LMFDB - Embedded newform 2007.1.v.a.91.1

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			91.1
Root			$-0.985616 + 0.169001i$ of defining polynomial
Character	$\chi$	$=$	2007.91
Dual form			2007.1.v.a.397.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.967733 - 0.251978i) q^{4} +(1.03825 - 1.40380i) q^{7} +O(q^{10})$	$q+(0.967733 - 0.251978i) q^{4} +(1.03825 - 1.40380i) q^{7} +(0.618234 + 1.74977i) q^{13} +(0.873014 - 0.487695i) q^{16} +(-0.552192 + 1.80310i) q^{19} +(-0.721956 - 0.691939i) q^{25} +(0.651018 - 1.62012i) q^{28} +(-1.50992 + 0.682528i) q^{31} +(-0.383954 - 1.78155i) q^{37} +(0.422810 - 0.405231i) q^{43} +(-0.599881 - 1.95882i) q^{49} +(1.03919 + 1.53753i) q^{52} +(-0.535687 - 1.51614i) q^{61} +(0.721956 - 0.691939i) q^{64} +(-0.0427385 - 0.164139i) q^{67} +(-1.60355 + 1.08381i) q^{73} +(-0.0800337 + 1.88406i) q^{76} +(-0.233876 + 1.36397i) q^{79} +(3.09821 + 0.948816i) q^{91} +(0.313628 - 0.126026i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2007\mathbb{Z}\right)^\times$.

$n$	$226$	$893$
$\chi(n)$	$e\left(\frac{45}{74}\right)$	$1$

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		0.991900	−	0.127018i	$-0.0405405\pi$
$2$		0			0		−0.991900	+	0.127018i	$0.959459\pi$
$3$		0			0
$3$		0			0
$4$	0.967733	−	0.251978i	0.967733	−	0.251978i
$5$		0			0		0.372856	−	0.927889i	$-0.378378\pi$
$5$		0			0		−0.372856	+	0.927889i	$0.621622\pi$
$6$		0			0
$7$	1.03825	−	1.40380i	1.03825	−	1.40380i	0.127018	−	0.991900i	$-0.459459\pi$
$7$	1.03825	−	1.40380i	1.03825	−	1.40380i	0.911228	−	0.411901i	$-0.135135\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.210679	−	0.977555i	$-0.432432\pi$
$11$		0			0		−0.210679	+	0.977555i	$0.567568\pi$
$12$		0			0
$13$	0.618234	+	1.74977i	0.618234	+	1.74977i	0.660675	+	0.750672i	$0.270270\pi$
$13$	0.618234	+	1.74977i	0.618234	+	1.74977i	−0.0424412	+	0.999099i	$0.513514\pi$
$14$		0			0
$15$		0			0
$16$	0.873014	−	0.487695i	0.873014	−	0.487695i
$17$		0			0		−0.977555	−	0.210679i	$-0.932432\pi$
$17$		0			0		0.977555	+	0.210679i	$0.0675676\pi$
$18$		0			0
$19$	−0.552192	+	1.80310i	−0.552192	+	1.80310i	0.0424412	+	0.999099i	$0.486486\pi$
$19$	−0.552192	+	1.80310i	−0.552192	+	1.80310i	−0.594633	+	0.803997i	$0.702703\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.372856	−	0.927889i	$-0.378378\pi$
$23$		0			0		−0.372856	+	0.927889i	$0.621622\pi$
$24$		0			0
$25$	−0.721956	−	0.691939i	−0.721956	−	0.691939i
$26$		0			0
$27$		0			0
$28$	0.651018	−	1.62012i	0.651018	−	1.62012i
$29$		0			0		−0.487695	−	0.873014i	$-0.662162\pi$
$29$		0			0		0.487695	+	0.873014i	$0.337838\pi$
$30$		0			0
$31$	−1.50992	+	0.682528i	−1.50992	+	0.682528i	−0.985616	−	0.169001i	$-0.945946\pi$
$31$	−1.50992	+	0.682528i	−1.50992	+	0.682528i	−0.524307	+	0.851529i	$0.675676\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.383954	−	1.78155i	−0.383954	−	1.78155i	−0.594633	−	0.803997i	$-0.702703\pi$
$37$	−0.383954	−	1.78155i	−0.383954	−	1.78155i	0.210679	−	0.977555i	$-0.432432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.169001	−	0.985616i	$-0.445946\pi$
$41$		0			0		−0.169001	+	0.985616i	$0.554054\pi$
$42$		0			0
$43$	0.422810	−	0.405231i	0.422810	−	0.405231i	−0.450204	−	0.892926i	$-0.648649\pi$
$43$	0.422810	−	0.405231i	0.422810	−	0.405231i	0.873014	+	0.487695i	$0.162162\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.956167	−	0.292823i	$-0.0945946\pi$
$47$		0			0		−0.956167	+	0.292823i	$0.905405\pi$
$48$		0			0
$49$	−0.599881	−	1.95882i	−0.599881	−	1.95882i
$50$		0			0
$51$		0			0
$52$	1.03919	+	1.53753i	1.03919	+	1.53753i
$53$		0			0		−0.892926	−	0.450204i	$-0.851351\pi$
$53$		0			0		0.892926	+	0.450204i	$0.148649\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.996397	−	0.0848059i	$-0.0270270\pi$
$59$		0			0		−0.996397	+	0.0848059i	$0.972973\pi$
$60$		0			0
$61$	−0.535687	−	1.51614i	−0.535687	−	1.51614i	−0.828510	−	0.559975i	$-0.810811\pi$
$61$	−0.535687	−	1.51614i	−0.535687	−	1.51614i	0.292823	−	0.956167i	$-0.405405\pi$
$62$		0			0
$63$		0			0
$64$	0.721956	−	0.691939i	0.721956	−	0.691939i
$65$		0			0
$66$		0			0
$67$	−0.0427385	−	0.164139i	−0.0427385	−	0.164139i	0.942877	−	0.333140i	$-0.108108\pi$
$67$	−0.0427385	−	0.164139i	−0.0427385	−	0.164139i	−0.985616	+	0.169001i	$0.945946\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.996397	−	0.0848059i	$-0.0270270\pi$
$71$		0			0		−0.996397	+	0.0848059i	$0.972973\pi$
$72$		0			0
$73$	−1.60355	+	1.08381i	−1.60355	+	1.08381i	−0.660675	+	0.750672i	$0.729730\pi$
