LMFDB - Embedded newform 2007.1.v.a.118.1

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			118.1
Root			$-0.524307 - 0.851529i$ of defining polynomial
Character	$\chi$	$=$	2007.118
Dual form			2007.1.v.a.1990.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.0424412 - 0.999099i) q^{4} +(1.55047 - 1.25191i) q^{7} +O(q^{10})$	$q+(-0.0424412 - 0.999099i) q^{4} +(1.55047 - 1.25191i) q^{7} +(1.34059 + 0.675911i) q^{13} +(-0.996397 + 0.0848059i) q^{16} +(-0.189697 - 0.880198i) q^{19} +(-0.127018 + 0.991900i) q^{25} +(-1.31659 - 1.49594i) q^{28} +(-1.50992 + 1.02053i) q^{31} +(0.485213 + 1.58439i) q^{37} +(-0.0535200 - 0.417946i) q^{43} +(0.625977 - 2.90454i) q^{49} +(0.618406 - 1.36807i) q^{52} +(-1.12191 - 0.565654i) q^{61} +(0.127018 + 0.991900i) q^{64} +(-0.974511 + 0.0413967i) q^{67} +(0.0773473 + 0.0349632i) q^{73} +(-0.871354 + 0.226883i) q^{76} +(-1.68926 - 1.04012i) q^{79} +(2.92472 - 0.630325i) q^{91} +(1.27844 + 1.12517i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2007\mathbb{Z}\right)^\times$.

$n$	$226$	$893$
$\chi(n)$	$e\left(\frac{63}{74}\right)$	$1$

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		0.691939	−	0.721956i	$-0.256757\pi$
$2$		0			0		−0.691939	+	0.721956i	$0.743243\pi$
$3$		0			0
$3$		0			0
$4$	−0.0424412	−	0.999099i	−0.0424412	−	0.999099i
$5$		0			0		−0.660675	−	0.750672i	$-0.729730\pi$
$5$		0			0		0.660675	+	0.750672i	$0.270270\pi$
$6$		0			0
$7$	1.55047	−	1.25191i	1.55047	−	1.25191i	0.721956	−	0.691939i	$-0.243243\pi$
$7$	1.55047	−	1.25191i	1.55047	−	1.25191i	0.828510	−	0.559975i	$-0.189189\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.292823	−	0.956167i	$-0.405405\pi$
$11$		0			0		−0.292823	+	0.956167i	$0.594595\pi$
$12$		0			0
$13$	1.34059	+	0.675911i	1.34059	+	0.675911i	0.967733	−	0.251978i	$-0.0810811\pi$
$13$	1.34059	+	0.675911i	1.34059	+	0.675911i	0.372856	+	0.927889i	$0.378378\pi$
$14$		0			0
$15$		0			0
$16$	−0.996397	+	0.0848059i	−0.996397	+	0.0848059i
$17$		0			0		−0.956167	−	0.292823i	$-0.905405\pi$
$17$		0			0		0.956167	+	0.292823i	$0.0945946\pi$
$18$		0			0
$19$	−0.189697	−	0.880198i	−0.189697	−	0.880198i	−0.967733	−	0.251978i	$-0.918919\pi$
$19$	−0.189697	−	0.880198i	−0.189697	−	0.880198i	0.778036	−	0.628220i	$-0.216216\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.660675	−	0.750672i	$-0.729730\pi$
$23$		0			0		0.660675	+	0.750672i	$0.270270\pi$
$24$		0			0
$25$	−0.127018	+	0.991900i	−0.127018	+	0.991900i
$26$		0			0
$27$		0			0
$28$	−1.31659	−	1.49594i	−1.31659	−	1.49594i
$29$		0			0		−0.0848059	−	0.996397i	$-0.527027\pi$
$29$		0			0		0.0848059	+	0.996397i	$0.472973\pi$
$30$		0			0
$31$	−1.50992	+	1.02053i	−1.50992	+	1.02053i	−0.524307	+	0.851529i	$0.675676\pi$
$31$	−1.50992	+	1.02053i	−1.50992	+	1.02053i	−0.985616	+	0.169001i	$0.945946\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.485213	+	1.58439i	0.485213	+	1.58439i	0.778036	+	0.628220i	$0.216216\pi$
$37$	0.485213	+	1.58439i	0.485213	+	1.58439i	−0.292823	+	0.956167i	$0.594595\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.851529	−	0.524307i	$-0.824324\pi$
$41$		0			0		0.851529	+	0.524307i	$0.175676\pi$
$42$		0			0
$43$	−0.0535200	−	0.417946i	−0.0535200	−	0.417946i	−0.996397	−	0.0848059i	$-0.972973\pi$
$43$	−0.0535200	−	0.417946i	−0.0535200	−	0.417946i	0.942877	−	0.333140i	$-0.108108\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.977555	−	0.210679i	$-0.932432\pi$
$47$		0			0		0.977555	+	0.210679i	$0.0675676\pi$
$48$		0			0
$49$	0.625977	−	2.90454i	0.625977	−	2.90454i
$50$		0			0
$51$		0			0
$52$	0.618406	−	1.36807i	0.618406	−	1.36807i
$53$		0			0		0.333140	−	0.942877i	$-0.391892\pi$
$53$		0			0		−0.333140	+	0.942877i	$0.608108\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.873014	−	0.487695i	$-0.837838\pi$
$59$		0			0		0.873014	+	0.487695i	$0.162162\pi$
$60$		0			0
$61$	−1.12191	−	0.565654i	−1.12191	−	0.565654i	−0.210679	−	0.977555i	$-0.567568\pi$
$61$	−1.12191	−	0.565654i	−1.12191	−	0.565654i	−0.911228	+	0.411901i	$0.864865\pi$
$62$		0			0
$63$		0			0
$64$	0.127018	+	0.991900i	0.127018	+	0.991900i
$65$		0			0
$66$		0			0
$67$	−0.974511	+	0.0413967i	−0.974511	+	0.0413967i	−0.524307	−	0.851529i	$-0.675676\pi$
$67$	−0.974511	+	0.0413967i	−0.974511	+	0.0413967i	−0.450204	+	0.892926i	$0.648649\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.873014	−	0.487695i	$-0.837838\pi$
$71$		0			0		0.873014	+	0.487695i	$0.162162\pi$
$72$		0			0
$73$	0.0773473	+	0.0349632i	0.0773473	+	0.0349632i	0.450204	−	0.892926i	$-0.351351\pi$
