LMFDB - Embedded newform 2352.4.a.bf.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 42)
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)

$f(q)$

$=$

$q+3.00000 q^{3} +6.00000 q^{5} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} +6.00000 q^{5} +9.00000 q^{9} +30.0000 q^{11} -53.0000 q^{13} +18.0000 q^{15} +84.0000 q^{17} -97.0000 q^{19} -84.0000 q^{23} -89.0000 q^{25} +27.0000 q^{27} -180.000 q^{29} +179.000 q^{31} +90.0000 q^{33} -145.000 q^{37} -159.000 q^{39} -126.000 q^{41} +325.000 q^{43} +54.0000 q^{45} -366.000 q^{47} +252.000 q^{51} -768.000 q^{53} +180.000 q^{55} -291.000 q^{57} -264.000 q^{59} -818.000 q^{61} -318.000 q^{65} +523.000 q^{67} -252.000 q^{69} +342.000 q^{71} +43.0000 q^{73} -267.000 q^{75} +1171.00 q^{79} +81.0000 q^{81} -810.000 q^{83} +504.000 q^{85} -540.000 q^{87} +600.000 q^{89} +537.000 q^{93} -582.000 q^{95} -386.000 q^{97} +270.000 q^{99} +O(q^{100})$

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
$5$	6.00000	0.536656	0.268328	−	0.963328i	$-0.413529\pi$
$5$	6.00000	0.536656	0.268328	+	0.963328i	$0.413529\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	30.0000	0.822304	0.411152	−	0.911567i	$-0.365127\pi$
$11$	30.0000	0.822304	0.411152	+	0.911567i	$0.365127\pi$
$12$	0	0
$13$	−53.0000	−1.13074	−0.565368	−	0.824839i	$-0.691266\pi$
$13$	−53.0000	−1.13074	−0.565368	+	0.824839i	$0.691266\pi$
$14$	0	0
$15$	18.0000	0.309839
$16$	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
$19$	−97.0000	−1.17123	−0.585614	−	0.810590i	$-0.699146\pi$
$19$	−97.0000	−1.17123	−0.585614	+	0.810590i	$0.699146\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−84.0000	−0.761531	−0.380765	−	0.924672i	$-0.624339\pi$
$23$	−84.0000	−0.761531	−0.380765	+	0.924672i	$0.624339\pi$
$24$	0	0
$25$	−89.0000	−0.712000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−180.000	−1.15259	−0.576296	−	0.817241i	$-0.695502\pi$
$29$	−180.000	−1.15259	−0.576296	+	0.817241i	$0.695502\pi$
$30$	0	0
$31$	179.000	1.03708	0.518538	−	0.855055i	$-0.326477\pi$
$31$	179.000	1.03708	0.518538	+	0.855055i	$0.326477\pi$
$32$	0	0
$33$	90.0000	0.474757
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−145.000	−0.644266	−0.322133	−	0.946694i	$-0.604400\pi$
$37$	−145.000	−0.644266	−0.322133	+	0.946694i	$0.604400\pi$
$38$	0	0
$39$	−159.000	−0.652830
$40$	0	0
$41$	−126.000	−0.479949	−0.239974	−	0.970779i	$-0.577139\pi$
$41$	−126.000	−0.479949	−0.239974	+	0.970779i	$0.577139\pi$
$42$	0	0
$43$	325.000	1.15261	0.576303	−	0.817236i	$-0.304495\pi$
$43$	325.000	1.15261	0.576303	+	0.817236i	$0.304495\pi$
$44$	0	0
$45$	54.0000	0.178885
$46$	0	0
$47$	−366.000	−1.13588	−0.567942	−	0.823068i	$-0.692260\pi$
$47$	−366.000	−1.13588	−0.567942	+	0.823068i	$0.692260\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	252.000	0.691903
$52$	0	0
$53$	−768.000	−1.99043	−0.995216	−	0.0976975i	$-0.968852\pi$
$53$	−768.000	−1.99043	−0.995216	+	0.0976975i	$0.968852\pi$
$54$	0	0
$55$	180.000	0.441294
$56$	0	0
$57$	−291.000	−0.676209
$58$	0	0
$59$	−264.000	−0.582540	−0.291270	−	0.956641i	$-0.594078\pi$
$59$	−264.000	−0.582540	−0.291270	+	0.956641i	$0.594078\pi$
$60$	0	0
$61$	−818.000	−1.71695	−0.858477	−	0.512852i	$-0.828589\pi$
$61$	−818.000	−1.71695	−0.858477	+	0.512852i	$0.828589\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−318.000	−0.606816
$66$	0	0
$67$	523.000	0.953651	0.476826	−	0.878998i	$-0.341787\pi$
$67$	523.000	0.953651	0.476826	+	0.878998i	$0.341787\pi$
$68$	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
$73$	43.0000	0.0689420	0.0344710	−	0.999406i	$-0.489025\pi$
$73$	43.0000	0.0689420	0.0344710	+	0.999406i	$0.489025\pi$
$74$	0	0
$75$	−267.000	−0.411073
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1171.00	1.66769	0.833847	−	0.551996i	$-0.186134\pi$
$79$	1171.00	1.66769	0.833847	+	0.551996i	$0.186134\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−810.000	−1.07119	−0.535597	−	0.844474i	$-0.679913\pi$
$83$	−810.000	−1.07119	−0.535597	+	0.844474i	$0.679913\pi$
$84$	0	0
$85$	504.000	0.643135
$86$	0	0
$87$	−540.000	−0.665449
$88$	0	0
$89$	600.000	0.714605	0.357303	−	0.933989i	$-0.383696\pi$
$89$	600.000	0.714605	0.357303	+	0.933989i	$0.383696\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	537.000	0.598756
$94$	0	0
$95$	−582.000	−0.628547
$96$	0	0
$97$	−386.000	−0.404045	−0.202022	−	0.979381i	$-0.564751\pi$
