LMFDB - Embedded newform 7644.2.e.o.4705.9

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			4705.9
Root			$3.99442i$ of defining polynomial
Character	$\chi$	$=$	7644.4705
Dual form			7644.2.e.o.4705.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.99442i q^{5} +1.00000 q^{9} +5.96039i q^{11} +(3.53216 - 0.723785i) q^{13} +3.99442i q^{15} -2.84934 q^{17} +8.14855i q^{19} -5.21498 q^{23} -10.9554 q^{25} +1.00000 q^{27} -6.35580 q^{29} -2.72746i q^{31} +5.96039i q^{33} +1.80626i q^{37} +(3.53216 - 0.723785i) q^{39} -10.6409i q^{41} +1.84934 q^{43} +3.99442i q^{45} +6.88159i q^{47} -2.84934 q^{51} +8.83033 q^{53} -23.8083 q^{55} +8.14855i q^{57} +2.88716i q^{59} -11.2793 q^{61} +(2.89110 + 14.1089i) q^{65} +0.900292i q^{67} -5.21498 q^{69} -2.56776i q^{71} -13.5696i q^{73} -10.9554 q^{75} +2.61536 q^{79} +1.00000 q^{81} +0.740592i q^{83} -11.3815i q^{85} -6.35580 q^{87} -0.921202i q^{89} -2.72746i q^{93} -32.5487 q^{95} -5.09624i q^{97} +5.96039i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$			3.99442i			1.78636i	0.449699	+	0.893180i	$0.351531\pi$
$5$			3.99442i			1.78636i	−0.449699	+	0.893180i	$0.648469\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			5.96039i			1.79712i	0.438847	+	0.898562i	$0.355387\pi$
$11$			5.96039i			1.79712i	−0.438847	+	0.898562i	$0.644613\pi$
$12$		0			0
$13$	3.53216	−	0.723785i	0.979644	−	0.200742i
$13$	3.53216	−	0.723785i	0.979644	−	0.200742i
$14$		0			0
$15$			3.99442i			1.03136i
$16$		0			0
$17$	−2.84934			−0.691066			−0.345533	−	0.938407i	$-0.612302\pi$
$17$	−2.84934			−0.691066			−0.345533	+	0.938407i	$0.612302\pi$
$18$		0			0
$19$			8.14855i			1.86940i	0.355431	+	0.934702i	$0.384334\pi$
$19$			8.14855i			1.86940i	−0.355431	+	0.934702i	$0.615666\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−5.21498			−1.08740			−0.543699	−	0.839280i	$-0.682977\pi$
$23$	−5.21498			−1.08740			−0.543699	+	0.839280i	$0.682977\pi$
$24$		0			0
$25$	−10.9554			−2.19108
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−6.35580			−1.18024			−0.590121	−	0.807315i	$-0.700920\pi$
$29$	−6.35580			−1.18024			−0.590121	+	0.807315i	$0.700920\pi$
$30$		0			0
$31$		−	2.72746i		−	0.489867i	−0.969540	−	0.244934i	$-0.921234\pi$
$31$		−	2.72746i		−	0.489867i	0.969540	−	0.244934i	$-0.0787661\pi$
$32$		0			0
$33$			5.96039i			1.03757i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			1.80626i			0.296948i	0.988916	+	0.148474i	$0.0474361\pi$
$37$			1.80626i			0.296948i	−0.988916	+	0.148474i	$0.952564\pi$
$38$		0			0
$39$	3.53216	−	0.723785i	0.565598	−	0.115898i
$40$		0			0
$41$		−	10.6409i		−	1.66183i	−0.556403	−	0.830913i	$-0.687819\pi$
$41$		−	10.6409i		−	1.66183i	0.556403	−	0.830913i	$-0.312181\pi$
$42$		0			0
$43$	1.84934			0.282022			0.141011	−	0.990008i	$-0.454965\pi$
$43$	1.84934			0.282022			0.141011	+	0.990008i	$0.454965\pi$
$44$		0			0
$45$			3.99442i			0.595453i
$46$		0			0
$47$			6.88159i			1.00378i	0.864931	+	0.501891i	$0.167362\pi$
$47$			6.88159i			1.00378i	−0.864931	+	0.501891i	$0.832638\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−2.84934			−0.398987
$52$		0			0
$53$	8.83033			1.21294			0.606470	−	0.795107i	$-0.292585\pi$
$53$	8.83033			1.21294			0.606470	+	0.795107i	$0.292585\pi$
$54$		0			0
$55$	−23.8083			−3.21031
$56$		0			0
$57$			8.14855i			1.07930i
$58$		0			0
$59$			2.88716i			0.375877i	0.982181	+	0.187938i	$0.0601805\pi$
$59$			2.88716i			0.375877i	−0.982181	+	0.187938i	$0.939819\pi$
$60$		0			0
$61$	−11.2793			−1.44417			−0.722083	−	0.691807i	$-0.756815\pi$
$61$	−11.2793			−1.44417			−0.722083	+	0.691807i	$0.756815\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.89110	+	14.1089i	0.358597	+	1.75000i
$66$		0			0
$67$			0.900292i			0.109988i	0.998487	+	0.0549941i	$0.0175140\pi$
$67$			0.900292i			0.109988i	−0.998487	+	0.0549941i	$0.982486\pi$
$68$		0			0
$69$	−5.21498			−0.627809
$70$		0			0
$71$		−	2.56776i		−	0.304738i	−0.988324	−	0.152369i	$-0.951310\pi$
$71$		−	2.56776i		−	0.304738i	0.988324	−	0.152369i	$-0.0486901\pi$
$72$		0			0
$73$		−	13.5696i		−	1.58820i	−0.607784	−	0.794102i	$-0.707941\pi$
$73$		−	13.5696i		−	1.58820i	0.607784	−	0.794102i	$-0.292059\pi$
$74$		0			0
$75$	−10.9554			−1.26502
$76$		0			0
$77$		0			0
$78$		0			0
$79$	2.61536			0.294251			0.147125	−	0.989118i	$-0.452998\pi$
$79$	2.61536			0.294251			0.147125	+	0.989118i	$0.452998\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			0.740592i			0.0812905i	0.999174	+	0.0406452i	$0.0129413\pi$
$83$			0.740592i			0.0812905i	−0.999174	+	0.0406452i	$0.987059\pi$
$84$		0			0
$85$		−	11.3815i		−	1.23449i
$86$		0			0
$87$	−6.35580			−0.681413
$88$		0			0
$89$		−	0.921202i		−	0.0976473i	−0.998807	−	0.0488236i	$-0.984453\pi$
$89$		−	0.921202i		−	0.0976473i	0.998807	−	0.0488236i	$-0.0155472\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	2.72746i		−	0.282825i
$94$		0			0
$95$	−32.5487			−3.33943
$96$		0			0
$97$		−	5.09624i		−	0.517444i	−0.965952	−	0.258722i	$-0.916699\pi$
$97$		−	5.09624i		−	0.517444i	0.965952	−	0.258722i	$-0.0833014\pi$
$98$		0			0
$99$			5.96039i			0.599041i
