LMFDB - Embedded newform 2352.4.a.i.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 21)
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)

$f(q)$

$=$

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

$q-3.00000 q^{3} -3.00000 q^{5} +9.00000 q^{9} +15.0000 q^{11} -64.0000 q^{13} +9.00000 q^{15} +84.0000 q^{17} +16.0000 q^{19} +84.0000 q^{23} -116.000 q^{25} -27.0000 q^{27} -297.000 q^{29} +253.000 q^{31} -45.0000 q^{33} -316.000 q^{37} +192.000 q^{39} +360.000 q^{41} -26.0000 q^{43} -27.0000 q^{45} +30.0000 q^{47} -252.000 q^{51} +363.000 q^{53} -45.0000 q^{55} -48.0000 q^{57} +15.0000 q^{59} -118.000 q^{61} +192.000 q^{65} +370.000 q^{67} -252.000 q^{69} +342.000 q^{71} +362.000 q^{73} +348.000 q^{75} -467.000 q^{79} +81.0000 q^{81} -477.000 q^{83} -252.000 q^{85} +891.000 q^{87} +906.000 q^{89} -759.000 q^{93} -48.0000 q^{95} +503.000 q^{97} +135.000 q^{99} +O(q^{100})$

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
$5$	−3.00000	−0.268328	−0.134164	−	0.990959i	$-0.542835\pi$
$5$	−3.00000	−0.268328	−0.134164	+	0.990959i	$0.542835\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	15.0000	0.411152	0.205576	−	0.978641i	$-0.434093\pi$
$11$	15.0000	0.411152	0.205576	+	0.978641i	$0.434093\pi$
$12$	0	0
$13$	−64.0000	−1.36542	−0.682708	−	0.730691i	$-0.739198\pi$
$13$	−64.0000	−1.36542	−0.682708	+	0.730691i	$0.739198\pi$
$14$	0	0
$15$	9.00000	0.154919
$16$	0	0
$17$	84.0000	1.19841	0.599206	−	0.800595i	$-0.295483\pi$
$17$	84.0000	1.19841	0.599206	+	0.800595i	$0.295483\pi$
$18$	0	0
$19$	16.0000	0.193192	0.0965961	−	0.995324i	$-0.469204\pi$
$19$	16.0000	0.193192	0.0965961	+	0.995324i	$0.469204\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	84.0000	0.761531	0.380765	−	0.924672i	$-0.375661\pi$
$23$	84.0000	0.761531	0.380765	+	0.924672i	$0.375661\pi$
$24$	0	0
$25$	−116.000	−0.928000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−297.000	−1.90178	−0.950888	−	0.309535i	$-0.899827\pi$
$29$	−297.000	−1.90178	−0.950888	+	0.309535i	$0.899827\pi$
$30$	0	0
$31$	253.000	1.46581	0.732906	−	0.680330i	$-0.238164\pi$
$31$	253.000	1.46581	0.732906	+	0.680330i	$0.238164\pi$
$32$	0	0
$33$	−45.0000	−0.237379
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−316.000	−1.40406	−0.702028	−	0.712149i	$-0.747722\pi$
$37$	−316.000	−1.40406	−0.702028	+	0.712149i	$0.747722\pi$
$38$	0	0
$39$	192.000	0.788323
$40$	0	0
$41$	360.000	1.37128	0.685641	−	0.727940i	$-0.259522\pi$
$41$	360.000	1.37128	0.685641	+	0.727940i	$0.259522\pi$
$42$	0	0
$43$	−26.0000	−0.0922084	−0.0461042	−	0.998937i	$-0.514681\pi$
$43$	−26.0000	−0.0922084	−0.0461042	+	0.998937i	$0.514681\pi$
$44$	0	0
$45$	−27.0000	−0.0894427
$46$	0	0
$47$	30.0000	0.0931053	0.0465527	−	0.998916i	$-0.485176\pi$
$47$	30.0000	0.0931053	0.0465527	+	0.998916i	$0.485176\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−252.000	−0.691903
$52$	0	0
$53$	363.000	0.940790	0.470395	−	0.882456i	$-0.344111\pi$
$53$	363.000	0.940790	0.470395	+	0.882456i	$0.344111\pi$
$54$	0	0
$55$	−45.0000	−0.110324
$56$	0	0
$57$	−48.0000	−0.111540
$58$	0	0
$59$	15.0000	0.0330989	0.0165494	−	0.999863i	$-0.494732\pi$
$59$	15.0000	0.0330989	0.0165494	+	0.999863i	$0.494732\pi$
$60$	0	0
$61$	−118.000	−0.247678	−0.123839	−	0.992302i	$-0.539521\pi$
$61$	−118.000	−0.247678	−0.123839	+	0.992302i	$0.539521\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	192.000	0.366380
$66$	0	0
$67$	370.000	0.674667	0.337334	−	0.941385i	$-0.390475\pi$
$67$	370.000	0.674667	0.337334	+	0.941385i	$0.390475\pi$
$68$	0	0
$69$	−252.000	−0.439670
$70$	0	0
$71$	342.000	0.571661	0.285831	−	0.958280i	$-0.407731\pi$
$71$	342.000	0.571661	0.285831	+	0.958280i	$0.407731\pi$
$72$	0	0
$73$	362.000	0.580396	0.290198	−	0.956967i	$-0.406279\pi$
$73$	362.000	0.580396	0.290198	+	0.956967i	$0.406279\pi$
$74$	0	0
$75$	348.000	0.535781
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−467.000	−0.665084	−0.332542	−	0.943089i	$-0.607906\pi$
$79$	−467.000	−0.665084	−0.332542	+	0.943089i	$0.607906\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−477.000	−0.630814	−0.315407	−	0.948957i	$-0.602141\pi$
$83$	−477.000	−0.630814	−0.315407	+	0.948957i	$0.602141\pi$
$84$	0	0
$85$	−252.000	−0.321568
$86$	0	0
$87$	891.000	1.09799
$88$	0	0
$89$	906.000	1.07905	0.539527	−	0.841968i	$-0.318603\pi$
$89$	906.000	1.07905	0.539527	+	0.841968i	$0.318603\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−759.000	−0.846286
