LMFDB - Embedded newform 8512.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8512,2,Mod(1,8512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8512, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.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:	$67.9686622005$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 532)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	8512.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{3} +4.23607 q^{5} -1.00000 q^{7} -0.381966 q^{9} +2.61803 q^{11} -3.47214 q^{13} -6.85410 q^{15} -5.85410 q^{17} -1.00000 q^{19} +1.61803 q^{21} -8.23607 q^{23} +12.9443 q^{25} +5.47214 q^{27} +4.61803 q^{29} -3.85410 q^{31} -4.23607 q^{33} -4.23607 q^{35} +8.23607 q^{37} +5.61803 q^{39} +11.5623 q^{41} -4.47214 q^{43} -1.61803 q^{45} +7.47214 q^{47} +1.00000 q^{49} +9.47214 q^{51} -6.09017 q^{53} +11.0902 q^{55} +1.61803 q^{57} -11.0000 q^{59} +4.23607 q^{61} +0.381966 q^{63} -14.7082 q^{65} +7.56231 q^{67} +13.3262 q^{69} -3.76393 q^{71} +3.61803 q^{73} -20.9443 q^{75} -2.61803 q^{77} -4.47214 q^{79} -7.70820 q^{81} -10.5623 q^{83} -24.7984 q^{85} -7.47214 q^{87} -15.7082 q^{89} +3.47214 q^{91} +6.23607 q^{93} -4.23607 q^{95} -3.47214 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 4 q^{5} - 2 q^{7} - 3 q^{9} + 3 q^{11} + 2 q^{13} - 7 q^{15} - 5 q^{17} - 2 q^{19} + q^{21} - 12 q^{23} + 8 q^{25} + 2 q^{27} + 7 q^{29} - q^{31} - 4 q^{33} - 4 q^{35} + 12 q^{37} + 9 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) $ 2 * q - q^3 + 4 * q^5 - 2 * q^7 - 3 * q^9 + 3 * q^11 + 2 * q^13 - 7 * q^15 - 5 * q^17 - 2 * q^19 + q^21 - 12 * q^23 + 8 * q^25 + 2 * q^27 + 7 * q^29 - q^31 - 4 * q^33 - 4 * q^35 + 12 * q^37 + 9 * q^39 + 3 * q^41 - q^45 + 6 * q^47 + 2 * q^49 + 10 * q^51 - q^53 + 11 * q^55 + q^57 - 22 * q^59 + 4 * q^61 + 3 * q^63 - 16 * q^65 - 5 * q^67 + 11 * q^69 - 12 * q^71 + 5 * q^73 - 24 * q^75 - 3 * q^77 - 2 * q^81 - q^83 - 25 * q^85 - 6 * q^87 - 18 * q^89 - 2 * q^91 + 8 * q^93 - 4 * q^95 + 2 * q^97 - 2 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.61803	−0.934172	−0.467086	−	0.884212i	$-0.654696\pi$
$3$	−1.61803	−0.934172	−0.467086	+	0.884212i	$0.654696\pi$
$4$	0	0
$5$	4.23607	1.89443	0.947214	−	0.320603i	$-0.103886\pi$
$5$	4.23607	1.89443	0.947214	+	0.320603i	$0.103886\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	2.61803	0.789367	0.394683	−	0.918817i	$-0.370854\pi$
$11$	2.61803	0.789367	0.394683	+	0.918817i	$0.370854\pi$
$12$	0	0
$13$	−3.47214	−0.962997	−0.481499	−	0.876447i	$-0.659907\pi$
$13$	−3.47214	−0.962997	−0.481499	+	0.876447i	$0.659907\pi$
$14$	0	0
$15$	−6.85410	−1.76972
$16$	0	0
$17$	−5.85410	−1.41983	−0.709914	−	0.704288i	$-0.751266\pi$
$17$	−5.85410	−1.41983	−0.709914	+	0.704288i	$0.751266\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.61803	0.353084
$22$	0	0
$23$	−8.23607	−1.71734	−0.858669	−	0.512530i	$-0.828708\pi$
$23$	−8.23607	−1.71734	−0.858669	+	0.512530i	$0.828708\pi$
$24$	0	0
$25$	12.9443	2.58885
$26$	0	0
$27$	5.47214	1.05311
$28$	0	0
$29$	4.61803	0.857547	0.428774	−	0.903412i	$-0.358946\pi$
$29$	4.61803	0.857547	0.428774	+	0.903412i	$0.358946\pi$
$30$	0	0
$31$	−3.85410	−0.692217	−0.346109	−	0.938194i	$-0.612497\pi$
$31$	−3.85410	−0.692217	−0.346109	+	0.938194i	$0.612497\pi$
$32$	0	0
$33$	−4.23607	−0.737405
$34$	0	0
$35$	−4.23607	−0.716026
$36$	0	0
$37$	8.23607	1.35400	0.677001	−	0.735982i	$-0.263279\pi$
$37$	8.23607	1.35400	0.677001	+	0.735982i	$0.263279\pi$
$38$	0	0
$39$	5.61803	0.899605
$40$	0	0
$41$	11.5623	1.80573	0.902864	−	0.429925i	$-0.141460\pi$
$41$	11.5623	1.80573	0.902864	+	0.429925i	$0.141460\pi$
$42$	0	0
$43$	−4.47214	−0.681994	−0.340997	−	0.940064i	$-0.610765\pi$
$43$	−4.47214	−0.681994	−0.340997	+	0.940064i	$0.610765\pi$
$44$	0	0
$45$	−1.61803	−0.241202
$46$	0	0
$47$	7.47214	1.08992	0.544962	−	0.838461i	$-0.316544\pi$
$47$	7.47214	1.08992	0.544962	+	0.838461i	$0.316544\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	9.47214	1.32636
$52$	0	0
$53$	−6.09017	−0.836549	−0.418275	−	0.908321i	$-0.637365\pi$
$53$	−6.09017	−0.836549	−0.418275	+	0.908321i	$0.637365\pi$
$54$	0	0
$55$	11.0902	1.49540
$56$	0	0
$57$	1.61803	0.214314
$58$	0	0
$59$	−11.0000	−1.43208	−0.716039	−	0.698060i	$-0.754047\pi$
$59$	−11.0000	−1.43208	−0.716039	+	0.698060i	$0.754047\pi$
$60$	0	0
$61$	4.23607	0.542373	0.271186	−	0.962527i	$-0.412584\pi$
$61$	4.23607	0.542373	0.271186	+	0.962527i	$0.412584\pi$
$62$	0	0
$63$	0.381966	0.0481232
$64$	0	0
$65$	−14.7082	−1.82433
$66$	0	0
$67$	7.56231	0.923883	0.461941	−	0.886910i	$-0.347153\pi$
$67$	7.56231	0.923883	0.461941	+	0.886910i	$0.347153\pi$
$68$	0	0
$69$	13.3262	1.60429
