LMFDB - Embedded newform 1511.1.b.b.1510.5

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			1510.5
Root			$-1.98358$ 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)

$f(q)$	$=$	$q-1.52289 q^{2} -0.445042 q^{3} +1.31920 q^{4} +1.98358 q^{5} +0.677751 q^{6} -1.96312 q^{7} -0.486107 q^{8} -0.801938 q^{9} +O(q^{10})$	\(q-1.52289 q^{2} -0.445042 q^{3} +1.31920 q^{4} +1.98358 q^{5} +0.677751 q^{6} -1.96312 q^{7} -0.486107 q^{8} -0.801938 q^{9} -3.02078 q^{10} -0.587099 q^{12} -1.14423 q^{13} +2.98962 q^{14} -0.882776 q^{15} -0.578912 q^{16} +1.43670 q^{17} +1.22126 q^{18} +0.569055 q^{19} +2.61674 q^{20} +0.873670 q^{21} +0.216338 q^{24} +2.93459 q^{25} +1.74254 q^{26} +0.801938 q^{27} -2.58975 q^{28} +1.34437 q^{30} +1.74264 q^{31} +1.36773 q^{32} -2.18794 q^{34} -3.89400 q^{35} -1.05792 q^{36} -0.866610 q^{38} +0.509232 q^{39} -0.964232 q^{40} -1.33050 q^{42} -1.59071 q^{45} +0.257640 q^{48} +2.85383 q^{49} -4.46906 q^{50} -0.639391 q^{51} -1.50947 q^{52} -1.22126 q^{54} +0.954285 q^{56} -0.253253 q^{57} -1.16456 q^{60} +1.03679 q^{61} -2.65385 q^{62} +1.57430 q^{63} -1.50399 q^{64} -2.26968 q^{65} +1.89529 q^{68} +5.93014 q^{70} +0.389827 q^{72} -1.30602 q^{75} +0.750697 q^{76} -0.775505 q^{78} -1.14832 q^{80} +0.445042 q^{81} +1.15255 q^{84} +2.84981 q^{85} +1.60283 q^{89} +2.42248 q^{90} +2.24627 q^{91} -0.775547 q^{93} +1.12877 q^{95} -0.608696 q^{96} -4.34608 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.52289	−1.52289	−0.761446	−	0.648228i	$-0.775510\pi$
$2$	−1.52289	−1.52289	−0.761446	+	0.648228i	$0.775510\pi$
$3$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$3$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$4$	1.31920	1.31920
$5$	1.98358	1.98358	0.991790	−	0.127877i	$-0.0408163\pi$
$5$	1.98358	1.98358	0.991790	+	0.127877i	$0.0408163\pi$
$6$	0.677751	0.677751
$7$	−1.96312	−1.96312	−0.981559	−	0.191159i	$-0.938776\pi$
$7$	−1.96312	−1.96312	−0.981559	+	0.191159i	$0.938776\pi$
$8$	−0.486107	−0.486107
$9$	−0.801938	−0.801938
$10$	−3.02078	−3.02078
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−0.587099	−0.587099
$13$	−1.14423	−1.14423	−0.572117	−	0.820172i	$-0.693878\pi$
$13$	−1.14423	−1.14423	−0.572117	+	0.820172i	$0.693878\pi$
$14$	2.98962	2.98962
$15$	−0.882776	−0.882776
$16$	−0.578912	−0.578912
$17$	1.43670	1.43670	0.718349	−	0.695683i	$-0.244898\pi$
$17$	1.43670	1.43670	0.718349	+	0.695683i	$0.244898\pi$
$18$	1.22126	1.22126
$19$	0.569055	0.569055	0.284528	−	0.958668i	$-0.408163\pi$
$19$	0.569055	0.569055	0.284528	+	0.958668i	$0.408163\pi$
$20$	2.61674	2.61674
$21$	0.873670	0.873670
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0.216338	0.216338
$25$	2.93459	2.93459
$26$	1.74254	1.74254
$27$	0.801938	0.801938
$28$	−2.58975	−2.58975
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	1.34437	1.34437
$31$	1.74264	1.74264	0.871319	−	0.490718i	$-0.163265\pi$
$31$	1.74264	1.74264	0.871319	+	0.490718i	$0.163265\pi$
$32$	1.36773	1.36773
$33$	0	0
$34$	−2.18794	−2.18794
$35$	−3.89400	−3.89400
$36$	−1.05792	−1.05792
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−0.866610	−0.866610
$39$	0.509232	0.509232
$40$	−0.964232	−0.964232
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	−1.33050	−1.33050
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	−1.59071	−1.59071
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0.257640	0.257640
$49$	2.85383	2.85383
$50$	−4.46906	−4.46906
$51$	−0.639391	−0.639391
$52$	−1.50947	−1.50947
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−1.22126	−1.22126
$55$	0	0
$56$	0.954285	0.954285
$57$	−0.253253	−0.253253
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−1.16456	−1.16456
$61$	1.03679	1.03679	0.518393	−	0.855143i	$-0.326531\pi$
$61$	1.03679	1.03679	0.518393	+	0.855143i	$0.326531\pi$
