LMFDB - Embedded newform 2352.4.a.o.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)

$f(q)$

$=$

$q-3.00000 q^{3} +11.0000 q^{5} +9.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} +11.0000 q^{5} +9.00000 q^{9} -39.0000 q^{11} +32.0000 q^{13} -33.0000 q^{15} -12.0000 q^{17} -88.0000 q^{19} +92.0000 q^{23} -4.00000 q^{25} -27.0000 q^{27} +255.000 q^{29} -35.0000 q^{31} +117.000 q^{33} -4.00000 q^{37} -96.0000 q^{39} -16.0000 q^{41} +330.000 q^{43} +99.0000 q^{45} -298.000 q^{47} +36.0000 q^{51} -717.000 q^{53} -429.000 q^{55} +264.000 q^{57} -217.000 q^{59} -386.000 q^{61} +352.000 q^{65} -906.000 q^{67} -276.000 q^{69} +34.0000 q^{71} +838.000 q^{73} +12.0000 q^{75} -1325.00 q^{79} +81.0000 q^{81} +1163.00 q^{83} -132.000 q^{85} -765.000 q^{87} +54.0000 q^{89} +105.000 q^{93} -968.000 q^{95} -7.00000 q^{97} -351.000 q^{99} +O(q^{100})$

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
$5$	11.0000	0.983870	0.491935	−	0.870632i	$-0.336290\pi$
$5$	11.0000	0.983870	0.491935	+	0.870632i	$0.336290\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−39.0000	−1.06899	−0.534497	−	0.845170i	$-0.679499\pi$
$11$	−39.0000	−1.06899	−0.534497	+	0.845170i	$0.679499\pi$
$12$	0	0
$13$	32.0000	0.682708	0.341354	−	0.939935i	$-0.389115\pi$
$13$	32.0000	0.682708	0.341354	+	0.939935i	$0.389115\pi$
$14$	0	0
$15$	−33.0000	−0.568038
$16$	0	0
$17$	−12.0000	−0.171202	−0.0856008	−	0.996330i	$-0.527281\pi$
$17$	−12.0000	−0.171202	−0.0856008	+	0.996330i	$0.527281\pi$
$18$	0	0
$19$	−88.0000	−1.06256	−0.531279	−	0.847197i	$-0.678288\pi$
$19$	−88.0000	−1.06256	−0.531279	+	0.847197i	$0.678288\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	92.0000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	92.0000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−4.00000	−0.0320000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	255.000	1.63284	0.816419	−	0.577460i	$-0.195956\pi$
$29$	255.000	1.63284	0.816419	+	0.577460i	$0.195956\pi$
$30$	0	0
$31$	−35.0000	−0.202780	−0.101390	−	0.994847i	$-0.532329\pi$
$31$	−35.0000	−0.202780	−0.101390	+	0.994847i	$0.532329\pi$
$32$	0	0
$33$	117.000	0.617184
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.00000	−0.0177729	−0.00888643	−	0.999961i	$-0.502829\pi$
$37$	−4.00000	−0.0177729	−0.00888643	+	0.999961i	$0.502829\pi$
$38$	0	0
$39$	−96.0000	−0.394162
$40$	0	0
$41$	−16.0000	−0.0609459	−0.0304729	−	0.999536i	$-0.509701\pi$
$41$	−16.0000	−0.0609459	−0.0304729	+	0.999536i	$0.509701\pi$
$42$	0	0
$43$	330.000	1.17034	0.585169	−	0.810911i	$-0.301028\pi$
$43$	330.000	1.17034	0.585169	+	0.810911i	$0.301028\pi$
$44$	0	0
$45$	99.0000	0.327957
$46$	0	0
$47$	−298.000	−0.924846	−0.462423	−	0.886659i	$-0.653020\pi$
$47$	−298.000	−0.924846	−0.462423	+	0.886659i	$0.653020\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	36.0000	0.0988433
$52$	0	0
$53$	−717.000	−1.85826	−0.929128	−	0.369759i	$-0.879440\pi$
$53$	−717.000	−1.85826	−0.929128	+	0.369759i	$0.879440\pi$
$54$	0	0
$55$	−429.000	−1.05175
$56$	0	0
$57$	264.000	0.613468
$58$	0	0
$59$	−217.000	−0.478830	−0.239415	−	0.970917i	$-0.576956\pi$
$59$	−217.000	−0.478830	−0.239415	+	0.970917i	$0.576956\pi$
$60$	0	0
$61$	−386.000	−0.810201	−0.405100	−	0.914272i	$-0.632763\pi$
$61$	−386.000	−0.810201	−0.405100	+	0.914272i	$0.632763\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	352.000	0.671696
$66$	0	0
$67$	−906.000	−1.65202	−0.826011	−	0.563654i	$-0.809395\pi$
$67$	−906.000	−1.65202	−0.826011	+	0.563654i	$0.809395\pi$
$68$	0	0
$69$	−276.000	−0.481543
$70$	0	0
$71$	34.0000	0.0568318	0.0284159	−	0.999596i	$-0.490954\pi$
$71$	34.0000	0.0568318	0.0284159	+	0.999596i	$0.490954\pi$
$72$	0	0
$73$	838.000	1.34357	0.671784	−	0.740747i	$-0.265528\pi$
$73$	838.000	1.34357	0.671784	+	0.740747i	$0.265528\pi$
$74$	0	0
$75$	12.0000	0.0184752
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1325.00	−1.88701	−0.943507	−	0.331352i	$-0.892495\pi$
$79$	−1325.00	−1.88701	−0.943507	+	0.331352i	$0.892495\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1163.00	1.53802	0.769011	−	0.639235i	$-0.220749\pi$
$83$	1163.00	1.53802	0.769011	+	0.639235i	$0.220749\pi$
$84$	0	0
$85$	−132.000	−0.168440
$86$	0	0
$87$	−765.000	−0.942720
$88$	0	0
$89$	54.0000	0.0643145	0.0321572	−	0.999483i	$-0.489762\pi$
