LMFDB - Embedded newform 919.1.b.a.918.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [919,1,Mod(918,919)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(919, 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("919.918");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 919 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	919.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:	$0.458640746610$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\Q(\zeta_{38})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 8x^{7} + 7x^{6} + 21x^{5} - 15x^{4} - 20x^{3} + 10x^{2} + 5x - 1 $ x^9 - x^8 - 8x^7 + 7x^6 + 21x^5 - 15x^4 - 20x^3 + 10x^2 + 5*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{19}$
Projective field:	Galois closure of 19.1.467562425055097089773569879.1

Embedding invariants

Embedding label			918.5
Root			$1.97272$ of defining polynomial
Character	$\chi$	$=$	919.918

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.165159 q^{2} -0.972723 q^{4} -1.75895 q^{5} +0.325812 q^{8} +1.00000 q^{9} +O(q^{10})$	$q-0.165159 q^{2} -0.972723 q^{4} -1.75895 q^{5} +0.325812 q^{8} +1.00000 q^{9} +0.290505 q^{10} -1.35456 q^{11} +1.57828 q^{13} +0.918912 q^{16} +1.89163 q^{17} -0.165159 q^{18} +1.71097 q^{20} +0.223718 q^{22} -0.803391 q^{23} +2.09390 q^{25} -0.260667 q^{26} +0.490971 q^{29} -0.477579 q^{32} -0.312420 q^{34} -0.972723 q^{36} -0.573087 q^{40} +1.31761 q^{44} -1.75895 q^{45} +0.132687 q^{46} +1.09390 q^{47} +1.00000 q^{49} -0.345825 q^{50} -1.53523 q^{52} +2.38261 q^{55} -0.0810881 q^{58} -1.97272 q^{59} -0.803391 q^{61} -0.840036 q^{64} -2.77611 q^{65} +0.490971 q^{67} -1.84004 q^{68} +1.57828 q^{71} +0.325812 q^{72} -1.61632 q^{80} +1.00000 q^{81} +1.09390 q^{83} -3.32729 q^{85} -0.441333 q^{88} +0.290505 q^{90} +0.781476 q^{92} -0.180666 q^{94} -0.165159 q^{98} -1.35456 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{2} + 8 q^{4} - q^{5} - 2 q^{8} + 9 q^{9}+O(q^{10}) $ 9 * q - q^2 + 8 * q^4 - q^5 - 2 * q^8 + 9 * q^9	$ 9 q - q^{2} + 8 q^{4} - q^{5} - 2 q^{8} + 9 q^{9} - 2 q^{10} - q^{11} - q^{13} + 7 q^{16} - q^{17} - q^{18} - 3 q^{20} - 2 q^{22} - q^{23} + 8 q^{25} - 2 q^{26} - q^{29} - 3 q^{32} - 2 q^{34} + 8 q^{36} - 4 q^{40} - 3 q^{44} - q^{45} - 2 q^{46} - q^{47} + 9 q^{49} - 3 q^{50} - 3 q^{52} - 2 q^{55} - 2 q^{58} - q^{59} - q^{61} + 6 q^{64} - 2 q^{65} - q^{67} - 3 q^{68} - q^{71} - 2 q^{72} - 5 q^{80} + 9 q^{81} - q^{83} - 2 q^{85} - 4 q^{88} - 2 q^{90} - 3 q^{92} - 2 q^{94} - q^{98} - q^{99}+O(q^{100}) $ 9 * q - q^2 + 8 * q^4 - q^5 - 2 * q^8 + 9 * q^9 - 2 * q^10 - q^11 - q^13 + 7 * q^16 - q^17 - q^18 - 3 * q^20 - 2 * q^22 - q^23 + 8 * q^25 - 2 * q^26 - q^29 - 3 * q^32 - 2 * q^34 + 8 * q^36 - 4 * q^40 - 3 * q^44 - q^45 - 2 * q^46 - q^47 + 9 * q^49 - 3 * q^50 - 3 * q^52 - 2 * q^55 - 2 * q^58 - q^59 - q^61 + 6 * q^64 - 2 * q^65 - q^67 - 3 * q^68 - q^71 - 2 * q^72 - 5 * q^80 + 9 * q^81 - q^83 - 2 * q^85 - 4 * q^88 - 2 * q^90 - 3 * q^92 - 2 * q^94 - q^98 - q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/919\mathbb{Z}\right)^\times$.

