LMFDB - Embedded newform 531.8.a.c.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,8,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("531.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	531.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:	$165.876448532$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + \cdots + 24\!\cdots\!16 $ x^17 - 2x^16 - 1669x^15 + 2385x^14 + 1108684x^13 - 848131x^12 - 377920980x^11 + 12724944x^10 + 71331230512x^9 + 50741131904x^8 - 7480805165760x^7 - 10751966150272x^6 + 413177144536320x^5 + 886760582981376x^4 - 10454479722123264x^3 - 29180140031461376x^2 + 79787300207378432x + 248723246810300416
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	multiple of $ 2^{10}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$-20.9857$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+20.9857 q^{2} +312.401 q^{4} +465.631 q^{5} +286.579 q^{7} +3869.79 q^{8} +O(q^{10})$	\(q+20.9857 q^{2} +312.401 q^{4} +465.631 q^{5} +286.579 q^{7} +3869.79 q^{8} +9771.60 q^{10} +3497.66 q^{11} -4677.13 q^{13} +6014.07 q^{14} +41223.0 q^{16} +1899.70 q^{17} +6396.17 q^{19} +145463. q^{20} +73400.9 q^{22} -9375.09 q^{23} +138687. q^{25} -98153.0 q^{26} +89527.5 q^{28} +160070. q^{29} -272299. q^{31} +369763. q^{32} +39866.7 q^{34} +133440. q^{35} -35341.8 q^{37} +134228. q^{38} +1.80189e6 q^{40} +727812. q^{41} -712974. q^{43} +1.09267e6 q^{44} -196743. q^{46} +476612. q^{47} -741416. q^{49} +2.91045e6 q^{50} -1.46114e6 q^{52} +1.49764e6 q^{53} +1.62862e6 q^{55} +1.10900e6 q^{56} +3.35919e6 q^{58} +205379. q^{59} -1.04757e6 q^{61} -5.71439e6 q^{62} +2.48319e6 q^{64} -2.17782e6 q^{65} -2.56238e6 q^{67} +593470. q^{68} +2.80033e6 q^{70} -3.16847e6 q^{71} +1.62960e6 q^{73} -741674. q^{74} +1.99817e6 q^{76} +1.00235e6 q^{77} -2.11177e6 q^{79} +1.91947e7 q^{80} +1.52737e7 q^{82} -5.43782e6 q^{83} +884561. q^{85} -1.49623e7 q^{86} +1.35352e7 q^{88} +2.27504e6 q^{89} -1.34037e6 q^{91} -2.92879e6 q^{92} +1.00021e7 q^{94} +2.97825e6 q^{95} +1.00026e7 q^{97} -1.55591e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - 2 q^{2} + 1166 q^{4} + 318 q^{5} + 3145 q^{7} - 2355 q^{8}+O(q^{10}) $ 17 * q - 2 * q^2 + 1166 * q^4 + 318 * q^5 + 3145 * q^7 - 2355 * q^8	\( 17 q - 2 q^{2} + 1166 q^{4} + 318 q^{5} + 3145 q^{7} - 2355 q^{8} + 6521 q^{10} + 1764 q^{11} + 18192 q^{13} + 7827 q^{14} + 139226 q^{16} + 15507 q^{17} + 52083 q^{19} - 721 q^{20} - 234434 q^{22} - 63823 q^{23} + 202153 q^{25} + 367956 q^{26} + 182306 q^{28} + 502955 q^{29} + 347531 q^{31} + 243908 q^{32} - 330872 q^{34} - 92641 q^{35} + 447615 q^{37} - 775669 q^{38} + 2203270 q^{40} - 940335 q^{41} + 478562 q^{43} + 596924 q^{44} - 3078663 q^{46} - 703121 q^{47} + 1895082 q^{49} + 876967 q^{50} + 6278296 q^{52} + 1005974 q^{53} + 5212846 q^{55} - 3425294 q^{56} + 6710166 q^{58} + 3491443 q^{59} + 11510749 q^{61} - 5996234 q^{62} + 29496941 q^{64} - 11094180 q^{65} + 14007144 q^{67} - 19688159 q^{68} + 30909708 q^{70} - 5229074 q^{71} + 5452211 q^{73} - 12819662 q^{74} + 41929340 q^{76} - 9930777 q^{77} + 15275654 q^{79} - 36576105 q^{80} + 32025935 q^{82} - 7826609 q^{83} + 11836945 q^{85} - 51649136 q^{86} + 30223741 q^{88} + 6436185 q^{89} + 11633535 q^{91} - 43357972 q^{92} - 4494252 q^{94} - 23741055 q^{95} + 26377540 q^{97} - 26517816 q^{98}+O(q^{100}) \) 17 * q - 2 * q^2 + 1166 * q^4 + 318 * q^5 + 3145 * q^7 - 2355 * q^8 + 6521 * q^10 + 1764 * q^11 + 18192 * q^13 + 7827 * q^14 + 139226 * q^16 + 15507 * q^17 + 52083 * q^19 - 721 * q^20 - 234434 * q^22 - 63823 * q^23 + 202153 * q^25 + 367956 * q^26 + 182306 * q^28 + 502955 * q^29 + 347531 * q^31 + 243908 * q^32 - 330872 * q^34 - 92641 * q^35 + 447615 * q^37 - 775669 * q^38 + 2203270 * q^40 - 940335 * q^41 + 478562 * q^43 + 596924 * q^44 - 3078663 * q^46 - 703121 * q^47 + 1895082 * q^49 + 876967 * q^50 + 6278296 * q^52 + 1005974 * q^53 + 5212846 * q^55 - 3425294 * q^56 + 6710166 * q^58 + 3491443 * q^59 + 11510749 * q^61 - 5996234 * q^62 + 29496941 * q^64 - 11094180 * q^65 + 14007144 * q^67 - 19688159 * q^68 + 30909708 * q^70 - 5229074 * q^71 + 5452211 * q^73 - 12819662 * q^74 + 41929340 * q^76 - 9930777 * q^77 + 15275654 * q^79 - 36576105 * q^80 + 32025935 * q^82 - 7826609 * q^83 + 11836945 * q^85 - 51649136 * q^86 + 30223741 * q^88 + 6436185 * q^89 + 11633535 * q^91 - 43357972 * q^92 - 4494252 * q^94 - 23741055 * q^95 + 26377540 * q^97 - 26517816 * q^98

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 $

$2$	20.9857	1.85489	0.927447	−	0.373954i	$-0.121998\pi$
$2$	20.9857	1.85489	0.927447	+	0.373954i	$0.121998\pi$
$3$	0	0
$3$	0	0
$4$	312.401	2.44063
$5$	465.631	1.66589	0.832945	−	0.553355i	$-0.186653\pi$
$5$	465.631	1.66589	0.832945	+	0.553355i	$0.186653\pi$
$6$	0	0
$7$	286.579	0.315792	0.157896	−	0.987456i	$-0.449529\pi$
$7$	286.579	0.315792	0.157896	+	0.987456i	$0.449529\pi$
$8$	3869.79	2.67222
$9$	0	0
$10$	9771.60	3.09005
$11$	3497.66	0.792324	0.396162	−	0.918181i	$-0.370342\pi$
$11$	3497.66	0.792324	0.396162	+	0.918181i	$0.370342\pi$
$12$	0	0
$13$	−4677.13	−0.590443	−0.295221	−	0.955429i	$-0.595393\pi$
$13$	−4677.13	−0.590443	−0.295221	+	0.955429i	$0.595393\pi$
$14$	6014.07	0.585761
$15$	0	0
$16$	41223.0	2.51605
$17$	1899.70	0.0937810	0.0468905	−	0.998900i	$-0.485069\pi$
$17$	1899.70	0.0937810	0.0468905	+	0.998900i	$0.485069\pi$
$18$	0	0
$19$	6396.17	0.213935	0.106968	−	0.994263i	$-0.465886\pi$
$19$	6396.17	0.213935	0.106968	+	0.994263i	$0.465886\pi$
$20$	145463.	4.06583
$21$	0	0
$22$	73400.9	1.46968
$23$	−9375.09	−0.160668	−0.0803338	−	0.996768i	$-0.525599\pi$
$23$	−9375.09	−0.160668	−0.0803338	+	0.996768i	$0.525599\pi$
$24$	0	0
$25$	138687.	1.77519