$89$	54.0000	0.0643145	0.0321572	+	0.999483i	$0.489762\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	105.000	0.117075
$94$	0	0
$95$	−968.000	−1.04542
$96$	0	0
$97$	−7.00000	−0.00732724	−0.00366362	−	0.999993i	$-0.501166\pi$
$97$	−7.00000	−0.00732724	−0.00366362	+	0.999993i	$0.501166\pi$
$98$	0	0
$99$	−351.000	−0.356332
$100$	0	0
$101$	46.0000	0.0453185	0.0226593	−	0.999743i	$-0.492787\pi$
$101$	46.0000	0.0453185	0.0226593	+	0.999743i	$0.492787\pi$
$102$	0	0
$103$	1656.00	1.58418	0.792090	−	0.610404i	$-0.208993\pi$
$103$	1656.00	1.58418	0.792090	+	0.610404i	$0.208993\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	367.000	0.331582	0.165791	−	0.986161i	$-0.446982\pi$
$107$	367.000	0.331582	0.165791	+	0.986161i	$0.446982\pi$
$108$	0	0
$109$	1622.00	1.42532	0.712658	−	0.701512i	$-0.247491\pi$
$109$	1622.00	1.42532	0.712658	+	0.701512i	$0.247491\pi$
$110$	0	0
$111$	12.0000	0.0102612
$112$	0	0
$113$	−736.000	−0.612717	−0.306359	−	0.951916i	$-0.599111\pi$
$113$	−736.000	−0.612717	−0.306359	+	0.951916i	$0.599111\pi$
$114$	0	0
$115$	1012.00	0.820604
$116$	0	0
$117$	288.000	0.227569
$118$	0	0
$119$	0	0
$120$	0	0
$121$	190.000	0.142750
$122$	0	0
$123$	48.0000	0.0351871
$124$	0	0
$125$	−1419.00	−1.01535
$126$	0	0
$127$	−1175.00	−0.820979	−0.410490	−	0.911865i	$-0.634642\pi$
$127$	−1175.00	−0.820979	−0.410490	+	0.911865i	$0.634642\pi$
$128$	0	0
$129$	−990.000	−0.675695
$130$	0	0
$131$	−2541.00	−1.69472	−0.847360	−	0.531020i	$-0.821809\pi$
$131$	−2541.00	−1.69472	−0.847360	+	0.531020i	$0.821809\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−297.000	−0.189346
$136$	0	0
$137$	−802.000	−0.500142	−0.250071	−	0.968227i	$-0.580454\pi$
$137$	−802.000	−0.500142	−0.250071	+	0.968227i	$0.580454\pi$
$138$	0	0
$139$	3110.00	1.89775	0.948873	−	0.315657i	$-0.102225\pi$
$139$	3110.00	1.89775	0.948873	+	0.315657i	$0.102225\pi$
$140$	0	0
$141$	894.000	0.533960
$142$	0	0
$143$	−1248.00	−0.729811
$144$	0	0
$145$	2805.00	1.60650
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2598.00	1.42843	0.714216	−	0.699925i	$-0.246783\pi$
$149$	2598.00	1.42843	0.714216	+	0.699925i	$0.246783\pi$
$150$	0	0
$151$	−1333.00	−0.718397	−0.359199	−	0.933261i	$-0.616950\pi$
$151$	−1333.00	−0.718397	−0.359199	+	0.933261i	$0.616950\pi$
$152$	0	0
$153$	−108.000	−0.0570672
$154$	0	0
$155$	−385.000	−0.199509
$156$	0	0
$157$	−1580.00	−0.803170	−0.401585	−	0.915822i	$-0.631541\pi$
$157$	−1580.00	−0.803170	−0.401585	+	0.915822i	$0.631541\pi$
$158$	0	0
$159$	2151.00	1.07286
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−1268.00	−0.609309	−0.304655	−	0.952463i	$-0.598541\pi$
$163$	−1268.00	−0.609309	−0.304655	+	0.952463i	$0.598541\pi$
$164$	0	0
$165$	1287.00	0.607229
$166$	0	0
$167$	878.000	0.406836	0.203418	−	0.979092i	$-0.434795\pi$
$167$	878.000	0.406836	0.203418	+	0.979092i	$0.434795\pi$
$168$	0	0
$169$	−1173.00	−0.533910
$170$	0	0
$171$	−792.000	−0.354186
$172$	0	0
$173$	2882.00	1.26656	0.633279	−	0.773924i	$-0.281709\pi$
$173$	2882.00	1.26656	0.633279	+	0.773924i	$0.281709\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	651.000	0.276453
$178$	0	0
$179$	−1476.00	−0.616321	−0.308160	−	0.951334i	$-0.599713\pi$
$179$	−1476.00	−0.616321	−0.308160	+	0.951334i	$0.599713\pi$
$180$	0	0
$181$	−4368.00	−1.79376	−0.896881	−	0.442272i	$-0.854173\pi$
$181$	−4368.00	−1.79376	−0.896881	+	0.442272i	$0.854173\pi$
$182$	0	0
$183$	1158.00	0.467770
$184$	0	0
$185$	−44.0000	−0.0174862
$186$	0	0
$187$	468.000	0.183014
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3016.00	−1.14257	−0.571283	−	0.820753i	$-0.693554\pi$
$191$	−3016.00	−1.14257	−0.571283	+	0.820753i	$0.693554\pi$
$192$	0	0
$193$	−1563.00	−0.582939	−0.291469	−	0.956580i	$-0.594144\pi$
$193$	−1563.00	−0.582939	−0.291469	+	0.956580i	$0.594144\pi$
$194$	0	0
$195$	−1056.00	−0.387804
$196$	0	0
$197$	−1846.00	−0.667625	−0.333812	−	0.942640i	$-0.608335\pi$
$197$	−1846.00	−0.667625	−0.333812	+	0.942640i	$0.608335\pi$
$198$	0	0
$199$	−3996.00	−1.42346	−0.711731	−	0.702452i	$-0.752088\pi$
$199$	−3996.00	−1.42346	−0.711731	+	0.702452i	$0.752088\pi$
$200$	0	0
$201$	2718.00	0.953796
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−176.000	−0.0599628
$206$	0	0
$207$	828.000	0.278019
$208$	0	0
$209$	3432.00	1.13587
$210$	0	0
$211$	−4182.00	−1.36446	−0.682229	−	0.731138i	$-0.738989\pi$
$211$	−4182.00	−1.36446	−0.682229	+	0.731138i	$0.738989\pi$
