LMFDB - Embedded newform 2352.3.f.k.97.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2352,3,Mod(97,2352)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [8,0,0,0,0,0,0,0,-24,0,28,0,0,0,12,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$64.0873581775$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.126303473664.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 2x^{5} + 92x^{4} + 14x^{3} - 441x^{2} - 686x + 2401 $ x^8 - 2x^7 - 9x^6 + 2x^5 + 92x^4 + 14x^3 - 441x^2 - 686*x + 2401
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.5
Root			$-1.43536 + 2.22255i$ of defining polynomial
Character	$\chi$	$=$	2352.97
Dual form			2352.3.f.k.97.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -8.99938i q^{5} -3.00000 q^{9} +4.63934 q^{11} +15.7769i q^{13} +15.5874 q^{15} -9.25507i q^{17} +27.5480i q^{19} -5.65685 q^{23} -55.9889 q^{25} -5.19615i q^{27} -14.6969 q^{29} -9.97185i q^{31} +8.03556i q^{33} -50.8491 q^{37} -27.3264 q^{39} -70.2028i q^{41} +49.9937 q^{43} +26.9981i q^{45} -16.5310i q^{47} +16.0302 q^{51} -8.56275 q^{53} -41.7512i q^{55} -47.7146 q^{57} -98.1740i q^{59} +34.6410i q^{61} +141.983 q^{65} -10.3809 q^{67} -9.79796i q^{69} -93.3690 q^{71} +124.553i q^{73} -96.9756i q^{75} -110.741 q^{79} +9.00000 q^{81} +136.246i q^{83} -83.2899 q^{85} -25.4557i q^{87} +160.317i q^{89} +17.2717 q^{93} +247.915 q^{95} +129.419i q^{97} -13.9180 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9} + 28 q^{11} + 12 q^{15} + 12 q^{25} - 4 q^{29} - 100 q^{37} - 12 q^{39} + 20 q^{43} - 24 q^{51} + 100 q^{53} - 156 q^{57} + 296 q^{65} + 68 q^{67} - 424 q^{71} - 80 q^{79} + 72 q^{81} - 232 q^{85}+ \cdots - 84 q^{99}+O(q^{100}) $ 8 * q - 24 * q^9 + 28 * q^11 + 12 * q^15 + 12 * q^25 - 4 * q^29 - 100 * q^37 - 12 * q^39 + 20 * q^43 - 24 * q^51 + 100 * q^53 - 156 * q^57 + 296 * q^65 + 68 * q^67 - 424 * q^71 - 80 * q^79 + 72 * q^81 - 232 * q^85 - 48 * q^93 + 288 * q^95 - 84 * q^99

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
$5$	− 8.99938i	− 1.79988i	−0.436017	−	0.899938i	$-0.643611\pi$
$5$	− 8.99938i	− 1.79988i	0.436017	−	0.899938i	$-0.356389\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	4.63934	0.421758	0.210879	−	0.977512i	$-0.432367\pi$
$11$	4.63934	0.421758	0.210879	+	0.977512i	$0.432367\pi$
$12$	0	0
$13$	15.7769i	1.21361i	0.794851	+	0.606805i	$0.207549\pi$
$13$	15.7769i	1.21361i	−0.794851	+	0.606805i	$0.792451\pi$
$14$	0	0
$15$	15.5874	1.03916
$16$	0	0
$17$	− 9.25507i	− 0.544416i	−0.962238	−	0.272208i	$-0.912246\pi$
$17$	− 9.25507i	− 0.544416i	0.962238	−	0.272208i	$-0.0877538\pi$
$18$	0	0
$19$	27.5480i	1.44990i	0.688803	+	0.724948i	$0.258136\pi$
$19$	27.5480i	1.44990i	−0.688803	+	0.724948i	$0.741864\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.65685	−0.245950	−0.122975	−	0.992410i	$-0.539244\pi$
$23$	−5.65685	−0.245950	−0.122975	+	0.992410i	$0.539244\pi$
$24$	0	0
$25$	−55.9889	−2.23956
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	−14.6969	−0.506789	−0.253395	−	0.967363i	$-0.581547\pi$
$29$	−14.6969	−0.506789	−0.253395	+	0.967363i	$0.581547\pi$
$30$	0	0
$31$	− 9.97185i	− 0.321672i	−0.986981	−	0.160836i	$-0.948581\pi$
$31$	− 9.97185i	− 0.321672i	0.986981	−	0.160836i	$-0.0514191\pi$
$32$	0	0
$33$	8.03556i	0.243502i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−50.8491	−1.37430	−0.687151	−	0.726515i	$-0.741139\pi$
$37$	−50.8491	−1.37430	−0.687151	+	0.726515i	$0.741139\pi$
$38$	0	0
$39$	−27.3264	−0.700678
$40$	0	0
$41$	− 70.2028i	− 1.71226i	−0.516758	−	0.856132i	$-0.672861\pi$
$41$	− 70.2028i	− 1.71226i	0.516758	−	0.856132i	$-0.327139\pi$
$42$	0	0
$43$	49.9937	1.16264	0.581322	−	0.813674i	$-0.302536\pi$
$43$	49.9937	1.16264	0.581322	+	0.813674i	$0.302536\pi$
$44$	0	0
$45$	26.9981i	0.599959i
$46$	0	0
$47$	− 16.5310i	− 0.351724i	−0.984415	−	0.175862i	$-0.943729\pi$
$47$	− 16.5310i	− 0.351724i	0.984415	−	0.175862i	$-0.0562713\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	16.0302	0.314318
$52$	0	0
$53$	−8.56275	−0.161561	−0.0807806	−	0.996732i	$-0.525741\pi$
$53$	−8.56275	−0.161561	−0.0807806	+	0.996732i	$0.525741\pi$
$54$	0	0
$55$	− 41.7512i	− 0.759112i
$56$	0	0
$57$	−47.7146	−0.837098
$58$	0	0
$59$	− 98.1740i	− 1.66397i	−0.554801	−	0.831983i	$-0.687206\pi$
$59$	− 98.1740i	− 1.66397i	0.554801	−	0.831983i	$-0.312794\pi$
$60$	0	0
$61$	34.6410i	0.567886i	0.958841	+	0.283943i	$0.0916426\pi$
$61$	34.6410i	0.567886i	−0.958841	+	0.283943i	$0.908357\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	141.983	2.18435
$66$	0	0
$67$	−10.3809	−0.154939	−0.0774696	−	0.996995i	$-0.524684\pi$
$67$	−10.3809	−0.154939	−0.0774696	+	0.996995i	$0.524684\pi$
$68$	0	0
$69$	− 9.79796i	− 0.141999i
$70$	0	0
$71$	−93.3690	−1.31506	−0.657528	−	0.753430i	$-0.728398\pi$
$71$	−93.3690	−1.31506	−0.657528	+	0.753430i	$0.728398\pi$
