LMFDB - Embedded newform 7644.2.e.q.4705.7

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 = [16,0,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0] 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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 46x^{14} + 825x^{12} + 7500x^{10} + 37308x^{8} + 101640x^{6} + 144580x^{4} + 100432x^{2} + 26896 $ x^16 + 46x^14 + 825x^12 + 7500x^10 + 37308x^8 + 101640x^6 + 144580x^4 + 100432*x^2 + 26896
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4705.7
Root			$-1.03273i$ of defining polynomial
Character	$\chi$	$=$	7644.4705
Dual form			7644.2.e.q.4705.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.03273i q^{5} +1.00000 q^{9} +1.53941i q^{11} +(-3.57628 + 0.458467i) q^{13} -1.03273i q^{15} -2.59762 q^{17} +5.93315i q^{19} -1.96832 q^{23} +3.93348 q^{25} +1.00000 q^{27} +3.63330 q^{29} -7.07872i q^{31} +1.53941i q^{33} -4.06240i q^{37} +(-3.57628 + 0.458467i) q^{39} -4.76561i q^{41} +0.849674 q^{43} -1.03273i q^{45} -11.6987i q^{47} -2.59762 q^{51} +10.9201 q^{53} +1.58979 q^{55} +5.93315i q^{57} +10.3335i q^{59} +9.40087 q^{61} +(0.473471 + 3.69332i) q^{65} -15.5105i q^{67} -1.96832 q^{69} +6.38275i q^{71} +8.34045i q^{73} +3.93348 q^{75} +0.841893 q^{79} +1.00000 q^{81} +15.7121i q^{83} +2.68263i q^{85} +3.63330 q^{87} +16.8796i q^{89} -7.07872i q^{93} +6.12732 q^{95} +9.90578i q^{97} +1.53941i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{3} + 16 q^{9} - 4 q^{23} - 12 q^{25} + 16 q^{27} - 4 q^{29} - 12 q^{43} - 12 q^{53} + 16 q^{61} - 24 q^{65} - 4 q^{69} - 12 q^{75} - 20 q^{79} + 16 q^{81} - 4 q^{87} + 12 q^{95}+O(q^{100}) $ 16 * q + 16 * q^3 + 16 * q^9 - 4 * q^23 - 12 * q^25 + 16 * q^27 - 4 * q^29 - 12 * q^43 - 12 * q^53 + 16 * q^61 - 24 * q^65 - 4 * q^69 - 12 * q^75 - 20 * q^79 + 16 * q^81 - 4 * q^87 + 12 * 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
$5$		−	1.03273i		−	0.461849i	−0.972972	−	0.230925i	$-0.925825\pi$
$5$		−	1.03273i		−	0.461849i	0.972972	−	0.230925i	$-0.0741751\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			1.53941i			0.464150i	0.972698	+	0.232075i	$0.0745516\pi$
$11$			1.53941i			0.464150i	−0.972698	+	0.232075i	$0.925448\pi$
$12$		0			0
$13$	−3.57628	+	0.458467i	−0.991883	+	0.127156i
$13$	−3.57628	+	0.458467i	−0.991883	+	0.127156i
$14$		0			0
$15$		−	1.03273i		−	0.266649i
$16$		0			0
$17$	−2.59762			−0.630016			−0.315008	−	0.949089i	$-0.602007\pi$
$17$	−2.59762			−0.630016			−0.315008	+	0.949089i	$0.602007\pi$
$18$		0			0
$19$			5.93315i			1.36116i	0.732675	+	0.680579i	$0.238272\pi$
$19$			5.93315i			1.36116i	−0.732675	+	0.680579i	$0.761728\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.96832			−0.410424			−0.205212	−	0.978718i	$-0.565788\pi$
$23$	−1.96832			−0.410424			−0.205212	+	0.978718i	$0.565788\pi$
$24$		0			0
$25$	3.93348			0.786695
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	3.63330			0.674688			0.337344	−	0.941381i	$-0.390471\pi$
$29$	3.63330			0.674688			0.337344	+	0.941381i	$0.390471\pi$
$30$		0			0
$31$		−	7.07872i		−	1.27138i	−0.771946	−	0.635688i	$-0.780717\pi$
$31$		−	7.07872i		−	1.27138i	0.771946	−	0.635688i	$-0.219283\pi$
$32$		0			0
$33$			1.53941i			0.267977i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	4.06240i		−	0.667855i	−0.942599	−	0.333928i	$-0.891626\pi$
$37$		−	4.06240i		−	0.667855i	0.942599	−	0.333928i	$-0.108374\pi$
$38$		0			0
$39$	−3.57628	+	0.458467i	−0.572664	+	0.0734135i
$40$		0			0
$41$		−	4.76561i		−	0.744263i	−0.928180	−	0.372131i	$-0.878627\pi$
$41$		−	4.76561i		−	0.744263i	0.928180	−	0.372131i	$-0.121373\pi$
$42$		0			0
$43$	0.849674			0.129574			0.0647870	−	0.997899i	$-0.479363\pi$
$43$	0.849674			0.129574			0.0647870	+	0.997899i	$0.479363\pi$
$44$		0			0
$45$		−	1.03273i		−	0.153950i
$46$		0			0
$47$		−	11.6987i		−	1.70643i	−0.521557	−	0.853217i	$-0.674648\pi$
$47$		−	11.6987i		−	1.70643i	0.521557	−	0.853217i	$-0.325352\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−2.59762			−0.363740
$52$		0			0
$53$	10.9201			1.50000			0.749998	−	0.661440i	$-0.230054\pi$
$53$	10.9201			1.50000			0.749998	+	0.661440i	$0.230054\pi$
$54$		0			0
$55$	1.58979			0.214368
$56$		0			0
$57$			5.93315i			0.785865i
$58$		0			0
$59$			10.3335i			1.34531i	0.739955	+	0.672657i	$0.234847\pi$
$59$			10.3335i			1.34531i	−0.739955	+	0.672657i	$0.765153\pi$
$60$		0			0
$61$	9.40087			1.20366			0.601829	−	0.798625i	$-0.294439\pi$
$61$	9.40087			1.20366			0.601829	+	0.798625i	$0.294439\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.473471	+	3.69332i	0.0587269	+	0.458100i
$66$		0			0
$67$		−	15.5105i		−	1.89491i	−0.319887	−	0.947456i	$-0.603645\pi$
$67$		−	15.5105i		−	1.89491i	0.319887	−	0.947456i	$-0.396355\pi$
$68$		0			0
$69$	−1.96832			−0.236958
$70$		0			0
$71$			6.38275i			0.757493i	0.925500	+	0.378747i	$0.123645\pi$
$71$			6.38275i			0.757493i	−0.925500	+	0.378747i	$0.876355\pi$
$72$		0			0
$73$			8.34045i			0.976176i	0.872794	+	0.488088i	$0.162306\pi$
$73$			8.34045i			0.976176i	−0.872794	+	0.488088i	$0.837694\pi$
$74$		0			0
$75$	3.93348			0.454199
