LMFDB - Embedded newform 900.3.f.a.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,3,Mod(199,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("900.199");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	900.f (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:	no
Analytic conductor:	$24.5232237924$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			199.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	900.199
Dual form			900.3.f.a.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -4.00000 q^{4} +8.00000i q^{8} -10.0000i q^{13} +16.0000 q^{16} -16.0000i q^{17} -20.0000 q^{26} -40.0000 q^{29} -32.0000i q^{32} -32.0000 q^{34} +70.0000i q^{37} -80.0000 q^{41} -49.0000 q^{49} +40.0000i q^{52} -56.0000i q^{53} +80.0000i q^{58} -22.0000 q^{61} -64.0000 q^{64} +64.0000i q^{68} +110.000i q^{73} +140.000 q^{74} +160.000i q^{82} -160.000 q^{89} +130.000i q^{97} +98.0000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} + 32 q^{16} - 40 q^{26} - 80 q^{29} - 64 q^{34} - 160 q^{41} - 98 q^{49} - 44 q^{61} - 128 q^{64} + 280 q^{74} - 320 q^{89}+O(q^{100}) $ 2 * q - 8 * q^4 + 32 * q^16 - 40 * q^26 - 80 * q^29 - 64 * q^34 - 160 * q^41 - 98 * q^49 - 44 * q^61 - 128 * q^64 + 280 * q^74 - 320 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	− 2.00000i	− 1.00000i
$2$	− 2.00000i	− 1.00000i
$3$	0	0
$3$	0	0
$4$	−4.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	8.00000i	1.00000i
$9$	0	0
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	− 10.0000i	− 0.769231i	−0.923077	−	0.384615i	$-0.874334\pi$
$13$	− 10.0000i	− 0.769231i	0.923077	−	0.384615i	$-0.125666\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	1.00000
$17$	− 16.0000i	− 0.941176i	−0.882353	−	0.470588i	$-0.844042\pi$
$17$	− 16.0000i	− 0.941176i	0.882353	−	0.470588i	$-0.155958\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	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
$26$	−20.0000	−0.769231
$27$	0	0
$28$	0	0
$29$	−40.0000	−1.37931	−0.689655	−	0.724138i	$-0.742238\pi$
$29$	−40.0000	−1.37931	−0.689655	+	0.724138i	$0.742238\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	− 32.0000i	− 1.00000i
$33$	0	0
$34$	−32.0000	−0.941176
$35$	0	0
$36$	0	0
$37$	70.0000i	1.89189i	0.324324	+	0.945946i	$0.394863\pi$
$37$	70.0000i	1.89189i	−0.324324	+	0.945946i	$0.605137\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−80.0000	−1.95122	−0.975610	−	0.219512i	$-0.929553\pi$
$41$	−80.0000	−1.95122	−0.975610	+	0.219512i	$0.929553\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	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
$49$	−49.0000	−1.00000
$50$	0	0
$51$	0	0
$52$	40.0000i	0.769231i
$53$	− 56.0000i	− 1.05660i	−0.849057	−	0.528302i	$-0.822829\pi$
$53$	− 56.0000i	− 1.05660i	0.849057	−	0.528302i	$-0.177171\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	80.0000i	1.37931i
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	−22.0000	−0.360656	−0.180328	−	0.983607i	$-0.557716\pi$
$61$	−22.0000	−0.360656	−0.180328	+	0.983607i	$0.557716\pi$
$62$	0	0
$63$	0	0
$64$	−64.0000	−1.00000
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	64.0000i	0.941176i
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	110.000i	1.50685i	0.657534	+	0.753425i	$0.271599\pi$
$73$	110.000i	1.50685i	−0.657534	+	0.753425i	$0.728401\pi$
$74$	140.000	1.89189
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0	0
$82$	160.000i	1.95122i
$83$	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
$89$	−160.000	−1.79775	−0.898876	−	0.438202i	$-0.855615\pi$
$89$	−160.000	−1.79775	−0.898876	+	0.438202i	$0.855615\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	130.000i	1.34021i	0.742268	+	0.670103i	$0.233750\pi$