$100$		0			0
$101$	6.72144			0.668808			0.334404	−	0.942430i	$-0.391465\pi$
$101$	6.72144			0.668808			0.334404	+	0.942430i	$0.391465\pi$
$102$		0			0
$103$	2.40740			0.237208			0.118604	−	0.992942i	$-0.462158\pi$
$103$	2.40740			0.237208			0.118604	+	0.992942i	$0.462158\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.91365			0.668368			0.334184	−	0.942508i	$-0.391539\pi$
$107$	6.91365			0.668368			0.334184	+	0.942508i	$0.391539\pi$
$108$		0			0
$109$			1.56315i			0.149723i	0.997194	+	0.0748614i	$0.0238514\pi$
$109$			1.56315i			0.149723i	−0.997194	+	0.0748614i	$0.976149\pi$
$110$		0			0
$111$			1.80626i			0.171443i
$112$		0			0
$113$	14.2375			1.33935			0.669677	−	0.742653i	$-0.266433\pi$
$113$	14.2375			1.33935			0.669677	+	0.742653i	$0.266433\pi$
$114$		0			0
$115$		−	20.8308i		−	1.94248i
$116$		0			0
$117$	3.53216	−	0.723785i	0.326548	−	0.0669139i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−24.5262			−2.22965
$122$		0			0
$123$		−	10.6409i		−	0.959455i
$124$		0			0
$125$		−	23.7885i		−	2.12770i
$126$		0			0
$127$	−1.85918			−0.164975			−0.0824876	−	0.996592i	$-0.526286\pi$
$127$	−1.85918			−0.164975			−0.0824876	+	0.996592i	$0.526286\pi$
$128$		0			0
$129$	1.84934			0.162825
$130$		0			0
$131$	7.87189			0.687770			0.343885	−	0.939012i	$-0.388257\pi$
$131$	7.87189			0.687770			0.343885	+	0.939012i	$0.388257\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$			3.99442i			0.343785i
$136$		0			0
$137$			11.3815i			0.972384i	0.873852	+	0.486192i	$0.161615\pi$
$137$			11.3815i			0.972384i	−0.873852	+	0.486192i	$0.838385\pi$
$138$		0			0
$139$	10.3976			0.881910			0.440955	−	0.897529i	$-0.354640\pi$
$139$	10.3976			0.881910			0.440955	+	0.897529i	$0.354640\pi$
$140$		0			0
$141$			6.88159i			0.579534i
$142$		0			0
$143$	4.31404	+	21.0530i	0.360758	+	1.76054i
$144$		0			0
$145$		−	25.3878i		−	2.10834i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			5.89933i			0.483292i	0.970365	+	0.241646i	$0.0776872\pi$
$149$			5.89933i			0.483292i	−0.970365	+	0.241646i	$0.922313\pi$
$150$		0			0
$151$		−	0.264875i		−	0.0215552i	−0.999942	−	0.0107776i	$-0.996569\pi$
$151$		−	0.264875i		−	0.0215552i	0.999942	−	0.0107776i	$-0.00343069\pi$
$152$		0			0
$153$	−2.84934			−0.230355
$154$		0			0
$155$	10.8946			0.875079
$156$		0			0
$157$	15.0962			1.20481			0.602406	−	0.798190i	$-0.294209\pi$
$157$	15.0962			1.20481			0.602406	+	0.798190i	$0.294209\pi$
$158$		0			0
$159$	8.83033			0.700291
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	18.5455i		−	1.45259i	−0.687382	−	0.726296i	$-0.741240\pi$
$163$		−	18.5455i		−	1.45259i	0.687382	−	0.726296i	$-0.258760\pi$
$164$		0			0
$165$	−23.8083			−1.85347
$166$		0			0
$167$			10.3135i			0.798083i	0.916933	+	0.399041i	$0.130657\pi$
$167$			10.3135i			0.798083i	−0.916933	+	0.399041i	$0.869343\pi$
$168$		0			0
$169$	11.9523	−	5.11304i	0.919405	−	0.393311i
$170$		0			0
$171$			8.14855i			0.623135i
$172$		0			0
$173$	−8.79559			−0.668716			−0.334358	−	0.942446i	$-0.608519\pi$
$173$	−8.79559			−0.668716			−0.334358	+	0.942446i	$0.608519\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			2.88716i			0.217013i
$178$		0			0
$179$	17.9456			1.34132			0.670658	−	0.741767i	$-0.266012\pi$
$179$	17.9456			1.34132			0.670658	+	0.741767i	$0.266012\pi$
$180$		0			0
$181$	−15.4044			−1.14500			−0.572499	−	0.819905i	$-0.694026\pi$
$181$	−15.4044			−1.14500			−0.572499	+	0.819905i	$0.694026\pi$
$182$		0			0
$183$	−11.2793			−0.833789
$184$		0			0
$185$	−7.21498			−0.530456
$186$		0			0
$187$		−	16.9832i		−	1.24193i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	17.9207			1.29669			0.648347	−	0.761345i	$-0.275461\pi$
$191$	17.9207			1.29669			0.648347	+	0.761345i	$0.275461\pi$
$192$		0			0
$193$			17.8602i			1.28561i	0.766030	+	0.642804i	$0.222229\pi$
$193$			17.8602i			1.28561i	−0.766030	+	0.642804i	$0.777771\pi$
$194$		0			0
$195$	2.89110	+	14.1089i	0.207036	+	1.01036i
$196$		0			0
$197$			10.0744i			0.717769i	0.933382	+	0.358884i	$0.116843\pi$
$197$			10.0744i			0.717769i	−0.933382	+	0.358884i	$0.883157\pi$
$198$		0			0
$199$	15.3626			1.08903			0.544513	−	0.838752i	$-0.316714\pi$
$199$	15.3626			1.08903			0.544513	+	0.838752i	$0.316714\pi$
$200$		0			0
$201$			0.900292i			0.0635017i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	42.5042			2.96862
$206$		0			0
$207$	−5.21498			−0.362466
$208$		0			0
$209$	−48.5685			−3.35955
$210$		0			0
$211$	−0.656915			−0.0452239			−0.0226120	−	0.999744i	$-0.507198\pi$
$211$	−0.656915			−0.0452239			−0.0226120	+	0.999744i	$0.507198\pi$
$212$		0			0
$213$		−	2.56776i		−	0.175940i
$214$		0			0
$215$			7.38704i			0.503792i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		−	13.5696i		−	0.916951i
$220$		0			0
$221$	−10.0643	+	2.06231i	−0.676999	+	0.138726i
$222$		0			0
$223$		−	6.16821i		−	0.413054i	−0.978441	−	0.206527i	$-0.933784\pi$
$223$		−	6.16821i		−	0.413054i	0.978441	−	0.206527i	$-0.0662161\pi$
$224$		0			0
$225$	−10.9554			−0.730361
$226$		0			0
$227$			9.75897i			0.647725i	0.946104	+	0.323863i	$0.104982\pi$
