LMFDB - Embedded newform 1136.3.h.b.993.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1136,3,Mod(993,1136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1136, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1136.993");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1136 = 2^{4} \cdot 71 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1136.h (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:	yes
Analytic conductor:	$30.9537580313$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	7.7.294755098673.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 14x^{5} + 56x^{3} - 56x - 21 $ x^7 - 14x^5 + 56x^3 - 56*x - 21
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 71)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			993.2
Root			$1.88136$ of defining polynomial
Character	$\chi$	$=$	1136.993

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.97864 q^{3} +6.09921 q^{5} +6.82955 q^{9} +O(q^{10})$	$q-3.97864 q^{3} +6.09921 q^{5} +6.82955 q^{9} -24.2666 q^{15} +37.9534 q^{19} +12.2004 q^{25} +8.63542 q^{27} -57.9522 q^{29} +58.3438 q^{37} -81.9128 q^{43} +41.6549 q^{45} +49.0000 q^{49} -151.003 q^{57} +71.0000 q^{71} +46.6290 q^{73} -48.5411 q^{75} +34.4305 q^{79} -95.8232 q^{81} -80.4470 q^{83} +230.571 q^{87} +172.993 q^{89} +231.486 q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 63 q^{9}+O(q^{10}) $ 7 * q + 63 * q^9	$ 7 q + 63 q^{9} + 175 q^{25} + 343 q^{49} + 497 q^{71} + 938 q^{75} + 567 q^{81} + 770 q^{87}+O(q^{100}) $ 7 * q + 63 * q^9 + 175 * q^25 + 343 * q^49 + 497 * q^71 + 938 * q^75 + 567 * q^81 + 770 * q^87

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1136\mathbb{Z}\right)^\times$.

$n$	$143$	$433$	$853$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	−3.97864	−1.32621	−0.663106	−	0.748525i	$-0.730762\pi$
$3$	−3.97864	−1.32621	−0.663106	+	0.748525i	$0.730762\pi$
$4$	0	0
$5$	6.09921	1.21984	0.609921	−	0.792462i	$-0.291201\pi$
$5$	6.09921	1.21984	0.609921	+	0.792462i	$0.291201\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	6.82955	0.758839
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	−24.2666	−1.61777
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	37.9534	1.99755	0.998773	−	0.0495195i	$-0.0157690\pi$
$19$	37.9534	1.99755	0.998773	+	0.0495195i	$0.0157690\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	12.2004	0.488017
$26$	0	0
$27$	8.63542	0.319830
$28$	0	0
$29$	−57.9522	−1.99835	−0.999176	−	0.0405812i	$-0.987079\pi$
$29$	−57.9522	−1.99835	−0.999176	+	0.0405812i	$0.987079\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	58.3438	1.57686	0.788430	−	0.615125i	$-0.210894\pi$
$37$	58.3438	1.57686	0.788430	+	0.615125i	$0.210894\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	−81.9128	−1.90495	−0.952474	−	0.304620i	$-0.901471\pi$
$43$	−81.9128	−1.90495	−0.952474	+	0.304620i	$0.901471\pi$
$44$	0	0
$45$	41.6549	0.925665
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	49.0000	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−151.003	−2.64917
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	71.0000	1.00000
$71$	71.0000	1.00000
$72$	0	0
$73$	46.6290	0.638753	0.319376	−	0.947628i	$-0.396527\pi$
$73$	46.6290	0.638753	0.319376	+	0.947628i	$0.396527\pi$
$74$	0	0
$75$	−48.5411	−0.647214
$76$	0	0
$77$	0	0
$78$	0	0
$79$	34.4305	0.435830	0.217915	−	0.975968i	$-0.430075\pi$
$79$	34.4305	0.435830	0.217915	+	0.975968i	$0.430075\pi$
$80$	0	0
$81$	−95.8232	−1.18300
$82$	0	0
$83$	−80.4470	−0.969241	−0.484621	−	0.874724i	$-0.661042\pi$
