LMFDB - Embedded newform 1895.1.d.i.1894.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1895,1,Mod(1894,1895)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1895, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1895.1894");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1895 = 5 \cdot 379 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1895.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:	$0.945728198940$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{48})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 20x^{4} - 16x^{2} + 1 $ x^8 - 8x^6 + 20x^4 - 16*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{24}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{24} - \cdots)$

Embedding invariants

Embedding label			1894.5
Root			$-0.261052$ of defining polynomial
Character	$\chi$	$=$	1895.1894

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.261052 q^{2} -1.21752 q^{3} -0.931852 q^{4} -1.00000 q^{5} -0.317837 q^{6} +1.98289 q^{7} -0.504314 q^{8} +0.482362 q^{9} +O(q^{10})$	\(q+0.261052 q^{2} -1.21752 q^{3} -0.931852 q^{4} -1.00000 q^{5} -0.317837 q^{6} +1.98289 q^{7} -0.504314 q^{8} +0.482362 q^{9} -0.261052 q^{10} +1.13455 q^{12} -1.58671 q^{13} +0.517638 q^{14} +1.21752 q^{15} +0.800199 q^{16} -1.84776 q^{17} +0.125922 q^{18} +1.00000 q^{19} +0.931852 q^{20} -2.41421 q^{21} +0.614014 q^{24} +1.00000 q^{25} -0.414214 q^{26} +0.630236 q^{27} -1.84776 q^{28} +0.317837 q^{30} +0.713208 q^{32} -0.482362 q^{34} -1.98289 q^{35} -0.449490 q^{36} +0.261052 q^{38} +1.93185 q^{39} +0.504314 q^{40} +1.73205 q^{41} -0.630236 q^{42} +1.58671 q^{43} -0.482362 q^{45} +0.765367 q^{47} -0.974261 q^{48} +2.93185 q^{49} +0.261052 q^{50} +2.24969 q^{51} +1.47858 q^{52} -0.261052 q^{53} +0.164525 q^{54} -1.00000 q^{56} -1.21752 q^{57} -1.13455 q^{60} +0.956470 q^{63} -0.614014 q^{64} +1.58671 q^{65} +1.72184 q^{68} -0.517638 q^{70} -0.243262 q^{72} -0.765367 q^{73} -1.21752 q^{75} -0.931852 q^{76} +0.504314 q^{78} +1.41421 q^{79} -0.800199 q^{80} -1.24969 q^{81} +0.452156 q^{82} +2.24969 q^{84} +1.84776 q^{85} +0.414214 q^{86} -0.125922 q^{90} -3.14626 q^{91} +0.199801 q^{94} -1.00000 q^{95} -0.868348 q^{96} +0.765367 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{4} - 8 q^{5} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^4 - 8 * q^5 + 8 * q^9	$ 8 q + 8 q^{4} - 8 q^{5} + 8 q^{9} + 8 q^{16} + 8 q^{19} - 8 q^{20} - 8 q^{21} - 8 q^{24} + 8 q^{25} + 8 q^{26} - 8 q^{34} + 16 q^{36} - 8 q^{45} + 8 q^{49} + 8 q^{54} - 8 q^{56} + 8 q^{64} + 8 q^{76} - 8 q^{80} + 8 q^{81} - 8 q^{86} - 8 q^{95} - 24 q^{96}+O(q^{100}) $ 8 * q + 8 * q^4 - 8 * q^5 + 8 * q^9 + 8 * q^16 + 8 * q^19 - 8 * q^20 - 8 * q^21 - 8 * q^24 + 8 * q^25 + 8 * q^26 - 8 * q^34 + 16 * q^36 - 8 * q^45 + 8 * q^49 + 8 * q^54 - 8 * q^56 + 8 * q^64 + 8 * q^76 - 8 * q^80 + 8 * q^81 - 8 * q^86 - 8 * q^95 - 24 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1895\mathbb{Z}\right)^\times$.