$26$	−98153.0	−1.09521
$27$	0	0
$28$	89527.5	0.770732
$29$	160070.	1.21876	0.609379	−	0.792879i	$-0.291419\pi$
$29$	160070.	1.21876	0.609379	+	0.792879i	$0.291419\pi$
$30$	0	0
$31$	−272299.	−1.64165	−0.820823	−	0.571183i	$-0.806485\pi$
$31$	−272299.	−1.64165	−0.820823	+	0.571183i	$0.806485\pi$
$32$	369763.	1.99479
$33$	0	0
$34$	39866.7	0.173954
$35$	133440.	0.526075
$36$	0	0
$37$	−35341.8	−0.114705	−0.0573525	−	0.998354i	$-0.518266\pi$
$37$	−35341.8	−0.114705	−0.0573525	+	0.998354i	$0.518266\pi$
$38$	134228.	0.396827
$39$	0	0
$40$	1.80189e6	4.45163
$41$	727812.	1.64921	0.824605	−	0.565709i	$-0.191397\pi$
$41$	727812.	1.64921	0.824605	+	0.565709i	$0.191397\pi$
$42$	0	0
$43$	−712974.	−1.36752	−0.683761	−	0.729706i	$-0.739657\pi$
$43$	−712974.	−1.36752	−0.683761	+	0.729706i	$0.739657\pi$
$44$	1.09267e6	1.93377
$45$	0	0
$46$	−196743.	−0.298021
$47$	476612.	0.669611	0.334805	−	0.942287i	$-0.391329\pi$
$47$	476612.	0.669611	0.334805	+	0.942287i	$0.391329\pi$
$48$	0	0
$49$	−741416.	−0.900275
$50$	2.91045e6	3.29279
$51$	0	0
$52$	−1.46114e6	−1.44105
$53$	1.49764e6	1.38179	0.690894	−	0.722956i	$-0.257217\pi$
$53$	1.49764e6	1.38179	0.690894	+	0.722956i	$0.257217\pi$
$54$	0	0
$55$	1.62862e6	1.31993
$56$	1.10900e6	0.843866
$57$	0	0
$58$	3.35919e6	2.26067
$59$	205379.	0.130189
$59$	205379.	0.130189
$60$	0	0
$61$	−1.04757e6	−0.590919	−0.295460	−	0.955355i	$-0.595473\pi$
$61$	−1.04757e6	−0.590919	−0.295460	+	0.955355i	$0.595473\pi$
$62$	−5.71439e6	−3.04508
$63$	0	0
$64$	2.48319e6	1.18408
$65$	−2.17782e6	−0.983613
$66$	0	0
$67$	−2.56238e6	−1.04083	−0.520416	−	0.853913i	$-0.674223\pi$
$67$	−2.56238e6	−1.04083	−0.520416	+	0.853913i	$0.674223\pi$
$68$	593470.	0.228885
$69$	0	0
$70$	2.80033e6	0.975813
$71$	−3.16847e6	−1.05062	−0.525309	−	0.850912i	$-0.676050\pi$
$71$	−3.16847e6	−1.05062	−0.525309	+	0.850912i	$0.676050\pi$
$72$	0	0
$73$	1.62960e6	0.490287	0.245144	−	0.969487i	$-0.421165\pi$
$73$	1.62960e6	0.490287	0.245144	+	0.969487i	$0.421165\pi$
$74$	−741674.	−0.212766
$75$	0	0
$76$	1.99817e6	0.522138
$77$	1.00235e6	0.250210
$78$	0	0
$79$	−2.11177e6	−0.481893	−0.240947	−	0.970538i	$-0.577458\pi$
$79$	−2.11177e6	−0.481893	−0.240947	+	0.970538i	$0.577458\pi$
$80$	1.91947e7	4.19147
$81$	0	0
$82$	1.52737e7	3.05911
$83$	−5.43782e6	−1.04388	−0.521941	−	0.852982i	$-0.674792\pi$
$83$	−5.43782e6	−1.04388	−0.521941	+	0.852982i	$0.674792\pi$
$84$	0	0
$85$	884561.	0.156229
$86$	−1.49623e7	−2.53661
$87$	0	0
$88$	1.35352e7	2.11727
$89$	2.27504e6	0.342078	0.171039	−	0.985264i	$-0.445288\pi$
$89$	2.27504e6	0.342078	0.171039	+	0.985264i	$0.445288\pi$
$90$	0	0
$91$	−1.34037e6	−0.186457
$92$	−2.92879e6	−0.392130
$93$	0	0
$94$	1.00021e7	1.24206
$95$	2.97825e6	0.356393
$96$	0	0
$97$	1.00026e7	1.11279	0.556395	−	0.830918i	$-0.312184\pi$
$97$	1.00026e7	1.11279	0.556395	+	0.830918i	$0.312184\pi$
$98$	−1.55591e7	−1.66992
$99$	0	0
$100$	4.33259e7	4.33259
$101$	4.08392e6	0.394414	0.197207	−	0.980362i	$-0.436813\pi$
$101$	4.08392e6	0.394414	0.197207	+	0.980362i	$0.436813\pi$
$102$	0	0
$103$	−1.63687e7	−1.47599	−0.737996	−	0.674805i	$-0.764228\pi$
$103$	−1.63687e7	−1.47599	−0.737996	+	0.674805i	$0.764228\pi$
$104$	−1.80995e7	−1.57779
$105$	0	0
$106$	3.14290e7	2.56307
$107$	1.43241e7	1.13038	0.565189	−	0.824961i	$-0.308803\pi$
$107$	1.43241e7	1.13038	0.565189	+	0.824961i	$0.308803\pi$
$108$	0	0
$109$	−666192.	−0.0492728	−0.0246364	−	0.999696i	$-0.507843\pi$
$109$	−666192.	−0.0492728	−0.0246364	+	0.999696i	$0.507843\pi$
$110$	3.41777e7	2.44832
$111$	0	0
$112$	1.18137e7	0.794550
$113$	−2.37986e7	−1.55159	−0.775795	−	0.630986i	$-0.782651\pi$
$113$	−2.37986e7	−1.55159	−0.775795	+	0.630986i	$0.782651\pi$
$114$	0	0
$115$	−4.36533e6	−0.267655
$116$	5.00060e7	2.97454
$117$	0	0
$118$	4.31003e6	0.241487
$119$	544415.	0.0296153
$120$	0	0
$121$	−7.25356e6	−0.372222
$122$	−2.19840e7	−1.09609
$123$	0	0
$124$	−8.50663e7	−4.00665
$125$	2.81995e7	1.29139
$126$	0	0
$127$	7.78497e6	0.337244	0.168622	−	0.985681i	$-0.446068\pi$
$127$	7.78497e6	0.337244	0.168622	+	0.985681i	$0.446068\pi$
$128$	4.78197e6	0.201545
$129$	0	0
$130$	−4.57031e7	−1.82450
$131$	2.68679e7	1.04420	0.522101	−	0.852884i	$-0.325148\pi$
$131$	2.68679e7	1.04420	0.522101	+	0.852884i	$0.325148\pi$
$132$	0	0
$133$	1.83301e6	0.0675590
$134$	−5.37733e7	−1.93063
$135$	0	0
$136$	7.35146e6	0.250604
$137$	5.02442e7	1.66941	0.834707	−	0.550695i	$-0.185637\pi$
$137$	5.02442e7	1.66941	0.834707	+	0.550695i	$0.185637\pi$
$138$	0	0
$139$	−5.46930e7	−1.72735	−0.863675	−	0.504049i	$-0.831843\pi$
$139$	−5.46930e7	−1.72735	−0.863675	+	0.504049i	$0.831843\pi$
$140$	4.16868e7	1.28396
$141$	0	0
$142$	−6.64926e7	−1.94879
$143$	−1.63590e7	−0.467822
$144$	0	0
$145$	7.45335e7	2.03032
$146$	3.41983e7	0.909431
$147$	0	0
$148$	−1.10408e7	−0.279953
$149$	−3.71390e7	−0.919769	−0.459884	−	0.887979i	$-0.652109\pi$
$149$	−3.71390e7	−0.919769	−0.459884	+	0.887979i	$0.652109\pi$
$150$	0	0
$151$	1.36501e7	0.322639	0.161319	−	0.986902i	$-0.448425\pi$
$151$	1.36501e7	0.322639	0.161319	+	0.986902i	$0.448425\pi$
$152$	2.47518e7	0.571682
$153$	0	0
$154$	2.10352e7	0.464112
$155$	−1.26791e8	−2.73480
$156$	0	0
$157$	3.77615e7	0.778755	0.389378	−	0.921078i	$-0.372690\pi$
$157$	3.77615e7	0.778755	0.389378	+	0.921078i	$0.372690\pi$
$158$	−4.43169e7	−0.893861
$159$	0	0
$160$	1.72173e8	3.32311
$161$	−2.68670e6	−0.0507375
$162$	0	0
$163$	4.92529e7	0.890790	0.445395	−	0.895334i	$-0.353063\pi$
$163$	4.92529e7	0.890790	0.445395	+	0.895334i	$0.353063\pi$
$164$	2.27369e8	4.02512
$165$	0	0
$166$	−1.14117e8	−1.93629
$167$	−1.04186e8	−1.73102	−0.865508	−	0.500895i	$-0.833004\pi$
$167$	−1.04186e8	−1.73102	−0.865508	+	0.500895i	$0.833004\pi$
$168$	0	0
