LMFDB - Embedded newform 8910.2.a.cd.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8910,2,Mod(1,8910)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8910, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8910.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8910 = 2 \cdot 3^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8910.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,7,0,7,7,0,5,7,0,7,7,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$71.1467082010$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 13x^{5} + 31x^{4} + 21x^{3} - 37x^{2} - 15x + 6 $ x^7 - 3x^6 - 13x^5 + 31x^4 + 21x^3 - 37x^2 - 15x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 990)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.729749$ of defining polynomial
Character	$\chi$	$=$	8910.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.74624 q^{7} +1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +0.902331 q^{13} +4.74624 q^{14} +1.00000 q^{16} -4.57073 q^{17} +4.76727 q^{19} +1.00000 q^{20} +1.00000 q^{22} -4.01374 q^{23} +1.00000 q^{25} +0.902331 q^{26} +4.74624 q^{28} -5.51350 q^{29} +6.62116 q^{31} +1.00000 q^{32} -4.57073 q^{34} +4.74624 q^{35} -3.67903 q^{37} +4.76727 q^{38} +1.00000 q^{40} +5.11141 q^{41} +7.37532 q^{43} +1.00000 q^{44} -4.01374 q^{46} -7.25024 q^{47} +15.5268 q^{49} +1.00000 q^{50} +0.902331 q^{52} +2.77789 q^{53} +1.00000 q^{55} +4.74624 q^{56} -5.51350 q^{58} +14.5733 q^{59} +11.5897 q^{61} +6.62116 q^{62} +1.00000 q^{64} +0.902331 q^{65} -11.8465 q^{67} -4.57073 q^{68} +4.74624 q^{70} -7.93501 q^{71} -11.7843 q^{73} -3.67903 q^{74} +4.76727 q^{76} +4.74624 q^{77} +9.63130 q^{79} +1.00000 q^{80} +5.11141 q^{82} -14.4184 q^{83} -4.57073 q^{85} +7.37532 q^{86} +1.00000 q^{88} +9.10413 q^{89} +4.28268 q^{91} -4.01374 q^{92} -7.25024 q^{94} +4.76727 q^{95} +3.26319 q^{97} +15.5268 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} + 7 q^{4} + 7 q^{5} + 5 q^{7} + 7 q^{8} + 7 q^{10} + 7 q^{11} + 5 q^{13} + 5 q^{14} + 7 q^{16} + 3 q^{17} + 11 q^{19} + 7 q^{20} + 7 q^{22} + 3 q^{23} + 7 q^{25} + 5 q^{26} + 5 q^{28} + 12 q^{29}+ \cdots + 12 q^{98}+O(q^{100}) $ 7 * q + 7 * q^2 + 7 * q^4 + 7 * q^5 + 5 * q^7 + 7 * q^8 + 7 * q^10 + 7 * q^11 + 5 * q^13 + 5 * q^14 + 7 * q^16 + 3 * q^17 + 11 * q^19 + 7 * q^20 + 7 * q^22 + 3 * q^23 + 7 * q^25 + 5 * q^26 + 5 * q^28 + 12 * q^29 + 5 * q^31 + 7 * q^32 + 3 * q^34 + 5 * q^35 + 2 * q^37 + 11 * q^38 + 7 * q^40 + 6 * q^41 + 17 * q^43 + 7 * q^44 + 3 * q^46 - 3 * q^47 + 12 * q^49 + 7 * q^50 + 5 * q^52 + 3 * q^53 + 7 * q^55 + 5 * q^56 + 12 * q^58 + 18 * q^59 + 11 * q^61 + 5 * q^62 + 7 * q^64 + 5 * q^65 + 20 * q^67 + 3 * q^68 + 5 * q^70 + 12 * q^71 + 2 * q^73 + 2 * q^74 + 11 * q^76 + 5 * q^77 - 7 * q^79 + 7 * q^80 + 6 * q^82 + 3 * q^85 + 17 * q^86 + 7 * q^88 - 3 * q^89 + 7 * q^91 + 3 * q^92 - 3 * q^94 + 11 * q^95 + 23 * q^97 + 12 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.74624	1.79391	0.896955	−	0.442123i	$-0.145774\pi$
$7$	4.74624	1.79391	0.896955	+	0.442123i	$0.145774\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0.902331	0.250261	0.125131	−	0.992140i	$-0.460065\pi$
$13$	0.902331	0.250261	0.125131	+	0.992140i	$0.460065\pi$
$14$	4.74624	1.26849
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.57073	−1.10857	−0.554283	−	0.832328i	$-0.687007\pi$
$17$	−4.57073	−1.10857	−0.554283	+	0.832328i	$0.687007\pi$
$18$	0	0
$19$	4.76727	1.09369	0.546843	−	0.837235i	$-0.315829\pi$
$19$	4.76727	1.09369	0.546843	+	0.837235i	$0.315829\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	1.00000	0.213201
$23$	−4.01374	−0.836923	−0.418462	−	0.908234i	$-0.637431\pi$
$23$	−4.01374	−0.836923	−0.418462	+	0.908234i	$0.637431\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.902331	0.176962
$27$	0	0
$28$	4.74624	0.896955
$29$	−5.51350	−1.02383	−0.511916	−	0.859035i	$-0.671064\pi$
$29$	−5.51350	−1.02383	−0.511916	+	0.859035i	$0.671064\pi$
$30$	0	0
$31$	6.62116	1.18919	0.594597	−	0.804024i	$-0.297311\pi$
$31$	6.62116	1.18919	0.594597	+	0.804024i	$0.297311\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.57073	−0.783875
$35$	4.74624	0.802261
$36$	0	0
$37$	−3.67903	−0.604828	−0.302414	−	0.953177i	$-0.597793\pi$
$37$	−3.67903	−0.604828	−0.302414	+	0.953177i	$0.597793\pi$
$38$	4.76727	0.773353
$39$	0	0
$40$	1.00000	0.158114
$41$	5.11141	0.798269	0.399134	−	0.916892i	$-0.369311\pi$
$41$	5.11141	0.798269	0.399134	+	0.916892i	$0.369311\pi$
$42$	0	0
$43$	7.37532	1.12473	0.562363	−	0.826891i	$-0.309892\pi$
$43$	7.37532	1.12473	0.562363	+	0.826891i	$0.309892\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−4.01374	−0.591794
$47$	−7.25024	−1.05756	−0.528778	−	0.848760i	$-0.677350\pi$
$47$	−7.25024	−1.05756	−0.528778	+	0.848760i	$0.677350\pi$
$48$	0	0
$49$	15.5268	2.21811
$50$	1.00000	0.141421
$51$	0	0
$52$	0.902331	0.125131
$53$	2.77789	0.381573	0.190786	−	0.981632i	$-0.438896\pi$
$53$	2.77789	0.381573	0.190786	+	0.981632i	$0.438896\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	4.74624	0.634243
$57$	0	0
$58$	−5.51350	−0.723959
$59$	14.5733	1.89728	0.948641	−	0.316354i	$-0.102459\pi$