$n$	$381$	$1517$
$\chi(n)$	$-1$	$-1$

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 $

$2$	0.261052	0.261052	0.130526	−	0.991445i	$-0.458333\pi$
$2$	0.261052	0.261052	0.130526	+	0.991445i	$0.458333\pi$
$3$	−1.21752	−1.21752	−0.608761	−	0.793353i	$-0.708333\pi$
$3$	−1.21752	−1.21752	−0.608761	+	0.793353i	$0.708333\pi$
$4$	−0.931852	−0.931852
$5$	−1.00000	−1.00000
$5$	−1.00000	−1.00000
$6$	−0.317837	−0.317837
$7$	1.98289	1.98289	0.991445	−	0.130526i	$-0.0416667\pi$
$7$	1.98289	1.98289	0.991445	+	0.130526i	$0.0416667\pi$
$8$	−0.504314	−0.504314
$9$	0.482362	0.482362
$10$	−0.261052	−0.261052
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	1.13455	1.13455
$13$	−1.58671	−1.58671	−0.793353	−	0.608761i	$-0.791667\pi$
$13$	−1.58671	−1.58671	−0.793353	+	0.608761i	$0.791667\pi$
$14$	0.517638	0.517638
$15$	1.21752	1.21752
$16$	0.800199	0.800199
$17$	−1.84776	−1.84776	−0.923880	−	0.382683i	$-0.875000\pi$
$17$	−1.84776	−1.84776	−0.923880	+	0.382683i	$0.875000\pi$
$18$	0.125922	0.125922
$19$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$19$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$20$	0.931852	0.931852
$21$	−2.41421	−2.41421
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0.614014	0.614014
$25$	1.00000	1.00000
$26$	−0.414214	−0.414214
$27$	0.630236	0.630236
$28$	−1.84776	−1.84776
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0.317837	0.317837
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0.713208	0.713208
$33$	0	0
$34$	−0.482362	−0.482362
$35$	−1.98289	−1.98289
$36$	−0.449490	−0.449490
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0.261052	0.261052
$39$	1.93185	1.93185
$40$	0.504314	0.504314
$41$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$41$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$42$	−0.630236	−0.630236
$43$	1.58671	1.58671	0.793353	−	0.608761i	$-0.208333\pi$
$43$	1.58671	1.58671	0.793353	+	0.608761i	$0.208333\pi$
$44$	0	0
$45$	−0.482362	−0.482362
$46$	0	0
$47$	0.765367	0.765367	0.382683	−	0.923880i	$-0.375000\pi$
$47$	0.765367	0.765367	0.382683	+	0.923880i	$0.375000\pi$
$48$	−0.974261	−0.974261
$49$	2.93185	2.93185
$50$	0.261052	0.261052
$51$	2.24969	2.24969
$52$	1.47858	1.47858
$53$	−0.261052	−0.261052	−0.130526	−	0.991445i	$-0.541667\pi$
$53$	−0.261052	−0.261052	−0.130526	+	0.991445i	$0.541667\pi$
$54$	0.164525	0.164525
$55$	0	0
$56$	−1.00000	−1.00000
$57$	−1.21752	−1.21752
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−1.13455	−1.13455
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0.956470	0.956470
$64$	−0.614014	−0.614014
$65$	1.58671	1.58671
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	1.72184	1.72184
$69$	0	0
$70$	−0.517638	−0.517638
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−0.243262	−0.243262
$73$	−0.765367	−0.765367	−0.382683	−	0.923880i	$-0.625000\pi$
$73$	−0.765367	−0.765367	−0.382683	+	0.923880i	$0.625000\pi$
$74$	0	0
$75$	−1.21752	−1.21752
$76$	−0.931852	−0.931852
$77$	0	0
$78$	0.504314	0.504314
$79$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$79$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$80$	−0.800199	−0.800199
$81$	−1.24969	−1.24969
$82$	0.452156	0.452156
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	2.24969	2.24969
$85$	1.84776	1.84776
$86$	0.414214	0.414214
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−0.125922	−0.125922
$91$	−3.14626	−3.14626
$92$	0	0
$93$	0	0
$94$	0.199801	0.199801
$95$	−1.00000	−1.00000
$96$	−0.868348	−0.868348
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0.765367	0.765367
$99$	0	0
$100$	−0.931852	−0.931852
$101$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$101$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$102$	0.587287	0.587287
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0.800199	0.800199
$105$	2.41421	2.41421
$106$	−0.0681483	−0.0681483
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−0.587287	−0.587287
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	1.58671	1.58671
$113$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$113$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$114$	−0.317837	−0.317837