$73$	0.0773473	+	0.0349632i	0.0773473	+	0.0349632i	−0.372856	+	0.927889i	$0.621622\pi$
$74$		0			0
$75$		0			0
$76$	−0.871354	+	0.226883i	−0.871354	+	0.226883i
$77$		0			0
$78$		0			0
$79$	−1.68926	−	1.04012i	−1.68926	−	1.04012i	−0.911228	−	0.411901i	$-0.864865\pi$
$79$	−1.68926	−	1.04012i	−1.68926	−	1.04012i	−0.778036	−	0.628220i	$-0.783784\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.991900	−	0.127018i	$-0.959459\pi$
$83$		0			0		0.991900	+	0.127018i	$0.0405405\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.750672	−	0.660675i	$-0.770270\pi$
$89$		0			0		0.750672	+	0.660675i	$0.229730\pi$
$90$		0			0
$91$	2.92472	−	0.630325i	2.92472	−	0.630325i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.27844	+	1.12517i	1.27844	+	1.12517i	0.985616	+	0.169001i	$0.0540541\pi$
$97$	1.27844	+	1.12517i	1.27844	+	1.12517i	0.292823	+	0.956167i	$0.405405\pi$
$98$		0			0
$99$		0			0
$100$	0.996397	+	0.0848059i	0.996397	+	0.0848059i
$101$		0			0		−0.333140	−	0.942877i	$-0.608108\pi$
$101$		0			0		0.333140	+	0.942877i	$0.391892\pi$
$102$		0			0
$103$	0.594938	−	0.299962i	0.594938	−	0.299962i	−0.127018	−	0.991900i	$-0.540541\pi$
$103$	0.594938	−	0.299962i	0.594938	−	0.299962i	0.721956	+	0.691939i	$0.243243\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.127018	−	0.991900i	$-0.540541\pi$
$107$		0			0		0.127018	+	0.991900i	$0.459459\pi$
$108$		0			0
$109$	1.92849	+	0.164139i	1.92849	+	0.164139i	0.985616	−	0.169001i	$-0.0540541\pi$
$109$	1.92849	+	0.164139i	1.92849	+	0.164139i	0.942877	+	0.333140i	$0.108108\pi$
$110$		0			0
$111$		0			0
$112$	−1.43871	+	1.37889i	−1.43871	+	1.37889i
$113$		0			0		−0.942877	−	0.333140i	$-0.891892\pi$
$113$		0			0		0.942877	+	0.333140i	$0.108108\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.828510	−	0.559975i	−0.828510	−	0.559975i
$122$		0			0
$123$		0			0
$124$	1.08369	+	1.46525i	1.08369	+	1.46525i
$125$		0			0
$126$		0			0
$127$	0.123383	−	0.572501i	0.123383	−	0.572501i	−0.873014	−	0.487695i	$-0.837838\pi$
$127$	0.123383	−	0.572501i	0.123383	−	0.572501i	0.996397	−	0.0848059i	$-0.0270270\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.691939	−	0.721956i	$-0.743243\pi$
$131$		0			0		0.691939	+	0.721956i	$0.256757\pi$
$132$		0			0
$133$	−1.39605	−	1.12723i	−1.39605	−	1.12723i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.594633	−	0.803997i	$-0.297297\pi$
$137$		0			0		−0.594633	+	0.803997i	$0.702703\pi$
$138$		0			0
$139$	−0.443426	+	1.10351i	−0.443426	+	1.10351i	0.524307	+	0.851529i	$0.324324\pi$
$139$	−0.443426	+	1.10351i	−0.443426	+	1.10351i	−0.967733	+	0.251978i	$0.918919\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	1.56237	−	0.552019i	1.56237	−	0.552019i
$149$		0			0		0.450204	−	0.892926i	$-0.351351\pi$
$149$		0			0		−0.450204	+	0.892926i	$0.648649\pi$
$150$		0			0
$151$	−1.81413	+	0.390974i	−1.81413	+	0.390974i	−0.985616	−	0.169001i	$-0.945946\pi$
$151$	−1.81413	+	0.390974i	−1.81413	+	0.390974i	−0.828510	+	0.559975i	$0.810811\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.282203	+	0.417533i	−0.282203	+	0.417533i	−0.942877	−	0.333140i	$-0.891892\pi$
$157$	−0.282203	+	0.417533i	−0.282203	+	0.417533i	0.660675	+	0.750672i	$0.270270\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.637074	+	1.80310i	−0.637074	+	1.80310i	−0.0424412	+	0.999099i	$0.513514\pi$
$163$	−0.637074	+	1.80310i	−0.637074	+	1.80310i	−0.594633	+	0.803997i	$0.702703\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.0424412	−	0.999099i	$-0.486486\pi$
$167$		0			0		−0.0424412	+	0.999099i	$0.513514\pi$
$168$		0			0
$169$	0.745691	+	1.00824i	0.745691	+	1.00824i
$170$		0			0
$171$		0			0
$172$	−0.415298	+	0.0712099i	−0.415298	+	0.0712099i
$173$		0			0		0.778036	−	0.628220i	$-0.216216\pi$
$173$		0			0		−0.778036	+	0.628220i	$0.783784\pi$
$174$		0			0
$175$	1.04484	+	1.69692i	1.04484	+	1.69692i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.251978	−	0.967733i	$-0.418919\pi$
$179$		0			0		−0.251978	+	0.967733i	$0.581081\pi$
$180$		0			0
$181$	0.703116	+	1.74977i	0.703116	+	1.74977i	0.660675	+	0.750672i	$0.270270\pi$
$181$	0.703116	+	1.74977i	0.703116	+	1.74977i	0.0424412	+	0.999099i	$0.486486\pi$
$182$		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		−0.778036	−	0.628220i	$-0.783784\pi$
$191$		0			0		0.778036	+	0.628220i	$0.216216\pi$
$192$		0			0
$193$	−0.823060	−	1.82081i	−0.823060	−	1.82081i	−0.450204	−	0.892926i	$-0.648649\pi$
$193$	−0.823060	−	1.82081i	−0.823060	−	1.82081i	−0.372856	−	0.927889i	$-0.621622\pi$
