LMFDB - Embedded newform 4009.2.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$0$
Dimension:	$83$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.67158 q^{2} -3.12490 q^{3} +5.13731 q^{4} +1.64338 q^{5} +8.34840 q^{6} -0.166545 q^{7} -8.38157 q^{8} +6.76498 q^{9} +O(q^{10})$	\(q-2.67158 q^{2} -3.12490 q^{3} +5.13731 q^{4} +1.64338 q^{5} +8.34840 q^{6} -0.166545 q^{7} -8.38157 q^{8} +6.76498 q^{9} -4.39041 q^{10} +3.58072 q^{11} -16.0536 q^{12} -2.65893 q^{13} +0.444938 q^{14} -5.13539 q^{15} +12.1174 q^{16} -5.38022 q^{17} -18.0732 q^{18} -1.00000 q^{19} +8.44255 q^{20} +0.520437 q^{21} -9.56617 q^{22} -5.51103 q^{23} +26.1915 q^{24} -2.29931 q^{25} +7.10353 q^{26} -11.7652 q^{27} -0.855596 q^{28} +1.03127 q^{29} +13.7196 q^{30} -6.29172 q^{31} -15.6093 q^{32} -11.1894 q^{33} +14.3737 q^{34} -0.273697 q^{35} +34.7538 q^{36} -5.34837 q^{37} +2.67158 q^{38} +8.30888 q^{39} -13.7741 q^{40} -9.81313 q^{41} -1.39039 q^{42} +6.39265 q^{43} +18.3953 q^{44} +11.1174 q^{45} +14.7231 q^{46} -0.640569 q^{47} -37.8655 q^{48} -6.97226 q^{49} +6.14278 q^{50} +16.8126 q^{51} -13.6598 q^{52} +13.0990 q^{53} +31.4315 q^{54} +5.88448 q^{55} +1.39591 q^{56} +3.12490 q^{57} -2.75512 q^{58} -2.33086 q^{59} -26.3821 q^{60} -3.83664 q^{61} +16.8088 q^{62} -1.12668 q^{63} +17.4667 q^{64} -4.36963 q^{65} +29.8933 q^{66} +5.51744 q^{67} -27.6399 q^{68} +17.2214 q^{69} +0.731202 q^{70} +4.58568 q^{71} -56.7011 q^{72} +0.823429 q^{73} +14.2886 q^{74} +7.18510 q^{75} -5.13731 q^{76} -0.596352 q^{77} -22.1978 q^{78} -9.68159 q^{79} +19.9134 q^{80} +16.4700 q^{81} +26.2165 q^{82} +9.69150 q^{83} +2.67365 q^{84} -8.84174 q^{85} -17.0784 q^{86} -3.22262 q^{87} -30.0121 q^{88} +13.2518 q^{89} -29.7010 q^{90} +0.442832 q^{91} -28.3119 q^{92} +19.6610 q^{93} +1.71133 q^{94} -1.64338 q^{95} +48.7775 q^{96} -2.16213 q^{97} +18.6269 q^{98} +24.2235 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9}+O(q^{10}) $ 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9	\( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9} + 9 q^{10} + 56 q^{11} - 2 q^{12} - 5 q^{13} + 6 q^{14} + 19 q^{15} + 123 q^{16} + 19 q^{17} + 40 q^{18} - 83 q^{19} + 49 q^{20} + 9 q^{21} + 18 q^{22} + 74 q^{23} + 38 q^{24} + 98 q^{25} + 28 q^{26} + 6 q^{27} + 50 q^{28} + 16 q^{29} + 56 q^{30} + 24 q^{31} + 81 q^{32} + 13 q^{33} + 9 q^{34} + 71 q^{35} + 156 q^{36} - 6 q^{37} - 11 q^{38} + 126 q^{39} + q^{40} - q^{42} + 34 q^{43} + 140 q^{44} + 42 q^{45} + 34 q^{46} + 53 q^{47} + 16 q^{48} + 118 q^{49} + 51 q^{50} + 57 q^{51} + 32 q^{52} + q^{53} + 53 q^{54} + 60 q^{55} - 2 q^{56} - 2 q^{58} + 44 q^{59} - 9 q^{60} + 21 q^{61} + 28 q^{62} + 83 q^{63} + 154 q^{64} + 44 q^{65} + 17 q^{66} + 5 q^{67} + 63 q^{68} - 36 q^{69} - 48 q^{70} + 193 q^{71} + 135 q^{72} + 54 q^{73} + 127 q^{74} + 5 q^{75} - 95 q^{76} + 54 q^{77} + 45 q^{78} + 54 q^{79} + 45 q^{80} + 147 q^{81} - 35 q^{82} + 84 q^{83} + 12 q^{84} + 28 q^{85} + 60 q^{86} + 51 q^{87} + 23 q^{88} - 24 q^{89} + 31 q^{90} + 28 q^{91} + 108 q^{92} + 39 q^{93} - 49 q^{94} - 15 q^{95} + 25 q^{96} - 22 q^{97} - 67 q^{98} + 132 q^{99}+O(q^{100}) \) 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9 + 9 * q^10 + 56 * q^11 - 2 * q^12 - 5 * q^13 + 6 * q^14 + 19 * q^15 + 123 * q^16 + 19 * q^17 + 40 * q^18 - 83 * q^19 + 49 * q^20 + 9 * q^21 + 18 * q^22 + 74 * q^23 + 38 * q^24 + 98 * q^25 + 28 * q^26 + 6 * q^27 + 50 * q^28 + 16 * q^29 + 56 * q^30 + 24 * q^31 + 81 * q^32 + 13 * q^33 + 9 * q^34 + 71 * q^35 + 156 * q^36 - 6 * q^37 - 11 * q^38 + 126 * q^39 + q^40 - q^42 + 34 * q^43 + 140 * q^44 + 42 * q^45 + 34 * q^46 + 53 * q^47 + 16 * q^48 + 118 * q^49 + 51 * q^50 + 57 * q^51 + 32 * q^52 + q^53 + 53 * q^54 + 60 * q^55 - 2 * q^56 - 2 * q^58 + 44 * q^59 - 9 * q^60 + 21 * q^61 + 28 * q^62 + 83 * q^63 + 154 * q^64 + 44 * q^65 + 17 * q^66 + 5 * q^67 + 63 * q^68 - 36 * q^69 - 48 * q^70 + 193 * q^71 + 135 * q^72 + 54 * q^73 + 127 * q^74 + 5 * q^75 - 95 * q^76 + 54 * q^77 + 45 * q^78 + 54 * q^79 + 45 * q^80 + 147 * q^81 - 35 * q^82 + 84 * q^83 + 12 * q^84 + 28 * q^85 + 60 * q^86 + 51 * q^87 + 23 * q^88 - 24 * q^89 + 31 * q^90 + 28 * q^91 + 108 * q^92 + 39 * q^93 - 49 * q^94 - 15 * q^95 + 25 * q^96 - 22 * q^97 - 67 * q^98 + 132 * 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 $

$2$	−2.67158	−1.88909	−0.944544	−	0.328384i	$-0.893496\pi$
$2$	−2.67158	−1.88909	−0.944544	+	0.328384i	$0.893496\pi$
$3$	−3.12490	−1.80416	−0.902080	−	0.431569i	$-0.857960\pi$
$3$	−3.12490	−1.80416	−0.902080	+	0.431569i	$0.857960\pi$
$4$	5.13731	2.56866
$5$	1.64338	0.734941	0.367470	−	0.930035i	$-0.380224\pi$
$5$	1.64338	0.734941	0.367470	+	0.930035i	$0.380224\pi$
$6$	8.34840	3.40822
$7$	−0.166545	−0.0629482	−0.0314741	−	0.999505i	$-0.510020\pi$
$7$	−0.166545	−0.0629482	−0.0314741	+	0.999505i	$0.510020\pi$
$8$	−8.38157	−2.96333
$9$	6.76498	2.25499
$10$	−4.39041	−1.38837
$11$	3.58072	1.07963	0.539814	−	0.841784i	$-0.318495\pi$
$11$	3.58072	1.07963	0.539814	+	0.841784i	$0.318495\pi$
$12$	−16.0536	−4.63427
$13$	−2.65893	−0.737454	−0.368727	−	0.929538i	$-0.620206\pi$
$13$	−2.65893	−0.737454	−0.368727	+	0.929538i	$0.620206\pi$
$14$	0.444938	0.118915
$15$	−5.13539	−1.32595
$16$	12.1174	3.02934
$17$	−5.38022	−1.30490	−0.652448	−	0.757834i	$-0.726258\pi$
$17$	−5.38022	−1.30490	−0.652448	+	0.757834i	$0.726258\pi$
$18$	−18.0732	−4.25988
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	8.44255	1.88781
$21$	0.520437	0.113569
$22$	−9.56617	−2.03951
$23$	−5.51103	−1.14913	−0.574564	−	0.818459i	$-0.694829\pi$
$23$	−5.51103	−1.14913	−0.574564	+	0.818459i	$0.694829\pi$
$24$	26.1915	5.34633
$25$	−2.29931	−0.459862
$26$	7.10353	1.39312
$27$	−11.7652	−2.26421
$28$	−0.855596	−0.161692
$29$	1.03127	0.191502	0.0957512	−	0.995405i	$-0.469475\pi$
$29$	1.03127	0.191502	0.0957512	+	0.995405i	$0.469475\pi$
$30$	13.7196	2.50484
$31$	−6.29172	−1.13003	−0.565013	−	0.825082i	$-0.691129\pi$
$31$	−6.29172	−1.13003	−0.565013	+	0.825082i	$0.691129\pi$
$32$	−15.6093	−2.75936
$33$	−11.1894	−1.94782
$34$	14.3737	2.46506
$35$	−0.273697	−0.0462632
