LMFDB - Embedded newform 207.8.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,8,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, 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("207.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	207.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:	$64.6637002752$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 466x^{4} + 540x^{3} + 48973x^{2} - 77282x - 1061812 $ x^6 - 2x^5 - 466x^4 + 540x^3 + 48973x^2 - 77282*x - 1061812
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.35108$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.35108 q^{2} -116.770 q^{4} -0.849066 q^{5} +405.736 q^{7} +820.245 q^{8} +O(q^{10})$	\(q-3.35108 q^{2} -116.770 q^{4} -0.849066 q^{5} +405.736 q^{7} +820.245 q^{8} +2.84529 q^{10} -2663.62 q^{11} -9506.50 q^{13} -1359.65 q^{14} +12197.9 q^{16} +36191.1 q^{17} -35806.9 q^{19} +99.1457 q^{20} +8925.99 q^{22} +12167.0 q^{23} -78124.3 q^{25} +31857.0 q^{26} -47377.9 q^{28} -128430. q^{29} -32645.9 q^{31} -145867. q^{32} -121279. q^{34} -344.497 q^{35} -583405. q^{37} +119992. q^{38} -696.442 q^{40} +756639. q^{41} +769827. q^{43} +311031. q^{44} -40772.6 q^{46} +84751.0 q^{47} -658922. q^{49} +261801. q^{50} +1.11008e6 q^{52} +52581.8 q^{53} +2261.59 q^{55} +332803. q^{56} +430380. q^{58} -2.26778e6 q^{59} +1.62312e6 q^{61} +109399. q^{62} -1.07252e6 q^{64} +8071.65 q^{65} +2.11923e6 q^{67} -4.22604e6 q^{68} +1154.44 q^{70} +4.91652e6 q^{71} +3.45746e6 q^{73} +1.95504e6 q^{74} +4.18118e6 q^{76} -1.08072e6 q^{77} +2.91026e6 q^{79} -10356.8 q^{80} -2.53556e6 q^{82} -1.88250e6 q^{83} -30728.6 q^{85} -2.57975e6 q^{86} -2.18482e6 q^{88} +234356. q^{89} -3.85713e6 q^{91} -1.42074e6 q^{92} -284007. q^{94} +30402.4 q^{95} +1.52882e7 q^{97} +2.20810e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{2} + 178 q^{4} + 372 q^{5} - 1104 q^{7} + 1956 q^{8}+O(q^{10}) $ 6 * q + 8 * q^2 + 178 * q^4 + 372 * q^5 - 1104 * q^7 + 1956 * q^8	\( 6 q + 8 q^{2} + 178 q^{4} + 372 q^{5} - 1104 q^{7} + 1956 q^{8} - 13042 q^{10} + 14824 q^{11} - 756 q^{13} + 3926 q^{14} - 13022 q^{16} + 69484 q^{17} - 43864 q^{19} - 78886 q^{20} + 98204 q^{22} + 73002 q^{23} + 228018 q^{25} + 311956 q^{26} - 545442 q^{28} + 311100 q^{29} - 245248 q^{31} + 390156 q^{32} + 235834 q^{34} + 1331256 q^{35} - 630044 q^{37} - 80910 q^{38} - 2153982 q^{40} + 969204 q^{41} - 1770208 q^{43} + 1749140 q^{44} + 97336 q^{46} + 1400024 q^{47} + 1985598 q^{49} + 956660 q^{50} + 3217272 q^{52} + 1573516 q^{53} - 431296 q^{55} - 7740702 q^{56} + 5987188 q^{58} + 1410320 q^{59} - 942172 q^{61} - 3334412 q^{62} + 1996866 q^{64} + 420944 q^{65} - 452072 q^{67} + 9258254 q^{68} + 21981136 q^{70} - 122928 q^{71} + 16490716 q^{73} + 600104 q^{74} + 7428658 q^{76} - 7239696 q^{77} + 2458408 q^{79} - 19440230 q^{80} + 20510784 q^{82} + 7566456 q^{83} + 5817744 q^{85} + 669666 q^{86} + 14775668 q^{88} + 20368036 q^{89} + 8815576 q^{91} + 2165726 q^{92} + 16952576 q^{94} - 5143832 q^{95} + 12586972 q^{97} + 39164812 q^{98}+O(q^{100}) \) 6 * q + 8 * q^2 + 178 * q^4 + 372 * q^5 - 1104 * q^7 + 1956 * q^8 - 13042 * q^10 + 14824 * q^11 - 756 * q^13 + 3926 * q^14 - 13022 * q^16 + 69484 * q^17 - 43864 * q^19 - 78886 * q^20 + 98204 * q^22 + 73002 * q^23 + 228018 * q^25 + 311956 * q^26 - 545442 * q^28 + 311100 * q^29 - 245248 * q^31 + 390156 * q^32 + 235834 * q^34 + 1331256 * q^35 - 630044 * q^37 - 80910 * q^38 - 2153982 * q^40 + 969204 * q^41 - 1770208 * q^43 + 1749140 * q^44 + 97336 * q^46 + 1400024 * q^47 + 1985598 * q^49 + 956660 * q^50 + 3217272 * q^52 + 1573516 * q^53 - 431296 * q^55 - 7740702 * q^56 + 5987188 * q^58 + 1410320 * q^59 - 942172 * q^61 - 3334412 * q^62 + 1996866 * q^64 + 420944 * q^65 - 452072 * q^67 + 9258254 * q^68 + 21981136 * q^70 - 122928 * q^71 + 16490716 * q^73 + 600104 * q^74 + 7428658 * q^76 - 7239696 * q^77 + 2458408 * q^79 - 19440230 * q^80 + 20510784 * q^82 + 7566456 * q^83 + 5817744 * q^85 + 669666 * q^86 + 14775668 * q^88 + 20368036 * q^89 + 8815576 * q^91 + 2165726 * q^92 + 16952576 * q^94 - 5143832 * q^95 + 12586972 * q^97 + 39164812 * 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 $

$2$	−3.35108	−0.296196	−0.148098	−	0.988973i	$-0.547315\pi$
$2$	−3.35108	−0.296196	−0.148098	+	0.988973i	$0.547315\pi$
$3$	0	0
$3$	0	0
$4$	−116.770	−0.912268
$5$	−0.849066	−0.00303771	−0.00151886	−	0.999999i	$-0.500483\pi$
$5$	−0.849066	−0.00303771	−0.00151886	+	0.999999i	$0.500483\pi$
$6$	0	0
$7$	405.736	0.447095	0.223548	−	0.974693i	$-0.428236\pi$
$7$	405.736	0.447095	0.223548	+	0.974693i	$0.428236\pi$
$8$	820.245	0.566407
$9$	0	0
$10$	2.84529	0.000899759 0
$11$	−2663.62	−0.603389	−0.301695	−	0.953405i	$-0.597552\pi$
$11$	−2663.62	−0.603389	−0.301695	+	0.953405i	$0.597552\pi$
$12$	0	0
$13$	−9506.50	−1.20010	−0.600052	−	0.799961i	$-0.704854\pi$
$13$	−9506.50	−1.20010	−0.600052	+	0.799961i	$0.704854\pi$
$14$	−1359.65	−0.132428
$15$	0	0
$16$	12197.9	0.744500
$17$	36191.1	1.78661	0.893306	−	0.449448i	$-0.148379\pi$
$17$	36191.1	1.78661	0.893306	+	0.449448i	$0.148379\pi$
$18$	0	0
$19$	−35806.9	−1.19765	−0.598824	−	0.800881i	$-0.704365\pi$
$19$	−35806.9	−1.19765	−0.598824	+	0.800881i	$0.704365\pi$
$20$	99.1457	0.00277121
$21$	0	0
$22$	8925.99	0.178722
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	0	0
$25$	−78124.3	−0.999991
$26$	31857.0	0.355467
$27$	0	0
$28$	−47377.9	−0.407871
$29$	−128430.	−0.977855	−0.488927	−	0.872325i	$-0.662612\pi$
$29$	−128430.	−0.977855	−0.488927	+	0.872325i	$0.662612\pi$
$30$	0	0
$31$	−32645.9	−0.196817	−0.0984085	−	0.995146i	$-0.531375\pi$
$31$	−32645.9	−0.196817	−0.0984085	+	0.995146i	$0.531375\pi$
$32$	−145867.	−0.786925
$33$	0	0
