LMFDB - Embedded newform 531.8.a.c.1.1

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			1.1
Root			$21.8139$ 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)

$f(q)$	$=$	$q-21.8139 q^{2} +347.844 q^{4} -424.937 q^{5} +1192.89 q^{7} -4795.65 q^{8} +O(q^{10})$	\(q-21.8139 q^{2} +347.844 q^{4} -424.937 q^{5} +1192.89 q^{7} -4795.65 q^{8} +9269.52 q^{10} +3093.76 q^{11} +1043.61 q^{13} -26021.4 q^{14} +60087.5 q^{16} +16626.4 q^{17} +25587.3 q^{19} -147812. q^{20} -67486.8 q^{22} +98700.2 q^{23} +102447. q^{25} -22765.2 q^{26} +414938. q^{28} +241457. q^{29} -117145. q^{31} -696897. q^{32} -362686. q^{34} -506902. q^{35} +73455.9 q^{37} -558158. q^{38} +2.03785e6 q^{40} +333204. q^{41} +608971. q^{43} +1.07615e6 q^{44} -2.15303e6 q^{46} +923874. q^{47} +599434. q^{49} -2.23475e6 q^{50} +363014. q^{52} -617858. q^{53} -1.31465e6 q^{55} -5.72066e6 q^{56} -5.26710e6 q^{58} +205379. q^{59} +3.02318e6 q^{61} +2.55538e6 q^{62} +7.51081e6 q^{64} -443469. q^{65} +4.24635e6 q^{67} +5.78339e6 q^{68} +1.10575e7 q^{70} -809917. q^{71} +1.31862e6 q^{73} -1.60236e6 q^{74} +8.90040e6 q^{76} +3.69050e6 q^{77} +5.69982e6 q^{79} -2.55334e7 q^{80} -7.26847e6 q^{82} +3.55052e6 q^{83} -7.06517e6 q^{85} -1.32840e7 q^{86} -1.48366e7 q^{88} +1.13503e7 q^{89} +1.24491e6 q^{91} +3.43323e7 q^{92} -2.01532e7 q^{94} -1.08730e7 q^{95} -3.73640e6 q^{97} -1.30760e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−21.8139	−1.92809	−0.964045	−	0.265738i	$-0.914384\pi$
$2$	−21.8139	−1.92809	−0.964045	+	0.265738i	$0.914384\pi$
$3$	0	0
$3$	0	0
$4$	347.844	2.71753
$5$	−424.937	−1.52030	−0.760151	−	0.649747i	$-0.774875\pi$
$5$	−424.937	−1.52030	−0.760151	+	0.649747i	$0.774875\pi$
$6$	0	0
$7$	1192.89	1.31449	0.657243	−	0.753679i	$-0.271723\pi$
$7$	1192.89	1.31449	0.657243	+	0.753679i	$0.271723\pi$
$8$	−4795.65	−3.31156
$9$	0	0
$10$	9269.52	2.93128
$11$	3093.76	0.700829	0.350415	−	0.936595i	$-0.386041\pi$
$11$	3093.76	0.700829	0.350415	+	0.936595i	$0.386041\pi$
$12$	0	0
$13$	1043.61	0.131746	0.0658729	−	0.997828i	$-0.479017\pi$
$13$	1043.61	0.131746	0.0658729	+	0.997828i	$0.479017\pi$
$14$	−26021.4	−2.53445
$15$	0	0
$16$	60087.5	3.66745
$17$	16626.4	0.820780	0.410390	−	0.911910i	$-0.365393\pi$
$17$	16626.4	0.820780	0.410390	+	0.911910i	$0.365393\pi$
$18$	0	0
$19$	25587.3	0.855829	0.427915	−	0.903819i	$-0.359248\pi$
$19$	25587.3	0.855829	0.427915	+	0.903819i	$0.359248\pi$
$20$	−147812.	−4.13147
$21$	0	0
$22$	−67486.8	−1.35126
$23$	98700.2	1.69149	0.845747	−	0.533584i	$-0.179155\pi$
$23$	98700.2	1.69149	0.845747	+	0.533584i	$0.179155\pi$
$24$	0	0
$25$	102447.	1.31132
$26$	−22765.2	−0.254018
$27$	0	0
$28$	414938.	3.57216
$29$	241457.	1.83842	0.919212	−	0.393762i	$-0.128827\pi$
$29$	241457.	1.83842	0.919212	+	0.393762i	$0.128827\pi$
$30$	0	0
$31$	−117145.	−0.706248	−0.353124	−	0.935576i	$-0.614881\pi$
$31$	−117145.	−0.706248	−0.353124	+	0.935576i	$0.614881\pi$
$32$	−696897.	−3.75962
$33$	0	0
$34$	−362686.	−1.58254
$35$	−506902.	−1.99841
$36$	0	0
$37$	73455.9	0.238408	0.119204	−	0.992870i	$-0.461966\pi$
$37$	73455.9	0.238408	0.119204	+	0.992870i	$0.461966\pi$
$38$	−558158.	−1.65012
$39$	0	0
$40$	2.03785e6	5.03457
$41$	333204.	0.755035	0.377517	−	0.926003i	$-0.376778\pi$
$41$	333204.	0.755035	0.377517	+	0.926003i	$0.376778\pi$
$42$	0	0
$43$	608971.	1.16804	0.584019	−	0.811740i	$-0.301479\pi$
$43$	608971.	1.16804	0.584019	+	0.811740i	$0.301479\pi$
$44$	1.07615e6	1.90453
$45$	0	0
$46$	−2.15303e6	−3.26135
$47$	923874.	1.29799	0.648993	−	0.760794i	$-0.275190\pi$
$47$	923874.	1.29799	0.648993	+	0.760794i	$0.275190\pi$
$48$	0	0
$49$	599434.	0.727872
$50$	−2.23475e6	−2.52834
$51$	0	0
$52$	363014.	0.358024
$53$	−617858.	−0.570064	−0.285032	−	0.958518i	$-0.592004\pi$
$53$	−617858.	−0.570064	−0.285032	+	0.958518i	$0.592004\pi$
$54$	0	0
$55$	−1.31465e6	−1.06547
$56$	−5.72066e6	−4.35299
$57$	0	0
$58$	−5.26710e6	−3.54465
$59$	205379.	0.130189
$59$	205379.	0.130189
$60$	0	0
$61$	3.02318e6	1.70533	0.852667	−	0.522455i	$-0.174984\pi$
$61$	3.02318e6	1.70533	0.852667	+	0.522455i	$0.174984\pi$
$62$	2.55538e6	1.36171
$63$	0	0
$64$	7.51081e6	3.58143
$65$	−443469.	−0.200293
$66$	0	0
$67$	4.24635e6	1.72486	0.862429	−	0.506177i	$-0.168942\pi$
$67$	4.24635e6	1.72486	0.862429	+	0.506177i	$0.168942\pi$
$68$	5.78339e6	2.23050
$69$	0	0
$70$	1.10575e7	3.85312
$71$	−809917.	−0.268557	−0.134278	−	0.990944i	$-0.542872\pi$
$71$	−809917.	−0.268557	−0.134278	+	0.990944i	$0.542872\pi$
$72$	0	0
$73$	1.31862e6	0.396726	0.198363	−	0.980129i	$-0.436437\pi$
$73$	1.31862e6	0.396726	0.198363	+	0.980129i	$0.436437\pi$
$74$	−1.60236e6	−0.459672
$75$	0	0
$76$	8.90040e6	2.32574
$77$	3.69050e6	0.921230
$78$	0	0
$79$	5.69982e6	1.30067	0.650334	−	0.759649i	$-0.274629\pi$
$79$	5.69982e6	1.30067	0.650334	+	0.759649i	$0.274629\pi$
$80$	−2.55334e7	−5.57563
$81$	0	0
$82$	−7.26847e6	−1.45577
$83$	3.55052e6	0.681583	0.340791	−	0.940139i	$-0.389305\pi$
$83$	3.55052e6	0.681583	0.340791	+	0.940139i	$0.389305\pi$
$84$	0	0
$85$	−7.06517e6	−1.24783
