LMFDB - Embedded newform 9280.2.a.ck.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9280,2,Mod(1,9280)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9280.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9280 = 2^{6} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9280.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,1,0,5,0,7,0,12,0,10,0,-3,0,1,0,3,0,-2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	yes
Analytic conductor:	$74.1011730757$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.3145252.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 11x^{3} + 9x^{2} + 22x - 11 $ x^5 - 2x^4 - 11x^3 + 9x^2 + 22x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1160)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.90462$ of defining polynomial
Character	$\chi$	$=$	9280.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.08906 q^{3} +1.00000 q^{5} -3.25238 q^{7} +1.36419 q^{9} +5.61657 q^{11} +2.90462 q^{13} +2.08906 q^{15} +5.25238 q^{17} +4.99368 q^{19} -6.79443 q^{21} +7.72018 q^{23} +1.00000 q^{25} -3.41732 q^{27} -1.00000 q^{29} -4.73499 q^{31} +11.7334 q^{33} -3.25238 q^{35} +6.62480 q^{37} +6.06794 q^{39} -4.43213 q^{41} -4.30286 q^{43} +1.36419 q^{45} -2.19267 q^{47} +3.57799 q^{49} +10.9726 q^{51} -5.81394 q^{53} +5.61657 q^{55} +10.4321 q^{57} +8.90462 q^{59} -13.2313 q^{61} -4.43686 q^{63} +2.90462 q^{65} -12.1800 q^{67} +16.1279 q^{69} -1.31840 q^{71} +5.17974 q^{73} +2.08906 q^{75} -18.2672 q^{77} +11.5294 q^{79} -11.2316 q^{81} -2.05809 q^{83} +5.25238 q^{85} -2.08906 q^{87} +2.17813 q^{89} -9.44694 q^{91} -9.89169 q^{93} +4.99368 q^{95} +12.9627 q^{97} +7.66205 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} + 5 q^{5} + 7 q^{7} + 12 q^{9} + 10 q^{11} - 3 q^{13} + q^{15} + 3 q^{17} - 2 q^{19} - 4 q^{21} + 13 q^{23} + 5 q^{25} + 4 q^{27} - 5 q^{29} + 7 q^{31} + 12 q^{33} + 7 q^{35} - 10 q^{37} - q^{39}+ \cdots + 96 q^{99}+O(q^{100}) $ 5 * q + q^3 + 5 * q^5 + 7 * q^7 + 12 * q^9 + 10 * q^11 - 3 * q^13 + q^15 + 3 * q^17 - 2 * q^19 - 4 * q^21 + 13 * q^23 + 5 * q^25 + 4 * q^27 - 5 * q^29 + 7 * q^31 + 12 * q^33 + 7 * q^35 - 10 * q^37 - q^39 + 4 * q^41 - 17 * q^43 + 12 * q^45 + 6 * q^47 + 28 * q^49 + 6 * q^51 - 5 * q^53 + 10 * q^55 + 26 * q^57 + 27 * q^59 - 21 * q^61 + 54 * q^63 - 3 * q^65 + 10 * q^67 - 25 * q^69 - 4 * q^71 + 23 * q^73 + q^75 - 2 * q^77 - 3 * q^79 + 57 * q^81 + 12 * q^83 + 3 * q^85 - q^87 - 8 * q^89 - 16 * q^91 - 46 * q^93 - 2 * q^95 + 25 * q^97 + 96 * q^99

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
$3$	2.08906	1.20612	0.603061	−	0.797695i	$-0.293948\pi$
$3$	2.08906	1.20612	0.603061	+	0.797695i	$0.293948\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.25238	−1.22928	−0.614642	−	0.788806i	$-0.710700\pi$
$7$	−3.25238	−1.22928	−0.614642	+	0.788806i	$0.710700\pi$
$8$	0	0
$9$	1.36419	0.454729
$10$	0	0
$11$	5.61657	1.69346	0.846730	−	0.532023i	$-0.178568\pi$
$11$	5.61657	1.69346	0.846730	+	0.532023i	$0.178568\pi$
$12$	0	0
$13$	2.90462	0.805597	0.402798	−	0.915289i	$-0.368038\pi$
$13$	2.90462	0.805597	0.402798	+	0.915289i	$0.368038\pi$
$14$	0	0
$15$	2.08906	0.539394
$16$	0	0
$17$	5.25238	1.27389	0.636945	−	0.770909i	$-0.280198\pi$
$17$	5.25238	1.27389	0.636945	+	0.770909i	$0.280198\pi$
$18$	0	0
$19$	4.99368	1.14563	0.572815	−	0.819685i	$-0.305851\pi$
$19$	4.99368	1.14563	0.572815	+	0.819685i	$0.305851\pi$
$20$	0	0
$21$	−6.79443	−1.48267
$22$	0	0
$23$	7.72018	1.60977	0.804884	−	0.593432i	$-0.202227\pi$
$23$	7.72018	1.60977	0.804884	+	0.593432i	$0.202227\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−3.41732	−0.657663
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−4.73499	−0.850429	−0.425215	−	0.905093i	$-0.639801\pi$
$31$	−4.73499	−0.850429	−0.425215	+	0.905093i	$0.639801\pi$
$32$	0	0
$33$	11.7334	2.04252
$34$	0	0
$35$	−3.25238	−0.549753
$36$	0	0
$37$	6.62480	1.08911	0.544555	−	0.838725i	$-0.316698\pi$
$37$	6.62480	1.08911	0.544555	+	0.838725i	$0.316698\pi$
$38$	0	0
$39$	6.06794	0.971648
$40$	0	0
$41$	−4.43213	−0.692182	−0.346091	−	0.938201i	$-0.612491\pi$
$41$	−4.43213	−0.692182	−0.346091	+	0.938201i	$0.612491\pi$
$42$	0	0
$43$	−4.30286	−0.656180	−0.328090	−	0.944646i	$-0.606405\pi$
$43$	−4.30286	−0.656180	−0.328090	+	0.944646i	$0.606405\pi$
$44$	0	0
$45$	1.36419	0.203361
$46$	0	0
$47$	−2.19267	−0.319834	−0.159917	−	0.987130i	$-0.551123\pi$
$47$	−2.19267	−0.319834	−0.159917	+	0.987130i	$0.551123\pi$
$48$	0	0
$49$	3.57799	0.511141
$50$	0	0
$51$	10.9726	1.53647
$52$	0	0
$53$	−5.81394	−0.798606	−0.399303	−	0.916819i	$-0.630748\pi$
$53$	−5.81394	−0.798606	−0.399303	+	0.916819i	$0.630748\pi$
$54$	0	0
$55$	5.61657	0.757338
$56$	0	0
$57$	10.4321	1.38177
$58$	0	0
