LMFDB - Embedded newform 431.1.b.c.430.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [431,1,Mod(430,431)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(431, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("431.430");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 431 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	431.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$0.215097020445$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{21})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 6x^{4} + 6x^{3} + 8x^{2} - 8x + 1 $ x^6 - x^5 - 6x^4 + 6x^3 + 8x^2 - 8x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{21}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{21} - \cdots)$

Embedding invariants

Embedding label			430.1
Root			$1.65248$ of defining polynomial
Character	$\chi$	$=$	431.430

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.80194 q^{2} -1.46610 q^{3} +2.24698 q^{4} +1.65248 q^{5} +2.64183 q^{6} -2.24698 q^{8} +1.14946 q^{9} +O(q^{10})$	$q-1.80194 q^{2} -1.46610 q^{3} +2.24698 q^{4} +1.65248 q^{5} +2.64183 q^{6} -2.24698 q^{8} +1.14946 q^{9} -2.97766 q^{10} -1.00000 q^{11} -3.29431 q^{12} -2.42270 q^{15} +1.80194 q^{16} -2.07126 q^{18} +1.91115 q^{19} +3.71308 q^{20} +1.80194 q^{22} +0.730682 q^{23} +3.29431 q^{24} +1.73068 q^{25} -0.219124 q^{27} -1.97766 q^{29} +4.36556 q^{30} -1.00000 q^{32} +1.46610 q^{33} +2.58281 q^{36} -3.44377 q^{38} -3.71308 q^{40} +1.24698 q^{41} -2.24698 q^{44} +1.89946 q^{45} -1.31664 q^{46} -2.64183 q^{48} +1.00000 q^{49} -3.11858 q^{50} +1.91115 q^{53} +0.394848 q^{54} -1.65248 q^{55} -2.80194 q^{57} +3.56362 q^{58} +0.149460 q^{59} -5.44377 q^{60} -0.445042 q^{61} -2.64183 q^{66} -1.07126 q^{69} -2.58281 q^{72} -2.53736 q^{75} +4.29431 q^{76} +2.97766 q^{80} -0.828201 q^{81} -2.24698 q^{82} +2.89946 q^{87} +2.24698 q^{88} -3.42270 q^{90} +1.64183 q^{92} +3.15813 q^{95} +1.46610 q^{96} -1.97766 q^{97} -1.80194 q^{98} -1.14946 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} + q^{3} + 4 q^{4} + q^{5} + 2 q^{6} - 4 q^{8} + 7 q^{9}+O(q^{10}) $ 6 * q - 2 * q^2 + q^3 + 4 * q^4 + q^5 + 2 * q^6 - 4 * q^8 + 7 * q^9	$ 6 q - 2 q^{2} + q^{3} + 4 q^{4} + q^{5} + 2 q^{6} - 4 q^{8} + 7 q^{9} - 5 q^{10} - 6 q^{11} + 3 q^{12} - q^{15} + 2 q^{16} - 7 q^{18} + q^{19} + 3 q^{20} + 2 q^{22} + q^{23} - 3 q^{24} + 7 q^{25} - q^{27} + q^{29} - 2 q^{30} - 6 q^{32} - q^{33} + 7 q^{36} + 2 q^{38} - 3 q^{40} - 2 q^{41} - 4 q^{44} - 7 q^{45} + 2 q^{46} - 2 q^{48} + 6 q^{49} + q^{53} - 2 q^{54} - q^{55} - 8 q^{57} + 2 q^{58} + q^{59} - 10 q^{60} - 2 q^{61} - 2 q^{66} - q^{69} - 7 q^{72} + 3 q^{76} + 5 q^{80} + 8 q^{81} - 4 q^{82} - q^{87} + 4 q^{88} - 7 q^{90} - 4 q^{92} - q^{95} - q^{96} + q^{97} - 2 q^{98} - 7 q^{99}+O(q^{100}) $ 6 * q - 2 * q^2 + q^3 + 4 * q^4 + q^5 + 2 * q^6 - 4 * q^8 + 7 * q^9 - 5 * q^10 - 6 * q^11 + 3 * q^12 - q^15 + 2 * q^16 - 7 * q^18 + q^19 + 3 * q^20 + 2 * q^22 + q^23 - 3 * q^24 + 7 * q^25 - q^27 + q^29 - 2 * q^30 - 6 * q^32 - q^33 + 7 * q^36 + 2 * q^38 - 3 * q^40 - 2 * q^41 - 4 * q^44 - 7 * q^45 + 2 * q^46 - 2 * q^48 + 6 * q^49 + q^53 - 2 * q^54 - q^55 - 8 * q^57 + 2 * q^58 + q^59 - 10 * q^60 - 2 * q^61 - 2 * q^66 - q^69 - 7 * q^72 + 3 * q^76 + 5 * q^80 + 8 * q^81 - 4 * q^82 - q^87 + 4 * q^88 - 7 * q^90 - 4 * q^92 - q^95 - q^96 + q^97 - 2 * q^98 - 7 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/431\mathbb{Z}\right)^\times$.