$83$	−80.4470	−0.969241	−0.484621	+	0.874724i	$0.661042\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	230.571	2.65024
$88$	0	0
$89$	172.993	1.94374	0.971871	−	0.235515i	$-0.0756776\pi$
$89$	172.993	1.94374	0.971871	+	0.235515i	$0.0756776\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	231.486	2.43669
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	176.744	1.74994	0.874971	−	0.484175i	$-0.160880\pi$
$101$	176.744	1.74994	0.874971	+	0.484175i	$0.160880\pi$
$102$	0	0
$103$	198.592	1.92808	0.964040	−	0.265759i	$-0.0856224\pi$
$103$	198.592	1.92808	0.964040	+	0.265759i	$0.0856224\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	70.0000	0.654206	0.327103	−	0.944989i	$-0.393928\pi$
$107$	70.0000	0.654206	0.327103	+	0.944989i	$0.393928\pi$
$108$	0	0
$109$	−52.7702	−0.484130	−0.242065	−	0.970260i	$-0.577825\pi$
$109$	−52.7702	−0.484130	−0.242065	+	0.970260i	$0.577825\pi$
$110$	0	0
$111$	−232.129	−2.09125
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−78.0674	−0.624539
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	325.901	2.52637
$130$	0	0
$131$	216.095	1.64958	0.824792	−	0.565436i	$-0.191292\pi$
$131$	216.095	1.64958	0.824792	+	0.565436i	$0.191292\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	52.6693	0.390143
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−353.463	−2.43768
$146$	0	0
$147$	−194.953	−1.32621
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	−167.811	−1.11133	−0.555666	−	0.831406i	$-0.687536\pi$
$151$	−167.811	−1.11133	−0.555666	+	0.831406i	$0.687536\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	313.633	1.99766	0.998831	−	0.0483388i	$-0.0153927\pi$
$157$	313.633	1.99766	0.998831	+	0.0483388i	$0.0153927\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	332.439	1.99065	0.995326	−	0.0965745i	$-0.0307886\pi$
$167$	332.439	1.99065	0.995326	+	0.0965745i	$0.0307886\pi$
$168$	0	0
$169$	169.000	1.00000
$170$	0	0
$171$	259.205	1.51582
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−162.972	−0.910460	−0.455230	−	0.890374i	$-0.650443\pi$
$179$	−162.972	−0.910460	−0.455230	+	0.890374i	$0.650443\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	355.851	1.92352
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	283.715	1.48542	0.742711	−	0.669613i	$-0.233540\pi$
$191$	283.715	1.48542	0.742711	+	0.669613i	$0.233540\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	−121.486	−0.610483	−0.305242	−	0.952275i	$-0.598737\pi$
$199$	−121.486	−0.610483	−0.305242	+	0.952275i	$0.598737\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	−282.483	−1.32621
$214$	0	0
$215$	−499.604	−2.32374
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−185.520	−0.847122
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−113.602	−0.509425	−0.254713	−	0.967017i	$-0.581981\pi$
$223$	−113.602	−0.509425	−0.254713	+	0.967017i	$0.581981\pi$
$224$	0	0
$225$	83.3234	0.370326
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	−210.850	−0.920740	−0.460370	−	0.887727i	$-0.652283\pi$
$229$	−210.850	−0.920740	−0.460370	+	0.887727i	$0.652283\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−456.297	−1.95836	−0.979178	−	0.203003i	$-0.934930\pi$
$233$	−456.297	−1.95836	−0.979178	+	0.203003i	$0.934930\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−136.987	−0.578003
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	303.527	1.24908
$244$	0	0
$245$	298.862	1.21984
$246$	0	0
$247$	0	0
$248$	0	0
$249$	320.070	1.28542
$250$	0	0
