LMFDB - Embedded newform 8512.2.a.cj.1.7

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:	$7$
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} - 13x^{5} - 9x^{4} + 37x^{3} + 41x^{2} - 5 $ x^7 - 13x^5 - 9x^4 + 37x^3 + 41x^2 - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4256)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.37843$ 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)

$f(q)$	$=$	$q+3.37843 q^{3} -4.36754 q^{5} -1.00000 q^{7} +8.41379 q^{9} +O(q^{10})$	$q+3.37843 q^{3} -4.36754 q^{5} -1.00000 q^{7} +8.41379 q^{9} -1.86688 q^{11} +3.09599 q^{13} -14.7554 q^{15} -5.38539 q^{17} +1.00000 q^{19} -3.37843 q^{21} +0.672482 q^{23} +14.0754 q^{25} +18.2901 q^{27} -7.68437 q^{29} +6.26545 q^{31} -6.30711 q^{33} +4.36754 q^{35} +2.63811 q^{37} +10.4596 q^{39} -9.32726 q^{41} -11.1282 q^{43} -36.7476 q^{45} -3.60289 q^{47} +1.00000 q^{49} -18.1941 q^{51} -5.04940 q^{53} +8.15365 q^{55} +3.37843 q^{57} -5.12440 q^{59} -4.42684 q^{61} -8.41379 q^{63} -13.5219 q^{65} +2.37810 q^{67} +2.27193 q^{69} +6.67728 q^{71} +1.69478 q^{73} +47.5527 q^{75} +1.86688 q^{77} -17.3813 q^{79} +36.5505 q^{81} -0.104382 q^{83} +23.5209 q^{85} -25.9611 q^{87} +13.3438 q^{89} -3.09599 q^{91} +21.1674 q^{93} -4.36754 q^{95} +6.00099 q^{97} -15.7075 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} - 5 q^{5} - 7 q^{7} + 12 q^{9}+O(q^{10}) $ 7 * q + 7 * q^3 - 5 * q^5 - 7 * q^7 + 12 * q^9	$ 7 q + 7 q^{3} - 5 q^{5} - 7 q^{7} + 12 q^{9} - 3 q^{11} - 12 q^{13} - 6 q^{15} - 16 q^{17} + 7 q^{19} - 7 q^{21} - 10 q^{23} + 28 q^{25} + 16 q^{27} - 7 q^{29} - 2 q^{31} + 5 q^{33} + 5 q^{35} - 7 q^{37} + 3 q^{41} - 15 q^{43} - 32 q^{45} + 5 q^{47} + 7 q^{49} - 13 q^{53} + 14 q^{55} + 7 q^{57} + 23 q^{59} - 33 q^{61} - 12 q^{63} - 4 q^{65} - 10 q^{67} - 10 q^{69} - 13 q^{71} + 18 q^{73} + 28 q^{75} + 3 q^{77} - 21 q^{79} + 11 q^{81} + 2 q^{83} - 14 q^{85} - 64 q^{87} - 17 q^{89} + 12 q^{91} - 26 q^{93} - 5 q^{95} + q^{97} - 55 q^{99}+O(q^{100}) $ 7 * q + 7 * q^3 - 5 * q^5 - 7 * q^7 + 12 * q^9 - 3 * q^11 - 12 * q^13 - 6 * q^15 - 16 * q^17 + 7 * q^19 - 7 * q^21 - 10 * q^23 + 28 * q^25 + 16 * q^27 - 7 * q^29 - 2 * q^31 + 5 * q^33 + 5 * q^35 - 7 * q^37 + 3 * q^41 - 15 * q^43 - 32 * q^45 + 5 * q^47 + 7 * q^49 - 13 * q^53 + 14 * q^55 + 7 * q^57 + 23 * q^59 - 33 * q^61 - 12 * q^63 - 4 * q^65 - 10 * q^67 - 10 * q^69 - 13 * q^71 + 18 * q^73 + 28 * q^75 + 3 * q^77 - 21 * q^79 + 11 * q^81 + 2 * q^83 - 14 * q^85 - 64 * q^87 - 17 * q^89 + 12 * q^91 - 26 * q^93 - 5 * q^95 + q^97 - 55 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.37843	1.95054	0.975269	−	0.221022i	$-0.0709392\pi$
$3$	3.37843	1.95054	0.975269	+	0.221022i	$0.0709392\pi$
$4$	0	0
$5$	−4.36754	−1.95322	−0.976611	−	0.215014i	$-0.931020\pi$
$5$	−4.36754	−1.95322	−0.976611	+	0.215014i	$0.931020\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	8.41379	2.80460
$10$	0	0
$11$	−1.86688	−0.562885	−0.281442	−	0.959578i	$-0.590813\pi$
$11$	−1.86688	−0.562885	−0.281442	+	0.959578i	$0.590813\pi$
$12$	0	0
$13$	3.09599	0.858673	0.429337	−	0.903145i	$-0.358747\pi$
$13$	3.09599	0.858673	0.429337	+	0.903145i	$0.358747\pi$
$14$	0	0
$15$	−14.7554	−3.80983
$16$	0	0
$17$	−5.38539	−1.30615	−0.653074	−	0.757294i	$-0.726521\pi$
$17$	−5.38539	−1.30615	−0.653074	+	0.757294i	$0.726521\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−3.37843	−0.737234
$22$	0	0
$23$	0.672482	0.140222	0.0701111	−	0.997539i	$-0.477665\pi$
$23$	0.672482	0.140222	0.0701111	+	0.997539i	$0.477665\pi$
$24$	0	0
$25$	14.0754	2.81508
$26$	0	0
$27$	18.2901	3.51994
$28$	0	0
$29$	−7.68437	−1.42695	−0.713475	−	0.700680i	$-0.752880\pi$
$29$	−7.68437	−1.42695	−0.713475	+	0.700680i	$0.752880\pi$
$30$	0	0
$31$	6.26545	1.12531	0.562654	−	0.826693i	$-0.309780\pi$
$31$	6.26545	1.12531	0.562654	+	0.826693i	$0.309780\pi$
$32$	0	0
$33$	−6.30711	−1.09793
$34$	0	0
$35$	4.36754	0.738249
$36$	0	0
$37$	2.63811	0.433703	0.216851	−	0.976205i	$-0.430421\pi$
$37$	2.63811	0.433703	0.216851	+	0.976205i	$0.430421\pi$
$38$	0	0
$39$	10.4596	1.67487
$40$	0	0
$41$	−9.32726	−1.45667	−0.728336	−	0.685220i	$-0.759706\pi$
$41$	−9.32726	−1.45667	−0.728336	+	0.685220i	$0.759706\pi$
$42$	0	0
$43$	−11.1282	−1.69704	−0.848518	−	0.529167i	$-0.822504\pi$
$43$	−11.1282	−1.69704	−0.848518	+	0.529167i	$0.822504\pi$
$44$	0	0
$45$	−36.7476	−5.47800
$46$	0	0
$47$	−3.60289	−0.525535	−0.262768	−	0.964859i	$-0.584635\pi$
$47$	−3.60289	−0.525535	−0.262768	+	0.964859i	$0.584635\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−18.1941	−2.54769
$52$	0	0
$53$	−5.04940	−0.693589	−0.346795	−	0.937941i	$-0.612730\pi$
$53$	−5.04940	−0.693589	−0.346795	+	0.937941i	$0.612730\pi$
$54$	0	0
$55$	8.15365	1.09944
$56$	0	0
$57$	3.37843	0.447484
$58$	0	0