$72$	0	0
$73$	124.553i	1.70621i	0.521742	+	0.853103i	$0.325282\pi$
$73$	124.553i	1.70621i	−0.521742	+	0.853103i	$0.674718\pi$
$74$	0	0
$75$	− 96.9756i	− 1.29301i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−110.741	−1.40179	−0.700893	−	0.713267i	$-0.747215\pi$
$79$	−110.741	−1.40179	−0.700893	+	0.713267i	$0.747215\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	136.246i	1.64151i	0.571277	+	0.820757i	$0.306448\pi$
$83$	136.246i	1.64151i	−0.571277	+	0.820757i	$0.693552\pi$
$84$	0	0
$85$	−83.2899	−0.979881
$86$	0	0
$87$	− 25.4557i	− 0.292595i
$88$	0	0
$89$	160.317i	1.80132i	0.434529	+	0.900658i	$0.356915\pi$
$89$	160.317i	1.80132i	−0.434529	+	0.900658i	$0.643085\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	17.2717	0.185718
$94$	0	0
$95$	247.915	2.60963
$96$	0	0
$97$	129.419i	1.33421i	0.744962	+	0.667107i	$0.232468\pi$
$97$	129.419i	1.33421i	−0.744962	+	0.667107i	$0.767532\pi$
$98$	0	0
$99$	−13.9180	−0.140586
$100$	0	0
$101$	37.4159i	0.370455i	0.982696	+	0.185227i	$0.0593022\pi$
$101$	37.4159i	0.370455i	−0.982696	+	0.185227i	$0.940698\pi$
$102$	0	0
$103$	151.626i	1.47210i	0.676929	+	0.736049i	$0.263310\pi$
$103$	151.626i	1.47210i	−0.676929	+	0.736049i	$0.736690\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−68.9690	−0.644570	−0.322285	−	0.946643i	$-0.604451\pi$
$107$	−68.9690	−0.644570	−0.322285	+	0.946643i	$0.604451\pi$
$108$	0	0
$109$	−58.1140	−0.533156	−0.266578	−	0.963813i	$-0.585893\pi$
$109$	−58.1140	−0.533156	−0.266578	+	0.963813i	$0.585893\pi$
$110$	0	0
$111$	− 88.0733i	− 0.793453i
$112$	0	0
$113$	−123.136	−1.08970	−0.544849	−	0.838534i	$-0.683413\pi$
$113$	−123.136	−1.08970	−0.544849	+	0.838534i	$0.683413\pi$
$114$	0	0
$115$	50.9082i	0.442680i
$116$	0	0
$117$	− 47.3308i	− 0.404536i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−99.4766	−0.822120
$122$	0	0
$123$	121.595	0.988576
$124$	0	0
$125$	278.881i	2.23105i
$126$	0	0
$127$	34.1002	0.268505	0.134253	−	0.990947i	$-0.457137\pi$
$127$	34.1002	0.268505	0.134253	+	0.990947i	$0.457137\pi$
$128$	0	0
$129$	86.5916i	0.671253i
$130$	0	0
$131$	49.6164i	0.378751i	0.981905	+	0.189376i	$0.0606464\pi$
$131$	49.6164i	0.378751i	−0.981905	+	0.189376i	$0.939354\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−46.7622	−0.346386
$136$	0	0
$137$	105.668	0.771302	0.385651	−	0.922645i	$-0.373977\pi$
$137$	105.668	0.771302	0.385651	+	0.922645i	$0.373977\pi$
$138$	0	0
$139$	108.092i	0.777644i	0.921313	+	0.388822i	$0.127118\pi$
$139$	108.092i	0.777644i	−0.921313	+	0.388822i	$0.872882\pi$
$140$	0	0
$141$	28.6326	0.203068
$142$	0	0
$143$	73.1944i	0.511849i
$144$	0	0
$145$	132.263i	0.912158i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−55.1974	−0.370452	−0.185226	−	0.982696i	$-0.559302\pi$
$149$	−55.1974	−0.370452	−0.185226	+	0.982696i	$0.559302\pi$
$150$	0	0
$151$	108.082	0.715774	0.357887	−	0.933765i	$-0.383497\pi$
$151$	108.082	0.715774	0.357887	+	0.933765i	$0.383497\pi$
$152$	0	0
$153$	27.7652i	0.181472i
$154$	0	0
$155$	−89.7405	−0.578971
$156$	0	0
$157$	− 168.812i	− 1.07524i	−0.843188	−	0.537618i	$-0.819324\pi$
$157$	− 168.812i	− 1.07524i	0.843188	−	0.537618i	$-0.180676\pi$
$158$	0	0
$159$	− 14.8311i	− 0.0932774i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	134.911	0.827677	0.413838	−	0.910350i	$-0.364188\pi$
$163$	134.911	0.827677	0.413838	+	0.910350i	$0.364188\pi$
$164$	0	0
$165$	72.3151	0.438273
$166$	0	0
$167$	− 36.1722i	− 0.216600i	−0.994118	−	0.108300i	$-0.965459\pi$
$167$	− 36.1722i	− 0.216600i	0.994118	−	0.108300i	$-0.0345407\pi$
$168$	0	0
$169$	−79.9113	−0.472848
$170$	0	0
$171$	− 82.6441i	− 0.483299i
$172$	0	0
$173$	− 259.516i	− 1.50009i	−0.661385	−	0.750047i	$-0.730031\pi$
$173$	− 259.516i	− 1.50009i	0.661385	−	0.750047i	$-0.269969\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	170.042	0.960691
$178$	0	0
$179$	29.9781	0.167476	0.0837378	−	0.996488i	$-0.473314\pi$
$179$	29.9781	0.167476	0.0837378	+	0.996488i	$0.473314\pi$
$180$	0	0
$181$	61.8216i	0.341556i	0.985310	+	0.170778i	$0.0546281\pi$
$181$	61.8216i	0.341556i	−0.985310	+	0.170778i	$0.945372\pi$
$182$	0	0
$183$	−60.0000	−0.327869
$184$	0	0
$185$	457.611i	2.47357i
$186$	0	0
$187$	− 42.9374i	− 0.229612i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	87.3714	0.457442	0.228721	−	0.973492i	$-0.426546\pi$
$191$	87.3714	0.457442	0.228721	+	0.973492i	$0.426546\pi$
$192$	0	0
$193$	8.38336	0.0434371	0.0217185	−	0.999764i	$-0.493086\pi$
$193$	8.38336	0.0434371	0.0217185	+	0.999764i	$0.493086\pi$
$194$	0	0
$195$	245.921i	1.26113i
$196$	0	0
$197$	−253.870	−1.28868	−0.644340	−	0.764739i	$-0.722868\pi$
$197$	−253.870	−1.28868	−0.644340	+	0.764739i	$0.722868\pi$
$198$	0	0
$199$	207.405i	1.04224i	0.853485	+	0.521118i	$0.174485\pi$
$199$	207.405i	1.04224i	−0.853485	+	0.521118i	$0.825515\pi$
$200$	0	0