$169$	−4.08730e7	−0.651377
$170$	1.85632e7	0.289788
$171$	0	0
$172$	−2.22734e8	−3.33762
$173$	−9.60679e7	−1.41064	−0.705321	−	0.708888i	$-0.749197\pi$
$173$	−9.60679e7	−1.41064	−0.705321	+	0.708888i	$0.749197\pi$
$174$	0	0
$175$	3.97447e7	0.560591
$176$	1.44184e8	1.99353
$177$	0	0
$178$	4.77435e7	0.634518
$179$	8.54665e7	1.11381	0.556904	−	0.830577i	$-0.311989\pi$
$179$	8.54665e7	1.11381	0.556904	+	0.830577i	$0.311989\pi$
$180$	0	0
$181$	4.14050e7	0.519012	0.259506	−	0.965741i	$-0.416440\pi$
$181$	4.14050e7	0.519012	0.259506	+	0.965741i	$0.416440\pi$
$182$	−2.81286e7	−0.345858
$183$	0	0
$184$	−3.62796e7	−0.429339
$185$	−1.64562e7	−0.191086
$186$	0	0
$187$	6.64452e6	0.0743050
$188$	1.48894e8	1.63427
$189$	0	0
$190$	6.25008e7	0.661071
$191$	8.00346e7	0.831114	0.415557	−	0.909567i	$-0.363587\pi$
$191$	8.00346e7	0.831114	0.415557	+	0.909567i	$0.363587\pi$
$192$	0	0
$193$	1.36187e8	1.36360	0.681798	−	0.731540i	$-0.261198\pi$
$193$	1.36187e8	1.36360	0.681798	+	0.731540i	$0.261198\pi$
$194$	2.09913e8	2.06411
$195$	0	0
$196$	−2.31619e8	−2.19724
$197$	1.36747e8	1.27434	0.637171	−	0.770723i	$-0.280105\pi$
$197$	1.36747e8	1.27434	0.637171	+	0.770723i	$0.280105\pi$
$198$	0	0
$199$	−4.28301e7	−0.385268	−0.192634	−	0.981271i	$-0.561703\pi$
$199$	−4.28301e7	−0.385268	−0.192634	+	0.981271i	$0.561703\pi$
$200$	5.36689e8	4.74371
$201$	0	0
$202$	8.57040e7	0.731596
$203$	4.58727e7	0.384874
$204$	0	0
$205$	3.38892e8	2.74740
$206$	−3.43509e8	−2.73781
$207$	0	0
$208$	−1.92806e8	−1.48559
$209$	2.23716e7	0.169506
$210$	0	0
$211$	−4.75756e7	−0.348655	−0.174327	−	0.984688i	$-0.555775\pi$
$211$	−4.75756e7	−0.348655	−0.174327	+	0.984688i	$0.555775\pi$
$212$	4.67864e8	3.37244
$213$	0	0
$214$	3.00602e8	2.09673
$215$	−3.31982e8	−2.27814
$216$	0	0
$217$	−7.80350e7	−0.518419
$218$	−1.39805e7	−0.0913958
$219$	0	0
$220$	5.08781e8	3.22145
$221$	−8.88517e6	−0.0553723
$222$	0	0
$223$	1.14043e8	0.688652	0.344326	−	0.938850i	$-0.388107\pi$
$223$	1.14043e8	0.688652	0.344326	+	0.938850i	$0.388107\pi$
$224$	1.05966e8	0.629940
$225$	0	0
$226$	−4.99431e8	−2.87803
$227$	−5.49344e7	−0.311712	−0.155856	−	0.987780i	$-0.549814\pi$
$227$	−5.49344e7	−0.311712	−0.155856	+	0.987780i	$0.549814\pi$
$228$	0	0
$229$	−5.39847e7	−0.297062	−0.148531	−	0.988908i	$-0.547454\pi$
$229$	−5.39847e7	−0.297062	−0.148531	+	0.988908i	$0.547454\pi$
$230$	−9.16096e7	−0.496471
$231$	0	0
$232$	6.19437e8	3.25679
$233$	2.85964e8	1.48104	0.740518	−	0.672036i	$-0.234580\pi$
$233$	2.85964e8	1.48104	0.740518	+	0.672036i	$0.234580\pi$
$234$	0	0
$235$	2.21925e8	1.11550
$236$	6.41606e7	0.317743
$237$	0	0
$238$	1.14250e7	0.0549332
$239$	8.27309e7	0.391990	0.195995	−	0.980605i	$-0.437206\pi$
$239$	8.27309e7	0.391990	0.195995	+	0.980605i	$0.437206\pi$
$240$	0	0
$241$	−2.15524e8	−0.991825	−0.495912	−	0.868373i	$-0.665166\pi$
$241$	−2.15524e8	−0.991825	−0.495912	+	0.868373i	$0.665166\pi$
$242$	−1.52221e8	−0.690433
$243$	0	0
$244$	−3.27262e8	−1.44222
$245$	−3.45226e8	−1.49976
$246$	0	0
$247$	−2.99157e7	−0.126317
$248$	−1.05374e9	−4.38684
$249$	0	0
$250$	5.91786e8	2.39538
$251$	3.78187e8	1.50955	0.754776	−	0.655982i	$-0.227745\pi$
$251$	3.78187e8	1.50955	0.754776	+	0.655982i	$0.227745\pi$
$252$	0	0
$253$	−3.27909e7	−0.127301
$254$	1.63373e8	0.625551
$255$	0	0
$256$	−2.17495e8	−0.810233
$257$	−2.36018e8	−0.867320	−0.433660	−	0.901076i	$-0.642778\pi$
$257$	−2.36018e8	−0.867320	−0.433660	+	0.901076i	$0.642778\pi$
$258$	0	0
$259$	−1.01282e7	−0.0362229
$260$	−6.80352e8	−2.40064
$261$	0	0
$262$	5.63843e8	1.93688
$263$	3.99714e8	1.35489	0.677445	−	0.735573i	$-0.263087\pi$
$263$	3.99714e8	1.35489	0.677445	+	0.735573i	$0.263087\pi$
$264$	0	0
$265$	6.97346e8	2.30191
$266$	3.84670e7	0.125315
$267$	0	0
$268$	−8.00489e8	−2.54029
$269$	−3.97589e8	−1.24538	−0.622689	−	0.782470i	$-0.713960\pi$
$269$	−3.97589e8	−1.24538	−0.622689	+	0.782470i	$0.713960\pi$
$270$	0	0
$271$	5.14541e8	1.57046	0.785232	−	0.619202i	$-0.212544\pi$
$271$	5.14541e8	1.57046	0.785232	+	0.619202i	$0.212544\pi$
$272$	7.83116e7	0.235958
$273$	0	0
$274$	1.05441e9	3.09658
$275$	4.85079e8	1.40653
$276$	0	0
$277$	−2.94631e8	−0.832911	−0.416455	−	0.909156i	$-0.636728\pi$
$277$	−2.94631e8	−0.832911	−0.416455	+	0.909156i	$0.636728\pi$
$278$	−1.14777e9	−3.20405
$279$	0	0
$280$	5.16384e8	1.40579
$281$	−6.15936e8	−1.65601	−0.828006	−	0.560719i	$-0.810525\pi$
$281$	−6.15936e8	−1.65601	−0.828006	+	0.560719i	$0.810525\pi$
$282$	0	0
$283$	−4.48859e6	−0.0117722	−0.00588609	−	0.999983i	$-0.501874\pi$
$283$	−4.48859e6	−0.0117722	−0.00588609	+	0.999983i	$0.501874\pi$
$284$	−9.89832e8	−2.56417
$285$	0	0
$286$	−3.43306e8	−0.867761
$287$	2.08576e8	0.520807
$288$	0	0
$289$	−4.06730e8	−0.991205
$290$	1.56414e9	3.76602
$291$	0	0
$292$	5.09088e8	1.19661
$293$	−3.57368e8	−0.830002	−0.415001	−	0.909821i	$-0.636219\pi$
$293$	−3.57368e8	−0.830002	−0.415001	+	0.909821i	$0.636219\pi$
$294$	0	0
$295$	9.56307e7	0.216880
$296$	−1.36765e8	−0.306517
$297$	0	0
$298$	−7.79390e8	−1.70607
$299$	4.38485e7	0.0948650
$300$	0	0
$301$	−2.04323e8	−0.431852
$302$	2.86457e8	0.598461
$303$	0	0
$304$	2.63670e8	0.538273
$305$	−4.87780e8	−0.984407
$306$	0	0
$307$	−4.17291e8	−0.823105	−0.411552	−	0.911386i	$-0.635013\pi$
$307$	−4.17291e8	−0.823105	−0.411552	+	0.911386i	$0.635013\pi$
$308$	3.13137e8	0.610670
$309$	0	0
$310$	−2.66079e9	−5.07277
$311$	−1.57129e8	−0.296207	−0.148103	−	0.988972i	$-0.547317\pi$
$311$	−1.57129e8	−0.296207	−0.148103	+	0.988972i	$0.547317\pi$
$312$	0	0
$313$	8.52228e8	1.57091	0.785454	−	0.618920i	$-0.212430\pi$
$313$	8.52228e8	1.57091	0.785454	+	0.618920i	$0.212430\pi$
$314$	7.92454e8	1.44451
$315$	0	0
$316$	−6.59718e8	−1.17612
$317$	7.40371e8	1.30539	0.652697	−	0.757619i	$-0.273637\pi$
