LMFDB - Embedded newform 720.4.a.y.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [720,4,Mod(1,720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("720.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	720.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,0,0,5,0,4,0,0,0,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$42.4813752041$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	720.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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
$7$	4.00000	0.215980	0.107990	−	0.994152i	$-0.465559\pi$
$7$	4.00000	0.215980	0.107990	+	0.994152i	$0.465559\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−48.0000	−1.31569	−0.657843	−	0.753155i	$-0.728531\pi$
$11$	−48.0000	−1.31569	−0.657843	+	0.753155i	$0.728531\pi$
$12$	0	0
$13$	2.00000	0.0426692	0.0213346	−	0.999772i	$-0.493208\pi$
$13$	2.00000	0.0426692	0.0213346	+	0.999772i	$0.493208\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	114.000	1.62642	0.813208	−	0.581974i	$-0.197719\pi$
$17$	114.000	1.62642	0.813208	+	0.581974i	$0.197719\pi$
$18$	0	0
$19$	−140.000	−1.69043	−0.845216	−	0.534425i	$-0.820528\pi$
$19$	−140.000	−1.69043	−0.845216	+	0.534425i	$0.820528\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	72.0000	0.652741	0.326370	−	0.945242i	$-0.394174\pi$
$23$	72.0000	0.652741	0.326370	+	0.945242i	$0.394174\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−210.000	−1.34469	−0.672345	−	0.740238i	$-0.734713\pi$
$29$	−210.000	−1.34469	−0.672345	+	0.740238i	$0.734713\pi$
$30$	0	0
$31$	−272.000	−1.57589	−0.787946	−	0.615745i	$-0.788855\pi$
$31$	−272.000	−1.57589	−0.787946	+	0.615745i	$0.788855\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	20.0000	0.0965891
$36$	0	0
$37$	−334.000	−1.48403	−0.742017	−	0.670381i	$-0.766131\pi$
$37$	−334.000	−1.48403	−0.742017	+	0.670381i	$0.766131\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	198.000	0.754205	0.377102	−	0.926172i	$-0.376920\pi$
$41$	198.000	0.754205	0.377102	+	0.926172i	$0.376920\pi$
$42$	0	0
$43$	268.000	0.950456	0.475228	−	0.879863i	$-0.342366\pi$
$43$	268.000	0.950456	0.475228	+	0.879863i	$0.342366\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	216.000	0.670358	0.335179	−	0.942154i	$-0.391203\pi$
$47$	216.000	0.670358	0.335179	+	0.942154i	$0.391203\pi$
$48$	0	0
$49$	−327.000	−0.953353
$50$	0	0
$51$	0	0
$52$	0	0
$53$	78.0000	0.202153	0.101077	−	0.994879i	$-0.467771\pi$
$53$	78.0000	0.202153	0.101077	+	0.994879i	$0.467771\pi$
$54$	0	0
$55$	−240.000	−0.588393
$56$	0	0
$57$	0	0
$58$	0	0
$59$	240.000	0.529582	0.264791	−	0.964306i	$-0.414697\pi$
$59$	240.000	0.529582	0.264791	+	0.964306i	$0.414697\pi$
$60$	0	0
$61$	302.000	0.633888	0.316944	−	0.948444i	$-0.397343\pi$
$61$	302.000	0.633888	0.316944	+	0.948444i	$0.397343\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	10.0000	0.0190823
$66$	0	0
$67$	−596.000	−1.08676	−0.543381	−	0.839487i	$-0.682856\pi$
$67$	−596.000	−1.08676	−0.543381	+	0.839487i	$0.682856\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−768.000	−1.28373	−0.641865	−	0.766818i	$-0.721839\pi$
$71$	−768.000	−1.28373	−0.641865	+	0.766818i	$0.721839\pi$
$72$	0	0
$73$	−478.000	−0.766379	−0.383190	−	0.923670i	$-0.625174\pi$
$73$	−478.000	−0.766379	−0.383190	+	0.923670i	$0.625174\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−192.000	−0.284161
$78$	0	0
$79$	640.000	0.911464	0.455732	−	0.890117i	$-0.349378\pi$
$79$	640.000	0.911464	0.455732	+	0.890117i	$0.349378\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−348.000	−0.460216	−0.230108	−	0.973165i	$-0.573908\pi$
$83$	−348.000	−0.460216	−0.230108	+	0.973165i	$0.573908\pi$
$84$	0	0
$85$	570.000	0.727355
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−210.000	−0.250112	−0.125056	−	0.992150i	$-0.539911\pi$
$89$	−210.000	−0.250112	−0.125056	+	0.992150i	$0.539911\pi$
$90$	0	0
$91$	8.00000	0.00921569
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−700.000	−0.755984
$96$	0	0
$97$	−1534.00	−1.60571	−0.802856	−	0.596173i	$-0.796687\pi$
$97$	−1534.00	−1.60571	−0.802856	+	0.596173i	$0.796687\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1722.00	−1.69649	−0.848245	−	0.529605i	$-0.822340\pi$
$101$	−1722.00	−1.69649	−0.848245	+	0.529605i	$0.822340\pi$
$102$	0	0
$103$	−1052.00	−1.00638	−0.503188	−	0.864177i	$-0.667840\pi$
