LMFDB - Embedded newform 751.1.b.b.750.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [751,1,Mod(750,751)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(751, 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("751.750");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 751 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	751.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:	no
Analytic conductor:	$0.374797824487$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.751.1
Artin image:	$\GL(2,3)$
Artin field:	Galois closure of 8.2.423564751.1

Embedding invariants

Embedding label			750.1
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	751.750
Dual form			751.1.b.b.750.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.41421i q^{3} +1.41421i q^{6} +1.41421i q^{7} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{2} -1.41421i q^{3} +1.41421i q^{6} +1.41421i q^{7} +1.00000 q^{8} -1.00000 q^{9} +1.41421i q^{11} +1.00000 q^{13} -1.41421i q^{14} -1.00000 q^{16} +1.41421i q^{17} +1.00000 q^{18} +1.00000 q^{19} +2.00000 q^{21} -1.41421i q^{22} -1.00000 q^{23} -1.41421i q^{24} -1.00000 q^{25} -1.00000 q^{26} +2.00000 q^{33} -1.41421i q^{34} +1.00000 q^{37} -1.00000 q^{38} -1.41421i q^{39} -1.41421i q^{41} -2.00000 q^{42} +1.00000 q^{46} +1.00000 q^{47} +1.41421i q^{48} -1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} -1.00000 q^{53} +1.41421i q^{56} -1.41421i q^{57} -1.00000 q^{59} +1.00000 q^{61} -1.41421i q^{63} +1.00000 q^{64} -2.00000 q^{66} +1.41421i q^{67} +1.41421i q^{69} -1.00000 q^{72} -1.00000 q^{74} +1.41421i q^{75} -2.00000 q^{77} +1.41421i q^{78} -1.41421i q^{79} -1.00000 q^{81} +1.41421i q^{82} +1.41421i q^{83} +1.41421i q^{88} +1.00000 q^{89} +1.41421i q^{91} -1.00000 q^{94} +1.00000 q^{97} +1.00000 q^{98} -1.41421i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^8 - 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{8} - 2 q^{9} + 2 q^{13} - 2 q^{16} + 2 q^{18} + 2 q^{19} + 4 q^{21} - 2 q^{23} - 2 q^{25} - 2 q^{26} + 4 q^{33} + 2 q^{37} - 2 q^{38} - 4 q^{42} + 2 q^{46} + 2 q^{47} - 2 q^{49} + 2 q^{50} + 4 q^{51} - 2 q^{53} - 2 q^{59} + 2 q^{61} + 2 q^{64} - 4 q^{66} - 2 q^{72} - 2 q^{74} - 4 q^{77} - 2 q^{81} + 2 q^{89} - 2 q^{94} + 2 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^8 - 2 * q^9 + 2 * q^13 - 2 * q^16 + 2 * q^18 + 2 * q^19 + 4 * q^21 - 2 * q^23 - 2 * q^25 - 2 * q^26 + 4 * q^33 + 2 * q^37 - 2 * q^38 - 4 * q^42 + 2 * q^46 + 2 * q^47 - 2 * q^49 + 2 * q^50 + 4 * q^51 - 2 * q^53 - 2 * q^59 + 2 * q^61 + 2 * q^64 - 4 * q^66 - 2 * q^72 - 2 * q^74 - 4 * q^77 - 2 * q^81 + 2 * q^89 - 2 * q^94 + 2 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/751\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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 $

$2$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$2$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$3$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$3$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	1.41421i	1.41421i
$7$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$7$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$8$	1.00000	1.00000
$9$	−1.00000	−1.00000
$10$	0	0
$11$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$11$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$12$	0	0
$13$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$13$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$14$	− 1.41421i	− 1.41421i
$15$	0	0
$16$	−1.00000	−1.00000
$17$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$17$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$18$	1.00000	1.00000
$19$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$19$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$20$	0	0
$21$	2.00000	2.00000
$22$	− 1.41421i	− 1.41421i
$23$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$23$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$24$	− 1.41421i	− 1.41421i
$25$	−1.00000	−1.00000
$26$	−1.00000	−1.00000
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	2.00000	2.00000
$34$	− 1.41421i	− 1.41421i
$35$	0	0
$36$	0	0
$37$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$37$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$38$	−1.00000	−1.00000
$39$	− 1.41421i	− 1.41421i
$40$	0	0
$41$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$41$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$42$	−2.00000	−2.00000
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	1.00000	1.00000
