LMFDB - Embedded newform 2300.1.k.c.2207.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,1,Mod(643,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2300.643");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2300.k (of order $4$, degree $2$, 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:	$1.14784952906$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{72})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{12} + 1 $ x^24 - x^12 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{18}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{18} - \cdots)$

Embedding invariants

Embedding label			2207.8
Root			$0.906308 + 0.422618i$ of defining polynomial
Character	$\chi$	$=$	2300.2207
Dual form			2300.1.k.c.643.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.422618 - 0.906308i) q^{2} +(-1.08335 - 1.08335i) q^{3} +(-0.642788 - 0.766044i) q^{4} +(-1.43969 + 0.524005i) q^{6} +(-0.965926 + 0.258819i) q^{8} +1.34730i q^{9} +O(q^{10})$	\(q+(0.422618 - 0.906308i) q^{2} +(-1.08335 - 1.08335i) q^{3} +(-0.642788 - 0.766044i) q^{4} +(-1.43969 + 0.524005i) q^{6} +(-0.965926 + 0.258819i) q^{8} +1.34730i q^{9} +(-0.133530 + 1.52626i) q^{12} +(-1.39273 + 1.39273i) q^{13} +(-0.173648 + 0.984808i) q^{16} +(1.22107 + 0.569392i) q^{18} +(-0.707107 - 0.707107i) q^{23} +(1.32683 + 0.766044i) q^{24} +(0.673648 + 1.85083i) q^{26} +(0.376244 - 0.376244i) q^{27} +1.87939i q^{29} +0.684040i q^{31} +(0.819152 + 0.573576i) q^{32} +(1.03209 - 0.866025i) q^{36} +3.01763 q^{39} -1.53209 q^{41} +(-0.939693 + 0.342020i) q^{46} +(-0.245576 + 0.245576i) q^{47} +(1.25501 - 0.878770i) q^{48} -1.00000i q^{49} +(1.96212 + 0.171663i) q^{52} +(-0.181985 - 0.500000i) q^{54} +(1.70330 + 0.794263i) q^{58} -1.73205 q^{59} +(0.619951 + 0.289088i) q^{62} +(0.866025 - 0.500000i) q^{64} +1.53209i q^{69} -1.96962i q^{71} +(-0.348706 - 1.30139i) q^{72} +(0.483690 - 0.483690i) q^{73} +(1.27530 - 2.73490i) q^{78} +0.532089 q^{81} +(-0.647489 + 1.38854i) q^{82} +(2.03603 - 2.03603i) q^{87} +(-0.0871557 + 0.996195i) q^{92} +(0.741055 - 0.741055i) q^{93} +(0.118782 + 0.326352i) q^{94} +(-0.266044 - 1.50881i) q^{96} +(-0.906308 - 0.422618i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 12 q^{6}+O(q^{10}) $ 24 * q - 12 * q^6	$ 24 q - 12 q^{6} + 12 q^{26} - 12 q^{36} - 24 q^{81} + 12 q^{96}+O(q^{100}) $ 24 * q - 12 * q^6 + 12 * q^26 - 12 * q^36 - 24 * q^81 + 12 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$-1$

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 $

$2$	0.422618	−	0.906308i	0.422618	−	0.906308i
$2$	0.422618	−	0.906308i	0.422618	−	0.906308i
$3$	−1.08335	−	1.08335i	−1.08335	−	1.08335i	−0.996195	−	0.0871557i	$-0.972222\pi$
$3$	−1.08335	−	1.08335i	−1.08335	−	1.08335i	−0.0871557	−	0.996195i	$-0.527778\pi$
$4$	−0.642788	−	0.766044i	−0.642788	−	0.766044i
$5$		0			0
$5$		0			0
$6$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$9$			1.34730i			1.34730i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−0.133530	+	1.52626i	−0.133530	+	1.52626i
$13$	−1.39273	+	1.39273i	−1.39273	+	1.39273i	−0.573576	+	0.819152i	$0.694444\pi$
$13$	−1.39273	+	1.39273i	−1.39273	+	1.39273i	−0.819152	+	0.573576i	$0.805556\pi$
$14$		0			0
$15$		0			0
$16$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$	1.22107	+	0.569392i	1.22107	+	0.569392i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$23$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$24$	1.32683	+	0.766044i	1.32683	+	0.766044i
$25$		0			0
$26$	0.673648	+	1.85083i	0.673648	+	1.85083i
$27$	0.376244	−	0.376244i	0.376244	−	0.376244i
$28$		0			0
$29$			1.87939i			1.87939i	0.342020	+	0.939693i	$0.388889\pi$
$29$			1.87939i			1.87939i	−0.342020	+	0.939693i	$0.611111\pi$
$30$		0			0
$31$			0.684040i			0.684040i	0.939693	+	0.342020i	$0.111111\pi$
$31$			0.684040i			0.684040i	−0.939693	+	0.342020i	$0.888889\pi$
$32$	0.819152	+	0.573576i	0.819152	+	0.573576i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	1.03209	−	0.866025i	1.03209	−	0.866025i
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$	3.01763			3.01763
$40$		0			0
$41$	−1.53209			−1.53209			−0.766044	−	0.642788i	$-0.777778\pi$
$41$	−1.53209			−1.53209			−0.766044	+	0.642788i	$0.777778\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$		0			0
$46$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$47$	−0.245576	+	0.245576i	−0.245576	+	0.245576i	−0.819152	−	0.573576i	$-0.805556\pi$
$47$	−0.245576	+	0.245576i	−0.245576	+	0.245576i	0.573576	+	0.819152i	$0.305556\pi$