$97$	−386.000	−0.404045	−0.202022	+	0.979381i	$0.564751\pi$
$98$	0	0
$99$	270.000	0.274101
$100$	0	0
$101$	−618.000	−0.608845	−0.304422	−	0.952537i	$-0.598463\pi$
$101$	−618.000	−0.608845	−0.304422	+	0.952537i	$0.598463\pi$
$102$	0	0
$103$	1475.00	1.41103	0.705515	−	0.708695i	$-0.250716\pi$
$103$	1475.00	1.41103	0.705515	+	0.708695i	$0.250716\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1884.00	−1.70218	−0.851090	−	0.525021i	$-0.824058\pi$
$107$	−1884.00	−1.70218	−0.851090	+	0.525021i	$0.824058\pi$
$108$	0	0
$109$	413.000	0.362920	0.181460	−	0.983398i	$-0.441918\pi$
$109$	413.000	0.362920	0.181460	+	0.983398i	$0.441918\pi$
$110$	0	0
$111$	−435.000	−0.371967
$112$	0	0
$113$	−882.000	−0.734262	−0.367131	−	0.930169i	$-0.619660\pi$
$113$	−882.000	−0.734262	−0.367131	+	0.930169i	$0.619660\pi$
$114$	0	0
$115$	−504.000	−0.408680
$116$	0	0
$117$	−477.000	−0.376912
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−431.000	−0.323817
$122$	0	0
$123$	−378.000	−0.277098
$124$	0	0
$125$	−1284.00	−0.918756
$126$	0	0
$127$	−2483.00	−1.73489	−0.867443	−	0.497536i	$-0.834238\pi$
$127$	−2483.00	−1.73489	−0.867443	+	0.497536i	$0.834238\pi$
$128$	0	0
$129$	975.000	0.665457
$130$	0	0
$131$	2118.00	1.41260	0.706300	−	0.707913i	$-0.250363\pi$
$131$	2118.00	1.41260	0.706300	+	0.707913i	$0.250363\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	162.000	0.103280
$136$	0	0
$137$	3012.00	1.87834	0.939170	−	0.343453i	$-0.111597\pi$
$137$	3012.00	1.87834	0.939170	+	0.343453i	$0.111597\pi$
$138$	0	0
$139$	−37.0000	−0.0225777	−0.0112888	−	0.999936i	$-0.503593\pi$
$139$	−37.0000	−0.0225777	−0.0112888	+	0.999936i	$0.503593\pi$
$140$	0	0
$141$	−1098.00	−0.655803
$142$	0	0
$143$	−1590.00	−0.929808
$144$	0	0
$145$	−1080.00	−0.618546
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1644.00	−0.903904	−0.451952	−	0.892042i	$-0.649272\pi$
$149$	−1644.00	−0.903904	−0.451952	+	0.892042i	$0.649272\pi$
$150$	0	0
$151$	−1088.00	−0.586359	−0.293179	−	0.956057i	$-0.594713\pi$
$151$	−1088.00	−0.586359	−0.293179	+	0.956057i	$0.594713\pi$
$152$	0	0
$153$	756.000	0.399470
$154$	0	0
$155$	1074.00	0.556553
$156$	0	0
$157$	−506.000	−0.257218	−0.128609	−	0.991695i	$-0.541051\pi$
$157$	−506.000	−0.257218	−0.128609	+	0.991695i	$0.541051\pi$
$158$	0	0
$159$	−2304.00	−1.14918
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−1844.00	−0.886093	−0.443047	−	0.896499i	$-0.646102\pi$
$163$	−1844.00	−0.886093	−0.443047	+	0.896499i	$0.646102\pi$
$164$	0	0
$165$	540.000	0.254781
$166$	0	0
$167$	162.000	0.0750655	0.0375327	−	0.999295i	$-0.488050\pi$
$167$	162.000	0.0750655	0.0375327	+	0.999295i	$0.488050\pi$
$168$	0	0
$169$	612.000	0.278562
$170$	0	0
$171$	−873.000	−0.390409
$172$	0	0
$173$	2724.00	1.19712	0.598560	−	0.801078i	$-0.295740\pi$
$173$	2724.00	1.19712	0.598560	+	0.801078i	$0.295740\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−792.000	−0.336330
$178$	0	0
$179$	1254.00	0.523622	0.261811	−	0.965119i	$-0.415680\pi$
$179$	1254.00	0.523622	0.261811	+	0.965119i	$0.415680\pi$
$180$	0	0
$181$	1807.00	0.742062	0.371031	−	0.928620i	$-0.379004\pi$
$181$	1807.00	0.742062	0.371031	+	0.928620i	$0.379004\pi$
$182$	0	0
$183$	−2454.00	−0.991284
$184$	0	0
$185$	−870.000	−0.345750
$186$	0	0
$187$	2520.00	0.985458
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−714.000	−0.270488	−0.135244	−	0.990812i	$-0.543182\pi$
$191$	−714.000	−0.270488	−0.135244	+	0.990812i	$0.543182\pi$
$192$	0	0
$193$	−3709.00	−1.38331	−0.691657	−	0.722226i	$-0.743119\pi$
$193$	−3709.00	−1.38331	−0.691657	+	0.722226i	$0.743119\pi$
$194$	0	0
$195$	−954.000	−0.350345
$196$	0	0
$197$	−1044.00	−0.377573	−0.188787	−	0.982018i	$-0.560455\pi$
$197$	−1044.00	−0.377573	−0.188787	+	0.982018i	$0.560455\pi$
$198$	0	0
$199$	−136.000	−0.0484462	−0.0242231	−	0.999707i	$-0.507711\pi$
$199$	−136.000	−0.0484462	−0.0242231	+	0.999707i	$0.507711\pi$
$200$	0	0
$201$	1569.00	0.550591
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−756.000	−0.257567
$206$	0	0
$207$	−756.000	−0.253844
$208$	0	0
$209$	−2910.00	−0.963105
$210$	0	0
$211$	−1484.00	−0.484184	−0.242092	−	0.970253i	$-0.577834\pi$
$211$	−1484.00	−0.484184	−0.242092	+	0.970253i	$0.577834\pi$
$212$	0	0
$213$	1026.00	0.330049
$214$	0	0
$215$	1950.00	0.618553
$216$	0	0
$217$	0	0
$218$	0	0
$219$	129.000	0.0398037
$220$	0	0
$221$	−4452.00	−1.35509
$222$	0	0
