LMFDB - Embedded newform 23.5.b.a.22.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,5,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, 5, names="a")

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

chi := DirichletCharacter("23.22");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 5 $
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:	$2.37750915093$
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			$2.66908$ 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)

$f(q)$	$=$	$q+1.75927 q^{2} +15.5596 q^{3} -12.9050 q^{4} +27.3735 q^{6} -50.8517 q^{8} +161.100 q^{9} +O(q^{10})$	$q+1.75927 q^{2} +15.5596 q^{3} -12.9050 q^{4} +27.3735 q^{6} -50.8517 q^{8} +161.100 q^{9} -200.796 q^{12} -333.361 q^{13} +117.018 q^{16} +283.419 q^{18} +529.000 q^{23} -791.230 q^{24} +625.000 q^{25} -586.472 q^{26} +1246.33 q^{27} -1511.14 q^{29} +1910.11 q^{31} +1019.49 q^{32} -2078.99 q^{36} -5186.95 q^{39} +1021.99 q^{41} +930.654 q^{46} -1818.39 q^{47} +1820.74 q^{48} +2401.00 q^{49} +1099.54 q^{50} +4302.01 q^{52} +2192.62 q^{54} -2658.50 q^{58} -6286.00 q^{59} +3360.41 q^{62} -78.7184 q^{64} +8231.01 q^{69} -7024.57 q^{71} -8192.22 q^{72} -2905.70 q^{73} +9724.73 q^{75} -9125.26 q^{78} +6343.17 q^{81} +1797.96 q^{82} -23512.6 q^{87} -6826.73 q^{92} +29720.5 q^{93} -3199.04 q^{94} +15862.9 q^{96} +4224.01 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 48 q^{4} + 147 q^{6} - 237 q^{8} + 243 q^{9}+O(q^{10}) $ 3 * q + 48 * q^4 + 147 * q^6 - 237 * q^8 + 243 * q^9	$ 3 q + 48 q^{4} + 147 q^{6} - 237 q^{8} + 243 q^{9} - 861 q^{12} + 768 q^{16} - 1437 q^{18} + 1587 q^{23} + 2352 q^{24} + 1875 q^{25} - 1533 q^{26} - 42 q^{27} - 3792 q^{32} + 3315 q^{36} - 8538 q^{39} - 11613 q^{48} + 7203 q^{49} + 15603 q^{52} + 11907 q^{54} + 4659 q^{58} - 18858 q^{59} + 7539 q^{62} + 6435 q^{64} - 42189 q^{72} + 16563 q^{78} + 19683 q^{81} - 34797 q^{82} - 25242 q^{87} + 25392 q^{92} + 47478 q^{93} - 51933 q^{94} + 68019 q^{96}+O(q^{100}) $ 3 * q + 48 * q^4 + 147 * q^6 - 237 * q^8 + 243 * q^9 - 861 * q^12 + 768 * q^16 - 1437 * q^18 + 1587 * q^23 + 2352 * q^24 + 1875 * q^25 - 1533 * q^26 - 42 * q^27 - 3792 * q^32 + 3315 * q^36 - 8538 * q^39 - 11613 * q^48 + 7203 * q^49 + 15603 * q^52 + 11907 * q^54 + 4659 * q^58 - 18858 * q^59 + 7539 * q^62 + 6435 * q^64 - 42189 * q^72 + 16563 * q^78 + 19683 * q^81 - 34797 * q^82 - 25242 * q^87 + 25392 * q^92 + 47478 * q^93 - 51933 * q^94 + 68019 * 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 $

$2$	1.75927	0.439818	0.219909	−	0.975520i	$-0.429424\pi$
$2$	1.75927	0.439818	0.219909	+	0.975520i	$0.429424\pi$
$3$	15.5596	1.72884	0.864421	−	0.502769i	$-0.167686\pi$
$3$	15.5596	1.72884	0.864421	+	0.502769i	$0.167686\pi$
$4$	−12.9050	−0.806560
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	27.3735	0.760375
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−50.8517	−0.794557
$9$	161.100	1.98889
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−200.796	−1.39441
$13$	−333.361	−1.97255	−0.986275	−	0.165110i	$-0.947202\pi$
$13$	−333.361	−1.97255	−0.986275	+	0.165110i	$0.947202\pi$
$14$	0	0
$15$	0	0
$16$	117.018	0.457100
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	283.419	0.874750
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	529.000	1.00000
$23$	529.000	1.00000
$24$	−791.230	−1.37366
$25$	625.000	1.00000
$26$	−586.472	−0.867563
$27$	1246.33	1.70964
$28$	0	0
$29$	−1511.14	−1.79683	−0.898416	−	0.439146i	$-0.855281\pi$
$29$	−1511.14	−1.79683	−0.898416	+	0.439146i	$0.855281\pi$
$30$	0	0
$31$	1910.11	1.98763	0.993816	−	0.111042i	$-0.0354190\pi$
$31$	1910.11	1.98763	0.993816	+	0.111042i	$0.0354190\pi$
