LMFDB - Embedded newform 5184.2.c.m.5183.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5184,2,Mod(5183,5184)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5184, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5184.5183");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5184 = 2^{6} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5184.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$41.3944484078$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5183.4
Character	$\chi$	$=$	5184.5183
Dual form			5184.2.c.m.5183.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.47695i q^{5} +3.60624i q^{7} +O(q^{10})$	$q-3.47695i q^{5} +3.60624i q^{7} +2.64859 q^{11} +4.72607 q^{13} -1.79223i q^{17} +4.55459i q^{19} +0.755051 q^{23} -7.08920 q^{25} -8.30671i q^{29} +2.24311i q^{31} +12.5387 q^{35} +3.98496 q^{37} +6.44089i q^{41} +8.78266i q^{43} +2.75638 q^{47} -6.00495 q^{49} +4.41211i q^{53} -9.20903i q^{55} -2.72163 q^{59} +2.38312 q^{61} -16.4323i q^{65} +10.1458i q^{67} -0.0730340 q^{71} +13.3207 q^{73} +9.55145i q^{77} -5.20903i q^{79} -0.489692 q^{83} -6.23151 q^{85} +4.41211i q^{89} +17.0433i q^{91} +15.8361 q^{95} -14.4371 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 24 q^{25} - 24 q^{49} + 24 q^{73} - 24 q^{97}+O(q^{100}) $ 24 * q - 24 * q^25 - 24 * q^49 + 24 * q^73 - 24 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 3.47695i	− 1.55494i	−0.628920	−	0.777470i	$-0.716503\pi$
$5$	− 3.47695i	− 1.55494i	0.628920	−	0.777470i	$-0.283497\pi$
$6$	0	0
$7$	3.60624i	1.36303i	0.731804	+	0.681515i	$0.238679\pi$
$7$	3.60624i	1.36303i	−0.731804	+	0.681515i	$0.761321\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.64859	0.798581	0.399290	−	0.916825i	$-0.369257\pi$
$11$	2.64859	0.798581	0.399290	+	0.916825i	$0.369257\pi$
$12$	0	0
$13$	4.72607	1.31078	0.655388	−	0.755292i	$-0.272505\pi$
$13$	4.72607	1.31078	0.655388	+	0.755292i	$0.272505\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.79223i	− 0.434681i	−0.976096	−	0.217340i	$-0.930262\pi$
$17$	− 1.79223i	− 0.434681i	0.976096	−	0.217340i	$-0.0697381\pi$
$18$	0	0
$19$	4.55459i	1.04490i	0.852671	+	0.522448i	$0.174981\pi$
$19$	4.55459i	1.04490i	−0.852671	+	0.522448i	$0.825019\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.755051	0.157439	0.0787195	−	0.996897i	$-0.474917\pi$
$23$	0.755051	0.157439	0.0787195	+	0.996897i	$0.474917\pi$
$24$	0	0
$25$	−7.08920	−1.41784
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 8.30671i	− 1.54252i	−0.636522	−	0.771259i	$-0.719627\pi$
$29$	− 8.30671i	− 1.54252i	0.636522	−	0.771259i	$-0.280373\pi$
$30$	0	0
$31$	2.24311i	0.402875i	0.979501	+	0.201437i	$0.0645613\pi$
$31$	2.24311i	0.402875i	−0.979501	+	0.201437i	$0.935439\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	12.5387	2.11943
$36$	0	0
$37$	3.98496	0.655123	0.327561	−	0.944830i	$-0.393773\pi$
$37$	3.98496	0.655123	0.327561	+	0.944830i	$0.393773\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.44089i	1.00590i	0.864316	+	0.502949i	$0.167752\pi$
$41$	6.44089i	1.00590i	−0.864316	+	0.502949i	$0.832248\pi$
$42$	0	0
$43$	8.78266i	1.33934i	0.742657	+	0.669672i	$0.233565\pi$
$43$	8.78266i	1.33934i	−0.742657	+	0.669672i	$0.766435\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.75638	0.402060	0.201030	−	0.979585i	$-0.435571\pi$
$47$	2.75638	0.402060	0.201030	+	0.979585i	$0.435571\pi$
$48$	0	0
$49$	−6.00495	−0.857850
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.41211i	0.606050i	0.952983	+	0.303025i	$0.0979966\pi$
$53$	4.41211i	0.606050i	−0.952983	+	0.303025i	$0.902003\pi$
$54$	0	0
$55$	− 9.20903i	− 1.24175i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.72163	−0.354326	−0.177163	−	0.984182i	$-0.556692\pi$
$59$	−2.72163	−0.354326	−0.177163	+	0.984182i	$0.556692\pi$
$60$	0	0
$61$	2.38312	0.305127	0.152563	−	0.988294i	$-0.451247\pi$
$61$	2.38312	0.305127	0.152563	+	0.988294i	$0.451247\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 16.4323i	− 2.03818i
$66$	0	0
$67$	10.1458i	1.23951i	0.784797	+	0.619753i	$0.212767\pi$
$67$	10.1458i	1.23951i	−0.784797	+	0.619753i	$0.787233\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.0730340	−0.00866754	−0.00433377	−	0.999991i	$-0.501379\pi$
$71$	−0.0730340	−0.00866754	−0.00433377	+	0.999991i	$0.501379\pi$
$72$	0	0
$73$	13.3207	1.55907	0.779536	−	0.626358i	$-0.215455\pi$
$73$	13.3207	1.55907	0.779536	+	0.626358i	$0.215455\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9.55145i	1.08849i
$78$	0	0
$79$	− 5.20903i	− 0.586062i	−0.956103	−	0.293031i	$-0.905336\pi$
$79$	− 5.20903i	− 0.586062i	0.956103	−	0.293031i	$-0.0946639\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.489692	−0.0537507	−0.0268753	−	0.999639i	$-0.508556\pi$