$94$	0	0
$95$	−48.0000	−0.0518389
$96$	0	0
$97$	503.000	0.526515	0.263257	−	0.964726i	$-0.415203\pi$
$97$	503.000	0.526515	0.263257	+	0.964726i	$0.415203\pi$
$98$	0	0
$99$	135.000	0.137051
$100$	0	0
$101$	−1086.00	−1.06991	−0.534956	−	0.844880i	$-0.679672\pi$
$101$	−1086.00	−1.06991	−0.534956	+	0.844880i	$0.679672\pi$
$102$	0	0
$103$	−1736.00	−1.66071	−0.830355	−	0.557235i	$-0.811862\pi$
$103$	−1736.00	−1.66071	−0.830355	+	0.557235i	$0.811862\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1353.00	1.22242	0.611212	−	0.791467i	$-0.290682\pi$
$107$	1353.00	1.22242	0.611212	+	0.791467i	$0.290682\pi$
$108$	0	0
$109$	−370.000	−0.325134	−0.162567	−	0.986698i	$-0.551977\pi$
$109$	−370.000	−0.325134	−0.162567	+	0.986698i	$0.551977\pi$
$110$	0	0
$111$	948.000	0.810632
$112$	0	0
$113$	−648.000	−0.539458	−0.269729	−	0.962936i	$-0.586934\pi$
$113$	−648.000	−0.539458	−0.269729	+	0.962936i	$0.586934\pi$
$114$	0	0
$115$	−252.000	−0.204340
$116$	0	0
$117$	−576.000	−0.455139
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1106.00	−0.830954
$122$	0	0
$123$	−1080.00	−0.791710
$124$	0	0
$125$	723.000	0.517337
$126$	0	0
$127$	−377.000	−0.263412	−0.131706	−	0.991289i	$-0.542046\pi$
$127$	−377.000	−0.263412	−0.131706	+	0.991289i	$0.542046\pi$
$128$	0	0
$129$	78.0000	0.0532366
$130$	0	0
$131$	651.000	0.434184	0.217092	−	0.976151i	$-0.430343\pi$
$131$	651.000	0.434184	0.217092	+	0.976151i	$0.430343\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	81.0000	0.0516398
$136$	0	0
$137$	−1770.00	−1.10381	−0.551903	−	0.833909i	$-0.686098\pi$
$137$	−1770.00	−1.10381	−0.551903	+	0.833909i	$0.686098\pi$
$138$	0	0
$139$	1558.00	0.950704	0.475352	−	0.879796i	$-0.342321\pi$
$139$	1558.00	0.950704	0.475352	+	0.879796i	$0.342321\pi$
$140$	0	0
$141$	−90.0000	−0.0537544
$142$	0	0
$143$	−960.000	−0.561393
$144$	0	0
$145$	891.000	0.510300
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2454.00	1.34926	0.674629	−	0.738157i	$-0.264304\pi$
$149$	2454.00	1.34926	0.674629	+	0.738157i	$0.264304\pi$
$150$	0	0
$151$	−1259.00	−0.678516	−0.339258	−	0.940693i	$-0.610176\pi$
$151$	−1259.00	−0.678516	−0.339258	+	0.940693i	$0.610176\pi$
$152$	0	0
$153$	756.000	0.399470
$154$	0	0
$155$	−759.000	−0.393318
$156$	0	0
$157$	−196.000	−0.0996338	−0.0498169	−	0.998758i	$-0.515864\pi$
$157$	−196.000	−0.0996338	−0.0498169	+	0.998758i	$0.515864\pi$
$158$	0	0
$159$	−1089.00	−0.543166
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1252.00	0.601621	0.300810	−	0.953684i	$-0.402743\pi$
$163$	1252.00	0.601621	0.300810	+	0.953684i	$0.402743\pi$
$164$	0	0
$165$	135.000	0.0636954
$166$	0	0
$167$	2646.00	1.22607	0.613035	−	0.790056i	$-0.289949\pi$
$167$	2646.00	1.22607	0.613035	+	0.790056i	$0.289949\pi$
$168$	0	0
$169$	1899.00	0.864360
$170$	0	0
$171$	144.000	0.0643974
$172$	0	0
$173$	−786.000	−0.345425	−0.172712	−	0.984972i	$-0.555253\pi$
$173$	−786.000	−0.345425	−0.172712	+	0.984972i	$0.555253\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−45.0000	−0.0191096
$178$	0	0
$179$	−2892.00	−1.20759	−0.603794	−	0.797140i	$-0.706345\pi$
$179$	−2892.00	−1.20759	−0.603794	+	0.797140i	$0.706345\pi$
$180$	0	0
$181$	1352.00	0.555212	0.277606	−	0.960695i	$-0.410459\pi$
$181$	1352.00	0.555212	0.277606	+	0.960695i	$0.410459\pi$
$182$	0	0
$183$	354.000	0.142997
$184$	0	0
$185$	948.000	0.376748
$186$	0	0
$187$	1260.00	0.492729
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3912.00	−1.48200	−0.741001	−	0.671504i	$-0.765649\pi$
$191$	−3912.00	−1.48200	−0.741001	+	0.671504i	$0.765649\pi$
$192$	0	0
$193$	1493.00	0.556832	0.278416	−	0.960461i	$-0.410191\pi$
$193$	1493.00	0.556832	0.278416	+	0.960461i	$0.410191\pi$
$194$	0	0
$195$	−576.000	−0.211529
$196$	0	0
$197$	−4086.00	−1.47774	−0.738872	−	0.673846i	$-0.764641\pi$
$197$	−4086.00	−1.47774	−0.738872	+	0.673846i	$0.764641\pi$
$198$	0	0
$199$	3556.00	1.26672	0.633362	−	0.773855i	$-0.281674\pi$
$199$	3556.00	1.26672	0.633362	+	0.773855i	$0.281674\pi$
$200$	0	0
$201$	−1110.00	−0.389519
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1080.00	−0.367954
$206$	0	0
$207$	756.000	0.253844
$208$	0	0
$209$	240.000	0.0794313
$210$	0	0
$211$	−1250.00	−0.407837	−0.203918	−	0.978988i	$-0.565368\pi$
$211$	−1250.00	−0.407837	−0.203918	+	0.978988i	$0.565368\pi$
$212$	0	0
$213$	−1026.00	−0.330049
$214$	0	0
$215$	78.0000	0.0247421