$227$			9.75897i			0.647725i	−0.946104	+	0.323863i	$0.895018\pi$
$228$		0			0
$229$		−	14.2405i		−	0.941036i	−0.882390	−	0.470518i	$-0.844067\pi$
$229$		−	14.2405i		−	0.941036i	0.882390	−	0.470518i	$-0.155933\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.36282			−0.613379			−0.306689	−	0.951810i	$-0.599221\pi$
$233$	−9.36282			−0.613379			−0.306689	+	0.951810i	$0.599221\pi$
$234$		0			0
$235$	−27.4880			−1.79312
$236$		0			0
$237$	2.61536			0.169886
$238$		0			0
$239$			7.28210i			0.471040i	0.971870	+	0.235520i	$0.0756793\pi$
$239$			7.28210i			0.471040i	−0.971870	+	0.235520i	$0.924321\pi$
$240$		0			0
$241$		−	3.67502i		−	0.236729i	−0.992970	−	0.118365i	$-0.962235\pi$
$241$		−	3.67502i		−	0.236729i	0.992970	−	0.118365i	$-0.0377651\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$		0			0
$246$		0			0
$247$	5.89779	+	28.7819i	0.375268	+	1.83135i
$248$		0			0
$249$			0.740592i			0.0469331i
$250$		0			0
$251$	−11.1894			−0.706269			−0.353134	−	0.935573i	$-0.614884\pi$
$251$	−11.1894			−0.706269			−0.353134	+	0.935573i	$0.614884\pi$
$252$		0			0
$253$		−	31.0833i		−	1.95419i
$254$		0			0
$255$		−	11.3815i		−	0.712735i
$256$		0			0
$257$	−14.1634			−0.883487			−0.441744	−	0.897141i	$-0.645640\pi$
$257$	−14.1634			−0.883487			−0.441744	+	0.897141i	$0.645640\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−6.35580			−0.393414
$262$		0			0
$263$	10.2567			0.632458			0.316229	−	0.948683i	$-0.397583\pi$
$263$	10.2567			0.632458			0.316229	+	0.948683i	$0.397583\pi$
$264$		0			0
$265$			35.2721i			2.16675i
$266$		0			0
$267$		−	0.921202i		−	0.0563767i
$268$		0			0
$269$	−20.4398			−1.24624			−0.623120	−	0.782126i	$-0.714135\pi$
$269$	−20.4398			−1.24624			−0.623120	+	0.782126i	$0.714135\pi$
$270$		0			0
$271$		−	22.4381i		−	1.36302i	−0.731810	−	0.681508i	$-0.761324\pi$
$271$		−	22.4381i		−	1.36302i	0.731810	−	0.681508i	$-0.238676\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	65.2985i		−	3.93765i
$276$		0			0
$277$	7.45486			0.447919			0.223960	−	0.974598i	$-0.428102\pi$
$277$	7.45486			0.447919			0.223960	+	0.974598i	$0.428102\pi$
$278$		0			0
$279$		−	2.72746i		−	0.163289i
$280$		0			0
$281$			18.9000i			1.12748i	0.825952	+	0.563741i	$0.190638\pi$
$281$			18.9000i			1.12748i	−0.825952	+	0.563741i	$0.809362\pi$
$282$		0			0
$283$	−30.6994			−1.82489			−0.912445	−	0.409199i	$-0.865808\pi$
$283$	−30.6994			−1.82489			−0.912445	+	0.409199i	$0.865808\pi$
$284$		0			0
$285$	−32.5487			−1.92802
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.88127			−0.522427
$290$		0			0
$291$		−	5.09624i		−	0.298747i
$292$		0			0
$293$			11.0693i			0.646677i	0.946283	+	0.323338i	$0.104805\pi$
$293$			11.0693i			0.646677i	−0.946283	+	0.323338i	$0.895195\pi$
$294$		0			0
$295$	−11.5326			−0.671451
$296$		0			0
$297$			5.96039i			0.345857i
$298$		0			0
$299$	−18.4201	+	3.77452i	−1.06526	+	0.218286i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	6.72144			0.386136
$304$		0			0
$305$		−	45.0543i		−	2.57980i
$306$		0			0
$307$		−	12.4977i		−	0.713279i	−0.934242	−	0.356640i	$-0.883922\pi$
$307$		−	12.4977i		−	0.713279i	0.934242	−	0.356640i	$-0.116078\pi$
$308$		0			0
$309$	2.40740			0.136952
$310$		0			0
$311$	−12.5933			−0.714102			−0.357051	−	0.934085i	$-0.616218\pi$
$311$	−12.5933			−0.714102			−0.357051	+	0.934085i	$0.616218\pi$
$312$		0			0
$313$	−8.49708			−0.480284			−0.240142	−	0.970738i	$-0.577194\pi$
$313$	−8.49708			−0.480284			−0.240142	+	0.970738i	$0.577194\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			15.1746i			0.852291i	0.904655	+	0.426145i	$0.140129\pi$
$317$			15.1746i			0.852291i	−0.904655	+	0.426145i	$0.859871\pi$
$318$		0			0
$319$		−	37.8830i		−	2.12104i
$320$		0			0
$321$	6.91365			0.385883
$322$		0			0
$323$		−	23.2180i		−	1.29188i
$324$		0			0
$325$	−38.6963	+	7.92936i	−2.14648	+	0.439842i
$326$		0			0
$327$			1.56315i			0.0864425i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			2.81317i			0.154626i	0.997007	+	0.0773129i	$0.0246340\pi$
$331$			2.81317i			0.154626i	−0.997007	+	0.0773129i	$0.975366\pi$
$332$		0			0
$333$			1.80626i			0.0989826i
$334$		0			0
$335$	−3.59615			−0.196478
$336$		0			0
$337$	−3.94840			−0.215083			−0.107542	−	0.994201i	$-0.534298\pi$
$337$	−3.94840			−0.215083			−0.107542	+	0.994201i	$0.534298\pi$
$338$		0			0
$339$	14.2375			0.773276
$340$		0			0
$341$	16.2567			0.880352
$342$		0			0
$343$		0			0
$344$		0			0
$345$		−	20.8308i		−	1.12149i
$346$		0			0
$347$	17.3338			0.930525			0.465263	−	0.885173i	$-0.345960\pi$
$347$	17.3338			0.930525			0.465263	+	0.885173i	$0.345960\pi$
$348$		0			0
$349$			27.4475i			1.46923i	0.678483	+	0.734616i	$0.262638\pi$
$349$			27.4475i			1.46923i	−0.678483	+	0.734616i	$0.737362\pi$
$350$		0			0
$351$	3.53216	−	0.723785i	0.188533	−	0.0386328i
$352$		0			0
$353$			20.9968i			1.11755i	0.829320	+	0.558774i	$0.188728\pi$
$353$			20.9968i			1.11755i	−0.829320	+	0.558774i	$0.811272\pi$
$354$		0			0
$355$	10.2567			0.544371
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	11.8420i		−	0.624994i	−0.949919	−	0.312497i	$-0.898835\pi$