$201$	− 17.9803i	− 0.0894542i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−631.782	−3.08186
$206$	0	0
$207$	16.9706	0.0819834
$208$	0	0
$209$	127.805i	0.611505i
$210$	0	0
$211$	155.940	0.739051	0.369525	−	0.929221i	$-0.379520\pi$
$211$	155.940	0.739051	0.369525	+	0.929221i	$0.379520\pi$
$212$	0	0
$213$	− 161.720i	− 0.759248i
$214$	0	0
$215$	− 449.912i	− 2.09262i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−215.732	−0.985079
$220$	0	0
$221$	146.016	0.660708
$222$	0	0
$223$	241.512i	1.08301i	0.840696	+	0.541507i	$0.182146\pi$
$223$	241.512i	1.08301i	−0.840696	+	0.541507i	$0.817854\pi$
$224$	0	0
$225$	167.967	0.746518
$226$	0	0
$227$	142.668i	0.628496i	0.949341	+	0.314248i	$0.101752\pi$
$227$	142.668i	0.628496i	−0.949341	+	0.314248i	$0.898248\pi$
$228$	0	0
$229$	− 322.861i	− 1.40988i	−0.709269	−	0.704938i	$-0.750975\pi$
$229$	− 322.861i	− 1.40988i	0.709269	−	0.704938i	$-0.249025\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	64.7342	0.277829	0.138915	−	0.990304i	$-0.455639\pi$
$233$	64.7342	0.277829	0.138915	+	0.990304i	$0.455639\pi$
$234$	0	0
$235$	−148.769	−0.633060
$236$	0	0
$237$	− 191.809i	− 0.809321i
$238$	0	0
$239$	81.7863	0.342202	0.171101	−	0.985253i	$-0.445268\pi$
$239$	81.7863	0.342202	0.171101	+	0.985253i	$0.445268\pi$
$240$	0	0
$241$	31.9918i	0.132746i	0.997795	+	0.0663730i	$0.0211427\pi$
$241$	31.9918i	0.132746i	−0.997795	+	0.0663730i	$0.978857\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−434.623	−1.75961
$248$	0	0
$249$	−235.985	−0.947729
$250$	0	0
$251$	251.044i	1.00018i	0.865975	+	0.500088i	$0.166699\pi$
$251$	251.044i	1.00018i	−0.865975	+	0.500088i	$0.833301\pi$
$252$	0	0
$253$	−26.2440	−0.103731
$254$	0	0
$255$	− 144.262i	− 0.565734i
$256$	0	0
$257$	136.106i	0.529595i	0.964304	+	0.264797i	$0.0853050\pi$
$257$	136.106i	0.529595i	−0.964304	+	0.264797i	$0.914695\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	44.0907	0.168930
$262$	0	0
$263$	−157.131	−0.597457	−0.298728	−	0.954338i	$-0.596562\pi$
$263$	−157.131	−0.597457	−0.298728	+	0.954338i	$0.596562\pi$
$264$	0	0
$265$	77.0594i	0.290790i
$266$	0	0
$267$	−277.677	−1.03999
$268$	0	0
$269$	46.8077i	0.174006i	0.996208	+	0.0870032i	$0.0277290\pi$
$269$	46.8077i	0.174006i	−0.996208	+	0.0870032i	$0.972271\pi$
$270$	0	0
$271$	354.021i	1.30635i	0.757207	+	0.653175i	$0.226563\pi$
$271$	354.021i	1.30635i	−0.757207	+	0.653175i	$0.773437\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−259.751	−0.944550
$276$	0	0
$277$	−127.766	−0.461248	−0.230624	−	0.973043i	$-0.574077\pi$
$277$	−127.766	−0.461248	−0.230624	+	0.973043i	$0.574077\pi$
$278$	0	0
$279$	29.9155i	0.107224i
$280$	0	0
$281$	−23.9203	−0.0851256	−0.0425628	−	0.999094i	$-0.513552\pi$
$281$	−23.9203	−0.0851256	−0.0425628	+	0.999094i	$0.513552\pi$
$282$	0	0
$283$	− 109.248i	− 0.386034i	−0.981195	−	0.193017i	$-0.938173\pi$
$283$	− 109.248i	− 0.386034i	0.981195	−	0.193017i	$-0.0618273\pi$
$284$	0	0
$285$	429.402i	1.50667i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	203.344	0.703612
$290$	0	0
$291$	−224.160	−0.770309
$292$	0	0
$293$	233.340i	0.796384i	0.917302	+	0.398192i	$0.130362\pi$
$293$	233.340i	0.796384i	−0.917302	+	0.398192i	$0.869638\pi$
$294$	0	0
$295$	−883.505	−2.99493
$296$	0	0
$297$	− 24.1067i	− 0.0811673i
$298$	0	0
$299$	− 89.2478i	− 0.298487i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−64.8063	−0.213882
$304$	0	0
$305$	311.748	1.02212
$306$	0	0
$307$	− 355.416i	− 1.15771i	−0.815432	−	0.578854i	$-0.803500\pi$
$307$	− 355.416i	− 1.15771i	0.815432	−	0.578854i	$-0.196500\pi$
$308$	0	0
$309$	−262.624	−0.849916
$310$	0	0
$311$	− 266.053i	− 0.855475i	−0.903903	−	0.427737i	$-0.859311\pi$
$311$	− 266.053i	− 0.855475i	0.903903	−	0.427737i	$-0.140689\pi$
$312$	0	0
$313$	− 256.244i	− 0.818669i	−0.912384	−	0.409335i	$-0.865761\pi$
$313$	− 256.244i	− 0.818669i	0.912384	−	0.409335i	$-0.134239\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−357.398	−1.12744	−0.563719	−	0.825966i	$-0.690630\pi$
$317$	−357.398	−1.12744	−0.563719	+	0.825966i	$0.690630\pi$
$318$	0	0
$319$	−68.1838	−0.213742
$320$	0	0
$321$	− 119.458i	− 0.372143i
$322$	0	0
$323$	254.959	0.789346
$324$	0	0
$325$	− 883.332i	− 2.71795i
$326$	0	0
$327$	− 100.656i	− 0.307818i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	473.599	1.43081	0.715407	−	0.698708i	$-0.246241\pi$
$331$	473.599	1.43081	0.715407	+	0.698708i	$0.246241\pi$
$332$	0	0
$333$	152.547	0.458100
$334$	0	0
$335$	93.4219i	0.278871i
$336$	0	0
$337$	−401.285	−1.19076	−0.595378	−	0.803446i	$-0.702998\pi$
$337$	−401.285	−1.19076	−0.595378	+	0.803446i	$0.702998\pi$
$338$	0	0
$339$	− 213.277i	− 0.629137i
$340$	0	0
$341$	− 46.2627i	− 0.135668i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−88.1756	−0.255581
$346$	0	0
$347$	156.073	0.449777	0.224889	−	0.974385i	$-0.427798\pi$
