LMFDB - Embedded newform 1620.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(1,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1620.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.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:	$95.5830942093$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.244785.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 90x - 240 $ x^3 - x^2 - 90*x - 240
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$11.0789$ of defining polynomial
Character	$\chi$	$=$	1620.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000 q^{5} -26.8314 q^{7} +O(q^{10})$	$q-5.00000 q^{5} -26.8314 q^{7} +46.6419 q^{11} +1.21044 q^{13} +87.2838 q^{17} -125.136 q^{19} -7.16859 q^{23} +25.0000 q^{25} +43.8314 q^{29} +140.347 q^{31} +134.157 q^{35} -187.030 q^{37} -239.905 q^{41} +457.556 q^{43} -25.6106 q^{47} +376.925 q^{49} -82.8314 q^{53} -233.210 q^{55} +739.574 q^{59} -28.6306 q^{61} -6.05219 q^{65} -126.209 q^{67} -611.336 q^{71} +983.531 q^{73} -1251.47 q^{77} +372.737 q^{79} +693.280 q^{83} -436.419 q^{85} -873.848 q^{89} -32.4778 q^{91} +625.681 q^{95} -82.7538 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 15 q^{5} - 3 q^{7}+O(q^{10}) $ 3 * q - 15 * q^5 - 3 * q^7	$ 3 q - 15 q^{5} - 3 q^{7} + 24 q^{11} + 3 q^{13} + 30 q^{17} - 27 q^{19} - 99 q^{23} + 75 q^{25} + 54 q^{29} + 72 q^{31} + 15 q^{35} - 18 q^{37} - 411 q^{41} + 444 q^{43} + 75 q^{47} + 204 q^{49} - 171 q^{53} - 120 q^{55} - 297 q^{59} + 684 q^{61} - 15 q^{65} + 12 q^{67} - 642 q^{71} - 66 q^{73} - 1044 q^{77} + 1122 q^{79} - 90 q^{83} - 150 q^{85} - 756 q^{89} - 1653 q^{91} + 135 q^{95} + 1536 q^{97}+O(q^{100}) $ 3 * q - 15 * q^5 - 3 * q^7 + 24 * q^11 + 3 * q^13 + 30 * q^17 - 27 * q^19 - 99 * q^23 + 75 * q^25 + 54 * q^29 + 72 * q^31 + 15 * q^35 - 18 * q^37 - 411 * q^41 + 444 * q^43 + 75 * q^47 + 204 * q^49 - 171 * q^53 - 120 * q^55 - 297 * q^59 + 684 * q^61 - 15 * q^65 + 12 * q^67 - 642 * q^71 - 66 * q^73 - 1044 * q^77 + 1122 * q^79 - 90 * q^83 - 150 * q^85 - 756 * q^89 - 1653 * q^91 + 135 * q^95 + 1536 * 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
$7$	−26.8314	−1.44876	−0.724380	−	0.689401i	$-0.757874\pi$
$7$	−26.8314	−1.44876	−0.724380	+	0.689401i	$0.757874\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	46.6419	1.27846	0.639230	−	0.769016i	$-0.279253\pi$
$11$	46.6419	1.27846	0.639230	+	0.769016i	$0.279253\pi$
$12$	0	0
$13$	1.21044	0.0258242	0.0129121	−	0.999917i	$-0.495890\pi$
$13$	1.21044	0.0258242	0.0129121	+	0.999917i	$0.495890\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	87.2838	1.24526	0.622630	−	0.782516i	$-0.286064\pi$
$17$	87.2838	1.24526	0.622630	+	0.782516i	$0.286064\pi$
$18$	0	0
$19$	−125.136	−1.51096	−0.755479	−	0.655173i	$-0.772596\pi$
$19$	−125.136	−1.51096	−0.755479	+	0.655173i	$0.772596\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.16859	−0.0649893	−0.0324946	−	0.999472i	$-0.510345\pi$
$23$	−7.16859	−0.0649893	−0.0324946	+	0.999472i	$0.510345\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	43.8314	0.280665	0.140333	−	0.990104i	$-0.455183\pi$
$29$	43.8314	0.280665	0.140333	+	0.990104i	$0.455183\pi$
$30$	0	0
$31$	140.347	0.813129	0.406564	−	0.913622i	$-0.366727\pi$
$31$	140.347	0.813129	0.406564	+	0.913622i	$0.366727\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	134.157	0.647905
$36$	0	0
$37$	−187.030	−0.831016	−0.415508	−	0.909589i	$-0.636396\pi$
$37$	−187.030	−0.831016	−0.415508	+	0.909589i	$0.636396\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−239.905	−0.913825	−0.456913	−	0.889512i	$-0.651045\pi$
$41$	−239.905	−0.913825	−0.456913	+	0.889512i	$0.651045\pi$
$42$	0	0
$43$	457.556	1.62271	0.811356	−	0.584552i	$-0.198730\pi$
$43$	457.556	1.62271	0.811356	+	0.584552i	$0.198730\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−25.6106	−0.0794829	−0.0397414	−	0.999210i	$-0.512653\pi$
$47$	−25.6106	−0.0794829	−0.0397414	+	0.999210i	$0.512653\pi$
$48$	0	0
$49$	376.925	1.09891
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−82.8314	−0.214675	−0.107337	−	0.994223i	$-0.534233\pi$
$53$	−82.8314	−0.214675	−0.107337	+	0.994223i	$0.534233\pi$
$54$	0	0
$55$	−233.210	−0.571745
$56$	0	0
$57$	0	0
$58$	0	0
$59$	739.574	1.63194	0.815969	−	0.578096i	$-0.196204\pi$
$59$	739.574	1.63194	0.815969	+	0.578096i	$0.196204\pi$
$60$	0	0
$61$	−28.6306	−0.0600947	−0.0300474	−	0.999548i	$-0.509566\pi$
$61$	−28.6306	−0.0600947	−0.0300474	+	0.999548i	$0.509566\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.05219	−0.0115490
$66$	0	0
$67$	−126.209	−0.230133	−0.115067	−	0.993358i	$-0.536708\pi$