$223$	−2032.00	−0.610192	−0.305096	−	0.952322i	$-0.598689\pi$
$223$	−2032.00	−0.610192	−0.305096	+	0.952322i	$0.598689\pi$
$224$	0	0
$225$	−801.000	−0.237333
$226$	0	0
$227$	6198.00	1.81223	0.906114	−	0.423034i	$-0.139035\pi$
$227$	6198.00	1.81223	0.906114	+	0.423034i	$0.139035\pi$
$228$	0	0
$229$	4591.00	1.32481	0.662406	−	0.749145i	$-0.269536\pi$
$229$	4591.00	1.32481	0.662406	+	0.749145i	$0.269536\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4530.00	1.27369	0.636846	−	0.770991i	$-0.280239\pi$
$233$	4530.00	1.27369	0.636846	+	0.770991i	$0.280239\pi$
$234$	0	0
$235$	−2196.00	−0.609580
$236$	0	0
$237$	3513.00	0.962843
$238$	0	0
$239$	−1530.00	−0.414090	−0.207045	−	0.978331i	$-0.566385\pi$
$239$	−1530.00	−0.414090	−0.207045	+	0.978331i	$0.566385\pi$
$240$	0	0
$241$	−5534.00	−1.47915	−0.739577	−	0.673072i	$-0.764975\pi$
$241$	−5534.00	−1.47915	−0.739577	+	0.673072i	$0.764975\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5141.00	1.32435
$248$	0	0
$249$	−2430.00	−0.618454
$250$	0	0
$251$	−468.000	−0.117689	−0.0588444	−	0.998267i	$-0.518742\pi$
$251$	−468.000	−0.117689	−0.0588444	+	0.998267i	$0.518742\pi$
$252$	0	0
$253$	−2520.00	−0.626210
$254$	0	0
$255$	1512.00	0.371314
$256$	0	0
$257$	2490.00	0.604365	0.302183	−	0.953250i	$-0.402285\pi$
$257$	2490.00	0.604365	0.302183	+	0.953250i	$0.402285\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1620.00	−0.384197
$262$	0	0
$263$	−1572.00	−0.368569	−0.184285	−	0.982873i	$-0.558997\pi$
$263$	−1572.00	−0.368569	−0.184285	+	0.982873i	$0.558997\pi$
$264$	0	0
$265$	−4608.00	−1.06818
$266$	0	0
$267$	1800.00	0.412578
$268$	0	0
$269$	−1806.00	−0.409345	−0.204672	−	0.978831i	$-0.565613\pi$
$269$	−1806.00	−0.409345	−0.204672	+	0.978831i	$0.565613\pi$
$270$	0	0
$271$	−6112.00	−1.37003	−0.685014	−	0.728530i	$-0.740204\pi$
$271$	−6112.00	−1.37003	−0.685014	+	0.728530i	$0.740204\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2670.00	−0.585480
$276$	0	0
$277$	−4231.00	−0.917748	−0.458874	−	0.888501i	$-0.651747\pi$
$277$	−4231.00	−0.917748	−0.458874	+	0.888501i	$0.651747\pi$
$278$	0	0
$279$	1611.00	0.345692
$280$	0	0
$281$	−3816.00	−0.810119	−0.405060	−	0.914290i	$-0.632749\pi$
$281$	−3816.00	−0.810119	−0.405060	+	0.914290i	$0.632749\pi$
$282$	0	0
$283$	−3997.00	−0.839565	−0.419783	−	0.907625i	$-0.637894\pi$
$283$	−3997.00	−0.839565	−0.419783	+	0.907625i	$0.637894\pi$
$284$	0	0
$285$	−1746.00	−0.362892
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2143.00	0.436190
$290$	0	0
$291$	−1158.00	−0.233275
$292$	0	0
$293$	−4608.00	−0.918779	−0.459389	−	0.888235i	$-0.651932\pi$
$293$	−4608.00	−0.918779	−0.459389	+	0.888235i	$0.651932\pi$
$294$	0	0
$295$	−1584.00	−0.312624
$296$	0	0
$297$	810.000	0.158252
$298$	0	0
$299$	4452.00	0.861090
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1854.00	−0.351517
$304$	0	0
$305$	−4908.00	−0.921414
$306$	0	0
$307$	−631.000	−0.117306	−0.0586532	−	0.998278i	$-0.518681\pi$
$307$	−631.000	−0.117306	−0.0586532	+	0.998278i	$0.518681\pi$
$308$	0	0
$309$	4425.00	0.814658
$310$	0	0
$311$	3894.00	0.709995	0.354998	−	0.934867i	$-0.384482\pi$
$311$	3894.00	0.709995	0.354998	+	0.934867i	$0.384482\pi$
$312$	0	0
$313$	2185.00	0.394580	0.197290	−	0.980345i	$-0.436786\pi$
$313$	2185.00	0.394580	0.197290	+	0.980345i	$0.436786\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3504.00	0.620834	0.310417	−	0.950601i	$-0.399531\pi$
$317$	3504.00	0.620834	0.310417	+	0.950601i	$0.399531\pi$
$318$	0	0
$319$	−5400.00	−0.947780
$320$	0	0
$321$	−5652.00	−0.982754
$322$	0	0
$323$	−8148.00	−1.40361
$324$	0	0
$325$	4717.00	0.805083
$326$	0	0
$327$	1239.00	0.209532
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2945.00	−0.489039	−0.244519	−	0.969644i	$-0.578630\pi$
$331$	−2945.00	−0.489039	−0.244519	+	0.969644i	$0.578630\pi$
$332$	0	0
$333$	−1305.00	−0.214755
$334$	0	0
$335$	3138.00	0.511783
$336$	0	0
$337$	4277.00	0.691344	0.345672	−	0.938355i	$-0.387651\pi$
$337$	4277.00	0.691344	0.345672	+	0.938355i	$0.387651\pi$
$338$	0	0
$339$	−2646.00	−0.423926
$340$	0	0
$341$	5370.00	0.852791
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1512.00	−0.235952
$346$	0	0
$347$	−7188.00	−1.11202	−0.556012	−	0.831175i	$-0.687669\pi$
$347$	−7188.00	−1.11202	−0.556012	+	0.831175i	$0.687669\pi$
$348$	0	0
$349$	9406.00	1.44267	0.721335	−	0.692587i	$-0.243529\pi$
