LMFDB - Embedded newform 2352.4.a.bg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, 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("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.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:	$138.772492334$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +7.00000 q^{5} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} +7.00000 q^{5} +9.00000 q^{9} -7.00000 q^{11} -52.0000 q^{13} +21.0000 q^{15} +72.0000 q^{17} -20.0000 q^{19} +48.0000 q^{23} -76.0000 q^{25} +27.0000 q^{27} -243.000 q^{29} -95.0000 q^{31} -21.0000 q^{33} +352.000 q^{37} -156.000 q^{39} -296.000 q^{41} -158.000 q^{43} +63.0000 q^{45} +142.000 q^{47} +216.000 q^{51} -375.000 q^{53} -49.0000 q^{55} -60.0000 q^{57} -279.000 q^{59} +246.000 q^{61} -364.000 q^{65} +730.000 q^{67} +144.000 q^{69} -338.000 q^{71} -542.000 q^{73} -228.000 q^{75} +305.000 q^{79} +81.0000 q^{81} -1123.00 q^{83} +504.000 q^{85} -729.000 q^{87} -426.000 q^{89} -285.000 q^{93} -140.000 q^{95} -369.000 q^{97} -63.0000 q^{99} +O(q^{100})$

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	7.00000	0.626099	0.313050	−	0.949737i	$-0.398649\pi$
$5$	7.00000	0.626099	0.313050	+	0.949737i	$0.398649\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−7.00000	−0.191871	−0.0959354	−	0.995388i	$-0.530584\pi$
$11$	−7.00000	−0.191871	−0.0959354	+	0.995388i	$0.530584\pi$
$12$	0	0
$13$	−52.0000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−52.0000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	21.0000	0.361478
$16$	0	0
$17$	72.0000	1.02721	0.513605	−	0.858027i	$-0.328310\pi$
$17$	72.0000	1.02721	0.513605	+	0.858027i	$0.328310\pi$
$18$	0	0
$19$	−20.0000	−0.241490	−0.120745	−	0.992684i	$-0.538528\pi$
$19$	−20.0000	−0.241490	−0.120745	+	0.992684i	$0.538528\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	48.0000	0.435161	0.217580	−	0.976042i	$-0.430184\pi$
$23$	48.0000	0.435161	0.217580	+	0.976042i	$0.430184\pi$
$24$	0	0
$25$	−76.0000	−0.608000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−243.000	−1.55600	−0.777999	−	0.628265i	$-0.783765\pi$
$29$	−243.000	−1.55600	−0.777999	+	0.628265i	$0.783765\pi$
$30$	0	0
$31$	−95.0000	−0.550403	−0.275202	−	0.961387i	$-0.588745\pi$
$31$	−95.0000	−0.550403	−0.275202	+	0.961387i	$0.588745\pi$
$32$	0	0
$33$	−21.0000	−0.110777
$34$	0	0
$35$	0	0
$36$	0	0
$37$	352.000	1.56401	0.782006	−	0.623271i	$-0.214197\pi$
$37$	352.000	1.56401	0.782006	+	0.623271i	$0.214197\pi$
$38$	0	0
$39$	−156.000	−0.640513
$40$	0	0
$41$	−296.000	−1.12750	−0.563749	−	0.825946i	$-0.690642\pi$
$41$	−296.000	−1.12750	−0.563749	+	0.825946i	$0.690642\pi$
$42$	0	0
$43$	−158.000	−0.560344	−0.280172	−	0.959950i	$-0.590391\pi$
$43$	−158.000	−0.560344	−0.280172	+	0.959950i	$0.590391\pi$
$44$	0	0
$45$	63.0000	0.208700
$46$	0	0
$47$	142.000	0.440698	0.220349	−	0.975421i	$-0.429280\pi$
$47$	142.000	0.440698	0.220349	+	0.975421i	$0.429280\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	216.000	0.593060
$52$	0	0
$53$	−375.000	−0.971891	−0.485945	−	0.873989i	$-0.661525\pi$
$53$	−375.000	−0.971891	−0.485945	+	0.873989i	$0.661525\pi$
$54$	0	0
$55$	−49.0000	−0.120130
$56$	0	0
$57$	−60.0000	−0.139424
$58$	0	0
$59$	−279.000	−0.615639	−0.307820	−	0.951445i	$-0.599599\pi$
$59$	−279.000	−0.615639	−0.307820	+	0.951445i	$0.599599\pi$
$60$	0	0
$61$	246.000	0.516345	0.258173	−	0.966099i	$-0.416880\pi$
$61$	246.000	0.516345	0.258173	+	0.966099i	$0.416880\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−364.000	−0.694595
$66$	0	0
$67$	730.000	1.33110	0.665550	−	0.746353i	$-0.268197\pi$
$67$	730.000	1.33110	0.665550	+	0.746353i	$0.268197\pi$
$68$	0	0
$69$	144.000	0.251240
$70$	0	0
$71$	−338.000	−0.564975	−0.282487	−	0.959271i	$-0.591160\pi$
$71$	−338.000	−0.564975	−0.282487	+	0.959271i	$0.591160\pi$
$72$	0	0
$73$	−542.000	−0.868990	−0.434495	−	0.900674i	$-0.643073\pi$
$73$	−542.000	−0.868990	−0.434495	+	0.900674i	$0.643073\pi$
$74$	0	0
$75$	−228.000	−0.351029
$76$	0	0
$77$	0	0
$78$	0	0
$79$	305.000	0.434369	0.217185	−	0.976131i	$-0.430313\pi$
$79$	305.000	0.434369	0.217185	+	0.976131i	$0.430313\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1123.00	−1.48512	−0.742562	−	0.669778i	$-0.766389\pi$
$83$	−1123.00	−1.48512	−0.742562	+	0.669778i	$0.766389\pi$
$84$	0	0
$85$	504.000	0.643135
$86$	0	0
$87$	−729.000	−0.898356