$103$	−1052.00	−1.00638	−0.503188	+	0.864177i	$0.667840\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−564.000	−0.509570	−0.254785	−	0.966998i	$-0.582005\pi$
$107$	−564.000	−0.509570	−0.254785	+	0.966998i	$0.582005\pi$
$108$	0	0
$109$	−610.000	−0.536031	−0.268016	−	0.963415i	$-0.586368\pi$
$109$	−610.000	−0.536031	−0.268016	+	0.963415i	$0.586368\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1302.00	−1.08391	−0.541955	−	0.840407i	$-0.682316\pi$
$113$	−1302.00	−1.08391	−0.541955	+	0.840407i	$0.682316\pi$
$114$	0	0
$115$	360.000	0.291915
$116$	0	0
$117$	0	0
$118$	0	0
$119$	456.000	0.351273
$120$	0	0
$121$	973.000	0.731029
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	124.000	0.0866395	0.0433198	−	0.999061i	$-0.486207\pi$
$127$	124.000	0.0866395	0.0433198	+	0.999061i	$0.486207\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	192.000	0.128054	0.0640272	−	0.997948i	$-0.479606\pi$
$131$	192.000	0.128054	0.0640272	+	0.997948i	$0.479606\pi$
$132$	0	0
$133$	−560.000	−0.365099
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2514.00	1.56778	0.783889	−	0.620901i	$-0.213233\pi$
$137$	2514.00	1.56778	0.783889	+	0.620901i	$0.213233\pi$
$138$	0	0
$139$	−1340.00	−0.817679	−0.408839	−	0.912606i	$-0.634066\pi$
$139$	−1340.00	−0.817679	−0.408839	+	0.912606i	$0.634066\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−96.0000	−0.0561393
$144$	0	0
$145$	−1050.00	−0.601364
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1410.00	−0.775246	−0.387623	−	0.921818i	$-0.626704\pi$
$149$	−1410.00	−0.775246	−0.387623	+	0.921818i	$0.626704\pi$
$150$	0	0
$151$	2128.00	1.14685	0.573424	−	0.819258i	$-0.305615\pi$
$151$	2128.00	1.14685	0.573424	+	0.819258i	$0.305615\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1360.00	−0.704760
$156$	0	0
$157$	3026.00	1.53822	0.769112	−	0.639114i	$-0.220699\pi$
$157$	3026.00	1.53822	0.769112	+	0.639114i	$0.220699\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	288.000	0.140979
$162$	0	0
$163$	−2612.00	−1.25514	−0.627569	−	0.778561i	$-0.715950\pi$
$163$	−2612.00	−1.25514	−0.627569	+	0.778561i	$0.715950\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−24.0000	−0.0111208	−0.00556041	−	0.999985i	$-0.501770\pi$
$167$	−24.0000	−0.0111208	−0.00556041	+	0.999985i	$0.501770\pi$
$168$	0	0
$169$	−2193.00	−0.998179
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1962.00	−0.862243	−0.431122	−	0.902294i	$-0.641882\pi$
$173$	−1962.00	−0.862243	−0.431122	+	0.902294i	$0.641882\pi$
$174$	0	0
$175$	100.000	0.0431959
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−120.000	−0.0501074	−0.0250537	−	0.999686i	$-0.507976\pi$
$179$	−120.000	−0.0501074	−0.0250537	+	0.999686i	$0.507976\pi$
$180$	0	0
$181$	902.000	0.370415	0.185208	−	0.982699i	$-0.440704\pi$
$181$	902.000	0.370415	0.185208	+	0.982699i	$0.440704\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1670.00	−0.663680
$186$	0	0
$187$	−5472.00	−2.13985
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−168.000	−0.0636443	−0.0318221	−	0.999494i	$-0.510131\pi$
$191$	−168.000	−0.0636443	−0.0318221	+	0.999494i	$0.510131\pi$
$192$	0	0
$193$	−1318.00	−0.491563	−0.245782	−	0.969325i	$-0.579045\pi$
$193$	−1318.00	−0.491563	−0.245782	+	0.969325i	$0.579045\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4014.00	1.45170	0.725852	−	0.687851i	$-0.241446\pi$
$197$	4014.00	1.45170	0.725852	+	0.687851i	$0.241446\pi$
$198$	0	0
$199$	−2000.00	−0.712443	−0.356222	−	0.934401i	$-0.615935\pi$
$199$	−2000.00	−0.712443	−0.356222	+	0.934401i	$0.615935\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−840.000	−0.290426
$204$	0	0
$205$	990.000	0.337291
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6720.00	2.22408
$210$	0	0
$211$	3868.00	1.26201	0.631005	−	0.775779i	$-0.282643\pi$
$211$	3868.00	1.26201	0.631005	+	0.775779i	$0.282643\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1340.00	0.425057
$216$	0	0
$217$	−1088.00	−0.340361
$218$	0	0
$219$	0	0
$220$	0	0
$221$	228.000	0.0693979
$222$	0	0
$223$	3148.00	0.945317	0.472658	−	0.881246i	$-0.343294\pi$
$223$	3148.00	0.945317	0.472658	+	0.881246i	$0.343294\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2556.00	0.747347	0.373673	−	0.927560i	$-0.378098\pi$
$227$	2556.00	0.747347	0.373673	+	0.927560i	$0.378098\pi$
$228$	0	0
