 15309 q^{6} + 268912 q^{7} - 187965 q^{8} + 944784 q^{9}+O(q^{10}) $ 16 * q - 63 * q^2 - 3888 * q^3 + 20761 * q^4 - 3168 * q^5 + 15309 * q^6 + 268912 * q^7 - 187965 * q^8 + 944784 * q^9	\( 16 q - 63 q^{2} - 3888 q^{3} + 20761 q^{4} - 3168 q^{5} + 15309 q^{6} + 268912 q^{7} - 187965 q^{8} + 944784 q^{9} + 1056711 q^{10} - 466884 q^{11} - 5044923 q^{12} - 5940688 q^{13} - 1058841 q^{14} + 769824 q^{15} + 40058265 q^{16} + 1753452 q^{17} - 3720087 q^{18} + 4237800 q^{19} - 20710503 q^{20} - 65345616 q^{21} - 37479661 q^{22} - 150481440 q^{23} + 45675495 q^{24} + 150983176 q^{25} + 23391459 q^{26} - 229582512 q^{27} + 348930127 q^{28} - 111411432 q^{29} - 256780773 q^{30} - 90536236 q^{31} - 609941313 q^{32} + 113452812 q^{33} + 443745577 q^{34} - 53244576 q^{35} + 1225916289 q^{36} - 1756337900 q^{37} - 4378868973 q^{38} + 1443587184 q^{39} + 57535383 q^{40} + 649575720 q^{41} + 257298363 q^{42} - 1889554520 q^{43} - 4979587689 q^{44} - 187067232 q^{45} - 3204000867 q^{46} - 4036082940 q^{47} - 9734158395 q^{48} + 4519603984 q^{49} - 15928886862 q^{50} - 426088836 q^{51} - 7708413973 q^{52} + 511144020 q^{53} + 903981141 q^{54} - 8519365572 q^{55} - 3159127755 q^{56} - 1029785400 q^{57} - 7851149577 q^{58} + 3728287296 q^{59} + 5032652229 q^{60} + 26227205052 q^{61} + 2996618490 q^{62} + 15878984688 q^{63} + 51104408465 q^{64} + 1176256224 q^{65} + 9107557623 q^{66} - 5295680024 q^{67} + 7919540325 q^{68} + 36566989920 q^{69} + 17760141777 q^{70} + 17082800928 q^{71} - 11099145285 q^{72} + 21076301488 q^{73} + 52794333675 q^{74} - 36688911768 q^{75} + 81459179177 q^{76} - 7846919388 q^{77} - 5684124537 q^{78} - 83677977852 q^{79} - 163988465427 q^{80} + 55788550416 q^{81} - 61543728760 q^{82} - 72857072340 q^{83} - 84790020861 q^{84} - 12608565060 q^{85} - 20177818011 q^{86} + 27072977976 q^{87} - 202213679643 q^{88} + 32238476676 q^{89} + 62397727839 q^{90} - 99845143216 q^{91} - 487057197033 q^{92} + 22000305348 q^{93} + 183904776602 q^{94} - 143022915504 q^{95} + 148215739059 q^{96} + 6973535140 q^{97} - 17795940687 q^{98} - 27569033316 q^{99}+O(q^{100}) \) 16 * q - 63 * q^2 - 3888 * q^3 + 20761 * q^4 - 3168 * q^5 + 15309 * q^6 + 268912 * q^7 - 187965 * q^8 + 944784 * q^9 + 1056711 * q^10 - 466884 * q^11 - 5044923 * q^12 - 5940688 * q^13 - 1058841 * q^14 + 769824 * q^15 + 40058265 * q^16 + 1753452 * q^17 - 3720087 * q^18 + 4237800 * q^19 - 20710503 * q^20 - 65345616 * q^21 - 37479661 * q^22 - 150481440 * q^23 + 45675495 * q^24 + 150983176 * q^25 + 23391459 * q^26 - 229582512 * q^27 + 348930127 * q^28 - 111411432 * q^29 - 256780773 * q^30 - 90536236 * q^31 - 609941313 * q^32 + 113452812 * q^33 + 443745577 * q^34 - 53244576 * q^35 + 1225916289 * q^36 - 1756337900 * q^37 - 4378868973 * q^38 + 1443587184 * q^39 + 57535383 * q^40 + 649575720 * q^41 + 257298363 * q^42 - 1889554520 * q^43 - 4979587689 * q^44 - 187067232 * q^45 - 3204000867 * q^46 - 4036082940 * q^47 - 9734158395 * q^48 + 4519603984 * q^49 - 15928886862 * q^50 - 426088836 * q^51 - 7708413973 * q^52 + 511144020 * q^53 + 903981141 * q^54 - 8519365572 * q^55 - 3159127755 * q^56 - 1029785400 * q^57 - 7851149577 * q^58 + 3728287296 * q^59 + 5032652229 * q^60 + 26227205052 * q^61 + 2996618490 * q^62 + 15878984688 * q^63 + 51104408465 * q^64 + 1176256224 * q^65 + 9107557623 * q^66 - 5295680024 * q^67 + 7919540325 * q^68 + 36566989920 * q^69 + 17760141777 * q^70 + 17082800928 * q^71 - 11099145285 * q^72 + 21076301488 * q^73 + 52794333675 * q^74 - 36688911768 * q^75 + 81459179177 * q^76 - 7846919388 * q^77 - 5684124537 * q^78 - 83677977852 * q^79 - 163988465427 * q^80 + 55788550416 * q^81 - 61543728760 * q^82 - 72857072340 * q^83 - 84790020861 * q^84 - 12608565060 * q^85 - 20177818011 * q^86 + 27072977976 * q^87 - 202213679643 * q^88 + 32238476676 * q^89 + 62397727839 * q^90 - 99845143216 * q^91 - 487057197033 * q^92 + 22000305348 * q^93 + 183904776602 * q^94 - 143022915504 * q^95 + 148215739059 * q^96 + 6973535140 * q^97 - 17795940687 * q^98 - 27569033316 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−27.9940	−0.618587	−0.309293	−	0.950967i	$-0.600093\pi$
$2$	−27.9940	−0.618587	−0.309293	+	0.950967i	$0.600093\pi$
$3$	−243.000	−0.577350
$3$	−243.000	−0.577350
$4$	−1264.33	−0.617350
$5$	−7765.11	−1.11125	−0.555626	−	0.831432i	$-0.687521\pi$
$5$	−7765.11	−1.11125	−0.555626	+	0.831432i	$0.687521\pi$
$6$	6802.55	0.357141
$7$	16807.0	0.377964
$7$	16807.0	0.377964
$8$	92725.6	1.00047
$9$	59049.0	0.333333
$10$	217377.	0.687406
$11$	67931.3	0.127177	0.0635887	−	0.997976i	$-0.479745\pi$
$11$	67931.3	0.127177	0.0635887	+	0.997976i	$0.479745\pi$
$12$	307233.	0.356427
$13$	−371293.	−0.277350
$13$	−371293.	−0.277350
$14$	−470496.	−0.233804
$15$	1.88692e6	0.641581
$16$	−6408.51	−0.00152791
$17$	−3.74683e6	−0.640022	−0.320011	−	0.947414i	$-0.603687\pi$
$17$	−3.74683e6	−0.640022	−0.320011	+	0.947414i	$0.603687\pi$
$18$	−1.65302e6	−0.206196
$19$	−8.63030e6	−0.799615	−0.399808	−	0.916599i	$-0.630923\pi$
$19$	−8.63030e6	−0.799615	−0.399808	+	0.916599i	$0.630923\pi$
$20$	9.81769e6	0.686032
$21$	−4.08410e6	−0.218218
$22$	−1.90167e6	−0.0786702
$23$	−4.83626e6	−0.156677	−0.0783387	−	0.996927i	$-0.524962\pi$
$23$	−4.83626e6	−0.156677	−0.0783387	+	0.996927i	$0.524962\pi$
$24$	−2.25323e7	−0.577622
$25$	1.14688e7	0.234880
$26$	1.03940e7	0.171565
$27$	−1.43489e7	−0.192450
$28$	−2.12497e7	−0.233337
$29$	−9.95736e7	−0.901478	−0.450739	−	0.892656i	$-0.648840\pi$
$29$	−9.95736e7	−0.901478	−0.450739	+	0.892656i	$0.648840\pi$
$30$	−5.28225e7	−0.396874
$31$	−2.31224e7	−0.145058	−0.0725292	−	0.997366i	$-0.523107\pi$
$31$	−2.31224e7	−0.145058	−0.0725292	+	0.997366i	$0.523107\pi$
$32$	−1.89723e8	−0.999526
$33$	−1.65073e7	−0.0734259
$34$	1.04889e8	0.395909
$35$	−1.30508e8	−0.420014
$36$	−7.46576e7	−0.205783
$37$	1.11079e8	0.263344	0.131672	−	0.991293i	$-0.457965\pi$
$37$	1.11079e8	0.263344	0.131672	+	0.991293i	$0.457965\pi$
$38$	2.41597e8	0.494631
$39$	9.02242e7	0.160128
$40$	−7.20024e8	−1.11178
$41$	−5.91074e8	−0.796766	−0.398383	−	0.917219i	$-0.630429\pi$
$41$	−5.91074e8	−0.796766	−0.398383	+	0.917219i	$0.630429\pi$
$42$	1.14330e8	0.134987
$43$	−1.14229e8	−0.118495	−0.0592477	−	0.998243i	$-0.518870\pi$
$43$	−1.14229e8	−0.118495	−0.0592477	+	0.998243i	$0.518870\pi$
$44$	−8.58878e7	−0.0785130
$45$	−4.58522e8	−0.370417
$46$	1.35386e8	0.0969185
$47$	3.04238e9	1.93498	0.967488	−	0.252916i	$-0.0813898\pi$
$47$	3.04238e9	1.93498	0.967488	+	0.252916i	$0.0813898\pi$
$48$	1.55727e6	0.000882138 0
$49$	2.82475e8	0.142857
$50$	−3.21057e8	−0.145294
$51$	9.10480e8	0.369517
$52$	4.69438e8	0.171222
$53$	2.10800e9	0.692394	0.346197	−	0.938162i	$-0.387473\pi$
$53$	2.10800e9	0.692394	0.346197	+	0.938162i	$0.387473\pi$
$54$	4.01684e8	0.119047
$55$	−5.27494e8	−0.141326
$56$	1.55844e9	0.378143
