LMFDB - Embedded newform 882.3.c.b.685.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,3,Mod(685,882)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("882.685");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	882.c (of order $2$, degree $1$, 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:	$24.0327593166$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			685.4
Root			$0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	882.685
Dual form			882.3.c.b.685.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} +1.43488i q^{5} +2.82843 q^{8} +O(q^{10})$	$q+1.41421 q^{2} +2.00000 q^{4} +1.43488i q^{5} +2.82843 q^{8} +2.02922i q^{10} -6.00000 q^{11} -21.3280i q^{13} +4.00000 q^{16} -8.95743i q^{17} -7.22538i q^{19} +2.86976i q^{20} -8.48528 q^{22} +37.4558 q^{23} +22.9411 q^{25} -30.1623i q^{26} +33.9411 q^{29} -44.1418i q^{31} +5.65685 q^{32} -12.6677i q^{34} -27.9706 q^{37} -10.2182i q^{38} +4.05845i q^{40} +54.8313i q^{41} -1.48528 q^{43} -12.0000 q^{44} +52.9706 q^{46} +43.0041i q^{47} +32.4437 q^{50} -42.6559i q^{52} +85.4558 q^{53} -8.60927i q^{55} +48.0000 q^{58} -41.2211i q^{59} -1.18869i q^{61} -62.4259i q^{62} +8.00000 q^{64} +30.6030 q^{65} +4.39697 q^{67} -17.9149i q^{68} -137.397 q^{71} -78.9270i q^{73} -39.5563 q^{74} -14.4508i q^{76} +98.3381 q^{79} +5.73951i q^{80} +77.5431i q^{82} -110.401i q^{83} +12.8528 q^{85} -2.10051 q^{86} -16.9706 q^{88} +20.7846i q^{89} +74.9117 q^{92} +60.8170i q^{94} +10.3675 q^{95} -10.9867i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4}+O(q^{10}) $ 4 * q + 8 * q^4	$ 4 q + 8 q^{4} - 24 q^{11} + 16 q^{16} + 48 q^{23} - 44 q^{25} - 44 q^{37} + 28 q^{43} - 48 q^{44} + 144 q^{46} + 192 q^{50} + 240 q^{53} + 192 q^{58} + 32 q^{64} + 360 q^{65} - 220 q^{67} - 312 q^{71} - 96 q^{74} + 20 q^{79} - 288 q^{85} - 48 q^{86} + 96 q^{92} - 264 q^{95}+O(q^{100}) $ 4 * q + 8 * q^4 - 24 * q^11 + 16 * q^16 + 48 * q^23 - 44 * q^25 - 44 * q^37 + 28 * q^43 - 48 * q^44 + 144 * q^46 + 192 * q^50 + 240 * q^53 + 192 * q^58 + 32 * q^64 + 360 * q^65 - 220 * q^67 - 312 * q^71 - 96 * q^74 + 20 * q^79 - 288 * q^85 - 48 * q^86 + 96 * q^92 - 264 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.41421	0.707107
$2$	1.41421	0.707107
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	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
$8$	2.82843	0.353553
$9$	0	0
$10$	2.02922i	0.202922i
$11$	−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	$-0.693803\pi$
$13$	− 21.3280i	− 1.64061i	0.571924	−	0.820306i	$-0.306197\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	0.250000
$17$	− 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	$-0.0608947\pi$
$20$	2.86976i	0.143488i
$21$	0	0
$22$	−8.48528	−0.385695
$23$	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
$26$	− 30.1623i	− 1.16009i
$27$	0	0
$28$	0	0
$29$	33.9411	1.17038	0.585192	−	0.810895i	$-0.301019\pi$
$29$	33.9411	1.17038	0.585192	+	0.810895i	$0.301019\pi$
$30$	0	0
$31$	− 44.1418i	− 1.42393i	−0.702215	−	0.711965i	$-0.747806\pi$
$31$	− 44.1418i	− 1.42393i	0.702215	−	0.711965i	$-0.252194\pi$
$32$	5.65685	0.176777
$33$	0	0
$34$	− 12.6677i	− 0.372580i
$35$	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	$0.623381\pi$
$38$	− 10.2182i	− 0.268901i
$39$	0	0
$40$	4.05845i	0.101461i
$41$	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	$-0.505498\pi$
$43$	−1.48528	−0.0345414	−0.0172707	+	0.999851i	$0.505498\pi$
$44$	−12.0000	−0.272727
$45$	0	0
$46$	52.9706	1.15153
$47$	43.0041i	0.914981i	0.889215	+	0.457490i	$0.151252\pi$
$47$	43.0041i	0.914981i	−0.889215	+	0.457490i	$0.848748\pi$
$48$	0	0
$49$	0	0
$50$	32.4437	0.648873
$51$	0	0
$52$	− 42.6559i	− 0.820306i
$53$	85.4558	1.61237	0.806187	−	0.591661i	$-0.201527\pi$
$53$	85.4558	1.61237	0.806187	+	0.591661i	$0.201527\pi$
$54$	0	0
$55$	− 8.60927i	− 0.156532i
$56$	0	0
$57$	0	0
$58$	48.0000	0.827586
$59$	− 41.2211i	− 0.698662i	−0.936999	−	0.349331i	$-0.886409\pi$
$59$	− 41.2211i	− 0.698662i	0.936999	−	0.349331i	$-0.113591\pi$
$60$	0	0
$61$	− 1.18869i	− 0.0194867i	−0.999953	−	0.00974337i	$-0.996899\pi$
$61$	− 1.18869i	− 0.0194867i	0.999953	−	0.00974337i	$-0.00310146\pi$
$62$	− 62.4259i	− 1.00687i
$63$	0	0
$64$	8.00000	0.125000
$65$	30.6030	0.470816
$66$	0	0