$59$	8.90462	1.15928	0.579641	−	0.814872i	$-0.303193\pi$
$59$	8.90462	1.15928	0.579641	+	0.814872i	$0.303193\pi$
$60$	0	0
$61$	−13.2313	−1.69409	−0.847044	−	0.531522i	$-0.821620\pi$
$61$	−13.2313	−1.69409	−0.847044	+	0.531522i	$0.821620\pi$
$62$	0	0
$63$	−4.43686	−0.558992
$64$	0	0
$65$	2.90462	0.360274
$66$	0	0
$67$	−12.1800	−1.48803	−0.744015	−	0.668163i	$-0.767081\pi$
$67$	−12.1800	−1.48803	−0.744015	+	0.668163i	$0.767081\pi$
$68$	0	0
$69$	16.1279	1.94158
$70$	0	0
$71$	−1.31840	−0.156466	−0.0782329	−	0.996935i	$-0.524928\pi$
$71$	−1.31840	−0.156466	−0.0782329	+	0.996935i	$0.524928\pi$
$72$	0	0
$73$	5.17974	0.606243	0.303122	−	0.952952i	$-0.401971\pi$
$73$	5.17974	0.606243	0.303122	+	0.952952i	$0.401971\pi$
$74$	0	0
$75$	2.08906	0.241224
$76$	0	0
$77$	−18.2672	−2.08174
$78$	0	0
$79$	11.5294	1.29716	0.648581	−	0.761146i	$-0.275363\pi$
$79$	11.5294	1.29716	0.648581	+	0.761146i	$0.275363\pi$
$80$	0	0
$81$	−11.2316	−1.24795
$82$	0	0
$83$	−2.05809	−0.225905	−0.112952	−	0.993600i	$-0.536031\pi$
$83$	−2.05809	−0.225905	−0.112952	+	0.993600i	$0.536031\pi$
$84$	0	0
$85$	5.25238	0.569701
$86$	0	0
$87$	−2.08906	−0.223971
$88$	0	0
$89$	2.17813	0.230881	0.115441	−	0.993314i	$-0.463172\pi$
$89$	2.17813	0.230881	0.115441	+	0.993314i	$0.463172\pi$
$90$	0	0
$91$	−9.44694	−0.990308
$92$	0	0
$93$	−9.89169	−1.02572
$94$	0	0
$95$	4.99368	0.512341
$96$	0	0
$97$	12.9627	1.31616	0.658082	−	0.752946i	$-0.271368\pi$
$97$	12.9627	1.31616	0.658082	+	0.752946i	$0.271368\pi$
$98$	0	0
$99$	7.66205	0.770065
$100$	0	0
$101$	2.28174	0.227041	0.113521	−	0.993536i	$-0.463787\pi$
$101$	2.28174	0.227041	0.113521	+	0.993536i	$0.463787\pi$
$102$	0	0
$103$	8.56156	0.843595	0.421798	−	0.906690i	$-0.361399\pi$
$103$	8.56156	0.843595	0.421798	+	0.906690i	$0.361399\pi$
$104$	0	0
$105$	−6.79443	−0.663069
$106$	0	0
$107$	16.0746	1.55399	0.776993	−	0.629509i	$-0.216744\pi$
$107$	16.0746	1.55399	0.776993	+	0.629509i	$0.216744\pi$
$108$	0	0
$109$	−5.55524	−0.532096	−0.266048	−	0.963960i	$-0.585718\pi$
$109$	−5.55524	−0.532096	−0.266048	+	0.963960i	$0.585718\pi$
$110$	0	0
$111$	13.8396	1.31360
$112$	0	0
$113$	−5.02466	−0.472680	−0.236340	−	0.971670i	$-0.575948\pi$
$113$	−5.02466	−0.472680	−0.236340	+	0.971670i	$0.575948\pi$
$114$	0	0
$115$	7.72018	0.719910
$116$	0	0
$117$	3.96245	0.366328
$118$	0	0
$119$	−17.0827	−1.56597
$120$	0	0
$121$	20.5458	1.86780
$122$	0	0
$123$	−9.25899	−0.834855
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−22.2901	−1.97792	−0.988962	−	0.148171i	$-0.952661\pi$
$127$	−22.2901	−1.97792	−0.988962	+	0.148171i	$0.952661\pi$
$128$	0	0
$129$	−8.98895	−0.791433
$130$	0	0
$131$	2.27794	0.199025	0.0995123	−	0.995036i	$-0.468272\pi$
$131$	2.27794	0.199025	0.0995123	+	0.995036i	$0.468272\pi$
$132$	0	0
$133$	−16.2414	−1.40831
$134$	0	0
$135$	−3.41732	−0.294116
$136$	0	0
$137$	−13.4535	−1.14941	−0.574707	−	0.818359i	$-0.694884\pi$
$137$	−13.4535	−1.14941	−0.574707	+	0.818359i	$0.694884\pi$
$138$	0	0
$139$	21.1784	1.79633	0.898165	−	0.439658i	$-0.144900\pi$
$139$	21.1784	1.79633	0.898165	+	0.439658i	$0.144900\pi$
$140$	0	0
$141$	−4.58063	−0.385759
$142$	0	0
$143$	16.3140	1.36425
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	7.47464	0.616498
$148$	0	0
$149$	2.20722	0.180822	0.0904112	−	0.995905i	$-0.471182\pi$
$149$	2.20722	0.180822	0.0904112	+	0.995905i	$0.471182\pi$
$150$	0	0
$151$	23.4839	1.91109	0.955546	−	0.294841i	$-0.0952666\pi$
$151$	23.4839	1.91109	0.955546	+	0.294841i	$0.0952666\pi$
$152$	0	0
$153$	7.16523	0.579275
$154$	0	0
$155$	−4.73499	−0.380323
$156$	0	0
$157$	21.0278	1.67820	0.839102	−	0.543974i	$-0.183081\pi$
$157$	21.0278	1.67820	0.839102	+	0.543974i	$0.183081\pi$
$158$	0	0
$159$	−12.1457	−0.963216
$160$	0	0
$161$	−25.1090	−1.97886
$162$	0	0
$163$	−14.5489	−1.13956	−0.569780	−	0.821797i	$-0.692972\pi$
$163$	−14.5489	−1.13956	−0.569780	+	0.821797i	$0.692972\pi$
$164$	0	0
$165$	11.7334	0.913442
$166$	0	0
$167$	−4.57079	−0.353698	−0.176849	−	0.984238i	$-0.556590\pi$
$167$	−4.57079	−0.353698	−0.176849	+	0.984238i	$0.556590\pi$
$168$	0	0
$169$	−4.56318	−0.351014
$170$	0	0
$171$	6.81232	0.520951
$172$	0	0
$173$	−17.1599	−1.30464	−0.652322	−	0.757942i	$-0.726205\pi$
$173$	−17.1599	−1.30464	−0.652322	+	0.757942i	$0.726205\pi$
$174$	0	0
$175$	−3.25238	−0.245857
$176$	0	0
$177$	18.6023	1.39824
$178$	0	0
$179$	−13.5445	−1.01236	−0.506182	−	0.862427i	$-0.668943\pi$
$179$	−13.5445	−1.01236	−0.506182	+	0.862427i	$0.668943\pi$
$180$	0	0
$181$	1.55524	0.115600	0.0578002	−	0.998328i	$-0.481591\pi$
$181$	1.55524	0.115600	0.0578002	+	0.998328i	$0.481591\pi$
$182$	0	0
$183$	−27.6409	−2.04328
$184$	0	0
$185$	6.62480	0.487065
$186$	0	0
$187$	29.5004	2.15728
$188$	0	0
$189$	11.1144	0.808455
$190$	0	0
$191$	6.05339	0.438008	0.219004	−	0.975724i	$-0.429719\pi$