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1086.00	−0.335092
$220$	0	0
$221$	−5376.00	−1.63633
$222$	0	0
$223$	−425.000	−0.127624	−0.0638119	−	0.997962i	$-0.520326\pi$
$223$	−425.000	−0.127624	−0.0638119	+	0.997962i	$0.520326\pi$
$224$	0	0
$225$	−1044.00	−0.309333
$226$	0	0
$227$	−3855.00	−1.12716	−0.563580	−	0.826061i	$-0.690576\pi$
$227$	−3855.00	−1.12716	−0.563580	+	0.826061i	$0.690576\pi$
$228$	0	0
$229$	−2188.00	−0.631385	−0.315692	−	0.948862i	$-0.602237\pi$
$229$	−2188.00	−0.631385	−0.315692	+	0.948862i	$0.602237\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	852.000	0.239555	0.119778	−	0.992801i	$-0.461782\pi$
$233$	852.000	0.239555	0.119778	+	0.992801i	$0.461782\pi$
$234$	0	0
$235$	−90.0000	−0.0249828
$236$	0	0
$237$	1401.00	0.383986
$238$	0	0
$239$	−5508.00	−1.49072	−0.745362	−	0.666660i	$-0.767723\pi$
$239$	−5508.00	−1.49072	−0.745362	+	0.666660i	$0.767723\pi$
$240$	0	0
$241$	791.000	0.211422	0.105711	−	0.994397i	$-0.466288\pi$
$241$	791.000	0.211422	0.105711	+	0.994397i	$0.466288\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1024.00	−0.263788
$248$	0	0
$249$	1431.00	0.364201
$250$	0	0
$251$	−5265.00	−1.32400	−0.662000	−	0.749504i	$-0.730292\pi$
$251$	−5265.00	−1.32400	−0.662000	+	0.749504i	$0.730292\pi$
$252$	0	0
$253$	1260.00	0.313105
$254$	0	0
$255$	756.000	0.185657
$256$	0	0
$257$	−6870.00	−1.66747	−0.833733	−	0.552168i	$-0.813801\pi$
$257$	−6870.00	−1.66747	−0.833733	+	0.552168i	$0.813801\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2673.00	−0.633925
$262$	0	0
$263$	222.000	0.0520498	0.0260249	−	0.999661i	$-0.491715\pi$
$263$	222.000	0.0520498	0.0260249	+	0.999661i	$0.491715\pi$
$264$	0	0
$265$	−1089.00	−0.252441
$266$	0	0
$267$	−2718.00	−0.622992
$268$	0	0
$269$	7851.00	1.77949	0.889747	−	0.456454i	$-0.150881\pi$
$269$	7851.00	1.77949	0.889747	+	0.456454i	$0.150881\pi$
$270$	0	0
$271$	−5183.00	−1.16179	−0.580895	−	0.813979i	$-0.697297\pi$
$271$	−5183.00	−1.16179	−0.580895	+	0.813979i	$0.697297\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1740.00	−0.381549
$276$	0	0
$277$	−4960.00	−1.07588	−0.537938	−	0.842985i	$-0.680796\pi$
$277$	−4960.00	−1.07588	−0.537938	+	0.842985i	$0.680796\pi$
$278$	0	0
$279$	2277.00	0.488604
$280$	0	0
$281$	−774.000	−0.164317	−0.0821583	−	0.996619i	$-0.526181\pi$
$281$	−774.000	−0.164317	−0.0821583	+	0.996619i	$0.526181\pi$
$282$	0	0
$283$	−3698.00	−0.776761	−0.388380	−	0.921499i	$-0.626965\pi$
$283$	−3698.00	−0.776761	−0.388380	+	0.921499i	$0.626965\pi$
$284$	0	0
$285$	144.000	0.0299292
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2143.00	0.436190
$290$	0	0
$291$	−1509.00	−0.303983
$292$	0	0
$293$	−6273.00	−1.25076	−0.625380	−	0.780321i	$-0.715056\pi$
$293$	−6273.00	−1.25076	−0.625380	+	0.780321i	$0.715056\pi$
$294$	0	0
$295$	−45.0000	−0.00888136
$296$	0	0
$297$	−405.000	−0.0791262
$298$	0	0
$299$	−5376.00	−1.03981
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3258.00	0.617714
$304$	0	0
$305$	354.000	0.0664590
$306$	0	0
$307$	1684.00	0.313065	0.156533	−	0.987673i	$-0.449968\pi$
$307$	1684.00	0.313065	0.156533	+	0.987673i	$0.449968\pi$
$308$	0	0
$309$	5208.00	0.958812
$310$	0	0
$311$	1320.00	0.240676	0.120338	−	0.992733i	$-0.461602\pi$
$311$	1320.00	0.240676	0.120338	+	0.992733i	$0.461602\pi$
$312$	0	0
$313$	−8503.00	−1.53552	−0.767760	−	0.640737i	$-0.778629\pi$
$313$	−8503.00	−1.53552	−0.767760	+	0.640737i	$0.778629\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2577.00	−0.456589	−0.228295	−	0.973592i	$-0.573315\pi$
$317$	−2577.00	−0.456589	−0.228295	+	0.973592i	$0.573315\pi$
$318$	0	0
$319$	−4455.00	−0.781919
$320$	0	0
$321$	−4059.00	−0.705767
$322$	0	0
$323$	1344.00	0.231524
$324$	0	0
$325$	7424.00	1.26711
$326$	0	0
$327$	1110.00	0.187716
$328$	0	0
$329$	0	0
$330$	0	0
$331$	484.000	0.0803717	0.0401859	−	0.999192i	$-0.487205\pi$
$331$	484.000	0.0803717	0.0401859	+	0.999192i	$0.487205\pi$
$332$	0	0
$333$	−2844.00	−0.468019
$334$	0	0
$335$	−1110.00	−0.181032
$336$	0	0
$337$	−8359.00	−1.35117	−0.675584	−	0.737283i	$-0.736109\pi$
$337$	−8359.00	−1.35117	−0.675584	+	0.737283i	$0.736109\pi$
$338$	0	0
$339$	1944.00	0.311456
$340$	0	0
$341$	3795.00	0.602671
$342$	0	0
$343$	0	0
$344$	0	0
$345$	756.000	0.117976
$346$	0	0
$347$	1860.00	0.287752	0.143876	−	0.989596i	$-0.454043\pi$
$347$	1860.00	0.287752	0.143876	+	0.989596i	$0.454043\pi$
$348$	0	0