$347$	156.073	0.449777	0.224889	+	0.974385i	$0.427798\pi$
$348$	0	0
$349$	− 123.443i	− 0.353706i	−0.984237	−	0.176853i	$-0.943408\pi$
$349$	− 123.443i	− 0.353706i	0.984237	−	0.176853i	$-0.0565917\pi$
$350$	0	0
$351$	81.9793	0.233559
$352$	0	0
$353$	− 109.204i	− 0.309359i	−0.987965	−	0.154680i	$-0.950565\pi$
$353$	− 109.204i	− 0.309359i	0.987965	−	0.154680i	$-0.0494346\pi$
$354$	0	0
$355$	840.263i	2.36694i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−294.107	−0.819238	−0.409619	−	0.912257i	$-0.634338\pi$
$359$	−294.107	−0.819238	−0.409619	+	0.912257i	$0.634338\pi$
$360$	0	0
$361$	−397.894	−1.10220
$362$	0	0
$363$	− 172.298i	− 0.474651i
$364$	0	0
$365$	1120.90	3.07096
$366$	0	0
$367$	274.314i	0.747450i	0.927540	+	0.373725i	$0.121920\pi$
$367$	274.314i	0.747450i	−0.927540	+	0.373725i	$0.878080\pi$
$368$	0	0
$369$	210.608i	0.570754i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	83.6511	0.224266	0.112133	−	0.993693i	$-0.464232\pi$
$373$	83.6511	0.224266	0.112133	+	0.993693i	$0.464232\pi$
$374$	0	0
$375$	−483.036	−1.28810
$376$	0	0
$377$	− 231.872i	− 0.615044i
$378$	0	0
$379$	−77.2239	−0.203757	−0.101878	−	0.994797i	$-0.532485\pi$
$379$	−77.2239	−0.203757	−0.101878	+	0.994797i	$0.532485\pi$
$380$	0	0
$381$	59.0633i	0.155022i
$382$	0	0
$383$	3.65557i	0.00954456i	0.999989	+	0.00477228i	$0.00151907\pi$
$383$	3.65557i	0.00954456i	−0.999989	+	0.00477228i	$0.998481\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−149.981	−0.387548
$388$	0	0
$389$	286.398	0.736241	0.368120	−	0.929778i	$-0.380001\pi$
$389$	286.398	0.736241	0.368120	+	0.929778i	$0.380001\pi$
$390$	0	0
$391$	52.3546i	0.133899i
$392$	0	0
$393$	−85.9381	−0.218672
$394$	0	0
$395$	996.601i	2.52304i
$396$	0	0
$397$	− 199.550i	− 0.502646i	−0.967903	−	0.251323i	$-0.919134\pi$
$397$	− 199.550i	− 0.502646i	0.967903	−	0.251323i	$-0.0808656\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−519.657	−1.29590	−0.647951	−	0.761682i	$-0.724374\pi$
$401$	−519.657	−1.29590	−0.647951	+	0.761682i	$0.724374\pi$
$402$	0	0
$403$	157.325	0.390385
$404$	0	0
$405$	− 80.9944i	− 0.199986i
$406$	0	0
$407$	−235.906	−0.579622
$408$	0	0
$409$	− 559.784i	− 1.36866i	−0.729171	−	0.684332i	$-0.760094\pi$
$409$	− 559.784i	− 1.36866i	0.729171	−	0.684332i	$-0.239906\pi$
$410$	0	0
$411$	183.023i	0.445311i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1226.13	2.95452
$416$	0	0
$417$	−187.222	−0.448973
$418$	0	0
$419$	659.099i	1.57303i	0.617572	+	0.786514i	$0.288116\pi$
$419$	659.099i	1.57303i	−0.617572	+	0.786514i	$0.711884\pi$
$420$	0	0
$421$	−90.2042	−0.214262	−0.107131	−	0.994245i	$-0.534166\pi$
$421$	−90.2042	−0.214262	−0.107131	+	0.994245i	$0.534166\pi$
$422$	0	0
$423$	49.5931i	0.117241i
$424$	0	0
$425$	518.181i	1.21925i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−126.776	−0.295516
$430$	0	0
$431$	−651.726	−1.51213	−0.756063	−	0.654499i	$-0.772879\pi$
$431$	−651.726	−1.51213	−0.756063	+	0.654499i	$0.772879\pi$
$432$	0	0
$433$	320.857i	0.741008i	0.928831	+	0.370504i	$0.120815\pi$
$433$	320.857i	0.741008i	−0.928831	+	0.370504i	$0.879185\pi$
$434$	0	0
$435$	−229.086	−0.526635
$436$	0	0
$437$	− 155.835i	− 0.356602i
$438$	0	0
$439$	− 334.246i	− 0.761380i	−0.924703	−	0.380690i	$-0.875686\pi$
$439$	− 334.246i	− 0.761380i	0.924703	−	0.380690i	$-0.124314\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−112.420	−0.253770	−0.126885	−	0.991917i	$-0.540498\pi$
$443$	−112.420	−0.253770	−0.126885	+	0.991917i	$0.540498\pi$
$444$	0	0
$445$	1442.75	3.24215
$446$	0	0
$447$	− 95.6047i	− 0.213881i
$448$	0	0
$449$	−496.635	−1.10609	−0.553045	−	0.833151i	$-0.686534\pi$
$449$	−496.635	−1.10609	−0.553045	+	0.833151i	$0.686534\pi$
$450$	0	0
$451$	− 325.694i	− 0.722160i
$452$	0	0
$453$	187.203i	0.413252i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−788.224	−1.72478	−0.862390	−	0.506245i	$-0.831033\pi$
$457$	−788.224	−1.72478	−0.862390	+	0.506245i	$0.831033\pi$
$458$	0	0
$459$	−48.0907	−0.104773
$460$	0	0
$461$	− 422.078i	− 0.915570i	−0.889063	−	0.457785i	$-0.848643\pi$
$461$	− 422.078i	− 0.915570i	0.889063	−	0.457785i	$-0.151357\pi$
$462$	0	0
$463$	7.47288	0.0161401	0.00807006	−	0.999967i	$-0.497431\pi$
$463$	7.47288	0.0161401	0.00807006	+	0.999967i	$0.497431\pi$
$464$	0	0
$465$	− 155.435i	− 0.334269i
$466$	0	0
$467$	434.047i	0.929437i	0.885458	+	0.464719i	$0.153845\pi$
$467$	434.047i	0.929437i	−0.885458	+	0.464719i	$0.846155\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	292.391	0.620788
$472$	0	0
$473$	231.937	0.490354
$474$	0	0
$475$	− 1542.38i	− 3.24712i
$476$	0	0
$477$	25.6882	0.0538537
$478$	0	0
$479$	538.407i	1.12402i	0.827129	+	0.562012i	$0.189972\pi$
$479$	538.407i	1.12402i	−0.827129	+	0.562012i	$0.810028\pi$
$480$	0	0
$481$	− 802.243i	− 1.66787i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1164.69	2.40142
$486$	0	0