$86$	−1.32840e7	−2.25208
$87$	0	0
$88$	−1.48366e7	−2.32084
$89$	1.13503e7	1.70664	0.853318	−	0.521390i	$-0.174586\pi$
$89$	1.13503e7	1.70664	0.853318	+	0.521390i	$0.174586\pi$
$90$	0	0
$91$	1.24491e6	0.173178
$92$	3.43323e7	4.59669
$93$	0	0
$94$	−2.01532e7	−2.50264
$95$	−1.08730e7	−1.30112
$96$	0	0
$97$	−3.73640e6	−0.415673	−0.207837	−	0.978164i	$-0.566642\pi$
$97$	−3.73640e6	−0.415673	−0.207837	+	0.978164i	$0.566642\pi$
$98$	−1.30760e7	−1.40340
$99$	0	0
$100$	3.56354e7	3.56354
$101$	−1.81426e6	−0.175216	−0.0876080	−	0.996155i	$-0.527922\pi$
$101$	−1.81426e6	−0.175216	−0.0876080	+	0.996155i	$0.527922\pi$
$102$	0	0
$103$	−2.08845e7	−1.88318	−0.941592	−	0.336755i	$-0.890671\pi$
$103$	−2.08845e7	−1.88318	−0.941592	+	0.336755i	$0.890671\pi$
$104$	−5.00479e6	−0.436284
$105$	0	0
$106$	1.34779e7	1.09913
$107$	−3.03824e6	−0.239761	−0.119881	−	0.992788i	$-0.538251\pi$
$107$	−3.03824e6	−0.239761	−0.119881	+	0.992788i	$0.538251\pi$
$108$	0	0
$109$	−7.65394e6	−0.566099	−0.283049	−	0.959105i	$-0.591346\pi$
$109$	−7.65394e6	−0.566099	−0.283049	+	0.959105i	$0.591346\pi$
$110$	2.86777e7	2.05433
$111$	0	0
$112$	7.16775e7	4.82081
$113$	−1.47494e6	−0.0961611	−0.0480805	−	0.998843i	$-0.515310\pi$
$113$	−1.47494e6	−0.0961611	−0.0480805	+	0.998843i	$0.515310\pi$
$114$	0	0
$115$	−4.19414e7	−2.57158
$116$	8.39892e7	4.99598
$117$	0	0
$118$	−4.48011e6	−0.251016
$119$	1.98334e7	1.07890
$120$	0	0
$121$	−9.91582e6	−0.508838
$122$	−6.59472e7	−3.28804
$123$	0	0
$124$	−4.07482e7	−1.91925
$125$	−1.03351e7	−0.473294
$126$	0	0
$127$	−3.33858e7	−1.44627	−0.723134	−	0.690708i	$-0.757299\pi$
$127$	−3.33858e7	−1.44627	−0.723134	+	0.690708i	$0.757299\pi$
$128$	−7.46368e7	−3.14571
$129$	0	0
$130$	9.67377e6	0.386184
$131$	−4.37782e7	−1.70141	−0.850704	−	0.525645i	$-0.823824\pi$
$131$	−4.37782e7	−1.70141	−0.850704	+	0.525645i	$0.823824\pi$
$132$	0	0
$133$	3.05228e7	1.12498
$134$	−9.26292e7	−3.32568
$135$	0	0
$136$	−7.97343e7	−2.71806
$137$	2.96908e7	0.986507	0.493254	−	0.869886i	$-0.335807\pi$
$137$	2.96908e7	0.986507	0.493254	+	0.869886i	$0.335807\pi$
$138$	0	0
$139$	−6.45869e6	−0.203983	−0.101991	−	0.994785i	$-0.532521\pi$
$139$	−6.45869e6	−0.203983	−0.101991	+	0.994785i	$0.532521\pi$
$140$	−1.76323e8	−5.43076
$141$	0	0
$142$	1.76674e7	0.517802
$143$	3.22868e6	0.0923314
$144$	0	0
$145$	−1.02604e8	−2.79496
$146$	−2.87643e7	−0.764924
$147$	0	0
$148$	2.55512e7	0.647881
$149$	−3.19641e6	−0.0791608	−0.0395804	−	0.999216i	$-0.512602\pi$
$149$	−3.19641e6	−0.0791608	−0.0395804	+	0.999216i	$0.512602\pi$
$150$	0	0
$151$	−1.51144e7	−0.357250	−0.178625	−	0.983917i	$-0.557165\pi$
$151$	−1.51144e7	−0.357250	−0.178625	+	0.983917i	$0.557165\pi$
$152$	−1.22708e8	−2.83413
$153$	0	0
$154$	−8.05041e7	−1.77621
$155$	4.97792e7	1.07371
$156$	0	0
$157$	6.70787e7	1.38336	0.691681	−	0.722203i	$-0.256870\pi$
$157$	6.70787e7	1.38336	0.691681	+	0.722203i	$0.256870\pi$
$158$	−1.24335e8	−2.50780
$159$	0	0
$160$	2.96137e8	5.71575
$161$	1.17738e8	2.22344
$162$	0	0
$163$	−7.17787e7	−1.29819	−0.649096	−	0.760707i	$-0.724853\pi$
$163$	−7.17787e7	−1.29819	−0.649096	+	0.760707i	$0.724853\pi$
$164$	1.15903e8	2.05183
$165$	0	0
$166$	−7.74505e7	−1.31415
$167$	−6.81851e7	−1.13287	−0.566437	−	0.824105i	$-0.691679\pi$
$167$	−6.81851e7	−1.13287	−0.566437	+	0.824105i	$0.691679\pi$
$168$	0	0
$169$	−6.16594e7	−0.982643
$170$	1.54119e8	2.40593
$171$	0	0
$172$	2.11827e8	3.17418
$173$	−5.64546e7	−0.828968	−0.414484	−	0.910057i	$-0.636038\pi$
$173$	−5.64546e7	−0.828968	−0.414484	+	0.910057i	$0.636038\pi$
$174$	0	0
$175$	1.22207e8	1.72371
$176$	1.85896e8	2.57026
$177$	0	0
$178$	−2.47593e8	−3.29055
$179$	−6.74593e7	−0.879136	−0.439568	−	0.898209i	$-0.644868\pi$
$179$	−6.74593e7	−0.879136	−0.439568	+	0.898209i	$0.644868\pi$
$180$	0	0
$181$	2.62245e7	0.328724	0.164362	−	0.986400i	$-0.447443\pi$
$181$	2.62245e7	0.328724	0.164362	+	0.986400i	$0.447443\pi$
$182$	−2.71563e7	−0.333903
$183$	0	0
$184$	−4.73331e8	−5.60148
$185$	−3.12141e7	−0.362452
$186$	0	0
$187$	5.14381e7	0.575227
$188$	3.21364e8	3.52732
$189$	0	0
$190$	2.37182e8	2.50867
$191$	−1.01202e8	−1.05092	−0.525461	−	0.850818i	$-0.676107\pi$
$191$	−1.01202e8	−1.05092	−0.525461	+	0.850818i	$0.676107\pi$
$192$	0	0
$193$	9.77806e7	0.979044	0.489522	−	0.871991i	$-0.337171\pi$
$193$	9.77806e7	0.979044	0.489522	+	0.871991i	$0.337171\pi$
$194$	8.15052e7	0.801455
$195$	0	0
$196$	2.08510e8	1.97802
$197$	2.53181e7	0.235939	0.117969	−	0.993017i	$-0.462362\pi$
$197$	2.53181e7	0.235939	0.117969	+	0.993017i	$0.462362\pi$
$198$	0	0
$199$	1.47608e8	1.32778	0.663888	−	0.747832i	$-0.268905\pi$
$199$	1.47608e8	1.32778	0.663888	+	0.747832i	$0.268905\pi$
$200$	−4.91298e8	−4.34250
$201$	0	0
$202$	3.95759e7	0.337832
$203$	2.88030e8	2.41658
$204$	0	0
$205$	−1.41591e8	−1.14788
$206$	4.55571e8	3.63095
$207$	0	0
$208$	6.27080e7	0.483171
$209$	7.91610e7	0.599790
$210$	0	0
$211$	−1.08855e7	−0.0797736	−0.0398868	−	0.999204i	$-0.512700\pi$
$211$	−1.08855e7	−0.0797736	−0.0398868	+	0.999204i	$0.512700\pi$
$212$	−2.14918e8	−1.54917
$213$	0	0