$359$		−	11.8420i		−	0.624994i	0.949919	−	0.312497i	$-0.101165\pi$
$360$		0			0
$361$	−47.3988			−2.49467
$362$		0			0
$363$	−24.5262			−1.28729
$364$		0			0
$365$	54.2028			2.83711
$366$		0			0
$367$	−4.62453			−0.241398			−0.120699	−	0.992689i	$-0.538514\pi$
$367$	−4.62453			−0.241398			−0.120699	+	0.992689i	$0.538514\pi$
$368$		0			0
$369$		−	10.6409i		−	0.553942i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−7.73127			−0.400310			−0.200155	−	0.979764i	$-0.564145\pi$
$373$	−7.73127			−0.400310			−0.200155	+	0.979764i	$0.564145\pi$
$374$		0			0
$375$		−	23.7885i		−	1.22843i
$376$		0			0
$377$	−22.4497	+	4.60023i	−1.15622	+	0.236924i
$378$		0			0
$379$			30.9806i			1.59137i	0.605712	+	0.795684i	$0.292888\pi$
$379$			30.9806i			1.59137i	−0.605712	+	0.795684i	$0.707112\pi$
$380$		0			0
$381$	−1.85918			−0.0952485
$382$		0			0
$383$			19.5630i			0.999623i	0.866134	+	0.499811i	$0.166597\pi$
$383$			19.5630i			0.999623i	−0.866134	+	0.499811i	$0.833403\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1.84934			0.0940072
$388$		0			0
$389$	12.7700			0.647465			0.323733	−	0.946149i	$-0.395062\pi$
$389$	12.7700			0.647465			0.323733	+	0.946149i	$0.395062\pi$
$390$		0			0
$391$	14.8592			0.751464
$392$		0			0
$393$	7.87189			0.397084
$394$		0			0
$395$			10.4468i			0.525638i
$396$		0			0
$397$		−	28.5661i		−	1.43369i	−0.697231	−	0.716847i	$-0.745585\pi$
$397$		−	28.5661i		−	1.43369i	0.697231	−	0.716847i	$-0.254415\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			23.5101i			1.17404i	0.809574	+	0.587018i	$0.199698\pi$
$401$			23.5101i			1.17404i	−0.809574	+	0.587018i	$0.800302\pi$
$402$		0			0
$403$	−1.97410	−	9.63383i	−0.0983368	−	0.479895i
$404$		0			0
$405$			3.99442i			0.198484i
$406$		0			0
$407$	−10.7660			−0.533652
$408$		0			0
$409$			34.4125i			1.70159i	0.525498	+	0.850795i	$0.323879\pi$
$409$			34.4125i			1.70159i	−0.525498	+	0.850795i	$0.676121\pi$
$410$		0			0
$411$			11.3815i			0.561406i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−2.95824			−0.145214
$416$		0			0
$417$	10.3976			0.509171
$418$		0			0
$419$	20.0998			0.981939			0.490969	−	0.871177i	$-0.336643\pi$
$419$	20.0998			0.981939			0.490969	+	0.871177i	$0.336643\pi$
$420$		0			0
$421$		−	2.91208i		−	0.141926i	−0.997479	−	0.0709630i	$-0.977393\pi$
$421$		−	2.91208i		−	0.141926i	0.997479	−	0.0709630i	$-0.0226072\pi$
$422$		0			0
$423$			6.88159i			0.334594i
$424$		0			0
$425$	31.2157			1.51418
$426$		0			0
$427$		0			0
$428$		0			0
$429$	4.31404	+	21.0530i	0.208284	+	1.01645i
$430$		0			0
$431$		−	30.1348i		−	1.45154i	−0.687935	−	0.725772i	$-0.741483\pi$
$431$		−	30.1348i		−	1.45154i	0.687935	−	0.725772i	$-0.258517\pi$
$432$		0			0
$433$	−13.5835			−0.652781			−0.326390	−	0.945235i	$-0.605832\pi$
$433$	−13.5835			−0.652781			−0.326390	+	0.945235i	$0.605832\pi$
$434$		0			0
$435$		−	25.3878i		−	1.21725i
$436$		0			0
$437$		−	42.4945i		−	2.03279i
$438$		0			0
$439$	−31.7090			−1.51339			−0.756695	−	0.653768i	$-0.773187\pi$
$439$	−31.7090			−1.51339			−0.756695	+	0.653768i	$0.773187\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	23.3626			1.10999			0.554996	−	0.831853i	$-0.312720\pi$
$443$	23.3626			1.10999			0.554996	+	0.831853i	$0.312720\pi$
$444$		0			0
$445$	3.67967			0.174433
$446$		0			0
$447$			5.89933i			0.279029i
$448$		0			0
$449$			8.37327i			0.395159i	0.980287	+	0.197580i	$0.0633081\pi$
$449$			8.37327i			0.395159i	−0.980287	+	0.197580i	$0.936692\pi$
$450$		0			0
$451$	63.4237			2.98651
$452$		0			0
$453$		−	0.264875i		−	0.0124449i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			11.5884i			0.542084i	0.962568	+	0.271042i	$0.0873682\pi$
$457$			11.5884i			0.542084i	−0.962568	+	0.271042i	$0.912632\pi$
$458$		0			0
$459$	−2.84934			−0.132996
$460$		0			0
$461$			4.75569i			0.221495i	0.993849	+	0.110747i	$0.0353244\pi$
$461$			4.75569i			0.221495i	−0.993849	+	0.110747i	$0.964676\pi$
$462$		0			0
$463$		−	12.3181i		−	0.572472i	−0.958159	−	0.286236i	$-0.907596\pi$
$463$		−	12.3181i		−	0.572472i	0.958159	−	0.286236i	$-0.0924042\pi$
$464$		0			0
$465$	10.8946			0.505227
$466$		0			0
$467$	14.1188			0.653340			0.326670	−	0.945138i	$-0.394073\pi$
$467$	14.1188			0.653340			0.326670	+	0.945138i	$0.394073\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	15.0962			0.695598
$472$		0			0
$473$			11.0228i			0.506828i
$474$		0			0
$475$		−	89.2707i		−	4.09602i
$476$		0			0
$477$	8.83033			0.404313
$478$		0			0
$479$		−	12.5443i		−	0.573165i	−0.958056	−	0.286583i	$-0.907481\pi$
$479$		−	12.5443i		−	0.573165i	0.958056	−	0.286583i	$-0.0925193\pi$
$480$		0			0
$481$	1.30734	+	6.38000i	0.0596098	+	0.290903i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	20.3565			0.924342
$486$		0			0
$487$		−	13.8970i		−	0.629733i	−0.949136	−	0.314867i	$-0.898040\pi$
$487$		−	13.8970i		−	0.629733i	0.949136	−	0.314867i	$-0.101960\pi$
$488$		0			0
$489$		−	18.5455i		−	0.838655i
$490$		0			0
$491$	−29.6443			−1.33783			−0.668913	−	0.743340i	$-0.733240\pi$
$491$	−29.6443			−1.33783			−0.668913	+	0.743340i	$0.733240\pi$