$73$	−1.60355	+	1.08381i	−1.60355	+	1.08381i	−0.942877	+	0.333140i	$0.891892\pi$
$74$		0			0
$75$		0			0
$76$	−0.0800337	+	1.88406i	−0.0800337	+	1.88406i
$77$		0			0
$78$		0			0
$79$	−0.233876	+	1.36397i	−0.233876	+	1.36397i	0.594633	+	0.803997i	$0.297297\pi$
$79$	−0.233876	+	1.36397i	−0.233876	+	1.36397i	−0.828510	+	0.559975i	$0.810811\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.691939	−	0.721956i	$-0.256757\pi$
$83$		0			0		−0.691939	+	0.721956i	$0.743243\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.927889	−	0.372856i	$-0.121622\pi$
$89$		0			0		−0.927889	+	0.372856i	$0.878378\pi$
$90$		0			0
$91$	3.09821	+	0.948816i	3.09821	+	0.948816i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.313628	−	0.126026i	0.313628	−	0.126026i	−0.210679	−	0.977555i	$-0.567568\pi$
$97$	0.313628	−	0.126026i	0.313628	−	0.126026i	0.524307	+	0.851529i	$0.324324\pi$
$98$		0			0
$99$		0			0
$100$	−0.873014	−	0.487695i	−0.873014	−	0.487695i
$101$		0			0		0.892926	−	0.450204i	$-0.148649\pi$
$101$		0			0		−0.892926	+	0.450204i	$0.851351\pi$
$102$		0			0
$103$	−0.594938	+	1.68384i	−0.594938	+	1.68384i	0.127018	+	0.991900i	$0.459459\pi$
$103$	−0.594938	+	1.68384i	−0.594938	+	1.68384i	−0.721956	+	0.691939i	$0.756757\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.721956	−	0.691939i	$-0.243243\pi$
$107$		0			0		−0.721956	+	0.691939i	$0.756757\pi$
$108$		0			0
$109$	0.0741035	+	0.0413967i	0.0741035	+	0.0413967i	0.524307	−	0.851529i	$-0.324324\pi$
$109$	0.0741035	+	0.0413967i	0.0741035	+	0.0413967i	−0.450204	+	0.892926i	$0.648649\pi$
$110$		0			0
$111$		0			0
$112$	0.221777	−	1.73189i	0.221777	−	1.73189i
$113$		0			0		0.450204	−	0.892926i	$-0.351351\pi$
$113$		0			0		−0.450204	+	0.892926i	$0.648649\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.911228	−	0.411901i	−0.911228	−	0.411901i
$122$		0			0
$123$		0			0
$124$	−1.28922	+	1.04097i	−1.28922	+	1.04097i
$125$		0			0
$126$		0			0
$127$	0.123383	+	0.402889i	0.123383	+	0.402889i	0.996397	−	0.0848059i	$-0.0270270\pi$
$127$	0.123383	+	0.402889i	0.123383	+	0.402889i	−0.873014	+	0.487695i	$0.837838\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.991900	−	0.127018i	$-0.959459\pi$
$131$		0			0		0.991900	+	0.127018i	$0.0405405\pi$
$132$		0			0
$133$	1.95788	+	2.64723i	1.95788	+	2.64723i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.778036	−	0.628220i	$-0.783784\pi$
$137$		0			0		0.778036	+	0.628220i	$0.216216\pi$
$138$		0			0
$139$	1.02806	−	1.16810i	1.02806	−	1.16810i	0.0424412	−	0.999099i	$-0.486486\pi$
$139$	1.02806	−	1.16810i	1.02806	−	1.16810i	0.985616	−	0.169001i	$-0.0540541\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−0.820477	−	1.62732i	−0.820477	−	1.62732i
$149$		0			0		0.942877	−	0.333140i	$-0.108108\pi$
$149$		0			0		−0.942877	+	0.333140i	$0.891892\pi$
$150$		0			0
$151$	−1.43554	−	0.439628i	−1.43554	−	0.439628i	−0.524307	−	0.851529i	$-0.675676\pi$
$151$	−1.43554	−	0.439628i	−1.43554	−	0.439628i	−0.911228	+	0.411901i	$0.864865\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.823060	−	1.82081i	0.823060	−	1.82081i	0.372856	−	0.927889i	$-0.378378\pi$
$157$	0.823060	−	1.82081i	0.823060	−	1.82081i	0.450204	−	0.892926i	$-0.351351\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.74577	+	0.880198i	1.74577	+	0.880198i	0.967733	+	0.251978i	$0.0810811\pi$
$163$	1.74577	+	0.880198i	1.74577	+	0.880198i	0.778036	+	0.628220i	$0.216216\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.967733	−	0.251978i	$-0.918919\pi$
$167$		0			0		0.967733	+	0.251978i	$0.0810811\pi$
$168$		0			0
$169$	−1.90145	+	1.53531i	−1.90145	+	1.53531i
$170$		0			0
$171$		0			0
$172$	0.307058	−	0.498694i	0.307058	−	0.498694i
$173$		0			0		0.594633	−	0.803997i	$-0.297297\pi$
$173$		0			0		−0.594633	+	0.803997i	$0.702703\pi$
$174$		0			0
$175$	−1.72091	+	0.295080i	−1.72091	+	0.295080i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.999099	−	0.0424412i	$-0.0135135\pi$
$179$		0			0		−0.999099	+	0.0424412i	$0.986486\pi$
$180$		0			0
$181$	−0.594876	−	0.675911i	−0.594876	−	0.675911i	0.372856	−	0.927889i	$-0.378378\pi$
$181$	−0.594876	−	0.675911i	−0.594876	−	0.675911i	−0.967733	+	0.251978i	$0.918919\pi$
$182$		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		−0.594633	−	0.803997i	$-0.702703\pi$
$191$		0			0		0.594633	+	0.803997i	$0.297297\pi$
$192$		0			0
$193$	0.282203	−	0.417533i	0.282203	−	0.417533i	−0.660675	−	0.750672i	$-0.729730\pi$
$193$	0.282203	−	0.417533i	0.282203	−	0.417533i	0.942877	+	0.333140i	$0.108108\pi$
