LMFDB - Embedded newform 23.3.b.a.22.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,3,Mod(22,23)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(23, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("23.22");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	23.b (of order $2$, degree $1$, 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:	yes
Analytic conductor:	$0.626704608029$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			22.2
Root			$-2.14510$ of defining polynomial
Character	$\chi$	$=$	23.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.601466 q^{2} +1.54364 q^{3} -3.63824 q^{4} +0.928445 q^{6} -4.59414 q^{8} -6.61718 q^{9} +O(q^{10})$	$q+0.601466 q^{2} +1.54364 q^{3} -3.63824 q^{4} +0.928445 q^{6} -4.59414 q^{8} -6.61718 q^{9} -5.61612 q^{12} +23.5162 q^{13} +11.7897 q^{16} -3.98001 q^{18} -23.0000 q^{23} -7.09168 q^{24} +25.0000 q^{25} +14.1442 q^{26} -24.1073 q^{27} -42.4015 q^{29} -27.9663 q^{31} +25.4677 q^{32} +24.0749 q^{36} +36.3005 q^{39} +74.9986 q^{41} -13.8337 q^{46} -93.8839 q^{47} +18.1991 q^{48} +49.0000 q^{49} +15.0366 q^{50} -85.5575 q^{52} -14.4997 q^{54} -25.5030 q^{58} +26.0000 q^{59} -16.8208 q^{62} -31.8410 q^{64} -35.5037 q^{69} +140.916 q^{71} +30.4003 q^{72} -56.8366 q^{73} +38.5909 q^{75} +21.8335 q^{78} +22.3418 q^{81} +45.1091 q^{82} -65.4525 q^{87} +83.6795 q^{92} -43.1698 q^{93} -56.4679 q^{94} +39.3128 q^{96} +29.4718 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 12 q^{4} - 33 q^{6} - 21 q^{8} + 27 q^{9}+O(q^{10}) $ 3 * q + 12 * q^4 - 33 * q^6 - 21 * q^8 + 27 * q^9	$ 3 q + 12 q^{4} - 33 q^{6} - 21 q^{8} + 27 q^{9} + 3 q^{12} + 48 q^{16} + 39 q^{18} - 69 q^{23} - 132 q^{24} + 75 q^{25} + 87 q^{26} - 114 q^{27} - 84 q^{32} + 255 q^{36} - 42 q^{39} + 231 q^{48} + 147 q^{49} - 309 q^{52} - 297 q^{54} - 273 q^{58} + 78 q^{59} + 303 q^{62} - 45 q^{64} - 33 q^{72} + 399 q^{78} + 243 q^{81} - 129 q^{82} + 246 q^{87} - 276 q^{92} - 546 q^{93} - 57 q^{94} - 21 q^{96}+O(q^{100}) $ 3 * q + 12 * q^4 - 33 * q^6 - 21 * q^8 + 27 * q^9 + 3 * q^12 + 48 * q^16 + 39 * q^18 - 69 * q^23 - 132 * q^24 + 75 * q^25 + 87 * q^26 - 114 * q^27 - 84 * q^32 + 255 * q^36 - 42 * q^39 + 231 * q^48 + 147 * q^49 - 309 * q^52 - 297 * q^54 - 273 * q^58 + 78 * q^59 + 303 * q^62 - 45 * q^64 - 33 * q^72 + 399 * q^78 + 243 * q^81 - 129 * q^82 + 246 * q^87 - 276 * q^92 - 546 * q^93 - 57 * q^94 - 21 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/23\mathbb{Z}\right)^\times$.