$47$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$47$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$48$	1.41421i	1.41421i
$49$	−1.00000	−1.00000
$50$	1.00000	1.00000
$51$	2.00000	2.00000
$52$	0	0
$53$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$53$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$54$	0	0
$55$	0	0
$56$	1.41421i	1.41421i
$57$	− 1.41421i	− 1.41421i
$58$	0	0
$59$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$59$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$60$	0	0
$61$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$61$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$62$	0	0
$63$	− 1.41421i	− 1.41421i
$64$	1.00000	1.00000
$65$	0	0
$66$	−2.00000	−2.00000
$67$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$67$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$68$	0	0
$69$	1.41421i	1.41421i
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−1.00000
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	−1.00000	−1.00000
$75$	1.41421i	1.41421i
$76$	0	0
$77$	−2.00000	−2.00000
$78$	1.41421i	1.41421i
$79$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$79$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$80$	0	0
$81$	−1.00000	−1.00000
$82$	1.41421i	1.41421i
$83$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$83$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	1.41421i	1.41421i
$89$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$89$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$90$	0	0
$91$	1.41421i	1.41421i
$92$	0	0
$93$	0	0
$94$	−1.00000	−1.00000
$95$	0	0
$96$	0	0
$97$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$97$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$98$	1.00000	1.00000
$99$	− 1.41421i	− 1.41421i
$100$	0	0
$101$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$101$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$102$	−2.00000	−2.00000
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	1.00000	1.00000
$105$	0	0
$106$	1.00000	1.00000
$107$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$107$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$108$	0	0
$109$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$109$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$110$	0	0
$111$	− 1.41421i	− 1.41421i
$112$	− 1.41421i	− 1.41421i
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	1.41421i	1.41421i
$115$	0	0
$116$	0	0
$117$	−1.00000	−1.00000
$118$	1.00000	1.00000
$119$	−2.00000	−2.00000
$120$	0	0
$121$	−1.00000	−1.00000
$122$	−1.00000	−1.00000
$123$	−2.00000	−2.00000
$124$	0	0
$125$	0	0
$126$	1.41421i	1.41421i
$127$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$127$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$128$	−1.00000	−1.00000
$129$	0	0
$130$	0	0
$131$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$131$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$132$	0	0
$133$	1.41421i	1.41421i
$134$	− 1.41421i	− 1.41421i
$135$	0	0
$136$	1.41421i	1.41421i
$137$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$137$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$138$	− 1.41421i	− 1.41421i
$139$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$139$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$140$	0	0
$141$	− 1.41421i	− 1.41421i
$142$	0	0
$143$	1.41421i	1.41421i
$144$	1.00000	1.00000
$145$	0	0
$146$	0	0
$147$	1.41421i	1.41421i
$148$	0	0
$149$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$149$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$150$	− 1.41421i	− 1.41421i
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	1.00000	1.00000
$153$	− 1.41421i	− 1.41421i
$154$	2.00000	2.00000
$155$	0	0
$156$	0	0
$157$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$157$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$158$	1.41421i	1.41421i
$159$	1.41421i	1.41421i
$160$	0	0
$161$	− 1.41421i	− 1.41421i
$162$	1.00000	1.00000
$163$	−2.00000	−2.00000	−1.00000			$\pi$
$163$	−2.00000	−2.00000	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	− 1.41421i	− 1.41421i
$167$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$167$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$168$	2.00000	2.00000
$169$	0	0
$170$	0	0
$171$	−1.00000	−1.00000
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	− 1.41421i	− 1.41421i
$176$	− 1.41421i	− 1.41421i
$177$	1.41421i	1.41421i
$178$	−1.00000	−1.00000
$179$	2.00000	2.00000	1.00000			$0$
$179$	2.00000	2.00000	1.00000			$0$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	− 1.41421i	− 1.41421i
$183$	− 1.41421i	− 1.41421i
$184$	−1.00000	−1.00000
$185$	0	0
$186$	0	0
