LMFDB - Embedded newform 7644.2.e.o.4705.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7644,2,Mod(4705,7644)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7644, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7644.4705"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7644 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7644.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,10,0,0,0,0,0,10,0,0,0,1,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$61.0376473051$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 37x^{8} + 408x^{6} + 1219x^{4} + 1072x^{2} + 144 $ x^10 + 37x^8 + 408x^6 + 1219x^4 + 1072x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1092)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4705.10
Root			$4.12696i$ of defining polynomial
Character	$\chi$	$=$	7644.4705
Dual form			7644.2.e.o.4705.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.12696i q^{5} +1.00000 q^{9} -4.83063i q^{11} +(-1.36787 - 3.33601i) q^{13} +4.12696i q^{15} +0.193191 q^{17} -0.814515i q^{19} +1.54255 q^{23} -12.0318 q^{25} +1.00000 q^{27} +0.154256 q^{29} +4.84786i q^{31} -4.83063i q^{33} +0.110845i q^{37} +(-1.36787 - 3.33601i) q^{39} -1.85860i q^{41} -1.19319 q^{43} +4.12696i q^{45} -9.78934i q^{47} +0.193191 q^{51} -12.2712 q^{53} +19.9359 q^{55} -0.814515i q^{57} -13.9163i q^{59} +5.27829 q^{61} +(13.7676 - 5.64515i) q^{65} -11.7243i q^{67} +1.54255 q^{69} -4.22059i q^{71} -3.21883i q^{73} -12.0318 q^{75} -11.7286 q^{79} +1.00000 q^{81} -2.65589i q^{83} +0.797294i q^{85} +0.154256 q^{87} +4.95870i q^{89} +4.84786i q^{93} +3.36147 q^{95} +3.13454i q^{97} -4.83063i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{3} + 10 q^{9} + q^{13} - 4 q^{17} - 8 q^{23} - 24 q^{25} + 10 q^{27} - 2 q^{29} + q^{39} - 6 q^{43} - 4 q^{51} + 24 q^{53} - 12 q^{55} + 12 q^{65} - 8 q^{69} - 24 q^{75} + 6 q^{79} + 10 q^{81}+ \cdots - 58 q^{95}+O(q^{100}) $ 10 * q + 10 * q^3 + 10 * q^9 + q^13 - 4 * q^17 - 8 * q^23 - 24 * q^25 + 10 * q^27 - 2 * q^29 + q^39 - 6 * q^43 - 4 * q^51 + 24 * q^53 - 12 * q^55 + 12 * q^65 - 8 * q^69 - 24 * q^75 + 6 * q^79 + 10 * q^81 - 2 * q^87 - 58 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7644\mathbb{Z}\right)^\times$.

$n$	$2549$	$3433$	$3823$	$5293$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$			4.12696i			1.84563i	0.385238	+	0.922817i	$0.374119\pi$
$5$			4.12696i			1.84563i	−0.385238	+	0.922817i	$0.625881\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	4.83063i		−	1.45649i	−0.685316	−	0.728246i	$-0.740336\pi$
$11$		−	4.83063i		−	1.45649i	0.685316	−	0.728246i	$-0.259664\pi$
$12$		0			0
$13$	−1.36787	−	3.33601i	−0.379379	−	0.925241i
$13$	−1.36787	−	3.33601i	−0.379379	−	0.925241i
$14$		0			0
$15$			4.12696i			1.06558i
$16$		0			0
$17$	0.193191			0.0468558			0.0234279	−	0.999726i	$-0.492542\pi$
$17$	0.193191			0.0468558			0.0234279	+	0.999726i	$0.492542\pi$
$18$		0			0
$19$		−	0.814515i		−	0.186863i	−0.995626	−	0.0934313i	$-0.970216\pi$
$19$		−	0.814515i		−	0.186863i	0.995626	−	0.0934313i	$-0.0297835\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.54255			0.321643			0.160822	−	0.986983i	$-0.448586\pi$
$23$	1.54255			0.321643			0.160822	+	0.986983i	$0.448586\pi$
$24$		0			0
$25$	−12.0318			−2.40637
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	0.154256			0.0286447			0.0143223	−	0.999897i	$-0.495441\pi$
$29$	0.154256			0.0286447			0.0143223	+	0.999897i	$0.495441\pi$
$30$		0			0
$31$			4.84786i			0.870701i	0.900261	+	0.435350i	$0.143375\pi$
$31$			4.84786i			0.870701i	−0.900261	+	0.435350i	$0.856625\pi$
$32$		0			0
$33$		−	4.83063i		−	0.840906i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			0.110845i			0.0182228i	0.999958	+	0.00911138i	$0.00290028\pi$
$37$			0.110845i			0.0182228i	−0.999958	+	0.00911138i	$0.997100\pi$
$38$		0			0
$39$	−1.36787	−	3.33601i	−0.219034	−	0.534188i
$40$		0			0
$41$		−	1.85860i		−	0.290264i	−0.989412	−	0.145132i	$-0.953639\pi$
$41$		−	1.85860i		−	0.290264i	0.989412	−	0.145132i	$-0.0463607\pi$
$42$		0			0
$43$	−1.19319			−0.181960			−0.0909800	−	0.995853i	$-0.529000\pi$
$43$	−1.19319			−0.181960			−0.0909800	+	0.995853i	$0.529000\pi$
$44$		0			0
$45$			4.12696i			0.615212i
$46$		0			0
$47$		−	9.78934i		−	1.42792i	−0.700186	−	0.713961i	$-0.746900\pi$
$47$		−	9.78934i		−	1.42792i	0.700186	−	0.713961i	$-0.253100\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	0.193191			0.0270522
$52$		0			0
$53$	−12.2712			−1.68558			−0.842789	−	0.538244i	$-0.819088\pi$
$53$	−12.2712			−1.68558			−0.842789	+	0.538244i	$0.819088\pi$
$54$		0			0
$55$	19.9359			2.68815
$56$		0			0
$57$		−	0.814515i		−	0.107885i
$58$		0			0
$59$		−	13.9163i		−	1.81175i	−0.423547	−	0.905874i	$-0.639215\pi$
$59$		−	13.9163i		−	1.81175i	0.423547	−	0.905874i	$-0.360785\pi$
$60$		0			0
$61$	5.27829			0.675815			0.337908	−	0.941179i	$-0.390281\pi$
$61$	5.27829			0.675815			0.337908	+	0.941179i	$0.390281\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	13.7676	−	5.64515i	1.70766	−	0.700195i
$66$		0			0
$67$		−	11.7243i		−	1.43236i	−0.697918	−	0.716178i	$-0.745890\pi$
$67$		−	11.7243i		−	1.43236i	0.697918	−	0.716178i	$-0.254110\pi$
$68$		0			0
$69$	1.54255			0.185701
$70$		0			0
$71$		−	4.22059i		−	0.500892i	−0.968131	−	0.250446i	$-0.919423\pi$
