LMFDB - Embedded newform 1176.4.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1176,4,Mod(1,1176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1176, 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("1176.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1176.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:	$69.3862461668$
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	$\chi$	$=$	1176.1

$q$-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	$-0.601351\pi$
$5$	−7.00000	−0.626099	−0.313050	+	0.949737i	$0.601351\pi$
$6$	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	$-0.469416\pi$
$11$	7.00000	0.191871	0.0959354	+	0.995388i	$0.469416\pi$
$12$	0	0
$13$	52.0000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	52.0000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−21.0000	−0.361478
$16$	0	0
$17$	−72.0000	−1.02721	−0.513605	−	0.858027i	$-0.671690\pi$
$17$	−72.0000	−1.02721	−0.513605	+	0.858027i	$0.671690\pi$
$18$	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	$-0.569816\pi$
$23$	−48.0000	−0.435161	−0.217580	+	0.976042i	$0.569816\pi$
$24$	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	$-0.309358\pi$
$41$	296.000	1.12750	0.563749	+	0.825946i	$0.309358\pi$
$42$	0	0
$43$	158.000	0.560344	0.280172	−	0.959950i	$-0.409609\pi$
$43$	158.000	0.560344	0.280172	+	0.959950i	$0.409609\pi$
$44$	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	$-0.583120\pi$
$61$	−246.000	−0.516345	−0.258173	+	0.966099i	$0.583120\pi$
$62$	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	$-0.731803\pi$
$67$	−730.000	−1.33110	−0.665550	+	0.746353i	$0.731803\pi$
$68$	0	0
$69$	−144.000	−0.251240
$70$	0	0
$71$	338.000	0.564975	0.282487	−	0.959271i	$-0.408840\pi$
$71$	338.000	0.564975	0.282487	+	0.959271i	$0.408840\pi$
$72$	0	0
$73$	542.000	0.868990	0.434495	−	0.900674i	$-0.356927\pi$
$73$	542.000	0.868990	0.434495	+	0.900674i	$0.356927\pi$
$74$	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	$-0.569687\pi$
$79$	−305.000	−0.434369	−0.217185	+	0.976131i	$0.569687\pi$
$80$	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	$-0.418357\pi$
$89$	426.000	0.507370	0.253685	+	0.967287i	$0.418357\pi$
$90$	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	$-0.438138\pi$
$97$	369.000	0.386250	0.193125	+	0.981174i	$0.438138\pi$
$98$	0	0
$99$	63.0000	0.0639570
$100$	0	0
$101$	−1270.00	−1.25119	−0.625593	−	0.780150i	$-0.715143\pi$
$101$	−1270.00	−1.25119	−0.625593	+	0.780150i	$0.715143\pi$
$102$	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	$-0.649038\pi$
$107$	−999.000	−0.902589	−0.451294	+	0.892375i	$0.649038\pi$
$108$	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	$-0.735684\pi$
$127$	−1931.00	−1.34920	−0.674601	+	0.738183i	$0.735684\pi$
$128$	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	$-0.308225\pi$
$151$	2103.00	1.13338	0.566688	+	0.823933i	$0.308225\pi$
$152$	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	$-0.616150\pi$
$157$	−1404.00	−0.713703	−0.356852	+	0.934161i	$0.616150\pi$
$158$	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	$-0.620315\pi$
$163$	−1536.00	−0.738091	−0.369045	+	0.929411i	$0.620315\pi$
$164$	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	$-0.560373\pi$
$173$	−858.000	−0.377067	−0.188533	+	0.982067i	$0.560373\pi$
$174$	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	$-0.668600\pi$