$34$	−121279.	−0.529188
$35$	−344.497	−0.00135815
$36$	0	0
$37$	−583405.	−1.89349	−0.946747	−	0.321978i	$-0.895652\pi$
$37$	−583405.	−1.89349	−0.946747	+	0.321978i	$0.895652\pi$
$38$	119992.	0.354739
$39$	0	0
$40$	−696.442	−0.00172058
$41$	756639.	1.71453	0.857266	−	0.514874i	$-0.172161\pi$
$41$	756639.	1.71453	0.857266	+	0.514874i	$0.172161\pi$
$42$	0	0
$43$	769827.	1.47657	0.738284	−	0.674490i	$-0.235636\pi$
$43$	769827.	1.47657	0.738284	+	0.674490i	$0.235636\pi$
$44$	311031.	0.550452
$45$	0	0
$46$	−40772.6	−0.0617612
$47$	84751.0	0.119070	0.0595350	−	0.998226i	$-0.481038\pi$
$47$	84751.0	0.119070	0.0595350	+	0.998226i	$0.481038\pi$
$48$	0	0
$49$	−658922.	−0.800106
$50$	261801.	0.296194
$51$	0	0
$52$	1.11008e6	1.09482
$53$	52581.8	0.0485143	0.0242572	−	0.999706i	$-0.492278\pi$
$53$	52581.8	0.0485143	0.0242572	+	0.999706i	$0.492278\pi$
$54$	0	0
$55$	2261.59	0.00183292
$56$	332803.	0.253238
$57$	0	0
$58$	430380.	0.289637
$59$	−2.26778e6	−1.43753	−0.718767	−	0.695251i	$-0.755293\pi$
$59$	−2.26778e6	−1.43753	−0.718767	+	0.695251i	$0.755293\pi$
$60$	0	0
$61$	1.62312e6	0.915578	0.457789	−	0.889061i	$-0.348642\pi$
$61$	1.62312e6	0.915578	0.457789	+	0.889061i	$0.348642\pi$
$62$	109399.	0.0582965
$63$	0	0
$64$	−1.07252e6	−0.511416
$65$	8071.65	0.00364557
$66$	0	0
$67$	2.11923e6	0.860827	0.430414	−	0.902632i	$-0.358368\pi$
$67$	2.11923e6	0.860827	0.430414	+	0.902632i	$0.358368\pi$
$68$	−4.22604e6	−1.62987
$69$	0	0
$70$	1154.44	0.000402278 0
$71$	4.91652e6	1.63025	0.815124	−	0.579286i	$-0.196669\pi$
$71$	4.91652e6	1.63025	0.815124	+	0.579286i	$0.196669\pi$
$72$	0	0
$73$	3.45746e6	1.04022	0.520112	−	0.854098i	$-0.325890\pi$
$73$	3.45746e6	1.04022	0.520112	+	0.854098i	$0.325890\pi$
$74$	1.95504e6	0.560846
$75$	0	0
$76$	4.18118e6	1.09258
$77$	−1.08072e6	−0.269772
$78$	0	0
$79$	2.91026e6	0.664106	0.332053	−	0.943261i	$-0.392259\pi$
$79$	2.91026e6	0.664106	0.332053	+	0.943261i	$0.392259\pi$
$80$	−10356.8	−0.00226158
$81$	0	0
$82$	−2.53556e6	−0.507838
$83$	−1.88250e6	−0.361379	−0.180689	−	0.983540i	$-0.557833\pi$
$83$	−1.88250e6	−0.361379	−0.180689	+	0.983540i	$0.557833\pi$
$84$	0	0
$85$	−30728.6	−0.00542721
$86$	−2.57975e6	−0.437354
$87$	0	0
$88$	−2.18482e6	−0.341764
$89$	234356.	0.0352380	0.0176190	−	0.999845i	$-0.494391\pi$
$89$	234356.	0.0352380	0.0176190	+	0.999845i	$0.494391\pi$
$90$	0	0
$91$	−3.85713e6	−0.536561
$92$	−1.42074e6	−0.190221
$93$	0	0
$94$	−284007.	−0.0352681
$95$	30402.4	0.00363811
$96$	0	0
$97$	1.52882e7	1.70081	0.850405	−	0.526128i	$-0.176357\pi$
$97$	1.52882e7	1.70081	0.850405	+	0.526128i	$0.176357\pi$
$98$	2.20810e6	0.236988
$99$	0	0
$100$	9.12259e6	0.912259
$101$	1.36904e7	1.32218	0.661090	−	0.750307i	$-0.270094\pi$
$101$	1.36904e7	1.32218	0.661090	+	0.750307i	$0.270094\pi$
$102$	0	0
$103$	5.79120e6	0.522202	0.261101	−	0.965311i	$-0.415914\pi$
$103$	5.79120e6	0.522202	0.261101	+	0.965311i	$0.415914\pi$
$104$	−7.79766e6	−0.679747
$105$	0	0
$106$	−176206.	−0.0143698
$107$	−7.59691e6	−0.599506	−0.299753	−	0.954017i	$-0.596904\pi$
$107$	−7.59691e6	−0.599506	−0.299753	+	0.954017i	$0.596904\pi$
$108$	0	0
$109$	6.28572e6	0.464903	0.232452	−	0.972608i	$-0.425325\pi$
$109$	6.28572e6	0.464903	0.232452	+	0.972608i	$0.425325\pi$
$110$	−7578.76	−0.000542905 0
$111$	0	0
$112$	4.94912e6	0.332862
$113$	2.80396e7	1.82809	0.914044	−	0.405615i	$-0.132943\pi$
$113$	2.80396e7	1.82809	0.914044	+	0.405615i	$0.132943\pi$
$114$	0	0
$115$	−10330.6	−0.000633407 0
$116$	1.49968e7	0.892065
$117$	0	0
$118$	7.59950e6	0.425792
$119$	1.46840e7	0.798786
$120$	0	0
$121$	−1.23923e7	−0.635922
$122$	−5.43919e6	−0.271191
$123$	0	0
$124$	3.81207e6	0.179550
$125$	132666.	0.00607540
$126$	0	0
$127$	−3.71325e7	−1.60858	−0.804288	−	0.594239i	$-0.797453\pi$
$127$	−3.71325e7	−1.60858	−0.804288	+	0.594239i	$0.797453\pi$
$128$	2.22651e7	0.938404
$129$	0	0
$130$	−27048.7	−0.00107981
$131$	−4.87971e6	−0.189646	−0.0948232	−	0.995494i	$-0.530229\pi$
$131$	−4.87971e6	−0.189646	−0.0948232	+	0.995494i	$0.530229\pi$
$132$	0	0
$133$	−1.45281e7	−0.535463
$134$	−7.10170e6	−0.254974
$135$	0	0
$136$	2.96855e7	1.01195
$137$	3.10114e7	1.03038	0.515192	−	0.857075i	$-0.327721\pi$
$137$	3.10114e7	1.03038	0.515192	+	0.857075i	$0.327721\pi$
$138$	0	0
$139$	4.11634e7	1.30005	0.650024	−	0.759914i	$-0.274759\pi$
$139$	4.11634e7	1.30005	0.650024	+	0.759914i	$0.274759\pi$
$140$	40227.0	0.00123899
$141$	0	0
$142$	−1.64757e7	−0.482874
$143$	2.53217e7	0.724130
$144$	0	0
$145$	109046.	0.00297044
$146$	−1.15862e7	−0.308110
$147$	0	0
$148$	6.81243e7	1.72737
$149$	5.31056e7	1.31519	0.657594	−	0.753372i	$-0.271574\pi$
$149$	5.31056e7	1.31519	0.657594	+	0.753372i	$0.271574\pi$
$150$	0	0
$151$	4.43931e7	1.04929	0.524646	−	0.851320i	$-0.324198\pi$
$151$	4.43931e7	1.04929	0.524646	+	0.851320i	$0.324198\pi$
$152$	−2.93704e7	−0.678356
$153$	0	0
$154$	3.62159e6	0.0799056
$155$	27718.5	0.000597873 0
$156$	0	0
$157$	−8.18134e7	−1.68723	−0.843617	−	0.536945i	$-0.819578\pi$
$157$	−8.18134e7	−1.68723	−0.843617	+	0.536945i	$0.819578\pi$
$158$	−9.75252e6	−0.196706
$159$	0	0
$160$	123851.	0.00239045
$161$	4.93659e6	0.0932258
$162$	0	0
$163$	8.21430e7	1.48564	0.742820	−	0.669491i	$-0.233488\pi$
$163$	8.21430e7	1.48564	0.742820	+	0.669491i	$0.233488\pi$
$164$	−8.83530e7	−1.56411
$165$	0	0
$166$	6.30842e6	0.107039
$167$	−7.89460e7	−1.31166	−0.655831	−	0.754907i	$-0.727682\pi$
$167$	−7.89460e7	−1.31166	−0.655831	+	0.754907i	$0.727682\pi$
$168$	0	0
$169$	2.76251e7	0.440251
$170$	102974.	0.00160752
$171$	0	0
$172$	−8.98929e7	−1.34703
$173$	−9.10823e7	−1.33744	−0.668718	−	0.743517i	$-0.733156\pi$
$173$	−9.10823e7	−1.33744	−0.668718	+	0.743517i	$0.733156\pi$
$174$	0	0
$175$	−3.16978e7	−0.447091
$176$	−3.24905e7	−0.449223