$n$	$7$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$2$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	−0.972723	−0.972723
$5$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$5$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0.325812	0.325812
$9$	1.00000	1.00000
$10$	0.290505	0.290505
$11$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$11$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$12$	0	0
$13$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$13$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$14$	0	0
$15$	0	0
$16$	0.918912	0.918912
$17$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$17$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$18$	−0.165159	−0.165159
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	1.71097	1.71097
$21$	0	0
$22$	0.223718	0.223718
$23$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$23$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$24$	0	0
$25$	2.09390	2.09390
$26$	−0.260667	−0.260667
$27$	0	0
$28$	0	0
$29$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$29$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−0.477579	−0.477579
$33$	0	0
$34$	−0.312420	−0.312420
$35$	0	0
$36$	−0.972723	−0.972723
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	−0.573087	−0.573087
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	1.31761	1.31761
$45$	−1.75895	−1.75895
$46$	0.132687	0.132687
$47$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$47$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$48$	0	0
$49$	1.00000	1.00000
$50$	−0.345825	−0.345825
$51$	0	0
$52$	−1.53523	−1.53523
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	2.38261	2.38261
$56$	0	0
$57$	0	0
$58$	−0.0810881	−0.0810881
$59$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$59$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$60$	0	0
$61$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$61$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$62$	0	0
$63$	0	0
$64$	−0.840036	−0.840036
$65$	−2.77611	−2.77611
$66$	0	0
$67$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$67$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$68$	−1.84004	−1.84004
$69$	0	0
$70$	0	0
$71$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$71$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$72$	0.325812	0.325812
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−1.61632	−1.61632
$81$	1.00000	1.00000
$82$	0	0
$83$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$83$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$84$	0	0
$85$	−3.32729	−3.32729
$86$	0	0
$87$	0	0
$88$	−0.441333	−0.441333
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0.290505	0.290505
$91$	0	0
$92$	0.781476	0.781476
$93$	0	0
$94$	−0.180666	−0.180666
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−0.165159	−0.165159
$99$	−1.35456	−1.35456
$100$	−2.03678	−2.03678
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$103$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$104$	0.514223	0.514223
$105$	0	0
$106$	0	0
$107$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$107$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	−0.393508	−0.393508
$111$	0	0
$112$	0	0
$113$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$113$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$114$	0	0
$115$	1.41312	1.41312
$116$	−0.477579	−0.477579
$117$	1.57828	1.57828
$118$	0.325812	0.325812
$119$	0	0
$120$	0	0
$121$	0.834841	0.834841
$122$	0.132687	0.132687
$123$	0	0
$124$	0	0
$125$	−1.92411	−1.92411
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0.616318	0.616318
$129$	0	0
$130$	0.458499	0.458499
$131$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$131$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$132$	0	0
$133$	0	0
$134$	−0.0810881	−0.0810881
$135$	0	0
$136$	0.616318	0.616318
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$139$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$140$	0	0
$141$	0	0
$142$	−0.260667	−0.260667
$143$	−2.13788	−2.13788
$144$	0.918912	0.918912
$145$	−0.863592	−0.863592
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$149$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$150$	0	0
$151$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$151$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$152$	0	0
$153$	1.89163	1.89163
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0.840036	0.840036
$161$	0	0
$162$	−0.165159	−0.165159
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	−0.180666	−0.180666
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	1.49097	1.49097
$170$	0.549530	0.549530
$171$	0	0
$172$	0	0
$173$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$173$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$174$	0	0
$175$	0	0
$176$	−1.24472	−1.24472
$177$	0	0
$178$	0	0
