LMFDB - Embedded newform 1344.2.j.d.1247.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,2,Mod(1247,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1344.1247");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1344.j (of order $2$, degree $1$, 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:	$10.7318940317$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1247.4
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1344.1247
Dual form			1344.2.j.d.1247.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 1.22474i) q^{3} +2.44949 q^{5} +1.00000i q^{7} +3.00000i q^{9} +O(q^{10})$	$q+(1.22474 + 1.22474i) q^{3} +2.44949 q^{5} +1.00000i q^{7} +3.00000i q^{9} +2.44949i q^{13} +(3.00000 + 3.00000i) q^{15} +6.00000i q^{17} -7.34847 q^{19} +(-1.22474 + 1.22474i) q^{21} +1.00000 q^{25} +(-3.67423 + 3.67423i) q^{27} +4.89898 q^{29} -8.00000i q^{31} +2.44949i q^{35} +4.89898i q^{37} +(-3.00000 + 3.00000i) q^{39} -6.00000i q^{41} +4.89898 q^{43} +7.34847i q^{45} +12.0000 q^{47} -1.00000 q^{49} +(-7.34847 + 7.34847i) q^{51} +9.79796 q^{53} +(-9.00000 - 9.00000i) q^{57} +2.44949i q^{59} -2.44949i q^{61} -3.00000 q^{63} +6.00000i q^{65} -9.79796 q^{67} -6.00000 q^{71} +10.0000 q^{73} +(1.22474 + 1.22474i) q^{75} -8.00000i q^{79} -9.00000 q^{81} +7.34847i q^{83} +14.6969i q^{85} +(6.00000 + 6.00000i) q^{87} -18.0000i q^{89} -2.44949 q^{91} +(9.79796 - 9.79796i) q^{93} -18.0000 q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 12 q^{15} + 4 q^{25} - 12 q^{39} + 48 q^{47} - 4 q^{49} - 36 q^{57} - 12 q^{63} - 24 q^{71} + 40 q^{73} - 36 q^{81} + 24 q^{87} - 72 q^{95} + 8 q^{97}+O(q^{100}) $ 4 * q + 12 * q^15 + 4 * q^25 - 12 * q^39 + 48 * q^47 - 4 * q^49 - 36 * q^57 - 12 * q^63 - 24 * q^71 + 40 * q^73 - 36 * q^81 + 24 * q^87 - 72 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.22474	+	1.22474i	0.707107	+	0.707107i
$3$	1.22474	+	1.22474i	0.707107	+	0.707107i
$4$		0			0
$5$	2.44949			1.09545			0.547723	−	0.836660i	$-0.315495\pi$
$5$	2.44949			1.09545			0.547723	+	0.836660i	$0.315495\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$			3.00000i			1.00000i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$			2.44949i			0.679366i	0.940540	+	0.339683i	$0.110320\pi$
$13$			2.44949i			0.679366i	−0.940540	+	0.339683i	$0.889680\pi$
$14$		0			0
$15$	3.00000	+	3.00000i	0.774597	+	0.774597i
$16$		0			0
$17$			6.00000i			1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$			6.00000i			1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$		0			0
$19$	−7.34847			−1.68585			−0.842927	−	0.538028i	$-0.819170\pi$
$19$	−7.34847			−1.68585			−0.842927	+	0.538028i	$0.819170\pi$
$20$		0			0
$21$	−1.22474	+	1.22474i	−0.267261	+	0.267261i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−3.67423	+	3.67423i	−0.707107	+	0.707107i
$28$		0			0
$29$	4.89898			0.909718			0.454859	−	0.890564i	$-0.349690\pi$
$29$	4.89898			0.909718			0.454859	+	0.890564i	$0.349690\pi$
$30$		0			0
$31$		−	8.00000i		−	1.43684i	−0.695608	−	0.718421i	$-0.744865\pi$
$31$		−	8.00000i		−	1.43684i	0.695608	−	0.718421i	$-0.255135\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			2.44949i			0.414039i
$36$		0			0
$37$			4.89898i			0.805387i	0.915335	+	0.402694i	$0.131926\pi$
$37$			4.89898i			0.805387i	−0.915335	+	0.402694i	$0.868074\pi$
$38$		0			0
$39$	−3.00000	+	3.00000i	−0.480384	+	0.480384i
$40$		0			0
$41$		−	6.00000i		−	0.937043i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$		−	6.00000i		−	0.937043i	0.883452	−	0.468521i	$-0.155213\pi$
$42$		0			0
$43$	4.89898			0.747087			0.373544	−	0.927613i	$-0.378143\pi$
$43$	4.89898			0.747087			0.373544	+	0.927613i	$0.378143\pi$
$44$		0			0
$45$			7.34847i			1.09545i
$46$		0			0
$47$	12.0000			1.75038			0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000			1.75038			0.875190	+	0.483779i	$0.160736\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−7.34847	+	7.34847i	−1.02899	+	1.02899i
$52$		0			0
$53$	9.79796			1.34585			0.672927	−	0.739709i	$-0.265037\pi$
$53$	9.79796			1.34585			0.672927	+	0.739709i	$0.265037\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−9.00000	−	9.00000i	−1.19208	−	1.19208i
$58$		0			0
$59$			2.44949i			0.318896i	0.987206	+	0.159448i	$0.0509715\pi$
$59$			2.44949i			0.318896i	−0.987206	+	0.159448i	$0.949029\pi$
$60$		0			0
$61$		−	2.44949i		−	0.313625i	−0.987628	−	0.156813i	$-0.949878\pi$
$61$		−	2.44949i		−	0.313625i	0.987628	−	0.156813i	$-0.0501218\pi$
$62$		0			0
$63$	−3.00000			−0.377964
$64$		0			0
$65$			6.00000i			0.744208i
$66$		0			0
$67$	−9.79796			−1.19701			−0.598506	−	0.801119i	$-0.704239\pi$
$67$	−9.79796			−1.19701			−0.598506	+	0.801119i	$0.704239\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$	10.0000			1.17041			0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000			1.17041			0.585206	+	0.810885i	$0.301014\pi$
$74$		0			0
$75$	1.22474	+	1.22474i	0.141421	+	0.141421i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	8.00000i		−	0.900070i	−0.893011	−	0.450035i	$-0.851411\pi$
$79$		−	8.00000i		−	0.900070i	0.893011	−	0.450035i	$-0.148589\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$			7.34847i			0.806599i	0.915068	+	0.403300i	$0.132137\pi$