$59$	−5.12440	−0.667140	−0.333570	−	0.942725i	$-0.608253\pi$
$59$	−5.12440	−0.667140	−0.333570	+	0.942725i	$0.608253\pi$
$60$	0	0
$61$	−4.42684	−0.566799	−0.283400	−	0.959002i	$-0.591462\pi$
$61$	−4.42684	−0.566799	−0.283400	+	0.959002i	$0.591462\pi$
$62$	0	0
$63$	−8.41379	−1.06004
$64$	0	0
$65$	−13.5219	−1.67718
$66$	0	0
$67$	2.37810	0.290531	0.145266	−	0.989393i	$-0.453596\pi$
$67$	2.37810	0.290531	0.145266	+	0.989393i	$0.453596\pi$
$68$	0	0
$69$	2.27193	0.273509
$70$	0	0
$71$	6.67728	0.792447	0.396224	−	0.918154i	$-0.370321\pi$
$71$	6.67728	0.792447	0.396224	+	0.918154i	$0.370321\pi$
$72$	0	0
$73$	1.69478	0.198359	0.0991797	−	0.995070i	$-0.468378\pi$
$73$	1.69478	0.198359	0.0991797	+	0.995070i	$0.468378\pi$
$74$	0	0
$75$	47.5527	5.49091
$76$	0	0
$77$	1.86688	0.212750
$78$	0	0
$79$	−17.3813	−1.95555	−0.977775	−	0.209655i	$-0.932766\pi$
$79$	−17.3813	−1.95555	−0.977775	+	0.209655i	$0.932766\pi$
$80$	0	0
$81$	36.5505	4.06117
$82$	0	0
$83$	−0.104382	−0.0114574	−0.00572872	−	0.999984i	$-0.501824\pi$
$83$	−0.104382	−0.0114574	−0.00572872	+	0.999984i	$0.501824\pi$
$84$	0	0
$85$	23.5209	2.55120
$86$	0	0
$87$	−25.9611	−2.78332
$88$	0	0
$89$	13.3438	1.41444	0.707220	−	0.706994i	$-0.249949\pi$
$89$	13.3438	1.41444	0.707220	+	0.706994i	$0.249949\pi$
$90$	0	0
$91$	−3.09599	−0.324548
$92$	0	0
$93$	21.1674	2.19496
$94$	0	0
$95$	−4.36754	−0.448100
$96$	0	0
$97$	6.00099	0.609308	0.304654	−	0.952463i	$-0.401459\pi$
$97$	6.00099	0.609308	0.304654	+	0.952463i	$0.401459\pi$
$98$	0	0
$99$	−15.7075	−1.57866
$100$	0	0
$101$	7.40257	0.736583	0.368292	−	0.929710i	$-0.379943\pi$
$101$	7.40257	0.736583	0.368292	+	0.929710i	$0.379943\pi$
$102$	0	0
$103$	−11.2489	−1.10839	−0.554195	−	0.832387i	$-0.686974\pi$
$103$	−11.2489	−1.10839	−0.554195	+	0.832387i	$0.686974\pi$
$104$	0	0
$105$	14.7554	1.43998
$106$	0	0
$107$	2.14716	0.207574	0.103787	−	0.994600i	$-0.466904\pi$
$107$	2.14716	0.207574	0.103787	+	0.994600i	$0.466904\pi$
$108$	0	0
$109$	−1.74426	−0.167070	−0.0835348	−	0.996505i	$-0.526621\pi$
$109$	−1.74426	−0.167070	−0.0835348	+	0.996505i	$0.526621\pi$
$110$	0	0
$111$	8.91267	0.845953
$112$	0	0
$113$	−14.8183	−1.39399	−0.696996	−	0.717075i	$-0.745481\pi$
$113$	−14.8183	−1.39399	−0.696996	+	0.717075i	$0.745481\pi$
$114$	0	0
$115$	−2.93709	−0.273885
$116$	0	0
$117$	26.0490	2.40823
$118$	0	0
$119$	5.38539	0.493677
$120$	0	0
$121$	−7.51477	−0.683161
$122$	0	0
$123$	−31.5115	−2.84129
$124$	0	0
$125$	−39.6371	−3.54525
$126$	0	0
$127$	14.8525	1.31795	0.658974	−	0.752166i	$-0.270991\pi$
$127$	14.8525	1.31795	0.658974	+	0.752166i	$0.270991\pi$
$128$	0	0
$129$	−37.5959	−3.31013
$130$	0	0
$131$	−14.1185	−1.23354	−0.616769	−	0.787144i	$-0.711559\pi$
$131$	−14.1185	−1.23354	−0.616769	+	0.787144i	$0.711559\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	−79.8828	−6.87522
$136$	0	0
$137$	1.69308	0.144650	0.0723250	−	0.997381i	$-0.476958\pi$
$137$	1.69308	0.144650	0.0723250	+	0.997381i	$0.476958\pi$
$138$	0	0
$139$	−9.15281	−0.776331	−0.388165	−	0.921590i	$-0.626891\pi$
$139$	−9.15281	−0.776331	−0.388165	+	0.921590i	$0.626891\pi$
$140$	0	0
$141$	−12.1721	−1.02508
$142$	0	0
$143$	−5.77983	−0.483334
$144$	0	0
$145$	33.5618	2.78715
$146$	0	0
$147$	3.37843	0.278648
$148$	0	0
$149$	6.23731	0.510981	0.255490	−	0.966812i	$-0.417763\pi$
$149$	6.23731	0.510981	0.255490	+	0.966812i	$0.417763\pi$
$150$	0	0
$151$	10.1502	0.826011	0.413005	−	0.910729i	$-0.364479\pi$
$151$	10.1502	0.826011	0.413005	+	0.910729i	$0.364479\pi$
$152$	0	0
$153$	−45.3115	−3.66322
$154$	0	0
$155$	−27.3646	−2.19798
$156$	0	0
$157$	1.13016	0.0901970	0.0450985	−	0.998983i	$-0.485640\pi$
$157$	1.13016	0.0901970	0.0450985	+	0.998983i	$0.485640\pi$
$158$	0	0
$159$	−17.0591	−1.35287
$160$	0	0
$161$	−0.672482	−0.0529990
$162$	0	0
$163$	−9.79997	−0.767593	−0.383796	−	0.923418i	$-0.625384\pi$
$163$	−9.79997	−0.767593	−0.383796	+	0.923418i	$0.625384\pi$
$164$	0	0
$165$	27.5466	2.14450
$166$	0	0
$167$	−13.6163	−1.05366	−0.526832	−	0.849970i	$-0.676620\pi$
$167$	−13.6163	−1.05366	−0.526832	+	0.849970i	$0.676620\pi$
$168$	0	0
$169$	−3.41484	−0.262680
$170$	0	0
$171$	8.41379	0.643419
$172$	0	0
$173$	−9.95013	−0.756494	−0.378247	−	0.925705i	$-0.623473\pi$
$173$	−9.95013	−0.756494	−0.378247	+	0.925705i	$0.623473\pi$
$174$	0	0
$175$	−14.0754	−1.06400
$176$	0	0
$177$	−17.3124	−1.30128
$178$	0	0
$179$	−12.5562	−0.938492	−0.469246	−	0.883067i	$-0.655474\pi$
$179$	−12.5562	−0.938492	−0.469246	+	0.883067i	$0.655474\pi$
$180$	0	0
$181$	8.61606	0.640427	0.320213	−	0.947345i	$-0.396245\pi$
$181$	8.61606	0.640427	0.320213	+	0.947345i	$0.396245\pi$
$182$	0	0
$183$	−14.9558	−1.10556
$184$	0	0
$185$	−11.5220	−0.847118
$186$	0	0
$187$	10.0539	0.735210
$188$	0	0
$189$	−18.2901	−1.33041
$190$	0	0