$487$	−62.9049	−0.129168	−0.0645841	−	0.997912i	$-0.520572\pi$
$487$	−62.9049	−0.129168	−0.0645841	+	0.997912i	$0.520572\pi$
$488$	0	0
$489$	233.673i	0.477859i
$490$	0	0
$491$	−824.426	−1.67908	−0.839538	−	0.543301i	$-0.817174\pi$
$491$	−824.426	−1.67908	−0.839538	+	0.543301i	$0.817174\pi$
$492$	0	0
$493$	136.021i	0.275904i
$494$	0	0
$495$	125.253i	0.253037i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	244.055	0.489088	0.244544	−	0.969638i	$-0.421362\pi$
$499$	244.055	0.489088	0.244544	+	0.969638i	$0.421362\pi$
$500$	0	0
$501$	62.6521	0.125054
$502$	0	0
$503$	152.196i	0.302577i	0.988490	+	0.151289i	$0.0483423\pi$
$503$	152.196i	0.302577i	−0.988490	+	0.151289i	$0.951658\pi$
$504$	0	0
$505$	336.720	0.666773
$506$	0	0
$507$	− 138.410i	− 0.272999i
$508$	0	0
$509$	− 706.727i	− 1.38846i	−0.719752	−	0.694231i	$-0.755745\pi$
$509$	− 706.727i	− 1.38846i	0.719752	−	0.694231i	$-0.244255\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	143.144	0.279033
$514$	0	0
$515$	1364.54	2.64959
$516$	0	0
$517$	− 76.6931i	− 0.148342i
$518$	0	0
$519$	449.495	0.866079
$520$	0	0
$521$	888.948i	1.70623i	0.521719	+	0.853117i	$0.325291\pi$
$521$	888.948i	1.70623i	−0.521719	+	0.853117i	$0.674709\pi$
$522$	0	0
$523$	− 941.066i	− 1.79936i	−0.436548	−	0.899681i	$-0.643799\pi$
$523$	− 941.066i	− 1.79936i	0.436548	−	0.899681i	$-0.356201\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−92.2901	−0.175124
$528$	0	0
$529$	−497.000	−0.939509
$530$	0	0
$531$	294.522i	0.554655i
$532$	0	0
$533$	1107.58	2.07802
$534$	0	0
$535$	620.679i	1.16015i
$536$	0	0
$537$	51.9236i	0.0966921i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	530.128	0.979903	0.489951	−	0.871750i	$-0.337014\pi$
$541$	530.128	0.979903	0.489951	+	0.871750i	$0.337014\pi$
$542$	0	0
$543$	−107.078	−0.197197
$544$	0	0
$545$	522.990i	0.959614i
$546$	0	0
$547$	1063.97	1.94510	0.972548	−	0.232701i	$-0.0747562\pi$
$547$	1063.97	1.94510	0.972548	+	0.232701i	$0.0747562\pi$
$548$	0	0
$549$	− 103.923i	− 0.189295i
$550$	0	0
$551$	− 404.870i	− 0.734792i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−792.605	−1.42812
$556$	0	0
$557$	−23.6589	−0.0424755	−0.0212378	−	0.999774i	$-0.506761\pi$
$557$	−23.6589	−0.0424755	−0.0212378	+	0.999774i	$0.506761\pi$
$558$	0	0
$559$	788.746i	1.41100i
$560$	0	0
$561$	74.3697	0.132566
$562$	0	0
$563$	− 61.5033i	− 0.109242i	−0.998507	−	0.0546210i	$-0.982605\pi$
$563$	− 61.5033i	− 0.109242i	0.998507	−	0.0546210i	$-0.0173951\pi$
$564$	0	0
$565$	1108.15i	1.96132i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	681.471	1.19766	0.598832	−	0.800875i	$-0.295632\pi$
$569$	681.471	1.19766	0.598832	+	0.800875i	$0.295632\pi$
$570$	0	0
$571$	−364.947	−0.639137	−0.319569	−	0.947563i	$-0.603538\pi$
$571$	−364.947	−0.639137	−0.319569	+	0.947563i	$0.603538\pi$
$572$	0	0
$573$	151.332i	0.264104i
$574$	0	0
$575$	316.721	0.550819
$576$	0	0
$577$	314.606i	0.545245i	0.962121	+	0.272622i	$0.0878910\pi$
$577$	314.606i	0.545245i	−0.962121	+	0.272622i	$0.912109\pi$
$578$	0	0
$579$	14.5204i	0.0250784i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−39.7254	−0.0681397
$584$	0	0
$585$	−425.948	−0.728116
$586$	0	0
$587$	190.516i	0.324560i	0.986745	+	0.162280i	$0.0518847\pi$
$587$	190.516i	0.324560i	−0.986745	+	0.162280i	$0.948115\pi$
$588$	0	0
$589$	274.705	0.466392
$590$	0	0
$591$	− 439.716i	− 0.744019i
$592$	0	0
$593$	− 133.324i	− 0.224830i	−0.993661	−	0.112415i	$-0.964141\pi$
$593$	− 133.324i	− 0.224830i	0.993661	−	0.112415i	$-0.0358587\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−359.236	−0.601735
$598$	0	0
$599$	272.041	0.454159	0.227080	−	0.973876i	$-0.427082\pi$
$599$	272.041	0.454159	0.227080	+	0.973876i	$0.427082\pi$
$600$	0	0
$601$	− 661.949i	− 1.10141i	−0.834699	−	0.550706i	$-0.814358\pi$
$601$	− 661.949i	− 1.10141i	0.834699	−	0.550706i	$-0.185642\pi$
$602$	0	0
$603$	31.1428	0.0516464
$604$	0	0
$605$	895.228i	1.47972i
$606$	0	0
$607$	72.7843i	0.119908i	0.998201	+	0.0599541i	$0.0190954\pi$
$607$	72.7843i	0.119908i	−0.998201	+	0.0599541i	$0.980905\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	260.809	0.426856
$612$	0	0
$613$	−92.5299	−0.150946	−0.0754730	−	0.997148i	$-0.524047\pi$
$613$	−92.5299	−0.150946	−0.0754730	+	0.997148i	$0.524047\pi$
$614$	0	0
$615$	− 1094.28i	− 1.77931i
$616$	0	0
$617$	128.000	0.207455	0.103728	−	0.994606i	$-0.466923\pi$
$617$	128.000	0.207455	0.103728	+	0.994606i	$0.466923\pi$
$618$	0	0
$619$	242.933i	0.392461i	0.980558	+	0.196230i	$0.0628700\pi$
$619$	242.933i	0.392461i	−0.980558	+	0.196230i	$0.937130\pi$
$620$	0	0
$621$	29.3939i	0.0473331i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1110.03	1.77605
$626$	0	0
$627$	−221.364	−0.353053
$628$	0	0
$629$	470.612i	0.748191i
$630$	0	0
$631$	179.943	0.285171	0.142586	−	0.989782i	$-0.454458\pi$