$70$	0	0
$71$	−3.76393	−0.446697	−0.223348	−	0.974739i	$-0.571699\pi$
$71$	−3.76393	−0.446697	−0.223348	+	0.974739i	$0.571699\pi$
$72$	0	0
$73$	3.61803	0.423459	0.211729	−	0.977328i	$-0.432090\pi$
$73$	3.61803	0.423459	0.211729	+	0.977328i	$0.432090\pi$
$74$	0	0
$75$	−20.9443	−2.41844
$76$	0	0
$77$	−2.61803	−0.298353
$78$	0	0
$79$	−4.47214	−0.503155	−0.251577	−	0.967837i	$-0.580949\pi$
$79$	−4.47214	−0.503155	−0.251577	+	0.967837i	$0.580949\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	−10.5623	−1.15936	−0.579682	−	0.814843i	$-0.696823\pi$
$83$	−10.5623	−1.15936	−0.579682	+	0.814843i	$0.696823\pi$
$84$	0	0
$85$	−24.7984	−2.68976
$86$	0	0
$87$	−7.47214	−0.801097
$88$	0	0
$89$	−15.7082	−1.66507	−0.832533	−	0.553975i	$-0.813110\pi$
$89$	−15.7082	−1.66507	−0.832533	+	0.553975i	$0.813110\pi$
$90$	0	0
$91$	3.47214	0.363979
$92$	0	0
$93$	6.23607	0.646650
$94$	0	0
$95$	−4.23607	−0.434611
$96$	0	0
$97$	−3.47214	−0.352542	−0.176271	−	0.984342i	$-0.556404\pi$
$97$	−3.47214	−0.352542	−0.176271	+	0.984342i	$0.556404\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	14.4164	1.43449	0.717243	−	0.696823i	$-0.245404\pi$
$101$	14.4164	1.43449	0.717243	+	0.696823i	$0.245404\pi$
$102$	0	0
$103$	6.52786	0.643210	0.321605	−	0.946874i	$-0.395778\pi$
$103$	6.52786	0.643210	0.321605	+	0.946874i	$0.395778\pi$
$104$	0	0
$105$	6.85410	0.668892
$106$	0	0
$107$	−15.1803	−1.46754	−0.733769	−	0.679399i	$-0.762241\pi$
$107$	−15.1803	−1.46754	−0.733769	+	0.679399i	$0.762241\pi$
$108$	0	0
$109$	−0.527864	−0.0505602	−0.0252801	−	0.999680i	$-0.508048\pi$
$109$	−0.527864	−0.0505602	−0.0252801	+	0.999680i	$0.508048\pi$
$110$	0	0
$111$	−13.3262	−1.26487
$112$	0	0
$113$	−16.5623	−1.55805	−0.779025	−	0.626992i	$-0.784286\pi$
$113$	−16.5623	−1.55805	−0.779025	+	0.626992i	$0.784286\pi$
$114$	0	0
$115$	−34.8885	−3.25337
$116$	0	0
$117$	1.32624	0.122611
$118$	0	0
$119$	5.85410	0.536645
$120$	0	0
$121$	−4.14590	−0.376900
$122$	0	0
$123$	−18.7082	−1.68686
$124$	0	0
$125$	33.6525	3.00997
$126$	0	0
$127$	−15.7082	−1.39388	−0.696939	−	0.717131i	$-0.745455\pi$
$127$	−15.7082	−1.39388	−0.696939	+	0.717131i	$0.745455\pi$
$128$	0	0
$129$	7.23607	0.637100
$130$	0	0
$131$	−4.90983	−0.428974	−0.214487	−	0.976727i	$-0.568808\pi$
$131$	−4.90983	−0.428974	−0.214487	+	0.976727i	$0.568808\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	23.1803	1.99505
$136$	0	0
$137$	5.41641	0.462755	0.231377	−	0.972864i	$-0.425677\pi$
$137$	5.41641	0.462755	0.231377	+	0.972864i	$0.425677\pi$
$138$	0	0
$139$	−2.52786	−0.214411	−0.107205	−	0.994237i	$-0.534190\pi$
$139$	−2.52786	−0.214411	−0.107205	+	0.994237i	$0.534190\pi$
$140$	0	0
$141$	−12.0902	−1.01818
$142$	0	0
$143$	−9.09017	−0.760158
$144$	0	0
$145$	19.5623	1.62456
$146$	0	0
$147$	−1.61803	−0.133453
$148$	0	0
$149$	−10.4164	−0.853345	−0.426673	−	0.904406i	$-0.640314\pi$
$149$	−10.4164	−0.853345	−0.426673	+	0.904406i	$0.640314\pi$
$150$	0	0
$151$	1.09017	0.0887168	0.0443584	−	0.999016i	$-0.485876\pi$
$151$	1.09017	0.0887168	0.0443584	+	0.999016i	$0.485876\pi$
$152$	0	0
$153$	2.23607	0.180775
$154$	0	0
$155$	−16.3262	−1.31135
$156$	0	0
$157$	11.5623	0.922772	0.461386	−	0.887199i	$-0.347352\pi$
$157$	11.5623	0.922772	0.461386	+	0.887199i	$0.347352\pi$
$158$	0	0
$159$	9.85410	0.781481
$160$	0	0
$161$	8.23607	0.649093
$162$	0	0
$163$	3.14590	0.246406	0.123203	−	0.992382i	$-0.460683\pi$
$163$	3.14590	0.246406	0.123203	+	0.992382i	$0.460683\pi$
$164$	0	0
$165$	−17.9443	−1.39696
$166$	0	0
$167$	−6.23607	−0.482561	−0.241281	−	0.970455i	$-0.577567\pi$
$167$	−6.23607	−0.482561	−0.241281	+	0.970455i	$0.577567\pi$
$168$	0	0
$169$	−0.944272	−0.0726363
$170$	0	0
$171$	0.381966	0.0292097
$172$	0	0
$173$	−7.70820	−0.586044	−0.293022	−	0.956106i	$-0.594661\pi$
$173$	−7.70820	−0.586044	−0.293022	+	0.956106i	$0.594661\pi$
$174$	0	0
$175$	−12.9443	−0.978495
$176$	0	0
$177$	17.7984	1.33781
$178$	0	0
$179$	6.79837	0.508134	0.254067	−	0.967187i	$-0.418232\pi$
$179$	6.79837	0.508134	0.254067	+	0.967187i	$0.418232\pi$
$180$	0	0
$181$	−10.7984	−0.802637	−0.401318	−	0.915939i	$-0.631448\pi$
$181$	−10.7984	−0.802637	−0.401318	+	0.915939i	$0.631448\pi$
$182$	0	0
$183$	−6.85410	−0.506670
$184$	0	0
$185$	34.8885	2.56506
$186$	0	0
$187$	−15.3262	−1.12077
$188$	0	0
$189$	−5.47214	−0.398039
$190$	0	0
$191$	−16.0902	−1.16424	−0.582122	−	0.813102i	$-0.697777\pi$
$191$	−16.0902	−1.16424	−0.582122	+	0.813102i	$0.697777\pi$
$192$	0	0
$193$	−9.27051	−0.667306	−0.333653	−	0.942696i	$-0.608281\pi$
$193$	−9.27051	−0.667306	−0.333653	+	0.942696i	$0.608281\pi$
$194$	0	0
$195$	23.7984	1.70424
$196$	0	0
$197$	9.90983	0.706046	0.353023	−	0.935615i	$-0.385154\pi$
