LMFDB - Embedded newform 2848.1.cc.a.943.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2848,1,Mod(47,2848)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2848, base_ring=CyclotomicField(44))

chi = DirichletCharacter(H, H._module([22, 22, 27]))

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

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

chi := DirichletCharacter("2848.47");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2848 = 2^{5} \cdot 89 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2848.cc (of order $44$, degree $20$, not 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:	$1.42133715598$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{44})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 712)
Projective image:	$D_{44}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{44} - \cdots)$

Embedding invariants

Embedding label			943.1
Root			$-0.755750 - 0.654861i$ of defining polynomial
Character	$\chi$	$=$	2848.943
Dual form			2848.1.cc.a.1199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.898064 - 0.334961i) q^{3} +(-0.0614286 + 0.0532282i) q^{9} +O(q^{10})$	$q+(0.898064 - 0.334961i) q^{3} +(-0.0614286 + 0.0532282i) q^{9} +(0.281733 + 1.95949i) q^{11} +(0.909632 + 1.41542i) q^{17} +(-1.59700 - 0.114220i) q^{19} +(0.959493 + 0.281733i) q^{25} +(-0.496697 + 0.909632i) q^{27} +(0.909367 + 1.66538i) q^{33} +(0.494217 - 1.32505i) q^{41} +(0.254771 - 0.340335i) q^{43} +(0.281733 - 0.959493i) q^{49} +(1.29102 + 0.966443i) q^{51} +(-1.47247 + 0.432356i) q^{57} +(-1.83107 - 0.682956i) q^{59} +(0.627899 - 1.37491i) q^{67} +(-0.368991 + 0.425839i) q^{73} +(0.956056 - 0.0683785i) q^{75} +(-0.129807 + 0.902828i) q^{81} +(1.83107 + 0.398326i) q^{83} +(0.959493 - 0.281733i) q^{89} +(0.186393 - 1.29639i) q^{97} +(-0.121607 - 0.105373i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{3}+O(q^{10}) $ 20 * q + 2 * q^3	$ 20 q + 2 q^{3} + 2 q^{19} + 2 q^{25} + 22 q^{27} + 2 q^{41} - 2 q^{43} - 4 q^{51} - 4 q^{57} + 2 q^{59} - 2 q^{75} - 2 q^{81} - 2 q^{83} + 2 q^{89} - 4 q^{97}+O(q^{100}) $ 20 * q + 2 * q^3 + 2 * q^19 + 2 * q^25 + 22 * q^27 + 2 * q^41 - 2 * q^43 - 4 * q^51 - 4 * q^57 + 2 * q^59 - 2 * q^75 - 2 * q^81 - 2 * q^83 + 2 * q^89 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2848\mathbb{Z}\right)^\times$.

$n$	$357$	$1247$	$1249$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{39}{44}\right)$

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
$2$		0			0
$3$	0.898064	−	0.334961i	0.898064	−	0.334961i	0.142315	−	0.989821i	$-0.454545\pi$
$3$	0.898064	−	0.334961i	0.898064	−	0.334961i	0.755750	+	0.654861i	$0.227273\pi$
$4$		0			0
$5$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$5$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$6$		0			0
$7$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$7$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$8$		0			0
$9$	−0.0614286	+	0.0532282i	−0.0614286	+	0.0532282i
$10$		0			0
$11$	0.281733	+	1.95949i	0.281733	+	1.95949i	0.281733	+	0.959493i	$0.409091\pi$
$11$	0.281733	+	1.95949i	0.281733	+	1.95949i			1.00000i	$0.500000\pi$
$12$		0			0
$13$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$13$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.909632	+	1.41542i	0.909632	+	1.41542i	0.909632	+	0.415415i	$0.136364\pi$
$17$	0.909632	+	1.41542i	0.909632	+	1.41542i			1.00000i	$0.5\pi$
$18$		0			0
$19$	−1.59700	−	0.114220i	−1.59700	−	0.114220i	−0.755750	−	0.654861i	$-0.772727\pi$
$19$	−1.59700	−	0.114220i	−1.59700	−	0.114220i	−0.841254	+	0.540641i	$0.818182\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$23$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$24$		0			0
$25$	0.959493	+	0.281733i	0.959493	+	0.281733i
$26$		0			0
$27$	−0.496697	+	0.909632i	−0.496697	+	0.909632i
$28$		0			0
$29$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$29$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$30$		0			0
$31$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$31$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$32$		0			0
$33$	0.909367	+	1.66538i	0.909367	+	1.66538i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.494217	−	1.32505i	0.494217	−	1.32505i	−0.415415	−	0.909632i	$-0.636364\pi$
$41$	0.494217	−	1.32505i	0.494217	−	1.32505i	0.909632	−	0.415415i	$-0.136364\pi$