$187$	−2.00000	−2.00000
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$191$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$192$	− 1.41421i	− 1.41421i
$193$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$193$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$194$	−1.00000	−1.00000
$195$	0	0
$196$	0	0
$197$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$197$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$198$	1.41421i	1.41421i
$199$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$199$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$200$	−1.00000	−1.00000
$201$	2.00000	2.00000
$202$	1.41421i	1.41421i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.00000	1.00000
$208$	−1.00000	−1.00000
$209$	1.41421i	1.41421i
$210$	0	0
$211$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$211$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$212$	0	0
$213$	0	0
$214$	1.00000	1.00000
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−1.00000	−1.00000
$219$	0	0
$220$	0	0
$221$	1.41421i	1.41421i
$222$	1.41421i	1.41421i
$223$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$223$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$224$	0	0
$225$	1.00000	1.00000
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$229$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$230$	0	0
$231$	2.82843i	2.82843i
$232$	0	0
$233$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$233$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$234$	1.00000	1.00000
$235$	0	0
$236$	0	0
$237$	−2.00000	−2.00000
$238$	2.00000	2.00000
$239$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$239$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	1.00000	1.00000
$243$	1.41421i	1.41421i
$244$	0	0
$245$	0	0
$246$	2.00000	2.00000
$247$	1.00000	1.00000
$248$	0	0
$249$	2.00000	2.00000
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	− 1.41421i	− 1.41421i
$254$	1.00000	1.00000
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	1.41421i	1.41421i
$260$	0	0
$261$	0	0
$262$	1.00000	1.00000
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	2.00000	2.00000
$265$	0	0
$266$	− 1.41421i	− 1.41421i
$267$	− 1.41421i	− 1.41421i
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	− 1.41421i	− 1.41421i
$273$	2.00000	2.00000
$274$	1.41421i	1.41421i
$275$	− 1.41421i	− 1.41421i
$276$	0	0
$277$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$277$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$278$	−1.00000	−1.00000
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	1.41421i	1.41421i
$283$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$283$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$284$	0	0
$285$	0	0
$286$	− 1.41421i	− 1.41421i
$287$	2.00000	2.00000
$288$	0	0
$289$	−1.00000	−1.00000
$290$	0	0
$291$	− 1.41421i	− 1.41421i
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	− 1.41421i	− 1.41421i
$295$	0	0
$296$	1.00000	1.00000
$297$	0	0
$298$	−1.00000	−1.00000
$299$	−1.00000	−1.00000
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−2.00000	−2.00000
$304$	−1.00000	−1.00000
$305$	0	0
$306$	1.41421i	1.41421i
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$311$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$312$	− 1.41421i	− 1.41421i
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	1.41421i	1.41421i
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	− 1.41421i	− 1.41421i
$319$	0	0
$320$	0	0
$321$	1.41421i	1.41421i
$322$	1.41421i	1.41421i
$323$	1.41421i	1.41421i
$324$	0	0
$325$	−1.00000	−1.00000
$326$	2.00000	2.00000
$327$	− 1.41421i	− 1.41421i
$328$	− 1.41421i	− 1.41421i
$329$	1.41421i	1.41421i
$330$	0	0
$331$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$331$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$332$	0	0
$333$	−1.00000	−1.00000
$334$	1.00000	1.00000
$335$	0	0
$336$	−2.00000	−2.00000
$337$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$337$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	1.00000	1.00000
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$347$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$348$	0	0
$349$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$349$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$350$	1.41421i	1.41421i
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	− 1.41421i	− 1.41421i
$355$	0	0
$356$	0	0
$357$	2.82843i	2.82843i
$358$	−2.00000	−2.00000
$359$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$359$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	1.41421i	1.41421i
$364$	0	0
$365$	0	0