$212$	0	0
$213$	−102.000	−0.0328119
$214$	0	0
$215$	3630.00	1.15146
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−2514.00	−0.775709
$220$	0	0
$221$	−384.000	−0.116881
$222$	0	0
$223$	1159.00	0.348038	0.174019	−	0.984742i	$-0.444325\pi$
$223$	1159.00	0.348038	0.174019	+	0.984742i	$0.444325\pi$
$224$	0	0
$225$	−36.0000	−0.0106667
$226$	0	0
$227$	1785.00	0.521915	0.260957	−	0.965350i	$-0.415962\pi$
$227$	1785.00	0.521915	0.260957	+	0.965350i	$0.415962\pi$
$228$	0	0
$229$	−588.000	−0.169677	−0.0848387	−	0.996395i	$-0.527038\pi$
$229$	−588.000	−0.169677	−0.0848387	+	0.996395i	$0.527038\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6828.00	1.91982	0.959908	−	0.280315i	$-0.0904389\pi$
$233$	6828.00	1.91982	0.959908	+	0.280315i	$0.0904389\pi$
$234$	0	0
$235$	−3278.00	−0.909928
$236$	0	0
$237$	3975.00	1.08947
$238$	0	0
$239$	−2508.00	−0.678783	−0.339391	−	0.940645i	$-0.610221\pi$
$239$	−2508.00	−0.678783	−0.339391	+	0.940645i	$0.610221\pi$
$240$	0	0
$241$	4297.00	1.14852	0.574262	−	0.818672i	$-0.305289\pi$
$241$	4297.00	1.14852	0.574262	+	0.818672i	$0.305289\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2816.00	−0.725416
$248$	0	0
$249$	−3489.00	−0.887977
$250$	0	0
$251$	−2489.00	−0.625913	−0.312957	−	0.949767i	$-0.601319\pi$
$251$	−2489.00	−0.625913	−0.312957	+	0.949767i	$0.601319\pi$
$252$	0	0
$253$	−3588.00	−0.891603
$254$	0	0
$255$	396.000	0.0972489
$256$	0	0
$257$	−1786.00	−0.433493	−0.216746	−	0.976228i	$-0.569544\pi$
$257$	−1786.00	−0.433493	−0.216746	+	0.976228i	$0.569544\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2295.00	0.544279
$262$	0	0
$263$	−2318.00	−0.543475	−0.271738	−	0.962371i	$-0.587598\pi$
$263$	−2318.00	−0.543475	−0.271738	+	0.962371i	$0.587598\pi$
$264$	0	0
$265$	−7887.00	−1.82828
$266$	0	0
$267$	−162.000	−0.0371320
$268$	0	0
$269$	2749.00	0.623084	0.311542	−	0.950232i	$-0.399155\pi$
$269$	2749.00	0.623084	0.311542	+	0.950232i	$0.399155\pi$
$270$	0	0
$271$	1825.00	0.409081	0.204540	−	0.978858i	$-0.434430\pi$
$271$	1825.00	0.409081	0.204540	+	0.978858i	$0.434430\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	156.000	0.0342078
$276$	0	0
$277$	−304.000	−0.0659408	−0.0329704	−	0.999456i	$-0.510497\pi$
$277$	−304.000	−0.0659408	−0.0329704	+	0.999456i	$0.510497\pi$
$278$	0	0
$279$	−315.000	−0.0675934
$280$	0	0
$281$	4578.00	0.971888	0.485944	−	0.873990i	$-0.338476\pi$
$281$	4578.00	0.971888	0.485944	+	0.873990i	$0.338476\pi$
$282$	0	0
$283$	4302.00	0.903630	0.451815	−	0.892112i	$-0.350777\pi$
$283$	4302.00	0.903630	0.451815	+	0.892112i	$0.350777\pi$
$284$	0	0
$285$	2904.00	0.603572
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4769.00	−0.970690
$290$	0	0
$291$	21.0000	0.00423038
$292$	0	0
$293$	−6727.00	−1.34128	−0.670641	−	0.741782i	$-0.733981\pi$
$293$	−6727.00	−1.34128	−0.670641	+	0.741782i	$0.733981\pi$
$294$	0	0
$295$	−2387.00	−0.471107
$296$	0	0
$297$	1053.00	0.205728
$298$	0	0
$299$	2944.00	0.569418
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−138.000	−0.0261647
$304$	0	0
$305$	−4246.00	−0.797132
$306$	0	0
$307$	116.000	0.0215650	0.0107825	−	0.999942i	$-0.496568\pi$
$307$	116.000	0.0215650	0.0107825	+	0.999942i	$0.496568\pi$
$308$	0	0
$309$	−4968.00	−0.914627
$310$	0	0
$311$	−3792.00	−0.691397	−0.345699	−	0.938346i	$-0.612358\pi$
$311$	−3792.00	−0.691397	−0.345699	+	0.938346i	$0.612358\pi$
$312$	0	0
$313$	3687.00	0.665820	0.332910	−	0.942959i	$-0.391970\pi$
$313$	3687.00	0.665820	0.332910	+	0.942959i	$0.391970\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8281.00	−1.46722	−0.733608	−	0.679573i	$-0.762165\pi$
$317$	−8281.00	−1.46722	−0.733608	+	0.679573i	$0.762165\pi$
$318$	0	0
$319$	−9945.00	−1.74550
$320$	0	0
$321$	−1101.00	−0.191439
$322$	0	0
$323$	1056.00	0.181911
$324$	0	0
$325$	−128.000	−0.0218467
$326$	0	0
$327$	−4866.00	−0.822906
$328$	0	0
$329$	0	0
$330$	0	0
$331$	6668.00	1.10727	0.553635	−	0.832759i	$-0.313240\pi$
$331$	6668.00	1.10727	0.553635	+	0.832759i	$0.313240\pi$
$332$	0	0
$333$	−36.0000	−0.00592429
$334$	0	0
$335$	−9966.00	−1.62538
$336$	0	0
$337$	−8679.00	−1.40289	−0.701447	−	0.712722i	$-0.747462\pi$
$337$	−8679.00	−1.40289	−0.701447	+	0.712722i	$0.747462\pi$
$338$	0	0
$339$	2208.00	0.353753
$340$	0	0
$341$	1365.00	0.216771
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3036.00	−0.473776