$67$	4.39697	0.0656264	0.0328132	−	0.999462i	$-0.489553\pi$
$67$	4.39697	0.0656264	0.0328132	+	0.999462i	$0.489553\pi$
$68$	− 17.9149i	− 0.263454i
$69$	0	0
$70$	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	$-0.818199\pi$
$73$	− 78.9270i	− 1.08119i	0.841282	−	0.540596i	$-0.181801\pi$
$74$	−39.5563	−0.534545
$75$	0	0
$76$	− 14.4508i	− 0.190141i
$77$	0	0
$78$	0	0
$79$	98.3381	1.24479	0.622393	−	0.782705i	$-0.286161\pi$
$79$	98.3381	1.24479	0.622393	+	0.782705i	$0.286161\pi$
$80$	5.73951i	0.0717439i
$81$	0	0
$82$	77.5431i	0.945648i
$83$	− 110.401i	− 1.33013i	−0.746784	−	0.665067i	$-0.768403\pi$
$83$	− 110.401i	− 1.33013i	0.746784	−	0.665067i	$-0.231597\pi$
$84$	0	0
$85$	12.8528	0.151210
$86$	−2.10051	−0.0244245
$87$	0	0
$88$	−16.9706	−0.192847
$89$	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
$92$	74.9117	0.814257
$93$	0	0
$94$	60.8170i	0.646989i
$95$	10.3675	0.109132
$96$	0	0
$97$	− 10.9867i	− 0.113264i	−0.998395	−	0.0566322i	$-0.981964\pi$
$97$	− 10.9867i	− 0.113264i	0.998395	−	0.0566322i	$-0.0180362\pi$
$98$	0	0
$99$	0	0
$100$	45.8823	0.458823
$101$	− 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	$0.829385\pi$
$104$	− 60.3246i	− 0.580044i
$105$	0	0
$106$	120.853	1.14012
$107$	−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	$0.329851\pi$
$110$	− 12.1753i	− 0.110685i
$111$	0	0
$112$	0	0
$113$	−101.397	−0.897318	−0.448659	−	0.893703i	$-0.648098\pi$
$113$	−101.397	−0.897318	−0.448659	+	0.893703i	$0.648098\pi$
$114$	0	0
$115$	53.7446i	0.467344i
$116$	67.8823	0.585192
$117$	0	0
$118$	− 58.2954i	− 0.494029i
$119$	0	0
$120$	0	0
$121$	−85.0000	−0.702479
$122$	− 1.68106i	− 0.0137792i
$123$	0	0
$124$	− 88.2836i	− 0.711965i
$125$	68.7897i	0.550317i
$126$	0	0
$127$	82.5736	0.650186	0.325093	−	0.945682i	$-0.394604\pi$
$127$	82.5736	0.650186	0.325093	+	0.945682i	$0.394604\pi$
$128$	11.3137	0.0883883
$129$	0	0
$130$	43.2792	0.332917
$131$	60.5708i	0.462372i	0.972910	+	0.231186i	$0.0742607\pi$
$131$	60.5708i	0.462372i	−0.972910	+	0.231186i	$0.925739\pi$
$132$	0	0
$133$	0	0
$134$	6.21825	0.0464049
$135$	0	0
$136$	− 25.3354i	− 0.186290i
$137$	−67.0294	−0.489266	−0.244633	−	0.969616i	$-0.578667\pi$
$137$	−67.0294	−0.489266	−0.244633	+	0.969616i	$0.578667\pi$
$138$	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
$141$	0	0
$142$	−194.309	−1.36837
$143$	127.968i	0.894880i
$144$	0	0
$145$	48.7014i	0.335872i
$146$	− 111.620i	− 0.764518i
$147$	0	0
$148$	−55.9411	−0.377981
$149$	−81.0883	−0.544217	−0.272108	−	0.962267i	$-0.587721\pi$
$149$	−81.0883	−0.544217	−0.272108	+	0.962267i	$0.587721\pi$
$150$	0	0
$151$	−51.2061	−0.339113	−0.169556	−	0.985520i	$-0.554234\pi$
$151$	−51.2061	−0.339113	−0.169556	+	0.985520i	$0.554234\pi$
$152$	− 20.4364i	− 0.134450i
$153$	0	0
$154$	0	0
$155$	63.3381	0.408633
$156$	0	0
$157$	− 187.061i	− 1.19147i	−0.803179	−	0.595737i	$-0.796860\pi$
$157$	− 187.061i	− 1.19147i	0.803179	−	0.595737i	$-0.203140\pi$
$158$	139.071	0.880197
$159$	0	0
$160$	8.11689i	0.0507306i
$161$	0	0
$162$	0	0
$163$	83.9411	0.514976	0.257488	−	0.966281i	$-0.417105\pi$
$163$	83.9411	0.514976	0.257488	+	0.966281i	$0.417105\pi$
$164$	109.663i	0.668674i
$165$	0	0
$166$	− 156.131i	− 0.940547i
$167$	127.620i	0.764190i	0.924123	+	0.382095i	$0.124797\pi$
$167$	127.620i	0.764190i	−0.924123	+	0.382095i	$0.875203\pi$
$168$	0	0
$169$	−285.882	−1.69161
$170$	18.1766	0.106921
$171$	0	0
$172$	−2.97056	−0.0172707
$173$	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
$176$	−24.0000	−0.136364
$177$	0	0
$178$	29.3939i	0.165134i
$179$	−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	$-0.803043\pi$
$181$	− 209.969i	− 1.16005i	0.814600	−	0.580024i	$-0.196957\pi$
$182$	0	0
$183$	0	0
$184$	105.941	0.575767
$185$	− 40.1343i	− 0.216942i
$186$	0	0
$187$	53.7446i	0.287404i
$188$	86.0082i	0.457490i
$189$	0	0
$190$	14.6619	0.0771679
$191$	−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	$0.508077\pi$
$194$	− 15.5375i	− 0.0800901i
$195$	0	0
$196$	0	0
$197$	267.161	1.35615	0.678075	−	0.734993i	$-0.262815\pi$
$197$	267.161	1.35615	0.678075	+	0.734993i	$0.262815\pi$
$198$	0	0
$199$	130.940i	0.657988i	0.944332	+	0.328994i	$0.106710\pi$
$199$	130.940i	0.657988i	−0.944332	+	0.328994i	$0.893290\pi$
$200$	64.8873	0.324437
$201$	0	0
