LMFDB - Embedded newform 1511.1.b.b.1510.19

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			1510.19
Root			$-1.60283$ 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)

$f(q)$	$=$	$q+1.85383 q^{2} -1.80194 q^{3} +2.43670 q^{4} +1.60283 q^{5} -3.34049 q^{6} -1.14423 q^{7} +2.66340 q^{8} +2.24698 q^{9} +O(q^{10})$	\(q+1.85383 q^{2} -1.80194 q^{3} +2.43670 q^{4} +1.60283 q^{5} -3.34049 q^{6} -1.14423 q^{7} +2.66340 q^{8} +2.24698 q^{9} +2.97137 q^{10} -4.39078 q^{12} -0.192046 q^{13} -2.12122 q^{14} -2.88819 q^{15} +2.50080 q^{16} -1.52289 q^{17} +4.16553 q^{18} +1.98358 q^{19} +3.90561 q^{20} +2.06184 q^{21} -4.79928 q^{24} +1.56906 q^{25} -0.356021 q^{26} -2.24698 q^{27} -2.78815 q^{28} -5.35423 q^{30} -1.67618 q^{31} +1.97267 q^{32} -2.82319 q^{34} -1.83401 q^{35} +5.47521 q^{36} +3.67723 q^{38} +0.346055 q^{39} +4.26897 q^{40} +3.82230 q^{42} +3.60152 q^{45} -4.50629 q^{48} +0.309270 q^{49} +2.90877 q^{50} +2.74416 q^{51} -0.467958 q^{52} -4.16553 q^{54} -3.04755 q^{56} -3.57429 q^{57} -7.03766 q^{60} +0.809567 q^{61} -3.10735 q^{62} -2.57107 q^{63} +1.15620 q^{64} -0.307817 q^{65} -3.71083 q^{68} -3.39995 q^{70} +5.98461 q^{72} -2.82734 q^{75} +4.83339 q^{76} +0.641528 q^{78} +4.00835 q^{80} +1.80194 q^{81} +5.02408 q^{84} -2.44093 q^{85} -1.99589 q^{89} +6.67662 q^{90} +0.219745 q^{91} +3.02037 q^{93} +3.17934 q^{95} -3.55463 q^{96} +0.573335 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.85383	1.85383	0.926917	−	0.375267i	$-0.122449\pi$
$2$	1.85383	1.85383	0.926917	+	0.375267i	$0.122449\pi$
$3$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$3$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$4$	2.43670	2.43670
$5$	1.60283	1.60283	0.801414	−	0.598111i	$-0.204082\pi$
$5$	1.60283	1.60283	0.801414	+	0.598111i	$0.204082\pi$
$6$	−3.34049	−3.34049
$7$	−1.14423	−1.14423	−0.572117	−	0.820172i	$-0.693878\pi$
$7$	−1.14423	−1.14423	−0.572117	+	0.820172i	$0.693878\pi$
$8$	2.66340	2.66340
$9$	2.24698	2.24698
$10$	2.97137	2.97137
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−4.39078	−4.39078
$13$	−0.192046	−0.192046	−0.0960230	−	0.995379i	$-0.530612\pi$
$13$	−0.192046	−0.192046	−0.0960230	+	0.995379i	$0.530612\pi$
$14$	−2.12122	−2.12122
$15$	−2.88819	−2.88819
$16$	2.50080	2.50080
$17$	−1.52289	−1.52289	−0.761446	−	0.648228i	$-0.775510\pi$
$17$	−1.52289	−1.52289	−0.761446	+	0.648228i	$0.775510\pi$
$18$	4.16553	4.16553
$19$	1.98358	1.98358	0.991790	−	0.127877i	$-0.0408163\pi$
$19$	1.98358	1.98358	0.991790	+	0.127877i	$0.0408163\pi$
$20$	3.90561	3.90561
$21$	2.06184	2.06184
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−4.79928	−4.79928
$25$	1.56906	1.56906
$26$	−0.356021	−0.356021
$27$	−2.24698	−2.24698
$28$	−2.78815	−2.78815
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−5.35423	−5.35423
$31$	−1.67618	−1.67618	−0.838088	−	0.545535i	$-0.816327\pi$
$31$	−1.67618	−1.67618	−0.838088	+	0.545535i	$0.816327\pi$
$32$	1.97267	1.97267
$33$	0	0
$34$	−2.82319	−2.82319
$35$	−1.83401	−1.83401
$36$	5.47521	5.47521
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	3.67723	3.67723
$39$	0.346055	0.346055
$40$	4.26897	4.26897
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	3.82230	3.82230
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	3.60152	3.60152
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−4.50629	−4.50629
$49$	0.309270	0.309270
$50$	2.90877	2.90877
$51$	2.74416	2.74416
$52$	−0.467958	−0.467958
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−4.16553	−4.16553
$55$	0	0
$56$	−3.04755	−3.04755
$57$	−3.57429	−3.57429
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−7.03766	−7.03766
$61$	0.809567	0.809567	0.404783	−	0.914413i	$-0.367347\pi$
$61$	0.809567	0.809567	0.404783	+	0.914413i	$0.367347\pi$