$67$	−126.209	−0.230133	−0.115067	+	0.993358i	$0.536708\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−611.336	−1.02186	−0.510931	−	0.859622i	$-0.670699\pi$
$71$	−611.336	−1.02186	−0.510931	+	0.859622i	$0.670699\pi$
$72$	0	0
$73$	983.531	1.57690	0.788449	−	0.615100i	$-0.210884\pi$
$73$	983.531	1.57690	0.788449	+	0.615100i	$0.210884\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1251.47	−1.85218
$78$	0	0
$79$	372.737	0.530838	0.265419	−	0.964133i	$-0.414490\pi$
$79$	372.737	0.530838	0.265419	+	0.964133i	$0.414490\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	693.280	0.916835	0.458418	−	0.888737i	$-0.348416\pi$
$83$	693.280	0.916835	0.458418	+	0.888737i	$0.348416\pi$
$84$	0	0
$85$	−436.419	−0.556898
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−873.848	−1.04076	−0.520380	−	0.853935i	$-0.674210\pi$
$89$	−873.848	−1.04076	−0.520380	+	0.853935i	$0.674210\pi$
$90$	0	0
$91$	−32.4778	−0.0374131
$92$	0	0
$93$	0	0
$94$	0	0
$95$	625.681	0.675721
$96$	0	0
$97$	−82.7538	−0.0866225	−0.0433112	−	0.999062i	$-0.513791\pi$
$97$	−82.7538	−0.0866225	−0.0433112	+	0.999062i	$0.513791\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1326.31	−1.30666	−0.653331	−	0.757072i	$-0.726629\pi$
$101$	−1326.31	−1.30666	−0.653331	+	0.757072i	$0.726629\pi$
$102$	0	0
$103$	−1668.61	−1.59625	−0.798124	−	0.602494i	$-0.794174\pi$
$103$	−1668.61	−1.59625	−0.798124	+	0.602494i	$0.794174\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−192.253	−0.173699	−0.0868495	−	0.996221i	$-0.527680\pi$
$107$	−192.253	−0.173699	−0.0868495	+	0.996221i	$0.527680\pi$
$108$	0	0
$109$	−1645.59	−1.44605	−0.723025	−	0.690822i	$-0.757249\pi$
$109$	−1645.59	−1.44605	−0.723025	+	0.690822i	$0.757249\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1556.96	−1.29616	−0.648081	−	0.761571i	$-0.724428\pi$
$113$	−1556.96	−1.29616	−0.648081	+	0.761571i	$0.724428\pi$
$114$	0	0
$115$	35.8429	0.0290641
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2341.95	−1.80408
$120$	0	0
$121$	844.467	0.634461
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−409.005	−0.285774	−0.142887	−	0.989739i	$-0.545639\pi$
$127$	−409.005	−0.285774	−0.142887	+	0.989739i	$0.545639\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2906.32	−1.93837	−0.969186	−	0.246330i	$-0.920775\pi$
$131$	−2906.32	−1.93837	−0.969186	+	0.246330i	$0.920775\pi$
$132$	0	0
$133$	3357.58	2.18902
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1890.00	−1.17864	−0.589319	−	0.807900i	$-0.700604\pi$
$137$	−1890.00	−1.17864	−0.589319	+	0.807900i	$0.700604\pi$
$138$	0	0
$139$	−2709.54	−1.65339	−0.826693	−	0.562654i	$-0.809780\pi$
$139$	−2709.54	−1.65339	−0.826693	+	0.562654i	$0.809780\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	56.4572	0.0330153
$144$	0	0
$145$	−219.157	−0.125517
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1364.39	0.750167	0.375084	−	0.926991i	$-0.377614\pi$
$149$	1364.39	0.750167	0.375084	+	0.926991i	$0.377614\pi$
$150$	0	0
$151$	882.893	0.475820	0.237910	−	0.971287i	$-0.423538\pi$
$151$	882.893	0.475820	0.237910	+	0.971287i	$0.423538\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−701.733	−0.363642
$156$	0	0
$157$	1196.34	0.608141	0.304071	−	0.952650i	$-0.401654\pi$
$157$	1196.34	0.608141	0.304071	+	0.952650i	$0.401654\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	192.343	0.0941539
$162$	0	0
$163$	−2328.91	−1.11910	−0.559552	−	0.828795i	$-0.689027\pi$
$163$	−2328.91	−1.11910	−0.559552	+	0.828795i	$0.689027\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1549.69	0.718077	0.359038	−	0.933323i	$-0.383105\pi$
$167$	1549.69	0.718077	0.359038	+	0.933323i	$0.383105\pi$
$168$	0	0
$169$	−2195.53	−0.999333
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2406.13	1.05743	0.528714	−	0.848800i	$-0.322674\pi$
$173$	2406.13	1.05743	0.528714	+	0.848800i	$0.322674\pi$
$174$	0	0
$175$	−670.785	−0.289752
$176$	0	0
$177$	0	0
$178$	0	0
$179$	115.757	0.0483358	0.0241679	−	0.999708i	$-0.492306\pi$
$179$	115.757	0.0483358	0.0241679	+	0.999708i	$0.492306\pi$
$180$	0	0
$181$	−2214.21	−0.909285	−0.454643	−	0.890674i	$-0.650233\pi$
$181$	−2214.21	−0.909285	−0.454643	+	0.890674i	$0.650233\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	935.152	0.371642
$186$	0	0
$187$	4071.08	1.59202
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2710.81	1.02695	0.513474	−	0.858105i	$-0.328358\pi$
$191$	2710.81	1.02695	0.513474	+	0.858105i	$0.328358\pi$
$192$	0	0
$193$	4336.75	1.61744	0.808721	−	0.588193i	$-0.200160\pi$
$193$	4336.75	1.61744	0.808721	+	0.588193i	$0.200160\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4840.88	−1.75075	−0.875376	−	0.483443i	$-0.839386\pi$
