LMFDB - Embedded newform 2160.4.a.bl.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,4,Mod(1,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2160.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2160.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:	$127.444125612$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.697.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 7x - 5 $ x^3 - 7*x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1080)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.16601$ of defining polynomial
Character	$\chi$	$=$	2160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000 q^{5} +21.1457 q^{7} +O(q^{10})$	$q-5.00000 q^{5} +21.1457 q^{7} +67.4294 q^{11} +2.55493 q^{13} +126.114 q^{17} +103.539 q^{19} +200.434 q^{23} +25.0000 q^{25} +71.4215 q^{29} +158.130 q^{31} -105.729 q^{35} +7.18163 q^{37} -347.052 q^{41} -189.318 q^{43} -585.443 q^{47} +104.142 q^{49} +77.2791 q^{53} -337.147 q^{55} -200.790 q^{59} +681.137 q^{61} -12.7746 q^{65} +810.751 q^{67} -515.476 q^{71} +385.577 q^{73} +1425.84 q^{77} +209.405 q^{79} -887.189 q^{83} -630.572 q^{85} -548.890 q^{89} +54.0258 q^{91} -517.696 q^{95} -1523.96 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 15 q^{5} + 24 q^{7}+O(q^{10}) $ 3 * q - 15 * q^5 + 24 * q^7	$ 3 q - 15 q^{5} + 24 q^{7} + 6 q^{11} + 48 q^{13} + 27 q^{17} + 195 q^{19} - 27 q^{23} + 75 q^{25} - 60 q^{29} + 279 q^{31} - 120 q^{35} - 138 q^{37} - 66 q^{41} - 222 q^{43} - 264 q^{47} - 237 q^{49} + 507 q^{53} - 30 q^{55} - 960 q^{59} + 543 q^{61} - 240 q^{65} + 1086 q^{67} - 1818 q^{71} + 1362 q^{73} + 1776 q^{77} + 129 q^{79} - 1569 q^{83} - 135 q^{85} + 1770 q^{89} - 1488 q^{91} - 975 q^{95} - 336 q^{97}+O(q^{100}) $ 3 * q - 15 * q^5 + 24 * q^7 + 6 * q^11 + 48 * q^13 + 27 * q^17 + 195 * q^19 - 27 * q^23 + 75 * q^25 - 60 * q^29 + 279 * q^31 - 120 * q^35 - 138 * q^37 - 66 * q^41 - 222 * q^43 - 264 * q^47 - 237 * q^49 + 507 * q^53 - 30 * q^55 - 960 * q^59 + 543 * q^61 - 240 * q^65 + 1086 * q^67 - 1818 * q^71 + 1362 * q^73 + 1776 * q^77 + 129 * q^79 - 1569 * q^83 - 135 * q^85 + 1770 * q^89 - 1488 * q^91 - 975 * q^95 - 336 * q^97

Display 100 coefficients 10 coefficients

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	21.1457	1.14176	0.570881	−	0.821033i	$-0.306602\pi$
$7$	21.1457	1.14176	0.570881	+	0.821033i	$0.306602\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	67.4294	1.84825	0.924124	−	0.382093i	$-0.124797\pi$
$11$	67.4294	1.84825	0.924124	+	0.382093i	$0.124797\pi$
$12$	0	0
$13$	2.55493	0.0545084	0.0272542	−	0.999629i	$-0.491324\pi$
$13$	2.55493	0.0545084	0.0272542	+	0.999629i	$0.491324\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	126.114	1.79925	0.899624	−	0.436665i	$-0.143840\pi$
$17$	126.114	1.79925	0.899624	+	0.436665i	$0.143840\pi$
$18$	0	0
$19$	103.539	1.25019	0.625093	−	0.780550i	$-0.285061\pi$
$19$	103.539	1.25019	0.625093	+	0.780550i	$0.285061\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	200.434	1.81710	0.908551	−	0.417774i	$-0.137190\pi$
$23$	200.434	1.81710	0.908551	+	0.417774i	$0.137190\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	71.4215	0.457333	0.228666	−	0.973505i	$-0.426564\pi$
$29$	71.4215	0.457333	0.228666	+	0.973505i	$0.426564\pi$
$30$	0	0
$31$	158.130	0.916161	0.458081	−	0.888911i	$-0.348537\pi$
$31$	158.130	0.916161	0.458081	+	0.888911i	$0.348537\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−105.729	−0.510612
$36$	0	0
$37$	7.18163	0.0319095	0.0159548	−	0.999873i	$-0.494921\pi$
$37$	7.18163	0.0319095	0.0159548	+	0.999873i	$0.494921\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−347.052	−1.32196	−0.660980	−	0.750404i	$-0.729859\pi$
$41$	−347.052	−1.32196	−0.660980	+	0.750404i	$0.729859\pi$
$42$	0	0
$43$	−189.318	−0.671413	−0.335707	−	0.941967i	$-0.608975\pi$
$43$	−189.318	−0.671413	−0.335707	+	0.941967i	$0.608975\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−585.443	−1.81693	−0.908464	−	0.417963i	$-0.862744\pi$
$47$	−585.443	−1.81693	−0.908464	+	0.417963i	$0.862744\pi$
$48$	0	0
$49$	104.142	0.303622
$50$	0	0
$51$	0	0
$52$	0	0
$53$	77.2791	0.200285	0.100143	−	0.994973i	$-0.468070\pi$
$53$	77.2791	0.200285	0.100143	+	0.994973i	$0.468070\pi$
$54$	0	0
$55$	−337.147	−0.826561
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−200.790	−0.443062	−0.221531	−	0.975153i	$-0.571105\pi$
$59$	−200.790	−0.443062	−0.221531	+	0.975153i	$0.571105\pi$
$60$	0	0
$61$	681.137	1.42968	0.714841	−	0.699287i	$-0.246499\pi$
$61$	681.137	1.42968	0.714841	+	0.699287i	$0.246499\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−12.7746	−0.0243769
$66$	0	0
