LMFDB - Embedded newform 23.3.b.a.22.1

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			22.1
Root			$-0.523976$ 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)

$f(q)$	$=$	$q-3.72545 q^{2} +4.24943 q^{3} +9.87897 q^{4} -15.8310 q^{6} -21.9018 q^{8} +9.05761 q^{9} +O(q^{10})$	$q-3.72545 q^{2} +4.24943 q^{3} +9.87897 q^{4} -15.8310 q^{6} -21.9018 q^{8} +9.05761 q^{9} +41.9799 q^{12} -21.3624 q^{13} +42.0781 q^{16} -33.7437 q^{18} -23.0000 q^{23} -93.0700 q^{24} +25.0000 q^{25} +79.5844 q^{26} +0.244826 q^{27} +55.4730 q^{29} -33.9378 q^{31} -69.1528 q^{32} +89.4799 q^{36} -90.7777 q^{39} -8.78692 q^{41} +85.6853 q^{46} +42.8975 q^{47} +178.808 q^{48} +49.0000 q^{49} -93.1362 q^{50} -211.038 q^{52} -0.912087 q^{54} -206.662 q^{58} +26.0000 q^{59} +126.433 q^{62} +89.3126 q^{64} -97.7368 q^{69} -85.6223 q^{71} -198.378 q^{72} +144.884 q^{73} +106.236 q^{75} +338.188 q^{78} -80.4782 q^{81} +32.7352 q^{82} +235.728 q^{87} -227.216 q^{92} -144.216 q^{93} -159.813 q^{94} -293.860 q^{96} -182.547 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.72545	−1.86272	−0.931362	−	0.364094i	$-0.881379\pi$
$2$	−3.72545	−1.86272	−0.931362	+	0.364094i	$0.881379\pi$
$3$	4.24943	1.41648	0.708238	−	0.705974i	$-0.249491\pi$
$3$	4.24943	1.41648	0.708238	+	0.705974i	$0.249491\pi$
$4$	9.87897	2.46974
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−15.8310	−2.63850
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−21.9018	−2.73772
$9$	9.05761	1.00640
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	41.9799	3.49833
$13$	−21.3624	−1.64326	−0.821629	−	0.570023i	$-0.806934\pi$
$13$	−21.3624	−1.64326	−0.821629	+	0.570023i	$0.806934\pi$
$14$	0	0
$15$	0	0
$16$	42.0781	2.62988
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−33.7437	−1.87465
$19$	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
$24$	−93.0700	−3.87792
$25$	25.0000	1.00000
$26$	79.5844	3.06094
$27$	0.244826	0.00906764
$28$	0	0
$29$	55.4730	1.91286	0.956431	−	0.291959i	$-0.0943072\pi$
$29$	55.4730	1.91286	0.956431	+	0.291959i	$0.0943072\pi$
$30$	0	0
$31$	−33.9378	−1.09477	−0.547384	−	0.836882i	$-0.684376\pi$
$31$	−33.9378	−1.09477	−0.547384	+	0.836882i	$0.684376\pi$
$32$	−69.1528	−2.16102
$33$	0	0
$34$	0	0
$35$	0	0
$36$	89.4799	2.48555
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−90.7777	−2.32763
$40$	0	0
$41$	−8.78692	−0.214315	−0.107158	−	0.994242i	$-0.534175\pi$
$41$	−8.78692	−0.214315	−0.107158	+	0.994242i	$0.534175\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	85.6853	1.86272
$47$	42.8975	0.912714	0.456357	−	0.889797i	$-0.349154\pi$
$47$	42.8975	0.912714	0.456357	+	0.889797i	$0.349154\pi$
$48$	178.808	3.72516
$49$	49.0000	1.00000
$50$	−93.1362	−1.86272
$51$	0	0
$52$	−211.038	−4.05842
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−0.912087	−0.0168905
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−206.662	−3.56313
$59$	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			$\pi$
$62$	126.433	2.03925
$63$	0	0
$64$	89.3126	1.39551
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	−97.7368	−1.41648
$70$	0	0
$71$	−85.6223	−1.20595	−0.602974	−	0.797761i	$-0.706018\pi$
$71$	−85.6223	−1.20595	−0.602974	+	0.797761i	$0.706018\pi$
$72$	−198.378	−2.75525
$73$	144.884	1.98471	0.992354	−	0.123420i	$-0.0393864\pi$
$73$	144.884	1.98471	0.992354	+	0.123420i	$0.0393864\pi$
$74$	0	0
$75$	106.236	1.41648
$76$	0	0
$77$	0	0
$78$	338.188	4.33574
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	−80.4782	−0.993557
$82$	32.7352	0.399210
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	235.728	2.70952
