LMFDB - Embedded newform 225.4.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	225.a (trivial)

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:	$13.2754297513$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{2} -3.00000 q^{4} -24.5967 q^{8} +O(q^{10})$	$q+2.23607 q^{2} -3.00000 q^{4} -24.5967 q^{8} -31.0000 q^{16} -138.636 q^{17} -164.000 q^{19} +98.3870 q^{23} -232.000 q^{31} +127.456 q^{32} -310.000 q^{34} -366.715 q^{38} +220.000 q^{46} +545.601 q^{47} -343.000 q^{49} +621.627 q^{53} -358.000 q^{61} -518.768 q^{62} +533.000 q^{64} +415.909 q^{68} +492.000 q^{76} +304.000 q^{79} -1270.09 q^{83} -295.161 q^{92} +1220.00 q^{94} -766.971 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{4}+O(q^{10}) $ 2 * q - 6 * q^4	$ 2 q - 6 q^{4} - 62 q^{16} - 328 q^{19} - 464 q^{31} - 620 q^{34} + 440 q^{46} - 686 q^{49} - 716 q^{61} + 1066 q^{64} + 984 q^{76} + 608 q^{79} + 2440 q^{94}+O(q^{100}) $ 2 * q - 6 * q^4 - 62 * q^16 - 328 * q^19 - 464 * q^31 - 620 * q^34 + 440 * q^46 - 686 * q^49 - 716 * q^61 + 1066 * q^64 + 984 * q^76 + 608 * q^79 + 2440 * q^94

Display 100 coefficients 10 coefficients

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 $

$2$	2.23607	0.790569	0.395285	−	0.918559i	$-0.370646\pi$
$2$	2.23607	0.790569	0.395285	+	0.918559i	$0.370646\pi$
$3$	0	0
$3$	0	0
$4$	−3.00000	−0.375000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−24.5967	−1.08703
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	−31.0000	−0.484375
$17$	−138.636	−1.97790	−0.988948	−	0.148265i	$-0.952631\pi$
$17$	−138.636	−1.97790	−0.988948	+	0.148265i	$0.952631\pi$
$18$	0	0
$19$	−164.000	−1.98022	−0.990110	−	0.140293i	$-0.955195\pi$
$19$	−164.000	−1.98022	−0.990110	+	0.140293i	$0.955195\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	98.3870	0.891961	0.445981	−	0.895043i	$-0.352855\pi$
$23$	98.3870	0.891961	0.445981	+	0.895043i	$0.352855\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−232.000	−1.34414	−0.672071	−	0.740486i	$-0.734595\pi$
$31$	−232.000	−1.34414	−0.672071	+	0.740486i	$0.734595\pi$
$32$	127.456	0.704101
$33$	0	0
$34$	−310.000	−1.56366
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−366.715	−1.56550
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	220.000	0.705157
$47$	545.601	1.69328	0.846639	−	0.532168i	$-0.178623\pi$
$47$	545.601	1.69328	0.846639	+	0.532168i	$0.178623\pi$
$48$	0	0
$49$	−343.000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	621.627	1.61108	0.805538	−	0.592544i	$-0.201876\pi$
$53$	621.627	1.61108	0.805538	+	0.592544i	$0.201876\pi$
$54$	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
$61$	−358.000	−0.751430	−0.375715	−	0.926735i	$-0.622603\pi$
$61$	−358.000	−0.751430	−0.375715	+	0.926735i	$0.622603\pi$
$62$	−518.768	−1.06264
$63$	0	0
$64$	533.000	1.04102
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	415.909	0.741711
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	492.000	0.742583
$77$	0	0
$78$	0	0
$79$	304.000	0.432945	0.216473	−	0.976289i	$-0.430545\pi$
$79$	304.000	0.432945	0.216473	+	0.976289i	$0.430545\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1270.09	−1.67964	−0.839820	−	0.542865i	$-0.817340\pi$
$83$	−1270.09	−1.67964	−0.839820	+	0.542865i	$0.817340\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	−295.161	−0.334485
$93$	0	0
$94$	1220.00	1.33865
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	−766.971	−0.790569
$99$	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		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	1390.00	1.27367
$107$	−17.8885	−0.0161622	−0.00808108	−	0.999967i	$-0.502572\pi$
$107$	−17.8885	−0.0161622	−0.00808108	+	0.999967i	$0.502572\pi$
$108$	0	0
$109$	1834.00	1.61161	0.805804	−	0.592182i	$-0.201733\pi$