$197$	−4840.88	−1.75075	−0.875376	+	0.483443i	$0.839386\pi$
$198$	0	0
$199$	−4580.41	−1.63164	−0.815820	−	0.578306i	$-0.803714\pi$
$199$	−4580.41	−1.63164	−0.815820	+	0.578306i	$0.803714\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1176.06	−0.406616
$204$	0	0
$205$	1199.52	0.408675
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5836.59	−1.93170
$210$	0	0
$211$	1970.43	0.642891	0.321445	−	0.946928i	$-0.395831\pi$
$211$	1970.43	0.642891	0.321445	+	0.946928i	$0.395831\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2287.78	−0.725699
$216$	0	0
$217$	−3765.70	−1.17803
$218$	0	0
$219$	0	0
$220$	0	0
$221$	105.652	0.0321579
$222$	0	0
$223$	5128.20	1.53995	0.769977	−	0.638072i	$-0.220268\pi$
$223$	5128.20	1.53995	0.769977	+	0.638072i	$0.220268\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3163.63	0.925010	0.462505	−	0.886617i	$-0.346951\pi$
$227$	3163.63	0.925010	0.462505	+	0.886617i	$0.346951\pi$
$228$	0	0
$229$	−3955.14	−1.14132	−0.570662	−	0.821185i	$-0.693313\pi$
$229$	−3955.14	−1.14132	−0.570662	+	0.821185i	$0.693313\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−111.366	−0.0313126	−0.0156563	−	0.999877i	$-0.504984\pi$
$233$	−111.366	−0.0313126	−0.0156563	+	0.999877i	$0.504984\pi$
$234$	0	0
$235$	128.053	0.0355458
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2618.34	−0.708645	−0.354322	−	0.935123i	$-0.615288\pi$
$239$	−2618.34	−0.708645	−0.354322	+	0.935123i	$0.615288\pi$
$240$	0	0
$241$	5486.56	1.46647	0.733237	−	0.679973i	$-0.238008\pi$
$241$	5486.56	1.46647	0.733237	+	0.679973i	$0.238008\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1884.62	−0.491446
$246$	0	0
$247$	−151.470	−0.0390194
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−517.287	−0.130083	−0.0650416	−	0.997883i	$-0.520718\pi$
$251$	−517.287	−0.130083	−0.0650416	+	0.997883i	$0.520718\pi$
$252$	0	0
$253$	−334.356	−0.0830862
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4374.00	−1.06164	−0.530822	−	0.847483i	$-0.678117\pi$
$257$	−4374.00	−1.06164	−0.530822	+	0.847483i	$0.678117\pi$
$258$	0	0
$259$	5018.29	1.20394
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3005.82	−0.704742	−0.352371	−	0.935860i	$-0.614624\pi$
$263$	−3005.82	−0.704742	−0.352371	+	0.935860i	$0.614624\pi$
$264$	0	0
$265$	414.157	0.0960055
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3265.82	−0.740224	−0.370112	−	0.928987i	$-0.620681\pi$
$269$	−3265.82	−0.740224	−0.370112	+	0.928987i	$0.620681\pi$
$270$	0	0
$271$	−6523.97	−1.46237	−0.731186	−	0.682178i	$-0.761033\pi$
$271$	−6523.97	−1.46237	−0.731186	+	0.682178i	$0.761033\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1166.05	0.255692
$276$	0	0
$277$	4017.17	0.871365	0.435683	−	0.900100i	$-0.356507\pi$
$277$	4017.17	0.871365	0.435683	+	0.900100i	$0.356507\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4408.32	−0.935866	−0.467933	−	0.883764i	$-0.655001\pi$
$281$	−4408.32	−0.935866	−0.467933	+	0.883764i	$0.655001\pi$
$282$	0	0
$283$	5793.69	1.21696	0.608479	−	0.793570i	$-0.291780\pi$
$283$	5793.69	1.21696	0.608479	+	0.793570i	$0.291780\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6436.98	1.32391
$288$	0	0
$289$	2705.46	0.550674
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−7035.28	−1.40275	−0.701375	−	0.712793i	$-0.747430\pi$
$293$	−7035.28	−1.40275	−0.701375	+	0.712793i	$0.747430\pi$
$294$	0	0
$295$	−3697.87	−0.729825
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8.67713	−0.00167830
$300$	0	0
$301$	−12276.9	−2.35092
$302$	0	0
$303$	0	0
$304$	0	0
$305$	143.153	0.0268752
$306$	0	0
$307$	2744.60	0.510236	0.255118	−	0.966910i	$-0.417886\pi$
$307$	2744.60	0.510236	0.255118	+	0.966910i	$0.417886\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7637.57	−1.39256	−0.696281	−	0.717769i	$-0.745163\pi$
$311$	−7637.57	−1.39256	−0.696281	+	0.717769i	$0.745163\pi$
$312$	0	0
$313$	−3125.86	−0.564486	−0.282243	−	0.959343i	$-0.591078\pi$
$313$	−3125.86	−0.564486	−0.282243	+	0.959343i	$0.591078\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4800.58	−0.850560	−0.425280	−	0.905062i	$-0.639824\pi$
$317$	−4800.58	−0.850560	−0.425280	+	0.905062i	$0.639824\pi$
$318$	0	0
$319$	2044.38	0.358819
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−10922.4	−1.88154
$324$	0	0
$325$	30.2610	0.00516485
$326$	0	0
$327$	0	0
$328$	0	0
$329$	687.170	0.115152
$330$	0	0
$331$	2068.76	0.343532	0.171766	−	0.985138i	$-0.445053\pi$
$331$	2068.76	0.343532	0.171766	+	0.985138i	$0.445053\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	631.046	0.102919