$42$		0			0
$43$	0.254771	−	0.340335i	0.254771	−	0.340335i	−0.654861	−	0.755750i	$-0.727273\pi$
$43$	0.254771	−	0.340335i	0.254771	−	0.340335i	0.909632	+	0.415415i	$0.136364\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$47$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$48$		0			0
$49$	0.281733	−	0.959493i	0.281733	−	0.959493i
$50$		0			0
$51$	1.29102	+	0.966443i	1.29102	+	0.966443i
$52$		0			0
$53$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$53$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.47247	+	0.432356i	−1.47247	+	0.432356i
$58$		0			0
$59$	−1.83107	−	0.682956i	−1.83107	−	0.682956i	−0.989821	−	0.142315i	$-0.954545\pi$
$59$	−1.83107	−	0.682956i	−1.83107	−	0.682956i	−0.841254	−	0.540641i	$-0.818182\pi$
$60$		0			0
$61$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$61$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	0.627899	−	1.37491i	0.627899	−	1.37491i	−0.281733	−	0.959493i	$-0.590909\pi$
$67$	0.627899	−	1.37491i	0.627899	−	1.37491i	0.909632	−	0.415415i	$-0.136364\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$71$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$72$		0			0
$73$	−0.368991	+	0.425839i	−0.368991	+	0.425839i	−0.909632	−	0.415415i	$-0.863636\pi$
$73$	−0.368991	+	0.425839i	−0.368991	+	0.425839i	0.540641	+	0.841254i	$0.318182\pi$
$74$		0			0
$75$	0.956056	−	0.0683785i	0.956056	−	0.0683785i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$79$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$80$		0			0
$81$	−0.129807	+	0.902828i	−0.129807	+	0.902828i
$82$		0			0
$83$	1.83107	+	0.398326i	1.83107	+	0.398326i	0.989821	−	0.142315i	$-0.0454545\pi$
$83$	1.83107	+	0.398326i	1.83107	+	0.398326i	0.841254	+	0.540641i	$0.181818\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.959493	−	0.281733i	0.959493	−	0.281733i
$89$	0.959493	−	0.281733i	0.959493	−	0.281733i
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.186393	−	1.29639i	0.186393	−	1.29639i	−0.654861	−	0.755750i	$-0.727273\pi$
$97$	0.186393	−	1.29639i	0.186393	−	1.29639i	0.841254	−	0.540641i	$-0.181818\pi$
$98$		0			0
$99$	−0.121607	−	0.105373i	−0.121607	−	0.105373i
$100$		0			0
$101$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$101$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$102$		0			0
$103$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$103$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$107$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$108$		0			0
$109$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$109$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.398326	−	1.83107i	−0.398326	−	1.83107i	−0.540641	−	0.841254i	$-0.681818\pi$
$113$	−0.398326	−	1.83107i	−0.398326	−	1.83107i	0.142315	−	0.989821i	$-0.454545\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.80075	+	0.822373i	−2.80075	+	0.822373i
$122$		0			0
$123$		−	1.35552i		−	1.35552i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$127$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$128$		0			0
$129$	0.114802	−	0.390981i	0.114802	−	0.390981i
$130$		0			0
$131$	1.74557	−	0.797176i	1.74557	−	0.797176i	0.755750	−	0.654861i	$-0.227273\pi$
$131$	1.74557	−	0.797176i	1.74557	−	0.797176i	0.989821	−	0.142315i	$-0.0454545\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.682956	+	1.83107i	−0.682956	+	1.83107i	−0.142315	+	0.989821i	$0.545455\pi$
$137$	−0.682956	+	1.83107i	−0.682956	+	1.83107i	−0.540641	+	0.841254i	$0.681818\pi$
$138$		0			0
$139$	0.186393	+	0.215109i	0.186393	+	0.215109i	0.841254	−	0.540641i	$-0.181818\pi$
$139$	0.186393	+	0.215109i	0.186393	+	0.215109i	−0.654861	+	0.755750i	$0.727273\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−0.0683785	−	0.956056i	−0.0683785	−	0.956056i
$148$		0			0
$149$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$149$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$150$		0			0
$151$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$151$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$152$		0			0
$153$	−0.131217	−	0.0385289i	−0.131217	−	0.0385289i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$157$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.71524	+	0.373128i	−1.71524	+	0.373128i	−0.959493	−	0.281733i	$-0.909091\pi$