$36$	34.7538	5.79230
$37$	−5.34837	−0.879267	−0.439633	−	0.898177i	$-0.644892\pi$
$37$	−5.34837	−0.879267	−0.439633	+	0.898177i	$0.644892\pi$
$38$	2.67158	0.433387
$39$	8.30888	1.33049
$40$	−13.7741	−2.17787
$41$	−9.81313	−1.53255	−0.766277	−	0.642510i	$-0.777893\pi$
$41$	−9.81313	−1.53255	−0.766277	+	0.642510i	$0.777893\pi$
$42$	−1.39039	−0.214541
$43$	6.39265	0.974869	0.487435	−	0.873159i	$-0.337933\pi$
$43$	6.39265	0.974869	0.487435	+	0.873159i	$0.337933\pi$
$44$	18.3953	2.77319
$45$	11.1174	1.65729
$46$	14.7231	2.17081
$47$	−0.640569	−0.0934366	−0.0467183	−	0.998908i	$-0.514876\pi$
$47$	−0.640569	−0.0934366	−0.0467183	+	0.998908i	$0.514876\pi$
$48$	−37.8655	−5.46542
$49$	−6.97226	−0.996038
$50$	6.14278	0.868720
$51$	16.8126	2.35424
$52$	−13.6598	−1.89427
$53$	13.0990	1.79929	0.899646	−	0.436619i	$-0.143824\pi$
$53$	13.0990	1.79929	0.899646	+	0.436619i	$0.143824\pi$
$54$	31.4315	4.27729
$55$	5.88448	0.793463
$56$	1.39591	0.186536
$57$	3.12490	0.413903
$58$	−2.75512	−0.361765
$59$	−2.33086	−0.303452	−0.151726	−	0.988423i	$-0.548483\pi$
$59$	−2.33086	−0.303452	−0.151726	+	0.988423i	$0.548483\pi$
$60$	−26.3821	−3.40591
$61$	−3.83664	−0.491232	−0.245616	−	0.969367i	$-0.578990\pi$
$61$	−3.83664	−0.491232	−0.245616	+	0.969367i	$0.578990\pi$
$62$	16.8088	2.13472
$63$	−1.12668	−0.141948
$64$	17.4667	2.18334
$65$	−4.36963	−0.541985
$66$	29.8933	3.67961
$67$	5.51744	0.674063	0.337031	−	0.941493i	$-0.390577\pi$
$67$	5.51744	0.674063	0.337031	+	0.941493i	$0.390577\pi$
$68$	−27.6399	−3.35183
$69$	17.2214	2.07321
$70$	0.731202	0.0873953
$71$	4.58568	0.544220	0.272110	−	0.962266i	$-0.412279\pi$
$71$	4.58568	0.544220	0.272110	+	0.962266i	$0.412279\pi$
$72$	−56.7011	−6.68229
$73$	0.823429	0.0963751	0.0481876	−	0.998838i	$-0.484655\pi$
$73$	0.823429	0.0963751	0.0481876	+	0.998838i	$0.484655\pi$
$74$	14.2886	1.66101
$75$	7.18510	0.829664
$76$	−5.13731	−0.589290
$77$	−0.596352	−0.0679607
$78$	−22.1978	−2.51341
$79$	−9.68159	−1.08926	−0.544632	−	0.838675i	$-0.683331\pi$
$79$	−9.68159	−1.08926	−0.544632	+	0.838675i	$0.683331\pi$
$80$	19.9134	2.22639
$81$	16.4700	1.83000
$82$	26.2165	2.89513
$83$	9.69150	1.06378	0.531890	−	0.846813i	$-0.321482\pi$
$83$	9.69150	1.06378	0.531890	+	0.846813i	$0.321482\pi$
$84$	2.67365	0.291719
$85$	−8.84174	−0.959021
$86$	−17.0784	−1.84161
$87$	−3.22262	−0.345501
$88$	−30.0121	−3.19930
$89$	13.2518	1.40469	0.702346	−	0.711835i	$-0.252136\pi$
$89$	13.2518	1.40469	0.702346	+	0.711835i	$0.252136\pi$
$90$	−29.7010	−3.13076
$91$	0.442832	0.0464214
$92$	−28.3119	−2.95172
$93$	19.6610	2.03875
$94$	1.71133	0.176510
$95$	−1.64338	−0.168607
$96$	48.7775	4.97833
$97$	−2.16213	−0.219531	−0.109766	−	0.993957i	$-0.535010\pi$
$97$	−2.16213	−0.219531	−0.109766	+	0.993957i	$0.535010\pi$
$98$	18.6269	1.88160
$99$	24.2235	2.43455
$100$	−11.8123	−1.18123
$101$	9.36880	0.932230	0.466115	−	0.884724i	$-0.345653\pi$
$101$	9.36880	0.932230	0.466115	+	0.884724i	$0.345653\pi$
$102$	−44.9162	−4.44737
$103$	8.48833	0.836380	0.418190	−	0.908360i	$-0.362665\pi$
$103$	8.48833	0.836380	0.418190	+	0.908360i	$0.362665\pi$
$104$	22.2860	2.18532
$105$	0.855274	0.0834662
$106$	−34.9951	−3.39902
$107$	−5.60601	−0.541953	−0.270977	−	0.962586i	$-0.587347\pi$
$107$	−5.60601	−0.541953	−0.270977	+	0.962586i	$0.587347\pi$
$108$	−60.4414	−5.81597
$109$	12.7494	1.22117	0.610587	−	0.791949i	$-0.290934\pi$
$109$	12.7494	1.22117	0.610587	+	0.791949i	$0.290934\pi$
$110$	−15.7208	−1.49892
$111$	16.7131	1.58634
$112$	−2.01809	−0.190692
$113$	−11.8938	−1.11887	−0.559437	−	0.828873i	$-0.688982\pi$
$113$	−11.8938	−1.11887	−0.559437	+	0.828873i	$0.688982\pi$
$114$	−8.34840	−0.781899
$115$	−9.05670	−0.844542
$116$	5.29797	0.491904
$117$	−17.9876	−1.66295
$118$	6.22706	0.573247
$119$	0.896051	0.0821408
$120$	43.0426	3.92923
$121$	1.82157	0.165597
$122$	10.2499	0.927980
$123$	30.6650	2.76497
$124$	−32.3225	−2.90265
$125$	−11.9955	−1.07291
$126$	3.01000	0.268152
$127$	6.08771	0.540197	0.270098	−	0.962833i	$-0.412944\pi$
$127$	6.08771	0.540197	0.270098	+	0.962833i	$0.412944\pi$
$128$	−15.4450	−1.36516
$129$	−19.9764	−1.75882
$130$	11.6738	1.02386
$131$	7.99019	0.698106	0.349053	−	0.937103i	$-0.386503\pi$
$131$	7.99019	0.698106	0.349053	+	0.937103i	$0.386503\pi$
$132$	−57.4834	−5.00329
$133$	0.166545	0.0144413
$134$	−14.7403	−1.27336
$135$	−19.3346	−1.66406
$136$	45.0947	3.86684
$137$	−13.7213	−1.17229	−0.586144	−	0.810207i	$-0.699355\pi$
$137$	−13.7213	−1.17229	−0.586144	+	0.810207i	$0.699355\pi$
$138$	−46.0082	−3.91648
$139$	7.63640	0.647711	0.323855	−	0.946107i	$-0.395021\pi$
$139$	7.63640	0.647711	0.323855	+	0.946107i	$0.395021\pi$
$140$	−1.40607	−0.118834
$141$	2.00171	0.168575
$142$	−12.2510	−1.02808
$143$	−9.52089	−0.796176
$144$	81.9737	6.83114
$145$	1.69477	0.140743
$146$	−2.19985	−0.182061
$147$	21.7876	1.79701
$148$	−27.4763	−2.25853
$149$	6.03014	0.494008	0.247004	−	0.969014i	$-0.420554\pi$
$149$	6.03014	0.494008	0.247004	+	0.969014i	$0.420554\pi$
$150$	−19.1955	−1.56731
$151$	−22.1234	−1.80037	−0.900187	−	0.435503i	$-0.856570\pi$
$151$	−22.1234	−1.80037	−0.900187	+	0.435503i	$0.856570\pi$
$152$	8.38157	0.679835
$153$	−36.3971	−2.94253
$154$	1.59320	0.128384
$155$	−10.3397	−0.830502
$156$	42.6853	3.41756
$157$	−21.5678	−1.72130	−0.860650	−	0.509197i	$-0.829942\pi$
$157$	−21.5678	−1.72130	−0.860650	+	0.509197i	$0.829942\pi$
$158$	25.8651	2.05772
$159$	−40.9332	−3.24621
$160$	−25.6520	−2.02797
$161$	0.917836	0.0723356
$162$	−44.0009	−3.45703
$163$	−7.27368	−0.569718	−0.284859	−	0.958569i	$-0.591947\pi$
$163$	−7.27368	−0.569718	−0.284859	+	0.958569i	$0.591947\pi$
$164$	−50.4131	−3.93661
$165$	−18.3884	−1.43153
$166$	−25.8916	−2.00958
$167$	19.6431	1.52003	0.760016	−	0.649904i	$-0.225191\pi$
$167$	19.6431	1.52003	0.760016	+	0.649904i	$0.225191\pi$
$168$	−4.36208	−0.336542
$169$	−5.93009	−0.456161
$170$	23.6214	1.81168
$171$	−6.76498	−0.517331
$172$	32.8410	2.50410
$173$	5.44330	0.413847	0.206923	−	0.978357i	$-0.433655\pi$
$173$	5.44330	0.413847	0.206923	+	0.978357i	$0.433655\pi$
$174$	8.60947	0.652682
$175$	0.382939	0.0289475