$179$	−2420.00	−1.01050	−0.505249	+	0.862973i	$0.668600\pi$
$180$	0	0
$181$	2672.00	1.09728	0.548641	−	0.836058i	$-0.315145\pi$
$181$	2672.00	1.09728	0.548641	+	0.836058i	$0.315145\pi$
$182$	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	$-0.605123\pi$
$191$	−1712.00	−0.648565	−0.324283	+	0.945960i	$0.605123\pi$
$192$	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	$-0.325296\pi$
$211$	3198.00	1.04341	0.521705	+	0.853126i	$0.325296\pi$
$212$	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	$-0.121992\pi$
$229$	6428.00	1.85491	0.927455	+	0.373936i	$0.121992\pi$
$230$	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	$-0.475509\pi$
$239$	568.000	0.153727	0.0768637	+	0.997042i	$0.475509\pi$
$240$	0	0
$241$	−3535.00	−0.944852	−0.472426	−	0.881370i	$-0.656622\pi$
$241$	−3535.00	−0.944852	−0.472426	+	0.881370i	$0.656622\pi$
$242$	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	$-0.642805\pi$
$257$	−3574.00	−0.867471	−0.433735	+	0.901040i	$0.642805\pi$
$258$	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	$-0.278613\pi$
$263$	5466.00	1.28155	0.640776	+	0.767728i	$0.278613\pi$
$264$	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	$-0.270976\pi$
$269$	5815.00	1.31802	0.659009	+	0.752135i	$0.270976\pi$
$270$	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	$-0.597379\pi$
$293$	−3021.00	−0.602351	−0.301175	+	0.953569i	$0.597379\pi$
$294$	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	$-0.597423\pi$
$313$	−3337.00	−0.602615	−0.301307	+	0.953527i	$0.597423\pi$
$314$	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	$-0.448504\pi$
$331$	1940.00	0.322151	0.161076	+	0.986942i	$0.448504\pi$
$332$	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	$-0.234919\pi$
$347$	9564.00	1.47960	0.739802	+	0.672825i	$0.234919\pi$
$348$	0	0
$349$	918.000	0.140801	0.0704003	−	0.997519i	$-0.477572\pi$
$349$	918.000	0.140801	0.0704003	+	0.997519i	$0.477572\pi$
$350$	0	0
$351$	1404.00	0.213504
$352$	0	0
$353$	−11480.0	−1.73093	−0.865466	−	0.500968i	$-0.832977\pi$
$353$	−11480.0	−1.73093	−0.865466	+	0.500968i	$0.832977\pi$
$354$	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	$-0.413441\pi$
$359$	3654.00	0.537189	0.268594	+	0.963253i	$0.413441\pi$
$360$	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	$-0.475472\pi$
$379$	1136.00	0.153964	0.0769821	+	0.997032i	$0.475472\pi$
$380$	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	$-0.303936\pi$
$397$	9140.00	1.15547	0.577737	+	0.816223i	$0.303936\pi$
$398$	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	$-0.293355\pi$
$409$	10001.0	1.20909	0.604545	+	0.796571i	$0.293355\pi$
$410$	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	$-0.257413\pi$
$431$	12356.0	1.38090	0.690450	+	0.723380i	$0.257413\pi$
$432$	0	0
$433$	−862.000	−0.0956699	−0.0478350	−	0.998855i	$-0.515232\pi$
$433$	−862.000	−0.0956699	−0.0478350	+	0.998855i	$0.515232\pi$
$434$	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	$-0.341028\pi$
$443$	8931.00	0.957843	0.478922	+	0.877858i	$0.341028\pi$
$444$	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	$-0.885720\pi$
$461$	−18534.0	−1.87248	−0.936241	+	0.351358i	$0.885720\pi$
$462$	0	0
$463$	17096.0	1.71602	0.858011	−	0.513631i	$-0.171700\pi$
$463$	17096.0	1.71602	0.858011	+	0.513631i	$0.171700\pi$
$464$	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	$-0.851029\pi$
