LMFDB - Embedded newform 1136.3.h.b.993.6

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			993.6
Root			$1.64039$ 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)

$f(q)$	$=$	$q+5.26388 q^{3} +9.99851 q^{5} +18.7084 q^{9} +O(q^{10})$	$q+5.26388 q^{3} +9.99851 q^{5} +18.7084 q^{9} +52.6310 q^{15} -35.0113 q^{19} +74.9703 q^{25} +51.1040 q^{27} +10.6009 q^{29} -72.3160 q^{37} -30.5899 q^{43} +187.056 q^{45} +49.0000 q^{49} -184.295 q^{57} +71.0000 q^{71} -79.0966 q^{73} +394.635 q^{75} -99.0936 q^{79} +100.629 q^{81} +9.47851 q^{83} +55.8018 q^{87} +2.37599 q^{89} -350.061 q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	5.26388	1.75463	0.877313	−	0.479918i	$-0.159334\pi$
$3$	5.26388	1.75463	0.877313	+	0.479918i	$0.159334\pi$
$4$	0	0
$5$	9.99851	1.99970	0.999851	−	0.0172361i	$-0.00548668\pi$
$5$	9.99851	1.99970	0.999851	+	0.0172361i	$0.00548668\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	18.7084	2.07871
$10$	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
$15$	52.6310	3.50873
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	−35.0113	−1.84270	−0.921349	−	0.388736i	$-0.872912\pi$
$19$	−35.0113	−1.84270	−0.921349	+	0.388736i	$0.872912\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	74.9703	2.99881
$26$	0	0
$27$	51.1040	1.89274
$28$	0	0
$29$	10.6009	0.365548	0.182774	−	0.983155i	$-0.441492\pi$
$29$	10.6009	0.365548	0.182774	+	0.983155i	$0.441492\pi$
$30$	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
$37$	−72.3160	−1.95449	−0.977243	−	0.212121i	$-0.931963\pi$
$37$	−72.3160	−1.95449	−0.977243	+	0.212121i	$0.931963\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	−30.5899	−0.711393	−0.355697	−	0.934601i	$-0.615756\pi$
$43$	−30.5899	−0.711393	−0.355697	+	0.934601i	$0.615756\pi$
$44$	0	0
$45$	187.056	4.15681
$46$	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
$57$	−184.295	−3.23325
$58$	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
$73$	−79.0966	−1.08351	−0.541757	−	0.840535i	$-0.682241\pi$
$73$	−79.0966	−1.08351	−0.541757	+	0.840535i	$0.682241\pi$
$74$	0	0
$75$	394.635	5.26179
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−99.0936	−1.25435	−0.627175	−	0.778879i	$-0.715789\pi$
$79$	−99.0936	−1.25435	−0.627175	+	0.778879i	$0.715789\pi$
$80$	0	0
$81$	100.629	1.24234
$82$	0	0
$83$	9.47851	0.114199	0.0570995	−	0.998368i	$-0.481815\pi$
$83$	9.47851	0.114199	0.0570995	+	0.998368i	$0.481815\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	55.8018	0.641400
$88$	0	0
$89$	2.37599	0.0266965	0.0133483	−	0.999911i	$-0.495751\pi$
$89$	2.37599	0.0266965	0.0133483	+	0.999911i	$0.495751\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−350.061	−3.68485
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−116.806	−1.15649	−0.578246	−	0.815862i	$-0.696263\pi$
$101$	−116.806	−1.15649	−0.578246	+	0.815862i	$0.696263\pi$
$102$	0	0
$103$	−97.5646	−0.947229	−0.473615	−	0.880732i	$-0.657051\pi$
$103$	−97.5646	−0.947229	−0.473615	+	0.880732i	$0.657051\pi$
$104$	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
$109$	−194.471	−1.78414	−0.892069	−	0.451899i	$-0.850747\pi$
$109$	−194.471	−1.78414	−0.892069	+	0.451899i	$0.850747\pi$
$110$	0	0
$111$	−380.663	−3.42939
$112$	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
$125$	499.629	3.99703
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	−161.022	−1.24823
$130$	0	0
$131$	250.557	1.91265	0.956325	−	0.292305i	$-0.0944221\pi$
$131$	250.557	1.91265	0.956325	+	0.292305i	$0.0944221\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	510.964	3.78492
$136$	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
$145$	105.993	0.730987
$146$	0	0
$147$	257.930	1.75463
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	−207.448	−1.37383	−0.686914	−	0.726739i	$-0.741035\pi$