$179$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$179$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$180$	1.71097	1.71097
$181$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$181$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$182$	0	0
$183$	0	0
$184$	−0.261755	−0.261755
$185$	0	0
$186$	0	0
$187$	−2.56234	−2.56234
$188$	−1.06406	−1.06406
$189$	0	0
$190$	0	0
$191$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$191$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$192$	0	0
$193$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$193$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$194$	0	0
$195$	0	0
$196$	−0.972723	−0.972723
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0.223718	0.223718
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0.682217	0.682217
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0.325812	0.325812
$207$	−0.803391	−0.803391
$208$	1.45030	1.45030
$209$	0	0
$210$	0	0
$211$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$211$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$212$	0	0
$213$	0	0
$214$	−0.312420	−0.312420
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	−2.31761	−2.31761
$221$	2.98553	2.98553
$222$	0	0
$223$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$223$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$224$	0	0
$225$	2.09390	2.09390
$226$	0.223718	0.223718
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$229$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$230$	−0.233389	−0.233389
$231$	0	0
$232$	0.159964	0.159964
$233$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$233$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$234$	−0.260667	−0.260667
$235$	−1.92411	−1.92411
$236$	1.91891	1.91891
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$241$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$242$	−0.137881	−0.137881
$243$	0	0
$244$	0.781476	0.781476
$245$	−1.75895	−1.75895
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0.317783	0.317783
$251$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$251$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$252$	0	0
$253$	1.08824	1.08824
$254$	0	0
$255$	0	0
$256$	0.738245	0.738245
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	2.70039	2.70039
$261$	0.490971	0.490971
$262$	−0.260667	−0.260667
$263$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$263$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−0.477579	−0.477579
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	1.73825	1.73825
$273$	0	0
$274$	0	0
$275$	−2.83631	−2.83631
$276$	0	0
$277$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$277$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$278$	−0.312420	−0.312420
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	−1.53523	−1.53523
$285$	0	0
$286$	0.353090	0.353090
$287$	0	0
$288$	−0.477579	−0.477579
$289$	2.57828	2.57828
$290$	0.142630	0.142630
$291$	0	0
$292$	0	0
$293$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$293$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$294$	0	0
$295$	3.46992	3.46992
$296$	0	0
$297$	0	0
$298$	−0.180666	−0.180666
$299$	−1.26798	−1.26798
$300$	0	0
$301$	0	0
$302$	0.0272774	0.0272774
$303$	0	0
$304$	0	0
$305$	1.41312	1.41312
$306$	−0.312420	−0.312420
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$311$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	−0.665051	−0.665051
$320$	1.47758	1.47758
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−0.972723	−0.972723
$325$	3.30476	3.30476
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$331$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$332$	−1.06406	−1.06406
$333$	0	0
$334$	0	0
$335$	−0.863592	−0.863592
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−0.246247	−0.246247
$339$	0	0
$340$	3.23653	3.23653
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0.325812	0.325812
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0.646910	0.646910
$353$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$353$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$354$	0	0
$355$	−2.77611	−2.77611
$356$	0	0
$357$	0	0
$358$	0.132687	0.132687
$359$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$359$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$360$	−0.573087	−0.573087
$361$	1.00000	1.00000
$362$	0.0272774	0.0272774
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$367$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$368$	−0.738245	−0.738245
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0.423192	0.423192
$375$	0	0
$376$	0.356405	0.356405
$377$	0.774890	0.774890
$378$	0	0
$379$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$379$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$380$	0	0
$381$	0	0
$382$	0.290505	0.290505
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0.0272774	0.0272774
$387$	0	0
$388$	0	0
