LMFDB - Embedded newform 531.8.a.e.1.18

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			1.18
Root			$-20.9501$ 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)

$f(q)$	$=$	$q+19.9501 q^{2} +270.005 q^{4} -505.865 q^{5} +222.970 q^{7} +2833.01 q^{8} +O(q^{10})$	\(q+19.9501 q^{2} +270.005 q^{4} -505.865 q^{5} +222.970 q^{7} +2833.01 q^{8} -10092.0 q^{10} +1831.62 q^{11} +688.848 q^{13} +4448.27 q^{14} +21958.1 q^{16} -2843.94 q^{17} +34416.2 q^{19} -136586. q^{20} +36540.9 q^{22} +3272.29 q^{23} +177775. q^{25} +13742.6 q^{26} +60203.0 q^{28} -128507. q^{29} -16509.0 q^{31} +75439.5 q^{32} -56736.8 q^{34} -112793. q^{35} -424491. q^{37} +686606. q^{38} -1.43312e6 q^{40} +12500.6 q^{41} +338393. q^{43} +494546. q^{44} +65282.4 q^{46} -560303. q^{47} -773827. q^{49} +3.54662e6 q^{50} +185992. q^{52} -2.04079e6 q^{53} -926551. q^{55} +631676. q^{56} -2.56371e6 q^{58} -205379. q^{59} -2.42610e6 q^{61} -329355. q^{62} -1.30561e6 q^{64} -348464. q^{65} -2.63298e6 q^{67} -767879. q^{68} -2.25022e6 q^{70} -2.62379e6 q^{71} +3.21761e6 q^{73} -8.46862e6 q^{74} +9.29256e6 q^{76} +408396. q^{77} +3.42920e6 q^{79} -1.11078e7 q^{80} +249387. q^{82} -1.20365e6 q^{83} +1.43865e6 q^{85} +6.75097e6 q^{86} +5.18899e6 q^{88} +2.50843e6 q^{89} +153592. q^{91} +883534. q^{92} -1.11781e7 q^{94} -1.74100e7 q^{95} +8.73888e6 q^{97} -1.54379e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	19.9501	1.76335	0.881677	−	0.471854i	$-0.156415\pi$
$2$	19.9501	1.76335	0.881677	+	0.471854i	$0.156415\pi$
$3$	0	0
$3$	0	0
$4$	270.005	2.10941
$5$	−505.865	−1.80984	−0.904919	−	0.425583i	$-0.860069\pi$
$5$	−505.865	−1.80984	−0.904919	+	0.425583i	$0.860069\pi$
$6$	0	0
$7$	222.970	0.245699	0.122849	−	0.992425i	$-0.460797\pi$
$7$	222.970	0.245699	0.122849	+	0.992425i	$0.460797\pi$
$8$	2833.01	1.95629
$9$	0	0
$10$	−10092.0	−3.19138
$11$	1831.62	0.414916	0.207458	−	0.978244i	$-0.433481\pi$
$11$	1831.62	0.414916	0.207458	+	0.978244i	$0.433481\pi$
$12$	0	0
$13$	688.848	0.0869604	0.0434802	−	0.999054i	$-0.486155\pi$
$13$	688.848	0.0869604	0.0434802	+	0.999054i	$0.486155\pi$
$14$	4448.27	0.433254
$15$	0	0
$16$	21958.1	1.34021
$17$	−2843.94	−0.140394	−0.0701972	−	0.997533i	$-0.522363\pi$
$17$	−2843.94	−0.140394	−0.0701972	+	0.997533i	$0.522363\pi$
$18$	0	0
$19$	34416.2	1.15113	0.575567	−	0.817755i	$-0.304781\pi$
$19$	34416.2	1.15113	0.575567	+	0.817755i	$0.304781\pi$
$20$	−136586.	−3.81770
$21$	0	0
$22$	36540.9	0.731643
$23$	3272.29	0.0560795	0.0280398	−	0.999607i	$-0.491073\pi$
$23$	3272.29	0.0560795	0.0280398	+	0.999607i	$0.491073\pi$
$24$	0	0
$25$	177775.	2.27552
$26$	13742.6	0.153342
$27$	0	0
$28$	60203.0	0.518281
$29$	−128507.	−0.978436	−0.489218	−	0.872162i	$-0.662718\pi$
$29$	−128507.	−0.978436	−0.489218	+	0.872162i	$0.662718\pi$
$30$	0	0
$31$	−16509.0	−0.0995301	−0.0497651	−	0.998761i	$-0.515847\pi$
$31$	−16509.0	−0.0995301	−0.0497651	+	0.998761i	$0.515847\pi$
$32$	75439.5	0.406981
$33$	0	0
$34$	−56736.8	−0.247565
$35$	−112793.	−0.444675
$36$	0	0
$37$	−424491.	−1.37772	−0.688862	−	0.724892i	$-0.741889\pi$
$37$	−424491.	−1.37772	−0.688862	+	0.724892i	$0.741889\pi$
$38$	686606.	2.02986
$39$	0	0
$40$	−1.43312e6	−3.54057
$41$	12500.6	0.0283261	0.0141630	−	0.999900i	$-0.495492\pi$
$41$	12500.6	0.0283261	0.0141630	+	0.999900i	$0.495492\pi$
$42$	0	0
$43$	338393.	0.649056	0.324528	−	0.945876i	$-0.394795\pi$
$43$	338393.	0.649056	0.324528	+	0.945876i	$0.394795\pi$
$44$	494546.	0.875230
$45$	0	0
$46$	65282.4	0.0988880
$47$	−560303.	−0.787192	−0.393596	−	0.919284i	$-0.628769\pi$
$47$	−560303.	−0.787192	−0.393596	+	0.919284i	$0.628769\pi$
$48$	0	0
$49$	−773827.	−0.939632
$50$	3.54662e6	4.01254
$51$	0	0
$52$	185992.	0.183436
$53$	−2.04079e6	−1.88293	−0.941463	−	0.337116i	$-0.890548\pi$
$53$	−2.04079e6	−1.88293	−0.941463	+	0.337116i	$0.890548\pi$
$54$	0	0
$55$	−926551.	−0.750931
$56$	631676.	0.480658
$57$	0	0
$58$	−2.56371e6	−1.72533
$59$	−205379.	−0.130189
$59$	−205379.	−0.130189
$60$	0	0
$61$	−2.42610e6	−1.36853	−0.684265	−	0.729233i	$-0.739877\pi$
$61$	−2.42610e6	−1.36853	−0.684265	+	0.729233i	$0.739877\pi$
$62$	−329355.	−0.175507
$63$	0	0
$64$	−1.30561e6	−0.622563
$65$	−348464.	−0.157384
$66$	0	0
$67$	−2.63298e6	−1.06951	−0.534757	−	0.845006i	$-0.679597\pi$
$67$	−2.63298e6	−1.06951	−0.534757	+	0.845006i	$0.679597\pi$
$68$	−767879.	−0.296150
$69$	0	0
$70$	−2.25022e6	−0.784120
$71$	−2.62379e6	−0.870013	−0.435006	−	0.900427i	$-0.643254\pi$
$71$	−2.62379e6	−0.870013	−0.435006	+	0.900427i	$0.643254\pi$
$72$	0	0
$73$	3.21761e6	0.968062	0.484031	−	0.875051i	$-0.339172\pi$
$73$	3.21761e6	0.968062	0.484031	+	0.875051i	$0.339172\pi$
$74$	−8.46862e6	−2.42941
$75$	0	0
$76$	9.29256e6	2.42822
$77$	408396.	0.101944
$78$	0	0
$79$	3.42920e6	0.782525	0.391263	−	0.920279i	$-0.372038\pi$
$79$	3.42920e6	0.782525	0.391263	+	0.920279i	$0.372038\pi$
$80$	−1.11078e7	−2.42557
$81$	0	0
$82$	249387.	0.0499489
$83$	−1.20365e6	−0.231060	−0.115530	−	0.993304i	$-0.536857\pi$
$83$	−1.20365e6	−0.231060	−0.115530	+	0.993304i	$0.536857\pi$
