LMFDB - Embedded newform 1344.2.k.d.1217.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,2,Mod(1217,1344)]

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([0, 0, 1, 1]))

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

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

chi := DirichletCharacter("1344.1217");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1344.k (of order $2$, degree $1$, 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:	$10.7318940317$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1217.4
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1344.1217
Dual form			1344.2.k.d.1217.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 1.22474i) q^{3} +2.44949 q^{5} +(1.00000 + 2.44949i) q^{7} +3.00000i q^{9} +O(q^{10})$	$q+(1.22474 + 1.22474i) q^{3} +2.44949 q^{5} +(1.00000 + 2.44949i) q^{7} +3.00000i q^{9} -2.44949i q^{13} +(3.00000 + 3.00000i) q^{15} +4.89898 q^{17} -2.44949i q^{19} +(-1.77526 + 4.22474i) q^{21} +6.00000i q^{23} +1.00000 q^{25} +(-3.67423 + 3.67423i) q^{27} -6.00000i q^{29} +(2.44949 + 6.00000i) q^{35} +2.00000 q^{37} +(3.00000 - 3.00000i) q^{39} -4.89898 q^{41} +4.00000 q^{43} +7.34847i q^{45} -4.89898 q^{47} +(-5.00000 + 4.89898i) q^{49} +(6.00000 + 6.00000i) q^{51} +6.00000i q^{53} +(3.00000 - 3.00000i) q^{57} -12.2474 q^{59} -12.2474i q^{61} +(-7.34847 + 3.00000i) q^{63} -6.00000i q^{65} +8.00000 q^{67} +(-7.34847 + 7.34847i) q^{69} -9.79796i q^{73} +(1.22474 + 1.22474i) q^{75} +10.0000 q^{79} -9.00000 q^{81} +2.44949 q^{83} +12.0000 q^{85} +(7.34847 - 7.34847i) q^{87} +(6.00000 - 2.44949i) q^{91} -6.00000i q^{95} -4.89898i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^7	$ 4 q + 4 q^{7} + 12 q^{15} - 12 q^{21} + 4 q^{25} + 8 q^{37} + 12 q^{39} + 16 q^{43} - 20 q^{49} + 24 q^{51} + 12 q^{57} + 32 q^{67} + 40 q^{79} - 36 q^{81} + 48 q^{85} + 24 q^{91}+O(q^{100}) $ 4 * q + 4 * q^7 + 12 * q^15 - 12 * q^21 + 4 * q^25 + 8 * q^37 + 12 * q^39 + 16 * q^43 - 20 * q^49 + 24 * q^51 + 12 * q^57 + 32 * q^67 + 40 * q^79 - 36 * q^81 + 48 * q^85 + 24 * q^91

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
$7$	1.00000	+	2.44949i	0.377964	+	0.925820i
$7$	1.00000	+	2.44949i	0.377964	+	0.925820i
$8$		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	$-0.889680\pi$
$13$		−	2.44949i		−	0.679366i	0.940540	−	0.339683i	$-0.110320\pi$
$14$		0			0
$15$	3.00000	+	3.00000i	0.774597	+	0.774597i
$16$		0			0
$17$	4.89898			1.18818			0.594089	−	0.804400i	$-0.297513\pi$
$17$	4.89898			1.18818			0.594089	+	0.804400i	$0.297513\pi$
$18$		0			0
$19$		−	2.44949i		−	0.561951i	−0.959715	−	0.280976i	$-0.909342\pi$
$19$		−	2.44949i		−	0.561951i	0.959715	−	0.280976i	$-0.0906580\pi$
$20$		0			0
$21$	−1.77526	+	4.22474i	−0.387392	+	0.921915i
$22$		0			0
$23$			6.00000i			1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$			6.00000i			1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−3.67423	+	3.67423i	−0.707107	+	0.707107i
$28$		0			0
$29$		−	6.00000i		−	1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$		−	6.00000i		−	1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.44949	+	6.00000i	0.414039	+	1.01419i
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$	3.00000	−	3.00000i	0.480384	−	0.480384i
$40$		0			0
$41$	−4.89898			−0.765092			−0.382546	−	0.923936i	$-0.624953\pi$
$41$	−4.89898			−0.765092			−0.382546	+	0.923936i	$0.624953\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$			7.34847i			1.09545i
$46$		0			0
$47$	−4.89898			−0.714590			−0.357295	−	0.933992i	$-0.616301\pi$
$47$	−4.89898			−0.714590			−0.357295	+	0.933992i	$0.616301\pi$
$48$		0			0
$49$	−5.00000	+	4.89898i	−0.714286	+	0.699854i
$50$		0			0
$51$	6.00000	+	6.00000i	0.840168	+	0.840168i
$52$		0			0
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	3.00000	−	3.00000i	0.397360	−	0.397360i
$58$		0			0
$59$	−12.2474			−1.59448			−0.797241	−	0.603661i	$-0.793708\pi$
$59$	−12.2474			−1.59448			−0.797241	+	0.603661i	$0.793708\pi$
$60$		0			0
$61$		−	12.2474i		−	1.56813i	−0.620682	−	0.784063i	$-0.713144\pi$
$61$		−	12.2474i		−	1.56813i	0.620682	−	0.784063i	$-0.286856\pi$
$62$		0			0
$63$	−7.34847	+	3.00000i	−0.925820	+	0.377964i
$64$		0			0
$65$		−	6.00000i		−	0.744208i
$66$		0			0
$67$	8.00000			0.977356			0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000			0.977356			0.488678	+	0.872464i	$0.337479\pi$
$68$		0			0
$69$	−7.34847	+	7.34847i	−0.884652	+	0.884652i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	9.79796i		−	1.14676i	−0.819288	−	0.573382i	$-0.805631\pi$
$73$		−	9.79796i		−	1.14676i	0.819288	−	0.573382i	$-0.194369\pi$
$74$		0			0
$75$	1.22474	+	1.22474i	0.141421	+	0.141421i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	10.0000			1.12509			0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000			1.12509			0.562544	+	0.826767i	$0.309823\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$	2.44949			0.268866			0.134433	−	0.990923i	$-0.457079\pi$
$83$	2.44949			0.268866			0.134433	+	0.990923i	$0.457079\pi$
$84$		0			0
$85$	12.0000			1.30158
$86$		0			0
$87$	7.34847	−	7.34847i	0.787839	−	0.787839i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	6.00000	−	2.44949i	0.628971	−	0.256776i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	6.00000i		−	0.615587i
$96$		0			0
$97$		−	4.89898i		−	0.497416i	−0.968579	−	0.248708i	$-0.919994\pi$