$194$		0			0
$195$		0			0
$196$	−2.92849	−	0.502141i	−2.92849	−	0.502141i
$197$		0			0		−0.333140	−	0.942877i	$-0.608108\pi$
$197$		0			0		0.333140	+	0.942877i	$0.391892\pi$
$198$		0			0
$199$	1.39732	+	0.363834i	1.39732	+	0.363834i	0.873014	−	0.487695i	$-0.162162\pi$
$199$	1.39732	+	0.363834i	1.39732	+	0.363834i	0.524307	+	0.851529i	$0.324324\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−1.39308	−	0.559786i	−1.39308	−	0.559786i
$209$		0			0
$210$		0			0
$211$	1.26056	−	0.704189i	1.26056	−	0.704189i	0.292823	−	0.956167i	$-0.405405\pi$
$211$	1.26056	−	0.704189i	1.26056	−	0.704189i	0.967733	+	0.251978i	$0.0810811\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.06347	+	3.47259i	−1.06347	+	3.47259i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.127018	−	0.991900i	0.127018	−	0.991900i
$223$	0.127018	−	0.991900i	0.127018	−	0.991900i
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.999099	−	0.0424412i	$-0.0135135\pi$
$227$		0			0		−0.999099	+	0.0424412i	$0.986486\pi$
$228$		0			0
$229$	−1.82851	−	0.559975i	−1.82851	−	0.559975i	−0.828510	−	0.559975i	$-0.810811\pi$
$229$	−1.82851	−	0.559975i	−1.82851	−	0.559975i	−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.0424412	−	0.999099i	$-0.486486\pi$
$233$		0			0		−0.0424412	+	0.999099i	$0.513514\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.0848059	−	0.996397i	$-0.527027\pi$
$239$		0			0		0.0848059	+	0.996397i	$0.472973\pi$
$240$		0			0
$241$	−1.53369	+	0.262977i	−1.53369	+	0.262977i	−0.873014	−	0.487695i	$-0.837838\pi$
$241$	−1.53369	+	0.262977i	−1.53369	+	0.262977i	−0.660675	+	0.750672i	$0.729730\pi$
$242$		0			0
$243$		0			0
$244$	−0.517529	+	1.14490i	−0.517529	+	1.14490i
$245$		0			0
$246$		0			0
$247$	0.340629	−	1.30820i	0.340629	−	1.30820i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.750672	−	0.660675i	$-0.229730\pi$
$251$		0			0		−0.750672	+	0.660675i	$0.770270\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.985616	−	0.169001i	0.985616	−	0.169001i
$257$		0			0		−0.487695	−	0.873014i	$-0.662162\pi$
$257$		0			0		0.487695	+	0.873014i	$0.337838\pi$
$258$		0			0
$259$	2.73582	+	1.84909i	2.73582	+	1.84909i
$260$		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
$268$	0.0827188	+	0.971876i	0.0827188	+	0.971876i
$269$		0			0		0.0424412	−	0.999099i	$-0.486486\pi$
$269$		0			0		−0.0424412	+	0.999099i	$0.513514\pi$
$270$		0			0
$271$	−1.70152	+	1.04767i	−1.70152	+	1.04767i	−0.828510	+	0.559975i	$0.810811\pi$
$271$	−1.70152	+	1.04767i	−1.70152	+	1.04767i	−0.873014	+	0.487695i	$0.837838\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.271752	−	0.200987i	0.271752	−	0.200987i	−0.450204	−	0.892926i	$-0.648649\pi$
$277$	0.271752	−	0.200987i	0.271752	−	0.200987i	0.721956	+	0.691939i	$0.243243\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.628220	−	0.778036i	$-0.283784\pi$
$281$		0			0		−0.628220	+	0.778036i	$0.716216\pi$
$282$		0			0
$283$	0.703116	+	0.248427i	0.703116	+	0.248427i	0.660675	−	0.750672i	$-0.270270\pi$
$283$	0.703116	+	0.248427i	0.703116	+	0.248427i	0.0424412	+	0.999099i	$0.486486\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.828510	+	0.559975i	0.828510	+	0.559975i
$290$		0			0
$291$		0			0
$292$	0.0316490	−	0.0787615i	0.0316490	−	0.0787615i
$293$		0			0		−0.594633	−	0.803997i	$-0.702703\pi$
$293$		0			0		0.594633	+	0.803997i	$0.297297\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−0.606213	−	0.581008i	−0.606213	−	0.581008i
$302$		0			0
$303$		0			0
$304$	0.263660	+	0.860940i	0.263660	+	0.860940i
$305$		0			0
$306$		0			0
$307$	0.313628	+	0.126026i	0.313628	+	0.126026i	0.524307	−	0.851529i	$-0.324324\pi$
$307$	0.313628	+	0.126026i	0.313628	+	0.126026i	−0.210679	+	0.977555i	$0.567568\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.524307	−	0.851529i	$-0.675676\pi$
$311$		0			0		0.524307	+	0.851529i	$0.324324\pi$
$312$		0			0
$313$	−1.22824	+	1.52115i	−1.22824	+	1.52115i	−0.450204	+	0.892926i	$0.648649\pi$
$313$	−1.22824	+	1.52115i	−1.22824	+	1.52115i	−0.778036	+	0.628220i	$0.783784\pi$
$314$		0			0
$315$		0			0
$316$	−0.967490	+	1.73189i	−0.967490	+	1.73189i
$317$		0			0		−0.559975	−	0.828510i	$-0.689189\pi$
$317$		0			0		0.559975	+	0.828510i	$0.310811\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−0.840715	+	1.24388i	−0.840715	+	1.24388i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.93927	+	0.248334i	1.93927	+	0.248334i	0.996397	−	0.0848059i	$-0.0270270\pi$