$163$	−1.71524	+	0.373128i	−1.71524	+	0.373128i	−0.755750	+	0.654861i	$0.772727\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$167$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$168$		0			0
$169$	−0.755750	+	0.654861i	−0.755750	+	0.654861i
$170$		0			0
$171$	0.104181	−	0.0779892i	0.104181	−	0.0779892i
$172$		0			0
$173$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$173$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−1.87319			−1.87319
$178$		0			0
$179$	1.30972			1.30972			0.654861	−	0.755750i	$-0.272727\pi$
$179$	1.30972			1.30972			0.654861	+	0.755750i	$0.272727\pi$
$180$		0			0
$181$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$181$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−2.51722	+	2.18119i	−2.51722	+	2.18119i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$191$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$192$		0			0
$193$	0.139418	−	0.0303285i	0.139418	−	0.0303285i	−0.142315	−	0.989821i	$-0.545455\pi$
$193$	0.139418	−	0.0303285i	0.139418	−	0.0303285i	0.281733	+	0.959493i	$0.409091\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$197$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$198$		0			0
$199$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$199$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$200$		0			0
$201$	0.103354	−	1.44508i	0.103354	−	1.44508i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.226115	−	3.16150i	−0.226115	−	3.16150i
$210$		0			0
$211$	0.203743	+	0.373128i	0.203743	+	0.373128i	0.959493	−	0.281733i	$-0.0909091\pi$
$211$	0.203743	+	0.373128i	0.203743	+	0.373128i	−0.755750	+	0.654861i	$0.772727\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−0.188739	+	0.506028i	−0.188739	+	0.506028i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$223$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$224$		0			0
$225$	−0.0739364	+	0.0337656i	−0.0739364	+	0.0337656i
$226$		0			0
$227$	0.158746	−	0.540641i	0.158746	−	0.540641i	−0.841254	−	0.540641i	$-0.818182\pi$
$227$	0.158746	−	0.540641i	0.158746	−	0.540641i	1.00000			$0$
$228$		0			0
$229$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$229$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			0.830830i			0.830830i	0.909632	+	0.415415i	$0.136364\pi$
$233$			0.830830i			0.830830i	−0.909632	+	0.415415i	$0.863636\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$239$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$240$		0			0
$241$	0.148568	−	0.682956i	0.148568	−	0.682956i	−0.841254	−	0.540641i	$-0.818182\pi$
$241$	0.148568	−	0.682956i	0.148568	−	0.682956i	0.989821	−	0.142315i	$-0.0454545\pi$
$242$		0			0
$243$	−0.0344673	−	0.158443i	−0.0344673	−	0.158443i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	1.77785	−	0.255616i	1.77785	−	0.255616i
$250$		0			0
$251$	0.544078	−	0.627899i	0.544078	−	0.627899i	−0.415415	−	0.909632i	$-0.636364\pi$
$251$	0.544078	−	0.627899i	0.544078	−	0.627899i	0.959493	+	0.281733i	$0.0909091\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	0.215109	+	0.186393i	0.215109	+	0.186393i	0.755750	−	0.654861i	$-0.227273\pi$
$257$	0.215109	+	0.186393i	0.215109	+	0.186393i	−0.540641	+	0.841254i	$0.681818\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$263$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	0.767317	−	0.574406i	0.767317	−	0.574406i
$268$		0			0
$269$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$269$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$270$		0			0
$271$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$271$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.281733	+	1.95949i	−0.281733	+	1.95949i
$276$		0			0
$277$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$277$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.19550	−	0.0855040i	1.19550	−	0.0855040i	0.540641	−	0.841254i	$-0.318182\pi$
$281$	1.19550	−	0.0855040i	1.19550	−	0.0855040i	0.654861	+	0.755750i	$0.272727\pi$
$282$		0			0
$283$	−1.19136	+	1.37491i	−1.19136	+	1.37491i	−0.281733	+	0.959493i	$0.590909\pi$
$283$	−1.19136	+	1.37491i	−1.19136	+	1.37491i	−0.909632	+	0.415415i	$0.863636\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.760554	+	1.66538i	−0.760554	+	1.66538i
$290$		0			0
$291$	−0.266847	−	1.22668i	−0.266847	−	1.22668i
$292$		0			0
$293$		0			0		0.212565	−	0.977147i	$-0.431818\pi$