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	−227.216	−2.46974
$93$	−144.216	−1.55071
$94$	−159.813	−1.70013
$95$	0	0
$96$	−293.860	−3.06104
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−182.547	−1.86272
$99$	0	0
$100$	246.974	2.46974
$101$	−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			$\pi$
$104$	467.874	4.49879
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	2.41863	0.0223947
$109$	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
$116$	548.016	4.72427
$117$	−193.492	−1.65378
$118$	−96.8617	−0.820862
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	0	0
$123$	−37.3394	−0.303572
$124$	−335.270	−2.70379
$125$	0	0
$126$	0	0
$127$	−60.4714	−0.476153	−0.238076	−	0.971246i	$-0.576517\pi$
$127$	−60.4714	−0.476153	−0.238076	+	0.971246i	$0.576517\pi$
$128$	−56.1183	−0.438424
$129$	0	0
$130$	0	0
$131$	−71.6641	−0.547054	−0.273527	−	0.961864i	$-0.588190\pi$
$131$	−71.6641	−0.547054	−0.273527	+	0.961864i	$0.588190\pi$
$132$	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			$\pi$
$138$	364.113	2.63850
$139$	−277.019	−1.99294	−0.996472	−	0.0839253i	$-0.973254\pi$
$139$	−277.019	−1.99294	−0.996472	+	0.0839253i	$0.973254\pi$
$140$	0	0
$141$	182.290	1.29284
$142$	318.981	2.24635
$143$	0	0
$144$	381.128	2.64672
$145$	0	0
$146$	−539.757	−3.69697
$147$	208.222	1.41648
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−395.775	−2.63850
$151$	298.554	1.97718	0.988591	−	0.150626i	$-0.0481289\pi$
$151$	298.554	1.97718	0.988591	+	0.150626i	$0.0481289\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−896.790	−5.74866
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	299.817	1.85072
$163$	82.0066	0.503108	0.251554	−	0.967843i	$-0.419058\pi$
$163$	82.0066	0.503108	0.251554	+	0.967843i	$0.419058\pi$
$164$	−86.8057	−0.529303
$165$	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
$169$	287.350	1.70030
$170$	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	$0.520253\pi$
$174$	−878.194	−5.04709
$175$	0	0
$176$	0	0
$177$	110.485	0.624209
$178$	0	0
$179$	−328.704	−1.83633	−0.918167	−	0.396194i	$-0.870331\pi$
$179$	−328.704	−1.83633	−0.918167	+	0.396194i	$0.870331\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	503.741	2.73772
$185$	0	0
$186$	537.270	2.88855
$187$	0	0
$188$	423.784	2.25417
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	379.527	1.97670
$193$	−314.746	−1.63081	−0.815403	−	0.578894i	$-0.803485\pi$
$193$	−314.746	−1.63081	−0.815403	+	0.578894i	$0.803485\pi$
$194$	0	0
$195$	0	0
$196$	484.069	2.46974
$197$	−173.650	−0.881474	−0.440737	−	0.897636i	$-0.645283\pi$
$197$	−173.650	−0.881474	−0.440737	+	0.897636i	$0.645283\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−547.545	−2.73772
$201$	0	0
$202$	618.424	3.06151
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−208.325	−1.00640
$208$	−898.888	−4.32158
$209$	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
$213$	−363.845	−1.70819
$214$	0	0
$215$	0	0
$216$	−5.36213	−0.0248247
$217$	0	0
$218$	0	0
$219$	615.673	2.81129
$220$	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
$225$	226.440	1.00640
$226$	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
$232$	−1214.96	−5.23689
$233$	425.691	1.82700	0.913501	−	0.406836i	$-0.133368\pi$
$233$	425.691	1.82700	0.913501	+	0.406836i	$0.133368\pi$
$234$	720.844	3.08053
$235$	0	0
$236$	256.853	1.08836
$237$	0	0
$238$	0	0
$239$	477.376	1.99739	0.998694	−	0.0510816i	$-0.0162668\pi$
$239$	477.376	1.99739	0.998694	+	0.0510816i	$0.0162668\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−450.779	−1.86272
$243$	−344.189	−1.41642