$151$	−207.448	−1.37383	−0.686914	+	0.726739i	$0.741035\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−84.5877	−0.538775	−0.269388	−	0.963032i	$-0.586821\pi$
$157$	−84.5877	−0.538775	−0.269388	+	0.963032i	$0.586821\pi$
$158$	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
$167$	182.054	1.09014	0.545070	−	0.838390i	$-0.316503\pi$
$167$	182.054	1.09014	0.545070	+	0.838390i	$0.316503\pi$
$168$	0	0
$169$	169.000	1.00000
$170$	0	0
$171$	−655.006	−3.83044
$172$	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
$179$	147.600	0.824583	0.412291	−	0.911052i	$-0.364729\pi$
$179$	147.600	0.824583	0.412291	+	0.911052i	$0.364729\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−723.053	−3.90839
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	186.246	0.975111	0.487555	−	0.873092i	$-0.337889\pi$
$191$	186.246	0.975111	0.487555	+	0.873092i	$0.337889\pi$
$192$	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
$199$	273.900	1.37638	0.688190	−	0.725531i	$-0.258406\pi$
$199$	273.900	1.37638	0.688190	+	0.725531i	$0.258406\pi$
$200$	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
$213$	373.735	1.75463
$214$	0	0
$215$	−305.854	−1.42257
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−416.355	−1.90116
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−84.7777	−0.380169	−0.190085	−	0.981768i	$-0.560876\pi$
$223$	−84.7777	−0.380169	−0.190085	+	0.981768i	$0.560876\pi$
$224$	0	0
$225$	1402.58	6.23367
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	−449.339	−1.96218	−0.981089	−	0.193557i	$-0.937998\pi$
$229$	−449.339	−1.96218	−0.981089	+	0.193557i	$0.937998\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	370.064	1.58826	0.794129	−	0.607749i	$-0.207927\pi$
$233$	370.064	1.58826	0.794129	+	0.607749i	$0.207927\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−521.617	−2.20091
$238$	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
$243$	69.7652	0.287100
$244$	0	0
$245$	489.927	1.99970
$246$	0	0
$247$	0	0
$248$	0	0
$249$	49.8937	0.200376
$250$	0	0
$251$	501.625	1.99851	0.999253	−	0.0386396i	$-0.0123024\pi$
$251$	501.625	1.99851	0.999253	+	0.0386396i	$0.0123024\pi$
$252$	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
$261$	198.326	0.759869
$262$	0	0
$263$	−23.8604	−0.0907239	−0.0453619	−	0.998971i	$-0.514444\pi$
$263$	−23.8604	−0.0907239	−0.0453619	+	0.998971i	$0.514444\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	12.5069	0.0468425
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	391.318	1.44398	0.721989	−	0.691904i	$-0.243228\pi$
$271$	391.318	1.44398	0.721989	+	0.691904i	$0.243228\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	527.158	1.90310	0.951549	−	0.307499i	$-0.0994920\pi$
$277$	527.158	1.90310	0.951549	+	0.307499i	$0.0994920\pi$
$278$	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
$285$	−1842.68	−6.46553
$286$	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
$303$	−614.852	−2.02921
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	−513.568	−1.66203
$310$	0	0
$311$	−414.554	−1.33297	−0.666485	−	0.745518i	$-0.732202\pi$
$311$	−414.554	−1.33297	−0.666485	+	0.745518i	$0.732202\pi$
$312$	0	0
$313$	−451.859	−1.44364	−0.721819	−	0.692082i	$-0.756694\pi$
$313$	−451.859	−1.44364	−0.721819	+	0.692082i	$0.756694\pi$
$314$	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
$321$	368.472	1.14789
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1023.67	−3.13050
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	−1352.92	−4.06282
$334$	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
$355$	709.895	1.99970
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−194.918	−0.542948	−0.271474	−	0.962446i	$-0.587511\pi$
$359$	−194.918	−0.542948	−0.271474	+	0.962446i	$0.587511\pi$
$360$	0	0
$361$	864.789	2.39554
$362$	0	0
$363$	636.929	1.75463
$364$	0	0
$365$	−790.848	−2.16671
$366$	0	0
$367$	−681.979	−1.85825	−0.929127	−	0.369761i	$-0.879440\pi$