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.841893			0.0947204			0.0473602	−	0.998878i	$-0.484919\pi$
$79$	0.841893			0.0947204			0.0473602	+	0.998878i	$0.484919\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			15.7121i			1.72463i	0.506372	+	0.862315i	$0.330986\pi$
$83$			15.7121i			1.72463i	−0.506372	+	0.862315i	$0.669014\pi$
$84$		0			0
$85$			2.68263i			0.290973i
$86$		0			0
$87$	3.63330			0.389531
$88$		0			0
$89$			16.8796i			1.78924i	0.446832	+	0.894618i	$0.352552\pi$
$89$			16.8796i			1.78924i	−0.446832	+	0.894618i	$0.647448\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	7.07872i		−	0.734029i
$94$		0			0
$95$	6.12732			0.628649
$96$		0			0
$97$			9.90578i			1.00578i	0.864351	+	0.502890i	$0.167730\pi$
$97$			9.90578i			1.00578i	−0.864351	+	0.502890i	$0.832270\pi$
$98$		0			0
$99$			1.53941i			0.154717i
$100$		0			0
$101$	−9.98961			−0.994003			−0.497002	−	0.867750i	$-0.665566\pi$
$101$	−9.98961			−0.994003			−0.497002	+	0.867750i	$0.665566\pi$
$102$		0			0
$103$	19.5720			1.92849			0.964243	−	0.265018i	$-0.0853779\pi$
$103$	19.5720			1.92849			0.964243	+	0.265018i	$0.0853779\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.86479			0.663644			0.331822	−	0.943342i	$-0.392337\pi$
$107$	6.86479			0.663644			0.331822	+	0.943342i	$0.392337\pi$
$108$		0			0
$109$			15.5038i			1.48500i	0.669846	+	0.742500i	$0.266360\pi$
$109$			15.5038i			1.48500i	−0.669846	+	0.742500i	$0.733640\pi$
$110$		0			0
$111$		−	4.06240i		−	0.385586i
$112$		0			0
$113$	0.556572			0.0523579			0.0261790	−	0.999657i	$-0.491666\pi$
$113$	0.556572			0.0523579			0.0261790	+	0.999657i	$0.491666\pi$
$114$		0			0
$115$			2.03274i			0.189554i
$116$		0			0
$117$	−3.57628	+	0.458467i	−0.330628	+	0.0423853i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	8.63021			0.784564
$122$		0			0
$123$		−	4.76561i		−	0.429700i
$124$		0			0
$125$		−	9.22583i		−	0.825184i
$126$		0			0
$127$	5.78315			0.513172			0.256586	−	0.966521i	$-0.417402\pi$
$127$	5.78315			0.513172			0.256586	+	0.966521i	$0.417402\pi$
$128$		0			0
$129$	0.849674			0.0748096
$130$		0			0
$131$	19.0922			1.66810			0.834049	−	0.551691i	$-0.186017\pi$
$131$	19.0922			1.66810			0.834049	+	0.551691i	$0.186017\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		−	1.03273i		−	0.0888829i
$136$		0			0
$137$			8.75302i			0.747821i	0.927465	+	0.373911i	$0.121983\pi$
$137$			8.75302i			0.747821i	−0.927465	+	0.373911i	$0.878017\pi$
$138$		0			0
$139$	−5.26687			−0.446730			−0.223365	−	0.974735i	$-0.571704\pi$
$139$	−5.26687			−0.446730			−0.223365	+	0.974735i	$0.571704\pi$
$140$		0			0
$141$		−	11.6987i		−	0.985210i
$142$		0			0
$143$	−0.705771	−	5.50538i	−0.0590195	−	0.460383i
$144$		0			0
$145$		−	3.75221i		−	0.311604i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	1.49151i		−	0.122189i	−0.998132	−	0.0610945i	$-0.980541\pi$
$149$		−	1.49151i		−	0.122189i	0.998132	−	0.0610945i	$-0.0194591\pi$
$150$		0			0
$151$			8.35531i			0.679945i	0.940435	+	0.339973i	$0.110418\pi$
$151$			8.35531i			0.679945i	−0.940435	+	0.339973i	$0.889582\pi$
$152$		0			0
$153$	−2.59762			−0.210005
$154$		0			0
$155$	−7.31038			−0.587184
$156$		0			0
$157$	7.10455			0.567005			0.283502	−	0.958972i	$-0.408504\pi$
$157$	7.10455			0.567005			0.283502	+	0.958972i	$0.408504\pi$
$158$		0			0
$159$	10.9201			0.866023
$160$		0			0
$161$		0			0
$162$		0			0
$163$			12.5870i			0.985891i	0.870060	+	0.492946i	$0.164080\pi$
$163$			12.5870i			0.985891i	−0.870060	+	0.492946i	$0.835920\pi$
$164$		0			0
$165$	1.58979			0.123765
$166$		0			0
$167$		−	19.1612i		−	1.48273i	−0.671100	−	0.741367i	$-0.734178\pi$
$167$		−	19.1612i		−	1.48273i	0.671100	−	0.741367i	$-0.265822\pi$
$168$		0			0
$169$	12.5796	−	3.27922i	0.967663	−	0.252248i
$170$		0			0
$171$			5.93315i			0.453719i
$172$		0			0
$173$	9.66620			0.734907			0.367454	−	0.930042i	$-0.380230\pi$
$173$	9.66620			0.734907			0.367454	+	0.930042i	$0.380230\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			10.3335i			0.776717i
$178$		0			0
$179$	0.694977			0.0519450			0.0259725	−	0.999663i	$-0.491732\pi$
$179$	0.694977			0.0519450			0.0259725	+	0.999663i	$0.491732\pi$
$180$		0			0
$181$	−7.81108			−0.580593			−0.290297	−	0.956937i	$-0.593754\pi$
$181$	−7.81108			−0.580593			−0.290297	+	0.956937i	$0.593754\pi$
$182$		0			0
$183$	9.40087			0.694933
$184$		0			0
$185$	−4.19535			−0.308448
$186$		0			0
$187$		−	3.99882i		−	0.292422i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	24.5453			1.77604			0.888018	−	0.459808i	$-0.152082\pi$
$191$	24.5453			1.77604			0.888018	+	0.459808i	$0.152082\pi$
$192$		0			0
$193$		−	20.9887i		−	1.51080i	−0.655263	−	0.755401i	$-0.727442\pi$
$193$		−	20.9887i		−	1.51080i	0.655263	−	0.755401i	$-0.272558\pi$
$194$		0			0
$195$	0.473471	+	3.69332i	0.0339060	+	0.264484i
$196$		0			0
$197$			2.07654i			0.147947i	0.997260	+	0.0739737i	$0.0235681\pi$
$197$			2.07654i			0.147947i	−0.997260	+	0.0739737i	$0.976432\pi$
$198$		0			0
$199$	6.29595			0.446308			0.223154	−	0.974783i	$-0.428365\pi$
$199$	6.29595			0.446308			0.223154	+	0.974783i	$0.428365\pi$
$200$		0			0
$201$		−	15.5105i		−	1.09403i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−4.92157			−0.343737