$115$	0	0
$116$	0	0
$117$	−0.765367	−0.765367
$118$	0	0
$119$	−3.66390	−3.66390
$120$	−0.614014	−0.614014
$121$	1.00000	1.00000
$122$	0	0
$123$	−2.10881	−2.10881
$124$	0	0
$125$	−1.00000	−1.00000
$126$	0.249689	0.249689
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−0.873498	−0.873498
$129$	−1.93185	−1.93185
$130$	0.414214	0.414214
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	1.98289	1.98289
$134$	0	0
$135$	−0.630236	−0.630236
$136$	0.931852	0.931852
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$139$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$140$	1.84776	1.84776
$141$	−0.931852	−0.931852
$142$	0	0
$143$	0	0
$144$	0.385986	0.385986
$145$	0	0
$146$	−0.199801	−0.199801
$147$	−3.56960	−3.56960
$148$	0	0
$149$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$149$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$150$	−0.317837	−0.317837
$151$	0.517638	0.517638	0.258819	−	0.965926i	$-0.416667\pi$
$151$	0.517638	0.517638	0.258819	+	0.965926i	$0.416667\pi$
$152$	−0.504314	−0.504314
$153$	−0.891289	−0.891289
$154$	0	0
$155$	0	0
$156$	−1.80020	−1.80020
$157$	−1.98289	−1.98289	−0.991445	−	0.130526i	$-0.958333\pi$
$157$	−1.98289	−1.98289	−0.991445	+	0.130526i	$0.958333\pi$
$158$	0.369184	0.369184
$159$	0.317837	0.317837
$160$	−0.713208	−0.713208
$161$	0	0
$162$	−0.326234	−0.326234
$163$	0.765367	0.765367	0.382683	−	0.923880i	$-0.375000\pi$
$163$	0.765367	0.765367	0.382683	+	0.923880i	$0.375000\pi$
$164$	−1.61401	−1.61401
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	1.21752	1.21752
$169$	1.51764	1.51764
$170$	0.482362	0.482362
$171$	0.482362	0.482362
$172$	−1.47858	−1.47858
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	1.98289	1.98289
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$179$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$180$	0.449490	0.449490
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	−0.821340	−0.821340
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−0.713208	−0.713208
$189$	1.24969	1.24969
$190$	−0.261052	−0.261052
$191$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$191$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$192$	0.747577	0.747577
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	−1.93185	−1.93185
$196$	−2.73205	−2.73205
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−0.504314	−0.504314
$201$	0	0
$202$	0.369184	0.369184
$203$	0	0
$204$	−2.09638	−2.09638
$205$	−1.73205	−1.73205
$206$	0	0
$207$	0	0
$208$	−1.26968	−1.26968
$209$	0	0
$210$	0.630236	0.630236
$211$	−0.517638	−0.517638	−0.258819	−	0.965926i	$-0.583333\pi$
$211$	−0.517638	−0.517638	−0.258819	+	0.965926i	$0.583333\pi$
$212$	0.243262	0.243262
$213$	0	0
$214$	0	0
$215$	−1.58671	−1.58671
$216$	−0.317837	−0.317837
$217$	0	0
$218$	0	0
$219$	0.931852	0.931852
$220$	0	0
$221$	2.93185	2.93185
$222$	0	0
$223$	−1.98289	−1.98289	−0.991445	−	0.130526i	$-0.958333\pi$
$223$	−1.98289	−1.98289	−0.991445	+	0.130526i	$0.958333\pi$
$224$	1.41421	1.41421
$225$	0.482362	0.482362
$226$	0.482362	0.482362
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	1.13455	1.13455
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.21752	1.21752	0.608761	−	0.793353i	$-0.291667\pi$
$233$	1.21752	1.21752	0.608761	+	0.793353i	$0.291667\pi$
$234$	−0.199801	−0.199801
$235$	−0.765367	−0.765367
$236$	0	0
$237$	−1.72184	−1.72184
$238$	−0.956470	−0.956470
$239$	−0.517638	−0.517638	−0.258819	−	0.965926i	$-0.583333\pi$
$239$	−0.517638	−0.517638	−0.258819	+	0.965926i	$0.583333\pi$
$240$	0.974261	0.974261
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0.261052	0.261052
$243$	0.891289	0.891289
$244$	0	0
$245$	−2.93185	−2.93185
$246$	−0.550510	−0.550510
$247$	−1.58671	−1.58671
$248$	0	0
$249$	0	0
$250$	−0.261052	−0.261052
$251$	1.93185	1.93185	0.965926	−	0.258819i	$-0.0833333\pi$
$251$	1.93185	1.93185	0.965926	+	0.258819i	$0.0833333\pi$
$252$	−0.891289	−0.891289
$253$	0	0
$254$	0	0
$255$	−2.24969	−2.24969
$256$	0.385986	0.385986
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	−0.504314	−0.504314
$259$	0	0
$260$	−1.47858	−1.47858
