LMFDB - Embedded newform 1452.3.f.a.241.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1452,3,Mod(241,1452)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1452, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1452.241"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1452 = 2^{2} \cdot 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1452.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$39.5641343851$
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} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.4
Root			$0.517638i$ of defining polynomial
Character	$\chi$	$=$	1452.241
Dual form			1452.3.f.a.241.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} +0.267949 q^{5} +2.44949i q^{7} +3.00000 q^{9} +7.48717i q^{13} +0.464102 q^{15} +18.4219i q^{17} +7.72741i q^{19} +4.24264i q^{21} -23.5167 q^{23} -24.9282 q^{25} +5.19615 q^{27} -8.52245i q^{29} -8.53590 q^{31} +0.656339i q^{35} -7.48334 q^{37} +12.9682i q^{39} +3.69780i q^{41} +4.24264i q^{43} +0.803848 q^{45} -24.5359 q^{47} +43.0000 q^{49} +31.9077i q^{51} -17.0000 q^{53} +13.3843i q^{57} -11.8038 q^{59} +101.038i q^{61} +7.34847i q^{63} +2.00618i q^{65} -65.0859 q^{67} -40.7321 q^{69} +79.2679 q^{71} +75.1763i q^{73} -43.1769 q^{75} -31.4644i q^{79} +9.00000 q^{81} +97.1275i q^{83} +4.93614i q^{85} -14.7613i q^{87} -112.124 q^{89} -18.3397 q^{91} -14.7846 q^{93} +2.07055i q^{95} -99.3397 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5} + 12 q^{9} - 12 q^{15} - 4 q^{23} - 72 q^{25} - 48 q^{31} - 120 q^{37} + 24 q^{45} - 112 q^{47} + 172 q^{49} - 68 q^{53} - 68 q^{59} - 4 q^{67} - 156 q^{69} + 324 q^{71} - 48 q^{75} + 36 q^{81}+ \cdots - 432 q^{97}+O(q^{100}) $ 4 * q + 8 * q^5 + 12 * q^9 - 12 * q^15 - 4 * q^23 - 72 * q^25 - 48 * q^31 - 120 * q^37 + 24 * q^45 - 112 * q^47 + 172 * q^49 - 68 * q^53 - 68 * q^59 - 4 * q^67 - 156 * q^69 + 324 * q^71 - 48 * q^75 + 36 * q^81 - 400 * q^89 - 108 * q^91 + 24 * q^93 - 432 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1452\mathbb{Z}\right)^\times$.

$n$	$485$	$727$	$1333$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.73205	0.577350
$3$	1.73205	0.577350
$4$	0	0
$5$	0.267949	0.0535898	0.0267949	−	0.999641i	$-0.491470\pi$
$5$	0.267949	0.0535898	0.0267949	+	0.999641i	$0.491470\pi$
$6$	0	0
$7$	2.44949i	0.349927i	0.984575	+	0.174964i	$0.0559808\pi$
$7$	2.44949i	0.349927i	−0.984575	+	0.174964i	$0.944019\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	7.48717i	0.575936i	0.957640	+	0.287968i	$0.0929797\pi$