$206$		0			0
$207$	−1.96832			−0.136808
$208$		0			0
$209$	−9.13356			−0.631782
$210$		0			0
$211$	13.9093			0.957555			0.478777	−	0.877936i	$-0.341080\pi$
$211$	13.9093			0.957555			0.478777	+	0.877936i	$0.341080\pi$
$212$		0			0
$213$			6.38275i			0.437339i
$214$		0			0
$215$		−	0.877480i		−	0.0598437i
$216$		0			0
$217$		0			0
$218$		0			0
$219$			8.34045i			0.563596i
$220$		0			0
$221$	9.28984	−	1.19093i	0.624902	−	0.0801104i
$222$		0			0
$223$		−	1.99158i		−	0.133366i	−0.997774	−	0.0666829i	$-0.978758\pi$
$223$		−	1.99158i		−	0.133366i	0.997774	−	0.0666829i	$-0.0212416\pi$
$224$		0			0
$225$	3.93348			0.262232
$226$		0			0
$227$		−	16.1480i		−	1.07178i	−0.844287	−	0.535891i	$-0.819976\pi$
$227$		−	16.1480i		−	1.07178i	0.844287	−	0.535891i	$-0.180024\pi$
$228$		0			0
$229$			23.0670i			1.52431i	0.647395	+	0.762155i	$0.275858\pi$
$229$			23.0670i			1.52431i	−0.647395	+	0.762155i	$0.724142\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.26648			0.0829701			0.0414851	−	0.999139i	$-0.486791\pi$
$233$	1.26648			0.0829701			0.0414851	+	0.999139i	$0.486791\pi$
$234$		0			0
$235$	−12.0816			−0.788115
$236$		0			0
$237$	0.841893			0.0546868
$238$		0			0
$239$		−	7.46395i		−	0.482803i	−0.970425	−	0.241402i	$-0.922393\pi$
$239$		−	7.46395i		−	0.482803i	0.970425	−	0.241402i	$-0.0776071\pi$
$240$		0			0
$241$		−	25.2319i		−	1.62533i	−0.582730	−	0.812666i	$-0.698016\pi$
$241$		−	25.2319i		−	1.62533i	0.582730	−	0.812666i	$-0.301984\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−2.72015	−	21.2186i	−0.173079	−	1.35011i
$248$		0			0
$249$			15.7121i			0.995716i
$250$		0			0
$251$	−25.4021			−1.60337			−0.801683	−	0.597750i	$-0.796062\pi$
$251$	−25.4021			−1.60337			−0.801683	+	0.597750i	$0.796062\pi$
$252$		0			0
$253$		−	3.03006i		−	0.190499i
$254$		0			0
$255$			2.68263i			0.167993i
$256$		0			0
$257$	9.29097			0.579555			0.289777	−	0.957094i	$-0.406419\pi$
$257$	9.29097			0.579555			0.289777	+	0.957094i	$0.406419\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	3.63330			0.224896
$262$		0			0
$263$	−7.53559			−0.464664			−0.232332	−	0.972637i	$-0.574636\pi$
$263$	−7.53559			−0.464664			−0.232332	+	0.972637i	$0.574636\pi$
$264$		0			0
$265$		−	11.2775i		−	0.692772i
$266$		0			0
$267$			16.8796i			1.03302i
$268$		0			0
$269$	−16.4290			−1.00169			−0.500846	−	0.865536i	$-0.666978\pi$
$269$	−16.4290			−1.00169			−0.500846	+	0.865536i	$0.666978\pi$
$270$		0			0
$271$		−	3.37238i		−	0.204857i	−0.994740	−	0.102429i	$-0.967339\pi$
$271$		−	3.37238i		−	0.204857i	0.994740	−	0.102429i	$-0.0326613\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			6.05524i			0.365145i
$276$		0			0
$277$	18.1400			1.08992			0.544962	−	0.838461i	$-0.316544\pi$
$277$	18.1400			1.08992			0.544962	+	0.838461i	$0.316544\pi$
$278$		0			0
$279$		−	7.07872i		−	0.423792i
$280$		0			0
$281$			3.31382i			0.197686i	0.995103	+	0.0988431i	$0.0315142\pi$
$281$			3.31382i			0.197686i	−0.995103	+	0.0988431i	$0.968486\pi$
$282$		0			0
$283$	−18.0808			−1.07479			−0.537395	−	0.843330i	$-0.680592\pi$
$283$	−18.0808			−1.07479			−0.537395	+	0.843330i	$0.680592\pi$
$284$		0			0
$285$	6.12732			0.362951
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−10.2523			−0.603079
$290$		0			0
$291$			9.90578i			0.580687i
$292$		0			0
$293$			22.9860i			1.34286i	0.741070	+	0.671428i	$0.234319\pi$
$293$			22.9860i			1.34286i	−0.741070	+	0.671428i	$0.765681\pi$
$294$		0			0
$295$	10.6717			0.621332
$296$		0			0
$297$			1.53941i			0.0893258i
$298$		0			0
$299$	7.03929	−	0.902413i	0.407093	−	0.0521879i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−9.98961			−0.573888
$304$		0			0
$305$		−	9.70853i		−	0.555909i
$306$		0			0
$307$		−	2.75982i		−	0.157511i	−0.996894	−	0.0787556i	$-0.974905\pi$
$307$		−	2.75982i		−	0.157511i	0.996894	−	0.0787556i	$-0.0250947\pi$
$308$		0			0
$309$	19.5720			1.11341
$310$		0			0
$311$	−22.6438			−1.28401			−0.642007	−	0.766698i	$-0.721898\pi$
$311$	−22.6438			−1.28401			−0.642007	+	0.766698i	$0.721898\pi$
$312$		0			0
$313$	3.18533			0.180045			0.0900227	−	0.995940i	$-0.471306\pi$
$313$	3.18533			0.180045			0.0900227	+	0.995940i	$0.471306\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			14.9006i			0.836903i	0.908239	+	0.418451i	$0.137427\pi$
$317$			14.9006i			0.836903i	−0.908239	+	0.418451i	$0.862573\pi$
$318$		0			0
$319$			5.59316i			0.313157i
$320$		0			0
$321$	6.86479			0.383155
$322$		0			0
$323$		−	15.4121i		−	0.857551i
$324$		0			0
$325$	−14.0672	+	1.80337i	−0.780310	+	0.100033i
$326$		0			0
$327$			15.5038i			0.857365i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			0.223213i			0.0122689i	0.999981	+	0.00613444i	$0.00195267\pi$
$331$			0.223213i			0.0122689i	−0.999981	+	0.00613444i	$0.998047\pi$
$332$		0			0
$333$		−	4.06240i		−	0.222618i
$334$		0			0
$335$	−16.0181			−0.875163
$336$		0			0
$337$	28.4641			1.55054			0.775269	−	0.631631i	$-0.217614\pi$
$337$	28.4641			1.55054			0.775269	+	0.631631i	$0.217614\pi$
$338$		0			0
$339$	0.556572			0.0302289
$340$		0			0
$341$	10.8971			0.590110
$342$		0			0
$343$		0			0
$344$		0			0