$194$		0			0
$195$		0			0
$196$	−1.07410	−	1.74445i	−1.07410	−	1.74445i
$197$		0			0		0.892926	−	0.450204i	$-0.148649\pi$
$197$		0			0		−0.892926	+	0.450204i	$0.851351\pi$
$198$		0			0
$199$	−0.0107816	−	0.253807i	−0.0107816	−	0.253807i	−0.996397	−	0.0848059i	$-0.972973\pi$
$199$	−0.0107816	−	0.253807i	−0.0107816	−	0.253807i	0.985616	−	0.169001i	$-0.0540541\pi$
$200$		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	+	1.22607i	1.39308	+	1.22607i
$209$		0			0
$210$		0			0
$211$	−0.253120	−	0.0215437i	−0.253120	−	0.0215437i	−0.0424412	−	0.999099i	$-0.513514\pi$
$211$	−0.253120	−	0.0215437i	−0.253120	−	0.0215437i	−0.210679	+	0.977555i	$0.567568\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.609538	+	2.82827i	−0.609538	+	2.82827i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.721956	+	0.691939i	0.721956	+	0.691939i
$223$	0.721956	+	0.691939i	0.721956	+	0.691939i
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.251978	−	0.967733i	$-0.581081\pi$
$227$		0			0		0.251978	+	0.967733i	$0.418919\pi$
$228$		0			0
$229$	−1.91123	−	0.411901i	−1.91123	−	0.411901i	−0.911228	−	0.411901i	$-0.864865\pi$
$229$	−1.91123	−	0.411901i	−1.91123	−	0.411901i	−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.967733	−	0.251978i	$-0.918919\pi$
$233$		0			0		0.967733	+	0.251978i	$0.0810811\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.487695	−	0.873014i	$-0.662162\pi$
$239$		0			0		0.487695	+	0.873014i	$0.337838\pi$
$240$		0			0
$241$	0.623541	−	1.01269i	0.623541	−	1.01269i	−0.372856	−	0.927889i	$-0.621622\pi$
$241$	0.623541	−	1.01269i	0.623541	−	1.01269i	0.996397	−	0.0848059i	$-0.0270270\pi$
$242$		0			0
$243$		0			0
$244$	−0.900436	−	1.33224i	−0.900436	−	1.33224i
$245$		0			0
$246$		0			0
$247$	−3.49639	+	0.148525i	−3.49639	+	0.148525i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.927889	−	0.372856i	$-0.878378\pi$
$251$		0			0		0.927889	+	0.372856i	$0.121622\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.524307	−	0.851529i	0.524307	−	0.851529i
$257$		0			0		0.0848059	−	0.996397i	$-0.472973\pi$
$257$		0			0		−0.0848059	+	0.996397i	$0.527027\pi$
$258$		0			0
$259$	−2.89958	−	1.31070i	−2.89958	−	1.31070i
$260$		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	−	0.148074i	−0.0827188	−	0.148074i
$269$		0			0		−0.967733	−	0.251978i	$-0.918919\pi$
$269$		0			0		0.967733	+	0.251978i	$0.0810811\pi$
$270$		0			0
$271$	0.0851690	+	0.496707i	0.0851690	+	0.496707i	0.996397	+	0.0848059i	$0.0270270\pi$
$271$	0.0851690	+	0.496707i	0.0851690	+	0.496707i	−0.911228	+	0.411901i	$0.864865\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.06990	+	1.32504i	1.06990	+	1.32504i	0.942877	+	0.333140i	$0.108108\pi$
$277$	1.06990	+	1.32504i	1.06990	+	1.32504i	0.127018	+	0.991900i	$0.459459\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.803997	−	0.594633i	$-0.202703\pi$
$281$		0			0		−0.803997	+	0.594633i	$0.797297\pi$
$282$		0			0
$283$	−0.594876	+	1.17987i	−0.594876	+	1.17987i	0.372856	+	0.927889i	$0.378378\pi$
$283$	−0.594876	+	1.17987i	−0.594876	+	1.17987i	−0.967733	+	0.251978i	$0.918919\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.911228	+	0.411901i	0.911228	+	0.411901i
$290$		0			0
$291$		0			0
$292$	−1.27871	+	1.45290i	−1.27871	+	1.45290i
$293$		0			0		0.778036	−	0.628220i	$-0.216216\pi$
$293$		0			0		−0.778036	+	0.628220i	$0.783784\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−0.129883	−	1.01427i	−0.129883	−	1.01427i
$302$		0			0
$303$		0			0
$304$	0.397289	+	1.84343i	0.397289	+	1.84343i
$305$		0			0
$306$		0			0
$307$	1.27844	+	1.12517i	1.27844	+	1.12517i	0.985616	+	0.169001i	$0.0540541\pi$
$307$	1.27844	+	1.12517i	1.27844	+	1.12517i	0.292823	+	0.956167i	$0.405405\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.985616	−	0.169001i	$-0.0540541\pi$
$311$		0			0		−0.985616	+	0.169001i	$0.945946\pi$
$312$		0			0
$313$	1.53751	−	1.13714i	1.53751	−	1.13714i	0.594633	−	0.803997i	$-0.297297\pi$
$313$	1.53751	−	1.13714i	1.53751	−	1.13714i	0.942877	−	0.333140i	$-0.108108\pi$
$314$		0			0
$315$		0			0
$316$	0.117361	+	1.37889i	0.117361	+	1.37889i
$317$		0			0		−0.411901	−	0.911228i	$-0.635135\pi$
$317$		0			0		0.411901	+	0.911228i	$0.364865\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	0.764397	−	1.69104i	0.764397	−	1.69104i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.32322	+	1.38062i	−1.32322	+	1.38062i	−0.450204	+	0.892926i	$0.648649\pi$