$84$	0	0
$85$	1.43865e6	0.254091
$86$	6.75097e6	1.14451
$87$	0	0
$88$	5.18899e6	0.811695
$89$	2.50843e6	0.377170	0.188585	−	0.982057i	$-0.439610\pi$
$89$	2.50843e6	0.377170	0.188585	+	0.982057i	$0.439610\pi$
$90$	0	0
$91$	153592.	0.0213661
$92$	883534.	0.118295
$93$	0	0
$94$	−1.11781e7	−1.38810
$95$	−1.74100e7	−2.08337
$96$	0	0
$97$	8.73888e6	0.972198	0.486099	−	0.873904i	$-0.338420\pi$
$97$	8.73888e6	0.972198	0.486099	+	0.873904i	$0.338420\pi$
$98$	−1.54379e7	−1.65690
$99$	0	0
$100$	4.80000e7	4.80000
$101$	814288.	0.0786418	0.0393209	−	0.999227i	$-0.487481\pi$
$101$	814288.	0.0786418	0.0393209	+	0.999227i	$0.487481\pi$
$102$	0	0
$103$	−1.34032e7	−1.20859	−0.604296	−	0.796760i	$-0.706546\pi$
$103$	−1.34032e7	−1.20859	−0.604296	+	0.796760i	$0.706546\pi$
$104$	1.95151e6	0.170120
$105$	0	0
$106$	−4.07139e7	−3.32026
$107$	5.36199e6	0.423139	0.211569	−	0.977363i	$-0.432143\pi$
$107$	5.36199e6	0.423139	0.211569	+	0.977363i	$0.432143\pi$
$108$	0	0
$109$	2.36393e6	0.174841	0.0874203	−	0.996172i	$-0.472138\pi$
$109$	2.36393e6	0.174841	0.0874203	+	0.996172i	$0.472138\pi$
$110$	−1.84848e7	−1.32416
$111$	0	0
$112$	4.89599e6	0.329289
$113$	−1.26518e7	−0.824855	−0.412428	−	0.910990i	$-0.635319\pi$
$113$	−1.26518e7	−0.824855	−0.412428	+	0.910990i	$0.635319\pi$
$114$	0	0
$115$	−1.65534e6	−0.101495
$116$	−3.46974e7	−2.06393
$117$	0	0
$118$	−4.09732e6	−0.229569
$119$	−634114.	−0.0344947
$120$	0	0
$121$	−1.61324e7	−0.827845
$122$	−4.84009e7	−2.41320
$123$	0	0
$124$	−4.45751e6	−0.209950
$125$	−5.04093e7	−2.30848
$126$	0	0
$127$	−2.94888e7	−1.27745	−0.638726	−	0.769434i	$-0.720538\pi$
$127$	−2.94888e7	−1.27745	−0.638726	+	0.769434i	$0.720538\pi$
$128$	−3.57032e7	−1.50478
$129$	0	0
$130$	−6.95188e6	−0.277524
$131$	−2.80740e7	−1.09107	−0.545537	−	0.838087i	$-0.683674\pi$
$131$	−2.80740e7	−1.09107	−0.545537	+	0.838087i	$0.683674\pi$
$132$	0	0
$133$	7.67379e6	0.282832
$134$	−5.25282e7	−1.88593
$135$	0	0
$136$	−8.05691e6	−0.274652
$137$	5.12815e7	1.70388	0.851939	−	0.523641i	$-0.175427\pi$
$137$	5.12815e7	1.70388	0.851939	+	0.523641i	$0.175427\pi$
$138$	0	0
$139$	3.88050e6	0.122556	0.0612782	−	0.998121i	$-0.480482\pi$
$139$	3.88050e6	0.122556	0.0612782	+	0.998121i	$0.480482\pi$
$140$	−3.04546e7	−0.938005
$141$	0	0
$142$	−5.23449e7	−1.53414
$143$	1.26171e6	0.0360813
$144$	0	0
$145$	6.50070e7	1.77081
$146$	6.41915e7	1.70704
$147$	0	0
$148$	−1.14615e8	−2.90619
$149$	3.38987e6	0.0839520	0.0419760	−	0.999119i	$-0.486635\pi$
$149$	3.38987e6	0.0839520	0.0419760	+	0.999119i	$0.486635\pi$
$150$	0	0
$151$	4.99988e7	1.18179	0.590895	−	0.806748i	$-0.298775\pi$
$151$	4.99988e7	1.18179	0.590895	+	0.806748i	$0.298775\pi$
$152$	9.75015e7	2.25195
$153$	0	0
$154$	8.14752e6	0.179764
$155$	8.35132e6	0.180133
$156$	0	0
$157$	−4.76455e7	−0.982591	−0.491295	−	0.870993i	$-0.663476\pi$
$157$	−4.76455e7	−0.982591	−0.491295	+	0.870993i	$0.663476\pi$
$158$	6.84128e7	1.37987
$159$	0	0
$160$	−3.81622e7	−0.736569
$161$	729622.	0.0137787
$162$	0	0
$163$	7.64073e7	1.38191	0.690953	−	0.722900i	$-0.257191\pi$
$163$	7.64073e7	1.38191	0.690953	+	0.722900i	$0.257191\pi$
$164$	3.37522e6	0.0597514
$165$	0	0
$166$	−2.40128e7	−0.407441
$167$	6.91463e6	0.114885	0.0574423	−	0.998349i	$-0.481705\pi$
$167$	6.91463e6	0.114885	0.0574423	+	0.998349i	$0.481705\pi$
$168$	0	0
$169$	−6.22740e7	−0.992438
$170$	2.87012e7	0.448052
$171$	0	0
$172$	9.13679e7	1.36913
$173$	−9.54381e7	−1.40139	−0.700697	−	0.713459i	$-0.747128\pi$
$173$	−9.54381e7	−1.40139	−0.700697	+	0.713459i	$0.747128\pi$
$174$	0	0
$175$	3.96384e7	0.559092
$176$	4.02187e7	0.556076
$177$	0	0
$178$	5.00434e7	0.665085
$179$	−3.26295e7	−0.425231	−0.212616	−	0.977136i	$-0.568198\pi$
$179$	−3.26295e7	−0.425231	−0.212616	+	0.977136i	$0.568198\pi$
$180$	0	0
$181$	−8.74090e7	−1.09567	−0.547837	−	0.836585i	$-0.684548\pi$
$181$	−8.74090e7	−1.09567	−0.547837	+	0.836585i	$0.684548\pi$
$182$	3.06418e6	0.0376760
$183$	0	0
$184$	9.27042e6	0.109708
$185$	2.14735e8	2.49346
$186$	0	0
$187$	−5.20901e6	−0.0582519
$188$	−1.51285e8	−1.66051
$189$	0	0
$190$	−3.47330e8	−3.67371
$191$	−2.62452e7	−0.272542	−0.136271	−	0.990672i	$-0.543512\pi$
$191$	−2.62452e7	−0.272542	−0.136271	+	0.990672i	$0.543512\pi$
$192$	0	0
$193$	3.00614e7	0.300995	0.150497	−	0.988610i	$-0.451912\pi$
$193$	3.00614e7	0.300995	0.150497	+	0.988610i	$0.451912\pi$
$194$	1.74341e8	1.71433
$195$	0	0
$196$	−2.08937e8	−1.98207
$197$	6.36038e7	0.592722	0.296361	−	0.955076i	$-0.404227\pi$
$197$	6.36038e7	0.592722	0.296361	+	0.955076i	$0.404227\pi$
$198$	0	0
$199$	1.34497e8	1.20984	0.604919	−	0.796287i	$-0.293205\pi$
$199$	1.34497e8	1.20984	0.604919	+	0.796287i	$0.293205\pi$
$200$	5.03637e8	4.45157
$201$	0	0
$202$	1.62451e7	0.138673
$203$	−2.86531e7	−0.240401
$204$	0	0
$205$	−6.32361e6	−0.0512656
$206$	−2.67396e8	−2.13117
$207$	0	0
$208$	1.51258e7	0.116545
$209$	6.30374e7	0.477624
$210$	0	0
$211$	1.72469e8	1.26393	0.631965	−	0.774997i	$-0.282249\pi$
$211$	1.72469e8	1.26393	0.631965	+	0.774997i	$0.282249\pi$
$212$	−5.51024e8	−3.97187
$213$	0	0
