LMFDB - Embedded newform 9680.2.a.cv.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9680,2,Mod(1,9680)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

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

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

chi := DirichletCharacter("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9680 = 2^{4} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9680.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$77.2951891566$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2525.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 5 $ x^4 - 2x^3 - 4x^2 + 5*x + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.46673$ of defining polynomial
Character	$\chi$	$=$	9680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.46673 q^{3} +1.00000 q^{5} +4.52452 q^{7} -0.848698 q^{9} +1.14256 q^{13} -1.46673 q^{15} -3.37322 q^{17} +6.08477 q^{19} -6.63626 q^{21} +5.45258 q^{23} +1.00000 q^{25} +5.64501 q^{27} -3.32083 q^{29} -1.79091 q^{31} +4.52452 q^{35} +1.48881 q^{37} -1.67583 q^{39} +1.74726 q^{41} -0.263041 q^{43} -0.848698 q^{45} -6.92472 q^{47} +13.4713 q^{49} +4.94761 q^{51} -1.43976 q^{53} -8.92472 q^{57} +7.06810 q^{59} -2.50245 q^{61} -3.83995 q^{63} +1.14256 q^{65} +0.516598 q^{67} -7.99748 q^{69} +10.7303 q^{71} +5.68123 q^{73} -1.46673 q^{75} -11.3033 q^{79} -5.73362 q^{81} +4.48088 q^{83} -3.37322 q^{85} +4.87077 q^{87} +13.2676 q^{89} +5.16953 q^{91} +2.62678 q^{93} +6.08477 q^{95} -3.35655 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 4 q^{5} + 11 q^{7} - 7 q^{13} + 2 q^{15} - 3 q^{17} + 12 q^{19} + 6 q^{21} + 9 q^{23} + 4 q^{25} + 5 q^{27} + 8 q^{29} - 3 q^{31} + 11 q^{35} - 3 q^{37} - 3 q^{39} + 7 q^{41} + 21 q^{43}+ \cdots + 6 q^{97}+O(q^{100}) $ 4 * q + 2 * q^3 + 4 * q^5 + 11 * q^7 - 7 * q^13 + 2 * q^15 - 3 * q^17 + 12 * q^19 + 6 * q^21 + 9 * q^23 + 4 * q^25 + 5 * q^27 + 8 * q^29 - 3 * q^31 + 11 * q^35 - 3 * q^37 - 3 * q^39 + 7 * q^41 + 21 * q^43 + 3 * q^47 + 15 * q^49 + 9 * q^51 - 11 * q^53 - 5 * q^57 + 7 * q^59 - 4 * q^61 + 3 * q^63 - 7 * q^65 + q^67 - 28 * q^69 + 15 * q^71 + 9 * q^73 + 2 * q^75 + 6 * q^79 - 20 * q^81 + 15 * q^83 - 3 * q^85 + 15 * q^87 - 4 * q^91 + 21 * q^93 + 12 * q^95 + 6 * q^97

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
$3$	−1.46673	−0.846818	−0.423409	−	0.905939i	$-0.639167\pi$
$3$	−1.46673	−0.846818	−0.423409	+	0.905939i	$0.639167\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.52452	1.71011	0.855055	−	0.518538i	$-0.173523\pi$
$7$	4.52452	1.71011	0.855055	+	0.518538i	$0.173523\pi$
$8$	0	0
$9$	−0.848698	−0.282899
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.14256	0.316889	0.158444	−	0.987368i	$-0.449352\pi$
$13$	1.14256	0.316889	0.158444	+	0.987368i	$0.449352\pi$
$14$	0	0
$15$	−1.46673	−0.378709
$16$	0	0
$17$	−3.37322	−0.818126	−0.409063	−	0.912506i	$-0.634144\pi$
$17$	−3.37322	−0.818126	−0.409063	+	0.912506i	$0.634144\pi$
$18$	0	0
$19$	6.08477	1.39594	0.697971	−	0.716127i	$-0.254087\pi$
$19$	6.08477	1.39594	0.697971	+	0.716127i	$0.254087\pi$
$20$	0	0
$21$	−6.63626	−1.44815
$22$	0	0
$23$	5.45258	1.13694	0.568471	−	0.822703i	$-0.307535\pi$
$23$	5.45258	1.13694	0.568471	+	0.822703i	$0.307535\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.64501	1.08638
$28$	0	0
$29$	−3.32083	−0.616663	−0.308332	−	0.951279i	$-0.599771\pi$
$29$	−3.32083	−0.616663	−0.308332	+	0.951279i	$0.599771\pi$
$30$	0	0
$31$	−1.79091	−0.321656	−0.160828	−	0.986982i	$-0.551416\pi$
$31$	−1.79091	−0.321656	−0.160828	+	0.986982i	$0.551416\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.52452	0.764784
$36$	0	0
$37$	1.48881	0.244758	0.122379	−	0.992483i	$-0.460948\pi$
$37$	1.48881	0.244758	0.122379	+	0.992483i	$0.460948\pi$
$38$	0	0
$39$	−1.67583	−0.268347
$40$	0	0
$41$	1.74726	0.272876	0.136438	−	0.990649i	$-0.456434\pi$
$41$	1.74726	0.272876	0.136438	+	0.990649i	$0.456434\pi$
$42$	0	0
$43$	−0.263041	−0.0401134	−0.0200567	−	0.999799i	$-0.506385\pi$
$43$	−0.263041	−0.0401134	−0.0200567	+	0.999799i	$0.506385\pi$
$44$	0	0
$45$	−0.848698	−0.126516
$46$	0	0
$47$	−6.92472	−1.01007	−0.505037	−	0.863098i	$-0.668521\pi$
$47$	−6.92472	−1.01007	−0.505037	+	0.863098i	$0.668521\pi$
$48$	0	0
$49$	13.4713	1.92447
$50$	0	0
$51$	4.94761	0.692804
$52$	0	0
$53$	−1.43976	−0.197766	−0.0988830	−	0.995099i	$-0.531527\pi$
$53$	−1.43976	−0.197766	−0.0988830	+	0.995099i	$0.531527\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−8.92472	−1.18211
$58$	0	0
$59$	7.06810	0.920188	0.460094	−	0.887870i	$-0.347816\pi$
$59$	7.06810	0.920188	0.460094	+	0.887870i	$0.347816\pi$
$60$	0	0
$61$	−2.50245	−0.320406	−0.160203	−	0.987084i	$-0.551215\pi$
$61$	−2.50245	−0.320406	−0.160203	+	0.987084i	$0.551215\pi$
$62$	0	0
$63$	−3.83995	−0.483789
$64$	0	0
$65$	1.14256	0.141717
$66$	0	0