$62$	−2.65385	−2.65385
$63$	1.57430	1.57430
$64$	−1.50399	−1.50399
$65$	−2.26968	−2.26968
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	1.89529	1.89529
$69$	0	0
$70$	5.93014	5.93014
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0.389827	0.389827
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−1.30602	−1.30602
$76$	0.750697	0.750697
$77$	0	0
$78$	−0.775505	−0.775505
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−1.14832	−1.14832
$81$	0.445042	0.445042
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	1.15255	1.15255
$85$	2.84981	2.84981
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.60283	1.60283	0.801414	−	0.598111i	$-0.204082\pi$
$89$	1.60283	1.60283	0.801414	+	0.598111i	$0.204082\pi$
$90$	2.42248	2.42248
$91$	2.24627	2.24627
$92$	0	0
$93$	−0.775547	−0.775547
$94$	0	0
$95$	1.12877	1.12877
$96$	−0.608696	−0.608696
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−4.34608	−4.34608
$99$	0	0
$100$	3.87131	3.87131
$101$	−0.690730	−0.690730	−0.345365	−	0.938468i	$-0.612245\pi$
$101$	−0.690730	−0.690730	−0.345365	+	0.938468i	$0.612245\pi$
$102$	0.973723	0.973723
$103$	−0.925077	−0.925077	−0.462538	−	0.886599i	$-0.653061\pi$
$103$	−0.925077	−0.925077	−0.462538	+	0.886599i	$0.653061\pi$
$104$	0.556220	0.556220
$105$	1.73299	1.73299
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	1.05792	1.05792
$109$	−1.99589	−1.99589	−0.997945	−	0.0640702i	$-0.979592\pi$
$109$	−1.99589	−1.99589	−0.997945	+	0.0640702i	$0.979592\pi$
$110$	0	0
$111$	0	0
$112$	1.13647	1.13647
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0.385678	0.385678
$115$	0	0
$116$	0	0
$117$	0.917604	0.917604
$118$	0	0
$119$	−2.82041	−2.82041
$120$	0.429123	0.429123
$121$	1.00000	1.00000
$122$	−1.57891	−1.57891
$123$	0	0
$124$	2.29889	2.29889
$125$	3.83741	3.83741
$126$	−2.39749	−2.39749
$127$	0.0641032	0.0641032	0.0320516	−	0.999486i	$-0.489796\pi$
$127$	0.0641032	0.0641032	0.0320516	+	0.999486i	$0.489796\pi$
$128$	0.922685	0.922685
$129$	0	0
$130$	3.45647	3.45647
$131$	1.93459	1.93459	0.967295	−	0.253655i	$-0.0816327\pi$
$131$	1.93459	1.93459	0.967295	+	0.253655i	$0.0816327\pi$
$132$	0	0
$133$	−1.11712	−1.11712
$134$	0	0
$135$	1.59071	1.59071
$136$	−0.698389	−0.698389
$137$	−1.67618	−1.67618	−0.838088	−	0.545535i	$-0.816327\pi$
$137$	−1.67618	−1.67618	−0.838088	+	0.545535i	$0.816327\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	−5.13697	−5.13697
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0.464251	0.464251
$145$	0	0
$146$	0	0
$147$	−1.27008	−1.27008
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	1.98892	1.98892
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	−0.276622	−0.276622
$153$	−1.15214	−1.15214
$154$	0	0
$155$	3.45666	3.45666
$156$	0.671778	0.671778
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	2.71300	2.71300
$161$	0	0
$162$	−0.677751	−0.677751
$163$	1.60283	1.60283	0.801414	−	0.598111i	$-0.204082\pi$
$163$	1.60283	1.60283	0.801414	+	0.598111i	$0.204082\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	−0.424697	−0.424697
$169$	0.309270	0.309270
$170$	−4.33995	−4.33995
$171$	−0.456347	−0.456347
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	−5.76095	−5.76095
$176$	0	0
$177$	0	0
$178$	−2.44093	−2.44093
$179$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$179$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$180$	−2.09846	−2.09846
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	−3.42082	−3.42082
$183$	−0.461413	−0.461413
$184$	0	0
$185$	0	0
$186$	1.18107	1.18107
$187$	0	0
$188$	0	0
$189$	−1.57430	−1.57430
$190$	−1.71899	−1.71899
$191$	0.0641032	0.0641032	0.0320516	−	0.999486i	$-0.489796\pi$
$191$	0.0641032	0.0641032	0.0320516	+	0.999486i	$0.489796\pi$
$192$	0.669338	0.669338
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	1.01010	1.01010