$214$	6.62757e7	0.462281
$215$	−2.58774e8	−1.77577
$216$	0	0
$217$	−1.39740e8	−0.928353
$218$	1.66962e8	1.09149
$219$	0	0
$220$	−4.57295e8	−2.89545
$221$	1.73515e7	0.108134
$222$	0	0
$223$	2.34198e8	1.41421	0.707107	−	0.707106i	$-0.250000\pi$
$223$	2.34198e8	1.41421	0.707107	+	0.707106i	$0.250000\pi$
$224$	−8.31319e8	−4.94196
$225$	0	0
$226$	3.21741e7	0.185407
$227$	−2.70843e8	−1.53683	−0.768417	−	0.639949i	$-0.778955\pi$
$227$	−2.70843e8	−1.53683	−0.768417	+	0.639949i	$0.778955\pi$
$228$	0	0
$229$	7.25435e7	0.399185	0.199592	−	0.979879i	$-0.436038\pi$
$229$	7.25435e7	0.399185	0.199592	+	0.979879i	$0.436038\pi$
$230$	9.14903e8	4.95824
$231$	0	0
$232$	−1.15794e9	−6.08805
$233$	−6.86130e7	−0.355354	−0.177677	−	0.984089i	$-0.556858\pi$
$233$	−6.86130e7	−0.355354	−0.177677	+	0.984089i	$0.556858\pi$
$234$	0	0
$235$	−3.92588e8	−1.97333
$236$	7.14399e7	0.353793
$237$	0	0
$238$	−4.32643e8	−2.08022
$239$	1.66795e8	0.790298	0.395149	−	0.918617i	$-0.370693\pi$
$239$	1.66795e8	0.790298	0.395149	+	0.918617i	$0.370693\pi$
$240$	0	0
$241$	−1.09566e8	−0.504213	−0.252107	−	0.967699i	$-0.581123\pi$
$241$	−1.09566e8	−0.504213	−0.252107	+	0.967699i	$0.581123\pi$
$242$	2.16302e8	0.981086
$243$	0	0
$244$	1.05160e9	4.63430
$245$	−2.54722e8	−1.10659
$246$	0	0
$247$	2.67032e7	0.112752
$248$	5.61786e8	2.33878
$249$	0	0
$250$	2.25449e8	0.912554
$251$	2.24011e8	0.894154	0.447077	−	0.894496i	$-0.352465\pi$
$251$	2.24011e8	0.894154	0.447077	+	0.894496i	$0.352465\pi$
$252$	0	0
$253$	3.05355e8	1.18545
$254$	7.28273e8	2.78853
$255$	0	0
$256$	6.66733e8	2.48378
$257$	−4.92524e8	−1.80993	−0.904964	−	0.425489i	$-0.860102\pi$
$257$	−4.92524e8	−1.80993	−0.904964	+	0.425489i	$0.860102\pi$
$258$	0	0
$259$	8.76245e7	0.313384
$260$	−1.54258e8	−0.544304
$261$	0	0
$262$	9.54971e8	3.28047
$263$	4.27801e8	1.45010	0.725049	−	0.688698i	$-0.241817\pi$
$263$	4.27801e8	1.45010	0.725049	+	0.688698i	$0.241817\pi$
$264$	0	0
$265$	2.62551e8	0.866669
$266$	−6.65819e8	−2.16905
$267$	0	0
$268$	1.47707e9	4.68736
$269$	−1.72598e8	−0.540633	−0.270316	−	0.962772i	$-0.587128\pi$
$269$	−1.72598e8	−0.540633	−0.270316	+	0.962772i	$0.587128\pi$
$270$	0	0
$271$	2.54060e8	0.775431	0.387716	−	0.921779i	$-0.373264\pi$
$271$	2.54060e8	0.775431	0.387716	+	0.921779i	$0.373264\pi$
$272$	9.99038e8	3.01017
$273$	0	0
$274$	−6.47671e8	−1.90208
$275$	3.16945e8	0.919009
$276$	0	0
$277$	−3.42400e8	−0.967954	−0.483977	−	0.875081i	$-0.660808\pi$
$277$	−3.42400e8	−0.967954	−0.483977	+	0.875081i	$0.660808\pi$
$278$	1.40889e8	0.393297
$279$	0	0
$280$	2.43092e9	6.61786
$281$	6.48885e8	1.74460	0.872300	−	0.488971i	$-0.162628\pi$
$281$	6.48885e8	1.74460	0.872300	+	0.488971i	$0.162628\pi$
$282$	0	0
$283$	−1.70974e8	−0.448413	−0.224206	−	0.974542i	$-0.571979\pi$
$283$	−1.70974e8	−0.448413	−0.224206	+	0.974542i	$0.571979\pi$
$284$	−2.81725e8	−0.729812
$285$	0	0
$286$	−7.04300e7	−0.178023
$287$	3.97475e8	0.992482
$288$	0	0
$289$	−1.33902e8	−0.326320
$290$	2.23818e9	5.38894
$291$	0	0
$292$	4.58676e8	1.07812
$293$	7.65359e8	1.77758	0.888789	−	0.458316i	$-0.151547\pi$
$293$	7.65359e8	1.77758	0.888789	+	0.458316i	$0.151547\pi$
$294$	0	0
$295$	−8.72732e7	−0.197926
$296$	−3.52268e8	−0.789501
$297$	0	0
$298$	6.97260e7	0.152629
$299$	1.03005e8	0.222847
$300$	0	0
$301$	7.26433e8	1.53537
$302$	3.29704e8	0.688811
$303$	0	0
$304$	1.53748e9	3.13871
$305$	−1.28466e9	−2.59262
$306$	0	0
$307$	1.89113e8	0.373024	0.186512	−	0.982453i	$-0.440282\pi$
$307$	1.89113e8	0.373024	0.186512	+	0.982453i	$0.440282\pi$
$308$	1.28372e9	2.50347
$309$	0	0
$310$	−1.08588e9	−2.07021
$311$	−2.41634e8	−0.455508	−0.227754	−	0.973719i	$-0.573138\pi$
$311$	−2.41634e8	−0.455508	−0.227754	+	0.973719i	$0.573138\pi$
$312$	0	0
$313$	−4.22149e8	−0.778145	−0.389073	−	0.921207i	$-0.627204\pi$
$313$	−4.22149e8	−0.778145	−0.389073	+	0.921207i	$0.627204\pi$
$314$	−1.46325e9	−2.66725
$315$	0	0
$316$	1.98265e9	3.53461
$317$	−1.02142e9	−1.80093	−0.900467	−	0.434925i	$-0.856775\pi$
$317$	−1.02142e9	−1.80093	−0.900467	+	0.434925i	$0.856775\pi$
$318$	0	0
$319$	7.47008e8	1.28842
$320$	−3.19162e9	−5.44486
$321$	0	0
$322$	−2.56832e9	−4.28700
$323$	4.25425e8	0.702448
$324$	0	0
$325$	1.06914e8	0.172760
$326$	1.56577e9	2.50303
$327$	0	0
$328$	−1.59793e9	−2.50034
$329$	1.10208e9	1.70618
$330$	0	0
$331$	8.62048e8	1.30657	0.653286	−	0.757111i	$-0.273390\pi$
$331$	8.62048e8	1.30657	0.653286	+	0.757111i	$0.273390\pi$
$332$	1.23503e9	1.85222
$333$	0	0
$334$	1.48738e9	2.18428
$335$	−1.80443e9	−2.62231
$336$	0	0
$337$	−9.09270e8	−1.29416	−0.647080	−	0.762422i	$-0.724010\pi$
$337$	−9.09270e8	−1.29416	−0.647080	+	0.762422i	$0.724010\pi$
$338$	1.34503e9	1.89462
$339$	0	0
$340$	−2.45758e9	−3.39103
$341$	−3.62418e8	−0.494959
$342$	0	0
$343$	−2.67336e8	−0.357708
$344$	−2.92041e9	−3.86803
$345$	0	0
$346$	1.23149e9	1.59833
$347$	−4.58759e8	−0.589428	−0.294714	−	0.955585i	$-0.595224\pi$
$347$	−4.58759e8	−0.589428	−0.294714	+	0.955585i	$0.595224\pi$
$348$	0	0
$349$	−9.57438e8	−1.20565	−0.602826	−	0.797873i	$-0.705959\pi$