$331$	1.93927	+	0.248334i	1.93927	+	0.248334i	0.942877	+	0.333140i	$0.108108\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.0949784	−	1.11591i	0.0949784	−	1.11591i	−0.778036	−	0.628220i	$-0.783784\pi$
$337$	0.0949784	−	1.11591i	0.0949784	−	1.11591i	0.873014	−	0.487695i	$-0.162162\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.76852	−	3.50765i	−1.76852	−	3.50765i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.803997	−	0.594633i	$-0.202703\pi$
$347$		0			0		−0.803997	+	0.594633i	$0.797297\pi$
$348$		0			0
$349$	0.692793	−	0.787166i	0.692793	−	0.787166i	−0.292823	−	0.956167i	$-0.594595\pi$
$349$	0.692793	−	0.787166i	0.692793	−	0.787166i	0.985616	+	0.169001i	$0.0540541\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.487695	−	0.873014i	$-0.337838\pi$
$353$		0			0		−0.487695	+	0.873014i	$0.662162\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.851529	−	0.524307i	$-0.175676\pi$
$359$		0			0		−0.851529	+	0.524307i	$0.824324\pi$
$360$		0			0
$361$	0.172465	−	0.0779590i	0.172465	−	0.0779590i
$362$		0			0
$363$		0			0
$364$	−0.753886	−	2.89533i	−0.753886	−	2.89533i
$365$		0			0
$366$		0			0
$367$	0.307058	−	0.498694i	0.307058	−	0.498694i	−0.660675	−	0.750672i	$-0.729730\pi$
$367$	0.307058	−	0.498694i	0.307058	−	0.498694i	0.967733	+	0.251978i	$0.0810811\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.274441	+	0.607133i	−0.274441	+	0.607133i	−0.996397	−	0.0848059i	$-0.972973\pi$
$373$	−0.274441	+	0.607133i	−0.274441	+	0.607133i	0.721956	+	0.691939i	$0.243243\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.0316490	−	0.745041i	−0.0316490	−	0.745041i	−0.942877	−	0.333140i	$-0.891892\pi$
$379$	−0.0316490	−	0.745041i	−0.0316490	−	0.745041i	0.911228	−	0.411901i	$-0.135135\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.0424412	−	0.999099i	$-0.513514\pi$
$383$		0			0		0.0424412	+	0.999099i	$0.486486\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	1.06990	−	1.32504i	1.06990	−	1.32504i
$389$		0			0		−0.628220	−	0.778036i	$-0.716216\pi$
$389$		0			0		0.628220	+	0.778036i	$0.283784\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.81413	+	0.390974i	1.81413	+	0.390974i	0.985616	−	0.169001i	$-0.0540541\pi$
$397$	1.81413	+	0.390974i	1.81413	+	0.390974i	0.828510	+	0.559975i	$0.189189\pi$
$398$		0			0
$399$		0			0
$400$	0.0424412	−	0.999099i	0.0424412	−	0.999099i
$401$		0			0		−0.0848059	−	0.996397i	$-0.527027\pi$
$401$		0			0		0.0848059	+	0.996397i	$0.472973\pi$
$402$		0			0
$403$	−2.71397	+	0.347538i	−2.71397	+	0.347538i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.637074	−	0.195102i	0.637074	−	0.195102i	0.0424412	−	0.999099i	$-0.486486\pi$
$409$	0.637074	−	0.195102i	0.637074	−	0.195102i	0.594633	+	0.803997i	$0.297297\pi$
$410$		0			0
$411$		0			0
$412$	−0.324941	−	0.581671i	−0.324941	−	0.581671i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.169001	−	0.985616i	$-0.445946\pi$
$419$		0			0		−0.169001	+	0.985616i	$0.554054\pi$
$420$		0			0
$421$	1.68926	+	0.216319i	1.68926	+	0.216319i	0.911228	−	0.411901i	$-0.135135\pi$
$421$	1.68926	+	0.216319i	1.68926	+	0.216319i	0.778036	+	0.628220i	$0.216216\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−2.44763	+	0.527505i	−2.44763	+	0.527505i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.372856	−	0.927889i	$-0.378378\pi$
$431$		0			0		−0.372856	+	0.927889i	$0.621622\pi$
$432$		0			0
$433$	−0.871354	+	1.72823i	−0.871354	+	1.72823i	−0.210679	+	0.977555i	$0.567568\pi$
$433$	−0.871354	+	1.72823i	−0.871354	+	1.72823i	−0.660675	+	0.750672i	$0.729730\pi$
$434$		0			0
$435$		0			0
$436$	0.0821435	−	1.93372i	0.0821435	−	1.93372i
$437$		0			0
$438$		0			0
$439$	0.869380	+	1.07671i	0.869380	+	1.07671i	0.996397	+	0.0848059i	$0.0270270\pi$
$439$	0.869380	+	1.07671i	0.869380	+	1.07671i	−0.127018	+	0.991900i	$0.540541\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.892926	−	0.450204i	$-0.148649\pi$
$443$		0			0		−0.892926	+	0.450204i	$0.851351\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	1.43871	+	1.37889i	1.43871	+	1.37889i
$449$		0			0		−0.450204	−	0.892926i	$-0.648649\pi$
$449$		0			0		0.450204	+	0.892926i	$0.351351\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.53751	−	0.470857i	−1.53751	−	0.470857i	−0.594633	−	0.803997i	$-0.702703\pi$
$457$	−1.53751	−	0.470857i	−1.53751	−	0.470857i	−0.942877	+	0.333140i	$0.891892\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.927889	−	0.372856i	$-0.878378\pi$
$461$		0			0		0.927889	+	0.372856i	$0.121622\pi$
$462$		0			0