$13$	7.48717i	0.575936i	−0.957640	+	0.287968i	$0.907020\pi$
$14$	0	0
$15$	0.464102	0.0309401
$16$	0	0
$17$	18.4219i	1.08364i	0.840493	+	0.541822i	$0.182265\pi$
$17$	18.4219i	1.08364i	−0.840493	+	0.541822i	$0.817735\pi$
$18$	0	0
$19$	7.72741i	0.406706i	0.979106	+	0.203353i	$0.0651838\pi$
$19$	7.72741i	0.406706i	−0.979106	+	0.203353i	$0.934816\pi$
$20$	0	0
$21$	4.24264i	0.202031i
$22$	0	0
$23$	−23.5167	−1.02246	−0.511232	−	0.859443i	$-0.670811\pi$
$23$	−23.5167	−1.02246	−0.511232	+	0.859443i	$0.670811\pi$
$24$	0	0
$25$	−24.9282	−0.997128
$26$	0	0
$27$	5.19615	0.192450
$28$	0	0
$29$	− 8.52245i	− 0.293877i	−0.989146	−	0.146939i	$-0.953058\pi$
$29$	− 8.52245i	− 0.293877i	0.989146	−	0.146939i	$-0.0469420\pi$
$30$	0	0
$31$	−8.53590	−0.275352	−0.137676	−	0.990477i	$-0.543963\pi$
$31$	−8.53590	−0.275352	−0.137676	+	0.990477i	$0.543963\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.656339i	0.0187525i
$36$	0	0
$37$	−7.48334	−0.202252	−0.101126	−	0.994874i	$-0.532245\pi$
$37$	−7.48334	−0.202252	−0.101126	+	0.994874i	$0.532245\pi$
$38$	0	0
$39$	12.9682i	0.332517i
$40$	0	0
$41$	3.69780i	0.0901901i	0.998983	+	0.0450951i	$0.0143591\pi$
$41$	3.69780i	0.0901901i	−0.998983	+	0.0450951i	$0.985641\pi$
$42$	0	0
$43$	4.24264i	0.0986661i	0.998782	+	0.0493330i	$0.0157096\pi$
$43$	4.24264i	0.0986661i	−0.998782	+	0.0493330i	$0.984290\pi$
$44$	0	0
$45$	0.803848	0.0178633
$46$	0	0
$47$	−24.5359	−0.522040	−0.261020	−	0.965333i	$-0.584059\pi$
$47$	−24.5359	−0.522040	−0.261020	+	0.965333i	$0.584059\pi$
$48$	0	0
$49$	43.0000	0.877551
$50$	0	0
$51$	31.9077i	0.625642i
$52$	0	0
$53$	−17.0000	−0.320755	−0.160377	−	0.987056i	$-0.551271\pi$
$53$	−17.0000	−0.320755	−0.160377	+	0.987056i	$0.551271\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	13.3843i	0.234812i
$58$	0	0
$59$	−11.8038	−0.200065	−0.100033	−	0.994984i	$-0.531895\pi$
$59$	−11.8038	−0.200065	−0.100033	+	0.994984i	$0.531895\pi$
$60$	0	0
$61$	101.038i	1.65637i	0.560458	+	0.828183i	$0.310625\pi$
$61$	101.038i	1.65637i	−0.560458	+	0.828183i	$0.689375\pi$
$62$	0	0
$63$	7.34847i	0.116642i
$64$	0	0
$65$	2.00618i	0.0308643i
$66$	0	0
$67$	−65.0859	−0.971431	−0.485716	−	0.874117i	$-0.661441\pi$
$67$	−65.0859	−0.971431	−0.485716	+	0.874117i	$0.661441\pi$
$68$	0	0
$69$	−40.7321	−0.590320
$70$	0	0
$71$	79.2679	1.11645	0.558225	−	0.829690i	$-0.311483\pi$
$71$	79.2679	1.11645	0.558225	+	0.829690i	$0.311483\pi$
$72$	0	0
$73$	75.1763i	1.02981i	0.857247	+	0.514906i	$0.172173\pi$
$73$	75.1763i	1.02981i	−0.857247	+	0.514906i	$0.827827\pi$
$74$	0	0
