LMFDB - Embedded newform 9216.2.a.br.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9216,2,Mod(1,9216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9216, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9216 = 2^{10} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9216.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:	$73.5901305028$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.3288334336.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 12x^{6} - 8x^{5} + 24x^{4} + 8x^{3} - 16x^{2} + 2 $ x^8 - 12x^6 - 8x^5 + 24x^4 + 8x^3 - 16*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 4608)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.08509$ of defining polynomial
Character	$\chi$	$=$	9216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.97127 q^{5} -2.27411 q^{7} +O(q^{10})$	$q-2.97127 q^{5} -2.27411 q^{7} -3.69552 q^{11} -3.41421 q^{13} +1.74053 q^{17} +7.76429 q^{19} +2.16478 q^{23} +3.82843 q^{25} +5.43275 q^{29} -8.70626 q^{31} +6.75699 q^{35} -0.585786 q^{37} +10.1445 q^{41} +1.33214 q^{43} +12.6173 q^{47} -1.82843 q^{49} -5.43275 q^{53} +10.9804 q^{55} +4.32957 q^{59} -7.41421 q^{61} +10.1445 q^{65} -6.43215 q^{67} +6.12293 q^{71} +10.4853 q^{73} +8.40401 q^{77} +2.27411 q^{79} -9.81845 q^{83} -5.17157 q^{85} +11.8851 q^{89} +7.76429 q^{91} -23.0698 q^{95} -15.3137 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 16 q^{13} + 8 q^{25} - 16 q^{37} + 8 q^{49} - 48 q^{61} + 16 q^{73} - 64 q^{85} - 32 q^{97}+O(q^{100}) $ 8 * q - 16 * q^13 + 8 * q^25 - 16 * q^37 + 8 * q^49 - 48 * q^61 + 16 * q^73 - 64 * q^85 - 32 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−2.97127	−1.32879	−0.664395	−	0.747381i	$-0.731311\pi$
$5$	−2.97127	−1.32879	−0.664395	+	0.747381i	$0.731311\pi$
$6$	0	0
$7$	−2.27411	−0.859533	−0.429766	−	0.902940i	$-0.641404\pi$
$7$	−2.27411	−0.859533	−0.429766	+	0.902940i	$0.641404\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.69552	−1.11424	−0.557120	−	0.830432i	$-0.688094\pi$
$11$	−3.69552	−1.11424	−0.557120	+	0.830432i	$0.688094\pi$
$12$	0	0
$13$	−3.41421	−0.946932	−0.473466	−	0.880812i	$-0.656997\pi$
$13$	−3.41421	−0.946932	−0.473466	+	0.880812i	$0.656997\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.74053	0.422140	0.211070	−	0.977471i	$-0.432305\pi$
$17$	1.74053	0.422140	0.211070	+	0.977471i	$0.432305\pi$
$18$	0	0
$19$	7.76429	1.78125	0.890626	−	0.454737i	$-0.150267\pi$
$19$	7.76429	1.78125	0.890626	+	0.454737i	$0.150267\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.16478	0.451389	0.225694	−	0.974198i	$-0.427535\pi$
$23$	2.16478	0.451389	0.225694	+	0.974198i	$0.427535\pi$
$24$	0	0
$25$	3.82843	0.765685
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.43275	1.00884	0.504418	−	0.863460i	$-0.331707\pi$
$29$	5.43275	1.00884	0.504418	+	0.863460i	$0.331707\pi$
$30$	0	0
$31$	−8.70626	−1.56369	−0.781845	−	0.623472i	$-0.785721\pi$
$31$	−8.70626	−1.56369	−0.781845	+	0.623472i	$0.785721\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	6.75699	1.14214
$36$	0	0
$37$	−0.585786	−0.0963027	−0.0481513	−	0.998840i	$-0.515333\pi$
$37$	−0.585786	−0.0963027	−0.0481513	+	0.998840i	$0.515333\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.1445	1.58431	0.792155	−	0.610319i	$-0.208959\pi$
$41$	10.1445	1.58431	0.792155	+	0.610319i	$0.208959\pi$
$42$	0	0
$43$	1.33214	0.203150	0.101575	−	0.994828i	$-0.467612\pi$
$43$	1.33214	0.203150	0.101575	+	0.994828i	$0.467612\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.6173	1.84042	0.920210	−	0.391424i	$-0.128018\pi$
$47$	12.6173	1.84042	0.920210	+	0.391424i	$0.128018\pi$
$48$	0	0
$49$	−1.82843	−0.261204
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.43275	−0.746245	−0.373122	−	0.927782i	$-0.621713\pi$
$53$	−5.43275	−0.746245	−0.373122	+	0.927782i	$0.621713\pi$
$54$	0	0
$55$	10.9804	1.48059
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.32957	0.563662	0.281831	−	0.959464i	$-0.409058\pi$
$59$	4.32957	0.563662	0.281831	+	0.959464i	$0.409058\pi$
$60$	0	0
$61$	−7.41421	−0.949293	−0.474646	−	0.880177i	$-0.657424\pi$
$61$	−7.41421	−0.949293	−0.474646	+	0.880177i	$0.657424\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	10.1445	1.25828
$66$	0	0
$67$	−6.43215	−0.785812	−0.392906	−	0.919579i	$-0.628530\pi$
$67$	−6.43215	−0.785812	−0.392906	+	0.919579i	$0.628530\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.12293	0.726659	0.363329	−	0.931661i	$-0.381640\pi$
$71$	6.12293	0.726659	0.363329	+	0.931661i	$0.381640\pi$
$72$	0	0
$73$	10.4853	1.22721	0.613605	−	0.789613i	$-0.289719\pi$