$244$	0	0
$245$	0	0
$246$	139.106	0.565471
$247$	0	0
$248$	743.298	2.99717
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	225.283	0.886941
$255$	0	0
$256$	−148.185	−0.578846
$257$	−494.950	−1.92587	−0.962937	−	0.269725i	$-0.913067\pi$
$257$	−494.950	−1.92587	−0.962937	+	0.269725i	$0.913067\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	502.453	1.92511
$262$	266.981	1.01901
$263$	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
$269$	232.912	0.865843	0.432922	−	0.901432i	$-0.357483\pi$
$269$	232.912	0.865843	0.432922	+	0.901432i	$0.357483\pi$
$270$	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
$276$	−965.539	−3.49833
$277$	210.526	0.760023	0.380012	−	0.924982i	$-0.375920\pi$
$277$	210.526	0.760023	0.380012	+	0.924982i	$0.375920\pi$
$278$	1032.02	3.71231
$279$	−307.395	−1.10178
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−679.112	−2.40820
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	−845.860	−2.97838
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−626.359	−2.17486
$289$	289.000	1.00000
$290$	0	0
$291$	0	0
$292$	1431.30	4.90172
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−775.720	−2.63850
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	491.334	1.64326
$300$	1049.50	3.49833
$301$	0	0
$302$	−1112.25	−3.68294
$303$	−705.405	−2.32807
$304$	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
$311$	91.8165	0.295230	0.147615	−	0.989045i	$-0.452840\pi$
$311$	91.8165	0.295230	0.147615	+	0.989045i	$0.452840\pi$
$312$	1988.20	6.37242
$313$	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
$324$	−795.041	−2.45383
$325$	−534.059	−1.64326
$326$	−305.511	−0.937151
$327$	0	0
$328$	192.449	0.586736
$329$	0	0
$330$	0	0
$331$	−330.086	−0.997240	−0.498620	−	0.866821i	$-0.666160\pi$
$331$	−330.086	−0.997240	−0.498620	+	0.866821i	$0.666160\pi$
$332$	0	0
$333$	0	0
$334$	−901.559	−2.69928
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−1070.51	−3.16718
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	81.9599	0.236878
$347$	602.000	1.73487	0.867435	−	0.497550i	$-0.165767\pi$
$347$	602.000	1.73487	0.867435	+	0.497550i	$0.165767\pi$
$348$	2328.75	6.69182
$349$	617.088	1.76816	0.884081	−	0.467334i	$-0.154785\pi$
$349$	617.088	1.76816	0.884081	+	0.467334i	$0.154785\pi$
$350$	0	0
$351$	−5.23006	−0.0149005
$352$	0	0
$353$	453.608	1.28501	0.642504	−	0.766282i	$-0.277896\pi$
$353$	453.608	1.28501	0.642504	+	0.766282i	$0.277896\pi$
$354$	−411.606	−1.16273
$355$	0	0
$356$	0	0
$357$	0	0
$358$	1224.57	3.42058
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	0	0
$363$	514.180	1.41648
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	−967.797	−2.62988
$369$	−79.5885	−0.215687
$370$	0	0
$371$	0	0
$372$	−1424.71	−3.82986
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	−939.533	−2.49876
$377$	−1185.03	−3.14332
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−256.969	−0.674458
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−238.470	−0.621017
$385$	0	0
$386$	1172.57	3.03774
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−1073.19	−2.73772
$393$	−304.531	−0.774888
$394$	646.925	1.64194
$395$	0	0
$396$	0	0
$397$	−249.103	−0.627463	−0.313732	−	0.949512i	$-0.601579\pi$
$397$	−249.103	−0.627463	−0.313732	+	0.949512i	$0.601579\pi$
$398$	0	0
$399$	0	0
$400$	1051.95	2.62988
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	724.991	1.79899
$404$	−1639.91	−4.05918
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−800.908	−1.95821	−0.979106	−	0.203352i	$-0.934816\pi$