$317$	7.40371e8	1.30539	0.652697	+	0.757619i	$0.273637\pi$
$318$	0	0
$319$	5.59870e8	0.965651
$320$	1.15625e9	1.97254
$321$	0	0
$322$	−5.63824e7	−0.0941127
$323$	1.21508e7	0.0200631
$324$	0	0
$325$	−6.48657e8	−1.04815
$326$	1.03361e9	1.65232
$327$	0	0
$328$	2.81648e9	4.40705
$329$	1.36587e8	0.211458
$330$	0	0
$331$	−2.16470e8	−0.328095	−0.164047	−	0.986452i	$-0.552455\pi$
$331$	−2.16470e8	−0.328095	−0.164047	+	0.986452i	$0.552455\pi$
$332$	−1.69878e9	−2.54773
$333$	0	0
$334$	−2.18642e9	−3.21085
$335$	−1.19312e9	−1.73391
$336$	0	0
$337$	−1.73965e8	−0.247604	−0.123802	−	0.992307i	$-0.539509\pi$
$337$	−1.73965e8	−0.247604	−0.123802	+	0.992307i	$0.539509\pi$
$338$	−8.57749e8	−1.20824
$339$	0	0
$340$	2.76338e8	0.381297
$341$	−9.52407e8	−1.30072
$342$	0	0
$343$	−4.48484e8	−0.600092
$344$	−2.75906e9	−3.65432
$345$	0	0
$346$	−2.01606e9	−2.61659
$347$	7.38996e7	0.0949486	0.0474743	−	0.998872i	$-0.484883\pi$
$347$	7.38996e7	0.0949486	0.0474743	+	0.998872i	$0.484883\pi$
$348$	0	0
$349$	5.44309e8	0.685419	0.342710	−	0.939441i	$-0.388655\pi$
$349$	5.44309e8	0.685419	0.342710	+	0.939441i	$0.388655\pi$
$350$	8.34072e8	1.03984
$351$	0	0
$352$	1.29330e9	1.58052
$353$	−1.58939e9	−1.92317	−0.961587	−	0.274499i	$-0.911488\pi$
$353$	−1.58939e9	−1.92317	−0.961587	+	0.274499i	$0.911488\pi$
$354$	0	0
$355$	−1.47533e9	−1.75021
$356$	7.10726e8	0.834886
$357$	0	0
$358$	1.79358e9	2.06599
$359$	−1.69409e9	−1.93244	−0.966221	−	0.257713i	$-0.917031\pi$
$359$	−1.69409e9	−1.93244	−0.966221	+	0.257713i	$0.917031\pi$
$360$	0	0
$361$	−8.52961e8	−0.954232
$362$	8.68914e8	0.962713
$363$	0	0
$364$	−4.18732e8	−0.455073
$365$	7.58791e8	0.816765
$366$	0	0
$367$	−1.50784e9	−1.59230	−0.796151	−	0.605098i	$-0.793134\pi$
$367$	−1.50784e9	−1.59230	−0.796151	+	0.605098i	$0.793134\pi$
$368$	−3.86470e8	−0.404248
$369$	0	0
$370$	−3.45346e8	−0.354445
$371$	4.29192e8	0.436358
$372$	0	0
$373$	−9.09574e8	−0.907522	−0.453761	−	0.891123i	$-0.649918\pi$
$373$	−9.09574e8	−0.907522	−0.453761	+	0.891123i	$0.649918\pi$
$374$	1.39440e8	0.137828
$375$	0	0
$376$	1.84439e9	1.78935
$377$	−7.48669e8	−0.719606
$378$	0	0
$379$	1.38013e9	1.30221	0.651106	−	0.758987i	$-0.274305\pi$
$379$	1.38013e9	1.30221	0.651106	+	0.758987i	$0.274305\pi$
$380$	9.30409e8	0.869824
$381$	0	0
$382$	1.67958e9	1.54163
$383$	1.20160e9	1.09286	0.546430	−	0.837505i	$-0.315987\pi$
$383$	1.20160e9	1.09286	0.546430	+	0.837505i	$0.315987\pi$
$384$	0	0
$385$	4.66727e8	0.416822
$386$	2.85799e9	2.52933
$387$	0	0
$388$	3.12483e9	2.71591
$389$	−1.11547e9	−0.960802	−0.480401	−	0.877049i	$-0.659509\pi$
$389$	−1.11547e9	−0.960802	−0.480401	+	0.877049i	$0.659509\pi$
$390$	0	0
$391$	−1.78099e7	−0.0150676
$392$	−2.86912e9	−2.40574
$393$	0	0
$394$	2.86973e9	2.36377
$395$	−9.83303e8	−0.802782
$396$	0	0
$397$	−1.21715e9	−0.976286	−0.488143	−	0.872764i	$-0.662325\pi$
$397$	−1.21715e9	−0.976286	−0.488143	+	0.872764i	$0.662325\pi$
$398$	−8.98820e8	−0.714631
$399$	0	0
$400$	5.71709e9	4.46648
$401$	9.38458e8	0.726791	0.363396	−	0.931635i	$-0.381617\pi$
$401$	9.38458e8	0.726791	0.363396	+	0.931635i	$0.381617\pi$
$402$	0	0
$403$	1.27358e9	0.969298
$404$	1.27582e9	0.962620
$405$	0	0
$406$	9.62672e8	0.713900
$407$	−1.23614e8	−0.0908836
$408$	0	0
$409$	−2.00800e9	−1.45122	−0.725608	−	0.688108i	$-0.758441\pi$
$409$	−2.00800e9	−1.45122	−0.725608	+	0.688108i	$0.758441\pi$
$410$	7.11189e9	5.09614
$411$	0	0
$412$	−5.11360e9	−3.60235
$413$	5.88573e7	0.0411126
$414$	0	0
$415$	−2.53201e9	−1.73899
$416$	−1.72943e9	−1.17781
$417$	0	0
$418$	4.69485e8	0.314416
$419$	2.40847e9	1.59953	0.799765	−	0.600313i	$-0.204957\pi$
$419$	2.40847e9	1.59953	0.799765	+	0.600313i	$0.204957\pi$
$420$	0	0
$421$	−5.54976e8	−0.362482	−0.181241	−	0.983439i	$-0.558011\pi$
$421$	−5.54976e8	−0.362482	−0.181241	+	0.983439i	$0.558011\pi$
$422$	−9.98409e8	−0.646718
$423$	0	0
$424$	5.79555e9	3.69244
$425$	2.63464e8	0.166479
$426$	0	0
$427$	−3.00211e8	−0.186608
$428$	4.47486e9	2.75884
$429$	0	0
$430$	−6.96690e9	−4.22571
$431$	2.19055e9	1.31790	0.658951	−	0.752186i	$-0.271001\pi$
$431$	2.19055e9	1.31790	0.658951	+	0.752186i	$0.271001\pi$
$432$	0	0
$433$	2.56663e8	0.151934	0.0759672	−	0.997110i	$-0.475796\pi$
$433$	2.56663e8	0.151934	0.0759672	+	0.997110i	$0.475796\pi$
$434$	−1.63762e9	−0.961611
$435$	0	0
$436$	−2.08119e8	−0.120257
$437$	−5.99647e7	−0.0343725
$438$	0	0
$439$	2.12343e9	1.19787	0.598937	−	0.800796i	$-0.295590\pi$
$439$	2.12343e9	1.19787	0.598937	+	0.800796i	$0.295590\pi$
$440$	6.30240e9	3.52713
$441$	0	0
$442$	−1.86462e8	−0.102710
$443$	−3.50552e9	−1.91575	−0.957875	−	0.287185i	$-0.907281\pi$
$443$	−3.50552e9	−1.91575	−0.957875	+	0.287185i	$0.907281\pi$
$444$	0	0
$445$	1.05933e9	0.569864
$446$	2.39327e9	1.27738
$447$	0	0
$448$	7.11630e8	0.373922
$449$	−1.44417e9	−0.752933	−0.376467	−	0.926430i	$-0.622861\pi$
$449$	−1.44417e9	−0.752933	−0.376467	+	0.926430i	$0.622861\pi$
$450$	0	0
$451$	2.54564e9	1.30671
$452$	−7.43470e9	−3.78686
$453$	0	0
$454$	−1.15284e9	−0.578193
$455$	−6.24116e8	−0.310617
$456$	0	0
$457$	−3.55204e7	−0.0174089	−0.00870445	−	0.999962i	$-0.502771\pi$
$457$	−3.55204e7	−0.0174089	−0.00870445	+	0.999962i	$0.502771\pi$
$458$	−1.13291e9	−0.551018
$459$	0	0
$460$	−1.36373e9	−0.653246
$461$	1.47773e9	0.702491	0.351245	−	0.936283i	$-0.385758\pi$
$461$	1.47773e9	0.702491	0.351245	+	0.936283i	$0.385758\pi$
$462$	0	0
$463$	−1.25958e9	−0.589782	−0.294891	−	0.955531i	$-0.595283\pi$
$463$	−1.25958e9	−0.589782	−0.294891	+	0.955531i	$0.595283\pi$
$464$	6.59858e9	3.06646
$465$	0	0
$466$	6.00117e9	2.74717
$467$	−1.25254e9	−0.569090	−0.284545	−	0.958663i	$-0.591843\pi$
$467$	−1.25254e9	−0.569090	−0.284545	+	0.958663i	$0.591843\pi$
$468$	0	0