$336$	0	0
$337$	−9923.85	−1.60411	−0.802057	−	0.597248i	$-0.796261\pi$
$337$	−9923.85	−1.60411	−0.802057	+	0.597248i	$0.796261\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6546.03	1.03955
$342$	0	0
$343$	−910.250	−0.143291
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5631.64	−0.871245	−0.435623	−	0.900129i	$-0.643472\pi$
$347$	−5631.64	−0.871245	−0.435623	+	0.900129i	$0.643472\pi$
$348$	0	0
$349$	2866.57	0.439668	0.219834	−	0.975537i	$-0.429448\pi$
$349$	2866.57	0.439668	0.219834	+	0.975537i	$0.429448\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2476.54	−0.373408	−0.186704	−	0.982416i	$-0.559781\pi$
$353$	−2476.54	−0.373408	−0.186704	+	0.982416i	$0.559781\pi$
$354$	0	0
$355$	3056.68	0.456990
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5526.01	−0.812400	−0.406200	−	0.913784i	$-0.633146\pi$
$359$	−5526.01	−0.812400	−0.406200	+	0.913784i	$0.633146\pi$
$360$	0	0
$361$	8800.05	1.28299
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4917.66	−0.705211
$366$	0	0
$367$	−2946.58	−0.419101	−0.209551	−	0.977798i	$-0.567200\pi$
$367$	−2946.58	−0.419101	−0.209551	+	0.977798i	$0.567200\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2222.48	0.311012
$372$	0	0
$373$	−4898.58	−0.679997	−0.339998	−	0.940426i	$-0.610427\pi$
$373$	−4898.58	−0.679997	−0.339998	+	0.940426i	$0.610427\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	53.0552	0.00724797
$378$	0	0
$379$	−4680.57	−0.634366	−0.317183	−	0.948364i	$-0.602737\pi$
$379$	−4680.57	−0.634366	−0.317183	+	0.948364i	$0.602737\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	9032.91	1.20512	0.602559	−	0.798075i	$-0.294148\pi$
$383$	9032.91	1.20512	0.602559	+	0.798075i	$0.294148\pi$
$384$	0	0
$385$	6257.34	0.828321
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8600.27	1.12095	0.560477	−	0.828170i	$-0.310618\pi$
$389$	8600.27	1.12095	0.560477	+	0.828170i	$0.310618\pi$
$390$	0	0
$391$	−625.701	−0.0809286
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1863.69	−0.237398
$396$	0	0
$397$	−12355.8	−1.56201	−0.781006	−	0.624523i	$-0.785293\pi$
$397$	−12355.8	−1.56201	−0.781006	+	0.624523i	$0.785293\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4492.21	0.559428	0.279714	−	0.960083i	$-0.409760\pi$
$401$	4492.21	0.559428	0.279714	+	0.960083i	$0.409760\pi$
$402$	0	0
$403$	169.881	0.0209984
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8723.45	−1.06242
$408$	0	0
$409$	6090.53	0.736326	0.368163	−	0.929761i	$-0.379987\pi$
$409$	6090.53	0.736326	0.368163	+	0.929761i	$0.379987\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−19843.8	−2.36429
$414$	0	0
$415$	−3466.40	−0.410021
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2182.76	0.254498	0.127249	−	0.991871i	$-0.459385\pi$
$419$	2182.76	0.254498	0.127249	+	0.991871i	$0.459385\pi$
$420$	0	0
$421$	3416.77	0.395541	0.197771	−	0.980248i	$-0.436630\pi$
$421$	3416.77	0.395541	0.197771	+	0.980248i	$0.436630\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2182.10	0.249052
$426$	0	0
$427$	768.200	0.0870628
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1697.09	−0.189665	−0.0948327	−	0.995493i	$-0.530232\pi$
$431$	−1697.09	−0.189665	−0.0948327	+	0.995493i	$0.530232\pi$
$432$	0	0
$433$	8934.22	0.991574	0.495787	−	0.868444i	$-0.334880\pi$
$433$	8934.22	0.991574	0.495787	+	0.868444i	$0.334880\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	897.049	0.0981961
$438$	0	0
$439$	11480.7	1.24817	0.624084	−	0.781357i	$-0.285472\pi$
$439$	11480.7	1.24817	0.624084	+	0.781357i	$0.285472\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−11601.5	−1.24425	−0.622125	−	0.782918i	$-0.713730\pi$
$443$	−11601.5	−1.24425	−0.622125	+	0.782918i	$0.713730\pi$
$444$	0	0
$445$	4369.24	0.465442
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11918.6	1.25273	0.626364	−	0.779531i	$-0.284543\pi$
$449$	11918.6	1.25273	0.626364	+	0.779531i	$0.284543\pi$
$450$	0	0
$451$	−11189.6	−1.16829
$452$	0	0
$453$	0	0
$454$	0	0
$455$	162.389	0.0167317
$456$	0	0
$457$	−3070.12	−0.314254	−0.157127	−	0.987578i	$-0.550223\pi$
$457$	−3070.12	−0.314254	−0.157127	+	0.987578i	$0.550223\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5957.26	−0.601860	−0.300930	−	0.953646i	$-0.597297\pi$
$461$	−5957.26	−0.601860	−0.300930	+	0.953646i	$0.597297\pi$
$462$	0	0
$463$	7991.99	0.802201	0.401101	−	0.916034i	$-0.368628\pi$
$463$	7991.99	0.802201	0.401101	+	0.916034i	$0.368628\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13084.9	1.29657	0.648283	−	0.761400i	$-0.275487\pi$
$467$	13084.9	1.29657	0.648283	+	0.761400i	$0.275487\pi$
