LMFDB - Embedded newform 531.8.a.e.1.15

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:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + \cdots + 51\!\cdots\!48 $ x^18 - 6x^17 - 1798x^16 + 11087x^15 + 1326765x^14 - 8403720x^13 - 518334228x^12 + 3375594921x^11 + 115310342333x^10 - 774932111214x^9 - 14600047830166x^8 + 102185027148481x^7 + 988557475638619x^6 - 7379206238519716x^5 - 30152342836849520x^4 + 260578770749067175x^3 + 182609347488069978x^2 - 3481290425710753600*x + 5164646074739714048
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$-16.0158$ 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)

$f(q)$	$=$	$q+15.0158 q^{2} +97.4757 q^{4} -6.55078 q^{5} +1466.16 q^{7} -458.348 q^{8} +O(q^{10})$	\(q+15.0158 q^{2} +97.4757 q^{4} -6.55078 q^{5} +1466.16 q^{7} -458.348 q^{8} -98.3656 q^{10} -4599.84 q^{11} +15226.3 q^{13} +22015.6 q^{14} -19359.4 q^{16} -38992.1 q^{17} -499.547 q^{19} -638.542 q^{20} -69070.4 q^{22} -58259.9 q^{23} -78082.1 q^{25} +228636. q^{26} +142915. q^{28} -211255. q^{29} +61846.8 q^{31} -232029. q^{32} -585500. q^{34} -9604.48 q^{35} -62940.4 q^{37} -7501.12 q^{38} +3002.54 q^{40} +619876. q^{41} -356192. q^{43} -448372. q^{44} -874822. q^{46} +953515. q^{47} +1.32607e6 q^{49} -1.17247e6 q^{50} +1.48419e6 q^{52} +1.97760e6 q^{53} +30132.5 q^{55} -672010. q^{56} -3.17218e6 q^{58} -205379. q^{59} -2.98506e6 q^{61} +928683. q^{62} -1.00611e6 q^{64} -99744.2 q^{65} -1.04862e6 q^{67} -3.80078e6 q^{68} -144219. q^{70} -4.10180e6 q^{71} -260342. q^{73} -945104. q^{74} -48693.7 q^{76} -6.74408e6 q^{77} -1.33466e6 q^{79} +126819. q^{80} +9.30797e6 q^{82} -8.30823e6 q^{83} +255429. q^{85} -5.34852e6 q^{86} +2.10832e6 q^{88} -3.49331e6 q^{89} +2.23242e7 q^{91} -5.67893e6 q^{92} +1.43178e7 q^{94} +3272.42 q^{95} -1.00211e7 q^{97} +1.99121e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8}+O(q^{10}) $ 18 * q - 24 * q^2 + 1358 * q^4 - 678 * q^5 + 3081 * q^7 - 4107 * q^8	\( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8} + 3609 q^{10} - 15070 q^{11} + 13662 q^{13} - 20861 q^{14} + 60482 q^{16} - 71919 q^{17} + 56231 q^{19} - 143053 q^{20} + 274198 q^{22} - 150029 q^{23} + 399672 q^{25} - 182846 q^{26} + 434150 q^{28} - 591285 q^{29} + 426733 q^{31} - 1205630 q^{32} + 403548 q^{34} - 912879 q^{35} + 7703 q^{37} + 417859 q^{38} + 618020 q^{40} - 770959 q^{41} + 793050 q^{43} - 2591274 q^{44} - 4068019 q^{46} - 1410373 q^{47} + 1637427 q^{49} - 1021549 q^{50} - 3749190 q^{52} - 1037934 q^{53} + 331974 q^{55} + 391748 q^{56} + 653724 q^{58} - 3696822 q^{59} - 1374623 q^{61} - 5251718 q^{62} + 5077197 q^{64} - 3257170 q^{65} - 2436904 q^{67} - 14119909 q^{68} + 5185580 q^{70} - 14289172 q^{71} + 5482515 q^{73} - 14934154 q^{74} + 3822912 q^{76} - 23157109 q^{77} + 19786414 q^{79} - 31978143 q^{80} + 9749509 q^{82} - 30227337 q^{83} + 9946981 q^{85} - 44295864 q^{86} + 39970897 q^{88} - 31061677 q^{89} + 26377785 q^{91} - 4719698 q^{92} + 44488296 q^{94} - 15534599 q^{95} + 12084118 q^{97} - 42274744 q^{98}+O(q^{100}) \) 18 * q - 24 * q^2 + 1358 * q^4 - 678 * q^5 + 3081 * q^7 - 4107 * q^8 + 3609 * q^10 - 15070 * q^11 + 13662 * q^13 - 20861 * q^14 + 60482 * q^16 - 71919 * q^17 + 56231 * q^19 - 143053 * q^20 + 274198 * q^22 - 150029 * q^23 + 399672 * q^25 - 182846 * q^26 + 434150 * q^28 - 591285 * q^29 + 426733 * q^31 - 1205630 * q^32 + 403548 * q^34 - 912879 * q^35 + 7703 * q^37 + 417859 * q^38 + 618020 * q^40 - 770959 * q^41 + 793050 * q^43 - 2591274 * q^44 - 4068019 * q^46 - 1410373 * q^47 + 1637427 * q^49 - 1021549 * q^50 - 3749190 * q^52 - 1037934 * q^53 + 331974 * q^55 + 391748 * q^56 + 653724 * q^58 - 3696822 * q^59 - 1374623 * q^61 - 5251718 * q^62 + 5077197 * q^64 - 3257170 * q^65 - 2436904 * q^67 - 14119909 * q^68 + 5185580 * q^70 - 14289172 * q^71 + 5482515 * q^73 - 14934154 * q^74 + 3822912 * q^76 - 23157109 * q^77 + 19786414 * q^79 - 31978143 * q^80 + 9749509 * q^82 - 30227337 * q^83 + 9946981 * q^85 - 44295864 * q^86 + 39970897 * q^88 - 31061677 * q^89 + 26377785 * q^91 - 4719698 * q^92 + 44488296 * q^94 - 15534599 * q^95 + 12084118 * q^97 - 42274744 * 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 $

$2$	15.0158	1.32723	0.663613	−	0.748076i	$-0.269022\pi$
$2$	15.0158	1.32723	0.663613	+	0.748076i	$0.269022\pi$
$3$	0	0
$3$	0	0
$4$	97.4757	0.761529
$5$	−6.55078	−0.0234368	−0.0117184	−	0.999931i	$-0.503730\pi$
$5$	−6.55078	−0.0234368	−0.0117184	+	0.999931i	$0.503730\pi$
$6$	0	0
$7$	1466.16	1.61561	0.807807	−	0.589448i	$-0.200655\pi$
$7$	1466.16	1.61561	0.807807	+	0.589448i	$0.200655\pi$
$8$	−458.348	−0.316505
$9$	0	0
$10$	−98.3656	−0.0311059
$11$	−4599.84	−1.04200	−0.521000	−	0.853556i	$-0.674441\pi$
$11$	−4599.84	−1.04200	−0.521000	+	0.853556i	$0.674441\pi$
$12$	0	0
$13$	15226.3	1.92217	0.961087	−	0.276246i	$-0.0890903\pi$
$13$	15226.3	1.92217	0.961087	+	0.276246i	$0.0890903\pi$
$14$	22015.6	2.14428
$15$	0	0
$16$	−19359.4	−1.18160
$17$	−38992.1	−1.92489	−0.962444	−	0.271481i	$-0.912487\pi$
$17$	−38992.1	−1.92489	−0.962444	+	0.271481i	$0.912487\pi$
$18$	0	0
$19$	−499.547	−0.0167085	−0.00835427	−	0.999965i	$-0.502659\pi$
$19$	−499.547	−0.0167085	−0.00835427	+	0.999965i	$0.502659\pi$
$20$	−638.542	−0.0178478
$21$	0	0
$22$	−69070.4	−1.38297
$23$	−58259.9	−0.998441	−0.499220	−	0.866475i	$-0.666380\pi$
$23$	−58259.9	−0.998441	−0.499220	+	0.866475i	$0.666380\pi$
$24$	0	0
$25$	−78082.1	−0.999451
$26$	228636.	2.55116
$27$	0	0
$28$	142915.	1.23034
$29$	−211255.	−1.60848	−0.804238	−	0.594307i	$-0.797426\pi$
$29$	−211255.	−1.60848	−0.804238	+	0.594307i	$0.797426\pi$
$30$	0	0
$31$	61846.8	0.372865	0.186432	−	0.982468i	$-0.440307\pi$