$48$	1.25501	−	0.878770i	1.25501	−	0.878770i
$49$		−	1.00000i		−	1.00000i
$50$		0			0
$51$		0			0
$52$	1.96212	+	0.171663i	1.96212	+	0.171663i
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$	−0.181985	−	0.500000i	−0.181985	−	0.500000i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	1.70330	+	0.794263i	1.70330	+	0.794263i
$59$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$59$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	0.619951	+	0.289088i	0.619951	+	0.289088i
$63$		0			0
$64$	0.866025	−	0.500000i	0.866025	−	0.500000i
$65$		0			0
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$		0			0
$69$			1.53209i			1.53209i
$70$		0			0
$71$		−	1.96962i		−	1.96962i	−0.173648	−	0.984808i	$-0.555556\pi$
$71$		−	1.96962i		−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$72$	−0.348706	−	1.30139i	−0.348706	−	1.30139i
$73$	0.483690	−	0.483690i	0.483690	−	0.483690i	−0.422618	−	0.906308i	$-0.638889\pi$
$73$	0.483690	−	0.483690i	0.483690	−	0.483690i	0.906308	+	0.422618i	$0.138889\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$	1.27530	−	2.73490i	1.27530	−	2.73490i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	0.532089			0.532089
$82$	−0.647489	+	1.38854i	−0.647489	+	1.38854i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	2.03603	−	2.03603i	2.03603	−	2.03603i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	−0.0871557	+	0.996195i	−0.0871557	+	0.996195i
$93$	0.741055	−	0.741055i	0.741055	−	0.741055i
$94$	0.118782	+	0.326352i	0.118782	+	0.326352i
$95$		0			0
$96$	−0.266044	−	1.50881i	−0.266044	−	1.50881i
$97$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$97$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$98$	−0.906308	−	0.422618i	−0.906308	−	0.422618i
$99$		0			0
$100$		0			0
$101$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$101$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$	0.984808	−	1.70574i	0.984808	−	1.70574i
$105$		0			0
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$	−0.530064	−	0.0463746i	−0.530064	−	0.0463746i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		0			0
$116$	1.43969	−	1.20805i	1.43969	−	1.20805i
$117$	−1.87642	−	1.87642i	−1.87642	−	1.87642i
$118$	−0.731996	+	1.56977i	−0.731996	+	1.56977i
$119$		0			0
$120$		0			0
$121$	−1.00000			−1.00000
$122$		0			0
$123$	1.65979	+	1.65979i	1.65979	+	1.65979i
$124$	0.524005	−	0.439693i	0.524005	−	0.439693i
$125$		0			0
$126$		0			0
$127$	−1.32893	+	1.32893i	−1.32893	+	1.32893i	−0.422618	+	0.906308i	$0.638889\pi$
$127$	−1.32893	+	1.32893i	−1.32893	+	1.32893i	−0.906308	+	0.422618i	$0.861111\pi$
$128$	−0.0871557	−	0.996195i	−0.0871557	−	0.996195i
$129$		0			0
$130$		0			0
$131$			1.28558i			1.28558i	0.766044	+	0.642788i	$0.222222\pi$
$131$			1.28558i			1.28558i	−0.766044	+	0.642788i	$0.777778\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$137$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$138$	1.38854	+	0.647489i	1.38854	+	0.647489i
$139$	−1.96962			−1.96962			−0.984808	−	0.173648i	$-0.944444\pi$
$139$	−1.96962			−1.96962			−0.984808	+	0.173648i	$0.944444\pi$
$140$		0			0
$141$	0.532089			0.532089
$142$	−1.78508	−	0.832395i	−1.78508	−	0.832395i
$143$		0			0
$144$	−1.32683	−	0.233956i	−1.32683	−	0.233956i
$145$		0			0
$146$	−0.233956	−	0.642788i	−0.233956	−	0.642788i
$147$	−1.08335	+	1.08335i	−1.08335	+	1.08335i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$			1.96962i			1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$151$			1.96962i			1.96962i	−0.173648	+	0.984808i	$0.555556\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.93969	−	2.31164i	−1.93969	−	2.31164i
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.224870	−	0.482236i	0.224870	−	0.482236i
$163$	1.32893	+	1.32893i	1.32893	+	1.32893i	0.906308	+	0.422618i	$0.138889\pi$
$163$	1.32893	+	1.32893i	1.32893	+	1.32893i	0.422618	+	0.906308i	$0.361111\pi$
$164$	0.984808	+	1.17365i	0.984808	+	1.17365i
$165$		0			0
$166$		0			0
$167$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$167$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$168$		0			0
$169$		−	2.87939i		−	2.87939i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.22474	+	1.22474i	−1.22474	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$173$	−1.22474	+	1.22474i	−1.22474	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$174$	−0.984808	−	2.70574i	−0.984808	−	2.70574i
$175$		0			0
$176$		0			0
$177$	1.87642	+	1.87642i	1.87642	+	1.87642i
$178$		0			0
$179$	−0.684040			−0.684040			−0.342020	−	0.939693i	$-0.611111\pi$