$409$	−800.908	−1.95821	−0.979106	+	0.203352i	$0.934816\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	776.105	1.87465
$415$	0	0
$416$	1477.27	3.55112
$417$	−1177.17	−2.82296
$418$	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			$\pi$
$422$	1512.53	3.58420
$423$	388.549	0.918557
$424$	0	0
$425$	0	0
$426$	1355.49	3.18189
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	10.3018	0.0238468
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	−2293.66	−5.23666
$439$	218.954	0.498755	0.249378	−	0.968406i	$-0.419774\pi$
$439$	218.954	0.498755	0.249378	+	0.968406i	$0.419774\pi$
$440$	0	0
$441$	443.823	1.00640
$442$	0	0
$443$	−279.785	−0.631568	−0.315784	−	0.948831i	$-0.602268\pi$
$443$	−279.785	−0.631568	−0.315784	+	0.948831i	$0.602268\pi$
$444$	0	0
$445$	0	0
$446$	1423.12	3.19086
$447$	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	$0.720734\pi$
$450$	−843.592	−1.87465
$451$	0	0
$452$	0	0
$453$	1268.68	2.80063
$454$	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
$461$	−916.853	−1.98883	−0.994417	−	0.105519i	$-0.966350\pi$
$461$	−916.853	−1.98883	−0.994417	+	0.105519i	$0.966350\pi$
$462$	0	0
$463$	98.0000	0.211663	0.105832	−	0.994384i	$-0.466250\pi$
$463$	98.0000	0.211663	0.105832	+	0.994384i	$0.466250\pi$
$464$	2334.20	5.03060
$465$	0	0
$466$	−1585.89	−3.40320
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−1911.50	−4.08440
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−569.447	−1.20645
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−1778.44	−3.72059
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	1195.36	2.46974
$485$	0	0
$486$	1282.26	2.63839
$487$	937.005	1.92404	0.962018	−	0.272987i	$-0.0880116\pi$
$487$	937.005	1.92404	0.962018	+	0.272987i	$0.0880116\pi$
$488$	0	0
$489$	348.481	0.712640
$490$	0	0
$491$	−658.430	−1.34100	−0.670499	−	0.741910i	$-0.733920\pi$
$491$	−658.430	−1.34100	−0.670499	+	0.741910i	$0.733920\pi$
$492$	−368.874	−0.749745
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−1428.04	−2.87911
$497$	0	0
$498$	0	0
$499$	−863.786	−1.73103	−0.865517	−	0.500880i	$-0.833010\pi$
$499$	−863.786	−1.73103	−0.865517	+	0.500880i	$0.833010\pi$
$500$	0	0
$501$	1028.36	2.05262
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1221.07	2.40843
$508$	−597.395	−1.17597
$509$	288.744	0.567278	0.283639	−	0.958931i	$-0.408458\pi$
$509$	288.744	0.567278	0.283639	+	0.958931i	$0.408458\pi$
$510$	0	0
$511$	0	0
$512$	776.527	1.51665
$513$	0	0
$514$	1843.91	3.58737
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−93.4874	−0.180130
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−1871.86	−3.58594
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−707.967	−1.35108
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	529.000	1.00000
$530$	0	0
$531$	235.498	0.443499
$532$	0	0
$533$	187.709	0.352175
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1396.80	−2.60112
$538$	−867.701	−1.61283
$539$	0	0
$540$	0	0
$541$	564.021	1.04255	0.521277	−	0.853388i	$-0.325456\pi$
$541$	564.021	1.04255	0.521277	+	0.853388i	$0.325456\pi$
$542$	1065.48	1.96583
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−145.734	−0.266424	−0.133212	−	0.991088i	$-0.542529\pi$
$547$	−145.734	−0.266424	−0.133212	+	0.991088i	$0.542529\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	2140.61	3.87792
$553$	0	0
$554$	−784.305	−1.41571
$555$	0	0
$556$	−2736.66	−4.92206
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	1145.19	2.05230
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	1800.84	3.19297
$565$	0	0
$566$	0	0
$567$	0	0
$568$	1875.28	3.30155
$569$	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
