LMFDB - Embedded newform 3087.2.c.c.3086.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3087,2,Mod(3086,3087)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3087, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3087.3086");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3087 = 3^{2} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3087.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$24.6498191040$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3086.11
Character	$\chi$	$=$	3087.3086
Dual form			3087.2.c.c.3086.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.88777i q^{2} -1.56366 q^{4} +1.89903 q^{5} -0.823703i q^{8} +O(q^{10})$	$q-1.88777i q^{2} -1.56366 q^{4} +1.89903 q^{5} -0.823703i q^{8} -3.58493i q^{10} +1.25838i q^{11} -6.80085i q^{13} -4.68228 q^{16} -7.12478 q^{17} +7.22610i q^{19} -2.96945 q^{20} +2.37553 q^{22} -3.66613i q^{23} -1.39367 q^{25} -12.8384 q^{26} -6.95237i q^{29} +4.28108i q^{31} +7.19165i q^{32} +13.4499i q^{34} -9.49868 q^{37} +13.6412 q^{38} -1.56424i q^{40} -2.61917 q^{41} +2.15791 q^{43} -1.96769i q^{44} -6.92080 q^{46} +3.68959 q^{47} +2.63093i q^{50} +10.6342i q^{52} +2.36985i q^{53} +2.38971i q^{55} -13.1245 q^{58} +4.94485 q^{59} -7.92225i q^{61} +8.08169 q^{62} +4.21160 q^{64} -12.9150i q^{65} -11.1413 q^{67} +11.1408 q^{68} -8.63318i q^{71} -9.20466i q^{73} +17.9313i q^{74} -11.2992i q^{76} -12.6363 q^{79} -8.89182 q^{80} +4.94438i q^{82} +15.5711 q^{83} -13.5302 q^{85} -4.07363i q^{86} +1.03653 q^{88} +1.57854 q^{89} +5.73259i q^{92} -6.96508i q^{94} +13.7226i q^{95} +1.36110i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 40 q^{16} + 64 q^{22} + 136 q^{25} + 32 q^{37} + 32 q^{43} + 32 q^{46} - 128 q^{58} + 32 q^{64} + 40 q^{67} + 8 q^{79} + 64 q^{85} + 16 q^{88}+O(q^{100}) $ 24 * q - 40 * q^16 + 64 * q^22 + 136 * q^25 + 32 * q^37 + 32 * q^43 + 32 * q^46 - 128 * q^58 + 32 * q^64 + 40 * q^67 + 8 * q^79 + 64 * q^85 + 16 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3087\mathbb{Z}\right)^\times$.

$n$	$344$	$2404$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$	− 1.88777i	− 1.33485i	−0.744676	−	0.667426i	$-0.767396\pi$
$2$	− 1.88777i	− 1.33485i	0.744676	−	0.667426i	$-0.232604\pi$
$3$	0	0
$3$	0	0
$4$	−1.56366	−0.781831
$5$	1.89903	0.849274	0.424637	−	0.905364i	$-0.360402\pi$
$5$	1.89903	0.849274	0.424637	+	0.905364i	$0.360402\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 0.823703i	− 0.291223i
$9$	0	0
$10$	− 3.58493i	− 1.13366i
$11$	1.25838i	0.379417i	0.981840	+	0.189708i	$0.0607542\pi$
$11$	1.25838i	0.379417i	−0.981840	+	0.189708i	$0.939246\pi$
$12$	0	0
$13$	− 6.80085i	− 1.88622i	−0.332487	−	0.943108i	$-0.607888\pi$
$13$	− 6.80085i	− 1.88622i	0.332487	−	0.943108i	$-0.392112\pi$
$14$	0	0
$15$	0	0
$16$	−4.68228	−1.17057
$17$	−7.12478	−1.72801	−0.864007	−	0.503480i	$-0.832053\pi$
$17$	−7.12478	−1.72801	−0.864007	+	0.503480i	$0.832053\pi$
$18$	0	0
$19$	7.22610i	1.65778i	0.559411	+	0.828890i	$0.311027\pi$
$19$	7.22610i	1.65778i	−0.559411	+	0.828890i	$0.688973\pi$
$20$	−2.96945	−0.663989
$21$	0	0
$22$	2.37553	0.506465
$23$	− 3.66613i	− 0.764441i	−0.924071	−	0.382221i	$-0.875159\pi$
$23$	− 3.66613i	− 0.764441i	0.924071	−	0.382221i	$-0.124841\pi$
$24$	0	0
$25$	−1.39367	−0.278734
$26$	−12.8384	−2.51782
$27$	0	0
$28$	0	0
$29$	− 6.95237i	− 1.29102i	−0.763750	−	0.645512i	$-0.776644\pi$
$29$	− 6.95237i	− 1.29102i	0.763750	−	0.645512i	$-0.223356\pi$
$30$	0	0
$31$	4.28108i	0.768905i	0.923145	+	0.384453i	$0.125610\pi$
$31$	4.28108i	0.768905i	−0.923145	+	0.384453i	$0.874390\pi$
$32$	7.19165i	1.27132i
$33$	0	0
$34$	13.4499i	2.30664i
$35$	0	0
$36$	0	0
$37$	−9.49868	−1.56157	−0.780787	−	0.624798i	$-0.785181\pi$
$37$	−9.49868	−1.56157	−0.780787	+	0.624798i	$0.785181\pi$
$38$	13.6412	2.21289
$39$	0	0
$40$	− 1.56424i	− 0.247328i
$41$	−2.61917	−0.409046	−0.204523	−	0.978862i	$-0.565564\pi$
$41$	−2.61917	−0.409046	−0.204523	+	0.978862i	$0.565564\pi$
$42$	0	0
$43$	2.15791	0.329078	0.164539	−	0.986371i	$-0.447386\pi$
$43$	2.15791	0.329078	0.164539	+	0.986371i	$0.447386\pi$
$44$	− 1.96769i	− 0.296640i
$45$	0	0
$46$	−6.92080	−1.02042
$47$	3.68959	0.538181	0.269091	−	0.963115i	$-0.413277\pi$
$47$	3.68959	0.538181	0.269091	+	0.963115i	$0.413277\pi$
$48$	0	0
$49$	0	0
$50$	2.63093i	0.372069i
$51$	0	0
$52$	10.6342i	1.47470i
$53$	2.36985i	0.325523i	0.986665	+	0.162762i	$0.0520402\pi$
$53$	2.36985i	0.325523i	−0.986665	+	0.162762i	$0.947960\pi$
$54$	0	0
$55$	2.38971i	0.322229i
$56$	0	0
$57$	0	0
$58$	−13.1245	−1.72333
$59$	4.94485	0.643764	0.321882	−	0.946780i	$-0.395685\pi$
$59$	4.94485	0.643764	0.321882	+	0.946780i	$0.395685\pi$
$60$	0	0
$61$	− 7.92225i	− 1.01434i	−0.861846	−	0.507170i	$-0.830692\pi$
$61$	− 7.92225i	− 1.01434i	0.861846	−	0.507170i	$-0.169308\pi$
$62$	8.08169	1.02638
$63$	0	0
$64$	4.21160	0.526450
$65$	− 12.9150i	− 1.60191i
$66$	0	0
$67$	−11.1413	−1.36112	−0.680562	−	0.732691i	$-0.738264\pi$
