LMFDB - Embedded newform 95.5.d.b.94.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [95,5,Mod(94,95)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(95, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("95.94");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 95 = 5 \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	95.d (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:	$9.82014649297$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{19}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 19 $ x^2 - 19
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			94.2
Root			$4.35890$ of defining polynomial
Character	$\chi$	$=$	95.94

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.35890 q^{2} -17.4356 q^{3} +3.00000 q^{4} +25.0000 q^{5} -76.0000 q^{6} -56.6657 q^{8} +223.000 q^{9} +108.972 q^{10} +62.0000 q^{11} -52.3068 q^{12} +331.276 q^{13} -435.890 q^{15} -295.000 q^{16} +972.034 q^{18} +361.000 q^{19} +75.0000 q^{20} +270.252 q^{22} +988.000 q^{24} +625.000 q^{25} +1444.00 q^{26} -2475.85 q^{27} -1900.00 q^{30} -379.224 q^{32} -1081.01 q^{33} +669.000 q^{36} -714.859 q^{37} +1573.56 q^{38} -5776.00 q^{39} -1416.64 q^{40} +186.000 q^{44} +5575.00 q^{45} +5143.50 q^{48} +2401.00 q^{49} +2724.31 q^{50} +993.829 q^{52} +5561.96 q^{53} -10792.0 q^{54} +1550.00 q^{55} -6294.25 q^{57} -1307.67 q^{60} -7138.00 q^{61} +3067.00 q^{64} +8281.91 q^{65} -4712.00 q^{66} +6608.09 q^{67} -12636.4 q^{72} -3116.00 q^{74} -10897.2 q^{75} +1083.00 q^{76} -25177.0 q^{78} -7375.00 q^{80} +25105.0 q^{81} -3513.27 q^{88} +24300.9 q^{90} +9025.00 q^{95} +6612.00 q^{96} +10792.6 q^{97} +10465.7 q^{98} +13826.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{4} + 50 q^{5} - 152 q^{6} + 446 q^{9} + 124 q^{11} - 590 q^{16} + 722 q^{19} + 150 q^{20} + 1976 q^{24} + 1250 q^{25} + 2888 q^{26} - 3800 q^{30} + 1338 q^{36} - 11552 q^{39} + 372 q^{44} + 11150 q^{45}+ \cdots + 27652 q^{99}+O(q^{100}) $ 2 * q + 6 * q^4 + 50 * q^5 - 152 * q^6 + 446 * q^9 + 124 * q^11 - 590 * q^16 + 722 * q^19 + 150 * q^20 + 1976 * q^24 + 1250 * q^25 + 2888 * q^26 - 3800 * q^30 + 1338 * q^36 - 11552 * q^39 + 372 * q^44 + 11150 * q^45 + 4802 * q^49 - 21584 * q^54 + 3100 * q^55 - 14276 * q^61 + 6134 * q^64 - 9424 * q^66 - 6232 * q^74 + 2166 * q^76 - 14750 * q^80 + 50210 * q^81 + 18050 * q^95 + 13224 * q^96 + 27652 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/95\mathbb{Z}\right)^\times$.

$n$	$21$	$77$
$\chi(n)$	$-1$	$-1$

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 $

$2$	4.35890	1.08972	0.544862	−	0.838525i	$-0.316582\pi$
$2$	4.35890	1.08972	0.544862	+	0.838525i	$0.316582\pi$
$3$	−17.4356	−1.93729	−0.968644	−	0.248452i	$-0.920078\pi$
$3$	−17.4356	−1.93729	−0.968644	+	0.248452i	$0.920078\pi$
$4$	3.00000	0.187500
$5$	25.0000	1.00000
$5$	25.0000	1.00000
$6$	−76.0000	−2.11111
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−56.6657	−0.885401
$9$	223.000	2.75309
$10$	108.972	1.08972
$11$	62.0000	0.512397	0.256198	−	0.966624i	$-0.417530\pi$
$11$	62.0000	0.512397	0.256198	+	0.966624i	$0.417530\pi$
$12$	−52.3068	−0.363242
$13$	331.276	1.96021	0.980107	−	0.198468i	$-0.0635965\pi$
$13$	331.276	1.96021	0.980107	+	0.198468i	$0.0635965\pi$
$14$	0	0
$15$	−435.890	−1.93729
$16$	−295.000	−1.15234
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	972.034	3.00011
$19$	361.000	1.00000
$19$	361.000	1.00000
$20$	75.0000	0.187500
$21$	0	0
$22$	270.252	0.558371
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	988.000	1.71528
$25$	625.000	1.00000
$26$	1444.00	2.13609
$27$	−2475.85	−3.39623
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−1900.00	−2.11111
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−379.224	−0.370336
$33$	−1081.01	−0.992660
$34$	0	0
$35$	0	0
$36$	669.000	0.516204
$37$	−714.859	−0.522176	−0.261088	−	0.965315i	$-0.584081\pi$
$37$	−714.859	−0.522176	−0.261088	+	0.965315i	$0.584081\pi$