$251$	−443.533	−1.76706	−0.883531	−	0.468373i	$-0.844840\pi$
$251$	−443.533	−1.76706	−0.883531	+	0.468373i	$0.844840\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−395.788	−1.51643
$262$	0	0
$263$	−425.697	−1.61862	−0.809309	−	0.587383i	$-0.800158\pi$
$263$	−425.697	−1.61862	−0.809309	+	0.587383i	$0.800158\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−688.276	−2.57781
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	278.533	1.02780	0.513899	−	0.857850i	$-0.328200\pi$
$271$	278.533	1.02780	0.513899	+	0.857850i	$0.328200\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−401.039	−1.44779	−0.723897	−	0.689908i	$-0.757651\pi$
$277$	−401.039	−1.44779	−0.723897	+	0.689908i	$0.757651\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	−920.998	−3.23157
$286$	0	0
$287$	0	0
$288$	0	0
$289$	289.000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−550.000	−1.87713	−0.938567	−	0.345098i	$-0.887846\pi$
$293$	−550.000	−1.87713	−0.938567	+	0.345098i	$0.887846\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−703.201	−2.32080
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	−790.126	−2.55704
$310$	0	0
$311$	574.697	1.84790	0.923951	−	0.382511i	$-0.124941\pi$
$311$	574.697	1.84790	0.923951	+	0.382511i	$0.124941\pi$
$312$	0	0
$313$	595.088	1.90124	0.950619	−	0.310359i	$-0.100449\pi$
$313$	595.088	1.90124	0.950619	+	0.310359i	$0.100449\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−278.505	−0.867616
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	209.954	0.642060
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	398.462	1.19658
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	433.044	1.21984
$356$	0	0
$357$	0	0
$358$	0	0
$359$	418.744	1.16642	0.583209	−	0.812322i	$-0.301797\pi$
$359$	418.744	1.16642	0.583209	+	0.812322i	$0.301797\pi$
$360$	0	0
$361$	1079.46	2.99019
$362$	0	0
$363$	−481.415	−1.32621
$364$	0	0
$365$	284.400	0.779178
$366$	0	0
$367$	−637.400	−1.73678	−0.868392	−	0.495879i	$-0.834846\pi$
$367$	−637.400	−1.73678	−0.868392	+	0.495879i	$0.834846\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−676.223	−1.81293	−0.906465	−	0.422281i	$-0.861229\pi$
$373$	−676.223	−1.81293	−0.906465	+	0.422281i	$0.861229\pi$
$374$	0	0
$375$	310.602	0.828271
$376$	0	0
$377$	0	0
$378$	0	0
$379$	535.389	1.41264	0.706318	−	0.707895i	$-0.250355\pi$
$379$	535.389	1.41264	0.706318	+	0.707895i	$0.250355\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−559.428	−1.44555
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−859.766	−2.18770
$394$	0	0
$395$	209.999	0.531644
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−584.446	−1.44308
$406$	0	0
$407$	0	0
$408$	0	0
$409$	193.721	0.473645	0.236823	−	0.971553i	$-0.423894\pi$
$409$	193.721	0.473645	0.236823	+	0.971553i	$0.423894\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−490.664	−1.18232
$416$	0	0
$417$	0	0
$418$	0	0
$419$	78.6802	0.187781	0.0938905	−	0.995583i	$-0.470070\pi$
$419$	78.6802	0.187781	0.0938905	+	0.995583i	$0.470070\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−602.972	−1.39901	−0.699504	−	0.714629i	$-0.746595\pi$
$431$	−602.972	−1.39901	−0.699504	+	0.714629i	$0.746595\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	1406.30	3.23288
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	334.648	0.758839
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	1055.12	2.37106
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	667.659	1.47386
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	345.781	0.746827	0.373413	−	0.927665i	$-0.378187\pi$
