LMFDB - Embedded newform 2007.1.v.a.1108.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2007,1,Mod(91,2007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2007, base_ring=CyclotomicField(74))

chi = DirichletCharacter(H, H._module([0, 45]))

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

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

chi := DirichletCharacter("2007.91");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2007 = 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2007.v (of order $74$, degree $36$, 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:	$1.00162348035$
Analytic rank:	$0$
Dimension:	$36$
Coefficient field:	$\Q(\zeta_{74})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{36} - x^{35} + x^{34} - x^{33} + x^{32} - x^{31} + x^{30} - x^{29} + x^{28} - x^{27} + x^{26} + \cdots + 1 $ x^36 - x^35 + x^34 - x^33 + x^32 - x^31 + x^30 - x^29 + x^28 - x^27 + x^26 - x^25 + x^24 - x^23 + x^22 - x^21 + x^20 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{74}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{74} - \cdots)$

Embedding invariants

Embedding label			1108.1
Root			$0.996397 + 0.0848059i$ of defining polynomial
Character	$\chi$	$=$	2007.1108
Dual form			2007.1.v.a.1306.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.127018 - 0.991900i) q^{4} +(-0.871354 + 1.72823i) q^{7} +O(q^{10})$	$q+(0.127018 - 0.991900i) q^{4} +(-0.871354 + 1.72823i) q^{7} +(-0.189272 + 1.10384i) q^{13} +(-0.967733 - 0.251978i) q^{16} +(-1.17216 + 1.58486i) q^{19} +(0.372856 + 0.927889i) q^{25} +(1.60355 + 1.08381i) q^{28} +(0.123383 - 0.572501i) q^{31} +(0.327832 - 0.264706i) q^{37} +(-0.443426 + 1.10351i) q^{43} +(-1.63288 - 2.20780i) q^{49} +(1.07086 + 0.327947i) q^{52} +(0.301810 - 1.76016i) q^{61} +(-0.372856 + 0.927889i) q^{64} +(1.98201 + 0.253807i) q^{67} +(-0.0743874 + 0.242900i) q^{73} +(1.42314 + 1.36397i) q^{76} +(0.157381 + 1.84909i) q^{79} +(-1.74276 - 1.28894i) q^{91} +(0.0949784 + 0.140525i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + q^{4} + 2 q^{7}+O(q^{10}) $ 36 * q + q^4 + 2 * q^7	$ 36 q + q^{4} + 2 q^{7} - q^{16} - 2 q^{19} - q^{25} - 2 q^{28} + 2 q^{31} - 2 q^{37} - 2 q^{43} - 3 q^{49} + q^{64} + 2 q^{73} + 2 q^{76}+O(q^{100}) $ 36 * q + q^4 + 2 * q^7 - q^16 - 2 * q^19 - q^25 - 2 * q^28 + 2 * q^31 - 2 * q^37 - 2 * q^43 - 3 * q^49 + q^64 + 2 * q^73 + 2 * q^76

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2007\mathbb{Z}\right)^\times$.

$n$	$226$	$893$
$\chi(n)$	$e\left(\frac{33}{74}\right)$	$1$

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			0		0.750672	−	0.660675i	$-0.229730\pi$
$2$		0			0		−0.750672	+	0.660675i	$0.770270\pi$
$3$		0			0
$3$		0			0
$4$	0.127018	−	0.991900i	0.127018	−	0.991900i
$5$		0			0		−0.828510	−	0.559975i	$-0.810811\pi$
$5$		0			0		0.828510	+	0.559975i	$0.189189\pi$
$6$		0			0
$7$	−0.871354	+	1.72823i	−0.871354	+	1.72823i	−0.210679	+	0.977555i	$0.567568\pi$
$7$	−0.871354	+	1.72823i	−0.871354	+	1.72823i	−0.660675	+	0.750672i	$0.729730\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.778036	−	0.628220i	$-0.783784\pi$
$11$		0			0		0.778036	+	0.628220i	$0.216216\pi$
$12$		0			0
$13$	−0.189272	+	1.10384i	−0.189272	+	1.10384i	0.721956	+	0.691939i	$0.243243\pi$
$13$	−0.189272	+	1.10384i	−0.189272	+	1.10384i	−0.911228	+	0.411901i	$0.864865\pi$
$14$		0			0
$15$		0			0
$16$	−0.967733	−	0.251978i	−0.967733	−	0.251978i
$17$		0			0		0.628220	−	0.778036i	$-0.283784\pi$
$17$		0			0		−0.628220	+	0.778036i	$0.716216\pi$
$18$		0			0
$19$	−1.17216	+	1.58486i	−1.17216	+	1.58486i	−0.450204	+	0.892926i	$0.648649\pi$
$19$	−1.17216	+	1.58486i	−1.17216	+	1.58486i	−0.721956	+	0.691939i	$0.756757\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.828510	−	0.559975i	$-0.810811\pi$
$23$		0			0		0.828510	+	0.559975i	$0.189189\pi$
$24$		0			0
$25$	0.372856	+	0.927889i	0.372856	+	0.927889i
$26$		0			0
$27$		0			0
$28$	1.60355	+	1.08381i	1.60355	+	1.08381i
$29$		0			0		0.251978	−	0.967733i	$-0.418919\pi$
$29$		0			0		−0.251978	+	0.967733i	$0.581081\pi$
$30$		0			0
$31$	0.123383	−	0.572501i	0.123383	−	0.572501i	−0.873014	−	0.487695i	$-0.837838\pi$
$31$	0.123383	−	0.572501i	0.123383	−	0.572501i	0.996397	−	0.0848059i	$-0.0270270\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.327832	−	0.264706i	0.327832	−	0.264706i	−0.450204	−	0.892926i	$-0.648649\pi$
$37$	0.327832	−	0.264706i	0.327832	−	0.264706i	0.778036	+	0.628220i	$0.216216\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.0848059	−	0.996397i	$-0.527027\pi$
$41$		0			0		0.0848059	+	0.996397i	$0.472973\pi$
$42$		0			0
$43$	−0.443426	+	1.10351i	−0.443426	+	1.10351i	0.524307	+	0.851529i	$0.324324\pi$