$71$		−	4.22059i		−	0.500892i	0.968131	−	0.250446i	$-0.0805772\pi$
$72$		0			0
$73$		−	3.21883i		−	0.376735i	−0.982099	−	0.188368i	$-0.939680\pi$
$73$		−	3.21883i		−	0.376735i	0.982099	−	0.188368i	$-0.0603196\pi$
$74$		0			0
$75$	−12.0318			−1.38932
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−11.7286			−1.31958			−0.659788	−	0.751452i	$-0.729354\pi$
$79$	−11.7286			−1.31958			−0.659788	+	0.751452i	$0.729354\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	2.65589i		−	0.291522i	−0.989320	−	0.145761i	$-0.953437\pi$
$83$		−	2.65589i		−	0.291522i	0.989320	−	0.145761i	$-0.0465630\pi$
$84$		0			0
$85$			0.797294i			0.0864787i
$86$		0			0
$87$	0.154256			0.0165380
$88$		0			0
$89$			4.95870i			0.525621i	0.964847	+	0.262811i	$0.0846494\pi$
$89$			4.95870i			0.525621i	−0.964847	+	0.262811i	$0.915351\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$			4.84786i			0.502699i
$94$		0			0
$95$	3.36147			0.344880
$96$		0			0
$97$			3.13454i			0.318265i	0.987257	+	0.159132i	$0.0508697\pi$
$97$			3.13454i			0.318265i	−0.987257	+	0.159132i	$0.949130\pi$
$98$		0			0
$99$		−	4.83063i		−	0.485497i
$100$		0			0
$101$	−3.50361			−0.348622			−0.174311	−	0.984691i	$-0.555770\pi$
$101$	−3.50361			−0.348622			−0.174311	+	0.984691i	$0.555770\pi$
$102$		0			0
$103$	12.6114			1.24264			0.621320	−	0.783557i	$-0.286597\pi$
$103$	12.6114			1.24264			0.621320	+	0.783557i	$0.286597\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.92893			−0.573171			−0.286586	−	0.958055i	$-0.592520\pi$
$107$	−5.92893			−0.573171			−0.286586	+	0.958055i	$0.592520\pi$
$108$		0			0
$109$		−	19.6285i		−	1.88007i	−0.341078	−	0.940035i	$-0.610792\pi$
$109$		−	19.6285i		−	1.88007i	0.341078	−	0.940035i	$-0.389208\pi$
$110$		0			0
$111$			0.110845i			0.0105209i
$112$		0			0
$113$	−16.2391			−1.52764			−0.763821	−	0.645428i	$-0.776679\pi$
$113$	−16.2391			−1.52764			−0.763821	+	0.645428i	$0.776679\pi$
$114$		0			0
$115$			6.36604i			0.593636i
$116$		0			0
$117$	−1.36787	−	3.33601i	−0.126460	−	0.308414i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−12.3350			−1.12137
$122$		0			0
$123$		−	1.85860i		−	0.167584i
$124$		0			0
$125$		−	29.0201i		−	2.59564i
$126$		0			0
$127$	−1.61171			−0.143016			−0.0715080	−	0.997440i	$-0.522781\pi$
$127$	−1.61171			−0.143016			−0.0715080	+	0.997440i	$0.522781\pi$
$128$		0			0
$129$	−1.19319			−0.105055
$130$		0			0
$131$	−18.8897			−1.65040			−0.825200	−	0.564840i	$-0.808938\pi$
$131$	−18.8897			−1.65040			−0.825200	+	0.564840i	$0.808938\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$			4.12696i			0.355193i
$136$		0			0
$137$		−	0.797294i		−	0.0681174i	−0.999420	−	0.0340587i	$-0.989157\pi$
$137$		−	0.797294i		−	0.0681174i	0.999420	−	0.0340587i	$-0.0108433\pi$
$138$		0			0
$139$	17.8065			1.51033			0.755164	−	0.655536i	$-0.227557\pi$
$139$	17.8065			1.51033			0.755164	+	0.655536i	$0.227557\pi$
$140$		0			0
$141$		−	9.78934i		−	0.824411i
$142$		0			0
$143$	−16.1150	+	6.60768i	−1.34761	+	0.552562i
$144$		0			0
$145$			0.636610i			0.0528676i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			16.2517i			1.33139i	0.746222	+	0.665697i	$0.231866\pi$
$149$			16.2517i			1.33139i	−0.746222	+	0.665697i	$0.768134\pi$
$150$		0			0
$151$			12.9187i			1.05131i	0.850697	+	0.525656i	$0.176180\pi$
$151$			12.9187i			1.05131i	−0.850697	+	0.525656i	$0.823820\pi$
$152$		0			0
$153$	0.193191			0.0156186
$154$		0			0
$155$	−20.0069			−1.60700
$156$		0			0
$157$	16.4201			1.31047			0.655234	−	0.755426i	$-0.272570\pi$
$157$	16.4201			1.31047			0.655234	+	0.755426i	$0.272570\pi$
$158$		0			0
$159$	−12.2712			−0.973169
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	20.7284i		−	1.62358i	−0.583951	−	0.811789i	$-0.698494\pi$
$163$		−	20.7284i		−	1.62358i	0.583951	−	0.811789i	$-0.301506\pi$
$164$		0			0
$165$	19.9359			1.55200
$166$		0			0
$167$		−	7.26484i		−	0.562170i	−0.959683	−	0.281085i	$-0.909306\pi$
$167$		−	7.26484i		−	0.562170i	0.959683	−	0.281085i	$-0.0906943\pi$
$168$		0			0
$169$	−9.25787	+	9.12644i	−0.712143	+	0.702034i
$170$		0			0
$171$		−	0.814515i		−	0.0622875i
$172$		0			0
$173$	8.43445			0.641260			0.320630	−	0.947205i	$-0.396105\pi$
$173$	8.43445			0.641260			0.320630	+	0.947205i	$0.396105\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	13.9163i		−	1.04601i
$178$		0			0
$179$	16.2269			1.21286			0.606429	−	0.795138i	$-0.292601\pi$
$179$	16.2269			1.21286			0.606429	+	0.795138i	$0.292601\pi$
$180$		0			0
$181$	−21.0247			−1.56276			−0.781378	−	0.624058i	$-0.785483\pi$
$181$	−21.0247			−1.56276			−0.781378	+	0.624058i	$0.785483\pi$
$182$		0			0
$183$	5.27829			0.390182
$184$		0			0
$185$	−0.457452			−0.0336325
$186$		0			0
$187$		−	0.933237i		−	0.0682451i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	22.8686			1.65471			0.827356	−	0.561678i	$-0.189844\pi$
$191$	22.8686			1.65471			0.827356	+	0.561678i	$0.189844\pi$
$192$		0			0
$193$		−	21.2575i		−	1.53015i	−0.643940	−	0.765076i	$-0.722702\pi$
$193$		−	21.2575i		−	1.53015i	0.643940	−	0.765076i	$-0.277298\pi$
$194$		0			0
$195$	13.7676	−	5.64515i	0.985917	−	0.404258i
$196$		0			0
$197$			18.0759i			1.28785i	0.765087	+	0.643927i	$0.222696\pi$
$197$			18.0759i			1.28785i	−0.765087	+	0.643927i	$0.777304\pi$
$198$		0			0
$199$	7.06397			0.500752			0.250376	−	0.968149i	$-0.419446\pi$
