LMFDB - Embedded newform 675.4.a.k.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,4,Mod(1,675)] 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("675.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-9,0,27,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.8262892539$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	675.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.38197 q^{2} +3.43769 q^{4} +15.4296 q^{8} -79.6838 q^{16} -87.3181 q^{17} +102.125 q^{19} -121.807 q^{23} +337.371 q^{31} +146.051 q^{32} +295.307 q^{34} -345.382 q^{38} +411.945 q^{46} -545.601 q^{47} -343.000 q^{49} +706.813 q^{53} +943.735 q^{61} -1140.98 q^{62} +143.530 q^{64} -300.173 q^{68} +351.073 q^{76} -1339.35 q^{79} -1346.04 q^{83} -418.734 q^{92} +1845.20 q^{94} +1160.01 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 9 q^{2} + 27 q^{4} - 72 q^{8} + 223 q^{16} - 36 q^{17} + 164 q^{19} - 342 q^{23} + 232 q^{31} - 855 q^{32} + 7 q^{34} - 693 q^{38} + 1649 q^{46} - 686 q^{49} + 792 q^{53} + 358 q^{61} - 549 q^{62}+ \cdots + 3087 q^{98}+O(q^{100}) $ 2 * q - 9 * q^2 + 27 * q^4 - 72 * q^8 + 223 * q^16 - 36 * q^17 + 164 * q^19 - 342 * q^23 + 232 * q^31 - 855 * q^32 + 7 * q^34 - 693 * q^38 + 1649 * q^46 - 686 * q^49 + 792 * q^53 + 358 * q^61 - 549 * q^62 + 3346 * q^64 + 909 * q^68 + 1809 * q^76 - 304 * q^79 - 1422 * q^83 - 5607 * q^92 - 1220 * q^94 + 3087 * q^98

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 $

$2$	−3.38197	−1.19571	−0.597853	−	0.801606i	$-0.703979\pi$
$2$	−3.38197	−1.19571	−0.597853	+	0.801606i	$0.703979\pi$
$3$	0	0
$3$	0	0
$4$	3.43769	0.429712
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	15.4296	0.681897
$9$	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
$16$	−79.6838	−1.24506
$17$	−87.3181	−1.24575	−0.622875	−	0.782321i	$-0.714036\pi$
$17$	−87.3181	−1.24575	−0.622875	+	0.782321i	$0.714036\pi$
$18$	0	0
$19$	102.125	1.23310	0.616552	−	0.787314i	$-0.288529\pi$
$19$	102.125	1.23310	0.616552	+	0.787314i	$0.288529\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−121.807	−1.10428	−0.552139	−	0.833752i	$-0.686188\pi$
$23$	−121.807	−1.10428	−0.552139	+	0.833752i	$0.686188\pi$
$24$	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
$31$	337.371	1.95463	0.977316	−	0.211788i	$-0.0679286\pi$
$31$	337.371	1.95463	0.977316	+	0.211788i	$0.0679286\pi$
$32$	146.051	0.806828
$33$	0	0
$34$	295.307	1.48955
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−345.382	−1.47443
$39$	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
$46$	411.945	1.32039
$47$	−545.601	−1.69328	−0.846639	−	0.532168i	$-0.821377\pi$
$47$	−545.601	−1.69328	−0.846639	+	0.532168i	$0.821377\pi$
$48$	0	0
$49$	−343.000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	706.813	1.83185	0.915927	−	0.401344i	$-0.131457\pi$
$53$	706.813	1.83185	0.915927	+	0.401344i	$0.131457\pi$
$54$	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
$61$	943.735	1.98087	0.990434	−	0.137989i	$-0.0440639\pi$
$61$	943.735	1.98087	0.990434	+	0.137989i	$0.0440639\pi$
$62$	−1140.98	−2.33716
$63$	0	0
$64$	143.530	0.280331
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−300.173	−0.535313
$69$	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
$76$	351.073	0.529880
$77$	0	0
$78$	0	0
$79$	−1339.35	−1.90745	−0.953727	−	0.300674i	$-0.902789\pi$
$79$	−1339.35	−1.90745	−0.953727	+	0.300674i	$0.902789\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1346.04	−1.78009	−0.890045	−	0.455873i	$-0.849327\pi$
$83$	−1346.04	−1.78009	−0.890045	+	0.455873i	$0.849327\pi$