$43$	−0.443426	+	1.10351i	−0.443426	+	1.10351i	−0.967733	+	0.251978i	$0.918919\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.803997	−	0.594633i	$-0.202703\pi$
$47$		0			0		−0.803997	+	0.594633i	$0.797297\pi$
$48$		0			0
$49$	−1.63288	−	2.20780i	−1.63288	−	2.20780i
$50$		0			0
$51$		0			0
$52$	1.07086	+	0.327947i	1.07086	+	0.327947i
$53$		0			0		−0.851529	−	0.524307i	$-0.824324\pi$
$53$		0			0		0.851529	+	0.524307i	$0.175676\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.0424412	−	0.999099i	$-0.486486\pi$
$59$		0			0		−0.0424412	+	0.999099i	$0.513514\pi$
$60$		0			0
$61$	0.301810	−	1.76016i	0.301810	−	1.76016i	−0.292823	−	0.956167i	$-0.594595\pi$
$61$	0.301810	−	1.76016i	0.301810	−	1.76016i	0.594633	−	0.803997i	$-0.297297\pi$
$62$		0			0
$63$		0			0
$64$	−0.372856	+	0.927889i	−0.372856	+	0.927889i
$65$		0			0
$66$		0			0
$67$	1.98201	+	0.253807i	1.98201	+	0.253807i	0.996397	+	0.0848059i	$0.0270270\pi$
$67$	1.98201	+	0.253807i	1.98201	+	0.253807i	0.985616	+	0.169001i	$0.0540541\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.0424412	−	0.999099i	$-0.486486\pi$
$71$		0			0		−0.0424412	+	0.999099i	$0.513514\pi$
$72$		0			0
$73$	−0.0743874	+	0.242900i	−0.0743874	+	0.242900i	−0.985616	−	0.169001i	$-0.945946\pi$
$73$	−0.0743874	+	0.242900i	−0.0743874	+	0.242900i	0.911228	+	0.411901i	$0.135135\pi$
$74$		0			0
$75$		0			0
$76$	1.42314	+	1.36397i	1.42314	+	1.36397i
$77$		0			0
$78$		0			0
$79$	0.157381	+	1.84909i	0.157381	+	1.84909i	0.450204	+	0.892926i	$0.351351\pi$
$79$	0.157381	+	1.84909i	0.157381	+	1.84909i	−0.292823	+	0.956167i	$0.594595\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.927889	−	0.372856i	$-0.121622\pi$
$83$		0			0		−0.927889	+	0.372856i	$0.878378\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.559975	−	0.828510i	$-0.689189\pi$
$89$		0			0		0.559975	+	0.828510i	$0.310811\pi$
$90$		0			0
$91$	−1.74276	−	1.28894i	−1.74276	−	1.28894i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.0949784	+	0.140525i	0.0949784	+	0.140525i	0.873014	−	0.487695i	$-0.162162\pi$
$97$	0.0949784	+	0.140525i	0.0949784	+	0.140525i	−0.778036	+	0.628220i	$0.783784\pi$
$98$		0			0
$99$		0			0
$100$	0.967733	−	0.251978i	0.967733	−	0.251978i
$101$		0			0		0.851529	−	0.524307i	$-0.175676\pi$
$101$		0			0		−0.851529	+	0.524307i	$0.824324\pi$
$102$		0			0
$103$	−0.287818	−	1.67856i	−0.287818	−	1.67856i	−0.660675	−	0.750672i	$-0.729730\pi$
$103$	−0.287818	−	1.67856i	−0.287818	−	1.67856i	0.372856	−	0.927889i	$-0.378378\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.372856	−	0.927889i	$-0.378378\pi$
$107$		0			0		−0.372856	+	0.927889i	$0.621622\pi$
$108$		0			0
$109$	1.39732	−	0.363834i	1.39732	−	0.363834i	0.524307	−	0.851529i	$-0.324324\pi$
$109$	1.39732	−	0.363834i	1.39732	−	0.363834i	0.873014	+	0.487695i	$0.162162\pi$
$110$		0			0
$111$		0			0
$112$	1.27871	−	1.45290i	1.27871	−	1.45290i
$113$		0			0		0.524307	−	0.851529i	$-0.324324\pi$
$113$		0			0		−0.524307	+	0.851529i	$0.675676\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.210679	+	0.977555i	0.210679	+	0.977555i
$122$		0			0
$123$		0			0
$124$	−0.552192	−	0.195102i	−0.552192	−	0.195102i
$125$		0			0
$126$		0			0
$127$	0.925292	+	1.25108i	0.925292	+	1.25108i	0.967733	+	0.251978i	$0.0810811\pi$
$127$	0.925292	+	1.25108i	0.925292	+	1.25108i	−0.0424412	+	0.999099i	$0.513514\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.750672	−	0.660675i	$-0.770270\pi$
$131$		0			0		0.750672	+	0.660675i	$0.229730\pi$
$132$		0			0
$133$	−1.71764	−	3.40674i	−1.71764	−	3.40674i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.942877	−	0.333140i	$-0.108108\pi$
$137$		0			0		−0.942877	+	0.333140i	$0.891892\pi$
$138$		0			0
$139$	−1.71835	−	0.776745i	−1.71835	−	0.776745i	−0.996397	−	0.0848059i	$-0.972973\pi$
$139$	−1.71835	−	0.776745i	−1.71835	−	0.776745i	−0.721956	−	0.691939i	$-0.756757\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−0.220921	−	0.358799i	−0.220921	−	0.358799i
$149$		0			0		−0.985616	−	0.169001i	$-0.945946\pi$
$149$		0			0		0.985616	+	0.169001i	$0.0540541\pi$
$150$		0			0
$151$	−0.662335	−	0.489860i	−0.662335	−	0.489860i	0.210679	−	0.977555i	$-0.432432\pi$
$151$	−0.662335	−	0.489860i	−0.662335	−	0.489860i	−0.873014	+	0.487695i	$0.837838\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.35282	+	0.291554i	−1.35282	+	0.291554i	−0.828510	−	0.559975i	$-0.810811\pi$
$157$	−1.35282	+	0.291554i	−1.35282	+	0.291554i	−0.524307	+	0.851529i	$0.675676\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.06990	+	0.658761i	1.06990	+	0.658761i	0.942877	−	0.333140i	$-0.108108\pi$
$163$	1.06990	+	0.658761i	1.06990	+	0.658761i	0.127018	+	0.991900i	$0.459459\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.127018	−	0.991900i	$-0.540541\pi$