$57$	2.09716e9	0.461658
$58$	2.78747e9	0.557643
$59$	1.98209e8	0.0360941	0.0180471	−	0.999837i	$-0.494255\pi$
$59$	1.98209e8	0.0360941	0.0180471	+	0.999837i	$0.494255\pi$
$60$	−2.38570e9	−0.396081
$61$	−2.68281e9	−0.406701	−0.203350	−	0.979106i	$-0.565183\pi$
$61$	−2.68281e9	−0.406701	−0.203350	+	0.979106i	$0.565183\pi$
$62$	6.47289e8	0.0897313
$63$	9.92437e8	0.125988
$64$	5.32423e9	0.619822
$65$	2.88313e9	0.308206
$66$	4.62106e8	0.0454203
$67$	9.22410e9	0.834665	0.417333	−	0.908754i	$-0.362965\pi$
$67$	9.22410e9	0.834665	0.417333	+	0.908754i	$0.362965\pi$
$68$	4.73725e9	0.395118
$69$	1.17521e9	0.0904577
$70$	3.65345e9	0.259815
$71$	1.63258e10	1.07387	0.536936	−	0.843623i	$-0.319582\pi$
$71$	1.63258e10	1.07387	0.536936	+	0.843623i	$0.319582\pi$
$72$	5.47535e9	0.333491
$73$	2.98009e10	1.68249	0.841246	−	0.540653i	$-0.181823\pi$
$73$	2.98009e10	1.68249	0.841246	+	0.540653i	$0.181823\pi$
$74$	−3.10956e9	−0.162901
$75$	−2.78691e9	−0.135608
$76$	1.09116e10	0.493643
$77$	1.14172e9	0.0480685
$78$	−2.52574e9	−0.0990531
$79$	−1.17864e10	−0.430956	−0.215478	−	0.976509i	$-0.569131\pi$
$79$	−1.17864e10	−0.430956	−0.215478	+	0.976509i	$0.569131\pi$
$80$	4.97628e7	0.00169789
$81$	3.48678e9	0.111111
$82$	1.65466e10	0.492869
$83$	1.06107e10	0.295676	0.147838	−	0.989012i	$-0.452769\pi$
$83$	1.06107e10	0.295676	0.147838	+	0.989012i	$0.452769\pi$
$84$	5.16367e9	0.134717
$85$	2.90945e10	0.711226
$86$	3.19774e9	0.0732996
$87$	2.41964e10	0.520469
$88$	6.29897e9	0.127237
$89$	−8.40634e10	−1.59574	−0.797870	−	0.602829i	$-0.794040\pi$
$89$	−8.40634e10	−1.59574	−0.797870	+	0.602829i	$0.794040\pi$
$90$	1.28359e10	0.229135
$91$	−6.24032e9	−0.104828
$92$	6.11464e9	0.0967248
$93$	5.61874e9	0.0837496
$94$	−8.51686e10	−1.19695
$95$	6.70152e10	0.888574
$96$	4.61026e10	0.577077
$97$	−2.50839e10	−0.296586	−0.148293	−	0.988943i	$-0.547378\pi$
$97$	−2.50839e10	−0.296586	−0.148293	+	0.988943i	$0.547378\pi$
$98$	−7.90762e9	−0.0883695
$99$	4.01127e9	0.0423925
$100$	−1.45003e10	−0.145003
$101$	1.47040e11	1.39209	0.696045	−	0.717999i	$-0.254942\pi$
$101$	1.47040e11	1.39209	0.696045	+	0.717999i	$0.254942\pi$
$102$	−2.54880e10	−0.228578
$103$	1.40697e11	1.19586	0.597930	−	0.801549i	$-0.295990\pi$
$103$	1.40697e11	1.19586	0.597930	+	0.801549i	$0.295990\pi$
$104$	−3.44284e10	−0.277481
$105$	3.17135e10	0.242495
$106$	−5.90115e10	−0.428306
$107$	−1.38266e11	−0.953028	−0.476514	−	0.879167i	$-0.658100\pi$
$107$	−1.38266e11	−0.953028	−0.476514	+	0.879167i	$0.658100\pi$
$108$	1.81418e10	0.118809
$109$	−1.00269e11	−0.624199	−0.312099	−	0.950049i	$-0.601032\pi$
$109$	−1.00269e11	−0.624199	−0.312099	+	0.950049i	$0.601032\pi$
$110$	1.47667e10	0.0874224
$111$	−2.69923e10	−0.152042
$112$	−1.07708e8	−0.000577495 0
$113$	−1.65178e11	−0.843373	−0.421686	−	0.906742i	$-0.638562\pi$
$113$	−1.65178e11	−0.843373	−0.421686	+	0.906742i	$0.638562\pi$
$114$	−5.87080e10	−0.285576
$115$	3.75540e10	0.174108
$116$	1.25894e11	0.556528
$117$	−2.19245e10	−0.0924500
$118$	−5.54866e9	−0.0223274
$119$	−6.29730e10	−0.241906
$120$	1.74966e11	0.641884
$121$	−2.80697e11	−0.983826
$122$	7.51026e10	0.251580
$123$	1.43631e11	0.460013
$124$	2.92344e10	0.0895519
$125$	2.90099e11	0.850241
$126$	−2.77823e10	−0.0779346
$127$	3.91680e11	1.05199	0.525994	−	0.850488i	$-0.323693\pi$
$127$	3.91680e11	1.05199	0.525994	+	0.850488i	$0.323693\pi$
$128$	2.39505e11	0.616113
$129$	2.77577e10	0.0684133
$130$	−8.07105e10	−0.190652
$131$	−3.75036e11	−0.849339	−0.424670	−	0.905348i	$-0.639610\pi$
$131$	−3.75036e11	−0.849339	−0.424670	+	0.905348i	$0.639610\pi$
$132$	2.08707e10	0.0453295
$133$	−1.45049e11	−0.302226
$134$	−2.58220e11	−0.516313
$135$	1.11421e11	0.213860
$136$	−3.47427e11	−0.640324
$137$	4.98564e10	0.0882588	0.0441294	−	0.999026i	$-0.485949\pi$
$137$	4.98564e10	0.0882588	0.0441294	+	0.999026i	$0.485949\pi$
$138$	−3.28989e10	−0.0559559
$139$	1.03385e12	1.68995	0.844976	−	0.534805i	$-0.179615\pi$
$139$	1.03385e12	1.68995	0.844976	+	0.534805i	$0.179615\pi$
$140$	1.65006e11	0.259296
$141$	−7.39299e11	−1.11716
$142$	−4.57024e11	−0.664283
$143$	−2.52224e10	−0.0352727
$144$	−3.78416e8	−0.000509303 0
$145$	7.73200e11	1.00177
$146$	−8.34246e11	−1.04077
$147$	−6.86415e10	−0.0824786
$148$	−1.40441e11	−0.162576
$149$	1.10842e12	1.23646	0.618230	−	0.785997i	$-0.287850\pi$
$149$	1.10842e12	1.23646	0.618230	+	0.785997i	$0.287850\pi$
$150$	7.80169e10	0.0838854
$151$	−5.67818e11	−0.588622	−0.294311	−	0.955710i	$-0.595090\pi$
$151$	−5.67818e11	−0.588622	−0.294311	+	0.955710i	$0.595090\pi$
$152$	−8.00250e11	−0.799992
$153$	−2.21247e11	−0.213341
$154$	−3.19614e10	−0.0297346
$155$	1.79548e11	0.161196
$156$	−1.14074e11	−0.0988552
$157$	1.07354e12	0.898197	0.449099	−	0.893482i	$-0.351745\pi$
$157$	1.07354e12	0.898197	0.449099	+	0.893482i	$0.351745\pi$
$158$	3.29949e11	0.266584
$159$	−5.12244e11	−0.399754
$160$	1.47322e12	1.11073
$161$	−8.12830e10	−0.0592185
$162$	−9.76092e10	−0.0687319
$163$	−8.28938e10	−0.0564274	−0.0282137	−	0.999602i	$-0.508982\pi$
$163$	−8.28938e10	−0.0564274	−0.0282137	+	0.999602i	$0.508982\pi$
$164$	7.47315e11	0.491884
$165$	1.28181e11	0.0815947
$166$	−2.97037e11	−0.182901
$167$	2.35920e12	1.40548	0.702740	−	0.711447i	$-0.251960\pi$
$167$	2.35920e12	1.40548	0.702740	+	0.711447i	$0.251960\pi$
$168$	−3.78701e11	−0.218321
$169$	1.37858e11	0.0769231
$170$	−8.14474e11	−0.439955
$171$	−5.09610e11	−0.266538
$172$	1.44424e11	0.0731531
$173$	−7.44443e11	−0.365240	−0.182620	−	0.983184i	$-0.558458\pi$
$173$	−7.44443e11	−0.365240	−0.182620	+	0.983184i	$0.558458\pi$
$174$	−6.77355e11	−0.321955
$175$	1.92756e11	0.0887764
$176$	−4.35338e8	−0.000194315 0
$177$	−4.81647e10	−0.0208390
$178$	2.35328e12	0.987104
$179$	−1.08843e12	−0.442700	−0.221350	−	0.975194i	$-0.571046\pi$
$179$	−1.08843e12	−0.442700	−0.221350	+	0.975194i	$0.571046\pi$
$180$	5.79725e11	0.228677
$181$	2.55567e12	0.977852	0.488926	−	0.872325i	$-0.337389\pi$
$181$	2.55567e12	0.977852	0.488926	+	0.872325i	$0.337389\pi$
$182$	1.74692e11	0.0648455
$183$	6.51922e11	0.234809
$184$	−4.48445e11	−0.156751
$185$	−8.62542e11	−0.292642
$186$	−1.57291e11	−0.0518064
$187$	−2.54527e11	−0.0813964
$188$	−3.84659e12	−1.19456
$189$	−2.41162e11	−0.0727393
$190$	−1.87603e12	−0.549660
$191$	5.18397e12	1.47564	0.737818	−	0.675000i	$-0.235856\pi$
$191$	5.18397e12	1.47564	0.737818	+	0.675000i	$0.235856\pi$
$192$	−1.29379e12	−0.357854
$193$	3.26259e12	0.876997	0.438498	−	0.898732i	$-0.355511\pi$
$193$	3.26259e12	0.876997	0.438498	+	0.898732i	$0.355511\pi$
$194$	7.02199e11	0.183464
$195$	−7.00601e11	−0.177943
$196$	−3.57143e11	−0.0881929
$197$	−3.73924e12	−0.897882	−0.448941	−	0.893561i	$-0.648199\pi$
$197$	−3.73924e12	−0.897882	−0.448941	+	0.893561i	$0.648199\pi$
$198$	−1.12292e11	−0.0262234
$199$	−7.90284e12	−1.79511	−0.897556	−	0.440901i	$-0.854659\pi$