$n$	$5$
$\chi(n)$	$-1$

Coefficient data

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

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

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

$2$	0.601466	0.300733	0.150366	−	0.988630i	$-0.451955\pi$
$2$	0.601466	0.300733	0.150366	+	0.988630i	$0.451955\pi$
$3$	1.54364	0.514546	0.257273	−	0.966339i	$-0.417176\pi$
$3$	1.54364	0.514546	0.257273	+	0.966339i	$0.417176\pi$
$4$	−3.63824	−0.909560
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0.928445	0.154741
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−4.59414	−0.574267
$9$	−6.61718	−0.735243
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−5.61612	−0.468010
$13$	23.5162	1.80894	0.904469	−	0.426540i	$-0.140268\pi$
$13$	23.5162	1.80894	0.904469	+	0.426540i	$0.140268\pi$
$14$	0	0
$15$	0	0
$16$	11.7897	0.736859
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−3.98001	−0.221112
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−23.0000	−1.00000
$23$	−23.0000	−1.00000
$24$	−7.09168	−0.295487
$25$	25.0000	1.00000
$26$	14.1442	0.544007
$27$	−24.1073	−0.892862
$28$	0	0
$29$	−42.4015	−1.46212	−0.731060	−	0.682314i	$-0.760974\pi$
$29$	−42.4015	−1.46212	−0.731060	+	0.682314i	$0.760974\pi$
$30$	0	0
$31$	−27.9663	−0.902138	−0.451069	−	0.892489i	$-0.648957\pi$
$31$	−27.9663	−0.902138	−0.451069	+	0.892489i	$0.648957\pi$
$32$	25.4677	0.795865
$33$	0	0
$34$	0	0
$35$	0	0
$36$	24.0749	0.668747
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	36.3005	0.930781
$40$	0	0
$41$	74.9986	1.82924	0.914618	−	0.404320i	$-0.132492\pi$
$41$	74.9986	1.82924	0.914618	+	0.404320i	$0.132492\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	−13.8337	−0.300733
$47$	−93.8839	−1.99753	−0.998765	−	0.0496817i	$-0.984179\pi$
$47$	−93.8839	−1.99753	−0.998765	+	0.0496817i	$0.984179\pi$
$48$	18.1991	0.379148
$49$	49.0000	1.00000
$50$	15.0366	0.300733
$51$	0	0
$52$	−85.5575	−1.64534
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−14.4997	−0.268513
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−25.5030	−0.439707
$59$	26.0000	0.440678	0.220339	−	0.975423i	$-0.429284\pi$
$59$	26.0000	0.440678	0.220339	+	0.975423i	$0.429284\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−16.8208	−0.271303
$63$	0	0
$64$	−31.8410	−0.497516
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	−35.5037	−0.514546
$70$	0	0
$71$	140.916	1.98474	0.992368	−	0.123310i	$-0.0393509\pi$
$71$	140.916	1.98474	0.992368	+	0.123310i	$0.0393509\pi$
$72$	30.4003	0.422226
$73$	−56.8366	−0.778584	−0.389292	−	0.921114i	$-0.627280\pi$
$73$	−56.8366	−0.778584	−0.389292	+	0.921114i	$0.627280\pi$
$74$	0	0
$75$	38.5909	0.514546
$76$	0	0
$77$	0	0
$78$	21.8335	0.279916
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	22.3418	0.275825
$82$	45.1091	0.550111
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−65.4525	−0.752327
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	83.6795	0.909560
$93$	−43.1698	−0.464191
$94$	−56.4679	−0.600723
$95$	0	0
$96$	39.3128	0.409509
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	29.4718	0.300733
$99$	0	0
$100$	−90.9560	−0.909560
$101$	−166.000	−1.64356	−0.821782	−	0.569802i	$-0.807020\pi$
$101$	−166.000	−1.64356	−0.821782	+	0.569802i	$0.807020\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−108.037	−1.03881
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	87.7080	0.812111
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	154.267	1.32988
$117$	−155.611	−1.33001
$118$	15.6381	0.132526
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	0	0
$123$	115.771	0.941225
$124$	101.748	0.820549
$125$	0	0
$126$	0	0
$127$	243.881	1.92032	0.960162	−	0.279443i	$-0.0901498\pi$
$127$	243.881	1.92032	0.960162	+	0.279443i	$0.0901498\pi$
$128$	−121.022	−0.945484
$129$	0	0
$130$	0	0
$131$	−182.414	−1.39247	−0.696235	−	0.717814i	$-0.745143\pi$