$62$	−3.10735	−3.10735
$63$	−2.57107	−2.57107
$64$	1.15620	1.15620
$65$	−0.307817	−0.307817
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−3.71083	−3.71083
$69$	0	0
$70$	−3.39995	−3.39995
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	5.98461	5.98461
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−2.82734	−2.82734
$76$	4.83339	4.83339
$77$	0	0
$78$	0.641528	0.641528
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	4.00835	4.00835
$81$	1.80194	1.80194
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	5.02408	5.02408
$85$	−2.44093	−2.44093
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.99589	−1.99589	−0.997945	−	0.0640702i	$-0.979592\pi$
$89$	−1.99589	−1.99589	−0.997945	+	0.0640702i	$0.979592\pi$
$90$	6.67662	6.67662
$91$	0.219745	0.219745
$92$	0	0
$93$	3.02037	3.02037
$94$	0	0
$95$	3.17934	3.17934
$96$	−3.55463	−3.55463
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0.573335	0.573335
$99$	0	0
$100$	3.82331	3.82331
$101$	−1.96312	−1.96312	−0.981559	−	0.191159i	$-0.938776\pi$
$101$	−1.96312	−1.96312	−0.981559	+	0.191159i	$0.938776\pi$
$102$	5.08721	5.08721
$103$	−1.34460	−1.34460	−0.672301	−	0.740278i	$-0.734694\pi$
$103$	−1.34460	−1.34460	−0.672301	+	0.740278i	$0.734694\pi$
$104$	−0.511495	−0.511495
$105$	3.30477	3.30477
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−5.47521	−5.47521
$109$	−1.89811	−1.89811	−0.949056	−	0.315108i	$-0.897959\pi$
$109$	−1.89811	−1.89811	−0.949056	+	0.315108i	$0.897959\pi$
$110$	0	0
$111$	0	0
$112$	−2.86150	−2.86150
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	−6.62613	−6.62613
$115$	0	0
$116$	0	0
$117$	−0.431524	−0.431524
$118$	0	0
$119$	1.74254	1.74254
$120$	−7.69242	−7.69242
$121$	1.00000	1.00000
$122$	1.50080	1.50080
$123$	0	0
$124$	−4.08434	−4.08434
$125$	0.912097	0.912097
$126$	−4.76633	−4.76633
$127$	0.319200	0.319200	0.159600	−	0.987182i	$-0.448980\pi$
$127$	0.319200	0.319200	0.159600	+	0.987182i	$0.448980\pi$
$128$	0.170732	0.170732
$129$	0	0
$130$	−0.570641	−0.570641
$131$	0.569055	0.569055	0.284528	−	0.958668i	$-0.408163\pi$
$131$	0.569055	0.569055	0.284528	+	0.958668i	$0.408163\pi$
$132$	0	0
$133$	−2.26968	−2.26968
$134$	0	0
$135$	−3.60152	−3.60152
$136$	−4.05607	−4.05607
$137$	1.93459	1.93459	0.967295	−	0.253655i	$-0.0816327\pi$
$137$	1.93459	1.93459	0.967295	+	0.253655i	$0.0816327\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	−4.46893	−4.46893
$141$	0	0
$142$	0	0
$143$	0	0
$144$	5.61925	5.61925
$145$	0	0
$146$	0	0
$147$	−0.557285	−0.557285
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−5.24142	−5.24142
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	5.28307	5.28307
$153$	−3.42191	−3.42191
$154$	0	0
$155$	−2.68662	−2.68662
$156$	0.843232	0.843232
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	3.16185	3.16185
$161$	0	0
$162$	3.34049	3.34049
$163$	−1.99589	−1.99589	−0.997945	−	0.0640702i	$-0.979592\pi$
$163$	−1.99589	−1.99589	−0.997945	+	0.0640702i	$0.979592\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	5.49150	5.49150
$169$	−0.963118	−0.963118
$170$	−4.52508	−4.52508
$171$	4.45706	4.45706
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	−1.79537	−1.79537
$176$	0	0
$177$	0	0
$178$	−3.70005	−3.70005
$179$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$179$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$180$	8.77582	8.77582
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0.407372	0.407372
$183$	−1.45879	−1.45879
$184$	0	0
$185$	0	0
$186$	5.59925	5.59925
$187$	0	0
$188$	0	0
$189$	2.57107	2.57107
$190$	5.89396	5.89396
$191$	0.319200	0.319200	0.159600	−	0.987182i	$-0.448980\pi$
$191$	0.319200	0.319200	0.159600	+	0.987182i	$0.448980\pi$
$192$	−2.08340	−2.08340