$191$	6.05339	0.438008	0.219004	+	0.975724i	$0.429719\pi$
$192$	0	0
$193$	−18.1397	−1.30572	−0.652861	−	0.757478i	$-0.726432\pi$
$193$	−18.1397	−1.30572	−0.652861	+	0.757478i	$0.726432\pi$
$194$	0	0
$195$	6.06794	0.434534
$196$	0	0
$197$	−4.35026	−0.309943	−0.154972	−	0.987919i	$-0.549529\pi$
$197$	−4.35026	−0.309943	−0.154972	+	0.987919i	$0.549529\pi$
$198$	0	0
$199$	−3.40997	−0.241726	−0.120863	−	0.992669i	$-0.538566\pi$
$199$	−3.40997	−0.241726	−0.120863	+	0.992669i	$0.538566\pi$
$200$	0	0
$201$	−25.4449	−1.79474
$202$	0	0
$203$	3.25238	0.228272
$204$	0	0
$205$	−4.43213	−0.309553
$206$	0	0
$207$	10.5318	0.732009
$208$	0	0
$209$	28.0474	1.94008
$210$	0	0
$211$	−18.8371	−1.29680	−0.648399	−	0.761301i	$-0.724561\pi$
$211$	−18.8371	−1.29680	−0.648399	+	0.761301i	$0.724561\pi$
$212$	0	0
$213$	−2.75423	−0.188717
$214$	0	0
$215$	−4.30286	−0.293453
$216$	0	0
$217$	15.4000	1.04542
$218$	0	0
$219$	10.8208	0.731203
$220$	0	0
$221$	15.2562	1.02624
$222$	0	0
$223$	24.5844	1.64630	0.823148	−	0.567827i	$-0.192216\pi$
$223$	24.5844	1.64630	0.823148	+	0.567827i	$0.192216\pi$
$224$	0	0
$225$	1.36419	0.0909458
$226$	0	0
$227$	16.5193	1.09643	0.548213	−	0.836339i	$-0.315308\pi$
$227$	16.5193	1.09643	0.548213	+	0.836339i	$0.315308\pi$
$228$	0	0
$229$	6.08057	0.401815	0.200908	−	0.979610i	$-0.435611\pi$
$229$	6.08057	0.401815	0.200908	+	0.979610i	$0.435611\pi$
$230$	0	0
$231$	−38.1614	−2.51084
$232$	0	0
$233$	−1.78902	−0.117202	−0.0586012	−	0.998281i	$-0.518664\pi$
$233$	−1.78902	−0.117202	−0.0586012	+	0.998281i	$0.518664\pi$
$234$	0	0
$235$	−2.19267	−0.143034
$236$	0	0
$237$	24.0857	1.56453
$238$	0	0
$239$	−8.79725	−0.569047	−0.284523	−	0.958669i	$-0.591835\pi$
$239$	−8.79725	−0.569047	−0.284523	+	0.958669i	$0.591835\pi$
$240$	0	0
$241$	3.13840	0.202162	0.101081	−	0.994878i	$-0.467770\pi$
$241$	3.13840	0.202162	0.101081	+	0.994878i	$0.467770\pi$
$242$	0	0
$243$	−13.2115	−0.847517
$244$	0	0
$245$	3.57799	0.228589
$246$	0	0
$247$	14.5048	0.922916
$248$	0	0
$249$	−4.29948	−0.272469
$250$	0	0
$251$	−9.27730	−0.585578	−0.292789	−	0.956177i	$-0.594583\pi$
$251$	−9.27730	−0.585578	−0.292789	+	0.956177i	$0.594583\pi$
$252$	0	0
$253$	43.3609	2.72608
$254$	0	0
$255$	10.9726	0.687128
$256$	0	0
$257$	6.80101	0.424235	0.212118	−	0.977244i	$-0.431964\pi$
$257$	6.80101	0.424235	0.212118	+	0.977244i	$0.431964\pi$
$258$	0	0
$259$	−21.5464	−1.33883
$260$	0	0
$261$	−1.36419	−0.0844411
$262$	0	0
$263$	7.10481	0.438101	0.219051	−	0.975713i	$-0.429704\pi$
$263$	7.10481	0.438101	0.219051	+	0.975713i	$0.429704\pi$
$264$	0	0
$265$	−5.81394	−0.357147
$266$	0	0
$267$	4.55025	0.278471
$268$	0	0
$269$	20.4968	1.24971	0.624857	−	0.780739i	$-0.285157\pi$
$269$	20.4968	1.24971	0.624857	+	0.780739i	$0.285157\pi$
$270$	0	0
$271$	14.6646	0.890810	0.445405	−	0.895329i	$-0.353060\pi$
$271$	14.6646	0.890810	0.445405	+	0.895329i	$0.353060\pi$
$272$	0	0
$273$	−19.7353	−1.19443
$274$	0	0
$275$	5.61657	0.338692
$276$	0	0
$277$	16.2710	0.977629	0.488814	−	0.872388i	$-0.337429\pi$
$277$	16.2710	0.977629	0.488814	+	0.872388i	$0.337429\pi$
$278$	0	0
$279$	−6.45941	−0.386715
$280$	0	0
$281$	8.89804	0.530813	0.265406	−	0.964137i	$-0.414494\pi$
$281$	8.89804	0.530813	0.265406	+	0.964137i	$0.414494\pi$
$282$	0	0
$283$	3.61657	0.214983	0.107491	−	0.994206i	$-0.465718\pi$
$283$	3.61657	0.214983	0.107491	+	0.994206i	$0.465718\pi$
$284$	0	0
$285$	10.4321	0.617946
$286$	0	0
$287$	14.4150	0.850889
$288$	0	0
$289$	10.5875	0.622795
$290$	0	0
$291$	27.0799	1.58745
$292$	0	0
$293$	10.0442	0.586786	0.293393	−	0.955992i	$-0.405215\pi$
$293$	10.0442	0.586786	0.293393	+	0.955992i	$0.405215\pi$
$294$	0	0
$295$	8.90462	0.518447
$296$	0	0
$297$	−19.1936	−1.11373
$298$	0	0
$299$	22.4242	1.29682
$300$	0	0
$301$	13.9945	0.806632
$302$	0	0
$303$	4.76669	0.273839
$304$	0	0
$305$	−13.2313	−0.757620
$306$	0	0
$307$	−7.79470	−0.444867	−0.222433	−	0.974948i	$-0.571400\pi$
$307$	−7.79470	−0.444867	−0.222433	+	0.974948i	$0.571400\pi$
$308$	0	0
$309$	17.8856	1.01748
$310$	0	0
$311$	−22.2110	−1.25947	−0.629736	−	0.776810i	$-0.716837\pi$
$311$	−22.2110	−1.25947	−0.629736	+	0.776810i	$0.716837\pi$
$312$	0	0
$313$	−0.173597	−0.00981231	−0.00490615	−	0.999988i	$-0.501562\pi$
$313$	−0.173597	−0.00981231	−0.00490615	+	0.999988i	$0.501562\pi$
$314$	0	0
$315$	−4.43686	−0.249989
$316$	0	0
$317$	−9.81186	−0.551089	−0.275544	−	0.961288i	$-0.588858\pi$
$317$	−9.81186	−0.551089	−0.275544	+	0.961288i	$0.588858\pi$
$318$	0	0
$319$	−5.61657	−0.314467
$320$	0	0
$321$	33.5808	1.87430
$322$	0	0
$323$	26.2287	1.45941
$324$	0	0
$325$	2.90462	0.161119
$326$	0	0
$327$	−11.6053	−0.641772
$328$	0	0
$329$	7.13141	0.393167
$330$	0	0
$331$	−8.41435	−0.462494	−0.231247	−	0.972895i	$-0.574281\pi$