$88$	0	0
$89$	−426.000	−0.507370	−0.253685	−	0.967287i	$-0.581643\pi$
$89$	−426.000	−0.507370	−0.253685	+	0.967287i	$0.581643\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−285.000	−0.317776
$94$	0	0
$95$	−140.000	−0.151197
$96$	0	0
$97$	−369.000	−0.386250	−0.193125	−	0.981174i	$-0.561862\pi$
$97$	−369.000	−0.386250	−0.193125	+	0.981174i	$0.561862\pi$
$98$	0	0
$99$	−63.0000	−0.0639570
$100$	0	0
$101$	1270.00	1.25119	0.625593	−	0.780150i	$-0.284857\pi$
$101$	1270.00	1.25119	0.625593	+	0.780150i	$0.284857\pi$
$102$	0	0
$103$	−1832.00	−1.75255	−0.876273	−	0.481814i	$-0.839978\pi$
$103$	−1832.00	−1.75255	−0.876273	+	0.481814i	$0.839978\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	999.000	0.902589	0.451294	−	0.892375i	$-0.350962\pi$
$107$	999.000	0.902589	0.451294	+	0.892375i	$0.350962\pi$
$108$	0	0
$109$	−1666.00	−1.46398	−0.731990	−	0.681315i	$-0.761408\pi$
$109$	−1666.00	−1.46398	−0.731990	+	0.681315i	$0.761408\pi$
$110$	0	0
$111$	1056.00	0.902983
$112$	0	0
$113$	−1832.00	−1.52513	−0.762567	−	0.646910i	$-0.776061\pi$
$113$	−1832.00	−1.52513	−0.762567	+	0.646910i	$0.776061\pi$
$114$	0	0
$115$	336.000	0.272454
$116$	0	0
$117$	−468.000	−0.369800
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1282.00	−0.963186
$122$	0	0
$123$	−888.000	−0.650961
$124$	0	0
$125$	−1407.00	−1.00677
$126$	0	0
$127$	1931.00	1.34920	0.674601	−	0.738183i	$-0.264316\pi$
$127$	1931.00	1.34920	0.674601	+	0.738183i	$0.264316\pi$
$128$	0	0
$129$	−474.000	−0.323515
$130$	0	0
$131$	−979.000	−0.652944	−0.326472	−	0.945207i	$-0.605860\pi$
$131$	−979.000	−0.652944	−0.326472	+	0.945207i	$0.605860\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	189.000	0.120493
$136$	0	0
$137$	−486.000	−0.303079	−0.151539	−	0.988451i	$-0.548423\pi$
$137$	−486.000	−0.303079	−0.151539	+	0.988451i	$0.548423\pi$
$138$	0	0
$139$	2630.00	1.60485	0.802423	−	0.596755i	$-0.203544\pi$
$139$	2630.00	1.60485	0.802423	+	0.596755i	$0.203544\pi$
$140$	0	0
$141$	426.000	0.254437
$142$	0	0
$143$	364.000	0.212862
$144$	0	0
$145$	−1701.00	−0.974209
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1878.00	−1.03256	−0.516281	−	0.856419i	$-0.672684\pi$
$149$	−1878.00	−1.03256	−0.516281	+	0.856419i	$0.672684\pi$
$150$	0	0
$151$	−2103.00	−1.13338	−0.566688	−	0.823933i	$-0.691775\pi$
$151$	−2103.00	−1.13338	−0.566688	+	0.823933i	$0.691775\pi$
$152$	0	0
$153$	648.000	0.342403
$154$	0	0
$155$	−665.000	−0.344607
$156$	0	0
$157$	1404.00	0.713703	0.356852	−	0.934161i	$-0.383850\pi$
$157$	1404.00	0.713703	0.356852	+	0.934161i	$0.383850\pi$
$158$	0	0
$159$	−1125.00	−0.561121
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1536.00	0.738091	0.369045	−	0.929411i	$-0.379685\pi$
$163$	1536.00	0.738091	0.369045	+	0.929411i	$0.379685\pi$
$164$	0	0
$165$	−147.000	−0.0693572
$166$	0	0
$167$	3634.00	1.68388	0.841938	−	0.539574i	$-0.181415\pi$
$167$	3634.00	1.68388	0.841938	+	0.539574i	$0.181415\pi$
$168$	0	0
$169$	507.000	0.230769
$170$	0	0
$171$	−180.000	−0.0804967
$172$	0	0
$173$	858.000	0.377067	0.188533	−	0.982067i	$-0.439627\pi$
$173$	858.000	0.377067	0.188533	+	0.982067i	$0.439627\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−837.000	−0.355439
$178$	0	0
$179$	2420.00	1.01050	0.505249	−	0.862973i	$-0.331400\pi$
$179$	2420.00	1.01050	0.505249	+	0.862973i	$0.331400\pi$
$180$	0	0
$181$	−2672.00	−1.09728	−0.548641	−	0.836058i	$-0.684855\pi$
$181$	−2672.00	−1.09728	−0.548641	+	0.836058i	$0.684855\pi$
$182$	0	0
$183$	738.000	0.298112
$184$	0	0
$185$	2464.00	0.979226
$186$	0	0
$187$	−504.000	−0.197092
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1712.00	0.648565	0.324283	−	0.945960i	$-0.394877\pi$
$191$	1712.00	0.648565	0.324283	+	0.945960i	$0.394877\pi$
$192$	0	0
$193$	−3395.00	−1.26620	−0.633102	−	0.774068i	$-0.718219\pi$
$193$	−3395.00	−1.26620	−0.633102	+	0.774068i	$0.718219\pi$
$194$	0	0
$195$	−1092.00	−0.401024
$196$	0	0
$197$	510.000	0.184447	0.0922233	−	0.995738i	$-0.470603\pi$
$197$	510.000	0.184447	0.0922233	+	0.995738i	$0.470603\pi$
$198$	0	0
$199$	−276.000	−0.0983172	−0.0491586	−	0.998791i	$-0.515654\pi$
$199$	−276.000	−0.0983172	−0.0491586	+	0.998791i	$0.515654\pi$
$200$	0	0
$201$	2190.00	0.768511
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2072.00	−0.705926
$206$	0	0
$207$	432.000	0.145054
$208$	0	0
$209$	140.000	0.0463349
$210$	0	0
