LMFDB - Embedded newform 2352.3.f.f.97.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,3,Mod(97,2352)]

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, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2352.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2352.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$64.0873581775$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{65})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 17x^{2} + 16x + 256 $ x^4 - x^3 + 17x^2 + 16x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.4
Root			$2.26556 + 3.92407i$ of defining polynomial
Character	$\chi$	$=$	2352.97
Dual form			2352.3.f.f.97.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +9.58020i q^{5} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} +9.58020i q^{5} -3.00000 q^{9} -4.59339 q^{11} +7.84815i q^{13} -16.5934 q^{15} -19.1604i q^{17} -16.4006i q^{19} +24.0000 q^{23} -66.7802 q^{25} -5.19615i q^{27} -10.5934 q^{29} -55.7491i q^{31} -7.95598i q^{33} -24.4066 q^{37} -13.5934 q^{39} -48.7131i q^{41} +18.7802 q^{43} -28.7406i q^{45} -12.0165i q^{47} +33.1868 q^{51} -37.4066 q^{53} -44.0055i q^{55} +28.4066 q^{57} -63.1659i q^{59} +111.283i q^{61} -75.1868 q^{65} -63.9669 q^{67} +41.5692i q^{69} -21.5603 q^{71} -61.8652i q^{73} -115.667i q^{75} +3.81323 q^{79} +9.00000 q^{81} -100.510i q^{83} +183.560 q^{85} -18.3483i q^{87} -99.0504i q^{89} +96.5603 q^{93} +157.121 q^{95} +63.5973i q^{97} +13.7802 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} + 30 q^{11} - 18 q^{15} + 96 q^{23} - 122 q^{25} + 6 q^{29} - 146 q^{37} - 6 q^{39} - 70 q^{43} + 36 q^{51} - 198 q^{53} + 162 q^{57} - 204 q^{65} - 14 q^{67} + 204 q^{71} + 112 q^{79} + 36 q^{81} + 444 q^{85} + 96 q^{93} + 48 q^{95} - 90 q^{99}+O(q^{100}) $ 4 * q - 12 * q^9 + 30 * q^11 - 18 * q^15 + 96 * q^23 - 122 * q^25 + 6 * q^29 - 146 * q^37 - 6 * q^39 - 70 * q^43 + 36 * q^51 - 198 * q^53 + 162 * q^57 - 204 * q^65 - 14 * q^67 + 204 * q^71 + 112 * q^79 + 36 * q^81 + 444 * q^85 + 96 * q^93 + 48 * q^95 - 90 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	9.58020i	1.91604i	0.286703	+	0.958020i	$0.407441\pi$
$5$	9.58020i	1.91604i	−0.286703	+	0.958020i	$0.592559\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	−4.59339	−0.417581	−0.208790	−	0.977960i	$-0.566953\pi$
$11$	−4.59339	−0.417581	−0.208790	+	0.977960i	$0.566953\pi$
$12$	0	0
$13$	7.84815i	0.603703i	0.953355	+	0.301852i	$0.0976048\pi$
$13$	7.84815i	0.603703i	−0.953355	+	0.301852i	$0.902395\pi$
$14$	0	0
$15$	−16.5934	−1.10623
$16$	0	0
$17$	− 19.1604i	− 1.12708i	−0.826088	−	0.563541i	$-0.809439\pi$
$17$	− 19.1604i	− 1.12708i	0.826088	−	0.563541i	$-0.190561\pi$
$18$	0	0
$19$	− 16.4006i	− 0.863188i	−0.902068	−	0.431594i	$-0.857951\pi$
$19$	− 16.4006i	− 0.863188i	0.902068	−	0.431594i	$-0.142049\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	24.0000	1.04348	0.521739	−	0.853105i	$-0.325283\pi$
$23$	24.0000	1.04348	0.521739	+	0.853105i	$0.325283\pi$
$24$	0	0
$25$	−66.7802	−2.67121
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	−10.5934	−0.365289	−0.182645	−	0.983179i	$-0.558466\pi$
$29$	−10.5934	−0.365289	−0.182645	+	0.983179i	$0.558466\pi$
$30$	0	0
$31$	− 55.7491i	− 1.79836i	−0.437580	−	0.899179i	$-0.644164\pi$
$31$	− 55.7491i	− 1.79836i	0.437580	−	0.899179i	$-0.355836\pi$
$32$	0	0
$33$	− 7.95598i	− 0.241090i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−24.4066	−0.659638	−0.329819	−	0.944044i	$-0.606988\pi$
$37$	−24.4066	−0.659638	−0.329819	+	0.944044i	$0.606988\pi$
$38$	0	0
$39$	−13.5934	−0.348548
$40$	0	0
$41$	− 48.7131i	− 1.18812i	−0.804419	−	0.594062i	$-0.797523\pi$
$41$	− 48.7131i	− 1.18812i	0.804419	−	0.594062i	$-0.202477\pi$
$42$	0	0
$43$	18.7802	0.436748	0.218374	−	0.975865i	$-0.429925\pi$
$43$	18.7802	0.436748	0.218374	+	0.975865i	$0.429925\pi$
$44$	0	0
$45$	− 28.7406i	− 0.638680i
$46$	0	0
$47$	− 12.0165i	− 0.255671i	−0.991795	−	0.127835i	$-0.959197\pi$
$47$	− 12.0165i	− 0.255671i	0.991795	−	0.127835i	$-0.0408029\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	33.1868	0.650721
$52$	0	0
$53$	−37.4066	−0.705785	−0.352893	−	0.935664i	$-0.614802\pi$
$53$	−37.4066	−0.705785	−0.352893	+	0.935664i	$0.614802\pi$
$54$	0	0
$55$	− 44.0055i	− 0.800101i
$56$	0	0
$57$	28.4066	0.498362
$58$	0	0
$59$	− 63.1659i	− 1.07061i	−0.844659	−	0.535305i	$-0.820197\pi$
$59$	− 63.1659i	− 1.07061i	0.844659	−	0.535305i	$-0.179803\pi$
$60$	0	0
$61$	111.283i	1.82430i	0.409851	+	0.912152i	$0.365581\pi$
$61$	111.283i	1.82430i	−0.409851	+	0.912152i	$0.634419\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−75.1868	−1.15672
$66$	0	0
$67$	−63.9669	−0.954730	−0.477365	−	0.878705i	$-0.658408\pi$
$67$	−63.9669	−0.954730	−0.477365	+	0.878705i	$0.658408\pi$
$68$	0	0
$69$	41.5692i	0.602452i
$70$	0	0
$71$	−21.5603	−0.303666	−0.151833	−	0.988406i	$-0.548518\pi$
$71$	−21.5603	−0.303666	−0.151833	+	0.988406i	$0.548518\pi$
