LMFDB - Embedded newform 825.4.c.r.199.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$48.6765757547$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 83x^{8} + 2275x^{6} + 24517x^{4} + 87636x^{2} + 40000 $ x^10 + 83x^8 + 2275x^6 + 24517x^4 + 87636x^2 + 40000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.10
Root			$4.58039i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.4.c.r.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.58039i q^{2} -3.00000i q^{3} -23.1407 q^{4} +16.7412 q^{6} -34.4181i q^{7} -84.4912i q^{8} -9.00000 q^{9} +O(q^{10})$	\(q+5.58039i q^{2} -3.00000i q^{3} -23.1407 q^{4} +16.7412 q^{6} -34.4181i q^{7} -84.4912i q^{8} -9.00000 q^{9} +11.0000 q^{11} +69.4222i q^{12} -71.2887i q^{13} +192.066 q^{14} +286.368 q^{16} +22.3593i q^{17} -50.2235i q^{18} -88.1755 q^{19} -103.254 q^{21} +61.3843i q^{22} +21.5477i q^{23} -253.474 q^{24} +397.819 q^{26} +27.0000i q^{27} +796.460i q^{28} -118.280 q^{29} -33.5268 q^{31} +922.115i q^{32} -33.0000i q^{33} -124.774 q^{34} +208.267 q^{36} -364.750i q^{37} -492.054i q^{38} -213.866 q^{39} +48.9166 q^{41} -576.199i q^{42} +95.8534i q^{43} -254.548 q^{44} -120.244 q^{46} +132.383i q^{47} -859.104i q^{48} -841.605 q^{49} +67.0780 q^{51} +1649.67i q^{52} +300.685i q^{53} -150.671 q^{54} -2908.03 q^{56} +264.526i q^{57} -660.050i q^{58} +654.281 q^{59} -772.721 q^{61} -187.092i q^{62} +309.763i q^{63} -2854.82 q^{64} +184.153 q^{66} +112.087i q^{67} -517.412i q^{68} +64.6430 q^{69} +548.159 q^{71} +760.421i q^{72} +559.178i q^{73} +2035.45 q^{74} +2040.45 q^{76} -378.599i q^{77} -1193.46i q^{78} -48.5332 q^{79} +81.0000 q^{81} +272.974i q^{82} +447.844i q^{83} +2389.38 q^{84} -534.899 q^{86} +354.841i q^{87} -929.403i q^{88} +552.419 q^{89} -2453.62 q^{91} -498.629i q^{92} +100.580i q^{93} -738.746 q^{94} +2766.34 q^{96} +413.909i q^{97} -4696.48i q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 92 q^{4} + 24 q^{6} - 90 q^{9}+O(q^{10}) $ 10 * q - 92 * q^4 + 24 * q^6 - 90 * q^9	$ 10 q - 92 q^{4} + 24 q^{6} - 90 q^{9} + 110 q^{11} - 110 q^{14} + 1268 q^{16} + 674 q^{19} - 228 q^{21} - 432 q^{24} + 718 q^{26} + 306 q^{29} + 526 q^{31} - 1034 q^{34} + 828 q^{36} - 258 q^{39} - 176 q^{41} - 1012 q^{44} + 872 q^{46} - 4762 q^{49} + 900 q^{51} - 216 q^{54} + 422 q^{56} - 820 q^{59} - 2260 q^{61} - 6340 q^{64} + 264 q^{66} + 1206 q^{69} + 2498 q^{71} + 5970 q^{74} - 5112 q^{76} - 4516 q^{79} + 810 q^{81} + 2646 q^{84} - 2870 q^{86} - 694 q^{89} - 1224 q^{91} - 1814 q^{94} + 1680 q^{96} - 990 q^{99}+O(q^{100}) $ 10 * q - 92 * q^4 + 24 * q^6 - 90 * q^9 + 110 * q^11 - 110 * q^14 + 1268 * q^16 + 674 * q^19 - 228 * q^21 - 432 * q^24 + 718 * q^26 + 306 * q^29 + 526 * q^31 - 1034 * q^34 + 828 * q^36 - 258 * q^39 - 176 * q^41 - 1012 * q^44 + 872 * q^46 - 4762 * q^49 + 900 * q^51 - 216 * q^54 + 422 * q^56 - 820 * q^59 - 2260 * q^61 - 6340 * q^64 + 264 * q^66 + 1206 * q^69 + 2498 * q^71 + 5970 * q^74 - 5112 * q^76 - 4516 * q^79 + 810 * q^81 + 2646 * q^84 - 2870 * q^86 - 694 * q^89 - 1224 * q^91 - 1814 * q^94 + 1680 * q^96 - 990 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	5.58039i	1.97297i	0.163865	+	0.986483i	$0.447604\pi$
$2$	5.58039i	1.97297i	−0.163865	+	0.986483i	$0.552396\pi$
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	−23.1407	−2.89259
$5$	0	0
$5$	0	0
$6$	16.7412	1.13909
$7$	− 34.4181i	− 1.85840i	−0.369575	−	0.929201i	$-0.620497\pi$
$7$	− 34.4181i	− 1.85840i	0.369575	−	0.929201i	$-0.379503\pi$
$8$	− 84.4912i	− 3.73402i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	69.4222i	1.67004i
$13$	− 71.2887i	− 1.52092i	−0.649386	−	0.760459i	$-0.724974\pi$
$13$	− 71.2887i	− 1.52092i	0.649386	−	0.760459i	$-0.275026\pi$
$14$	192.066	3.66656
$15$	0	0
$16$	286.368	4.47450
$17$	22.3593i	0.318996i	0.987198	+	0.159498i	$0.0509876\pi$
$17$	22.3593i	0.318996i	−0.987198	+	0.159498i	$0.949012\pi$
$18$	− 50.2235i	− 0.657655i
$19$	−88.1755	−1.06468	−0.532338	−	0.846532i	$-0.678686\pi$
$19$	−88.1755	−1.06468	−0.532338	+	0.846532i	$0.678686\pi$
$20$	0	0
$21$	−103.254	−1.07295
$22$	61.3843i	0.594871i
$23$	21.5477i	0.195348i	0.995218	+	0.0976738i	$0.0311402\pi$
$23$	21.5477i	0.195348i	−0.995218	+	0.0976738i	$0.968860\pi$
$24$	−253.474	−2.15584
$25$	0	0
$26$	397.819	3.00072
$27$	27.0000i	0.192450i
$28$	796.460i	5.37560i
$29$	−118.280	−0.757383	−0.378691	−	0.925523i	$-0.623626\pi$
$29$	−118.280	−0.757383	−0.378691	+	0.925523i	$0.623626\pi$
$30$	0	0
$31$	−33.5268	−0.194245	−0.0971223	−	0.995272i	$-0.530964\pi$
$31$	−33.5268	−0.194245	−0.0971223	+	0.995272i	$0.530964\pi$
$32$	922.115i	5.09401i
$33$	− 33.0000i	− 0.174078i
$34$	−124.774	−0.629369
$35$	0	0
$36$	208.267	0.964198
$37$	− 364.750i	− 1.62066i	−0.585970	−	0.810332i	$-0.699287\pi$
$37$	− 364.750i	− 1.62066i	0.585970	−	0.810332i	$-0.300713\pi$
$38$	− 492.054i	− 2.10057i
$39$	−213.866	−0.878102
$40$	0	0
$41$	48.9166	0.186329	0.0931645	−	0.995651i	$-0.470302\pi$
$41$	48.9166	0.186329	0.0931645	+	0.995651i	$0.470302\pi$
$42$	− 576.199i	− 2.11689i
$43$	95.8534i	0.339942i	0.985449	+	0.169971i	$0.0543674\pi$
$43$	95.8534i	0.339942i	−0.985449	+	0.169971i	$0.945633\pi$
$44$	−254.548	−0.872149
$45$	0	0
$46$	−120.244	−0.385414
$47$	132.383i	0.410851i	0.978673	+	0.205425i	$0.0658578\pi$
$47$	132.383i	0.410851i	−0.978673	+	0.205425i	$0.934142\pi$
$48$	− 859.104i	− 2.58335i
$49$	−841.605	−2.45366
$50$	0	0
$51$	67.0780	0.184173
$52$	1649.67i	4.39939i
$53$	300.685i	0.779288i	0.920966	+	0.389644i	$0.127402\pi$
$53$	300.685i	0.779288i	−0.920966	+	0.389644i	$0.872598\pi$
$54$	−150.671	−0.379697
$55$	0	0
$56$	−2908.03	−6.93931
$57$	264.526i	0.614691i
$58$	− 660.050i	− 1.49429i