$67$	0.516598	0.0631124	0.0315562	−	0.999502i	$-0.489954\pi$
$67$	0.516598	0.0631124	0.0315562	+	0.999502i	$0.489954\pi$
$68$	0	0
$69$	−7.99748	−0.962783
$70$	0	0
$71$	10.7303	1.27345	0.636725	−	0.771091i	$-0.280289\pi$
$71$	10.7303	1.27345	0.636725	+	0.771091i	$0.280289\pi$
$72$	0	0
$73$	5.68123	0.664938	0.332469	−	0.943114i	$-0.392118\pi$
$73$	5.68123	0.664938	0.332469	+	0.943114i	$0.392118\pi$
$74$	0	0
$75$	−1.46673	−0.169364
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−11.3033	−1.27173	−0.635863	−	0.771802i	$-0.719356\pi$
$79$	−11.3033	−1.27173	−0.635863	+	0.771802i	$0.719356\pi$
$80$	0	0
$81$	−5.73362	−0.637069
$82$	0	0
$83$	4.48088	0.491840	0.245920	−	0.969290i	$-0.420910\pi$
$83$	4.48088	0.491840	0.245920	+	0.969290i	$0.420910\pi$
$84$	0	0
$85$	−3.37322	−0.365877
$86$	0	0
$87$	4.87077	0.522202
$88$	0	0
$89$	13.2676	1.40637	0.703183	−	0.711009i	$-0.251762\pi$
$89$	13.2676	1.40637	0.703183	+	0.711009i	$0.251762\pi$
$90$	0	0
$91$	5.16953	0.541914
$92$	0	0
$93$	2.62678	0.272384
$94$	0	0
$95$	6.08477	0.624284
$96$	0	0
$97$	−3.35655	−0.340806	−0.170403	−	0.985374i	$-0.554507\pi$
$97$	−3.35655	−0.340806	−0.170403	+	0.985374i	$0.554507\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.33498	0.928865	0.464433	−	0.885608i	$-0.346258\pi$
$101$	9.33498	0.928865	0.464433	+	0.885608i	$0.346258\pi$
$102$	0	0
$103$	13.9160	1.37118	0.685591	−	0.727987i	$-0.259544\pi$
$103$	13.9160	1.37118	0.685591	+	0.727987i	$0.259544\pi$
$104$	0	0
$105$	−6.63626	−0.647633
$106$	0	0
$107$	−16.7883	−1.62299	−0.811493	−	0.584362i	$-0.801345\pi$
$107$	−16.7883	−1.62299	−0.811493	+	0.584362i	$0.801345\pi$
$108$	0	0
$109$	3.65293	0.349888	0.174944	−	0.984578i	$-0.444026\pi$
$109$	3.65293	0.349888	0.174944	+	0.984578i	$0.444026\pi$
$110$	0	0
$111$	−2.18368	−0.207266
$112$	0	0
$113$	−11.9023	−1.11968	−0.559839	−	0.828602i	$-0.689137\pi$
$113$	−11.9023	−1.11968	−0.559839	+	0.828602i	$0.689137\pi$
$114$	0	0
$115$	5.45258	0.508456
$116$	0	0
$117$	−0.969687	−0.0896476
$118$	0	0
$119$	−15.2622	−1.39909
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−2.56276	−0.231077
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	19.7627	1.75365	0.876826	−	0.480808i	$-0.159656\pi$
$127$	19.7627	1.75365	0.876826	+	0.480808i	$0.159656\pi$
$128$	0	0
$129$	0.385811	0.0339688
$130$	0	0
$131$	−1.93479	−0.169043	−0.0845215	−	0.996422i	$-0.526936\pi$
$131$	−1.93479	−0.169043	−0.0845215	+	0.996422i	$0.526936\pi$
$132$	0	0
$133$	27.5307	2.38721
$134$	0	0
$135$	5.64501	0.485845
$136$	0	0
$137$	−12.5353	−1.07097	−0.535483	−	0.844546i	$-0.679870\pi$
$137$	−12.5353	−1.07097	−0.535483	+	0.844546i	$0.679870\pi$
$138$	0	0
$139$	−11.2450	−0.953793	−0.476896	−	0.878960i	$-0.658238\pi$
$139$	−11.2450	−0.953793	−0.476896	+	0.878960i	$0.658238\pi$
$140$	0	0
$141$	10.1567	0.855349
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−3.32083	−0.275780
$146$	0	0
$147$	−19.7588	−1.62968
$148$	0	0
$149$	17.3337	1.42003	0.710014	−	0.704187i	$-0.248688\pi$
$149$	17.3337	1.42003	0.710014	+	0.704187i	$0.248688\pi$
$150$	0	0
$151$	−0.00540415	−0.000439784 0	−0.000219892	−	1.00000i	$-0.500070\pi$
$151$	−0.00540415	−0.000439784 0	−0.000219892		1.00000i	$0.500070\pi$
$152$	0	0
$153$	2.86285	0.231447
$154$	0	0
$155$	−1.79091	−0.143849
$156$	0	0
$157$	0.554838	0.0442809	0.0221404	−	0.999755i	$-0.492952\pi$
$157$	0.554838	0.0442809	0.0221404	+	0.999755i	$0.492952\pi$
$158$	0	0
$159$	2.11174	0.167472
$160$	0	0
$161$	24.6703	1.94430
$162$	0	0
$163$	7.96428	0.623811	0.311905	−	0.950113i	$-0.399033\pi$
$163$	7.96428	0.623811	0.311905	+	0.950113i	$0.399033\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.42643	0.265145	0.132572	−	0.991173i	$-0.457676\pi$
$167$	3.42643	0.265145	0.132572	+	0.991173i	$0.457676\pi$
$168$	0	0
$169$	−11.6946	−0.899582
$170$	0	0
$171$	−5.16413	−0.394911
$172$	0	0
$173$	−20.5669	−1.56367	−0.781836	−	0.623484i	$-0.785717\pi$
$173$	−20.5669	−1.56367	−0.781836	+	0.623484i	$0.785717\pi$
$174$	0	0
$175$	4.52452	0.342022
$176$	0	0
$177$	−10.3670	−0.779231
$178$	0	0
$179$	2.56432	0.191666	0.0958332	−	0.995397i	$-0.469448\pi$
$179$	2.56432	0.191666	0.0958332	+	0.995397i	$0.469448\pi$
$180$	0	0
$181$	13.4169	0.997268	0.498634	−	0.866813i	$-0.333835\pi$
$181$	13.4169	0.997268	0.498634	+	0.866813i	$0.333835\pi$
$182$	0	0
$183$	3.67042	0.271325
$184$	0	0
$185$	1.48881	0.109459
$186$	0	0
$187$	0	0
$188$	0	0
$189$	25.5410	1.85783
$190$	0	0
$191$	−18.1898	−1.31617	−0.658085	−	0.752944i	$-0.728633\pi$
$191$	−18.1898	−1.31617	−0.658085	+	0.752944i	$0.728633\pi$
$192$	0	0
$193$	−15.6887	−1.12929	−0.564647	−	0.825333i	$-0.690988\pi$
$193$	−15.6887	−1.12929	−0.564647	+	0.825333i	$0.690988\pi$
$194$	0	0
$195$	−1.67583	−0.120008
$196$	0	0
$197$	21.8486	1.55665	0.778325	−	0.627862i	$-0.216070\pi$
$197$	21.8486	1.55665	0.778325	+	0.627862i	$0.216070\pi$
$198$	0	0