$72$	0	0
$73$	− 61.8652i	− 0.847469i	−0.905786	−	0.423734i	$-0.860719\pi$
$73$	− 61.8652i	− 0.847469i	0.905786	−	0.423734i	$-0.139281\pi$
$74$	0	0
$75$	− 115.667i	− 1.54222i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.81323	0.0482687	0.0241343	−	0.999709i	$-0.492317\pi$
$79$	3.81323	0.0482687	0.0241343	+	0.999709i	$0.492317\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 100.510i	− 1.21096i	−0.795861	−	0.605479i	$-0.792982\pi$
$83$	− 100.510i	− 1.21096i	0.795861	−	0.605479i	$-0.207018\pi$
$84$	0	0
$85$	183.560	2.15953
$86$	0	0
$87$	− 18.3483i	− 0.210900i
$88$	0	0
$89$	− 99.0504i	− 1.11293i	−0.830872	−	0.556463i	$-0.812158\pi$
$89$	− 99.0504i	− 1.11293i	0.830872	−	0.556463i	$-0.187842\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	96.5603	1.03828
$94$	0	0
$95$	157.121	1.65390
$96$	0	0
$97$	63.5973i	0.655642i	0.944740	+	0.327821i	$0.106314\pi$
$97$	63.5973i	0.655642i	−0.944740	+	0.327821i	$0.893686\pi$
$98$	0	0
$99$	13.7802	0.139194
$100$	0	0
$101$	120.482i	1.19289i	0.802654	+	0.596446i	$0.203421\pi$
$101$	120.482i	1.19289i	−0.802654	+	0.596446i	$0.796579\pi$
$102$	0	0
$103$	18.8875i	0.183373i	0.995788	+	0.0916867i	$0.0292258\pi$
$103$	18.8875i	0.183373i	−0.995788	+	0.0916867i	$0.970774\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.9669	−0.158569	−0.0792847	−	0.996852i	$-0.525264\pi$
$107$	−16.9669	−0.158569	−0.0792847	+	0.996852i	$0.525264\pi$
$108$	0	0
$109$	56.7802	0.520919	0.260459	−	0.965485i	$-0.416126\pi$
$109$	56.7802	0.520919	0.260459	+	0.965485i	$0.416126\pi$
$110$	0	0
$111$	− 42.2735i	− 0.380842i
$112$	0	0
$113$	−156.374	−1.38384	−0.691918	−	0.721976i	$-0.743234\pi$
$113$	−156.374	−1.38384	−0.691918	+	0.721976i	$0.743234\pi$
$114$	0	0
$115$	229.925i	1.99935i
$116$	0	0
$117$	− 23.5444i	− 0.201234i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−99.9008	−0.825626
$122$	0	0
$123$	84.3735	0.685964
$124$	0	0
$125$	− 400.262i	− 3.20210i
$126$	0	0
$127$	25.0000	0.196850	0.0984252	−	0.995144i	$-0.468619\pi$
$127$	25.0000	0.196850	0.0984252	+	0.995144i	$0.468619\pi$
$128$	0	0
$129$	32.5282i	0.252157i
$130$	0	0
$131$	100.840i	0.769769i	0.922965	+	0.384884i	$0.125759\pi$
$131$	100.840i	0.769769i	−0.922965	+	0.384884i	$0.874241\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	49.7802	0.368742
$136$	0	0
$137$	43.1206	0.314749	0.157375	−	0.987539i	$-0.449697\pi$
$137$	43.1206	0.314749	0.157375	+	0.987539i	$0.449697\pi$
$138$	0	0
$139$	117.938i	0.848474i	0.905551	+	0.424237i	$0.139458\pi$
$139$	117.938i	0.848474i	−0.905551	+	0.424237i	$0.860542\pi$
$140$	0	0
$141$	20.8132	0.147612
$142$	0	0
$143$	− 36.0496i	− 0.252095i
$144$	0	0
$145$	− 101.487i	− 0.699908i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	177.187	1.18917	0.594586	−	0.804032i	$-0.297316\pi$
$149$	177.187	1.18917	0.594586	+	0.804032i	$0.297316\pi$
$150$	0	0
$151$	−182.967	−1.21170	−0.605851	−	0.795578i	$-0.707167\pi$
$151$	−182.967	−1.21170	−0.605851	+	0.795578i	$0.707167\pi$
$152$	0	0
$153$	57.4812i	0.375694i
$154$	0	0
$155$	534.088	3.44573
$156$	0	0
$157$	84.4324i	0.537786i	0.963170	+	0.268893i	$0.0866578\pi$
$157$	84.4324i	0.537786i	−0.963170	+	0.268893i	$0.913342\pi$
$158$	0	0
$159$	− 64.7902i	− 0.407485i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−143.934	−0.883030	−0.441515	−	0.897254i	$-0.645559\pi$
$163$	−143.934	−0.883030	−0.441515	+	0.897254i	$0.645559\pi$
$164$	0	0
$165$	76.2198	0.461938
$166$	0	0
$167$	123.730i	0.740901i	0.928852	+	0.370450i	$0.120797\pi$
$167$	123.730i	0.740901i	−0.928852	+	0.370450i	$0.879203\pi$
$168$	0	0
$169$	107.407	0.635542
$170$	0	0
$171$	49.2017i	0.287729i
$172$	0	0
$173$	− 134.453i	− 0.777185i	−0.921410	−	0.388592i	$-0.872961\pi$
$173$	− 134.453i	− 0.777185i	0.921410	−	0.388592i	$-0.127039\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	109.407	0.618116
$178$	0	0
$179$	−138.000	−0.770950	−0.385475	−	0.922718i	$-0.625962\pi$
$179$	−138.000	−0.770950	−0.385475	+	0.922718i	$0.625962\pi$
$180$	0	0
$181$	98.5618i	0.544540i	0.962221	+	0.272270i	$0.0877744\pi$
$181$	98.5618i	0.544540i	−0.962221	+	0.272270i	$0.912226\pi$
$182$	0	0
$183$	−192.747	−1.05326
$184$	0	0
$185$	− 233.820i	− 1.26389i
$186$	0	0
$187$	88.0111i	0.470648i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	166.307	0.870719	0.435360	−	0.900257i	$-0.356621\pi$
$191$	166.307	0.870719	0.435360	+	0.900257i	$0.356621\pi$
$192$	0	0
$193$	−63.3074	−0.328018	−0.164009	−	0.986459i	$-0.552443\pi$
$193$	−63.3074	−0.328018	−0.164009	+	0.986459i	$0.552443\pi$
$194$	0	0
$195$	− 130.227i	− 0.667832i
$196$	0	0
$197$	184.307	0.935571	0.467785	−	0.883842i	$-0.345052\pi$
$197$	184.307	0.935571	0.467785	+	0.883842i	$0.345052\pi$
$198$	0	0
$199$	− 35.5037i	− 0.178410i	−0.996013	−	0.0892052i	$-0.971567\pi$
$199$	− 35.5037i	− 0.178410i	0.996013	−	0.0892052i	$-0.0284327\pi$
$200$	0	0
$201$	− 110.794i	− 0.551214i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	466.681	2.27649