$229$	−610.000	−0.176026	−0.0880130	−	0.996119i	$-0.528052\pi$
$229$	−610.000	−0.176026	−0.0880130	+	0.996119i	$0.528052\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2058.00	0.578644	0.289322	−	0.957232i	$-0.406570\pi$
$233$	2058.00	0.578644	0.289322	+	0.957232i	$0.406570\pi$
$234$	0	0
$235$	1080.00	0.299793
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4920.00	1.33158	0.665792	−	0.746138i	$-0.268094\pi$
$239$	4920.00	1.33158	0.665792	+	0.746138i	$0.268094\pi$
$240$	0	0
$241$	−1438.00	−0.384356	−0.192178	−	0.981360i	$-0.561555\pi$
$241$	−1438.00	−0.384356	−0.192178	+	0.981360i	$0.561555\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1635.00	−0.426352
$246$	0	0
$247$	−280.000	−0.0721294
$248$	0	0
$249$	0	0
$250$	0	0
$251$	792.000	0.199166	0.0995829	−	0.995029i	$-0.468249\pi$
$251$	792.000	0.199166	0.0995829	+	0.995029i	$0.468249\pi$
$252$	0	0
$253$	−3456.00	−0.858802
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2166.00	−0.525725	−0.262863	−	0.964833i	$-0.584667\pi$
$257$	−2166.00	−0.525725	−0.262863	+	0.964833i	$0.584667\pi$
$258$	0	0
$259$	−1336.00	−0.320521
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3192.00	0.748392	0.374196	−	0.927350i	$-0.377919\pi$
$263$	3192.00	0.748392	0.374196	+	0.927350i	$0.377919\pi$
$264$	0	0
$265$	390.000	0.0904057
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5490.00	−1.24435	−0.622177	−	0.782877i	$-0.713752\pi$
$269$	−5490.00	−1.24435	−0.622177	+	0.782877i	$0.713752\pi$
$270$	0	0
$271$	6328.00	1.41845	0.709223	−	0.704985i	$-0.249046\pi$
$271$	6328.00	1.41845	0.709223	+	0.704985i	$0.249046\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1200.00	−0.263137
$276$	0	0
$277$	−574.000	−0.124507	−0.0622533	−	0.998060i	$-0.519829\pi$
$277$	−574.000	−0.124507	−0.0622533	+	0.998060i	$0.519829\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4242.00	−0.900557	−0.450278	−	0.892888i	$-0.648675\pi$
$281$	−4242.00	−0.900557	−0.450278	+	0.892888i	$0.648675\pi$
$282$	0	0
$283$	628.000	0.131911	0.0659553	−	0.997823i	$-0.478991\pi$
$283$	628.000	0.131911	0.0659553	+	0.997823i	$0.478991\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	792.000	0.162893
$288$	0	0
$289$	8083.00	1.64523
$290$	0	0
$291$	0	0
$292$	0	0
$293$	558.000	0.111258	0.0556292	−	0.998451i	$-0.482284\pi$
$293$	558.000	0.111258	0.0556292	+	0.998451i	$0.482284\pi$
$294$	0	0
$295$	1200.00	0.236836
$296$	0	0
$297$	0	0
$298$	0	0
$299$	144.000	0.0278520
$300$	0	0
$301$	1072.00	0.205279
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1510.00	0.283483
$306$	0	0
$307$	6964.00	1.29465	0.647323	−	0.762216i	$-0.275888\pi$
$307$	6964.00	1.29465	0.647323	+	0.762216i	$0.275888\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2832.00	0.516360	0.258180	−	0.966097i	$-0.416877\pi$
$311$	2832.00	0.516360	0.258180	+	0.966097i	$0.416877\pi$
$312$	0	0
$313$	8642.00	1.56062	0.780311	−	0.625392i	$-0.215061\pi$
$313$	8642.00	1.56062	0.780311	+	0.625392i	$0.215061\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2214.00	0.392273	0.196137	−	0.980577i	$-0.437160\pi$
$317$	2214.00	0.392273	0.196137	+	0.980577i	$0.437160\pi$
$318$	0	0
$319$	10080.0	1.76919
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−15960.0	−2.74934
$324$	0	0
$325$	50.0000	0.00853385
$326$	0	0
$327$	0	0
$328$	0	0
$329$	864.000	0.144784
$330$	0	0
$331$	−10772.0	−1.78877	−0.894385	−	0.447299i	$-0.852386\pi$
$331$	−10772.0	−1.78877	−0.894385	+	0.447299i	$0.852386\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2980.00	−0.486014
$336$	0	0
$337$	−1654.00	−0.267356	−0.133678	−	0.991025i	$-0.542679\pi$
$337$	−1654.00	−0.267356	−0.133678	+	0.991025i	$0.542679\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	13056.0	2.07338
$342$	0	0
$343$	−2680.00	−0.421885
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2196.00	0.339733	0.169867	−	0.985467i	$-0.445666\pi$
$347$	2196.00	0.339733	0.169867	+	0.985467i	$0.445666\pi$
$348$	0	0
$349$	8270.00	1.26843	0.634216	−	0.773156i	$-0.281323\pi$
$349$	8270.00	1.26843	0.634216	+	0.773156i	$0.281323\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10302.0	−1.55331	−0.776657	−	0.629923i	$-0.783086\pi$
$353$	−10302.0	−1.55331	−0.776657	+	0.629923i	$0.783086\pi$
$354$	0	0
$355$	−3840.00	−0.574102
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2280.00	−0.335192	−0.167596	−	0.985856i	$-0.553600\pi$