$179$	−0.684040			−0.684040			−0.342020	+	0.939693i	$0.611111\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	0.866025	+	0.500000i	0.866025	+	0.500000i
$185$		0			0
$186$	−0.358441	−	0.984808i	−0.358441	−	0.984808i
$187$		0			0
$188$	0.345975	+	0.0302689i	0.345975	+	0.0302689i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−1.47988	−	0.396534i	−1.47988	−	0.396534i
$193$	−0.909039	+	0.909039i	−0.909039	+	0.909039i	−0.996195	−	0.0871557i	$-0.972222\pi$
$193$	−0.909039	+	0.909039i	−0.909039	+	0.909039i	0.0871557	+	0.996195i	$0.472222\pi$
$194$		0			0
$195$		0			0
$196$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$197$	−0.909039	−	0.909039i	−0.909039	−	0.909039i	0.0871557	−	0.996195i	$-0.472222\pi$
$197$	−0.909039	−	0.909039i	−0.909039	−	0.909039i	−0.996195	+	0.0871557i	$0.972222\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$	0.422618	−	0.906308i	0.422618	−	0.906308i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.952682	−	0.952682i	0.952682	−	0.952682i
$208$	−1.12973	−	1.61341i	−1.12973	−	1.61341i
$209$		0			0
$210$		0			0
$211$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$211$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$212$		0			0
$213$	−2.13378	+	2.13378i	−2.13378	+	2.13378i
$214$		0			0
$215$		0			0
$216$	−0.266044	+	0.460802i	−0.266044	+	0.460802i
$217$		0			0
$218$		0			0
$219$	−1.04801			−1.04801
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$223$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$	−0.486421	−	1.81535i	−0.486421	−	1.81535i
$233$	1.39273	−	1.39273i	1.39273	−	1.39273i	0.573576	−	0.819152i	$-0.305556\pi$
$233$	1.39273	−	1.39273i	1.39273	−	1.39273i	0.819152	−	0.573576i	$-0.194444\pi$
$234$	−2.49362	+	0.907604i	−2.49362	+	0.907604i
$235$		0			0
$236$	1.11334	+	1.32683i	1.11334	+	1.32683i
$237$		0			0
$238$		0			0
$239$	−1.28558			−1.28558			−0.642788	−	0.766044i	$-0.722222\pi$
$239$	−1.28558			−1.28558			−0.642788	+	0.766044i	$0.722222\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−0.422618	+	0.906308i	−0.422618	+	0.906308i
$243$	−0.952682	−	0.952682i	−0.952682	−	0.952682i
$244$		0			0
$245$		0			0
$246$	2.20574	−	0.802823i	2.20574	−	0.802823i
$247$		0			0
$248$	−0.177043	−	0.660732i	−0.177043	−	0.660732i
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$	0.642788	+	1.76604i	0.642788	+	1.76604i
$255$		0			0
$256$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$257$	−0.909039	−	0.909039i	−0.909039	−	0.909039i	0.0871557	−	0.996195i	$-0.472222\pi$
$257$	−0.909039	−	0.909039i	−0.909039	−	0.909039i	−0.996195	+	0.0871557i	$0.972222\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−2.53209			−2.53209
$262$	1.16513	+	0.543308i	1.16513	+	0.543308i
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	0.347296i		−	0.347296i	−0.984808	−	0.173648i	$-0.944444\pi$
$269$		−	0.347296i		−	0.347296i	0.984808	−	0.173648i	$-0.0555556\pi$
$270$		0			0
$271$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$271$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	1.17365	−	0.984808i	1.17365	−	0.984808i
$277$	0.909039	+	0.909039i	0.909039	+	0.909039i	0.996195	−	0.0871557i	$-0.0277778\pi$
$277$	0.909039	+	0.909039i	0.909039	+	0.909039i	−0.0871557	+	0.996195i	$0.527778\pi$
$278$	−0.832395	+	1.78508i	−0.832395	+	1.78508i
$279$	−0.921605			−0.921605
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	0.224870	−	0.482236i	0.224870	−	0.482236i
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$	−1.50881	+	1.26604i	−1.50881	+	1.26604i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.772777	+	1.10364i	−0.772777	+	1.10364i
$289$			1.00000i			1.00000i
$290$		0			0
$291$		0			0
$292$	−0.681437	−	0.0596180i	−0.681437	−	0.0596180i
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$	0.524005	+	1.43969i	0.524005	+	1.43969i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.96962			1.96962
$300$		0			0
$301$		0			0
$302$	1.78508	+	0.832395i	1.78508	+	0.832395i
$303$	−1.08335	−	1.08335i	−1.08335	−	1.08335i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$307$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	0.684040i		−	0.684040i	−0.939693	−	0.342020i	$-0.888889\pi$
$311$		−	0.684040i		−	0.684040i	0.939693	−	0.342020i	$-0.111111\pi$
$312$	−2.91480	+	0.781019i	−2.91480	+	0.781019i
$313$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.342020	−	0.407604i	−0.342020	−	0.407604i
$325$		0			0
$326$	1.76604	−	0.642788i	1.76604	−	0.642788i
$327$		0			0
$328$	1.47988	−	0.396534i	1.47988	−	0.396534i