$468$	0	0
$469$	3386.37	0.333408
$470$	0	0
$471$	0	0
$472$	0	0
$473$	21341.3	2.07457
$474$	0	0
$475$	−3128.40	−0.302192
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6749.17	0.643795	0.321897	−	0.946775i	$-0.395679\pi$
$479$	6749.17	0.643795	0.321897	+	0.946775i	$0.395679\pi$
$480$	0	0
$481$	−226.389	−0.0214604
$482$	0	0
$483$	0	0
$484$	0	0
$485$	413.769	0.0387387
$486$	0	0
$487$	1764.54	0.164187	0.0820936	−	0.996625i	$-0.473839\pi$
$487$	1764.54	0.164187	0.0820936	+	0.996625i	$0.473839\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5428.14	0.498918	0.249459	−	0.968385i	$-0.419747\pi$
$491$	5428.14	0.498918	0.249459	+	0.968385i	$0.419747\pi$
$492$	0	0
$493$	3825.77	0.349501
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16403.0	1.48043
$498$	0	0
$499$	17686.0	1.58664	0.793321	−	0.608803i	$-0.208350\pi$
$499$	17686.0	1.58664	0.793321	+	0.608803i	$0.208350\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4258.96	0.377530	0.188765	−	0.982022i	$-0.439552\pi$
$503$	4258.96	0.377530	0.188765	+	0.982022i	$0.439552\pi$
$504$	0	0
$505$	6631.55	0.584357
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−17019.5	−1.48208	−0.741038	−	0.671463i	$-0.765666\pi$
$509$	−17019.5	−1.48208	−0.741038	+	0.671463i	$0.765666\pi$
$510$	0	0
$511$	−26389.5	−2.28455
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8343.07	0.713863
$516$	0	0
$517$	−1194.53	−0.101616
$518$	0	0
$519$	0	0
$520$	0	0
$521$	17688.5	1.48742	0.743712	−	0.668500i	$-0.233063\pi$
$521$	17688.5	1.48742	0.743712	+	0.668500i	$0.233063\pi$
$522$	0	0
$523$	−8952.32	−0.748485	−0.374242	−	0.927331i	$-0.622097\pi$
$523$	−8952.32	−0.748485	−0.374242	+	0.927331i	$0.622097\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12250.0	1.01256
$528$	0	0
$529$	−12115.6	−0.995776
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−290.390	−0.0235988
$534$	0	0
$535$	961.265	0.0776806
$536$	0	0
$537$	0	0
$538$	0	0
$539$	17580.5	1.40491
$540$	0	0
$541$	−10010.9	−0.795570	−0.397785	−	0.917479i	$-0.630221\pi$
$541$	−10010.9	−0.795570	−0.397785	+	0.917479i	$0.630221\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8227.97	0.646693
$546$	0	0
$547$	−12031.4	−0.940446	−0.470223	−	0.882548i	$-0.655827\pi$
$547$	−12031.4	−0.940446	−0.470223	+	0.882548i	$0.655827\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5484.89	−0.424073
$552$	0	0
$553$	−10001.1	−0.769057
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19557.6	−1.48776	−0.743878	−	0.668315i	$-0.767016\pi$
$557$	−19557.6	−1.48776	−0.743878	+	0.668315i	$0.767016\pi$
$558$	0	0
$559$	553.843	0.0419053
$560$	0	0
$561$	0	0
$562$	0	0
$563$	22299.4	1.66928	0.834642	−	0.550794i	$-0.185675\pi$
$563$	22299.4	1.66928	0.834642	+	0.550794i	$0.185675\pi$
$564$	0	0
$565$	7784.79	0.579662
$566$	0	0
$567$	0	0
$568$	0	0
$569$	19389.3	1.42854	0.714271	−	0.699869i	$-0.246758\pi$
$569$	19389.3	1.42854	0.714271	+	0.699869i	$0.246758\pi$
$570$	0	0
$571$	−17788.9	−1.30375	−0.651877	−	0.758325i	$-0.726018\pi$
$571$	−17788.9	−1.30375	−0.651877	+	0.758325i	$0.726018\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−179.215	−0.0129979
$576$	0	0
$577$	13034.2	0.940420	0.470210	−	0.882555i	$-0.344178\pi$
$577$	13034.2	0.940420	0.470210	+	0.882555i	$0.344178\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−18601.7	−1.32827
$582$	0	0
$583$	−3863.41	−0.274453
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2897.19	−0.203713	−0.101857	−	0.994799i	$-0.532478\pi$
$587$	−2897.19	−0.203713	−0.101857	+	0.994799i	$0.532478\pi$
$588$	0	0
$589$	−17562.4	−1.22860
$590$	0	0
$591$	0	0
$592$	0	0
$593$	9064.89	0.627741	0.313871	−	0.949466i	$-0.398374\pi$
$593$	9064.89	0.627741	0.313871	+	0.949466i	$0.398374\pi$
$594$	0	0
$595$	11709.7	0.806811
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−26128.5	−1.78228	−0.891138	−	0.453732i	$-0.850092\pi$
$599$	−26128.5	−1.78228	−0.891138	+	0.453732i	$0.850092\pi$
$600$	0	0
$601$	−21542.4	−1.46212	−0.731060	−	0.682313i	$-0.760974\pi$
$601$	−21542.4	−1.46212	−0.731060	+	0.682313i	$0.760974\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4222.33	−0.283739
$606$	0	0
$607$	19188.2	1.28308	0.641538	−	0.767091i	$-0.278297\pi$
$607$	19188.2	1.28308	0.641538	+	0.767091i	$0.278297\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−31.0001	−0.00205259
$612$	0	0
$613$	19212.5	1.26588	0.632942	−	0.774199i	$-0.281847\pi$
$613$	19212.5	1.26588	0.632942	+	0.774199i	$0.281847\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−22425.9	−1.46326	−0.731630	−	0.681702i	$-0.761240\pi$
$617$	−22425.9	−1.46326	−0.731630	+	0.681702i	$0.761240\pi$