$211$	−3198.00	−1.04341	−0.521705	−	0.853126i	$-0.674704\pi$
$211$	−3198.00	−1.04341	−0.521705	+	0.853126i	$0.674704\pi$
$212$	0	0
$213$	−1014.00	−0.326188
$214$	0	0
$215$	−1106.00	−0.350831
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1626.00	−0.501712
$220$	0	0
$221$	−3744.00	−1.13959
$222$	0	0
$223$	5091.00	1.52878	0.764391	−	0.644752i	$-0.223040\pi$
$223$	5091.00	1.52878	0.764391	+	0.644752i	$0.223040\pi$
$224$	0	0
$225$	−684.000	−0.202667
$226$	0	0
$227$	3895.00	1.13886	0.569428	−	0.822041i	$-0.307165\pi$
$227$	3895.00	1.13886	0.569428	+	0.822041i	$0.307165\pi$
$228$	0	0
$229$	−6428.00	−1.85491	−0.927455	−	0.373936i	$-0.878008\pi$
$229$	−6428.00	−1.85491	−0.927455	+	0.373936i	$0.878008\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3824.00	−1.07519	−0.537593	−	0.843204i	$-0.680666\pi$
$233$	−3824.00	−1.07519	−0.537593	+	0.843204i	$0.680666\pi$
$234$	0	0
$235$	994.000	0.275921
$236$	0	0
$237$	915.000	0.250783
$238$	0	0
$239$	−568.000	−0.153727	−0.0768637	−	0.997042i	$-0.524491\pi$
$239$	−568.000	−0.153727	−0.0768637	+	0.997042i	$0.524491\pi$
$240$	0	0
$241$	3535.00	0.944852	0.472426	−	0.881370i	$-0.343378\pi$
$241$	3535.00	0.944852	0.472426	+	0.881370i	$0.343378\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1040.00	0.267909
$248$	0	0
$249$	−3369.00	−0.857437
$250$	0	0
$251$	−4335.00	−1.09013	−0.545065	−	0.838394i	$-0.683495\pi$
$251$	−4335.00	−1.09013	−0.545065	+	0.838394i	$0.683495\pi$
$252$	0	0
$253$	−336.000	−0.0834946
$254$	0	0
$255$	1512.00	0.371314
$256$	0	0
$257$	3574.00	0.867471	0.433735	−	0.901040i	$-0.357195\pi$
$257$	3574.00	0.867471	0.433735	+	0.901040i	$0.357195\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2187.00	−0.518666
$262$	0	0
$263$	−5466.00	−1.28155	−0.640776	−	0.767728i	$-0.721387\pi$
$263$	−5466.00	−1.28155	−0.640776	+	0.767728i	$0.721387\pi$
$264$	0	0
$265$	−2625.00	−0.608500
$266$	0	0
$267$	−1278.00	−0.292930
$268$	0	0
$269$	−5815.00	−1.31802	−0.659009	−	0.752135i	$-0.729024\pi$
$269$	−5815.00	−1.31802	−0.659009	+	0.752135i	$0.729024\pi$
$270$	0	0
$271$	3237.00	0.725586	0.362793	−	0.931870i	$-0.381823\pi$
$271$	3237.00	0.725586	0.362793	+	0.931870i	$0.381823\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	532.000	0.116657
$276$	0	0
$277$	−3176.00	−0.688907	−0.344454	−	0.938803i	$-0.611936\pi$
$277$	−3176.00	−0.688907	−0.344454	+	0.938803i	$0.611936\pi$
$278$	0	0
$279$	−855.000	−0.183468
$280$	0	0
$281$	−3282.00	−0.696753	−0.348377	−	0.937355i	$-0.613267\pi$
$281$	−3282.00	−0.696753	−0.348377	+	0.937355i	$0.613267\pi$
$282$	0	0
$283$	−2182.00	−0.458327	−0.229163	−	0.973388i	$-0.573599\pi$
$283$	−2182.00	−0.458327	−0.229163	+	0.973388i	$0.573599\pi$
$284$	0	0
$285$	−420.000	−0.0872935
$286$	0	0
$287$	0	0
$288$	0	0
$289$	271.000	0.0551598
$290$	0	0
$291$	−1107.00	−0.223002
$292$	0	0
$293$	3021.00	0.602351	0.301175	−	0.953569i	$-0.402621\pi$
$293$	3021.00	0.602351	0.301175	+	0.953569i	$0.402621\pi$
$294$	0	0
$295$	−1953.00	−0.385451
$296$	0	0
$297$	−189.000	−0.0369256
$298$	0	0
$299$	−2496.00	−0.482767
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3810.00	0.722372
$304$	0	0
$305$	1722.00	0.323283
$306$	0	0
$307$	1304.00	0.242421	0.121210	−	0.992627i	$-0.461322\pi$
$307$	1304.00	0.242421	0.121210	+	0.992627i	$0.461322\pi$
$308$	0	0
$309$	−5496.00	−1.01183
$310$	0	0
$311$	−6208.00	−1.13191	−0.565954	−	0.824437i	$-0.691492\pi$
$311$	−6208.00	−1.13191	−0.565954	+	0.824437i	$0.691492\pi$
$312$	0	0
$313$	3337.00	0.602615	0.301307	−	0.953527i	$-0.402577\pi$
$313$	3337.00	0.602615	0.301307	+	0.953527i	$0.402577\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8637.00	1.53029	0.765146	−	0.643857i	$-0.222667\pi$
$317$	8637.00	1.53029	0.765146	+	0.643857i	$0.222667\pi$
$318$	0	0
$319$	1701.00	0.298551
$320$	0	0
$321$	2997.00	0.521110
$322$	0	0
$323$	−1440.00	−0.248061
$324$	0	0
$325$	3952.00	0.674515
$326$	0	0
$327$	−4998.00	−0.845229
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1940.00	−0.322151	−0.161076	−	0.986942i	$-0.551496\pi$
$331$	−1940.00	−0.322151	−0.161076	+	0.986942i	$0.551496\pi$
$332$	0	0
$333$	3168.00	0.521337
$334$	0	0
$335$	5110.00	0.833400
$336$	0	0
$337$	−5527.00	−0.893397	−0.446699	−	0.894684i	$-0.647400\pi$
$337$	−5527.00	−0.893397	−0.446699	+	0.894684i	$0.647400\pi$
$338$	0	0
$339$	−5496.00	−0.880536
$340$	0	0
$341$	665.000	0.105606
