LMFDB - Embedded newform 431.1.b.c.430.5

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			430.5
Root			$0.730682$ 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)

$f(q)$	$=$	$q+1.24698 q^{2} +0.149460 q^{3} +0.554958 q^{4} +0.730682 q^{5} +0.186374 q^{6} -0.554958 q^{8} -0.977662 q^{9} +O(q^{10})$	$q+1.24698 q^{2} +0.149460 q^{3} +0.554958 q^{4} +0.730682 q^{5} +0.186374 q^{6} -0.554958 q^{8} -0.977662 q^{9} +0.911146 q^{10} -1.00000 q^{11} +0.0829441 q^{12} +0.109208 q^{15} -1.24698 q^{16} -1.21912 q^{18} +1.65248 q^{19} +0.405498 q^{20} -1.24698 q^{22} -1.46610 q^{23} -0.0829441 q^{24} -0.466104 q^{25} -0.295582 q^{27} +1.91115 q^{29} +0.136180 q^{30} -1.00000 q^{32} -0.149460 q^{33} -0.542561 q^{36} +2.06061 q^{38} -0.405498 q^{40} -0.445042 q^{41} -0.554958 q^{44} -0.714360 q^{45} -1.82820 q^{46} -0.186374 q^{48} +1.00000 q^{49} -0.581222 q^{50} +1.65248 q^{53} -0.368584 q^{54} -0.730682 q^{55} +0.246980 q^{57} +2.38316 q^{58} -1.97766 q^{59} +0.0606058 q^{60} -1.80194 q^{61} -0.186374 q^{66} -0.219124 q^{69} +0.542561 q^{72} -0.0696640 q^{75} +0.917056 q^{76} -0.911146 q^{80} +0.933484 q^{81} -0.554958 q^{82} +0.285640 q^{87} +0.554958 q^{88} -0.890792 q^{90} -0.813626 q^{92} +1.20744 q^{95} -0.149460 q^{96} +1.91115 q^{97} +1.24698 q^{98} +0.977662 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$2$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$3$	0.149460	0.149460	0.0747301	−	0.997204i	$-0.476190\pi$
$3$	0.149460	0.149460	0.0747301	+	0.997204i	$0.476190\pi$
$4$	0.554958	0.554958
$5$	0.730682	0.730682	0.365341	−	0.930874i	$-0.380952\pi$
$5$	0.730682	0.730682	0.365341	+	0.930874i	$0.380952\pi$
$6$	0.186374	0.186374
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−0.554958	−0.554958
$9$	−0.977662	−0.977662
$10$	0.911146	0.911146
$11$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$11$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$12$	0.0829441	0.0829441
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0.109208	0.109208
$16$	−1.24698	−1.24698
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−1.21912	−1.21912
$19$	1.65248	1.65248	0.826239	−	0.563320i	$-0.190476\pi$
$19$	1.65248	1.65248	0.826239	+	0.563320i	$0.190476\pi$
$20$	0.405498	0.405498
$21$	0	0
$22$	−1.24698	−1.24698
$23$	−1.46610	−1.46610	−0.733052	−	0.680173i	$-0.761905\pi$
$23$	−1.46610	−1.46610	−0.733052	+	0.680173i	$0.761905\pi$
$24$	−0.0829441	−0.0829441
$25$	−0.466104	−0.466104
$26$	0	0
$27$	−0.295582	−0.295582
$28$	0	0
$29$	1.91115	1.91115	0.955573	−	0.294755i	$-0.0952381\pi$
$29$	1.91115	1.91115	0.955573	+	0.294755i	$0.0952381\pi$
$30$	0.136180	0.136180
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−1.00000	−1.00000
$33$	−0.149460	−0.149460
$34$	0	0
$35$	0	0
$36$	−0.542561	−0.542561
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	2.06061	2.06061
$39$	0	0
$40$	−0.405498	−0.405498
$41$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$41$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	−0.554958	−0.554958
$45$	−0.714360	−0.714360
$46$	−1.82820	−1.82820
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−0.186374	−0.186374
$49$	1.00000	1.00000
$50$	−0.581222	−0.581222
$51$	0	0
$52$	0	0
$53$	1.65248	1.65248	0.826239	−	0.563320i	$-0.190476\pi$
$53$	1.65248	1.65248	0.826239	+	0.563320i	$0.190476\pi$
$54$	−0.368584	−0.368584
$55$	−0.730682	−0.730682
$56$	0	0
$57$	0.246980	0.246980
$58$	2.38316	2.38316
$59$	−1.97766	−1.97766	−0.988831	−	0.149042i	$-0.952381\pi$