$176$	43.3889	3.27056
$177$	7.28369	0.547476
$178$	−35.4033	−2.65359
$179$	3.82251	0.285708	0.142854	−	0.989744i	$-0.454372\pi$
$179$	3.82251	0.285708	0.142854	+	0.989744i	$0.454372\pi$
$180$	57.1137	4.25700
$181$	13.0807	0.972283	0.486141	−	0.873880i	$-0.338404\pi$
$181$	13.0807	0.972283	0.486141	+	0.873880i	$0.338404\pi$
$182$	−1.18306	−0.0876942
$183$	11.9891	0.886260
$184$	46.1911	3.40525
$185$	−8.78939	−0.646209
$186$	−52.5257	−3.85137
$187$	−19.2651	−1.40880
$188$	−3.29080	−0.240007
$189$	1.95943	0.142528
$190$	4.39041	0.318514
$191$	12.9673	0.938280	0.469140	−	0.883124i	$-0.344564\pi$
$191$	12.9673	0.938280	0.469140	+	0.883124i	$0.344564\pi$
$192$	−54.5817	−3.93910
$193$	3.01410	0.216960	0.108480	−	0.994099i	$-0.465402\pi$
$193$	3.01410	0.216960	0.108480	+	0.994099i	$0.465402\pi$
$194$	5.77630	0.414714
$195$	13.6546	0.977828
$196$	−35.8187	−2.55848
$197$	−12.5510	−0.894224	−0.447112	−	0.894478i	$-0.647547\pi$
$197$	−12.5510	−0.894224	−0.447112	+	0.894478i	$0.647547\pi$
$198$	−64.7149	−4.59909
$199$	−8.05425	−0.570951	−0.285475	−	0.958386i	$-0.592152\pi$
$199$	−8.05425	−0.570951	−0.285475	+	0.958386i	$0.592152\pi$
$200$	19.2718	1.36272
$201$	−17.2414	−1.21612
$202$	−25.0294	−1.76107
$203$	−0.171754	−0.0120547
$204$	86.3718	6.04724
$205$	−16.1267	−1.12634
$206$	−22.6772	−1.58000
$207$	−37.2820	−2.59128
$208$	−32.2192	−2.23400
$209$	−3.58072	−0.247684
$210$	−2.28493	−0.157675
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	67.2939	4.62177
$213$	−14.3298	−0.981860
$214$	14.9769	1.02380
$215$	10.5055	0.716471
$216$	98.6106	6.70960
$217$	1.04786	0.0711331
$218$	−34.0611	−2.30691
$219$	−2.57313	−0.173876
$220$	30.2304	2.03813
$221$	14.3056	0.962301
$222$	−44.6503	−2.99673
$223$	6.72303	0.450207	0.225104	−	0.974335i	$-0.427728\pi$
$223$	6.72303	0.450207	0.225104	+	0.974335i	$0.427728\pi$
$224$	2.59966	0.173697
$225$	−15.5548	−1.03699
$226$	31.7751	2.11365
$227$	29.6815	1.97003	0.985016	−	0.172464i	$-0.0551729\pi$
$227$	29.6815	1.97003	0.985016	+	0.172464i	$0.0551729\pi$
$228$	16.0536	1.06317
$229$	9.42401	0.622756	0.311378	−	0.950286i	$-0.399209\pi$
$229$	9.42401	0.622756	0.311378	+	0.950286i	$0.399209\pi$
$230$	24.1957	1.59541
$231$	1.86354	0.122612
$232$	−8.64368	−0.567486
$233$	25.2666	1.65527	0.827636	−	0.561265i	$-0.189685\pi$
$233$	25.2666	1.65527	0.827636	+	0.561265i	$0.189685\pi$
$234$	48.0552	3.14147
$235$	−1.05270	−0.0686704
$236$	−11.9743	−0.779464
$237$	30.2540	1.96521
$238$	−2.39387	−0.155171
$239$	4.51624	0.292131	0.146066	−	0.989275i	$-0.453339\pi$
$239$	4.51624	0.292131	0.146066	+	0.989275i	$0.453339\pi$
$240$	−62.2274	−4.01676
$241$	−23.5491	−1.51693	−0.758464	−	0.651715i	$-0.774050\pi$
$241$	−23.5491	−1.51693	−0.758464	+	0.651715i	$0.774050\pi$
$242$	−4.86645	−0.312827
$243$	−16.1716	−1.03741
$244$	−19.7100	−1.26181
$245$	−11.4581	−0.732029
$246$	−81.9239	−5.22328
$247$	2.65893	0.169184
$248$	52.7345	3.34864
$249$	−30.2849	−1.91923
$250$	32.0469	2.02683
$251$	−30.0258	−1.89521	−0.947606	−	0.319441i	$-0.896505\pi$
$251$	−30.0258	−1.89521	−0.947606	+	0.319441i	$0.896505\pi$
$252$	−5.78809	−0.364615
$253$	−19.7335	−1.24063
$254$	−16.2638	−1.02048
$255$	27.6295	1.73023
$256$	6.32913	0.395571
$257$	16.5942	1.03512	0.517558	−	0.855648i	$-0.326841\pi$
$257$	16.5942	1.03512	0.517558	+	0.855648i	$0.326841\pi$
$258$	53.3683	3.32257
$259$	0.890746	0.0553483
$260$	−22.4481	−1.39217
$261$	6.97654	0.431837
$262$	−21.3464	−1.31879
$263$	23.7595	1.46507	0.732536	−	0.680728i	$-0.238336\pi$
$263$	23.7595	1.46507	0.732536	+	0.680728i	$0.238336\pi$
$264$	93.7846	5.77204
$265$	21.5267	1.32237
$266$	−0.444938	−0.0272809
$267$	−41.4107	−2.53429
$268$	28.3448	1.73144
$269$	−12.6856	−0.773452	−0.386726	−	0.922195i	$-0.626394\pi$
$269$	−12.6856	−0.773452	−0.386726	+	0.922195i	$0.626394\pi$
$270$	51.6539	3.14356
$271$	3.85154	0.233964	0.116982	−	0.993134i	$-0.462678\pi$
$271$	3.85154	0.233964	0.116982	+	0.993134i	$0.462678\pi$
$272$	−65.1941	−3.95297
$273$	−1.38380	−0.0837517
$274$	36.6574	2.21456
$275$	−8.23318	−0.496480
$276$	88.4717	5.32537
$277$	11.5596	0.694547	0.347274	−	0.937764i	$-0.387107\pi$
$277$	11.5596	0.694547	0.347274	+	0.937764i	$0.387107\pi$
$278$	−20.4012	−1.22358
$279$	−42.5633	−2.54820
$280$	2.29401	0.137093
$281$	−3.15560	−0.188247	−0.0941237	−	0.995561i	$-0.530005\pi$
$281$	−3.15560	−0.188247	−0.0941237	+	0.995561i	$0.530005\pi$
$282$	−5.34772	−0.318452
$283$	10.2909	0.611731	0.305865	−	0.952075i	$-0.401054\pi$
$283$	10.2909	0.611731	0.305865	+	0.952075i	$0.401054\pi$
$284$	23.5581	1.39791
$285$	5.13539	0.304194
$286$	25.4358	1.50405
$287$	1.63433	0.0964715
$288$	−105.597	−6.22234
$289$	11.9468	0.702753
$290$	−4.52771	−0.265876
$291$	6.75644	0.396070
$292$	4.23021	0.247555
$293$	−17.9678	−1.04969	−0.524846	−	0.851197i	$-0.675877\pi$
$293$	−17.9678	−1.04969	−0.524846	+	0.851197i	$0.675877\pi$
$294$	−58.2072	−3.39471
$295$	−3.83048	−0.223019
$296$	44.8277	2.60556
$297$	−42.1278	−2.44450
$298$	−16.1100	−0.933225
$299$	14.6534	0.847430
$300$	36.9121	2.13112
$301$	−1.06467	−0.0613663
$302$	59.1042	3.40107
$303$	−29.2765	−1.68189
$304$	−12.1174	−0.694979
$305$	−6.30505	−0.361026
$306$	97.2376	5.55870
$307$	28.7590	1.64136	0.820681	−	0.571387i	$-0.193594\pi$
$307$	28.7590	1.64136	0.820681	+	0.571387i	$0.193594\pi$
$308$	−3.06365	−0.174568
$309$	−26.5252	−1.50896
$310$	27.6232	1.56889
$311$	−11.0091	−0.624267	−0.312133	−	0.950038i	$-0.601044\pi$
$311$	−11.0091	−0.624267	−0.312133	+	0.950038i	$0.601044\pi$
$312$	−69.6415	−3.94267
$313$	14.0054	0.791630	0.395815	−	0.918330i	$-0.370462\pi$
$313$	14.0054	0.791630	0.395815	+	0.918330i	$0.370462\pi$
$314$	57.6201	3.25169
$315$	−1.85155	−0.104323
$316$	−49.7374	−2.79795
$317$	22.5695	1.26763	0.633814	−	0.773486i	$-0.281489\pi$
$317$	22.5695	1.26763	0.633814	+	0.773486i	$0.281489\pi$
$318$	109.356	6.13238
$319$	3.69270	0.206751
$320$	28.7044	1.60463
$321$	17.5182	0.977771
$322$	−2.45207	−0.136648
$323$	5.38022	0.299364
$324$	84.6116	4.70064
$325$	6.11370	0.339127
$326$	19.4322	1.07625
$327$	−39.8406	−2.20319