$59$	654.281	1.44373	0.721865	−	0.692033i	$-0.243285\pi$
$59$	654.281	1.44373	0.721865	+	0.692033i	$0.243285\pi$
$60$	0	0
$61$	−772.721	−1.62191	−0.810957	−	0.585106i	$-0.801053\pi$
$61$	−772.721	−1.62191	−0.810957	+	0.585106i	$0.801053\pi$
$62$	− 187.092i	− 0.383238i
$63$	309.763i	0.619467i
$64$	−2854.82	−5.57581
$65$	0	0
$66$	184.153	0.343449
$67$	112.087i	0.204382i	0.994765	+	0.102191i	$0.0325853\pi$
$67$	112.087i	0.204382i	−0.994765	+	0.102191i	$0.967415\pi$
$68$	− 517.412i	− 0.922726i
$69$	64.6430	0.112784
$70$	0	0
$71$	548.159	0.916261	0.458130	−	0.888885i	$-0.348519\pi$
$71$	548.159	0.916261	0.458130	+	0.888885i	$0.348519\pi$
$72$	760.421i	1.24467i
$73$	559.178i	0.896531i	0.893900	+	0.448266i	$0.147958\pi$
$73$	559.178i	0.896531i	−0.893900	+	0.448266i	$0.852042\pi$
$74$	2035.45	3.19752
$75$	0	0
$76$	2040.45	3.07967
$77$	− 378.599i	− 0.560329i
$78$	− 1193.46i	− 1.73247i
$79$	−48.5332	−0.0691192	−0.0345596	−	0.999403i	$-0.511003\pi$
$79$	−48.5332	−0.0691192	−0.0345596	+	0.999403i	$0.511003\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	272.974i	0.367621i
$83$	447.844i	0.592256i	0.955148	+	0.296128i	$0.0956956\pi$
$83$	447.844i	0.592256i	−0.955148	+	0.296128i	$0.904304\pi$
$84$	2389.38	3.10360
$85$	0	0
$86$	−534.899	−0.670694
$87$	354.841i	0.437275i
$88$	− 929.403i	− 1.12585i
$89$	552.419	0.657936	0.328968	−	0.944341i	$-0.393299\pi$
$89$	552.419	0.657936	0.328968	+	0.944341i	$0.393299\pi$
$90$	0	0
$91$	−2453.62	−2.82648
$92$	− 498.629i	− 0.565061i
$93$	100.580i	0.112147i
$94$	−738.746	−0.810594
$95$	0	0
$96$	2766.34	2.94103
$97$	413.909i	0.433259i	0.976254	+	0.216630i	$0.0695064\pi$
$97$	413.909i	0.433259i	−0.976254	+	0.216630i	$0.930494\pi$
$98$	− 4696.48i	− 4.84098i
$99$	−99.0000	−0.100504
$100$	0	0
$101$	−439.536	−0.433025	−0.216512	−	0.976280i	$-0.569468\pi$
$101$	−439.536	−0.433025	−0.216512	+	0.976280i	$0.569468\pi$
$102$	374.322i	0.363366i
$103$	− 1112.25i	− 1.06401i	−0.846741	−	0.532005i	$-0.821439\pi$
$103$	− 1112.25i	− 1.06401i	0.846741	−	0.532005i	$-0.178561\pi$
$104$	−6023.27	−5.67914
$105$	0	0
$106$	−1677.94	−1.53751
$107$	− 863.930i	− 0.780554i	−0.920698	−	0.390277i	$-0.872379\pi$
$107$	− 863.930i	− 0.780554i	0.920698	−	0.390277i	$-0.127621\pi$
$108$	− 624.800i	− 0.556680i
$109$	1040.80	0.914588	0.457294	−	0.889316i	$-0.348819\pi$
$109$	1040.80	0.914588	0.457294	+	0.889316i	$0.348819\pi$
$110$	0	0
$111$	−1094.25	−0.935691
$112$	− 9856.24i	− 8.31542i
$113$	1826.14i	1.52025i	0.649774	+	0.760127i	$0.274863\pi$
$113$	1826.14i	1.52025i	−0.649774	+	0.760127i	$0.725137\pi$
$114$	−1476.16	−1.21276
$115$	0	0
$116$	2737.09	2.19080
$117$	641.598i	0.506972i
$118$	3651.14i	2.84843i
$119$	769.566	0.592823
$120$	0	0
$121$	121.000	0.0909091
$122$	− 4312.08i	− 3.19998i
$123$	− 146.750i	− 0.107577i
$124$	775.834	0.561871
$125$	0	0
$126$	−1728.60	−1.22219
$127$	574.281i	0.401254i	0.979668	+	0.200627i	$0.0642979\pi$
$127$	574.281i	0.401254i	−0.979668	+	0.200627i	$0.935702\pi$
$128$	− 8554.06i	− 5.90687i
$129$	287.560	0.196266
$130$	0	0
$131$	−2684.56	−1.79047	−0.895234	−	0.445596i	$-0.852992\pi$
$131$	−2684.56	−1.79047	−0.895234	+	0.445596i	$0.852992\pi$
$132$	763.644i	0.503536i
$133$	3034.83i	1.97860i
$134$	−625.489	−0.403239
$135$	0	0
$136$	1889.17	1.19114
$137$	− 3013.45i	− 1.87925i	−0.342212	−	0.939623i	$-0.611176\pi$
$137$	− 3013.45i	− 1.87925i	0.342212	−	0.939623i	$-0.388824\pi$
$138$	360.733i	0.222519i
$139$	−277.411	−0.169278	−0.0846391	−	0.996412i	$-0.526974\pi$
$139$	−277.411	−0.169278	−0.0846391	+	0.996412i	$0.526974\pi$
$140$	0	0
$141$	397.148	0.237205
$142$	3058.94i	1.80775i
$143$	− 784.176i	− 0.458574i
$144$	−2577.31	−1.49150
$145$	0	0
$146$	−3120.43	−1.76883
$147$	2524.81i	1.41662i
$148$	8440.59i	4.68792i
$149$	−1594.62	−0.876755	−0.438378	−	0.898791i	$-0.644447\pi$
$149$	−1594.62	−0.876755	−0.438378	+	0.898791i	$0.644447\pi$
$150$	0	0
$151$	2213.29	1.19281	0.596407	−	0.802682i	$-0.296594\pi$
$151$	2213.29	1.19281	0.596407	+	0.802682i	$0.296594\pi$
$152$	7450.06i	3.97552i
$153$	− 201.234i	− 0.106332i
$154$	2112.73	1.10551
$155$	0	0
$156$	4949.02	2.53999
$157$	− 1145.63i	− 0.582367i	−0.956667	−	0.291183i	$-0.905951\pi$
$157$	− 1145.63i	− 0.582367i	0.956667	−	0.291183i	$-0.0940490\pi$
$158$	− 270.834i	− 0.136370i
$159$	902.055	0.449922
$160$	0	0
$161$	741.629	0.363034
$162$	452.012i	0.219218i
$163$	594.732i	0.285785i	0.989738	+	0.142893i	$0.0456404\pi$
$163$	594.732i	0.285785i	−0.989738	+	0.142893i	$0.954360\pi$
$164$	−1131.97	−0.538974
$165$	0	0
$166$	−2499.14	−1.16850
$167$	2927.43i	1.35648i	0.734842	+	0.678239i	$0.237256\pi$
$167$	2927.43i	1.35648i	−0.734842	+	0.678239i	$0.762744\pi$
$168$	8724.08i	4.00641i
$169$	−2885.08	−1.31319
$170$	0	0
$171$	793.579	0.354892
$172$	− 2218.12i	− 0.983314i
$173$	1274.26i	0.560000i	0.960000	+	0.280000i	$0.0903344\pi$
$173$	1274.26i	0.560000i	−0.960000	+	0.280000i	$0.909666\pi$
$174$	−1980.15	−0.862729
$175$	0	0
$176$	3150.05	1.34911
$177$	− 1962.84i	− 0.833538i
$178$	3082.71i	1.29809i
$179$	−1626.12	−0.679005	−0.339503	−	0.940605i	$-0.610259\pi$
$179$	−1626.12	−0.679005	−0.339503	+	0.940605i	$0.610259\pi$
$180$	0	0
$181$	1638.75	0.672970	0.336485	−	0.941689i	$-0.390762\pi$
$181$	1638.75	0.672970	0.336485	+	0.941689i	$0.390762\pi$
$182$	− 13692.2i	− 5.57654i
$183$	2318.16i	0.936412i
$184$	1820.59	0.729432
$185$	0	0
$186$	−561.277	−0.221263
$187$	245.953i	0.0961810i
$188$	− 3063.43i	− 1.18842i
$189$	929.288	0.357650
$190$	0	0
$191$	3460.92	1.31112	0.655559	−	0.755144i	$-0.272433\pi$
$191$	3460.92	1.31112	0.655559	+	0.755144i	$0.272433\pi$
$192$	8564.45i	3.21920i
$193$	− 1725.78i	− 0.643650i	−0.946799	−	0.321825i	$-0.895704\pi$