$167$		0			0		0.127018	+	0.991900i	$0.459459\pi$
$168$		0			0
$169$	−0.239762	−	0.0847131i	−0.239762	−	0.0847131i
$170$		0			0
$171$		0			0
$172$	1.03825	+	0.579999i	1.03825	+	0.579999i
$173$		0			0		0.450204	−	0.892926i	$-0.351351\pi$
$173$		0			0		−0.450204	+	0.892926i	$0.648649\pi$
$174$		0			0
$175$	−1.92849	−	0.164139i	−1.92849	−	0.164139i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.691939	−	0.721956i	$-0.743243\pi$
$179$		0			0		0.691939	+	0.721956i	$0.256757\pi$
$180$		0			0
$181$	−0.955527	+	0.431926i	−0.955527	+	0.431926i	−0.828510	−	0.559975i	$-0.810811\pi$
$181$	−0.955527	+	0.431926i	−0.955527	+	0.431926i	−0.127018	+	0.991900i	$0.540541\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.450204	−	0.892926i	$-0.648649\pi$
$191$		0			0		0.450204	+	0.892926i	$0.351351\pi$
$192$		0			0
$193$	1.89684	−	0.580902i	1.89684	−	0.580902i	0.911228	−	0.411901i	$-0.135135\pi$
$193$	1.89684	−	0.580902i	1.89684	−	0.580902i	0.985616	−	0.169001i	$-0.0540541\pi$
$194$		0			0
$195$		0			0
$196$	−2.39732	+	1.33922i	−2.39732	+	1.33922i
$197$		0			0		0.851529	−	0.524307i	$-0.175676\pi$
$197$		0			0		−0.851529	+	0.524307i	$0.824324\pi$
$198$		0			0
$199$	−0.953956	+	0.914293i	−0.953956	+	0.914293i	−0.996397	−	0.0848059i	$-0.972973\pi$
$199$	−0.953956	+	0.914293i	−0.953956	+	0.914293i	0.0424412	+	0.999099i	$0.486486\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.461309	−	1.02053i	0.461309	−	1.02053i
$209$		0			0
$210$		0			0
$211$	−0.0560797	−	1.32016i	−0.0560797	−	1.32016i	−0.778036	−	0.628220i	$-0.783784\pi$
$211$	−0.0560797	−	1.32016i	−0.0560797	−	1.32016i	0.721956	−	0.691939i	$-0.243243\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.881901	+	0.712085i	0.881901	+	0.712085i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.372856	−	0.927889i	−0.372856	−	0.927889i
$223$	−0.372856	−	0.927889i	−0.372856	−	0.927889i
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.991900	−	0.127018i	$-0.959459\pi$
$227$		0			0		0.991900	+	0.127018i	$0.0405405\pi$
$228$		0			0
$229$	−0.789321	+	0.977555i	−0.789321	+	0.977555i	0.210679	+	0.977555i	$0.432432\pi$
$229$	−0.789321	+	0.977555i	−0.789321	+	0.977555i	−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.127018	−	0.991900i	$-0.540541\pi$
$233$		0			0		0.127018	+	0.991900i	$0.459459\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.251978	−	0.967733i	$-0.418919\pi$
$239$		0			0		−0.251978	+	0.967733i	$0.581081\pi$
$240$		0			0
$241$	0.786068	+	0.439124i	0.786068	+	0.439124i	0.828510	−	0.559975i	$-0.189189\pi$
$241$	0.786068	+	0.439124i	0.786068	+	0.439124i	−0.0424412	+	0.999099i	$0.513514\pi$
$242$		0			0
$243$		0			0
$244$	−1.70757	−	0.522938i	−1.70757	−	0.522938i
$245$		0			0
$246$		0			0
$247$	−1.52758	−	1.59385i	−1.52758	−	1.59385i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.559975	−	0.828510i	$-0.310811\pi$
$251$		0			0		−0.559975	+	0.828510i	$0.689189\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.873014	+	0.487695i	0.873014	+	0.487695i
$257$		0			0		0.999099	−	0.0424412i	$-0.0135135\pi$
$257$		0			0		−0.999099	+	0.0424412i	$0.986486\pi$
$258$		0			0
$259$	0.171814	+	0.797221i	0.171814	+	0.797221i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.503502	−	1.93372i	0.503502	−	1.93372i
$269$		0			0		−0.127018	−	0.991900i	$-0.540541\pi$
$269$		0			0		0.127018	+	0.991900i	$0.459459\pi$
$270$		0			0
$271$	0.168238	−	1.97665i	0.168238	−	1.97665i	−0.0424412	−	0.999099i	$-0.513514\pi$
$271$	0.168238	−	1.97665i	0.168238	−	1.97665i	0.210679	−	0.977555i	$-0.432432\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.324941	−	0.919673i	0.324941	−	0.919673i	−0.660675	−	0.750672i	$-0.729730\pi$
$277$	0.324941	−	0.919673i	0.324941	−	0.919673i	0.985616	−	0.169001i	$-0.0540541\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.892926	−	0.450204i	$-0.148649\pi$
$281$		0			0		−0.892926	+	0.450204i	$0.851351\pi$
$282$		0			0
$283$	−0.955527	+	1.55188i	−0.955527	+	1.55188i	−0.127018	+	0.991900i	$0.540541\pi$
$283$	−0.955527	+	1.55188i	−0.955527	+	1.55188i	−0.828510	+	0.559975i	$0.810811\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.210679	−	0.977555i	−0.210679	−	0.977555i
$290$		0			0
$291$		0			0
$292$	0.231485	+	0.104638i	0.231485	+	0.104638i
$293$		0			0		−0.942877	−	0.333140i	$-0.891892\pi$
$293$		0			0		0.942877	+	0.333140i	$0.108108\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−1.52073	−	1.72789i	−1.52073	−	1.72789i
$302$		0			0
$303$		0			0
$304$	1.53369	−	1.23837i	1.53369	−	1.23837i
$305$		0			0
$306$		0			0
$307$	−0.401764	+	0.888803i	−0.401764	+	0.888803i	0.594633	+	0.803997i	$0.297297\pi$