$359$	−2280.00	−0.335192	−0.167596	+	0.985856i	$0.553600\pi$
$360$	0	0
$361$	12741.0	1.85756
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2390.00	−0.342735
$366$	0	0
$367$	8764.00	1.24653	0.623266	−	0.782010i	$-0.285805\pi$
$367$	8764.00	1.24653	0.623266	+	0.782010i	$0.285805\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	312.000	0.0436610
$372$	0	0
$373$	−1318.00	−0.182958	−0.0914792	−	0.995807i	$-0.529159\pi$
$373$	−1318.00	−0.182958	−0.0914792	+	0.995807i	$0.529159\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−420.000	−0.0573769
$378$	0	0
$379$	−1100.00	−0.149085	−0.0745425	−	0.997218i	$-0.523750\pi$
$379$	−1100.00	−0.149085	−0.0745425	+	0.997218i	$0.523750\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3528.00	−0.470685	−0.235343	−	0.971912i	$-0.575621\pi$
$383$	−3528.00	−0.470685	−0.235343	+	0.971912i	$0.575621\pi$
$384$	0	0
$385$	−960.000	−0.127081
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9630.00	1.25517	0.627584	−	0.778549i	$-0.284044\pi$
$389$	9630.00	1.25517	0.627584	+	0.778549i	$0.284044\pi$
$390$	0	0
$391$	8208.00	1.06163
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3200.00	0.407619
$396$	0	0
$397$	−3094.00	−0.391142	−0.195571	−	0.980690i	$-0.562656\pi$
$397$	−3094.00	−0.391142	−0.195571	+	0.980690i	$0.562656\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1638.00	0.203985	0.101992	−	0.994785i	$-0.467478\pi$
$401$	1638.00	0.203985	0.101992	+	0.994785i	$0.467478\pi$
$402$	0	0
$403$	−544.000	−0.0672421
$404$	0	0
$405$	0	0
$406$	0	0
$407$	16032.0	1.95252
$408$	0	0
$409$	−13750.0	−1.66233	−0.831166	−	0.556024i	$-0.812326\pi$
$409$	−13750.0	−1.66233	−0.831166	+	0.556024i	$0.812326\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	960.000	0.114379
$414$	0	0
$415$	−1740.00	−0.205815
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12480.0	−1.45510	−0.727551	−	0.686053i	$-0.759342\pi$
$419$	−12480.0	−1.45510	−0.727551	+	0.686053i	$0.759342\pi$
$420$	0	0
$421$	7262.00	0.840685	0.420342	−	0.907366i	$-0.361910\pi$
$421$	7262.00	0.840685	0.420342	+	0.907366i	$0.361910\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2850.00	0.325283
$426$	0	0
$427$	1208.00	0.136907
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9792.00	1.09435	0.547174	−	0.837019i	$-0.315704\pi$
$431$	9792.00	1.09435	0.547174	+	0.837019i	$0.315704\pi$
$432$	0	0
$433$	1802.00	0.199997	0.0999984	−	0.994988i	$-0.468116\pi$
$433$	1802.00	0.199997	0.0999984	+	0.994988i	$0.468116\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−10080.0	−1.10341
$438$	0	0
$439$	2320.00	0.252227	0.126113	−	0.992016i	$-0.459750\pi$
$439$	2320.00	0.252227	0.126113	+	0.992016i	$0.459750\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11172.0	1.19819	0.599095	−	0.800678i	$-0.295527\pi$
$443$	11172.0	1.19819	0.599095	+	0.800678i	$0.295527\pi$
$444$	0	0
$445$	−1050.00	−0.111853
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6810.00	−0.715777	−0.357888	−	0.933764i	$-0.616503\pi$
$449$	−6810.00	−0.715777	−0.357888	+	0.933764i	$0.616503\pi$
$450$	0	0
$451$	−9504.00	−0.992297
$452$	0	0
$453$	0	0
$454$	0	0
$455$	40.0000	0.00412138
$456$	0	0
$457$	17066.0	1.74686	0.873429	−	0.486952i	$-0.161891\pi$
$457$	17066.0	1.74686	0.873429	+	0.486952i	$0.161891\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18918.0	1.91128	0.955639	−	0.294541i	$-0.0951667\pi$
$461$	18918.0	1.91128	0.955639	+	0.294541i	$0.0951667\pi$
$462$	0	0
$463$	−1052.00	−0.105595	−0.0527976	−	0.998605i	$-0.516814\pi$
$463$	−1052.00	−0.105595	−0.0527976	+	0.998605i	$0.516814\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11076.0	1.09751	0.548754	−	0.835984i	$-0.315102\pi$
$467$	11076.0	1.09751	0.548754	+	0.835984i	$0.315102\pi$
$468$	0	0
$469$	−2384.00	−0.234718
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−12864.0	−1.25050
$474$	0	0
$475$	−3500.00	−0.338086
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9000.00	−0.858498	−0.429249	−	0.903186i	$-0.641222\pi$
$479$	−9000.00	−0.858498	−0.429249	+	0.903186i	$0.641222\pi$
$480$	0	0
$481$	−668.000	−0.0633226
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7670.00	−0.718096
$486$	0	0
$487$	8764.00	0.815472	0.407736	−	0.913100i	$-0.366318\pi$