$389$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$389$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$390$	0	0
$391$	−1.51972	−1.51972
$392$	0.325812	0.325812
$393$	0	0
$394$	0	0
$395$	0	0
$396$	1.31761	1.31761
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	1.92411	1.92411
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.75895	−1.75895
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$409$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$410$	0	0
$411$	0	0
$412$	1.91891	1.91891
$413$	0	0
$414$	0.132687	0.132687
$415$	−1.92411	−1.92411
$416$	−0.753753	−0.753753
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$421$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$422$	−0.180666	−0.180666
$423$	1.09390	1.09390
$424$	0	0
$425$	3.96089	3.96089
$426$	0	0
$427$	0	0
$428$	−1.84004	−1.84004
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$433$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$439$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$440$	0.776282	0.776282
$441$	1.00000	1.00000
$442$	−0.493086	−0.493086
$443$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$443$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$444$	0	0
$445$	0	0
$446$	0.290505	0.290505
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−0.345825	−0.345825
$451$	0	0
$452$	1.31761	1.31761
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0.223718	0.223718
$459$	0	0
$460$	−1.37458	−1.37458
$461$	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			$\pi$
$464$	0.451159	0.451159
$465$	0	0
$466$	0.325812	0.325812
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−1.53523	−1.53523
$469$	0	0
$470$	0.317783	0.317783
$471$	0	0
$472$	−0.642737	−0.642737
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0.132687	0.132687
$483$	0	0
$484$	−0.812069	−0.812069
$485$	0	0
$486$	0	0
$487$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$487$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$488$	−0.261755	−0.261755
$489$	0	0
$490$	0.290505	0.290505
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0.928738	0.928738
$494$	0	0
$495$	2.38261	2.38261
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	1.87162	1.87162
$501$	0	0
$502$	0.0272774	0.0272774
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	−0.179733	−0.179733
$507$	0	0
$508$	0	0
$509$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$509$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$510$	0	0
$511$	0	0
$512$	−0.738245	−0.738245
$513$	0	0
$514$	0	0
$515$	3.46992	3.46992
$516$	0	0
$517$	−1.48175	−1.48175
$518$	0	0
$519$	0	0
$520$	−0.904492	−0.904492
$521$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$521$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$522$	−0.0810881	−0.0810881
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−1.53523	−1.53523
$525$	0	0
$526$	−0.260667	−0.260667
$527$	0	0
$528$	0	0
$529$	−0.354563	−0.354563
$530$	0	0
$531$	−1.97272	−1.97272
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−3.32729	−3.32729
$536$	0.159964	0.159964
$537$	0	0
$538$	0	0
$539$	−1.35456	−1.35456
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	−0.903404	−0.903404
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	−0.803391	−0.803391
$550$	0.468442	0.468442
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−0.180666	−0.180666
$555$	0	0
$556$	−1.84004	−1.84004
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$563$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$564$	0	0
$565$	2.38261	2.38261
$566$	0	0
$567$	0	0
$568$	0.514223	0.514223
$569$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$569$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$570$	0	0
$571$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$571$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$572$	2.07957	2.07957
$573$	0	0
$574$	0	0
$575$	−1.68222	−1.68222
$576$	−0.840036	−0.840036
$577$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$577$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$578$	−0.425826	−0.425826
$579$	0	0
$580$	0.840036	0.840036
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−2.77611	−2.77611
$586$	0.223718	0.223718
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	−0.573087	−0.573087
$591$	0	0
$592$	0	0
$593$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$593$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$594$	0	0
$595$	0	0
$596$	−1.06406	−1.06406
$597$	0	0
$598$	0.209417	0.209417
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0.490971	0.490971
$604$	0.160654	0.160654
$605$	−1.46844	−1.46844
$606$	0	0
$607$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$607$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$608$	0	0
$609$	0	0
$610$	−0.233389	−0.233389
$611$	1.72648	1.72648
$612$	−1.84004	−1.84004