$59$	14.5733	1.89728	0.948641	+	0.316354i	$0.102459\pi$
$60$	0	0
$61$	11.5897	1.48391	0.741953	−	0.670452i	$-0.233900\pi$
$61$	11.5897	1.48391	0.741953	+	0.670452i	$0.233900\pi$
$62$	6.62116	0.840888
$63$	0	0
$64$	1.00000	0.125000
$65$	0.902331	0.111920
$66$	0	0
$67$	−11.8465	−1.44729	−0.723643	−	0.690174i	$-0.757534\pi$
$67$	−11.8465	−1.44729	−0.723643	+	0.690174i	$0.757534\pi$
$68$	−4.57073	−0.554283
$69$	0	0
$70$	4.74624	0.567284
$71$	−7.93501	−0.941712	−0.470856	−	0.882210i	$-0.656055\pi$
$71$	−7.93501	−0.941712	−0.470856	+	0.882210i	$0.656055\pi$
$72$	0	0
$73$	−11.7843	−1.37925	−0.689627	−	0.724165i	$-0.742225\pi$
$73$	−11.7843	−1.37925	−0.689627	+	0.724165i	$0.742225\pi$
$74$	−3.67903	−0.427678
$75$	0	0
$76$	4.76727	0.546843
$77$	4.74624	0.540884
$78$	0	0
$79$	9.63130	1.08361	0.541803	−	0.840506i	$-0.317742\pi$
$79$	9.63130	1.08361	0.541803	+	0.840506i	$0.317742\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	5.11141	0.564461
$83$	−14.4184	−1.58263	−0.791313	−	0.611412i	$-0.790602\pi$
$83$	−14.4184	−1.58263	−0.791313	+	0.611412i	$0.790602\pi$
$84$	0	0
$85$	−4.57073	−0.495766
$86$	7.37532	0.795301
$87$	0	0
$88$	1.00000	0.106600
$89$	9.10413	0.965036	0.482518	−	0.875886i	$-0.339722\pi$
$89$	9.10413	0.965036	0.482518	+	0.875886i	$0.339722\pi$
$90$	0	0
$91$	4.28268	0.448946
$92$	−4.01374	−0.418462
$93$	0	0
$94$	−7.25024	−0.747805
$95$	4.76727	0.489111
$96$	0	0
$97$	3.26319	0.331327	0.165663	−	0.986182i	$-0.447023\pi$
$97$	3.26319	0.331327	0.165663	+	0.986182i	$0.447023\pi$
$98$	15.5268	1.56844
$99$	0	0
$100$	1.00000	0.100000
$101$	11.1922	1.11367	0.556835	−	0.830623i	$-0.312016\pi$
$101$	11.1922	1.11367	0.556835	+	0.830623i	$0.312016\pi$
$102$	0	0
$103$	0.973785	0.0959498	0.0479749	−	0.998849i	$-0.484723\pi$
$103$	0.973785	0.0959498	0.0479749	+	0.998849i	$0.484723\pi$
$104$	0.902331	0.0884808
$105$	0	0
$106$	2.77789	0.269813
$107$	11.0164	1.06499	0.532497	−	0.846432i	$-0.321254\pi$
$107$	11.0164	1.06499	0.532497	+	0.846432i	$0.321254\pi$
$108$	0	0
$109$	10.9050	1.04451	0.522254	−	0.852790i	$-0.325091\pi$
$109$	10.9050	1.04451	0.522254	+	0.852790i	$0.325091\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	4.74624	0.448477
$113$	8.20932	0.772268	0.386134	−	0.922443i	$-0.373810\pi$
$113$	8.20932	0.772268	0.386134	+	0.922443i	$0.373810\pi$
$114$	0	0
$115$	−4.01374	−0.374284
$116$	−5.51350	−0.511916
$117$	0	0
$118$	14.5733	1.34158
$119$	−21.6938	−1.98867
$120$	0	0
$121$	1.00000	0.0909091
$122$	11.5897	1.04928
$123$	0	0
$124$	6.62116	0.594597
$125$	1.00000	0.0894427
$126$	0	0
$127$	−10.6046	−0.941007	−0.470503	−	0.882398i	$-0.655928\pi$
$127$	−10.6046	−0.941007	−0.470503	+	0.882398i	$0.655928\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0.902331	0.0791396
$131$	−6.53972	−0.571378	−0.285689	−	0.958322i	$-0.592222\pi$
$131$	−6.53972	−0.571378	−0.285689	+	0.958322i	$0.592222\pi$
$132$	0	0
$133$	22.6266	1.96197
$134$	−11.8465	−1.02339
$135$	0	0
$136$	−4.57073	−0.391937
$137$	20.5462	1.75538	0.877689	−	0.479230i	$-0.159084\pi$
$137$	20.5462	1.75538	0.877689	+	0.479230i	$0.159084\pi$
$138$	0	0
$139$	−6.29325	−0.533786	−0.266893	−	0.963726i	$-0.585997\pi$
$139$	−6.29325	−0.533786	−0.266893	+	0.963726i	$0.585997\pi$
$140$	4.74624	0.401130
$141$	0	0
$142$	−7.93501	−0.665891
$143$	0.902331	0.0754567
$144$	0	0
$145$	−5.51350	−0.457872
$146$	−11.7843	−0.975280
$147$	0	0
$148$	−3.67903	−0.302414
$149$	−7.78011	−0.637371	−0.318686	−	0.947860i	$-0.603241\pi$
$149$	−7.78011	−0.637371	−0.318686	+	0.947860i	$0.603241\pi$
$150$	0	0
$151$	5.05466	0.411343	0.205671	−	0.978621i	$-0.434062\pi$
$151$	5.05466	0.411343	0.205671	+	0.978621i	$0.434062\pi$
$152$	4.76727	0.386676
$153$	0	0
$154$	4.74624	0.382463
$155$	6.62116	0.531824
$156$	0	0
$157$	14.5734	1.16309	0.581543	−	0.813516i	$-0.302449\pi$
$157$	14.5734	1.16309	0.581543	+	0.813516i	$0.302449\pi$
$158$	9.63130	0.766225
$159$	0	0
$160$	1.00000	0.0790569
$161$	−19.0502	−1.50136
$162$	0	0
$163$	−16.7404	−1.31121	−0.655604	−	0.755105i	$-0.727586\pi$
$163$	−16.7404	−1.31121	−0.655604	+	0.755105i	$0.727586\pi$
$164$	5.11141	0.399134
$165$	0	0
$166$	−14.4184	−1.11909
$167$	−16.0736	−1.24381	−0.621907	−	0.783091i	$-0.713642\pi$
$167$	−16.0736	−1.24381	−0.621907	+	0.783091i	$0.713642\pi$
$168$	0	0
$169$	−12.1858	−0.937369
$170$	−4.57073	−0.350559
$171$	0	0
$172$	7.37532	0.562363
$173$	−1.50662	−0.114546	−0.0572732	−	0.998359i	$-0.518241\pi$
$173$	−1.50662	−0.114546	−0.0572732	+	0.998359i	$0.518241\pi$
$174$	0	0
$175$	4.74624	0.358782
$176$	1.00000	0.0753778
$177$	0	0
$178$	9.10413	0.682383
$179$	−15.9536	−1.19243	−0.596216	−	0.802824i	$-0.703330\pi$
$179$	−15.9536	−1.19243	−0.596216	+	0.802824i	$0.703330\pi$
$180$	0	0
$181$	−16.0426	−1.19244	−0.596219	−	0.802822i	$-0.703331\pi$
$181$	−16.0426	−1.19244	−0.596219	+	0.802822i	$0.703331\pi$
$182$	4.28268	0.317453
$183$	0	0
$184$	−4.01374	−0.295897
$185$	−3.67903	−0.270487
$186$	0	0
$187$	−4.57073	−0.334245
$188$	−7.25024	−0.528778
$189$	0	0
$190$	4.76727	0.345854
$191$	15.1478	1.09605	0.548027	−	0.836460i	$-0.315379\pi$
$191$	15.1478	1.09605	0.548027	+	0.836460i	$0.315379\pi$