$487$	8764.00	0.815472	0.407736	+	0.913100i	$0.366318\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5592.00	0.513978	0.256989	−	0.966414i	$-0.417270\pi$
$491$	5592.00	0.513978	0.256989	+	0.966414i	$0.417270\pi$
$492$	0	0
$493$	−23940.0	−2.18703
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3072.00	−0.277260
$498$	0	0
$499$	−4700.00	−0.421645	−0.210823	−	0.977524i	$-0.567614\pi$
$499$	−4700.00	−0.421645	−0.210823	+	0.977524i	$0.567614\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−11808.0	−1.04671	−0.523353	−	0.852116i	$-0.675319\pi$
$503$	−11808.0	−1.04671	−0.523353	+	0.852116i	$0.675319\pi$
$504$	0	0
$505$	−8610.00	−0.758693
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1170.00	−0.101885	−0.0509424	−	0.998702i	$-0.516222\pi$
$509$	−1170.00	−0.101885	−0.0509424	+	0.998702i	$0.516222\pi$
$510$	0	0
$511$	−1912.00	−0.165522
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5260.00	−0.450065
$516$	0	0
$517$	−10368.0	−0.881981
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16638.0	1.39909	0.699543	−	0.714590i	$-0.253387\pi$
$521$	16638.0	1.39909	0.699543	+	0.714590i	$0.253387\pi$
$522$	0	0
$523$	−15692.0	−1.31198	−0.655988	−	0.754771i	$-0.727748\pi$
$523$	−15692.0	−1.31198	−0.655988	+	0.754771i	$0.727748\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−31008.0	−2.56305
$528$	0	0
$529$	−6983.00	−0.573929
$530$	0	0
$531$	0	0
$532$	0	0
$533$	396.000	0.0321814
$534$	0	0
$535$	−2820.00	−0.227886
$536$	0	0
$537$	0	0
$538$	0	0
$539$	15696.0	1.25431
$540$	0	0
$541$	−22018.0	−1.74977	−0.874887	−	0.484327i	$-0.839064\pi$
$541$	−22018.0	−1.74977	−0.874887	+	0.484327i	$0.839064\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3050.00	−0.239720
$546$	0	0
$547$	4564.00	0.356751	0.178375	−	0.983963i	$-0.442916\pi$
$547$	4564.00	0.356751	0.178375	+	0.983963i	$0.442916\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	29400.0	2.27311
$552$	0	0
$553$	2560.00	0.196858
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7734.00	0.588331	0.294165	−	0.955755i	$-0.404958\pi$
$557$	7734.00	0.588331	0.294165	+	0.955755i	$0.404958\pi$
$558$	0	0
$559$	536.000	0.0405552
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−20148.0	−1.50824	−0.754118	−	0.656739i	$-0.771935\pi$
$563$	−20148.0	−1.50824	−0.754118	+	0.656739i	$0.771935\pi$
$564$	0	0
$565$	−6510.00	−0.484739
$566$	0	0
$567$	0	0
$568$	0	0
$569$	24030.0	1.77046	0.885228	−	0.465156i	$-0.154002\pi$
$569$	24030.0	1.77046	0.885228	+	0.465156i	$0.154002\pi$
$570$	0	0
$571$	−2372.00	−0.173844	−0.0869222	−	0.996215i	$-0.527703\pi$
$571$	−2372.00	−0.173844	−0.0869222	+	0.996215i	$0.527703\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1800.00	0.130548
$576$	0	0
$577$	8546.00	0.616594	0.308297	−	0.951290i	$-0.400241\pi$
$577$	8546.00	0.616594	0.308297	+	0.951290i	$0.400241\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1392.00	−0.0993974
$582$	0	0
$583$	−3744.00	−0.265970
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15444.0	−1.08593	−0.542966	−	0.839755i	$-0.682699\pi$
$587$	−15444.0	−1.08593	−0.542966	+	0.839755i	$0.682699\pi$
$588$	0	0
$589$	38080.0	2.66394
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−18342.0	−1.27018	−0.635089	−	0.772439i	$-0.719037\pi$
$593$	−18342.0	−1.27018	−0.635089	+	0.772439i	$0.719037\pi$
$594$	0	0
$595$	2280.00	0.157094
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24600.0	1.67801	0.839006	−	0.544123i	$-0.183137\pi$
$599$	24600.0	1.67801	0.839006	+	0.544123i	$0.183137\pi$
$600$	0	0
$601$	−8998.00	−0.610709	−0.305354	−	0.952239i	$-0.598775\pi$
$601$	−8998.00	−0.610709	−0.305354	+	0.952239i	$0.598775\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4865.00	0.326926
$606$	0	0
$607$	−4076.00	−0.272553	−0.136277	−	0.990671i	$-0.543514\pi$
$607$	−4076.00	−0.272553	−0.136277	+	0.990671i	$0.543514\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	432.000	0.0286037
$612$	0	0
$613$	−4078.00	−0.268693	−0.134347	−	0.990934i	$-0.542894\pi$
$613$	−4078.00	−0.268693	−0.134347	+	0.990934i	$0.542894\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−10086.0	−0.658099	−0.329049	−	0.944313i	$-0.606728\pi$
$617$	−10086.0	−0.658099	−0.329049	+	0.944313i	$0.606728\pi$
$618$	0	0
$619$	−8780.00	−0.570110	−0.285055	−	0.958511i	$-0.592012\pi$
$619$	−8780.00	−0.570110	−0.285055	+	0.958511i	$0.592012\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−840.000	−0.0540191