$331$	−8.41435	−0.462494	−0.231247	+	0.972895i	$0.574281\pi$
$332$	0	0
$333$	9.03747	0.495250
$334$	0	0
$335$	−12.1800	−0.665467
$336$	0	0
$337$	30.9215	1.68440	0.842200	−	0.539165i	$-0.181260\pi$
$337$	30.9215	1.68440	0.842200	+	0.539165i	$0.181260\pi$
$338$	0	0
$339$	−10.4968	−0.570110
$340$	0	0
$341$	−26.5944	−1.44017
$342$	0	0
$343$	11.1297	0.600947
$344$	0	0
$345$	16.1279	0.868299
$346$	0	0
$347$	−28.1719	−1.51235	−0.756175	−	0.654370i	$-0.772934\pi$
$347$	−28.1719	−1.51235	−0.756175	+	0.654370i	$0.772934\pi$
$348$	0	0
$349$	−20.5124	−1.09800	−0.549001	−	0.835822i	$-0.684992\pi$
$349$	−20.5124	−1.09800	−0.549001	+	0.835822i	$0.684992\pi$
$350$	0	0
$351$	−9.92601	−0.529811
$352$	0	0
$353$	8.13135	0.432788	0.216394	−	0.976306i	$-0.430570\pi$
$353$	8.13135	0.432788	0.216394	+	0.976306i	$0.430570\pi$
$354$	0	0
$355$	−1.31840	−0.0699737
$356$	0	0
$357$	−35.6870	−1.88875
$358$	0	0
$359$	19.7802	1.04396	0.521979	−	0.852958i	$-0.325194\pi$
$359$	19.7802	1.04396	0.521979	+	0.852958i	$0.325194\pi$
$360$	0	0
$361$	5.93689	0.312468
$362$	0	0
$363$	42.9216	2.25280
$364$	0	0
$365$	5.17974	0.271120
$366$	0	0
$367$	23.6166	1.23277	0.616387	−	0.787443i	$-0.288596\pi$
$367$	23.6166	1.23277	0.616387	+	0.787443i	$0.288596\pi$
$368$	0	0
$369$	−6.04625	−0.314755
$370$	0	0
$371$	18.9092	0.981714
$372$	0	0
$373$	−14.8165	−0.767171	−0.383586	−	0.923505i	$-0.625311\pi$
$373$	−14.8165	−0.767171	−0.383586	+	0.923505i	$0.625311\pi$
$374$	0	0
$375$	2.08906	0.107879
$376$	0	0
$377$	−2.90462	−0.149596
$378$	0	0
$379$	17.3803	0.892767	0.446384	−	0.894842i	$-0.352712\pi$
$379$	17.3803	0.892767	0.446384	+	0.894842i	$0.352712\pi$
$380$	0	0
$381$	−46.5654	−2.38562
$382$	0	0
$383$	35.7445	1.82646	0.913230	−	0.407445i	$-0.133580\pi$
$383$	35.7445	1.82646	0.913230	+	0.407445i	$0.133580\pi$
$384$	0	0
$385$	−18.2672	−0.930984
$386$	0	0
$387$	−5.86991	−0.298384
$388$	0	0
$389$	−31.5332	−1.59880	−0.799399	−	0.600801i	$-0.794849\pi$
$389$	−31.5332	−1.59880	−0.799399	+	0.600801i	$0.794849\pi$
$390$	0	0
$391$	40.5493	2.05067
$392$	0	0
$393$	4.75876	0.240048
$394$	0	0
$395$	11.5294	0.580108
$396$	0	0
$397$	10.4538	0.524662	0.262331	−	0.964978i	$-0.415509\pi$
$397$	10.4538	0.524662	0.262331	+	0.964978i	$0.415509\pi$
$398$	0	0
$399$	−33.9293	−1.69859
$400$	0	0
$401$	−16.2243	−0.810204	−0.405102	−	0.914272i	$-0.632764\pi$
$401$	−16.2243	−0.810204	−0.405102	+	0.914272i	$0.632764\pi$
$402$	0	0
$403$	−13.7533	−0.685103
$404$	0	0
$405$	−11.2316	−0.558100
$406$	0	0
$407$	37.2086	1.84436
$408$	0	0
$409$	−30.2452	−1.49553	−0.747764	−	0.663964i	$-0.768873\pi$
$409$	−30.2452	−1.49553	−0.747764	+	0.663964i	$0.768873\pi$
$410$	0	0
$411$	−28.1053	−1.38633
$412$	0	0
$413$	−28.9612	−1.42509
$414$	0	0
$415$	−2.05809	−0.101028
$416$	0	0
$417$	44.2431	2.16659
$418$	0	0
$419$	38.8437	1.89764	0.948819	−	0.315820i	$-0.102279\pi$
$419$	38.8437	1.89764	0.948819	+	0.315820i	$0.102279\pi$
$420$	0	0
$421$	−11.6747	−0.568988	−0.284494	−	0.958678i	$-0.591826\pi$
$421$	−11.6747	−0.568988	−0.284494	+	0.958678i	$0.591826\pi$
$422$	0	0
$423$	−2.99122	−0.145438
$424$	0	0
$425$	5.25238	0.254778
$426$	0	0
$427$	43.0331	2.08252
$428$	0	0
$429$	34.0810	1.64545
$430$	0	0
$431$	−12.2943	−0.592196	−0.296098	−	0.955158i	$-0.595686\pi$
$431$	−12.2943	−0.592196	−0.296098	+	0.955158i	$0.595686\pi$
$432$	0	0
$433$	−26.9426	−1.29478	−0.647389	−	0.762160i	$-0.724139\pi$
$433$	−26.9426	−1.29478	−0.647389	+	0.762160i	$0.724139\pi$
$434$	0	0
$435$	−2.08906	−0.100163
$436$	0	0
$437$	38.5521	1.84420
$438$	0	0
$439$	28.5976	1.36489	0.682445	−	0.730937i	$-0.260917\pi$
$439$	28.5976	1.36489	0.682445	+	0.730937i	$0.260917\pi$
$440$	0	0
$441$	4.88104	0.232431
$442$	0	0
$443$	7.75303	0.368358	0.184179	−	0.982893i	$-0.441037\pi$
$443$	7.75303	0.368358	0.184179	+	0.982893i	$0.441037\pi$
$444$	0	0
$445$	2.17813	0.103253
$446$	0	0
$447$	4.61102	0.218094
$448$	0	0
$449$	−28.4360	−1.34198	−0.670988	−	0.741468i	$-0.734130\pi$
$449$	−28.4360	−1.34198	−0.670988	+	0.741468i	$0.734130\pi$
$450$	0	0
$451$	−24.8933	−1.17218
$452$	0	0
$453$	49.0594	2.30501
$454$	0	0
$455$	−9.44694	−0.442879
$456$	0	0
$457$	34.3153	1.60520	0.802601	−	0.596516i	$-0.203449\pi$
$457$	34.3153	1.60520	0.802601	+	0.596516i	$0.203449\pi$
$458$	0	0
$459$	−17.9491	−0.837790
$460$	0	0
$461$	−9.15257	−0.426278	−0.213139	−	0.977022i	$-0.568369\pi$
$461$	−9.15257	−0.426278	−0.213139	+	0.977022i	$0.568369\pi$
$462$	0	0
$463$	38.4353	1.78624	0.893121	−	0.449817i	$-0.148511\pi$
$463$	38.4353	1.78624	0.893121	+	0.449817i	$0.148511\pi$
$464$	0	0
$465$	−9.89169	−0.458716
$466$	0	0
$467$	27.1639	1.25699	0.628497	−	0.777812i	$-0.283670\pi$
$467$	27.1639	1.25699	0.628497	+	0.777812i	$0.283670\pi$
$468$	0	0