$199$	7.06397			0.500752			0.250376	+	0.968149i	$0.419446\pi$
$200$		0			0
$201$		−	11.7243i		−	0.826971i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	7.67036			0.535721
$206$		0			0
$207$	1.54255			0.107214
$208$		0			0
$209$	−3.93462			−0.272164
$210$		0			0
$211$	19.3472			1.33191			0.665956	−	0.745991i	$-0.268024\pi$
$211$	19.3472			1.33191			0.665956	+	0.745991i	$0.268024\pi$
$212$		0			0
$213$		−	4.22059i		−	0.289190i
$214$		0			0
$215$		−	4.92426i		−	0.335832i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		−	3.21883i		−	0.217508i
$220$		0			0
$221$	−0.264261	−	0.644488i	−0.0177761	−	0.0433529i
$222$		0			0
$223$			14.4868i			0.970105i	0.874485	+	0.485052i	$0.161199\pi$
$223$			14.4868i			0.970105i	−0.874485	+	0.485052i	$0.838801\pi$
$224$		0			0
$225$	−12.0318			−0.802123
$226$		0			0
$227$			18.3930i			1.22079i	0.792098	+	0.610394i	$0.208989\pi$
$227$			18.3930i			1.22079i	−0.792098	+	0.610394i	$0.791011\pi$
$228$		0			0
$229$		−	22.5750i		−	1.49180i	−0.666060	−	0.745898i	$-0.732021\pi$
$229$		−	22.5750i		−	1.49180i	0.666060	−	0.745898i	$-0.267979\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−20.6432			−1.35238			−0.676192	−	0.736725i	$-0.736371\pi$
$233$	−20.6432			−1.35238			−0.676192	+	0.736725i	$0.736371\pi$
$234$		0			0
$235$	40.4002			2.63542
$236$		0			0
$237$	−11.7286			−0.761857
$238$		0			0
$239$		−	2.81923i		−	0.182361i	−0.995834	−	0.0911805i	$-0.970936\pi$
$239$		−	2.81923i		−	0.182361i	0.995834	−	0.0911805i	$-0.0290640\pi$
$240$		0			0
$241$		−	22.2639i		−	1.43414i	−0.697001	−	0.717070i	$-0.745483\pi$
$241$		−	22.2639i		−	1.43414i	0.697001	−	0.717070i	$-0.254517\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−2.71723	+	1.11415i	−0.172893	+	0.0708917i
$248$		0			0
$249$		−	2.65589i		−	0.168310i
$250$		0			0
$251$	−23.5673			−1.48755			−0.743777	−	0.668428i	$-0.766968\pi$
$251$	−23.5673			−1.48755			−0.743777	+	0.668428i	$0.766968\pi$
$252$		0			0
$253$		−	7.45149i		−	0.468471i
$254$		0			0
$255$			0.797294i			0.0499285i
$256$		0			0
$257$	9.30822			0.580631			0.290315	−	0.956931i	$-0.406240\pi$
$257$	9.30822			0.580631			0.290315	+	0.956931i	$0.406240\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.154256			0.00954822
$262$		0			0
$263$	17.4182			1.07405			0.537027	−	0.843565i	$-0.319547\pi$
$263$	17.4182			1.07405			0.537027	+	0.843565i	$0.319547\pi$
$264$		0			0
$265$		−	50.6428i		−	3.11096i
$266$		0			0
$267$			4.95870i			0.303468i
$268$		0			0
$269$	4.59390			0.280095			0.140047	−	0.990145i	$-0.455274\pi$
$269$	4.59390			0.280095			0.140047	+	0.990145i	$0.455274\pi$
$270$		0			0
$271$			8.76247i			0.532282i	0.963934	+	0.266141i	$0.0857487\pi$
$271$			8.76247i			0.532282i	−0.963934	+	0.266141i	$0.914251\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			58.1214i			3.50485i
$276$		0			0
$277$	−12.7267			−0.764675			−0.382338	−	0.924023i	$-0.624881\pi$
$277$	−12.7267			−0.764675			−0.382338	+	0.924023i	$0.624881\pi$
$278$		0			0
$279$			4.84786i			0.290234i
$280$		0			0
$281$			22.4986i			1.34215i	0.741388	+	0.671076i	$0.234168\pi$
$281$			22.4986i			1.34215i	−0.741388	+	0.671076i	$0.765832\pi$
$282$		0			0
$283$	2.16828			0.128891			0.0644455	−	0.997921i	$-0.479472\pi$
$283$	2.16828			0.128891			0.0644455	+	0.997921i	$0.479472\pi$
$284$		0			0
$285$	3.36147			0.199117
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−16.9627			−0.997805
$290$		0			0
$291$			3.13454i			0.183750i
$292$		0			0
$293$		−	26.7146i		−	1.56068i	−0.625353	−	0.780342i	$-0.715045\pi$
$293$		−	26.7146i		−	1.56068i	0.625353	−	0.780342i	$-0.284955\pi$
$294$		0			0
$295$	57.4321			3.34382
$296$		0			0
$297$		−	4.83063i		−	0.280302i
$298$		0			0
$299$	−2.11000	−	5.14595i	−0.122025	−	0.297598i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−3.50361			−0.201277
$304$		0			0
$305$			21.7833i			1.24731i
$306$		0			0
$307$		−	14.5710i		−	0.831613i	−0.909453	−	0.415806i	$-0.863499\pi$
$307$		−	14.5710i		−	0.831613i	0.909453	−	0.415806i	$-0.136501\pi$
$308$		0			0
$309$	12.6114			0.717438
$310$		0			0
$311$	24.3933			1.38322			0.691609	−	0.722272i	$-0.256902\pi$
$311$	24.3933			1.38322			0.691609	+	0.722272i	$0.256902\pi$
$312$		0			0
$313$	29.8134			1.68516			0.842578	−	0.538575i	$-0.181037\pi$
$313$	29.8134			1.68516			0.842578	+	0.538575i	$0.181037\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	2.87841i		−	0.161668i	−0.996728	−	0.0808339i	$-0.974242\pi$
$317$		−	2.87841i		−	0.161668i	0.996728	−	0.0808339i	$-0.0257583\pi$
$318$		0			0
$319$		−	0.745156i		−	0.0417207i
$320$		0			0
$321$	−5.92893			−0.330921
$322$		0			0
$323$		−	0.157357i		−	0.00875559i
$324$		0			0
$325$	16.4580	+	40.1383i	0.912925	+	2.22647i
$326$		0			0
$327$		−	19.6285i		−	1.08546i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			27.6607i			1.52037i	0.649705	+	0.760186i	$0.274892\pi$
$331$			27.6607i			1.52037i	−0.649705	+	0.760186i	$0.725108\pi$
$332$		0			0
$333$			0.110845i			0.00607425i
$334$		0			0
$335$	48.3859			2.64361
$336$		0			0
$337$	12.7657			0.695390			0.347695	−	0.937608i	$-0.386964\pi$
$337$	12.7657			0.695390			0.347695	+	0.937608i	$0.386964\pi$
$338$		0			0
$339$	−16.2391			−0.881984
$340$		0			0
$341$	23.4182			1.26817
$342$		0			0
$343$		0			0
$344$		0			0
$345$			6.36604i			0.342736i
