LMFDB - Embedded newform 95.5.d.c.94.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [95,5,Mod(94,95)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,0,58] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$9.82014649297$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			94.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q-6.70820 q^{2} -4.47214 q^{3} +29.0000 q^{4} +25.0000 q^{5} +30.0000 q^{6} -87.2067 q^{8} -61.0000 q^{9} -167.705 q^{10} -62.0000 q^{11} -129.692 q^{12} +67.0820 q^{13} -111.803 q^{15} +121.000 q^{16} +409.200 q^{18} +361.000 q^{19} +725.000 q^{20} +415.909 q^{22} +390.000 q^{24} +625.000 q^{25} -450.000 q^{26} +635.043 q^{27} +750.000 q^{30} +583.614 q^{32} +277.272 q^{33} -1769.00 q^{36} +2643.03 q^{37} -2421.66 q^{38} -300.000 q^{39} -2180.17 q^{40} -1798.00 q^{44} -1525.00 q^{45} -541.128 q^{48} +2401.00 q^{49} -4192.63 q^{50} +1945.38 q^{52} -791.568 q^{53} -4260.00 q^{54} -1550.00 q^{55} -1614.44 q^{57} -3242.30 q^{60} +7138.00 q^{61} -5851.00 q^{64} +1677.05 q^{65} -1860.00 q^{66} +6077.63 q^{67} +5319.61 q^{72} -17730.0 q^{74} -2795.08 q^{75} +10469.0 q^{76} +2012.46 q^{78} +3025.00 q^{80} +2101.00 q^{81} +5406.81 q^{88} +10230.0 q^{90} +9025.00 q^{95} -2610.00 q^{96} +15415.5 q^{97} -16106.4 q^{98} +3782.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 58 q^{4} + 50 q^{5} + 60 q^{6} - 122 q^{9} - 124 q^{11} + 242 q^{16} + 722 q^{19} + 1450 q^{20} + 780 q^{24} + 1250 q^{25} - 900 q^{26} + 1500 q^{30} - 3538 q^{36} - 600 q^{39} - 3596 q^{44} - 3050 q^{45}+ \cdots + 7564 q^{99}+O(q^{100}) $ 2 * q + 58 * q^4 + 50 * q^5 + 60 * q^6 - 122 * q^9 - 124 * q^11 + 242 * q^16 + 722 * q^19 + 1450 * q^20 + 780 * q^24 + 1250 * q^25 - 900 * q^26 + 1500 * q^30 - 3538 * q^36 - 600 * q^39 - 3596 * q^44 - 3050 * q^45 + 4802 * q^49 - 8520 * q^54 - 3100 * q^55 + 14276 * q^61 - 11702 * q^64 - 3720 * q^66 - 35460 * q^74 + 20938 * q^76 + 6050 * q^80 + 4202 * q^81 + 18050 * q^95 - 5220 * q^96 + 7564 * 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 $

$2$	−6.70820	−1.67705	−0.838525	−	0.544862i	$-0.816582\pi$
$2$	−6.70820	−1.67705	−0.838525	+	0.544862i	$0.816582\pi$
$3$	−4.47214	−0.496904	−0.248452	−	0.968644i	$-0.579922\pi$
$3$	−4.47214	−0.496904	−0.248452	+	0.968644i	$0.579922\pi$
$4$	29.0000	1.81250
$5$	25.0000	1.00000
$5$	25.0000	1.00000
$6$	30.0000	0.833333
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−87.2067	−1.36260
$9$	−61.0000	−0.753086
$10$	−167.705	−1.67705
$11$	−62.0000	−0.512397	−0.256198	−	0.966624i	$-0.582470\pi$
$11$	−62.0000	−0.512397	−0.256198	+	0.966624i	$0.582470\pi$
$12$	−129.692	−0.900638
$13$	67.0820	0.396935	0.198468	−	0.980107i	$-0.436404\pi$
$13$	67.0820	0.396935	0.198468	+	0.980107i	$0.436404\pi$
$14$	0	0
$15$	−111.803	−0.496904
$16$	121.000	0.472656
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	409.200	1.26296
$19$	361.000	1.00000
$19$	361.000	1.00000
$20$	725.000	1.81250
$21$	0	0
$22$	415.909	0.859315
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	390.000	0.677083
$25$	625.000	1.00000
$26$	−450.000	−0.665680
$27$	635.043	0.871116
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	750.000	0.833333
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	583.614	0.569935
$33$	277.272	0.254612
$34$	0	0
$35$	0	0
$36$	−1769.00	−1.36497
$37$	2643.03	1.93063	0.965315	−	0.261088i	$-0.0840813\pi$
$37$	2643.03	1.93063	0.965315	+	0.261088i	$0.0840813\pi$
$38$	−2421.66	−1.67705
$39$	−300.000	−0.197239
$40$	−2180.17	−1.36260
$41$	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			$\pi$
$44$	−1798.00	−0.928719
$45$	−1525.00	−0.753086