$177$	0	0
$178$	−785346.	−0.0104374
$179$	1.15252e8	1.50197	0.750985	−	0.660319i	$-0.229579\pi$
$179$	1.15252e8	1.50197	0.750985	+	0.660319i	$0.229579\pi$
$180$	0	0
$181$	4.18769e6	0.0524927	0.0262464	−	0.999656i	$-0.491645\pi$
$181$	4.18769e6	0.0524927	0.0262464	+	0.999656i	$0.491645\pi$
$182$	1.29255e7	0.158927
$183$	0	0
$184$	9.97992e6	0.118104
$185$	495349.	0.00575189
$186$	0	0
$187$	−9.63992e7	−1.07802
$188$	−9.89639e6	−0.108624
$189$	0	0
$190$	−101881.	−0.00107759
$191$	1.27500e7	0.132402	0.0662010	−	0.997806i	$-0.478912\pi$
$191$	1.27500e7	0.132402	0.0662010	+	0.997806i	$0.478912\pi$
$192$	0	0
$193$	−3.88747e7	−0.389240	−0.194620	−	0.980879i	$-0.562347\pi$
$193$	−3.88747e7	−0.389240	−0.194620	+	0.980879i	$0.562347\pi$
$194$	−5.12321e7	−0.503774
$195$	0	0
$196$	7.69424e7	0.729911
$197$	−7.01957e7	−0.654152	−0.327076	−	0.944998i	$-0.606063\pi$
$197$	−7.01957e7	−0.654152	−0.327076	+	0.944998i	$0.606063\pi$
$198$	0	0
$199$	1.64934e7	0.148362	0.0741811	−	0.997245i	$-0.476366\pi$
$199$	1.64934e7	0.148362	0.0741811	+	0.997245i	$0.476366\pi$
$200$	−6.40810e7	−0.566402
$201$	0	0
$202$	−4.58775e7	−0.391625
$203$	−5.21087e7	−0.437194
$204$	0	0
$205$	−642437.	−0.00520825
$206$	−1.94068e7	−0.154674
$207$	0	0
$208$	−1.15959e8	−0.893478
$209$	9.53759e7	0.722648
$210$	0	0
$211$	−1.48167e8	−1.08583	−0.542917	−	0.839787i	$-0.682680\pi$
$211$	−1.48167e8	−1.08583	−0.542917	+	0.839787i	$0.682680\pi$
$212$	−6.13999e6	−0.0442581
$213$	0	0
$214$	2.54578e7	0.177571
$215$	−653634.	−0.00448539
$216$	0	0
$217$	−1.32456e7	−0.0879959
$218$	−2.10640e7	−0.137703
$219$	0	0
$220$	−264086.	−0.00167212
$221$	−3.44051e8	−2.14412
$222$	0	0
$223$	1.24032e8	0.748972	0.374486	−	0.927233i	$-0.377819\pi$
$223$	1.24032e8	0.748972	0.374486	+	0.927233i	$0.377819\pi$
$224$	−5.91836e7	−0.351830
$225$	0	0
$226$	−9.39629e7	−0.541473
$227$	3.44311e8	1.95371	0.976855	−	0.213902i	$-0.0686173\pi$
$227$	3.44311e8	1.95371	0.976855	+	0.213902i	$0.0686173\pi$
$228$	0	0
$229$	−1.13249e8	−0.623177	−0.311589	−	0.950217i	$-0.600861\pi$
$229$	−1.13249e8	−0.623177	−0.311589	+	0.950217i	$0.600861\pi$
$230$	34618.6	0.000187613 0
$231$	0	0
$232$	−1.05344e8	−0.553863
$233$	1.91098e8	0.989717	0.494859	−	0.868973i	$-0.335220\pi$
$233$	1.91098e8	0.989717	0.494859	+	0.868973i	$0.335220\pi$
$234$	0	0
$235$	−71959.2	−0.000361700 0
$236$	2.64809e8	1.31142
$237$	0	0
$238$	−4.92073e7	−0.236598
$239$	−3.38240e7	−0.160263	−0.0801313	−	0.996784i	$-0.525534\pi$
$239$	−3.38240e7	−0.160263	−0.0801313	+	0.996784i	$0.525534\pi$
$240$	0	0
$241$	−1.53606e8	−0.706883	−0.353441	−	0.935457i	$-0.614989\pi$
$241$	−1.53606e8	−0.706883	−0.353441	+	0.935457i	$0.614989\pi$
$242$	4.15276e7	0.188358
$243$	0	0
$244$	−1.89532e8	−0.835252
$245$	559468.	0.00243049
$246$	0	0
$247$	3.40398e8	1.43730
$248$	−2.67776e7	−0.111478
$249$	0	0
$250$	−444574.	−0.00179951
$251$	−1.31841e8	−0.526250	−0.263125	−	0.964762i	$-0.584753\pi$
$251$	−1.31841e8	−0.526250	−0.263125	+	0.964762i	$0.584753\pi$
$252$	0	0
$253$	−3.24082e7	−0.125815
$254$	1.24434e8	0.476454
$255$	0	0
$256$	6.26700e7	0.233464
$257$	−1.01425e8	−0.372718	−0.186359	−	0.982482i	$-0.559669\pi$
$257$	−1.01425e8	−0.372718	−0.186359	+	0.982482i	$0.559669\pi$
$258$	0	0
$259$	−2.36708e8	−0.846572
$260$	−942529.	−0.00332574
$261$	0	0
$262$	1.63523e7	0.0561726
$263$	−3.14970e8	−1.06764	−0.533819	−	0.845599i	$-0.679244\pi$
$263$	−3.14970e8	−1.06764	−0.533819	+	0.845599i	$0.679244\pi$
$264$	0	0
$265$	−44645.5	−0.000147373 0
$266$	4.86850e7	0.158602
$267$	0	0
$268$	−2.47463e8	−0.785305
$269$	−2.25334e8	−0.705819	−0.352909	−	0.935657i	$-0.614808\pi$
$269$	−2.25334e8	−0.705819	−0.352909	+	0.935657i	$0.614808\pi$
$270$	0	0
$271$	3.42841e7	0.104641	0.0523204	−	0.998630i	$-0.483338\pi$
$271$	3.42841e7	0.104641	0.0523204	+	0.998630i	$0.483338\pi$
$272$	4.41455e8	1.33013
$273$	0	0
$274$	−1.03922e8	−0.305196
$275$	2.08093e8	0.603384
$276$	0	0
$277$	−3.88720e8	−1.09890	−0.549449	−	0.835527i	$-0.685162\pi$
$277$	−3.88720e8	−1.09890	−0.549449	+	0.835527i	$0.685162\pi$
$278$	−1.37942e8	−0.385070
$279$	0	0
$280$	−282571.	−0.000769263 0
$281$	2.65787e8	0.714597	0.357299	−	0.933990i	$-0.383698\pi$
$281$	2.65787e8	0.714597	0.357299	+	0.933990i	$0.383698\pi$
$282$	0	0
$283$	9.51931e7	0.249662	0.124831	−	0.992178i	$-0.460161\pi$
$283$	9.51931e7	0.249662	0.124831	+	0.992178i	$0.460161\pi$
$284$	−5.74104e8	−1.48722
$285$	0	0
$286$	−8.48550e7	−0.214485
$287$	3.06996e8	0.766559
$288$	0	0
$289$	8.99456e8	2.19198
$290$	−365421.	−0.000879834 0
$291$	0	0
$292$	−4.03728e8	−0.948962
$293$	2.36355e8	0.548944	0.274472	−	0.961595i	$-0.411497\pi$
$293$	2.36355e8	0.548944	0.274472	+	0.961595i	$0.411497\pi$
$294$	0	0
$295$	1.92549e6	0.00436682
$296$	−4.78535e8	−1.07249
$297$	0	0
$298$	−1.77961e8	−0.389554
$299$	−1.15666e8	−0.250239
$300$	0	0
$301$	3.12346e8	0.660167
$302$	−1.48765e8	−0.310797
$303$	0	0
$304$	−4.36769e8	−0.891649
$305$	−1.37813e6	−0.00278126
$306$	0	0
$307$	−4.36850e8	−0.861684	−0.430842	−	0.902427i	$-0.641783\pi$
$307$	−4.36850e8	−0.861684	−0.430842	+	0.902427i	$0.641783\pi$
$308$	1.26197e8	0.246105
$309$	0	0
$310$	−92886.9	−0.000177088 0
$311$	1.43706e8	0.270904	0.135452	−	0.990784i	$-0.456751\pi$
$311$	1.43706e8	0.270904	0.135452	+	0.990784i	$0.456751\pi$
$312$	0	0
$313$	7.98141e8	1.47121	0.735605	−	0.677411i	$-0.236898\pi$
$313$	7.98141e8	1.47121	0.735605	+	0.677411i	$0.236898\pi$
$314$	2.74163e8	0.499753
$315$	0	0
$316$	−3.39832e8	−0.605842
$317$	−1.58206e8	−0.278943	−0.139472	−	0.990226i	$-0.544540\pi$
$317$	−1.58206e8	−0.278943	−0.139472	+	0.990226i	$0.544540\pi$
$318$	0	0
$319$	3.42089e8	0.590027
$320$	910638.	0.00155353
$321$	0	0
$322$	−1.65429e7	−0.0276131