$329$		0			0
$330$		0			0
$331$		−	1.28558i		−	1.28558i	−0.766044	−	0.642788i	$-0.777778\pi$
$331$		−	1.28558i		−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$332$		0			0
$333$		0			0
$334$	−0.342020	−	0.939693i	−0.342020	−	0.939693i
$335$		0			0
$336$		0			0
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$	−2.60961	−	1.21688i	−2.60961	−	1.21688i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	0.592396	+	1.62760i	0.592396	+	1.62760i
$347$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$347$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$348$	−2.86843	−	0.250955i	−2.86843	−	0.250955i
$349$		−	0.347296i		−	0.347296i	−0.984808	−	0.173648i	$-0.944444\pi$
$349$		−	0.347296i		−	0.347296i	0.984808	−	0.173648i	$-0.0555556\pi$
$350$		0			0
$351$			1.04801i			1.04801i
$352$		0			0
$353$	0.909039	−	0.909039i	0.909039	−	0.909039i	−0.0871557	−	0.996195i	$-0.527778\pi$
$353$	0.909039	−	0.909039i	0.909039	−	0.909039i	0.996195	+	0.0871557i	$0.0277778\pi$
$354$	2.49362	−	0.907604i	2.49362	−	0.907604i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−0.289088	+	0.619951i	−0.289088	+	0.619951i
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	1.00000			1.00000
$362$		0			0
$363$	1.08335	+	1.08335i	1.08335	+	1.08335i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$	0.819152	−	0.573576i	0.819152	−	0.573576i
$369$		−	2.06418i		−	2.06418i
$370$		0			0
$371$		0			0
$372$	−1.04402	−	0.0913401i	−1.04402	−	0.0913401i
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$		0			0
$376$	0.173648	−	0.300767i	0.173648	−	0.300767i
$377$	−2.61747	−	2.61747i	−2.61747	−	2.61747i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	2.87939			2.87939
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$	−0.984808	+	1.17365i	−0.984808	+	1.17365i
$385$		0			0
$386$	0.439693	+	1.20805i	0.439693	+	1.20805i
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	0.258819	+	0.965926i	0.258819	+	0.965926i
$393$	1.39273	−	1.39273i	1.39273	−	1.39273i
$394$	−1.20805	+	0.439693i	−1.20805	+	0.439693i
$395$		0			0
$396$		0			0
$397$	0.483690	+	0.483690i	0.483690	+	0.483690i	0.906308	−	0.422618i	$-0.138889\pi$
$397$	0.483690	+	0.483690i	0.483690	+	0.483690i	−0.422618	+	0.906308i	$0.638889\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	−0.952682	−	0.952682i	−0.952682	−	0.952682i
$404$	−0.642788	−	0.766044i	−0.642788	−	0.766044i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			1.53209i			1.53209i	0.642788	+	0.766044i	$0.277778\pi$
$409$			1.53209i			1.53209i	−0.642788	+	0.766044i	$0.722222\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−0.460802	−	1.26604i	−0.460802	−	1.26604i
$415$		0			0
$416$	−1.93969	+	0.342020i	−1.93969	+	0.342020i
$417$	2.13378	+	2.13378i	2.13378	+	2.13378i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$	−1.56977	−	0.731996i	−1.56977	−	0.731996i
$423$	−0.330863	−	0.330863i	−0.330863	−	0.330863i
$424$		0			0
$425$		0			0
$426$	1.03209	+	2.83564i	1.03209	+	2.83564i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	0.305194	+	0.435862i	0.305194	+	0.435862i
$433$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−0.442908	+	0.949820i	−0.442908	+	0.949820i
$439$	−1.28558			−1.28558			−0.642788	−	0.766044i	$-0.722222\pi$
$439$	−1.28558			−1.28558			−0.642788	+	0.766044i	$0.722222\pi$
$440$		0			0
$441$	1.34730			1.34730
$442$		0			0
$443$	−0.245576	−	0.245576i	−0.245576	−	0.245576i	0.573576	−	0.819152i	$-0.305556\pi$
$443$	−0.245576	−	0.245576i	−0.245576	−	0.245576i	−0.819152	+	0.573576i	$0.805556\pi$
$444$		0			0
$445$		0			0
$446$	0.939693	−	0.342020i	0.939693	−	0.342020i
$447$		0			0
$448$		0			0
$449$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$449$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	2.13378	−	2.13378i	2.13378	−	2.13378i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$457$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−0.347296			−0.347296			−0.173648	−	0.984808i	$-0.555556\pi$
$461$	−0.347296			−0.347296			−0.173648	+	0.984808i	$0.555556\pi$
$462$		0			0
$463$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$463$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$464$	−1.85083	−	0.326352i	−1.85083	−	0.326352i
$465$		0			0
$466$	−0.673648	−	1.85083i	−0.673648	−	1.85083i
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	−0.231281	+	2.64356i	−0.231281	+	2.64356i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	1.67303	−	0.448288i	1.67303	−	0.448288i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−0.543308	+	1.16513i	−0.543308	+	1.16513i