$199$	4.55200	0.322683	0.161341	−	0.986899i	$-0.448418\pi$
$199$	4.55200	0.322683	0.161341	+	0.986899i	$0.448418\pi$
$200$	0	0
$201$	−0.757710	−0.0534448
$202$	0	0
$203$	−15.0252	−1.05456
$204$	0	0
$205$	1.74726	0.122034
$206$	0	0
$207$	−4.62760	−0.321640
$208$	0	0
$209$	0	0
$210$	0	0
$211$	18.9604	1.30529	0.652645	−	0.757664i	$-0.273659\pi$
$211$	18.9604	1.30529	0.652645	+	0.757664i	$0.273659\pi$
$212$	0	0
$213$	−15.7384	−1.07838
$214$	0	0
$215$	−0.263041	−0.0179393
$216$	0	0
$217$	−8.10300	−0.550067
$218$	0	0
$219$	−8.33284	−0.563081
$220$	0	0
$221$	−3.85410	−0.259255
$222$	0	0
$223$	−4.87952	−0.326757	−0.163378	−	0.986563i	$-0.552239\pi$
$223$	−4.87952	−0.326757	−0.163378	+	0.986563i	$0.552239\pi$
$224$	0	0
$225$	−0.848698	−0.0565799
$226$	0	0
$227$	16.3229	1.08339	0.541693	−	0.840577i	$-0.317784\pi$
$227$	16.3229	1.08339	0.541693	+	0.840577i	$0.317784\pi$
$228$	0	0
$229$	−4.83167	−0.319286	−0.159643	−	0.987175i	$-0.551034\pi$
$229$	−4.83167	−0.319286	−0.159643	+	0.987175i	$0.551034\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.41975	0.551596	0.275798	−	0.961216i	$-0.411058\pi$
$233$	8.41975	0.551596	0.275798	+	0.961216i	$0.411058\pi$
$234$	0	0
$235$	−6.92472	−0.451719
$236$	0	0
$237$	16.5790	1.07692
$238$	0	0
$239$	−22.6928	−1.46787	−0.733937	−	0.679218i	$-0.762319\pi$
$239$	−22.6928	−1.46787	−0.733937	+	0.679218i	$0.762319\pi$
$240$	0	0
$241$	−11.6065	−0.747638	−0.373819	−	0.927502i	$-0.621952\pi$
$241$	−11.6065	−0.747638	−0.373819	+	0.927502i	$0.621952\pi$
$242$	0	0
$243$	−8.52534	−0.546901
$244$	0	0
$245$	13.4713	0.860651
$246$	0	0
$247$	6.95220	0.442358
$248$	0	0
$249$	−6.57225	−0.416499
$250$	0	0
$251$	−3.31305	−0.209118	−0.104559	−	0.994519i	$-0.533343\pi$
$251$	−3.31305	−0.209118	−0.104559	+	0.994519i	$0.533343\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	4.94761	0.309831
$256$	0	0
$257$	−26.8466	−1.67465	−0.837323	−	0.546709i	$-0.815880\pi$
$257$	−26.8466	−1.67465	−0.837323	+	0.546709i	$0.815880\pi$
$258$	0	0
$259$	6.73614	0.418563
$260$	0	0
$261$	2.81838	0.174454
$262$	0	0
$263$	12.1682	0.750324	0.375162	−	0.926959i	$-0.377587\pi$
$263$	12.1682	0.750324	0.375162	+	0.926959i	$0.377587\pi$
$264$	0	0
$265$	−1.43976	−0.0884436
$266$	0	0
$267$	−19.4601	−1.19094
$268$	0	0
$269$	−2.09351	−0.127644	−0.0638218	−	0.997961i	$-0.520329\pi$
$269$	−2.09351	−0.127644	−0.0638218	+	0.997961i	$0.520329\pi$
$270$	0	0
$271$	15.1428	0.919859	0.459930	−	0.887955i	$-0.347875\pi$
$271$	15.1428	0.919859	0.459930	+	0.887955i	$0.347875\pi$
$272$	0	0
$273$	−7.58232	−0.458903
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−8.27066	−0.496936	−0.248468	−	0.968640i	$-0.579927\pi$
$277$	−8.27066	−0.496936	−0.248468	+	0.968640i	$0.579927\pi$
$278$	0	0
$279$	1.51994	0.0909963
$280$	0	0
$281$	−2.44723	−0.145989	−0.0729947	−	0.997332i	$-0.523256\pi$
$281$	−2.44723	−0.145989	−0.0729947	+	0.997332i	$0.523256\pi$
$282$	0	0
$283$	26.0948	1.55117	0.775586	−	0.631242i	$-0.217454\pi$
$283$	26.0948	1.55117	0.775586	+	0.631242i	$0.217454\pi$
$284$	0	0
$285$	−8.92472	−0.528655
$286$	0	0
$287$	7.90553	0.466648
$288$	0	0
$289$	−5.62137	−0.330669
$290$	0	0
$291$	4.92316	0.288601
$292$	0	0
$293$	−13.4529	−0.785929	−0.392965	−	0.919554i	$-0.628551\pi$
$293$	−13.4529	−0.785929	−0.392965	+	0.919554i	$0.628551\pi$
$294$	0	0
$295$	7.06810	0.411520
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.22989	0.360284
$300$	0	0
$301$	−1.19014	−0.0685984
$302$	0	0
$303$	−13.6919	−0.786580
$304$	0	0
$305$	−2.50245	−0.143290
$306$	0	0
$307$	27.1844	1.55150	0.775748	−	0.631042i	$-0.217373\pi$
$307$	27.1844	1.55150	0.775748	+	0.631042i	$0.217373\pi$
$308$	0	0
$309$	−20.4110	−1.16114
$310$	0	0
$311$	13.1990	0.748448	0.374224	−	0.927338i	$-0.377909\pi$
$311$	13.1990	0.748448	0.374224	+	0.927338i	$0.377909\pi$
$312$	0	0
$313$	16.1719	0.914090	0.457045	−	0.889443i	$-0.348908\pi$
$313$	16.1719	0.914090	0.457045	+	0.889443i	$0.348908\pi$
$314$	0	0
$315$	−3.83995	−0.216357
$316$	0	0
$317$	−5.38115	−0.302235	−0.151118	−	0.988516i	$-0.548287\pi$
$317$	−5.38115	−0.302235	−0.151118	+	0.988516i	$0.548287\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	24.6239	1.37437
$322$	0	0
$323$	−20.5253	−1.14206
$324$	0	0
$325$	1.14256	0.0633777
$326$	0	0
$327$	−5.35787	−0.296291
$328$	0	0
$329$	−31.3311	−1.72734
$330$	0	0
$331$	18.5702	1.02071	0.510356	−	0.859963i	$-0.329514\pi$
$331$	18.5702	1.02071	0.510356	+	0.859963i	$0.329514\pi$
$332$	0	0
$333$	−1.26355	−0.0692419
$334$	0	0
$335$	0.516598	0.0282247
$336$	0	0
$337$	−9.00615	−0.490596	−0.245298	−	0.969448i	$-0.578886\pi$
$337$	−9.00615	−0.490596	−0.245298	+	0.969448i	$0.578886\pi$
$338$	0	0
$339$	17.4575	0.948163
$340$	0	0
$341$	0	0
$342$	0	0
$343$	29.2796	1.58095
$344$	0	0
$345$	−7.99748	−0.430570
$346$	0	0
$347$	10.2710	0.551374	0.275687	−	0.961247i	$-0.411095\pi$