$75$	−43.1769	−0.575692
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 31.4644i	− 0.398284i	−0.979971	−	0.199142i	$-0.936185\pi$
$79$	− 31.4644i	− 0.398284i	0.979971	−	0.199142i	$-0.0638155\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	97.1275i	1.17021i	0.810957	+	0.585105i	$0.198947\pi$
$83$	97.1275i	1.17021i	−0.810957	+	0.585105i	$0.801053\pi$
$84$	0	0
$85$	4.93614i	0.0580723i
$86$	0	0
$87$	− 14.7613i	− 0.169670i
$88$	0	0
$89$	−112.124	−1.25982	−0.629912	−	0.776666i	$-0.716909\pi$
$89$	−112.124	−1.25982	−0.629912	+	0.776666i	$0.716909\pi$
$90$	0	0
$91$	−18.3397	−0.201536
$92$	0	0
$93$	−14.7846	−0.158974
$94$	0	0
$95$	2.07055i	0.0217953i
$96$	0	0
$97$	−99.3397	−1.02412	−0.512061	−	0.858949i	$-0.671118\pi$
$97$	−99.3397	−1.02412	−0.512061	+	0.858949i	$0.671118\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 124.424i	− 1.23192i	−0.787779	−	0.615958i	$-0.788769\pi$
$101$	− 124.424i	− 1.23192i	0.787779	−	0.615958i	$-0.211231\pi$
$102$	0	0
$103$	98.9038	0.960231	0.480116	−	0.877205i	$-0.340595\pi$
$103$	98.9038	0.960231	0.480116	+	0.877205i	$0.340595\pi$
$104$	0	0
$105$	1.13681i	0.0108268i
$106$	0	0
$107$	180.606i	1.68791i	0.536417	+	0.843953i	$0.319777\pi$
$107$	180.606i	1.68791i	−0.536417	+	0.843953i	$0.680223\pi$
$108$	0	0
$109$	2.22917i	0.0204511i	0.999948	+	0.0102255i	$0.00325495\pi$
$109$	2.22917i	0.0204511i	−0.999948	+	0.0102255i	$0.996745\pi$
$110$	0	0
$111$	−12.9615	−0.116770
$112$	0	0
$113$	−33.9282	−0.300250	−0.150125	−	0.988667i	$-0.547968\pi$
$113$	−33.9282	−0.300250	−0.150125	+	0.988667i	$0.547968\pi$
$114$	0	0
$115$	−6.30127	−0.0547937
$116$	0	0
$117$	22.4615i	0.191979i
$118$	0	0
$119$	−45.1244	−0.379196
$120$	0	0
$121$	0	0
$122$	0	0
$123$	6.40477i	0.0520713i
$124$	0	0
$125$	−13.3782	−0.107026
$126$	0	0
$127$	65.1282i	0.512820i	0.966568	+	0.256410i	$0.0825397\pi$
$127$	65.1282i	0.512820i	−0.966568	+	0.256410i	$0.917460\pi$
$128$	0	0
$129$	7.34847i	0.0569649i
$130$	0	0
$131$	− 178.089i	− 1.35946i	−0.733462	−	0.679730i	$-0.762097\pi$
$131$	− 178.089i	− 1.35946i	0.733462	−	0.679730i	$-0.237903\pi$
$132$	0	0
$133$	−18.9282	−0.142317
$134$	0	0
$135$	1.39230	0.0103134
$136$	0	0
$137$	215.205	1.57084	0.785420	−	0.618963i	$-0.212447\pi$
$137$	215.205	1.57084	0.785420	+	0.618963i	$0.212447\pi$
$138$	0	0
$139$	168.108i	1.20941i	0.796449	+	0.604706i	$0.206709\pi$
$139$	168.108i	1.20941i	−0.796449	+	0.604706i	$0.793291\pi$
$140$	0	0
$141$	−42.4974	−0.301400
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 2.28358i	− 0.0157488i
$146$	0	0