$n$	$7$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$2$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$3$	−1.46610	−1.46610	−0.733052	−	0.680173i	$-0.761905\pi$
$3$	−1.46610	−1.46610	−0.733052	+	0.680173i	$0.761905\pi$
$4$	2.24698	2.24698
$5$	1.65248	1.65248	0.826239	−	0.563320i	$-0.190476\pi$
$5$	1.65248	1.65248	0.826239	+	0.563320i	$0.190476\pi$
$6$	2.64183	2.64183
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−2.24698	−2.24698
$9$	1.14946	1.14946
$10$	−2.97766	−2.97766
$11$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$11$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$12$	−3.29431	−3.29431
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	−2.42270	−2.42270
$16$	1.80194	1.80194
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−2.07126	−2.07126
$19$	1.91115	1.91115	0.955573	−	0.294755i	$-0.0952381\pi$
$19$	1.91115	1.91115	0.955573	+	0.294755i	$0.0952381\pi$
$20$	3.71308	3.71308
$21$	0	0
$22$	1.80194	1.80194
$23$	0.730682	0.730682	0.365341	−	0.930874i	$-0.380952\pi$
$23$	0.730682	0.730682	0.365341	+	0.930874i	$0.380952\pi$
$24$	3.29431	3.29431
$25$	1.73068	1.73068
$26$	0	0
$27$	−0.219124	−0.219124
$28$	0	0
$29$	−1.97766	−1.97766	−0.988831	−	0.149042i	$-0.952381\pi$
$29$	−1.97766	−1.97766	−0.988831	+	0.149042i	$0.952381\pi$
$30$	4.36556	4.36556
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−1.00000	−1.00000
$33$	1.46610	1.46610
$34$	0	0
$35$	0	0
$36$	2.58281	2.58281
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−3.44377	−3.44377
$39$	0	0
$40$	−3.71308	−3.71308
$41$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$41$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	−2.24698	−2.24698
$45$	1.89946	1.89946
$46$	−1.31664	−1.31664
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−2.64183	−2.64183
$49$	1.00000	1.00000
$50$	−3.11858	−3.11858
$51$	0	0
$52$	0	0
$53$	1.91115	1.91115	0.955573	−	0.294755i	$-0.0952381\pi$
$53$	1.91115	1.91115	0.955573	+	0.294755i	$0.0952381\pi$
$54$	0.394848	0.394848
$55$	−1.65248	−1.65248
$56$	0	0
$57$	−2.80194	−2.80194
$58$	3.56362	3.56362
$59$	0.149460	0.149460	0.0747301	−	0.997204i	$-0.476190\pi$
$59$	0.149460	0.149460	0.0747301	+	0.997204i	$0.476190\pi$
$60$	−5.44377	−5.44377
$61$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$61$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	−2.64183	−2.64183
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	−1.07126	−1.07126
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−2.58281	−2.58281
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−2.53736	−2.53736
$76$	4.29431	4.29431
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	2.97766	2.97766
$81$	−0.828201	−0.828201
$82$	−2.24698	−2.24698
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.89946	2.89946
$88$	2.24698	2.24698
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−3.42270	−3.42270
$91$	0	0
$92$	1.64183	1.64183
$93$	0	0
$94$	0	0
$95$	3.15813	3.15813
$96$	1.46610	1.46610
$97$	−1.97766	−1.97766	−0.988831	−	0.149042i	$-0.952381\pi$
$97$	−1.97766	−1.97766	−0.988831	+	0.149042i	$0.952381\pi$
$98$	−1.80194	−1.80194
$99$	−1.14946	−1.14946
$100$	3.88881	3.88881
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	−3.44377	−3.44377
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−0.492367	−0.492367
$109$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$109$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$110$	2.97766	2.97766
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	5.04892	5.04892
$115$	1.20744	1.20744
$116$	−4.44377	−4.44377
$117$	0	0
$118$	−0.269318	−0.269318
$119$	0	0
$120$	5.44377	5.44377
$121$	0	0
$122$	0.801938	0.801938
$123$	−1.82820	−1.82820
$124$	0	0
$125$	1.20744	1.20744
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.00000	1.00000
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	3.29431	3.29431
$133$	0	0
$134$	0	0
$135$	−0.362098	−0.362098
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	1.93034	1.93034
