LMFDB - Embedded newform 177.14.a.b.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,14,Mod(1,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 14, names="a")

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

chi := DirichletCharacter("177.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	177.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:	$189.798744245$
Analytic rank:	$1$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-149.360 q^{2} -729.000 q^{3} +14116.3 q^{4} -48711.9 q^{5} +108883. q^{6} -300158. q^{7} -884852. q^{8} +531441. q^{9} +O(q^{10})$	\(q-149.360 q^{2} -729.000 q^{3} +14116.3 q^{4} -48711.9 q^{5} +108883. q^{6} -300158. q^{7} -884852. q^{8} +531441. q^{9} +7.27559e6 q^{10} +2.91688e6 q^{11} -1.02908e7 q^{12} +4.39145e6 q^{13} +4.48315e7 q^{14} +3.55110e7 q^{15} +1.65205e7 q^{16} -6.68228e7 q^{17} -7.93758e7 q^{18} -3.05356e8 q^{19} -6.87632e8 q^{20} +2.18815e8 q^{21} -4.35664e8 q^{22} -4.79121e8 q^{23} +6.45057e8 q^{24} +1.15215e9 q^{25} -6.55906e8 q^{26} -3.87420e8 q^{27} -4.23712e9 q^{28} +1.34762e9 q^{29} -5.30391e9 q^{30} +8.39044e9 q^{31} +4.78122e9 q^{32} -2.12641e9 q^{33} +9.98063e9 q^{34} +1.46213e10 q^{35} +7.50198e9 q^{36} -2.42362e10 q^{37} +4.56079e10 q^{38} -3.20137e9 q^{39} +4.31028e10 q^{40} -2.76354e10 q^{41} -3.26821e10 q^{42} -7.37570e10 q^{43} +4.11756e10 q^{44} -2.58875e10 q^{45} +7.15614e10 q^{46} -5.77097e10 q^{47} -1.20434e10 q^{48} -6.79429e9 q^{49} -1.72084e11 q^{50} +4.87138e10 q^{51} +6.19911e10 q^{52} +1.52339e11 q^{53} +5.78650e10 q^{54} -1.42087e11 q^{55} +2.65595e11 q^{56} +2.22605e11 q^{57} -2.01281e11 q^{58} -4.21805e10 q^{59} +5.01284e11 q^{60} +6.51411e11 q^{61} -1.25319e12 q^{62} -1.59516e11 q^{63} -8.49457e11 q^{64} -2.13916e11 q^{65} +3.17599e11 q^{66} -1.13534e12 q^{67} -9.43291e11 q^{68} +3.49279e11 q^{69} -2.18383e12 q^{70} -9.22926e10 q^{71} -4.70247e11 q^{72} +2.04726e12 q^{73} +3.61991e12 q^{74} -8.39915e11 q^{75} -4.31050e12 q^{76} -8.75525e11 q^{77} +4.78155e11 q^{78} +1.06213e12 q^{79} -8.04744e11 q^{80} +2.82430e11 q^{81} +4.12761e12 q^{82} +1.85336e12 q^{83} +3.08886e12 q^{84} +3.25506e12 q^{85} +1.10163e13 q^{86} -9.82418e11 q^{87} -2.58101e12 q^{88} +2.98879e12 q^{89} +3.86655e12 q^{90} -1.31813e12 q^{91} -6.76342e12 q^{92} -6.11663e12 q^{93} +8.61950e12 q^{94} +1.48745e13 q^{95} -3.48551e12 q^{96} -1.02877e13 q^{97} +1.01479e12 q^{98} +1.55015e12 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q - 52 q^{2} - 22599 q^{3} + 126886 q^{4} + 33486 q^{5} + 37908 q^{6} - 1135539 q^{7} - 1519749 q^{8} + 16474671 q^{9}+O(q^{10}) $ 31 * q - 52 * q^2 - 22599 * q^3 + 126886 * q^4 + 33486 * q^5 + 37908 * q^6 - 1135539 * q^7 - 1519749 * q^8 + 16474671 * q^9	\( 31 q - 52 q^{2} - 22599 q^{3} + 126886 q^{4} + 33486 q^{5} + 37908 q^{6} - 1135539 q^{7} - 1519749 q^{8} + 16474671 q^{9} - 3854663 q^{10} + 3943968 q^{11} - 92499894 q^{12} - 48510022 q^{13} - 51427459 q^{14} - 24411294 q^{15} + 370110498 q^{16} + 83288419 q^{17} - 27634932 q^{18} - 180425297 q^{19} + 753620445 q^{20} + 827807931 q^{21} + 2300196142 q^{22} - 1305810279 q^{23} + 1107897021 q^{24} + 8070954867 q^{25} + 464550322 q^{26} - 12010035159 q^{27} - 9887169562 q^{28} + 6248352277 q^{29} + 2810049327 q^{30} - 26730150789 q^{31} - 24001343230 q^{32} - 2875152672 q^{33} - 36571033348 q^{34} + 10255900979 q^{35} + 67432422726 q^{36} - 43284776933 q^{37} - 36293696947 q^{38} + 35363806038 q^{39} - 105980683856 q^{40} - 9961079285 q^{41} + 37490617611 q^{42} - 51755851288 q^{43} - 59623729442 q^{44} + 17795833326 q^{45} - 202287132683 q^{46} - 82747063727 q^{47} - 269810553042 q^{48} + 535277836542 q^{49} + 526974390461 q^{50} - 60717257451 q^{51} + 544982341446 q^{52} + 561701818494 q^{53} + 20145865428 q^{54} - 521861534450 q^{55} - 228056576664 q^{56} + 131530041513 q^{57} + 10555409160 q^{58} - 1307596542871 q^{59} - 549389304405 q^{60} + 618193248201 q^{61} - 1486611437386 q^{62} - 603471981699 q^{63} + 679062548045 q^{64} - 1130583307122 q^{65} - 1676842987518 q^{66} - 4137387490592 q^{67} - 3901389300295 q^{68} + 951935693391 q^{69} - 819291947844 q^{70} - 3766439869810 q^{71} - 807656928309 q^{72} - 2386775553523 q^{73} + 3060770694642 q^{74} - 5883726098043 q^{75} - 847741068784 q^{76} + 1650423006137 q^{77} - 338657184738 q^{78} + 787155757766 q^{79} + 13999832121779 q^{80} + 8755315630911 q^{81} + 10083281915577 q^{82} + 8743877051639 q^{83} + 7207746610698 q^{84} + 15373177520565 q^{85} + 18939443838984 q^{86} - 4555048809933 q^{87} + 39713314506713 q^{88} + 11026795445259 q^{89} - 2048525959383 q^{90} + 23285721962531 q^{91} + 40411079823254 q^{92} + 19486279925181 q^{93} + 35237377585624 q^{94} + 13730236994039 q^{95} + 17496979214670 q^{96} + 10134565481560 q^{97} + 70916776240976 q^{98} + 2095986297888 q^{99}+O(q^{100}) \) 31 * q - 52 * q^2 - 22599 * q^3 + 126886 * q^4 + 33486 * q^5 + 37908 * q^6 - 1135539 * q^7 - 1519749 * q^8 + 16474671 * q^9 - 3854663 * q^10 + 3943968 * q^11 - 92499894 * q^12 - 48510022 * q^13 - 51427459 * q^14 - 24411294 * q^15 + 370110498 * q^16 + 83288419 * q^17 - 27634932 * q^18 - 180425297 * q^19 + 753620445 * q^20 + 827807931 * q^21 + 2300196142 * q^22 - 1305810279 * q^23 + 1107897021 * q^24 + 8070954867 * q^25 + 464550322 * q^26 - 12010035159 * q^27 - 9887169562 * q^28 + 6248352277 * q^29 + 2810049327 * q^30 - 26730150789 * q^31 - 24001343230 * q^32 - 2875152672 * q^33 - 36571033348 * q^34 + 10255900979 * q^35 + 67432422726 * q^36 - 43284776933 * q^37 - 36293696947 * q^38 + 35363806038 * q^39 - 105980683856 * q^40 - 9961079285 * q^41 + 37490617611 * q^42 - 51755851288 * q^43 - 59623729442 * q^44 + 17795833326 * q^45 - 202287132683 * q^46 - 82747063727 * q^47 - 269810553042 * q^48 + 535277836542 * q^49 + 526974390461 * q^50 - 60717257451 * q^51 + 544982341446 * q^52 + 561701818494 * q^53 + 20145865428 * q^54 - 521861534450 * q^55 - 228056576664 * q^56 + 131530041513 * q^57 + 10555409160 * q^58 - 1307596542871 * q^59 - 549389304405 * q^60 + 618193248201 * q^61 - 1486611437386 * q^62 - 603471981699 * q^63 + 679062548045 * q^64 - 1130583307122 * q^65 - 1676842987518 * q^66 - 4137387490592 * q^67 - 3901389300295 * q^68 + 951935693391 * q^69 - 819291947844 * q^70 - 3766439869810 * q^71 - 807656928309 * q^72 - 2386775553523 * q^73 + 3060770694642 * q^74 - 5883726098043 * q^75 - 847741068784 * q^76 + 1650423006137 * q^77 - 338657184738 * q^78 + 787155757766 * q^79 + 13999832121779 * q^80 + 8755315630911 * q^81 + 10083281915577 * q^82 + 8743877051639 * q^83 + 7207746610698 * q^84 + 15373177520565 * q^85 + 18939443838984 * q^86 - 4555048809933 * q^87 + 39713314506713 * q^88 + 11026795445259 * q^89 - 2048525959383 * q^90 + 23285721962531 * q^91 + 40411079823254 * q^92 + 19486279925181 * q^93 + 35237377585624 * q^94 + 13730236994039 * q^95 + 17496979214670 * q^96 + 10134565481560 * q^97 + 70916776240976 * q^98 + 2095986297888 * 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 $

$2$	−149.360	−1.65021	−0.825103	−	0.564982i	$-0.808883\pi$
$2$	−149.360	−1.65021	−0.825103	+	0.564982i	$0.808883\pi$
$3$	−729.000	−0.577350
$3$	−729.000	−0.577350
$4$	14116.3	1.72318