$206$	0	0
$207$	−72.0000	−0.347826
$208$	0	0
$209$	75.3341i	0.360450i
$210$	0	0
$211$	9.69259	0.0459364	0.0229682	−	0.999736i	$-0.492688\pi$
$211$	9.69259	0.0459364	0.0229682	+	0.999736i	$0.492688\pi$
$212$	0	0
$213$	− 37.3436i	− 0.175322i
$214$	0	0
$215$	179.918i	0.836826i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	107.154	0.489286
$220$	0	0
$221$	150.374	0.680423
$222$	0	0
$223$	− 430.462i	− 1.93032i	−0.261657	−	0.965161i	$-0.584269\pi$
$223$	− 430.462i	− 1.93032i	0.261657	−	0.965161i	$-0.415731\pi$
$224$	0	0
$225$	200.340	0.890402
$226$	0	0
$227$	− 190.145i	− 0.837642i	−0.908069	−	0.418821i	$-0.862443\pi$
$227$	− 190.145i	− 0.837642i	0.908069	−	0.418821i	$-0.137557\pi$
$228$	0	0
$229$	115.667i	0.505094i	0.967585	+	0.252547i	$0.0812683\pi$
$229$	115.667i	0.505094i	−0.967585	+	0.252547i	$0.918732\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	415.494	1.78324	0.891618	−	0.452788i	$-0.149570\pi$
$233$	415.494	1.78324	0.891618	+	0.452788i	$0.149570\pi$
$234$	0	0
$235$	115.121	0.489875
$236$	0	0
$237$	6.60470i	0.0278679i
$238$	0	0
$239$	−82.3074	−0.344382	−0.172191	−	0.985064i	$-0.555085\pi$
$239$	−82.3074	−0.344382	−0.172191	+	0.985064i	$0.555085\pi$
$240$	0	0
$241$	201.730i	0.837054i	0.908204	+	0.418527i	$0.137453\pi$
$241$	201.730i	0.837054i	−0.908204	+	0.418527i	$0.862547\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	128.714	0.521109
$248$	0	0
$249$	174.088	0.699147
$250$	0	0
$251$	117.068i	0.466408i	0.972428	+	0.233204i	$0.0749210\pi$
$251$	117.068i	0.466408i	−0.972428	+	0.233204i	$0.925079\pi$
$252$	0	0
$253$	−110.241	−0.435736
$254$	0	0
$255$	317.936i	1.24681i
$256$	0	0
$257$	− 261.102i	− 1.01596i	−0.861369	−	0.507980i	$-0.830392\pi$
$257$	− 261.102i	− 1.01596i	0.861369	−	0.507980i	$-0.169608\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	31.7802	0.121763
$262$	0	0
$263$	−454.681	−1.72882	−0.864412	−	0.502783i	$-0.832309\pi$
$263$	−454.681	−1.72882	−0.864412	+	0.502783i	$0.832309\pi$
$264$	0	0
$265$	− 358.363i	− 1.35231i
$266$	0	0
$267$	171.560	0.642548
$268$	0	0
$269$	− 139.161i	− 0.517325i	−0.965968	−	0.258663i	$-0.916718\pi$
$269$	− 139.161i	− 0.517325i	0.965968	−	0.258663i	$-0.0832818\pi$
$270$	0	0
$271$	− 58.7246i	− 0.216696i	−0.994113	−	0.108348i	$-0.965444\pi$
$271$	− 58.7246i	− 0.216696i	0.994113	−	0.108348i	$-0.0345561\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	306.747	1.11544
$276$	0	0
$277$	−44.7802	−0.161661	−0.0808306	−	0.996728i	$-0.525757\pi$
$277$	−44.7802	−0.161661	−0.0808306	+	0.996728i	$0.525757\pi$
$278$	0	0
$279$	167.247i	0.599453i
$280$	0	0
$281$	−325.494	−1.15834	−0.579171	−	0.815206i	$-0.696624\pi$
$281$	−325.494	−1.15834	−0.579171	+	0.815206i	$0.696624\pi$
$282$	0	0
$283$	− 333.143i	− 1.17719i	−0.808430	−	0.588593i	$-0.799682\pi$
$283$	− 333.143i	− 1.17719i	0.808430	−	0.588593i	$-0.200318\pi$
$284$	0	0
$285$	272.141i	0.954880i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−78.1206	−0.270314
$290$	0	0
$291$	−110.154	−0.378535
$292$	0	0
$293$	− 405.135i	− 1.38271i	−0.722514	−	0.691356i	$-0.757014\pi$
$293$	− 405.135i	− 1.38271i	0.722514	−	0.691356i	$-0.242986\pi$
$294$	0	0
$295$	605.142	2.05133
$296$	0	0
$297$	23.8679i	0.0803634i
$298$	0	0
$299$	188.355i	0.629951i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−208.681	−0.688716
$304$	0	0
$305$	−1066.11	−3.49544
$306$	0	0
$307$	24.9530i	0.0812801i	0.999174	+	0.0406400i	$0.0129397\pi$
$307$	24.9530i	0.0812801i	−0.999174	+	0.0406400i	$0.987060\pi$
$308$	0	0
$309$	−32.7140	−0.105871
$310$	0	0
$311$	− 361.776i	− 1.16327i	−0.813451	−	0.581634i	$-0.802414\pi$
$311$	− 361.776i	− 1.16327i	0.813451	−	0.581634i	$-0.197586\pi$
$312$	0	0
$313$	136.401i	0.435785i	0.975973	+	0.217892i	$0.0699182\pi$
$313$	136.401i	0.435785i	−0.975973	+	0.217892i	$0.930082\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−543.648	−1.71498	−0.857489	−	0.514503i	$-0.827977\pi$
$317$	−543.648	−1.71498	−0.857489	+	0.514503i	$0.827977\pi$
$318$	0	0
$319$	48.6595	0.152538
$320$	0	0
$321$	− 29.3876i	− 0.0915501i
$322$	0	0
$323$	−314.241	−0.972883
$324$	0	0
$325$	− 524.100i	− 1.61262i
$326$	0	0
$327$	98.3461i	0.300753i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	140.099	0.423260	0.211630	−	0.977350i	$-0.432123\pi$
$331$	140.099	0.423260	0.211630	+	0.977350i	$0.432123\pi$
$332$	0	0
$333$	73.2198	0.219879
$334$	0	0
$335$	− 612.816i	− 1.82930i
$336$	0	0
$337$	311.681	0.924869	0.462435	−	0.886653i	$-0.346976\pi$
$337$	311.681	0.924869	0.462435	+	0.886653i	$0.346976\pi$
$338$	0	0
$339$	− 270.847i	− 0.798958i
$340$	0	0
$341$	256.077i	0.750960i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−398.241	−1.15432
$346$	0	0
$347$	−112.307	−0.323652	−0.161826	−	0.986819i	$-0.551738\pi$
$347$	−112.307	−0.323652	−0.161826	+	0.986819i	$0.551738\pi$
$348$	0	0
$349$	− 211.526i	− 0.606091i	−0.952976	−	0.303046i	$-0.901997\pi$