$38$	1573.56	1.08972
$39$	−5776.00	−3.79750
$40$	−1416.64	−0.885401
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	186.000	0.0960744
$45$	5575.00	2.75309
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	5143.50	2.23242
$49$	2401.00	1.00000
$50$	2724.31	1.08972
$51$	0	0
$52$	993.829	0.367540
$53$	5561.96	1.98005	0.990024	−	0.140899i	$-0.0449991\pi$
$53$	5561.96	1.98005	0.990024	+	0.140899i	$0.0449991\pi$
$54$	−10792.0	−3.70096
$55$	1550.00	0.512397
$56$	0	0
$57$	−6294.25	−1.93729
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−1307.67	−0.363242
$61$	−7138.00	−1.91830	−0.959151	−	0.282895i	$-0.908705\pi$
$61$	−7138.00	−1.91830	−0.959151	+	0.282895i	$0.908705\pi$
$62$	0	0
$63$	0	0
$64$	3067.00	0.748779
$65$	8281.91	1.96021
$66$	−4712.00	−1.08173
$67$	6608.09	1.47206	0.736031	−	0.676947i	$-0.236698\pi$
$67$	6608.09	1.47206	0.736031	+	0.676947i	$0.236698\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−12636.4	−2.43759
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−3116.00	−0.569028
$75$	−10897.2	−1.93729
$76$	1083.00	0.187500
$77$	0	0
$78$	−25177.0	−4.13823
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−7375.00	−1.15234
$81$	25105.0	3.82640
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	−3513.27	−0.453677
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	24300.9	3.00011
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	9025.00	1.00000
$96$	6612.00	0.717448
$97$	10792.6	1.14705	0.573527	−	0.819187i	$-0.305575\pi$
$97$	10792.6	1.14705	0.573527	+	0.819187i	$0.305575\pi$
$98$	10465.7	1.08972
$99$	13826.0	1.41067
$100$	1875.00	0.187500
$101$	−20098.0	−1.97020	−0.985100	−	0.171985i	$-0.944982\pi$
$101$	−20098.0	−1.97020	−0.985100	+	0.171985i	$0.944982\pi$
$102$	0	0
$103$	−15360.8	−1.44790	−0.723949	−	0.689853i	$-0.757675\pi$
$103$	−15360.8	−1.44790	−0.723949	+	0.689853i	$0.757675\pi$
$104$	−18772.0	−1.73558
$105$	0	0
$106$	24244.0	2.15771
$107$	14977.2	1.30816	0.654082	−	0.756423i	$-0.273055\pi$
$107$	14977.2	1.30816	0.654082	+	0.756423i	$0.273055\pi$
$108$	−7427.56	−0.636794
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	6756.29	0.558371
$111$	12464.0	1.01161
$112$	0	0
$113$	−9083.95	−0.711406	−0.355703	−	0.934599i	$-0.615759\pi$
$113$	−9083.95	−0.711406	−0.355703	+	0.934599i	$0.615759\pi$
$114$	−27436.0	−2.11111
$115$	0	0
$116$	0	0
$117$	73874.6	5.39664
$118$	0	0
$119$	0	0
$120$	24700.0	1.71528
$121$	−10797.0	−0.737450
$122$	−31113.8	−2.09042
$123$	0	0
$124$	0	0
$125$	15625.0	1.00000
$126$	0	0
$127$	−28960.5	−1.79556	−0.897778	−	0.440448i	$-0.854820\pi$
$127$	−28960.5	−1.79556	−0.897778	+	0.440448i	$0.854820\pi$
$128$	19436.3	1.18630
$129$	0	0
$130$	36100.0	2.13609
$131$	−20398.0	−1.18863	−0.594313	−	0.804234i	$-0.702576\pi$
$131$	−20398.0	−1.18863	−0.594313	+	0.804234i	$0.702576\pi$
$132$	−3243.02	−0.186124
$133$	0	0
$134$	28804.0	1.60414
$135$	−61896.4	−3.39623
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−1858.00	−0.0961648	−0.0480824	−	0.998843i	$-0.515311\pi$
$139$	−1858.00	−0.0961648	−0.0480824	+	0.998843i	$0.515311\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	20539.1	1.00441
$144$	−65785.0	−3.17250
$145$	0	0
$146$	0	0
$147$	−41862.9	−1.93729
$148$	−2144.58	−0.0979081
$149$	−7618.00	−0.343138	−0.171569	−	0.985172i	$-0.554884\pi$
$149$	−7618.00	−0.343138	−0.171569	+	0.985172i	$0.554884\pi$
$150$	−47500.0	−2.11111
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	−20456.3	−0.885401
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−17328.0	−0.712032
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	−96976.0	−3.83592
$160$	−9480.61	−0.370336
$161$	0	0
$162$	109430.	4.16972
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	−27025.2	−0.992660
$166$	0	0
$167$	12884.9	0.462007	0.231003	−	0.972953i	$-0.425799\pi$
$167$	12884.9	0.462007	0.231003	+	0.972953i	$0.425799\pi$
$168$	0	0
$169$	81183.0	2.84244