$73$	10.4853	1.22721	0.613605	+	0.789613i	$0.289719\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.40401	0.957726
$78$	0	0
$79$	2.27411	0.255857	0.127929	−	0.991783i	$-0.459167\pi$
$79$	2.27411	0.255857	0.127929	+	0.991783i	$0.459167\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.81845	−1.07772	−0.538858	−	0.842397i	$-0.681144\pi$
$83$	−9.81845	−1.07772	−0.538858	+	0.842397i	$0.681144\pi$
$84$	0	0
$85$	−5.17157	−0.560936
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.8851	1.25981	0.629907	−	0.776670i	$-0.283093\pi$
$89$	11.8851	1.25981	0.629907	+	0.776670i	$0.283093\pi$
$90$	0	0
$91$	7.76429	0.813919
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−23.0698	−2.36691
$96$	0	0
$97$	−15.3137	−1.55487	−0.777436	−	0.628962i	$-0.783480\pi$
$97$	−15.3137	−1.55487	−0.777436	+	0.628962i	$0.783480\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.89422	−0.785505	−0.392752	−	0.919644i	$-0.628477\pi$
$101$	−7.89422	−0.785505	−0.392752	+	0.919644i	$0.628477\pi$
$102$	0	0
$103$	8.70626	0.857853	0.428927	−	0.903339i	$-0.358892\pi$
$103$	8.70626	0.857853	0.428927	+	0.903339i	$0.358892\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.7206	−1.13307	−0.566537	−	0.824036i	$-0.691717\pi$
$107$	−11.7206	−1.13307	−0.566537	+	0.824036i	$0.691717\pi$
$108$	0	0
$109$	1.07107	0.102590	0.0512948	−	0.998684i	$-0.483665\pi$
$109$	1.07107	0.102590	0.0512948	+	0.998684i	$0.483665\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.8080	1.58117	0.790583	−	0.612355i	$-0.209778\pi$
$113$	16.8080	1.58117	0.790583	+	0.612355i	$0.209778\pi$
$114$	0	0
$115$	−6.43215	−0.599801
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.95815	−0.362843
$120$	0	0
$121$	2.65685	0.241532
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.48106	0.311355
$126$	0	0
$127$	13.2545	1.17614	0.588072	−	0.808808i	$-0.299887\pi$
$127$	13.2545	1.17614	0.588072	+	0.808808i	$0.299887\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.5140	1.18072	0.590361	−	0.807140i	$-0.298986\pi$
$131$	13.5140	1.18072	0.590361	+	0.807140i	$0.298986\pi$
$132$	0	0
$133$	−17.6569	−1.53104
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.0675	−1.28730	−0.643652	−	0.765319i	$-0.722581\pi$
$137$	−15.0675	−1.28730	−0.643652	+	0.765319i	$0.722581\pi$
$138$	0	0
$139$	12.8643	1.09114	0.545568	−	0.838067i	$-0.316314\pi$
$139$	12.8643	1.09114	0.545568	+	0.838067i	$0.316314\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	12.6173	1.05511
$144$	0	0
$145$	−16.1421	−1.34053
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.91380	−0.730247	−0.365124	−	0.930959i	$-0.618973\pi$
$149$	−8.91380	−0.730247	−0.365124	+	0.930959i	$0.618973\pi$
$150$	0	0
$151$	−13.2545	−1.07863	−0.539317	−	0.842103i	$-0.681318\pi$
$151$	−13.2545	−1.07863	−0.539317	+	0.842103i	$0.681318\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	25.8686	2.07782
$156$	0	0
$157$	−5.75736	−0.459487	−0.229744	−	0.973251i	$-0.573789\pi$
$157$	−5.75736	−0.459487	−0.229744	+	0.973251i	$0.573789\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.92296	−0.387983
$162$	0	0
$163$	20.6286	1.61576	0.807878	−	0.589349i	$-0.200616\pi$
$163$	20.6286	1.61576	0.807878	+	0.589349i	$0.200616\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.2764	−1.64642	−0.823210	−	0.567738i	$-0.807819\pi$
$167$	−21.2764	−1.64642	−0.823210	+	0.567738i	$0.807819\pi$
$168$	0	0
$169$	−1.34315	−0.103319
$170$	0	0
$171$	0	0
$172$	0	0
$173$	11.3753	0.864846	0.432423	−	0.901671i	$-0.357659\pi$
$173$	11.3753	0.864846	0.432423	+	0.901671i	$0.357659\pi$
$174$	0	0
$175$	−8.70626	−0.658132
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−23.9665	−1.79134	−0.895669	−	0.444721i	$-0.853303\pi$
$179$	−23.9665	−1.79134	−0.895669	+	0.444721i	$0.853303\pi$
$180$	0	0
$181$	−26.7279	−1.98667	−0.993335	−	0.115260i	$-0.963230\pi$
$181$	−26.7279	−1.98667	−0.993335	+	0.115260i	$0.963230\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.74053	0.127966
$186$	0	0
$187$	−6.43215	−0.470366
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.9469	−1.22623	−0.613116	−	0.789993i	$-0.710084\pi$
$191$	−16.9469	−1.22623	−0.613116	+	0.789993i	$0.710084\pi$
$192$	0	0
$193$	−4.48528	−0.322858	−0.161429	−	0.986884i	$-0.551610\pi$
$193$	−4.48528	−0.322858	−0.161429	+	0.986884i	$0.551610\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.3753	0.810455	0.405228	−	0.914216i	$-0.367192\pi$
$197$	11.3753	0.810455	0.405228	+	0.914216i	$0.367192\pi$
$198$	0	0
$199$	0.390175	0.0276588	0.0138294	−	0.999904i	$-0.495598\pi$
$199$	0.390175	0.0276588	0.0138294	+	0.999904i	$0.495598\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−12.3547	−0.867127
$204$	0	0
$205$	−30.1421	−2.10522