$83$	−0.489692	−0.0537507	−0.0268753	+	0.999639i	$0.508556\pi$
$84$	0	0
$85$	−6.23151	−0.675902
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.41211i	0.467683i	0.972275	+	0.233841i	$0.0751297\pi$
$89$	4.41211i	0.467683i	−0.972275	+	0.233841i	$0.924870\pi$
$90$	0	0
$91$	17.0433i	1.78663i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	15.8361	1.62475
$96$	0	0
$97$	−14.4371	−1.46587	−0.732933	−	0.680301i	$-0.761849\pi$
$97$	−14.4371	−1.46587	−0.732933	+	0.680301i	$0.761849\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 2.17990i	− 0.216908i	−0.994101	−	0.108454i	$-0.965410\pi$
$101$	− 2.17990i	− 0.216908i	0.994101	−	0.108454i	$-0.0345901\pi$
$102$	0	0
$103$	− 3.84591i	− 0.378948i	−0.981886	−	0.189474i	$-0.939322\pi$
$103$	− 3.84591i	− 0.378948i	0.981886	−	0.189474i	$-0.0606783\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−19.4071	−1.87615	−0.938077	−	0.346426i	$-0.887395\pi$
$107$	−19.4071	−1.87615	−0.938077	+	0.346426i	$0.887395\pi$
$108$	0	0
$109$	13.0941	1.25419	0.627096	−	0.778942i	$-0.284243\pi$
$109$	13.0941	1.25419	0.627096	+	0.778942i	$0.284243\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.42584i	0.134132i	0.997749	+	0.0670660i	$0.0213638\pi$
$113$	1.42584i	0.134132i	−0.997749	+	0.0670660i	$0.978636\pi$
$114$	0	0
$115$	− 2.62528i	− 0.244808i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.46322	0.592483
$120$	0	0
$121$	−3.98496	−0.362269
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.26404i	0.649715i
$126$	0	0
$127$	13.5554i	1.20285i	0.798929	+	0.601425i	$0.205400\pi$
$127$	13.5554i	1.20285i	−0.798929	+	0.601425i	$0.794600\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.7635	1.37726	0.688632	−	0.725111i	$-0.258211\pi$
$131$	15.7635	1.37726	0.688632	+	0.725111i	$0.258211\pi$
$132$	0	0
$133$	−16.4249	−1.42422
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.73794i	0.661097i	0.943789	+	0.330549i	$0.107234\pi$
$137$	7.73794i	0.661097i	−0.943789	+	0.330549i	$0.892766\pi$
$138$	0	0
$139$	0.566195i	0.0480240i	0.999712	+	0.0240120i	$0.00764399\pi$
$139$	0.566195i	0.0480240i	−0.999712	+	0.0240120i	$0.992356\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	12.5174	1.04676
$144$	0	0
$145$	−28.8820	−2.39852
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.45715i	0.774760i	0.921920	+	0.387380i	$0.126620\pi$
$149$	9.45715i	0.774760i	−0.921920	+	0.387380i	$0.873380\pi$
$150$	0	0
$151$	− 19.0825i	− 1.55292i	−0.630169	−	0.776458i	$-0.717015\pi$
$151$	− 19.0825i	− 1.55292i	0.630169	−	0.776458i	$-0.282985\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.79920	0.626446
$156$	0	0
$157$	21.1360	1.68683	0.843417	−	0.537259i	$-0.180540\pi$
$157$	21.1360	1.68683	0.843417	+	0.537259i	$0.180540\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.72289i	0.214594i
$162$	0	0
$163$	− 22.6646i	− 1.77523i	−0.460586	−	0.887615i	$-0.652361\pi$
$163$	− 22.6646i	− 1.77523i	0.460586	−	0.887615i	$-0.347639\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.85424	−0.453015	−0.226507	−	0.974009i	$-0.572731\pi$
$167$	−5.85424	−0.453015	−0.226507	+	0.974009i	$0.572731\pi$
$168$	0	0
$169$	9.33575	0.718135
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.91099i	0.449404i	0.974428	+	0.224702i	$0.0721409\pi$
$173$	5.91099i	0.449404i	−0.974428	+	0.224702i	$0.927859\pi$
$174$	0	0
$175$	− 25.5653i	− 1.93256i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.8823	0.962871	0.481436	−	0.876481i	$-0.340116\pi$
$179$	12.8823	0.962871	0.481436	+	0.876481i	$0.340116\pi$
$180$	0	0
$181$	10.1934	0.757672	0.378836	−	0.925464i	$-0.376324\pi$
$181$	10.1934	0.757672	0.378836	+	0.925464i	$0.376324\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 13.8555i	− 1.01868i
$186$	0	0
$187$	− 4.74690i	− 0.347128i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.5921	1.63471	0.817353	−	0.576138i	$-0.195441\pi$
$191$	22.5921	1.63471	0.817353	+	0.576138i	$0.195441\pi$
$192$	0	0
$193$	2.25871	0.162585	0.0812926	−	0.996690i	$-0.474095\pi$
$193$	2.25871	0.162585	0.0812926	+	0.996690i	$0.474095\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5.65685i	− 0.403034i	−0.979485	−	0.201517i	$-0.935413\pi$
$197$	− 5.65685i	− 0.403034i	0.979485	−	0.201517i	$-0.0645872\pi$
$198$	0	0
$199$	− 13.8193i	− 0.979621i	−0.871829	−	0.489811i	$-0.837066\pi$
$199$	− 13.8193i	− 0.979621i	0.871829	−	0.489811i	$-0.162934\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	29.9560	2.10250
$204$	0	0
$205$	22.3947	1.56411
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.0633i	0.834434i
$210$	0	0
$211$	6.37872i	0.439129i	0.975598	+	0.219565i	$0.0704637\pi$
$211$	6.37872i	0.439129i	−0.975598	+	0.219565i	$0.929536\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	30.5369	2.08260
$216$	0	0