$328$	82.2495	4.54147
$329$	0.106684	0.00588167
$330$	49.1260	2.70430
$331$	19.3140	1.06159	0.530797	−	0.847499i	$-0.321893\pi$
$331$	19.3140	1.06159	0.530797	+	0.847499i	$0.321893\pi$
$332$	49.7883	2.73249
$333$	−36.1816	−1.98274
$334$	−52.4781	−2.87148
$335$	9.06724	0.495396
$336$	6.30632	0.344038
$337$	5.81625	0.316831	0.158416	−	0.987373i	$-0.449361\pi$
$337$	5.81625	0.316831	0.158416	+	0.987373i	$0.449361\pi$
$338$	15.8427	0.861729
$339$	37.1669	2.01863
$340$	−45.4228	−2.46340
$341$	−22.5289	−1.22001
$342$	18.0732	0.977284
$343$	2.32701	0.125647
$344$	−53.5804	−2.88886
$345$	28.3013	1.52369
$346$	−14.5422	−0.781793
$347$	15.9540	0.856455	0.428228	−	0.903671i	$-0.359138\pi$
$347$	15.9540	0.856455	0.428228	+	0.903671i	$0.359138\pi$
$348$	−16.5556	−0.887474
$349$	−35.3986	−1.89484	−0.947421	−	0.319990i	$-0.896320\pi$
$349$	−35.3986	−1.89484	−0.947421	+	0.319990i	$0.896320\pi$
$350$	−1.02305	−0.0546844
$351$	31.2828	1.66975
$352$	−55.8926	−2.97909
$353$	17.7615	0.945351	0.472676	−	0.881237i	$-0.343288\pi$
$353$	17.7615	0.945351	0.472676	+	0.881237i	$0.343288\pi$
$354$	−19.4589	−1.03423
$355$	7.53600	0.399970
$356$	68.0789	3.60817
$357$	−2.80007	−0.148195
$358$	−10.2121	−0.539727
$359$	35.0335	1.84900	0.924498	−	0.381186i	$-0.124484\pi$
$359$	35.0335	1.84900	0.924498	+	0.381186i	$0.124484\pi$
$360$	−93.1814	−4.91109
$361$	1.00000	0.0526316
$362$	−34.9461	−1.83673
$363$	−5.69221	−0.298763
$364$	2.27497	0.119241
$365$	1.35321	0.0708300
$366$	−32.0298	−1.67422
$367$	26.2041	1.36784	0.683921	−	0.729556i	$-0.260273\pi$
$367$	26.2041	1.36784	0.683921	+	0.729556i	$0.260273\pi$
$368$	−66.7791	−3.48110
$369$	−66.3856	−3.45590
$370$	23.4815	1.22075
$371$	−2.18159	−0.113262
$372$	101.005	5.23684
$373$	−32.9958	−1.70846	−0.854229	−	0.519897i	$-0.825970\pi$
$373$	−32.9958	−1.70846	−0.854229	+	0.519897i	$0.825970\pi$
$374$	51.4681	2.66135
$375$	37.4848	1.93571
$376$	5.36897	0.276884
$377$	−2.74208	−0.141224
$378$	−5.23478	−0.269248
$379$	−4.98391	−0.256006	−0.128003	−	0.991774i	$-0.540857\pi$
$379$	−4.98391	−0.256006	−0.128003	+	0.991774i	$0.540857\pi$
$380$	−8.44255	−0.433094
$381$	−19.0235	−0.974601
$382$	−34.6431	−1.77249
$383$	−31.4193	−1.60545	−0.802727	−	0.596347i	$-0.796618\pi$
$383$	−31.4193	−1.60545	−0.802727	+	0.596347i	$0.796618\pi$
$384$	48.2642	2.46297
$385$	−0.980032	−0.0499471
$386$	−8.05239	−0.409856
$387$	43.2461	2.19832
$388$	−11.1076	−0.563901
$389$	−12.4386	−0.630663	−0.315332	−	0.948982i	$-0.602116\pi$
$389$	−12.4386	−0.630663	−0.315332	+	0.948982i	$0.602116\pi$
$390$	−36.4794	−1.84720
$391$	29.6506	1.49949
$392$	58.4385	2.95159
$393$	−24.9685	−1.25950
$394$	33.5310	1.68927
$395$	−15.9105	−0.800545
$396$	124.444	6.25353
$397$	−20.6151	−1.03464	−0.517322	−	0.855791i	$-0.673071\pi$
$397$	−20.6151	−1.03464	−0.517322	+	0.855791i	$0.673071\pi$
$398$	21.5175	1.07858
$399$	−0.520437	−0.0260544
$400$	−27.8616	−1.39308
$401$	14.2491	0.711565	0.355783	−	0.934569i	$-0.384214\pi$
$401$	14.2491	0.711565	0.355783	+	0.934569i	$0.384214\pi$
$402$	46.0618	2.29735
$403$	16.7292	0.833342
$404$	48.1304	2.39458
$405$	27.0664	1.34494
$406$	0.458853	0.0227725
$407$	−19.1510	−0.949281
$408$	−140.916	−6.97640
$409$	3.62202	0.179098	0.0895488	−	0.995982i	$-0.471458\pi$
$409$	3.62202	0.179098	0.0895488	+	0.995982i	$0.471458\pi$
$410$	43.0837	2.12775
$411$	42.8776	2.11499
$412$	43.6072	2.14837
$413$	0.388193	0.0191017
$414$	99.6016	4.89515
$415$	15.9268	0.781816
$416$	41.5041	2.03490
$417$	−23.8629	−1.16857
$418$	9.56617	0.467897
$419$	4.04751	0.197734	0.0988668	−	0.995101i	$-0.468478\pi$
$419$	4.04751	0.197734	0.0988668	+	0.995101i	$0.468478\pi$
$420$	4.39381	0.214396
$421$	13.7355	0.669429	0.334715	−	0.942320i	$-0.391360\pi$
$421$	13.7355	0.669429	0.334715	+	0.942320i	$0.391360\pi$
$422$	−2.67158	−0.130050
$423$	−4.33344	−0.210699
$424$	−109.791	−5.33190
$425$	12.3708	0.600072
$426$	38.2831	1.85482
$427$	0.638975	0.0309222
$428$	−28.7998	−1.39209
$429$	29.7518	1.43643
$430$	−28.0663	−1.35348
$431$	37.7988	1.82070	0.910352	−	0.413835i	$-0.135811\pi$
$431$	37.7988	1.82070	0.910352	+	0.413835i	$0.135811\pi$
$432$	−142.563	−6.85906
$433$	−31.8963	−1.53284	−0.766419	−	0.642341i	$-0.777963\pi$
$433$	−31.8963	−1.53284	−0.766419	+	0.642341i	$0.777963\pi$
$434$	−2.79943	−0.134377
$435$	−5.29598	−0.253923
$436$	65.4978	3.13678
$437$	5.51103	0.263628
$438$	6.87431	0.328467
$439$	−4.26445	−0.203531	−0.101765	−	0.994808i	$-0.532449\pi$
$439$	−4.26445	−0.203531	−0.101765	+	0.994808i	$0.532449\pi$
$440$	−49.3212	−2.35129
$441$	−47.1672	−2.24606
$442$	−38.2186	−1.81787
$443$	−14.8472	−0.705413	−0.352706	−	0.935734i	$-0.614739\pi$
$443$	−14.8472	−0.705413	−0.352706	+	0.935734i	$0.614739\pi$
$444$	85.8605	4.07476
$445$	21.7778	1.03237
$446$	−17.9611	−0.850481
$447$	−18.8435	−0.891269
$448$	−2.90900	−0.137437
$449$	33.7441	1.59248	0.796240	−	0.604980i	$-0.206819\pi$
$449$	33.7441	1.59248	0.796240	+	0.604980i	$0.206819\pi$
$450$	41.5558	1.95896
$451$	−35.1381	−1.65459
$452$	−61.1021	−2.87400
$453$	69.1332	3.24816
$454$	−79.2964	−3.72156
$455$	0.727741	0.0341170
$456$	−26.1915	−1.22653
$457$	6.74620	0.315574	0.157787	−	0.987473i	$-0.449564\pi$
$457$	6.74620	0.315574	0.157787	+	0.987473i	$0.449564\pi$
$458$	−25.1769	−1.17644
$459$	63.2992	2.95456
$460$	−46.5271	−2.16934
$461$	−12.5703	−0.585459	−0.292730	−	0.956195i	$-0.594564\pi$
$461$	−12.5703	−0.585459	−0.292730	+	0.956195i	$0.594564\pi$
$462$	−4.97859	−0.231625
$463$	19.3864	0.900964	0.450482	−	0.892785i	$-0.351252\pi$
$463$	19.3864	0.900964	0.450482	+	0.892785i	$0.351252\pi$
$464$	12.4963	0.580126
$465$	32.3104	1.49836
$466$	−67.5017	−3.12696
$467$	23.8610	1.10416	0.552078	−	0.833792i	$-0.313835\pi$
$467$	23.8610	1.10416	0.552078	+	0.833792i	$0.313835\pi$
$468$	−92.4080	−4.27156
$469$	−0.918904	−0.0424311
$470$	2.81236	0.129724
$471$	67.3972	3.10550
$472$	19.5362	0.899229
$473$	22.8903	1.05250
$474$	−80.8258	−3.71245
$475$	2.29931	0.105500
$476$	4.60329	0.210992
$477$	88.6148	4.05739
$478$	−12.0655	−0.551861
$479$	7.99568	0.365332	0.182666	−	0.983175i	$-0.441527\pi$