$97$	130.000i	1.34021i	−0.742268	+	0.670103i	$0.766250\pi$
$98$	98.0000i	1.00000i
$99$	0	0
$100$	0	0
$101$	40.0000	0.396040	0.198020	−	0.980198i	$-0.436549\pi$
$101$	40.0000	0.396040	0.198020	+	0.980198i	$0.436549\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	80.0000	0.769231
$105$	0	0
$106$	−112.000	−1.05660
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−182.000	−1.66972	−0.834862	−	0.550459i	$-0.814453\pi$
$109$	−182.000	−1.66972	−0.834862	+	0.550459i	$0.814453\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 224.000i	− 1.98230i	−0.132743	−	0.991150i	$-0.542379\pi$
$113$	− 224.000i	− 1.98230i	0.132743	−	0.991150i	$-0.457621\pi$
$114$	0	0
$115$	0	0
$116$	160.000	1.37931
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	44.0000i	0.360656i
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	128.000i	1.00000i
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	128.000	0.941176
$137$	176.000i	1.28467i	0.766423	+	0.642336i	$0.222035\pi$
$137$	176.000i	1.28467i	−0.766423	+	0.642336i	$0.777965\pi$
$138$	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
$145$	0	0
$146$	220.000	1.50685
$147$	0	0
$148$	− 280.000i	− 1.89189i
$149$	−280.000	−1.87919	−0.939597	−	0.342282i	$-0.888800\pi$
$149$	−280.000	−1.87919	−0.939597	+	0.342282i	$0.888800\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 170.000i	− 1.08280i	−0.840764	−	0.541401i	$-0.817894\pi$
$157$	− 170.000i	− 1.08280i	0.840764	−	0.541401i	$-0.182106\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	320.000	1.95122
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	69.0000	0.408284
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 104.000i	− 0.601156i	−0.953757	−	0.300578i	$-0.902820\pi$
$173$	− 104.000i	− 0.601156i	0.953757	−	0.300578i	$-0.0971796\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	320.000i	1.79775i
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	38.0000	0.209945	0.104972	−	0.994475i	$-0.466525\pi$
$181$	38.0000	0.209945	0.104972	+	0.994475i	$0.466525\pi$
$182$	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
$193$	− 190.000i	− 0.984456i	−0.870466	−	0.492228i	$-0.836183\pi$
$193$	− 190.000i	− 0.984456i	0.870466	−	0.492228i	$-0.163817\pi$
$194$	260.000	1.34021
$195$	0	0
$196$	196.000	1.00000
$197$	56.0000i	0.284264i	0.989848	+	0.142132i	$0.0453957\pi$
$197$	56.0000i	0.284264i	−0.989848	+	0.142132i	$0.954604\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	− 80.0000i	− 0.396040i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	− 160.000i	− 0.769231i
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	224.000i	1.05660i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	364.000i	1.66972i
$219$	0	0
$220$	0	0
$221$	−160.000	−0.723982
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	0	0
$226$	−448.000	−1.98230
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	442.000	1.93013	0.965066	−	0.262009i	$-0.0843849\pi$
$229$	442.000	1.93013	0.965066	+	0.262009i	$0.0843849\pi$
$230$	0	0
$231$	0	0
$232$	− 320.000i	− 1.37931i
$233$	− 416.000i	− 1.78541i	−0.450644	−	0.892704i	$-0.648806\pi$
$233$	− 416.000i	− 1.78541i	0.450644	−	0.892704i	$-0.351194\pi$
$234$	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
$241$	−418.000	−1.73444	−0.867220	−	0.497925i	$-0.834095\pi$
$241$	−418.000	−1.73444	−0.867220	+	0.497925i	$0.834095\pi$
$242$	− 242.000i	− 1.00000i
$243$	0	0
$244$	88.0000	0.360656
$245$	0	0
$246$	0	0
$247$	0	0
$248$	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
$256$	256.000	1.00000
$257$	− 64.0000i	− 0.249027i	−0.992218	−	0.124514i	$-0.960263\pi$