$345$			2.03274i			0.109439i
$346$		0			0
$347$	6.22869			0.334373			0.167187	−	0.985925i	$-0.446532\pi$
$347$	6.22869			0.334373			0.167187	+	0.985925i	$0.446532\pi$
$348$		0			0
$349$		−	5.41696i		−	0.289963i	−0.989434	−	0.144981i	$-0.953688\pi$
$349$		−	5.41696i		−	0.289963i	0.989434	−	0.144981i	$-0.0463123\pi$
$350$		0			0
$351$	−3.57628	+	0.458467i	−0.190888	+	0.0244712i
$352$		0			0
$353$		−	5.96971i		−	0.317736i	−0.987300	−	0.158868i	$-0.949216\pi$
$353$		−	5.96971i		−	0.317736i	0.987300	−	0.158868i	$-0.0507844\pi$
$354$		0			0
$355$	6.59163			0.349848
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	13.4673i		−	0.710776i	−0.934719	−	0.355388i	$-0.884349\pi$
$359$		−	13.4673i		−	0.710776i	0.934719	−	0.355388i	$-0.115651\pi$
$360$		0			0
$361$	−16.2022			−0.852749
$362$		0			0
$363$	8.63021			0.452968
$364$		0			0
$365$	8.61340			0.450846
$366$		0			0
$367$	15.3952			0.803623			0.401812	−	0.915722i	$-0.368381\pi$
$367$	15.3952			0.803623			0.401812	+	0.915722i	$0.368381\pi$
$368$		0			0
$369$		−	4.76561i		−	0.248088i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	28.9376			1.49833			0.749165	−	0.662383i	$-0.230455\pi$
$373$	28.9376			1.49833			0.749165	+	0.662383i	$0.230455\pi$
$374$		0			0
$375$		−	9.22583i		−	0.476420i
$376$		0			0
$377$	−12.9937	+	1.66575i	−0.669211	+	0.0857906i
$378$		0			0
$379$			1.43062i			0.0734860i	0.999325	+	0.0367430i	$0.0116983\pi$
$379$			1.43062i			0.0734860i	−0.999325	+	0.0367430i	$0.988302\pi$
$380$		0			0
$381$	5.78315			0.296280
$382$		0			0
$383$			10.4343i			0.533168i	0.963812	+	0.266584i	$0.0858950\pi$
$383$			10.4343i			0.533168i	−0.963812	+	0.266584i	$0.914105\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.849674			0.0431913
$388$		0			0
$389$	16.7064			0.847047			0.423524	−	0.905885i	$-0.360793\pi$
$389$	16.7064			0.847047			0.423524	+	0.905885i	$0.360793\pi$
$390$		0			0
$391$	5.11297			0.258574
$392$		0			0
$393$	19.0922			0.963077
$394$		0			0
$395$		−	0.869445i		−	0.0437465i
$396$		0			0
$397$		−	20.8252i		−	1.04519i	−0.852582	−	0.522594i	$-0.824964\pi$
$397$		−	20.8252i		−	1.04519i	0.852582	−	0.522594i	$-0.175036\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	20.4129i		−	1.01937i	−0.860361	−	0.509685i	$-0.829762\pi$
$401$		−	20.4129i		−	1.01937i	0.860361	−	0.509685i	$-0.170238\pi$
$402$		0			0
$403$	3.24536	+	25.3155i	0.161663	+	1.26106i
$404$		0			0
$405$		−	1.03273i		−	0.0513166i
$406$		0			0
$407$	6.25372			0.309985
$408$		0			0
$409$			15.2757i			0.755334i	0.925942	+	0.377667i	$0.123274\pi$
$409$			15.2757i			0.755334i	−0.925942	+	0.377667i	$0.876726\pi$
$410$		0			0
$411$			8.75302i			0.431755i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	16.2263			0.796519
$416$		0			0
$417$	−5.26687			−0.257920
$418$		0			0
$419$	−30.8019			−1.50477			−0.752386	−	0.658723i	$-0.771097\pi$
$419$	−30.8019			−1.50477			−0.752386	+	0.658723i	$0.771097\pi$
$420$		0			0
$421$			25.6176i			1.24853i	0.781215	+	0.624263i	$0.214601\pi$
$421$			25.6176i			1.24853i	−0.781215	+	0.624263i	$0.785399\pi$
$422$		0			0
$423$		−	11.6987i		−	0.568811i
$424$		0			0
$425$	−10.2177			−0.495631
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−0.705771	−	5.50538i	−0.0340749	−	0.265802i
$430$		0			0
$431$		−	4.98233i		−	0.239991i	−0.992774	−	0.119995i	$-0.961712\pi$
$431$		−	4.98233i		−	0.239991i	0.992774	−	0.119995i	$-0.0382879\pi$
$432$		0			0
$433$	4.59730			0.220932			0.110466	−	0.993880i	$-0.464766\pi$
$433$	4.59730			0.220932			0.110466	+	0.993880i	$0.464766\pi$
$434$		0			0
$435$		−	3.75221i		−	0.179905i
$436$		0			0
$437$		−	11.6784i		−	0.558652i
$438$		0			0
$439$	32.8118			1.56602			0.783011	−	0.622008i	$-0.213683\pi$
$439$	32.8118			1.56602			0.783011	+	0.622008i	$0.213683\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	41.0763			1.95159			0.975797	−	0.218680i	$-0.0701751\pi$
$443$	41.0763			1.95159			0.975797	+	0.218680i	$0.0701751\pi$
$444$		0			0
$445$	17.4320			0.826357
$446$		0			0
$447$		−	1.49151i		−	0.0705459i
$448$		0			0
$449$		−	5.29494i		−	0.249884i	−0.992164	−	0.124942i	$-0.960126\pi$
$449$		−	5.29494i		−	0.249884i	0.992164	−	0.124942i	$-0.0398744\pi$
$450$		0			0
$451$	7.33624			0.345450
$452$		0			0
$453$			8.35531i			0.392566i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			8.50567i			0.397879i	0.980012	+	0.198939i	$0.0637497\pi$
$457$			8.50567i			0.397879i	−0.980012	+	0.198939i	$0.936250\pi$
$458$		0			0
$459$	−2.59762			−0.121247
$460$		0			0
$461$		−	27.3100i		−	1.27195i	−0.771708	−	0.635977i	$-0.780597\pi$
$461$		−	27.3100i		−	1.27195i	0.771708	−	0.635977i	$-0.219403\pi$
$462$		0			0
$463$		−	15.3373i		−	0.712783i	−0.934337	−	0.356392i	$-0.884007\pi$
$463$		−	15.3373i		−	0.712783i	0.934337	−	0.356392i	$-0.115993\pi$
$464$		0			0
$465$	−7.31038			−0.339011
$466$		0			0
$467$	6.20400			0.287087			0.143543	−	0.989644i	$-0.454150\pi$
$467$	6.20400			0.287087			0.143543	+	0.989644i	$0.454150\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	7.10455			0.327360
$472$		0			0
$473$			1.30800i			0.0601418i
$474$		0			0
$475$			23.3379i			1.07082i
$476$		0			0