$32$	1019.49	0.995598
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−2078.99	−1.60416
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−5186.95	−3.41023
$40$	0	0
$41$	1021.99	0.607967	0.303984	−	0.952677i	$-0.401683\pi$
$41$	1021.99	0.607967	0.303984	+	0.952677i	$0.401683\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	930.654	0.439818
$47$	−1818.39	−0.823173	−0.411587	−	0.911371i	$-0.635025\pi$
$47$	−1818.39	−0.823173	−0.411587	+	0.911371i	$0.635025\pi$
$48$	1820.74	0.790253
$49$	2401.00	1.00000
$50$	1099.54	0.439818
$51$	0	0
$52$	4302.01	1.59098
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	2192.62	0.751929
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−2658.50	−0.790278
$59$	−6286.00	−1.80580	−0.902901	−	0.429848i	$-0.858567\pi$
$59$	−6286.00	−1.80580	−0.902901	+	0.429848i	$0.858567\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	3360.41	0.874195
$63$	0	0
$64$	−78.7184	−0.0192184
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	8231.01	1.72884
$70$	0	0
$71$	−7024.57	−1.39349	−0.696744	−	0.717320i	$-0.745369\pi$
$71$	−7024.57	−1.39349	−0.696744	+	0.717320i	$0.745369\pi$
$72$	−8192.22	−1.58029
$73$	−2905.70	−0.545263	−0.272631	−	0.962119i	$-0.587894\pi$
$73$	−2905.70	−0.545263	−0.272631	+	0.962119i	$0.587894\pi$
$74$	0	0
$75$	9724.73	1.72884
$76$	0	0
$77$	0	0
$78$	−9125.26	−1.49988
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	6343.17	0.966799
$82$	1797.96	0.267395
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−23512.6	−3.10644
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	−6826.73	−0.806560
$93$	29720.5	3.43630
$94$	−3199.04	−0.362046
$95$	0	0
$96$	15862.9	1.72123
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	4224.01	0.439818
$99$	0	0
$100$	−8065.60	−0.806560
$101$	7154.00	0.701304	0.350652	−	0.936506i	$-0.385960\pi$
$101$	7154.00	0.701304	0.350652	+	0.936506i	$0.385960\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	16952.0	1.56730
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−16083.8	−1.37893
$109$	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
$116$	19501.1	1.44925
$117$	−53704.5	−3.92319
$118$	−11058.8	−0.794224
$119$	0	0
$120$	0	0
$121$	14641.0	1.00000
$122$	0	0
$123$	15901.8	1.05108
$124$	−24650.0	−1.60314
$125$	0	0
$126$	0	0
$127$	1381.16	0.0856323	0.0428162	−	0.999083i	$-0.486367\pi$
$127$	1381.16	0.0856323	0.0428162	+	0.999083i	$0.486367\pi$
$128$	−16450.4	−1.00405
$129$	0	0
$130$	0	0
$131$	30233.5	1.76176	0.880878	−	0.473343i	$-0.156953\pi$
$131$	30233.5	1.76176	0.880878	+	0.473343i	$0.156953\pi$
$132$	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			$\pi$
$138$	14480.6	0.760375
$139$	−13451.5	−0.696213	−0.348106	−	0.937455i	$-0.613175\pi$
$139$	−13451.5	−0.696213	−0.348106	+	0.937455i	$0.613175\pi$
$140$	0	0
$141$	−28293.4	−1.42314
$142$	−12358.1	−0.612880
$143$	0	0
$144$	18851.6	0.909122
$145$	0	0
$146$	−5111.92	−0.239816
$147$	37358.5	1.72884
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	17108.4	0.760375
$151$	−33527.8	−1.47045	−0.735227	−	0.677821i	$-0.762925\pi$
$151$	−33527.8	−1.47045	−0.735227	+	0.677821i	$0.762925\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	66937.5	2.75055
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	11159.3	0.425215
$163$	798.517	0.0300545	0.0150272	−	0.999887i	$-0.495217\pi$
$163$	798.517	0.0300545	0.0150272	+	0.999887i	$0.495217\pi$
$164$	−13188.8	−0.490362
$165$	0	0
$166$	0	0
$167$	2786.00	0.0998960	0.0499480	−	0.998752i	$-0.484094\pi$
$167$	2786.00	0.0998960	0.0499480	+	0.998752i	$0.484094\pi$
$168$	0	0
$169$	82568.6	2.89096
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−59374.0	−1.98383	−0.991914	−	0.126910i	$-0.959494\pi$
