LMFDB - Embedded newform 2352.3.f.e.97.2

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:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.2
Root			$0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2352.97
Dual form			2352.3.f.e.97.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +1.43488i q^{5} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} +1.43488i q^{5} -3.00000 q^{9} -6.00000 q^{11} +21.3280i q^{13} +2.48528 q^{15} -8.95743i q^{17} -7.22538i q^{19} +37.4558 q^{23} +22.9411 q^{25} +5.19615i q^{27} -33.9411 q^{29} -44.1418i q^{31} +10.3923i q^{33} -27.9706 q^{37} +36.9411 q^{39} +54.8313i q^{41} +1.48528 q^{43} -4.30463i q^{45} -43.0041i q^{47} -15.5147 q^{51} -85.4558 q^{53} -8.60927i q^{55} -12.5147 q^{57} +41.2211i q^{59} +1.18869i q^{61} -30.6030 q^{65} -4.39697 q^{67} -64.8754i q^{69} -137.397 q^{71} +78.9270i q^{73} -39.7352i q^{75} -98.3381 q^{79} +9.00000 q^{81} +110.401i q^{83} +12.8528 q^{85} +58.7878i q^{87} +20.7846i q^{89} -76.4558 q^{93} +10.3675 q^{95} +10.9867i q^{97} +18.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} - 24 q^{11} - 24 q^{15} + 48 q^{23} - 44 q^{25} - 44 q^{37} + 12 q^{39} - 28 q^{43} - 96 q^{51} - 240 q^{53} - 84 q^{57} - 360 q^{65} + 220 q^{67} - 312 q^{71} - 20 q^{79} + 36 q^{81} - 288 q^{85} - 204 q^{93} - 264 q^{95} + 72 q^{99}+O(q^{100}) $ 4 * q - 12 * q^9 - 24 * q^11 - 24 * q^15 + 48 * q^23 - 44 * q^25 - 44 * q^37 + 12 * q^39 - 28 * q^43 - 96 * q^51 - 240 * q^53 - 84 * q^57 - 360 * q^65 + 220 * q^67 - 312 * q^71 - 20 * q^79 + 36 * q^81 - 288 * q^85 - 204 * q^93 - 264 * q^95 + 72 * 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
$5$	1.43488i	0.286976i	0.989652	+	0.143488i	$0.0458318\pi$
$5$	1.43488i	0.286976i	−0.989652	+	0.143488i	$0.954168\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	−6.00000	−0.545455	−0.272727	−	0.962091i	$-0.587926\pi$
$11$	−6.00000	−0.545455	−0.272727	+	0.962091i	$0.587926\pi$
$12$	0	0
$13$	21.3280i	1.64061i	0.571924	+	0.820306i	$0.306197\pi$
$13$	21.3280i	1.64061i	−0.571924	+	0.820306i	$0.693803\pi$
$14$	0	0
$15$	2.48528	0.165685
$16$	0	0
$17$	− 8.95743i	− 0.526907i	−0.964672	−	0.263454i	$-0.915138\pi$
$17$	− 8.95743i	− 0.526907i	0.964672	−	0.263454i	$-0.0848616\pi$
$18$	0	0
$19$	− 7.22538i	− 0.380283i	−0.981757	−	0.190141i	$-0.939105\pi$
$19$	− 7.22538i	− 0.380283i	0.981757	−	0.190141i	$-0.0608947\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	37.4558	1.62851	0.814257	−	0.580504i	$-0.197144\pi$
$23$	37.4558	1.62851	0.814257	+	0.580504i	$0.197144\pi$
$24$	0	0
$25$	22.9411	0.917645
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	−33.9411	−1.17038	−0.585192	−	0.810895i	$-0.698981\pi$
$29$	−33.9411	−1.17038	−0.585192	+	0.810895i	$0.698981\pi$
$30$	0	0
$31$	− 44.1418i	− 1.42393i	−0.702215	−	0.711965i	$-0.747806\pi$
$31$	− 44.1418i	− 1.42393i	0.702215	−	0.711965i	$-0.252194\pi$
$32$	0	0
$33$	10.3923i	0.314918i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−27.9706	−0.755961	−0.377981	−	0.925814i	$-0.623381\pi$
$37$	−27.9706	−0.755961	−0.377981	+	0.925814i	$0.623381\pi$
$38$	0	0
$39$	36.9411	0.947208
$40$	0	0
$41$	54.8313i	1.33735i	0.743556	+	0.668674i	$0.233138\pi$
$41$	54.8313i	1.33735i	−0.743556	+	0.668674i	$0.766862\pi$
$42$	0	0
$43$	1.48528	0.0345414	0.0172707	−	0.999851i	$-0.494502\pi$
$43$	1.48528	0.0345414	0.0172707	+	0.999851i	$0.494502\pi$
$44$	0	0
$45$	− 4.30463i	− 0.0956585i
$46$	0	0
$47$	− 43.0041i	− 0.914981i	−0.889215	−	0.457490i	$-0.848748\pi$
$47$	− 43.0041i	− 0.914981i	0.889215	−	0.457490i	$-0.151252\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−15.5147	−0.304210
$52$	0	0
$53$	−85.4558	−1.61237	−0.806187	−	0.591661i	$-0.798473\pi$
$53$	−85.4558	−1.61237	−0.806187	+	0.591661i	$0.798473\pi$
$54$	0	0
$55$	− 8.60927i	− 0.156532i
$56$	0	0
$57$	−12.5147	−0.219556
$58$	0	0
$59$	41.2211i	0.698662i	0.936999	+	0.349331i	$0.113591\pi$
$59$	41.2211i	0.698662i	−0.936999	+	0.349331i	$0.886409\pi$
$60$	0	0
$61$	1.18869i	0.0194867i	0.999953	+	0.00974337i	$0.00310146\pi$
$61$	1.18869i	0.0194867i	−0.999953	+	0.00974337i	$0.996899\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−30.6030	−0.470816
$66$	0	0
$67$	−4.39697	−0.0656264	−0.0328132	−	0.999462i	$-0.510447\pi$
$67$	−4.39697	−0.0656264	−0.0328132	+	0.999462i	$0.510447\pi$
$68$	0	0
$69$	− 64.8754i	− 0.940224i
$70$	0	0
$71$	−137.397	−1.93517	−0.967584	−	0.252548i	$-0.918731\pi$
$71$	−137.397	−1.93517	−0.967584	+	0.252548i	$0.918731\pi$
$72$	0	0
$73$	78.9270i	1.08119i	0.841282	+	0.540596i	$0.181801\pi$
$73$	78.9270i	1.08119i	−0.841282	+	0.540596i	$0.818199\pi$
$74$	0	0
$75$	− 39.7352i	− 0.529803i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−98.3381	−1.24479	−0.622393	−	0.782705i	$-0.713839\pi$
$79$	−98.3381	−1.24479	−0.622393	+	0.782705i	$0.713839\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	110.401i	1.33013i	0.746784	+	0.665067i	$0.231597\pi$