$477$	10.9201			0.499999
$478$		0			0
$479$			28.5211i			1.30316i	0.758579	+	0.651582i	$0.225894\pi$
$479$			28.5211i			1.30316i	−0.758579	+	0.651582i	$0.774106\pi$
$480$		0			0
$481$	1.86248	+	14.5283i	0.0849218	+	0.662434i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	10.2300			0.464518
$486$		0			0
$487$		−	21.6837i		−	0.982584i	−0.870995	−	0.491292i	$-0.836525\pi$
$487$		−	21.6837i		−	0.982584i	0.870995	−	0.491292i	$-0.163475\pi$
$488$		0			0
$489$			12.5870i			0.569205i
$490$		0			0
$491$	−22.7589			−1.02709			−0.513547	−	0.858061i	$-0.671669\pi$
$491$	−22.7589			−1.02709			−0.513547	+	0.858061i	$0.671669\pi$
$492$		0			0
$493$	−9.43796			−0.425064
$494$		0			0
$495$	1.58979			0.0714558
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	5.56425i		−	0.249090i	−0.992214	−	0.124545i	$-0.960253\pi$
$499$		−	5.56425i		−	0.249090i	0.992214	−	0.124545i	$-0.0397471\pi$
$500$		0			0
$501$		−	19.1612i		−	0.856057i
$502$		0			0
$503$	−24.1810			−1.07818			−0.539089	−	0.842249i	$-0.681231\pi$
$503$	−24.1810			−1.07818			−0.539089	+	0.842249i	$0.681231\pi$
$504$		0			0
$505$			10.3165i			0.459080i
$506$		0			0
$507$	12.5796	−	3.27922i	0.558680	−	0.145635i
$508$		0			0
$509$		−	26.4547i		−	1.17258i	−0.810100	−	0.586292i	$-0.800587\pi$
$509$		−	26.4547i		−	1.17258i	0.810100	−	0.586292i	$-0.199413\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$			5.93315i			0.261955i
$514$		0			0
$515$		−	20.2125i		−	0.890670i
$516$		0			0
$517$	18.0092			0.792042
$518$		0			0
$519$	9.66620			0.424299
$520$		0			0
$521$	−36.5080			−1.59945			−0.799723	−	0.600369i	$-0.795020\pi$
$521$	−36.5080			−1.59945			−0.799723	+	0.600369i	$0.795020\pi$
$522$		0			0
$523$	−8.26328			−0.361328			−0.180664	−	0.983545i	$-0.557825\pi$
$523$	−8.26328			−0.361328			−0.180664	+	0.983545i	$0.557825\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			18.3879i			0.800988i
$528$		0			0
$529$	−19.1257			−0.831552
$530$		0			0
$531$			10.3335i			0.448438i
$532$		0			0
$533$	2.18488	+	17.0432i	0.0946375	+	0.738221i
$534$		0			0
$535$		−	7.08944i		−	0.306503i
$536$		0			0
$537$	0.694977			0.0299905
$538$		0			0
$539$		0			0
$540$		0			0
$541$			2.56309i			0.110196i	0.998481	+	0.0550980i	$0.0175471\pi$
$541$			2.56309i			0.110196i	−0.998481	+	0.0550980i	$0.982453\pi$
$542$		0			0
$543$	−7.81108			−0.335206
$544$		0			0
$545$	16.0112			0.685846
$546$		0			0
$547$	5.54538			0.237103			0.118552	−	0.992948i	$-0.462175\pi$
$547$	5.54538			0.237103			0.118552	+	0.992948i	$0.462175\pi$
$548$		0			0
$549$	9.40087			0.401220
$550$		0			0
$551$			21.5569i			0.918356i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−4.19535			−0.178083
$556$		0			0
$557$			39.2095i			1.66136i	0.556749	+	0.830681i	$0.312049\pi$
$557$			39.2095i			1.66136i	−0.556749	+	0.830681i	$0.687951\pi$
$558$		0			0
$559$	−3.03867	+	0.389548i	−0.128522	+	0.0164761i
$560$		0			0
$561$		−	3.99882i		−	0.168830i
$562$		0			0
$563$	−25.4640			−1.07318			−0.536590	−	0.843843i	$-0.680288\pi$
$563$	−25.4640			−1.07318			−0.536590	+	0.843843i	$0.680288\pi$
$564$		0			0
$565$		−	0.574787i		−	0.0241815i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−14.7708			−0.619224			−0.309612	−	0.950863i	$-0.600199\pi$
$569$	−14.7708			−0.619224			−0.309612	+	0.950863i	$0.600199\pi$
$570$		0			0
$571$	−22.8985			−0.958271			−0.479135	−	0.877741i	$-0.659050\pi$
$571$	−22.8985			−0.958271			−0.479135	+	0.877741i	$0.659050\pi$
$572$		0			0
$573$	24.5453			1.02540
$574$		0			0
$575$	−7.74236			−0.322879
$576$		0			0
$577$			3.38812i			0.141049i	0.997510	+	0.0705247i	$0.0224673\pi$
$577$			3.38812i			0.141049i	−0.997510	+	0.0705247i	$0.977533\pi$
$578$		0			0
$579$		−	20.9887i		−	0.872262i
$580$		0			0
$581$		0			0
$582$		0			0
$583$			16.8106i			0.696224i
$584$		0			0
$585$	0.473471	+	3.69332i	0.0195756	+	0.152700i
$586$		0			0
$587$		−	24.6516i		−	1.01748i	−0.860920	−	0.508741i	$-0.830111\pi$
$587$		−	24.6516i		−	1.01748i	0.860920	−	0.508741i	$-0.169889\pi$
$588$		0			0
$589$	41.9991			1.73054
$590$		0			0
$591$			2.07654i			0.0854175i
$592$		0			0
$593$		−	16.8200i		−	0.690714i	−0.938471	−	0.345357i	$-0.887758\pi$
$593$		−	16.8200i		−	0.690714i	0.938471	−	0.345357i	$-0.112242\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	6.29595			0.257676
$598$		0			0
$599$	−36.3058			−1.48342			−0.741708	−	0.670723i	$-0.765984\pi$
$599$	−36.3058			−1.48342			−0.741708	+	0.670723i	$0.765984\pi$
$600$		0			0
$601$	−10.5378			−0.429848			−0.214924	−	0.976631i	$-0.568950\pi$
$601$	−10.5378			−0.429848			−0.214924	+	0.976631i	$0.568950\pi$
$602$		0			0
$603$		−	15.5105i		−	0.631637i
$604$		0			0
$605$		−	8.91264i		−	0.362350i
$606$		0			0
$607$	21.3240			0.865515			0.432758	−	0.901510i	$-0.357541\pi$
$607$	21.3240			0.865515			0.432758	+	0.901510i	$0.357541\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.36348	+	41.8379i	0.216983	+	1.69258i
$612$		0			0
$613$			10.3189i			0.416775i	0.978046	+	0.208387i	$0.0668214\pi$
$613$			10.3189i			0.416775i	−0.978046	+	0.208387i	$0.933179\pi$
$614$		0			0
$615$	−4.92157			−0.198457
$616$		0			0
$617$		−	33.6554i		−	1.35492i	−0.735561	−	0.677458i	$-0.763081\pi$