$131$	−182.414	−1.39247	−0.696235	+	0.717814i	$0.745143\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	−21.3542	−0.154741
$139$	118.304	0.851109	0.425555	−	0.904933i	$-0.360079\pi$
$139$	118.304	0.851109	0.425555	+	0.904933i	$0.360079\pi$
$140$	0	0
$141$	−144.923	−1.02782
$142$	84.7563	0.596875
$143$	0	0
$144$	−78.0149	−0.541770
$145$	0	0
$146$	−34.1853	−0.234146
$147$	75.6382	0.514546
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	23.2111	0.154741
$151$	−188.672	−1.24948	−0.624741	−	0.780832i	$-0.714796\pi$
$151$	−188.672	−1.24948	−0.624741	+	0.780832i	$0.714796\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−132.070	−0.846601
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	13.4378	0.0829495
$163$	−314.249	−1.92791	−0.963954	−	0.266070i	$-0.914275\pi$
$163$	−314.249	−1.92791	−0.963954	+	0.266070i	$0.914275\pi$
$164$	−272.863	−1.66380
$165$	0	0
$166$	0	0
$167$	242.000	1.44910	0.724551	−	0.689221i	$-0.242047\pi$
$167$	242.000	1.44910	0.724551	+	0.689221i	$0.242047\pi$
$168$	0	0
$169$	384.011	2.27225
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−22.0000	−0.127168	−0.0635838	−	0.997977i	$-0.520253\pi$
$173$	−22.0000	−0.127168	−0.0635838	+	0.997977i	$0.520253\pi$
$174$	−39.3674	−0.226249
$175$	0	0
$176$	0	0
$177$	40.1346	0.226749
$178$	0	0
$179$	287.187	1.60440	0.802198	−	0.597059i	$-0.203664\pi$
$179$	287.187	1.60440	0.802198	+	0.597059i	$0.203664\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	105.665	0.574267
$185$	0	0
$186$	−25.9651	−0.139598
$187$	0	0
$188$	341.572	1.81687
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−49.1510	−0.255995
$193$	−36.1432	−0.187271	−0.0936353	−	0.995607i	$-0.529849\pi$
$193$	−36.1432	−0.187271	−0.0936353	+	0.995607i	$0.529849\pi$
$194$	0	0
$195$	0	0
$196$	−178.274	−0.909560
$197$	−219.461	−1.11402	−0.557008	−	0.830507i	$-0.688051\pi$
$197$	−219.461	−1.11402	−0.557008	+	0.830507i	$0.688051\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−114.853	−0.574267
$201$	0	0
$202$	−99.8433	−0.494274
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	152.195	0.735243
$208$	277.250	1.33293
$209$	0	0
$210$	0	0
$211$	−406.000	−1.92417	−0.962085	−	0.272749i	$-0.912067\pi$
$211$	−406.000	−1.92417	−0.962085	+	0.272749i	$0.912067\pi$
$212$	0	0
$213$	217.524	1.02124
$214$	0	0
$215$	0	0
$216$	110.752	0.512741
$217$	0	0
$218$	0	0
$219$	−87.7351	−0.400617
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−382.000	−1.71300	−0.856502	−	0.516143i	$-0.827367\pi$
$223$	−382.000	−1.71300	−0.856502	+	0.516143i	$0.827367\pi$
$224$	0	0
$225$	−165.430	−0.735243
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	194.798	0.839647
$233$	−48.6597	−0.208840	−0.104420	−	0.994533i	$-0.533299\pi$
$233$	−48.6597	−0.208840	−0.104420	+	0.994533i	$0.533299\pi$
$234$	−93.5946	−0.399977
$235$	0	0
$236$	−94.5942	−0.400823
$237$	0	0
$238$	0	0
$239$	−217.542	−0.910219	−0.455109	−	0.890436i	$-0.650400\pi$
$239$	−217.542	−0.910219	−0.455109	+	0.890436i	$0.650400\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	72.7773	0.300733
$243$	251.453	1.03479
$244$	0	0
$245$	0	0
$246$	69.6321	0.283057
$247$	0	0
$248$	128.481	0.518068
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	146.686	0.577505
$255$	0	0
$256$	54.5736	0.213178
$257$	367.540	1.43011	0.715057	−	0.699066i	$-0.246400\pi$
$257$	367.540	1.43011	0.715057	+	0.699066i	$0.246400\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	280.578	1.07501
$262$	−109.716	−0.418762
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	303.541	1.12840	0.564202	−	0.825637i	$-0.309184\pi$
$269$	303.541	1.12840	0.564202	+	0.825637i	$0.309184\pi$
$270$	0	0
$271$	−286.000	−1.05535	−0.527675	−	0.849446i	$-0.676936\pi$
$271$	−286.000	−1.05535	−0.527675	+	0.849446i	$0.676936\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	129.171	0.468010
$277$	−549.049	−1.98213	−0.991063	−	0.133391i	$-0.957413\pi$
$277$	−549.049	−1.98213	−0.991063	+	0.133391i	$0.957413\pi$