$293$		0			0		−0.212565	+	0.977147i	$0.568182\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−1.92235	−	0.717001i	−1.92235	−	0.717001i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−0.234072	+	0.797176i	−0.234072	+	0.797176i	0.755750	+	0.654861i	$0.227273\pi$
$307$	−0.234072	+	0.797176i	−0.234072	+	0.797176i	−0.989821	+	0.142315i	$0.954545\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$311$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$312$		0			0
$313$	−0.0855040	+	0.114220i	−0.0855040	+	0.114220i	−0.841254	−	0.540641i	$-0.818182\pi$
$313$	−0.0855040	+	0.114220i	−0.0855040	+	0.114220i	0.755750	+	0.654861i	$0.227273\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$317$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.29102	−	2.36432i	−1.29102	−	2.36432i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.273100	+	0.0801894i	0.273100	+	0.0801894i	0.415415	−	0.909632i	$-0.363636\pi$
$331$	0.273100	+	0.0801894i	0.273100	+	0.0801894i	−0.142315	+	0.989821i	$0.545455\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.19550	+	0.0855040i	1.19550	+	0.0855040i	0.654861	−	0.755750i	$-0.272727\pi$
$337$	1.19550	+	0.0855040i	1.19550	+	0.0855040i	0.540641	+	0.841254i	$0.318182\pi$
$338$		0			0
$339$	−0.971061	−	1.51100i	−0.971061	−	1.51100i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.37491	−	1.19136i	1.37491	−	1.19136i	0.415415	−	0.909632i	$-0.363636\pi$
$347$	1.37491	−	1.19136i	1.37491	−	1.19136i	0.959493	−	0.281733i	$-0.0909091\pi$
$348$		0			0
$349$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$349$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.133682	+	0.0498610i	−0.133682	+	0.0498610i	−0.415415	−	0.909632i	$-0.636364\pi$
$353$	−0.133682	+	0.0498610i	−0.133682	+	0.0498610i	0.281733	+	0.959493i	$0.409091\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$359$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$360$		0			0
$361$	1.54755	+	0.222504i	1.54755	+	0.222504i
$362$		0			0
$363$	−2.23979	+	1.67668i	−2.23979	+	1.67668i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$367$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$368$		0			0
$369$	0.0401708	+	0.107702i	0.0401708	+	0.107702i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$373$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.142315	−	1.98982i	0.142315	−	1.98982i		−	1.00000i	$-0.5\pi$
$379$	0.142315	−	1.98982i	0.142315	−	1.98982i	0.142315	−	0.989821i	$-0.454545\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$383$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.00246515	+	0.0344673i	0.00246515	+	0.0344673i
$388$		0			0
$389$		0			0		−0.479249	−	0.877679i	$-0.659091\pi$
$389$		0			0		0.479249	+	0.877679i	$0.340909\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	1.30061	−	1.30061i	1.30061	−	1.30061i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$397$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.27155	+	0.817178i	−1.27155	+	0.817178i	−0.989821	−	0.142315i	$-0.954545\pi$
$401$	−1.27155	+	0.817178i	−1.27155	+	0.817178i	−0.281733	+	0.959493i	$0.590909\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.19136	−	0.544078i	−1.19136	−	0.544078i	−0.281733	−	0.959493i	$-0.590909\pi$
$409$	−1.19136	−	0.544078i	−1.19136	−	0.544078i	−0.909632	+	0.415415i	$0.863636\pi$
$410$		0			0
$411$			1.87319i			1.87319i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.239446	+	0.130747i	0.239446	+	0.130747i
$418$		0			0
$419$	0.0303285	−	0.139418i	0.0303285	−	0.139418i	−0.959493	−	0.281733i	$-0.909091\pi$
$419$	0.0303285	−	0.139418i	0.0303285	−	0.139418i	0.989821	+	0.142315i	$0.0454545\pi$
$420$		0			0
$421$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$421$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.474017	+	1.61435i	0.474017	+	1.61435i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$431$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$432$		0			0
$433$	−1.24123	−	1.24123i	−1.24123	−	1.24123i	−0.959493	−	0.281733i	$-0.909091\pi$
$433$	−1.24123	−	1.24123i	−1.24123	−	1.24123i	−0.281733	−	0.959493i	$-0.590909\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$439$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$440$		0			0
$441$	0.0337656	+	0.0739364i	0.0337656	+	0.0739364i
$442$		0			0
$443$	−1.53046	−	0.983568i	−1.53046	−	0.983568i	−0.989821	−	0.142315i	$-0.954545\pi$