$173$	−59374.0	−1.98383	−0.991914	+	0.126910i	$0.959494\pi$
$174$	−41365.1	−1.36627
$175$	0	0
$176$	0	0
$177$	−97807.5	−3.12195
$178$	0	0
$179$	−62358.3	−1.94620	−0.973102	−	0.230373i	$-0.926005\pi$
$179$	−62358.3	−1.94620	−0.973102	+	0.230373i	$0.926005\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	−26900.5	−0.794557
$185$	0	0
$186$	52286.5	1.51135
$187$	0	0
$188$	23466.3	0.663939
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−1224.82	−0.0332255
$193$	48624.9	1.30540	0.652701	−	0.757616i	$-0.273636\pi$
$193$	48624.9	1.30540	0.652701	+	0.757616i	$0.273636\pi$
$194$	0	0
$195$	0	0
$196$	−30984.8	−0.806560
$197$	76918.5	1.98197	0.990987	−	0.133955i	$-0.0427678\pi$
$197$	76918.5	1.98197	0.990987	+	0.133955i	$0.0427678\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−31782.3	−0.794557
$201$	0	0
$202$	12585.8	0.308446
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	85222.0	1.98889
$208$	−39009.1	−0.901653
$209$	0	0
$210$	0	0
$211$	75794.0	1.70243	0.851216	−	0.524815i	$-0.175866\pi$
$211$	75794.0	1.70243	0.851216	+	0.524815i	$0.175866\pi$
$212$	0	0
$213$	−109299.	−2.40912
$214$	0	0
$215$	0	0
$216$	−63377.7	−1.35840
$217$	0	0
$218$	0	0
$219$	−45211.5	−0.942672
$220$	0	0
$221$	0	0
$222$	0	0
$223$	46466.0	0.934384	0.467192	−	0.884156i	$-0.345266\pi$
$223$	46466.0	0.934384	0.467192	+	0.884156i	$0.345266\pi$
$224$	0	0
$225$	100688.	1.98889
$226$	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
$232$	76843.7	1.42769
$233$	33575.0	0.618449	0.309225	−	0.950989i	$-0.399931\pi$
$233$	33575.0	0.618449	0.309225	+	0.950989i	$0.399931\pi$
$234$	−94480.8	−1.72549
$235$	0	0
$236$	81120.6	1.45649
$237$	0	0
$238$	0	0
$239$	−46728.4	−0.818061	−0.409030	−	0.912521i	$-0.634133\pi$
$239$	−46728.4	−0.818061	−0.409030	+	0.912521i	$0.634133\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	25757.5	0.439818
$243$	−2255.40	−0.0381955
$244$	0	0
$245$	0	0
$246$	27975.5	0.462283
$247$	0	0
$248$	−97132.5	−1.57929
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	2429.84	0.0376626
$255$	0	0
$256$	−27681.2	−0.422381
$257$	−115865.	−1.75422	−0.877111	−	0.480287i	$-0.840532\pi$
$257$	−115865.	−1.75422	−0.877111	+	0.480287i	$0.840532\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−243444.	−3.57370
$262$	53188.9	0.774852
$263$	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
$269$	143059.	1.97702	0.988510	−	0.151153i	$-0.0482988\pi$
$269$	143059.	1.97702	0.988510	+	0.151153i	$0.0482988\pi$
$270$	0	0
$271$	−65086.0	−0.886235	−0.443118	−	0.896463i	$-0.646128\pi$
$271$	−65086.0	−0.886235	−0.443118	+	0.896463i	$0.646128\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	−106221.	−1.39441
$277$	−38860.3	−0.506462	−0.253231	−	0.967406i	$-0.581493\pi$
$277$	−38860.3	−0.506462	−0.253231	+	0.967406i	$0.581493\pi$
$278$	−23664.9	−0.306207
$279$	307720.	3.95318
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−49775.7	−0.625920
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	90651.8	1.12393
$285$	0	0
$286$	0	0
$287$	0	0
$288$	164240.	1.98014
$289$	83521.0	1.00000
$290$	0	0
$291$	0	0
$292$	37498.0	0.439787
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	65723.8	0.760375
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−176348.	−1.97255
$300$	−125497.	−1.39441
$301$	0	0
$302$	−58984.5	−0.646732
$303$	111313.	1.21244
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−142702.	−1.51410	−0.757048	−	0.653359i	$-0.773359\pi$
$307$	−142702.	−1.51410	−0.757048	+	0.653359i	$0.773359\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	141423.	1.46217	0.731086	−	0.682286i	$-0.239014\pi$
$311$	141423.	1.46217	0.731086	+	0.682286i	$0.239014\pi$