$463$	0.348244	−	1.13714i	0.348244	−	1.13714i	−0.594633	−	0.803997i	$-0.702703\pi$
$463$	0.348244	−	1.13714i	0.348244	−	1.13714i	0.942877	−	0.333140i	$-0.108108\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.911228	−	0.411901i	$-0.864865\pi$
$467$		0			0		0.911228	+	0.411901i	$0.135135\pi$
$468$		0			0
$469$	−1.45912	+	1.28419i	−1.45912	+	1.28419i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.897164	−	0.0763599i	0.897164	−	0.0763599i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.169001	−	0.985616i	$-0.554054\pi$
$479$		0			0		0.169001	+	0.985616i	$0.445946\pi$
$480$		0			0
$481$	−0.420433	+	2.45197i	−0.420433	+	2.45197i
$482$		0			0
$483$		0			0
$484$	−0.524307	+	0.851529i	−0.524307	+	0.851529i
$485$		0			0
$486$		0			0
$487$	0.988715	−	1.60578i	0.988715	−	1.60578i	0.210679	−	0.977555i	$-0.432432\pi$
$487$	0.988715	−	1.60578i	0.988715	−	1.60578i	0.778036	−	0.628220i	$-0.216216\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.996397	−	0.0848059i	$-0.0270270\pi$
$491$		0			0		−0.996397	+	0.0848059i	$0.972973\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1.41794	−	1.14490i	1.41794	−	1.14490i
$497$		0			0
$498$		0			0
$499$	0.988715	+	0.349335i	0.988715	+	0.349335i	0.778036	−	0.628220i	$-0.216216\pi$
$499$	0.988715	+	0.349335i	0.988715	+	0.349335i	0.210679	+	0.977555i	$0.432432\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.778036	−	0.628220i	$-0.216216\pi$
$503$		0			0		−0.778036	+	0.628220i	$0.783784\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−0.577222	−	0.0989746i	−0.577222	−	0.0989746i
$509$		0			0		0.999099	−	0.0424412i	$-0.0135135\pi$
$509$		0			0		−0.999099	+	0.0424412i	$0.986486\pi$
$510$		0			0
$511$	0.163695	−	0.0426229i	0.163695	−	0.0426229i
$512$		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		0.942877	−	0.333140i	$-0.108108\pi$
$521$		0			0		−0.942877	+	0.333140i	$0.891892\pi$
$522$		0			0
$523$	−0.764397	+	0.307160i	−0.764397	+	0.307160i	−0.721956	−	0.691939i	$-0.756757\pi$
$523$	−0.764397	+	0.307160i	−0.764397	+	0.307160i	−0.0424412	+	0.999099i	$0.513514\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.127018	+	0.991900i	−0.127018	+	0.991900i
$530$		0			0
$531$		0			0
$532$	−1.06697	+	1.44263i	−1.06697	+	1.44263i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.271752	+	0.200987i	0.271752	+	0.200987i	0.721956	−	0.691939i	$-0.243243\pi$
$541$	0.271752	+	0.200987i	0.271752	+	0.200987i	−0.450204	+	0.892926i	$0.648649\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.36143	−	0.481025i	1.36143	−	0.481025i	0.450204	−	0.892926i	$-0.351351\pi$
$547$	1.36143	−	0.481025i	1.36143	−	0.481025i	0.911228	+	0.411901i	$0.135135\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−3.92129	+	0.502141i	−3.92129	+	0.502141i
$554$		0			0
$555$		0			0
$556$	1.12133	+	0.396192i	1.12133	+	0.396192i
$557$		0			0		0.721956	−	0.691939i	$-0.243243\pi$
$557$		0			0		−0.721956	+	0.691939i	$0.756757\pi$
$558$		0			0
$559$	0.210746	−	0.596468i	0.210746	−	0.596468i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.450204	−	0.892926i	$-0.648649\pi$
$563$		0			0		0.450204	+	0.892926i	$0.351351\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.996397	−	0.0848059i	$-0.972973\pi$
$569$		0			0		0.996397	+	0.0848059i	$0.0270270\pi$
$570$		0			0
$571$	−0.953670	−	1.41100i	−0.953670	−	1.41100i	−0.911228	−	0.411901i	$-0.864865\pi$
$571$	−0.953670	−	1.41100i	−0.953670	−	1.41100i	−0.0424412	−	0.999099i	$-0.513514\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.26056	−	1.20814i	1.26056	−	1.20814i	0.292823	−	0.956167i	$-0.405405\pi$
$577$	1.26056	−	1.20814i	1.26056	−	1.20814i	0.967733	−	0.251978i	$-0.0810811\pi$
$578$		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		0.967733	−	0.251978i	$-0.0810811\pi$
$587$		0			0		−0.967733	+	0.251978i	$0.918919\pi$
$588$		0			0
$589$	1.18470	+	1.13544i	1.18470	+	1.13544i
$590$		0			0
$591$		0			0
$592$	−0.617830	−	1.53753i	−0.617830	−	1.53753i
$593$		0			0		0.967733	−	0.251978i	$-0.0810811\pi$
$593$		0			0		−0.967733	+	0.251978i	$0.918919\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.803997	−	0.594633i	$-0.202703\pi$
$599$		0			0		−0.803997	+	0.594633i	$0.797297\pi$
$600$		0			0
$601$	0.245777	−	0.943917i	0.245777	−	0.943917i	−0.721956	−	0.691939i	$-0.756757\pi$
$601$	0.245777	−	0.943917i	0.245777	−	0.943917i	0.967733	−	0.251978i	$-0.0810811\pi$
$602$		0			0
$603$		0			0
$604$	0.467615	+	1.79590i	0.467615	+	1.79590i
$605$		0			0
$606$		0			0