$59$	−1.97766	−1.97766	−0.988831	+	0.149042i	$0.952381\pi$
$60$	0.0606058	0.0606058
$61$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$61$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	−0.186374	−0.186374
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	−0.219124	−0.219124
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0.542561	0.542561
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−0.0696640	−0.0696640
$76$	0.917056	0.917056
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−0.911146	−0.911146
$81$	0.933484	0.933484
$82$	−0.554958	−0.554958
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.285640	0.285640
$88$	0.554958	0.554958
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−0.890792	−0.890792
$91$	0	0
$92$	−0.813626	−0.813626
$93$	0	0
$94$	0	0
$95$	1.20744	1.20744
$96$	−0.149460	−0.149460
$97$	1.91115	1.91115	0.955573	−	0.294755i	$-0.0952381\pi$
$97$	1.91115	1.91115	0.955573	+	0.294755i	$0.0952381\pi$
$98$	1.24698	1.24698
$99$	0.977662	0.977662
$100$	−0.258668	−0.258668
$101$	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
$106$	2.06061	2.06061
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−0.164035	−0.164035
$109$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$109$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$110$	−0.911146	−0.911146
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0.307979	0.307979
$115$	−1.07126	−1.07126
$116$	1.06061	1.06061
$117$	0	0
$118$	−2.46610	−2.46610
$119$	0	0
$120$	−0.0606058	−0.0606058
$121$	0	0
$122$	−2.24698	−2.24698
$123$	−0.0665160	−0.0665160
$124$	0	0
$125$	−1.07126	−1.07126
$126$	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			$\pi$
$132$	−0.0829441	−0.0829441
$133$	0	0
$134$	0	0
$135$	−0.215976	−0.215976
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	−0.273243	−0.273243
$139$	1.91115	1.91115	0.955573	−	0.294755i	$-0.0952381\pi$
$139$	1.91115	1.91115	0.955573	+	0.294755i	$0.0952381\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	1.21912	1.21912
$145$	1.39644	1.39644
$146$	0	0
$147$	0.149460	0.149460
$148$	0	0
$149$	0.149460	0.149460	0.0747301	−	0.997204i	$-0.476190\pi$
$149$	0.149460	0.149460	0.0747301	+	0.997204i	$0.476190\pi$
$150$	−0.0868695	−0.0868695
$151$	0.730682	0.730682	0.365341	−	0.930874i	$-0.380952\pi$
$151$	0.730682	0.730682	0.365341	+	0.930874i	$0.380952\pi$
$152$	−0.917056	−0.917056
$153$	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
$159$	0.246980	0.246980
$160$	−0.730682	−0.730682
$161$	0	0
$162$	1.16404	1.16404
$163$	−1.46610	−1.46610	−0.733052	−	0.680173i	$-0.761905\pi$
$163$	−1.46610	−1.46610	−0.733052	+	0.680173i	$0.761905\pi$
$164$	−0.246980	−0.246980
$165$	−0.109208	−0.109208
$166$	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
$171$	−1.61556	−1.61556
$172$	0	0
$173$	−1.46610	−1.46610	−0.733052	−	0.680173i	$-0.761905\pi$
$173$	−1.46610	−1.46610	−0.733052	+	0.680173i	$0.761905\pi$
$174$	0.356187	0.356187
$175$	0	0
$176$	1.24698	1.24698
$177$	−0.295582	−0.295582
$178$	0	0
$179$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$179$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$180$	−0.396440	−0.396440
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	−0.269318	−0.269318
$184$	0.813626	0.813626
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	1.50565	1.50565
$191$	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			$\pi$
$194$	2.38316	2.38316
$195$	0	0
$196$	0.554958	0.554958
