LMFDB - Embedded newform 1511.1.b.b.1510.13

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			1510.13
Root			$0.925077$ 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)

$f(q)$	$=$	$q+0.569055 q^{2} -1.80194 q^{3} -0.676176 q^{4} -0.925077 q^{5} -1.02540 q^{6} -1.99589 q^{7} -0.953837 q^{8} +2.24698 q^{9} +O(q^{10})$	\(q+0.569055 q^{2} -1.80194 q^{3} -0.676176 q^{4} -0.925077 q^{5} -1.02540 q^{6} -1.99589 q^{7} -0.953837 q^{8} +2.24698 q^{9} -0.526420 q^{10} +1.21843 q^{12} -1.89811 q^{13} -1.13577 q^{14} +1.66693 q^{15} +0.133390 q^{16} +1.93459 q^{17} +1.27866 q^{18} -0.192046 q^{19} +0.625515 q^{20} +3.59647 q^{21} +1.71875 q^{24} -0.144233 q^{25} -1.08013 q^{26} -2.24698 q^{27} +1.34957 q^{28} +0.948575 q^{30} -0.690730 q^{31} +1.02974 q^{32} +1.10089 q^{34} +1.84635 q^{35} -1.51935 q^{36} -0.109285 q^{38} +3.42028 q^{39} +0.882372 q^{40} +2.04659 q^{42} -2.07863 q^{45} -0.240361 q^{48} +2.98358 q^{49} -0.0820767 q^{50} -3.48601 q^{51} +1.28346 q^{52} -1.27866 q^{54} +1.90375 q^{56} +0.346055 q^{57} -1.12714 q^{60} -1.52289 q^{61} -0.393064 q^{62} -4.48473 q^{63} +0.452590 q^{64} +1.75590 q^{65} -1.30812 q^{68} +1.05068 q^{70} -2.14325 q^{72} +0.259899 q^{75} +0.129857 q^{76} +1.94633 q^{78} -0.123396 q^{80} +1.80194 q^{81} -2.43185 q^{84} -1.78964 q^{85} -1.34460 q^{89} -1.18285 q^{90} +3.78842 q^{91} +1.24465 q^{93} +0.177657 q^{95} -1.85553 q^{96} +1.69782 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.569055	0.569055	0.284528	−	0.958668i	$-0.408163\pi$
$2$	0.569055	0.569055	0.284528	+	0.958668i	$0.408163\pi$
$3$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$3$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$4$	−0.676176	−0.676176
$5$	−0.925077	−0.925077	−0.462538	−	0.886599i	$-0.653061\pi$
$5$	−0.925077	−0.925077	−0.462538	+	0.886599i	$0.653061\pi$
$6$	−1.02540	−1.02540
$7$	−1.99589	−1.99589	−0.997945	−	0.0640702i	$-0.979592\pi$
$7$	−1.99589	−1.99589	−0.997945	+	0.0640702i	$0.979592\pi$
$8$	−0.953837	−0.953837
$9$	2.24698	2.24698
$10$	−0.526420	−0.526420
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	1.21843	1.21843
$13$	−1.89811	−1.89811	−0.949056	−	0.315108i	$-0.897959\pi$
$13$	−1.89811	−1.89811	−0.949056	+	0.315108i	$0.897959\pi$
$14$	−1.13577	−1.13577
$15$	1.66693	1.66693
$16$	0.133390	0.133390
$17$	1.93459	1.93459	0.967295	−	0.253655i	$-0.0816327\pi$
$17$	1.93459	1.93459	0.967295	+	0.253655i	$0.0816327\pi$
$18$	1.27866	1.27866
$19$	−0.192046	−0.192046	−0.0960230	−	0.995379i	$-0.530612\pi$
$19$	−0.192046	−0.192046	−0.0960230	+	0.995379i	$0.530612\pi$
$20$	0.625515	0.625515
$21$	3.59647	3.59647
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	1.71875	1.71875
$25$	−0.144233	−0.144233
$26$	−1.08013	−1.08013
$27$	−2.24698	−2.24698
$28$	1.34957	1.34957
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0.948575	0.948575
$31$	−0.690730	−0.690730	−0.345365	−	0.938468i	$-0.612245\pi$
$31$	−0.690730	−0.690730	−0.345365	+	0.938468i	$0.612245\pi$
$32$	1.02974	1.02974
$33$	0	0
$34$	1.10089	1.10089
$35$	1.84635	1.84635
$36$	−1.51935	−1.51935
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−0.109285	−0.109285
$39$	3.42028	3.42028
$40$	0.882372	0.882372
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	2.04659	2.04659
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	−2.07863	−2.07863
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−0.240361	−0.240361
$49$	2.98358	2.98358
$50$	−0.0820767	−0.0820767
$51$	−3.48601	−3.48601
$52$	1.28346	1.28346
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−1.27866	−1.27866
$55$	0	0
$56$	1.90375	1.90375
$57$	0.346055	0.346055
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−1.12714	−1.12714
$61$	−1.52289	−1.52289	−0.761446	−	0.648228i	$-0.775510\pi$