$349$	−1918.00	−0.294178	−0.147089	−	0.989123i	$-0.546990\pi$
$349$	−1918.00	−0.294178	−0.147089	+	0.989123i	$0.546990\pi$
$350$	0	0
$351$	1728.00	0.262774
$352$	0	0
$353$	−3048.00	−0.459571	−0.229786	−	0.973241i	$-0.573803\pi$
$353$	−3048.00	−0.459571	−0.229786	+	0.973241i	$0.573803\pi$
$354$	0	0
$355$	−1026.00	−0.153393
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.0000	0.00441042	0.00220521	−	0.999998i	$-0.499298\pi$
$359$	30.0000	0.00441042	0.00220521	+	0.999998i	$0.499298\pi$
$360$	0	0
$361$	−6603.00	−0.962677
$362$	0	0
$363$	3318.00	0.479752
$364$	0	0
$365$	−1086.00	−0.155737
$366$	0	0
$367$	11311.0	1.60880	0.804400	−	0.594088i	$-0.202487\pi$
$367$	11311.0	1.60880	0.804400	+	0.594088i	$0.202487\pi$
$368$	0	0
$369$	3240.00	0.457094
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1208.00	0.167689	0.0838443	−	0.996479i	$-0.473280\pi$
$373$	1208.00	0.167689	0.0838443	+	0.996479i	$0.473280\pi$
$374$	0	0
$375$	−2169.00	−0.298684
$376$	0	0
$377$	19008.0	2.59672
$378$	0	0
$379$	−7640.00	−1.03546	−0.517731	−	0.855543i	$-0.673223\pi$
$379$	−7640.00	−1.03546	−0.517731	+	0.855543i	$0.673223\pi$
$380$	0	0
$381$	1131.00	0.152081
$382$	0	0
$383$	−12750.0	−1.70103	−0.850515	−	0.525951i	$-0.823710\pi$
$383$	−12750.0	−1.70103	−0.850515	+	0.525951i	$0.823710\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−234.000	−0.0307361
$388$	0	0
$389$	3126.00	0.407441	0.203720	−	0.979029i	$-0.434697\pi$
$389$	3126.00	0.407441	0.203720	+	0.979029i	$0.434697\pi$
$390$	0	0
$391$	7056.00	0.912627
$392$	0	0
$393$	−1953.00	−0.250676
$394$	0	0
$395$	1401.00	0.178461
$396$	0	0
$397$	−5932.00	−0.749921	−0.374960	−	0.927041i	$-0.622344\pi$
$397$	−5932.00	−0.749921	−0.374960	+	0.927041i	$0.622344\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1608.00	0.200249	0.100124	−	0.994975i	$-0.468076\pi$
$401$	1608.00	0.200249	0.100124	+	0.994975i	$0.468076\pi$
$402$	0	0
$403$	−16192.0	−2.00144
$404$	0	0
$405$	−243.000	−0.0298142
$406$	0	0
$407$	−4740.00	−0.577280
$408$	0	0
$409$	−4465.00	−0.539805	−0.269902	−	0.962888i	$-0.586991\pi$
$409$	−4465.00	−0.539805	−0.269902	+	0.962888i	$0.586991\pi$
$410$	0	0
$411$	5310.00	0.637282
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1431.00	0.169265
$416$	0	0
$417$	−4674.00	−0.548889
$418$	0	0
$419$	1584.00	0.184686	0.0923430	−	0.995727i	$-0.470564\pi$
$419$	1584.00	0.184686	0.0923430	+	0.995727i	$0.470564\pi$
$420$	0	0
$421$	−1330.00	−0.153967	−0.0769837	−	0.997032i	$-0.524529\pi$
$421$	−1330.00	−0.153967	−0.0769837	+	0.997032i	$0.524529\pi$
$422$	0	0
$423$	270.000	0.0310351
$424$	0	0
$425$	−9744.00	−1.11213
$426$	0	0
$427$	0	0
$428$	0	0
$429$	2880.00	0.324121
$430$	0	0
$431$	−9588.00	−1.07155	−0.535775	−	0.844361i	$-0.679980\pi$
$431$	−9588.00	−1.07155	−0.535775	+	0.844361i	$0.679980\pi$
$432$	0	0
$433$	494.000	0.0548271	0.0274135	−	0.999624i	$-0.491273\pi$
$433$	494.000	0.0548271	0.0274135	+	0.999624i	$0.491273\pi$
$434$	0	0
$435$	−2673.00	−0.294622
$436$	0	0
$437$	1344.00	0.147122
$438$	0	0
$439$	16009.0	1.74047	0.870237	−	0.492634i	$-0.163966\pi$
$439$	16009.0	1.74047	0.870237	+	0.492634i	$0.163966\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7773.00	−0.833649	−0.416824	−	0.908987i	$-0.636857\pi$
$443$	−7773.00	−0.833649	−0.416824	+	0.908987i	$0.636857\pi$
$444$	0	0
$445$	−2718.00	−0.289541
$446$	0	0
$447$	−7362.00	−0.778995
$448$	0	0
$449$	864.000	0.0908122	0.0454061	−	0.998969i	$-0.485542\pi$
$449$	864.000	0.0908122	0.0454061	+	0.998969i	$0.485542\pi$
$450$	0	0
$451$	5400.00	0.563805
$452$	0	0
$453$	3777.00	0.391742
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2519.00	0.257842	0.128921	−	0.991655i	$-0.458849\pi$
$457$	2519.00	0.257842	0.128921	+	0.991655i	$0.458849\pi$
$458$	0	0
$459$	−2268.00	−0.230634
$460$	0	0
$461$	−342.000	−0.0345521	−0.0172761	−	0.999851i	$-0.505499\pi$
$461$	−342.000	−0.0345521	−0.0172761	+	0.999851i	$0.505499\pi$
$462$	0	0
$463$	4336.00	0.435229	0.217614	−	0.976035i	$-0.430172\pi$
$463$	4336.00	0.435229	0.217614	+	0.976035i	$0.430172\pi$
$464$	0	0
$465$	2277.00	0.227082
$466$	0	0
$467$	−18636.0	−1.84662	−0.923310	−	0.384056i	$-0.874527\pi$
$467$	−18636.0	−1.84662	−0.923310	+	0.384056i	$0.874527\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	588.000	0.0575236
$472$	0	0
$473$	−390.000	−0.0379117
$474$	0	0
$475$	−1856.00	−0.179282
$476$	0	0
$477$	3267.00	0.313597
$478$	0	0