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−28.6931	−1.98474
$210$	0	0
$211$	−18.1929	−1.25245	−0.626225	−	0.779643i	$-0.715401\pi$
$211$	−18.1929	−1.25245	−0.626225	+	0.779643i	$0.715401\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.95815	−0.269944
$216$	0	0
$217$	19.7990	1.34404
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5.94253	−0.399738
$222$	0	0
$223$	10.5902	0.709172	0.354586	−	0.935023i	$-0.384622\pi$
$223$	10.5902	0.709172	0.354586	+	0.935023i	$0.384622\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.42742	0.161113	0.0805567	−	0.996750i	$-0.474330\pi$
$227$	2.42742	0.161113	0.0805567	+	0.996750i	$0.474330\pi$
$228$	0	0
$229$	−10.9289	−0.722204	−0.361102	−	0.932526i	$-0.617599\pi$
$229$	−10.9289	−0.722204	−0.361102	+	0.932526i	$0.617599\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.7310	1.42364	0.711822	−	0.702360i	$-0.247870\pi$
$233$	21.7310	1.42364	0.711822	+	0.702360i	$0.247870\pi$
$234$	0	0
$235$	−37.4893	−2.44553
$236$	0	0
$237$	0	0
$238$	0	0
$239$	18.7402	1.21220	0.606102	−	0.795387i	$-0.292732\pi$
$239$	18.7402	1.21220	0.606102	+	0.795387i	$0.292732\pi$
$240$	0	0
$241$	−18.1421	−1.16864	−0.584319	−	0.811524i	$-0.698638\pi$
$241$	−18.1421	−1.16864	−0.584319	+	0.811524i	$0.698638\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5.43275	0.347085
$246$	0	0
$247$	−26.5090	−1.68672
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.4776	1.16630	0.583148	−	0.812366i	$-0.301821\pi$
$251$	18.4776	1.16630	0.583148	+	0.812366i	$0.301821\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.92296	0.307086	0.153543	−	0.988142i	$-0.450932\pi$
$257$	4.92296	0.307086	0.153543	+	0.988142i	$0.450932\pi$
$258$	0	0
$259$	1.33214	0.0827753
$260$	0	0
$261$	0	0
$262$	0	0
$263$	16.5754	1.02209	0.511043	−	0.859555i	$-0.329259\pi$
$263$	16.5754	1.02209	0.511043	+	0.859555i	$0.329259\pi$
$264$	0	0
$265$	16.1421	0.991604
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−25.7218	−1.56829	−0.784144	−	0.620579i	$-0.786897\pi$
$269$	−25.7218	−1.56829	−0.784144	+	0.620579i	$0.786897\pi$
$270$	0	0
$271$	−11.3705	−0.690711	−0.345356	−	0.938472i	$-0.612242\pi$
$271$	−11.3705	−0.690711	−0.345356	+	0.938472i	$0.612242\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−14.1480	−0.853158
$276$	0	0
$277$	8.38478	0.503792	0.251896	−	0.967754i	$-0.418946\pi$
$277$	8.38478	0.503792	0.251896	+	0.967754i	$0.418946\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−18.8472	−1.12433	−0.562164	−	0.827026i	$-0.690031\pi$
$281$	−18.8472	−1.12433	−0.562164	+	0.827026i	$0.690031\pi$
$282$	0	0
$283$	−24.6250	−1.46381	−0.731903	−	0.681409i	$-0.761368\pi$
$283$	−24.6250	−1.46381	−0.731903	+	0.681409i	$0.761368\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−23.0698	−1.36177
$288$	0	0
$289$	−13.9706	−0.821798
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.41317	0.257820	0.128910	−	0.991656i	$-0.458852\pi$
$293$	4.41317	0.257820	0.128910	+	0.991656i	$0.458852\pi$
$294$	0	0
$295$	−12.8643	−0.748989
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.39104	−0.427435
$300$	0	0
$301$	−3.02944	−0.174614
$302$	0	0
$303$	0	0
$304$	0	0
$305$	22.0296	1.26141
$306$	0	0
$307$	−6.43215	−0.367102	−0.183551	−	0.983010i	$-0.558759\pi$
$307$	−6.43215	−0.367102	−0.183551	+	0.983010i	$0.558759\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4.32957	0.245507	0.122754	−	0.992437i	$-0.460827\pi$
$311$	4.32957	0.245507	0.122754	+	0.992437i	$0.460827\pi$
$312$	0	0
$313$	18.1421	1.02545	0.512727	−	0.858552i	$-0.328635\pi$
$313$	18.1421	1.02545	0.512727	+	0.858552i	$0.328635\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.93338	−0.557914	−0.278957	−	0.960304i	$-0.589989\pi$
$317$	−9.93338	−0.557914	−0.278957	+	0.960304i	$0.589989\pi$
$318$	0	0
$319$	−20.0768	−1.12409
$320$	0	0
$321$	0	0
$322$	0	0
$323$	13.5140	0.751937
$324$	0	0
$325$	−13.0711	−0.725052
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−28.6931	−1.58190
$330$	0	0
$331$	12.8643	0.707086	0.353543	−	0.935418i	$-0.384977\pi$
$331$	12.8643	0.707086	0.353543	+	0.935418i	$0.384977\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	19.1116	1.04418
$336$	0	0
$337$	−2.48528	−0.135382	−0.0676910	−	0.997706i	$-0.521563\pi$
$337$	−2.48528	−0.135382	−0.0676910	+	0.997706i	$0.521563\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	32.1741	1.74233
$342$	0	0
$343$	20.0768	1.08405
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−23.3324	−1.25255	−0.626275	−	0.779602i	$-0.715421\pi$
$347$	−23.3324	−1.25255	−0.626275	+	0.779602i	$0.715421\pi$
$348$	0	0
$349$	−30.2426	−1.61885	−0.809426	−	0.587222i	$-0.800221\pi$