$469$	39.6141	1.82921
$470$	0	0
$471$	43.9285	2.02412
$472$	0	0
$473$	−24.1673	−1.11121
$474$	0	0
$475$	4.99368	0.229126
$476$	0	0
$477$	−7.93130	−0.363149
$478$	0	0
$479$	−32.2993	−1.47579	−0.737896	−	0.674914i	$-0.764181\pi$
$479$	−32.2993	−1.47579	−0.737896	+	0.674914i	$0.764181\pi$
$480$	0	0
$481$	19.2425	0.877384
$482$	0	0
$483$	−52.4542	−2.38675
$484$	0	0
$485$	12.9627	0.588606
$486$	0	0
$487$	6.60516	0.299308	0.149654	−	0.988738i	$-0.452184\pi$
$487$	6.60516	0.299308	0.149654	+	0.988738i	$0.452184\pi$
$488$	0	0
$489$	−30.3936	−1.37445
$490$	0	0
$491$	12.6703	0.571802	0.285901	−	0.958259i	$-0.407707\pi$
$491$	12.6703	0.571802	0.285901	+	0.958259i	$0.407707\pi$
$492$	0	0
$493$	−5.25238	−0.236555
$494$	0	0
$495$	7.66205	0.344384
$496$	0	0
$497$	4.28796	0.192341
$498$	0	0
$499$	−26.7827	−1.19896	−0.599480	−	0.800390i	$-0.704626\pi$
$499$	−26.7827	−1.19896	−0.599480	+	0.800390i	$0.704626\pi$
$500$	0	0
$501$	−9.54866	−0.426603
$502$	0	0
$503$	−12.8042	−0.570912	−0.285456	−	0.958392i	$-0.592145\pi$
$503$	−12.8042	−0.570912	−0.285456	+	0.958392i	$0.592145\pi$
$504$	0	0
$505$	2.28174	0.101536
$506$	0	0
$507$	−9.53277	−0.423365
$508$	0	0
$509$	−35.9280	−1.59248	−0.796239	−	0.604982i	$-0.793180\pi$
$509$	−35.9280	−1.59248	−0.796239	+	0.604982i	$0.793180\pi$
$510$	0	0
$511$	−16.8465	−0.745246
$512$	0	0
$513$	−17.0650	−0.753438
$514$	0	0
$515$	8.56156	0.377267
$516$	0	0
$517$	−12.3153	−0.541626
$518$	0	0
$519$	−35.8482	−1.57356
$520$	0	0
$521$	−39.1375	−1.71464	−0.857322	−	0.514781i	$-0.827873\pi$
$521$	−39.1375	−1.71464	−0.857322	+	0.514781i	$0.827873\pi$
$522$	0	0
$523$	−0.0271758	−0.00118831	−0.000594157	−	1.00000i	$-0.500189\pi$
$523$	−0.0271758	−0.00118831	−0.000594157		1.00000i	$0.500189\pi$
$524$	0	0
$525$	−6.79443	−0.296533
$526$	0	0
$527$	−24.8700	−1.08335
$528$	0	0
$529$	36.6012	1.59135
$530$	0	0
$531$	12.1476	0.527160
$532$	0	0
$533$	−12.8736	−0.557620
$534$	0	0
$535$	16.0746	0.694963
$536$	0	0
$537$	−28.2953	−1.22103
$538$	0	0
$539$	20.0960	0.865596
$540$	0	0
$541$	0.693359	0.0298098	0.0149049	−	0.999889i	$-0.495255\pi$
$541$	0.693359	0.0298098	0.0149049	+	0.999889i	$0.495255\pi$
$542$	0	0
$543$	3.24900	0.139428
$544$	0	0
$545$	−5.55524	−0.237960
$546$	0	0
$547$	0.812323	0.0347324	0.0173662	−	0.999849i	$-0.494472\pi$
$547$	0.812323	0.0347324	0.0173662	+	0.999849i	$0.494472\pi$
$548$	0	0
$549$	−18.0499	−0.770352
$550$	0	0
$551$	−4.99368	−0.212738
$552$	0	0
$553$	−37.4981	−1.59458
$554$	0	0
$555$	13.8396	0.587460
$556$	0	0
$557$	−17.0190	−0.721117	−0.360559	−	0.932737i	$-0.617414\pi$
$557$	−17.0190	−0.721117	−0.360559	+	0.932737i	$0.617414\pi$
$558$	0	0
$559$	−12.4982	−0.528617
$560$	0	0
$561$	61.6281	2.60194
$562$	0	0
$563$	−42.3393	−1.78439	−0.892195	−	0.451651i	$-0.850835\pi$
$563$	−42.3393	−1.78439	−0.892195	+	0.451651i	$0.850835\pi$
$564$	0	0
$565$	−5.02466	−0.211389
$566$	0	0
$567$	36.5293	1.53409
$568$	0	0
$569$	31.4910	1.32017	0.660085	−	0.751191i	$-0.270520\pi$
$569$	31.4910	1.32017	0.660085	+	0.751191i	$0.270520\pi$
$570$	0	0
$571$	6.64123	0.277927	0.138963	−	0.990298i	$-0.455623\pi$
$571$	6.64123	0.277927	0.138963	+	0.990298i	$0.455623\pi$
$572$	0	0
$573$	12.6459	0.528291
$574$	0	0
$575$	7.72018	0.321954
$576$	0	0
$577$	−23.7281	−0.987812	−0.493906	−	0.869515i	$-0.664431\pi$
$577$	−23.7281	−0.987812	−0.493906	+	0.869515i	$0.664431\pi$
$578$	0	0
$579$	−37.8949	−1.57486
$580$	0	0
$581$	6.69370	0.277701
$582$	0	0
$583$	−32.6544	−1.35241
$584$	0	0
$585$	3.96245	0.163827
$586$	0	0
$587$	−2.15042	−0.0887575	−0.0443787	−	0.999015i	$-0.514131\pi$
$587$	−2.15042	−0.0887575	−0.0443787	+	0.999015i	$0.514131\pi$
$588$	0	0
$589$	−23.6450	−0.974277
$590$	0	0
$591$	−9.08797	−0.373829
$592$	0	0
$593$	4.42649	0.181774	0.0908872	−	0.995861i	$-0.471030\pi$
$593$	4.42649	0.181774	0.0908872	+	0.995861i	$0.471030\pi$
$594$	0	0
$595$	−17.0827	−0.700324
$596$	0	0
$597$	−7.12365	−0.291551
$598$	0	0
$599$	35.5803	1.45377	0.726886	−	0.686758i	$-0.240967\pi$
$599$	35.5803	1.45377	0.726886	+	0.686758i	$0.240967\pi$
$600$	0	0
$601$	−46.3708	−1.89150	−0.945752	−	0.324890i	$-0.894673\pi$
$601$	−46.3708	−1.89150	−0.945752	+	0.324890i	$0.894673\pi$
$602$	0	0
$603$	−16.6159	−0.676650
$604$	0	0
$605$	20.5458	0.835308
$606$	0	0
$607$	10.9595	0.444834	0.222417	−	0.974952i	$-0.428605\pi$
$607$	10.9595	0.444834	0.222417	+	0.974952i	$0.428605\pi$
$608$	0	0
$609$	6.79443	0.275324
$610$	0	0
$611$	−6.36889	−0.257658
$612$	0	0
$613$	−20.8227	−0.841021	−0.420510	−	0.907288i	$-0.638149\pi$
$613$	−20.8227	−0.841021	−0.420510	+	0.907288i	$0.638149\pi$
$614$	0	0
$615$	−9.25899	−0.373359
$616$	0	0
$617$	−40.8665	−1.64522	−0.822612	−	0.568603i	$-0.807484\pi$
$617$	−40.8665	−1.64522	−0.822612	+	0.568603i	$0.807484\pi$