$331$	−1.32322	+	1.38062i	−1.32322	+	1.38062i	−0.873014	+	0.487695i	$0.837838\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.401764	+	0.719191i	−0.401764	+	0.719191i	−0.996397	−	0.0848059i	$-0.972973\pi$
$337$	−0.401764	+	0.719191i	−0.401764	+	0.719191i	0.594633	+	0.803997i	$0.297297\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.72632	−	0.609949i	−1.72632	−	0.609949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.628220	−	0.778036i	$-0.716216\pi$
$347$		0			0		0.628220	+	0.778036i	$0.283784\pi$
$348$		0			0
$349$	0.734987	+	1.82908i	0.734987	+	1.82908i	0.524307	+	0.851529i	$0.324324\pi$
$349$	0.734987	+	1.82908i	0.734987	+	1.82908i	0.210679	+	0.977555i	$0.432432\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.0848059	−	0.996397i	$-0.527027\pi$
$353$		0			0		0.0848059	+	0.996397i	$0.472973\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.169001	−	0.985616i	$-0.554054\pi$
$359$		0			0		0.169001	+	0.985616i	$0.445946\pi$
$360$		0			0
$361$	−2.11773	−	1.43134i	−2.11773	−	1.43134i
$362$		0			0
$363$		0			0
$364$	3.23732	+	0.137520i	3.23732	+	0.137520i
$365$		0			0
$366$		0			0
$367$	−0.415298	−	0.0712099i	−0.415298	−	0.0712099i	−0.0424412	−	0.999099i	$-0.513514\pi$
$367$	−0.415298	−	0.0712099i	−0.415298	−	0.0712099i	−0.372856	+	0.927889i	$0.621622\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.00003	+	1.47960i	1.00003	+	1.47960i	0.873014	+	0.487695i	$0.162162\pi$
$373$	1.00003	+	1.47960i	1.00003	+	1.47960i	0.127018	+	0.991900i	$0.459459\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.27871	−	0.332951i	1.27871	−	0.332951i	0.450204	−	0.892926i	$-0.351351\pi$
$379$	1.27871	−	0.332951i	1.27871	−	0.332951i	0.828510	+	0.559975i	$0.189189\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.967733	−	0.251978i	$-0.0810811\pi$
$383$		0			0		−0.967733	+	0.251978i	$0.918919\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.271752	−	0.200987i	0.271752	−	0.200987i
$389$		0			0		−0.803997	−	0.594633i	$-0.797297\pi$
$389$		0			0		0.803997	+	0.594633i	$0.202703\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.43554	−	0.439628i	1.43554	−	0.439628i	0.524307	−	0.851529i	$-0.324324\pi$
$397$	1.43554	−	0.439628i	1.43554	−	0.439628i	0.911228	+	0.411901i	$0.135135\pi$
$398$		0			0
$399$		0			0
$400$	−0.967733	−	0.251978i	−0.967733	−	0.251978i
$401$		0			0		−0.487695	−	0.873014i	$-0.662162\pi$
$401$		0			0		0.487695	+	0.873014i	$0.337838\pi$
$402$		0			0
$403$	−2.12775	−	2.22006i	−2.12775	−	2.22006i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.74577	+	0.376242i	−1.74577	+	0.376242i	−0.967733	−	0.251978i	$-0.918919\pi$
$409$	−1.74577	+	0.376242i	−1.74577	+	0.376242i	−0.778036	+	0.628220i	$0.783784\pi$
$410$		0			0
$411$		0			0
$412$	−0.151451	+	1.77942i	−0.151451	+	1.77942i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.851529	−	0.524307i	$-0.175676\pi$
$419$		0			0		−0.851529	+	0.524307i	$0.824324\pi$
$420$		0			0
$421$	0.233876	−	0.244022i	0.233876	−	0.244022i	−0.594633	−	0.803997i	$-0.702703\pi$
$421$	0.233876	−	0.244022i	0.233876	−	0.244022i	0.828510	+	0.559975i	$0.189189\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−2.68454	−	0.822130i	−2.68454	−	0.822130i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.660675	−	0.750672i	$-0.270270\pi$
$431$		0			0		−0.660675	+	0.750672i	$0.729730\pi$
$432$		0			0
$433$	−0.0800337	+	0.0282777i	−0.0800337	+	0.0282777i	−0.372856	−	0.927889i	$-0.621622\pi$
$433$	−0.0800337	+	0.0282777i	−0.0800337	+	0.0282777i	0.292823	+	0.956167i	$0.405405\pi$
$434$		0			0
$435$		0			0
$436$	0.0821435	+	0.0213885i	0.0821435	+	0.0213885i
$437$		0			0
$438$		0			0
$439$	−1.59497	−	1.17963i	−1.59497	−	1.17963i	−0.873014	−	0.487695i	$-0.837838\pi$
$439$	−1.59497	−	1.17963i	−1.59497	−	1.17963i	−0.721956	−	0.691939i	$-0.756757\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.333140	−	0.942877i	$-0.391892\pi$
$443$		0			0		−0.333140	+	0.942877i	$0.608108\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.221777	−	1.73189i	−0.221777	−	1.73189i
$449$		0			0		−0.942877	−	0.333140i	$-0.891892\pi$
$449$		0			0		0.942877	+	0.333140i	$0.108108\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.22824	+	0.264706i	1.22824	+	0.264706i	0.778036	−	0.628220i	$-0.216216\pi$
$457$	1.22824	+	0.264706i	1.22824	+	0.264706i	0.450204	+	0.892926i	$0.351351\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.750672	−	0.660675i	$-0.770270\pi$
$461$		0			0		0.750672	+	0.660675i	$0.229730\pi$
$462$		0			0