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1008.00	0.157301
$346$	0	0
$347$	−9564.00	−1.47960	−0.739802	−	0.672825i	$-0.765081\pi$
$347$	−9564.00	−1.47960	−0.739802	+	0.672825i	$0.765081\pi$
$348$	0	0
$349$	−918.000	−0.140801	−0.0704003	−	0.997519i	$-0.522428\pi$
$349$	−918.000	−0.140801	−0.0704003	+	0.997519i	$0.522428\pi$
$350$	0	0
$351$	−1404.00	−0.213504
$352$	0	0
$353$	11480.0	1.73093	0.865466	−	0.500968i	$-0.167023\pi$
$353$	11480.0	1.73093	0.865466	+	0.500968i	$0.167023\pi$
$354$	0	0
$355$	−2366.00	−0.353730
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3654.00	−0.537189	−0.268594	−	0.963253i	$-0.586559\pi$
$359$	−3654.00	−0.537189	−0.268594	+	0.963253i	$0.586559\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	0	0
$363$	−3846.00	−0.556095
$364$	0	0
$365$	−3794.00	−0.544074
$366$	0	0
$367$	10067.0	1.43186	0.715931	−	0.698171i	$-0.246003\pi$
$367$	10067.0	1.43186	0.715931	+	0.698171i	$0.246003\pi$
$368$	0	0
$369$	−2664.00	−0.375833
$370$	0	0
$371$	0	0
$372$	0	0
$373$	8800.00	1.22157	0.610786	−	0.791795i	$-0.290853\pi$
$373$	8800.00	1.22157	0.610786	+	0.791795i	$0.290853\pi$
$374$	0	0
$375$	−4221.00	−0.581257
$376$	0	0
$377$	12636.0	1.72623
$378$	0	0
$379$	−1136.00	−0.153964	−0.0769821	−	0.997032i	$-0.524528\pi$
$379$	−1136.00	−0.153964	−0.0769821	+	0.997032i	$0.524528\pi$
$380$	0	0
$381$	5793.00	0.778962
$382$	0	0
$383$	−4290.00	−0.572347	−0.286173	−	0.958178i	$-0.592383\pi$
$383$	−4290.00	−0.572347	−0.286173	+	0.958178i	$0.592383\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1422.00	−0.186781
$388$	0	0
$389$	−862.000	−0.112353	−0.0561763	−	0.998421i	$-0.517891\pi$
$389$	−862.000	−0.112353	−0.0561763	+	0.998421i	$0.517891\pi$
$390$	0	0
$391$	3456.00	0.447001
$392$	0	0
$393$	−2937.00	−0.376977
$394$	0	0
$395$	2135.00	0.271958
$396$	0	0
$397$	−9140.00	−1.15547	−0.577737	−	0.816223i	$-0.696064\pi$
$397$	−9140.00	−1.15547	−0.577737	+	0.816223i	$0.696064\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14232.0	−1.77235	−0.886175	−	0.463351i	$-0.846647\pi$
$401$	−14232.0	−1.77235	−0.886175	+	0.463351i	$0.846647\pi$
$402$	0	0
$403$	4940.00	0.610618
$404$	0	0
$405$	567.000	0.0695666
$406$	0	0
$407$	−2464.00	−0.300088
$408$	0	0
$409$	−10001.0	−1.20909	−0.604545	−	0.796571i	$-0.706645\pi$
$409$	−10001.0	−1.20909	−0.604545	+	0.796571i	$0.706645\pi$
$410$	0	0
$411$	−1458.00	−0.174983
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−7861.00	−0.929834
$416$	0	0
$417$	7890.00	0.926559
$418$	0	0
$419$	10256.0	1.19580	0.597898	−	0.801572i	$-0.296003\pi$
$419$	10256.0	1.19580	0.597898	+	0.801572i	$0.296003\pi$
$420$	0	0
$421$	−4502.00	−0.521174	−0.260587	−	0.965450i	$-0.583916\pi$
$421$	−4502.00	−0.521174	−0.260587	+	0.965450i	$0.583916\pi$
$422$	0	0
$423$	1278.00	0.146899
$424$	0	0
$425$	−5472.00	−0.624544
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1092.00	0.122896
$430$	0	0
$431$	−12356.0	−1.38090	−0.690450	−	0.723380i	$-0.742587\pi$
$431$	−12356.0	−1.38090	−0.690450	+	0.723380i	$0.742587\pi$
$432$	0	0
$433$	862.000	0.0956699	0.0478350	−	0.998855i	$-0.484768\pi$
$433$	862.000	0.0956699	0.0478350	+	0.998855i	$0.484768\pi$
$434$	0	0
$435$	−5103.00	−0.562460
$436$	0	0
$437$	−960.000	−0.105087
$438$	0	0
$439$	1365.00	0.148401	0.0742003	−	0.997243i	$-0.476360\pi$
$439$	1365.00	0.148401	0.0742003	+	0.997243i	$0.476360\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8931.00	−0.957843	−0.478922	−	0.877858i	$-0.658972\pi$
$443$	−8931.00	−0.957843	−0.478922	+	0.877858i	$0.658972\pi$
$444$	0	0
$445$	−2982.00	−0.317664
$446$	0	0
$447$	−5634.00	−0.596150
$448$	0	0
$449$	11228.0	1.18014	0.590069	−	0.807353i	$-0.299100\pi$
$449$	11228.0	1.18014	0.590069	+	0.807353i	$0.299100\pi$
$450$	0	0
$451$	2072.00	0.216334
$452$	0	0
$453$	−6309.00	−0.654355
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12151.0	1.24376	0.621882	−	0.783111i	$-0.286368\pi$
$457$	12151.0	1.24376	0.621882	+	0.783111i	$0.286368\pi$
$458$	0	0
$459$	1944.00	0.197687
$460$	0	0
$461$	18534.0	1.87248	0.936241	−	0.351358i	$-0.114280\pi$
$461$	18534.0	1.87248	0.936241	+	0.351358i	$0.114280\pi$
$462$	0	0
$463$	−17096.0	−1.71602	−0.858011	−	0.513631i	$-0.828300\pi$
$463$	−17096.0	−1.71602	−0.858011	+	0.513631i	$0.828300\pi$
$464$	0	0
$465$	−1995.00	−0.198959
$466$	0	0
$467$	2140.00	0.212050	0.106025	−	0.994363i	$-0.466188\pi$