$83$	110.401i	1.33013i	−0.746784	+	0.665067i	$0.768403\pi$
$84$	0	0
$85$	12.8528	0.151210
$86$	0	0
$87$	58.7878i	0.675721i
$88$	0	0
$89$	20.7846i	0.233535i	0.993159	+	0.116767i	$0.0372532\pi$
$89$	20.7846i	0.233535i	−0.993159	+	0.116767i	$0.962747\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−76.4558	−0.822106
$94$	0	0
$95$	10.3675	0.109132
$96$	0	0
$97$	10.9867i	0.113264i	0.998395	+	0.0566322i	$0.0180362\pi$
$97$	10.9867i	0.113264i	−0.998395	+	0.0566322i	$0.981964\pi$
$98$	0	0
$99$	18.0000	0.181818
$100$	0	0
$101$	− 107.183i	− 1.06122i	−0.847616	−	0.530610i	$-0.821963\pi$
$101$	− 107.183i	− 1.06122i	0.847616	−	0.530610i	$-0.178037\pi$
$102$	0	0
$103$	105.205i	1.02141i	0.859757	+	0.510704i	$0.170615\pi$
$103$	105.205i	1.02141i	−0.859757	+	0.510704i	$0.829385\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−118.544	−1.10789	−0.553945	−	0.832553i	$-0.686878\pi$
$107$	−118.544	−1.10789	−0.553945	+	0.832553i	$0.686878\pi$
$108$	0	0
$109$	111.059	1.01889	0.509444	−	0.860504i	$-0.329851\pi$
$109$	111.059	1.01889	0.509444	+	0.860504i	$0.329851\pi$
$110$	0	0
$111$	48.4464i	0.436454i
$112$	0	0
$113$	101.397	0.897318	0.448659	−	0.893703i	$-0.351902\pi$
$113$	101.397	0.897318	0.448659	+	0.893703i	$0.351902\pi$
$114$	0	0
$115$	53.7446i	0.467344i
$116$	0	0
$117$	− 63.9839i	− 0.546871i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−85.0000	−0.702479
$122$	0	0
$123$	94.9706	0.772118
$124$	0	0
$125$	68.7897i	0.550317i
$126$	0	0
$127$	−82.5736	−0.650186	−0.325093	−	0.945682i	$-0.605396\pi$
$127$	−82.5736	−0.650186	−0.325093	+	0.945682i	$0.605396\pi$
$128$	0	0
$129$	− 2.57258i	− 0.0199425i
$130$	0	0
$131$	− 60.5708i	− 0.462372i	−0.972910	−	0.231186i	$-0.925739\pi$
$131$	− 60.5708i	− 0.462372i	0.972910	−	0.231186i	$-0.0742607\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−7.45584	−0.0552285
$136$	0	0
$137$	67.0294	0.489266	0.244633	−	0.969616i	$-0.421333\pi$
$137$	67.0294	0.489266	0.244633	+	0.969616i	$0.421333\pi$
$138$	0	0
$139$	91.5525i	0.658651i	0.944216	+	0.329326i	$0.106821\pi$
$139$	91.5525i	0.658651i	−0.944216	+	0.329326i	$0.893179\pi$
$140$	0	0
$141$	−74.4853	−0.528264
$142$	0	0
$143$	− 127.968i	− 0.894880i
$144$	0	0
$145$	− 48.7014i	− 0.335872i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	81.0883	0.544217	0.272108	−	0.962267i	$-0.412279\pi$
$149$	81.0883	0.544217	0.272108	+	0.962267i	$0.412279\pi$
$150$	0	0
$151$	51.2061	0.339113	0.169556	−	0.985520i	$-0.445766\pi$
$151$	51.2061	0.339113	0.169556	+	0.985520i	$0.445766\pi$
$152$	0	0
$153$	26.8723i	0.175636i
$154$	0	0
$155$	63.3381	0.408633
$156$	0	0
$157$	187.061i	1.19147i	0.803179	+	0.595737i	$0.203140\pi$
$157$	187.061i	1.19147i	−0.803179	+	0.595737i	$0.796860\pi$
$158$	0	0
$159$	148.014i	0.930905i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−83.9411	−0.514976	−0.257488	−	0.966281i	$-0.582895\pi$
$163$	−83.9411	−0.514976	−0.257488	+	0.966281i	$0.582895\pi$
$164$	0	0
$165$	−14.9117	−0.0903739
$166$	0	0
$167$	− 127.620i	− 0.764190i	−0.924123	−	0.382095i	$-0.875203\pi$
$167$	− 127.620i	− 0.764190i	0.924123	−	0.382095i	$-0.124797\pi$
$168$	0	0
$169$	−285.882	−1.69161
$170$	0	0
$171$	21.6761i	0.126761i
$172$	0	0
$173$	142.971i	0.826420i	0.910636	+	0.413210i	$0.135592\pi$
$173$	142.971i	0.826420i	−0.910636	+	0.413210i	$0.864408\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	71.3970	0.403373
$178$	0	0
$179$	−169.279	−0.945694	−0.472847	−	0.881145i	$-0.656774\pi$
$179$	−169.279	−0.945694	−0.472847	+	0.881145i	$0.656774\pi$
$180$	0	0
$181$	209.969i	1.16005i	0.814600	+	0.580024i	$0.196957\pi$
$181$	209.969i	1.16005i	−0.814600	+	0.580024i	$0.803043\pi$
$182$	0	0
$183$	2.05887	0.0112507
$184$	0	0
$185$	− 40.1343i	− 0.216942i
$186$	0	0
$187$	53.7446i	0.287404i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−66.6030	−0.348707	−0.174353	−	0.984683i	$-0.555784\pi$
$191$	−66.6030	−0.348707	−0.174353	+	0.984683i	$0.555784\pi$
$192$	0	0
$193$	−9.79394	−0.0507458	−0.0253729	−	0.999678i	$-0.508077\pi$
$193$	−9.79394	−0.0507458	−0.0253729	+	0.999678i	$0.508077\pi$
$194$	0	0
$195$	53.0060i	0.271826i
$196$	0	0
$197$	−267.161	−1.35615	−0.678075	−	0.734993i	$-0.737185\pi$
$197$	−267.161	−1.35615	−0.678075	+	0.734993i	$0.737185\pi$
$198$	0	0
$199$	130.940i	0.657988i	0.944332	+	0.328994i	$0.106710\pi$
$199$	130.940i	0.657988i	−0.944332	+	0.328994i	$0.893290\pi$
$200$	0	0
$201$	7.61577i	0.0378894i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−78.6762	−0.383786
$206$	0	0
$207$	−112.368	−0.542838
$208$	0	0
$209$	43.3523i	0.207427i
$210$	0	0
$211$	23.0883	0.109423	0.0547116	−	0.998502i	$-0.482576\pi$
$211$	23.0883	0.109423	0.0547116	+	0.998502i	$0.482576\pi$
$212$	0	0
$213$	237.979i	1.11727i
$214$	0	0
$215$	2.13120i	0.00991255i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	136.706	0.624227