$278$	71.1559	0.255956
$279$	185.058	0.663290
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−87.1660	−0.309099
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	−512.687	−1.80524
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−168.524	−0.585154
$289$	289.000	1.00000
$290$	0	0
$291$	0	0
$292$	206.785	0.708169
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	45.4938	0.154741
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−540.872	−1.80894
$300$	−140.403	−0.468010
$301$	0	0
$302$	−113.480	−0.375760
$303$	−256.244	−0.845689
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−214.000	−0.697068	−0.348534	−	0.937296i	$-0.613320\pi$
$307$	−214.000	−0.697068	−0.348534	+	0.937296i	$0.613320\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	486.858	1.56546	0.782730	−	0.622361i	$-0.213826\pi$
$311$	486.858	1.56546	0.782730	+	0.622361i	$0.213826\pi$
$312$	−166.769	−0.534517
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	266.000	0.839117	0.419558	−	0.907728i	$-0.362185\pi$
$317$	266.000	0.839117	0.419558	+	0.907728i	$0.362185\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−81.2848	−0.250879
$325$	587.905	1.80894
$326$	−189.010	−0.579785
$327$	0	0
$328$	−344.554	−1.05047
$329$	0	0
$330$	0	0
$331$	661.999	2.00000	0.999999	−	0.00159277i	$-0.000506994\pi$
$331$	661.999	2.00000	0.999999	+	0.00159277i	$0.000506994\pi$
$332$	0	0
$333$	0	0
$334$	145.555	0.435792
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	230.969	0.683341
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−13.2322	−0.0382435
$347$	602.000	1.73487	0.867435	−	0.497550i	$-0.165767\pi$
$347$	602.000	1.73487	0.867435	+	0.497550i	$0.165767\pi$
$348$	238.132	0.684287
$349$	−26.0476	−0.0746349	−0.0373174	−	0.999303i	$-0.511881\pi$
$349$	−26.0476	−0.0746349	−0.0373174	+	0.999303i	$0.511881\pi$
$350$	0	0
$351$	−566.911	−1.61513
$352$	0	0
$353$	−695.320	−1.96974	−0.984872	−	0.173284i	$-0.944562\pi$
$353$	−695.320	−1.96974	−0.984872	+	0.173284i	$0.944562\pi$
$354$	24.1396	0.0681908
$355$	0	0
$356$	0	0
$357$	0	0
$358$	172.733	0.482494
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	0	0
$363$	186.780	0.514546
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	−271.164	−0.736859
$369$	−496.280	−1.34493
$370$	0	0
$371$	0	0
$372$	157.062	0.422210
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	431.316	1.14712
$377$	−997.120	−2.64488
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	376.464	0.988095
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−186.814	−0.486495
$385$	0	0
$386$	−21.7389	−0.0563184
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−225.113	−0.574267
$393$	−281.581	−0.716490
$394$	−131.998	−0.335021
$395$	0	0
$396$	0	0
$397$	−528.356	−1.33087	−0.665435	−	0.746455i	$-0.731754\pi$
$397$	−528.356	−1.33087	−0.665435	+	0.746455i	$0.731754\pi$
$398$	0	0
$399$	0	0
$400$	294.744	0.736859
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	−657.660	−1.63191
$404$	603.948	1.49492
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	256.398	0.626889	0.313445	−	0.949606i	$-0.398517\pi$
$409$	256.398	0.626889	0.313445	+	0.949606i	$0.398517\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	91.5402	0.221112
$415$	0	0
$416$	598.902	1.43967
$417$	182.619	0.437935
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−244.195	−0.578661
$423$	621.247	1.46867
$424$	0	0
$425$	0	0
$426$	130.833	0.307120
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−284.218	−0.657913
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	−52.7697	−0.120479
$439$	626.871	1.42795	0.713976	−	0.700171i	$-0.246893\pi$
$439$	626.871	1.42795	0.713976	+	0.700171i	$0.246893\pi$
$440$	0	0
$441$	−324.242	−0.735243
$442$	0	0
$443$	867.929	1.95921	0.979604	−	0.200938i	$-0.0643991\pi$
$443$	867.929	1.95921	0.979604	+	0.200938i	$0.0643991\pi$
$444$	0	0
$445$	0	0
$446$	−229.760	−0.515157
