LMFDB - Embedded newform 23.3.b.a.22.3

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			22.3
Root			$2.66908$ 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)

$f(q)$	$=$	$q+3.12398 q^{2} -5.79306 q^{3} +5.75927 q^{4} -18.0974 q^{6} +5.49593 q^{8} +24.5596 q^{9} +O(q^{10})$	$q+3.12398 q^{2} -5.79306 q^{3} +5.75927 q^{4} -18.0974 q^{6} +5.49593 q^{8} +24.5596 q^{9} -33.3638 q^{12} -2.15383 q^{13} -5.86788 q^{16} +76.7237 q^{18} -23.0000 q^{23} -31.8383 q^{24} +25.0000 q^{25} -6.72853 q^{26} -90.1376 q^{27} -13.0715 q^{29} +61.9041 q^{31} -40.3149 q^{32} +141.445 q^{36} +12.4773 q^{39} -66.2117 q^{41} -71.8516 q^{46} +50.9864 q^{47} +33.9930 q^{48} +49.0000 q^{49} +78.0996 q^{50} -12.4045 q^{52} -281.588 q^{54} -40.8352 q^{58} +26.0000 q^{59} +193.387 q^{62} -102.472 q^{64} +133.240 q^{69} -55.2940 q^{71} +134.978 q^{72} -88.0471 q^{73} -144.827 q^{75} +38.9788 q^{78} +301.136 q^{81} -206.844 q^{82} +75.7242 q^{87} -132.463 q^{92} -358.614 q^{93} +159.281 q^{94} +233.547 q^{96} +153.075 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	3.12398	1.56199	0.780996	−	0.624536i	$-0.214712\pi$
$2$	3.12398	1.56199	0.780996	+	0.624536i	$0.214712\pi$
$3$	−5.79306	−1.93102	−0.965510	−	0.260365i	$-0.916157\pi$
$3$	−5.79306	−1.93102	−0.965510	+	0.260365i	$0.916157\pi$
$4$	5.75927	1.43982
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−18.0974	−3.01624
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	5.49593	0.686992
$9$	24.5596	2.72884
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−33.3638	−2.78032
$13$	−2.15383	−0.165679	−0.0828396	−	0.996563i	$-0.526399\pi$
$13$	−2.15383	−0.165679	−0.0828396	+	0.996563i	$0.526399\pi$
$14$	0	0
$15$	0	0
$16$	−5.86788	−0.366743
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	76.7237	4.26243
$19$	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
$24$	−31.8383	−1.32660
$25$	25.0000	1.00000
$26$	−6.72853	−0.258790
$27$	−90.1376	−3.33843
$28$	0	0
$29$	−13.0715	−0.450742	−0.225371	−	0.974273i	$-0.572359\pi$
$29$	−13.0715	−0.450742	−0.225371	+	0.974273i	$0.572359\pi$
$30$	0	0
$31$	61.9041	1.99691	0.998453	−	0.0556073i	$-0.0177095\pi$
$31$	61.9041	1.99691	0.998453	+	0.0556073i	$0.0177095\pi$
$32$	−40.3149	−1.25984
$33$	0	0
$34$	0	0
$35$	0	0
$36$	141.445	3.92903
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	12.4773	0.319930
$40$	0	0
$41$	−66.2117	−1.61492	−0.807460	−	0.589922i	$-0.799158\pi$
$41$	−66.2117	−1.61492	−0.807460	+	0.589922i	$0.799158\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	−71.8516	−1.56199
$47$	50.9864	1.08482	0.542408	−	0.840115i	$-0.317513\pi$
$47$	50.9864	1.08482	0.542408	+	0.840115i	$0.317513\pi$
$48$	33.9930	0.708188
$49$	49.0000	1.00000
$50$	78.0996	1.56199
$51$	0	0
$52$	−12.4045	−0.238548
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−281.588	−5.21460
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−40.8352	−0.704056
$59$	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			$\pi$
$62$	193.387	3.11915
$63$	0	0
$64$	−102.472	−1.60112
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	133.240	1.93102
$70$	0	0
$71$	−55.2940	−0.778789	−0.389395	−	0.921071i	$-0.627316\pi$
$71$	−55.2940	−0.778789	−0.389395	+	0.921071i	$0.627316\pi$
$72$	134.978	1.87469
$73$	−88.0471	−1.20612	−0.603062	−	0.797694i	$-0.706053\pi$
$73$	−88.0471	−1.20612	−0.603062	+	0.797694i	$0.706053\pi$
$74$	0	0
$75$	−144.827	−1.93102
$76$	0	0
$77$	0	0
$78$	38.9788	0.499728
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	301.136	3.71773
$82$	−206.844	−2.52249
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	75.7242	0.870393