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$619$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$620$	0	0
$621$	0	0
$622$	−0.0810881	−0.0810881
$623$	0	0
$624$	0	0
$625$	1.29051	1.29051
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.57828	1.57828
$638$	0.109839	0.109839
$639$	1.57828	1.57828
$640$	−1.08407	−1.08407
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$643$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0.325812	0.325812
$649$	2.67218	2.67218
$650$	−0.545809	−0.545809
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	−2.77611	−2.77611
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−0.180666	−0.180666
$663$	0	0
$664$	0.356405	0.356405
$665$	0	0
$666$	0	0
$667$	−0.394442	−0.394442
$668$	0	0
$669$	0	0
$670$	0.142630	0.142630
$671$	1.08824	1.08824
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	−1.45030	−1.45030
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	−1.08407	−1.08407
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	1.91891	1.91891
$693$	0	0
$694$	0	0
$695$	−3.32729	−3.32729
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$701$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$702$	0	0
$703$	0	0
$704$	1.13788	1.13788
$705$	0	0
$706$	−0.0810881	−0.0810881
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0.458499	0.458499
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	3.76042	3.76042
$716$	0.781476	0.781476
$717$	0	0
$718$	−0.312420	−0.312420
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−1.61632	−1.61632
$721$	0	0
$722$	−0.165159	−0.165159
$723$	0	0
$724$	0.160654	0.160654
$725$	1.02804	1.02804
$726$	0	0
$727$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$727$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$728$	0	0
$729$	1.00000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0.223718	0.223718
$735$	0	0
$736$	0.383682	0.383682
$737$	−0.665051	−0.665051
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−1.92411	−1.92411
$746$	0	0
$747$	1.09390	1.09390
$748$	2.49244	2.49244
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	1.00519	1.00519
$753$	0	0
$754$	−0.127980	−0.127980
$755$	0.290505	0.290505
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0.132687	0.132687
$759$	0	0
$760$	0	0
$761$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$761$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$762$	0	0
$763$	0	0
$764$	1.71097	1.71097
$765$	−3.32729	−3.32729
$766$	0	0
$767$	−3.11351	−3.11351
$768$	0	0
$769$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$769$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$770$	0	0
$771$	0	0
$772$	0.160654	0.160654
$773$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$773$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0.132687	0.132687
$779$	0	0
$780$	0	0
$781$	−2.13788	−2.13788
$782$	0.250995	0.250995
$783$	0	0
$784$	0.918912	0.918912
$785$	0	0
$786$	0	0
$787$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$787$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−0.441333	−0.441333
$793$	−1.26798	−1.26798
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	2.06925	2.06925
$800$	−1.00000	−1.00000
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0.290505	0.290505
$811$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$811$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0.325812	0.325812
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$823$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$824$	−0.642737	−0.642737
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0.781476	0.781476
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0.317783	0.317783
$831$	0	0
$832$	−1.32581	−1.32581
$833$	1.89163	1.89163
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	−0.758948	−0.758948
$842$	−0.312420	−0.312420
$843$	0	0
$844$	−1.06406	−1.06406
$845$	−2.62254	−2.62254
$846$	−0.180666	−0.180666
$847$	0	0
$848$	0	0
$849$	0	0
$850$	−0.654175	−0.654175
$851$	0	0
$852$	0	0
$853$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$853$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$854$	0	0
$855$	0	0
$856$	0.616318	0.616318
$857$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$857$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$858$	0	0
$859$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$859$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$863$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$864$	0	0
$865$	3.46992	3.46992
$866$	0.223718	0.223718
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0.774890	0.774890
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	−0.180666	−0.180666
$879$	0	0
$880$	2.18940	2.18940
$881$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$881$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$882$	−0.165159	−0.165159
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	−2.90409	−2.90409
$885$	0	0
$886$	0.290505	0.290505