$307$	−0.401764	+	0.888803i	−0.401764	+	0.888803i	−0.996397	+	0.0848059i	$0.972973\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.996397	−	0.0848059i	$-0.972973\pi$
$311$		0			0		0.996397	+	0.0848059i	$0.0270270\pi$
$312$		0			0
$313$	1.43582	−	0.723925i	1.43582	−	0.723925i	0.450204	−	0.892926i	$-0.351351\pi$
$313$	1.43582	−	0.723925i	1.43582	−	0.723925i	0.985616	+	0.169001i	$0.0540541\pi$
$314$		0			0
$315$		0			0
$316$	1.85411	+	0.0787615i	1.85411	+	0.0787615i
$317$		0			0		−0.977555	−	0.210679i	$-0.932432\pi$
$317$		0			0		0.977555	+	0.210679i	$0.0675676\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−1.09481	+	0.235950i	−1.09481	+	0.235950i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.49204	−	0.599551i	1.49204	−	0.599551i	0.524307	−	0.851529i	$-0.324324\pi$
$331$	1.49204	−	0.599551i	1.49204	−	0.599551i	0.967733	+	0.251978i	$0.0810811\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.492645	+	1.89202i	0.492645	+	1.89202i	0.450204	+	0.892926i	$0.351351\pi$
$337$	0.492645	+	1.89202i	0.492645	+	1.89202i	0.0424412	+	0.999099i	$0.486486\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	3.33077	−	0.571118i	3.33077	−	0.571118i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.333140	−	0.942877i	$-0.391892\pi$
$347$		0			0		−0.333140	+	0.942877i	$0.608108\pi$
$348$		0			0
$349$	1.65105	−	1.11591i	1.65105	−	1.11591i	0.778036	−	0.628220i	$-0.216216\pi$
$349$	1.65105	−	1.11591i	1.65105	−	1.11591i	0.873014	−	0.487695i	$-0.162162\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.999099	−	0.0424412i	$-0.986486\pi$
$353$		0			0		0.999099	+	0.0424412i	$0.0135135\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.0848059	−	0.996397i	$-0.472973\pi$
$359$		0			0		−0.0848059	+	0.996397i	$0.527027\pi$
$360$		0			0
$361$	−0.845015	−	2.75926i	−0.845015	−	2.75926i
$362$		0			0
$363$		0			0
$364$	−1.49986	+	1.56493i	−1.49986	+	1.56493i
$365$		0			0
$366$		0			0
$367$	1.55047	−	0.131964i	1.55047	−	0.131964i	0.721956	−	0.691939i	$-0.243243\pi$
$367$	1.55047	−	0.131964i	1.55047	−	0.131964i	0.828510	+	0.559975i	$0.189189\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1.62841	−	0.498694i	−1.62841	−	0.498694i	−0.660675	−	0.750672i	$-0.729730\pi$
$373$	−1.62841	−	0.498694i	−1.62841	−	0.498694i	−0.967733	+	0.251978i	$0.918919\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.231485	+	1.80770i	−0.231485	+	1.80770i	0.292823	+	0.956167i	$0.405405\pi$
$379$	−0.231485	+	1.80770i	−0.231485	+	1.80770i	−0.524307	+	0.851529i	$0.675676\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.127018	−	0.991900i	$-0.459459\pi$
$383$		0			0		−0.127018	+	0.991900i	$0.540541\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.151451	−	0.0763599i	0.151451	−	0.0763599i
$389$		0			0		−0.892926	−	0.450204i	$-0.851351\pi$
$389$		0			0		0.892926	+	0.450204i	$0.148649\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.662335	−	0.489860i	0.662335	−	0.489860i	−0.210679	−	0.977555i	$-0.567568\pi$
$397$	0.662335	−	0.489860i	0.662335	−	0.489860i	0.873014	+	0.487695i	$0.162162\pi$
$398$		0			0
$399$		0			0
$400$	−0.127018	−	0.991900i	−0.127018	−	0.991900i
$401$		0			0		0.251978	−	0.967733i	$-0.418919\pi$
$401$		0			0		−0.251978	+	0.967733i	$0.581081\pi$
$402$		0			0
$403$	0.608596	+	0.244554i	0.608596	+	0.244554i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.06990	−	1.32504i	−1.06990	−	1.32504i	−0.942877	−	0.333140i	$-0.891892\pi$
$409$	−1.06990	−	1.32504i	−1.06990	−	1.32504i	−0.127018	−	0.991900i	$-0.540541\pi$
$410$		0			0
$411$		0			0
$412$	−1.70152	+	0.0722798i	−1.70152	+	0.0722798i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.487695	−	0.873014i	$-0.662162\pi$
$419$		0			0		0.487695	+	0.873014i	$0.337838\pi$
$420$		0			0
$421$	−0.157381	+	0.0632409i	−0.157381	+	0.0632409i	−0.450204	−	0.892926i	$-0.648649\pi$
$421$	−0.157381	+	0.0632409i	−0.157381	+	0.0632409i	0.292823	+	0.956167i	$0.405405\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	2.77898	+	2.05532i	2.77898	+	2.05532i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.911228	−	0.411901i	$-0.864865\pi$
$431$		0			0		0.911228	+	0.411901i	$0.135135\pi$
$432$		0			0
$433$	1.42314	+	0.244022i	1.42314	+	0.244022i	0.828510	−	0.559975i	$-0.189189\pi$
$433$	1.42314	+	0.244022i	1.42314	+	0.244022i	0.594633	+	0.803997i	$0.297297\pi$
$434$		0			0
$435$		0			0
$436$	−0.183403	−	1.43222i	−0.183403	−	1.43222i
$437$		0			0
$438$		0			0
$439$	1.34059	+	0.675911i	1.34059	+	0.675911i	0.967733	−	0.251978i	$-0.0810811\pi$
$439$	1.34059	+	0.675911i	1.34059	+	0.675911i	0.372856	+	0.927889i	$0.378378\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.169001	−	0.985616i	$-0.554054\pi$
$443$		0			0		0.169001	+	0.985616i	$0.445946\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−1.27871	−	1.45290i	−1.27871	−	1.45290i