$366$	1.41421i	1.41421i
$367$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$367$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$368$	1.00000	1.00000
$369$	1.41421i	1.41421i
$370$	0	0
$371$	− 1.41421i	− 1.41421i
$372$	0	0
$373$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$373$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$374$	2.00000	2.00000
$375$	0	0
$376$	1.00000	1.00000
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	1.41421i	1.41421i
$382$	−1.00000	−1.00000
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	1.41421i	1.41421i
$385$	0	0
$386$	−1.00000	−1.00000
$387$	0	0
$388$	0	0
$389$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$389$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$390$	0	0
$391$	− 1.41421i	− 1.41421i
$392$	−1.00000	−1.00000
$393$	1.41421i	1.41421i
$394$	1.00000	1.00000
$395$	0	0
$396$	0	0
$397$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$397$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$398$	1.00000	1.00000
$399$	2.00000	2.00000
$400$	1.00000	1.00000
$401$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$401$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$402$	−2.00000	−2.00000
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.41421i	1.41421i
$408$	2.00000	2.00000
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	−2.00000	−2.00000
$412$	0	0
$413$	− 1.41421i	− 1.41421i
$414$	−1.00000	−1.00000
$415$	0	0
$416$	0	0
$417$	− 1.41421i	− 1.41421i
$418$	− 1.41421i	− 1.41421i
$419$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$419$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$420$	0	0
$421$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$421$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$422$	−1.00000	−1.00000
$423$	−1.00000	−1.00000
$424$	−1.00000	−1.00000
$425$	− 1.41421i	− 1.41421i
$426$	0	0
$427$	1.41421i	1.41421i
$428$	0	0
$429$	2.00000	2.00000
$430$	0	0
$431$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$431$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$432$	0	0
$433$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$433$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.00000	−1.00000
$438$	0	0
$439$	−2.00000	−2.00000	−1.00000			$\pi$
$439$	−2.00000	−2.00000	−1.00000			$\pi$
$440$	0	0
$441$	1.00000	1.00000
$442$	− 1.41421i	− 1.41421i
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	1.41421i	1.41421i
$447$	− 1.41421i	− 1.41421i
$448$	1.41421i	1.41421i
$449$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$449$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$450$	−1.00000	−1.00000
$451$	2.00000	2.00000
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	− 1.41421i	− 1.41421i
$457$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$457$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$458$	1.00000	1.00000
$459$	0	0
$460$	0	0
$461$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$461$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$462$	− 2.82843i	− 2.82843i
$463$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$463$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$464$	0	0
$465$	0	0
$466$	−1.00000	−1.00000
$467$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$467$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$468$	0	0
$469$	−2.00000	−2.00000
$470$	0	0
$471$	−2.00000	−2.00000
$472$	−1.00000	−1.00000
$473$	0	0
$474$	2.00000	2.00000
$475$	−1.00000	−1.00000
$476$	0	0
$477$	1.00000	1.00000
$478$	− 1.41421i	− 1.41421i
$479$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$479$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$480$	0	0
$481$	1.00000	1.00000
$482$	0	0
$483$	−2.00000	−2.00000
$484$	0	0
$485$	0	0
$486$	− 1.41421i	− 1.41421i
$487$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$487$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$488$	1.00000	1.00000
$489$	2.82843i	2.82843i
$490$	0	0
$491$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$491$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$492$	0	0
$493$	0	0
$494$	−1.00000	−1.00000
$495$	0	0
$496$	0	0
$497$	0	0
$498$	−2.00000	−2.00000
$499$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$499$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$500$	0	0
$501$	1.41421i	1.41421i
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	− 1.41421i	− 1.41421i
$505$	0	0
$506$	1.41421i	1.41421i
$507$	0	0
$508$	0	0
$509$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$509$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	1.00000
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.41421i	1.41421i