$67$	810.751	1.47834	0.739172	−	0.673517i	$-0.235217\pi$
$67$	810.751	1.47834	0.739172	+	0.673517i	$0.235217\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−515.476	−0.861631	−0.430816	−	0.902440i	$-0.641774\pi$
$71$	−515.476	−0.861631	−0.430816	+	0.902440i	$0.641774\pi$
$72$	0	0
$73$	385.577	0.618197	0.309099	−	0.951030i	$-0.399973\pi$
$73$	385.577	0.618197	0.309099	+	0.951030i	$0.399973\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1425.84	2.11026
$78$	0	0
$79$	209.405	0.298226	0.149113	−	0.988820i	$-0.452358\pi$
$79$	209.405	0.298226	0.149113	+	0.988820i	$0.452358\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−887.189	−1.17327	−0.586637	−	0.809850i	$-0.699548\pi$
$83$	−887.189	−1.17327	−0.586637	+	0.809850i	$0.699548\pi$
$84$	0	0
$85$	−630.572	−0.804648
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−548.890	−0.653733	−0.326867	−	0.945071i	$-0.605993\pi$
$89$	−548.890	−0.653733	−0.326867	+	0.945071i	$0.605993\pi$
$90$	0	0
$91$	54.0258	0.0622357
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−517.696	−0.559100
$96$	0	0
$97$	−1523.96	−1.59520	−0.797600	−	0.603186i	$-0.793897\pi$
$97$	−1523.96	−1.59520	−0.797600	+	0.603186i	$0.793897\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	347.825	0.342672	0.171336	−	0.985213i	$-0.445192\pi$
$101$	347.825	0.342672	0.171336	+	0.985213i	$0.445192\pi$
$102$	0	0
$103$	1955.17	1.87037	0.935187	−	0.354155i	$-0.115232\pi$
$103$	1955.17	1.87037	0.935187	+	0.354155i	$0.115232\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1576.16	−1.42405	−0.712026	−	0.702153i	$-0.752222\pi$
$107$	−1576.16	−1.42405	−0.712026	+	0.702153i	$0.752222\pi$
$108$	0	0
$109$	−158.232	−0.139045	−0.0695224	−	0.997580i	$-0.522148\pi$
$109$	−158.232	−0.139045	−0.0695224	+	0.997580i	$0.522148\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	583.856	0.486058	0.243029	−	0.970019i	$-0.421859\pi$
$113$	583.856	0.486058	0.243029	+	0.970019i	$0.421859\pi$
$114$	0	0
$115$	−1002.17	−0.812633
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2666.78	2.05431
$120$	0	0
$121$	3215.72	2.41602
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−2572.68	−1.79754	−0.898772	−	0.438416i	$-0.855540\pi$
$127$	−2572.68	−1.79754	−0.898772	+	0.438416i	$0.855540\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1793.93	−1.19646	−0.598229	−	0.801325i	$-0.704129\pi$
$131$	−1793.93	−1.19646	−0.598229	+	0.801325i	$0.704129\pi$
$132$	0	0
$133$	2189.41	1.42742
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1729.59	−1.07860	−0.539302	−	0.842112i	$-0.681312\pi$
$137$	−1729.59	−1.07860	−0.539302	+	0.842112i	$0.681312\pi$
$138$	0	0
$139$	−690.713	−0.421479	−0.210739	−	0.977542i	$-0.567587\pi$
$139$	−690.713	−0.421479	−0.210739	+	0.977542i	$0.567587\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	172.277	0.100745
$144$	0	0
$145$	−357.108	−0.204525
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2876.27	1.58143	0.790715	−	0.612185i	$-0.209709\pi$
$149$	2876.27	1.58143	0.790715	+	0.612185i	$0.209709\pi$
$150$	0	0
$151$	−1204.10	−0.648930	−0.324465	−	0.945898i	$-0.605184\pi$
$151$	−1204.10	−0.648930	−0.324465	+	0.945898i	$0.605184\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−790.650	−0.409720
$156$	0	0
$157$	−790.488	−0.401833	−0.200917	−	0.979608i	$-0.564392\pi$
$157$	−790.488	−0.401833	−0.200917	+	0.979608i	$0.564392\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4238.32	2.07470
$162$	0	0
$163$	2202.61	1.05842	0.529209	−	0.848492i	$-0.322489\pi$
$163$	2202.61	1.05842	0.529209	+	0.848492i	$0.322489\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1833.20	0.849445	0.424722	−	0.905324i	$-0.360372\pi$
$167$	1833.20	0.849445	0.424722	+	0.905324i	$0.360372\pi$
$168$	0	0
$169$	−2190.47	−0.997029
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2686.62	1.18069	0.590347	−	0.807150i	$-0.298991\pi$
$173$	2686.62	1.18069	0.590347	+	0.807150i	$0.298991\pi$
$174$	0	0
$175$	528.644	0.228353
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4500.41	−1.87920	−0.939598	−	0.342280i	$-0.888801\pi$
$179$	−4500.41	−1.87920	−0.939598	+	0.342280i	$0.888801\pi$
$180$	0	0
$181$	−1600.82	−0.657390	−0.328695	−	0.944436i	$-0.606609\pi$
$181$	−1600.82	−0.657390	−0.328695	+	0.944436i	$0.606609\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−35.9081	−0.0142704
$186$	0	0
$187$	8503.81	3.32546
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−862.280	−0.326662	−0.163331	−	0.986571i	$-0.552224\pi$
$191$	−862.280	−0.326662	−0.163331	+	0.986571i	$0.552224\pi$
$192$	0	0
$193$	−768.319	−0.286554	−0.143277	−	0.989683i	$-0.545764\pi$