$192$	0	0
$193$	0.108369	0.00780058	0.00390029	−	0.999992i	$-0.498758\pi$
$193$	0.108369	0.00780058	0.00390029	+	0.999992i	$0.498758\pi$
$194$	3.26319	0.234283
$195$	0	0
$196$	15.5268	1.10906
$197$	−18.1938	−1.29625	−0.648127	−	0.761532i	$-0.724448\pi$
$197$	−18.1938	−1.29625	−0.648127	+	0.761532i	$0.724448\pi$
$198$	0	0
$199$	−14.5370	−1.03050	−0.515249	−	0.857040i	$-0.672301\pi$
$199$	−14.5370	−1.03050	−0.515249	+	0.857040i	$0.672301\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	11.1922	0.787483
$203$	−26.1684	−1.83666
$204$	0	0
$205$	5.11141	0.356997
$206$	0.973785	0.0678468
$207$	0	0
$208$	0.902331	0.0625654
$209$	4.76727	0.329759
$210$	0	0
$211$	−6.87778	−0.473486	−0.236743	−	0.971572i	$-0.576080\pi$
$211$	−6.87778	−0.473486	−0.236743	+	0.971572i	$0.576080\pi$
$212$	2.77789	0.190786
$213$	0	0
$214$	11.0164	0.753065
$215$	7.37532	0.502993
$216$	0	0
$217$	31.4256	2.13331
$218$	10.9050	0.738578
$219$	0	0
$220$	1.00000	0.0674200
$221$	−4.12431	−0.277431
$222$	0	0
$223$	27.7666	1.85939	0.929695	−	0.368331i	$-0.120071\pi$
$223$	27.7666	1.85939	0.929695	+	0.368331i	$0.120071\pi$
$224$	4.74624	0.317121
$225$	0	0
$226$	8.20932	0.546076
$227$	6.60000	0.438057	0.219029	−	0.975718i	$-0.429711\pi$
$227$	6.60000	0.438057	0.219029	+	0.975718i	$0.429711\pi$
$228$	0	0
$229$	−15.4162	−1.01873	−0.509366	−	0.860550i	$-0.670120\pi$
$229$	−15.4162	−1.01873	−0.509366	+	0.860550i	$0.670120\pi$
$230$	−4.01374	−0.264658
$231$	0	0
$232$	−5.51350	−0.361979
$233$	−17.6519	−1.15641	−0.578207	−	0.815890i	$-0.696248\pi$
$233$	−17.6519	−1.15641	−0.578207	+	0.815890i	$0.696248\pi$
$234$	0	0
$235$	−7.25024	−0.472953
$236$	14.5733	0.948641
$237$	0	0
$238$	−21.6938	−1.40620
$239$	3.17998	0.205696	0.102848	−	0.994697i	$-0.467204\pi$
$239$	3.17998	0.205696	0.102848	+	0.994697i	$0.467204\pi$
$240$	0	0
$241$	1.58689	0.102220	0.0511101	−	0.998693i	$-0.483724\pi$
$241$	1.58689	0.102220	0.0511101	+	0.998693i	$0.483724\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	11.5897	0.741953
$245$	15.5268	0.991969
$246$	0	0
$247$	4.30165	0.273707
$248$	6.62116	0.420444
$249$	0	0
$250$	1.00000	0.0632456
$251$	−15.1957	−0.959141	−0.479571	−	0.877503i	$-0.659208\pi$
$251$	−15.1957	−0.959141	−0.479571	+	0.877503i	$0.659208\pi$
$252$	0	0
$253$	−4.01374	−0.252342
$254$	−10.6046	−0.665392
$255$	0	0
$256$	1.00000	0.0625000
$257$	15.8537	0.988927	0.494463	−	0.869198i	$-0.335365\pi$
$257$	15.8537	0.988927	0.494463	+	0.869198i	$0.335365\pi$
$258$	0	0
$259$	−17.4615	−1.08501
$260$	0.902331	0.0559602
$261$	0	0
$262$	−6.53972	−0.404025
$263$	−24.3795	−1.50330	−0.751651	−	0.659561i	$-0.770742\pi$
$263$	−24.3795	−1.50330	−0.751651	+	0.659561i	$0.770742\pi$
$264$	0	0
$265$	2.77789	0.170644
$266$	22.6266	1.38732
$267$	0	0
$268$	−11.8465	−0.723643
$269$	13.5591	0.826712	0.413356	−	0.910570i	$-0.364357\pi$
$269$	13.5591	0.826712	0.413356	+	0.910570i	$0.364357\pi$
$270$	0	0
$271$	20.2337	1.22911	0.614554	−	0.788874i	$-0.289336\pi$
$271$	20.2337	1.22911	0.614554	+	0.788874i	$0.289336\pi$
$272$	−4.57073	−0.277142
$273$	0	0
$274$	20.5462	1.24124
$275$	1.00000	0.0603023
$276$	0	0
$277$	−15.1332	−0.909266	−0.454633	−	0.890679i	$-0.650230\pi$
$277$	−15.1332	−0.909266	−0.454633	+	0.890679i	$0.650230\pi$
$278$	−6.29325	−0.377444
$279$	0	0
$280$	4.74624	0.283642
$281$	16.3497	0.975338	0.487669	−	0.873029i	$-0.337847\pi$
$281$	16.3497	0.975338	0.487669	+	0.873029i	$0.337847\pi$
$282$	0	0
$283$	−16.7738	−0.997100	−0.498550	−	0.866861i	$-0.666134\pi$
$283$	−16.7738	−0.997100	−0.498550	+	0.866861i	$0.666134\pi$
$284$	−7.93501	−0.470856
$285$	0	0
$286$	0.902331	0.0533559
$287$	24.2600	1.43202
$288$	0	0
$289$	3.89162	0.228919
$290$	−5.51350	−0.323764
$291$	0	0
$292$	−11.7843	−0.689627
$293$	−5.43829	−0.317708	−0.158854	−	0.987302i	$-0.550780\pi$
$293$	−5.43829	−0.317708	−0.158854	+	0.987302i	$0.550780\pi$
$294$	0	0
$295$	14.5733	0.848491
$296$	−3.67903	−0.213839
$297$	0	0
$298$	−7.78011	−0.450690
$299$	−3.62172	−0.209450
$300$	0	0
$301$	35.0050	2.01766
$302$	5.05466	0.290863
$303$	0	0
$304$	4.76727	0.273421
$305$	11.5897	0.663622
$306$	0	0
$307$	9.37652	0.535146	0.267573	−	0.963538i	$-0.413778\pi$
$307$	9.37652	0.535146	0.267573	+	0.963538i	$0.413778\pi$
$308$	4.74624	0.270442
$309$	0	0
$310$	6.62116	0.376056
$311$	12.6466	0.717122	0.358561	−	0.933506i	$-0.383267\pi$
$311$	12.6466	0.717122	0.358561	+	0.933506i	$0.383267\pi$
$312$	0	0
$313$	−11.6705	−0.659658	−0.329829	−	0.944041i	$-0.606991\pi$
$313$	−11.6705	−0.659658	−0.329829	+	0.944041i	$0.606991\pi$
$314$	14.5734	0.822426
$315$	0	0
$316$	9.63130	0.541803
$317$	25.4187	1.42766	0.713828	−	0.700321i	$-0.246960\pi$
$317$	25.4187	1.42766	0.713828	+	0.700321i	$0.246960\pi$
$318$	0	0
$319$	−5.51350	−0.308697
$320$	1.00000	0.0559017
$321$	0	0
$322$	−19.0502	−1.06163
$323$	−21.7899	−1.21242
$324$	0	0
$325$	0.902331	0.0500523
$326$	−16.7404	−0.927164
$327$	0	0
$328$	5.11141	0.282231
$329$	−34.4114	−1.89716
$330$	0	0
$331$	−15.5359	−0.853928	−0.426964	−	0.904269i	$-0.640417\pi$
$331$	−15.5359	−0.853928	−0.426964	+	0.904269i	$0.640417\pi$
$332$	−14.4184	−0.791313
$333$	0	0
$334$	−16.0736	−0.879509