$312$	263765.	2.70962
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−130222.	−1.29588	−0.647942	−	0.761690i	$-0.724370\pi$
$317$	−130222.	−1.29588	−0.647942	+	0.761690i	$0.724370\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−81858.3	−0.779781
$325$	−208351.	−1.97255
$326$	1404.81	0.0132185
$327$	0	0
$328$	−51970.0	−0.483065
$329$	0	0
$330$	0	0
$331$	−108956.	−0.994477	−0.497239	−	0.867614i	$-0.665653\pi$
$331$	−108956.	−0.994477	−0.497239	+	0.867614i	$0.665653\pi$
$332$	0	0
$333$	0	0
$334$	4901.33	0.0439360
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	145260.	1.27149
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−104455.	−0.872523
$347$	121586.	1.00978	0.504888	−	0.863185i	$-0.331534\pi$
$347$	121586.	1.00978	0.504888	+	0.863185i	$0.331534\pi$
$348$	303429.	2.50553
$349$	105727.	0.868033	0.434017	−	0.900905i	$-0.357096\pi$
$349$	105727.	0.868033	0.434017	+	0.900905i	$0.357096\pi$
$350$	0	0
$351$	−415476.	−3.37234
$352$	0	0
$353$	−190793.	−1.53114	−0.765568	−	0.643355i	$-0.777542\pi$
$353$	−190793.	−1.53114	−0.765568	+	0.643355i	$0.777542\pi$
$354$	−172070.	−1.37309
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−109705.	−0.855975
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	130321.	1.00000
$362$	0	0
$363$	227808.	1.72884
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	61902.3	0.457100
$369$	164643.	1.20918
$370$	0	0
$371$	0	0
$372$	−383543.	−2.77158
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	92468.2	0.654058
$377$	503754.	3.54434
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	21490.3	0.148045
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−255961.	−1.73584
$385$	0	0
$386$	85544.4	0.574139
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−122095.	−0.794557
$393$	470420.	3.04580
$394$	135320.	0.871708
$395$	0	0
$396$	0	0
$397$	289224.	1.83507	0.917536	−	0.397652i	$-0.130175\pi$
$397$	289224.	1.83507	0.917536	+	0.397652i	$0.130175\pi$
$398$	0	0
$399$	0	0
$400$	73136.0	0.457100
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	−636757.	−3.92070
$404$	−92322.1	−0.565644
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−38070.1	−0.227581	−0.113791	−	0.993505i	$-0.536299\pi$
$409$	−38070.1	−0.227581	−0.113791	+	0.993505i	$0.536299\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	149929.	0.874750
$415$	0	0
$416$	−339859.	−1.96387
$417$	−209300.	−1.20364
$418$	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			$\pi$
$422$	133342.	0.748760
$423$	−292943.	−1.63720
$424$	0	0
$425$	0	0
$426$	−192287.	−1.05957
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	145842.	0.781475
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	−79539.3	−0.414604
$439$	329977.	1.71220	0.856099	−	0.516812i	$-0.172881\pi$
$439$	329977.	1.71220	0.856099	+	0.516812i	$0.172881\pi$
$440$	0	0
$441$	386802.	1.98889
$442$	0	0
$443$	−46584.3	−0.237373	−0.118687	−	0.992932i	$-0.537868\pi$
$443$	−46584.3	−0.237373	−0.118687	+	0.992932i	$0.537868\pi$
$444$	0	0
$445$	0	0
$446$	81746.3	0.410959
$447$	0	0
$448$	0	0
$449$	−73726.0	−0.365703	−0.182851	−	0.983141i	$-0.558533\pi$
$449$	−73726.0	−0.365703	−0.182851	+	0.983141i	$0.558533\pi$
$450$	177137.	0.874750
$451$	0	0
$452$	0	0
$453$	−521679.	−2.54218
$454$	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
$461$	−285037.	−1.34122	−0.670609	−	0.741811i	$-0.733967\pi$
$461$	−285037.	−1.34122	−0.670609	+	0.741811i	$0.733967\pi$
$462$	0	0
$463$	−419134.	−1.95520	−0.977599	−	0.210474i	$-0.932499\pi$
$463$	−419134.	−1.95520	−0.977599	+	0.210474i	$0.932499\pi$
$464$	−176829.	−0.821332
$465$	0	0
$466$	59067.5	0.272005