$349$	−9.57438e8	−1.20565	−0.602826	+	0.797873i	$0.705959\pi$
$350$	−2.66581e9	−3.32346
$351$	0	0
$352$	−2.15603e9	−2.63485
$353$	−3.43926e8	−0.416154	−0.208077	−	0.978112i	$-0.566720\pi$
$353$	−3.43926e8	−0.416154	−0.208077	+	0.978112i	$0.566720\pi$
$354$	0	0
$355$	3.44164e8	0.408287
$356$	3.94812e9	4.63784
$357$	0	0
$358$	1.47155e9	1.69505
$359$	9.03951e8	1.03113	0.515566	−	0.856850i	$-0.327582\pi$
$359$	9.03951e8	1.03113	0.515566	+	0.856850i	$0.327582\pi$
$360$	0	0
$361$	−2.39161e8	−0.267556
$362$	−5.72057e8	−0.633810
$363$	0	0
$364$	4.33034e8	0.470617
$365$	−5.60332e8	−0.603144
$366$	0	0
$367$	1.55115e9	1.63804	0.819019	−	0.573767i	$-0.194518\pi$
$367$	1.55115e9	1.63804	0.819019	+	0.573767i	$0.194518\pi$
$368$	5.93065e9	6.20347
$369$	0	0
$370$	6.80900e8	0.698840
$371$	−7.37035e8	−0.749341
$372$	0	0
$373$	−1.60756e9	−1.60393	−0.801967	−	0.597368i	$-0.796213\pi$
$373$	−1.60756e9	−1.60393	−0.801967	+	0.597368i	$0.796213\pi$
$374$	−1.12206e9	−1.10909
$375$	0	0
$376$	−4.43057e9	−4.29836
$377$	2.51987e8	0.242205
$378$	0	0
$379$	5.29474e8	0.499583	0.249791	−	0.968300i	$-0.419638\pi$
$379$	5.29474e8	0.499583	0.249791	+	0.968300i	$0.419638\pi$
$380$	−3.78211e9	−3.53583
$381$	0	0
$382$	2.20760e9	2.02627
$383$	1.13814e9	1.03515	0.517573	−	0.855639i	$-0.326836\pi$
$383$	1.13814e9	1.03515	0.517573	+	0.855639i	$0.326836\pi$
$384$	0	0
$385$	−1.56823e9	−1.40055
$386$	−2.13297e9	−1.88769
$387$	0	0
$388$	−1.29968e9	−1.12961
$389$	8.63651e8	0.743900	0.371950	−	0.928253i	$-0.378689\pi$
$389$	8.63651e8	0.743900	0.371950	+	0.928253i	$0.378689\pi$
$390$	0	0
$391$	1.64103e9	1.38834
$392$	−2.87467e9	−2.41039
$393$	0	0
$394$	−5.52285e8	−0.454911
$395$	−2.42206e9	−1.97741
$396$	0	0
$397$	3.54481e7	0.0284332	0.0142166	−	0.999899i	$-0.495475\pi$
$397$	3.54481e7	0.0284332	0.0142166	+	0.999899i	$0.495475\pi$
$398$	−3.21991e9	−2.56007
$399$	0	0
$400$	6.15576e9	4.80919
$401$	−2.17381e8	−0.168351	−0.0841756	−	0.996451i	$-0.526826\pi$
$401$	−2.17381e8	−0.168351	−0.0841756	+	0.996451i	$0.526826\pi$
$402$	0	0
$403$	−1.22254e8	−0.0930453
$404$	−6.31078e8	−0.476155
$405$	0	0
$406$	−6.28305e9	−4.65939
$407$	2.27255e8	0.167083
$408$	0	0
$409$	2.43499e9	1.75981	0.879904	−	0.475151i	$-0.157607\pi$
$409$	2.43499e9	1.75981	0.879904	+	0.475151i	$0.157607\pi$
$410$	3.08864e9	2.21322
$411$	0	0
$412$	−7.26454e9	−5.11762
$413$	2.44994e8	0.171131
$414$	0	0
$415$	−1.50875e9	−1.03621
$416$	−7.27290e8	−0.495314
$417$	0	0
$418$	−1.72681e9	−1.15645
$419$	4.11950e8	0.273587	0.136794	−	0.990600i	$-0.456320\pi$
$419$	4.11950e8	0.273587	0.136794	+	0.990600i	$0.456320\pi$
$420$	0	0
$421$	−1.54001e9	−1.00585	−0.502927	−	0.864329i	$-0.667744\pi$
$421$	−1.54001e9	−1.00585	−0.502927	+	0.864329i	$0.667744\pi$
$422$	2.37454e8	0.153811
$423$	0	0
$424$	2.96303e9	1.88780
$425$	1.70332e9	1.07630
$426$	0	0
$427$	3.60631e9	2.24164
$428$	−1.05683e9	−0.651559
$429$	0	0
$430$	5.64487e9	3.42385
$431$	−2.03052e9	−1.22162	−0.610811	−	0.791776i	$-0.709157\pi$
$431$	−2.03052e9	−1.22162	−0.610811	+	0.791776i	$0.709157\pi$
$432$	0	0
$433$	6.40897e8	0.379385	0.189693	−	0.981844i	$-0.439251\pi$
$433$	6.40897e8	0.379385	0.189693	+	0.981844i	$0.439251\pi$
$434$	3.04828e9	1.78995
$435$	0	0
$436$	−2.66238e9	−1.53839
$437$	2.52547e9	1.44763
$438$	0	0
$439$	−4.52255e8	−0.255128	−0.127564	−	0.991830i	$-0.540716\pi$
$439$	−4.52255e8	−0.255128	−0.127564	+	0.991830i	$0.540716\pi$
$440$	6.30462e9	3.52837
$441$	0	0
$442$	−3.78503e8	−0.208493
$443$	−1.95166e9	−1.06657	−0.533287	−	0.845935i	$-0.679043\pi$
$443$	−1.95166e9	−1.06657	−0.533287	+	0.845935i	$0.679043\pi$
$444$	0	0
$445$	−4.82315e9	−2.59460
$446$	−5.10875e9	−2.72673
$447$	0	0
$448$	8.95954e9	4.70774
$449$	7.36502e8	0.383983	0.191991	−	0.981397i	$-0.438505\pi$
$449$	7.36502e8	0.383983	0.191991	+	0.981397i	$0.438505\pi$
$450$	0	0
$451$	1.03085e9	0.529150
$452$	−5.13049e8	−0.261321
$453$	0	0
$454$	5.90813e9	2.96316
$455$	−5.29008e8	−0.263283
$456$	0	0
$457$	8.44943e8	0.414115	0.207057	−	0.978329i	$-0.433611\pi$
$457$	8.44943e8	0.414115	0.207057	+	0.978329i	$0.433611\pi$
$458$	−1.58245e9	−0.769664
$459$	0	0
$460$	−1.45891e10	−6.98835
$461$	−2.59377e8	−0.123305	−0.0616523	−	0.998098i	$-0.519637\pi$
$461$	−2.59377e8	−0.123305	−0.0616523	+	0.998098i	$0.519637\pi$
$462$	0	0
$463$	4.01914e8	0.188191	0.0940957	−	0.995563i	$-0.470004\pi$
$463$	4.01914e8	0.188191	0.0940957	+	0.995563i	$0.470004\pi$
$464$	1.45085e10	6.74233
$465$	0	0
$466$	1.49671e9	0.685154
$467$	3.10097e9	1.40893	0.704463	−	0.709740i	$-0.251188\pi$
$467$	3.10097e9	1.40893	0.704463	+	0.709740i	$0.251188\pi$
$468$	0	0
$469$	5.06541e9	2.26730
$470$	8.56386e9	3.80476
$471$	0	0
$472$	−9.84925e8	−0.431128
$473$	1.88401e9	0.818596
$474$	0	0
$475$	2.62133e9	1.12226
$476$	6.89893e9	2.93195
$477$	0	0
$478$	−3.63844e9	−1.52377
$479$	−3.79632e9	−1.57829	−0.789147	−	0.614204i	$-0.789477\pi$
$479$	−3.79632e9	−1.57829	−0.789147	+	0.614204i	$0.789477\pi$
$480$	0	0
$481$	7.66594e7	0.0314092
$482$	2.39005e9	0.972169
$483$	0	0
$484$	−3.44916e9	−1.38278
$485$	1.58773e9	0.631948