$196$	3.76478	3.76478
$197$	−0.192046	−0.192046	−0.0960230	−	0.995379i	$-0.530612\pi$
$197$	−0.192046	−0.192046	−0.0960230	+	0.995379i	$0.530612\pi$
$198$	0	0
$199$	−1.52289	−1.52289	−0.761446	−	0.648228i	$-0.775510\pi$
$199$	−1.52289	−1.52289	−0.761446	+	0.648228i	$0.775510\pi$
$200$	−1.42652	−1.42652
$201$	0	0
$202$	1.05191	1.05191
$203$	0	0
$204$	−0.843485	−0.843485
$205$	0	0
$206$	1.40879	1.40879
$207$	0	0
$208$	0.662410	0.662410
$209$	0	0
$210$	−2.63916	−2.63916
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	−0.389827	−0.389827
$217$	−3.42100	−3.42100
$218$	3.03953	3.03953
$219$	0	0
$220$	0	0
$221$	−1.64392	−1.64392
$222$	0	0
$223$	1.03679	1.03679	0.518393	−	0.855143i	$-0.326531\pi$
$223$	1.03679	1.03679	0.518393	+	0.855143i	$0.326531\pi$
$224$	−2.68501	−2.68501
$225$	−2.35336	−2.35336
$226$	0	0
$227$	−1.34460	−1.34460	−0.672301	−	0.740278i	$-0.734694\pi$
$227$	−1.34460	−1.34460	−0.672301	+	0.740278i	$0.734694\pi$
$228$	−0.334092	−0.334092
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.569055	0.569055	0.284528	−	0.958668i	$-0.408163\pi$
$233$	0.569055	0.569055	0.284528	+	0.958668i	$0.408163\pi$
$234$	−1.39741	−1.39741
$235$	0	0
$236$	0	0
$237$	0	0
$238$	4.29518	4.29518
$239$	−0.925077	−0.925077	−0.462538	−	0.886599i	$-0.653061\pi$
$239$	−0.925077	−0.925077	−0.462538	+	0.886599i	$0.653061\pi$
$240$	0.511049	0.511049
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−1.52289	−1.52289
$243$	−1.00000	−1.00000
$244$	1.36773	1.36773
$245$	5.66081	5.66081
$246$	0	0
$247$	−0.651132	−0.651132
$248$	−0.847108	−0.847108
$249$	0	0
$250$	−5.84397	−5.84397
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	2.07681	2.07681
$253$	0	0
$254$	−0.0976222	−0.0976222
$255$	−1.26828	−1.26828
$256$	0.0988390	0.0988390
$257$	0.319200	0.319200	0.159600	−	0.987182i	$-0.448980\pi$
$257$	0.319200	0.319200	0.159600	+	0.987182i	$0.448980\pi$
$258$	0	0
$259$	0	0
$260$	−2.99416	−2.99416
$261$	0	0
$262$	−2.94617	−2.94617
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	1.70126	1.70126
$267$	−0.713325	−0.713325
$268$	0	0
$269$	0.569055	0.569055	0.284528	−	0.958668i	$-0.408163\pi$
$269$	0.569055	0.569055	0.284528	+	0.958668i	$0.408163\pi$
$270$	−2.42248	−2.42248
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	−0.831722	−0.831722
$273$	−0.999682	−0.999682
$274$	2.55264	2.55264
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	−1.39749	−1.39749
$280$	1.89290	1.89290
$281$	−0.192046	−0.192046	−0.0960230	−	0.995379i	$-0.530612\pi$
$281$	−0.192046	−0.192046	−0.0960230	+	0.995379i	$0.530612\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	−0.502348	−0.502348
$286$	0	0
$287$	0	0
$288$	−1.09683	−1.09683
$289$	1.06410	1.06410
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$293$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$294$	1.93419	1.93419
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	−1.72290	−1.72290
$301$	0	0
$302$	0	0
$303$	0.307404	0.307404
$304$	−0.329433	−0.329433
$305$	2.05655	2.05655
$306$	1.75459	1.75459
$307$	1.85383	1.85383	0.926917	−	0.375267i	$-0.122449\pi$
$307$	1.85383	1.85383	0.926917	+	0.375267i	$0.122449\pi$
$308$	0	0
$309$	0.411698	0.411698
$310$	−5.26412	−5.26412
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	−0.247541	−0.247541
$313$	0.809567	0.809567	0.404783	−	0.914413i	$-0.367347\pi$
$313$	0.809567	0.809567	0.404783	+	0.914413i	$0.367347\pi$
$314$	0	0
$315$	3.12275	3.12275
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	−2.98328	−2.98328
$321$	0	0
$322$	0	0
$323$	0.817561	0.817561
$324$	0.587099	0.587099
$325$	−3.35786	−3.35786
$326$	−2.44093	−2.44093
$327$	0.888255	0.888255
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.98358	1.98358	0.991790	−	0.127877i	$-0.0408163\pi$