$349$	9406.00	1.44267	0.721335	+	0.692587i	$0.243529\pi$
$350$	0	0
$351$	−1431.00	−0.217610
$352$	0	0
$353$	−3390.00	−0.511137	−0.255569	−	0.966791i	$-0.582263\pi$
$353$	−3390.00	−0.511137	−0.255569	+	0.966791i	$0.582263\pi$
$354$	0	0
$355$	2052.00	0.306785
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4812.00	0.707431	0.353715	−	0.935353i	$-0.384918\pi$
$359$	4812.00	0.707431	0.353715	+	0.935353i	$0.384918\pi$
$360$	0	0
$361$	2550.00	0.371774
$362$	0	0
$363$	−1293.00	−0.186956
$364$	0	0
$365$	258.000	0.0369982
$366$	0	0
$367$	−7099.00	−1.00971	−0.504857	−	0.863203i	$-0.668455\pi$
$367$	−7099.00	−1.00971	−0.504857	+	0.863203i	$0.668455\pi$
$368$	0	0
$369$	−1134.00	−0.159983
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2963.00	0.411309	0.205655	−	0.978625i	$-0.434068\pi$
$373$	2963.00	0.411309	0.205655	+	0.978625i	$0.434068\pi$
$374$	0	0
$375$	−3852.00	−0.530444
$376$	0	0
$377$	9540.00	1.30328
$378$	0	0
$379$	11899.0	1.61269	0.806346	−	0.591444i	$-0.201442\pi$
$379$	11899.0	1.61269	0.806346	+	0.591444i	$0.201442\pi$
$380$	0	0
$381$	−7449.00	−1.00164
$382$	0	0
$383$	2568.00	0.342607	0.171304	−	0.985218i	$-0.445202\pi$
$383$	2568.00	0.342607	0.171304	+	0.985218i	$0.445202\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2925.00	0.384202
$388$	0	0
$389$	−10146.0	−1.32242	−0.661212	−	0.750199i	$-0.729957\pi$
$389$	−10146.0	−1.32242	−0.661212	+	0.750199i	$0.729957\pi$
$390$	0	0
$391$	−7056.00	−0.912627
$392$	0	0
$393$	6354.00	0.815565
$394$	0	0
$395$	7026.00	0.894978
$396$	0	0
$397$	6229.00	0.787467	0.393734	−	0.919225i	$-0.371183\pi$
$397$	6229.00	0.787467	0.393734	+	0.919225i	$0.371183\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2472.00	−0.307845	−0.153922	−	0.988083i	$-0.549191\pi$
$401$	−2472.00	−0.307845	−0.153922	+	0.988083i	$0.549191\pi$
$402$	0	0
$403$	−9487.00	−1.17266
$404$	0	0
$405$	486.000	0.0596285
$406$	0	0
$407$	−4350.00	−0.529783
$408$	0	0
$409$	7075.00	0.855345	0.427673	−	0.903934i	$-0.359334\pi$
$409$	7075.00	0.855345	0.427673	+	0.903934i	$0.359334\pi$
$410$	0	0
$411$	9036.00	1.08446
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4860.00	−0.574863
$416$	0	0
$417$	−111.000	−0.0130352
$418$	0	0
$419$	−4158.00	−0.484801	−0.242400	−	0.970176i	$-0.577935\pi$
$419$	−4158.00	−0.484801	−0.242400	+	0.970176i	$0.577935\pi$
$420$	0	0
$421$	−6595.00	−0.763469	−0.381735	−	0.924272i	$-0.624673\pi$
$421$	−6595.00	−0.763469	−0.381735	+	0.924272i	$0.624673\pi$
$422$	0	0
$423$	−3294.00	−0.378628
$424$	0	0
$425$	−7476.00	−0.853269
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−4770.00	−0.536825
$430$	0	0
$431$	−1518.00	−0.169651	−0.0848254	−	0.996396i	$-0.527033\pi$
$431$	−1518.00	−0.169651	−0.0848254	+	0.996396i	$0.527033\pi$
$432$	0	0
$433$	−8567.00	−0.950817	−0.475408	−	0.879765i	$-0.657700\pi$
$433$	−8567.00	−0.950817	−0.475408	+	0.879765i	$0.657700\pi$
$434$	0	0
$435$	−3240.00	−0.357117
$436$	0	0
$437$	8148.00	0.891926
$438$	0	0
$439$	10640.0	1.15676	0.578382	−	0.815766i	$-0.303684\pi$
$439$	10640.0	1.15676	0.578382	+	0.815766i	$0.303684\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7032.00	−0.754177	−0.377088	−	0.926177i	$-0.623075\pi$
$443$	−7032.00	−0.754177	−0.377088	+	0.926177i	$0.623075\pi$
$444$	0	0
$445$	3600.00	0.383497
$446$	0	0
$447$	−4932.00	−0.521869
$448$	0	0
$449$	−14814.0	−1.55705	−0.778525	−	0.627613i	$-0.784032\pi$
$449$	−14814.0	−1.55705	−0.778525	+	0.627613i	$0.784032\pi$
$450$	0	0
$451$	−3780.00	−0.394664
$452$	0	0
$453$	−3264.00	−0.338534
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−11251.0	−1.15164	−0.575820	−	0.817576i	$-0.695317\pi$
$457$	−11251.0	−1.15164	−0.575820	+	0.817576i	$0.695317\pi$
$458$	0	0
$459$	2268.00	0.230634
$460$	0	0
$461$	3852.00	0.389166	0.194583	−	0.980886i	$-0.437665\pi$
$461$	3852.00	0.389166	0.194583	+	0.980886i	$0.437665\pi$
$462$	0	0
$463$	475.000	0.0476784	0.0238392	−	0.999716i	$-0.492411\pi$
$463$	475.000	0.0476784	0.0238392	+	0.999716i	$0.492411\pi$
$464$	0	0
$465$	3222.00	0.321326
$466$	0	0
$467$	5934.00	0.587993	0.293997	−	0.955806i	$-0.405015\pi$
$467$	5934.00	0.587993	0.293997	+	0.955806i	$0.405015\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1518.00	−0.148505
$472$	0	0
$473$	9750.00	0.947792
$474$	0	0
$475$	8633.00	0.833914
$476$	0	0
$477$	−6912.00	−0.663477
$478$	0	0
$479$	−13368.0	−1.27516	−0.637578	−	0.770386i	$-0.720064\pi$