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−541.128	−0.234865
$49$	2401.00	1.00000
$50$	−4192.63	−1.67705
$51$	0	0
$52$	1945.38	0.719445
$53$	−791.568	−0.281797	−0.140899	−	0.990024i	$-0.544999\pi$
$53$	−791.568	−0.281797	−0.140899	+	0.990024i	$0.544999\pi$
$54$	−4260.00	−1.46091
$55$	−1550.00	−0.512397
$56$	0	0
$57$	−1614.44	−0.496904
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−3242.30	−0.900638
$61$	7138.00	1.91830	0.959151	−	0.282895i	$-0.0912949\pi$
$61$	7138.00	1.91830	0.959151	+	0.282895i	$0.0912949\pi$
$62$	0	0
$63$	0	0
$64$	−5851.00	−1.42847
$65$	1677.05	0.396935
$66$	−1860.00	−0.426997
$67$	6077.63	1.35389	0.676947	−	0.736031i	$-0.263302\pi$
$67$	6077.63	1.35389	0.676947	+	0.736031i	$0.263302\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	5319.61	1.02616
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−17730.0	−3.23776
$75$	−2795.08	−0.496904
$76$	10469.0	1.81250
$77$	0	0
$78$	2012.46	0.330779
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	3025.00	0.472656
$81$	2101.00	0.320226
$82$	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
$88$	5406.81	0.698194
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	10230.0	1.26296
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	9025.00	1.00000
$96$	−2610.00	−0.283203
$97$	15415.5	1.63837	0.819187	−	0.573527i	$-0.194425\pi$
$97$	15415.5	1.63837	0.819187	+	0.573527i	$0.194425\pi$
$98$	−16106.4	−1.67705
$99$	3782.00	0.385879
$100$	18125.0	1.81250
$101$	20098.0	1.97020	0.985100	−	0.171985i	$-0.0550182\pi$
$101$	20098.0	1.97020	0.985100	+	0.171985i	$0.0550182\pi$
$102$	0	0
$103$	−14637.3	−1.37971	−0.689853	−	0.723949i	$-0.742325\pi$
$103$	−14637.3	−1.37971	−0.689853	+	0.723949i	$0.742325\pi$
$104$	−5850.00	−0.540865
$105$	0	0
$106$	5310.00	0.472588
$107$	−17320.6	−1.51285	−0.756423	−	0.654082i	$-0.773055\pi$
$107$	−17320.6	−1.51285	−0.756423	+	0.654082i	$0.773055\pi$
$108$	18416.3	1.57890
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	10397.7	0.859315
$111$	−11820.0	−0.959338
$112$	0	0
$113$	−23867.8	−1.86920	−0.934599	−	0.355703i	$-0.884241\pi$
$113$	−23867.8	−1.86920	−0.934599	+	0.355703i	$0.884241\pi$
$114$	10830.0	0.833333
$115$	0	0
$116$	0	0
$117$	−4092.00	−0.298926
$118$	0	0
$119$	0	0
$120$	9750.00	0.677083
$121$	−10797.0	−0.737450
$122$	−47883.2	−3.21709
$123$	0	0
$124$	0	0
$125$	15625.0	1.00000
$126$	0	0
$127$	−14208.0	−0.880896	−0.440448	−	0.897778i	$-0.645180\pi$
$127$	−14208.0	−0.880896	−0.440448	+	0.897778i	$0.645180\pi$
$128$	29911.9	1.82568
$129$	0	0
$130$	−11250.0	−0.665680
$131$	−20398.0	−1.18863	−0.594313	−	0.804234i	$-0.702576\pi$
$131$	−20398.0	−1.18863	−0.594313	+	0.804234i	$0.702576\pi$
$132$	8040.90	0.461484
$133$	0	0
$134$	−40770.0	−2.27055
$135$	15876.1	0.871116
$136$	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	$-0.484689\pi$
$139$	1858.00	0.0961648	0.0480824	+	0.998843i	$0.484689\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4159.09	−0.203388
$144$	−7381.00	−0.355951
$145$	0	0
$146$	0	0
$147$	−10737.6	−0.496904
$148$	76647.9	3.49927
$149$	7618.00	0.343138	0.171569	−	0.985172i	$-0.445116\pi$
$149$	7618.00	0.343138	0.171569	+	0.985172i	$0.445116\pi$
$150$	18750.0	0.833333
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	−31481.6	−1.36260
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−8700.00	−0.357495
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	3540.00	0.140026
$160$	14590.3	0.569935
$161$	0	0
$162$	−14093.9	−0.537035
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	6931.81	0.254612
$166$	0	0
$167$	54269.4	1.94591	0.972953	−	0.231003i	$-0.0742008\pi$