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	−132.463	−1.43982
$93$	−358.614	−3.85607
$94$	159.281	1.69447
$95$	0	0
$96$	233.547	2.43278
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	153.075	1.56199
$99$	0	0
$100$	143.982	1.43982
$101$	−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			$\pi$
$104$	−11.8373	−0.113820
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−519.127	−4.80673
$109$	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
$116$	−75.2825	−0.648987
$117$	−52.8971	−0.452112
$118$	81.2236	0.688335
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	0	0
$123$	383.569	3.11844
$124$	356.522	2.87518
$125$	0	0
$126$	0	0
$127$	−183.410	−1.44417	−0.722086	−	0.691803i	$-0.756816\pi$
$127$	−183.410	−1.44417	−0.722086	+	0.691803i	$0.756816\pi$
$128$	−158.860	−1.24109
$129$	0	0
$130$	0	0
$131$	254.078	1.93952	0.969762	−	0.244051i	$-0.0784764\pi$
$131$	254.078	1.93952	0.969762	+	0.244051i	$0.0784764\pi$
$132$	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			$\pi$
$138$	416.241	3.01624
$139$	158.715	1.14183	0.570917	−	0.821007i	$-0.306588\pi$
$139$	158.715	1.14183	0.570917	+	0.821007i	$0.306588\pi$
$140$	0	0
$141$	−295.367	−2.09480
$142$	−172.738	−1.21646
$143$	0	0
$144$	−144.113	−1.00078
$145$	0	0
$146$	−275.058	−1.88396
$147$	−283.860	−1.93102
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−452.436	−3.01624
$151$	−109.883	−0.727699	−0.363849	−	0.931458i	$-0.618538\pi$
$151$	−109.883	−0.727699	−0.363849	+	0.931458i	$0.618538\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	71.8600	0.460641
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	940.745	5.80707
$163$	232.242	1.42480	0.712400	−	0.701774i	$-0.247608\pi$
$163$	232.242	1.42480	0.712400	+	0.701774i	$0.247608\pi$
$164$	−381.331	−2.32519
$165$	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
$169$	−164.361	−0.972550
$170$	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	$0.520253\pi$
$174$	236.561	1.35955
$175$	0	0
$176$	0	0
$177$	−150.620	−0.850958
$178$	0	0
$179$	41.5170	0.231938	0.115969	−	0.993253i	$-0.463003\pi$
$179$	41.5170	0.231938	0.115969	+	0.993253i	$0.463003\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	−126.406	−0.686992
$185$	0	0
$186$	−1120.30	−6.02314
$187$	0	0
$188$	293.644	1.56194
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	593.624	3.09179
$193$	350.889	1.81808	0.909038	−	0.416713i	$-0.136818\pi$
$193$	350.889	1.81808	0.909038	+	0.416713i	$0.136818\pi$
$194$	0	0
$195$	0	0
$196$	282.204	1.43982
$197$	393.111	1.99549	0.997744	−	0.0671289i	$-0.0213839\pi$
$197$	393.111	1.99549	0.997744	+	0.0671289i	$0.0213839\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	137.398	0.686992
$201$	0	0
$202$	−518.581	−2.56723
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−564.870	−2.72884
$208$	12.6384	0.0607616
$209$	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
$213$	320.322	1.50386
$214$	0	0
$215$	0	0
$216$	−495.390	−2.29347
$217$	0	0
$218$	0	0
$219$	510.062	2.32905
$220$	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
$225$	613.989	2.72884
$226$	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
$232$	−71.8402	−0.309656
$233$	−377.032	−1.61816	−0.809081	−	0.587697i	$-0.800035\pi$
$233$	−377.032	−1.61816	−0.809081	+	0.587697i	$0.800035\pi$
$234$	−165.250	−0.706195
$235$	0	0
$236$	149.741	0.634496
$237$	0	0
$238$	0	0
$239$	−259.834	−1.08717	−0.543585	−	0.839354i	$-0.682934\pi$
$239$	−259.834	−1.08717	−0.543585	+	0.839354i	$0.682934\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	378.002	1.56199