$447$	0	0
$448$	0	0
$449$	−574.000	−1.27840	−0.639198	−	0.769042i	$-0.720734\pi$
$449$	−574.000	−1.27840	−0.639198	+	0.769042i	$0.720734\pi$
$450$	−99.5002	−0.221112
$451$	0	0
$452$	0	0
$453$	−291.241	−0.642916
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	542.680	1.17718	0.588590	−	0.808431i	$-0.299683\pi$
$461$	542.680	1.17718	0.588590	+	0.808431i	$0.299683\pi$
$462$	0	0
$463$	98.0000	0.211663	0.105832	−	0.994384i	$-0.466250\pi$
$463$	98.0000	0.211663	0.105832	+	0.994384i	$0.466250\pi$
$464$	−499.902	−1.07738
$465$	0	0
$466$	−29.2671	−0.0628050
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	566.150	1.20972
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−119.448	−0.253067
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−130.844	−0.273733
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−440.227	−0.909560
$485$	0	0
$486$	151.240	0.311194
$487$	−238.236	−0.489190	−0.244595	−	0.969625i	$-0.578655\pi$
$487$	−238.236	−0.489190	−0.244595	+	0.969625i	$0.578655\pi$
$488$	0	0
$489$	−485.086	−0.991997
$490$	0	0
$491$	−301.732	−0.614526	−0.307263	−	0.951625i	$-0.599413\pi$
$491$	−301.732	−0.614526	−0.307263	+	0.951625i	$0.599413\pi$
$492$	−421.201	−0.856101
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−329.715	−0.664748
$497$	0	0
$498$	0	0
$499$	−1.01465	−0.00203337	−0.00101668	−	0.999999i	$-0.500324\pi$
$499$	−1.01465	−0.00203337	−0.00101668	+	0.999999i	$0.500324\pi$
$500$	0	0
$501$	373.560	0.745629
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	592.773	1.16918
$508$	−887.298	−1.74665
$509$	−989.779	−1.94456	−0.972278	−	0.233827i	$-0.924875\pi$
$509$	−989.779	−1.94456	−0.972278	+	0.233827i	$0.924875\pi$
$510$	0	0
$511$	0	0
$512$	516.912	1.00959
$513$	0	0
$514$	221.062	0.430082
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−33.9600	−0.0654336
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	168.758	0.323291
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	663.665	1.26654
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	529.000	1.00000
$530$	0	0
$531$	−172.047	−0.324005
$532$	0	0
$533$	1763.68	3.30897
$534$	0	0
$535$	0	0
$536$	0	0
$537$	443.312	0.825535
$538$	182.569	0.339348
$539$	0	0
$540$	0	0
$541$	517.647	0.956834	0.478417	−	0.878133i	$-0.341211\pi$
$541$	517.647	0.956834	0.478417	+	0.878133i	$0.341211\pi$
$542$	−172.019	−0.317379
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−866.121	−1.58340	−0.791701	−	0.610909i	$-0.790804\pi$
$547$	−866.121	−1.58340	−0.791701	+	0.610909i	$0.790804\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	163.109	0.295487
$553$	0	0
$554$	−330.234	−0.596091
$555$	0	0
$556$	−430.419	−0.774135
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	111.306	0.199473
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	527.263	0.934864
$565$	0	0
$566$	0	0
$567$	0	0
$568$	−647.389	−1.13977
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−575.000	−1.00000
$576$	210.698	0.365795
$577$	−950.813	−1.64786	−0.823928	−	0.566694i	$-0.808222\pi$
$577$	−950.813	−1.64786	−0.823928	+	0.566694i	$0.808222\pi$
$578$	173.824	0.300733
$579$	−55.7920	−0.0963593
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	261.115	0.447115
$585$	0	0
$586$	0	0
$587$	637.468	1.08598	0.542988	−	0.839740i	$-0.317293\pi$
$587$	637.468	1.08598	0.542988	+	0.839740i	$0.317293\pi$
$588$	−275.190	−0.468010
$589$	0	0
$590$	0	0
$591$	−338.768	−0.573212
$592$	0	0
$593$	−286.000	−0.482293	−0.241147	−	0.970489i	$-0.577524\pi$
$593$	−286.000	−0.482293	−0.241147	+	0.970489i	$0.577524\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	−325.316	−0.544007
$599$	1106.00	1.84641	0.923205	−	0.384307i	$-0.125560\pi$
$599$	1106.00	1.84641	0.923205	+	0.384307i	$0.125560\pi$
$600$	−177.292	−0.295487
$601$	−1201.97	−1.99995	−0.999973	−	0.00737547i	$-0.997652\pi$
$601$	−1201.97	−1.99995	−0.999973	+	0.00737547i	$0.997652\pi$
$602$	0	0
$603$	0	0