$31$	61846.8	0.372865	0.186432	+	0.982468i	$0.440307\pi$
$32$	−232029.	−1.25175
$33$	0	0
$34$	−585500.	−2.55476
$35$	−9604.48	−0.0378648
$36$	0	0
$37$	−62940.4	−0.204279	−0.102139	−	0.994770i	$-0.532569\pi$
$37$	−62940.4	−0.204279	−0.102139	+	0.994770i	$0.532569\pi$
$38$	−7501.12	−0.0221760
$39$	0	0
$40$	3002.54	0.00741785
$41$	619876.	1.40463	0.702314	−	0.711867i	$-0.252150\pi$
$41$	619876.	1.40463	0.702314	+	0.711867i	$0.252150\pi$
$42$	0	0
$43$	−356192.	−0.683194	−0.341597	−	0.939846i	$-0.610968\pi$
$43$	−356192.	−0.683194	−0.341597	+	0.939846i	$0.610968\pi$
$44$	−448372.	−0.793514
$45$	0	0
$46$	−874822.	−1.32516
$47$	953515.	1.33963	0.669815	−	0.742528i	$-0.266373\pi$
$47$	953515.	1.33963	0.669815	+	0.742528i	$0.266373\pi$
$48$	0	0
$49$	1.32607e6	1.61021
$50$	−1.17247e6	−1.32650
$51$	0	0
$52$	1.48419e6	1.46379
$53$	1.97760e6	1.82462	0.912310	−	0.409501i	$-0.134297\pi$
$53$	1.97760e6	1.82462	0.912310	+	0.409501i	$0.134297\pi$
$54$	0	0
$55$	30132.5	0.0244211
$56$	−672010.	−0.511349
$57$	0	0
$58$	−3.17218e6	−2.13481
$59$	−205379.	−0.130189
$59$	−205379.	−0.130189
$60$	0	0
$61$	−2.98506e6	−1.68383	−0.841917	−	0.539607i	$-0.818573\pi$
$61$	−2.98506e6	−1.68383	−0.841917	+	0.539607i	$0.818573\pi$
$62$	928683.	0.494876
$63$	0	0
$64$	−1.00611e6	−0.479751
$65$	−99744.2	−0.0450496
$66$	0	0
$67$	−1.04862e6	−0.425948	−0.212974	−	0.977058i	$-0.568315\pi$
$67$	−1.04862e6	−0.425948	−0.212974	+	0.977058i	$0.568315\pi$
$68$	−3.80078e6	−1.46586
$69$	0	0
$70$	−144219.	−0.0502551
$71$	−4.10180e6	−1.36010	−0.680049	−	0.733166i	$-0.738042\pi$
$71$	−4.10180e6	−1.36010	−0.680049	+	0.733166i	$0.738042\pi$
$72$	0	0
$73$	−260342.	−0.0783273	−0.0391637	−	0.999233i	$-0.512469\pi$
$73$	−260342.	−0.0783273	−0.0391637	+	0.999233i	$0.512469\pi$
$74$	−945104.	−0.271124
$75$	0	0
$76$	−48693.7	−0.0127240
$77$	−6.74408e6	−1.68347
$78$	0	0
$79$	−1.33466e6	−0.304562	−0.152281	−	0.988337i	$-0.548662\pi$
$79$	−1.33466e6	−0.304562	−0.152281	+	0.988337i	$0.548662\pi$
$80$	126819.	0.0276930
$81$	0	0
$82$	9.30797e6	1.86426
$83$	−8.30823e6	−1.59491	−0.797454	−	0.603380i	$-0.793820\pi$
$83$	−8.30823e6	−1.59491	−0.797454	+	0.603380i	$0.793820\pi$
$84$	0	0
$85$	255429.	0.0451132
$86$	−5.34852e6	−0.906753
$87$	0	0
$88$	2.10832e6	0.329798
$89$	−3.49331e6	−0.525257	−0.262628	−	0.964897i	$-0.584589\pi$
$89$	−3.49331e6	−0.525257	−0.262628	+	0.964897i	$0.584589\pi$
$90$	0	0
$91$	2.23242e7	3.10549
$92$	−5.67893e6	−0.760342
$93$	0	0
$94$	1.43178e7	1.77799
$95$	3272.42	0.000391594 0
$96$	0	0
$97$	−1.00211e7	−1.11484	−0.557420	−	0.830230i	$-0.688209\pi$
$97$	−1.00211e7	−1.11484	−0.557420	+	0.830230i	$0.688209\pi$
$98$	1.99121e7	2.13711
$99$	0	0
$100$	−7.61111e6	−0.761111
$101$	−9.44535e6	−0.912206	−0.456103	−	0.889927i	$-0.650755\pi$
$101$	−9.44535e6	−0.912206	−0.456103	+	0.889927i	$0.650755\pi$
$102$	0	0
$103$	−4.07761e6	−0.367684	−0.183842	−	0.982956i	$-0.558854\pi$
$103$	−4.07761e6	−0.367684	−0.183842	+	0.982956i	$0.558854\pi$
$104$	−6.97894e6	−0.608377
$105$	0	0
$106$	2.96953e7	2.42168
$107$	2.17048e7	1.71282	0.856410	−	0.516296i	$-0.172690\pi$
$107$	2.17048e7	1.71282	0.856410	+	0.516296i	$0.172690\pi$
$108$	0	0
$109$	−4.43849e6	−0.328278	−0.164139	−	0.986437i	$-0.552485\pi$
$109$	−4.43849e6	−0.328278	−0.164139	+	0.986437i	$0.552485\pi$
$110$	452465.	0.0324124
$111$	0	0
$112$	−2.83839e7	−1.90901
$113$	−7.13404e6	−0.465116	−0.232558	−	0.972583i	$-0.574710\pi$
$113$	−7.13404e6	−0.465116	−0.232558	+	0.972583i	$0.574710\pi$
$114$	0	0
$115$	381648.	0.0234002
$116$	−2.05923e7	−1.22490
$117$	0	0
$118$	−3.08394e6	−0.172790
$119$	−5.71686e7	−3.10987
$120$	0	0
$121$	1.67132e6	0.0857652
$122$	−4.48233e7	−2.23483
$123$	0	0
$124$	6.02857e6	0.283948
$125$	1.02328e6	0.0468607
$126$	0	0
$127$	−1.23488e7	−0.534948	−0.267474	−	0.963565i	$-0.586189\pi$
$127$	−1.23488e7	−0.534948	−0.267474	+	0.963565i	$0.586189\pi$
$128$	1.45921e7	0.615010
$129$	0	0
$130$	−1.49774e6	−0.0597910
$131$	1.76969e7	0.687779	0.343889	−	0.939010i	$-0.388256\pi$
$131$	1.76969e7	0.687779	0.343889	+	0.939010i	$0.388256\pi$
$132$	0	0
$133$	−732414.	−0.0269945
$134$	−1.57459e7	−0.565329
$135$	0	0
$136$	1.78719e7	0.609236
$137$	2.34535e6	0.0779265	0.0389632	−	0.999241i	$-0.487594\pi$
$137$	2.34535e6	0.0779265	0.0389632	+	0.999241i	$0.487594\pi$
$138$	0	0
$139$	−1.84954e7	−0.584134	−0.292067	−	0.956398i	$-0.594343\pi$
$139$	−1.84954e7	−0.584134	−0.292067	+	0.956398i	$0.594343\pi$
$140$	−936203.	−0.0288351
$141$	0	0
$142$	−6.15920e7	−1.80516
$143$	−7.00385e7	−2.00291
$144$	0	0
$145$	1.38389e6	0.0376975
$146$	−3.90925e6	−0.103958
$147$	0	0
$148$	−6.13516e6	−0.155564
$149$	1.87066e7	0.463278	0.231639	−	0.972802i	$-0.425591\pi$
$149$	1.87066e7	0.463278	0.231639	+	0.972802i	$0.425591\pi$
$150$	0	0
$151$	−3.23257e7	−0.764062	−0.382031	−	0.924150i	$-0.624775\pi$
$151$	−3.23257e7	−0.764062	−0.382031	+	0.924150i	$0.624775\pi$
$152$	228966.	0.00528833
$153$	0	0
$154$	−1.01268e8	−2.23435
$155$	−405145.	−0.00873876
$156$	0	0
$157$	−5.46991e7	−1.12806	−0.564029	−	0.825755i	$-0.690749\pi$
$157$	−5.46991e7	−1.12806	−0.564029	+	0.825755i	$0.690749\pi$
$158$	−2.00411e7	−0.404223
$159$	0	0
$160$	1.51997e6	0.0293370
$161$	−8.54182e7	−1.61309
$162$	0	0
$163$	−4.22142e7	−0.763487	−0.381744	−	0.924268i	$-0.624676\pi$
$163$	−4.22142e7	−0.763487	−0.381744	+	0.924268i	$0.624676\pi$
$164$	6.04229e7	1.06967
$165$	0	0
$166$	−1.24755e8	−2.11680
$167$	8.74223e6	0.145250	0.0726248	−	0.997359i	$-0.476862\pi$
$167$	8.74223e6	0.145250	0.0726248	+	0.997359i	$0.476862\pi$
$168$	0	0
$169$	1.69092e8	2.69475
$170$	3.83548e6	0.0598754
$171$	0	0
$172$	−3.47201e7	−0.520272
$173$	−4.41610e7	−0.648451	−0.324225	−	0.945980i	$-0.605104\pi$
