LMFDB - Embedded newform 2700.3.b.c.1349.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2700,3,Mod(1349,2700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2700, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2700.1349"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2700.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-22,0,0,0,0,0,0,0,0,0,0, 0,-26] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	no
Analytic conductor:	$73.5696713773$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1349.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2700.1349
Dual form			2700.3.b.c.1349.1

$q$-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 $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	13.0000i	1.85714i	0.371154	+	0.928571i	$0.378962\pi$
$7$	13.0000i	1.85714i	−0.371154	+	0.928571i	$0.621038\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	− 22.0000i	− 1.69231i	−0.532939	−	0.846154i	$-0.678912\pi$
$13$	− 22.0000i	− 1.69231i	0.532939	−	0.846154i	$-0.321088\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−11.0000	−0.578947	−0.289474	−	0.957186i	$-0.593480\pi$
$19$	−11.0000	−0.578947	−0.289474	+	0.957186i	$0.593480\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−13.0000	−0.419355	−0.209677	−	0.977771i	$-0.567241\pi$
$31$	−13.0000	−0.419355	−0.209677	+	0.977771i	$0.567241\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	73.0000i	1.97297i	0.163843	+	0.986486i	$0.447611\pi$
$37$	73.0000i	1.97297i	−0.163843	+	0.986486i	$0.552389\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	− 61.0000i	− 1.41860i	−0.704904	−	0.709302i	$-0.749010\pi$
$43$	− 61.0000i	− 1.41860i	0.704904	−	0.709302i	$-0.250990\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−120.000	−2.44898
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	−121.000	−1.98361	−0.991803	−	0.127774i	$-0.959217\pi$
$61$	−121.000	−1.98361	−0.991803	+	0.127774i	$0.959217\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 122.000i	− 1.82090i	−0.413624	−	0.910448i	$-0.635737\pi$
$67$	− 122.000i	− 1.82090i	0.413624	−	0.910448i	$-0.364263\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	143.000i	1.95890i	0.201677	+	0.979452i	$0.435361\pi$
$73$	143.000i	1.95890i	−0.201677	+	0.979452i	$0.564639\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−11.0000	−0.139241	−0.0696203	−	0.997574i	$-0.522179\pi$
$79$	−11.0000	−0.139241	−0.0696203	+	0.997574i	$0.522179\pi$
$80$	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
$91$	286.000	3.14286
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 167.000i	− 1.72165i	−0.508902	−	0.860825i	$-0.669948\pi$
$97$	− 167.000i	− 1.72165i	0.508902	−	0.860825i	$-0.330052\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	− 157.000i	− 1.52427i	−0.647417	−	0.762136i	$-0.724151\pi$
$103$	− 157.000i	− 1.52427i	0.647417	−	0.762136i	$-0.275849\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−143.000	−1.31193	−0.655963	−	0.754793i	$-0.727737\pi$
$109$	−143.000	−1.31193	−0.655963	+	0.754793i	$0.727737\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	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
$125$	0	0
$126$	0	0
$127$	− 146.000i	− 1.14961i	−0.818292	−	0.574803i	$-0.805079\pi$
$127$	− 146.000i	− 1.14961i	0.818292	−	0.574803i	$-0.194921\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	− 143.000i	− 1.07519i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	229.000	1.64748	0.823741	−	0.566966i	$-0.191883\pi$
$139$	229.000	1.64748	0.823741	+	0.566966i	$0.191883\pi$
$140$	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
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	59.0000	0.390728	0.195364	−	0.980731i	$-0.437411\pi$
$151$	59.0000	0.390728	0.195364	+	0.980731i	$0.437411\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 311.000i	− 1.98089i	−0.137902	−	0.990446i	$-0.544036\pi$
$157$	− 311.000i	− 1.98089i	0.137902	−	0.990446i	$-0.455964\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 262.000i	− 1.60736i	−0.595060	−	0.803681i	$-0.702872\pi$
$163$	− 262.000i	− 1.60736i	0.595060	−	0.803681i	$-0.297128\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−315.000	−1.86391
$170$	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
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	314.000	1.73481	0.867403	−	0.497606i	$-0.165787\pi$
$181$	314.000	1.73481	0.867403	+	0.497606i	$0.165787\pi$
$182$	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
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	143.000i	0.740933i	0.928846	+	0.370466i	$0.120802\pi$
$193$	143.000i	0.740933i	−0.928846	+	0.370466i	$0.879198\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−386.000	−1.93970	−0.969849	−	0.243706i	$-0.921637\pi$