$197$	−1.97766	−1.97766	−0.988831	−	0.149042i	$-0.952381\pi$
$197$	−1.97766	−1.97766	−0.988831	+	0.149042i	$0.952381\pi$
$198$	1.21912	1.21912
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0.258668	0.258668
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−0.325184	−0.325184
$206$	0	0
$207$	1.43335	1.43335
$208$	0	0
$209$	−1.65248	−1.65248
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0.917056	0.917056
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0.164035	0.164035
$217$	0	0
$218$	1.55496	1.55496
$219$	0	0
$220$	−0.405498	−0.405498
$221$	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
$225$	0.455692	0.455692
$226$	0	0
$227$	1.91115	1.91115	0.955573	−	0.294755i	$-0.0952381\pi$
$227$	1.91115	1.91115	0.955573	+	0.294755i	$0.0952381\pi$
$228$	0.137063	0.137063
$229$	0.149460	0.149460	0.0747301	−	0.997204i	$-0.476190\pi$
$229$	0.149460	0.149460	0.0747301	+	0.997204i	$0.476190\pi$
$230$	−1.33583	−1.33583
$231$	0	0
$232$	−1.06061	−1.06061
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	−1.09752	−1.09752
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	−0.136180	−0.136180
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0.435100	0.435100
$244$	−1.00000	−1.00000
$245$	0.730682	0.730682
$246$	−0.0829441	−0.0829441
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−1.33583	−1.33583
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	1.46610	1.46610
$254$	0	0
$255$	0	0
$256$	1.24698	1.24698
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.86845	−1.86845
$262$	0	0
$263$	−1.97766	−1.97766	−0.988831	−	0.149042i	$-0.952381\pi$
$263$	−1.97766	−1.97766	−0.988831	+	0.149042i	$0.952381\pi$
$264$	0.0829441	0.0829441
$265$	1.20744	1.20744
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	−0.269318	−0.269318
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.466104	0.466104
$276$	−0.121605	−0.121605
$277$	0.149460	0.149460	0.0747301	−	0.997204i	$-0.476190\pi$
$277$	0.149460	0.149460	0.0747301	+	0.997204i	$0.476190\pi$
$278$	2.38316	2.38316
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	−1.97766	−1.97766	−0.988831	−	0.149042i	$-0.952381\pi$
$283$	−1.97766	−1.97766	−0.988831	+	0.149042i	$0.952381\pi$
$284$	0	0
$285$	0.180464	0.180464
$286$	0	0
$287$	0	0
$288$	0.977662	0.977662
$289$	1.00000	1.00000
$290$	1.74133	1.74133
$291$	0.285640	0.285640
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0.186374	0.186374
$295$	−1.44504	−1.44504
$296$	0	0
$297$	0.295582	0.295582
$298$	0.186374	0.186374
$299$	0	0
$300$	−0.0386606	−0.0386606
$301$	0	0
$302$	0.911146	0.911146
$303$	0	0
$304$	−2.06061	−2.06061
$305$	−1.31664	−1.31664
$306$	0	0
$307$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$307$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$308$	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			$\pi$
$314$	−1.24698	−1.24698
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0.307979	0.307979
$319$	−1.91115	−1.91115
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0.518045	0.518045
$325$	0	0
$326$	−1.82820	−1.82820
$327$	0.186374	0.186374
$328$	0.246980	0.246980
$329$	0	0
$330$	−0.136180	−0.136180
$331$	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
$337$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$337$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$338$	1.24698	1.24698
$339$	0	0
$340$	0	0
$341$	0	0
$342$	−2.01458	−2.01458
$343$	0	0
$344$	0	0
$345$	−0.160110	−0.160110
$346$	−1.82820	−1.82820