$349$	−30.2426	−1.61885	−0.809426	+	0.587222i	$0.800221\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−35.6552	−1.89773	−0.948867	−	0.315675i	$-0.897769\pi$
$353$	−35.6552	−1.89773	−0.948867	+	0.315675i	$0.897769\pi$
$354$	0	0
$355$	−18.1929	−0.965577
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.4525	0.551662	0.275831	−	0.961206i	$-0.411047\pi$
$359$	10.4525	0.551662	0.275831	+	0.961206i	$0.411047\pi$
$360$	0	0
$361$	41.2843	2.17286
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−31.1546	−1.63070
$366$	0	0
$367$	26.1188	1.36339	0.681695	−	0.731637i	$-0.261243\pi$
$367$	26.1188	1.36339	0.681695	+	0.731637i	$0.261243\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	12.3547	0.641422
$372$	0	0
$373$	−30.7279	−1.59103	−0.795516	−	0.605933i	$-0.792800\pi$
$373$	−30.7279	−1.59103	−0.795516	+	0.605933i	$0.792800\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−18.5486	−0.955299
$378$	0	0
$379$	14.1964	0.729222	0.364611	−	0.931160i	$-0.381202\pi$
$379$	14.1964	0.729222	0.364611	+	0.931160i	$0.381202\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.95815	−0.202252	−0.101126	−	0.994874i	$-0.532245\pi$
$383$	−3.95815	−0.202252	−0.101126	+	0.994874i	$0.532245\pi$
$384$	0	0
$385$	−24.9706	−1.27262
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−17.3178	−0.878048	−0.439024	−	0.898475i	$-0.644676\pi$
$389$	−17.3178	−0.878048	−0.439024	+	0.898475i	$0.644676\pi$
$390$	0	0
$391$	3.76787	0.190549
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6.75699	−0.339981
$396$	0	0
$397$	−20.3848	−1.02308	−0.511541	−	0.859259i	$-0.670925\pi$
$397$	−20.3848	−1.02308	−0.511541	+	0.859259i	$0.670925\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−35.3566	−1.76562	−0.882812	−	0.469727i	$-0.844352\pi$
$401$	−35.3566	−1.76562	−0.882812	+	0.469727i	$0.844352\pi$
$402$	0	0
$403$	29.7250	1.48071
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.16478	0.107304
$408$	0	0
$409$	9.65685	0.477501	0.238750	−	0.971081i	$-0.423262\pi$
$409$	9.65685	0.477501	0.238750	+	0.971081i	$0.423262\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−9.84591	−0.484486
$414$	0	0
$415$	29.1732	1.43206
$416$	0	0
$417$	0	0
$418$	0	0
$419$	33.2597	1.62484	0.812420	−	0.583072i	$-0.198149\pi$
$419$	33.2597	1.62484	0.812420	+	0.583072i	$0.198149\pi$
$420$	0	0
$421$	8.38478	0.408649	0.204324	−	0.978903i	$-0.434500\pi$
$421$	8.38478	0.408649	0.204324	+	0.978903i	$0.434500\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.66348	0.323226
$426$	0	0
$427$	16.8607	0.815948
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.49435	0.312822	0.156411	−	0.987692i	$-0.450008\pi$
$431$	6.49435	0.312822	0.156411	+	0.987692i	$0.450008\pi$
$432$	0	0
$433$	−4.97056	−0.238870	−0.119435	−	0.992842i	$-0.538108\pi$
$433$	−4.97056	−0.238870	−0.119435	+	0.992842i	$0.538108\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	16.8080	0.804037
$438$	0	0
$439$	9.48661	0.452771	0.226386	−	0.974038i	$-0.427309\pi$
$439$	9.48661	0.452771	0.226386	+	0.974038i	$0.427309\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9.81845	0.466489	0.233244	−	0.972418i	$-0.425066\pi$
$443$	9.81845	0.466489	0.233244	+	0.972418i	$0.425066\pi$
$444$	0	0
$445$	−35.3137	−1.67403
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.66348	0.314469	0.157235	−	0.987561i	$-0.449742\pi$
$449$	6.66348	0.314469	0.157235	+	0.987561i	$0.449742\pi$
$450$	0	0
$451$	−37.4893	−1.76530
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−23.0698	−1.08153
$456$	0	0
$457$	7.31371	0.342121	0.171060	−	0.985261i	$-0.445281\pi$
$457$	7.31371	0.342121	0.171060	+	0.985261i	$0.445281\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	24.2799	1.13083	0.565414	−	0.824807i	$-0.308716\pi$
$461$	24.2799	1.13083	0.565414	+	0.824807i	$0.308716\pi$
$462$	0	0
$463$	31.4474	1.46148	0.730741	−	0.682655i	$-0.239175\pi$
$463$	31.4474	1.46148	0.730741	+	0.682655i	$0.239175\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.2095	0.796360	0.398180	−	0.917307i	$-0.369642\pi$
$467$	17.2095	0.796360	0.398180	+	0.917307i	$0.369642\pi$
$468$	0	0
$469$	14.6274	0.675431
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.92296	−0.226358
$474$	0	0
$475$	29.7250	1.36388
$476$	0	0
$477$	0	0
$478$	0	0
$479$	31.3575	1.43276	0.716381	−	0.697710i	$-0.245797\pi$
$479$	31.3575	1.43276	0.716381	+	0.697710i	$0.245797\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	45.5011	2.06610
$486$	0	0
$487$	−28.7831	−1.30429	−0.652143	−	0.758096i	$-0.726130\pi$
$487$	−28.7831	−1.30429	−0.652143	+	0.758096i	$0.726130\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7.39104	−0.333553	−0.166776	−	0.985995i	$-0.553336\pi$