$167$	54269.4	1.94591	0.972953	+	0.231003i	$0.0742008\pi$
$168$	0	0
$169$	−24061.0	−0.842442
$170$	0	0
$171$	−22021.0	−0.753086
$172$	0	0
$173$	29690.5	0.992031	0.496016	−	0.868314i	$-0.334796\pi$
$173$	29690.5	0.992031	0.496016	+	0.868314i	$0.334796\pi$
$174$	0	0
$175$	0	0
$176$	−7502.00	−0.242188
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−44225.0	−1.36497
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	−31922.1	−0.953212
$184$	0	0
$185$	66075.8	1.93063
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	−60541.5	−1.67705
$191$	18242.0	0.500041	0.250021	−	0.968241i	$-0.419563\pi$
$191$	18242.0	0.500041	0.250021	+	0.968241i	$0.419563\pi$
$192$	26166.5	0.709811
$193$	−70020.2	−1.87979	−0.939894	−	0.341466i	$-0.889077\pi$
$193$	−70020.2	−1.87979	−0.939894	+	0.341466i	$0.889077\pi$
$194$	−103410.	−2.74764
$195$	−7500.00	−0.197239
$196$	69629.0	1.81250
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−25370.4	−0.647139
$199$	24482.0	0.618217	0.309108	−	0.951027i	$-0.399969\pi$
$199$	24482.0	0.618217	0.309108	+	0.951027i	$0.399969\pi$
$200$	−54504.2	−1.36260
$201$	−27180.0	−0.672756
$202$	−134821.	−3.30412
$203$	0	0
$204$	0	0
$205$	0	0
$206$	98190.0	2.31384
$207$	0	0
$208$	8116.93	0.187614
$209$	−22382.0	−0.512397
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−22955.5	−0.510757
$213$	0	0
$214$	116190.	2.53712
$215$	0	0
$216$	−55380.0	−1.18699
$217$	0	0
$218$	0	0
$219$	0	0
$220$	−44950.0	−0.928719
$221$	0	0
$222$	79291.0	1.60886
$223$	49761.5	1.00065	0.500326	−	0.865837i	$-0.333213\pi$
$223$	49761.5	1.00065	0.500326	+	0.865837i	$0.333213\pi$
$224$	0	0
$225$	−38125.0	−0.753086
$226$	160110.	3.13474
$227$	92372.0	1.79262	0.896311	−	0.443427i	$-0.146237\pi$
$227$	92372.0	1.79262	0.896311	+	0.443427i	$0.146237\pi$
$228$	−46818.8	−0.900638
$229$	68098.0	1.29856	0.649282	−	0.760548i	$-0.275069\pi$
$229$	68098.0	1.29856	0.649282	+	0.760548i	$0.275069\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	27450.0	0.501315
$235$	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	$0.868548\pi$
$240$	−13528.2	−0.234865
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	72428.5	1.23674
$243$	−60834.5	−1.03024
$244$	207002.	3.47692
$245$	60025.0	1.00000
$246$	0	0
$247$	24216.6	0.396935
$248$	0	0
$249$	0	0
$250$	−104816.	−1.67705
$251$	−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
$254$	95310.0	1.47731
$255$	0	0
$256$	−107039.	−1.63329
$257$	122317.	1.85192	0.925959	−	0.377623i	$-0.123258\pi$
$257$	122317.	1.85192	0.925959	+	0.377623i	$0.123258\pi$
$258$	0	0
$259$	0	0
$260$	48634.5	0.719445
$261$	0	0
$262$	136834.	1.99339
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−24180.0	−0.346935
$265$	−19789.2	−0.281797
$266$	0	0
$267$	0	0
$268$	176251.	2.45393
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−106500.	−1.46091
$271$	−145262.	−1.97794	−0.988971	−	0.148111i	$-0.952681\pi$
$271$	−145262.	−1.97794	−0.988971	+	0.148111i	$0.952681\pi$
$272$	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			$\pi$
$278$	−12463.8	−0.161273
$279$	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
$285$	−40361.0	−0.496904
$286$	27900.0	0.341092
$287$	0	0
$288$	−35600.4	−0.429211
$289$	83521.0	1.00000
$290$	0	0
$291$	−68940.0	−0.814114
$292$	0	0
$293$	−171663.	−1.99959	−0.999796	−	0.0202081i	$-0.993567\pi$
$293$	−171663.	−1.99959	−0.999796	+	0.0202081i	$0.993567\pi$
$294$	72030.0	0.833333
$295$	0	0
$296$	−230490.	−2.63068
$297$	−39372.7	−0.446357
$298$	−51103.1	−0.575459
$299$	0	0
$300$	−81057.5	−0.900638
$301$	0	0
$302$	0	0
$303$	−89881.0	−0.979000
$304$	43681.0	0.472656
$305$	178450.	1.91830
$306$	0	0