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0.261052	0.261052
$266$	0.517638	0.517638
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−0.164525	−0.164525
$271$	1.73205	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$271$	1.73205	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$272$	−1.47858	−1.47858
$273$	3.83065	3.83065
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−0.261052	−0.261052
$279$	0	0
$280$	1.00000	1.00000
$281$	−1.93185	−1.93185	−0.965926	−	0.258819i	$-0.916667\pi$
$281$	−1.93185	−1.93185	−0.965926	+	0.258819i	$0.916667\pi$
$282$	−0.243262	−0.243262
$283$	−1.84776	−1.84776	−0.923880	−	0.382683i	$-0.875000\pi$
$283$	−1.84776	−1.84776	−0.923880	+	0.382683i	$0.875000\pi$
$284$	0	0
$285$	1.21752	1.21752
$286$	0	0
$287$	3.43447	3.43447
$288$	0.344025	0.344025
$289$	2.41421	2.41421
$290$	0	0
$291$	0	0
$292$	0.713208	0.713208
$293$	1.21752	1.21752	0.608761	−	0.793353i	$-0.291667\pi$
$293$	1.21752	1.21752	0.608761	+	0.793353i	$0.291667\pi$
$294$	−0.931852	−0.931852
$295$	0	0
$296$	0	0
$297$	0	0
$298$	−0.452156	−0.452156
$299$	0	0
$300$	1.13455	1.13455
$301$	3.14626	3.14626
$302$	0.135131	0.135131
$303$	−1.72184	−1.72184
$304$	0.800199	0.800199
$305$	0	0
$306$	−0.232673	−0.232673
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0.517638	0.517638	0.258819	−	0.965926i	$-0.416667\pi$
$311$	0.517638	0.517638	0.258819	+	0.965926i	$0.416667\pi$
$312$	−0.974261	−0.974261
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	−0.517638	−0.517638
$315$	−0.956470	−0.956470
$316$	−1.31784	−1.31784
$317$	1.98289	1.98289	0.991445	−	0.130526i	$-0.0416667\pi$
$317$	1.98289	1.98289	0.991445	+	0.130526i	$0.0416667\pi$
$318$	0.0829722	0.0829722
$319$	0	0
$320$	0.614014	0.614014
$321$	0	0
$322$	0	0
$323$	−1.84776	−1.84776
$324$	1.16452	1.16452
$325$	−1.58671	−1.58671
$326$	0.199801	0.199801
$327$	0	0
$328$	−0.873498	−0.873498
$329$	1.51764	1.51764
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	−1.93185	−1.93185
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0.396183	0.396183
$339$	−2.24969	−2.24969
$340$	−1.72184	−1.72184
$341$	0	0
$342$	0.125922	0.125922
$343$	3.83065	3.83065
$344$	−0.800199	−0.800199
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0.517638	0.517638
$351$	−1.00000	−1.00000
$352$	0	0
$353$	1.98289	1.98289	0.991445	−	0.130526i	$-0.0416667\pi$
$353$	1.98289	1.98289	0.991445	+	0.130526i	$0.0416667\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.46088	4.46088
$358$	0.369184	0.369184
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0.243262	0.243262
$361$	0	0
$362$	0	0
$363$	−1.21752	−1.21752
$364$	2.93185	2.93185
$365$	0.765367	0.765367
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0.835475	0.835475
$370$	0	0
$371$	−0.517638	−0.517638
$372$	0	0
$373$	0.261052	0.261052	0.130526	−	0.991445i	$-0.458333\pi$
$373$	0.261052	0.261052	0.130526	+	0.991445i	$0.458333\pi$
$374$	0	0
$375$	1.21752	1.21752
$376$	−0.385986	−0.385986
$377$	0	0
$378$	0.326234	0.326234
$379$	−1.00000	−1.00000
$379$	−1.00000	−1.00000
$380$	0.931852	0.931852
$381$	0	0
$382$	−0.369184	−0.369184
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	1.06350	1.06350
$385$	0	0
$386$	0	0
$387$	0.765367	0.765367
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	−0.504314	−0.504314
$391$	0	0
$392$	−1.47858	−1.47858
$393$	0	0
$394$	0	0
$395$	−1.41421	−1.41421
$396$	0	0
$397$	1.58671	1.58671	0.793353	−	0.608761i	$-0.208333\pi$
$397$	1.58671	1.58671	0.793353	+	0.608761i	$0.208333\pi$
$398$	0	0
$399$	−2.41421	−2.41421
$400$	0.800199	0.800199
$401$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$401$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$402$	0	0
$403$	0	0
$404$	−1.31784	−1.31784
$405$	1.24969	1.24969
$406$	0	0
$407$	0	0
$408$	−1.13455	−1.13455
$409$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$409$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$410$	−0.452156	−0.452156
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−1.13165	−1.13165
$417$	1.21752	1.21752
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	−2.24969	−2.24969