$463$	0.327832	−	1.52115i	0.327832	−	1.52115i	−0.450204	−	0.892926i	$-0.648649\pi$
$463$	0.327832	−	1.52115i	0.327832	−	1.52115i	0.778036	−	0.628220i	$-0.216216\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.828510	−	0.559975i	$-0.189189\pi$
$467$		0			0		−0.828510	+	0.559975i	$0.810811\pi$
$468$		0			0
$469$	−0.274792	−	0.110420i	−0.274792	−	0.110420i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	1.64629	−	0.919673i	1.64629	−	0.919673i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.851529	−	0.524307i	$-0.824324\pi$
$479$		0			0		0.851529	+	0.524307i	$0.175676\pi$
$480$		0			0
$481$	2.87994	−	1.77325i	2.87994	−	1.77325i
$482$		0			0
$483$		0			0
$484$	−0.985616	−	0.169001i	−0.985616	−	0.169001i
$485$		0			0
$486$		0			0
$487$	−0.887456	−	0.152170i	−0.887456	−	0.152170i	−0.292823	−	0.956167i	$-0.594595\pi$
$487$	−0.887456	−	0.152170i	−0.887456	−	0.152170i	−0.594633	+	0.803997i	$0.702703\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.873014	−	0.487695i	$-0.162162\pi$
$491$		0			0		−0.873014	+	0.487695i	$0.837838\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.985319	+	1.33224i	−0.985319	+	1.33224i
$497$		0			0
$498$		0			0
$499$	−0.887456	+	1.76016i	−0.887456	+	1.76016i	−0.292823	+	0.956167i	$0.594595\pi$
$499$	−0.887456	+	1.76016i	−0.887456	+	1.76016i	−0.594633	+	0.803997i	$0.702703\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.594633	−	0.803997i	$-0.297297\pi$
$503$		0			0		−0.594633	+	0.803997i	$0.702703\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	0.220921	+	0.358799i	0.220921	+	0.358799i
$509$		0			0		−0.251978	−	0.967733i	$-0.581081\pi$
$509$		0			0		0.251978	+	0.967733i	$0.418919\pi$
$510$		0			0
$511$	−0.143425	+	3.37633i	−0.143425	+	3.37633i
$512$		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		−0.450204	−	0.892926i	$-0.648649\pi$
$521$		0			0		0.450204	+	0.892926i	$0.351351\pi$
$522$		0			0
$523$	0.840715	−	0.739922i	0.840715	−	0.739922i	−0.127018	−	0.991900i	$-0.540541\pi$
$523$	0.840715	−	0.739922i	0.840715	−	0.739922i	0.967733	+	0.251978i	$0.0810811\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.721956	−	0.691939i	−0.721956	−	0.691939i
$530$		0			0
$531$		0			0
$532$	2.56175	+	2.06847i	2.56175	+	2.06847i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.06990	−	1.32504i	1.06990	−	1.32504i	0.127018	−	0.991900i	$-0.459459\pi$
$541$	1.06990	−	1.32504i	1.06990	−	1.32504i	0.942877	−	0.333140i	$-0.108108\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.114368	−	0.226835i	−0.114368	−	0.226835i	0.828510	−	0.559975i	$-0.189189\pi$
$547$	−0.114368	−	0.226835i	−0.114368	−	0.226835i	−0.942877	+	0.333140i	$0.891892\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	1.67192	+	1.74445i	1.67192	+	1.74445i
$554$		0			0
$555$		0			0
$556$	0.700549	−	1.38946i	0.700549	−	1.38946i
$557$		0			0		0.127018	−	0.991900i	$-0.459459\pi$
$557$		0			0		−0.127018	+	0.991900i	$0.540541\pi$
$558$		0			0
$559$	0.970457	+	0.489294i	0.970457	+	0.489294i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.942877	−	0.333140i	$-0.891892\pi$
$563$		0			0		0.942877	+	0.333140i	$0.108108\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.873014	−	0.487695i	$-0.837838\pi$
$569$		0			0		0.873014	+	0.487695i	$0.162162\pi$
$570$		0			0
$571$	0.139223	+	0.307997i	0.139223	+	0.307997i	0.967733	−	0.251978i	$-0.0810811\pi$
$571$	0.139223	+	0.307997i	0.139223	+	0.307997i	−0.828510	+	0.559975i	$0.810811\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−0.253120	+	1.97665i	−0.253120	+	1.97665i	−0.0424412	+	0.999099i	$0.513514\pi$
$577$	−0.253120	+	1.97665i	−0.253120	+	1.97665i	−0.210679	+	0.977555i	$0.567568\pi$
$578$		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		0.0424412	−	0.999099i	$-0.486486\pi$
$587$		0			0		−0.0424412	+	0.999099i	$0.513514\pi$
$588$		0			0
$589$	−0.396897	−	3.09942i	−0.396897	−	3.09942i
$590$		0			0
$591$		0			0
$592$	−1.20405	−	1.36807i	−1.20405	−	1.36807i
$593$		0			0		0.0424412	−	0.999099i	$-0.486486\pi$
$593$		0			0		−0.0424412	+	0.999099i	$0.513514\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.628220	−	0.778036i	$-0.716216\pi$
$599$		0			0		0.628220	+	0.778036i	$0.283784\pi$
$600$		0			0
$601$	−0.169459	+	0.00719853i	−0.169459	+	0.00719853i	−0.127018	−	0.991900i	$-0.540541\pi$
$601$	−0.169459	+	0.00719853i	−0.169459	+	0.00719853i	−0.0424412	+	0.999099i	$0.513514\pi$
$602$		0			0
$603$		0			0
$604$	−1.49999	−	0.0637189i	−1.49999	−	0.0637189i
$605$		0			0
$606$		0			0