$220$	0	0
$221$	191.044	0.864451
$222$	0	0
$223$	228.631i	1.02525i	0.858613	+	0.512625i	$0.171327\pi$
$223$	228.631i	1.02525i	−0.858613	+	0.512625i	$0.828673\pi$
$224$	0	0
$225$	−68.8234	−0.305882
$226$	0	0
$227$	− 65.6140i	− 0.289048i	−0.989501	−	0.144524i	$-0.953835\pi$
$227$	− 65.6140i	− 0.289048i	0.989501	−	0.144524i	$-0.0461652\pi$
$228$	0	0
$229$	− 93.4798i	− 0.408209i	−0.978949	−	0.204104i	$-0.934572\pi$
$229$	− 93.4798i	− 0.408209i	0.978949	−	0.204104i	$-0.0654282\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−237.515	−1.01938	−0.509688	−	0.860359i	$-0.670239\pi$
$233$	−237.515	−1.01938	−0.509688	+	0.860359i	$0.670239\pi$
$234$	0	0
$235$	61.7056	0.262577
$236$	0	0
$237$	170.327i	0.718678i
$238$	0	0
$239$	−366.853	−1.53495	−0.767475	−	0.641079i	$-0.778487\pi$
$239$	−366.853	−1.53495	−0.767475	+	0.641079i	$0.778487\pi$
$240$	0	0
$241$	− 421.024i	− 1.74699i	−0.486836	−	0.873493i	$-0.661849\pi$
$241$	− 421.024i	− 1.74699i	0.486836	−	0.873493i	$-0.338151\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	154.103	0.623897
$248$	0	0
$249$	191.220	0.767953
$250$	0	0
$251$	− 146.621i	− 0.584148i	−0.956396	−	0.292074i	$-0.905655\pi$
$251$	− 146.621i	− 0.584148i	0.956396	−	0.292074i	$-0.0943454\pi$
$252$	0	0
$253$	−224.735	−0.888281
$254$	0	0
$255$	− 22.2617i	− 0.0873009i
$256$	0	0
$257$	− 25.0892i	− 0.0976235i	−0.998808	−	0.0488118i	$-0.984457\pi$
$257$	− 25.0892i	− 0.0976235i	0.998808	−	0.0488118i	$-0.0155434\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	101.823	0.390128
$262$	0	0
$263$	90.6762	0.344776	0.172388	−	0.985029i	$-0.444852\pi$
$263$	90.6762	0.344776	0.172388	+	0.985029i	$0.444852\pi$
$264$	0	0
$265$	− 122.619i	− 0.462712i
$266$	0	0
$267$	36.0000	0.134831
$268$	0	0
$269$	− 68.3993i	− 0.254272i	−0.991885	−	0.127136i	$-0.959421\pi$
$269$	− 68.3993i	− 0.254272i	0.991885	−	0.127136i	$-0.0405785\pi$
$270$	0	0
$271$	− 123.519i	− 0.455790i	−0.973686	−	0.227895i	$-0.926816\pi$
$271$	− 123.519i	− 0.455790i	0.973686	−	0.227895i	$-0.0731842\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−137.647	−0.500534
$276$	0	0
$277$	−272.882	−0.985134	−0.492567	−	0.870274i	$-0.663941\pi$
$277$	−272.882	−0.985134	−0.492567	+	0.870274i	$0.663941\pi$
$278$	0	0
$279$	132.425i	0.474643i
$280$	0	0
$281$	133.103	0.473675	0.236837	−	0.971549i	$-0.423889\pi$
$281$	133.103	0.473675	0.236837	+	0.971549i	$0.423889\pi$
$282$	0	0
$283$	128.757i	0.454973i	0.973781	+	0.227486i	$0.0730508\pi$
$283$	128.757i	0.454973i	−0.973781	+	0.227486i	$0.926949\pi$
$284$	0	0
$285$	− 17.9571i	− 0.0630073i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	208.765	0.722369
$290$	0	0
$291$	19.0294	0.0653933
$292$	0	0
$293$	308.984i	1.05455i	0.849694	+	0.527276i	$0.176787\pi$
$293$	308.984i	1.05455i	−0.849694	+	0.527276i	$0.823213\pi$
$294$	0	0
$295$	−59.1472	−0.200499
$296$	0	0
$297$	− 31.1769i	− 0.104973i
$298$	0	0
$299$	798.857i	2.67176i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−185.647	−0.612696
$304$	0	0
$305$	−1.70563	−0.00559222
$306$	0	0
$307$	606.090i	1.97423i	0.160003	+	0.987117i	$0.448850\pi$
$307$	606.090i	1.97423i	−0.160003	+	0.987117i	$0.551150\pi$
$308$	0	0
$309$	182.220	0.589710
$310$	0	0
$311$	203.278i	0.653626i	0.945089	+	0.326813i	$0.105975\pi$
$311$	203.278i	0.653626i	−0.945089	+	0.326813i	$0.894025\pi$
$312$	0	0
$313$	405.639i	1.29597i	0.761652	+	0.647986i	$0.224389\pi$
$313$	405.639i	1.29597i	−0.761652	+	0.647986i	$0.775611\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.0589	−0.0822047	−0.0411023	−	0.999155i	$-0.513087\pi$
$317$	−26.0589	−0.0822047	−0.0411023	+	0.999155i	$0.513087\pi$
$318$	0	0
$319$	203.647	0.638391
$320$	0	0
$321$	205.325i	0.639640i
$322$	0	0
$323$	−64.7208	−0.200374
$324$	0	0
$325$	489.288i	1.50550i
$326$	0	0
$327$	− 192.360i	− 0.588256i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	108.632	0.328195	0.164097	−	0.986444i	$-0.447529\pi$
$331$	108.632	0.328195	0.164097	+	0.986444i	$0.447529\pi$
$332$	0	0
$333$	83.9117	0.251987
$334$	0	0
$335$	− 6.30911i	− 0.0188332i
$336$	0	0
$337$	441.735	1.31079	0.655393	−	0.755288i	$-0.272503\pi$
$337$	441.735	1.31079	0.655393	+	0.755288i	$0.272503\pi$
$338$	0	0
$339$	− 175.625i	− 0.518067i
$340$	0	0
$341$	264.851i	0.776689i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	93.0883	0.269821
$346$	0	0
$347$	−34.1909	−0.0985329	−0.0492664	−	0.998786i	$-0.515688\pi$
$347$	−34.1909	−0.0985329	−0.0492664	+	0.998786i	$0.515688\pi$
$348$	0	0
$349$	− 221.787i	− 0.635493i	−0.948176	−	0.317746i	$-0.897074\pi$
$349$	− 221.787i	− 0.635493i	0.948176	−	0.317746i	$-0.102926\pi$
$350$	0	0
$351$	−110.823	−0.315736
$352$	0	0
$353$	447.386i	1.26738i	0.773586	+	0.633692i	$0.218461\pi$
$353$	447.386i	1.26738i	−0.773586	+	0.633692i	$0.781539\pi$
$354$	0	0
$355$	− 197.148i	− 0.555346i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−291.765	−0.812714	−0.406357	−	0.913714i	$-0.633201\pi$