$347$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$347$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$348$	0.158518	0.158518
$349$	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	$0.666667\pi$
$354$	−0.368584	−0.368584
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−2.24698	−2.24698
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0.396440	0.396440
$361$	1.73068	1.73068
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	−0.335834	−0.335834
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	1.82820	1.82820
$369$	0.435100	0.435100
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−0.160110	−0.160110
$376$	0	0
$377$	0	0
$378$	0	0
$379$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$379$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$380$	0.670076	0.670076
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0.149460	0.149460
$385$	0	0
$386$	0	0
$387$	0	0
$388$	1.06061	1.06061
$389$	−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
$392$	−0.554958	−0.554958
$393$	0	0
$394$	−2.46610	−2.46610
$395$	0	0
$396$	0.542561	0.542561
$397$	−1.97766	−1.97766	−0.988831	−	0.149042i	$-0.952381\pi$
$397$	−1.97766	−1.97766	−0.988831	+	0.149042i	$0.952381\pi$
$398$	0	0
$399$	0	0
$400$	0.581222	0.581222
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0.682080	0.682080
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	−0.405498	−0.405498
$411$	0	0
$412$	0	0
$413$	0	0
$414$	1.78736	1.78736
$415$	0	0
$416$	0	0
$417$	0.285640	0.285640
$418$	−2.06061	−2.06061
$419$	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
$424$	−0.917056	−0.917056
$425$	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
$432$	0.368584	0.368584
$433$	0.730682	0.730682	0.365341	−	0.930874i	$-0.380952\pi$
$433$	0.730682	0.730682	0.365341	+	0.930874i	$0.380952\pi$
$434$	0	0
$435$	0.208712	0.208712
$436$	0.692021	0.692021
$437$	−2.42270	−2.42270
$438$	0	0
$439$	0.730682	0.730682	0.365341	−	0.930874i	$-0.380952\pi$
$439$	0.730682	0.730682	0.365341	+	0.930874i	$0.380952\pi$
$440$	0.405498	0.405498
$441$	−0.977662	−0.977662
$442$	0	0
$443$	1.65248	1.65248	0.826239	−	0.563320i	$-0.190476\pi$
$443$	1.65248	1.65248	0.826239	+	0.563320i	$0.190476\pi$
$444$	0	0
$445$	0	0
$446$	−1.24698	−1.24698
$447$	0.0223383	0.0223383
$448$	0	0
$449$	1.65248	1.65248	0.826239	−	0.563320i	$-0.190476\pi$
$449$	1.65248	1.65248	0.826239	+	0.563320i	$0.190476\pi$
$450$	0.568238	0.568238
$451$	0.445042	0.445042
$452$	0	0
$453$	0.109208	0.109208
$454$	2.38316	2.38316
$455$	0	0
$456$	−0.137063	−0.137063
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0.186374	0.186374
$459$	0	0
$460$	−0.594502	−0.594502
$461$	−1.46610	−1.46610	−0.733052	−	0.680173i	$-0.761905\pi$
$461$	−1.46610	−1.46610	−0.733052	+	0.680173i	$0.761905\pi$
$462$	0	0
$463$	1.91115	1.91115	0.955573	−	0.294755i	$-0.0952381\pi$
$463$	1.91115	1.91115	0.955573	+	0.294755i	$0.0952381\pi$
$464$	−2.38316	−2.38316
$465$	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
$471$	−0.149460	−0.149460
$472$	1.09752	1.09752
$473$	0	0
$474$	0	0
$475$	−0.770226	−0.770226
$476$	0	0
$477$	−1.61556	−1.61556
$478$	0	0
$479$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$479$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$480$	−0.109208	−0.109208
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.39644	1.39644
$486$	0.542561	0.542561
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	1.00000	1.00000
$489$	−0.219124	−0.219124
$490$	0.911146	0.911146