$607$	0.330415	−	1.92699i	0.330415	−	1.92699i	−0.0424412	−	0.999099i	$-0.513514\pi$
$607$	0.330415	−	1.92699i	0.330415	−	1.92699i	0.372856	−	0.927889i	$-0.378378\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.98201	−	0.0841949i	−1.98201	−	0.0841949i	−0.985616	−	0.169001i	$-0.945946\pi$
$613$	−1.98201	−	0.0841949i	−1.98201	−	0.0841949i	−0.996397	+	0.0848059i	$0.972973\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.411901	−	0.911228i	$-0.364865\pi$
$617$		0			0		−0.411901	+	0.911228i	$0.635135\pi$
$618$		0			0
$619$	0.789321	+	0.977555i	0.789321	+	0.977555i	1.00000			$0$
$619$	0.789321	+	0.977555i	0.789321	+	0.977555i	−0.210679	+	0.977555i	$0.567568\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.967733	−	0.251978i	−0.967733	−	0.251978i
$626$		0			0
$627$		0			0
$628$	0.429133	+	0.264228i	0.429133	+	0.264228i
$629$		0			0
$630$		0			0
$631$	0.287818	+	0.177217i	0.287818	+	0.177217i	0.660675	−	0.750672i	$-0.270270\pi$
$631$	0.287818	+	0.177217i	0.287818	+	0.177217i	−0.372856	+	0.927889i	$0.621622\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	2.80239	−	3.47070i	2.80239	−	3.47070i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.660675	−	0.750672i	$-0.729730\pi$
$641$		0			0		0.660675	+	0.750672i	$0.270270\pi$
$642$		0			0
$643$	−1.46718	+	0.518389i	−1.46718	+	0.518389i	−0.942877	−	0.333140i	$-0.891892\pi$
$643$	−1.46718	+	0.518389i	−1.46718	+	0.518389i	−0.524307	+	0.851529i	$0.675676\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.487695	−	0.873014i	$-0.662162\pi$
$647$		0			0		0.487695	+	0.873014i	$0.337838\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	1.82851	+	0.559975i	1.82851	+	0.559975i
$653$		0			0		0.996397	−	0.0848059i	$-0.0270270\pi$
$653$		0			0		−0.996397	+	0.0848059i	$0.972973\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.999099	−	0.0424412i	$-0.986486\pi$
$659$		0			0		0.999099	+	0.0424412i	$0.0135135\pi$
$660$		0			0
$661$	1.34059	+	1.17987i	1.34059	+	1.17987i	0.967733	+	0.251978i	$0.0810811\pi$
$661$	1.34059	+	1.17987i	1.34059	+	1.17987i	0.372856	+	0.927889i	$0.378378\pi$
$662$		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
$673$	−0.0445045	−	1.04767i	−0.0445045	−	1.04767i	−0.873014	−	0.487695i	$-0.837838\pi$
$673$	−0.0445045	−	1.04767i	−0.0445045	−	1.04767i	0.828510	−	0.559975i	$-0.189189\pi$
$674$		0			0
$675$		0			0
$676$	0.975685	−	0.787810i	0.975685	−	0.787810i
$677$		0			0		−0.750672	−	0.660675i	$-0.770270\pi$
$677$		0			0		0.750672	+	0.660675i	$0.229730\pi$
$678$		0			0
$679$	3.39079	+	0.144039i	3.39079	+	0.144039i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.0848059	−	0.996397i	$-0.472973\pi$
$683$		0			0		−0.0848059	+	0.996397i	$0.527027\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	0.0887715	+	0.411901i	0.0887715	+	0.411901i
$689$		0			0
$690$		0			0
$691$	−0.967490	−	1.73189i	−0.967490	−	1.73189i	−0.594633	−	0.803997i	$-0.702703\pi$
$691$	−0.967490	−	1.73189i	−0.967490	−	1.73189i	−0.372856	−	0.927889i	$-0.621622\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.65105	−	1.11591i	1.65105	−	1.11591i
$701$		0			0		0.628220	−	0.778036i	$-0.283784\pi$
$701$		0			0		−0.628220	+	0.778036i	$0.716216\pi$
$702$		0			0
$703$	1.30253	−	0.727637i	1.30253	−	0.727637i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	1.99820i		−	1.99820i	−0.0424412	−	0.999099i	$-0.513514\pi$
$709$		−	1.99820i		−	1.99820i	0.0424412	−	0.999099i	$-0.486486\pi$
$710$		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		−0.628220	−	0.778036i	$-0.716216\pi$
$719$		0			0		0.628220	+	0.778036i	$0.283784\pi$
$720$		0			0
$721$	0.546905	−	1.20989i	0.546905	−	1.20989i
$722$		0			0
$723$		0			0
$724$	1.71835	−	0.776745i	1.71835	−	0.776745i
$725$		0			0
$726$		0			0
$727$	−0.925292	−	0.747121i	−0.925292	−	0.747121i	0.0424412	−	0.999099i	$-0.486486\pi$
$727$	−0.925292	−	0.747121i	−0.925292	−	0.747121i	−0.967733	+	0.251978i	$0.918919\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	0.183403	+	1.43222i	0.183403	+	1.43222i	0.778036	+	0.628220i	$0.216216\pi$
$733$	0.183403	+	1.43222i	0.183403	+	1.43222i	−0.594633	+	0.803997i	$0.702703\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.900436	−	0.665959i	0.900436	−	0.665959i	−0.0424412	−	0.999099i	$-0.513514\pi$
$739$	0.900436	−	0.665959i	0.900436	−	0.665959i	0.942877	+	0.333140i	$0.108108\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.892926	−	0.450204i	$-0.851351\pi$
$743$		0			0		0.892926	+	0.450204i	$0.148649\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.68969	+	0.439961i	−1.68969	+	0.439961i	−0.967733	−	0.251978i	$-0.918919\pi$