$5$	−48711.9	−1.39422	−0.697108	−	0.716966i	$-0.745530\pi$
$5$	−48711.9	−1.39422	−0.697108	+	0.716966i	$0.745530\pi$
$6$	108883.	0.952747
$7$	−300158.	−0.964301	−0.482150	−	0.876089i	$-0.660144\pi$
$7$	−300158.	−0.964301	−0.482150	+	0.876089i	$0.660144\pi$
$8$	−884852.	−1.19340
$9$	531441.	0.333333
$10$	7.27559e6	2.30074
$11$	2.91688e6	0.496440	0.248220	−	0.968704i	$-0.420154\pi$
$11$	2.91688e6	0.496440	0.248220	+	0.968704i	$0.420154\pi$
$12$	−1.02908e7	−0.994880
$13$	4.39145e6	0.252334	0.126167	−	0.992009i	$-0.459732\pi$
$13$	4.39145e6	0.252334	0.126167	+	0.992009i	$0.459732\pi$
$14$	4.48315e7	1.59130
$15$	3.55110e7	0.804951
$16$	1.65205e7	0.246174
$17$	−6.68228e7	−0.671439	−0.335719	−	0.941962i	$-0.608979\pi$
$17$	−6.68228e7	−0.671439	−0.335719	+	0.941962i	$0.608979\pi$
$18$	−7.93758e7	−0.550069
$19$	−3.05356e8	−1.48905	−0.744524	−	0.667596i	$-0.767323\pi$
$19$	−3.05356e8	−1.48905	−0.744524	+	0.667596i	$0.767323\pi$
$20$	−6.87632e8	−2.40249
$21$	2.18815e8	0.556739
$22$	−4.35664e8	−0.819228
$23$	−4.79121e8	−0.674861	−0.337431	−	0.941350i	$-0.609558\pi$
$23$	−4.79121e8	−0.674861	−0.337431	+	0.941350i	$0.609558\pi$
$24$	6.45057e8	0.689010
$25$	1.15215e9	0.943838
$26$	−6.55906e8	−0.416404
$27$	−3.87420e8	−0.192450
$28$	−4.23712e9	−1.66167
$29$	1.34762e9	0.420709	0.210354	−	0.977625i	$-0.432538\pi$
$29$	1.34762e9	0.420709	0.210354	+	0.977625i	$0.432538\pi$
$30$	−5.30391e9	−1.32834
$31$	8.39044e9	1.69799	0.848993	−	0.528405i	$-0.177210\pi$
$31$	8.39044e9	1.69799	0.848993	+	0.528405i	$0.177210\pi$
$32$	4.78122e9	0.787161
$33$	−2.12641e9	−0.286620
$34$	9.98063e9	1.10801
$35$	1.46213e10	1.34444
$36$	7.50198e9	0.574394
$37$	−2.42362e10	−1.55293	−0.776467	−	0.630158i	$-0.782990\pi$
$37$	−2.42362e10	−1.55293	−0.776467	+	0.630158i	$0.782990\pi$
$38$	4.56079e10	2.45724
$39$	−3.20137e9	−0.145685
$40$	4.31028e10	1.66386
$41$	−2.76354e10	−0.908594	−0.454297	−	0.890850i	$-0.650110\pi$
$41$	−2.76354e10	−0.908594	−0.454297	+	0.890850i	$0.650110\pi$
$42$	−3.26821e10	−0.918735
$43$	−7.37570e10	−1.77934	−0.889668	−	0.456608i	$-0.849064\pi$
$43$	−7.37570e10	−1.77934	−0.889668	+	0.456608i	$0.849064\pi$
$44$	4.11756e10	0.855456
$45$	−2.58875e10	−0.464739
$46$	7.15614e10	1.11366
$47$	−5.77097e10	−0.780931	−0.390465	−	0.920618i	$-0.627686\pi$
$47$	−5.77097e10	−0.780931	−0.390465	+	0.920618i	$0.627686\pi$
$48$	−1.20434e10	−0.142129
$49$	−6.79429e9	−0.0701245
$50$	−1.72084e11	−1.55753
$51$	4.87138e10	0.387655
$52$	6.19911e10	0.434818
$53$	1.52339e11	0.944101	0.472051	−	0.881571i	$-0.343514\pi$
$53$	1.52339e11	0.944101	0.472051	+	0.881571i	$0.343514\pi$
$54$	5.78650e10	0.317582
$55$	−1.42087e11	−0.692144
$56$	2.65595e11	1.15080
$57$	2.22605e11	0.859702
$58$	−2.01281e11	−0.694256
$59$	−4.21805e10	−0.130189
$59$	−4.21805e10	−0.130189
$60$	5.01284e11	1.38708
$61$	6.51411e11	1.61887	0.809433	−	0.587212i	$-0.199774\pi$
$61$	6.51411e11	1.61887	0.809433	+	0.587212i	$0.199774\pi$
$62$	−1.25319e12	−2.80203
$63$	−1.59516e11	−0.321434
$64$	−8.49457e11	−1.54515
$65$	−2.13916e11	−0.351809
$66$	3.17599e11	0.472981
$67$	−1.13534e12	−1.53334	−0.766672	−	0.642039i	$-0.778089\pi$
$67$	−1.13534e12	−1.53334	−0.766672	+	0.642039i	$0.778089\pi$
$68$	−9.43291e11	−1.15701
$69$	3.49279e11	0.389631
$70$	−2.18383e12	−2.21861
$71$	−9.22926e10	−0.0855042	−0.0427521	−	0.999086i	$-0.513613\pi$
$71$	−9.22926e10	−0.0855042	−0.0427521	+	0.999086i	$0.513613\pi$
$72$	−4.70247e11	−0.397800
$73$	2.04726e12	1.58334	0.791672	−	0.610946i	$-0.209211\pi$
$73$	2.04726e12	1.58334	0.791672	+	0.610946i	$0.209211\pi$
$74$	3.61991e12	2.56266
$75$	−8.39915e11	−0.544925
$76$	−4.31050e12	−2.56590
$77$	−8.75525e11	−0.478717
$78$	4.78155e11	0.240411
$79$	1.06213e12	0.491590	0.245795	−	0.969322i	$-0.420951\pi$
$79$	1.06213e12	0.491590	0.245795	+	0.969322i	$0.420951\pi$
$80$	−8.04744e11	−0.343220
$81$	2.82430e11	0.111111
$82$	4.12761e12	1.49937
$83$	1.85336e12	0.622232	0.311116	−	0.950372i	$-0.399297\pi$
$83$	1.85336e12	0.622232	0.311116	+	0.950372i	$0.399297\pi$
$84$	3.08886e12	0.959363
$85$	3.25506e12	0.936131
$86$	1.10163e13	2.93627
$87$	−9.82418e11	−0.242896
$88$	−2.58101e12	−0.592451
$89$	2.98879e12	0.637471	0.318736	−	0.947844i	$-0.396742\pi$
$89$	2.98879e12	0.637471	0.318736	+	0.947844i	$0.396742\pi$
$90$	3.86655e12	0.766915
$91$	−1.31813e12	−0.243326
$92$	−6.76342e12	−1.16291
$93$	−6.11663e12	−0.980332
$94$	8.61950e12	1.28870
$95$	1.48745e13	2.07605
$96$	−3.48551e12	−0.454468
$97$	−1.02877e13	−1.25401	−0.627006	−	0.779015i	$-0.715720\pi$
$97$	−1.02877e13	−1.25401	−0.627006	+	0.779015i	$0.715720\pi$
$98$	1.01479e12	0.115720
$99$	1.55015e12	0.165480
$100$	1.62641e13	1.62641
$101$	1.89626e13	1.77750	0.888748	−	0.458396i	$-0.151576\pi$
$101$	1.89626e13	1.77750	0.888748	+	0.458396i	$0.151576\pi$
$102$	−7.27588e12	−0.639712
$103$	−6.09533e12	−0.502986	−0.251493	−	0.967859i	$-0.580921\pi$
$103$	−6.09533e12	−0.502986	−0.251493	+	0.967859i	$0.580921\pi$
$104$	−3.88579e12	−0.301136
$105$	−1.06589e13	−0.776215
$106$	−2.27533e13	−1.55796
$107$	2.28388e13	1.47123	0.735613	−	0.677402i	$-0.236894\pi$
$107$	2.28388e13	1.47123	0.735613	+	0.677402i	$0.236894\pi$
$108$	−5.46895e12	−0.331627
$109$	−7.12166e12	−0.406733	−0.203366	−	0.979103i	$-0.565188\pi$
$109$	−7.12166e12	−0.406733	−0.203366	+	0.979103i	$0.565188\pi$
$110$	2.12220e13	1.14218
$111$	1.76682e13	0.896587
$112$	−4.95875e12	−0.237386
$113$	5.34147e11	0.0241352	0.0120676	−	0.999927i	$-0.496159\pi$
$113$	5.34147e11	0.0241352	0.0120676	+	0.999927i	$0.496159\pi$
$114$	−3.32482e13	−1.41869
$115$	2.33389e13	0.940902
$116$	1.90235e13	0.724958
$117$	2.33380e12	0.0841114
$118$	6.30007e12	0.214839
$119$	2.00574e13	0.647469
$120$	−3.14220e13	−0.960628
$121$	−2.60145e13	−0.753548
$122$	−9.72945e13	−2.67146
$123$	2.01462e13	0.524577
$124$	1.18442e14	2.92594
$125$	3.33952e12	0.0783014
$126$	2.38253e13	0.530432
$127$	7.02815e13	1.48633	0.743167	−	0.669106i	$-0.233323\pi$
$127$	7.02815e13	1.48633	0.743167	+	0.669106i	$0.233323\pi$
$128$	8.77069e13	1.76266
$129$	5.37688e13	1.02730
$130$	3.19504e13	0.580557
$131$	2.04972e13	0.354350	0.177175	−	0.984179i	$-0.443304\pi$
$131$	2.04972e13	0.354350	0.177175	+	0.984179i	$0.443304\pi$
$132$	−3.00170e13	−0.493898
$133$	9.16551e13	1.43589
$134$	1.69574e14	2.53033
$135$	1.88720e13	0.268317
$136$	5.91283e13	0.801295
$137$	4.51513e13	0.583428	0.291714	−	0.956506i	$-0.405775\pi$
$137$	4.51513e13	0.583428	0.291714	+	0.956506i	$0.405775\pi$
$138$	−5.21682e13	−0.642972
$139$	−8.65933e12	−0.101833	−0.0509165	−	0.998703i	$-0.516214\pi$
$139$	−8.65933e12	−0.101833	−0.0509165	+	0.998703i	$0.516214\pi$
$140$	2.06398e14	2.31672
$141$	4.20703e13	0.450871
$142$	1.37848e13	0.141100
$143$	1.28093e13	0.125269
$144$	8.77966e12	0.0820581
$145$	−6.56453e13	−0.586559
$146$	−3.05779e14	−2.61284
$147$	4.95304e12	0.0404864
$148$	−3.42125e14	−2.67599
$149$	1.90864e14	1.42893	0.714467	−	0.699669i	$-0.246669\pi$
$149$	1.90864e14	1.42893	0.714467	+	0.699669i	$0.246669\pi$
$150$	1.25449e14	0.899239
$151$	2.80301e14	1.92431	0.962154	−	0.272507i	$-0.0878527\pi$
$151$	2.80301e14	1.92431	0.962154	+	0.272507i	$0.0878527\pi$
$152$	2.70195e14	1.77703
$153$	−3.55124e13	−0.223813