$346$	0	0
$347$	−9092.00	−1.40658	−0.703291	−	0.710902i	$-0.748287\pi$
$347$	−9092.00	−1.40658	−0.703291	+	0.710902i	$0.748287\pi$
$348$	0	0
$349$	−11642.0	−1.78562	−0.892811	−	0.450432i	$-0.851270\pi$
$349$	−11642.0	−1.78562	−0.892811	+	0.450432i	$0.851270\pi$
$350$	0	0
$351$	−864.000	−0.131387
$352$	0	0
$353$	−6808.00	−1.02650	−0.513248	−	0.858240i	$-0.671558\pi$
$353$	−6808.00	−1.02650	−0.513248	+	0.858240i	$0.671558\pi$
$354$	0	0
$355$	374.000	0.0559151
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1110.00	−0.163185	−0.0815927	−	0.996666i	$-0.526001\pi$
$359$	−1110.00	−0.163185	−0.0815927	+	0.996666i	$0.526001\pi$
$360$	0	0
$361$	885.000	0.129028
$362$	0	0
$363$	−570.000	−0.0824166
$364$	0	0
$365$	9218.00	1.32190
$366$	0	0
$367$	959.000	0.136402	0.0682008	−	0.997672i	$-0.478274\pi$
$367$	959.000	0.136402	0.0682008	+	0.997672i	$0.478274\pi$
$368$	0	0
$369$	−144.000	−0.0203153
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.0000	0.00222104	0.00111052	−	0.999999i	$-0.499647\pi$
$373$	16.0000	0.00222104	0.00111052	+	0.999999i	$0.499647\pi$
$374$	0	0
$375$	4257.00	0.586215
$376$	0	0
$377$	8160.00	1.11475
$378$	0	0
$379$	736.000	0.0997514	0.0498757	−	0.998755i	$-0.484117\pi$
$379$	736.000	0.0997514	0.0498757	+	0.998755i	$0.484117\pi$
$380$	0	0
$381$	3525.00	0.473993
$382$	0	0
$383$	−8414.00	−1.12255	−0.561273	−	0.827631i	$-0.689688\pi$
$383$	−8414.00	−1.12255	−0.561273	+	0.827631i	$0.689688\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2970.00	0.390113
$388$	0	0
$389$	694.000	0.0904555	0.0452278	−	0.998977i	$-0.485599\pi$
$389$	694.000	0.0904555	0.0452278	+	0.998977i	$0.485599\pi$
$390$	0	0
$391$	−1104.00	−0.142792
$392$	0	0
$393$	7623.00	0.978447
$394$	0	0
$395$	−14575.0	−1.85658
$396$	0	0
$397$	−11476.0	−1.45079	−0.725395	−	0.688332i	$-0.758343\pi$
$397$	−11476.0	−1.45079	−0.725395	+	0.688332i	$0.758343\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12040.0	−1.49937	−0.749687	−	0.661793i	$-0.769796\pi$
$401$	−12040.0	−1.49937	−0.749687	+	0.661793i	$0.769796\pi$
$402$	0	0
$403$	−1120.00	−0.138440
$404$	0	0
$405$	891.000	0.109319
$406$	0	0
$407$	156.000	0.0189991
$408$	0	0
$409$	−13711.0	−1.65762	−0.828808	−	0.559532i	$-0.810981\pi$
$409$	−13711.0	−1.65762	−0.828808	+	0.559532i	$0.810981\pi$
$410$	0	0
$411$	2406.00	0.288757
$412$	0	0
$413$	0	0
$414$	0	0
$415$	12793.0	1.51321
$416$	0	0
$417$	−9330.00	−1.09566
$418$	0	0
$419$	−4128.00	−0.481303	−0.240652	−	0.970612i	$-0.577361\pi$
$419$	−4128.00	−0.481303	−0.240652	+	0.970612i	$0.577361\pi$
$420$	0	0
$421$	−8442.00	−0.977287	−0.488644	−	0.872483i	$-0.662508\pi$
$421$	−8442.00	−0.977287	−0.488644	+	0.872483i	$0.662508\pi$
$422$	0	0
$423$	−2682.00	−0.308282
$424$	0	0
$425$	48.0000	0.00547845
$426$	0	0
$427$	0	0
$428$	0	0
$429$	3744.00	0.421357
$430$	0	0
$431$	9564.00	1.06887	0.534433	−	0.845211i	$-0.320525\pi$
$431$	9564.00	1.06887	0.534433	+	0.845211i	$0.320525\pi$
$432$	0	0
$433$	13394.0	1.48655	0.743273	−	0.668988i	$-0.233272\pi$
$433$	13394.0	1.48655	0.743273	+	0.668988i	$0.233272\pi$
$434$	0	0
$435$	−8415.00	−0.927513
$436$	0	0
$437$	−8096.00	−0.886234
$438$	0	0
$439$	15513.0	1.68655	0.843275	−	0.537483i	$-0.180625\pi$
$439$	15513.0	1.68655	0.843275	+	0.537483i	$0.180625\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12379.0	−1.32764	−0.663819	−	0.747893i	$-0.731066\pi$
$443$	−12379.0	−1.32764	−0.663819	+	0.747893i	$0.731066\pi$
$444$	0	0
$445$	594.000	0.0632771
$446$	0	0
$447$	−7794.00	−0.824706
$448$	0	0
$449$	−8368.00	−0.879533	−0.439767	−	0.898112i	$-0.644939\pi$
$449$	−8368.00	−0.879533	−0.439767	+	0.898112i	$0.644939\pi$
$450$	0	0
$451$	624.000	0.0651508
$452$	0	0
$453$	3999.00	0.414767
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1577.00	−0.161420	−0.0807100	−	0.996738i	$-0.525719\pi$
$457$	−1577.00	−0.161420	−0.0807100	+	0.996738i	$0.525719\pi$
$458$	0	0
$459$	324.000	0.0329478
$460$	0	0
$461$	−8202.00	−0.828645	−0.414322	−	0.910130i	$-0.635981\pi$
$461$	−8202.00	−0.828645	−0.414322	+	0.910130i	$0.635981\pi$
$462$	0	0
$463$	1936.00	0.194327	0.0971637	−	0.995268i	$-0.469023\pi$
$463$	1936.00	0.194327	0.0971637	+	0.995268i	$0.469023\pi$
$464$	0	0
$465$	1155.00	0.115187
$466$	0	0
$467$	13156.0	1.30361	0.651806	−	0.758385i	$-0.274012\pi$