$217$	−8.08920	−0.549130
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 8.47023i	− 0.569769i
$222$	0	0
$223$	− 2.60224i	− 0.174259i	−0.996197	−	0.0871296i	$-0.972231\pi$
$223$	− 2.60224i	− 0.174259i	0.996197	−	0.0871296i	$-0.0277694\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	21.4385	1.42292	0.711461	−	0.702726i	$-0.248034\pi$
$227$	21.4385	1.42292	0.711461	+	0.702726i	$0.248034\pi$
$228$	0	0
$229$	−0.741116	−0.0489743	−0.0244872	−	0.999700i	$-0.507795\pi$
$229$	−0.741116	−0.0489743	−0.0244872	+	0.999700i	$0.507795\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.3832i	1.33535i	0.744454	+	0.667673i	$0.232710\pi$
$233$	20.3832i	1.33535i	−0.744454	+	0.667673i	$0.767290\pi$
$234$	0	0
$235$	− 9.58381i	− 0.625179i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	9.57927	0.619632	0.309816	−	0.950797i	$-0.399733\pi$
$239$	9.57927	0.619632	0.309816	+	0.950797i	$0.399733\pi$
$240$	0	0
$241$	−6.35575	−0.409410	−0.204705	−	0.978824i	$-0.565623\pi$
$241$	−6.35575	−0.409410	−0.204705	+	0.978824i	$0.565623\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	20.8789i	1.33391i
$246$	0	0
$247$	21.5253i	1.36962i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	27.0938	1.71014	0.855072	−	0.518509i	$-0.173513\pi$
$251$	27.0938	1.71014	0.855072	+	0.518509i	$0.173513\pi$
$252$	0	0
$253$	1.99982	0.125728
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.83014i	0.238917i	0.992839	+	0.119459i	$0.0381159\pi$
$257$	3.83014i	0.238917i	−0.992839	+	0.119459i	$0.961884\pi$
$258$	0	0
$259$	14.3707i	0.892952i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−32.1977	−1.98539	−0.992697	−	0.120632i	$-0.961508\pi$
$263$	−32.1977	−1.98539	−0.992697	+	0.120632i	$0.961508\pi$
$264$	0	0
$265$	15.3407	0.942372
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 27.2601i	− 1.66208i	−0.556214	−	0.831039i	$-0.687747\pi$
$269$	− 27.2601i	− 1.66208i	0.556214	−	0.831039i	$-0.312253\pi$
$270$	0	0
$271$	− 8.48623i	− 0.515501i	−0.966211	−	0.257751i	$-0.917019\pi$
$271$	− 8.48623i	− 0.515501i	0.966211	−	0.257751i	$-0.0829813\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−18.7764	−1.13226
$276$	0	0
$277$	7.81032	0.469277	0.234638	−	0.972083i	$-0.424609\pi$
$277$	7.81032	0.469277	0.234638	+	0.972083i	$0.424609\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.98894i	0.416925i	0.978030	+	0.208462i	$0.0668459\pi$
$281$	6.98894i	0.416925i	−0.978030	+	0.208462i	$0.933154\pi$
$282$	0	0
$283$	− 28.0635i	− 1.66820i	−0.551611	−	0.834101i	$-0.685987\pi$
$283$	− 28.0635i	− 1.66820i	0.551611	−	0.834101i	$-0.314013\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−23.2274	−1.37107
$288$	0	0
$289$	13.7879	0.811053
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8.71616i	0.509203i	0.967046	+	0.254602i	$0.0819444\pi$
$293$	8.71616i	0.509203i	−0.967046	+	0.254602i	$0.918056\pi$
$294$	0	0
$295$	9.46297i	0.550955i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.56842	0.206367
$300$	0	0
$301$	−31.6724	−1.82556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 8.28599i	− 0.474454i
$306$	0	0
$307$	− 9.62380i	− 0.549259i	−0.961550	−	0.274630i	$-0.911445\pi$
$307$	− 9.62380i	− 0.549259i	0.961550	−	0.274630i	$-0.0885552\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−16.6337	−0.943211	−0.471606	−	0.881810i	$-0.656325\pi$
$311$	−16.6337	−0.943211	−0.471606	+	0.881810i	$0.656325\pi$
$312$	0	0
$313$	24.6305	1.39220	0.696100	−	0.717945i	$-0.254917\pi$
$313$	24.6305	1.39220	0.696100	+	0.717945i	$0.254917\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 24.0843i	− 1.35271i	−0.736576	−	0.676354i	$-0.763559\pi$
$317$	− 24.0843i	− 1.35271i	0.736576	−	0.676354i	$-0.236441\pi$
$318$	0	0
$319$	− 22.0011i	− 1.23183i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.16290	0.454196
$324$	0	0
$325$	−33.5041	−1.85847
$326$	0	0
$327$	0	0
$328$	0	0
$329$	9.94017i	0.548020i
$330$	0	0
$331$	− 12.8269i	− 0.705028i	−0.935806	−	0.352514i	$-0.885327\pi$
$331$	− 12.8269i	− 0.705028i	0.935806	−	0.352514i	$-0.114673\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	35.2764	1.92736
$336$	0	0
$337$	−20.1508	−1.09769	−0.548843	−	0.835925i	$-0.684932\pi$
$337$	−20.1508	−1.09769	−0.548843	+	0.835925i	$0.684932\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.94109i	0.321728i
$342$	0	0
$343$	3.58839i	0.193755i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7.00785	−0.376201	−0.188100	−	0.982150i	$-0.560233\pi$
$347$	−7.00785	−0.376201	−0.188100	+	0.982150i	$0.560233\pi$
$348$	0	0
$349$	−4.95758	−0.265373	−0.132687	−	0.991158i	$-0.542360\pi$
$349$	−4.95758	−0.265373	−0.132687	+	0.991158i	$0.542360\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 3.53321i	− 0.188054i	−0.995570	−	0.0940270i	$-0.970026\pi$