$518$	− 1.41421i	− 1.41421i
$519$	0	0
$520$	0	0
$521$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$521$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$522$	0	0
$523$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$523$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$524$	0	0
$525$	−2.00000	−2.00000
$526$	0	0
$527$	0	0
$528$	−2.00000	−2.00000
$529$	0	0
$530$	0	0
$531$	1.00000	1.00000
$532$	0	0
$533$	− 1.41421i	− 1.41421i
$534$	1.41421i	1.41421i
$535$	0	0
$536$	1.41421i	1.41421i
$537$	− 2.82843i	− 2.82843i
$538$	0	0
$539$	− 1.41421i	− 1.41421i
$540$	0	0
$541$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$541$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	−2.00000	−2.00000
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	−1.00000	−1.00000
$550$	1.41421i	1.41421i
$551$	0	0
$552$	1.41421i	1.41421i
$553$	2.00000	2.00000
$554$	1.41421i	1.41421i
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	2.82843i	2.82843i
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	− 1.41421i	− 1.41421i
$567$	− 1.41421i	− 1.41421i
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	− 1.41421i	− 1.41421i
$574$	−2.00000	−2.00000
$575$	1.00000	1.00000
$576$	−1.00000	−1.00000
$577$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$577$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$578$	1.00000	1.00000
$579$	− 1.41421i	− 1.41421i
$580$	0	0
$581$	−2.00000	−2.00000
$582$	1.41421i	1.41421i
$583$	− 1.41421i	− 1.41421i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.00000	2.00000	1.00000			$0$
$587$	2.00000	2.00000	1.00000			$0$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	1.41421i	1.41421i
$592$	−1.00000	−1.00000
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.41421i	1.41421i
$598$	1.00000	1.00000
$599$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$599$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$600$	1.41421i	1.41421i
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	− 1.41421i	− 1.41421i
$604$	0	0
$605$	0	0
$606$	2.00000	2.00000
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.00000	1.00000
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	−2.00000	−2.00000
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$619$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$620$	0	0
$621$	0	0
$622$	−1.00000	−1.00000
$623$	1.41421i	1.41421i
$624$	1.41421i	1.41421i
$625$	1.00000	1.00000
$626$	0	0
$627$	2.00000	2.00000
$628$	0	0
$629$	1.41421i	1.41421i
$630$	0	0
$631$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$631$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$632$	− 1.41421i	− 1.41421i
$633$	− 1.41421i	− 1.41421i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.00000	−1.00000
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$641$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$642$	− 1.41421i	− 1.41421i
$643$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$643$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$644$	0	0
$645$	0	0
$646$	− 1.41421i	− 1.41421i
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−1.00000	−1.00000
$649$	− 1.41421i	− 1.41421i
$650$	1.00000	1.00000
$651$	0	0
$652$	0	0
$653$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$653$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$654$	1.41421i	1.41421i
$655$	0	0
$656$	1.41421i	1.41421i
$657$	0	0
$658$	− 1.41421i	− 1.41421i
$659$	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			$\pi$
$662$	1.41421i	1.41421i
$663$	2.00000	2.00000
$664$	1.41421i	1.41421i
$665$	0	0
$666$	1.00000	1.00000
$667$	0	0
$668$	0	0
$669$	−2.00000	−2.00000
$670$	0	0
$671$	1.41421i	1.41421i
$672$	0	0
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	− 1.41421i	− 1.41421i
$675$	0	0
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	1.41421i	1.41421i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$683$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.41421i	1.41421i
$688$	0	0
$689$	−1.00000	−1.00000
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	2.00000	2.00000
$694$	−1.00000	−1.00000
$695$	0	0
$696$	0	0
$697$	2.00000	2.00000
$698$	1.41421i	1.41421i
$699$	− 1.41421i	− 1.41421i
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	1.00000	1.00000
$704$	1.41421i	1.41421i
$705$	0	0
$706$	0	0
$707$	2.00000	2.00000
$708$	0	0
$709$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$709$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$710$	0	0
$711$	1.41421i	1.41421i
$712$	1.00000	1.00000
$713$	0	0
$714$	− 2.82843i	− 2.82843i
$715$	0	0
$716$	0	0