$199$	−7.90284e12	−1.79511	−0.897556	+	0.440901i	$0.854659\pi$
$200$	1.06345e12	0.234991
$201$	−2.24146e12	−0.481894
$202$	−4.11624e12	−0.861128
$203$	−1.67353e12	−0.340727
$204$	−1.15115e12	−0.228121
$205$	4.58976e12	0.885408
$206$	−3.93868e12	−0.739743
$207$	−2.85576e11	−0.0522258
$208$	2.37943e9	0.000423765 0
$209$	−5.86267e11	−0.101693
$210$	−8.87788e11	−0.150004
$211$	−1.99562e12	−0.328492	−0.164246	−	0.986419i	$-0.552519\pi$
$211$	−1.99562e12	−0.328492	−0.164246	+	0.986419i	$0.552519\pi$
$212$	−2.66522e12	−0.427450
$213$	−3.96716e12	−0.620001
$214$	3.87063e12	0.589530
$215$	8.87003e11	0.131678
$216$	−1.33051e12	−0.192541
$217$	−3.88618e11	−0.0548270
$218$	2.80695e12	0.386121
$219$	−7.24161e12	−0.971387
$220$	6.66928e11	0.0872477
$221$	1.39117e12	0.177510
$222$	7.55622e11	0.0940510
$223$	8.67408e12	1.05329	0.526643	−	0.850086i	$-0.323450\pi$
$223$	8.67408e12	1.05329	0.526643	+	0.850086i	$0.323450\pi$
$224$	−3.18867e12	−0.377785
$225$	6.77219e11	0.0782934
$226$	4.62399e12	0.521699
$227$	7.15539e12	0.787936	0.393968	−	0.919124i	$-0.371102\pi$
$227$	7.15539e12	0.787936	0.393968	+	0.919124i	$0.371102\pi$
$228$	−2.65151e12	−0.285005
$229$	4.63925e12	0.486802	0.243401	−	0.969926i	$-0.421737\pi$
$229$	4.63925e12	0.486802	0.243401	+	0.969926i	$0.421737\pi$
$230$	−1.05129e12	−0.107701
$231$	−2.77438e11	−0.0277524
$232$	−9.23302e12	−0.901903
$233$	−1.04445e13	−0.996391	−0.498195	−	0.867065i	$-0.666004\pi$
$233$	−1.04445e13	−0.996391	−0.498195	+	0.867065i	$0.666004\pi$
$234$	6.13755e11	0.0571884
$235$	−2.36244e13	−2.15025
$236$	−2.50602e11	−0.0222827
$237$	2.86410e12	0.248812
$238$	1.76287e12	0.149640
$239$	−3.37665e12	−0.280090	−0.140045	−	0.990145i	$-0.544725\pi$
$239$	−3.37665e12	−0.280090	−0.140045	+	0.990145i	$0.544725\pi$
$240$	−1.20924e10	−0.000980277 0
$241$	−2.32949e12	−0.184573	−0.0922864	−	0.995733i	$-0.529418\pi$
$241$	−2.32949e12	−0.184573	−0.0922864	+	0.995733i	$0.529418\pi$
$242$	7.85784e12	0.608582
$243$	−8.47289e11	−0.0641500
$244$	3.39196e12	0.251077
$245$	−2.19345e12	−0.158750
$246$	−4.02081e12	−0.284558
$247$	3.20437e12	0.221773
$248$	−2.14404e12	−0.145127
$249$	−2.57841e12	−0.170709
$250$	−8.12106e12	−0.525948
$251$	1.21561e13	0.770176	0.385088	−	0.922880i	$-0.374171\pi$
$251$	1.21561e13	0.770176	0.385088	+	0.922880i	$0.374171\pi$
$252$	−1.25477e12	−0.0777788
$253$	−3.28533e11	−0.0199258
$254$	−1.09647e13	−0.650746
$255$	−7.06997e12	−0.410626
$256$	−1.76087e13	−1.00094
$257$	−3.25784e13	−1.81258	−0.906291	−	0.422653i	$-0.861099\pi$
$257$	−3.25784e13	−1.81258	−0.906291	+	0.422653i	$0.861099\pi$
$258$	−7.77051e11	−0.0423196
$259$	1.86691e12	0.0995347
$260$	−3.64524e12	−0.190271
$261$	−5.87972e12	−0.300493
$262$	1.04988e13	0.525390
$263$	−1.74701e13	−0.856126	−0.428063	−	0.903749i	$-0.640804\pi$
$263$	−1.74701e13	−0.856126	−0.428063	+	0.903749i	$0.640804\pi$
$264$	−1.53065e12	−0.0734605
$265$	−1.63689e13	−0.769424
$266$	4.06052e12	0.186953
$267$	2.04274e13	0.921301
$268$	−1.16623e13	−0.515281
$269$	2.58489e12	0.111893	0.0559466	−	0.998434i	$-0.482182\pi$
$269$	2.58489e12	0.111893	0.0559466	+	0.998434i	$0.482182\pi$
$270$	−3.11912e12	−0.132291
$271$	−3.46494e12	−0.144001	−0.0720003	−	0.997405i	$-0.522938\pi$
$271$	−3.46494e12	−0.144001	−0.0720003	+	0.997405i	$0.522938\pi$
$272$	2.40116e10	0.000977895 0
$273$	1.51640e12	0.0605228
$274$	−1.39568e12	−0.0545957
$275$	7.79088e11	0.0298715
$276$	−1.48586e12	−0.0558441
$277$	4.70036e12	0.173178	0.0865890	−	0.996244i	$-0.472403\pi$
$277$	4.70036e12	0.173178	0.0865890	+	0.996244i	$0.472403\pi$
$278$	−2.89415e13	−1.04538
$279$	−1.36535e12	−0.0483528
$280$	−1.21014e13	−0.420212
$281$	6.61980e12	0.225403	0.112702	−	0.993629i	$-0.464050\pi$
$281$	6.61980e12	0.225403	0.112702	+	0.993629i	$0.464050\pi$
$282$	2.06960e13	0.691060
$283$	4.57343e13	1.49767	0.748836	−	0.662755i	$-0.230613\pi$
$283$	4.57343e13	1.49767	0.748836	+	0.662755i	$0.230613\pi$
$284$	−2.06412e13	−0.662956
$285$	−1.62847e13	−0.513018
$286$	7.06077e11	0.0218192
$287$	−9.93419e12	−0.301149
$288$	−1.12029e13	−0.333175
$289$	−2.02332e13	−0.590372
$290$	−2.16450e13	−0.619681
$291$	6.09539e12	0.171234
$292$	−3.76782e13	−1.03869
$293$	4.26811e13	1.15469	0.577343	−	0.816502i	$-0.304090\pi$
$293$	4.26811e13	1.15469	0.577343	+	0.816502i	$0.304090\pi$
$294$	1.92155e12	0.0510202
$295$	−1.53911e12	−0.0401097
$296$	1.02999e13	0.263468
$297$	−9.74739e11	−0.0244753
$298$	−3.10292e13	−0.764858
$299$	1.79567e12	0.0434545
$300$	3.52358e12	0.0837178
$301$	−1.91985e12	−0.0447870
$302$	1.58955e13	0.364114
$303$	−3.57306e13	−0.803723
$304$	5.53073e10	0.00122174
$305$	2.08323e13	0.451947
$306$	6.19359e12	0.131970
$307$	−2.14159e13	−0.448204	−0.224102	−	0.974566i	$-0.571945\pi$
$307$	−2.14159e13	−0.448204	−0.224102	+	0.974566i	$0.571945\pi$
$308$	−1.44352e12	−0.0296751
$309$	−3.41894e13	−0.690430
$310$	−5.02627e12	−0.0997140
$311$	4.49138e13	0.875382	0.437691	−	0.899126i	$-0.355796\pi$
$311$	4.49138e13	0.875382	0.437691	+	0.899126i	$0.355796\pi$
$312$	8.36609e12	0.160204
$313$	−3.74928e13	−0.705430	−0.352715	−	0.935731i	$-0.614741\pi$
$313$	−3.74928e13	−0.705430	−0.352715	+	0.935731i	$0.614741\pi$
$314$	−3.00528e13	−0.555613
$315$	−7.70638e12	−0.140005
$316$	1.49020e13	0.266051
$317$	−5.54250e13	−0.972478	−0.486239	−	0.873826i	$-0.661631\pi$
$317$	−5.54250e13	−0.972478	−0.486239	+	0.873826i	$0.661631\pi$
$318$	1.43398e13	0.247283
$319$	−6.76416e12	−0.114648
$320$	−4.13432e13	−0.688778
$321$	3.35987e13	0.550231
$322$	2.27544e12	0.0366318
$323$	3.23363e13	0.511771
$324$	−4.40846e12	−0.0685945
$325$	−4.25827e12	−0.0651441
$326$	2.32053e12	0.0349052
$327$	2.43655e13	0.360381
$328$	−5.48077e13	−0.797142
$329$	5.11333e13	0.731352
$330$	−3.58830e12	−0.0504734
$331$	−7.81881e13	−1.08165	−0.540825	−	0.841135i	$-0.681888\pi$
$331$	−7.81881e13	−1.08165	−0.540825	+	0.841135i	$0.681888\pi$
$332$	−1.34155e13	−0.182536
$333$	6.55912e12	0.0877814
$334$	−6.60436e13	−0.869411
$335$	−7.16261e13	−0.927523
$336$	2.61730e10	0.000333417 0
$337$	−5.18100e13	−0.649305	−0.324653	−	0.945833i	$-0.605247\pi$
$337$	−5.18100e13	−0.649305	−0.324653	+	0.945833i	$0.605247\pi$
$338$	−3.85922e12	−0.0475836
$339$	4.01381e13	0.486922
$340$	−3.67852e13	−0.439076
$341$	−1.57073e12	−0.0184482
$342$	1.42661e13	0.164877
$343$	4.74756e12	0.0539949
$344$	−1.05920e13	−0.118551
$345$	−9.12563e12	−0.100521
$346$	2.08400e13	0.225932
$347$	1.12897e13	0.120468	0.0602339	−	0.998184i	$-0.480815\pi$
$347$	1.12897e13	0.120468	0.0602339	+	0.998184i	$0.480815\pi$
$348$	−3.05923e13	−0.321312
$349$	−6.61474e13	−0.683869	−0.341935	−	0.939724i	$-0.611082\pi$
$349$	−6.61474e13	−0.683869	−0.341935	+	0.939724i	$0.611082\pi$
$350$	−5.39601e12	−0.0549159
$351$	5.32765e12	0.0533761
$352$	−1.28881e13	−0.127117
$353$	−1.82480e13	−0.177197	−0.0885983	−	0.996067i	$-0.528239\pi$