$607$	1.62841	−	1.00265i	1.62841	−	1.00265i	0.660675	−	0.750672i	$-0.270270\pi$
$607$	1.62841	−	1.00265i	1.62841	−	1.00265i	0.967733	−	0.251978i	$-0.0810811\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.348707	−	1.33922i	0.348707	−	1.33922i	−0.524307	−	0.851529i	$-0.675676\pi$
$613$	0.348707	−	1.33922i	0.348707	−	1.33922i	0.873014	−	0.487695i	$-0.162162\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.559975	−	0.828510i	$-0.689189\pi$
$617$		0			0		0.559975	+	0.828510i	$0.310811\pi$
$618$		0			0
$619$	1.29282	+	0.956167i	1.29282	+	0.956167i	1.00000			$0$
$619$	1.29282	+	0.956167i	1.29282	+	0.956167i	0.292823	+	0.956167i	$0.405405\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.0424412	+	0.999099i	0.0424412	+	0.999099i
$626$		0			0
$627$		0			0
$628$	0.337697	−	1.96946i	0.337697	−	1.96946i
$629$		0			0
$630$		0			0
$631$	−0.287818	+	1.67856i	−0.287818	+	1.67856i	0.372856	+	0.927889i	$0.378378\pi$
$631$	−0.287818	+	1.67856i	−0.287818	+	1.67856i	−0.660675	+	0.750672i	$0.729730\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	3.05662	−	2.26066i	3.05662	−	2.26066i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.372856	−	0.927889i	$-0.378378\pi$
$641$		0			0		−0.372856	+	0.927889i	$0.621622\pi$
$642$		0			0
$643$	−0.535412	−	1.06193i	−0.535412	−	1.06193i	−0.985616	−	0.169001i	$-0.945946\pi$
$643$	−0.535412	−	1.06193i	−0.535412	−	1.06193i	0.450204	−	0.892926i	$-0.351351\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.0848059	−	0.996397i	$-0.472973\pi$
$647$		0			0		−0.0848059	+	0.996397i	$0.527027\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	1.91123	+	0.411901i	1.91123	+	0.411901i
$653$		0			0		0.873014	−	0.487695i	$-0.162162\pi$
$653$		0			0		−0.873014	+	0.487695i	$0.837838\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.251978	−	0.967733i	$-0.418919\pi$
$659$		0			0		−0.251978	+	0.967733i	$0.581081\pi$
$660$		0			0
$661$	0.618234	−	0.248427i	0.618234	−	0.248427i	−0.0424412	−	0.999099i	$-0.513514\pi$
$661$	0.618234	−	0.248427i	0.618234	−	0.248427i	0.660675	+	0.750672i	$0.270270\pi$
$662$		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
$673$	1.90763	−	0.496707i	1.90763	−	0.496707i	0.911228	−	0.411901i	$-0.135135\pi$
$673$	1.90763	−	0.496707i	1.90763	−	0.496707i	0.996397	−	0.0848059i	$-0.0270270\pi$
$674$		0			0
$675$		0			0
$676$	−1.45323	+	1.96490i	−1.45323	+	1.96490i
$677$		0			0		0.927889	−	0.372856i	$-0.121622\pi$
$677$		0			0		−0.927889	+	0.372856i	$0.878378\pi$
$678$		0			0
$679$	0.148707	−	0.571118i	0.148707	−	0.571118i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.487695	−	0.873014i	$-0.337838\pi$
$683$		0			0		−0.487695	+	0.873014i	$0.662162\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	0.171490	−	0.559975i	0.171490	−	0.559975i
$689$		0			0
$690$		0			0
$691$	0.117361	−	1.37889i	0.117361	−	1.37889i	−0.660675	−	0.750672i	$-0.729730\pi$
$691$	0.117361	−	1.37889i	0.117361	−	1.37889i	0.778036	−	0.628220i	$-0.216216\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.59103	+	0.719191i	−1.59103	+	0.719191i
$701$		0			0		0.803997	−	0.594633i	$-0.202703\pi$
$701$		0			0		−0.803997	+	0.594633i	$0.797297\pi$
$702$		0			0
$703$	3.42433	+	0.291453i	3.42433	+	0.291453i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	0.503956i		−	0.503956i	−0.967733	−	0.251978i	$-0.918919\pi$
$709$		−	0.503956i		−	0.503956i	0.967733	−	0.251978i	$-0.0810811\pi$
$710$		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		−0.803997	−	0.594633i	$-0.797297\pi$
$719$		0			0		0.803997	+	0.594633i	$0.202703\pi$
$720$		0			0
$721$	1.74608	+	2.58342i	1.74608	+	2.58342i
$722$		0			0
$723$		0			0
$724$	−0.745996	−	0.504205i	−0.745996	−	0.504205i
$725$		0			0
$726$		0			0
$727$	−0.925292	−	1.25108i	−0.925292	−	1.25108i	−0.967733	−	0.251978i	$-0.918919\pi$
$727$	−0.925292	−	1.25108i	−0.925292	−	1.25108i	0.0424412	−	0.999099i	$-0.486486\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	0.183403	−	0.175777i	0.183403	−	0.175777i	−0.594633	−	0.803997i	$-0.702703\pi$
$733$	0.183403	−	0.175777i	0.183403	−	0.175777i	0.778036	+	0.628220i	$0.216216\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.517529	+	0.640948i	0.517529	+	0.640948i	0.967733	−	0.251978i	$-0.0810811\pi$
$739$	0.517529	+	0.640948i	0.517529	+	0.640948i	−0.450204	+	0.892926i	$0.648649\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.333140	−	0.942877i	$-0.608108\pi$
$743$		0			0		0.333140	+	0.942877i	$0.391892\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.0845766	+	1.99100i	−0.0845766	+	1.99100i	0.0424412	+	0.999099i	$0.486486\pi$