$170$	0	0
$171$	80503.0	2.75309
$172$	0	0
$173$	−51975.5	−1.73663	−0.868314	−	0.496016i	$-0.834796\pi$
$173$	−51975.5	−1.73663	−0.868314	+	0.496016i	$0.834796\pi$
$174$	0	0
$175$	0	0
$176$	−18290.0	−0.590457
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	16725.0	0.516204
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	124455.	3.71630
$184$	0	0
$185$	−17871.5	−0.522176
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	39339.1	1.08972
$191$	18242.0	0.500041	0.250021	−	0.968241i	$-0.419563\pi$
$191$	18242.0	0.500041	0.250021	+	0.968241i	$0.419563\pi$
$192$	−53475.0	−1.45060
$193$	25438.5	0.682932	0.341466	−	0.939894i	$-0.389077\pi$
$193$	25438.5	0.682932	0.341466	+	0.939894i	$0.389077\pi$
$194$	47044.0	1.24997
$195$	−144400.	−3.79750
$196$	7203.00	0.187500
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	60266.1	1.53724
$199$	24482.0	0.618217	0.309108	−	0.951027i	$-0.399969\pi$
$199$	24482.0	0.618217	0.309108	+	0.951027i	$0.399969\pi$
$200$	−35416.1	−0.885401
$201$	−115216.	−2.85181
$202$	−87605.2	−2.14697
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−66956.0	−1.57781
$207$	0	0
$208$	−97726.5	−2.25884
$209$	22382.0	0.512397
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	16685.9	0.371259
$213$	0	0
$214$	65284.0	1.42554
$215$	0	0
$216$	140296.	3.00703
$217$	0	0
$218$	0	0
$219$	0	0
$220$	4650.00	0.0960744
$221$	0	0
$222$	54329.3	1.10237
$223$	86114.4	1.73167	0.865837	−	0.500326i	$-0.166787\pi$
$223$	86114.4	1.73167	0.865837	+	0.500326i	$0.166787\pi$
$224$	0	0
$225$	139375.	2.75309
$226$	−39596.0	−0.775237
$227$	−45698.7	−0.886854	−0.443427	−	0.896311i	$-0.646237\pi$
$227$	−45698.7	−0.886854	−0.443427	+	0.896311i	$0.646237\pi$
$228$	−18882.8	−0.363242
$229$	−68098.0	−1.29856	−0.649282	−	0.760548i	$-0.724931\pi$
$229$	−68098.0	−1.29856	−0.649282	+	0.760548i	$0.724931\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	322012.	5.88085
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−104638.	−1.83187	−0.915933	−	0.401332i	$-0.868548\pi$
$239$	−104638.	−1.83187	−0.915933	+	0.401332i	$0.868548\pi$
$240$	128588.	2.23242
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−47063.0	−0.803617
$243$	−237176.	−4.01660
$244$	−21414.0	−0.359682
$245$	60025.0	1.00000
$246$	0	0
$247$	119591.	1.96021
$248$	0	0
$249$	0	0
$250$	68107.8	1.08972
$251$	−92878.0	−1.47423	−0.737115	−	0.675767i	$-0.763813\pi$
$251$	−92878.0	−1.47423	−0.737115	+	0.675767i	$0.763813\pi$
$252$	0	0
$253$	0	0
$254$	−126236.	−1.95666
$255$	0	0
$256$	35649.0	0.543961
$257$	−49883.2	−0.755246	−0.377623	−	0.925959i	$-0.623258\pi$
$257$	−49883.2	−0.755246	−0.377623	+	0.925959i	$0.623258\pi$
$258$	0	0
$259$	0	0
$260$	24845.7	0.367540
$261$	0	0
$262$	−88912.8	−1.29527
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	61256.0	0.878903
$265$	139049.	1.98005
$266$	0	0
$267$	0	0
$268$	19824.3	0.276012
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−269800.	−3.70096
$271$	145262.	1.97794	0.988971	−	0.148111i	$-0.0473193\pi$
$271$	145262.	1.97794	0.988971	+	0.148111i	$0.0473193\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	38750.0	0.512397
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−8098.83	−0.104793
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	−157356.	−1.93729
$286$	89528.0	1.09453
$287$	0	0
$288$	−84567.0	−1.01957
$289$	83521.0	1.00000
$290$	0	0
$291$	−188176.	−2.22217
$292$	0	0
$293$	3469.68	0.0404161	0.0202081	−	0.999796i	$-0.493567\pi$
$293$	3469.68	0.0404161	0.0202081	+	0.999796i	$0.493567\pi$
$294$	−182476.	−2.11111
$295$	0	0
$296$	40508.0	0.462336
$297$	−153503.	−1.74022
$298$	−33206.1	−0.373926
$299$	0	0
$300$	−32691.7	−0.363242
$301$	0	0
$302$	0	0
$303$	350421.	3.81684
$304$	−106495.	−1.15234
$305$	−178450.	−1.91830
$306$	0	0
$307$	−28960.5	−0.307277	−0.153638	−	0.988127i	$-0.549099\pi$
$307$	−28960.5	−0.307277	−0.153638	+	0.988127i	$0.549099\pi$
$308$	0	0
$309$	267824.	2.80500