$202$	− 151.580i	− 0.750396i
$203$	0	0
$204$	0	0
$205$	−78.6762	−0.383786
$206$	148.782i	0.722244i
$207$	0	0
$208$	− 85.3119i	− 0.410153i
$209$	43.3523i	0.207427i
$210$	0	0
$211$	−23.0883	−0.109423	−0.0547116	−	0.998502i	$-0.517424\pi$
$211$	−23.0883	−0.109423	−0.0547116	+	0.998502i	$0.517424\pi$
$212$	170.912	0.806187
$213$	0	0
$214$	−167.647	−0.783396
$215$	− 2.13120i	− 0.00991255i
$216$	0	0
$217$	0	0
$218$	157.061	0.720463
$219$	0	0
$220$	− 17.2185i	− 0.0782661i
$221$	−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
$225$	0	0
$226$	−143.397	−0.634500
$227$	65.6140i	0.289048i	0.989501	+	0.144524i	$0.0461652\pi$
$227$	65.6140i	0.289048i	−0.989501	+	0.144524i	$0.953835\pi$
$228$	0	0
$229$	93.4798i	0.408209i	0.978949	+	0.204104i	$0.0654282\pi$
$229$	93.4798i	0.408209i	−0.978949	+	0.204104i	$0.934572\pi$
$230$	76.0063i	0.330462i
$231$	0	0
$232$	96.0000	0.413793
$233$	237.515	1.01938	0.509688	−	0.860359i	$-0.329761\pi$
$233$	237.515	1.01938	0.509688	+	0.860359i	$0.329761\pi$
$234$	0	0
$235$	−61.7056	−0.262577
$236$	− 82.4421i	− 0.349331i
$237$	0	0
$238$	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	$0.338151\pi$
$241$	421.024i	1.74699i	−0.486836	+	0.873493i	$0.661849\pi$
$242$	−120.208	−0.496728
$243$	0	0
$244$	− 2.37738i	− 0.00974337i
$245$	0	0
$246$	0	0
$247$	−154.103	−0.623897
$248$	− 124.852i	− 0.503435i
$249$	0	0
$250$	97.2833i	0.389133i
$251$	146.621i	0.584148i	0.956396	+	0.292074i	$0.0943454\pi$
$251$	146.621i	0.584148i	−0.956396	+	0.292074i	$0.905655\pi$
$252$	0	0
$253$	−224.735	−0.888281
$254$	116.777	0.459751
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 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
$260$	61.2061	0.235408
$261$	0	0
$262$	85.6600i	0.326947i
$263$	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
$267$	0	0
$268$	8.79394	0.0328132
$269$	− 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	$-0.0731842\pi$
$272$	− 35.8297i	− 0.131727i
$273$	0	0
$274$	−94.7939	−0.345963
$275$	−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	$0.663941\pi$
$278$	129.475i	0.465737i
$279$	0	0
$280$	0	0
$281$	−133.103	−0.473675	−0.236837	−	0.971549i	$-0.576111\pi$
$281$	−133.103	−0.473675	−0.236837	+	0.971549i	$0.576111\pi$
$282$	0	0
$283$	128.757i	0.454973i	0.973781	+	0.227486i	$0.0730508\pi$
$283$	128.757i	0.454973i	−0.973781	+	0.227486i	$0.926949\pi$
$284$	−274.794	−0.967584
$285$	0	0
$286$	180.974i	0.632776i
$287$	0	0
$288$	0	0
$289$	208.765	0.722369
$290$	68.8741i	0.237497i
$291$	0	0
$292$	− 157.854i	− 0.540596i
$293$	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
$296$	−79.1127	−0.267273
$297$	0	0
$298$	−114.676	−0.384819
$299$	− 798.857i	− 2.67176i
$300$	0	0
$301$	0	0
$302$	−72.4163	−0.239789
$303$	0	0
$304$	− 28.9015i	− 0.0950707i
$305$	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
$309$	0	0
$310$	89.5736	0.288947
$311$	− 203.278i	− 0.653626i	−0.945089	−	0.326813i	$-0.894025\pi$
$311$	− 203.278i	− 0.653626i	0.945089	−	0.326813i	$-0.105975\pi$
$312$	0	0
$313$	− 405.639i	− 1.29597i	−0.761652	−	0.647986i	$-0.775611\pi$
$313$	− 405.639i	− 1.29597i	0.761652	−	0.647986i	$-0.224389\pi$
$314$	− 264.545i	− 0.842500i
$315$	0	0
$316$	196.676	0.622393
$317$	26.0589	0.0822047	0.0411023	−	0.999155i	$-0.486913\pi$
$317$	26.0589	0.0822047	0.0411023	+	0.999155i	$0.486913\pi$
$318$	0	0
$319$	−203.647	−0.638391
$320$	11.4790i	0.0358719i
$321$	0	0
$322$	0	0
$323$	−64.7208	−0.200374
$324$	0	0
$325$	− 489.288i	− 1.50550i
$326$	118.711	0.364143
$327$	0	0
$328$	155.086i	0.472824i
$329$	0	0
$330$	0	0
$331$	−108.632	−0.328195	−0.164097	−	0.986444i	$-0.552471\pi$
$331$	−108.632	−0.328195	−0.164097	+	0.986444i	$0.552471\pi$
$332$	− 220.802i	− 0.665067i
$333$	0	0
$334$	180.481i	0.540364i
$335$	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	$0.272503\pi$
$338$	−404.299	−1.19615
$339$	0	0
$340$	25.7056	0.0756048
$341$	264.851i	0.776689i
$342$	0	0
$343$	0	0
$344$	−4.20101	−0.0122122
$345$	0	0
$346$	202.191i	0.584367i
$347$	−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	$0.102926\pi$
$349$	221.787i	0.635493i	−0.948176	+	0.317746i	$0.897074\pi$
$350$	0	0
$351$	0	0
$352$	−33.9411	−0.0964237
$353$	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
$356$	41.5692i	0.116767i
$357$	0	0