$197$	9.90983	0.706046	0.353023	+	0.935615i	$0.385154\pi$
$198$	0	0
$199$	−7.00000	−0.496217	−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000	−0.496217	−0.248108	+	0.968732i	$0.579809\pi$
$200$	0	0
$201$	−12.2361	−0.863066
$202$	0	0
$203$	−4.61803	−0.324122
$204$	0	0
$205$	48.9787	3.42082
$206$	0	0
$207$	3.14590	0.218655
$208$	0	0
$209$	−2.61803	−0.181093
$210$	0	0
$211$	0.381966	0.0262956	0.0131478	−	0.999914i	$-0.495815\pi$
$211$	0.381966	0.0262956	0.0131478	+	0.999914i	$0.495815\pi$
$212$	0	0
$213$	6.09017	0.417292
$214$	0	0
$215$	−18.9443	−1.29199
$216$	0	0
$217$	3.85410	0.261633
$218$	0	0
$219$	−5.85410	−0.395584
$220$	0	0
$221$	20.3262	1.36729
$222$	0	0
$223$	−21.7639	−1.45742	−0.728710	−	0.684822i	$-0.759880\pi$
$223$	−21.7639	−1.45742	−0.728710	+	0.684822i	$0.759880\pi$
$224$	0	0
$225$	−4.94427	−0.329618
$226$	0	0
$227$	−8.14590	−0.540662	−0.270331	−	0.962767i	$-0.587133\pi$
$227$	−8.14590	−0.540662	−0.270331	+	0.962767i	$0.587133\pi$
$228$	0	0
$229$	−24.4721	−1.61716	−0.808582	−	0.588383i	$-0.799765\pi$
$229$	−24.4721	−1.61716	−0.808582	+	0.588383i	$0.799765\pi$
$230$	0	0
$231$	4.23607	0.278713
$232$	0	0
$233$	−27.0902	−1.77474	−0.887368	−	0.461062i	$-0.847469\pi$
$233$	−27.0902	−1.77474	−0.887368	+	0.461062i	$0.847469\pi$
$234$	0	0
$235$	31.6525	2.06478
$236$	0	0
$237$	7.23607	0.470033
$238$	0	0
$239$	15.6525	1.01247	0.506237	−	0.862394i	$-0.331036\pi$
$239$	15.6525	1.01247	0.506237	+	0.862394i	$0.331036\pi$
$240$	0	0
$241$	−11.2361	−0.723779	−0.361889	−	0.932221i	$-0.617868\pi$
$241$	−11.2361	−0.723779	−0.361889	+	0.932221i	$0.617868\pi$
$242$	0	0
$243$	−3.94427	−0.253025
$244$	0	0
$245$	4.23607	0.270632
$246$	0	0
$247$	3.47214	0.220927
$248$	0	0
$249$	17.0902	1.08305
$250$	0	0
$251$	7.85410	0.495747	0.247873	−	0.968792i	$-0.420268\pi$
$251$	7.85410	0.495747	0.247873	+	0.968792i	$0.420268\pi$
$252$	0	0
$253$	−21.5623	−1.35561
$254$	0	0
$255$	40.1246	2.51270
$256$	0	0
$257$	−9.90983	−0.618158	−0.309079	−	0.951036i	$-0.600021\pi$
$257$	−9.90983	−0.618158	−0.309079	+	0.951036i	$0.600021\pi$
$258$	0	0
$259$	−8.23607	−0.511764
$260$	0	0
$261$	−1.76393	−0.109185
$262$	0	0
$263$	−5.43769	−0.335303	−0.167651	−	0.985846i	$-0.553618\pi$
$263$	−5.43769	−0.335303	−0.167651	+	0.985846i	$0.553618\pi$
$264$	0	0
$265$	−25.7984	−1.58478
$266$	0	0
$267$	25.4164	1.55546
$268$	0	0
$269$	7.32624	0.446689	0.223344	−	0.974740i	$-0.428303\pi$
$269$	7.32624	0.446689	0.223344	+	0.974740i	$0.428303\pi$
$270$	0	0
$271$	21.2705	1.29209	0.646046	−	0.763299i	$-0.276422\pi$
$271$	21.2705	1.29209	0.646046	+	0.763299i	$0.276422\pi$
$272$	0	0
$273$	−5.61803	−0.340019
$274$	0	0
$275$	33.8885	2.04356
$276$	0	0
$277$	8.18034	0.491509	0.245754	−	0.969332i	$-0.420964\pi$
$277$	8.18034	0.491509	0.245754	+	0.969332i	$0.420964\pi$
$278$	0	0
$279$	1.47214	0.0881345
$280$	0	0
$281$	−16.4164	−0.979321	−0.489660	−	0.871913i	$-0.662879\pi$
$281$	−16.4164	−0.979321	−0.489660	+	0.871913i	$0.662879\pi$
$282$	0	0
$283$	8.09017	0.480911	0.240455	−	0.970660i	$-0.422703\pi$
$283$	8.09017	0.480911	0.240455	+	0.970660i	$0.422703\pi$
$284$	0	0
$285$	6.85410	0.406002
$286$	0	0
$287$	−11.5623	−0.682501
$288$	0	0
$289$	17.2705	1.01591
$290$	0	0
$291$	5.61803	0.329335
$292$	0	0
$293$	−0.236068	−0.0137912	−0.00689562	−	0.999976i	$-0.502195\pi$
$293$	−0.236068	−0.0137912	−0.00689562	+	0.999976i	$0.502195\pi$
$294$	0	0
$295$	−46.5967	−2.71297
$296$	0	0
$297$	14.3262	0.831293
$298$	0	0
$299$	28.5967	1.65379
$300$	0	0
$301$	4.47214	0.257770
$302$	0	0
$303$	−23.3262	−1.34006
$304$	0	0
$305$	17.9443	1.02749
$306$	0	0
$307$	−19.8541	−1.13313	−0.566567	−	0.824016i	$-0.691729\pi$
$307$	−19.8541	−1.13313	−0.566567	+	0.824016i	$0.691729\pi$
$308$	0	0
$309$	−10.5623	−0.600869
$310$	0	0
$311$	13.7426	0.779274	0.389637	−	0.920969i	$-0.372600\pi$
$311$	13.7426	0.779274	0.389637	+	0.920969i	$0.372600\pi$
$312$	0	0
$313$	−7.23607	−0.409007	−0.204503	−	0.978866i	$-0.565558\pi$
$313$	−7.23607	−0.409007	−0.204503	+	0.978866i	$0.565558\pi$
$314$	0	0
$315$	1.61803	0.0911659
$316$	0	0
$317$	11.5279	0.647469	0.323735	−	0.946148i	$-0.395061\pi$
$317$	11.5279	0.647469	0.323735	+	0.946148i	$0.395061\pi$
$318$	0	0
$319$	12.0902	0.676920
$320$	0	0
$321$	24.5623	1.37093
$322$	0	0
$323$	5.85410	0.325731
$324$	0	0
$325$	−44.9443	−2.49306
$326$	0	0
$327$	0.854102	0.0472319
$328$	0	0
$329$	−7.47214	−0.411952
$330$	0	0
$331$	11.0344	0.606508	0.303254	−	0.952910i	$-0.401927\pi$
$331$	11.0344	0.606508	0.303254	+	0.952910i	$0.401927\pi$
$332$	0	0
$333$	−3.14590	−0.172394
$334$	0	0
$335$	32.0344	1.75023
$336$	0	0
$337$	−1.38197	−0.0752805	−0.0376402	−	0.999291i	$-0.511984\pi$