$173$	−4.41610e7	−0.648451	−0.324225	+	0.945980i	$0.605104\pi$
$174$	0	0
$175$	−1.14481e8	−1.61473
$176$	8.90500e7	1.23123
$177$	0	0
$178$	−5.24550e7	−0.697135
$179$	3.77776e7	0.492322	0.246161	−	0.969229i	$-0.420831\pi$
$179$	3.77776e7	0.492322	0.246161	+	0.969229i	$0.420831\pi$
$180$	0	0
$181$	1.26649e8	1.58755	0.793774	−	0.608213i	$-0.208113\pi$
$181$	1.26649e8	1.58755	0.793774	+	0.608213i	$0.208113\pi$
$182$	3.35216e8	4.12169
$183$	0	0
$184$	2.67033e7	0.316011
$185$	412309.	0.00478764
$186$	0	0
$187$	1.79357e8	2.00573
$188$	9.29445e7	1.02017
$189$	0	0
$190$	49138.2	0.000519734 0
$191$	2.15055e7	0.223323	0.111661	−	0.993746i	$-0.464383\pi$
$191$	2.15055e7	0.223323	0.111661	+	0.993746i	$0.464383\pi$
$192$	0	0
$193$	2.18519e7	0.218796	0.109398	−	0.993998i	$-0.465108\pi$
$193$	2.18519e7	0.218796	0.109398	+	0.993998i	$0.465108\pi$
$194$	−1.50475e8	−1.47965
$195$	0	0
$196$	1.29260e8	1.22622
$197$	8.73312e7	0.813837	0.406919	−	0.913464i	$-0.366603\pi$
$197$	8.73312e7	0.813837	0.406919	+	0.913464i	$0.366603\pi$
$198$	0	0
$199$	−2.03499e8	−1.83053	−0.915266	−	0.402850i	$-0.868020\pi$
$199$	−2.03499e8	−1.83053	−0.915266	+	0.402850i	$0.868020\pi$
$200$	3.57888e7	0.316331
$201$	0	0
$202$	−1.41830e8	−1.21070
$203$	−3.09734e8	−2.59868
$204$	0	0
$205$	−4.06067e6	−0.0329200
$206$	−6.12288e7	−0.488000
$207$	0	0
$208$	−2.94772e8	−2.27125
$209$	2.29783e6	0.0174103
$210$	0	0
$211$	−7.64544e7	−0.560291	−0.280146	−	0.959958i	$-0.590383\pi$
$211$	−7.64544e7	−0.560291	−0.280146	+	0.959958i	$0.590383\pi$
$212$	1.92768e8	1.38950
$213$	0	0
$214$	3.25915e8	2.27330
$215$	2.33333e6	0.0160119
$216$	0	0
$217$	9.06772e7	0.602406
$218$	−6.66477e7	−0.435700
$219$	0	0
$220$	2.93719e6	0.0185974
$221$	−5.93706e8	−3.69997
$222$	0	0
$223$	−5.24257e7	−0.316575	−0.158288	−	0.987393i	$-0.550597\pi$
$223$	−5.24257e7	−0.316575	−0.158288	+	0.987393i	$0.550597\pi$
$224$	−3.40191e8	−2.02234
$225$	0	0
$226$	−1.07124e8	−0.617314
$227$	−3.76491e6	−0.0213631	−0.0106816	−	0.999943i	$-0.503400\pi$
$227$	−3.76491e6	−0.0213631	−0.0106816	+	0.999943i	$0.503400\pi$
$228$	0	0
$229$	−1.48786e8	−0.818723	−0.409362	−	0.912372i	$-0.634249\pi$
$229$	−1.48786e8	−0.818723	−0.409362	+	0.912372i	$0.634249\pi$
$230$	5.73077e6	0.0310574
$231$	0	0
$232$	9.68284e7	0.509090
$233$	−1.35403e8	−0.701265	−0.350632	−	0.936513i	$-0.614033\pi$
$233$	−1.35403e8	−0.701265	−0.350632	+	0.936513i	$0.614033\pi$
$234$	0	0
$235$	−6.24627e6	−0.0313966
$236$	−2.00195e7	−0.0991427
$237$	0	0
$238$	−8.58434e8	−4.12751
$239$	3.35966e8	1.59185	0.795926	−	0.605394i	$-0.206984\pi$
$239$	3.35966e8	1.59185	0.795926	+	0.605394i	$0.206984\pi$
$240$	0	0
$241$	−2.48239e8	−1.14238	−0.571189	−	0.820818i	$-0.693518\pi$
$241$	−2.48239e8	−1.14238	−0.571189	+	0.820818i	$0.693518\pi$
$242$	2.50963e7	0.113830
$243$	0	0
$244$	−2.90971e8	−1.28229
$245$	−8.68682e6	−0.0377381
$246$	0	0
$247$	−7.60625e6	−0.0321167
$248$	−2.83474e7	−0.118014
$249$	0	0
$250$	1.53654e7	0.0621948
$251$	−9.71369e7	−0.387727	−0.193864	−	0.981028i	$-0.562102\pi$
$251$	−9.71369e7	−0.387727	−0.193864	+	0.981028i	$0.562102\pi$
$252$	0	0
$253$	2.67986e8	1.04038
$254$	−1.85428e8	−0.709997
$255$	0	0
$256$	3.47895e8	1.29601
$257$	−5.01702e6	−0.0184366	−0.00921829	−	0.999958i	$-0.502934\pi$
$257$	−5.01702e6	−0.0184366	−0.00921829	+	0.999958i	$0.502934\pi$
$258$	0	0
$259$	−9.22806e7	−0.330036
$260$	−9.72264e6	−0.0343066
$261$	0	0
$262$	2.65735e8	0.912838
$263$	3.81564e8	1.29337	0.646685	−	0.762757i	$-0.276155\pi$
$263$	3.81564e8	1.29337	0.646685	+	0.762757i	$0.276155\pi$
$264$	0	0
$265$	−1.29548e7	−0.0427632
$266$	−1.09978e7	−0.0358278
$267$	0	0
$268$	−1.02215e8	−0.324372
$269$	2.82835e8	0.885931	0.442965	−	0.896539i	$-0.353926\pi$
$269$	2.82835e8	0.885931	0.442965	+	0.896539i	$0.353926\pi$
$270$	0	0
$271$	2.76884e7	0.0845095	0.0422547	−	0.999107i	$-0.486546\pi$
$271$	2.76884e7	0.0845095	0.0422547	+	0.999107i	$0.486546\pi$
$272$	7.54863e8	2.27445
$273$	0	0
$274$	3.52174e7	0.103426
$275$	3.59165e8	1.04143
$276$	0	0
$277$	1.63786e7	0.0463018	0.0231509	−	0.999732i	$-0.492630\pi$
$277$	1.63786e7	0.0463018	0.0231509	+	0.999732i	$0.492630\pi$
$278$	−2.77725e8	−0.775278
$279$	0	0
$280$	4.40219e6	0.0119844
$281$	1.44402e8	0.388240	0.194120	−	0.980978i	$-0.437815\pi$
$281$	1.44402e8	0.388240	0.194120	+	0.980978i	$0.437815\pi$
$282$	0	0
$283$	1.29642e8	0.340011	0.170005	−	0.985443i	$-0.445622\pi$
$283$	1.29642e8	0.340011	0.170005	+	0.985443i	$0.445622\pi$
$284$	−3.99826e8	−1.03575
$285$	0	0
$286$	−1.05169e9	−2.65831
$287$	9.08836e8	2.26934
$288$	0	0
$289$	1.11005e9	2.70519
$290$	2.07802e7	0.0500331
$291$	0	0
$292$	−2.53770e7	−0.0596486
$293$	5.48210e8	1.27324	0.636620	−	0.771178i	$-0.280332\pi$
$293$	5.48210e8	1.27324	0.636620	+	0.771178i	$0.280332\pi$
$294$	0	0
$295$	1.34539e6	0.00305121
$296$	2.88486e7	0.0646553
$297$	0	0
$298$	2.80895e8	0.614875
$299$	−8.87083e8	−1.91918
$300$	0	0
$301$	−5.22233e8	−1.10378
$302$	−4.85398e8	−1.01408
$303$	0	0
$304$	9.67091e6	0.0197428
$305$	1.95545e7	0.0394637
$306$	0	0
$307$	7.57326e8	1.49382	0.746910	−	0.664925i	$-0.231536\pi$
$307$	7.57326e8	1.49382	0.746910	+	0.664925i	$0.231536\pi$
$308$	−6.57384e8	−1.28201
$309$	0	0
$310$	−6.08360e6	−0.0115983
$311$	−3.48341e8	−0.656665	−0.328332	−	0.944562i	$-0.606487\pi$
$311$	−3.48341e8	−0.656665	−0.328332	+	0.944562i	$0.606487\pi$
$312$	0	0
$313$	−7.49639e8	−1.38180	−0.690902	−	0.722948i	$-0.742787\pi$
$313$	−7.49639e8	−1.38180	−0.690902	+	0.722948i	$0.742787\pi$
$314$	−8.21354e8	−1.49719
$315$	0	0
$316$	−1.30097e8	−0.231933
$317$	−7.39103e7	−0.130316	−0.0651580	−	0.997875i	$-0.520755\pi$
$317$	−7.39103e7	−0.130316	−0.0651580	+	0.997875i	$0.520755\pi$
$318$	0	0