$154$	1.30768e14	0.789982
$155$	−4.08715e14	−2.36736
$156$	−4.51915e13	−0.251042
$157$	−2.96112e14	−1.57800	−0.789002	−	0.614391i	$-0.789402\pi$
$157$	−2.96112e14	−1.57800	−0.789002	+	0.614391i	$0.789402\pi$
$158$	−1.58640e14	−0.811225
$159$	−1.11055e14	−0.545077
$160$	−2.32902e14	−1.09747
$161$	1.43812e14	0.650769
$162$	−4.21836e13	−0.183356
$163$	2.75231e14	1.14942	0.574710	−	0.818357i	$-0.305115\pi$
$163$	2.75231e14	1.14942	0.574710	+	0.818357i	$0.305115\pi$
$164$	−3.90109e14	−1.56567
$165$	1.03581e14	0.399610
$166$	−2.76817e14	−1.02681
$167$	2.97487e14	1.06123	0.530617	−	0.847611i	$-0.321960\pi$
$167$	2.97487e14	1.06123	0.530617	+	0.847611i	$0.321960\pi$
$168$	−1.93619e14	−0.664412
$169$	−2.83590e14	−0.936327
$170$	−4.86175e14	−1.54481
$171$	−1.62279e14	−0.496349
$172$	−1.04118e15	−3.06612
$173$	2.89857e14	0.822025	0.411012	−	0.911630i	$-0.365175\pi$
$173$	2.89857e14	0.822025	0.411012	+	0.911630i	$0.365175\pi$
$174$	1.46734e14	0.400829
$175$	−3.45826e14	−0.910144
$176$	4.81883e13	0.122211
$177$	3.07496e13	0.0751646
$178$	−4.46405e14	−1.05196
$179$	−1.26450e14	−0.287327	−0.143663	−	0.989627i	$-0.545888\pi$
$179$	−1.26450e14	−0.287327	−0.143663	+	0.989627i	$0.545888\pi$
$180$	−3.65436e14	−0.800829
$181$	1.48782e13	0.0314513	0.0157257	−	0.999876i	$-0.494994\pi$
$181$	1.48782e13	0.0314513	0.0157257	+	0.999876i	$0.494994\pi$
$182$	1.96875e14	0.401538
$183$	−4.74879e14	−0.934653
$184$	4.23952e14	0.805379
$185$	1.18059e15	2.16513
$186$	9.13578e14	1.61775
$187$	−1.94914e14	−0.333329
$188$	−8.14647e14	−1.34569
$189$	1.16287e14	0.185580
$190$	−2.22165e15	−3.42592
$191$	−1.19462e14	−0.178038	−0.0890192	−	0.996030i	$-0.528373\pi$
$191$	−1.19462e14	−0.178038	−0.0890192	+	0.996030i	$0.528373\pi$
$192$	6.19254e14	0.892095
$193$	9.05529e14	1.26119	0.630594	−	0.776113i	$-0.282811\pi$
$193$	9.05529e14	1.26119	0.630594	+	0.776113i	$0.282811\pi$
$194$	1.53657e15	2.06938
$195$	1.55945e14	0.203117
$196$	−9.59103e13	−0.120837
$197$	3.02457e14	0.368666	0.184333	−	0.982864i	$-0.440987\pi$
$197$	3.02457e14	0.368666	0.184333	+	0.982864i	$0.440987\pi$
$198$	−2.31530e14	−0.273076
$199$	−1.44303e15	−1.64714	−0.823572	−	0.567212i	$-0.808022\pi$
$199$	−1.44303e15	−1.64714	−0.823572	+	0.567212i	$0.808022\pi$
$200$	−1.01948e15	−1.12638
$201$	8.27663e14	0.885277
$202$	−2.83225e15	−2.93324
$203$	−4.04500e14	−0.405690
$204$	6.87659e14	0.668001
$205$	1.34617e15	1.26678
$206$	9.10397e14	0.830030
$207$	−2.54625e14	−0.224954
$208$	7.25489e13	0.0621182
$209$	−8.90688e14	−0.739222
$210$	1.59201e15	1.28091
$211$	1.25818e15	0.981540	0.490770	−	0.871289i	$-0.336716\pi$
$211$	1.25818e15	0.981540	0.490770	+	0.871289i	$0.336716\pi$
$212$	2.15047e15	1.62686
$213$	6.72813e13	0.0493659
$214$	−3.41120e15	−2.42783
$215$	3.59284e15	2.48078
$216$	3.42810e14	0.229670
$217$	−2.51846e15	−1.63737
$218$	1.06369e15	0.671193
$219$	−1.49246e15	−0.914144
$220$	−2.00574e15	−1.19269
$221$	−2.93449e14	−0.169427
$222$	−2.63891e15	−1.47955
$223$	−2.51120e15	−1.36741	−0.683706	−	0.729758i	$-0.739633\pi$
$223$	−2.51120e15	−1.36741	−0.683706	+	0.729758i	$0.739633\pi$
$224$	−1.43512e15	−0.759060
$225$	6.12298e14	0.314613
$226$	−7.97800e13	−0.0398281
$227$	−2.31766e15	−1.12430	−0.562149	−	0.827036i	$-0.690025\pi$
$227$	−2.31766e15	−1.12430	−0.562149	+	0.827036i	$0.690025\pi$
$228$	3.14236e15	1.48142
$229$	2.23588e15	1.02451	0.512257	−	0.858832i	$-0.328810\pi$
$229$	2.23588e15	1.02451	0.512257	+	0.858832i	$0.328810\pi$
$230$	−3.48589e15	−1.55268
$231$	6.38258e14	0.276387
$232$	−1.19245e15	−0.502074
$233$	3.42937e15	1.40411	0.702055	−	0.712123i	$-0.252266\pi$
$233$	3.42937e15	1.40411	0.702055	+	0.712123i	$0.252266\pi$
$234$	−3.48575e14	−0.138801
$235$	2.81115e15	1.08879
$236$	−5.95433e14	−0.224339
$237$	−7.74295e14	−0.283819
$238$	−2.99576e15	−1.06846
$239$	5.23983e14	0.181857	0.0909286	−	0.995857i	$-0.471016\pi$
$239$	5.23983e14	0.181857	0.0909286	+	0.995857i	$0.471016\pi$
$240$	5.86658e14	0.198158
$241$	−1.64381e14	−0.0540431	−0.0270215	−	0.999635i	$-0.508602\pi$
$241$	−1.64381e14	−0.0540431	−0.0270215	+	0.999635i	$0.508602\pi$
$242$	3.88552e15	1.24351
$243$	−2.05891e14	−0.0641500
$244$	9.19552e15	2.78960
$245$	3.30963e14	0.0977686
$246$	−3.00903e15	−0.865661
$247$	−1.34096e15	−0.375738
$248$	−7.42431e15	−2.02638
$249$	−1.35110e15	−0.359246
$250$	−4.98790e14	−0.129214
$251$	−4.20818e15	−1.06222	−0.531111	−	0.847303i	$-0.678225\pi$
$251$	−4.20818e15	−1.06222	−0.531111	+	0.847303i	$0.678225\pi$
$252$	−2.25178e15	−0.553888
$253$	−1.39754e15	−0.335028
$254$	−1.04972e16	−2.45276
$255$	−2.37294e15	−0.540475
$256$	−6.14111e15	−1.36360
$257$	5.28008e15	1.14308	0.571538	−	0.820576i	$-0.306347\pi$
$257$	5.28008e15	1.14308	0.571538	+	0.820576i	$0.306347\pi$
$258$	−8.03089e15	−1.69526
$259$	7.27468e15	1.49750
$260$	−3.01970e15	−0.606230
$261$	7.16182e14	0.140236
$262$	−3.06146e15	−0.584750
$263$	−4.56695e15	−0.850970	−0.425485	−	0.904965i	$-0.639896\pi$
$263$	−4.56695e15	−0.850970	−0.425485	+	0.904965i	$0.639896\pi$
$264$	1.88156e15	0.342052
$265$	−7.42073e15	−1.31628
$266$	−1.36896e16	−2.36951
$267$	−2.17883e15	−0.368044
$268$	−1.60268e16	−2.64223
$269$	4.32339e15	0.695720	0.347860	−	0.937546i	$-0.386908\pi$
$269$	4.32339e15	0.695720	0.347860	+	0.937546i	$0.386908\pi$
$270$	−2.81871e15	−0.442779
$271$	1.48315e15	0.227449	0.113725	−	0.993512i	$-0.463722\pi$
$271$	1.48315e15	0.227449	0.113725	+	0.993512i	$0.463722\pi$
$272$	−1.10394e15	−0.165291
$273$	9.60916e14	0.140484
$274$	−6.74379e15	−0.962776
$275$	3.36068e15	0.468559
$276$	4.93053e15	0.671406
$277$	−5.04924e15	−0.671595	−0.335798	−	0.941934i	$-0.609006\pi$
$277$	−5.04924e15	−0.671595	−0.335798	+	0.941934i	$0.609006\pi$
$278$	1.29336e15	0.168045
$279$	4.45903e15	0.565995
$280$	−1.29377e16	−1.60446
$281$	−1.25712e16	−1.52331	−0.761653	−	0.647986i	$-0.775612\pi$
$281$	−1.25712e16	−1.52331	−0.761653	+	0.647986i	$0.775612\pi$
$282$	−6.28361e15	−0.744030
$283$	−5.69495e15	−0.658989	−0.329495	−	0.944157i	$-0.606878\pi$
$283$	−5.69495e15	−0.658989	−0.329495	+	0.944157i	$0.606878\pi$
$284$	−1.30283e15	−0.147339
$285$	−1.08435e16	−1.19861
$286$	−1.91320e15	−0.206719
$287$	8.29497e15	0.876158
$288$	2.54094e15	0.262387
$289$	−5.43930e15	−0.549170
$290$	9.80476e15	0.967943
$291$	7.49973e15	0.724004
$292$	2.88998e16	2.72839
$293$	−1.44892e16	−1.33785	−0.668923	−	0.743332i	$-0.733244\pi$
$293$	−1.44892e16	−1.33785	−0.668923	+	0.743332i	$0.733244\pi$
$294$	−7.39784e14	−0.0668109
$295$	2.05469e15	0.181511
$296$	2.14454e16	1.85327
$297$	−1.13006e15	−0.0955398
$298$	−2.85073e16	−2.35804
$299$	−2.10404e15	−0.170291
$300$	−1.18565e16	−0.939006
$301$	2.21387e16	1.71581
$302$	−4.18657e16	−3.17551
$303$	−1.38237e16	−1.02624
$304$	−5.04463e15	−0.366565
$305$	−3.17315e16	−2.25705
$306$	5.30411e15	0.369338
$307$	7.50929e15	0.511917	0.255958	−	0.966688i	$-0.417609\pi$
$307$	7.50929e15	0.511917	0.255958	+	0.966688i	$0.417609\pi$
$308$	−1.23592e16	−0.824916
$309$	4.44350e15	0.290399
$310$	6.10455e16	3.90663
$311$	1.11208e16	0.696940	0.348470	−	0.937320i	$-0.386701\pi$
$311$	1.11208e16	0.696940	0.348470	+	0.937320i	$0.386701\pi$
$312$	2.83274e15	0.173861