$467$	2140.00	0.212050	0.106025	+	0.994363i	$0.466188\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	4212.00	0.412057
$472$	0	0
$473$	1106.00	0.107514
$474$	0	0
$475$	1520.00	0.146826
$476$	0	0
$477$	−3375.00	−0.323964
$478$	0	0
$479$	13266.0	1.26543	0.632713	−	0.774386i	$-0.281941\pi$
$479$	13266.0	1.26543	0.632713	+	0.774386i	$0.281941\pi$
$480$	0	0
$481$	−18304.0	−1.73512
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2583.00	−0.241831
$486$	0	0
$487$	19183.0	1.78494	0.892469	−	0.451109i	$-0.148971\pi$
$487$	19183.0	1.78494	0.892469	+	0.451109i	$0.148971\pi$
$488$	0	0
$489$	4608.00	0.426137
$490$	0	0
$491$	21693.0	1.99387	0.996936	−	0.0782185i	$-0.0249232\pi$
$491$	21693.0	1.99387	0.996936	+	0.0782185i	$0.0249232\pi$
$492$	0	0
$493$	−17496.0	−1.59834
$494$	0	0
$495$	−441.000	−0.0400434
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−19658.0	−1.76355	−0.881776	−	0.471667i	$-0.843652\pi$
$499$	−19658.0	−1.76355	−0.881776	+	0.471667i	$0.843652\pi$
$500$	0	0
$501$	10902.0	0.972187
$502$	0	0
$503$	−19436.0	−1.72288	−0.861440	−	0.507860i	$-0.830437\pi$
$503$	−19436.0	−1.72288	−0.861440	+	0.507860i	$0.830437\pi$
$504$	0	0
$505$	8890.00	0.783366
$506$	0	0
$507$	1521.00	0.133235
$508$	0	0
$509$	3089.00	0.268993	0.134497	−	0.990914i	$-0.457058\pi$
$509$	3089.00	0.268993	0.134497	+	0.990914i	$0.457058\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−540.000	−0.0464748
$514$	0	0
$515$	−12824.0	−1.09727
$516$	0	0
$517$	−994.000	−0.0845572
$518$	0	0
$519$	2574.00	0.217700
$520$	0	0
$521$	1950.00	0.163975	0.0819876	−	0.996633i	$-0.473873\pi$
$521$	1950.00	0.163975	0.0819876	+	0.996633i	$0.473873\pi$
$522$	0	0
$523$	−13132.0	−1.09794	−0.548970	−	0.835842i	$-0.684980\pi$
$523$	−13132.0	−1.09794	−0.548970	+	0.835842i	$0.684980\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6840.00	−0.565380
$528$	0	0
$529$	−9863.00	−0.810635
$530$	0	0
$531$	−2511.00	−0.205213
$532$	0	0
$533$	15392.0	1.25085
$534$	0	0
$535$	6993.00	0.565110
$536$	0	0
$537$	7260.00	0.583412
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2930.00	0.232848	0.116424	−	0.993200i	$-0.462857\pi$
$541$	2930.00	0.232848	0.116424	+	0.993200i	$0.462857\pi$
$542$	0	0
$543$	−8016.00	−0.633517
$544$	0	0
$545$	−11662.0	−0.916597
$546$	0	0
$547$	−19824.0	−1.54957	−0.774783	−	0.632227i	$-0.782141\pi$
$547$	−19824.0	−1.54957	−0.774783	+	0.632227i	$0.782141\pi$
$548$	0	0
$549$	2214.00	0.172115
$550$	0	0
$551$	4860.00	0.375759
$552$	0	0
$553$	0	0
$554$	0	0
$555$	7392.00	0.565357
$556$	0	0
$557$	−19169.0	−1.45820	−0.729099	−	0.684408i	$-0.760061\pi$
$557$	−19169.0	−1.45820	−0.729099	+	0.684408i	$0.760061\pi$
$558$	0	0
$559$	8216.00	0.621645
$560$	0	0
$561$	−1512.00	−0.113791
$562$	0	0
$563$	−7267.00	−0.543992	−0.271996	−	0.962298i	$-0.587684\pi$
$563$	−7267.00	−0.543992	−0.271996	+	0.962298i	$0.587684\pi$
$564$	0	0
$565$	−12824.0	−0.954884
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4892.00	−0.360428	−0.180214	−	0.983627i	$-0.557679\pi$
$569$	−4892.00	−0.360428	−0.180214	+	0.983627i	$0.557679\pi$
$570$	0	0
$571$	−22126.0	−1.62162	−0.810809	−	0.585310i	$-0.800973\pi$
$571$	−22126.0	−1.62162	−0.810809	+	0.585310i	$0.800973\pi$
$572$	0	0
$573$	5136.00	0.374449
$574$	0	0
$575$	−3648.00	−0.264578
$576$	0	0
$577$	−1097.00	−0.0791485	−0.0395743	−	0.999217i	$-0.512600\pi$
$577$	−1097.00	−0.0791485	−0.0395743	+	0.999217i	$0.512600\pi$
$578$	0	0
$579$	−10185.0	−0.731043
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2625.00	0.186478
$584$	0	0
$585$	−3276.00	−0.231532
$586$	0	0
$587$	−9969.00	−0.700962	−0.350481	−	0.936570i	$-0.613982\pi$
$587$	−9969.00	−0.700962	−0.350481	+	0.936570i	$0.613982\pi$
$588$	0	0
$589$	1900.00	0.132917
$590$	0	0
$591$	1530.00	0.106490
$592$	0	0
$593$	−6424.00	−0.444860	−0.222430	−	0.974949i	$-0.571399\pi$
$593$	−6424.00	−0.444860	−0.222430	+	0.974949i	$0.571399\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−828.000	−0.0567635
$598$	0	0
$599$	−17046.0	−1.16274	−0.581370	−	0.813640i	$-0.697483\pi$
$599$	−17046.0	−1.16274	−0.581370	+	0.813640i	$0.697483\pi$
$600$	0	0
$601$	−18877.0	−1.28121	−0.640606	−	0.767869i	$-0.721317\pi$
$601$	−18877.0	−1.28121	−0.640606	+	0.767869i	$0.721317\pi$
$602$	0	0
$603$	6570.00	0.443700
$604$	0	0
$605$	−8974.00	−0.603050
$606$	0	0
$607$	13783.0	0.921639	0.460819	−	0.887494i	$-0.347556\pi$