$307$	−186260.	−1.97625	−0.988127	−	0.153638i	$-0.950901\pi$
$307$	−186260.	−1.97625	−0.988127	+	0.153638i	$0.950901\pi$
$308$	0	0
$309$	65460.0	0.685581
$310$	0	0
$311$	−62222.0	−0.643314	−0.321657	−	0.946856i	$-0.604240\pi$
$311$	−62222.0	−0.643314	−0.321657	+	0.946856i	$0.604240\pi$
$312$	26162.0	0.268758
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	65324.5	0.650066	0.325033	−	0.945703i	$-0.394625\pi$
$317$	65324.5	0.650066	0.325033	+	0.945703i	$0.394625\pi$
$318$	−23747.0	−0.234831
$319$	0	0
$320$	−146275.	−1.42847
$321$	77460.0	0.751740
$322$	0	0
$323$	0	0
$324$	60929.0	0.580409
$325$	41926.3	0.396935
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	−46500.0	−0.426997
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	−161225.	−1.45393
$334$	−364050.	−3.26338
$335$	151941.	1.35389
$336$	0	0
$337$	−223504.	−1.96800	−0.984001	−	0.178165i	$-0.942984\pi$
$337$	−223504.	−1.96800	−0.984001	+	0.178165i	$0.942984\pi$
$338$	161406.	1.41282
$339$	106740.	0.928812
$340$	0	0
$341$	0	0
$342$	147721.	1.26296
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−199170.	−1.66369
$347$	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
$351$	42600.0	0.345776
$352$	−36184.1	−0.292033
$353$	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	$-0.940434\pi$
$359$	−253262.	−1.96508	−0.982542	+	0.186041i	$0.940434\pi$
$360$	132990.	1.02616
$361$	130321.	1.00000
$362$	0	0
$363$	48285.7	0.366442
$364$	0	0
$365$	0	0
$366$	214140.	1.59858
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	−443250.	−3.23776
$371$	0	0
$372$	0	0
$373$	223316.	1.60510	0.802551	−	0.596584i	$-0.203476\pi$
$373$	223316.	1.60510	0.802551	+	0.596584i	$0.203476\pi$
$374$	0	0
$375$	−69877.1	−0.496904
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	261725.	1.81250
$381$	63540.0	0.437721
$382$	−122371.	−0.838594
$383$	276660.	1.88603	0.943015	−	0.332751i	$-0.107977\pi$
$383$	276660.	1.88603	0.943015	+	0.332751i	$0.107977\pi$
$384$	−133770.	−0.907186
$385$	0	0
$386$	469710.	3.15250
$387$	0	0
$388$	447048.	2.96955
$389$	247922.	1.63838	0.819192	−	0.573519i	$-0.194422\pi$
$389$	247922.	1.63838	0.819192	+	0.573519i	$0.194422\pi$
$390$	50311.5	0.330779
$391$	0	0
$392$	−209383.	−1.36260
$393$	91222.6	0.590633
$394$	0	0
$395$	0	0
$396$	109678.	0.699406
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	−164230.	−1.03678
$399$	0	0
$400$	75625.0	0.472656
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	182329.	1.12825
$403$	0	0
$404$	582842.	3.57099
$405$	52525.0	0.320226
$406$	0	0
$407$	−163868.	−0.989248
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	−424482.	−2.50072
$413$	0	0
$414$	0	0
$415$	0	0
$416$	39150.0	0.226227
$417$	−8309.23	−0.0477847
$418$	150143.	0.859315
$419$	−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
$424$	69030.0	0.383978
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−502297.	−2.74203
$429$	18600.0	0.101064
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	76840.2	0.411738
$433$	86898.1	0.463484	0.231742	−	0.972777i	$-0.425558\pi$
$433$	86898.1	0.463484	0.231742	+	0.972777i	$0.425558\pi$
$434$	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			$\pi$
$440$	135170.	0.698194
$441$	−146461.	−0.753086
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	−342780.	−1.73880
$445$	0	0
$446$	−333810.	−1.67815
$447$	−34068.7	−0.170506
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	255750.	1.26296
$451$	0	0
$452$	−692166.	−3.38792
$453$	0	0
$454$	−619650.	−3.00632
$455$	0	0
$456$	140790.	0.677083
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−456815.	−2.17776