$346$		0			0
$347$	−11.8189			−0.634473			−0.317237	−	0.948346i	$-0.602755\pi$
$347$	−11.8189			−0.634473			−0.317237	+	0.948346i	$0.602755\pi$
$348$		0			0
$349$		−	3.23665i		−	0.173254i	−0.996241	−	0.0866268i	$-0.972391\pi$
$349$		−	3.23665i		−	0.173254i	0.996241	−	0.0866268i	$-0.0276088\pi$
$350$		0			0
$351$	−1.36787	−	3.33601i	−0.0730115	−	0.178063i
$352$		0			0
$353$		−	29.5534i		−	1.57297i	−0.617609	−	0.786485i	$-0.711899\pi$
$353$		−	29.5534i		−	1.57297i	0.617609	−	0.786485i	$-0.288101\pi$
$354$		0			0
$355$	17.4182			0.924463
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	26.2594i		−	1.38592i	−0.720976	−	0.692960i	$-0.756306\pi$
$359$		−	26.2594i		−	1.38592i	0.720976	−	0.692960i	$-0.243694\pi$
$360$		0			0
$361$	18.3366			0.965082
$362$		0			0
$363$	−12.3350			−0.647421
$364$		0			0
$365$	13.2840			0.695315
$366$		0			0
$367$	−5.54446			−0.289418			−0.144709	−	0.989474i	$-0.546225\pi$
$367$	−5.54446			−0.289418			−0.144709	+	0.989474i	$0.546225\pi$
$368$		0			0
$369$		−	1.85860i		−	0.0967547i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.301287			−0.0156001			−0.00780003	−	0.999970i	$-0.502483\pi$
$373$	−0.301287			−0.0156001			−0.00780003	+	0.999970i	$0.502483\pi$
$374$		0			0
$375$		−	29.0201i		−	1.49859i
$376$		0			0
$377$	−0.211002	−	0.514600i	−0.0108672	−	0.0265032i
$378$		0			0
$379$			13.2573i			0.680983i	0.940248	+	0.340491i	$0.110593\pi$
$379$			13.2573i			0.680983i	−0.940248	+	0.340491i	$0.889407\pi$
$380$		0			0
$381$	−1.61171			−0.0825703
$382$		0			0
$383$			24.0829i			1.23058i	0.788302	+	0.615288i	$0.210960\pi$
$383$			24.0829i			1.23058i	−0.788302	+	0.615288i	$0.789040\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.19319			−0.0606533
$388$		0			0
$389$	14.6754			0.744071			0.372036	−	0.928218i	$-0.378660\pi$
$389$	14.6754			0.744071			0.372036	+	0.928218i	$0.378660\pi$
$390$		0			0
$391$	0.298007			0.0150709
$392$		0			0
$393$	−18.8897			−0.952859
$394$		0			0
$395$		−	48.4037i		−	2.43545i
$396$		0			0
$397$		−	4.59877i		−	0.230806i	−0.993319	−	0.115403i	$-0.963184\pi$
$397$		−	4.59877i		−	0.230806i	0.993319	−	0.115403i	$-0.0368159\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	20.1147i		−	1.00448i	−0.864728	−	0.502240i	$-0.832510\pi$
$401$		−	20.1147i		−	1.00448i	0.864728	−	0.502240i	$-0.167490\pi$
$402$		0			0
$403$	16.1725	−	6.63123i	0.805608	−	0.330325i
$404$		0			0
$405$			4.12696i			0.205071i
$406$		0			0
$407$	0.535450			0.0265413
$408$		0			0
$409$			15.7818i			0.780361i	0.920738	+	0.390180i	$0.127587\pi$
$409$			15.7818i			0.780361i	−0.920738	+	0.390180i	$0.872413\pi$
$410$		0			0
$411$		−	0.797294i		−	0.0393276i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	10.9608			0.538043
$416$		0			0
$417$	17.8065			0.871989
$418$		0			0
$419$	−20.3544			−0.994376			−0.497188	−	0.867643i	$-0.665634\pi$
$419$	−20.3544			−0.994376			−0.497188	+	0.867643i	$0.665634\pi$
$420$		0			0
$421$			24.9704i			1.21698i	0.793560	+	0.608492i	$0.208225\pi$
$421$			24.9704i			1.21698i	−0.793560	+	0.608492i	$0.791775\pi$
$422$		0			0
$423$		−	9.78934i		−	0.475974i
$424$		0			0
$425$	−2.32445			−0.112752
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−16.1150	+	6.60768i	−0.778041	+	0.319022i
$430$		0			0
$431$		−	19.7635i		−	0.951974i	−0.879452	−	0.475987i	$-0.842091\pi$
$431$		−	19.7635i		−	0.951974i	0.879452	−	0.475987i	$-0.157909\pi$
$432$		0			0
$433$	26.1982			1.25901			0.629503	−	0.776998i	$-0.283259\pi$
$433$	26.1982			1.25901			0.629503	+	0.776998i	$0.283259\pi$
$434$		0			0
$435$			0.636610i			0.0305231i
$436$		0			0
$437$		−	1.25643i		−	0.0601031i
$438$		0			0
$439$	17.9427			0.856356			0.428178	−	0.903694i	$-0.359156\pi$
$439$	17.9427			0.856356			0.428178	+	0.903694i	$0.359156\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	15.0640			0.715711			0.357855	−	0.933777i	$-0.383508\pi$
$443$	15.0640			0.715711			0.357855	+	0.933777i	$0.383508\pi$
$444$		0			0
$445$	−20.4644			−0.970105
$446$		0			0
$447$			16.2517i			0.768680i
$448$		0			0
$449$			14.1148i			0.666116i	0.942906	+	0.333058i	$0.108081\pi$
$449$			14.1148i			0.666116i	−0.942906	+	0.333058i	$0.891919\pi$
$450$		0			0
$451$	−8.97820			−0.422767
$452$		0			0
$453$			12.9187i			0.606975i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			28.9703i			1.35517i	0.735443	+	0.677587i	$0.236974\pi$
$457$			28.9703i			1.35517i	−0.735443	+	0.677587i	$0.763026\pi$
$458$		0			0
$459$	0.193191			0.00901740
$460$		0			0
$461$			9.98197i			0.464907i	0.972608	+	0.232453i	$0.0746753\pi$
$461$			9.98197i			0.464907i	−0.972608	+	0.232453i	$0.925325\pi$
$462$		0			0
$463$			24.2951i			1.12909i	0.825402	+	0.564545i	$0.190948\pi$
$463$			24.2951i			1.12909i	−0.825402	+	0.564545i	$0.809052\pi$
$464$		0			0
$465$	−20.0069			−0.927799
$466$		0			0
$467$	−8.27638			−0.382985			−0.191493	−	0.981494i	$-0.561333\pi$
$467$	−8.27638			−0.382985			−0.191493	+	0.981494i	$0.561333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	16.4201			0.756600
$472$		0			0
$473$			5.76387i			0.265023i
$474$		0			0
$475$			9.80011i			0.449660i
$476$		0			0
$477$	−12.2712			−0.561859
$478$		0			0
$479$			29.1412i			1.33149i	0.746178	+	0.665747i	$0.231887\pi$
$479$			29.1412i			1.33149i	−0.746178	+	0.665747i	$0.768113\pi$
$480$		0			0