$604$	686.434	1.13648
$605$	0	0
$606$	−154.122	−0.254326
$607$	386.000	0.635914	0.317957	−	0.948105i	$-0.397003\pi$
$607$	386.000	0.635914	0.317957	+	0.948105i	$0.397003\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2207.79	−3.61341
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−128.714	−0.209631
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	554.467	0.892862
$622$	292.829	0.470785
$623$	0	0
$624$	427.973	0.685854
$625$	625.000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	−626.717	−0.990074
$634$	159.990	0.252350
$635$	0	0
$636$	0	0
$637$	1152.29	1.80894
$638$	0	0
$639$	−932.469	−1.45926
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−230.059	−0.355578	−0.177789	−	0.984069i	$-0.556894\pi$
$647$	−230.059	−0.355578	−0.177789	+	0.984069i	$0.556894\pi$
$648$	−102.641	−0.158397
$649$	0	0
$650$	353.604	0.544007
$651$	0	0
$652$	1143.31	1.75355
$653$	56.2240	0.0861010	0.0430505	−	0.999073i	$-0.486292\pi$
$653$	56.2240	0.0861010	0.0430505	+	0.999073i	$0.486292\pi$
$654$	0	0
$655$	0	0
$656$	884.215	1.34789
$657$	376.098	0.572448
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	398.170	0.601465
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	975.233	1.46212
$668$	−880.454	−1.31804
$669$	−589.669	−0.881419
$670$	0	0
$671$	0	0
$672$	0	0
$673$	1323.09	1.96596	0.982982	−	0.183700i	$-0.0588075\pi$
$673$	1323.09	1.96596	0.982982	+	0.183700i	$0.0588075\pi$
$674$	0	0
$675$	−602.682	−0.892862
$676$	−1397.12	−2.06675
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−788.189	−1.15401	−0.577005	−	0.816741i	$-0.695779\pi$
$683$	−788.189	−1.15401	−0.577005	+	0.816741i	$0.695779\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	554.000	0.801737	0.400868	−	0.916136i	$-0.368708\pi$
$691$	554.000	0.801737	0.400868	+	0.916136i	$0.368708\pi$
$692$	80.0413	0.115667
$693$	0	0
$694$	362.082	0.521732
$695$	0	0
$696$	300.698	0.432037
$697$	0	0
$698$	−15.6667	−0.0224451
$699$	−75.1129	−0.107458
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	−340.977	−0.485723
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−418.211	−0.592367
$707$	0	0
$708$	−146.019	−0.206242
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	643.224	0.902138
$714$	0	0
$715$	0	0
$716$	−1044.85	−1.45929
$717$	−335.806	−0.468349
$718$	0	0
$719$	−862.000	−1.19889	−0.599444	−	0.800417i	$-0.704611\pi$
$719$	−862.000	−1.19889	−0.599444	+	0.800417i	$0.704611\pi$
$720$	0	0
$721$	0	0
$722$	217.129	0.300733
$723$	0	0
$724$	0	0
$725$	−1060.04	−1.46212
$726$	112.342	0.154741
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	187.076	0.256620
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	−585.757	−0.795865
$737$	0	0
$738$	−298.495	−0.404465
$739$	1316.84	1.78192	0.890958	−	0.454086i	$-0.150034\pi$
$739$	1316.84	1.78192	0.890958	+	0.454086i	$0.150034\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	198.328	0.266570
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−1106.87	−1.47190
$753$	0	0
$754$	−599.734	−0.795403
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1475.62	1.93906	0.969529	−	0.244977i	$-0.0787804\pi$
$761$	1475.62	1.93906	0.969529	+	0.244977i	$0.0787804\pi$
$762$	226.430	0.297152
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	611.421	0.797159
$768$	84.2418	0.109690
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	567.348	0.735860
$772$	131.498	0.170334
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	−699.157	−0.902138
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	1022.18	1.30547
$784$	577.697	0.736859
$785$	0	0
$786$	−169.361	−0.215472
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	798.451	1.01326
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	−317.788	−0.400237
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	636.692	0.795865
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−395.560	−0.490769
$807$	468.557	0.580615