$83$			7.34847i			0.806599i	−0.915068	+	0.403300i	$0.867863\pi$
$84$		0			0
$85$			14.6969i			1.59411i
$86$		0			0
$87$	6.00000	+	6.00000i	0.643268	+	0.643268i
$88$		0			0
$89$		−	18.0000i		−	1.90800i	−0.299813	−	0.953998i	$-0.596924\pi$
$89$		−	18.0000i		−	1.90800i	0.299813	−	0.953998i	$-0.403076\pi$
$90$		0			0
$91$	−2.44949			−0.256776
$92$		0			0
$93$	9.79796	−	9.79796i	1.01600	−	1.01600i
$94$		0			0
$95$	−18.0000			−1.84676
$96$		0			0
$97$	2.00000			0.203069			0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000			0.203069			0.101535	+	0.994832i	$0.467625\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	12.2474			1.21867			0.609333	−	0.792914i	$-0.291437\pi$
$101$	12.2474			1.21867			0.609333	+	0.792914i	$0.291437\pi$
$102$		0			0
$103$		−	8.00000i		−	0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$		−	8.00000i		−	0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$		0			0
$105$	−3.00000	+	3.00000i	−0.292770	+	0.292770i
$106$		0			0
$107$		−	4.89898i		−	0.473602i	−0.971558	−	0.236801i	$-0.923901\pi$
$107$		−	4.89898i		−	0.473602i	0.971558	−	0.236801i	$-0.0760990\pi$
$108$		0			0
$109$			4.89898i			0.469237i	0.972088	+	0.234619i	$0.0753841\pi$
$109$			4.89898i			0.469237i	−0.972088	+	0.234619i	$0.924616\pi$
$110$		0			0
$111$	−6.00000	+	6.00000i	−0.569495	+	0.569495i
$112$		0			0
$113$		−	6.00000i		−	0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$		−	6.00000i		−	0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−7.34847			−0.679366
$118$		0			0
$119$	−6.00000			−0.550019
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$	7.34847	−	7.34847i	0.662589	−	0.662589i
$124$		0			0
$125$	−9.79796			−0.876356
$126$		0			0
$127$			2.00000i			0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$			2.00000i			0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$		0			0
$129$	6.00000	+	6.00000i	0.528271	+	0.528271i
$130$		0			0
$131$			22.0454i			1.92612i	0.269294	+	0.963058i	$0.413210\pi$
$131$			22.0454i			1.92612i	−0.269294	+	0.963058i	$0.586790\pi$
$132$		0			0
$133$		−	7.34847i		−	0.637193i
$134$		0			0
$135$	−9.00000	+	9.00000i	−0.774597	+	0.774597i
$136$		0			0
$137$			12.0000i			1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$			12.0000i			1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$		0			0
$139$	−2.44949			−0.207763			−0.103882	−	0.994590i	$-0.533126\pi$
$139$	−2.44949			−0.207763			−0.103882	+	0.994590i	$0.533126\pi$
$140$		0			0
$141$	14.6969	+	14.6969i	1.23771	+	1.23771i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	12.0000			0.996546
$146$		0			0
$147$	−1.22474	−	1.22474i	−0.101015	−	0.101015i
$148$		0			0
$149$	−19.5959			−1.60536			−0.802680	−	0.596410i	$-0.796593\pi$
$149$	−19.5959			−1.60536			−0.802680	+	0.596410i	$0.796593\pi$
$150$		0			0
$151$			2.00000i			0.162758i	0.996683	+	0.0813788i	$0.0259324\pi$
$151$			2.00000i			0.162758i	−0.996683	+	0.0813788i	$0.974068\pi$
$152$		0			0
$153$	−18.0000			−1.45521
$154$		0			0
$155$		−	19.5959i		−	1.57398i
$156$		0			0
$157$			7.34847i			0.586472i	0.956040	+	0.293236i	$0.0947321\pi$
$157$			7.34847i			0.586472i	−0.956040	+	0.293236i	$0.905268\pi$
$158$		0			0
$159$	12.0000	+	12.0000i	0.951662	+	0.951662i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−4.89898			−0.383718			−0.191859	−	0.981423i	$-0.561452\pi$
$163$	−4.89898			−0.383718			−0.191859	+	0.981423i	$0.561452\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	7.00000			0.538462
$170$		0			0
$171$		−	22.0454i		−	1.68585i
$172$		0			0
$173$	−12.2474			−0.931156			−0.465578	−	0.885007i	$-0.654154\pi$
$173$	−12.2474			−0.931156			−0.465578	+	0.885007i	$0.654154\pi$
$174$		0			0
$175$			1.00000i			0.0755929i
$176$		0			0
$177$	−3.00000	+	3.00000i	−0.225494	+	0.225494i
$178$		0			0
$179$			4.89898i			0.366167i	0.983097	+	0.183083i	$0.0586079\pi$
$179$			4.89898i			0.366167i	−0.983097	+	0.183083i	$0.941392\pi$
$180$		0			0
$181$		−	22.0454i		−	1.63862i	−0.573349	−	0.819311i	$-0.694356\pi$
$181$		−	22.0454i		−	1.63862i	0.573349	−	0.819311i	$-0.305644\pi$
$182$		0			0
$183$	3.00000	−	3.00000i	0.221766	−	0.221766i
$184$		0			0
$185$			12.0000i			0.882258i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−3.67423	−	3.67423i	−0.267261	−	0.267261i
$190$		0			0
$191$	−6.00000			−0.434145			−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000			−0.434145			−0.217072	+	0.976156i	$0.569651\pi$
$192$		0			0
$193$	8.00000			0.575853			0.287926	−	0.957653i	$-0.407034\pi$
$193$	8.00000			0.575853			0.287926	+	0.957653i	$0.407034\pi$
$194$		0			0
$195$	−7.34847	+	7.34847i	−0.526235	+	0.526235i
$196$		0			0
$197$	9.79796			0.698076			0.349038	−	0.937109i	$-0.386508\pi$
$197$	9.79796			0.698076			0.349038	+	0.937109i	$0.386508\pi$
$198$		0			0
$199$		−	20.0000i		−	1.41776i	−0.705328	−	0.708881i	$-0.749200\pi$
$199$		−	20.0000i		−	1.41776i	0.705328	−	0.708881i	$-0.250800\pi$
$200$		0			0
$201$	−12.0000	−	12.0000i	−0.846415	−	0.846415i
$202$		0			0
$203$			4.89898i			0.343841i
$204$		0			0
$205$		−	14.6969i		−	1.02648i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	19.5959			1.34904			0.674519	−	0.738257i	$-0.264351\pi$