$443$	−1.53046	−	0.983568i	−1.53046	−	0.983568i	−0.540641	−	0.841254i	$-0.681818\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.755750	+	1.65486i	0.755750	+	1.65486i	0.755750	+	0.654861i	$0.227273\pi$
$449$	0.755750	+	1.65486i	0.755750	+	1.65486i			1.00000i	$0.5\pi$
$450$		0			0
$451$	2.73566	+	0.595106i	2.73566	+	0.595106i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.41061	−	1.41061i	−1.41061	−	1.41061i	−0.755750	−	0.654861i	$-0.772727\pi$
$457$	−1.41061	−	1.41061i	−1.41061	−	1.41061i	−0.654861	−	0.755750i	$-0.727273\pi$
$458$		0			0
$459$	−1.73932	+	0.124398i	−1.73932	+	0.124398i
$460$		0			0
$461$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$461$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$462$		0			0
$463$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$463$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.698939	−	1.53046i	0.698939	−	1.53046i	−0.142315	−	0.989821i	$-0.545455\pi$
$467$	0.698939	−	1.53046i	0.698939	−	1.53046i	0.841254	−	0.540641i	$-0.181818\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.738661	+	0.403339i	0.738661	+	0.403339i
$474$		0			0
$475$	−1.50013	−	0.559521i	−1.50013	−	0.559521i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$487$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$488$		0			0
$489$	−1.41542	+	0.909632i	−1.41542	+	0.909632i
$490$		0			0
$491$	−0.718267	+	0.959493i	−0.718267	+	0.959493i	0.281733	+	0.959493i	$0.409091\pi$
$491$	−0.718267	+	0.959493i	−0.718267	+	0.959493i	−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−0.613435	+	0.334961i	−0.613435	+	0.334961i	−0.755750	−	0.654861i	$-0.772727\pi$
$499$	−0.613435	+	0.334961i	−0.613435	+	0.334961i	0.142315	+	0.989821i	$0.454545\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$503$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.459359	+	0.841254i	−0.459359	+	0.841254i
$508$		0			0
$509$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$509$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	0.897124	−	1.39595i	0.897124	−	1.39595i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.148568	+	0.398326i	0.148568	+	0.398326i	0.989821	−	0.142315i	$-0.0454545\pi$
$521$	0.148568	+	0.398326i	0.148568	+	0.398326i	−0.841254	+	0.540641i	$0.818182\pi$
$522$		0			0
$523$	0.0801894	+	0.557730i	0.0801894	+	0.557730i	0.989821	+	0.142315i	$0.0454545\pi$
$523$	0.0801894	+	0.557730i	0.0801894	+	0.557730i	−0.909632	+	0.415415i	$0.863636\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.989821	−	0.142315i	−0.989821	−	0.142315i
$530$		0			0
$531$	0.148833	−	0.0555118i	0.148833	−	0.0555118i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	1.17621	−	0.438705i	1.17621	−	0.438705i
$538$		0			0
$539$	1.95949	+	0.281733i	1.95949	+	0.281733i
$540$		0			0
$541$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$541$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.418852	+	1.12299i	0.418852	+	1.12299i	0.959493	+	0.281733i	$0.0909091\pi$
$547$	0.418852	+	1.12299i	0.418852	+	1.12299i	−0.540641	+	0.841254i	$0.681818\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$557$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−1.53002	+	2.80202i	−1.53002	+	2.80202i
$562$		0			0
$563$	−0.718267	−	0.959493i	−0.718267	−	0.959493i	0.281733	−	0.959493i	$-0.409091\pi$
$563$	−0.718267	−	0.959493i	−0.718267	−	0.959493i	−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.75089	−	0.956056i	1.75089	−	0.956056i	0.841254	−	0.540641i	$-0.181818\pi$
$569$	1.75089	−	0.956056i	1.75089	−	0.956056i	0.909632	−	0.415415i	$-0.136364\pi$
$570$		0			0
$571$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$571$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	0.959493	−	1.28173i	0.959493	−	1.28173i		−	1.00000i	$-0.5\pi$
$577$	0.959493	−	1.28173i	0.959493	−	1.28173i	0.959493	−	0.281733i	$-0.0909091\pi$
$578$		0			0
$579$	0.115047	−	0.0739364i	0.115047	−	0.0739364i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.258908	−	0.118239i	−0.258908	−	0.118239i	0.281733	−	0.959493i	$-0.409091\pi$
$587$	−0.258908	−	0.118239i	−0.258908	−	0.118239i	−0.540641	+	0.841254i	$0.681818\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.86912	−	0.697148i	−1.86912	−	0.697148i	−0.959493	−	0.281733i	$-0.909091\pi$
$593$	−1.86912	−	0.697148i	−1.86912	−	0.697148i	−0.909632	−	0.415415i	$-0.863636\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$599$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$600$		0			0