$479$	−15078.0	−1.43827	−0.719135	−	0.694870i	$-0.755462\pi$
$479$	−15078.0	−1.43827	−0.719135	+	0.694870i	$0.755462\pi$
$480$	0	0
$481$	20224.0	1.91712
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1509.00	−0.141279
$486$	0	0
$487$	−6221.00	−0.578851	−0.289425	−	0.957201i	$-0.593464\pi$
$487$	−6221.00	−0.578851	−0.289425	+	0.957201i	$0.593464\pi$
$488$	0	0
$489$	−3756.00	−0.347346
$490$	0	0
$491$	7371.00	0.677492	0.338746	−	0.940878i	$-0.389997\pi$
$491$	7371.00	0.677492	0.338746	+	0.940878i	$0.389997\pi$
$492$	0	0
$493$	−24948.0	−2.27911
$494$	0	0
$495$	−405.000	−0.0367745
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−4274.00	−0.383428	−0.191714	−	0.981451i	$-0.561405\pi$
$499$	−4274.00	−0.383428	−0.191714	+	0.981451i	$0.561405\pi$
$500$	0	0
$501$	−7938.00	−0.707872
$502$	0	0
$503$	2520.00	0.223382	0.111691	−	0.993743i	$-0.464373\pi$
$503$	2520.00	0.223382	0.111691	+	0.993743i	$0.464373\pi$
$504$	0	0
$505$	3258.00	0.287087
$506$	0	0
$507$	−5697.00	−0.499039
$508$	0	0
$509$	−14277.0	−1.24326	−0.621628	−	0.783313i	$-0.713528\pi$
$509$	−14277.0	−1.24326	−0.621628	+	0.783313i	$0.713528\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−432.000	−0.0371799
$514$	0	0
$515$	5208.00	0.445615
$516$	0	0
$517$	450.000	0.0382804
$518$	0	0
$519$	2358.00	0.199431
$520$	0	0
$521$	−6306.00	−0.530270	−0.265135	−	0.964211i	$-0.585417\pi$
$521$	−6306.00	−0.530270	−0.265135	+	0.964211i	$0.585417\pi$
$522$	0	0
$523$	−8072.00	−0.674883	−0.337442	−	0.941346i	$-0.609562\pi$
$523$	−8072.00	−0.674883	−0.337442	+	0.941346i	$0.609562\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	21252.0	1.75664
$528$	0	0
$529$	−5111.00	−0.420071
$530$	0	0
$531$	135.000	0.0110330
$532$	0	0
$533$	−23040.0	−1.87237
$534$	0	0
$535$	−4059.00	−0.328011
$536$	0	0
$537$	8676.00	0.697201
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−22858.0	−1.81653	−0.908264	−	0.418396i	$-0.862592\pi$
$541$	−22858.0	−1.81653	−0.908264	+	0.418396i	$0.862592\pi$
$542$	0	0
$543$	−4056.00	−0.320552
$544$	0	0
$545$	1110.00	0.0872425
$546$	0	0
$547$	24724.0	1.93258	0.966291	−	0.257454i	$-0.0828835\pi$
$547$	24724.0	1.93258	0.966291	+	0.257454i	$0.0828835\pi$
$548$	0	0
$549$	−1062.00	−0.0825593
$550$	0	0
$551$	−4752.00	−0.367408
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−2844.00	−0.217515
$556$	0	0
$557$	−9843.00	−0.748764	−0.374382	−	0.927275i	$-0.622145\pi$
$557$	−9843.00	−0.748764	−0.374382	+	0.927275i	$0.622145\pi$
$558$	0	0
$559$	1664.00	0.125903
$560$	0	0
$561$	−3780.00	−0.284477
$562$	0	0
$563$	13371.0	1.00092	0.500462	−	0.865758i	$-0.333163\pi$
$563$	13371.0	1.00092	0.500462	+	0.865758i	$0.333163\pi$
$564$	0	0
$565$	1944.00	0.144752
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5232.00	−0.385478	−0.192739	−	0.981250i	$-0.561737\pi$
$569$	−5232.00	−0.385478	−0.192739	+	0.981250i	$0.561737\pi$
$570$	0	0
$571$	14398.0	1.05523	0.527616	−	0.849483i	$-0.323086\pi$
$571$	14398.0	1.05523	0.527616	+	0.849483i	$0.323086\pi$
$572$	0	0
$573$	11736.0	0.855634
$574$	0	0
$575$	−9744.00	−0.706701
$576$	0	0
$577$	19871.0	1.43369	0.716846	−	0.697231i	$-0.245585\pi$
$577$	19871.0	1.43369	0.716846	+	0.697231i	$0.245585\pi$
$578$	0	0
$579$	−4479.00	−0.321487
$580$	0	0
$581$	0	0
$582$	0	0
$583$	5445.00	0.386808
$584$	0	0
$585$	1728.00	0.122127
$586$	0	0
$587$	16137.0	1.13466	0.567330	−	0.823491i	$-0.307976\pi$
$587$	16137.0	1.13466	0.567330	+	0.823491i	$0.307976\pi$
$588$	0	0
$589$	4048.00	0.283183
$590$	0	0
$591$	12258.0	0.853176
$592$	0	0
$593$	−21324.0	−1.47668	−0.738340	−	0.674428i	$-0.764390\pi$
$593$	−21324.0	−1.47668	−0.738340	+	0.674428i	$0.764390\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−10668.0	−0.731344
$598$	0	0
$599$	8646.00	0.589760	0.294880	−	0.955534i	$-0.404720\pi$
$599$	8646.00	0.589760	0.294880	+	0.955534i	$0.404720\pi$
$600$	0	0
$601$	11195.0	0.759823	0.379911	−	0.925023i	$-0.375954\pi$
$601$	11195.0	0.759823	0.379911	+	0.925023i	$0.375954\pi$
$602$	0	0
$603$	3330.00	0.224889
$604$	0	0
$605$	3318.00	0.222968
$606$	0	0
$607$	8971.00	0.599871	0.299935	−	0.953959i	$-0.403035\pi$
$607$	8971.00	0.599871	0.299935	+	0.953959i	$0.403035\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1920.00	−0.127127
$612$	0	0
$613$	−12772.0	−0.841527	−0.420764	−	0.907170i	$-0.638238\pi$
$613$	−12772.0	−0.841527	−0.420764	+	0.907170i	$0.638238\pi$
$614$	0	0