$479$	7.99568	0.365332	0.182666	+	0.983175i	$0.441527\pi$
$480$	80.1599	3.65878
$481$	14.2209	0.648419
$482$	62.9131	2.86561
$483$	−2.86814	−0.130505
$484$	9.35796	0.425362
$485$	−3.55320	−0.161343
$486$	43.2035	1.95975
$487$	1.43254	0.0649148	0.0324574	−	0.999473i	$-0.489667\pi$
$487$	1.43254	0.0649148	0.0324574	+	0.999473i	$0.489667\pi$
$488$	32.1571	1.45568
$489$	22.7295	1.02786
$490$	30.6111	1.38287
$491$	−6.63121	−0.299262	−0.149631	−	0.988742i	$-0.547809\pi$
$491$	−6.63121	−0.299262	−0.149631	+	0.988742i	$0.547809\pi$
$492$	157.536	7.10227
$493$	−5.54848	−0.249891
$494$	−7.10353	−0.319603
$495$	39.8084	1.78925
$496$	−76.2390	−3.42323
$497$	−0.763723	−0.0342577
$498$	80.9085	3.62559
$499$	21.4786	0.961512	0.480756	−	0.876854i	$-0.340362\pi$
$499$	21.4786	0.961512	0.480756	+	0.876854i	$0.340362\pi$
$500$	−61.6248	−2.75594
$501$	−61.3828	−2.74238
$502$	80.2162	3.58022
$503$	33.6892	1.50213	0.751064	−	0.660229i	$-0.229541\pi$
$503$	33.6892	1.50213	0.751064	+	0.660229i	$0.229541\pi$
$504$	9.44331	0.420638
$505$	15.3965	0.685134
$506$	52.7194	2.34366
$507$	18.5309	0.822988
$508$	31.2745	1.38758
$509$	−39.6757	−1.75859	−0.879297	−	0.476273i	$-0.841987\pi$
$509$	−39.6757	−1.75859	−0.879297	+	0.476273i	$0.841987\pi$
$510$	−73.8143	−3.26855
$511$	−0.137138	−0.00606664
$512$	13.9813	0.617893
$513$	11.7652	0.519445
$514$	−44.3326	−1.95543
$515$	13.9495	0.614690
$516$	−102.625	−4.51781
$517$	−2.29370	−0.100877
$518$	−2.37969	−0.104558
$519$	−17.0098	−0.746646
$520$	36.6243	1.60608
$521$	4.14589	0.181635	0.0908174	−	0.995868i	$-0.471052\pi$
$521$	4.14589	0.181635	0.0908174	+	0.995868i	$0.471052\pi$
$522$	−18.6383	−0.815778
$523$	−9.02330	−0.394561	−0.197281	−	0.980347i	$-0.563211\pi$
$523$	−9.02330	−0.394561	−0.197281	+	0.980347i	$0.563211\pi$
$524$	41.0481	1.79320
$525$	−1.19665	−0.0522259
$526$	−63.4752	−2.76765
$527$	33.8508	1.47457
$528$	−135.586	−5.90062
$529$	7.37143	0.320497
$530$	−57.5102	−2.49808
$531$	−15.7682	−0.684282
$532$	0.855596	0.0370948
$533$	26.0924	1.13019
$534$	110.632	4.78750
$535$	−9.21279	−0.398304
$536$	−46.2448	−1.99747
$537$	−11.9449	−0.515462
$538$	33.8904	1.46112
$539$	−24.9657	−1.07535
$540$	−99.3280	−4.27440
$541$	−16.2230	−0.697483	−0.348742	−	0.937219i	$-0.613391\pi$
$541$	−16.2230	−0.697483	−0.348742	+	0.937219i	$0.613391\pi$
$542$	−10.2897	−0.441979
$543$	−40.8759	−1.75415
$544$	83.9816	3.60068
$545$	20.9521	0.897491
$546$	3.69694	0.158214
$547$	−1.14322	−0.0488807	−0.0244404	−	0.999701i	$-0.507780\pi$
$547$	−1.14322	−0.0488807	−0.0244404	+	0.999701i	$0.507780\pi$
$548$	−70.4905	−3.01120
$549$	−25.9548	−1.10772
$550$	21.9956	0.937894
$551$	−1.03127	−0.0439337
$552$	−144.342	−6.14362
$553$	1.61242	0.0685673
$554$	−30.8823	−1.31206
$555$	27.4659	1.16586
$556$	39.2306	1.66375
$557$	15.6208	0.661877	0.330938	−	0.943652i	$-0.392635\pi$
$557$	15.6208	0.661877	0.330938	+	0.943652i	$0.392635\pi$
$558$	113.711	4.81378
$559$	−16.9976	−0.718922
$560$	−3.31649	−0.140147
$561$	60.2014	2.54170
$562$	8.43043	0.355616
$563$	−28.0081	−1.18040	−0.590200	−	0.807257i	$-0.700951\pi$
$563$	−28.0081	−1.18040	−0.590200	+	0.807257i	$0.700951\pi$
$564$	10.2834	0.433010
$565$	−19.5460	−0.822306
$566$	−27.4929	−1.15561
$567$	−2.74300	−0.115195
$568$	−38.4352	−1.61270
$569$	11.0611	0.463704	0.231852	−	0.972751i	$-0.425521\pi$
$569$	11.0611	0.463704	0.231852	+	0.972751i	$0.425521\pi$
$570$	−13.7196	−0.574650
$571$	1.48061	0.0619616	0.0309808	−	0.999520i	$-0.490137\pi$
$571$	1.48061	0.0619616	0.0309808	+	0.999520i	$0.490137\pi$
$572$	−48.9118	−2.04510
$573$	−40.5214	−1.69281
$574$	−4.36624	−0.182243
$575$	12.6716	0.528440
$576$	118.162	4.92342
$577$	−1.75687	−0.0731396	−0.0365698	−	0.999331i	$-0.511643\pi$
$577$	−1.75687	−0.0731396	−0.0365698	+	0.999331i	$0.511643\pi$
$578$	−31.9168	−1.32756
$579$	−9.41874	−0.391430
$580$	8.70657	0.361521
$581$	−1.61407	−0.0669631
$582$	−18.0503	−0.748211
$583$	46.9040	1.94257
$584$	−6.90163	−0.285591
$585$	−29.5604	−1.22217
$586$	48.0024	1.98296
$587$	−32.3949	−1.33708	−0.668541	−	0.743675i	$-0.733081\pi$
$587$	−32.3949	−1.33708	−0.668541	+	0.743675i	$0.733081\pi$
$588$	111.930	4.61590
$589$	6.29172	0.259246
$590$	10.2334	0.421303
$591$	39.2207	1.61332
$592$	−64.8082	−2.66360
$593$	−41.8615	−1.71905	−0.859523	−	0.511098i	$-0.829239\pi$
$593$	−41.8615	−1.71905	−0.859523	+	0.511098i	$0.829239\pi$
$594$	112.548	4.61788
$595$	1.47255	0.0603687
$596$	30.9787	1.26894
$597$	25.1687	1.03009
$598$	−39.1477	−1.60087
$599$	22.6866	0.926949	0.463475	−	0.886110i	$-0.346603\pi$
$599$	22.6866	0.926949	0.463475	+	0.886110i	$0.346603\pi$
$600$	−60.2224	−2.45857
$601$	23.7668	0.969470	0.484735	−	0.874661i	$-0.338916\pi$
$601$	23.7668	0.969470	0.484735	+	0.874661i	$0.338916\pi$
$602$	2.84433	0.115926
$603$	37.3254	1.52001
$604$	−113.655	−4.62454
$605$	2.99352	0.121704
$606$	78.2144	3.17724
$607$	−47.3201	−1.92066	−0.960331	−	0.278862i	$-0.910043\pi$
$607$	−47.3201	−1.92066	−0.960331	+	0.278862i	$0.910043\pi$
$608$	15.6093	0.633041
$609$	0.536712	0.0217487
$610$	16.8444	0.682011
$611$	1.70323	0.0689052
$612$	−186.983	−7.55835
$613$	−1.62158	−0.0654949	−0.0327474	−	0.999464i	$-0.510426\pi$
$613$	−1.62158	−0.0654949	−0.0327474	+	0.999464i	$0.510426\pi$
$614$	−76.8318	−3.10068
$615$	50.3942	2.03209
$616$	4.99837	0.201390
$617$	14.7052	0.592010	0.296005	−	0.955186i	$-0.404345\pi$
$617$	14.7052	0.592010	0.296005	+	0.955186i	$0.404345\pi$
$618$	70.8640	2.85057
$619$	−25.3831	−1.02023	−0.510117	−	0.860105i	$-0.670398\pi$
$619$	−25.3831	−1.02023	−0.510117	+	0.860105i	$0.670398\pi$
$620$	−53.1181	−2.13328
$621$	64.8382	2.60187
$622$	29.4116	1.17930
$623$	−2.20703	−0.0884229
$624$	100.682	4.03049
$625$	−8.21663	−0.328665
$626$	−37.4164	−1.49546
$627$	11.1894	0.446861
$628$	−110.801	−4.42143
$629$	28.7754	1.14735
$630$	4.94657	0.197076
$631$	−8.23487	−0.327825	−0.163912	−	0.986475i	$-0.552411\pi$
$631$	−8.23487	−0.327825	−0.163912	+	0.986475i	$0.552411\pi$
$632$	81.1470	3.22785
$633$	−3.12490	−0.124203
$634$	−60.2960	−2.39466
$635$	10.0044	0.397013
$636$	−210.287	−8.33840
