LMFDB - Embedded newform 675.4.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,4,Mod(1,675)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.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:	$39.8262892539$
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			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q-5.61803 q^{2} +23.5623 q^{4} -87.4296 q^{8} +O(q^{10})$	$q-5.61803 q^{2} +23.5623 q^{4} -87.4296 q^{8} +302.684 q^{16} +51.3181 q^{17} +61.8754 q^{19} -220.193 q^{23} -105.371 q^{31} -1001.05 q^{32} -288.307 q^{34} -347.618 q^{38} +1237.05 q^{46} +545.601 q^{47} -343.000 q^{49} +85.1866 q^{53} -585.735 q^{61} +591.976 q^{62} +3202.47 q^{64} +1209.17 q^{68} +1457.93 q^{76} +1035.35 q^{79} -75.9567 q^{83} -5188.27 q^{92} -3065.20 q^{94} +1926.99 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 9 q^{2} + 27 q^{4} - 72 q^{8}+O(q^{10}) $ 2 * q - 9 * q^2 + 27 * q^4 - 72 * q^8	$ 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}+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

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 $

$2$	−5.61803	−1.98627	−0.993137	−	0.116953i	$-0.962687\pi$
$2$	−5.61803	−1.98627	−0.993137	+	0.116953i	$0.962687\pi$
$3$	0	0
$3$	0	0
$4$	23.5623	2.94529
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−87.4296	−3.86388
$9$	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
$16$	302.684	4.72943
$17$	51.3181	0.732145	0.366073	−	0.930586i	$-0.380702\pi$
$17$	51.3181	0.732145	0.366073	+	0.930586i	$0.380702\pi$
$18$	0	0
$19$	61.8754	0.747115	0.373558	−	0.927607i	$-0.378138\pi$
$19$	61.8754	0.747115	0.373558	+	0.927607i	$0.378138\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−220.193	−1.99624	−0.998120	−	0.0612908i	$-0.980478\pi$
$23$	−220.193	−1.99624	−0.998120	+	0.0612908i	$0.980478\pi$
$24$	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
$31$	−105.371	−0.610488	−0.305244	−	0.952274i	$-0.598738\pi$
$31$	−105.371	−0.610488	−0.305244	+	0.952274i	$0.598738\pi$
$32$	−1001.05	−5.53008
$33$	0	0
$34$	−288.307	−1.45424
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−347.618	−1.48398
$39$	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
$46$	1237.05	3.96508
$47$	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
$53$	85.1866	0.220779	0.110389	−	0.993888i	$-0.464790\pi$
$53$	85.1866	0.220779	0.110389	+	0.993888i	$0.464790\pi$
$54$	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
$61$	−585.735	−1.22944	−0.614719	−	0.788746i	$-0.710731\pi$
$61$	−585.735	−1.22944	−0.614719	+	0.788746i	$0.710731\pi$
$62$	591.976	1.21260
$63$	0	0
$64$	3202.47	6.25483
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	1209.17	2.15638
$69$	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
$76$	1457.93	2.20047
$77$	0	0
$78$	0	0
$79$	1035.35	1.47451	0.737255	−	0.675615i	$-0.236122\pi$
$79$	1035.35	1.47451	0.737255	+	0.675615i	$0.236122\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−75.9567	−0.100450	−0.0502249	−	0.998738i	$-0.515994\pi$
$83$	−75.9567	−0.100450	−0.0502249	+	0.998738i	$0.515994\pi$
$84$	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
$92$	−5188.27	−5.87950
$93$	0	0
$94$	−3065.20	−3.36331
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	1926.99	1.98627
$99$	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
$106$	−478.581	−0.438527
$107$	−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
$109$	−2084.23	−1.83149	−0.915747	−	0.401756i	$-0.868400\pi$
$109$	−2084.23	−1.83149	−0.915747	+	0.401756i	$0.868400\pi$
$110$	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
$122$	3290.68	2.44200
$123$	0	0
$124$	−2482.78	−1.79806
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−9983.18	−6.89372
$129$	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
$136$	−4486.72	−2.82892
$137$	−1182.95	−0.737710	−0.368855	−	0.929487i	$-0.620250\pi$