$139$	−1.97766	−1.97766	−0.988831	−	0.149042i	$-0.952381\pi$
$139$	−1.97766	−1.97766	−0.988831	+	0.149042i	$0.952381\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	2.07126	2.07126
$145$	−3.26804	−3.26804
$146$	0	0
$147$	−1.46610	−1.46610
$148$	0	0
$149$	−1.46610	−1.46610	−0.733052	−	0.680173i	$-0.761905\pi$
$149$	−1.46610	−1.46610	−0.733052	+	0.680173i	$0.761905\pi$
$150$	4.57216	4.57216
$151$	1.65248	1.65248	0.826239	−	0.563320i	$-0.190476\pi$
$151$	1.65248	1.65248	0.826239	+	0.563320i	$0.190476\pi$
$152$	−4.29431	−4.29431
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$157$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$158$	0	0
$159$	−2.80194	−2.80194
$160$	−1.65248	−1.65248
$161$	0	0
$162$	1.49237	1.49237
$163$	0.730682	0.730682	0.365341	−	0.930874i	$-0.380952\pi$
$163$	0.730682	0.730682	0.365341	+	0.930874i	$0.380952\pi$
$164$	2.80194	2.80194
$165$	2.42270	2.42270
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	1.00000	1.00000
$170$	0	0
$171$	2.19679	2.19679
$172$	0	0
$173$	0.730682	0.730682	0.365341	−	0.930874i	$-0.380952\pi$
$173$	0.730682	0.730682	0.365341	+	0.930874i	$0.380952\pi$
$174$	−5.22464	−5.22464
$175$	0	0
$176$	−1.80194	−1.80194
$177$	−0.219124	−0.219124
$178$	0	0
$179$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$179$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$180$	4.26804	4.26804
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0.652478	0.652478
$184$	−1.64183	−1.64183
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	−5.69074	−5.69074
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	3.56362	3.56362
$195$	0	0
$196$	2.24698	2.24698
$197$	0.149460	0.149460	0.0747301	−	0.997204i	$-0.476190\pi$
$197$	0.149460	0.149460	0.0747301	+	0.997204i	$0.476190\pi$
$198$	2.07126	2.07126
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−3.88881	−3.88881
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.06061	2.06061
$206$	0	0
$207$	0.839890	0.839890
$208$	0	0
$209$	−1.91115	−1.91115
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	4.29431	4.29431
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0.492367	0.492367
$217$	0	0
$218$	3.24698	3.24698
$219$	0	0
$220$	−3.71308	−3.71308
$221$	0	0
$222$	0	0
$223$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$223$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$224$	0	0
$225$	1.98935	1.98935
$226$	0	0
$227$	−1.97766	−1.97766	−0.988831	−	0.149042i	$-0.952381\pi$
$227$	−1.97766	−1.97766	−0.988831	+	0.149042i	$0.952381\pi$
$228$	−6.29590	−6.29590
$229$	−1.46610	−1.46610	−0.733052	−	0.680173i	$-0.761905\pi$
$229$	−1.46610	−1.46610	−0.733052	+	0.680173i	$0.761905\pi$
$230$	−2.17572	−2.17572
$231$	0	0
$232$	4.44377	4.44377
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0.335834	0.335834
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	−4.36556	−4.36556
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	1.43335	1.43335
$244$	−1.00000	−1.00000
$245$	1.65248	1.65248
$246$	3.29431	3.29431
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−2.17572	−2.17572
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	−0.730682	−0.730682
$254$	0	0
$255$	0	0
$256$	−1.80194	−1.80194
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2.27324	−2.27324
$262$	0	0
$263$	0.149460	0.149460	0.0747301	−	0.997204i	$-0.476190\pi$
$263$	0.149460	0.149460	0.0747301	+	0.997204i	$0.476190\pi$
$264$	−3.29431	−3.29431
$265$	3.15813	3.15813
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0.652478	0.652478
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.73068	−1.73068
$276$	−2.40709	−2.40709
$277$	−1.46610	−1.46610	−0.733052	−	0.680173i	$-0.761905\pi$
$277$	−1.46610	−1.46610	−0.733052	+	0.680173i	$0.761905\pi$
$278$	3.56362	3.56362
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0.149460	0.149460	0.0747301	−	0.997204i	$-0.476190\pi$
$283$	0.149460	0.149460	0.0747301	+	0.997204i	$0.476190\pi$
$284$	0	0
$285$	−4.63014	−4.63014
$286$	0	0
$287$	0	0
$288$	−1.14946	−1.14946
$289$	1.00000	1.00000