$199$	−386.000	−1.93970	−0.969849	+	0.243706i	$0.921637\pi$
$200$	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
$211$	−166.000	−0.786730	−0.393365	−	0.919382i	$-0.628689\pi$
$211$	−166.000	−0.786730	−0.393365	+	0.919382i	$0.628689\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 169.000i	− 0.778802i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 421.000i	− 1.88789i	−0.330099	−	0.943946i	$-0.607082\pi$
$223$	− 421.000i	− 1.88789i	0.330099	−	0.943946i	$-0.392918\pi$
$224$	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
$229$	−383.000	−1.67249	−0.836245	−	0.548357i	$-0.815254\pi$
$229$	−383.000	−1.67249	−0.836245	+	0.548357i	$0.815254\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−286.000	−1.18672	−0.593361	−	0.804936i	$-0.702199\pi$
$241$	−286.000	−1.18672	−0.593361	+	0.804936i	$0.702199\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	242.000i	0.979757i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−949.000	−3.66409
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	−541.000	−1.99631	−0.998155	−	0.0607176i	$-0.980661\pi$
$271$	−541.000	−1.99631	−0.998155	+	0.0607176i	$0.980661\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	529.000i	1.90975i	0.297012	+	0.954874i	$0.404010\pi$
$277$	529.000i	1.90975i	−0.297012	+	0.954874i	$0.595990\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	59.0000i	0.208481i	0.994552	+	0.104240i	$0.0332411\pi$
$283$	59.0000i	0.208481i	−0.994552	+	0.104240i	$0.966759\pi$
$284$	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
$301$	793.000	2.63455
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 611.000i	− 1.99023i	−0.0987325	−	0.995114i	$-0.531479\pi$
$307$	− 611.000i	− 1.99023i	0.0987325	−	0.995114i	$-0.468521\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	− 142.000i	− 0.453674i	−0.973933	−	0.226837i	$-0.927162\pi$
$313$	− 142.000i	− 0.453674i	0.973933	−	0.226837i	$-0.0728385\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	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
$331$	299.000	0.903323	0.451662	−	0.892189i	$-0.350831\pi$
$331$	299.000	0.903323	0.451662	+	0.892189i	$0.350831\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 482.000i	− 1.43027i	−0.698988	−	0.715134i	$-0.746366\pi$
$337$	− 482.000i	− 1.43027i	0.698988	−	0.715134i	$-0.253634\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 923.000i	− 2.69096i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	169.000	0.484241	0.242120	−	0.970246i	$-0.422157\pi$
$349$	169.000	0.484241	0.242120	+	0.970246i	$0.422157\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−240.000	−0.664820
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	718.000i	1.95640i	0.207657	+	0.978202i	$0.433416\pi$
$367$	718.000i	1.95640i	−0.207657	+	0.978202i	$0.566584\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 577.000i	− 1.54692i	−0.633847	−	0.773458i	$-0.718525\pi$
$373$	− 577.000i	− 1.54692i	0.633847	−	0.773458i	$-0.281475\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	694.000	1.83113	0.915567	−	0.402165i	$-0.131742\pi$
$379$	694.000	1.83113	0.915567	+	0.402165i	$0.131742\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 431.000i	− 1.08564i	−0.839848	−	0.542821i	$-0.817356\pi$
$397$	− 431.000i	− 1.08564i	0.839848	−	0.542821i	$-0.182644\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	286.000i	0.709677i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−626.000	−1.53056	−0.765281	−	0.643696i	$-0.777400\pi$
$409$	−626.000	−1.53056	−0.765281	+	0.643696i	$0.777400\pi$
$410$	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	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	839.000	1.99287	0.996437	−	0.0843398i	$-0.0268781\pi$
$421$	839.000	1.99287	0.996437	+	0.0843398i	$0.0268781\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1573.00i	− 3.68384i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	359.000i	0.829099i	0.910027	+	0.414550i	$0.136061\pi$
$433$	359.000i	0.829099i	−0.910027	+	0.414550i	$0.863939\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−803.000	−1.82916	−0.914579	−	0.404408i	$-0.867478\pi$
$439$	−803.000	−1.82916	−0.914579	+	0.404408i	$0.867478\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	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
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	814.000i	1.78118i	0.454805	+	0.890591i	$0.349709\pi$
$457$	814.000i	1.78118i	−0.454805	+	0.890591i	$0.650291\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	923.000i	1.99352i	0.0804300	+	0.996760i	$0.474371\pi$
$463$	923.000i	1.99352i	−0.0804300	+	0.996760i	$0.525629\pi$
$464$	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
$469$	1586.00	3.38166