$481$	0.369778	−	0.151621i	0.0168604	−	0.00691333i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−12.9362			−0.587401
$486$		0			0
$487$		−	12.3423i		−	0.559281i	−0.960105	−	0.279641i	$-0.909785\pi$
$487$		−	12.3423i		−	0.559281i	0.960105	−	0.279641i	$-0.0902153\pi$
$488$		0			0
$489$		−	20.7284i		−	0.937373i
$490$		0			0
$491$	−21.8406			−0.985650			−0.492825	−	0.870128i	$-0.664036\pi$
$491$	−21.8406			−0.985650			−0.492825	+	0.870128i	$0.664036\pi$
$492$		0			0
$493$	0.0298010			0.00134217
$494$		0			0
$495$	19.9359			0.896050
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	17.8681i		−	0.799885i	−0.916540	−	0.399943i	$-0.869030\pi$
$499$		−	17.8681i		−	0.799885i	0.916540	−	0.399943i	$-0.130970\pi$
$500$		0			0
$501$		−	7.26484i		−	0.324569i
$502$		0			0
$503$	−3.63823			−0.162221			−0.0811103	−	0.996705i	$-0.525847\pi$
$503$	−3.63823			−0.162221			−0.0811103	+	0.996705i	$0.525847\pi$
$504$		0			0
$505$		−	14.4593i		−	0.643430i
$506$		0			0
$507$	−9.25787	+	9.12644i	−0.411156	+	0.405319i
$508$		0			0
$509$			17.7192i			0.785391i	0.919668	+	0.392696i	$0.128457\pi$
$509$			17.7192i			0.785391i	−0.919668	+	0.392696i	$0.871543\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		−	0.814515i		−	0.0359617i
$514$		0			0
$515$			52.0469i			2.29346i
$516$		0			0
$517$	−47.2887			−2.07975
$518$		0			0
$519$	8.43445			0.370231
$520$		0			0
$521$	−9.63823			−0.422258			−0.211129	−	0.977458i	$-0.567714\pi$
$521$	−9.63823			−0.422258			−0.211129	+	0.977458i	$0.567714\pi$
$522$		0			0
$523$	29.2144			1.27746			0.638729	−	0.769432i	$-0.279460\pi$
$523$	29.2144			1.27746			0.638729	+	0.769432i	$0.279460\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			0.936564i			0.0407974i
$528$		0			0
$529$	−20.6205			−0.896545
$530$		0			0
$531$		−	13.9163i		−	0.603916i
$532$		0			0
$533$	−6.20029	+	2.54232i	−0.268564	+	0.110120i
$534$		0			0
$535$		−	24.4685i		−	1.05786i
$536$		0			0
$537$	16.2269			0.700244
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	12.5235i		−	0.538425i	−0.963081	−	0.269213i	$-0.913237\pi$
$541$		−	12.5235i		−	0.538425i	0.963081	−	0.269213i	$-0.0867634\pi$
$542$		0			0
$543$	−21.0247			−0.902258
$544$		0			0
$545$	81.0062			3.46992
$546$		0			0
$547$	−25.0456			−1.07087			−0.535436	−	0.844576i	$-0.679853\pi$
$547$	−25.0456			−1.07087			−0.535436	+	0.844576i	$0.679853\pi$
$548$		0			0
$549$	5.27829			0.225272
$550$		0			0
$551$		−	0.125644i		−	0.00535261i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−0.457452			−0.0194178
$556$		0			0
$557$		−	23.9883i		−	1.01642i	−0.861234	−	0.508209i	$-0.830308\pi$
$557$		−	23.9883i		−	1.01642i	0.861234	−	0.508209i	$-0.169692\pi$
$558$		0			0
$559$	1.63213	+	3.98049i	0.0690318	+	0.168357i
$560$		0			0
$561$		−	0.933237i		−	0.0394013i
$562$		0			0
$563$	−9.97888			−0.420559			−0.210280	−	0.977641i	$-0.567437\pi$
$563$	−9.97888			−0.420559			−0.210280	+	0.977641i	$0.567437\pi$
$564$		0			0
$565$		−	67.0180i		−	2.81947i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11.6474			−0.488283			−0.244141	−	0.969740i	$-0.578506\pi$
$569$	−11.6474			−0.488283			−0.244141	+	0.969740i	$0.578506\pi$
$570$		0			0
$571$	11.7887			0.493342			0.246671	−	0.969099i	$-0.420663\pi$
$571$	11.7887			0.493342			0.246671	+	0.969099i	$0.420663\pi$
$572$		0			0
$573$	22.8686			0.955348
$574$		0			0
$575$	−18.5597			−0.773992
$576$		0			0
$577$		−	30.5253i		−	1.27078i	−0.772189	−	0.635392i	$-0.780838\pi$
$577$		−	30.5253i		−	1.27078i	0.772189	−	0.635392i	$-0.219162\pi$
$578$		0			0
$579$		−	21.2575i		−	0.883433i
$580$		0			0
$581$		0			0
$582$		0			0
$583$			59.2776i			2.45503i
$584$		0			0
$585$	13.7676	−	5.64515i	0.569219	−	0.233398i
$586$		0			0
$587$		−	9.85549i		−	0.406780i	−0.979098	−	0.203390i	$-0.934804\pi$
$587$		−	9.85549i		−	0.406780i	0.979098	−	0.203390i	$-0.0651959\pi$
$588$		0			0
$589$	3.94865			0.162701
$590$		0			0
$591$			18.0759i			0.743542i
$592$		0			0
$593$			5.73731i			0.235603i	0.993037	+	0.117802i	$0.0375847\pi$
$593$			5.73731i			0.235603i	−0.993037	+	0.117802i	$0.962415\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	7.06397			0.289109
$598$		0			0
$599$	−34.9923			−1.42975			−0.714873	−	0.699254i	$-0.753516\pi$
$599$	−34.9923			−1.42975			−0.714873	+	0.699254i	$0.753516\pi$
$600$		0			0
$601$	−12.9037			−0.526354			−0.263177	−	0.964748i	$-0.584770\pi$
$601$	−12.9037			−0.526354			−0.263177	+	0.964748i	$0.584770\pi$
$602$		0			0
$603$		−	11.7243i		−	0.477452i
$604$		0			0
$605$		−	50.9062i		−	2.06963i
$606$		0			0
$607$	−19.0867			−0.774704			−0.387352	−	0.921932i	$-0.626610\pi$
$607$	−19.0867			−0.774704			−0.387352	+	0.921932i	$0.626610\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−32.6573	+	13.3905i	−1.32117	+	0.541723i
$612$		0			0
$613$			5.60592i			0.226421i	0.993571	+	0.113210i	$0.0361134\pi$
$613$			5.60592i			0.226421i	−0.993571	+	0.113210i	$0.963887\pi$
$614$		0			0
$615$	7.67036			0.309299
$616$		0			0
$617$		−	28.9162i		−	1.16412i	−0.813146	−	0.582060i	$-0.802247\pi$
$617$		−	28.9162i		−	1.16412i	0.813146	−	0.582060i	$-0.197753\pi$
$618$		0			0
$619$		−	14.8927i		−	0.598589i	−0.954161	−	0.299295i	$-0.903249\pi$
$619$		−	14.8927i		−	0.598589i	0.954161	−	0.299295i	$-0.0967513\pi$
$620$		0			0
$621$	1.54255			0.0619003
$622$		0			0
$623$		0			0