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−0.135131	−0.135131
$423$	0.369184	0.369184
$424$	0.131652	0.131652
$425$	−1.84776	−1.84776
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−0.414214	−0.414214
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0.504314	0.504314
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0.243262	0.243262
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	1.41421	1.41421
$442$	0.765367	0.765367
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	−0.517638	−0.517638
$447$	2.10881	2.10881
$448$	−1.21752	−1.21752
$449$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$449$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$450$	0.125922	0.125922
$451$	0	0
$452$	−1.72184	−1.72184
$453$	−0.630236	−0.630236
$454$	0	0
$455$	3.14626	3.14626
$456$	0.614014	0.614014
$457$	−0.261052	−0.261052	−0.130526	−	0.991445i	$-0.541667\pi$
$457$	−0.261052	−0.261052	−0.130526	+	0.991445i	$0.541667\pi$
$458$	0	0
$459$	−1.16452	−1.16452
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0.317837	0.317837
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0.713208	0.713208
$469$	0	0
$470$	−0.199801	−0.199801
$471$	2.41421	2.41421
$472$	0	0
$473$	0	0
$474$	−0.449490	−0.449490
$475$	1.00000	1.00000
$476$	3.41421	3.41421
$477$	−0.125922	−0.125922
$478$	−0.135131	−0.135131
$479$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$479$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$480$	0.868348	0.868348
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−0.931852	−0.931852
$485$	0	0
$486$	0.232673	0.232673
$487$	−0.765367	−0.765367	−0.382683	−	0.923880i	$-0.625000\pi$
$487$	−0.765367	−0.765367	−0.382683	+	0.923880i	$0.625000\pi$
$488$	0	0
$489$	−0.931852	−0.931852
$490$	−0.765367	−0.765367
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	1.96510	1.96510
$493$	0	0
$494$	−0.414214	−0.414214
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	1.93185	1.93185	0.965926	−	0.258819i	$-0.0833333\pi$
$499$	1.93185	1.93185	0.965926	+	0.258819i	$0.0833333\pi$
$500$	0.931852	0.931852
$501$	0	0
$502$	0.504314	0.504314
$503$	−0.261052	−0.261052	−0.130526	−	0.991445i	$-0.541667\pi$
$503$	−0.261052	−0.261052	−0.130526	+	0.991445i	$0.541667\pi$
$504$	−0.482362	−0.482362
$505$	−1.41421	−1.41421
$506$	0	0
$507$	−1.84776	−1.84776
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	−0.587287	−0.587287
$511$	−1.51764	−1.51764
$512$	0.974261	0.974261
$513$	0.630236	0.630236
$514$	0	0
$515$	0	0
$516$	1.80020	1.80020
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−0.800199	−0.800199
$521$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$521$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	−2.41421	−2.41421
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	1.00000
$530$	0.0681483	0.0681483
$531$	0	0
$532$	−1.84776	−1.84776
$533$	−2.74826	−2.74826
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.72184	−1.72184
$538$	0	0
$539$	0	0
$540$	0.587287	0.587287
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0.452156	0.452156
$543$	0	0
$544$	−1.31784	−1.31784
$545$	0	0
$546$	1.00000	1.00000
$547$	1.21752	1.21752	0.608761	−	0.793353i	$-0.291667\pi$
$547$	1.21752	1.21752	0.608761	+	0.793353i	$0.291667\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	2.80423	2.80423
$554$	0	0
$555$	0	0
$556$	0.931852	0.931852
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	−2.51764	−2.51764
$560$	−1.58671	−1.58671
$561$	0	0
$562$	−0.504314	−0.504314
$563$	−0.765367	−0.765367	−0.382683	−	0.923880i	$-0.625000\pi$
$563$	−0.765367	−0.765367	−0.382683	+	0.923880i	$0.625000\pi$
$564$	0.868348	0.868348
$565$	−1.84776	−1.84776
$566$	−0.482362	−0.482362
$567$	−2.47800	−2.47800
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0.317837	0.317837
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	1.72184	1.72184
$574$	0.896575	0.896575
$575$	0	0
$576$	−0.296177	−0.296177
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0.630236	0.630236
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0.385986	0.385986
$585$	0.765367	0.765367
$586$	0.317837	0.317837