$84$	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
$92$	−418.734	−0.474522
$93$	0	0
$94$	1845.20	2.02466
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	1160.01	1.19571
$99$	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
$106$	−2390.42	−2.19036
$107$	17.8885	0.0161622	0.00808108	−	0.999967i	$-0.497428\pi$
$107$	17.8885	0.0161622	0.00808108	+	0.999967i	$0.497428\pi$
$108$	0	0
$109$	250.227	0.219885	0.109942	−	0.993938i	$-0.464933\pi$
$109$	250.227	0.219885	0.109942	+	0.993938i	$0.464933\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1426.61	−1.18765	−0.593824	−	0.804595i	$-0.702383\pi$
$113$	−1426.61	−1.18765	−0.593824	+	0.804595i	$0.702383\pi$
$114$	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
$122$	−3191.68	−2.36853
$123$	0	0
$124$	1159.78	0.839928
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1653.82	−1.14202
$129$	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
$136$	−1347.28	−0.849473
$137$	−3173.05	−1.97877	−0.989387	−	0.145306i	$-0.953583\pi$
$137$	−3173.05	−1.97877	−0.989387	+	0.145306i	$0.953583\pi$
$138$	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	$0.816613\pi$
$152$	1575.74	0.840850
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	4529.64	2.28075
$159$	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
$166$	4552.27	2.12846
$167$	55.7692	0.0258416	0.0129208	−	0.999917i	$-0.495887\pi$
$167$	55.7692	0.0258416	0.0129208	+	0.999917i	$0.495887\pi$
$168$	0	0
$169$	−2197.00	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4136.43	−1.81784	−0.908921	−	0.416968i	$-0.863093\pi$
$173$	−4136.43	−1.81784	−0.908921	+	0.416968i	$0.863093\pi$
$174$	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
$181$	−4714.17	−1.93592	−0.967960	−	0.251103i	$-0.919207\pi$
$181$	−4714.17	−1.93592	−0.967960	+	0.251103i	$0.919207\pi$
$182$	0	0
$183$	0	0
$184$	−1879.42	−0.753004
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−1875.61	−0.727621
$189$	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
$196$	−1179.13	−0.429712
$197$	2971.21	1.07457	0.537283	−	0.843402i	$-0.319451\pi$
$197$	2971.21	1.07457	0.537283	+	0.843402i	$0.319451\pi$
$198$	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
$211$	5957.80	1.94385	0.971924	−	0.235295i	$-0.0756056\pi$
$211$	5957.80	1.94385	0.971924	+	0.235295i	$0.0756056\pi$
$212$	2429.81	0.787169
$213$	0	0
$214$	−60.4984	−0.0193252
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−846.261	−0.262918
$219$	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
$226$	4824.75	1.42008
$227$	−1275.88	−0.373054	−0.186527	−	0.982450i	$-0.559723\pi$
$227$	−1275.88	−0.373054	−0.186527	+	0.982450i	$0.559723\pi$
$228$	0	0
$229$	5854.13	1.68931	0.844655	−	0.535311i	$-0.179805\pi$
$229$	5854.13	1.68931	0.844655	+	0.535311i	$0.179805\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4449.78	1.25114	0.625568	−	0.780170i	$-0.284867\pi$
$233$	4449.78	1.25114	0.625568	+	0.780170i	$0.284867\pi$
$234$	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
$241$	−5640.38	−1.50759	−0.753794	−	0.657111i	$-0.771778\pi$
$241$	−5640.38	−1.50759	−0.753794	+	0.657111i	$0.771778\pi$
$242$	4501.40	1.19571
$243$	0	0
$244$	3244.27	0.851202
$245$	0	0
$246$	0	0
$247$	0	0
$248$	5205.48	1.33286
$249$	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
$256$	4444.94	1.08519
$257$	−1066.63	−0.258890	−0.129445	−	0.991587i	$-0.541320\pi$
$257$	−1066.63	−0.258890	−0.129445	+	0.991587i	$0.541320\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6341.49	1.48682	0.743409	−	0.668837i	$-0.233208\pi$
$263$	6341.49	1.48682	0.743409	+	0.668837i	$0.233208\pi$