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.642788	+	0.766044i	0.642788	+	0.766044i
$485$		0			0
$486$	−1.26604	+	0.460802i	−1.26604	+	0.460802i
$487$	0.245576	−	0.245576i	0.245576	−	0.245576i	−0.573576	−	0.819152i	$-0.694444\pi$
$487$	0.245576	−	0.245576i	0.245576	−	0.245576i	0.819152	+	0.573576i	$0.194444\pi$
$488$		0			0
$489$		−	2.87939i		−	2.87939i
$490$		0			0
$491$		−	1.96962i		−	1.96962i	−0.173648	−	0.984808i	$-0.555556\pi$
$491$		−	1.96962i		−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$492$	0.204580	−	2.33836i	0.204580	−	2.33836i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.673648	−	0.118782i	−0.673648	−	0.118782i
$497$		0			0
$498$		0			0
$499$	1.28558			1.28558			0.642788	−	0.766044i	$-0.277778\pi$
$499$	1.28558			1.28558			0.642788	+	0.766044i	$0.277778\pi$
$500$		0			0
$501$	−1.53209			−1.53209
$502$		0			0
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−3.11938	+	3.11938i	−3.11938	+	3.11938i
$508$	1.87223	+	0.163799i	1.87223	+	0.163799i
$509$		−	1.53209i		−	1.53209i	−0.642788	−	0.766044i	$-0.722222\pi$
$509$		−	1.53209i		−	1.53209i	0.642788	−	0.766044i	$-0.277778\pi$
$510$		0			0
$511$		0			0
$512$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$513$		0			0
$514$	−1.20805	+	0.439693i	−1.20805	+	0.439693i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	2.65366			2.65366
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	−1.07011	+	2.29485i	−1.07011	+	2.29485i
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$	0.984808	−	0.826352i	0.984808	−	0.826352i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			1.00000i			1.00000i
$530$		0			0
$531$		−	2.33359i		−	2.33359i
$532$		0			0
$533$	2.13378	−	2.13378i	2.13378	−	2.13378i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.741055	+	0.741055i	0.741055	+	0.741055i
$538$	−0.314757	−	0.146774i	−0.314757	−	0.146774i
$539$		0			0
$540$		0			0
$541$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$541$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$542$	1.56977	+	0.731996i	1.56977	+	0.731996i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.245576	−	0.245576i	0.245576	−	0.245576i	−0.573576	−	0.819152i	$-0.694444\pi$
$547$	0.245576	−	0.245576i	0.245576	−	0.245576i	0.819152	+	0.573576i	$0.194444\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	−0.396534	−	1.47988i	−0.396534	−	1.47988i
$553$		0			0
$554$	1.20805	−	0.439693i	1.20805	−	0.439693i
$555$		0			0
$556$	1.26604	+	1.50881i	1.26604	+	1.50881i
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$	−0.389487	+	0.835258i	−0.389487	+	0.835258i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$	−0.342020	−	0.407604i	−0.342020	−	0.407604i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	0.509774	+	1.90250i	0.509774	+	1.90250i
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.673648	+	1.16679i	0.673648	+	1.16679i
$577$	0.909039	+	0.909039i	0.909039	+	0.909039i	0.996195	−	0.0871557i	$-0.0277778\pi$
$577$	0.909039	+	0.909039i	0.909039	+	0.909039i	−0.0871557	+	0.996195i	$0.527778\pi$
$578$	0.906308	+	0.422618i	0.906308	+	0.422618i
$579$	1.96962			1.96962
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.342020	+	0.592396i	−0.342020	+	0.592396i
$585$		0			0
$586$		0			0
$587$	−0.245576	+	0.245576i	−0.245576	+	0.245576i	−0.819152	−	0.573576i	$-0.805556\pi$
$587$	−0.245576	+	0.245576i	−0.245576	+	0.245576i	0.573576	+	0.819152i	$0.305556\pi$
$588$	1.52626	+	0.133530i	1.52626	+	0.133530i
$589$		0			0
$590$		0			0
$591$			1.96962i			1.96962i
$592$		0			0
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	0.832395	−	1.78508i	0.832395	−	1.78508i
$599$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$599$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$601$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$602$		0			0
$603$		0			0
$604$	1.50881	−	1.26604i	1.50881	−	1.26604i
$605$		0			0
$606$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$607$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$607$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	0.684040i		−	0.684040i
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$	0.684040	+	1.87939i	0.684040	+	1.87939i
$615$		0			0
$616$		0			0
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$	−0.532089			−0.532089
$622$	−0.619951	−	0.289088i	−0.619951	−	0.289088i
$623$		0			0
$624$	−0.524005	+	2.97178i	−0.524005	+	2.97178i