$191$	7.96897	0.576615	0.288307	−	0.957538i	$-0.406908\pi$
$191$	7.96897	0.576615	0.288307	+	0.957538i	$0.406908\pi$
$192$	0	0
$193$	−8.18757	−0.589354	−0.294677	−	0.955597i	$-0.595212\pi$
$193$	−8.18757	−0.589354	−0.294677	+	0.955597i	$0.595212\pi$
$194$	0	0
$195$	−45.6826	−3.27140
$196$	0	0
$197$	−14.1811	−1.01036	−0.505180	−	0.863014i	$-0.668574\pi$
$197$	−14.1811	−1.01036	−0.505180	+	0.863014i	$0.668574\pi$
$198$	0	0
$199$	1.14648	0.0812721	0.0406361	−	0.999174i	$-0.487062\pi$
$199$	1.14648	0.0812721	0.0406361	+	0.999174i	$0.487062\pi$
$200$	0	0
$201$	8.03425	0.566692
$202$	0	0
$203$	7.68437	0.539337
$204$	0	0
$205$	40.7371	2.84521
$206$	0	0
$207$	5.65813	0.393267
$208$	0	0
$209$	−1.86688	−0.129135
$210$	0	0
$211$	−8.04396	−0.553769	−0.276884	−	0.960903i	$-0.589302\pi$
$211$	−8.04396	−0.553769	−0.276884	+	0.960903i	$0.589302\pi$
$212$	0	0
$213$	22.5587	1.54570
$214$	0	0
$215$	48.6028	3.31469
$216$	0	0
$217$	−6.26545	−0.425326
$218$	0	0
$219$	5.72571	0.386907
$220$	0	0
$221$	−16.6731	−1.12155
$222$	0	0
$223$	1.36295	0.0912696	0.0456348	−	0.998958i	$-0.485469\pi$
$223$	1.36295	0.0912696	0.0456348	+	0.998958i	$0.485469\pi$
$224$	0	0
$225$	118.427	7.89516
$226$	0	0
$227$	−23.9494	−1.58958	−0.794788	−	0.606888i	$-0.792418\pi$
$227$	−23.9494	−1.58958	−0.794788	+	0.606888i	$0.792418\pi$
$228$	0	0
$229$	7.69573	0.508548	0.254274	−	0.967132i	$-0.418164\pi$
$229$	7.69573	0.508548	0.254274	+	0.967132i	$0.418164\pi$
$230$	0	0
$231$	6.30711	0.414978
$232$	0	0
$233$	5.12460	0.335723	0.167862	−	0.985811i	$-0.446314\pi$
$233$	5.12460	0.335723	0.167862	+	0.985811i	$0.446314\pi$
$234$	0	0
$235$	15.7357	1.02649
$236$	0	0
$237$	−58.7216	−3.81438
$238$	0	0
$239$	−10.0003	−0.646863	−0.323431	−	0.946252i	$-0.604836\pi$
$239$	−10.0003	−0.646863	−0.323431	+	0.946252i	$0.604836\pi$
$240$	0	0
$241$	−3.35875	−0.216356	−0.108178	−	0.994132i	$-0.534502\pi$
$241$	−3.35875	−0.216356	−0.108178	+	0.994132i	$0.534502\pi$
$242$	0	0
$243$	68.6131	4.40153
$244$	0	0
$245$	−4.36754	−0.279032
$246$	0	0
$247$	3.09599	0.196993
$248$	0	0
$249$	−0.352648	−0.0223482
$250$	0	0
$251$	−29.1859	−1.84220	−0.921100	−	0.389326i	$-0.872708\pi$
$251$	−29.1859	−1.84220	−0.921100	+	0.389326i	$0.872708\pi$
$252$	0	0
$253$	−1.25544	−0.0789289
$254$	0	0
$255$	79.4636	4.97621
$256$	0	0
$257$	13.3076	0.830107	0.415054	−	0.909797i	$-0.363763\pi$
$257$	13.3076	0.830107	0.415054	+	0.909797i	$0.363763\pi$
$258$	0	0
$259$	−2.63811	−0.163924
$260$	0	0
$261$	−64.6547	−4.00202
$262$	0	0
$263$	16.8752	1.04057	0.520284	−	0.853994i	$-0.325826\pi$
$263$	16.8752	1.04057	0.520284	+	0.853994i	$0.325826\pi$
$264$	0	0
$265$	22.0535	1.35473
$266$	0	0
$267$	45.0811	2.75892
$268$	0	0
$269$	−15.9369	−0.971693	−0.485846	−	0.874044i	$-0.661489\pi$
$269$	−15.9369	−0.971693	−0.485846	+	0.874044i	$0.661489\pi$
$270$	0	0
$271$	−20.6742	−1.25587	−0.627934	−	0.778266i	$-0.716099\pi$
$271$	−20.6742	−1.25587	−0.627934	+	0.778266i	$0.716099\pi$
$272$	0	0
$273$	−10.4596	−0.633043
$274$	0	0
$275$	−26.2770	−1.58456
$276$	0	0
$277$	−26.0975	−1.56805	−0.784024	−	0.620731i	$-0.786836\pi$
$277$	−26.0975	−1.56805	−0.784024	+	0.620731i	$0.786836\pi$
$278$	0	0
$279$	52.7162	3.15604
$280$	0	0
$281$	−13.9269	−0.830810	−0.415405	−	0.909637i	$-0.636360\pi$
$281$	−13.9269	−0.830810	−0.415405	+	0.909637i	$0.636360\pi$
$282$	0	0
$283$	−28.9027	−1.71809	−0.859044	−	0.511902i	$-0.828941\pi$
$283$	−28.9027	−1.71809	−0.859044	+	0.511902i	$0.828941\pi$
$284$	0	0
$285$	−14.7554	−0.874036
$286$	0	0
$287$	9.32726	0.550570
$288$	0	0
$289$	12.0024	0.706022
$290$	0	0
$291$	20.2739	1.18848
$292$	0	0
$293$	−5.66153	−0.330750	−0.165375	−	0.986231i	$-0.552883\pi$
$293$	−5.66153	−0.330750	−0.165375	+	0.986231i	$0.552883\pi$
$294$	0	0
$295$	22.3810	1.30307
$296$	0	0
$297$	−34.1454	−1.98132
$298$	0	0
$299$	2.08200	0.120405
$300$	0	0
$301$	11.1282	0.641419
$302$	0	0
$303$	25.0091	1.43673
$304$	0	0
$305$	19.3344	1.10708
$306$	0	0
$307$	11.0435	0.630289	0.315144	−	0.949044i	$-0.397947\pi$
$307$	11.0435	0.630289	0.315144	+	0.949044i	$0.397947\pi$
$308$	0	0
$309$	−38.0037	−2.16196
$310$	0	0
$311$	−3.31782	−0.188136	−0.0940681	−	0.995566i	$-0.529987\pi$
$311$	−3.31782	−0.188136	−0.0940681	+	0.995566i	$0.529987\pi$
$312$	0	0
$313$	3.57579	0.202116	0.101058	−	0.994881i	$-0.467777\pi$
$313$	3.57579	0.202116	0.101058	+	0.994881i	$0.467777\pi$
$314$	0	0
$315$	36.7476	2.07049
$316$	0	0
$317$	6.44784	0.362147	0.181073	−	0.983470i	$-0.442043\pi$
$317$	6.44784	0.362147	0.181073	+	0.983470i	$0.442043\pi$
$318$	0	0
$319$	14.3458	0.803209
$320$	0	0
$321$	7.25402	0.404880
$322$	0	0
$323$	−5.38539	−0.299651
$324$	0	0
$325$	43.5773	2.41723
$326$	0	0
$327$	−5.89286	−0.325876
$328$	0	0
$329$	3.60289	0.198634
$330$	0	0
$331$	10.0308	0.551344	0.275672	−	0.961252i	$-0.411100\pi$