$463$	345.781	0.746827	0.373413	+	0.927665i	$0.378187\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1247.83	−2.64932
$472$	0	0
$473$	0	0
$474$	0	0
$475$	463.047	0.974836
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	981.733	1.96740	0.983701	−	0.179815i	$-0.0575498\pi$
$499$	981.733	1.96740	0.983701	+	0.179815i	$0.0575498\pi$
$500$	0	0
$501$	−1322.65	−2.64003
$502$	0	0
$503$	172.682	0.343304	0.171652	−	0.985158i	$-0.445089\pi$
$503$	172.682	0.343304	0.171652	+	0.985158i	$0.445089\pi$
$504$	0	0
$505$	1078.00	2.13466
$506$	0	0
$507$	−672.390	−1.32621
$508$	0	0
$509$	−118.000	−0.231827	−0.115914	−	0.993259i	$-0.536980\pi$
$509$	−118.000	−0.231827	−0.115914	+	0.993259i	$0.536980\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	327.743	0.638876
$514$	0	0
$515$	1211.26	2.35195
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−387.269	−0.743318	−0.371659	−	0.928369i	$-0.621211\pi$
$521$	−387.269	−0.743318	−0.371659	+	0.928369i	$0.621211\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	529.000	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	426.945	0.798028
$536$	0	0
$537$	648.408	1.20746
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−321.857	−0.590563
$546$	0	0
$547$	−1043.42	−1.90753	−0.953765	−	0.300553i	$-0.902829\pi$
$547$	−1043.42	−1.90753	−0.953765	+	0.300553i	$0.902829\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2199.48	−3.99180
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1415.80	−2.55100
$556$	0	0
$557$	339.543	0.609592	0.304796	−	0.952418i	$-0.401412\pi$
$557$	339.543	0.609592	0.304796	+	0.952418i	$0.401412\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−8.20078	−0.0144126	−0.00720631	−	0.999974i	$-0.502294\pi$
$569$	−8.20078	−0.0144126	−0.00720631	+	0.999974i	$0.502294\pi$
$570$	0	0
$571$	−1035.53	−1.81355	−0.906773	−	0.421619i	$-0.861462\pi$
$571$	−1035.53	−1.81355	−0.906773	+	0.421619i	$0.861462\pi$
$572$	0	0
$573$	−1128.80	−1.96998
$574$	0	0
$575$	0	0
$576$	0	0
$577$	20.3992	0.0353539	0.0176770	−	0.999844i	$-0.494373\pi$
$577$	20.3992	0.0353539	0.0176770	+	0.999844i	$0.494373\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1066.86	1.81747	0.908736	−	0.417371i	$-0.137048\pi$
$587$	1066.86	1.81747	0.908736	+	0.417371i	$0.137048\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−867.003	−1.46206	−0.731031	−	0.682344i	$-0.760961\pi$
$593$	−867.003	−1.46206	−0.731031	+	0.682344i	$0.760961\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	483.349	0.809630
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	738.005	1.21984
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1092.77	−1.78265	−0.891327	−	0.453362i	$-0.850225\pi$
$613$	−1092.77	−1.78265	−0.891327	+	0.453362i	$0.850225\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1231.71	−1.99629	−0.998144	−	0.0608999i	$-0.980603\pi$
$617$	−1231.71	−1.99629	−0.998144	+	0.0608999i	$0.980603\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−781.160	−1.24986
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	484.898	0.758839
$640$	0	0
$641$	−952.002	−1.48518	−0.742591	−	0.669745i	$-0.766403\pi$
$641$	−952.002	−1.48518	−0.742591	+	0.669745i	$0.766403\pi$
$642$	0	0
$643$	1270.00	1.97512	0.987558	−	0.157253i	$-0.0502639\pi$
$643$	1270.00	1.97512	0.987558	+	0.157253i	$0.0502639\pi$
$644$	0	0
$645$	1987.74	3.08177
$646$	0	0
$647$	−1010.00	−1.56105	−0.780526	−	0.625124i	$-0.785048\pi$