$290$	5.88881	5.88881
$291$	2.89946	2.89946
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	2.64183	2.64183
$295$	0.246980	0.246980
$296$	0	0
$297$	0.219124	0.219124
$298$	2.64183	2.64183
$299$	0	0
$300$	−5.70139	−5.70139
$301$	0	0
$302$	−2.97766	−2.97766
$303$	0	0
$304$	3.44377	3.44377
$305$	−0.735422	−0.735422
$306$	0	0
$307$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$307$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	1.80194	1.80194
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	5.04892	5.04892
$319$	1.97766	1.97766
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−1.86095	−1.86095
$325$	0	0
$326$	−1.31664	−1.31664
$327$	2.64183	2.64183
$328$	−2.80194	−2.80194
$329$	0	0
$330$	−4.36556	−4.36556
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$337$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$338$	−1.80194	−1.80194
$339$	0	0
$340$	0	0
$341$	0	0
$342$	−3.95847	−3.95847
$343$	0	0
$344$	0	0
$345$	−1.77023	−1.77023
$346$	−1.31664	−1.31664
$347$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$347$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$348$	6.51502	6.51502
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	1.00000	1.00000
$353$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$353$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$354$	0.394848	0.394848
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0.801938	0.801938
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−4.26804	−4.26804
$361$	2.65248	2.65248
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	−1.17572	−1.17572
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	1.31664	1.31664
$369$	1.43335	1.43335
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−1.77023	−1.77023
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$379$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$380$	7.09624	7.09624
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−1.46610	−1.46610
$385$	0	0
$386$	0	0
$387$	0	0
$388$	−4.44377	−4.44377
$389$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$389$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$390$	0	0
$391$	0	0
$392$	−2.24698	−2.24698
$393$	0	0
$394$	−0.269318	−0.269318
$395$	0	0
$396$	−2.58281	−2.58281
$397$	0.149460	0.149460	0.0747301	−	0.997204i	$-0.476190\pi$
$397$	0.149460	0.149460	0.0747301	+	0.997204i	$0.476190\pi$
$398$	0	0
$399$	0	0
$400$	3.11858	3.11858
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.36858	−1.36858
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	−3.71308	−3.71308
$411$	0	0
$412$	0	0
$413$	0	0
$414$	−1.51343	−1.51343
$415$	0	0
$416$	0	0
$417$	2.89946	2.89946
$418$	3.44377	3.44377
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	−4.29431	−4.29431
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.00000	1.00000
$431$	1.00000	1.00000
$432$	−0.394848	−0.394848
$433$	1.65248	1.65248	0.826239	−	0.563320i	$-0.190476\pi$
$433$	1.65248	1.65248	0.826239	+	0.563320i	$0.190476\pi$
$434$	0	0
$435$	4.79129	4.79129
$436$	−4.04892	−4.04892
$437$	1.39644	1.39644
$438$	0	0
$439$	1.65248	1.65248	0.826239	−	0.563320i	$-0.190476\pi$
$439$	1.65248	1.65248	0.826239	+	0.563320i	$0.190476\pi$
$440$	3.71308	3.71308
$441$	1.14946	1.14946
$442$	0	0
$443$	1.91115	1.91115	0.955573	−	0.294755i	$-0.0952381\pi$
$443$	1.91115	1.91115	0.955573	+	0.294755i	$0.0952381\pi$
$444$	0	0
$445$	0	0
$446$	1.80194	1.80194
$447$	2.14946	2.14946
$448$	0	0
$449$	1.91115	1.91115	0.955573	−	0.294755i	$-0.0952381\pi$
$449$	1.91115	1.91115	0.955573	+	0.294755i	$0.0952381\pi$
$450$	−3.58469	−3.58469
$451$	−1.24698	−1.24698
$452$	0	0
$453$	−2.42270	−2.42270
$454$	3.56362	3.56362
$455$	0	0
$456$	6.29590	6.29590
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	2.64183	2.64183
$459$	0	0
$460$	2.71308	2.71308
$461$	0.730682	0.730682	0.365341	−	0.930874i	$-0.380952\pi$
$461$	0.730682	0.730682	0.365341	+	0.930874i	$0.380952\pi$
$462$	0	0
$463$	−1.97766	−1.97766	−0.988831	−	0.149042i	$-0.952381\pi$