$353$	− 3.53321i	− 0.188054i	0.995570	−	0.0940270i	$-0.0299740\pi$
$354$	0	0
$355$	0.253936i	0.0134775i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.2054	0.802511	0.401255	−	0.915966i	$-0.368574\pi$
$359$	15.2054	0.802511	0.401255	+	0.915966i	$0.368574\pi$
$360$	0	0
$361$	−1.74433	−0.0918070
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 46.3155i	− 2.42426i
$366$	0	0
$367$	3.73560i	0.194997i	0.995236	+	0.0974983i	$0.0310841\pi$
$367$	3.73560i	0.194997i	−0.995236	+	0.0974983i	$0.968916\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−15.9111	−0.826064
$372$	0	0
$373$	1.52049	0.0787277	0.0393639	−	0.999225i	$-0.487467\pi$
$373$	1.52049	0.0787277	0.0393639	+	0.999225i	$0.487467\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 39.2581i	− 2.02190i
$378$	0	0
$379$	23.7948i	1.22226i	0.791531	+	0.611129i	$0.209284\pi$
$379$	23.7948i	1.22226i	−0.791531	+	0.611129i	$0.790716\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.96998	0.151759	0.0758796	−	0.997117i	$-0.475824\pi$
$383$	2.96998	0.151759	0.0758796	+	0.997117i	$0.475824\pi$
$384$	0	0
$385$	33.2100	1.69254
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.10567i	0.208166i	0.994569	+	0.104083i	$0.0331907\pi$
$389$	4.10567i	0.208166i	−0.994569	+	0.104083i	$0.966809\pi$
$390$	0	0
$391$	− 1.35323i	− 0.0684357i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−18.1116	−0.911291
$396$	0	0
$397$	−16.4643	−0.826319	−0.413160	−	0.910659i	$-0.635575\pi$
$397$	−16.4643	−0.826319	−0.413160	+	0.910659i	$0.635575\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 30.9991i	− 1.54802i	−0.633171	−	0.774012i	$-0.718247\pi$
$401$	− 30.9991i	− 1.54802i	0.633171	−	0.774012i	$-0.281753\pi$
$402$	0	0
$403$	10.6011i	0.528079i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.5545	0.523168
$408$	0	0
$409$	−4.48223	−0.221632	−0.110816	−	0.993841i	$-0.535346\pi$
$409$	−4.48223	−0.221632	−0.110816	+	0.993841i	$0.535346\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 9.81483i	− 0.482956i
$414$	0	0
$415$	1.70263i	0.0835791i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12.9714	0.633695	0.316848	−	0.948476i	$-0.397376\pi$
$419$	12.9714	0.633695	0.316848	+	0.948476i	$0.397376\pi$
$420$	0	0
$421$	18.4251	0.897986	0.448993	−	0.893535i	$-0.351783\pi$
$421$	18.4251	0.897986	0.448993	+	0.893535i	$0.351783\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	12.7055i	0.616307i
$426$	0	0
$427$	8.59409i	0.415897i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−22.3743	−1.07773	−0.538867	−	0.842391i	$-0.681148\pi$
$431$	−22.3743	−1.07773	−0.538867	+	0.842391i	$0.681148\pi$
$432$	0	0
$433$	0.857684	0.0412177	0.0206088	−	0.999788i	$-0.493440\pi$
$433$	0.857684	0.0412177	0.0206088	+	0.999788i	$0.493440\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.43895i	0.164507i
$438$	0	0
$439$	11.9037i	0.568134i	0.958804	+	0.284067i	$0.0916839\pi$
$439$	11.9037i	0.568134i	−0.958804	+	0.284067i	$0.908316\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	17.9955	0.854990	0.427495	−	0.904018i	$-0.359396\pi$
$443$	17.9955	0.854990	0.427495	+	0.904018i	$0.359396\pi$
$444$	0	0
$445$	15.3407	0.727219
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 37.0489i	− 1.74845i	−0.485524	−	0.874223i	$-0.661371\pi$
$449$	− 37.0489i	− 1.74845i	0.485524	−	0.874223i	$-0.338629\pi$
$450$	0	0
$451$	17.0593i	0.803291i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	59.2589	2.77810
$456$	0	0
$457$	−8.33080	−0.389699	−0.194849	−	0.980833i	$-0.562422\pi$
$457$	−8.33080	−0.389699	−0.194849	+	0.980833i	$0.562422\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	23.3385i	1.08698i	0.839414	+	0.543492i	$0.182898\pi$
$461$	23.3385i	1.08698i	−0.839414	+	0.543492i	$0.817102\pi$
$462$	0	0
$463$	34.6399i	1.60985i	0.593375	+	0.804926i	$0.297795\pi$
$463$	34.6399i	1.60985i	−0.593375	+	0.804926i	$0.702205\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.1281	0.792595	0.396298	−	0.918122i	$-0.370295\pi$
$467$	17.1281	0.792595	0.396298	+	0.918122i	$0.370295\pi$
$468$	0	0
$469$	−36.5881	−1.68948
$470$	0	0
$471$	0	0
$472$	0	0
$473$	23.2617i	1.06957i
$474$	0	0
$475$	− 32.2884i	− 1.48149i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−36.2999	−1.65859	−0.829293	−	0.558815i	$-0.811256\pi$
$479$	−36.2999	−1.65859	−0.829293	+	0.558815i	$0.811256\pi$
$480$	0	0
$481$	18.8332	0.858719
$482$	0	0
$483$	0	0
$484$	0	0
$485$	50.1971i	2.27933i
$486$	0	0
$487$	1.00757i	0.0456575i	0.999739	+	0.0228287i	$0.00726724\pi$
$487$	1.00757i	0.0456575i	−0.999739	+	0.0228287i	$0.992733\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−10.4083	−0.469721	−0.234860	−	0.972029i	$-0.575463\pi$
$491$	−10.4083	−0.469721	−0.234860	+	0.972029i	$0.575463\pi$
$492$	0	0