$492$		0			0
$493$	18.1098			0.815626
$494$		0			0
$495$	−23.8083			−1.07010
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	18.9064i		−	0.846369i	−0.906044	−	0.423184i	$-0.860912\pi$
$499$		−	18.9064i		−	0.846369i	0.906044	−	0.423184i	$-0.139088\pi$
$500$		0			0
$501$			10.3135i			0.460773i
$502$		0			0
$503$	−29.0970			−1.29737			−0.648685	−	0.761057i	$-0.724681\pi$
$503$	−29.0970			−1.29737			−0.648685	+	0.761057i	$0.724681\pi$
$504$		0			0
$505$			26.8483i			1.19473i
$506$		0			0
$507$	11.9523	−	5.11304i	0.530819	−	0.227078i
$508$		0			0
$509$		−	6.35752i		−	0.281792i	−0.990024	−	0.140896i	$-0.955002\pi$
$509$		−	6.35752i		−	0.281792i	0.990024	−	0.140896i	$-0.0449983\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$			8.14855i			0.359767i
$514$		0			0
$515$			9.61617i			0.423739i
$516$		0			0
$517$	−41.0169			−1.80392
$518$		0			0
$519$	−8.79559			−0.386083
$520$		0			0
$521$	−35.0970			−1.53763			−0.768813	−	0.639474i	$-0.779152\pi$
$521$	−35.0970			−1.53763			−0.768813	+	0.639474i	$0.779152\pi$
$522$		0			0
$523$	−20.6358			−0.902341			−0.451171	−	0.892438i	$-0.648993\pi$
$523$	−20.6358			−0.902341			−0.451171	+	0.892438i	$0.648993\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			7.77147i			0.338531i
$528$		0			0
$529$	4.19597			0.182433
$530$		0			0
$531$			2.88716i			0.125292i
$532$		0			0
$533$	−7.70170	−	37.5852i	−0.333598	−	1.62800i
$534$		0			0
$535$			27.6161i			1.19395i
$536$		0			0
$537$	17.9456			0.774409
$538$		0			0
$539$		0			0
$540$		0			0
$541$			14.7830i			0.635572i	0.948163	+	0.317786i	$0.102939\pi$
$541$			14.7830i			0.635572i	−0.948163	+	0.317786i	$0.897061\pi$
$542$		0			0
$543$	−15.4044			−0.661065
$544$		0			0
$545$	−6.24389			−0.267459
$546$		0			0
$547$	12.8400			0.548997			0.274499	−	0.961587i	$-0.411488\pi$
$547$	12.8400			0.548997			0.274499	+	0.961587i	$0.411488\pi$
$548$		0			0
$549$	−11.2793			−0.481388
$550$		0			0
$551$		−	51.7905i		−	2.20635i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−7.21498			−0.306259
$556$		0			0
$557$			16.5500i			0.701246i	0.936517	+	0.350623i	$0.114030\pi$
$557$			16.5500i			0.701246i	−0.936517	+	0.350623i	$0.885970\pi$
$558$		0			0
$559$	6.53216	−	1.33852i	0.276281	−	0.0566135i
$560$		0			0
$561$		−	16.9832i		−	0.717030i
$562$		0			0
$563$	−31.7926			−1.33990			−0.669948	−	0.742408i	$-0.733684\pi$
$563$	−31.7926			−1.33990			−0.669948	+	0.742408i	$0.733684\pi$
$564$		0			0
$565$			56.8707i			2.39257i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−22.4936			−0.942981			−0.471490	−	0.881871i	$-0.656284\pi$
$569$	−22.4936			−0.942981			−0.471490	+	0.881871i	$0.656284\pi$
$570$		0			0
$571$	−20.9015			−0.874699			−0.437349	−	0.899292i	$-0.644083\pi$
$571$	−20.9015			−0.874699			−0.437349	+	0.899292i	$0.644083\pi$
$572$		0			0
$573$	17.9207			0.748647
$574$		0			0
$575$	57.1322			2.38258
$576$		0			0
$577$			5.92739i			0.246761i	0.992359	+	0.123380i	$0.0393735\pi$
$577$			5.92739i			0.246761i	−0.992359	+	0.123380i	$0.960627\pi$
$578$		0			0
$579$			17.8602i			0.742246i
$580$		0			0
$581$		0			0
$582$		0			0
$583$			52.6322i			2.17980i
$584$		0			0
$585$	2.89110	+	14.1089i	0.119532	+	0.583333i
$586$		0			0
$587$			28.9883i			1.19648i	0.801319	+	0.598238i	$0.204132\pi$
$587$			28.9883i			1.19648i	−0.801319	+	0.598238i	$0.795868\pi$
$588$		0			0
$589$	22.2249			0.915760
$590$		0			0
$591$			10.0744i			0.414404i
$592$		0			0
$593$			27.8824i			1.14499i	0.819908	+	0.572496i	$0.194025\pi$
$593$			27.8824i			1.14499i	−0.819908	+	0.572496i	$0.805975\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	15.3626			0.628750
$598$		0			0
$599$	−9.54540			−0.390015			−0.195007	−	0.980802i	$-0.562473\pi$
$599$	−9.54540			−0.390015			−0.195007	+	0.980802i	$0.562473\pi$
$600$		0			0
$601$	40.2155			1.64042			0.820212	−	0.572059i	$-0.193855\pi$
$601$	40.2155			1.64042			0.820212	+	0.572059i	$0.193855\pi$
$602$		0			0
$603$			0.900292i			0.0366627i
$604$		0			0
$605$		−	97.9680i		−	3.98297i
$606$		0			0
$607$	−40.9214			−1.66095			−0.830474	−	0.557057i	$-0.811930\pi$
$607$	−40.9214			−1.66095			−0.830474	+	0.557057i	$0.811930\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.98079	+	24.3069i	0.201501	+	0.983350i
$612$		0			0
$613$			32.0549i			1.29468i	0.762199	+	0.647342i	$0.224120\pi$
$613$			32.0549i			1.29468i	−0.762199	+	0.647342i	$0.775880\pi$
$614$		0			0
$615$	42.5042			1.71393
$616$		0			0
$617$			6.94517i			0.279602i	0.990180	+	0.139801i	$0.0446463\pi$
$617$			6.94517i			0.279602i	−0.990180	+	0.139801i	$0.955354\pi$
$618$		0			0
$619$		−	44.7194i		−	1.79742i	−0.438539	−	0.898712i	$-0.644504\pi$
$619$		−	44.7194i		−	1.79742i	0.438539	−	0.898712i	$-0.355496\pi$
$620$		0			0
$621$	−5.21498			−0.209270
$622$		0			0
$623$		0			0
$624$		0			0
$625$	40.2441			1.60976
$626$		0			0
$627$	−48.5685			−1.93964
$628$		0			0
$629$		−	5.14665i		−	0.205211i
$630$		0			0
$631$			8.73344i			0.347673i	0.984775	+	0.173836i	$0.0556164\pi$
$631$			8.73344i			0.347673i	−0.984775	+	0.173836i	$0.944384\pi$
$632$		0			0