$359$	−291.765	−0.812714	−0.406357	+	0.913714i	$0.633201\pi$
$360$	0	0
$361$	308.794	0.855385
$362$	0	0
$363$	147.224i	0.405577i
$364$	0	0
$365$	−113.251	−0.310276
$366$	0	0
$367$	419.351i	1.14265i	0.820725	+	0.571324i	$0.193570\pi$
$367$	419.351i	1.14265i	−0.820725	+	0.571324i	$0.806430\pi$
$368$	0	0
$369$	− 164.494i	− 0.445783i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	31.3818	0.0841336	0.0420668	−	0.999115i	$-0.486606\pi$
$373$	31.3818	0.0841336	0.0420668	+	0.999115i	$0.486606\pi$
$374$	0	0
$375$	119.147	0.317726
$376$	0	0
$377$	− 723.895i	− 1.92015i
$378$	0	0
$379$	−206.779	−0.545590	−0.272795	−	0.962072i	$-0.587948\pi$
$379$	−206.779	−0.545590	−0.272795	+	0.962072i	$0.587948\pi$
$380$	0	0
$381$	143.022i	0.375385i
$382$	0	0
$383$	− 498.567i	− 1.30174i	−0.759189	−	0.650871i	$-0.774404\pi$
$383$	− 498.567i	− 1.30174i	0.759189	−	0.650871i	$-0.225596\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.45584	−0.0115138
$388$	0	0
$389$	−648.426	−1.66691	−0.833453	−	0.552590i	$-0.813639\pi$
$389$	−648.426	−1.66691	−0.833453	+	0.552590i	$0.813639\pi$
$390$	0	0
$391$	− 335.508i	− 0.858077i
$392$	0	0
$393$	−104.912	−0.266951
$394$	0	0
$395$	− 141.103i	− 0.357223i
$396$	0	0
$397$	− 75.7514i	− 0.190809i	−0.995439	−	0.0954047i	$-0.969585\pi$
$397$	− 75.7514i	− 0.190809i	0.995439	−	0.0954047i	$-0.0304145\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−564.250	−1.40711	−0.703553	−	0.710642i	$-0.748404\pi$
$401$	−564.250	−1.40711	−0.703553	+	0.710642i	$0.748404\pi$
$402$	0	0
$403$	941.455	2.33612
$404$	0	0
$405$	12.9139i	0.0318862i
$406$	0	0
$407$	167.823	0.412342
$408$	0	0
$409$	− 357.448i	− 0.873955i	−0.899472	−	0.436978i	$-0.856049\pi$
$409$	− 357.448i	− 0.873955i	0.899472	−	0.436978i	$-0.143951\pi$
$410$	0	0
$411$	− 116.098i	− 0.282478i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−158.412	−0.381716
$416$	0	0
$417$	158.574	0.380272
$418$	0	0
$419$	− 502.175i	− 1.19851i	−0.800559	−	0.599254i	$-0.795464\pi$
$419$	− 502.175i	− 1.19851i	0.800559	−	0.599254i	$-0.204536\pi$
$420$	0	0
$421$	33.7939	0.0802706	0.0401353	−	0.999194i	$-0.487221\pi$
$421$	33.7939	0.0802706	0.0401353	+	0.999194i	$0.487221\pi$
$422$	0	0
$423$	129.012i	0.304994i
$424$	0	0
$425$	− 205.493i	− 0.483514i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−221.647	−0.516659
$430$	0	0
$431$	−503.720	−1.16872	−0.584362	−	0.811493i	$-0.698655\pi$
$431$	−503.720	−1.16872	−0.584362	+	0.811493i	$0.698655\pi$
$432$	0	0
$433$	837.548i	1.93429i	0.254224	+	0.967145i	$0.418180\pi$
$433$	837.548i	1.93429i	−0.254224	+	0.967145i	$0.581820\pi$
$434$	0	0
$435$	−84.3532	−0.193916
$436$	0	0
$437$	− 270.633i	− 0.619296i
$438$	0	0
$439$	− 190.016i	− 0.432838i	−0.976301	−	0.216419i	$-0.930562\pi$
$439$	− 190.016i	− 0.432838i	0.976301	−	0.216419i	$-0.0694377\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	169.456	0.382519	0.191259	−	0.981540i	$-0.438743\pi$
$443$	169.456	0.382519	0.191259	+	0.981540i	$0.438743\pi$
$444$	0	0
$445$	−29.8234	−0.0670188
$446$	0	0
$447$	− 140.449i	− 0.314204i
$448$	0	0
$449$	18.1035	0.0403195	0.0201598	−	0.999797i	$-0.493583\pi$
$449$	18.1035	0.0403195	0.0201598	+	0.999797i	$0.493583\pi$
$450$	0	0
$451$	− 328.988i	− 0.729463i
$452$	0	0
$453$	− 88.6915i	− 0.195787i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−328.823	−0.719526	−0.359763	−	0.933044i	$-0.617142\pi$
$457$	−328.823	−0.719526	−0.359763	+	0.933044i	$0.617142\pi$
$458$	0	0
$459$	46.5442	0.101403
$460$	0	0
$461$	− 794.331i	− 1.72306i	−0.507706	−	0.861530i	$-0.669506\pi$
$461$	− 794.331i	− 1.72306i	0.507706	−	0.861530i	$-0.330494\pi$
$462$	0	0
$463$	403.396	0.871266	0.435633	−	0.900124i	$-0.356525\pi$
$463$	403.396	0.871266	0.435633	+	0.900124i	$0.356525\pi$
$464$	0	0
$465$	− 109.705i	− 0.235924i
$466$	0	0
$467$	− 2.82752i	− 0.00605464i	−0.999995	−	0.00302732i	$-0.999036\pi$
$467$	− 2.82752i	− 0.00605464i	0.999995	−	0.00302732i	$-0.000963627\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	324.000	0.687898
$472$	0	0
$473$	−8.91169	−0.0188408
$474$	0	0
$475$	− 165.758i	− 0.348965i
$476$	0	0
$477$	256.368	0.537458
$478$	0	0
$479$	− 379.514i	− 0.792305i	−0.918185	−	0.396153i	$-0.870345\pi$
$479$	− 379.514i	− 0.792305i	0.918185	−	0.396153i	$-0.129655\pi$
$480$	0	0
$481$	− 596.555i	− 1.24024i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−15.7645	−0.0325041
$486$	0	0
$487$	−575.514	−1.18175	−0.590877	−	0.806762i	$-0.701218\pi$
$487$	−575.514	−1.18175	−0.590877	+	0.806762i	$0.701218\pi$
$488$	0	0
$489$	145.390i	0.297322i
$490$	0	0
$491$	−238.441	−0.485623	−0.242811	−	0.970074i	$-0.578070\pi$
$491$	−238.441	−0.485623	−0.242811	+	0.970074i	$0.578070\pi$
$492$	0	0
$493$	304.025i	0.616684i
$494$	0	0
$495$	25.8278i	0.0521774i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	286.574	0.574296	0.287148	−	0.957886i	$-0.407293\pi$