$486$	0	0
$487$	7.82358e6	0.00306940	0.00153470	−	0.999999i	$-0.499511\pi$
$487$	7.82358e6	0.00306940	0.00153470	+	0.999999i	$0.499511\pi$
$488$	−1.44981e10	−5.64731
$489$	0	0
$490$	5.55646e9	2.13360
$491$	2.48479e9	0.947336	0.473668	−	0.880704i	$-0.342930\pi$
$491$	2.48479e9	0.947336	0.473668	+	0.880704i	$0.342930\pi$
$492$	0	0
$493$	4.01455e9	1.50894
$494$	−5.82500e8	−0.217396
$495$	0	0
$496$	−7.03894e9	−2.59013
$497$	−9.66138e8	−0.353014
$498$	0	0
$499$	3.32153e9	1.19670	0.598352	−	0.801234i	$-0.295823\pi$
$499$	3.32153e9	1.19670	0.598352	+	0.801234i	$0.295823\pi$
$500$	−3.59502e9	−1.28619
$501$	0	0
$502$	−4.88655e9	−1.72401
$503$	−2.45772e9	−0.861082	−0.430541	−	0.902571i	$-0.641677\pi$
$503$	−2.45772e9	−0.861082	−0.430541	+	0.902571i	$0.641677\pi$
$504$	0	0
$505$	7.70945e8	0.266381
$506$	−6.66096e9	−2.28565
$507$	0	0
$508$	−1.16131e10	−3.93028
$509$	4.91205e8	0.165101	0.0825507	−	0.996587i	$-0.473693\pi$
$509$	4.91205e8	0.165101	0.0825507	+	0.996587i	$0.473693\pi$
$510$	0	0
$511$	1.57297e9	0.521491
$512$	−4.99051e9	−1.64324
$513$	0	0
$514$	1.07438e10	3.48970
$515$	8.87458e9	2.86301
$516$	0	0
$517$	2.85824e9	0.909667
$518$	−1.91143e9	−0.604232
$519$	0	0
$520$	2.12672e9	0.663283
$521$	−3.12235e9	−0.967275	−0.483638	−	0.875268i	$-0.660685\pi$
$521$	−3.12235e9	−0.967275	−0.483638	+	0.875268i	$0.660685\pi$
$522$	0	0
$523$	1.24953e9	0.381938	0.190969	−	0.981596i	$-0.438837\pi$
$523$	1.24953e9	0.381938	0.190969	+	0.981596i	$0.438837\pi$
$524$	−1.52280e10	−4.62363
$525$	0	0
$526$	−9.33200e9	−2.79592
$527$	−1.94770e9	−0.579674
$528$	0	0
$529$	6.33690e9	1.86115
$530$	−5.72725e9	−1.67102
$531$	0	0
$532$	1.06172e10	3.05716
$533$	3.47736e8	0.0994727
$534$	0	0
$535$	1.29106e9	0.364509
$536$	−2.03640e10	−5.71197
$537$	0	0
$538$	3.76502e9	1.04239
$539$	1.85451e9	0.510114
$540$	0	0
$541$	8.58513e8	0.233108	0.116554	−	0.993184i	$-0.462815\pi$
$541$	8.58513e8	0.233108	0.116554	+	0.993184i	$0.462815\pi$
$542$	−5.54202e9	−1.49510
$543$	0	0
$544$	−1.15869e10	−3.08582
$545$	3.25244e9	0.860641
$546$	0	0
$547$	6.72132e9	1.75590	0.877948	−	0.478755i	$-0.158912\pi$
$547$	6.72132e9	1.75590	0.877948	+	0.478755i	$0.158912\pi$
$548$	1.03278e10	2.68087
$549$	0	0
$550$	−6.91379e9	−1.77193
$551$	6.17823e9	1.57338
$552$	0	0
$553$	6.79923e9	1.70971
$554$	7.46907e9	1.86630
$555$	0	0
$556$	−2.24662e9	−0.554329
$557$	2.12425e9	0.520850	0.260425	−	0.965494i	$-0.416137\pi$
$557$	2.12425e9	0.520850	0.260425	+	0.965494i	$0.416137\pi$
$558$	0	0
$559$	6.35529e8	0.153884
$560$	−3.04584e10	−7.32908
$561$	0	0
$562$	−1.41547e10	−3.36375
$563$	−1.69196e9	−0.399587	−0.199793	−	0.979838i	$-0.564027\pi$
$563$	−1.69196e9	−0.399587	−0.199793	+	0.979838i	$0.564027\pi$
$564$	0	0
$565$	6.26756e8	0.146194
$566$	3.72961e9	0.864580
$567$	0	0
$568$	3.88407e9	0.889341
$569$	7.54396e8	0.171675	0.0858374	−	0.996309i	$-0.472643\pi$
$569$	7.54396e8	0.171675	0.0858374	+	0.996309i	$0.472643\pi$
$570$	0	0
$571$	−3.56448e9	−0.801254	−0.400627	−	0.916241i	$-0.631208\pi$
$571$	−3.56448e9	−0.801254	−0.400627	+	0.916241i	$0.631208\pi$
$572$	1.12308e9	0.250913
$573$	0	0
$574$	−8.67045e9	−1.91360
$575$	1.01115e10	2.21808
$576$	0	0
$577$	−6.20845e9	−1.34545	−0.672725	−	0.739892i	$-0.734876\pi$
$577$	−6.20845e9	−1.34545	−0.672725	+	0.739892i	$0.734876\pi$
$578$	2.92091e9	0.629175
$579$	0	0
$580$	−3.56901e10	−7.59539
$581$	4.23537e9	0.895931
$582$	0	0
$583$	−1.91151e9	−0.399517
$584$	−6.32366e9	−1.31378
$585$	0	0
$586$	−1.66954e10	−3.42733
$587$	1.55164e9	0.316635	0.158317	−	0.987388i	$-0.449393\pi$
$587$	1.55164e9	0.316635	0.158317	+	0.987388i	$0.449393\pi$
$588$	0	0
$589$	−2.99742e9	−0.604428
$590$	1.90376e9	0.381620
$591$	0	0
$592$	4.41378e9	0.874349
$593$	−6.42351e9	−1.26497	−0.632486	−	0.774572i	$-0.717965\pi$
$593$	−6.42351e9	−1.26497	−0.632486	+	0.774572i	$0.717965\pi$
$594$	0	0
$595$	−8.42794e9	−1.64026
$596$	−1.11185e9	−0.215122
$597$	0	0
$598$	−2.24693e9	−0.429670
$599$	3.92988e9	0.747111	0.373556	−	0.927608i	$-0.378139\pi$
$599$	3.92988e9	0.747111	0.373556	+	0.927608i	$0.378139\pi$
$600$	0	0
$601$	−5.05280e9	−0.949448	−0.474724	−	0.880135i	$-0.657452\pi$
$601$	−5.05280e9	−0.949448	−0.474724	+	0.880135i	$0.657452\pi$
$602$	−1.58463e10	−2.96033
$603$	0	0
$604$	−5.25747e9	−0.970839
$605$	4.21360e9	0.773588
$606$	0	0
$607$	−7.19598e9	−1.30596	−0.652979	−	0.757376i	$-0.726481\pi$
$607$	−7.19598e9	−1.30596	−0.652979	+	0.757376i	$0.726481\pi$
$608$	−1.78317e10	−3.21759
$609$	0	0
$610$	2.80234e10	4.99881
$611$	9.64165e8	0.171004
$612$	0	0
$613$	−4.01935e9	−0.704765	−0.352383	−	0.935856i	$-0.614628\pi$
$613$	−4.01935e9	−0.704765	−0.352383	+	0.935856i	$0.614628\pi$
$614$	−4.12528e9	−0.719223
$615$	0	0
$616$	−1.76984e10	−3.05071
$617$	−2.76235e9	−0.473458	−0.236729	−	0.971576i	$-0.576075\pi$
$617$	−2.76235e9	−0.473458	−0.236729	+	0.971576i	$0.576075\pi$
$618$	0	0
$619$	6.94825e9	1.17749	0.588746	−	0.808318i	$-0.299622\pi$
$619$	6.94825e9	1.17749	0.588746	+	0.808318i	$0.299622\pi$
$620$	1.73154e10	2.91784
$621$	0	0