$147$	74.4782	0.506654
$148$	0	0
$149$	281.837i	1.89153i	0.324858	+	0.945763i	$0.394683\pi$
$149$	281.837i	1.89153i	−0.324858	+	0.945763i	$0.605317\pi$
$150$	0	0
$151$	177.324i	1.17433i	0.809466	+	0.587166i	$0.199757\pi$
$151$	177.324i	1.17433i	−0.809466	+	0.587166i	$0.800243\pi$
$152$	0	0
$153$	55.2658i	0.361215i
$154$	0	0
$155$	−2.28719	−0.0147560
$156$	0	0
$157$	−215.033	−1.36964	−0.684819	−	0.728713i	$-0.740119\pi$
$157$	−215.033	−1.36964	−0.684819	+	0.728713i	$0.740119\pi$
$158$	0	0
$159$	−29.4449	−0.185188
$160$	0	0
$161$	− 57.6038i	− 0.357788i
$162$	0	0
$163$	−249.899	−1.53312	−0.766560	−	0.642172i	$-0.778033\pi$
$163$	−249.899	−1.53312	−0.766560	+	0.642172i	$0.778033\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 235.768i	− 1.41178i	−0.708321	−	0.705891i	$-0.750547\pi$
$167$	− 235.768i	− 1.41178i	0.708321	−	0.705891i	$-0.249453\pi$
$168$	0	0
$169$	112.942	0.668298
$170$	0	0
$171$	23.1822i	0.135569i
$172$	0	0
$173$	− 35.6128i	− 0.205854i	−0.994689	−	0.102927i	$-0.967179\pi$
$173$	− 35.6128i	− 0.205854i	0.994689	−	0.102927i	$-0.0328209\pi$
$174$	0	0
$175$	− 61.0614i	− 0.348922i
$176$	0	0
$177$	−20.4449	−0.115508
$178$	0	0
$179$	267.310	1.49335	0.746677	−	0.665187i	$-0.231648\pi$
$179$	267.310	1.49335	0.746677	+	0.665187i	$0.231648\pi$
$180$	0	0
$181$	53.9808	0.298236	0.149118	−	0.988819i	$-0.452357\pi$
$181$	53.9808	0.298236	0.149118	+	0.988819i	$0.452357\pi$
$182$	0	0
$183$	175.003i	0.956303i
$184$	0	0
$185$	−2.00515	−0.0108387
$186$	0	0
$187$	0	0
$188$	0	0
$189$	12.7279i	0.0673435i
$190$	0	0
$191$	−15.4256	−0.0807624	−0.0403812	−	0.999184i	$-0.512857\pi$
$191$	−15.4256	−0.0807624	−0.0403812	+	0.999184i	$0.512857\pi$
$192$	0	0
$193$	23.7271i	0.122938i	0.998109	+	0.0614691i	$0.0195786\pi$
$193$	23.7271i	0.122938i	−0.998109	+	0.0614691i	$0.980421\pi$
$194$	0	0
$195$	3.47481i	0.0178195i
$196$	0	0
$197$	− 56.8161i	− 0.288406i	−0.989548	−	0.144203i	$-0.953938\pi$
$197$	− 56.8161i	− 0.288406i	0.989548	−	0.144203i	$-0.0460619\pi$
$198$	0	0
$199$	−64.0807	−0.322014	−0.161007	−	0.986953i	$-0.551474\pi$
$199$	−64.0807	−0.322014	−0.161007	+	0.986953i	$0.551474\pi$
$200$	0	0
$201$	−112.732	−0.560856
$202$	0	0
$203$	20.8756	0.102836
$204$	0	0
$205$	0.990821i	0.00483328i
$206$	0	0
$207$	−70.5500	−0.340821
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 192.454i	− 0.912107i	−0.889952	−	0.456053i	$-0.849263\pi$
$211$	− 192.454i	− 0.912107i	0.889952	−	0.456053i	$-0.150737\pi$
$212$	0	0
$213$	137.296	0.644583
$214$	0	0
$215$	1.13681i	0.00528750i
$216$	0	0