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$	−1.87642	+	1.87642i	−1.87642	+	1.87642i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.39273	+	1.39273i	1.39273	+	1.39273i
$638$		0			0
$639$	2.65366			2.65366
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.32893	+	1.32893i	−1.32893	+	1.32893i	−0.422618	+	0.906308i	$0.638889\pi$
$647$	−1.32893	+	1.32893i	−1.32893	+	1.32893i	−0.906308	+	0.422618i	$0.861111\pi$
$648$	−0.513958	+	0.137715i	−0.513958	+	0.137715i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.163799	−	1.87223i	0.163799	−	1.87223i
$653$	−0.483690	+	0.483690i	−0.483690	+	0.483690i	−0.906308	−	0.422618i	$-0.861111\pi$
$653$	−0.483690	+	0.483690i	−0.483690	+	0.483690i	0.422618	+	0.906308i	$0.361111\pi$
$654$		0			0
$655$		0			0
$656$	0.266044	−	1.50881i	0.266044	−	1.50881i
$657$	0.651673	+	0.651673i	0.651673	+	0.651673i
$658$		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			$\pi$
$662$	−1.16513	−	0.543308i	−1.16513	−	0.543308i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1.32893	−	1.32893i	1.32893	−	1.32893i
$668$	−0.996195	−	0.0871557i	−0.996195	−	0.0871557i
$669$		−	1.53209i		−	1.53209i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.909039	−	0.909039i	0.909039	−	0.909039i	−0.0871557	−	0.996195i	$-0.527778\pi$
$673$	0.909039	−	0.909039i	0.909039	−	0.909039i	0.996195	+	0.0871557i	$0.0277778\pi$
$674$		0			0
$675$		0			0
$676$	−2.20574	+	1.85083i	−2.20574	+	1.85083i
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.32893	−	1.32893i	−1.32893	−	1.32893i	−0.906308	−	0.422618i	$-0.861111\pi$
$683$	−1.32893	−	1.32893i	−1.32893	−	1.32893i	−0.422618	−	0.906308i	$-0.638889\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$691$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$692$	1.72546	+	0.150958i	1.72546	+	0.150958i
$693$		0			0
$694$	0.342020	+	0.939693i	0.342020	+	0.939693i
$695$		0			0
$696$	−1.43969	+	2.49362i	−1.43969	+	2.49362i
$697$		0			0
$698$	−0.314757	−	0.146774i	−0.314757	−	0.146774i
$699$	−3.01763			−3.01763
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	0.949820	+	0.442908i	0.949820	+	0.442908i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.439693	−	1.20805i	−0.439693	−	1.20805i
$707$		0			0
$708$	0.231281	−	2.64356i	0.231281	−	2.64356i
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.483690	−	0.483690i	0.483690	−	0.483690i
$714$		0			0
$715$		0			0
$716$	0.439693	+	0.524005i	0.439693	+	0.524005i
$717$	1.39273	+	1.39273i	1.39273	+	1.39273i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	0.422618	−	0.906308i	0.422618	−	0.906308i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	1.43969	−	0.524005i	1.43969	−	0.524005i
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$			1.53209i			1.53209i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$		0			0
$735$		0			0
$736$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$737$		0			0
$738$	−1.87078	−	0.872359i	−1.87078	−	0.872359i
$739$	0.684040			0.684040			0.342020	−	0.939693i	$-0.388889\pi$
$739$	0.684040			0.684040			0.342020	+	0.939693i	$0.388889\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$	−0.524005	+	0.907604i	−0.524005	+	0.907604i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$	−0.199201	−	0.284489i	−0.199201	−	0.284489i
$753$		0			0
$754$	−3.47843	+	1.26604i	−3.47843	+	1.26604i
$755$		0			0
$756$		0			0
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$761$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$762$	1.21688	−	2.60961i	1.21688	−	2.60961i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.41228	−	2.41228i	2.41228	−	2.41228i
$768$	0.647489	+	1.38854i	0.647489	+	1.38854i
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$			1.96962i			1.96962i
$772$	1.28068	+	0.112045i	1.28068	+	0.112045i
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	0.707107	+	0.707107i	0.707107	+	0.707107i
$784$	0.984808	+	0.173648i	0.984808	+	0.173648i
$785$		0			0
$786$	−0.673648	−	1.85083i	−0.673648	−	1.85083i
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	−0.112045	+	1.28068i	−0.112045	+	1.28068i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	0.642788	−	0.233956i	0.642788	−	0.233956i
$795$		0			0
$796$		0			0
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	−1.26604	+	0.460802i	−1.26604	+	0.460802i
$807$	−0.376244	+	0.376244i	−0.376244	+	0.376244i
$808$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$809$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$809$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$810$		0			0