$618$	0	0
$619$	−726.094	−0.0471473	−0.0235736	−	0.999722i	$-0.507504\pi$
$619$	−726.094	−0.0471473	−0.0235736	+	0.999722i	$0.507504\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	23446.6	1.50781
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−16324.7	−1.03483
$630$	0	0
$631$	−11350.8	−0.716116	−0.358058	−	0.933699i	$-0.616561\pi$
$631$	−11350.8	−0.716116	−0.358058	+	0.933699i	$0.616561\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2045.02	0.127802
$636$	0	0
$637$	456.244	0.0283784
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−13397.8	−0.825557	−0.412779	−	0.910831i	$-0.635442\pi$
$641$	−13397.8	−0.825557	−0.412779	+	0.910831i	$0.635442\pi$
$642$	0	0
$643$	−3270.54	−0.200587	−0.100294	−	0.994958i	$-0.531978\pi$
$643$	−3270.54	−0.200587	−0.100294	+	0.994958i	$0.531978\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24464.4	1.48655	0.743273	−	0.668989i	$-0.233273\pi$
$647$	24464.4	1.48655	0.743273	+	0.668989i	$0.233273\pi$
$648$	0	0
$649$	34495.1	2.08637
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27993.0	1.67757	0.838783	−	0.544466i	$-0.183268\pi$
$653$	27993.0	1.67757	0.838783	+	0.544466i	$0.183268\pi$
$654$	0	0
$655$	14531.6	0.866866
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−13331.8	−0.788062	−0.394031	−	0.919097i	$-0.628920\pi$
$659$	−13331.8	−0.788062	−0.394031	+	0.919097i	$0.628920\pi$
$660$	0	0
$661$	22157.7	1.30383	0.651917	−	0.758290i	$-0.273965\pi$
$661$	22157.7	1.30383	0.651917	+	0.758290i	$0.273965\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−16787.9	−0.978958
$666$	0	0
$667$	−314.209	−0.0182402
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1335.39	−0.0768287
$672$	0	0
$673$	−6289.35	−0.360233	−0.180116	−	0.983645i	$-0.557647\pi$
$673$	−6289.35	−0.360233	−0.180116	+	0.983645i	$0.557647\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9563.14	−0.542897	−0.271448	−	0.962453i	$-0.587503\pi$
$677$	−9563.14	−0.542897	−0.271448	+	0.962453i	$0.587503\pi$
$678$	0	0
$679$	2220.40	0.125495
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−10285.5	−0.576226	−0.288113	−	0.957596i	$-0.593028\pi$
$683$	−10285.5	−0.576226	−0.288113	+	0.957596i	$0.593028\pi$
$684$	0	0
$685$	9449.99	0.527103
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−100.262	−0.00554382
$690$	0	0
$691$	−13755.2	−0.757268	−0.378634	−	0.925546i	$-0.623606\pi$
$691$	−13755.2	−0.757268	−0.378634	+	0.925546i	$0.623606\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13547.7	0.739416
$696$	0	0
$697$	−20939.8	−1.13795
$698$	0	0
$699$	0	0
$700$	0	0
$701$	24564.4	1.32352	0.661758	−	0.749717i	$-0.269810\pi$
$701$	24564.4	1.32352	0.661758	+	0.749717i	$0.269810\pi$
$702$	0	0
$703$	23404.3	1.25563
$704$	0	0
$705$	0	0
$706$	0	0
$707$	35586.8	1.89304
$708$	0	0
$709$	−33743.2	−1.78738	−0.893689	−	0.448687i	$-0.851892\pi$
$709$	−33743.2	−1.78738	−0.893689	+	0.448687i	$0.851892\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1006.09	−0.0528447
$714$	0	0
$715$	−282.286	−0.0147649
$716$	0	0
$717$	0	0
$718$	0	0
$719$	20159.9	1.04567	0.522837	−	0.852433i	$-0.324874\pi$
$719$	20159.9	1.04567	0.522837	+	0.852433i	$0.324874\pi$
$720$	0	0
$721$	44771.3	2.31258
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1095.79	0.0561330
$726$	0	0
$727$	−16749.2	−0.854461	−0.427230	−	0.904143i	$-0.640511\pi$
$727$	−16749.2	−0.854461	−0.427230	+	0.904143i	$0.640511\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	39937.2	2.02070
$732$	0	0
$733$	8143.13	0.410332	0.205166	−	0.978727i	$-0.434227\pi$
$733$	8143.13	0.410332	0.205166	+	0.978727i	$0.434227\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5886.64	−0.294216
$738$	0	0
$739$	26837.1	1.33588	0.667942	−	0.744213i	$-0.267175\pi$
$739$	26837.1	1.33588	0.667942	+	0.744213i	$0.267175\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16798.2	0.829430	0.414715	−	0.909951i	$-0.363881\pi$
$743$	16798.2	0.829430	0.414715	+	0.909951i	$0.363881\pi$
$744$	0	0
$745$	−6821.94	−0.335485
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5158.42	0.251648
$750$	0	0
$751$	9903.73	0.481215	0.240607	−	0.970623i	$-0.422653\pi$
$751$	9903.73	0.481215	0.240607	+	0.970623i	$0.422653\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4414.46	−0.212793
$756$	0	0
$757$	20197.0	0.969713	0.484856	−	0.874594i	$-0.338872\pi$
$757$	20197.0	0.969713	0.484856	+	0.874594i	$0.338872\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−28183.2	−1.34250	−0.671248	−	0.741233i	$-0.734242\pi$
$761$	−28183.2	−1.34250	−0.671248	+	0.741233i	$0.734242\pi$
$762$	0	0
$763$	44153.6	2.09498
$764$	0	0
$765$	0	0
$766$	0	0