$67$	−11.1413	−1.36112	−0.680562	+	0.732691i	$0.738264\pi$
$68$	11.1408	1.35102
$69$	0	0
$70$	0	0
$71$	− 8.63318i	− 1.02457i	−0.858816	−	0.512285i	$-0.828799\pi$
$71$	− 8.63318i	− 1.02457i	0.858816	−	0.512285i	$-0.171201\pi$
$72$	0	0
$73$	− 9.20466i	− 1.07732i	−0.842522	−	0.538662i	$-0.818930\pi$
$73$	− 9.20466i	− 1.07732i	0.842522	−	0.538662i	$-0.181070\pi$
$74$	17.9313i	2.08447i
$75$	0	0
$76$	− 11.2992i	− 1.29610i
$77$	0	0
$78$	0	0
$79$	−12.6363	−1.42170	−0.710848	−	0.703345i	$-0.751689\pi$
$79$	−12.6363	−1.42170	−0.710848	+	0.703345i	$0.751689\pi$
$80$	−8.89182	−0.994135
$81$	0	0
$82$	4.94438i	0.546016i
$83$	15.5711	1.70915	0.854575	−	0.519328i	$-0.173818\pi$
$83$	15.5711	1.70915	0.854575	+	0.519328i	$0.173818\pi$
$84$	0	0
$85$	−13.5302	−1.46756
$86$	− 4.07363i	− 0.439271i
$87$	0	0
$88$	1.03653	0.110495
$89$	1.57854	0.167325	0.0836626	−	0.996494i	$-0.473338\pi$
$89$	1.57854	0.167325	0.0836626	+	0.996494i	$0.473338\pi$
$90$	0	0
$91$	0	0
$92$	5.73259i	0.597664i
$93$	0	0
$94$	− 6.96508i	− 0.718393i
$95$	13.7226i	1.40791i
$96$	0	0
$97$	1.36110i	0.138198i	0.997610	+	0.0690992i	$0.0220125\pi$
$97$	1.36110i	0.138198i	−0.997610	+	0.0690992i	$0.977987\pi$
$98$	0	0
$99$	0	0
$100$	2.17923	0.217923
$101$	−3.73507	−0.371654	−0.185827	−	0.982582i	$-0.559496\pi$
$101$	−3.73507	−0.371654	−0.185827	+	0.982582i	$0.559496\pi$
$102$	0	0
$103$	7.59882i	0.748734i	0.927281	+	0.374367i	$0.122140\pi$
$103$	7.59882i	0.748734i	−0.927281	+	0.374367i	$0.877860\pi$
$104$	−5.60188	−0.549309
$105$	0	0
$106$	4.47372	0.434526
$107$	− 14.8163i	− 1.43235i	−0.697922	−	0.716174i	$-0.745892\pi$
$107$	− 14.8163i	− 1.43235i	0.697922	−	0.716174i	$-0.254108\pi$
$108$	0	0
$109$	4.64221	0.444643	0.222321	−	0.974973i	$-0.428637\pi$
$109$	4.64221	0.444643	0.222321	+	0.974973i	$0.428637\pi$
$110$	4.51122	0.430128
$111$	0	0
$112$	0	0
$113$	− 0.0919008i	− 0.00864530i	−0.999991	−	0.00432265i	$-0.998624\pi$
$113$	− 0.0919008i	− 0.00864530i	0.999991	−	0.00432265i	$-0.00137595\pi$
$114$	0	0
$115$	− 6.96211i	− 0.649220i
$116$	10.8712i	1.00936i
$117$	0	0
$118$	− 9.33472i	− 0.859330i
$119$	0	0
$120$	0	0
$121$	9.41647	0.856043
$122$	−14.9554	−1.35399
$123$	0	0
$124$	− 6.69417i	− 0.601154i
$125$	−12.1418	−1.08600
$126$	0	0
$127$	−3.86495	−0.342959	−0.171480	−	0.985188i	$-0.554855\pi$
$127$	−3.86495	−0.342959	−0.171480	+	0.985188i	$0.554855\pi$
$128$	6.43280i	0.568584i
$129$	0	0
$130$	−24.3806	−2.13832
$131$	−17.2936	−1.51095	−0.755476	−	0.655177i	$-0.772594\pi$
$131$	−17.2936	−1.51095	−0.755476	+	0.655177i	$0.772594\pi$
$132$	0	0
$133$	0	0
$134$	21.0321i	1.81690i
$135$	0	0
$136$	5.86870i	0.503237i
$137$	− 5.32978i	− 0.455354i	−0.973737	−	0.227677i	$-0.926887\pi$
$137$	− 5.32978i	− 0.455354i	0.973737	−	0.227677i	$-0.0731131\pi$
$138$	0	0
$139$	9.13506i	0.774826i	0.921906	+	0.387413i	$0.126631\pi$
$139$	9.13506i	0.774826i	−0.921906	+	0.387413i	$0.873369\pi$
$140$	0	0
$141$	0	0
$142$	−16.2974	−1.36765
$143$	8.55807	0.715662
$144$	0	0
$145$	− 13.2028i	− 1.09643i
$146$	−17.3763	−1.43807
$147$	0	0
$148$	14.8527	1.22089
$149$	− 11.1193i	− 0.910932i	−0.890253	−	0.455466i	$-0.849473\pi$
$149$	− 11.1193i	− 0.910932i	0.890253	−	0.455466i	$-0.150527\pi$
$150$	0	0
$151$	−9.13983	−0.743789	−0.371894	−	0.928275i	$-0.621292\pi$
$151$	−9.13983	−0.743789	−0.371894	+	0.928275i	$0.621292\pi$
$152$	5.95215	0.482783
$153$	0	0
$154$	0	0
$155$	8.12992i	0.653011i
$156$	0	0
$157$	10.7518i	0.858089i	0.903283	+	0.429045i	$0.141150\pi$
$157$	10.7518i	0.858089i	−0.903283	+	0.429045i	$0.858850\pi$
$158$	23.8544i	1.89776i
$159$	0	0
$160$	13.6572i	1.07970i
$161$	0	0
$162$	0	0
$163$	8.96253	0.702000	0.351000	−	0.936376i	$-0.385842\pi$
$163$	8.96253	0.702000	0.351000	+	0.936376i	$0.385842\pi$
$164$	4.09550	0.319805
$165$	0	0
$166$	− 29.3946i	− 2.28146i
$167$	12.3488	0.955577	0.477789	−	0.878475i	$-0.341438\pi$
$167$	12.3488	0.955577	0.477789	+	0.878475i	$0.341438\pi$
$168$	0	0
$169$	−33.2515	−2.55781
$170$	25.5419i	1.95897i
$171$	0	0
$172$	−3.37424	−0.257284
$173$	14.4748	1.10050	0.550250	−	0.835000i	$-0.314532\pi$
$173$	14.4748	1.10050	0.550250	+	0.835000i	$0.314532\pi$
$174$	0	0
$175$	0	0
$176$	− 5.89210i	− 0.444134i
$177$	0	0
$178$	− 2.97992i	− 0.223354i
$179$	− 7.62946i	− 0.570253i	−0.958490	−	0.285126i	$-0.907964\pi$
$179$	− 7.62946i	− 0.570253i	0.958490	−	0.285126i	$-0.0920356\pi$
$180$	0	0
$181$	− 21.5295i	− 1.60027i	−0.599817	−	0.800137i	$-0.704760\pi$
$181$	− 21.5295i	− 1.60027i	0.599817	−	0.800137i	$-0.295240\pi$
$182$	0	0
$183$	0	0
$184$	−3.01980	−0.222623
$185$	−18.0383	−1.32620
$186$	0	0
$187$	− 8.96570i	− 0.655637i
$188$	−5.76927	−0.420767
$189$	0	0
$190$	25.9051	1.87935
$191$	18.4581i	1.33558i	0.744350	+	0.667790i	$0.232760\pi$
$191$	18.4581i	1.33558i	−0.744350	+	0.667790i	$0.767240\pi$
$192$	0	0
$193$	−0.487234	−0.0350719	−0.0175359	−	0.999846i	$-0.505582\pi$
$193$	−0.487234	−0.0350719	−0.0175359	+	0.999846i	$0.505582\pi$
$194$	2.56943	0.184475
$195$	0	0
$196$	0	0