$467$	13156.0	1.30361	0.651806	+	0.758385i	$0.274012\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	4740.00	0.463711
$472$	0	0
$473$	−12870.0	−1.25109
$474$	0	0
$475$	352.000	0.0340018
$476$	0	0
$477$	−6453.00	−0.619418
$478$	0	0
$479$	−6630.00	−0.632427	−0.316213	−	0.948688i	$-0.602412\pi$
$479$	−6630.00	−0.632427	−0.316213	+	0.948688i	$0.602412\pi$
$480$	0	0
$481$	−128.000	−0.0121337
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−77.0000	−0.00720905
$486$	0	0
$487$	−4195.00	−0.390336	−0.195168	−	0.980770i	$-0.562525\pi$
$487$	−4195.00	−0.390336	−0.195168	+	0.980770i	$0.562525\pi$
$488$	0	0
$489$	3804.00	0.351785
$490$	0	0
$491$	−1107.00	−0.101748	−0.0508739	−	0.998705i	$-0.516201\pi$
$491$	−1107.00	−0.101748	−0.0508739	+	0.998705i	$0.516201\pi$
$492$	0	0
$493$	−3060.00	−0.279545
$494$	0	0
$495$	−3861.00	−0.350584
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−13630.0	−1.22277	−0.611385	−	0.791333i	$-0.709387\pi$
$499$	−13630.0	−1.22277	−0.611385	+	0.791333i	$0.709387\pi$
$500$	0	0
$501$	−2634.00	−0.234887
$502$	0	0
$503$	−11760.0	−1.04245	−0.521225	−	0.853419i	$-0.674525\pi$
$503$	−11760.0	−1.04245	−0.521225	+	0.853419i	$0.674525\pi$
$504$	0	0
$505$	506.000	0.0445875
$506$	0	0
$507$	3519.00	0.308253
$508$	0	0
$509$	19101.0	1.66333	0.831667	−	0.555275i	$-0.187387\pi$
$509$	19101.0	1.66333	0.831667	+	0.555275i	$0.187387\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2376.00	0.204489
$514$	0	0
$515$	18216.0	1.55863
$516$	0	0
$517$	11622.0	0.988656
$518$	0	0
$519$	−8646.00	−0.731247
$520$	0	0
$521$	−10590.0	−0.890511	−0.445256	−	0.895404i	$-0.646887\pi$
$521$	−10590.0	−0.890511	−0.445256	+	0.895404i	$0.646887\pi$
$522$	0	0
$523$	17400.0	1.45478	0.727389	−	0.686225i	$-0.240734\pi$
$523$	17400.0	1.45478	0.727389	+	0.686225i	$0.240734\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	420.000	0.0347163
$528$	0	0
$529$	−3703.00	−0.304348
$530$	0	0
$531$	−1953.00	−0.159610
$532$	0	0
$533$	−512.000	−0.0416082
$534$	0	0
$535$	4037.00	0.326233
$536$	0	0
$537$	4428.00	0.355833
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−6410.00	−0.509404	−0.254702	−	0.967020i	$-0.581977\pi$
$541$	−6410.00	−0.509404	−0.254702	+	0.967020i	$0.581977\pi$
$542$	0	0
$543$	13104.0	1.03563
$544$	0	0
$545$	17842.0	1.40233
$546$	0	0
$547$	24020.0	1.87755	0.938776	−	0.344528i	$-0.111961\pi$
$547$	24020.0	1.87755	0.938776	+	0.344528i	$0.111961\pi$
$548$	0	0
$549$	−3474.00	−0.270067
$550$	0	0
$551$	−22440.0	−1.73498
$552$	0	0
$553$	0	0
$554$	0	0
$555$	132.000	0.0100957
$556$	0	0
$557$	−26139.0	−1.98841	−0.994206	−	0.107496i	$-0.965717\pi$
$557$	−26139.0	−1.98841	−0.994206	+	0.107496i	$0.965717\pi$
$558$	0	0
$559$	10560.0	0.798999
$560$	0	0
$561$	−1404.00	−0.105663
$562$	0	0
$563$	9171.00	0.686521	0.343261	−	0.939240i	$-0.388469\pi$
$563$	9171.00	0.686521	0.343261	+	0.939240i	$0.388469\pi$
$564$	0	0
$565$	−8096.00	−0.602834
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−19608.0	−1.44466	−0.722329	−	0.691550i	$-0.756928\pi$
$569$	−19608.0	−1.44466	−0.722329	+	0.691550i	$0.756928\pi$
$570$	0	0
$571$	3674.00	0.269268	0.134634	−	0.990895i	$-0.457014\pi$
$571$	3674.00	0.269268	0.134634	+	0.990895i	$0.457014\pi$
$572$	0	0
$573$	9048.00	0.659661
$574$	0	0
$575$	−368.000	−0.0266898
$576$	0	0
$577$	3873.00	0.279437	0.139718	−	0.990191i	$-0.455380\pi$
$577$	3873.00	0.279437	0.139718	+	0.990191i	$0.455380\pi$
$578$	0	0
$579$	4689.00	0.336560
$580$	0	0
$581$	0	0
$582$	0	0
$583$	27963.0	1.98647
$584$	0	0
$585$	3168.00	0.223899
$586$	0	0
$587$	−13743.0	−0.966328	−0.483164	−	0.875530i	$-0.660512\pi$
$587$	−13743.0	−0.966328	−0.483164	+	0.875530i	$0.660512\pi$
$588$	0	0
$589$	3080.00	0.215466
$590$	0	0
$591$	5538.00	0.385453
$592$	0	0
$593$	−16356.0	−1.13265	−0.566324	−	0.824183i	$-0.691635\pi$
$593$	−16356.0	−1.13265	−0.566324	+	0.824183i	$0.691635\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	11988.0	0.821836
$598$	0	0
$599$	−10782.0	−0.735460	−0.367730	−	0.929933i	$-0.619865\pi$
$599$	−10782.0	−0.735460	−0.367730	+	0.929933i	$0.619865\pi$
$600$	0	0
$601$	−25547.0	−1.73392	−0.866958	−	0.498381i	$-0.833928\pi$
$601$	−25547.0	−1.73392	−0.866958	+	0.498381i	$0.833928\pi$
$602$	0	0
$603$	−8154.00	−0.550674
$604$	0	0
$605$	2090.00	0.140447
$606$	0	0
$607$	14299.0	0.956143	0.478071	−	0.878321i	$-0.341336\pi$