$193$	−768.319	−0.286554	−0.143277	+	0.989683i	$0.545764\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3090.06	−1.11755	−0.558776	−	0.829318i	$-0.688729\pi$
$197$	−3090.06	−1.11755	−0.558776	+	0.829318i	$0.688729\pi$
$198$	0	0
$199$	−3255.41	−1.15965	−0.579824	−	0.814742i	$-0.696879\pi$
$199$	−3255.41	−1.15965	−0.579824	+	0.814742i	$0.696879\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1510.26	0.522165
$204$	0	0
$205$	1735.26	0.591198
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6981.59	2.31065
$210$	0	0
$211$	2891.42	0.943383	0.471691	−	0.881764i	$-0.343644\pi$
$211$	2891.42	0.943383	0.471691	+	0.881764i	$0.343644\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	946.592	0.300265
$216$	0	0
$217$	3343.78	1.04604
$218$	0	0
$219$	0	0
$220$	0	0
$221$	322.213	0.0980742
$222$	0	0
$223$	1345.31	0.403983	0.201992	−	0.979387i	$-0.435259\pi$
$223$	1345.31	0.403983	0.201992	+	0.979387i	$0.435259\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3643.36	−1.06528	−0.532639	−	0.846343i	$-0.678800\pi$
$227$	−3643.36	−1.06528	−0.532639	+	0.846343i	$0.678800\pi$
$228$	0	0
$229$	314.732	0.0908213	0.0454106	−	0.998968i	$-0.485540\pi$
$229$	314.732	0.0908213	0.0454106	+	0.998968i	$0.485540\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1962.28	0.551732	0.275866	−	0.961196i	$-0.411035\pi$
$233$	1962.28	0.551732	0.275866	+	0.961196i	$0.411035\pi$
$234$	0	0
$235$	2927.21	0.812555
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3128.78	−0.846795	−0.423397	−	0.905944i	$-0.639163\pi$
$239$	−3128.78	−0.846795	−0.423397	+	0.905944i	$0.639163\pi$
$240$	0	0
$241$	1538.75	0.411284	0.205642	−	0.978627i	$-0.434072\pi$
$241$	1538.75	0.411284	0.205642	+	0.978627i	$0.434072\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−520.712	−0.135784
$246$	0	0
$247$	264.535	0.0681456
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5699.82	1.43334	0.716672	−	0.697410i	$-0.245664\pi$
$251$	5699.82	1.43334	0.716672	+	0.697410i	$0.245664\pi$
$252$	0	0
$253$	13515.1	3.35845
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1406.28	0.341327	0.170664	−	0.985329i	$-0.445409\pi$
$257$	1406.28	0.341327	0.170664	+	0.985329i	$0.445409\pi$
$258$	0	0
$259$	151.861	0.0364331
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7658.46	−1.79559	−0.897796	−	0.440412i	$-0.854833\pi$
$263$	−7658.46	−1.79559	−0.897796	+	0.440412i	$0.854833\pi$
$264$	0	0
$265$	−386.396	−0.0895702
$266$	0	0
$267$	0	0
$268$	0	0
$269$	808.242	0.183195	0.0915974	−	0.995796i	$-0.470803\pi$
$269$	808.242	0.183195	0.0915974	+	0.995796i	$0.470803\pi$
$270$	0	0
$271$	4146.22	0.929390	0.464695	−	0.885471i	$-0.346164\pi$
$271$	4146.22	0.929390	0.464695	+	0.885471i	$0.346164\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1685.73	0.369649
$276$	0	0
$277$	1948.21	0.422588	0.211294	−	0.977423i	$-0.432232\pi$
$277$	1948.21	0.422588	0.211294	+	0.977423i	$0.432232\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5888.41	−1.25008	−0.625041	−	0.780592i	$-0.714918\pi$
$281$	−5888.41	−1.25008	−0.625041	+	0.780592i	$0.714918\pi$
$282$	0	0
$283$	−826.285	−0.173560	−0.0867801	−	0.996227i	$-0.527658\pi$
$283$	−826.285	−0.173560	−0.0867801	+	0.996227i	$0.527658\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7338.66	−1.50936
$288$	0	0
$289$	10991.8	2.23729
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1817.15	0.362318	0.181159	−	0.983454i	$-0.442015\pi$
$293$	1817.15	0.362318	0.181159	+	0.983454i	$0.442015\pi$
$294$	0	0
$295$	1003.95	0.198143
$296$	0	0
$297$	0	0
$298$	0	0
$299$	512.094	0.0990474
$300$	0	0
$301$	−4003.28	−0.766595
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3405.68	−0.639373
$306$	0	0
$307$	7055.67	1.31169	0.655845	−	0.754896i	$-0.272313\pi$
$307$	7055.67	1.31169	0.655845	+	0.754896i	$0.272313\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1144.46	0.208669	0.104335	−	0.994542i	$-0.466729\pi$
$311$	1144.46	0.208669	0.104335	+	0.994542i	$0.466729\pi$
$312$	0	0
$313$	−5868.73	−1.05981	−0.529905	−	0.848057i	$-0.677772\pi$
$313$	−5868.73	−1.05981	−0.529905	+	0.848057i	$0.677772\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9215.18	−1.63273	−0.816366	−	0.577534i	$-0.804015\pi$
$317$	−9215.18	−1.63273	−0.816366	+	0.577534i	$0.804015\pi$
$318$	0	0
$319$	4815.91	0.845264
$320$	0	0
$321$	0	0
$322$	0	0
$323$	13057.8	2.24939
$324$	0	0
$325$	63.8732	0.0109017
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−12379.6	−2.07450
$330$	0	0
$331$	−4947.10	−0.821502	−0.410751	−	0.911748i	$-0.634733\pi$
$331$	−4947.10	−0.821502	−0.410751	+	0.911748i	$0.634733\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4053.76	−0.661135