$137$	−1182.95	−0.737710	−0.368855	+	0.929487i	$0.620250\pi$
$138$	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	$0.816613\pi$
$152$	−5409.74	−2.88676
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	−5816.64	−2.92878
$159$	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
$166$	426.727	0.199521
$167$	−3709.77	−1.71899	−0.859493	−	0.511148i	$-0.829220\pi$
$167$	−3709.77	−1.71899	−0.859493	+	0.511148i	$0.829220\pi$
$168$	0	0
$169$	−2197.00	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3711.57	−1.63113	−0.815566	−	0.578665i	$-0.803574\pi$
$173$	−3711.57	−1.63113	−0.815566	+	0.578665i	$0.803574\pi$
$174$	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
$181$	3416.17	1.40288	0.701442	−	0.712726i	$-0.252540\pi$
$181$	3416.17	1.40288	0.701442	+	0.712726i	$0.252540\pi$
$182$	0	0
$183$	0	0
$184$	19251.4	7.71323
$185$	0	0
$186$	0	0
$187$	0	0
$188$	12855.6	4.98719
$189$	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
$196$	−8081.87	−2.94529
$197$	5524.79	1.99810	0.999049	−	0.0436000i	$-0.0138827\pi$
$197$	5524.79	1.99810	0.999049	+	0.0436000i	$0.0138827\pi$
$198$	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
$211$	−1729.80	−0.564381	−0.282191	−	0.959358i	$-0.591061\pi$
$211$	−1729.80	−0.564381	−0.282191	+	0.959358i	$0.591061\pi$
$212$	2007.19	0.650257
$213$	0	0
$214$	100.498	0.0321025
$215$	0	0
$216$	0	0
$217$	0	0
$218$	11709.3	3.63785
$219$	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
$226$	−8014.75	−2.35900
$227$	5181.88	1.51513	0.757563	−	0.652762i	$-0.226390\pi$
$227$	5181.88	1.51513	0.757563	+	0.652762i	$0.226390\pi$
$228$	0	0
$229$	−6140.13	−1.77184	−0.885920	−	0.463837i	$-0.846472\pi$
$229$	−6140.13	−1.77184	−0.885920	+	0.463837i	$0.846472\pi$
$230$	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
$241$	7078.38	1.89194	0.945972	−	0.324249i	$-0.105112\pi$
$241$	7078.38	1.89194	0.945972	+	0.324249i	$0.105112\pi$
$242$	7477.60	1.98627
$243$	0	0
$244$	−13801.3	−3.62105
$245$	0	0
$246$	0	0
$247$	0	0
$248$	9212.52	2.35885
$249$	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
$256$	30466.1	7.43800
$257$	−7609.37	−1.84692	−0.923462	−	0.383691i	$-0.874653\pi$
$257$	−7609.37	−1.84692	−0.923462	+	0.383691i	$0.874653\pi$
$258$	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
$271$	−8427.28	−1.88901	−0.944503	−	0.328503i	$-0.893456\pi$
$271$	−8427.28	−1.88901	−0.944503	+	0.328503i	$0.893456\pi$
$272$	15533.2	3.46263
$273$	0	0
$274$	6645.85	1.46529
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	9011.33	1.94411
$279$	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
$289$	−2279.45	−0.463963
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6663.10	−1.32854	−0.664270	−	0.747492i	$-0.731258\pi$
$293$	−6663.10	−1.32854	−0.664270	+	0.747492i	$0.731258\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	17483.3	3.33130
$303$	0	0
$304$	18728.7	3.53343
$305$	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
$316$	24395.3	4.34285
$317$	−9434.33	−1.67156	−0.835781	−	0.549063i	$-0.814985\pi$
$317$	−9434.33	−1.67156	−0.835781	+	0.549063i	$0.814985\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3175.33	0.546997
$324$	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	$0.338498\pi$
$332$	−1789.71	−0.295854
$333$	0	0
$334$	20841.6	3.41438
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	12342.8	1.98627
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	20851.7	3.23988
$347$	−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
$349$	−12680.0	−1.94483	−0.972417	−	0.233249i	$-0.925064\pi$
$349$	−12680.0	−1.94483	−0.972417	+	0.233249i	$0.925064\pi$
$350$	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
$361$	−3030.44	−0.441819
$362$	−19192.2	−2.78651
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	−66649.0	−9.44109