$109$	1834.00	1.61161	0.805804	+	0.592182i	$0.201733\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1426.61	1.18765	0.593824	−	0.804595i	$-0.297617\pi$
$113$	1426.61	1.18765	0.593824	+	0.804595i	$0.297617\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1331.00	−1.00000
$122$	−800.512	−0.594057
$123$	0	0
$124$	696.000	0.504054
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	172.177	0.118894
$129$	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
$136$	3410.00	2.15004
$137$	−1990.10	−1.24106	−0.620532	−	0.784181i	$-0.713083\pi$
$137$	−1990.10	−1.24106	−0.620532	+	0.784181i	$0.713083\pi$
$138$	0	0
$139$	−1604.00	−0.978773	−0.489387	−	0.872067i	$-0.662779\pi$
$139$	−1604.00	−0.978773	−0.489387	+	0.872067i	$0.662779\pi$
$140$	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		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−3112.00	−1.67716	−0.838579	−	0.544779i	$-0.816613\pi$
$151$	−3112.00	−1.67716	−0.838579	+	0.544779i	$0.816613\pi$
$152$	4033.87	2.15256
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	679.765	0.342273
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	−2840.00	−1.32787
$167$	3765.54	1.74483	0.872414	−	0.488769i	$-0.162554\pi$
$167$	3765.54	1.74483	0.872414	+	0.488769i	$0.162554\pi$
$168$	0	0
$169$	−2197.00	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−424.853	−0.186711	−0.0933554	−	0.995633i	$-0.529759\pi$
$173$	−424.853	−0.186711	−0.0933554	+	0.995633i	$0.529759\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	1298.00	0.533036	0.266518	−	0.963830i	$-0.414127\pi$
$181$	1298.00	0.533036	0.266518	+	0.963830i	$0.414127\pi$
$182$	0	0
$183$	0	0
$184$	−2420.00	−0.969591
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−1636.80	−0.634979
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	1029.00	0.375000
$197$	−2553.59	−0.923532	−0.461766	−	0.887002i	$-0.652784\pi$
$197$	−2553.59	−0.923532	−0.461766	+	0.887002i	$0.652784\pi$
$198$	0	0
$199$	−5456.00	−1.94355	−0.971773	−	0.235919i	$-0.924190\pi$
$199$	−5456.00	−1.94355	−0.971773	+	0.235919i	$0.924190\pi$
$200$	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
$211$	−4228.00	−1.37947	−0.689733	−	0.724063i	$-0.742272\pi$
$211$	−4228.00	−1.37947	−0.689733	+	0.724063i	$0.742272\pi$
$212$	−1864.88	−0.604153
$213$	0	0
$214$	−40.0000	−0.0127773
$215$	0	0
$216$	0	0
$217$	0	0
$218$	4100.95	1.27409
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	0	0
$226$	3190.00	0.938919
$227$	−6457.76	−1.88818	−0.944090	−	0.329688i	$-0.893057\pi$
$227$	−6457.76	−1.88818	−0.944090	+	0.329688i	$0.893057\pi$
$228$	0	0
$229$	286.000	0.0825302	0.0412651	−	0.999148i	$-0.486861\pi$
$229$	286.000	0.0825302	0.0412651	+	0.999148i	$0.486861\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4449.78	−1.25114	−0.625568	−	0.780170i	$-0.715133\pi$
$233$	−4449.78	−1.25114	−0.625568	+	0.780170i	$0.715133\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−1438.00	−0.384356	−0.192178	−	0.981360i	$-0.561555\pi$
$241$	−1438.00	−0.384356	−0.192178	+	0.981360i	$0.561555\pi$
$242$	−2976.21	−0.790569
$243$	0	0
$244$	1074.00	0.281786
$245$	0	0
$246$	0	0
$247$	0	0
$248$	5706.45	1.46113
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−3879.00	−0.947021
$257$	6542.73	1.58803	0.794017	−	0.607896i	$-0.207986\pi$
$257$	6542.73	1.58803	0.794017	+	0.607896i	$0.207986\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6341.49	−1.48682	−0.743409	−	0.668837i	$-0.766792\pi$
$263$	−6341.49	−1.48682	−0.743409	+	0.668837i	$0.766792\pi$
$264$	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
$271$	6752.00	1.51349	0.756743	−	0.653712i	$-0.226789\pi$
$271$	6752.00	1.51349	0.756743	+	0.653712i	$0.226789\pi$
$272$	4297.72	0.958043
$273$	0	0
$274$	−4450.00	−0.981147
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	−3586.65	−0.773788