$61$	−1.52289	−1.52289	−0.761446	+	0.648228i	$0.775510\pi$
$62$	−0.393064	−0.393064
$63$	−4.48473	−4.48473
$64$	0.452590	0.452590
$65$	1.75590	1.75590
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−1.30812	−1.30812
$69$	0	0
$70$	1.05068	1.05068
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−2.14325	−2.14325
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0.259899	0.259899
$76$	0.129857	0.129857
$77$	0	0
$78$	1.94633	1.94633
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−0.123396	−0.123396
$81$	1.80194	1.80194
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−2.43185	−2.43185
$85$	−1.78964	−1.78964
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.34460	−1.34460	−0.672301	−	0.740278i	$-0.734694\pi$
$89$	−1.34460	−1.34460	−0.672301	+	0.740278i	$0.734694\pi$
$90$	−1.18285	−1.18285
$91$	3.78842	3.78842
$92$	0	0
$93$	1.24465	1.24465
$94$	0	0
$95$	0.177657	0.177657
$96$	−1.85553	−1.85553
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	1.69782	1.69782
$99$	0	0
$100$	0.0975271	0.0975271
$101$	1.60283	1.60283	0.801414	−	0.598111i	$-0.204082\pi$
$101$	1.60283	1.60283	0.801414	+	0.598111i	$0.204082\pi$
$102$	−1.98373	−1.98373
$103$	0.319200	0.319200	0.159600	−	0.987182i	$-0.448980\pi$
$103$	0.319200	0.319200	0.159600	+	0.987182i	$0.448980\pi$
$104$	1.81049	1.81049
$105$	−3.32701	−3.32701
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	1.51935	1.51935
$109$	1.03679	1.03679	0.518393	−	0.855143i	$-0.326531\pi$
$109$	1.03679	1.03679	0.518393	+	0.855143i	$0.326531\pi$
$110$	0	0
$111$	0	0
$112$	−0.266233	−0.266233
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0.196924	0.196924
$115$	0	0
$116$	0	0
$117$	−4.26502	−4.26502
$118$	0	0
$119$	−3.86123	−3.86123
$120$	−1.58998	−1.58998
$121$	1.00000	1.00000
$122$	−0.866610	−0.866610
$123$	0	0
$124$	0.467055	0.467055
$125$	1.05850	1.05850
$126$	−2.55206	−2.55206
$127$	1.74264	1.74264	0.871319	−	0.490718i	$-0.163265\pi$
$127$	1.74264	1.74264	0.871319	+	0.490718i	$0.163265\pi$
$128$	−0.772194	−0.772194
$129$	0	0
$130$	0.999203	0.999203
$131$	−1.14423	−1.14423	−0.572117	−	0.820172i	$-0.693878\pi$
$131$	−1.14423	−1.14423	−0.572117	+	0.820172i	$0.693878\pi$
$132$	0	0
$133$	0.383303	0.383303
$134$	0	0
$135$	2.07863	2.07863
$136$	−1.84528	−1.84528
$137$	−1.96312	−1.96312	−0.981559	−	0.191159i	$-0.938776\pi$
$137$	−1.96312	−1.96312	−0.981559	+	0.191159i	$0.938776\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	−1.24846	−1.24846
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0.299726	0.299726
$145$	0	0
$146$	0	0
$147$	−5.37623	−5.37623
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0.147897	0.147897
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0.183181	0.183181
$153$	4.34698	4.34698
$154$	0	0
$155$	0.638978	0.638978
$156$	−2.31271	−2.31271
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	−0.952591	−0.952591
$161$	0	0
$162$	1.02540	1.02540
$163$	−1.34460	−1.34460	−0.672301	−	0.740278i	$-0.734694\pi$
$163$	−1.34460	−1.34460	−0.672301	+	0.740278i	$0.734694\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	−3.43045	−3.43045
$169$	2.60283	2.60283
$170$	−1.01841	−1.01841
$171$	−0.431524	−0.431524
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0.287874	0.287874
$176$	0	0
$177$	0	0
$178$	−0.765153	−0.765153
$179$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$179$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$180$	1.40552	1.40552
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	2.15582	2.15582
$183$	2.74416	2.74416
$184$	0	0
$185$	0	0
$186$	0.708276	0.708276
$187$	0	0
$188$	0	0
$189$	4.48473	4.48473
$190$	0.101097	0.101097
$191$	1.74264	1.74264	0.871319	−	0.490718i	$-0.163265\pi$
$191$	1.74264	1.74264	0.871319	+	0.490718i	$0.163265\pi$
$192$	−0.815539	−0.815539