$499$	286.574	0.574296	0.287148	+	0.957886i	$0.407293\pi$
$500$	0	0
$501$	−221.044	−0.441205
$502$	0	0
$503$	− 25.4374i	− 0.0505714i	−0.999680	−	0.0252857i	$-0.991950\pi$
$503$	− 25.4374i	− 0.0505714i	0.999680	−	0.0252857i	$-0.00804954\pi$
$504$	0	0
$505$	153.795	0.304544
$506$	0	0
$507$	495.163i	0.976652i
$508$	0	0
$509$	805.852i	1.58321i	0.611035	+	0.791603i	$0.290753\pi$
$509$	805.852i	1.58321i	−0.611035	+	0.791603i	$0.709247\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	37.5442	0.0731855
$514$	0	0
$515$	−150.956	−0.293119
$516$	0	0
$517$	258.025i	0.499080i
$518$	0	0
$519$	247.632	0.477134
$520$	0	0
$521$	− 764.072i	− 1.46655i	−0.679933	−	0.733274i	$-0.737991\pi$
$521$	− 764.072i	− 1.46655i	0.679933	−	0.733274i	$-0.262009\pi$
$522$	0	0
$523$	− 176.780i	− 0.338011i	−0.985615	−	0.169006i	$-0.945944\pi$
$523$	− 176.780i	− 0.338011i	0.985615	−	0.169006i	$-0.0540556\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−395.397	−0.750279
$528$	0	0
$529$	873.940	1.65206
$530$	0	0
$531$	− 123.663i	− 0.232887i
$532$	0	0
$533$	−1169.44	−2.19407
$534$	0	0
$535$	− 170.096i	− 0.317937i
$536$	0	0
$537$	293.200i	0.545997i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	17.1766	0.0317498	0.0158749	−	0.999874i	$-0.494947\pi$
$541$	17.1766	0.0317498	0.0158749	+	0.999874i	$0.494947\pi$
$542$	0	0
$543$	363.676	0.669754
$544$	0	0
$545$	159.356i	0.292396i
$546$	0	0
$547$	−212.676	−0.388805	−0.194402	−	0.980922i	$-0.562277\pi$
$547$	−212.676	−0.388805	−0.194402	+	0.980922i	$0.562277\pi$
$548$	0	0
$549$	− 3.56608i	− 0.00649558i
$550$	0	0
$551$	245.237i	0.445077i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−69.5147	−0.125252
$556$	0	0
$557$	−881.647	−1.58285	−0.791424	−	0.611267i	$-0.790660\pi$
$557$	−881.647	−1.58285	−0.791424	+	0.611267i	$0.790660\pi$
$558$	0	0
$559$	31.6780i	0.0566691i
$560$	0	0
$561$	93.0883	0.165933
$562$	0	0
$563$	− 767.068i	− 1.36247i	−0.732067	−	0.681233i	$-0.761444\pi$
$563$	− 767.068i	− 1.36247i	0.732067	−	0.681233i	$-0.238556\pi$
$564$	0	0
$565$	145.492i	0.257508i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−29.2935	−0.0514824	−0.0257412	−	0.999669i	$-0.508195\pi$
$569$	−29.2935	−0.0514824	−0.0257412	+	0.999669i	$0.508195\pi$
$570$	0	0
$571$	−965.043	−1.69009	−0.845046	−	0.534693i	$-0.820427\pi$
$571$	−965.043	−1.69009	−0.845046	+	0.534693i	$0.820427\pi$
$572$	0	0
$573$	115.360i	0.201326i
$574$	0	0
$575$	859.279	1.49440
$576$	0	0
$577$	263.136i	0.456042i	0.973656	+	0.228021i	$0.0732255\pi$
$577$	263.136i	0.456042i	−0.973656	+	0.228021i	$0.926775\pi$
$578$	0	0
$579$	16.9636i	0.0292981i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	512.735	0.879477
$584$	0	0
$585$	91.8091	0.156939
$586$	0	0
$587$	436.477i	0.743572i	0.928318	+	0.371786i	$0.121254\pi$
$587$	436.477i	0.743572i	−0.928318	+	0.371786i	$0.878746\pi$
$588$	0	0
$589$	−318.941	−0.541496
$590$	0	0
$591$	462.737i	0.782973i
$592$	0	0
$593$	− 696.981i	− 1.17535i	−0.809098	−	0.587673i	$-0.800044\pi$
$593$	− 696.981i	− 1.17535i	0.809098	−	0.587673i	$-0.199956\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	226.794	0.379889
$598$	0	0
$599$	−398.412	−0.665129	−0.332564	−	0.943081i	$-0.607914\pi$
$599$	−398.412	−0.665129	−0.332564	+	0.943081i	$0.607914\pi$
$600$	0	0
$601$	− 36.1691i	− 0.0601816i	−0.999547	−	0.0300908i	$-0.990420\pi$
$601$	− 36.1691i	− 0.0601816i	0.999547	−	0.0300908i	$-0.00957964\pi$
$602$	0	0
$603$	13.1909	0.0218755
$604$	0	0
$605$	− 121.965i	− 0.201594i
$606$	0	0
$607$	31.5761i	0.0520199i	0.999662	+	0.0260099i	$0.00828015\pi$
$607$	31.5761i	0.0520199i	−0.999662	+	0.0260099i	$0.991720\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	917.190	1.50113
$612$	0	0
$613$	−409.265	−0.667643	−0.333821	−	0.942636i	$-0.608338\pi$
$613$	−409.265	−0.667643	−0.333821	+	0.942636i	$0.608338\pi$
$614$	0	0
$615$	136.271i	0.221579i
$616$	0	0
$617$	1227.38	1.98927	0.994636	−	0.103436i	$-0.0329837\pi$
$617$	1227.38	1.98927	0.994636	+	0.103436i	$0.0329837\pi$
$618$	0	0
$619$	− 475.762i	− 0.768598i	−0.923209	−	0.384299i	$-0.874443\pi$
$619$	− 475.762i	− 0.768598i	0.923209	−	0.384299i	$-0.125557\pi$
$620$	0	0
$621$	194.626i	0.313408i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	474.823	0.759717
$626$	0	0
$627$	75.0883	0.119758
$628$	0	0
$629$	250.544i	0.398322i
$630$	0	0
$631$	54.9420	0.0870713	0.0435357	−	0.999052i	$-0.486138\pi$
$631$	54.9420	0.0870713	0.0435357	+	0.999052i	$0.486138\pi$
$632$	0	0
$633$	− 39.9901i	− 0.0631756i
$634$	0	0
$635$	− 118.483i	− 0.186587i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	412.191	0.645056
$640$	0	0
$641$	−229.103	−0.357414	−0.178707	−	0.983902i	$-0.557191\pi$
$641$	−229.103	−0.357414	−0.178707	+	0.983902i	$0.557191\pi$
$642$	0	0
$643$	− 854.640i	− 1.32914i	−0.747224	−	0.664572i	$-0.768614\pi$