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−38076.0	−2.41366
$630$	0	0
$631$	−2792.00	−0.176145	−0.0880727	−	0.996114i	$-0.528071\pi$
$631$	−2792.00	−0.176145	−0.0880727	+	0.996114i	$0.528071\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	620.000	0.0387464
$636$	0	0
$637$	−654.000	−0.0406788
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−7602.00	−0.468426	−0.234213	−	0.972185i	$-0.575251\pi$
$641$	−7602.00	−0.468426	−0.234213	+	0.972185i	$0.575251\pi$
$642$	0	0
$643$	−24212.0	−1.48496	−0.742479	−	0.669869i	$-0.766350\pi$
$643$	−24212.0	−1.48496	−0.742479	+	0.669869i	$0.766350\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9456.00	0.574581	0.287290	−	0.957844i	$-0.407246\pi$
$647$	9456.00	0.574581	0.287290	+	0.957844i	$0.407246\pi$
$648$	0	0
$649$	−11520.0	−0.696764
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9558.00	0.572792	0.286396	−	0.958111i	$-0.407543\pi$
$653$	9558.00	0.572792	0.286396	+	0.958111i	$0.407543\pi$
$654$	0	0
$655$	960.000	0.0572676
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−29280.0	−1.73078	−0.865392	−	0.501095i	$-0.832931\pi$
$659$	−29280.0	−1.73078	−0.865392	+	0.501095i	$0.832931\pi$
$660$	0	0
$661$	−29098.0	−1.71223	−0.856113	−	0.516789i	$-0.827127\pi$
$661$	−29098.0	−1.71223	−0.856113	+	0.516789i	$0.827127\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2800.00	−0.163277
$666$	0	0
$667$	−15120.0	−0.877734
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−14496.0	−0.833997
$672$	0	0
$673$	−11638.0	−0.666585	−0.333293	−	0.942823i	$-0.608160\pi$
$673$	−11638.0	−0.666585	−0.333293	+	0.942823i	$0.608160\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3426.00	−0.194493	−0.0972466	−	0.995260i	$-0.531004\pi$
$677$	−3426.00	−0.194493	−0.0972466	+	0.995260i	$0.531004\pi$
$678$	0	0
$679$	−6136.00	−0.346801
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−20148.0	−1.12876	−0.564379	−	0.825516i	$-0.690884\pi$
$683$	−20148.0	−1.12876	−0.564379	+	0.825516i	$0.690884\pi$
$684$	0	0
$685$	12570.0	0.701131
$686$	0	0
$687$	0	0
$688$	0	0
$689$	156.000	0.00862573
$690$	0	0
$691$	29428.0	1.62011	0.810053	−	0.586356i	$-0.199438\pi$
$691$	29428.0	1.62011	0.810053	+	0.586356i	$0.199438\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6700.00	−0.365677
$696$	0	0
$697$	22572.0	1.22665
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−16242.0	−0.875110	−0.437555	−	0.899192i	$-0.644155\pi$
$701$	−16242.0	−0.875110	−0.437555	+	0.899192i	$0.644155\pi$
$702$	0	0
$703$	46760.0	2.50866
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6888.00	−0.366407
$708$	0	0
$709$	2030.00	0.107529	0.0537646	−	0.998554i	$-0.482878\pi$
$709$	2030.00	0.107529	0.0537646	+	0.998554i	$0.482878\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−19584.0	−1.02865
$714$	0	0
$715$	−480.000	−0.0251063
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6960.00	0.361007	0.180504	−	0.983574i	$-0.442227\pi$
$719$	6960.00	0.361007	0.180504	+	0.983574i	$0.442227\pi$
$720$	0	0
$721$	−4208.00	−0.217357
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5250.00	−0.268938
$726$	0	0
$727$	−18596.0	−0.948676	−0.474338	−	0.880343i	$-0.657313\pi$
$727$	−18596.0	−0.948676	−0.474338	+	0.880343i	$0.657313\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	30552.0	1.54584
$732$	0	0
$733$	21242.0	1.07038	0.535192	−	0.844731i	$-0.320239\pi$
$733$	21242.0	1.07038	0.535192	+	0.844731i	$0.320239\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	28608.0	1.42984
$738$	0	0
$739$	340.000	0.0169244	0.00846218	−	0.999964i	$-0.497306\pi$
$739$	340.000	0.0169244	0.00846218	+	0.999964i	$0.497306\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−21888.0	−1.08074	−0.540372	−	0.841426i	$-0.681716\pi$
$743$	−21888.0	−1.08074	−0.540372	+	0.841426i	$0.681716\pi$
$744$	0	0
$745$	−7050.00	−0.346701
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2256.00	−0.110057
$750$	0	0
$751$	−17792.0	−0.864500	−0.432250	−	0.901754i	$-0.642280\pi$
$751$	−17792.0	−0.864500	−0.432250	+	0.901754i	$0.642280\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	10640.0	0.512886
$756$	0	0
$757$	37346.0	1.79308	0.896541	−	0.442960i	$-0.146072\pi$
$757$	37346.0	1.79308	0.896541	+	0.442960i	$0.146072\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11358.0	0.541034	0.270517	−	0.962715i	$-0.412805\pi$