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.35456	−1.35456
$892$	1.71097	1.71097
$893$	0	0
$894$	0	0
$895$	1.41312	1.41312
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−2.03678	−2.03678
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−0.441333	−0.441333
$905$	0.290505	0.290505
$906$	0	0
$907$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$907$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−1.48175	−1.48175
$914$	0	0
$915$	0	0
$916$	1.31761	1.31761
$917$	0	0
$918$	0	0
$919$	1.00000	1.00000
$919$	1.00000	1.00000
$920$	0.460413	0.460413
$921$	0	0
$922$	0	0
$923$	2.49097	2.49097
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.97272	−1.97272
$928$	−0.234477	−0.234477
$929$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$929$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$930$	0	0
$931$	0	0
$932$	1.91891	1.91891
$933$	0	0
$934$	0	0
$935$	4.50702	4.50702
$936$	0.514223	0.514223
$937$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$937$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$938$	0	0
$939$	0	0
$940$	1.87162	1.87162
$941$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$941$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$942$	0	0
$943$	0	0
$944$	−1.81276	−1.81276
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$953$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$954$	0	0
$955$	3.09390	3.09390
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.00000	1.00000
$962$	0	0
$963$	1.89163	1.89163
$964$	0.781476	0.781476
$965$	0.290505	0.290505
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0.272002	0.272002
$969$	0	0
$970$	0	0
$971$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$971$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$972$	0	0
$973$	0	0
$974$	0.223718	0.223718
$975$	0	0
$976$	−0.738245	−0.738245
$977$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$977$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$978$	0	0
$979$	0	0
$980$	1.71097	1.71097
$981$	0	0
$982$	0	0
$983$	2.00000	2.00000	1.00000			$0$
$983$	2.00000	2.00000	1.00000			$0$
$984$	0	0
$985$	0	0
$986$	−0.153389	−0.153389
$987$	0	0
$988$	0	0
$989$	0	0
$990$	−0.393508	−0.393508
$991$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$991$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	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	919.1.b.a.918.5	✓	9
919.918	odd	2	CM	919.1.b.a.918.5	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
919.1.b.a.918.5	✓	9	1.1	even	1	trivial
919.1.b.a.918.5	✓	9	919.918	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:	\( N \)	\(=\)	\( 919 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	919.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.165159 q^{2} -0.972723 q^{4} -1.75895 q^{5} +0.325812 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.165159 q^{2} -0.972723 q^{4} -1.75895 q^{5} +0.325812 q^{8} +1.00000 q^{9} +0.290505 q^{10} -1.35456 q^{11} +1.57828 q^{13} +0.918912 q^{16} +1.89163 q^{17} -0.165159 q^{18} +1.71097 q^{20} +0.223718 q^{22} -0.803391 q^{23} +2.09390 q^{25} -0.260667 q^{26} +0.490971 q^{29} -0.477579 q^{32} -0.312420 q^{34} -0.972723 q^{36} -0.573087 q^{40} +1.31761 q^{44} -1.75895 q^{45} +0.132687 q^{46} +1.09390 q^{47} +1.00000 q^{49} -0.345825 q^{50} -1.53523 q^{52} +2.38261 q^{55} -0.0810881 q^{58} -1.97272 q^{59} -0.803391 q^{61} -0.840036 q^{64} -2.77611 q^{65} +0.490971 q^{67} -1.84004 q^{68} +1.57828 q^{71} +0.325812 q^{72} -1.61632 q^{80} +1.00000 q^{81} +1.09390 q^{83} -3.32729 q^{85} -0.441333 q^{88} +0.290505 q^{90} +0.781476 q^{92} -0.180666 q^{94} -0.165159 q^{98} -1.35456 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{2} + 8 q^{4} - q^{5} - 2 q^{8} + 9 q^{9}+O(q^{10}) \) 9 * q - q^2 + 8 * q^4 - q^5 - 2 * q^8 + 9 * q^9	\( 9 q - q^{2} + 8 q^{4} - q^{5} - 2 q^{8} + 9 q^{9} - 2 q^{10} - q^{11} - q^{13} + 7 q^{16} - q^{17} - q^{18} - 3 q^{20} - 2 q^{22} - q^{23} + 8 q^{25} - 2 q^{26} - q^{29} - 3 q^{32} - 2 q^{34} + 8 q^{36} - 4 q^{40} - 3 q^{44} - q^{45} - 2 q^{46} - q^{47} + 9 q^{49} - 3 q^{50} - 3 q^{52} - 2 q^{55} - 2 q^{58} - q^{59} - q^{61} + 6 q^{64} - 2 q^{65} - q^{67} - 3 q^{68} - q^{71} - 2 q^{72} - 5 q^{80} + 9 q^{81} - q^{83} - 2 q^{85} - 4 q^{88} - 2 q^{90} - 3 q^{92} - 2 q^{94} - q^{98} - q^{99}+O(q^{100}) \) 9 * q - q^2 + 8 * q^4 - q^5 - 2 * q^8 + 9 * q^9 - 2 * q^10 - q^11 - q^13 + 7 * q^16 - q^17 - q^18 - 3 * q^20 - 2 * q^22 - q^23 + 8 * q^25 - 2 * q^26 - q^29 - 3 * q^32 - 2 * q^34 + 8 * q^36 - 4 * q^40 - 3 * q^44 - q^45 - 2 * q^46 - q^47 + 9 * q^49 - 3 * q^50 - 3 * q^52 - 2 * q^55 - 2 * q^58 - q^59 - q^61 + 6 * q^64 - 2 * q^65 - q^67 - 3 * q^68 - q^71 - 2 * q^72 - 5 * q^80 + 9 * q^81 - q^83 - 2 * q^85 - 4 * q^88 - 2 * q^90 - 3 * q^92 - 2 * q^94 - q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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