$347$	10.2710	0.551374	0.275687	+	0.961247i	$0.411095\pi$
$348$	0	0
$349$	−17.2613	−0.923974	−0.461987	−	0.886887i	$-0.652863\pi$
$349$	−17.2613	−0.923974	−0.461987	+	0.886887i	$0.652863\pi$
$350$	0	0
$351$	6.44975	0.344262
$352$	0	0
$353$	−22.8096	−1.21403	−0.607017	−	0.794689i	$-0.707634\pi$
$353$	−22.8096	−1.21403	−0.607017	+	0.794689i	$0.707634\pi$
$354$	0	0
$355$	10.7303	0.569504
$356$	0	0
$357$	22.3856	1.18477
$358$	0	0
$359$	−16.0747	−0.848390	−0.424195	−	0.905571i	$-0.639443\pi$
$359$	−16.0747	−0.848390	−0.424195	+	0.905571i	$0.639443\pi$
$360$	0	0
$361$	18.0244	0.948651
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.68123	0.297369
$366$	0	0
$367$	−21.9975	−1.14826	−0.574129	−	0.818765i	$-0.694659\pi$
$367$	−21.9975	−1.14826	−0.574129	+	0.818765i	$0.694659\pi$
$368$	0	0
$369$	−1.48290	−0.0771965
$370$	0	0
$371$	−6.51422	−0.338202
$372$	0	0
$373$	20.2604	1.04905	0.524523	−	0.851396i	$-0.324244\pi$
$373$	20.2604	1.04905	0.524523	+	0.851396i	$0.324244\pi$
$374$	0	0
$375$	−1.46673	−0.0757417
$376$	0	0
$377$	−3.79425	−0.195414
$378$	0	0
$379$	3.72771	0.191480	0.0957398	−	0.995406i	$-0.469478\pi$
$379$	3.72771	0.191480	0.0957398	+	0.995406i	$0.469478\pi$
$380$	0	0
$381$	−28.9865	−1.48502
$382$	0	0
$383$	10.9974	0.561941	0.280970	−	0.959716i	$-0.409344\pi$
$383$	10.9974	0.561941	0.280970	+	0.959716i	$0.409344\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.223243	0.0113481
$388$	0	0
$389$	9.38220	0.475697	0.237848	−	0.971302i	$-0.423558\pi$
$389$	9.38220	0.475697	0.237848	+	0.971302i	$0.423558\pi$
$390$	0	0
$391$	−18.3928	−0.930163
$392$	0	0
$393$	2.83781	0.143149
$394$	0	0
$395$	−11.3033	−0.568733
$396$	0	0
$397$	22.3136	1.11989	0.559945	−	0.828530i	$-0.310822\pi$
$397$	22.3136	1.11989	0.559945	+	0.828530i	$0.310822\pi$
$398$	0	0
$399$	−40.3801	−2.02153
$400$	0	0
$401$	−24.6822	−1.23257	−0.616285	−	0.787523i	$-0.711363\pi$
$401$	−24.6822	−1.23257	−0.616285	+	0.787523i	$0.711363\pi$
$402$	0	0
$403$	−2.04621	−0.101929
$404$	0	0
$405$	−5.73362	−0.284906
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−29.5056	−1.45896	−0.729478	−	0.684004i	$-0.760237\pi$
$409$	−29.5056	−1.45896	−0.729478	+	0.684004i	$0.760237\pi$
$410$	0	0
$411$	18.3860	0.906913
$412$	0	0
$413$	31.9798	1.57362
$414$	0	0
$415$	4.48088	0.219958
$416$	0	0
$417$	16.4935	0.807689
$418$	0	0
$419$	−9.03564	−0.441420	−0.220710	−	0.975339i	$-0.570837\pi$
$419$	−9.03564	−0.441420	−0.220710	+	0.975339i	$0.570837\pi$
$420$	0	0
$421$	14.2201	0.693047	0.346524	−	0.938041i	$-0.387362\pi$
$421$	14.2201	0.693047	0.346524	+	0.938041i	$0.387362\pi$
$422$	0	0
$423$	5.87699	0.285749
$424$	0	0
$425$	−3.37322	−0.163625
$426$	0	0
$427$	−11.3224	−0.547929
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.73155	−0.0834060	−0.0417030	−	0.999130i	$-0.513278\pi$
$431$	−1.73155	−0.0834060	−0.0417030	+	0.999130i	$0.513278\pi$
$432$	0	0
$433$	18.4476	0.886535	0.443268	−	0.896389i	$-0.353819\pi$
$433$	18.4476	0.886535	0.443268	+	0.896389i	$0.353819\pi$
$434$	0	0
$435$	4.87077	0.233536
$436$	0	0
$437$	33.1777	1.58710
$438$	0	0
$439$	−17.1704	−0.819499	−0.409750	−	0.912198i	$-0.634384\pi$
$439$	−17.1704	−0.819499	−0.409750	+	0.912198i	$0.634384\pi$
$440$	0	0
$441$	−11.4331	−0.544432
$442$	0	0
$443$	−36.5992	−1.73888	−0.869441	−	0.494037i	$-0.835521\pi$
$443$	−36.5992	−1.73888	−0.869441	+	0.494037i	$0.835521\pi$
$444$	0	0
$445$	13.2676	0.628946
$446$	0	0
$447$	−25.4238	−1.20251
$448$	0	0
$449$	−16.6029	−0.783540	−0.391770	−	0.920063i	$-0.628137\pi$
$449$	−16.6029	−0.783540	−0.391770	+	0.920063i	$0.628137\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0.00792644	0.000372417 0
$454$	0	0
$455$	5.16953	0.242351
$456$	0	0
$457$	31.6653	1.48124	0.740620	−	0.671924i	$-0.234532\pi$
$457$	31.6653	1.48124	0.740620	+	0.671924i	$0.234532\pi$
$458$	0	0
$459$	−19.0419	−0.888798
$460$	0	0
$461$	25.4351	1.18463	0.592315	−	0.805706i	$-0.298214\pi$
$461$	25.4351	1.18463	0.592315	+	0.805706i	$0.298214\pi$
$462$	0	0
$463$	16.3319	0.759007	0.379503	−	0.925190i	$-0.376095\pi$
$463$	16.3319	0.759007	0.379503	+	0.925190i	$0.376095\pi$
$464$	0	0
$465$	2.62678	0.121814
$466$	0	0
$467$	−8.52911	−0.394680	−0.197340	−	0.980335i	$-0.563230\pi$
$467$	−8.52911	−0.394680	−0.197340	+	0.980335i	$0.563230\pi$
$468$	0	0
$469$	2.33736	0.107929
$470$	0	0
$471$	−0.813798	−0.0374978
$472$	0	0
$473$	0	0
$474$	0	0
$475$	6.08477	0.279188
$476$	0	0
$477$	1.22192	0.0559479
$478$	0	0
$479$	29.9478	1.36835	0.684175	−	0.729318i	$-0.260162\pi$
$479$	29.9478	1.36835	0.684175	+	0.729318i	$0.260162\pi$
$480$	0	0
$481$	1.70105	0.0775611
$482$	0	0
$483$	−36.1848	−1.64646
$484$	0	0
$485$	−3.35655	−0.152413
$486$	0	0
$487$	19.6031	0.888300	0.444150	−	0.895952i	$-0.353506\pi$
$487$	19.6031	0.888300	0.444150	+	0.895952i	$0.353506\pi$
$488$	0	0
$489$	−11.6815	−0.528254
$490$	0	0