$217$	− 20.9086i	− 0.0963530i
$218$	0	0
$219$	130.209i	0.594562i
$220$	0	0
$221$	−137.928	−0.624110
$222$	0	0
$223$	−15.0436	−0.0674602	−0.0337301	−	0.999431i	$-0.510739\pi$
$223$	−15.0436	−0.0674602	−0.0337301	+	0.999431i	$0.510739\pi$
$224$	0	0
$225$	−74.7846	−0.332376
$226$	0	0
$227$	− 263.611i	− 1.16128i	−0.814159	−	0.580641i	$-0.802802\pi$
$227$	− 263.611i	− 1.16128i	0.814159	−	0.580641i	$-0.197198\pi$
$228$	0	0
$229$	78.5936	0.343204	0.171602	−	0.985166i	$-0.445106\pi$
$229$	78.5936	0.343204	0.171602	+	0.985166i	$0.445106\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	87.7854i	0.376762i	0.982096	+	0.188381i	$0.0603239\pi$
$233$	87.7854i	0.376762i	−0.982096	+	0.188381i	$0.939676\pi$
$234$	0	0
$235$	−6.57437	−0.0279761
$236$	0	0
$237$	− 54.4980i	− 0.229949i
$238$	0	0
$239$	55.7762i	0.233373i	0.993169	+	0.116687i	$0.0372273\pi$
$239$	55.7762i	0.233373i	−0.993169	+	0.116687i	$0.962773\pi$
$240$	0	0
$241$	180.789i	0.750162i	0.926992	+	0.375081i	$0.122385\pi$
$241$	180.789i	0.750162i	−0.926992	+	0.375081i	$0.877615\pi$
$242$	0	0
$243$	15.5885	0.0641500
$244$	0	0
$245$	11.5218	0.0470278
$246$	0	0
$247$	−57.8564	−0.234236
$248$	0	0
$249$	168.230i	0.675621i
$250$	0	0
$251$	22.1474	0.0882365	0.0441183	−	0.999026i	$-0.485952\pi$
$251$	22.1474	0.0882365	0.0441183	+	0.999026i	$0.485952\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	8.54965i	0.0335280i
$256$	0	0
$257$	−373.252	−1.45234	−0.726172	−	0.687513i	$-0.758702\pi$
$257$	−373.252	−1.45234	−0.726172	+	0.687513i	$0.758702\pi$
$258$	0	0
$259$	− 18.3304i	− 0.0707736i
$260$	0	0
$261$	− 25.5673i	− 0.0979592i
$262$	0	0
$263$	− 248.664i	− 0.945491i	−0.881199	−	0.472745i	$-0.843263\pi$
$263$	− 248.664i	− 0.945491i	0.881199	−	0.472745i	$-0.156737\pi$
$264$	0	0
$265$	−4.55514	−0.0171892
$266$	0	0
$267$	−194.205	−0.727360
$268$	0	0
$269$	454.664	1.69020	0.845100	−	0.534608i	$-0.179541\pi$
$269$	454.664	1.69020	0.845100	+	0.534608i	$0.179541\pi$
$270$	0	0
$271$	363.582i	1.34163i	0.741625	+	0.670815i	$0.234055\pi$
$271$	363.582i	1.34163i	−0.741625	+	0.670815i	$0.765945\pi$
$272$	0	0
$273$	−31.7654	−0.116357
$274$	0	0
$275$	0	0
$276$	0	0
$277$	397.486i	1.43497i	0.696576	+	0.717483i	$0.254706\pi$
$277$	397.486i	1.43497i	−0.696576	+	0.717483i	$0.745294\pi$
$278$	0	0
$279$	−25.6077	−0.0917839
$280$	0	0
$281$	227.193i	0.808517i	0.914645	+	0.404259i	$0.132470\pi$
$281$	227.193i	0.808517i	−0.914645	+	0.404259i	$0.867530\pi$
$282$	0	0
$283$	− 71.6643i	− 0.253231i	−0.991952	−	0.126615i	$-0.959589\pi$