$353$	−1.82480e13	−0.177197	−0.0885983	+	0.996067i	$0.528239\pi$
$354$	1.34832e12	0.0128907
$355$	−1.26771e14	−1.19334
$356$	1.06284e14	0.985131
$357$	1.53024e13	0.139664
$358$	3.04696e13	0.273848
$359$	−7.01666e13	−0.621028	−0.310514	−	0.950569i	$-0.600501\pi$
$359$	−7.01666e13	−0.621028	−0.310514	+	0.950569i	$0.600501\pi$
$360$	−4.25167e13	−0.370592
$361$	−4.20082e13	−0.360616
$362$	−7.15436e13	−0.604886
$363$	6.82094e13	0.568012
$364$	7.88985e12	0.0647159
$365$	−2.31407e14	−1.86967
$366$	−1.82499e13	−0.145250
$367$	8.52793e13	0.668622	0.334311	−	0.942463i	$-0.391496\pi$
$367$	8.52793e13	0.668622	0.334311	+	0.942463i	$0.391496\pi$
$368$	3.09932e10	0.000239389 0
$369$	−3.49024e13	−0.265589
$370$	2.41460e13	0.181024
$371$	3.54292e13	0.261700
$372$	−7.10396e12	−0.0517028
$373$	−2.11449e13	−0.151638	−0.0758189	−	0.997122i	$-0.524157\pi$
$373$	−2.11449e13	−0.151638	−0.0758189	+	0.997122i	$0.524157\pi$
$374$	7.12524e12	0.0503507
$375$	−7.04942e13	−0.490887
$376$	2.82107e14	1.93589
$377$	3.69710e13	0.250025
$378$	6.75110e12	0.0449956
$379$	1.08166e14	0.710521	0.355260	−	0.934767i	$-0.384392\pi$
$379$	1.08166e14	0.710521	0.355260	+	0.934767i	$0.384392\pi$
$380$	−8.47296e13	−0.548561
$381$	−9.51782e13	−0.607365
$382$	−1.45120e14	−0.912809
$383$	−2.30663e14	−1.43016	−0.715079	−	0.699044i	$-0.753609\pi$
$383$	−2.30663e14	−1.43016	−0.715079	+	0.699044i	$0.753609\pi$
$384$	−5.81998e13	−0.355713
$385$	−8.86558e12	−0.0534162
$386$	−9.13332e13	−0.542498
$387$	−6.74513e12	−0.0394984
$388$	3.17144e13	0.183097
$389$	−5.14434e13	−0.292824	−0.146412	−	0.989224i	$-0.546773\pi$
$389$	−5.14434e13	−0.292824	−0.146412	+	0.989224i	$0.546773\pi$
$390$	1.96126e13	0.110073
$391$	1.81206e13	0.100277
$392$	2.61927e13	0.142925
$393$	9.11338e13	0.490366
$394$	1.04676e14	0.555418
$395$	9.15228e13	0.478900
$396$	−5.07159e12	−0.0261710
$397$	2.59388e14	1.32009	0.660043	−	0.751228i	$-0.270538\pi$
$397$	2.59388e14	1.32009	0.660043	+	0.751228i	$0.270538\pi$
$398$	2.21232e14	1.11043
$399$	3.52470e13	0.174490
$400$	−7.34977e10	−0.000358875 0
$401$	−3.31150e14	−1.59489	−0.797446	−	0.603391i	$-0.793816\pi$
$401$	−3.31150e14	−1.59489	−0.797446	+	0.603391i	$0.793816\pi$
$402$	6.27474e13	0.298093
$403$	8.58518e12	0.0402320
$404$	−1.85907e14	−0.859407
$405$	−2.70753e13	−0.123472
$406$	4.68490e13	0.210769
$407$	7.54576e12	0.0334914
$408$	8.44248e13	0.369691
$409$	1.24312e13	0.0537075	0.0268538	−	0.999639i	$-0.491451\pi$
$409$	1.24312e13	0.0537075	0.0268538	+	0.999639i	$0.491451\pi$
$410$	−1.28486e14	−0.547702
$411$	−1.21151e13	−0.0509563
$412$	−1.77888e14	−0.738264
$413$	3.33129e12	0.0136423
$414$	7.99443e12	0.0323062
$415$	−8.23935e13	−0.328570
$416$	7.04427e13	0.277219
$417$	−2.51224e14	−0.975694
$418$	1.64120e13	0.0629059
$419$	−1.69756e13	−0.0642166	−0.0321083	−	0.999484i	$-0.510222\pi$
$419$	−1.69756e13	−0.0642166	−0.0321083	+	0.999484i	$0.510222\pi$
$420$	−4.00964e13	−0.149704
$421$	2.93834e14	1.08280	0.541402	−	0.840764i	$-0.317894\pi$
$421$	2.93834e14	1.08280	0.541402	+	0.840764i	$0.317894\pi$
$422$	5.58655e13	0.203201
$423$	1.79650e14	0.644992
$424$	1.95466e14	0.692721
$425$	−4.29715e13	−0.150329
$426$	1.11057e14	0.383524
$427$	−4.50899e13	−0.153719
$428$	1.74815e14	0.588352
$429$	6.12904e12	0.0203647
$430$	−2.48308e13	−0.0814543
$431$	−1.10394e14	−0.357536	−0.178768	−	0.983891i	$-0.557211\pi$
$431$	−1.10394e14	−0.357536	−0.178768	+	0.983891i	$0.557211\pi$
$432$	9.19551e10	0.000294046 0
$433$	3.97720e14	1.25572	0.627862	−	0.778324i	$-0.283930\pi$
$433$	3.97720e14	1.25572	0.627862	+	0.778324i	$0.283930\pi$
$434$	1.08790e13	0.0339152
$435$	−1.87888e14	−0.578372
$436$	1.26774e14	0.385349
$437$	4.17383e13	0.125282
$438$	2.02722e14	0.600887
$439$	1.19617e14	0.350138	0.175069	−	0.984556i	$-0.443985\pi$
$439$	1.19617e14	0.350138	0.175069	+	0.984556i	$0.443985\pi$
$440$	−4.89122e13	−0.141393
$441$	1.66799e13	0.0476190
$442$	−3.89445e13	−0.109805
$443$	9.98452e13	0.278040	0.139020	−	0.990290i	$-0.455605\pi$
$443$	9.98452e13	0.278040	0.139020	+	0.990290i	$0.455605\pi$
$444$	3.41272e13	0.0938631
$445$	6.52762e14	1.77327
$446$	−2.42822e14	−0.651549
$447$	−2.69346e14	−0.713871
$448$	8.94843e13	0.234271
$449$	−6.18671e14	−1.59994	−0.799972	−	0.600038i	$-0.795152\pi$
$449$	−6.18671e14	−1.59994	−0.799972	+	0.600038i	$0.795152\pi$
$450$	−1.89581e13	−0.0484313
$451$	−4.01524e13	−0.101331
$452$	2.08840e14	0.520657
$453$	1.37980e14	0.339841
$454$	−2.00308e14	−0.487407
$455$	4.84568e13	0.116491
$456$	1.94461e14	0.461876
$457$	2.85106e14	0.669063	0.334532	−	0.942385i	$-0.391422\pi$
$457$	2.85106e14	0.669063	0.334532	+	0.942385i	$0.391422\pi$
$458$	−1.29871e14	−0.301130
$459$	5.37629e13	0.123172
$460$	−4.74809e13	−0.107486
$461$	−4.77342e14	−1.06776	−0.533881	−	0.845559i	$-0.679267\pi$
$461$	−4.77342e14	−1.06776	−0.533881	+	0.845559i	$0.679267\pi$
$462$	7.76661e12	0.0171673
$463$	4.89216e14	1.06858	0.534288	−	0.845303i	$-0.320580\pi$
$463$	4.89216e14	1.06858	0.534288	+	0.845303i	$0.320580\pi$
$464$	6.38118e11	0.00137738
$465$	−4.36301e13	−0.0930668
$466$	2.92384e14	0.616354
$467$	1.61631e14	0.336731	0.168365	−	0.985725i	$-0.446151\pi$
$467$	1.61631e14	0.336731	0.168365	+	0.985725i	$0.446151\pi$
$468$	2.77199e13	0.0570741
$469$	1.55029e14	0.315474
$470$	6.61343e14	1.33011
$471$	−2.60871e14	−0.518575
$472$	1.83790e13	0.0361112
$473$	−7.75974e12	−0.0150699
$474$	−8.01777e13	−0.153912
$475$	−9.89789e13	−0.187814
$476$	7.96189e13	0.149341
$477$	1.24475e14	0.230798
$478$	9.45260e13	0.173260
$479$	7.26758e14	1.31688	0.658438	−	0.752635i	$-0.271218\pi$
$479$	7.26758e14	1.31688	0.658438	+	0.752635i	$0.271218\pi$
$480$	−3.57992e14	−0.641278
$481$	−4.12430e13	−0.0730385
$482$	6.52119e13	0.114174
$483$	1.97518e13	0.0341898
$484$	3.54895e14	0.607365
$485$	1.94779e14	0.329581
$486$	2.37190e13	0.0396824
$487$	3.05280e14	0.504997	0.252499	−	0.967597i	$-0.418748\pi$
$487$	3.05280e14	0.504997	0.252499	+	0.967597i	$0.418748\pi$
$488$	−2.48765e14	−0.406893
$489$	2.01432e13	0.0325784
$490$	6.14035e13	0.0982008
$491$	5.25752e14	0.831443	0.415721	−	0.909492i	$-0.363529\pi$
$491$	5.25752e14	0.831443	0.415721	+	0.909492i	$0.363529\pi$
$492$	−1.81598e14	−0.283989
$493$	3.73086e14	0.576966
$494$	−8.97032e13	−0.137186
$495$	−3.11480e13	−0.0471087
$496$	1.48180e11	0.000221636 0
$497$	2.74387e14	0.405886
$498$	7.21801e13	0.105598
$499$	6.16879e14	0.892579	0.446290	−	0.894889i	$-0.352745\pi$
$499$	6.16879e14	0.892579	0.446290	+	0.894889i	$0.352745\pi$
$500$	−3.66783e14	−0.524896
$501$	−5.73286e14	−0.811454
$502$	−3.40299e14	−0.476420
$503$	−3.24552e14	−0.449428	−0.224714	−	0.974425i	$-0.572145\pi$
$503$	−3.24552e14	−0.449428	−0.224714	+	0.974425i	$0.572145\pi$
$504$	9.20243e13	0.126048
$505$	−1.14178e15	−1.54696
$506$	9.19697e12	0.0123258