$257$	− 64.0000i	− 0.249027i	0.992218	−	0.124514i	$-0.0397370\pi$
$258$	0	0
$259$	0	0
$260$	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
$269$	−520.000	−1.93309	−0.966543	−	0.256506i	$-0.917429\pi$
$269$	−520.000	−1.93309	−0.966543	+	0.256506i	$0.917429\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	− 256.000i	− 0.941176i
$273$	0	0
$274$	352.000	1.28467
$275$	0	0
$276$	0	0
$277$	− 230.000i	− 0.830325i	−0.909747	−	0.415162i	$-0.863725\pi$
$277$	− 230.000i	− 0.830325i	0.909747	−	0.415162i	$-0.136275\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−320.000	−1.13879	−0.569395	−	0.822064i	$-0.692822\pi$
$281$	−320.000	−1.13879	−0.569395	+	0.822064i	$0.692822\pi$
$282$	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
$289$	33.0000	0.114187
$290$	0	0
$291$	0	0
$292$	− 440.000i	− 1.50685i
$293$	136.000i	0.464164i	0.972696	+	0.232082i	$0.0745537\pi$
$293$	136.000i	0.464164i	−0.972696	+	0.232082i	$0.925446\pi$
$294$	0	0
$295$	0	0
$296$	−560.000	−1.89189
$297$	0	0
$298$	560.000i	1.87919i
$299$	0	0
$300$	0	0
$301$	0	0
$302$	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
$313$	50.0000i	0.159744i	0.996805	+	0.0798722i	$0.0254512\pi$
$313$	50.0000i	0.159744i	−0.996805	+	0.0798722i	$0.974549\pi$
$314$	−340.000	−1.08280
$315$	0	0
$316$	0	0
$317$	− 616.000i	− 1.94322i	−0.236593	−	0.971609i	$-0.576031\pi$
$317$	− 616.000i	− 1.94322i	0.236593	−	0.971609i	$-0.423969\pi$
$318$	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
$328$	− 640.000i	− 1.95122i
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 350.000i	− 1.03858i	−0.854599	−	0.519288i	$-0.826197\pi$
$337$	− 350.000i	− 1.03858i	0.854599	−	0.519288i	$-0.173803\pi$
$338$	− 138.000i	− 0.408284i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−208.000	−0.601156
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	598.000	1.71347	0.856734	−	0.515759i	$-0.172490\pi$
$349$	598.000	1.71347	0.856734	+	0.515759i	$0.172490\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	544.000i	1.54108i	0.637394	+	0.770538i	$0.280012\pi$
$353$	544.000i	1.54108i	−0.637394	+	0.770538i	$0.719988\pi$
$354$	0	0
$355$	0	0
$356$	640.000	1.79775
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	− 76.0000i	− 0.209945i
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 550.000i	− 1.47453i	−0.675603	−	0.737265i	$-0.736117\pi$
$373$	− 550.000i	− 1.47453i	0.675603	−	0.737265i	$-0.263883\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	400.000i	1.06101i
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	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
$386$	−380.000	−0.984456
$387$	0	0
$388$	− 520.000i	− 1.34021i
$389$	680.000	1.74807	0.874036	−	0.485861i	$-0.161494\pi$
$389$	680.000	1.74807	0.874036	+	0.485861i	$0.161494\pi$
$390$	0	0
$391$	0	0
$392$	− 392.000i	− 1.00000i
$393$	0	0
$394$	112.000	0.284264
$395$	0	0
$396$	0	0
$397$	− 650.000i	− 1.63728i	−0.574307	−	0.818640i	$-0.694729\pi$
$397$	− 650.000i	− 1.63728i	0.574307	−	0.818640i	$-0.305271\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−80.0000	−0.199501	−0.0997506	−	0.995012i	$-0.531805\pi$
$401$	−80.0000	−0.199501	−0.0997506	+	0.995012i	$0.531805\pi$
$402$	0	0
$403$	0	0
$404$	−160.000	−0.396040
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−782.000	−1.91198	−0.955990	−	0.293399i	$-0.905214\pi$
$409$	−782.000	−1.91198	−0.955990	+	0.293399i	$0.905214\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−320.000	−0.769231
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−58.0000	−0.137767	−0.0688836	−	0.997625i	$-0.521944\pi$
$421$	−58.0000	−0.137767	−0.0688836	+	0.997625i	$0.521944\pi$