$493$	−14.8876	−0.670503
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 0.263378i	− 0.0118141i
$498$	0	0
$499$	3.75547i	0.168118i	0.996461	+	0.0840590i	$0.0267884\pi$
$499$	3.75547i	0.168118i	−0.996461	+	0.0840590i	$0.973212\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	20.5980	0.918417	0.459209	−	0.888328i	$-0.348133\pi$
$503$	20.5980	0.918417	0.459209	+	0.888328i	$0.348133\pi$
$504$	0	0
$505$	−7.57942	−0.337280
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 12.4689i	− 0.552674i	−0.961061	−	0.276337i	$-0.910879\pi$
$509$	− 12.4689i	− 0.552674i	0.961061	−	0.276337i	$-0.0891206\pi$
$510$	0	0
$511$	48.0376i	2.12506i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−13.3720	−0.589242
$516$	0	0
$517$	7.30054	0.321077
$518$	0	0
$519$	0	0
$520$	0	0
$521$	15.9193i	0.697438i	0.937227	+	0.348719i	$0.113383\pi$
$521$	15.9193i	0.697438i	−0.937227	+	0.348719i	$0.886617\pi$
$522$	0	0
$523$	3.71297i	0.162357i	0.996700	+	0.0811784i	$0.0258683\pi$
$523$	3.71297i	0.162357i	−0.996700	+	0.0811784i	$0.974132\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.02018	0.175122
$528$	0	0
$529$	−22.4299	−0.975213
$530$	0	0
$531$	0	0
$532$	0	0
$533$	30.4401i	1.31851i
$534$	0	0
$535$	67.4775i	2.91731i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−15.9047	−0.685062
$540$	0	0
$541$	−19.1191	−0.821994	−0.410997	−	0.911637i	$-0.634819\pi$
$541$	−19.1191	−0.821994	−0.410997	+	0.911637i	$0.634819\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 45.5277i	− 1.95019i
$546$	0	0
$547$	− 26.4275i	− 1.12996i	−0.825106	−	0.564978i	$-0.808884\pi$
$547$	− 26.4275i	− 1.12996i	0.825106	−	0.564978i	$-0.191116\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	37.8337	1.61177
$552$	0	0
$553$	18.7850	0.798820
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 24.9370i	− 1.05662i	−0.849053	−	0.528308i	$-0.822827\pi$
$557$	− 24.9370i	− 1.05662i	0.849053	−	0.528308i	$-0.177173\pi$
$558$	0	0
$559$	41.5075i	1.75558i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−30.5197	−1.28625	−0.643127	−	0.765760i	$-0.722363\pi$
$563$	−30.5197	−1.28625	−0.643127	+	0.765760i	$0.722363\pi$
$564$	0	0
$565$	4.95758	0.208567
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 10.3223i	− 0.432733i	−0.976312	−	0.216366i	$-0.930579\pi$
$569$	− 10.3223i	− 0.432733i	0.976312	−	0.216366i	$-0.0694206\pi$
$570$	0	0
$571$	− 8.72841i	− 0.365273i	−0.983181	−	0.182636i	$-0.941537\pi$
$571$	− 8.72841i	− 0.365273i	0.983181	−	0.182636i	$-0.0584631\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.35270	−0.223223
$576$	0	0
$577$	34.4398	1.43375	0.716874	−	0.697203i	$-0.245572\pi$
$577$	34.4398	1.43375	0.716874	+	0.697203i	$0.245572\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 1.76594i	− 0.0732637i
$582$	0	0
$583$	11.6859i	0.483980i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.63395	0.191264	0.0956318	−	0.995417i	$-0.469513\pi$
$587$	4.63395	0.191264	0.0956318	+	0.995417i	$0.469513\pi$
$588$	0	0
$589$	−10.2165	−0.420962
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.49543i	0.0614098i	0.999528	+	0.0307049i	$0.00977521\pi$
$593$	1.49543i	0.0614098i	−0.999528	+	0.0307049i	$0.990225\pi$
$594$	0	0
$595$	− 22.4723i	− 0.921275i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−31.6189	−1.29191	−0.645957	−	0.763374i	$-0.723541\pi$
$599$	−31.6189	−1.29191	−0.645957	+	0.763374i	$0.723541\pi$
$600$	0	0
$601$	−14.1659	−0.577839	−0.288919	−	0.957353i	$-0.593296\pi$
$601$	−14.1659	−0.577839	−0.288919	+	0.957353i	$0.593296\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	13.8555i	0.563306i
$606$	0	0
$607$	8.38404i	0.340298i	0.985418	+	0.170149i	$0.0544248\pi$
$607$	8.38404i	0.340298i	−0.985418	+	0.170149i	$0.945575\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	13.0269	0.527011
$612$	0	0
$613$	20.9574	0.846462	0.423231	−	0.906022i	$-0.360896\pi$
$613$	20.9574	0.846462	0.423231	+	0.906022i	$0.360896\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	29.5045i	1.18781i	0.804537	+	0.593903i	$0.202414\pi$
$617$	29.5045i	1.18781i	−0.804537	+	0.593903i	$0.797586\pi$
$618$	0	0
$619$	37.8445i	1.52110i	0.649280	+	0.760549i	$0.275070\pi$
$619$	37.8445i	1.52110i	−0.649280	+	0.760549i	$0.724930\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−15.9111	−0.637466
$624$	0	0
$625$	−10.1893	−0.407571
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 7.14197i	− 0.284769i
$630$	0	0
$631$	15.9867i	0.636420i	0.948020	+	0.318210i	$0.103082\pi$
$631$	15.9867i	0.636420i	−0.948020	+	0.318210i	$0.896918\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	47.1316	1.87036
$636$	0	0
$637$	−28.3798	−1.12445
$638$	0	0
$639$	0	0
$640$	0	0
$641$	32.7827i	1.29484i	0.762133	+	0.647420i	$0.224152\pi$