$336$	0	0
$337$	1879.56	0.303816	0.151908	−	0.988395i	$-0.451458\pi$
$337$	1879.56	0.303816	0.151908	+	0.988395i	$0.451458\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	10662.6	1.69329
$342$	0	0
$343$	−5050.82	−0.795098
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2086.38	0.322775	0.161388	−	0.986891i	$-0.448403\pi$
$347$	2086.38	0.322775	0.161388	+	0.986891i	$0.448403\pi$
$348$	0	0
$349$	7986.00	1.22487	0.612437	−	0.790519i	$-0.290189\pi$
$349$	7986.00	1.22487	0.612437	+	0.790519i	$0.290189\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1443.39	0.217631	0.108815	−	0.994062i	$-0.465294\pi$
$353$	1443.39	0.217631	0.108815	+	0.994062i	$0.465294\pi$
$354$	0	0
$355$	2577.38	0.385333
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9598.40	1.41110	0.705549	−	0.708661i	$-0.250700\pi$
$359$	9598.40	1.41110	0.705549	+	0.708661i	$0.250700\pi$
$360$	0	0
$361$	3861.37	0.562964
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1927.89	−0.276466
$366$	0	0
$367$	−1237.82	−0.176059	−0.0880296	−	0.996118i	$-0.528057\pi$
$367$	−1237.82	−0.176059	−0.0880296	+	0.996118i	$0.528057\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1634.12	0.228678
$372$	0	0
$373$	−3832.31	−0.531982	−0.265991	−	0.963976i	$-0.585699\pi$
$373$	−3832.31	−0.531982	−0.265991	+	0.963976i	$0.585699\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	182.477	0.0249285
$378$	0	0
$379$	−8252.24	−1.11844	−0.559220	−	0.829019i	$-0.688899\pi$
$379$	−8252.24	−1.11844	−0.559220	+	0.829019i	$0.688899\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5080.20	−0.677771	−0.338885	−	0.940828i	$-0.610050\pi$
$383$	−5080.20	−0.677771	−0.338885	+	0.940828i	$0.610050\pi$
$384$	0	0
$385$	−7129.22	−0.943737
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7382.79	−0.962268	−0.481134	−	0.876647i	$-0.659775\pi$
$389$	−7382.79	−0.962268	−0.481134	+	0.876647i	$0.659775\pi$
$390$	0	0
$391$	25277.6	3.26942
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1047.02	−0.133371
$396$	0	0
$397$	7108.65	0.898673	0.449336	−	0.893363i	$-0.351660\pi$
$397$	7108.65	0.898673	0.449336	+	0.893363i	$0.351660\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5347.35	−0.665920	−0.332960	−	0.942941i	$-0.608047\pi$
$401$	−5347.35	−0.665920	−0.332960	+	0.942941i	$0.608047\pi$
$402$	0	0
$403$	404.011	0.0499385
$404$	0	0
$405$	0	0
$406$	0	0
$407$	484.253	0.0589767
$408$	0	0
$409$	−223.839	−0.0270615	−0.0135307	−	0.999908i	$-0.504307\pi$
$409$	−223.839	−0.0270615	−0.0135307	+	0.999908i	$0.504307\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4245.86	−0.505872
$414$	0	0
$415$	4435.95	0.524704
$416$	0	0
$417$	0	0
$418$	0	0
$419$	384.631	0.0448459	0.0224230	−	0.999749i	$-0.492862\pi$
$419$	384.631	0.0448459	0.0224230	+	0.999749i	$0.492862\pi$
$420$	0	0
$421$	4125.76	0.477618	0.238809	−	0.971067i	$-0.423243\pi$
$421$	4125.76	0.477618	0.238809	+	0.971067i	$0.423243\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3152.86	0.359850
$426$	0	0
$427$	14403.1	1.63236
$428$	0	0
$429$	0	0
$430$	0	0
$431$	457.020	0.0510762	0.0255381	−	0.999674i	$-0.491870\pi$
$431$	457.020	0.0510762	0.0255381	+	0.999674i	$0.491870\pi$
$432$	0	0
$433$	−11085.8	−1.23037	−0.615187	−	0.788382i	$-0.710919\pi$
$433$	−11085.8	−1.23037	−0.615187	+	0.788382i	$0.710919\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	20752.8	2.27172
$438$	0	0
$439$	4082.70	0.443865	0.221933	−	0.975062i	$-0.428764\pi$
$439$	4082.70	0.443865	0.221933	+	0.975062i	$0.428764\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14975.7	1.60613	0.803067	−	0.595888i	$-0.203200\pi$
$443$	14975.7	1.60613	0.803067	+	0.595888i	$0.203200\pi$
$444$	0	0
$445$	2744.45	0.292358
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1863.53	0.195870	0.0979349	−	0.995193i	$-0.468776\pi$
$449$	1863.53	0.195870	0.0979349	+	0.995193i	$0.468776\pi$
$450$	0	0
$451$	−23401.5	−2.44331
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−270.129	−0.0278326
$456$	0	0
$457$	−4029.90	−0.412496	−0.206248	−	0.978500i	$-0.566125\pi$
$457$	−4029.90	−0.412496	−0.206248	+	0.978500i	$0.566125\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−9320.82	−0.941679	−0.470839	−	0.882219i	$-0.656049\pi$
$461$	−9320.82	−0.941679	−0.470839	+	0.882219i	$0.656049\pi$
$462$	0	0
$463$	−9948.81	−0.998619	−0.499309	−	0.866424i	$-0.666413\pi$
$463$	−9948.81	−0.998619	−0.499309	+	0.866424i	$0.666413\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7946.35	0.787395	0.393697	−	0.919240i	$-0.371196\pi$
$467$	7946.35	0.787395	0.393697	+	0.919240i	$0.371196\pi$
$468$	0	0
$469$	17143.9	1.68792