$331$	10.0308	0.551344	0.275672	+	0.961252i	$0.411100\pi$
$332$	0	0
$333$	22.1965	1.21636
$334$	0	0
$335$	−10.3864	−0.567472
$336$	0	0
$337$	−27.0204	−1.47190	−0.735948	−	0.677038i	$-0.763263\pi$
$337$	−27.0204	−1.47190	−0.735948	+	0.677038i	$0.763263\pi$
$338$	0	0
$339$	−50.0628	−2.71904
$340$	0	0
$341$	−11.6968	−0.633418
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−9.92276	−0.534223
$346$	0	0
$347$	−6.44424	−0.345945	−0.172973	−	0.984927i	$-0.555337\pi$
$347$	−6.44424	−0.345945	−0.172973	+	0.984927i	$0.555337\pi$
$348$	0	0
$349$	11.3595	0.608062	0.304031	−	0.952662i	$-0.401667\pi$
$349$	11.3595	0.608062	0.304031	+	0.952662i	$0.401667\pi$
$350$	0	0
$351$	56.6260	3.02247
$352$	0	0
$353$	−1.33909	−0.0712725	−0.0356363	−	0.999365i	$-0.511346\pi$
$353$	−1.33909	−0.0712725	−0.0356363	+	0.999365i	$0.511346\pi$
$354$	0	0
$355$	−29.1633	−1.54783
$356$	0	0
$357$	18.1941	0.962937
$358$	0	0
$359$	−24.4903	−1.29255	−0.646274	−	0.763106i	$-0.723674\pi$
$359$	−24.4903	−1.29255	−0.646274	+	0.763106i	$0.723674\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−25.3881	−1.33253
$364$	0	0
$365$	−7.40203	−0.387440
$366$	0	0
$367$	28.2792	1.47616	0.738081	−	0.674712i	$-0.235732\pi$
$367$	28.2792	1.47616	0.738081	+	0.674712i	$0.235732\pi$
$368$	0	0
$369$	−78.4776	−4.08538
$370$	0	0
$371$	5.04940	0.262152
$372$	0	0
$373$	6.74799	0.349398	0.174699	−	0.984622i	$-0.444105\pi$
$373$	6.74799	0.349398	0.174699	+	0.984622i	$0.444105\pi$
$374$	0	0
$375$	−133.911	−6.91514
$376$	0	0
$377$	−23.7907	−1.22528
$378$	0	0
$379$	6.85416	0.352075	0.176037	−	0.984383i	$-0.443672\pi$
$379$	6.85416	0.352075	0.176037	+	0.984383i	$0.443672\pi$
$380$	0	0
$381$	50.1782	2.57071
$382$	0	0
$383$	35.7508	1.82678	0.913391	−	0.407083i	$-0.133454\pi$
$383$	35.7508	1.82678	0.913391	+	0.407083i	$0.133454\pi$
$384$	0	0
$385$	−8.15365	−0.415549
$386$	0	0
$387$	−93.6304	−4.75950
$388$	0	0
$389$	−10.4471	−0.529691	−0.264845	−	0.964291i	$-0.585321\pi$
$389$	−10.4471	−0.529691	−0.264845	+	0.964291i	$0.585321\pi$
$390$	0	0
$391$	−3.62158	−0.183151
$392$	0	0
$393$	−47.6983	−2.40606
$394$	0	0
$395$	75.9136	3.81963
$396$	0	0
$397$	−22.7284	−1.14070	−0.570352	−	0.821400i	$-0.693193\pi$
$397$	−22.7284	−1.14070	−0.570352	+	0.821400i	$0.693193\pi$
$398$	0	0
$399$	−3.37843	−0.169133
$400$	0	0
$401$	30.3469	1.51545	0.757725	−	0.652574i	$-0.226311\pi$
$401$	30.3469	1.51545	0.757725	+	0.652574i	$0.226311\pi$
$402$	0	0
$403$	19.3978	0.966272
$404$	0	0
$405$	−159.636	−7.93237
$406$	0	0
$407$	−4.92503	−0.244125
$408$	0	0
$409$	35.5067	1.75569	0.877847	−	0.478941i	$-0.158979\pi$
$409$	35.5067	1.75569	0.877847	+	0.478941i	$0.158979\pi$
$410$	0	0
$411$	5.71997	0.282145
$412$	0	0
$413$	5.12440	0.252155
$414$	0	0
$415$	0.455893	0.0223789
$416$	0	0
$417$	−30.9221	−1.51426
$418$	0	0
$419$	18.9917	0.927805	0.463902	−	0.885886i	$-0.346449\pi$
$419$	18.9917	0.927805	0.463902	+	0.885886i	$0.346449\pi$
$420$	0	0
$421$	−27.5839	−1.34436	−0.672179	−	0.740389i	$-0.734642\pi$
$421$	−27.5839	−1.34436	−0.672179	+	0.740389i	$0.734642\pi$
$422$	0	0
$423$	−30.3139	−1.47391
$424$	0	0
$425$	−75.8014	−3.67691
$426$	0	0
$427$	4.42684	0.214230
$428$	0	0
$429$	−19.5268	−0.942761
$430$	0	0
$431$	−18.1122	−0.872435	−0.436217	−	0.899841i	$-0.643682\pi$
$431$	−18.1122	−0.872435	−0.436217	+	0.899841i	$0.643682\pi$
$432$	0	0
$433$	−15.3302	−0.736723	−0.368362	−	0.929683i	$-0.620081\pi$
$433$	−15.3302	−0.736723	−0.368362	+	0.929683i	$0.620081\pi$
$434$	0	0
$435$	113.386	5.43645
$436$	0	0
$437$	0.672482	0.0321692
$438$	0	0
$439$	−22.0939	−1.05449	−0.527243	−	0.849714i	$-0.676774\pi$
$439$	−22.0939	−1.05449	−0.527243	+	0.849714i	$0.676774\pi$
$440$	0	0
$441$	8.41379	0.400657
$442$	0	0
$443$	8.61576	0.409347	0.204674	−	0.978830i	$-0.434387\pi$
$443$	8.61576	0.409347	0.204674	+	0.978830i	$0.434387\pi$
$444$	0	0
$445$	−58.2795	−2.76271
$446$	0	0
$447$	21.0723	0.996687
$448$	0	0
$449$	35.3183	1.66677	0.833386	−	0.552691i	$-0.186399\pi$
$449$	35.3183	1.66677	0.833386	+	0.552691i	$0.186399\pi$
$450$	0	0
$451$	17.4128	0.819938
$452$	0	0
$453$	34.2917	1.61117
$454$	0	0
$455$	13.5219	0.633914
$456$	0	0
$457$	7.04291	0.329453	0.164727	−	0.986339i	$-0.447326\pi$
$457$	7.04291	0.329453	0.164727	+	0.986339i	$0.447326\pi$
$458$	0	0
$459$	−98.4994	−4.59756
$460$	0	0
$461$	23.1407	1.07777	0.538885	−	0.842380i	$-0.318846\pi$
$461$	23.1407	1.07777	0.538885	+	0.842380i	$0.318846\pi$
$462$	0	0
$463$	3.66355	0.170260	0.0851298	−	0.996370i	$-0.472870\pi$
$463$	3.66355	0.170260	0.0851298	+	0.996370i	$0.472870\pi$
$464$	0	0
$465$	−92.4493	−4.28724
$466$	0	0
$467$	25.5221	1.18102	0.590510	−	0.807030i	$-0.298927\pi$
$467$	25.5221	1.18102	0.590510	+	0.807030i	$0.298927\pi$
$468$	0	0
$469$	−2.37810	−0.109810
$470$	0	0
$471$	3.81818	0.175933