$479$	−13368.0	−1.27516	−0.637578	+	0.770386i	$0.720064\pi$
$480$	0	0
$481$	7685.00	0.728494
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2316.00	−0.216833
$486$	0	0
$487$	−6653.00	−0.619048	−0.309524	−	0.950892i	$-0.600170\pi$
$487$	−6653.00	−0.619048	−0.309524	+	0.950892i	$0.600170\pi$
$488$	0	0
$489$	−5532.00	−0.511586
$490$	0	0
$491$	−15444.0	−1.41951	−0.709754	−	0.704450i	$-0.751194\pi$
$491$	−15444.0	−1.41951	−0.709754	+	0.704450i	$0.751194\pi$
$492$	0	0
$493$	−15120.0	−1.38128
$494$	0	0
$495$	1620.00	0.147098
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−683.000	−0.0612731	−0.0306366	−	0.999531i	$-0.509753\pi$
$499$	−683.000	−0.0612731	−0.0306366	+	0.999531i	$0.509753\pi$
$500$	0	0
$501$	486.000	0.0433391
$502$	0	0
$503$	9882.00	0.875977	0.437989	−	0.898980i	$-0.355691\pi$
$503$	9882.00	0.875977	0.437989	+	0.898980i	$0.355691\pi$
$504$	0	0
$505$	−3708.00	−0.326740
$506$	0	0
$507$	1836.00	0.160828
$508$	0	0
$509$	−4206.00	−0.366263	−0.183131	−	0.983088i	$-0.558623\pi$
$509$	−4206.00	−0.366263	−0.183131	+	0.983088i	$0.558623\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2619.00	−0.225403
$514$	0	0
$515$	8850.00	0.757238
$516$	0	0
$517$	−10980.0	−0.934042
$518$	0	0
$519$	8172.00	0.691158
$520$	0	0
$521$	−9060.00	−0.761854	−0.380927	−	0.924605i	$-0.624395\pi$
$521$	−9060.00	−0.761854	−0.380927	+	0.924605i	$0.624395\pi$
$522$	0	0
$523$	−15679.0	−1.31089	−0.655444	−	0.755243i	$-0.727519\pi$
$523$	−15679.0	−1.31089	−0.655444	+	0.755243i	$0.727519\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	15036.0	1.24284
$528$	0	0
$529$	−5111.00	−0.420071
$530$	0	0
$531$	−2376.00	−0.194180
$532$	0	0
$533$	6678.00	0.542695
$534$	0	0
$535$	−11304.0	−0.913485
$536$	0	0
$537$	3762.00	0.302313
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−7711.00	−0.612794	−0.306397	−	0.951904i	$-0.599124\pi$
$541$	−7711.00	−0.612794	−0.306397	+	0.951904i	$0.599124\pi$
$542$	0	0
$543$	5421.00	0.428430
$544$	0	0
$545$	2478.00	0.194763
$546$	0	0
$547$	−4292.00	−0.335489	−0.167745	−	0.985830i	$-0.553648\pi$
$547$	−4292.00	−0.335489	−0.167745	+	0.985830i	$0.553648\pi$
$548$	0	0
$549$	−7362.00	−0.572318
$550$	0	0
$551$	17460.0	1.34995
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−2610.00	−0.199619
$556$	0	0
$557$	−9858.00	−0.749905	−0.374952	−	0.927044i	$-0.622341\pi$
$557$	−9858.00	−0.749905	−0.374952	+	0.927044i	$0.622341\pi$
$558$	0	0
$559$	−17225.0	−1.30329
$560$	0	0
$561$	7560.00	0.568954
$562$	0	0
$563$	−13890.0	−1.03978	−0.519888	−	0.854235i	$-0.674026\pi$
$563$	−13890.0	−1.03978	−0.519888	+	0.854235i	$0.674026\pi$
$564$	0	0
$565$	−5292.00	−0.394046
$566$	0	0
$567$	0	0
$568$	0	0
$569$	19038.0	1.40266	0.701331	−	0.712836i	$-0.252590\pi$
$569$	19038.0	1.40266	0.701331	+	0.712836i	$0.252590\pi$
$570$	0	0
$571$	8053.00	0.590206	0.295103	−	0.955465i	$-0.404646\pi$
$571$	8053.00	0.590206	0.295103	+	0.955465i	$0.404646\pi$
$572$	0	0
$573$	−2142.00	−0.156166
$574$	0	0
$575$	7476.00	0.542210
$576$	0	0
$577$	17137.0	1.23643	0.618217	−	0.786007i	$-0.287855\pi$
$577$	17137.0	1.23643	0.618217	+	0.786007i	$0.287855\pi$
$578$	0	0
$579$	−11127.0	−0.798657
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−23040.0	−1.63674
$584$	0	0
$585$	−2862.00	−0.202272
$586$	0	0
$587$	18144.0	1.27578	0.637890	−	0.770127i	$-0.279807\pi$
$587$	18144.0	1.27578	0.637890	+	0.770127i	$0.279807\pi$
$588$	0	0
$589$	−17363.0	−1.21465
$590$	0	0
$591$	−3132.00	−0.217992
$592$	0	0
$593$	24702.0	1.71061	0.855303	−	0.518128i	$-0.173371\pi$
$593$	24702.0	1.71061	0.855303	+	0.518128i	$0.173371\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−408.000	−0.0279704
$598$	0	0
$599$	2172.00	0.148156	0.0740781	−	0.997252i	$-0.476399\pi$
$599$	2172.00	0.148156	0.0740781	+	0.997252i	$0.476399\pi$
$600$	0	0
$601$	−4175.00	−0.283364	−0.141682	−	0.989912i	$-0.545251\pi$
$601$	−4175.00	−0.283364	−0.141682	+	0.989912i	$0.545251\pi$
$602$	0	0
$603$	4707.00	0.317884
$604$	0	0
$605$	−2586.00	−0.173778
$606$	0	0
$607$	2261.00	0.151188	0.0755940	−	0.997139i	$-0.475915\pi$
$607$	2261.00	0.151188	0.0755940	+	0.997139i	$0.475915\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	19398.0	1.28438
$612$	0	0
$613$	−16318.0	−1.07517	−0.537584	−	0.843210i	$-0.680663\pi$
$613$	−16318.0	−1.07517	−0.537584	+	0.843210i	$0.680663\pi$
$614$	0	0
$615$	−2268.00	−0.148707
$616$	0	0