$243$	−933.264	−3.84059
$244$	0	0
$245$	0	0
$246$	1198.26	4.87098
$247$	0	0
$248$	340.221	1.37186
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	−572.969	−2.25578
$255$	0	0
$256$	−86.3891	−0.337457
$257$	127.410	0.495760	0.247880	−	0.968791i	$-0.420266\pi$
$257$	127.410	0.495760	0.247880	+	0.968791i	$0.420266\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−321.031	−1.23000
$262$	793.735	3.02952
$263$	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
$269$	−536.452	−1.99425	−0.997123	−	0.0757948i	$-0.975851\pi$
$269$	−536.452	−1.99425	−0.997123	+	0.0757948i	$0.975851\pi$
$270$	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
$276$	767.368	2.78032
$277$	338.523	1.22210	0.611052	−	0.791591i	$-0.290747\pi$
$277$	338.523	1.22210	0.611052	+	0.791591i	$0.290747\pi$
$278$	495.823	1.78354
$279$	1520.34	5.44924
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−922.722	−3.27206
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	−318.453	−1.12131
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−990.117	−3.43790
$289$	289.000	1.00000
$290$	0	0
$291$	0	0
$292$	−507.087	−1.73660
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−886.774	−3.01624
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	49.5381	0.165679
$300$	−834.095	−2.78032
$301$	0	0
$302$	−343.271	−1.13666
$303$	961.648	3.17376
$304$	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
$311$	−578.675	−1.86069	−0.930346	−	0.366684i	$-0.880493\pi$
$311$	−578.675	−1.86069	−0.930346	+	0.366684i	$0.880493\pi$
$312$	68.5742	0.219789
$313$	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
$324$	1734.33	5.35286
$325$	−53.8457	−0.165679
$326$	725.521	2.22553
$327$	0	0
$328$	−363.895	−1.10944
$329$	0	0
$330$	0	0
$331$	−331.913	−1.00276	−0.501379	−	0.865228i	$-0.667174\pi$
$331$	−331.913	−1.00276	−0.501379	+	0.865228i	$0.667174\pi$
$332$	0	0
$333$	0	0
$334$	756.004	2.26348
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−513.461	−1.51912
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−68.7276	−0.198635
$347$	602.000	1.73487	0.867435	−	0.497550i	$-0.165767\pi$
$347$	602.000	1.73487	0.867435	+	0.497550i	$0.165767\pi$
$348$	436.116	1.25321
$349$	−591.041	−1.69353	−0.846763	−	0.531970i	$-0.821452\pi$
$349$	−591.041	−1.69353	−0.846763	+	0.531970i	$0.821452\pi$
$350$	0	0
$351$	194.141	0.553108
$352$	0	0
$353$	241.712	0.684736	0.342368	−	0.939566i	$-0.388771\pi$
$353$	241.712	0.684736	0.342368	+	0.939566i	$0.388771\pi$
$354$	−470.533	−1.32919
$355$	0	0
$356$	0	0
$357$	0	0
$358$	129.698	0.362286
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	0	0
$363$	−700.961	−1.93102
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	134.961	0.366743
$369$	−1626.13	−4.40686
$370$	0	0
$371$	0	0
$372$	−2065.36	−5.55203
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	280.218	0.745260
$377$	28.1538	0.0746786
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	1062.50	2.78873
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	920.284	2.39657
$385$	0	0
$386$	1096.17	2.83982
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	269.301	0.686992
$393$	−1471.89	−3.74526
$394$	1228.07	3.11694
$395$	0	0
$396$	0	0
$397$	777.459	1.95833	0.979167	−	0.203056i	$-0.0650875\pi$
$397$	777.459	1.95833	0.979167	+	0.203056i	$0.0650875\pi$
$398$	0	0
$399$	0	0
$400$	−146.697	−0.366743
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	−133.331	−0.330846
$404$	−956.039	−2.36643
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	544.511	1.33132	0.665661	−	0.746254i	$-0.268150\pi$