$491$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$491$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$492$	−0.0369136	−0.0369136
$493$	0	0
$494$	0	0
$495$	0.714360	0.714360
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−0.594502	−0.594502
$501$	0	0
$502$	0	0
$503$	−1.97766	−1.97766	−0.988831	−	0.149042i	$-0.952381\pi$
$503$	−1.97766	−1.97766	−0.988831	+	0.149042i	$0.952381\pi$
$504$	0	0
$505$	0	0
$506$	1.82820	1.82820
$507$	0.149460	0.149460
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0.554958	0.554958
$513$	−0.488442	−0.488442
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−0.219124	−0.219124
$520$	0	0
$521$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$521$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$522$	−2.32992	−2.32992
$523$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$523$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$524$	0	0
$525$	0	0
$526$	−2.46610	−2.46610
$527$	0	0
$528$	0.186374	0.186374
$529$	1.14946	1.14946
$530$	1.50565	1.50565
$531$	1.93348	1.93348
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−0.269318	−0.269318
$538$	0	0
$539$	−1.00000	−1.00000
$540$	−0.119858	−0.119858
$541$	1.65248	1.65248	0.826239	−	0.563320i	$-0.190476\pi$
$541$	1.65248	1.65248	0.826239	+	0.563320i	$0.190476\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0.911146	0.911146
$546$	0	0
$547$	0.730682	0.730682	0.365341	−	0.930874i	$-0.380952\pi$
$547$	0.730682	0.730682	0.365341	+	0.930874i	$0.380952\pi$
$548$	0	0
$549$	1.76169	1.76169
$550$	0.581222	0.581222
$551$	3.15813	3.15813
$552$	0.121605	0.121605
$553$	0	0
$554$	0.186374	0.186374
$555$	0	0
$556$	1.06061	1.06061
$557$	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
$563$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$563$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$564$	0	0
$565$	0	0
$566$	−2.46610	−2.46610
$567$	0	0
$568$	0	0
$569$	1.65248	1.65248	0.826239	−	0.563320i	$-0.190476\pi$
$569$	1.65248	1.65248	0.826239	+	0.563320i	$0.190476\pi$
$570$	0.225034	0.225034
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.683356	0.683356
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	1.24698	1.24698
$579$	0	0
$580$	0.774966	0.774966
$581$	0	0
$582$	0.356187	0.356187
$583$	−1.65248	−1.65248
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0.0829441	0.0829441
$589$	0	0
$590$	−1.80194	−1.80194
$591$	−0.295582	−0.295582
$592$	0	0
$593$	1.91115	1.91115	0.955573	−	0.294755i	$-0.0952381\pi$
$593$	1.91115	1.91115	0.955573	+	0.294755i	$0.0952381\pi$
$594$	0.368584	0.368584
$595$	0	0
$596$	0.0829441	0.0829441
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0.0386606	0.0386606
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0.405498	0.405498
$605$	0	0
$606$	0	0
$607$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$607$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$608$	−1.65248	−1.65248
$609$	0	0
$610$	−1.64183	−1.64183
$611$	0	0
$612$	0	0
$613$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$613$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$614$	1.55496	1.55496
$615$	−0.0486021	−0.0486021
$616$	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
$621$	0.433353	0.433353
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−0.316644	−0.316644
$626$	0	0
$627$	−0.246980	−0.246980
$628$	−0.554958	−0.554958
$629$	0	0
$630$	0	0
$631$	0.149460	0.149460	0.0747301	−	0.997204i	$-0.476190\pi$