$767$	895.209	0.0421436
$768$	0	0
$769$	−11839.3	−0.555183	−0.277591	−	0.960699i	$-0.589536\pi$
$769$	−11839.3	−0.555183	−0.277591	+	0.960699i	$0.589536\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−38537.8	−1.79316	−0.896578	−	0.442886i	$-0.853955\pi$
$773$	−38537.8	−1.79316	−0.896578	+	0.442886i	$0.853955\pi$
$774$	0	0
$775$	3508.66	0.162626
$776$	0	0
$777$	0	0
$778$	0	0
$779$	30020.8	1.38075
$780$	0	0
$781$	−28513.9	−1.30641
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5981.69	−0.271969
$786$	0	0
$787$	17994.1	0.815021	0.407511	−	0.913200i	$-0.366397\pi$
$787$	17994.1	0.815021	0.407511	+	0.913200i	$0.366397\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	41775.4	1.87783
$792$	0	0
$793$	−34.6556	−0.00155190
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25248.2	−1.12213	−0.561065	−	0.827772i	$-0.689608\pi$
$797$	−25248.2	−1.12213	−0.561065	+	0.827772i	$0.689608\pi$
$798$	0	0
$799$	−2235.39	−0.0989769
$800$	0	0
$801$	0	0
$802$	0	0
$803$	45873.8	2.01600
$804$	0	0
$805$	−961.716	−0.0421069
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1766.26	−0.0767595	−0.0383798	−	0.999263i	$-0.512220\pi$
$809$	−1766.26	−0.0767595	−0.0383798	+	0.999263i	$0.512220\pi$
$810$	0	0
$811$	−41618.0	−1.80198	−0.900990	−	0.433840i	$-0.857158\pi$
$811$	−41618.0	−1.80198	−0.900990	+	0.433840i	$0.857158\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	11644.5	0.500478
$816$	0	0
$817$	−57256.8	−2.45185
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−45001.8	−1.91300	−0.956501	−	0.291729i	$-0.905769\pi$
$821$	−45001.8	−1.91300	−0.956501	+	0.291729i	$0.905769\pi$
$822$	0	0
$823$	40613.8	1.72018	0.860091	−	0.510141i	$-0.170407\pi$
$823$	40613.8	1.72018	0.860091	+	0.510141i	$0.170407\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25678.5	1.07972	0.539861	−	0.841754i	$-0.318477\pi$
$827$	25678.5	1.07972	0.539861	+	0.841754i	$0.318477\pi$
$828$	0	0
$829$	−28161.7	−1.17985	−0.589925	−	0.807458i	$-0.700843\pi$
$829$	−28161.7	−1.17985	−0.589925	+	0.807458i	$0.700843\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	32899.4	1.36842
$834$	0	0
$835$	−7748.46	−0.321134
$836$	0	0
$837$	0	0
$838$	0	0
$839$	4005.02	0.164802	0.0824008	−	0.996599i	$-0.473741\pi$
$839$	4005.02	0.164802	0.0824008	+	0.996599i	$0.473741\pi$
$840$	0	0
$841$	−22467.8	−0.921227
$842$	0	0
$843$	0	0
$844$	0	0
$845$	10977.7	0.446915
$846$	0	0
$847$	−22658.2	−0.919181
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1340.74	0.0540071
$852$	0	0
$853$	−7815.74	−0.313723	−0.156862	−	0.987621i	$-0.550138\pi$
$853$	−7815.74	−0.313723	−0.156862	+	0.987621i	$0.550138\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	43052.6	1.71604	0.858021	−	0.513615i	$-0.171694\pi$
$857$	43052.6	1.71604	0.858021	+	0.513615i	$0.171694\pi$
$858$	0	0
$859$	11579.7	0.459945	0.229973	−	0.973197i	$-0.426136\pi$
$859$	11579.7	0.459945	0.229973	+	0.973197i	$0.426136\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−9555.34	−0.376903	−0.188452	−	0.982082i	$-0.560347\pi$
$863$	−9555.34	−0.376903	−0.188452	+	0.982082i	$0.560347\pi$
$864$	0	0
$865$	−12030.7	−0.472896
$866$	0	0
$867$	0	0
$868$	0	0
$869$	17385.2	0.678656
$870$	0	0
$871$	−152.769	−0.00594301
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3353.93	0.129581
$876$	0	0
$877$	−20891.9	−0.804412	−0.402206	−	0.915549i	$-0.631756\pi$
$877$	−20891.9	−0.804412	−0.402206	+	0.915549i	$0.631756\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	11671.3	0.446327	0.223164	−	0.974781i	$-0.428362\pi$
$881$	11671.3	0.446327	0.223164	+	0.974781i	$0.428362\pi$
$882$	0	0
$883$	34959.3	1.33236	0.666181	−	0.745790i	$-0.267928\pi$
$883$	34959.3	1.33236	0.666181	+	0.745790i	$0.267928\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−35705.5	−1.35160	−0.675802	−	0.737083i	$-0.736203\pi$
$887$	−35705.5	−1.35160	−0.675802	+	0.737083i	$0.736203\pi$
$888$	0	0
$889$	10974.2	0.414018
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3204.82	0.120095
$894$	0	0
$895$	−578.787	−0.0216164
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6151.59	0.228217
$900$	0	0
$901$	−7229.84	−0.267326
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11071.0	0.406645
$906$	0	0
$907$	10921.6	0.399829	0.199914	−	0.979813i	$-0.435934\pi$
$907$	10921.6	0.399829	0.199914	+	0.979813i	$0.435934\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	45690.5	1.66168	0.830842	−	0.556508i	$-0.187859\pi$
$911$	45690.5	1.66168	0.830842	+	0.556508i	$0.187859\pi$
$912$	0	0
$913$	32335.9	1.17214
$914$	0	0
$915$	0	0
$916$	0	0
$917$	77980.8	2.80824
$918$	0	0