$472$	0	0
$473$	20.7750	0.955235
$474$	0	0
$475$	14.0754	0.645823
$476$	0	0
$477$	−42.4846	−1.94524
$478$	0	0
$479$	−11.2904	−0.515869	−0.257935	−	0.966162i	$-0.583042\pi$
$479$	−11.2904	−0.515869	−0.257935	+	0.966162i	$0.583042\pi$
$480$	0	0
$481$	8.16757	0.372409
$482$	0	0
$483$	−2.27193	−0.103377
$484$	0	0
$485$	−26.2095	−1.19011
$486$	0	0
$487$	22.4172	1.01582	0.507910	−	0.861410i	$-0.330418\pi$
$487$	22.4172	1.01582	0.507910	+	0.861410i	$0.330418\pi$
$488$	0	0
$489$	−33.1085	−1.49722
$490$	0	0
$491$	13.9118	0.627832	0.313916	−	0.949451i	$-0.398359\pi$
$491$	13.9118	0.627832	0.313916	+	0.949451i	$0.398359\pi$
$492$	0	0
$493$	41.3833	1.86381
$494$	0	0
$495$	68.6032	3.08348
$496$	0	0
$497$	−6.67728	−0.299517
$498$	0	0
$499$	−12.7724	−0.571771	−0.285886	−	0.958264i	$-0.592288\pi$
$499$	−12.7724	−0.571771	−0.285886	+	0.958264i	$0.592288\pi$
$500$	0	0
$501$	−46.0018	−2.05521
$502$	0	0
$503$	6.15261	0.274331	0.137166	−	0.990548i	$-0.456201\pi$
$503$	6.15261	0.274331	0.137166	+	0.990548i	$0.456201\pi$
$504$	0	0
$505$	−32.3310	−1.43871
$506$	0	0
$507$	−11.5368	−0.512368
$508$	0	0
$509$	−8.58379	−0.380470	−0.190235	−	0.981739i	$-0.560925\pi$
$509$	−8.58379	−0.380470	−0.190235	+	0.981739i	$0.560925\pi$
$510$	0	0
$511$	−1.69478	−0.0749728
$512$	0	0
$513$	18.2901	0.807529
$514$	0	0
$515$	49.1301	2.16493
$516$	0	0
$517$	6.72615	0.295816
$518$	0	0
$519$	−33.6158	−1.47557
$520$	0	0
$521$	35.8157	1.56911	0.784557	−	0.620056i	$-0.212890\pi$
$521$	35.8157	1.56911	0.784557	+	0.620056i	$0.212890\pi$
$522$	0	0
$523$	18.8221	0.823032	0.411516	−	0.911402i	$-0.364999\pi$
$523$	18.8221	0.823032	0.411516	+	0.911402i	$0.364999\pi$
$524$	0	0
$525$	−47.5527	−2.07537
$526$	0	0
$527$	−33.7419	−1.46982
$528$	0	0
$529$	−22.5478	−0.980338
$530$	0	0
$531$	−43.1156	−1.87106
$532$	0	0
$533$	−28.8771	−1.25081
$534$	0	0
$535$	−9.37779	−0.405437
$536$	0	0
$537$	−42.4202	−1.83056
$538$	0	0
$539$	−1.86688	−0.0804121
$540$	0	0
$541$	14.3914	0.618736	0.309368	−	0.950942i	$-0.399883\pi$
$541$	14.3914	0.618736	0.309368	+	0.950942i	$0.399883\pi$
$542$	0	0
$543$	29.1088	1.24918
$544$	0	0
$545$	7.61811	0.326324
$546$	0	0
$547$	−12.1980	−0.521550	−0.260775	−	0.965400i	$-0.583978\pi$
$547$	−12.1980	−0.521550	−0.260775	+	0.965400i	$0.583978\pi$
$548$	0	0
$549$	−37.2466	−1.58964
$550$	0	0
$551$	−7.68437	−0.327365
$552$	0	0
$553$	17.3813	0.739129
$554$	0	0
$555$	−38.9264	−1.65234
$556$	0	0
$557$	4.75415	0.201440	0.100720	−	0.994915i	$-0.467885\pi$
$557$	4.75415	0.201440	0.100720	+	0.994915i	$0.467885\pi$
$558$	0	0
$559$	−34.4528	−1.45720
$560$	0	0
$561$	33.9662	1.43406
$562$	0	0
$563$	16.1426	0.680331	0.340165	−	0.940366i	$-0.389517\pi$
$563$	16.1426	0.680331	0.340165	+	0.940366i	$0.389517\pi$
$564$	0	0
$565$	64.7197	2.72278
$566$	0	0
$567$	−36.5505	−1.53498
$568$	0	0
$569$	26.4767	1.10996	0.554981	−	0.831863i	$-0.312725\pi$
$569$	26.4767	1.10996	0.554981	+	0.831863i	$0.312725\pi$
$570$	0	0
$571$	−36.1733	−1.51381	−0.756903	−	0.653527i	$-0.773288\pi$
$571$	−36.1733	−1.51381	−0.756903	+	0.653527i	$0.773288\pi$
$572$	0	0
$573$	26.9226	1.12471
$574$	0	0
$575$	9.46545	0.394736
$576$	0	0
$577$	−35.5350	−1.47934	−0.739670	−	0.672969i	$-0.765019\pi$
$577$	−35.5350	−1.47934	−0.739670	+	0.672969i	$0.765019\pi$
$578$	0	0
$579$	−27.6611	−1.14956
$580$	0	0
$581$	0.104382	0.00433050
$582$	0	0
$583$	9.42662	0.390411
$584$	0	0
$585$	−113.770	−4.70381
$586$	0	0
$587$	19.3485	0.798599	0.399300	−	0.916821i	$-0.369253\pi$
$587$	19.3485	0.798599	0.399300	+	0.916821i	$0.369253\pi$
$588$	0	0
$589$	6.26545	0.258163
$590$	0	0
$591$	−47.9098	−1.97074
$592$	0	0
$593$	36.8882	1.51482	0.757409	−	0.652940i	$-0.226465\pi$
$593$	36.8882	1.51482	0.757409	+	0.652940i	$0.226465\pi$
$594$	0	0
$595$	−23.5209	−0.964262
$596$	0	0
$597$	3.87332	0.158524
$598$	0	0
$599$	43.1827	1.76440	0.882198	−	0.470879i	$-0.156063\pi$
$599$	43.1827	1.76440	0.882198	+	0.470879i	$0.156063\pi$
$600$	0	0
$601$	21.0423	0.858334	0.429167	−	0.903225i	$-0.358807\pi$
$601$	21.0423	0.858334	0.429167	+	0.903225i	$0.358807\pi$
$602$	0	0
$603$	20.0088	0.814823
$604$	0	0
$605$	32.8210	1.33437
$606$	0	0
$607$	2.68496	0.108979	0.0544897	−	0.998514i	$-0.482647\pi$
$607$	2.68496	0.108979	0.0544897	+	0.998514i	$0.482647\pi$
$608$	0	0
$609$	25.9611	1.05200
$610$	0	0
$611$	−11.1545	−0.451263
$612$	0	0
$613$	−40.6347	−1.64122	−0.820611	−	0.571488i	$-0.806366\pi$
$613$	−40.6347	−1.64122	−0.820611	+	0.571488i	$0.806366\pi$
$614$	0	0
$615$	137.628	5.54968
$616$	0	0
$617$	−17.3927	−0.700202	−0.350101	−	0.936712i	$-0.613853\pi$
$617$	−17.3927	−0.700202	−0.350101	+	0.936712i	$0.613853\pi$
$618$	0	0
$619$	18.6257	0.748631	0.374316	−	0.927301i	$-0.377878\pi$
$619$	18.6257	0.748631	0.374316	+	0.927301i	$0.377878\pi$
$620$	0	0