$643$	− 854.640i	− 1.32914i	0.747224	−	0.664572i	$-0.231386\pi$
$644$	0	0
$645$	3.69134	0.00572301
$646$	0	0
$647$	− 1003.01i	− 1.55025i	−0.631809	−	0.775124i	$-0.717687\pi$
$647$	− 1003.01i	− 1.55025i	0.631809	−	0.775124i	$-0.282313\pi$
$648$	0	0
$649$	− 247.326i	− 0.381088i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1270.76	1.94604	0.973020	−	0.230722i	$-0.0741089\pi$
$653$	1270.76	1.94604	0.973020	+	0.230722i	$0.0741089\pi$
$654$	0	0
$655$	86.9117	0.132690
$656$	0	0
$657$	− 236.781i	− 0.360397i
$658$	0	0
$659$	783.308	1.18863	0.594315	−	0.804232i	$-0.297423\pi$
$659$	783.308	1.18863	0.594315	+	0.804232i	$0.297423\pi$
$660$	0	0
$661$	83.7838i	0.126753i	0.997990	+	0.0633765i	$0.0201869\pi$
$661$	83.7838i	0.126753i	−0.997990	+	0.0633765i	$0.979813\pi$
$662$	0	0
$663$	− 330.897i	− 0.499091i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1271.29	−1.90599
$668$	0	0
$669$	396.000	0.591928
$670$	0	0
$671$	− 7.13215i	− 0.0106291i
$672$	0	0
$673$	415.676	0.617647	0.308823	−	0.951119i	$-0.400065\pi$
$673$	415.676	0.617647	0.308823	+	0.951119i	$0.400065\pi$
$674$	0	0
$675$	119.206i	0.176601i
$676$	0	0
$677$	− 791.292i	− 1.16882i	−0.811458	−	0.584411i	$-0.801326\pi$
$677$	− 791.292i	− 1.16882i	0.811458	−	0.584411i	$-0.198674\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−113.647	−0.166882
$682$	0	0
$683$	328.161	0.480469	0.240235	−	0.970715i	$-0.422776\pi$
$683$	328.161	0.480469	0.240235	+	0.970715i	$0.422776\pi$
$684$	0	0
$685$	96.1791i	0.140407i
$686$	0	0
$687$	−161.912	−0.235679
$688$	0	0
$689$	− 1822.60i	− 2.64528i
$690$	0	0
$691$	1010.57i	1.46248i	0.682120	+	0.731240i	$0.261058\pi$
$691$	1010.57i	1.46248i	−0.682120	+	0.731240i	$0.738942\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−131.367	−0.189017
$696$	0	0
$697$	491.147	0.704659
$698$	0	0
$699$	411.388i	0.588537i
$700$	0	0
$701$	−0.103464	−0.000147594 0	−7.37972e−5	−	1.00000i	$-0.500023\pi$
$701$	−0.103464	−0.000147594 0	−7.37972e−5		1.00000i	$0.500023\pi$
$702$	0	0
$703$	202.098i	0.287479i
$704$	0	0
$705$	− 106.877i	− 0.151599i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1205.18	1.69982	0.849912	−	0.526924i	$-0.176655\pi$
$709$	1205.18	1.69982	0.849912	+	0.526924i	$0.176655\pi$
$710$	0	0
$711$	295.014	0.414929
$712$	0	0
$713$	− 1653.37i	− 2.31889i
$714$	0	0
$715$	183.618	0.256809
$716$	0	0
$717$	635.408i	0.886203i
$718$	0	0
$719$	982.564i	1.36657i	0.730152	+	0.683285i	$0.239449\pi$
$719$	982.564i	1.36657i	−0.730152	+	0.683285i	$0.760551\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−729.235	−1.00862
$724$	0	0
$725$	−778.648	−1.07400
$726$	0	0
$727$	− 630.440i	− 0.867181i	−0.901110	−	0.433590i	$-0.857247\pi$
$727$	− 630.440i	− 0.867181i	0.901110	−	0.433590i	$-0.142753\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 13.3043i	− 0.0182001i
$732$	0	0
$733$	− 298.474i	− 0.407195i	−0.979055	−	0.203597i	$-0.934737\pi$
$733$	− 298.474i	− 0.407195i	0.979055	−	0.203597i	$-0.0652633\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	26.3818	0.0357962
$738$	0	0
$739$	−345.368	−0.467344	−0.233672	−	0.972315i	$-0.575074\pi$
$739$	−345.368	−0.467344	−0.233672	+	0.972315i	$0.575074\pi$
$740$	0	0
$741$	− 266.914i	− 0.360207i
$742$	0	0
$743$	−683.616	−0.920076	−0.460038	−	0.887899i	$-0.652164\pi$
$743$	−683.616	−0.920076	−0.460038	+	0.887899i	$0.652164\pi$
$744$	0	0
$745$	116.352i	0.156177i
$746$	0	0
$747$	− 331.203i	− 0.443378i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	578.338	0.770091	0.385045	−	0.922898i	$-0.374186\pi$
$751$	578.338	0.770091	0.385045	+	0.922898i	$0.374186\pi$
$752$	0	0
$753$	−253.955	−0.337258
$754$	0	0
$755$	73.4744i	0.0973171i
$756$	0	0
$757$	1204.82	1.59158	0.795788	−	0.605576i	$-0.207057\pi$
$757$	1204.82	1.59158	0.795788	+	0.605576i	$0.207057\pi$
$758$	0	0
$759$	389.253i	0.512849i
$760$	0	0
$761$	234.022i	0.307519i	0.988108	+	0.153760i	$0.0491381\pi$
$761$	234.022i	0.307519i	−0.988108	+	0.153760i	$0.950862\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−38.5584	−0.0504032
$766$	0	0
$767$	−879.161	−1.14623
$768$	0	0
$769$	− 1290.16i	− 1.67771i	−0.544358	−	0.838853i	$-0.683226\pi$
$769$	− 1290.16i	− 1.67771i	0.544358	−	0.838853i	$-0.316774\pi$
$770$	0	0
$771$	−43.4558	−0.0563630
$772$	0	0
$773$	399.117i	0.516323i	0.966102	+	0.258161i	$0.0831166\pi$
$773$	399.117i	0.516323i	−0.966102	+	0.258161i	$0.916883\pi$
$774$	0	0
$775$	− 1012.66i	− 1.30666i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	396.177	0.508571
$780$	0	0
$781$	824.382	1.05555
$782$	0	0
$783$	− 176.363i	− 0.225240i
$784$	0	0
$785$	−268.410	−0.341924
$786$	0	0
$787$	1556.72i	1.97805i	0.147763	+	0.989023i	$0.452793\pi$
$787$	1556.72i	1.97805i	−0.147763	+	0.989023i	$0.547207\pi$
$788$	0	0
$789$	− 157.056i	− 0.199057i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−25.3524	−0.0319702
$794$	0	0
$795$	−212.382	−0.267147