$369$	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
$376$	−47701.6	−6.54262
$377$	0	0
$378$	0	0
$379$	14417.4	1.95401	0.977007	−	0.213206i	$-0.0683904\pi$
$379$	14417.4	1.95401	0.977007	+	0.213206i	$0.0683904\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5218.29	−0.696193	−0.348097	−	0.937459i	$-0.613172\pi$
$383$	−5218.29	−0.696193	−0.348097	+	0.937459i	$0.613172\pi$
$384$	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
$391$	−11299.9	−1.46154
$392$	29988.3	3.86388
$393$	0	0
$394$	−31038.5	−3.96877
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	30652.0	3.86042
$399$	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
$409$	1.36180	0.000164637 0	8.23187e−5	−	1.00000i	$-0.499974\pi$
$409$	1.36180	0.000164637 0	8.23187e−5		1.00000i	$0.499974\pi$
$410$	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
$421$	10286.0	1.19076	0.595378	−	0.803446i	$-0.297002\pi$
$421$	10286.0	1.19076	0.595378	+	0.803446i	$0.297002\pi$
$422$	9718.08	1.12102
$423$	0	0
$424$	−7447.82	−0.853062
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−421.495	−0.0476022
$429$	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
$436$	−49109.2	−5.39428
$437$	−13624.6	−1.49142
$438$	0	0
$439$	14626.0	1.59012	0.795058	−	0.606533i	$-0.207440\pi$
$439$	14626.0	1.59012	0.795058	+	0.606533i	$0.207440\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13083.8	−1.40323	−0.701613	−	0.712559i	$-0.747536\pi$
$443$	−13083.8	−1.40323	−0.701613	+	0.712559i	$0.747536\pi$
$444$	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
$452$	33614.3	3.49797
$453$	0	0
$454$	−29112.0	−3.00946
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	34495.5	3.51936
$459$	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
$466$	24999.0	2.48510
$467$	14908.5	1.47726	0.738632	−	0.674109i	$-0.235472\pi$
$467$	14908.5	1.47726	0.738632	+	0.674109i	$0.235472\pi$
$468$	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
$482$	−39766.6	−3.75792
$483$	0	0
$484$	−31361.4	−2.94529
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	51210.6	4.75040
$489$	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
$496$	−31894.0	−2.88727
$497$	0	0
$498$	0	0
$499$	19297.7	1.73123	0.865614	−	0.500711i	$-0.166928\pi$
$499$	19297.7	1.73123	0.865614	+	0.500711i	$0.166928\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6488.09	0.575129	0.287564	−	0.957761i	$-0.407154\pi$
$503$	6488.09	0.575129	0.287564	+	0.957761i	$0.407154\pi$
$504$	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
$512$	−91294.0	−7.88020
$513$	0	0
$514$	42749.7	3.66850
$515$	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
$526$	35626.7	2.95323
$527$	−5407.43	−0.446966
$528$	0	0
$529$	36318.2	2.98497
$530$	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	$0.541068\pi$
$542$	47344.7	3.75209
$543$	0	0
$544$	−51372.1	−4.04882
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	−27873.0	−2.17277
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−37793.9	−2.88277
$557$	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
$571$	−5861.76	−0.429610	−0.214805	−	0.976657i	$-0.568912\pi$
$571$	−5861.76	−0.429610	−0.214805	+	0.976657i	$0.568912\pi$
$572$	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	$0.5\pi$
$578$	12806.0	0.921559
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	37433.5	2.63885
$587$	2475.70	0.174076	0.0870382	−	0.996205i	$-0.472260\pi$
$587$	2475.70	0.174076	0.0870382	+	0.996205i	$0.472260\pi$
$588$	0	0
$589$	−6519.85	−0.456105
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−27999.2	−1.93894	−0.969470	−	0.245211i	$-0.921143\pi$
$593$	−27999.2	−1.93894	−0.969470	+	0.245211i	$0.921143\pi$
$594$	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
$601$	−15799.1	−1.07231	−0.536154	−	0.844120i	$-0.680123\pi$
$601$	−15799.1	−1.07231	−0.536154	+	0.844120i	$0.680123\pi$