$622$	5.27096e9	0.878261
$623$	1.35396e10	2.24335
$624$	0	0
$625$	−3.61185e9	−0.591766
$626$	9.20870e9	1.50033
$627$	0	0
$628$	2.33329e10	3.75933
$629$	1.22131e9	0.195680
$630$	0	0
$631$	4.84723e9	0.768052	0.384026	−	0.923322i	$-0.374537\pi$
$631$	4.84723e9	0.768052	0.384026	+	0.923322i	$0.374537\pi$
$632$	−2.73343e10	−4.30723
$633$	0	0
$634$	2.22811e10	3.47236
$635$	1.41869e10	2.19876
$636$	0	0
$637$	6.25576e8	0.0958942
$638$	−1.62951e10	−2.48419
$639$	0	0
$640$	3.17160e10	4.78242
$641$	1.40292e9	0.210393	0.105196	−	0.994451i	$-0.466453\pi$
$641$	1.40292e9	0.210393	0.105196	+	0.994451i	$0.466453\pi$
$642$	0	0
$643$	−8.62484e9	−1.27942	−0.639709	−	0.768617i	$-0.720945\pi$
$643$	−8.62484e9	−1.27942	−0.639709	+	0.768617i	$0.720945\pi$
$644$	4.09545e10	6.04228
$645$	0	0
$646$	−9.28015e9	−1.35438
$647$	2.56483e9	0.372300	0.186150	−	0.982521i	$-0.440399\pi$
$647$	2.56483e9	0.372300	0.186150	+	0.982521i	$0.440399\pi$
$648$	0	0
$649$	6.35393e8	0.0912402
$650$	−2.33221e9	−0.333098
$651$	0	0
$652$	−2.49678e10	−3.52788
$653$	2.05512e9	0.288829	0.144414	−	0.989517i	$-0.453870\pi$
$653$	2.05512e9	0.288829	0.144414	+	0.989517i	$0.453870\pi$
$654$	0	0
$655$	1.86030e10	2.58665
$656$	2.00214e10	2.76905
$657$	0	0
$658$	−2.40405e10	−3.28968
$659$	7.82229e9	1.06472	0.532360	−	0.846518i	$-0.321305\pi$
$659$	7.82229e9	1.06472	0.532360	+	0.846518i	$0.321305\pi$
$660$	0	0
$661$	−4.59162e8	−0.0618388	−0.0309194	−	0.999522i	$-0.509844\pi$
$661$	−4.59162e8	−0.0618388	−0.0309194	+	0.999522i	$0.509844\pi$
$662$	−1.88046e10	−2.51919
$663$	0	0
$664$	−1.70270e10	−2.25710
$665$	−1.29703e10	−1.71030
$666$	0	0
$667$	2.38318e10	3.10968
$668$	−2.37178e10	−3.07862
$669$	0	0
$670$	3.93616e10	5.05604
$671$	9.35299e9	1.19515
$672$	0	0
$673$	−6.96032e9	−0.880190	−0.440095	−	0.897951i	$-0.645055\pi$
$673$	−6.96032e9	−0.880190	−0.440095	+	0.897951i	$0.645055\pi$
$674$	1.98347e10	2.49526
$675$	0	0
$676$	−2.14479e10	−2.67036
$677$	−1.09451e10	−1.35568	−0.677842	−	0.735208i	$-0.737085\pi$
$677$	−1.09451e10	−1.35568	−0.677842	+	0.735208i	$0.737085\pi$
$678$	0	0
$679$	−4.45710e9	−0.546396
$680$	3.38821e10	4.13227
$681$	0	0
$682$	7.90574e9	0.954326
$683$	−2.98731e9	−0.358763	−0.179382	−	0.983780i	$-0.557410\pi$
$683$	−2.98731e9	−0.358763	−0.179382	+	0.983780i	$0.557410\pi$
$684$	0	0
$685$	−1.26167e10	−1.49979
$686$	5.83164e9	0.689693
$687$	0	0
$688$	3.65916e10	4.28372
$689$	−6.44804e8	−0.0751036
$690$	0	0
$691$	5.37053e9	0.619219	0.309609	−	0.950864i	$-0.399802\pi$
$691$	5.37053e9	0.619219	0.309609	+	0.950864i	$0.399802\pi$
$692$	−1.96374e10	−2.25275
$693$	0	0
$694$	1.00073e10	1.13647
$695$	2.74454e9	0.310115
$696$	0	0
$697$	5.53998e9	0.619717
$698$	2.08854e10	2.32461
$699$	0	0
$700$	4.25090e10	4.68423
$701$	−6.64856e9	−0.728979	−0.364489	−	0.931208i	$-0.618756\pi$
$701$	−6.64856e9	−0.728979	−0.364489	+	0.931208i	$0.618756\pi$
$702$	0	0
$703$	1.87954e9	0.204036
$704$	2.32366e10	2.50997
$705$	0	0
$706$	7.50236e9	0.802382
$707$	−2.16420e9	−0.230319
$708$	0	0
$709$	5.41047e8	0.0570129	0.0285064	−	0.999594i	$-0.490925\pi$
$709$	5.41047e8	0.0570129	0.0285064	+	0.999594i	$0.490925\pi$
$710$	−7.50753e9	−0.787215
$711$	0	0
$712$	−5.44319e10	−5.65163
$713$	−1.15622e10	−1.19461
$714$	0	0
$715$	−1.37199e9	−0.140372
$716$	−2.34653e10	−2.38908
$717$	0	0
$718$	−1.97186e10	−1.98812
$719$	−9.95307e9	−0.998633	−0.499316	−	0.866420i	$-0.666415\pi$
$719$	−9.95307e9	−0.998633	−0.499316	+	0.866420i	$0.666415\pi$
$720$	0	0
$721$	−2.49128e10	−2.47542
$722$	5.21702e9	0.515872
$723$	0	0
$724$	9.12203e9	0.893319
$725$	2.47364e10	2.41076
$726$	0	0
$727$	−1.32542e10	−1.27933	−0.639665	−	0.768654i	$-0.720927\pi$
$727$	−1.32542e10	−1.27933	−0.639665	+	0.768654i	$0.720927\pi$
$728$	−5.97015e9	−0.573489
$729$	0	0
$730$	1.22230e10	1.16292
$731$	1.01250e10	0.958703
$732$	0	0
$733$	6.98110e8	0.0654727	0.0327363	−	0.999464i	$-0.489578\pi$
$733$	6.98110e8	0.0654727	0.0327363	+	0.999464i	$0.489578\pi$
$734$	−3.38367e10	−3.15828
$735$	0	0
$736$	−6.87838e10	−6.35937
$737$	1.31372e10	1.20883
$738$	0	0
$739$	5.01583e9	0.457180	0.228590	−	0.973523i	$-0.426588\pi$
$739$	5.01583e9	0.457180	0.228590	+	0.973523i	$0.426588\pi$
$740$	−1.08576e10	−0.984974
$741$	0	0
$742$	1.60776e10	1.44480
$743$	−1.65206e10	−1.47763	−0.738814	−	0.673910i	$-0.764614\pi$
$743$	−1.65206e10	−1.47763	−0.738814	+	0.673910i	$0.764614\pi$
$744$	0	0
$745$	1.35827e9	0.120348
$746$	3.50671e10	3.09253
$747$	0	0
$748$	1.78924e10	1.56320
$749$	−3.62427e9	−0.315162
$750$	0	0
$751$	1.65574e10	1.42643	0.713216	−	0.700944i	$-0.247238\pi$
$751$	1.65574e10	1.42643	0.713216	+	0.700944i	$0.247238\pi$
$752$	5.55133e10	4.76030
$753$	0	0
$754$	−5.49680e9	−0.466993
$755$	6.42269e9	0.543128
$756$	0	0
$757$	−8.37645e9	−0.701818	−0.350909	−	0.936410i	$-0.614127\pi$
$757$	−8.37645e9	−0.701818	−0.350909	+	0.936410i	$0.614127\pi$
$758$	−1.15499e10	−0.963240
$759$	0	0
$760$	5.21431e10	4.30873
$761$	−2.25820e9	−0.185745	−0.0928725	−	0.995678i	$-0.529605\pi$
$761$	−2.25820e9	−0.185745	−0.0928725	+	0.995678i	$0.529605\pi$