$618$	0	0
$619$	−20.7890	−0.835581	−0.417790	−	0.908544i	$-0.637195\pi$
$619$	−20.7890	−0.835581	−0.417790	+	0.908544i	$0.637195\pi$
$620$	0	0
$621$	−26.3823	−1.05869
$622$	0	0
$623$	−7.08410	−0.283819
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	58.5928	2.33997
$628$	0	0
$629$	34.7960	1.38741
$630$	0	0
$631$	26.6350	1.06032	0.530161	−	0.847897i	$-0.322131\pi$
$631$	26.6350	1.06032	0.530161	+	0.847897i	$0.322131\pi$
$632$	0	0
$633$	−39.3519	−1.56410
$634$	0	0
$635$	−22.2901	−0.884554
$636$	0	0
$637$	10.3927	0.411773
$638$	0	0
$639$	−1.79855	−0.0711496
$640$	0	0
$641$	22.8301	0.901735	0.450867	−	0.892591i	$-0.351115\pi$
$641$	22.8301	0.901735	0.450867	+	0.892591i	$0.351115\pi$
$642$	0	0
$643$	−18.5395	−0.731128	−0.365564	−	0.930786i	$-0.619124\pi$
$643$	−18.5395	−0.731128	−0.365564	+	0.930786i	$0.619124\pi$
$644$	0	0
$645$	−8.98895	−0.353940
$646$	0	0
$647$	23.0121	0.904701	0.452350	−	0.891840i	$-0.350586\pi$
$647$	23.0121	0.904701	0.452350	+	0.891840i	$0.350586\pi$
$648$	0	0
$649$	50.0134	1.96320
$650$	0	0
$651$	32.1716	1.26090
$652$	0	0
$653$	21.1467	0.827535	0.413768	−	0.910383i	$-0.364213\pi$
$653$	21.1467	0.827535	0.413768	+	0.910383i	$0.364213\pi$
$654$	0	0
$655$	2.27794	0.0890065
$656$	0	0
$657$	7.06614	0.275677
$658$	0	0
$659$	−27.9257	−1.08783	−0.543916	−	0.839140i	$-0.683059\pi$
$659$	−27.9257	−1.08783	−0.543916	+	0.839140i	$0.683059\pi$
$660$	0	0
$661$	−3.64305	−0.141698	−0.0708490	−	0.997487i	$-0.522571\pi$
$661$	−3.64305	−0.141698	−0.0708490	+	0.997487i	$0.522571\pi$
$662$	0	0
$663$	31.8711	1.23777
$664$	0	0
$665$	−16.2414	−0.629813
$666$	0	0
$667$	−7.72018	−0.298927
$668$	0	0
$669$	51.3584	1.98563
$670$	0	0
$671$	−74.3143	−2.86887
$672$	0	0
$673$	34.5805	1.33298	0.666489	−	0.745514i	$-0.267796\pi$
$673$	34.5805	1.33298	0.666489	+	0.745514i	$0.267796\pi$
$674$	0	0
$675$	−3.41732	−0.131533
$676$	0	0
$677$	−30.8756	−1.18664	−0.593322	−	0.804965i	$-0.702184\pi$
$677$	−30.8756	−1.18664	−0.593322	+	0.804965i	$0.702184\pi$
$678$	0	0
$679$	−42.1597	−1.61794
$680$	0	0
$681$	34.5099	1.32242
$682$	0	0
$683$	−40.4442	−1.54755	−0.773777	−	0.633458i	$-0.781635\pi$
$683$	−40.4442	−1.54755	−0.773777	+	0.633458i	$0.781635\pi$
$684$	0	0
$685$	−13.4535	−0.514034
$686$	0	0
$687$	12.7027	0.484638
$688$	0	0
$689$	−16.8873	−0.643355
$690$	0	0
$691$	28.7243	1.09272	0.546361	−	0.837550i	$-0.316013\pi$
$691$	28.7243	1.09272	0.546361	+	0.837550i	$0.316013\pi$
$692$	0	0
$693$	−24.9199	−0.946630
$694$	0	0
$695$	21.1784	0.803343
$696$	0	0
$697$	−23.2792	−0.881763
$698$	0	0
$699$	−3.73737	−0.141360
$700$	0	0
$701$	−28.1207	−1.06210	−0.531052	−	0.847339i	$-0.678203\pi$
$701$	−28.1207	−1.06210	−0.531052	+	0.847339i	$0.678203\pi$
$702$	0	0
$703$	33.0822	1.24772
$704$	0	0
$705$	−4.58063	−0.172517
$706$	0	0
$707$	−7.42108	−0.279098
$708$	0	0
$709$	22.4908	0.844661	0.422331	−	0.906442i	$-0.361212\pi$
$709$	22.4908	0.844661	0.422331	+	0.906442i	$0.361212\pi$
$710$	0	0
$711$	15.7283	0.589857
$712$	0	0
$713$	−36.5550	−1.36899
$714$	0	0
$715$	16.3140	0.610109
$716$	0	0
$717$	−18.3780	−0.686340
$718$	0	0
$719$	30.6526	1.14315	0.571574	−	0.820551i	$-0.306333\pi$
$719$	30.6526	1.14315	0.571574	+	0.820551i	$0.306333\pi$
$720$	0	0
$721$	−27.8455	−1.03702
$722$	0	0
$723$	6.55631	0.243832
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	30.9437	1.14764	0.573819	−	0.818982i	$-0.305461\pi$
$727$	30.9437	1.14764	0.573819	+	0.818982i	$0.305461\pi$
$728$	0	0
$729$	6.09503	0.225742
$730$	0	0
$731$	−22.6003	−0.835901
$732$	0	0
$733$	−25.0013	−0.923443	−0.461722	−	0.887025i	$-0.652768\pi$
$733$	−25.0013	−0.923443	−0.461722	+	0.887025i	$0.652768\pi$
$734$	0	0
$735$	7.47464	0.275706
$736$	0	0
$737$	−68.4101	−2.51992
$738$	0	0
$739$	−19.6569	−0.723090	−0.361545	−	0.932355i	$-0.617751\pi$
$739$	−19.6569	−0.723090	−0.361545	+	0.932355i	$0.617751\pi$
$740$	0	0
$741$	30.3014	1.11315
$742$	0	0
$743$	−9.06572	−0.332589	−0.166295	−	0.986076i	$-0.553180\pi$
$743$	−9.06572	−0.332589	−0.166295	+	0.986076i	$0.553180\pi$
$744$	0	0
$745$	2.20722	0.0808662
$746$	0	0
$747$	−2.80762	−0.102726
$748$	0	0
$749$	−52.2806	−1.91029
$750$	0	0
$751$	−38.6602	−1.41073	−0.705366	−	0.708843i	$-0.749217\pi$
$751$	−38.6602	−1.41073	−0.705366	+	0.708843i	$0.749217\pi$
$752$	0	0
$753$	−19.3809	−0.706278
$754$	0	0
$755$	23.4839	0.854667
$756$	0	0
$757$	−22.1706	−0.805806	−0.402903	−	0.915243i	$-0.631999\pi$
$757$	−22.1706	−0.805806	−0.402903	+	0.915243i	$0.631999\pi$
$758$	0	0
$759$	90.5837	3.28798
$760$	0	0
$761$	−1.14874	−0.0416417	−0.0208209	−	0.999783i	$-0.506628\pi$
$761$	−1.14874	−0.0416417	−0.0208209	+	0.999783i	$0.506628\pi$
$762$	0	0
$763$	18.0678	0.654097
$764$	0	0
$765$	7.16523	0.259060
$766$	0	0
$767$	25.8646	0.933915