$358$	−239.397	−0.668707
$359$	−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
$362$	− 296.940i	− 0.820277i
$363$	0	0
$364$	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	$0.806430\pi$
$368$	149.823	0.407129
$369$	0	0
$370$	− 56.7585i	− 0.153401i
$371$	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	$0.486606\pi$
$374$	76.0063i	0.203225i
$375$	0	0
$376$	121.634i	0.323495i
$377$	− 723.895i	− 1.92015i
$378$	0	0
$379$	206.779	0.545590	0.272795	−	0.962072i	$-0.412052\pi$
$379$	206.779	0.545590	0.272795	+	0.962072i	$0.412052\pi$
$380$	20.7351	0.0545660
$381$	0	0
$382$	−94.1909	−0.246573
$383$	498.567i	1.30174i	0.759189	+	0.650871i	$0.225596\pi$
$383$	498.567i	1.30174i	−0.759189	+	0.650871i	$0.774404\pi$
$384$	0	0
$385$	0	0
$386$	−13.8507	−0.0358827
$387$	0	0
$388$	− 21.9733i	− 0.0566322i
$389$	648.426	1.66691	0.833453	−	0.552590i	$-0.186361\pi$
$389$	648.426	1.66691	0.833453	+	0.552590i	$0.186361\pi$
$390$	0	0
$391$	− 335.508i	− 0.858077i
$392$	0	0
$393$	0	0
$394$	377.823	0.958943
$395$	141.103i	0.357223i
$396$	0	0
$397$	75.7514i	0.190809i	0.995439	+	0.0954047i	$0.0304145\pi$
$397$	75.7514i	0.190809i	−0.995439	+	0.0954047i	$0.969585\pi$
$398$	185.176i	0.465268i
$399$	0	0
$400$	91.7645	0.229411
$401$	564.250	1.40711	0.703553	−	0.710642i	$-0.251596\pi$
$401$	564.250	1.40711	0.703553	+	0.710642i	$0.251596\pi$
$402$	0	0
$403$	−941.455	−2.33612
$404$	− 214.366i	− 0.530610i
$405$	0	0
$406$	0	0
$407$	167.823	0.412342
$408$	0	0
$409$	357.448i	0.873955i	0.899472	+	0.436978i	$0.143951\pi$
$409$	357.448i	0.873955i	−0.899472	+	0.436978i	$0.856049\pi$
$410$	−111.265	−0.271378
$411$	0	0
$412$	210.410i	0.510704i
$413$	0	0
$414$	0	0
$415$	158.412	0.381716
$416$	− 120.649i	− 0.290022i
$417$	0	0
$418$	61.3093i	0.146673i
$419$	502.175i	1.19851i	0.800559	+	0.599254i	$0.204536\pi$
$419$	502.175i	1.19851i	−0.800559	+	0.599254i	$0.795464\pi$
$420$	0	0
$421$	33.7939	0.0802706	0.0401353	−	0.999194i	$-0.487221\pi$
$421$	33.7939	0.0802706	0.0401353	+	0.999194i	$0.487221\pi$
$422$	−32.6518	−0.0773739
$423$	0	0
$424$	241.706	0.570060
$425$	− 205.493i	− 0.483514i
$426$	0	0
$427$	0	0
$428$	−237.088	−0.553945
$429$	0	0
$430$	− 3.01397i	− 0.00700923i
$431$	−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	$-0.581820\pi$
$433$	− 837.548i	− 1.93429i	0.254224	−	0.967145i	$-0.418180\pi$
$434$	0	0
$435$	0	0
$436$	222.118	0.509444
$437$	− 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	$-0.0694377\pi$
$440$	− 24.3507i	− 0.0553425i
$441$	0	0
$442$	−270.177	−0.611259
$443$	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
$446$	323.333i	0.724961i
$447$	0	0
$448$	0	0
$449$	−18.1035	−0.0403195	−0.0201598	−	0.999797i	$-0.506417\pi$
$449$	−18.1035	−0.0403195	−0.0201598	+	0.999797i	$0.506417\pi$
$450$	0	0
$451$	− 328.988i	− 0.729463i
$452$	−202.794	−0.448659
$453$	0	0
$454$	92.7922i	0.204388i
$455$	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	$0.617142\pi$
$458$	132.200i	0.288647i
$459$	0	0
$460$	107.489i	0.233672i
$461$	− 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	$-0.643475\pi$
$463$	−403.396	−0.871266	−0.435633	+	0.900124i	$0.643475\pi$
$464$	135.765	0.292596
$465$	0	0
$466$	335.897	0.720808
$467$	2.82752i	0.00605464i	0.999995	+	0.00302732i	$0.000963627\pi$
$467$	2.82752i	0.00605464i	−0.999995	+	0.00302732i	$0.999036\pi$
$468$	0	0
$469$	0	0
$470$	−87.2649	−0.185670
$471$	0	0
$472$	− 116.591i	− 0.247014i
$473$	8.91169	0.0188408
$474$	0	0
$475$	− 165.758i	− 0.348965i
$476$	0	0
$477$	0	0
$478$	−518.808	−1.08537
$479$	379.514i	0.792305i	0.918185	+	0.396153i	$0.129655\pi$
$479$	379.514i	0.792305i	−0.918185	+	0.396153i	$0.870345\pi$
$480$	0	0
$481$	596.555i	1.24024i
$482$	595.418i	1.23531i
$483$	0	0
$484$	−170.000	−0.351240
$485$	15.7645	0.0325041
$486$	0	0
$487$	575.514	1.18175	0.590877	−	0.806762i	$-0.298782\pi$
$487$	575.514	1.18175	0.590877	+	0.806762i	$0.298782\pi$
$488$	− 3.36213i	− 0.00688961i
$489$	0	0
$490$	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
$494$	−217.934	−0.441162
$495$	0	0
$496$	− 176.567i	− 0.355982i
$497$	0	0
$498$	0	0
$499$	−286.574	−0.574296	−0.287148	−	0.957886i	$-0.592707\pi$
$499$	−286.574	−0.574296	−0.287148	+	0.957886i	$0.592707\pi$
$500$	137.579i	0.275159i
$501$	0	0
$502$	207.354i	0.413055i