$751$	−0.0845766	+	1.99100i	−0.0845766	+	1.99100i	−0.127018	+	0.991900i	$0.540541\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.112602	−	0.656696i	−0.112602	−	0.656696i	−0.985616	−	0.169001i	$-0.945946\pi$
$757$	−0.112602	−	0.656696i	−0.112602	−	0.656696i	0.873014	−	0.487695i	$-0.162162\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.127018	−	0.991900i	$-0.459459\pi$
$761$		0			0		−0.127018	+	0.991900i	$0.540541\pi$
$762$		0			0
$763$	0.135051	−	0.0610467i	0.135051	−	0.0610467i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.221777	+	0.123892i	0.221777	+	0.123892i	0.594633	−	0.803997i	$-0.297297\pi$
$769$	0.221777	+	0.123892i	0.221777	+	0.123892i	−0.372856	+	0.927889i	$0.621622\pi$
$770$		0			0
$771$		0			0
$772$	0.167888	−	0.475169i	0.167888	−	0.475169i
$773$		0			0		−0.660675	−	0.750672i	$-0.729730\pi$
$773$		0			0		0.660675	+	0.750672i	$0.270270\pi$
$774$		0			0
$775$	1.56237	+	0.552019i	1.56237	+	0.552019i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1.47901	−	1.41752i	−1.47901	−	1.41752i
$785$		0			0
$786$		0			0
$787$	−0.492645	+	0.106173i	−0.492645	+	0.106173i	−0.450204	−	0.892926i	$-0.648649\pi$
$787$	−0.492645	+	0.106173i	−0.492645	+	0.106173i	−0.0424412	+	0.999099i	$0.513514\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	2.32172	−	1.87466i	2.32172	−	1.87466i
$794$		0			0
$795$		0			0
$796$	−0.0743874	−	0.242900i	−0.0743874	−	0.242900i
$797$		0			0		0.628220	−	0.778036i	$-0.283784\pi$
$797$		0			0		−0.628220	+	0.778036i	$0.716216\pi$
$798$		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		−0.721956	−	0.691939i	$-0.756757\pi$
$809$		0			0		0.721956	+	0.691939i	$0.243243\pi$
$810$		0			0
$811$	0.651325	−	0.140371i	0.651325	−	0.140371i	0.127018	−	0.991900i	$-0.459459\pi$
$811$	0.651325	−	0.140371i	0.651325	−	0.140371i	0.524307	+	0.851529i	$0.324324\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.497198	+	0.986133i	0.497198	+	0.986133i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.0848059	−	0.996397i	$-0.527027\pi$
$821$		0			0		0.0848059	+	0.996397i	$0.472973\pi$
$822$		0			0
$823$	−0.618406	−	0.544265i	−0.618406	−	0.544265i	0.292823	−	0.956167i	$-0.405405\pi$
$823$	−0.618406	−	0.544265i	−0.618406	−	0.544265i	−0.911228	+	0.411901i	$0.864865\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.0424412	−	0.999099i	$-0.486486\pi$
$827$		0			0		−0.0424412	+	0.999099i	$0.513514\pi$
$828$		0			0
$829$	−0.282203	−	1.08381i	−0.282203	−	1.08381i	−0.942877	−	0.333140i	$-0.891892\pi$
$829$	−0.282203	−	1.08381i	−0.282203	−	1.08381i	0.660675	−	0.750672i	$-0.270270\pi$
$830$		0			0
$831$		0			0
$832$	1.65707	+	0.835478i	1.65707	+	0.835478i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.450204	−	0.892926i	$-0.351351\pi$
$839$		0			0		−0.450204	+	0.892926i	$0.648649\pi$
$840$		0			0
$841$	−0.524307	+	0.851529i	−0.524307	+	0.851529i
$842$		0			0
$843$		0			0
$844$	−0.250382	+	0.0429322i	−0.250382	+	0.0429322i
$845$		0			0
$846$		0			0
$847$	−1.52431	+	0.851529i	−1.52431	+	0.851529i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$			1.50134i			1.50134i	0.660675	+	0.750672i	$0.270270\pi$
$853$			1.50134i			1.50134i	−0.660675	+	0.750672i	$0.729730\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.851529	−	0.524307i	$-0.175676\pi$
$857$		0			0		−0.851529	+	0.524307i	$0.824324\pi$
$858$		0			0
$859$	−1.66483	−	1.02508i	−1.66483	−	1.02508i	−0.942877	−	0.333140i	$-0.891892\pi$
$859$	−1.66483	−	1.02508i	−1.66483	−	1.02508i	−0.721956	−	0.691939i	$-0.756757\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.873014	−	0.487695i	$-0.162162\pi$
$863$		0			0		−0.873014	+	0.487695i	$0.837838\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	0.122791	+	2.89060i	0.122791	+	2.89060i
$869$		0			0
$870$		0			0
$871$	0.260783	−	0.176259i	0.260783	−	0.176259i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.127323	+	0.112058i	0.127323	+	0.112058i	0.721956	−	0.691939i	$-0.243243\pi$
$877$	0.127323	+	0.112058i	0.127323	+	0.112058i	−0.594633	+	0.803997i	$0.702703\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.977555	−	0.210679i	$-0.932432\pi$
$881$		0			0		0.977555	+	0.210679i	$0.0675676\pi$
$882$		0			0
$883$	−0.789321	+	0.977555i	−0.789321	+	0.977555i	0.210679	+	0.977555i	$0.432432\pi$
$883$	−0.789321	+	0.977555i	−0.789321	+	0.977555i	−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.927889	−	0.372856i	$-0.878378\pi$