$449$		0			0		0.985616	−	0.169001i	$-0.0540541\pi$
$449$		0			0		−0.985616	+	0.169001i	$0.945946\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.418570	−	0.518389i	0.418570	−	0.518389i	−0.524307	−	0.851529i	$-0.675676\pi$
$457$	0.418570	−	0.518389i	0.418570	−	0.518389i	0.942877	+	0.333140i	$0.108108\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.411901	−	0.911228i	$-0.364865\pi$
$461$		0			0		−0.411901	+	0.911228i	$0.635135\pi$
$462$		0			0
$463$	1.46718	+	1.18467i	1.46718	+	1.18467i	0.942877	+	0.333140i	$0.108108\pi$
$463$	1.46718	+	1.18467i	1.46718	+	1.18467i	0.524307	+	0.851529i	$0.324324\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.292823	−	0.956167i	$-0.405405\pi$
$467$		0			0		−0.292823	+	0.956167i	$0.594595\pi$
$468$		0			0
$469$	−2.16567	+	3.20421i	−2.16567	+	3.20421i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.90763	−	0.496707i	−1.90763	−	0.496707i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.487695	−	0.873014i	$-0.337838\pi$
$479$		0			0		−0.487695	+	0.873014i	$0.662162\pi$
$480$		0			0
$481$	0.230143	+	0.411976i	0.230143	+	0.411976i
$482$		0			0
$483$		0			0
$484$	0.996397	−	0.0848059i	0.996397	−	0.0848059i
$485$		0			0
$486$		0			0
$487$	−1.04484	+	0.0889287i	−1.04484	+	0.0889287i	−0.594633	−	0.803997i	$-0.702703\pi$
$487$	−1.04484	+	0.0889287i	−1.04484	+	0.0889287i	−0.450204	+	0.892926i	$0.648649\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.967733	−	0.251978i	$-0.918919\pi$
$491$		0			0		0.967733	+	0.251978i	$0.0810811\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.263660	+	0.522938i	−0.263660	+	0.522938i
$497$		0			0
$498$		0			0
$499$	−1.04484	+	1.69692i	−1.04484	+	1.69692i	−0.450204	+	0.892926i	$0.648649\pi$
$499$	−1.04484	+	1.69692i	−1.04484	+	1.69692i	−0.594633	+	0.803997i	$0.702703\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.450204	−	0.892926i	$-0.351351\pi$
$503$		0			0		−0.450204	+	0.892926i	$0.648649\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	1.35847	−	0.758888i	1.35847	−	0.758888i
$509$		0			0		−0.991900	−	0.127018i	$-0.959459\pi$
$509$		0			0		0.991900	+	0.127018i	$0.0405405\pi$
$510$		0			0
$511$	−0.354969	−	0.340211i	−0.354969	−	0.340211i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.524307	−	0.851529i	$-0.675676\pi$
$521$		0			0		0.524307	+	0.851529i	$0.324324\pi$
$522$		0			0
$523$	0.787693	+	1.74257i	0.787693	+	1.74257i	0.660675	+	0.750672i	$0.270270\pi$
$523$	0.787693	+	1.74257i	0.787693	+	1.74257i	0.127018	+	0.991900i	$0.459459\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.372856	+	0.927889i	0.372856	+	0.927889i
$530$		0			0
$531$		0			0
$532$	−3.59732	+	1.27101i	−3.59732	+	1.27101i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.324941	+	0.919673i	0.324941	+	0.919673i	0.985616	+	0.169001i	$0.0540541\pi$
$541$	0.324941	+	0.919673i	0.324941	+	0.919673i	−0.660675	+	0.750672i	$0.729730\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.692793	−	1.12517i	−0.692793	−	1.12517i	−0.985616	−	0.169001i	$-0.945946\pi$
$547$	−0.692793	−	1.12517i	−0.692793	−	1.12517i	0.292823	−	0.956167i	$-0.405405\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−3.33279	−	1.33922i	−3.33279	−	1.33922i
$554$		0			0
$555$		0			0
$556$	−0.988715	+	1.60578i	−0.988715	+	1.60578i
$557$		0			0		0.660675	−	0.750672i	$-0.270270\pi$
$557$		0			0		−0.660675	+	0.750672i	$0.729730\pi$
$558$		0			0
$559$	−1.13417	−	0.698334i	−1.13417	−	0.698334i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.985616	−	0.169001i	$-0.0540541\pi$
$563$		0			0		−0.985616	+	0.169001i	$0.945946\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.967733	−	0.251978i	$-0.0810811\pi$
$569$		0			0		−0.967733	+	0.251978i	$0.918919\pi$
$570$		0			0
$571$	−0.165805	−	0.0357337i	−0.165805	−	0.0357337i	0.127018	−	0.991900i	$-0.459459\pi$
$571$	−0.165805	−	0.0357337i	−0.165805	−	0.0357337i	−0.292823	+	0.956167i	$0.594595\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−0.0560797	+	0.0637189i	−0.0560797	+	0.0637189i	−0.778036	−	0.628220i	$-0.783784\pi$
$577$	−0.0560797	+	0.0637189i	−0.0560797	+	0.0637189i	0.721956	+	0.691939i	$0.243243\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.721956	−	0.691939i	$-0.756757\pi$
$587$		0			0		0.721956	+	0.691939i	$0.243243\pi$
$588$		0			0
$589$	0.762711	+	0.866608i	0.762711	+	0.866608i
$590$		0			0
$591$		0			0
$592$	−0.383954	+	0.173558i	−0.383954	+	0.173558i
$593$		0			0		−0.721956	−	0.691939i	$-0.756757\pi$
$593$		0			0		0.721956	+	0.691939i	$0.243243\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.333140	−	0.942877i	$-0.391892\pi$
$599$		0			0		−0.333140	+	0.942877i	$0.608108\pi$
$600$		0			0