$422$	0	0
$423$	0	0
$424$	448.000	1.05660
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	290.000i	0.669746i	0.942263	+	0.334873i	$0.108693\pi$
$433$	290.000i	0.669746i	−0.942263	+	0.334873i	$0.891307\pi$
$434$	0	0
$435$	0	0
$436$	728.000	1.66972
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	320.000i	0.723982i
$443$	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
$449$	560.000	1.24722	0.623608	−	0.781737i	$-0.285666\pi$
$449$	560.000	1.24722	0.623608	+	0.781737i	$0.285666\pi$
$450$	0	0
$451$	0	0
$452$	896.000i	1.98230i
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	850.000i	1.85996i	0.367615	+	0.929978i	$0.380174\pi$
$457$	850.000i	1.85996i	−0.367615	+	0.929978i	$0.619826\pi$
$458$	− 884.000i	− 1.93013i
$459$	0	0
$460$	0	0
$461$	760.000	1.64859	0.824295	−	0.566161i	$-0.191572\pi$
$461$	760.000	1.64859	0.824295	+	0.566161i	$0.191572\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	−640.000	−1.37931
$465$	0	0
$466$	−832.000	−1.78541
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	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
$481$	700.000	1.45530
$482$	836.000i	1.73444i
$483$	0	0
$484$	−484.000	−1.00000
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	− 176.000i	− 0.360656i
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	640.000i	1.29817i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	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
$509$	440.000	0.864440	0.432220	−	0.901768i	$-0.357730\pi$
$509$	440.000	0.864440	0.432220	+	0.901768i	$0.357730\pi$
$510$	0	0
$511$	0	0
$512$	− 512.000i	− 1.00000i
$513$	0	0
$514$	−128.000	−0.249027
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	880.000	1.68906	0.844530	−	0.535509i	$-0.179880\pi$
$521$	880.000	1.68906	0.844530	+	0.535509i	$0.179880\pi$
$522$	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
$529$	−529.000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	800.000i	1.50094i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	1040.00i	1.93309i
$539$	0	0
$540$	0	0
$541$	−682.000	−1.26063	−0.630314	−	0.776340i	$-0.717074\pi$
$541$	−682.000	−1.26063	−0.630314	+	0.776340i	$0.717074\pi$
$542$	0	0
$543$	0	0
$544$	−512.000	−0.941176
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	− 704.000i	− 1.28467i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−460.000	−0.830325
$555$	0	0
$556$	0	0
$557$	1064.00i	1.91023i	0.296230	+	0.955117i	$0.404271\pi$
$557$	1064.00i	1.91023i	−0.296230	+	0.955117i	$0.595729\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	640.000i	1.13879i
$563$	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
$569$	1040.00	1.82777	0.913884	−	0.405975i	$-0.133068\pi$
$569$	1040.00	1.82777	0.913884	+	0.405975i	$0.133068\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1150.00i	1.99307i	0.0831889	+	0.996534i	$0.473490\pi$
$577$	1150.00i	1.99307i	−0.0831889	+	0.996534i	$0.526510\pi$
$578$	− 66.0000i	− 0.114187i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−880.000	−1.50685
$585$	0	0
$586$	272.000	0.464164
$587$	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
$592$	1120.00i	1.89189i
$593$	736.000i	1.24115i	0.784148	+	0.620573i	$0.213100\pi$
$593$	736.000i	1.24115i	−0.784148	+	0.620573i	$0.786900\pi$
$594$	0	0
$595$	0	0
$596$	1120.00	1.87919
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−1102.00	−1.83361	−0.916805	−	0.399334i	$-0.869241\pi$
$601$	−1102.00	−1.83361	−0.916805	+	0.399334i	$0.869241\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 70.0000i	− 0.114192i	−0.998369	−	0.0570962i	$-0.981816\pi$