$617$		−	33.6554i		−	1.35492i	0.735561	−	0.677458i	$-0.236919\pi$
$618$		0			0
$619$		−	33.6509i		−	1.35254i	−0.736652	−	0.676272i	$-0.763594\pi$
$619$		−	33.6509i		−	1.35254i	0.736652	−	0.676272i	$-0.236406\pi$
$620$		0			0
$621$	−1.96832			−0.0789861
$622$		0			0
$623$		0			0
$624$		0			0
$625$	10.1396			0.405585
$626$		0			0
$627$	−9.13356			−0.364759
$628$		0			0
$629$			10.5526i			0.420760i
$630$		0			0
$631$		−	8.16711i		−	0.325128i	−0.986698	−	0.162564i	$-0.948024\pi$
$631$		−	8.16711i		−	0.325128i	0.986698	−	0.162564i	$-0.0519763\pi$
$632$		0			0
$633$	13.9093			0.552845
$634$		0			0
$635$		−	5.97241i		−	0.237008i
$636$		0			0
$637$		0			0
$638$		0			0
$639$			6.38275i			0.252498i
$640$		0			0
$641$	18.5538			0.732831			0.366416	−	0.930451i	$-0.380585\pi$
$641$	18.5538			0.732831			0.366416	+	0.930451i	$0.380585\pi$
$642$		0			0
$643$		−	42.1637i		−	1.66277i	−0.555694	−	0.831387i	$-0.687547\pi$
$643$		−	42.1637i		−	1.66277i	0.555694	−	0.831387i	$-0.312453\pi$
$644$		0			0
$645$		−	0.877480i		−	0.0345507i
$646$		0			0
$647$	−16.3130			−0.641329			−0.320665	−	0.947193i	$-0.603906\pi$
$647$	−16.3130			−0.641329			−0.320665	+	0.947193i	$0.603906\pi$
$648$		0			0
$649$	−15.9076			−0.624428
$650$		0			0
$651$		0			0
$652$		0			0
$653$	11.0675			0.433103			0.216551	−	0.976271i	$-0.430519\pi$
$653$	11.0675			0.433103			0.216551	+	0.976271i	$0.430519\pi$
$654$		0			0
$655$		−	19.7171i		−	0.770409i
$656$		0			0
$657$			8.34045i			0.325392i
$658$		0			0
$659$	−20.4824			−0.797882			−0.398941	−	0.916977i	$-0.630622\pi$
$659$	−20.4824			−0.797882			−0.398941	+	0.916977i	$0.630622\pi$
$660$		0			0
$661$			1.61115i			0.0626665i	0.999509	+	0.0313333i	$0.00997532\pi$
$661$			1.61115i			0.0626665i	−0.999509	+	0.0313333i	$0.990025\pi$
$662$		0			0
$663$	9.28984	−	1.19093i	0.360788	−	0.0462517i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−7.15152			−0.276908
$668$		0			0
$669$		−	1.99158i		−	0.0769987i
$670$		0			0
$671$			14.4718i			0.558679i
$672$		0			0
$673$	26.8874			1.03643			0.518217	−	0.855249i	$-0.326596\pi$
$673$	26.8874			1.03643			0.518217	+	0.855249i	$0.326596\pi$
$674$		0			0
$675$	3.93348			0.151400
$676$		0			0
$677$	13.3811			0.514277			0.257138	−	0.966375i	$-0.417220\pi$
$677$	13.3811			0.514277			0.257138	+	0.966375i	$0.417220\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		−	16.1480i		−	0.618793i
$682$		0			0
$683$			37.0003i			1.41578i	0.706325	+	0.707888i	$0.250352\pi$
$683$			37.0003i			1.41578i	−0.706325	+	0.707888i	$0.749648\pi$
$684$		0			0
$685$	9.03948			0.345381
$686$		0			0
$687$			23.0670i			0.880060i
$688$		0			0
$689$	−39.0535	+	5.00653i	−1.48782	+	0.190733i
$690$		0			0
$691$		−	0.284680i		−	0.0108297i	−0.999985	−	0.00541487i	$-0.998276\pi$
$691$		−	0.284680i		−	0.0108297i	0.999985	−	0.00541487i	$-0.00172361\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			5.43923i			0.206322i
$696$		0			0
$697$			12.3793i			0.468898i
$698$		0			0
$699$	1.26648			0.0479028
$700$		0			0
$701$	−39.2818			−1.48365			−0.741826	−	0.670592i	$-0.766040\pi$
$701$	−39.2818			−1.48365			−0.741826	+	0.670592i	$0.766040\pi$
$702$		0			0
$703$	24.1028			0.909056
$704$		0			0
$705$	−12.0816			−0.455018
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	48.8608i		−	1.83500i	−0.397731	−	0.917502i	$-0.630202\pi$
$709$		−	48.8608i		−	1.83500i	0.397731	−	0.917502i	$-0.369798\pi$
$710$		0			0
$711$	0.841893			0.0315735
$712$		0			0
$713$			13.9332i			0.521803i
$714$		0			0
$715$	−5.68555	+	0.728868i	−0.212627	+	0.0272581i
$716$		0			0
$717$		−	7.46395i		−	0.278746i
$718$		0			0
$719$	−33.7989			−1.26049			−0.630243	−	0.776398i	$-0.717045\pi$
$719$	−33.7989			−1.26049			−0.630243	+	0.776398i	$0.717045\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	25.2319i		−	0.938385i
$724$		0			0
$725$	14.2915			0.530774
$726$		0			0
$727$	−35.4765			−1.31575			−0.657875	−	0.753127i	$-0.728545\pi$
$727$	−35.4765			−1.31575			−0.657875	+	0.753127i	$0.728545\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−2.20713			−0.0816337
$732$		0			0
$733$			11.6407i			0.429958i	0.976619	+	0.214979i	$0.0689683\pi$
$733$			11.6407i			0.429958i	−0.976619	+	0.214979i	$0.931032\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	23.8771			0.879524
$738$		0			0
$739$		−	4.59842i		−	0.169156i	−0.996417	−	0.0845779i	$-0.973046\pi$
$739$		−	4.59842i		−	0.169156i	0.996417	−	0.0845779i	$-0.0269542\pi$
$740$		0			0
$741$	−2.72015	−	21.2186i	−0.0999274	−	0.779485i
$742$		0			0
$743$			35.6187i			1.30672i	0.757045	+	0.653362i	$0.226642\pi$
$743$			35.6187i			1.30672i	−0.757045	+	0.653362i	$0.773358\pi$
$744$		0			0
$745$	−1.54032			−0.0564329
$746$		0			0
$747$			15.7121i			0.574877i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	13.0712			0.476974			0.238487	−	0.971146i	$-0.423349\pi$
$751$	13.0712			0.476974			0.238487	+	0.971146i	$0.423349\pi$
$752$		0			0
$753$	−25.4021			−0.925704
$754$		0			0
$755$	8.62874			0.314032
$756$		0			0
$757$	49.9763			1.81642			0.908210	−	0.418515i	$-0.137449\pi$
$757$	49.9763			1.81642			0.908210	+	0.418515i	$0.137449\pi$