$503$	25.4374i	0.0505714i	0.999680	+	0.0252857i	$0.00804954\pi$
$503$	25.4374i	0.0505714i	−0.999680	+	0.0252857i	$0.991950\pi$
$504$	0	0
$505$	153.795	0.304544
$506$	−317.823	−0.628109
$507$	0	0
$508$	165.147	0.325093
$509$	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
$512$	22.6274	0.0441942
$513$	0	0
$514$	− 35.4815i	− 0.0690302i
$515$	−150.956	−0.293119
$516$	0	0
$517$	− 258.025i	− 0.499080i
$518$	0	0
$519$	0	0
$520$	86.5584	0.166459
$521$	− 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	$-0.0540556\pi$
$524$	121.142i	0.231186i
$525$	0	0
$526$	128.235	0.243794
$527$	−395.397	−0.750279
$528$	0	0
$529$	873.940	1.65206
$530$	173.409i	0.327187i
$531$	0	0
$532$	0	0
$533$	1169.44	2.19407
$534$	0	0
$535$	− 170.096i	− 0.317937i
$536$	12.4365	0.0232024
$537$	0	0
$538$	− 96.7312i	− 0.179798i
$539$	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	$0.494947\pi$
$542$	− 174.682i	− 0.322292i
$543$	0	0
$544$	− 50.6709i	− 0.0931450i
$545$	159.356i	0.292396i
$546$	0	0
$547$	212.676	0.388805	0.194402	−	0.980922i	$-0.437723\pi$
$547$	212.676	0.388805	0.194402	+	0.980922i	$0.437723\pi$
$548$	−134.059	−0.244633
$549$	0	0
$550$	−194.662	−0.353931
$551$	− 245.237i	− 0.445077i
$552$	0	0
$553$	0	0
$554$	−385.914	−0.696595
$555$	0	0
$556$	183.105i	0.329326i
$557$	881.647	1.58285	0.791424	−	0.611267i	$-0.209340\pi$
$557$	881.647	1.58285	0.791424	+	0.611267i	$0.209340\pi$
$558$	0	0
$559$	31.6780i	0.0566691i
$560$	0	0
$561$	0	0
$562$	−188.235	−0.334939
$563$	767.068i	1.36247i	0.732067	+	0.681233i	$0.238556\pi$
$563$	767.068i	1.36247i	−0.732067	+	0.681233i	$0.761444\pi$
$564$	0	0
$565$	− 145.492i	− 0.257508i
$566$	182.090i	0.321714i
$567$	0	0
$568$	−388.617	−0.684185
$569$	29.2935	0.0514824	0.0257412	−	0.999669i	$-0.491805\pi$
$569$	29.2935	0.0514824	0.0257412	+	0.999669i	$0.491805\pi$
$570$	0	0
$571$	965.043	1.69009	0.845046	−	0.534693i	$-0.179573\pi$
$571$	965.043	1.69009	0.845046	+	0.534693i	$0.179573\pi$
$572$	255.936i	0.447440i
$573$	0	0
$574$	0	0
$575$	859.279	1.49440
$576$	0	0
$577$	− 263.136i	− 0.456042i	−0.973656	−	0.228021i	$-0.926775\pi$
$577$	− 263.136i	− 0.456042i	0.973656	−	0.228021i	$-0.0732255\pi$
$578$	295.238	0.510792
$579$	0	0
$580$	97.4027i	0.167936i
$581$	0	0
$582$	0	0
$583$	−512.735	−0.879477
$584$	− 223.239i	− 0.382259i
$585$	0	0
$586$	436.969i	0.745681i
$587$	− 436.477i	− 0.743572i	−0.928318	−	0.371786i	$-0.878746\pi$
$587$	− 436.477i	− 0.743572i	0.928318	−	0.371786i	$-0.121254\pi$
$588$	0	0
$589$	−318.941	−0.541496
$590$	83.6468	0.141774
$591$	0	0
$592$	−111.882	−0.188990
$593$	− 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
$596$	−162.177	−0.272108
$597$	0	0
$598$	− 1129.75i	− 1.88922i
$599$	−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	$0.00957964\pi$
$601$	36.1691i	0.0601816i	−0.999547	+	0.0300908i	$0.990420\pi$
$602$	0	0
$603$	0	0
$604$	−102.412	−0.169556
$605$	− 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	$0.991720\pi$
$608$	− 40.8729i	− 0.0672252i
$609$	0	0
$610$	2.41212	0.00395430
$611$	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	$0.608338\pi$
$614$	857.140i	1.39599i
$615$	0	0
$616$	0	0
$617$	−1227.38	−1.98927	−0.994636	−	0.103436i	$-0.967016\pi$
$617$	−1227.38	−1.98927	−0.994636	+	0.103436i	$0.967016\pi$
$618$	0	0
$619$	− 475.762i	− 0.768598i	−0.923209	−	0.384299i	$-0.874443\pi$
$619$	− 475.762i	− 0.768598i	0.923209	−	0.384299i	$-0.125557\pi$
$620$	126.676	0.204316
$621$	0	0
$622$	− 287.478i	− 0.462184i
$623$	0	0
$624$	0	0
$625$	474.823	0.759717
$626$	− 573.661i	− 0.916391i
$627$	0	0
$628$	− 374.123i	− 0.595737i
$629$	250.544i	0.398322i
$630$	0	0
$631$	−54.9420	−0.0870713	−0.0435357	−	0.999052i	$-0.513862\pi$
$631$	−54.9420	−0.0870713	−0.0435357	+	0.999052i	$0.513862\pi$
$632$	278.142	0.440098
$633$	0	0
$634$	36.8528	0.0581275
$635$	118.483i	0.186587i
$636$	0	0
$637$	0	0
$638$	−288.000	−0.451411
$639$	0	0
$640$	16.2338i	0.0253653i
$641$	229.103	0.357414	0.178707	−	0.983902i	$-0.442809\pi$
$641$	229.103	0.357414	0.178707	+	0.983902i	$0.442809\pi$
$642$	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
$645$	0	0
$646$	−91.5290	−0.141686
$647$	1003.01i	1.55025i	0.631809	+	0.775124i	$0.282313\pi$
$647$	1003.01i	1.55025i	−0.631809	+	0.775124i	$0.717687\pi$