$768$	0	0
$769$	−32.1111	−1.15796	−0.578979	−	0.815343i	$-0.696549\pi$
$769$	−32.1111	−1.15796	−0.578979	+	0.815343i	$0.696549\pi$
$770$	0	0
$771$	14.2077	0.511680
$772$	0	0
$773$	−8.68727	−0.312460	−0.156230	−	0.987721i	$-0.549934\pi$
$773$	−8.68727	−0.312460	−0.156230	+	0.987721i	$0.549934\pi$
$774$	0	0
$775$	−4.73499	−0.170086
$776$	0	0
$777$	−45.0118	−1.61479
$778$	0	0
$779$	−22.1326	−0.792984
$780$	0	0
$781$	−7.40491	−0.264969
$782$	0	0
$783$	3.41732	0.122125
$784$	0	0
$785$	21.0278	0.750516
$786$	0	0
$787$	−54.9945	−1.96034	−0.980171	−	0.198155i	$-0.936505\pi$
$787$	−54.9945	−1.96034	−0.980171	+	0.198155i	$0.936505\pi$
$788$	0	0
$789$	14.8424	0.528403
$790$	0	0
$791$	16.3421	0.581058
$792$	0	0
$793$	−38.4318	−1.36475
$794$	0	0
$795$	−12.1457	−0.430763
$796$	0	0
$797$	9.95960	0.352787	0.176394	−	0.984320i	$-0.443557\pi$
$797$	9.95960	0.352787	0.176394	+	0.984320i	$0.443557\pi$
$798$	0	0
$799$	−11.5168	−0.407434
$800$	0	0
$801$	2.97137	0.104988
$802$	0	0
$803$	29.0924	1.02665
$804$	0	0
$805$	−25.1090	−0.884975
$806$	0	0
$807$	42.8192	1.50731
$808$	0	0
$809$	−5.59503	−0.196711	−0.0983553	−	0.995151i	$-0.531358\pi$
$809$	−5.59503	−0.196711	−0.0983553	+	0.995151i	$0.531358\pi$
$810$	0	0
$811$	19.0351	0.668414	0.334207	−	0.942500i	$-0.391531\pi$
$811$	19.0351	0.668414	0.334207	+	0.942500i	$0.391531\pi$
$812$	0	0
$813$	30.6352	1.07443
$814$	0	0
$815$	−14.5489	−0.509627
$816$	0	0
$817$	−21.4871	−0.751740
$818$	0	0
$819$	−12.8874	−0.450322
$820$	0	0
$821$	−41.4720	−1.44738	−0.723692	−	0.690123i	$-0.757556\pi$
$821$	−41.4720	−1.44738	−0.723692	+	0.690123i	$0.757556\pi$
$822$	0	0
$823$	−46.8732	−1.63390	−0.816949	−	0.576710i	$-0.804336\pi$
$823$	−46.8732	−1.63390	−0.816949	+	0.576710i	$0.804336\pi$
$824$	0	0
$825$	11.7334	0.408504
$826$	0	0
$827$	39.8556	1.38592	0.692958	−	0.720978i	$-0.256307\pi$
$827$	39.8556	1.38592	0.692958	+	0.720978i	$0.256307\pi$
$828$	0	0
$829$	−12.8097	−0.444900	−0.222450	−	0.974944i	$-0.571405\pi$
$829$	−12.8097	−0.444900	−0.222450	+	0.974944i	$0.571405\pi$
$830$	0	0
$831$	33.9911	1.17914
$832$	0	0
$833$	18.7929	0.651137
$834$	0	0
$835$	−4.57079	−0.158179
$836$	0	0
$837$	16.1810	0.559296
$838$	0	0
$839$	11.3442	0.391644	0.195822	−	0.980639i	$-0.437262\pi$
$839$	11.3442	0.391644	0.195822	+	0.980639i	$0.437262\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	18.5886	0.640225
$844$	0	0
$845$	−4.56318	−0.156978
$846$	0	0
$847$	−66.8229	−2.29606
$848$	0	0
$849$	7.55524	0.259295
$850$	0	0
$851$	51.1446	1.75322
$852$	0	0
$853$	30.1206	1.03131	0.515655	−	0.856796i	$-0.327548\pi$
$853$	30.1206	1.03131	0.515655	+	0.856796i	$0.327548\pi$
$854$	0	0
$855$	6.81232	0.232977
$856$	0	0
$857$	−29.0771	−0.993256	−0.496628	−	0.867964i	$-0.665429\pi$
$857$	−29.0771	−0.993256	−0.496628	+	0.867964i	$0.665429\pi$
$858$	0	0
$859$	41.3102	1.40949	0.704743	−	0.709462i	$-0.251062\pi$
$859$	41.3102	1.40949	0.704743	+	0.709462i	$0.251062\pi$
$860$	0	0
$861$	30.1138	1.02628
$862$	0	0
$863$	9.60206	0.326858	0.163429	−	0.986555i	$-0.447745\pi$
$863$	9.60206	0.326858	0.163429	+	0.986555i	$0.447745\pi$
$864$	0	0
$865$	−17.1599	−0.583455
$866$	0	0
$867$	22.1180	0.751166
$868$	0	0
$869$	64.7558	2.19669
$870$	0	0
$871$	−35.3784	−1.19875
$872$	0	0
$873$	17.6836	0.598498
$874$	0	0
$875$	−3.25238	−0.109951
$876$	0	0
$877$	−29.7562	−1.00480	−0.502398	−	0.864636i	$-0.667549\pi$
$877$	−29.7562	−1.00480	−0.502398	+	0.864636i	$0.667549\pi$
$878$	0	0
$879$	20.9829	0.707736
$880$	0	0
$881$	2.34996	0.0791720	0.0395860	−	0.999216i	$-0.487396\pi$
$881$	2.34996	0.0791720	0.0395860	+	0.999216i	$0.487396\pi$
$882$	0	0
$883$	−42.9628	−1.44581	−0.722907	−	0.690946i	$-0.757194\pi$
$883$	−42.9628	−1.44581	−0.722907	+	0.690946i	$0.757194\pi$
$884$	0	0
$885$	18.6023	0.625310
$886$	0	0
$887$	12.3872	0.415921	0.207961	−	0.978137i	$-0.433317\pi$
$887$	12.3872	0.415921	0.207961	+	0.978137i	$0.433317\pi$
$888$	0	0
$889$	72.4958	2.43143
$890$	0	0
$891$	−63.0828	−2.11335
$892$	0	0
$893$	−10.9495	−0.366412
$894$	0	0
$895$	−13.5445	−0.452743
$896$	0	0
$897$	46.8456	1.56413
$898$	0	0
$899$	4.73499	0.157921
$900$	0	0
$901$	−30.5370	−1.01734
$902$	0	0
$903$	29.2355	0.972897
$904$	0	0
$905$	1.55524	0.0516980
$906$	0	0
$907$	46.4672	1.54292	0.771459	−	0.636279i	$-0.219527\pi$
$907$	46.4672	1.54292	0.771459	+	0.636279i	$0.219527\pi$
$908$	0	0
$909$	3.11272	0.103242
$910$	0	0
$911$	−9.63348	−0.319171	−0.159586	−	0.987184i	$-0.551016\pi$
$911$	−9.63348	−0.319171	−0.159586	+	0.987184i	$0.551016\pi$
$912$	0	0
$913$	−11.5594	−0.382561
$914$	0	0
$915$	−27.6409	−0.913781
$916$	0	0
$917$	−7.40873	−0.244658
$918$	0	0
$919$	30.5201	1.00677	0.503383	−	0.864063i	$-0.332089\pi$
$919$	30.5201	1.00677	0.503383	+	0.864063i	$0.332089\pi$