$633$	−0.656915			−0.0261100
$634$		0			0
$635$		−	7.42634i		−	0.294705i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		−	2.56776i		−	0.101579i
$640$		0			0
$641$	36.4305			1.43892			0.719459	−	0.694535i	$-0.244390\pi$
$641$	36.4305			1.43892			0.719459	+	0.694535i	$0.244390\pi$
$642$		0			0
$643$		−	13.3836i		−	0.527797i	−0.964551	−	0.263898i	$-0.914992\pi$
$643$		−	13.3836i		−	0.527797i	0.964551	−	0.263898i	$-0.0850083\pi$
$644$		0			0
$645$			7.38704i			0.290865i
$646$		0			0
$647$	−15.0319			−0.590966			−0.295483	−	0.955348i	$-0.595481\pi$
$647$	−15.0319			−0.590966			−0.295483	+	0.955348i	$0.595481\pi$
$648$		0			0
$649$	−17.2086			−0.675497
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−14.6668			−0.573954			−0.286977	−	0.957937i	$-0.592650\pi$
$653$	−14.6668			−0.573954			−0.286977	+	0.957937i	$0.592650\pi$
$654$		0			0
$655$			31.4437i			1.22861i
$656$		0			0
$657$		−	13.5696i		−	0.529402i
$658$		0			0
$659$	43.1415			1.68056			0.840278	−	0.542156i	$-0.182392\pi$
$659$	43.1415			1.68056			0.840278	+	0.542156i	$0.182392\pi$
$660$		0			0
$661$			36.8487i			1.43325i	0.697460	+	0.716623i	$0.254313\pi$
$661$			36.8487i			1.43325i	−0.697460	+	0.716623i	$0.745687\pi$
$662$		0			0
$663$	−10.0643	+	2.06231i	−0.390866	+	0.0800934i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	33.1453			1.28339
$668$		0			0
$669$		−	6.16821i		−	0.238477i
$670$		0			0
$671$		−	67.2289i		−	2.59534i
$672$		0			0
$673$	19.2669			0.742684			0.371342	−	0.928496i	$-0.378898\pi$
$673$	19.2669			0.742684			0.371342	+	0.928496i	$0.378898\pi$
$674$		0			0
$675$	−10.9554			−0.421674
$676$		0			0
$677$	−48.9078			−1.87968			−0.939840	−	0.341615i	$-0.889026\pi$
$677$	−48.9078			−1.87968			−0.939840	+	0.341615i	$0.889026\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$			9.75897i			0.373964i
$682$		0			0
$683$			47.4912i			1.81720i	0.417667	+	0.908600i	$0.362848\pi$
$683$			47.4912i			1.81720i	−0.417667	+	0.908600i	$0.637152\pi$
$684$		0			0
$685$	−45.4624			−1.73703
$686$		0			0
$687$		−	14.2405i		−	0.543308i
$688$		0			0
$689$	31.1901	−	6.39126i	1.18825	−	0.243488i
$690$		0			0
$691$		−	11.2941i		−	0.429646i	−0.976653	−	0.214823i	$-0.931083\pi$
$691$		−	11.2941i		−	0.429646i	0.976653	−	0.214823i	$-0.0689175\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			41.5323i			1.57541i
$696$		0			0
$697$			30.3195i			1.14843i
$698$		0			0
$699$	−9.36282			−0.354134
$700$		0			0
$701$	−20.0906			−0.758812			−0.379406	−	0.925230i	$-0.623872\pi$
$701$	−20.0906			−0.758812			−0.379406	+	0.925230i	$0.623872\pi$
$702$		0			0
$703$	−14.7184			−0.555115
$704$		0			0
$705$	−27.4880			−1.03526
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	40.4649i		−	1.51969i	−0.650103	−	0.759846i	$-0.725274\pi$
$709$		−	40.4649i		−	1.51969i	0.650103	−	0.759846i	$-0.274726\pi$
$710$		0			0
$711$	2.61536			0.0980836
$712$		0			0
$713$			14.2237i			0.532680i
$714$		0			0
$715$	−84.0947	+	17.2321i	−3.14496	+	0.644443i
$716$		0			0
$717$			7.28210i			0.271955i
$718$		0			0
$719$	11.1223			0.414791			0.207396	−	0.978257i	$-0.433501\pi$
$719$	11.1223			0.414791			0.207396	+	0.978257i	$0.433501\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	3.67502i		−	0.136676i
$724$		0			0
$725$	69.6304			2.58601
$726$		0			0
$727$	30.3823			1.12682			0.563409	−	0.826178i	$-0.309489\pi$
$727$	30.3823			1.12682			0.563409	+	0.826178i	$0.309489\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−5.26940			−0.194896
$732$		0			0
$733$			44.3375i			1.63764i	0.574048	+	0.818821i	$0.305372\pi$
$733$			44.3375i			1.63764i	−0.574048	+	0.818821i	$0.694628\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.36608			−0.197662
$738$		0			0
$739$		−	10.6498i		−	0.391759i	−0.980628	−	0.195880i	$-0.937244\pi$
$739$		−	10.6498i		−	0.391759i	0.980628	−	0.195880i	$-0.0627562\pi$
$740$		0			0
$741$	5.89779	+	28.7819i	0.216661	+	1.05733i
$742$		0			0
$743$			18.6409i			0.683870i	0.939724	+	0.341935i	$0.111082\pi$
$743$			18.6409i			0.683870i	−0.939724	+	0.341935i	$0.888918\pi$
$744$		0			0
$745$	−23.5644			−0.863333
$746$		0			0
$747$			0.740592i			0.0270968i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−37.3763			−1.36388			−0.681939	−	0.731409i	$-0.738863\pi$
$751$	−37.3763			−1.36388			−0.681939	+	0.731409i	$0.738863\pi$
$752$		0			0
$753$	−11.1894			−0.407764
$754$		0			0
$755$	1.05802			0.0385054
$756$		0			0
$757$	−13.7918			−0.501273			−0.250636	−	0.968081i	$-0.580640\pi$
$757$	−13.7918			−0.501273			−0.250636	+	0.968081i	$0.580640\pi$
$758$		0			0
$759$		−	31.0833i		−	1.12825i
$760$		0			0
$761$		−	31.0687i		−	1.12624i	−0.826376	−	0.563119i	$-0.809601\pi$
$761$		−	31.0687i		−	1.12624i	0.826376	−	0.563119i	$-0.190399\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		−	11.3815i		−	0.411498i
$766$		0			0
$767$	2.08969	+	10.1979i	0.0754542	+	0.368225i
$768$		0			0
$769$			1.35266i			0.0487783i	0.999703	+	0.0243892i	$0.00776408\pi$
$769$			1.35266i			0.0487783i	−0.999703	+	0.0243892i	$0.992236\pi$
$770$		0			0
$771$	−14.1634			−0.510081
$772$		0			0