$919$	−46123.5	−1.65558	−0.827788	−	0.561041i	$-0.810401\pi$
$919$	−46123.5	−1.65558	−0.827788	+	0.561041i	$0.810401\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−739.984	−0.0263888
$924$	0	0
$925$	−4675.76	−0.166203
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−41652.4	−1.47101	−0.735506	−	0.677518i	$-0.763056\pi$
$929$	−41652.4	−1.47101	−0.735506	+	0.677518i	$0.763056\pi$
$930$	0	0
$931$	−47166.9	−1.66040
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−20355.4	−0.711971
$936$	0	0
$937$	−26210.8	−0.913841	−0.456920	−	0.889508i	$-0.651048\pi$
$937$	−26210.8	−0.913841	−0.456920	+	0.889508i	$0.651048\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−8569.04	−0.296857	−0.148429	−	0.988923i	$-0.547422\pi$
$941$	−8569.04	−0.296857	−0.148429	+	0.988923i	$0.547422\pi$
$942$	0	0
$943$	1719.78	0.0593888
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−34387.2	−1.17997	−0.589986	−	0.807413i	$-0.700867\pi$
$947$	−34387.2	−1.17997	−0.589986	+	0.807413i	$0.700867\pi$
$948$	0	0
$949$	1190.50	0.0407222
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−48075.9	−1.63414	−0.817068	−	0.576541i	$-0.804402\pi$
$953$	−48075.9	−1.63414	−0.817068	+	0.576541i	$0.804402\pi$
$954$	0	0
$955$	−13554.0	−0.459266
$956$	0	0
$957$	0	0
$958$	0	0
$959$	50711.3	1.70756
$960$	0	0
$961$	−10093.8	−0.338822
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−21683.8	−0.723342
$966$	0	0
$967$	−15669.5	−0.521094	−0.260547	−	0.965461i	$-0.583903\pi$
$967$	−15669.5	−0.521094	−0.260547	+	0.965461i	$0.583903\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	42128.0	1.39233	0.696165	−	0.717881i	$-0.254888\pi$
$971$	42128.0	1.39233	0.696165	+	0.717881i	$0.254888\pi$
$972$	0	0
$973$	72700.9	2.39536
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4856.99	−0.159047	−0.0795236	−	0.996833i	$-0.525340\pi$
$977$	−4856.99	−0.159047	−0.0795236	+	0.996833i	$0.525340\pi$
$978$	0	0
$979$	−40757.9	−1.33057
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4604.65	−0.149405	−0.0747026	−	0.997206i	$-0.523801\pi$
$983$	−4604.65	−0.149405	−0.0747026	+	0.997206i	$0.523801\pi$
$984$	0	0
$985$	24204.4	0.782960
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3280.03	−0.105459
$990$	0	0
$991$	4082.46	0.130861	0.0654307	−	0.997857i	$-0.479158\pi$
$991$	4082.46	0.130861	0.0654307	+	0.997857i	$0.479158\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	22902.0	0.729692
$996$	0	0
$997$	34069.6	1.08224	0.541121	−	0.840945i	$-0.318000\pi$
$997$	34069.6	1.08224	0.541121	+	0.840945i	$0.318000\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.a.c.1.1	✓	3
3.2	odd	2		1620.4.a.e.1.1	yes	3
9.2	odd	6		1620.4.i.t.1081.3		6
9.4	even	3		1620.4.i.v.541.3		6
9.5	odd	6		1620.4.i.t.541.3		6
9.7	even	3		1620.4.i.v.1081.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.4.a.c.1.1	✓	3	1.1	even	1	trivial
1620.4.a.e.1.1	yes	3	3.2	odd	2
1620.4.i.t.541.3		6	9.5	odd	6
1620.4.i.t.1081.3		6	9.2	odd	6
1620.4.i.v.541.3		6	9.4	even	3
1620.4.i.v.1081.3		6	9.7	even	3

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} -26.8314 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} -26.8314 q^{7} +46.6419 q^{11} +1.21044 q^{13} +87.2838 q^{17} -125.136 q^{19} -7.16859 q^{23} +25.0000 q^{25} +43.8314 q^{29} +140.347 q^{31} +134.157 q^{35} -187.030 q^{37} -239.905 q^{41} +457.556 q^{43} -25.6106 q^{47} +376.925 q^{49} -82.8314 q^{53} -233.210 q^{55} +739.574 q^{59} -28.6306 q^{61} -6.05219 q^{65} -126.209 q^{67} -611.336 q^{71} +983.531 q^{73} -1251.47 q^{77} +372.737 q^{79} +693.280 q^{83} -436.419 q^{85} -873.848 q^{89} -32.4778 q^{91} +625.681 q^{95} -82.7538 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 15 q^{5} - 3 q^{7}+O(q^{10}) \) 3 * q - 15 * q^5 - 3 * q^7	\( 3 q - 15 q^{5} - 3 q^{7} + 24 q^{11} + 3 q^{13} + 30 q^{17} - 27 q^{19} - 99 q^{23} + 75 q^{25} + 54 q^{29} + 72 q^{31} + 15 q^{35} - 18 q^{37} - 411 q^{41} + 444 q^{43} + 75 q^{47} + 204 q^{49} - 171 q^{53} - 120 q^{55} - 297 q^{59} + 684 q^{61} - 15 q^{65} + 12 q^{67} - 642 q^{71} - 66 q^{73} - 1044 q^{77} + 1122 q^{79} - 90 q^{83} - 150 q^{85} - 756 q^{89} - 1653 q^{91} + 135 q^{95} + 1536 q^{97}+O(q^{100}) \) 3 * q - 15 * q^5 - 3 * q^7 + 24 * q^11 + 3 * q^13 + 30 * q^17 - 27 * q^19 - 99 * q^23 + 75 * q^25 + 54 * q^29 + 72 * q^31 + 15 * q^35 - 18 * q^37 - 411 * q^41 + 444 * q^43 + 75 * q^47 + 204 * q^49 - 171 * q^53 - 120 * q^55 - 297 * q^59 + 684 * q^61 - 15 * q^65 + 12 * q^67 - 642 * q^71 - 66 * q^73 - 1044 * q^77 + 1122 * q^79 - 90 * q^83 - 150 * q^85 - 756 * q^89 - 1653 * q^91 + 135 * q^95 + 1536 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more