$197$	10.6599i	0.759490i	0.925091	+	0.379745i	$0.123988\pi$
$197$	10.6599i	0.759490i	−0.925091	+	0.379745i	$0.876012\pi$
$198$	0	0
$199$	− 7.65135i	− 0.542390i	−0.962524	−	0.271195i	$-0.912581\pi$
$199$	− 7.65135i	− 0.542390i	0.962524	−	0.271195i	$-0.0874187\pi$
$200$	1.14797i	0.0811737i
$201$	0	0
$202$	7.05095i	0.496103i
$203$	0	0
$204$	0	0
$205$	−4.97389	−0.347392
$206$	14.3448	0.999449
$207$	0	0
$208$	31.8435i	2.20795i
$209$	−9.09319	−0.628989
$210$	0	0
$211$	−21.4157	−1.47432	−0.737159	−	0.675719i	$-0.763833\pi$
$211$	−21.4157	−1.47432	−0.737159	+	0.675719i	$0.763833\pi$
$212$	− 3.70564i	− 0.254504i
$213$	0	0
$214$	−27.9698	−1.91197
$215$	4.09794	0.279478
$216$	0	0
$217$	0	0
$218$	− 8.76341i	− 0.593533i
$219$	0	0
$220$	− 3.73670i	− 0.251928i
$221$	48.4546i	3.25941i
$222$	0	0
$223$	− 6.16876i	− 0.413090i	−0.978437	−	0.206545i	$-0.933778\pi$
$223$	− 6.16876i	− 0.413090i	0.978437	−	0.206545i	$-0.0662221\pi$
$224$	0	0
$225$	0	0
$226$	−0.173487	−0.0115402
$227$	28.9777	1.92332	0.961659	−	0.274249i	$-0.0884294\pi$
$227$	28.9777	1.92332	0.961659	+	0.274249i	$0.0884294\pi$
$228$	0	0
$229$	6.25018i	0.413023i	0.978444	+	0.206512i	$0.0662111\pi$
$229$	6.25018i	0.413023i	−0.978444	+	0.206512i	$0.933789\pi$
$230$	−13.1428	−0.866613
$231$	0	0
$232$	−5.72669	−0.375976
$233$	− 4.85343i	− 0.317958i	−0.987282	−	0.158979i	$-0.949180\pi$
$233$	− 4.85343i	− 0.317958i	0.987282	−	0.158979i	$-0.0508203\pi$
$234$	0	0
$235$	7.00665	0.457063
$236$	−7.73207	−0.503315
$237$	0	0
$238$	0	0
$239$	− 14.9406i	− 0.966427i	−0.875502	−	0.483214i	$-0.839469\pi$
$239$	− 14.9406i	− 0.966427i	0.875502	−	0.483214i	$-0.160531\pi$
$240$	0	0
$241$	10.3402i	0.666071i	0.942914	+	0.333035i	$0.108073\pi$
$241$	10.3402i	0.666071i	−0.942914	+	0.333035i	$0.891927\pi$
$242$	− 17.7761i	− 1.14269i
$243$	0	0
$244$	12.3877i	0.793043i
$245$	0	0
$246$	0	0
$247$	49.1436	3.12693
$248$	3.52634	0.223923
$249$	0	0
$250$	22.9209i	1.44964i
$251$	15.0408	0.949366	0.474683	−	0.880157i	$-0.342563\pi$
$251$	15.0408	0.949366	0.474683	+	0.880157i	$0.342563\pi$
$252$	0	0
$253$	4.61340	0.290042
$254$	7.29613i	0.457800i
$255$	0	0
$256$	20.5668	1.28543
$257$	−17.1988	−1.07283	−0.536415	−	0.843954i	$-0.680222\pi$
$257$	−17.1988	−1.07283	−0.536415	+	0.843954i	$0.680222\pi$
$258$	0	0
$259$	0	0
$260$	20.1948i	1.25243i
$261$	0	0
$262$	32.6463i	2.01690i
$263$	− 10.0939i	− 0.622419i	−0.950341	−	0.311210i	$-0.899266\pi$
$263$	− 10.0939i	− 0.622419i	0.950341	−	0.311210i	$-0.100734\pi$
$264$	0	0
$265$	4.50042i	0.276459i
$266$	0	0
$267$	0	0
$268$	17.4212	1.06417
$269$	22.5855	1.37706	0.688530	−	0.725207i	$-0.258256\pi$
$269$	22.5855	1.37706	0.688530	+	0.725207i	$0.258256\pi$
$270$	0	0
$271$	9.66164i	0.586903i	0.955974	+	0.293451i	$0.0948038\pi$
$271$	9.66164i	0.586903i	−0.955974	+	0.293451i	$0.905196\pi$
$272$	33.3603	2.02276
$273$	0	0
$274$	−10.0614	−0.607831
$275$	− 1.75377i	− 0.105756i
$276$	0	0
$277$	−4.80351	−0.288615	−0.144307	−	0.989533i	$-0.546095\pi$
$277$	−4.80351	−0.288615	−0.144307	+	0.989533i	$0.546095\pi$
$278$	17.2449	1.03428
$279$	0	0
$280$	0	0
$281$	2.06852i	0.123398i	0.998095	+	0.0616988i	$0.0196518\pi$
$281$	2.06852i	0.123398i	−0.998095	+	0.0616988i	$0.980348\pi$
$282$	0	0
$283$	− 1.45624i	− 0.0865646i	−0.999063	−	0.0432823i	$-0.986219\pi$
$283$	− 1.45624i	− 0.0865646i	0.999063	−	0.0432823i	$-0.0137815\pi$
$284$	13.4994i	0.801041i
$285$	0	0
$286$	− 16.1556i	− 0.955303i
$287$	0	0
$288$	0	0
$289$	33.7625	1.98603
$290$	−24.9238	−1.46358
$291$	0	0
$292$	14.3930i	0.842286i
$293$	24.2665	1.41766	0.708831	−	0.705379i	$-0.249223\pi$
$293$	24.2665	1.41766	0.708831	+	0.705379i	$0.249223\pi$
$294$	0	0
$295$	9.39043	0.546732
$296$	7.82409i	0.454766i
$297$	0	0
$298$	−20.9907	−1.21596
$299$	−24.9328	−1.44190
$300$	0	0
$301$	0	0
$302$	17.2539i	0.992848i
$303$	0	0
$304$	− 33.8346i	− 1.94055i
$305$	− 15.0446i	− 0.861452i
$306$	0	0
$307$	0.425249i	0.0242702i	0.999926	+	0.0121351i	$0.00386282\pi$
$307$	0.425249i	0.0242702i	−0.999926	+	0.0121351i	$0.996137\pi$
$308$	0	0
$309$	0	0
$310$	15.3474	0.871674
$311$	−8.54154	−0.484346	−0.242173	−	0.970233i	$-0.577860\pi$
$311$	−8.54154	−0.484346	−0.242173	+	0.970233i	$0.577860\pi$
$312$	0	0
$313$	1.24775i	0.0705271i	0.999378	+	0.0352636i	$0.0112271\pi$
$313$	1.24775i	0.0705271i	−0.999378	+	0.0352636i	$0.988773\pi$
$314$	20.2969	1.14542
$315$	0	0
$316$	19.7589	1.11153
$317$	15.7455i	0.884354i	0.896928	+	0.442177i	$0.145794\pi$
$317$	15.7455i	0.884354i	−0.896928	+	0.442177i	$0.854206\pi$
$318$	0	0
$319$	8.74875	0.489836
$320$	7.99797	0.447100
$321$	0	0
$322$	0	0
$323$	− 51.4844i	− 2.86467i
$324$	0	0
$325$	9.47814i	0.525753i
$326$	− 16.9192i	− 0.937066i
$327$	0	0
$328$	2.15742i	0.119123i
$329$	0	0
$330$	0	0
$331$	−5.54477	−0.304768	−0.152384	−	0.988321i	$-0.548695\pi$
$331$	−5.54477	−0.304768	−0.152384	+	0.988321i	$0.548695\pi$
$332$	−24.3479	−1.33627
$333$	0	0
$334$	− 23.3116i	− 1.27555i
$335$	−21.1577	−1.15597
$336$	0	0