$617$	−26550.0	−1.73235	−0.866177	−	0.499737i	$-0.833430\pi$
$617$	−26550.0	−1.73235	−0.866177	+	0.499737i	$0.833430\pi$
$618$	0	0
$619$	19925.0	1.29379	0.646893	−	0.762581i	$-0.276068\pi$
$619$	19925.0	1.29379	0.646893	+	0.762581i	$0.276068\pi$
$620$	0	0
$621$	−2268.00	−0.146557
$622$	0	0
$623$	0	0
$624$	0	0
$625$	3421.00	0.218944
$626$	0	0
$627$	−8730.00	−0.556049
$628$	0	0
$629$	−12180.0	−0.772096
$630$	0	0
$631$	6832.00	0.431026	0.215513	−	0.976501i	$-0.430858\pi$
$631$	6832.00	0.431026	0.215513	+	0.976501i	$0.430858\pi$
$632$	0	0
$633$	−4452.00	−0.279544
$634$	0	0
$635$	−14898.0	−0.931038
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3078.00	0.190554
$640$	0	0
$641$	10212.0	0.629251	0.314625	−	0.949216i	$-0.398121\pi$
$641$	10212.0	0.629251	0.314625	+	0.949216i	$0.398121\pi$
$642$	0	0
$643$	3779.00	0.231772	0.115886	−	0.993263i	$-0.463029\pi$
$643$	3779.00	0.231772	0.115886	+	0.993263i	$0.463029\pi$
$644$	0	0
$645$	5850.00	0.357122
$646$	0	0
$647$	16998.0	1.03286	0.516430	−	0.856329i	$-0.327261\pi$
$647$	16998.0	1.03286	0.516430	+	0.856329i	$0.327261\pi$
$648$	0	0
$649$	−7920.00	−0.479025
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−21750.0	−1.30344	−0.651718	−	0.758462i	$-0.725951\pi$
$653$	−21750.0	−1.30344	−0.651718	+	0.758462i	$0.725951\pi$
$654$	0	0
$655$	12708.0	0.758080
$656$	0	0
$657$	387.000	0.0229807
$658$	0	0
$659$	10944.0	0.646916	0.323458	−	0.946243i	$-0.395155\pi$
$659$	10944.0	0.646916	0.323458	+	0.946243i	$0.395155\pi$
$660$	0	0
$661$	−10955.0	−0.644630	−0.322315	−	0.946633i	$-0.604461\pi$
$661$	−10955.0	−0.644630	−0.322315	+	0.946633i	$0.604461\pi$
$662$	0	0
$663$	−13356.0	−0.782359
$664$	0	0
$665$	0	0
$666$	0	0
$667$	15120.0	0.877734
$668$	0	0
$669$	−6096.00	−0.352294
$670$	0	0
$671$	−24540.0	−1.41186
$672$	0	0
$673$	25103.0	1.43782	0.718908	−	0.695106i	$-0.244642\pi$
$673$	25103.0	1.43782	0.718908	+	0.695106i	$0.244642\pi$
$674$	0	0
$675$	−2403.00	−0.137024
$676$	0	0
$677$	5604.00	0.318138	0.159069	−	0.987267i	$-0.449151\pi$
$677$	5604.00	0.318138	0.159069	+	0.987267i	$0.449151\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	18594.0	1.04629
$682$	0	0
$683$	−10968.0	−0.614464	−0.307232	−	0.951635i	$-0.599403\pi$
$683$	−10968.0	−0.614464	−0.307232	+	0.951635i	$0.599403\pi$
$684$	0	0
$685$	18072.0	1.00802
$686$	0	0
$687$	13773.0	0.764880
$688$	0	0
$689$	40704.0	2.25065
$690$	0	0
$691$	8405.00	0.462723	0.231361	−	0.972868i	$-0.425682\pi$
$691$	8405.00	0.462723	0.231361	+	0.972868i	$0.425682\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−222.000	−0.0121165
$696$	0	0
$697$	−10584.0	−0.575176
$698$	0	0
$699$	13590.0	0.735366
$700$	0	0
$701$	468.000	0.0252156	0.0126078	−	0.999921i	$-0.495987\pi$
$701$	468.000	0.0252156	0.0126078	+	0.999921i	$0.495987\pi$
$702$	0	0
$703$	14065.0	0.754583
$704$	0	0
$705$	−6588.00	−0.351941
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−25066.0	−1.32775	−0.663874	−	0.747844i	$-0.731089\pi$
$709$	−25066.0	−1.32775	−0.663874	+	0.747844i	$0.731089\pi$
$710$	0	0
$711$	10539.0	0.555898
$712$	0	0
$713$	−15036.0	−0.789765
$714$	0	0
$715$	−9540.00	−0.498987
$716$	0	0
$717$	−4590.00	−0.239075
$718$	0	0
$719$	11082.0	0.574811	0.287405	−	0.957809i	$-0.407207\pi$
$719$	11082.0	0.574811	0.287405	+	0.957809i	$0.407207\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−16602.0	−0.853990
$724$	0	0
$725$	16020.0	0.820645
$726$	0	0
$727$	13481.0	0.687734	0.343867	−	0.939018i	$-0.388263\pi$
$727$	13481.0	0.687734	0.343867	+	0.939018i	$0.388263\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	27300.0	1.38130
$732$	0	0
$733$	−24317.0	−1.22533	−0.612666	−	0.790342i	$-0.709903\pi$
$733$	−24317.0	−1.22533	−0.612666	+	0.790342i	$0.709903\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	15690.0	0.784191
$738$	0	0
$739$	18217.0	0.906797	0.453399	−	0.891308i	$-0.350211\pi$
$739$	18217.0	0.906797	0.453399	+	0.891308i	$0.350211\pi$
$740$	0	0
$741$	15423.0	0.764613
$742$	0	0
$743$	−19782.0	−0.976758	−0.488379	−	0.872632i	$-0.662412\pi$
$743$	−19782.0	−0.976758	−0.488379	+	0.872632i	$0.662412\pi$
$744$	0	0
$745$	−9864.00	−0.485086
$746$	0	0
$747$	−7290.00	−0.357064
$748$	0	0
$749$	0	0
$750$	0	0
$751$	4921.00	0.239108	0.119554	−	0.992828i	$-0.461854\pi$
$751$	4921.00	0.239108	0.119554	+	0.992828i	$0.461854\pi$
$752$	0	0
$753$	−1404.00	−0.0679477
$754$	0	0
$755$	−6528.00	−0.314673
$756$	0	0
