LMFDB - Embedded newform 1511.1.b.b.1510.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1511,1,Mod(1510,1511)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1511, 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("1511.1510");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1511 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1511.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:	$0.754087234088$
Analytic rank:	$0$
Dimension:	$21$
Coefficient field:	$\Q(\zeta_{98})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{21} - 21 x^{19} + 189 x^{17} - 952 x^{15} - x^{14} + 2940 x^{13} + 14 x^{12} - 5733 x^{11} - 77 x^{10} + \cdots + 1 $ x^21 - 21x^19 + 189x^17 - 952x^15 - x^14 + 2940x^13 + 14x^12 - 5733x^11 - 77x^10 + 7007x^9 + 210x^8 - 5147x^7 - 294x^6 + 2072x^5 + 196x^4 - 371x^3 - 49x^2 + 14x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{49}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{49} - \cdots)$

Embedding invariants

Embedding label			1510.12
Root			$-1.93459$ of defining polynomial
Character	$\chi$	$=$	1511.1510

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.319200 q^{2} -1.80194 q^{3} -0.898111 q^{4} +1.93459 q^{5} -0.575178 q^{6} +1.85383 q^{7} -0.605877 q^{8} +2.24698 q^{9} +O(q^{10})$	\(q+0.319200 q^{2} -1.80194 q^{3} -0.898111 q^{4} +1.93459 q^{5} -0.575178 q^{6} +1.85383 q^{7} -0.605877 q^{8} +2.24698 q^{9} +0.617521 q^{10} +1.61834 q^{12} -0.690730 q^{13} +0.591743 q^{14} -3.48601 q^{15} +0.704716 q^{16} +0.0641032 q^{17} +0.717235 q^{18} -1.67618 q^{19} -1.73748 q^{20} -3.34049 q^{21} +1.09175 q^{24} +2.74264 q^{25} -0.220481 q^{26} -2.24698 q^{27} -1.66495 q^{28} -1.11273 q^{30} +1.03679 q^{31} +0.830822 q^{32} +0.0204617 q^{34} +3.58641 q^{35} -2.01804 q^{36} -0.535035 q^{38} +1.24465 q^{39} -1.17212 q^{40} -1.06628 q^{42} +4.34698 q^{45} -1.26985 q^{48} +2.43670 q^{49} +0.875449 q^{50} -0.115510 q^{51} +0.620353 q^{52} -0.717235 q^{54} -1.12319 q^{56} +3.02037 q^{57} +3.13083 q^{60} -0.925077 q^{61} +0.330942 q^{62} +4.16553 q^{63} -0.439518 q^{64} -1.33628 q^{65} -0.0575718 q^{68} +1.14478 q^{70} -1.36139 q^{72} -4.94206 q^{75} +1.50539 q^{76} +0.397293 q^{78} +1.36334 q^{80} +1.80194 q^{81} +3.00013 q^{84} +0.124013 q^{85} +0.569055 q^{89} +1.38756 q^{90} -1.28050 q^{91} -1.86822 q^{93} -3.24271 q^{95} -1.49709 q^{96} +0.777794 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - 7 q^{3} + 21 q^{4} + 14 q^{9}+O(q^{10}) $ 21 * q - 7 * q^3 + 21 * q^4 + 14 * q^9	$ 21 q - 7 q^{3} + 21 q^{4} + 14 q^{9} - 7 q^{12} - 7 q^{14} + 21 q^{16} + 21 q^{25} - 14 q^{27} + 14 q^{36} - 7 q^{40} - 14 q^{42} - 7 q^{48} + 21 q^{49} - 7 q^{50} - 7 q^{52} - 7 q^{56} + 21 q^{64} - 7 q^{75} - 7 q^{76} - 7 q^{80} + 7 q^{81} - 7 q^{85}+O(q^{100}) $ 21 * q - 7 * q^3 + 21 * q^4 + 14 * q^9 - 7 * q^12 - 7 * q^14 + 21 * q^16 + 21 * q^25 - 14 * q^27 + 14 * q^36 - 7 * q^40 - 14 * q^42 - 7 * q^48 + 21 * q^49 - 7 * q^50 - 7 * q^52 - 7 * q^56 + 21 * q^64 - 7 * q^75 - 7 * q^76 - 7 * q^80 + 7 * q^81 - 7 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1511\mathbb{Z}\right)^\times$.