$613$	− 70.0000i	− 0.114192i	0.998369	−	0.0570962i	$-0.0181842\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1216.00i	− 1.97083i	−0.170178	−	0.985413i	$-0.554434\pi$
$617$	− 1216.00i	− 1.97083i	0.170178	−	0.985413i	$-0.445566\pi$
$618$	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
$625$	0	0
$626$	100.000	0.159744
$627$	0	0
$628$	680.000i	1.08280i
$629$	1120.00	1.78060
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	−1232.00	−1.94322
$635$	0	0
$636$	0	0
$637$	490.000i	0.769231i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	400.000	0.624025	0.312012	−	0.950078i	$-0.398997\pi$
$641$	400.000	0.624025	0.312012	+	0.950078i	$0.398997\pi$
$642$	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		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1144.00i	1.75191i	0.482389	+	0.875957i	$0.339769\pi$
$653$	1144.00i	1.75191i	−0.482389	+	0.875957i	$0.660231\pi$
$654$	0	0
$655$	0	0
$656$	−1280.00	−1.95122
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	1178.00	1.78215	0.891074	−	0.453858i	$-0.149953\pi$
$661$	1178.00	1.78215	0.891074	+	0.453858i	$0.149953\pi$
$662$	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
$673$	770.000i	1.14413i	0.820208	+	0.572065i	$0.193858\pi$
$673$	770.000i	1.14413i	−0.820208	+	0.572065i	$0.806142\pi$
$674$	−700.000	−1.03858
$675$	0	0
$676$	−276.000	−0.408284
$677$	104.000i	0.153619i	0.997046	+	0.0768095i	$0.0244733\pi$
$677$	104.000i	0.153619i	−0.997046	+	0.0768095i	$0.975527\pi$
$678$	0	0
$679$	0	0
$680$	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
$689$	−560.000	−0.812772
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	416.000i	0.601156i
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1280.00i	1.83644i
$698$	− 1196.00i	− 1.71347i
$699$	0	0
$700$	0	0
$701$	520.000	0.741797	0.370899	−	0.928673i	$-0.379050\pi$
$701$	520.000	0.741797	0.370899	+	0.928673i	$0.379050\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	1088.00	1.54108
$707$	0	0
$708$	0	0
$709$	−518.000	−0.730606	−0.365303	−	0.930889i	$-0.619035\pi$
$709$	−518.000	−0.730606	−0.365303	+	0.930889i	$0.619035\pi$
$710$	0	0
$711$	0	0
$712$	− 1280.00i	− 1.79775i
$713$	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
$722$	− 722.000i	− 1.00000i
$723$	0	0
$724$	−152.000	−0.209945
$725$	0	0
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 1450.00i	− 1.97817i	−0.147340	−	0.989086i	$-0.547071\pi$
$733$	− 1450.00i	− 1.97817i	0.147340	−	0.989086i	$-0.452929\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	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
$746$	−1100.00	−1.47453
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	800.000	1.06101
$755$	0	0
$756$	0	0
$757$	− 1190.00i	− 1.57199i	−0.618230	−	0.785997i	$-0.712150\pi$
$757$	− 1190.00i	− 1.57199i	0.618230	−	0.785997i	$-0.287850\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1520.00	−1.99737	−0.998686	−	0.0512484i	$-0.983680\pi$
$761$	−1520.00	−1.99737	−0.998686	+	0.0512484i	$0.983680\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−962.000	−1.25098	−0.625488	−	0.780234i	$-0.715100\pi$
$769$	−962.000	−1.25098	−0.625488	+	0.780234i	$0.715100\pi$
$770$	0	0
$771$	0	0
$772$	760.000i	0.984456i
$773$	− 1496.00i	− 1.93532i	−0.252264	−	0.967658i	$-0.581175\pi$
$773$	− 1496.00i	− 1.93532i	0.252264	−	0.967658i	$-0.418825\pi$
$774$	0	0
$775$	0	0
$776$	−1040.00	−1.34021
$777$	0	0
$778$	− 1360.00i	− 1.74807i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−784.000	−1.00000
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	− 224.000i	− 0.284264i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	220.000i	0.277427i
$794$	−1300.00	−1.63728
$795$	0	0
$796$	0	0