$641$	32.7827i	1.29484i	−0.762133	+	0.647420i	$0.775848\pi$
$642$	0	0
$643$	5.24115i	0.206691i	0.994646	+	0.103345i	$0.0329547\pi$
$643$	5.24115i	0.206691i	−0.994646	+	0.103345i	$0.967045\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−30.4906	−1.19871	−0.599354	−	0.800484i	$-0.704576\pi$
$647$	−30.4906	−1.19871	−0.599354	+	0.800484i	$0.704576\pi$
$648$	0	0
$649$	−7.20848	−0.282958
$650$	0	0
$651$	0	0
$652$	0	0
$653$	25.3755i	0.993019i	0.868031	+	0.496510i	$0.165385\pi$
$653$	25.3755i	0.993019i	−0.868031	+	0.496510i	$0.834615\pi$
$654$	0	0
$655$	− 54.8090i	− 2.14156i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−33.2253	−1.29427	−0.647137	−	0.762374i	$-0.724034\pi$
$659$	−33.2253	−1.29427	−0.647137	+	0.762374i	$0.724034\pi$
$660$	0	0
$661$	−31.9986	−1.24460	−0.622301	−	0.782778i	$-0.713802\pi$
$661$	−31.9986	−1.24460	−0.622301	+	0.782778i	$0.713802\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	57.1088i	2.21458i
$666$	0	0
$667$	− 6.27199i	− 0.242852i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6.31191	0.243669
$672$	0	0
$673$	50.3742	1.94178	0.970891	−	0.239523i	$-0.0769909\pi$
$673$	50.3742	1.94178	0.970891	+	0.239523i	$0.0769909\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 33.6365i	− 1.29276i	−0.763017	−	0.646379i	$-0.776283\pi$
$677$	− 33.6365i	− 1.29276i	0.763017	−	0.646379i	$-0.223717\pi$
$678$	0	0
$679$	− 52.0636i	− 1.99802i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	12.2382	0.468280	0.234140	−	0.972203i	$-0.424773\pi$
$683$	12.2382	0.468280	0.234140	+	0.972203i	$0.424773\pi$
$684$	0	0
$685$	26.9045	1.02797
$686$	0	0
$687$	0	0
$688$	0	0
$689$	20.8520i	0.794396i
$690$	0	0
$691$	− 33.8166i	− 1.28644i	−0.765680	−	0.643222i	$-0.777597\pi$
$691$	− 33.8166i	− 1.28644i	0.765680	−	0.643222i	$-0.222403\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.96863	0.0746745
$696$	0	0
$697$	11.5436	0.437245
$698$	0	0
$699$	0	0
$700$	0	0
$701$	40.9032i	1.54489i	0.635079	+	0.772447i	$0.280967\pi$
$701$	40.9032i	1.54489i	−0.635079	+	0.772447i	$0.719033\pi$
$702$	0	0
$703$	18.1499i	0.684535i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.86124	0.295653
$708$	0	0
$709$	0.572620	0.0215052	0.0107526	−	0.999942i	$-0.496577\pi$
$709$	0.572620	0.0215052	0.0107526	+	0.999942i	$0.496577\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.69366i	0.0634282i
$714$	0	0
$715$	− 43.5225i	− 1.62765i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−0.517752	−0.0193089	−0.00965445	−	0.999953i	$-0.503073\pi$
$719$	−0.517752	−0.0193089	−0.00965445	+	0.999953i	$0.503073\pi$
$720$	0	0
$721$	13.8692	0.516518
$722$	0	0
$723$	0	0
$724$	0	0
$725$	58.8879i	2.18704i
$726$	0	0
$727$	8.40958i	0.311894i	0.987765	+	0.155947i	$0.0498429\pi$
$727$	8.40958i	0.311894i	−0.987765	+	0.155947i	$0.950157\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	15.7406	0.582187
$732$	0	0
$733$	−13.1189	−0.484558	−0.242279	−	0.970207i	$-0.577895\pi$
$733$	−13.1189	−0.484558	−0.242279	+	0.970207i	$0.577895\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	26.8721i	0.989845i
$738$	0	0
$739$	− 9.78098i	− 0.359799i	−0.983685	−	0.179900i	$-0.942423\pi$
$739$	− 9.78098i	− 0.359799i	0.983685	−	0.179900i	$-0.0575773\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.40641	0.0515964	0.0257982	−	0.999667i	$-0.491787\pi$
$743$	1.40641	0.0515964	0.0257982	+	0.999667i	$0.491787\pi$
$744$	0	0
$745$	32.8820	1.20470
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 69.9866i	− 2.55725i
$750$	0	0
$751$	7.49415i	0.273466i	0.990608	+	0.136733i	$0.0436602\pi$
$751$	7.49415i	0.273466i	−0.990608	+	0.136733i	$0.956340\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−66.3491	−2.41469
$756$	0	0
$757$	36.6513	1.33211	0.666057	−	0.745901i	$-0.267981\pi$
$757$	36.6513	1.33211	0.666057	+	0.745901i	$0.267981\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	35.2777i	1.27881i	0.768869	+	0.639407i	$0.220820\pi$
$761$	35.2777i	1.27881i	−0.768869	+	0.639407i	$0.779180\pi$
$762$	0	0
$763$	47.2206i	1.70950i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.8626	−0.464442
$768$	0	0
$769$	38.9387	1.40417	0.702083	−	0.712095i	$-0.252253\pi$
$769$	38.9387	1.40417	0.702083	+	0.712095i	$0.252253\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 47.4524i	− 1.70674i	−0.521303	−	0.853372i	$-0.674554\pi$
$773$	− 47.4524i	− 1.70674i	0.521303	−	0.853372i	$-0.325446\pi$
$774$	0	0
$775$	− 15.9019i	− 0.571212i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−29.3357	−1.05106
$780$	0	0
$781$	−0.193437	−0.00692173
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 73.4888i	− 2.62293i
$786$	0	0
$787$	22.0301i	0.785287i	0.919691	+	0.392643i	$0.128439\pi$
$787$	22.0301i	0.785287i	−0.919691	+	0.392643i	$0.871561\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.14192	−0.182826