$335$	−11.8465	−0.647246
$336$	0	0
$337$	−21.4931	−1.17081	−0.585403	−	0.810742i	$-0.699064\pi$
$337$	−21.4931	−1.17081	−0.585403	+	0.810742i	$0.699064\pi$
$338$	−12.1858	−0.662820
$339$	0	0
$340$	−4.57073	−0.247883
$341$	6.62116	0.358556
$342$	0	0
$343$	40.4701	2.18518
$344$	7.37532	0.397651
$345$	0	0
$346$	−1.50662	−0.0809965
$347$	30.6326	1.64444	0.822221	−	0.569168i	$-0.192735\pi$
$347$	30.6326	1.64444	0.822221	+	0.569168i	$0.192735\pi$
$348$	0	0
$349$	−2.07686	−0.111172	−0.0555858	−	0.998454i	$-0.517703\pi$
$349$	−2.07686	−0.111172	−0.0555858	+	0.998454i	$0.517703\pi$
$350$	4.74624	0.253697
$351$	0	0
$352$	1.00000	0.0533002
$353$	−15.3883	−0.819036	−0.409518	−	0.912302i	$-0.634303\pi$
$353$	−15.3883	−0.819036	−0.409518	+	0.912302i	$0.634303\pi$
$354$	0	0
$355$	−7.93501	−0.421147
$356$	9.10413	0.482518
$357$	0	0
$358$	−15.9536	−0.843176
$359$	2.63989	0.139328	0.0696640	−	0.997571i	$-0.477807\pi$
$359$	2.63989	0.139328	0.0696640	+	0.997571i	$0.477807\pi$
$360$	0	0
$361$	3.72683	0.196149
$362$	−16.0426	−0.843181
$363$	0	0
$364$	4.28268	0.224473
$365$	−11.7843	−0.616821
$366$	0	0
$367$	12.8038	0.668351	0.334175	−	0.942511i	$-0.391542\pi$
$367$	12.8038	0.668351	0.334175	+	0.942511i	$0.391542\pi$
$368$	−4.01374	−0.209231
$369$	0	0
$370$	−3.67903	−0.191264
$371$	13.1845	0.684507
$372$	0	0
$373$	−9.70724	−0.502622	−0.251311	−	0.967906i	$-0.580862\pi$
$373$	−9.70724	−0.502622	−0.251311	+	0.967906i	$0.580862\pi$
$374$	−4.57073	−0.236347
$375$	0	0
$376$	−7.25024	−0.373903
$377$	−4.97500	−0.256226
$378$	0	0
$379$	−19.7800	−1.01603	−0.508015	−	0.861348i	$-0.669620\pi$
$379$	−19.7800	−1.01603	−0.508015	+	0.861348i	$0.669620\pi$
$380$	4.76727	0.244556
$381$	0	0
$382$	15.1478	0.775028
$383$	2.00397	0.102398	0.0511990	−	0.998688i	$-0.483696\pi$
$383$	2.00397	0.102398	0.0511990	+	0.998688i	$0.483696\pi$
$384$	0	0
$385$	4.74624	0.241891
$386$	0.108369	0.00551584
$387$	0	0
$388$	3.26319	0.165663
$389$	−4.44981	−0.225615	−0.112807	−	0.993617i	$-0.535984\pi$
$389$	−4.44981	−0.225615	−0.112807	+	0.993617i	$0.535984\pi$
$390$	0	0
$391$	18.3458	0.927785
$392$	15.5268	0.784220
$393$	0	0
$394$	−18.1938	−0.916590
$395$	9.63130	0.484603
$396$	0	0
$397$	−38.5665	−1.93560	−0.967800	−	0.251722i	$-0.919003\pi$
$397$	−38.5665	−1.93560	−0.967800	+	0.251722i	$0.919003\pi$
$398$	−14.5370	−0.728672
$399$	0	0
$400$	1.00000	0.0500000
$401$	−1.46851	−0.0733338	−0.0366669	−	0.999328i	$-0.511674\pi$
$401$	−1.46851	−0.0733338	−0.0366669	+	0.999328i	$0.511674\pi$
$402$	0	0
$403$	5.97447	0.297610
$404$	11.1922	0.556835
$405$	0	0
$406$	−26.1684	−1.29872
$407$	−3.67903	−0.182363
$408$	0	0
$409$	7.17759	0.354909	0.177455	−	0.984129i	$-0.443214\pi$
$409$	7.17759	0.354909	0.177455	+	0.984129i	$0.443214\pi$
$410$	5.11141	0.252435
$411$	0	0
$412$	0.973785	0.0479749
$413$	69.1684	3.40355
$414$	0	0
$415$	−14.4184	−0.707772
$416$	0.902331	0.0442404
$417$	0	0
$418$	4.76727	0.233175
$419$	5.68512	0.277736	0.138868	−	0.990311i	$-0.455654\pi$
$419$	5.68512	0.277736	0.138868	+	0.990311i	$0.455654\pi$
$420$	0	0
$421$	7.36149	0.358777	0.179389	−	0.983778i	$-0.442588\pi$
$421$	7.36149	0.358777	0.179389	+	0.983778i	$0.442588\pi$
$422$	−6.87778	−0.334805
$423$	0	0
$424$	2.77789	0.134906
$425$	−4.57073	−0.221713
$426$	0	0
$427$	55.0073	2.66199
$428$	11.0164	0.532497
$429$	0	0
$430$	7.37532	0.355670
$431$	−2.27955	−0.109802	−0.0549011	−	0.998492i	$-0.517484\pi$
$431$	−2.27955	−0.109802	−0.0549011	+	0.998492i	$0.517484\pi$
$432$	0	0
$433$	12.0353	0.578381	0.289190	−	0.957272i	$-0.406614\pi$
$433$	12.0353	0.578381	0.289190	+	0.957272i	$0.406614\pi$
$434$	31.4256	1.50848
$435$	0	0
$436$	10.9050	0.522254
$437$	−19.1346	−0.915331
$438$	0	0
$439$	3.78394	0.180598	0.0902989	−	0.995915i	$-0.471218\pi$
$439$	3.78394	0.180598	0.0902989	+	0.995915i	$0.471218\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	−4.12431	−0.196174
$443$	−28.1658	−1.33820	−0.669100	−	0.743173i	$-0.733320\pi$
$443$	−28.1658	−1.33820	−0.669100	+	0.743173i	$0.733320\pi$
$444$	0	0
$445$	9.10413	0.431577
$446$	27.7666	1.31479
$447$	0	0
$448$	4.74624	0.224239
$449$	6.77168	0.319575	0.159788	−	0.987151i	$-0.448919\pi$
$449$	6.77168	0.319575	0.159788	+	0.987151i	$0.448919\pi$
$450$	0	0
$451$	5.11141	0.240687
$452$	8.20932	0.386134
$453$	0	0
$454$	6.60000	0.309753
$455$	4.28268	0.200775
$456$	0	0
$457$	17.7551	0.830549	0.415274	−	0.909696i	$-0.363686\pi$
$457$	17.7551	0.830549	0.415274	+	0.909696i	$0.363686\pi$
$458$	−15.4162	−0.720353
$459$	0	0
$460$	−4.01374	−0.187142
$461$	33.2182	1.54713	0.773563	−	0.633719i	$-0.218472\pi$
$461$	33.2182	1.54713	0.773563	+	0.633719i	$0.218472\pi$
$462$	0	0
$463$	−19.4191	−0.902480	−0.451240	−	0.892403i	$-0.649018\pi$
$463$	−19.4191	−0.902480	−0.451240	+	0.892403i	$0.649018\pi$
$464$	−5.51350	−0.255958
$465$	0	0
$466$	−17.6519	−0.817709
$467$	22.8797	1.05875	0.529373	−	0.848389i	$-0.322427\pi$
$467$	22.8797	1.05875	0.529373	+	0.848389i	$0.322427\pi$
$468$	0	0
$469$	−56.2265	−2.59630
$470$	−7.25024	−0.334429
$471$	0	0
$472$	14.5733	0.670791
$473$	7.37532	0.339118
$474$	0	0
$475$	4.76727	0.218737
$476$	−21.6938	−0.994333
$477$	0	0