$331$	1.98358	1.98358	0.991790	+	0.127877i	$0.0408163\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	−0.505778	−0.505778
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−0.470985	−0.470985
$339$	0	0
$340$	3.75946	3.75946
$341$	0	0
$342$	0.694967	0.694967
$343$	−3.63929	−3.63929
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.0641032	0.0641032	0.0320516	−	0.999486i	$-0.489796\pi$
$347$	0.0641032	0.0641032	0.0320516	+	0.999486i	$0.489796\pi$
$348$	0	0
$349$	−1.34460	−1.34460	−0.672301	−	0.740278i	$-0.734694\pi$
$349$	−1.34460	−1.34460	−0.672301	+	0.740278i	$0.734694\pi$
$350$	8.77330	8.77330
$351$	−0.917604	−0.917604
$352$	0	0
$353$	0.0641032	0.0641032	0.0320516	−	0.999486i	$-0.489796\pi$
$353$	0.0641032	0.0641032	0.0320516	+	0.999486i	$0.489796\pi$
$354$	0	0
$355$	0	0
$356$	2.11445	2.11445
$357$	1.25520	1.25520
$358$	2.74416	2.74416
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0.773254	0.773254
$361$	−0.676176	−0.676176
$362$	0	0
$363$	−0.445042	−0.445042
$364$	2.96327	2.96327
$365$	0	0
$366$	0.702682	0.702682
$367$	1.98358	1.98358	0.991790	−	0.127877i	$-0.0408163\pi$
$367$	1.98358	1.98358	0.991790	+	0.127877i	$0.0408163\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	−1.02310	−1.02310
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−1.70781	−1.70781
$376$	0	0
$377$	0	0
$378$	2.39749	2.39749
$379$	−1.52289	−1.52289	−0.761446	−	0.648228i	$-0.775510\pi$
$379$	−1.52289	−1.52289	−0.761446	+	0.648228i	$0.775510\pi$
$380$	1.48907	1.48907
$381$	−0.0285286	−0.0285286
$382$	−0.0976222	−0.0976222
$383$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$383$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$384$	−0.410633	−0.410633
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.99589	−1.99589	−0.997945	−	0.0640702i	$-0.979592\pi$
$389$	−1.99589	−1.99589	−0.997945	+	0.0640702i	$0.979592\pi$
$390$	−1.53828	−1.53828
$391$	0	0
$392$	−1.38727	−1.38727
$393$	−0.860973	−0.860973
$394$	0.292465	0.292465
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	2.31920	2.31920
$399$	0.497166	0.497166
$400$	−1.69887	−1.69887
$401$	−1.96312	−1.96312	−0.981559	−	0.191159i	$-0.938776\pi$
$401$	−1.96312	−1.96312	−0.981559	+	0.191159i	$0.938776\pi$
$402$	0	0
$403$	−1.99398	−1.99398
$404$	−0.911211	−0.911211
$405$	0.882776	0.882776
$406$	0	0
$407$	0	0
$408$	0.310812	0.310812
$409$	−0.192046	−0.192046	−0.0960230	−	0.995379i	$-0.530612\pi$
$409$	−0.192046	−0.192046	−0.0960230	+	0.995379i	$0.530612\pi$
$410$	0	0
$411$	0.745969	0.745969
$412$	−1.22036	−1.22036
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−1.56500	−1.56500
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	2.28617	2.28617
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.21612	4.21612
$426$	0	0
$427$	−2.03533	−2.03533
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−0.464251	−0.464251
$433$	−1.34460	−1.34460	−0.672301	−	0.740278i	$-0.734694\pi$
$433$	−1.34460	−1.34460	−0.672301	+	0.740278i	$0.734694\pi$
$434$	5.20982	5.20982
$435$	0	0
$436$	−2.63298	−2.63298
$437$	0	0
$438$	0	0
$439$	1.60283	1.60283	0.801414	−	0.598111i	$-0.204082\pi$
$439$	1.60283	1.60283	0.801414	+	0.598111i	$0.204082\pi$
$440$	0	0
$441$	−2.28860	−2.28860
$442$	2.50351	2.50351
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	3.17934	3.17934
$446$	−1.57891	−1.57891
$447$	0	0
$448$	2.95251	2.95251
$449$	−1.89811	−1.89811	−0.949056	−	0.315108i	$-0.897959\pi$
$449$	−1.89811	−1.89811	−0.949056	+	0.315108i	$0.897959\pi$
$450$	3.58391	3.58391
$451$	0	0
$452$	0	0
$453$	0	0
$454$	2.04768	2.04768
$455$	4.45565	4.45565
$456$	0.123108	0.123108
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	1.15214	1.15214
$460$	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
$465$	−1.53836	−1.53836
$466$	−0.866610	−0.866610