$607$	14299.0	0.956143	0.478071	+	0.878321i	$0.341336\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9536.00	−0.631400
$612$	0	0
$613$	23900.0	1.57473	0.787367	−	0.616485i	$-0.211444\pi$
$613$	23900.0	1.57473	0.787367	+	0.616485i	$0.211444\pi$
$614$	0	0
$615$	528.000	0.0346195
$616$	0	0
$617$	−9246.00	−0.603290	−0.301645	−	0.953420i	$-0.597536\pi$
$617$	−9246.00	−0.603290	−0.301645	+	0.953420i	$0.597536\pi$
$618$	0	0
$619$	13950.0	0.905812	0.452906	−	0.891558i	$-0.350387\pi$
$619$	13950.0	0.905812	0.452906	+	0.891558i	$0.350387\pi$
$620$	0	0
$621$	−2484.00	−0.160514
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−15109.0	−0.966976
$626$	0	0
$627$	−10296.0	−0.655794
$628$	0	0
$629$	48.0000	0.00304274
$630$	0	0
$631$	−15787.0	−0.995991	−0.497996	−	0.867180i	$-0.665930\pi$
$631$	−15787.0	−0.995991	−0.497996	+	0.867180i	$0.665930\pi$
$632$	0	0
$633$	12546.0	0.787771
$634$	0	0
$635$	−12925.0	−0.807737
$636$	0	0
$637$	0	0
$638$	0	0
$639$	306.000	0.0189439
$640$	0	0
$641$	−8438.00	−0.519939	−0.259970	−	0.965617i	$-0.583713\pi$
$641$	−8438.00	−0.519939	−0.259970	+	0.965617i	$0.583713\pi$
$642$	0	0
$643$	22522.0	1.38131	0.690654	−	0.723185i	$-0.257323\pi$
$643$	22522.0	1.38131	0.690654	+	0.723185i	$0.257323\pi$
$644$	0	0
$645$	−10890.0	−0.664796
$646$	0	0
$647$	17138.0	1.04137	0.520683	−	0.853750i	$-0.325677\pi$
$647$	17138.0	1.04137	0.520683	+	0.853750i	$0.325677\pi$
$648$	0	0
$649$	8463.00	0.511867
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9963.00	−0.597063	−0.298532	−	0.954400i	$-0.596497\pi$
$653$	−9963.00	−0.597063	−0.298532	+	0.954400i	$0.596497\pi$
$654$	0	0
$655$	−27951.0	−1.66738
$656$	0	0
$657$	7542.00	0.447856
$658$	0	0
$659$	8664.00	0.512142	0.256071	−	0.966658i	$-0.417572\pi$
$659$	8664.00	0.512142	0.256071	+	0.966658i	$0.417572\pi$
$660$	0	0
$661$	−2450.00	−0.144166	−0.0720832	−	0.997399i	$-0.522965\pi$
$661$	−2450.00	−0.144166	−0.0720832	+	0.997399i	$0.522965\pi$
$662$	0	0
$663$	1152.00	0.0674811
$664$	0	0
$665$	0	0
$666$	0	0
$667$	23460.0	1.36188
$668$	0	0
$669$	−3477.00	−0.200940
$670$	0	0
$671$	15054.0	0.866100
$672$	0	0
$673$	−6643.00	−0.380489	−0.190244	−	0.981737i	$-0.560928\pi$
$673$	−6643.00	−0.380489	−0.190244	+	0.981737i	$0.560928\pi$
$674$	0	0
$675$	108.000	0.00615840
$676$	0	0
$677$	−15593.0	−0.885211	−0.442605	−	0.896717i	$-0.645946\pi$
$677$	−15593.0	−0.885211	−0.442605	+	0.896717i	$0.645946\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−5355.00	−0.301328
$682$	0	0
$683$	371.000	0.0207847	0.0103923	−	0.999946i	$-0.496692\pi$
$683$	371.000	0.0207847	0.0103923	+	0.999946i	$0.496692\pi$
$684$	0	0
$685$	−8822.00	−0.492075
$686$	0	0
$687$	1764.00	0.0979633
$688$	0	0
$689$	−22944.0	−1.26865
$690$	0	0
$691$	8232.00	0.453198	0.226599	−	0.973988i	$-0.427239\pi$
$691$	8232.00	0.453198	0.226599	+	0.973988i	$0.427239\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	34210.0	1.86714
$696$	0	0
$697$	192.000	0.0104340
$698$	0	0
$699$	−20484.0	−1.10841
$700$	0	0
$701$	30751.0	1.65685	0.828423	−	0.560103i	$-0.189238\pi$
$701$	30751.0	1.65685	0.828423	+	0.560103i	$0.189238\pi$
$702$	0	0
$703$	352.000	0.0188847
$704$	0	0
$705$	9834.00	0.525347
$706$	0	0
$707$	0	0
$708$	0	0
$709$	12318.0	0.652485	0.326243	−	0.945286i	$-0.394217\pi$
$709$	12318.0	0.652485	0.326243	+	0.945286i	$0.394217\pi$
$710$	0	0
$711$	−11925.0	−0.629005
$712$	0	0
$713$	−3220.00	−0.169130
$714$	0	0
$715$	−13728.0	−0.718039
$716$	0	0
$717$	7524.00	0.391895
$718$	0	0
$719$	18022.0	0.934781	0.467390	−	0.884051i	$-0.345194\pi$
$719$	18022.0	0.934781	0.467390	+	0.884051i	$0.345194\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−12891.0	−0.663100
$724$	0	0
$725$	−1020.00	−0.0522508
$726$	0	0
$727$	−1279.00	−0.0652483	−0.0326241	−	0.999468i	$-0.510386\pi$
$727$	−1279.00	−0.0652483	−0.0326241	+	0.999468i	$0.510386\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−3960.00	−0.200364
$732$	0	0
$733$	36910.0	1.85989	0.929947	−	0.367694i	$-0.119853\pi$
$733$	36910.0	1.85989	0.929947	+	0.367694i	$0.119853\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	35334.0	1.76600
$738$	0	0
$739$	31226.0	1.55435	0.777177	−	0.629283i	$-0.216651\pi$
$739$	31226.0	1.55435	0.777177	+	0.629283i	$0.216651\pi$
$740$	0	0
$741$	8448.00	0.418819
$742$	0	0
$743$	17758.0	0.876821	0.438410	−	0.898775i	$-0.355542\pi$
$743$	17758.0	0.876821	0.438410	+	0.898775i	$0.355542\pi$