$887$		0			0		0.927889	+	0.372856i	$0.121622\pi$
$888$		0			0
$889$	0.693679	+	0.245092i	0.693679	+	0.245092i
$890$		0			0
$891$		0			0
$892$	0.873014	+	0.487695i	0.873014	+	0.487695i
$893$		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
$907$	−0.953956	+	1.08390i	−0.953956	+	1.08390i	0.0424412	+	0.999099i	$0.486486\pi$
$907$	−0.953956	+	1.08390i	−0.953956	+	1.08390i	−0.996397	+	0.0848059i	$0.972973\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.956167	−	0.292823i	$-0.905405\pi$
$911$		0			0		0.956167	+	0.292823i	$0.0945946\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.95335	+	0.0829772i	−1.95335	+	0.0829772i
$917$		0			0
$918$		0			0
$919$	1.52070	−	0.936335i	1.52070	−	0.936335i	0.524307	−	0.851529i	$-0.324324\pi$
$919$	1.52070	−	0.936335i	1.52070	−	0.936335i	0.996397	−	0.0848059i	$-0.0270270\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.955527	+	1.55188i	−0.955527	+	1.55188i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.977555	−	0.210679i	$-0.0675676\pi$
$929$		0			0		−0.977555	+	0.210679i	$0.932432\pi$
$930$		0			0
$931$	3.86318			3.86318
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.164842	−	0.295080i	−0.164842	−	0.295080i	0.778036	−	0.628220i	$-0.216216\pi$
$937$	−0.164842	−	0.295080i	−0.164842	−	0.295080i	−0.942877	+	0.333140i	$0.891892\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.956167	−	0.292823i	$-0.0945946\pi$
$941$		0			0		−0.956167	+	0.292823i	$0.905405\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.559975	−	0.828510i	$-0.310811\pi$
$947$		0			0		−0.559975	+	0.828510i	$0.689189\pi$
$948$		0			0
$949$	−2.88779	−	2.13580i	−2.88779	−	2.13580i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.942877	−	0.333140i	$-0.108108\pi$
$953$		0			0		−0.942877	+	0.333140i	$0.891892\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1.15335	−	1.31046i	1.15335	−	1.31046i
$962$		0			0
$963$		0			0
$964$	0.348244	−	1.13714i	0.348244	−	1.13714i
$965$		0			0
$966$		0			0
$967$	0.594938	−	0.299962i	0.594938	−	0.299962i	−0.127018	−	0.991900i	$-0.540541\pi$
$967$	0.594938	−	0.299962i	0.594938	−	0.299962i	0.721956	+	0.691939i	$0.243243\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.985616	−	0.169001i	$-0.945946\pi$
$971$		0			0		0.985616	+	0.169001i	$0.0540541\pi$
$972$		0			0
$973$	−0.572404	−	2.65596i	−0.572404	−	2.65596i
$974$		0			0
$975$		0			0
$976$	−1.20708	−	1.06236i	−1.20708	−	1.06236i
$977$		0			0		−0.828510	−	0.559975i	$-0.810811\pi$
$977$		0			0		0.828510	+	0.559975i	$0.189189\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.292823	−	0.956167i	$-0.405405\pi$
$983$		0			0		−0.292823	+	0.956167i	$0.594595\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−3.34615	+	1.02475i	−3.34615	+	1.02475i
$989$		0			0
$990$		0			0
$991$	−1.25531	−	1.55467i	−1.25531	−	1.55467i	−0.660675	−	0.750672i	$-0.729730\pi$
$991$	−1.25531	−	1.55467i	−1.25531	−	1.55467i	−0.594633	−	0.803997i	$-0.702703\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.220921	−	1.02508i	−0.220921	−	1.02508i	−0.942877	−	0.333140i	$-0.891892\pi$
$997$	−0.220921	−	1.02508i	−0.220921	−	1.02508i	0.721956	−	0.691939i	$-0.243243\pi$
$998$		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.91.1	✓	36
3.2	odd	2	CM	2007.1.v.a.91.1	✓	36
223.174	odd	74	inner	2007.1.v.a.397.1	yes	36
669.620	even	74	inner	2007.1.v.a.397.1	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2007.1.v.a.91.1	✓	36	1.1	even	1	trivial
2007.1.v.a.91.1	✓	36	3.2	odd	2	CM
2007.1.v.a.397.1	yes	36	223.174	odd	74	inner
2007.1.v.a.397.1	yes	36	669.620	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)

\(f(q)\)	\(=\)	\(q+(0.967733 - 0.251978i) q^{4} +(1.03825 - 1.40380i) q^{7} +O(q^{10})\)	\(q+(0.967733 - 0.251978i) q^{4} +(1.03825 - 1.40380i) q^{7} +(0.618234 + 1.74977i) q^{13} +(0.873014 - 0.487695i) q^{16} +(-0.552192 + 1.80310i) q^{19} +(-0.721956 - 0.691939i) q^{25} +(0.651018 - 1.62012i) q^{28} +(-1.50992 + 0.682528i) q^{31} +(-0.383954 - 1.78155i) q^{37} +(0.422810 - 0.405231i) q^{43} +(-0.599881 - 1.95882i) q^{49} +(1.03919 + 1.53753i) q^{52} +(-0.535687 - 1.51614i) q^{61} +(0.721956 - 0.691939i) q^{64} +(-0.0427385 - 0.164139i) q^{67} +(-1.60355 + 1.08381i) q^{73} +(-0.0800337 + 1.88406i) q^{76} +(-0.233876 + 1.36397i) q^{79} +(3.09821 + 0.948816i) q^{91} +(0.313628 - 0.126026i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + q^{4} + 2 q^{7}+O(q^{10}) \) 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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