$587$	1.21752	1.21752	0.608761	−	0.793353i	$-0.291667\pi$
$587$	1.21752	1.21752	0.608761	+	0.793353i	$0.291667\pi$
$588$	3.32633	3.32633
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0.765367	0.765367	0.382683	−	0.923880i	$-0.375000\pi$
$593$	0.765367	0.765367	0.382683	+	0.923880i	$0.375000\pi$
$594$	0	0
$595$	3.66390	3.66390
$596$	1.61401	1.61401
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0.614014	0.614014
$601$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$601$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$602$	0.821340	0.821340
$603$	0	0
$604$	−0.482362	−0.482362
$605$	−1.00000	−1.00000
$606$	−0.449490	−0.449490
$607$	−0.261052	−0.261052	−0.130526	−	0.991445i	$-0.541667\pi$
$607$	−0.261052	−0.261052	−0.130526	+	0.991445i	$0.541667\pi$
$608$	0.713208	0.713208
$609$	0	0
$610$	0	0
$611$	−1.21441	−1.21441
$612$	0.830549	0.830549
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	2.10881	2.10881
$616$	0	0
$617$	−1.58671	−1.58671	−0.793353	−	0.608761i	$-0.791667\pi$
$617$	−1.58671	−1.58671	−0.793353	+	0.608761i	$0.791667\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0.135131	0.135131
$623$	0	0
$624$	1.54587	1.54587
$625$	1.00000	1.00000
$626$	0	0
$627$	0	0
$628$	1.84776	1.84776
$629$	0	0
$630$	−0.249689	−0.249689
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	−0.713208	−0.713208
$633$	0.630236	0.630236
$634$	0.517638	0.517638
$635$	0	0
$636$	−0.296177	−0.296177
$637$	−4.65199	−4.65199
$638$	0	0
$639$	0	0
$640$	0.873498	0.873498
$641$	1.93185	1.93185	0.965926	−	0.258819i	$-0.0833333\pi$
$641$	1.93185	1.93185	0.965926	+	0.258819i	$0.0833333\pi$
$642$	0	0
$643$	0.261052	0.261052	0.130526	−	0.991445i	$-0.458333\pi$
$643$	0.261052	0.261052	0.130526	+	0.991445i	$0.458333\pi$
$644$	0	0
$645$	1.93185	1.93185
$646$	−0.482362	−0.482362
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0.630236	0.630236
$649$	0	0
$650$	−0.414214	−0.414214
$651$	0	0
$652$	−0.713208	−0.713208
$653$	−1.21752	−1.21752	−0.608761	−	0.793353i	$-0.708333\pi$
$653$	−1.21752	−1.21752	−0.608761	+	0.793353i	$0.708333\pi$
$654$	0	0
$655$	0	0
$656$	1.38599	1.38599
$657$	−0.369184	−0.369184
$658$	0.396183	0.396183
$659$	−1.93185	−1.93185	−0.965926	−	0.258819i	$-0.916667\pi$
$659$	−1.93185	−1.93185	−0.965926	+	0.258819i	$0.916667\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	−3.56960	−3.56960
$664$	0	0
$665$	−1.98289	−1.98289
$666$	0	0
$667$	0	0
$668$	0	0
$669$	2.41421	2.41421
$670$	0	0
$671$	0	0
$672$	−1.72184	−1.72184
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0.630236	0.630236
$676$	−1.41421	−1.41421
$677$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$677$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$678$	−0.587287	−0.587287
$679$	0	0
$680$	−0.931852	−0.931852
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	−0.449490	−0.449490
$685$	0	0
$686$	1.00000	1.00000
$687$	0	0
$688$	1.26968	1.26968
$689$	0.414214	0.414214
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.00000	1.00000
$696$	0	0
$697$	−3.20041	−3.20041
$698$	0	0
$699$	−1.48236	−1.48236
$700$	−1.84776	−1.84776
$701$	0.517638	0.517638	0.258819	−	0.965926i	$-0.416667\pi$
$701$	0.517638	0.517638	0.258819	+	0.965926i	$0.416667\pi$
$702$	−0.261052	−0.261052
$703$	0	0
$704$	0	0
$705$	0.931852	0.931852
$706$	0.517638	0.517638
$707$	2.80423	2.80423
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0.682163	0.682163
$712$	0	0
$713$	0	0
$714$	1.16452	1.16452
$715$	0	0
$716$	−1.31784	−1.31784
$717$	0.630236	0.630236
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−0.385986	−0.385986
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	−0.317837	−0.317837
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	1.58671	1.58671
$729$	0.164525	0.164525
$730$	0.199801	0.199801
$731$	−2.93185	−2.93185
$732$	0	0
$733$	0.261052	0.261052	0.130526	−	0.991445i	$-0.458333\pi$
$733$	0.261052	0.261052	0.130526	+	0.991445i	$0.458333\pi$
$734$	0	0
$735$	3.56960	3.56960
$736$	0	0
$737$	0	0
$738$	0.218103	0.218103
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	1.93185	1.93185
$742$	−0.135131	−0.135131