$601$	−0.797176	+	1.74557i	−0.797176	+	1.74557i	−0.142315	+	0.989821i	$0.545455\pi$
$601$	−0.797176	+	1.74557i	−0.797176	+	1.74557i	−0.654861	+	0.755750i	$0.727273\pi$
$602$		0			0
$603$	0.0346129	+	0.117881i	0.0346129	+	0.117881i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$607$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$613$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.71524	−	0.373128i	−1.71524	−	0.373128i	−0.755750	−	0.654861i	$-0.772727\pi$
$617$	−1.71524	−	0.373128i	−1.71524	−	0.373128i	−0.959493	+	0.281733i	$0.909091\pi$
$618$		0			0
$619$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.654861	−	0.755750i	$-0.727273\pi$
$619$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.142315	−	0.989821i	$-0.545455\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.841254	+	0.540641i	0.841254	+	0.540641i
$626$		0			0
$627$	−1.26204	−	2.76349i	−1.26204	−	2.76349i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$631$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$632$		0			0
$633$	0.307958	+	0.266847i	0.307958	+	0.266847i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.89945	+	0.273100i	−1.89945	+	0.273100i	−0.989821	−	0.142315i	$-0.954545\pi$
$641$	−1.89945	+	0.273100i	−1.89945	+	0.273100i	−0.909632	+	0.415415i	$0.863636\pi$
$642$		0			0
$643$	−0.368991	−	1.25667i	−0.368991	−	1.25667i	−0.909632	−	0.415415i	$-0.863636\pi$
$643$	−0.368991	−	1.25667i	−0.368991	−	1.25667i	0.540641	−	0.841254i	$-0.318182\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$647$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$648$		0			0
$649$	0.822373	−	3.78039i	0.822373	−	3.78039i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$653$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	0.0457994i		−	0.0457994i
$658$		0			0
$659$	1.74557	+	0.797176i	1.74557	+	0.797176i	0.989821	+	0.142315i	$0.0454545\pi$
$659$	1.74557	+	0.797176i	1.74557	+	0.797176i	0.755750	+	0.654861i	$0.227273\pi$
$660$		0			0
$661$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$661$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$662$		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			0		0.755750	−	0.654861i	$-0.227273\pi$
$673$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$674$		0			0
$675$	−0.732850	+	0.732850i	−0.732850	+	0.732850i
$676$		0			0
$677$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$677$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−0.0385289	−	0.538704i	−0.0385289	−	0.538704i
$682$		0			0
$683$	1.05195	+	1.40524i	1.05195	+	1.40524i	0.909632	+	0.415415i	$0.136364\pi$
$683$	1.05195	+	1.40524i	1.05195	+	1.40524i	0.142315	+	0.989821i	$0.454545\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−0.449181	+	0.698939i	−0.449181	+	0.698939i	−0.989821	−	0.142315i	$-0.954545\pi$
$691$	−0.449181	+	0.698939i	−0.449181	+	0.698939i	0.540641	+	0.841254i	$0.318182\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	2.32505	−	0.505783i	2.32505	−	0.505783i
$698$		0			0
$699$	0.278295	+	0.746139i	0.278295	+	0.746139i
$700$		0			0
$701$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$701$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$709$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$710$		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		0.800541	−	0.599278i	$-0.204545\pi$
$719$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−0.0953398	−	0.663103i	−0.0953398	−	0.663103i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$727$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$728$		0			0
$729$	−0.577151	−	0.898064i	−0.577151	−	0.898064i
$730$		0			0
$731$	0.713463	+	0.0510279i	0.713463	+	0.0510279i
$732$		0			0
$733$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$733$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.87102	+	0.843008i	2.87102	+	0.843008i
$738$		0			0
$739$	−0.0683785	+	0.125226i	−0.0683785	+	0.125226i	−0.909632	−	0.415415i	$-0.863636\pi$
$739$	−0.0683785	+	0.125226i	−0.0683785	+	0.125226i	0.841254	+	0.540641i	$0.181818\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$743$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−0.133682	+	0.0729961i	−0.133682	+	0.0729961i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$751$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$752$		0			0