$717$	2.00000	2.00000
$718$	1.41421i	1.41421i
$719$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$719$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	− 1.41421i	− 1.41421i
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	1.41421i	1.41421i
$729$	1.00000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	1.00000	1.00000
$735$	0	0
$736$	0	0
$737$	−2.00000	−2.00000
$738$	− 1.41421i	− 1.41421i
$739$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$739$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$740$	0	0
$741$	− 1.41421i	− 1.41421i
$742$	1.41421i	1.41421i
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	− 1.41421i	− 1.41421i
$747$	− 1.41421i	− 1.41421i
$748$	0	0
$749$	− 1.41421i	− 1.41421i
$750$	0	0
$751$	1.00000	1.00000
$751$	1.00000	1.00000
$752$	−1.00000	−1.00000
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	−2.00000	−2.00000
$760$	0	0
$761$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$761$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$762$	− 1.41421i	− 1.41421i
$763$	1.41421i	1.41421i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.00000	−1.00000
$768$	0	0
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	0	0
$776$	1.00000	1.00000
$777$	2.00000	2.00000
$778$	1.41421i	1.41421i
$779$	− 1.41421i	− 1.41421i
$780$	0	0
$781$	0	0
$782$	1.41421i	1.41421i
$783$	0	0
$784$	1.00000	1.00000
$785$	0	0
$786$	− 1.41421i	− 1.41421i
$787$	−2.00000	−2.00000	−1.00000			$\pi$
$787$	−2.00000	−2.00000	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	− 1.41421i	− 1.41421i
$793$	1.00000	1.00000
$794$	1.00000	1.00000
$795$	0	0
$796$	0	0
$797$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$797$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$798$	−2.00000	−2.00000
$799$	1.41421i	1.41421i
$800$	0	0
$801$	−1.00000	−1.00000
$802$	1.00000	1.00000
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	− 1.41421i	− 1.41421i
$809$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$809$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	− 1.41421i	− 1.41421i
$815$	0	0
$816$	−2.00000	−2.00000
$817$	0	0
$818$	0	0
$819$	− 1.41421i	− 1.41421i
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	2.00000	2.00000
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	−2.00000	−2.00000
$826$	1.41421i	1.41421i
$827$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$827$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$828$	0	0
$829$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$829$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$830$	0	0
$831$	−2.00000	−2.00000
$832$	1.00000	1.00000
$833$	− 1.41421i	− 1.41421i
$834$	1.41421i	1.41421i
$835$	0	0
$836$	0	0
$837$	0	0
$838$	1.00000	1.00000
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	1.41421i	1.41421i
$843$	0	0
$844$	0	0
$845$	0	0
$846$	1.00000	1.00000
$847$	− 1.41421i	− 1.41421i
$848$	1.00000	1.00000
$849$	2.00000	2.00000
$850$	1.41421i	1.41421i
$851$	−1.00000	−1.00000
$852$	0	0
$853$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$853$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$854$	− 1.41421i	− 1.41421i
$855$	0	0
$856$	−1.00000	−1.00000
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	−2.00000	−2.00000
$859$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$859$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$860$	0	0
$861$	− 2.82843i	− 2.82843i
$862$	1.41421i	1.41421i
$863$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$863$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$864$	0	0
$865$	0	0
$866$	−1.00000	−1.00000
$867$	1.41421i	1.41421i
$868$	0	0
$869$	2.00000	2.00000
$870$	0	0
$871$	1.41421i	1.41421i
$872$	1.00000	1.00000
$873$	−1.00000	−1.00000
$874$	1.00000	1.00000
$875$	0	0
$876$	0	0
$877$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$877$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$878$	2.00000	2.00000
$879$	0	0
$880$	0	0
$881$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$881$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$882$	−1.00000	−1.00000
$883$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$883$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	− 1.41421i	− 1.41421i
$889$	− 1.41421i	− 1.41421i
$890$	0	0
$891$	− 1.41421i	− 1.41421i
$892$	0	0
$893$	1.00000	1.00000
$894$	1.41421i	1.41421i
$895$	0	0
$896$	− 1.41421i	− 1.41421i
$897$	1.41421i	1.41421i
$898$	− 1.41421i	− 1.41421i
$899$	0	0
$900$	0	0
$901$	− 1.41421i	− 1.41421i
$902$	−2.00000	−2.00000
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	1.41421i	1.41421i
$910$	0	0