$762$	0	0
$763$	−9.13028e9	−0.744129
$764$	−3.52024e10	−2.85592
$765$	0	0
$766$	−2.48273e10	−1.99585
$767$	2.14336e8	0.0171519
$768$	0	0
$769$	5.84510e9	0.463500	0.231750	−	0.972775i	$-0.425555\pi$
$769$	5.84510e9	0.463500	0.231750	+	0.972775i	$0.425555\pi$
$770$	3.42092e10	2.70038
$771$	0	0
$772$	3.40124e10	2.66058
$773$	4.05810e9	0.316005	0.158003	−	0.987439i	$-0.449495\pi$
$773$	4.05810e9	0.316005	0.158003	+	0.987439i	$0.449495\pi$
$774$	0	0
$775$	−1.20011e10	−0.926115
$776$	1.79184e10	1.37653
$777$	0	0
$778$	−1.88396e10	−1.43431
$779$	8.52580e9	0.646181
$780$	0	0
$781$	−2.50569e9	−0.188212
$782$	−3.57971e10	−2.67685
$783$	0	0
$784$	3.60185e10	2.66943
$785$	−2.85042e10	−2.10313
$786$	0	0
$787$	−1.54824e10	−1.13221	−0.566106	−	0.824333i	$-0.691551\pi$
$787$	−1.54824e10	−1.13221	−0.566106	+	0.824333i	$0.691551\pi$
$788$	8.80675e9	0.641171
$789$	0	0
$790$	5.28346e10	3.81262
$791$	−1.75943e9	−0.126402
$792$	0	0
$793$	3.15502e9	0.224671
$794$	−7.73260e8	−0.0548218
$795$	0	0
$796$	5.13447e10	3.60828
$797$	−2.07763e10	−1.45366	−0.726831	−	0.686816i	$-0.759008\pi$
$797$	−2.07763e10	−1.45366	−0.726831	+	0.686816i	$0.759008\pi$
$798$	0	0
$799$	1.53607e10	1.06536
$800$	−7.13947e10	−4.93005
$801$	0	0
$802$	4.74192e9	0.324596
$803$	4.07951e9	0.278037
$804$	0	0
$805$	−5.00313e10	−3.38031
$806$	2.66682e9	0.179400
$807$	0	0
$808$	8.70053e9	0.580238
$809$	−1.24188e10	−0.824628	−0.412314	−	0.911042i	$-0.635279\pi$
$809$	−1.24188e10	−0.824628	−0.412314	+	0.911042i	$0.635279\pi$
$810$	0	0
$811$	−2.86382e10	−1.88526	−0.942632	−	0.333835i	$-0.891657\pi$
$811$	−2.86382e10	−1.88526	−0.942632	+	0.333835i	$0.891657\pi$
$812$	1.00190e11	6.56714
$813$	0	0
$814$	−4.95730e9	−0.322151
$815$	3.05014e10	1.97364
$816$	0	0
$817$	1.55819e10	0.999642
$818$	−5.31165e10	−3.39307
$819$	0	0
$820$	−4.92515e10	−3.11940
$821$	−6.53708e9	−0.412271	−0.206135	−	0.978523i	$-0.566089\pi$
$821$	−6.53708e9	−0.412271	−0.206135	+	0.978523i	$0.566089\pi$
$822$	0	0
$823$	−6.03479e8	−0.0377366	−0.0188683	−	0.999822i	$-0.506006\pi$
$823$	−6.03479e8	−0.0377366	−0.0188683	+	0.999822i	$0.506006\pi$
$824$	1.00155e11	6.23628
$825$	0	0
$826$	−5.34426e9	−0.329957
$827$	1.88036e10	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$827$	1.88036e10	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$828$	0	0
$829$	1.67240e10	1.01953	0.509763	−	0.860315i	$-0.329733\pi$
$829$	1.67240e10	1.01953	0.509763	+	0.860315i	$0.329733\pi$
$830$	3.29116e10	1.99791
$831$	0	0
$832$	7.83836e9	0.471839
$833$	9.96642e9	0.597423
$834$	0	0
$835$	2.89744e10	1.72231
$836$	2.75357e10	1.62995
$837$	0	0
$838$	−8.98622e9	−0.527501
$839$	−2.23616e10	−1.30718	−0.653592	−	0.756847i	$-0.726739\pi$
$839$	−2.23616e10	−1.30718	−0.653592	+	0.756847i	$0.726739\pi$
$840$	0	0
$841$	4.10514e10	2.37981
$842$	3.35935e10	1.93938
$843$	0	0
$844$	−3.78645e9	−0.216787
$845$	2.62014e10	1.49391
$846$	0	0
$847$	−1.18284e10	−0.668861
$848$	−3.71256e10	−2.09068
$849$	0	0
$850$	−3.71559e10	−2.07521
$851$	7.25011e9	0.403265
$852$	0	0
$853$	−1.44921e9	−0.0799483	−0.0399742	−	0.999201i	$-0.512728\pi$
$853$	−1.44921e9	−0.0799483	−0.0399742	+	0.999201i	$0.512728\pi$
$854$	−7.86675e10	−4.32208
$855$	0	0
$856$	1.45703e10	0.793983
$857$	8.02175e9	0.435347	0.217674	−	0.976022i	$-0.430153\pi$
$857$	8.02175e9	0.435347	0.217674	+	0.976022i	$0.430153\pi$
$858$	0	0
$859$	1.19930e10	0.645584	0.322792	−	0.946470i	$-0.395379\pi$
$859$	1.19930e10	0.645584	0.322792	+	0.946470i	$0.395379\pi$
$860$	−9.00132e10	−4.82571
$861$	0	0
$862$	4.42935e10	2.35540
$863$	2.67440e10	1.41641	0.708205	−	0.706007i	$-0.249505\pi$
$863$	2.67440e10	1.41641	0.708205	+	0.706007i	$0.249505\pi$
$864$	0	0
$865$	2.39896e10	1.26028
$866$	−1.39804e10	−0.731489
$867$	0	0
$868$	−4.86079e10	−2.52283
$869$	1.76339e10	0.911546
$870$	0	0
$871$	4.43153e9	0.227243
$872$	3.67056e10	1.87467
$873$	0	0
$874$	−5.50903e10	−2.79116
$875$	−1.23286e10	−0.622139
$876$	0	0
$877$	−6.68932e8	−0.0334875	−0.0167438	−	0.999860i	$-0.505330\pi$
$877$	−6.68932e8	−0.0334875	−0.0167438	+	0.999860i	$0.505330\pi$
$878$	9.86542e9	0.491909
$879$	0	0
$880$	−7.89942e10	−3.90756
$881$	−2.73982e10	−1.34992	−0.674958	−	0.737856i	$-0.735838\pi$
$881$	−2.73982e10	−1.34992	−0.674958	+	0.737856i	$0.735838\pi$
$882$	0	0
$883$	1.56749e9	0.0766198	0.0383099	−	0.999266i	$-0.487803\pi$
$883$	1.56749e9	0.0766198	0.0383099	+	0.999266i	$0.487803\pi$
$884$	6.03561e9	0.293859
$885$	0	0
$886$	4.25732e10	2.05645
$887$	−6.07648e9	−0.292361	−0.146181	−	0.989258i	$-0.546698\pi$
$887$	−6.07648e9	−0.292361	−0.146181	+	0.989258i	$0.546698\pi$
$888$	0	0
$889$	−3.98254e10	−1.90110
$890$	1.05211e11	5.00263
$891$	0	0
$892$	8.14643e10	3.84317
$893$	2.36395e10	1.11086
$894$	0	0
$895$	2.86659e10	1.33655
$896$	−8.90332e10	−4.13499
$897$	0	0
$898$	−1.60659e10	−0.740354
$899$	−2.82854e10	−1.29838
$900$	0	0
$901$	−1.02728e10	−0.467897
$902$	−2.24869e10	−1.02025
$903$	0	0
$904$	7.07329e9	0.318443
$905$	−1.11438e10	−0.499760
$906$	0	0
$907$	−3.19626e10	−1.42238	−0.711191	−	0.702999i	$-0.751844\pi$