$753$	0.278295	−	0.746139i	0.278295	−	0.746139i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$757$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.557730	−	1.89945i	0.557730	−	1.89945i	0.142315	−	0.989821i	$-0.454545\pi$
$761$	0.557730	−	1.89945i	0.557730	−	1.89945i	0.415415	−	0.909632i	$-0.363636\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.74557	−	0.512546i	1.74557	−	0.512546i	0.755750	−	0.654861i	$-0.227273\pi$
$769$	1.74557	−	0.512546i	1.74557	−	0.512546i	0.989821	+	0.142315i	$0.0454545\pi$
$770$		0			0
$771$	0.255616	+	0.0953398i	0.255616	+	0.0953398i
$772$		0			0
$773$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$773$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.940613	+	2.05965i	−0.940613	+	2.05965i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	0.697148	−	0.0498610i	0.697148	−	0.0498610i	0.281733	−	0.959493i	$-0.409091\pi$
$787$	0.697148	−	0.0498610i	0.697148	−	0.0498610i	0.415415	+	0.909632i	$0.363636\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$797$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−0.0439442	+	0.0683785i	−0.0439442	+	0.0683785i
$802$		0			0
$803$	−0.938384	−	0.603063i	−0.938384	−	0.603063i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.239446	+	1.66538i	−0.239446	+	1.66538i	0.415415	+	0.909632i	$0.363636\pi$
$809$	−0.239446	+	1.66538i	−0.239446	+	1.66538i	−0.654861	+	0.755750i	$0.727273\pi$
$810$		0			0
$811$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$811$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.445743	+	0.514415i	−0.445743	+	0.514415i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$821$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$822$		0			0
$823$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$823$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$824$		0			0
$825$	0.403339	+	1.85412i	0.403339	+	1.85412i
$826$		0			0
$827$	−0.148568	+	0.682956i	−0.148568	+	0.682956i	0.841254	+	0.540641i	$0.181818\pi$
$827$	−0.148568	+	0.682956i	−0.148568	+	0.682956i	−0.989821	+	0.142315i	$0.954545\pi$
$828$		0			0
$829$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$829$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.61435	−	0.474017i	1.61435	−	0.474017i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$839$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$840$		0			0
$841$	−0.281733	+	0.959493i	−0.281733	+	0.959493i
$842$		0			0
$843$	1.04500	−	0.477234i	1.04500	−	0.477234i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−0.609382	+	1.63382i	−0.609382	+	1.63382i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.767317	−	1.40524i	−0.767317	−	1.40524i	−0.909632	−	0.415415i	$-0.863636\pi$
$857$	−0.767317	−	1.40524i	−0.767317	−	1.40524i	0.142315	−	0.989821i	$-0.454545\pi$
$858$		0			0
$859$	−0.133682	−	1.86912i	−0.133682	−	1.86912i	−0.415415	−	0.909632i	$-0.636364\pi$
$859$	−0.133682	−	1.86912i	−0.133682	−	1.86912i	0.281733	−	0.959493i	$-0.409091\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$863$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.125189	+	1.75038i	−0.125189	+	1.75038i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.0575546	+	0.0895567i	0.0575546	+	0.0895567i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$877$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.817178	+	0.708089i	−0.817178	+	0.708089i	−0.959493	−	0.281733i	$-0.909091\pi$
$881$	−0.817178	+	0.708089i	−0.817178	+	0.708089i	0.142315	+	0.989821i	$0.454545\pi$
$882$		0			0
$883$	0.559521	−	0.418852i	0.559521	−	0.418852i	−0.281733	−	0.959493i	$-0.590909\pi$
$883$	0.559521	−	0.418852i	0.559521	−	0.418852i	0.841254	+	0.540641i	$0.181818\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$887$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−1.80566			−1.80566
$892$		0			0
$893$		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
$907$	0.584585	+	0.909632i	0.584585	+	0.909632i	1.00000			$0$
$907$	0.584585	+	0.909632i	0.584585	+	0.909632i	−0.415415	+	0.909632i	$0.636364\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$911$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$912$		0			0
$913$	−0.264644	+	3.70020i	−0.264644	+	3.70020i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$919$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$920$		0			0