$323$	−1.29589e9	−2.13973
$324$	0	0
$325$	7.42689e8	1.20009
$326$	−2.75268e8	−0.440041
$327$	0	0
$328$	6.20629e8	0.971122
$329$	3.43865e7	0.0532356
$330$	0	0
$331$	5.55118e8	0.841370	0.420685	−	0.907207i	$-0.361790\pi$
$331$	5.55118e8	0.841370	0.420685	+	0.907207i	$0.361790\pi$
$332$	2.19821e8	0.329674
$333$	0	0
$334$	2.64554e8	0.388510
$335$	−1.79937e6	−0.00261495
$336$	0	0
$337$	−2.24145e8	−0.319025	−0.159512	−	0.987196i	$-0.550992\pi$
$337$	−2.24145e8	−0.319025	−0.159512	+	0.987196i	$0.550992\pi$
$338$	−9.25738e7	−0.130401
$339$	0	0
$340$	3.58819e6	0.00495107
$341$	8.69561e7	0.118757
$342$	0	0
$343$	−6.01489e8	−0.804819
$344$	6.31447e8	0.836339
$345$	0	0
$346$	3.05224e8	0.396143
$347$	−3.33457e7	−0.0428437	−0.0214219	−	0.999771i	$-0.506819\pi$
$347$	−3.33457e7	−0.0428437	−0.0214219	+	0.999771i	$0.506819\pi$
$348$	0	0
$349$	9.08225e8	1.14368	0.571840	−	0.820365i	$-0.306230\pi$
$349$	9.08225e8	1.14368	0.571840	+	0.820365i	$0.306230\pi$
$350$	1.06222e8	0.132427
$351$	0	0
$352$	3.88535e8	0.474822
$353$	−3.91774e8	−0.474051	−0.237025	−	0.971503i	$-0.576172\pi$
$353$	−3.91774e8	−0.474051	−0.237025	+	0.971503i	$0.576172\pi$
$354$	0	0
$355$	−4.17445e6	−0.00495222
$356$	−2.73658e7	−0.0321465
$357$	0	0
$358$	−3.86217e8	−0.444878
$359$	1.07880e9	1.23058	0.615288	−	0.788302i	$-0.289040\pi$
$359$	1.07880e9	1.23058	0.615288	+	0.788302i	$0.289040\pi$
$360$	0	0
$361$	3.88263e8	0.434361
$362$	−1.40333e7	−0.0155482
$363$	0	0
$364$	4.50398e8	0.489487
$365$	−2.93561e6	−0.00315990
$366$	0	0
$367$	1.24645e9	1.31626	0.658131	−	0.752903i	$-0.271347\pi$
$367$	1.24645e9	1.31626	0.658131	+	0.752903i	$0.271347\pi$
$368$	1.48412e8	0.155239
$369$	0	0
$370$	−1.65996e6	−0.00170369
$371$	2.13343e7	0.0216905
$372$	0	0
$373$	4.12114e8	0.411184	0.205592	−	0.978638i	$-0.434088\pi$
$373$	4.12114e8	0.411184	0.205592	+	0.978638i	$0.434088\pi$
$374$	3.23041e8	0.319306
$375$	0	0
$376$	6.95165e7	0.0674420
$377$	1.22092e9	1.17353
$378$	0	0
$379$	−1.27834e9	−1.20617	−0.603085	−	0.797677i	$-0.706062\pi$
$379$	−1.27834e9	−1.20617	−0.603085	+	0.797677i	$0.706062\pi$
$380$	−3.55010e6	−0.00331893
$381$	0	0
$382$	−4.27264e7	−0.0392170
$383$	9.34297e8	0.849746	0.424873	−	0.905253i	$-0.360319\pi$
$383$	9.34297e8	0.849746	0.424873	+	0.905253i	$0.360319\pi$
$384$	0	0
$385$	917607.	0.000819491 0
$386$	1.30272e8	0.115291
$387$	0	0
$388$	−1.78521e9	−1.55159
$389$	−3.03608e8	−0.261511	−0.130755	−	0.991415i	$-0.541740\pi$
$389$	−3.03608e8	−0.261511	−0.130755	+	0.991415i	$0.541740\pi$
$390$	0	0
$391$	4.40337e8	0.372534
$392$	−5.40477e8	−0.453185
$393$	0	0
$394$	2.35231e8	0.193757
$395$	−2.47100e6	−0.00201736
$396$	0	0
$397$	−4.05735e8	−0.325443	−0.162722	−	0.986672i	$-0.552027\pi$
$397$	−4.05735e8	−0.325443	−0.162722	+	0.986672i	$0.552027\pi$
$398$	−5.52706e7	−0.0439444
$399$	0	0
$400$	−9.52951e8	−0.744493
$401$	8.42816e8	0.652721	0.326360	−	0.945245i	$-0.394178\pi$
$401$	8.42816e8	0.652721	0.326360	+	0.945245i	$0.394178\pi$
$402$	0	0
$403$	3.10348e8	0.236201
$404$	−1.59863e9	−1.20618
$405$	0	0
$406$	1.74621e8	0.129495
$407$	1.55397e9	1.14251
$408$	0	0
$409$	9.52820e8	0.688619	0.344310	−	0.938856i	$-0.388113\pi$
$409$	9.52820e8	0.688619	0.344310	+	0.938856i	$0.388113\pi$
$410$	2.15286e6	0.00154267
$411$	0	0
$412$	−6.76240e8	−0.476388
$413$	−9.20118e8	−0.642715
$414$	0	0
$415$	1.59837e6	0.00109776
$416$	1.38669e9	0.944392
$417$	0	0
$418$	−3.19612e8	−0.214046
$419$	1.41758e9	0.941454	0.470727	−	0.882279i	$-0.343992\pi$
$419$	1.41758e9	0.941454	0.470727	+	0.882279i	$0.343992\pi$
$420$	0	0
$421$	1.59148e9	1.03948	0.519738	−	0.854326i	$-0.326030\pi$
$421$	1.59148e9	1.03948	0.519738	+	0.854326i	$0.326030\pi$
$422$	4.96520e8	0.321620
$423$	0	0
$424$	4.31300e7	0.0274788
$425$	−2.82740e9	−1.78660
$426$	0	0
$427$	6.58556e8	0.409351
$428$	8.87093e8	0.546910
$429$	0	0
$430$	2.19038e6	0.00132856
$431$	2.04772e9	1.23197	0.615984	−	0.787759i	$-0.288759\pi$
$431$	2.04772e9	1.23197	0.615984	+	0.787759i	$0.288759\pi$
$432$	0	0
$433$	1.15795e9	0.685458	0.342729	−	0.939434i	$-0.388649\pi$
$433$	1.15795e9	0.685458	0.342729	+	0.939434i	$0.388649\pi$
$434$	4.43870e7	0.0260641
$435$	0	0
$436$	−7.33985e8	−0.424116
$437$	−4.35663e8	−0.249727
$438$	0	0
$439$	1.71302e8	0.0966356	0.0483178	−	0.998832i	$-0.484614\pi$
$439$	1.71302e8	0.0966356	0.0483178	+	0.998832i	$0.484614\pi$
$440$	1.85506e6	0.00103818
$441$	0	0
$442$	1.15294e9	0.635081
$443$	2.01064e9	1.09881	0.549404	−	0.835557i	$-0.314855\pi$
$443$	2.01064e9	1.09881	0.549404	+	0.835557i	$0.314855\pi$
$444$	0	0
$445$	−198984.	−0.000107043 0
$446$	−4.15640e8	−0.221843
$447$	0	0
$448$	−4.35158e8	−0.228652
$449$	−1.66574e9	−0.868451	−0.434225	−	0.900804i	$-0.642978\pi$
$449$	−1.66574e9	−0.868451	−0.434225	+	0.900804i	$0.642978\pi$
$450$	0	0
$451$	−2.01540e9	−1.03453
$452$	−3.27419e9	−1.66771
$453$	0	0
$454$	−1.15381e9	−0.578682
$455$	3.27496e6	0.00162992
$456$	0	0
$457$	−1.76044e9	−0.862811	−0.431405	−	0.902158i	$-0.641982\pi$
$457$	−1.76044e9	−0.862811	−0.431405	+	0.902158i	$0.641982\pi$
$458$	3.79508e8	0.184583
$459$	0	0
$460$	1.20631e6	0.000577837 0
$461$	2.02454e9	0.962440	0.481220	−	0.876600i	$-0.340194\pi$
$461$	2.02454e9	0.962440	0.481220	+	0.876600i	$0.340194\pi$
$462$	0	0
$463$	−2.65240e9	−1.24196	−0.620978	−	0.783828i	$-0.713264\pi$
$463$	−2.65240e9	−1.24196	−0.620978	+	0.783828i	$0.713264\pi$
$464$	−1.56658e9	−0.728013
$465$	0	0
$466$	−6.40386e8	−0.293151
$467$	−2.57500e9	−1.16995	−0.584977	−	0.811050i	$-0.698897\pi$
$467$	−2.57500e9	−1.16995	−0.584977	+	0.811050i	$0.698897\pi$
$468$	0	0
$469$	8.59847e8	0.384872
$470$	241141.	0.000107134 0
$471$	0	0
$472$	−1.86013e9	−0.814229
$473$	−2.05053e9	−0.890946
$474$	0	0
$475$	2.79739e9	1.19764