$631$	179.943	0.285171	0.142586	+	0.989782i	$0.454458\pi$
$632$	0	0
$633$	270.095i	0.426691i
$634$	0	0
$635$	− 306.881i	− 0.483277i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	280.107	0.438352
$640$	0	0
$641$	−260.932	−0.407071	−0.203535	−	0.979068i	$-0.565243\pi$
$641$	−260.932	−0.407071	−0.203535	+	0.979068i	$0.565243\pi$
$642$	0	0
$643$	845.112i	1.31433i	0.753748	+	0.657163i	$0.228244\pi$
$643$	845.112i	1.31433i	−0.753748	+	0.657163i	$0.771756\pi$
$644$	0	0
$645$	779.271	1.20817
$646$	0	0
$647$	49.0859i	0.0758669i	0.999280	+	0.0379334i	$0.0120775\pi$
$647$	49.0859i	0.0758669i	−0.999280	+	0.0379334i	$0.987923\pi$
$648$	0	0
$649$	− 455.462i	− 0.701791i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	537.256	0.822751	0.411376	−	0.911466i	$-0.365048\pi$
$653$	537.256	0.822751	0.411376	+	0.911466i	$0.365048\pi$
$654$	0	0
$655$	446.517	0.681705
$656$	0	0
$657$	− 373.659i	− 0.568735i
$658$	0	0
$659$	687.009	1.04250	0.521251	−	0.853403i	$-0.325465\pi$
$659$	687.009	1.04250	0.521251	+	0.853403i	$0.325465\pi$
$660$	0	0
$661$	17.5276i	0.0265168i	0.999912	+	0.0132584i	$0.00422040\pi$
$661$	17.5276i	0.0265168i	−0.999912	+	0.0132584i	$0.995780\pi$
$662$	0	0
$663$	252.908i	0.381460i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	83.1381	0.124645
$668$	0	0
$669$	−418.312	−0.625279
$670$	0	0
$671$	160.711i	0.239510i
$672$	0	0
$673$	−238.854	−0.354909	−0.177455	−	0.984129i	$-0.556786\pi$
$673$	−238.854	−0.354909	−0.177455	+	0.984129i	$0.556786\pi$
$674$	0	0
$675$	290.927i	0.431003i
$676$	0	0
$677$	690.562i	1.02003i	0.860165	+	0.510016i	$0.170361\pi$
$677$	690.562i	1.02003i	−0.860165	+	0.510016i	$0.829639\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−247.109	−0.362862
$682$	0	0
$683$	−727.545	−1.06522	−0.532609	−	0.846361i	$-0.678789\pi$
$683$	−727.545	−1.06522	−0.532609	+	0.846361i	$0.678789\pi$
$684$	0	0
$685$	− 950.950i	− 1.38825i
$686$	0	0
$687$	559.212	0.813992
$688$	0	0
$689$	− 135.094i	− 0.196072i
$690$	0	0
$691$	− 297.944i	− 0.431179i	−0.976484	−	0.215589i	$-0.930833\pi$
$691$	− 297.944i	− 0.431179i	0.976484	−	0.215589i	$-0.0691672\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	972.765	1.39966
$696$	0	0
$697$	−649.731	−0.932183
$698$	0	0
$699$	112.123i	0.160405i
$700$	0	0
$701$	−621.118	−0.886046	−0.443023	−	0.896510i	$-0.646094\pi$
$701$	−621.118	−0.886046	−0.443023	+	0.896510i	$0.646094\pi$
$702$	0	0
$703$	− 1400.79i	− 1.99259i
$704$	0	0
$705$	− 257.676i	− 0.365498i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−428.490	−0.604358	−0.302179	−	0.953251i	$-0.597714\pi$
$709$	−428.490	−0.604358	−0.302179	+	0.953251i	$0.597714\pi$
$710$	0	0
$711$	332.223	0.467262
$712$	0	0
$713$	56.4093i	0.0791154i
$714$	0	0
$715$	658.705	0.921265
$716$	0	0
$717$	141.658i	0.197570i
$718$	0	0
$719$	172.307i	0.239648i	0.992795	+	0.119824i	$0.0382330\pi$
$719$	172.307i	0.239648i	−0.992795	+	0.119824i	$0.961767\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−55.4114	−0.0766409
$724$	0	0
$725$	822.862	1.13498
$726$	0	0
$727$	− 1320.20i	− 1.81596i	−0.419010	−	0.907981i	$-0.637623\pi$
$727$	− 1320.20i	− 1.81596i	0.419010	−	0.907981i	$-0.362377\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 462.695i	− 0.632961i
$732$	0	0
$733$	− 559.051i	− 0.762689i	−0.924433	−	0.381344i	$-0.875461\pi$
$733$	− 559.051i	− 0.762689i	0.924433	−	0.381344i	$-0.124539\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−48.1606	−0.0653468
$738$	0	0
$739$	−1142.28	−1.54572	−0.772858	−	0.634579i	$-0.781174\pi$
$739$	−1142.28	−1.54572	−0.772858	+	0.634579i	$0.781174\pi$
$740$	0	0
$741$	− 752.790i	− 1.01591i
$742$	0	0
$743$	1392.95	1.87477	0.937383	−	0.348301i	$-0.113241\pi$
$743$	1392.95	1.87477	0.937383	+	0.348301i	$0.113241\pi$
$744$	0	0
$745$	496.742i	0.666768i
$746$	0	0
$747$	− 408.737i	− 0.547172i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1391.46	−1.85282	−0.926408	−	0.376522i	$-0.877120\pi$
$751$	−1391.46	−1.85282	−0.926408	+	0.376522i	$0.877120\pi$
$752$	0	0
$753$	−434.821	−0.577452
$754$	0	0
$755$	− 972.670i	− 1.28830i
$756$	0	0
$757$	139.507	0.184289	0.0921446	−	0.995746i	$-0.470628\pi$
$757$	139.507	0.184289	0.0921446	+	0.995746i	$0.470628\pi$
$758$	0	0
$759$	− 45.4560i	− 0.0598894i
$760$	0	0
$761$	− 743.028i	− 0.976384i	−0.872736	−	0.488192i	$-0.837657\pi$
$761$	− 743.028i	− 0.976384i	0.872736	−	0.488192i	$-0.162343\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	249.870	0.326627
$766$	0	0
$767$	1548.88	2.01940
$768$	0	0
$769$	− 157.528i	− 0.204848i	−0.994741	−	0.102424i	$-0.967340\pi$
$769$	− 157.528i	− 0.204848i	0.994741	−	0.102424i	$-0.0326598\pi$
$770$	0	0
$771$	−235.742	−0.305762
$772$	0	0
$773$	− 410.753i	− 0.531375i	−0.964059	−	0.265688i	$-0.914401\pi$
$773$	− 410.753i	− 0.531375i	0.964059	−	0.265688i	$-0.0855990\pi$
$774$	0	0