$337$	−1.38197	−0.0752805	−0.0376402	+	0.999291i	$0.511984\pi$
$338$	0	0
$339$	26.7984	1.45549
$340$	0	0
$341$	−10.0902	−0.546413
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	56.4508	3.03921
$346$	0	0
$347$	−19.9787	−1.07251	−0.536257	−	0.844055i	$-0.680162\pi$
$347$	−19.9787	−1.07251	−0.536257	+	0.844055i	$0.680162\pi$
$348$	0	0
$349$	1.43769	0.0769580	0.0384790	−	0.999259i	$-0.487749\pi$
$349$	1.43769	0.0769580	0.0384790	+	0.999259i	$0.487749\pi$
$350$	0	0
$351$	−19.0000	−1.01414
$352$	0	0
$353$	10.9787	0.584338	0.292169	−	0.956367i	$-0.405623\pi$
$353$	10.9787	0.584338	0.292169	+	0.956367i	$0.405623\pi$
$354$	0	0
$355$	−15.9443	−0.846234
$356$	0	0
$357$	−9.47214	−0.501319
$358$	0	0
$359$	27.2705	1.43928	0.719641	−	0.694346i	$-0.244306\pi$
$359$	27.2705	1.43928	0.719641	+	0.694346i	$0.244306\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	6.70820	0.352089
$364$	0	0
$365$	15.3262	0.802212
$366$	0	0
$367$	−22.7082	−1.18536	−0.592679	−	0.805439i	$-0.701930\pi$
$367$	−22.7082	−1.18536	−0.592679	+	0.805439i	$0.701930\pi$
$368$	0	0
$369$	−4.41641	−0.229909
$370$	0	0
$371$	6.09017	0.316186
$372$	0	0
$373$	−23.7984	−1.23223	−0.616117	−	0.787655i	$-0.711295\pi$
$373$	−23.7984	−1.23223	−0.616117	+	0.787655i	$0.711295\pi$
$374$	0	0
$375$	−54.4508	−2.81183
$376$	0	0
$377$	−16.0344	−0.825816
$378$	0	0
$379$	13.0000	0.667765	0.333883	−	0.942615i	$-0.391641\pi$
$379$	13.0000	0.667765	0.333883	+	0.942615i	$0.391641\pi$
$380$	0	0
$381$	25.4164	1.30212
$382$	0	0
$383$	14.4164	0.736644	0.368322	−	0.929698i	$-0.379932\pi$
$383$	14.4164	0.736644	0.368322	+	0.929698i	$0.379932\pi$
$384$	0	0
$385$	−11.0902	−0.565207
$386$	0	0
$387$	1.70820	0.0868329
$388$	0	0
$389$	−2.43769	−0.123596	−0.0617980	−	0.998089i	$-0.519683\pi$
$389$	−2.43769	−0.123596	−0.0617980	+	0.998089i	$0.519683\pi$
$390$	0	0
$391$	48.2148	2.43833
$392$	0	0
$393$	7.94427	0.400736
$394$	0	0
$395$	−18.9443	−0.953190
$396$	0	0
$397$	3.70820	0.186109	0.0930547	−	0.995661i	$-0.470337\pi$
$397$	3.70820	0.186109	0.0930547	+	0.995661i	$0.470337\pi$
$398$	0	0
$399$	−1.61803	−0.0810030
$400$	0	0
$401$	2.72949	0.136304	0.0681521	−	0.997675i	$-0.478290\pi$
$401$	2.72949	0.136304	0.0681521	+	0.997675i	$0.478290\pi$
$402$	0	0
$403$	13.3820	0.666603
$404$	0	0
$405$	−32.6525	−1.62251
$406$	0	0
$407$	21.5623	1.06880
$408$	0	0
$409$	14.8541	0.734488	0.367244	−	0.930125i	$-0.380301\pi$
$409$	14.8541	0.734488	0.367244	+	0.930125i	$0.380301\pi$
$410$	0	0
$411$	−8.76393	−0.432293
$412$	0	0
$413$	11.0000	0.541275
$414$	0	0
$415$	−44.7426	−2.19633
$416$	0	0
$417$	4.09017	0.200296
$418$	0	0
$419$	−20.5279	−1.00285	−0.501426	−	0.865201i	$-0.667191\pi$
$419$	−20.5279	−1.00285	−0.501426	+	0.865201i	$0.667191\pi$
$420$	0	0
$421$	10.1246	0.493443	0.246722	−	0.969086i	$-0.420647\pi$
$421$	10.1246	0.493443	0.246722	+	0.969086i	$0.420647\pi$
$422$	0	0
$423$	−2.85410	−0.138771
$424$	0	0
$425$	−75.7771	−3.67573
$426$	0	0
$427$	−4.23607	−0.204998
$428$	0	0
$429$	14.7082	0.710119
$430$	0	0
$431$	−10.0000	−0.481683	−0.240842	−	0.970564i	$-0.577423\pi$
$431$	−10.0000	−0.481683	−0.240842	+	0.970564i	$0.577423\pi$
$432$	0	0
$433$	−36.5967	−1.75873	−0.879364	−	0.476151i	$-0.842032\pi$
$433$	−36.5967	−1.75873	−0.879364	+	0.476151i	$0.842032\pi$
$434$	0	0
$435$	−31.6525	−1.51762
$436$	0	0
$437$	8.23607	0.393985
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	−0.381966	−0.0181889
$442$	0	0
$443$	−10.5623	−0.501830	−0.250915	−	0.968009i	$-0.580732\pi$
$443$	−10.5623	−0.501830	−0.250915	+	0.968009i	$0.580732\pi$
$444$	0	0
$445$	−66.5410	−3.15435
$446$	0	0
$447$	16.8541	0.797172
$448$	0	0
$449$	−6.27051	−0.295924	−0.147962	−	0.988993i	$-0.547271\pi$
$449$	−6.27051	−0.295924	−0.147962	+	0.988993i	$0.547271\pi$
$450$	0	0
$451$	30.2705	1.42538
$452$	0	0
$453$	−1.76393	−0.0828768
$454$	0	0
$455$	14.7082	0.689531
$456$	0	0
$457$	−27.7984	−1.30035	−0.650177	−	0.759783i	$-0.725305\pi$
$457$	−27.7984	−1.30035	−0.650177	+	0.759783i	$0.725305\pi$
$458$	0	0
$459$	−32.0344	−1.49524
$460$	0	0
$461$	−31.2148	−1.45382	−0.726909	−	0.686734i	$-0.759044\pi$
$461$	−31.2148	−1.45382	−0.726909	+	0.686734i	$0.759044\pi$
$462$	0	0
$463$	−4.05573	−0.188486	−0.0942428	−	0.995549i	$-0.530043\pi$
$463$	−4.05573	−0.188486	−0.0942428	+	0.995549i	$0.530043\pi$
$464$	0	0
$465$	26.4164	1.22503
$466$	0	0
$467$	40.4508	1.87184	0.935921	−	0.352210i	$-0.114570\pi$
$467$	40.4508	1.87184	0.935921	+	0.352210i	$0.114570\pi$
$468$	0	0
$469$	−7.56231	−0.349195
$470$	0	0
$471$	−18.7082	−0.862029
$472$	0	0
$473$	−11.7082	−0.538344
$474$	0	0
$475$	−12.9443	−0.593924
$476$	0	0
$477$	2.32624	0.106511
$478$	0	0