$615$	3240.00	0.212438
$616$	0	0
$617$	12762.0	0.832705	0.416352	−	0.909203i	$-0.363308\pi$
$617$	12762.0	0.832705	0.416352	+	0.909203i	$0.363308\pi$
$618$	0	0
$619$	−12842.0	−0.833867	−0.416933	−	0.908937i	$-0.636895\pi$
$619$	−12842.0	−0.833867	−0.416933	+	0.908937i	$0.636895\pi$
$620$	0	0
$621$	−2268.00	−0.146557
$622$	0	0
$623$	0	0
$624$	0	0
$625$	12331.0	0.789184
$626$	0	0
$627$	−720.000	−0.0458597
$628$	0	0
$629$	−26544.0	−1.68264
$630$	0	0
$631$	−21365.0	−1.34790	−0.673952	−	0.738775i	$-0.735404\pi$
$631$	−21365.0	−1.34790	−0.673952	+	0.738775i	$0.735404\pi$
$632$	0	0
$633$	3750.00	0.235465
$634$	0	0
$635$	1131.00	0.0706809
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3078.00	0.190554
$640$	0	0
$641$	8274.00	0.509834	0.254917	−	0.966963i	$-0.417952\pi$
$641$	8274.00	0.509834	0.254917	+	0.966963i	$0.417952\pi$
$642$	0	0
$643$	−27998.0	−1.71716	−0.858580	−	0.512680i	$-0.828653\pi$
$643$	−27998.0	−1.71716	−0.858580	+	0.512680i	$0.828653\pi$
$644$	0	0
$645$	−234.000	−0.0142849
$646$	0	0
$647$	17466.0	1.06130	0.530649	−	0.847592i	$-0.321948\pi$
$647$	17466.0	1.06130	0.530649	+	0.847592i	$0.321948\pi$
$648$	0	0
$649$	225.000	0.0136087
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2157.00	0.129265	0.0646324	−	0.997909i	$-0.479413\pi$
$653$	2157.00	0.129265	0.0646324	+	0.997909i	$0.479413\pi$
$654$	0	0
$655$	−1953.00	−0.116504
$656$	0	0
$657$	3258.00	0.193465
$658$	0	0
$659$	−19944.0	−1.17892	−0.589460	−	0.807798i	$-0.700659\pi$
$659$	−19944.0	−1.17892	−0.589460	+	0.807798i	$0.700659\pi$
$660$	0	0
$661$	27506.0	1.61855	0.809273	−	0.587432i	$-0.199861\pi$
$661$	27506.0	1.61855	0.809273	+	0.587432i	$0.199861\pi$
$662$	0	0
$663$	16128.0	0.944735
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−24948.0	−1.44826
$668$	0	0
$669$	1275.00	0.0736836
$670$	0	0
$671$	−1770.00	−0.101833
$672$	0	0
$673$	−19123.0	−1.09530	−0.547650	−	0.836707i	$-0.684478\pi$
$673$	−19123.0	−1.09530	−0.547650	+	0.836707i	$0.684478\pi$
$674$	0	0
$675$	3132.00	0.178594
$676$	0	0
$677$	13857.0	0.786658	0.393329	−	0.919398i	$-0.371323\pi$
$677$	13857.0	0.786658	0.393329	+	0.919398i	$0.371323\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	11565.0	0.650766
$682$	0	0
$683$	22245.0	1.24624	0.623120	−	0.782127i	$-0.285865\pi$
$683$	22245.0	1.24624	0.623120	+	0.782127i	$0.285865\pi$
$684$	0	0
$685$	5310.00	0.296182
$686$	0	0
$687$	6564.00	0.364530
$688$	0	0
$689$	−23232.0	−1.28457
$690$	0	0
$691$	640.000	0.0352341	0.0176170	−	0.999845i	$-0.494392\pi$
$691$	640.000	0.0352341	0.0176170	+	0.999845i	$0.494392\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4674.00	−0.255101
$696$	0	0
$697$	30240.0	1.64336
$698$	0	0
$699$	−2556.00	−0.138307
$700$	0	0
$701$	−15561.0	−0.838418	−0.419209	−	0.907890i	$-0.637693\pi$
$701$	−15561.0	−0.838418	−0.419209	+	0.907890i	$0.637693\pi$
$702$	0	0
$703$	−5056.00	−0.271253
$704$	0	0
$705$	270.000	0.0144238
$706$	0	0
$707$	0	0
$708$	0	0
$709$	5534.00	0.293136	0.146568	−	0.989201i	$-0.453177\pi$
$709$	5534.00	0.293136	0.146568	+	0.989201i	$0.453177\pi$
$710$	0	0
$711$	−4203.00	−0.221695
$712$	0	0
$713$	21252.0	1.11626
$714$	0	0
$715$	2880.00	0.150638
$716$	0	0
$717$	16524.0	0.860670
$718$	0	0
$719$	21846.0	1.13313	0.566564	−	0.824018i	$-0.308273\pi$
$719$	21846.0	1.13313	0.566564	+	0.824018i	$0.308273\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−2373.00	−0.122065
$724$	0	0
$725$	34452.0	1.76485
$726$	0	0
$727$	11089.0	0.565706	0.282853	−	0.959163i	$-0.408719\pi$
$727$	11089.0	0.565706	0.282853	+	0.959163i	$0.408719\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−2184.00	−0.110504
$732$	0	0
$733$	11762.0	0.592687	0.296343	−	0.955081i	$-0.404233\pi$
$733$	11762.0	0.592687	0.296343	+	0.955081i	$0.404233\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5550.00	0.277391
$738$	0	0
$739$	22726.0	1.13124	0.565622	−	0.824665i	$-0.308636\pi$
$739$	22726.0	1.13124	0.565622	+	0.824665i	$0.308636\pi$
$740$	0	0
$741$	3072.00	0.152298
$742$	0	0
$743$	−6678.00	−0.329734	−0.164867	−	0.986316i	$-0.552719\pi$
$743$	−6678.00	−0.329734	−0.164867	+	0.986316i	$0.552719\pi$
$744$	0	0
$745$	−7362.00	−0.362044
$746$	0	0
$747$	−4293.00	−0.210271
$748$	0	0
$749$	0	0
$750$	0	0
$751$	19987.0	0.971153	0.485577	−	0.874194i	$-0.338610\pi$
$751$	19987.0	0.971153	0.485577	+	0.874194i	$0.338610\pi$
$752$	0	0
$753$	15795.0	0.764411
$754$	0	0
$755$	3777.00	0.182065