$637$	18.5388	0.734532
$638$	−9.86532	−0.390572
$639$	31.0220	1.22721
$640$	−25.3820	−1.00331
$641$	11.8794	0.469206	0.234603	−	0.972091i	$-0.424621\pi$
$641$	11.8794	0.469206	0.234603	+	0.972091i	$0.424621\pi$
$642$	−46.8012	−1.84710
$643$	−14.9957	−0.591373	−0.295687	−	0.955285i	$-0.595548\pi$
$643$	−14.9957	−0.591373	−0.295687	+	0.955285i	$0.595548\pi$
$644$	4.71521	0.185805
$645$	−32.8287	−1.29263
$646$	−14.3737	−0.565524
$647$	−19.6440	−0.772286	−0.386143	−	0.922439i	$-0.626193\pi$
$647$	−19.6440	−0.772286	−0.386143	+	0.922439i	$0.626193\pi$
$648$	−138.045	−5.42290
$649$	−8.34615	−0.327615
$650$	−16.3332	−0.640641
$651$	−3.27444	−0.128335
$652$	−37.3672	−1.46341
$653$	−2.21179	−0.0865539	−0.0432770	−	0.999063i	$-0.513780\pi$
$653$	−2.21179	−0.0865539	−0.0432770	+	0.999063i	$0.513780\pi$
$654$	106.437	4.16203
$655$	13.1309	0.513067
$656$	−118.909	−4.64263
$657$	5.57048	0.217325
$658$	−0.285014	−0.0111110
$659$	25.3835	0.988799	0.494400	−	0.869235i	$-0.335388\pi$
$659$	25.3835	0.988799	0.494400	+	0.869235i	$0.335388\pi$
$660$	−94.4669	−3.67712
$661$	−38.6692	−1.50406	−0.752029	−	0.659130i	$-0.770925\pi$
$661$	−38.6692	−1.50406	−0.752029	+	0.659130i	$0.770925\pi$
$662$	−51.5988	−2.00544
$663$	−44.7036	−1.73614
$664$	−81.2300	−3.15233
$665$	0.273697	0.0106135
$666$	96.6619	3.74557
$667$	−5.68337	−0.220061
$668$	100.913	3.90444
$669$	−21.0088	−0.812246
$670$	−24.2238	−0.935848
$671$	−13.7379	−0.530348
$672$	−8.12366	−0.313377
$673$	−27.4794	−1.05925	−0.529627	−	0.848231i	$-0.677668\pi$
$673$	−27.4794	−1.05925	−0.529627	+	0.848231i	$0.677668\pi$
$674$	−15.5386	−0.598523
$675$	27.0518	1.04122
$676$	−30.4648	−1.17172
$677$	−10.5375	−0.404989	−0.202495	−	0.979283i	$-0.564905\pi$
$677$	−10.5375	−0.404989	−0.202495	+	0.979283i	$0.564905\pi$
$678$	−99.2940	−3.81336
$679$	0.360093	0.0138191
$680$	74.1077	2.84190
$681$	−92.7517	−3.55425
$682$	60.1876	2.30470
$683$	15.2453	0.583345	0.291673	−	0.956518i	$-0.405788\pi$
$683$	15.2453	0.583345	0.291673	+	0.956518i	$0.405788\pi$
$684$	−34.7538	−1.32885
$685$	−22.5492	−0.861562
$686$	−6.21679	−0.237358
$687$	−29.4490	−1.12355
$688$	77.4620	2.95321
$689$	−34.8294	−1.32690
$690$	−75.6089	−2.87838
$691$	45.9577	1.74831	0.874157	−	0.485644i	$-0.161415\pi$
$691$	45.9577	1.74831	0.874157	+	0.485644i	$0.161415\pi$
$692$	27.9640	1.06303
$693$	−4.03431	−0.153251
$694$	−42.6223	−1.61792
$695$	12.5495	0.476029
$696$	27.0106	1.02383
$697$	52.7968	1.99982
$698$	94.5699	3.57952
$699$	−78.9556	−2.98638
$700$	1.96728	0.0743561
$701$	35.0263	1.32292	0.661462	−	0.749979i	$-0.269936\pi$
$701$	35.0263	1.32292	0.661462	+	0.749979i	$0.269936\pi$
$702$	−83.5742	−3.15431
$703$	5.34837	0.201718
$704$	62.5435	2.35720
$705$	3.28957	0.123892
$706$	−47.4513	−1.78585
$707$	−1.56033	−0.0586822
$708$	37.4186	1.40628
$709$	−3.24133	−0.121731	−0.0608653	−	0.998146i	$-0.519386\pi$
$709$	−3.24133	−0.121731	−0.0608653	+	0.998146i	$0.519386\pi$
$710$	−20.1330	−0.755578
$711$	−65.4958	−2.45628
$712$	−111.071	−4.16257
$713$	34.6738	1.29854
$714$	7.48059	0.279954
$715$	−15.6464	−0.585143
$716$	19.6374	0.733885
$717$	−14.1128	−0.527051
$718$	−93.5946	−3.49292
$719$	9.58408	0.357426	0.178713	−	0.983901i	$-0.442807\pi$
$719$	9.58408	0.357426	0.178713	+	0.983901i	$0.442807\pi$
$720$	134.714	5.02049
$721$	−1.41369	−0.0526486
$722$	−2.67158	−0.0994257
$723$	73.5884	2.73678
$724$	67.1998	2.49746
$725$	−2.37121	−0.0880647
$726$	15.2072	0.564391
$727$	−13.2950	−0.493083	−0.246542	−	0.969132i	$-0.579294\pi$
$727$	−13.2950	−0.493083	−0.246542	+	0.969132i	$0.579294\pi$
$728$	−3.71163	−0.137562
$729$	1.12442	0.0416450
$730$	−3.61519	−0.133804
$731$	−34.3939	−1.27210
$732$	61.5918	2.27650
$733$	−15.1404	−0.559224	−0.279612	−	0.960113i	$-0.590206\pi$
$733$	−15.1404	−0.559224	−0.279612	+	0.960113i	$0.590206\pi$
$734$	−70.0061	−2.58397
$735$	35.8053	1.32070
$736$	86.0234	3.17086
$737$	19.7564	0.727737
$738$	177.354	6.52850
$739$	23.0991	0.849712	0.424856	−	0.905261i	$-0.360325\pi$
$739$	23.0991	0.849712	0.424856	+	0.905261i	$0.360325\pi$
$740$	−45.1539	−1.65989
$741$	−8.30888	−0.305234
$742$	5.82827	0.213962
$743$	−35.6640	−1.30838	−0.654192	−	0.756329i	$-0.726991\pi$
$743$	−35.6640	−1.30838	−0.654192	+	0.756329i	$0.726991\pi$
$744$	−164.790	−6.04149
$745$	9.90979	0.363067
$746$	88.1507	3.22743
$747$	65.5628	2.39882
$748$	−98.9708	−3.61873
$749$	0.933655	0.0341150
$750$	−100.143	−3.65672
$751$	10.8134	0.394585	0.197293	−	0.980345i	$-0.436785\pi$
$751$	10.8134	0.394585	0.197293	+	0.980345i	$0.436785\pi$
$752$	−7.76201	−0.283051
$753$	93.8275	3.41927
$754$	7.32567	0.266785
$755$	−36.3571	−1.32317
$756$	10.0662	0.366105
$757$	47.4641	1.72511	0.862557	−	0.505960i	$-0.168862\pi$
$757$	47.4641	1.72511	0.862557	+	0.505960i	$0.168862\pi$
$758$	13.3149	0.483619
$759$	61.6650	2.23830
$760$	13.7741	0.499639
$761$	−0.160414	−0.00581500	−0.00290750	−	0.999996i	$-0.500925\pi$
$761$	−0.160414	−0.00581500	−0.00290750	+	0.999996i	$0.500925\pi$
$762$	50.8226	1.84111
$763$	−2.12336	−0.0768707
$764$	66.6170	2.41012
$765$	−59.8142	−2.16259
$766$	83.9391	3.03284
$767$	6.19759	0.223782
$768$	−19.7779	−0.713673
$769$	12.0948	0.436148	0.218074	−	0.975932i	$-0.430023\pi$
$769$	12.0948	0.436148	0.218074	+	0.975932i	$0.430023\pi$
$770$	2.61823	0.0943545
$771$	−51.8551	−1.86751
$772$	15.4844	0.557295
$773$	−10.3243	−0.371339	−0.185670	−	0.982612i	$-0.559445\pi$
$773$	−10.3243	−0.371339	−0.185670	+	0.982612i	$0.559445\pi$
$774$	−115.535	−4.15283
$775$	14.4666	0.519656
$776$	18.1221	0.650544
$777$	−2.78349	−0.0998571
$778$	33.2307	1.19138
$779$	9.81313	0.351592
$780$	70.1481	2.51171
$781$	16.4200	0.587555
$782$	−79.2137	−2.83268
$783$	−12.1331	−0.433602
$784$	−84.4855	−3.01734
$785$	−35.4441	−1.26505
$786$	66.7053	2.37930
$787$	−33.2274	−1.18443	−0.592215	−	0.805780i	$-0.701747\pi$
$787$	−33.2274	−1.18443	−0.592215	+	0.805780i	$0.701747\pi$
$788$	−64.4786	−2.29695
$789$	−74.2459	−2.64322
$790$	42.5061	1.51230
$791$	1.98085	0.0704311
$792$	−203.031	−7.21439
$793$	10.2014	0.362261
$794$	55.0748	1.95453
$795$	−67.2687	−2.38577