$648$	0	0
$649$	247.326i	0.381088i
$650$	− 691.957i	− 1.06455i
$651$	0	0
$652$	167.882	0.257488
$653$	−1270.76	−1.94604	−0.973020	−	0.230722i	$-0.925891\pi$
$653$	−1270.76	−1.94604	−0.973020	+	0.230722i	$0.925891\pi$
$654$	0	0
$655$	−86.9117	−0.132690
$656$	219.325i	0.334337i
$657$	0	0
$658$	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	$-0.979813\pi$
$661$	− 83.7838i	− 0.126753i	0.997990	−	0.0633765i	$-0.0201869\pi$
$662$	−153.630	−0.232069
$663$	0	0
$664$	− 312.262i	− 0.470273i
$665$	0	0
$666$	0	0
$667$	1271.29	1.90599
$668$	255.239i	0.382095i
$669$	0	0
$670$	8.92244i	0.0133171i
$671$	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	$0.400065\pi$
$674$	624.708	0.926866
$675$	0	0
$676$	−571.765	−0.845805
$677$	− 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
$680$	36.3532	0.0534607
$681$	0	0
$682$	374.556i	0.549202i
$683$	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
$687$	0	0
$688$	−5.94113	−0.00863536
$689$	− 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	$0.738942\pi$
$692$	285.941i	0.413210i
$693$	0	0
$694$	−48.3532	−0.0696733
$695$	−131.367	−0.189017
$696$	0	0
$697$	491.147	0.704659
$698$	313.654i	0.449361i
$699$	0	0
$700$	0	0
$701$	0.103464	0.000147594 0	7.37972e−5	−	1.00000i	$-0.499977\pi$
$701$	0.103464	0.000147594 0	7.37972e−5		1.00000i	$0.499977\pi$
$702$	0	0
$703$	202.098i	0.287479i
$704$	−48.0000	−0.0681818
$705$	0	0
$706$	632.700i	0.896175i
$707$	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	$0.176655\pi$
$710$	− 278.809i	− 0.392689i
$711$	0	0
$712$	58.7878i	0.0825671i
$713$	− 1653.37i	− 2.31889i
$714$	0	0
$715$	−183.618	−0.256809
$716$	−338.558	−0.472847
$717$	0	0
$718$	−412.617	−0.574676
$719$	− 982.564i	− 1.36657i	−0.730152	−	0.683285i	$-0.760551\pi$
$719$	− 982.564i	− 1.36657i	0.730152	−	0.683285i	$-0.239449\pi$
$720$	0	0
$721$	0	0
$722$	436.701	0.604848
$723$	0	0
$724$	− 419.937i	− 0.580024i
$725$	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
$729$	0	0
$730$	160.161	0.219398
$731$	13.3043i	0.0182001i
$732$	0	0
$733$	298.474i	0.407195i	0.979055	+	0.203597i	$0.0652633\pi$
$733$	298.474i	0.407195i	−0.979055	+	0.203597i	$0.934737\pi$
$734$	593.053i	0.807974i
$735$	0	0
$736$	211.882	0.287883
$737$	−26.3818	−0.0357962
$738$	0	0
$739$	345.368	0.467344	0.233672	−	0.972315i	$-0.424926\pi$
$739$	345.368	0.467344	0.233672	+	0.972315i	$0.424926\pi$
$740$	− 80.2687i	− 0.108471i
$741$	0	0
$742$	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
$746$	44.3806	0.0594914
$747$	0	0
$748$	107.489i	0.143702i
$749$	0	0
$750$	0	0
$751$	−578.338	−0.770091	−0.385045	−	0.922898i	$-0.625814\pi$
$751$	−578.338	−0.770091	−0.385045	+	0.922898i	$0.625814\pi$
$752$	172.016i	0.228745i
$753$	0	0
$754$	− 1023.74i	− 1.35775i
$755$	− 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	$0.207057\pi$
$758$	292.429	0.385791
$759$	0	0
$760$	29.3238	0.0385840
$761$	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
$764$	−133.206	−0.174353
$765$	0	0
$766$	705.080i	0.920470i
$767$	−879.161	−1.14623
$768$	0	0
$769$	1290.16i	1.67771i	0.544358	+	0.838853i	$0.316774\pi$
$769$	1290.16i	1.67771i	−0.544358	+	0.838853i	$0.683226\pi$
$770$	0	0
$771$	0	0
$772$	−19.5879	−0.0253729
$773$	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
$776$	− 31.0749i	− 0.0400450i
$777$	0	0
$778$	917.013	1.17868
$779$	396.177	0.508571
$780$	0	0
$781$	824.382	1.05555
$782$	− 474.480i	− 0.606752i
$783$	0	0
$784$	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	$0.547207\pi$
$788$	534.323	0.678075
$789$	0	0
$790$	199.550i	0.252595i
$791$	0	0
$792$	0	0
$793$	−25.3524	−0.0319702
$794$	107.129i	0.134923i
$795$	0	0
$796$	261.879i	0.328994i
$797$	− 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
$800$	129.775	0.162218
$801$	0	0
$802$	797.970	0.994975
$803$	473.562i	0.589741i
$804$	0	0
$805$	0	0
$806$	−1331.42	−1.65188
$807$	0	0
$808$	− 303.160i	− 0.375198i
$809$	229.279	0.283411	0.141705	−	0.989909i	$-0.454741\pi$
$809$	229.279	0.283411	0.141705	+	0.989909i	$0.454741\pi$
$810$	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
$813$	0	0
$814$	237.338	0.291570
$815$	120.445i	0.147786i
$816$	0	0
$817$	10.7317i	0.0131355i
$818$	505.508i	0.617980i
$819$	0	0
$820$	−157.352	−0.191893