$921$	0.0568109	+	0.794320i	0.0568109	+	0.794320i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.544078	+	0.627899i	0.544078	+	0.627899i	0.959493	−	0.281733i	$-0.0909091\pi$
$929$	0.544078	+	0.627899i	0.544078	+	0.627899i	−0.415415	+	0.909632i	$0.636364\pi$
$930$		0			0
$931$	−0.559521	+	1.50013i	−0.559521	+	1.50013i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.983568	−	0.449181i	0.983568	−	0.449181i	0.142315	−	0.989821i	$-0.454545\pi$
$937$	0.983568	−	0.449181i	0.983568	−	0.449181i	0.841254	+	0.540641i	$0.181818\pi$
$938$		0			0
$939$	−0.0385289	+	0.131217i	−0.0385289	+	0.131217i
$940$		0			0
$941$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$941$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.45027	−	0.425839i	1.45027	−	0.425839i	0.540641	−	0.841254i	$-0.318182\pi$
$947$	1.45027	−	0.425839i	1.45027	−	0.425839i	0.909632	+	0.415415i	$0.136364\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.373128	+	1.71524i	−0.373128	+	1.71524i	0.281733	+	0.959493i	$0.409091\pi$
$953$	−0.373128	+	1.71524i	−0.373128	+	1.71524i	−0.654861	+	0.755750i	$0.727273\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.989821	+	0.142315i	−0.989821	+	0.142315i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$		0			0
$969$	−1.95137	−	1.69087i	−1.95137	−	1.69087i
$970$		0			0
$971$	−0.186393	+	1.29639i	−0.186393	+	1.29639i	0.654861	+	0.755750i	$0.272727\pi$
$971$	−0.186393	+	1.29639i	−0.186393	+	1.29639i	−0.841254	+	0.540641i	$0.818182\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.474017	−	0.304632i	−0.474017	−	0.304632i	0.281733	−	0.959493i	$-0.409091\pi$
$977$	−0.474017	−	0.304632i	−0.474017	−	0.304632i	−0.755750	+	0.654861i	$0.772727\pi$
$978$		0			0
$979$	0.822373	+	1.80075i	0.822373	+	1.80075i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$983$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$991$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$992$		0			0
$993$	0.272122	−	0.0194625i	0.272122	−	0.0194625i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$997$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2848.1.cc.a.943.1		20
4.3	odd	2		712.1.y.a.587.1	yes	20
8.3	odd	2	CM	2848.1.cc.a.943.1		20
8.5	even	2		712.1.y.a.587.1	yes	20
89.42	even	44	inner	2848.1.cc.a.1199.1		20
356.131	odd	44		712.1.y.a.131.1	✓	20
712.131	odd	44	inner	2848.1.cc.a.1199.1		20
712.309	even	44		712.1.y.a.131.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
712.1.y.a.131.1	✓	20	356.131	odd	44
712.1.y.a.131.1	✓	20	712.309	even	44
712.1.y.a.587.1	yes	20	4.3	odd	2
712.1.y.a.587.1	yes	20	8.5	even	2
2848.1.cc.a.943.1		20	1.1	even	1	trivial
2848.1.cc.a.943.1		20	8.3	odd	2	CM
2848.1.cc.a.1199.1		20	89.42	even	44	inner
2848.1.cc.a.1199.1		20	712.131	odd	44	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:	\( N \)	\(=\)	\( 2848 = 2^{5} \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2848.cc (of order \(44\), degree \(20\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.898064 - 0.334961i) q^{3} +(-0.0614286 + 0.0532282i) q^{9} +O(q^{10})\)	\(q+(0.898064 - 0.334961i) q^{3} +(-0.0614286 + 0.0532282i) q^{9} +(0.281733 + 1.95949i) q^{11} +(0.909632 + 1.41542i) q^{17} +(-1.59700 - 0.114220i) q^{19} +(0.959493 + 0.281733i) q^{25} +(-0.496697 + 0.909632i) q^{27} +(0.909367 + 1.66538i) q^{33} +(0.494217 - 1.32505i) q^{41} +(0.254771 - 0.340335i) q^{43} +(0.281733 - 0.959493i) q^{49} +(1.29102 + 0.966443i) q^{51} +(-1.47247 + 0.432356i) q^{57} +(-1.83107 - 0.682956i) q^{59} +(0.627899 - 1.37491i) q^{67} +(-0.368991 + 0.425839i) q^{73} +(0.956056 - 0.0683785i) q^{75} +(-0.129807 + 0.902828i) q^{81} +(1.83107 + 0.398326i) q^{83} +(0.959493 - 0.281733i) q^{89} +(0.186393 - 1.29639i) q^{97} +(-0.121607 - 0.105373i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{3}+O(q^{10}) \) 20 * q + 2 * q^3	\( 20 q + 2 q^{3} + 2 q^{19} + 2 q^{25} + 22 q^{27} + 2 q^{41} - 2 q^{43} - 4 q^{51} - 4 q^{57} + 2 q^{59} - 2 q^{75} - 2 q^{81} - 2 q^{83} + 2 q^{89} - 4 q^{97}+O(q^{100}) \) 20 * q + 2 * q^3 + 2 * q^19 + 2 * q^25 + 22 * q^27 + 2 * q^41 - 2 * q^43 - 4 * q^51 - 4 * q^57 + 2 * q^59 - 2 * q^75 - 2 * q^81 - 2 * q^83 + 2 * q^89 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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