$n$	$11$
$\chi(n)$	$-1$

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 $

$2$	0.319200	0.319200	0.159600	−	0.987182i	$-0.448980\pi$
$2$	0.319200	0.319200	0.159600	+	0.987182i	$0.448980\pi$
$3$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$3$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$4$	−0.898111	−0.898111
$5$	1.93459	1.93459	0.967295	−	0.253655i	$-0.0816327\pi$
$5$	1.93459	1.93459	0.967295	+	0.253655i	$0.0816327\pi$
$6$	−0.575178	−0.575178
$7$	1.85383	1.85383	0.926917	−	0.375267i	$-0.122449\pi$
$7$	1.85383	1.85383	0.926917	+	0.375267i	$0.122449\pi$
$8$	−0.605877	−0.605877
$9$	2.24698	2.24698
$10$	0.617521	0.617521
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	1.61834	1.61834
$13$	−0.690730	−0.690730	−0.345365	−	0.938468i	$-0.612245\pi$
$13$	−0.690730	−0.690730	−0.345365	+	0.938468i	$0.612245\pi$
$14$	0.591743	0.591743
$15$	−3.48601	−3.48601
$16$	0.704716	0.704716
$17$	0.0641032	0.0641032	0.0320516	−	0.999486i	$-0.489796\pi$
$17$	0.0641032	0.0641032	0.0320516	+	0.999486i	$0.489796\pi$
$18$	0.717235	0.717235
$19$	−1.67618	−1.67618	−0.838088	−	0.545535i	$-0.816327\pi$
$19$	−1.67618	−1.67618	−0.838088	+	0.545535i	$0.816327\pi$
$20$	−1.73748	−1.73748
$21$	−3.34049	−3.34049
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	1.09175	1.09175
$25$	2.74264	2.74264
$26$	−0.220481	−0.220481
$27$	−2.24698	−2.24698
$28$	−1.66495	−1.66495
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−1.11273	−1.11273
$31$	1.03679	1.03679	0.518393	−	0.855143i	$-0.326531\pi$
$31$	1.03679	1.03679	0.518393	+	0.855143i	$0.326531\pi$
$32$	0.830822	0.830822
$33$	0	0
$34$	0.0204617	0.0204617
$35$	3.58641	3.58641
$36$	−2.01804	−2.01804
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−0.535035	−0.535035
$39$	1.24465	1.24465
$40$	−1.17212	−1.17212
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	−1.06628	−1.06628
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	4.34698	4.34698
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−1.26985	−1.26985
$49$	2.43670	2.43670
$50$	0.875449	0.875449
$51$	−0.115510	−0.115510
$52$	0.620353	0.620353
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−0.717235	−0.717235
$55$	0	0
$56$	−1.12319	−1.12319
$57$	3.02037	3.02037
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	3.13083	3.13083
$61$	−0.925077	−0.925077	−0.462538	−	0.886599i	$-0.653061\pi$
$61$	−0.925077	−0.925077	−0.462538	+	0.886599i	$0.653061\pi$
$62$	0.330942	0.330942
$63$	4.16553	4.16553
$64$	−0.439518	−0.439518
$65$	−1.33628	−1.33628
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−0.0575718	−0.0575718
$69$	0	0
$70$	1.14478	1.14478
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−1.36139	−1.36139
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−4.94206	−4.94206
$76$	1.50539	1.50539
$77$	0	0
$78$	0.397293	0.397293
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	1.36334	1.36334
$81$	1.80194	1.80194
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	3.00013	3.00013
$85$	0.124013	0.124013
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0.569055	0.569055	0.284528	−	0.958668i	$-0.408163\pi$
$89$	0.569055	0.569055	0.284528	+	0.958668i	$0.408163\pi$
$90$	1.38756	1.38756
$91$	−1.28050	−1.28050
$92$	0	0
$93$	−1.86822	−1.86822
$94$	0	0
$95$	−3.24271	−3.24271
$96$	−1.49709	−1.49709
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0.777794	0.777794
$99$	0	0
$100$	−2.46319	−2.46319
$101$	−1.52289	−1.52289	−0.761446	−	0.648228i	$-0.775510\pi$
$101$	−1.52289	−1.52289	−0.761446	+	0.648228i	$0.775510\pi$
$102$	−0.0368707	−0.0368707
$103$	−1.14423	−1.14423	−0.572117	−	0.820172i	$-0.693878\pi$
$103$	−1.14423	−1.14423	−0.572117	+	0.820172i	$0.693878\pi$
$104$	0.418497	0.418497
$105$	−6.46248	−6.46248
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	2.01804	2.01804
$109$	1.98358	1.98358	0.991790	−	0.127877i	$-0.0408163\pi$
$109$	1.98358	1.98358	0.991790	+	0.127877i	$0.0408163\pi$
$110$	0	0
$111$	0	0
$112$	1.30643	1.30643
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0.964100	0.964100
$115$	0	0
$116$	0	0
$117$	−1.55206	−1.55206
$118$	0	0
$119$	0.118837	0.118837
$120$	2.11209	2.11209
$121$	1.00000	1.00000
$122$	−0.295284	−0.295284
$123$	0	0
$124$	−0.931149	−0.931149
$125$	3.37129	3.37129