$491$	−7.39104	−0.333553	−0.166776	+	0.985995i	$0.553336\pi$
$492$	0	0
$493$	9.45584	0.425870
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13.9242	−0.624587
$498$	0	0
$499$	6.43215	0.287943	0.143971	−	0.989582i	$-0.454013\pi$
$499$	6.43215	0.287943	0.143971	+	0.989582i	$0.454013\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	10.4525	0.466054	0.233027	−	0.972470i	$-0.425137\pi$
$503$	10.4525	0.466054	0.233027	+	0.972470i	$0.425137\pi$
$504$	0	0
$505$	23.4558	1.04377
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6.45232	0.285994	0.142997	−	0.989723i	$-0.454326\pi$
$509$	6.45232	0.285994	0.142997	+	0.989723i	$0.454326\pi$
$510$	0	0
$511$	−23.8447	−1.05483
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−25.8686	−1.13991
$516$	0	0
$517$	−46.6274	−2.05067
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−23.4715	−1.02831	−0.514153	−	0.857699i	$-0.671894\pi$
$521$	−23.4715	−1.02831	−0.514153	+	0.857699i	$0.671894\pi$
$522$	0	0
$523$	−5.10001	−0.223008	−0.111504	−	0.993764i	$-0.535567\pi$
$523$	−5.10001	−0.223008	−0.111504	+	0.993764i	$0.535567\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.1535	−0.660096
$528$	0	0
$529$	−18.3137	−0.796248
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−34.6356	−1.50024
$534$	0	0
$535$	34.8250	1.50562
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.75699	0.291044
$540$	0	0
$541$	18.7279	0.805176	0.402588	−	0.915381i	$-0.368111\pi$
$541$	18.7279	0.805176	0.402588	+	0.915381i	$0.368111\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.18243	−0.136320
$546$	0	0
$547$	3.99643	0.170875	0.0854374	−	0.996344i	$-0.472771\pi$
$547$	3.99643	0.170875	0.0854374	+	0.996344i	$0.472771\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	42.1814	1.79699
$552$	0	0
$553$	−5.17157	−0.219918
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.97127	0.125897	0.0629483	−	0.998017i	$-0.479950\pi$
$557$	2.97127	0.125897	0.0629483	+	0.998017i	$0.479950\pi$
$558$	0	0
$559$	−4.54822	−0.192369
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−13.6228	−0.574131	−0.287065	−	0.957911i	$-0.592680\pi$
$563$	−13.6228	−0.574131	−0.287065	+	0.957911i	$0.592680\pi$
$564$	0	0
$565$	−49.9411	−2.10104
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13.6256	−0.571215	−0.285607	−	0.958347i	$-0.592195\pi$
$569$	−13.6256	−0.571215	−0.285607	+	0.958347i	$0.592195\pi$
$570$	0	0
$571$	−24.6250	−1.03053	−0.515263	−	0.857032i	$-0.672306\pi$
$571$	−24.6250	−1.03053	−0.515263	+	0.857032i	$0.672306\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.28772	0.345622
$576$	0	0
$577$	−25.1127	−1.04546	−0.522728	−	0.852500i	$-0.675086\pi$
$577$	−25.1127	−1.04546	−0.522728	+	0.852500i	$0.675086\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	22.3282	0.926331
$582$	0	0
$583$	20.0768	0.831496
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.12293	−0.252721	−0.126360	−	0.991984i	$-0.540330\pi$
$587$	−6.12293	−0.252721	−0.126360	+	0.991984i	$0.540330\pi$
$588$	0	0
$589$	−67.5980	−2.78533
$590$	0	0
$591$	0	0
$592$	0	0
$593$	16.8080	0.690223	0.345111	−	0.938562i	$-0.387841\pi$
$593$	16.8080	0.690223	0.345111	+	0.938562i	$0.387841\pi$
$594$	0	0
$595$	11.7607	0.482143
$596$	0	0
$597$	0	0
$598$	0	0
$599$	18.7402	0.765705	0.382852	−	0.923810i	$-0.374942\pi$
$599$	18.7402	0.765705	0.382852	+	0.923810i	$0.374942\pi$
$600$	0	0
$601$	−15.1716	−0.618861	−0.309431	−	0.950922i	$-0.600138\pi$
$601$	−15.1716	−0.618861	−0.309431	+	0.950922i	$0.600138\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−7.89422	−0.320946
$606$	0	0
$607$	−39.7634	−1.61395	−0.806974	−	0.590587i	$-0.798896\pi$
$607$	−39.7634	−1.61395	−0.806974	+	0.590587i	$0.798896\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−43.0781	−1.74275
$612$	0	0
$613$	−0.100505	−0.00405936	−0.00202968	−	0.999998i	$-0.500646\pi$
$613$	−0.100505	−0.00405936	−0.00202968	+	0.999998i	$0.500646\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	18.1929	0.731233	0.365617	−	0.930766i	$-0.380858\pi$
$619$	18.1929	0.731233	0.365617	+	0.930766i	$0.380858\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−27.0279	−1.08285
$624$	0	0
$625$	−29.4853	−1.17941
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.01958	−0.0406532
$630$	0	0
$631$	−3.05446	−0.121596	−0.0607981	−	0.998150i	$-0.519365\pi$
$631$	−3.05446	−0.121596	−0.0607981	+	0.998150i	$0.519365\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−39.3826	−1.56285
$636$	0	0
$637$	6.24264	0.247342
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.74053	0.0687467	0.0343734	−	0.999409i	$-0.489056\pi$