$467$	−0.192046	−0.192046	−0.0960230	−	0.995379i	$-0.530612\pi$
$467$	−0.192046	−0.192046	−0.0960230	+	0.995379i	$0.530612\pi$
$468$	1.21050	1.21050
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	1.66994	1.66994
$476$	−3.72068	−3.72068
$477$	0	0
$478$	1.40879	1.40879
$479$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$479$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$480$	−1.20740	−1.20740
$481$	0	0
$482$	0	0
$483$	0	0
$484$	1.31920	1.31920
$485$	0	0
$486$	1.52289	1.52289
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	−0.503988	−0.503988
$489$	−0.713325	−0.713325
$490$	−8.62080	−8.62080
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0.991603	0.991603
$495$	0	0
$496$	−1.00883	−1.00883
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	5.06232	5.06232
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−0.765277	−0.765277
$505$	−1.37012	−1.37012
$506$	0	0
$507$	−0.137638	−0.137638
$508$	0.0845649	0.0845649
$509$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$509$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$510$	1.93146	1.93146
$511$	0	0
$512$	−1.07321	−1.07321
$513$	0.456347	0.456347
$514$	−0.486107	−0.486107
$515$	−1.83496	−1.83496
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	1.10331	1.10331
$521$	−1.14423	−1.14423	−0.572117	−	0.820172i	$-0.693878\pi$
$521$	−1.14423	−1.14423	−0.572117	+	0.820172i	$0.693878\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	2.55211	2.55211
$525$	2.56386	2.56386
$526$	0	0
$527$	2.50364	2.50364
$528$	0	0
$529$	1.00000	1.00000
$530$	0	0
$531$	0	0
$532$	−1.47371	−1.47371
$533$	0	0
$534$	1.08632	1.08632
$535$	0	0
$536$	0	0
$537$	0.801938	0.801938
$538$	−0.866610	−0.866610
$539$	0	0
$540$	2.09846	2.09846
$541$	−1.89811	−1.89811	−0.949056	−	0.315108i	$-0.897959\pi$
$541$	−1.89811	−1.89811	−0.949056	+	0.315108i	$0.897959\pi$
$542$	0	0
$543$	0	0
$544$	1.96501	1.96501
$545$	−3.95901	−3.95901
$546$	1.52241	1.52241
$547$	−1.67618	−1.67618	−0.838088	−	0.545535i	$-0.816327\pi$
$547$	−1.67618	−1.67618	−0.838088	+	0.545535i	$0.816327\pi$
$548$	−2.21121	−2.21121
$549$	−0.831437	−0.831437
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.03679	1.03679	0.518393	−	0.855143i	$-0.326531\pi$
$557$	1.03679	1.03679	0.518393	+	0.855143i	$0.326531\pi$
$558$	2.12822	2.12822
$559$	0	0
$560$	2.25428	2.25428
$561$	0	0
$562$	0.292465	0.292465
$563$	−0.690730	−0.690730	−0.345365	−	0.938468i	$-0.612245\pi$
$563$	−0.690730	−0.690730	−0.345365	+	0.938468i	$0.612245\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−0.873670	−0.873670
$568$	0	0
$569$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$569$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$570$	0.765022	0.765022
$571$	1.93459	1.93459	0.967295	−	0.253655i	$-0.0816327\pi$
$571$	1.93459	1.93459	0.967295	+	0.253655i	$0.0816327\pi$
$572$	0	0
$573$	−0.0285286	−0.0285286
$574$	0	0
$575$	0	0
$576$	1.20610	1.20610
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−1.62051	−1.62051
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	1.82014	1.82014
$586$	−1.89902	−1.89902
$587$	0.809567	0.809567	0.404783	−	0.914413i	$-0.367347\pi$
$587$	0.809567	0.809567	0.404783	+	0.914413i	$0.367347\pi$
$588$	−1.67548	−1.67548
$589$	0.991657	0.991657
$590$	0	0
$591$	0.0854685	0.0854685
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	−5.59451	−5.59451
$596$	0	0
$597$	0.677751	0.677751
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0.634863	0.634863
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.98358	1.98358
$606$	−0.468143	−0.468143
$607$	0.809567	0.809567	0.404783	−	0.914413i	$-0.367347\pi$
$607$	0.809567	0.809567	0.404783	+	0.914413i	$0.367347\pi$
$608$	0.778312	0.778312
$609$	0	0
$610$	−3.13190	−3.13190
$611$	0	0
$612$	−1.51991	−1.51991