$631$	0.149460	0.149460	0.0747301	+	0.997204i	$0.476190\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0.137063	0.137063
$637$	0	0
$638$	−2.38316	−2.38316
$639$	0	0
$640$	0.730682	0.730682
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$643$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.91115	1.91115	0.955573	−	0.294755i	$-0.0952381\pi$
$647$	1.91115	1.91115	0.955573	+	0.294755i	$0.0952381\pi$
$648$	−0.518045	−0.518045
$649$	1.97766	1.97766
$650$	0	0
$651$	0	0
$652$	−0.813626	−0.813626
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0.232404	0.232404
$655$	0	0
$656$	0.554958	0.554958
$657$	0	0
$658$	0	0
$659$	1.65248	1.65248	0.826239	−	0.563320i	$-0.190476\pi$
$659$	1.65248	1.65248	0.826239	+	0.563320i	$0.190476\pi$
$660$	−0.0606058	−0.0606058
$661$	−1.97766	−1.97766	−0.988831	−	0.149042i	$-0.952381\pi$
$661$	−1.97766	−1.97766	−0.988831	+	0.149042i	$0.952381\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.80194	−2.80194
$668$	0	0
$669$	−0.149460	−0.149460
$670$	0	0
$671$	1.80194	1.80194
$672$	0	0
$673$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$673$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$674$	−2.24698	−2.24698
$675$	0.137772	0.137772
$676$	0.554958	0.554958
$677$	0.149460	0.149460	0.0747301	−	0.997204i	$-0.476190\pi$
$677$	0.149460	0.149460	0.0747301	+	0.997204i	$0.476190\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0.285640	0.285640
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	−0.896570	−0.896570
$685$	0	0
$686$	0	0
$687$	0.0223383	0.0223383
$688$	0	0
$689$	0	0
$690$	−0.199654	−0.199654
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	−0.813626	−0.813626
$693$	0	0
$694$	−2.24698	−2.24698
$695$	1.39644	1.39644
$696$	−0.158518	−0.158518
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$701$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−1.24698	−1.24698
$707$	0	0
$708$	−0.164035	−0.164035
$709$	−1.46610	−1.46610	−0.733052	−	0.680173i	$-0.761905\pi$
$709$	−1.46610	−1.46610	−0.733052	+	0.680173i	$0.761905\pi$
$710$	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
$719$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$719$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$720$	0.890792	0.890792
$721$	0	0
$722$	2.15813	2.15813
$723$	0	0
$724$	0	0
$725$	−0.890792	−0.890792
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−0.868454	−0.868454
$730$	0	0
$731$	0	0
$732$	−0.149460	−0.149460
$733$	−1.46610	−1.46610	−0.733052	−	0.680173i	$-0.761905\pi$
$733$	−1.46610	−1.46610	−0.733052	+	0.680173i	$0.761905\pi$
$734$	0	0
$735$	0.109208	0.109208
$736$	1.46610	1.46610
$737$	0	0
$738$	0.542561	0.542561
$739$	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
$745$	0.109208	0.109208
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	−0.199654	−0.199654
$751$	0.149460	0.149460	0.0747301	−	0.997204i	$-0.476190\pi$
$751$	0.149460	0.149460	0.0747301	+	0.997204i	$0.476190\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.533896	0.533896
$756$	0	0
$757$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$757$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$758$	1.55496	1.55496
$759$	0.219124	0.219124
$760$	−0.670076	−0.670076
$761$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$761$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0.186374	0.186374
$769$	−1.97766	−1.97766	−0.988831	−	0.149042i	$-0.952381\pi$
$769$	−1.97766	−1.97766	−0.988831	+	0.149042i	$0.952381\pi$
$770$	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