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	693055.	3.16429
$469$	0	0
$470$	0	0
$471$	0	0
$472$	319654.	1.43481
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−82208.0	−0.359798
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−188942.	−0.806560
$485$	0	0
$486$	−3967.87	−0.0167990
$487$	13940.9	0.0587806	0.0293903	−	0.999568i	$-0.490643\pi$
$487$	13940.9	0.0587806	0.0293903	+	0.999568i	$0.490643\pi$
$488$	0	0
$489$	12424.6	0.0519594
$490$	0	0
$491$	439751.	1.82408	0.912040	−	0.410102i	$-0.134507\pi$
$491$	439751.	1.82408	0.912040	+	0.410102i	$0.134507\pi$
$492$	−205212.	−0.847758
$493$	0	0
$494$	0	0
$495$	0	0
$496$	223517.	0.908546
$497$	0	0
$498$	0	0
$499$	249877.	1.00352	0.501760	−	0.865007i	$-0.332686\pi$
$499$	249877.	1.00352	0.501760	+	0.865007i	$0.332686\pi$
$500$	0	0
$501$	43349.0	0.172704
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.28473e6	4.99800
$508$	−17823.9	−0.0690677
$509$	−26712.3	−0.103104	−0.0515520	−	0.998670i	$-0.516417\pi$
$509$	−26712.3	−0.103104	−0.0515520	+	0.998670i	$0.516417\pi$
$510$	0	0
$511$	0	0
$512$	214507.	0.818280
$513$	0	0
$514$	−203837.	−0.771538
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−923834.	−3.42972
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−428284.	−1.57178
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−390162.	−1.42096
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	279841.	1.00000
$530$	0	0
$531$	−1.01268e6	−3.59155
$532$	0	0
$533$	−340693.	−1.19925
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−970269.	−3.36468
$538$	251680.	0.869529
$539$	0	0
$540$	0	0
$541$	584645.	1.99755	0.998775	−	0.0494746i	$-0.0157547\pi$
$541$	584645.	1.99755	0.998775	+	0.0494746i	$0.0157547\pi$
$542$	−114504.	−0.389782
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	425432.	1.42186	0.710928	−	0.703265i	$-0.248275\pi$
$547$	425432.	1.42186	0.710928	+	0.703265i	$0.248275\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	−418561.	−1.37366
$553$	0	0
$554$	−68365.9	−0.222751
$555$	0	0
$556$	173592.	0.561538
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	541362.	1.73868
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	365125.	1.14785
$565$	0	0
$566$	0	0
$567$	0	0
$568$	357211.	1.10721
$569$	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
$575$	330625.	1.00000
$576$	−12681.6	−0.0382232
$577$	−657587.	−1.97516	−0.987579	−	0.157125i	$-0.949778\pi$
$577$	−657587.	−1.97516	−0.987579	+	0.157125i	$0.949778\pi$
$578$	146936.	0.439818
$579$	756583.	2.25683
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	147760.	0.433242
$585$	0	0
$586$	0	0
$587$	−402869.	−1.16920	−0.584598	−	0.811323i	$-0.698748\pi$
$587$	−402869.	−1.16920	−0.584598	+	0.811323i	$0.698748\pi$
$588$	−482111.	−1.39441
$589$	0	0
$590$	0	0
$591$	1.19682e6	3.42652
$592$	0	0
$593$	−621502.	−1.76739	−0.883697	−	0.468060i	$-0.844953\pi$
$593$	−621502.	−1.76739	−0.883697	+	0.468060i	$0.844953\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	−310244.	−0.867563
$599$	505634.	1.40923	0.704616	−	0.709589i	$-0.251119\pi$
$599$	505634.	1.40923	0.704616	+	0.709589i	$0.251119\pi$
$600$	−494519.	−1.37366
$601$	−370390.	−1.02544	−0.512720	−	0.858556i	$-0.671362\pi$
$601$	−370390.	−1.02544	−0.512720	+	0.858556i	$0.671362\pi$
$602$	0	0
$603$	0	0
$604$	432676.	1.18601
$605$	0	0
$606$	195830.	0.533254
$607$	−587902.	−1.59561	−0.797806	−	0.602914i	$-0.794006\pi$
$607$	−587902.	−1.59561	−0.797806	+	0.602914i	$0.794006\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	606180.	1.62375
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−251051.	−0.665926
$615$	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