$367$	−681.979	−1.85825	−0.929127	+	0.369761i	$0.879440\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−175.324	−0.470038	−0.235019	−	0.971991i	$-0.575515\pi$
$373$	−175.324	−0.470038	−0.235019	+	0.971991i	$0.575515\pi$
$374$	0	0
$375$	2629.99	7.01329
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−642.266	−1.69463	−0.847317	−	0.531087i	$-0.821784\pi$
$379$	−642.266	−1.69463	−0.847317	+	0.531087i	$0.821784\pi$
$380$	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
$387$	−572.289	−1.47878
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	1318.90	3.35599
$394$	0	0
$395$	−990.789	−2.50833
$396$	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
$405$	1006.14	2.48431
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−817.912	−1.99978	−0.999892	−	0.0146943i	$-0.995323\pi$
$409$	−817.912	−1.99978	−0.999892	+	0.0146943i	$0.995323\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	94.7710	0.228364
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−830.889	−1.98303	−0.991514	−	0.130001i	$-0.958502\pi$
$419$	−830.889	−1.98303	−0.991514	+	0.130001i	$0.958502\pi$
$420$	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
$431$	810.536	1.88059	0.940297	−	0.340355i	$-0.110547\pi$
$431$	810.536	1.88059	0.940297	+	0.340355i	$0.110547\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	557.935	1.28261
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	916.713	2.07871
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	23.7564	0.0533852
$446$	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
$453$	−1091.98	−2.41055
$454$	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
$463$	−684.252	−1.47787	−0.738933	−	0.673779i	$-0.764670\pi$
$463$	−684.252	−1.47787	−0.738933	+	0.673779i	$0.764670\pi$
$464$	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
$471$	−445.260	−0.945349
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−2624.81	−5.52591
$476$	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
$499$	−43.5005	−0.0871753	−0.0435876	−	0.999050i	$-0.513879\pi$
$499$	−43.5005	−0.0871753	−0.0435876	+	0.999050i	$0.513879\pi$
$500$	0	0
$501$	958.308	1.91279
$502$	0	0
$503$	927.795	1.84452	0.922261	−	0.386567i	$-0.126339\pi$
$503$	927.795	1.84452	0.922261	+	0.386567i	$0.126339\pi$
$504$	0	0
$505$	−1167.88	−2.31264
$506$	0	0
$507$	889.596	1.75463
$508$	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
$513$	−1789.22	−3.48775
$514$	0	0
$515$	−975.501	−1.89418
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−997.771	−1.91511	−0.957554	−	0.288254i	$-0.906925\pi$
$521$	−997.771	−1.91511	−0.957554	+	0.288254i	$0.906925\pi$
$522$	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
$535$	699.896	1.30822
$536$	0	0
$537$	776.950	1.44683
$538$	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
$545$	−1944.42	−3.56775
$546$	0	0
$547$	1082.75	1.97944	0.989718	−	0.143034i	$-0.0456858\pi$
$547$	1082.75	1.97944	0.989718	+	0.143034i	$0.0456858\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−371.151	−0.673594
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−3806.06	−6.85777
$556$	0	0
$557$	−1109.95	−1.99272	−0.996362	−	0.0852214i	$-0.972840\pi$
$557$	−1109.95	−1.99272	−0.996362	+	0.0852214i	$0.972840\pi$
$558$	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
$569$	−894.814	−1.57261	−0.786304	−	0.617839i	$-0.788008\pi$
$569$	−894.814	−1.57261	−0.786304	+	0.617839i	$0.788008\pi$
$570$	0	0
$571$	724.074	1.26808	0.634040	−	0.773300i	$-0.281395\pi$
$571$	724.074	1.26808	0.634040	+	0.773300i	$0.281395\pi$
$572$	0	0
$573$	980.377	1.71096
$574$	0	0
$575$	0	0
$576$	0	0
$577$	914.811	1.58546	0.792731	−	0.609572i	$-0.208659\pi$
$577$	914.811	1.58546	0.792731	+	0.609572i	$0.208659\pi$
$578$	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
$587$	240.310	0.409387	0.204694	−	0.978826i	$-0.434380\pi$