$796$	0	0
$797$	− 600.232i	− 0.753114i	−0.926393	−	0.376557i	$-0.877108\pi$
$797$	− 600.232i	− 0.753114i	0.926393	−	0.376557i	$-0.122892\pi$
$798$	0	0
$799$	−385.206	−0.482110
$800$	0	0
$801$	− 62.3538i	− 0.0778450i
$802$	0	0
$803$	− 473.562i	− 0.589741i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−118.471	−0.146804
$808$	0	0
$809$	−229.279	−0.283411	−0.141705	−	0.989909i	$-0.545259\pi$
$809$	−229.279	−0.283411	−0.141705	+	0.989909i	$0.545259\pi$
$810$	0	0
$811$	529.955i	0.653459i	0.945118	+	0.326729i	$0.105947\pi$
$811$	529.955i	0.653459i	−0.945118	+	0.326729i	$0.894053\pi$
$812$	0	0
$813$	−213.941	−0.263150
$814$	0	0
$815$	− 120.445i	− 0.147786i
$816$	0	0
$817$	− 10.7317i	− 0.0131355i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−303.338	−0.369474	−0.184737	−	0.982788i	$-0.559143\pi$
$821$	−303.338	−0.369474	−0.184737	+	0.982788i	$0.559143\pi$
$822$	0	0
$823$	1129.91	1.37292	0.686459	−	0.727169i	$-0.259164\pi$
$823$	1129.91	1.37292	0.686459	+	0.727169i	$0.259164\pi$
$824$	0	0
$825$	238.411i	0.288983i
$826$	0	0
$827$	161.604	0.195410	0.0977049	−	0.995215i	$-0.468850\pi$
$827$	161.604	0.195410	0.0977049	+	0.995215i	$0.468850\pi$
$828$	0	0
$829$	1530.35i	1.84602i	0.384776	+	0.923010i	$0.374279\pi$
$829$	1530.35i	1.84602i	−0.384776	+	0.923010i	$0.625721\pi$
$830$	0	0
$831$	472.646i	0.568768i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	183.119	0.219304
$836$	0	0
$837$	229.368	0.274035
$838$	0	0
$839$	− 218.629i	− 0.260583i	−0.991476	−	0.130291i	$-0.958409\pi$
$839$	− 218.629i	− 0.260583i	0.991476	−	0.130291i	$-0.0415913\pi$
$840$	0	0
$841$	311.000	0.369798
$842$	0	0
$843$	− 230.540i	− 0.273476i
$844$	0	0
$845$	− 410.206i	− 0.485451i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	223.014	0.262679
$850$	0	0
$851$	−1047.66	−1.23109
$852$	0	0
$853$	− 762.730i	− 0.894174i	−0.894491	−	0.447087i	$-0.852461\pi$
$853$	− 762.730i	− 0.894174i	0.894491	−	0.447087i	$-0.147539\pi$
$854$	0	0
$855$	−31.1026	−0.0363773
$856$	0	0
$857$	− 918.004i	− 1.07118i	−0.844477	−	0.535592i	$-0.820089\pi$
$857$	− 918.004i	− 1.07118i	0.844477	−	0.535592i	$-0.179911\pi$
$858$	0	0
$859$	− 879.150i	− 1.02346i	−0.859147	−	0.511729i	$-0.829005\pi$
$859$	− 879.150i	− 1.02346i	0.859147	−	0.511729i	$-0.170995\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−350.589	−0.406244	−0.203122	−	0.979153i	$-0.565109\pi$
$863$	−350.589	−0.406244	−0.203122	+	0.979153i	$0.565109\pi$
$864$	0	0
$865$	−205.145	−0.237162
$866$	0	0
$867$	− 361.591i	− 0.417060i
$868$	0	0
$869$	590.029	0.678974
$870$	0	0
$871$	− 93.7784i	− 0.107668i
$872$	0	0
$873$	− 32.9600i	− 0.0377548i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.55931	0.00405850	0.00202925	−	0.999998i	$-0.499354\pi$
$877$	3.55931	0.00405850	0.00202925	+	0.999998i	$0.499354\pi$
$878$	0	0
$879$	535.176	0.608846
$880$	0	0
$881$	488.565i	0.554557i	0.960790	+	0.277279i	$0.0894325\pi$
$881$	488.565i	0.554557i	−0.960790	+	0.277279i	$0.910567\pi$
$882$	0	0
$883$	1162.16	1.31615	0.658075	−	0.752953i	$-0.271371\pi$
$883$	1162.16	1.31615	0.658075	+	0.752953i	$0.271371\pi$
$884$	0	0
$885$	102.446i	0.115758i
$886$	0	0
$887$	86.9260i	0.0980000i	0.998799	+	0.0490000i	$0.0156034\pi$
$887$	86.9260i	0.0980000i	−0.998799	+	0.0490000i	$0.984397\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−54.0000	−0.0606061
$892$	0	0
$893$	−310.721	−0.347952
$894$	0	0
$895$	− 242.895i	− 0.271391i
$896$	0	0
$897$	1383.66	1.54254
$898$	0	0
$899$	1498.22i	1.66654i
$900$	0	0
$901$	765.464i	0.849572i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−301.279	−0.332905
$906$	0	0
$907$	460.897	0.508155	0.254077	−	0.967184i	$-0.418228\pi$
$907$	460.897	0.508155	0.254077	+	0.967184i	$0.418228\pi$
$908$	0	0
$909$	321.550i	0.353740i
$910$	0	0
$911$	1184.28	1.29998	0.649988	−	0.759944i	$-0.274774\pi$
$911$	1184.28	1.29998	0.649988	+	0.759944i	$0.274774\pi$
$912$	0	0
$913$	− 662.407i	− 0.725528i
$914$	0	0
$915$	2.95423i	0.00322867i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	541.839	0.589596	0.294798	−	0.955560i	$-0.404748\pi$
$919$	541.839	0.589596	0.294798	+	0.955560i	$0.404748\pi$
$920$	0	0
$921$	1049.78	1.13982
$922$	0	0
$923$	− 2930.40i	− 3.17486i
$924$	0	0
$925$	−641.676	−0.693704
$926$	0	0
$927$	− 315.615i	− 0.340469i
$928$	0	0
$929$	900.216i	0.969016i	0.874787	+	0.484508i	$0.161001\pi$
$929$	900.216i	0.969016i	−0.874787	+	0.484508i	$0.838999\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	352.087	0.377371
$934$	0	0
$935$	−77.1169	−0.0824779
$936$	0	0
$937$	− 233.964i	− 0.249695i	−0.992176	−	0.124847i	$-0.960156\pi$
$937$	− 233.964i	− 0.249695i	0.992176	−	0.124847i	$-0.0398441\pi$
$938$	0	0
$939$	702.588	0.748230
$940$	0	0
$941$	1307.69i	1.38968i	0.719164	+	0.694840i	$0.244525\pi$
$941$	1307.69i	1.38968i	−0.719164	+	0.694840i	$0.755475\pi$
$942$	0	0