$776$	−1.06061	−1.06061
$777$	0	0
$778$	−1.24698	−1.24698
$779$	−0.735422	−0.735422
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−0.564900	−0.564900
$784$	−1.24698	−1.24698
$785$	−0.730682	−0.730682
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−1.09752	−1.09752
$789$	−0.295582	−0.295582
$790$	0	0
$791$	0	0
$792$	−0.542561	−0.542561
$793$	0	0
$794$	−2.46610	−2.46610
$795$	0.180464	0.180464
$796$	0	0
$797$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$797$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$798$	0	0
$799$	0	0
$800$	0.466104	0.466104
$801$	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			$\pi$
$810$	0.850540	0.850540
$811$	−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
$815$	−1.07126	−1.07126
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	−0.180464	−0.180464
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	1.65248	1.65248	0.826239	−	0.563320i	$-0.190476\pi$
$823$	1.65248	1.65248	0.826239	+	0.563320i	$0.190476\pi$
$824$	0	0
$825$	0.0696640	0.0696640
$826$	0	0
$827$	0.730682	0.730682	0.365341	−	0.930874i	$-0.380952\pi$
$827$	0.730682	0.730682	0.365341	+	0.930874i	$0.380952\pi$
$828$	0.795451	0.795451
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0.0223383	0.0223383
$832$	0	0
$833$	0	0
$834$	0.356187	0.356187
$835$	0	0
$836$	−0.917056	−0.917056
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	2.65248	2.65248
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0.730682	0.730682
$846$	0	0
$847$	0	0
$848$	−2.06061	−2.06061
$849$	−0.295582	−0.295582
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	−1.18046	−1.18046
$856$	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
$862$	1.24698	1.24698
$863$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$863$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$864$	0.295582	0.295582
$865$	−1.07126	−1.07126
$866$	0.911146	0.911146
$867$	0.149460	0.149460
$868$	0	0
$869$	0	0
$870$	0.260260	0.260260
$871$	0	0
$872$	−0.692021	−0.692021
$873$	−1.86845	−1.86845
$874$	−3.02106	−3.02106
$875$	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	$0.666667\pi$
$878$	0.911146	0.911146
$879$	0	0
$880$	0.911146	0.911146
$881$	−1.46610	−1.46610	−0.733052	−	0.680173i	$-0.761905\pi$
$881$	−1.46610	−1.46610	−0.733052	+	0.680173i	$0.761905\pi$
$882$	−1.21912	−1.21912
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	−0.215976	−0.215976
$886$	2.06061	2.06061
$887$	1.65248	1.65248	0.826239	−	0.563320i	$-0.190476\pi$
$887$	1.65248	1.65248	0.826239	+	0.563320i	$0.190476\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−0.933484	−0.933484
$892$	−0.554958	−0.554958
$893$	0	0
$894$	0.0278555	0.0278555
$895$	−1.31664	−1.31664
$896$	0	0
$897$	0	0
$898$	2.06061	2.06061
$899$	0	0
$900$	0.252890	0.252890
$901$	0	0
$902$	0.554958	0.554958
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0.136180	0.136180
$907$	1.91115	1.91115	0.955573	−	0.294755i	$-0.0952381\pi$
$907$	1.91115	1.91115	0.955573	+	0.294755i	$0.0952381\pi$
$908$	1.06061	1.06061
$909$	0	0
$910$	0	0
$911$	0.730682	0.730682	0.365341	−	0.930874i	$-0.380952\pi$
$911$	0.730682	0.730682	0.365341	+	0.930874i	$0.380952\pi$
$912$	−0.307979	−0.307979
$913$	0	0
$914$	0	0
$915$	−0.196786	−0.196786
$916$	0.0829441	0.0829441
$917$	0	0
$918$	0	0
$919$	1.91115	1.91115	0.955573	−	0.294755i	$-0.0952381\pi$
$919$	1.91115	1.91115	0.955573	+	0.294755i	$0.0952381\pi$
$920$	0.594502	0.594502
$921$	0.186374	0.186374
$922$	−1.82820	−1.82820