$264$	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
$271$	1675.28	0.375520	0.187760	−	0.982215i	$-0.439877\pi$
$271$	1675.28	0.375520	0.187760	+	0.982215i	$0.439877\pi$
$272$	6957.84	1.55103
$273$	0	0
$274$	10731.1	2.36603
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	5424.67	1.17032
$279$	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
$289$	2711.45	0.551893
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9824.90	−1.95897	−0.979483	−	0.201529i	$-0.935409\pi$
$293$	−9824.90	−1.95897	−0.979483	+	0.201529i	$0.935409\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	10524.7	2.00539
$303$	0	0
$304$	−8137.68	−1.53529
$305$	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
$316$	−4604.28	−0.819656
$317$	650.333	0.115225	0.0576125	−	0.998339i	$-0.481651\pi$
$317$	650.333	0.115225	0.0576125	+	0.998339i	$0.481651\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−8917.33	−1.53614
$324$	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	$0.338498\pi$
$332$	−4627.29	−0.764925
$333$	0	0
$334$	−188.610	−0.0308990
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	7430.18	1.19571
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	13989.3	2.17360
$347$	8550.72	1.32284	0.661422	−	0.750014i	$-0.269953\pi$
$347$	8550.72	1.32284	0.661422	+	0.750014i	$0.269953\pi$
$348$	0	0
$349$	8974.04	1.37642	0.688208	−	0.725513i	$-0.258398\pi$
$349$	8974.04	1.37642	0.688208	+	0.725513i	$0.258398\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2473.09	−0.372888	−0.186444	−	0.982466i	$-0.559696\pi$
$353$	−2473.09	−0.372888	−0.186444	+	0.982466i	$0.559696\pi$
$354$	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
$361$	3570.44	0.520548
$362$	15943.2	2.31479
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	9706.01	1.37489
$369$	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
$376$	−8418.38	−1.15464
$377$	0	0
$378$	0	0
$379$	−9933.39	−1.34629	−0.673145	−	0.739510i	$-0.735057\pi$
$379$	−9933.39	−1.34629	−0.673145	+	0.739510i	$0.735057\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−14779.7	−1.97182	−0.985911	−	0.167269i	$-0.946505\pi$
$383$	−14779.7	−1.97182	−0.985911	+	0.167269i	$0.946505\pi$
$384$	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
$391$	10635.9	1.37566
$392$	−5292.34	−0.681897
$393$	0	0
$394$	−10048.5	−1.28487
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	18452.0	2.32391
$399$	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
$409$	−14327.4	−1.73213	−0.866067	−	0.499929i	$-0.833360\pi$
$409$	−14327.4	−1.73213	−0.866067	+	0.499929i	$0.833360\pi$
$410$	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
$421$	−17164.0	−1.98699	−0.993493	−	0.113890i	$-0.963669\pi$
$421$	−17164.0	−1.98699	−0.993493	+	0.113890i	$0.963669\pi$
$422$	−20149.1	−2.32427
$423$	0	0
$424$	10905.8	1.24914
$425$	0	0
$426$	0	0
$427$	0	0
$428$	61.4953	0.00694507
$429$	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
$436$	860.206	0.0944871
$437$	−12439.4	−1.36169
$438$	0	0
$439$	2349.99	0.255488	0.127744	−	0.991807i	$-0.459226\pi$
$439$	2349.99	0.255488	0.127744	+	0.991807i	$0.459226\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4965.77	0.532575	0.266288	−	0.963894i	$-0.414203\pi$
$443$	4965.77	0.532575	0.266288	+	0.963894i	$0.414203\pi$
$444$	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
$452$	−4904.25	−0.510347
$453$	0	0
$454$	4314.99	0.446063
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	−19798.5	−2.01992
$459$	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
$466$	−15049.0	−1.49599