$469$	−7.34323e8	−0.328687
$470$	4.65726e9	2.06913
$471$	0	0
$472$	7.94773e8	0.347894
$473$	−2.49374e9	−1.08352
$474$	0	0
$475$	8.87065e8	0.379776
$476$	1.70076e8	0.0722800
$477$	0	0
$478$	1.73617e9	0.727100
$479$	−3.28427e9	−1.36541	−0.682707	−	0.730693i	$-0.739197\pi$
$479$	−3.28427e9	−1.36541	−0.682707	+	0.730693i	$0.739197\pi$
$480$	0	0
$481$	1.65298e8	0.0677268
$482$	−4.52292e9	−1.83973
$483$	0	0
$484$	−2.26602e9	−0.908458
$485$	4.65753e9	1.85379
$486$	0	0
$487$	1.74837e8	0.0685935	0.0342968	−	0.999412i	$-0.489081\pi$
$487$	1.74837e8	0.0685935	0.0342968	+	0.999412i	$0.489081\pi$
$488$	−4.05387e9	−1.57907
$489$	0	0
$490$	−7.24482e9	−2.78190
$491$	1.05796e9	0.403353	0.201676	−	0.979452i	$-0.435361\pi$
$491$	1.05796e9	0.403353	0.201676	+	0.979452i	$0.435361\pi$
$492$	0	0
$493$	3.04086e8	0.114296
$494$	−6.27804e8	−0.234304
$495$	0	0
$496$	−1.12250e10	−4.13047
$497$	−9.08015e8	−0.331777
$498$	0	0
$499$	−7.20840e8	−0.259709	−0.129854	−	0.991533i	$-0.541451\pi$
$499$	−7.20840e8	−0.259709	−0.129854	+	0.991533i	$0.541451\pi$
$500$	8.80954e9	3.15180
$501$	0	0
$502$	7.93653e9	2.80006
$503$	−5.82049e8	−0.203925	−0.101963	−	0.994788i	$-0.532512\pi$
$503$	−5.82049e8	−0.203925	−0.101963	+	0.994788i	$0.532512\pi$
$504$	0	0
$505$	1.90160e9	0.657051
$506$	−6.88140e8	−0.236129
$507$	0	0
$508$	2.43203e9	0.823088
$509$	5.21497e9	1.75283	0.876414	−	0.481558i	$-0.159929\pi$
$509$	5.21497e9	1.75283	0.876414	+	0.481558i	$0.159929\pi$
$510$	0	0
$511$	4.67009e8	0.154829
$512$	−5.17639e9	−1.70444
$513$	0	0
$514$	−4.95301e9	−1.60879
$515$	−7.62177e9	−2.45884
$516$	0	0
$517$	1.66703e9	0.530549
$518$	−2.12548e8	−0.0671897
$519$	0	0
$520$	−8.42769e9	−2.62843
$521$	−4.72585e8	−0.146402	−0.0732011	−	0.997317i	$-0.523321\pi$
$521$	−4.72585e8	−0.146402	−0.0732011	+	0.997317i	$0.523321\pi$
$522$	0	0
$523$	8.79553e8	0.268847	0.134424	−	0.990924i	$-0.457082\pi$
$523$	8.79553e8	0.268847	0.134424	+	0.990924i	$0.457082\pi$
$524$	8.39357e9	2.54851
$525$	0	0
$526$	8.38829e9	2.51318
$527$	−5.17287e8	−0.153955
$528$	0	0
$529$	−3.31693e9	−0.974186
$530$	1.46343e10	4.26980
$531$	0	0
$532$	5.72633e8	0.164887
$533$	−3.40407e9	−0.973764
$534$	0	0
$535$	6.66974e9	1.88309
$536$	−9.91585e9	−2.78134
$537$	0	0
$538$	−8.34369e9	−2.31004
$539$	−2.59322e9	−0.713310
$540$	0	0
$541$	−1.83033e8	−0.0496981	−0.0248491	−	0.999691i	$-0.507911\pi$
$541$	−1.83033e8	−0.0496981	−0.0248491	+	0.999691i	$0.507911\pi$
$542$	1.07980e10	2.91304
$543$	0	0
$544$	7.02440e8	0.187074
$545$	−3.10200e8	−0.0820830
$546$	0	0
$547$	−2.64278e9	−0.690408	−0.345204	−	0.938528i	$-0.612190\pi$
$547$	−2.64278e9	−0.690408	−0.345204	+	0.938528i	$0.612190\pi$
$548$	1.56963e10	4.07442
$549$	0	0
$550$	1.01797e10	2.60896
$551$	1.02384e9	0.260735
$552$	0	0
$553$	−6.05187e8	−0.152178
$554$	−6.18304e9	−1.54496
$555$	0	0
$556$	−1.70862e10	−4.21583
$557$	−2.28927e9	−0.561313	−0.280656	−	0.959808i	$-0.590552\pi$
$557$	−2.28927e9	−0.561313	−0.280656	+	0.959808i	$0.590552\pi$
$558$	0	0
$559$	3.33467e9	0.807443
$560$	5.50080e9	1.32363
$561$	0	0
$562$	−1.29259e10	−3.07173
$563$	−1.07781e9	−0.254543	−0.127271	−	0.991868i	$-0.540622\pi$
$563$	−1.07781e9	−0.254543	−0.127271	+	0.991868i	$0.540622\pi$
$564$	0	0
$565$	−1.10814e10	−2.58478
$566$	−9.41963e7	−0.0218362
$567$	0	0
$568$	−1.22613e10	−2.80748
$569$	1.08728e9	0.247429	0.123714	−	0.992318i	$-0.460519\pi$
$569$	1.08728e9	0.247429	0.123714	+	0.992318i	$0.460519\pi$
$570$	0	0
$571$	2.15499e8	0.0484416	0.0242208	−	0.999707i	$-0.492290\pi$
$571$	2.15499e8	0.0484416	0.0242208	+	0.999707i	$0.492290\pi$
$572$	−5.11057e9	−1.14178
$573$	0	0
$574$	4.37711e9	0.966042
$575$	−1.30020e9	−0.285216
$576$	0	0
$577$	7.86457e9	1.70435	0.852176	−	0.523255i	$-0.175282\pi$
$577$	7.86457e9	1.70435	0.852176	+	0.523255i	$0.175282\pi$
$578$	−8.53552e9	−1.83858
$579$	0	0
$580$	2.32843e10	4.95526
$581$	−1.55836e9	−0.329649
$582$	0	0
$583$	5.23823e9	1.09482
$584$	6.30620e9	1.31016
$585$	0	0
$586$	−7.49963e9	−1.53957
$587$	−7.86924e9	−1.60583	−0.802914	−	0.596094i	$-0.796718\pi$
$587$	−7.86924e9	−1.60583	−0.802914	+	0.596094i	$0.796718\pi$
$588$	0	0
$589$	−1.74167e9	−0.351206
$590$	2.00688e9	0.402290
$591$	0	0
$592$	−1.45690e9	−0.288604
$593$	9.64856e9	1.90008	0.950038	−	0.312134i	$-0.101044\pi$
$593$	9.64856e9	1.90008	0.950038	+	0.312134i	$0.101044\pi$
$594$	0	0
$595$	2.53496e8	0.0493358
$596$	−1.16023e10	−2.24482
$597$	0	0
$598$	9.20194e8	0.175965
$599$	−5.43654e9	−1.03354	−0.516772	−	0.856123i	$-0.672866\pi$
$599$	−5.43654e9	−1.03354	−0.516772	+	0.856123i	$0.672866\pi$
$600$	0	0
$601$	−2.28199e9	−0.428798	−0.214399	−	0.976746i	$-0.568779\pi$
$601$	−2.28199e9	−0.428798	−0.214399	+	0.976746i	$0.568779\pi$
$602$	−4.28787e9	−0.801040
$603$	0	0
$604$	4.26431e9	0.787443
$605$	−3.37748e9	−0.620082
$606$	0	0
$607$	−2.22820e9	−0.404383	−0.202191	−	0.979346i	$-0.564806\pi$
$607$	−2.22820e9	−0.404383	−0.202191	+	0.979346i	$0.564806\pi$
$608$	2.36507e9	0.426757
$609$	0	0
$610$	−1.02364e10	−1.82597
$611$	−2.22918e9	−0.395367
$612$	0	0
$613$	9.10899e8	0.159720	0.0798599	−	0.996806i	$-0.474553\pi$
$613$	9.10899e8	0.159720	0.0798599	+	0.996806i	$0.474553\pi$
$614$	−8.75716e9	−1.52677
$615$	0	0
$616$	3.87890e9	0.668615
$617$	−6.17621e9	−1.05858	−0.529290	−	0.848441i	$-0.677542\pi$
$617$	−6.17621e9	−1.05858	−0.529290	+	0.848441i	$0.677542\pi$
$618$	0	0
$619$	−9.92262e9	−1.68155	−0.840774	−	0.541387i	$-0.817900\pi$
$619$	−9.92262e9	−1.68155	−0.840774	+	0.541387i	$0.817900\pi$
$620$	−3.96095e10	−6.67465
$621$	0	0
$622$	−3.29747e9	−0.549432
$623$	6.51980e8	0.108025
$624$	0	0
$625$	2.29562e9	0.376114
$626$	1.78846e10	2.91387
$627$	0	0
$628$	1.17967e10	1.90066
$629$	−6.71390e7	−0.0107572
$630$	0	0