$214$	1.06972e8	0.746143
$215$	−1.71181e8	−1.17469
$216$	0	0
$217$	−3.68101e6	−0.0244544
$218$	4.71606e7	0.308306
$219$	0	0
$220$	−2.50173e8	−1.58402
$221$	−1.95904e6	−0.0122088
$222$	0	0
$223$	1.98662e8	1.19963	0.599815	−	0.800138i	$-0.295241\pi$
$223$	1.98662e8	1.19963	0.599815	+	0.800138i	$0.295241\pi$
$224$	1.68207e7	0.0999947
$225$	0	0
$226$	−2.52404e8	−1.45451
$227$	1.97341e8	1.11977	0.559884	−	0.828571i	$-0.310846\pi$
$227$	1.97341e8	1.11977	0.559884	+	0.828571i	$0.310846\pi$
$228$	0	0
$229$	2.77784e8	1.52856	0.764281	−	0.644883i	$-0.223094\pi$
$229$	2.77784e8	1.52856	0.764281	+	0.644883i	$0.223094\pi$
$230$	−3.30241e7	−0.178971
$231$	0	0
$232$	−3.64060e8	−1.91410
$233$	−3.29662e8	−1.70735	−0.853677	−	0.520803i	$-0.825633\pi$
$233$	−3.29662e8	−1.70735	−0.853677	+	0.520803i	$0.825633\pi$
$234$	0	0
$235$	2.83438e8	1.42469
$236$	−5.54534e7	−0.274622
$237$	0	0
$238$	−1.26506e7	−0.0608264
$239$	−3.52106e8	−1.66833	−0.834163	−	0.551518i	$-0.814049\pi$
$239$	−3.52106e8	−1.66833	−0.834163	+	0.551518i	$0.814049\pi$
$240$	0	0
$241$	−1.57129e8	−0.723097	−0.361549	−	0.932353i	$-0.617752\pi$
$241$	−1.57129e8	−0.723097	−0.361549	+	0.932353i	$0.617752\pi$
$242$	−3.21841e8	−1.45978
$243$	0	0
$244$	−6.55060e8	−2.88680
$245$	3.91452e8	1.70058
$246$	0	0
$247$	2.37076e7	0.100103
$248$	−4.67701e7	−0.194710
$249$	0	0
$250$	−1.00567e9	−4.07066
$251$	2.31577e8	0.924353	0.462177	−	0.886788i	$-0.347069\pi$
$251$	2.31577e8	0.924353	0.462177	+	0.886788i	$0.347069\pi$
$252$	0	0
$253$	5.99358e6	0.0232683
$254$	−5.88304e8	−2.25260
$255$	0	0
$256$	−5.45164e8	−2.03089
$257$	−3.21230e8	−1.18046	−0.590228	−	0.807237i	$-0.700962\pi$
$257$	−3.21230e8	−1.18046	−0.590228	+	0.807237i	$0.700962\pi$
$258$	0	0
$259$	−9.46487e7	−0.338505
$260$	−9.40871e7	−0.331989
$261$	0	0
$262$	−5.60078e8	−1.92395
$263$	−3.68323e8	−1.24849	−0.624243	−	0.781230i	$-0.714592\pi$
$263$	−3.68323e8	−1.24849	−0.624243	+	0.781230i	$0.714592\pi$
$264$	0	0
$265$	1.03237e9	3.40779
$266$	1.53093e8	0.498733
$267$	0	0
$268$	−7.10919e8	−2.25605
$269$	2.17400e8	0.680967	0.340484	−	0.940250i	$-0.389409\pi$
$269$	2.17400e8	0.680967	0.340484	+	0.940250i	$0.389409\pi$
$270$	0	0
$271$	1.69294e8	0.516713	0.258356	−	0.966050i	$-0.416819\pi$
$271$	1.69294e8	0.516713	0.258356	+	0.966050i	$0.416819\pi$
$272$	−6.24474e7	−0.188158
$273$	0	0
$274$	1.02307e9	3.00454
$275$	3.25615e8	0.944148
$276$	0	0
$277$	5.18960e6	0.0146708	0.00733541	−	0.999973i	$-0.497665\pi$
$277$	5.18960e6	0.0146708	0.00733541	+	0.999973i	$0.497665\pi$
$278$	7.74162e7	0.216110
$279$	0	0
$280$	−3.19543e8	−0.869913
$281$	4.92517e8	1.32419	0.662093	−	0.749421i	$-0.269668\pi$
$281$	4.92517e8	1.32419	0.662093	+	0.749421i	$0.269668\pi$
$282$	0	0
$283$	−5.84814e8	−1.53379	−0.766893	−	0.641775i	$-0.778198\pi$
$283$	−5.84814e8	−1.53379	−0.766893	+	0.641775i	$0.778198\pi$
$284$	−7.08438e8	−1.83522
$285$	0	0
$286$	2.51711e7	0.0636240
$287$	2.78725e6	0.00695969
$288$	0	0
$289$	−4.02251e8	−0.980289
$290$	1.29689e9	3.12256
$291$	0	0
$292$	8.68770e8	2.04204
$293$	−6.94020e8	−1.61189	−0.805945	−	0.591990i	$-0.798342\pi$
$293$	−6.94020e8	−1.61189	−0.805945	+	0.591990i	$0.798342\pi$
$294$	0	0
$295$	1.03894e8	0.235621
$296$	−1.20259e9	−2.69523
$297$	0	0
$298$	6.76281e7	0.148037
$299$	2.25411e6	0.00487670
$300$	0	0
$301$	7.54515e7	0.159472
$302$	9.97480e8	2.08391
$303$	0	0
$304$	7.55714e8	1.54276
$305$	1.22728e9	2.47682
$306$	0	0
$307$	6.33917e8	1.25040	0.625199	−	0.780465i	$-0.285018\pi$
$307$	6.33917e8	1.25040	0.625199	+	0.780465i	$0.285018\pi$
$308$	1.10269e8	0.215043
$309$	0	0
$310$	1.66609e8	0.317639
$311$	2.89601e8	0.545932	0.272966	−	0.962024i	$-0.411995\pi$
$311$	2.89601e8	0.545932	0.272966	+	0.962024i	$0.411995\pi$
$312$	0	0
$313$	5.90653e8	1.08875	0.544374	−	0.838843i	$-0.316767\pi$
$313$	5.90653e8	1.08875	0.544374	+	0.838843i	$0.316767\pi$
$314$	−9.50530e8	−1.73265
$315$	0	0
$316$	9.25902e8	1.65067
$317$	9.69791e8	1.70990	0.854950	−	0.518710i	$-0.173587\pi$
$317$	9.69791e8	1.70990	0.854950	+	0.518710i	$0.173587\pi$
$318$	0	0
$319$	−2.35375e8	−0.405969
$320$	6.60462e8	1.12674
$321$	0	0
$322$	1.45560e7	0.0242967
$323$	−9.78778e7	−0.161613
$324$	0	0
$325$	1.22460e8	0.197880
$326$	1.52433e9	2.43679
$327$	0	0
$328$	3.54142e7	0.0554140
$329$	−1.24931e8	−0.193412
$330$	0	0
$331$	−1.23241e9	−1.86792	−0.933959	−	0.357381i	$-0.883670\pi$
$331$	−1.23241e9	−1.86792	−0.933959	+	0.357381i	$0.883670\pi$
$332$	−3.24991e8	−0.487402
$333$	0	0
$334$	1.37947e8	0.202582
$335$	1.33194e9	1.93565
$336$	0	0
$337$	3.81254e8	0.542638	0.271319	−	0.962490i	$-0.412540\pi$
$337$	3.81254e8	0.542638	0.271319	+	0.962490i	$0.412540\pi$
$338$	−1.24237e9	−1.75002
$339$	0	0
$340$	3.88443e8	0.535983
$341$	−3.02381e7	−0.0412966
$342$	0	0
$343$	−3.56166e8	−0.476566
$344$	9.58671e8	1.26974
$345$	0	0
$346$	−1.90400e9	−2.47115
$347$	2.23058e7	0.0286592	0.0143296	−	0.999897i	$-0.495439\pi$
$347$	2.23058e7	0.0286592	0.0143296	+	0.999897i	$0.495439\pi$
$348$	0	0
$349$	−1.22256e9	−1.53951	−0.769755	−	0.638340i	$-0.779622\pi$