$126$	1.32964	1.32964
$127$	−1.99589	−1.99589	−0.997945	−	0.0640702i	$-0.979592\pi$
$127$	−1.99589	−1.99589	−0.997945	+	0.0640702i	$0.979592\pi$
$128$	−0.971116	−0.971116
$129$	0	0
$130$	−0.426540	−0.426540
$131$	1.74264	1.74264	0.871319	−	0.490718i	$-0.163265\pi$
$131$	1.74264	1.74264	0.871319	+	0.490718i	$0.163265\pi$
$132$	0	0
$133$	−3.10735	−3.10735
$134$	0	0
$135$	−4.34698	−4.34698
$136$	−0.0388386	−0.0388386
$137$	0.809567	0.809567	0.404783	−	0.914413i	$-0.367347\pi$
$137$	0.809567	0.809567	0.404783	+	0.914413i	$0.367347\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	−3.22099	−3.22099
$141$	0	0
$142$	0	0
$143$	0	0
$144$	1.58348	1.58348
$145$	0	0
$146$	0	0
$147$	−4.39078	−4.39078
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−1.57751	−1.57751
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	1.01556	1.01556
$153$	0.144038	0.144038
$154$	0	0
$155$	2.00575	2.00575
$156$	−1.11784	−1.11784
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	1.60730	1.60730
$161$	0	0
$162$	0.575178	0.575178
$163$	0.569055	0.569055	0.284528	−	0.958668i	$-0.408163\pi$
$163$	0.569055	0.569055	0.284528	+	0.958668i	$0.408163\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	2.02393	2.02393
$169$	−0.522892	−0.522892
$170$	0.0395850	0.0395850
$171$	−3.76633	−3.76633
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	5.08439	5.08439
$176$	0	0
$177$	0	0
$178$	0.181642	0.181642
$179$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$179$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$180$	−3.90408	−3.90408
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	−0.408735	−0.408735
$183$	1.66693	1.66693
$184$	0	0
$185$	0	0
$186$	−0.596336	−0.596336
$187$	0	0
$188$	0	0
$189$	−4.16553	−4.16553
$190$	−1.03507	−1.03507
$191$	−1.99589	−1.99589	−0.997945	−	0.0640702i	$-0.979592\pi$
$191$	−1.99589	−1.99589	−0.997945	+	0.0640702i	$0.979592\pi$
$192$	0.791983	0.791983
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	2.40789	2.40789
$196$	−2.18843	−2.18843
$197$	−1.96312	−1.96312	−0.981559	−	0.191159i	$-0.938776\pi$
$197$	−1.96312	−1.96312	−0.981559	+	0.191159i	$0.938776\pi$
$198$	0	0
$199$	0.319200	0.319200	0.159600	−	0.987182i	$-0.448980\pi$
$199$	0.319200	0.319200	0.159600	+	0.987182i	$0.448980\pi$
$200$	−1.66170	−1.66170
$201$	0	0
$202$	−0.486107	−0.486107
$203$	0	0
$204$	0.103741	0.103741
$205$	0	0
$206$	−0.365239	−0.365239
$207$	0	0
$208$	−0.486768	−0.486768
$209$	0	0
$210$	−2.06282	−2.06282
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	1.36139	1.36139
$217$	1.92203	1.92203
$218$	0.633158	0.633158
$219$	0	0
$220$	0	0
$221$	−0.0442780	−0.0442780
$222$	0	0
$223$	−0.925077	−0.925077	−0.462538	−	0.886599i	$-0.653061\pi$
$223$	−0.925077	−0.925077	−0.462538	+	0.886599i	$0.653061\pi$
$224$	1.54021	1.54021
$225$	6.16265	6.16265
$226$	0	0
$227$	−0.192046	−0.192046	−0.0960230	−	0.995379i	$-0.530612\pi$
$227$	−0.192046	−0.192046	−0.0960230	+	0.995379i	$0.530612\pi$
$228$	−2.71262	−2.71262
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.67618	−1.67618	−0.838088	−	0.545535i	$-0.816327\pi$
$233$	−1.67618	−1.67618	−0.838088	+	0.545535i	$0.816327\pi$
$234$	−0.495416	−0.495416
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0.0379326	0.0379326
$239$	−1.14423	−1.14423	−0.572117	−	0.820172i	$-0.693878\pi$
$239$	−1.14423	−1.14423	−0.572117	+	0.820172i	$0.693878\pi$
$240$	−2.45665	−2.45665
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0.319200	0.319200
$243$	−1.00000	−1.00000
$244$	0.830822	0.830822
$245$	4.71401	4.71401
$246$	0	0
$247$	1.15779	1.15779
$248$	−0.628164	−0.628164
$249$	0	0
$250$	1.07611	1.07611
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−3.74111	−3.74111
$253$	0	0
$254$	−0.637088	−0.637088
$255$	−0.223464	−0.223464
$256$	0.129538	0.129538
$257$	−1.89811	−1.89811	−0.949056	−	0.315108i	$-0.897959\pi$
$257$	−1.89811	−1.89811	−0.949056	+	0.315108i	$0.897959\pi$
$258$	0	0
$259$	0	0
$260$	1.20013	1.20013
$261$	0	0
$262$	0.556249	0.556249
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	−0.991866	−0.991866
$267$	−1.02540	−1.02540
$268$	0	0
$269$	−1.67618	−1.67618	−0.838088	−	0.545535i	$-0.816327\pi$
$269$	−1.67618	−1.67618	−0.838088	+	0.545535i	$0.816327\pi$
$270$	−1.38756	−1.38756