$775$	558.313i	0.720403i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1933.95	2.48260
$780$	0	0
$781$	−433.170	−0.554635
$782$	0	0
$783$	76.3672i	0.0975316i
$784$	0	0
$785$	−1519.20	−1.93529
$786$	0	0
$787$	− 1082.57i	− 1.37557i	−0.725916	−	0.687783i	$-0.758584\pi$
$787$	− 1082.57i	− 1.37557i	0.725916	−	0.687783i	$-0.241416\pi$
$788$	0	0
$789$	− 272.159i	− 0.344942i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−546.529	−0.689191
$794$	0	0
$795$	−133.471	−0.167888
$796$	0	0
$797$	− 569.370i	− 0.714392i	−0.934029	−	0.357196i	$-0.883733\pi$
$797$	− 569.370i	− 0.714392i	0.934029	−	0.357196i	$-0.116267\pi$
$798$	0	0
$799$	−152.996	−0.191484
$800$	0	0
$801$	− 480.951i	− 0.600438i
$802$	0	0
$803$	577.843i	0.719606i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−81.0734	−0.100463
$808$	0	0
$809$	−1078.48	−1.33310	−0.666548	−	0.745462i	$-0.732229\pi$
$809$	−1078.48	−1.33310	−0.666548	+	0.745462i	$0.732229\pi$
$810$	0	0
$811$	− 1159.86i	− 1.43016i	−0.699041	−	0.715082i	$-0.746390\pi$
$811$	− 1159.86i	− 1.43016i	0.699041	−	0.715082i	$-0.253610\pi$
$812$	0	0
$813$	−613.182	−0.754221
$814$	0	0
$815$	− 1214.12i	− 1.48972i
$816$	0	0
$817$	1377.23i	1.68571i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−837.038	−1.01953	−0.509767	−	0.860312i	$-0.670269\pi$
$821$	−837.038	−1.01953	−0.509767	+	0.860312i	$0.670269\pi$
$822$	0	0
$823$	−544.531	−0.661642	−0.330821	−	0.943693i	$-0.607326\pi$
$823$	−544.531	−0.661642	−0.330821	+	0.943693i	$0.607326\pi$
$824$	0	0
$825$	− 449.902i	− 0.545336i
$826$	0	0
$827$	−888.956	−1.07492	−0.537458	−	0.843290i	$-0.680615\pi$
$827$	−888.956	−1.07492	−0.537458	+	0.843290i	$0.680615\pi$
$828$	0	0
$829$	− 896.491i	− 1.08141i	−0.841211	−	0.540707i	$-0.818157\pi$
$829$	− 896.491i	− 1.08141i	0.841211	−	0.540707i	$-0.181843\pi$
$830$	0	0
$831$	− 221.297i	− 0.266301i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−325.528	−0.389853
$836$	0	0
$837$	−51.8152	−0.0619059
$838$	0	0
$839$	− 183.658i	− 0.218901i	−0.993992	−	0.109450i	$-0.965091\pi$
$839$	− 183.658i	− 0.218901i	0.993992	−	0.109450i	$-0.0349091\pi$
$840$	0	0
$841$	−625.002	−0.743165
$842$	0	0
$843$	− 41.4312i	− 0.0491473i
$844$	0	0
$845$	719.152i	0.851068i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	189.223	0.222877
$850$	0	0
$851$	287.646	0.338010
$852$	0	0
$853$	− 1013.43i	− 1.18808i	−0.804435	−	0.594041i	$-0.797532\pi$
$853$	− 1013.43i	− 1.18808i	0.804435	−	0.594041i	$-0.202468\pi$
$854$	0	0
$855$	−743.746	−0.869878
$856$	0	0
$857$	− 416.997i	− 0.486577i	−0.969954	−	0.243289i	$-0.921774\pi$
$857$	− 416.997i	− 0.486577i	0.969954	−	0.243289i	$-0.0782262\pi$
$858$	0	0
$859$	508.903i	0.592436i	0.955120	+	0.296218i	$0.0957255\pi$
$859$	508.903i	0.592436i	−0.955120	+	0.296218i	$0.904275\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−295.153	−0.342008	−0.171004	−	0.985270i	$-0.554701\pi$
$863$	−295.153	−0.342008	−0.171004	+	0.985270i	$0.554701\pi$
$864$	0	0
$865$	−2335.49	−2.69998
$866$	0	0
$867$	352.202i	0.406230i
$868$	0	0
$869$	−513.765	−0.591214
$870$	0	0
$871$	− 163.779i	− 0.188036i
$872$	0	0
$873$	− 388.256i	− 0.444738i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	308.608	0.351890	0.175945	−	0.984400i	$-0.443702\pi$
$877$	308.608	0.351890	0.175945	+	0.984400i	$0.443702\pi$
$878$	0	0
$879$	−404.158	−0.459792
$880$	0	0
$881$	1262.47i	1.43300i	0.697589	+	0.716499i	$0.254256\pi$
$881$	1262.47i	1.43300i	−0.697589	+	0.716499i	$0.745744\pi$
$882$	0	0
$883$	−441.944	−0.500502	−0.250251	−	0.968181i	$-0.580513\pi$
$883$	−441.944	−0.500502	−0.250251	+	0.968181i	$0.580513\pi$
$884$	0	0
$885$	− 1530.28i	− 1.72913i
$886$	0	0
$887$	− 227.001i	− 0.255920i	−0.991779	−	0.127960i	$-0.959157\pi$
$887$	− 227.001i	− 0.255920i	0.991779	−	0.127960i	$-0.0408429\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	41.7540	0.0468620
$892$	0	0
$893$	455.398	0.509964
$894$	0	0
$895$	− 269.785i	− 0.301435i
$896$	0	0
$897$	154.582	0.172332
$898$	0	0
$899$	146.555i	0.163020i
$900$	0	0
$901$	79.2488i	0.0879565i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	556.356	0.614758
$906$	0	0
$907$	1312.78	1.44739	0.723695	−	0.690120i	$-0.242442\pi$
$907$	1312.78	1.44739	0.723695	+	0.690120i	$0.242442\pi$
$908$	0	0
$909$	− 112.248i	− 0.123485i
$910$	0	0
$911$	1305.51	1.43305	0.716527	−	0.697560i	$-0.245731\pi$
$911$	1305.51	1.43305	0.716527	+	0.697560i	$0.245731\pi$
$912$	0	0
$913$	632.090i	0.692322i
$914$	0	0
$915$	539.963i	0.590123i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1395.94	−1.51898	−0.759489	−	0.650521i	$-0.774551\pi$
$919$	−1395.94	−1.51898	−0.759489	+	0.650521i	$0.774551\pi$
$920$	0	0
$921$	615.599	0.668403
$922$	0	0
$923$	− 1473.08i	− 1.59596i
$924$	0	0
$925$	2846.99	3.07782
$926$	0	0
$927$	− 454.878i	− 0.490699i
$928$	0	0
$929$	− 849.251i	− 0.914156i	−0.889427	−	0.457078i	$-0.848896\pi$