$624$		0			0
$625$	59.6059			2.38424
$626$		0			0
$627$	−3.93462			−0.157134
$628$		0			0
$629$			0.0214142i			0.000853842i
$630$		0			0
$631$		−	12.2217i		−	0.486539i	−0.969959	−	0.243269i	$-0.921780\pi$
$631$		−	12.2217i		−	0.486539i	0.969959	−	0.243269i	$-0.0782199\pi$
$632$		0			0
$633$	19.3472			0.768980
$634$		0			0
$635$		−	6.65146i		−	0.263955i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		−	4.22059i		−	0.166964i
$640$		0			0
$641$	−23.4463			−0.926072			−0.463036	−	0.886339i	$-0.653240\pi$
$641$	−23.4463			−0.926072			−0.463036	+	0.886339i	$0.653240\pi$
$642$		0			0
$643$			19.7831i			0.780171i	0.920779	+	0.390085i	$0.127555\pi$
$643$			19.7831i			0.780171i	−0.920779	+	0.390085i	$0.872445\pi$
$644$		0			0
$645$		−	4.92426i		−	0.193893i
$646$		0			0
$647$	−26.1559			−1.02829			−0.514147	−	0.857702i	$-0.671891\pi$
$647$	−26.1559			−1.02829			−0.514147	+	0.857702i	$0.671891\pi$
$648$		0			0
$649$	−67.2246			−2.63879
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.54225			0.0994860			0.0497430	−	0.998762i	$-0.484160\pi$
$653$	2.54225			0.0994860			0.0497430	+	0.998762i	$0.484160\pi$
$654$		0			0
$655$		−	77.9571i		−	3.04604i
$656$		0			0
$657$		−	3.21883i		−	0.125578i
$658$		0			0
$659$	16.6064			0.646893			0.323447	−	0.946246i	$-0.395158\pi$
$659$	16.6064			0.646893			0.323447	+	0.946246i	$0.395158\pi$
$660$		0			0
$661$		−	17.7062i		−	0.688691i	−0.938843	−	0.344345i	$-0.888101\pi$
$661$		−	17.7062i		−	0.688691i	0.938843	−	0.344345i	$-0.111899\pi$
$662$		0			0
$663$	−0.264261	−	0.644488i	−0.0102630	−	0.0250298i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	0.237948			0.00921337
$668$		0			0
$669$			14.4868i			0.560090i
$670$		0			0
$671$		−	25.4975i		−	0.984319i
$672$		0			0
$673$	−36.7173			−1.41535			−0.707675	−	0.706538i	$-0.750256\pi$
$673$	−36.7173			−1.41535			−0.707675	+	0.706538i	$0.750256\pi$
$674$		0			0
$675$	−12.0318			−0.463106
$676$		0			0
$677$	−46.4770			−1.78626			−0.893128	−	0.449802i	$-0.851494\pi$
$677$	−46.4770			−1.78626			−0.893128	+	0.449802i	$0.851494\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$			18.3930i			0.704822i
$682$		0			0
$683$			25.5317i			0.976943i	0.872580	+	0.488471i	$0.162445\pi$
$683$			25.5317i			0.976943i	−0.872580	+	0.488471i	$0.837555\pi$
$684$		0			0
$685$	3.29040			0.125720
$686$		0			0
$687$		−	22.5750i		−	0.861289i
$688$		0			0
$689$	16.7854	+	40.9368i	0.639472	+	1.55957i
$690$		0			0
$691$			11.7853i			0.448335i	0.974551	+	0.224168i	$0.0719664\pi$
$691$			11.7853i			0.448335i	−0.974551	+	0.224168i	$0.928034\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			73.4868i			2.78751i
$696$		0			0
$697$		−	0.359065i		−	0.0136006i
$698$		0			0
$699$	−20.6432			−0.780799
$700$		0			0
$701$	35.6275			1.34563			0.672816	−	0.739810i	$-0.265085\pi$
$701$	35.6275			1.34563			0.672816	+	0.739810i	$0.265085\pi$
$702$		0			0
$703$	0.0902846			0.00340515
$704$		0			0
$705$	40.4002			1.52156
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	29.3408i		−	1.10192i	−0.834532	−	0.550959i	$-0.814262\pi$
$709$		−	29.3408i		−	1.10192i	0.834532	−	0.550959i	$-0.185738\pi$
$710$		0			0
$711$	−11.7286			−0.439858
$712$		0			0
$713$			7.47805i			0.280055i
$714$		0			0
$715$	−27.2697	−	66.5061i	−1.01983	−	2.48719i
$716$		0			0
$717$		−	2.81923i		−	0.105286i
$718$		0			0
$719$	−38.7372			−1.44465			−0.722327	−	0.691552i	$-0.756927\pi$
$719$	−38.7372			−1.44465			−0.722327	+	0.691552i	$0.756927\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	22.2639i		−	0.828002i
$724$		0			0
$725$	−1.85599			−0.0689296
$726$		0			0
$727$	13.3601			0.495498			0.247749	−	0.968824i	$-0.420309\pi$
$727$	13.3601			0.495498			0.247749	+	0.968824i	$0.420309\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−0.230514			−0.00852588
$732$		0			0
$733$			10.9874i			0.405831i	0.979196	+	0.202915i	$0.0650416\pi$
$733$			10.9874i			0.405831i	−0.979196	+	0.202915i	$0.934958\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−56.6360			−2.08621
$738$		0			0
$739$			0.816338i			0.0300295i	0.999887	+	0.0150147i	$0.00477952\pi$
$739$			0.816338i			0.0300295i	−0.999887	+	0.0150147i	$0.995220\pi$
$740$		0			0
$741$	−2.71723	+	1.11415i	−0.0998198	+	0.0409293i
$742$		0			0
$743$			11.1384i			0.408628i	0.978905	+	0.204314i	$0.0654964\pi$
$743$			11.1384i			0.408628i	−0.978905	+	0.204314i	$0.934504\pi$
$744$		0			0
$745$	−67.0703			−2.45727
$746$		0			0
$747$		−	2.65589i		−	0.0971740i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	4.64006			0.169318			0.0846591	−	0.996410i	$-0.473020\pi$
$751$	4.64006			0.169318			0.0846591	+	0.996410i	$0.473020\pi$
$752$		0			0
$753$	−23.5673			−0.858840
$754$		0			0
$755$	−53.3151			−1.94034
$756$		0			0
$757$	−18.7608			−0.681872			−0.340936	−	0.940087i	$-0.610744\pi$
$757$	−18.7608			−0.681872			−0.340936	+	0.940087i	$0.610744\pi$
$758$		0			0
$759$		−	7.45149i		−	0.270472i
$760$		0			0
$761$			13.4331i			0.486949i	0.969907	+	0.243475i	$0.0782873\pi$
$761$			13.4331i			0.486949i	−0.969907	+	0.243475i	$0.921713\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$			0.797294i			0.0288262i
$766$		0			0
$767$	−46.4249	+	19.0357i	−1.67630	+	0.687339i
$768$		0			0
$769$			41.4834i			1.49593i	0.663738	+	0.747965i	$0.268969\pi$
$769$			41.4834i			1.49593i	−0.663738	+	0.747965i	$0.731031\pi$