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−12765.6	−1.24094
$474$	0	0
$475$	2588.48	0.250037
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4166.01	0.397390	0.198695	−	0.980061i	$-0.436330\pi$
$479$	4166.01	0.397390	0.198695	+	0.980061i	$0.436330\pi$
$480$	0	0
$481$	18.3485	0.00173934
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7619.79	0.713395
$486$	0	0
$487$	−4662.27	−0.433815	−0.216907	−	0.976192i	$-0.569597\pi$
$487$	−4662.27	−0.433815	−0.216907	+	0.976192i	$0.569597\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−9609.89	−0.883275	−0.441638	−	0.897194i	$-0.645602\pi$
$491$	−9609.89	−0.883275	−0.441638	+	0.897194i	$0.645602\pi$
$492$	0	0
$493$	9007.28	0.822855
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10900.1	−0.983778
$498$	0	0
$499$	5439.00	0.487942	0.243971	−	0.969782i	$-0.421550\pi$
$499$	5439.00	0.487942	0.243971	+	0.969782i	$0.421550\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	10407.2	0.922537	0.461268	−	0.887261i	$-0.347395\pi$
$503$	10407.2	0.922537	0.461268	+	0.887261i	$0.347395\pi$
$504$	0	0
$505$	−1739.13	−0.153248
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−8982.15	−0.782174	−0.391087	−	0.920354i	$-0.627901\pi$
$509$	−8982.15	−0.782174	−0.391087	+	0.920354i	$0.627901\pi$
$510$	0	0
$511$	8153.32	0.705835
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−9775.84	−0.836456
$516$	0	0
$517$	−39476.0	−3.35813
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−871.580	−0.0732910	−0.0366455	−	0.999328i	$-0.511667\pi$
$521$	−871.580	−0.0732910	−0.0366455	+	0.999328i	$0.511667\pi$
$522$	0	0
$523$	−16215.9	−1.35578	−0.677888	−	0.735165i	$-0.737105\pi$
$523$	−16215.9	−1.35578	−0.677888	+	0.735165i	$0.737105\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	19942.5	1.64840
$528$	0	0
$529$	28006.7	2.30186
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−886.691	−0.0720579
$534$	0	0
$535$	7880.82	0.636856
$536$	0	0
$537$	0	0
$538$	0	0
$539$	7022.26	0.561169
$540$	0	0
$541$	7195.89	0.571859	0.285929	−	0.958251i	$-0.407698\pi$
$541$	7195.89	0.571859	0.285929	+	0.958251i	$0.407698\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	791.161	0.0621828
$546$	0	0
$547$	12847.2	1.00422	0.502109	−	0.864804i	$-0.332558\pi$
$547$	12847.2	1.00422	0.502109	+	0.864804i	$0.332558\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7394.93	0.571751
$552$	0	0
$553$	4428.02	0.340504
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1916.53	−0.145792	−0.0728958	−	0.997340i	$-0.523224\pi$
$557$	−1916.53	−0.145792	−0.0728958	+	0.997340i	$0.523224\pi$
$558$	0	0
$559$	−483.695	−0.0365977
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5124.14	0.383582	0.191791	−	0.981436i	$-0.438570\pi$
$563$	5124.14	0.383582	0.191791	+	0.981436i	$0.438570\pi$
$564$	0	0
$565$	−2919.28	−0.217372
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−17498.6	−1.28924	−0.644620	−	0.764503i	$-0.722985\pi$
$569$	−17498.6	−1.28924	−0.644620	+	0.764503i	$0.722985\pi$
$570$	0	0
$571$	−9253.34	−0.678179	−0.339090	−	0.940754i	$-0.610119\pi$
$571$	−9253.34	−0.678179	−0.339090	+	0.940754i	$0.610119\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5010.85	0.363420
$576$	0	0
$577$	−17420.7	−1.25690	−0.628451	−	0.777849i	$-0.716311\pi$
$577$	−17420.7	−1.25690	−0.628451	+	0.777849i	$0.716311\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−18760.3	−1.33960
$582$	0	0
$583$	5210.88	0.370176
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7839.38	−0.551219	−0.275610	−	0.961270i	$-0.588880\pi$
$587$	−7839.38	−0.551219	−0.275610	+	0.961270i	$0.588880\pi$
$588$	0	0
$589$	16372.7	1.14537
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1611.03	0.111564	0.0557818	−	0.998443i	$-0.482235\pi$
$593$	1611.03	0.111564	0.0557818	+	0.998443i	$0.482235\pi$
$594$	0	0
$595$	−13333.9	−0.918717
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−17274.0	−1.17829	−0.589146	−	0.808026i	$-0.700536\pi$
$599$	−17274.0	−1.17829	−0.589146	+	0.808026i	$0.700536\pi$
$600$	0	0
$601$	−26908.5	−1.82632	−0.913161	−	0.407599i	$-0.866366\pi$
$601$	−26908.5	−1.82632	−0.913161	+	0.407599i	$0.866366\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−16078.6	−1.08048
$606$	0	0
$607$	5500.73	0.367822	0.183911	−	0.982943i	$-0.441124\pi$
$607$	5500.73	0.367822	0.183911	+	0.982943i	$0.441124\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1495.76	−0.0990379
$612$	0	0
$613$	17362.0	1.14396	0.571978	−	0.820269i	$-0.306176\pi$
$613$	17362.0	1.14396	0.571978	+	0.820269i	$0.306176\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2946.26	0.192240	0.0961198	−	0.995370i	$-0.469357\pi$