$507$	−3.34996e13	−0.0444116
$508$	−4.95214e14	−0.649445
$509$	−6.77260e14	−0.878633	−0.439316	−	0.898332i	$-0.644779\pi$
$509$	−6.77260e14	−0.878633	−0.439316	+	0.898332i	$0.644779\pi$
$510$	1.97917e14	0.254008
$511$	5.00863e14	0.635922
$512$	2.43283e12	0.00305581
$513$	1.23835e14	0.153886
$514$	9.12002e14	1.12124
$515$	−1.09253e15	−1.32890
$516$	−3.50950e13	−0.0422350
$517$	2.06673e14	0.246085
$518$	−5.22623e13	−0.0615708
$519$	1.80900e14	0.210871
$520$	2.67340e14	0.308351
$521$	−9.63071e14	−1.09913	−0.549567	−	0.835449i	$-0.685207\pi$
$521$	−9.63071e14	−1.09913	−0.549567	+	0.835449i	$0.685207\pi$
$522$	1.64597e14	0.185881
$523$	−1.59177e15	−1.77878	−0.889388	−	0.457153i	$-0.848869\pi$
$523$	−1.59177e15	−1.77878	−0.889388	+	0.457153i	$0.848869\pi$
$524$	4.74171e14	0.524340
$525$	−4.68396e13	−0.0512551
$526$	4.89057e14	0.529588
$527$	8.66356e13	0.0928407
$528$	1.05787e11	0.000112188 0
$529$	−9.29420e14	−0.975452
$530$	4.58230e14	0.475956
$531$	1.17040e13	0.0120314
$532$	1.83391e14	0.186579
$533$	2.19462e14	0.220983
$534$	−5.71846e14	−0.569905
$535$	1.07365e15	1.05905
$536$	8.55310e14	0.835059
$537$	2.64489e14	0.255593
$538$	−7.23614e13	−0.0692157
$539$	1.91889e13	0.0181682
$540$	−1.40873e14	−0.132027
$541$	1.24912e15	1.15883	0.579413	−	0.815034i	$-0.303282\pi$
$541$	1.24912e15	1.15883	0.579413	+	0.815034i	$0.303282\pi$
$542$	9.69975e13	0.0890768
$543$	−6.21029e14	−0.564563
$544$	7.10859e14	0.639719
$545$	7.78603e14	0.693642
$546$	−4.24501e13	−0.0374386
$547$	3.97885e14	0.347398	0.173699	−	0.984799i	$-0.444428\pi$
$547$	3.97885e14	0.347398	0.173699	+	0.984799i	$0.444428\pi$
$548$	−6.30352e13	−0.0544866
$549$	−1.58417e14	−0.135567
$550$	−2.18098e13	−0.0184781
$551$	8.59350e14	0.720836
$552$	1.08972e14	0.0905004
$553$	−1.98094e14	−0.162886
$554$	−1.31582e14	−0.107126
$555$	2.09598e14	0.168957
$556$	−1.30713e15	−1.04329
$557$	−1.20186e15	−0.949840	−0.474920	−	0.880029i	$-0.657523\pi$
$557$	−1.20186e15	−0.949840	−0.474920	+	0.880029i	$0.657523\pi$
$558$	3.82217e13	0.0299104
$559$	4.24126e13	0.0328647
$560$	8.36363e11	0.000641742 0
$561$	6.18501e13	0.0469942
$562$	−1.85315e14	−0.139431
$563$	−1.47161e15	−1.09647	−0.548236	−	0.836324i	$-0.684700\pi$
$563$	−1.47161e15	−1.09647	−0.548236	+	0.836324i	$0.684700\pi$
$564$	9.34721e14	0.689679
$565$	1.28262e15	0.937200
$566$	−1.28029e15	−0.926440
$567$	5.86024e13	0.0419961
$568$	1.51382e15	1.07438
$569$	−1.19731e15	−0.841565	−0.420782	−	0.907162i	$-0.638244\pi$
$569$	−1.19731e15	−0.841565	−0.420782	+	0.907162i	$0.638244\pi$
$570$	4.55874e14	0.317346
$571$	8.71655e14	0.600960	0.300480	−	0.953788i	$-0.402853\pi$
$571$	8.71655e14	0.600960	0.300480	+	0.953788i	$0.402853\pi$
$572$	3.18895e13	0.0217756
$573$	−1.25971e15	−0.851959
$574$	2.78098e14	0.186287
$575$	−5.54659e13	−0.0368004
$576$	3.14390e14	0.206607
$577$	5.38004e14	0.350202	0.175101	−	0.984550i	$-0.443975\pi$
$577$	5.38004e14	0.350202	0.175101	+	0.984550i	$0.443975\pi$
$578$	5.66408e14	0.365196
$579$	−7.92811e14	−0.506334
$580$	−9.77583e14	−0.618443
$581$	1.78335e14	0.111755
$582$	−1.70634e14	−0.105923
$583$	1.43199e14	0.0880569
$584$	2.76330e15	1.68329
$585$	1.70246e14	0.102735
$586$	−1.19482e15	−0.714274
$587$	2.12875e15	1.26071	0.630354	−	0.776308i	$-0.282910\pi$
$587$	2.12875e15	1.26071	0.630354	+	0.776308i	$0.282910\pi$
$588$	8.67857e13	0.0509182
$589$	1.99553e14	0.115991
$590$	4.30860e13	0.0248113
$591$	9.08636e14	0.518392
$592$	−7.11853e11	−0.000402365 0
$593$	1.29899e15	0.727453	0.363726	−	0.931506i	$-0.381504\pi$
$593$	1.29899e15	0.727453	0.363726	+	0.931506i	$0.381504\pi$
$594$	2.72869e13	0.0151401
$595$	4.88992e14	0.268818
$596$	−1.40141e15	−0.763329
$597$	1.92039e15	1.03641
$598$	−5.02680e13	−0.0268804
$599$	−5.05943e14	−0.268073	−0.134037	−	0.990976i	$-0.542794\pi$
$599$	−5.05943e14	−0.268073	−0.134037	+	0.990976i	$0.542794\pi$
$600$	−2.58418e14	−0.135672
$601$	5.06529e14	0.263509	0.131754	−	0.991282i	$-0.457939\pi$
$601$	5.06529e14	0.263509	0.131754	+	0.991282i	$0.457939\pi$
$602$	5.37444e13	0.0277047
$603$	5.44674e14	0.278222
$604$	7.17912e14	0.363386
$605$	2.17964e15	1.09328
$606$	1.00025e15	0.497172
$607$	1.81640e15	0.894694	0.447347	−	0.894360i	$-0.352369\pi$
$607$	1.81640e15	0.894694	0.447347	+	0.894360i	$0.352369\pi$
$608$	1.63736e15	0.799236
$609$	4.06669e14	0.196719
$610$	−5.83180e14	−0.279568
$611$	−1.12962e15	−0.536666
$612$	2.79730e14	0.131706
$613$	2.97298e15	1.38727	0.693633	−	0.720328i	$-0.256009\pi$
$613$	2.97298e15	1.38727	0.693633	+	0.720328i	$0.256009\pi$
$614$	5.99518e14	0.277253
$615$	−1.11531e15	−0.511190
$616$	1.05867e14	0.0480912
$617$	−7.56701e14	−0.340687	−0.170344	−	0.985385i	$-0.554488\pi$
$617$	−7.56701e14	−0.340687	−0.170344	+	0.985385i	$0.554488\pi$
$618$	9.57098e14	0.427091
$619$	−1.46268e15	−0.646920	−0.323460	−	0.946242i	$-0.604846\pi$
$619$	−1.46268e15	−0.646920	−0.323460	+	0.946242i	$0.604846\pi$
$620$	−2.27008e14	−0.0995147
$621$	6.93950e13	0.0301526
$622$	−1.25732e15	−0.541499
$623$	−1.41285e15	−0.603133
$624$	−5.78203e11	−0.000244661 0
$625$	−2.81265e15	−1.17971
$626$	1.04957e15	0.436369
$627$	1.42463e14	0.0587125
$628$	−1.35732e15	−0.554503
$629$	−4.16195e14	−0.168546
$630$	2.15733e14	0.0866050
$631$	−4.16872e15	−1.65898	−0.829491	−	0.558520i	$-0.811369\pi$
$631$	−4.16872e15	−1.65898	−0.829491	+	0.558520i	$0.811369\pi$
$632$	−1.09290e15	−0.431159
$633$	4.84936e14	0.189655
$634$	1.55157e15	0.601562
$635$	−3.04144e15	−1.16902
$636$	6.47648e14	0.246788
$637$	−1.04881e14	−0.0396214
$638$	1.89356e14	0.0709195
$639$	9.64020e14	0.357958
$640$	−1.85978e15	−0.684657
$641$	2.73106e15	0.996808	0.498404	−	0.866945i	$-0.333920\pi$
$641$	2.73106e15	0.996808	0.498404	+	0.866945i	$0.333920\pi$
$642$	−9.40563e14	−0.340365
$643$	8.25785e14	0.296283	0.148142	−	0.988966i	$-0.452671\pi$
$643$	8.25785e14	0.296283	0.148142	+	0.988966i	$0.452671\pi$
$644$	1.02769e14	0.0365586
$645$	−2.15542e14	−0.0760244
$646$	−9.05223e14	−0.316575
$647$	−2.56430e15	−0.889192	−0.444596	−	0.895731i	$-0.646653\pi$
$647$	−2.56430e15	−0.889192	−0.444596	+	0.895731i	$0.646653\pi$
$648$	3.23314e14	0.111164
$649$	1.34646e13	0.00459036
$650$	1.19206e14	0.0402973
$651$	9.44341e13	0.0316544
$652$	1.04805e14	0.0348355
$653$	−2.22874e15	−0.734576	−0.367288	−	0.930107i	$-0.619714\pi$
$653$	−2.22874e15	−0.734576	−0.367288	+	0.930107i	$0.619714\pi$
$654$	−6.82088e14	−0.222927
$655$	2.91220e15	0.943830
$656$	3.78791e12	0.00121739
$657$	1.75971e15	0.560831
$658$	−1.43143e15	−0.452405
$659$	−2.68766e15	−0.842373	−0.421186	−	0.906974i	$-0.638386\pi$
$659$	−2.68766e15	−0.842373	−0.421186	+	0.906974i	$0.638386\pi$
$660$	−1.62063e14	−0.0503725
$661$	4.81438e15	1.48400	0.741998	−	0.670403i	$-0.233879\pi$
$661$	4.81438e15	1.48400	0.741998	+	0.670403i	$0.233879\pi$
$662$	2.18880e15	0.669095
$663$	−3.38055e14	−0.102486