$601$	1.38263	+	1.44261i	1.38263	+	1.44261i	0.721956	+	0.691939i	$0.243243\pi$
$601$	1.38263	+	1.44261i	1.38263	+	1.44261i	0.660675	+	0.750672i	$0.270270\pi$
$602$		0			0
$603$		0			0
$604$	−0.570021	+	0.594749i	−0.570021	+	0.594749i
$605$		0			0
$606$		0			0
$607$	−0.784211	−	1.40380i	−0.784211	−	1.40380i	−0.911228	−	0.411901i	$-0.864865\pi$
$607$	−0.784211	−	1.40380i	−0.784211	−	1.40380i	0.127018	−	0.991900i	$-0.459459\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.84075	+	0.235717i	−1.84075	+	0.235717i	−0.967733	−	0.251978i	$-0.918919\pi$
$613$	−1.84075	+	0.235717i	−1.84075	+	0.235717i	−0.873014	+	0.487695i	$0.837838\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.956167	−	0.292823i	$-0.905405\pi$
$617$		0			0		0.956167	+	0.292823i	$0.0945946\pi$
$618$		0			0
$619$	1.59463	+	0.803997i	1.59463	+	0.803997i	1.00000			$0$
$619$	1.59463	+	0.803997i	1.59463	+	0.803997i	0.594633	+	0.803997i	$0.297297\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.721956	+	0.691939i	−0.721956	+	0.691939i
$626$		0			0
$627$		0			0
$628$	0.117361	+	1.37889i	0.117361	+	1.37889i
$629$		0			0
$630$		0			0
$631$	0.0827188	+	0.971876i	0.0827188	+	0.971876i	0.911228	+	0.411901i	$0.135135\pi$
$631$	0.0827188	+	0.971876i	0.0827188	+	0.971876i	−0.828510	+	0.559975i	$0.810811\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	2.74612	−	1.38456i	2.74612	−	1.38456i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.828510	−	0.559975i	$-0.810811\pi$
$641$		0			0		0.828510	+	0.559975i	$0.189189\pi$
$642$		0			0
$643$	0.472090	+	0.766723i	0.472090	+	0.766723i	0.996397	−	0.0848059i	$-0.0270270\pi$
$643$	0.472090	+	0.766723i	0.472090	+	0.766723i	−0.524307	+	0.851529i	$0.675676\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.999099	−	0.0424412i	$-0.0135135\pi$
$647$		0			0		−0.999099	+	0.0424412i	$0.986486\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.789321	−	0.977555i	0.789321	−	0.977555i
$653$		0			0		−0.967733	−	0.251978i	$-0.918919\pi$
$653$		0			0		0.967733	+	0.251978i	$0.0810811\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.991900	−	0.127018i	$-0.0405405\pi$
$659$		0			0		−0.991900	+	0.127018i	$0.959459\pi$
$660$		0			0
$661$	−0.189272	−	0.280038i	−0.189272	−	0.280038i	0.721956	−	0.691939i	$-0.243243\pi$
$661$	−0.189272	−	0.280038i	−0.189272	−	0.280038i	−0.911228	+	0.411901i	$0.864865\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.253120	+	1.97665i	−0.253120	+	1.97665i	−0.0424412	+	0.999099i	$0.513514\pi$
$673$	−0.253120	+	1.97665i	−0.253120	+	1.97665i	−0.210679	+	0.977555i	$0.567568\pi$
$674$		0			0
$675$		0			0
$676$	−0.114481	+	0.227059i	−0.114481	+	0.227059i
$677$		0			0		−0.559975	−	0.828510i	$-0.689189\pi$
$677$		0			0		0.559975	+	0.828510i	$0.310811\pi$
$678$		0			0
$679$	−0.325619	+	0.0416971i	−0.325619	+	0.0416971i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.251978	−	0.967733i	$-0.581081\pi$
$683$		0			0		0.251978	+	0.967733i	$0.418919\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	0.707177	−	0.956167i	0.707177	−	0.956167i
$689$		0			0
$690$		0			0
$691$	1.85411	−	0.0787615i	1.85411	−	0.0787615i	0.911228	−	0.411901i	$-0.135135\pi$
$691$	1.85411	−	0.0787615i	1.85411	−	0.0787615i	0.942877	+	0.333140i	$0.108108\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.407763	+	1.89202i	−0.407763	+	1.89202i
$701$		0			0		0.892926	−	0.450204i	$-0.148649\pi$
$701$		0			0		−0.892926	+	0.450204i	$0.851351\pi$
$702$		0			0
$703$	0.0352515	+	0.829847i	0.0352515	+	0.829847i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	1.98380i		−	1.98380i	−0.127018	−	0.991900i	$-0.540541\pi$
$709$		−	1.98380i		−	1.98380i	0.127018	−	0.991900i	$-0.459459\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.892926	−	0.450204i	$-0.851351\pi$
$719$		0			0		0.892926	+	0.450204i	$0.148649\pi$
$720$		0			0
$721$	3.15173	+	0.965206i	3.15173	+	0.965206i
$722$		0			0
$723$		0			0
$724$	0.307058	+	1.00265i	0.307058	+	1.00265i
$725$		0			0
$726$		0			0
$727$	−0.848974	−	1.68384i	−0.848974	−	1.68384i	−0.721956	−	0.691939i	$-0.756757\pi$
$727$	−0.848974	−	1.68384i	−0.848974	−	1.68384i	−0.127018	−	0.991900i	$-0.540541\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	0.492674	−	1.22607i	0.492674	−	1.22607i	−0.450204	−	0.892926i	$-0.648649\pi$
$733$	0.492674	−	1.22607i	0.492674	−	1.22607i	0.942877	−	0.333140i	$-0.108108\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.651325	−	1.84343i	0.651325	−	1.84343i	0.127018	−	0.991900i	$-0.459459\pi$
$739$	0.651325	−	1.84343i	0.651325	−	1.84343i	0.524307	−	0.851529i	$-0.324324\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.169001	−	0.985616i	$-0.445946\pi$