$211$	19.5959			1.34904			0.674519	+	0.738257i	$0.264351\pi$
$212$		0			0
$213$	−7.34847	−	7.34847i	−0.503509	−	0.503509i
$214$		0			0
$215$	12.0000			0.818393
$216$		0			0
$217$	8.00000			0.543075
$218$		0			0
$219$	12.2474	+	12.2474i	0.827606	+	0.827606i
$220$		0			0
$221$	−14.6969			−0.988623
$222$		0			0
$223$		−	4.00000i		−	0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$		−	4.00000i		−	0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$		0			0
$225$			3.00000i			0.200000i
$226$		0			0
$227$		−	2.44949i		−	0.162578i	−0.996691	−	0.0812892i	$-0.974096\pi$
$227$		−	2.44949i		−	0.162578i	0.996691	−	0.0812892i	$-0.0259037\pi$
$228$		0			0
$229$		−	17.1464i		−	1.13307i	−0.824038	−	0.566534i	$-0.808284\pi$
$229$		−	17.1464i		−	1.13307i	0.824038	−	0.566534i	$-0.191716\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			12.0000i			0.786146i	0.919507	+	0.393073i	$0.128588\pi$
$233$			12.0000i			0.786146i	−0.919507	+	0.393073i	$0.871412\pi$
$234$		0			0
$235$	29.3939			1.91745
$236$		0			0
$237$	9.79796	−	9.79796i	0.636446	−	0.636446i
$238$		0			0
$239$	24.0000			1.55243			0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000			1.55243			0.776215	+	0.630468i	$0.217137\pi$
$240$		0			0
$241$	−26.0000			−1.67481			−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000			−1.67481			−0.837404	+	0.546585i	$0.815928\pi$
$242$		0			0
$243$	−11.0227	−	11.0227i	−0.707107	−	0.707107i
$244$		0			0
$245$	−2.44949			−0.156492
$246$		0			0
$247$		−	18.0000i		−	1.14531i
$248$		0			0
$249$	−9.00000	+	9.00000i	−0.570352	+	0.570352i
$250$		0			0
$251$		−	17.1464i		−	1.08227i	−0.840935	−	0.541136i	$-0.817994\pi$
$251$		−	17.1464i		−	1.08227i	0.840935	−	0.541136i	$-0.182006\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−18.0000	+	18.0000i	−1.12720	+	1.12720i
$256$		0			0
$257$			6.00000i			0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$			6.00000i			0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$		0			0
$259$	−4.89898			−0.304408
$260$		0			0
$261$			14.6969i			0.909718i
$262$		0			0
$263$	18.0000			1.10993			0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000			1.10993			0.554964	+	0.831875i	$0.312732\pi$
$264$		0			0
$265$	24.0000			1.47431
$266$		0			0
$267$	22.0454	−	22.0454i	1.34916	−	1.34916i
$268$		0			0
$269$	17.1464			1.04544			0.522718	−	0.852506i	$-0.324918\pi$
$269$	17.1464			1.04544			0.522718	+	0.852506i	$0.324918\pi$
$270$		0			0
$271$		−	16.0000i		−	0.971931i	−0.873978	−	0.485965i	$-0.838468\pi$
$271$		−	16.0000i		−	0.971931i	0.873978	−	0.485965i	$-0.161532\pi$
$272$		0			0
$273$	−3.00000	−	3.00000i	−0.181568	−	0.181568i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			19.5959i			1.17740i	0.808350	+	0.588702i	$0.200361\pi$
$277$			19.5959i			1.17740i	−0.808350	+	0.588702i	$0.799639\pi$
$278$		0			0
$279$	24.0000			1.43684
$280$		0			0
$281$		−	12.0000i		−	0.715860i	−0.933748	−	0.357930i	$-0.883483\pi$
$281$		−	12.0000i		−	0.715860i	0.933748	−	0.357930i	$-0.116517\pi$
$282$		0			0
$283$	31.8434			1.89289			0.946446	−	0.322863i	$-0.104645\pi$
$283$	31.8434			1.89289			0.946446	+	0.322863i	$0.104645\pi$
$284$		0			0
$285$	−22.0454	−	22.0454i	−1.30586	−	1.30586i
$286$		0			0
$287$	6.00000			0.354169
$288$		0			0
$289$	−19.0000			−1.11765
$290$		0			0
$291$	2.44949	+	2.44949i	0.143592	+	0.143592i
$292$		0			0
$293$	−31.8434			−1.86031			−0.930155	−	0.367168i	$-0.880327\pi$
$293$	−31.8434			−1.86031			−0.930155	+	0.367168i	$0.880327\pi$
$294$		0			0
$295$			6.00000i			0.349334i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$			4.89898i			0.282372i
$302$		0			0
$303$	15.0000	+	15.0000i	0.861727	+	0.861727i
$304$		0			0
$305$		−	6.00000i		−	0.343559i
$306$		0			0
$307$	−26.9444			−1.53780			−0.768899	−	0.639371i	$-0.779195\pi$
$307$	−26.9444			−1.53780			−0.768899	+	0.639371i	$0.779195\pi$
$308$		0			0
$309$	9.79796	−	9.79796i	0.557386	−	0.557386i
$310$		0			0
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$		0			0
$313$	−14.0000			−0.791327			−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000			−0.791327			−0.395663	+	0.918396i	$0.629485\pi$
$314$		0			0
$315$	−7.34847			−0.414039
$316$		0			0
$317$	−29.3939			−1.65092			−0.825462	−	0.564457i	$-0.809085\pi$
$317$	−29.3939			−1.65092			−0.825462	+	0.564457i	$0.809085\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	6.00000	−	6.00000i	0.334887	−	0.334887i
$322$		0			0
$323$		−	44.0908i		−	2.45328i
$324$		0			0
$325$			2.44949i			0.135873i
$326$		0			0
$327$	−6.00000	+	6.00000i	−0.331801	+	0.331801i
$328$		0			0
$329$			12.0000i			0.661581i
$330$		0			0
$331$	−14.6969			−0.807817			−0.403908	−	0.914799i	$-0.632349\pi$
$331$	−14.6969			−0.807817			−0.403908	+	0.914799i	$0.632349\pi$
$332$		0			0
$333$	−14.6969			−0.805387
$334$		0			0
$335$	−24.0000			−1.31126
$336$		0			0
$337$	20.0000			1.08947			0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000			1.08947			0.544735	+	0.838608i	$0.316630\pi$
$338$		0			0
$339$	7.34847	−	7.34847i	0.399114	−	0.399114i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	9.79796i		−	0.525982i	−0.964798	−	0.262991i	$-0.915291\pi$
$347$		−	9.79796i		−	0.525982i	0.964798	−	0.262991i	$-0.0847090\pi$
$348$		0			0