$193$	− 1725.78i	− 0.643650i	0.946799	−	0.321825i	$-0.104296\pi$
$194$	−2309.78	−0.854805
$195$	0	0
$196$	19475.4	7.09743
$197$	− 405.275i	− 0.146572i	−0.997311	−	0.0732859i	$-0.976651\pi$
$197$	− 405.275i	− 0.146572i	0.997311	−	0.0732859i	$-0.0233486\pi$
$198$	− 552.459i	− 0.198290i
$199$	−3801.45	−1.35416	−0.677079	−	0.735910i	$-0.736755\pi$
$199$	−3801.45	−1.35416	−0.677079	+	0.735910i	$0.736755\pi$
$200$	0	0
$201$	336.261	0.118000
$202$	− 2452.78i	− 0.854343i
$203$	4070.98i	1.40752i
$204$	−1552.24	−0.532736
$205$	0	0
$206$	6206.78	2.09926
$207$	− 193.929i	− 0.0651159i
$208$	− 20414.8i	− 6.80534i
$209$	−969.930	−0.321012
$210$	0	0
$211$	−1538.57	−0.501987	−0.250993	−	0.967989i	$-0.580757\pi$
$211$	−1538.57	−0.501987	−0.250993	+	0.967989i	$0.580757\pi$
$212$	− 6958.07i	− 2.25416i
$213$	− 1644.48i	− 0.529003i
$214$	4821.07	1.54001
$215$	0	0
$216$	2281.26	0.718612
$217$	1153.93i	0.360985i
$218$	5808.04i	1.80445i
$219$	1677.53	0.517613
$220$	0	0
$221$	1593.97	0.485167
$222$	− 6106.35i	− 1.84609i
$223$	− 2558.19i	− 0.768201i	−0.923291	−	0.384101i	$-0.874512\pi$
$223$	− 2558.19i	− 0.768201i	0.923291	−	0.384101i	$-0.125488\pi$
$224$	31737.4	9.46672
$225$	0	0
$226$	−10190.6	−2.99941
$227$	5834.09i	1.70582i	0.522055	+	0.852912i	$0.325166\pi$
$227$	5834.09i	1.70582i	−0.522055	+	0.852912i	$0.674834\pi$
$228$	− 6121.34i	− 1.77805i
$229$	1308.94	0.377718	0.188859	−	0.982004i	$-0.439521\pi$
$229$	1308.94	0.377718	0.188859	+	0.982004i	$0.439521\pi$
$230$	0	0
$231$	−1135.80	−0.323506
$232$	9993.65i	2.82808i
$233$	2360.58i	0.663719i	0.943329	+	0.331859i	$0.107676\pi$
$233$	2360.58i	0.663719i	−0.943329	+	0.331859i	$0.892324\pi$
$234$	−3580.37	−1.00024
$235$	0	0
$236$	−15140.5	−4.17613
$237$	145.600i	0.0399060i
$238$	4294.48i	1.16962i
$239$	−5766.38	−1.56065	−0.780327	−	0.625372i	$-0.784947\pi$
$239$	−5766.38	−1.56065	−0.780327	+	0.625372i	$0.784947\pi$
$240$	0	0
$241$	−2644.73	−0.706896	−0.353448	−	0.935454i	$-0.614991\pi$
$241$	−2644.73	−0.706896	−0.353448	+	0.935454i	$0.614991\pi$
$242$	675.227i	0.179360i
$243$	− 243.000i	− 0.0641500i
$244$	17881.3	4.69154
$245$	0	0
$246$	818.921	0.212246
$247$	6285.92i	1.61928i
$248$	2832.72i	0.725313i
$249$	1343.53	0.341939
$250$	0	0
$251$	−2435.59	−0.612483	−0.306241	−	0.951954i	$-0.599071\pi$
$251$	−2435.59	−0.612483	−0.306241	+	0.951954i	$0.599071\pi$
$252$	− 7168.14i	− 1.79187i
$253$	237.024i	0.0588995i
$254$	−3204.71	−0.791660
$255$	0	0
$256$	24896.5	6.07824
$257$	− 5522.51i	− 1.34041i	−0.742177	−	0.670204i	$-0.766207\pi$
$257$	− 5522.51i	− 1.34041i	0.742177	−	0.670204i	$-0.233793\pi$
$258$	1604.70i	0.387225i
$259$	−12554.0	−3.01185
$260$	0	0
$261$	1064.52	0.252461
$262$	− 14980.9i	− 3.53253i
$263$	− 2699.88i	− 0.633011i	−0.948591	−	0.316505i	$-0.897490\pi$
$263$	− 2699.88i	− 0.633011i	0.948591	−	0.316505i	$-0.102510\pi$
$264$	−2788.21	−0.650009
$265$	0	0
$266$	−16935.5	−3.90370
$267$	− 1657.26i	− 0.379860i
$268$	− 2593.77i	− 0.591194i
$269$	−1593.94	−0.361279	−0.180639	−	0.983549i	$-0.557817\pi$
$269$	−1593.94	−0.361279	−0.180639	+	0.983549i	$0.557817\pi$
$270$	0	0
$271$	1826.60	0.409440	0.204720	−	0.978821i	$-0.434372\pi$
$271$	1826.60	0.409440	0.204720	+	0.978821i	$0.434372\pi$
$272$	6403.00i	1.42735i
$273$	7360.86i	1.63187i
$274$	16816.2	3.70769
$275$	0	0
$276$	−1495.89	−0.326238
$277$	2702.41i	0.586181i	0.956085	+	0.293091i	$0.0946838\pi$
$277$	2702.41i	0.586181i	−0.956085	+	0.293091i	$0.905316\pi$
$278$	− 1548.06i	− 0.333980i
$279$	301.741	0.0647482
$280$	0	0
$281$	−2961.96	−0.628810	−0.314405	−	0.949289i	$-0.601805\pi$
$281$	−2961.96	−0.628810	−0.314405	+	0.949289i	$0.601805\pi$
$282$	2216.24i	0.467997i
$283$	3504.45i	0.736105i	0.929805	+	0.368052i	$0.119975\pi$
$283$	3504.45i	0.736105i	−0.929805	+	0.368052i	$0.880025\pi$
$284$	−12684.8	−2.65037
$285$	0	0
$286$	4376.01	0.904750
$287$	− 1683.62i	− 0.346274i
$288$	− 8299.03i	− 1.69800i
$289$	4413.06	0.898241
$290$	0	0
$291$	1241.73	0.250142
$292$	− 12939.8i	− 2.59330i
$293$	1004.75i	0.200335i	0.994971	+	0.100168i	$0.0319379\pi$
$293$	1004.75i	0.200335i	−0.994971	+	0.100168i	$0.968062\pi$
$294$	−14089.4	−2.79494
$295$	0	0
$296$	−30818.2	−6.05159
$297$	297.000i	0.0580259i
$298$	− 8898.61i	− 1.72981i
$299$	1536.10	0.297108
$300$	0	0
$301$	3299.09	0.631749
$302$	12351.0i	2.35338i
$303$	1318.61i	0.250007i
$304$	−25250.6	−4.76389
$305$	0	0
$306$	1122.96	0.209790
$307$	− 6690.01i	− 1.24371i	−0.783132	−	0.621856i	$-0.786379\pi$
$307$	− 6690.01i	− 1.24371i	0.783132	−	0.621856i	$-0.213621\pi$
$308$	8761.06i	1.62080i
$309$	−3336.74	−0.614307
$310$	0	0
$311$	−1700.37	−0.310030	−0.155015	−	0.987912i	$-0.549543\pi$
$311$	−1700.37	−0.310030	−0.155015	+	0.987912i	$0.549543\pi$
$312$	18069.8i	3.27885i
$313$	439.046i	0.0792855i	0.999214	+	0.0396428i	$0.0126220\pi$
$313$	439.046i	0.0792855i	−0.999214	+	0.0396428i	$0.987378\pi$
$314$	6393.09	1.14899
$315$	0	0
$316$	1123.10	0.199934
$317$	5152.91i	0.912985i	0.889727	+	0.456492i	$0.150894\pi$
$317$	5152.91i	0.912985i	−0.889727	+	0.456492i	$0.849106\pi$
$318$	5033.82i	0.887681i
$319$	−1301.08	−0.228360
$320$	0	0
$321$	−2591.79	−0.450653
$322$	4138.58i	0.716255i
$323$	− 1971.55i	− 0.339628i
$324$	−1874.40	−0.321399
$325$	0	0
$326$	−3318.84	−0.563844
$327$	− 3122.39i	− 0.528038i
$328$	− 4133.02i	− 0.695756i
$329$	4556.36	0.763526
$330$	0	0
$331$	−3246.99	−0.539187	−0.269593	−	0.962974i	$-0.586889\pi$
$331$	−3246.99	−0.539187	−0.269593	+	0.962974i	$0.586889\pi$
$332$	− 10363.4i	− 1.71316i
$333$	3282.75i	0.540222i
$334$	−16336.2	−2.67628
$335$	0	0
$336$	−29568.7	−4.80091
$337$	− 1338.15i	− 0.216301i	−0.994135	−	0.108151i	$-0.965507\pi$