$337$	22.4442	1.22261	0.611307	−	0.791394i	$-0.290644\pi$
$337$	22.4442	1.22261	0.611307	+	0.791394i	$0.290644\pi$
$338$	62.7711i	3.41430i
$339$	0	0
$340$	21.1567	1.14738
$341$	−5.38724	−0.291735
$342$	0	0
$343$	0	0
$344$	− 1.77748i	− 0.0958351i
$345$	0	0
$346$	− 27.3251i	− 1.46901i
$347$	7.94305i	0.426405i	0.977008	+	0.213203i	$0.0683895\pi$
$347$	7.94305i	0.426405i	−0.977008	+	0.213203i	$0.931611\pi$
$348$	0	0
$349$	− 16.3404i	− 0.874679i	−0.899296	−	0.437340i	$-0.855921\pi$
$349$	− 16.3404i	− 0.874679i	0.899296	−	0.437340i	$-0.144079\pi$
$350$	0	0
$351$	0	0
$352$	−9.04985	−0.482359
$353$	−26.2057	−1.39479	−0.697394	−	0.716688i	$-0.745657\pi$
$353$	−26.2057	−1.39479	−0.697394	+	0.716688i	$0.745657\pi$
$354$	0	0
$355$	− 16.3947i	− 0.870140i
$356$	−2.46831	−0.130820
$357$	0	0
$358$	−14.4026	−0.761203
$359$	− 19.3618i	− 1.02188i	−0.859617	−	0.510939i	$-0.829298\pi$
$359$	− 19.3618i	− 1.02188i	0.859617	−	0.510939i	$-0.170702\pi$
$360$	0	0
$361$	−33.2165	−1.74824
$362$	−40.6427	−2.13613
$363$	0	0
$364$	0	0
$365$	− 17.4800i	− 0.914943i
$366$	0	0
$367$	− 4.96867i	− 0.259363i	−0.991556	−	0.129681i	$-0.958605\pi$
$367$	− 4.96867i	− 0.259363i	0.991556	−	0.129681i	$-0.0413954\pi$
$368$	17.1659i	0.894833i
$369$	0	0
$370$	34.0521i	1.77029i
$371$	0	0
$372$	0	0
$373$	10.7580	0.557026	0.278513	−	0.960432i	$-0.410158\pi$
$373$	10.7580	0.557026	0.278513	+	0.960432i	$0.410158\pi$
$374$	−16.9252	−0.875179
$375$	0	0
$376$	− 3.03912i	− 0.156731i
$377$	−47.2820	−2.43515
$378$	0	0
$379$	−16.9101	−0.868613	−0.434307	−	0.900765i	$-0.643007\pi$
$379$	−16.9101	−0.868613	−0.434307	+	0.900765i	$0.643007\pi$
$380$	− 21.4575i	− 1.10075i
$381$	0	0
$382$	34.8446	1.78280
$383$	22.0195	1.12514	0.562571	−	0.826749i	$-0.309812\pi$
$383$	22.0195	1.12514	0.562571	+	0.826749i	$0.309812\pi$
$384$	0	0
$385$	0	0
$386$	0.919784i	0.0468158i
$387$	0	0
$388$	− 2.12830i	− 0.108048i
$389$	− 29.0082i	− 1.47077i	−0.677648	−	0.735386i	$-0.737001\pi$
$389$	− 29.0082i	− 1.47077i	0.677648	−	0.735386i	$-0.262999\pi$
$390$	0	0
$391$	26.1204i	1.32096i
$392$	0	0
$393$	0	0
$394$	20.1235	1.01381
$395$	−23.9968	−1.20741
$396$	0	0
$397$	− 14.9862i	− 0.752134i	−0.926592	−	0.376067i	$-0.877276\pi$
$397$	− 14.9862i	− 0.752134i	0.926592	−	0.376067i	$-0.122724\pi$
$398$	−14.4440	−0.724010
$399$	0	0
$400$	6.52556	0.326278
$401$	− 5.84746i	− 0.292008i	−0.989284	−	0.146004i	$-0.953359\pi$
$401$	− 5.84746i	− 0.292008i	0.989284	−	0.146004i	$-0.0466412\pi$
$402$	0	0
$403$	29.1150	1.45032
$404$	5.84040	0.290571
$405$	0	0
$406$	0	0
$407$	− 11.9530i	− 0.592487i
$408$	0	0
$409$	17.2266i	0.851800i	0.904770	+	0.425900i	$0.140042\pi$
$409$	17.2266i	0.851800i	−0.904770	+	0.425900i	$0.859958\pi$
$410$	9.38955i	0.463717i
$411$	0	0
$412$	− 11.8820i	− 0.585383i
$413$	0	0
$414$	0	0
$415$	29.5700	1.45154
$416$	48.9094	2.39798
$417$	0	0
$418$	17.1658i	0.839608i
$419$	−11.0644	−0.540531	−0.270266	−	0.962786i	$-0.587111\pi$
$419$	−11.0644	−0.540531	−0.270266	+	0.962786i	$0.587111\pi$
$420$	0	0
$421$	−19.3493	−0.943028	−0.471514	−	0.881858i	$-0.656292\pi$
$421$	−19.3493	−0.943028	−0.471514	+	0.881858i	$0.656292\pi$
$422$	40.4279i	1.96800i
$423$	0	0
$424$	1.95205	0.0947999
$425$	9.92960	0.481656
$426$	0	0
$427$	0	0
$428$	23.1677i	1.11985i
$429$	0	0
$430$	− 7.73596i	− 0.373061i
$431$	− 11.8075i	− 0.568747i	−0.958714	−	0.284374i	$-0.908214\pi$
$431$	− 11.8075i	− 0.568747i	0.958714	−	0.284374i	$-0.0917856\pi$
$432$	0	0
$433$	32.1847i	1.54670i	0.633979	+	0.773350i	$0.281420\pi$
$433$	32.1847i	1.54670i	−0.633979	+	0.773350i	$0.718580\pi$
$434$	0	0
$435$	0	0
$436$	−7.25885	−0.347636
$437$	26.4918	1.26728
$438$	0	0
$439$	24.2718i	1.15843i	0.815176	+	0.579214i	$0.196640\pi$
$439$	24.2718i	1.15843i	−0.815176	+	0.579214i	$0.803360\pi$
$440$	1.96841	0.0938403
$441$	0	0
$442$	91.4709	4.35083
$443$	− 1.20946i	− 0.0574632i	−0.999587	−	0.0287316i	$-0.990853\pi$
$443$	− 1.20946i	− 0.0574632i	0.999587	−	0.0287316i	$-0.00914681\pi$
$444$	0	0
$445$	2.99771	0.142105
$446$	−11.6452	−0.551415
$447$	0	0
$448$	0	0
$449$	11.1441i	0.525922i	0.964806	+	0.262961i	$0.0846991\pi$
$449$	11.1441i	0.525922i	−0.964806	+	0.262961i	$0.915301\pi$
$450$	0	0
$451$	− 3.29592i	− 0.155199i
$452$	0.143702i	0.00675917i
$453$	0	0
$454$	− 54.7031i	− 2.56734i
$455$	0	0
$456$	0	0
$457$	−5.02386	−0.235006	−0.117503	−	0.993073i	$-0.537489\pi$
$457$	−5.02386	−0.235006	−0.117503	+	0.993073i	$0.537489\pi$
$458$	11.7989	0.551325
$459$	0	0
$460$	10.8864i	0.507581i
$461$	−3.37908	−0.157379	−0.0786897	−	0.996899i	$-0.525074\pi$
$461$	−3.37908	−0.157379	−0.0786897	+	0.996899i	$0.525074\pi$
$462$	0	0
$463$	−10.7132	−0.497884	−0.248942	−	0.968518i	$-0.580083\pi$
$463$	−10.7132	−0.497884	−0.248942	+	0.968518i	$0.580083\pi$
$464$	32.5530i	1.51123i
$465$	0	0
$466$	−9.16214	−0.424428
$467$	7.94750	0.367767	0.183883	−	0.982948i	$-0.441133\pi$
$467$	7.94750	0.367767	0.183883	+	0.982948i	$0.441133\pi$
$468$	0	0
$469$	0	0