$349$	−1.22256e9	−1.53951	−0.769755	+	0.638340i	$0.779622\pi$
$350$	7.90789e8	0.985876
$351$	0	0
$352$	1.38176e8	0.168863
$353$	1.53975e9	1.86311	0.931557	−	0.363595i	$-0.118451\pi$
$353$	1.53975e9	1.86311	0.931557	+	0.363595i	$0.118451\pi$
$354$	0	0
$355$	1.32729e9	1.57458
$356$	6.77290e8	0.795609
$357$	0	0
$358$	−6.50961e8	−0.749832
$359$	6.36027e7	0.0725513	0.0362757	−	0.999342i	$-0.488451\pi$
$359$	6.36027e7	0.0725513	0.0362757	+	0.999342i	$0.488451\pi$
$360$	0	0
$361$	2.90606e8	0.325110
$362$	−1.74381e9	−1.93206
$363$	0	0
$364$	4.14707e7	0.0450699
$365$	−1.62768e9	−1.75204
$366$	0	0
$367$	2.43871e8	0.257531	0.128765	−	0.991675i	$-0.458899\pi$
$367$	2.43871e8	0.257531	0.128765	+	0.991675i	$0.458899\pi$
$368$	7.18531e7	0.0751585
$369$	0	0
$370$	4.28398e9	4.39685
$371$	−4.55035e8	−0.462633
$372$	0	0
$373$	−5.36181e8	−0.534972	−0.267486	−	0.963562i	$-0.586193\pi$
$373$	−5.36181e8	−0.534972	−0.267486	+	0.963562i	$0.586193\pi$
$374$	−1.03920e8	−0.102719
$375$	0	0
$376$	−1.58734e9	−1.53997
$377$	−8.85215e7	−0.0850852
$378$	0	0
$379$	−9.43440e8	−0.890178	−0.445089	−	0.895486i	$-0.646828\pi$
$379$	−9.43440e8	−0.890178	−0.445089	+	0.895486i	$0.646828\pi$
$380$	−4.70078e9	−4.39468
$381$	0	0
$382$	−5.23593e8	−0.480587
$383$	−7.47419e8	−0.679780	−0.339890	−	0.940465i	$-0.610390\pi$
$383$	−7.47419e8	−0.679780	−0.339890	+	0.940465i	$0.610390\pi$
$384$	0	0
$385$	−2.06593e8	−0.184503
$386$	5.99727e8	0.530760
$387$	0	0
$388$	2.35954e9	2.05077
$389$	−5.07334e8	−0.436989	−0.218494	−	0.975838i	$-0.570115\pi$
$389$	−5.07334e8	−0.436989	−0.218494	+	0.975838i	$0.570115\pi$
$390$	0	0
$391$	−9.30620e6	−0.00787324
$392$	−2.19226e9	−1.83819
$393$	0	0
$394$	1.26890e9	1.04518
$395$	−1.73471e9	−1.41624
$396$	0	0
$397$	3.30439e8	0.265048	0.132524	−	0.991180i	$-0.457692\pi$
$397$	3.30439e8	0.265048	0.132524	+	0.991180i	$0.457692\pi$
$398$	2.68323e9	2.13337
$399$	0	0
$400$	3.90359e9	3.04968
$401$	8.76898e8	0.679116	0.339558	−	0.940585i	$-0.389723\pi$
$401$	8.76898e8	0.679116	0.339558	+	0.940585i	$0.389723\pi$
$402$	0	0
$403$	−1.13722e7	−0.00865518
$404$	2.19862e8	0.165888
$405$	0	0
$406$	−5.71631e8	−0.423911
$407$	−7.77505e8	−0.571640
$408$	0	0
$409$	1.20367e9	0.869915	0.434958	−	0.900451i	$-0.356763\pi$
$409$	1.20367e9	0.869915	0.434958	+	0.900451i	$0.356763\pi$
$410$	−1.26156e8	−0.0903994
$411$	0	0
$412$	−3.61894e9	−2.54942
$413$	−4.57934e7	−0.0319873
$414$	0	0
$415$	6.08883e8	0.418182
$416$	5.19663e7	0.0353912
$417$	0	0
$418$	1.25760e9	0.842220
$419$	−1.30342e9	−0.865638	−0.432819	−	0.901481i	$-0.642481\pi$
$419$	−1.30342e9	−0.865638	−0.432819	+	0.901481i	$0.642481\pi$
$420$	0	0
$421$	−2.37970e9	−1.55430	−0.777150	−	0.629315i	$-0.783336\pi$
$421$	−2.37970e9	−1.55430	−0.777150	+	0.629315i	$0.783336\pi$
$422$	3.44077e9	2.22875
$423$	0	0
$424$	−5.78158e9	−3.68355
$425$	−5.05581e8	−0.319470
$426$	0	0
$427$	−5.40948e8	−0.336247
$428$	1.44776e9	0.892574
$429$	0	0
$430$	−3.41508e9	−2.07139
$431$	−1.38543e9	−0.833515	−0.416758	−	0.909018i	$-0.636834\pi$
$431$	−1.38543e9	−0.833515	−0.416758	+	0.909018i	$0.636834\pi$
$432$	0	0
$433$	2.85632e9	1.69082	0.845412	−	0.534114i	$-0.179355\pi$
$433$	2.85632e9	1.69082	0.845412	+	0.534114i	$0.179355\pi$
$434$	−7.34364e7	−0.0431218
$435$	0	0
$436$	6.38274e8	0.368811
$437$	1.12620e8	0.0645550
$438$	0	0
$439$	−3.08827e9	−1.74217	−0.871083	−	0.491137i	$-0.836582\pi$
$439$	−3.08827e9	−1.74217	−0.871083	+	0.491137i	$0.836582\pi$
$440$	−2.62493e9	−1.46904
$441$	0	0
$442$	−3.90830e7	−0.0215283
$443$	−2.10701e9	−1.15147	−0.575737	−	0.817635i	$-0.695285\pi$
$443$	−2.10701e9	−1.15147	−0.575737	+	0.817635i	$0.695285\pi$
$444$	0	0
$445$	−1.26893e9	−0.682618
$446$	3.96332e9	2.11537
$447$	0	0
$448$	−2.91112e8	−0.152963
$449$	5.27169e8	0.274845	0.137422	−	0.990513i	$-0.456118\pi$
$449$	5.27169e8	0.274845	0.137422	+	0.990513i	$0.456118\pi$
$450$	0	0
$451$	2.28963e7	0.0117529
$452$	−3.41605e9	−1.73996
$453$	0	0
$454$	3.93697e9	1.97454
$455$	−7.76971e7	−0.0386692
$456$	0	0
$457$	1.47995e9	0.725337	0.362668	−	0.931918i	$-0.381866\pi$
$457$	1.47995e9	0.725337	0.362668	+	0.931918i	$0.381866\pi$
$458$	5.54181e9	2.69540
$459$	0	0
$460$	−4.46949e8	−0.214095
$461$	3.37266e9	1.60332	0.801658	−	0.597783i	$-0.203952\pi$
$461$	3.37266e9	1.60332	0.801658	+	0.597783i	$0.203952\pi$
$462$	0	0
$463$	−2.58669e9	−1.21119	−0.605593	−	0.795774i	$-0.707064\pi$
$463$	−2.58669e9	−1.21119	−0.605593	+	0.795774i	$0.707064\pi$
$464$	−2.82175e9	−1.31131
$465$	0	0
$466$	−6.57678e9	−3.01067
$467$	7.12728e8	0.323828	0.161914	−	0.986805i	$-0.448233\pi$
$467$	7.12728e8	0.323828	0.161914	+	0.986805i	$0.448233\pi$
$468$	0	0
$469$	−5.87077e8	−0.262778
$470$	5.65460e9	2.51223
$471$	0	0
$472$	−5.81840e8	−0.254687
$473$	6.19807e8	0.269304
$474$	0	0
$475$	6.11834e9	2.61942
$476$	−1.71214e8	−0.0727637
$477$	0	0
$478$	−7.02454e9	−2.94185
$479$	1.28719e9	0.535140	0.267570	−	0.963538i	$-0.413779\pi$
$479$	1.28719e9	0.535140	0.267570	+	0.963538i	$0.413779\pi$
$480$	0	0
$481$	−2.92410e8	−0.119807
$482$	−3.13473e9	−1.27508
$483$	0	0
$484$	−4.35582e9	−1.74627