$641$	1.74053	0.0687467	0.0343734	+	0.999409i	$0.489056\pi$
$642$	0	0
$643$	23.2929	0.918582	0.459291	−	0.888286i	$-0.348104\pi$
$643$	23.2929	0.918582	0.459291	+	0.888286i	$0.348104\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−35.3157	−1.38840	−0.694201	−	0.719781i	$-0.744242\pi$
$647$	−35.3157	−1.38840	−0.694201	+	0.719781i	$0.744242\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−13.4144	−0.524947	−0.262474	−	0.964939i	$-0.584538\pi$
$653$	−13.4144	−0.524947	−0.262474	+	0.964939i	$0.584538\pi$
$654$	0	0
$655$	−40.1536	−1.56893
$656$	0	0
$657$	0	0
$658$	0	0
$659$	7.39104	0.287914	0.143957	−	0.989584i	$-0.454017\pi$
$659$	7.39104	0.287914	0.143957	+	0.989584i	$0.454017\pi$
$660$	0	0
$661$	−3.61522	−0.140616	−0.0703080	−	0.997525i	$-0.522398\pi$
$661$	−3.61522	−0.140616	−0.0703080	+	0.997525i	$0.522398\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	52.4632	2.03444
$666$	0	0
$667$	11.7607	0.455377
$668$	0	0
$669$	0	0
$670$	0	0
$671$	27.3994	1.05774
$672$	0	0
$673$	17.6569	0.680622	0.340311	−	0.940313i	$-0.389468\pi$
$673$	17.6569	0.680622	0.340311	+	0.940313i	$0.389468\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−42.5298	−1.63455	−0.817277	−	0.576244i	$-0.804518\pi$
$677$	−42.5298	−1.63455	−0.817277	+	0.576244i	$0.804518\pi$
$678$	0	0
$679$	34.8250	1.33646
$680$	0	0
$681$	0	0
$682$	0	0
$683$	40.6507	1.55546	0.777728	−	0.628601i	$-0.216372\pi$
$683$	40.6507	1.55546	0.777728	+	0.628601i	$0.216372\pi$
$684$	0	0
$685$	44.7696	1.71056
$686$	0	0
$687$	0	0
$688$	0	0
$689$	18.5486	0.706644
$690$	0	0
$691$	7.76429	0.295368	0.147684	−	0.989035i	$-0.452818\pi$
$691$	7.76429	0.295368	0.147684	+	0.989035i	$0.452818\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−38.2233	−1.44989
$696$	0	0
$697$	17.6569	0.668801
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−32.6839	−1.23446	−0.617228	−	0.786785i	$-0.711744\pi$
$701$	−32.6839	−1.23446	−0.617228	+	0.786785i	$0.711744\pi$
$702$	0	0
$703$	−4.54822	−0.171539
$704$	0	0
$705$	0	0
$706$	0	0
$707$	17.9523	0.675167
$708$	0	0
$709$	−18.7279	−0.703342	−0.351671	−	0.936124i	$-0.614386\pi$
$709$	−18.7279	−0.703342	−0.351671	+	0.936124i	$0.614386\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−18.8472	−0.705832
$714$	0	0
$715$	−37.4893	−1.40202
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−14.7821	−0.551278	−0.275639	−	0.961261i	$-0.588889\pi$
$719$	−14.7821	−0.551278	−0.275639	+	0.961261i	$0.588889\pi$
$720$	0	0
$721$	−19.7990	−0.737353
$722$	0	0
$723$	0	0
$724$	0	0
$725$	20.7989	0.772451
$726$	0	0
$727$	−30.6670	−1.13738	−0.568688	−	0.822553i	$-0.692549\pi$
$727$	−30.6670	−1.13738	−0.568688	+	0.822553i	$0.692549\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.31863	0.0857577
$732$	0	0
$733$	1.55635	0.0574851	0.0287425	−	0.999587i	$-0.490850\pi$
$733$	1.55635	0.0574851	0.0287425	+	0.999587i	$0.490850\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	23.7701	0.875584
$738$	0	0
$739$	24.6250	0.905846	0.452923	−	0.891550i	$-0.350381\pi$
$739$	24.6250	0.905846	0.452923	+	0.891550i	$0.350381\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−18.7402	−0.687512	−0.343756	−	0.939059i	$-0.611699\pi$
$743$	−18.7402	−0.687512	−0.343756	+	0.939059i	$0.611699\pi$
$744$	0	0
$745$	26.4853	0.970346
$746$	0	0
$747$	0	0
$748$	0	0
$749$	26.6539	0.973914
$750$	0	0
$751$	−40.8670	−1.49126	−0.745629	−	0.666361i	$-0.767851\pi$
$751$	−40.8670	−1.49126	−0.745629	+	0.666361i	$0.767851\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	39.3826	1.43328
$756$	0	0
$757$	1.95837	0.0711781	0.0355891	−	0.999367i	$-0.488669\pi$
$757$	1.95837	0.0711781	0.0355891	+	0.999367i	$0.488669\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	23.4715	0.850842	0.425421	−	0.904996i	$-0.360126\pi$
$761$	23.4715	0.850842	0.425421	+	0.904996i	$0.360126\pi$
$762$	0	0
$763$	−2.43573	−0.0881792
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−14.7821	−0.533750
$768$	0	0
$769$	−30.8284	−1.11170	−0.555851	−	0.831282i	$-0.687607\pi$
$769$	−30.8284	−1.11170	−0.555851	+	0.831282i	$0.687607\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−21.8184	−0.784755	−0.392377	−	0.919804i	$-0.628347\pi$
$773$	−21.8184	−0.784755	−0.392377	+	0.919804i	$0.628347\pi$
$774$	0	0
$775$	−33.3313	−1.19730
$776$	0	0
$777$	0	0
$778$	0	0
$779$	78.7652	2.82206
$780$	0	0
$781$	−22.6274	−0.809673
$782$	0	0
$783$	0	0
$784$	0	0
$785$	17.1067	0.610563
$786$	0	0
$787$	−14.1964	−0.506049	−0.253024	−	0.967460i	$-0.581425\pi$
$787$	−14.1964	−0.506049	−0.253024	+	0.967460i	$0.581425\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−38.2233	−1.35906