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	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
$481$	1606.00	3.33888
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 962.000i	− 1.97536i	−0.156489	−	0.987680i	$-0.550018\pi$
$487$	− 962.000i	− 1.97536i	0.156489	−	0.987680i	$-0.449982\pi$
$488$	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
$499$	877.000	1.75752	0.878758	−	0.477269i	$-0.158373\pi$
$499$	877.000	1.75752	0.878758	+	0.477269i	$0.158373\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	−1859.00	−3.63796
$512$	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
$523$	803.000i	1.53537i	0.640826	+	0.767686i	$0.278592\pi$
$523$	803.000i	1.53537i	−0.640826	+	0.767686i	$0.721408\pi$
$524$	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
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−241.000	−0.445471	−0.222736	−	0.974879i	$-0.571499\pi$
$541$	−241.000	−0.445471	−0.222736	+	0.974879i	$0.571499\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 587.000i	− 1.07313i	−0.843860	−	0.536563i	$-0.819722\pi$
$547$	− 587.000i	− 1.07313i	0.843860	−	0.536563i	$-0.180278\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 143.000i	− 0.258590i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−1342.00	−2.40072
$560$	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
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	−181.000	−0.316988	−0.158494	−	0.987360i	$-0.550664\pi$
$571$	−181.000	−0.316988	−0.158494	+	0.987360i	$0.550664\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1033.00i	1.79029i	0.445770	+	0.895147i	$0.352930\pi$
$577$	1033.00i	1.79029i	−0.445770	+	0.895147i	$0.647070\pi$
$578$	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
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	143.000	0.242784
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−673.000	−1.11980	−0.559900	−	0.828560i	$-0.689161\pi$
$601$	−673.000	−1.11980	−0.559900	+	0.828560i	$0.689161\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	373.000i	0.614498i	0.951629	+	0.307249i	$0.0994084\pi$
$607$	373.000i	0.614498i	−0.951629	+	0.307249i	$0.900592\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	143.000i	0.233279i	0.993174	+	0.116639i	$0.0372122\pi$
$613$	143.000i	0.233279i	−0.993174	+	0.116639i	$0.962788\pi$
$614$	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
$619$	949.000	1.53312	0.766559	−	0.642174i	$-0.221967\pi$
$619$	949.000	1.53312	0.766559	+	0.642174i	$0.221967\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	674.000	1.06815	0.534073	−	0.845438i	$-0.320661\pi$
$631$	674.000	1.06815	0.534073	+	0.845438i	$0.320661\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2640.00i	4.14443i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	314.000i	0.488336i	0.969733	+	0.244168i	$0.0785148\pi$
$643$	314.000i	0.488336i	−0.969733	+	0.244168i	$0.921485\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	−1201.00	−1.81694	−0.908472	−	0.417946i	$-0.862750\pi$
$661$	−1201.00	−1.81694	−0.908472	+	0.417946i	$0.862750\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 1177.00i	− 1.74889i	−0.485129	−	0.874443i	$-0.661227\pi$
$673$	− 1177.00i	− 1.74889i	0.485129	−	0.874443i	$-0.338773\pi$
$674$	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
$679$	2171.00	3.19735
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−1318.00	−1.90738	−0.953690	−	0.300790i	$-0.902750\pi$
$691$	−1318.00	−1.90738	−0.953690	+	0.300790i	$0.902750\pi$
$692$	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
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	− 803.000i	− 1.14225i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1391.00	−1.96192	−0.980959	−	0.194214i	$-0.937784\pi$
$709$	−1391.00	−1.96192	−0.980959	+	0.194214i	$0.937784\pi$
$710$	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
$721$	2041.00	2.83079
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1429.00i	1.96561i	0.184641	+	0.982806i	$0.440888\pi$
$727$	1429.00i	1.96561i	−0.184641	+	0.982806i	$0.559112\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1034.00i	1.41064i	0.708888	+	0.705321i	$0.249197\pi$
$733$	1034.00i	1.41064i	−0.708888	+	0.705321i	$0.750803\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	1222.00	1.65359	0.826793	−	0.562506i	$-0.190163\pi$
$739$	1222.00	1.65359	0.826793	+	0.562506i	$0.190163\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1381.00	−1.83888	−0.919441	−	0.393229i	$-0.871358\pi$
$751$	−1381.00	−1.83888	−0.919441	+	0.393229i	$0.871358\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	673.000i	0.889036i	0.895770	+	0.444518i	$0.146625\pi$
$757$	673.000i	0.889036i	−0.895770	+	0.444518i	$0.853375\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	− 1859.00i	− 2.43644i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	1534.00	1.99480	0.997399	−	0.0720749i	$-0.0229621\pi$