$758$		0			0
$759$		−	3.03006i		−	0.109984i
$760$		0			0
$761$			51.8461i			1.87942i	0.341973	+	0.939710i	$0.388905\pi$
$761$			51.8461i			1.87942i	−0.341973	+	0.939710i	$0.611095\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$			2.68263i			0.0969908i
$766$		0			0
$767$	−4.73760	−	36.9557i	−0.171065	−	1.33439i
$768$		0			0
$769$			13.8806i			0.500546i	0.968175	+	0.250273i	$0.0805203\pi$
$769$			13.8806i			0.500546i	−0.968175	+	0.250273i	$0.919480\pi$
$770$		0			0
$771$	9.29097			0.334606
$772$		0			0
$773$			43.0185i			1.54727i	0.633633	+	0.773633i	$0.281563\pi$
$773$			43.0185i			1.54727i	−0.633633	+	0.773633i	$0.718437\pi$
$774$		0			0
$775$		−	27.8440i		−	1.00019i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	28.2751			1.01306
$780$		0			0
$781$	−9.82569			−0.351591
$782$		0			0
$783$	3.63330			0.129844
$784$		0			0
$785$		−	7.33705i		−	0.261871i
$786$		0			0
$787$			46.1860i			1.64635i	0.567785	+	0.823177i	$0.307800\pi$
$787$			46.1860i			1.64635i	−0.567785	+	0.823177i	$0.692200\pi$
$788$		0			0
$789$	−7.53559			−0.268274
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−33.6202	+	4.30999i	−1.19389	+	0.153052i
$794$		0			0
$795$		−	11.2775i		−	0.399972i
$796$		0			0
$797$	−44.2564			−1.56764			−0.783821	−	0.620987i	$-0.786732\pi$
$797$	−44.2564			−1.56764			−0.783821	+	0.620987i	$0.786732\pi$
$798$		0			0
$799$			30.3889i			1.07508i
$800$		0			0
$801$			16.8796i			0.596412i
$802$		0			0
$803$	−12.8394			−0.453093
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−16.4290			−0.578327
$808$		0			0
$809$	40.5315			1.42501			0.712506	−	0.701666i	$-0.247560\pi$
$809$	40.5315			1.42501			0.712506	+	0.701666i	$0.247560\pi$
$810$		0			0
$811$			21.6526i			0.760324i	0.924920	+	0.380162i	$0.124132\pi$
$811$			21.6526i			0.760324i	−0.924920	+	0.380162i	$0.875868\pi$
$812$		0			0
$813$		−	3.37238i		−	0.118274i
$814$		0			0
$815$	12.9989			0.455333
$816$		0			0
$817$			5.04124i			0.176371i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			9.40284i			0.328161i	0.986447	+	0.164081i	$0.0524657\pi$
$821$			9.40284i			0.328161i	−0.986447	+	0.164081i	$0.947534\pi$
$822$		0			0
$823$	−27.8277			−0.970014			−0.485007	−	0.874510i	$-0.661183\pi$
$823$	−27.8277			−0.970014			−0.485007	+	0.874510i	$0.661183\pi$
$824$		0			0
$825$			6.05524i			0.210817i
$826$		0			0
$827$			43.0432i			1.49676i	0.663270	+	0.748380i	$0.269168\pi$
$827$			43.0432i			1.49676i	−0.663270	+	0.748380i	$0.730832\pi$
$828$		0			0
$829$	8.09336			0.281094			0.140547	−	0.990074i	$-0.455114\pi$
$829$	8.09336			0.281094			0.140547	+	0.990074i	$0.455114\pi$
$830$		0			0
$831$	18.1400			0.629268
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−19.7882			−0.684800
$836$		0			0
$837$		−	7.07872i		−	0.244676i
$838$		0			0
$839$		−	21.3057i		−	0.735556i	−0.929914	−	0.367778i	$-0.880119\pi$
$839$		−	21.3057i		−	0.735556i	0.929914	−	0.367778i	$-0.119881\pi$
$840$		0			0
$841$	−15.7991			−0.544796
$842$		0			0
$843$			3.31382i			0.114134i
$844$		0			0
$845$	−3.38654	−	12.9913i	−0.116500	−	0.446914i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−18.0808			−0.620531
$850$		0			0
$851$			7.99613i			0.274104i
$852$		0			0
$853$			20.9324i			0.716713i	0.933585	+	0.358356i	$0.116663\pi$
$853$			20.9324i			0.716713i	−0.933585	+	0.358356i	$0.883337\pi$
$854$		0			0
$855$	6.12732			0.209550
$856$		0			0
$857$	−26.1151			−0.892073			−0.446037	−	0.895015i	$-0.647165\pi$
$857$	−26.1151			−0.892073			−0.446037	+	0.895015i	$0.647165\pi$
$858$		0			0
$859$	38.7926			1.32359			0.661793	−	0.749687i	$-0.269796\pi$
$859$	38.7926			1.32359			0.661793	+	0.749687i	$0.269796\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	28.6994i		−	0.976940i	−0.872581	−	0.488470i	$-0.837555\pi$
$863$		−	28.6994i		−	0.976940i	0.872581	−	0.488470i	$-0.162445\pi$
$864$		0			0
$865$		−	9.98253i		−	0.339416i
$866$		0			0
$867$	−10.2523			−0.348188
$868$		0			0
$869$			1.29602i			0.0439645i
$870$		0			0
$871$	7.11107	+	55.4700i	0.240949	+	1.87953i
$872$		0			0
$873$			9.90578i			0.335260i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			35.5972i			1.20203i	0.799238	+	0.601015i	$0.205237\pi$
$877$			35.5972i			1.20203i	−0.799238	+	0.601015i	$0.794763\pi$
$878$		0			0
$879$			22.9860i			0.775298i
$880$		0			0
$881$	15.6969			0.528840			0.264420	−	0.964408i	$-0.414819\pi$
$881$	15.6969			0.528840			0.264420	+	0.964408i	$0.414819\pi$
$882$		0			0
$883$	−26.4371			−0.889681			−0.444840	−	0.895610i	$-0.646740\pi$
$883$	−26.4371			−0.889681			−0.444840	+	0.895610i	$0.646740\pi$
$884$		0			0
$885$	10.6717			0.358726
$886$		0			0
$887$	−31.3875			−1.05389			−0.526944	−	0.849900i	$-0.676662\pi$
$887$	−31.3875			−1.05389			−0.526944	+	0.849900i	$0.676662\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$			1.53941i			0.0515723i
$892$		0			0
$893$	69.4102			2.32272
$894$		0			0
$895$		−	0.717721i		−	0.0239908i
$896$		0			0
$897$	7.03929	−	0.902413i	0.235035	−	0.0301307i
$898$		0			0
$899$		−	25.7192i		−	0.857782i
$900$		0			0
$901$	−28.3664			−0.945022
$902$		0			0
$903$		0			0
$904$		0			0
$905$			8.06671i			0.268146i