$792$	0	0
$793$	25.3137	0.898916
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.7597	0.664503	0.332252	−	0.943191i	$-0.392192\pi$
$797$	18.7597	0.664503	0.332252	+	0.943191i	$0.392192\pi$
$798$	0	0
$799$	21.9607	0.776915
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−38.7485	−1.36741
$804$	0	0
$805$	14.6274	0.515549
$806$	0	0
$807$	0	0
$808$	0	0
$809$	37.3957	1.31476	0.657382	−	0.753558i	$-0.271664\pi$
$809$	37.3957	1.31476	0.657382	+	0.753558i	$0.271664\pi$
$810$	0	0
$811$	24.3965	0.856676	0.428338	−	0.903619i	$-0.359099\pi$
$811$	24.3965	0.856676	0.428338	+	0.903619i	$0.359099\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−61.2931	−2.14700
$816$	0	0
$817$	10.3431	0.361861
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−30.6448	−1.06951	−0.534755	−	0.845007i	$-0.679596\pi$
$821$	−30.6448	−1.06951	−0.534755	+	0.845007i	$0.679596\pi$
$822$	0	0
$823$	2.27411	0.0792705	0.0396352	−	0.999214i	$-0.487380\pi$
$823$	2.27411	0.0792705	0.0396352	+	0.999214i	$0.487380\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−25.2346	−0.877492	−0.438746	−	0.898611i	$-0.644577\pi$
$827$	−25.2346	−0.877492	−0.438746	+	0.898611i	$0.644577\pi$
$828$	0	0
$829$	−21.5563	−0.748683	−0.374341	−	0.927291i	$-0.622131\pi$
$829$	−21.5563	−0.748683	−0.374341	+	0.927291i	$0.622131\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.18243	−0.110265
$834$	0	0
$835$	63.2179	2.18775
$836$	0	0
$837$	0	0
$838$	0	0
$839$	12.9887	0.448420	0.224210	−	0.974541i	$-0.428020\pi$
$839$	12.9887	0.448420	0.224210	+	0.974541i	$0.428020\pi$
$840$	0	0
$841$	0.514719	0.0177489
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.99084	0.137289
$846$	0	0
$847$	−6.04198	−0.207605
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.26810	−0.0434700
$852$	0	0
$853$	14.5269	0.497392	0.248696	−	0.968582i	$-0.419998\pi$
$853$	14.5269	0.497392	0.248696	+	0.968582i	$0.419998\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.5486	−0.633606	−0.316803	−	0.948491i	$-0.602609\pi$
$857$	−18.5486	−0.633606	−0.316803	+	0.948491i	$0.602609\pi$
$858$	0	0
$859$	−20.6286	−0.703839	−0.351919	−	0.936030i	$-0.614471\pi$
$859$	−20.6286	−0.703839	−0.351919	+	0.936030i	$0.614471\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	33.5223	1.14111	0.570556	−	0.821259i	$-0.306728\pi$
$863$	33.5223	1.14111	0.570556	+	0.821259i	$0.306728\pi$
$864$	0	0
$865$	−33.7990	−1.14920
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8.40401	−0.285087
$870$	0	0
$871$	21.9607	0.744111
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−7.91630	−0.267620
$876$	0	0
$877$	45.8406	1.54793	0.773964	−	0.633230i	$-0.218271\pi$
$877$	45.8406	1.54793	0.773964	+	0.633230i	$0.218271\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−14.7689	−0.497576	−0.248788	−	0.968558i	$-0.580032\pi$
$881$	−14.7689	−0.497576	−0.248788	+	0.968558i	$0.580032\pi$
$882$	0	0
$883$	−16.8607	−0.567409	−0.283704	−	0.958912i	$-0.591563\pi$
$883$	−16.8607	−0.567409	−0.283704	+	0.958912i	$0.591563\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	43.6034	1.46406	0.732029	−	0.681273i	$-0.238573\pi$
$887$	43.6034	1.46406	0.732029	+	0.681273i	$0.238573\pi$
$888$	0	0
$889$	−30.1421	−1.01093
$890$	0	0
$891$	0	0
$892$	0	0
$893$	97.9643	3.27825
$894$	0	0
$895$	71.2108	2.38031
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−47.2989	−1.57751
$900$	0	0
$901$	−9.45584	−0.315020
$902$	0	0
$903$	0	0
$904$	0	0
$905$	79.4158	2.63987
$906$	0	0
$907$	10.4286	0.346275	0.173138	−	0.984898i	$-0.444609\pi$
$907$	10.4286	0.346275	0.173138	+	0.984898i	$0.444609\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−26.6565	−0.883170	−0.441585	−	0.897219i	$-0.645584\pi$
$911$	−26.6565	−0.883170	−0.441585	+	0.897219i	$0.645584\pi$
$912$	0	0
$913$	36.2843	1.20083
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−30.7322	−1.01487
$918$	0	0
$919$	−59.8402	−1.97395	−0.986974	−	0.160881i	$-0.948566\pi$
$919$	−59.8402	−1.97395	−0.986974	+	0.160881i	$0.948566\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−20.9050	−0.688097
$924$	0	0
$925$	−2.24264	−0.0737376
$926$	0	0
$927$	0	0
$928$	0	0
$929$	3.18243	0.104412	0.0522060	−	0.998636i	$-0.483375\pi$
$929$	3.18243	0.104412	0.0522060	+	0.998636i	$0.483375\pi$
$930$	0	0
$931$	−14.1964	−0.465270
$932$	0	0
$933$	0	0
$934$	0	0
$935$	19.1116	0.625018
$936$	0	0
$937$	15.9411	0.520774	0.260387	−	0.965504i	$-0.416150\pi$
$937$	15.9411	0.520774	0.260387	+	0.965504i	$0.416150\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	31.2420	1.01846	0.509231	−	0.860630i	$-0.329930\pi$
$941$	31.2420	1.01846	0.509231	+	0.860630i	$0.329930\pi$