$664$	9.83887e14	0.295815
$665$	1.12632e15	0.335849
$666$	−1.83616e14	−0.0543004
$667$	4.81564e14	0.141241
$668$	−2.98282e15	−0.867674
$669$	−2.10780e15	−0.608115
$670$	2.00510e15	0.573754
$671$	−1.82246e14	−0.0517232
$672$	7.74846e14	0.218115
$673$	−8.34965e14	−0.233123	−0.116562	−	0.993183i	$-0.537187\pi$
$673$	−8.34965e14	−0.233123	−0.116562	+	0.993183i	$0.537187\pi$
$674$	1.45037e15	0.401652
$675$	−1.64564e14	−0.0452027
$676$	−1.74299e14	−0.0474885
$677$	−3.08465e15	−0.833621	−0.416810	−	0.908994i	$-0.636852\pi$
$677$	−3.08465e15	−0.833621	−0.416810	+	0.908994i	$0.636852\pi$
$678$	−1.12363e15	−0.301203
$679$	−4.21585e14	−0.112099
$680$	2.69781e15	0.711561
$681$	−1.73876e15	−0.454915
$682$	4.39711e13	0.0114118
$683$	2.62617e13	0.00676097	0.00338048	−	0.999994i	$-0.498924\pi$
$683$	2.62617e13	0.00676097	0.00338048	+	0.999994i	$0.498924\pi$
$684$	6.44318e14	0.164548
$685$	−3.87141e14	−0.0980778
$686$	−1.32903e14	−0.0334005
$687$	−1.12734e15	−0.281056
$688$	7.32040e11	0.000181050 0
$689$	−7.82686e14	−0.192036
$690$	2.55463e14	0.0621811
$691$	7.29739e15	1.76213	0.881065	−	0.472994i	$-0.156827\pi$
$691$	7.29739e15	1.76213	0.881065	+	0.472994i	$0.156827\pi$
$692$	9.41224e14	0.225481
$693$	6.74175e13	0.0160228
$694$	−3.16045e14	−0.0745198
$695$	−8.02792e15	−1.87796
$696$	2.24362e15	0.520714
$697$	2.21466e15	0.509948
$698$	1.85173e15	0.423032
$699$	2.53801e15	0.575267
$700$	−2.43707e14	−0.0548062
$701$	−7.58115e15	−1.69155	−0.845777	−	0.533537i	$-0.820863\pi$
$701$	−7.58115e15	−1.69155	−0.845777	+	0.533537i	$0.820863\pi$
$702$	−1.49142e14	−0.0330177
$703$	−9.58647e14	−0.210574
$704$	3.61681e14	0.0788273
$705$	5.74074e15	1.24144
$706$	5.10836e14	0.109611
$707$	2.47130e15	0.526160
$708$	6.08963e13	0.0128649
$709$	−3.79628e15	−0.795800	−0.397900	−	0.917429i	$-0.630261\pi$
$709$	−3.79628e15	−0.795800	−0.397900	+	0.917429i	$0.630261\pi$
$710$	3.54884e15	0.738186
$711$	−6.95976e14	−0.143652
$712$	−7.79483e15	−1.59649
$713$	1.11826e14	0.0227274
$714$	−4.28377e14	−0.0863945
$715$	1.95855e14	0.0391968
$716$	1.37614e15	0.273301
$717$	8.20525e14	0.161710
$718$	1.96425e15	0.384160
$719$	2.14797e15	0.416887	0.208444	−	0.978034i	$-0.433160\pi$
$719$	2.14797e15	0.416887	0.208444	+	0.978034i	$0.433160\pi$
$720$	2.93844e12	0.000565963 0
$721$	2.36469e15	0.451992
$722$	1.17598e15	0.223072
$723$	5.66067e14	0.106563
$724$	−3.23122e15	−0.603677
$725$	−1.14199e15	−0.211739
$726$	−1.90946e15	−0.351365
$727$	3.74079e13	0.00683162	0.00341581	−	0.999994i	$-0.498913\pi$
$727$	3.74079e13	0.00683162	0.00341581	+	0.999994i	$0.498913\pi$
$728$	−5.78638e14	−0.104878
$729$	2.05891e14	0.0370370
$730$	6.47801e15	1.15655
$731$	4.27998e14	0.0758396
$732$	−8.24247e14	−0.144959
$733$	−4.72650e15	−0.825027	−0.412513	−	0.910952i	$-0.635349\pi$
$733$	−4.72650e15	−0.825027	−0.412513	+	0.910952i	$0.635349\pi$
$734$	−2.38731e15	−0.413601
$735$	5.33008e14	0.0916545
$736$	9.17547e14	0.156603
$737$	6.26605e14	0.106151
$738$	9.77058e14	0.164290
$739$	7.50006e15	1.25176	0.625879	−	0.779920i	$-0.284740\pi$
$739$	7.50006e15	1.25176	0.625879	+	0.779920i	$0.284740\pi$
$740$	1.09054e15	0.180662
$741$	−7.78662e14	−0.128041
$742$	−9.91806e14	−0.161884
$743$	6.96566e15	1.12856	0.564279	−	0.825584i	$-0.309154\pi$
$743$	6.96566e15	1.12856	0.564279	+	0.825584i	$0.309154\pi$
$744$	5.21001e14	0.0837890
$745$	−8.60701e15	−1.37402
$746$	5.91932e14	0.0938011
$747$	6.26553e14	0.0985587
$748$	3.21807e14	0.0502501
$749$	−2.32384e15	−0.360211
$750$	1.97342e15	0.303656
$751$	−2.43127e15	−0.371376	−0.185688	−	0.982609i	$-0.559451\pi$
$751$	−2.43127e15	−0.371376	−0.185688	+	0.982609i	$0.559451\pi$
$752$	−1.94971e13	−0.00295647
$753$	−2.95394e15	−0.444661
$754$	−1.03497e15	−0.154662
$755$	4.40917e15	0.654107
$756$	3.04909e14	0.0449056
$757$	−1.57249e15	−0.229911	−0.114956	−	0.993371i	$-0.536673\pi$
$757$	−1.57249e15	−0.229911	−0.114956	+	0.993371i	$0.536673\pi$
$758$	−3.02801e15	−0.439519
$759$	7.98335e13	0.0115042
$760$	6.21402e15	0.888993
$761$	−7.74938e15	−1.10066	−0.550328	−	0.834949i	$-0.685497\pi$
$761$	−7.74938e15	−1.10066	−0.550328	+	0.834949i	$0.685497\pi$
$762$	2.66442e15	0.375708
$763$	−1.68523e15	−0.235925
$764$	−6.55427e15	−0.910984
$765$	1.71800e15	0.237075
$766$	6.45718e15	0.884677
$767$	−7.35935e13	−0.0100107
$768$	4.27892e15	0.577894
$769$	−4.55118e15	−0.610280	−0.305140	−	0.952307i	$-0.598703\pi$
$769$	−4.55118e15	−0.610280	−0.305140	+	0.952307i	$0.598703\pi$
$770$	2.48183e14	0.0330426
$771$	7.91656e15	1.04650
$772$	−4.12501e15	−0.541414
$773$	−5.36624e15	−0.699331	−0.349666	−	0.936875i	$-0.613705\pi$
$773$	−5.36624e15	−0.699331	−0.349666	+	0.936875i	$0.613705\pi$
$774$	1.88823e14	0.0244332
$775$	−2.65185e14	−0.0340714
$776$	−2.32592e15	−0.296726
$777$	−4.53659e14	−0.0574664
$778$	1.44011e15	0.181137
$779$	5.10115e15	0.637106
$780$	8.85793e14	0.109853
$781$	1.10903e15	0.136572
$782$	−5.07270e14	−0.0620300
$783$	1.42877e15	0.173490
$784$	−1.81025e12	−0.000218273 0
$785$	−8.33618e15	−0.998123
$786$	−2.55120e15	−0.303334
$787$	−6.89130e15	−0.813655	−0.406827	−	0.913505i	$-0.633365\pi$
$787$	−6.89130e15	−0.813655	−0.406827	+	0.913505i	$0.633365\pi$
$788$	4.72765e15	0.554308
$789$	4.24522e15	0.494285
$790$	−2.56209e15	−0.296241
$791$	−2.77614e15	−0.318765
$792$	3.71948e14	0.0424125
$793$	9.96107e14	0.112799
$794$	−7.26132e15	−0.816587
$795$	3.97763e15	0.444227
$796$	9.99183e15	1.10821
$797$	1.78138e15	0.196217	0.0981084	−	0.995176i	$-0.468721\pi$
$797$	1.78138e15	0.196217	0.0981084	+	0.995176i	$0.468721\pi$
$798$	−9.86706e14	−0.107937
$799$	−1.13993e16	−1.23843
$800$	−2.17588e15	−0.234769
$801$	−4.96386e15	−0.531913
$802$	9.27023e15	0.986578
$803$	2.02441e15	0.213975
$804$	2.83395e15	0.297498
$805$	6.31171e14	0.0658066
$806$	−2.40334e14	−0.0248870
$807$	−6.28127e14	−0.0646016
$808$	1.36343e16	1.39275
$809$	−8.21287e15	−0.833255	−0.416628	−	0.909077i	$-0.636788\pi$
$809$	−8.21287e15	−0.833255	−0.416628	+	0.909077i	$0.636788\pi$
$810$	7.57946e14	0.0763784
$811$	−1.86004e16	−1.86169	−0.930847	−	0.365410i	$-0.880929\pi$
$811$	−1.86004e16	−1.86169	−0.930847	+	0.365410i	$0.880929\pi$
$812$	2.11591e15	0.210348
$813$	8.41979e14	0.0831387
$814$	−2.11236e14	−0.0207173
$815$	6.43679e14	0.0627051
$816$	−5.83482e12	−0.000564588 0
$817$	9.85833e14	0.0947507
$818$	−3.48000e14	−0.0332227
$819$	−3.68485e14	−0.0349428
$820$	−5.80298e15	−0.546607
$821$	−1.06487e16	−0.996347	−0.498173	−	0.867077i	$-0.665996\pi$
$821$	−1.06487e16	−0.996347	−0.498173	+	0.867077i	$0.665996\pi$
$822$	3.39151e14	0.0315209
$823$	1.53701e16	1.41899	0.709493	−	0.704713i	$-0.248924\pi$
$823$	1.53701e16	1.41899	0.709493	+	0.704713i	$0.248924\pi$
$824$	1.30462e16	1.19642
$825$	−1.89318e14	−0.0172463
$826$	−9.32564e13	−0.00843895
$827$	3.15024e14	0.0283180	0.0141590	−	0.999900i	$-0.495493\pi$