$97$		−	4.89898i		−	0.497416i	0.968579	−	0.248708i	$-0.0800060\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.34847			−0.731200			−0.365600	−	0.930772i	$-0.619136\pi$
$101$	−7.34847			−0.731200			−0.365600	+	0.930772i	$0.619136\pi$
$102$		0			0
$103$			9.79796i			0.965422i	0.875780	+	0.482711i	$0.160348\pi$
$103$			9.79796i			0.965422i	−0.875780	+	0.482711i	$0.839652\pi$
$104$		0			0
$105$	−4.34847	+	10.3485i	−0.424367	+	1.00991i
$106$		0			0
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$		0			0
$111$	2.44949	+	2.44949i	0.232495	+	0.232495i
$112$		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
$115$			14.6969i			1.37050i
$116$		0			0
$117$	7.34847			0.679366
$118$		0			0
$119$	4.89898	+	12.0000i	0.449089	+	1.10004i
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$	−6.00000	−	6.00000i	−0.541002	−	0.541002i
$124$		0			0
$125$	−9.79796			−0.876356
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$	4.89898	+	4.89898i	0.431331	+	0.431331i
$130$		0			0
$131$	7.34847			0.642039			0.321019	−	0.947073i	$-0.395975\pi$
$131$	7.34847			0.642039			0.321019	+	0.947073i	$0.395975\pi$
$132$		0			0
$133$	6.00000	−	2.44949i	0.520266	−	0.212398i
$134$		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
$139$		−	2.44949i		−	0.207763i	−0.994590	−	0.103882i	$-0.966874\pi$
$139$		−	2.44949i		−	0.207763i	0.994590	−	0.103882i	$-0.0331263\pi$
$140$		0			0
$141$	−6.00000	−	6.00000i	−0.505291	−	0.505291i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		−	14.6969i		−	1.22051i
$146$		0			0
$147$	−12.1237	−	0.123724i	−0.999948	−	0.0102046i
$148$		0			0
$149$		−	6.00000i		−	0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$		−	6.00000i		−	0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$			14.6969i			1.18818i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	7.34847i		−	0.586472i	−0.956040	−	0.293236i	$-0.905268\pi$
$157$		−	7.34847i		−	0.586472i	0.956040	−	0.293236i	$-0.0947321\pi$
$158$		0			0
$159$	−7.34847	+	7.34847i	−0.582772	+	0.582772i
$160$		0			0
$161$	−14.6969	+	6.00000i	−1.15828	+	0.472866i
$162$		0			0
$163$	−16.0000			−1.25322			−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000			−1.25322			−0.626608	+	0.779334i	$0.715557\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.89898			−0.379094			−0.189547	−	0.981872i	$-0.560702\pi$
$167$	−4.89898			−0.379094			−0.189547	+	0.981872i	$0.560702\pi$
$168$		0			0
$169$	7.00000			0.538462
$170$		0			0
$171$	7.34847			0.561951
$172$		0			0
$173$	22.0454			1.67608			0.838041	−	0.545608i	$-0.183701\pi$
$173$	22.0454			1.67608			0.838041	+	0.545608i	$0.183701\pi$
$174$		0			0
$175$	1.00000	+	2.44949i	0.0755929	+	0.185164i
$176$		0			0
$177$	−15.0000	−	15.0000i	−1.12747	−	1.12747i
$178$		0			0
$179$		−	24.0000i		−	1.79384i	−0.442189	−	0.896922i	$-0.645798\pi$
$179$		−	24.0000i		−	1.79384i	0.442189	−	0.896922i	$-0.354202\pi$
$180$		0			0
$181$			12.2474i			0.910346i	0.890403	+	0.455173i	$0.150423\pi$
$181$			12.2474i			0.910346i	−0.890403	+	0.455173i	$0.849577\pi$
$182$		0			0
$183$	15.0000	−	15.0000i	1.10883	−	1.10883i
$184$		0			0
$185$	4.89898			0.360180
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−12.6742	−	5.32577i	−0.921915	−	0.387392i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	4.00000			0.287926			0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000			0.287926			0.143963	+	0.989583i	$0.454015\pi$
$194$		0			0
$195$	7.34847	−	7.34847i	0.526235	−	0.526235i
$196$		0			0
$197$			18.0000i			1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$			18.0000i			1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$		0			0
$199$		−	9.79796i		−	0.694559i	−0.937762	−	0.347279i	$-0.887106\pi$
$199$		−	9.79796i		−	0.694559i	0.937762	−	0.347279i	$-0.112894\pi$
$200$		0			0
$201$	9.79796	+	9.79796i	0.691095	+	0.691095i
$202$		0			0
$203$	14.6969	−	6.00000i	1.03152	−	0.421117i
$204$		0			0
$205$	−12.0000			−0.838116
$206$		0			0
$207$	−18.0000			−1.25109
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−8.00000			−0.550743			−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000			−0.550743			−0.275371	+	0.961338i	$0.588801\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	9.79796			0.668215
$216$		0			0
$217$		0			0
$218$		0			0
$219$	12.0000	−	12.0000i	0.810885	−	0.810885i
$220$		0			0
$221$		−	12.0000i		−	0.807207i
$222$		0			0
$223$		−	14.6969i		−	0.984180i	−0.870544	−	0.492090i	$-0.836233\pi$
$223$		−	14.6969i		−	0.984180i	0.870544	−	0.492090i	$-0.163767\pi$
$224$		0			0
$225$			3.00000i			0.200000i
$226$		0			0
$227$	−7.34847			−0.487735			−0.243868	−	0.969809i	$-0.578416\pi$
$227$	−7.34847			−0.487735			−0.243868	+	0.969809i	$0.578416\pi$
$228$		0			0
$229$		−	22.0454i		−	1.45680i	−0.685151	−	0.728401i	$-0.740264\pi$
$229$		−	22.0454i		−	1.45680i	0.685151	−	0.728401i	$-0.259736\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			24.0000i			1.57229i	0.618041	+	0.786146i	$0.287927\pi$
$233$			24.0000i			1.57229i	−0.618041	+	0.786146i	$0.712073\pi$
$234$		0			0
$235$	−12.0000			−0.782794
$236$		0			0
$237$	12.2474	+	12.2474i	0.795557	+	0.795557i
$238$		0			0
$239$		−	6.00000i		−	0.388108i	−0.980991	−	0.194054i	$-0.937836\pi$
$239$		−	6.00000i		−	0.388108i	0.980991	−	0.194054i	$-0.0621637\pi$
$240$		0			0
$241$		−	24.4949i		−	1.57786i	−0.614486	−	0.788928i	$-0.710637\pi$
$241$		−	24.4949i		−	1.57786i	0.614486	−	0.788928i	$-0.289363\pi$
$242$		0			0
$243$	−11.0227	−	11.0227i	−0.707107	−	0.707107i