$487$	−19183.0	−1.78494	−0.892469	+	0.451109i	$0.851029\pi$
$488$	0	0
$489$	−4608.00	−0.426137
$490$	0	0
$491$	−21693.0	−1.99387	−0.996936	−	0.0782185i	$-0.975077\pi$
$491$	−21693.0	−1.99387	−0.996936	+	0.0782185i	$0.975077\pi$
$492$	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	$-0.156348\pi$
$499$	19658.0	1.76355	0.881776	+	0.471667i	$0.156348\pi$
$500$	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	$-0.542942\pi$
$509$	−3089.00	−0.268993	−0.134497	+	0.990914i	$0.542942\pi$
$510$	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	$-0.526127\pi$
$521$	−1950.00	−0.163975	−0.0819876	+	0.996633i	$0.526127\pi$
$522$	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	$-0.217859\pi$
$547$	19824.0	1.54957	0.774783	+	0.632227i	$0.217859\pi$
$548$	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	$-0.199027\pi$
$571$	22126.0	1.62162	0.810809	+	0.585310i	$0.199027\pi$
$572$	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	$-0.487400\pi$
$577$	1097.00	0.0791485	0.0395743	+	0.999217i	$0.487400\pi$
$578$	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	$-0.428601\pi$
$593$	6424.00	0.444860	0.222430	+	0.974949i	$0.428601\pi$
$594$	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	$-0.302517\pi$
$599$	17046.0	1.16274	0.581370	+	0.813640i	$0.302517\pi$
$600$	0	0
$601$	18877.0	1.28121	0.640606	−	0.767869i	$-0.278683\pi$
$601$	18877.0	1.28121	0.640606	+	0.767869i	$0.278683\pi$
$602$	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	$-0.471677\pi$
$631$	2817.00	0.177723	0.0888613	+	0.996044i	$0.471677\pi$
$632$	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	$-0.815291\pi$
$659$	−28296.0	−1.67262	−0.836309	+	0.548258i	$0.815291\pi$
$660$	0	0
$661$	−21254.0	−1.25066	−0.625329	−	0.780361i	$-0.715035\pi$
$661$	−21254.0	−1.25066	−0.625329	+	0.780361i	$0.715035\pi$
$662$	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	$-0.857933\pi$
$677$	−31779.0	−1.80409	−0.902043	+	0.431646i	$0.857933\pi$
$678$	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	$-0.588026\pi$
$683$	−9747.00	−0.546059	−0.273030	+	0.962006i	$0.588026\pi$
$684$	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	$-0.511278\pi$
$733$	−1406.00	−0.0708483	−0.0354241	+	0.999372i	$0.511278\pi$
$734$	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	$-0.446430\pi$
$739$	6730.00	0.335003	0.167501	+	0.985872i	$0.446430\pi$
$740$	0	0
$741$	−3120.00	−0.154678
$742$	0	0
$743$	−1166.00	−0.0575725	−0.0287863	−	0.999586i	$-0.509164\pi$
$743$	−1166.00	−0.0575725	−0.0287863	+	0.999586i	$0.509164\pi$
$744$	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	$-0.635242\pi$
$751$	−16967.0	−0.824414	−0.412207	+	0.911090i	$0.635242\pi$
$752$	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	$-0.725560\pi$
$761$	−27324.0	−1.30157	−0.650785	+	0.759262i	$0.725560\pi$
$762$	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	$-0.409440\pi$
$769$	11971.0	0.561359	0.280680	+	0.959802i	$0.409440\pi$
$770$	0	0
$771$	−10722.0	−0.500834
$772$	0	0
$773$	338.000	0.0157271	0.00786353	−	0.999969i	$-0.497497\pi$
$773$	338.000	0.0157271	0.00786353	+	0.999969i	$0.497497\pi$
$774$	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	$-0.511058\pi$
$797$	−1563.00	−0.0694659	−0.0347329	+	0.999397i	$0.511058\pi$
$798$	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	$-0.715171\pi$
$823$	−29544.0	−1.25132	−0.625662	+	0.780095i	$0.715171\pi$
$824$	0	0