$761$	11358.0	0.541034	0.270517	+	0.962715i	$0.412805\pi$
$762$	0	0
$763$	−2440.00	−0.115772
$764$	0	0
$765$	0	0
$766$	0	0
$767$	480.000	0.0225969
$768$	0	0
$769$	−34270.0	−1.60703	−0.803516	−	0.595283i	$-0.797040\pi$
$769$	−34270.0	−1.60703	−0.803516	+	0.595283i	$0.797040\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	13278.0	0.617822	0.308911	−	0.951091i	$-0.400035\pi$
$773$	13278.0	0.617822	0.308911	+	0.951091i	$0.400035\pi$
$774$	0	0
$775$	−6800.00	−0.315178
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−27720.0	−1.27493
$780$	0	0
$781$	36864.0	1.68899
$782$	0	0
$783$	0	0
$784$	0	0
$785$	15130.0	0.687914
$786$	0	0
$787$	11164.0	0.505659	0.252829	−	0.967511i	$-0.418639\pi$
$787$	11164.0	0.505659	0.252829	+	0.967511i	$0.418639\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5208.00	−0.234103
$792$	0	0
$793$	604.000	0.0270475
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5094.00	0.226397	0.113199	−	0.993572i	$-0.463890\pi$
$797$	5094.00	0.226397	0.113199	+	0.993572i	$0.463890\pi$
$798$	0	0
$799$	24624.0	1.09028
$800$	0	0
$801$	0	0
$802$	0	0
$803$	22944.0	1.00831
$804$	0	0
$805$	1440.00	0.0630476
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8790.00	0.382002	0.191001	−	0.981590i	$-0.438827\pi$
$809$	8790.00	0.382002	0.191001	+	0.981590i	$0.438827\pi$
$810$	0	0
$811$	−5852.00	−0.253380	−0.126690	−	0.991942i	$-0.540435\pi$
$811$	−5852.00	−0.253380	−0.126690	+	0.991942i	$0.540435\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−13060.0	−0.561315
$816$	0	0
$817$	−37520.0	−1.60668
$818$	0	0
$819$	0	0
$820$	0	0
$821$	29478.0	1.25309	0.626546	−	0.779384i	$-0.284468\pi$
$821$	29478.0	1.25309	0.626546	+	0.779384i	$0.284468\pi$
$822$	0	0
$823$	−39332.0	−1.66589	−0.832945	−	0.553356i	$-0.813347\pi$
$823$	−39332.0	−1.66589	−0.832945	+	0.553356i	$0.813347\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6756.00	0.284074	0.142037	−	0.989861i	$-0.454635\pi$
$827$	6756.00	0.284074	0.142037	+	0.989861i	$0.454635\pi$
$828$	0	0
$829$	3950.00	0.165488	0.0827438	−	0.996571i	$-0.473632\pi$
$829$	3950.00	0.165488	0.0827438	+	0.996571i	$0.473632\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−37278.0	−1.55055
$834$	0	0
$835$	−120.000	−0.00497338
$836$	0	0
$837$	0	0
$838$	0	0
$839$	12360.0	0.508599	0.254300	−	0.967126i	$-0.418155\pi$
$839$	12360.0	0.508599	0.254300	+	0.967126i	$0.418155\pi$
$840$	0	0
$841$	19711.0	0.808192
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10965.0	−0.446399
$846$	0	0
$847$	3892.00	0.157887
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−24048.0	−0.968690
$852$	0	0
$853$	−35998.0	−1.44496	−0.722478	−	0.691394i	$-0.756997\pi$
$853$	−35998.0	−1.44496	−0.722478	+	0.691394i	$0.756997\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21594.0	0.860720	0.430360	−	0.902657i	$-0.358387\pi$
$857$	21594.0	0.860720	0.430360	+	0.902657i	$0.358387\pi$
$858$	0	0
$859$	−9260.00	−0.367808	−0.183904	−	0.982944i	$-0.558874\pi$
$859$	−9260.00	−0.367808	−0.183904	+	0.982944i	$0.558874\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31632.0	1.24770	0.623850	−	0.781544i	$-0.285567\pi$
$863$	31632.0	1.24770	0.623850	+	0.781544i	$0.285567\pi$
$864$	0	0
$865$	−9810.00	−0.385607
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−30720.0	−1.19920
$870$	0	0
$871$	−1192.00	−0.0463713
$872$	0	0
$873$	0	0
$874$	0	0
$875$	500.000	0.0193178
$876$	0	0
$877$	−39694.0	−1.52836	−0.764180	−	0.645003i	$-0.776856\pi$
$877$	−39694.0	−1.52836	−0.764180	+	0.645003i	$0.776856\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1242.00	−0.0474961	−0.0237480	−	0.999718i	$-0.507560\pi$
$881$	−1242.00	−0.0474961	−0.0237480	+	0.999718i	$0.507560\pi$
$882$	0	0
$883$	2668.00	0.101682	0.0508411	−	0.998707i	$-0.483810\pi$
$883$	2668.00	0.101682	0.0508411	+	0.998707i	$0.483810\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4344.00	−0.164439	−0.0822194	−	0.996614i	$-0.526201\pi$
$887$	−4344.00	−0.164439	−0.0822194	+	0.996614i	$0.526201\pi$
$888$	0	0
$889$	496.000	0.0187124
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−30240.0	−1.13319
$894$	0	0
$895$	−600.000	−0.0224087
$896$	0	0
$897$	0	0
$898$	0	0
$899$	57120.0	2.11909
$900$	0	0
$901$	8892.00	0.328785
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4510.00	0.165655