$244$		0			0
$245$	−12.2474	+	12.0000i	−0.782461	+	0.766652i
$246$		0			0
$247$	−6.00000			−0.381771
$248$		0			0
$249$	3.00000	+	3.00000i	0.190117	+	0.190117i
$250$		0			0
$251$	−17.1464			−1.08227			−0.541136	−	0.840935i	$-0.682006\pi$
$251$	−17.1464			−1.08227			−0.541136	+	0.840935i	$0.682006\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	14.6969	+	14.6969i	0.920358	+	0.920358i
$256$		0			0
$257$	29.3939			1.83354			0.916770	−	0.399416i	$-0.130787\pi$
$257$	29.3939			1.83354			0.916770	+	0.399416i	$0.130787\pi$
$258$		0			0
$259$	2.00000	+	4.89898i	0.124274	+	0.304408i
$260$		0			0
$261$	18.0000			1.11417
$262$		0			0
$263$		−	24.0000i		−	1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$		−	24.0000i		−	1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$		0			0
$265$			14.6969i			0.902826i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	12.2474			0.746740			0.373370	−	0.927682i	$-0.378202\pi$
$269$	12.2474			0.746740			0.373370	+	0.927682i	$0.378202\pi$
$270$		0			0
$271$		−	24.4949i		−	1.48796i	−0.668202	−	0.743980i	$-0.732936\pi$
$271$		−	24.4949i		−	1.48796i	0.668202	−	0.743980i	$-0.267064\pi$
$272$		0			0
$273$	10.3485	+	4.34847i	0.626318	+	0.263181i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	22.0000			1.32185			0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000			1.32185			0.660926	+	0.750451i	$0.270164\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		−	22.0454i		−	1.31046i	−0.755428	−	0.655232i	$-0.772571\pi$
$283$		−	22.0454i		−	1.31046i	0.755428	−	0.655232i	$-0.227429\pi$
$284$		0			0
$285$	7.34847	−	7.34847i	0.435286	−	0.435286i
$286$		0			0
$287$	−4.89898	−	12.0000i	−0.289178	−	0.708338i
$288$		0			0
$289$	7.00000			0.411765
$290$		0			0
$291$	6.00000	−	6.00000i	0.351726	−	0.351726i
$292$		0			0
$293$	−2.44949			−0.143101			−0.0715504	−	0.997437i	$-0.522795\pi$
$293$	−2.44949			−0.143101			−0.0715504	+	0.997437i	$0.522795\pi$
$294$		0			0
$295$	−30.0000			−1.74667
$296$		0			0
$297$		0			0
$298$		0			0
$299$	14.6969			0.849946
$300$		0			0
$301$	4.00000	+	9.79796i	0.230556	+	0.564745i
$302$		0			0
$303$	−9.00000	−	9.00000i	−0.517036	−	0.517036i
$304$		0			0
$305$		−	30.0000i		−	1.71780i
$306$		0			0
$307$			7.34847i			0.419399i	0.977766	+	0.209700i	$0.0672486\pi$
$307$			7.34847i			0.419399i	−0.977766	+	0.209700i	$0.932751\pi$
$308$		0			0
$309$	−12.0000	+	12.0000i	−0.682656	+	0.682656i
$310$		0			0
$311$	−19.5959			−1.11118			−0.555591	−	0.831456i	$-0.687508\pi$
$311$	−19.5959			−1.11118			−0.555591	+	0.831456i	$0.687508\pi$
$312$		0			0
$313$		−	34.2929i		−	1.93835i	−0.246380	−	0.969173i	$-0.579241\pi$
$313$		−	34.2929i		−	1.93835i	0.246380	−	0.969173i	$-0.420759\pi$
$314$		0			0
$315$	−18.0000	+	7.34847i	−1.01419	+	0.414039i
$316$		0			0
$317$			18.0000i			1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$			18.0000i			1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−14.6969	+	14.6969i	−0.820303	+	0.820303i
$322$		0			0
$323$		−	12.0000i		−	0.667698i
$324$		0			0
$325$		−	2.44949i		−	0.135873i
$326$		0			0
$327$	−12.2474	−	12.2474i	−0.677285	−	0.677285i
$328$		0			0
$329$	−4.89898	−	12.0000i	−0.270089	−	0.661581i
$330$		0			0
$331$	−8.00000			−0.439720			−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000			−0.439720			−0.219860	+	0.975531i	$0.570560\pi$
$332$		0			0
$333$			6.00000i			0.328798i
$334$		0			0
$335$	19.5959			1.07064
$336$		0			0
$337$	−32.0000			−1.74315			−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−32.0000			−1.74315			−0.871576	+	0.490261i	$0.836901\pi$
$338$		0			0
$339$	7.34847	−	7.34847i	0.399114	−	0.399114i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−17.0000	−	7.34847i	−0.917914	−	0.396780i
$344$		0			0
$345$	−18.0000	+	18.0000i	−0.969087	+	0.969087i
$346$		0			0
$347$			12.0000i			0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$			12.0000i			0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$		0			0
$349$			2.44949i			0.131118i	0.997849	+	0.0655591i	$0.0208831\pi$
$349$			2.44949i			0.131118i	−0.997849	+	0.0655591i	$0.979117\pi$
$350$		0			0
$351$	9.00000	+	9.00000i	0.480384	+	0.480384i
$352$		0			0
$353$	−9.79796			−0.521493			−0.260746	−	0.965407i	$-0.583969\pi$
$353$	−9.79796			−0.521493			−0.260746	+	0.965407i	$0.583969\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−8.69694	+	20.6969i	−0.460291	+	1.09540i
$358$		0			0
$359$		−	6.00000i		−	0.316668i	−0.987386	−	0.158334i	$-0.949388\pi$
$359$		−	6.00000i		−	0.316668i	0.987386	−	0.158334i	$-0.0506123\pi$
$360$		0			0
$361$	13.0000			0.684211
$362$		0			0
$363$	13.4722	+	13.4722i	0.707107	+	0.707107i
$364$		0			0
$365$		−	24.0000i		−	1.25622i
$366$		0			0
$367$			4.89898i			0.255725i	0.991792	+	0.127862i	$0.0408116\pi$
$367$			4.89898i			0.255725i	−0.991792	+	0.127862i	$0.959188\pi$
$368$		0			0
$369$		−	14.6969i		−	0.765092i
$370$		0			0
$371$	−14.6969	+	6.00000i	−0.763027	+	0.311504i
$372$		0			0
$373$	−14.0000			−0.724893			−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000			−0.724893			−0.362446	+	0.932005i	$0.618058\pi$
$374$		0			0
$375$	−12.0000	−	12.0000i	−0.619677	−	0.619677i
$376$		0			0
$377$	−14.6969			−0.756931
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$		0			0
$381$	−9.79796	−	9.79796i	−0.501965	−	0.501965i
$382$		0			0
$383$	34.2929			1.75228			0.876142	−	0.482054i	$-0.160109\pi$