$463$	−1.97766	−1.97766	−0.988831	+	0.149042i	$0.952381\pi$
$464$	−3.56362	−3.56362
$465$	0	0
$466$	0	0
$467$	2.00000	2.00000	1.00000			$0$
$467$	2.00000	2.00000	1.00000			$0$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	1.46610	1.46610
$472$	−0.335834	−0.335834
$473$	0	0
$474$	0	0
$475$	3.30759	3.30759
$476$	0	0
$477$	2.19679	2.19679
$478$	0	0
$479$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$479$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$480$	2.42270	2.42270
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.26804	−3.26804
$486$	−2.58281	−2.58281
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	1.00000	1.00000
$489$	−1.07126	−1.07126
$490$	−2.97766	−2.97766
$491$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$491$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$492$	−4.10793	−4.10793
$493$	0	0
$494$	0	0
$495$	−1.89946	−1.89946
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	2.71308	2.71308
$501$	0	0
$502$	0	0
$503$	0.149460	0.149460	0.0747301	−	0.997204i	$-0.476190\pi$
$503$	0.149460	0.149460	0.0747301	+	0.997204i	$0.476190\pi$
$504$	0	0
$505$	0	0
$506$	1.31664	1.31664
$507$	−1.46610	−1.46610
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	2.24698	2.24698
$513$	−0.418778	−0.418778
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−1.07126	−1.07126
$520$	0	0
$521$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$521$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$522$	4.09624	4.09624
$523$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$523$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$524$	0	0
$525$	0	0
$526$	−0.269318	−0.269318
$527$	0	0
$528$	2.64183	2.64183
$529$	−0.466104	−0.466104
$530$	−5.69074	−5.69074
$531$	0.171799	0.171799
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.652478	0.652478
$538$	0	0
$539$	−1.00000	−1.00000
$540$	−0.813626	−0.813626
$541$	1.91115	1.91115	0.955573	−	0.294755i	$-0.0952381\pi$
$541$	1.91115	1.91115	0.955573	+	0.294755i	$0.0952381\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.97766	−2.97766
$546$	0	0
$547$	1.65248	1.65248	0.826239	−	0.563320i	$-0.190476\pi$
$547$	1.65248	1.65248	0.826239	+	0.563320i	$0.190476\pi$
$548$	0	0
$549$	−0.511558	−0.511558
$550$	3.11858	3.11858
$551$	−3.77960	−3.77960
$552$	2.40709	2.40709
$553$	0	0
$554$	2.64183	2.64183
$555$	0	0
$556$	−4.44377	−4.44377
$557$	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
$563$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$563$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$564$	0	0
$565$	0	0
$566$	−0.269318	−0.269318
$567$	0	0
$568$	0	0
$569$	1.91115	1.91115	0.955573	−	0.294755i	$-0.0952381\pi$
$569$	1.91115	1.91115	0.955573	+	0.294755i	$0.0952381\pi$
$570$	8.34322	8.34322
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.26458	1.26458
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−1.80194	−1.80194
$579$	0	0
$580$	−7.34322	−7.34322
$581$	0	0
$582$	−5.22464	−5.22464
$583$	−1.91115	−1.91115
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−3.29431	−3.29431
$589$	0	0
$590$	−0.445042	−0.445042
$591$	−0.219124	−0.219124
$592$	0	0
$593$	−1.97766	−1.97766	−0.988831	−	0.149042i	$-0.952381\pi$
$593$	−1.97766	−1.97766	−0.988831	+	0.149042i	$0.952381\pi$
$594$	−0.394848	−0.394848
$595$	0	0
$596$	−3.29431	−3.29431
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	5.70139	5.70139
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	3.71308	3.71308
$605$	0	0
$606$	0	0
$607$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$607$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$608$	−1.91115	−1.91115
$609$	0	0
$610$	1.32518	1.32518
$611$	0	0
$612$	0	0
$613$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$613$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$614$	3.24698	3.24698
$615$	−3.02106	−3.02106
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	−0.160110	−0.160110
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0.264578	0.264578
$626$	0	0