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0.554666	0.554666
$196$	0.753598	0.753598
$197$	−0.925077	−0.925077	−0.462538	−	0.886599i	$-0.653061\pi$
$197$	−0.925077	−0.925077	−0.462538	+	0.886599i	$0.653061\pi$
$198$	0	0
$199$	1.85383	1.85383	0.926917	−	0.375267i	$-0.122449\pi$
$199$	1.85383	1.85383	0.926917	+	0.375267i	$0.122449\pi$
$200$	4.17902	4.17902
$201$	0	0
$202$	−3.63929	−3.63929
$203$	0	0
$204$	6.68668	6.68668
$205$	0	0
$206$	−2.49267	−2.49267
$207$	0	0
$208$	−0.480269	−0.480269
$209$	0	0
$210$	6.12649	6.12649
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	−5.98461	−5.98461
$217$	1.91794	1.91794
$218$	−3.51878	−3.51878
$219$	0	0
$220$	0	0
$221$	0.292465	0.292465
$222$	0	0
$223$	0.809567	0.809567	0.404783	−	0.914413i	$-0.367347\pi$
$223$	0.809567	0.809567	0.404783	+	0.914413i	$0.367347\pi$
$224$	−2.25719	−2.25719
$225$	3.52563	3.52563
$226$	0	0
$227$	1.03679	1.03679	0.518393	−	0.855143i	$-0.326531\pi$
$227$	1.03679	1.03679	0.518393	+	0.855143i	$0.326531\pi$
$228$	−8.70946	−8.70946
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.98358	1.98358	0.991790	−	0.127877i	$-0.0408163\pi$
$233$	1.98358	1.98358	0.991790	+	0.127877i	$0.0408163\pi$
$234$	−0.799973	−0.799973
$235$	0	0
$236$	0	0
$237$	0	0
$238$	3.23039	3.23039
$239$	−1.34460	−1.34460	−0.672301	−	0.740278i	$-0.734694\pi$
$239$	−1.34460	−1.34460	−0.672301	+	0.740278i	$0.734694\pi$
$240$	−7.22280	−7.22280
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	1.85383	1.85383
$243$	−1.00000	−1.00000
$244$	1.97267	1.97267
$245$	0.495706	0.495706
$246$	0	0
$247$	−0.380939	−0.380939
$248$	−4.46433	−4.46433
$249$	0	0
$250$	1.69088	1.69088
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−6.26492	−6.26492
$253$	0	0
$254$	0.591743	0.591743
$255$	4.39841	4.39841
$256$	−0.839691	−0.839691
$257$	1.43670	1.43670	0.718349	−	0.695683i	$-0.244898\pi$
$257$	1.43670	1.43670	0.718349	+	0.695683i	$0.244898\pi$
$258$	0	0
$259$	0	0
$260$	−0.750056	−0.750056
$261$	0	0
$262$	1.05493	1.05493
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	−4.20761	−4.20761
$267$	3.59647	3.59647
$268$	0	0
$269$	1.98358	1.98358	0.991790	−	0.127877i	$-0.0408163\pi$
$269$	1.98358	1.98358	0.991790	+	0.127877i	$0.0408163\pi$
$270$	−6.67662	−6.67662
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	−3.80845	−3.80845
$273$	−0.395968	−0.395968
$274$	3.58641	3.58641
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	−3.76633	−3.76633
$280$	−4.88470	−4.88470
$281$	−0.925077	−0.925077	−0.462538	−	0.886599i	$-0.653061\pi$
$281$	−0.925077	−0.925077	−0.462538	+	0.886599i	$0.653061\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	−5.72897	−5.72897
$286$	0	0
$287$	0	0
$288$	4.43255	4.43255
$289$	1.31920	1.31920
$290$	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	$0.571429\pi$
$294$	−1.03311	−1.03311
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	−6.88938	−6.88938
$301$	0	0
$302$	0	0
$303$	3.53742	3.53742
$304$	4.96054	4.96054
$305$	1.29760	1.29760
$306$	−6.34365	−6.34365
$307$	−0.690730	−0.690730	−0.345365	−	0.938468i	$-0.612245\pi$
$307$	−0.690730	−0.690730	−0.345365	+	0.938468i	$0.612245\pi$
$308$	0	0
$309$	2.42289	2.42289
$310$	−4.98055	−4.98055
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0.921683	0.921683
$313$	1.74264	1.74264	0.871319	−	0.490718i	$-0.163265\pi$
$313$	1.74264	1.74264	0.871319	+	0.490718i	$0.163265\pi$
$314$	0	0
$315$	−4.12098	−4.12098
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	1.85319	1.85319
$321$	0	0
$322$	0	0
$323$	−3.02078	−3.02078
$324$	4.39078	4.39078
$325$	−0.301331	−0.301331
$326$	−3.70005	−3.70005
$327$	3.42028	3.42028
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.60283	1.60283	0.801414	−	0.598111i	$-0.204082\pi$