$349$		−	12.2474i		−	0.655591i	−0.944749	−	0.327795i	$-0.893694\pi$
$349$		−	12.2474i		−	0.655591i	0.944749	−	0.327795i	$-0.106306\pi$
$350$		0			0
$351$	−9.00000	−	9.00000i	−0.480384	−	0.480384i
$352$		0			0
$353$		−	30.0000i		−	1.59674i	−0.602168	−	0.798369i	$-0.705696\pi$
$353$		−	30.0000i		−	1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$		0			0
$355$	−14.6969			−0.780033
$356$		0			0
$357$	−7.34847	−	7.34847i	−0.388922	−	0.388922i
$358$		0			0
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	35.0000			1.84211
$362$		0			0
$363$	13.4722	+	13.4722i	0.707107	+	0.707107i
$364$		0			0
$365$	24.4949			1.28212
$366$		0			0
$367$			4.00000i			0.208798i	0.994535	+	0.104399i	$0.0332919\pi$
$367$			4.00000i			0.208798i	−0.994535	+	0.104399i	$0.966708\pi$
$368$		0			0
$369$	18.0000			0.937043
$370$		0			0
$371$			9.79796i			0.508685i
$372$		0			0
$373$		−	29.3939i		−	1.52196i	−0.648777	−	0.760979i	$-0.724719\pi$
$373$		−	29.3939i		−	1.52196i	0.648777	−	0.760979i	$-0.275281\pi$
$374$		0			0
$375$	−12.0000	−	12.0000i	−0.619677	−	0.619677i
$376$		0			0
$377$			12.0000i			0.618031i
$378$		0			0
$379$	34.2929			1.76151			0.880753	−	0.473576i	$-0.157037\pi$
$379$	34.2929			1.76151			0.880753	+	0.473576i	$0.157037\pi$
$380$		0			0
$381$	−2.44949	+	2.44949i	−0.125491	+	0.125491i
$382$		0			0
$383$	−24.0000			−1.22634			−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000			−1.22634			−0.613171	+	0.789950i	$0.710106\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			14.6969i			0.747087i
$388$		0			0
$389$	24.4949			1.24194			0.620970	−	0.783834i	$-0.286739\pi$
$389$	24.4949			1.24194			0.620970	+	0.783834i	$0.286739\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−27.0000	+	27.0000i	−1.36197	+	1.36197i
$394$		0			0
$395$		−	19.5959i		−	0.985978i
$396$		0			0
$397$		−	2.44949i		−	0.122936i	−0.998109	−	0.0614682i	$-0.980422\pi$
$397$		−	2.44949i		−	0.122936i	0.998109	−	0.0614682i	$-0.0195783\pi$
$398$		0			0
$399$	9.00000	−	9.00000i	0.450564	−	0.450564i
$400$		0			0
$401$		−	30.0000i		−	1.49813i	−0.662497	−	0.749064i	$-0.730503\pi$
$401$		−	30.0000i		−	1.49813i	0.662497	−	0.749064i	$-0.269497\pi$
$402$		0			0
$403$	19.5959			0.976142
$404$		0			0
$405$	−22.0454			−1.09545
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−14.0000			−0.692255			−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000			−0.692255			−0.346128	+	0.938187i	$0.612504\pi$
$410$		0			0
$411$	−14.6969	+	14.6969i	−0.724947	+	0.724947i
$412$		0			0
$413$	−2.44949			−0.120532
$414$		0			0
$415$			18.0000i			0.883585i
$416$		0			0
$417$	−3.00000	−	3.00000i	−0.146911	−	0.146911i
$418$		0			0
$419$			22.0454i			1.07699i	0.842629	+	0.538494i	$0.181007\pi$
$419$			22.0454i			1.07699i	−0.842629	+	0.538494i	$0.818993\pi$
$420$		0			0
$421$			29.3939i			1.43257i	0.697808	+	0.716285i	$0.254159\pi$
$421$			29.3939i			1.43257i	−0.697808	+	0.716285i	$0.745841\pi$
$422$		0			0
$423$			36.0000i			1.75038i
$424$		0			0
$425$			6.00000i			0.291043i
$426$		0			0
$427$	2.44949			0.118539
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−2.00000			−0.0961139			−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000			−0.0961139			−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$	14.6969	+	14.6969i	0.704664	+	0.704664i
$436$		0			0
$437$		0			0
$438$		0			0
$439$			28.0000i			1.33637i	0.743996	+	0.668184i	$0.232928\pi$
$439$			28.0000i			1.33637i	−0.743996	+	0.668184i	$0.767072\pi$
$440$		0			0
$441$		−	3.00000i		−	0.142857i
$442$		0			0
$443$		−	24.4949i		−	1.16379i	−0.813265	−	0.581894i	$-0.802312\pi$
$443$		−	24.4949i		−	1.16379i	0.813265	−	0.581894i	$-0.197688\pi$
$444$		0			0
$445$		−	44.0908i		−	2.09011i
$446$		0			0
$447$	−24.0000	−	24.0000i	−1.13516	−	1.13516i
$448$		0			0
$449$			24.0000i			1.13263i	0.824189	+	0.566315i	$0.191631\pi$
$449$			24.0000i			1.13263i	−0.824189	+	0.566315i	$0.808369\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−2.44949	+	2.44949i	−0.115087	+	0.115087i
$454$		0			0
$455$	−6.00000			−0.281284
$456$		0			0
$457$	−8.00000			−0.374224			−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000			−0.374224			−0.187112	+	0.982339i	$0.559913\pi$
$458$		0			0
$459$	−22.0454	−	22.0454i	−1.02899	−	1.02899i
$460$		0			0
$461$	12.2474			0.570421			0.285210	−	0.958465i	$-0.407937\pi$
$461$	12.2474			0.570421			0.285210	+	0.958465i	$0.407937\pi$
$462$		0			0
$463$		−	16.0000i		−	0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$		−	16.0000i		−	0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$		0			0
$465$	24.0000	−	24.0000i	1.11297	−	1.11297i
$466$		0			0
$467$			26.9444i			1.24684i	0.781888	+	0.623419i	$0.214257\pi$
$467$			26.9444i			1.24684i	−0.781888	+	0.623419i	$0.785743\pi$
$468$		0			0
$469$		−	9.79796i		−	0.452428i
$470$		0			0
$471$	−9.00000	+	9.00000i	−0.414698	+	0.414698i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−7.34847			−0.337171
$476$		0			0
$477$			29.3939i			1.34585i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−12.0000			−0.547153
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.89898			0.222451
$486$		0			0
$487$		−	10.0000i		−	0.453143i	−0.973995	−	0.226572i	$-0.927248\pi$