$283$	− 71.6643i	− 0.253231i	0.991952	−	0.126615i	$-0.0404114\pi$
$284$	0	0
$285$	3.58630i	0.0125835i
$286$	0	0
$287$	−9.05771	−0.0315600
$288$	0	0
$289$	−50.3679	−0.174283
$290$	0	0
$291$	−172.061	−0.591277
$292$	0	0
$293$	− 400.401i	− 1.36656i	−0.730158	−	0.683278i	$-0.760554\pi$
$293$	− 400.401i	− 1.36656i	0.730158	−	0.683278i	$-0.239446\pi$
$294$	0	0
$295$	−3.16283	−0.0107215
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 176.073i	− 0.588874i
$300$	0	0
$301$	−10.3923	−0.0345259
$302$	0	0
$303$	− 215.508i	− 0.711247i
$304$	0	0
$305$	27.0731i	0.0887644i
$306$	0	0
$307$	80.5159i	0.262267i	0.991365	+	0.131133i	$0.0418616\pi$
$307$	80.5159i	0.262267i	−0.991365	+	0.131133i	$0.958138\pi$
$308$	0	0
$309$	171.306	0.554390
$310$	0	0
$311$	299.895	0.964292	0.482146	−	0.876091i	$-0.339857\pi$
$311$	299.895	0.964292	0.482146	+	0.876091i	$0.339857\pi$
$312$	0	0
$313$	−368.061	−1.17592	−0.587958	−	0.808892i	$-0.700068\pi$
$313$	−368.061	−1.17592	−0.587958	+	0.808892i	$0.700068\pi$
$314$	0	0
$315$	1.96902i	0.00625085i
$316$	0	0
$317$	383.990	1.21132	0.605662	−	0.795722i	$-0.292908\pi$
$317$	383.990	1.21132	0.605662	+	0.795722i	$0.292908\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	312.819i	0.974513i
$322$	0	0
$323$	−142.354	−0.440724
$324$	0	0
$325$	− 186.642i	− 0.574282i
$326$	0	0
$327$	3.86103i	0.0118074i
$328$	0	0
$329$	− 60.1004i	− 0.182676i
$330$	0	0
$331$	233.895	0.706631	0.353316	−	0.935504i	$-0.385054\pi$
$331$	233.895	0.706631	0.353316	+	0.935504i	$0.385054\pi$
$332$	0	0
$333$	−22.4500	−0.0674175
$334$	0	0
$335$	−17.4397	−0.0520588
$336$	0	0
$337$	15.9452i	0.0473153i	0.999720	+	0.0236576i	$0.00753116\pi$
$337$	15.9452i	0.0473153i	−0.999720	+	0.0236576i	$0.992469\pi$
$338$	0	0
$339$	−58.7654	−0.173349
$340$	0	0
$341$	0	0
$342$	0	0
$343$	225.353i	0.657006i
$344$	0	0
$345$	−10.9141	−0.0316351
$346$	0	0
$347$	− 219.845i	− 0.633559i	−0.948499	−	0.316779i	$-0.897399\pi$
$347$	− 219.845i	− 0.633559i	0.948499	−	0.316779i	$-0.102601\pi$
$348$	0	0
$349$	− 444.992i	− 1.27505i	−0.770430	−	0.637525i	$-0.779958\pi$
$349$	− 444.992i	− 1.27505i	0.770430	−	0.637525i	$-0.220042\pi$
$350$	0	0
$351$	38.9045i	0.110839i
$352$	0	0
$353$	−156.068	−0.442119	−0.221060	−	0.975260i	$-0.570952\pi$
$353$	−156.068	−0.442119	−0.221060	+	0.975260i	$0.570952\pi$
$354$	0	0
$355$	21.2398	0.0598304
$356$	0	0
$357$	−78.1577	−0.218929
$358$	0	0
$359$	− 661.999i	− 1.84401i	−0.387181	−	0.922004i	$-0.626551\pi$
$359$	− 661.999i	− 1.84401i	0.387181	−	0.922004i	$-0.373449\pi$
$360$	0	0
$361$	301.287	0.834591