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−2.82319	−2.82319
$615$	0	0
$616$	0	0
$617$	−0.690730	−0.690730	−0.345365	−	0.938468i	$-0.612245\pi$
$617$	−0.690730	−0.690730	−0.345365	+	0.938468i	$0.612245\pi$
$618$	−0.626971	−0.626971
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	4.56003	4.56003
$621$	0	0
$622$	0	0
$623$	−3.14654	−3.14654
$624$	−0.294800	−0.294800
$625$	4.67723	4.67723
$626$	−1.23288	−1.23288
$627$	0	0
$628$	0	0
$629$	0	0
$630$	−4.75561	−4.75561
$631$	1.43670	1.43670	0.718349	−	0.695683i	$-0.244898\pi$
$631$	1.43670	1.43670	0.718349	+	0.695683i	$0.244898\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0.127154	0.127154
$636$	0	0
$637$	−3.26545	−3.26545
$638$	0	0
$639$	0	0
$640$	1.83022	1.83022
$641$	−1.89811	−1.89811	−0.949056	−	0.315108i	$-0.897959\pi$
$641$	−1.89811	−1.89811	−0.949056	+	0.315108i	$0.897959\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	−1.24506	−1.24506
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−0.216338	−0.216338
$649$	0	0
$650$	5.11365	5.11365
$651$	1.52249	1.52249
$652$	2.11445	2.11445
$653$	1.60283	1.60283	0.801414	−	0.598111i	$-0.204082\pi$
$653$	1.60283	1.60283	0.801414	+	0.598111i	$0.204082\pi$
$654$	−1.35272	−1.35272
$655$	3.83741	3.83741
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.14423	−1.14423	−0.572117	−	0.820172i	$-0.693878\pi$
$659$	−1.14423	−1.14423	−0.572117	+	0.820172i	$0.693878\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−3.02078	−3.02078
$663$	0.731613	0.731613
$664$	0	0
$665$	−2.21590	−2.21590
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−0.461413	−0.461413
$670$	0	0
$671$	0	0
$672$	1.19494	1.19494
$673$	1.60283	1.60283	0.801414	−	0.598111i	$-0.204082\pi$
$673$	1.60283	1.60283	0.801414	+	0.598111i	$0.204082\pi$
$674$	0	0
$675$	2.35336	2.35336
$676$	0.407989	0.407989
$677$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$677$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$678$	0	0
$679$	0	0
$680$	−1.38531	−1.38531
$681$	0.598404	0.598404
$682$	0	0
$683$	1.74264	1.74264	0.871319	−	0.490718i	$-0.163265\pi$
$683$	1.74264	1.74264	0.871319	+	0.490718i	$0.163265\pi$
$684$	−0.602013	−0.602013
$685$	−3.32483	−3.32483
$686$	5.54225	5.54225
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−1.89811	−1.89811	−0.949056	−	0.315108i	$-0.897959\pi$
$691$	−1.89811	−1.89811	−0.949056	+	0.315108i	$0.897959\pi$
$692$	0	0
$693$	0	0
$694$	−0.0976222	−0.0976222
$695$	0	0
$696$	0	0
$697$	0	0
$698$	2.04768	2.04768
$699$	−0.253253	−0.253253
$700$	−7.59984	−7.59984
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	1.39741	1.39741
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−0.0976222	−0.0976222
$707$	1.35598	1.35598
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	−0.779145	−0.779145
$713$	0	0
$714$	−1.91153	−1.91153
$715$	0	0
$716$	−2.37712	−2.37712
$717$	0.411698	0.411698
$718$	0	0
$719$	1.03679	1.03679	0.518393	−	0.855143i	$-0.326531\pi$
$719$	1.03679	1.03679	0.518393	+	0.855143i	$0.326531\pi$
$720$	0.920879	0.920879
$721$	1.81603	1.81603
$722$	1.02974	1.02974
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0.677751	0.677751
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	−1.09192	−1.09192
$729$	0	0
$730$	0	0
$731$	0	0
$732$	−0.608696	−0.608696
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	−3.02078	−3.02078
$735$	−2.51930	−2.51930
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−0.690730	−0.690730	−0.345365	−	0.938468i	$-0.612245\pi$
$739$	−0.690730	−0.690730	−0.345365	+	0.938468i	$0.612245\pi$
$740$	0	0
$741$	0.289781	0.289781
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0.376998	0.376998
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	2.60081	2.60081
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	−2.07681	−2.07681