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	−3.16402	−3.16402
$196$	−2.01743	−2.01743
$197$	0.0641032	0.0641032	0.0320516	−	0.999486i	$-0.489796\pi$
$197$	0.0641032	0.0641032	0.0320516	+	0.999486i	$0.489796\pi$
$198$	0	0
$199$	0.569055	0.569055	0.284528	−	0.958668i	$-0.408163\pi$
$199$	0.569055	0.569055	0.284528	+	0.958668i	$0.408163\pi$
$200$	0.137575	0.137575
$201$	0	0
$202$	0.912097	0.912097
$203$	0	0
$204$	2.35716	2.35716
$205$	0	0
$206$	0.181642	0.181642
$207$	0	0
$208$	−0.253190	−0.253190
$209$	0	0
$210$	−1.89325	−1.89325
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	2.14325	2.14325
$217$	1.37862	1.37862
$218$	0.589988	0.589988
$219$	0	0
$220$	0	0
$221$	−3.67207	−3.67207
$222$	0	0
$223$	−1.52289	−1.52289	−0.761446	−	0.648228i	$-0.775510\pi$
$223$	−1.52289	−1.52289	−0.761446	+	0.648228i	$0.775510\pi$
$224$	−2.05526	−2.05526
$225$	−0.324089	−0.324089
$226$	0	0
$227$	1.43670	1.43670	0.718349	−	0.695683i	$-0.244898\pi$
$227$	1.43670	1.43670	0.718349	+	0.695683i	$0.244898\pi$
$228$	−0.233994	−0.233994
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−0.192046	−0.192046	−0.0960230	−	0.995379i	$-0.530612\pi$
$233$	−0.192046	−0.192046	−0.0960230	+	0.995379i	$0.530612\pi$
$234$	−2.42703	−2.42703
$235$	0	0
$236$	0	0
$237$	0	0
$238$	−2.19725	−2.19725
$239$	0.319200	0.319200	0.159600	−	0.987182i	$-0.448980\pi$
$239$	0.319200	0.319200	0.159600	+	0.987182i	$0.448980\pi$
$240$	0.222353	0.222353
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0.569055	0.569055
$243$	−1.00000	−1.00000
$244$	1.02974	1.02974
$245$	−2.76004	−2.76004
$246$	0	0
$247$	0.364525	0.364525
$248$	0.658844	0.658844
$249$	0	0
$250$	0.602347	0.602347
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	3.03246	3.03246
$253$	0	0
$254$	0.991657	0.991657
$255$	3.22483	3.22483
$256$	−0.892012	−0.892012
$257$	−1.67618	−1.67618	−0.838088	−	0.545535i	$-0.816327\pi$
$257$	−1.67618	−1.67618	−0.838088	+	0.545535i	$0.816327\pi$
$258$	0	0
$259$	0	0
$260$	−1.18730	−1.18730
$261$	0	0
$262$	−0.651132	−0.651132
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0.218121	0.218121
$267$	2.42289	2.42289
$268$	0	0
$269$	−0.192046	−0.192046	−0.0960230	−	0.995379i	$-0.530612\pi$
$269$	−0.192046	−0.192046	−0.0960230	+	0.995379i	$0.530612\pi$
$270$	1.18285	1.18285
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0.258056	0.258056
$273$	−6.82650	−6.82650
$274$	−1.11712	−1.11712
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	−1.55206	−1.55206
$280$	−1.76112	−1.76112
$281$	0.0641032	0.0641032	0.0320516	−	0.999486i	$-0.489796\pi$
$281$	0.0641032	0.0641032	0.0320516	+	0.999486i	$0.489796\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	−0.320127	−0.320127
$286$	0	0
$287$	0	0
$288$	2.31381	2.31381
$289$	2.74264	2.74264
$290$	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	$0.571429\pi$
$294$	−3.05937	−3.05937
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	−0.175738	−0.175738
$301$	0	0
$302$	0	0
$303$	−2.88819	−2.88819
$304$	−0.0256171	−0.0256171
$305$	1.40879	1.40879
$306$	2.47367	2.47367
$307$	1.98358	1.98358	0.991790	−	0.127877i	$-0.0408163\pi$
$307$	1.98358	1.98358	0.991790	+	0.127877i	$0.0408163\pi$
$308$	0	0
$309$	−0.575178	−0.575178
$310$	0.363614	0.363614
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	−3.26239	−3.26239
$313$	1.85383	1.85383	0.926917	−	0.375267i	$-0.122449\pi$
$313$	1.85383	1.85383	0.926917	+	0.375267i	$0.122449\pi$
$314$	0	0
$315$	4.14871	4.14871
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	−0.418681	−0.418681
$321$	0	0
$322$	0	0
$323$	−0.371530	−0.371530
$324$	−1.21843	−1.21843
$325$	0.273771	0.273771
$326$	−0.765153	−0.765153
$327$	−1.86822	−1.86822
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−0.925077	−0.925077	−0.462538	−	0.886599i	$-0.653061\pi$