$313$	4.58215e15	0.275443	0.137721	−	0.990471i	$-0.456022\pi$
$313$	4.58215e15	0.275443	0.137721	+	0.990471i	$0.456022\pi$
$314$	4.42272e16	2.60403
$315$	7.77034e15	0.448148
$316$	1.49934e16	0.847099
$317$	−7.82999e15	−0.433387	−0.216694	−	0.976240i	$-0.569527\pi$
$317$	−7.82999e15	−0.433387	−0.216694	+	0.976240i	$0.569527\pi$
$318$	1.65872e16	0.899490
$319$	3.93086e15	0.208856
$320$	4.13787e16	2.15428
$321$	−1.66495e16	−0.849413
$322$	−2.14797e16	−1.07390
$323$	2.04048e16	0.999804
$324$	3.98686e15	0.191465
$325$	5.05960e15	0.238163
$326$	−4.11085e16	−1.89678
$327$	5.19169e15	0.234827
$328$	2.44532e16	1.08432
$329$	1.73220e16	0.753052
$330$	−1.54709e16	−0.659438
$331$	−1.80009e16	−0.752336	−0.376168	−	0.926552i	$-0.622758\pi$
$331$	−1.80009e16	−0.752336	−0.376168	+	0.926552i	$0.622758\pi$
$332$	2.61626e16	1.07222
$333$	−1.28801e16	−0.517645
$334$	−4.44326e16	−1.75126
$335$	5.53046e16	2.13781
$336$	3.61493e15	0.137055
$337$	2.28670e16	0.850385	0.425193	−	0.905103i	$-0.360206\pi$
$337$	2.28670e16	0.850385	0.425193	+	0.905103i	$0.360206\pi$
$338$	4.23569e16	1.54513
$339$	−3.89393e14	−0.0139345
$340$	4.59495e16	1.61312
$341$	2.44739e16	0.842947
$342$	2.42379e16	0.819079
$343$	3.11214e16	1.03192
$344$	6.52640e16	2.12346
$345$	−1.70141e16	−0.543230
$346$	−4.32930e16	−1.35651
$347$	−1.97866e16	−0.608458	−0.304229	−	0.952599i	$-0.598399\pi$
$347$	−1.97866e16	−0.608458	−0.304229	+	0.952599i	$0.598399\pi$
$348$	−1.38681e16	−0.418555
$349$	−3.56274e16	−1.05540	−0.527702	−	0.849430i	$-0.676946\pi$
$349$	−3.56274e16	−1.05540	−0.527702	+	0.849430i	$0.676946\pi$
$350$	5.16524e16	1.50193
$351$	−1.70134e15	−0.0485618
$352$	1.39463e16	0.390778
$353$	1.28305e16	0.352946	0.176473	−	0.984305i	$-0.443531\pi$
$353$	1.28305e16	0.352946	0.176473	+	0.984305i	$0.443531\pi$
$354$	−4.59275e15	−0.124037
$355$	4.49575e15	0.119211
$356$	4.21907e16	1.09848
$357$	−1.46218e16	−0.373816
$358$	1.88866e16	0.474148
$359$	−5.94927e16	−1.46673	−0.733364	−	0.679836i	$-0.762051\pi$
$359$	−5.94927e16	−1.46673	−0.733364	+	0.679836i	$0.762051\pi$
$360$	2.29066e16	0.554619
$361$	5.11895e16	1.21726
$362$	−2.22220e15	−0.0519012
$363$	1.89646e16	0.435061
$364$	−1.86071e16	−0.419295
$365$	−9.97261e16	−2.20752
$366$	7.09277e16	1.54237
$367$	1.35344e16	0.289142	0.144571	−	0.989494i	$-0.453820\pi$
$367$	1.35344e16	0.289142	0.144571	+	0.989494i	$0.453820\pi$
$368$	−7.91531e15	−0.166133
$369$	−1.46866e16	−0.302865
$370$	−1.76333e17	−3.57290
$371$	−4.57258e16	−0.910397
$372$	−8.63443e16	−1.68929
$373$	−2.07601e16	−0.399138	−0.199569	−	0.979884i	$-0.563954\pi$
$373$	−2.07601e16	−0.399138	−0.199569	+	0.979884i	$0.563954\pi$
$374$	2.91123e16	0.550061
$375$	−2.43451e15	−0.0452073
$376$	5.10645e16	0.931962
$377$	5.91802e15	0.106159
$378$	−1.73686e16	−0.306245
$379$	−1.53191e16	−0.265509	−0.132755	−	0.991149i	$-0.542382\pi$
$379$	−1.53191e16	−0.265509	−0.132755	+	0.991149i	$0.542382\pi$
$380$	2.09973e17	3.57742
$381$	−5.12352e16	−0.858136
$382$	1.78428e16	0.293800
$383$	−3.45080e16	−0.558635	−0.279318	−	0.960199i	$-0.590108\pi$
$383$	−3.45080e16	−0.558635	−0.279318	+	0.960199i	$0.590108\pi$
$384$	−6.39383e16	−1.01767
$385$	4.26485e16	0.667435
$386$	−1.35250e17	−2.08122
$387$	−3.91975e16	−0.593112
$388$	−1.45224e17	−2.16089
$389$	6.06404e16	0.887339	0.443670	−	0.896190i	$-0.353676\pi$
$389$	6.06404e16	0.887339	0.443670	+	0.896190i	$0.353676\pi$
$390$	−2.32919e16	−0.335185
$391$	3.20162e16	0.453128
$392$	6.01194e15	0.0836865
$393$	−1.49425e16	−0.204584
$394$	−4.51749e16	−0.608375
$395$	−5.17385e16	−0.685382
$396$	2.18824e16	0.285152
$397$	6.28868e16	0.806160	0.403080	−	0.915165i	$-0.367940\pi$
$397$	6.28868e16	0.806160	0.403080	+	0.915165i	$0.367940\pi$
$398$	2.15531e17	2.71813
$399$	−6.68166e16	−0.829011
$400$	1.90340e16	0.232349
$401$	4.82766e16	0.579827	0.289914	−	0.957053i	$-0.406373\pi$
$401$	4.82766e16	0.579827	0.289914	+	0.957053i	$0.406373\pi$
$402$	−1.23619e17	−1.46089
$403$	3.68462e16	0.428460
$404$	2.67682e17	3.06295
$405$	−1.37577e16	−0.154913
$406$	6.04159e16	0.669472
$407$	−7.06941e16	−0.770938
$408$	−4.31045e16	−0.462628
$409$	−1.15320e17	−1.21816	−0.609081	−	0.793108i	$-0.708461\pi$
$409$	−1.15320e17	−1.21816	−0.609081	+	0.793108i	$0.708461\pi$
$410$	−2.01064e17	−2.09044
$411$	−3.29153e16	−0.336842
$412$	−8.60436e16	−0.866736
$413$	1.26608e16	0.125541
$414$	3.80306e16	0.371220
$415$	−9.02808e16	−0.867526
$416$	2.09965e16	0.198628
$417$	6.31265e15	0.0587933
$418$	1.33033e17	1.21987
$419$	1.81294e17	1.63679	0.818395	−	0.574656i	$-0.194864\pi$
$419$	1.81294e17	1.63679	0.818395	+	0.574656i	$0.194864\pi$
$420$	−1.50464e17	−1.33756
$421$	7.01686e16	0.614199	0.307100	−	0.951677i	$-0.400641\pi$
$421$	7.01686e16	0.614199	0.307100	+	0.951677i	$0.400641\pi$
$422$	−1.87922e17	−1.61974
$423$	−3.06693e16	−0.260310
$424$	−1.34798e17	−1.12669
$425$	−7.69896e16	−0.633730
$426$	−1.00491e16	−0.0814639
$427$	−1.95526e17	−1.56107
$428$	3.22400e17	2.53519
$429$	−9.33801e15	−0.0723239
$430$	−5.36626e17	−4.09380
$431$	−1.51237e17	−1.13646	−0.568230	−	0.822869i	$-0.692372\pi$
$431$	−1.51237e17	−1.13646	−0.568230	+	0.822869i	$0.692372\pi$
$432$	−6.40037e15	−0.0473763
$433$	−2.18201e16	−0.159105	−0.0795526	−	0.996831i	$-0.525349\pi$
$433$	−2.18201e16	−0.159105	−0.0795526	+	0.996831i	$0.525349\pi$
$434$	3.76156e17	2.70200
$435$	4.78554e16	0.338650
$436$	−1.00532e17	−0.700875
$437$	1.46303e17	1.00490
$438$	2.22913e17	1.50853
$439$	5.33810e15	0.0355932	0.0177966	−	0.999842i	$-0.494335\pi$
$439$	5.33810e15	0.0355932	0.0177966	+	0.999842i	$0.494335\pi$
$440$	1.25726e17	0.826004
$441$	−3.61076e15	−0.0233748
$442$	4.38294e16	0.279590
$443$	−8.52324e16	−0.535773	−0.267886	−	0.963451i	$-0.586325\pi$
$443$	−8.52324e16	−0.535773	−0.267886	+	0.963451i	$0.586325\pi$
$444$	2.49409e17	1.54498
$445$	−1.45590e17	−0.888772
$446$	3.75072e17	2.25651
$447$	−1.39140e17	−0.824995
$448$	2.54971e17	1.48999
$449$	1.74520e17	1.00518	0.502590	−	0.864525i	$-0.332381\pi$
$449$	1.74520e17	1.00518	0.502590	+	0.864525i	$0.332381\pi$
$450$	−9.14526e16	−0.519176
$451$	−8.06091e16	−0.451062
$452$	7.54018e15	0.0415893
$453$	−2.04339e17	−1.11100
$454$	3.46165e17	1.85532
$455$	6.42086e16	0.339249
$456$	−1.96972e17	−1.02597
$457$	4.87787e16	0.250481	0.125241	−	0.992126i	$-0.460030\pi$
$457$	4.87787e16	0.250481	0.125241	+	0.992126i	$0.460030\pi$
$458$	−3.33950e17	−1.69066
$459$	2.58885e16	0.129218
$460$	3.29459e17	1.62135
$461$	2.05231e17	0.995832	0.497916	−	0.867225i	$-0.334099\pi$
$461$	2.05231e17	0.995832	0.497916	+	0.867225i	$0.334099\pi$
$462$	−9.53299e16	−0.456096
$463$	3.12712e17	1.47526	0.737628	−	0.675207i	$-0.235946\pi$
$463$	3.12712e17	1.47526	0.737628	+	0.675207i	$0.235946\pi$
$464$	2.22634e16	0.103568
$465$	2.97953e17	1.36680
$466$	−5.12210e17	−2.31707
$467$	−1.40290e17	−0.625844	−0.312922	−	0.949779i	$-0.601308\pi$
$467$	−1.40290e17	−0.625844	−0.312922	+	0.949779i	$0.601308\pi$
$468$	3.29446e16	0.144939
$469$	3.40781e17	1.47860
$470$	−4.19872e17	−1.79672
$471$	2.15866e17	0.911061
$472$	3.73235e16	0.155367
$473$	−2.15140e17	−0.883333
$474$	1.15648e17	0.468361
$475$	−3.51815e17	−1.40542
$476$	2.83136e17	1.11571
$477$	8.09593e16	0.314700
$478$	−7.82619e16	−0.300102