$751$	−1.68969	+	0.439961i	−1.68969	+	0.439961i	−0.721956	+	0.691939i	$0.756757\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.52070	+	0.936335i	−1.52070	+	0.936335i	−0.524307	+	0.851529i	$0.675676\pi$
$757$	−1.52070	+	0.936335i	−1.52070	+	0.936335i	−0.996397	+	0.0848059i	$0.972973\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.721956	−	0.691939i	$-0.243243\pi$
$761$		0			0		−0.721956	+	0.691939i	$0.756757\pi$
$762$		0			0
$763$	3.19555	−	2.15982i	3.19555	−	2.15982i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1.43871	−	0.122452i	−1.43871	−	0.122452i	−0.660675	−	0.750672i	$-0.729730\pi$
$769$	−1.43871	−	0.122452i	−1.43871	−	0.122452i	−0.778036	+	0.628220i	$0.783784\pi$
$770$		0			0
$771$		0			0
$772$	−1.78424	+	0.899596i	−1.78424	+	0.899596i
$773$		0			0		−0.372856	−	0.927889i	$-0.621622\pi$
$773$		0			0		0.372856	+	0.927889i	$0.378378\pi$
$774$		0			0
$775$	−0.820477	−	1.62732i	−0.820477	−	1.62732i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.377399	+	2.94717i	−0.377399	+	2.94717i
$785$		0			0
$786$		0			0
$787$	1.91061	−	0.585118i	1.91061	−	0.585118i	0.942877	−	0.333140i	$-0.108108\pi$
$787$	1.91061	−	0.585118i	1.91061	−	0.585118i	0.967733	−	0.251978i	$-0.0810811\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1.12169	−	1.51662i	−1.12169	−	1.51662i
$794$		0			0
$795$		0			0
$796$	0.304202	−	1.41150i	0.304202	−	1.41150i
$797$		0			0		−0.803997	−	0.594633i	$-0.797297\pi$
$797$		0			0		0.803997	+	0.594633i	$0.202703\pi$
$798$		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		0.127018	−	0.991900i	$-0.459459\pi$
$809$		0			0		−0.127018	+	0.991900i	$0.540541\pi$
$810$		0			0
$811$	1.70757	−	0.522938i	1.70757	−	0.522938i	0.721956	−	0.691939i	$-0.243243\pi$
$811$	1.70757	−	0.522938i	1.70757	−	0.522938i	0.985616	+	0.169001i	$0.0540541\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.357722	+	0.126391i	−0.357722	+	0.126391i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.487695	−	0.873014i	$-0.337838\pi$
$821$		0			0		−0.487695	+	0.873014i	$0.662162\pi$
$822$		0			0
$823$	−1.03919	−	0.417580i	−1.03919	−	0.417580i	−0.210679	−	0.977555i	$-0.567568\pi$
$823$	−1.03919	−	0.417580i	−1.03919	−	0.417580i	−0.828510	+	0.559975i	$0.810811\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.967733	−	0.251978i	$-0.0810811\pi$
$827$		0			0		−0.967733	+	0.251978i	$0.918919\pi$
$828$		0			0
$829$	0.823060	−	0.0349632i	0.823060	−	0.0349632i	0.372856	−	0.927889i	$-0.378378\pi$
$829$	0.823060	−	0.0349632i	0.823060	−	0.0349632i	0.450204	+	0.892926i	$0.351351\pi$
$830$		0			0
$831$		0			0
$832$	−0.500158	+	1.41558i	−0.500158	+	1.41558i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.942877	−	0.333140i	$-0.891892\pi$
$839$		0			0		0.942877	+	0.333140i	$0.108108\pi$
$840$		0			0
$841$	−0.985616	+	0.169001i	−0.985616	+	0.169001i
$842$		0			0
$843$		0			0
$844$	−0.757054	−	1.22953i	−0.757054	−	1.22953i
$845$		0			0
$846$		0			0
$847$	−1.98562	+	0.169001i	−1.98562	+	0.169001i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$			1.85578i			1.85578i	0.372856	+	0.927889i	$0.378378\pi$
$853$			1.85578i			1.85578i	−0.372856	+	0.927889i	$0.621622\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.169001	−	0.985616i	$-0.445946\pi$
$857$		0			0		−0.169001	+	0.985616i	$0.554054\pi$
$858$		0			0
$859$	0.323186	+	1.88483i	0.323186	+	1.88483i	0.450204	+	0.892926i	$0.351351\pi$
$859$	0.323186	+	1.88483i	0.323186	+	1.88483i	−0.127018	+	0.991900i	$0.540541\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.996397	−	0.0848059i	$-0.0270270\pi$
$863$		0			0		−0.996397	+	0.0848059i	$0.972973\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	3.51460	+	0.915130i	3.51460	+	0.915130i
$869$		0			0
$870$		0			0
$871$	−1.33440	−	0.603187i	−1.33440	−	0.603187i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.905054	+	0.363680i	0.905054	+	0.363680i	0.778036	−	0.628220i	$-0.216216\pi$
$877$	0.905054	+	0.363680i	0.905054	+	0.363680i	0.127018	+	0.991900i	$0.459459\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.956167	−	0.292823i	$-0.905405\pi$
$881$		0			0		0.956167	+	0.292823i	$0.0945946\pi$
$882$		0			0
$883$	−1.29282	−	0.956167i	−1.29282	−	0.956167i	−0.292823	−	0.956167i	$-0.594595\pi$
$883$	−1.29282	−	0.956167i	−1.29282	−	0.956167i	−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.750672	−	0.660675i	$-0.229730\pi$
$887$		0			0		−0.750672	+	0.660675i	$0.770270\pi$