$825$	−1596.00	−0.0673522
$826$	0	0
$827$	24163.0	1.01600	0.507999	−	0.861358i	$-0.330385\pi$
$827$	24163.0	1.01600	0.507999	+	0.861358i	$0.330385\pi$
$828$	0	0
$829$	39196.0	1.64214	0.821070	−	0.570828i	$-0.193378\pi$
$829$	39196.0	1.64214	0.821070	+	0.570828i	$0.193378\pi$
$830$	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	$-0.506683\pi$
$853$	−1046.00	−0.0419864	−0.0209932	+	0.999780i	$0.506683\pi$
$854$	0	0
$855$	1260.00	0.0503989
$856$	0	0
$857$	−32910.0	−1.31177	−0.655883	−	0.754862i	$-0.727704\pi$
$857$	−32910.0	−1.31177	−0.655883	+	0.754862i	$0.727704\pi$
$858$	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	$-0.546939\pi$
$863$	−7450.00	−0.293860	−0.146930	+	0.989147i	$0.546939\pi$
$864$	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	$-0.282980\pi$
$881$	32958.0	1.26037	0.630183	+	0.776446i	$0.282980\pi$
$882$	0	0
$883$	−19784.0	−0.754003	−0.377001	−	0.926213i	$-0.623045\pi$
$883$	−19784.0	−0.754003	−0.377001	+	0.926213i	$0.623045\pi$
$884$	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	$-0.811744\pi$
$907$	−45352.0	−1.66030	−0.830148	+	0.557543i	$0.811744\pi$
$908$	0	0
$909$	−11430.0	−0.417062
$910$	0	0
$911$	−39906.0	−1.45131	−0.725656	−	0.688058i	$-0.758463\pi$
$911$	−39906.0	−1.45131	−0.725656	+	0.688058i	$0.758463\pi$
$912$	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	$-0.398137\pi$
$919$	17528.0	0.629157	0.314579	+	0.949231i	$0.398137\pi$
$920$	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	$-0.597439\pi$
$929$	−17066.0	−0.602710	−0.301355	+	0.953512i	$0.597439\pi$
$930$	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	$-0.319449\pi$
$937$	30821.0	1.07458	0.537288	+	0.843399i	$0.319449\pi$
$938$	0	0
$939$	−10011.0	−0.347920
$940$	0	0
$941$	−4179.00	−0.144773	−0.0723866	−	0.997377i	$-0.523062\pi$
$941$	−4179.00	−0.144773	−0.0723866	+	0.997377i	$0.523062\pi$
$942$	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	$-0.446357\pi$
$947$	9776.00	0.335457	0.167728	+	0.985833i	$0.446357\pi$
$948$	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	$-0.453475\pi$
$967$	8759.00	0.291283	0.145641	+	0.989337i	$0.453475\pi$
$968$	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	$-0.754930\pi$
$991$	−44797.0	−1.43595	−0.717974	+	0.696070i	$0.754930\pi$
$992$	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	$-0.554123\pi$
$997$	−10654.0	−0.338431	−0.169215	+	0.985579i	$0.554123\pi$
$998$	0	0
$999$	9504.00	0.300994

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.4.a.i.1.1		1
4.3	odd	2		2352.4.a.c.1.1		1
7.3	odd	6		168.4.q.a.121.1	yes	2
7.5	odd	6		168.4.q.a.25.1	✓	2
7.6	odd	2		1176.4.a.f.1.1		1
21.5	even	6		504.4.s.e.361.1		2
21.17	even	6		504.4.s.e.289.1		2
28.3	even	6		336.4.q.a.289.1		2
28.19	even	6		336.4.q.a.193.1		2
28.27	even	2		2352.4.a.bg.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.q.a.25.1	✓	2	7.5	odd	6
168.4.q.a.121.1	yes	2	7.3	odd	6
336.4.q.a.193.1		2	28.19	even	6
336.4.q.a.289.1		2	28.3	even	6
504.4.s.e.289.1		2	21.17	even	6
504.4.s.e.361.1		2	21.5	even	6
1176.4.a.f.1.1		1	7.6	odd	2
1176.4.a.i.1.1		1	1.1	even	1	trivial
2352.4.a.c.1.1		1	4.3	odd	2
2352.4.a.bg.1.1		1	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1176.a (trivial)

Introduction

L-functions

Modular forms

Varieties

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Representations

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