$576$	808.959	1.40444
$577$	1041.76	1.80547	0.902736	−	0.430196i	$-0.141555\pi$
$577$	1041.76	1.80547	0.902736	+	0.430196i	$0.141555\pi$
$578$	−1076.65	−1.86272
$579$	−1337.49	−2.31000
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−3173.21	−5.43359
$585$	0	0
$586$	0	0
$587$	−1172.51	−1.99746	−0.998731	−	0.0503716i	$-0.983959\pi$
$587$	−1172.51	−1.99746	−0.998731	+	0.0503716i	$0.983959\pi$
$588$	2057.02	3.49833
$589$	0	0
$590$	0	0
$591$	−737.914	−1.24859
$592$	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
$598$	−1830.44	−3.06094
$599$	1106.00	1.84641	0.923205	−	0.384307i	$-0.125560\pi$
$599$	1106.00	1.84641	0.923205	+	0.384307i	$0.125560\pi$
$600$	−2326.75	−3.87792
$601$	608.661	1.01275	0.506374	−	0.862314i	$-0.330986\pi$
$601$	608.661	1.01275	0.506374	+	0.862314i	$0.330986\pi$
$602$	0	0
$603$	0	0
$604$	2949.41	4.88313
$605$	0	0
$606$	2627.95	4.33655
$607$	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
$611$	−916.393	−1.49982
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	797.246	1.29845
$615$	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
$621$	−5.63100	−0.00906764
$622$	−342.058	−0.549932
$623$	0	0
$624$	−3819.76	−6.12141
$625$	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
$633$	−1725.27	−2.72554
$634$	−990.969	−1.56304
$635$	0	0
$636$	0	0
$637$	−1046.76	−1.64326
$638$	0	0
$639$	−775.533	−1.21367
$640$	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
$647$	1217.81	1.88225	0.941123	−	0.338065i	$-0.109772\pi$
$647$	1217.81	1.88225	0.941123	+	0.338065i	$0.109772\pi$
$648$	1762.62	2.72009
$649$	0	0
$650$	1989.61	3.06094
$651$	0	0
$652$	810.140	1.24255
$653$	1101.87	1.68739	0.843697	−	0.536819i	$-0.180374\pi$
$653$	1101.87	1.68739	0.843697	+	0.536819i	$0.180374\pi$
$654$	0	0
$655$	0	0
$656$	−369.737	−0.563624
$657$	1312.30	1.99741
$658$	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			$\pi$
$662$	1229.72	1.85758
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1275.88	−1.91286
$668$	2390.71	3.57891
$669$	−1623.28	−2.42643
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−447.414	−0.664805	−0.332402	−	0.943138i	$-0.607859\pi$
$673$	−447.414	−0.664805	−0.332402	+	0.943138i	$0.607859\pi$
$674$	0	0
$675$	6.12065	0.00906764
$676$	2838.72	4.19929
$677$	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
$683$	1360.29	1.99164	0.995821	−	0.0913307i	$-0.0291120\pi$
$683$	1360.29	1.99164	0.995821	+	0.0913307i	$0.0291120\pi$
$684$	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	$0.368708\pi$
$692$	−217.337	−0.314071
$693$	0	0
$694$	−2242.72	−3.23159
$695$	0	0
$696$	−5162.87	−7.41792
$697$	0	0
$698$	−2298.93	−3.29360
$699$	1808.94	2.58790
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	19.4843	0.0277555
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−1689.89	−2.39362
$707$	0	0
$708$	1091.48	1.54164
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	780.569	1.09477
$714$	0	0
$715$	0	0
$716$	−3247.25	−4.53527
$717$	2028.57	2.82925
$718$	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
$722$	−1344.89	−1.86272
$723$	0	0
$724$	0	0
$725$	1386.82	1.91286
$726$	−1915.55	−2.63850
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−738.303	−1.01276
$730$	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
$736$	1590.51	2.16102
$737$	0	0
$738$	296.503	0.401766
$739$	−77.1950	−0.104459	−0.0522294	−	0.998635i	$-0.516633\pi$
$739$	−77.1950	−0.104459	−0.0522294	+	0.998635i	$0.516633\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	3158.59	4.24542
$745$	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			$\pi$