$888$		0			0
$889$	−0.525420	−	1.04211i	−0.525420	−	1.04211i
$890$		0			0
$891$		0			0
$892$	−0.996397	−	0.0848059i	−0.996397	−	0.0848059i
$893$		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
$907$	−0.0947188	+	0.235717i	−0.0947188	+	0.235717i	−0.967733	−	0.251978i	$-0.918919\pi$
$907$	−0.0947188	+	0.235717i	−0.0947188	+	0.235717i	0.873014	+	0.487695i	$0.162162\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.977555	−	0.210679i	$-0.0675676\pi$
$911$		0			0		−0.977555	+	0.210679i	$0.932432\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.481866	+	1.85063i	−0.481866	+	1.85063i
$917$		0			0
$918$		0			0
$919$	0.112602	−	0.656696i	0.112602	−	0.656696i	−0.873014	−	0.487695i	$-0.837838\pi$
$919$	0.112602	−	0.656696i	0.112602	−	0.656696i	0.985616	−	0.169001i	$-0.0540541\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.63318	+	0.280038i	−1.63318	+	0.280038i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.956167	−	0.292823i	$-0.0945946\pi$
$929$		0			0		−0.956167	+	0.292823i	$0.905405\pi$
$930$		0			0
$931$	−2.67532			−2.67532
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.144429	−	1.69692i	−0.144429	−	1.69692i	−0.594633	−	0.803997i	$-0.702703\pi$
$937$	−0.144429	−	1.69692i	−0.144429	−	1.69692i	0.450204	−	0.892926i	$-0.351351\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.977555	−	0.210679i	$-0.932432\pi$
$941$		0			0		0.977555	+	0.210679i	$0.0675676\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.411901	−	0.911228i	$-0.635135\pi$
$947$		0			0		0.411901	+	0.911228i	$0.364865\pi$
$948$		0			0
$949$	0.0800589	+	0.0991511i	0.0800589	+	0.0991511i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.450204	−	0.892926i	$-0.351351\pi$
$953$		0			0		−0.450204	+	0.892926i	$0.648649\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.865530	−	2.15395i	0.865530	−	2.15395i
$962$		0			0
$963$		0			0
$964$	0.327832	+	1.52115i	0.327832	+	1.52115i
$965$		0			0
$966$		0			0
$967$	−0.594938	−	1.68384i	−0.594938	−	1.68384i	−0.721956	−	0.691939i	$-0.756757\pi$
$967$	−0.594938	−	1.68384i	−0.594938	−	1.68384i	0.127018	−	0.991900i	$-0.459459\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.524307	−	0.851529i	$-0.324324\pi$
$971$		0			0		−0.524307	+	0.851529i	$0.675676\pi$
$972$		0			0
$973$	0.693979	+	2.26608i	0.693979	+	2.26608i
$974$		0			0
$975$		0			0
$976$	1.16584	+	0.468472i	1.16584	+	0.468472i
$977$		0			0		0.911228	−	0.411901i	$-0.135135\pi$
$977$		0			0		−0.911228	+	0.411901i	$0.864865\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.210679	−	0.977555i	$-0.567568\pi$
$983$		0			0		0.210679	+	0.977555i	$0.432432\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−1.32148	−	0.284801i	−1.32148	−	0.284801i
$989$		0			0
$990$		0			0
$991$	0.405179	−	0.299669i	0.405179	−	0.299669i	−0.372856	−	0.927889i	$-0.621622\pi$
$991$	0.405179	−	0.299669i	0.405179	−	0.299669i	0.778036	+	0.628220i	$0.216216\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.577222	+	1.88483i	0.577222	+	1.88483i	0.450204	+	0.892926i	$0.351351\pi$
$997$	0.577222	+	1.88483i	0.577222	+	1.88483i	0.127018	+	0.991900i	$0.459459\pi$
$998$		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.118.1	✓	36
3.2	odd	2	CM	2007.1.v.a.118.1	✓	36
223.206	odd	74	inner	2007.1.v.a.1990.1	yes	36
669.206	even	74	inner	2007.1.v.a.1990.1	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2007.1.v.a.118.1	✓	36	1.1	even	1	trivial
2007.1.v.a.118.1	✓	36	3.2	odd	2	CM
2007.1.v.a.1990.1	yes	36	223.206	odd	74	inner
2007.1.v.a.1990.1	yes	36	669.206	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)

\(f(q)\)	\(=\)	\(q+(-0.0424412 - 0.999099i) q^{4} +(1.55047 - 1.25191i) q^{7} +O(q^{10})\)	\(q+(-0.0424412 - 0.999099i) q^{4} +(1.55047 - 1.25191i) q^{7} +(1.34059 + 0.675911i) q^{13} +(-0.996397 + 0.0848059i) q^{16} +(-0.189697 - 0.880198i) q^{19} +(-0.127018 + 0.991900i) q^{25} +(-1.31659 - 1.49594i) q^{28} +(-1.50992 + 1.02053i) q^{31} +(0.485213 + 1.58439i) q^{37} +(-0.0535200 - 0.417946i) q^{43} +(0.625977 - 2.90454i) q^{49} +(0.618406 - 1.36807i) q^{52} +(-1.12191 - 0.565654i) q^{61} +(0.127018 + 0.991900i) q^{64} +(-0.974511 + 0.0413967i) q^{67} +(0.0773473 + 0.0349632i) q^{73} +(-0.871354 + 0.226883i) q^{76} +(-1.68926 - 1.04012i) q^{79} +(2.92472 - 0.630325i) q^{91} +(1.27844 + 1.12517i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + q^{4} + 2 q^{7}+O(q^{10}) \) 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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