$349$	− 211.526i	− 0.606091i	0.952976	−	0.303046i	$-0.0980035\pi$
$350$	0	0
$351$	40.7802	0.116183
$352$	0	0
$353$	− 91.9065i	− 0.260358i	−0.991490	−	0.130179i	$-0.958445\pi$
$353$	− 91.9065i	− 0.260358i	0.991490	−	0.130179i	$-0.0415553\pi$
$354$	0	0
$355$	− 206.552i	− 0.581837i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−470.988	−1.31195	−0.655973	−	0.754785i	$-0.727741\pi$
$359$	−470.988	−1.31195	−0.655973	+	0.754785i	$0.727741\pi$
$360$	0	0
$361$	92.0214	0.254907
$362$	0	0
$363$	− 173.033i	− 0.476676i
$364$	0	0
$365$	592.681	1.62378
$366$	0	0
$367$	446.862i	1.21761i	0.793320	+	0.608804i	$0.208351\pi$
$367$	446.862i	1.21761i	−0.793320	+	0.608804i	$0.791649\pi$
$368$	0	0
$369$	146.139i	0.396041i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−53.2198	−0.142681	−0.0713403	−	0.997452i	$-0.522728\pi$
$373$	−53.2198	−0.142681	−0.0713403	+	0.997452i	$0.522728\pi$
$374$	0	0
$375$	693.274	1.84873
$376$	0	0
$377$	− 83.1384i	− 0.220526i
$378$	0	0
$379$	−700.076	−1.84717	−0.923583	−	0.383398i	$-0.874754\pi$
$379$	−700.076	−1.84717	−0.923583	+	0.383398i	$0.874754\pi$
$380$	0	0
$381$	43.3013i	0.113652i
$382$	0	0
$383$	− 623.208i	− 1.62718i	−0.581442	−	0.813588i	$-0.697511\pi$
$383$	− 623.208i	− 1.62718i	0.581442	−	0.813588i	$-0.302489\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−56.3405	−0.145583
$388$	0	0
$389$	−397.868	−1.02280	−0.511398	−	0.859344i	$-0.670872\pi$
$389$	−397.868	−1.02280	−0.511398	+	0.859344i	$0.670872\pi$
$390$	0	0
$391$	− 459.849i	− 1.17609i
$392$	0	0
$393$	−174.660	−0.444426
$394$	0	0
$395$	36.5315i	0.0924847i
$396$	0	0
$397$	− 338.131i	− 0.851715i	−0.904790	−	0.425857i	$-0.859973\pi$
$397$	− 338.131i	− 0.851715i	0.904790	−	0.425857i	$-0.140027\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	19.8677	0.0495455	0.0247727	−	0.999693i	$-0.492114\pi$
$401$	19.8677	0.0495455	0.0247727	+	0.999693i	$0.492114\pi$
$402$	0	0
$403$	437.527	1.08568
$404$	0	0
$405$	86.2218i	0.212893i
$406$	0	0
$407$	112.109	0.275452
$408$	0	0
$409$	410.280i	1.00313i	0.865120	+	0.501565i	$0.167242\pi$
$409$	410.280i	1.00313i	−0.865120	+	0.501565i	$0.832758\pi$
$410$	0	0
$411$	74.6871i	0.181721i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	962.901	2.32024
$416$	0	0
$417$	−204.274	−0.489867
$418$	0	0
$419$	− 564.750i	− 1.34785i	−0.738799	−	0.673926i	$-0.764607\pi$
$419$	− 564.750i	− 1.34785i	0.738799	−	0.673926i	$-0.235393\pi$
$420$	0	0
$421$	384.582	0.913496	0.456748	−	0.889596i	$-0.349014\pi$
$421$	384.582	0.913496	0.456748	+	0.889596i	$0.349014\pi$
$422$	0	0
$423$	36.0496i	0.0852236i
$424$	0	0
$425$	1279.53i	3.01067i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	62.4397	0.145547
$430$	0	0
$431$	−553.868	−1.28508	−0.642538	−	0.766254i	$-0.722119\pi$
$431$	−553.868	−1.28508	−0.642538	+	0.766254i	$0.722119\pi$
$432$	0	0
$433$	221.760i	0.512147i	0.966657	+	0.256074i	$0.0824290\pi$
$433$	221.760i	0.512147i	−0.966657	+	0.256074i	$0.917571\pi$
$434$	0	0
$435$	175.780	0.404092
$436$	0	0
$437$	− 393.614i	− 0.900718i
$438$	0	0
$439$	− 555.169i	− 1.26462i	−0.774714	−	0.632311i	$-0.782106\pi$
$439$	− 555.169i	− 1.26462i	0.774714	−	0.632311i	$-0.217894\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	385.582	0.870388	0.435194	−	0.900337i	$-0.356680\pi$
$443$	385.582	0.870388	0.435194	+	0.900337i	$0.356680\pi$
$444$	0	0
$445$	948.922	2.13241
$446$	0	0
$447$	306.896i	0.686569i
$448$	0	0
$449$	−107.802	−0.240093	−0.120046	−	0.992768i	$-0.538304\pi$
$449$	−107.802	−0.240093	−0.120046	+	0.992768i	$0.538304\pi$
$450$	0	0
$451$	223.758i	0.496138i
$452$	0	0
$453$	− 316.908i	− 0.699576i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−11.0000	−0.0240700	−0.0120350	−	0.999928i	$-0.503831\pi$
$457$	−11.0000	−0.0240700	−0.0120350	+	0.999928i	$0.503831\pi$
$458$	0	0
$459$	−99.5603	−0.216907
$460$	0	0
$461$	636.532i	1.38076i	0.723445	+	0.690382i	$0.242557\pi$
$461$	636.532i	1.38076i	−0.723445	+	0.690382i	$0.757443\pi$
$462$	0	0
$463$	−179.660	−0.388034	−0.194017	−	0.980998i	$-0.562152\pi$
$463$	−179.660	−0.388034	−0.194017	+	0.980998i	$0.562152\pi$
$464$	0	0
$465$	925.067i	1.98939i
$466$	0	0
$467$	45.7949i	0.0980618i	0.998797	+	0.0490309i	$0.0156133\pi$
$467$	45.7949i	0.0980618i	−0.998797	+	0.0490309i	$0.984387\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−146.241	−0.310491
$472$	0	0
$473$	−86.2645	−0.182377
$474$	0	0
$475$	1095.23i	2.30575i
$476$	0	0
$477$	112.220	0.235262
$478$	0	0
$479$	− 162.065i	− 0.338340i	−0.985587	−	0.169170i	$-0.945891\pi$
$479$	− 162.065i	− 0.338340i	0.985587	−	0.169170i	$-0.0541086\pi$
$480$	0	0
$481$	− 191.547i	− 0.398226i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−609.274	−1.25624
$486$	0	0
$487$	664.802	1.36510	0.682548	−	0.730841i	$-0.260872\pi$
$487$	664.802	1.36510	0.682548	+	0.730841i	$0.260872\pi$
$488$	0	0
$489$	− 249.301i	− 0.509818i
$490$	0	0
$491$	379.955	0.773840	0.386920	−	0.922113i	$-0.373539\pi$