$587$	240.310	0.409387	0.204694	+	0.978826i	$0.434380\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	981.896	1.65581	0.827906	−	0.560867i	$-0.189532\pi$
$593$	981.896	1.65581	0.827906	+	0.560867i	$0.189532\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1441.77	2.41503
$598$	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
$605$	1209.82	1.99970
$606$	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
$613$	785.049	1.28067	0.640334	−	0.768097i	$-0.278796\pi$
$613$	785.049	1.28067	0.640334	+	0.768097i	$0.278796\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−826.713	−1.33989	−0.669946	−	0.742410i	$-0.733683\pi$
$617$	−826.713	−1.33989	−0.669946	+	0.742410i	$0.733683\pi$
$618$	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
$625$	3121.29	4.99406
$626$	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
$639$	1328.30	2.07871
$640$	0	0
$641$	485.186	0.756920	0.378460	−	0.925618i	$-0.376454\pi$
$641$	485.186	0.756920	0.378460	+	0.925618i	$0.376454\pi$
$642$	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
$645$	−1609.98	−2.49609
$646$	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
$655$	2505.20	3.82473
$656$	0	0
$657$	−1479.77	−2.25232
$658$	0	0
$659$	−797.989	−1.21091	−0.605454	−	0.795880i	$-0.707009\pi$
$659$	−797.989	−1.21091	−0.605454	+	0.795880i	$0.707009\pi$
$660$	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
$669$	−446.260	−0.667055
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	3831.28	5.67597
$676$	0	0
$677$	1319.77	1.94944	0.974720	−	0.223429i	$-0.0717250\pi$
$677$	1319.77	1.94944	0.974720	+	0.223429i	$0.0717250\pi$
$678$	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
$687$	−2365.27	−3.44289
$688$	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
$699$	1947.97	2.78680
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	2531.88	3.60153
$704$	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
$711$	−1853.89	−2.60743
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−506.292	−0.704162	−0.352081	−	0.935970i	$-0.614526\pi$
$719$	−506.292	−0.704162	−0.352081	+	0.935970i	$0.614526\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	794.752	1.09621
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−538.429	−0.738586
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	2578.92	3.50873
$736$	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
$747$	177.328	0.237387
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	2640.49	3.50663
$754$	0	0
$755$	−2074.17	−2.74725
$756$	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
$783$	541.748	0.691887
$784$	0	0
$785$	−845.751	−1.07739
$786$	0	0
$787$	−702.800	−0.893012	−0.446506	−	0.894781i	$-0.647332\pi$
$787$	−702.800	−0.893012	−0.446506	+	0.894781i	$0.647332\pi$
$788$	0	0
$789$	−125.598	−0.159187
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1315.14	1.65012	0.825059	−	0.565046i	$-0.191142\pi$
$797$	1315.14	1.65012	0.825059	+	0.565046i	$0.191142\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	44.4511	0.0554945
$802$	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
$811$	1141.51	1.40753	0.703766	−	0.710432i	$-0.251500\pi$
$811$	1141.51	1.40753	0.703766	+	0.710432i	$0.251500\pi$
$812$	0	0
$813$	2059.85	2.53364
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1070.99	1.31088
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1352.45	−1.64732	−0.823660	−	0.567085i	$-0.808071\pi$
$821$	−1352.45	−1.64732	−0.823660	+	0.567085i	$0.808071\pi$
$822$	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
$829$	−208.544	−0.251561	−0.125781	−	0.992058i	$-0.540144\pi$
$829$	−208.544	−0.251561	−0.125781	+	0.992058i	$0.540144\pi$
$830$	0	0
$831$	2774.90	3.33922
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1820.26	2.17996
$836$	0	0
$837$	0	0
$838$	0	0