$491$	−15.7891	−0.712553	−0.356277	−	0.934381i	$-0.615954\pi$
$491$	−15.7891	−0.712553	−0.356277	+	0.934381i	$0.615954\pi$
$492$	0	0
$493$	11.2019	0.504509
$494$	0	0
$495$	0	0
$496$	0	0
$497$	48.5494	2.17774
$498$	0	0
$499$	−11.2081	−0.501745	−0.250872	−	0.968020i	$-0.580717\pi$
$499$	−11.2081	−0.501745	−0.250872	+	0.968020i	$0.580717\pi$
$500$	0	0
$501$	−5.02565	−0.224530
$502$	0	0
$503$	−0.342908	−0.0152895	−0.00764477	−	0.999971i	$-0.502433\pi$
$503$	−0.342908	−0.0152895	−0.00764477	+	0.999971i	$0.502433\pi$
$504$	0	0
$505$	9.33498	0.415401
$506$	0	0
$507$	17.1528	0.761782
$508$	0	0
$509$	−19.5626	−0.867098	−0.433549	−	0.901130i	$-0.642739\pi$
$509$	−19.5626	−0.867098	−0.433549	+	0.901130i	$0.642739\pi$
$510$	0	0
$511$	25.7049	1.13712
$512$	0	0
$513$	34.3485	1.51653
$514$	0	0
$515$	13.9160	0.613211
$516$	0	0
$517$	0	0
$518$	0	0
$519$	30.1661	1.32415
$520$	0	0
$521$	42.0264	1.84121	0.920606	−	0.390493i	$-0.127695\pi$
$521$	42.0264	1.84121	0.920606	+	0.390493i	$0.127695\pi$
$522$	0	0
$523$	30.0105	1.31227	0.656135	−	0.754644i	$-0.272190\pi$
$523$	30.0105	1.31227	0.656135	+	0.754644i	$0.272190\pi$
$524$	0	0
$525$	−6.63626	−0.289630
$526$	0	0
$527$	6.04112	0.263155
$528$	0	0
$529$	6.73067	0.292638
$530$	0	0
$531$	−5.99868	−0.260320
$532$	0	0
$533$	1.99635	0.0864714
$534$	0	0
$535$	−16.7883	−0.725822
$536$	0	0
$537$	−3.76117	−0.162307
$538$	0	0
$539$	0	0
$540$	0	0
$541$	10.3236	0.443846	0.221923	−	0.975064i	$-0.428767\pi$
$541$	10.3236	0.443846	0.221923	+	0.975064i	$0.428767\pi$
$542$	0	0
$543$	−19.6789	−0.844504
$544$	0	0
$545$	3.65293	0.156474
$546$	0	0
$547$	41.8168	1.78796	0.893979	−	0.448108i	$-0.147902\pi$
$547$	41.8168	1.78796	0.893979	+	0.448108i	$0.147902\pi$
$548$	0	0
$549$	2.12382	0.0906426
$550$	0	0
$551$	−20.2065	−0.860826
$552$	0	0
$553$	−51.1423	−2.17479
$554$	0	0
$555$	−2.18368	−0.0926920
$556$	0	0
$557$	38.4828	1.63057	0.815285	−	0.579060i	$-0.196580\pi$
$557$	38.4828	1.63057	0.815285	+	0.579060i	$0.196580\pi$
$558$	0	0
$559$	−0.300540	−0.0127115
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.6864	1.29328	0.646639	−	0.762796i	$-0.276174\pi$
$563$	30.6864	1.29328	0.646639	+	0.762796i	$0.276174\pi$
$564$	0	0
$565$	−11.9023	−0.500735
$566$	0	0
$567$	−25.9419	−1.08946
$568$	0	0
$569$	−13.0485	−0.547020	−0.273510	−	0.961869i	$-0.588185\pi$
$569$	−13.0485	−0.547020	−0.273510	+	0.961869i	$0.588185\pi$
$570$	0	0
$571$	16.1300	0.675018	0.337509	−	0.941322i	$-0.390416\pi$
$571$	16.1300	0.675018	0.337509	+	0.941322i	$0.390416\pi$
$572$	0	0
$573$	26.6796	1.11456
$574$	0	0
$575$	5.45258	0.227388
$576$	0	0
$577$	−14.5714	−0.606617	−0.303308	−	0.952892i	$-0.598091\pi$
$577$	−14.5714	−0.606617	−0.303308	+	0.952892i	$0.598091\pi$
$578$	0	0
$579$	23.0110	0.956306
$580$	0	0
$581$	20.2738	0.841101
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−0.969687	−0.0400916
$586$	0	0
$587$	27.8742	1.15049	0.575246	−	0.817981i	$-0.304906\pi$
$587$	27.8742	1.15049	0.575246	+	0.817981i	$0.304906\pi$
$588$	0	0
$589$	−10.8972	−0.449013
$590$	0	0
$591$	−32.0461	−1.31820
$592$	0	0
$593$	−15.1037	−0.620236	−0.310118	−	0.950698i	$-0.600369\pi$
$593$	−15.1037	−0.620236	−0.310118	+	0.950698i	$0.600369\pi$
$594$	0	0
$595$	−15.2622	−0.625690
$596$	0	0
$597$	−6.67657	−0.273254
$598$	0	0
$599$	25.8757	1.05725	0.528626	−	0.848855i	$-0.322708\pi$
$599$	25.8757	1.05725	0.528626	+	0.848855i	$0.322708\pi$
$600$	0	0
$601$	47.0268	1.91826	0.959132	−	0.282959i	$-0.0913160\pi$
$601$	47.0268	1.91826	0.959132	+	0.282959i	$0.0913160\pi$
$602$	0	0
$603$	−0.438435	−0.0178545
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−35.2239	−1.42969	−0.714847	−	0.699281i	$-0.753504\pi$
$607$	−35.2239	−1.42969	−0.714847	+	0.699281i	$0.753504\pi$
$608$	0	0
$609$	22.0379	0.893022
$610$	0	0
$611$	−7.91189	−0.320081
$612$	0	0
$613$	23.4117	0.945590	0.472795	−	0.881173i	$-0.343245\pi$
$613$	23.4117	0.945590	0.472795	+	0.881173i	$0.343245\pi$
$614$	0	0
$615$	−2.56276	−0.103341
$616$	0	0
$617$	22.8910	0.921557	0.460778	−	0.887515i	$-0.347570\pi$
$617$	22.8910	0.921557	0.460778	+	0.887515i	$0.347570\pi$
$618$	0	0
$619$	−2.12752	−0.0855124	−0.0427562	−	0.999086i	$-0.513614\pi$
$619$	−2.12752	−0.0855124	−0.0427562	+	0.999086i	$0.513614\pi$
$620$	0	0
$621$	30.7799	1.23515
$622$	0	0
$623$	60.0297	2.40504
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.02207	−0.200243
$630$	0	0
$631$	−15.4588	−0.615404	−0.307702	−	0.951483i	$-0.599560\pi$
$631$	−15.4588	−0.615404	−0.307702	+	0.951483i	$0.599560\pi$
$632$	0	0
$633$	−27.8099	−1.10534
$634$	0	0
$635$	19.7627	0.784257
$636$	0	0
$637$	15.3918	0.609844
$638$	0	0
$639$	−9.10676	−0.360258
$640$	0	0
$641$	14.1805	0.560096	0.280048	−	0.959986i	$-0.409650\pi$
$641$	14.1805	0.560096	0.280048	+	0.959986i	$0.409650\pi$
$642$	0	0
$643$	12.3941	0.488774	0.244387	−	0.969678i	$-0.421413\pi$