$943$	2053.75i	2.17789i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1127.76	1.19088	0.595440	−	0.803400i	$-0.296978\pi$
$947$	1127.76	1.19088	0.595440	+	0.803400i	$0.296978\pi$
$948$	0	0
$949$	−1683.35	−1.77382
$950$	0	0
$951$	45.1353i	0.0474609i
$952$	0	0
$953$	−91.4255	−0.0959345	−0.0479672	−	0.998849i	$-0.515274\pi$
$953$	−91.4255	−0.0959345	−0.0479672	+	0.998849i	$0.515274\pi$
$954$	0	0
$955$	− 95.5672i	− 0.100070i
$956$	0	0
$957$	− 352.727i	− 0.368575i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−987.499	−1.02757
$962$	0	0
$963$	355.632	0.369296
$964$	0	0
$965$	− 14.0531i	− 0.0145628i
$966$	0	0
$967$	1098.19	1.13567	0.567834	−	0.823143i	$-0.307782\pi$
$967$	1098.19	1.13567	0.567834	+	0.823143i	$0.307782\pi$
$968$	0	0
$969$	112.100i	0.115686i
$970$	0	0
$971$	132.103i	0.136049i	0.997684	+	0.0680245i	$0.0216696\pi$
$971$	132.103i	0.136049i	−0.997684	+	0.0680245i	$0.978330\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	847.471	0.869201
$976$	0	0
$977$	448.234	0.458786	0.229393	−	0.973334i	$-0.426326\pi$
$977$	448.234	0.458786	0.229393	+	0.973334i	$0.426326\pi$
$978$	0	0
$979$	− 124.708i	− 0.127383i
$980$	0	0
$981$	−333.177	−0.339630
$982$	0	0
$983$	− 1646.76i	− 1.67524i	−0.546251	−	0.837621i	$-0.683946\pi$
$983$	− 1646.76i	− 1.67524i	0.546251	−	0.837621i	$-0.316054\pi$
$984$	0	0
$985$	− 383.344i	− 0.389182i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	55.6325	0.0562512
$990$	0	0
$991$	628.897	0.634608	0.317304	−	0.948324i	$-0.397222\pi$
$991$	628.897	0.634608	0.317304	+	0.948324i	$0.397222\pi$
$992$	0	0
$993$	− 188.157i	− 0.189483i
$994$	0	0
$995$	−187.882	−0.188826
$996$	0	0
$997$	− 1004.18i	− 1.00720i	−0.863936	−	0.503601i	$-0.832008\pi$
$997$	− 1004.18i	− 1.00720i	0.863936	−	0.503601i	$-0.167992\pi$
$998$	0	0
$999$	− 145.339i	− 0.145485i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.3.f.e.97.2		4
4.3	odd	2		294.3.c.a.97.2		4
7.4	even	3		336.3.bh.e.145.1		4
7.5	odd	6		336.3.bh.e.241.1		4
7.6	odd	2	inner	2352.3.f.e.97.3		4
12.11	even	2		882.3.c.b.685.3		4
21.5	even	6		1008.3.cg.h.577.2		4
21.11	odd	6		1008.3.cg.h.145.2		4
28.3	even	6		294.3.g.a.19.2		4
28.11	odd	6		42.3.g.a.19.2	✓	4
28.19	even	6		42.3.g.a.31.2	yes	4
28.23	odd	6		294.3.g.a.31.2		4
28.27	even	2		294.3.c.a.97.1		4
84.11	even	6		126.3.n.a.19.1		4
84.23	even	6		882.3.n.e.325.1		4
84.47	odd	6		126.3.n.a.73.1		4
84.59	odd	6		882.3.n.e.19.1		4
84.83	odd	2		882.3.c.b.685.4		4
140.19	even	6		1050.3.p.a.451.1		4
140.39	odd	6		1050.3.p.a.901.1		4
140.47	odd	12		1050.3.q.a.199.2		8
140.67	even	12		1050.3.q.a.649.3		8
140.103	odd	12		1050.3.q.a.199.3		8
140.123	even	12		1050.3.q.a.649.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.3.g.a.19.2	✓	4	28.11	odd	6
42.3.g.a.31.2	yes	4	28.19	even	6
126.3.n.a.19.1		4	84.11	even	6
126.3.n.a.73.1		4	84.47	odd	6
294.3.c.a.97.1		4	28.27	even	2
294.3.c.a.97.2		4	4.3	odd	2
294.3.g.a.19.2		4	28.3	even	6
294.3.g.a.31.2		4	28.23	odd	6
336.3.bh.e.145.1		4	7.4	even	3
336.3.bh.e.241.1		4	7.5	odd	6
882.3.c.b.685.3		4	12.11	even	2
882.3.c.b.685.4		4	84.83	odd	2
882.3.n.e.19.1		4	84.59	odd	6
882.3.n.e.325.1		4	84.23	even	6
1008.3.cg.h.145.2		4	21.11	odd	6
1008.3.cg.h.577.2		4	21.5	even	6
1050.3.p.a.451.1		4	140.19	even	6
1050.3.p.a.901.1		4	140.39	odd	6
1050.3.q.a.199.2		8	140.47	odd	12
1050.3.q.a.199.3		8	140.103	odd	12
1050.3.q.a.649.2		8	140.123	even	12
1050.3.q.a.649.3		8	140.67	even	12
2352.3.f.e.97.2		4	1.1	even	1	trivial
2352.3.f.e.97.3		4	7.6	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +1.43488i q^{5} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} +1.43488i q^{5} -3.00000 q^{9} -6.00000 q^{11} +21.3280i q^{13} +2.48528 q^{15} -8.95743i q^{17} -7.22538i q^{19} +37.4558 q^{23} +22.9411 q^{25} +5.19615i q^{27} -33.9411 q^{29} -44.1418i q^{31} +10.3923i q^{33} -27.9706 q^{37} +36.9411 q^{39} +54.8313i q^{41} +1.48528 q^{43} -4.30463i q^{45} -43.0041i q^{47} -15.5147 q^{51} -85.4558 q^{53} -8.60927i q^{55} -12.5147 q^{57} +41.2211i q^{59} +1.18869i q^{61} -30.6030 q^{65} -4.39697 q^{67} -64.8754i q^{69} -137.397 q^{71} +78.9270i q^{73} -39.7352i q^{75} -98.3381 q^{79} +9.00000 q^{81} +110.401i q^{83} +12.8528 q^{85} +58.7878i q^{87} +20.7846i q^{89} -76.4558 q^{93} +10.3675 q^{95} +10.9867i q^{97} +18.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} - 24 q^{11} - 24 q^{15} + 48 q^{23} - 44 q^{25} - 44 q^{37} + 12 q^{39} - 28 q^{43} - 96 q^{51} - 240 q^{53} - 84 q^{57} - 360 q^{65} + 220 q^{67} - 312 q^{71} - 20 q^{79} + 36 q^{81} - 288 q^{85} - 204 q^{93} - 264 q^{95} + 72 q^{99}+O(q^{100}) \) 4 * q - 12 * q^9 - 24 * q^11 - 24 * q^15 + 48 * q^23 - 44 * q^25 - 44 * q^37 + 12 * q^39 - 28 * q^43 - 96 * q^51 - 240 * q^53 - 84 * q^57 - 360 * q^65 + 220 * q^67 - 312 * q^71 - 20 * q^79 + 36 * q^81 - 288 * q^85 - 204 * q^93 - 264 * q^95 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more