$310$	0	0
$311$	62222.0	0.643314	0.321657	−	0.946856i	$-0.395760\pi$
$311$	62222.0	0.643314	0.321657	+	0.946856i	$0.395760\pi$
$312$	327301.	3.36231
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−190065.	−1.89141	−0.945703	−	0.325033i	$-0.894625\pi$
$317$	−190065.	−1.89141	−0.945703	+	0.325033i	$0.894625\pi$
$318$	−422709.	−4.18010
$319$	0	0
$320$	76675.0	0.748779
$321$	−261136.	−2.53429
$322$	0	0
$323$	0	0
$324$	75315.0	0.717450
$325$	207048.	1.96021
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	−117800.	−1.08173
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	−159414.	−1.43760
$334$	56164.0	0.503460
$335$	165202.	1.47206
$336$	0	0
$337$	−40468.0	−0.356330	−0.178165	−	0.984001i	$-0.557016\pi$
$337$	−40468.0	−0.356330	−0.178165	+	0.984001i	$0.557016\pi$
$338$	353868.	3.09748
$339$	158384.	1.37820
$340$	0	0
$341$	0	0
$342$	350904.	3.00011
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−226556.	−1.89245
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	24722.0	0.202970	0.101485	−	0.994837i	$-0.467641\pi$
$349$	24722.0	0.202970	0.101485	+	0.994837i	$0.467641\pi$
$350$	0	0
$351$	−820192.	−6.65735
$352$	−23511.9	−0.189759
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	253262.	1.96508	0.982542	−	0.186041i	$-0.0595656\pi$
$359$	253262.	1.96508	0.982542	+	0.186041i	$0.0595656\pi$
$360$	−315911.	−2.43759
$361$	130321.	1.00000
$362$	0	0
$363$	188252.	1.42865
$364$	0	0
$365$	0	0
$366$	542488.	4.04975
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	−77900.0	−0.569028
$371$	0	0
$372$	0	0
$373$	−166004.	−1.19317	−0.596584	−	0.802551i	$-0.703476\pi$
$373$	−166004.	−1.19317	−0.596584	+	0.802551i	$0.703476\pi$
$374$	0	0
$375$	−272431.	−1.93729
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	27075.0	0.187500
$381$	504944.	3.47851
$382$	79515.0	0.544907
$383$	97621.9	0.665503	0.332751	−	0.943015i	$-0.392023\pi$
$383$	97621.9	0.665503	0.332751	+	0.943015i	$0.392023\pi$
$384$	−338884.	−2.29820
$385$	0	0
$386$	110884.	0.744208
$387$	0	0
$388$	32377.9	0.215073
$389$	247922.	1.63838	0.819192	−	0.573519i	$-0.194422\pi$
$389$	247922.	1.63838	0.819192	+	0.573519i	$0.194422\pi$
$390$	−629425.	−4.13823
$391$	0	0
$392$	−136054.	−0.885401
$393$	355651.	2.30271
$394$	0	0
$395$	0	0
$396$	41478.0	0.264501
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	106715.	0.673686
$399$	0	0
$400$	−184375.	−1.15234
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	−502215.	−3.10769
$403$	0	0
$404$	−60294.0	−0.369412
$405$	627625.	3.82640
$406$	0	0
$407$	−44321.3	−0.267561
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	−46082.3	−0.271481
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−125628.	−0.725938
$417$	32395.3	0.186299
$418$	97560.9	0.558371
$419$	−141358.	−0.805179	−0.402589	−	0.915381i	$-0.631890\pi$
$419$	−141358.	−0.805179	−0.402589	+	0.915381i	$0.631890\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	−315172.	−1.75314
$425$	0	0
$426$	0	0
$427$	0	0
$428$	44931.5	0.245281
$429$	−358112.	−1.94583
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	730377.	3.91363
$433$	−364770.	−1.94555	−0.972777	−	0.231742i	$-0.925558\pi$
$433$	−364770.	−1.94555	−0.972777	+	0.231742i	$0.925558\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	−87831.8	−0.453677
$441$	535423.	2.75309
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	37392.0	0.189676
$445$	0	0
$446$	375364.	1.88705
$447$	132824.	0.664757
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	607522.	3.00011
$451$	0	0
$452$	−27251.8	−0.133389
$453$	0	0
$454$	−199196.	−0.966427
$455$	0	0
$456$	356668.	1.71528
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−296832.	−1.41508
$459$	0	0
$460$	0	0
$461$	−67438.0	−0.317324	−0.158662	−	0.987333i	$-0.550718\pi$
$461$	−67438.0	−0.317324	−0.158662	+	0.987333i	$0.550718\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	221624.	1.01187