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0.0451745	0.0451745
$273$	2.30738	2.30738
$274$	0.258414	0.258414
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	2.32964	2.32964
$280$	−2.17292	−2.17292
$281$	−1.96312	−1.96312	−0.981559	−	0.191159i	$-0.938776\pi$
$281$	−1.96312	−1.96312	−0.981559	+	0.191159i	$0.938776\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	5.84317	5.84317
$286$	0	0
$287$	0	0
$288$	1.86684	1.86684
$289$	−0.995891	−0.995891
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$293$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$294$	−1.40154	−1.40154
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	4.43852	4.43852
$301$	0	0
$302$	0	0
$303$	2.74416	2.74416
$304$	−1.18123	−1.18123
$305$	−1.78964	−1.78964
$306$	0.0459771	0.0459771
$307$	1.43670	1.43670	0.718349	−	0.695683i	$-0.244898\pi$
$307$	1.43670	1.43670	0.718349	+	0.695683i	$0.244898\pi$
$308$	0	0
$309$	2.06184	2.06184
$310$	0.640236	0.640236
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	−0.754106	−0.754106
$313$	−1.34460	−1.34460	−0.672301	−	0.740278i	$-0.734694\pi$
$313$	−1.34460	−1.34460	−0.672301	+	0.740278i	$0.734694\pi$
$314$	0	0
$315$	8.05858	8.05858
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	−0.850286	−0.850286
$321$	0	0
$322$	0	0
$323$	−0.107448	−0.107448
$324$	−1.61834	−1.61834
$325$	−1.89442	−1.89442
$326$	0.181642	0.181642
$327$	−3.57429	−3.57429
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.93459	1.93459	0.967295	−	0.253655i	$-0.0816327\pi$
$331$	1.93459	1.93459	0.967295	+	0.253655i	$0.0816327\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	−2.35410	−2.35410
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−0.166907	−0.166907
$339$	0	0
$340$	−0.111378	−0.111378
$341$	0	0
$342$	−1.20221	−1.20221
$343$	2.66340	2.66340
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.99589	−1.99589	−0.997945	−	0.0640702i	$-0.979592\pi$
$347$	−1.99589	−1.99589	−0.997945	+	0.0640702i	$0.979592\pi$
$348$	0	0
$349$	−0.192046	−0.192046	−0.0960230	−	0.995379i	$-0.530612\pi$
$349$	−0.192046	−0.192046	−0.0960230	+	0.995379i	$0.530612\pi$
$350$	1.62294	1.62294
$351$	1.55206	1.55206
$352$	0	0
$353$	−1.99589	−1.99589	−0.997945	−	0.0640702i	$-0.979592\pi$
$353$	−1.99589	−1.99589	−0.997945	+	0.0640702i	$0.979592\pi$
$354$	0	0
$355$	0	0
$356$	−0.511075	−0.511075
$357$	−0.214136	−0.214136
$358$	0.398036	0.398036
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−2.63374	−2.63374
$361$	1.80957	1.80957
$362$	0	0
$363$	−1.80194	−1.80194
$364$	1.15003	1.15003
$365$	0	0
$366$	0.532084	0.532084
$367$	1.93459	1.93459	0.967295	−	0.253655i	$-0.0816327\pi$
$367$	1.93459	1.93459	0.967295	+	0.253655i	$0.0816327\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	1.67787	1.67787
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−6.07485	−6.07485
$376$	0	0
$377$	0	0
$378$	−1.32964	−1.32964
$379$	0.319200	0.319200	0.159600	−	0.987182i	$-0.448980\pi$
$379$	0.319200	0.319200	0.159600	+	0.987182i	$0.448980\pi$
$380$	2.91232	2.91232
$381$	3.59647	3.59647
$382$	−0.637088	−0.637088
$383$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$383$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$384$	1.74989	1.74989
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.98358	1.98358	0.991790	−	0.127877i	$-0.0408163\pi$
$389$	1.98358	1.98358	0.991790	+	0.127877i	$0.0408163\pi$
$390$	0.768599	0.768599
$391$	0	0
$392$	−1.47634	−1.47634
$393$	−3.14012	−3.14012
$394$	−0.626627	−0.626627
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0.101889	0.101889
$399$	5.59925	5.59925
$400$	1.93278	1.93278
$401$	1.85383	1.85383	0.926917	−	0.375267i	$-0.122449\pi$
$401$	1.85383	1.85383	0.926917	+	0.375267i	$0.122449\pi$
$402$	0	0
$403$	−0.716139	−0.716139
$404$	1.36773	1.36773
$405$	3.48601	3.48601
$406$	0	0
$407$	0	0
$408$	0.0699848	0.0699848
$409$	−1.96312	−1.96312	−0.981559	−	0.191159i	$-0.938776\pi$
$409$	−1.96312	−1.96312	−0.981559	+	0.191159i	$0.938776\pi$
$410$	0	0
$411$	−1.45879	−1.45879
$412$	1.02765	1.02765
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−0.573874	−0.573874
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	5.80403	5.80403
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.175812	0.175812
$426$	0	0
$427$	−1.71494	−1.71494