$476$	−1.71466e9	−0.728707
$477$	0	0
$478$	1.13347e8	0.0474692
$479$	8.23862e8	0.342515	0.171258	−	0.985226i	$-0.445217\pi$
$479$	8.23862e8	0.342515	0.171258	+	0.985226i	$0.445217\pi$
$480$	0	0
$481$	5.54614e9	2.27239
$482$	5.14745e8	0.209376
$483$	0	0
$484$	1.44705e9	0.580131
$485$	−1.29807e7	−0.00516657
$486$	0	0
$487$	2.67396e9	1.04907	0.524533	−	0.851390i	$-0.324240\pi$
$487$	2.67396e9	1.04907	0.524533	+	0.851390i	$0.324240\pi$
$488$	1.33135e9	0.518589
$489$	0	0
$490$	−1.87482e6	−0.000719903 0
$491$	1.23865e9	0.472239	0.236120	−	0.971724i	$-0.424124\pi$
$491$	1.23865e9	0.472239	0.236120	+	0.971724i	$0.424124\pi$
$492$	0	0
$493$	−4.64803e9	−1.74705
$494$	−1.14070e9	−0.425724
$495$	0	0
$496$	−3.98211e8	−0.146530
$497$	1.99481e9	0.728876
$498$	0	0
$499$	−2.12672e9	−0.766229	−0.383115	−	0.923701i	$-0.625149\pi$
$499$	−2.12672e9	−0.766229	−0.383115	+	0.923701i	$0.625149\pi$
$500$	−1.54914e7	−0.00554239
$501$	0	0
$502$	4.41809e8	0.155873
$503$	−2.18140e8	−0.0764271	−0.0382135	−	0.999270i	$-0.512167\pi$
$503$	−2.18140e8	−0.0764271	−0.0382135	+	0.999270i	$0.512167\pi$
$504$	0	0
$505$	−1.16240e7	−0.00401640
$506$	1.08603e8	0.0372660
$507$	0	0
$508$	4.33598e9	1.46745
$509$	−2.84720e9	−0.956985	−0.478492	−	0.878092i	$-0.658817\pi$
$509$	−2.84720e9	−0.956985	−0.478492	+	0.878092i	$0.658817\pi$
$510$	0	0
$511$	1.40281e9	0.465079
$512$	−3.05995e9	−1.00756
$513$	0	0
$514$	3.39884e8	0.110398
$515$	−4.91712e6	−0.00158630
$516$	0	0
$517$	−2.25744e8	−0.0718455
$518$	7.93228e8	0.250752
$519$	0	0
$520$	6.62073e6	0.00206488
$521$	3.03891e9	0.941426	0.470713	−	0.882286i	$-0.343997\pi$
$521$	3.03891e9	0.941426	0.470713	+	0.882286i	$0.343997\pi$
$522$	0	0
$523$	−3.95904e8	−0.121013	−0.0605067	−	0.998168i	$-0.519272\pi$
$523$	−3.95904e8	−0.121013	−0.0605067	+	0.998168i	$0.519272\pi$
$524$	5.69805e8	0.173008
$525$	0	0
$526$	1.05549e9	0.316230
$527$	−1.18149e9	−0.351636
$528$	0	0
$529$	1.48036e8	0.0434783
$530$	149610.	4.36512e−5 0
$531$	0	0
$532$	1.69645e9	0.488485
$533$	−7.19299e9	−2.05762
$534$	0	0
$535$	6.45028e6	0.00182113
$536$	1.73829e9	0.487578
$537$	0	0
$538$	7.55111e8	0.209061
$539$	1.75512e9	0.482775
$540$	0	0
$541$	−5.18940e9	−1.40905	−0.704525	−	0.709679i	$-0.748840\pi$
$541$	−5.18940e9	−1.40905	−0.704525	+	0.709679i	$0.748840\pi$
$542$	−1.14889e8	−0.0309942
$543$	0	0
$544$	−5.27910e9	−1.40593
$545$	−5.33700e6	−0.00141224
$546$	0	0
$547$	−7.01245e9	−1.83195	−0.915976	−	0.401232i	$-0.868582\pi$
$547$	−7.01245e9	−1.83195	−0.915976	+	0.401232i	$0.868582\pi$
$548$	−3.62121e9	−0.939987
$549$	0	0
$550$	−6.97337e8	−0.178720
$551$	4.59869e9	1.17113
$552$	0	0
$553$	1.18080e9	0.296919
$554$	1.30263e9	0.325490
$555$	0	0
$556$	−4.80666e9	−1.18599
$557$	2.56612e9	0.629193	0.314597	−	0.949225i	$-0.398131\pi$
$557$	2.56612e9	0.629193	0.314597	+	0.949225i	$0.398131\pi$
$558$	0	0
$559$	−7.31836e9	−1.77204
$560$	−4.20213e6	−0.00101114
$561$	0	0
$562$	−8.90673e8	−0.211661
$563$	8.98840e8	0.212277	0.106139	−	0.994351i	$-0.466151\pi$
$563$	8.98840e8	0.212277	0.106139	+	0.994351i	$0.466151\pi$
$564$	0	0
$565$	−2.38075e7	−0.00555320
$566$	−3.18999e8	−0.0739490
$567$	0	0
$568$	4.03275e9	0.923384
$569$	3.86463e9	0.879458	0.439729	−	0.898130i	$-0.355074\pi$
$569$	3.86463e9	0.879458	0.439729	+	0.898130i	$0.355074\pi$
$570$	0	0
$571$	4.00586e9	0.900471	0.450235	−	0.892910i	$-0.351340\pi$
$571$	4.00586e9	0.900471	0.450235	+	0.892910i	$0.351340\pi$
$572$	−2.95682e9	−0.660600
$573$	0	0
$574$	−1.02877e9	−0.227052
$575$	−9.50538e8	−0.208512
$576$	0	0
$577$	−5.38283e9	−1.16653	−0.583264	−	0.812282i	$-0.698225\pi$
$577$	−5.38283e9	−1.16653	−0.583264	+	0.812282i	$0.698225\pi$
$578$	−3.01415e9	−0.649258
$579$	0	0
$580$	−1.27333e7	−0.00270984
$581$	−7.63799e8	−0.161571
$582$	0	0
$583$	−1.40058e8	−0.0292730
$584$	2.83596e9	0.589190
$585$	0	0
$586$	−7.92044e8	−0.162595
$587$	−4.39683e9	−0.897234	−0.448617	−	0.893724i	$-0.648083\pi$
$587$	−4.39683e9	−0.897234	−0.448617	+	0.893724i	$0.648083\pi$
$588$	0	0
$589$	1.16895e9	0.235717
$590$	−6.45248e6	−0.00129343
$591$	0	0
$592$	−7.11631e9	−1.40971
$593$	1.78104e9	0.350737	0.175368	−	0.984503i	$-0.443888\pi$
$593$	1.78104e9	0.350737	0.175368	+	0.984503i	$0.443888\pi$
$594$	0	0
$595$	−1.24677e7	−0.00242648
$596$	−6.20115e9	−1.19980
$597$	0	0
$598$	3.87605e8	0.0741199
$599$	3.87905e9	0.737449	0.368724	−	0.929539i	$-0.379795\pi$
$599$	3.87905e9	0.737449	0.368724	+	0.929539i	$0.379795\pi$
$600$	0	0
$601$	3.87683e9	0.728478	0.364239	−	0.931305i	$-0.381329\pi$
$601$	3.87683e9	0.728478	0.364239	+	0.931305i	$0.381329\pi$
$602$	−1.04670e9	−0.195539
$603$	0	0
$604$	−5.18380e9	−0.957236
$605$	1.05219e7	0.00193175
$606$	0	0
$607$	−6.73366e9	−1.22205	−0.611027	−	0.791610i	$-0.709243\pi$
$607$	−6.73366e9	−1.22205	−0.611027	+	0.791610i	$0.709243\pi$
$608$	5.22306e9	0.942459
$609$	0	0
$610$	4.61824e6	0.000823800 0
$611$	−8.05685e8	−0.142896
$612$	0	0
$613$	6.28531e9	1.10208	0.551042	−	0.834477i	$-0.314230\pi$
$613$	6.28531e9	1.10208	0.551042	+	0.834477i	$0.314230\pi$
$614$	1.46392e9	0.255228
$615$	0	0
$616$	−8.86459e8	−0.152801
$617$	6.71956e9	1.15171	0.575854	−	0.817552i	$-0.304670\pi$
$617$	6.71956e9	1.15171	0.575854	+	0.817552i	$0.304670\pi$
$618$	0	0
$619$	1.62081e9	0.274672	0.137336	−	0.990525i	$-0.456146\pi$
$619$	1.62081e9	0.274672	0.137336	+	0.990525i	$0.456146\pi$
$620$	−3.23670e6	−0.000545420 0
$621$	0	0
$622$	−4.81572e8	−0.0802407
$623$	9.50866e7	0.0157547
$624$	0	0
$625$	6.10335e9	0.999972
$626$	−2.67463e9	−0.435767
$627$	0	0
$628$	9.55337e9	1.53921
$629$	−2.11141e10	−3.38294
$630$	0	0
$631$	−8.43752e9	−1.33694	−0.668470	−	0.743739i	$-0.733051\pi$
$631$	−8.43752e9	−1.33694	−0.668470	+	0.743739i	$0.733051\pi$