$752$	1805.05	2.40033
$753$	0	0
$754$	4414.78	5.85515
$755$	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
$761$	−1060.71	−1.39384	−0.696921	−	0.717148i	$-0.745447\pi$
$761$	−1060.71	−1.39384	−0.696921	+	0.717148i	$0.745447\pi$
$762$	957.323	1.25633
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−555.421	−0.724148
$768$	−629.699	−0.819921
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	−2103.25	−2.72795
$772$	−3109.36	−4.02767
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	−848.445	−1.09477
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	13.5812	0.0173451
$784$	2061.83	2.62988
$785$	0	0
$786$	1134.52	1.44340
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−1715.49	−2.17701
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	928.020	1.16879
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−1728.82	−2.16102
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−2700.92	−3.35101
$807$	989.741	1.22645
$808$	3635.70	4.49963
$809$	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
$811$	878.276	1.08295	0.541477	−	0.840715i	$-0.317865\pi$
$811$	878.276	1.08295	0.541477	+	0.840715i	$0.317865\pi$
$812$	0	0
$813$	−1215.34	−1.49488
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	2983.74	3.64761
$819$	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
$823$	−1593.03	−1.93564	−0.967819	−	0.251648i	$-0.919028\pi$
$823$	−1593.03	−1.93564	−0.967819	+	0.251648i	$0.919028\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−2058.04	−2.48555
$829$	−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
$831$	894.616	1.07655
$832$	−1907.93	−2.29318
$833$	0	0
$834$	4385.50	5.25839
$835$	0	0
$836$	0	0
$837$	−8.30886	−0.00992695
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	2236.25	2.65904
$842$	0	0
$843$	0	0
$844$	−4010.86	−4.75221
$845$	0	0
$846$	−1447.52	−1.71102
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	−3594.42	−4.21880
$853$	−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
$857$	1350.48	1.57582	0.787912	−	0.615788i	$-0.211162\pi$
$857$	1350.48	1.57582	0.787912	+	0.615788i	$0.211162\pi$
$858$	0	0
$859$	1717.93	1.99992	0.999962	−	0.00876272i	$-0.00278930\pi$
$859$	1717.93	1.99992	0.999962	+	0.00876272i	$0.00278930\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−182.078	−0.210982	−0.105491	−	0.994420i	$-0.533641\pi$
$863$	−182.078	−0.210982	−0.105491	+	0.994420i	$0.533641\pi$
$864$	−16.9304	−0.0195954
$865$	0	0
$866$	0	0
$867$	1228.08	1.41648
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	6082.21	6.94316
$877$	−1558.00	−1.77651	−0.888255	−	0.459350i	$-0.848082\pi$
$877$	−1558.00	−1.77651	−0.888255	+	0.459350i	$0.848082\pi$
$878$	−815.700	−0.929044
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−1653.44	−1.87465
$883$	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
$886$	1042.32	1.17644
$887$	−1773.23	−1.99914	−0.999568	−	0.0293802i	$-0.990647\pi$
$887$	−1773.23	−1.99914	−0.999568	+	0.0293802i	$0.990647\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−3773.77	−4.23068
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2087.89	2.32763
$898$	2138.41	2.38130
$899$	−1882.63	−2.09414
$900$	2237.00	2.48555
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	−4726.42	−5.21680
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	−1503.56	−1.65409
$910$	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
$921$	−909.377	−0.987380
$922$	3415.69	3.70465
$923$	1829.09	1.98168
$924$	0	0
$925$	0	0
$926$	−365.094	−0.394270
$927$	0	0
$928$	−3836.11	−4.13374
$929$	−1411.44	−1.51931	−0.759657	−	0.650324i	$-0.774633\pi$