$631$	3.99548e9	0.633091	0.316546	−	0.948577i	$-0.397477\pi$
$631$	3.99548e9	0.633091	0.316546	+	0.948577i	$0.397477\pi$
$632$	−8.17209e9	−1.28773
$633$	0	0
$634$	1.55372e10	2.42137
$635$	3.62492e9	0.561811
$636$	0	0
$637$	3.46770e9	0.531561
$638$	1.17493e10	1.79118
$639$	0	0
$640$	2.22663e9	0.335752
$641$	4.03744e9	0.605485	0.302742	−	0.953072i	$-0.402098\pi$
$641$	4.03744e9	0.605485	0.302742	+	0.953072i	$0.402098\pi$
$642$	0	0
$643$	−8.64628e9	−1.28260	−0.641299	−	0.767291i	$-0.721604\pi$
$643$	−8.64628e9	−1.28260	−0.641299	+	0.767291i	$0.721604\pi$
$644$	−8.39329e8	−0.123832
$645$	0	0
$646$	2.54994e8	0.0372149
$647$	−7.61025e9	−1.10467	−0.552337	−	0.833621i	$-0.686264\pi$
$647$	−7.61025e9	−1.10467	−0.552337	+	0.833621i	$0.686264\pi$
$648$	0	0
$649$	7.18346e8	0.103152
$650$	−1.36125e10	−1.94421
$651$	0	0
$652$	1.53867e10	2.17409
$653$	1.04681e10	1.47120	0.735599	−	0.677417i	$-0.236901\pi$
$653$	1.04681e10	1.47120	0.735599	+	0.677417i	$0.236901\pi$
$654$	0	0
$655$	1.25105e10	1.73953
$656$	3.00026e10	4.14950
$657$	0	0
$658$	2.86638e9	0.392232
$659$	−3.31882e9	−0.451737	−0.225868	−	0.974158i	$-0.572522\pi$
$659$	−3.31882e9	−0.451737	−0.225868	+	0.974158i	$0.572522\pi$
$660$	0	0
$661$	7.95737e9	1.07168	0.535839	−	0.844320i	$-0.319995\pi$
$661$	7.95737e9	1.07168	0.535839	+	0.844320i	$0.319995\pi$
$662$	−4.54278e9	−0.608581
$663$	0	0
$664$	−2.10432e10	−2.78948
$665$	8.53504e8	0.112546
$666$	0	0
$667$	−1.50067e9	−0.195815
$668$	−3.25478e10	−4.22477
$669$	0	0
$670$	−2.50385e10	−3.21623
$671$	−3.66404e9	−0.468200
$672$	0	0
$673$	−7.14650e9	−0.903734	−0.451867	−	0.892085i	$-0.649242\pi$
$673$	−7.14650e9	−0.903734	−0.451867	+	0.892085i	$0.649242\pi$
$674$	−3.65079e9	−0.459279
$675$	0	0
$676$	−1.27687e10	−1.58977
$677$	7.35419e9	0.910908	0.455454	−	0.890259i	$-0.349477\pi$
$677$	7.35419e9	0.910908	0.455454	+	0.890259i	$0.349477\pi$
$678$	0	0
$679$	2.86654e9	0.351410
$680$	3.42306e9	0.417478
$681$	0	0
$682$	−1.99870e10	−2.41269
$683$	−4.04823e9	−0.486175	−0.243087	−	0.970004i	$-0.578160\pi$
$683$	−4.04823e9	−0.486175	−0.243087	+	0.970004i	$0.578160\pi$
$684$	0	0
$685$	2.33952e10	2.78106
$686$	−9.41177e9	−1.11311
$687$	0	0
$688$	−2.93910e10	−3.44076
$689$	−7.00465e9	−0.815867
$690$	0	0
$691$	−1.70309e9	−0.196365	−0.0981823	−	0.995168i	$-0.531303\pi$
$691$	−1.70309e9	−0.196365	−0.0981823	+	0.995168i	$0.531303\pi$
$692$	−3.00117e10	−3.44286
$693$	0	0
$694$	1.55084e9	0.176120
$695$	−2.54668e10	−2.87758
$696$	0	0
$697$	1.38263e9	0.154665
$698$	1.14227e10	1.27138
$699$	0	0
$700$	1.24163e10	1.36820
$701$	−7.38709e9	−0.809954	−0.404977	−	0.914327i	$-0.632721\pi$
$701$	−7.38709e9	−0.809954	−0.404977	+	0.914327i	$0.632721\pi$
$702$	0	0
$703$	−2.26052e8	−0.0245395
$704$	8.68536e9	0.938174
$705$	0	0
$706$	−3.33545e10	−3.56729
$707$	1.17037e9	0.124553
$708$	0	0
$709$	−3.00194e9	−0.316330	−0.158165	−	0.987413i	$-0.550558\pi$
$709$	−3.00194e9	−0.316330	−0.158165	+	0.987413i	$0.550558\pi$
$710$	−3.09610e10	−3.24646
$711$	0	0
$712$	8.80394e9	0.914108
$713$	2.55282e9	0.263759
$714$	0	0
$715$	−7.61726e9	−0.779341
$716$	2.66998e10	2.71839
$717$	0	0
$718$	−3.55518e10	−3.58448
$719$	−3.86807e9	−0.388100	−0.194050	−	0.980992i	$-0.562162\pi$
$719$	−3.86807e9	−0.388100	−0.194050	+	0.980992i	$0.562162\pi$
$720$	0	0
$721$	−4.69092e9	−0.466106
$722$	−1.79000e10	−1.77000
$723$	0	0
$724$	1.29350e10	1.26672
$725$	2.21996e10	2.16353
$726$	0	0
$727$	1.88812e10	1.82247	0.911235	−	0.411887i	$-0.135130\pi$
$727$	1.88812e10	1.82247	0.911235	+	0.411887i	$0.135130\pi$
$728$	−5.18694e9	−0.498255
$729$	0	0
$730$	1.59238e10	1.51501
$731$	−1.35444e9	−0.128247
$732$	0	0
$733$	−2.89464e9	−0.271476	−0.135738	−	0.990745i	$-0.543340\pi$
$733$	−2.89464e9	−0.271476	−0.135738	+	0.990745i	$0.543340\pi$
$734$	−3.16432e10	−2.95355
$735$	0	0
$736$	−3.46656e9	−0.320499
$737$	−8.96231e9	−0.824677
$738$	0	0
$739$	−4.83541e9	−0.440735	−0.220368	−	0.975417i	$-0.570726\pi$
$739$	−4.83541e9	−0.440735	−0.220368	+	0.975417i	$0.570726\pi$
$740$	−5.14094e9	−0.466371
$741$	0	0
$742$	9.00690e9	0.809397
$743$	1.65827e10	1.48318	0.741591	−	0.670852i	$-0.234071\pi$
$743$	1.65827e10	1.48318	0.741591	+	0.670852i	$0.234071\pi$
$744$	0	0
$745$	−1.72931e10	−1.53223
$746$	−1.90881e10	−1.68336
$747$	0	0
$748$	2.07575e9	0.181351
$749$	4.10498e9	0.356965
$750$	0	0
$751$	1.94126e10	1.67242	0.836208	−	0.548413i	$-0.184768\pi$
$751$	1.94126e10	1.67242	0.836208	+	0.548413i	$0.184768\pi$
$752$	1.96474e10	1.68478
$753$	0	0
$754$	−1.57114e10	−1.33479
$755$	6.35591e9	0.537481
$756$	0	0
$757$	1.23613e10	1.03569	0.517843	−	0.855475i	$-0.326735\pi$
$757$	1.23613e10	1.03569	0.517843	+	0.855475i	$0.326735\pi$
$758$	2.89630e10	2.41547
$759$	0	0
$760$	1.15252e10	0.952361
$761$	1.27885e10	1.05189	0.525947	−	0.850517i	$-0.323711\pi$
$761$	1.27885e10	1.05189	0.525947	+	0.850517i	$0.323711\pi$
$762$	0	0
$763$	−1.90917e8	−0.0155599
$764$	2.50029e10	2.02845
$765$	0	0
$766$	2.52165e10	2.02714
$767$	−9.60585e8	−0.0768691
$768$	0	0
$769$	−6.74352e9	−0.534742	−0.267371	−	0.963594i	$-0.586155\pi$
$769$	−6.74352e9	−0.534742	−0.267371	+	0.963594i	$0.586155\pi$
$770$	9.79461e9	0.773160
$771$	0	0
$772$	4.25450e10	3.32804
$773$	−3.73822e9	−0.291096	−0.145548	−	0.989351i	$-0.546495\pi$
$773$	−3.73822e9	−0.291096	−0.145548	+	0.989351i	$0.546495\pi$
$774$	0	0
$775$	−3.77642e10	−2.91424
$776$	3.87081e10	2.97362
$777$	0	0
$778$	−2.34089e10	−1.78219
$779$	4.65521e9	0.352824
$780$	0	0
$781$	−1.10822e10	−0.832430
$782$	−3.73754e8	−0.0279487
$783$	0	0
$784$	−3.05634e10	−2.26514
$785$	1.75829e10	1.29732
$786$	0	0
$787$	−1.47464e10	−1.07839	−0.539193	−	0.842182i	$-0.681271\pi$
$787$	−1.47464e10	−1.07839	−0.539193	+	0.842182i	$0.681271\pi$