$485$	−4.42069e9	−1.75952
$486$	0	0
$487$	−4.87502e8	−0.191260	−0.0956301	−	0.995417i	$-0.530487\pi$
$487$	−4.87502e8	−0.191260	−0.0956301	+	0.995417i	$0.530487\pi$
$488$	−6.87317e9	−2.67724
$489$	0	0
$490$	7.80950e9	2.99873
$491$	5.36881e8	0.204688	0.102344	−	0.994749i	$-0.467366\pi$
$491$	5.36881e8	0.204688	0.102344	+	0.994749i	$0.467366\pi$
$492$	0	0
$493$	3.65465e8	0.137367
$494$	4.72967e8	0.176517
$495$	0	0
$496$	−3.62505e8	−0.133392
$497$	−5.85028e8	−0.213761
$498$	0	0
$499$	−4.34284e8	−0.156467	−0.0782334	−	0.996935i	$-0.524928\pi$
$499$	−4.34284e8	−0.156467	−0.0782334	+	0.996935i	$0.524928\pi$
$500$	−1.36108e10	−4.86954
$501$	0	0
$502$	4.61998e9	1.62996
$503$	3.59458e9	1.25939	0.629696	−	0.776842i	$-0.283180\pi$
$503$	3.59458e9	1.25939	0.629696	+	0.776842i	$0.283180\pi$
$504$	0	0
$505$	−4.11920e8	−0.142329
$506$	1.19572e8	0.0410302
$507$	0	0
$508$	−7.96213e9	−2.69467
$509$	3.79038e9	1.27400	0.637002	−	0.770862i	$-0.280174\pi$
$509$	3.79038e9	1.27400	0.637002	+	0.770862i	$0.280174\pi$
$510$	0	0
$511$	7.17430e8	0.237852
$512$	−6.30604e9	−2.07640
$513$	0	0
$514$	−6.40855e9	−2.08156
$515$	6.78024e9	2.18736
$516$	0	0
$517$	−1.02626e9	−0.326619
$518$	−1.88825e9	−0.596905
$519$	0	0
$520$	−9.87202e8	−0.307889
$521$	−2.93658e9	−0.909724	−0.454862	−	0.890562i	$-0.650312\pi$
$521$	−2.93658e9	−0.909724	−0.454862	+	0.890562i	$0.650312\pi$
$522$	0	0
$523$	2.13933e8	0.0653917	0.0326959	−	0.999465i	$-0.489591\pi$
$523$	2.13933e8	0.0653917	0.0326959	+	0.999465i	$0.489591\pi$
$524$	−7.58011e9	−2.30153
$525$	0	0
$526$	−7.34806e9	−2.20152
$527$	4.69506e7	0.0139735
$528$	0	0
$529$	−3.39412e9	−0.996855
$530$	2.05958e10	6.00914
$531$	0	0
$532$	2.07196e9	0.596611
$533$	8.61100e6	0.00246325
$534$	0	0
$535$	−2.71244e9	−0.765813
$536$	−7.45927e9	−2.09228
$537$	0	0
$538$	4.33714e9	1.20079
$539$	−1.41736e9	−0.389868
$540$	0	0
$541$	2.06559e9	0.560860	0.280430	−	0.959874i	$-0.409523\pi$
$541$	2.06559e9	0.560860	0.280430	+	0.959874i	$0.409523\pi$
$542$	3.37743e9	0.911147
$543$	0	0
$544$	−2.14545e8	−0.0571378
$545$	−1.19583e9	−0.316433
$546$	0	0
$547$	2.12658e9	0.555555	0.277777	−	0.960645i	$-0.410402\pi$
$547$	2.12658e9	0.555555	0.277777	+	0.960645i	$0.410402\pi$
$548$	1.38463e10	3.59418
$549$	0	0
$550$	6.49604e9	1.66487
$551$	−4.42271e9	−1.12631
$552$	0	0
$553$	7.64609e8	0.192266
$554$	1.03533e8	0.0258698
$555$	0	0
$556$	1.04775e9	0.258522
$557$	−3.52559e9	−0.864448	−0.432224	−	0.901766i	$-0.642271\pi$
$557$	−3.52559e9	−0.864448	−0.432224	+	0.901766i	$0.642271\pi$
$558$	0	0
$559$	2.33101e8	0.0564422
$560$	−2.47671e9	−0.595960
$561$	0	0
$562$	9.82575e9	2.33501
$563$	−3.62920e9	−0.857100	−0.428550	−	0.903518i	$-0.640975\pi$
$563$	−3.62920e9	−0.857100	−0.428550	+	0.903518i	$0.640975\pi$
$564$	0	0
$565$	6.40011e9	1.49285
$566$	−1.16671e10	−2.70461
$567$	0	0
$568$	−7.43323e9	−1.70200
$569$	1.16789e9	0.265772	0.132886	−	0.991131i	$-0.457576\pi$
$569$	1.16789e9	0.265772	0.132886	+	0.991131i	$0.457576\pi$
$570$	0	0
$571$	4.02884e8	0.0905635	0.0452818	−	0.998974i	$-0.485581\pi$
$571$	4.02884e8	0.0905635	0.0452818	+	0.998974i	$0.485581\pi$
$572$	3.40667e8	0.0761103
$573$	0	0
$574$	5.56059e7	0.0122724
$575$	5.81730e8	0.127610
$576$	0	0
$577$	3.29893e9	0.714921	0.357461	−	0.933928i	$-0.383643\pi$
$577$	3.29893e9	0.714921	0.357461	+	0.933928i	$0.383643\pi$
$578$	−8.02493e9	−1.72860
$579$	0	0
$580$	1.75522e10	3.73537
$581$	−2.68377e8	−0.0567713
$582$	0	0
$583$	−3.73795e9	−0.781256
$584$	9.11551e9	1.89381
$585$	0	0
$586$	−1.38457e10	−2.84233
$587$	−4.64184e9	−0.947233	−0.473617	−	0.880731i	$-0.657052\pi$
$587$	−4.64184e9	−0.947233	−0.473617	+	0.880731i	$0.657052\pi$
$588$	0	0
$589$	−5.68177e8	−0.114572
$590$	2.07269e9	0.415483
$591$	0	0
$592$	−9.32099e9	−1.84644
$593$	4.01516e7	0.00790699	0.00395349	−	0.999992i	$-0.498742\pi$
$593$	4.01516e7	0.00790699	0.00395349	+	0.999992i	$0.498742\pi$
$594$	0	0
$595$	3.20776e8	0.0624299
$596$	9.15281e8	0.177089
$597$	0	0
$598$	4.49696e7	0.00859934
$599$	−6.08823e9	−1.15744	−0.578719	−	0.815527i	$-0.696447\pi$
$599$	−6.08823e9	−1.15744	−0.578719	+	0.815527i	$0.696447\pi$
$600$	0	0
$601$	8.96979e9	1.68547	0.842736	−	0.538327i	$-0.180944\pi$
$601$	8.96979e9	1.68547	0.842736	+	0.538327i	$0.180944\pi$
$602$	1.50526e9	0.281206
$603$	0	0
$604$	1.34999e10	2.49289
$605$	8.16080e9	1.49827
$606$	0	0
$607$	2.86102e9	0.519230	0.259615	−	0.965712i	$-0.416404\pi$
$607$	2.86102e9	0.519230	0.259615	+	0.965712i	$0.416404\pi$
$608$	2.59634e9	0.468489
$609$	0	0
$610$	2.44843e10	4.36751
$611$	−3.85964e8	−0.0684545
$612$	0	0
$613$	−2.84576e9	−0.498984	−0.249492	−	0.968377i	$-0.580264\pi$
$613$	−2.84576e9	−0.498984	−0.249492	+	0.968377i	$0.580264\pi$
$614$	1.26467e10	2.20489
$615$	0	0
$616$	1.15699e9	0.199433
$617$	9.53360e9	1.63403	0.817013	−	0.576619i	$-0.195628\pi$
$617$	9.53360e9	1.63403	0.817013	+	0.576619i	$0.195628\pi$
$618$	0	0
$619$	4.09983e9	0.694783	0.347391	−	0.937720i	$-0.387068\pi$
$619$	4.09983e9	0.694783	0.347391	+	0.937720i	$0.387068\pi$
$620$	2.25490e9	0.379976
$621$	0	0