$808$	762.627	0.943845
$809$	146.000	0.180470	0.0902349	−	0.995921i	$-0.471238\pi$
$809$	146.000	0.180470	0.0902349	+	0.995921i	$0.471238\pi$
$810$	0	0
$811$	−1620.09	−1.99764	−0.998820	−	0.0485755i	$-0.984532\pi$
$811$	−1620.09	−1.99764	−0.998820	+	0.0485755i	$0.984532\pi$
$812$	0	0
$813$	−441.480	−0.543026
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	154.214	0.188526
$819$	0	0
$820$	0	0
$821$	1274.00	1.55177	0.775883	−	0.630877i	$-0.217305\pi$
$821$	1274.00	1.55177	0.775883	+	0.630877i	$0.217305\pi$
$822$	0	0
$823$	437.797	0.531952	0.265976	−	0.963980i	$-0.414306\pi$
$823$	437.797	0.531952	0.265976	+	0.963980i	$0.414306\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−553.723	−0.668747
$829$	−1654.00	−1.99517	−0.997587	−	0.0694210i	$-0.977885\pi$
$829$	−1654.00	−1.99517	−0.997587	+	0.0694210i	$0.977885\pi$
$830$	0	0
$831$	−847.533	−1.01989
$832$	−748.780	−0.899976
$833$	0	0
$834$	109.839	0.131701
$835$	0	0
$836$	0	0
$837$	674.191	0.805485
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	956.883	1.13779
$842$	0	0
$843$	0	0
$844$	1477.13	1.75015
$845$	0	0
$846$	373.659	0.441677
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	−791.403	−0.928877
$853$	−1606.00	−1.88277	−0.941383	−	0.337339i	$-0.890473\pi$
$853$	−1606.00	−1.88277	−0.941383	+	0.337339i	$0.890473\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1589.30	−1.85449	−0.927244	−	0.374458i	$-0.877829\pi$
$857$	−1589.30	−1.85449	−0.927244	+	0.374458i	$0.877829\pi$
$858$	0	0
$859$	−845.930	−0.984784	−0.492392	−	0.870374i	$-0.663877\pi$
$859$	−845.930	−0.984784	−0.492392	+	0.870374i	$0.663877\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1395.38	−1.61690	−0.808448	−	0.588568i	$-0.799692\pi$
$863$	−1395.38	−1.61690	−0.808448	+	0.588568i	$0.799692\pi$
$864$	−613.956	−0.710597
$865$	0	0
$866$	0	0
$867$	446.111	0.514546
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	319.201	0.364385
$877$	−1558.00	−1.77651	−0.888255	−	0.459350i	$-0.848082\pi$
$877$	−1558.00	−1.77651	−0.888255	+	0.459350i	$0.848082\pi$
$878$	377.041	0.429432
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−195.020	−0.221112
$883$	938.000	1.06229	0.531144	−	0.847282i	$-0.321762\pi$
$883$	938.000	1.06229	0.531144	+	0.847282i	$0.321762\pi$
$884$	0	0
$885$	0	0
$886$	522.029	0.589198
$887$	841.479	0.948680	0.474340	−	0.880342i	$-0.342687\pi$
$887$	841.479	0.948680	0.474340	+	0.880342i	$0.342687\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	1389.81	1.55808
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−834.910	−0.930781
$898$	−345.241	−0.384456
$899$	1185.81	1.31903
$900$	601.872	0.668747
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	−175.171	−0.193346
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	1098.45	1.20842
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−330.338	−0.358674
$922$	326.404	0.354017
$923$	3313.81	3.59026
$924$	0	0
$925$	0	0
$926$	58.9436	0.0636540
$927$	0	0
$928$	−1079.87	−1.16365
$929$	−340.699	−0.366737	−0.183368	−	0.983044i	$-0.558700\pi$
$929$	−340.699	−0.366737	−0.183368	+	0.983044i	$0.558700\pi$
$930$	0	0
$931$	0	0
$932$	177.036	0.189952
$933$	751.533	0.805501
$934$	0	0
$935$	0	0
$936$	714.898	0.763780
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	−1724.97	−1.82924
$944$	306.533	0.324717
$945$	0	0
$946$	0	0
$947$	−1846.71	−1.95006	−0.975031	−	0.222069i	$-0.928719\pi$
$947$	−1846.71	−1.95006	−0.975031	+	0.222069i	$0.928719\pi$
$948$	0	0
$949$	−1336.58	−1.40841
$950$	0	0
$951$	410.607	0.431764
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	791.471	0.827898
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−178.887	−0.186147
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1121.00	1.15926	0.579629	−	0.814881i	$-0.303198\pi$
$967$	1121.00	1.15926	0.579629	+	0.814881i	$0.303198\pi$
$968$	−555.891	−0.574267
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−914.846	−0.941200