$479$	−12.3262	−0.563200	−0.281600	−	0.959532i	$-0.590865\pi$
$479$	−12.3262	−0.563200	−0.281600	+	0.959532i	$0.590865\pi$
$480$	0	0
$481$	−28.5967	−1.30390
$482$	0	0
$483$	−13.3262	−0.606365
$484$	0	0
$485$	−14.7082	−0.667865
$486$	0	0
$487$	33.1803	1.50354	0.751772	−	0.659423i	$-0.229199\pi$
$487$	33.1803	1.50354	0.751772	+	0.659423i	$0.229199\pi$
$488$	0	0
$489$	−5.09017	−0.230185
$490$	0	0
$491$	23.4164	1.05677	0.528384	−	0.849006i	$-0.322798\pi$
$491$	23.4164	1.05677	0.528384	+	0.849006i	$0.322798\pi$
$492$	0	0
$493$	−27.0344	−1.21757
$494$	0	0
$495$	−4.23607	−0.190397
$496$	0	0
$497$	3.76393	0.168835
$498$	0	0
$499$	−35.6180	−1.59448	−0.797241	−	0.603661i	$-0.793708\pi$
$499$	−35.6180	−1.59448	−0.797241	+	0.603661i	$0.793708\pi$
$500$	0	0
$501$	10.0902	0.450796
$502$	0	0
$503$	−12.9443	−0.577157	−0.288578	−	0.957456i	$-0.593183\pi$
$503$	−12.9443	−0.577157	−0.288578	+	0.957456i	$0.593183\pi$
$504$	0	0
$505$	61.0689	2.71753
$506$	0	0
$507$	1.52786	0.0678548
$508$	0	0
$509$	−32.8328	−1.45529	−0.727644	−	0.685954i	$-0.759385\pi$
$509$	−32.8328	−1.45529	−0.727644	+	0.685954i	$0.759385\pi$
$510$	0	0
$511$	−3.61803	−0.160052
$512$	0	0
$513$	−5.47214	−0.241601
$514$	0	0
$515$	27.6525	1.21851
$516$	0	0
$517$	19.5623	0.860349
$518$	0	0
$519$	12.4721	0.547466
$520$	0	0
$521$	39.1803	1.71652	0.858261	−	0.513214i	$-0.171545\pi$
$521$	39.1803	1.71652	0.858261	+	0.513214i	$0.171545\pi$
$522$	0	0
$523$	−34.5967	−1.51281	−0.756405	−	0.654103i	$-0.773046\pi$
$523$	−34.5967	−1.51281	−0.756405	+	0.654103i	$0.773046\pi$
$524$	0	0
$525$	20.9443	0.914083
$526$	0	0
$527$	22.5623	0.982829
$528$	0	0
$529$	44.8328	1.94925
$530$	0	0
$531$	4.20163	0.182335
$532$	0	0
$533$	−40.1459	−1.73891
$534$	0	0
$535$	−64.3050	−2.78015
$536$	0	0
$537$	−11.0000	−0.474685
$538$	0	0
$539$	2.61803	0.112767
$540$	0	0
$541$	26.3607	1.13333	0.566667	−	0.823947i	$-0.308233\pi$
$541$	26.3607	1.13333	0.566667	+	0.823947i	$0.308233\pi$
$542$	0	0
$543$	17.4721	0.749801
$544$	0	0
$545$	−2.23607	−0.0957826
$546$	0	0
$547$	−35.3951	−1.51339	−0.756693	−	0.653770i	$-0.773186\pi$
$547$	−35.3951	−1.51339	−0.756693	+	0.653770i	$0.773186\pi$
$548$	0	0
$549$	−1.61803	−0.0690560
$550$	0	0
$551$	−4.61803	−0.196735
$552$	0	0
$553$	4.47214	0.190175
$554$	0	0
$555$	−56.4508	−2.39621
$556$	0	0
$557$	18.3262	0.776508	0.388254	−	0.921552i	$-0.373078\pi$
$557$	18.3262	0.776508	0.388254	+	0.921552i	$0.373078\pi$
$558$	0	0
$559$	15.5279	0.656759
$560$	0	0
$561$	24.7984	1.04699
$562$	0	0
$563$	37.8328	1.59446	0.797232	−	0.603674i	$-0.206297\pi$
$563$	37.8328	1.59446	0.797232	+	0.603674i	$0.206297\pi$
$564$	0	0
$565$	−70.1591	−2.95161
$566$	0	0
$567$	7.70820	0.323714
$568$	0	0
$569$	−38.0132	−1.59359	−0.796797	−	0.604247i	$-0.793474\pi$
$569$	−38.0132	−1.59359	−0.796797	+	0.604247i	$0.793474\pi$
$570$	0	0
$571$	−0.527864	−0.0220904	−0.0110452	−	0.999939i	$-0.503516\pi$
$571$	−0.527864	−0.0220904	−0.0110452	+	0.999939i	$0.503516\pi$
$572$	0	0
$573$	26.0344	1.08760
$574$	0	0
$575$	−106.610	−4.44594
$576$	0	0
$577$	−23.6869	−0.986099	−0.493050	−	0.870001i	$-0.664118\pi$
$577$	−23.6869	−0.986099	−0.493050	+	0.870001i	$0.664118\pi$
$578$	0	0
$579$	15.0000	0.623379
$580$	0	0
$581$	10.5623	0.438198
$582$	0	0
$583$	−15.9443	−0.660344
$584$	0	0
$585$	5.61803	0.232277
$586$	0	0
$587$	−10.5836	−0.436832	−0.218416	−	0.975856i	$-0.570089\pi$
$587$	−10.5836	−0.436832	−0.218416	+	0.975856i	$0.570089\pi$
$588$	0	0
$589$	3.85410	0.158806
$590$	0	0
$591$	−16.0344	−0.659569
$592$	0	0
$593$	−26.5279	−1.08937	−0.544684	−	0.838641i	$-0.683351\pi$
$593$	−26.5279	−1.08937	−0.544684	+	0.838641i	$0.683351\pi$
$594$	0	0
$595$	24.7984	1.01663
$596$	0	0
$597$	11.3262	0.463552
$598$	0	0
$599$	12.5623	0.513282	0.256641	−	0.966507i	$-0.417384\pi$
$599$	12.5623	0.513282	0.256641	+	0.966507i	$0.417384\pi$
$600$	0	0
$601$	−3.14590	−0.128324	−0.0641619	−	0.997940i	$-0.520437\pi$
$601$	−3.14590	−0.128324	−0.0641619	+	0.997940i	$0.520437\pi$
$602$	0	0
$603$	−2.88854	−0.117631
$604$	0	0
$605$	−17.5623	−0.714009
$606$	0	0
$607$	3.00000	0.121766	0.0608831	−	0.998145i	$-0.480608\pi$
$607$	3.00000	0.121766	0.0608831	+	0.998145i	$0.480608\pi$
$608$	0	0
$609$	7.47214	0.302786
$610$	0	0
$611$	−25.9443	−1.04959
$612$	0	0
$613$	14.0902	0.569097	0.284548	−	0.958662i	$-0.408156\pi$
$613$	14.0902	0.569097	0.284548	+	0.958662i	$0.408156\pi$
$614$	0	0
$615$	−79.2492	−3.19564
$616$	0	0
$617$	16.4377	0.661757	0.330878	−	0.943673i	$-0.392655\pi$
$617$	16.4377	0.661757	0.330878	+	0.943673i	$0.392655\pi$
$618$	0	0
$619$	36.3820	1.46231	0.731157	−	0.682209i	$-0.238981\pi$
$619$	36.3820	1.46231	0.731157	+	0.682209i	$0.238981\pi$
$620$	0	0