$331$	1.60283	1.60283	0.801414	+	0.598111i	$0.204082\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	5.15625	5.15625
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−1.78546	−1.78546
$339$	0	0
$340$	−5.94782	−5.94782
$341$	0	0
$342$	8.26265	8.26265
$343$	0.790356	0.790356
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.319200	0.319200	0.159600	−	0.987182i	$-0.448980\pi$
$347$	0.319200	0.319200	0.159600	+	0.987182i	$0.448980\pi$
$348$	0	0
$349$	1.03679	1.03679	0.518393	−	0.855143i	$-0.326531\pi$
$349$	1.03679	1.03679	0.518393	+	0.855143i	$0.326531\pi$
$350$	−3.32831	−3.32831
$351$	0.431524	0.431524
$352$	0	0
$353$	0.319200	0.319200	0.159600	−	0.987182i	$-0.448980\pi$
$353$	0.319200	0.319200	0.159600	+	0.987182i	$0.448980\pi$
$354$	0	0
$355$	0	0
$356$	−4.86338	−4.86338
$357$	−3.13996	−3.13996
$358$	2.31169	2.31169
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	9.59229	9.59229
$361$	2.93459	2.93459
$362$	0	0
$363$	−1.80194	−1.80194
$364$	0.535454	0.535454
$365$	0	0
$366$	−2.70435	−2.70435
$367$	1.60283	1.60283	0.801414	−	0.598111i	$-0.204082\pi$
$367$	1.60283	1.60283	0.801414	+	0.598111i	$0.204082\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	7.35972	7.35972
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−1.64354	−1.64354
$376$	0	0
$377$	0	0
$378$	4.76633	4.76633
$379$	1.85383	1.85383	0.926917	−	0.375267i	$-0.122449\pi$
$379$	1.85383	1.85383	0.926917	+	0.375267i	$0.122449\pi$
$380$	7.74708	7.74708
$381$	−0.575178	−0.575178
$382$	0.591743	0.591743
$383$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$383$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$384$	−0.307649	−0.307649
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.89811	−1.89811	−0.949056	−	0.315108i	$-0.897959\pi$
$389$	−1.89811	−1.89811	−0.949056	+	0.315108i	$0.897959\pi$
$390$	1.02826	1.02826
$391$	0	0
$392$	0.823709	0.823709
$393$	−1.02540	−1.02540
$394$	−1.71494	−1.71494
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	3.43670	3.43670
$399$	4.08982	4.08982
$400$	3.92390	3.92390
$401$	−1.14423	−1.14423	−0.572117	−	0.820172i	$-0.693878\pi$
$401$	−1.14423	−1.14423	−0.572117	+	0.820172i	$0.693878\pi$
$402$	0	0
$403$	0.321903	0.321903
$404$	−4.78353	−4.78353
$405$	2.88819	2.88819
$406$	0	0
$407$	0	0
$408$	7.30879	7.30879
$409$	−0.925077	−0.925077	−0.462538	−	0.886599i	$-0.653061\pi$
$409$	−0.925077	−0.925077	−0.462538	+	0.886599i	$0.653061\pi$
$410$	0	0
$411$	−3.48601	−3.48601
$412$	−3.27639	−3.27639
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−0.378844	−0.378844
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	8.05273	8.05273
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.38950	−2.38950
$426$	0	0
$427$	−0.926333	−0.926333
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−5.61925	−5.61925
$433$	1.03679	1.03679	0.518393	−	0.855143i	$-0.326531\pi$
$433$	1.03679	1.03679	0.518393	+	0.855143i	$0.326531\pi$
$434$	3.55554	3.55554
$435$	0	0
$436$	−4.62513	−4.62513
$437$	0	0
$438$	0	0
$439$	−1.99589	−1.99589	−0.997945	−	0.0640702i	$-0.979592\pi$
$439$	−1.99589	−1.99589	−0.997945	+	0.0640702i	$0.979592\pi$
$440$	0	0
$441$	0.694923	0.694923
$442$	0.542182	0.542182
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	−3.19907	−3.19907
$446$	1.50080	1.50080
$447$	0	0
$448$	−1.32296	−1.32296
$449$	0.0641032	0.0641032	0.0320516	−	0.999486i	$-0.489796\pi$
$449$	0.0641032	0.0641032	0.0320516	+	0.999486i	$0.489796\pi$
$450$	6.53594	6.53594
$451$	0	0
$452$	0	0
$453$	0	0
$454$	1.92203	1.92203
$455$	0.352214	0.352214
$456$	−9.51976	−9.51976
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	3.42191	3.42191
$460$	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
$465$	4.84112	4.84112
$466$	3.67723	3.67723