$337$	− 1338.15i	− 0.216301i	0.994135	−	0.108151i	$-0.0344929\pi$
$338$	− 16099.9i	− 2.59088i
$339$	5478.42	0.877719
$340$	0	0
$341$	−368.794	−0.0585670
$342$	4428.48i	0.700190i
$343$	17161.0i	2.70148i
$344$	8098.77	1.26935
$345$	0	0
$346$	−7110.85	−1.10486
$347$	− 3421.40i	− 0.529309i	−0.964343	−	0.264655i	$-0.914742\pi$
$347$	− 3421.40i	− 0.529309i	0.964343	−	0.264655i	$-0.0852579\pi$
$348$	− 8211.28i	− 1.26486i
$349$	7537.81	1.15613	0.578066	−	0.815990i	$-0.303808\pi$
$349$	7537.81	1.15613	0.578066	+	0.815990i	$0.303808\pi$
$350$	0	0
$351$	1924.79	0.292701
$352$	10143.3i	1.53590i
$353$	5803.16i	0.874988i	0.899221	+	0.437494i	$0.144134\pi$
$353$	5803.16i	0.874988i	−0.899221	+	0.437494i	$0.855866\pi$
$354$	10953.4	1.64454
$355$	0	0
$356$	−12783.4	−1.90314
$357$	− 2308.70i	− 0.342267i
$358$	− 9074.39i	− 1.33965i
$359$	−5558.49	−0.817175	−0.408587	−	0.912719i	$-0.633978\pi$
$359$	−5558.49	−0.817175	−0.408587	+	0.912719i	$0.633978\pi$
$360$	0	0
$361$	915.919	0.133535
$362$	9144.88i	1.32775i
$363$	− 363.000i	− 0.0524864i
$364$	56778.6	8.17584
$365$	0	0
$366$	−12936.2	−1.84751
$367$	− 2853.78i	− 0.405902i	−0.979189	−	0.202951i	$-0.934947\pi$
$367$	− 2853.78i	− 0.405902i	0.979189	−	0.202951i	$-0.0650532\pi$
$368$	6170.56i	0.874083i
$369$	−440.249	−0.0621097
$370$	0	0
$371$	10349.0	1.44823
$372$	− 2327.50i	− 0.324396i
$373$	7500.89i	1.04124i	0.853789	+	0.520618i	$0.174299\pi$
$373$	7500.89i	1.04124i	−0.853789	+	0.520618i	$0.825701\pi$
$374$	−1372.51	−0.189762
$375$	0	0
$376$	11185.2	1.53412
$377$	8432.05i	1.15192i
$378$	5185.79i	0.705630i
$379$	−12965.1	−1.75718	−0.878591	−	0.477575i	$-0.841516\pi$
$379$	−12965.1	−1.75718	−0.878591	+	0.477575i	$0.841516\pi$
$380$	0	0
$381$	1722.84	0.231664
$382$	19313.3i	2.58679i
$383$	4928.95i	0.657591i	0.944401	+	0.328796i	$0.106643\pi$
$383$	4928.95i	0.657591i	−0.944401	+	0.328796i	$0.893357\pi$
$384$	−25662.2	−3.41033
$385$	0	0
$386$	9630.53	1.26990
$387$	− 862.681i	− 0.113314i
$388$	− 9578.17i	− 1.25324i
$389$	−13182.9	−1.71825	−0.859123	−	0.511769i	$-0.828990\pi$
$389$	−13182.9	−1.71825	−0.859123	+	0.511769i	$0.828990\pi$
$390$	0	0
$391$	−481.791	−0.0623152
$392$	71108.2i	9.16201i
$393$	8053.69i	1.03373i
$394$	2261.59	0.289181
$395$	0	0
$396$	2290.93	0.290716
$397$	− 13815.5i	− 1.74655i	−0.487223	−	0.873277i	$-0.661990\pi$
$397$	− 13815.5i	− 1.74655i	0.487223	−	0.873277i	$-0.338010\pi$
$398$	− 21213.6i	− 2.67171i
$399$	9104.50	1.14234
$400$	0	0
$401$	4812.84	0.599357	0.299678	−	0.954040i	$-0.403121\pi$
$401$	4812.84	0.599357	0.299678	+	0.954040i	$0.403121\pi$
$402$	1876.47i	0.232810i
$403$	2390.08i	0.295430i
$404$	10171.2	1.25256
$405$	0	0
$406$	−22717.7	−2.77699
$407$	− 4012.25i	− 0.488649i
$408$	− 5667.51i	− 0.687704i
$409$	9776.38	1.18193	0.590967	−	0.806696i	$-0.298746\pi$
$409$	9776.38	1.18193	0.590967	+	0.806696i	$0.298746\pi$
$410$	0	0
$411$	−9040.36	−1.08498
$412$	25738.2i	3.07775i
$413$	− 22519.1i	− 2.68303i
$414$	1082.20	0.128471
$415$	0	0
$416$	65736.4	7.74757
$417$	832.233i	0.0977329i
$418$	− 5412.59i	− 0.633345i
$419$	−14077.8	−1.64139	−0.820697	−	0.571363i	$-0.806415\pi$
$419$	−14077.8	−1.64139	−0.820697	+	0.571363i	$0.806415\pi$
$420$	0	0
$421$	17100.8	1.97967	0.989837	−	0.142204i	$-0.0454188\pi$
$421$	17100.8	1.97967	0.989837	+	0.142204i	$0.0454188\pi$
$422$	− 8585.79i	− 0.990402i
$423$	− 1191.44i	− 0.136950i
$424$	25405.2	2.90988
$425$	0	0
$426$	9176.82	1.04371
$427$	26595.6i	3.01417i
$428$	19992.0i	2.25782i
$429$	−2352.53	−0.264758
$430$	0	0
$431$	−4374.76	−0.488920	−0.244460	−	0.969659i	$-0.578611\pi$
$431$	−4374.76	−0.488920	−0.244460	+	0.969659i	$0.578611\pi$
$432$	7731.93i	0.861118i
$433$	− 5388.62i	− 0.598061i	−0.954244	−	0.299031i	$-0.903337\pi$
$433$	− 5388.62i	− 0.598061i	0.954244	−	0.299031i	$-0.0966633\pi$
$434$	−6439.36	−0.712210
$435$	0	0
$436$	−24084.8	−2.64553
$437$	− 1899.98i	− 0.207982i
$438$	9361.29i	1.02123i
$439$	−5378.90	−0.584786	−0.292393	−	0.956298i	$-0.594451\pi$
$439$	−5378.90	−0.584786	−0.292393	+	0.956298i	$0.594451\pi$
$440$	0	0
$441$	7574.44	0.817886
$442$	8894.96i	0.957218i
$443$	− 6872.36i	− 0.737055i	−0.929617	−	0.368528i	$-0.879862\pi$
$443$	− 6872.36i	− 0.737055i	0.929617	−	0.368528i	$-0.120138\pi$
$444$	25321.8	2.70657
$445$	0	0
$446$	14275.7	1.51563
$447$	4783.87i	0.506195i
$448$	98257.3i	10.3621i
$449$	−8196.97	−0.861557	−0.430778	−	0.902458i	$-0.641761\pi$
$449$	−8196.97	−0.861557	−0.430778	+	0.902458i	$0.641761\pi$
$450$	0	0
$451$	538.083	0.0561803
$452$	− 42258.2i	− 4.39748i
$453$	− 6639.87i	− 0.688672i
$454$	−32556.5	−3.36553
$455$	0	0
$456$	22350.2	2.29527
$457$	3727.32i	0.381525i	0.981636	+	0.190762i	$0.0610960\pi$
$457$	3727.32i	0.381525i	−0.981636	+	0.190762i	$0.938904\pi$
$458$	7304.42i	0.745225i
$459$	−603.702	−0.0613909
$460$	0	0
$461$	5801.06	0.586079	0.293039	−	0.956100i	$-0.405333\pi$
$461$	5801.06	0.586079	0.293039	+	0.956100i	$0.405333\pi$
$462$	− 6338.19i	− 0.638267i
$463$	12402.6i	1.24492i	0.782651	+	0.622461i	$0.213867\pi$
$463$	12402.6i	1.24492i	−0.782651	+	0.622461i	$0.786133\pi$
$464$	−33871.7	−3.38891
$465$	0	0
$466$	−13172.9	−1.30949
$467$	− 11343.7i	− 1.12404i	−0.827125	−	0.562018i	$-0.810025\pi$
$467$	− 11343.7i	− 1.12404i	0.827125	−	0.562018i	$-0.189975\pi$
$468$	− 14847.1i	− 1.46646i
$469$	3857.82	0.379824
$470$	0	0
$471$	−3436.90	−0.336230
$472$	− 55281.0i	− 5.39092i
$473$	1054.39i	0.102496i
$474$	−812.503	−0.0787331
$475$	0	0
$476$	−17808.3	−1.71480
$477$	− 2706.16i	− 0.259763i
$478$	− 32178.6i	− 3.07912i
$479$	2383.99	0.227405	0.113703	−	0.993515i	$-0.463729\pi$
$479$	2383.99	0.227405	0.113703	+	0.993515i	$0.463729\pi$
$480$	0	0