$607$	13783.0	0.921639	0.460819	+	0.887494i	$0.347556\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7384.00	−0.488911
$612$	0	0
$613$	18192.0	1.19864	0.599321	−	0.800509i	$-0.295437\pi$
$613$	18192.0	1.19864	0.599321	+	0.800509i	$0.295437\pi$
$614$	0	0
$615$	−6216.00	−0.407566
$616$	0	0
$617$	−13054.0	−0.851757	−0.425879	−	0.904780i	$-0.640035\pi$
$617$	−13054.0	−0.851757	−0.425879	+	0.904780i	$0.640035\pi$
$618$	0	0
$619$	−7246.00	−0.470503	−0.235251	−	0.971935i	$-0.575591\pi$
$619$	−7246.00	−0.470503	−0.235251	+	0.971935i	$0.575591\pi$
$620$	0	0
$621$	1296.00	0.0837467
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−349.000	−0.0223360
$626$	0	0
$627$	420.000	0.0267515
$628$	0	0
$629$	25344.0	1.60657
$630$	0	0
$631$	−2817.00	−0.177723	−0.0888613	−	0.996044i	$-0.528323\pi$
$631$	−2817.00	−0.177723	−0.0888613	+	0.996044i	$0.528323\pi$
$632$	0	0
$633$	−9594.00	−0.602413
$634$	0	0
$635$	13517.0	0.844734
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−3042.00	−0.188325
$640$	0	0
$641$	15786.0	0.972714	0.486357	−	0.873760i	$-0.338326\pi$
$641$	15786.0	0.972714	0.486357	+	0.873760i	$0.338326\pi$
$642$	0	0
$643$	17426.0	1.06876	0.534381	−	0.845244i	$-0.320545\pi$
$643$	17426.0	1.06876	0.534381	+	0.845244i	$0.320545\pi$
$644$	0	0
$645$	−3318.00	−0.202552
$646$	0	0
$647$	25834.0	1.56977	0.784884	−	0.619643i	$-0.212723\pi$
$647$	25834.0	1.56977	0.784884	+	0.619643i	$0.212723\pi$
$648$	0	0
$649$	1953.00	0.118123
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27983.0	1.67697	0.838483	−	0.544927i	$-0.183443\pi$
$653$	27983.0	1.67697	0.838483	+	0.544927i	$0.183443\pi$
$654$	0	0
$655$	−6853.00	−0.408807
$656$	0	0
$657$	−4878.00	−0.289663
$658$	0	0
$659$	28296.0	1.67262	0.836309	−	0.548258i	$-0.184709\pi$
$659$	28296.0	1.67262	0.836309	+	0.548258i	$0.184709\pi$
$660$	0	0
$661$	21254.0	1.25066	0.625329	−	0.780361i	$-0.284965\pi$
$661$	21254.0	1.25066	0.625329	+	0.780361i	$0.284965\pi$
$662$	0	0
$663$	−11232.0	−0.657941
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−11664.0	−0.677109
$668$	0	0
$669$	15273.0	0.882643
$670$	0	0
$671$	−1722.00	−0.0990716
$672$	0	0
$673$	−28259.0	−1.61858	−0.809290	−	0.587409i	$-0.800148\pi$
$673$	−28259.0	−1.61858	−0.809290	+	0.587409i	$0.800148\pi$
$674$	0	0
$675$	−2052.00	−0.117010
$676$	0	0
$677$	31779.0	1.80409	0.902043	−	0.431646i	$-0.142067\pi$
$677$	31779.0	1.80409	0.902043	+	0.431646i	$0.142067\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	11685.0	0.657519
$682$	0	0
$683$	9747.00	0.546059	0.273030	−	0.962006i	$-0.411974\pi$
$683$	9747.00	0.546059	0.273030	+	0.962006i	$0.411974\pi$
$684$	0	0
$685$	−3402.00	−0.189757
$686$	0	0
$687$	−19284.0	−1.07093
$688$	0	0
$689$	19500.0	1.07822
$690$	0	0
$691$	24512.0	1.34947	0.674733	−	0.738062i	$-0.264259\pi$
$691$	24512.0	1.34947	0.674733	+	0.738062i	$0.264259\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	18410.0	1.00479
$696$	0	0
$697$	−21312.0	−1.15818
$698$	0	0
$699$	−11472.0	−0.620760
$700$	0	0
$701$	14325.0	0.771823	0.385911	−	0.922536i	$-0.373887\pi$
$701$	14325.0	0.771823	0.385911	+	0.922536i	$0.373887\pi$
$702$	0	0
$703$	−7040.00	−0.377694
$704$	0	0
$705$	2982.00	0.159303
$706$	0	0
$707$	0	0
$708$	0	0
$709$	12078.0	0.639773	0.319886	−	0.947456i	$-0.396355\pi$
$709$	12078.0	0.639773	0.319886	+	0.947456i	$0.396355\pi$
$710$	0	0
$711$	2745.00	0.144790
$712$	0	0
$713$	−4560.00	−0.239514
$714$	0	0
$715$	2548.00	0.133272
$716$	0	0
$717$	−1704.00	−0.0887546
$718$	0	0
$719$	10658.0	0.552818	0.276409	−	0.961040i	$-0.410856\pi$
$719$	10658.0	0.552818	0.276409	+	0.961040i	$0.410856\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	10605.0	0.545511
$724$	0	0
$725$	18468.0	0.946047
$726$	0	0
$727$	−19099.0	−0.974337	−0.487168	−	0.873308i	$-0.661970\pi$
$727$	−19099.0	−0.974337	−0.487168	+	0.873308i	$0.661970\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−11376.0	−0.575590
$732$	0	0
$733$	1406.00	0.0708483	0.0354241	−	0.999372i	$-0.488722\pi$
$733$	1406.00	0.0708483	0.0354241	+	0.999372i	$0.488722\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5110.00	−0.255399
$738$	0	0
$739$	−6730.00	−0.335003	−0.167501	−	0.985872i	$-0.553570\pi$
$739$	−6730.00	−0.335003	−0.167501	+	0.985872i	$0.553570\pi$
$740$	0	0
$741$	3120.00	0.154678
$742$	0	0
$743$	1166.00	0.0575725	0.0287863	−	0.999586i	$-0.490836\pi$