$602$	0	0
$603$	0	0
$604$	−73325.9	−4.93972
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	−61940.4	−4.13161
$609$	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
$617$	3177.55	0.207331	0.103666	−	0.994612i	$-0.466943\pi$
$617$	3177.55	0.207331	0.103666	+	0.994612i	$0.466943\pi$
$618$	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
$631$	27085.6	1.70881	0.854407	−	0.519604i	$-0.173921\pi$
$631$	27085.6	1.70881	0.854407	+	0.519604i	$0.173921\pi$
$632$	−90520.4	−5.69732
$633$	0	0
$634$	53002.4	3.32018
$635$	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
$646$	−17839.1	−1.08649
$647$	27093.7	1.64631	0.823156	−	0.567815i	$-0.192211\pi$
$647$	27093.7	1.64631	0.823156	+	0.567815i	$0.192211\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−32543.1	−1.95024	−0.975122	−	0.221667i	$-0.928850\pi$
$653$	−32543.1	−1.95024	−0.975122	+	0.221667i	$0.928850\pi$
$654$	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	$0.0778130\pi$
$662$	−32876.7	−1.93020
$663$	0	0
$664$	6640.86	0.388126
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−87410.7	−5.06291
$669$	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
$676$	−51766.4	−2.94529
$677$	−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
$683$	33457.3	1.87439	0.937194	−	0.348807i	$-0.113413\pi$
$683$	33457.3	1.87439	0.937194	+	0.348807i	$0.113413\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	2902.39	0.159786	0.0798929	−	0.996803i	$-0.474542\pi$
$691$	2902.39	0.159786	0.0798929	+	0.996803i	$0.474542\pi$
$692$	−87453.2	−4.80415
$693$	0	0
$694$	48038.3	2.62753
$695$	0	0
$696$	0	0
$697$	0	0
$698$	71236.9	3.86297
$699$	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
$706$	−13893.9	−0.740658
$707$	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
$713$	23201.9	1.21868
$714$	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
$722$	17025.1	0.877574
$723$	0	0
$724$	80492.9	4.13190
$725$	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
$736$	220425.	11.0394
$737$	0	0
$738$	0	0
$739$	−29548.2	−1.47084	−0.735419	−	0.677613i	$-0.763014\pi$
$739$	−29548.2	−1.47084	−0.735419	+	0.677613i	$0.763014\pi$
$740$	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
$751$	41015.8	1.99292	0.996462	−	0.0840433i	$-0.0267834\pi$
$751$	41015.8	1.99292	0.996462	+	0.0840433i	$0.0267834\pi$
$752$	165144.	8.00824
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	−80997.4	−3.88121
$759$	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
$766$	29316.5	1.38283
$767$	0	0
$768$	0	0
$769$	40536.8	1.90090	0.950452	−	0.310873i	$-0.100621\pi$
$769$	40536.8	1.90090	0.950452	+	0.310873i	$0.100621\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	31003.3	1.44257	0.721287	−	0.692636i	$-0.243551\pi$
$773$	31003.3	1.44257	0.721287	+	0.692636i	$0.243551\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	63483.3	2.90302
$783$	0	0
$784$	−103821.	−4.72943
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	130177.	5.88497
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−128556.	−5.72430
$797$	−38238.7	−1.69948	−0.849739	−	0.527204i	$-0.823241\pi$
$797$	−38238.7	−1.69948	−0.849739	+	0.527204i	$0.823241\pi$
$798$	0	0
$799$	27999.2	1.23972
$800$	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
$818$	−7.65064	−0.000327015 0
$819$	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
$827$	−3817.14	−0.160502	−0.0802509	−	0.996775i	$-0.525572\pi$
$827$	−3817.14	−0.160502	−0.0802509	+	0.996775i	$0.525572\pi$
$828$	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
$833$	−17602.1	−0.732145
$834$	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
$842$	−57787.0	−2.36517
$843$	0	0
$844$	−40758.1	−1.66226
$845$	0	0
$846$	0	0
$847$	0	0
$848$	25784.6	1.04416
$849$	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
$856$	1563.99	0.0624486