$744$	0	0
$745$	28578.0	1.40539
$746$	0	0
$747$	10467.0	0.512674
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−2019.00	−0.0981017	−0.0490508	−	0.998796i	$-0.515620\pi$
$751$	−2019.00	−0.0981017	−0.0490508	+	0.998796i	$0.515620\pi$
$752$	0	0
$753$	7467.00	0.361371
$754$	0	0
$755$	−14663.0	−0.706810
$756$	0	0
$757$	−2398.00	−0.115134	−0.0575672	−	0.998342i	$-0.518334\pi$
$757$	−2398.00	−0.115134	−0.0575672	+	0.998342i	$0.518334\pi$
$758$	0	0
$759$	10764.0	0.514767
$760$	0	0
$761$	−10440.0	−0.497306	−0.248653	−	0.968593i	$-0.579988\pi$
$761$	−10440.0	−0.497306	−0.248653	+	0.968593i	$0.579988\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1188.00	−0.0561467
$766$	0	0
$767$	−6944.00	−0.326901
$768$	0	0
$769$	4739.00	0.222227	0.111114	−	0.993808i	$-0.464558\pi$
$769$	4739.00	0.222227	0.111114	+	0.993808i	$0.464558\pi$
$770$	0	0
$771$	5358.00	0.250277
$772$	0	0
$773$	34030.0	1.58341	0.791704	−	0.610905i	$-0.209194\pi$
$773$	34030.0	1.58341	0.791704	+	0.610905i	$0.209194\pi$
$774$	0	0
$775$	140.000	0.00648897
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1408.00	0.0647585
$780$	0	0
$781$	−1326.00	−0.0607529
$782$	0	0
$783$	−6885.00	−0.314240
$784$	0	0
$785$	−17380.0	−0.790215
$786$	0	0
$787$	7226.00	0.327292	0.163646	−	0.986519i	$-0.447674\pi$
$787$	7226.00	0.327292	0.163646	+	0.986519i	$0.447674\pi$
$788$	0	0
$789$	6954.00	0.313776
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−12352.0	−0.553130
$794$	0	0
$795$	23661.0	1.05556
$796$	0	0
$797$	−4881.00	−0.216931	−0.108465	−	0.994100i	$-0.534594\pi$
$797$	−4881.00	−0.216931	−0.108465	+	0.994100i	$0.534594\pi$
$798$	0	0
$799$	3576.00	0.158335
$800$	0	0
$801$	486.000	0.0214382
$802$	0	0
$803$	−32682.0	−1.43627
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−8247.00	−0.359737
$808$	0	0
$809$	−13112.0	−0.569831	−0.284916	−	0.958553i	$-0.591966\pi$
$809$	−13112.0	−0.569831	−0.284916	+	0.958553i	$0.591966\pi$
$810$	0	0
$811$	40982.0	1.77444	0.887221	−	0.461344i	$-0.152633\pi$
$811$	40982.0	1.77444	0.887221	+	0.461344i	$0.152633\pi$
$812$	0	0
$813$	−5475.00	−0.236183
$814$	0	0
$815$	−13948.0	−0.599481
$816$	0	0
$817$	−29040.0	−1.24355
$818$	0	0
$819$	0	0
$820$	0	0
$821$	39843.0	1.69370	0.846852	−	0.531829i	$-0.178495\pi$
$821$	39843.0	1.69370	0.846852	+	0.531829i	$0.178495\pi$
$822$	0	0
$823$	−7256.00	−0.307325	−0.153662	−	0.988123i	$-0.549107\pi$
$823$	−7256.00	−0.307325	−0.153662	+	0.988123i	$0.549107\pi$
$824$	0	0
$825$	−468.000	−0.0197499
$826$	0	0
$827$	23005.0	0.967306	0.483653	−	0.875260i	$-0.339310\pi$
$827$	23005.0	0.967306	0.483653	+	0.875260i	$0.339310\pi$
$828$	0	0
$829$	−368.000	−0.0154176	−0.00770879	−	0.999970i	$-0.502454\pi$
$829$	−368.000	−0.0154176	−0.00770879	+	0.999970i	$0.502454\pi$
$830$	0	0
$831$	912.000	0.0380709
$832$	0	0
$833$	0	0
$834$	0	0
$835$	9658.00	0.400274
$836$	0	0
$837$	945.000	0.0390251
$838$	0	0
$839$	−5556.00	−0.228623	−0.114311	−	0.993445i	$-0.536466\pi$
$839$	−5556.00	−0.228623	−0.114311	+	0.993445i	$0.536466\pi$
$840$	0	0
$841$	40636.0	1.66616
$842$	0	0
$843$	−13734.0	−0.561120
$844$	0	0
$845$	−12903.0	−0.525298
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−12906.0	−0.521711
$850$	0	0
$851$	−368.000	−0.0148236
$852$	0	0
$853$	5674.00	0.227754	0.113877	−	0.993495i	$-0.463673\pi$
$853$	5674.00	0.227754	0.113877	+	0.993495i	$0.463673\pi$
$854$	0	0
$855$	−8712.00	−0.348473
$856$	0	0
$857$	7826.00	0.311938	0.155969	−	0.987762i	$-0.450150\pi$
$857$	7826.00	0.311938	0.155969	+	0.987762i	$0.450150\pi$
$858$	0	0
$859$	3370.00	0.133857	0.0669284	−	0.997758i	$-0.478680\pi$
$859$	3370.00	0.133857	0.0669284	+	0.997758i	$0.478680\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−27734.0	−1.09395	−0.546974	−	0.837150i	$-0.684220\pi$
$863$	−27734.0	−1.09395	−0.546974	+	0.837150i	$0.684220\pi$
$864$	0	0
$865$	31702.0	1.24613
$866$	0	0
$867$	14307.0	0.560428
$868$	0	0
$869$	51675.0	2.01721
$870$	0	0
$871$	−28992.0	−1.12785
$872$	0	0
$873$	−63.0000	−0.00244241
$874$	0	0
$875$	0	0
$876$	0	0
$877$	45280.0	1.74344	0.871720	−	0.490004i	$-0.163005\pi$
$877$	45280.0	1.74344	0.871720	+	0.490004i	$0.163005\pi$
$878$	0	0
$879$	20181.0	0.774389
$880$	0	0
$881$	47610.0	1.82068	0.910341	−	0.413858i	$-0.135819\pi$
$881$	47610.0	1.82068	0.910341	+	0.413858i	$0.135819\pi$
$882$	0	0