$467$	19237.5	1.90622	0.953111	−	0.302620i	$-0.0978613\pi$
$467$	19237.5	1.90622	0.953111	+	0.302620i	$0.0978613\pi$
$468$	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
$482$	19075.6	1.80263
$483$	0	0
$484$	−4575.57	−0.429712
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	14561.4	1.35075
$489$	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
$496$	−26883.0	−2.43363
$497$	0	0
$498$	0	0
$499$	18.3109	0.00164271	0.000821353	−	1.00000i	$-0.499739\pi$
$499$	18.3109	0.00164271	0.000821353		1.00000i	$0.499739\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−15470.1	−1.37133	−0.685663	−	0.727919i	$-0.740488\pi$
$503$	−15470.1	−1.37133	−0.685663	+	0.727919i	$0.740488\pi$
$504$	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
$512$	−1802.04	−0.155547
$513$	0	0
$514$	3607.32	0.309556
$515$	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
$526$	−21446.7	−1.77780
$527$	−29458.6	−2.43498
$528$	0	0
$529$	2669.82	0.219432
$530$	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	$0.541068\pi$
$542$	−5665.73	−0.449011
$543$	0	0
$544$	−12752.9	−1.00511
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	−10908.0	−0.850302
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−5514.06	−0.420590
$557$	−26184.4	−1.99186	−0.995931	−	0.0901226i	$-0.971274\pi$
$557$	−26184.4	−1.99186	−0.995931	+	0.0901226i	$0.971274\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14632.8	1.09538	0.547691	−	0.836681i	$-0.315507\pi$
$563$	14632.8	1.09538	0.547691	+	0.836681i	$0.315507\pi$
$564$	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
$571$	−20150.2	−1.47681	−0.738407	−	0.674355i	$-0.764422\pi$
$571$	−20150.2	−1.47681	−0.738407	+	0.674355i	$0.764422\pi$
$572$	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	$0.5\pi$
$578$	−9170.04	−0.659902
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	33227.5	2.34235
$587$	−23301.7	−1.63844	−0.819220	−	0.573480i	$-0.805593\pi$
$587$	−23301.7	−1.63844	−0.819220	+	0.573480i	$0.805593\pi$
$588$	0	0
$589$	34453.9	2.41027
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−20132.8	−1.39419	−0.697094	−	0.716980i	$-0.745524\pi$
$593$	−20132.8	−1.39419	−0.697094	+	0.716980i	$0.745524\pi$
$594$	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
$601$	29441.1	1.99821	0.999107	−	0.0422630i	$-0.0134567\pi$
$601$	29441.1	1.99821	0.999107	+	0.0422630i	$0.0134567\pi$
$602$	0	0
$603$	0	0
$604$	−10698.1	−0.720695
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	14915.4	0.994903
$609$	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
$617$	−24813.5	−1.61905	−0.809527	−	0.587083i	$-0.800276\pi$
$617$	−24813.5	−1.61905	−0.809527	+	0.587083i	$0.800276\pi$
$618$	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
$631$	722.379	0.0455744	0.0227872	−	0.999740i	$-0.492746\pi$
$631$	722.379	0.0455744	0.0227872	+	0.999740i	$0.492746\pi$
$632$	−20665.6	−1.30069
$633$	0	0
$634$	−2199.41	−0.137775
$635$	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
$646$	30158.1	1.83677
$647$	29732.3	1.80664	0.903320	−	0.428967i	$-0.141122\pi$
$647$	29732.3	1.80664	0.903320	+	0.428967i	$0.141122\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9864.90	−0.591184	−0.295592	−	0.955314i	$-0.595517\pi$
$653$	−9864.90	−0.591184	−0.295592	+	0.955314i	$0.595517\pi$
$654$	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	$0.0778130\pi$
$662$	−19791.3	−1.16195
$663$	0	0
$664$	−20768.9	−1.21384
$665$	0	0
$666$	0	0
$667$	0	0
$668$	191.718	0.0111045
$669$	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
$676$	−7552.61	−0.429712