$827$	3.15024e14	0.0283180	0.0141590	+	0.999900i	$0.495493\pi$
$828$	3.61063e14	0.0322416
$829$	−1.21809e16	−1.08052	−0.540258	−	0.841499i	$-0.681673\pi$
$829$	−1.21809e16	−1.08052	−0.540258	+	0.841499i	$0.681673\pi$
$830$	2.30653e15	0.203249
$831$	−1.14219e15	−0.0999843
$832$	−1.97685e15	−0.171908
$833$	−1.05839e15	−0.0914317
$834$	7.03279e15	0.603551
$835$	−1.83195e16	−1.56184
$836$	7.41237e14	0.0627802
$837$	3.31781e14	0.0279165
$838$	4.75215e14	0.0397235
$839$	−1.07354e16	−0.891509	−0.445754	−	0.895155i	$-0.647065\pi$
$839$	−1.07354e16	−0.891509	−0.445754	+	0.895155i	$0.647065\pi$
$840$	2.94065e15	0.242609
$841$	−2.28560e15	−0.187337
$842$	−8.22559e15	−0.669808
$843$	−1.60861e15	−0.130137
$844$	2.52313e15	0.202794
$845$	−1.07049e15	−0.0854809
$846$	−5.02912e15	−0.398984
$847$	−4.71767e15	−0.371851
$848$	−1.35091e13	−0.00105791
$849$	−1.11134e16	−0.864682
$850$	1.20295e15	0.0929913
$851$	−5.37208e14	−0.0412600
$852$	5.01582e15	0.382758
$853$	−6.55116e15	−0.496705	−0.248352	−	0.968670i	$-0.579889\pi$
$853$	−6.55116e15	−0.496705	−0.248352	+	0.968670i	$0.579889\pi$
$854$	1.26225e15	0.0950882
$855$	3.95718e15	0.296191
$856$	−1.28208e16	−0.953477
$857$	−2.31638e16	−1.71165	−0.855824	−	0.517267i	$-0.826949\pi$
$857$	−2.31638e16	−1.71165	−0.855824	+	0.517267i	$0.826949\pi$
$858$	−1.71577e14	−0.0125973
$859$	−2.08082e15	−0.151800	−0.0759001	−	0.997115i	$-0.524183\pi$
$859$	−2.08082e15	−0.151800	−0.0759001	+	0.997115i	$0.524183\pi$
$860$	−1.12147e15	−0.0812916
$861$	2.41401e15	0.173869
$862$	3.09037e15	0.221167
$863$	−2.45246e16	−1.74399	−0.871993	−	0.489518i	$-0.837173\pi$
$863$	−2.45246e16	−1.74399	−0.871993	+	0.489518i	$0.837173\pi$
$864$	2.72231e15	0.192359
$865$	5.78068e15	0.405873
$866$	−1.11338e16	−0.776775
$867$	4.91666e15	0.340851
$868$	4.91343e14	0.0338474
$869$	−8.00666e14	−0.0548078
$870$	5.25973e15	0.357773
$871$	−3.42484e15	−0.231495
$872$	−9.29755e15	−0.624493
$873$	−1.48118e15	−0.0988619
$874$	−1.16842e15	−0.0774975
$875$	4.87570e15	0.321361
$876$	9.15581e15	0.599686
$877$	1.25450e16	0.816529	0.408265	−	0.912864i	$-0.366134\pi$
$877$	1.25450e16	0.816529	0.408265	+	0.912864i	$0.366134\pi$
$878$	−3.34857e15	−0.216591
$879$	−1.03715e16	−0.666658
$880$	3.38045e12	0.000215933 0
$881$	−1.92064e15	−0.121921	−0.0609605	−	0.998140i	$-0.519416\pi$
$881$	−1.92064e15	−0.121921	−0.0609605	+	0.998140i	$0.519416\pi$
$882$	−4.66937e14	−0.0294565
$883$	1.07504e16	0.673969	0.336985	−	0.941510i	$-0.390593\pi$
$883$	1.07504e16	0.673969	0.336985	+	0.941510i	$0.390593\pi$
$884$	−1.75891e15	−0.109586
$885$	3.74004e14	0.0231573
$886$	−2.79507e15	−0.171992
$887$	−1.44522e16	−0.883799	−0.441899	−	0.897065i	$-0.645695\pi$
$887$	−1.44522e16	−0.883799	−0.441899	+	0.897065i	$0.645695\pi$
$888$	−2.50287e15	−0.152113
$889$	6.58296e15	0.397614
$890$	−1.82734e16	−1.09692
$891$	2.36862e14	0.0141308
$892$	−1.09669e16	−0.650247
$893$	−2.62567e16	−1.54724
$894$	7.54009e15	0.441591
$895$	8.45178e15	0.491951
$896$	4.02537e15	0.232869
$897$	−4.36347e14	−0.0250885
$898$	1.73191e16	0.989704
$899$	2.30238e15	0.130767
$900$	−8.56231e14	−0.0483345
$901$	−7.89832e15	−0.443148
$902$	1.12403e15	0.0626818
$903$	4.66524e14	0.0258578
$904$	−1.53162e16	−0.843771
$905$	−1.98451e16	−1.08664
$906$	−3.86261e15	−0.210221
$907$	6.27309e15	0.339345	0.169672	−	0.985501i	$-0.445729\pi$
$907$	6.27309e15	0.339345	0.169672	+	0.985501i	$0.445729\pi$
$908$	−9.04680e15	−0.486433
$909$	8.68255e15	0.464030
$910$	−1.35650e15	−0.0720597
$911$	−9.36431e15	−0.494452	−0.247226	−	0.968958i	$-0.579519\pi$
$911$	−9.36431e15	−0.494452	−0.247226	+	0.968958i	$0.579519\pi$
$912$	−1.34397e13	−0.000705371 0
$913$	7.20801e14	0.0376033
$914$	−7.98127e15	−0.413874
$915$	−5.06224e15	−0.260932
$916$	−5.86556e15	−0.300528
$917$	−6.30324e15	−0.321020
$918$	−1.50504e15	−0.0761928
$919$	6.96370e15	0.350433	0.175216	−	0.984530i	$-0.443937\pi$
$919$	6.96370e15	0.350433	0.175216	+	0.984530i	$0.443937\pi$
$920$	3.48222e15	0.174190
$921$	5.20406e15	0.258770
$922$	1.33627e16	0.660504
$923$	−6.06164e15	−0.297839
$924$	3.50774e14	0.0171329
$925$	1.27394e15	0.0618543
$926$	−1.36951e16	−0.661007
$927$	8.30802e15	0.398620
$928$	1.88914e16	0.901051
$929$	−6.73919e15	−0.319537	−0.159769	−	0.987154i	$-0.551075\pi$
$929$	−6.73919e15	−0.319537	−0.159769	+	0.987154i	$0.551075\pi$
$930$	1.22138e15	0.0575699
$931$	−2.43785e15	−0.114231
$932$	1.32053e16	0.615122
$933$	−1.09140e16	−0.505402
$934$	−4.52471e15	−0.208297
$935$	1.97643e15	0.0904518
$936$	−2.03296e15	−0.0924936
$937$	−1.49538e16	−0.676370	−0.338185	−	0.941080i	$-0.609813\pi$
$937$	−1.49538e16	−0.676370	−0.338185	+	0.941080i	$0.609813\pi$
$938$	−4.33990e15	−0.195148
$939$	9.11074e15	0.407280
$940$	2.98692e16	1.32746
$941$	−3.10870e16	−1.37352	−0.686761	−	0.726883i	$-0.740968\pi$
$941$	−3.10870e16	−1.37352	−0.686761	+	0.726883i	$0.740968\pi$
$942$	7.30284e15	0.320783
$943$	2.85859e15	0.124835
$944$	−1.27022e12	−5.51485e−5 0
$945$	1.87265e15	0.0808317
$946$	2.17227e14	0.00932206
$947$	3.45691e15	0.147490	0.0737451	−	0.997277i	$-0.476505\pi$
$947$	3.45691e15	0.147490	0.0737451	+	0.997277i	$0.476505\pi$
$948$	−3.62118e15	−0.153605
$949$	−1.10648e16	−0.466639
$950$	2.77082e15	0.116179
$951$	1.34683e16	0.561460
$952$	−5.83921e15	−0.242020
$953$	−1.81821e16	−0.749262	−0.374631	−	0.927174i	$-0.622231\pi$
$953$	−1.81821e16	−0.749262	−0.374631	+	0.927174i	$0.622231\pi$
$954$	−3.48457e15	−0.142769
$955$	−4.02541e16	−1.63980
$956$	4.26921e15	0.172913
$957$	1.64369e15	0.0661919
$958$	−2.03449e16	−0.814602
$959$	8.37937e14	0.0333587
$960$	1.00464e16	0.397666
$961$	−2.48738e16	−0.978958
$962$	1.15456e15	0.0451806
$963$	−8.16448e15	−0.317676
$964$	2.94526e15	0.113946
$965$	−2.53344e16	−0.974564
$966$	−5.52932e14	−0.0211494
$967$	3.36513e16	1.27984	0.639920	−	0.768442i	$-0.278967\pi$
$967$	3.36513e16	1.27984	0.639920	+	0.768442i	$0.278967\pi$
$968$	−2.60278e16	−0.984290
$969$	−7.85771e15	−0.295471
$970$	−5.45265e15	−0.203875
$971$	6.64836e15	0.247177	0.123589	−	0.992334i	$-0.460560\pi$
$971$	6.64836e15	0.247177	0.123589	+	0.992334i	$0.460560\pi$
$972$	1.07126e15	0.0396031
$973$	1.73758e16	0.638742
$974$	−8.54602e15	−0.312384
$975$	1.03476e15	0.0376109
$976$	1.71928e13	0.000621402 0
$977$	3.31540e16	1.19156	0.595780	−	0.803148i	$-0.296843\pi$
$977$	3.31540e16	1.19156	0.595780	+	0.803148i	$0.296843\pi$
$978$	−5.63889e14	−0.0201526
$979$	−5.71054e15	−0.202942
$980$	2.77325e15	0.0980045
$981$	−5.92081e15	−0.208066
$982$	−1.47179e16	−0.514319
$983$	−2.80970e16	−0.976371	−0.488186	−	0.872740i	$-0.662341\pi$
$983$	−2.80970e16	−0.976371	−0.488186	+	0.872740i	$0.662341\pi$
$984$	1.33183e16	0.460230
$985$	2.90356e16	0.997773
$986$	−1.04442e16	−0.356904
$987$	−1.24254e16	−0.422246
$988$	−4.05139e15	−0.136912
$989$	5.52442e14	0.0185655
$990$	8.71957e14	0.0291408