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	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
$289$	14307.0	2.91207
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3161.80	−0.630424	−0.315212	−	0.949021i	$-0.602076\pi$
$293$	−3161.80	−0.630424	−0.315212	+	0.949021i	$0.602076\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	−6958.64	−1.32591
$303$	0	0
$304$	5084.00	0.959169
$305$	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		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	−912.000	−0.162354
$317$	10084.7	1.78679	0.893393	−	0.449276i	$-0.148318\pi$
$317$	10084.7	1.78679	0.893393	+	0.449276i	$0.148318\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	22736.3	3.91667
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	5852.00	0.971767	0.485884	−	0.874023i	$-0.338498\pi$
$331$	5852.00	0.971767	0.485884	+	0.874023i	$0.338498\pi$
$332$	3810.26	0.629865
$333$	0	0
$334$	8420.00	1.37941
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	−4912.64	−0.790569
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−950.000	−0.147608
$347$	−8550.72	−1.32284	−0.661422	−	0.750014i	$-0.730047\pi$
$347$	−8550.72	−1.32284	−0.661422	+	0.750014i	$0.730047\pi$
$348$	0	0
$349$	3706.00	0.568417	0.284209	−	0.958762i	$-0.408269\pi$
$349$	3706.00	0.568417	0.284209	+	0.958762i	$0.408269\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2473.09	0.372888	0.186444	−	0.982466i	$-0.440304\pi$
$353$	2473.09	0.372888	0.186444	+	0.982466i	$0.440304\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	20037.0	2.92127
$362$	2902.42	0.421402
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	−3050.00	−0.432044
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	−13420.0	−1.84065
$377$	0	0
$378$	0	0
$379$	−4484.00	−0.607725	−0.303862	−	0.952716i	$-0.598276\pi$
$379$	−4484.00	−0.607725	−0.303862	+	0.952716i	$0.598276\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9561.43	−1.27563	−0.637815	−	0.770190i	$-0.720161\pi$
$383$	−9561.43	−1.27563	−0.637815	+	0.770190i	$0.720161\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	−13640.0	−1.76421
$392$	8436.68	1.08703
$393$	0	0
$394$	−5710.00	−0.730116
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	−12200.0	−1.53651
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	14326.0	1.73197	0.865984	−	0.500071i	$-0.166693\pi$
$409$	14326.0	1.73197	0.865984	+	0.500071i	$0.166693\pi$
$410$	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
$421$	6878.00	0.796231	0.398115	−	0.917335i	$-0.369664\pi$
$421$	6878.00	0.796231	0.398115	+	0.917335i	$0.369664\pi$
$422$	−9454.10	−1.09056
$423$	0	0
$424$	−15290.0	−1.75129
$425$	0	0
$426$	0	0
$427$	0	0
$428$	53.6656	0.00606081
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	−5502.00	−0.604353
$437$	−16135.5	−1.76628
$438$	0	0
$439$	−16976.0	−1.84560	−0.922802	−	0.385274i	$-0.874107\pi$
$439$	−16976.0	−1.84560	−0.922802	+	0.385274i	$0.874107\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18049.5	1.93580	0.967900	−	0.251335i	$-0.0808696\pi$
$443$	18049.5	1.93580	0.967900	+	0.251335i	$0.0808696\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	−4279.83	−0.445368
$453$	0	0
$454$	−14440.0	−1.49274
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	639.515	0.0652458
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	−9950.00	−0.989109
$467$	4329.03	0.428958	0.214479	−	0.976729i	$-0.431195\pi$
$467$	4329.03	0.428958	0.214479	+	0.976729i	$0.431195\pi$
$468$	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		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	−3215.47	−0.303860
$483$	0	0
$484$	3993.00	0.375000
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	8805.64	0.816829
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	7192.00	0.651069
$497$	0	0
$498$	0	0