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−1.58348	−1.58348
$433$	−0.192046	−0.192046	−0.0960230	−	0.995379i	$-0.530612\pi$
$433$	−0.192046	−0.192046	−0.0960230	+	0.995379i	$0.530612\pi$
$434$	0.613511	0.613511
$435$	0	0
$436$	−1.78148	−1.78148
$437$	0	0
$438$	0	0
$439$	0.569055	0.569055	0.284528	−	0.958668i	$-0.408163\pi$
$439$	0.569055	0.569055	0.284528	+	0.958668i	$0.408163\pi$
$440$	0	0
$441$	5.47521	5.47521
$442$	−0.0141335	−0.0141335
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	1.10089	1.10089
$446$	−0.295284	−0.295284
$447$	0	0
$448$	−0.814792	−0.814792
$449$	1.60283	1.60283	0.801414	−	0.598111i	$-0.204082\pi$
$449$	1.60283	1.60283	0.801414	+	0.598111i	$0.204082\pi$
$450$	1.96712	1.96712
$451$	0	0
$452$	0	0
$453$	0	0
$454$	−0.0613011	−0.0613011
$455$	−2.47724	−2.47724
$456$	−1.82997	−1.82997
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	−0.144038	−0.144038
$460$	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
$465$	−3.61424	−3.61424
$466$	−0.535035	−0.535035
$467$	−1.96312	−1.96312	−0.981559	−	0.191159i	$-0.938776\pi$
$467$	−1.96312	−1.96312	−0.981559	+	0.191159i	$0.938776\pi$
$468$	1.39392	1.39392
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−4.59714	−4.59714
$476$	−0.106728	−0.106728
$477$	0	0
$478$	−0.365239	−0.365239
$479$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$479$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$480$	−2.89625	−2.89625
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−0.898111	−0.898111
$485$	0	0
$486$	−0.319200	−0.319200
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0.560482	0.560482
$489$	−1.02540	−1.02540
$490$	1.50471	1.50471
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0.369565	0.369565
$495$	0	0
$496$	0.730639	0.730639
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−3.02779	−3.02779
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−2.52380	−2.52380
$505$	−2.94617	−2.94617
$506$	0	0
$507$	0.942219	0.942219
$508$	1.79253	1.79253
$509$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$509$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$510$	−0.0713297	−0.0713297
$511$	0	0
$512$	1.01246	1.01246
$513$	3.76633	3.76633
$514$	−0.605877	−0.605877
$515$	−2.21362	−2.21362
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0.809621	0.809621
$521$	−0.690730	−0.690730	−0.345365	−	0.938468i	$-0.612245\pi$
$521$	−0.690730	−0.690730	−0.345365	+	0.938468i	$0.612245\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−1.56508	−1.56508
$525$	−9.16176	−9.16176
$526$	0	0
$527$	0.0664612	0.0664612
$528$	0	0
$529$	1.00000	1.00000
$530$	0	0
$531$	0	0
$532$	2.79075	2.79075
$533$	0	0
$534$	−0.327308	−0.327308
$535$	0	0
$536$	0	0
$537$	−2.24698	−2.24698
$538$	−0.535035	−0.535035
$539$	0	0
$540$	3.90408	3.90408
$541$	1.60283	1.60283	0.801414	−	0.598111i	$-0.204082\pi$
$541$	1.60283	1.60283	0.801414	+	0.598111i	$0.204082\pi$
$542$	0	0
$543$	0	0
$544$	0.0532583	0.0532583
$545$	3.83741	3.83741
$546$	0.736515	0.736515
$547$	0.809567	0.809567	0.404783	−	0.914413i	$-0.367347\pi$
$547$	0.809567	0.809567	0.404783	+	0.914413i	$0.367347\pi$
$548$	−0.727081	−0.727081
$549$	−2.07863	−2.07863
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−0.925077	−0.925077	−0.462538	−	0.886599i	$-0.653061\pi$
$557$	−0.925077	−0.925077	−0.462538	+	0.886599i	$0.653061\pi$
$558$	0.743619	0.743619
$559$	0	0
$560$	2.52740	2.52740
$561$	0	0
$562$	−0.626627	−0.626627
$563$	−1.52289	−1.52289	−0.761446	−	0.648228i	$-0.775510\pi$
$563$	−1.52289	−1.52289	−0.761446	+	0.648228i	$0.775510\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.34049	3.34049
$568$	0	0
$569$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$569$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$570$	1.86514	1.86514
$571$	1.74264	1.74264	0.871319	−	0.490718i	$-0.163265\pi$
$571$	1.74264	1.74264	0.871319	+	0.490718i	$0.163265\pi$
$572$	0	0
$573$	3.59647	3.59647
$574$	0	0
$575$	0	0
$576$	−0.987587	−0.987587
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−0.317888	−0.317888
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−3.00259	−3.00259
$586$	−0.142057	−0.142057
$587$	−1.34460	−1.34460	−0.672301	−	0.740278i	$-0.734694\pi$
$587$	−1.34460	−1.34460	−0.672301	+	0.740278i	$0.734694\pi$