$627$	2.80194	2.80194
$628$	−2.24698	−2.24698
$629$	0	0
$630$	0	0
$631$	−1.46610	−1.46610	−0.733052	−	0.680173i	$-0.761905\pi$
$631$	−1.46610	−1.46610	−0.733052	+	0.680173i	$0.761905\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−6.29590	−6.29590
$637$	0	0
$638$	−3.56362	−3.56362
$639$	0	0
$640$	1.65248	1.65248
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$643$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.97766	−1.97766	−0.988831	−	0.149042i	$-0.952381\pi$
$647$	−1.97766	−1.97766	−0.988831	+	0.149042i	$0.952381\pi$
$648$	1.86095	1.86095
$649$	−0.149460	−0.149460
$650$	0	0
$651$	0	0
$652$	1.64183	1.64183
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	−4.76041	−4.76041
$655$	0	0
$656$	2.24698	2.24698
$657$	0	0
$658$	0	0
$659$	1.91115	1.91115	0.955573	−	0.294755i	$-0.0952381\pi$
$659$	1.91115	1.91115	0.955573	+	0.294755i	$0.0952381\pi$
$660$	5.44377	5.44377
$661$	0.149460	0.149460	0.0747301	−	0.997204i	$-0.476190\pi$
$661$	0.149460	0.149460	0.0747301	+	0.997204i	$0.476190\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.44504	−1.44504
$668$	0	0
$669$	1.46610	1.46610
$670$	0	0
$671$	0.445042	0.445042
$672$	0	0
$673$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$673$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$674$	0.801938	0.801938
$675$	−0.379234	−0.379234
$676$	2.24698	2.24698
$677$	−1.46610	−1.46610	−0.733052	−	0.680173i	$-0.761905\pi$
$677$	−1.46610	−1.46610	−0.733052	+	0.680173i	$0.761905\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	2.89946	2.89946
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	4.93613	4.93613
$685$	0	0
$686$	0	0
$687$	2.14946	2.14946
$688$	0	0
$689$	0	0
$690$	3.18984	3.18984
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	1.64183	1.64183
$693$	0	0
$694$	0.801938	0.801938
$695$	−3.26804	−3.26804
$696$	−6.51502	−6.51502
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$701$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	1.80194	1.80194
$707$	0	0
$708$	−0.492367	−0.492367
$709$	0.730682	0.730682	0.365341	−	0.930874i	$-0.380952\pi$
$709$	0.730682	0.730682	0.365341	+	0.930874i	$0.380952\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−1.00000	−1.00000
$717$	0	0
$718$	0	0
$719$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$719$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$720$	3.42270	3.42270
$721$	0	0
$722$	−4.77960	−4.77960
$723$	0	0
$724$	0	0
$725$	−3.42270	−3.42270
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−1.27324	−1.27324
$730$	0	0
$731$	0	0
$732$	1.46610	1.46610
$733$	0.730682	0.730682	0.365341	−	0.930874i	$-0.380952\pi$
$733$	0.730682	0.730682	0.365341	+	0.930874i	$0.380952\pi$
$734$	0	0
$735$	−2.42270	−2.42270
$736$	−0.730682	−0.730682
$737$	0	0
$738$	−2.58281	−2.58281
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−2.42270	−2.42270
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	3.18984	3.18984
$751$	−1.46610	−1.46610	−0.733052	−	0.680173i	$-0.761905\pi$
$751$	−1.46610	−1.46610	−0.733052	+	0.680173i	$0.761905\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.73068	2.73068
$756$	0	0
$757$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$757$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$758$	3.24698	3.24698
$759$	1.07126	1.07126
$760$	−7.09624	−7.09624
$761$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$761$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	2.64183	2.64183
$769$	0.149460	0.149460	0.0747301	−	0.997204i	$-0.476190\pi$
$769$	0.149460	0.149460	0.0747301	+	0.997204i	$0.476190\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$773$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$774$	0	0
$775$	0	0
$776$	4.44377	4.44377
$777$	0	0
$778$	1.80194	1.80194
$779$	2.38316	2.38316
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0.433353	0.433353
$784$	1.80194	1.80194
$785$	−1.65248	−1.65248
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0.335834	0.335834
$789$	−0.219124	−0.219124
$790$	0	0
