LMFDB - Embedded newform 2016.3.l.e.433.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2016,3,Mod(433,2016)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2016.433"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2016, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2016.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	no
Analytic conductor:	$54.9320212950$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{7})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 3x^{2} + 4 $ x^4 - 3*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			433.4
Root			$-1.32288 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2016.433
Dual form			2016.3.l.e.433.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$1$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	7.00000	1.00000
$7$	7.00000	1.00000
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.00000i	0.545455i	0.962091	+	0.272727i	$0.0879257\pi$
$11$	6.00000i	0.545455i	−0.962091	+	0.272727i	$0.912074\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	42.3320	1.84052	0.920261	−	0.391304i	$-0.127976\pi$
$23$	42.3320	1.84052	0.920261	+	0.391304i	$0.127976\pi$
$24$	0	0
$25$	−25.0000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	54.0000i	1.86207i	0.364931	+	0.931034i	$0.381093\pi$
$29$	54.0000i	1.86207i	−0.364931	+	0.931034i	$0.618907\pi$
$30$	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
$37$	− 63.4980i	− 1.71616i	−0.513514	−	0.858082i	$-0.671656\pi$
$37$	− 63.4980i	− 1.71616i	0.513514	−	0.858082i	$-0.328344\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	63.4980i	1.47670i	0.674419	+	0.738349i	$0.264394\pi$
$43$	63.4980i	1.47670i	−0.674419	+	0.738349i	$0.735606\pi$
$44$	0	0
$45$	0	0
$46$	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
$53$	6.00000i	0.113208i	0.998397	+	0.0566038i	$0.0180272\pi$
$53$	6.00000i	0.113208i	−0.998397	+	0.0566038i	$0.981973\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	63.4980i	0.947732i	0.880597	+	0.473866i	$0.157142\pi$
$67$	63.4980i	0.947732i	−0.880597	+	0.473866i	$0.842858\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	84.6640	1.19245	0.596226	−	0.802817i	$-0.296666\pi$
$71$	84.6640	1.19245	0.596226	+	0.802817i	$0.296666\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	42.0000i	0.545455i
$78$	0	0
$79$	−94.0000	−1.18987	−0.594937	−	0.803773i	$-0.702823\pi$
$79$	−94.0000	−1.18987	−0.594937	+	0.803773i	$0.702823\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	186.000i	1.73832i	0.494533	+	0.869159i	$0.335339\pi$
$107$	186.000i	1.73832i	−0.494533	+	0.869159i	$0.664661\pi$
$108$	0	0
$109$	− 190.494i	− 1.74765i	−0.486239	−	0.873826i	$-0.661631\pi$
$109$	− 190.494i	− 1.74765i	0.486239	−	0.873826i	$-0.338369\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	42.3320	0.374620	0.187310	−	0.982301i	$-0.440023\pi$
$113$	42.3320	0.374620	0.187310	+	0.982301i	$0.440023\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	85.0000	0.702479
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2.00000	−0.0157480	−0.00787402	−	0.999969i	$-0.502506\pi$
$127$	−2.00000	−0.0157480	−0.00787402	+	0.999969i	$0.502506\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	211.660	1.54496	0.772482	−	0.635036i	$-0.219015\pi$
$137$	211.660	1.54496	0.772482	+	0.635036i	$0.219015\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	186.000i	1.24832i	0.781296	+	0.624161i	$0.214559\pi$
$149$	186.000i	1.24832i	−0.781296	+	0.624161i	$0.785441\pi$
$150$	0	0
$151$	274.000	1.81457	0.907285	−	0.420517i	$-0.138151\pi$
$151$	274.000	1.81457	0.907285	+	0.420517i	$0.138151\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	296.324	1.84052
$162$	0	0
$163$	317.490i	1.94779i	0.226994	+	0.973896i	$0.427110\pi$