$743$	1166.00	0.0575725	0.0287863	+	0.999586i	$0.490836\pi$
$744$	0	0
$745$	−13146.0	−0.646486
$746$	0	0
$747$	−10107.0	−0.495041
$748$	0	0
$749$	0	0
$750$	0	0
$751$	16967.0	0.824414	0.412207	−	0.911090i	$-0.364758\pi$
$751$	16967.0	0.824414	0.412207	+	0.911090i	$0.364758\pi$
$752$	0	0
$753$	−13005.0	−0.629387
$754$	0	0
$755$	−14721.0	−0.709605
$756$	0	0
$757$	11878.0	0.570295	0.285147	−	0.958484i	$-0.407957\pi$
$757$	11878.0	0.570295	0.285147	+	0.958484i	$0.407957\pi$
$758$	0	0
$759$	−1008.00	−0.0482056
$760$	0	0
$761$	27324.0	1.30157	0.650785	−	0.759262i	$-0.274440\pi$
$761$	27324.0	1.30157	0.650785	+	0.759262i	$0.274440\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	4536.00	0.214378
$766$	0	0
$767$	14508.0	0.682990
$768$	0	0
$769$	−11971.0	−0.561359	−0.280680	−	0.959802i	$-0.590560\pi$
$769$	−11971.0	−0.561359	−0.280680	+	0.959802i	$0.590560\pi$
$770$	0	0
$771$	10722.0	0.500834
$772$	0	0
$773$	−338.000	−0.0157271	−0.00786353	−	0.999969i	$-0.502503\pi$
$773$	−338.000	−0.0157271	−0.00786353	+	0.999969i	$0.502503\pi$
$774$	0	0
$775$	7220.00	0.334645
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5920.00	0.272280
$780$	0	0
$781$	2366.00	0.108402
$782$	0	0
$783$	−6561.00	−0.299452
$784$	0	0
$785$	9828.00	0.446849
$786$	0	0
$787$	−14114.0	−0.639275	−0.319638	−	0.947540i	$-0.603561\pi$
$787$	−14114.0	−0.639275	−0.319638	+	0.947540i	$0.603561\pi$
$788$	0	0
$789$	−16398.0	−0.739904
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−12792.0	−0.572834
$794$	0	0
$795$	−7875.00	−0.351318
$796$	0	0
$797$	1563.00	0.0694659	0.0347329	−	0.999397i	$-0.488942\pi$
$797$	1563.00	0.0694659	0.0347329	+	0.999397i	$0.488942\pi$
$798$	0	0
$799$	10224.0	0.452690
$800$	0	0
$801$	−3834.00	−0.169123
$802$	0	0
$803$	3794.00	0.166734
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−17445.0	−0.760958
$808$	0	0
$809$	−31568.0	−1.37191	−0.685953	−	0.727646i	$-0.740614\pi$
$809$	−31568.0	−1.37191	−0.685953	+	0.727646i	$0.740614\pi$
$810$	0	0
$811$	−2626.00	−0.113701	−0.0568504	−	0.998383i	$-0.518106\pi$
$811$	−2626.00	−0.113701	−0.0568504	+	0.998383i	$0.518106\pi$
$812$	0	0
$813$	9711.00	0.418917
$814$	0	0
$815$	10752.0	0.462118
$816$	0	0
$817$	3160.00	0.135318
$818$	0	0
$819$	0	0
$820$	0	0
$821$	361.000	0.0153459	0.00767295	−	0.999971i	$-0.497558\pi$
$821$	361.000	0.0153459	0.00767295	+	0.999971i	$0.497558\pi$
$822$	0	0
$823$	29544.0	1.25132	0.625662	−	0.780095i	$-0.284829\pi$
$823$	29544.0	1.25132	0.625662	+	0.780095i	$0.284829\pi$
$824$	0	0
$825$	1596.00	0.0673522
$826$	0	0
$827$	−24163.0	−1.01600	−0.507999	−	0.861358i	$-0.669615\pi$
$827$	−24163.0	−1.01600	−0.507999	+	0.861358i	$0.669615\pi$
$828$	0	0
$829$	−39196.0	−1.64214	−0.821070	−	0.570828i	$-0.806622\pi$
$829$	−39196.0	−1.64214	−0.821070	+	0.570828i	$0.806622\pi$
$830$	0	0
$831$	−9528.00	−0.397741
$832$	0	0
$833$	0	0
$834$	0	0
$835$	25438.0	1.05427
$836$	0	0
$837$	−2565.00	−0.105925
$838$	0	0
$839$	−6528.00	−0.268619	−0.134310	−	0.990939i	$-0.542882\pi$
$839$	−6528.00	−0.268619	−0.134310	+	0.990939i	$0.542882\pi$
$840$	0	0
$841$	34660.0	1.42113
$842$	0	0
$843$	−9846.00	−0.402271
$844$	0	0
$845$	3549.00	0.144484
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−6546.00	−0.264615
$850$	0	0
$851$	16896.0	0.680596
$852$	0	0
$853$	1046.00	0.0419864	0.0209932	−	0.999780i	$-0.493317\pi$
$853$	1046.00	0.0419864	0.0209932	+	0.999780i	$0.493317\pi$
$854$	0	0
$855$	−1260.00	−0.0503989
$856$	0	0
$857$	32910.0	1.31177	0.655883	−	0.754862i	$-0.272296\pi$
$857$	32910.0	1.31177	0.655883	+	0.754862i	$0.272296\pi$
$858$	0	0
$859$	−22682.0	−0.900931	−0.450466	−	0.892794i	$-0.648742\pi$
$859$	−22682.0	−0.900931	−0.450466	+	0.892794i	$0.648742\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7450.00	0.293860	0.146930	−	0.989147i	$-0.453061\pi$
$863$	7450.00	0.293860	0.146930	+	0.989147i	$0.453061\pi$
$864$	0	0
$865$	6006.00	0.236081
$866$	0	0
$867$	813.000	0.0318465
$868$	0	0
$869$	−2135.00	−0.0833428
$870$	0	0
$871$	−37960.0	−1.47672
$872$	0	0
$873$	−3321.00	−0.128750
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10080.0	0.388116	0.194058	−	0.980990i	$-0.437835\pi$
$877$	10080.0	0.388116	0.194058	+	0.980990i	$0.437835\pi$
$878$	0	0
$879$	9063.00	0.347767
$880$	0	0
$881$	−32958.0	−1.26037	−0.630183	−	0.776446i	$-0.717020\pi$
$881$	−32958.0	−1.26037	−0.630183	+	0.776446i	$0.717020\pi$
$882$	0	0