$362$	0	0
$363$	0	0
$364$	0	0
$365$	20.1434i	0.0551875i
$366$	0	0
$367$	384.627	1.04803	0.524015	−	0.851709i	$-0.324434\pi$
$367$	384.627	1.04803	0.524015	+	0.851709i	$0.324434\pi$
$368$	0	0
$369$	11.0934i	0.0300634i
$370$	0	0
$371$	− 41.6413i	− 0.112241i
$372$	0	0
$373$	537.082i	1.43990i	0.694027	+	0.719949i	$0.255835\pi$
$373$	537.082i	1.43990i	−0.694027	+	0.719949i	$0.744165\pi$
$374$	0	0
$375$	−23.1718	−0.0617914
$376$	0	0
$377$	63.8090	0.169255
$378$	0	0
$379$	−313.611	−0.827471	−0.413735	−	0.910397i	$-0.635776\pi$
$379$	−313.611	−0.827471	−0.413735	+	0.910397i	$0.635776\pi$
$380$	0	0
$381$	112.805i	0.296077i
$382$	0	0
$383$	667.458	1.74271	0.871355	−	0.490654i	$-0.163242\pi$
$383$	667.458	1.74271	0.871355	+	0.490654i	$0.163242\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	12.7279i	0.0328887i
$388$	0	0
$389$	−494.878	−1.27218	−0.636090	−	0.771615i	$-0.719449\pi$
$389$	−494.878	−1.27218	−0.636090	+	0.771615i	$0.719449\pi$
$390$	0	0
$391$	− 433.223i	− 1.10799i
$392$	0	0
$393$	− 308.460i	− 0.784885i
$394$	0	0
$395$	− 8.43087i	− 0.0213440i
$396$	0	0
$397$	−12.1384	−0.0305754	−0.0152877	−	0.999883i	$-0.504866\pi$
$397$	−12.1384	−0.0305754	−0.0152877	+	0.999883i	$0.504866\pi$
$398$	0	0
$399$	−32.7846	−0.0821669
$400$	0	0
$401$	98.1281	0.244709	0.122354	−	0.992486i	$-0.460956\pi$
$401$	98.1281	0.244709	0.122354	+	0.992486i	$0.460956\pi$
$402$	0	0
$403$	− 63.9097i	− 0.158585i
$404$	0	0
$405$	2.41154	0.00595443
$406$	0	0
$407$	0	0
$408$	0	0
$409$	467.939i	1.14410i	0.820217	+	0.572052i	$0.193853\pi$
$409$	467.939i	1.14410i	−0.820217	+	0.572052i	$0.806147\pi$
$410$	0	0
$411$	372.746	0.906925
$412$	0	0
$413$	− 28.9134i	− 0.0700082i
$414$	0	0
$415$	26.0252i	0.0627114i
$416$	0	0
$417$	291.172i	0.698254i
$418$	0	0
$419$	428.922	1.02368	0.511840	−	0.859081i	$-0.328964\pi$
$419$	428.922	1.02368	0.511840	+	0.859081i	$0.328964\pi$
$420$	0	0
$421$	−252.818	−0.600518	−0.300259	−	0.953858i	$-0.597073\pi$
$421$	−252.818	−0.600518	−0.300259	+	0.953858i	$0.597073\pi$
$422$	0	0
$423$	−73.6077	−0.174013
$424$	0	0
$425$	− 459.226i	− 1.08053i
$426$	0	0
$427$	−247.492	−0.579607
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 430.300i	− 0.998376i	−0.866494	−	0.499188i	$-0.833632\pi$
$431$	− 430.300i	− 0.998376i	0.866494	−	0.499188i	$-0.166368\pi$
$432$	0	0
$433$	−440.142	−1.01649	−0.508247	−	0.861211i	$-0.669706\pi$
$433$	−440.142	−1.01649	−0.508247	+	0.861211i	$0.669706\pi$
$434$	0	0
$435$	− 3.95528i	− 0.00909260i
$436$	0	0
$437$	− 181.723i	− 0.415842i
$438$	0	0