$677$	32901.5	1.86781	0.933905	−	0.357521i	$-0.116378\pi$
$677$	32901.5	1.86781	0.933905	+	0.357521i	$0.116378\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5944.71	0.333042	0.166521	−	0.986038i	$-0.446747\pi$
$683$	5944.71	0.333042	0.166521	+	0.986038i	$0.446747\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	29909.6	1.64662	0.823311	−	0.567591i	$-0.192124\pi$
$691$	29909.6	1.64662	0.823311	+	0.567591i	$0.192124\pi$
$692$	−14219.8	−0.781148
$693$	0	0
$694$	−28918.3	−1.58173
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−30349.9	−1.64579
$699$	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
$706$	8363.91	0.445864
$707$	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
$713$	−41093.9	−2.15846
$714$	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
$722$	−12075.1	−0.622422
$723$	0	0
$724$	−16205.9	−0.831888
$725$	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
$736$	−17790.0	−0.890963
$737$	0	0
$738$	0	0
$739$	38352.2	1.90908	0.954539	−	0.298085i	$-0.0963478\pi$
$739$	38352.2	1.90908	0.954539	+	0.298085i	$0.0963478\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10241.2	−0.505670	−0.252835	−	0.967509i	$-0.581363\pi$
$743$	−10241.2	−0.505670	−0.252835	+	0.967509i	$0.581363\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−23503.8	−1.14203	−0.571015	−	0.820940i	$-0.693450\pi$
$751$	−23503.8	−1.14203	−0.571015	+	0.820940i	$0.693450\pi$
$752$	43475.5	2.10823
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	33594.4	1.60977
$759$	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
$766$	49984.5	2.35772
$767$	0	0
$768$	0	0
$769$	−31750.8	−1.48890	−0.744449	−	0.667679i	$-0.767288\pi$
$769$	−31750.8	−1.48890	−0.744449	+	0.667679i	$0.767288\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	41284.7	1.92097	0.960484	−	0.278335i	$-0.0897827\pi$
$773$	41284.7	1.92097	0.960484	+	0.278335i	$0.0897827\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	−35970.3	−1.64488
$783$	0	0
$784$	27331.5	1.24506
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	10214.1	0.461754
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−18756.1	−0.835164
$797$	−39665.3	−1.76288	−0.881441	−	0.472294i	$-0.843426\pi$
$797$	−39665.3	−1.76288	−0.881441	+	0.472294i	$0.843426\pi$
$798$	0	0
$799$	47640.8	2.10940
$800$	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
$818$	48454.7	2.07112
$819$	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
$827$	39151.1	1.64621	0.823107	−	0.567887i	$-0.192239\pi$
$827$	39151.1	1.64621	0.823107	+	0.567887i	$0.192239\pi$
$828$	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
$833$	29950.1	1.24575
$834$	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
$842$	58048.0	2.37585
$843$	0	0
$844$	20481.1	0.835294
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−56321.6	−2.28077
$849$	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
$856$	276.012	0.0110209
$857$	8378.73	0.333969	0.166985	−	0.985959i	$-0.446597\pi$
$857$	8378.73	0.333969	0.166985	+	0.985959i	$0.446597\pi$
$858$	0	0
$859$	22021.7	0.874703	0.437352	−	0.899291i	$-0.355917\pi$
$859$	22021.7	0.874703	0.437352	+	0.899291i	$0.355917\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	50691.4	1.99949	0.999743	−	0.0226629i	$-0.00721446\pi$
$863$	50691.4	1.99949	0.999743	+	0.0226629i	$0.00721446\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	3860.90	0.149939
$873$	0	0
$874$	42069.8	1.62818
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	−7947.60	−0.305488
$879$	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
$886$	−16794.1	−0.636803