$588$	3.94341	3.94341
$589$	−1.73783	−1.73783
$590$	0	0
$591$	3.53742	3.53742
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0.229900	0.229900
$596$	0	0
$597$	−0.575178	−0.575178
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	2.99428	2.99428
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.93459	1.93459
$606$	0.875934	0.875934
$607$	−1.34460	−1.34460	−0.672301	−	0.740278i	$-0.734694\pi$
$607$	−1.34460	−1.34460	−0.672301	+	0.740278i	$0.734694\pi$
$608$	−1.39260	−1.39260
$609$	0	0
$610$	−0.571254	−0.571254
$611$	0	0
$612$	−0.129363	−0.129363
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0.458594	0.458594
$615$	0	0
$616$	0	0
$617$	−1.52289	−1.52289	−0.761446	−	0.648228i	$-0.775510\pi$
$617$	−1.52289	−1.52289	−0.761446	+	0.648228i	$0.775510\pi$
$618$	0.658138	0.658138
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	−1.80139	−1.80139
$621$	0	0
$622$	0	0
$623$	1.05493	1.05493
$624$	0.877126	0.877126
$625$	3.77942	3.77942
$626$	−0.429197	−0.429197
$627$	0	0
$628$	0	0
$629$	0	0
$630$	2.57230	2.57230
$631$	0.0641032	0.0641032	0.0320516	−	0.999486i	$-0.489796\pi$
$631$	0.0641032	0.0641032	0.0320516	+	0.999486i	$0.489796\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.86123	−3.86123
$636$	0	0
$637$	−1.68310	−1.68310
$638$	0	0
$639$	0	0
$640$	−1.87871	−1.87871
$641$	1.60283	1.60283	0.801414	−	0.598111i	$-0.204082\pi$
$641$	1.60283	1.60283	0.801414	+	0.598111i	$0.204082\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	−0.0342974	−0.0342974
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−1.09175	−1.09175
$649$	0	0
$650$	−0.604699	−0.604699
$651$	−3.46337	−3.46337
$652$	−0.511075	−0.511075
$653$	0.569055	0.569055	0.284528	−	0.958668i	$-0.408163\pi$
$653$	0.569055	0.569055	0.284528	+	0.958668i	$0.408163\pi$
$654$	−1.14091	−1.14091
$655$	3.37129	3.37129
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−0.690730	−0.690730	−0.345365	−	0.938468i	$-0.612245\pi$
$659$	−0.690730	−0.690730	−0.345365	+	0.938468i	$0.612245\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0.617521	0.617521
$663$	0.0797862	0.0797862
$664$	0	0
$665$	−6.01145	−6.01145
$666$	0	0
$667$	0	0
$668$	0	0
$669$	1.66693	1.66693
$670$	0	0
$671$	0	0
$672$	−2.77535	−2.77535
$673$	0.569055	0.569055	0.284528	−	0.958668i	$-0.408163\pi$
$673$	0.569055	0.569055	0.284528	+	0.958668i	$0.408163\pi$
$674$	0	0
$675$	−6.16265	−6.16265
$676$	0.469615	0.469615
$677$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$677$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$678$	0	0
$679$	0	0
$680$	−0.0751368	−0.0751368
$681$	0.346055	0.346055
$682$	0	0
$683$	1.03679	1.03679	0.518393	−	0.855143i	$-0.326531\pi$
$683$	1.03679	1.03679	0.518393	+	0.855143i	$0.326531\pi$
$684$	3.38259	3.38259
$685$	1.56618	1.56618
$686$	0.850157	0.850157
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	1.60283	1.60283	0.801414	−	0.598111i	$-0.204082\pi$
$691$	1.60283	1.60283	0.801414	+	0.598111i	$0.204082\pi$
$692$	0	0
$693$	0	0
$694$	−0.637088	−0.637088
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−0.0613011	−0.0613011
$699$	3.02037	3.02037
$700$	−4.56635	−4.56635
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0.495416	0.495416
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−0.637088	−0.637088
$707$	−2.82319	−2.82319
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	−0.344777	−0.344777
$713$	0	0
$714$	−0.0683522	−0.0683522
$715$	0	0
$716$	−1.11993	−1.11993
$717$	2.06184	2.06184
$718$	0	0
$719$	−0.925077	−0.925077	−0.462538	−	0.886599i	$-0.653061\pi$
$719$	−0.925077	−0.925077	−0.462538	+	0.886599i	$0.653061\pi$
$720$	3.06339	3.06339
$721$	−2.12122	−2.12122
$722$	0.577613	0.577613
$723$	0	0
$724$	0	0
$725$	0	0
$726$	−0.575178	−0.575178
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0.775824	0.775824
$729$	0	0
$730$	0	0
$731$	0	0
$732$	−1.49709	−1.49709
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0.617521	0.617521
$735$	−8.49436	−8.49436
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−1.52289	−1.52289	−0.761446	−	0.648228i	$-0.775510\pi$
$739$	−1.52289	−1.52289	−0.761446	+	0.648228i	$0.775510\pi$
$740$	0	0
$741$	−2.08626	−2.08626
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	1.13191	1.13191