$481$	−26002.6	−2.46490
$482$	− 14758.6i	− 1.39468i
$483$	− 2224.89i	− 0.209598i
$484$	−2800.03	−0.262963
$485$	0	0
$486$	1356.03	0.126566
$487$	− 11186.7i	− 1.04090i	−0.853891	−	0.520451i	$-0.825764\pi$
$487$	− 11186.7i	− 1.04090i	0.853891	−	0.520451i	$-0.174236\pi$
$488$	65288.1i	6.05626i
$489$	1784.20	0.164998
$490$	0	0
$491$	−9326.27	−0.857207	−0.428604	−	0.903493i	$-0.640994\pi$
$491$	−9326.27	−0.857207	−0.428604	+	0.903493i	$0.640994\pi$
$492$	3395.90i	0.311177i
$493$	− 2644.67i	− 0.241602i
$494$	−35077.9	−3.19479
$495$	0	0
$496$	−9600.99	−0.869148
$497$	− 18866.6i	− 1.70278i
$498$	7497.43i	0.674635i
$499$	−19118.7	−1.71517	−0.857587	−	0.514339i	$-0.828037\pi$
$499$	−19118.7	−1.71517	−0.857587	+	0.514339i	$0.828037\pi$
$500$	0	0
$501$	8782.30	0.783163
$502$	− 13591.5i	− 1.20841i
$503$	8523.10i	0.755519i	0.925904	+	0.377760i	$0.123305\pi$
$503$	8523.10i	0.755519i	−0.925904	+	0.377760i	$0.876695\pi$
$504$	26172.2	2.31310
$505$	0	0
$506$	−1322.69	−0.116207
$507$	8655.24i	0.758171i
$508$	− 13289.3i	− 1.16066i
$509$	10064.2	0.876397	0.438199	−	0.898878i	$-0.355617\pi$
$509$	10064.2	0.876397	0.438199	+	0.898878i	$0.355617\pi$
$510$	0	0
$511$	19245.8	1.66612
$512$	70499.5i	6.08529i
$513$	− 2380.74i	− 0.204897i
$514$	30817.7	2.64458
$515$	0	0
$516$	−6654.36	−0.567716
$517$	1456.21i	0.123876i
$518$	− 70056.3i	− 5.94227i
$519$	3822.77	0.323316
$520$	0	0
$521$	20495.7	1.72348	0.861742	−	0.507347i	$-0.169374\pi$
$521$	20495.7	1.72348	0.861742	+	0.507347i	$0.169374\pi$
$522$	5940.45i	0.498097i
$523$	8079.94i	0.675547i	0.941227	+	0.337773i	$0.109674\pi$
$523$	8079.94i	0.675547i	−0.941227	+	0.337773i	$0.890326\pi$
$524$	62122.8	5.17910
$525$	0	0
$526$	15066.4	1.24891
$527$	− 749.637i	− 0.0619633i
$528$	− 9450.14i	− 0.778910i
$529$	11702.7	0.961839
$530$	0	0
$531$	−5888.53	−0.481244
$532$	− 70228.3i	− 5.72327i
$533$	− 3487.20i	− 0.283391i
$534$	9248.14	0.749450
$535$	0	0
$536$	9470.36	0.763167
$537$	4878.36i	0.392024i
$538$	− 8894.78i	− 0.712791i
$539$	−9257.65	−0.739806
$540$	0	0
$541$	−17460.5	−1.38759	−0.693793	−	0.720174i	$-0.744062\pi$
$541$	−17460.5	−1.38759	−0.693793	+	0.720174i	$0.744062\pi$
$542$	10193.2i	0.807811i
$543$	− 4916.26i	− 0.388539i
$544$	−20617.9	−1.62497
$545$	0	0
$546$	−41076.5	−3.21962
$547$	6652.51i	0.520001i	0.965608	+	0.260001i	$0.0837228\pi$
$547$	6652.51i	0.520001i	−0.965608	+	0.260001i	$0.916277\pi$
$548$	69733.5i	5.43589i
$549$	6954.49	0.540638
$550$	0	0
$551$	10429.4	0.806368
$552$	− 5461.76i	− 0.421138i
$553$	1670.42i	0.128451i
$554$	−15080.5	−1.15652
$555$	0	0
$556$	6419.49	0.489653
$557$	− 5887.04i	− 0.447831i	−0.974609	−	0.223916i	$-0.928116\pi$
$557$	− 5887.04i	− 0.447831i	0.974609	−	0.223916i	$-0.0718840\pi$
$558$	1683.83i	0.127746i
$559$	6833.26	0.517024
$560$	0	0
$561$	737.858	0.0555301
$562$	− 16528.9i	− 1.24062i
$563$	− 3208.71i	− 0.240197i	−0.992762	−	0.120099i	$-0.961679\pi$
$563$	− 3208.71i	− 0.240197i	0.992762	−	0.120099i	$-0.0383210\pi$
$564$	−9190.29	−0.686137
$565$	0	0
$566$	−19556.2	−1.45231
$567$	− 2787.86i	− 0.206489i
$568$	− 46314.6i	− 3.42134i
$569$	−17083.4	−1.25865	−0.629326	−	0.777142i	$-0.716669\pi$
$569$	−17083.4	−1.25865	−0.629326	+	0.777142i	$0.716669\pi$
$570$	0	0
$571$	−16469.8	−1.20707	−0.603536	−	0.797336i	$-0.706242\pi$
$571$	−16469.8	−1.20707	−0.603536	+	0.797336i	$0.706242\pi$
$572$	18146.4i	1.32647i
$573$	− 10382.8i	− 0.756974i
$574$	9395.23	0.683187
$575$	0	0
$576$	25693.3	1.85860
$577$	12436.4i	0.897287i	0.893711	+	0.448643i	$0.148093\pi$
$577$	12436.4i	0.897287i	−0.893711	+	0.448643i	$0.851907\pi$
$578$	24626.6i	1.77220i
$579$	−5177.34	−0.371611
$580$	0	0
$581$	15413.9	1.10065
$582$	6929.33i	0.493522i
$583$	3307.53i	0.234964i
$584$	47245.6	3.34767
$585$	0	0
$586$	−5606.91	−0.395255
$587$	− 15445.5i	− 1.08604i	−0.839720	−	0.543019i	$-0.817281\pi$
$587$	− 15445.5i	− 1.08604i	0.839720	−	0.543019i	$-0.182719\pi$
$588$	− 58426.1i	− 4.09770i
$589$	2956.24	0.206808
$590$	0	0
$591$	−1215.83	−0.0846233
$592$	− 104453.i	− 7.25166i
$593$	22999.6i	1.59272i	0.604824	+	0.796360i	$0.293244\pi$
$593$	22999.6i	1.59272i	−0.604824	+	0.796360i	$0.706756\pi$
$594$	−1657.38	−0.114483
$595$	0	0
$596$	36900.7	2.53610
$597$	11404.3i	0.781824i
$598$	8572.06i	0.586183i
$599$	−5960.87	−0.406602	−0.203301	−	0.979116i	$-0.565167\pi$
$599$	−5960.87	−0.406602	−0.203301	+	0.979116i	$0.565167\pi$
$600$	0	0
$601$	−4282.39	−0.290653	−0.145326	−	0.989384i	$-0.546423\pi$
$601$	−4282.39	−0.290653	−0.145326	+	0.989384i	$0.546423\pi$
$602$	18410.2i	1.24642i
$603$	− 1008.78i	− 0.0681273i
$604$	−51217.2	−3.45033
$605$	0	0
$606$	−7358.35	−0.493255
$607$	− 8197.97i	− 0.548180i	−0.961704	−	0.274090i	$-0.911623\pi$
$607$	− 8197.97i	− 0.548180i	0.961704	−	0.274090i	$-0.0883767\pi$
$608$	− 81307.9i	− 5.42347i
$609$	12212.9	0.812633
$610$	0	0
$611$	9437.38	0.624870
$612$	4656.71i	0.307575i
$613$	− 4185.87i	− 0.275801i	−0.990446	−	0.137900i	$-0.955965\pi$
$613$	− 4185.87i	− 0.275801i	0.990446	−	0.137900i	$-0.0440354\pi$
$614$	37332.9	2.45380
$615$	0	0
$616$	−31988.3	−2.09228
$617$	21993.7i	1.43506i	0.696528	+	0.717530i	$0.254727\pi$
$617$	21993.7i	1.43506i	−0.696528	+	0.717530i	$0.745273\pi$
$618$	− 18620.3i	− 1.21201i
$619$	25378.3	1.64788	0.823941	−	0.566676i	$-0.191771\pi$
$619$	25378.3	1.64788	0.823941	+	0.566676i	$0.191771\pi$
$620$	0	0
$621$	−581.787	−0.0375947
$622$	− 9488.75i	− 0.611679i
$623$	− 19013.2i	− 1.22271i
$624$	−61244.4	−3.92907
$625$	0	0
$626$	−2450.05	−0.156428
$627$	2909.79i	0.185336i
$628$	26510.8i	1.68455i
$629$	8155.58	0.516986
$630$	0	0
$631$	17172.2	1.08338	0.541691	−	0.840578i	$-0.317784\pi$
$631$	17172.2	1.08338	0.541691	+	0.840578i	$0.317784\pi$