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	225625.	1.00000
$476$	0	0
$477$	1.24032e6	5.45124
$478$	−456106.	−1.99623
$479$	−349138.	−1.52169	−0.760845	−	0.648934i	$-0.775215\pi$
$479$	−349138.	−1.52169	−0.760845	+	0.648934i	$0.775215\pi$
$480$	165300.	0.717448
$481$	−236816.	−1.02358
$482$	0	0
$483$	0	0
$484$	−32391.0	−0.138272
$485$	269816.	1.14705
$486$	−1.03383e6	−4.37699
$487$	−63483.0	−0.267670	−0.133835	−	0.991004i	$-0.542729\pi$
$487$	−63483.0	−0.267670	−0.133835	+	0.991004i	$0.542729\pi$
$488$	404480.	1.69847
$489$	0	0
$490$	261643.	1.08972
$491$	−393358.	−1.63164	−0.815821	−	0.578304i	$-0.803715\pi$
$491$	−393358.	−1.63164	−0.815821	+	0.578304i	$0.803715\pi$
$492$	0	0
$493$	0	0
$494$	521284.	2.13609
$495$	345650.	1.41067
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−461218.	−1.85227	−0.926137	−	0.377188i	$-0.876891\pi$
$499$	−461218.	−1.85227	−0.926137	+	0.377188i	$0.876891\pi$
$500$	46875.0	0.187500
$501$	−224656.	−0.895040
$502$	−404846.	−1.60651
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	−502450.	−1.97020
$506$	0	0
$507$	−1.41547e6	−5.50663
$508$	−86881.6	−0.336667
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−155591.	−0.593532
$513$	−893784.	−3.39623
$514$	−217436.	−0.823010
$515$	−384019.	−1.44790
$516$	0	0
$517$	0	0
$518$	0	0
$519$	906224.	3.36435
$520$	−469300.	−1.73558
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	−490306.	−1.79252	−0.896260	−	0.443528i	$-0.853727\pi$
$523$	−490306.	−1.79252	−0.896260	+	0.443528i	$0.853727\pi$
$524$	−61194.0	−0.222867
$525$	0	0
$526$	0	0
$527$	0	0
$528$	318897.	1.14389
$529$	279841.	1.00000
$530$	606100.	2.15771
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	374429.	1.30816
$536$	−374452.	−1.30337
$537$	0	0
$538$	0	0
$539$	148862.	0.512397
$540$	−185689.	−0.636794
$541$	472862.	1.61562	0.807811	−	0.589441i	$-0.200652\pi$
$541$	472862.	1.61562	0.807811	+	0.589441i	$0.200652\pi$
$542$	633182.	2.15541
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−130436.	−0.435935	−0.217968	−	0.975956i	$-0.569943\pi$
$547$	−130436.	−0.435935	−0.217968	+	0.975956i	$0.569943\pi$
$548$	0	0
$549$	−1.59177e6	−5.28125
$550$	168907.	0.558371
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	311600.	1.01161
$556$	−5574.00	−0.0180309
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−322925.	−1.01879	−0.509395	−	0.860533i	$-0.670131\pi$
$563$	−322925.	−1.01879	−0.509395	+	0.860533i	$0.670131\pi$
$564$	0	0
$565$	−227099.	−0.711406
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	−685900.	−2.11111
$571$	479102.	1.46945	0.734727	−	0.678363i	$-0.237310\pi$
$571$	479102.	1.46945	0.734727	+	0.678363i	$0.237310\pi$
$572$	61617.4	0.188326
$573$	−318060.	−0.968724
$574$	0	0
$575$	0	0
$576$	683941.	2.06145
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	364060.	1.08972
$579$	−443536.	−1.32304
$580$	0	0
$581$	0	0
$582$	−820240.	−2.42156
$583$	344841.	1.01457
$584$	0	0
$585$	1.84687e6	5.39664
$586$	15124.0	0.0440424
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−125589.	−0.363242
$589$	0	0
$590$	0	0
$591$	0	0
$592$	210884.	0.601727
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	−669104.	−1.89636
$595$	0	0
$596$	−22854.0	−0.0643383
$597$	−426858.	−1.19766
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	617500.	1.71528
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	1.47360e6	4.05272
$604$	0	0
$605$	−269925.	−0.737450
$606$	1.52745e6	4.15931
$607$	668812.	1.81521	0.907605	−	0.419826i	$-0.137909\pi$
$607$	668812.	1.81521	0.907605	+	0.419826i	$0.137909\pi$
$608$	−136900.	−0.370336
$609$	0	0
$610$	−777846.	−2.09042
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−126236.	−0.334847
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	1.16742e6	3.05668
$619$	766142.	1.99953	0.999765	−	0.0216731i	$-0.00689929\pi$
$619$	766142.	1.99953	0.999765	+	0.0216731i	$0.00689929\pi$
$620$	0	0
$621$	0	0