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	1.73205	1.73205
$746$	0.0681483	0.0681483
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0.317837	0.317837
$751$	−1.73205	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$751$	−1.73205	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$752$	0.612446	0.612446
$753$	−2.35207	−2.35207
$754$	0	0
$755$	−0.517638	−0.517638
$756$	−1.16452	−1.16452
$757$	−0.261052	−0.261052	−0.130526	−	0.991445i	$-0.541667\pi$
$757$	−0.261052	−0.261052	−0.130526	+	0.991445i	$0.541667\pi$
$758$	−0.261052	−0.261052
$759$	0	0
$760$	0.504314	0.504314
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	1.31784	1.31784
$765$	0.891289	0.891289
$766$	0	0
$767$	0	0
$768$	−0.469946	−0.469946
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.98289	−1.98289	−0.991445	−	0.130526i	$-0.958333\pi$
$773$	−1.98289	−1.98289	−0.991445	+	0.130526i	$0.958333\pi$
$774$	0.199801	0.199801
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.73205	1.73205
$780$	1.80020	1.80020
$781$	0	0
$782$	0	0
$783$	0	0
$784$	2.34607	2.34607
$785$	1.98289	1.98289
$786$	0	0
$787$	1.21752	1.21752	0.608761	−	0.793353i	$-0.291667\pi$
$787$	1.21752	1.21752	0.608761	+	0.793353i	$0.291667\pi$
$788$	0	0
$789$	0	0
$790$	−0.369184	−0.369184
$791$	3.66390	3.66390
$792$	0	0
$793$	0	0
$794$	0.414214	0.414214
$795$	−0.317837	−0.317837
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	−0.630236	−0.630236
$799$	−1.41421	−1.41421
$800$	0.713208	0.713208
$801$	0	0
$802$	−0.369184	−0.369184
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−0.713208	−0.713208
$809$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$809$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$810$	0.326234	0.326234
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	−2.10881	−2.10881
$814$	0	0
$815$	−0.765367	−0.765367
$816$	1.80020	1.80020
$817$	1.58671	1.58671
$818$	−0.452156	−0.452156
$819$	−1.51764	−1.51764
$820$	1.61401	1.61401
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$823$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−0.765367	−0.765367	−0.382683	−	0.923880i	$-0.625000\pi$
$827$	−0.765367	−0.765367	−0.382683	+	0.923880i	$0.625000\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0.974261	0.974261
$833$	−5.41736	−5.41736
$834$	0.317837	0.317837
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	−1.21752	−1.21752
$841$	1.00000	1.00000
$842$	0	0
$843$	2.35207	2.35207
$844$	0.482362	0.482362
$845$	−1.51764	−1.51764
$846$	0.0963763	0.0963763
$847$	1.98289	1.98289
$848$	−0.208894	−0.208894
$849$	2.24969	2.24969
$850$	−0.482362	−0.482362
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	−0.482362	−0.482362
$856$	0	0
$857$	−1.21752	−1.21752	−0.608761	−	0.793353i	$-0.708333\pi$
$857$	−1.21752	−1.21752	−0.608761	+	0.793353i	$0.708333\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	1.47858	1.47858
$861$	−4.18154	−4.18154
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0.449490	0.449490
$865$	0	0
$866$	0	0
$867$	−2.93936	−2.93936
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.98289	−1.98289
$876$	−0.868348	−0.868348
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	−1.48236	−1.48236
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0.369184	0.369184
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	−2.73205	−2.73205
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	1.84776	1.84776
$893$	0.765367	0.765367
$894$	0.550510	0.550510
$895$	−1.41421	−1.41421
$896$	−1.73205	−1.73205
$897$	0	0
$898$	−0.261052	−0.261052
$899$	0	0
$900$	−0.449490	−0.449490
$901$	0.482362	0.482362
$902$	0	0
$903$	−3.83065	−3.83065
$904$	−0.931852	−0.931852
$905$	0	0
$906$	−0.164525	−0.164525
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0.682163	0.682163
$910$	0.821340	0.821340
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−0.974261	−0.974261
$913$	0	0
$914$	−0.0681483	−0.0681483
$915$	0	0
$916$	0	0
$917$	0	0
$918$	−0.304002	−0.304002
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0.517638	0.517638	0.258819	−	0.965926i	$-0.416667\pi$
$929$	0.517638	0.517638	0.258819	+	0.965926i	$0.416667\pi$