$467$	−0.925077	−0.925077	−0.462538	−	0.886599i	$-0.653061\pi$
$467$	−0.925077	−0.925077	−0.462538	+	0.886599i	$0.653061\pi$
$468$	−1.05149	−1.05149
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	3.11235	3.11235
$476$	4.24605	4.24605
$477$	0	0
$478$	−2.49267	−2.49267
$479$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$479$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$480$	−5.69746	−5.69746
$481$	0	0
$482$	0	0
$483$	0	0
$484$	2.43670	2.43670
$485$	0	0
$486$	−1.85383	−1.85383
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	2.15620	2.15620
$489$	3.59647	3.59647
$490$	0.918957	0.918957
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	−0.706197	−0.706197
$495$	0	0
$496$	−4.19178	−4.19178
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	2.22251	2.22251
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−6.84779	−6.84779
$505$	−3.14654	−3.14654
$506$	0	0
$507$	1.73548	1.73548
$508$	0.777794	0.777794
$509$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$509$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$510$	8.15392	8.15392
$511$	0	0
$512$	−1.72738	−1.72738
$513$	−4.45706	−4.45706
$514$	2.66340	2.66340
$515$	−2.15516	−2.15516
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−0.819839	−0.819839
$521$	−0.192046	−0.192046	−0.0960230	−	0.995379i	$-0.530612\pi$
$521$	−0.192046	−0.192046	−0.0960230	+	0.995379i	$0.530612\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	1.38662	1.38662
$525$	3.23514	3.23514
$526$	0	0
$527$	2.55264	2.55264
$528$	0	0
$529$	1.00000	1.00000
$530$	0	0
$531$	0	0
$532$	−5.53052	−5.53052
$533$	0	0
$534$	6.66726	6.66726
$535$	0	0
$536$	0	0
$537$	−2.24698	−2.24698
$538$	3.67723	3.67723
$539$	0	0
$540$	−8.77582	−8.77582
$541$	0.0641032	0.0641032	0.0320516	−	0.999486i	$-0.489796\pi$
$541$	0.0641032	0.0641032	0.0320516	+	0.999486i	$0.489796\pi$
$542$	0	0
$543$	0	0
$544$	−3.00416	−3.00416
$545$	−3.04234	−3.04234
$546$	−0.734058	−0.734058
$547$	1.93459	1.93459	0.967295	−	0.253655i	$-0.0816327\pi$
$547$	1.93459	1.93459	0.967295	+	0.253655i	$0.0816327\pi$
$548$	4.71401	4.71401
$549$	1.81908	1.81908
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.809567	0.809567	0.404783	−	0.914413i	$-0.367347\pi$
$557$	0.809567	0.809567	0.404783	+	0.914413i	$0.367347\pi$
$558$	−6.98216	−6.98216
$559$	0	0
$560$	−4.58649	−4.58649
$561$	0	0
$562$	−1.71494	−1.71494
$563$	−1.96312	−1.96312	−0.981559	−	0.191159i	$-0.938776\pi$
$563$	−1.96312	−1.96312	−0.981559	+	0.191159i	$0.938776\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.06184	−2.06184
$568$	0	0
$569$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$569$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$570$	−10.6205	−10.6205
$571$	0.569055	0.569055	0.284528	−	0.958668i	$-0.408163\pi$
$571$	0.569055	0.569055	0.284528	+	0.958668i	$0.408163\pi$
$572$	0	0
$573$	−0.575178	−0.575178
$574$	0	0
$575$	0	0
$576$	2.59796	2.59796
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	2.44558	2.44558
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−0.691658	−0.691658
$586$	−0.825034	−0.825034
$587$	1.74264	1.74264	0.871319	−	0.490718i	$-0.163265\pi$
$587$	1.74264	1.74264	0.871319	+	0.490718i	$0.163265\pi$
$588$	−1.35794	−1.35794
$589$	−3.32483	−3.32483
$590$	0	0
$591$	1.66693	1.66693
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	2.79300	2.79300
$596$	0	0
$597$	−3.34049	−3.34049
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−7.53034	−7.53034
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.60283	1.60283
$606$	6.55778	6.55778
$607$	1.74264	1.74264	0.871319	−	0.490718i	$-0.163265\pi$
$607$	1.74264	1.74264	0.871319	+	0.490718i	$0.163265\pi$
$608$	3.91295	3.91295
$609$	0	0
$610$	2.40553	2.40553
$611$	0	0
$612$	−8.33816	−8.33816