$906$	0	0
$907$	−4436.00	−0.162398	−0.0811990	−	0.996698i	$-0.525875\pi$
$907$	−4436.00	−0.162398	−0.0811990	+	0.996698i	$0.525875\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	22752.0	0.827450	0.413725	−	0.910402i	$-0.364227\pi$
$911$	22752.0	0.827450	0.413725	+	0.910402i	$0.364227\pi$
$912$	0	0
$913$	16704.0	0.605500
$914$	0	0
$915$	0	0
$916$	0	0
$917$	768.000	0.0276571
$918$	0	0
$919$	27160.0	0.974892	0.487446	−	0.873153i	$-0.337929\pi$
$919$	27160.0	0.974892	0.487446	+	0.873153i	$0.337929\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1536.00	−0.0547758
$924$	0	0
$925$	−8350.00	−0.296807
$926$	0	0
$927$	0	0
$928$	0	0
$929$	33030.0	1.16650	0.583250	−	0.812292i	$-0.301781\pi$
$929$	33030.0	1.16650	0.583250	+	0.812292i	$0.301781\pi$
$930$	0	0
$931$	45780.0	1.61158
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−27360.0	−0.956971
$936$	0	0
$937$	−29974.0	−1.04505	−0.522523	−	0.852625i	$-0.675009\pi$
$937$	−29974.0	−1.04505	−0.522523	+	0.852625i	$0.675009\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−13962.0	−0.483686	−0.241843	−	0.970315i	$-0.577752\pi$
$941$	−13962.0	−0.483686	−0.241843	+	0.970315i	$0.577752\pi$
$942$	0	0
$943$	14256.0	0.492300
$944$	0	0
$945$	0	0
$946$	0	0
$947$	35196.0	1.20773	0.603863	−	0.797088i	$-0.293627\pi$
$947$	35196.0	1.20773	0.603863	+	0.797088i	$0.293627\pi$
$948$	0	0
$949$	−956.000	−0.0327008
$950$	0	0
$951$	0	0
$952$	0	0
$953$	28338.0	0.963230	0.481615	−	0.876383i	$-0.340050\pi$
$953$	28338.0	0.963230	0.481615	+	0.876383i	$0.340050\pi$
$954$	0	0
$955$	−840.000	−0.0284626
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10056.0	0.338608
$960$	0	0
$961$	44193.0	1.48343
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6590.00	−0.219834
$966$	0	0
$967$	17524.0	0.582765	0.291383	−	0.956607i	$-0.405885\pi$
$967$	17524.0	0.582765	0.291383	+	0.956607i	$0.405885\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−26808.0	−0.886004	−0.443002	−	0.896521i	$-0.646087\pi$
$971$	−26808.0	−0.886004	−0.443002	+	0.896521i	$0.646087\pi$
$972$	0	0
$973$	−5360.00	−0.176602
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10914.0	0.357390	0.178695	−	0.983905i	$-0.442813\pi$
$977$	10914.0	0.357390	0.178695	+	0.983905i	$0.442813\pi$
$978$	0	0
$979$	10080.0	0.329069
$980$	0	0
$981$	0	0
$982$	0	0
$983$	22272.0	0.722652	0.361326	−	0.932440i	$-0.382324\pi$
$983$	22272.0	0.722652	0.361326	+	0.932440i	$0.382324\pi$
$984$	0	0
$985$	20070.0	0.649222
$986$	0	0
$987$	0	0
$988$	0	0
$989$	19296.0	0.620402
$990$	0	0
$991$	−14072.0	−0.451071	−0.225536	−	0.974235i	$-0.572413\pi$
$991$	−14072.0	−0.451071	−0.225536	+	0.974235i	$0.572413\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−10000.0	−0.318614
$996$	0	0
$997$	4826.00	0.153301	0.0766504	−	0.997058i	$-0.475577\pi$
$997$	4826.00	0.153301	0.0766504	+	0.997058i	$0.475577\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.4.a.y.1.1		1
3.2	odd	2		240.4.a.b.1.1		1
4.3	odd	2		90.4.a.c.1.1		1
12.11	even	2		30.4.a.b.1.1	✓	1
15.2	even	4		1200.4.f.r.49.2		2
15.8	even	4		1200.4.f.r.49.1		2
15.14	odd	2		1200.4.a.ba.1.1		1
20.3	even	4		450.4.c.j.199.2		2
20.7	even	4		450.4.c.j.199.1		2
20.19	odd	2		450.4.a.r.1.1		1
24.5	odd	2		960.4.a.bg.1.1		1
24.11	even	2		960.4.a.n.1.1		1
36.7	odd	6		810.4.e.p.271.1		2
36.11	even	6		810.4.e.i.271.1		2
36.23	even	6		810.4.e.i.541.1		2
36.31	odd	6		810.4.e.p.541.1		2
60.23	odd	4		150.4.c.c.49.1		2
60.47	odd	4		150.4.c.c.49.2		2
60.59	even	2		150.4.a.b.1.1		1
84.83	odd	2		1470.4.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.4.a.b.1.1	✓	1	12.11	even	2
90.4.a.c.1.1		1	4.3	odd	2
150.4.a.b.1.1		1	60.59	even	2
150.4.c.c.49.1		2	60.23	odd	4
150.4.c.c.49.2		2	60.47	odd	4
240.4.a.b.1.1		1	3.2	odd	2
450.4.a.r.1.1		1	20.19	odd	2
450.4.c.j.199.1		2	20.7	even	4
450.4.c.j.199.2		2	20.3	even	4
720.4.a.y.1.1		1	1.1	even	1	trivial
810.4.e.i.271.1		2	36.11	even	6
810.4.e.i.541.1		2	36.23	even	6
810.4.e.p.271.1		2	36.7	odd	6
810.4.e.p.541.1		2	36.31	odd	6
960.4.a.n.1.1		1	24.11	even	2
960.4.a.bg.1.1		1	24.5	odd	2
1200.4.a.ba.1.1		1	15.14	odd	2
1200.4.f.r.49.1		2	15.8	even	4
1200.4.f.r.49.2		2	15.2	even	4
1470.4.a.r.1.1		1	84.83	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	720.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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