$163$	317.490i	1.94779i	−0.226994	+	0.973896i	$0.572890\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−169.000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	−175.000	−1.00000
$176$	0	0
$177$	0	0
$178$	0	0
$179$	342.000i	1.91061i	0.295615	+	0.955307i	$0.404476\pi$
$179$	342.000i	1.91061i	−0.295615	+	0.955307i	$0.595524\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	211.660	1.10817	0.554084	−	0.832461i	$-0.313069\pi$
$191$	211.660	1.10817	0.554084	+	0.832461i	$0.313069\pi$
$192$	0	0
$193$	−62.0000	−0.321244	−0.160622	−	0.987016i	$-0.551350\pi$
$193$	−62.0000	−0.321244	−0.160622	+	0.987016i	$0.551350\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	282.000i	1.43147i	0.698371	+	0.715736i	$0.253909\pi$
$197$	282.000i	1.43147i	−0.698371	+	0.715736i	$0.746091\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	378.000i	1.86207i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 317.490i	− 1.50469i	−0.658768	−	0.752346i	$-0.728922\pi$
$211$	− 317.490i	− 1.50469i	0.658768	−	0.752346i	$-0.271078\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	465.652	1.99851	0.999254	−	0.0386266i	$-0.0122983\pi$
$233$	465.652	1.99851	0.999254	+	0.0386266i	$0.0122983\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	423.320	1.77121	0.885607	−	0.464435i	$-0.153743\pi$
$239$	423.320	1.77121	0.885607	+	0.464435i	$0.153743\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	253.992i	1.00392i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	− 444.486i	− 1.71616i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−169.328	−0.643833	−0.321917	−	0.946768i	$-0.604327\pi$
$263$	−169.328	−0.643833	−0.321917	+	0.946768i	$0.604327\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 150.000i	− 0.545455i
$276$	0	0
$277$	317.490i	1.14617i	0.819495	+	0.573087i	$0.194254\pi$
$277$	317.490i	1.14617i	−0.819495	+	0.573087i	$0.805746\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−550.316	−1.95842	−0.979210	−	0.202847i	$-0.934981\pi$
$281$	−550.316	−1.95842	−0.979210	+	0.202847i	$0.934981\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	289.000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	444.486i	1.47670i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	522.000i	1.64669i	0.567543	+	0.823344i	$0.307894\pi$
$317$	522.000i	1.64669i	−0.567543	+	0.823344i	$0.692106\pi$
$318$	0	0
$319$	−324.000	−1.01567
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 190.494i	− 0.575511i	−0.957704	−	0.287755i	$-0.907091\pi$
$331$	− 190.494i	− 0.575511i	0.957704	−	0.287755i	$-0.0929090\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	226.000	0.670623	0.335312	−	0.942107i	$-0.391158\pi$
$337$	226.000	0.670623	0.335312	+	0.942107i	$0.391158\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	343.000	1.00000
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 678.000i	− 1.95389i	−0.213490	−	0.976945i	$-0.568483\pi$
$347$	− 678.000i	− 1.95389i	0.213490	−	0.976945i	$-0.431517\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	296.324	0.825415	0.412708	−	0.910864i	$-0.364583\pi$
$359$	296.324	0.825415	0.412708	+	0.910864i	$0.364583\pi$
$360$	0	0
$361$	−361.000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	42.0000i	0.113208i
$372$	0	0
$373$	− 698.478i	− 1.87260i	−0.351206	−	0.936298i	$-0.614228\pi$
$373$	− 698.478i	− 1.87260i	0.351206	−	0.936298i	$-0.385772\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 444.486i	− 1.17279i	−0.810026	−	0.586393i	$-0.800547\pi$
$379$	− 444.486i	− 1.17279i	0.810026	−	0.586393i	$-0.199453\pi$
$380$	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
$387$	0	0
$388$	0	0
$389$	− 666.000i	− 1.71208i	−0.516908	−	0.856041i	$-0.672917\pi$