$773$			8.90752i			0.320381i	0.987086	+	0.160191i	$0.0512109\pi$
$773$			8.90752i			0.320381i	−0.987086	+	0.160191i	$0.948789\pi$
$774$		0			0
$775$			29.8805i			1.07334i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	86.7077			3.10662
$780$		0			0
$781$	15.3049			0.547651
$782$		0			0
$783$	−6.35580			−0.227138
$784$		0			0
$785$			60.3008i			2.15223i
$786$		0			0
$787$		−	21.8338i		−	0.778289i	−0.921177	−	0.389145i	$-0.872771\pi$
$787$		−	21.8338i		−	0.778289i	0.921177	−	0.389145i	$-0.127229\pi$
$788$		0			0
$789$	10.2567			0.365150
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−39.8402	+	8.16378i	−1.41477	+	0.289904i
$794$		0			0
$795$			35.2721i			1.25097i
$796$		0			0
$797$	−0.593993			−0.0210403			−0.0105202	−	0.999945i	$-0.503349\pi$
$797$	−0.593993			−0.0210403			−0.0105202	+	0.999945i	$0.503349\pi$
$798$		0			0
$799$		−	19.6080i		−	0.693680i
$800$		0			0
$801$		−	0.921202i		−	0.0325491i
$802$		0			0
$803$	80.8802			2.85420
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−20.4398			−0.719517
$808$		0			0
$809$	−22.4013			−0.787588			−0.393794	−	0.919199i	$-0.628838\pi$
$809$	−22.4013			−0.787588			−0.393794	+	0.919199i	$0.628838\pi$
$810$		0			0
$811$			2.68043i			0.0941226i	0.998892	+	0.0470613i	$0.0149856\pi$
$811$			2.68043i			0.0941226i	−0.998892	+	0.0470613i	$0.985014\pi$
$812$		0			0
$813$		−	22.4381i		−	0.786938i
$814$		0			0
$815$	74.0784			2.59485
$816$		0			0
$817$			15.0694i			0.527213i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	14.4154i		−	0.503101i	−0.967844	−	0.251550i	$-0.919060\pi$
$821$		−	14.4154i		−	0.503101i	0.967844	−	0.251550i	$-0.0809404\pi$
$822$		0			0
$823$	−36.5000			−1.27231			−0.636154	−	0.771562i	$-0.719476\pi$
$823$	−36.5000			−1.27231			−0.636154	+	0.771562i	$0.719476\pi$
$824$		0			0
$825$		−	65.2985i		−	2.27340i
$826$		0			0
$827$		−	35.5929i		−	1.23769i	−0.785514	−	0.618844i	$-0.787601\pi$
$827$		−	35.5929i		−	1.23769i	0.785514	−	0.618844i	$-0.212399\pi$
$828$		0			0
$829$	−25.8566			−0.898038			−0.449019	−	0.893522i	$-0.648226\pi$
$829$	−25.8566			−0.898038			−0.449019	+	0.893522i	$0.648226\pi$
$830$		0			0
$831$	7.45486			0.258606
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−41.1965			−1.42566
$836$		0			0
$837$		−	2.72746i		−	0.0942750i
$838$		0			0
$839$			38.5611i			1.33128i	0.746274	+	0.665639i	$0.231841\pi$
$839$			38.5611i			1.33128i	−0.746274	+	0.665639i	$0.768159\pi$
$840$		0			0
$841$	11.3962			0.392972
$842$		0			0
$843$			18.9000i			0.650952i
$844$		0			0
$845$	20.4237	+	47.7424i	0.702595	+	1.64239i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−30.6994			−1.05360
$850$		0			0
$851$		−	9.41961i		−	0.322900i
$852$		0			0
$853$			28.4825i			0.975222i	0.873061	+	0.487611i	$0.162132\pi$
$853$			28.4825i			0.975222i	−0.873061	+	0.487611i	$0.837868\pi$
$854$		0			0
$855$	−32.5487			−1.11314
$856$		0			0
$857$	29.2049			0.997622			0.498811	−	0.866711i	$-0.333770\pi$
$857$	29.2049			0.997622			0.498811	+	0.866711i	$0.333770\pi$
$858$		0			0
$859$	42.3223			1.44402			0.722009	−	0.691884i	$-0.243219\pi$
$859$	42.3223			1.44402			0.722009	+	0.691884i	$0.243219\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	10.6071i		−	0.361071i	−0.983568	−	0.180535i	$-0.942217\pi$
$863$		−	10.6071i		−	0.361071i	0.983568	−	0.180535i	$-0.0577831\pi$
$864$		0			0
$865$		−	35.1333i		−	1.19457i
$866$		0			0
$867$	−8.88127			−0.301624
$868$		0			0
$869$			15.5885i			0.528805i
$870$		0			0
$871$	0.651617	+	3.17997i	0.0220792	+	0.107749i
$872$		0			0
$873$		−	5.09624i		−	0.172481i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	3.70342i		−	0.125056i	−0.998043	−	0.0625279i	$-0.980084\pi$
$877$		−	3.70342i		−	0.125056i	0.998043	−	0.0625279i	$-0.0199162\pi$
$878$		0			0
$879$			11.0693i			0.373359i
$880$		0			0
$881$	40.4655			1.36332			0.681658	−	0.731671i	$-0.261259\pi$
$881$	40.4655			1.36332			0.681658	+	0.731671i	$0.261259\pi$
$882$		0			0
$883$	13.8212			0.465119			0.232560	−	0.972582i	$-0.425290\pi$
$883$	13.8212			0.465119			0.232560	+	0.972582i	$0.425290\pi$
$884$		0			0
$885$	−11.5326			−0.387663
$886$		0			0
$887$	42.8501			1.43876			0.719382	−	0.694614i	$-0.244425\pi$
$887$	42.8501			1.43876			0.719382	+	0.694614i	$0.244425\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$			5.96039i			0.199680i
$892$		0			0
$893$	−56.0749			−1.87648
$894$		0			0
$895$			71.6822i			2.39607i
$896$		0			0
$897$	−18.4201	+	3.77452i	−0.615030	+	0.126028i
$898$		0			0
$899$			17.3352i			0.578162i
$900$		0			0
$901$	−25.1606			−0.838222
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	61.5316i		−	2.04538i
$906$		0			0
$907$	19.5100			0.647819			0.323910	−	0.946088i	$-0.395003\pi$
$907$	19.5100			0.647819			0.323910	+	0.946088i	$0.395003\pi$
$908$		0			0
$909$	6.72144			0.222936
$910$		0			0
$911$	−17.9461			−0.594581			−0.297290	−	0.954787i	$-0.596083\pi$
$911$	−17.9461			−0.594581			−0.297290	+	0.954787i	$0.596083\pi$
$912$		0			0
$913$	−4.41421			−0.146089
$914$		0			0
$915$		−	45.0543i		−	1.48945i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	45.7928			1.51057			0.755283	−	0.655399i	$-0.227499\pi$