$331$	−0.925077	−0.925077	−0.462538	+	0.886599i	$0.653061\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0.479735	0.479735
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	1.48115	1.48115
$339$	0	0
$340$	1.21011	1.21011
$341$	0	0
$342$	−0.245561	−0.245561
$343$	−3.95901	−3.95901
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.74264	1.74264	0.871319	−	0.490718i	$-0.163265\pi$
$347$	1.74264	1.74264	0.871319	+	0.490718i	$0.163265\pi$
$348$	0	0
$349$	1.43670	1.43670	0.718349	−	0.695683i	$-0.244898\pi$
$349$	1.43670	1.43670	0.718349	+	0.695683i	$0.244898\pi$
$350$	0.163816	0.163816
$351$	4.26502	4.26502
$352$	0	0
$353$	1.74264	1.74264	0.871319	−	0.490718i	$-0.163265\pi$
$353$	1.74264	1.74264	0.871319	+	0.490718i	$0.163265\pi$
$354$	0	0
$355$	0	0
$356$	0.909188	0.909188
$357$	6.95770	6.95770
$358$	0.709600	0.709600
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	1.98267	1.98267
$361$	−0.963118	−0.963118
$362$	0	0
$363$	−1.80194	−1.80194
$364$	−2.56164	−2.56164
$365$	0	0
$366$	1.56158	1.56158
$367$	−0.925077	−0.925077	−0.462538	−	0.886599i	$-0.653061\pi$
$367$	−0.925077	−0.925077	−0.462538	+	0.886599i	$0.653061\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	−0.841605	−0.841605
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−1.90736	−1.90736
$376$	0	0
$377$	0	0
$378$	2.55206	2.55206
$379$	0.569055	0.569055	0.284528	−	0.958668i	$-0.408163\pi$
$379$	0.569055	0.569055	0.284528	+	0.958668i	$0.408163\pi$
$380$	−0.120128	−0.120128
$381$	−3.14012	−3.14012
$382$	0.991657	0.991657
$383$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$383$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$384$	1.39145	1.39145
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.03679	1.03679	0.518393	−	0.855143i	$-0.326531\pi$
$389$	1.03679	1.03679	0.518393	+	0.855143i	$0.326531\pi$
$390$	−1.80050	−1.80050
$391$	0	0
$392$	−2.84585	−2.84585
$393$	2.06184	2.06184
$394$	0.0364782	0.0364782
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0.323824	0.323824
$399$	−0.690688	−0.690688
$400$	−0.0192394	−0.0192394
$401$	−1.99589	−1.99589	−0.997945	−	0.0640702i	$-0.979592\pi$
$401$	−1.99589	−1.99589	−0.997945	+	0.0640702i	$0.979592\pi$
$402$	0	0
$403$	1.31108	1.31108
$404$	−1.08379	−1.08379
$405$	−1.66693	−1.66693
$406$	0	0
$407$	0	0
$408$	3.32508	3.32508
$409$	0.0641032	0.0641032	0.0320516	−	0.999486i	$-0.489796\pi$
$409$	0.0641032	0.0641032	0.0320516	+	0.999486i	$0.489796\pi$
$410$	0	0
$411$	3.53742	3.53742
$412$	−0.215835	−0.215835
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−1.95457	−1.95457
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	2.24965	2.24965
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.279032	−0.279032
$426$	0	0
$427$	3.03953	3.03953
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−0.299726	−0.299726
$433$	1.43670	1.43670	0.718349	−	0.695683i	$-0.244898\pi$
$433$	1.43670	1.43670	0.718349	+	0.695683i	$0.244898\pi$
$434$	0.784512	0.784512
$435$	0	0
$436$	−0.701049	−0.701049
$437$	0	0
$438$	0	0
$439$	−1.34460	−1.34460	−0.672301	−	0.740278i	$-0.734694\pi$
$439$	−1.34460	−1.34460	−0.672301	+	0.740278i	$0.734694\pi$
$440$	0	0
$441$	6.70404	6.70404
$442$	−2.08961	−2.08961
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	1.24386	1.24386
$446$	−0.866610	−0.866610
$447$	0	0
$448$	−0.903321	−0.903321
$449$	0.809567	0.809567	0.404783	−	0.914413i	$-0.367347\pi$
$449$	0.809567	0.809567	0.404783	+	0.914413i	$0.367347\pi$
$450$	−0.184425	−0.184425
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0.817561	0.817561
$455$	−3.50458	−3.50458
$456$	−0.330080	−0.330080
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	−4.34698	−4.34698
$460$	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
$465$	−1.15140	−1.15140
$466$	−0.109285	−0.109285