$811$			1.96962i			1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$811$			1.96962i			1.96962i	−0.173648	+	0.984808i	$0.555556\pi$
$812$		0			0
$813$	1.87642	−	1.87642i	1.87642	−	1.87642i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.38854	+	0.647489i	1.38854	+	0.647489i
$819$		0			0
$820$		0			0
$821$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$821$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	−1.08335	−	1.08335i	−1.08335	−	1.08335i	−0.996195	−	0.0871557i	$-0.972222\pi$
$823$	−1.08335	−	1.08335i	−1.08335	−	1.08335i	−0.0871557	−	0.996195i	$-0.527778\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$	−1.34217	−	0.117425i	−1.34217	−	0.117425i
$829$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$829$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$830$		0			0
$831$		−	1.96962i		−	1.96962i
$832$	−0.509774	+	1.90250i	−0.509774	+	1.90250i
$833$		0			0
$834$	2.83564	−	1.03209i	2.83564	−	1.03209i
$835$		0			0
$836$		0			0
$837$	0.257366	+	0.257366i	0.257366	+	0.257366i
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	−2.53209			−2.53209
$842$		0			0
$843$		0			0
$844$	−1.32683	+	1.11334i	−1.32683	+	1.11334i
$845$		0			0
$846$	−0.439693	+	0.160035i	−0.439693	+	0.160035i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	3.00614	+	0.263003i	3.00614	+	0.263003i
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.483690	−	0.483690i	−0.483690	−	0.483690i	0.422618	−	0.906308i	$-0.361111\pi$
$857$	−0.483690	−	0.483690i	−0.483690	−	0.483690i	−0.906308	+	0.422618i	$0.861111\pi$
$858$		0			0
$859$	−0.684040			−0.684040			−0.342020	−	0.939693i	$-0.611111\pi$
$859$	−0.684040			−0.684040			−0.342020	+	0.939693i	$0.611111\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.08335	+	1.08335i	1.08335	+	1.08335i	0.996195	+	0.0871557i	$0.0277778\pi$
$863$	1.08335	+	1.08335i	1.08335	+	1.08335i	0.0871557	+	0.996195i	$0.472222\pi$
$864$	0.524005	−	0.0923963i	0.524005	−	0.0923963i
$865$		0			0
$866$		0			0
$867$	1.08335	−	1.08335i	1.08335	−	1.08335i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	0.673648	+	0.802823i	0.673648	+	0.802823i
$877$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$877$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$878$	−0.543308	+	1.16513i	−0.543308	+	1.16513i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	0.569392	−	1.22107i	0.569392	−	1.22107i
$883$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$883$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$884$		0			0
$885$		0			0
$886$	−0.326352	+	0.118782i	−0.326352	+	0.118782i
$887$	1.08335	−	1.08335i	1.08335	−	1.08335i	0.0871557	−	0.996195i	$-0.472222\pi$
$887$	1.08335	−	1.08335i	1.08335	−	1.08335i	0.996195	−	0.0871557i	$-0.0277778\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	0.0871557	−	0.996195i	0.0871557	−	0.996195i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−2.13378	−	2.13378i	−2.13378	−	2.13378i
$898$	−0.906308	−	0.422618i	−0.906308	−	0.422618i
$899$	−1.28558			−1.28558
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	−1.03209	−	2.83564i	−1.03209	−	2.83564i
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$			1.34730i			1.34730i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	3.06418			3.06418
$922$	−0.146774	+	0.314757i	−0.146774	+	0.314757i
$923$	2.74314	+	2.74314i	2.74314	+	2.74314i
$924$		0			0
$925$		0			0
$926$	0.939693	−	0.342020i	0.939693	−	0.342020i
$927$		0			0
$928$	−1.07797	+	1.53950i	−1.07797	+	1.53950i
$929$			1.87939i			1.87939i	0.342020	+	0.939693i	$0.388889\pi$
$929$			1.87939i			1.87939i	−0.342020	+	0.939693i	$0.611111\pi$
$930$		0			0
$931$		0			0
$932$	−1.96212	−	0.171663i	−1.96212	−	0.171663i
$933$	−0.741055	+	0.741055i	−0.741055	+	0.741055i
$934$		0			0
$935$		0			0
$936$	2.29813	+	1.32683i	2.29813	+	1.32683i
$937$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$937$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$	1.08335	+	1.08335i	1.08335	+	1.08335i
$944$	0.300767	−	1.70574i	0.300767	−	1.70574i
$945$		0			0
$946$		0			0
$947$	1.32893	−	1.32893i	1.32893	−	1.32893i	0.422618	−	0.906308i	$-0.361111\pi$
$947$	1.32893	−	1.32893i	1.32893	−	1.32893i	0.906308	−	0.422618i	$-0.138889\pi$
$948$		0			0
$949$			1.34730i			1.34730i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$953$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$954$		0			0
$955$		0			0
$956$	0.826352	+	0.984808i	0.826352	+	0.984808i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.532089			0.532089