$647$	−1010.00	−1.56105	−0.780526	+	0.625124i	$0.785048\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	1318.01	2.01223
$656$	0	0
$657$	318.455	0.484711
$658$	0	0
$659$	−845.101	−1.28240	−0.641199	−	0.767374i	$-0.721563\pi$
$659$	−845.101	−1.28240	−0.641199	+	0.767374i	$0.721563\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	451.981	0.675606
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	105.356	0.156083
$676$	0	0
$677$	1.26105	0.00186271	0.000931353	−	1.00000i	$-0.499704\pi$
$677$	1.26105	0.00186271	0.000931353		1.00000i	$0.499704\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	838.894	1.22110
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	1815.44	2.59720
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	2214.34	3.14985
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	235.145	0.330725
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−127.821	−0.177776	−0.0888879	−	0.996042i	$-0.528331\pi$
$719$	−127.821	−0.177776	−0.0888879	+	0.996042i	$0.528331\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−707.042	−0.975230
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−345.215	−0.473545
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	−1189.06	−1.61777
$736$	0	0
$737$	0	0
$738$	0	0
$739$	1078.00	1.45873	0.729364	−	0.684126i	$-0.239816\pi$
$739$	1078.00	1.45873	0.729364	+	0.684126i	$0.239816\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−549.417	−0.735498
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	1764.66	2.34350
$754$	0	0
$755$	−1023.52	−1.35565
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−500.442	−0.639134
$784$	0	0
$785$	1912.91	2.43683
$786$	0	0
$787$	−1216.69	−1.54598	−0.772990	−	0.634419i	$-0.781240\pi$
$787$	−1216.69	−1.54598	−0.772990	+	0.634419i	$0.781240\pi$
$788$	0	0
$789$	1693.69	2.14663
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−794.112	−0.996376	−0.498188	−	0.867069i	$-0.666001\pi$
$797$	−794.112	−0.996376	−0.498188	+	0.867069i	$0.666001\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	1181.46	1.47499
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	−189.202	−0.233295	−0.116647	−	0.993173i	$-0.537215\pi$
$811$	−189.202	−0.233295	−0.116647	+	0.993173i	$0.537215\pi$
$812$	0	0
$813$	−1108.18	−1.36308
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−3108.87	−3.80522
$818$	0	0
$819$	0	0
$820$	0	0
$821$	814.502	0.992086	0.496043	−	0.868298i	$-0.334786\pi$
$821$	814.502	0.992086	0.496043	+	0.868298i	$0.334786\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−525.774	−0.634227	−0.317113	−	0.948388i	$-0.602714\pi$
$829$	−525.774	−0.634227	−0.317113	+	0.948388i	$0.602714\pi$
$830$	0	0
$831$	1595.59	1.92008
$832$	0	0
$833$	0	0
$834$	0	0
$835$	2027.62	2.42828
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1305.49	−1.55601	−0.778004	−	0.628260i	$-0.783768\pi$
$839$	−1305.49	−1.55601	−0.778004	+	0.628260i	$0.783768\pi$
$840$	0	0
$841$	2517.46	2.99341
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1030.77	1.21984
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	739.249	0.866646	0.433323	−	0.901239i	$-0.357341\pi$
$853$	739.249	0.866646	0.433323	+	0.901239i	$0.357341\pi$
$854$	0	0
$855$	1580.94	1.84906
$856$	0	0
$857$	1494.27	1.74360	0.871802	−	0.489858i	$-0.162951\pi$
$857$	1494.27	1.74360	0.871802	+	0.489858i	$0.162951\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1149.83	−1.32621
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−702.453	−0.800972	−0.400486	−	0.916303i	$-0.631159\pi$
$877$	−702.453	−0.800972	−0.400486	+	0.916303i	$0.631159\pi$