$929$	− 849.251i	− 0.914156i	0.889427	−	0.457078i	$-0.151104\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	460.817	0.493909
$934$	0	0
$935$	−386.410	−0.413272
$936$	0	0
$937$	522.446i	0.557573i	0.960353	+	0.278787i	$0.0899322\pi$
$937$	522.446i	0.557573i	−0.960353	+	0.278787i	$0.910068\pi$
$938$	0	0
$939$	443.827	0.472659
$940$	0	0
$941$	− 1287.19i	− 1.36790i	−0.729530	−	0.683949i	$-0.760261\pi$
$941$	− 1287.19i	− 1.36790i	0.729530	−	0.683949i	$-0.239739\pi$
$942$	0	0
$943$	397.127i	0.421131i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−253.671	−0.267868	−0.133934	−	0.990990i	$-0.542761\pi$
$947$	−253.671	−0.267868	−0.133934	+	0.990990i	$0.542761\pi$
$948$	0	0
$949$	−1965.06	−2.07067
$950$	0	0
$951$	− 619.032i	− 0.650927i
$952$	0	0
$953$	338.890	0.355603	0.177801	−	0.984066i	$-0.443102\pi$
$953$	338.890	0.355603	0.177801	+	0.984066i	$0.443102\pi$
$954$	0	0
$955$	− 786.289i	− 0.823339i
$956$	0	0
$957$	− 118.098i	− 0.123404i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	861.562	0.896527
$962$	0	0
$963$	206.907	0.214857
$964$	0	0
$965$	− 75.4450i	− 0.0781814i
$966$	0	0
$967$	−283.286	−0.292954	−0.146477	−	0.989214i	$-0.546793\pi$
$967$	−283.286	−0.292954	−0.146477	+	0.989214i	$0.546793\pi$
$968$	0	0
$969$	441.602i	0.455729i
$970$	0	0
$971$	− 1540.99i	− 1.58701i	−0.608563	−	0.793506i	$-0.708254\pi$
$971$	− 1540.99i	− 1.58701i	0.608563	−	0.793506i	$-0.291746\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1529.98	1.56921
$976$	0	0
$977$	1081.53	1.10699	0.553494	−	0.832853i	$-0.313294\pi$
$977$	1081.53	1.10699	0.553494	+	0.832853i	$0.313294\pi$
$978$	0	0
$979$	743.765i	0.759719i
$980$	0	0
$981$	174.342	0.177719
$982$	0	0
$983$	− 40.3811i	− 0.0410794i	−0.999789	−	0.0205397i	$-0.993462\pi$
$983$	− 40.3811i	− 0.0410794i	0.999789	−	0.0205397i	$-0.00653845\pi$
$984$	0	0
$985$	2284.67i	2.31946i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−282.807	−0.285952
$990$	0	0
$991$	658.100	0.664076	0.332038	−	0.943266i	$-0.392264\pi$
$991$	658.100	0.664076	0.332038	+	0.943266i	$0.392264\pi$
$992$	0	0
$993$	820.298i	0.826081i
$994$	0	0
$995$	1866.52	1.87589
$996$	0	0
$997$	952.015i	0.954880i	0.878664	+	0.477440i	$0.158435\pi$
$997$	952.015i	0.954880i	−0.878664	+	0.477440i	$0.841565\pi$
$998$	0	0
$999$	264.220i	0.264484i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.3.f.k.97.5		8
4.3	odd	2		1176.3.f.a.97.1		8
7.4	even	3		336.3.bh.h.145.4		8
7.5	odd	6		336.3.bh.h.241.4		8
7.6	odd	2	inner	2352.3.f.k.97.4		8
12.11	even	2		3528.3.f.f.2449.8		8
21.5	even	6		1008.3.cg.n.577.1		8
21.11	odd	6		1008.3.cg.n.145.1		8
28.3	even	6		1176.3.z.d.313.1		8
28.11	odd	6		168.3.z.a.145.4	yes	8
28.19	even	6		168.3.z.a.73.4	✓	8
28.23	odd	6		1176.3.z.d.913.1		8
28.27	even	2		1176.3.f.a.97.8		8
84.11	even	6		504.3.by.a.145.1		8
84.47	odd	6		504.3.by.a.73.1		8
84.83	odd	2		3528.3.f.f.2449.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.3.z.a.73.4	✓	8	28.19	even	6
168.3.z.a.145.4	yes	8	28.11	odd	6
336.3.bh.h.145.4		8	7.4	even	3
336.3.bh.h.241.4		8	7.5	odd	6
504.3.by.a.73.1		8	84.47	odd	6
504.3.by.a.145.1		8	84.11	even	6
1008.3.cg.n.145.1		8	21.11	odd	6
1008.3.cg.n.577.1		8	21.5	even	6
1176.3.f.a.97.1		8	4.3	odd	2
1176.3.f.a.97.8		8	28.27	even	2
1176.3.z.d.313.1		8	28.3	even	6
1176.3.z.d.913.1		8	28.23	odd	6
2352.3.f.k.97.4		8	7.6	odd	2	inner
2352.3.f.k.97.5		8	1.1	even	1	trivial
3528.3.f.f.2449.1		8	84.83	odd	2
3528.3.f.f.2449.8		8	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -8.99938i q^{5} -3.00000 q^{9} +4.63934 q^{11} +15.7769i q^{13} +15.5874 q^{15} -9.25507i q^{17} +27.5480i q^{19} -5.65685 q^{23} -55.9889 q^{25} -5.19615i q^{27} -14.6969 q^{29} -9.97185i q^{31} +8.03556i q^{33} -50.8491 q^{37} -27.3264 q^{39} -70.2028i q^{41} +49.9937 q^{43} +26.9981i q^{45} -16.5310i q^{47} +16.0302 q^{51} -8.56275 q^{53} -41.7512i q^{55} -47.7146 q^{57} -98.1740i q^{59} +34.6410i q^{61} +141.983 q^{65} -10.3809 q^{67} -9.79796i q^{69} -93.3690 q^{71} +124.553i q^{73} -96.9756i q^{75} -110.741 q^{79} +9.00000 q^{81} +136.246i q^{83} -83.2899 q^{85} -25.4557i q^{87} +160.317i q^{89} +17.2717 q^{93} +247.915 q^{95} +129.419i q^{97} -13.9180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9} + 28 q^{11} + 12 q^{15} + 12 q^{25} - 4 q^{29} - 100 q^{37} - 12 q^{39} + 20 q^{43} - 24 q^{51} + 100 q^{53} - 156 q^{57} + 296 q^{65} + 68 q^{67} - 424 q^{71} - 80 q^{79} + 72 q^{81} - 232 q^{85}+ \cdots - 84 q^{99}+O(q^{100}) \) 8 * q - 24 * q^9 + 28 * q^11 + 12 * q^15 + 12 * q^25 - 4 * q^29 - 100 * q^37 - 12 * q^39 + 20 * q^43 - 24 * q^51 + 100 * q^53 - 156 * q^57 + 296 * q^65 + 68 * q^67 - 424 * q^71 - 80 * q^79 + 72 * q^81 - 232 * q^85 - 48 * q^93 + 288 * q^95 - 84 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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