$887$	52699.6	1.99490	0.997451	−	0.0713489i	$-0.0227304\pi$
$887$	52699.6	1.99490	0.997451	+	0.0713489i	$0.0227304\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−55719.2	−2.08799
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−61717.6	−2.28203
$902$	0	0
$903$	0	0
$904$	−22012.0	−0.809854
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	−4386.09	−0.160306
$909$	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
$916$	20124.7	0.725917
$917$	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
$931$	−35028.7	−1.23310
$932$	15297.0	0.537627
$933$	0	0
$934$	−65060.6	−2.27928
$935$	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
$947$	9428.49	0.323532	0.161766	−	0.986829i	$-0.448281\pi$
$947$	9428.49	0.323532	0.161766	+	0.986829i	$0.448281\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	30289.8	1.02957	0.514786	−	0.857319i	$-0.327871\pi$
$953$	30289.8	1.02957	0.514786	+	0.857319i	$0.327871\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	84028.0	2.82058
$962$	0	0
$963$	0	0
$964$	−19389.9	−0.647828
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	−20536.7	−0.681897
$969$	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
$976$	−75200.4	−2.46630
$977$	13018.4	0.426300	0.213150	−	0.977019i	$-0.431628\pi$
$977$	13018.4	0.426300	0.213150	+	0.977019i	$0.431628\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	21586.1	0.700397	0.350198	−	0.936676i	$-0.386114\pi$
$983$	21586.1	0.700397	0.350198	+	0.936676i	$0.386114\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	32733.6	1.04926	0.524630	−	0.851330i	$-0.324204\pi$
$991$	32733.6	1.04926	0.524630	+	0.851330i	$0.324204\pi$
$992$	49273.5	1.57705
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	−61.9270	−0.00196419
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.k.1.2		2
3.2	odd	2		675.4.a.o.1.1		2
5.2	odd	4		135.4.b.a.109.2	✓	4
5.3	odd	4		135.4.b.a.109.3	yes	4
5.4	even	2		675.4.a.o.1.1		2
15.2	even	4		135.4.b.a.109.3	yes	4
15.8	even	4		135.4.b.a.109.2	✓	4
15.14	odd	2	CM	675.4.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.b.a.109.2	✓	4	5.2	odd	4
135.4.b.a.109.2	✓	4	15.8	even	4
135.4.b.a.109.3	yes	4	5.3	odd	4
135.4.b.a.109.3	yes	4	15.2	even	4
675.4.a.k.1.2		2	1.1	even	1	trivial
675.4.a.k.1.2		2	15.14	odd	2	CM
675.4.a.o.1.1		2	3.2	odd	2
675.4.a.o.1.1		2	5.4	even	2

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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

\(f(q)\)	\(=\)	\(q-3.38197 q^{2} +3.43769 q^{4} +15.4296 q^{8} -79.6838 q^{16} -87.3181 q^{17} +102.125 q^{19} -121.807 q^{23} +337.371 q^{31} +146.051 q^{32} +295.307 q^{34} -345.382 q^{38} +411.945 q^{46} -545.601 q^{47} -343.000 q^{49} +706.813 q^{53} +943.735 q^{61} -1140.98 q^{62} +143.530 q^{64} -300.173 q^{68} +351.073 q^{76} -1339.35 q^{79} -1346.04 q^{83} -418.734 q^{92} +1845.20 q^{94} +1160.01 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9 q^{2} + 27 q^{4} - 72 q^{8} + 223 q^{16} - 36 q^{17} + 164 q^{19} - 342 q^{23} + 232 q^{31} - 855 q^{32} + 7 q^{34} - 693 q^{38} + 1649 q^{46} - 686 q^{49} + 792 q^{53} + 358 q^{61} - 549 q^{62}+ \cdots + 3087 q^{98}+O(q^{100}) \) 2 * q - 9 * q^2 + 27 * q^4 - 72 * q^8 + 223 * q^16 - 36 * q^17 + 164 * q^19 - 342 * q^23 + 232 * q^31 - 855 * q^32 + 7 * q^34 - 693 * q^38 + 1649 * q^46 - 686 * q^49 + 792 * q^53 + 358 * q^61 - 549 * q^62 + 3346 * q^64 + 909 * q^68 + 1809 * q^76 - 304 * q^79 - 1422 * q^83 - 5607 * q^92 - 1220 * q^94 + 3087 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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