$643$	12.3941	0.488774	0.244387	+	0.969678i	$0.421413\pi$
$644$	0	0
$645$	0.385811	0.0151913
$646$	0	0
$647$	−33.2465	−1.30705	−0.653527	−	0.756903i	$-0.726711\pi$
$647$	−33.2465	−1.30705	−0.653527	+	0.756903i	$0.726711\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	11.8849	0.465807
$652$	0	0
$653$	−34.8066	−1.36209	−0.681043	−	0.732243i	$-0.738474\pi$
$653$	−34.8066	−1.36209	−0.681043	+	0.732243i	$0.738474\pi$
$654$	0	0
$655$	−1.93479	−0.0755984
$656$	0	0
$657$	−4.82165	−0.188110
$658$	0	0
$659$	−34.4953	−1.34375	−0.671873	−	0.740666i	$-0.734510\pi$
$659$	−34.4953	−1.34375	−0.671873	+	0.740666i	$0.734510\pi$
$660$	0	0
$661$	1.77827	0.0691668	0.0345834	−	0.999402i	$-0.488990\pi$
$661$	1.77827	0.0691668	0.0345834	+	0.999402i	$0.488990\pi$
$662$	0	0
$663$	5.65293	0.219542
$664$	0	0
$665$	27.5307	1.06759
$666$	0	0
$667$	−18.1071	−0.701111
$668$	0	0
$669$	7.15694	0.276703
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−19.2964	−0.743823	−0.371911	−	0.928268i	$-0.621297\pi$
$673$	−19.2964	−0.743823	−0.371911	+	0.928268i	$0.621297\pi$
$674$	0	0
$675$	5.64501	0.217276
$676$	0	0
$677$	16.9101	0.649907	0.324953	−	0.945730i	$-0.394651\pi$
$677$	16.9101	0.649907	0.324953	+	0.945730i	$0.394651\pi$
$678$	0	0
$679$	−15.1868	−0.582816
$680$	0	0
$681$	−23.9412	−0.917430
$682$	0	0
$683$	−4.14018	−0.158420	−0.0792098	−	0.996858i	$-0.525240\pi$
$683$	−4.14018	−0.158420	−0.0792098	+	0.996858i	$0.525240\pi$
$684$	0	0
$685$	−12.5353	−0.478950
$686$	0	0
$687$	7.08676	0.270377
$688$	0	0
$689$	−1.64501	−0.0626698
$690$	0	0
$691$	46.3520	1.76331	0.881657	−	0.471891i	$-0.156429\pi$
$691$	46.3520	1.76331	0.881657	+	0.471891i	$0.156429\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.2450	−0.426549
$696$	0	0
$697$	−5.89390	−0.223247
$698$	0	0
$699$	−12.3495	−0.467101
$700$	0	0
$701$	45.4553	1.71682	0.858412	−	0.512961i	$-0.171452\pi$
$701$	45.4553	1.71682	0.858412	+	0.512961i	$0.171452\pi$
$702$	0	0
$703$	9.05904	0.341668
$704$	0	0
$705$	10.1567	0.382524
$706$	0	0
$707$	42.2364	1.58846
$708$	0	0
$709$	−14.0241	−0.526688	−0.263344	−	0.964702i	$-0.584825\pi$
$709$	−14.0241	−0.526688	−0.263344	+	0.964702i	$0.584825\pi$
$710$	0	0
$711$	9.59312	0.359770
$712$	0	0
$713$	−9.76506	−0.365704
$714$	0	0
$715$	0	0
$716$	0	0
$717$	33.2842	1.24302
$718$	0	0
$719$	−13.1384	−0.489980	−0.244990	−	0.969526i	$-0.578785\pi$
$719$	−13.1384	−0.489980	−0.244990	+	0.969526i	$0.578785\pi$
$720$	0	0
$721$	62.9632	2.34487
$722$	0	0
$723$	17.0236	0.633113
$724$	0	0
$725$	−3.32083	−0.123333
$726$	0	0
$727$	18.3635	0.681063	0.340532	−	0.940233i	$-0.389393\pi$
$727$	18.3635	0.681063	0.340532	+	0.940233i	$0.389393\pi$
$728$	0	0
$729$	29.7052	1.10019
$730$	0	0
$731$	0.887297	0.0328179
$732$	0	0
$733$	36.9977	1.36654	0.683270	−	0.730166i	$-0.260557\pi$
$733$	36.9977	1.36654	0.683270	+	0.730166i	$0.260557\pi$
$734$	0	0
$735$	−19.7588	−0.728815
$736$	0	0
$737$	0	0
$738$	0	0
$739$	9.85033	0.362350	0.181175	−	0.983451i	$-0.442010\pi$
$739$	9.85033	0.362350	0.181175	+	0.983451i	$0.442010\pi$
$740$	0	0
$741$	−10.1970	−0.374597
$742$	0	0
$743$	27.4471	1.00694	0.503468	−	0.864014i	$-0.332057\pi$
$743$	27.4471	1.00694	0.503468	+	0.864014i	$0.332057\pi$
$744$	0	0
$745$	17.3337	0.635056
$746$	0	0
$747$	−3.80291	−0.139141
$748$	0	0
$749$	−75.9591	−2.77549
$750$	0	0
$751$	−13.7284	−0.500957	−0.250479	−	0.968122i	$-0.580588\pi$
$751$	−13.7284	−0.500957	−0.250479	+	0.968122i	$0.580588\pi$
$752$	0	0
$753$	4.85936	0.177085
$754$	0	0
$755$	−0.00540415	−0.000196677 0
$756$	0	0
$757$	−22.9299	−0.833401	−0.416701	−	0.909044i	$-0.636814\pi$
$757$	−22.9299	−0.833401	−0.416701	+	0.909044i	$0.636814\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	20.6818	0.749716	0.374858	−	0.927082i	$-0.377692\pi$
$761$	20.6818	0.749716	0.374858	+	0.927082i	$0.377692\pi$
$762$	0	0
$763$	16.5278	0.598346
$764$	0	0
$765$	2.86285	0.103506
$766$	0	0
$767$	8.07571	0.291597
$768$	0	0
$769$	−38.4306	−1.38584	−0.692922	−	0.721013i	$-0.743677\pi$
$769$	−38.4306	−1.38584	−0.692922	+	0.721013i	$0.743677\pi$
$770$	0	0
$771$	39.3768	1.41812
$772$	0	0
$773$	49.9406	1.79624	0.898120	−	0.439751i	$-0.144933\pi$
$773$	49.9406	1.79624	0.898120	+	0.439751i	$0.144933\pi$
$774$	0	0
$775$	−1.79091	−0.0643312
$776$	0	0
$777$	−9.88011	−0.354447
$778$	0	0
$779$	10.6317	0.380919
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−18.7461	−0.669932
$784$	0	0
$785$	0.554838	0.0198030
$786$	0	0
$787$	−15.5540	−0.554440	−0.277220	−	0.960806i	$-0.589413\pi$
$787$	−15.5540	−0.554440	−0.277220	+	0.960806i	$0.589413\pi$
$788$	0	0
$789$	−17.8475	−0.635388
$790$	0	0
$791$	−53.8524	−1.91477
$792$	0	0
$793$	−2.85919	−0.101533
$794$	0	0
$795$	2.11174	0.0748957
$796$	0	0
$797$	46.1068	1.63319	0.816593	−	0.577214i	$-0.195860\pi$
$797$	46.1068	1.63319	0.816593	+	0.577214i	$0.195860\pi$
$798$	0	0