$907$	−3.19626e10	−1.42238	−0.711191	+	0.702999i	$0.751844\pi$
$908$	−9.42112e10	−4.17640
$909$	0	0
$910$	1.15397e10	0.507633
$911$	−3.33824e10	−1.46286	−0.731430	−	0.681916i	$-0.761147\pi$
$911$	−3.33824e10	−1.46286	−0.731430	+	0.681916i	$0.761147\pi$
$912$	0	0
$913$	1.09845e10	0.477673
$914$	−1.84315e10	−0.798451
$915$	0	0
$916$	2.52338e10	1.08480
$917$	−5.22224e10	−2.23648
$918$	0	0
$919$	−8.98427e9	−0.381837	−0.190919	−	0.981606i	$-0.561147\pi$
$919$	−8.98427e9	−0.381837	−0.190919	+	0.981606i	$0.561147\pi$
$920$	2.01136e11	8.51594
$921$	0	0
$922$	5.65802e9	0.237742
$923$	−8.45238e8	−0.0353812
$924$	0	0
$925$	7.52530e9	0.312628
$926$	−8.76730e9	−0.362850
$927$	0	0
$928$	−1.68270e11	−6.91177
$929$	4.29286e10	1.75668	0.878339	−	0.478039i	$-0.158652\pi$
$929$	4.29286e10	1.75668	0.878339	+	0.478039i	$0.158652\pi$
$930$	0	0
$931$	1.53379e10	0.622934
$932$	−2.38666e10	−0.965685
$933$	0	0
$934$	−6.76441e10	−2.71654
$935$	−2.18579e10	−0.874518
$936$	0	0
$937$	−2.92971e9	−0.116342	−0.0581710	−	0.998307i	$-0.518527\pi$
$937$	−2.92971e9	−0.116342	−0.0581710	+	0.998307i	$0.518527\pi$
$938$	−1.10496e11	−4.37156
$939$	0	0
$940$	−1.36560e11	−5.36259
$941$	−1.23877e10	−0.484650	−0.242325	−	0.970195i	$-0.577910\pi$
$941$	−1.23877e10	−0.484650	−0.242325	+	0.970195i	$0.577910\pi$
$942$	0	0
$943$	3.28873e10	1.27714
$944$	1.23407e10	0.477461
$945$	0	0
$946$	−4.10975e10	−1.57833
$947$	−5.27697e9	−0.201911	−0.100955	−	0.994891i	$-0.532190\pi$
$947$	−5.27697e9	−0.201911	−0.100955	+	0.994891i	$0.532190\pi$
$948$	0	0
$949$	1.37613e9	0.0522671
$950$	−5.71814e10	−2.16382
$951$	0	0
$952$	−9.51139e10	−3.57285
$953$	1.49617e10	0.559959	0.279980	−	0.960006i	$-0.409672\pi$
$953$	1.49617e10	0.559959	0.279980	+	0.960006i	$0.409672\pi$
$954$	0	0
$955$	4.30043e10	1.59772
$956$	5.80187e10	2.14766
$957$	0	0
$958$	8.28123e10	3.04309
$959$	3.54178e10	1.29675
$960$	0	0
$961$	−1.37897e10	−0.501213
$962$	−1.67224e9	−0.0605599
$963$	0	0
$964$	−3.81117e10	−1.37022
$965$	−4.15506e10	−1.48844
$966$	0	0
$967$	3.49771e10	1.24392	0.621958	−	0.783051i	$-0.286337\pi$
$967$	3.49771e10	1.24392	0.621958	+	0.783051i	$0.286337\pi$
$968$	4.75528e10	1.68505
$969$	0	0
$970$	−3.46346e10	−1.21845
$971$	−4.12202e10	−1.44492	−0.722459	−	0.691414i	$-0.756988\pi$
$971$	−4.12202e10	−1.44492	−0.722459	+	0.691414i	$0.756988\pi$
$972$	0	0
$973$	−7.70449e9	−0.268132
$974$	−1.70662e8	−0.00591809
$975$	0	0
$976$	1.81655e11	6.25423
$977$	3.74910e10	1.28616	0.643081	−	0.765798i	$-0.277656\pi$
$977$	3.74910e10	1.28616	0.643081	+	0.765798i	$0.277656\pi$
$978$	0	0
$979$	3.51150e10	1.19606
$980$	−8.86035e10	−3.00718
$981$	0	0
$982$	−5.42028e10	−1.82655
$983$	−5.07263e9	−0.170332	−0.0851660	−	0.996367i	$-0.527142\pi$
$983$	−5.07263e9	−0.170332	−0.0851660	+	0.996367i	$0.527142\pi$
$984$	0	0
$985$	−1.07586e10	−0.358698
$986$	−8.75728e10	−2.90938
$987$	0	0
$988$	9.28856e9	0.306407
$989$	6.01056e10	1.97573
$990$	0	0
$991$	1.45130e10	0.473694	0.236847	−	0.971547i	$-0.423886\pi$
$991$	1.45130e10	0.473694	0.236847	+	0.971547i	$0.423886\pi$
$992$	8.16379e10	2.65522
$993$	0	0
$994$	2.10752e10	0.680643
$995$	−6.27243e10	−2.01862
$996$	0	0
$997$	1.97848e10	0.632265	0.316132	−	0.948715i	$-0.397616\pi$
$997$	1.97848e10	0.632265	0.316132	+	0.948715i	$0.397616\pi$
$998$	−7.24554e10	−2.30735
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.8.a.c.1.1		17
3.2	odd	2		177.8.a.c.1.17	✓	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.c.1.17	✓	17	3.2	odd	2
531.8.a.c.1.1		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)

\(f(q)\)	\(=\)	\(q-21.8139 q^{2} +347.844 q^{4} -424.937 q^{5} +1192.89 q^{7} -4795.65 q^{8} +O(q^{10})\)	\(q-21.8139 q^{2} +347.844 q^{4} -424.937 q^{5} +1192.89 q^{7} -4795.65 q^{8} +9269.52 q^{10} +3093.76 q^{11} +1043.61 q^{13} -26021.4 q^{14} +60087.5 q^{16} +16626.4 q^{17} +25587.3 q^{19} -147812. q^{20} -67486.8 q^{22} +98700.2 q^{23} +102447. q^{25} -22765.2 q^{26} +414938. q^{28} +241457. q^{29} -117145. q^{31} -696897. q^{32} -362686. q^{34} -506902. q^{35} +73455.9 q^{37} -558158. q^{38} +2.03785e6 q^{40} +333204. q^{41} +608971. q^{43} +1.07615e6 q^{44} -2.15303e6 q^{46} +923874. q^{47} +599434. q^{49} -2.23475e6 q^{50} +363014. q^{52} -617858. q^{53} -1.31465e6 q^{55} -5.72066e6 q^{56} -5.26710e6 q^{58} +205379. q^{59} +3.02318e6 q^{61} +2.55538e6 q^{62} +7.51081e6 q^{64} -443469. q^{65} +4.24635e6 q^{67} +5.78339e6 q^{68} +1.10575e7 q^{70} -809917. q^{71} +1.31862e6 q^{73} -1.60236e6 q^{74} +8.90040e6 q^{76} +3.69050e6 q^{77} +5.69982e6 q^{79} -2.55334e7 q^{80} -7.26847e6 q^{82} +3.55052e6 q^{83} -7.06517e6 q^{85} -1.32840e7 q^{86} -1.48366e7 q^{88} +1.13503e7 q^{89} +1.24491e6 q^{91} +3.43323e7 q^{92} -2.01532e7 q^{94} -1.08730e7 q^{95} -3.73640e6 q^{97} -1.30760e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - 2 q^{2} + 1166 q^{4} + 318 q^{5} + 3145 q^{7} - 2355 q^{8}+O(q^{10}) \) 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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