$319$	9.71740e8	1.67603
$320$	6.59082e6	0.0112438
$321$	0	0
$322$	−1.28263e9	−2.14094
$323$	1.94784e7	0.0321621
$324$	0	0
$325$	−1.18890e9	−1.92112
$326$	−6.33882e8	−1.01332
$327$	0	0
$328$	−2.84119e8	−0.444572
$329$	1.39800e9	2.16432
$330$	0	0
$331$	−6.07136e8	−0.920212	−0.460106	−	0.887864i	$-0.652189\pi$
$331$	−6.07136e8	−0.920212	−0.460106	+	0.887864i	$0.652189\pi$
$332$	−8.09851e8	−1.21457
$333$	0	0
$334$	1.31272e8	0.192779
$335$	6.86928e6	0.00998285
$336$	0	0
$337$	1.21856e9	1.73437	0.867186	−	0.497985i	$-0.165926\pi$
$337$	1.21856e9	1.73437	0.867186	+	0.497985i	$0.165926\pi$
$338$	2.53906e9	3.57655
$339$	0	0
$340$	2.48981e7	0.0343550
$341$	−2.84485e8	−0.388526
$342$	0	0
$343$	7.36790e8	0.985858
$344$	1.63260e8	0.216234
$345$	0	0
$346$	−6.63114e8	−0.860641
$347$	−8.27617e8	−1.06335	−0.531675	−	0.846948i	$-0.678437\pi$
$347$	−8.27617e8	−1.06335	−0.531675	+	0.846948i	$0.678437\pi$
$348$	0	0
$349$	−9.88709e8	−1.24503	−0.622515	−	0.782608i	$-0.713889\pi$
$349$	−9.88709e8	−1.24503	−0.622515	+	0.782608i	$0.713889\pi$
$350$	−1.71902e9	−2.14311
$351$	0	0
$352$	1.06730e9	1.30432
$353$	−2.48535e7	−0.0300730	−0.0150365	−	0.999887i	$-0.504786\pi$
$353$	−2.48535e7	−0.0300730	−0.0150365	+	0.999887i	$0.504786\pi$
$354$	0	0
$355$	2.68700e7	0.0318763
$356$	−3.40513e8	−0.399998
$357$	0	0
$358$	5.67263e8	0.653423
$359$	−6.66922e8	−0.760755	−0.380377	−	0.924831i	$-0.624206\pi$
$359$	−6.66922e8	−0.760755	−0.380377	+	0.924831i	$0.624206\pi$
$360$	0	0
$361$	−8.93622e8	−0.999721
$362$	1.90174e9	2.10703
$363$	0	0
$364$	2.17606e9	2.36492
$365$	1.70544e6	0.00183574
$366$	0	0
$367$	4.68476e8	0.494716	0.247358	−	0.968924i	$-0.420438\pi$
$367$	4.68476e8	0.494716	0.247358	+	0.968924i	$0.420438\pi$
$368$	1.12788e9	1.17976
$369$	0	0
$370$	6.19117e6	0.00635429
$371$	2.89947e9	2.94788
$372$	0	0
$373$	5.89698e8	0.588368	0.294184	−	0.955749i	$-0.404952\pi$
$373$	5.89698e8	0.588368	0.294184	+	0.955749i	$0.404952\pi$
$374$	2.69320e9	2.66206
$375$	0	0
$376$	−4.37041e8	−0.423999
$377$	−3.21664e9	−3.09177
$378$	0	0
$379$	7.55066e8	0.712439	0.356220	−	0.934402i	$-0.384066\pi$
$379$	7.55066e8	0.712439	0.356220	+	0.934402i	$0.384066\pi$
$380$	318982.	0.000298211 0
$381$	0	0
$382$	3.22924e8	0.296400
$383$	5.16218e8	0.469502	0.234751	−	0.972056i	$-0.424573\pi$
$383$	5.16218e8	0.469502	0.234751	+	0.972056i	$0.424573\pi$
$384$	0	0
$385$	4.41790e7	0.0394551
$386$	3.28125e8	0.290392
$387$	0	0
$388$	−9.76811e8	−0.848984
$389$	−2.02594e9	−1.74503	−0.872514	−	0.488589i	$-0.837512\pi$
$389$	−2.02594e9	−1.74503	−0.872514	+	0.488589i	$0.837512\pi$
$390$	0	0
$391$	2.27168e9	1.92189
$392$	−6.07803e8	−0.509638
$393$	0	0
$394$	1.31135e9	1.08015
$395$	8.74307e6	0.00713796
$396$	0	0
$397$	1.11905e9	0.897601	0.448801	−	0.893632i	$-0.351851\pi$
$397$	1.11905e9	0.897601	0.448801	+	0.893632i	$0.351851\pi$
$398$	−3.05572e9	−2.42953
$399$	0	0
$400$	1.51162e9	1.18095
$401$	2.75487e8	0.213352	0.106676	−	0.994294i	$-0.465979\pi$
$401$	2.75487e8	0.213352	0.106676	+	0.994294i	$0.465979\pi$
$402$	0	0
$403$	9.41699e8	0.716711
$404$	−9.20692e8	−0.694672
$405$	0	0
$406$	−4.65091e9	−3.44903
$407$	2.89516e8	0.212859
$408$	0	0
$409$	−7.49535e8	−0.541702	−0.270851	−	0.962621i	$-0.587305\pi$
$409$	−7.49535e8	−0.541702	−0.270851	+	0.962621i	$0.587305\pi$
$410$	−6.09745e7	−0.0436923
$411$	0	0
$412$	−3.97468e8	−0.280002
$413$	−3.01118e8	−0.210335
$414$	0	0
$415$	5.44254e7	0.0373795
$416$	−3.53294e9	−2.40608
$417$	0	0
$418$	3.45039e7	0.0231074
$419$	−2.09732e9	−1.39289	−0.696445	−	0.717611i	$-0.745236\pi$
$419$	−2.09732e9	−1.39289	−0.696445	+	0.717611i	$0.745236\pi$
$420$	0	0
$421$	8.22045e8	0.536919	0.268459	−	0.963291i	$-0.413486\pi$
$421$	8.22045e8	0.536919	0.268459	+	0.963291i	$0.413486\pi$
$422$	−1.14803e9	−0.743633
$423$	0	0
$424$	−9.06427e8	−0.577501
$425$	3.04458e9	1.92383
$426$	0	0
$427$	−4.37657e9	−2.72042
$428$	2.11569e9	1.30436
$429$	0	0
$430$	3.50370e7	0.0212514
$431$	9.73123e8	0.585460	0.292730	−	0.956195i	$-0.405436\pi$
$431$	9.73123e8	0.585460	0.292730	+	0.956195i	$0.405436\pi$
$432$	0	0
$433$	2.68877e9	1.59165	0.795823	−	0.605529i	$-0.207039\pi$
$433$	2.68877e9	1.59165	0.795823	+	0.605529i	$0.207039\pi$
$434$	1.36160e9	0.799528
$435$	0	0
$436$	−4.32645e8	−0.249994
$437$	2.91035e7	0.0166825
$438$	0	0
$439$	2.20359e9	1.24309	0.621547	−	0.783377i	$-0.286504\pi$
$439$	2.20359e9	1.24309	0.621547	+	0.783377i	$0.286504\pi$
$440$	−1.38112e7	−0.00772941
$441$	0	0
$442$	−8.91499e9	−4.91070
$443$	1.04965e9	0.573630	0.286815	−	0.957986i	$-0.407403\pi$
$443$	1.04965e9	0.573630	0.286815	+	0.957986i	$0.407403\pi$
$444$	0	0
$445$	2.28839e7	0.0123103
$446$	−7.87216e8	−0.420167
$447$	0	0
$448$	−1.47512e9	−0.775093
$449$	2.02286e9	1.05464	0.527320	−	0.849667i	$-0.323197\pi$
$449$	2.02286e9	1.05464	0.527320	+	0.849667i	$0.323197\pi$
$450$	0	0
$451$	−2.85133e9	−1.46362
$452$	−6.95396e8	−0.354199
$453$	0	0
$454$	−5.65334e7	−0.0283537
$455$	−1.46241e8	−0.0727827
$456$	0	0
$457$	−3.01761e9	−1.47896	−0.739480	−	0.673178i	$-0.764929\pi$
$457$	−3.01761e9	−1.47896	−0.739480	+	0.673178i	$0.764929\pi$
$458$	−2.23414e9	−1.08663
$459$	0	0
$460$	3.72014e7	0.0178200
$461$	−3.13368e9	−1.48971	−0.744854	−	0.667228i	$-0.767481\pi$
$461$	−3.13368e9	−1.48971	−0.744854	+	0.667228i	$0.767481\pi$
$462$	0	0
$463$	−2.30769e9	−1.08055	−0.540275	−	0.841489i	$-0.681680\pi$
$463$	−2.30769e9	−1.08055	−0.540275	+	0.841489i	$0.681680\pi$
$464$	4.08977e9	1.90058
$465$	0	0
$466$	−2.03319e9	−0.930737
$467$	3.23147e9	1.46822	0.734110	−	0.679030i	$-0.237600\pi$
$467$	3.23147e9	1.46822	0.734110	+	0.679030i	$0.237600\pi$
$468$	0	0
$469$	−1.53744e9	−0.688167
$470$	−9.37930e7	−0.0416704
$471$	0	0