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−1.28050	−1.28050
$615$	0	0
$616$	0	0
$617$	−1.96312	−1.96312	−0.981559	−	0.191159i	$-0.938776\pi$
$617$	−1.96312	−1.96312	−0.981559	+	0.191159i	$0.938776\pi$
$618$	4.49163	4.49163
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	−6.54649	−6.54649
$621$	0	0
$622$	0	0
$623$	2.28376	2.28376
$624$	0.865415	0.865415
$625$	−0.107121	−0.107121
$626$	3.23056	3.23056
$627$	0	0
$628$	0	0
$629$	0	0
$630$	−7.63961	−7.63961
$631$	−1.52289	−1.52289	−0.761446	−	0.648228i	$-0.775510\pi$
$631$	−1.52289	−1.52289	−0.761446	+	0.648228i	$0.775510\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0.511622	0.511622
$636$	0	0
$637$	−0.0593941	−0.0593941
$638$	0	0
$639$	0	0
$640$	0.273654	0.273654
$641$	0.0641032	0.0641032	0.0320516	−	0.999486i	$-0.489796\pi$
$641$	0.0641032	0.0641032	0.0320516	+	0.999486i	$0.489796\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	−5.60002	−5.60002
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	4.79928	4.79928
$649$	0	0
$650$	−0.558617	−0.558617
$651$	−3.45600	−3.45600
$652$	−4.86338	−4.86338
$653$	−1.99589	−1.99589	−0.997945	−	0.0640702i	$-0.979592\pi$
$653$	−1.99589	−1.99589	−0.997945	+	0.0640702i	$0.979592\pi$
$654$	6.34063	6.34063
$655$	0.912097	0.912097
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−0.192046	−0.192046	−0.0960230	−	0.995379i	$-0.530612\pi$
$659$	−0.192046	−0.192046	−0.0960230	+	0.995379i	$0.530612\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	2.97137	2.97137
$663$	−0.527004	−0.527004
$664$	0	0
$665$	−3.63790	−3.63790
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−1.45879	−1.45879
$670$	0	0
$671$	0	0
$672$	4.06732	4.06732
$673$	−1.99589	−1.99589	−0.997945	−	0.0640702i	$-0.979592\pi$
$673$	−1.99589	−1.99589	−0.997945	+	0.0640702i	$0.979592\pi$
$674$	0	0
$675$	−3.52563	−3.52563
$676$	−2.34683	−2.34683
$677$	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
$680$	−6.50118	−6.50118
$681$	−1.86822	−1.86822
$682$	0	0
$683$	−1.67618	−1.67618	−0.838088	−	0.545535i	$-0.816327\pi$
$683$	−1.67618	−1.67618	−0.838088	+	0.545535i	$0.816327\pi$
$684$	10.8605	10.8605
$685$	3.10081	3.10081
$686$	1.46519	1.46519
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0.0641032	0.0641032	0.0320516	−	0.999486i	$-0.489796\pi$
$691$	0.0641032	0.0641032	0.0320516	+	0.999486i	$0.489796\pi$
$692$	0	0
$693$	0	0
$694$	0.591743	0.591743
$695$	0	0
$696$	0	0
$697$	0	0
$698$	1.92203	1.92203
$699$	−3.57429	−3.57429
$700$	−4.37476	−4.37476
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0.799973	0.799973
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0.591743	0.591743
$707$	2.24627	2.24627
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	−5.31586	−5.31586
$713$	0	0
$714$	−5.82095	−5.82095
$715$	0	0
$716$	3.03851	3.03851
$717$	2.42289	2.42289
$718$	0	0
$719$	0.809567	0.809567	0.404783	−	0.914413i	$-0.367347\pi$
$719$	0.809567	0.809567	0.404783	+	0.914413i	$0.367347\pi$
$720$	9.00669	9.00669
$721$	1.53854	1.53854
$722$	5.44024	5.44024
$723$	0	0
$724$	0	0
$725$	0	0
$726$	−3.34049	−3.34049
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0.585270	0.585270
$729$	0	0
$730$	0	0
$731$	0	0
$732$	−3.55463	−3.55463
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	2.97137	2.97137
$735$	−0.893232	−0.893232
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−1.96312	−1.96312	−0.981559	−	0.191159i	$-0.938776\pi$
$739$	−1.96312	−1.96312	−0.981559	+	0.191159i	$0.938776\pi$
$740$	0	0
$741$	0.686428	0.686428
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	8.04444	8.04444
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	−3.04685	−3.04685
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	6.26492	6.26492
$757$	1.98358	1.98358	0.991790	−	0.127877i	$-0.0408163\pi$