$499$	−19316.0	−1.73287	−0.866436	−	0.499289i	$-0.833595\pi$
$499$	−19316.0	−1.73287	−0.866436	+	0.499289i	$0.833595\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−21958.2	−1.94646	−0.973228	−	0.229843i	$-0.926179\pi$
$503$	−21958.2	−1.94646	−0.973228	+	0.229843i	$0.926179\pi$
$504$	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
$512$	−10051.1	−0.867580
$513$	0	0
$514$	14630.0	1.25545
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	−14180.0	−1.17543
$527$	32163.6	2.65857
$528$	0	0
$529$	−2487.00	−0.204405
$530$	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
$541$	−3238.00	−0.257324	−0.128662	−	0.991688i	$-0.541068\pi$
$541$	−3238.00	−0.257324	−0.128662	+	0.991688i	$0.541068\pi$
$542$	15097.9	1.19652
$543$	0	0
$544$	−17670.0	−1.39264
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	5970.30	0.465399
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	4812.00	0.367040
$557$	26184.4	1.99186	0.995931	−	0.0901226i	$-0.0287259\pi$
$557$	26184.4	1.99186	0.995931	+	0.0901226i	$0.0287259\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14632.8	−1.09538	−0.547691	−	0.836681i	$-0.684493\pi$
$563$	−14632.8	−1.09538	−0.547691	+	0.836681i	$0.684493\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	26012.0	1.90642	0.953212	−	0.302302i	$-0.0977551\pi$
$571$	26012.0	1.90642	0.953212	+	0.302302i	$0.0977551\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	31991.4	2.30219
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−7070.00	−0.498394
$587$	−25777.4	−1.81252	−0.906258	−	0.422725i	$-0.861074\pi$
$587$	−25777.4	−1.81252	−0.906258	+	0.422725i	$0.861074\pi$
$588$	0	0
$589$	38048.0	2.66170
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7866.49	0.544752	0.272376	−	0.962191i	$-0.412191\pi$
$593$	7866.49	0.544752	0.272376	+	0.962191i	$0.412191\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−13642.0	−0.925905	−0.462952	−	0.886383i	$-0.653210\pi$
$601$	−13642.0	−0.925905	−0.462952	+	0.886383i	$0.653210\pi$
$602$	0	0
$603$	0	0
$604$	9336.00	0.628935
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	−20902.8	−1.39427
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−27991.1	−1.82638	−0.913192	−	0.407529i	$-0.866391\pi$
$617$	−27991.1	−1.82638	−0.913192	+	0.407529i	$0.866391\pi$
$618$	0	0
$619$	−3476.00	−0.225706	−0.112853	−	0.993612i	$-0.535999\pi$
$619$	−3476.00	−0.225706	−0.112853	+	0.993612i	$0.535999\pi$
$620$	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
$631$	−27808.0	−1.75439	−0.877194	−	0.480136i	$-0.840587\pi$
$631$	−27808.0	−1.75439	−0.877194	+	0.480136i	$0.840587\pi$
$632$	−7477.41	−0.470626
$633$	0	0
$634$	22550.0	1.41258
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	50840.0	3.09640
$647$	2638.56	0.160328	0.0801642	−	0.996782i	$-0.474456\pi$
$647$	2638.56	0.160328	0.0801642	+	0.996782i	$0.474456\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22678.2	1.35906	0.679530	−	0.733647i	$-0.262184\pi$
$653$	22678.2	1.35906	0.679530	+	0.733647i	$0.262184\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	32978.0	1.94054	0.970269	−	0.242029i	$-0.0778130\pi$
$661$	32978.0	1.94054	0.970269	+	0.242029i	$0.0778130\pi$
$662$	13085.5	0.768250
$663$	0	0
$664$	31240.0	1.82582
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−11296.6	−0.654310
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	6591.00	0.375000
$677$	−32901.5	−1.86781	−0.933905	−	0.357521i	$-0.883622\pi$
$677$	−32901.5	−1.86781	−0.933905	+	0.357521i	$0.883622\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−27512.6	−1.54135	−0.770673	−	0.637231i	$-0.780080\pi$
$683$	−27512.6	−1.54135	−0.770673	+	0.637231i	$0.780080\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−32812.0	−1.80641	−0.903204	−	0.429212i	$-0.858791\pi$
$691$	−32812.0	−1.80641	−0.903204	+	0.429212i	$0.858791\pi$