$389$	− 666.000i	− 1.71208i	0.516908	−	0.856041i	$-0.327083\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−719.644	−1.79462	−0.897312	−	0.441397i	$-0.854483\pi$
$401$	−719.644	−1.79462	−0.897312	+	0.441397i	$0.854483\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	380.988	0.936089
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	825.474i	1.96075i	0.197150	+	0.980373i	$0.436832\pi$
$421$	825.474i	1.96075i	−0.197150	+	0.980373i	$0.563168\pi$
$422$	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
$431$	−846.640	−1.96436	−0.982181	−	0.187935i	$-0.939821\pi$
$431$	−846.640	−1.96436	−0.982181	+	0.187935i	$0.939821\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	486.000i	1.09707i	0.836129	+	0.548533i	$0.184813\pi$
$443$	486.000i	1.09707i	−0.836129	+	0.548533i	$0.815187\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	84.6640	0.188561	0.0942807	−	0.995546i	$-0.469945\pi$
$449$	84.6640	0.188561	0.0942807	+	0.995546i	$0.469945\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−878.000	−1.92123	−0.960613	−	0.277891i	$-0.910365\pi$
$457$	−878.000	−1.92123	−0.960613	+	0.277891i	$0.910365\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	674.000	1.45572	0.727862	−	0.685724i	$-0.240514\pi$
$463$	674.000	1.45572	0.727862	+	0.685724i	$0.240514\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	444.486i	0.947732i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−380.988	−0.805472
$474$	0	0
$475$	0	0
$476$	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
$487$	−398.000	−0.817248	−0.408624	−	0.912703i	$-0.633991\pi$
$487$	−398.000	−0.817248	−0.408624	+	0.912703i	$0.633991\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 954.000i	− 1.94297i	−0.237094	−	0.971487i	$-0.576195\pi$
$491$	− 954.000i	− 1.94297i	0.237094	−	0.971487i	$-0.423805\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	592.648	1.19245
$498$	0	0
$499$	952.470i	1.90876i	0.298597	+	0.954379i	$0.403481\pi$
$499$	952.470i	1.90876i	−0.298597	+	0.954379i	$0.596519\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1263.00	2.38752
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	294.000i	0.545455i
$540$	0	0
$541$	− 1079.47i	− 1.99532i	−0.0683919	−	0.997659i	$-0.521787\pi$
$541$	− 1079.47i	− 1.99532i	0.0683919	−	0.997659i	$-0.478213\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 698.478i	− 1.27693i	−0.769653	−	0.638463i	$-0.779571\pi$
$547$	− 698.478i	− 1.27693i	0.769653	−	0.638463i	$-0.220429\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−658.000	−1.18987
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1002.00i	− 1.79892i	−0.437000	−	0.899461i	$-0.643959\pi$
$557$	− 1002.00i	− 1.79892i	0.437000	−	0.899461i	$-0.356041\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	931.304	1.63674	0.818369	−	0.574692i	$-0.194878\pi$
$569$	931.304	1.63674	0.818369	+	0.574692i	$0.194878\pi$
$570$	0	0
$571$	190.494i	0.333615i	0.985989	+	0.166807i	$0.0533458\pi$
$571$	190.494i	0.333615i	−0.985989	+	0.166807i	$0.946654\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1058.30	−1.84052
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−36.0000	−0.0617496
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1185.30	−1.97879	−0.989396	−	0.145242i	$-0.953604\pi$
$599$	−1185.30	−1.97879	−0.989396	+	0.145242i	$0.953604\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	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
$613$	− 1206.46i	− 1.96813i	−0.177814	−	0.984064i	$-0.556903\pi$
$613$	− 1206.46i	− 1.96813i	0.177814	−	0.984064i	$-0.443097\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1100.63	−1.78385	−0.891923	−	0.452188i	$-0.850644\pi$