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.245576	+	0.245576i	−0.245576	+	0.245576i	−0.819152	−	0.573576i	$-0.805556\pi$
$967$	−0.245576	+	0.245576i	−0.245576	+	0.245576i	0.573576	+	0.819152i	$0.305556\pi$
$968$	0.965926	−	0.258819i	0.965926	−	0.258819i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	−0.117425	+	1.34217i	−0.117425	+	1.34217i
$973$		0			0
$974$	−0.118782	−	0.326352i	−0.118782	−	0.326352i
$975$		0			0
$976$		0			0
$977$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$977$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$978$	−2.60961	−	1.21688i	−2.60961	−	1.21688i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−1.78508	−	0.832395i	−1.78508	−	0.832395i
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$	−2.03282	−	1.17365i	−2.03282	−	1.17365i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$991$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$992$	−0.392349	+	0.560333i	−0.392349	+	0.560333i
$993$	−1.39273	+	1.39273i	−1.39273	+	1.39273i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$997$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$998$	0.543308	−	1.16513i	0.543308	−	1.16513i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.1.k.c.2207.8	yes	24
4.3	odd	2	inner	2300.1.k.c.2207.2	yes	24
5.2	odd	4	inner	2300.1.k.c.643.11	yes	24
5.3	odd	4	inner	2300.1.k.c.643.2	✓	24
5.4	even	2	inner	2300.1.k.c.2207.5	yes	24
20.3	even	4	inner	2300.1.k.c.643.8	yes	24
20.7	even	4	inner	2300.1.k.c.643.5	yes	24
20.19	odd	2	inner	2300.1.k.c.2207.11	yes	24
23.22	odd	2	CM	2300.1.k.c.2207.8	yes	24
92.91	even	2	inner	2300.1.k.c.2207.2	yes	24
115.22	even	4	inner	2300.1.k.c.643.11	yes	24
115.68	even	4	inner	2300.1.k.c.643.2	✓	24
115.114	odd	2	inner	2300.1.k.c.2207.5	yes	24
460.183	odd	4	inner	2300.1.k.c.643.8	yes	24
460.367	odd	4	inner	2300.1.k.c.643.5	yes	24
460.459	even	2	inner	2300.1.k.c.2207.11	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2300.1.k.c.643.2	✓	24	5.3	odd	4	inner
2300.1.k.c.643.2	✓	24	115.68	even	4	inner
2300.1.k.c.643.5	yes	24	20.7	even	4	inner
2300.1.k.c.643.5	yes	24	460.367	odd	4	inner
2300.1.k.c.643.8	yes	24	20.3	even	4	inner
2300.1.k.c.643.8	yes	24	460.183	odd	4	inner
2300.1.k.c.643.11	yes	24	5.2	odd	4	inner
2300.1.k.c.643.11	yes	24	115.22	even	4	inner
2300.1.k.c.2207.2	yes	24	4.3	odd	2	inner
2300.1.k.c.2207.2	yes	24	92.91	even	2	inner
2300.1.k.c.2207.5	yes	24	5.4	even	2	inner
2300.1.k.c.2207.5	yes	24	115.114	odd	2	inner
2300.1.k.c.2207.8	yes	24	1.1	even	1	trivial
2300.1.k.c.2207.8	yes	24	23.22	odd	2	CM
2300.1.k.c.2207.11	yes	24	20.19	odd	2	inner
2300.1.k.c.2207.11	yes	24	460.459	even	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:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2300.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.422618 - 0.906308i) q^{2} +(-1.08335 - 1.08335i) q^{3} +(-0.642788 - 0.766044i) q^{4} +(-1.43969 + 0.524005i) q^{6} +(-0.965926 + 0.258819i) q^{8} +1.34730i q^{9} +O(q^{10})\)	\(q+(0.422618 - 0.906308i) q^{2} +(-1.08335 - 1.08335i) q^{3} +(-0.642788 - 0.766044i) q^{4} +(-1.43969 + 0.524005i) q^{6} +(-0.965926 + 0.258819i) q^{8} +1.34730i q^{9} +(-0.133530 + 1.52626i) q^{12} +(-1.39273 + 1.39273i) q^{13} +(-0.173648 + 0.984808i) q^{16} +(1.22107 + 0.569392i) q^{18} +(-0.707107 - 0.707107i) q^{23} +(1.32683 + 0.766044i) q^{24} +(0.673648 + 1.85083i) q^{26} +(0.376244 - 0.376244i) q^{27} +1.87939i q^{29} +0.684040i q^{31} +(0.819152 + 0.573576i) q^{32} +(1.03209 - 0.866025i) q^{36} +3.01763 q^{39} -1.53209 q^{41} +(-0.939693 + 0.342020i) q^{46} +(-0.245576 + 0.245576i) q^{47} +(1.25501 - 0.878770i) q^{48} -1.00000i q^{49} +(1.96212 + 0.171663i) q^{52} +(-0.181985 - 0.500000i) q^{54} +(1.70330 + 0.794263i) q^{58} -1.73205 q^{59} +(0.619951 + 0.289088i) q^{62} +(0.866025 - 0.500000i) q^{64} +1.53209i q^{69} -1.96962i q^{71} +(-0.348706 - 1.30139i) q^{72} +(0.483690 - 0.483690i) q^{73} +(1.27530 - 2.73490i) q^{78} +0.532089 q^{81} +(-0.647489 + 1.38854i) q^{82} +(2.03603 - 2.03603i) q^{87} +(-0.0871557 + 0.996195i) q^{92} +(0.741055 - 0.741055i) q^{93} +(0.118782 + 0.326352i) q^{94} +(-0.266044 - 1.50881i) q^{96} +(-0.906308 - 0.422618i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 12 q^{6}+O(q^{10}) \) 24 * q - 12 * q^6	\( 24 q - 12 q^{6} + 12 q^{26} - 12 q^{36} - 24 q^{81} + 12 q^{96}+O(q^{100}) \) 24 * q - 12 * q^6 + 12 * q^26 - 12 * q^36 - 24 * q^81 + 12 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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