$479$	3.43199e16	0.129827	0.0649135	−	0.997891i	$-0.479323\pi$
$479$	3.43199e16	0.129827	0.0649135	+	0.997891i	$0.479323\pi$
$480$	1.69786e17	0.633626
$481$	−1.06432e17	−0.391859
$482$	2.45519e16	0.0891823
$483$	−1.04839e17	−0.375722
$484$	−3.67229e17	−1.29850
$485$	5.01133e17	1.74836
$486$	3.07518e16	0.105861
$487$	−5.94598e16	−0.201969	−0.100985	−	0.994888i	$-0.532199\pi$
$487$	−5.94598e16	−0.201969	−0.100985	+	0.994888i	$0.532199\pi$
$488$	−5.76403e17	−1.93196
$489$	−2.00644e17	−0.663618
$490$	−4.94325e16	−0.161338
$491$	4.54593e17	1.46417	0.732087	−	0.681211i	$-0.238546\pi$
$491$	4.54593e17	1.46417	0.732087	+	0.681211i	$0.238546\pi$
$492$	2.84390e17	0.903942
$493$	−9.00519e16	−0.282480
$494$	2.00285e17	0.620045
$495$	−7.55108e16	−0.230715
$496$	1.38614e17	0.418000
$497$	2.77023e16	0.0824517
$498$	2.01800e17	0.592830
$499$	−1.80055e17	−0.522097	−0.261049	−	0.965326i	$-0.584068\pi$
$499$	−1.80055e17	−0.522097	−0.261049	+	0.965326i	$0.584068\pi$
$500$	4.71418e16	0.134928
$501$	−2.16868e17	−0.612704
$502$	6.28532e17	1.75288
$503$	2.50098e17	0.688523	0.344261	−	0.938874i	$-0.388129\pi$
$503$	2.50098e17	0.688523	0.344261	+	0.938874i	$0.388129\pi$
$504$	1.41148e17	0.383599
$505$	−9.23704e17	−2.47821
$506$	2.08736e17	0.552865
$507$	2.06737e17	0.540589
$508$	9.92115e17	2.56123
$509$	−6.69517e16	−0.170646	−0.0853229	−	0.996353i	$-0.527192\pi$
$509$	−6.69517e16	−0.170646	−0.0853229	+	0.996353i	$0.527192\pi$
$510$	3.54422e17	0.891896
$511$	−6.14502e17	−1.52682
$512$	1.98740e17	0.487563
$513$	1.18301e17	0.286567
$514$	−7.88630e17	−1.88631
$515$	2.96915e17	0.701271
$516$	7.59017e17	1.77023
$517$	−1.68332e17	−0.387685
$518$	−1.08654e18	−2.47118
$519$	−2.11306e17	−0.474596
$520$	1.89284e17	0.419848
$521$	−4.25805e17	−0.932750	−0.466375	−	0.884587i	$-0.654440\pi$
$521$	−4.25805e17	−0.932750	−0.466375	+	0.884587i	$0.654440\pi$
$522$	−1.06969e17	−0.231419
$523$	8.03111e17	1.71599	0.857994	−	0.513659i	$-0.171710\pi$
$523$	8.03111e17	1.71599	0.857994	+	0.513659i	$0.171710\pi$
$524$	2.89345e17	0.610609
$525$	2.52107e17	0.525472
$526$	6.82119e17	1.40428
$527$	−5.60673e17	−1.14009
$528$	−3.51293e16	−0.0705583
$529$	−2.74479e17	−0.544563
$530$	1.10836e18	2.17214
$531$	−2.24165e16	−0.0433963
$532$	1.29383e18	2.47430
$533$	−1.21359e17	−0.229269
$534$	3.25429e17	0.607349
$535$	−1.11252e18	−2.05121
$536$	1.00461e18	1.82989
$537$	9.21824e16	0.165888
$538$	−6.45740e17	−1.14808
$539$	−1.98181e16	−0.0348126
$540$	2.66403e17	0.462359
$541$	3.74227e17	0.641730	0.320865	−	0.947125i	$-0.396026\pi$
$541$	3.74227e17	0.641730	0.320865	+	0.947125i	$0.396026\pi$
$542$	−2.21522e17	−0.375338
$543$	−1.08462e16	−0.0181584
$544$	−3.19494e17	−0.528531
$545$	3.46910e17	0.567073
$546$	−1.43522e17	−0.231828
$547$	−6.31631e17	−1.00820	−0.504099	−	0.863646i	$-0.668175\pi$
$547$	−6.31631e17	−1.00820	−0.504099	+	0.863646i	$0.668175\pi$
$548$	6.37370e17	1.00535
$549$	3.46186e17	0.539622
$550$	−5.01949e17	−0.773219
$551$	−4.11505e17	−0.626455
$552$	−3.09061e17	−0.464986
$553$	−3.18808e17	−0.474040
$554$	7.54153e17	1.10827
$555$	−8.60650e17	−1.25004
$556$	−1.22238e17	−0.175477
$557$	−1.17208e18	−1.66303	−0.831514	−	0.555504i	$-0.812525\pi$
$557$	−1.17208e18	−1.66303	−0.831514	+	0.555504i	$0.812525\pi$
$558$	−6.65999e17	−0.934009
$559$	−3.23900e17	−0.448988
$560$	2.41550e17	0.330967
$561$	1.42092e17	0.192447
$562$	1.87764e18	2.51377
$563$	3.53756e17	0.468165	0.234083	−	0.972217i	$-0.424791\pi$
$563$	3.53756e17	0.468165	0.234083	+	0.972217i	$0.424791\pi$
$564$	5.93878e17	0.776932
$565$	−2.60193e16	−0.0336497
$566$	8.50596e17	1.08747
$567$	−8.47734e16	−0.107145
$568$	8.16653e16	0.102041
$569$	1.31518e18	1.62463	0.812317	−	0.583216i	$-0.198206\pi$
$569$	1.31518e18	1.62463	0.812317	+	0.583216i	$0.198206\pi$
$570$	1.61958e18	1.97795
$571$	1.16295e18	1.40419	0.702097	−	0.712081i	$-0.252247\pi$
$571$	1.16295e18	1.40419	0.702097	+	0.712081i	$0.252247\pi$
$572$	1.80821e17	0.215861
$573$	8.70880e16	0.102791
$574$	−1.23893e18	−1.44584
$575$	−5.52018e17	−0.636960
$576$	−4.51436e17	−0.515051
$577$	−7.79934e17	−0.879863	−0.439931	−	0.898031i	$-0.644997\pi$
$577$	−7.79934e17	−0.879863	−0.439931	+	0.898031i	$0.644997\pi$
$578$	8.12411e17	0.906244
$579$	−6.60131e17	−0.728148
$580$	−9.26669e17	−1.01075
$581$	−5.56301e17	−0.600019
$582$	−1.12016e18	−1.19476
$583$	4.44356e17	0.468689
$584$	−1.81153e18	−1.88956
$585$	−1.13684e17	−0.117270
$586$	2.16411e18	2.20772
$587$	2.89189e16	0.0291766	0.0145883	−	0.999894i	$-0.495356\pi$
$587$	2.89189e16	0.0291766	0.0145883	+	0.999894i	$0.495356\pi$
$588$	6.99186e16	0.0697654
$589$	−2.56208e18	−2.52838
$590$	−3.06888e17	−0.299531
$591$	−2.20491e17	−0.212849
$592$	−4.00393e17	−0.382292
$593$	−3.41632e17	−0.322629	−0.161315	−	0.986903i	$-0.551573\pi$
$593$	−3.41632e17	−0.322629	−0.161315	+	0.986903i	$0.551573\pi$
$594$	1.68785e17	0.157660
$595$	−9.77033e17	−0.902712
$596$	2.69429e18	2.46231
$597$	1.05197e18	0.950979
$598$	3.14258e17	0.281015
$599$	1.77459e18	1.56972	0.784862	−	0.619671i	$-0.212734\pi$
$599$	1.77459e18	1.56972	0.784862	+	0.619671i	$0.212734\pi$
$600$	7.43201e17	0.650314
$601$	2.66906e16	0.0231033	0.0115517	−	0.999933i	$-0.496323\pi$
$601$	2.66906e16	0.0231033	0.0115517	+	0.999933i	$0.496323\pi$
$602$	−3.30663e18	−2.83145
$603$	−6.03366e17	−0.511115
$604$	3.95682e18	3.31593
$605$	1.26722e18	1.05061
$606$	2.06471e18	1.69350
$607$	−1.50590e18	−1.22200	−0.610999	−	0.791632i	$-0.709232\pi$
$607$	−1.50590e18	−1.22200	−0.610999	+	0.791632i	$0.709232\pi$
$608$	−1.45998e18	−1.17212
$609$	2.94880e17	0.234225
$610$	4.73940e18	3.72460
$611$	−2.53429e17	−0.197056
$612$	−5.01303e17	−0.385670
$613$	1.10960e18	0.844641	0.422320	−	0.906447i	$-0.361216\pi$
$613$	1.10960e18	0.844641	0.422320	+	0.906447i	$0.361216\pi$
$614$	−1.12158e18	−0.844768
$615$	−9.81358e17	−0.731374
$616$	7.74710e17	0.571301
$617$	−1.50847e18	−1.10073	−0.550367	−	0.834923i	$-0.685512\pi$
$617$	−1.50847e18	−1.10073	−0.550367	+	0.834923i	$0.685512\pi$
$618$	−6.63679e17	−0.479218
$619$	1.81904e18	1.29973	0.649866	−	0.760049i	$-0.274825\pi$
$619$	1.81904e18	1.29973	0.649866	+	0.760049i	$0.274825\pi$
$620$	−5.76954e18	−4.07939
$621$	1.85621e17	0.129877
$622$	−1.66101e18	−1.15009
$623$	−8.97109e17	−0.614714
$624$	−5.28881e16	−0.0358640
$625$	−1.56910e18	−1.05301
$626$	−6.84389e17	−0.454537
$627$	6.49312e17	0.426790
$628$	−4.18001e18	−2.71919
$629$	1.61953e18	1.04270
$630$	−1.16057e18	−0.739536
$631$	2.37161e18	1.49573	0.747865	−	0.663851i	$-0.231079\pi$
$631$	2.37161e18	1.49573	0.747865	+	0.663851i	$0.231079\pi$
$632$	−9.39831e17	−0.586663
$633$	−9.17216e17	−0.566692
$634$	1.16948e18	0.715178
$635$	−3.42354e18	−2.07227
$636$	−1.56769e18	−0.939267
$637$	−2.98368e16	−0.0176948
$638$	−5.87112e17	−0.344656
$639$	−4.90480e16	−0.0285014
$640$	−4.27237e18	−2.45753
$641$	1.05065e18	0.598247	0.299124	−	0.954214i	$-0.403306\pi$
$641$	1.05065e18	0.598247	0.299124	+	0.954214i	$0.403306\pi$
$642$	2.48676e18	1.40171
$643$	−1.89488e18	−1.05733	−0.528666	−	0.848830i	$-0.677308\pi$
$643$	−1.89488e18	−1.05733	−0.528666	+	0.848830i	$0.677308\pi$
$644$	2.03009e18	1.12139
$645$	−2.61918e18	−1.43228
$646$	−3.04765e18	−1.64988