$857$	−38654.7	−1.54075	−0.770374	−	0.637593i	$-0.779930\pi$
$857$	−38654.7	−1.54075	−0.770374	+	0.637593i	$0.779930\pi$
$858$	0	0
$859$	−50225.7	−1.99497	−0.997484	−	0.0708878i	$-0.977417\pi$
$859$	−50225.7	−1.99497	−0.997484	+	0.0708878i	$0.977417\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24350.6	0.960490	0.480245	−	0.877134i	$-0.340548\pi$
$863$	24350.6	0.960490	0.480245	+	0.877134i	$0.340548\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	182223.	7.07667
$873$	0	0
$874$	76543.2	2.96237
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	−82169.4	−3.15841
$879$	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
$886$	73505.1	2.78719
$887$	29614.4	1.12103	0.560516	−	0.828144i	$-0.310603\pi$
$887$	29614.4	1.12103	0.560516	+	0.828144i	$0.310603\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	33759.2	1.26507
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	4371.61	0.161642
$902$	0	0
$903$	0	0
$904$	−124728.	−4.58893
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	122097.	4.46248
$909$	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
$916$	−144676.	−5.21858
$917$	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
$931$	−21223.3	−0.747115
$932$	−104847.	−3.68495
$933$	0	0
$934$	−83756.4	−2.93425
$935$	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
$947$	54525.5	1.87100	0.935502	−	0.353321i	$-0.114948\pi$
$947$	54525.5	1.87100	0.935502	+	0.353321i	$0.114948\pi$
$948$	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
$961$	−18688.0	−0.627304
$962$	0	0
$963$	0	0
$964$	166783.	5.57232
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	116369.	3.86388
$969$	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
$976$	−177293.	−5.81455
$977$	−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
$983$	−39208.1	−1.27217	−0.636086	−	0.771618i	$-0.719448\pi$
$983$	−39208.1	−1.27217	−0.636086	+	0.771618i	$0.719448\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	29634.4	0.949917	0.474959	−	0.880008i	$-0.342463\pi$
$991$	29634.4	0.949917	0.474959	+	0.880008i	$0.342463\pi$
$992$	105482.	3.37605
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	−108415.	−3.43870
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.k.1.1		2
3.2	odd	2		675.4.a.o.1.2		2
5.2	odd	4		135.4.b.a.109.1	✓	4
5.3	odd	4		135.4.b.a.109.4	yes	4
5.4	even	2		675.4.a.o.1.2		2
15.2	even	4		135.4.b.a.109.4	yes	4
15.8	even	4		135.4.b.a.109.1	✓	4
15.14	odd	2	CM	675.4.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.b.a.109.1	✓	4	5.2	odd	4
135.4.b.a.109.1	✓	4	15.8	even	4
135.4.b.a.109.4	yes	4	5.3	odd	4
135.4.b.a.109.4	yes	4	15.2	even	4
675.4.a.k.1.1		2	1.1	even	1	trivial
675.4.a.k.1.1		2	15.14	odd	2	CM
675.4.a.o.1.2		2	3.2	odd	2
675.4.a.o.1.2		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-5.61803 q^{2} +23.5623 q^{4} -87.4296 q^{8} +O(q^{10})\)	\(q-5.61803 q^{2} +23.5623 q^{4} -87.4296 q^{8} +302.684 q^{16} +51.3181 q^{17} +61.8754 q^{19} -220.193 q^{23} -105.371 q^{31} -1001.05 q^{32} -288.307 q^{34} -347.618 q^{38} +1237.05 q^{46} +545.601 q^{47} -343.000 q^{49} +85.1866 q^{53} -585.735 q^{61} +591.976 q^{62} +3202.47 q^{64} +1209.17 q^{68} +1457.93 q^{76} +1035.35 q^{79} -75.9567 q^{83} -5188.27 q^{92} -3065.20 q^{94} +1926.99 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9 q^{2} + 27 q^{4} - 72 q^{8}+O(q^{10}) \) 2 * q - 9 * q^2 + 27 * q^4 - 72 * q^8	\( 2 q - 9 q^{2} + 27 q^{4} - 72 q^{8} + 223 q^{16} - 36 q^{17} + 164 q^{19} - 342 q^{23} + 232 q^{31} - 855 q^{32} + 7 q^{34} - 693 q^{38} + 1649 q^{46} - 686 q^{49} + 792 q^{53} + 358 q^{61} - 549 q^{62} + 3346 q^{64} + 909 q^{68} + 1809 q^{76} - 304 q^{79} - 1422 q^{83} - 5607 q^{92} - 1220 q^{94} + 3087 q^{98}+O(q^{100}) \) 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

Properties

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