$409$	544.511	1.33132	0.665661	+	0.746254i	$0.268150\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	−1764.64	−4.26243
$415$	0	0
$416$	86.8314	0.208729
$417$	−919.446	−2.20491
$418$	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			$\pi$
$422$	−1268.34	−3.00554
$423$	1252.20	2.96029
$424$	0	0
$425$	0	0
$426$	1000.68	2.34901
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	528.917	1.22434
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	1593.43	3.63796
$439$	−845.824	−1.92671	−0.963353	−	0.268236i	$-0.913559\pi$
$439$	−845.824	−1.92671	−0.963353	+	0.268236i	$0.913559\pi$
$440$	0	0
$441$	1203.42	2.72884
$442$	0	0
$443$	−588.144	−1.32764	−0.663820	−	0.747893i	$-0.731066\pi$
$443$	−588.144	−1.32764	−0.663820	+	0.747893i	$0.731066\pi$
$444$	0	0
$445$	0	0
$446$	−1193.36	−2.67570
$447$	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	$0.720734\pi$
$450$	1918.09	4.26243
$451$	0	0
$452$	0	0
$453$	636.556	1.40520
$454$	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
$461$	374.172	0.811654	0.405827	−	0.913950i	$-0.366984\pi$
$461$	374.172	0.811654	0.405827	+	0.913950i	$0.366984\pi$
$462$	0	0
$463$	98.0000	0.211663	0.105832	−	0.994384i	$-0.466250\pi$
$463$	98.0000	0.211663	0.105832	+	0.994384i	$0.466250\pi$
$464$	76.7022	0.165306
$465$	0	0
$466$	−1177.84	−2.52756
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−304.649	−0.650959
$469$	0	0
$470$	0	0
$471$	0	0
$472$	142.894	0.302742
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−811.716	−1.69815
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	696.872	1.43982
$485$	0	0
$486$	−2915.50	−5.99897
$487$	−698.770	−1.43485	−0.717423	−	0.696638i	$-0.754678\pi$
$487$	−698.770	−1.43485	−0.717423	+	0.696638i	$0.754678\pi$
$488$	0	0
$489$	−1345.39	−2.75132
$490$	0	0
$491$	960.163	1.95553	0.977763	−	0.209715i	$-0.0672535\pi$
$491$	960.163	1.95553	0.977763	+	0.209715i	$0.0672535\pi$
$492$	2209.08	4.48999
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−363.246	−0.732350
$497$	0	0
$498$	0	0
$499$	864.800	1.73307	0.866533	−	0.499119i	$-0.166343\pi$
$499$	864.800	1.73307	0.866533	+	0.499119i	$0.166343\pi$
$500$	0	0
$501$	−1401.92	−2.79825
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	952.154	1.87802
$508$	−1056.31	−2.07934
$509$	701.035	1.37728	0.688639	−	0.725104i	$-0.258208\pi$
$509$	701.035	1.37728	0.688639	+	0.725104i	$0.258208\pi$
$510$	0	0
$511$	0	0
$512$	365.561	0.713986
$513$	0	0
$514$	398.028	0.774373
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	127.447	0.245563
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−1002.90	−1.92126
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	1463.30	2.79256
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	529.000	1.00000
$530$	0	0
$531$	638.549	1.20254
$532$	0	0
$533$	142.609	0.267559
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−240.510	−0.447878
$538$	−1675.87	−3.11500
$539$	0	0
$540$	0	0
$541$	−1081.67	−1.99939	−0.999694	−	0.0247449i	$-0.992123\pi$
$541$	−1081.67	−1.99939	−0.999694	+	0.0247449i	$0.992123\pi$
$542$	−893.459	−1.64845
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1011.85	1.84983	0.924913	−	0.380179i	$-0.124138\pi$
$547$	1011.85	1.84983	0.924913	+	0.380179i	$0.124138\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	732.280	1.32660
$553$	0	0
$554$	1057.54	1.90892
$555$	0	0
$556$	914.083	1.64403
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	4749.51	8.51166
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	−1701.10	−3.01613
$565$	0	0
$566$	0	0
$567$	0	0
$568$	−303.892	−0.535022
$569$	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