$920$	0	0
$921$	−16.2836	−0.536563
$922$	0	0
$923$	−3.82947	−0.126048
$924$	0	0
$925$	6.62480	0.217822
$926$	0	0
$927$	11.6796	0.383607
$928$	0	0
$929$	21.9036	0.718633	0.359316	−	0.933216i	$-0.383010\pi$
$929$	21.9036	0.718633	0.359316	+	0.933216i	$0.383010\pi$
$930$	0	0
$931$	17.8673	0.585578
$932$	0	0
$933$	−46.4002	−1.51908
$934$	0	0
$935$	29.5004	0.964765
$936$	0	0
$937$	−35.7562	−1.16810	−0.584052	−	0.811716i	$-0.698534\pi$
$937$	−35.7562	−1.16810	−0.584052	+	0.811716i	$0.698534\pi$
$938$	0	0
$939$	−0.362656	−0.0118348
$940$	0	0
$941$	−23.2756	−0.758764	−0.379382	−	0.925240i	$-0.623863\pi$
$941$	−23.2756	−0.758764	−0.379382	+	0.925240i	$0.623863\pi$
$942$	0	0
$943$	−34.2168	−1.11425
$944$	0	0
$945$	11.1144	0.361552
$946$	0	0
$947$	−21.0531	−0.684134	−0.342067	−	0.939676i	$-0.611127\pi$
$947$	−21.0531	−0.684134	−0.342067	+	0.939676i	$0.611127\pi$
$948$	0	0
$949$	15.0452	0.488388
$950$	0	0
$951$	−20.4976	−0.664680
$952$	0	0
$953$	26.0026	0.842307	0.421153	−	0.906989i	$-0.361625\pi$
$953$	26.0026	0.842307	0.421153	+	0.906989i	$0.361625\pi$
$954$	0	0
$955$	6.05339	0.195883
$956$	0	0
$957$	−11.7334	−0.379286
$958$	0	0
$959$	43.7561	1.41296
$960$	0	0
$961$	−8.57988	−0.276770
$962$	0	0
$963$	21.9287	0.706643
$964$	0	0
$965$	−18.1397	−0.583937
$966$	0	0
$967$	5.91411	0.190185	0.0950925	−	0.995468i	$-0.469685\pi$
$967$	5.91411	0.190185	0.0950925	+	0.995468i	$0.469685\pi$
$968$	0	0
$969$	54.7935	1.76022
$970$	0	0
$971$	−18.3368	−0.588457	−0.294229	−	0.955735i	$-0.595063\pi$
$971$	−18.3368	−0.588457	−0.294229	+	0.955735i	$0.595063\pi$
$972$	0	0
$973$	−68.8803	−2.20820
$974$	0	0
$975$	6.06794	0.194330
$976$	0	0
$977$	10.0172	0.320477	0.160239	−	0.987078i	$-0.448774\pi$
$977$	10.0172	0.320477	0.160239	+	0.987078i	$0.448774\pi$
$978$	0	0
$979$	12.2336	0.390988
$980$	0	0
$981$	−7.57839	−0.241959
$982$	0	0
$983$	−14.7655	−0.470948	−0.235474	−	0.971881i	$-0.575664\pi$
$983$	−14.7655	−0.470948	−0.235474	+	0.971881i	$0.575664\pi$
$984$	0	0
$985$	−4.35026	−0.138611
$986$	0	0
$987$	14.8980	0.474208
$988$	0	0
$989$	−33.2189	−1.05630
$990$	0	0
$991$	−15.9863	−0.507821	−0.253910	−	0.967228i	$-0.581717\pi$
$991$	−15.9863	−0.507821	−0.253910	+	0.967228i	$0.581717\pi$
$992$	0	0
$993$	−17.5781	−0.557824
$994$	0	0
$995$	−3.40997	−0.108103
$996$	0	0
$997$	3.85897	0.122215	0.0611074	−	0.998131i	$-0.480537\pi$
$997$	3.85897	0.122215	0.0611074	+	0.998131i	$0.480537\pi$
$998$	0	0
$999$	−22.6390	−0.716267

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9280.2.a.ck.1.4		5
4.3	odd	2		9280.2.a.ci.1.2		5
8.3	odd	2		2320.2.a.v.1.4		5
8.5	even	2		1160.2.a.h.1.2	✓	5
40.29	even	2		5800.2.a.u.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.2.a.h.1.2	✓	5	8.5	even	2
2320.2.a.v.1.4		5	8.3	odd	2
5800.2.a.u.1.4		5	40.29	even	2
9280.2.a.ci.1.2		5	4.3	odd	2
9280.2.a.ck.1.4		5	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:	\( N \)	\(=\)	\( 9280 = 2^{6} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9280.a (trivial)

\(f(q)\)	\(=\)	\(q+2.08906 q^{3} +1.00000 q^{5} -3.25238 q^{7} +1.36419 q^{9} +5.61657 q^{11} +2.90462 q^{13} +2.08906 q^{15} +5.25238 q^{17} +4.99368 q^{19} -6.79443 q^{21} +7.72018 q^{23} +1.00000 q^{25} -3.41732 q^{27} -1.00000 q^{29} -4.73499 q^{31} +11.7334 q^{33} -3.25238 q^{35} +6.62480 q^{37} +6.06794 q^{39} -4.43213 q^{41} -4.30286 q^{43} +1.36419 q^{45} -2.19267 q^{47} +3.57799 q^{49} +10.9726 q^{51} -5.81394 q^{53} +5.61657 q^{55} +10.4321 q^{57} +8.90462 q^{59} -13.2313 q^{61} -4.43686 q^{63} +2.90462 q^{65} -12.1800 q^{67} +16.1279 q^{69} -1.31840 q^{71} +5.17974 q^{73} +2.08906 q^{75} -18.2672 q^{77} +11.5294 q^{79} -11.2316 q^{81} -2.05809 q^{83} +5.25238 q^{85} -2.08906 q^{87} +2.17813 q^{89} -9.44694 q^{91} -9.89169 q^{93} +4.99368 q^{95} +12.9627 q^{97} +7.66205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} + 5 q^{5} + 7 q^{7} + 12 q^{9} + 10 q^{11} - 3 q^{13} + q^{15} + 3 q^{17} - 2 q^{19} - 4 q^{21} + 13 q^{23} + 5 q^{25} + 4 q^{27} - 5 q^{29} + 7 q^{31} + 12 q^{33} + 7 q^{35} - 10 q^{37} - q^{39}+ \cdots + 96 q^{99}+O(q^{100}) \) 5 * q + q^3 + 5 * q^5 + 7 * q^7 + 12 * q^9 + 10 * q^11 - 3 * q^13 + q^15 + 3 * q^17 - 2 * q^19 - 4 * q^21 + 13 * q^23 + 5 * q^25 + 4 * q^27 - 5 * q^29 + 7 * q^31 + 12 * q^33 + 7 * q^35 - 10 * q^37 - q^39 + 4 * q^41 - 17 * q^43 + 12 * q^45 + 6 * q^47 + 28 * q^49 + 6 * q^51 - 5 * q^53 + 10 * q^55 + 26 * q^57 + 27 * q^59 - 21 * q^61 + 54 * q^63 - 3 * q^65 + 10 * q^67 - 25 * q^69 - 4 * q^71 + 23 * q^73 + q^75 - 2 * q^77 - 3 * q^79 + 57 * q^81 + 12 * q^83 + 3 * q^85 - q^87 - 8 * q^89 - 16 * q^91 - 46 * q^93 - 2 * q^95 + 25 * q^97 + 96 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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