$756$	0	0
$757$	314.000	0.0150760	0.00753799	−	0.999972i	$-0.497601\pi$
$757$	314.000	0.0150760	0.00753799	+	0.999972i	$0.497601\pi$
$758$	0	0
$759$	−3780.00	−0.180771
$760$	0	0
$761$	−11496.0	−0.547608	−0.273804	−	0.961786i	$-0.588282\pi$
$761$	−11496.0	−0.547608	−0.273804	+	0.961786i	$0.588282\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2268.00	−0.107189
$766$	0	0
$767$	−960.000	−0.0451937
$768$	0	0
$769$	2765.00	0.129660	0.0648299	−	0.997896i	$-0.479350\pi$
$769$	2765.00	0.129660	0.0648299	+	0.997896i	$0.479350\pi$
$770$	0	0
$771$	20610.0	0.962712
$772$	0	0
$773$	−14046.0	−0.653557	−0.326778	−	0.945101i	$-0.605963\pi$
$773$	−14046.0	−0.653557	−0.326778	+	0.945101i	$0.605963\pi$
$774$	0	0
$775$	−29348.0	−1.36027
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5760.00	0.264921
$780$	0	0
$781$	5130.00	0.235039
$782$	0	0
$783$	8019.00	0.365997
$784$	0	0
$785$	588.000	0.0267345
$786$	0	0
$787$	18514.0	0.838568	0.419284	−	0.907855i	$-0.362281\pi$
$787$	18514.0	0.838568	0.419284	+	0.907855i	$0.362281\pi$
$788$	0	0
$789$	−666.000	−0.0300510
$790$	0	0
$791$	0	0
$792$	0	0
$793$	7552.00	0.338183
$794$	0	0
$795$	3267.00	0.145747
$796$	0	0
$797$	−27495.0	−1.22199	−0.610993	−	0.791636i	$-0.709230\pi$
$797$	−27495.0	−1.22199	−0.610993	+	0.791636i	$0.709230\pi$
$798$	0	0
$799$	2520.00	0.111578
$800$	0	0
$801$	8154.00	0.359685
$802$	0	0
$803$	5430.00	0.238631
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−23553.0	−1.02739
$808$	0	0
$809$	−7944.00	−0.345236	−0.172618	−	0.984989i	$-0.555223\pi$
$809$	−7944.00	−0.345236	−0.172618	+	0.984989i	$0.555223\pi$
$810$	0	0
$811$	28942.0	1.25313	0.626567	−	0.779368i	$-0.284460\pi$
$811$	28942.0	1.25313	0.626567	+	0.779368i	$0.284460\pi$
$812$	0	0
$813$	15549.0	0.670759
$814$	0	0
$815$	−3756.00	−0.161432
$816$	0	0
$817$	−416.000	−0.0178140
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8187.00	0.348025	0.174012	−	0.984743i	$-0.444327\pi$
$821$	8187.00	0.348025	0.174012	+	0.984743i	$0.444327\pi$
$822$	0	0
$823$	280.000	0.0118593	0.00592964	−	0.999982i	$-0.498113\pi$
$823$	280.000	0.0118593	0.00592964	+	0.999982i	$0.498113\pi$
$824$	0	0
$825$	5220.00	0.220287
$826$	0	0
$827$	−25317.0	−1.06452	−0.532260	−	0.846581i	$-0.678657\pi$
$827$	−25317.0	−1.06452	−0.532260	+	0.846581i	$0.678657\pi$
$828$	0	0
$829$	15320.0	0.641840	0.320920	−	0.947106i	$-0.396008\pi$
$829$	15320.0	0.641840	0.320920	+	0.947106i	$0.396008\pi$
$830$	0	0
$831$	14880.0	0.621157
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−7938.00	−0.328989
$836$	0	0
$837$	−6831.00	−0.282095
$838$	0	0
$839$	−34092.0	−1.40284	−0.701422	−	0.712746i	$-0.747451\pi$
$839$	−34092.0	−1.40284	−0.701422	+	0.712746i	$0.747451\pi$
$840$	0	0
$841$	63820.0	2.61675
$842$	0	0
$843$	2322.00	0.0948682
$844$	0	0
$845$	−5697.00	−0.231932
$846$	0	0
$847$	0	0
$848$	0	0
$849$	11094.0	0.448463
$850$	0	0
$851$	−26544.0	−1.06923
$852$	0	0
$853$	−7378.00	−0.296152	−0.148076	−	0.988976i	$-0.547308\pi$
$853$	−7378.00	−0.296152	−0.148076	+	0.988976i	$0.547308\pi$
$854$	0	0
$855$	−432.000	−0.0172796
$856$	0	0
$857$	−15594.0	−0.621565	−0.310782	−	0.950481i	$-0.600591\pi$
$857$	−15594.0	−0.621565	−0.310782	+	0.950481i	$0.600591\pi$
$858$	0	0
$859$	30538.0	1.21297	0.606486	−	0.795094i	$-0.292579\pi$
$859$	30538.0	1.21297	0.606486	+	0.795094i	$0.292579\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	822.000	0.0324232	0.0162116	−	0.999869i	$-0.494839\pi$
$863$	822.000	0.0324232	0.0162116	+	0.999869i	$0.494839\pi$
$864$	0	0
$865$	2358.00	0.0926872
$866$	0	0
$867$	−6429.00	−0.251834
$868$	0	0
$869$	−7005.00	−0.273450
$870$	0	0
$871$	−23680.0	−0.921201
$872$	0	0
$873$	4527.00	0.175505
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−41824.0	−1.61037	−0.805186	−	0.593022i	$-0.797935\pi$
$877$	−41824.0	−1.61037	−0.805186	+	0.593022i	$0.797935\pi$
$878$	0	0
$879$	18819.0	0.722126
$880$	0	0
$881$	−46098.0	−1.76286	−0.881431	−	0.472313i	$-0.843419\pi$
$881$	−46098.0	−1.76286	−0.881431	+	0.472313i	$0.843419\pi$
$882$	0	0
$883$	−21008.0	−0.800652	−0.400326	−	0.916373i	$-0.631103\pi$
$883$	−21008.0	−0.800652	−0.400326	+	0.916373i	$0.631103\pi$
$884$	0	0
$885$	135.000	0.00512766
$886$	0	0
$887$	−24036.0	−0.909865	−0.454932	−	0.890526i	$-0.650337\pi$
$887$	−24036.0	−0.909865	−0.454932	+	0.890526i	$0.650337\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1215.00	0.0456835