$622$	5.77756e9	0.962672
$623$	5.59306e8	0.0926704
$624$	0	0
$625$	1.16117e10	1.90246
$626$	1.17836e10	1.91985
$627$	0	0
$628$	−1.28645e10	−2.07269
$629$	1.20723e9	0.193425
$630$	0	0
$631$	−1.39182e9	−0.220536	−0.110268	−	0.993902i	$-0.535171\pi$
$631$	−1.39182e9	−0.220536	−0.110268	+	0.993902i	$0.535171\pi$
$632$	9.71496e9	1.53085
$633$	0	0
$634$	1.93474e10	3.01516
$635$	1.49174e10	2.31198
$636$	0	0
$637$	−5.33049e8	−0.0817108
$638$	−4.69574e9	−0.715866
$639$	0	0
$640$	1.80610e10	2.72341
$641$	5.13220e9	0.769663	0.384832	−	0.922987i	$-0.374260\pi$
$641$	5.13220e9	0.769663	0.384832	+	0.922987i	$0.374260\pi$
$642$	0	0
$643$	−3.56886e9	−0.529408	−0.264704	−	0.964330i	$-0.585274\pi$
$643$	−3.56886e9	−0.529408	−0.264704	+	0.964330i	$0.585274\pi$
$644$	1.97002e8	0.0290649
$645$	0	0
$646$	−1.95267e9	−0.284980
$647$	7.81292e9	1.13409	0.567046	−	0.823686i	$-0.308086\pi$
$647$	7.81292e9	1.13409	0.567046	+	0.823686i	$0.308086\pi$
$648$	0	0
$649$	−3.76176e8	−0.0540175
$650$	2.44308e9	0.348932
$651$	0	0
$652$	2.06304e10	2.91501
$653$	8.76103e9	1.23129	0.615643	−	0.788025i	$-0.288896\pi$
$653$	8.76103e9	1.23129	0.615643	+	0.788025i	$0.288896\pi$
$654$	0	0
$655$	1.42016e10	1.97467
$656$	2.74488e8	0.0379630
$657$	0	0
$658$	−2.49238e9	−0.341054
$659$	1.26819e10	1.72618	0.863089	−	0.505052i	$-0.168527\pi$
$659$	1.26819e10	1.72618	0.863089	+	0.505052i	$0.168527\pi$
$660$	0	0
$661$	−1.96990e9	−0.265301	−0.132651	−	0.991163i	$-0.542349\pi$
$661$	−1.96990e9	−0.265301	−0.132651	+	0.991163i	$0.542349\pi$
$662$	−2.45867e10	−3.29380
$663$	0	0
$664$	−3.40994e9	−0.452021
$665$	−3.88190e9	−0.511881
$666$	0	0
$667$	−4.20511e8	−0.0548702
$668$	1.86699e9	0.242339
$669$	0	0
$670$	2.65722e10	3.41323
$671$	−4.44369e9	−0.567825
$672$	0	0
$673$	8.38158e9	1.05992	0.529960	−	0.848023i	$-0.322207\pi$
$673$	8.38158e9	1.05992	0.529960	+	0.848023i	$0.322207\pi$
$674$	7.60605e9	0.956862
$675$	0	0
$676$	−1.68143e10	−2.09346
$677$	−8.10465e9	−1.00386	−0.501931	−	0.864908i	$-0.667377\pi$
$677$	−8.10465e9	−1.00386	−0.501931	+	0.864908i	$0.667377\pi$
$678$	0	0
$679$	1.94851e9	0.238868
$680$	4.07571e9	0.497075
$681$	0	0
$682$	−6.03253e8	−0.0728206
$683$	−8.52346e9	−1.02363	−0.511816	−	0.859095i	$-0.671027\pi$
$683$	−8.52346e9	−1.02363	−0.511816	+	0.859095i	$0.671027\pi$
$684$	0	0
$685$	−2.59415e10	−3.08374
$686$	−7.10553e9	−0.840353
$687$	0	0
$688$	7.43046e9	0.869873
$689$	−1.40580e9	−0.163740
$690$	0	0
$691$	−6.80331e9	−0.784417	−0.392208	−	0.919876i	$-0.628289\pi$
$691$	−6.80331e9	−0.784417	−0.392208	+	0.919876i	$0.628289\pi$
$692$	−2.57688e10	−2.95612
$693$	0	0
$694$	4.45001e8	0.0505362
$695$	−1.96301e9	−0.221807
$696$	0	0
$697$	−3.55509e7	−0.00397682
$698$	−2.43902e10	−2.71470
$699$	0	0
$700$	1.07026e10	1.17936
$701$	1.06363e9	0.116621	0.0583106	−	0.998298i	$-0.481429\pi$
$701$	1.06363e9	0.116621	0.0583106	+	0.998298i	$0.481429\pi$
$702$	0	0
$703$	−1.46094e10	−1.58595
$704$	−2.39138e9	−0.258311
$705$	0	0
$706$	3.07182e10	3.28533
$707$	1.81562e8	0.0193222
$708$	0	0
$709$	1.77993e10	1.87561	0.937803	−	0.347168i	$-0.112857\pi$
$709$	1.77993e10	1.87561	0.937803	+	0.347168i	$0.112857\pi$
$710$	2.64794e10	2.77655
$711$	0	0
$712$	7.10641e9	0.737854
$713$	−5.40222e7	−0.00558160
$714$	0	0
$715$	−6.38253e8	−0.0653013
$716$	−8.81013e9	−0.896988
$717$	0	0
$718$	1.26888e9	0.127934
$719$	1.95303e10	1.95956	0.979780	−	0.200078i	$-0.0641197\pi$
$719$	1.95303e10	1.95956	0.979780	+	0.200078i	$0.0641197\pi$
$720$	0	0
$721$	−2.98852e9	−0.296950
$722$	5.79761e9	0.573283
$723$	0	0
$724$	−2.36009e10	−2.31123
$725$	−2.28452e10	−2.22645
$726$	0	0
$727$	−1.51589e10	−1.46317	−0.731587	−	0.681748i	$-0.761220\pi$
$727$	−1.51589e10	−1.46317	−0.731587	+	0.681748i	$0.761220\pi$
$728$	4.35129e8	0.0417982
$729$	0	0
$730$	−3.24723e10	−3.08946
$731$	−9.62371e8	−0.0911238
$732$	0	0
$733$	1.19810e7	0.00112365	0.000561823	−	1.00000i	$-0.499821\pi$
$733$	1.19810e7	0.00112365	0.000561823		1.00000i	$0.499821\pi$
$734$	4.86524e9	0.454118
$735$	0	0
$736$	2.46860e8	0.0228233
$737$	−4.82262e9	−0.443758
$738$	0	0
$739$	−1.13380e10	−1.03343	−0.516715	−	0.856158i	$-0.672845\pi$
$739$	−1.13380e10	−1.03343	−0.516715	+	0.856158i	$0.672845\pi$
$740$	5.79796e10	5.25974
$741$	0	0
$742$	−9.07799e9	−0.815785
$743$	2.12227e10	1.89819	0.949096	−	0.314987i	$-0.102000\pi$
$743$	2.12227e10	1.89819	0.949096	+	0.314987i	$0.102000\pi$
$744$	0	0
$745$	−1.71482e9	−0.151939
$746$	−1.06969e10	−0.943344
$747$	0	0
$748$	−1.40646e9	−0.122877
$749$	1.19556e9	0.103965
$750$	0	0
$751$	−2.06229e10	−1.77669	−0.888343	−	0.459181i	$-0.848143\pi$
$751$	−2.06229e10	−1.77669	−0.888343	+	0.459181i	$0.848143\pi$
$752$	−1.23032e10	−1.05500
$753$	0	0
$754$	−1.76601e9	−0.150035
$755$	−2.52927e10	−2.13885
$756$	0	0
$757$	−2.12910e10	−1.78386	−0.891929	−	0.452175i	$-0.850648\pi$
$757$	−2.12910e10	−1.78386	−0.891929	+	0.452175i	$0.850648\pi$
$758$	−1.88217e10	−1.56970
$759$	0	0
$760$	−4.93226e10	−4.07567
$761$	4.78297e9	0.393415	0.196708	−	0.980462i	$-0.436975\pi$
$761$	4.78297e9	0.393415	0.196708	+	0.980462i	$0.436975\pi$