$491$	379.955	0.773840	0.386920	+	0.922113i	$0.373539\pi$
$492$	0	0
$493$	202.973i	0.411711i
$494$	0	0
$495$	132.017i	0.266700i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−526.099	−1.05431	−0.527154	−	0.849770i	$-0.676741\pi$
$499$	−526.099	−1.05431	−0.527154	+	0.849770i	$0.676741\pi$
$500$	0	0
$501$	−214.307	−0.427759
$502$	0	0
$503$	573.504i	1.14017i	0.821587	+	0.570084i	$0.193089\pi$
$503$	573.504i	1.14017i	−0.821587	+	0.570084i	$0.806911\pi$
$504$	0	0
$505$	−1154.24	−2.28563
$506$	0	0
$507$	186.034i	0.366930i
$508$	0	0
$509$	− 116.091i	− 0.228077i	−0.993476	−	0.114039i	$-0.963621\pi$
$509$	− 116.091i	− 0.228077i	0.993476	−	0.114039i	$-0.0363787\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−85.2198	−0.166121
$514$	0	0
$515$	−180.945	−0.351350
$516$	0	0
$517$	55.1965i	0.106763i
$518$	0	0
$519$	232.879	0.448708
$520$	0	0
$521$	164.653i	0.316032i	0.987437	+	0.158016i	$0.0505098\pi$
$521$	164.653i	0.316032i	−0.987437	+	0.158016i	$0.949490\pi$
$522$	0	0
$523$	503.316i	0.962363i	0.876621	+	0.481181i	$0.159792\pi$
$523$	503.316i	0.962363i	−0.876621	+	0.481181i	$0.840208\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1068.18	−2.02690
$528$	0	0
$529$	47.0000	0.0888469
$530$	0	0
$531$	189.498i	0.356870i
$532$	0	0
$533$	382.307	0.717275
$534$	0	0
$535$	− 162.547i	− 0.303825i
$536$	0	0
$537$	− 239.023i	− 0.445108i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−562.099	−1.03900	−0.519500	−	0.854470i	$-0.673882\pi$
$541$	−562.099	−1.03900	−0.519500	+	0.854470i	$0.673882\pi$
$542$	0	0
$543$	−170.714	−0.314390
$544$	0	0
$545$	543.965i	0.998101i
$546$	0	0
$547$	−200.374	−0.366314	−0.183157	−	0.983084i	$-0.558632\pi$
$547$	−200.374	−0.366314	−0.183157	+	0.983084i	$0.558632\pi$
$548$	0	0
$549$	− 333.848i	− 0.608102i
$550$	0	0
$551$	173.738i	0.315313i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	404.988	0.729709
$556$	0	0
$557$	436.593	0.783830	0.391915	−	0.920001i	$-0.371813\pi$
$557$	436.593	0.783830	0.391915	+	0.920001i	$0.371813\pi$
$558$	0	0
$559$	147.389i	0.263666i
$560$	0	0
$561$	−152.440	−0.271728
$562$	0	0
$563$	− 109.594i	− 0.194661i	−0.995252	−	0.0973307i	$-0.968970\pi$
$563$	− 109.594i	− 0.194661i	0.995252	−	0.0973307i	$-0.0310305\pi$
$564$	0	0
$565$	− 1498.09i	− 2.65149i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	530.615	0.932539	0.466270	−	0.884643i	$-0.345598\pi$
$569$	530.615	0.932539	0.466270	+	0.884643i	$0.345598\pi$
$570$	0	0
$571$	877.263	1.53636	0.768181	−	0.640233i	$-0.221162\pi$
$571$	877.263	1.53636	0.768181	+	0.640233i	$0.221162\pi$
$572$	0	0
$573$	288.053i	0.502710i
$574$	0	0
$575$	−1602.72	−2.78735
$576$	0	0
$577$	610.969i	1.05887i	0.848350	+	0.529436i	$0.177596\pi$
$577$	610.969i	1.05887i	−0.848350	+	0.529436i	$0.822404\pi$
$578$	0	0
$579$	− 109.652i	− 0.189381i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	171.823	0.294722
$584$	0	0
$585$	225.560	0.385573
$586$	0	0
$587$	− 352.196i	− 0.599993i	−0.953940	−	0.299997i	$-0.903014\pi$
$587$	− 352.196i	− 0.599993i	0.953940	−	0.299997i	$-0.0969856\pi$
$588$	0	0
$589$	−914.317	−1.55232
$590$	0	0
$591$	319.230i	0.540152i
$592$	0	0
$593$	331.577i	0.559151i	0.960124	+	0.279575i	$0.0901937\pi$
$593$	331.577i	0.559151i	−0.960124	+	0.279575i	$0.909806\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	61.4942	0.103005
$598$	0	0
$599$	365.230	0.609732	0.304866	−	0.952395i	$-0.401388\pi$
$599$	365.230	0.609732	0.304866	+	0.952395i	$0.401388\pi$
$600$	0	0
$601$	749.102i	1.24643i	0.782052	+	0.623213i	$0.214173\pi$
$601$	749.102i	1.24643i	−0.782052	+	0.623213i	$0.785827\pi$
$602$	0	0
$603$	191.901	0.318243
$604$	0	0
$605$	− 957.069i	− 1.58193i
$606$	0	0
$607$	− 1058.69i	− 1.74413i	−0.489389	−	0.872066i	$-0.662780\pi$
$607$	− 1058.69i	− 1.74413i	0.489389	−	0.872066i	$-0.337220\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	94.3074	0.154349
$612$	0	0
$613$	218.572	0.356561	0.178281	−	0.983980i	$-0.442947\pi$
$613$	218.572	0.356561	0.178281	+	0.983980i	$0.442947\pi$
$614$	0	0
$615$	808.315i	1.31433i
$616$	0	0
$617$	559.669	0.907082	0.453541	−	0.891236i	$-0.350161\pi$
$617$	559.669	0.907082	0.453541	+	0.891236i	$0.350161\pi$
$618$	0	0
$619$	− 531.345i	− 0.858393i	−0.903211	−	0.429197i	$-0.858797\pi$
$619$	− 531.345i	− 0.858393i	0.903211	−	0.429197i	$-0.141203\pi$
$620$	0	0
$621$	− 124.708i	− 0.200817i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2165.09	3.46414
$626$	0	0
$627$	−130.483	−0.208106
$628$	0	0
$629$	467.640i	0.743466i
$630$	0	0
$631$	1086.53	1.72191	0.860957	−	0.508678i	$-0.169866\pi$
$631$	1086.53	1.72191	0.860957	+	0.508678i	$0.169866\pi$
$632$	0	0
$633$	16.7881i	0.0265214i
$634$	0	0
$635$	239.505i	0.377173i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	64.6810	0.101222
$640$	0	0
$641$	463.868	0.723663	0.361831	−	0.932244i	$-0.382152\pi$
$641$	463.868	0.723663	0.361831	+	0.932244i	$0.382152\pi$
$642$	0	0