$467$	0.0641032	0.0641032	0.0320516	−	0.999486i	$-0.489796\pi$
$467$	0.0641032	0.0641032	0.0320516	+	0.999486i	$0.489796\pi$
$468$	2.88390	2.88390
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0.0276994	0.0276994
$476$	2.61087	2.61087
$477$	0	0
$478$	0.181642	0.181642
$479$	−0.445042	−0.445042	−0.222521	−	0.974928i	$-0.571429\pi$
$479$	−0.445042	−0.445042	−0.222521	+	0.974928i	$0.571429\pi$
$480$	1.71651	1.71651
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−0.676176	−0.676176
$485$	0	0
$486$	−0.569055	−0.569055
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	1.45259	1.45259
$489$	2.42289	2.42289
$490$	−1.57062	−1.57062
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0.207435	0.207435
$495$	0	0
$496$	−0.0921368	−0.0921368
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−0.715735	−0.715735
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	4.27770	4.27770
$505$	−1.48274	−1.48274
$506$	0	0
$507$	−4.69013	−4.69013
$508$	−1.17833	−1.17833
$509$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$509$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$510$	1.83510	1.83510
$511$	0	0
$512$	0.264591	0.264591
$513$	0.431524	0.431524
$514$	−0.953837	−0.953837
$515$	−0.295284	−0.295284
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−1.67484	−1.67484
$521$	−1.89811	−1.89811	−0.949056	−	0.315108i	$-0.897959\pi$
$521$	−1.89811	−1.89811	−0.949056	+	0.315108i	$0.897959\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0.773703	0.773703
$525$	−0.518731	−0.518731
$526$	0	0
$527$	−1.33628	−1.33628
$528$	0	0
$529$	1.00000	1.00000
$530$	0	0
$531$	0	0
$532$	−0.259180	−0.259180
$533$	0	0
$534$	1.37876	1.37876
$535$	0	0
$536$	0	0
$537$	−2.24698	−2.24698
$538$	−0.109285	−0.109285
$539$	0	0
$540$	−1.40552	−1.40552
$541$	0.809567	0.809567	0.404783	−	0.914413i	$-0.367347\pi$
$541$	0.809567	0.809567	0.404783	+	0.914413i	$0.367347\pi$
$542$	0	0
$543$	0	0
$544$	1.99213	1.99213
$545$	−0.959106	−0.959106
$546$	−3.88466	−3.88466
$547$	−1.96312	−1.96312	−0.981559	−	0.191159i	$-0.938776\pi$
$547$	−1.96312	−1.96312	−0.981559	+	0.191159i	$0.938776\pi$
$548$	1.32741	1.32741
$549$	−3.42191	−3.42191
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.52289	−1.52289	−0.761446	−	0.648228i	$-0.775510\pi$
$557$	−1.52289	−1.52289	−0.761446	+	0.648228i	$0.775510\pi$
$558$	−0.883206	−0.883206
$559$	0	0
$560$	0.246286	0.246286
$561$	0	0
$562$	0.0364782	0.0364782
$563$	1.60283	1.60283	0.801414	−	0.598111i	$-0.204082\pi$
$563$	1.60283	1.60283	0.801414	+	0.598111i	$0.204082\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−3.59647	−3.59647
$568$	0	0
$569$	−1.80194	−1.80194	−0.900969	−	0.433884i	$-0.857143\pi$
$569$	−1.80194	−1.80194	−0.900969	+	0.433884i	$0.857143\pi$
$570$	−0.182170	−0.182170
$571$	−1.14423	−1.14423	−0.572117	−	0.820172i	$-0.693878\pi$
$571$	−1.14423	−1.14423	−0.572117	+	0.820172i	$0.693878\pi$
$572$	0	0
$573$	−3.14012	−3.14012
$574$	0	0
$575$	0	0
$576$	1.01696	1.01696
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	1.56071	1.56071
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	3.94547	3.94547
$586$	−0.253253	−0.253253
$587$	1.85383	1.85383	0.926917	−	0.375267i	$-0.122449\pi$
$587$	1.85383	1.85383	0.926917	+	0.375267i	$0.122449\pi$
$588$	3.63528	3.63528
$589$	0.132652	0.132652
$590$	0	0
$591$	−0.115510	−0.115510
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	3.57193	3.57193
$596$	0	0
$597$	−1.02540	−1.02540
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−0.247902	−0.247902
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−0.925077	−0.925077
$606$	−1.64354	−1.64354
$607$	1.85383	1.85383	0.926917	−	0.375267i	$-0.122449\pi$
$607$	1.85383	1.85383	0.926917	+	0.375267i	$0.122449\pi$
$608$	−0.197758	−0.197758
$609$	0	0
$610$	0.801680	0.801680