$621$	12.2998	0.493573
$622$	0	0
$623$	−13.3438	−0.534608
$624$	0	0
$625$	102.739	4.10958
$626$	0	0
$627$	−6.30711	−0.251882
$628$	0	0
$629$	−14.2072	−0.566480
$630$	0	0
$631$	9.77563	0.389162	0.194581	−	0.980886i	$-0.437665\pi$
$631$	9.77563	0.389162	0.194581	+	0.980886i	$0.437665\pi$
$632$	0	0
$633$	−27.1760	−1.08015
$634$	0	0
$635$	−64.8689	−2.57425
$636$	0	0
$637$	3.09599	0.122668
$638$	0	0
$639$	56.1812	2.22250
$640$	0	0
$641$	−13.5365	−0.534659	−0.267329	−	0.963605i	$-0.586141\pi$
$641$	−13.5365	−0.534659	−0.267329	+	0.963605i	$0.586141\pi$
$642$	0	0
$643$	3.57530	0.140996	0.0704981	−	0.997512i	$-0.477541\pi$
$643$	3.57530	0.140996	0.0704981	+	0.997512i	$0.477541\pi$
$644$	0	0
$645$	164.201	6.46542
$646$	0	0
$647$	−24.2142	−0.951959	−0.475980	−	0.879456i	$-0.657906\pi$
$647$	−24.2142	−0.951959	−0.475980	+	0.879456i	$0.657906\pi$
$648$	0	0
$649$	9.56662	0.375523
$650$	0	0
$651$	−21.1674	−0.829615
$652$	0	0
$653$	21.0220	0.822654	0.411327	−	0.911488i	$-0.365065\pi$
$653$	21.0220	0.822654	0.411327	+	0.911488i	$0.365065\pi$
$654$	0	0
$655$	61.6630	2.40937
$656$	0	0
$657$	14.2596	0.556318
$658$	0	0
$659$	−34.1902	−1.33186	−0.665931	−	0.746013i	$-0.731965\pi$
$659$	−34.1902	−1.33186	−0.665931	+	0.746013i	$0.731965\pi$
$660$	0	0
$661$	−33.7592	−1.31308	−0.656541	−	0.754290i	$-0.727981\pi$
$661$	−33.7592	−1.31308	−0.656541	+	0.754290i	$0.727981\pi$
$662$	0	0
$663$	−56.3289	−2.18763
$664$	0	0
$665$	4.36754	0.169366
$666$	0	0
$667$	−5.16760	−0.200090
$668$	0	0
$669$	4.60462	0.178025
$670$	0	0
$671$	8.26437	0.319043
$672$	0	0
$673$	−0.781214	−0.0301136	−0.0150568	−	0.999887i	$-0.504793\pi$
$673$	−0.781214	−0.0301136	−0.0150568	+	0.999887i	$0.504793\pi$
$674$	0	0
$675$	257.440	9.90889
$676$	0	0
$677$	11.4264	0.439151	0.219575	−	0.975596i	$-0.429533\pi$
$677$	11.4264	0.439151	0.219575	+	0.975596i	$0.429533\pi$
$678$	0	0
$679$	−6.00099	−0.230297
$680$	0	0
$681$	−80.9113	−3.10053
$682$	0	0
$683$	31.1653	1.19251	0.596253	−	0.802796i	$-0.296655\pi$
$683$	31.1653	1.19251	0.596253	+	0.802796i	$0.296655\pi$
$684$	0	0
$685$	−7.39461	−0.282533
$686$	0	0
$687$	25.9995	0.991942
$688$	0	0
$689$	−15.6329	−0.595566
$690$	0	0
$691$	37.0517	1.40951	0.704757	−	0.709449i	$-0.251056\pi$
$691$	37.0517	1.40951	0.704757	+	0.709449i	$0.251056\pi$
$692$	0	0
$693$	15.7075	0.596679
$694$	0	0
$695$	39.9752	1.51635
$696$	0	0
$697$	50.2309	1.90263
$698$	0	0
$699$	17.3131	0.654841
$700$	0	0
$701$	36.5921	1.38206	0.691032	−	0.722825i	$-0.257157\pi$
$701$	36.5921	1.38206	0.691032	+	0.722825i	$0.257157\pi$
$702$	0	0
$703$	2.63811	0.0994982
$704$	0	0
$705$	53.1621	2.00220
$706$	0	0
$707$	−7.40257	−0.278402
$708$	0	0
$709$	−33.1456	−1.24481	−0.622405	−	0.782696i	$-0.713844\pi$
$709$	−33.1456	−1.24481	−0.622405	+	0.782696i	$0.713844\pi$
$710$	0	0
$711$	−146.243	−5.48453
$712$	0	0
$713$	4.21340	0.157793
$714$	0	0
$715$	25.2436	0.944058
$716$	0	0
$717$	−33.7852	−1.26173
$718$	0	0
$719$	44.4860	1.65905	0.829525	−	0.558470i	$-0.188611\pi$
$719$	44.4860	1.65905	0.829525	+	0.558470i	$0.188611\pi$
$720$	0	0
$721$	11.2489	0.418932
$722$	0	0
$723$	−11.3473	−0.422011
$724$	0	0
$725$	−108.160	−4.01698
$726$	0	0
$727$	48.3774	1.79422	0.897110	−	0.441807i	$-0.145663\pi$
$727$	48.3774	1.79422	0.897110	+	0.441807i	$0.145663\pi$
$728$	0	0
$729$	122.153	4.52418
$730$	0	0
$731$	59.9297	2.21658
$732$	0	0
$733$	−12.1171	−0.447554	−0.223777	−	0.974640i	$-0.571839\pi$
$733$	−12.1171	−0.447554	−0.223777	+	0.974640i	$0.571839\pi$
$734$	0	0
$735$	−14.7554	−0.544262
$736$	0	0
$737$	−4.43962	−0.163535
$738$	0	0
$739$	−20.9790	−0.771726	−0.385863	−	0.922556i	$-0.626096\pi$
$739$	−20.9790	−0.771726	−0.385863	+	0.922556i	$0.626096\pi$
$740$	0	0
$741$	10.4596	0.384243
$742$	0	0
$743$	−7.16040	−0.262690	−0.131345	−	0.991337i	$-0.541930\pi$
$743$	−7.16040	−0.262690	−0.131345	+	0.991337i	$0.541930\pi$
$744$	0	0
$745$	−27.2417	−0.998059
$746$	0	0
$747$	−0.878250	−0.0321335
$748$	0	0
$749$	−2.14716	−0.0784555
$750$	0	0
$751$	−44.1869	−1.61240	−0.806201	−	0.591642i	$-0.798480\pi$
$751$	−44.1869	−1.61240	−0.806201	+	0.591642i	$0.798480\pi$
$752$	0	0
$753$	−98.6027	−3.59328
$754$	0	0
$755$	−44.3313	−1.61338
$756$	0	0
$757$	−20.9688	−0.762123	−0.381062	−	0.924550i	$-0.624441\pi$
$757$	−20.9688	−0.762123	−0.381062	+	0.924550i	$0.624441\pi$
$758$	0	0
$759$	−4.24142	−0.153954
$760$	0	0
$761$	−19.7690	−0.716624	−0.358312	−	0.933602i	$-0.616648\pi$
$761$	−19.7690	−0.716624	−0.358312	+	0.933602i	$0.616648\pi$
$762$	0	0
$763$	1.74426	0.0631464
$764$	0	0
$765$	197.900	7.15508
$766$	0	0
$767$	−15.8651	−0.572855
$768$	0	0
$769$	21.5736	0.777966	0.388983	−	0.921245i	$-0.372827\pi$
$769$	21.5736	0.777966	0.388983	+	0.921245i	$0.372827\pi$
$770$	0	0
$771$	44.9589	1.61916
$772$	0	0