$632$	4100.63i	0.258092i
$633$	4615.70i	0.289822i
$634$	−28755.2	−1.80129
$635$	0	0
$636$	−20874.2	−1.30144
$637$	59996.9i	3.73181i
$638$	− 7260.55i	− 0.450545i
$639$	−4933.43	−0.305420
$640$	0	0
$641$	31171.2	1.92073	0.960364	−	0.278749i	$-0.0899198\pi$
$641$	31171.2	1.92073	0.960364	+	0.278749i	$0.0899198\pi$
$642$	− 14463.2i	− 0.889123i
$643$	− 20847.3i	− 1.27860i	−0.768959	−	0.639298i	$-0.779225\pi$
$643$	− 20847.3i	− 1.27860i	0.768959	−	0.639298i	$-0.220775\pi$
$644$	−17161.8	−1.05011
$645$	0	0
$646$	11002.0	0.670074
$647$	− 5380.69i	− 0.326950i	−0.986547	−	0.163475i	$-0.947730\pi$
$647$	− 5380.69i	− 0.326950i	0.986547	−	0.163475i	$-0.0522703\pi$
$648$	− 6843.79i	− 0.414891i
$649$	7197.09	0.435301
$650$	0	0
$651$	3461.78	0.208415
$652$	− 13762.5i	− 0.826660i
$653$	− 26506.6i	− 1.58849i	−0.607597	−	0.794245i	$-0.707867\pi$
$653$	− 26506.6i	− 1.58849i	0.607597	−	0.794245i	$-0.292133\pi$
$654$	17424.1	1.04180
$655$	0	0
$656$	14008.1	0.833729
$657$	− 5032.60i	− 0.298844i
$658$	25426.2i	1.50641i
$659$	25165.6	1.48758	0.743789	−	0.668414i	$-0.233027\pi$
$659$	25165.6	1.48758	0.743789	+	0.668414i	$0.233027\pi$
$660$	0	0
$661$	2365.09	0.139170	0.0695849	−	0.997576i	$-0.477833\pi$
$661$	2365.09	0.139170	0.0695849	+	0.997576i	$0.477833\pi$
$662$	− 18119.5i	− 1.06380i
$663$	− 4781.91i	− 0.280111i
$664$	37838.9	2.21150
$665$	0	0
$666$	−18319.0	−1.06584
$667$	− 2548.66i	− 0.147953i
$668$	− 67743.0i	− 3.92374i
$669$	−7674.56	−0.443521
$670$	0	0
$671$	−8499.93	−0.489025
$672$	− 95212.3i	− 5.46561i
$673$	− 9349.93i	− 0.535532i	−0.963484	−	0.267766i	$-0.913715\pi$
$673$	− 9349.93i	− 0.535532i	0.963484	−	0.267766i	$-0.0862854\pi$
$674$	7467.38	0.426755
$675$	0	0
$676$	66762.9	3.79852
$677$	767.511i	0.0435714i	0.999763	+	0.0217857i	$0.00693515\pi$
$677$	767.511i	0.0435714i	−0.999763	+	0.0217857i	$0.993065\pi$
$678$	30571.7i	1.73171i
$679$	14246.0	0.805170
$680$	0	0
$681$	17502.3	0.984858
$682$	− 2058.02i	− 0.115551i
$683$	− 6514.91i	− 0.364987i	−0.983207	−	0.182494i	$-0.941583\pi$
$683$	− 6514.91i	− 0.364987i	0.983207	−	0.182494i	$-0.0584169\pi$
$684$	−18364.0	−1.02656
$685$	0	0
$686$	−95765.1	−5.32993
$687$	− 3926.83i	− 0.218076i
$688$	27449.3i	1.52107i
$689$	21435.4	1.18523
$690$	0	0
$691$	30296.0	1.66789	0.833946	−	0.551846i	$-0.186076\pi$
$691$	30296.0	1.66789	0.833946	+	0.551846i	$0.186076\pi$
$692$	− 29487.2i	− 1.61985i
$693$	3407.39i	0.186776i
$694$	19092.7	1.04431
$695$	0	0
$696$	29980.9	1.63279
$697$	1093.74i	0.0594383i
$698$	42063.9i	2.28101i
$699$	7081.73	0.383198
$700$	0	0
$701$	24513.3	1.32076	0.660381	−	0.750931i	$-0.270395\pi$
$701$	24513.3	1.32076	0.660381	+	0.750931i	$0.270395\pi$
$702$	10741.1i	0.577488i
$703$	32162.1i	1.72548i
$704$	−31403.0	−1.68117
$705$	0	0
$706$	−32383.9	−1.72632
$707$	15128.0i	0.804734i
$708$	45421.6i	2.41109i
$709$	36554.1	1.93628	0.968138	−	0.250419i	$-0.0805683\pi$
$709$	36554.1	1.93628	0.968138	+	0.250419i	$0.0805683\pi$
$710$	0	0
$711$	436.799	0.0230397
$712$	− 46674.6i	− 2.45675i
$713$	− 722.423i	− 0.0379452i
$714$	12883.4	0.675280
$715$	0	0
$716$	37629.6	1.96409
$717$	17299.1i	0.901044i
$718$	− 31018.5i	− 1.61226i
$719$	24860.8	1.28950	0.644752	−	0.764392i	$-0.276961\pi$
$719$	24860.8	1.28950	0.644752	+	0.764392i	$0.276961\pi$
$720$	0	0
$721$	−38281.5	−1.97736
$722$	5111.18i	0.263461i
$723$	7934.18i	0.408126i
$724$	−37922.0	−1.94663
$725$	0	0
$726$	2025.68	0.103554
$727$	11414.7i	0.582324i	0.956674	+	0.291162i	$0.0940418\pi$
$727$	11414.7i	0.582324i	−0.956674	+	0.291162i	$0.905958\pi$
$728$	207309.i	10.5541i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−2143.22	−0.108440
$732$	− 53644.0i	− 2.70866i
$733$	11746.4i	0.591902i	0.955203	+	0.295951i	$0.0956365\pi$
$733$	11746.4i	0.591902i	−0.955203	+	0.295951i	$0.904364\pi$
$734$	15925.2	0.800831
$735$	0	0
$736$	−19869.4	−0.995103
$737$	1232.96i	0.0616235i
$738$	− 2456.76i	− 0.122540i
$739$	5642.98	0.280894	0.140447	−	0.990088i	$-0.455146\pi$
$739$	5642.98	0.280894	0.140447	+	0.990088i	$0.455146\pi$
$740$	0	0
$741$	18857.7	0.934894
$742$	57751.5i	2.85731i
$743$	− 3718.47i	− 0.183604i	−0.995777	−	0.0918018i	$-0.970737\pi$
$743$	− 3718.47i	− 0.183604i	0.995777	−	0.0918018i	$-0.0292626\pi$
$744$	8498.15	0.418760
$745$	0	0
$746$	−41857.9	−2.05432
$747$	− 4030.60i	− 0.197419i
$748$	− 5691.53i	− 0.278212i
$749$	−29734.8	−1.45058
$750$	0	0
$751$	−11732.6	−0.570080	−0.285040	−	0.958516i	$-0.592007\pi$
$751$	−11732.6	−0.570080	−0.285040	+	0.958516i	$0.592007\pi$
$752$	37910.1i	1.83835i
$753$	7306.77i	0.353617i
$754$	−47054.1	−2.27269
$755$	0	0
$756$	−21504.4	−1.03453
$757$	− 11048.3i	− 0.530457i	−0.964186	−	0.265229i	$-0.914553\pi$
$757$	− 11048.3i	− 0.530457i	0.964186	−	0.265229i	$-0.0854474\pi$
$758$	− 72350.2i	− 3.46686i
$759$	711.073	0.0340057
$760$	0	0
$761$	16924.6	0.806198	0.403099	−	0.915156i	$-0.367933\pi$
$761$	16924.6	0.806198	0.403099	+	0.915156i	$0.367933\pi$
$762$	9614.14i	0.457065i
$763$	− 35822.2i	− 1.69967i
$764$	−80088.3	−3.79253
$765$	0	0
$766$	−27505.4	−1.29740
$767$	− 46642.8i	− 2.19580i
$768$	− 74689.4i	− 3.50927i
$769$	−26448.0	−1.24023	−0.620116	−	0.784510i	$-0.712914\pi$
$769$	−26448.0	−1.24023	−0.620116	+	0.784510i	$0.712914\pi$
$770$	0	0
$771$	−16567.5	−0.773884
$772$	39935.8i	1.86182i
$773$	− 33612.4i	− 1.56398i	−0.623294	−	0.781988i	$-0.714206\pi$
$773$	− 33612.4i	− 1.56398i	0.623294	−	0.781988i	$-0.285794\pi$
$774$	4814.09	0.223565
$775$	0	0
$776$	34971.7	1.61780
$777$	37662.0i	1.73889i
$778$	− 73565.5i	− 3.39004i
$779$	−4313.25	−0.198380
$780$	0	0
$781$	6029.75	0.276263
$782$	− 2688.58i	− 0.122946i
$783$	− 3193.57i	− 0.145758i