$799$	23.3586	0.826368
$800$	0	0
$801$	−11.2602	−0.397860
$802$	0	0
$803$	0	0
$804$	0	0
$805$	24.6703	0.869515
$806$	0	0
$807$	3.07062	0.108091
$808$	0	0
$809$	37.5394	1.31981	0.659907	−	0.751347i	$-0.270596\pi$
$809$	37.5394	1.31981	0.659907	+	0.751347i	$0.270596\pi$
$810$	0	0
$811$	7.20547	0.253018	0.126509	−	0.991965i	$-0.459623\pi$
$811$	7.20547	0.253018	0.126509	+	0.991965i	$0.459623\pi$
$812$	0	0
$813$	−22.2104	−0.778953
$814$	0	0
$815$	7.96428	0.278977
$816$	0	0
$817$	−1.60055	−0.0559960
$818$	0	0
$819$	−4.38737	−0.153307
$820$	0	0
$821$	8.63405	0.301331	0.150665	−	0.988585i	$-0.451858\pi$
$821$	8.63405	0.301331	0.150665	+	0.988585i	$0.451858\pi$
$822$	0	0
$823$	25.0121	0.871865	0.435933	−	0.899979i	$-0.356419\pi$
$823$	25.0121	0.871865	0.435933	+	0.899979i	$0.356419\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	48.8725	1.69946	0.849732	−	0.527216i	$-0.176764\pi$
$827$	48.8725	1.69946	0.849732	+	0.527216i	$0.176764\pi$
$828$	0	0
$829$	5.29170	0.183788	0.0918941	−	0.995769i	$-0.470708\pi$
$829$	5.29170	0.183788	0.0918941	+	0.995769i	$0.470708\pi$
$830$	0	0
$831$	12.1308	0.420814
$832$	0	0
$833$	−45.4417	−1.57446
$834$	0	0
$835$	3.42643	0.118576
$836$	0	0
$837$	−10.1097	−0.349441
$838$	0	0
$839$	13.2419	0.457160	0.228580	−	0.973525i	$-0.426592\pi$
$839$	13.2419	0.457160	0.228580	+	0.973525i	$0.426592\pi$
$840$	0	0
$841$	−17.9721	−0.619726
$842$	0	0
$843$	3.58943	0.123626
$844$	0	0
$845$	−11.6946	−0.402305
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−38.2740	−1.31356
$850$	0	0
$851$	8.11784	0.278276
$852$	0	0
$853$	−2.21885	−0.0759720	−0.0379860	−	0.999278i	$-0.512094\pi$
$853$	−2.21885	−0.0759720	−0.0379860	+	0.999278i	$0.512094\pi$
$854$	0	0
$855$	−5.16413	−0.176609
$856$	0	0
$857$	−31.4625	−1.07474	−0.537368	−	0.843348i	$-0.680582\pi$
$857$	−31.4625	−1.07474	−0.537368	+	0.843348i	$0.680582\pi$
$858$	0	0
$859$	−9.07676	−0.309695	−0.154848	−	0.987938i	$-0.549489\pi$
$859$	−9.07676	−0.309695	−0.154848	+	0.987938i	$0.549489\pi$
$860$	0	0
$861$	−11.5953	−0.395166
$862$	0	0
$863$	−42.4842	−1.44618	−0.723089	−	0.690755i	$-0.757279\pi$
$863$	−42.4842	−1.44618	−0.723089	+	0.690755i	$0.757279\pi$
$864$	0	0
$865$	−20.5669	−0.699295
$866$	0	0
$867$	8.24505	0.280017
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0.590243	0.0199996
$872$	0	0
$873$	2.84870	0.0964138
$874$	0	0
$875$	4.52452	0.152957
$876$	0	0
$877$	31.3101	1.05727	0.528634	−	0.848850i	$-0.322704\pi$
$877$	31.3101	1.05727	0.528634	+	0.848850i	$0.322704\pi$
$878$	0	0
$879$	19.7319	0.665539
$880$	0	0
$881$	21.5189	0.724990	0.362495	−	0.931986i	$-0.381925\pi$
$881$	21.5189	0.724990	0.362495	+	0.931986i	$0.381925\pi$
$882$	0	0
$883$	0.652552	0.0219601	0.0109801	−	0.999940i	$-0.496505\pi$
$883$	0.652552	0.0219601	0.0109801	+	0.999940i	$0.496505\pi$
$884$	0	0
$885$	−10.3670	−0.348483
$886$	0	0
$887$	−14.6462	−0.491771	−0.245886	−	0.969299i	$-0.579079\pi$
$887$	−14.6462	−0.491771	−0.245886	+	0.969299i	$0.579079\pi$
$888$	0	0
$889$	89.4166	2.99894
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−42.1353	−1.41000
$894$	0	0
$895$	2.56432	0.0857159
$896$	0	0
$897$	−9.13758	−0.305095
$898$	0	0
$899$	5.94730	0.198354
$900$	0	0
$901$	4.85662	0.161798
$902$	0	0
$903$	1.74561	0.0580903
$904$	0	0
$905$	13.4169	0.445992
$906$	0	0
$907$	−28.7450	−0.954461	−0.477231	−	0.878778i	$-0.658359\pi$
$907$	−28.7450	−0.954461	−0.477231	+	0.878778i	$0.658359\pi$
$908$	0	0
$909$	−7.92258	−0.262775
$910$	0	0
$911$	−17.2819	−0.572573	−0.286287	−	0.958144i	$-0.592421\pi$
$911$	−17.2819	−0.572573	−0.286287	+	0.958144i	$0.592421\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	3.67042	0.121340
$916$	0	0
$917$	−8.75399	−0.289082
$918$	0	0
$919$	−2.19845	−0.0725203	−0.0362601	−	0.999342i	$-0.511544\pi$
$919$	−2.19845	−0.0725203	−0.0362601	+	0.999342i	$0.511544\pi$
$920$	0	0
$921$	−39.8723	−1.31384
$922$	0	0
$923$	12.2600	0.403542
$924$	0	0
$925$	1.48881	0.0489517
$926$	0	0
$927$	−11.8105	−0.387906
$928$	0	0
$929$	24.2104	0.794317	0.397158	−	0.917750i	$-0.369996\pi$
$929$	24.2104	0.794317	0.397158	+	0.917750i	$0.369996\pi$
$930$	0	0
$931$	81.9698	2.68645
$932$	0	0
$933$	−19.3594	−0.633799
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.8227	0.680249	0.340125	−	0.940380i	$-0.389531\pi$
$937$	20.8227	0.680249	0.340125	+	0.940380i	$0.389531\pi$
$938$	0	0
$939$	−23.7199	−0.774068
$940$	0	0
$941$	−34.2128	−1.11530	−0.557652	−	0.830075i	$-0.688298\pi$
$941$	−34.2128	−1.11530	−0.557652	+	0.830075i	$0.688298\pi$
$942$	0	0
$943$	9.52709	0.310245
$944$	0	0
$945$	25.5410	0.830848
$946$	0	0
$947$	−33.8128	−1.09877	−0.549383	−	0.835570i	$-0.685137\pi$
$947$	−33.8128	−1.09877	−0.549383	+	0.835570i	$0.685137\pi$
$948$	0	0
$949$	6.49114	0.210711
$950$	0	0
$951$	7.89270	0.255938
$952$	0	0
$953$	−25.5982	−0.829206	−0.414603	−	0.910002i	$-0.636080\pi$