$621$	−45.0689	−1.80855
$622$	0	0
$623$	15.7082	0.629336
$624$	0	0
$625$	77.8328	3.11331
$626$	0	0
$627$	4.23607	0.169172
$628$	0	0
$629$	−48.2148	−1.92245
$630$	0	0
$631$	12.8885	0.513085	0.256542	−	0.966533i	$-0.417417\pi$
$631$	12.8885	0.513085	0.256542	+	0.966533i	$0.417417\pi$
$632$	0	0
$633$	−0.618034	−0.0245646
$634$	0	0
$635$	−66.5410	−2.64060
$636$	0	0
$637$	−3.47214	−0.137571
$638$	0	0
$639$	1.43769	0.0568743
$640$	0	0
$641$	−23.6738	−0.935057	−0.467529	−	0.883978i	$-0.654856\pi$
$641$	−23.6738	−0.935057	−0.467529	+	0.883978i	$0.654856\pi$
$642$	0	0
$643$	−23.1803	−0.914143	−0.457072	−	0.889430i	$-0.651102\pi$
$643$	−23.1803	−0.914143	−0.457072	+	0.889430i	$0.651102\pi$
$644$	0	0
$645$	30.6525	1.20694
$646$	0	0
$647$	4.41641	0.173627	0.0868135	−	0.996225i	$-0.472332\pi$
$647$	4.41641	0.173627	0.0868135	+	0.996225i	$0.472332\pi$
$648$	0	0
$649$	−28.7984	−1.13044
$650$	0	0
$651$	−6.23607	−0.244411
$652$	0	0
$653$	−18.1246	−0.709271	−0.354635	−	0.935005i	$-0.615395\pi$
$653$	−18.1246	−0.709271	−0.354635	+	0.935005i	$0.615395\pi$
$654$	0	0
$655$	−20.7984	−0.812660
$656$	0	0
$657$	−1.38197	−0.0539156
$658$	0	0
$659$	−18.6180	−0.725256	−0.362628	−	0.931934i	$-0.618120\pi$
$659$	−18.6180	−0.725256	−0.362628	+	0.931934i	$0.618120\pi$
$660$	0	0
$661$	25.0557	0.974555	0.487277	−	0.873247i	$-0.337990\pi$
$661$	25.0557	0.974555	0.487277	+	0.873247i	$0.337990\pi$
$662$	0	0
$663$	−32.8885	−1.27729
$664$	0	0
$665$	4.23607	0.164268
$666$	0	0
$667$	−38.0344	−1.47270
$668$	0	0
$669$	35.2148	1.36148
$670$	0	0
$671$	11.0902	0.428131
$672$	0	0
$673$	−12.5623	−0.484241	−0.242121	−	0.970246i	$-0.577843\pi$
$673$	−12.5623	−0.484241	−0.242121	+	0.970246i	$0.577843\pi$
$674$	0	0
$675$	70.8328	2.72636
$676$	0	0
$677$	30.3820	1.16767	0.583837	−	0.811871i	$-0.301551\pi$
$677$	30.3820	1.16767	0.583837	+	0.811871i	$0.301551\pi$
$678$	0	0
$679$	3.47214	0.133248
$680$	0	0
$681$	13.1803	0.505072
$682$	0	0
$683$	10.2361	0.391672	0.195836	−	0.980637i	$-0.437258\pi$
$683$	10.2361	0.391672	0.195836	+	0.980637i	$0.437258\pi$
$684$	0	0
$685$	22.9443	0.876656
$686$	0	0
$687$	39.5967	1.51071
$688$	0	0
$689$	21.1459	0.805595
$690$	0	0
$691$	14.2918	0.543686	0.271843	−	0.962342i	$-0.412367\pi$
$691$	14.2918	0.543686	0.271843	+	0.962342i	$0.412367\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	−10.7082	−0.406185
$696$	0	0
$697$	−67.6869	−2.56382
$698$	0	0
$699$	43.8328	1.65791
$700$	0	0
$701$	−20.3607	−0.769012	−0.384506	−	0.923122i	$-0.625628\pi$
$701$	−20.3607	−0.769012	−0.384506	+	0.923122i	$0.625628\pi$
$702$	0	0
$703$	−8.23607	−0.310629
$704$	0	0
$705$	−51.2148	−1.92886
$706$	0	0
$707$	−14.4164	−0.542185
$708$	0	0
$709$	−35.2918	−1.32541	−0.662706	−	0.748880i	$-0.730592\pi$
$709$	−35.2918	−1.32541	−0.662706	+	0.748880i	$0.730592\pi$
$710$	0	0
$711$	1.70820	0.0640627
$712$	0	0
$713$	31.7426	1.18877
$714$	0	0
$715$	−38.5066	−1.44006
$716$	0	0
$717$	−25.3262	−0.945826
$718$	0	0
$719$	−16.9443	−0.631915	−0.315957	−	0.948773i	$-0.602326\pi$
$719$	−16.9443	−0.631915	−0.315957	+	0.948773i	$0.602326\pi$
$720$	0	0
$721$	−6.52786	−0.243110
$722$	0	0
$723$	18.1803	0.676134
$724$	0	0
$725$	59.7771	2.22007
$726$	0	0
$727$	37.0689	1.37481	0.687404	−	0.726275i	$-0.258750\pi$
$727$	37.0689	1.37481	0.687404	+	0.726275i	$0.258750\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	26.1803	0.968315
$732$	0	0
$733$	32.0000	1.18195	0.590973	−	0.806691i	$-0.298744\pi$
$733$	32.0000	1.18195	0.590973	+	0.806691i	$0.298744\pi$
$734$	0	0
$735$	−6.85410	−0.252817
$736$	0	0
$737$	19.7984	0.729282
$738$	0	0
$739$	−3.29180	−0.121091	−0.0605453	−	0.998165i	$-0.519284\pi$
$739$	−3.29180	−0.121091	−0.0605453	+	0.998165i	$0.519284\pi$
$740$	0	0
$741$	−5.61803	−0.206384
$742$	0	0
$743$	41.5410	1.52399	0.761996	−	0.647582i	$-0.224220\pi$
$743$	41.5410	1.52399	0.761996	+	0.647582i	$0.224220\pi$
$744$	0	0
$745$	−44.1246	−1.61660
$746$	0	0
$747$	4.03444	0.147613
$748$	0	0
$749$	15.1803	0.554678
$750$	0	0
$751$	10.0902	0.368196	0.184098	−	0.982908i	$-0.441064\pi$
$751$	10.0902	0.368196	0.184098	+	0.982908i	$0.441064\pi$
$752$	0	0
$753$	−12.7082	−0.463113
$754$	0	0
$755$	4.61803	0.168067
$756$	0	0
$757$	−7.52786	−0.273605	−0.136802	−	0.990598i	$-0.543683\pi$
$757$	−7.52786	−0.273605	−0.136802	+	0.990598i	$0.543683\pi$
$758$	0	0
$759$	34.8885	1.26637
$760$	0	0
$761$	46.5967	1.68913	0.844565	−	0.535452i	$-0.179859\pi$
$761$	46.5967	1.68913	0.844565	+	0.535452i	$0.179859\pi$
$762$	0	0
$763$	0.527864	0.0191100
$764$	0	0
$765$	9.47214	0.342466
$766$	0	0
$767$	38.1935	1.37909
$768$	0	0
$769$	24.9443	0.899513	0.449757	−	0.893151i	$-0.351511\pi$
$769$	24.9443	0.899513	0.449757	+	0.893151i	$0.351511\pi$