$743$		0			0		−0.169001	+	0.985616i	$0.554054\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.0612814	−	0.0587334i	−0.0612814	−	0.0587334i	0.660675	−	0.750672i	$-0.270270\pi$
$751$	−0.0612814	−	0.0587334i	−0.0612814	−	0.0587334i	−0.721956	+	0.691939i	$0.756757\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.0286645	−	0.336784i	0.0286645	−	0.336784i	−0.967733	−	0.251978i	$-0.918919\pi$
$757$	0.0286645	−	0.336784i	0.0286645	−	0.336784i	0.996397	−	0.0848059i	$-0.0270270\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.660675	−	0.750672i	$-0.270270\pi$
$761$		0			0		−0.660675	+	0.750672i	$0.729730\pi$
$762$		0			0
$763$	−0.588773	+	2.73192i	−0.588773	+	2.73192i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.27871	−	0.332951i	1.27871	−	0.332951i	0.450204	−	0.892926i	$-0.351351\pi$
$769$	1.27871	−	0.332951i	1.27871	−	0.332951i	0.828510	+	0.559975i	$0.189189\pi$
$770$		0			0
$771$		0			0
$772$	−0.335264	−	1.95527i	−0.335264	−	1.95527i
$773$		0			0		0.911228	−	0.411901i	$-0.135135\pi$
$773$		0			0		−0.911228	+	0.411901i	$0.864865\pi$
$774$		0			0
$775$	0.577222	−	0.0989746i	0.577222	−	0.0989746i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.02387	+	2.54801i	1.02387	+	2.54801i
$785$		0			0
$786$		0			0
$787$	1.24626	+	1.54347i	1.24626	+	1.54347i	0.721956	+	0.691939i	$0.243243\pi$
$787$	1.24626	+	1.54347i	1.24626	+	1.54347i	0.524307	+	0.851529i	$0.324324\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.88582	+	0.666301i	1.88582	+	0.666301i
$794$		0			0
$795$		0			0
$796$	0.785718	+	1.06236i	0.785718	+	1.06236i
$797$		0			0		−0.333140	−	0.942877i	$-0.608108\pi$
$797$		0			0		0.333140	+	0.942877i	$0.391892\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.372856	−	0.927889i	$-0.621622\pi$
$809$		0			0		0.372856	+	0.927889i	$0.378378\pi$
$810$		0			0
$811$	0.212339	+	0.262977i	0.212339	+	0.262977i	0.873014	−	0.487695i	$-0.162162\pi$
$811$	0.212339	+	0.262977i	0.212339	+	0.262977i	−0.660675	+	0.750672i	$0.729730\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.22914	−	1.99626i	−1.22914	−	1.99626i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.999099	−	0.0424412i	$-0.986486\pi$
$821$		0			0		0.999099	+	0.0424412i	$0.0135135\pi$
$822$		0			0
$823$	0.805312	−	1.78155i	0.805312	−	1.78155i	0.210679	−	0.977555i	$-0.432432\pi$
$823$	0.805312	−	1.78155i	0.805312	−	1.78155i	0.594633	−	0.803997i	$-0.297297\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.721956	−	0.691939i	$-0.756757\pi$
$827$		0			0		0.721956	+	0.691939i	$0.243243\pi$
$828$		0			0
$829$	−1.89684	−	0.242900i	−1.89684	−	0.242900i	−0.911228	−	0.411901i	$-0.864865\pi$
$829$	−1.89684	−	0.242900i	−1.89684	−	0.242900i	−0.985616	+	0.169001i	$0.945946\pi$
$830$		0			0
$831$		0			0
$832$	−0.953670	−	0.587198i	−0.953670	−	0.587198i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.524307	−	0.851529i	$-0.324324\pi$
$839$		0			0		−0.524307	+	0.851529i	$0.675676\pi$
$840$		0			0
$841$	−0.873014	−	0.487695i	−0.873014	−	0.487695i
$842$		0			0
$843$		0			0
$844$	−1.31659	−	0.112058i	−1.31659	−	0.112058i
$845$		0			0
$846$		0			0
$847$	−1.87301	−	0.487695i	−1.87301	−	0.487695i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$			0.823802i			0.823802i	0.911228	+	0.411901i	$0.135135\pi$
$853$			0.823802i			0.823802i	−0.911228	+	0.411901i	$0.864865\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.487695	−	0.873014i	$-0.662162\pi$
$857$		0			0		0.487695	+	0.873014i	$0.337838\pi$
$858$		0			0
$859$	−0.612759	+	1.09689i	−0.612759	+	1.09689i	0.372856	+	0.927889i	$0.378378\pi$
$859$	−0.612759	+	1.09689i	−0.612759	+	1.09689i	−0.985616	+	0.169001i	$0.945946\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.967733	−	0.251978i	$-0.918919\pi$
$863$		0			0		0.967733	+	0.251978i	$0.0810811\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	0.818335	−	0.784311i	0.818335	−	0.784311i
$869$		0			0
$870$		0			0
$871$	−0.655302	+	2.13979i	−0.655302	+	2.13979i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.823060	+	1.82081i	−0.823060	+	1.82081i	−0.372856	+	0.927889i	$0.621622\pi$
$877$	−0.823060	+	1.82081i	−0.823060	+	1.82081i	−0.450204	+	0.892926i	$0.648649\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.628220	−	0.778036i	$-0.283784\pi$
$881$		0			0		−0.628220	+	0.778036i	$0.716216\pi$
$882$		0			0
$883$	−0.221964	−	0.628220i	−0.221964	−	0.628220i	0.778036	−	0.628220i	$-0.216216\pi$
$883$	−0.221964	−	0.628220i	−0.221964	−	0.628220i	−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.559975	−	0.828510i	$-0.310811\pi$
$887$		0			0		−0.559975	+	0.828510i	$0.689189\pi$
$888$		0			0
$889$	−2.96840	+	0.508984i	−2.96840	+	0.508984i
$890$		0			0
$891$		0			0
$892$	−0.967733	+	0.251978i	−0.967733	+	0.251978i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.679515	−	0.307160i	−0.679515	−	0.307160i	0.0424412	−	0.999099i	$-0.486486\pi$