$478$	3.17998	0.145449
$479$	−7.91106	−0.361466	−0.180733	−	0.983532i	$-0.557847\pi$
$479$	−7.91106	−0.361466	−0.180733	+	0.983532i	$0.557847\pi$
$480$	0	0
$481$	−3.31970	−0.151365
$482$	1.58689	0.0722807
$483$	0	0
$484$	1.00000	0.0454545
$485$	3.26319	0.148174
$486$	0	0
$487$	40.5746	1.83861	0.919306	−	0.393544i	$-0.128751\pi$
$487$	40.5746	1.83861	0.919306	+	0.393544i	$0.128751\pi$
$488$	11.5897	0.524640
$489$	0	0
$490$	15.5268	0.701428
$491$	3.55897	0.160614	0.0803071	−	0.996770i	$-0.474410\pi$
$491$	3.55897	0.160614	0.0803071	+	0.996770i	$0.474410\pi$
$492$	0	0
$493$	25.2008	1.13499
$494$	4.30165	0.193540
$495$	0	0
$496$	6.62116	0.297299
$497$	−37.6614	−1.68935
$498$	0	0
$499$	23.6158	1.05719	0.528594	−	0.848875i	$-0.322720\pi$
$499$	23.6158	1.05719	0.528594	+	0.848875i	$0.322720\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−15.1957	−0.678215
$503$	−40.2594	−1.79508	−0.897539	−	0.440934i	$-0.854647\pi$
$503$	−40.2594	−1.79508	−0.897539	+	0.440934i	$0.854647\pi$
$504$	0	0
$505$	11.1922	0.498048
$506$	−4.01374	−0.178433
$507$	0	0
$508$	−10.6046	−0.470503
$509$	−2.82837	−0.125365	−0.0626826	−	0.998034i	$-0.519966\pi$
$509$	−2.82837	−0.125365	−0.0626826	+	0.998034i	$0.519966\pi$
$510$	0	0
$511$	−55.9313	−2.47426
$512$	1.00000	0.0441942
$513$	0	0
$514$	15.8537	0.699277
$515$	0.973785	0.0429101
$516$	0	0
$517$	−7.25024	−0.318865
$518$	−17.4615	−0.767216
$519$	0	0
$520$	0.902331	0.0395698
$521$	−5.24617	−0.229839	−0.114919	−	0.993375i	$-0.536661\pi$
$521$	−5.24617	−0.229839	−0.114919	+	0.993375i	$0.536661\pi$
$522$	0	0
$523$	41.3842	1.80960	0.904802	−	0.425833i	$-0.140019\pi$
$523$	41.3842	1.80960	0.904802	+	0.425833i	$0.140019\pi$
$524$	−6.53972	−0.285689
$525$	0	0
$526$	−24.3795	−1.06300
$527$	−30.2636	−1.31830
$528$	0	0
$529$	−6.88986	−0.299559
$530$	2.77789	0.120664
$531$	0	0
$532$	22.6266	0.980987
$533$	4.61218	0.199776
$534$	0	0
$535$	11.0164	0.476280
$536$	−11.8465	−0.511693
$537$	0	0
$538$	13.5591	0.584573
$539$	15.5268	0.668785
$540$	0	0
$541$	43.4527	1.86818	0.934089	−	0.357039i	$-0.116214\pi$
$541$	43.4527	1.86818	0.934089	+	0.357039i	$0.116214\pi$
$542$	20.2337	0.869111
$543$	0	0
$544$	−4.57073	−0.195969
$545$	10.9050	0.467118
$546$	0	0
$547$	14.6453	0.626188	0.313094	−	0.949722i	$-0.398634\pi$
$547$	14.6453	0.626188	0.313094	+	0.949722i	$0.398634\pi$
$548$	20.5462	0.877689
$549$	0	0
$550$	1.00000	0.0426401
$551$	−26.2843	−1.11975
$552$	0	0
$553$	45.7124	1.94389
$554$	−15.1332	−0.642948
$555$	0	0
$556$	−6.29325	−0.266893
$557$	−42.4840	−1.80010	−0.900052	−	0.435782i	$-0.856472\pi$
$557$	−42.4840	−1.80010	−0.900052	+	0.435782i	$0.856472\pi$
$558$	0	0
$559$	6.65498	0.281476
$560$	4.74624	0.200565
$561$	0	0
$562$	16.3497	0.689668
$563$	1.76894	0.0745518	0.0372759	−	0.999305i	$-0.488132\pi$
$563$	1.76894	0.0745518	0.0372759	+	0.999305i	$0.488132\pi$
$564$	0	0
$565$	8.20932	0.345369
$566$	−16.7738	−0.705056
$567$	0	0
$568$	−7.93501	−0.332946
$569$	−27.3228	−1.14543	−0.572716	−	0.819754i	$-0.694110\pi$
$569$	−27.3228	−1.14543	−0.572716	+	0.819754i	$0.694110\pi$
$570$	0	0
$571$	−11.7047	−0.489828	−0.244914	−	0.969545i	$-0.578760\pi$
$571$	−11.7047	−0.489828	−0.244914	+	0.969545i	$0.578760\pi$
$572$	0.902331	0.0377283
$573$	0	0
$574$	24.2600	1.01259
$575$	−4.01374	−0.167385
$576$	0	0
$577$	44.8871	1.86867	0.934336	−	0.356392i	$-0.115993\pi$
$577$	44.8871	1.86867	0.934336	+	0.356392i	$0.115993\pi$
$578$	3.89162	0.161870
$579$	0	0
$580$	−5.51350	−0.228936
$581$	−68.4332	−2.83909
$582$	0	0
$583$	2.77789	0.115048
$584$	−11.7843	−0.487640
$585$	0	0
$586$	−5.43829	−0.224654
$587$	−21.5291	−0.888600	−0.444300	−	0.895878i	$-0.646548\pi$
$587$	−21.5291	−0.888600	−0.444300	+	0.895878i	$0.646548\pi$
$588$	0	0
$589$	31.5648	1.30061
$590$	14.5733	0.599973
$591$	0	0
$592$	−3.67903	−0.151207
$593$	−6.96602	−0.286060	−0.143030	−	0.989718i	$-0.545685\pi$
$593$	−6.96602	−0.286060	−0.143030	+	0.989718i	$0.545685\pi$
$594$	0	0
$595$	−21.6938	−0.889359
$596$	−7.78011	−0.318686
$597$	0	0
$598$	−3.62172	−0.148103
$599$	−17.7521	−0.725332	−0.362666	−	0.931919i	$-0.618133\pi$
$599$	−17.7521	−0.725332	−0.362666	+	0.931919i	$0.618133\pi$
$600$	0	0
$601$	−22.6035	−0.922015	−0.461008	−	0.887396i	$-0.652512\pi$
$601$	−22.6035	−0.922015	−0.461008	+	0.887396i	$0.652512\pi$
$602$	35.0050	1.42670
$603$	0	0
$604$	5.05466	0.205671
$605$	1.00000	0.0406558
$606$	0	0
$607$	−2.29881	−0.0933058	−0.0466529	−	0.998911i	$-0.514855\pi$
$607$	−2.29881	−0.0933058	−0.0466529	+	0.998911i	$0.514855\pi$
$608$	4.76727	0.193338
$609$	0	0
$610$	11.5897	0.469252
$611$	−6.54211	−0.264666
$612$	0	0
$613$	−25.1350	−1.01519	−0.507596	−	0.861595i	$-0.669466\pi$
$613$	−25.1350	−1.01519	−0.507596	+	0.861595i	$0.669466\pi$
$614$	9.37652	0.378405
$615$	0	0
$616$	4.74624	0.191231
$617$	−15.9144	−0.640690	−0.320345	−	0.947301i	$-0.603799\pi$
$617$	−15.9144	−0.640690	−0.320345	+	0.947301i	$0.603799\pi$
$618$	0	0
$619$	−47.1805	−1.89635	−0.948173	−	0.317754i	$-0.897071\pi$
$619$	−47.1805	−1.89635	−0.948173	+	0.317754i	$0.897071\pi$
$620$	6.62116	0.265912
$621$	0	0
$622$	12.6466	0.507082
$623$	43.2104	1.73119
$624$	0	0