$770$	0	0
$771$	16.0344	0.577466
$772$	0	0
$773$	29.7082	1.06853	0.534265	−	0.845317i	$-0.320588\pi$
$773$	29.7082	1.06853	0.534265	+	0.845317i	$0.320588\pi$
$774$	0	0
$775$	−49.8885	−1.79205
$776$	0	0
$777$	13.3262	0.478076
$778$	0	0
$779$	−11.5623	−0.414263
$780$	0	0
$781$	−9.85410	−0.352607
$782$	0	0
$783$	25.2705	0.903094
$784$	0	0
$785$	48.9787	1.74813
$786$	0	0
$787$	−31.4164	−1.11987	−0.559937	−	0.828535i	$-0.689175\pi$
$787$	−31.4164	−1.11987	−0.559937	+	0.828535i	$0.689175\pi$
$788$	0	0
$789$	8.79837	0.313230
$790$	0	0
$791$	16.5623	0.588888
$792$	0	0
$793$	−14.7082	−0.522304
$794$	0	0
$795$	41.7426	1.48046
$796$	0	0
$797$	22.9787	0.813948	0.406974	−	0.913440i	$-0.366584\pi$
$797$	22.9787	0.813948	0.406974	+	0.913440i	$0.366584\pi$
$798$	0	0
$799$	−43.7426	−1.54750
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	9.47214	0.334264
$804$	0	0
$805$	34.8885	1.22966
$806$	0	0
$807$	−11.8541	−0.417284
$808$	0	0
$809$	11.0689	0.389161	0.194581	−	0.980887i	$-0.437665\pi$
$809$	11.0689	0.389161	0.194581	+	0.980887i	$0.437665\pi$
$810$	0	0
$811$	48.8328	1.71475	0.857376	−	0.514691i	$-0.172093\pi$
$811$	48.8328	1.71475	0.857376	+	0.514691i	$0.172093\pi$
$812$	0	0
$813$	−34.4164	−1.20704
$814$	0	0
$815$	13.3262	0.466798
$816$	0	0
$817$	4.47214	0.156460
$818$	0	0
$819$	−1.32624	−0.0463425
$820$	0	0
$821$	−44.3607	−1.54820	−0.774099	−	0.633064i	$-0.781797\pi$
$821$	−44.3607	−1.54820	−0.774099	+	0.633064i	$0.781797\pi$
$822$	0	0
$823$	−8.76393	−0.305491	−0.152746	−	0.988266i	$-0.548812\pi$
$823$	−8.76393	−0.305491	−0.152746	+	0.988266i	$0.548812\pi$
$824$	0	0
$825$	−54.8328	−1.90903
$826$	0	0
$827$	−7.29180	−0.253561	−0.126780	−	0.991931i	$-0.540464\pi$
$827$	−7.29180	−0.253561	−0.126780	+	0.991931i	$0.540464\pi$
$828$	0	0
$829$	6.11146	0.212260	0.106130	−	0.994352i	$-0.466154\pi$
$829$	6.11146	0.212260	0.106130	+	0.994352i	$0.466154\pi$
$830$	0	0
$831$	−13.2361	−0.459154
$832$	0	0
$833$	−5.85410	−0.202833
$834$	0	0
$835$	−26.4164	−0.914177
$836$	0	0
$837$	−21.0902	−0.728983
$838$	0	0
$839$	34.4721	1.19011	0.595055	−	0.803685i	$-0.297130\pi$
$839$	34.4721	1.19011	0.595055	+	0.803685i	$0.297130\pi$
$840$	0	0
$841$	−7.67376	−0.264612
$842$	0	0
$843$	26.5623	0.914854
$844$	0	0
$845$	−4.00000	−0.137604
$846$	0	0
$847$	4.14590	0.142455
$848$	0	0
$849$	−13.0902	−0.449253
$850$	0	0
$851$	−67.8328	−2.32528
$852$	0	0
$853$	31.6180	1.08258	0.541290	−	0.840836i	$-0.317936\pi$
$853$	31.6180	1.08258	0.541290	+	0.840836i	$0.317936\pi$
$854$	0	0
$855$	1.61803	0.0553356
$856$	0	0
$857$	−49.9230	−1.70534	−0.852668	−	0.522453i	$-0.825017\pi$
$857$	−49.9230	−1.70534	−0.852668	+	0.522453i	$0.825017\pi$
$858$	0	0
$859$	19.4508	0.663654	0.331827	−	0.943340i	$-0.392335\pi$
$859$	19.4508	0.663654	0.331827	+	0.943340i	$0.392335\pi$
$860$	0	0
$861$	18.7082	0.637574
$862$	0	0
$863$	−28.1591	−0.958545	−0.479273	−	0.877666i	$-0.659099\pi$
$863$	−28.1591	−0.958545	−0.479273	+	0.877666i	$0.659099\pi$
$864$	0	0
$865$	−32.6525	−1.11022
$866$	0	0
$867$	−27.9443	−0.949037
$868$	0	0
$869$	−11.7082	−0.397174
$870$	0	0
$871$	−26.2574	−0.889697
$872$	0	0
$873$	1.32624	0.0448864
$874$	0	0
$875$	−33.6525	−1.13766
$876$	0	0
$877$	−54.5410	−1.84172	−0.920860	−	0.389894i	$-0.872512\pi$
$877$	−54.5410	−1.84172	−0.920860	+	0.389894i	$0.872512\pi$
$878$	0	0
$879$	0.381966	0.0128834
$880$	0	0
$881$	−5.21478	−0.175690	−0.0878452	−	0.996134i	$-0.527998\pi$
$881$	−5.21478	−0.175690	−0.0878452	+	0.996134i	$0.527998\pi$
$882$	0	0
$883$	45.0689	1.51669	0.758344	−	0.651854i	$-0.226009\pi$
$883$	45.0689	1.51669	0.758344	+	0.651854i	$0.226009\pi$
$884$	0	0
$885$	75.3951	2.53438
$886$	0	0
$887$	−16.1246	−0.541411	−0.270706	−	0.962662i	$-0.587257\pi$
$887$	−16.1246	−0.541411	−0.270706	+	0.962662i	$0.587257\pi$
$888$	0	0
$889$	15.7082	0.526836
$890$	0	0
$891$	−20.1803	−0.676067
$892$	0	0
$893$	−7.47214	−0.250045
$894$	0	0
$895$	28.7984	0.962623
$896$	0	0
$897$	−46.2705	−1.54493
$898$	0	0
$899$	−17.7984	−0.593609
$900$	0	0
$901$	35.6525	1.18776
$902$	0	0
$903$	−7.23607	−0.240801
$904$	0	0
$905$	−45.7426	−1.52054
$906$	0	0
$907$	−46.1935	−1.53383	−0.766915	−	0.641749i	$-0.778209\pi$
$907$	−46.1935	−1.53383	−0.766915	+	0.641749i	$0.778209\pi$
$908$	0	0
$909$	−5.50658	−0.182642
$910$	0	0
$911$	−6.58359	−0.218124	−0.109062	−	0.994035i	$-0.534785\pi$
$911$	−6.58359	−0.218124	−0.109062	+	0.994035i	$0.534785\pi$
$912$	0	0
$913$	−27.6525	−0.915163
$914$	0	0
$915$	−29.0344	−0.959849
$916$	0	0
$917$	4.90983	0.162137
$918$	0	0
$919$	2.63932	0.0870631	0.0435316	−	0.999052i	$-0.486139\pi$
$919$	2.63932	0.0870631	0.0435316	+	0.999052i	$0.486139\pi$