$621$	659306.	1.70964
$622$	248801.	0.643089
$623$	0	0
$624$	−606965.	−1.55881
$625$	390625.	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
$633$	1.17932e6	2.94324
$634$	−229096.	−0.569952
$635$	0	0
$636$	0	0
$637$	−800400.	−1.97255
$638$	0	0
$639$	−1.13166e6	−2.77150
$640$	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
$647$	138441.	0.330716	0.165358	−	0.986234i	$-0.447122\pi$
$647$	138441.	0.330716	0.165358	+	0.986234i	$0.447122\pi$
$648$	−322561.	−0.768177
$649$	0	0
$650$	−366545.	−0.867563
$651$	0	0
$652$	−10304.8	−0.0242407
$653$	488360.	1.14529	0.572643	−	0.819805i	$-0.305918\pi$
$653$	488360.	1.14529	0.572643	+	0.819805i	$0.305918\pi$
$654$	0	0
$655$	0	0
$656$	119591.	0.277902
$657$	−468110.	−1.08447
$658$	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			$\pi$
$662$	−191683.	−0.437389
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−799391.	−1.79683
$668$	−35953.2	−0.0805722
$669$	722991.	1.61540
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−139041.	−0.306983	−0.153491	−	0.988150i	$-0.549052\pi$
$673$	−139041.	−0.306983	−0.153491	+	0.988150i	$0.549052\pi$
$674$	0	0
$675$	778953.	1.70964
$676$	−1.06554e6	−2.33173
$677$	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
$683$	−605677.	−1.29837	−0.649187	−	0.760629i	$-0.724891\pi$
$683$	−605677.	−1.29837	−0.649187	+	0.760629i	$0.724891\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−648046.	−1.35722	−0.678609	−	0.734500i	$-0.737417\pi$
$691$	−648046.	−1.35722	−0.678609	+	0.734500i	$0.737417\pi$
$692$	766219.	1.60008
$693$	0	0
$694$	213903.	0.444117
$695$	0	0
$696$	1.19566e6	2.46824
$697$	0	0
$698$	186003.	0.381777
$699$	522412.	1.06920
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	−730935.	−1.48322
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−335657.	−0.673421
$707$	0	0
$708$	1.26220e6	2.51804
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.01045e6	1.98763
$714$	0	0
$715$	0	0
$716$	804732.	1.56973
$717$	−727074.	−1.41430
$718$	0	0
$719$	−290878.	−0.562669	−0.281335	−	0.959610i	$-0.590777\pi$
$719$	−290878.	−0.562669	−0.281335	+	0.959610i	$0.590777\pi$
$720$	0	0
$721$	0	0
$722$	229270.	0.439818
$723$	0	0
$724$	0	0
$725$	−944459.	−1.79683
$726$	400775.	0.760375
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−548890.	−1.03283
$730$	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
$736$	539311.	0.995598
$737$	0	0
$738$	289652.	0.531819
$739$	444468.	0.813863	0.406932	−	0.913459i	$-0.366599\pi$
$739$	444468.	0.813863	0.406932	+	0.913459i	$0.366599\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	−1.51134e6	−2.73034
$745$	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			$\pi$
$752$	−212784.	−0.376273
$753$	0	0
$754$	886239.	1.55886
$755$	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
$761$	−986092.	−1.70274	−0.851370	−	0.524566i	$-0.824228\pi$
$761$	−986092.	−1.70274	−0.851370	+	0.524566i	$0.824228\pi$
$762$	37807.3	0.0651127
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.09551e6	3.56204
$768$	−430707.	−0.730229
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	−1.80280e6	−3.03277
$772$	−627503.	−1.05289
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	1.19382e6	1.98763
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−1.88337e6	−3.07193
$784$	280959.	0.457100
$785$	0	0
$786$	827597.	1.33960
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−992630.	−1.59858
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	508823.	0.807097
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	637183.	0.995598
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−1.12023e6	−1.72439
$807$	2.22594e6	3.41795