$383$	34.2929			1.75228			0.876142	+	0.482054i	$0.160109\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			12.0000i			0.609994i
$388$		0			0
$389$		−	6.00000i		−	0.304212i	−0.988364	−	0.152106i	$-0.951394\pi$
$389$		−	6.00000i		−	0.304212i	0.988364	−	0.152106i	$-0.0486055\pi$
$390$		0			0
$391$			29.3939i			1.48651i
$392$		0			0
$393$	9.00000	+	9.00000i	0.453990	+	0.453990i
$394$		0			0
$395$	24.4949			1.23247
$396$		0			0
$397$		−	7.34847i		−	0.368809i	−0.982850	−	0.184405i	$-0.940964\pi$
$397$		−	7.34847i		−	0.368809i	0.982850	−	0.184405i	$-0.0590357\pi$
$398$		0			0
$399$	10.3485	+	4.34847i	0.518071	+	0.217696i
$400$		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
$403$		0			0
$404$		0			0
$405$	−22.0454			−1.09545
$406$		0			0
$407$		0			0
$408$		0			0
$409$			34.2929i			1.69567i	0.530258	+	0.847836i	$0.322095\pi$
$409$			34.2929i			1.69567i	−0.530258	+	0.847836i	$0.677905\pi$
$410$		0			0
$411$	−14.6969	+	14.6969i	−0.724947	+	0.724947i
$412$		0			0
$413$	−12.2474	−	30.0000i	−0.602658	−	1.47620i
$414$		0			0
$415$	6.00000			0.294528
$416$		0			0
$417$	3.00000	−	3.00000i	0.146911	−	0.146911i
$418$		0			0
$419$	12.2474			0.598327			0.299164	−	0.954202i	$-0.403292\pi$
$419$	12.2474			0.598327			0.299164	+	0.954202i	$0.403292\pi$
$420$		0			0
$421$	−2.00000			−0.0974740			−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000			−0.0974740			−0.0487370	+	0.998812i	$0.515520\pi$
$422$		0			0
$423$		−	14.6969i		−	0.714590i
$424$		0			0
$425$	4.89898			0.237635
$426$		0			0
$427$	30.0000	−	12.2474i	1.45180	−	0.592696i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			30.0000i			1.44505i	0.691345	+	0.722525i	$0.257018\pi$
$431$			30.0000i			1.44505i	−0.691345	+	0.722525i	$0.742982\pi$
$432$		0			0
$433$			14.6969i			0.706290i	0.935569	+	0.353145i	$0.114888\pi$
$433$			14.6969i			0.706290i	−0.935569	+	0.353145i	$0.885112\pi$
$434$		0			0
$435$	18.0000	−	18.0000i	0.863034	−	0.863034i
$436$		0			0
$437$	14.6969			0.703050
$438$		0			0
$439$			14.6969i			0.701447i	0.936479	+	0.350723i	$0.114064\pi$
$439$			14.6969i			0.701447i	−0.936479	+	0.350723i	$0.885936\pi$
$440$		0			0
$441$	−14.6969	−	15.0000i	−0.699854	−	0.714286i
$442$		0			0
$443$		−	36.0000i		−	1.71041i	−0.518289	−	0.855206i	$-0.673431\pi$
$443$		−	36.0000i		−	1.71041i	0.518289	−	0.855206i	$-0.326569\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	7.34847	−	7.34847i	0.347571	−	0.347571i
$448$		0			0
$449$			36.0000i			1.69895i	0.527633	+	0.849473i	$0.323080\pi$
$449$			36.0000i			1.69895i	−0.527633	+	0.849473i	$0.676920\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	9.79796	+	9.79796i	0.460348	+	0.460348i
$454$		0			0
$455$	14.6969	−	6.00000i	0.689003	−	0.281284i
$456$		0			0
$457$	28.0000			1.30978			0.654892	−	0.755722i	$-0.272714\pi$
$457$	28.0000			1.30978			0.654892	+	0.755722i	$0.272714\pi$
$458$		0			0
$459$	−18.0000	+	18.0000i	−0.840168	+	0.840168i
$460$		0			0
$461$	−31.8434			−1.48309			−0.741547	−	0.670901i	$-0.765907\pi$
$461$	−31.8434			−1.48309			−0.741547	+	0.670901i	$0.765907\pi$
$462$		0			0
$463$	−14.0000			−0.650635			−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000			−0.650635			−0.325318	+	0.945605i	$0.605471\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.34847			−0.340047			−0.170023	−	0.985440i	$-0.554384\pi$
$467$	−7.34847			−0.340047			−0.170023	+	0.985440i	$0.554384\pi$
$468$		0			0
$469$	8.00000	+	19.5959i	0.369406	+	0.904855i
$470$		0			0
$471$	9.00000	−	9.00000i	0.414698	−	0.414698i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	2.44949i		−	0.112390i
$476$		0			0
$477$	−18.0000			−0.824163
$478$		0			0
$479$	−24.4949			−1.11920			−0.559600	−	0.828763i	$-0.689045\pi$
$479$	−24.4949			−1.11920			−0.559600	+	0.828763i	$0.689045\pi$
$480$		0			0
$481$		−	4.89898i		−	0.223374i
$482$		0			0
$483$	−25.3485	−	10.6515i	−1.15340	−	0.484661i
$484$		0			0
$485$		−	12.0000i		−	0.544892i
$486$		0			0
$487$	32.0000			1.45006			0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000			1.45006			0.725029	+	0.688718i	$0.241826\pi$
$488$		0			0
$489$	−19.5959	−	19.5959i	−0.886158	−	0.886158i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		−	29.3939i		−	1.32383i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−20.0000			−0.895323			−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000			−0.895323			−0.447661	+	0.894203i	$0.647743\pi$
$500$		0			0
$501$	−6.00000	−	6.00000i	−0.268060	−	0.268060i
$502$		0			0
$503$	−39.1918			−1.74748			−0.873739	−	0.486395i	$-0.838311\pi$
$503$	−39.1918			−1.74748			−0.873739	+	0.486395i	$0.838311\pi$
$504$		0			0
$505$	−18.0000			−0.800989
$506$		0			0
$507$	8.57321	+	8.57321i	0.380750	+	0.380750i
$508$		0			0
$509$	12.2474			0.542859			0.271429	−	0.962458i	$-0.412504\pi$
$509$	12.2474			0.542859			0.271429	+	0.962458i	$0.412504\pi$
$510$		0			0
$511$	24.0000	−	9.79796i	1.06170	−	0.433436i
$512$		0			0
$513$	9.00000	+	9.00000i	0.397360	+	0.397360i
$514$		0			0
$515$			24.0000i			1.05757i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	27.0000	+	27.0000i	1.18517	+	1.18517i
$520$		0			0
$521$	−4.89898			−0.214628			−0.107314	−	0.994225i	$-0.534225\pi$
$521$	−4.89898			−0.214628			−0.107314	+	0.994225i	$0.534225\pi$
$522$		0			0
$523$			2.44949i			0.107109i	0.998565	+	0.0535544i	$0.0170550\pi$
$523$			2.44949i			0.107109i	−0.998565	+	0.0535544i	$0.982945\pi$
$524$		0			0
$525$	−1.77526	+	4.22474i	−0.0774785	+	0.184383i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$		−	36.7423i		−	1.59448i