$632$	2.38713e9	0.376154
$633$	0	0
$634$	5.30162e8	0.0826220
$635$	3.15280e7	0.00488639
$636$	0	0
$637$	6.26404e9	0.960211
$638$	−1.14637e9	−0.174764
$639$	0	0
$640$	−1.89046e7	−0.00285060
$641$	1.15180e10	1.72733	0.863663	−	0.504069i	$-0.168164\pi$
$641$	1.15180e10	1.72733	0.863663	+	0.504069i	$0.168164\pi$
$642$	0	0
$643$	−1.06978e10	−1.58692	−0.793461	−	0.608621i	$-0.791723\pi$
$643$	−1.06978e10	−1.58692	−0.793461	+	0.608621i	$0.791723\pi$
$644$	−5.76447e8	−0.0850469
$645$	0	0
$646$	4.34263e9	0.633781
$647$	−1.28732e10	−1.86862	−0.934311	−	0.356459i	$-0.883984\pi$
$647$	−1.28732e10	−1.86862	−0.934311	+	0.356459i	$0.883984\pi$
$648$	0	0
$649$	6.04049e9	0.867393
$650$	−2.48881e9	−0.355463
$651$	0	0
$652$	−9.59186e9	−1.35530
$653$	4.92927e9	0.692765	0.346383	−	0.938093i	$-0.387410\pi$
$653$	4.92927e9	0.692765	0.346383	+	0.938093i	$0.387410\pi$
$654$	0	0
$655$	4.14320e6	0.000576091 0
$656$	9.22940e9	1.27647
$657$	0	0
$658$	−1.15232e8	−0.0157682
$659$	7.40103e9	1.00738	0.503690	−	0.863885i	$-0.331976\pi$
$659$	7.40103e9	1.00738	0.503690	+	0.863885i	$0.331976\pi$
$660$	0	0
$661$	2.57426e9	0.346695	0.173348	−	0.984861i	$-0.444542\pi$
$661$	2.57426e9	0.346695	0.173348	+	0.984861i	$0.444542\pi$
$662$	−1.86024e9	−0.249211
$663$	0	0
$664$	−1.54411e9	−0.204687
$665$	1.23354e7	0.00162658
$666$	0	0
$667$	−1.56261e9	−0.203897
$668$	9.21854e9	1.19659
$669$	0	0
$670$	6.02982e6	0.000774537 0
$671$	−4.32336e9	−0.552450
$672$	0	0
$673$	1.33767e9	0.169159	0.0845795	−	0.996417i	$-0.473045\pi$
$673$	1.33767e9	0.169159	0.0845795	+	0.996417i	$0.473045\pi$
$674$	7.51127e8	0.0944939
$675$	0	0
$676$	−3.22579e9	−0.401626
$677$	−5.47938e9	−0.678689	−0.339345	−	0.940662i	$-0.610205\pi$
$677$	−5.47938e9	−0.678689	−0.339345	+	0.940662i	$0.610205\pi$
$678$	0	0
$679$	6.20298e9	0.760425
$680$	−2.52050e7	−0.00307401
$681$	0	0
$682$	−2.91397e8	−0.0351754
$683$	−9.90832e9	−1.18995	−0.594973	−	0.803745i	$-0.702837\pi$
$683$	−9.90832e9	−1.18995	−0.594973	+	0.803745i	$0.702837\pi$
$684$	0	0
$685$	−2.63307e7	−0.00313001
$686$	2.01564e9	0.238384
$687$	0	0
$688$	9.39027e9	1.09931
$689$	−4.99869e8	−0.0582223
$690$	0	0
$691$	−9.22230e9	−1.06332	−0.531662	−	0.846956i	$-0.678432\pi$
$691$	−9.22230e9	−1.06332	−0.531662	+	0.846956i	$0.678432\pi$
$692$	1.06357e10	1.22010
$693$	0	0
$694$	1.11744e8	0.0126902
$695$	−3.49505e7	−0.00394917
$696$	0	0
$697$	2.73836e10	3.06320
$698$	−3.04353e9	−0.338754
$699$	0	0
$700$	3.70136e9	0.407867
$701$	−1.26162e9	−0.138329	−0.0691646	−	0.997605i	$-0.522033\pi$
$701$	−1.26162e9	−0.138329	−0.0691646	+	0.997605i	$0.522033\pi$
$702$	0	0
$703$	2.08899e10	2.26774
$704$	2.85677e9	0.308583
$705$	0	0
$706$	1.31287e9	0.140412
$707$	5.55467e9	0.591140
$708$	0	0
$709$	5.11962e9	0.539481	0.269740	−	0.962933i	$-0.413062\pi$
$709$	5.11962e9	0.539481	0.269740	+	0.962933i	$0.413062\pi$
$710$	1.39889e7	0.00146683
$711$	0	0
$712$	1.92229e8	0.0199590
$713$	−3.97202e8	−0.0410392
$714$	0	0
$715$	−2.14998e7	−0.00219970
$716$	−1.34580e10	−1.37020
$717$	0	0
$718$	−3.61513e9	−0.364492
$719$	−3.58731e9	−0.359930	−0.179965	−	0.983673i	$-0.557598\pi$
$719$	−3.58731e9	−0.359930	−0.179965	+	0.983673i	$0.557598\pi$
$720$	0	0
$721$	2.34970e9	0.233474
$722$	−1.30110e9	−0.128656
$723$	0	0
$724$	−4.88997e8	−0.0478874
$725$	1.00335e10	0.977846
$726$	0	0
$727$	1.00951e10	0.974406	0.487203	−	0.873289i	$-0.338017\pi$
$727$	1.00951e10	0.974406	0.487203	+	0.873289i	$0.338017\pi$
$728$	−3.16379e9	−0.303912
$729$	0	0
$730$	9.83746e6	0.000935951 0
$731$	2.78609e10	2.63806
$732$	0	0
$733$	5.17030e9	0.484900	0.242450	−	0.970164i	$-0.422049\pi$
$733$	5.17030e9	0.484900	0.242450	+	0.970164i	$0.422049\pi$
$734$	−4.17694e9	−0.389872
$735$	0	0
$736$	−1.77477e9	−0.164085
$737$	−5.64482e9	−0.519414
$738$	0	0
$739$	1.13371e10	1.03335	0.516675	−	0.856182i	$-0.327170\pi$
$739$	1.13371e10	1.03335	0.516675	+	0.856182i	$0.327170\pi$
$740$	−5.78421e7	−0.00524726
$741$	0	0
$742$	−7.14930e7	−0.00642466
$743$	−7.35179e9	−0.657555	−0.328778	−	0.944407i	$-0.606637\pi$
$743$	−7.35179e9	−0.657555	−0.328778	+	0.944407i	$0.606637\pi$
$744$	0	0
$745$	−4.50901e7	−0.00399516
$746$	−1.38103e9	−0.121791
$747$	0	0
$748$	1.12566e10	0.983445
$749$	−3.08234e9	−0.268036
$750$	0	0
$751$	2.57204e8	0.0221584	0.0110792	−	0.999939i	$-0.496473\pi$
$751$	2.57204e8	0.0221584	0.0110792	+	0.999939i	$0.496473\pi$
$752$	1.03378e9	0.0886476
$753$	0	0
$754$	−4.09141e9	−0.347595
$755$	−3.76927e7	−0.00318745
$756$	0	0
$757$	2.62859e9	0.220236	0.110118	−	0.993919i	$-0.464877\pi$
$757$	2.62859e9	0.220236	0.110118	+	0.993919i	$0.464877\pi$
$758$	4.28381e9	0.357263
$759$	0	0
$760$	2.49374e7	0.00206065
$761$	3.73461e9	0.307184	0.153592	−	0.988134i	$-0.450916\pi$
$761$	3.73461e9	0.307184	0.153592	+	0.988134i	$0.450916\pi$
$762$	0	0
$763$	2.55034e9	0.207856
$764$	−1.48883e9	−0.120786
$765$	0	0
$766$	−3.13090e9	−0.251692
$767$	2.15586e10	1.72519
$768$	0	0
$769$	−1.96580e10	−1.55882	−0.779412	−	0.626512i	$-0.784482\pi$
$769$	−1.96580e10	−1.55882	−0.779412	+	0.626512i	$0.784482\pi$
$770$	−3.07497e6	−0.000242730 0
$771$	0	0
$772$	4.53941e9	0.355091
$773$	−1.22575e9	−0.0954497	−0.0477248	−	0.998861i	$-0.515197\pi$
$773$	−1.22575e9	−0.0954497	−0.0477248	+	0.998861i	$0.515197\pi$
$774$	0	0
$775$	2.55044e9	0.196815
$776$	1.25401e10	0.963351
$777$	0	0
$778$	1.01741e9	0.0774585
$779$	−2.70929e10	−2.05341
$780$	0	0
$781$	−1.30957e10	−0.983674
$782$	−1.47560e9	−0.110343
$783$	0	0
$784$	−8.03745e9	−0.595679
$785$	6.94650e7	0.00512533
$786$	0	0
$787$	−1.08127e10	−0.790721	−0.395361	−	0.918526i	$-0.629380\pi$
$787$	−1.08127e10	−0.790721	−0.395361	+	0.918526i	$0.629380\pi$
$788$	8.19677e9	0.596762
$789$	0	0
$790$	8.28053e6	0.000597535 0
$791$	1.13767e10	0.817329