$487$		−	10.0000i		−	0.453143i	0.973995	−	0.226572i	$-0.0727517\pi$
$488$		0			0
$489$	−6.00000	−	6.00000i	−0.271329	−	0.271329i
$490$		0			0
$491$			4.89898i			0.221088i	0.993871	+	0.110544i	$0.0352593\pi$
$491$			4.89898i			0.221088i	−0.993871	+	0.110544i	$0.964741\pi$
$492$		0			0
$493$			29.3939i			1.32383i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	6.00000i		−	0.269137i
$498$		0			0
$499$	−9.79796			−0.438617			−0.219308	−	0.975656i	$-0.570380\pi$
$499$	−9.79796			−0.438617			−0.219308	+	0.975656i	$0.570380\pi$
$500$		0			0
$501$	14.6969	+	14.6969i	0.656611	+	0.656611i
$502$		0			0
$503$	12.0000			0.535054			0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000			0.535054			0.267527	+	0.963550i	$0.413794\pi$
$504$		0			0
$505$	30.0000			1.33498
$506$		0			0
$507$	8.57321	+	8.57321i	0.380750	+	0.380750i
$508$		0			0
$509$	−22.0454			−0.977146			−0.488573	−	0.872523i	$-0.662482\pi$
$509$	−22.0454			−0.977146			−0.488573	+	0.872523i	$0.662482\pi$
$510$		0			0
$511$			10.0000i			0.442374i
$512$		0			0
$513$	27.0000	−	27.0000i	1.19208	−	1.19208i
$514$		0			0
$515$		−	19.5959i		−	0.863499i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−15.0000	−	15.0000i	−0.658427	−	0.658427i
$520$		0			0
$521$		−	30.0000i		−	1.31432i	−0.753749	−	0.657162i	$-0.771757\pi$
$521$		−	30.0000i		−	1.31432i	0.753749	−	0.657162i	$-0.228243\pi$
$522$		0			0
$523$	−2.44949			−0.107109			−0.0535544	−	0.998565i	$-0.517055\pi$
$523$	−2.44949			−0.107109			−0.0535544	+	0.998565i	$0.517055\pi$
$524$		0			0
$525$	−1.22474	+	1.22474i	−0.0534522	+	0.0534522i
$526$		0			0
$527$	48.0000			2.09091
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	−7.34847			−0.318896
$532$		0			0
$533$	14.6969			0.636595
$534$		0			0
$535$		−	12.0000i		−	0.518805i
$536$		0			0
$537$	−6.00000	+	6.00000i	−0.258919	+	0.258919i
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	27.0000	−	27.0000i	1.15868	−	1.15868i
$544$		0			0
$545$			12.0000i			0.514024i
$546$		0			0
$547$	−4.89898			−0.209465			−0.104733	−	0.994500i	$-0.533399\pi$
$547$	−4.89898			−0.209465			−0.104733	+	0.994500i	$0.533399\pi$
$548$		0			0
$549$	7.34847			0.313625
$550$		0			0
$551$	−36.0000			−1.53365
$552$		0			0
$553$	8.00000			0.340195
$554$		0			0
$555$	−14.6969	+	14.6969i	−0.623850	+	0.623850i
$556$		0			0
$557$	29.3939			1.24546			0.622729	−	0.782437i	$-0.286024\pi$
$557$	29.3939			1.24546			0.622729	+	0.782437i	$0.286024\pi$
$558$		0			0
$559$			12.0000i			0.507546i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	2.44949i		−	0.103234i	−0.998667	−	0.0516168i	$-0.983563\pi$
$563$		−	2.44949i		−	0.103234i	0.998667	−	0.0516168i	$-0.0164375\pi$
$564$		0			0
$565$		−	14.6969i		−	0.618305i
$566$		0			0
$567$		−	9.00000i		−	0.377964i
$568$		0			0
$569$			6.00000i			0.251533i	0.992060	+	0.125767i	$0.0401390\pi$
$569$			6.00000i			0.251533i	−0.992060	+	0.125767i	$0.959861\pi$
$570$		0			0
$571$	−44.0908			−1.84514			−0.922572	−	0.385826i	$-0.873917\pi$
$571$	−44.0908			−1.84514			−0.922572	+	0.385826i	$0.873917\pi$
$572$		0			0
$573$	−7.34847	−	7.34847i	−0.306987	−	0.306987i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−38.0000			−1.58196			−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000			−1.58196			−0.790980	+	0.611842i	$0.790429\pi$
$578$		0			0
$579$	9.79796	+	9.79796i	0.407189	+	0.407189i
$580$		0			0
$581$	−7.34847			−0.304866
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−18.0000			−0.744208
$586$		0			0
$587$			31.8434i			1.31432i	0.753753	+	0.657158i	$0.228242\pi$
$587$			31.8434i			1.31432i	−0.753753	+	0.657158i	$0.771758\pi$
$588$		0			0
$589$			58.7878i			2.42231i
$590$		0			0
$591$	12.0000	+	12.0000i	0.493614	+	0.493614i
$592$		0			0
$593$			30.0000i			1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$			30.0000i			1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$		0			0
$595$	−14.6969			−0.602516
$596$		0			0
$597$	24.4949	−	24.4949i	1.00251	−	1.00251i
$598$		0			0
$599$	30.0000			1.22577			0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000			1.22577			0.612883	+	0.790173i	$0.290010\pi$
$600$		0			0
$601$	−10.0000			−0.407909			−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$		0			0
$603$		−	29.3939i		−	1.19701i
$604$		0			0
$605$	26.9444			1.09545
$606$		0			0
$607$		−	16.0000i		−	0.649420i	−0.945814	−	0.324710i	$-0.894733\pi$
$607$		−	16.0000i		−	0.649420i	0.945814	−	0.324710i	$-0.105267\pi$
$608$		0			0
$609$	−6.00000	+	6.00000i	−0.243132	+	0.243132i
$610$		0			0
$611$			29.3939i			1.18915i
$612$		0			0
$613$			34.2929i			1.38508i	0.721382	+	0.692538i	$0.243507\pi$
$613$			34.2929i			1.38508i	−0.721382	+	0.692538i	$0.756493\pi$
$614$		0			0
$615$	18.0000	−	18.0000i	0.725830	−	0.725830i
$616$		0			0
$617$			6.00000i			0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$			6.00000i			0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$		0			0
$619$	−12.2474			−0.492267			−0.246133	−	0.969236i	$-0.579160\pi$
$619$	−12.2474			−0.492267			−0.246133	+	0.969236i	$0.579160\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	18.0000			0.721155
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−29.3939			−1.17201
$630$		0			0