$472$	9.41350e7	0.0412054
$473$	1.63842e9	0.711889
$474$	0	0
$475$	3.90056e7	0.0166994
$476$	−5.57255e9	−2.36826
$477$	0	0
$478$	5.04482e9	2.11275
$479$	−3.12158e9	−1.29778	−0.648888	−	0.760884i	$-0.724766\pi$
$479$	−3.12158e9	−1.29778	−0.648888	+	0.760884i	$0.724766\pi$
$480$	0	0
$481$	−9.58350e8	−0.392660
$482$	−3.72752e9	−1.51619
$483$	0	0
$484$	1.62913e8	0.0653127
$485$	6.56458e7	0.0261283
$486$	0	0
$487$	−1.98211e9	−0.777636	−0.388818	−	0.921315i	$-0.627116\pi$
$487$	−1.98211e9	−0.777636	−0.388818	+	0.921315i	$0.627116\pi$
$488$	1.36820e9	0.532941
$489$	0	0
$490$	−1.30440e8	−0.0500870
$491$	−2.49858e9	−0.952594	−0.476297	−	0.879285i	$-0.658021\pi$
$491$	−2.49858e9	−0.952594	−0.476297	+	0.879285i	$0.658021\pi$
$492$	0	0
$493$	8.23729e9	3.09614
$494$	−1.14214e8	−0.0426261
$495$	0	0
$496$	−1.19732e9	−0.440578
$497$	−6.01389e9	−2.19739
$498$	0	0
$499$	4.80008e9	1.72940	0.864702	−	0.502285i	$-0.167507\pi$
$499$	4.80008e9	1.72940	0.864702	+	0.502285i	$0.167507\pi$
$500$	9.97448e7	0.0356858
$501$	0	0
$502$	−1.45859e9	−0.514602
$503$	3.37167e9	1.18129	0.590645	−	0.806931i	$-0.298873\pi$
$503$	3.37167e9	1.18129	0.590645	+	0.806931i	$0.298873\pi$
$504$	0	0
$505$	6.18744e7	0.0213792
$506$	4.02404e9	1.38081
$507$	0	0
$508$	−1.20371e9	−0.407379
$509$	2.96439e9	0.996376	0.498188	−	0.867069i	$-0.333999\pi$
$509$	2.96439e9	0.996376	0.498188	+	0.867069i	$0.333999\pi$
$510$	0	0
$511$	−3.81702e8	−0.126547
$512$	3.35615e9	1.10509
$513$	0	0
$514$	−7.53349e7	−0.0244695
$515$	2.67115e7	0.00861734
$516$	0	0
$517$	−4.38601e9	−1.39590
$518$	−1.38567e9	−0.438032
$519$	0	0
$520$	4.57175e7	0.0142584
$521$	2.91409e9	0.902757	0.451378	−	0.892333i	$-0.350933\pi$
$521$	2.91409e9	0.902757	0.451378	+	0.892333i	$0.350933\pi$
$522$	0	0
$523$	−4.99881e9	−1.52796	−0.763978	−	0.645243i	$-0.776756\pi$
$523$	−4.99881e9	−1.52796	−0.763978	+	0.645243i	$0.776756\pi$
$524$	1.72502e9	0.523763
$525$	0	0
$526$	5.72951e9	1.71659
$527$	−2.41154e9	−0.717723
$528$	0	0
$529$	−1.06102e7	−0.00311622
$530$	−1.94527e8	−0.0567565
$531$	0	0
$532$	−7.13926e7	−0.0205571
$533$	9.43842e9	2.69994
$534$	0	0
$535$	−1.42183e8	−0.0401430
$536$	4.80633e8	0.134814
$537$	0	0
$538$	4.24700e9	1.17583
$539$	−6.09972e9	−1.67784
$540$	0	0
$541$	−1.32516e9	−0.359813	−0.179906	−	0.983684i	$-0.557580\pi$
$541$	−1.32516e9	−0.359813	−0.179906	+	0.983684i	$0.557580\pi$
$542$	4.15765e8	0.112163
$543$	0	0
$544$	9.04730e9	2.40948
$545$	2.90756e7	0.00769379
$546$	0	0
$547$	1.49203e9	0.389782	0.194891	−	0.980825i	$-0.437565\pi$
$547$	1.49203e9	0.389782	0.194891	+	0.980825i	$0.437565\pi$
$548$	2.28614e8	0.0593433
$549$	0	0
$550$	5.39316e9	1.38221
$551$	1.05532e8	0.0268753
$552$	0	0
$553$	−1.95682e9	−0.492055
$554$	2.45939e8	0.0614530
$555$	0	0
$556$	−1.80286e9	−0.444835
$557$	1.87638e9	0.460074	0.230037	−	0.973182i	$-0.426115\pi$
$557$	1.87638e9	0.460074	0.230037	+	0.973182i	$0.426115\pi$
$558$	0	0
$559$	−5.42348e9	−1.31322
$560$	1.85937e8	0.0447411
$561$	0	0
$562$	2.16831e9	0.515282
$563$	2.13285e9	0.503711	0.251855	−	0.967765i	$-0.418959\pi$
$563$	2.13285e9	0.503711	0.251855	+	0.967765i	$0.418959\pi$
$564$	0	0
$565$	4.67336e7	0.0109008
$566$	1.94668e9	0.451271
$567$	0	0
$568$	1.88005e9	0.430478
$569$	1.42082e9	0.323331	0.161666	−	0.986846i	$-0.448313\pi$
$569$	1.42082e9	0.323331	0.161666	+	0.986846i	$0.448313\pi$
$570$	0	0
$571$	1.63133e9	0.366703	0.183352	−	0.983047i	$-0.441305\pi$
$571$	1.63133e9	0.366703	0.183352	+	0.983047i	$0.441305\pi$
$572$	−6.82705e9	−1.52527
$573$	0	0
$574$	1.36469e10	3.01192
$575$	4.54905e9	0.997892
$576$	0	0
$577$	7.83244e9	1.69739	0.848695	−	0.528882i	$-0.177389\pi$
$577$	7.83244e9	1.69739	0.848695	+	0.528882i	$0.177389\pi$
$578$	1.66683e10	3.59040
$579$	0	0
$580$	1.34895e8	0.0287078
$581$	−1.21812e10	−2.57675
$582$	0	0
$583$	−9.09662e9	−1.90125
$584$	1.19327e8	0.0247910
$585$	0	0
$586$	8.23184e9	1.68988
$587$	−5.25336e9	−1.07202	−0.536011	−	0.844211i	$-0.680069\pi$
$587$	−5.25336e9	−1.07202	−0.536011	+	0.844211i	$0.680069\pi$
$588$	0	0
$589$	−3.08954e7	−0.00623003
$590$	2.02022e7	0.00404965
$591$	0	0
$592$	1.21849e9	0.241377
$593$	−3.96594e9	−0.781007	−0.390503	−	0.920601i	$-0.627699\pi$
$593$	−3.96594e9	−0.781007	−0.390503	+	0.920601i	$0.627699\pi$
$594$	0	0
$595$	3.74499e8	0.0728855
$596$	1.82344e9	0.352800
$597$	0	0
$598$	−1.33203e10	−2.54718
$599$	−4.82067e9	−0.916461	−0.458230	−	0.888833i	$-0.651517\pi$
$599$	−4.82067e9	−0.916461	−0.458230	+	0.888833i	$0.651517\pi$
$600$	0	0
$601$	1.29063e9	0.242516	0.121258	−	0.992621i	$-0.461307\pi$
$601$	1.29063e9	0.242516	0.121258	+	0.992621i	$0.461307\pi$
$602$	−7.84177e9	−1.46496
$603$	0	0
$604$	−3.15097e9	−0.581855
$605$	−1.09485e7	−0.00201006
$606$	0	0
$607$	7.37737e9	1.33888	0.669440	−	0.742866i	$-0.266534\pi$
$607$	7.37737e9	1.33888	0.669440	+	0.742866i	$0.266534\pi$
$608$	1.15909e8	0.0209149
$609$	0	0
$610$	2.93627e8	0.0523772
$611$	1.45185e10	2.57500
$612$	0	0
$613$	−8.76333e9	−1.53659	−0.768294	−	0.640097i	$-0.778894\pi$
$613$	−8.76333e9	−1.53659	−0.768294	+	0.640097i	$0.778894\pi$
$614$	1.13719e10	1.98264
$615$	0	0
$616$	3.09114e9	0.532826
$617$	−9.05070e9	−1.55126	−0.775629	−	0.631189i	$-0.782567\pi$
$617$	−9.05070e9	−1.55126	−0.775629	+	0.631189i	$0.782567\pi$
$618$	0	0
$619$	7.25987e9	1.23030	0.615151	−	0.788409i	$-0.289095\pi$
$619$	7.25987e9	1.23030	0.615151	+	0.788409i	$0.289095\pi$
$620$	−3.94918e7	−0.00665482
$621$	0	0
$622$	−5.23064e9	−0.871543
$623$	−5.12174e9	−0.848612
$624$	0	0
$625$	6.09346e9	0.998352
$626$	−1.12565e10	−1.83397
$627$	0	0
$628$	−5.33184e9	−0.859049
$629$	2.45418e9	0.393214
$630$	0	0
$631$	6.82668e9	1.08170	0.540850	−	0.841119i	$-0.318103\pi$