$883$	28328.0	1.07963	0.539815	−	0.841784i	$-0.318494\pi$
$883$	28328.0	1.07963	0.539815	+	0.841784i	$0.318494\pi$
$884$	0	0
$885$	7161.00	0.271994
$886$	0	0
$887$	−48612.0	−1.84017	−0.920085	−	0.391718i	$-0.871881\pi$
$887$	−48612.0	−1.84017	−0.920085	+	0.391718i	$0.871881\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−3159.00	−0.118777
$892$	0	0
$893$	26224.0	0.982702
$894$	0	0
$895$	−16236.0	−0.606379
$896$	0	0
$897$	−8832.00	−0.328754
$898$	0	0
$899$	−8925.00	−0.331107
$900$	0	0
$901$	8604.00	0.318136
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−48048.0	−1.76483
$906$	0	0
$907$	−4100.00	−0.150097	−0.0750487	−	0.997180i	$-0.523911\pi$
$907$	−4100.00	−0.150097	−0.0750487	+	0.997180i	$0.523911\pi$
$908$	0	0
$909$	414.000	0.0151062
$910$	0	0
$911$	13902.0	0.505591	0.252796	−	0.967520i	$-0.418650\pi$
$911$	13902.0	0.505591	0.252796	+	0.967520i	$0.418650\pi$
$912$	0	0
$913$	−45357.0	−1.64414
$914$	0	0
$915$	12738.0	0.460224
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−45824.0	−1.64483	−0.822413	−	0.568892i	$-0.807372\pi$
$919$	−45824.0	−1.64483	−0.822413	+	0.568892i	$0.807372\pi$
$920$	0	0
$921$	−348.000	−0.0124506
$922$	0	0
$923$	1088.00	0.0387995
$924$	0	0
$925$	16.0000	0.000568732 0
$926$	0	0
$927$	14904.0	0.528060
$928$	0	0
$929$	−38874.0	−1.37289	−0.686445	−	0.727182i	$-0.740830\pi$
$929$	−38874.0	−1.37289	−0.686445	+	0.727182i	$0.740830\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	11376.0	0.399178
$934$	0	0
$935$	5148.00	0.180062
$936$	0	0
$937$	1829.00	0.0637682	0.0318841	−	0.999492i	$-0.489849\pi$
$937$	1829.00	0.0637682	0.0318841	+	0.999492i	$0.489849\pi$
$938$	0	0
$939$	−11061.0	−0.384411
$940$	0	0
$941$	5855.00	0.202835	0.101417	−	0.994844i	$-0.467662\pi$
$941$	5855.00	0.202835	0.101417	+	0.994844i	$0.467662\pi$
$942$	0	0
$943$	−1472.00	−0.0508324
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−21120.0	−0.724718	−0.362359	−	0.932039i	$-0.618029\pi$
$947$	−21120.0	−0.724718	−0.362359	+	0.932039i	$0.618029\pi$
$948$	0	0
$949$	26816.0	0.917265
$950$	0	0
$951$	24843.0	0.847097
$952$	0	0
$953$	21650.0	0.735900	0.367950	−	0.929846i	$-0.380060\pi$
$953$	21650.0	0.735900	0.367950	+	0.929846i	$0.380060\pi$
$954$	0	0
$955$	−33176.0	−1.12414
$956$	0	0
$957$	29835.0	1.00776
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28566.0	−0.958880
$962$	0	0
$963$	3303.00	0.110527
$964$	0	0
$965$	−17193.0	−0.573536
$966$	0	0
$967$	−36597.0	−1.21704	−0.608521	−	0.793538i	$-0.708237\pi$
$967$	−36597.0	−1.21704	−0.608521	+	0.793538i	$0.708237\pi$
$968$	0	0
$969$	−3168.00	−0.105027
$970$	0	0
$971$	−54187.0	−1.79088	−0.895440	−	0.445183i	$-0.853139\pi$
$971$	−54187.0	−1.79088	−0.895440	+	0.445183i	$0.853139\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	384.000	0.0126132
$976$	0	0
$977$	36034.0	1.17997	0.589985	−	0.807415i	$-0.299134\pi$
$977$	36034.0	1.17997	0.589985	+	0.807415i	$0.299134\pi$
$978$	0	0
$979$	−2106.00	−0.0687518
$980$	0	0
$981$	14598.0	0.475105
$982$	0	0
$983$	−40628.0	−1.31824	−0.659121	−	0.752037i	$-0.729072\pi$
$983$	−40628.0	−1.31824	−0.659121	+	0.752037i	$0.729072\pi$
$984$	0	0
$985$	−20306.0	−0.656856
$986$	0	0
$987$	0	0
$988$	0	0
$989$	30360.0	0.976129
$990$	0	0
$991$	40431.0	1.29600	0.647999	−	0.761642i	$-0.275606\pi$
$991$	40431.0	1.29600	0.647999	+	0.761642i	$0.275606\pi$
$992$	0	0
$993$	−20004.0	−0.639283
$994$	0	0
$995$	−43956.0	−1.40050
$996$	0	0
$997$	33266.0	1.05671	0.528357	−	0.849022i	$-0.322808\pi$
$997$	33266.0	1.05671	0.528357	+	0.849022i	$0.322808\pi$
$998$	0	0
$999$	108.000	0.00342039

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.o.1.1		1
4.3	odd	2		1176.4.a.n.1.1		1
7.3	odd	6		336.4.q.c.289.1		2
7.5	odd	6		336.4.q.c.193.1		2
7.6	odd	2		2352.4.a.y.1.1		1
28.3	even	6		168.4.q.c.121.1	yes	2
28.19	even	6		168.4.q.c.25.1	✓	2
28.27	even	2		1176.4.a.c.1.1		1
84.47	odd	6		504.4.s.a.361.1		2
84.59	odd	6		504.4.s.a.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.q.c.25.1	✓	2	28.19	even	6
168.4.q.c.121.1	yes	2	28.3	even	6
336.4.q.c.193.1		2	7.5	odd	6
336.4.q.c.289.1		2	7.3	odd	6
504.4.s.a.289.1		2	84.59	odd	6
504.4.s.a.361.1		2	84.47	odd	6
1176.4.a.c.1.1		1	28.27	even	2
1176.4.a.n.1.1		1	4.3	odd	2
2352.4.a.o.1.1		1	1.1	even	1	trivial
2352.4.a.y.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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