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	−1.93909	−1.93909
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	3.74111	3.74111
$757$	−1.67618	−1.67618	−0.838088	−	0.545535i	$-0.816327\pi$
$757$	−1.67618	−1.67618	−0.838088	+	0.545535i	$0.816327\pi$
$758$	0.101889	0.101889
$759$	0	0
$760$	1.96468	1.96468
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	1.14799	1.14799
$763$	3.67723	3.67723
$764$	1.79253	1.79253
$765$	0.278655	0.278655
$766$	0.398036	0.398036
$767$	0	0
$768$	−0.233419	−0.233419
$769$	0.809567	0.809567	0.404783	−	0.914413i	$-0.367347\pi$
$769$	0.809567	0.809567	0.404783	+	0.914413i	$0.367347\pi$
$770$	0	0
$771$	3.42028	3.42028
$772$	0	0
$773$	1.60283	1.60283	0.801414	−	0.598111i	$-0.204082\pi$
$773$	1.60283	1.60283	0.801414	+	0.598111i	$0.204082\pi$
$774$	0	0
$775$	2.84353	2.84353
$776$	0	0
$777$	0	0
$778$	0.633158	0.633158
$779$	0	0
$780$	−2.16256	−2.16256
$781$	0	0
$782$	0	0
$783$	0	0
$784$	1.71718	1.71718
$785$	0	0
$786$	−1.00233	−1.00233
$787$	1.43670	1.43670	0.718349	−	0.695683i	$-0.244898\pi$
$787$	1.43670	1.43670	0.718349	+	0.695683i	$0.244898\pi$
$788$	1.76310	1.76310
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0.638978	0.638978
$794$	0	0
$795$	0	0
$796$	−0.286677	−0.286677
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	1.78728	1.78728
$799$	0	0
$800$	2.27864	2.27864
$801$	1.27866	1.27866
$802$	0.591743	0.591743
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−0.228591	−0.228591
$807$	3.02037	3.02037
$808$	0.922685	0.922685
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	1.11273	1.11273
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.10089	1.10089
$816$	−0.0814016	−0.0814016
$817$	0	0
$818$	−0.626627	−0.626627
$819$	−2.87725	−2.87725
$820$	0	0
$821$	0.809567	0.809567	0.404783	−	0.914413i	$-0.367347\pi$
$821$	0.809567	0.809567	0.404783	+	0.914413i	$0.367347\pi$
$822$	−0.465645	−0.465645
$823$	−0.192046	−0.192046	−0.0960230	−	0.995379i	$-0.530612\pi$
$823$	−0.192046	−0.192046	−0.0960230	+	0.995379i	$0.530612\pi$
$824$	0.693264	0.693264
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	1.03679	1.03679	0.518393	−	0.855143i	$-0.326531\pi$
$829$	1.03679	1.03679	0.518393	+	0.855143i	$0.326531\pi$
$830$	0	0
$831$	0	0
$832$	0.303588	0.303588
$833$	0.156200	0.156200
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.32964	−2.32964
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	3.91547	3.91547
$841$	1.00000	1.00000
$842$	0	0
$843$	3.53742	3.53742
$844$	0	0
$845$	−1.01158	−1.01158
$846$	0	0
$847$	1.85383	1.85383
$848$	0	0
$849$	0	0
$850$	0.0561191	0.0561191
$851$	0	0
$852$	0	0
$853$	−1.89811	−1.89811	−0.949056	−	0.315108i	$-0.897959\pi$
$853$	−1.89811	−1.89811	−0.949056	+	0.315108i	$0.897959\pi$
$854$	−0.547408	−0.547408
$855$	−7.28631	−7.28631
$856$	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
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−1.86684	−1.86684
$865$	0	0
$866$	−0.0613011	−0.0613011
$867$	1.79453	1.79453
$868$	−1.72619	−1.72619
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−1.20181	−1.20181
$873$	0	0
$874$	0	0
$875$	6.24981	6.24981
$876$	0	0
$877$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$877$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$878$	0.181642	0.181642
$879$	0.801938	0.801938
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	1.74769	1.74769
$883$	−1.89811	−1.89811	−0.949056	−	0.315108i	$-0.897959\pi$
$883$	−1.89811	−1.89811	−0.949056	+	0.315108i	$0.897959\pi$
$884$	0.0397666	0.0397666
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−3.70005	−3.70005
$890$	0.351403	0.351403
$891$	0	0
$892$	0.830822	0.830822
$893$	0	0
$894$	0	0
$895$	2.41239	2.41239
$896$	−1.80029	−1.80029
$897$	0	0
$898$	0.511622	0.511622
$899$	0	0
$900$	−5.53475	−5.53475
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.98358	1.98358	0.991790	−	0.127877i	$-0.0408163\pi$
$907$	1.98358	1.98358	0.991790	+	0.127877i	$0.0408163\pi$
$908$	0.172479	0.172479
$909$	−3.42191	−3.42191
$910$	−0.790734	−0.790734
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	2.12850	2.12850
$913$	0	0
$914$	0	0
$915$	3.22483	3.22483
$916$	0	0
$917$	3.23056	3.23056
$918$	−0.0459771	−0.0459771