$631$	6.82668e9	1.08170	0.540850	+	0.841119i	$0.318103\pi$
$632$	6.11739e8	0.0963954
$633$	0	0
$634$	−1.10983e9	−0.172959
$635$	8.08943e7	0.0125375
$636$	0	0
$637$	2.01912e10	3.09510
$638$	1.45915e10	2.22448
$639$	0	0
$640$	−9.55896e7	−0.0144139
$641$	−2.79576e9	−0.419273	−0.209637	−	0.977779i	$-0.567228\pi$
$641$	−2.79576e9	−0.419273	−0.209637	+	0.977779i	$0.567228\pi$
$642$	0	0
$643$	−2.72738e9	−0.404583	−0.202292	−	0.979325i	$-0.564839\pi$
$643$	−2.72738e9	−0.404583	−0.202292	+	0.979325i	$0.564839\pi$
$644$	−8.32620e9	−1.22842
$645$	0	0
$646$	2.92484e8	0.0426863
$647$	4.23656e9	0.614962	0.307481	−	0.951554i	$-0.400514\pi$
$647$	4.23656e9	0.614962	0.307481	+	0.951554i	$0.400514\pi$
$648$	0	0
$649$	9.44710e8	0.135657
$650$	−1.78524e10	−2.54976
$651$	0	0
$652$	−4.11486e9	−0.581418
$653$	5.22693e9	0.734599	0.367300	−	0.930103i	$-0.380282\pi$
$653$	5.22693e9	0.734599	0.367300	+	0.930103i	$0.380282\pi$
$654$	0	0
$655$	−1.15929e8	−0.0161193
$656$	−1.20004e10	−1.65971
$657$	0	0
$658$	2.09922e10	2.87255
$659$	−3.27514e9	−0.445791	−0.222896	−	0.974842i	$-0.571551\pi$
$659$	−3.27514e9	−0.445791	−0.222896	+	0.974842i	$0.571551\pi$
$660$	0	0
$661$	1.03921e10	1.39958	0.699788	−	0.714351i	$-0.253278\pi$
$661$	1.03921e10	1.39958	0.699788	+	0.714351i	$0.253278\pi$
$662$	−9.11666e9	−1.22133
$663$	0	0
$664$	3.80806e9	0.504796
$665$	4.79788e6	0.000632665 0
$666$	0	0
$667$	1.23077e10	1.60597
$668$	8.52156e8	0.110612
$669$	0	0
$670$	1.03148e8	0.0132495
$671$	1.37308e10	1.75456
$672$	0	0
$673$	1.08020e10	1.36601	0.683003	−	0.730416i	$-0.260674\pi$
$673$	1.08020e10	1.36601	0.683003	+	0.730416i	$0.260674\pi$
$674$	1.82977e10	2.30190
$675$	0	0
$676$	1.64823e10	2.05213
$677$	−1.40245e8	−0.0173711	−0.00868555	−	0.999962i	$-0.502765\pi$
$677$	−1.40245e8	−0.0173711	−0.00868555	+	0.999962i	$0.502765\pi$
$678$	0	0
$679$	−1.46925e10	−1.80115
$680$	−1.17075e8	−0.0142785
$681$	0	0
$682$	−4.27179e9	−0.515661
$683$	5.38006e9	0.646122	0.323061	−	0.946378i	$-0.395288\pi$
$683$	5.38006e9	0.646122	0.323061	+	0.946378i	$0.395288\pi$
$684$	0	0
$685$	−1.53639e7	−0.00182635
$686$	1.10635e10	1.30846
$687$	0	0
$688$	6.89565e9	0.807264
$689$	3.01115e10	3.50724
$690$	0	0
$691$	−8.51475e9	−0.981745	−0.490873	−	0.871231i	$-0.663322\pi$
$691$	−8.51475e9	−0.981745	−0.490873	+	0.871231i	$0.663322\pi$
$692$	−4.30462e9	−0.493814
$693$	0	0
$694$	−1.24274e10	−1.41131
$695$	1.21160e8	0.0136902
$696$	0	0
$697$	−2.41703e10	−2.70375
$698$	−1.48463e10	−1.65244
$699$	0	0
$700$	−1.11591e10	−1.22966
$701$	−1.29311e10	−1.41782	−0.708911	−	0.705298i	$-0.750813\pi$
$701$	−1.29311e10	−1.41782	−0.708911	+	0.705298i	$0.750813\pi$
$702$	0	0
$703$	3.14417e7	0.00341320
$704$	4.62795e9	0.499901
$705$	0	0
$706$	−3.73197e8	−0.0399137
$707$	−1.38484e10	−1.47377
$708$	0	0
$709$	5.39507e9	0.568506	0.284253	−	0.958749i	$-0.408254\pi$
$709$	5.39507e9	0.568506	0.284253	+	0.958749i	$0.408254\pi$
$710$	4.03476e8	0.0423071
$711$	0	0
$712$	1.60115e9	0.166246
$713$	−3.60319e9	−0.372284
$714$	0	0
$715$	4.58807e8	0.0469417
$716$	3.68240e9	0.374918
$717$	0	0
$718$	−1.00144e10	−1.00969
$719$	8.62160e9	0.865041	0.432521	−	0.901624i	$-0.357624\pi$
$719$	8.62160e9	0.865041	0.432521	+	0.901624i	$0.357624\pi$
$720$	0	0
$721$	−5.97842e9	−0.594036
$722$	−1.34185e10	−1.32686
$723$	0	0
$724$	1.23452e10	1.20896
$725$	1.64953e10	1.60759
$726$	0	0
$727$	−7.76855e9	−0.749841	−0.374921	−	0.927057i	$-0.622330\pi$
$727$	−7.76855e9	−0.749841	−0.374921	+	0.927057i	$0.622330\pi$
$728$	−1.02322e10	−0.982902
$729$	0	0
$730$	2.56086e7	0.00243644
$731$	1.38887e10	1.31507
$732$	0	0
$733$	−2.43007e9	−0.227906	−0.113953	−	0.993486i	$-0.536351\pi$
$733$	−2.43007e9	−0.227906	−0.113953	+	0.993486i	$0.536351\pi$
$734$	7.03456e9	0.656600
$735$	0	0
$736$	1.35180e10	1.24980
$737$	4.82348e9	0.443838
$738$	0	0
$739$	−7.09023e9	−0.646256	−0.323128	−	0.946355i	$-0.604734\pi$
$739$	−7.09023e9	−0.646256	−0.323128	+	0.946355i	$0.604734\pi$
$740$	4.01901e7	0.00364593
$741$	0	0
$742$	4.35380e10	3.91250
$743$	−6.11412e9	−0.546857	−0.273428	−	0.961892i	$-0.588158\pi$
$743$	−6.11412e9	−0.546857	−0.273428	+	0.961892i	$0.588158\pi$
$744$	0	0
$745$	−1.22543e8	−0.0108578
$746$	8.85482e9	0.780897
$747$	0	0
$748$	1.74830e10	1.52743
$749$	3.18226e10	2.76726
$750$	0	0
$751$	1.01871e10	0.877629	0.438815	−	0.898578i	$-0.355398\pi$
$751$	1.01871e10	0.877629	0.438815	+	0.898578i	$0.355398\pi$
$752$	−1.84594e10	−1.58291
$753$	0	0
$754$	−4.83005e10	−4.10348
$755$	2.11759e8	0.0179072
$756$	0	0
$757$	2.04917e10	1.71689	0.858446	−	0.512903i	$-0.171430\pi$
$757$	2.04917e10	1.71689	0.858446	+	0.512903i	$0.171430\pi$
$758$	1.13380e10	0.945568
$759$	0	0
$760$	−1.49991e6	−0.000123941 0
$761$	−4.69491e9	−0.386172	−0.193086	−	0.981182i	$-0.561850\pi$
$761$	−4.69491e9	−0.386172	−0.193086	+	0.981182i	$0.561850\pi$
$762$	0	0
$763$	−6.50752e9	−0.530371
$764$	2.09627e9	0.170067
$765$	0	0
$766$	7.75145e9	0.623135
$767$	−3.12716e9	−0.250246
$768$	0	0
$769$	1.90779e10	1.51282	0.756412	−	0.654095i	$-0.226951\pi$
$769$	1.90779e10	1.51282	0.756412	+	0.654095i	$0.226951\pi$
$770$	6.63386e8	0.0523659
$771$	0	0
$772$	2.13003e9	0.166619
$773$	1.43599e10	1.11821	0.559105	−	0.829097i	$-0.311145\pi$
$773$	1.43599e10	1.11821	0.559105	+	0.829097i	$0.311145\pi$
$774$	0	0
$775$	−4.82913e9	−0.372660
$776$	4.59313e9	0.352852
$777$	0	0
$778$	−3.04212e10	−2.31605
$779$	−3.09657e8	−0.0234693
$780$	0	0
$781$	1.88676e10	1.41722
$782$	3.41111e10	2.55078
$783$	0	0
$784$	−2.56720e10	−1.90262
$785$	3.58322e8	0.0264381
$786$	0	0
$787$	−4.56113e9	−0.333550	−0.166775	−	0.985995i	$-0.553335\pi$
$787$	−4.56113e9	−0.333550	−0.166775	+	0.985995i	$0.553335\pi$
$788$	8.51267e9	0.619761
$789$	0	0