$757$	1.98358	1.98358	0.991790	+	0.127877i	$0.0408163\pi$
$758$	3.43670	3.43670
$759$	0	0
$760$	8.46784	8.46784
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	−1.06628	−1.06628
$763$	2.17188	2.17188
$764$	0.777794	0.777794
$765$	−5.48473	−5.48473
$766$	2.31169	2.31169
$767$	0	0
$768$	1.51307	1.51307
$769$	1.93459	1.93459	0.967295	−	0.253655i	$-0.0816327\pi$
$769$	1.93459	1.93459	0.967295	+	0.253655i	$0.0816327\pi$
$770$	0	0
$771$	−2.58884	−2.58884
$772$	0	0
$773$	0.0641032	0.0641032	0.0320516	−	0.999486i	$-0.489796\pi$
$773$	0.0641032	0.0641032	0.0320516	+	0.999486i	$0.489796\pi$
$774$	0	0
$775$	−2.63001	−2.63001
$776$	0	0
$777$	0	0
$778$	−3.51878	−3.51878
$779$	0	0
$780$	1.35155	1.35155
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0.773423	0.773423
$785$	0	0
$786$	−1.90092	−1.90092
$787$	−0.690730	−0.690730	−0.345365	−	0.938468i	$-0.612245\pi$
$787$	−0.690730	−0.690730	−0.345365	+	0.938468i	$0.612245\pi$
$788$	−2.25413	−2.25413
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−0.155474	−0.155474
$794$	0	0
$795$	0	0
$796$	4.51723	4.51723
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	7.58184	7.58184
$799$	0	0
$800$	3.09523	3.09523
$801$	−4.48473	−4.48473
$802$	−2.12122	−2.12122
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0.596755	0.596755
$807$	−3.57429	−3.57429
$808$	−5.22857	−5.22857
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	5.35423	5.35423
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.19907	−3.19907
$816$	6.86259	6.86259
$817$	0	0
$818$	−1.71494	−1.71494
$819$	0.493764	0.493764
$820$	0	0
$821$	1.93459	1.93459	0.967295	−	0.253655i	$-0.0816327\pi$
$821$	1.93459	1.93459	0.967295	+	0.253655i	$0.0816327\pi$
$822$	−6.46248	−6.46248
$823$	1.03679	1.03679	0.518393	−	0.855143i	$-0.326531\pi$
$823$	1.03679	1.03679	0.518393	+	0.855143i	$0.326531\pi$
$824$	−3.58121	−3.58121
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−1.67618	−1.67618	−0.838088	−	0.545535i	$-0.816327\pi$
$829$	−1.67618	−1.67618	−0.838088	+	0.545535i	$0.816327\pi$
$830$	0	0
$831$	0	0
$832$	−0.222044	−0.222044
$833$	−0.470985	−0.470985
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.76633	3.76633
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	8.80192	8.80192
$841$	1.00000	1.00000
$842$	0	0
$843$	1.66693	1.66693
$844$	0	0
$845$	−1.54371	−1.54371
$846$	0	0
$847$	−1.14423	−1.14423
$848$	0	0
$849$	0	0
$850$	−4.42974	−4.42974
$851$	0	0
$852$	0	0
$853$	1.43670	1.43670	0.718349	−	0.695683i	$-0.244898\pi$
$853$	1.43670	1.43670	0.718349	+	0.695683i	$0.244898\pi$
$854$	−1.71727	−1.71727
$855$	7.14390	7.14390
$856$	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			$\pi$
$864$	−4.43255	−4.43255
$865$	0	0
$866$	1.92203	1.92203
$867$	−2.37712	−2.37712
$868$	4.67343	4.67343
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−5.05543	−5.05543
$873$	0	0
$874$	0	0
$875$	−1.04365	−1.04365
$876$	0	0
$877$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$877$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$878$	−3.70005	−3.70005
$879$	0.801938	0.801938
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	1.28827	1.28827
$883$	1.43670	1.43670	0.718349	−	0.695683i	$-0.244898\pi$
$883$	1.43670	1.43670	0.718349	+	0.695683i	$0.244898\pi$
$884$	0.712650	0.712650
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−0.365239	−0.365239
$890$	−5.93054	−5.93054
$891$	0	0
$892$	1.97267	1.97267
$893$	0	0
$894$	0	0
$895$	1.99869	1.99869
$896$	−0.195358	−0.195358
$897$	0	0
$898$	0.118837	0.118837
$899$	0	0
$900$	8.59091	8.59091
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.89811	−1.89811	−0.949056	−	0.315108i	$-0.897959\pi$
$907$	−1.89811	−1.89811	−0.949056	+	0.315108i	$0.897959\pi$