$617$	−1100.63	−1.78385	−0.891923	+	0.452188i	$0.850644\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	625.000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−1006.00	−1.59429	−0.797147	−	0.603785i	$-0.793659\pi$
$631$	−1006.00	−1.59429	−0.797147	+	0.603785i	$0.793659\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−973.636	−1.51893	−0.759467	−	0.650546i	$-0.774540\pi$
$641$	−973.636	−1.51893	−0.759467	+	0.650546i	$0.774540\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 1194.00i	− 1.82848i	−0.405169	−	0.914242i	$-0.632787\pi$
$653$	− 1194.00i	− 1.82848i	0.405169	−	0.914242i	$-0.367213\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 618.000i	− 0.937785i	−0.883255	−	0.468892i	$-0.844653\pi$
$659$	− 618.000i	− 0.937785i	0.883255	−	0.468892i	$-0.155347\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2285.93i	3.42718i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	446.000	0.662704	0.331352	−	0.943507i	$-0.392495\pi$
$673$	446.000	0.662704	0.331352	+	0.943507i	$0.392495\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1338.00i	− 1.95900i	−0.201433	−	0.979502i	$-0.564560\pi$
$683$	− 1338.00i	− 1.95900i	0.201433	−	0.979502i	$-0.435440\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 1398.00i	− 1.99429i	−0.0754851	−	0.997147i	$-0.524051\pi$
$701$	− 1398.00i	− 1.99429i	0.0754851	−	0.997147i	$-0.475949\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 317.490i	− 0.447800i	−0.974612	−	0.223900i	$-0.928121\pi$
$709$	− 317.490i	− 0.447800i	0.974612	−	0.223900i	$-0.0718789\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1350.00i	− 1.86207i
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−380.988	−0.516945
$738$	0	0
$739$	825.474i	1.11702i	0.829499	+	0.558508i	$0.188626\pi$
$739$	825.474i	1.11702i	−0.829499	+	0.558508i	$0.811374\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1481.62	−1.99411	−0.997053	−	0.0767160i	$-0.975557\pi$
$743$	−1481.62	−1.99411	−0.997053	+	0.0767160i	$0.975557\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1302.00i	1.73832i
$750$	0	0
$751$	802.000	1.06791	0.533955	−	0.845513i	$-0.320705\pi$
$751$	802.000	1.06791	0.533955	+	0.845513i	$0.320705\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 571.482i	− 0.754930i	−0.926024	−	0.377465i	$-0.876796\pi$
$757$	− 571.482i	− 0.754930i	0.926024	−	0.377465i	$-0.123204\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	− 1333.46i	− 1.74765i
$764$	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		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	507.984i	0.650428i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	296.324	0.374620
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1608.62	−1.98840	−0.994201	−	0.107540i	$-0.965703\pi$
$809$	−1608.62	−1.98840	−0.994201	+	0.107540i	$0.965703\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1158.00i	1.41048i	0.708971	+	0.705238i	$0.249160\pi$
$821$	1158.00i	1.41048i	−0.708971	+	0.705238i	$0.750840\pi$
$822$	0	0
$823$	−622.000	−0.755772	−0.377886	−	0.925852i	$-0.623349\pi$
$823$	−622.000	−0.755772	−0.377886	+	0.925852i	$0.623349\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	282.000i	0.340992i	0.985358	+	0.170496i	$0.0545369\pi$
$827$	282.000i	0.340992i	−0.985358	+	0.170496i	$0.945463\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	−2075.00	−2.46730
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	595.000	0.702479
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 2688.00i	− 3.15864i
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	465.652	0.539574	0.269787	−	0.962920i	$-0.413047\pi$
$863$	465.652	0.539574	0.269787	+	0.962920i	$0.413047\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 564.000i	− 0.649022i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1587.45i	− 1.81009i	−0.425314	−	0.905046i	$-0.639836\pi$