$797$	− 1144.00i	− 1.43538i	−0.696361	−	0.717691i	$-0.745199\pi$
$797$	− 1144.00i	− 1.43538i	0.696361	−	0.717691i	$-0.254801\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	160.000i	0.199501i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	320.000i	0.396040i
$809$	560.000	0.692213	0.346106	−	0.938195i	$-0.387504\pi$
$809$	560.000	0.692213	0.346106	+	0.938195i	$0.387504\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	1564.00i	1.91198i
$819$	0	0
$820$	0	0
$821$	−1400.00	−1.70524	−0.852619	−	0.522533i	$-0.824987\pi$
$821$	−1400.00	−1.70524	−0.852619	+	0.522533i	$0.824987\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	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
$829$	1258.00	1.51749	0.758745	−	0.651387i	$-0.225813\pi$
$829$	1258.00	1.51749	0.758745	+	0.651387i	$0.225813\pi$
$830$	0	0
$831$	0	0
$832$	640.000i	0.769231i
$833$	784.000i	0.941176i
$834$	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
$841$	759.000	0.902497
$842$	116.000i	0.137767i
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	− 896.000i	− 1.05660i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	410.000i	0.480657i	0.970692	+	0.240328i	$0.0772551\pi$
$853$	410.000i	0.480657i	−0.970692	+	0.240328i	$0.922745\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	464.000i	0.541424i	0.962660	+	0.270712i	$0.0872590\pi$
$857$	464.000i	0.541424i	−0.962660	+	0.270712i	$0.912741\pi$
$858$	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		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	580.000	0.669746
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	− 1456.00i	− 1.66972i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1610.00i	− 1.83580i	−0.396807	−	0.917902i	$-0.629882\pi$
$877$	− 1610.00i	− 1.83580i	0.396807	−	0.917902i	$-0.370118\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1600.00	1.81612	0.908059	−	0.418842i	$-0.137564\pi$
$881$	1600.00	1.81612	0.908059	+	0.418842i	$0.137564\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	640.000	0.723982
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	− 1120.00i	− 1.24722i
$899$	0	0
$900$	0	0
$901$	−896.000	−0.994451
$902$	0	0
$903$	0	0
$904$	1792.00	1.98230
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	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
$914$	1700.00	1.85996
$915$	0	0
$916$	−1768.00	−1.93013
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	− 1520.00i	− 1.64859i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	1280.00i	1.37931i
$929$	−1840.00	−1.98062	−0.990312	−	0.138859i	$-0.955657\pi$
$929$	−1840.00	−1.98062	−0.990312	+	0.138859i	$0.955657\pi$
$930$	0	0
$931$	0	0
$932$	1664.00i	1.78541i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	430.000i	0.458911i	0.973319	+	0.229456i	$0.0736946\pi$
$937$	430.000i	0.458911i	−0.973319	+	0.229456i	$0.926305\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1160.00	−1.23273	−0.616366	−	0.787460i	$-0.711396\pi$
$941$	−1160.00	−1.23273	−0.616366	+	0.787460i	$0.711396\pi$
$942$	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
$949$	1100.00	1.15911
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1456.00i	1.52781i	0.645331	+	0.763903i	$0.276720\pi$
$953$	1456.00i	1.52781i	−0.645331	+	0.763903i	$0.723280\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	− 1400.00i	− 1.45530i
$963$	0	0
$964$	1672.00	1.73444
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	968.000i	1.00000i
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−352.000	−0.360656
$977$	− 496.000i	− 0.507677i	−0.967247	−	0.253838i	$-0.918307\pi$
$977$	− 496.000i	− 0.507677i	0.967247	−	0.253838i	$-0.0816931\pi$