$647$	−9.36382e17	−0.501851	−0.250926	−	0.968006i	$-0.580735\pi$
$647$	−9.36382e17	−0.501851	−0.250926	+	0.968006i	$0.580735\pi$
$648$	−2.49908e17	−0.132600
$649$	−1.23036e17	−0.0646309
$650$	−7.55699e17	−0.393018
$651$	1.83596e18	0.945335
$652$	3.88525e18	1.98066
$653$	3.15369e17	0.159178	0.0795890	−	0.996828i	$-0.474639\pi$
$653$	3.15369e17	0.159178	0.0795890	+	0.996828i	$0.474639\pi$
$654$	−7.75429e17	−0.387514
$655$	−9.98460e17	−0.494040
$656$	−4.56549e17	−0.223672
$657$	1.08800e18	0.527781
$658$	−2.58721e18	−1.24269
$659$	2.25110e18	1.07063	0.535314	−	0.844653i	$-0.320193\pi$
$659$	2.25110e18	1.07063	0.535314	+	0.844653i	$0.320193\pi$
$660$	1.46219e18	0.688600
$661$	1.12700e18	0.525552	0.262776	−	0.964857i	$-0.415362\pi$
$661$	1.12700e18	0.525552	0.262776	+	0.964857i	$0.415362\pi$
$662$	2.68860e18	1.24151
$663$	2.13924e17	0.0978188
$664$	−1.63995e18	−0.742572
$665$	−4.46469e18	−2.00194
$666$	1.92377e18	0.854221
$667$	−6.45675e17	−0.283920
$668$	4.19942e18	1.82870
$669$	1.83066e18	0.789476
$670$	−8.26027e18	−3.52783
$671$	1.90009e18	0.803670
$672$	1.04620e18	0.438244
$673$	−1.25240e18	−0.519570	−0.259785	−	0.965666i	$-0.583652\pi$
$673$	−1.25240e18	−0.519570	−0.259785	+	0.965666i	$0.583652\pi$
$674$	−3.41541e18	−1.40331
$675$	−4.46365e17	−0.181642
$676$	−4.00325e18	−1.61346
$677$	1.57557e18	0.628944	0.314472	−	0.949267i	$-0.398173\pi$
$677$	1.57557e18	0.628944	0.314472	+	0.949267i	$0.398173\pi$
$678$	5.81596e16	0.0229947
$679$	3.08793e18	1.20924
$680$	−2.88025e18	−1.11718
$681$	1.68957e18	0.649113
$682$	−3.65542e18	−1.39104
$683$	−1.11606e18	−0.420681	−0.210341	−	0.977628i	$-0.567457\pi$
$683$	−1.11606e18	−0.420681	−0.210341	+	0.977628i	$0.567457\pi$
$684$	−2.29078e18	−0.855300
$685$	−2.19941e18	−0.813424
$686$	−4.64827e18	−1.70288
$687$	−1.62996e18	−0.591503
$688$	−1.21850e18	−0.438027
$689$	6.68990e17	0.238229
$690$	2.54121e18	0.896442
$691$	−2.19065e18	−0.765537	−0.382769	−	0.923844i	$-0.625029\pi$
$691$	−2.19065e18	−0.765537	−0.382769	+	0.923844i	$0.625029\pi$
$692$	4.09172e18	1.41650
$693$	−4.65290e17	−0.159572
$694$	2.95533e18	1.00408
$695$	4.21813e17	0.141977
$696$	8.69295e17	0.289872
$697$	1.84667e18	0.610065
$698$	5.32129e18	1.74163
$699$	−2.50001e18	−0.810663
$700$	−4.88178e18	−1.56834
$701$	−2.28198e18	−0.726347	−0.363173	−	0.931722i	$-0.618307\pi$
$701$	−2.28198e18	−0.726347	−0.363173	+	0.931722i	$0.618307\pi$
$702$	2.54111e17	0.0801369
$703$	7.40067e18	2.31239
$704$	−2.47777e18	−0.767075
$705$	−2.04933e18	−0.628611
$706$	−1.91636e18	−0.582434
$707$	−5.69177e18	−1.71404
$708$	4.34071e17	0.129522
$709$	−7.56300e17	−0.223611	−0.111806	−	0.993730i	$-0.535663\pi$
$709$	−7.56300e17	−0.223611	−0.111806	+	0.993730i	$0.535663\pi$
$710$	−6.71483e17	−0.196723
$711$	5.64461e17	0.163863
$712$	−2.64464e18	−0.760758
$713$	−4.02004e18	−1.14590
$714$	2.18391e18	0.616874
$715$	−6.23968e17	−0.174652
$716$	−1.78501e18	−0.495116
$717$	−3.81984e17	−0.104995
$718$	8.88580e18	2.42040
$719$	−3.17132e18	−0.856057	−0.428029	−	0.903765i	$-0.640792\pi$
$719$	−3.17132e18	−0.856057	−0.428029	+	0.903765i	$0.640792\pi$
$720$	−4.27674e17	−0.114407
$721$	1.82956e18	0.485029
$722$	−7.64564e18	−2.00873
$723$	1.19834e17	0.0312018
$724$	2.10025e17	0.0541964
$725$	1.55266e18	0.397081
$726$	−2.83254e18	−0.717941
$727$	−9.61106e17	−0.241434	−0.120717	−	0.992687i	$-0.538519\pi$
$727$	−9.61106e17	−0.241434	−0.120717	+	0.992687i	$0.538519\pi$
$728$	1.16635e18	0.290385
$729$	1.50095e17	0.0370370
$730$	1.48951e19	3.64287
$731$	4.92864e18	1.19472
$732$	−6.70353e18	−1.61058
$733$	−2.61777e18	−0.623384	−0.311692	−	0.950183i	$-0.600896\pi$
$733$	−2.61777e18	−0.623384	−0.311692	+	0.950183i	$0.600896\pi$
$734$	−2.02150e18	−0.477144
$735$	−2.41272e17	−0.0564468
$736$	−2.29078e18	−0.531225
$737$	−3.31165e18	−0.761213
$738$	2.19358e18	0.499789
$739$	1.34184e17	0.0303048	0.0151524	−	0.999885i	$-0.495177\pi$
$739$	1.34184e17	0.0303048	0.0151524	+	0.999885i	$0.495177\pi$
$740$	1.66656e19	3.73091
$741$	9.77558e17	0.216932
$742$	6.82959e18	1.50234
$743$	−5.76566e18	−1.25725	−0.628625	−	0.777709i	$-0.716382\pi$
$743$	−5.76566e18	−1.25725	−0.628625	+	0.777709i	$0.716382\pi$
$744$	5.41232e18	1.16993
$745$	−9.29733e18	−1.99224
$746$	3.10073e18	0.658660
$747$	9.84952e17	0.207411
$748$	−2.75147e18	−0.574386
$749$	−6.85525e18	−1.41870
$750$	3.63618e17	0.0746015
$751$	4.90445e18	0.997541	0.498770	−	0.866734i	$-0.333785\pi$
$751$	4.90445e18	0.997541	0.498770	+	0.866734i	$0.333785\pi$
$752$	−9.53391e17	−0.192245
$753$	3.06776e18	0.613274
$754$	−8.83914e17	−0.175185
$755$	−1.36540e19	−2.68290
$756$	1.64155e18	0.319788
$757$	−8.55799e18	−1.65291	−0.826454	−	0.563005i	$-0.809645\pi$
$757$	−8.55799e18	−1.65291	−0.826454	+	0.563005i	$0.809645\pi$
$758$	2.28806e18	0.438145
$759$	1.01881e18	0.193428
$760$	−1.31617e19	−2.47756
$761$	−5.91805e18	−1.10453	−0.552266	−	0.833668i	$-0.686237\pi$
$761$	−5.91805e18	−1.10453	−0.552266	+	0.833668i	$0.686237\pi$
$762$	7.65247e18	1.41610
$763$	2.13762e18	0.392213
$764$	−1.68637e18	−0.306793
$765$	1.72987e18	0.312044
$766$	5.15411e18	0.921863
$767$	−1.85234e17	−0.0328511
$768$	4.47687e18	0.787276
$769$	−9.66387e18	−1.68512	−0.842558	−	0.538605i	$-0.818952\pi$
$769$	−9.66387e18	−1.68512	−0.842558	+	0.538605i	$0.818952\pi$
$770$	−6.36996e18	−1.10141
$771$	−3.84918e18	−0.659955
$772$	1.27827e19	2.17326
$773$	−5.08802e18	−0.857792	−0.428896	−	0.903354i	$-0.641097\pi$
$773$	−5.08802e18	−0.857792	−0.428896	+	0.903354i	$0.641097\pi$
$774$	5.85452e18	0.978758
$775$	9.66702e18	1.60262
$776$	9.10309e18	1.49654
$777$	−5.30324e18	−0.864579
$778$	−9.05723e18	−1.46429
$779$	8.43863e18	1.35294
$780$	2.20136e18	0.350007
$781$	−2.69206e17	−0.0424477
$782$	−4.78193e18	−0.747755
$783$	−5.22097e17	−0.0809654
$784$	−1.12245e17	−0.0172628
$785$	1.44242e19	2.20008
$786$	2.23181e18	0.337606
$787$	1.66855e18	0.250325	0.125163	−	0.992136i	$-0.460055\pi$
$787$	1.66855e18	0.250325	0.125163	+	0.992136i	$0.460055\pi$
$788$	4.26958e18	0.635279
$789$	3.32931e18	0.491308
$790$	7.72765e18	1.13102
$791$	−1.60328e17	−0.0232736
$792$	−1.37165e18	−0.197484
$793$	2.86064e18	0.408496
$794$	−9.39276e18	−1.33033
$795$	5.40972e18	0.759955
$796$	−2.03703e19	−2.83833
$797$	8.13831e18	1.12475	0.562374	−	0.826883i	$-0.309888\pi$
$797$	8.13831e18	1.12475	0.562374	+	0.826883i	$0.309888\pi$
$798$	9.97970e18	1.36804
$799$	3.85632e18	0.524347
$800$	5.50866e18	0.742953
$801$	1.58837e18	0.212490
$802$	−7.21058e18	−0.956835
$803$	5.97163e18	0.786035
$804$	1.16835e19	1.52549
$805$	−7.00535e18	−0.907312
$806$	−5.50334e18	−0.707048
$807$	−3.15175e18	−0.401674
$808$	−1.67791e19	−2.12126
$809$	2.88364e18	0.361640	0.180820	−	0.983516i	$-0.442125\pi$
$809$	2.88364e18	0.361640	0.180820	+	0.983516i	$0.442125\pi$
$810$	2.05484e18	0.255638
$811$	1.21256e19	1.49647	0.748235	−	0.663434i	$-0.230902\pi$
$811$	1.21256e19	1.49647	0.748235	+	0.663434i	$0.230902\pi$
$812$	−5.71004e18	−0.699077
$813$	−1.08121e18	−0.131318
$814$	1.05588e19	1.27221
$815$	−1.34070e19	−1.60254
$816$	8.04775e17	0.0954308
$817$	2.25222e19	2.64952
$818$	1.72242e19	2.01022
$819$	−7.00508e17	−0.0811087
$820$	1.90030e19	2.18289