$470$	− 13.2269i	− 0.610112i
$471$	0	0
$472$	− 4.07308i	− 0.187479i
$473$	2.71548i	0.124858i
$474$	0	0
$475$	− 10.0708i	− 0.462080i
$476$	0	0
$477$	0	0
$478$	−28.2044	−1.29004
$479$	23.3621	1.06744	0.533720	−	0.845661i	$-0.320794\pi$
$479$	23.3621	1.06744	0.533720	+	0.845661i	$0.320794\pi$
$480$	0	0
$481$	64.5991i	2.94546i
$482$	19.5199	0.889106
$483$	0	0
$484$	−14.7242	−0.669281
$485$	2.58477i	0.117368i
$486$	0	0
$487$	23.3593	1.05851	0.529255	−	0.848463i	$-0.322471\pi$
$487$	23.3593	1.05851	0.529255	+	0.848463i	$0.322471\pi$
$488$	−6.52558	−0.295399
$489$	0	0
$490$	0	0
$491$	− 12.3820i	− 0.558792i	−0.960176	−	0.279396i	$-0.909866\pi$
$491$	− 12.3820i	− 0.558792i	0.960176	−	0.279396i	$-0.0901342\pi$
$492$	0	0
$493$	49.5342i	2.23091i
$494$	− 92.7716i	− 4.17399i
$495$	0	0
$496$	− 20.0453i	− 0.900058i
$497$	0	0
$498$	0	0
$499$	4.81473	0.215537	0.107769	−	0.994176i	$-0.465629\pi$
$499$	4.81473	0.215537	0.107769	+	0.994176i	$0.465629\pi$
$500$	18.9857	0.849065
$501$	0	0
$502$	− 28.3935i	− 1.26726i
$503$	−3.79387	−0.169160	−0.0845801	−	0.996417i	$-0.526955\pi$
$503$	−3.79387	−0.169160	−0.0845801	+	0.996417i	$0.526955\pi$
$504$	0	0
$505$	−7.09303	−0.315636
$506$	− 8.70901i	− 0.387163i
$507$	0	0
$508$	6.04348	0.268136
$509$	20.8868	0.925793	0.462896	−	0.886412i	$-0.346810\pi$
$509$	20.8868	0.925793	0.462896	+	0.886412i	$0.346810\pi$
$510$	0	0
$511$	0	0
$512$	− 25.9598i	− 1.14727i
$513$	0	0
$514$	32.4673i	1.43207i
$515$	14.4304i	0.635880i
$516$	0	0
$517$	4.64291i	0.204195i
$518$	0	0
$519$	0	0
$520$	−10.6382	−0.466514
$521$	−31.1967	−1.36675	−0.683377	−	0.730066i	$-0.739489\pi$
$521$	−31.1967	−1.36675	−0.683377	+	0.730066i	$0.739489\pi$
$522$	0	0
$523$	− 29.6393i	− 1.29604i	−0.761624	−	0.648019i	$-0.775598\pi$
$523$	− 29.6393i	− 1.29604i	0.761624	−	0.648019i	$-0.224402\pi$
$524$	27.0414	1.18131
$525$	0	0
$526$	−19.0550	−0.830838
$527$	− 30.5018i	− 1.32868i
$528$	0	0
$529$	9.55948	0.415630
$530$	8.49574	0.369031
$531$	0	0
$532$	0	0
$533$	17.8126i	0.771548i
$534$	0	0
$535$	− 28.1367i	− 1.21646i
$536$	9.17709i	0.396390i
$537$	0	0
$538$	− 42.6361i	− 1.83817i
$539$	0	0
$540$	0	0
$541$	35.3937	1.52170	0.760848	−	0.648930i	$-0.224783\pi$
$541$	35.3937	1.52170	0.760848	+	0.648930i	$0.224783\pi$
$542$	18.2389	0.783428
$543$	0	0
$544$	− 51.2390i	− 2.19685i
$545$	8.81571	0.377624
$546$	0	0
$547$	19.4407	0.831223	0.415611	−	0.909542i	$-0.363568\pi$
$547$	19.4407	0.831223	0.415611	+	0.909542i	$0.363568\pi$
$548$	8.33399i	0.356010i
$549$	0	0
$550$	−3.31071	−0.141169
$551$	50.2385	2.14023
$552$	0	0
$553$	0	0
$554$	9.06790i	0.385258i
$555$	0	0
$556$	− 14.2842i	− 0.605783i
$557$	− 38.6477i	− 1.63756i	−0.574110	−	0.818779i	$-0.694652\pi$
$557$	− 38.6477i	− 1.63756i	0.574110	−	0.818779i	$-0.305348\pi$
$558$	0	0
$559$	− 14.6756i	− 0.620713i
$560$	0	0
$561$	0	0
$562$	3.90488	0.164718
$563$	−27.0656	−1.14068	−0.570340	−	0.821409i	$-0.693188\pi$
$563$	−27.0656	−1.14068	−0.570340	+	0.821409i	$0.693188\pi$
$564$	0	0
$565$	− 0.174523i	− 0.00734223i
$566$	−2.74905	−0.115551
$567$	0	0
$568$	−7.11117	−0.298378
$569$	− 29.4673i	− 1.23533i	−0.786440	−	0.617666i	$-0.788078\pi$
$569$	− 29.4673i	− 1.23533i	0.786440	−	0.617666i	$-0.211922\pi$
$570$	0	0
$571$	−31.9378	−1.33655	−0.668277	−	0.743913i	$-0.732968\pi$
$571$	−31.9378	−1.33655	−0.668277	+	0.743913i	$0.732968\pi$
$572$	−13.3819	−0.559527
$573$	0	0
$574$	0	0
$575$	5.10938i	0.213076i
$576$	0	0
$577$	− 46.5345i	− 1.93726i	−0.248516	−	0.968628i	$-0.579943\pi$
$577$	− 46.5345i	− 1.93726i	0.248516	−	0.968628i	$-0.420057\pi$
$578$	− 63.7358i	− 2.65106i
$579$	0	0
$580$	20.6447i	0.857225i
$581$	0	0
$582$	0	0
$583$	−2.98217	−0.123509
$584$	−7.58191	−0.313741
$585$	0	0
$586$	− 45.8094i	− 1.89237i
$587$	−16.1578	−0.666903	−0.333451	−	0.942767i	$-0.608213\pi$
$587$	−16.1578	−0.666903	−0.333451	+	0.942767i	$0.608213\pi$
$588$	0	0
$589$	−30.9355	−1.27468
$590$	− 17.7269i	− 0.729807i
$591$	0	0
$592$	44.4755	1.82793
$593$	−9.71629	−0.399000	−0.199500	−	0.979898i	$-0.563932\pi$
$593$	−9.71629	−0.399000	−0.199500	+	0.979898i	$0.563932\pi$
$594$	0	0
$595$	0	0
$596$	17.3869i	0.712195i
$597$	0	0
$598$	47.0673i	1.92473i
$599$	20.8405i	0.851518i	0.904837	+	0.425759i	$0.139993\pi$
$599$	20.8405i	0.851518i	−0.904837	+	0.425759i	$0.860007\pi$
$600$	0	0
$601$	41.9456i	1.71100i	0.517805	+	0.855499i	$0.326749\pi$
$601$	41.9456i	1.71100i	−0.517805	+	0.855499i	$0.673251\pi$
$602$	0	0
$603$	0	0
$604$	14.2916	0.581517
$605$	17.8822	0.727015
$606$	0	0
$607$	− 24.1864i	− 0.981696i	−0.871245	−	0.490848i	$-0.836687\pi$
$607$	− 24.1864i	− 0.981696i	0.871245	−	0.490848i	$-0.163313\pi$
$608$	−51.9676	−2.10756
$609$	0	0
$610$	−28.4007	−1.14991
$611$	− 25.0923i	− 1.01513i
$612$	0	0
$613$	35.6461	1.43973	0.719867	−	0.694112i	$-0.244203\pi$
$613$	35.6461	1.43973	0.719867	+	0.694112i	$0.244203\pi$
$614$	0.802770	0.0323972
$615$	0	0
$616$	0	0
$617$	15.4560i	0.622234i	0.950372	+	0.311117i	$0.100703\pi$