$821$	303.338	0.369474	0.184737	−	0.982788i	$-0.440857\pi$
$821$	303.338	0.369474	0.184737	+	0.982788i	$0.440857\pi$
$822$	0	0
$823$	−1129.91	−1.37292	−0.686459	−	0.727169i	$-0.740836\pi$
$823$	−1129.91	−1.37292	−0.686459	+	0.727169i	$0.740836\pi$
$824$	297.565i	0.361122i
$825$	0	0
$826$	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	$-0.625721\pi$
$829$	− 1530.35i	− 1.84602i	0.384776	−	0.923010i	$-0.374279\pi$
$830$	224.029	0.269914
$831$	0	0
$832$	− 170.624i	− 0.205077i
$833$	0	0
$834$	0	0
$835$	−183.119	−0.219304
$836$	86.7045i	0.103714i
$837$	0	0
$838$	710.183i	0.847474i
$839$	218.629i	0.260583i	0.991476	+	0.130291i	$0.0415913\pi$
$839$	218.629i	0.260583i	−0.991476	+	0.130291i	$0.958409\pi$
$840$	0	0
$841$	311.000	0.369798
$842$	47.7918	0.0567599
$843$	0	0
$844$	−46.1766	−0.0547116
$845$	− 410.206i	− 0.485451i
$846$	0	0
$847$	0	0
$848$	341.823	0.403094
$849$	0	0
$850$	− 290.612i	− 0.341896i
$851$	−1047.66	−1.23109
$852$	0	0
$853$	762.730i	0.894174i	0.894491	+	0.447087i	$0.147539\pi$
$853$	762.730i	0.894174i	−0.894491	+	0.447087i	$0.852461\pi$
$854$	0	0
$855$	0	0
$856$	−335.294	−0.391698
$857$	− 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	$-0.170995\pi$
$860$	− 4.26239i	− 0.00495627i
$861$	0	0
$862$	−712.368	−0.826412
$863$	−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
$866$	− 1184.47i	− 1.36775i
$867$	0	0
$868$	0	0
$869$	−590.029	−0.678974
$870$	0	0
$871$	− 93.7784i	− 0.107668i
$872$	314.122	0.360232
$873$	0	0
$874$	− 382.732i	− 0.437909i
$875$	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	$0.499354\pi$
$878$	− 268.723i	− 0.306062i
$879$	0	0
$880$	− 34.4371i	− 0.0391330i
$881$	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	$-0.728629\pi$
$883$	−1162.16	−1.31615	−0.658075	+	0.752953i	$0.728629\pi$
$884$	−382.087	−0.432226
$885$	0	0
$886$	239.647	0.270482
$887$	− 86.9260i	− 0.0980000i	−0.998799	−	0.0490000i	$-0.984397\pi$
$887$	− 86.9260i	− 0.0980000i	0.998799	−	0.0490000i	$-0.0156034\pi$
$888$	0	0
$889$	0	0
$890$	−42.1766	−0.0473895
$891$	0	0
$892$	457.261i	0.512625i
$893$	310.721	0.347952
$894$	0	0
$895$	− 242.895i	− 0.271391i
$896$	0	0
$897$	0	0
$898$	−25.6022	−0.0285102
$899$	− 1498.22i	− 1.66654i
$900$	0	0
$901$	− 765.464i	− 0.849572i
$902$	− 465.259i	− 0.515808i
$903$	0	0
$904$	−286.794	−0.317250
$905$	301.279	0.332905
$906$	0	0
$907$	−460.897	−0.508155	−0.254077	−	0.967184i	$-0.581772\pi$
$907$	−460.897	−0.508155	−0.254077	+	0.967184i	$0.581772\pi$
$908$	131.228i	0.144524i
$909$	0	0
$910$	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
$914$	−465.026	−0.508782
$915$	0	0
$916$	186.960i	0.204104i
$917$	0	0
$918$	0	0
$919$	−541.839	−0.589596	−0.294798	−	0.955560i	$-0.595252\pi$
$919$	−541.839	−0.589596	−0.294798	+	0.955560i	$0.595252\pi$
$920$	152.013i	0.165231i
$921$	0	0
$922$	− 1123.35i	− 1.21839i
$923$	2930.40i	3.17486i
$924$	0	0
$925$	−641.676	−0.693704
$926$	−570.488	−0.616078
$927$	0	0
$928$	192.000	0.206897
$929$	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
$932$	475.029	0.509688
$933$	0	0
$934$	3.99871i	0.00428128i
$935$	−77.1169	−0.0824779
$936$	0	0
$937$	233.964i	0.249695i	0.992176	+	0.124847i	$0.0398441\pi$
$937$	233.964i	0.249695i	−0.992176	+	0.124847i	$0.960156\pi$
$938$	0	0
$939$	0	0
$940$	−123.411	−0.131289
$941$	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
$944$	− 164.884i	− 0.174666i
$945$	0	0
$946$	12.6030	0.0133224
$947$	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
$950$	− 234.418i	− 0.246755i
$951$	0	0
$952$	0	0
$953$	91.4255	0.0959345	0.0479672	−	0.998849i	$-0.484726\pi$
$953$	91.4255	0.0959345	0.0479672	+	0.998849i	$0.484726\pi$
$954$	0	0
$955$	− 95.5672i	− 0.100070i
$956$	−733.706	−0.767475
$957$	0	0
$958$	536.714i	0.560245i
$959$	0	0
$960$	0	0
$961$	−987.499	−1.02757
$962$	843.657i	0.876982i
$963$	0	0
$964$	842.048i	0.873493i
$965$	− 14.0531i	− 0.0145628i
$966$	0	0
$967$	−1098.19	−1.13567	−0.567834	−	0.823143i	$-0.692218\pi$
$967$	−1098.19	−1.13567	−0.567834	+	0.823143i	$0.692218\pi$
$968$	−240.416	−0.248364
$969$	0	0
$970$	22.2944	0.0229839
$971$	− 132.103i	− 0.136049i	−0.997684	−	0.0680245i	$-0.978330\pi$
$971$	− 132.103i	− 0.136049i	0.997684	−	0.0680245i	$-0.0216696\pi$
$972$	0	0
$973$	0	0
$974$	813.899	0.835626
$975$	0	0