$930$	0	0
$931$	2.93185	2.93185
$932$	−1.13455	−1.13455
$933$	−0.630236	−0.630236
$934$	0	0
$935$	0	0
$936$	0.385986	0.385986
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0.713208	0.713208
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0.630236	0.630236
$943$	0	0
$944$	0	0
$945$	−1.24969	−1.24969
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	1.60450	1.60450
$949$	1.21441	1.21441
$950$	0.261052	0.261052
$951$	−2.41421	−2.41421
$952$	1.84776	1.84776
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	−0.0328722	−0.0328722
$955$	1.41421	1.41421
$956$	0.482362	0.482362
$957$	0	0
$958$	0.261052	0.261052
$959$	0	0
$960$	−0.747577	−0.747577
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0.765367	0.765367	0.382683	−	0.923880i	$-0.375000\pi$
$967$	0.765367	0.765367	0.382683	+	0.923880i	$0.375000\pi$
$968$	−0.504314	−0.504314
$969$	2.24969	2.24969
$970$	0	0
$971$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$971$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$972$	−0.830549	−0.830549
$973$	−1.98289	−1.98289
$974$	−0.199801	−0.199801
$975$	1.93185	1.93185
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	−0.243262	−0.243262
$979$	0	0
$980$	2.73205	2.73205
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	1.06350	1.06350
$985$	0	0
$986$	0	0
$987$	−1.84776	−1.84776
$988$	1.47858	1.47858
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0.504314	0.504314
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1895.1.d.i.1894.5	yes	8
5.4	even	2	inner	1895.1.d.i.1894.4	✓	8
379.378	odd	2	inner	1895.1.d.i.1894.4	✓	8
1895.1894	odd	2	CM	1895.1.d.i.1894.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1895.1.d.i.1894.4	✓	8	5.4	even	2	inner
1895.1.d.i.1894.4	✓	8	379.378	odd	2	inner
1895.1.d.i.1894.5	yes	8	1.1	even	1	trivial
1895.1.d.i.1894.5	yes	8	1895.1894	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1895 = 5 \cdot 379 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1895.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.261052 q^{2} -1.21752 q^{3} -0.931852 q^{4} -1.00000 q^{5} -0.317837 q^{6} +1.98289 q^{7} -0.504314 q^{8} +0.482362 q^{9} +O(q^{10})\)	\(q+0.261052 q^{2} -1.21752 q^{3} -0.931852 q^{4} -1.00000 q^{5} -0.317837 q^{6} +1.98289 q^{7} -0.504314 q^{8} +0.482362 q^{9} -0.261052 q^{10} +1.13455 q^{12} -1.58671 q^{13} +0.517638 q^{14} +1.21752 q^{15} +0.800199 q^{16} -1.84776 q^{17} +0.125922 q^{18} +1.00000 q^{19} +0.931852 q^{20} -2.41421 q^{21} +0.614014 q^{24} +1.00000 q^{25} -0.414214 q^{26} +0.630236 q^{27} -1.84776 q^{28} +0.317837 q^{30} +0.713208 q^{32} -0.482362 q^{34} -1.98289 q^{35} -0.449490 q^{36} +0.261052 q^{38} +1.93185 q^{39} +0.504314 q^{40} +1.73205 q^{41} -0.630236 q^{42} +1.58671 q^{43} -0.482362 q^{45} +0.765367 q^{47} -0.974261 q^{48} +2.93185 q^{49} +0.261052 q^{50} +2.24969 q^{51} +1.47858 q^{52} -0.261052 q^{53} +0.164525 q^{54} -1.00000 q^{56} -1.21752 q^{57} -1.13455 q^{60} +0.956470 q^{63} -0.614014 q^{64} +1.58671 q^{65} +1.72184 q^{68} -0.517638 q^{70} -0.243262 q^{72} -0.765367 q^{73} -1.21752 q^{75} -0.931852 q^{76} +0.504314 q^{78} +1.41421 q^{79} -0.800199 q^{80} -1.24969 q^{81} +0.452156 q^{82} +2.24969 q^{84} +1.84776 q^{85} +0.414214 q^{86} -0.125922 q^{90} -3.14626 q^{91} +0.199801 q^{94} -1.00000 q^{95} -0.868348 q^{96} +0.765367 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4} - 8 q^{5} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^4 - 8 * q^5 + 8 * q^9	\( 8 q + 8 q^{4} - 8 q^{5} + 8 q^{9} + 8 q^{16} + 8 q^{19} - 8 q^{20} - 8 q^{21} - 8 q^{24} + 8 q^{25} + 8 q^{26} - 8 q^{34} + 16 q^{36} - 8 q^{45} + 8 q^{49} + 8 q^{54} - 8 q^{56} + 8 q^{64} + 8 q^{76} - 8 q^{80} + 8 q^{81} - 8 q^{86} - 8 q^{95} - 24 q^{96}+O(q^{100}) \) 8 * q + 8 * q^4 - 8 * q^5 + 8 * q^9 + 8 * q^16 + 8 * q^19 - 8 * q^20 - 8 * q^21 - 8 * q^24 + 8 * q^25 + 8 * q^26 - 8 * q^34 + 16 * q^36 - 8 * q^45 + 8 * q^49 + 8 * q^54 - 8 * q^56 + 8 * q^64 + 8 * q^76 - 8 * q^80 + 8 * q^81 - 8 * q^86 - 8 * q^95 - 24 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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