$770$		0			0
$771$	9.30822			0.335227
$772$		0			0
$773$			1.33968i			0.0481850i	0.999710	+	0.0240925i	$0.00766962\pi$
$773$			1.33968i			0.0481850i	−0.999710	+	0.0240925i	$0.992330\pi$
$774$		0			0
$775$		−	58.3286i		−	2.09523i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.51385			−0.0542395
$780$		0			0
$781$	−20.3881			−0.729544
$782$		0			0
$783$	0.154256			0.00551267
$784$		0			0
$785$			67.7653i			2.41865i
$786$		0			0
$787$			23.8962i			0.851808i	0.904768	+	0.425904i	$0.140044\pi$
$787$			23.8962i			0.851808i	−0.904768	+	0.425904i	$0.859956\pi$
$788$		0			0
$789$	17.4182			0.620105
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−7.22001	−	17.6084i	−0.256390	−	0.625292i
$794$		0			0
$795$		−	50.6428i		−	1.79611i
$796$		0			0
$797$	48.8615			1.73076			0.865382	−	0.501113i	$-0.167076\pi$
$797$	48.8615			1.73076			0.865382	+	0.501113i	$0.167076\pi$
$798$		0			0
$799$		−	1.89122i		−	0.0669064i
$800$		0			0
$801$			4.95870i			0.175207i
$802$		0			0
$803$	−15.5490			−0.548711
$804$		0			0
$805$		0			0
$806$		0			0
$807$	4.59390			0.161713
$808$		0			0
$809$	−7.61128			−0.267598			−0.133799	−	0.991008i	$-0.542718\pi$
$809$	−7.61128			−0.267598			−0.133799	+	0.991008i	$0.542718\pi$
$810$		0			0
$811$		−	45.4292i		−	1.59523i	−0.603165	−	0.797617i	$-0.706094\pi$
$811$		−	45.4292i		−	1.59523i	0.603165	−	0.797617i	$-0.293906\pi$
$812$		0			0
$813$			8.76247i			0.307313i
$814$		0			0
$815$	85.5456			2.99653
$816$		0			0
$817$			0.971872i			0.0340015i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			3.28735i			0.114729i	0.998353	+	0.0573647i	$0.0182698\pi$
$821$			3.28735i			0.114729i	−0.998353	+	0.0573647i	$0.981730\pi$
$822$		0			0
$823$	31.1197			1.08477			0.542383	−	0.840131i	$-0.317522\pi$
$823$	31.1197			1.08477			0.542383	+	0.840131i	$0.317522\pi$
$824$		0			0
$825$			58.1214i			2.02353i
$826$		0			0
$827$			25.6551i			0.892116i	0.895004	+	0.446058i	$0.147173\pi$
$827$			25.6551i			0.892116i	−0.895004	+	0.446058i	$0.852827\pi$
$828$		0			0
$829$	41.1629			1.42965			0.714824	−	0.699305i	$-0.246507\pi$
$829$	41.1629			1.42965			0.714824	+	0.699305i	$0.246507\pi$
$830$		0			0
$831$	−12.7267			−0.441485
$832$		0			0
$833$		0			0
$834$		0			0
$835$	29.9817			1.03756
$836$		0			0
$837$			4.84786i			0.167566i
$838$		0			0
$839$		−	23.9529i		−	0.826946i	−0.910516	−	0.413473i	$-0.864316\pi$
$839$		−	23.9529i		−	0.826946i	0.910516	−	0.413473i	$-0.135684\pi$
$840$		0			0
$841$	−28.9762			−0.999179
$842$		0			0
$843$			22.4986i			0.774892i
$844$		0			0
$845$	−37.6645	−	38.2069i	−1.29570	−	1.31436i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	2.16828			0.0744153
$850$		0			0
$851$			0.170983i			0.00586123i
$852$		0			0
$853$		−	22.4638i		−	0.769145i	−0.923095	−	0.384572i	$-0.874349\pi$
$853$		−	22.4638i		−	0.769145i	0.923095	−	0.384572i	$-0.125651\pi$
$854$		0			0
$855$	3.36147			0.114960
$856$		0			0
$857$	0.0732746			0.00250301			0.00125151	−	0.999999i	$-0.499602\pi$
$857$	0.0732746			0.00250301			0.00125151	+	0.999999i	$0.499602\pi$
$858$		0			0
$859$	−33.4606			−1.14166			−0.570830	−	0.821068i	$-0.693379\pi$
$859$	−33.4606			−1.14166			−0.570830	+	0.821068i	$0.693379\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	25.0761i		−	0.853601i	−0.904346	−	0.426800i	$-0.859641\pi$
$863$		−	25.0761i		−	0.853601i	0.904346	−	0.426800i	$-0.140359\pi$
$864$		0			0
$865$			34.8087i			1.18353i
$866$		0			0
$867$	−16.9627			−0.576083
$868$		0			0
$869$			56.6568i			1.92195i
$870$		0			0
$871$	−39.1124	+	16.0374i	−1.32527	+	0.543405i
$872$		0			0
$873$			3.13454i			0.106088i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			16.7700i			0.566282i	0.959078	+	0.283141i	$0.0913764\pi$
$877$			16.7700i			0.566282i	−0.959078	+	0.283141i	$0.908624\pi$
$878$		0			0
$879$		−	26.7146i		−	0.901061i
$880$		0			0
$881$	−18.0174			−0.607023			−0.303511	−	0.952828i	$-0.598159\pi$
$881$	−18.0174			−0.607023			−0.303511	+	0.952828i	$0.598159\pi$
$882$		0			0
$883$	−22.5443			−0.758676			−0.379338	−	0.925258i	$-0.623848\pi$
$883$	−22.5443			−0.758676			−0.379338	+	0.925258i	$0.623848\pi$
$884$		0			0
$885$	57.4321			1.93056
$886$		0			0
$887$	13.0249			0.437334			0.218667	−	0.975800i	$-0.429829\pi$
$887$	13.0249			0.437334			0.218667	+	0.975800i	$0.429829\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		−	4.83063i		−	0.161832i
$892$		0			0
$893$	−7.97356			−0.266825
$894$		0			0
$895$			66.9680i			2.23849i
$896$		0			0
$897$	−2.11000	−	5.14595i	−0.0704510	−	0.171818i
$898$		0			0
$899$			0.747812i			0.0249409i
$900$		0			0
$901$	−2.37069			−0.0789791
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	86.7684i		−	2.88428i
$906$		0			0
$907$	−25.7356			−0.854536			−0.427268	−	0.904125i	$-0.640524\pi$
$907$	−25.7356			−0.854536			−0.427268	+	0.904125i	$0.640524\pi$
$908$		0			0
$909$	−3.50361			−0.116207
$910$		0			0
$911$	30.1342			0.998392			0.499196	−	0.866489i	$-0.333629\pi$
$911$	30.1342			0.998392			0.499196	+	0.866489i	$0.333629\pi$
$912$		0			0
$913$	−12.8296			−0.424599
$914$		0			0
$915$			21.7833i			0.720134i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−57.7884			−1.90626			−0.953131	−	0.302558i	$-0.902160\pi$
$919$	−57.7884			−1.90626			−0.953131	+	0.302558i	$0.902160\pi$
$920$		0			0