$929$	−1411.44	−1.51931	−0.759657	+	0.650324i	$0.774633\pi$
$930$	0	0
$931$	0	0
$932$	4205.39	4.51222
$933$	390.168	0.418186
$934$	0	0
$935$	0	0
$936$	4237.82	4.52759
$937$	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
$943$	202.099	0.214315
$944$	1094.03	1.15893
$945$	0	0
$946$	0	0
$947$	1287.60	1.35967	0.679833	−	0.733367i	$-0.262052\pi$
$947$	1287.60	1.35967	0.679833	+	0.733367i	$0.262052\pi$
$948$	0	0
$949$	−3095.06	−3.26139
$950$	0	0
$951$	1130.35	1.18859
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	4715.98	4.93304
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	190.773	0.198515
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	804.337	0.831786	0.415893	−	0.909414i	$-0.363469\pi$
$967$	804.337	0.831786	0.415893	+	0.909414i	$0.363469\pi$
$968$	−2650.12	−2.73772
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−3400.24	−3.49818
$973$	0	0
$974$	−3490.77	−3.58395
$975$	−2269.44	−2.32763
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	−1298.25	−1.32745
$979$	0	0
$980$	0	0
$981$	0	0
$982$	2452.95	2.49791
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	817.799	0.831097
$985$	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	$0.302176\pi$
$992$	2346.89	2.36582
$993$	−1402.68	−1.41257
$994$	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	$0.729860\pi$
$998$	3217.99	3.22444
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.3.b.a.22.1	✓	3
3.2	odd	2		207.3.d.a.91.3		3
4.3	odd	2		368.3.f.a.321.1		3
5.2	odd	4		575.3.c.a.574.1		6
5.3	odd	4		575.3.c.a.574.6		6
5.4	even	2		575.3.d.b.551.3		3
8.3	odd	2		1472.3.f.b.321.3		3
8.5	even	2		1472.3.f.a.321.1		3
23.22	odd	2	CM	23.3.b.a.22.1	✓	3
69.68	even	2		207.3.d.a.91.3		3
92.91	even	2		368.3.f.a.321.1		3
115.22	even	4		575.3.c.a.574.1		6
115.68	even	4		575.3.c.a.574.6		6
115.114	odd	2		575.3.d.b.551.3		3
184.45	odd	2		1472.3.f.a.321.1		3
184.91	even	2		1472.3.f.b.321.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.3.b.a.22.1	✓	3	1.1	even	1	trivial
23.3.b.a.22.1	✓	3	23.22	odd	2	CM
207.3.d.a.91.3		3	3.2	odd	2
207.3.d.a.91.3		3	69.68	even	2
368.3.f.a.321.1		3	4.3	odd	2
368.3.f.a.321.1		3	92.91	even	2
575.3.c.a.574.1		6	5.2	odd	4
575.3.c.a.574.1		6	115.22	even	4
575.3.c.a.574.6		6	5.3	odd	4
575.3.c.a.574.6		6	115.68	even	4
575.3.d.b.551.3		3	5.4	even	2
575.3.d.b.551.3		3	115.114	odd	2
1472.3.f.a.321.1		3	8.5	even	2
1472.3.f.a.321.1		3	184.45	odd	2
1472.3.f.b.321.3		3	8.3	odd	2
1472.3.f.b.321.3		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)

\(f(q)\)	\(=\)	\(q-3.72545 q^{2} +4.24943 q^{3} +9.87897 q^{4} -15.8310 q^{6} -21.9018 q^{8} +9.05761 q^{9} +O(q^{10})\)	\(q-3.72545 q^{2} +4.24943 q^{3} +9.87897 q^{4} -15.8310 q^{6} -21.9018 q^{8} +9.05761 q^{9} +41.9799 q^{12} -21.3624 q^{13} +42.0781 q^{16} -33.7437 q^{18} -23.0000 q^{23} -93.0700 q^{24} +25.0000 q^{25} +79.5844 q^{26} +0.244826 q^{27} +55.4730 q^{29} -33.9378 q^{31} -69.1528 q^{32} +89.4799 q^{36} -90.7777 q^{39} -8.78692 q^{41} +85.6853 q^{46} +42.8975 q^{47} +178.808 q^{48} +49.0000 q^{49} -93.1362 q^{50} -211.038 q^{52} -0.912087 q^{54} -206.662 q^{58} +26.0000 q^{59} +126.433 q^{62} +89.3126 q^{64} -97.7368 q^{69} -85.6223 q^{71} -198.378 q^{72} +144.884 q^{73} +106.236 q^{75} +338.188 q^{78} -80.4782 q^{81} +32.7352 q^{82} +235.728 q^{87} -227.216 q^{92} -144.216 q^{93} -159.813 q^{94} -293.860 q^{96} -182.547 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 12 q^{4} - 33 q^{6} - 21 q^{8} + 27 q^{9}+O(q^{10}) \) 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

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