$920$	0	0
$921$	32.1246	1.05854
$922$	0	0
$923$	13.0689	0.430168
$924$	0	0
$925$	106.610	3.50531
$926$	0	0
$927$	−2.49342	−0.0818947
$928$	0	0
$929$	16.4508	0.539735	0.269867	−	0.962897i	$-0.413020\pi$
$929$	16.4508	0.539735	0.269867	+	0.962897i	$0.413020\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	−22.2361	−0.727976
$934$	0	0
$935$	−64.9230	−2.12321
$936$	0	0
$937$	−38.2148	−1.24842	−0.624211	−	0.781256i	$-0.714580\pi$
$937$	−38.2148	−1.24842	−0.624211	+	0.781256i	$0.714580\pi$
$938$	0	0
$939$	11.7082	0.382083
$940$	0	0
$941$	24.4721	0.797769	0.398884	−	0.917001i	$-0.369397\pi$
$941$	24.4721	0.797769	0.398884	+	0.917001i	$0.369397\pi$
$942$	0	0
$943$	−95.2279	−3.10105
$944$	0	0
$945$	−23.1803	−0.754057
$946$	0	0
$947$	26.4508	0.859537	0.429769	−	0.902939i	$-0.358595\pi$
$947$	26.4508	0.859537	0.429769	+	0.902939i	$0.358595\pi$
$948$	0	0
$949$	−12.5623	−0.407790
$950$	0	0
$951$	−18.6525	−0.604848
$952$	0	0
$953$	−31.4508	−1.01879	−0.509396	−	0.860532i	$-0.670131\pi$
$953$	−31.4508	−1.01879	−0.509396	+	0.860532i	$0.670131\pi$
$954$	0	0
$955$	−68.1591	−2.20558
$956$	0	0
$957$	−19.5623	−0.632360
$958$	0	0
$959$	−5.41641	−0.174905
$960$	0	0
$961$	−16.1459	−0.520835
$962$	0	0
$963$	5.79837	0.186850
$964$	0	0
$965$	−39.2705	−1.26416
$966$	0	0
$967$	−24.6869	−0.793878	−0.396939	−	0.917845i	$-0.629928\pi$
$967$	−24.6869	−0.793878	−0.396939	+	0.917845i	$0.629928\pi$
$968$	0	0
$969$	−9.47214	−0.304289
$970$	0	0
$971$	−50.8885	−1.63309	−0.816546	−	0.577281i	$-0.804114\pi$
$971$	−50.8885	−1.63309	−0.816546	+	0.577281i	$0.804114\pi$
$972$	0	0
$973$	2.52786	0.0810396
$974$	0	0
$975$	72.7214	2.32895
$976$	0	0
$977$	−40.7639	−1.30415	−0.652077	−	0.758153i	$-0.726102\pi$
$977$	−40.7639	−1.30415	−0.652077	+	0.758153i	$0.726102\pi$
$978$	0	0
$979$	−41.1246	−1.31435
$980$	0	0
$981$	0.201626	0.00643743
$982$	0	0
$983$	33.5836	1.07115	0.535575	−	0.844488i	$-0.320095\pi$
$983$	33.5836	1.07115	0.535575	+	0.844488i	$0.320095\pi$
$984$	0	0
$985$	41.9787	1.33755
$986$	0	0
$987$	12.0902	0.384834
$988$	0	0
$989$	36.8328	1.17122
$990$	0	0
$991$	−0.763932	−0.0242671	−0.0121336	−	0.999926i	$-0.503862\pi$
$991$	−0.763932	−0.0242671	−0.0121336	+	0.999926i	$0.503862\pi$
$992$	0	0
$993$	−17.8541	−0.566583
$994$	0	0
$995$	−29.6525	−0.940047
$996$	0	0
$997$	−22.6312	−0.716737	−0.358368	−	0.933580i	$-0.616667\pi$
$997$	−22.6312	−0.716737	−0.358368	+	0.933580i	$0.616667\pi$
$998$	0	0
$999$	45.0689	1.42592

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.s.1.1		2
4.3	odd	2		8512.2.a.z.1.2		2
8.3	odd	2		2128.2.a.e.1.1		2
8.5	even	2		532.2.a.d.1.2	✓	2
24.5	odd	2		4788.2.a.n.1.2		2
56.13	odd	2		3724.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
532.2.a.d.1.2	✓	2	8.5	even	2
2128.2.a.e.1.1		2	8.3	odd	2
3724.2.a.d.1.1		2	56.13	odd	2
4788.2.a.n.1.2		2	24.5	odd	2
8512.2.a.s.1.1		2	1.1	even	1	trivial
8512.2.a.z.1.2		2	4.3	odd	2

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:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{3} +4.23607 q^{5} -1.00000 q^{7} -0.381966 q^{9} +2.61803 q^{11} -3.47214 q^{13} -6.85410 q^{15} -5.85410 q^{17} -1.00000 q^{19} +1.61803 q^{21} -8.23607 q^{23} +12.9443 q^{25} +5.47214 q^{27} +4.61803 q^{29} -3.85410 q^{31} -4.23607 q^{33} -4.23607 q^{35} +8.23607 q^{37} +5.61803 q^{39} +11.5623 q^{41} -4.47214 q^{43} -1.61803 q^{45} +7.47214 q^{47} +1.00000 q^{49} +9.47214 q^{51} -6.09017 q^{53} +11.0902 q^{55} +1.61803 q^{57} -11.0000 q^{59} +4.23607 q^{61} +0.381966 q^{63} -14.7082 q^{65} +7.56231 q^{67} +13.3262 q^{69} -3.76393 q^{71} +3.61803 q^{73} -20.9443 q^{75} -2.61803 q^{77} -4.47214 q^{79} -7.70820 q^{81} -10.5623 q^{83} -24.7984 q^{85} -7.47214 q^{87} -15.7082 q^{89} +3.47214 q^{91} +6.23607 q^{93} -4.23607 q^{95} -3.47214 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 4 q^{5} - 2 q^{7} - 3 q^{9} + 3 q^{11} + 2 q^{13} - 7 q^{15} - 5 q^{17} - 2 q^{19} + q^{21} - 12 q^{23} + 8 q^{25} + 2 q^{27} + 7 q^{29} - q^{31} - 4 q^{33} - 4 q^{35} + 12 q^{37} + 9 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 2 * q - q^3 + 4 * q^5 - 2 * q^7 - 3 * q^9 + 3 * q^11 + 2 * q^13 - 7 * q^15 - 5 * q^17 - 2 * q^19 + q^21 - 12 * q^23 + 8 * q^25 + 2 * q^27 + 7 * q^29 - q^31 - 4 * q^33 - 4 * q^35 + 12 * q^37 + 9 * q^39 + 3 * q^41 - q^45 + 6 * q^47 + 2 * q^49 + 10 * q^51 - q^53 + 11 * q^55 + q^57 - 22 * q^59 + 4 * q^61 + 3 * q^63 - 16 * q^65 - 5 * q^67 + 11 * q^69 - 12 * q^71 + 5 * q^73 - 24 * q^75 - 3 * q^77 - 2 * q^81 - q^83 - 25 * q^85 - 6 * q^87 - 18 * q^89 - 2 * q^91 + 8 * q^93 - 4 * q^95 + 2 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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