$878$	0	0
$879$	2188.25	2.48948
$880$	0	0
$881$	−1550.70	−1.76016	−0.880080	−	0.474825i	$-0.842511\pi$
$881$	−1550.70	−1.76016	−0.880080	+	0.474825i	$0.842511\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−994.003	−1.11062
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	1207.08	1.32793
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	711.819	0.769534
$926$	0	0
$927$	1356.30	1.46310
$928$	0	0
$929$	−1427.15	−1.53623	−0.768113	−	0.640315i	$-0.778804\pi$
$929$	−1427.15	−1.53623	−0.768113	+	0.640315i	$0.778804\pi$
$930$	0	0
$931$	1859.72	1.99755
$932$	0	0
$933$	−2286.51	−2.45071
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	−2367.64	−2.52145
$940$	0	0
$941$	−1221.96	−1.29857	−0.649287	−	0.760544i	$-0.724932\pi$
$941$	−1221.96	−1.29857	−0.649287	+	0.760544i	$0.724932\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	182.250	0.192449	0.0962247	−	0.995360i	$-0.469323\pi$
$947$	182.250	0.192449	0.0962247	+	0.995360i	$0.469323\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1421.76	1.49188	0.745942	−	0.666011i	$-0.232000\pi$
$953$	1421.76	1.49188	0.745942	+	0.666011i	$0.232000\pi$
$954$	0	0
$955$	1730.44	1.81198
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	478.069	0.496437
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1658.00	−1.70752	−0.853759	−	0.520668i	$-0.825683\pi$
$971$	−1658.00	−1.70752	−0.853759	+	0.520668i	$0.825683\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1155.99	1.18320	0.591600	−	0.806231i	$-0.298496\pi$
$977$	1155.99	1.18320	0.591600	+	0.806231i	$0.298496\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−360.397	−0.367377
$982$	0	0
$983$	−877.645	−0.892823	−0.446411	−	0.894828i	$-0.647298\pi$
$983$	−877.645	−0.892823	−0.446411	+	0.894828i	$0.647298\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−740.970	−0.744694
$996$	0	0
$997$	−103.560	−0.103872	−0.0519359	−	0.998650i	$-0.516539\pi$
$997$	−103.560	−0.103872	−0.0519359	+	0.998650i	$0.516539\pi$
$998$	0	0
$999$	503.823	0.504328

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1136.3.h.b.993.2		7
4.3	odd	2		71.3.b.b.70.4	✓	7
12.11	even	2		639.3.d.b.496.4		7
71.70	odd	2	CM	1136.3.h.b.993.2		7
284.283	even	2		71.3.b.b.70.4	✓	7
852.851	odd	2		639.3.d.b.496.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
71.3.b.b.70.4	✓	7	4.3	odd	2
71.3.b.b.70.4	✓	7	284.283	even	2
639.3.d.b.496.4		7	12.11	even	2
639.3.d.b.496.4		7	852.851	odd	2
1136.3.h.b.993.2		7	1.1	even	1	trivial
1136.3.h.b.993.2		7	71.70	odd	2	CM

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:	\( N \)	\(=\)	\( 1136 = 2^{4} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1136.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.97864 q^{3} +6.09921 q^{5} +6.82955 q^{9} +O(q^{10})\)	\(q-3.97864 q^{3} +6.09921 q^{5} +6.82955 q^{9} -24.2666 q^{15} +37.9534 q^{19} +12.2004 q^{25} +8.63542 q^{27} -57.9522 q^{29} +58.3438 q^{37} -81.9128 q^{43} +41.6549 q^{45} +49.0000 q^{49} -151.003 q^{57} +71.0000 q^{71} +46.6290 q^{73} -48.5411 q^{75} +34.4305 q^{79} -95.8232 q^{81} -80.4470 q^{83} +230.571 q^{87} +172.993 q^{89} +231.486 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 63 q^{9}+O(q^{10}) \) 7 * q + 63 * q^9	\( 7 q + 63 q^{9} + 175 q^{25} + 343 q^{49} + 497 q^{71} + 938 q^{75} + 567 q^{81} + 770 q^{87}+O(q^{100}) \) 7 * q + 63 * q^9 + 175 * q^25 + 343 * q^49 + 497 * q^71 + 938 * q^75 + 567 * q^81 + 770 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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