$617$	15.4560i	0.622234i	−0.950372	+	0.311117i	$0.899297\pi$
$618$	0	0
$619$	− 2.14653i	− 0.0862763i	−0.999069	−	0.0431382i	$-0.986264\pi$
$619$	− 2.14653i	− 0.0862763i	0.999069	−	0.0431382i	$-0.0137356\pi$
$620$	− 12.7125i	− 0.510545i
$621$	0	0
$622$	16.1244i	0.646531i
$623$	0	0
$624$	0	0
$625$	−16.0893	−0.643573
$626$	2.35546	0.0941433
$627$	0	0
$628$	− 16.8122i	− 0.670881i
$629$	67.6760	2.69842
$630$	0	0
$631$	36.5641	1.45559	0.727797	−	0.685793i	$-0.240544\pi$
$631$	36.5641	1.45559	0.727797	+	0.685793i	$0.240544\pi$
$632$	10.4086i	0.414031i
$633$	0	0
$634$	29.7238	1.18048
$635$	−7.33968	−0.291266
$636$	0	0
$637$	0	0
$638$	− 16.5156i	− 0.653859i
$639$	0	0
$640$	12.2161i	0.482884i
$641$	− 12.9826i	− 0.512783i	−0.966573	−	0.256391i	$-0.917466\pi$
$641$	− 12.9826i	− 0.512783i	0.966573	−	0.256391i	$-0.0825336\pi$
$642$	0	0
$643$	34.3758i	1.35565i	0.735223	+	0.677825i	$0.237077\pi$
$643$	34.3758i	1.35565i	−0.735223	+	0.677825i	$0.762923\pi$
$644$	0	0
$645$	0	0
$646$	−97.1905	−3.82391
$647$	−15.9679	−0.627765	−0.313882	−	0.949462i	$-0.601630\pi$
$647$	−15.9679	−0.627765	−0.313882	+	0.949462i	$0.601630\pi$
$648$	0	0
$649$	6.22251i	0.244255i
$650$	17.8925	0.701803
$651$	0	0
$652$	−14.0144	−0.548845
$653$	− 26.8269i	− 1.04982i	−0.851159	−	0.524908i	$-0.824100\pi$
$653$	− 26.8269i	− 1.04982i	0.851159	−	0.524908i	$-0.175900\pi$
$654$	0	0
$655$	−32.8412	−1.28321
$656$	12.2637	0.478817
$657$	0	0
$658$	0	0
$659$	− 48.5701i	− 1.89202i	−0.324135	−	0.946011i	$-0.605073\pi$
$659$	− 48.5701i	− 1.89202i	0.324135	−	0.946011i	$-0.394927\pi$
$660$	0	0
$661$	8.33150i	0.324058i	0.986786	+	0.162029i	$0.0518038\pi$
$661$	8.33150i	0.324058i	−0.986786	+	0.162029i	$0.948196\pi$
$662$	10.4672i	0.406821i
$663$	0	0
$664$	− 12.8259i	− 0.497743i
$665$	0	0
$666$	0	0
$667$	−25.4883	−0.986912
$668$	−19.3093	−0.747100
$669$	0	0
$670$	39.9407i	1.54304i
$671$	9.96922	0.384857
$672$	0	0
$673$	−48.9372	−1.88639	−0.943195	−	0.332240i	$-0.892196\pi$
$673$	−48.9372	−1.88639	−0.943195	+	0.332240i	$0.892196\pi$
$674$	− 42.3694i	− 1.63201i
$675$	0	0
$676$	51.9942	1.99978
$677$	−14.5031	−0.557400	−0.278700	−	0.960378i	$-0.589904\pi$
$677$	−14.5031	−0.557400	−0.278700	+	0.960378i	$0.589904\pi$
$678$	0	0
$679$	0	0
$680$	11.1449i	0.427386i
$681$	0	0
$682$	10.1699i	0.389424i
$683$	− 40.1584i	− 1.53662i	−0.640078	−	0.768310i	$-0.721098\pi$
$683$	− 40.1584i	− 1.53662i	0.640078	−	0.768310i	$-0.278902\pi$
$684$	0	0
$685$	− 10.1214i	− 0.386720i
$686$	0	0
$687$	0	0
$688$	−10.1040	−0.385209
$689$	16.1170	0.614008
$690$	0	0
$691$	− 2.14628i	− 0.0816483i	−0.999166	−	0.0408242i	$-0.987002\pi$
$691$	− 2.14628i	− 0.0816483i	0.999166	−	0.0408242i	$-0.0129983\pi$
$692$	−22.6338	−0.860406
$693$	0	0
$694$	14.9946	0.569188
$695$	17.3478i	0.658039i
$696$	0	0
$697$	18.6610	0.706837
$698$	−30.8468	−1.16757
$699$	0	0
$700$	0	0
$701$	39.4532i	1.49013i	0.666993	+	0.745064i	$0.267581\pi$
$701$	39.4532i	1.49013i	−0.666993	+	0.745064i	$0.732419\pi$
$702$	0	0
$703$	− 68.6384i	− 2.58875i
$704$	5.29980i	0.199744i
$705$	0	0
$706$	49.4702i	1.86184i
$707$	0	0
$708$	0	0
$709$	14.7476	0.553857	0.276928	−	0.960891i	$-0.410684\pi$
$709$	14.7476	0.553857	0.276928	+	0.960891i	$0.410684\pi$
$710$	−30.9494	−1.16151
$711$	0	0
$712$	− 1.30025i	− 0.0487289i
$713$	15.6950	0.587783
$714$	0	0
$715$	16.2521	0.607793
$716$	11.9299i	0.445842i
$717$	0	0
$718$	−36.5506	−1.36406
$719$	−31.8119	−1.18638	−0.593191	−	0.805062i	$-0.702132\pi$
$719$	−31.8119	−1.18638	−0.593191	+	0.805062i	$0.702132\pi$
$720$	0	0
$721$	0	0
$722$	62.7050i	2.33364i
$723$	0	0
$724$	33.6649i	1.25114i
$725$	9.68932i	0.359852i
$726$	0	0
$727$	− 46.1204i	− 1.71051i	−0.518208	−	0.855255i	$-0.673401\pi$
$727$	− 46.1204i	− 1.71051i	0.518208	−	0.855255i	$-0.326599\pi$
$728$	0	0
$729$	0	0
$730$	−32.9981	−1.22131
$731$	−15.3746	−0.568652
$732$	0	0
$733$	− 14.6480i	− 0.541035i	−0.962715	−	0.270517i	$-0.912805\pi$
$733$	− 14.6480i	− 0.541035i	0.962715	−	0.270517i	$-0.0871947\pi$
$734$	−9.37970	−0.346211
$735$	0	0
$736$	26.3656	0.971847
$737$	− 14.0200i	− 0.516433i
$738$	0	0
$739$	13.6707	0.502885	0.251442	−	0.967872i	$-0.419095\pi$
$739$	13.6707	0.502885	0.251442	+	0.967872i	$0.419095\pi$
$740$	28.2058	1.03687
$741$	0	0
$742$	0	0
$743$	35.9213i	1.31782i	0.752220	+	0.658912i	$0.228983\pi$
$743$	35.9213i	1.31782i	−0.752220	+	0.658912i	$0.771017\pi$
$744$	0	0
$745$	− 21.1160i	− 0.773630i
$746$	− 20.3085i	− 0.743548i
$747$	0	0
$748$	14.0193i	0.512598i
$749$	0	0
$750$	0	0
$751$	−41.1079	−1.50005	−0.750024	−	0.661411i	$-0.769958\pi$
$751$	−41.1079	−1.50005	−0.750024	+	0.661411i	$0.769958\pi$
$752$	−17.2757	−0.629979
$753$	0	0
$754$	89.2575i	3.25057i
$755$	−17.3568	−0.631680
$756$	0	0
$757$	27.7714	1.00937	0.504684	−	0.863304i	$-0.331609\pi$
$757$	27.7714	1.00937	0.504684	+	0.863304i	$0.331609\pi$
$758$	31.9223i	1.15947i
$759$	0	0
$760$	11.3033	0.410015
$761$	−13.6834	−0.496021	−0.248011	−	0.968757i	$-0.579777\pi$
$761$	−13.6834	−0.496021	−0.248011	+	0.968757i	$0.579777\pi$