$953$	−25.5982	−0.829206	−0.414603	+	0.910002i	$0.636080\pi$
$954$	0	0
$955$	−18.1898	−0.588609
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−56.7164	−1.83147
$960$	0	0
$961$	−27.7927	−0.896537
$962$	0	0
$963$	14.2482	0.459142
$964$	0	0
$965$	−15.6887	−0.505036
$966$	0	0
$967$	43.8942	1.41154	0.705772	−	0.708439i	$-0.250600\pi$
$967$	43.8942	1.41154	0.705772	+	0.708439i	$0.250600\pi$
$968$	0	0
$969$	30.1051	0.967114
$970$	0	0
$971$	−36.3707	−1.16719	−0.583596	−	0.812044i	$-0.698355\pi$
$971$	−36.3707	−1.16719	−0.583596	+	0.812044i	$0.698355\pi$
$972$	0	0
$973$	−50.8785	−1.63109
$974$	0	0
$975$	−1.67583	−0.0536694
$976$	0	0
$977$	10.3516	0.331177	0.165589	−	0.986195i	$-0.447048\pi$
$977$	10.3516	0.331177	0.165589	+	0.986195i	$0.447048\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−3.10024	−0.0989830
$982$	0	0
$983$	22.1158	0.705385	0.352693	−	0.935739i	$-0.385266\pi$
$983$	22.1158	0.705385	0.352693	+	0.935739i	$0.385266\pi$
$984$	0	0
$985$	21.8486	0.696155
$986$	0	0
$987$	45.9543	1.46274
$988$	0	0
$989$	−1.43426	−0.0456067
$990$	0	0
$991$	55.5911	1.76591	0.882955	−	0.469457i	$-0.155550\pi$
$991$	55.5911	1.76591	0.882955	+	0.469457i	$0.155550\pi$
$992$	0	0
$993$	−27.2376	−0.864358
$994$	0	0
$995$	4.55200	0.144308
$996$	0	0
$997$	−10.7911	−0.341759	−0.170879	−	0.985292i	$-0.554661\pi$
$997$	−10.7911	−0.341759	−0.170879	+	0.985292i	$0.554661\pi$
$998$	0	0
$999$	8.40432	0.265901

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.cv.1.1		4
4.3	odd	2		605.2.a.i.1.3		4
11.2	odd	10		880.2.bo.e.81.1		8
11.6	odd	10		880.2.bo.e.641.1		8
11.10	odd	2		9680.2.a.cs.1.1		4
12.11	even	2		5445.2.a.bu.1.2		4
20.19	odd	2		3025.2.a.be.1.2		4
44.3	odd	10		605.2.g.g.251.2		8
44.7	even	10		605.2.g.j.511.1		8
44.15	odd	10		605.2.g.g.511.2		8
44.19	even	10		605.2.g.j.251.1		8
44.27	odd	10		605.2.g.n.366.1		8
44.31	odd	10		605.2.g.n.81.1		8
44.35	even	10		55.2.g.a.26.2	✓	8
44.39	even	10		55.2.g.a.36.2	yes	8
44.43	even	2		605.2.a.l.1.2		4
132.35	odd	10		495.2.n.f.136.1		8
132.83	odd	10		495.2.n.f.91.1		8
132.131	odd	2		5445.2.a.bg.1.3		4
220.39	even	10		275.2.h.b.201.1		8
220.79	even	10		275.2.h.b.26.1		8
220.83	odd	20		275.2.z.b.124.3		16
220.123	odd	20		275.2.z.b.224.2		16
220.127	odd	20		275.2.z.b.124.2		16
220.167	odd	20		275.2.z.b.224.3		16
220.219	even	2		3025.2.a.v.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.g.a.26.2	✓	8	44.35	even	10
55.2.g.a.36.2	yes	8	44.39	even	10
275.2.h.b.26.1		8	220.79	even	10
275.2.h.b.201.1		8	220.39	even	10
275.2.z.b.124.2		16	220.127	odd	20
275.2.z.b.124.3		16	220.83	odd	20
275.2.z.b.224.2		16	220.123	odd	20
275.2.z.b.224.3		16	220.167	odd	20
495.2.n.f.91.1		8	132.83	odd	10
495.2.n.f.136.1		8	132.35	odd	10
605.2.a.i.1.3		4	4.3	odd	2
605.2.a.l.1.2		4	44.43	even	2
605.2.g.g.251.2		8	44.3	odd	10
605.2.g.g.511.2		8	44.15	odd	10
605.2.g.j.251.1		8	44.19	even	10
605.2.g.j.511.1		8	44.7	even	10
605.2.g.n.81.1		8	44.31	odd	10
605.2.g.n.366.1		8	44.27	odd	10
880.2.bo.e.81.1		8	11.2	odd	10
880.2.bo.e.641.1		8	11.6	odd	10
3025.2.a.v.1.3		4	220.219	even	2
3025.2.a.be.1.2		4	20.19	odd	2
5445.2.a.bg.1.3		4	132.131	odd	2
5445.2.a.bu.1.2		4	12.11	even	2
9680.2.a.cs.1.1		4	11.10	odd	2
9680.2.a.cv.1.1		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9680.a (trivial)

\(f(q)\)	\(=\)	\(q-1.46673 q^{3} +1.00000 q^{5} +4.52452 q^{7} -0.848698 q^{9} +1.14256 q^{13} -1.46673 q^{15} -3.37322 q^{17} +6.08477 q^{19} -6.63626 q^{21} +5.45258 q^{23} +1.00000 q^{25} +5.64501 q^{27} -3.32083 q^{29} -1.79091 q^{31} +4.52452 q^{35} +1.48881 q^{37} -1.67583 q^{39} +1.74726 q^{41} -0.263041 q^{43} -0.848698 q^{45} -6.92472 q^{47} +13.4713 q^{49} +4.94761 q^{51} -1.43976 q^{53} -8.92472 q^{57} +7.06810 q^{59} -2.50245 q^{61} -3.83995 q^{63} +1.14256 q^{65} +0.516598 q^{67} -7.99748 q^{69} +10.7303 q^{71} +5.68123 q^{73} -1.46673 q^{75} -11.3033 q^{79} -5.73362 q^{81} +4.48088 q^{83} -3.37322 q^{85} +4.87077 q^{87} +13.2676 q^{89} +5.16953 q^{91} +2.62678 q^{93} +6.08477 q^{95} -3.35655 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 4 q^{5} + 11 q^{7} - 7 q^{13} + 2 q^{15} - 3 q^{17} + 12 q^{19} + 6 q^{21} + 9 q^{23} + 4 q^{25} + 5 q^{27} + 8 q^{29} - 3 q^{31} + 11 q^{35} - 3 q^{37} - 3 q^{39} + 7 q^{41} + 21 q^{43}+ \cdots + 6 q^{97}+O(q^{100}) \) 4 * q + 2 * q^3 + 4 * q^5 + 11 * q^7 - 7 * q^13 + 2 * q^15 - 3 * q^17 + 12 * q^19 + 6 * q^21 + 9 * q^23 + 4 * q^25 + 5 * q^27 + 8 * q^29 - 3 * q^31 + 11 * q^35 - 3 * q^37 - 3 * q^39 + 7 * q^41 + 21 * q^43 + 3 * q^47 + 15 * q^49 + 9 * q^51 - 11 * q^53 - 5 * q^57 + 7 * q^59 - 4 * q^61 + 3 * q^63 - 7 * q^65 + q^67 - 28 * q^69 + 15 * q^71 + 9 * q^73 + 2 * q^75 + 6 * q^79 - 20 * q^81 + 15 * q^83 - 3 * q^85 + 15 * q^87 - 4 * q^91 + 21 * q^93 + 12 * q^95 + 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more