$622$	271219.	0.701035
$623$	0	0
$624$	1.70392e6	4.37603
$625$	390625.	1.00000
$626$	0	0
$627$	−390244.	−0.992660
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−504178.	−1.26627	−0.633133	−	0.774043i	$-0.718232\pi$
$631$	−504178.	−1.26627	−0.633133	+	0.774043i	$0.718232\pi$
$632$	0	0
$633$	0	0
$634$	−828476.	−2.06111
$635$	−724013.	−1.79556
$636$	−290928.	−0.719236
$637$	795394.	1.96021
$638$	0	0
$639$	0	0
$640$	485908.	1.18630
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	−1.13827e6	−2.76168
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−1.42259e6	−3.38790
$649$	0	0
$650$	902500.	2.13609
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	−509950.	−1.18863
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	−81075.5	−0.186124
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−694868.	−1.56658
$667$	0	0
$668$	38654.7	0.0866263
$669$	−1.50146e6	−3.35475
$670$	720100.	1.60414
$671$	−442556.	−0.982931
$672$	0	0
$673$	−726733.	−1.60452	−0.802259	−	0.596976i	$-0.796369\pi$
$673$	−726733.	−1.60452	−0.802259	+	0.596976i	$0.796369\pi$
$674$	−176396.	−0.388301
$675$	−1.54741e6	−3.39623
$676$	243549.	0.532958
$677$	−745564.	−1.62670	−0.813350	−	0.581775i	$-0.802358\pi$
$677$	−745564.	−1.62670	−0.813350	+	0.581775i	$0.802358\pi$
$678$	690380.	1.50186
$679$	0	0
$680$	0	0
$681$	796784.	1.71809
$682$	0	0
$683$	677181.	1.45166	0.725828	−	0.687877i	$-0.241457\pi$
$683$	677181.	1.45166	0.725828	+	0.687877i	$0.241457\pi$
$684$	241509.	0.516204
$685$	0	0
$686$	0	0
$687$	1.18733e6	2.51569
$688$	0	0
$689$	1.84254e6	3.88132
$690$	0	0
$691$	−954658.	−1.99936	−0.999682	−	0.0252304i	$-0.991968\pi$
$691$	−954658.	−1.99936	−0.999682	+	0.0252304i	$0.991968\pi$
$692$	−155927.	−0.325618
$693$	0	0
$694$	0	0
$695$	−46450.0	−0.0961648
$696$	0	0
$697$	0	0
$698$	107761.	0.221182
$699$	0	0
$700$	0	0
$701$	75422.0	0.153484	0.0767418	−	0.997051i	$-0.475548\pi$
$701$	75422.0	0.153484	0.0767418	+	0.997051i	$0.475548\pi$
$702$	−3.57513e6	−7.25468
$703$	−258064.	−0.522176
$704$	190154.	0.383672
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−964558.	−1.91883	−0.959414	−	0.282003i	$-0.909001\pi$
$709$	−964558.	−1.91883	−0.959414	+	0.282003i	$0.909001\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	513478.	1.00441
$716$	0	0
$717$	1.82443e6	3.54885
$718$	1.10394e6	2.14140
$719$	−522898.	−1.01148	−0.505742	−	0.862685i	$-0.668781\pi$
$719$	−522898.	−1.01148	−0.505742	+	0.862685i	$0.668781\pi$
$720$	−1.64462e6	−3.17250
$721$	0	0
$722$	568056.	1.08972
$723$	0	0
$724$	0	0
$725$	0	0
$726$	820572.	1.55684
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	2.10181e6	3.95492
$730$	0	0
$731$	0	0
$732$	373366.	0.696807
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	−1.04657e6	−1.93729
$736$	0	0
$737$	409702.	0.754280
$738$	0	0
$739$	873362.	1.59921	0.799605	−	0.600527i	$-0.205042\pi$
$739$	873362.	1.59921	0.799605	+	0.600527i	$0.205042\pi$
$740$	−53614.5	−0.0979081
$741$	−2.08514e6	−3.79750
$742$	0	0
$743$	−131482.	−0.238171	−0.119085	−	0.992884i	$-0.537996\pi$
$743$	−131482.	−0.238171	−0.119085	+	0.992884i	$0.537996\pi$
$744$	0	0
$745$	−190450.	−0.343138
$746$	−723596.	−1.30022
$747$	0	0
$748$	0	0
$749$	0	0
$750$	−1.18750e6	−2.11111
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	1.61938e6	2.85601
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	−511408.	−0.885401
$761$	−902578.	−1.55853	−0.779265	−	0.626694i	$-0.784408\pi$
$761$	−902578.	−1.55853	−0.779265	+	0.626694i	$0.784408\pi$
$762$	2.20100e6	3.79062
$763$	0	0
$764$	54726.0	0.0937577
$765$	0	0
$766$	425524.	0.725215
$767$	0	0
$768$	−621562.	−1.05381
$769$	714542.	1.20830	0.604150	−	0.796870i	$-0.293513\pi$
$769$	714542.	1.20830	0.604150	+	0.796870i	$0.293513\pi$
$770$	0	0
$771$	869744.	1.46313
$772$	76315.6	0.128050
$773$	−970483.	−1.62416	−0.812080	−	0.583546i	$-0.801665\pi$
$773$	−970483.	−1.62416	−0.812080	+	0.583546i	$0.801665\pi$