$762$	0	0
$763$	0	0
$764$	− 28.8622i	− 1.04420i
$765$	0	0
$766$	− 41.5676i	− 1.50190i
$767$	− 33.6292i	− 1.21428i
$768$	0	0
$769$	9.59976i	0.346176i	0.984906	+	0.173088i	$0.0553745\pi$
$769$	9.59976i	0.346176i	−0.984906	+	0.173088i	$0.944625\pi$
$770$	0	0
$771$	0	0
$772$	0.761870	0.0274203
$773$	33.2003	1.19413	0.597066	−	0.802192i	$-0.296333\pi$
$773$	33.2003	1.19413	0.597066	+	0.802192i	$0.296333\pi$
$774$	0	0
$775$	− 5.96642i	− 0.214320i
$776$	1.12114	0.0402466
$777$	0	0
$778$	−54.7607	−1.96326
$779$	− 18.9264i	− 0.678108i
$780$	0	0
$781$	10.8638	0.388739
$782$	49.3092	1.76329
$783$	0	0
$784$	0	0
$785$	20.4181i	0.728753i
$786$	0	0
$787$	− 39.9657i	− 1.42462i	−0.701864	−	0.712311i	$-0.747648\pi$
$787$	− 39.9657i	− 1.42462i	0.701864	−	0.712311i	$-0.252352\pi$
$788$	− 16.6686i	− 0.593793i
$789$	0	0
$790$	45.3003i	1.61171i
$791$	0	0
$792$	0	0
$793$	−53.8780	−1.91326
$794$	−28.2904	−1.00399
$795$	0	0
$796$	11.9641i	0.424057i
$797$	3.14911	0.111547	0.0557736	−	0.998443i	$-0.482237\pi$
$797$	3.14911	0.111547	0.0557736	+	0.998443i	$0.482237\pi$
$798$	0	0
$799$	−26.2875	−0.929985
$800$	− 10.0228i	− 0.354359i
$801$	0	0
$802$	−11.0386	−0.389788
$803$	11.5830	0.408755
$804$	0	0
$805$	0	0
$806$	− 54.9623i	− 1.93597i
$807$	0	0
$808$	3.07659i	0.108234i
$809$	− 16.1113i	− 0.566443i	−0.959055	−	0.283221i	$-0.908597\pi$
$809$	− 16.1113i	− 0.566443i	0.959055	−	0.283221i	$-0.0914031\pi$
$810$	0	0
$811$	− 24.4068i	− 0.857038i	−0.903533	−	0.428519i	$-0.859035\pi$
$811$	− 24.4068i	− 0.857038i	0.903533	−	0.428519i	$-0.140965\pi$
$812$	0	0
$813$	0	0
$814$	−22.5644	−0.790883
$815$	17.0201	0.596190
$816$	0	0
$817$	15.5933i	0.545539i
$818$	32.5198	1.13703
$819$	0	0
$820$	7.77749	0.271602
$821$	− 24.0153i	− 0.838141i	−0.907954	−	0.419070i	$-0.862356\pi$
$821$	− 24.0153i	− 0.838141i	0.907954	−	0.419070i	$-0.137644\pi$
$822$	0	0
$823$	−4.45736	−0.155374	−0.0776869	−	0.996978i	$-0.524753\pi$
$823$	−4.45736	−0.155374	−0.0776869	+	0.996978i	$0.524753\pi$
$824$	6.25916	0.218048
$825$	0	0
$826$	0	0
$827$	49.2167i	1.71143i	0.517445	+	0.855716i	$0.326883\pi$
$827$	49.2167i	1.71143i	−0.517445	+	0.855716i	$0.673117\pi$
$828$	0	0
$829$	41.8552i	1.45369i	0.686800	+	0.726846i	$0.259015\pi$
$829$	41.8552i	1.45369i	−0.686800	+	0.726846i	$0.740985\pi$
$830$	− 55.8213i	− 1.93759i
$831$	0	0
$832$	− 28.6424i	− 0.992998i
$833$	0	0
$834$	0	0
$835$	23.4507	0.811547
$836$	14.2187	0.491764
$837$	0	0
$838$	20.8870i	0.721529i
$839$	−30.4461	−1.05112	−0.525559	−	0.850757i	$-0.676144\pi$
$839$	−30.4461	−1.05112	−0.525559	+	0.850757i	$0.676144\pi$
$840$	0	0
$841$	−19.3355	−0.666742
$842$	36.5270i	1.25880i
$843$	0	0
$844$	33.4869	1.15267
$845$	−63.1458	−2.17228
$846$	0	0
$847$	0	0
$848$	− 11.0963i	− 0.381048i
$849$	0	0
$850$	− 18.7448i	− 0.642940i
$851$	34.8234i	1.19373i
$852$	0	0
$853$	12.7434i	0.436324i	0.975913	+	0.218162i	$0.0700062\pi$
$853$	12.7434i	0.436324i	−0.975913	+	0.218162i	$0.929994\pi$
$854$	0	0
$855$	0	0
$856$	−12.2042	−0.417132
$857$	−44.0076	−1.50327	−0.751635	−	0.659579i	$-0.770735\pi$
$857$	−44.0076	−1.50327	−0.751635	+	0.659579i	$0.770735\pi$
$858$	0	0
$859$	− 44.8363i	− 1.52979i	−0.644153	−	0.764897i	$-0.722790\pi$
$859$	− 44.8363i	− 1.52979i	0.644153	−	0.764897i	$-0.277210\pi$
$860$	−6.40780	−0.218504
$861$	0	0
$862$	−22.2898	−0.759194
$863$	− 32.0552i	− 1.09117i	−0.838055	−	0.545586i	$-0.816307\pi$
$863$	− 32.0552i	− 1.09117i	0.838055	−	0.545586i	$-0.183693\pi$
$864$	0	0
$865$	27.4882	0.934627
$866$	60.7573	2.06462
$867$	0	0
$868$	0	0
$869$	− 15.9013i	− 0.539415i
$870$	0	0
$871$	75.7701i	2.56737i
$872$	− 3.82380i	− 0.129490i
$873$	0	0
$874$	− 50.0104i	− 1.69163i
$875$	0	0
$876$	0	0
$877$	−9.52107	−0.321504	−0.160752	−	0.986995i	$-0.551392\pi$
$877$	−9.52107	−0.321504	−0.160752	+	0.986995i	$0.551392\pi$
$878$	45.8194	1.54633
$879$	0	0
$880$	− 11.1893i	− 0.377191i
$881$	4.94353	0.166552	0.0832759	−	0.996527i	$-0.473462\pi$
$881$	4.94353	0.166552	0.0832759	+	0.996527i	$0.473462\pi$
$882$	0	0
$883$	47.2867	1.59132	0.795662	−	0.605741i	$-0.207123\pi$
$883$	47.2867	1.59132	0.795662	+	0.605741i	$0.207123\pi$
$884$	− 75.7666i	− 2.54831i
$885$	0	0
$886$	−2.28318	−0.0767049
$887$	50.5894	1.69863	0.849314	−	0.527889i	$-0.177016\pi$
$887$	50.5894	1.69863	0.849314	+	0.527889i	$0.177016\pi$
$888$	0	0
$889$	0	0
$890$	− 5.65897i	− 0.189689i
$891$	0	0
$892$	9.64586i	0.322967i
$893$	26.6613i	0.892186i
$894$	0	0
$895$	− 14.4886i	− 0.484301i
$896$	0	0
$897$	0	0
$898$	21.0374	0.702028
$899$	29.7637	0.992675
$900$	0	0
$901$	− 16.8846i	− 0.562509i
$902$	−6.22192	−0.207167
$903$	0	0
$904$	−0.0756990	−0.00251771
$905$	− 40.8852i	− 1.35907i
$906$	0	0
$907$	25.9072	0.860236	0.430118	−	0.902773i	$-0.358472\pi$
$907$	25.9072	0.860236	0.430118	+	0.902773i	$0.358472\pi$
$908$	−45.3113	−1.50371
$909$	0	0
$910$	0	0
$911$	11.1556i	0.369601i	0.982776	+	0.184801i	$0.0591639\pi$
$911$	11.1556i	0.369601i	−0.982776	+	0.184801i	$0.940836\pi$