$792$	0	0
$793$	11.2628	0.399953
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.93939i	0.316650i	0.987387	+	0.158325i	$0.0506093\pi$
$797$	8.93939i	0.316650i	−0.987387	+	0.158325i	$0.949391\pi$
$798$	0	0
$799$	− 4.94009i	− 0.174768i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	35.2811	1.24504
$804$	0	0
$805$	9.46737	0.333681
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 8.93421i	− 0.314110i	−0.987590	−	0.157055i	$-0.949800\pi$
$809$	− 8.93421i	− 0.314110i	0.987590	−	0.157055i	$-0.0502000\pi$
$810$	0	0
$811$	− 22.1717i	− 0.778552i	−0.921121	−	0.389276i	$-0.872725\pi$
$811$	− 22.1717i	− 0.778552i	0.921121	−	0.389276i	$-0.127275\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−78.8038	−2.76038
$816$	0	0
$817$	−40.0015	−1.39947
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 2.11804i	− 0.0739202i	−0.999317	−	0.0369601i	$-0.988233\pi$
$821$	− 2.11804i	− 0.0739202i	0.999317	−	0.0369601i	$-0.0117674\pi$
$822$	0	0
$823$	− 23.5336i	− 0.820331i	−0.912011	−	0.410166i	$-0.865471\pi$
$823$	− 23.5336i	− 0.820331i	0.912011	−	0.410166i	$-0.134529\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−15.3293	−0.533051	−0.266526	−	0.963828i	$-0.585876\pi$
$827$	−15.3293	−0.533051	−0.266526	+	0.963828i	$0.585876\pi$
$828$	0	0
$829$	6.88566	0.239149	0.119574	−	0.992825i	$-0.461847\pi$
$829$	6.88566	0.239149	0.119574	+	0.992825i	$0.461847\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	10.7623i	0.372891i
$834$	0	0
$835$	20.3549i	0.704411i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−34.3712	−1.18662	−0.593312	−	0.804972i	$-0.702180\pi$
$839$	−34.3712	−1.18662	−0.593312	+	0.804972i	$0.702180\pi$
$840$	0	0
$841$	−40.0015	−1.37936
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 32.4600i	− 1.11666i
$846$	0	0
$847$	− 14.3707i	− 0.493783i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.00884	0.103142
$852$	0	0
$853$	−56.9600	−1.95027	−0.975136	−	0.221608i	$-0.928869\pi$
$853$	−56.9600	−1.95027	−0.975136	+	0.221608i	$0.928869\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14.9881i	0.511984i	0.966679	+	0.255992i	$0.0824021\pi$
$857$	14.9881i	0.511984i	−0.966679	+	0.255992i	$0.917598\pi$
$858$	0	0
$859$	43.1766i	1.47317i	0.676348	+	0.736583i	$0.263562\pi$
$859$	43.1766i	1.47317i	−0.676348	+	0.736583i	$0.736438\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	48.9439	1.66607	0.833035	−	0.553220i	$-0.186601\pi$
$863$	48.9439	1.66607	0.833035	+	0.553220i	$0.186601\pi$
$864$	0	0
$865$	20.5522	0.698797
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 13.7966i	− 0.468018i
$870$	0	0
$871$	47.9497i	1.62471i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−26.1958	−0.885581
$876$	0	0
$877$	−28.8560	−0.974399	−0.487200	−	0.873291i	$-0.661982\pi$
$877$	−28.8560	−0.974399	−0.487200	+	0.873291i	$0.661982\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 10.9613i	− 0.369294i	−0.982805	−	0.184647i	$-0.940886\pi$
$881$	− 10.9613i	− 0.369294i	0.982805	−	0.184647i	$-0.0591143\pi$
$882$	0	0
$883$	23.8707i	0.803313i	0.915790	+	0.401657i	$0.131565\pi$
$883$	23.8707i	0.803313i	−0.915790	+	0.401657i	$0.868435\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−54.5529	−1.83171	−0.915854	−	0.401512i	$-0.868485\pi$
$887$	−54.5529	−1.83171	−0.915854	+	0.401512i	$0.868485\pi$
$888$	0	0
$889$	−48.8841	−1.63952
$890$	0	0
$891$	0	0
$892$	0	0
$893$	12.5542i	0.420111i
$894$	0	0
$895$	− 44.7913i	− 1.49721i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	18.6329	0.621442
$900$	0	0
$901$	7.90754	0.263438
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 35.4421i	− 1.17814i
$906$	0	0
$907$	− 51.9103i	− 1.72365i	−0.507203	−	0.861827i	$-0.669321\pi$
$907$	− 51.9103i	− 1.72365i	0.507203	−	0.861827i	$-0.330679\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−54.0281	−1.79003	−0.895016	−	0.446035i	$-0.852836\pi$
$911$	−54.0281	−1.79003	−0.895016	+	0.446035i	$0.852836\pi$
$912$	0	0
$913$	−1.29699	−0.0429242
$914$	0	0
$915$	0	0
$916$	0	0
$917$	56.8470i	1.87725i
$918$	0	0
$919$	− 26.0348i	− 0.858808i	−0.903113	−	0.429404i	$-0.858724\pi$
$919$	− 26.0348i	− 0.858808i	0.903113	−	0.429404i	$-0.141276\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−0.345164	−0.0113612
$924$	0	0
$925$	−28.2501	−0.928859
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 38.2295i	− 1.25427i	−0.778911	−	0.627135i	$-0.784228\pi$
$929$	− 38.2295i	− 1.25427i	0.778911	−	0.627135i	$-0.215772\pi$
$930$	0	0
$931$	− 27.3501i	− 0.896364i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−16.5047	−0.539763
$936$	0	0
$937$	−19.6872	−0.643154	−0.321577	−	0.946883i	$-0.604213\pi$