$791$	0	0
$792$	2.58281	2.58281
$793$	0	0
$794$	−0.269318	−0.269318
$795$	−4.63014	−4.63014
$796$	0	0
$797$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$797$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$798$	0	0
$799$	0	0
$800$	−1.73068	−1.73068
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	2.46610	2.46610
$811$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$811$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.20744	1.20744
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	4.63014	4.63014
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	1.91115	1.91115	0.955573	−	0.294755i	$-0.0952381\pi$
$823$	1.91115	1.91115	0.955573	+	0.294755i	$0.0952381\pi$
$824$	0	0
$825$	2.53736	2.53736
$826$	0	0
$827$	1.65248	1.65248	0.826239	−	0.563320i	$-0.190476\pi$
$827$	1.65248	1.65248	0.826239	+	0.563320i	$0.190476\pi$
$828$	1.88722	1.88722
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	2.14946	2.14946
$832$	0	0
$833$	0	0
$834$	−5.22464	−5.22464
$835$	0	0
$836$	−4.29431	−4.29431
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	2.91115	2.91115
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.65248	1.65248
$846$	0	0
$847$	0	0
$848$	3.44377	3.44377
$849$	−0.219124	−0.219124
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	3.63014	3.63014
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	−1.80194	−1.80194
$863$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$863$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$864$	0.219124	0.219124
$865$	1.20744	1.20744
$866$	−2.97766	−2.97766
$867$	−1.46610	−1.46610
$868$	0	0
$869$	0	0
$870$	−8.63360	−8.63360
$871$	0	0
$872$	4.04892	4.04892
$873$	−2.27324	−2.27324
$874$	−2.51630	−2.51630
$875$	0	0
$876$	0	0
$877$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$877$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$878$	−2.97766	−2.97766
$879$	0	0
$880$	−2.97766	−2.97766
$881$	0.730682	0.730682	0.365341	−	0.930874i	$-0.380952\pi$
$881$	0.730682	0.730682	0.365341	+	0.930874i	$0.380952\pi$
$882$	−2.07126	−2.07126
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	−0.362098	−0.362098
$886$	−3.44377	−3.44377
$887$	1.91115	1.91115	0.955573	−	0.294755i	$-0.0952381\pi$
$887$	1.91115	1.91115	0.955573	+	0.294755i	$0.0952381\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0.828201	0.828201
$892$	−2.24698	−2.24698
$893$	0	0
$894$	−3.87319	−3.87319
$895$	−0.735422	−0.735422
$896$	0	0
$897$	0	0
$898$	−3.44377	−3.44377
$899$	0	0
$900$	4.47003	4.47003
$901$	0	0
$902$	2.24698	2.24698
$903$	0	0
$904$	0	0
$905$	0	0
$906$	4.36556	4.36556
$907$	−1.97766	−1.97766	−0.988831	−	0.149042i	$-0.952381\pi$
$907$	−1.97766	−1.97766	−0.988831	+	0.149042i	$0.952381\pi$
$908$	−4.44377	−4.44377
$909$	0	0
$910$	0	0
$911$	1.65248	1.65248	0.826239	−	0.563320i	$-0.190476\pi$
$911$	1.65248	1.65248	0.826239	+	0.563320i	$0.190476\pi$
$912$	−5.04892	−5.04892
$913$	0	0
$914$	0	0
$915$	1.07820	1.07820
$916$	−3.29431	−3.29431
$917$	0	0
$918$	0	0
$919$	−1.97766	−1.97766	−0.988831	−	0.149042i	$-0.952381\pi$
$919$	−1.97766	−1.97766	−0.988831	+	0.149042i	$0.952381\pi$
$920$	−2.71308	−2.71308
$921$	2.64183	2.64183
$922$	−1.31664	−1.31664
$923$	0	0
$924$	0	0
$925$	0	0
$926$	3.56362	3.56362
$927$	0	0
$928$	1.97766	1.97766
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	1.91115	1.91115
$932$	0	0
$933$	0	0
$934$	−3.60388	−3.60388
$935$	0	0
$936$	0	0
$937$	1.91115	1.91115	0.955573	−	0.294755i	$-0.0952381\pi$
$937$	1.91115	1.91115	0.955573	+	0.294755i	$0.0952381\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	−2.64183	−2.64183
$943$	0.911146	0.911146
$944$	0.269318	0.269318
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	−5.96006	−5.96006
$951$	0	0
$952$	0	0
$953$	0.149460	0.149460	0.0747301	−	0.997204i	$-0.476190\pi$
$953$	0.149460	0.149460	0.0747301	+	0.997204i	$0.476190\pi$
$954$	−3.95847	−3.95847
$955$	0	0
$956$	0	0
$957$	−2.89946	−2.89946
$958$	0.801938	0.801938
$959$	0	0
$960$	0	0