$611$	0	0
$612$	−2.93933	−2.93933
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	1.12877	1.12877
$615$	0	0
$616$	0	0
$617$	1.60283	1.60283	0.801414	−	0.598111i	$-0.204082\pi$
$617$	1.60283	1.60283	0.801414	+	0.598111i	$0.204082\pi$
$618$	−0.327308	−0.327308
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	−0.432062	−0.432062
$621$	0	0
$622$	0	0
$623$	2.68368	2.68368
$624$	0.456233	0.456233
$625$	−0.834963	−0.834963
$626$	1.05493	1.05493
$627$	0	0
$628$	0	0
$629$	0	0
$630$	2.36085	2.36085
$631$	1.93459	1.93459	0.967295	−	0.253655i	$-0.0816327\pi$
$631$	1.93459	1.93459	0.967295	+	0.253655i	$0.0816327\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.61207	−1.61207
$636$	0	0
$637$	−5.66317	−5.66317
$638$	0	0
$639$	0	0
$640$	0.714339	0.714339
$641$	0.809567	0.809567	0.404783	−	0.914413i	$-0.367347\pi$
$641$	0.809567	0.809567	0.404783	+	0.914413i	$0.367347\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	−0.211421	−0.211421
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−1.71875	−1.71875
$649$	0	0
$650$	0.155791	0.155791
$651$	−2.48419	−2.48419
$652$	0.909188	0.909188
$653$	−1.34460	−1.34460	−0.672301	−	0.740278i	$-0.734694\pi$
$653$	−1.34460	−1.34460	−0.672301	+	0.740278i	$0.734694\pi$
$654$	−1.06312	−1.06312
$655$	1.05850	1.05850
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.89811	−1.89811	−0.949056	−	0.315108i	$-0.897959\pi$
$659$	−1.89811	−1.89811	−0.949056	+	0.315108i	$0.897959\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−0.526420	−0.526420
$663$	6.61684	6.61684
$664$	0	0
$665$	−0.354585	−0.354585
$666$	0	0
$667$	0	0
$668$	0	0
$669$	2.74416	2.74416
$670$	0	0
$671$	0	0
$672$	3.70344	3.70344
$673$	−1.34460	−1.34460	−0.672301	−	0.740278i	$-0.734694\pi$
$673$	−1.34460	−1.34460	−0.672301	+	0.740278i	$0.734694\pi$
$674$	0	0
$675$	0.324089	0.324089
$676$	−1.75997	−1.75997
$677$	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
$680$	1.70703	1.70703
$681$	−2.58884	−2.58884
$682$	0	0
$683$	−0.690730	−0.690730	−0.345365	−	0.938468i	$-0.612245\pi$
$683$	−0.690730	−0.690730	−0.345365	+	0.938468i	$0.612245\pi$
$684$	0.291786	0.291786
$685$	1.81603	1.81603
$686$	−2.25289	−2.25289
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0.809567	0.809567	0.404783	−	0.914413i	$-0.367347\pi$
$691$	0.809567	0.809567	0.404783	+	0.914413i	$0.367347\pi$
$692$	0	0
$693$	0	0
$694$	0.991657	0.991657
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0.817561	0.817561
$699$	0.346055	0.346055
$700$	−0.194654	−0.194654
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	2.42703	2.42703
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0.991657	0.991657
$707$	−3.19907	−3.19907
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	1.28253	1.28253
$713$	0	0
$714$	3.95931	3.95931
$715$	0	0
$716$	−0.843178	−0.843178
$717$	−0.575178	−0.575178
$718$	0	0
$719$	−1.52289	−1.52289	−0.761446	−	0.648228i	$-0.775510\pi$
$719$	−1.52289	−1.52289	−0.761446	+	0.648228i	$0.775510\pi$
$720$	−0.277269	−0.277269
$721$	−0.637088	−0.637088
$722$	−0.548067	−0.548067
$723$	0	0
$724$	0	0
$725$	0	0
$726$	−1.02540	−1.02540
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	−3.61354	−3.61354
$729$	0	0
$730$	0	0
$731$	0	0
$732$	−1.85553	−1.85553
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	−0.526420	−0.526420
$735$	4.97342	4.97342
$736$	0	0
$737$	0	0
$738$	0	0
$739$	1.60283	1.60283	0.801414	−	0.598111i	$-0.204082\pi$
$739$	1.60283	1.60283	0.801414	+	0.598111i	$0.204082\pi$
$740$	0	0
$741$	−0.656851	−0.656851
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	−1.18720	−1.18720
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	−1.08539	−1.08539
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	−3.03246	−3.03246
$757$	−0.192046	−0.192046	−0.0960230	−	0.995379i	$-0.530612\pi$