$919$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$919$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$920$	0	0
$921$	−2.58884	−2.58884
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2.57107	−2.57107
$928$	0	0
$929$	−0.925077	−0.925077	−0.462538	−	0.886599i	$-0.653061\pi$
$929$	−0.925077	−0.925077	−0.462538	+	0.886599i	$0.653061\pi$
$930$	−1.15367	−1.15367
$931$	−4.08434	−4.08434
$932$	1.50539	1.50539
$933$	0	0
$934$	−0.626627	−0.626627
$935$	0	0
$936$	0.940355	0.940355
$937$	−1.96312	−1.96312	−0.981559	−	0.191159i	$-0.938776\pi$
$937$	−1.96312	−1.96312	−0.981559	+	0.191159i	$0.938776\pi$
$938$	0	0
$939$	2.42289	2.42289
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−8.05858	−8.05858
$946$	0	0
$947$	0.0641032	0.0641032	0.0320516	−	0.999486i	$-0.489796\pi$
$947$	0.0641032	0.0641032	0.0320516	+	0.999486i	$0.489796\pi$
$948$	0	0
$949$	0	0
$950$	−1.46741	−1.46741
$951$	0	0
$952$	−0.0720003	−0.0720003
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	−3.86123	−3.86123
$956$	1.02765	1.02765
$957$	0	0
$958$	−0.142057	−0.142057
$959$	1.50080	1.50080
$960$	1.53216	1.53216
$961$	0.0749234	0.0749234
$962$	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			$\pi$
$968$	−0.605877	−0.605877
$969$	0.193615	0.193615
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0.898111	0.898111
$973$	0	0
$974$	0	0
$975$	3.41363	3.41363
$976$	−0.651916	−0.651916
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	−0.327308	−0.327308
$979$	0	0
$980$	−4.23371	−4.23371
$981$	4.45706	4.45706
$982$	0	0
$983$	1.43670	1.43670	0.718349	−	0.695683i	$-0.244898\pi$
$983$	1.43670	1.43670	0.718349	+	0.695683i	$0.244898\pi$
$984$	0	0
$985$	−3.79783	−3.79783
$986$	0	0
$987$	0	0
$988$	−1.03982	−1.03982
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0.861384	0.861384
$993$	−3.48601	−3.48601
$994$	0	0
$995$	0.617521	0.617521
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1511.1.b.b.1510.12	✓	21
1511.1510	odd	2	CM	1511.1.b.b.1510.12	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1511.1.b.b.1510.12	✓	21	1.1	even	1	trivial
1511.1.b.b.1510.12	✓	21	1511.1510	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:	\( N \)	\(=\)	\( 1511 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1511.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.319200 q^{2} -1.80194 q^{3} -0.898111 q^{4} +1.93459 q^{5} -0.575178 q^{6} +1.85383 q^{7} -0.605877 q^{8} +2.24698 q^{9} +O(q^{10})\)	\(q+0.319200 q^{2} -1.80194 q^{3} -0.898111 q^{4} +1.93459 q^{5} -0.575178 q^{6} +1.85383 q^{7} -0.605877 q^{8} +2.24698 q^{9} +0.617521 q^{10} +1.61834 q^{12} -0.690730 q^{13} +0.591743 q^{14} -3.48601 q^{15} +0.704716 q^{16} +0.0641032 q^{17} +0.717235 q^{18} -1.67618 q^{19} -1.73748 q^{20} -3.34049 q^{21} +1.09175 q^{24} +2.74264 q^{25} -0.220481 q^{26} -2.24698 q^{27} -1.66495 q^{28} -1.11273 q^{30} +1.03679 q^{31} +0.830822 q^{32} +0.0204617 q^{34} +3.58641 q^{35} -2.01804 q^{36} -0.535035 q^{38} +1.24465 q^{39} -1.17212 q^{40} -1.06628 q^{42} +4.34698 q^{45} -1.26985 q^{48} +2.43670 q^{49} +0.875449 q^{50} -0.115510 q^{51} +0.620353 q^{52} -0.717235 q^{54} -1.12319 q^{56} +3.02037 q^{57} +3.13083 q^{60} -0.925077 q^{61} +0.330942 q^{62} +4.16553 q^{63} -0.439518 q^{64} -1.33628 q^{65} -0.0575718 q^{68} +1.14478 q^{70} -1.36139 q^{72} -4.94206 q^{75} +1.50539 q^{76} +0.397293 q^{78} +1.36334 q^{80} +1.80194 q^{81} +3.00013 q^{84} +0.124013 q^{85} +0.569055 q^{89} +1.38756 q^{90} -1.28050 q^{91} -1.86822 q^{93} -3.24271 q^{95} -1.49709 q^{96} +0.777794 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - 7 q^{3} + 21 q^{4} + 14 q^{9}+O(q^{10}) \) 21 * q - 7 * q^3 + 21 * q^4 + 14 * q^9	\( 21 q - 7 q^{3} + 21 q^{4} + 14 q^{9} - 7 q^{12} - 7 q^{14} + 21 q^{16} + 21 q^{25} - 14 q^{27} + 14 q^{36} - 7 q^{40} - 14 q^{42} - 7 q^{48} + 21 q^{49} - 7 q^{50} - 7 q^{52} - 7 q^{56} + 21 q^{64} - 7 q^{75} - 7 q^{76} - 7 q^{80} + 7 q^{81} - 7 q^{85}+O(q^{100}) \) 21 * q - 7 * q^3 + 21 * q^4 + 14 * q^9 - 7 * q^12 - 7 * q^14 + 21 * q^16 + 21 * q^25 - 14 * q^27 + 14 * q^36 - 7 * q^40 - 14 * q^42 - 7 * q^48 + 21 * q^49 - 7 * q^50 - 7 * q^52 - 7 * q^56 + 21 * q^64 - 7 * q^75 - 7 * q^76 - 7 * q^80 + 7 * q^81 - 7 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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