$784$	−241009.	−10.9789
$785$	0	0
$786$	−44942.7	−2.03951
$787$	19267.2i	0.872684i	0.899781	+	0.436342i	$0.143726\pi$
$787$	19267.2i	0.872684i	−0.899781	+	0.436342i	$0.856274\pi$
$788$	9378.37i	0.423973i
$789$	−8099.64	−0.365469
$790$	0	0
$791$	62852.2	2.82524
$792$	8364.63i	0.375283i
$793$	55086.3i	2.46680i
$794$	77096.1	3.44589
$795$	0	0
$796$	87968.3	3.91703
$797$	31120.9i	1.38314i	0.722312	+	0.691568i	$0.243080\pi$
$797$	31120.9i	1.38314i	−0.722312	+	0.691568i	$0.756920\pi$
$798$	50806.6i	2.25380i
$799$	−2959.99	−0.131060
$800$	0	0
$801$	−4971.77	−0.219312
$802$	26857.5i	1.18251i
$803$	6150.95i	0.270314i
$804$	−7781.32	−0.341326
$805$	0	0
$806$	−13337.6	−0.582873
$807$	4781.81i	0.208584i
$808$	37136.9i	1.61692i
$809$	−18201.5	−0.791015	−0.395508	−	0.918463i	$-0.629431\pi$
$809$	−18201.5	−0.791015	−0.395508	+	0.918463i	$0.629431\pi$
$810$	0	0
$811$	−3221.28	−0.139475	−0.0697376	−	0.997565i	$-0.522216\pi$
$811$	−3221.28	−0.139475	−0.0697376	+	0.997565i	$0.522216\pi$
$812$	− 94205.5i	− 4.07139i
$813$	− 5479.81i	− 0.236390i
$814$	22389.9	0.964087
$815$	0	0
$816$	19209.0	0.824080
$817$	− 8451.92i	− 0.361928i
$818$	54556.0i	2.33191i
$819$	22082.6	0.942159
$820$	0	0
$821$	6884.62	0.292661	0.146331	−	0.989236i	$-0.453254\pi$
$821$	6884.62	0.292661	0.146331	+	0.989236i	$0.453254\pi$
$822$	− 50448.7i	− 2.14063i
$823$	− 19720.3i	− 0.835246i	−0.908620	−	0.417623i	$-0.862863\pi$
$823$	− 19720.3i	− 0.835246i	0.908620	−	0.417623i	$-0.137137\pi$
$824$	−93975.2	−3.97304
$825$	0	0
$826$	125665.	5.29353
$827$	− 17844.0i	− 0.750297i	−0.926965	−	0.375148i	$-0.877592\pi$
$827$	− 17844.0i	− 0.750297i	0.926965	−	0.375148i	$-0.122408\pi$
$828$	4487.66i	0.188354i
$829$	31143.1	1.30476	0.652379	−	0.757893i	$-0.273771\pi$
$829$	31143.1	1.30476	0.652379	+	0.757893i	$0.273771\pi$
$830$	0	0
$831$	8107.24	0.338432
$832$	203516.i	8.48035i
$833$	− 18817.7i	− 0.782708i
$834$	−4644.18	−0.192824
$835$	0	0
$836$	22444.9	0.928557
$837$	− 905.223i	− 0.0373824i
$838$	− 78559.5i	− 3.23842i
$839$	−33261.2	−1.36866	−0.684329	−	0.729174i	$-0.739905\pi$
$839$	−33261.2	−1.36866	−0.684329	+	0.729174i	$0.739905\pi$
$840$	0	0
$841$	−10398.8	−0.426371
$842$	95429.2i	3.90583i
$843$	8885.87i	0.363043i
$844$	35603.5	1.45204
$845$	0	0
$846$	6648.72	0.270198
$847$	− 4164.59i	− 0.168946i
$848$	86106.5i	3.48692i
$849$	10513.3	0.424990
$850$	0	0
$851$	7859.52	0.316593
$852$	38054.4i	1.53019i
$853$	− 419.167i	− 0.0168253i	−0.999965	−	0.00841266i	$-0.997322\pi$
$853$	− 419.167i	− 0.0168253i	0.999965	−	0.00841266i	$-0.00267786\pi$
$854$	−148414.	−5.94685
$855$	0	0
$856$	−72994.5	−2.91460
$857$	− 8915.18i	− 0.355352i	−0.984089	−	0.177676i	$-0.943142\pi$
$857$	− 8915.18i	− 0.355352i	0.984089	−	0.177676i	$-0.0568579\pi$
$858$	− 13128.0i	− 0.522358i
$859$	−4449.47	−0.176733	−0.0883667	−	0.996088i	$-0.528165\pi$
$859$	−4449.47	−0.176733	−0.0883667	+	0.996088i	$0.528165\pi$
$860$	0	0
$861$	−5050.85	−0.199922
$862$	− 24412.8i	− 0.964623i
$863$	− 25151.7i	− 0.992089i	−0.868297	−	0.496044i	$-0.834785\pi$
$863$	− 25151.7i	− 0.992089i	0.868297	−	0.496044i	$-0.165215\pi$
$864$	−24897.1	−0.980343
$865$	0	0
$866$	30070.6	1.17995
$867$	− 13239.2i	− 0.518600i
$868$	− 26702.7i	− 1.04418i
$869$	−533.866	−0.0208402
$870$	0	0
$871$	7990.53	0.310848
$872$	− 87938.0i	− 3.41509i
$873$	− 3725.18i	− 0.144420i
$874$	10602.6	0.410341
$875$	0	0
$876$	−38819.4	−1.49724
$877$	16936.9i	0.652129i	0.945348	+	0.326065i	$0.105723\pi$
$877$	16936.9i	0.652129i	−0.945348	+	0.326065i	$0.894277\pi$
$878$	− 30016.4i	− 1.15376i
$879$	3014.26	0.115664
$880$	0	0
$881$	−32920.0	−1.25891	−0.629456	−	0.777036i	$-0.716722\pi$
$881$	−32920.0	−1.25891	−0.629456	+	0.777036i	$0.716722\pi$
$882$	42268.3i	1.61366i
$883$	23658.9i	0.901683i	0.892604	+	0.450842i	$0.148876\pi$
$883$	23658.9i	0.901683i	−0.892604	+	0.450842i	$0.851124\pi$
$884$	−36885.6	−1.40339
$885$	0	0
$886$	38350.4	1.45418
$887$	− 19319.2i	− 0.731312i	−0.930750	−	0.365656i	$-0.880845\pi$
$887$	− 19319.2i	− 0.731312i	0.930750	−	0.365656i	$-0.119155\pi$
$888$	92454.6i	3.49389i
$889$	19765.7	0.745691
$890$	0	0
$891$	891.000	0.0335013
$892$	59198.3i	2.22209i
$893$	− 11672.9i	− 0.437423i
$894$	−26695.8	−0.998705
$895$	0	0
$896$	−294414.	−10.9773
$897$	− 4608.31i	− 0.171535i
$898$	− 45742.3i	− 1.69982i
$899$	3965.56	0.147118
$900$	0	0
$901$	−6723.12	−0.248590
$902$	3002.71i	0.110842i
$903$	− 9897.27i	− 0.364740i
$904$	154293.	5.67666
$905$	0	0
$906$	37053.1	1.35873
$907$	13020.2i	0.476657i	0.971185	+	0.238329i	$0.0765995\pi$
$907$	13020.2i	0.476657i	−0.971185	+	0.238329i	$0.923400\pi$
$908$	− 135005.i	− 4.93425i
$909$	3955.83	0.144342
$910$	0	0
$911$	260.135	0.00946065	0.00473032	−	0.999989i	$-0.498494\pi$
$911$	260.135	0.00946065	0.00473032	+	0.999989i	$0.498494\pi$
$912$	75751.9i	2.75043i
$913$	4926.29i	0.178572i
$914$	−20799.9	−0.752735
$915$	0	0
$916$	−30289.9	−1.09258
$917$	92397.5i	3.32741i
$918$	− 3368.89i	− 0.121122i
$919$	−23631.7	−0.848247	−0.424123	−	0.905604i	$-0.639418\pi$
$919$	−23631.7	−0.848247	−0.424123	+	0.905604i	$0.639418\pi$
$920$	0	0
$921$	−20070.0	−0.718057
$922$	32372.2i	1.15631i
$923$	− 39077.5i	− 1.39356i
$924$	26283.2	0.935772
$925$	0	0
$926$	−69211.4	−2.45619
$927$	10010.2i	0.354670i
$928$	− 109068.i	− 3.85812i
$929$	−19536.7	−0.689966	−0.344983	−	0.938609i	$-0.612115\pi$
$929$	−19536.7	−0.689966	−0.344983	+	0.938609i	$0.612115\pi$
$930$	0	0
$931$	74208.9	2.61235
$932$	− 54625.5i	− 1.91987i
$933$	5101.12i	0.178996i
$934$	63302.4	2.21769
$935$	0	0
$936$	54209.4	1.89305
$937$	− 17072.3i	− 0.595227i	−0.954686	−	0.297614i	$-0.903809\pi$
$937$	− 17072.3i	− 0.595227i	0.954686	−	0.297614i	$-0.0961907\pi$