$532$		0			0
$533$			12.0000i			0.519778i
$534$		0			0
$535$			29.3939i			1.27081i
$536$		0			0
$537$	29.3939	−	29.3939i	1.26844	−	1.26844i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$	−15.0000	+	15.0000i	−0.643712	+	0.643712i
$544$		0			0
$545$	−24.4949			−1.04925
$546$		0			0
$547$	8.00000			0.342055			0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000			0.342055			0.171028	+	0.985266i	$0.445291\pi$
$548$		0			0
$549$	36.7423			1.56813
$550$		0			0
$551$	−14.6969			−0.626111
$552$		0			0
$553$	10.0000	+	24.4949i	0.425243	+	1.04163i
$554$		0			0
$555$	6.00000	+	6.00000i	0.254686	+	0.254686i
$556$		0			0
$557$			18.0000i			0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$			18.0000i			0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$		0			0
$559$		−	9.79796i		−	0.414410i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−22.0454			−0.929103			−0.464552	−	0.885546i	$-0.653784\pi$
$563$	−22.0454			−0.929103			−0.464552	+	0.885546i	$0.653784\pi$
$564$		0			0
$565$		−	14.6969i		−	0.618305i
$566$		0			0
$567$	−9.00000	−	22.0454i	−0.377964	−	0.925820i
$568$		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
$571$	32.0000			1.33916			0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000			1.33916			0.669579	+	0.742741i	$0.266474\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			6.00000i			0.250217i
$576$		0			0
$577$			19.5959i			0.815789i	0.913029	+	0.407894i	$0.133737\pi$
$577$			19.5959i			0.815789i	−0.913029	+	0.407894i	$0.866263\pi$
$578$		0			0
$579$	4.89898	+	4.89898i	0.203595	+	0.203595i
$580$		0			0
$581$	2.44949	+	6.00000i	0.101622	+	0.248922i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	18.0000			0.744208
$586$		0			0
$587$	−7.34847			−0.303304			−0.151652	−	0.988434i	$-0.548459\pi$
$587$	−7.34847			−0.303304			−0.151652	+	0.988434i	$0.548459\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−22.0454	+	22.0454i	−0.906827	+	0.906827i
$592$		0			0
$593$	39.1918			1.60942			0.804708	−	0.593671i	$-0.202322\pi$
$593$	39.1918			1.60942			0.804708	+	0.593671i	$0.202322\pi$
$594$		0			0
$595$	12.0000	+	29.3939i	0.491952	+	1.20503i
$596$		0			0
$597$	12.0000	−	12.0000i	0.491127	−	0.491127i
$598$		0			0
$599$			24.0000i			0.980613i	0.871550	+	0.490307i	$0.163115\pi$
$599$			24.0000i			0.980613i	−0.871550	+	0.490307i	$0.836885\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$			24.0000i			0.977356i
$604$		0			0
$605$	26.9444			1.09545
$606$		0			0
$607$			4.89898i			0.198843i	0.995045	+	0.0994217i	$0.0316993\pi$
$607$			4.89898i			0.198843i	−0.995045	+	0.0994217i	$0.968301\pi$
$608$		0			0
$609$	25.3485	+	10.6515i	1.02717	+	0.431622i
$610$		0			0
$611$			12.0000i			0.485468i
$612$		0			0
$613$	−14.0000			−0.565455			−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000			−0.565455			−0.282727	+	0.959200i	$0.591239\pi$
$614$		0			0
$615$	−14.6969	−	14.6969i	−0.592638	−	0.592638i
$616$		0			0
$617$		−	18.0000i		−	0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$		−	18.0000i		−	0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$		0			0
$619$		−	26.9444i		−	1.08299i	−0.840705	−	0.541493i	$-0.817859\pi$
$619$		−	26.9444i		−	1.08299i	0.840705	−	0.541493i	$-0.182141\pi$
$620$		0			0
$621$	−22.0454	−	22.0454i	−0.884652	−	0.884652i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	9.79796			0.390670
$630$		0			0
$631$	−2.00000			−0.0796187			−0.0398094	−	0.999207i	$-0.512675\pi$
$631$	−2.00000			−0.0796187			−0.0398094	+	0.999207i	$0.512675\pi$
$632$		0			0
$633$	−9.79796	−	9.79796i	−0.389434	−	0.389434i
$634$		0			0
$635$	−19.5959			−0.777640
$636$		0			0
$637$	12.0000	+	12.2474i	0.475457	+	0.485262i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			30.0000i			1.18493i	0.805597	+	0.592464i	$0.201845\pi$
$641$			30.0000i			1.18493i	−0.805597	+	0.592464i	$0.798155\pi$
$642$		0			0
$643$		−	22.0454i		−	0.869386i	−0.900579	−	0.434693i	$-0.856857\pi$
$643$		−	22.0454i		−	0.869386i	0.900579	−	0.434693i	$-0.143143\pi$
$644$		0			0
$645$	12.0000	+	12.0000i	0.472500	+	0.472500i
$646$		0			0
$647$	44.0908			1.73339			0.866694	−	0.498839i	$-0.166240\pi$
$647$	44.0908			1.73339			0.866694	+	0.498839i	$0.166240\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$			6.00000i			0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$			6.00000i			0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$		0			0
$655$	18.0000			0.703318
$656$		0			0
$657$	29.3939			1.14676
$658$		0			0
$659$			36.0000i			1.40236i	0.712984	+	0.701180i	$0.247343\pi$
$659$			36.0000i			1.40236i	−0.712984	+	0.701180i	$0.752657\pi$
$660$		0			0
$661$			12.2474i			0.476371i	0.971220	+	0.238185i	$0.0765525\pi$
$661$			12.2474i			0.476371i	−0.971220	+	0.238185i	$0.923447\pi$
$662$		0			0
$663$	14.6969	−	14.6969i	0.570782	−	0.570782i
$664$		0			0
$665$	14.6969	−	6.00000i	0.569923	−	0.232670i
$666$		0			0
$667$	36.0000			1.39393
$668$		0			0
$669$	18.0000	−	18.0000i	0.695920	−	0.695920i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	34.0000			1.31060			0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000			1.31060			0.655302	+	0.755367i	$0.272541\pi$
$674$		0			0
$675$	−3.67423	+	3.67423i	−0.141421	+	0.141421i
$676$		0			0
$677$	7.34847			0.282425			0.141212	−	0.989979i	$-0.454900\pi$
$677$	7.34847			0.282425			0.141212	+	0.989979i	$0.454900\pi$
$678$		0			0