$912$	0	0
$913$	19.5944i	0.648480i
$914$	9.48388i	0.313699i
$915$	0	0
$916$	− 9.77317i	− 0.322915i
$917$	0	0
$918$	0	0
$919$	−2.87785	−0.0949314	−0.0474657	−	0.998873i	$-0.515114\pi$
$919$	−2.87785	−0.0949314	−0.0474657	+	0.998873i	$0.515114\pi$
$920$	−5.73471	−0.189068
$921$	0	0
$922$	6.37891i	0.210078i
$923$	−58.7129	−1.93256
$924$	0	0
$925$	13.2380	0.435264
$926$	20.2240i	0.664602i
$927$	0	0
$928$	49.9991	1.64130
$929$	−30.7374	−1.00846	−0.504230	−	0.863569i	$-0.668224\pi$
$929$	−30.7374	−1.00846	−0.504230	+	0.863569i	$0.668224\pi$
$930$	0	0
$931$	0	0
$932$	7.58912i	0.248590i
$933$	0	0
$934$	− 15.0030i	− 0.490914i
$935$	− 17.0262i	− 0.556815i
$936$	0	0
$937$	− 41.5614i	− 1.35775i	−0.734254	−	0.678875i	$-0.762468\pi$
$937$	− 41.5614i	− 1.35775i	0.734254	−	0.678875i	$-0.237532\pi$
$938$	0	0
$939$	0	0
$940$	−10.9560	−0.357346
$941$	5.43859	0.177293	0.0886464	−	0.996063i	$-0.471746\pi$
$941$	5.43859	0.177293	0.0886464	+	0.996063i	$0.471746\pi$
$942$	0	0
$943$	9.60222i	0.312691i
$944$	−23.1532	−0.753572
$945$	0	0
$946$	5.12619	0.166667
$947$	− 10.1552i	− 0.330000i	−0.986294	−	0.165000i	$-0.947238\pi$
$947$	− 10.1552i	− 0.330000i	0.986294	−	0.165000i	$-0.0527624\pi$
$948$	0	0
$949$	−62.5995	−2.03207
$950$	−19.0113	−0.616809
$951$	0	0
$952$	0	0
$953$	0.875628i	0.0283644i	0.999899	+	0.0141822i	$0.00451448\pi$
$953$	0.875628i	0.0283644i	−0.999899	+	0.0141822i	$0.995486\pi$
$954$	0	0
$955$	35.0525i	1.13427i
$956$	23.3621i	0.755583i
$957$	0	0
$958$	− 44.1022i	− 1.42488i
$959$	0	0
$960$	0	0
$961$	12.6723	0.408784
$962$	121.948	3.93176
$963$	0	0
$964$	− 16.1686i	− 0.520755i
$965$	−0.925274	−0.0297856
$966$	0	0
$967$	59.5474	1.91492	0.957458	−	0.288573i	$-0.0931809\pi$
$967$	59.5474	1.91492	0.957458	+	0.288573i	$0.0931809\pi$
$968$	− 7.75637i	− 0.249299i
$969$	0	0
$970$	4.87944	0.156669
$971$	−55.6091	−1.78458	−0.892291	−	0.451460i	$-0.850903\pi$
$971$	−55.6091	−1.78458	−0.892291	+	0.451460i	$0.850903\pi$
$972$	0	0
$973$	0	0
$974$	− 44.0969i	− 1.41296i
$975$	0	0
$976$	37.0942i	1.18736i
$977$	− 46.6341i	− 1.49196i	−0.665970	−	0.745979i	$-0.731982\pi$
$977$	− 46.6341i	− 1.49196i	0.665970	−	0.745979i	$-0.268018\pi$
$978$	0	0
$979$	1.98641i	0.0634860i
$980$	0	0
$981$	0	0
$982$	−23.3743	−0.745905
$983$	−12.6584	−0.403740	−0.201870	−	0.979412i	$-0.564702\pi$
$983$	−12.6584	−0.403740	−0.201870	+	0.979412i	$0.564702\pi$
$984$	0	0
$985$	20.2436i	0.645015i
$986$	93.5089	2.97793
$987$	0	0
$988$	−76.8440	−2.44473
$989$	− 7.91118i	− 0.251561i
$990$	0	0
$991$	−12.3516	−0.392360	−0.196180	−	0.980568i	$-0.562854\pi$
$991$	−12.3516	−0.392360	−0.196180	+	0.980568i	$0.562854\pi$
$992$	−30.7881	−0.977523
$993$	0	0
$994$	0	0
$995$	− 14.5302i	− 0.460637i
$996$	0	0
$997$	8.72257i	0.276247i	0.990415	+	0.138123i	$0.0441070\pi$
$997$	8.72257i	0.276247i	−0.990415	+	0.138123i	$0.955893\pi$
$998$	− 9.08910i	− 0.287710i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3087.2.c.c.3086.11	✓	24
3.2	odd	2	inner	3087.2.c.c.3086.14	yes	24
7.6	odd	2	inner	3087.2.c.c.3086.13	yes	24
21.20	even	2	inner	3087.2.c.c.3086.12	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3087.2.c.c.3086.11	✓	24	1.1	even	1	trivial
3087.2.c.c.3086.12	yes	24	21.20	even	2	inner
3087.2.c.c.3086.13	yes	24	7.6	odd	2	inner
3087.2.c.c.3086.14	yes	24	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 3087 = 3^{2} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3087.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.88777i q^{2} -1.56366 q^{4} +1.89903 q^{5} -0.823703i q^{8} +O(q^{10})\)	\(q-1.88777i q^{2} -1.56366 q^{4} +1.89903 q^{5} -0.823703i q^{8} -3.58493i q^{10} +1.25838i q^{11} -6.80085i q^{13} -4.68228 q^{16} -7.12478 q^{17} +7.22610i q^{19} -2.96945 q^{20} +2.37553 q^{22} -3.66613i q^{23} -1.39367 q^{25} -12.8384 q^{26} -6.95237i q^{29} +4.28108i q^{31} +7.19165i q^{32} +13.4499i q^{34} -9.49868 q^{37} +13.6412 q^{38} -1.56424i q^{40} -2.61917 q^{41} +2.15791 q^{43} -1.96769i q^{44} -6.92080 q^{46} +3.68959 q^{47} +2.63093i q^{50} +10.6342i q^{52} +2.36985i q^{53} +2.38971i q^{55} -13.1245 q^{58} +4.94485 q^{59} -7.92225i q^{61} +8.08169 q^{62} +4.21160 q^{64} -12.9150i q^{65} -11.1413 q^{67} +11.1408 q^{68} -8.63318i q^{71} -9.20466i q^{73} +17.9313i q^{74} -11.2992i q^{76} -12.6363 q^{79} -8.89182 q^{80} +4.94438i q^{82} +15.5711 q^{83} -13.5302 q^{85} -4.07363i q^{86} +1.03653 q^{88} +1.57854 q^{89} +5.73259i q^{92} -6.96508i q^{94} +13.7226i q^{95} +1.36110i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 40 q^{16} + 64 q^{22} + 136 q^{25} + 32 q^{37} + 32 q^{43} + 32 q^{46} - 128 q^{58} + 32 q^{64} + 40 q^{67} + 8 q^{79} + 64 q^{85} + 16 q^{88}+O(q^{100}) \) 24 * q - 40 * q^16 + 64 * q^22 + 136 * q^25 + 32 * q^37 + 32 * q^43 + 32 * q^46 - 128 * q^58 + 32 * q^64 + 40 * q^67 + 8 * q^79 + 64 * q^85 + 16 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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