$631$			32.0000i			1.27390i	0.770905	+	0.636950i	$0.219804\pi$
$631$			32.0000i			1.27390i	−0.770905	+	0.636950i	$0.780196\pi$
$632$		0			0
$633$	24.0000	+	24.0000i	0.953914	+	0.953914i
$634$		0			0
$635$			4.89898i			0.194410i
$636$		0			0
$637$		−	2.44949i		−	0.0970523i
$638$		0			0
$639$		−	18.0000i		−	0.712069i
$640$		0			0
$641$		−	18.0000i		−	0.710957i	−0.934684	−	0.355479i	$-0.884318\pi$
$641$		−	18.0000i		−	0.710957i	0.934684	−	0.355479i	$-0.115682\pi$
$642$		0			0
$643$	17.1464			0.676189			0.338095	−	0.941112i	$-0.390218\pi$
$643$	17.1464			0.676189			0.338095	+	0.941112i	$0.390218\pi$
$644$		0			0
$645$	14.6969	+	14.6969i	0.578691	+	0.578691i
$646$		0			0
$647$	24.0000			0.943537			0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000			0.943537			0.471769	+	0.881722i	$0.343616\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	9.79796	+	9.79796i	0.384012	+	0.384012i
$652$		0			0
$653$	4.89898			0.191712			0.0958559	−	0.995395i	$-0.469441\pi$
$653$	4.89898			0.191712			0.0958559	+	0.995395i	$0.469441\pi$
$654$		0			0
$655$			54.0000i			2.10995i
$656$		0			0
$657$			30.0000i			1.17041i
$658$		0			0
$659$		−	9.79796i		−	0.381674i	−0.981622	−	0.190837i	$-0.938880\pi$
$659$		−	9.79796i		−	0.381674i	0.981622	−	0.190837i	$-0.0611202\pi$
$660$		0			0
$661$			22.0454i			0.857467i	0.903431	+	0.428733i	$0.141040\pi$
$661$			22.0454i			0.857467i	−0.903431	+	0.428733i	$0.858960\pi$
$662$		0			0
$663$	−18.0000	−	18.0000i	−0.699062	−	0.699062i
$664$		0			0
$665$		−	18.0000i		−	0.698010i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	4.89898	−	4.89898i	0.189405	−	0.189405i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−14.0000			−0.539660			−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000			−0.539660			−0.269830	+	0.962908i	$0.586968\pi$
$674$		0			0
$675$	−3.67423	+	3.67423i	−0.141421	+	0.141421i
$676$		0			0
$677$	2.44949			0.0941415			0.0470708	−	0.998892i	$-0.485011\pi$
$677$	2.44949			0.0941415			0.0470708	+	0.998892i	$0.485011\pi$
$678$		0			0
$679$			2.00000i			0.0767530i
$680$		0			0
$681$	3.00000	−	3.00000i	0.114960	−	0.114960i
$682$		0			0
$683$			24.4949i			0.937271i	0.883392	+	0.468636i	$0.155254\pi$
$683$			24.4949i			0.937271i	−0.883392	+	0.468636i	$0.844746\pi$
$684$		0			0
$685$			29.3939i			1.12308i
$686$		0			0
$687$	21.0000	−	21.0000i	0.801200	−	0.801200i
$688$		0			0
$689$			24.0000i			0.914327i
$690$		0			0
$691$	−17.1464			−0.652281			−0.326140	−	0.945321i	$-0.605748\pi$
$691$	−17.1464			−0.652281			−0.326140	+	0.945321i	$0.605748\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.00000			−0.227593
$696$		0			0
$697$	36.0000			1.36360
$698$		0			0
$699$	−14.6969	+	14.6969i	−0.555889	+	0.555889i
$700$		0			0
$701$	24.4949			0.925160			0.462580	−	0.886578i	$-0.346924\pi$
$701$	24.4949			0.925160			0.462580	+	0.886578i	$0.346924\pi$
$702$		0			0
$703$		−	36.0000i		−	1.35777i
$704$		0			0
$705$	36.0000	+	36.0000i	1.35584	+	1.35584i
$706$		0			0
$707$			12.2474i			0.460613i
$708$		0			0
$709$			34.2929i			1.28790i	0.765070	+	0.643948i	$0.222705\pi$
$709$			34.2929i			1.28790i	−0.765070	+	0.643948i	$0.777295\pi$
$710$		0			0
$711$	24.0000			0.900070
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	29.3939	+	29.3939i	1.09773	+	1.09773i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	8.00000			0.297936
$722$		0			0
$723$	−31.8434	−	31.8434i	−1.18427	−	1.18427i
$724$		0			0
$725$	4.89898			0.181944
$726$		0			0
$727$			44.0000i			1.63187i	0.578144	+	0.815935i	$0.303777\pi$
$727$			44.0000i			1.63187i	−0.578144	+	0.815935i	$0.696223\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$			29.3939i			1.08717i
$732$		0			0
$733$		−	46.5403i		−	1.71901i	−0.511131	−	0.859503i	$-0.670773\pi$
$733$		−	46.5403i		−	1.71901i	0.511131	−	0.859503i	$-0.329227\pi$
$734$		0			0
$735$	−3.00000	−	3.00000i	−0.110657	−	0.110657i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	4.89898			0.180212			0.0901059	−	0.995932i	$-0.471279\pi$
$739$	4.89898			0.180212			0.0901059	+	0.995932i	$0.471279\pi$
$740$		0			0
$741$	22.0454	−	22.0454i	0.809858	−	0.809858i
$742$		0			0
$743$	−48.0000			−1.76095			−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000			−1.76095			−0.880475	+	0.474093i	$0.842776\pi$
$744$		0			0
$745$	−48.0000			−1.75858
$746$		0			0
$747$	−22.0454			−0.806599
$748$		0			0
$749$	4.89898			0.179005
$750$		0			0
$751$		−	38.0000i		−	1.38664i	−0.720630	−	0.693320i	$-0.756147\pi$
$751$		−	38.0000i		−	1.38664i	0.720630	−	0.693320i	$-0.243853\pi$
$752$		0			0
$753$	21.0000	−	21.0000i	0.765283	−	0.765283i
$754$		0			0
$755$			4.89898i			0.178292i
$756$		0			0
$757$		−	24.4949i		−	0.890282i	−0.895460	−	0.445141i	$-0.853154\pi$
$757$		−	24.4949i		−	0.890282i	0.895460	−	0.445141i	$-0.146846\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			30.0000i			1.08750i	0.839248	+	0.543750i	$0.182996\pi$
$761$			30.0000i			1.08750i	−0.839248	+	0.543750i	$0.817004\pi$
$762$		0			0
$763$	−4.89898			−0.177355
$764$		0			0
$765$	−44.0908			−1.59411
$766$		0			0
$767$	−6.00000			−0.216647
$768$		0			0
$769$	14.0000			0.504853			0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000			0.504853			0.252426	+	0.967616i	$0.418771\pi$