$617$	2946.26	0.192240	0.0961198	+	0.995370i	$0.469357\pi$
$618$	0	0
$619$	15311.6	0.994223	0.497112	−	0.867687i	$-0.334394\pi$
$619$	15311.6	0.994223	0.497112	+	0.867687i	$0.334394\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−11606.7	−0.746408
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	905.706	0.0574132
$630$	0	0
$631$	−1067.12	−0.0673241	−0.0336621	−	0.999433i	$-0.510717\pi$
$631$	−1067.12	−0.0673241	−0.0336621	+	0.999433i	$0.510717\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12863.4	0.803886
$636$	0	0
$637$	266.076	0.0165500
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8977.71	0.553195	0.276598	−	0.960986i	$-0.410793\pi$
$641$	8977.71	0.553195	0.276598	+	0.960986i	$0.410793\pi$
$642$	0	0
$643$	15935.5	0.977346	0.488673	−	0.872467i	$-0.337481\pi$
$643$	15935.5	0.977346	0.488673	+	0.872467i	$0.337481\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21284.1	1.29330	0.646648	−	0.762789i	$-0.276170\pi$
$647$	21284.1	1.29330	0.646648	+	0.762789i	$0.276170\pi$
$648$	0	0
$649$	−13539.2	−0.818889
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8173.21	0.489805	0.244902	−	0.969548i	$-0.421244\pi$
$653$	8173.21	0.489805	0.244902	+	0.969548i	$0.421244\pi$
$654$	0	0
$655$	8969.63	0.535072
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−32350.6	−1.91229	−0.956145	−	0.292895i	$-0.905381\pi$
$659$	−32350.6	−1.91229	−0.956145	+	0.292895i	$0.905381\pi$
$660$	0	0
$661$	26775.6	1.57557	0.787783	−	0.615953i	$-0.211229\pi$
$661$	26775.6	1.57557	0.787783	+	0.615953i	$0.211229\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−10947.1	−0.638360
$666$	0	0
$667$	14315.3	0.831020
$668$	0	0
$669$	0	0
$670$	0	0
$671$	45928.6	2.64241
$672$	0	0
$673$	13448.7	0.770297	0.385149	−	0.922855i	$-0.374150\pi$
$673$	13448.7	0.770297	0.385149	+	0.922855i	$0.374150\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−20767.3	−1.17896	−0.589478	−	0.807784i	$-0.700667\pi$
$677$	−20767.3	−1.17896	−0.589478	+	0.807784i	$0.700667\pi$
$678$	0	0
$679$	−32225.2	−1.82134
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−11894.6	−0.666373	−0.333187	−	0.942861i	$-0.608124\pi$
$683$	−11894.6	−0.666373	−0.333187	+	0.942861i	$0.608124\pi$
$684$	0	0
$685$	8647.94	0.482366
$686$	0	0
$687$	0	0
$688$	0	0
$689$	197.443	0.0109172
$690$	0	0
$691$	−14871.9	−0.818747	−0.409374	−	0.912367i	$-0.634253\pi$
$691$	−14871.9	−0.818747	−0.409374	+	0.912367i	$0.634253\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3453.57	0.188491
$696$	0	0
$697$	−43768.2	−2.37853
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2423.29	0.130566	0.0652828	−	0.997867i	$-0.479205\pi$
$701$	2423.29	0.130566	0.0652828	+	0.997867i	$0.479205\pi$
$702$	0	0
$703$	743.580	0.0398928
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7355.02	0.391250
$708$	0	0
$709$	8971.88	0.475241	0.237621	−	0.971358i	$-0.423632\pi$
$709$	8971.88	0.475241	0.237621	+	0.971358i	$0.423632\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	31694.6	1.66476
$714$	0	0
$715$	−861.386	−0.0450545
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−18757.9	−0.972951	−0.486475	−	0.873694i	$-0.661718\pi$
$719$	−18757.9	−0.972951	−0.486475	+	0.873694i	$0.661718\pi$
$720$	0	0
$721$	41343.5	2.13552
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1785.54	0.0914665
$726$	0	0
$727$	14205.6	0.724700	0.362350	−	0.932042i	$-0.381974\pi$
$727$	14205.6	0.724700	0.362350	+	0.932042i	$0.381974\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−23875.8	−1.20804
$732$	0	0
$733$	23776.1	1.19808	0.599038	−	0.800720i	$-0.295550\pi$
$733$	23776.1	1.19808	0.599038	+	0.800720i	$0.295550\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	54668.4	2.73234
$738$	0	0
$739$	−14175.5	−0.705623	−0.352811	−	0.935694i	$-0.614774\pi$
$739$	−14175.5	−0.705623	−0.352811	+	0.935694i	$0.614774\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7564.68	0.373514	0.186757	−	0.982406i	$-0.440202\pi$
$743$	7564.68	0.373514	0.186757	+	0.982406i	$0.440202\pi$
$744$	0	0
$745$	−14381.3	−0.707237
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−33329.2	−1.62593
$750$	0	0
$751$	−18602.1	−0.903863	−0.451932	−	0.892053i	$-0.649265\pi$
$751$	−18602.1	−0.903863	−0.451932	+	0.892053i	$0.649265\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6020.51	0.290210
$756$	0	0
$757$	−4665.12	−0.223985	−0.111993	−	0.993709i	$-0.535723\pi$
$757$	−4665.12	−0.223985	−0.111993	+	0.993709i	$0.535723\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	36063.6	1.71788	0.858938	−	0.512079i	$-0.171125\pi$
$761$	36063.6	1.71788	0.858938	+	0.512079i	$0.171125\pi$
$762$	0	0
$763$	−3345.94	−0.158756