$692$	1274.56	0.0700166
$693$	0	0
$694$	−19120.0	−1.04580
$695$	0	0
$696$	0	0
$697$	0	0
$698$	8286.87	0.449373
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	5530.00	0.294794
$707$	0	0
$708$	0	0
$709$	37726.0	1.99835	0.999175	−	0.0406201i	$-0.0129334\pi$
$709$	37726.0	1.99835	0.999175	+	0.0406201i	$0.0129334\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−22825.8	−1.19892
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	44804.1	2.30947
$723$	0	0
$724$	−3894.00	−0.199889
$725$	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
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	12540.0	0.628031
$737$	0	0
$738$	0	0
$739$	−8804.00	−0.438241	−0.219121	−	0.975698i	$-0.570319\pi$
$739$	−8804.00	−0.438241	−0.219121	+	0.975698i	$0.570319\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10241.2	0.505670	0.252835	−	0.967509i	$-0.418637\pi$
$743$	10241.2	0.505670	0.252835	+	0.967509i	$0.418637\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−17512.0	−0.850895	−0.425447	−	0.904983i	$-0.639883\pi$
$751$	−17512.0	−0.850895	−0.425447	+	0.904983i	$0.639883\pi$
$752$	−16913.6	−0.820181
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	−10026.5	−0.480448
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−21380.0	−1.00847
$767$	0	0
$768$	0	0
$769$	−8786.00	−0.412004	−0.206002	−	0.978552i	$-0.566045\pi$
$769$	−8786.00	−0.412004	−0.206002	+	0.978552i	$0.566045\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	10281.4	0.478393	0.239196	−	0.970971i	$-0.423116\pi$
$773$	10281.4	0.478393	0.239196	+	0.970971i	$0.423116\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	−30500.0	−1.39473
$783$	0	0
$784$	10633.0	0.484375
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	7660.77	0.346324
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	16368.0	0.728830
$797$	−1426.61	−0.0634042	−0.0317021	−	0.999497i	$-0.510093\pi$
$797$	−1426.61	−0.0634042	−0.0317021	+	0.999497i	$0.510093\pi$
$798$	0	0
$799$	−75640.0	−3.34912
$800$	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		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	−14092.0	−0.610157	−0.305078	−	0.952327i	$-0.598683\pi$
$811$	−14092.0	−0.610157	−0.305078	+	0.952327i	$0.598683\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	32033.9	1.36924
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	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
$827$	42968.3	1.80672	0.903358	−	0.428888i	$-0.141095\pi$
$827$	42968.3	1.80672	0.903358	+	0.428888i	$0.141095\pi$
$828$	0	0
$829$	−45254.0	−1.89594	−0.947971	−	0.318356i	$-0.896869\pi$
$829$	−45254.0	−1.89594	−0.947971	+	0.318356i	$0.896869\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	47552.2	1.97790
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−24389.0	−1.00000
$842$	15379.7	0.629476
$843$	0	0
$844$	12684.0	0.517300
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−19270.4	−0.780365
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	440.000	0.0175688
$857$	47033.5	1.87472	0.937358	−	0.348367i	$-0.113264\pi$
$857$	47033.5	1.87472	0.937358	+	0.348367i	$0.113264\pi$
$858$	0	0
$859$	28204.0	1.12027	0.560133	−	0.828403i	$-0.310750\pi$
$859$	28204.0	1.12027	0.560133	+	0.828403i	$0.310750\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26340.9	1.03900	0.519498	−	0.854472i	$-0.326119\pi$
$863$	26340.9	1.03900	0.519498	+	0.854472i	$0.326119\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−45110.4	−1.75187
$873$	0	0
$874$	−36080.0	−1.39637
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	−37959.5	−1.45908
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	40360.0	1.53038
$887$	23085.2	0.873871	0.436936	−	0.899493i	$-0.356064\pi$
$887$	23085.2	0.873871	0.436936	+	0.899493i	$0.356064\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−89478.5	−3.35306