$774$	0	0
$775$	0	0
$776$	−611572.	−1.01560
$777$	0	0
$778$	1.08067e6	1.78539
$779$	0	0
$780$	−433200.	−0.712032
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−708295.	−1.15234
$785$	0	0
$786$	1.55025e6	2.50932
$787$	1.17096e6	1.89056	0.945282	−	0.326253i	$-0.105786\pi$
$787$	1.17096e6	1.89056	0.945282	+	0.326253i	$0.105786\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−783460.	−1.24901
$793$	−2.36465e6	−3.76028
$794$	0	0
$795$	−2.42440e6	−3.83592
$796$	73446.0	0.115916
$797$	−1.06359e6	−1.67439	−0.837196	−	0.546903i	$-0.815807\pi$
$797$	−1.06359e6	−1.67439	−0.837196	+	0.546903i	$0.815807\pi$
$798$	0	0
$799$	0	0
$800$	−237015.	−0.370336
$801$	0	0
$802$	0	0
$803$	0	0
$804$	−345648.	−0.534714
$805$	0	0
$806$	0	0
$807$	0	0
$808$	1.13887e6	1.74442
$809$	−59038.0	−0.0902058	−0.0451029	−	0.998982i	$-0.514362\pi$
$809$	−59038.0	−0.0902058	−0.0451029	+	0.998982i	$0.514362\pi$
$810$	2.73575e6	4.16972
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	−2.53273e6	−3.83184
$814$	−193192.	−0.291568
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.33320e6	−1.97792	−0.988959	−	0.148189i	$-0.952656\pi$
$821$	−1.33320e6	−1.97792	−0.988959	+	0.148189i	$0.952656\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	870428.	1.28197
$825$	−675629.	−0.992660
$826$	0	0
$827$	1.04124e6	1.52243	0.761217	−	0.648498i	$-0.224602\pi$
$827$	1.04124e6	1.52243	0.761217	+	0.648498i	$0.224602\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	1.01602e6	1.46777
$833$	0	0
$834$	141208.	0.203015
$835$	322123.	0.462007
$836$	67146.0	0.0960744
$837$	0	0
$838$	−616165.	−0.877423
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	707281.	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.02958e6	2.84244
$846$	0	0
$847$	0	0
$848$	−1.64078e6	−2.28170
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	2.01257e6	2.75309
$856$	−848692.	−1.15825
$857$	−949560.	−1.29289	−0.646444	−	0.762962i	$-0.723745\pi$
$857$	−949560.	−1.29289	−0.646444	+	0.762962i	$0.723745\pi$
$858$	−1.56097e6	−2.12042
$859$	1.42104e6	1.92584	0.962921	−	0.269784i	$-0.0869523\pi$
$859$	1.42104e6	1.92584	0.962921	+	0.269784i	$0.0869523\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	372756.	0.500498	0.250249	−	0.968181i	$-0.419487\pi$
$863$	372756.	0.500498	0.250249	+	0.968181i	$0.419487\pi$
$864$	938904.	1.25775
$865$	−1.29939e6	−1.73663
$866$	−1.59000e6	−2.12012
$867$	−1.45624e6	−1.93729
$868$	0	0
$869$	0	0
$870$	0	0
$871$	2.18910e6	2.88556
$872$	0	0
$873$	2.40676e6	3.15794
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−78128.9	−0.101581	−0.0507905	−	0.998709i	$-0.516174\pi$
$877$	−78128.9	−0.101581	−0.0507905	+	0.998709i	$0.516174\pi$
$878$	0	0
$879$	−60496.0	−0.0782977
$880$	−457250.	−0.590457
$881$	−1.26018e6	−1.62360	−0.811802	−	0.583933i	$-0.801513\pi$
$881$	−1.26018e6	−1.62360	−0.811802	+	0.583933i	$0.801513\pi$
$882$	2.33385e6	3.00011
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.43458e6	1.82339	0.911693	−	0.410872i	$-0.134776\pi$
$887$	1.43458e6	1.82339	0.911693	+	0.410872i	$0.134776\pi$
$888$	−706281.	−0.895677
$889$	0	0
$890$	0	0
$891$	1.55651e6	1.96063
$892$	258343.	0.324689
$893$	0	0
$894$	578968.	0.724402
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	418125.	0.516204
$901$	0	0
$902$	0	0
$903$	0	0
$904$	514748.	0.629880
$905$	0	0
$906$	0	0
$907$	−1.50192e6	−1.82571	−0.912856	−	0.408282i	$-0.866128\pi$
$907$	−1.50192e6	−1.82571	−0.912856	+	0.408282i	$0.866128\pi$
$908$	−137096.	−0.166285
$909$	−4.48185e6	−5.42413
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	1.85680e6	2.23242
$913$	0	0
$914$	0	0
$915$	3.11138e6	3.71630
$916$	−204294.	−0.243481
$917$	0	0
$918$	0	0
$919$	813602.	0.963343	0.481672	−	0.876352i	$-0.340030\pi$
$919$	813602.	0.963343	0.481672	+	0.876352i	$0.340030\pi$
$920$	0	0
$921$	504944.	0.595284
$922$	−293955.	−0.345796
$923$	0	0
$924$	0	0
$925$	−446787.	−0.522176
$926$	0	0