$576$	−2516.66	−4.36920
$577$	−90.9437	−0.157615	−0.0788074	−	0.996890i	$-0.525111\pi$
$577$	−90.9437	−0.157615	−0.0788074	+	0.996890i	$0.525111\pi$
$578$	902.831	1.56199
$579$	−2032.72	−3.51074
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−483.901	−0.828598
$585$	0	0
$586$	0	0
$587$	535.041	0.911484	0.455742	−	0.890112i	$-0.349374\pi$
$587$	535.041	0.911484	0.455742	+	0.890112i	$0.349374\pi$
$588$	−1634.83	−2.78032
$589$	0	0
$590$	0	0
$591$	−2277.32	−3.85333
$592$	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
$598$	154.756	0.258790
$599$	1106.00	1.84641	0.923205	−	0.384307i	$-0.125560\pi$
$599$	1106.00	1.84641	0.923205	+	0.384307i	$0.125560\pi$
$600$	−795.957	−1.32660
$601$	593.306	0.987198	0.493599	−	0.869690i	$-0.335681\pi$
$601$	593.306	0.987198	0.493599	+	0.869690i	$0.335681\pi$
$602$	0	0
$603$	0	0
$604$	−632.843	−1.04775
$605$	0	0
$606$	3004.17	4.95738
$607$	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
$611$	−109.816	−0.179732
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−668.532	−1.08881
$615$	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
$621$	2073.16	3.33843
$622$	−1807.77	−2.90638
$623$	0	0
$624$	−73.2151	−0.117332
$625$	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
$633$	2351.98	3.71561
$634$	830.980	1.31069
$635$	0	0
$636$	0	0
$637$	−105.538	−0.165679
$638$	0	0
$639$	−1358.00	−2.12519
$640$	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
$647$	−987.754	−1.52667	−0.763334	−	0.646004i	$-0.776439\pi$
$647$	−987.754	−1.52667	−0.763334	+	0.646004i	$0.776439\pi$
$648$	1655.03	2.55405
$649$	0	0
$650$	−168.213	−0.258790
$651$	0	0
$652$	1337.55	2.05145
$653$	−1158.09	−1.77350	−0.886748	−	0.462254i	$-0.847041\pi$
$653$	−1158.09	−1.77350	−0.886748	+	0.462254i	$0.847041\pi$
$654$	0	0
$655$	0	0
$656$	388.523	0.592260
$657$	−2162.40	−3.29132
$658$	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			$\pi$
$662$	−1036.89	−1.56630
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	300.645	0.450742
$668$	1393.74	2.08644
$669$	2212.95	3.30785
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−875.681	−1.30116	−0.650580	−	0.759438i	$-0.725474\pi$
$673$	−875.681	−1.30116	−0.650580	+	0.759438i	$0.725474\pi$
$674$	0	0
$675$	−2253.44	−3.33843
$676$	−946.600	−1.40030
$677$	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
$683$	−572.102	−0.837631	−0.418816	−	0.908071i	$-0.637555\pi$
$683$	−572.102	−0.837631	−0.418816	+	0.908071i	$0.637555\pi$
$684$	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	$0.368708\pi$
$692$	−126.704	−0.183098
$693$	0	0
$694$	1880.64	2.70985
$695$	0	0
$696$	416.175	0.597953
$697$	0	0
$698$	−1846.40	−2.64527
$699$	2184.17	3.12470
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	606.493	0.863950
$703$	0	0
$704$	0	0
$705$	0	0
$706$	755.104	1.06955
$707$	0	0
$708$	−867.459	−1.22522
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1423.79	−1.99691
$714$	0	0
$715$	0	0
$716$	239.107	0.333949
$717$	1505.23	2.09935
$718$	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
$722$	1127.76	1.56199
$723$	0	0
$724$	0	0
$725$	−326.788	−0.450742
$726$	−2189.79	−3.01624
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	2696.23	3.69853
$730$	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
$736$	927.243	1.25984
$737$	0	0
$738$	−5080.01	−6.88348
$739$	−1239.64	−1.67746	−0.838729	−	0.544550i	$-0.816701\pi$
$739$	−1239.64	−1.67746	−0.838729	+	0.544550i	$0.816701\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	−1970.92	−2.64908
$745$	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			$\pi$