$679$	12.0000	−	4.89898i	0.460518	−	0.188006i
$680$		0			0
$681$	−9.00000	−	9.00000i	−0.344881	−	0.344881i
$682$		0			0
$683$			24.0000i			0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$			24.0000i			0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$		0			0
$685$			29.3939i			1.12308i
$686$		0			0
$687$	27.0000	−	27.0000i	1.03011	−	1.03011i
$688$		0			0
$689$	14.6969			0.559909
$690$		0			0
$691$			36.7423i			1.39774i	0.715246	+	0.698872i	$0.246314\pi$
$691$			36.7423i			1.39774i	−0.715246	+	0.698872i	$0.753686\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	6.00000i		−	0.227593i
$696$		0			0
$697$	−24.0000			−0.909065
$698$		0			0
$699$	−29.3939	+	29.3939i	−1.11178	+	1.11178i
$700$		0			0
$701$			30.0000i			1.13308i	0.824033	+	0.566542i	$0.191719\pi$
$701$			30.0000i			1.13308i	−0.824033	+	0.566542i	$0.808281\pi$
$702$		0			0
$703$		−	4.89898i		−	0.184769i
$704$		0			0
$705$	−14.6969	−	14.6969i	−0.553519	−	0.553519i
$706$		0			0
$707$	−7.34847	−	18.0000i	−0.276368	−	0.676960i
$708$		0			0
$709$	−10.0000			−0.375558			−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000			−0.375558			−0.187779	+	0.982211i	$0.560129\pi$
$710$		0			0
$711$			30.0000i			1.12509i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	7.34847	−	7.34847i	0.274434	−	0.274434i
$718$		0			0
$719$	−24.4949			−0.913506			−0.456753	−	0.889594i	$-0.650988\pi$
$719$	−24.4949			−0.913506			−0.456753	+	0.889594i	$0.650988\pi$
$720$		0			0
$721$	−24.0000	+	9.79796i	−0.893807	+	0.364895i
$722$		0			0
$723$	30.0000	−	30.0000i	1.11571	−	1.11571i
$724$		0			0
$725$		−	6.00000i		−	0.222834i
$726$		0			0
$727$			29.3939i			1.09016i	0.838385	+	0.545079i	$0.183500\pi$
$727$			29.3939i			1.09016i	−0.838385	+	0.545079i	$0.816500\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$	19.5959			0.724781
$732$		0			0
$733$			22.0454i			0.814266i	0.913369	+	0.407133i	$0.133471\pi$
$733$			22.0454i			0.814266i	−0.913369	+	0.407133i	$0.866529\pi$
$734$		0			0
$735$	−29.6969	−	0.303062i	−1.09539	−	0.0111786i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	20.0000			0.735712			0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000			0.735712			0.367856	+	0.929883i	$0.380092\pi$
$740$		0			0
$741$	−7.34847	−	7.34847i	−0.269953	−	0.269953i
$742$		0			0
$743$			6.00000i			0.220119i	0.993925	+	0.110059i	$0.0351041\pi$
$743$			6.00000i			0.220119i	−0.993925	+	0.110059i	$0.964896\pi$
$744$		0			0
$745$		−	14.6969i		−	0.538454i
$746$		0			0
$747$			7.34847i			0.268866i
$748$		0			0
$749$	−29.3939	+	12.0000i	−1.07403	+	0.438470i
$750$		0			0
$751$	8.00000			0.291924			0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000			0.291924			0.145962	+	0.989290i	$0.453372\pi$
$752$		0			0
$753$	−21.0000	−	21.0000i	−0.765283	−	0.765283i
$754$		0			0
$755$	19.5959			0.713168
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4.89898			−0.177588			−0.0887939	−	0.996050i	$-0.528301\pi$
$761$	−4.89898			−0.177588			−0.0887939	+	0.996050i	$0.528301\pi$
$762$		0			0
$763$	−10.0000	−	24.4949i	−0.362024	−	0.886775i
$764$		0			0
$765$			36.0000i			1.30158i
$766$		0			0
$767$			30.0000i			1.08324i
$768$		0			0
$769$			34.2929i			1.23663i	0.785930	+	0.618316i	$0.212185\pi$
$769$			34.2929i			1.23663i	−0.785930	+	0.618316i	$0.787815\pi$
$770$		0			0
$771$	36.0000	+	36.0000i	1.29651	+	1.29651i
$772$		0			0
$773$	−26.9444			−0.969122			−0.484561	−	0.874757i	$-0.661021\pi$
$773$	−26.9444			−0.969122			−0.484561	+	0.874757i	$0.661021\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−3.55051	+	8.44949i	−0.127374	+	0.303124i
$778$		0			0
$779$			12.0000i			0.429945i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	22.0454	+	22.0454i	0.787839	+	0.787839i
$784$		0			0
$785$		−	18.0000i		−	0.642448i
$786$		0			0
$787$			31.8434i			1.13509i	0.823341	+	0.567547i	$0.192107\pi$
$787$			31.8434i			1.13509i	−0.823341	+	0.567547i	$0.807893\pi$
$788$		0			0
$789$	29.3939	−	29.3939i	1.04645	−	1.04645i
$790$		0			0
$791$	14.6969	−	6.00000i	0.522563	−	0.213335i
$792$		0			0
$793$	−30.0000			−1.06533
$794$		0			0
$795$	−18.0000	+	18.0000i	−0.638394	+	0.638394i
$796$		0			0
$797$	7.34847			0.260296			0.130148	−	0.991495i	$-0.458455\pi$
$797$	7.34847			0.260296			0.130148	+	0.991495i	$0.458455\pi$
$798$		0			0
$799$	−24.0000			−0.849059
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−36.0000	+	14.6969i	−1.26883	+	0.517999i
$806$		0			0
$807$	15.0000	+	15.0000i	0.528025	+	0.528025i
$808$		0			0
$809$			6.00000i			0.210949i	0.994422	+	0.105474i	$0.0336361\pi$
$809$			6.00000i			0.210949i	−0.994422	+	0.105474i	$0.966364\pi$
$810$		0			0
$811$			36.7423i			1.29020i	0.764099	+	0.645099i	$0.223184\pi$
$811$			36.7423i			1.29020i	−0.764099	+	0.645099i	$0.776816\pi$
$812$		0			0
$813$	30.0000	−	30.0000i	1.05215	−	1.05215i
$814$		0			0
$815$	−39.1918			−1.37283
$816$		0			0
$817$		−	9.79796i		−	0.342787i
$818$		0			0
$819$	7.34847	+	18.0000i	0.256776	+	0.628971i
$820$		0			0
$821$		−	30.0000i		−	1.04701i	−0.852023	−	0.523504i	$-0.824625\pi$
$821$		−	30.0000i		−	1.04701i	0.852023	−	0.523504i	$-0.175375\pi$
$822$		0			0
$823$	−14.0000			−0.488009			−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000			−0.488009			−0.244005	+	0.969774i	$0.578461\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			12.0000i			0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$			12.0000i			0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$		0			0