$459$	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			$\pi$
$468$	−118668.	−0.541804
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	225625.	1.00000
$476$	0	0
$477$	48285.7	0.212218
$478$	701933.	3.07213
$479$	349138.	1.52169	0.760845	−	0.648934i	$-0.224785\pi$
$479$	349138.	1.52169	0.760845	+	0.648934i	$0.224785\pi$
$480$	−65250.0	−0.283203
$481$	177300.	0.766335
$482$	0	0
$483$	0	0
$484$	−313113.	−1.33663
$485$	385386.	1.63837
$486$	408090.	1.72776
$487$	470071.	1.98201	0.991004	−	0.133835i	$-0.0427292\pi$
$487$	470071.	1.98201	0.991004	+	0.133835i	$0.0427292\pi$
$488$	−622481.	−2.61389
$489$	0	0
$490$	−402660.	−1.67705
$491$	−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
$494$	−162450.	−0.665680
$495$	94550.0	0.385879
$496$	0	0
$497$	0	0
$498$	0	0
$499$	461218.	1.85227	0.926137	−	0.377188i	$-0.123109\pi$
$499$	461218.	1.85227	0.926137	+	0.377188i	$0.123109\pi$
$500$	453125.	1.81250
$501$	−242700.	−0.966928
$502$	623045.	2.47236
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	502450.	1.97020
$506$	0	0
$507$	107604.	0.418613
$508$	−412031.	−1.59662
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	239449.	0.913427
$513$	229251.	0.871116
$514$	−820530.	−3.10576
$515$	−365933.	−1.37971
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−132780.	−0.492944
$520$	−146250.	−0.540865
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	242636.	0.887057	0.443528	−	0.896260i	$-0.353727\pi$
$523$	242636.	0.887057	0.443528	+	0.896260i	$0.353727\pi$
$524$	−591542.	−2.15438
$525$	0	0
$526$	0	0
$527$	0	0
$528$	33550.0	0.120344
$529$	279841.	1.00000
$530$	132750.	0.472588
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−433015.	−1.51285
$536$	−530010.	−1.84482
$537$	0	0
$538$	0	0
$539$	−148862.	−0.512397
$540$	460406.	1.57890
$541$	−472862.	−1.61562	−0.807811	−	0.589441i	$-0.799348\pi$
$541$	−472862.	−1.61562	−0.807811	+	0.589441i	$0.799348\pi$
$542$	974447.	3.31711
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−584030.	−1.95191	−0.975956	−	0.217968i	$-0.930057\pi$
$547$	−584030.	−1.95191	−0.975956	+	0.217968i	$0.930057\pi$
$548$	0	0
$549$	−435418.	−1.44465
$550$	259943.	0.859315
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−295500.	−0.959338
$556$	53882.0	0.174299
$557$	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
$563$	545525.	1.72107	0.860533	−	0.509395i	$-0.170131\pi$
$563$	545525.	1.72107	0.860533	+	0.509395i	$0.170131\pi$
$564$	0	0
$565$	−596695.	−1.86920
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	270750.	0.833333
$571$	−479102.	−1.46945	−0.734727	−	0.678363i	$-0.762690\pi$
$571$	−479102.	−1.46945	−0.734727	+	0.678363i	$0.762690\pi$
$572$	−120614.	−0.368641
$573$	−81580.7	−0.248472
$574$	0	0
$575$	0	0
$576$	356911.	1.07576
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−560276.	−1.67705
$579$	313140.	0.934074
$580$	0	0
$581$	0	0
$582$	462464.	1.36531
$583$	49077.2	0.144392
$584$	0	0
$585$	−102300.	−0.298926
$586$	1.15155e6	3.35342
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−311390.	−0.900638
$589$	0	0
$590$	0	0
$591$	0	0
$592$	319807.	0.912524
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	264120.	0.748563
$595$	0	0
$596$	220922.	0.621937
$597$	−109487.	−0.307194
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	243750.	0.677083
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	−370736.	−1.01960
$604$	0	0
$605$	−269925.	−0.737450
$606$	602940.	1.64183
$607$	−309369.	−0.839652	−0.419826	−	0.907605i	$-0.637909\pi$
$607$	−309369.	−0.839652	−0.419826	+	0.907605i	$0.637909\pi$
$608$	210685.	0.569935
$609$	0	0
$610$	−1.19708e6	−3.21709