$883$	19784.0	0.754003	0.377001	−	0.926213i	$-0.376955\pi$
$883$	19784.0	0.754003	0.377001	+	0.926213i	$0.376955\pi$
$884$	0	0
$885$	−5859.00	−0.222540
$886$	0	0
$887$	11480.0	0.434567	0.217283	−	0.976109i	$-0.430280\pi$
$887$	11480.0	0.434567	0.217283	+	0.976109i	$0.430280\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−567.000	−0.0213190
$892$	0	0
$893$	−2840.00	−0.106424
$894$	0	0
$895$	16940.0	0.632672
$896$	0	0
$897$	−7488.00	−0.278726
$898$	0	0
$899$	23085.0	0.856427
$900$	0	0
$901$	−27000.0	−0.998336
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−18704.0	−0.687008
$906$	0	0
$907$	45352.0	1.66030	0.830148	−	0.557543i	$-0.188256\pi$
$907$	45352.0	1.66030	0.830148	+	0.557543i	$0.188256\pi$
$908$	0	0
$909$	11430.0	0.417062
$910$	0	0
$911$	39906.0	1.45131	0.725656	−	0.688058i	$-0.241537\pi$
$911$	39906.0	1.45131	0.725656	+	0.688058i	$0.241537\pi$
$912$	0	0
$913$	7861.00	0.284952
$914$	0	0
$915$	5166.00	0.186648
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−17528.0	−0.629157	−0.314579	−	0.949231i	$-0.601863\pi$
$919$	−17528.0	−0.629157	−0.314579	+	0.949231i	$0.601863\pi$
$920$	0	0
$921$	3912.00	0.139962
$922$	0	0
$923$	17576.0	0.626783
$924$	0	0
$925$	−26752.0	−0.950919
$926$	0	0
$927$	−16488.0	−0.584182
$928$	0	0
$929$	17066.0	0.602710	0.301355	−	0.953512i	$-0.402561\pi$
$929$	17066.0	0.602710	0.301355	+	0.953512i	$0.402561\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−18624.0	−0.653507
$934$	0	0
$935$	−3528.00	−0.123399
$936$	0	0
$937$	−30821.0	−1.07458	−0.537288	−	0.843399i	$-0.680551\pi$
$937$	−30821.0	−1.07458	−0.537288	+	0.843399i	$0.680551\pi$
$938$	0	0
$939$	10011.0	0.347920
$940$	0	0
$941$	4179.00	0.144773	0.0723866	−	0.997377i	$-0.476938\pi$
$941$	4179.00	0.144773	0.0723866	+	0.997377i	$0.476938\pi$
$942$	0	0
$943$	−14208.0	−0.490643
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−9776.00	−0.335457	−0.167728	−	0.985833i	$-0.553643\pi$
$947$	−9776.00	−0.335457	−0.167728	+	0.985833i	$0.553643\pi$
$948$	0	0
$949$	28184.0	0.964058
$950$	0	0
$951$	25911.0	0.883514
$952$	0	0
$953$	14778.0	0.502315	0.251158	−	0.967946i	$-0.419189\pi$
$953$	14778.0	0.502315	0.251158	+	0.967946i	$0.419189\pi$
$954$	0	0
$955$	11984.0	0.406066
$956$	0	0
$957$	5103.00	0.172368
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−20766.0	−0.697056
$962$	0	0
$963$	8991.00	0.300863
$964$	0	0
$965$	−23765.0	−0.792769
$966$	0	0
$967$	−8759.00	−0.291283	−0.145641	−	0.989337i	$-0.546525\pi$
$967$	−8759.00	−0.291283	−0.145641	+	0.989337i	$0.546525\pi$
$968$	0	0
$969$	−4320.00	−0.143218
$970$	0	0
$971$	33579.0	1.10979	0.554893	−	0.831922i	$-0.312759\pi$
$971$	33579.0	1.10979	0.554893	+	0.831922i	$0.312759\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	11856.0	0.389432
$976$	0	0
$977$	42634.0	1.39609	0.698046	−	0.716053i	$-0.254053\pi$
$977$	42634.0	1.39609	0.698046	+	0.716053i	$0.254053\pi$
$978$	0	0
$979$	2982.00	0.0973495
$980$	0	0
$981$	−14994.0	−0.487993
$982$	0	0
$983$	−29148.0	−0.945755	−0.472877	−	0.881128i	$-0.656785\pi$
$983$	−29148.0	−0.945755	−0.472877	+	0.881128i	$0.656785\pi$
$984$	0	0
$985$	3570.00	0.115482
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7584.00	−0.243839
$990$	0	0
$991$	44797.0	1.43595	0.717974	−	0.696070i	$-0.245070\pi$
$991$	44797.0	1.43595	0.717974	+	0.696070i	$0.245070\pi$
$992$	0	0
$993$	−5820.00	−0.185994
$994$	0	0
$995$	−1932.00	−0.0615563
$996$	0	0
$997$	10654.0	0.338431	0.169215	−	0.985579i	$-0.445877\pi$
$997$	10654.0	0.338431	0.169215	+	0.985579i	$0.445877\pi$
$998$	0	0
$999$	9504.00	0.300994

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.bg.1.1		1
4.3	odd	2		1176.4.a.f.1.1		1
7.2	even	3		336.4.q.a.193.1		2
7.4	even	3		336.4.q.a.289.1		2
7.6	odd	2		2352.4.a.c.1.1		1
28.11	odd	6		168.4.q.a.121.1	yes	2
28.23	odd	6		168.4.q.a.25.1	✓	2
28.27	even	2		1176.4.a.i.1.1		1
84.11	even	6		504.4.s.e.289.1		2
84.23	even	6		504.4.s.e.361.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.q.a.25.1	✓	2	28.23	odd	6
168.4.q.a.121.1	yes	2	28.11	odd	6
336.4.q.a.193.1		2	7.2	even	3
336.4.q.a.289.1		2	7.4	even	3
504.4.s.e.289.1		2	84.11	even	6
504.4.s.e.361.1		2	84.23	even	6
1176.4.a.f.1.1		1	4.3	odd	2
1176.4.a.i.1.1		1	28.27	even	2
2352.4.a.c.1.1		1	7.6	odd	2
2352.4.a.bg.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

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