$796$	−41.3772	−1.46658
$797$	45.4304	1.60923	0.804613	−	0.593799i	$-0.202373\pi$
$797$	45.4304	1.60923	0.804613	+	0.593799i	$0.202373\pi$
$798$	1.39039	0.0492191
$799$	3.44640	0.121925
$800$	35.8906	1.26893
$801$	89.6485	3.16757
$802$	−38.0675	−1.34421
$803$	2.94847	0.104049
$804$	−88.5747	−3.12379
$805$	1.50835	0.0531624
$806$	−44.6934	−1.57426
$807$	39.6411	1.39543
$808$	−78.5252	−2.76251
$809$	26.5444	0.933252	0.466626	−	0.884455i	$-0.345469\pi$
$809$	26.5444	0.933252	0.466626	+	0.884455i	$0.345469\pi$
$810$	−72.3100	−2.54072
$811$	36.0025	1.26422	0.632109	−	0.774880i	$-0.282190\pi$
$811$	36.0025	1.26422	0.632109	+	0.774880i	$0.282190\pi$
$812$	−0.882352	−0.0309645
$813$	−12.0357	−0.422109
$814$	51.1634	1.79328
$815$	−11.9534	−0.418709
$816$	203.725	7.13180
$817$	−6.39265	−0.223650
$818$	−9.67651	−0.338331
$819$	2.99575	0.104680
$820$	−82.8479	−2.89317
$821$	−25.9523	−0.905743	−0.452871	−	0.891576i	$-0.649600\pi$
$821$	−25.9523	−0.905743	−0.452871	+	0.891576i	$0.649600\pi$
$822$	−114.551	−3.99541
$823$	−33.8569	−1.18018	−0.590088	−	0.807339i	$-0.700907\pi$
$823$	−33.8569	−1.18018	−0.590088	+	0.807339i	$0.700907\pi$
$824$	−71.1456	−2.47847
$825$	25.7279	0.895729
$826$	−1.03709	−0.0360849
$827$	−12.7569	−0.443600	−0.221800	−	0.975092i	$-0.571193\pi$
$827$	−12.7569	−0.443600	−0.221800	+	0.975092i	$0.571193\pi$
$828$	−191.529	−6.65610
$829$	−12.4453	−0.432244	−0.216122	−	0.976366i	$-0.569341\pi$
$829$	−12.4453	−0.432244	−0.216122	+	0.976366i	$0.569341\pi$
$830$	−42.5496	−1.47692
$831$	−36.1225	−1.25307
$832$	−46.4428	−1.61011
$833$	37.5123	1.29973
$834$	63.7517	2.20754
$835$	32.2811	1.11713
$836$	−18.3953	−0.636214
$837$	74.0231	2.55861
$838$	−10.8132	−0.373536
$839$	−30.6781	−1.05912	−0.529562	−	0.848271i	$-0.677644\pi$
$839$	−30.6781	−1.05912	−0.529562	+	0.848271i	$0.677644\pi$
$840$	−7.16854	−0.247338
$841$	−27.9365	−0.963327
$842$	−36.6955	−1.26461
$843$	9.86093	0.339628
$844$	5.13731	0.176834
$845$	−9.74539	−0.335252
$846$	11.5771	0.398029
$847$	−0.303373	−0.0104240
$848$	158.726	5.45067
$849$	−32.1580	−1.10366
$850$	−33.0495	−1.13359
$851$	29.4750	1.01039
$852$	−73.6165	−2.52206
$853$	53.2139	1.82201	0.911004	−	0.412397i	$-0.135308\pi$
$853$	53.2139	1.82201	0.911004	+	0.412397i	$0.135308\pi$
$854$	−1.70707	−0.0584147
$855$	−11.1174	−0.380208
$856$	46.9872	1.60599
$857$	32.4027	1.10685	0.553427	−	0.832897i	$-0.313320\pi$
$857$	32.4027	1.10685	0.553427	+	0.832897i	$0.313320\pi$
$858$	−79.4841	−2.71354
$859$	29.5708	1.00894	0.504471	−	0.863429i	$-0.331688\pi$
$859$	29.5708	1.00894	0.504471	+	0.863429i	$0.331688\pi$
$860$	53.9702	1.84037
$861$	−5.10712	−0.174050
$862$	−100.982	−3.43947
$863$	−12.9991	−0.442494	−0.221247	−	0.975218i	$-0.571013\pi$
$863$	−12.9991	−0.442494	−0.221247	+	0.975218i	$0.571013\pi$
$864$	183.646	6.24777
$865$	8.94541	0.304153
$866$	85.2133	2.89567
$867$	−37.3325	−1.26788
$868$	5.38316	0.182717
$869$	−34.6671	−1.17600
$870$	14.1486	0.479683
$871$	−14.6705	−0.497091
$872$	−106.860	−3.61874
$873$	−14.6268	−0.495042
$874$	−14.7231	−0.498017
$875$	1.99780	0.0675379
$876$	−13.2190	−0.446628
$877$	52.6144	1.77666	0.888330	−	0.459205i	$-0.151866\pi$
$877$	52.6144	1.77666	0.888330	+	0.459205i	$0.151866\pi$
$878$	11.3928	0.384488
$879$	56.1476	1.89381
$880$	71.3044	2.40367
$881$	−8.55802	−0.288327	−0.144164	−	0.989554i	$-0.546049\pi$
$881$	−8.55802	−0.288327	−0.144164	+	0.989554i	$0.546049\pi$
$882$	126.011	4.24300
$883$	−22.4962	−0.757057	−0.378528	−	0.925590i	$-0.623570\pi$
$883$	−22.4962	−0.757057	−0.378528	+	0.925590i	$0.623570\pi$
$884$	73.4925	2.47182
$885$	11.9699	0.402362
$886$	39.6655	1.33259
$887$	7.77461	0.261046	0.130523	−	0.991445i	$-0.458334\pi$
$887$	7.77461	0.261046	0.130523	+	0.991445i	$0.458334\pi$
$888$	−140.082	−4.70085
$889$	−1.01388	−0.0340044
$890$	−58.1810	−1.95023
$891$	58.9745	1.97572
$892$	34.5383	1.15643
$893$	0.640569	0.0214358
$894$	50.3420	1.68369
$895$	6.28182	0.209978
$896$	2.57230	0.0859345
$897$	−45.7905	−1.52890
$898$	−90.1498	−3.00834
$899$	−6.48847	−0.216403
$900$	−79.9098	−2.66366
$901$	−70.4758	−2.34789
$902$	93.8741	3.12566
$903$	3.32697	0.110715
$904$	99.6886	3.31559
$905$	21.4966	0.714570
$906$	−184.695	−6.13607
$907$	−10.3233	−0.342780	−0.171390	−	0.985203i	$-0.554826\pi$
$907$	−10.3233	−0.342780	−0.171390	+	0.985203i	$0.554826\pi$
$908$	152.483	5.06034
$909$	63.3797	2.10217
$910$	−1.94421	−0.0644501
$911$	−7.22473	−0.239366	−0.119683	−	0.992812i	$-0.538188\pi$
$911$	−7.22473	−0.239366	−0.119683	+	0.992812i	$0.538188\pi$
$912$	37.8655	1.25385
$913$	34.7026	1.14849
$914$	−18.0230	−0.596147
$915$	19.7026	0.651349
$916$	48.4141	1.59965
$917$	−1.33073	−0.0439446
$918$	−169.109	−5.58142
$919$	29.7836	0.982470	0.491235	−	0.871027i	$-0.336546\pi$
$919$	29.7836	0.982470	0.491235	+	0.871027i	$0.336546\pi$
$920$	75.9094	2.50266
$921$	−89.8689	−2.96128
$922$	33.5826	1.10598
$923$	−12.1930	−0.401337
$924$	9.57359	0.314948
$925$	12.2976	0.404341
$926$	−51.7923	−1.70200
$927$	57.4234	1.88603
$928$	−16.0975	−0.528425
$929$	−29.8516	−0.979398	−0.489699	−	0.871891i	$-0.662893\pi$
$929$	−29.8516	−0.979398	−0.489699	+	0.871891i	$0.662893\pi$
$930$	−86.3196	−2.83053
$931$	6.97226	0.228507
$932$	129.803	4.25183
$933$	34.4022	1.12628
$934$	−63.7465	−2.08585
$935$	−31.6598	−1.03539
$936$	150.764	4.92789
$937$	−12.4308	−0.406098	−0.203049	−	0.979169i	$-0.565085\pi$
$937$	−12.4308	−0.406098	−0.203049	+	0.979169i	$0.565085\pi$
$938$	2.45492	0.0801560
$939$	−43.7653	−1.42823
$940$	−5.40803	−0.176391
$941$	57.7787	1.88353	0.941765	−	0.336271i	$-0.109166\pi$
$941$	57.7787	1.88353	0.941765	+	0.336271i	$0.109166\pi$
$942$	−180.057	−5.86657
$943$	54.0805	1.76110
$944$	−28.2439	−0.919259
$945$	3.22009	0.104750
$946$	−61.1531	−1.98826
$947$	−12.0844	−0.392689	−0.196344	−	0.980535i	$-0.562907\pi$
$947$	−12.0844	−0.392689	−0.196344	+	0.980535i	$0.562907\pi$
$948$	155.424	5.04794
$949$	−2.18944	−0.0710722
$950$	−6.14278	−0.199298
$951$	−70.5273	−2.28700
$952$	−7.51031	−0.243411
$953$	−1.63488	−0.0529590	−0.0264795	−	0.999649i	$-0.508430\pi$