$643$	1055.85i	1.64206i	0.570883	+	0.821032i	$0.306601\pi$
$643$	1055.85i	1.64206i	−0.570883	+	0.821032i	$0.693399\pi$
$644$	0	0
$645$	−311.626	−0.483142
$646$	0	0
$647$	788.178i	1.21820i	0.793092	+	0.609102i	$0.208470\pi$
$647$	788.178i	1.21820i	−0.793092	+	0.609102i	$0.791530\pi$
$648$	0	0
$649$	290.146i	0.447066i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1015.76	−1.55552	−0.777762	−	0.628559i	$-0.783645\pi$
$653$	−1015.76	−1.55552	−0.777762	+	0.628559i	$0.783645\pi$
$654$	0	0
$655$	−966.064	−1.47491
$656$	0	0
$657$	185.596i	0.282490i
$658$	0	0
$659$	−1254.55	−1.90372	−0.951858	−	0.306540i	$-0.900829\pi$
$659$	−1254.55	−1.90372	−0.951858	+	0.306540i	$0.900829\pi$
$660$	0	0
$661$	− 624.330i	− 0.944524i	−0.881458	−	0.472262i	$-0.843438\pi$
$661$	− 624.330i	− 0.944524i	0.881458	−	0.472262i	$-0.156562\pi$
$662$	0	0
$663$	260.455i	0.392843i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−254.241	−0.381171
$668$	0	0
$669$	745.582	1.11447
$670$	0	0
$671$	− 511.164i	− 0.761794i
$672$	0	0
$673$	41.4826	0.0616383	0.0308191	−	0.999525i	$-0.490188\pi$
$673$	41.4826	0.0616383	0.0308191	+	0.999525i	$0.490188\pi$
$674$	0	0
$675$	347.000i	0.514074i
$676$	0	0
$677$	− 721.446i	− 1.06565i	−0.846225	−	0.532826i	$-0.821130\pi$
$677$	− 721.446i	− 1.06565i	0.846225	−	0.532826i	$-0.178870\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	329.340	0.483613
$682$	0	0
$683$	−839.516	−1.22916	−0.614580	−	0.788855i	$-0.710674\pi$
$683$	−839.516	−1.22916	−0.614580	+	0.788855i	$0.710674\pi$
$684$	0	0
$685$	413.104i	0.603072i
$686$	0	0
$687$	−200.340	−0.291616
$688$	0	0
$689$	− 293.573i	− 0.426085i
$690$	0	0
$691$	− 555.911i	− 0.804502i	−0.915529	−	0.402251i	$-0.868228\pi$
$691$	− 555.911i	− 0.804502i	0.915529	−	0.402251i	$-0.131772\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1129.87	−1.62571
$696$	0	0
$697$	−933.362	−1.33911
$698$	0	0
$699$	719.657i	1.02955i
$700$	0	0
$701$	−994.769	−1.41907	−0.709535	−	0.704670i	$-0.751095\pi$
$701$	−994.769	−1.41907	−0.709535	+	0.704670i	$0.751095\pi$
$702$	0	0
$703$	400.282i	0.569392i
$704$	0	0
$705$	199.395i	0.282829i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−835.977	−1.17909	−0.589546	−	0.807735i	$-0.700693\pi$
$709$	−835.977	−1.17909	−0.589546	+	0.807735i	$0.700693\pi$
$710$	0	0
$711$	−11.4397	−0.0160896
$712$	0	0
$713$	− 1337.98i	− 1.87655i
$714$	0	0
$715$	345.362	0.483024
$716$	0	0
$717$	− 142.561i	− 0.198829i
$718$	0	0
$719$	479.327i	0.666657i	0.942811	+	0.333329i	$0.108172\pi$
$719$	479.327i	0.666657i	−0.942811	+	0.333329i	$0.891828\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−349.407	−0.483273
$724$	0	0
$725$	707.428	0.975763
$726$	0	0
$727$	949.487i	1.30603i	0.757343	+	0.653017i	$0.226497\pi$
$727$	949.487i	1.30603i	−0.757343	+	0.653017i	$0.773503\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 359.835i	− 0.492251i
$732$	0	0
$733$	448.106i	0.611331i	0.952139	+	0.305666i	$0.0988790\pi$
$733$	448.106i	0.611331i	−0.952139	+	0.305666i	$0.901121\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	293.825	0.398677
$738$	0	0
$739$	58.0992	0.0786187	0.0393093	−	0.999227i	$-0.487484\pi$
$739$	58.0992	0.0786187	0.0393093	+	0.999227i	$0.487484\pi$
$740$	0	0
$741$	222.939i	0.300863i
$742$	0	0
$743$	386.043	0.519573	0.259787	−	0.965666i	$-0.416348\pi$
$743$	386.043	0.519573	0.259787	+	0.965666i	$0.416348\pi$
$744$	0	0
$745$	1697.48i	2.27850i
$746$	0	0
$747$	301.529i	0.403653i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−550.362	−0.732839	−0.366419	−	0.930450i	$-0.619416\pi$
$751$	−550.362	−0.732839	−0.366419	+	0.930450i	$0.619416\pi$
$752$	0	0
$753$	−202.769	−0.269281
$754$	0	0
$755$	− 1752.86i	− 2.32167i
$756$	0	0
$757$	379.187	0.500907	0.250454	−	0.968129i	$-0.419420\pi$
$757$	379.187	0.500907	0.250454	+	0.968129i	$0.419420\pi$
$758$	0	0
$759$	− 190.943i	− 0.251572i
$760$	0	0
$761$	515.706i	0.677669i	0.940846	+	0.338835i	$0.110033\pi$
$761$	515.706i	0.677669i	−0.940846	+	0.338835i	$0.889967\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−550.681	−0.719844
$766$	0	0
$767$	495.735	0.646330
$768$	0	0
$769$	− 548.514i	− 0.713282i	−0.934241	−	0.356641i	$-0.883922\pi$
$769$	− 548.514i	− 0.713282i	0.934241	−	0.356641i	$-0.116078\pi$
$770$	0	0
$771$	452.241	0.586565
$772$	0	0
$773$	− 119.848i	− 0.155043i	−0.996991	−	0.0775216i	$-0.975299\pi$
$773$	− 119.848i	− 0.155043i	0.996991	−	0.0775216i	$-0.0247007\pi$
$774$	0	0
$775$	3722.94i	4.80379i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−798.922	−1.02557
$780$	0	0
$781$	99.0349	0.126805
$782$	0	0
$783$	55.0449i	0.0702999i
$784$	0	0
$785$	−808.879	−1.03042
$786$	0	0
$787$	1012.78i	1.28689i	0.765491	+	0.643447i	$0.222496\pi$
$787$	1012.78i	1.28689i	−0.765491	+	0.643447i	$0.777504\pi$
$788$	0	0
$789$	− 787.531i	− 0.998138i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−873.362	−1.10134
$794$	0	0
$795$	620.702	0.780758
$796$	0	0