$757$	18098.0	0.868934	0.434467	−	0.900688i	$-0.356937\pi$
$757$	18098.0	0.868934	0.434467	+	0.900688i	$0.356937\pi$
$758$	0	0
$759$	−7560.00	−0.361542
$760$	0	0
$761$	24468.0	1.16552	0.582762	−	0.812643i	$-0.301972\pi$
$761$	24468.0	1.16552	0.582762	+	0.812643i	$0.301972\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	4536.00	0.214378
$766$	0	0
$767$	13992.0	0.658699
$768$	0	0
$769$	−21719.0	−1.01847	−0.509237	−	0.860626i	$-0.670072\pi$
$769$	−21719.0	−1.01847	−0.509237	+	0.860626i	$0.670072\pi$
$770$	0	0
$771$	7470.00	0.348931
$772$	0	0
$773$	30306.0	1.41013	0.705065	−	0.709142i	$-0.250918\pi$
$773$	30306.0	1.41013	0.705065	+	0.709142i	$0.250918\pi$
$774$	0	0
$775$	−15931.0	−0.738398
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12222.0	0.562129
$780$	0	0
$781$	10260.0	0.470079
$782$	0	0
$783$	−4860.00	−0.221816
$784$	0	0
$785$	−3036.00	−0.138038
$786$	0	0
$787$	27296.0	1.23634	0.618169	−	0.786046i	$-0.287875\pi$
$787$	27296.0	1.23634	0.618169	+	0.786046i	$0.287875\pi$
$788$	0	0
$789$	−4716.00	−0.212793
$790$	0	0
$791$	0	0
$792$	0	0
$793$	43354.0	1.94142
$794$	0	0
$795$	−13824.0	−0.616713
$796$	0	0
$797$	35100.0	1.55998	0.779991	−	0.625791i	$-0.215224\pi$
$797$	35100.0	1.55998	0.779991	+	0.625791i	$0.215224\pi$
$798$	0	0
$799$	−30744.0	−1.36126
$800$	0	0
$801$	5400.00	0.238202
$802$	0	0
$803$	1290.00	0.0566913
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5418.00	−0.236335
$808$	0	0
$809$	44394.0	1.92931	0.964654	−	0.263520i	$-0.0848836\pi$
$809$	44394.0	1.92931	0.964654	+	0.263520i	$0.0848836\pi$
$810$	0	0
$811$	−8584.00	−0.371671	−0.185835	−	0.982581i	$-0.559499\pi$
$811$	−8584.00	−0.371671	−0.185835	+	0.982581i	$0.559499\pi$
$812$	0	0
$813$	−18336.0	−0.790986
$814$	0	0
$815$	−11064.0	−0.475528
$816$	0	0
$817$	−31525.0	−1.34996
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−9834.00	−0.418038	−0.209019	−	0.977912i	$-0.567027\pi$
$821$	−9834.00	−0.418038	−0.209019	+	0.977912i	$0.567027\pi$
$822$	0	0
$823$	−43856.0	−1.85750	−0.928751	−	0.370704i	$-0.879116\pi$
$823$	−43856.0	−1.85750	−0.928751	+	0.370704i	$0.879116\pi$
$824$	0	0
$825$	−8010.00	−0.338027
$826$	0	0
$827$	−13266.0	−0.557804	−0.278902	−	0.960320i	$-0.589970\pi$
$827$	−13266.0	−0.557804	−0.278902	+	0.960320i	$0.589970\pi$
$828$	0	0
$829$	−17453.0	−0.731204	−0.365602	−	0.930771i	$-0.619137\pi$
$829$	−17453.0	−0.731204	−0.365602	+	0.930771i	$0.619137\pi$
$830$	0	0
$831$	−12693.0	−0.529862
$832$	0	0
$833$	0	0
$834$	0	0
$835$	972.000	0.0402844
$836$	0	0
$837$	4833.00	0.199585
$838$	0	0
$839$	−35172.0	−1.44729	−0.723643	−	0.690175i	$-0.757534\pi$
$839$	−35172.0	−1.44729	−0.723643	+	0.690175i	$0.757534\pi$
$840$	0	0
$841$	8011.00	0.328468
$842$	0	0
$843$	−11448.0	−0.467722
$844$	0	0
$845$	3672.00	0.149492
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−11991.0	−0.484723
$850$	0	0
$851$	12180.0	0.490629
$852$	0	0
$853$	−3503.00	−0.140610	−0.0703051	−	0.997526i	$-0.522397\pi$
$853$	−3503.00	−0.140610	−0.0703051	+	0.997526i	$0.522397\pi$
$854$	0	0
$855$	−5238.00	−0.209516
$856$	0	0
$857$	−22848.0	−0.910703	−0.455352	−	0.890312i	$-0.650486\pi$
$857$	−22848.0	−0.910703	−0.455352	+	0.890312i	$0.650486\pi$
$858$	0	0
$859$	−13456.0	−0.534474	−0.267237	−	0.963631i	$-0.586111\pi$
$859$	−13456.0	−0.534474	−0.267237	+	0.963631i	$0.586111\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−40710.0	−1.60578	−0.802888	−	0.596130i	$-0.796704\pi$
$863$	−40710.0	−1.60578	−0.802888	+	0.596130i	$0.796704\pi$
$864$	0	0
$865$	16344.0	0.642442
$866$	0	0
$867$	6429.00	0.251834
$868$	0	0
$869$	35130.0	1.37135
$870$	0	0
$871$	−27719.0	−1.07833
$872$	0	0
$873$	−3474.00	−0.134682
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2906.00	0.111891	0.0559456	−	0.998434i	$-0.482183\pi$
$877$	2906.00	0.111891	0.0559456	+	0.998434i	$0.482183\pi$
$878$	0	0
$879$	−13824.0	−0.530457
$880$	0	0
$881$	19188.0	0.733780	0.366890	−	0.930264i	$-0.380423\pi$
$881$	19188.0	0.733780	0.366890	+	0.930264i	$0.380423\pi$
$882$	0	0
$883$	17251.0	0.657466	0.328733	−	0.944423i	$-0.393378\pi$
$883$	17251.0	0.657466	0.328733	+	0.944423i	$0.393378\pi$
$884$	0	0
$885$	−4752.00	−0.180493
$886$	0	0
$887$	−2094.00	−0.0792668	−0.0396334	−	0.999214i	$-0.512619\pi$
$887$	−2094.00	−0.0792668	−0.0396334	+	0.999214i	$0.512619\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2430.00	0.0913671
$892$	0	0