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−513.005	−0.0241506
$768$	0	0
$769$	−28921.5	−1.35623	−0.678113	−	0.734958i	$-0.737202\pi$
$769$	−28921.5	−1.35623	−0.678113	+	0.734958i	$0.737202\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	25489.9	1.18604	0.593019	−	0.805188i	$-0.297936\pi$
$773$	25489.9	1.18604	0.593019	+	0.805188i	$0.297936\pi$
$774$	0	0
$775$	3953.25	0.183232
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−35933.5	−1.65270
$780$	0	0
$781$	−34758.3	−1.59251
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3952.44	0.179705
$786$	0	0
$787$	−3627.06	−0.164283	−0.0821415	−	0.996621i	$-0.526176\pi$
$787$	−3627.06	−0.164283	−0.0821415	+	0.996621i	$0.526176\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12346.1	0.554962
$792$	0	0
$793$	1740.26	0.0779297
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15571.1	0.692041	0.346021	−	0.938227i	$-0.387533\pi$
$797$	15571.1	0.692041	0.346021	+	0.938227i	$0.387533\pi$
$798$	0	0
$799$	−73832.7	−3.26910
$800$	0	0
$801$	0	0
$802$	0	0
$803$	25999.2	1.14258
$804$	0	0
$805$	−21191.6	−0.927834
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7352.63	0.319536	0.159768	−	0.987155i	$-0.448925\pi$
$809$	7352.63	0.319536	0.159768	+	0.987155i	$0.448925\pi$
$810$	0	0
$811$	−19743.8	−0.854869	−0.427434	−	0.904046i	$-0.640583\pi$
$811$	−19743.8	−0.854869	−0.427434	+	0.904046i	$0.640583\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−11013.1	−0.473339
$816$	0	0
$817$	−19601.9	−0.839392
$818$	0	0
$819$	0	0
$820$	0	0
$821$	43315.1	1.84130	0.920651	−	0.390387i	$-0.127659\pi$
$821$	43315.1	1.84130	0.920651	+	0.390387i	$0.127659\pi$
$822$	0	0
$823$	23632.2	1.00093	0.500466	−	0.865756i	$-0.333162\pi$
$823$	23632.2	1.00093	0.500466	+	0.865756i	$0.333162\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24109.3	1.01374	0.506869	−	0.862023i	$-0.330803\pi$
$827$	24109.3	1.01374	0.506869	+	0.862023i	$0.330803\pi$
$828$	0	0
$829$	1330.02	0.0557218	0.0278609	−	0.999612i	$-0.491130\pi$
$829$	1330.02	0.0557218	0.0278609	+	0.999612i	$0.491130\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	13133.8	0.546292
$834$	0	0
$835$	−9166.00	−0.379883
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−16823.3	−0.692258	−0.346129	−	0.938187i	$-0.612504\pi$
$839$	−16823.3	−0.692258	−0.346129	+	0.938187i	$0.612504\pi$
$840$	0	0
$841$	−19288.0	−0.790847
$842$	0	0
$843$	0	0
$844$	0	0
$845$	10952.4	0.445885
$846$	0	0
$847$	67998.8	2.75852
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1439.44	0.0579829
$852$	0	0
$853$	−11732.7	−0.470951	−0.235475	−	0.971880i	$-0.575665\pi$
$853$	−11732.7	−0.470951	−0.235475	+	0.971880i	$0.575665\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25843.7	−1.03011	−0.515055	−	0.857157i	$-0.672229\pi$
$857$	−25843.7	−1.03011	−0.515055	+	0.857157i	$0.672229\pi$
$858$	0	0
$859$	−21463.9	−0.852549	−0.426275	−	0.904594i	$-0.640174\pi$
$859$	−21463.9	−0.852549	−0.426275	+	0.904594i	$0.640174\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11980.8	−0.472575	−0.236287	−	0.971683i	$-0.575931\pi$
$863$	−11980.8	−0.472575	−0.236287	+	0.971683i	$0.575931\pi$
$864$	0	0
$865$	−13433.1	−0.528022
$866$	0	0
$867$	0	0
$868$	0	0
$869$	14120.0	0.551196
$870$	0	0
$871$	2071.41	0.0805822
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2643.22	−0.102122
$876$	0	0
$877$	3662.20	0.141008	0.0705038	−	0.997512i	$-0.477539\pi$
$877$	3662.20	0.141008	0.0705038	+	0.997512i	$0.477539\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−27016.4	−1.03315	−0.516575	−	0.856242i	$-0.672793\pi$
$881$	−27016.4	−1.03315	−0.516575	+	0.856242i	$0.672793\pi$
$882$	0	0
$883$	−24437.6	−0.931359	−0.465679	−	0.884954i	$-0.654190\pi$
$883$	−24437.6	−0.931359	−0.465679	+	0.884954i	$0.654190\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−35067.7	−1.32746	−0.663731	−	0.747971i	$-0.731028\pi$
$887$	−35067.7	−1.32746	−0.663731	+	0.747971i	$0.731028\pi$
$888$	0	0
$889$	−54401.2	−2.05237
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−60616.3	−2.27150
$894$	0	0
$895$	22502.0	0.840402
$896$	0	0
$897$	0	0
$898$	0	0
$899$	11293.9	0.418990
$900$	0	0
$901$	9746.01	0.360363
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8004.08	0.293994
$906$	0	0
$907$	29567.8	1.08245	0.541225	−	0.840878i	$-0.317961\pi$
$907$	29567.8	1.08245	0.541225	+	0.840878i	$0.317961\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−37564.2	−1.36614	−0.683072	−	0.730351i	$-0.739356\pi$
$911$	−37564.2	−1.36614	−0.683072	+	0.730351i	$0.739356\pi$
$912$	0	0
$913$	−59822.6	−2.16850
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−37933.9	−1.36607