$911$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$911$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$912$	1.41421i	1.41421i
$913$	−2.00000	−2.00000
$914$	1.00000	1.00000
$915$	0	0
$916$	0	0
$917$	− 1.41421i	− 1.41421i
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	0	0
$922$	1.00000	1.00000
$923$	0	0
$924$	0	0
$925$	−1.00000	−1.00000
$926$	1.41421i	1.41421i
$927$	0	0
$928$	0	0
$929$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$929$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$930$	0	0
$931$	−1.00000	−1.00000
$932$	0	0
$933$	− 1.41421i	− 1.41421i
$934$	− 1.41421i	− 1.41421i
$935$	0	0
$936$	−1.00000	−1.00000
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	2.00000	2.00000
$939$	0	0
$940$	0	0
$941$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$941$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$942$	2.00000	2.00000
$943$	1.41421i	1.41421i
$944$	1.00000	1.00000
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	0	0
$950$	1.00000	1.00000
$951$	0	0
$952$	−2.00000	−2.00000
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	−1.00000	−1.00000
$955$	0	0
$956$	0	0
$957$	0	0
$958$	−1.00000	−1.00000
$959$	2.00000	2.00000
$960$	0	0
$961$	1.00000	1.00000
$962$	−1.00000	−1.00000
$963$	1.00000	1.00000
$964$	0	0
$965$	0	0
$966$	2.00000	2.00000
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−1.00000	−1.00000
$969$	2.00000	2.00000
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	1.41421i	1.41421i
$974$	1.41421i	1.41421i
$975$	1.41421i	1.41421i
$976$	−1.00000	−1.00000
$977$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$977$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$978$	− 2.82843i	− 2.82843i
$979$	1.41421i	1.41421i
$980$	0	0
$981$	−1.00000	−1.00000
$982$	1.41421i	1.41421i
$983$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$983$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$984$	−2.00000	−2.00000
$985$	0	0
$986$	0	0
$987$	2.00000	2.00000
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	−2.00000	−2.00000
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$997$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$998$	1.00000	1.00000
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	751.1.b.b.750.1	✓	2
751.750	odd	2	inner	751.1.b.b.750.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
751.1.b.b.750.1	✓	2	1.1	even	1	trivial
751.1.b.b.750.2	yes	2	751.750	odd	2	inner

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:	\( N \)	\(=\)	\( 751 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	751.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.41421i q^{3} +1.41421i q^{6} +1.41421i q^{7} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.41421i q^{3} +1.41421i q^{6} +1.41421i q^{7} +1.00000 q^{8} -1.00000 q^{9} +1.41421i q^{11} +1.00000 q^{13} -1.41421i q^{14} -1.00000 q^{16} +1.41421i q^{17} +1.00000 q^{18} +1.00000 q^{19} +2.00000 q^{21} -1.41421i q^{22} -1.00000 q^{23} -1.41421i q^{24} -1.00000 q^{25} -1.00000 q^{26} +2.00000 q^{33} -1.41421i q^{34} +1.00000 q^{37} -1.00000 q^{38} -1.41421i q^{39} -1.41421i q^{41} -2.00000 q^{42} +1.00000 q^{46} +1.00000 q^{47} +1.41421i q^{48} -1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} -1.00000 q^{53} +1.41421i q^{56} -1.41421i q^{57} -1.00000 q^{59} +1.00000 q^{61} -1.41421i q^{63} +1.00000 q^{64} -2.00000 q^{66} +1.41421i q^{67} +1.41421i q^{69} -1.00000 q^{72} -1.00000 q^{74} +1.41421i q^{75} -2.00000 q^{77} +1.41421i q^{78} -1.41421i q^{79} -1.00000 q^{81} +1.41421i q^{82} +1.41421i q^{83} +1.41421i q^{88} +1.00000 q^{89} +1.41421i q^{91} -1.00000 q^{94} +1.00000 q^{97} +1.00000 q^{98} -1.41421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^8 - 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{8} - 2 q^{9} + 2 q^{13} - 2 q^{16} + 2 q^{18} + 2 q^{19} + 4 q^{21} - 2 q^{23} - 2 q^{25} - 2 q^{26} + 4 q^{33} + 2 q^{37} - 2 q^{38} - 4 q^{42} + 2 q^{46} + 2 q^{47} - 2 q^{49} + 2 q^{50} + 4 q^{51} - 2 q^{53} - 2 q^{59} + 2 q^{61} + 2 q^{64} - 4 q^{66} - 2 q^{72} - 2 q^{74} - 4 q^{77} - 2 q^{81} + 2 q^{89} - 2 q^{94} + 2 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^8 - 2 * q^9 + 2 * q^13 - 2 * q^16 + 2 * q^18 + 2 * q^19 + 4 * q^21 - 2 * q^23 - 2 * q^25 - 2 * q^26 + 4 * q^33 + 2 * q^37 - 2 * q^38 - 4 * q^42 + 2 * q^46 + 2 * q^47 - 2 * q^49 + 2 * q^50 + 4 * q^51 - 2 * q^53 - 2 * q^59 + 2 * q^61 + 2 * q^64 - 4 * q^66 - 2 * q^72 - 2 * q^74 - 4 * q^77 - 2 * q^81 + 2 * q^89 - 2 * q^94 + 2 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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