$821$	1.46547e19	1.67011	0.835056	−	0.550165i	$-0.185435\pi$
$821$	1.46547e19	1.67011	0.835056	+	0.550165i	$0.185435\pi$
$822$	4.91622e18	0.555859
$823$	1.18816e18	0.133283	0.0666415	−	0.997777i	$-0.478772\pi$
$823$	1.18816e18	0.133283	0.0666415	+	0.997777i	$0.478772\pi$
$824$	5.39347e18	0.600263
$825$	−2.44993e18	−0.270522
$826$	−1.89102e18	−0.207169
$827$	−4.76881e18	−0.518351	−0.259175	−	0.965830i	$-0.583451\pi$
$827$	−4.76881e18	−0.518351	−0.259175	+	0.965830i	$0.583451\pi$
$828$	−3.59436e18	−0.387636
$829$	8.43096e18	0.902137	0.451069	−	0.892489i	$-0.351043\pi$
$829$	8.43096e18	0.902137	0.451069	+	0.892489i	$0.351043\pi$
$830$	1.34843e19	1.43160
$831$	3.68090e18	0.387746
$832$	−3.73035e18	−0.389895
$833$	4.54013e17	0.0470843
$834$	−9.42856e17	−0.0970211
$835$	−1.44912e19	−1.47959
$836$	−1.25732e19	−1.27381
$837$	−3.25063e18	−0.326777
$838$	−2.70781e19	−2.70104
$839$	1.54031e19	1.52460	0.762299	−	0.647225i	$-0.224070\pi$
$839$	1.54031e19	1.52460	0.762299	+	0.647225i	$0.224070\pi$
$840$	9.43155e18	0.926334
$841$	−8.44454e18	−0.823004
$842$	−1.04804e19	−1.01356
$843$	9.16443e18	0.879481
$844$	1.77609e19	1.69137
$845$	1.38142e19	1.30544
$846$	4.58075e18	0.429566
$847$	7.80846e18	0.726647
$848$	2.51672e18	0.232413
$849$	4.15162e18	0.380468
$850$	1.14991e19	1.04579
$851$	1.16121e19	1.04801
$852$	9.49763e17	0.0850664
$853$	−5.34495e18	−0.475089	−0.237544	−	0.971377i	$-0.576343\pi$
$853$	−5.34495e18	−0.475089	−0.237544	+	0.971377i	$0.576343\pi$
$854$	2.92037e19	2.57609
$855$	7.90491e18	0.692018
$856$	−2.02090e19	−1.75576
$857$	1.69458e19	1.46112	0.730559	−	0.682849i	$-0.239259\pi$
$857$	1.69458e19	1.46112	0.730559	+	0.682849i	$0.239259\pi$
$858$	1.39472e18	0.119349
$859$	−2.10751e18	−0.178984	−0.0894919	−	0.995988i	$-0.528524\pi$
$859$	−2.10751e18	−0.178984	−0.0894919	+	0.995988i	$0.528524\pi$
$860$	5.07177e19	4.27483
$861$	−6.04703e18	−0.505850
$862$	2.25886e19	1.87540
$863$	−2.02123e19	−1.66550	−0.832752	−	0.553646i	$-0.813236\pi$
$863$	−2.02123e19	−1.66550	−0.832752	+	0.553646i	$0.813236\pi$
$864$	−1.85234e18	−0.151489
$865$	−1.41195e19	−1.14608
$866$	3.25904e18	0.262557
$867$	3.96525e18	0.317063
$868$	−3.55513e19	−2.82148
$869$	3.09812e18	0.244045
$870$	−7.14767e18	−0.558842
$871$	−4.98579e18	−0.386915
$872$	6.30162e18	0.485395
$873$	−5.46730e18	−0.418004
$874$	−2.18517e19	−1.65829
$875$	−1.00238e18	−0.0755061
$876$	−2.10680e19	−1.57524
$877$	1.75816e19	1.30485	0.652424	−	0.757854i	$-0.273752\pi$
$877$	1.75816e19	1.30485	0.652424	+	0.757854i	$0.273752\pi$
$878$	−7.97297e17	−0.0587362
$879$	1.05627e19	0.772405
$880$	−2.34734e18	−0.170388
$881$	−5.02565e18	−0.362117	−0.181058	−	0.983472i	$-0.557952\pi$
$881$	−5.02565e18	−0.362117	−0.181058	+	0.983472i	$0.557952\pi$
$882$	5.39302e17	0.0385733
$883$	−1.33118e19	−0.945128	−0.472564	−	0.881296i	$-0.656671\pi$
$883$	−1.33118e19	−0.945128	−0.472564	+	0.881296i	$0.656671\pi$
$884$	−4.14242e18	−0.291954
$885$	−1.49787e18	−0.104796
$886$	1.27303e19	0.884136
$887$	2.19979e19	1.51662	0.758311	−	0.651893i	$-0.226025\pi$
$887$	2.19979e19	1.51662	0.758311	+	0.651893i	$0.226025\pi$
$888$	−1.56337e19	−1.06999
$889$	−2.10955e19	−1.43327
$890$	2.17452e19	1.46666
$891$	8.23814e17	0.0551600
$892$	−3.54488e19	−2.35630
$893$	1.76220e19	1.16284
$894$	2.07818e19	1.36141
$895$	6.15964e18	0.400595
$896$	−2.63259e19	−1.69973
$897$	1.53384e18	0.0983173
$898$	−2.60662e19	−1.65875
$899$	1.13072e19	0.714357
$900$	8.64338e18	0.542135
$901$	−1.01797e19	−0.633906
$902$	1.20397e19	0.744346
$903$	−1.61391e19	−0.990626
$904$	−4.72641e17	−0.0288029
$905$	−7.24745e17	−0.0438500
$906$	3.05201e19	1.83338
$907$	−2.32996e19	−1.38963	−0.694817	−	0.719187i	$-0.744515\pi$
$907$	−2.32996e19	−1.38963	−0.694817	+	0.719187i	$0.744515\pi$
$908$	−3.27168e19	−1.93737
$909$	1.00775e19	0.592499
$910$	−9.59017e18	−0.559831
$911$	1.00388e19	0.581852	0.290926	−	0.956746i	$-0.406037\pi$
$911$	1.00388e19	0.581852	0.290926	+	0.956746i	$0.406037\pi$
$912$	3.67754e18	0.211636
$913$	5.40604e18	0.308901
$914$	−7.28557e18	−0.413345
$915$	2.31322e19	1.30311
$916$	3.15624e19	1.76542
$917$	−6.15241e18	−0.341700
$918$	−3.86670e18	−0.213237
$919$	−1.66327e19	−0.910776	−0.455388	−	0.890293i	$-0.650499\pi$
$919$	−1.66327e19	−0.910776	−0.455388	+	0.890293i	$0.650499\pi$
$920$	−2.06515e19	−1.12287
$921$	−5.47427e18	−0.295555
$922$	−3.06532e19	−1.64333
$923$	−4.05298e17	−0.0215756
$924$	9.00984e18	0.476266
$925$	−2.79236e19	−1.46572
$926$	−4.67065e19	−2.43448
$927$	−3.23931e18	−0.167662
$928$	6.44328e18	0.331166
$929$	−1.50596e19	−0.768619	−0.384309	−	0.923204i	$-0.625560\pi$
$929$	−1.50596e19	−0.768619	−0.384309	+	0.923204i	$0.625560\pi$
$930$	−4.45021e19	−2.25549
$931$	2.07468e18	0.104419
$932$	4.84101e19	2.41954
$933$	−8.10710e18	−0.402378
$934$	2.09536e19	1.03277
$935$	9.49464e18	0.464732
$936$	−2.06507e18	−0.100379
$937$	6.15475e18	0.297100	0.148550	−	0.988905i	$-0.452539\pi$
$937$	6.15475e18	0.297100	0.148550	+	0.988905i	$0.452539\pi$
$938$	−5.08990e19	−2.44000
$939$	−3.34039e18	−0.159027
$940$	3.96830e19	1.87618
$941$	1.57758e19	0.740730	0.370365	−	0.928886i	$-0.379233\pi$
$941$	1.57758e19	0.740730	0.370365	+	0.928886i	$0.379233\pi$
$942$	−3.22416e19	−1.50344
$943$	1.32407e19	0.613175
$944$	−6.96842e17	−0.0320492
$945$	−5.66458e18	−0.258738
$946$	3.21333e19	1.45768
$947$	1.02390e19	0.461298	0.230649	−	0.973037i	$-0.425915\pi$
$947$	1.02390e19	0.461298	0.230649	+	0.973037i	$0.425915\pi$
$948$	−1.09302e19	−0.489073
$949$	8.99046e18	0.399532
$950$	5.25470e19	2.31923
$951$	5.70806e18	0.250216
$952$	−1.77478e19	−0.772689
$953$	−8.30879e18	−0.359281	−0.179640	−	0.983732i	$-0.557493\pi$
$953$	−8.30879e18	−0.359281	−0.179640	+	0.983732i	$0.557493\pi$
$954$	−1.20921e19	−0.519321
$955$	5.81923e18	0.248224
$956$	7.39670e18	0.313373
$957$	−2.86560e18	−0.120583
$958$	−5.12601e18	−0.214242
$959$	−1.35525e19	−0.562600
$960$	−3.01650e19	−1.24377
$961$	4.59820e19	1.88315
$962$	1.58966e19	0.646648
$963$	1.21375e19	0.490409
$964$	−2.32045e18	−0.0931261
$965$	−4.41101e19	−1.75837
$966$	1.56587e19	0.620018
$967$	1.27015e19	0.499556	0.249778	−	0.968303i	$-0.419642\pi$
$967$	1.27015e19	0.499556	0.249778	+	0.968303i	$0.419642\pi$
$968$	2.30190e19	0.899284
$969$	−1.48751e19	−0.577237
$970$	−7.48490e19	−2.88516
$971$	1.32520e19	0.507408	0.253704	−	0.967282i	$-0.418351\pi$
$971$	1.32520e19	0.507408	0.253704	+	0.967282i	$0.418351\pi$
$972$	−2.90642e18	−0.110542
$973$	2.59917e18	0.0981976
$974$	8.88090e18	0.333291
$975$	−3.68845e18	−0.137503
$976$	1.07616e19	0.398523
$977$	−4.57821e19	−1.68415	−0.842075	−	0.539360i	$-0.818666\pi$
$977$	−4.57821e19	−1.68415	−0.842075	+	0.539360i	$0.818666\pi$
$978$	2.99681e19	1.09511
$979$	8.71795e18	0.316466
$980$	4.67197e18	0.168473
$981$	−3.78474e18	−0.135578
$982$	−6.78978e19	−2.41619
$983$	1.72596e19	0.610143	0.305072	−	0.952329i	$-0.401320\pi$
$983$	1.72596e19	0.610143	0.305072	+	0.952329i	$0.401320\pi$
$984$	−1.78264e19	−0.626030
$985$	−1.47333e19	−0.514000
$986$	1.34501e19	0.466151
$987$	−1.26277e19	−0.434775
$988$	−1.89294e19	−0.647464
$989$	3.53385e19	1.20080
$990$	1.12783e19	0.380727
$991$	4.19067e19	1.40542	0.702708	−	0.711479i	$-0.251974\pi$