$921$		−	14.5710i		−	0.480132i
$922$		0			0
$923$	−14.0799	+	5.77322i	−0.463446	+	0.190028i
$924$		0			0
$925$		−	1.33367i		−	0.0438506i
$926$		0			0
$927$	12.6114			0.414213
$928$		0			0
$929$			3.97293i			0.130348i	0.997874	+	0.0651739i	$0.0207602\pi$
$929$			3.97293i			0.130348i	−0.997874	+	0.0651739i	$0.979240\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	24.3933			0.798601
$934$		0			0
$935$	3.85144			0.125955
$936$		0			0
$937$	−14.0852			−0.460144			−0.230072	−	0.973174i	$-0.573896\pi$
$937$	−14.0852			−0.460144			−0.230072	+	0.973174i	$0.573896\pi$
$938$		0			0
$939$	29.8134			0.972925
$940$		0			0
$941$			1.59542i			0.0520093i	0.999662	+	0.0260046i	$0.00827847\pi$
$941$			1.59542i			0.0520093i	−0.999662	+	0.0260046i	$0.991722\pi$
$942$		0			0
$943$		−	2.86697i		−	0.0933615i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			2.15606i			0.0700625i	0.999386	+	0.0350312i	$0.0111531\pi$
$947$			2.15606i			0.0700625i	−0.999386	+	0.0350312i	$0.988847\pi$
$948$		0			0
$949$	−10.7380	+	4.40293i	−0.348571	+	0.142925i
$950$		0			0
$951$		−	2.87841i		−	0.0933390i
$952$		0			0
$953$	−34.4024			−1.11440			−0.557202	−	0.830377i	$-0.688125\pi$
$953$	−34.4024			−1.11440			−0.557202	+	0.830377i	$0.688125\pi$
$954$		0			0
$955$			94.3778i			3.05399i
$956$		0			0
$957$		−	0.745156i		−	0.0240875i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	7.49830			0.241881
$962$		0			0
$963$	−5.92893			−0.191057
$964$		0			0
$965$	87.7291			2.82410
$966$		0			0
$967$		−	41.1646i		−	1.32376i	−0.749609	−	0.661881i	$-0.769758\pi$
$967$		−	41.1646i		−	1.32376i	0.749609	−	0.661881i	$-0.230242\pi$
$968$		0			0
$969$		−	0.157357i		−	0.00505504i
$970$		0			0
$971$	11.3335			0.363711			0.181855	−	0.983325i	$-0.441790\pi$
$971$	11.3335			0.363711			0.181855	+	0.983325i	$0.441790\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	16.4580	+	40.1383i	0.527077	+	1.28545i
$976$		0			0
$977$			12.5784i			0.402420i	0.979548	+	0.201210i	$0.0644874\pi$
$977$			12.5784i			0.402420i	−0.979548	+	0.201210i	$0.935513\pi$
$978$		0			0
$979$	23.9537			0.765563
$980$		0			0
$981$		−	19.6285i		−	0.626690i
$982$		0			0
$983$		−	6.84841i		−	0.218430i	−0.994018	−	0.109215i	$-0.965166\pi$
$983$		−	6.84841i		−	0.218430i	0.994018	−	0.109215i	$-0.0348338\pi$
$984$		0			0
$985$	−74.5985			−2.37691
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.84055			−0.0585262
$990$		0			0
$991$	18.0689			0.573977			0.286989	−	0.957934i	$-0.407346\pi$
$991$	18.0689			0.573977			0.286989	+	0.957934i	$0.407346\pi$
$992$		0			0
$993$			27.6607i			0.877787i
$994$		0			0
$995$			29.1528i			0.924205i
$996$		0			0
$997$	42.0210			1.33082			0.665409	−	0.746479i	$-0.268257\pi$
$997$	42.0210			1.33082			0.665409	+	0.746479i	$0.268257\pi$
$998$		0			0
$999$			0.110845i			0.00350697i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7644.2.e.o.4705.10		10
7.2	even	3		1092.2.cu.c.25.10	yes	20
7.4	even	3		1092.2.cu.c.961.1	yes	20
7.6	odd	2		7644.2.e.n.4705.1		10
13.12	even	2	inner	7644.2.e.o.4705.1		10
21.2	odd	6		3276.2.gv.g.1117.1		20
21.11	odd	6		3276.2.gv.g.2053.10		20
91.25	even	6		1092.2.cu.c.961.10	yes	20
91.51	even	6		1092.2.cu.c.25.1	✓	20
91.90	odd	2		7644.2.e.n.4705.10		10
273.116	odd	6		3276.2.gv.g.2053.1		20
273.233	odd	6		3276.2.gv.g.1117.10		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1092.2.cu.c.25.1	✓	20	91.51	even	6
1092.2.cu.c.25.10	yes	20	7.2	even	3
1092.2.cu.c.961.1	yes	20	7.4	even	3
1092.2.cu.c.961.10	yes	20	91.25	even	6
3276.2.gv.g.1117.1		20	21.2	odd	6
3276.2.gv.g.1117.10		20	273.233	odd	6
3276.2.gv.g.2053.1		20	273.116	odd	6
3276.2.gv.g.2053.10		20	21.11	odd	6
7644.2.e.n.4705.1		10	7.6	odd	2
7644.2.e.n.4705.10		10	91.90	odd	2
7644.2.e.o.4705.1		10	13.12	even	2	inner
7644.2.e.o.4705.10		10	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7644 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7644.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.12696i q^{5} +1.00000 q^{9} -4.83063i q^{11} +(-1.36787 - 3.33601i) q^{13} +4.12696i q^{15} +0.193191 q^{17} -0.814515i q^{19} +1.54255 q^{23} -12.0318 q^{25} +1.00000 q^{27} +0.154256 q^{29} +4.84786i q^{31} -4.83063i q^{33} +0.110845i q^{37} +(-1.36787 - 3.33601i) q^{39} -1.85860i q^{41} -1.19319 q^{43} +4.12696i q^{45} -9.78934i q^{47} +0.193191 q^{51} -12.2712 q^{53} +19.9359 q^{55} -0.814515i q^{57} -13.9163i q^{59} +5.27829 q^{61} +(13.7676 - 5.64515i) q^{65} -11.7243i q^{67} +1.54255 q^{69} -4.22059i q^{71} -3.21883i q^{73} -12.0318 q^{75} -11.7286 q^{79} +1.00000 q^{81} -2.65589i q^{83} +0.797294i q^{85} +0.154256 q^{87} +4.95870i q^{89} +4.84786i q^{93} +3.36147 q^{95} +3.13454i q^{97} -4.83063i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{3} + 10 q^{9} + q^{13} - 4 q^{17} - 8 q^{23} - 24 q^{25} + 10 q^{27} - 2 q^{29} + q^{39} - 6 q^{43} - 4 q^{51} + 24 q^{53} - 12 q^{55} + 12 q^{65} - 8 q^{69} - 24 q^{75} + 6 q^{79} + 10 q^{81}+ \cdots - 58 q^{95}+O(q^{100}) \) 10 * q + 10 * q^3 + 10 * q^9 + q^13 - 4 * q^17 - 8 * q^23 - 24 * q^25 + 10 * q^27 - 2 * q^29 + q^39 - 6 * q^43 - 4 * q^51 + 24 * q^53 - 12 * q^55 + 12 * q^65 - 8 * q^69 - 24 * q^75 + 6 * q^79 + 10 * q^81 - 2 * q^87 - 58 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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