$762$	0	0
$763$	5.27086e8	0.0429582
$764$	−7.08633e9	−0.574903
$765$	0	0
$766$	−1.49111e10	−1.19869
$767$	−1.41475e8	−0.0113213
$768$	0	0
$769$	−7.38239e9	−0.585403	−0.292702	−	0.956204i	$-0.594554\pi$
$769$	−7.38239e9	−0.585403	−0.292702	+	0.956204i	$0.594554\pi$
$770$	−4.12155e9	−0.325344
$771$	0	0
$772$	8.11673e9	0.634922
$773$	1.10993e10	0.864305	0.432153	−	0.901801i	$-0.357754\pi$
$773$	1.10993e10	0.864305	0.432153	+	0.901801i	$0.357754\pi$
$774$	0	0
$775$	−2.93488e9	−0.226482
$776$	2.47573e10	1.90190
$777$	0	0
$778$	−1.01213e10	−0.770566
$779$	4.30223e8	0.0326071
$780$	0	0
$781$	−4.80579e9	−0.360982
$782$	−1.85659e8	−0.0138833
$783$	0	0
$784$	−1.69917e10	−1.25931
$785$	2.41022e10	1.77833
$786$	0	0
$787$	2.48619e10	1.81812	0.909060	−	0.416666i	$-0.136801\pi$
$787$	2.48619e10	1.81812	0.909060	+	0.416666i	$0.136801\pi$
$788$	1.71733e10	1.25030
$789$	0	0
$790$	−3.46077e10	−2.49734
$791$	−2.82097e9	−0.202666
$792$	0	0
$793$	−1.67122e9	−0.119008
$794$	6.59229e9	0.467374
$795$	0	0
$796$	3.63149e10	2.55205
$797$	9.78058e9	0.684322	0.342161	−	0.939641i	$-0.388841\pi$
$797$	9.78058e9	0.684322	0.342161	+	0.939641i	$0.388841\pi$
$798$	0	0
$799$	1.59347e9	0.110517
$800$	1.34112e10	0.926091
$801$	0	0
$802$	1.74942e10	1.19752
$803$	5.89343e9	0.401665
$804$	0	0
$805$	−3.69091e8	−0.0249372
$806$	−2.26876e8	−0.0152621
$807$	0	0
$808$	2.30688e9	0.153846
$809$	4.39665e9	0.291945	0.145973	−	0.989289i	$-0.453369\pi$
$809$	4.39665e9	0.291945	0.145973	+	0.989289i	$0.453369\pi$
$810$	0	0
$811$	−9.64279e9	−0.634790	−0.317395	−	0.948293i	$-0.602808\pi$
$811$	−9.64279e9	−0.634790	−0.317395	+	0.948293i	$0.602808\pi$
$812$	−7.73648e9	−0.507104
$813$	0	0
$814$	−1.55113e10	−1.00800
$815$	−3.86518e10	−2.50103
$816$	0	0
$817$	1.16462e10	0.747150
$818$	2.40133e10	1.53397
$819$	0	0
$820$	−1.70741e9	−0.108140
$821$	1.47265e10	0.928750	0.464375	−	0.885639i	$-0.346279\pi$
$821$	1.47265e10	0.928750	0.464375	+	0.885639i	$0.346279\pi$
$822$	0	0
$823$	−1.30056e10	−0.813264	−0.406632	−	0.913592i	$-0.633297\pi$
$823$	−1.30056e10	−0.813264	−0.406632	+	0.913592i	$0.633297\pi$
$824$	−3.79715e10	−2.36435
$825$	0	0
$826$	−9.13580e8	−0.0564049
$827$	−2.38555e10	−1.46662	−0.733312	−	0.679892i	$-0.762026\pi$
$827$	−2.38555e10	−1.46662	−0.733312	+	0.679892i	$0.762026\pi$
$828$	0	0
$829$	2.02291e10	1.23321	0.616605	−	0.787273i	$-0.288508\pi$
$829$	2.02291e10	1.23321	0.616605	+	0.787273i	$0.288508\pi$
$830$	1.21473e10	0.737403
$831$	0	0
$832$	−8.99366e8	−0.0541383
$833$	2.20072e9	0.131919
$834$	0	0
$835$	−3.49787e9	−0.207922
$836$	1.70204e10	1.00751
$837$	0	0
$838$	−2.60034e10	−1.52643
$839$	1.07547e10	0.628684	0.314342	−	0.949310i	$-0.398216\pi$
$839$	1.07547e10	0.628684	0.314342	+	0.949310i	$0.398216\pi$
$840$	0	0
$841$	−7.35941e8	−0.0426635
$842$	−4.74752e10	−2.74078
$843$	0	0
$844$	4.65675e10	2.66615
$845$	3.15023e10	1.79615
$846$	0	0
$847$	−3.59703e9	−0.203401
$848$	−4.48118e10	−2.52352
$849$	0	0
$850$	−1.00864e10	−0.563338
$851$	−1.38906e9	−0.0772621
$852$	0	0
$853$	3.60150e10	1.98683	0.993417	−	0.114556i	$-0.0365445\pi$
$853$	3.60150e10	1.98683	0.993417	+	0.114556i	$0.0365445\pi$
$854$	−1.07919e10	−0.592922
$855$	0	0
$856$	1.51906e10	0.827781
$857$	7.43401e9	0.403450	0.201725	−	0.979442i	$-0.435345\pi$
$857$	7.43401e9	0.403450	0.201725	+	0.979442i	$0.435345\pi$
$858$	0	0
$859$	−2.19350e10	−1.18076	−0.590380	−	0.807125i	$-0.701022\pi$
$859$	−2.19350e10	−1.18076	−0.590380	+	0.807125i	$0.701022\pi$
$860$	−4.62198e10	−2.47790
$861$	0	0
$862$	−2.76394e10	−1.46978
$863$	−2.36338e10	−1.25169	−0.625843	−	0.779949i	$-0.715245\pi$
$863$	−2.36338e10	−1.25169	−0.625843	+	0.779949i	$0.715245\pi$
$864$	0	0
$865$	4.82788e10	2.53630
$866$	5.69837e10	2.98152
$867$	0	0
$868$	−9.93891e8	−0.0515845
$869$	6.28099e9	0.324682
$870$	0	0
$871$	−1.81373e9	−0.0930054
$872$	6.69704e9	0.342039
$873$	0	0
$874$	2.24677e9	0.113833
$875$	−1.12398e10	−0.567191
$876$	0	0
$877$	−6.10352e9	−0.305550	−0.152775	−	0.988261i	$-0.548821\pi$
$877$	−6.10352e9	−0.305550	−0.152775	+	0.988261i	$0.548821\pi$
$878$	−6.16112e10	−3.07205
$879$	0	0
$880$	−2.03453e10	−1.00641
$881$	−7.45828e9	−0.367471	−0.183735	−	0.982976i	$-0.558819\pi$
$881$	−7.45828e9	−0.367471	−0.183735	+	0.982976i	$0.558819\pi$
$882$	0	0
$883$	2.44182e10	1.19358	0.596789	−	0.802398i	$-0.296443\pi$
$883$	2.44182e10	1.19358	0.596789	+	0.802398i	$0.296443\pi$
$884$	−5.28952e8	−0.0257533
$885$	0	0
$886$	−4.20350e10	−2.03046
$887$	1.25593e10	0.604272	0.302136	−	0.953265i	$-0.402300\pi$
$887$	1.25593e10	0.604272	0.302136	+	0.953265i	$0.402300\pi$
$888$	0	0
$889$	−6.57512e9	−0.313869
$890$	−2.53152e10	−1.20370
$891$	0	0
$892$	5.36397e10	2.53052
$893$	−1.92835e10	−0.906163
$894$	0	0
$895$	1.65061e10	0.769600
$896$	−7.96075e9	−0.369723
$897$	0	0
$898$	1.05170e10	0.484648
$899$	2.12151e9	0.0973838
$900$	0	0
$901$	5.80390e9	0.264352
$902$	4.56782e8	0.0207246
$903$	0	0
$904$	−3.58426e10	−1.61365
$905$	4.42172e10	1.98299
$906$	0	0
$907$	−1.35865e10	−0.604618	−0.302309	−	0.953210i	$-0.597758\pi$