$908$	2.52633	2.52633
$909$	−4.41109	−4.41109
$910$	0.652946	0.652946
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−8.93859	−8.93859
$913$	0	0
$914$	0	0
$915$	−2.33819	−2.33819
$916$	0	0
$917$	−0.651132	−0.651132
$918$	6.34365	6.34365
$919$	−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
$921$	1.24465	1.24465
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−3.02129	−3.02129
$928$	0	0
$929$	0.809567	0.809567	0.404783	−	0.914413i	$-0.367347\pi$
$929$	0.809567	0.809567	0.404783	+	0.914413i	$0.367347\pi$
$930$	8.97464	8.97464
$931$	0.613462	0.613462
$932$	4.83339	4.83339
$933$	0	0
$934$	−1.71494	−1.71494
$935$	0	0
$936$	−1.14932	−1.14932
$937$	−0.925077	−0.925077	−0.462538	−	0.886599i	$-0.653061\pi$
$937$	−0.925077	−0.925077	−0.462538	+	0.886599i	$0.653061\pi$
$938$	0	0
$939$	−3.14012	−3.14012
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	4.12098	4.12098
$946$	0	0
$947$	−1.52289	−1.52289	−0.761446	−	0.648228i	$-0.775510\pi$
$947$	−1.52289	−1.52289	−0.761446	+	0.648228i	$0.775510\pi$
$948$	0	0
$949$	0	0
$950$	5.76977	5.76977
$951$	0	0
$952$	4.64109	4.64109
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0.511622	0.511622
$956$	−3.27639	−3.27639
$957$	0	0
$958$	−0.825034	−0.825034
$959$	−2.21362	−2.21362
$960$	−3.33933	−3.33933
$961$	1.80957	1.80957
$962$	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			$\pi$
$968$	2.66340	2.66340
$969$	5.44325	5.44325
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−2.43670	−2.43670
$973$	0	0
$974$	0	0
$975$	0.542979	0.542979
$976$	2.02457	2.02457
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	6.66726	6.66726
$979$	0	0
$980$	1.20789	1.20789
$981$	−4.26502	−4.26502
$982$	0	0
$983$	−0.690730	−0.690730	−0.345365	−	0.938468i	$-0.612245\pi$
$983$	−0.690730	−0.690730	−0.345365	+	0.938468i	$0.612245\pi$
$984$	0	0
$985$	−1.48274	−1.48274
$986$	0	0
$987$	0	0
$988$	−0.928233	−0.928233
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	−3.30654	−3.30654
$993$	−2.88819	−2.88819
$994$	0	0
$995$	2.97137	2.97137
$996$	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.1510.19	✓	21
1511.1510	odd	2	CM	1511.1.b.b.1510.19	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1511.1.b.b.1510.19	✓	21	1.1	even	1	trivial
1511.1.b.b.1510.19	✓	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)

\(f(q)\)	\(=\)	\(q+1.85383 q^{2} -1.80194 q^{3} +2.43670 q^{4} +1.60283 q^{5} -3.34049 q^{6} -1.14423 q^{7} +2.66340 q^{8} +2.24698 q^{9} +O(q^{10})\)	\(q+1.85383 q^{2} -1.80194 q^{3} +2.43670 q^{4} +1.60283 q^{5} -3.34049 q^{6} -1.14423 q^{7} +2.66340 q^{8} +2.24698 q^{9} +2.97137 q^{10} -4.39078 q^{12} -0.192046 q^{13} -2.12122 q^{14} -2.88819 q^{15} +2.50080 q^{16} -1.52289 q^{17} +4.16553 q^{18} +1.98358 q^{19} +3.90561 q^{20} +2.06184 q^{21} -4.79928 q^{24} +1.56906 q^{25} -0.356021 q^{26} -2.24698 q^{27} -2.78815 q^{28} -5.35423 q^{30} -1.67618 q^{31} +1.97267 q^{32} -2.82319 q^{34} -1.83401 q^{35} +5.47521 q^{36} +3.67723 q^{38} +0.346055 q^{39} +4.26897 q^{40} +3.82230 q^{42} +3.60152 q^{45} -4.50629 q^{48} +0.309270 q^{49} +2.90877 q^{50} +2.74416 q^{51} -0.467958 q^{52} -4.16553 q^{54} -3.04755 q^{56} -3.57429 q^{57} -7.03766 q^{60} +0.809567 q^{61} -3.10735 q^{62} -2.57107 q^{63} +1.15620 q^{64} -0.307817 q^{65} -3.71083 q^{68} -3.39995 q^{70} +5.98461 q^{72} -2.82734 q^{75} +4.83339 q^{76} +0.641528 q^{78} +4.00835 q^{80} +1.80194 q^{81} +5.02408 q^{84} -2.44093 q^{85} -1.99589 q^{89} +6.67662 q^{90} +0.219745 q^{91} +3.02037 q^{93} +3.17934 q^{95} -3.55463 q^{96} +0.573335 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - 7 q^{3} + 21 q^{4} + 14 q^{9}+O(q^{10}) \) 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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