$829$			26.9444i			0.935817i	0.883777	+	0.467909i	$0.154992\pi$
$829$			26.9444i			0.935817i	−0.883777	+	0.467909i	$0.845008\pi$
$830$		0			0
$831$	26.9444	+	26.9444i	0.934690	+	0.934690i
$832$		0			0
$833$	−24.4949	+	24.0000i	−0.848698	+	0.831551i
$834$		0			0
$835$	−12.0000			−0.415277
$836$		0			0
$837$		0			0
$838$		0			0
$839$	24.4949			0.845658			0.422829	−	0.906210i	$-0.361037\pi$
$839$	24.4949			0.845658			0.422829	+	0.906210i	$0.361037\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$	17.1464			0.589855
$846$		0			0
$847$	11.0000	+	26.9444i	0.377964	+	0.925820i
$848$		0			0
$849$	27.0000	−	27.0000i	0.926638	−	0.926638i
$850$		0			0
$851$			12.0000i			0.411355i
$852$		0			0
$853$		−	2.44949i		−	0.0838689i	−0.999120	−	0.0419345i	$-0.986648\pi$
$853$		−	2.44949i		−	0.0838689i	0.999120	−	0.0419345i	$-0.0133521\pi$
$854$		0			0
$855$	18.0000			0.615587
$856$		0			0
$857$	4.89898			0.167346			0.0836730	−	0.996493i	$-0.473335\pi$
$857$	4.89898			0.167346			0.0836730	+	0.996493i	$0.473335\pi$
$858$		0			0
$859$		−	26.9444i		−	0.919331i	−0.888092	−	0.459665i	$-0.847969\pi$
$859$		−	26.9444i		−	0.919331i	0.888092	−	0.459665i	$-0.152031\pi$
$860$		0			0
$861$	8.69694	−	20.6969i	0.296391	−	0.705350i
$862$		0			0
$863$		−	24.0000i		−	0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$		−	24.0000i		−	0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$		0			0
$865$	54.0000			1.83606
$866$		0			0
$867$	8.57321	+	8.57321i	0.291162	+	0.291162i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		−	19.5959i		−	0.663982i
$872$		0			0
$873$	14.6969			0.497416
$874$		0			0
$875$	−9.79796	−	24.0000i	−0.331231	−	0.811348i
$876$		0			0
$877$	22.0000			0.742887			0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000			0.742887			0.371444	+	0.928456i	$0.378863\pi$
$878$		0			0
$879$	−3.00000	−	3.00000i	−0.101187	−	0.101187i
$880$		0			0
$881$	−29.3939			−0.990305			−0.495152	−	0.868806i	$-0.664888\pi$
$881$	−29.3939			−0.990305			−0.495152	+	0.868806i	$0.664888\pi$
$882$		0			0
$883$	−56.0000			−1.88455			−0.942275	−	0.334840i	$-0.891318\pi$
$883$	−56.0000			−1.88455			−0.942275	+	0.334840i	$0.891318\pi$
$884$		0			0
$885$	−36.7423	−	36.7423i	−1.23508	−	1.23508i
$886$		0			0
$887$	−4.89898			−0.164492			−0.0822458	−	0.996612i	$-0.526209\pi$
$887$	−4.89898			−0.164492			−0.0822458	+	0.996612i	$0.526209\pi$
$888$		0			0
$889$	−8.00000	−	19.5959i	−0.268311	−	0.657226i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			12.0000i			0.401565i
$894$		0			0
$895$		−	58.7878i		−	1.96506i
$896$		0			0
$897$	18.0000	+	18.0000i	0.601003	+	0.601003i
$898$		0			0
$899$		0			0
$900$		0			0
$901$			29.3939i			0.979252i
$902$		0			0
$903$	−7.10102	+	16.8990i	−0.236307	+	0.562363i
$904$		0			0
$905$			30.0000i			0.997234i
$906$		0			0
$907$	28.0000			0.929725			0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000			0.929725			0.464862	+	0.885383i	$0.346104\pi$
$908$		0			0
$909$		−	22.0454i		−	0.731200i
$910$		0			0
$911$			30.0000i			0.993944i	0.867766	+	0.496972i	$0.165555\pi$
$911$			30.0000i			0.993944i	−0.867766	+	0.496972i	$0.834445\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	36.7423	−	36.7423i	1.21466	−	1.21466i
$916$		0			0
$917$	7.34847	+	18.0000i	0.242668	+	0.594412i
$918$		0			0
$919$	−10.0000			−0.329870			−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000			−0.329870			−0.164935	+	0.986304i	$0.552741\pi$
$920$		0			0
$921$	−9.00000	+	9.00000i	−0.296560	+	0.296560i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	2.00000			0.0657596
$926$		0			0
$927$	−29.3939			−0.965422
$928$		0			0
$929$	−24.4949			−0.803652			−0.401826	−	0.915716i	$-0.631624\pi$
$929$	−24.4949			−0.803652			−0.401826	+	0.915716i	$0.631624\pi$
$930$		0			0
$931$	12.0000	+	12.2474i	0.393284	+	0.401394i
$932$		0			0
$933$	−24.0000	−	24.0000i	−0.785725	−	0.785725i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			19.5959i			0.640171i	0.947389	+	0.320085i	$0.103712\pi$
$937$			19.5959i			0.640171i	−0.947389	+	0.320085i	$0.896288\pi$
$938$		0			0
$939$	42.0000	−	42.0000i	1.37062	−	1.37062i
$940$		0			0
$941$	−31.8434			−1.03806			−0.519032	−	0.854755i	$-0.673707\pi$
$941$	−31.8434			−1.03806			−0.519032	+	0.854755i	$0.673707\pi$
$942$		0			0
$943$		−	29.3939i		−	0.957196i
$944$		0			0
$945$	−31.0454	−	13.0454i	−1.00991	−	0.424367i
$946$		0			0
$947$			12.0000i			0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$			12.0000i			0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$		0			0
$949$	−24.0000			−0.779073
$950$		0			0
$951$	−22.0454	+	22.0454i	−0.714871	+	0.714871i
$952$		0			0
$953$		−	36.0000i		−	1.16615i	−0.812417	−	0.583077i	$-0.801849\pi$
$953$		−	36.0000i		−	1.16615i	0.812417	−	0.583077i	$-0.198151\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−29.3939	+	12.0000i	−0.949178	+	0.387500i
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	−36.0000			−1.16008
$964$		0			0
$965$	9.79796			0.315407
$966$		0			0
$967$	−8.00000			−0.257263			−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000			−0.257263			−0.128631	+	0.991692i	$0.541058\pi$
$968$		0			0
$969$	14.6969	−	14.6969i	0.472134	−	0.472134i
$970$		0			0
$971$	−41.6413			−1.33633			−0.668167	−	0.744011i	$-0.732921\pi$
$971$	−41.6413			−1.33633			−0.668167	+	0.744011i	$0.732921\pi$
$972$		0			0
$973$	6.00000	−	2.44949i	0.192351	−	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