$625$	1.00000	0.0400000
$626$	−11.6705	−0.466449
$627$	0	0
$628$	14.5734	0.581543
$629$	16.8159	0.670492
$630$	0	0
$631$	10.4051	0.414221	0.207111	−	0.978318i	$-0.433594\pi$
$631$	10.4051	0.414221	0.207111	+	0.978318i	$0.433594\pi$
$632$	9.63130	0.383113
$633$	0	0
$634$	25.4187	1.00951
$635$	−10.6046	−0.420831
$636$	0	0
$637$	14.0103	0.555108
$638$	−5.51350	−0.218282
$639$	0	0
$640$	1.00000	0.0395285
$641$	−19.7955	−0.781873	−0.390937	−	0.920418i	$-0.627849\pi$
$641$	−19.7955	−0.781873	−0.390937	+	0.920418i	$0.627849\pi$
$642$	0	0
$643$	25.7817	1.01673	0.508366	−	0.861141i	$-0.330250\pi$
$643$	25.7817	1.01673	0.508366	+	0.861141i	$0.330250\pi$
$644$	−19.0502	−0.750682
$645$	0	0
$646$	−21.7899	−0.857313
$647$	−4.40776	−0.173287	−0.0866435	−	0.996239i	$-0.527614\pi$
$647$	−4.40776	−0.173287	−0.0866435	+	0.996239i	$0.527614\pi$
$648$	0	0
$649$	14.5733	0.572052
$650$	0.902331	0.0353923
$651$	0	0
$652$	−16.7404	−0.655604
$653$	−1.30666	−0.0511335	−0.0255668	−	0.999673i	$-0.508139\pi$
$653$	−1.30666	−0.0511335	−0.0255668	+	0.999673i	$0.508139\pi$
$654$	0	0
$655$	−6.53972	−0.255528
$656$	5.11141	0.199567
$657$	0	0
$658$	−34.4114	−1.34149
$659$	47.5992	1.85420	0.927100	−	0.374813i	$-0.122293\pi$
$659$	47.5992	1.85420	0.927100	+	0.374813i	$0.122293\pi$
$660$	0	0
$661$	3.40376	0.132391	0.0661955	−	0.997807i	$-0.478914\pi$
$661$	3.40376	0.132391	0.0661955	+	0.997807i	$0.478914\pi$
$662$	−15.5359	−0.603818
$663$	0	0
$664$	−14.4184	−0.559543
$665$	22.6266	0.877421
$666$	0	0
$667$	22.1298	0.856869
$668$	−16.0736	−0.621907
$669$	0	0
$670$	−11.8465	−0.457672
$671$	11.5897	0.447414
$672$	0	0
$673$	−44.7962	−1.72677	−0.863384	−	0.504547i	$-0.831659\pi$
$673$	−44.7962	−1.72677	−0.863384	+	0.504547i	$0.831659\pi$
$674$	−21.4931	−0.827885
$675$	0	0
$676$	−12.1858	−0.468685
$677$	36.3059	1.39535	0.697674	−	0.716415i	$-0.254218\pi$
$677$	36.3059	1.39535	0.697674	+	0.716415i	$0.254218\pi$
$678$	0	0
$679$	15.4879	0.594370
$680$	−4.57073	−0.175280
$681$	0	0
$682$	6.62116	0.253537
$683$	22.7675	0.871174	0.435587	−	0.900147i	$-0.356541\pi$
$683$	22.7675	0.871174	0.435587	+	0.900147i	$0.356541\pi$
$684$	0	0
$685$	20.5462	0.785029
$686$	40.4701	1.54516
$687$	0	0
$688$	7.37532	0.281181
$689$	2.50658	0.0954929
$690$	0	0
$691$	2.41991	0.0920578	0.0460289	−	0.998940i	$-0.485343\pi$
$691$	2.41991	0.0920578	0.0460289	+	0.998940i	$0.485343\pi$
$692$	−1.50662	−0.0572732
$693$	0	0
$694$	30.6326	1.16280
$695$	−6.29325	−0.238716
$696$	0	0
$697$	−23.3629	−0.884934
$698$	−2.07686	−0.0786103
$699$	0	0
$700$	4.74624	0.179391
$701$	−3.44745	−0.130208	−0.0651042	−	0.997878i	$-0.520738\pi$
$701$	−3.44745	−0.130208	−0.0651042	+	0.997878i	$0.520738\pi$
$702$	0	0
$703$	−17.5389	−0.661492
$704$	1.00000	0.0376889
$705$	0	0
$706$	−15.3883	−0.579146
$707$	53.1210	1.99782
$708$	0	0
$709$	19.1432	0.718940	0.359470	−	0.933157i	$-0.382958\pi$
$709$	19.1432	0.718940	0.359470	+	0.933157i	$0.382958\pi$
$710$	−7.93501	−0.297796
$711$	0	0
$712$	9.10413	0.341192
$713$	−26.5756	−0.995265
$714$	0	0
$715$	0.902331	0.0337453
$716$	−15.9536	−0.596216
$717$	0	0
$718$	2.63989	0.0985198
$719$	24.1115	0.899208	0.449604	−	0.893228i	$-0.351565\pi$
$719$	24.1115	0.899208	0.449604	+	0.893228i	$0.351565\pi$
$720$	0	0
$721$	4.62181	0.172125
$722$	3.72683	0.138698
$723$	0	0
$724$	−16.0426	−0.596219
$725$	−5.51350	−0.204766
$726$	0	0
$727$	31.4145	1.16510	0.582549	−	0.812795i	$-0.302055\pi$
$727$	31.4145	1.16510	0.582549	+	0.812795i	$0.302055\pi$
$728$	4.28268	0.158727
$729$	0	0
$730$	−11.7843	−0.436158
$731$	−33.7106	−1.24683
$732$	0	0
$733$	−29.8552	−1.10273	−0.551363	−	0.834265i	$-0.685892\pi$
$733$	−29.8552	−1.10273	−0.551363	+	0.834265i	$0.685892\pi$
$734$	12.8038	0.472595
$735$	0	0
$736$	−4.01374	−0.147949
$737$	−11.8465	−0.436373
$738$	0	0
$739$	−4.30474	−0.158352	−0.0791762	−	0.996861i	$-0.525229\pi$
$739$	−4.30474	−0.158352	−0.0791762	+	0.996861i	$0.525229\pi$
$740$	−3.67903	−0.135244
$741$	0	0
$742$	13.1845	0.484019
$743$	−19.6056	−0.719260	−0.359630	−	0.933095i	$-0.617097\pi$
$743$	−19.6056	−0.719260	−0.359630	+	0.933095i	$0.617097\pi$
$744$	0	0
$745$	−7.78011	−0.285041
$746$	−9.70724	−0.355407
$747$	0	0
$748$	−4.57073	−0.167123
$749$	52.2864	1.91050
$750$	0	0
$751$	−6.58613	−0.240331	−0.120166	−	0.992754i	$-0.538343\pi$
$751$	−6.58613	−0.240331	−0.120166	+	0.992754i	$0.538343\pi$
$752$	−7.25024	−0.264389
$753$	0	0
$754$	−4.97500	−0.181179
$755$	5.05466	0.183958
$756$	0	0
$757$	−3.26658	−0.118726	−0.0593629	−	0.998236i	$-0.518907\pi$
$757$	−3.26658	−0.118726	−0.0593629	+	0.998236i	$0.518907\pi$
$758$	−19.7800	−0.718441
$759$	0	0
$760$	4.76727	0.172927
$761$	−21.6045	−0.783161	−0.391581	−	0.920144i	$-0.628072\pi$
$761$	−21.6045	−0.783161	−0.391581	+	0.920144i	$0.628072\pi$
$762$	0	0
$763$	51.7576	1.87375
$764$	15.1478	0.548027
$765$	0	0
$766$	2.00397	0.0724063
$767$	13.1499	0.474817
$768$	0	0
$769$	−8.79420	−0.317127	−0.158564	−	0.987349i	$-0.550686\pi$
$769$	−8.79420	−0.317127	−0.158564	+	0.987349i	$0.550686\pi$
$770$	4.74624	0.171043
$771$	0	0
$772$	0.108369	0.00390029
$773$	−21.3813	−0.769033	−0.384516	−	0.923118i	$-0.625632\pi$