$757$	0.569055	0.569055	0.284528	−	0.958668i	$-0.408163\pi$
$757$	0.569055	0.569055	0.284528	+	0.958668i	$0.408163\pi$
$758$	2.31920	2.31920
$759$	0	0
$760$	−0.548701	−0.548701
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0.0434460	0.0434460
$763$	3.91817	3.91817
$764$	0.0845649	0.0845649
$765$	−2.28537	−2.28537
$766$	2.74416	2.74416
$767$	0	0
$768$	−0.0439875	−0.0439875
$769$	−1.67618	−1.67618	−0.838088	−	0.545535i	$-0.816327\pi$
$769$	−1.67618	−1.67618	−0.838088	+	0.545535i	$0.816327\pi$
$770$	0	0
$771$	−0.142057	−0.142057
$772$	0	0
$773$	−1.89811	−1.89811	−0.949056	−	0.315108i	$-0.897959\pi$
$773$	−1.89811	−1.89811	−0.949056	+	0.315108i	$0.897959\pi$
$774$	0	0
$775$	5.11393	5.11393
$776$	0	0
$777$	0	0
$778$	3.03953	3.03953
$779$	0	0
$780$	1.33253	1.33253
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−1.65212	−1.65212
$785$	0	0
$786$	1.31117	1.31117
$787$	1.85383	1.85383	0.926917	−	0.375267i	$-0.122449\pi$
$787$	1.85383	1.85383	0.926917	+	0.375267i	$0.122449\pi$
$788$	−0.253347	−0.253347
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−1.18632	−1.18632
$794$	0	0
$795$	0	0
$796$	−2.00900	−2.00900
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	−0.757131	−0.757131
$799$	0	0
$800$	4.01372	4.01372
$801$	−1.28537	−1.28537
$802$	2.98962	2.98962
$803$	0	0
$804$	0	0
$805$	0	0
$806$	3.03662	3.03662
$807$	−0.253253	−0.253253
$808$	0.335769	0.335769
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	−1.34437	−1.34437
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.17934	3.17934
$816$	0.370151	0.370151
$817$	0	0
$818$	0.292465	0.292465
$819$	−1.80136	−1.80136
$820$	0	0
$821$	−1.67618	−1.67618	−0.838088	−	0.545535i	$-0.816327\pi$
$821$	−1.67618	−1.67618	−0.838088	+	0.545535i	$0.816327\pi$
$822$	−1.13603	−1.13603
$823$	−1.34460	−1.34460	−0.672301	−	0.740278i	$-0.734694\pi$
$823$	−1.34460	−1.34460	−0.672301	+	0.740278i	$0.734694\pi$
$824$	0.449686	0.449686
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	1.74264	1.74264	0.871319	−	0.490718i	$-0.163265\pi$
$829$	1.74264	1.74264	0.871319	+	0.490718i	$0.163265\pi$
$830$	0	0
$831$	0	0
$832$	1.72091	1.72091
$833$	4.10010	4.10010
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.39749	1.39749
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	−0.842420	−0.842420
$841$	1.00000	1.00000
$842$	0	0
$843$	0.0854685	0.0854685
$844$	0	0
$845$	0.613462	0.613462
$846$	0	0
$847$	−1.96312	−1.96312
$848$	0	0
$849$	0	0
$850$	−6.42070	−6.42070
$851$	0	0
$852$	0	0
$853$	0.319200	0.319200	0.159600	−	0.987182i	$-0.448980\pi$
$853$	0.319200	0.319200	0.159600	+	0.987182i	$0.448980\pi$
$854$	3.09959	3.09959
$855$	−0.905200	−0.905200
$856$	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			$\pi$
$864$	1.09683	1.09683
$865$	0	0
$866$	2.04768	2.04768
$867$	−0.473570	−0.473570
$868$	−4.51299	−4.51299
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0.970216	0.970216
$873$	0	0
$874$	0	0
$875$	−7.53330	−7.53330
$876$	0	0
$877$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$877$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$878$	−2.44093	−2.44093
$879$	−0.554958	−0.554958
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	3.48529	3.48529
$883$	0.319200	0.319200	0.159600	−	0.987182i	$-0.448980\pi$
$883$	0.319200	0.319200	0.159600	+	0.987182i	$0.448980\pi$
$884$	−2.16866	−2.16866
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−0.125842	−0.125842
$890$	−4.84179	−4.84179
$891$	0	0
$892$	1.36773	1.36773
$893$	0	0
$894$	0	0
$895$	−3.57429	−3.57429
$896$	−1.81134	−1.81134
$897$	0	0
$898$	2.89062	2.89062
$899$	0	0
$900$	−3.10455	−3.10455
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.99589	−1.99589	−0.997945	−	0.0640702i	$-0.979592\pi$
$907$	−1.99589	−1.99589	−0.997945	+	0.0640702i	$0.979592\pi$
$908$	−1.77380	−1.77380