$797$	− 1067.63i	− 1.33956i	−0.742561	−	0.669779i	$-0.766389\pi$
$797$	− 1067.63i	− 1.33956i	0.742561	−	0.669779i	$-0.233611\pi$
$798$	0	0
$799$	−230.241	−0.288162
$800$	0	0
$801$	297.151i	0.370975i
$802$	0	0
$803$	284.171i	0.353887i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	241.033	0.298678
$808$	0	0
$809$	−851.055	−1.05198	−0.525992	−	0.850490i	$-0.676306\pi$
$809$	−851.055	−1.05198	−0.525992	+	0.850490i	$0.676306\pi$
$810$	0	0
$811$	− 364.378i	− 0.449294i	−0.974440	−	0.224647i	$-0.927877\pi$
$811$	− 364.378i	− 0.449294i	0.974440	−	0.224647i	$-0.0721229\pi$
$812$	0	0
$813$	101.714	0.125110
$814$	0	0
$815$	− 1378.91i	− 1.69192i
$816$	0	0
$817$	− 308.005i	− 0.376995i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	43.0098	0.0523871	0.0261936	−	0.999657i	$-0.491661\pi$
$821$	43.0098	0.0523871	0.0261936	+	0.999657i	$0.491661\pi$
$822$	0	0
$823$	−663.494	−0.806190	−0.403095	−	0.915158i	$-0.632066\pi$
$823$	−663.494	−0.806190	−0.403095	+	0.915158i	$0.632066\pi$
$824$	0	0
$825$	531.302i	0.644002i
$826$	0	0
$827$	684.811	0.828067	0.414034	−	0.910262i	$-0.364120\pi$
$827$	684.811	0.828067	0.414034	+	0.910262i	$0.364120\pi$
$828$	0	0
$829$	− 1247.87i	− 1.50527i	−0.658438	−	0.752635i	$-0.728783\pi$
$829$	− 1247.87i	− 1.50527i	0.658438	−	0.752635i	$-0.271217\pi$
$830$	0	0
$831$	− 77.5615i	− 0.0933352i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1185.36	−1.41960
$836$	0	0
$837$	−289.681	−0.346094
$838$	0	0
$839$	1289.61i	1.53708i	0.639802	+	0.768540i	$0.279016\pi$
$839$	1289.61i	1.53708i	−0.639802	+	0.768540i	$0.720984\pi$
$840$	0	0
$841$	−728.780	−0.866564
$842$	0	0
$843$	− 563.772i	− 0.668769i
$844$	0	0
$845$	1028.98i	1.21772i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	577.021	0.679648
$850$	0	0
$851$	−585.759	−0.688318
$852$	0	0
$853$	141.122i	0.165442i	0.996573	+	0.0827208i	$0.0263610\pi$
$853$	141.122i	0.165442i	−0.996573	+	0.0827208i	$0.973639\pi$
$854$	0	0
$855$	−471.362	−0.551300
$856$	0	0
$857$	163.028i	0.190232i	0.995466	+	0.0951158i	$0.0303221\pi$
$857$	163.028i	0.190232i	−0.995466	+	0.0951158i	$0.969678\pi$
$858$	0	0
$859$	− 305.400i	− 0.355530i	−0.984073	−	0.177765i	$-0.943113\pi$
$859$	− 305.400i	− 0.355530i	0.984073	−	0.177765i	$-0.0568867\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	495.362	0.574000	0.287000	−	0.957931i	$-0.407342\pi$
$863$	495.362	0.574000	0.287000	+	0.957931i	$0.407342\pi$
$864$	0	0
$865$	1288.09	1.48912
$866$	0	0
$867$	− 135.309i	− 0.156066i
$868$	0	0
$869$	−17.5156	−0.0201561
$870$	0	0
$871$	− 502.022i	− 0.576374i
$872$	0	0
$873$	− 190.792i	− 0.218547i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−90.7900	−0.103523	−0.0517617	−	0.998659i	$-0.516484\pi$
$877$	−90.7900	−0.103523	−0.0517617	+	0.998659i	$0.516484\pi$
$878$	0	0
$879$	701.714	0.798309
$880$	0	0
$881$	1403.61i	1.59320i	0.604508	+	0.796599i	$0.293370\pi$
$881$	1403.61i	1.59320i	−0.604508	+	0.796599i	$0.706630\pi$
$882$	0	0
$883$	−1595.94	−1.80741	−0.903705	−	0.428155i	$-0.859164\pi$
$883$	−1595.94	−1.80741	−0.903705	+	0.428155i	$0.859164\pi$
$884$	0	0
$885$	1048.14i	1.18434i
$886$	0	0
$887$	− 205.548i	− 0.231734i	−0.993265	−	0.115867i	$-0.963035\pi$
$887$	− 205.548i	− 0.231734i	0.993265	−	0.115867i	$-0.0369646\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−41.3405	−0.0463978
$892$	0	0
$893$	−197.078	−0.220692
$894$	0	0
$895$	− 1322.07i	− 1.47717i
$896$	0	0
$897$	−326.241	−0.363703
$898$	0	0
$899$	590.572i	0.656921i
$900$	0	0
$901$	716.725i	0.795478i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−944.241	−1.04336
$906$	0	0
$907$	−410.757	−0.452874	−0.226437	−	0.974026i	$-0.572708\pi$
$907$	−410.757	−0.452874	−0.226437	+	0.974026i	$0.572708\pi$
$908$	0	0
$909$	− 361.446i	− 0.397630i
$910$	0	0
$911$	−800.218	−0.878395	−0.439198	−	0.898390i	$-0.644737\pi$
$911$	−800.218	−0.878395	−0.439198	+	0.898390i	$0.644737\pi$
$912$	0	0
$913$	461.679i	0.505673i
$914$	0	0
$915$	− 1846.55i	− 2.01809i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1432.14	1.55837	0.779185	−	0.626794i	$-0.215633\pi$
$919$	1432.14	1.55837	0.779185	+	0.626794i	$0.215633\pi$
$920$	0	0
$921$	−43.2198	−0.0469271
$922$	0	0
$923$	− 169.209i	− 0.183325i
$924$	0	0
$925$	1629.88	1.76203
$926$	0	0
$927$	− 56.6624i	− 0.0611244i
$928$	0	0
$929$	337.110i	0.362874i	0.983403	+	0.181437i	$0.0580748\pi$
$929$	337.110i	0.362874i	−0.983403	+	0.181437i	$0.941925\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	626.615	0.671613
$934$	0	0
$935$	−843.164	−0.901779
$936$	0	0
$937$	53.3634i	0.0569513i	0.999594	+	0.0284756i	$0.00906531\pi$
$937$	53.3634i	0.0569513i	−0.999594	+	0.0284756i	$0.990935\pi$
$938$	0	0
$939$	−236.253	−0.251601
$940$	0	0
$941$	− 1271.23i	− 1.35094i	−0.737388	−	0.675470i	$-0.763941\pi$
$941$	− 1271.23i	− 1.35094i	0.737388	−	0.675470i	$-0.236059\pi$
$942$	0	0
$943$	− 1169.11i	− 1.23978i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−532.856	−0.562678	−0.281339	−	0.959608i	$-0.590779\pi$