$808$	−363793.	−0.557226
$809$	−1.28765e6	−1.96743	−0.983715	−	0.179733i	$-0.942476\pi$
$809$	−1.28765e6	−1.96743	−0.983715	+	0.179733i	$0.942476\pi$
$810$	0	0
$811$	−765161.	−1.16335	−0.581676	−	0.813420i	$-0.697603\pi$
$811$	−765161.	−1.16335	−0.581676	+	0.813420i	$0.697603\pi$
$812$	0	0
$813$	−1.01271e6	−1.53216
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−66975.5	−0.100094
$819$	0	0
$820$	0	0
$821$	274994.	0.407978	0.203989	−	0.978973i	$-0.434609\pi$
$821$	274994.	0.407978	0.203989	+	0.978973i	$0.434609\pi$
$822$	0	0
$823$	−20094.2	−0.0296669	−0.0148334	−	0.999890i	$-0.504722\pi$
$823$	−20094.2	−0.0296669	−0.0148334	+	0.999890i	$0.504722\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−1.09979e6	−1.60416
$829$	1.36123e6	1.98072	0.990361	−	0.138507i	$-0.0442303\pi$
$829$	1.36123e6	1.98072	0.990361	+	0.138507i	$0.0442303\pi$
$830$	0	0
$831$	−604650.	−0.875593
$832$	26241.7	0.0379092
$833$	0	0
$834$	−368215.	−0.529383
$835$	0	0
$836$	0	0
$837$	2.38062e6	3.39813
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.57625e6	2.22860
$842$	0	0
$843$	0	0
$844$	−978119.	−1.37311
$845$	0	0
$846$	−515366.	−0.720071
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	1.41050e6	1.94310
$853$	1.12402e6	1.54481	0.772405	−	0.635130i	$-0.219053\pi$
$853$	1.12402e6	1.54481	0.772405	+	0.635130i	$0.219053\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.41187e6	−1.92235	−0.961173	−	0.275946i	$-0.911009\pi$
$857$	−1.41187e6	−1.92235	−0.961173	+	0.275946i	$0.911009\pi$
$858$	0	0
$859$	−715370.	−0.969493	−0.484746	−	0.874655i	$-0.661088\pi$
$859$	−715370.	−0.969493	−0.484746	+	0.874655i	$0.661088\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	998836.	1.34114	0.670568	−	0.741848i	$-0.266050\pi$
$863$	998836.	1.34114	0.670568	+	0.741848i	$0.266050\pi$
$864$	1.27062e6	1.70211
$865$	0	0
$866$	0	0
$867$	1.29955e6	1.72884
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	583453.	0.760322
$877$	889106.	1.15599	0.577995	−	0.816040i	$-0.303835\pi$
$877$	889106.	1.15599	0.577995	+	0.816040i	$0.303835\pi$
$878$	580518.	0.753055
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	680489.	0.874750
$883$	−679534.	−0.871545	−0.435772	−	0.900057i	$-0.643525\pi$
$883$	−679534.	−0.871545	−0.435772	+	0.900057i	$0.643525\pi$
$884$	0	0
$885$	0	0
$886$	−81954.3	−0.104401
$887$	−705371.	−0.896542	−0.448271	−	0.893898i	$-0.647960\pi$
$887$	−705371.	−0.896542	−0.448271	+	0.893898i	$0.647960\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−599642.	−0.753637
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.74390e6	−3.41023
$898$	−129704.	−0.160842
$899$	−2.88644e6	−3.57144
$900$	−1.29937e6	−1.60416
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	−917774.	−1.11810
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	1.15251e6	1.39482
$910$	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
$921$	−2.22038e6	−2.61763
$922$	−501457.	−0.589891
$923$	2.34172e6	2.74872
$924$	0	0
$925$	0	0
$926$	−737370.	−0.859931
$927$	0	0
$928$	−1.54059e6	−1.78892
$929$	1.34392e6	1.55719	0.778594	−	0.627528i	$-0.215933\pi$
$929$	1.34392e6	1.55719	0.778594	+	0.627528i	$0.215933\pi$
$930$	0	0
$931$	0	0
$932$	−433284.	−0.498817
$933$	2.20048e6	2.52786
$934$	0	0
$935$	0	0
$936$	2.73097e6	3.11720
$937$	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
$943$	540634.	0.607967
$944$	−735573.	−0.825433
$945$	0	0
$946$	0	0
$947$	−1.48102e6	−1.65143	−0.825716	−	0.564085i	$-0.809229\pi$
$947$	−1.48102e6	−1.65143	−0.825716	+	0.564085i	$0.809229\pi$
$948$	0	0
$949$	968648.	1.07556
$950$	0	0
$951$	−2.02620e6	−2.24038
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	603029.	0.659815