$942$	0	0
$943$	21.9607	0.715140
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.0894	0.977774	0.488887	−	0.872347i	$-0.337403\pi$
$947$	30.0894	0.977774	0.488887	+	0.872347i	$0.337403\pi$
$948$	0	0
$949$	−35.7990	−1.16208
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0.298627	0.00967349	0.00483674	−	0.999988i	$-0.498460\pi$
$953$	0.298627	0.00967349	0.00483674	+	0.999988i	$0.498460\pi$
$954$	0	0
$955$	50.3536	1.62941
$956$	0	0
$957$	0	0
$958$	0	0
$959$	34.2651	1.10648
$960$	0	0
$961$	44.7990	1.44513
$962$	0	0
$963$	0	0
$964$	0	0
$965$	13.3270	0.429010
$966$	0	0
$967$	34.1116	1.09696	0.548478	−	0.836165i	$-0.315207\pi$
$967$	34.1116	1.09696	0.548478	+	0.836165i	$0.315207\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−14.6733	−0.470888	−0.235444	−	0.971888i	$-0.575654\pi$
$971$	−14.6733	−0.470888	−0.235444	+	0.971888i	$0.575654\pi$
$972$	0	0
$973$	−29.2548	−0.937867
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17.1067	0.547290	0.273645	−	0.961831i	$-0.411771\pi$
$977$	17.1067	0.547290	0.273645	+	0.961831i	$0.411771\pi$
$978$	0	0
$979$	−43.9215	−1.40374
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−17.3183	−0.552367	−0.276184	−	0.961105i	$-0.589070\pi$
$983$	−17.3183	−0.552367	−0.276184	+	0.961105i	$0.589070\pi$
$984$	0	0
$985$	−33.7990	−1.07693
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.88380	0.0916995
$990$	0	0
$991$	−42.4277	−1.34776	−0.673881	−	0.738840i	$-0.735374\pi$
$991$	−42.4277	−1.34776	−0.673881	+	0.738840i	$0.735374\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.15932	−0.0367528
$996$	0	0
$997$	38.0416	1.20479	0.602395	−	0.798198i	$-0.294213\pi$
$997$	38.0416	1.20479	0.602395	+	0.798198i	$0.294213\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9216.2.a.br.1.1		8
3.2	odd	2	inner	9216.2.a.br.1.7		8
4.3	odd	2	inner	9216.2.a.br.1.2		8
8.3	odd	2		9216.2.a.bs.1.8		8
8.5	even	2		9216.2.a.bs.1.7		8
12.11	even	2	inner	9216.2.a.br.1.8		8
24.5	odd	2		9216.2.a.bs.1.1		8
24.11	even	2		9216.2.a.bs.1.2		8
32.3	odd	8		4608.2.k.bl.1153.7	yes	16
32.5	even	8		4608.2.k.bk.3457.1	yes	16
32.11	odd	8		4608.2.k.bl.3457.8	yes	16
32.13	even	8		4608.2.k.bk.1153.2	yes	16
32.19	odd	8		4608.2.k.bk.1153.1	✓	16
32.21	even	8		4608.2.k.bl.3457.7	yes	16
32.27	odd	8		4608.2.k.bk.3457.2	yes	16
32.29	even	8		4608.2.k.bl.1153.8	yes	16
96.5	odd	8		4608.2.k.bk.3457.7	yes	16
96.11	even	8		4608.2.k.bl.3457.2	yes	16
96.29	odd	8		4608.2.k.bl.1153.2	yes	16
96.35	even	8		4608.2.k.bl.1153.1	yes	16
96.53	odd	8		4608.2.k.bl.3457.1	yes	16
96.59	even	8		4608.2.k.bk.3457.8	yes	16
96.77	odd	8		4608.2.k.bk.1153.8	yes	16
96.83	even	8		4608.2.k.bk.1153.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4608.2.k.bk.1153.1	✓	16	32.19	odd	8
4608.2.k.bk.1153.2	yes	16	32.13	even	8
4608.2.k.bk.1153.7	yes	16	96.83	even	8
4608.2.k.bk.1153.8	yes	16	96.77	odd	8
4608.2.k.bk.3457.1	yes	16	32.5	even	8
4608.2.k.bk.3457.2	yes	16	32.27	odd	8
4608.2.k.bk.3457.7	yes	16	96.5	odd	8
4608.2.k.bk.3457.8	yes	16	96.59	even	8
4608.2.k.bl.1153.1	yes	16	96.35	even	8
4608.2.k.bl.1153.2	yes	16	96.29	odd	8
4608.2.k.bl.1153.7	yes	16	32.3	odd	8
4608.2.k.bl.1153.8	yes	16	32.29	even	8
4608.2.k.bl.3457.1	yes	16	96.53	odd	8
4608.2.k.bl.3457.2	yes	16	96.11	even	8
4608.2.k.bl.3457.7	yes	16	32.21	even	8
4608.2.k.bl.3457.8	yes	16	32.11	odd	8
9216.2.a.br.1.1		8	1.1	even	1	trivial
9216.2.a.br.1.2		8	4.3	odd	2	inner
9216.2.a.br.1.7		8	3.2	odd	2	inner
9216.2.a.br.1.8		8	12.11	even	2	inner
9216.2.a.bs.1.1		8	24.5	odd	2
9216.2.a.bs.1.2		8	24.11	even	2
9216.2.a.bs.1.7		8	8.5	even	2
9216.2.a.bs.1.8		8	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9216 = 2^{10} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9216.a (trivial)

\(f(q)\)	\(=\)	\(q-2.97127 q^{5} -2.27411 q^{7} +O(q^{10})\)	\(q-2.97127 q^{5} -2.27411 q^{7} -3.69552 q^{11} -3.41421 q^{13} +1.74053 q^{17} +7.76429 q^{19} +2.16478 q^{23} +3.82843 q^{25} +5.43275 q^{29} -8.70626 q^{31} +6.75699 q^{35} -0.585786 q^{37} +10.1445 q^{41} +1.33214 q^{43} +12.6173 q^{47} -1.82843 q^{49} -5.43275 q^{53} +10.9804 q^{55} +4.32957 q^{59} -7.41421 q^{61} +10.1445 q^{65} -6.43215 q^{67} +6.12293 q^{71} +10.4853 q^{73} +8.40401 q^{77} +2.27411 q^{79} -9.81845 q^{83} -5.17157 q^{85} +11.8851 q^{89} +7.76429 q^{91} -23.0698 q^{95} -15.3137 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 16 q^{13} + 8 q^{25} - 16 q^{37} + 8 q^{49} - 48 q^{61} + 16 q^{73} - 64 q^{85} - 32 q^{97}+O(q^{100}) \) 8 * q - 16 * q^13 + 8 * q^25 - 16 * q^37 + 8 * q^49 - 48 * q^61 + 16 * q^73 - 64 * q^85 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more