$947$	−532.856	−0.562678	−0.281339	+	0.959608i	$0.590779\pi$
$948$	0	0
$949$	485.527	0.511620
$950$	0	0
$951$	− 941.626i	− 0.990143i
$952$	0	0
$953$	−1665.54	−1.74768	−0.873839	−	0.486215i	$-0.838377\pi$
$953$	−1665.54	−1.74768	−0.873839	+	0.486215i	$0.838377\pi$
$954$	0	0
$955$	1593.26i	1.66833i
$956$	0	0
$957$	84.2808i	0.0880677i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2146.97	−2.23409
$962$	0	0
$963$	50.9008	0.0528565
$964$	0	0
$965$	− 606.497i	− 0.628495i
$966$	0	0
$967$	−1328.19	−1.37351	−0.686756	−	0.726888i	$-0.740966\pi$
$967$	−1328.19	−1.37351	−0.686756	+	0.726888i	$0.740966\pi$
$968$	0	0
$969$	− 544.282i	− 0.561694i
$970$	0	0
$971$	− 213.834i	− 0.220221i	−0.993919	−	0.110110i	$-0.964880\pi$
$971$	− 213.834i	− 0.220221i	0.993919	−	0.110110i	$-0.0351204\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	907.769	0.931045
$976$	0	0
$977$	−1016.24	−1.04017	−0.520083	−	0.854116i	$-0.674099\pi$
$977$	−1016.24	−1.04017	−0.520083	+	0.854116i	$0.674099\pi$
$978$	0	0
$979$	454.977i	0.464736i
$980$	0	0
$981$	−170.340	−0.173640
$982$	0	0
$983$	− 416.656i	− 0.423862i	−0.977285	−	0.211931i	$-0.932025\pi$
$983$	− 416.656i	− 0.423862i	0.977285	−	0.211931i	$-0.0679751\pi$
$984$	0	0
$985$	1765.70i	1.79259i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	450.724	0.455737
$990$	0	0
$991$	1246.43	1.25775	0.628874	−	0.777507i	$-0.283516\pi$
$991$	1246.43	1.25775	0.628874	+	0.777507i	$0.283516\pi$
$992$	0	0
$993$	242.659i	0.244370i
$994$	0	0
$995$	340.132	0.341841
$996$	0	0
$997$	− 709.666i	− 0.711801i	−0.934524	−	0.355901i	$-0.884174\pi$
$997$	− 709.666i	− 0.711801i	0.934524	−	0.355901i	$-0.115826\pi$
$998$	0	0
$999$	126.820i	0.126947i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.3.f.f.97.4		4
4.3	odd	2		588.3.d.b.97.2		4
7.4	even	3		336.3.bh.f.145.1		4
7.5	odd	6		336.3.bh.f.241.1		4
7.6	odd	2	inner	2352.3.f.f.97.1		4
12.11	even	2		1764.3.d.f.685.1		4
21.5	even	6		1008.3.cg.m.577.2		4
21.11	odd	6		1008.3.cg.m.145.2		4
28.3	even	6		588.3.m.d.313.2		4
28.11	odd	6		84.3.m.b.61.1	✓	4
28.19	even	6		84.3.m.b.73.1	yes	4
28.23	odd	6		588.3.m.d.325.2		4
28.27	even	2		588.3.d.b.97.3		4
84.11	even	6		252.3.z.e.145.2		4
84.23	even	6		1764.3.z.h.325.1		4
84.47	odd	6		252.3.z.e.73.2		4
84.59	odd	6		1764.3.z.h.901.1		4
84.83	odd	2		1764.3.d.f.685.4		4
140.19	even	6		2100.3.bd.f.1501.1		4
140.39	odd	6		2100.3.bd.f.901.2		4
140.47	odd	12		2100.3.be.d.1249.1		8
140.67	even	12		2100.3.be.d.649.4		8
140.103	odd	12		2100.3.be.d.1249.4		8
140.123	even	12		2100.3.be.d.649.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.3.m.b.61.1	✓	4	28.11	odd	6
84.3.m.b.73.1	yes	4	28.19	even	6
252.3.z.e.73.2		4	84.47	odd	6
252.3.z.e.145.2		4	84.11	even	6
336.3.bh.f.145.1		4	7.4	even	3
336.3.bh.f.241.1		4	7.5	odd	6
588.3.d.b.97.2		4	4.3	odd	2
588.3.d.b.97.3		4	28.27	even	2
588.3.m.d.313.2		4	28.3	even	6
588.3.m.d.325.2		4	28.23	odd	6
1008.3.cg.m.145.2		4	21.11	odd	6
1008.3.cg.m.577.2		4	21.5	even	6
1764.3.d.f.685.1		4	12.11	even	2
1764.3.d.f.685.4		4	84.83	odd	2
1764.3.z.h.325.1		4	84.23	even	6
1764.3.z.h.901.1		4	84.59	odd	6
2100.3.bd.f.901.2		4	140.39	odd	6
2100.3.bd.f.1501.1		4	140.19	even	6
2100.3.be.d.649.1		8	140.123	even	12
2100.3.be.d.649.4		8	140.67	even	12
2100.3.be.d.1249.1		8	140.47	odd	12
2100.3.be.d.1249.4		8	140.103	odd	12
2352.3.f.f.97.1		4	7.6	odd	2	inner
2352.3.f.f.97.4		4	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2352.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +9.58020i q^{5} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} +9.58020i q^{5} -3.00000 q^{9} -4.59339 q^{11} +7.84815i q^{13} -16.5934 q^{15} -19.1604i q^{17} -16.4006i q^{19} +24.0000 q^{23} -66.7802 q^{25} -5.19615i q^{27} -10.5934 q^{29} -55.7491i q^{31} -7.95598i q^{33} -24.4066 q^{37} -13.5934 q^{39} -48.7131i q^{41} +18.7802 q^{43} -28.7406i q^{45} -12.0165i q^{47} +33.1868 q^{51} -37.4066 q^{53} -44.0055i q^{55} +28.4066 q^{57} -63.1659i q^{59} +111.283i q^{61} -75.1868 q^{65} -63.9669 q^{67} +41.5692i q^{69} -21.5603 q^{71} -61.8652i q^{73} -115.667i q^{75} +3.81323 q^{79} +9.00000 q^{81} -100.510i q^{83} +183.560 q^{85} -18.3483i q^{87} -99.0504i q^{89} +96.5603 q^{93} +157.121 q^{95} +63.5973i q^{97} +13.7802 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} + 30 q^{11} - 18 q^{15} + 96 q^{23} - 122 q^{25} + 6 q^{29} - 146 q^{37} - 6 q^{39} - 70 q^{43} + 36 q^{51} - 198 q^{53} + 162 q^{57} - 204 q^{65} - 14 q^{67} + 204 q^{71} + 112 q^{79} + 36 q^{81} + 444 q^{85} + 96 q^{93} + 48 q^{95} - 90 q^{99}+O(q^{100}) \) 4 * q - 12 * q^9 + 30 * q^11 - 18 * q^15 + 96 * q^23 - 122 * q^25 + 6 * q^29 - 146 * q^37 - 6 * q^39 - 70 * q^43 + 36 * q^51 - 198 * q^53 + 162 * q^57 - 204 * q^65 - 14 * q^67 + 204 * q^71 + 112 * q^79 + 36 * q^81 + 444 * q^85 + 96 * q^93 + 48 * q^95 - 90 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more