$906$		0			0
$907$	−9.77516			−0.324579			−0.162289	−	0.986743i	$-0.551888\pi$
$907$	−9.77516			−0.324579			−0.162289	+	0.986743i	$0.551888\pi$
$908$		0			0
$909$	−9.98961			−0.331334
$910$		0			0
$911$	−55.2193			−1.82950			−0.914749	−	0.404024i	$-0.867611\pi$
$911$	−55.2193			−1.82950			−0.914749	+	0.404024i	$0.867611\pi$
$912$		0			0
$913$	−24.1875			−0.800488
$914$		0			0
$915$		−	9.70853i		−	0.320954i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−3.40993			−0.112483			−0.0562415	−	0.998417i	$-0.517912\pi$
$919$	−3.40993			−0.112483			−0.0562415	+	0.998417i	$0.517912\pi$
$920$		0			0
$921$		−	2.75982i		−	0.0909391i
$922$		0			0
$923$	−2.92628	−	22.8265i	−0.0963198	−	0.751345i
$924$		0			0
$925$		−	15.9794i		−	0.525398i
$926$		0			0
$927$	19.5720			0.642829
$928$		0			0
$929$		−	23.9446i		−	0.785598i	−0.919624	−	0.392799i	$-0.871507\pi$
$929$		−	23.9446i		−	0.785598i	0.919624	−	0.392799i	$-0.128493\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−22.6438			−0.741326
$934$		0			0
$935$	−4.12968			−0.135055
$936$		0			0
$937$	50.4540			1.64826			0.824130	−	0.566400i	$-0.191664\pi$
$937$	50.4540			1.64826			0.824130	+	0.566400i	$0.191664\pi$
$938$		0			0
$939$	3.18533			0.103949
$940$		0			0
$941$		−	31.5937i		−	1.02992i	−0.857213	−	0.514962i	$-0.827806\pi$
$941$		−	31.5937i		−	1.02992i	0.857213	−	0.514962i	$-0.172194\pi$
$942$		0			0
$943$			9.38026i			0.305463i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	42.9608i		−	1.39604i	−0.716079	−	0.698020i	$-0.754065\pi$
$947$		−	42.9608i		−	1.39604i	0.716079	−	0.698020i	$-0.245935\pi$
$948$		0			0
$949$	−3.82383	−	29.8278i	−0.124127	−	0.968252i
$950$		0			0
$951$			14.9006i			0.483186i
$952$		0			0
$953$	6.86388			0.222343			0.111171	−	0.993801i	$-0.464540\pi$
$953$	6.86388			0.222343			0.111171	+	0.993801i	$0.464540\pi$
$954$		0			0
$955$		−	25.3486i		−	0.820261i
$956$		0			0
$957$			5.59316i			0.180801i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−19.1083			−0.616397
$962$		0			0
$963$	6.86479			0.221215
$964$		0			0
$965$	−21.6756			−0.697763
$966$		0			0
$967$		−	13.3204i		−	0.428356i	−0.976795	−	0.214178i	$-0.931293\pi$
$967$		−	13.3204i		−	0.428356i	0.976795	−	0.214178i	$-0.0687073\pi$
$968$		0			0
$969$		−	15.4121i		−	0.495108i
$970$		0			0
$971$	23.5330			0.755210			0.377605	−	0.925967i	$-0.376748\pi$
$971$	23.5330			0.755210			0.377605	+	0.925967i	$0.376748\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−14.0672	+	1.80337i	−0.450512	+	0.0577541i
$976$		0			0
$977$		−	54.5764i		−	1.74605i	−0.487672	−	0.873027i	$-0.662154\pi$
$977$		−	54.5764i		−	1.74605i	0.487672	−	0.873027i	$-0.337846\pi$
$978$		0			0
$979$	−25.9847			−0.830474
$980$		0			0
$981$			15.5038i			0.495000i
$982$		0			0
$983$		−	23.4645i		−	0.748403i	−0.927347	−	0.374201i	$-0.877917\pi$
$983$		−	23.4645i		−	0.748403i	0.927347	−	0.374201i	$-0.122083\pi$
$984$		0			0
$985$	2.14450			0.0683294
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.67243			−0.0531803
$990$		0			0
$991$	2.83495			0.0900551			0.0450275	−	0.998986i	$-0.485662\pi$
$991$	2.83495			0.0900551			0.0450275	+	0.998986i	$0.485662\pi$
$992$		0			0
$993$			0.223213i			0.00708344i
$994$		0			0
$995$		−	6.50199i		−	0.206127i
$996$		0			0
$997$	50.1515			1.58832			0.794158	−	0.607712i	$-0.207912\pi$
$997$	50.1515			1.58832			0.794158	+	0.607712i	$0.207912\pi$
$998$		0			0
$999$		−	4.06240i		−	0.128529i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7644.2.e.q.4705.7	yes	16
7.6	odd	2		7644.2.e.p.4705.10	yes	16
13.12	even	2	inner	7644.2.e.q.4705.10	yes	16
91.90	odd	2		7644.2.e.p.4705.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7644.2.e.p.4705.7	✓	16	91.90	odd	2
7644.2.e.p.4705.10	yes	16	7.6	odd	2
7644.2.e.q.4705.7	yes	16	1.1	even	1	trivial
7644.2.e.q.4705.10	yes	16	13.12	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.03273i q^{5} +1.00000 q^{9} +1.53941i q^{11} +(-3.57628 + 0.458467i) q^{13} -1.03273i q^{15} -2.59762 q^{17} +5.93315i q^{19} -1.96832 q^{23} +3.93348 q^{25} +1.00000 q^{27} +3.63330 q^{29} -7.07872i q^{31} +1.53941i q^{33} -4.06240i q^{37} +(-3.57628 + 0.458467i) q^{39} -4.76561i q^{41} +0.849674 q^{43} -1.03273i q^{45} -11.6987i q^{47} -2.59762 q^{51} +10.9201 q^{53} +1.58979 q^{55} +5.93315i q^{57} +10.3335i q^{59} +9.40087 q^{61} +(0.473471 + 3.69332i) q^{65} -15.5105i q^{67} -1.96832 q^{69} +6.38275i q^{71} +8.34045i q^{73} +3.93348 q^{75} +0.841893 q^{79} +1.00000 q^{81} +15.7121i q^{83} +2.68263i q^{85} +3.63330 q^{87} +16.8796i q^{89} -7.07872i q^{93} +6.12732 q^{95} +9.90578i q^{97} +1.53941i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{3} + 16 q^{9} - 4 q^{23} - 12 q^{25} + 16 q^{27} - 4 q^{29} - 12 q^{43} - 12 q^{53} + 16 q^{61} - 24 q^{65} - 4 q^{69} - 12 q^{75} - 20 q^{79} + 16 q^{81} - 4 q^{87} + 12 q^{95}+O(q^{100}) \) 16 * q + 16 * q^3 + 16 * q^9 - 4 * q^23 - 12 * q^25 + 16 * q^27 - 4 * q^29 - 12 * q^43 - 12 * q^53 + 16 * q^61 - 24 * q^65 - 4 * q^69 - 12 * q^75 - 20 * q^79 + 16 * q^81 - 4 * q^87 + 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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