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2.72501e6	2.95068
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.83675e6	1.96425	0.982127	−	0.188219i	$-0.0602714\pi$
$967$	1.83675e6	1.96425	0.982127	+	0.188219i	$0.0602714\pi$
$968$	−744519.	−0.794557
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	29105.9	0.0308069
$973$	0	0
$974$	24525.9	0.0258528
$975$	−3.24185e6	−3.41023
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	21858.2	0.0228527
$979$	0	0
$980$	0	0
$981$	0	0
$982$	773641.	0.802262
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−808631.	−0.835142
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−632446.	−0.643986	−0.321993	−	0.946742i	$-0.604353\pi$
$991$	−632446.	−0.643986	−0.321993	+	0.946742i	$0.604353\pi$
$992$	1.94735e6	1.97888
$993$	−1.69531e6	−1.71929
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−250894.	−0.252406	−0.126203	−	0.992004i	$-0.540279\pi$
$997$	−250894.	−0.252406	−0.126203	+	0.992004i	$0.540279\pi$
$998$	439602.	0.441366
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.5.b.a.22.2	✓	3
3.2	odd	2		207.5.d.a.91.2		3
4.3	odd	2		368.5.f.a.321.1		3
23.22	odd	2	CM	23.5.b.a.22.2	✓	3
69.68	even	2		207.5.d.a.91.2		3
92.91	even	2		368.5.f.a.321.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.5.b.a.22.2	✓	3	1.1	even	1	trivial
23.5.b.a.22.2	✓	3	23.22	odd	2	CM
207.5.d.a.91.2		3	3.2	odd	2
207.5.d.a.91.2		3	69.68	even	2
368.5.f.a.321.1		3	4.3	odd	2
368.5.f.a.321.1		3	92.91	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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	23.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.75927 q^{2} +15.5596 q^{3} -12.9050 q^{4} +27.3735 q^{6} -50.8517 q^{8} +161.100 q^{9} +O(q^{10})\)	\(q+1.75927 q^{2} +15.5596 q^{3} -12.9050 q^{4} +27.3735 q^{6} -50.8517 q^{8} +161.100 q^{9} -200.796 q^{12} -333.361 q^{13} +117.018 q^{16} +283.419 q^{18} +529.000 q^{23} -791.230 q^{24} +625.000 q^{25} -586.472 q^{26} +1246.33 q^{27} -1511.14 q^{29} +1910.11 q^{31} +1019.49 q^{32} -2078.99 q^{36} -5186.95 q^{39} +1021.99 q^{41} +930.654 q^{46} -1818.39 q^{47} +1820.74 q^{48} +2401.00 q^{49} +1099.54 q^{50} +4302.01 q^{52} +2192.62 q^{54} -2658.50 q^{58} -6286.00 q^{59} +3360.41 q^{62} -78.7184 q^{64} +8231.01 q^{69} -7024.57 q^{71} -8192.22 q^{72} -2905.70 q^{73} +9724.73 q^{75} -9125.26 q^{78} +6343.17 q^{81} +1797.96 q^{82} -23512.6 q^{87} -6826.73 q^{92} +29720.5 q^{93} -3199.04 q^{94} +15862.9 q^{96} +4224.01 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 48 q^{4} + 147 q^{6} - 237 q^{8} + 243 q^{9}+O(q^{10}) \) 3 * q + 48 * q^4 + 147 * q^6 - 237 * q^8 + 243 * q^9	\( 3 q + 48 q^{4} + 147 q^{6} - 237 q^{8} + 243 q^{9} - 861 q^{12} + 768 q^{16} - 1437 q^{18} + 1587 q^{23} + 2352 q^{24} + 1875 q^{25} - 1533 q^{26} - 42 q^{27} - 3792 q^{32} + 3315 q^{36} - 8538 q^{39} - 11613 q^{48} + 7203 q^{49} + 15603 q^{52} + 11907 q^{54} + 4659 q^{58} - 18858 q^{59} + 7539 q^{62} + 6435 q^{64} - 42189 q^{72} + 16563 q^{78} + 19683 q^{81} - 34797 q^{82} - 25242 q^{87} + 25392 q^{92} + 47478 q^{93} - 51933 q^{94} + 68019 q^{96}+O(q^{100}) \) 3 * q + 48 * q^4 + 147 * q^6 - 237 * q^8 + 243 * q^9 - 861 * q^12 + 768 * q^16 - 1437 * q^18 + 1587 * q^23 + 2352 * q^24 + 1875 * q^25 - 1533 * q^26 - 42 * q^27 - 3792 * q^32 + 3315 * q^36 - 8538 * q^39 - 11613 * q^48 + 7203 * q^49 + 15603 * q^52 + 11907 * q^54 + 4659 * q^58 - 18858 * q^59 + 7539 * q^62 + 6435 * q^64 - 42189 * q^72 + 16563 * q^78 + 19683 * q^81 - 34797 * q^82 - 25242 * q^87 + 25392 * q^92 + 47478 * q^93 - 51933 * q^94 + 68019 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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