$892$	0	0
$893$	480.000	0.0179872
$894$	0	0
$895$	8676.00	0.324030
$896$	0	0
$897$	16128.0	0.600332
$898$	0	0
$899$	−75141.0	−2.78764
$900$	0	0
$901$	30492.0	1.12745
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4056.00	−0.148979
$906$	0	0
$907$	−13292.0	−0.486608	−0.243304	−	0.969950i	$-0.578231\pi$
$907$	−13292.0	−0.486608	−0.243304	+	0.969950i	$0.578231\pi$
$908$	0	0
$909$	−9774.00	−0.356637
$910$	0	0
$911$	9306.00	0.338443	0.169221	−	0.985578i	$-0.445875\pi$
$911$	9306.00	0.338443	0.169221	+	0.985578i	$0.445875\pi$
$912$	0	0
$913$	−7155.00	−0.259360
$914$	0	0
$915$	−1062.00	−0.0383701
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−16496.0	−0.592114	−0.296057	−	0.955170i	$-0.595672\pi$
$919$	−16496.0	−0.592114	−0.296057	+	0.955170i	$0.595672\pi$
$920$	0	0
$921$	−5052.00	−0.180748
$922$	0	0
$923$	−21888.0	−0.780555
$924$	0	0
$925$	36656.0	1.30296
$926$	0	0
$927$	−15624.0	−0.553570
$928$	0	0
$929$	14154.0	0.499868	0.249934	−	0.968263i	$-0.419591\pi$
$929$	14154.0	0.499868	0.249934	+	0.968263i	$0.419591\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−3960.00	−0.138955
$934$	0	0
$935$	−3780.00	−0.132213
$936$	0	0
$937$	−3781.00	−0.131825	−0.0659124	−	0.997825i	$-0.520996\pi$
$937$	−3781.00	−0.131825	−0.0659124	+	0.997825i	$0.520996\pi$
$938$	0	0
$939$	25509.0	0.886533
$940$	0	0
$941$	−25863.0	−0.895972	−0.447986	−	0.894041i	$-0.647859\pi$
$941$	−25863.0	−0.895972	−0.447986	+	0.894041i	$0.647859\pi$
$942$	0	0
$943$	30240.0	1.04427
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42384.0	1.45438	0.727188	−	0.686438i	$-0.240827\pi$
$947$	42384.0	1.45438	0.727188	+	0.686438i	$0.240827\pi$
$948$	0	0
$949$	−23168.0	−0.792482
$950$	0	0
$951$	7731.00	0.263612
$952$	0	0
$953$	10530.0	0.357923	0.178961	−	0.983856i	$-0.442726\pi$
$953$	10530.0	0.357923	0.178961	+	0.983856i	$0.442726\pi$
$954$	0	0
$955$	11736.0	0.397663
$956$	0	0
$957$	13365.0	0.451441
$958$	0	0
$959$	0	0
$960$	0	0
$961$	34218.0	1.14860
$962$	0	0
$963$	12177.0	0.407475
$964$	0	0
$965$	−4479.00	−0.149414
$966$	0	0
$967$	38341.0	1.27504	0.637520	−	0.770434i	$-0.279960\pi$
$967$	38341.0	1.27504	0.637520	+	0.770434i	$0.279960\pi$
$968$	0	0
$969$	−4032.00	−0.133670
$970$	0	0
$971$	−1923.00	−0.0635551	−0.0317776	−	0.999495i	$-0.510117\pi$
$971$	−1923.00	−0.0635551	−0.0317776	+	0.999495i	$0.510117\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−22272.0	−0.731564
$976$	0	0
$977$	57090.0	1.86947	0.934734	−	0.355347i	$-0.115637\pi$
$977$	57090.0	1.86947	0.934734	+	0.355347i	$0.115637\pi$
$978$	0	0
$979$	13590.0	0.443655
$980$	0	0
$981$	−3330.00	−0.108378
$982$	0	0
$983$	−5484.00	−0.177937	−0.0889687	−	0.996034i	$-0.528357\pi$
$983$	−5484.00	−0.177937	−0.0889687	+	0.996034i	$0.528357\pi$
$984$	0	0
$985$	12258.0	0.396520
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2184.00	−0.0702196
$990$	0	0
$991$	22465.0	0.720105	0.360053	−	0.932932i	$-0.382759\pi$
$991$	22465.0	0.720105	0.360053	+	0.932932i	$0.382759\pi$
$992$	0	0
$993$	−1452.00	−0.0464026
$994$	0	0
$995$	−10668.0	−0.339898
$996$	0	0
$997$	29366.0	0.932829	0.466415	−	0.884566i	$-0.345546\pi$
$997$	29366.0	0.932829	0.466415	+	0.884566i	$0.345546\pi$
$998$	0	0
$999$	8532.00	0.270211

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.i.1.1		1
4.3	odd	2		147.4.a.b.1.1		1
7.2	even	3		336.4.q.e.193.1		2
7.4	even	3		336.4.q.e.289.1		2
7.6	odd	2		2352.4.a.bd.1.1		1
12.11	even	2		441.4.a.l.1.1		1
28.3	even	6		147.4.e.h.79.1		2
28.11	odd	6		21.4.e.a.16.1	yes	2
28.19	even	6		147.4.e.h.67.1		2
28.23	odd	6		21.4.e.a.4.1	✓	2
28.27	even	2		147.4.a.a.1.1		1
84.11	even	6		63.4.e.a.37.1		2
84.23	even	6		63.4.e.a.46.1		2
84.47	odd	6		441.4.e.c.361.1		2
84.59	odd	6		441.4.e.c.226.1		2
84.83	odd	2		441.4.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.e.a.4.1	✓	2	28.23	odd	6
21.4.e.a.16.1	yes	2	28.11	odd	6
63.4.e.a.37.1		2	84.11	even	6
63.4.e.a.46.1		2	84.23	even	6
147.4.a.a.1.1		1	28.27	even	2
147.4.a.b.1.1		1	4.3	odd	2
147.4.e.h.67.1		2	28.19	even	6
147.4.e.h.79.1		2	28.3	even	6
336.4.q.e.193.1		2	7.2	even	3
336.4.q.e.289.1		2	7.4	even	3
441.4.a.k.1.1		1	84.83	odd	2
441.4.a.l.1.1		1	12.11	even	2
441.4.e.c.226.1		2	84.59	odd	6
441.4.e.c.361.1		2	84.47	odd	6
2352.4.a.i.1.1		1	1.1	even	1	trivial
2352.4.a.bd.1.1		1	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

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