$770$		0			0
$771$	−7.34847	+	7.34847i	−0.264649	+	0.264649i
$772$		0			0
$773$	−36.7423			−1.32153			−0.660765	−	0.750593i	$-0.729768\pi$
$773$	−36.7423			−1.32153			−0.660765	+	0.750593i	$0.729768\pi$
$774$		0			0
$775$		−	8.00000i		−	0.287368i
$776$		0			0
$777$	−6.00000	−	6.00000i	−0.215249	−	0.215249i
$778$		0			0
$779$			44.0908i			1.57972i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−18.0000	+	18.0000i	−0.643268	+	0.643268i
$784$		0			0
$785$			18.0000i			0.642448i
$786$		0			0
$787$	−2.44949			−0.0873149			−0.0436574	−	0.999047i	$-0.513901\pi$
$787$	−2.44949			−0.0873149			−0.0436574	+	0.999047i	$0.513901\pi$
$788$		0			0
$789$	22.0454	+	22.0454i	0.784837	+	0.784837i
$790$		0			0
$791$	6.00000			0.213335
$792$		0			0
$793$	6.00000			0.213066
$794$		0			0
$795$	29.3939	+	29.3939i	1.04249	+	1.04249i
$796$		0			0
$797$	22.0454			0.780888			0.390444	−	0.920627i	$-0.372321\pi$
$797$	22.0454			0.780888			0.390444	+	0.920627i	$0.372321\pi$
$798$		0			0
$799$			72.0000i			2.54718i
$800$		0			0
$801$	54.0000			1.90800
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	21.0000	+	21.0000i	0.739235	+	0.739235i
$808$		0			0
$809$		−	18.0000i		−	0.632846i	−0.948618	−	0.316423i	$-0.897518\pi$
$809$		−	18.0000i		−	0.632846i	0.948618	−	0.316423i	$-0.102482\pi$
$810$		0			0
$811$	−36.7423			−1.29020			−0.645099	−	0.764099i	$-0.723184\pi$
$811$	−36.7423			−1.29020			−0.645099	+	0.764099i	$0.723184\pi$
$812$		0			0
$813$	19.5959	−	19.5959i	0.687259	−	0.687259i
$814$		0			0
$815$	−12.0000			−0.420342
$816$		0			0
$817$	−36.0000			−1.25948
$818$		0			0
$819$		−	7.34847i		−	0.256776i
$820$		0			0
$821$	19.5959			0.683902			0.341951	−	0.939718i	$-0.388912\pi$
$821$	19.5959			0.683902			0.341951	+	0.939718i	$0.388912\pi$
$822$		0			0
$823$			32.0000i			1.11545i	0.830026	+	0.557725i	$0.188326\pi$
$823$			32.0000i			1.11545i	−0.830026	+	0.557725i	$0.811674\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			24.4949i			0.851771i	0.904777	+	0.425886i	$0.140037\pi$
$827$			24.4949i			0.851771i	−0.904777	+	0.425886i	$0.859963\pi$
$828$		0			0
$829$		−	22.0454i		−	0.765669i	−0.923817	−	0.382834i	$-0.874948\pi$
$829$		−	22.0454i		−	0.765669i	0.923817	−	0.382834i	$-0.125052\pi$
$830$		0			0
$831$	−24.0000	+	24.0000i	−0.832551	+	0.832551i
$832$		0			0
$833$		−	6.00000i		−	0.207888i
$834$		0			0
$835$	29.3939			1.01722
$836$		0			0
$837$	29.3939	+	29.3939i	1.01600	+	1.01600i
$838$		0			0
$839$	24.0000			0.828572			0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000			0.828572			0.414286	+	0.910147i	$0.364031\pi$
$840$		0			0
$841$	−5.00000			−0.172414
$842$		0			0
$843$	14.6969	−	14.6969i	0.506189	−	0.506189i
$844$		0			0
$845$	17.1464			0.589855
$846$		0			0
$847$			11.0000i			0.377964i
$848$		0			0
$849$	39.0000	+	39.0000i	1.33848	+	1.33848i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	36.7423i		−	1.25803i	−0.777392	−	0.629017i	$-0.783458\pi$
$853$		−	36.7423i		−	1.25803i	0.777392	−	0.629017i	$-0.216542\pi$
$854$		0			0
$855$		−	54.0000i		−	1.84676i
$856$		0			0
$857$			42.0000i			1.43469i	0.696717	+	0.717346i	$0.254643\pi$
$857$			42.0000i			1.43469i	−0.696717	+	0.717346i	$0.745357\pi$
$858$		0			0
$859$	−46.5403			−1.58793			−0.793967	−	0.607960i	$-0.791988\pi$
$859$	−46.5403			−1.58793			−0.793967	+	0.607960i	$0.791988\pi$
$860$		0			0
$861$	7.34847	+	7.34847i	0.250435	+	0.250435i
$862$		0			0
$863$	6.00000			0.204242			0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000			0.204242			0.102121	+	0.994772i	$0.467437\pi$
$864$		0			0
$865$	−30.0000			−1.02003
$866$		0			0
$867$	−23.2702	−	23.2702i	−0.790296	−	0.790296i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		−	24.0000i		−	0.813209i
$872$		0			0
$873$			6.00000i			0.203069i
$874$		0			0
$875$		−	9.79796i		−	0.331231i
$876$		0			0
$877$			24.4949i			0.827134i	0.910474	+	0.413567i	$0.135717\pi$
$877$			24.4949i			0.827134i	−0.910474	+	0.413567i	$0.864283\pi$
$878$		0			0
$879$	−39.0000	−	39.0000i	−1.31544	−	1.31544i
$880$		0			0
$881$		−	18.0000i		−	0.606435i	−0.952921	−	0.303218i	$-0.901939\pi$
$881$		−	18.0000i		−	0.606435i	0.952921	−	0.303218i	$-0.0980609\pi$
$882$		0			0
$883$	29.3939			0.989183			0.494591	−	0.869126i	$-0.335318\pi$
$883$	29.3939			0.989183			0.494591	+	0.869126i	$0.335318\pi$
$884$		0			0
$885$	−7.34847	+	7.34847i	−0.247016	+	0.247016i
$886$		0			0
$887$	−12.0000			−0.402921			−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000			−0.402921			−0.201460	+	0.979497i	$0.564569\pi$
$888$		0			0
$889$	−2.00000			−0.0670778
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−88.1816			−2.95089
$894$		0			0
$895$			12.0000i			0.401116i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	39.1918i		−	1.30712i
$900$		0			0
$901$			58.7878i			1.95850i
$902$		0			0
$903$	−6.00000	+	6.00000i	−0.199667	+	0.199667i
$904$		0			0
$905$		−	54.0000i		−	1.79502i
$906$		0			0
$907$	19.5959			0.650672			0.325336	−	0.945599i	$-0.394523\pi$
$907$	19.5959			0.650672			0.325336	+	0.945599i	$0.394523\pi$
$908$		0			0
$909$			36.7423i			1.21867i
$910$		0			0