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	1.24947e6	3.31428
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	−439119.	−1.14976
$619$	−766142.	−1.99953	−0.999765	−	0.0216731i	$-0.993101\pi$
$619$	−766142.	−1.99953	−0.999765	+	0.0216731i	$0.993101\pi$
$620$	0	0
$621$	0	0
$622$	417398.	1.07887
$623$	0	0
$624$	−36300.0	−0.0932261
$625$	390625.	1.00000
$626$	0	0
$627$	100095.	0.254612
$628$	0	0
$629$	0	0
$630$	0	0
$631$	504178.	1.26627	0.633133	−	0.774043i	$-0.281768\pi$
$631$	504178.	1.26627	0.633133	+	0.774043i	$0.281768\pi$
$632$	0	0
$633$	0	0
$634$	−438210.	−1.09019
$635$	−355199.	−0.880896
$636$	102660.	0.253797
$637$	161064.	0.396935
$638$	0	0
$639$	0	0
$640$	747797.	1.82568
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	−519617.	−1.26071
$643$	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			$\pi$
$648$	−183221.	−0.436341
$649$	0	0
$650$	−281250.	−0.665680
$651$	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			$\pi$
$660$	201023.	0.461484
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	1.08153e6	2.43832
$667$	0	0
$668$	1.57381e6	3.52695
$669$	−222540.	−0.497228
$670$	−1.01925e6	−2.27055
$671$	−442556.	−0.982931
$672$	0	0
$673$	−540775.	−1.19395	−0.596976	−	0.802259i	$-0.703631\pi$
$673$	−540775.	−1.19395	−0.596976	+	0.802259i	$0.703631\pi$
$674$	1.49931e6	3.30044
$675$	396902.	0.871116
$676$	−697769.	−1.52693
$677$	533289.	1.16355	0.581775	−	0.813350i	$-0.302358\pi$
$677$	533289.	1.16355	0.581775	+	0.813350i	$0.302358\pi$
$678$	−716034.	−1.55767
$679$	0	0
$680$	0	0
$681$	−413100.	−0.890761
$682$	0	0
$683$	−641774.	−1.37575	−0.687877	−	0.725828i	$-0.741457\pi$
$683$	−641774.	−1.37575	−0.687877	+	0.725828i	$0.741457\pi$
$684$	−638609.	−1.36497
$685$	0	0
$686$	0	0
$687$	−304544.	−0.645262
$688$	0	0
$689$	−53100.0	−0.111855
$690$	0	0
$691$	954658.	1.99936	0.999682	−	0.0252304i	$-0.00803194\pi$
$691$	954658.	1.99936	0.999682	+	0.0252304i	$0.00803194\pi$
$692$	861025.	1.79806
$693$	0	0
$694$	0	0
$695$	46450.0	0.0961648
$696$	0	0
$697$	0	0
$698$	−165840.	−0.340392
$699$	0	0
$700$	0	0
$701$	−75422.0	−0.153484	−0.0767418	−	0.997051i	$-0.524452\pi$
$701$	−75422.0	−0.153484	−0.0767418	+	0.997051i	$0.524452\pi$
$702$	−285769.	−0.579885
$703$	954135.	1.93063
$704$	362762.	0.731942
$705$	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
$715$	−103977.	−0.203388
$716$	0	0
$717$	467955.	0.910261
$718$	1.69893e6	3.29555
$719$	522898.	1.01148	0.505742	−	0.862685i	$-0.331219\pi$
$719$	522898.	1.01148	0.505742	+	0.862685i	$0.331219\pi$
$720$	−184525.	−0.355951
$721$	0	0
$722$	−874220.	−1.67705
$723$	0	0
$724$	0	0
$725$	0	0
$726$	−323910.	−0.614541
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	101879.	0.191703
$730$	0	0
$731$	0	0
$732$	−925741.	−1.72770
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	−268440.	−0.496904
$736$	0	0
$737$	−376813.	−0.693731
$738$	0	0
$739$	873362.	1.59921	0.799605	−	0.600527i	$-0.205042\pi$
$739$	873362.	1.59921	0.799605	+	0.600527i	$0.205042\pi$
$740$	1.91620e6	3.49927
$741$	−108300.	−0.197239
$742$	0	0
$743$	1.09624e6	1.98577	0.992884	−	0.119085i	$-0.0379962\pi$
$743$	1.09624e6	1.98577	0.992884	+	0.119085i	$0.0379962\pi$
$744$	0	0
$745$	190450.	0.343138
$746$	−1.49805e6	−2.69184
$747$	0	0
$748$	0	0
$749$	0	0
$750$	468750.	0.833333
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	415363.	0.732551
$754$	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
$760$	−787040.	−1.36260
$761$	902578.	1.55853	0.779265	−	0.626694i	$-0.215592\pi$
$761$	902578.	1.55853	0.779265	+	0.626694i	$0.215592\pi$
$762$	−426239.	−0.734080
$763$	0	0
$764$	529018.	0.906325