$773$	−21.3813	−0.769033	−0.384516	+	0.923118i	$0.625632\pi$
$774$	0	0
$775$	6.62116	0.237839
$776$	3.26319	0.117142
$777$	0	0
$778$	−4.44981	−0.159534
$779$	24.3675	0.873055
$780$	0	0
$781$	−7.93501	−0.283937
$782$	18.3458	0.656043
$783$	0	0
$784$	15.5268	0.554528
$785$	14.5734	0.520148
$786$	0	0
$787$	14.2727	0.508767	0.254384	−	0.967103i	$-0.418127\pi$
$787$	14.2727	0.508767	0.254384	+	0.967103i	$0.418127\pi$
$788$	−18.1938	−0.648127
$789$	0	0
$790$	9.63130	0.342666
$791$	38.9634	1.38538
$792$	0	0
$793$	10.4577	0.371364
$794$	−38.5665	−1.36868
$795$	0	0
$796$	−14.5370	−0.515249
$797$	−2.33265	−0.0826266	−0.0413133	−	0.999146i	$-0.513154\pi$
$797$	−2.33265	−0.0826266	−0.0413133	+	0.999146i	$0.513154\pi$
$798$	0	0
$799$	33.1389	1.17237
$800$	1.00000	0.0353553
$801$	0	0
$802$	−1.46851	−0.0518548
$803$	−11.7843	−0.415861
$804$	0	0
$805$	−19.0502	−0.671431
$806$	5.97447	0.210442
$807$	0	0
$808$	11.1922	0.393742
$809$	−5.00683	−0.176031	−0.0880154	−	0.996119i	$-0.528052\pi$
$809$	−5.00683	−0.176031	−0.0880154	+	0.996119i	$0.528052\pi$
$810$	0	0
$811$	22.3073	0.783315	0.391657	−	0.920111i	$-0.371902\pi$
$811$	22.3073	0.783315	0.391657	+	0.920111i	$0.371902\pi$
$812$	−26.1684	−0.918331
$813$	0	0
$814$	−3.67903	−0.128950
$815$	−16.7404	−0.586390
$816$	0	0
$817$	35.1601	1.23010
$818$	7.17759	0.250959
$819$	0	0
$820$	5.11141	0.178498
$821$	−4.85415	−0.169411	−0.0847055	−	0.996406i	$-0.526995\pi$
$821$	−4.85415	−0.169411	−0.0847055	+	0.996406i	$0.526995\pi$
$822$	0	0
$823$	−0.332656	−0.0115957	−0.00579783	−	0.999983i	$-0.501846\pi$
$823$	−0.332656	−0.0115957	−0.00579783	+	0.999983i	$0.501846\pi$
$824$	0.973785	0.0339234
$825$	0	0
$826$	69.1684	2.40668
$827$	−46.2224	−1.60731	−0.803655	−	0.595095i	$-0.797114\pi$
$827$	−46.2224	−1.60731	−0.803655	+	0.595095i	$0.797114\pi$
$828$	0	0
$829$	−34.3165	−1.19186	−0.595931	−	0.803036i	$-0.703217\pi$
$829$	−34.3165	−1.19186	−0.595931	+	0.803036i	$0.703217\pi$
$830$	−14.4184	−0.500470
$831$	0	0
$832$	0.902331	0.0312827
$833$	−70.9688	−2.45892
$834$	0	0
$835$	−16.0736	−0.556250
$836$	4.76727	0.164879
$837$	0	0
$838$	5.68512	0.196389
$839$	23.3675	0.806735	0.403367	−	0.915038i	$-0.367840\pi$
$839$	23.3675	0.806735	0.403367	+	0.915038i	$0.367840\pi$
$840$	0	0
$841$	1.39873	0.0482321
$842$	7.36149	0.253694
$843$	0	0
$844$	−6.87778	−0.236743
$845$	−12.1858	−0.419204
$846$	0	0
$847$	4.74624	0.163083
$848$	2.77789	0.0953931
$849$	0	0
$850$	−4.57073	−0.156775
$851$	14.7667	0.506195
$852$	0	0
$853$	9.99714	0.342296	0.171148	−	0.985245i	$-0.445252\pi$
$853$	9.99714	0.342296	0.171148	+	0.985245i	$0.445252\pi$
$854$	55.0073	1.88231
$855$	0	0
$856$	11.0164	0.376532
$857$	17.5743	0.600325	0.300163	−	0.953888i	$-0.402959\pi$
$857$	17.5743	0.600325	0.300163	+	0.953888i	$0.402959\pi$
$858$	0	0
$859$	−32.8079	−1.11939	−0.559695	−	0.828698i	$-0.689082\pi$
$859$	−32.8079	−1.11939	−0.559695	+	0.828698i	$0.689082\pi$
$860$	7.37532	0.251496
$861$	0	0
$862$	−2.27955	−0.0776418
$863$	23.7700	0.809140	0.404570	−	0.914507i	$-0.367421\pi$
$863$	23.7700	0.809140	0.404570	+	0.914507i	$0.367421\pi$
$864$	0	0
$865$	−1.50662	−0.0512267
$866$	12.0353	0.408977
$867$	0	0
$868$	31.4256	1.06665
$869$	9.63130	0.326719
$870$	0	0
$871$	−10.6895	−0.362200
$872$	10.9050	0.369289
$873$	0	0
$874$	−19.1346	−0.647237
$875$	4.74624	0.160452
$876$	0	0
$877$	37.8910	1.27949	0.639745	−	0.768587i	$-0.279040\pi$
$877$	37.8910	1.27949	0.639745	+	0.768587i	$0.279040\pi$
$878$	3.78394	0.127702
$879$	0	0
$880$	1.00000	0.0337100
$881$	−20.0800	−0.676512	−0.338256	−	0.941054i	$-0.609837\pi$
$881$	−20.0800	−0.676512	−0.338256	+	0.941054i	$0.609837\pi$
$882$	0	0
$883$	−56.9302	−1.91585	−0.957927	−	0.287011i	$-0.907339\pi$
$883$	−56.9302	−1.91585	−0.957927	+	0.287011i	$0.907339\pi$
$884$	−4.12431	−0.138716
$885$	0	0
$886$	−28.1658	−0.946250
$887$	−41.9429	−1.40831	−0.704153	−	0.710049i	$-0.748673\pi$
$887$	−41.9429	−1.40831	−0.704153	+	0.710049i	$0.748673\pi$
$888$	0	0
$889$	−50.3320	−1.68808
$890$	9.10413	0.305171
$891$	0	0
$892$	27.7666	0.929695
$893$	−34.5638	−1.15663
$894$	0	0
$895$	−15.9536	−0.533272
$896$	4.74624	0.158561
$897$	0	0
$898$	6.77168	0.225974
$899$	−36.5058	−1.21754
$900$	0	0
$901$	−12.6970	−0.422998
$902$	5.11141	0.170191
$903$	0	0
$904$	8.20932	0.273038
$905$	−16.0426	−0.533274
$906$	0	0
$907$	29.9519	0.994536	0.497268	−	0.867597i	$-0.334337\pi$
$907$	29.9519	0.994536	0.497268	+	0.867597i	$0.334337\pi$
$908$	6.60000	0.219029
$909$	0	0
$910$	4.28268	0.141969
$911$	−28.6779	−0.950142	−0.475071	−	0.879947i	$-0.657578\pi$
$911$	−28.6779	−0.950142	−0.475071	+	0.879947i	$0.657578\pi$
$912$	0	0
$913$	−14.4184	−0.477179
$914$	17.7551	0.587287
$915$	0	0
$916$	−15.4162	−0.509366
$917$	−31.0391	−1.02500
$918$	0	0
$919$	19.1234	0.630821	0.315411	−	0.948955i	$-0.397858\pi$
$919$	19.1234	0.630821	0.315411	+	0.948955i	$0.397858\pi$
$920$	−4.01374	−0.132329
$921$	0	0
$922$	33.2182	1.09398
$923$	−7.16000	−0.235674
$924$	0	0
$925$	−3.67903	−0.120966
$926$	−19.4191	−0.638150
$927$	0	0
$928$	−5.51350	−0.180990
$929$	−30.2329	−0.991910	−0.495955	−	0.868348i	$-0.665182\pi$