$976$	− 4.75477i	− 0.00487169i
$977$	−448.234	−0.458786	−0.229393	−	0.973334i	$-0.573674\pi$
$977$	−448.234	−0.458786	−0.229393	+	0.973334i	$0.573674\pi$
$978$	0	0
$979$	− 124.708i	− 0.127383i
$980$	0	0
$981$	0	0
$982$	−337.206	−0.343387
$983$	1646.76i	1.67524i	0.546251	+	0.837621i	$0.316054\pi$
$983$	1646.76i	1.67524i	−0.546251	+	0.837621i	$0.683946\pi$
$984$	0	0
$985$	383.344i	0.389182i
$986$	− 429.956i	− 0.436061i
$987$	0	0
$988$	−308.205	−0.311949
$989$	−55.6325	−0.0562512
$990$	0	0
$991$	−628.897	−0.634608	−0.317304	−	0.948324i	$-0.602778\pi$
$991$	−628.897	−0.634608	−0.317304	+	0.948324i	$0.602778\pi$
$992$	− 249.704i	− 0.251717i
$993$	0	0
$994$	0	0
$995$	−187.882	−0.188826
$996$	0	0
$997$	1004.18i	1.00720i	0.863936	+	0.503601i	$0.167992\pi$
$997$	1004.18i	1.00720i	−0.863936	+	0.503601i	$0.832008\pi$
$998$	−405.276	−0.406088
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.3.c.b.685.4		4
3.2	odd	2		294.3.c.a.97.1		4
7.2	even	3		126.3.n.a.73.1		4
7.3	odd	6		126.3.n.a.19.1		4
7.4	even	3		882.3.n.e.19.1		4
7.5	odd	6		882.3.n.e.325.1		4
7.6	odd	2	inner	882.3.c.b.685.3		4
12.11	even	2		2352.3.f.e.97.3		4
21.2	odd	6		42.3.g.a.31.2	yes	4
21.5	even	6		294.3.g.a.31.2		4
21.11	odd	6		294.3.g.a.19.2		4
21.17	even	6		42.3.g.a.19.2	✓	4
21.20	even	2		294.3.c.a.97.2		4
28.3	even	6		1008.3.cg.h.145.2		4
28.23	odd	6		1008.3.cg.h.577.2		4
84.23	even	6		336.3.bh.e.241.1		4
84.59	odd	6		336.3.bh.e.145.1		4
84.83	odd	2		2352.3.f.e.97.2		4
105.2	even	12		1050.3.q.a.199.2		8
105.17	odd	12		1050.3.q.a.649.3		8
105.23	even	12		1050.3.q.a.199.3		8
105.38	odd	12		1050.3.q.a.649.2		8
105.44	odd	6		1050.3.p.a.451.1		4
105.59	even	6		1050.3.p.a.901.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.3.g.a.19.2	✓	4	21.17	even	6
42.3.g.a.31.2	yes	4	21.2	odd	6
126.3.n.a.19.1		4	7.3	odd	6
126.3.n.a.73.1		4	7.2	even	3
294.3.c.a.97.1		4	3.2	odd	2
294.3.c.a.97.2		4	21.20	even	2
294.3.g.a.19.2		4	21.11	odd	6
294.3.g.a.31.2		4	21.5	even	6
336.3.bh.e.145.1		4	84.59	odd	6
336.3.bh.e.241.1		4	84.23	even	6
882.3.c.b.685.3		4	7.6	odd	2	inner
882.3.c.b.685.4		4	1.1	even	1	trivial
882.3.n.e.19.1		4	7.4	even	3
882.3.n.e.325.1		4	7.5	odd	6
1008.3.cg.h.145.2		4	28.3	even	6
1008.3.cg.h.577.2		4	28.23	odd	6
1050.3.p.a.451.1		4	105.44	odd	6
1050.3.p.a.901.1		4	105.59	even	6
1050.3.q.a.199.2		8	105.2	even	12
1050.3.q.a.199.3		8	105.23	even	12
1050.3.q.a.649.2		8	105.38	odd	12
1050.3.q.a.649.3		8	105.17	odd	12
2352.3.f.e.97.2		4	84.83	odd	2
2352.3.f.e.97.3		4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	882.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} +1.43488i q^{5} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} +1.43488i q^{5} +2.82843 q^{8} +2.02922i q^{10} -6.00000 q^{11} -21.3280i q^{13} +4.00000 q^{16} -8.95743i q^{17} -7.22538i q^{19} +2.86976i q^{20} -8.48528 q^{22} +37.4558 q^{23} +22.9411 q^{25} -30.1623i q^{26} +33.9411 q^{29} -44.1418i q^{31} +5.65685 q^{32} -12.6677i q^{34} -27.9706 q^{37} -10.2182i q^{38} +4.05845i q^{40} +54.8313i q^{41} -1.48528 q^{43} -12.0000 q^{44} +52.9706 q^{46} +43.0041i q^{47} +32.4437 q^{50} -42.6559i q^{52} +85.4558 q^{53} -8.60927i q^{55} +48.0000 q^{58} -41.2211i q^{59} -1.18869i q^{61} -62.4259i q^{62} +8.00000 q^{64} +30.6030 q^{65} +4.39697 q^{67} -17.9149i q^{68} -137.397 q^{71} -78.9270i q^{73} -39.5563 q^{74} -14.4508i q^{76} +98.3381 q^{79} +5.73951i q^{80} +77.5431i q^{82} -110.401i q^{83} +12.8528 q^{85} -2.10051 q^{86} -16.9706 q^{88} +20.7846i q^{89} +74.9117 q^{92} +60.8170i q^{94} +10.3675 q^{95} -10.9867i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4}+O(q^{10}) \) 4 * q + 8 * q^4	\( 4 q + 8 q^{4} - 24 q^{11} + 16 q^{16} + 48 q^{23} - 44 q^{25} - 44 q^{37} + 28 q^{43} - 48 q^{44} + 144 q^{46} + 192 q^{50} + 240 q^{53} + 192 q^{58} + 32 q^{64} + 360 q^{65} - 220 q^{67} - 312 q^{71} - 96 q^{74} + 20 q^{79} - 288 q^{85} - 48 q^{86} + 96 q^{92} - 264 q^{95}+O(q^{100}) \) 4 * q + 8 * q^4 - 24 * q^11 + 16 * q^16 + 48 * q^23 - 44 * q^25 - 44 * q^37 + 28 * q^43 - 48 * q^44 + 144 * q^46 + 192 * q^50 + 240 * q^53 + 192 * q^58 + 32 * q^64 + 360 * q^65 - 220 * q^67 - 312 * q^71 - 96 * q^74 + 20 * q^79 - 288 * q^85 - 48 * q^86 + 96 * q^92 - 264 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more