$991$	4.19067e19	1.40542	0.702708	+	0.711479i	$0.251974\pi$
$992$	4.01166e19	1.33659
$993$	1.31226e19	0.434361
$994$	−4.13761e18	−0.136062
$995$	7.02929e19	2.29648
$996$	−1.90725e19	−0.619046
$997$	1.09283e19	0.352399	0.176199	−	0.984354i	$-0.443620\pi$
$997$	1.09283e19	0.352399	0.176199	+	0.984354i	$0.443620\pi$
$998$	2.68929e19	0.861568
$999$	9.38959e18	0.298862

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.14.a.b.1.5	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.14.a.b.1.5	✓	31	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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q-149.360 q^{2} -729.000 q^{3} +14116.3 q^{4} -48711.9 q^{5} +108883. q^{6} -300158. q^{7} -884852. q^{8} +531441. q^{9} +O(q^{10})\)	\(q-149.360 q^{2} -729.000 q^{3} +14116.3 q^{4} -48711.9 q^{5} +108883. q^{6} -300158. q^{7} -884852. q^{8} +531441. q^{9} +7.27559e6 q^{10} +2.91688e6 q^{11} -1.02908e7 q^{12} +4.39145e6 q^{13} +4.48315e7 q^{14} +3.55110e7 q^{15} +1.65205e7 q^{16} -6.68228e7 q^{17} -7.93758e7 q^{18} -3.05356e8 q^{19} -6.87632e8 q^{20} +2.18815e8 q^{21} -4.35664e8 q^{22} -4.79121e8 q^{23} +6.45057e8 q^{24} +1.15215e9 q^{25} -6.55906e8 q^{26} -3.87420e8 q^{27} -4.23712e9 q^{28} +1.34762e9 q^{29} -5.30391e9 q^{30} +8.39044e9 q^{31} +4.78122e9 q^{32} -2.12641e9 q^{33} +9.98063e9 q^{34} +1.46213e10 q^{35} +7.50198e9 q^{36} -2.42362e10 q^{37} +4.56079e10 q^{38} -3.20137e9 q^{39} +4.31028e10 q^{40} -2.76354e10 q^{41} -3.26821e10 q^{42} -7.37570e10 q^{43} +4.11756e10 q^{44} -2.58875e10 q^{45} +7.15614e10 q^{46} -5.77097e10 q^{47} -1.20434e10 q^{48} -6.79429e9 q^{49} -1.72084e11 q^{50} +4.87138e10 q^{51} +6.19911e10 q^{52} +1.52339e11 q^{53} +5.78650e10 q^{54} -1.42087e11 q^{55} +2.65595e11 q^{56} +2.22605e11 q^{57} -2.01281e11 q^{58} -4.21805e10 q^{59} +5.01284e11 q^{60} +6.51411e11 q^{61} -1.25319e12 q^{62} -1.59516e11 q^{63} -8.49457e11 q^{64} -2.13916e11 q^{65} +3.17599e11 q^{66} -1.13534e12 q^{67} -9.43291e11 q^{68} +3.49279e11 q^{69} -2.18383e12 q^{70} -9.22926e10 q^{71} -4.70247e11 q^{72} +2.04726e12 q^{73} +3.61991e12 q^{74} -8.39915e11 q^{75} -4.31050e12 q^{76} -8.75525e11 q^{77} +4.78155e11 q^{78} +1.06213e12 q^{79} -8.04744e11 q^{80} +2.82430e11 q^{81} +4.12761e12 q^{82} +1.85336e12 q^{83} +3.08886e12 q^{84} +3.25506e12 q^{85} +1.10163e13 q^{86} -9.82418e11 q^{87} -2.58101e12 q^{88} +2.98879e12 q^{89} +3.86655e12 q^{90} -1.31813e12 q^{91} -6.76342e12 q^{92} -6.11663e12 q^{93} +8.61950e12 q^{94} +1.48745e13 q^{95} -3.48551e12 q^{96} -1.02877e13 q^{97} +1.01479e12 q^{98} +1.55015e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q - 52 q^{2} - 22599 q^{3} + 126886 q^{4} + 33486 q^{5} + 37908 q^{6} - 1135539 q^{7} - 1519749 q^{8} + 16474671 q^{9}+O(q^{10}) \) 31 * q - 52 * q^2 - 22599 * q^3 + 126886 * q^4 + 33486 * q^5 + 37908 * q^6 - 1135539 * q^7 - 1519749 * q^8 + 16474671 * q^9	\( 31 q - 52 q^{2} - 22599 q^{3} + 126886 q^{4} + 33486 q^{5} + 37908 q^{6} - 1135539 q^{7} - 1519749 q^{8} + 16474671 q^{9} - 3854663 q^{10} + 3943968 q^{11} - 92499894 q^{12} - 48510022 q^{13} - 51427459 q^{14} - 24411294 q^{15} + 370110498 q^{16} + 83288419 q^{17} - 27634932 q^{18} - 180425297 q^{19} + 753620445 q^{20} + 827807931 q^{21} + 2300196142 q^{22} - 1305810279 q^{23} + 1107897021 q^{24} + 8070954867 q^{25} + 464550322 q^{26} - 12010035159 q^{27} - 9887169562 q^{28} + 6248352277 q^{29} + 2810049327 q^{30} - 26730150789 q^{31} - 24001343230 q^{32} - 2875152672 q^{33} - 36571033348 q^{34} + 10255900979 q^{35} + 67432422726 q^{36} - 43284776933 q^{37} - 36293696947 q^{38} + 35363806038 q^{39} - 105980683856 q^{40} - 9961079285 q^{41} + 37490617611 q^{42} - 51755851288 q^{43} - 59623729442 q^{44} + 17795833326 q^{45} - 202287132683 q^{46} - 82747063727 q^{47} - 269810553042 q^{48} + 535277836542 q^{49} + 526974390461 q^{50} - 60717257451 q^{51} + 544982341446 q^{52} + 561701818494 q^{53} + 20145865428 q^{54} - 521861534450 q^{55} - 228056576664 q^{56} + 131530041513 q^{57} + 10555409160 q^{58} - 1307596542871 q^{59} - 549389304405 q^{60} + 618193248201 q^{61} - 1486611437386 q^{62} - 603471981699 q^{63} + 679062548045 q^{64} - 1130583307122 q^{65} - 1676842987518 q^{66} - 4137387490592 q^{67} - 3901389300295 q^{68} + 951935693391 q^{69} - 819291947844 q^{70} - 3766439869810 q^{71} - 807656928309 q^{72} - 2386775553523 q^{73} + 3060770694642 q^{74} - 5883726098043 q^{75} - 847741068784 q^{76} + 1650423006137 q^{77} - 338657184738 q^{78} + 787155757766 q^{79} + 13999832121779 q^{80} + 8755315630911 q^{81} + 10083281915577 q^{82} + 8743877051639 q^{83} + 7207746610698 q^{84} + 15373177520565 q^{85} + 18939443838984 q^{86} - 4555048809933 q^{87} + 39713314506713 q^{88} + 11026795445259 q^{89} - 2048525959383 q^{90} + 23285721962531 q^{91} + 40411079823254 q^{92} + 19486279925181 q^{93} + 35237377585624 q^{94} + 13730236994039 q^{95} + 17496979214670 q^{96} + 10134565481560 q^{97} + 70916776240976 q^{98} + 2095986297888 q^{99}+O(q^{100}) \) 31 * q - 52 * q^2 - 22599 * q^3 + 126886 * q^4 + 33486 * q^5 + 37908 * q^6 - 1135539 * q^7 - 1519749 * q^8 + 16474671 * q^9 - 3854663 * q^10 + 3943968 * q^11 - 92499894 * q^12 - 48510022 * q^13 - 51427459 * q^14 - 24411294 * q^15 + 370110498 * q^16 + 83288419 * q^17 - 27634932 * q^18 - 180425297 * q^19 + 753620445 * q^20 + 827807931 * q^21 + 2300196142 * q^22 - 1305810279 * q^23 + 1107897021 * q^24 + 8070954867 * q^25 + 464550322 * q^26 - 12010035159 * q^27 - 9887169562 * q^28 + 6248352277 * q^29 + 2810049327 * q^30 - 26730150789 * q^31 - 24001343230 * q^32 - 2875152672 * q^33 - 36571033348 * q^34 + 10255900979 * q^35 + 67432422726 * q^36 - 43284776933 * q^37 - 36293696947 * q^38 + 35363806038 * q^39 - 105980683856 * q^40 - 9961079285 * q^41 + 37490617611 * q^42 - 51755851288 * q^43 - 59623729442 * q^44 + 17795833326 * q^45 - 202287132683 * q^46 - 82747063727 * q^47 - 269810553042 * q^48 + 535277836542 * q^49 + 526974390461 * q^50 - 60717257451 * q^51 + 544982341446 * q^52 + 561701818494 * q^53 + 20145865428 * q^54 - 521861534450 * q^55 - 228056576664 * q^56 + 131530041513 * q^57 + 10555409160 * q^58 - 1307596542871 * q^59 - 549389304405 * q^60 + 618193248201 * q^61 - 1486611437386 * q^62 - 603471981699 * q^63 + 679062548045 * q^64 - 1130583307122 * q^65 - 1676842987518 * q^66 - 4137387490592 * q^67 - 3901389300295 * q^68 + 951935693391 * q^69 - 819291947844 * q^70 - 3766439869810 * q^71 - 807656928309 * q^72 - 2386775553523 * q^73 + 3060770694642 * q^74 - 5883726098043 * q^75 - 847741068784 * q^76 + 1650423006137 * q^77 - 338657184738 * q^78 + 787155757766 * q^79 + 13999832121779 * q^80 + 8755315630911 * q^81 + 10083281915577 * q^82 + 8743877051639 * q^83 + 7207746610698 * q^84 + 15373177520565 * q^85 + 18939443838984 * q^86 - 4555048809933 * q^87 + 39713314506713 * q^88 + 11026795445259 * q^89 - 2048525959383 * q^90 + 23285721962531 * q^91 + 40411079823254 * q^92 + 19486279925181 * q^93 + 35237377585624 * q^94 + 13730236994039 * q^95 + 17496979214670 * q^96 + 10134565481560 * q^97 + 70916776240976 * q^98 + 2095986297888 * q^99

Introduction

L-functions

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