$765$	0	0
$766$	−1.85589e6	−3.16297
$767$	0	0
$768$	478693.	0.811586
$769$	−714542.	−1.20830	−0.604150	−	0.796870i	$-0.706487\pi$
$769$	−714542.	−1.20830	−0.604150	+	0.796870i	$0.706487\pi$
$770$	0	0
$771$	−547020.	−0.920226
$772$	−2.03059e6	−3.40712
$773$	−697371.	−1.16709	−0.583546	−	0.812080i	$-0.698335\pi$
$773$	−697371.	−1.16709	−0.583546	+	0.812080i	$0.698335\pi$
$774$	0	0
$775$	0	0
$776$	−1.34433e6	−2.23245
$777$	0	0
$778$	−1.66311e6	−2.74765
$779$	0	0
$780$	−217500.	−0.357495
$781$	0	0
$782$	0	0
$783$	0	0
$784$	290521.	0.472656
$785$	0	0
$786$	−611940.	−0.990521
$787$	−404142.	−0.652507	−0.326253	−	0.945282i	$-0.605786\pi$
$787$	−404142.	−0.652507	−0.326253	+	0.945282i	$0.605786\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−329816.	−0.525800
$793$	478832.	0.761441
$794$	0	0
$795$	88500.0	0.140026
$796$	709978.	1.12052
$797$	−694796.	−1.09381	−0.546903	−	0.837196i	$-0.684193\pi$
$797$	−694796.	−1.09381	−0.546903	+	0.837196i	$0.684193\pi$
$798$	0	0
$799$	0	0
$800$	364759.	0.569935
$801$	0	0
$802$	0	0
$803$	0	0
$804$	−788220.	−1.21937
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−1.75268e6	−2.68460
$809$	−59038.0	−0.0902058	−0.0451029	−	0.998982i	$-0.514362\pi$
$809$	−59038.0	−0.0902058	−0.0451029	+	0.998982i	$0.514362\pi$
$810$	−352348.	−0.537035
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	649631.	0.982847
$814$	1.09926e6	1.65902
$815$	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			$\pi$
$824$	1.27647e6	1.87999
$825$	173295.	0.254612
$826$	0	0
$827$	887053.	1.29700	0.648498	−	0.761217i	$-0.275398\pi$
$827$	887053.	1.29700	0.648498	+	0.761217i	$0.275398\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	−392497.	−0.567009
$833$	0	0
$834$	55740.0	0.0801373
$835$	1.35673e6	1.94591
$836$	−649078.	−0.928719
$837$	0	0
$838$	948258.	1.35033
$839$	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
$845$	−601525.	−0.842442
$846$	0	0
$847$	0	0
$848$	−95779.7	−0.133193
$849$	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
$855$	−550525.	−0.753086
$856$	1.51047e6	2.06141
$857$	1.12071e6	1.52592	0.762962	−	0.646444i	$-0.223745\pi$
$857$	1.12071e6	1.52592	0.762962	+	0.646444i	$0.223745\pi$
$858$	−124773.	−0.169490
$859$	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
$863$	−1.44214e6	−1.93636	−0.968181	−	0.250249i	$-0.919487\pi$
$863$	−1.44214e6	−1.93636	−0.968181	+	0.250249i	$0.919487\pi$
$864$	370620.	0.496480
$865$	742263.	0.992031
$866$	−582930.	−0.777286
$867$	−373517.	−0.496904
$868$	0	0
$869$	0	0
$870$	0	0
$871$	407700.	0.537408
$872$	0	0
$873$	−940343.	−1.23384
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.53627e6	−1.99742	−0.998709	−	0.0507905i	$-0.983826\pi$
$877$	−1.53627e6	−1.99742	−0.998709	+	0.0507905i	$0.983826\pi$
$878$	0	0
$879$	767700.	0.993605
$880$	−187550.	−0.242188
$881$	1.26018e6	1.62360	0.811802	−	0.583933i	$-0.198487\pi$
$881$	1.26018e6	1.62360	0.811802	+	0.583933i	$0.198487\pi$
$882$	982490.	1.26296
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	646523.	0.821745	0.410872	−	0.911693i	$-0.365224\pi$
$887$	646523.	0.821745	0.410872	+	0.911693i	$0.365224\pi$
$888$	1.03078e6	1.30720
$889$	0	0
$890$	0	0
$891$	−130262.	−0.164083
$892$	1.44308e6	1.81368
$893$	0	0
$894$	228540.	0.285948
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−1.10562e6	−1.36497
$901$	0	0
$902$	0	0
$903$	0	0
$904$	2.08143e6	2.54698
$905$	0	0
$906$	0	0
$907$	671746.	0.816565	0.408282	−	0.912856i	$-0.366128\pi$
$907$	671746.	0.816565	0.408282	+	0.912856i	$0.366128\pi$
$908$	2.67879e6	3.24913
$909$	−1.22598e6	−1.48373
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−195347.	−0.234865