$919$	45.7928			1.51057			0.755283	+	0.655399i	$0.227499\pi$
$920$		0			0
$921$		−	12.4977i		−	0.411812i
$922$		0			0
$923$	−1.85851	−	9.06975i	−0.0611736	−	0.298534i
$924$		0			0
$925$		−	19.7884i		−	0.650637i
$926$		0			0
$927$	2.40740			0.0790694
$928$		0			0
$929$		−	28.4352i		−	0.932930i	−0.884540	−	0.466465i	$-0.845527\pi$
$929$		−	28.4352i		−	0.932930i	0.884540	−	0.466465i	$-0.154473\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−12.5933			−0.412287
$934$		0			0
$935$	67.8379			2.21854
$936$		0			0
$937$	24.4239			0.797893			0.398946	−	0.916974i	$-0.369376\pi$
$937$	24.4239			0.797893			0.398946	+	0.916974i	$0.369376\pi$
$938$		0			0
$939$	−8.49708			−0.277292
$940$		0			0
$941$		−	40.8095i		−	1.33035i	−0.746687	−	0.665175i	$-0.768357\pi$
$941$		−	40.8095i		−	1.33035i	0.746687	−	0.665175i	$-0.231643\pi$
$942$		0			0
$943$			55.4919i			1.80707i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			34.9653i			1.13622i	0.822953	+	0.568109i	$0.192325\pi$
$947$			34.9653i			1.13622i	−0.822953	+	0.568109i	$0.807675\pi$
$948$		0			0
$949$	−9.82149	−	47.9301i	−0.318819	−	1.55588i
$950$		0			0
$951$			15.1746i			0.492070i
$952$		0			0
$953$	17.1967			0.557056			0.278528	−	0.960428i	$-0.410154\pi$
$953$	17.1967			0.557056			0.278528	+	0.960428i	$0.410154\pi$
$954$		0			0
$955$			71.5827i			2.31636i
$956$		0			0
$957$		−	37.8830i		−	1.22458i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	23.5609			0.760030
$962$		0			0
$963$	6.91365			0.222789
$964$		0			0
$965$	−71.3414			−2.29656
$966$		0			0
$967$			17.8251i			0.573215i	0.958048	+	0.286608i	$0.0925276\pi$
$967$			17.8251i			0.573215i	−0.958048	+	0.286608i	$0.907472\pi$
$968$		0			0
$969$		−	23.2180i		−	0.745869i
$970$		0			0
$971$	3.14455			0.100914			0.0504568	−	0.998726i	$-0.483932\pi$
$971$	3.14455			0.100914			0.0504568	+	0.998726i	$0.483932\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−38.6963	+	7.92936i	−1.23927	+	0.253943i
$976$		0			0
$977$		−	14.2959i		−	0.457368i	−0.973501	−	0.228684i	$-0.926558\pi$
$977$		−	14.2959i		−	0.457368i	0.973501	−	0.228684i	$-0.0734422\pi$
$978$		0			0
$979$	5.49072			0.175484
$980$		0			0
$981$			1.56315i			0.0499076i
$982$		0			0
$983$		−	25.3972i		−	0.810045i	−0.914307	−	0.405022i	$-0.867264\pi$
$983$		−	25.3972i		−	0.810045i	0.914307	−	0.405022i	$-0.132736\pi$
$984$		0			0
$985$	−40.2413			−1.28219
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−9.64426			−0.306670
$990$		0			0
$991$	−25.3412			−0.804991			−0.402496	−	0.915422i	$-0.631857\pi$
$991$	−25.3412			−0.804991			−0.402496	+	0.915422i	$0.631857\pi$
$992$		0			0
$993$			2.81317i			0.0892733i
$994$		0			0
$995$			61.3648i			1.94539i
$996$		0			0
$997$	−15.2383			−0.482600			−0.241300	−	0.970451i	$-0.577574\pi$
$997$	−15.2383			−0.482600			−0.241300	+	0.970451i	$0.577574\pi$
$998$		0			0
$999$			1.80626i			0.0571476i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7644.2.e.o.4705.9		10
7.2	even	3		1092.2.cu.c.25.9	yes	20
7.4	even	3		1092.2.cu.c.961.2	yes	20
7.6	odd	2		7644.2.e.n.4705.2		10
13.12	even	2	inner	7644.2.e.o.4705.2		10
21.2	odd	6		3276.2.gv.g.1117.2		20
21.11	odd	6		3276.2.gv.g.2053.9		20
91.25	even	6		1092.2.cu.c.961.9	yes	20
91.51	even	6		1092.2.cu.c.25.2	✓	20
91.90	odd	2		7644.2.e.n.4705.9		10
273.116	odd	6		3276.2.gv.g.2053.2		20
273.233	odd	6		3276.2.gv.g.1117.9		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1092.2.cu.c.25.2	✓	20	91.51	even	6
1092.2.cu.c.25.9	yes	20	7.2	even	3
1092.2.cu.c.961.2	yes	20	7.4	even	3
1092.2.cu.c.961.9	yes	20	91.25	even	6
3276.2.gv.g.1117.2		20	21.2	odd	6
3276.2.gv.g.1117.9		20	273.233	odd	6
3276.2.gv.g.2053.2		20	273.116	odd	6
3276.2.gv.g.2053.9		20	21.11	odd	6
7644.2.e.n.4705.2		10	7.6	odd	2
7644.2.e.n.4705.9		10	91.90	odd	2
7644.2.e.o.4705.2		10	13.12	even	2	inner
7644.2.e.o.4705.9		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.99442i q^{5} +1.00000 q^{9} +5.96039i q^{11} +(3.53216 - 0.723785i) q^{13} +3.99442i q^{15} -2.84934 q^{17} +8.14855i q^{19} -5.21498 q^{23} -10.9554 q^{25} +1.00000 q^{27} -6.35580 q^{29} -2.72746i q^{31} +5.96039i q^{33} +1.80626i q^{37} +(3.53216 - 0.723785i) q^{39} -10.6409i q^{41} +1.84934 q^{43} +3.99442i q^{45} +6.88159i q^{47} -2.84934 q^{51} +8.83033 q^{53} -23.8083 q^{55} +8.14855i q^{57} +2.88716i q^{59} -11.2793 q^{61} +(2.89110 + 14.1089i) q^{65} +0.900292i q^{67} -5.21498 q^{69} -2.56776i q^{71} -13.5696i q^{73} -10.9554 q^{75} +2.61536 q^{79} +1.00000 q^{81} +0.740592i q^{83} -11.3815i q^{85} -6.35580 q^{87} -0.921202i q^{89} -2.72746i q^{93} -32.5487 q^{95} -5.09624i q^{97} +5.96039i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{3} + 10 q^{9} + q^{13} - 4 q^{17} - 8 q^{23} - 24 q^{25} + 10 q^{27} - 2 q^{29} + q^{39} - 6 q^{43} - 4 q^{51} + 24 q^{53} - 12 q^{55} + 12 q^{65} - 8 q^{69} - 24 q^{75} + 6 q^{79} + 10 q^{81}+ \cdots - 58 q^{95}+O(q^{100}) \) 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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