$918$	0	0
$919$	−42968.8	−1.54234	−0.771170	−	0.636629i	$-0.780328\pi$
$919$	−42968.8	−1.54234	−0.771170	+	0.636629i	$0.780328\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1317.00	−0.0469661
$924$	0	0
$925$	179.541	0.00638191
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−25931.4	−0.915804	−0.457902	−	0.889003i	$-0.651399\pi$
$929$	−25931.4	−0.915804	−0.457902	+	0.889003i	$0.651399\pi$
$930$	0	0
$931$	10782.8	0.379584
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−42519.1	−1.48719
$936$	0	0
$937$	−20940.0	−0.730075	−0.365037	−	0.930993i	$-0.618944\pi$
$937$	−20940.0	−0.730075	−0.365037	+	0.930993i	$0.618944\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	8528.77	0.295462	0.147731	−	0.989028i	$-0.452803\pi$
$941$	8528.77	0.295462	0.147731	+	0.989028i	$0.452803\pi$
$942$	0	0
$943$	−69560.9	−2.40214
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−18566.9	−0.637112	−0.318556	−	0.947904i	$-0.603198\pi$
$947$	−18566.9	−0.637112	−0.318556	+	0.947904i	$0.603198\pi$
$948$	0	0
$949$	985.122	0.0336970
$950$	0	0
$951$	0	0
$952$	0	0
$953$	8866.54	0.301380	0.150690	−	0.988581i	$-0.451850\pi$
$953$	8866.54	0.301380	0.150690	+	0.988581i	$0.451850\pi$
$954$	0	0
$955$	4311.40	0.146088
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−36573.4	−1.23151
$960$	0	0
$961$	−4785.89	−0.160649
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3841.60	0.128151
$966$	0	0
$967$	−38880.2	−1.29297	−0.646485	−	0.762927i	$-0.723762\pi$
$967$	−38880.2	−1.29297	−0.646485	+	0.762927i	$0.723762\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−33655.7	−1.11232	−0.556161	−	0.831075i	$-0.687726\pi$
$971$	−33655.7	−1.11232	−0.556161	+	0.831075i	$0.687726\pi$
$972$	0	0
$973$	−14605.6	−0.481228
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−6269.02	−0.205285	−0.102643	−	0.994718i	$-0.532730\pi$
$977$	−6269.02	−0.205285	−0.102643	+	0.994718i	$0.532730\pi$
$978$	0	0
$979$	−37011.3	−1.20826
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−39772.2	−1.29048	−0.645238	−	0.763982i	$-0.723242\pi$
$983$	−39772.2	−1.29048	−0.645238	+	0.763982i	$0.723242\pi$
$984$	0	0
$985$	15450.3	0.499785
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−37945.8	−1.22003
$990$	0	0
$991$	8799.28	0.282057	0.141028	−	0.990006i	$-0.454959\pi$
$991$	8799.28	0.282057	0.141028	+	0.990006i	$0.454959\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16277.1	0.518610
$996$	0	0
$997$	42122.5	1.33805	0.669024	−	0.743241i	$-0.266713\pi$
$997$	42122.5	1.33805	0.669024	+	0.743241i	$0.266713\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bl.1.3		3
3.2	odd	2		2160.4.a.bt.1.3		3
4.3	odd	2		1080.4.a.c.1.1	✓	3
12.11	even	2		1080.4.a.i.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.c.1.1	✓	3	4.3	odd	2
1080.4.a.i.1.1	yes	3	12.11	even	2
2160.4.a.bl.1.3		3	1.1	even	1	trivial
2160.4.a.bt.1.3		3	3.2	odd	2

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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2160.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} +21.1457 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} +21.1457 q^{7} +67.4294 q^{11} +2.55493 q^{13} +126.114 q^{17} +103.539 q^{19} +200.434 q^{23} +25.0000 q^{25} +71.4215 q^{29} +158.130 q^{31} -105.729 q^{35} +7.18163 q^{37} -347.052 q^{41} -189.318 q^{43} -585.443 q^{47} +104.142 q^{49} +77.2791 q^{53} -337.147 q^{55} -200.790 q^{59} +681.137 q^{61} -12.7746 q^{65} +810.751 q^{67} -515.476 q^{71} +385.577 q^{73} +1425.84 q^{77} +209.405 q^{79} -887.189 q^{83} -630.572 q^{85} -548.890 q^{89} +54.0258 q^{91} -517.696 q^{95} -1523.96 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 15 q^{5} + 24 q^{7}+O(q^{10}) \) 3 * q - 15 * q^5 + 24 * q^7	\( 3 q - 15 q^{5} + 24 q^{7} + 6 q^{11} + 48 q^{13} + 27 q^{17} + 195 q^{19} - 27 q^{23} + 75 q^{25} - 60 q^{29} + 279 q^{31} - 120 q^{35} - 138 q^{37} - 66 q^{41} - 222 q^{43} - 264 q^{47} - 237 q^{49} + 507 q^{53} - 30 q^{55} - 960 q^{59} + 543 q^{61} - 240 q^{65} + 1086 q^{67} - 1818 q^{71} + 1362 q^{73} + 1776 q^{77} + 129 q^{79} - 1569 q^{83} - 135 q^{85} + 1770 q^{89} - 1488 q^{91} - 975 q^{95} - 336 q^{97}+O(q^{100}) \) 3 * q - 15 * q^5 + 24 * q^7 + 6 * q^11 + 48 * q^13 + 27 * q^17 + 195 * q^19 - 27 * q^23 + 75 * q^25 - 60 * q^29 + 279 * q^31 - 120 * q^35 - 138 * q^37 - 66 * q^41 - 222 * q^43 - 264 * q^47 - 237 * q^49 + 507 * q^53 - 30 * q^55 - 960 * q^59 + 543 * q^61 - 240 * q^65 + 1086 * q^67 - 1818 * q^71 + 1362 * q^73 + 1776 * q^77 + 129 * q^79 - 1569 * q^83 - 135 * q^85 + 1770 * q^89 - 1488 * q^91 - 975 * q^95 - 336 * q^97

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