$773$	19.4016	0.697829	0.348914	−	0.937155i	$-0.386550\pi$
$773$	19.4016	0.697829	0.348914	+	0.937155i	$0.386550\pi$
$774$	0	0
$775$	88.1886	3.16783
$776$	0	0
$777$	−8.91267	−0.319740
$778$	0	0
$779$	−9.32726	−0.334184
$780$	0	0
$781$	−12.4657	−0.446056
$782$	0	0
$783$	−140.548	−5.02278
$784$	0	0
$785$	−4.93604	−0.176175
$786$	0	0
$787$	10.2327	0.364755	0.182377	−	0.983229i	$-0.441621\pi$
$787$	10.2327	0.364755	0.182377	+	0.983229i	$0.441621\pi$
$788$	0	0
$789$	57.0116	2.02967
$790$	0	0
$791$	14.8183	0.526880
$792$	0	0
$793$	−13.7055	−0.486695
$794$	0	0
$795$	74.5061	2.64246
$796$	0	0
$797$	2.42530	0.0859085	0.0429543	−	0.999077i	$-0.486323\pi$
$797$	2.42530	0.0859085	0.0429543	+	0.999077i	$0.486323\pi$
$798$	0	0
$799$	19.4029	0.686426
$800$	0	0
$801$	112.272	3.96693
$802$	0	0
$803$	−3.16395	−0.111653
$804$	0	0
$805$	2.93709	0.103519
$806$	0	0
$807$	−53.8419	−1.89532
$808$	0	0
$809$	−4.71310	−0.165704	−0.0828520	−	0.996562i	$-0.526403\pi$
$809$	−4.71310	−0.165704	−0.0828520	+	0.996562i	$0.526403\pi$
$810$	0	0
$811$	38.4601	1.35052	0.675258	−	0.737582i	$-0.264032\pi$
$811$	38.4601	1.35052	0.675258	+	0.737582i	$0.264032\pi$
$812$	0	0
$813$	−69.8464	−2.44962
$814$	0	0
$815$	42.8017	1.49928
$816$	0	0
$817$	−11.1282	−0.389327
$818$	0	0
$819$	−26.0490	−0.910227
$820$	0	0
$821$	35.6854	1.24543	0.622714	−	0.782450i	$-0.286030\pi$
$821$	35.6854	1.24543	0.622714	+	0.782450i	$0.286030\pi$
$822$	0	0
$823$	−25.6502	−0.894109	−0.447054	−	0.894507i	$-0.647527\pi$
$823$	−25.6502	−0.894109	−0.447054	+	0.894507i	$0.647527\pi$
$824$	0	0
$825$	−88.7750	−3.09075
$826$	0	0
$827$	48.9952	1.70373	0.851865	−	0.523761i	$-0.175471\pi$
$827$	48.9952	1.70373	0.851865	+	0.523761i	$0.175471\pi$
$828$	0	0
$829$	−7.98292	−0.277258	−0.138629	−	0.990344i	$-0.544270\pi$
$829$	−7.98292	−0.277258	−0.138629	+	0.990344i	$0.544270\pi$
$830$	0	0
$831$	−88.1686	−3.05854
$832$	0	0
$833$	−5.38539	−0.186593
$834$	0	0
$835$	59.4699	2.05804
$836$	0	0
$837$	114.596	3.96101
$838$	0	0
$839$	16.0172	0.552975	0.276487	−	0.961018i	$-0.410830\pi$
$839$	16.0172	0.552975	0.276487	+	0.961018i	$0.410830\pi$
$840$	0	0
$841$	30.0495	1.03619
$842$	0	0
$843$	−47.0511	−1.62053
$844$	0	0
$845$	14.9144	0.513073
$846$	0	0
$847$	7.51477	0.258211
$848$	0	0
$849$	−97.6458	−3.35120
$850$	0	0
$851$	1.77408	0.0608148
$852$	0	0
$853$	15.0057	0.513785	0.256892	−	0.966440i	$-0.417301\pi$
$853$	15.0057	0.513785	0.256892	+	0.966440i	$0.417301\pi$
$854$	0	0
$855$	−36.7476	−1.25674
$856$	0	0
$857$	25.6593	0.876505	0.438252	−	0.898852i	$-0.355598\pi$
$857$	25.6593	0.876505	0.438252	+	0.898852i	$0.355598\pi$
$858$	0	0
$859$	52.8484	1.80316	0.901581	−	0.432609i	$-0.142407\pi$
$859$	52.8484	1.80316	0.901581	+	0.432609i	$0.142407\pi$
$860$	0	0
$861$	31.5115	1.07391
$862$	0	0
$863$	19.7143	0.671084	0.335542	−	0.942025i	$-0.391081\pi$
$863$	19.7143	0.671084	0.335542	+	0.942025i	$0.391081\pi$
$864$	0	0
$865$	43.4576	1.47760
$866$	0	0
$867$	40.5492	1.37712
$868$	0	0
$869$	32.4488	1.10075
$870$	0	0
$871$	7.36258	0.249471
$872$	0	0
$873$	50.4911	1.70886
$874$	0	0
$875$	39.6371	1.33998
$876$	0	0
$877$	−6.96919	−0.235333	−0.117666	−	0.993053i	$-0.537541\pi$
$877$	−6.96919	−0.235333	−0.117666	+	0.993053i	$0.537541\pi$
$878$	0	0
$879$	−19.1271	−0.645141
$880$	0	0
$881$	−45.3500	−1.52788	−0.763940	−	0.645287i	$-0.776738\pi$
$881$	−45.3500	−1.52788	−0.763940	+	0.645287i	$0.776738\pi$
$882$	0	0
$883$	3.65870	0.123125	0.0615625	−	0.998103i	$-0.480392\pi$
$883$	3.65870	0.123125	0.0615625	+	0.998103i	$0.480392\pi$
$884$	0	0
$885$	75.6127	2.54169
$886$	0	0
$887$	19.8348	0.665987	0.332993	−	0.942929i	$-0.391941\pi$
$887$	19.8348	0.665987	0.332993	+	0.942929i	$0.391941\pi$
$888$	0	0
$889$	−14.8525	−0.498138
$890$	0	0
$891$	−68.2353	−2.28597
$892$	0	0
$893$	−3.60289	−0.120566
$894$	0	0
$895$	54.8395	1.83308
$896$	0	0
$897$	7.03389	0.234855
$898$	0	0
$899$	−48.1460	−1.60576
$900$	0	0
$901$	27.1930	0.905930
$902$	0	0
$903$	37.5959	1.25111
$904$	0	0
$905$	−37.6310	−1.25090
$906$	0	0
$907$	−9.09465	−0.301983	−0.150991	−	0.988535i	$-0.548247\pi$
$907$	−9.09465	−0.301983	−0.150991	+	0.988535i	$0.548247\pi$
$908$	0	0
$909$	62.2837	2.06582
$910$	0	0
$911$	−11.6360	−0.385519	−0.192759	−	0.981246i	$-0.561744\pi$
$911$	−11.6360	−0.385519	−0.192759	+	0.981246i	$0.561744\pi$
$912$	0	0
$913$	0.194869	0.00644921
$914$	0	0
$915$	65.3200	2.15941
$916$	0	0
$917$	14.1185	0.466234
$918$	0	0
$919$	12.9105	0.425878	0.212939	−	0.977065i	$-0.431696\pi$
$919$	12.9105	0.425878	0.212939	+	0.977065i	$0.431696\pi$
$920$	0	0
$921$	37.3099	1.22940
$922$	0	0
$923$	20.6728	0.680453
$924$	0	0
$925$	37.1324	1.22091
$926$	0	0
$927$	−94.6461	−3.10859
$928$	0	0
$929$	18.8871	0.619664	0.309832	−	0.950791i	$-0.399727\pi$