$927$	−3.42545e6	−3.98619
$928$	0	0
$929$	−955198.	−1.10678	−0.553391	−	0.832922i	$-0.686666\pi$
$929$	−955198.	−1.10678	−0.553391	+	0.832922i	$0.686666\pi$
$930$	0	0
$931$	866761.	1.00000
$932$	0	0
$933$	−1.08488e6	−1.24629
$934$	0	0
$935$	0	0
$936$	−4.18616e6	−4.77819
$937$	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
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	983477.	1.08972
$951$	3.31390e6	3.66420
$952$	0	0
$953$	667766.	0.735256	0.367628	−	0.929973i	$-0.380170\pi$
$953$	667766.	0.735256	0.367628	+	0.929973i	$0.380170\pi$
$954$	5.40641e6	5.94035
$955$	456050.	0.500041
$956$	−313914.	−0.343475
$957$	0	0
$958$	−1.52186e6	−1.65822
$959$	0	0
$960$	−1.33687e6	−1.45060
$961$	923521.	1.00000
$962$	−1.03226e6	−1.11542
$963$	3.33991e6	3.60149
$964$	0	0
$965$	635963.	0.682932
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	611819.	0.652939
$969$	0	0
$970$	1.17610e6	1.24997
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−711529.	−0.753113
$973$	0	0
$974$	−276716.	−0.291687
$975$	−3.61000e6	−3.79750
$976$	2.10571e6	2.21054
$977$	1.67938e6	1.75938	0.879690	−	0.475548i	$-0.157750\pi$
$977$	1.67938e6	1.75938	0.879690	+	0.475548i	$0.157750\pi$
$978$	0	0
$979$	0	0
$980$	180075.	0.187500
$981$	0	0
$982$	−1.71461e6	−1.77804
$983$	−23729.8	−0.0245577	−0.0122789	−	0.999925i	$-0.503909\pi$
$983$	−23729.8	−0.0245577	−0.0122789	+	0.999925i	$0.503909\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	358772.	0.367540
$989$	0	0
$990$	1.50665e6	1.53724
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	612050.	0.618217
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−2.01040e6	−2.01847
$999$	1.76989e6	1.77343

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	95.5.d.b.94.2	yes	2
5.4	even	2	inner	95.5.d.b.94.1	✓	2
19.18	odd	2	inner	95.5.d.b.94.1	✓	2
95.94	odd	2	CM	95.5.d.b.94.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.5.d.b.94.1	✓	2	5.4	even	2	inner
95.5.d.b.94.1	✓	2	19.18	odd	2	inner
95.5.d.b.94.2	yes	2	1.1	even	1	trivial
95.5.d.b.94.2	yes	2	95.94	odd	2	CM

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 95 = 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	95.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.35890 q^{2} -17.4356 q^{3} +3.00000 q^{4} +25.0000 q^{5} -76.0000 q^{6} -56.6657 q^{8} +223.000 q^{9} +108.972 q^{10} +62.0000 q^{11} -52.3068 q^{12} +331.276 q^{13} -435.890 q^{15} -295.000 q^{16} +972.034 q^{18} +361.000 q^{19} +75.0000 q^{20} +270.252 q^{22} +988.000 q^{24} +625.000 q^{25} +1444.00 q^{26} -2475.85 q^{27} -1900.00 q^{30} -379.224 q^{32} -1081.01 q^{33} +669.000 q^{36} -714.859 q^{37} +1573.56 q^{38} -5776.00 q^{39} -1416.64 q^{40} +186.000 q^{44} +5575.00 q^{45} +5143.50 q^{48} +2401.00 q^{49} +2724.31 q^{50} +993.829 q^{52} +5561.96 q^{53} -10792.0 q^{54} +1550.00 q^{55} -6294.25 q^{57} -1307.67 q^{60} -7138.00 q^{61} +3067.00 q^{64} +8281.91 q^{65} -4712.00 q^{66} +6608.09 q^{67} -12636.4 q^{72} -3116.00 q^{74} -10897.2 q^{75} +1083.00 q^{76} -25177.0 q^{78} -7375.00 q^{80} +25105.0 q^{81} -3513.27 q^{88} +24300.9 q^{90} +9025.00 q^{95} +6612.00 q^{96} +10792.6 q^{97} +10465.7 q^{98} +13826.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{4} + 50 q^{5} - 152 q^{6} + 446 q^{9} + 124 q^{11} - 590 q^{16} + 722 q^{19} + 150 q^{20} + 1976 q^{24} + 1250 q^{25} + 2888 q^{26} - 3800 q^{30} + 1338 q^{36} - 11552 q^{39} + 372 q^{44} + 11150 q^{45}+ \cdots + 27652 q^{99}+O(q^{100}) \) 2 * q + 6 * q^4 + 50 * q^5 - 152 * q^6 + 446 * q^9 + 124 * q^11 - 590 * q^16 + 722 * q^19 + 150 * q^20 + 1976 * q^24 + 1250 * q^25 + 2888 * q^26 - 3800 * q^30 + 1338 * q^36 - 11552 * q^39 + 372 * q^44 + 11150 * q^45 + 4802 * q^49 - 21584 * q^54 + 3100 * q^55 - 14276 * q^61 + 6134 * q^64 - 9424 * q^66 - 6232 * q^74 + 2166 * q^76 - 14750 * q^80 + 50210 * q^81 + 18050 * q^95 + 13224 * q^96 + 27652 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more