$792$	0	0
$793$	−1.54302e10	−1.09879
$794$	1.35965e9	0.0963951
$795$	0	0
$796$	−1.92593e9	−0.135346
$797$	1.95009e10	1.36443	0.682215	−	0.731151i	$-0.261017\pi$
$797$	1.95009e10	1.36443	0.682215	+	0.731151i	$0.261017\pi$
$798$	0	0
$799$	3.06723e9	0.212732
$800$	1.13958e10	0.786918
$801$	0	0
$802$	−2.82434e9	−0.193333
$803$	−9.20934e9	−0.627660
$804$	0	0
$805$	−4.19149e6	−0.000283193 0
$806$	−1.04000e9	−0.0699618
$807$	0	0
$808$	1.12294e10	0.748891
$809$	−2.59821e10	−1.72526	−0.862631	−	0.505833i	$-0.831185\pi$
$809$	−2.59821e10	−1.72526	−0.862631	+	0.505833i	$0.831185\pi$
$810$	0	0
$811$	2.97565e9	0.195888	0.0979441	−	0.995192i	$-0.468773\pi$
$811$	2.97565e9	0.195888	0.0979441	+	0.995192i	$0.468773\pi$
$812$	6.08475e9	0.398838
$813$	0	0
$814$	−5.20747e9	−0.338408
$815$	−6.97448e7	−0.00451295
$816$	0	0
$817$	−2.75651e10	−1.76841
$818$	−3.19297e9	−0.203966
$819$	0	0
$820$	7.50175e7	0.00475132
$821$	−3.94381e9	−0.248722	−0.124361	−	0.992237i	$-0.539688\pi$
$821$	−3.94381e9	−0.248722	−0.124361	+	0.992237i	$0.539688\pi$
$822$	0	0
$823$	−1.36446e10	−0.853218	−0.426609	−	0.904436i	$-0.640292\pi$
$823$	−1.36446e10	−0.853218	−0.426609	+	0.904436i	$0.640292\pi$
$824$	4.75020e9	0.295779
$825$	0	0
$826$	3.08339e9	0.190370
$827$	2.13778e10	1.31430	0.657149	−	0.753760i	$-0.271762\pi$
$827$	2.13778e10	1.31430	0.657149	+	0.753760i	$0.271762\pi$
$828$	0	0
$829$	1.14474e10	0.697856	0.348928	−	0.937150i	$-0.386546\pi$
$829$	1.14474e10	0.697856	0.348928	+	0.937150i	$0.386546\pi$
$830$	−5.35627e6	−0.000325154 0
$831$	0	0
$832$	1.01959e10	0.613752
$833$	−2.38471e10	−1.42948
$834$	0	0
$835$	6.70304e7	0.00398445
$836$	−1.11371e10	−0.659248
$837$	0	0
$838$	−4.75043e9	−0.278855
$839$	3.27466e10	1.91426	0.957128	−	0.289665i	$-0.0935440\pi$
$839$	3.27466e10	1.91426	0.957128	+	0.289665i	$0.0935440\pi$
$840$	0	0
$841$	−7.55549e8	−0.0438003
$842$	−5.33318e9	−0.307889
$843$	0	0
$844$	1.73015e10	0.990571
$845$	−2.34555e7	−0.00133735
$846$	0	0
$847$	−5.02800e9	−0.284318
$848$	6.41387e8	0.0361189
$849$	0	0
$850$	9.47485e9	0.529183
$851$	−7.09829e9	−0.394821
$852$	0	0
$853$	1.38359e10	0.763281	0.381641	−	0.924311i	$-0.375359\pi$
$853$	1.38359e10	0.763281	0.381641	+	0.924311i	$0.375359\pi$
$854$	−2.20687e9	−0.121248
$855$	0	0
$856$	−6.23132e9	−0.339564
$857$	1.99079e10	1.08042	0.540209	−	0.841531i	$-0.318345\pi$
$857$	1.99079e10	1.08042	0.540209	+	0.841531i	$0.318345\pi$
$858$	0	0
$859$	−2.25943e10	−1.21625	−0.608124	−	0.793842i	$-0.708078\pi$
$859$	−2.25943e10	−1.21625	−0.608124	+	0.793842i	$0.708078\pi$
$860$	7.63251e7	0.00409188
$861$	0	0
$862$	−6.86206e9	−0.364904
$863$	−2.03862e10	−1.07969	−0.539844	−	0.841765i	$-0.681517\pi$
$863$	−2.03862e10	−1.07969	−0.539844	+	0.841765i	$0.681517\pi$
$864$	0	0
$865$	7.73349e7	0.00406274
$866$	−3.88037e9	−0.203030
$867$	0	0
$868$	1.54669e9	0.0802758
$869$	−7.75182e9	−0.400714
$870$	0	0
$871$	−2.01465e10	−1.03308
$872$	5.15583e9	0.263324
$873$	0	0
$874$	1.45994e9	0.0739682
$875$	5.38273e7	0.00271628
$876$	0	0
$877$	−5.24578e8	−0.0262610	−0.0131305	−	0.999914i	$-0.504180\pi$
$877$	−5.24578e8	−0.0262610	−0.0131305	+	0.999914i	$0.504180\pi$
$878$	−5.74047e8	−0.0286231
$879$	0	0
$880$	2.75866e7	0.00136461
$881$	−3.10939e10	−1.53200	−0.766000	−	0.642840i	$-0.777756\pi$
$881$	−3.10939e10	−1.53200	−0.766000	+	0.642840i	$0.777756\pi$
$882$	0	0
$883$	2.38047e10	1.16359	0.581794	−	0.813336i	$-0.302351\pi$
$883$	2.38047e10	1.16359	0.581794	+	0.813336i	$0.302351\pi$
$884$	4.01749e10	1.95601
$885$	0	0
$886$	−6.73782e9	−0.325463
$887$	−3.21183e10	−1.54533	−0.772663	−	0.634817i	$-0.781075\pi$
$887$	−3.21183e10	−1.54533	−0.772663	+	0.634817i	$0.781075\pi$
$888$	0	0
$889$	−1.50660e10	−0.719187
$890$	666810.	3.17057e−5 0
$891$	0	0
$892$	−1.44832e10	−0.683263
$893$	−3.03467e9	−0.142604
$894$	0	0
$895$	−9.78562e7	−0.00456255
$896$	9.03375e9	0.419556
$897$	0	0
$898$	5.58203e9	0.257232
$899$	4.19272e9	0.192458
$900$	0	0
$901$	1.90299e9	0.0866763
$902$	6.75376e9	0.306424
$903$	0	0
$904$	2.29993e10	1.03544
$905$	−3.55562e6	−0.000159458 0
$906$	0	0
$907$	−4.96459e9	−0.220931	−0.110466	−	0.993880i	$-0.535234\pi$
$907$	−4.96459e9	−0.220931	−0.110466	+	0.993880i	$0.535234\pi$
$908$	−4.02053e10	−1.78231
$909$	0	0
$910$	−1.09746e7	−0.000482776 0
$911$	2.58578e10	1.13312	0.566562	−	0.824019i	$-0.308273\pi$
$911$	2.58578e10	1.13312	0.566562	+	0.824019i	$0.308273\pi$
$912$	0	0
$913$	5.01427e9	0.218052
$914$	5.89939e9	0.255561
$915$	0	0
$916$	1.32242e10	0.568505
$917$	−1.97987e9	−0.0847900
$918$	0	0
$919$	−4.44902e10	−1.89087	−0.945433	−	0.325818i	$-0.894360\pi$
$919$	−4.44902e10	−1.89087	−0.945433	+	0.325818i	$0.894360\pi$
$920$	−8.47361e6	−0.000358766 0
$921$	0	0
$922$	−6.78440e9	−0.285071
$923$	−4.67389e10	−1.95647
$924$	0	0
$925$	4.55781e10	1.89348
$926$	8.88841e9	0.367863
$927$	0	0
$928$	1.87338e10	0.769498
$929$	−1.94659e9	−0.0796562	−0.0398281	−	0.999207i	$-0.512681\pi$
$929$	−1.94659e9	−0.0796562	−0.0398281	+	0.999207i	$0.512681\pi$
$930$	0	0
$931$	2.35939e10	0.958245
$932$	−2.23146e10	−0.902887
$933$	0	0
$934$	8.62905e9	0.346536
$935$	8.18493e7	0.00327472
$936$	0	0
$937$	−3.37108e10	−1.33869	−0.669346	−	0.742951i	$-0.733426\pi$
$937$	−3.37108e10	−1.33869	−0.669346	+	0.742951i	$0.733426\pi$
$938$	−2.88141e9	−0.113998
$939$	0	0
$940$	8.40269e6	0.000329967 0
$941$	−1.04003e10	−0.406895	−0.203448	−	0.979086i	$-0.565215\pi$
$941$	−1.04003e10	−0.406895	−0.203448	+	0.979086i	$0.565215\pi$
$942$	0	0
$943$	9.20603e9	0.357504
$944$	−2.76621e10	−1.07024
$945$	0	0
$946$	6.87147e9	0.263895
$947$	1.40028e10	0.535783	0.267892	−	0.963449i	$-0.413673\pi$
$947$	1.40028e10	0.535783	0.267892	+	0.963449i	$0.413673\pi$
$948$	0	0