$752$	−299.182	−0.397848
$753$	0	0
$754$	87.9521	0.116647
$755$	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
$761$	−414.909	−0.545216	−0.272608	−	0.962125i	$-0.587886\pi$
$761$	−414.909	−0.545216	−0.272608	+	0.962125i	$0.587886\pi$
$762$	3319.25	4.35597
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−55.9996	−0.0730112
$768$	500.457	0.651637
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	−738.096	−0.957322
$772$	2020.86	2.61770
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	1547.60	1.99691
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	1178.24	1.50477
$784$	−287.526	−0.366743
$785$	0	0
$786$	−4598.15	−5.85007
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	2264.03	2.87314
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	2428.77	3.05890
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−1007.87	−1.25984
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−416.523	−0.516778
$807$	3107.70	3.85093
$808$	−912.325	−1.12911
$809$	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
$811$	741.809	0.914684	0.457342	−	0.889291i	$-0.348801\pi$
$811$	741.809	0.914684	0.457342	+	0.889291i	$0.348801\pi$
$812$	0	0
$813$	1656.82	2.03790
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	1701.04	2.07951
$819$	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
$823$	1155.23	1.40369	0.701843	−	0.712332i	$-0.252361\pi$
$823$	1155.23	1.40369	0.701843	+	0.712332i	$0.252361\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−3253.24	−3.92903
$829$	−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
$831$	−1961.08	−2.35991
$832$	220.706	0.265272
$833$	0	0
$834$	−2872.33	−3.44405
$835$	0	0
$836$	0	0
$837$	−5579.88	−6.66653
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	−670.135	−0.796831
$842$	0	0
$843$	0	0
$844$	−2338.26	−2.77045
$845$	0	0
$846$	3911.86	4.62395
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	1844.82	2.16528
$853$	−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
$857$	238.815	0.278664	0.139332	−	0.990246i	$-0.455504\pi$
$857$	238.815	0.278664	0.139332	+	0.990246i	$0.455504\pi$
$858$	0	0
$859$	−872.004	−1.01514	−0.507570	−	0.861611i	$-0.669456\pi$
$859$	−872.004	−1.01514	−0.507570	+	0.861611i	$0.669456\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1577.46	1.82788	0.913939	−	0.405852i	$-0.133025\pi$
$863$	1577.46	1.82788	0.913939	+	0.405852i	$0.133025\pi$
$864$	3633.89	4.20589
$865$	0	0
$866$	0	0
$867$	−1674.19	−1.93102
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	2937.59	3.35341
$877$	−1558.00	−1.77651	−0.888255	−	0.459350i	$-0.848082\pi$
$877$	−1558.00	−1.77651	−0.888255	+	0.459350i	$0.848082\pi$
$878$	−2642.34	−3.00950
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	3759.46	4.26243
$883$	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
$886$	−1837.35	−2.07376
$887$	931.755	1.05046	0.525228	−	0.850961i	$-0.323980\pi$
$887$	931.755	1.05046	0.525228	+	0.850961i	$0.323980\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−2200.04	−2.46641
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−286.977	−0.319930
$898$	−1793.17	−1.99684
$899$	−809.181	−0.900090
$900$	3536.13	3.92903
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	1988.59	2.19491
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	−4076.89	−4.48503
$910$	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
$921$	1239.72	1.34605
$922$	1168.91	1.26780
$923$	119.094	0.129029
$924$	0	0
$925$	0	0
$926$	306.150	0.330616
$927$	0	0
$928$	526.977	0.567863
$929$	1752.14	1.88605	0.943026	−	0.332720i	$-0.107967\pi$