$929$	−30.2329	−0.991910	−0.495955	+	0.868348i	$0.665182\pi$
$930$	0	0
$931$	74.0203	2.42592
$932$	−17.6519	−0.578207
$933$	0	0
$934$	22.8797	0.748647
$935$	−4.57073	−0.149479
$936$	0	0
$937$	7.62598	0.249130	0.124565	−	0.992211i	$-0.460246\pi$
$937$	7.62598	0.249130	0.124565	+	0.992211i	$0.460246\pi$
$938$	−56.2265	−1.83586
$939$	0	0
$940$	−7.25024	−0.236477
$941$	−15.1076	−0.492493	−0.246246	−	0.969207i	$-0.579197\pi$
$941$	−15.1076	−0.492493	−0.246246	+	0.969207i	$0.579197\pi$
$942$	0	0
$943$	−20.5159	−0.668090
$944$	14.5733	0.474321
$945$	0	0
$946$	7.37532	0.239792
$947$	−12.9844	−0.421937	−0.210968	−	0.977493i	$-0.567662\pi$
$947$	−12.9844	−0.421937	−0.210968	+	0.977493i	$0.567662\pi$
$948$	0	0
$949$	−10.6334	−0.345174
$950$	4.76727	0.154671
$951$	0	0
$952$	−21.6938	−0.703100
$953$	35.9475	1.16445	0.582227	−	0.813026i	$-0.302181\pi$
$953$	35.9475	1.16445	0.582227	+	0.813026i	$0.302181\pi$
$954$	0	0
$955$	15.1478	0.490171
$956$	3.17998	0.102848
$957$	0	0
$958$	−7.91106	−0.255595
$959$	97.5170	3.14899
$960$	0	0
$961$	12.8397	0.414184
$962$	−3.31970	−0.107031
$963$	0	0
$964$	1.58689	0.0511101
$965$	0.108369	0.00348853
$966$	0	0
$967$	22.0328	0.708528	0.354264	−	0.935145i	$-0.384731\pi$
$967$	22.0328	0.708528	0.354264	+	0.935145i	$0.384731\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	3.26319	0.104775
$971$	−27.6515	−0.887380	−0.443690	−	0.896180i	$-0.646331\pi$
$971$	−27.6515	−0.887380	−0.443690	+	0.896180i	$0.646331\pi$
$972$	0	0
$973$	−29.8692	−0.957564
$974$	40.5746	1.30009
$975$	0	0
$976$	11.5897	0.370976
$977$	−4.70788	−0.150618	−0.0753092	−	0.997160i	$-0.523994\pi$
$977$	−4.70788	−0.150618	−0.0753092	+	0.997160i	$0.523994\pi$
$978$	0	0
$979$	9.10413	0.290969
$980$	15.5268	0.495985
$981$	0	0
$982$	3.55897	0.113571
$983$	−3.76974	−0.120236	−0.0601180	−	0.998191i	$-0.519148\pi$
$983$	−3.76974	−0.120236	−0.0601180	+	0.998191i	$0.519148\pi$
$984$	0	0
$985$	−18.1938	−0.579703
$986$	25.2008	0.802556
$987$	0	0
$988$	4.30165	0.136854
$989$	−29.6026	−0.941309
$990$	0	0
$991$	−25.2153	−0.800991	−0.400496	−	0.916299i	$-0.631162\pi$
$991$	−25.2153	−0.800991	−0.400496	+	0.916299i	$0.631162\pi$
$992$	6.62116	0.210222
$993$	0	0
$994$	−37.6614	−1.19455
$995$	−14.5370	−0.460853
$996$	0	0
$997$	25.0474	0.793259	0.396629	−	0.917979i	$-0.370180\pi$
$997$	25.0474	0.793259	0.396629	+	0.917979i	$0.370180\pi$
$998$	23.6158	0.747544
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8910.2.a.cd.1.7		7
3.2	odd	2		8910.2.a.ca.1.7		7
9.2	odd	6		2970.2.i.k.1981.1		14
9.4	even	3		990.2.i.j.331.4	✓	14
9.5	odd	6		2970.2.i.k.991.1		14
9.7	even	3		990.2.i.j.661.4	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
990.2.i.j.331.4	✓	14	9.4	even	3
990.2.i.j.661.4	yes	14	9.7	even	3
2970.2.i.k.991.1		14	9.5	odd	6
2970.2.i.k.1981.1		14	9.2	odd	6
8910.2.a.ca.1.7		7	3.2	odd	2
8910.2.a.cd.1.7		7	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8910 = 2 \cdot 3^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8910.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.74624 q^{7} +1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +0.902331 q^{13} +4.74624 q^{14} +1.00000 q^{16} -4.57073 q^{17} +4.76727 q^{19} +1.00000 q^{20} +1.00000 q^{22} -4.01374 q^{23} +1.00000 q^{25} +0.902331 q^{26} +4.74624 q^{28} -5.51350 q^{29} +6.62116 q^{31} +1.00000 q^{32} -4.57073 q^{34} +4.74624 q^{35} -3.67903 q^{37} +4.76727 q^{38} +1.00000 q^{40} +5.11141 q^{41} +7.37532 q^{43} +1.00000 q^{44} -4.01374 q^{46} -7.25024 q^{47} +15.5268 q^{49} +1.00000 q^{50} +0.902331 q^{52} +2.77789 q^{53} +1.00000 q^{55} +4.74624 q^{56} -5.51350 q^{58} +14.5733 q^{59} +11.5897 q^{61} +6.62116 q^{62} +1.00000 q^{64} +0.902331 q^{65} -11.8465 q^{67} -4.57073 q^{68} +4.74624 q^{70} -7.93501 q^{71} -11.7843 q^{73} -3.67903 q^{74} +4.76727 q^{76} +4.74624 q^{77} +9.63130 q^{79} +1.00000 q^{80} +5.11141 q^{82} -14.4184 q^{83} -4.57073 q^{85} +7.37532 q^{86} +1.00000 q^{88} +9.10413 q^{89} +4.28268 q^{91} -4.01374 q^{92} -7.25024 q^{94} +4.76727 q^{95} +3.26319 q^{97} +15.5268 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} + 7 q^{4} + 7 q^{5} + 5 q^{7} + 7 q^{8} + 7 q^{10} + 7 q^{11} + 5 q^{13} + 5 q^{14} + 7 q^{16} + 3 q^{17} + 11 q^{19} + 7 q^{20} + 7 q^{22} + 3 q^{23} + 7 q^{25} + 5 q^{26} + 5 q^{28} + 12 q^{29}+ \cdots + 12 q^{98}+O(q^{100}) \) 7 * q + 7 * q^2 + 7 * q^4 + 7 * q^5 + 5 * q^7 + 7 * q^8 + 7 * q^10 + 7 * q^11 + 5 * q^13 + 5 * q^14 + 7 * q^16 + 3 * q^17 + 11 * q^19 + 7 * q^20 + 7 * q^22 + 3 * q^23 + 7 * q^25 + 5 * q^26 + 5 * q^28 + 12 * q^29 + 5 * q^31 + 7 * q^32 + 3 * q^34 + 5 * q^35 + 2 * q^37 + 11 * q^38 + 7 * q^40 + 6 * q^41 + 17 * q^43 + 7 * q^44 + 3 * q^46 - 3 * q^47 + 12 * q^49 + 7 * q^50 + 5 * q^52 + 3 * q^53 + 7 * q^55 + 5 * q^56 + 12 * q^58 + 18 * q^59 + 11 * q^61 + 5 * q^62 + 7 * q^64 + 5 * q^65 + 20 * q^67 + 3 * q^68 + 5 * q^70 + 12 * q^71 + 2 * q^73 + 2 * q^74 + 11 * q^76 + 5 * q^77 - 7 * q^79 + 7 * q^80 + 6 * q^82 + 3 * q^85 + 17 * q^86 + 7 * q^88 - 3 * q^89 + 7 * q^91 + 3 * q^92 - 3 * q^94 + 11 * q^95 + 23 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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