$949$	−3.28683e10	−1.24838
$950$	−9.37427e9	−0.354736
$951$	0	0
$952$	1.20445e10	0.452438
$953$	−4.94805e9	−0.185186	−0.0925931	−	0.995704i	$-0.529516\pi$
$953$	−4.94805e9	−0.185186	−0.0925931	+	0.995704i	$0.529516\pi$
$954$	0	0
$955$	−1.08256e7	−0.000402199 0
$956$	3.94964e9	0.146202
$957$	0	0
$958$	−2.76083e9	−0.101452
$959$	1.25824e10	0.460680
$960$	0	0
$961$	−2.64469e10	−0.961263
$962$	−1.85856e10	−0.673074
$963$	0	0
$964$	1.79366e10	0.644866
$965$	3.30072e7	0.00118240
$966$	0	0
$967$	−1.79902e10	−0.639798	−0.319899	−	0.947452i	$-0.603649\pi$
$967$	−1.79902e10	−0.639798	−0.319899	+	0.947452i	$0.603649\pi$
$968$	−1.01647e10	−0.360190
$969$	0	0
$970$	4.34994e7	0.00153032
$971$	1.54029e10	0.539929	0.269964	−	0.962870i	$-0.412988\pi$
$971$	1.54029e10	0.539929	0.269964	+	0.962870i	$0.412988\pi$
$972$	0	0
$973$	1.67015e10	0.581245
$974$	−8.96064e9	−0.310730
$975$	0	0
$976$	1.97986e10	0.681648
$977$	1.43057e10	0.490769	0.245384	−	0.969426i	$-0.421086\pi$
$977$	1.43057e10	0.490769	0.245384	+	0.969426i	$0.421086\pi$
$978$	0	0
$979$	−6.24235e8	−0.0212622
$980$	−6.53292e7	−0.00221726
$981$	0	0
$982$	−4.15080e9	−0.139876
$983$	−1.72656e10	−0.579756	−0.289878	−	0.957064i	$-0.593615\pi$
$983$	−1.72656e10	−0.579756	−0.289878	+	0.957064i	$0.593615\pi$
$984$	0	0
$985$	5.96008e7	0.00198712
$986$	1.55759e10	0.517469
$987$	0	0
$988$	−3.97484e10	−1.31120
$989$	9.36649e9	0.307886
$990$	0	0
$991$	−1.25900e10	−0.410929	−0.205464	−	0.978665i	$-0.565870\pi$
$991$	−1.25900e10	−0.410929	−0.205464	+	0.978665i	$0.565870\pi$
$992$	4.76197e9	0.154880
$993$	0	0
$994$	−6.68476e9	−0.215891
$995$	−1.40040e7	−0.000450682 0
$996$	0	0
$997$	−2.34721e10	−0.750100	−0.375050	−	0.927005i	$-0.622374\pi$
$997$	−2.34721e10	−0.750100	−0.375050	+	0.927005i	$0.622374\pi$
$998$	7.12681e9	0.226954
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.8.a.c.1.3		6
3.2	odd	2		69.8.a.b.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.a.b.1.4	✓	6	3.2	odd	2
207.8.a.c.1.3		6	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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q-3.35108 q^{2} -116.770 q^{4} -0.849066 q^{5} +405.736 q^{7} +820.245 q^{8} +O(q^{10})\)	\(q-3.35108 q^{2} -116.770 q^{4} -0.849066 q^{5} +405.736 q^{7} +820.245 q^{8} +2.84529 q^{10} -2663.62 q^{11} -9506.50 q^{13} -1359.65 q^{14} +12197.9 q^{16} +36191.1 q^{17} -35806.9 q^{19} +99.1457 q^{20} +8925.99 q^{22} +12167.0 q^{23} -78124.3 q^{25} +31857.0 q^{26} -47377.9 q^{28} -128430. q^{29} -32645.9 q^{31} -145867. q^{32} -121279. q^{34} -344.497 q^{35} -583405. q^{37} +119992. q^{38} -696.442 q^{40} +756639. q^{41} +769827. q^{43} +311031. q^{44} -40772.6 q^{46} +84751.0 q^{47} -658922. q^{49} +261801. q^{50} +1.11008e6 q^{52} +52581.8 q^{53} +2261.59 q^{55} +332803. q^{56} +430380. q^{58} -2.26778e6 q^{59} +1.62312e6 q^{61} +109399. q^{62} -1.07252e6 q^{64} +8071.65 q^{65} +2.11923e6 q^{67} -4.22604e6 q^{68} +1154.44 q^{70} +4.91652e6 q^{71} +3.45746e6 q^{73} +1.95504e6 q^{74} +4.18118e6 q^{76} -1.08072e6 q^{77} +2.91026e6 q^{79} -10356.8 q^{80} -2.53556e6 q^{82} -1.88250e6 q^{83} -30728.6 q^{85} -2.57975e6 q^{86} -2.18482e6 q^{88} +234356. q^{89} -3.85713e6 q^{91} -1.42074e6 q^{92} -284007. q^{94} +30402.4 q^{95} +1.52882e7 q^{97} +2.20810e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{2} + 178 q^{4} + 372 q^{5} - 1104 q^{7} + 1956 q^{8}+O(q^{10}) \) 6 * q + 8 * q^2 + 178 * q^4 + 372 * q^5 - 1104 * q^7 + 1956 * q^8	\( 6 q + 8 q^{2} + 178 q^{4} + 372 q^{5} - 1104 q^{7} + 1956 q^{8} - 13042 q^{10} + 14824 q^{11} - 756 q^{13} + 3926 q^{14} - 13022 q^{16} + 69484 q^{17} - 43864 q^{19} - 78886 q^{20} + 98204 q^{22} + 73002 q^{23} + 228018 q^{25} + 311956 q^{26} - 545442 q^{28} + 311100 q^{29} - 245248 q^{31} + 390156 q^{32} + 235834 q^{34} + 1331256 q^{35} - 630044 q^{37} - 80910 q^{38} - 2153982 q^{40} + 969204 q^{41} - 1770208 q^{43} + 1749140 q^{44} + 97336 q^{46} + 1400024 q^{47} + 1985598 q^{49} + 956660 q^{50} + 3217272 q^{52} + 1573516 q^{53} - 431296 q^{55} - 7740702 q^{56} + 5987188 q^{58} + 1410320 q^{59} - 942172 q^{61} - 3334412 q^{62} + 1996866 q^{64} + 420944 q^{65} - 452072 q^{67} + 9258254 q^{68} + 21981136 q^{70} - 122928 q^{71} + 16490716 q^{73} + 600104 q^{74} + 7428658 q^{76} - 7239696 q^{77} + 2458408 q^{79} - 19440230 q^{80} + 20510784 q^{82} + 7566456 q^{83} + 5817744 q^{85} + 669666 q^{86} + 14775668 q^{88} + 20368036 q^{89} + 8815576 q^{91} + 2165726 q^{92} + 16952576 q^{94} - 5143832 q^{95} + 12586972 q^{97} + 39164812 q^{98}+O(q^{100}) \) 6 * q + 8 * q^2 + 178 * q^4 + 372 * q^5 - 1104 * q^7 + 1956 * q^8 - 13042 * q^10 + 14824 * q^11 - 756 * q^13 + 3926 * q^14 - 13022 * q^16 + 69484 * q^17 - 43864 * q^19 - 78886 * q^20 + 98204 * q^22 + 73002 * q^23 + 228018 * q^25 + 311956 * q^26 - 545442 * q^28 + 311100 * q^29 - 245248 * q^31 + 390156 * q^32 + 235834 * q^34 + 1331256 * q^35 - 630044 * q^37 - 80910 * q^38 - 2153982 * q^40 + 969204 * q^41 - 1770208 * q^43 + 1749140 * q^44 + 97336 * q^46 + 1400024 * q^47 + 1985598 * q^49 + 956660 * q^50 + 3217272 * q^52 + 1573516 * q^53 - 431296 * q^55 - 7740702 * q^56 + 5987188 * q^58 + 1410320 * q^59 - 942172 * q^61 - 3334412 * q^62 + 1996866 * q^64 + 420944 * q^65 - 452072 * q^67 + 9258254 * q^68 + 21981136 * q^70 - 122928 * q^71 + 16490716 * q^73 + 600104 * q^74 + 7428658 * q^76 - 7239696 * q^77 + 2458408 * q^79 - 19440230 * q^80 + 20510784 * q^82 + 7566456 * q^83 + 5817744 * q^85 + 669666 * q^86 + 14775668 * q^88 + 20368036 * q^89 + 8815576 * q^91 + 2165726 * q^92 + 16952576 * q^94 - 5143832 * q^95 + 12586972 * q^97 + 39164812 * q^98

Introduction

L-functions

Modular forms

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