$929$	18.8871	0.619664	0.309832	+	0.950791i	$0.399727\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−11.2090	−0.366967
$934$	0	0
$935$	−43.9106	−1.43603
$936$	0	0
$937$	40.5424	1.32446	0.662232	−	0.749299i	$-0.269609\pi$
$937$	40.5424	1.32446	0.662232	+	0.749299i	$0.269609\pi$
$938$	0	0
$939$	12.0806	0.394234
$940$	0	0
$941$	21.3763	0.696849	0.348424	−	0.937337i	$-0.386717\pi$
$941$	21.3763	0.696849	0.348424	+	0.937337i	$0.386717\pi$
$942$	0	0
$943$	−6.27241	−0.204258
$944$	0	0
$945$	79.8828	2.59859
$946$	0	0
$947$	−57.0948	−1.85533	−0.927666	−	0.373411i	$-0.878188\pi$
$947$	−57.0948	−1.85533	−0.927666	+	0.373411i	$0.878188\pi$
$948$	0	0
$949$	5.24703	0.170326
$950$	0	0
$951$	21.7836	0.706381
$952$	0	0
$953$	33.3049	1.07885	0.539426	−	0.842033i	$-0.318641\pi$
$953$	33.3049	1.07885	0.539426	+	0.842033i	$0.318641\pi$
$954$	0	0
$955$	−34.8048	−1.12626
$956$	0	0
$957$	48.4662	1.56669
$958$	0	0
$959$	−1.69308	−0.0546725
$960$	0	0
$961$	8.25586	0.266318
$962$	0	0
$963$	18.0657	0.582160
$964$	0	0
$965$	35.7595	1.15114
$966$	0	0
$967$	−22.7537	−0.731709	−0.365855	−	0.930672i	$-0.619223\pi$
$967$	−22.7537	−0.731709	−0.365855	+	0.930672i	$0.619223\pi$
$968$	0	0
$969$	−18.1941	−0.584480
$970$	0	0
$971$	60.3702	1.93737	0.968686	−	0.248290i	$-0.0798686\pi$
$971$	60.3702	1.93737	0.968686	+	0.248290i	$0.0798686\pi$
$972$	0	0
$973$	9.15281	0.293425
$974$	0	0
$975$	147.223	4.71490
$976$	0	0
$977$	−17.5460	−0.561347	−0.280674	−	0.959803i	$-0.590558\pi$
$977$	−17.5460	−0.561347	−0.280674	+	0.959803i	$0.590558\pi$
$978$	0	0
$979$	−24.9112	−0.796166
$980$	0	0
$981$	−14.6758	−0.468563
$982$	0	0
$983$	58.6978	1.87217	0.936085	−	0.351774i	$-0.114421\pi$
$983$	58.6978	1.87217	0.936085	+	0.351774i	$0.114421\pi$
$984$	0	0
$985$	61.9364	1.97346
$986$	0	0
$987$	12.1721	0.387442
$988$	0	0
$989$	−7.48352	−0.237962
$990$	0	0
$991$	−22.9757	−0.729848	−0.364924	−	0.931037i	$-0.618905\pi$
$991$	−22.9757	−0.729848	−0.364924	+	0.931037i	$0.618905\pi$
$992$	0	0
$993$	33.8884	1.07542
$994$	0	0
$995$	−5.00731	−0.158742
$996$	0	0
$997$	−49.5706	−1.56992	−0.784958	−	0.619549i	$-0.787315\pi$
$997$	−49.5706	−1.56992	−0.784958	+	0.619549i	$0.787315\pi$
$998$	0	0
$999$	48.2514	1.52661

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.cj.1.7		7
4.3	odd	2		8512.2.a.cg.1.1		7
8.3	odd	2		4256.2.a.r.1.7	yes	7
8.5	even	2		4256.2.a.o.1.1	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4256.2.a.o.1.1	✓	7	8.5	even	2
4256.2.a.r.1.7	yes	7	8.3	odd	2
8512.2.a.cg.1.1		7	4.3	odd	2
8512.2.a.cj.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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+3.37843 q^{3} -4.36754 q^{5} -1.00000 q^{7} +8.41379 q^{9} +O(q^{10})\)	\(q+3.37843 q^{3} -4.36754 q^{5} -1.00000 q^{7} +8.41379 q^{9} -1.86688 q^{11} +3.09599 q^{13} -14.7554 q^{15} -5.38539 q^{17} +1.00000 q^{19} -3.37843 q^{21} +0.672482 q^{23} +14.0754 q^{25} +18.2901 q^{27} -7.68437 q^{29} +6.26545 q^{31} -6.30711 q^{33} +4.36754 q^{35} +2.63811 q^{37} +10.4596 q^{39} -9.32726 q^{41} -11.1282 q^{43} -36.7476 q^{45} -3.60289 q^{47} +1.00000 q^{49} -18.1941 q^{51} -5.04940 q^{53} +8.15365 q^{55} +3.37843 q^{57} -5.12440 q^{59} -4.42684 q^{61} -8.41379 q^{63} -13.5219 q^{65} +2.37810 q^{67} +2.27193 q^{69} +6.67728 q^{71} +1.69478 q^{73} +47.5527 q^{75} +1.86688 q^{77} -17.3813 q^{79} +36.5505 q^{81} -0.104382 q^{83} +23.5209 q^{85} -25.9611 q^{87} +13.3438 q^{89} -3.09599 q^{91} +21.1674 q^{93} -4.36754 q^{95} +6.00099 q^{97} -15.7075 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} - 5 q^{5} - 7 q^{7} + 12 q^{9}+O(q^{10}) \) 7 * q + 7 * q^3 - 5 * q^5 - 7 * q^7 + 12 * q^9	\( 7 q + 7 q^{3} - 5 q^{5} - 7 q^{7} + 12 q^{9} - 3 q^{11} - 12 q^{13} - 6 q^{15} - 16 q^{17} + 7 q^{19} - 7 q^{21} - 10 q^{23} + 28 q^{25} + 16 q^{27} - 7 q^{29} - 2 q^{31} + 5 q^{33} + 5 q^{35} - 7 q^{37} + 3 q^{41} - 15 q^{43} - 32 q^{45} + 5 q^{47} + 7 q^{49} - 13 q^{53} + 14 q^{55} + 7 q^{57} + 23 q^{59} - 33 q^{61} - 12 q^{63} - 4 q^{65} - 10 q^{67} - 10 q^{69} - 13 q^{71} + 18 q^{73} + 28 q^{75} + 3 q^{77} - 21 q^{79} + 11 q^{81} + 2 q^{83} - 14 q^{85} - 64 q^{87} - 17 q^{89} + 12 q^{91} - 26 q^{93} - 5 q^{95} + q^{97} - 55 q^{99}+O(q^{100}) \) 7 * q + 7 * q^3 - 5 * q^5 - 7 * q^7 + 12 * q^9 - 3 * q^11 - 12 * q^13 - 6 * q^15 - 16 * q^17 + 7 * q^19 - 7 * q^21 - 10 * q^23 + 28 * q^25 + 16 * q^27 - 7 * q^29 - 2 * q^31 + 5 * q^33 + 5 * q^35 - 7 * q^37 + 3 * q^41 - 15 * q^43 - 32 * q^45 + 5 * q^47 + 7 * q^49 - 13 * q^53 + 14 * q^55 + 7 * q^57 + 23 * q^59 - 33 * q^61 - 12 * q^63 - 4 * q^65 - 10 * q^67 - 10 * q^69 - 13 * q^71 + 18 * q^73 + 28 * q^75 + 3 * q^77 - 21 * q^79 + 11 * q^81 + 2 * q^83 - 14 * q^85 - 64 * q^87 - 17 * q^89 + 12 * q^91 - 26 * q^93 - 5 * q^95 + q^97 - 55 * q^99

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