$981$		−	30.0000i		−	0.957826i
$982$		0			0
$983$	34.2929			1.09377			0.546886	−	0.837207i	$-0.315813\pi$
$983$	34.2929			1.09377			0.546886	+	0.837207i	$0.315813\pi$
$984$		0			0
$985$			44.0908i			1.40485i
$986$		0			0
$987$	8.69694	−	20.6969i	0.276827	−	0.658791i
$988$		0			0
$989$			24.0000i			0.763156i
$990$		0			0
$991$	38.0000			1.20711			0.603555	−	0.797321i	$-0.293750\pi$
$991$	38.0000			1.20711			0.603555	+	0.797321i	$0.293750\pi$
$992$		0			0
$993$	−9.79796	−	9.79796i	−0.310929	−	0.310929i
$994$		0			0
$995$		−	24.0000i		−	0.760851i
$996$		0			0
$997$		−	7.34847i		−	0.232728i	−0.993207	−	0.116364i	$-0.962876\pi$
$997$		−	7.34847i		−	0.232728i	0.993207	−	0.116364i	$-0.0371240\pi$
$998$		0			0
$999$	−7.34847	+	7.34847i	−0.232495	+	0.232495i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.2.k.d.1217.4		4
3.2	odd	2	inner	1344.2.k.d.1217.2		4
4.3	odd	2		1344.2.k.c.1217.1		4
7.6	odd	2	inner	1344.2.k.d.1217.1		4
8.3	odd	2		42.2.d.a.41.2	yes	4
8.5	even	2		336.2.k.b.209.1		4
12.11	even	2		1344.2.k.c.1217.3		4
21.20	even	2	inner	1344.2.k.d.1217.3		4
24.5	odd	2		336.2.k.b.209.3		4
24.11	even	2		42.2.d.a.41.3	yes	4
28.27	even	2		1344.2.k.c.1217.4		4
40.3	even	4		1050.2.d.b.1049.2		4
40.19	odd	2		1050.2.b.b.251.3		4
40.27	even	4		1050.2.d.e.1049.3		4
56.3	even	6		294.2.f.b.215.3		8
56.11	odd	6		294.2.f.b.215.4		8
56.13	odd	2		336.2.k.b.209.4		4
56.19	even	6		294.2.f.b.227.2		8
56.27	even	2		42.2.d.a.41.1	✓	4
56.51	odd	6		294.2.f.b.227.1		8
72.11	even	6		1134.2.m.g.377.3		8
72.43	odd	6		1134.2.m.g.377.2		8
72.59	even	6		1134.2.m.g.755.1		8
72.67	odd	6		1134.2.m.g.755.4		8
84.83	odd	2		1344.2.k.c.1217.2		4
120.59	even	2		1050.2.b.b.251.2		4
120.83	odd	4		1050.2.d.e.1049.1		4
120.107	odd	4		1050.2.d.b.1049.4		4
168.11	even	6		294.2.f.b.215.2		8
168.59	odd	6		294.2.f.b.215.1		8
168.83	odd	2		42.2.d.a.41.4	yes	4
168.107	even	6		294.2.f.b.227.3		8
168.125	even	2		336.2.k.b.209.2		4
168.131	odd	6		294.2.f.b.227.4		8
280.27	odd	4		1050.2.d.e.1049.2		4
280.83	odd	4		1050.2.d.b.1049.3		4
280.139	even	2		1050.2.b.b.251.4		4
504.83	odd	6		1134.2.m.g.377.4		8
504.139	even	6		1134.2.m.g.755.3		8
504.419	odd	6		1134.2.m.g.755.2		8
504.475	even	6		1134.2.m.g.377.1		8
840.83	even	4		1050.2.d.e.1049.4		4
840.419	odd	2		1050.2.b.b.251.1		4
840.587	even	4		1050.2.d.b.1049.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.2.d.a.41.1	✓	4	56.27	even	2
42.2.d.a.41.2	yes	4	8.3	odd	2
42.2.d.a.41.3	yes	4	24.11	even	2
42.2.d.a.41.4	yes	4	168.83	odd	2
294.2.f.b.215.1		8	168.59	odd	6
294.2.f.b.215.2		8	168.11	even	6
294.2.f.b.215.3		8	56.3	even	6
294.2.f.b.215.4		8	56.11	odd	6
294.2.f.b.227.1		8	56.51	odd	6
294.2.f.b.227.2		8	56.19	even	6
294.2.f.b.227.3		8	168.107	even	6
294.2.f.b.227.4		8	168.131	odd	6
336.2.k.b.209.1		4	8.5	even	2
336.2.k.b.209.2		4	168.125	even	2
336.2.k.b.209.3		4	24.5	odd	2
336.2.k.b.209.4		4	56.13	odd	2
1050.2.b.b.251.1		4	840.419	odd	2
1050.2.b.b.251.2		4	120.59	even	2
1050.2.b.b.251.3		4	40.19	odd	2
1050.2.b.b.251.4		4	280.139	even	2
1050.2.d.b.1049.1		4	840.587	even	4
1050.2.d.b.1049.2		4	40.3	even	4
1050.2.d.b.1049.3		4	280.83	odd	4
1050.2.d.b.1049.4		4	120.107	odd	4
1050.2.d.e.1049.1		4	120.83	odd	4
1050.2.d.e.1049.2		4	280.27	odd	4
1050.2.d.e.1049.3		4	40.27	even	4
1050.2.d.e.1049.4		4	840.83	even	4
1134.2.m.g.377.1		8	504.475	even	6
1134.2.m.g.377.2		8	72.43	odd	6
1134.2.m.g.377.3		8	72.11	even	6
1134.2.m.g.377.4		8	504.83	odd	6
1134.2.m.g.755.1		8	72.59	even	6
1134.2.m.g.755.2		8	504.419	odd	6
1134.2.m.g.755.3		8	504.139	even	6
1134.2.m.g.755.4		8	72.67	odd	6
1344.2.k.c.1217.1		4	4.3	odd	2
1344.2.k.c.1217.2		4	84.83	odd	2
1344.2.k.c.1217.3		4	12.11	even	2
1344.2.k.c.1217.4		4	28.27	even	2
1344.2.k.d.1217.1		4	7.6	odd	2	inner
1344.2.k.d.1217.2		4	3.2	odd	2	inner
1344.2.k.d.1217.3		4	21.20	even	2	inner
1344.2.k.d.1217.4		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.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 + 1.22474i) q^{3} +2.44949 q^{5} +(1.00000 + 2.44949i) q^{7} +3.00000i q^{9} +O(q^{10})\)	\(q+(1.22474 + 1.22474i) q^{3} +2.44949 q^{5} +(1.00000 + 2.44949i) q^{7} +3.00000i q^{9} -2.44949i q^{13} +(3.00000 + 3.00000i) q^{15} +4.89898 q^{17} -2.44949i q^{19} +(-1.77526 + 4.22474i) q^{21} +6.00000i q^{23} +1.00000 q^{25} +(-3.67423 + 3.67423i) q^{27} -6.00000i q^{29} +(2.44949 + 6.00000i) q^{35} +2.00000 q^{37} +(3.00000 - 3.00000i) q^{39} -4.89898 q^{41} +4.00000 q^{43} +7.34847i q^{45} -4.89898 q^{47} +(-5.00000 + 4.89898i) q^{49} +(6.00000 + 6.00000i) q^{51} +6.00000i q^{53} +(3.00000 - 3.00000i) q^{57} -12.2474 q^{59} -12.2474i q^{61} +(-7.34847 + 3.00000i) q^{63} -6.00000i q^{65} +8.00000 q^{67} +(-7.34847 + 7.34847i) q^{69} -9.79796i q^{73} +(1.22474 + 1.22474i) q^{75} +10.0000 q^{79} -9.00000 q^{81} +2.44949 q^{83} +12.0000 q^{85} +(7.34847 - 7.34847i) q^{87} +(6.00000 - 2.44949i) q^{91} -6.00000i q^{95} -4.89898i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^7	\( 4 q + 4 q^{7} + 12 q^{15} - 12 q^{21} + 4 q^{25} + 8 q^{37} + 12 q^{39} + 16 q^{43} - 20 q^{49} + 24 q^{51} + 12 q^{57} + 32 q^{67} + 40 q^{79} - 36 q^{81} + 48 q^{85} + 24 q^{91}+O(q^{100}) \) 4 * q + 4 * q^7 + 12 * q^15 - 12 * q^21 + 4 * q^25 + 8 * q^37 + 12 * q^39 + 16 * q^43 - 20 * q^49 + 24 * q^51 + 12 * q^57 + 32 * q^67 + 40 * q^79 - 36 * q^81 + 48 * q^85 + 24 * q^91

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