$909$	0.553923	0.553923
$910$	−6.78547	−6.78547
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0.146611	0.146611
$913$	0	0
$914$	0	0
$915$	−0.915249	−0.915249
$916$	0	0
$917$	−3.79783	−3.79783
$918$	−1.75459	−1.75459
$919$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$919$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$920$	0	0
$921$	−0.825034	−0.825034
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.741854	0.741854
$928$	0	0
$929$	1.03679	1.03679	0.518393	−	0.855143i	$-0.326531\pi$
$929$	1.03679	1.03679	0.518393	+	0.855143i	$0.326531\pi$
$930$	2.34275	2.34275
$931$	1.62399	1.62399
$932$	0.750697	0.750697
$933$	0	0
$934$	0.292465	0.292465
$935$	0	0
$936$	−0.446053	−0.446053
$937$	−0.192046	−0.192046	−0.0960230	−	0.995379i	$-0.530612\pi$
$937$	−0.192046	−0.192046	−0.0960230	+	0.995379i	$0.530612\pi$
$938$	0	0
$939$	−0.360291	−0.360291
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−3.12275	−3.12275
$946$	0	0
$947$	1.43670	1.43670	0.718349	−	0.695683i	$-0.244898\pi$
$947$	1.43670	1.43670	0.718349	+	0.695683i	$0.244898\pi$
$948$	0	0
$949$	0	0
$950$	−2.54314	−2.54314
$951$	0	0
$952$	1.37102	1.37102
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0.127154	0.127154
$956$	−1.22036	−1.22036
$957$	0	0
$958$	−1.89902	−1.89902
$959$	3.29053	3.29053
$960$	1.32769	1.32769
$961$	2.03679	2.03679
$962$	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			$\pi$
$968$	−0.486107	−0.486107
$969$	−0.363849	−0.363849
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−1.31920	−1.31920
$973$	0	0
$974$	0	0
$975$	1.49439	1.49439
$976$	−0.600207	−0.600207
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	1.08632	1.08632
$979$	0	0
$980$	7.46774	7.46774
$981$	1.60058	1.60058
$982$	0	0
$983$	1.85383	1.85383	0.926917	−	0.375267i	$-0.122449\pi$
$983$	1.85383	1.85383	0.926917	+	0.375267i	$0.122449\pi$
$984$	0	0
$985$	−0.380939	−0.380939
$986$	0	0
$987$	0	0
$988$	−0.858973	−0.858973
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	2.38345	2.38345
$993$	−0.882776	−0.882776
$994$	0	0
$995$	−3.02078	−3.02078
$996$	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.1510.5	✓	21
1511.1510	odd	2	CM	1511.1.b.b.1510.5	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1511.1.b.b.1510.5	✓	21	1.1	even	1	trivial
1511.1.b.b.1510.5	✓	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)

\(f(q)\)	\(=\)	\(q-1.52289 q^{2} -0.445042 q^{3} +1.31920 q^{4} +1.98358 q^{5} +0.677751 q^{6} -1.96312 q^{7} -0.486107 q^{8} -0.801938 q^{9} +O(q^{10})\)	\(q-1.52289 q^{2} -0.445042 q^{3} +1.31920 q^{4} +1.98358 q^{5} +0.677751 q^{6} -1.96312 q^{7} -0.486107 q^{8} -0.801938 q^{9} -3.02078 q^{10} -0.587099 q^{12} -1.14423 q^{13} +2.98962 q^{14} -0.882776 q^{15} -0.578912 q^{16} +1.43670 q^{17} +1.22126 q^{18} +0.569055 q^{19} +2.61674 q^{20} +0.873670 q^{21} +0.216338 q^{24} +2.93459 q^{25} +1.74254 q^{26} +0.801938 q^{27} -2.58975 q^{28} +1.34437 q^{30} +1.74264 q^{31} +1.36773 q^{32} -2.18794 q^{34} -3.89400 q^{35} -1.05792 q^{36} -0.866610 q^{38} +0.509232 q^{39} -0.964232 q^{40} -1.33050 q^{42} -1.59071 q^{45} +0.257640 q^{48} +2.85383 q^{49} -4.46906 q^{50} -0.639391 q^{51} -1.50947 q^{52} -1.22126 q^{54} +0.954285 q^{56} -0.253253 q^{57} -1.16456 q^{60} +1.03679 q^{61} -2.65385 q^{62} +1.57430 q^{63} -1.50399 q^{64} -2.26968 q^{65} +1.89529 q^{68} +5.93014 q^{70} +0.389827 q^{72} -1.30602 q^{75} +0.750697 q^{76} -0.775505 q^{78} -1.14832 q^{80} +0.445042 q^{81} +1.15255 q^{84} +2.84981 q^{85} +1.60283 q^{89} +2.42248 q^{90} +2.24627 q^{91} -0.775547 q^{93} +1.12877 q^{95} -0.608696 q^{96} -4.34608 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - 7 q^{3} + 21 q^{4} + 14 q^{9}+O(q^{10}) \) 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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