$911$	−24.0000			−0.795155			−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000			−0.795155			−0.397578	+	0.917568i	$0.630149\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	7.34847	−	7.34847i	0.242933	−	0.242933i
$916$		0			0
$917$	−22.0454			−0.728003
$918$		0			0
$919$		−	16.0000i		−	0.527791i	−0.964551	−	0.263896i	$-0.914993\pi$
$919$		−	16.0000i		−	0.527791i	0.964551	−	0.263896i	$-0.0850075\pi$
$920$		0			0
$921$	−33.0000	−	33.0000i	−1.08739	−	1.08739i
$922$		0			0
$923$		−	14.6969i		−	0.483756i
$924$		0			0
$925$			4.89898i			0.161077i
$926$		0			0
$927$	24.0000			0.788263
$928$		0			0
$929$		−	18.0000i		−	0.590561i	−0.955411	−	0.295280i	$-0.904587\pi$
$929$		−	18.0000i		−	0.590561i	0.955411	−	0.295280i	$-0.0954131\pi$
$930$		0			0
$931$	7.34847			0.240836
$932$		0			0
$933$	−14.6969	−	14.6969i	−0.481156	−	0.481156i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$		0			0
$939$	−17.1464	−	17.1464i	−0.559553	−	0.559553i
$940$		0			0
$941$	36.7423			1.19777			0.598883	−	0.800836i	$-0.295611\pi$
$941$	36.7423			1.19777			0.598883	+	0.800836i	$0.295611\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−9.00000	−	9.00000i	−0.292770	−	0.292770i
$946$		0			0
$947$			19.5959i			0.636782i	0.947960	+	0.318391i	$0.103142\pi$
$947$			19.5959i			0.636782i	−0.947960	+	0.318391i	$0.896858\pi$
$948$		0			0
$949$			24.4949i			0.795138i
$950$		0			0
$951$	−36.0000	−	36.0000i	−1.16738	−	1.16738i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$	−14.6969			−0.475582
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−12.0000			−0.387500
$960$		0			0
$961$	−33.0000			−1.06452
$962$		0			0
$963$	14.6969			0.473602
$964$		0			0
$965$	19.5959			0.630815
$966$		0			0
$967$			22.0000i			0.707472i	0.935345	+	0.353736i	$0.115089\pi$
$967$			22.0000i			0.707472i	−0.935345	+	0.353736i	$0.884911\pi$
$968$		0			0
$969$	54.0000	−	54.0000i	1.73473	−	1.73473i
$970$		0			0
$971$			12.2474i			0.393039i	0.980500	+	0.196520i	$0.0629640\pi$
$971$			12.2474i			0.393039i	−0.980500	+	0.196520i	$0.937036\pi$
$972$		0			0
$973$		−	2.44949i		−	0.0785270i
$974$		0			0
$975$	−3.00000	+	3.00000i	−0.0960769	+	0.0960769i
$976$		0			0
$977$			12.0000i			0.383914i	0.981403	+	0.191957i	$0.0614834\pi$
$977$			12.0000i			0.383914i	−0.981403	+	0.191957i	$0.938517\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−14.6969			−0.469237
$982$		0			0
$983$	36.0000			1.14822			0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000			1.14822			0.574111	+	0.818778i	$0.305348\pi$
$984$		0			0
$985$	24.0000			0.764704
$986$		0			0
$987$	−14.6969	+	14.6969i	−0.467809	+	0.467809i
$988$		0			0
$989$		0			0
$990$		0			0
$991$			16.0000i			0.508257i	0.967170	+	0.254128i	$0.0817886\pi$
$991$			16.0000i			0.508257i	−0.967170	+	0.254128i	$0.918211\pi$
$992$		0			0
$993$	−18.0000	−	18.0000i	−0.571213	−	0.571213i
$994$		0			0
$995$		−	48.9898i		−	1.55308i
$996$		0			0
$997$		−	56.3383i		−	1.78425i	−0.451788	−	0.892125i	$-0.649214\pi$
$997$		−	56.3383i		−	1.78425i	0.451788	−	0.892125i	$-0.350786\pi$
$998$		0			0
$999$	−18.0000	−	18.0000i	−0.569495	−	0.569495i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.2.j.d.1247.4	yes	4
3.2	odd	2		1344.2.j.c.1247.3	yes	4
4.3	odd	2		1344.2.j.c.1247.1	✓	4
8.3	odd	2		1344.2.j.c.1247.4	yes	4
8.5	even	2	inner	1344.2.j.d.1247.1	yes	4
12.11	even	2	inner	1344.2.j.d.1247.2	yes	4
24.5	odd	2		1344.2.j.c.1247.2	yes	4
24.11	even	2	inner	1344.2.j.d.1247.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.2.j.c.1247.1	✓	4	4.3	odd	2
1344.2.j.c.1247.2	yes	4	24.5	odd	2
1344.2.j.c.1247.3	yes	4	3.2	odd	2
1344.2.j.c.1247.4	yes	4	8.3	odd	2
1344.2.j.d.1247.1	yes	4	8.5	even	2	inner
1344.2.j.d.1247.2	yes	4	12.11	even	2	inner
1344.2.j.d.1247.3	yes	4	24.11	even	2	inner
1344.2.j.d.1247.4	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1344.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 + 1.22474i) q^{3} +2.44949 q^{5} +1.00000i q^{7} +3.00000i q^{9} +O(q^{10})\)	\(q+(1.22474 + 1.22474i) q^{3} +2.44949 q^{5} +1.00000i q^{7} +3.00000i q^{9} +2.44949i q^{13} +(3.00000 + 3.00000i) q^{15} +6.00000i q^{17} -7.34847 q^{19} +(-1.22474 + 1.22474i) q^{21} +1.00000 q^{25} +(-3.67423 + 3.67423i) q^{27} +4.89898 q^{29} -8.00000i q^{31} +2.44949i q^{35} +4.89898i q^{37} +(-3.00000 + 3.00000i) q^{39} -6.00000i q^{41} +4.89898 q^{43} +7.34847i q^{45} +12.0000 q^{47} -1.00000 q^{49} +(-7.34847 + 7.34847i) q^{51} +9.79796 q^{53} +(-9.00000 - 9.00000i) q^{57} +2.44949i q^{59} -2.44949i q^{61} -3.00000 q^{63} +6.00000i q^{65} -9.79796 q^{67} -6.00000 q^{71} +10.0000 q^{73} +(1.22474 + 1.22474i) q^{75} -8.00000i q^{79} -9.00000 q^{81} +7.34847i q^{83} +14.6969i q^{85} +(6.00000 + 6.00000i) q^{87} -18.0000i q^{89} -2.44949 q^{91} +(9.79796 - 9.79796i) q^{93} -18.0000 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 12 q^{15} + 4 q^{25} - 12 q^{39} + 48 q^{47} - 4 q^{49} - 36 q^{57} - 12 q^{63} - 24 q^{71} + 40 q^{73} - 36 q^{81} + 24 q^{87} - 72 q^{95} + 8 q^{97}+O(q^{100}) \) 4 * q + 12 * q^15 + 4 * q^25 - 12 * q^39 + 48 * q^47 - 4 * q^49 - 36 * q^57 - 12 * q^63 - 24 * q^71 + 40 * q^73 - 36 * q^81 + 24 * q^87 - 72 * q^95 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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