$978$	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
$986$	1280.00	1.29817
$987$	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
$995$	0	0
$996$	0	0
$997$	− 1850.00i	− 1.85557i	−0.373119	−	0.927783i	$-0.621712\pi$
$997$	− 1850.00i	− 1.85557i	0.373119	−	0.927783i	$-0.378288\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.3.f.a.199.1		2
3.2	odd	2		900.3.f.b.199.2		2
4.3	odd	2	CM	900.3.f.a.199.1		2
5.2	odd	4		36.3.d.b.19.1	yes	1
5.3	odd	4		900.3.c.b.451.1		1
5.4	even	2	inner	900.3.f.a.199.2		2
12.11	even	2		900.3.f.b.199.2		2
15.2	even	4		36.3.d.a.19.1	✓	1
15.8	even	4		900.3.c.c.451.1		1
15.14	odd	2		900.3.f.b.199.1		2
20.3	even	4		900.3.c.b.451.1		1
20.7	even	4		36.3.d.b.19.1	yes	1
20.19	odd	2	inner	900.3.f.a.199.2		2
40.27	even	4		576.3.g.c.127.1		1
40.37	odd	4		576.3.g.c.127.1		1
45.2	even	12		324.3.f.f.271.1		2
45.7	odd	12		324.3.f.e.271.1		2
45.22	odd	12		324.3.f.e.55.1		2
45.32	even	12		324.3.f.f.55.1		2
60.23	odd	4		900.3.c.c.451.1		1
60.47	odd	4		36.3.d.a.19.1	✓	1
60.59	even	2		900.3.f.b.199.1		2
80.27	even	4		2304.3.b.e.127.1		2
80.37	odd	4		2304.3.b.e.127.1		2
80.67	even	4		2304.3.b.e.127.2		2
80.77	odd	4		2304.3.b.e.127.2		2
120.77	even	4		576.3.g.a.127.1		1
120.107	odd	4		576.3.g.a.127.1		1
180.7	even	12		324.3.f.e.271.1		2
180.47	odd	12		324.3.f.f.271.1		2
180.67	even	12		324.3.f.e.55.1		2
180.167	odd	12		324.3.f.f.55.1		2
240.77	even	4		2304.3.b.d.127.1		2
240.107	odd	4		2304.3.b.d.127.2		2
240.197	even	4		2304.3.b.d.127.2		2
240.227	odd	4		2304.3.b.d.127.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.3.d.a.19.1	✓	1	15.2	even	4
36.3.d.a.19.1	✓	1	60.47	odd	4
36.3.d.b.19.1	yes	1	5.2	odd	4
36.3.d.b.19.1	yes	1	20.7	even	4
324.3.f.e.55.1		2	45.22	odd	12
324.3.f.e.55.1		2	180.67	even	12
324.3.f.e.271.1		2	45.7	odd	12
324.3.f.e.271.1		2	180.7	even	12
324.3.f.f.55.1		2	45.32	even	12
324.3.f.f.55.1		2	180.167	odd	12
324.3.f.f.271.1		2	45.2	even	12
324.3.f.f.271.1		2	180.47	odd	12
576.3.g.a.127.1		1	120.77	even	4
576.3.g.a.127.1		1	120.107	odd	4
576.3.g.c.127.1		1	40.27	even	4
576.3.g.c.127.1		1	40.37	odd	4
900.3.c.b.451.1		1	5.3	odd	4
900.3.c.b.451.1		1	20.3	even	4
900.3.c.c.451.1		1	15.8	even	4
900.3.c.c.451.1		1	60.23	odd	4
900.3.f.a.199.1		2	1.1	even	1	trivial
900.3.f.a.199.1		2	4.3	odd	2	CM
900.3.f.a.199.2		2	5.4	even	2	inner
900.3.f.a.199.2		2	20.19	odd	2	inner
900.3.f.b.199.1		2	15.14	odd	2
900.3.f.b.199.1		2	60.59	even	2
900.3.f.b.199.2		2	3.2	odd	2
900.3.f.b.199.2		2	12.11	even	2
2304.3.b.d.127.1		2	240.77	even	4
2304.3.b.d.127.1		2	240.227	odd	4
2304.3.b.d.127.2		2	240.107	odd	4
2304.3.b.d.127.2		2	240.197	even	4
2304.3.b.e.127.1		2	80.27	even	4
2304.3.b.e.127.1		2	80.37	odd	4
2304.3.b.e.127.2		2	80.67	even	4
2304.3.b.e.127.2		2	80.77	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -4.00000 q^{4} +8.00000i q^{8} -10.0000i q^{13} +16.0000 q^{16} -16.0000i q^{17} -20.0000 q^{26} -40.0000 q^{29} -32.0000i q^{32} -32.0000 q^{34} +70.0000i q^{37} -80.0000 q^{41} -49.0000 q^{49} +40.0000i q^{52} -56.0000i q^{53} +80.0000i q^{58} -22.0000 q^{61} -64.0000 q^{64} +64.0000i q^{68} +110.000i q^{73} +140.000 q^{74} +160.000i q^{82} -160.000 q^{89} +130.000i q^{97} +98.0000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} + 32 q^{16} - 40 q^{26} - 80 q^{29} - 64 q^{34} - 160 q^{41} - 98 q^{49} - 44 q^{61} - 128 q^{64} + 280 q^{74} - 320 q^{89}+O(q^{100}) \) 2 * q - 8 * q^4 + 32 * q^16 - 40 * q^26 - 80 * q^29 - 64 * q^34 - 160 * q^41 - 98 * q^49 - 44 * q^61 - 128 * q^64 + 280 * q^74 - 320 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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