$757$	−0.192046	−0.192046	−0.0960230	+	0.995379i	$0.530612\pi$
$758$	0.323824	0.323824
$759$	0	0
$760$	−0.169456	−0.169456
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	−1.78690	−1.78690
$763$	−2.06931	−2.06931
$764$	−1.17833	−1.17833
$765$	−4.02129	−4.02129
$766$	0.709600	0.709600
$767$	0	0
$768$	1.60735	1.60735
$769$	−1.96312	−1.96312	−0.981559	−	0.191159i	$-0.938776\pi$
$769$	−1.96312	−1.96312	−0.981559	+	0.191159i	$0.938776\pi$
$770$	0	0
$771$	3.02037	3.02037
$772$	0	0
$773$	0.809567	0.809567	0.404783	−	0.914413i	$-0.367347\pi$
$773$	0.809567	0.809567	0.404783	+	0.914413i	$0.367347\pi$
$774$	0	0
$775$	0.0996263	0.0996263
$776$	0	0
$777$	0	0
$778$	0.589988	0.589988
$779$	0	0
$780$	2.13943	2.13943
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0.397981	0.397981
$785$	0	0
$786$	1.17330	1.17330
$787$	1.98358	1.98358	0.991790	−	0.127877i	$-0.0408163\pi$
$787$	1.98358	1.98358	0.991790	+	0.127877i	$0.0408163\pi$
$788$	−0.0433450	−0.0433450
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2.89062	2.89062
$794$	0	0
$795$	0	0
$796$	−0.384782	−0.384782
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	−0.393040	−0.393040
$799$	0	0
$800$	−0.148523	−0.148523
$801$	−3.02129	−3.02129
$802$	−1.13577	−1.13577
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0.746078	0.746078
$807$	0.346055	0.346055
$808$	−1.52884	−1.52884
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	−0.948575	−0.948575
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.24386	1.24386
$816$	−0.465001	−0.465001
$817$	0	0
$818$	0.0364782	0.0364782
$819$	8.51251	8.51251
$820$	0	0
$821$	−1.96312	−1.96312	−0.981559	−	0.191159i	$-0.938776\pi$
$821$	−1.96312	−1.96312	−0.981559	+	0.191159i	$0.938776\pi$
$822$	2.01299	2.01299
$823$	1.43670	1.43670	0.718349	−	0.695683i	$-0.244898\pi$
$823$	1.43670	1.43670	0.718349	+	0.695683i	$0.244898\pi$
$824$	−0.304464	−0.304464
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−0.690730	−0.690730	−0.345365	−	0.938468i	$-0.612245\pi$
$829$	−0.690730	−0.690730	−0.345365	+	0.938468i	$0.612245\pi$
$830$	0	0
$831$	0	0
$832$	−0.859067	−0.859067
$833$	5.77200	5.77200
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.55206	1.55206
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	3.17343	3.17343
$841$	1.00000	1.00000
$842$	0	0
$843$	−0.115510	−0.115510
$844$	0	0
$845$	−2.40781	−2.40781
$846$	0	0
$847$	−1.99589	−1.99589
$848$	0	0
$849$	0	0
$850$	−0.158785	−0.158785
$851$	0	0
$852$	0	0
$853$	−1.67618	−1.67618	−0.838088	−	0.545535i	$-0.816327\pi$
$853$	−1.67618	−1.67618	−0.838088	+	0.545535i	$0.816327\pi$
$854$	1.72966	1.72966
$855$	0.399192	0.399192
$856$	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			$\pi$
$864$	−2.31381	−2.31381
$865$	0	0
$866$	0.817561	0.817561
$867$	−4.94206	−4.94206
$868$	−0.932191	−0.932191
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−0.988924	−0.988924
$873$	0	0
$874$	0	0
$875$	−2.11266	−2.11266
$876$	0	0
$877$	1.24698	1.24698	0.623490	−	0.781831i	$-0.285714\pi$
$877$	1.24698	1.24698	0.623490	+	0.781831i	$0.285714\pi$
$878$	−0.765153	−0.765153
$879$	0.801938	0.801938
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	3.81497	3.81497
$883$	−1.67618	−1.67618	−0.838088	−	0.545535i	$-0.816327\pi$
$883$	−1.67618	−1.67618	−0.838088	+	0.545535i	$0.816327\pi$
$884$	2.48296	2.48296
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−3.47811	−3.47811
$890$	0.707825	0.707825
$891$	0	0
$892$	1.02974	1.02974
$893$	0	0
$894$	0	0
$895$	−1.15355	−1.15355
$896$	1.54122	1.54122
$897$	0	0
$898$	0.460688	0.460688
$899$	0	0
$900$	0.219141	0.219141
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.03679	1.03679	0.518393	−	0.855143i	$-0.326531\pi$
$907$	1.03679	1.03679	0.518393	+	0.855143i	$0.326531\pi$