$961$	1.00000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	0	0
$970$	5.88881	5.88881
$971$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$971$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$972$	3.22072	3.22072
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−0.801938	−0.801938
$977$	−1.46610	−1.46610	−0.733052	−	0.680173i	$-0.761905\pi$
$977$	−1.46610	−1.46610	−0.733052	+	0.680173i	$0.761905\pi$
$978$	1.93034	1.93034
$979$	0	0
$980$	3.71308	3.71308
$981$	−2.07126	−2.07126
$982$	3.24698	3.24698
$983$	−1.46610	−1.46610	−0.733052	−	0.680173i	$-0.761905\pi$
$983$	−1.46610	−1.46610	−0.733052	+	0.680173i	$0.761905\pi$
$984$	4.10793	4.10793
$985$	0.246980	0.246980
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	3.42270	3.42270
$991$	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
$997$	0.730682	0.730682	0.365341	−	0.930874i	$-0.380952\pi$
$997$	0.730682	0.730682	0.365341	+	0.930874i	$0.380952\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	431.1.b.c.430.1	✓	6
3.2	odd	2		3879.1.d.c.3016.5		6
431.430	odd	2	CM	431.1.b.c.430.1	✓	6
1293.1292	even	2		3879.1.d.c.3016.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
431.1.b.c.430.1	✓	6	1.1	even	1	trivial
431.1.b.c.430.1	✓	6	431.430	odd	2	CM
3879.1.d.c.3016.5		6	3.2	odd	2
3879.1.d.c.3016.5		6	1293.1292	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 431 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	431.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.80194 q^{2} -1.46610 q^{3} +2.24698 q^{4} +1.65248 q^{5} +2.64183 q^{6} -2.24698 q^{8} +1.14946 q^{9} +O(q^{10})\)	\(q-1.80194 q^{2} -1.46610 q^{3} +2.24698 q^{4} +1.65248 q^{5} +2.64183 q^{6} -2.24698 q^{8} +1.14946 q^{9} -2.97766 q^{10} -1.00000 q^{11} -3.29431 q^{12} -2.42270 q^{15} +1.80194 q^{16} -2.07126 q^{18} +1.91115 q^{19} +3.71308 q^{20} +1.80194 q^{22} +0.730682 q^{23} +3.29431 q^{24} +1.73068 q^{25} -0.219124 q^{27} -1.97766 q^{29} +4.36556 q^{30} -1.00000 q^{32} +1.46610 q^{33} +2.58281 q^{36} -3.44377 q^{38} -3.71308 q^{40} +1.24698 q^{41} -2.24698 q^{44} +1.89946 q^{45} -1.31664 q^{46} -2.64183 q^{48} +1.00000 q^{49} -3.11858 q^{50} +1.91115 q^{53} +0.394848 q^{54} -1.65248 q^{55} -2.80194 q^{57} +3.56362 q^{58} +0.149460 q^{59} -5.44377 q^{60} -0.445042 q^{61} -2.64183 q^{66} -1.07126 q^{69} -2.58281 q^{72} -2.53736 q^{75} +4.29431 q^{76} +2.97766 q^{80} -0.828201 q^{81} -2.24698 q^{82} +2.89946 q^{87} +2.24698 q^{88} -3.42270 q^{90} +1.64183 q^{92} +3.15813 q^{95} +1.46610 q^{96} -1.97766 q^{97} -1.80194 q^{98} -1.14946 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} + q^{3} + 4 q^{4} + q^{5} + 2 q^{6} - 4 q^{8} + 7 q^{9}+O(q^{10}) \) 6 * q - 2 * q^2 + q^3 + 4 * q^4 + q^5 + 2 * q^6 - 4 * q^8 + 7 * q^9	\( 6 q - 2 q^{2} + q^{3} + 4 q^{4} + q^{5} + 2 q^{6} - 4 q^{8} + 7 q^{9} - 5 q^{10} - 6 q^{11} + 3 q^{12} - q^{15} + 2 q^{16} - 7 q^{18} + q^{19} + 3 q^{20} + 2 q^{22} + q^{23} - 3 q^{24} + 7 q^{25} - q^{27} + q^{29} - 2 q^{30} - 6 q^{32} - q^{33} + 7 q^{36} + 2 q^{38} - 3 q^{40} - 2 q^{41} - 4 q^{44} - 7 q^{45} + 2 q^{46} - 2 q^{48} + 6 q^{49} + q^{53} - 2 q^{54} - q^{55} - 8 q^{57} + 2 q^{58} + q^{59} - 10 q^{60} - 2 q^{61} - 2 q^{66} - q^{69} - 7 q^{72} + 3 q^{76} + 5 q^{80} + 8 q^{81} - 4 q^{82} - q^{87} + 4 q^{88} - 7 q^{90} - 4 q^{92} - q^{95} - q^{96} + q^{97} - 2 q^{98} - 7 q^{99}+O(q^{100}) \) 6 * q - 2 * q^2 + q^3 + 4 * q^4 + q^5 + 2 * q^6 - 4 * q^8 + 7 * q^9 - 5 * q^10 - 6 * q^11 + 3 * q^12 - q^15 + 2 * q^16 - 7 * q^18 + q^19 + 3 * q^20 + 2 * q^22 + q^23 - 3 * q^24 + 7 * q^25 - q^27 + q^29 - 2 * q^30 - 6 * q^32 - q^33 + 7 * q^36 + 2 * q^38 - 3 * q^40 - 2 * q^41 - 4 * q^44 - 7 * q^45 + 2 * q^46 - 2 * q^48 + 6 * q^49 + q^53 - 2 * q^54 - q^55 - 8 * q^57 + 2 * q^58 + q^59 - 10 * q^60 - 2 * q^61 - 2 * q^66 - q^69 - 7 * q^72 + 3 * q^76 + 5 * q^80 + 8 * q^81 - 4 * q^82 - q^87 + 4 * q^88 - 7 * q^90 - 4 * q^92 - q^95 - q^96 + q^97 - 2 * q^98 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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