$973$	0	0
$974$	−143.291	−0.147116
$975$	907.511	0.930781
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	−291.763	−0.298326
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−181.482	−0.184808
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−531.867	−0.540515
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	1154.00	1.16448	0.582240	−	0.813017i	$-0.302176\pi$
$991$	1154.00	1.16448	0.582240	+	0.813017i	$0.302176\pi$
$992$	−712.236	−0.717980
$993$	1021.89	1.02909
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1318.00	−1.32197	−0.660983	−	0.750401i	$-0.729860\pi$
$997$	−1318.00	−1.32197	−0.660983	+	0.750401i	$0.729860\pi$
$998$	−0.610277	−0.000611500 0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.3.b.a.22.2	✓	3
3.2	odd	2		207.3.d.a.91.2		3
4.3	odd	2		368.3.f.a.321.2		3
5.2	odd	4		575.3.c.a.574.4		6
5.3	odd	4		575.3.c.a.574.3		6
5.4	even	2		575.3.d.b.551.2		3
8.3	odd	2		1472.3.f.b.321.2		3
8.5	even	2		1472.3.f.a.321.2		3
23.22	odd	2	CM	23.3.b.a.22.2	✓	3
69.68	even	2		207.3.d.a.91.2		3
92.91	even	2		368.3.f.a.321.2		3
115.22	even	4		575.3.c.a.574.4		6
115.68	even	4		575.3.c.a.574.3		6
115.114	odd	2		575.3.d.b.551.2		3
184.45	odd	2		1472.3.f.a.321.2		3
184.91	even	2		1472.3.f.b.321.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.3.b.a.22.2	✓	3	1.1	even	1	trivial
23.3.b.a.22.2	✓	3	23.22	odd	2	CM
207.3.d.a.91.2		3	3.2	odd	2
207.3.d.a.91.2		3	69.68	even	2
368.3.f.a.321.2		3	4.3	odd	2
368.3.f.a.321.2		3	92.91	even	2
575.3.c.a.574.3		6	5.3	odd	4
575.3.c.a.574.3		6	115.68	even	4
575.3.c.a.574.4		6	5.2	odd	4
575.3.c.a.574.4		6	115.22	even	4
575.3.d.b.551.2		3	5.4	even	2
575.3.d.b.551.2		3	115.114	odd	2
1472.3.f.a.321.2		3	8.5	even	2
1472.3.f.a.321.2		3	184.45	odd	2
1472.3.f.b.321.2		3	8.3	odd	2
1472.3.f.b.321.2		3	184.91	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:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	23.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.601466 q^{2} +1.54364 q^{3} -3.63824 q^{4} +0.928445 q^{6} -4.59414 q^{8} -6.61718 q^{9} +O(q^{10})\)	\(q+0.601466 q^{2} +1.54364 q^{3} -3.63824 q^{4} +0.928445 q^{6} -4.59414 q^{8} -6.61718 q^{9} -5.61612 q^{12} +23.5162 q^{13} +11.7897 q^{16} -3.98001 q^{18} -23.0000 q^{23} -7.09168 q^{24} +25.0000 q^{25} +14.1442 q^{26} -24.1073 q^{27} -42.4015 q^{29} -27.9663 q^{31} +25.4677 q^{32} +24.0749 q^{36} +36.3005 q^{39} +74.9986 q^{41} -13.8337 q^{46} -93.8839 q^{47} +18.1991 q^{48} +49.0000 q^{49} +15.0366 q^{50} -85.5575 q^{52} -14.4997 q^{54} -25.5030 q^{58} +26.0000 q^{59} -16.8208 q^{62} -31.8410 q^{64} -35.5037 q^{69} +140.916 q^{71} +30.4003 q^{72} -56.8366 q^{73} +38.5909 q^{75} +21.8335 q^{78} +22.3418 q^{81} +45.1091 q^{82} -65.4525 q^{87} +83.6795 q^{92} -43.1698 q^{93} -56.4679 q^{94} +39.3128 q^{96} +29.4718 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 12 q^{4} - 33 q^{6} - 21 q^{8} + 27 q^{9}+O(q^{10}) \) 3 * q + 12 * q^4 - 33 * q^6 - 21 * q^8 + 27 * q^9	\( 3 q + 12 q^{4} - 33 q^{6} - 21 q^{8} + 27 q^{9} + 3 q^{12} + 48 q^{16} + 39 q^{18} - 69 q^{23} - 132 q^{24} + 75 q^{25} + 87 q^{26} - 114 q^{27} - 84 q^{32} + 255 q^{36} - 42 q^{39} + 231 q^{48} + 147 q^{49} - 309 q^{52} - 297 q^{54} - 273 q^{58} + 78 q^{59} + 303 q^{62} - 45 q^{64} - 33 q^{72} + 399 q^{78} + 243 q^{81} - 129 q^{82} + 246 q^{87} - 276 q^{92} - 546 q^{93} - 57 q^{94} - 21 q^{96}+O(q^{100}) \) 3 * q + 12 * q^4 - 33 * q^6 - 21 * q^8 + 27 * q^9 + 3 * q^12 + 48 * q^16 + 39 * q^18 - 69 * q^23 - 132 * q^24 + 75 * q^25 + 87 * q^26 - 114 * q^27 - 84 * q^32 + 255 * q^36 - 42 * q^39 + 231 * q^48 + 147 * q^49 - 309 * q^52 - 297 * q^54 - 273 * q^58 + 78 * q^59 + 303 * q^62 - 45 * q^64 - 33 * q^72 + 399 * q^78 + 243 * q^81 - 129 * q^82 + 246 * q^87 - 276 * q^92 - 546 * q^93 - 57 * q^94 - 21 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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