$839$	718.797	0.856731	0.428366	−	0.903606i	$-0.359090\pi$
$839$	718.797	0.856731	0.428366	+	0.903606i	$0.359090\pi$
$840$	0	0
$841$	−728.621	−0.866375
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1689.75	1.99970
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1334.47	1.56444	0.782219	−	0.623003i	$-0.214088\pi$
$853$	1334.47	1.56444	0.782219	+	0.623003i	$0.214088\pi$
$854$	0	0
$855$	−6549.09	−7.65975
$856$	0	0
$857$	−1151.07	−1.34314	−0.671570	−	0.740941i	$-0.734380\pi$
$857$	−1151.07	−1.34314	−0.671570	+	0.740941i	$0.734380\pi$
$858$	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
$867$	1521.26	1.75463
$868$	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
$877$	818.584	0.933391	0.466695	−	0.884418i	$-0.345444\pi$
$877$	818.584	0.933391	0.466695	+	0.884418i	$0.345444\pi$
$878$	0	0
$879$	−2895.13	−3.29367
$880$	0	0
$881$	−312.734	−0.354976	−0.177488	−	0.984123i	$-0.556797\pi$
$881$	−312.734	−0.354976	−0.177488	+	0.984123i	$0.556797\pi$
$882$	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
$895$	1475.78	1.64892
$896$	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
$909$	−2185.25	−2.40402
$910$	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
$925$	−5421.55	−5.86114
$926$	0	0
$927$	−1825.28	−1.96902
$928$	0	0
$929$	1802.01	1.93974	0.969868	−	0.243632i	$-0.0783390\pi$
$929$	1802.01	1.93974	0.969868	+	0.243632i	$0.0783390\pi$
$930$	0	0
$931$	−1715.55	−1.84270
$932$	0	0
$933$	−2182.16	−2.33887
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	−2378.53	−2.53305
$940$	0	0
$941$	479.909	0.509999	0.255000	−	0.966941i	$-0.417925\pi$
$941$	479.909	0.509999	0.255000	+	0.966941i	$0.417925\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1360.29	−1.43642	−0.718208	−	0.695828i	$-0.755038\pi$
$947$	−1360.29	−1.43642	−0.718208	+	0.695828i	$0.755038\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−106.015	−0.111243	−0.0556215	−	0.998452i	$-0.517714\pi$
$953$	−106.015	−0.111243	−0.0556215	+	0.998452i	$0.517714\pi$
$954$	0	0
$955$	1862.18	1.94993
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	1309.59	1.35991
$964$	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
$977$	1278.65	1.30875	0.654374	−	0.756171i	$-0.272932\pi$
$977$	1278.65	1.30875	0.654374	+	0.756171i	$0.272932\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−3638.25	−3.70871
$982$	0	0
$983$	1554.03	1.58091	0.790454	−	0.612521i	$-0.209845\pi$
$983$	1554.03	1.58091	0.790454	+	0.612521i	$0.209845\pi$
$984$	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
$995$	2738.59	2.75235
$996$	0	0
$997$	−1621.44	−1.62632	−0.813158	−	0.582043i	$-0.802253\pi$
$997$	−1621.44	−1.62632	−0.813158	+	0.582043i	$0.802253\pi$
$998$	0	0
$999$	−3695.64	−3.69934

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1136.3.h.b.993.6		7
4.3	odd	2		71.3.b.b.70.3	✓	7
12.11	even	2		639.3.d.b.496.5		7
71.70	odd	2	CM	1136.3.h.b.993.6		7
284.283	even	2		71.3.b.b.70.3	✓	7
852.851	odd	2		639.3.d.b.496.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
71.3.b.b.70.3	✓	7	4.3	odd	2
71.3.b.b.70.3	✓	7	284.283	even	2
639.3.d.b.496.5		7	12.11	even	2
639.3.d.b.496.5		7	852.851	odd	2
1136.3.h.b.993.6		7	1.1	even	1	trivial
1136.3.h.b.993.6		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)

\(f(q)\)	\(=\)	\(q+5.26388 q^{3} +9.99851 q^{5} +18.7084 q^{9} +O(q^{10})\)	\(q+5.26388 q^{3} +9.99851 q^{5} +18.7084 q^{9} +52.6310 q^{15} -35.0113 q^{19} +74.9703 q^{25} +51.1040 q^{27} +10.6009 q^{29} -72.3160 q^{37} -30.5899 q^{43} +187.056 q^{45} +49.0000 q^{49} -184.295 q^{57} +71.0000 q^{71} -79.0966 q^{73} +394.635 q^{75} -99.0936 q^{79} +100.629 q^{81} +9.47851 q^{83} +55.8018 q^{87} +2.37599 q^{89} -350.061 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 63 q^{9}+O(q^{10}) \) 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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