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−86180.0	−3.18654
$902$	0	0
$903$	0	0
$904$	−35090.0	−1.29101
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	19373.3	0.708068
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	−858.000	−0.0309488
$917$	0	0
$918$	0	0
$919$	−21224.0	−0.761823	−0.380911	−	0.924612i	$-0.624390\pi$
$919$	−21224.0	−0.761823	−0.380911	+	0.924612i	$0.624390\pi$
$920$	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		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	56252.0	1.98022
$932$	13349.3	0.469176
$933$	0	0
$934$	9680.00	0.339121
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	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
$947$	−45097.0	−1.54747	−0.773736	−	0.633508i	$-0.781614\pi$
$947$	−45097.0	−1.54747	−0.773736	+	0.633508i	$0.781614\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−30289.8	−1.02957	−0.514786	−	0.857319i	$-0.672129\pi$
$953$	−30289.8	−1.02957	−0.514786	+	0.857319i	$0.672129\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	24033.0	0.806720
$962$	0	0
$963$	0	0
$964$	4314.00	0.144133
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	32738.3	1.08703
$969$	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
$976$	11098.0	0.363974
$977$	−13018.4	−0.426300	−0.213150	−	0.977019i	$-0.568372\pi$
$977$	−13018.4	−0.426300	−0.213150	+	0.977019i	$0.568372\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	60794.2	1.97257	0.986284	−	0.165057i	$-0.0527809\pi$
$983$	60794.2	1.97257	0.986284	+	0.165057i	$0.0527809\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−62368.0	−1.99918	−0.999589	−	0.0286779i	$-0.990870\pi$
$991$	−62368.0	−1.99918	−0.999589	+	0.0286779i	$0.990870\pi$
$992$	−29569.8	−0.946412
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	−43191.9	−1.36996
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.a.k.1.2		2
3.2	odd	2	inner	225.4.a.k.1.1		2
5.2	odd	4		45.4.b.a.19.2	yes	2
5.3	odd	4		45.4.b.a.19.1	✓	2
5.4	even	2	inner	225.4.a.k.1.1		2
15.2	even	4		45.4.b.a.19.1	✓	2
15.8	even	4		45.4.b.a.19.2	yes	2
15.14	odd	2	CM	225.4.a.k.1.2		2
20.3	even	4		720.4.f.d.289.1		2
20.7	even	4		720.4.f.d.289.2		2
60.23	odd	4		720.4.f.d.289.2		2
60.47	odd	4		720.4.f.d.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.b.a.19.1	✓	2	5.3	odd	4
45.4.b.a.19.1	✓	2	15.2	even	4
45.4.b.a.19.2	yes	2	5.2	odd	4
45.4.b.a.19.2	yes	2	15.8	even	4
225.4.a.k.1.1		2	3.2	odd	2	inner
225.4.a.k.1.1		2	5.4	even	2	inner
225.4.a.k.1.2		2	1.1	even	1	trivial
225.4.a.k.1.2		2	15.14	odd	2	CM
720.4.f.d.289.1		2	20.3	even	4
720.4.f.d.289.1		2	60.47	odd	4
720.4.f.d.289.2		2	20.7	even	4
720.4.f.d.289.2		2	60.23	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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{2} -3.00000 q^{4} -24.5967 q^{8} +O(q^{10})\)	\(q+2.23607 q^{2} -3.00000 q^{4} -24.5967 q^{8} -31.0000 q^{16} -138.636 q^{17} -164.000 q^{19} +98.3870 q^{23} -232.000 q^{31} +127.456 q^{32} -310.000 q^{34} -366.715 q^{38} +220.000 q^{46} +545.601 q^{47} -343.000 q^{49} +621.627 q^{53} -358.000 q^{61} -518.768 q^{62} +533.000 q^{64} +415.909 q^{68} +492.000 q^{76} +304.000 q^{79} -1270.09 q^{83} -295.161 q^{92} +1220.00 q^{94} -766.971 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{4}+O(q^{10}) \) 2 * q - 6 * q^4	\( 2 q - 6 q^{4} - 62 q^{16} - 328 q^{19} - 464 q^{31} - 620 q^{34} + 440 q^{46} - 686 q^{49} - 716 q^{61} + 1066 q^{64} + 984 q^{76} + 608 q^{79} + 2440 q^{94}+O(q^{100}) \) 2 * q - 6 * q^4 - 62 * q^16 - 328 * q^19 - 464 * q^31 - 620 * q^34 + 440 * q^46 - 686 * q^49 - 716 * q^61 + 1066 * q^64 + 984 * q^76 + 608 * q^79 + 2440 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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