$907$	−1.35865e10	−0.604618	−0.302309	+	0.953210i	$0.597758\pi$
$908$	5.32832e10	2.36205
$909$	0	0
$910$	−1.55006e9	−0.0681874
$911$	−2.70365e9	−0.118478	−0.0592389	−	0.998244i	$-0.518867\pi$
$911$	−2.70365e9	−0.118478	−0.0592389	+	0.998244i	$0.518867\pi$
$912$	0	0
$913$	−2.20462e9	−0.0958707
$914$	2.95251e10	1.27903
$915$	0	0
$916$	7.50031e10	3.22437
$917$	−6.25965e9	−0.268076
$918$	0	0
$919$	−1.61178e10	−0.685019	−0.342509	−	0.939514i	$-0.611277\pi$
$919$	−1.61178e10	−0.685019	−0.342509	+	0.939514i	$0.611277\pi$
$920$	−4.68958e9	−0.198553
$921$	0	0
$922$	6.72848e10	2.82721
$923$	−1.80740e9	−0.0756567
$924$	0	0
$925$	−7.54637e10	−3.13503
$926$	−5.16046e10	−2.13575
$927$	0	0
$928$	−9.69446e9	−0.398204
$929$	4.59174e10	1.87898	0.939491	−	0.342573i	$-0.111298\pi$
$929$	4.59174e10	1.87898	0.939491	+	0.342573i	$0.111298\pi$
$930$	0	0
$931$	−2.66322e10	−1.08164
$932$	−8.90104e10	−3.60152
$933$	0	0
$934$	1.42190e10	0.571023
$935$	2.63506e9	0.105426
$936$	0	0
$937$	−2.98124e10	−1.18388	−0.591942	−	0.805981i	$-0.701638\pi$
$937$	−2.98124e10	−1.18388	−0.591942	+	0.805981i	$0.701638\pi$
$938$	−1.17122e10	−0.463371
$939$	0	0
$940$	7.65297e10	3.00526
$941$	3.77260e10	1.47597	0.737983	−	0.674819i	$-0.235778\pi$
$941$	3.77260e10	1.47597	0.737983	+	0.674819i	$0.235778\pi$
$942$	0	0
$943$	4.09055e7	0.00158851
$944$	−4.50972e9	−0.174481
$945$	0	0
$946$	1.23652e10	0.474877
$947$	−2.08865e10	−0.799173	−0.399587	−	0.916695i	$-0.630846\pi$
$947$	−2.08865e10	−0.799173	−0.399587	+	0.916695i	$0.630846\pi$
$948$	0	0
$949$	2.21644e9	0.0841831
$950$	1.22061e11	4.61897
$951$	0	0
$952$	−1.79645e9	−0.0674817
$953$	−3.06639e9	−0.114763	−0.0573815	−	0.998352i	$-0.518275\pi$
$953$	−3.06639e9	−0.114763	−0.0573815	+	0.998352i	$0.518275\pi$
$954$	0	0
$955$	1.32765e10	0.493256
$956$	−9.50704e10	−3.51919
$957$	0	0
$958$	2.56795e10	0.943641
$959$	1.14342e10	0.418641
$960$	0	0
$961$	−2.72401e10	−0.990094
$962$	−5.83359e9	−0.211263
$963$	0	0
$964$	−4.24256e10	−1.52531
$965$	−1.52070e10	−0.544752
$966$	0	0
$967$	3.45519e10	1.22880	0.614398	−	0.788997i	$-0.289399\pi$
$967$	3.45519e10	1.22880	0.614398	+	0.788997i	$0.289399\pi$
$968$	−4.57031e10	−1.61950
$969$	0	0
$970$	−8.81931e10	−3.10266
$971$	−3.51892e10	−1.23351	−0.616754	−	0.787156i	$-0.711553\pi$
$971$	−3.51892e10	−1.23351	−0.616754	+	0.787156i	$0.711553\pi$
$972$	0	0
$973$	8.65235e8	0.0301120
$974$	−9.72569e9	−0.337259
$975$	0	0
$976$	−5.32725e10	−1.83412
$977$	6.56102e9	0.225082	0.112541	−	0.993647i	$-0.464101\pi$
$977$	6.56102e9	0.225082	0.112541	+	0.993647i	$0.464101\pi$
$978$	0	0
$979$	4.59449e9	0.156494
$980$	1.05694e11	3.58723
$981$	0	0
$982$	1.07108e10	0.360938
$983$	1.57970e9	0.0530442	0.0265221	−	0.999648i	$-0.491557\pi$
$983$	1.57970e9	0.0530442	0.0265221	+	0.999648i	$0.491557\pi$
$984$	0	0
$985$	−3.21749e10	−1.07273
$986$	7.29105e9	0.242226
$987$	0	0
$988$	6.40116e9	0.211159
$989$	1.10732e9	0.0363987
$990$	0	0
$991$	−5.75725e10	−1.87913	−0.939566	−	0.342369i	$-0.888771\pi$
$991$	−5.75725e10	−1.87913	−0.939566	+	0.342369i	$0.888771\pi$
$992$	−1.24543e9	−0.0405068
$993$	0	0
$994$	−1.16713e10	−0.376937
$995$	−6.80374e10	−2.18961
$996$	0	0
$997$	−3.86631e10	−1.23556	−0.617780	−	0.786351i	$-0.711968\pi$
$997$	−3.86631e10	−1.23556	−0.617780	+	0.786351i	$0.711968\pi$
$998$	−8.66400e9	−0.275906
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.8.a.e.1.18		18
3.2	odd	2		177.8.a.d.1.1	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.d.1.1	✓	18	3.2	odd	2
531.8.a.e.1.18		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)

\(f(q)\)	\(=\)	\(q+19.9501 q^{2} +270.005 q^{4} -505.865 q^{5} +222.970 q^{7} +2833.01 q^{8} +O(q^{10})\)	\(q+19.9501 q^{2} +270.005 q^{4} -505.865 q^{5} +222.970 q^{7} +2833.01 q^{8} -10092.0 q^{10} +1831.62 q^{11} +688.848 q^{13} +4448.27 q^{14} +21958.1 q^{16} -2843.94 q^{17} +34416.2 q^{19} -136586. q^{20} +36540.9 q^{22} +3272.29 q^{23} +177775. q^{25} +13742.6 q^{26} +60203.0 q^{28} -128507. q^{29} -16509.0 q^{31} +75439.5 q^{32} -56736.8 q^{34} -112793. q^{35} -424491. q^{37} +686606. q^{38} -1.43312e6 q^{40} +12500.6 q^{41} +338393. q^{43} +494546. q^{44} +65282.4 q^{46} -560303. q^{47} -773827. q^{49} +3.54662e6 q^{50} +185992. q^{52} -2.04079e6 q^{53} -926551. q^{55} +631676. q^{56} -2.56371e6 q^{58} -205379. q^{59} -2.42610e6 q^{61} -329355. q^{62} -1.30561e6 q^{64} -348464. q^{65} -2.63298e6 q^{67} -767879. q^{68} -2.25022e6 q^{70} -2.62379e6 q^{71} +3.21761e6 q^{73} -8.46862e6 q^{74} +9.29256e6 q^{76} +408396. q^{77} +3.42920e6 q^{79} -1.11078e7 q^{80} +249387. q^{82} -1.20365e6 q^{83} +1.43865e6 q^{85} +6.75097e6 q^{86} +5.18899e6 q^{88} +2.50843e6 q^{89} +153592. q^{91} +883534. q^{92} -1.11781e7 q^{94} -1.74100e7 q^{95} +8.73888e6 q^{97} -1.54379e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8}+O(q^{10}) \) 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

Modular forms

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Groups

Database

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