$907$	−0.679515	−	0.307160i	−0.679515	−	0.307160i	−0.721956	+	0.691939i	$0.756757\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.803997	−	0.594633i	$-0.797297\pi$
$911$		0			0		0.803997	+	0.594633i	$0.202703\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.869380	+	0.907095i	0.869380	+	0.907095i
$917$		0			0
$918$		0			0
$919$	0.830573	+	1.48679i	0.830573	+	1.48679i	0.873014	+	0.487695i	$0.162162\pi$
$919$	0.830573	+	1.48679i	0.830573	+	1.48679i	−0.0424412	+	0.999099i	$0.513514\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.367852	+	0.205494i	0.367852	+	0.205494i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.628220	−	0.778036i	$-0.716216\pi$
$929$		0			0		0.628220	+	0.778036i	$0.283784\pi$
$930$		0			0
$931$	5.41306			5.41306
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.0427385	+	0.164139i	−0.0427385	+	0.164139i	−0.985616	−	0.169001i	$-0.945946\pi$
$937$	−0.0427385	+	0.164139i	−0.0427385	+	0.164139i	0.942877	+	0.333140i	$0.108108\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.803997	−	0.594633i	$-0.202703\pi$
$941$		0			0		−0.803997	+	0.594633i	$0.797297\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.956167	−	0.292823i	$-0.0945946\pi$
$947$		0			0		−0.956167	+	0.292823i	$0.905405\pi$
$948$		0			0
$949$	−0.254044	−	0.128086i	−0.254044	−	0.128086i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.985616	−	0.169001i	$-0.945946\pi$
$953$		0			0		0.985616	+	0.169001i	$0.0540541\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.598695	+	0.270627i	0.598695	+	0.270627i
$962$		0			0
$963$		0			0
$964$	0.535412	−	0.723925i	0.535412	−	0.723925i
$965$		0			0
$966$		0			0
$967$	0.287818	−	0.177217i	0.287818	−	0.177217i	−0.372856	−	0.927889i	$-0.621622\pi$
$967$	0.287818	−	0.177217i	0.287818	−	0.177217i	0.660675	+	0.750672i	$0.270270\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.996397	−	0.0848059i	$-0.0270270\pi$
$971$		0			0		−0.996397	+	0.0848059i	$0.972973\pi$
$972$		0			0
$973$	2.83969	−	2.29289i	2.83969	−	2.29289i
$974$		0			0
$975$		0			0
$976$	−0.735595	+	1.62732i	−0.735595	+	1.62732i
$977$		0			0		−0.292823	−	0.956167i	$-0.594595\pi$
$977$		0			0		0.292823	+	0.956167i	$0.405405\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.594633	−	0.803997i	$-0.297297\pi$
$983$		0			0		−0.594633	+	0.803997i	$0.702703\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−1.77497	+	1.31276i	−1.77497	+	1.31276i
$989$		0			0
$990$		0			0
$991$	0.461025	−	1.30483i	0.461025	−	1.30483i	−0.450204	−	0.892926i	$-0.648649\pi$
$991$	0.461025	−	1.30483i	0.461025	−	1.30483i	0.911228	−	0.411901i	$-0.135135\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.35847	+	1.09689i	−1.35847	+	1.09689i	−0.372856	+	0.927889i	$0.621622\pi$
$997$	−1.35847	+	1.09689i	−1.35847	+	1.09689i	−0.985616	+	0.169001i	$0.945946\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2007.1.v.a.1108.1	✓	36
3.2	odd	2	CM	2007.1.v.a.1108.1	✓	36
223.191	odd	74	inner	2007.1.v.a.1306.1	yes	36
669.191	even	74	inner	2007.1.v.a.1306.1	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2007.1.v.a.1108.1	✓	36	1.1	even	1	trivial
2007.1.v.a.1108.1	✓	36	3.2	odd	2	CM
2007.1.v.a.1306.1	yes	36	223.191	odd	74	inner
2007.1.v.a.1306.1	yes	36	669.191	even	74	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:	\( N \)	\(=\)	\( 2007 = 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2007.v (of order \(74\), degree \(36\), minimal)

\(f(q)\)	\(=\)	\(q+(0.127018 - 0.991900i) q^{4} +(-0.871354 + 1.72823i) q^{7} +O(q^{10})\)	\(q+(0.127018 - 0.991900i) q^{4} +(-0.871354 + 1.72823i) q^{7} +(-0.189272 + 1.10384i) q^{13} +(-0.967733 - 0.251978i) q^{16} +(-1.17216 + 1.58486i) q^{19} +(0.372856 + 0.927889i) q^{25} +(1.60355 + 1.08381i) q^{28} +(0.123383 - 0.572501i) q^{31} +(0.327832 - 0.264706i) q^{37} +(-0.443426 + 1.10351i) q^{43} +(-1.63288 - 2.20780i) q^{49} +(1.07086 + 0.327947i) q^{52} +(0.301810 - 1.76016i) q^{61} +(-0.372856 + 0.927889i) q^{64} +(1.98201 + 0.253807i) q^{67} +(-0.0743874 + 0.242900i) q^{73} +(1.42314 + 1.36397i) q^{76} +(0.157381 + 1.84909i) q^{79} +(-1.74276 - 1.28894i) q^{91} +(0.0949784 + 0.140525i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + q^{4} + 2 q^{7}+O(q^{10}) \) 36 * q + q^4 + 2 * q^7	\( 36 q + q^{4} + 2 q^{7} - q^{16} - 2 q^{19} - q^{25} - 2 q^{28} + 2 q^{31} - 2 q^{37} - 2 q^{43} - 3 q^{49} + q^{64} + 2 q^{73} + 2 q^{76}+O(q^{100}) \) 36 * q + q^4 + 2 * q^7 - q^16 - 2 * q^19 - q^25 - 2 * q^28 + 2 * q^31 - 2 * q^37 - 2 * q^43 - 3 * q^49 + q^64 + 2 * q^73 + 2 * q^76

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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