$938$	21528.1i	0.749380i
$939$	1317.14	0.0457755
$940$	0	0
$941$	−24908.1	−0.862890	−0.431445	−	0.902139i	$-0.641996\pi$
$941$	−24908.1	−0.862890	−0.431445	+	0.902139i	$0.641996\pi$
$942$	− 19179.3i	− 0.663370i
$943$	1054.04i	0.0363989i
$944$	187365.	6.45997
$945$	0	0
$946$	−5883.89	−0.202222
$947$	42208.2i	1.44835i	0.689619	+	0.724173i	$0.257778\pi$
$947$	42208.2i	1.44835i	−0.689619	+	0.724173i	$0.742222\pi$
$948$	− 3369.29i	− 0.115432i
$949$	39863.0	1.36355
$950$	0	0
$951$	15458.7	0.527112
$952$	− 65021.6i	− 2.21361i
$953$	− 38261.2i	− 1.30053i	−0.759709	−	0.650263i	$-0.774659\pi$
$953$	− 38261.2i	− 1.30053i	0.759709	−	0.650263i	$-0.225341\pi$
$954$	15101.5	0.512503
$955$	0	0
$956$	133438.	4.51433
$957$	3903.25i	0.131843i
$958$	13303.6i	0.448663i
$959$	−103717.	−3.49239
$960$	0	0
$961$	−28667.0	−0.962269
$962$	− 145105.i	− 4.86316i
$963$	7775.37i	0.260185i
$964$	61200.9	2.04476
$965$	0	0
$966$	12415.7	0.413530
$967$	− 2841.88i	− 0.0945074i	−0.998883	−	0.0472537i	$-0.984953\pi$
$967$	− 2841.88i	− 0.0945074i	0.998883	−	0.0472537i	$-0.0150469\pi$
$968$	− 10223.4i	− 0.339456i
$969$	−5914.64	−0.196084
$970$	0	0
$971$	−9655.22	−0.319105	−0.159552	−	0.987189i	$-0.551005\pi$
$971$	−9655.22	−0.319105	−0.159552	+	0.987189i	$0.551005\pi$
$972$	5623.20i	0.185560i
$973$	9547.95i	0.314587i
$974$	62426.3	2.05366
$975$	0	0
$976$	−221282.	−7.25725
$977$	− 20818.5i	− 0.681722i	−0.940114	−	0.340861i	$-0.889281\pi$
$977$	− 20818.5i	− 0.681722i	0.940114	−	0.340861i	$-0.110719\pi$
$978$	9956.51i	0.325536i
$979$	6076.61	0.198375
$980$	0	0
$981$	−9367.16	−0.304863
$982$	− 52044.2i	− 1.69124i
$983$	− 5762.06i	− 0.186959i	−0.995621	−	0.0934797i	$-0.970201\pi$
$983$	− 5762.06i	− 0.186959i	0.995621	−	0.0934797i	$-0.0297990\pi$
$984$	−12399.1	−0.401695
$985$	0	0
$986$	14758.3	0.476673
$987$	− 13669.1i	− 0.440822i
$988$	− 145461.i	− 4.68393i
$989$	−2065.42	−0.0664069
$990$	0	0
$991$	48292.2	1.54798	0.773992	−	0.633195i	$-0.218257\pi$
$991$	48292.2	1.54798	0.773992	+	0.633195i	$0.218257\pi$
$992$	− 30915.5i	− 0.989485i
$993$	9740.98i	0.311300i
$994$	105283.	3.35953
$995$	0	0
$996$	−31090.3	−0.989091
$997$	− 42833.0i	− 1.36062i	−0.732926	−	0.680308i	$-0.761846\pi$
$997$	− 42833.0i	− 1.36062i	0.732926	−	0.680308i	$-0.238154\pi$
$998$	− 106690.i	− 3.38398i
$999$	9848.26	0.311897

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.c.r.199.10		10
5.2	odd	4		825.4.a.w.1.1	✓	5
5.3	odd	4		825.4.a.z.1.5	yes	5
5.4	even	2	inner	825.4.c.r.199.1		10
15.2	even	4		2475.4.a.bm.1.5		5
15.8	even	4		2475.4.a.bf.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.w.1.1	✓	5	5.2	odd	4
825.4.a.z.1.5	yes	5	5.3	odd	4
825.4.c.r.199.1		10	5.4	even	2	inner
825.4.c.r.199.10		10	1.1	even	1	trivial
2475.4.a.bf.1.1		5	15.8	even	4
2475.4.a.bm.1.5		5	15.2	even	4

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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.58039i q^{2} -3.00000i q^{3} -23.1407 q^{4} +16.7412 q^{6} -34.4181i q^{7} -84.4912i q^{8} -9.00000 q^{9} +O(q^{10})\)	\(q+5.58039i q^{2} -3.00000i q^{3} -23.1407 q^{4} +16.7412 q^{6} -34.4181i q^{7} -84.4912i q^{8} -9.00000 q^{9} +11.0000 q^{11} +69.4222i q^{12} -71.2887i q^{13} +192.066 q^{14} +286.368 q^{16} +22.3593i q^{17} -50.2235i q^{18} -88.1755 q^{19} -103.254 q^{21} +61.3843i q^{22} +21.5477i q^{23} -253.474 q^{24} +397.819 q^{26} +27.0000i q^{27} +796.460i q^{28} -118.280 q^{29} -33.5268 q^{31} +922.115i q^{32} -33.0000i q^{33} -124.774 q^{34} +208.267 q^{36} -364.750i q^{37} -492.054i q^{38} -213.866 q^{39} +48.9166 q^{41} -576.199i q^{42} +95.8534i q^{43} -254.548 q^{44} -120.244 q^{46} +132.383i q^{47} -859.104i q^{48} -841.605 q^{49} +67.0780 q^{51} +1649.67i q^{52} +300.685i q^{53} -150.671 q^{54} -2908.03 q^{56} +264.526i q^{57} -660.050i q^{58} +654.281 q^{59} -772.721 q^{61} -187.092i q^{62} +309.763i q^{63} -2854.82 q^{64} +184.153 q^{66} +112.087i q^{67} -517.412i q^{68} +64.6430 q^{69} +548.159 q^{71} +760.421i q^{72} +559.178i q^{73} +2035.45 q^{74} +2040.45 q^{76} -378.599i q^{77} -1193.46i q^{78} -48.5332 q^{79} +81.0000 q^{81} +272.974i q^{82} +447.844i q^{83} +2389.38 q^{84} -534.899 q^{86} +354.841i q^{87} -929.403i q^{88} +552.419 q^{89} -2453.62 q^{91} -498.629i q^{92} +100.580i q^{93} -738.746 q^{94} +2766.34 q^{96} +413.909i q^{97} -4696.48i q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 92 q^{4} + 24 q^{6} - 90 q^{9}+O(q^{10}) \) 10 * q - 92 * q^4 + 24 * q^6 - 90 * q^9	\( 10 q - 92 q^{4} + 24 q^{6} - 90 q^{9} + 110 q^{11} - 110 q^{14} + 1268 q^{16} + 674 q^{19} - 228 q^{21} - 432 q^{24} + 718 q^{26} + 306 q^{29} + 526 q^{31} - 1034 q^{34} + 828 q^{36} - 258 q^{39} - 176 q^{41} - 1012 q^{44} + 872 q^{46} - 4762 q^{49} + 900 q^{51} - 216 q^{54} + 422 q^{56} - 820 q^{59} - 2260 q^{61} - 6340 q^{64} + 264 q^{66} + 1206 q^{69} + 2498 q^{71} + 5970 q^{74} - 5112 q^{76} - 4516 q^{79} + 810 q^{81} + 2646 q^{84} - 2870 q^{86} - 694 q^{89} - 1224 q^{91} - 1814 q^{94} + 1680 q^{96} - 990 q^{99}+O(q^{100}) \) 10 * q - 92 * q^4 + 24 * q^6 - 90 * q^9 + 110 * q^11 - 110 * q^14 + 1268 * q^16 + 674 * q^19 - 228 * q^21 - 432 * q^24 + 718 * q^26 + 306 * q^29 + 526 * q^31 - 1034 * q^34 + 828 * q^36 - 258 * q^39 - 176 * q^41 - 1012 * q^44 + 872 * q^46 - 4762 * q^49 + 900 * q^51 - 216 * q^54 + 422 * q^56 - 820 * q^59 - 2260 * q^61 - 6340 * q^64 + 264 * q^66 + 1206 * q^69 + 2498 * q^71 + 5970 * q^74 - 5112 * q^76 - 4516 * q^79 + 810 * q^81 + 2646 * q^84 - 2870 * q^86 - 694 * q^89 - 1224 * q^91 - 1814 * q^94 + 1680 * q^96 - 990 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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