$991$	1.80241e16	0.599031	0.299515	−	0.954092i	$-0.403175\pi$
$991$	1.80241e16	0.599031	0.299515	+	0.954092i	$0.403175\pi$
$992$	4.38684e15	0.144990
$993$	1.89997e16	0.624491
$994$	−7.68121e15	−0.251076
$995$	6.13664e16	1.99482
$996$	3.25997e15	0.105387
$997$	3.94574e16	1.26854	0.634272	−	0.773110i	$-0.281300\pi$
$997$	3.94574e16	1.26854	0.634272	+	0.773110i	$0.281300\pi$
$998$	−1.72689e16	−0.552138
$999$	−1.59387e15	−0.0506806

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.12.a.c.1.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.12.a.c.1.7	✓	16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	273.a (trivial)

\(f(q)\)	\(=\)	\(q-27.9940 q^{2} -243.000 q^{3} -1264.33 q^{4} -7765.11 q^{5} +6802.55 q^{6} +16807.0 q^{7} +92725.6 q^{8} +59049.0 q^{9} +O(q^{10})\)	\(q-27.9940 q^{2} -243.000 q^{3} -1264.33 q^{4} -7765.11 q^{5} +6802.55 q^{6} +16807.0 q^{7} +92725.6 q^{8} +59049.0 q^{9} +217377. q^{10} +67931.3 q^{11} +307233. q^{12} -371293. q^{13} -470496. q^{14} +1.88692e6 q^{15} -6408.51 q^{16} -3.74683e6 q^{17} -1.65302e6 q^{18} -8.63030e6 q^{19} +9.81769e6 q^{20} -4.08410e6 q^{21} -1.90167e6 q^{22} -4.83626e6 q^{23} -2.25323e7 q^{24} +1.14688e7 q^{25} +1.03940e7 q^{26} -1.43489e7 q^{27} -2.12497e7 q^{28} -9.95736e7 q^{29} -5.28225e7 q^{30} -2.31224e7 q^{31} -1.89723e8 q^{32} -1.65073e7 q^{33} +1.04889e8 q^{34} -1.30508e8 q^{35} -7.46576e7 q^{36} +1.11079e8 q^{37} +2.41597e8 q^{38} +9.02242e7 q^{39} -7.20024e8 q^{40} -5.91074e8 q^{41} +1.14330e8 q^{42} -1.14229e8 q^{43} -8.58878e7 q^{44} -4.58522e8 q^{45} +1.35386e8 q^{46} +3.04238e9 q^{47} +1.55727e6 q^{48} +2.82475e8 q^{49} -3.21057e8 q^{50} +9.10480e8 q^{51} +4.69438e8 q^{52} +2.10800e9 q^{53} +4.01684e8 q^{54} -5.27494e8 q^{55} +1.55844e9 q^{56} +2.09716e9 q^{57} +2.78747e9 q^{58} +1.98209e8 q^{59} -2.38570e9 q^{60} -2.68281e9 q^{61} +6.47289e8 q^{62} +9.92437e8 q^{63} +5.32423e9 q^{64} +2.88313e9 q^{65} +4.62106e8 q^{66} +9.22410e9 q^{67} +4.73725e9 q^{68} +1.17521e9 q^{69} +3.65345e9 q^{70} +1.63258e10 q^{71} +5.47535e9 q^{72} +2.98009e10 q^{73} -3.10956e9 q^{74} -2.78691e9 q^{75} +1.09116e10 q^{76} +1.14172e9 q^{77} -2.52574e9 q^{78} -1.17864e10 q^{79} +4.97628e7 q^{80} +3.48678e9 q^{81} +1.65466e10 q^{82} +1.06107e10 q^{83} +5.16367e9 q^{84} +2.90945e10 q^{85} +3.19774e9 q^{86} +2.41964e10 q^{87} +6.29897e9 q^{88} -8.40634e10 q^{89} +1.28359e10 q^{90} -6.24032e9 q^{91} +6.11464e9 q^{92} +5.61874e9 q^{93} -8.51686e10 q^{94} +6.70152e10 q^{95} +4.61026e10 q^{96} -2.50839e10 q^{97} -7.90762e9 q^{98} +4.01127e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 63 q^{2} - 3888 q^{3} + 20761 q^{4} - 3168 q^{5} + 15309 q^{6} + 268912 q^{7} - 187965 q^{8} + 944784 q^{9}+O(q^{10}) \) 16 * q - 63 * q^2 - 3888 * q^3 + 20761 * q^4 - 3168 * q^5 + 15309 * q^6 + 268912 * q^7 - 187965 * q^8 + 944784 * q^9	\( 16 q - 63 q^{2} - 3888 q^{3} + 20761 q^{4} - 3168 q^{5} + 15309 q^{6} + 268912 q^{7} - 187965 q^{8} + 944784 q^{9} + 1056711 q^{10} - 466884 q^{11} - 5044923 q^{12} - 5940688 q^{13} - 1058841 q^{14} + 769824 q^{15} + 40058265 q^{16} + 1753452 q^{17} - 3720087 q^{18} + 4237800 q^{19} - 20710503 q^{20} - 65345616 q^{21} - 37479661 q^{22} - 150481440 q^{23} + 45675495 q^{24} + 150983176 q^{25} + 23391459 q^{26} - 229582512 q^{27} + 348930127 q^{28} - 111411432 q^{29} - 256780773 q^{30} - 90536236 q^{31} - 609941313 q^{32} + 113452812 q^{33} + 443745577 q^{34} - 53244576 q^{35} + 1225916289 q^{36} - 1756337900 q^{37} - 4378868973 q^{38} + 1443587184 q^{39} + 57535383 q^{40} + 649575720 q^{41} + 257298363 q^{42} - 1889554520 q^{43} - 4979587689 q^{44} - 187067232 q^{45} - 3204000867 q^{46} - 4036082940 q^{47} - 9734158395 q^{48} + 4519603984 q^{49} - 15928886862 q^{50} - 426088836 q^{51} - 7708413973 q^{52} + 511144020 q^{53} + 903981141 q^{54} - 8519365572 q^{55} - 3159127755 q^{56} - 1029785400 q^{57} - 7851149577 q^{58} + 3728287296 q^{59} + 5032652229 q^{60} + 26227205052 q^{61} + 2996618490 q^{62} + 15878984688 q^{63} + 51104408465 q^{64} + 1176256224 q^{65} + 9107557623 q^{66} - 5295680024 q^{67} + 7919540325 q^{68} + 36566989920 q^{69} + 17760141777 q^{70} + 17082800928 q^{71} - 11099145285 q^{72} + 21076301488 q^{73} + 52794333675 q^{74} - 36688911768 q^{75} + 81459179177 q^{76} - 7846919388 q^{77} - 5684124537 q^{78} - 83677977852 q^{79} - 163988465427 q^{80} + 55788550416 q^{81} - 61543728760 q^{82} - 72857072340 q^{83} - 84790020861 q^{84} - 12608565060 q^{85} - 20177818011 q^{86} + 27072977976 q^{87} - 202213679643 q^{88} + 32238476676 q^{89} + 62397727839 q^{90} - 99845143216 q^{91} - 487057197033 q^{92} + 22000305348 q^{93} + 183904776602 q^{94} - 143022915504 q^{95} + 148215739059 q^{96} + 6973535140 q^{97} - 17795940687 q^{98} - 27569033316 q^{99}+O(q^{100}) \) 16 * q - 63 * q^2 - 3888 * q^3 + 20761 * q^4 - 3168 * q^5 + 15309 * q^6 + 268912 * q^7 - 187965 * q^8 + 944784 * q^9 + 1056711 * q^10 - 466884 * q^11 - 5044923 * q^12 - 5940688 * q^13 - 1058841 * q^14 + 769824 * q^15 + 40058265 * q^16 + 1753452 * q^17 - 3720087 * q^18 + 4237800 * q^19 - 20710503 * q^20 - 65345616 * q^21 - 37479661 * q^22 - 150481440 * q^23 + 45675495 * q^24 + 150983176 * q^25 + 23391459 * q^26 - 229582512 * q^27 + 348930127 * q^28 - 111411432 * q^29 - 256780773 * q^30 - 90536236 * q^31 - 609941313 * q^32 + 113452812 * q^33 + 443745577 * q^34 - 53244576 * q^35 + 1225916289 * q^36 - 1756337900 * q^37 - 4378868973 * q^38 + 1443587184 * q^39 + 57535383 * q^40 + 649575720 * q^41 + 257298363 * q^42 - 1889554520 * q^43 - 4979587689 * q^44 - 187067232 * q^45 - 3204000867 * q^46 - 4036082940 * q^47 - 9734158395 * q^48 + 4519603984 * q^49 - 15928886862 * q^50 - 426088836 * q^51 - 7708413973 * q^52 + 511144020 * q^53 + 903981141 * q^54 - 8519365572 * q^55 - 3159127755 * q^56 - 1029785400 * q^57 - 7851149577 * q^58 + 3728287296 * q^59 + 5032652229 * q^60 + 26227205052 * q^61 + 2996618490 * q^62 + 15878984688 * q^63 + 51104408465 * q^64 + 1176256224 * q^65 + 9107557623 * q^66 - 5295680024 * q^67 + 7919540325 * q^68 + 36566989920 * q^69 + 17760141777 * q^70 + 17082800928 * q^71 - 11099145285 * q^72 + 21076301488 * q^73 + 52794333675 * q^74 - 36688911768 * q^75 + 81459179177 * q^76 - 7846919388 * q^77 - 5684124537 * q^78 - 83677977852 * q^79 - 163988465427 * q^80 + 55788550416 * q^81 - 61543728760 * q^82 - 72857072340 * q^83 - 84790020861 * q^84 - 12608565060 * q^85 - 20177818011 * q^86 + 27072977976 * q^87 - 202213679643 * q^88 + 32238476676 * q^89 + 62397727839 * q^90 - 99845143216 * q^91 - 487057197033 * q^92 + 22000305348 * q^93 + 183904776602 * q^94 - 143022915504 * q^95 + 148215739059 * q^96 + 6973535140 * q^97 - 17795940687 * q^98 - 27569033316 * q^99

Introduction

L-functions

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