$790$	1.31285e8	0.00947369
$791$	−1.04596e10	−0.751447
$792$	0	0
$793$	−4.54515e10	−3.23662
$794$	1.68035e10	1.19132
$795$	0	0
$796$	−1.98363e10	−1.39400
$797$	−1.19252e10	−0.834374	−0.417187	−	0.908821i	$-0.636984\pi$
$797$	−1.19252e10	−0.834374	−0.417187	+	0.908821i	$0.636984\pi$
$798$	0	0
$799$	−3.71795e10	−2.57864
$800$	1.81173e10	1.25106
$801$	0	0
$802$	4.13667e9	0.283166
$803$	1.19753e9	0.0816171
$804$	0	0
$805$	5.59556e8	0.0378057
$806$	1.41404e10	0.951238
$807$	0	0
$808$	4.32925e9	0.288718
$809$	1.14473e10	0.760121	0.380061	−	0.924962i	$-0.375903\pi$
$809$	1.14473e10	0.760121	0.380061	+	0.924962i	$0.375903\pi$
$810$	0	0
$811$	−1.28875e10	−0.848389	−0.424195	−	0.905571i	$-0.639443\pi$
$811$	−1.28875e10	−0.848389	−0.424195	+	0.905571i	$0.639443\pi$
$812$	−3.01915e10	−1.97897
$813$	0	0
$814$	4.34732e9	0.282512
$815$	2.76536e8	0.0178937
$816$	0	0
$817$	1.77934e8	0.0114152
$818$	−1.12549e10	−0.718961
$819$	0	0
$820$	−3.95817e8	−0.0250695
$821$	1.25024e10	0.788481	0.394240	−	0.919007i	$-0.371008\pi$
$821$	1.25024e10	0.788481	0.394240	+	0.919007i	$0.371008\pi$
$822$	0	0
$823$	1.32497e10	0.828528	0.414264	−	0.910157i	$-0.364039\pi$
$823$	1.32497e10	0.828528	0.414264	+	0.910157i	$0.364039\pi$
$824$	1.86896e9	0.116374
$825$	0	0
$826$	−4.52154e9	−0.279162
$827$	−3.06111e9	−0.188196	−0.0940980	−	0.995563i	$-0.529997\pi$
$827$	−3.06111e9	−0.188196	−0.0940980	+	0.995563i	$0.529997\pi$
$828$	0	0
$829$	−1.74744e10	−1.06527	−0.532637	−	0.846344i	$-0.678799\pi$
$829$	−1.74744e10	−1.06527	−0.532637	+	0.846344i	$0.678799\pi$
$830$	8.17244e8	0.0496111
$831$	0	0
$832$	−1.53194e10	−0.922166
$833$	−5.17064e10	−3.09947
$834$	0	0
$835$	−5.72685e7	−0.00340418
$836$	2.23983e8	0.0132585
$837$	0	0
$838$	−3.14931e10	−1.84868
$839$	3.10592e9	0.181562	0.0907808	−	0.995871i	$-0.471064\pi$
$839$	3.10592e9	0.181562	0.0907808	+	0.995871i	$0.471064\pi$
$840$	0	0
$841$	2.73789e10	1.58720
$842$	1.23437e10	0.712612
$843$	0	0
$844$	−7.45245e9	−0.426678
$845$	−1.10768e9	−0.0631564
$846$	0	0
$847$	2.45042e9	0.138563
$848$	−3.82850e10	−2.15597
$849$	0	0
$850$	4.57170e10	2.55336
$851$	3.66690e9	0.203960
$852$	0	0
$853$	1.91007e10	1.05372	0.526862	−	0.849951i	$-0.323368\pi$
$853$	1.91007e10	1.05372	0.526862	+	0.849951i	$0.323368\pi$
$854$	−6.57180e10	−3.61062
$855$	0	0
$856$	−9.94833e9	−0.542116
$857$	−3.07421e9	−0.166840	−0.0834202	−	0.996514i	$-0.526584\pi$
$857$	−3.07421e9	−0.166840	−0.0834202	+	0.996514i	$0.526584\pi$
$858$	0	0
$859$	−7.57219e9	−0.407610	−0.203805	−	0.979011i	$-0.565331\pi$
$859$	−7.57219e9	−0.407610	−0.203805	+	0.979011i	$0.565331\pi$
$860$	2.27443e8	0.0121935
$861$	0	0
$862$	1.46123e10	0.777038
$863$	−6.48477e9	−0.343445	−0.171722	−	0.985145i	$-0.554933\pi$
$863$	−6.48477e9	−0.343445	−0.171722	+	0.985145i	$0.554933\pi$
$864$	0	0
$865$	2.89289e8	0.0151976
$866$	4.03742e10	2.11247
$867$	0	0
$868$	8.83883e9	0.458749
$869$	6.13922e9	0.317354
$870$	0	0
$871$	−1.59666e10	−0.818746
$872$	2.03437e9	0.103902
$873$	0	0
$874$	4.37014e8	0.0221414
$875$	1.50029e9	0.0757088
$876$	0	0
$877$	−7.00447e9	−0.350652	−0.175326	−	0.984510i	$-0.556098\pi$
$877$	−7.00447e9	−0.350652	−0.175326	+	0.984510i	$0.556098\pi$
$878$	3.30887e10	1.64987
$879$	0	0
$880$	−5.83347e8	−0.0288561
$881$	1.76566e10	0.869944	0.434972	−	0.900444i	$-0.356758\pi$
$881$	1.76566e10	0.869944	0.434972	+	0.900444i	$0.356758\pi$
$882$	0	0
$883$	−1.27783e10	−0.624610	−0.312305	−	0.949982i	$-0.601101\pi$
$883$	−1.27783e10	−0.624610	−0.312305	+	0.949982i	$0.601101\pi$
$884$	−5.78719e10	−2.81763
$885$	0	0
$886$	1.57614e10	0.761337
$887$	−3.30434e10	−1.58983	−0.794917	−	0.606719i	$-0.792485\pi$
$887$	−3.30434e10	−1.58983	−0.794917	+	0.606719i	$0.792485\pi$
$888$	0	0
$889$	−1.81053e10	−0.864270
$890$	3.43621e8	0.0163386
$891$	0	0
$892$	−5.11023e9	−0.241081
$893$	−4.76325e8	−0.0223833
$894$	0	0
$895$	−2.47473e8	−0.0115384
$896$	2.13943e10	0.993619
$897$	0	0
$898$	3.03750e10	1.39975
$899$	−1.30655e10	−0.599744
$900$	0	0
$901$	−7.71107e10	−3.51219
$902$	−4.28151e10	−1.94256
$903$	0	0
$904$	3.26987e9	0.147211
$905$	−8.29650e8	−0.0372070
$906$	0	0
$907$	−1.42333e10	−0.633401	−0.316701	−	0.948526i	$-0.602575\pi$
$907$	−1.42333e10	−0.633401	−0.316701	+	0.948526i	$0.602575\pi$
$908$	−3.66988e8	−0.0162686
$909$	0	0
$910$	−2.19593e9	−0.0965991
$911$	−8.60227e9	−0.376963	−0.188482	−	0.982077i	$-0.560357\pi$
$911$	−8.60227e9	−0.376963	−0.188482	+	0.982077i	$0.560357\pi$
$912$	0	0
$913$	3.82165e10	1.66189
$914$	−4.53120e10	−1.96292
$915$	0	0
$916$	−1.45030e10	−0.623482
$917$	2.59465e10	1.11118
$918$	0	0
$919$	2.07267e10	0.880898	0.440449	−	0.897778i	$-0.354819\pi$
$919$	2.07267e10	0.880898	0.440449	+	0.897778i	$0.354819\pi$
$920$	−1.74927e8	−0.00740629
$921$	0	0
$922$	−4.70548e10	−1.97718
$923$	−6.24553e10	−2.61435
$924$	0	0
$925$	4.91452e9	0.204167
$926$	−3.46520e10	−1.43413
$927$	0	0
$928$	4.90174e10	2.01341
$929$	4.14549e10	1.69637	0.848186	−	0.529699i	$-0.177695\pi$
$929$	4.14549e10	1.69637	0.848186	+	0.529699i	$0.177695\pi$
$930$	0	0
$931$	−6.62436e8	−0.0269042
$932$	−1.31985e10	−0.534034
$933$	0	0
$934$	4.85233e10	1.94866
$935$	−1.17493e9	−0.0470080
$936$	0	0
$937$	−2.37792e10	−0.944296	−0.472148	−	0.881519i	$-0.656521\pi$
$937$	−2.37792e10	−0.944296	−0.472148	+	0.881519i	$0.656521\pi$
$938$	−2.30860e10	−0.913353
$939$	0	0
$940$	−6.08859e8	−0.0239094
$941$	1.90527e10	0.745407	0.372703	−	0.927951i	$-0.378431\pi$
$941$	1.90527e10	0.745407	0.372703	+	0.927951i	$0.378431\pi$
$942$	0	0
$943$	−3.61139e10	−1.40244
$944$	3.97601e9	0.153832
$945$	0	0
$946$	2.46023e10	0.944838
$947$	−2.17206e10	−0.831089	−0.415544	−	0.909573i	$-0.636409\pi$
$947$	−2.17206e10	−0.831089	−0.415544	+	0.909573i	$0.636409\pi$