$877$	− 1587.45i	− 1.81009i	0.425314	−	0.905046i	$-0.360164\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	698.478i	0.791029i	0.918460	+	0.395514i	$0.129434\pi$
$883$	698.478i	0.791029i	−0.918460	+	0.395514i	$0.870566\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−14.0000	−0.0157480
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	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
$907$	317.490i	0.350044i	0.984564	+	0.175022i	$0.0559997\pi$
$907$	317.490i	0.350044i	−0.984564	+	0.175022i	$0.944000\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	931.304	1.02229	0.511144	−	0.859495i	$-0.329222\pi$
$911$	931.304	1.02229	0.511144	+	0.859495i	$0.329222\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−466.000	−0.507073	−0.253536	−	0.967326i	$-0.581594\pi$
$919$	−466.000	−0.507073	−0.253536	+	0.967326i	$0.581594\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	1587.45i	1.71616i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1494.00i	1.57761i	0.614641	+	0.788807i	$0.289301\pi$
$947$	1494.00i	1.57761i	−0.614641	+	0.788807i	$0.710699\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1227.63	1.28817	0.644086	−	0.764953i	$-0.277238\pi$
$953$	1227.63	1.28817	0.644086	+	0.764953i	$0.277238\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1481.62	1.54496
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	334.000	0.345398	0.172699	−	0.984975i	$-0.444751\pi$
$967$	334.000	0.345398	0.172699	+	0.984975i	$0.444751\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1947.27	−1.99311	−0.996557	−	0.0829069i	$-0.973580\pi$
$977$	−1947.27	−1.99311	−0.996557	+	0.0829069i	$0.973580\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2688.00i	2.71790i
$990$	0	0
$991$	1406.00	1.41877	0.709384	−	0.704822i	$-0.248973\pi$
$991$	1406.00	1.41877	0.709384	+	0.704822i	$0.248973\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.3.l.e.433.4		4
3.2	odd	2	inner	2016.3.l.e.433.1		4
4.3	odd	2		504.3.l.e.181.3	yes	4
7.6	odd	2	CM	2016.3.l.e.433.4		4
8.3	odd	2		504.3.l.e.181.4	yes	4
8.5	even	2	inner	2016.3.l.e.433.2		4
12.11	even	2		504.3.l.e.181.2	yes	4
21.20	even	2	inner	2016.3.l.e.433.1		4
24.5	odd	2	inner	2016.3.l.e.433.3		4
24.11	even	2		504.3.l.e.181.1	✓	4
28.27	even	2		504.3.l.e.181.3	yes	4
56.13	odd	2	inner	2016.3.l.e.433.2		4
56.27	even	2		504.3.l.e.181.4	yes	4
84.83	odd	2		504.3.l.e.181.2	yes	4
168.83	odd	2		504.3.l.e.181.1	✓	4
168.125	even	2	inner	2016.3.l.e.433.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.3.l.e.181.1	✓	4	24.11	even	2
504.3.l.e.181.1	✓	4	168.83	odd	2
504.3.l.e.181.2	yes	4	12.11	even	2
504.3.l.e.181.2	yes	4	84.83	odd	2
504.3.l.e.181.3	yes	4	4.3	odd	2
504.3.l.e.181.3	yes	4	28.27	even	2
504.3.l.e.181.4	yes	4	8.3	odd	2
504.3.l.e.181.4	yes	4	56.27	even	2
2016.3.l.e.433.1		4	3.2	odd	2	inner
2016.3.l.e.433.1		4	21.20	even	2	inner
2016.3.l.e.433.2		4	8.5	even	2	inner
2016.3.l.e.433.2		4	56.13	odd	2	inner
2016.3.l.e.433.3		4	24.5	odd	2	inner
2016.3.l.e.433.3		4	168.125	even	2	inner
2016.3.l.e.433.4		4	1.1	even	1	trivial
2016.3.l.e.433.4		4	7.6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2016.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+7.00000 q^{7} +6.00000i q^{11} +42.3320 q^{23} -25.0000 q^{25} +54.0000i q^{29} -63.4980i q^{37} +63.4980i q^{43} +49.0000 q^{49} +6.00000i q^{53} +63.4980i q^{67} +84.6640 q^{71} +42.0000i q^{77} -94.0000 q^{79} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 28 q^{7} - 100 q^{25} + 196 q^{49} - 376 q^{79}+O(q^{100}) \) 4 * q + 28 * q^7 - 100 * q^25 + 196 * q^49 - 376 * q^79

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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