$769$	1534.00	1.99480	0.997399	+	0.0720749i	$0.0229621\pi$
$770$	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
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	613.000i	0.778907i	0.921046	+	0.389454i	$0.127336\pi$
$787$	613.000i	0.778907i	−0.921046	+	0.389454i	$0.872664\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2662.00i	3.35687i
$794$	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
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	−253.000	−0.311961	−0.155980	−	0.987760i	$-0.549854\pi$
$811$	−253.000	−0.311961	−0.155980	+	0.987760i	$0.549854\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	671.000i	0.821297i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	1058.00i	1.28554i	0.766059	+	0.642770i	$0.222215\pi$
$823$	1058.00i	1.28554i	−0.766059	+	0.642770i	$0.777785\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	−1151.00	−1.38842	−0.694210	−	0.719773i	$-0.744246\pi$
$829$	−1151.00	−1.38842	−0.694210	+	0.719773i	$0.744246\pi$
$830$	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
$841$	841.000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1573.00i	1.85714i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1658.00i	1.94373i	0.235543	+	0.971864i	$0.424313\pi$
$853$	1658.00i	1.94373i	−0.235543	+	0.971864i	$0.575687\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	−131.000	−0.152503	−0.0762515	−	0.997089i	$-0.524295\pi$
$859$	−131.000	−0.152503	−0.0762515	+	0.997089i	$0.524295\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−2684.00	−3.08152
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1727.00i	− 1.96921i	−0.174785	−	0.984607i	$-0.555923\pi$
$877$	− 1727.00i	− 1.96921i	0.174785	−	0.984607i	$-0.444077\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	443.000i	0.501699i	0.968026	+	0.250849i	$0.0807099\pi$
$883$	443.000i	0.501699i	−0.968026	+	0.250849i	$0.919290\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	1898.00	2.13498
$890$	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
$907$	− 1667.00i	− 1.83793i	−0.394342	−	0.918964i	$-0.629028\pi$
$907$	− 1667.00i	− 1.83793i	0.394342	−	0.918964i	$-0.370972\pi$
$908$	0	0
$909$	0	0
$910$	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
$919$	1837.00	1.99891	0.999456	−	0.0329825i	$-0.0105006\pi$
$919$	1837.00	1.99891	0.999456	+	0.0329825i	$0.0105006\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	1320.00	1.41783
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1847.00i	− 1.97118i	−0.169138	−	0.985592i	$-0.554098\pi$
$937$	− 1847.00i	− 1.97118i	0.169138	−	0.985592i	$-0.445902\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	3146.00	3.31507
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−792.000	−0.824142
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1787.00i	− 1.84798i	−0.382412	−	0.923992i	$-0.624907\pi$
$967$	− 1787.00i	− 1.84798i	0.382412	−	0.923992i	$-0.375093\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	2977.00i	3.05961i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	1739.00	1.75479	0.877397	−	0.479766i	$-0.159278\pi$
$991$	1739.00	1.75479	0.877397	+	0.479766i	$0.159278\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1894.00i	1.89970i	0.312707	+	0.949850i	$0.398764\pi$
$997$	1894.00i	1.89970i	−0.312707	+	0.949850i	$0.601236\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.3.b.c.1349.2		2
3.2	odd	2	CM	2700.3.b.c.1349.2		2
5.2	odd	4		2700.3.g.a.701.1	✓	1
5.3	odd	4		2700.3.g.g.701.1	yes	1
5.4	even	2	inner	2700.3.b.c.1349.1		2
15.2	even	4		2700.3.g.a.701.1	✓	1
15.8	even	4		2700.3.g.g.701.1	yes	1
15.14	odd	2	inner	2700.3.b.c.1349.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2700.3.b.c.1349.1		2	5.4	even	2	inner
2700.3.b.c.1349.1		2	15.14	odd	2	inner
2700.3.b.c.1349.2		2	1.1	even	1	trivial
2700.3.b.c.1349.2		2	3.2	odd	2	CM
2700.3.g.a.701.1	✓	1	5.2	odd	4
2700.3.g.a.701.1	✓	1	15.2	even	4
2700.3.g.g.701.1	yes	1	5.3	odd	4
2700.3.g.g.701.1	yes	1	15.8	even	4

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:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2700.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+13.0000i q^{7} -22.0000i q^{13} -11.0000 q^{19} -13.0000 q^{31} +73.0000i q^{37} -61.0000i q^{43} -120.000 q^{49} -121.000 q^{61} -122.000i q^{67} +143.000i q^{73} -11.0000 q^{79} +286.000 q^{91} -167.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 22 q^{19} - 26 q^{31} - 240 q^{49} - 242 q^{61} - 22 q^{79} + 572 q^{91}+O(q^{100}) \) 2 * q - 22 * q^19 - 26 * q^31 - 240 * q^49 - 242 * q^61 - 22 * q^79 + 572 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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