$788$	4.27199e10	3.11020
$789$	0	0
$790$	−2.06353e10	−1.48907
$791$	−6.82017e9	−0.489979
$792$	0	0
$793$	4.89962e9	0.348904
$794$	−2.55428e10	−1.81091
$795$	0	0
$796$	−1.33802e10	−0.940298
$797$	1.86115e10	1.30220	0.651099	−	0.758993i	$-0.274308\pi$
$797$	1.86115e10	1.30220	0.651099	+	0.758993i	$0.274308\pi$
$798$	0	0
$799$	9.05422e8	0.0627968
$800$	5.12812e10	3.54114
$801$	0	0
$802$	1.96942e10	1.34812
$803$	5.69978e9	0.388466
$804$	0	0
$805$	−1.25101e9	−0.0845231
$806$	2.67269e10	1.79795
$807$	0	0
$808$	1.58039e10	1.05396
$809$	−4.53515e9	−0.301143	−0.150571	−	0.988599i	$-0.548111\pi$
$809$	−4.53515e9	−0.301143	−0.150571	+	0.988599i	$0.548111\pi$
$810$	0	0
$811$	7.48809e9	0.492945	0.246472	−	0.969150i	$-0.420729\pi$
$811$	7.48809e9	0.492945	0.246472	+	0.969150i	$0.420729\pi$
$812$	1.43307e10	0.939335
$813$	0	0
$814$	−2.59412e9	−0.168579
$815$	2.29337e10	1.48396
$816$	0	0
$817$	−4.56030e9	−0.292561
$818$	−4.21393e10	−2.69185
$819$	0	0
$820$	1.05870e11	6.70540
$821$	1.66654e10	1.05103	0.525514	−	0.850785i	$-0.323873\pi$
$821$	1.66654e10	1.05103	0.525514	+	0.850785i	$0.323873\pi$
$822$	0	0
$823$	1.72880e10	1.08105	0.540524	−	0.841328i	$-0.318226\pi$
$823$	1.72880e10	1.08105	0.540524	+	0.841328i	$0.318226\pi$
$824$	−6.33434e10	−3.94418
$825$	0	0
$826$	1.23516e9	0.0762595
$827$	1.68089e10	1.03341	0.516703	−	0.856165i	$-0.327159\pi$
$827$	1.68089e10	1.03341	0.516703	+	0.856165i	$0.327159\pi$
$828$	0	0
$829$	3.99422e9	0.243495	0.121748	−	0.992561i	$-0.461150\pi$
$829$	3.99422e9	0.243495	0.121748	+	0.992561i	$0.461150\pi$
$830$	−5.31362e10	−3.22565
$831$	0	0
$832$	−1.16142e10	−0.699130
$833$	−1.40847e9	−0.0844287
$834$	0	0
$835$	−4.85121e10	−2.88368
$836$	6.98892e9	0.413702
$837$	0	0
$838$	5.05435e10	2.96696
$839$	1.32657e10	0.775467	0.387734	−	0.921771i	$-0.373258\pi$
$839$	1.32657e10	0.775467	0.387734	+	0.921771i	$0.373258\pi$
$840$	0	0
$841$	8.37255e9	0.485369
$842$	−1.16466e10	−0.672367
$843$	0	0
$844$	−1.48627e10	−0.850938
$845$	−1.90317e10	−1.08512
$846$	0	0
$847$	−2.07872e9	−0.117545
$848$	6.17372e10	3.47665
$849$	0	0
$850$	5.52899e9	0.308801
$851$	3.31333e8	0.0184294
$852$	0	0
$853$	3.06500e10	1.69087	0.845433	−	0.534082i	$-0.179343\pi$
$853$	3.06500e10	1.69087	0.845433	+	0.534082i	$0.179343\pi$
$854$	−6.30015e9	−0.346137
$855$	0	0
$856$	5.54312e10	3.02062
$857$	5.05197e9	0.274175	0.137087	−	0.990559i	$-0.456226\pi$
$857$	5.05197e9	0.274175	0.137087	+	0.990559i	$0.456226\pi$
$858$	0	0
$859$	−6.74004e7	−0.00362816	−0.00181408	−	0.999998i	$-0.500577\pi$
$859$	−6.74004e7	−0.00362816	−0.00181408	+	0.999998i	$0.500577\pi$
$860$	−1.03712e11	−5.56010
$861$	0	0
$862$	4.59703e10	2.44457
$863$	−3.32493e10	−1.76094	−0.880471	−	0.474101i	$-0.842773\pi$
$863$	−3.32493e10	−1.76094	−0.880471	+	0.474101i	$0.842773\pi$
$864$	0	0
$865$	−4.47322e10	−2.34998
$866$	5.38627e9	0.281822
$867$	0	0
$868$	−2.43782e10	−1.26527
$869$	−7.38623e9	−0.381816
$870$	0	0
$871$	1.19846e10	0.614552
$872$	−2.57802e9	−0.131668
$873$	0	0
$874$	−1.25840e9	−0.0637573
$875$	8.08137e9	0.407809
$876$	0	0
$877$	−3.04224e10	−1.52298	−0.761491	−	0.648176i	$-0.775532\pi$
$877$	−3.04224e10	−1.52298	−0.761491	+	0.648176i	$0.775532\pi$
$878$	4.45616e10	2.22193
$879$	0	0
$880$	6.71365e10	3.32101
$881$	−1.67047e10	−0.823044	−0.411522	−	0.911400i	$-0.635003\pi$
$881$	−1.67047e10	−0.823044	−0.411522	+	0.911400i	$0.635003\pi$
$882$	0	0
$883$	4.43529e9	0.216800	0.108400	−	0.994107i	$-0.465427\pi$
$883$	4.43529e9	0.216800	0.108400	+	0.994107i	$0.465427\pi$
$884$	−2.77574e9	−0.135144
$885$	0	0
$886$	−7.35658e10	−3.55351
$887$	−3.65741e10	−1.75971	−0.879855	−	0.475242i	$-0.842360\pi$
$887$	−3.65741e10	−1.75971	−0.879855	+	0.475242i	$0.842360\pi$
$888$	0	0
$889$	2.23101e9	0.106499
$890$	2.22308e10	1.05704
$891$	0	0
$892$	3.56270e10	1.68075
$893$	3.04849e9	0.143253
$894$	0	0
$895$	3.97958e10	1.85548
$896$	1.37041e9	0.0636463
$897$	0	0
$898$	−3.03070e10	−1.39661
$899$	−4.35869e10	−2.00077
$900$	0	0
$901$	2.84507e9	0.129585
$902$	5.34221e10	2.42381
$903$	0	0
$904$	−9.20955e10	−4.14619
$905$	1.92794e10	0.864618
$906$	0	0
$907$	1.91299e10	0.851307	0.425654	−	0.904886i	$-0.360044\pi$
$907$	1.91299e10	0.851307	0.425654	+	0.904886i	$0.360044\pi$
$908$	−1.71616e10	−0.760775
$909$	0	0
$910$	−1.30975e10	−0.576162
$911$	3.39426e10	1.48741	0.743704	−	0.668509i	$-0.233067\pi$
$911$	3.39426e10	1.48741	0.743704	+	0.668509i	$0.233067\pi$
$912$	0	0
$913$	−1.90196e10	−0.827093
$914$	−7.45422e8	−0.0322917
$915$	0	0
$916$	−1.68649e10	−0.725018
$917$	7.69978e9	0.329751
$918$	0	0
$919$	−3.53345e10	−1.50174	−0.750870	−	0.660450i	$-0.770366\pi$
$919$	−3.53345e10	−1.50174	−0.750870	+	0.660450i	$0.770366\pi$
$920$	−1.68929e10	−0.715232
$921$	0	0
$922$	3.10111e10	1.30305
$923$	1.48193e10	0.620330
$924$	0	0
$925$	−4.90144e9	−0.203624
$926$	−2.64332e10	−1.09398
$927$	0	0
$928$	5.91879e10	2.43117
$929$	3.23304e10	1.32299	0.661493	−	0.749951i	$-0.269923\pi$
$929$	3.23304e10	1.32299	0.661493	+	0.749951i	$0.269923\pi$
$930$	0	0
$931$	−4.74222e9	−0.192601
$932$	8.93354e10	3.61467
$933$	0	0
$934$	−2.62854e10	−1.05560
$935$	3.09389e9	0.123784
$936$	0	0
$937$	9.18986e9	0.364939	0.182469	−	0.983212i	$-0.441591\pi$
$937$	9.18986e9	0.364939	0.182469	+	0.983212i	$0.441591\pi$
$938$	−1.54103e10	−0.609679
$939$	0	0
$940$	6.93296e10	2.72252
$941$	1.09465e10	0.428265	0.214133	−	0.976805i	$-0.431307\pi$
$941$	1.09465e10	0.428265	0.214133	+	0.976805i	$0.431307\pi$
$942$	0	0
$943$	−6.82331e9	−0.264974
$944$	8.46635e9	0.327562
$945$	0	0
$946$	−5.23329e10	−2.00982
$947$	2.72786e10	1.04375	0.521875	−	0.853022i	$-0.325233\pi$
$947$	2.72786e10	1.04375	0.521875	+	0.853022i	$0.325233\pi$
$948$	0	0