$929$	1752.14	1.88605	0.943026	+	0.332720i	$0.107967\pi$
$930$	0	0
$931$	0	0
$932$	−2171.43	−2.32986
$933$	3352.30	3.59303
$934$	0	0
$935$	0	0
$936$	−290.719	−0.310597
$937$	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
$943$	1522.87	1.61492
$944$	−152.565	−0.161615
$945$	0	0
$946$	0	0
$947$	559.105	0.590396	0.295198	−	0.955436i	$-0.404614\pi$
$947$	559.105	0.590396	0.295198	+	0.955436i	$0.404614\pi$
$948$	0	0
$949$	189.638	0.199830
$950$	0	0
$951$	−1540.95	−1.62035
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	−1496.45	−1.56533
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2871.11	2.98763
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1925.34	−1.99104	−0.995522	−	0.0945328i	$-0.969864\pi$
$967$	−1925.34	−1.99104	−0.995522	+	0.0945328i	$0.969864\pi$
$968$	665.008	0.686992
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−5374.92	−5.52975
$973$	0	0
$974$	−2182.94	−2.24122
$975$	311.932	0.319930
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	−4202.99	−4.29754
$979$	0	0
$980$	0	0
$981$	0	0
$982$	2999.53	3.05451
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	2108.07	2.14235
$985$	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	$0.302176\pi$
$992$	−2495.66	−2.51578
$993$	1922.79	1.93635
$994$	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	$0.729860\pi$
$998$	2701.62	2.70704
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.3.b.a.22.3	✓	3
3.2	odd	2		207.3.d.a.91.1		3
4.3	odd	2		368.3.f.a.321.3		3
5.2	odd	4		575.3.c.a.574.5		6
5.3	odd	4		575.3.c.a.574.2		6
5.4	even	2		575.3.d.b.551.1		3
8.3	odd	2		1472.3.f.b.321.1		3
8.5	even	2		1472.3.f.a.321.3		3
23.22	odd	2	CM	23.3.b.a.22.3	✓	3
69.68	even	2		207.3.d.a.91.1		3
92.91	even	2		368.3.f.a.321.3		3
115.22	even	4		575.3.c.a.574.5		6
115.68	even	4		575.3.c.a.574.2		6
115.114	odd	2		575.3.d.b.551.1		3
184.45	odd	2		1472.3.f.a.321.3		3
184.91	even	2		1472.3.f.b.321.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.3.b.a.22.3	✓	3	1.1	even	1	trivial
23.3.b.a.22.3	✓	3	23.22	odd	2	CM
207.3.d.a.91.1		3	3.2	odd	2
207.3.d.a.91.1		3	69.68	even	2
368.3.f.a.321.3		3	4.3	odd	2
368.3.f.a.321.3		3	92.91	even	2
575.3.c.a.574.2		6	5.3	odd	4
575.3.c.a.574.2		6	115.68	even	4
575.3.c.a.574.5		6	5.2	odd	4
575.3.c.a.574.5		6	115.22	even	4
575.3.d.b.551.1		3	5.4	even	2
575.3.d.b.551.1		3	115.114	odd	2
1472.3.f.a.321.3		3	8.5	even	2
1472.3.f.a.321.3		3	184.45	odd	2
1472.3.f.b.321.1		3	8.3	odd	2
1472.3.f.b.321.1		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)

\(f(q)\)	\(=\)	\(q+3.12398 q^{2} -5.79306 q^{3} +5.75927 q^{4} -18.0974 q^{6} +5.49593 q^{8} +24.5596 q^{9} +O(q^{10})\)	\(q+3.12398 q^{2} -5.79306 q^{3} +5.75927 q^{4} -18.0974 q^{6} +5.49593 q^{8} +24.5596 q^{9} -33.3638 q^{12} -2.15383 q^{13} -5.86788 q^{16} +76.7237 q^{18} -23.0000 q^{23} -31.8383 q^{24} +25.0000 q^{25} -6.72853 q^{26} -90.1376 q^{27} -13.0715 q^{29} +61.9041 q^{31} -40.3149 q^{32} +141.445 q^{36} +12.4773 q^{39} -66.2117 q^{41} -71.8516 q^{46} +50.9864 q^{47} +33.9930 q^{48} +49.0000 q^{49} +78.0996 q^{50} -12.4045 q^{52} -281.588 q^{54} -40.8352 q^{58} +26.0000 q^{59} +193.387 q^{62} -102.472 q^{64} +133.240 q^{69} -55.2940 q^{71} +134.978 q^{72} -88.0471 q^{73} -144.827 q^{75} +38.9788 q^{78} +301.136 q^{81} -206.844 q^{82} +75.7242 q^{87} -132.463 q^{92} -358.614 q^{93} +159.281 q^{94} +233.547 q^{96} +153.075 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 12 q^{4} - 33 q^{6} - 21 q^{8} + 27 q^{9}+O(q^{10}) \) 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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