$948$	0	0
$949$	−3.96404e9	−0.150559
$950$	5.85703e8	0.0221638
$951$	0	0
$952$	2.62031e10	0.984290
$953$	2.72331e9	0.101923	0.0509614	−	0.998701i	$-0.483771\pi$
$953$	2.72331e9	0.101923	0.0509614	+	0.998701i	$0.483771\pi$
$954$	0	0
$955$	−1.40878e8	−0.00523397
$956$	3.27485e10	1.21224
$957$	0	0
$958$	−4.68732e10	−1.72244
$959$	3.43865e9	0.125899
$960$	0	0
$961$	−2.36876e10	−0.860972
$962$	−1.43904e10	−0.521148
$963$	0	0
$964$	−2.41973e10	−0.869955
$965$	−1.43147e8	−0.00512787
$966$	0	0
$967$	−1.19767e10	−0.425935	−0.212967	−	0.977059i	$-0.568313\pi$
$967$	−1.19767e10	−0.425935	−0.212967	+	0.977059i	$0.568313\pi$
$968$	−7.66047e8	−0.0271451
$969$	0	0
$970$	9.85728e8	0.0346781
$971$	−4.20855e9	−0.147525	−0.0737625	−	0.997276i	$-0.523501\pi$
$971$	−4.20855e9	−0.147525	−0.0737625	+	0.997276i	$0.523501\pi$
$972$	0	0
$973$	−2.71172e10	−0.943735
$974$	−2.97631e10	−1.03210
$975$	0	0
$976$	5.77890e10	1.98962
$977$	−3.92707e10	−1.34722	−0.673608	−	0.739089i	$-0.735256\pi$
$977$	−3.92707e10	−1.34722	−0.673608	+	0.739089i	$0.735256\pi$
$978$	0	0
$979$	1.60686e10	0.547318
$980$	−8.46754e8	−0.0287386
$981$	0	0
$982$	−3.75183e10	−1.26431
$983$	−2.85690e10	−0.959308	−0.479654	−	0.877458i	$-0.659238\pi$
$983$	−2.85690e10	−0.959308	−0.479654	+	0.877458i	$0.659238\pi$
$984$	0	0
$985$	−5.72087e8	−0.0190737
$986$	1.23690e11	4.10927
$987$	0	0
$988$	−7.41424e8	−0.0244578
$989$	2.07517e10	0.682129
$990$	0	0
$991$	2.53806e10	0.828408	0.414204	−	0.910184i	$-0.364060\pi$
$991$	2.53806e10	0.828408	0.414204	+	0.910184i	$0.364060\pi$
$992$	−1.43503e10	−0.466733
$993$	0	0
$994$	−9.03036e10	−2.91644
$995$	1.33308e9	0.0429018
$996$	0	0
$997$	9.36748e9	0.299357	0.149679	−	0.988735i	$-0.452176\pi$
$997$	9.36748e9	0.299357	0.149679	+	0.988735i	$0.452176\pi$
$998$	7.20773e10	2.29531
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.8.a.e.1.15		18
3.2	odd	2		177.8.a.d.1.4	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.d.1.4	✓	18	3.2	odd	2
531.8.a.e.1.15		18	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)

\(f(q)\)	\(=\)	\(q+15.0158 q^{2} +97.4757 q^{4} -6.55078 q^{5} +1466.16 q^{7} -458.348 q^{8} +O(q^{10})\)	\(q+15.0158 q^{2} +97.4757 q^{4} -6.55078 q^{5} +1466.16 q^{7} -458.348 q^{8} -98.3656 q^{10} -4599.84 q^{11} +15226.3 q^{13} +22015.6 q^{14} -19359.4 q^{16} -38992.1 q^{17} -499.547 q^{19} -638.542 q^{20} -69070.4 q^{22} -58259.9 q^{23} -78082.1 q^{25} +228636. q^{26} +142915. q^{28} -211255. q^{29} +61846.8 q^{31} -232029. q^{32} -585500. q^{34} -9604.48 q^{35} -62940.4 q^{37} -7501.12 q^{38} +3002.54 q^{40} +619876. q^{41} -356192. q^{43} -448372. q^{44} -874822. q^{46} +953515. q^{47} +1.32607e6 q^{49} -1.17247e6 q^{50} +1.48419e6 q^{52} +1.97760e6 q^{53} +30132.5 q^{55} -672010. q^{56} -3.17218e6 q^{58} -205379. q^{59} -2.98506e6 q^{61} +928683. q^{62} -1.00611e6 q^{64} -99744.2 q^{65} -1.04862e6 q^{67} -3.80078e6 q^{68} -144219. q^{70} -4.10180e6 q^{71} -260342. q^{73} -945104. q^{74} -48693.7 q^{76} -6.74408e6 q^{77} -1.33466e6 q^{79} +126819. q^{80} +9.30797e6 q^{82} -8.30823e6 q^{83} +255429. q^{85} -5.34852e6 q^{86} +2.10832e6 q^{88} -3.49331e6 q^{89} +2.23242e7 q^{91} -5.67893e6 q^{92} +1.43178e7 q^{94} +3272.42 q^{95} -1.00211e7 q^{97} +1.99121e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8}+O(q^{10}) \) 18 * q - 24 * q^2 + 1358 * q^4 - 678 * q^5 + 3081 * q^7 - 4107 * q^8	\( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8} + 3609 q^{10} - 15070 q^{11} + 13662 q^{13} - 20861 q^{14} + 60482 q^{16} - 71919 q^{17} + 56231 q^{19} - 143053 q^{20} + 274198 q^{22} - 150029 q^{23} + 399672 q^{25} - 182846 q^{26} + 434150 q^{28} - 591285 q^{29} + 426733 q^{31} - 1205630 q^{32} + 403548 q^{34} - 912879 q^{35} + 7703 q^{37} + 417859 q^{38} + 618020 q^{40} - 770959 q^{41} + 793050 q^{43} - 2591274 q^{44} - 4068019 q^{46} - 1410373 q^{47} + 1637427 q^{49} - 1021549 q^{50} - 3749190 q^{52} - 1037934 q^{53} + 331974 q^{55} + 391748 q^{56} + 653724 q^{58} - 3696822 q^{59} - 1374623 q^{61} - 5251718 q^{62} + 5077197 q^{64} - 3257170 q^{65} - 2436904 q^{67} - 14119909 q^{68} + 5185580 q^{70} - 14289172 q^{71} + 5482515 q^{73} - 14934154 q^{74} + 3822912 q^{76} - 23157109 q^{77} + 19786414 q^{79} - 31978143 q^{80} + 9749509 q^{82} - 30227337 q^{83} + 9946981 q^{85} - 44295864 q^{86} + 39970897 q^{88} - 31061677 q^{89} + 26377785 q^{91} - 4719698 q^{92} + 44488296 q^{94} - 15534599 q^{95} + 12084118 q^{97} - 42274744 q^{98}+O(q^{100}) \) 18 * q - 24 * q^2 + 1358 * q^4 - 678 * q^5 + 3081 * q^7 - 4107 * q^8 + 3609 * q^10 - 15070 * q^11 + 13662 * q^13 - 20861 * q^14 + 60482 * q^16 - 71919 * q^17 + 56231 * q^19 - 143053 * q^20 + 274198 * q^22 - 150029 * q^23 + 399672 * q^25 - 182846 * q^26 + 434150 * q^28 - 591285 * q^29 + 426733 * q^31 - 1205630 * q^32 + 403548 * q^34 - 912879 * q^35 + 7703 * q^37 + 417859 * q^38 + 618020 * q^40 - 770959 * q^41 + 793050 * q^43 - 2591274 * q^44 - 4068019 * q^46 - 1410373 * q^47 + 1637427 * q^49 - 1021549 * q^50 - 3749190 * q^52 - 1037934 * q^53 + 331974 * q^55 + 391748 * q^56 + 653724 * q^58 - 3696822 * q^59 - 1374623 * q^61 - 5251718 * q^62 + 5077197 * q^64 - 3257170 * q^65 - 2436904 * q^67 - 14119909 * q^68 + 5185580 * q^70 - 14289172 * q^71 + 5482515 * q^73 - 14934154 * q^74 + 3822912 * q^76 - 23157109 * q^77 + 19786414 * q^79 - 31978143 * q^80 + 9749509 * q^82 - 30227337 * q^83 + 9946981 * q^85 - 44295864 * q^86 + 39970897 * q^88 - 31061677 * q^89 + 26377785 * q^91 - 4719698 * q^92 + 44488296 * q^94 - 15534599 * q^95 + 12084118 * q^97 - 42274744 * q^98

Introduction

L-functions

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