$913$	0	0
$914$	0	0
$915$	−798053.	−0.953212
$916$	1.97484e6	2.35365
$917$	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
$921$	832980.	0.982009
$922$	452388.	0.532168
$923$	0	0
$924$	0	0
$925$	1.65190e6	1.93063
$926$	0	0
$927$	892875.	1.03904
$928$	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
$933$	278265.	0.319665
$934$	0	0
$935$	0	0
$936$	356850.	0.407318
$937$	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
$950$	−1.51354e6	−1.67705
$951$	−292140.	−0.323020
$952$	0	0
$953$	−1.68922e6	−1.85995	−0.929973	−	0.367628i	$-0.880170\pi$
$953$	−1.68922e6	−1.85995	−0.929973	+	0.367628i	$0.880170\pi$
$954$	−323910.	−0.355900
$955$	456050.	0.500041
$956$	−3.03450e6	−3.32026
$957$	0	0
$958$	−2.34209e6	−2.55195
$959$	0	0
$960$	654162.	0.709811
$961$	923521.	1.00000
$962$	−1.18936e6	−1.28518
$963$	1.05656e6	1.13930
$964$	0	0
$965$	−1.75051e6	−1.87979
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	941570.	1.00485
$969$	0	0
$970$	−2.58525e6	−2.74764
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−1.76420e6	−1.86730
$973$	0	0
$974$	−3.15333e6	−3.32393
$975$	−187500.	−0.197239
$976$	863698.	0.906697
$977$	−907848.	−0.951095	−0.475548	−	0.879690i	$-0.657750\pi$
$977$	−907848.	−0.951095	−0.475548	+	0.879690i	$0.657750\pi$
$978$	0	0
$979$	0	0
$980$	1.74072e6	1.81250
$981$	0	0
$982$	2.63873e6	2.73635
$983$	−1.93243e6	−1.99985	−0.999925	−	0.0122789i	$-0.996091\pi$
$983$	−1.93243e6	−1.99985	−0.999925	+	0.0122789i	$0.996091\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	702282.	0.719445
$989$	0	0
$990$	−634261.	−0.647139
$991$	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			$\pi$
$998$	−3.09394e6	−3.10636
$999$	1.67844e6	1.68180

Twists

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

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

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)

\(f(q)\)	\(=\)	\(q-6.70820 q^{2} -4.47214 q^{3} +29.0000 q^{4} +25.0000 q^{5} +30.0000 q^{6} -87.2067 q^{8} -61.0000 q^{9} -167.705 q^{10} -62.0000 q^{11} -129.692 q^{12} +67.0820 q^{13} -111.803 q^{15} +121.000 q^{16} +409.200 q^{18} +361.000 q^{19} +725.000 q^{20} +415.909 q^{22} +390.000 q^{24} +625.000 q^{25} -450.000 q^{26} +635.043 q^{27} +750.000 q^{30} +583.614 q^{32} +277.272 q^{33} -1769.00 q^{36} +2643.03 q^{37} -2421.66 q^{38} -300.000 q^{39} -2180.17 q^{40} -1798.00 q^{44} -1525.00 q^{45} -541.128 q^{48} +2401.00 q^{49} -4192.63 q^{50} +1945.38 q^{52} -791.568 q^{53} -4260.00 q^{54} -1550.00 q^{55} -1614.44 q^{57} -3242.30 q^{60} +7138.00 q^{61} -5851.00 q^{64} +1677.05 q^{65} -1860.00 q^{66} +6077.63 q^{67} +5319.61 q^{72} -17730.0 q^{74} -2795.08 q^{75} +10469.0 q^{76} +2012.46 q^{78} +3025.00 q^{80} +2101.00 q^{81} +5406.81 q^{88} +10230.0 q^{90} +9025.00 q^{95} -2610.00 q^{96} +15415.5 q^{97} -16106.4 q^{98} +3782.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 58 q^{4} + 50 q^{5} + 60 q^{6} - 122 q^{9} - 124 q^{11} + 242 q^{16} + 722 q^{19} + 1450 q^{20} + 780 q^{24} + 1250 q^{25} - 900 q^{26} + 1500 q^{30} - 3538 q^{36} - 600 q^{39} - 3596 q^{44} - 3050 q^{45}+ \cdots + 7564 q^{99}+O(q^{100}) \) 2 * q + 58 * q^4 + 50 * q^5 + 60 * q^6 - 122 * q^9 - 124 * q^11 + 242 * q^16 + 722 * q^19 + 1450 * q^20 + 780 * q^24 + 1250 * q^25 - 900 * q^26 + 1500 * q^30 - 3538 * q^36 - 600 * q^39 - 3596 * q^44 - 3050 * q^45 + 4802 * q^49 - 8520 * q^54 - 3100 * q^55 + 14276 * q^61 - 11702 * q^64 - 3720 * q^66 - 35460 * q^74 + 20938 * q^76 + 6050 * q^80 + 4202 * q^81 + 18050 * q^95 - 5220 * q^96 + 7564 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more