$953$	−1.63488	−0.0529590	−0.0264795	+	0.999649i	$0.508430\pi$
$954$	−236.741	−7.66478
$955$	21.3102	0.689580
$956$	23.2013	0.750384
$957$	−11.5393	−0.373013
$958$	−21.3611	−0.690144
$959$	2.28521	0.0737934
$960$	−89.6984	−2.89500
$961$	8.58570	0.276958
$962$	−37.9923	−1.22492
$963$	−37.9245	−1.22210
$964$	−120.979	−3.89647
$965$	4.95330	0.159452
$966$	7.66246	0.246536
$967$	−30.2353	−0.972303	−0.486151	−	0.873875i	$-0.661600\pi$
$967$	−30.2353	−0.972303	−0.486151	+	0.873875i	$0.661600\pi$
$968$	−15.2676	−0.490719
$969$	−16.8126	−0.540100
$970$	9.49265	0.304791
$971$	−51.5756	−1.65514	−0.827570	−	0.561362i	$-0.810277\pi$
$971$	−51.5756	−1.65514	−0.827570	+	0.561362i	$0.810277\pi$
$972$	−83.0783	−2.66474
$973$	−1.27181	−0.0407722
$974$	−3.82715	−0.122630
$975$	−19.1047	−0.611839
$976$	−46.4900	−1.48811
$977$	−37.1261	−1.18777	−0.593884	−	0.804550i	$-0.702406\pi$
$977$	−37.1261	−1.18777	−0.593884	+	0.804550i	$0.702406\pi$
$978$	−60.7235	−1.94172
$979$	47.4512	1.51655
$980$	−58.8637	−1.88033
$981$	86.2496	2.75374
$982$	17.7158	0.565333
$983$	4.98940	0.159137	0.0795686	−	0.996829i	$-0.474646\pi$
$983$	4.98940	0.159137	0.0795686	+	0.996829i	$0.474646\pi$
$984$	−257.021	−8.19353
$985$	−20.6261	−0.657202
$986$	14.8232	0.472066
$987$	−0.333376	−0.0106115
$988$	13.6598	0.434575
$989$	−35.2300	−1.12025
$990$	−106.351	−3.38006
$991$	50.1098	1.59179	0.795895	−	0.605434i	$-0.207000\pi$
$991$	50.1098	1.59179	0.795895	+	0.605434i	$0.207000\pi$
$992$	98.2094	3.11815
$993$	−60.3543	−1.91528
$994$	2.04034	0.0647158
$995$	−13.2362	−0.419615
$996$	−155.583	−4.92984
$997$	−16.8805	−0.534609	−0.267305	−	0.963612i	$-0.586133\pi$
$997$	−16.8805	−0.534609	−0.267305	+	0.963612i	$0.586133\pi$
$998$	−57.3816	−1.81638
$999$	62.9245	1.99084

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.f.1.3	✓	83

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.f.1.3	✓	83	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.67158 q^{2} -3.12490 q^{3} +5.13731 q^{4} +1.64338 q^{5} +8.34840 q^{6} -0.166545 q^{7} -8.38157 q^{8} +6.76498 q^{9} +O(q^{10})\)	\(q-2.67158 q^{2} -3.12490 q^{3} +5.13731 q^{4} +1.64338 q^{5} +8.34840 q^{6} -0.166545 q^{7} -8.38157 q^{8} +6.76498 q^{9} -4.39041 q^{10} +3.58072 q^{11} -16.0536 q^{12} -2.65893 q^{13} +0.444938 q^{14} -5.13539 q^{15} +12.1174 q^{16} -5.38022 q^{17} -18.0732 q^{18} -1.00000 q^{19} +8.44255 q^{20} +0.520437 q^{21} -9.56617 q^{22} -5.51103 q^{23} +26.1915 q^{24} -2.29931 q^{25} +7.10353 q^{26} -11.7652 q^{27} -0.855596 q^{28} +1.03127 q^{29} +13.7196 q^{30} -6.29172 q^{31} -15.6093 q^{32} -11.1894 q^{33} +14.3737 q^{34} -0.273697 q^{35} +34.7538 q^{36} -5.34837 q^{37} +2.67158 q^{38} +8.30888 q^{39} -13.7741 q^{40} -9.81313 q^{41} -1.39039 q^{42} +6.39265 q^{43} +18.3953 q^{44} +11.1174 q^{45} +14.7231 q^{46} -0.640569 q^{47} -37.8655 q^{48} -6.97226 q^{49} +6.14278 q^{50} +16.8126 q^{51} -13.6598 q^{52} +13.0990 q^{53} +31.4315 q^{54} +5.88448 q^{55} +1.39591 q^{56} +3.12490 q^{57} -2.75512 q^{58} -2.33086 q^{59} -26.3821 q^{60} -3.83664 q^{61} +16.8088 q^{62} -1.12668 q^{63} +17.4667 q^{64} -4.36963 q^{65} +29.8933 q^{66} +5.51744 q^{67} -27.6399 q^{68} +17.2214 q^{69} +0.731202 q^{70} +4.58568 q^{71} -56.7011 q^{72} +0.823429 q^{73} +14.2886 q^{74} +7.18510 q^{75} -5.13731 q^{76} -0.596352 q^{77} -22.1978 q^{78} -9.68159 q^{79} +19.9134 q^{80} +16.4700 q^{81} +26.2165 q^{82} +9.69150 q^{83} +2.67365 q^{84} -8.84174 q^{85} -17.0784 q^{86} -3.22262 q^{87} -30.0121 q^{88} +13.2518 q^{89} -29.7010 q^{90} +0.442832 q^{91} -28.3119 q^{92} +19.6610 q^{93} +1.71133 q^{94} -1.64338 q^{95} +48.7775 q^{96} -2.16213 q^{97} +18.6269 q^{98} +24.2235 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9}+O(q^{10}) \) 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9	\( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9} + 9 q^{10} + 56 q^{11} - 2 q^{12} - 5 q^{13} + 6 q^{14} + 19 q^{15} + 123 q^{16} + 19 q^{17} + 40 q^{18} - 83 q^{19} + 49 q^{20} + 9 q^{21} + 18 q^{22} + 74 q^{23} + 38 q^{24} + 98 q^{25} + 28 q^{26} + 6 q^{27} + 50 q^{28} + 16 q^{29} + 56 q^{30} + 24 q^{31} + 81 q^{32} + 13 q^{33} + 9 q^{34} + 71 q^{35} + 156 q^{36} - 6 q^{37} - 11 q^{38} + 126 q^{39} + q^{40} - q^{42} + 34 q^{43} + 140 q^{44} + 42 q^{45} + 34 q^{46} + 53 q^{47} + 16 q^{48} + 118 q^{49} + 51 q^{50} + 57 q^{51} + 32 q^{52} + q^{53} + 53 q^{54} + 60 q^{55} - 2 q^{56} - 2 q^{58} + 44 q^{59} - 9 q^{60} + 21 q^{61} + 28 q^{62} + 83 q^{63} + 154 q^{64} + 44 q^{65} + 17 q^{66} + 5 q^{67} + 63 q^{68} - 36 q^{69} - 48 q^{70} + 193 q^{71} + 135 q^{72} + 54 q^{73} + 127 q^{74} + 5 q^{75} - 95 q^{76} + 54 q^{77} + 45 q^{78} + 54 q^{79} + 45 q^{80} + 147 q^{81} - 35 q^{82} + 84 q^{83} + 12 q^{84} + 28 q^{85} + 60 q^{86} + 51 q^{87} + 23 q^{88} - 24 q^{89} + 31 q^{90} + 28 q^{91} + 108 q^{92} + 39 q^{93} - 49 q^{94} - 15 q^{95} + 25 q^{96} - 22 q^{97} - 67 q^{98} + 132 q^{99}+O(q^{100}) \) 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9 + 9 * q^10 + 56 * q^11 - 2 * q^12 - 5 * q^13 + 6 * q^14 + 19 * q^15 + 123 * q^16 + 19 * q^17 + 40 * q^18 - 83 * q^19 + 49 * q^20 + 9 * q^21 + 18 * q^22 + 74 * q^23 + 38 * q^24 + 98 * q^25 + 28 * q^26 + 6 * q^27 + 50 * q^28 + 16 * q^29 + 56 * q^30 + 24 * q^31 + 81 * q^32 + 13 * q^33 + 9 * q^34 + 71 * q^35 + 156 * q^36 - 6 * q^37 - 11 * q^38 + 126 * q^39 + q^40 - q^42 + 34 * q^43 + 140 * q^44 + 42 * q^45 + 34 * q^46 + 53 * q^47 + 16 * q^48 + 118 * q^49 + 51 * q^50 + 57 * q^51 + 32 * q^52 + q^53 + 53 * q^54 + 60 * q^55 - 2 * q^56 - 2 * q^58 + 44 * q^59 - 9 * q^60 + 21 * q^61 + 28 * q^62 + 83 * q^63 + 154 * q^64 + 44 * q^65 + 17 * q^66 + 5 * q^67 + 63 * q^68 - 36 * q^69 - 48 * q^70 + 193 * q^71 + 135 * q^72 + 54 * q^73 + 127 * q^74 + 5 * q^75 - 95 * q^76 + 54 * q^77 + 45 * q^78 + 54 * q^79 + 45 * q^80 + 147 * q^81 - 35 * q^82 + 84 * q^83 + 12 * q^84 + 28 * q^85 + 60 * q^86 + 51 * q^87 + 23 * q^88 - 24 * q^89 + 31 * q^90 + 28 * q^91 + 108 * q^92 + 39 * q^93 - 49 * q^94 - 15 * q^95 + 25 * q^96 - 22 * q^97 - 67 * q^98 + 132 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more