$937$	−19.6872	−0.643154	−0.321577	+	0.946883i	$0.604213\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 30.6427i	− 0.998925i	−0.866336	−	0.499462i	$-0.833531\pi$
$941$	− 30.6427i	− 0.998925i	0.866336	−	0.499462i	$-0.166469\pi$
$942$	0	0
$943$	4.86320i	0.158368i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−33.6980	−1.09504	−0.547519	−	0.836793i	$-0.684428\pi$
$947$	−33.6980	−1.09504	−0.547519	+	0.836793i	$0.684428\pi$
$948$	0	0
$949$	62.9546	2.04359
$950$	0	0
$951$	0	0
$952$	0	0
$953$	58.8965i	1.90784i	0.300054	+	0.953922i	$0.402995\pi$
$953$	58.8965i	1.90784i	−0.300054	+	0.953922i	$0.597005\pi$
$954$	0	0
$955$	− 78.5516i	− 2.54187i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−27.9049	−0.901095
$960$	0	0
$961$	25.9684	0.837692
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 7.85341i	− 0.252810i
$966$	0	0
$967$	0.176804i	0.00568562i	0.999996	+	0.00284281i	$0.000904896\pi$
$967$	0.176804i	0.00568562i	−0.999996	+	0.00284281i	$0.999095\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	8.94370	0.287017	0.143509	−	0.989649i	$-0.454162\pi$
$971$	8.94370	0.287017	0.143509	+	0.989649i	$0.454162\pi$
$972$	0	0
$973$	−2.04183	−0.0654581
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.86182i	− 0.0595648i	−0.999556	−	0.0297824i	$-0.990519\pi$
$977$	− 1.86182i	− 0.0595648i	0.999556	−	0.0297824i	$-0.00948144\pi$
$978$	0	0
$979$	11.6859i	0.373483i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	16.8363	0.536995	0.268498	−	0.963280i	$-0.413473\pi$
$983$	16.8363	0.536995	0.268498	+	0.963280i	$0.413473\pi$
$984$	0	0
$985$	−19.6686	−0.626694
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6.63136i	0.210865i
$990$	0	0
$991$	− 15.9182i	− 0.505658i	−0.967511	−	0.252829i	$-0.918639\pi$
$991$	− 15.9182i	− 0.505658i	0.967511	−	0.252829i	$-0.0813609\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−48.0489	−1.52325
$996$	0	0
$997$	−6.80385	−0.215480	−0.107740	−	0.994179i	$-0.534361\pi$
$997$	−6.80385	−0.215480	−0.107740	+	0.994179i	$0.534361\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.c.m.5183.4		24
3.2	odd	2	inner	5184.2.c.m.5183.22		24
4.3	odd	2	inner	5184.2.c.m.5183.3		24
8.3	odd	2		2592.2.c.c.2591.21		24
8.5	even	2		2592.2.c.c.2591.22		24
9.2	odd	6		1728.2.s.g.1151.2		24
9.4	even	3		1728.2.s.g.575.1		24
9.5	odd	6		576.2.s.g.191.4		24
9.7	even	3		576.2.s.g.383.9		24
12.11	even	2	inner	5184.2.c.m.5183.21		24
24.5	odd	2		2592.2.c.c.2591.4		24
24.11	even	2		2592.2.c.c.2591.3		24
36.7	odd	6		576.2.s.g.383.4		24
36.11	even	6		1728.2.s.g.1151.1		24
36.23	even	6		576.2.s.g.191.9		24
36.31	odd	6		1728.2.s.g.575.2		24
72.5	odd	6		288.2.s.a.191.9	yes	24
72.11	even	6		864.2.s.a.287.11		24
72.13	even	6		864.2.s.a.575.11		24
72.29	odd	6		864.2.s.a.287.12		24
72.43	odd	6		288.2.s.a.95.9	yes	24
72.59	even	6		288.2.s.a.191.4	yes	24
72.61	even	6		288.2.s.a.95.4	✓	24
72.67	odd	6		864.2.s.a.575.12		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.2.s.a.95.4	✓	24	72.61	even	6
288.2.s.a.95.9	yes	24	72.43	odd	6
288.2.s.a.191.4	yes	24	72.59	even	6
288.2.s.a.191.9	yes	24	72.5	odd	6
576.2.s.g.191.4		24	9.5	odd	6
576.2.s.g.191.9		24	36.23	even	6
576.2.s.g.383.4		24	36.7	odd	6
576.2.s.g.383.9		24	9.7	even	3
864.2.s.a.287.11		24	72.11	even	6
864.2.s.a.287.12		24	72.29	odd	6
864.2.s.a.575.11		24	72.13	even	6
864.2.s.a.575.12		24	72.67	odd	6
1728.2.s.g.575.1		24	9.4	even	3
1728.2.s.g.575.2		24	36.31	odd	6
1728.2.s.g.1151.1		24	36.11	even	6
1728.2.s.g.1151.2		24	9.2	odd	6
2592.2.c.c.2591.3		24	24.11	even	2
2592.2.c.c.2591.4		24	24.5	odd	2
2592.2.c.c.2591.21		24	8.3	odd	2
2592.2.c.c.2591.22		24	8.5	even	2
5184.2.c.m.5183.3		24	4.3	odd	2	inner
5184.2.c.m.5183.4		24	1.1	even	1	trivial
5184.2.c.m.5183.21		24	12.11	even	2	inner
5184.2.c.m.5183.22		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:	\( N \)	\(=\)	\( 5184 = 2^{6} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5184.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.47695i q^{5} +3.60624i q^{7} +O(q^{10})\)	\(q-3.47695i q^{5} +3.60624i q^{7} +2.64859 q^{11} +4.72607 q^{13} -1.79223i q^{17} +4.55459i q^{19} +0.755051 q^{23} -7.08920 q^{25} -8.30671i q^{29} +2.24311i q^{31} +12.5387 q^{35} +3.98496 q^{37} +6.44089i q^{41} +8.78266i q^{43} +2.75638 q^{47} -6.00495 q^{49} +4.41211i q^{53} -9.20903i q^{55} -2.72163 q^{59} +2.38312 q^{61} -16.4323i q^{65} +10.1458i q^{67} -0.0730340 q^{71} +13.3207 q^{73} +9.55145i q^{77} -5.20903i q^{79} -0.489692 q^{83} -6.23151 q^{85} +4.41211i q^{89} +17.0433i q^{91} +15.8361 q^{95} -14.4371 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 24 q^{25} - 24 q^{49} + 24 q^{73} - 24 q^{97}+O(q^{100}) \) 24 * q - 24 * q^25 - 24 * q^49 + 24 * q^73 - 24 * q^97

Introduction

L-functions

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