$908$	−0.971461	−0.971461
$909$	3.60152	3.60152
$910$	−1.99430	−1.99430
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0.0461604	0.0461604
$913$	0	0
$914$	0	0
$915$	−2.53855	−2.53855
$916$	0	0
$917$	2.28376	2.28376
$918$	−2.47367	−2.47367
$919$	−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
$921$	−3.57429	−3.57429
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.717235	0.717235
$928$	0	0
$929$	−1.52289	−1.52289	−0.761446	−	0.648228i	$-0.775510\pi$
$929$	−1.52289	−1.52289	−0.761446	+	0.648228i	$0.775510\pi$
$930$	−0.655210	−0.655210
$931$	−0.572985	−0.572985
$932$	0.129857	0.129857
$933$	0	0
$934$	0.0364782	0.0364782
$935$	0	0
$936$	4.06813	4.06813
$937$	0.0641032	0.0641032	0.0320516	−	0.999486i	$-0.489796\pi$
$937$	0.0641032	0.0641032	0.0320516	+	0.999486i	$0.489796\pi$
$938$	0	0
$939$	−3.34049	−3.34049
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−4.14871	−4.14871
$946$	0	0
$947$	1.93459	1.93459	0.967295	−	0.253655i	$-0.0816327\pi$
$947$	1.93459	1.93459	0.967295	+	0.253655i	$0.0816327\pi$
$948$	0	0
$949$	0	0
$950$	0.0157625	0.0157625
$951$	0	0
$952$	3.68298	3.68298
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	−1.61207	−1.61207
$956$	−0.215835	−0.215835
$957$	0	0
$958$	−0.253253	−0.253253
$959$	3.91817	3.91817
$960$	0.754436	0.754436
$961$	−0.522892	−0.522892
$962$	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			$\pi$
$968$	−0.953837	−0.953837
$969$	0.669475	0.669475
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0.676176	0.676176
$973$	0	0
$974$	0	0
$975$	−0.493318	−0.493318
$976$	−0.203139	−0.203139
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	1.37876	1.37876
$979$	0	0
$980$	1.86627	1.86627
$981$	2.32964	2.32964
$982$	0	0
$983$	1.98358	1.98358	0.991790	−	0.127877i	$-0.0408163\pi$
$983$	1.98358	1.98358	0.991790	+	0.127877i	$0.0408163\pi$
$984$	0	0
$985$	−0.0593003	−0.0593003
$986$	0	0
$987$	0	0
$988$	−0.246483	−0.246483
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	−0.711275	−0.711275
$993$	1.66693	1.66693
$994$	0	0
$995$	−0.526420	−0.526420
$996$	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.1510.13	✓	21
1511.1510	odd	2	CM	1511.1.b.b.1510.13	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1511.1.b.b.1510.13	✓	21	1.1	even	1	trivial
1511.1.b.b.1510.13	✓	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)

\(f(q)\)	\(=\)	\(q+0.569055 q^{2} -1.80194 q^{3} -0.676176 q^{4} -0.925077 q^{5} -1.02540 q^{6} -1.99589 q^{7} -0.953837 q^{8} +2.24698 q^{9} +O(q^{10})\)	\(q+0.569055 q^{2} -1.80194 q^{3} -0.676176 q^{4} -0.925077 q^{5} -1.02540 q^{6} -1.99589 q^{7} -0.953837 q^{8} +2.24698 q^{9} -0.526420 q^{10} +1.21843 q^{12} -1.89811 q^{13} -1.13577 q^{14} +1.66693 q^{15} +0.133390 q^{16} +1.93459 q^{17} +1.27866 q^{18} -0.192046 q^{19} +0.625515 q^{20} +3.59647 q^{21} +1.71875 q^{24} -0.144233 q^{25} -1.08013 q^{26} -2.24698 q^{27} +1.34957 q^{28} +0.948575 q^{30} -0.690730 q^{31} +1.02974 q^{32} +1.10089 q^{34} +1.84635 q^{35} -1.51935 q^{36} -0.109285 q^{38} +3.42028 q^{39} +0.882372 q^{40} +2.04659 q^{42} -2.07863 q^{45} -0.240361 q^{48} +2.98358 q^{49} -0.0820767 q^{50} -3.48601 q^{51} +1.28346 q^{52} -1.27866 q^{54} +1.90375 q^{56} +0.346055 q^{57} -1.12714 q^{60} -1.52289 q^{61} -0.393064 q^{62} -4.48473 q^{63} +0.452590 q^{64} +1.75590 q^{65} -1.30812 q^{68} +1.05068 q^{70} -2.14325 q^{72} +0.259899 q^{75} +0.129857 q^{76} +1.94633 q^{78} -0.123396 q^{80} +1.80194 q^{81} -2.43185 q^{84} -1.78964 q^{85} -1.34460 q^{89} -1.18285 q^{90} +3.78842 q^{91} +1.24465 q^{93} +0.177657 q^{95} -1.85553 q^{96} +1.69782 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - 7 q^{3} + 21 q^{4} + 14 q^{9}+O(q^{10}) \) 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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