$923$	0	0
$924$	0	0
$925$	0	0
$926$	2.38316	2.38316
$927$	0	0
$928$	−1.91115	−1.91115
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	1.65248	1.65248
$932$	0	0
$933$	0	0
$934$	2.49396	2.49396
$935$	0	0
$936$	0	0
$937$	1.65248	1.65248	0.826239	−	0.563320i	$-0.190476\pi$
$937$	1.65248	1.65248	0.826239	+	0.563320i	$0.190476\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	−0.186374	−0.186374
$943$	0.652478	0.652478
$944$	2.46610	2.46610
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	−0.960456	−0.960456
$951$	0	0
$952$	0	0
$953$	−1.97766	−1.97766	−0.988831	−	0.149042i	$-0.952381\pi$
$953$	−1.97766	−1.97766	−0.988831	+	0.149042i	$0.952381\pi$
$954$	−2.01458	−2.01458
$955$	0	0
$956$	0	0
$957$	−0.285640	−0.285640
$958$	−2.24698	−2.24698
$959$	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
$970$	1.74133	1.74133
$971$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$971$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$972$	0.241462	0.241462
$973$	0	0
$974$	0	0
$975$	0	0
$976$	2.24698	2.24698
$977$	0.149460	0.149460	0.0747301	−	0.997204i	$-0.476190\pi$
$977$	0.149460	0.149460	0.0747301	+	0.997204i	$0.476190\pi$
$978$	−0.273243	−0.273243
$979$	0	0
$980$	0.405498	0.405498
$981$	−1.21912	−1.21912
$982$	1.55496	1.55496
$983$	0.149460	0.149460	0.0747301	−	0.997204i	$-0.476190\pi$
$983$	0.149460	0.149460	0.0747301	+	0.997204i	$0.476190\pi$
$984$	0.0369136	0.0369136
$985$	−1.44504	−1.44504
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0.890792	0.890792
$991$	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
$997$	−1.46610	−1.46610	−0.733052	−	0.680173i	$-0.761905\pi$
$997$	−1.46610	−1.46610	−0.733052	+	0.680173i	$0.761905\pi$
$998$	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.430.5	✓	6
3.2	odd	2		3879.1.d.c.3016.1		6
431.430	odd	2	CM	431.1.b.c.430.5	✓	6
1293.1292	even	2		3879.1.d.c.3016.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
431.1.b.c.430.5	✓	6	1.1	even	1	trivial
431.1.b.c.430.5	✓	6	431.430	odd	2	CM
3879.1.d.c.3016.1		6	3.2	odd	2
3879.1.d.c.3016.1		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)

\(f(q)\)	\(=\)	\(q+1.24698 q^{2} +0.149460 q^{3} +0.554958 q^{4} +0.730682 q^{5} +0.186374 q^{6} -0.554958 q^{8} -0.977662 q^{9} +O(q^{10})\)	\(q+1.24698 q^{2} +0.149460 q^{3} +0.554958 q^{4} +0.730682 q^{5} +0.186374 q^{6} -0.554958 q^{8} -0.977662 q^{9} +0.911146 q^{10} -1.00000 q^{11} +0.0829441 q^{12} +0.109208 q^{15} -1.24698 q^{16} -1.21912 q^{18} +1.65248 q^{19} +0.405498 q^{20} -1.24698 q^{22} -1.46610 q^{23} -0.0829441 q^{24} -0.466104 q^{25} -0.295582 q^{27} +1.91115 q^{29} +0.136180 q^{30} -1.00000 q^{32} -0.149460 q^{33} -0.542561 q^{36} +2.06061 q^{38} -0.405498 q^{40} -0.445042 q^{41} -0.554958 q^{44} -0.714360 q^{45} -1.82820 q^{46} -0.186374 q^{48} +1.00000 q^{49} -0.581222 q^{50} +1.65248 q^{53} -0.368584 q^{54} -0.730682 q^{55} +0.246980 q^{57} +2.38316 q^{58} -1.97766 q^{59} +0.0606058 q^{60} -1.80194 q^{61} -0.186374 q^{66} -0.219124 q^{69} +0.542561 q^{72} -0.0696640 q^{75} +0.917056 q^{76} -0.911146 q^{80} +0.933484 q^{81} -0.554958 q^{82} +0.285640 q^{87} +0.554958 q^{88} -0.890792 q^{90} -0.813626 q^{92} +1.20744 q^{95} -0.149460 q^{96} +1.91115 q^{97} +1.24698 q^{98} +0.977662 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} + q^{3} + 4 q^{4} + q^{5} + 2 q^{6} - 4 q^{8} + 7 q^{9}+O(q^{10}) \) 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

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