$893$	35502.0	1.33038
$894$	0	0
$895$	7524.00	0.281005
$896$	0	0
$897$	13356.0	0.497150
$898$	0	0
$899$	−32220.0	−1.19532
$900$	0	0
$901$	−64512.0	−2.38536
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10842.0	0.398232
$906$	0	0
$907$	40267.0	1.47414	0.737069	−	0.675817i	$-0.236209\pi$
$907$	40267.0	1.47414	0.737069	+	0.675817i	$0.236209\pi$
$908$	0	0
$909$	−5562.00	−0.202948
$910$	0	0
$911$	−17604.0	−0.640227	−0.320113	−	0.947379i	$-0.603721\pi$
$911$	−17604.0	−0.640227	−0.320113	+	0.947379i	$0.603721\pi$
$912$	0	0
$913$	−24300.0	−0.880846
$914$	0	0
$915$	−14724.0	−0.531979
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−3509.00	−0.125953	−0.0629767	−	0.998015i	$-0.520059\pi$
$919$	−3509.00	−0.125953	−0.0629767	+	0.998015i	$0.520059\pi$
$920$	0	0
$921$	−1893.00	−0.0677269
$922$	0	0
$923$	−18126.0	−0.646397
$924$	0	0
$925$	12905.0	0.458718
$926$	0	0
$927$	13275.0	0.470343
$928$	0	0
$929$	34638.0	1.22329	0.611645	−	0.791133i	$-0.290508\pi$
$929$	34638.0	1.22329	0.611645	+	0.791133i	$0.290508\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	11682.0	0.409916
$934$	0	0
$935$	15120.0	0.528852
$936$	0	0
$937$	17353.0	0.605014	0.302507	−	0.953147i	$-0.402176\pi$
$937$	17353.0	0.605014	0.302507	+	0.953147i	$0.402176\pi$
$938$	0	0
$939$	6555.00	0.227811
$940$	0	0
$941$	46920.0	1.62545	0.812725	−	0.582648i	$-0.197983\pi$
$941$	46920.0	1.62545	0.812725	+	0.582648i	$0.197983\pi$
$942$	0	0
$943$	10584.0	0.365496
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−18354.0	−0.629804	−0.314902	−	0.949124i	$-0.601972\pi$
$947$	−18354.0	−0.629804	−0.314902	+	0.949124i	$0.601972\pi$
$948$	0	0
$949$	−2279.00	−0.0779552
$950$	0	0
$951$	10512.0	0.358438
$952$	0	0
$953$	35568.0	1.20898	0.604491	−	0.796612i	$-0.293376\pi$
$953$	35568.0	1.20898	0.604491	+	0.796612i	$0.293376\pi$
$954$	0	0
$955$	−4284.00	−0.145159
$956$	0	0
$957$	−16200.0	−0.547201
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2250.00	0.0755262
$962$	0	0
$963$	−16956.0	−0.567393
$964$	0	0
$965$	−22254.0	−0.742364
$966$	0	0
$967$	27343.0	0.909298	0.454649	−	0.890671i	$-0.349765\pi$
$967$	27343.0	0.909298	0.454649	+	0.890671i	$0.349765\pi$
$968$	0	0
$969$	−24444.0	−0.810376
$970$	0	0
$971$	51024.0	1.68634	0.843171	−	0.537645i	$-0.180686\pi$
$971$	51024.0	1.68634	0.843171	+	0.537645i	$0.180686\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	14151.0	0.464815
$976$	0	0
$977$	−2226.00	−0.0728926	−0.0364463	−	0.999336i	$-0.511604\pi$
$977$	−2226.00	−0.0728926	−0.0364463	+	0.999336i	$0.511604\pi$
$978$	0	0
$979$	18000.0	0.587623
$980$	0	0
$981$	3717.00	0.120973
$982$	0	0
$983$	35304.0	1.14550	0.572748	−	0.819731i	$-0.305877\pi$
$983$	35304.0	1.14550	0.572748	+	0.819731i	$0.305877\pi$
$984$	0	0
$985$	−6264.00	−0.202627
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−27300.0	−0.877745
$990$	0	0
$991$	2341.00	0.0750397	0.0375198	−	0.999296i	$-0.488054\pi$
$991$	2341.00	0.0750397	0.0375198	+	0.999296i	$0.488054\pi$
$992$	0	0
$993$	−8835.00	−0.282347
$994$	0	0
$995$	−816.000	−0.0259989
$996$	0	0
$997$	−29015.0	−0.921679	−0.460840	−	0.887483i	$-0.652452\pi$
$997$	−29015.0	−0.921679	−0.460840	+	0.887483i	$0.652452\pi$
$998$	0	0
$999$	−3915.00	−0.123989

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.bf.1.1		1
4.3	odd	2		294.4.a.c.1.1		1
7.3	odd	6		336.4.q.f.289.1		2
7.5	odd	6		336.4.q.f.193.1		2
7.6	odd	2		2352.4.a.f.1.1		1
12.11	even	2		882.4.a.l.1.1		1
28.3	even	6		42.4.e.a.37.1	yes	2
28.11	odd	6		294.4.e.i.79.1		2
28.19	even	6		42.4.e.a.25.1	✓	2
28.23	odd	6		294.4.e.i.67.1		2
28.27	even	2		294.4.a.d.1.1		1
84.11	even	6		882.4.g.g.667.1		2
84.23	even	6		882.4.g.g.361.1		2
84.47	odd	6		126.4.g.b.109.1		2
84.59	odd	6		126.4.g.b.37.1		2
84.83	odd	2		882.4.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.e.a.25.1	✓	2	28.19	even	6
42.4.e.a.37.1	yes	2	28.3	even	6
126.4.g.b.37.1		2	84.59	odd	6
126.4.g.b.109.1		2	84.47	odd	6
294.4.a.c.1.1		1	4.3	odd	2
294.4.a.d.1.1		1	28.27	even	2
294.4.e.i.67.1		2	28.23	odd	6
294.4.e.i.79.1		2	28.11	odd	6
336.4.q.f.193.1		2	7.5	odd	6
336.4.q.f.289.1		2	7.3	odd	6
882.4.a.l.1.1		1	12.11	even	2
882.4.a.o.1.1		1	84.83	odd	2
882.4.g.g.361.1		2	84.23	even	6
882.4.g.g.667.1		2	84.11	even	6
2352.4.a.f.1.1		1	7.6	odd	2
2352.4.a.bf.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

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