$949$	−7.62185e9	−0.289487
$950$	1.86157e10	0.704445
$951$	0	0
$952$	2.10677e9	0.0791386
$953$	2.27671e10	0.852085	0.426043	−	0.904703i	$-0.359907\pi$
$953$	2.27671e10	0.852085	0.426043	+	0.904703i	$0.359907\pi$
$954$	0	0
$955$	3.72665e10	1.38455
$956$	2.58452e10	0.956704
$957$	0	0
$958$	−6.89227e10	−2.53270
$959$	1.43989e10	0.527187
$960$	0	0
$961$	4.66339e10	1.69500
$962$	3.46891e9	0.125626
$963$	0	0
$964$	−6.73298e10	−2.42068
$965$	6.34129e10	2.27160
$966$	0	0
$967$	−1.38243e10	−0.491645	−0.245822	−	0.969315i	$-0.579058\pi$
$967$	−1.38243e10	−0.491645	−0.245822	+	0.969315i	$0.579058\pi$
$968$	−2.80697e10	−0.994660
$969$	0	0
$970$	9.77418e10	3.43858
$971$	−1.62886e10	−0.570973	−0.285487	−	0.958383i	$-0.592155\pi$
$971$	−1.62886e10	−0.570973	−0.285487	+	0.958383i	$0.592155\pi$
$972$	0	0
$973$	−1.56739e10	−0.545483
$974$	3.66909e9	0.127234
$975$	0	0
$976$	−4.31840e10	−1.48679
$977$	−1.93224e10	−0.662874	−0.331437	−	0.943477i	$-0.607533\pi$
$977$	−1.93224e10	−0.662874	−0.331437	+	0.943477i	$0.607533\pi$
$978$	0	0
$979$	7.95733e9	0.271037
$980$	−1.07849e11	−3.66036
$981$	0	0
$982$	2.22021e10	0.748177
$983$	−3.13862e10	−1.05391	−0.526953	−	0.849895i	$-0.676666\pi$
$983$	−3.13862e10	−1.05391	−0.526953	+	0.849895i	$0.676666\pi$
$984$	0	0
$985$	6.36735e10	2.12291
$986$	6.38146e9	0.212007
$987$	0	0
$988$	−9.34571e9	−0.308292
$989$	6.68420e9	0.219716
$990$	0	0
$991$	−2.26508e9	−0.0739308	−0.0369654	−	0.999317i	$-0.511769\pi$
$991$	−2.26508e9	−0.0739308	−0.0369654	+	0.999317i	$0.511769\pi$
$992$	−1.00686e11	−3.27475
$993$	0	0
$994$	−1.90554e10	−0.615411
$995$	−1.99430e10	−0.641814
$996$	0	0
$997$	4.99508e10	1.59628	0.798140	−	0.602472i	$-0.205817\pi$
$997$	4.99508e10	1.59628	0.798140	+	0.602472i	$0.205817\pi$
$998$	−1.51273e10	−0.481732
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.8.a.c.1.16		17
3.2	odd	2		177.8.a.c.1.2	✓	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.c.1.2	✓	17	3.2	odd	2
531.8.a.c.1.16		17	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q+20.9857 q^{2} +312.401 q^{4} +465.631 q^{5} +286.579 q^{7} +3869.79 q^{8} +O(q^{10})\)	\(q+20.9857 q^{2} +312.401 q^{4} +465.631 q^{5} +286.579 q^{7} +3869.79 q^{8} +9771.60 q^{10} +3497.66 q^{11} -4677.13 q^{13} +6014.07 q^{14} +41223.0 q^{16} +1899.70 q^{17} +6396.17 q^{19} +145463. q^{20} +73400.9 q^{22} -9375.09 q^{23} +138687. q^{25} -98153.0 q^{26} +89527.5 q^{28} +160070. q^{29} -272299. q^{31} +369763. q^{32} +39866.7 q^{34} +133440. q^{35} -35341.8 q^{37} +134228. q^{38} +1.80189e6 q^{40} +727812. q^{41} -712974. q^{43} +1.09267e6 q^{44} -196743. q^{46} +476612. q^{47} -741416. q^{49} +2.91045e6 q^{50} -1.46114e6 q^{52} +1.49764e6 q^{53} +1.62862e6 q^{55} +1.10900e6 q^{56} +3.35919e6 q^{58} +205379. q^{59} -1.04757e6 q^{61} -5.71439e6 q^{62} +2.48319e6 q^{64} -2.17782e6 q^{65} -2.56238e6 q^{67} +593470. q^{68} +2.80033e6 q^{70} -3.16847e6 q^{71} +1.62960e6 q^{73} -741674. q^{74} +1.99817e6 q^{76} +1.00235e6 q^{77} -2.11177e6 q^{79} +1.91947e7 q^{80} +1.52737e7 q^{82} -5.43782e6 q^{83} +884561. q^{85} -1.49623e7 q^{86} +1.35352e7 q^{88} +2.27504e6 q^{89} -1.34037e6 q^{91} -2.92879e6 q^{92} +1.00021e7 q^{94} +2.97825e6 q^{95} +1.00026e7 q^{97} -1.55591e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - 2 q^{2} + 1166 q^{4} + 318 q^{5} + 3145 q^{7} - 2355 q^{8}+O(q^{10}) \) 17 * q - 2 * q^2 + 1166 * q^4 + 318 * q^5 + 3145 * q^7 - 2355 * q^8	\( 17 q - 2 q^{2} + 1166 q^{4} + 318 q^{5} + 3145 q^{7} - 2355 q^{8} + 6521 q^{10} + 1764 q^{11} + 18192 q^{13} + 7827 q^{14} + 139226 q^{16} + 15507 q^{17} + 52083 q^{19} - 721 q^{20} - 234434 q^{22} - 63823 q^{23} + 202153 q^{25} + 367956 q^{26} + 182306 q^{28} + 502955 q^{29} + 347531 q^{31} + 243908 q^{32} - 330872 q^{34} - 92641 q^{35} + 447615 q^{37} - 775669 q^{38} + 2203270 q^{40} - 940335 q^{41} + 478562 q^{43} + 596924 q^{44} - 3078663 q^{46} - 703121 q^{47} + 1895082 q^{49} + 876967 q^{50} + 6278296 q^{52} + 1005974 q^{53} + 5212846 q^{55} - 3425294 q^{56} + 6710166 q^{58} + 3491443 q^{59} + 11510749 q^{61} - 5996234 q^{62} + 29496941 q^{64} - 11094180 q^{65} + 14007144 q^{67} - 19688159 q^{68} + 30909708 q^{70} - 5229074 q^{71} + 5452211 q^{73} - 12819662 q^{74} + 41929340 q^{76} - 9930777 q^{77} + 15275654 q^{79} - 36576105 q^{80} + 32025935 q^{82} - 7826609 q^{83} + 11836945 q^{85} - 51649136 q^{86} + 30223741 q^{88} + 6436185 q^{89} + 11633535 q^{91} - 43357972 q^{92} - 4494252 q^{94} - 23741055 q^{95} + 26377540 q^{97} - 26517816 q^{98}+O(q^{100}) \) 17 * q - 2 * q^2 + 1166 * q^4 + 318 * q^5 + 3145 * q^7 - 2355 * q^8 + 6521 * q^10 + 1764 * q^11 + 18192 * q^13 + 7827 * q^14 + 139226 * q^16 + 15507 * q^17 + 52083 * q^19 - 721 * q^20 - 234434 * q^22 - 63823 * q^23 + 202153 * q^25 + 367956 * q^26 + 182306 * q^28 + 502955 * q^29 + 347531 * q^31 + 243908 * q^32 - 330872 * q^34 - 92641 * q^35 + 447615 * q^37 - 775669 * q^38 + 2203270 * q^40 - 940335 * q^41 + 478562 * q^43 + 596924 * q^44 - 3078663 * q^46 - 703121 * q^47 + 1895082 * q^49 + 876967 * q^50 + 6278296 * q^52 + 1005974 * q^53 + 5212846 * q^55 - 3425294 * q^56 + 6710166 * q^58 + 3491443 * q^59 + 11510749 * q^61 - 5996234 * q^62 + 29496941 * q^64 - 11094180 * q^65 + 14007144 * q^67 - 19688159 * q^68 + 30909708 * q^70 - 5229074 * q^71 + 5452211 * q^73 - 12819662 * q^74 + 41929340 * q^76 - 9930777 * q^77 + 15275654 * q^79 - 36576105 * q^80 + 32025935 * q^82 - 7826609 * q^83 + 11836945 * q^85 - 51649136 * q^86 + 30223741 * q^88 + 6436185 * q^89 + 11633535 * q^91 - 43357972 * q^92 - 4494252 * q^94 - 23741055 * q^95 + 26377540 * q^97 - 26517816 * q^98

Introduction

L-functions

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