$439$	− 478.678i	− 1.09038i	−0.838311	−	0.545192i	$-0.816457\pi$
$439$	− 478.678i	− 1.09038i	0.838311	−	0.545192i	$-0.183543\pi$
$440$	0	0
$441$	129.000	0.292517
$442$	0	0
$443$	779.229	1.75898	0.879492	−	0.475915i	$-0.157883\pi$
$443$	779.229	1.75898	0.879492	+	0.475915i	$0.157883\pi$
$444$	0	0
$445$	−30.0436	−0.0675138
$446$	0	0
$447$	488.157i	1.09207i
$448$	0	0
$449$	542.976	1.20930	0.604650	−	0.796491i	$-0.293313\pi$
$449$	542.976	1.20930	0.604650	+	0.796491i	$0.293313\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	307.135i	0.678001i
$454$	0	0
$455$	−4.91412	−0.0108003
$456$	0	0
$457$	− 342.737i	− 0.749972i	−0.927030	−	0.374986i	$-0.877647\pi$
$457$	− 342.737i	− 0.749972i	0.927030	−	0.374986i	$-0.122353\pi$
$458$	0	0
$459$	95.7232i	0.208547i
$460$	0	0
$461$	− 30.1544i	− 0.0654109i	−0.999465	−	0.0327054i	$-0.989588\pi$
$461$	− 30.1544i	− 0.0654109i	0.999465	−	0.0327054i	$-0.0104123\pi$
$462$	0	0
$463$	844.424	1.82381	0.911905	−	0.410401i	$-0.134611\pi$
$463$	844.424	1.82381	0.911905	+	0.410401i	$0.134611\pi$
$464$	0	0
$465$	−3.96152	−0.00851941
$466$	0	0
$467$	−236.799	−0.507064	−0.253532	−	0.967327i	$-0.581592\pi$
$467$	−236.799	−0.507064	−0.253532	+	0.967327i	$0.581592\pi$
$468$	0	0
$469$	− 159.427i	− 0.339930i
$470$	0	0
$471$	−372.449	−0.790761
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 192.630i	− 0.405538i
$476$	0	0
$477$	−51.0000	−0.106918
$478$	0	0
$479$	− 706.877i	− 1.47573i	−0.674946	−	0.737867i	$-0.735833\pi$
$479$	− 706.877i	− 1.47573i	0.674946	−	0.737867i	$-0.264167\pi$
$480$	0	0
$481$	− 56.0290i	− 0.116484i
$482$	0	0
$483$	− 99.7727i	− 0.206569i
$484$	0	0
$485$	−26.6180	−0.0548825
$486$	0	0
$487$	326.760	0.670966	0.335483	−	0.942046i	$-0.391101\pi$
$487$	326.760	0.670966	0.335483	+	0.942046i	$0.391101\pi$
$488$	0	0
$489$	−432.837	−0.885148
$490$	0	0
$491$	− 505.630i	− 1.02980i	−0.857251	−	0.514898i	$-0.827830\pi$
$491$	− 505.630i	− 1.02980i	0.857251	−	0.514898i	$-0.172170\pi$
$492$	0	0
$493$	157.000	0.318458
$494$	0	0
$495$	0	0
$496$	0	0
$497$	194.166i	0.390676i
$498$	0	0
$499$	56.6551	0.113537	0.0567686	−	0.998387i	$-0.481920\pi$
$499$	56.6551	0.113537	0.0567686	+	0.998387i	$0.481920\pi$
$500$	0	0
$501$	− 408.361i	− 0.815092i
$502$	0	0
$503$	− 458.118i	− 0.910772i	−0.890294	−	0.455386i	$-0.849501\pi$
$503$	− 458.118i	− 0.910772i	0.890294	−	0.455386i	$-0.150499\pi$
$504$	0	0
$505$	− 33.3392i	− 0.0660182i
$506$	0	0
$507$	195.622	0.385842
$508$	0	0
$509$	203.167	0.399149	0.199574	−	0.979883i	$-0.436044\pi$
$509$	203.167	0.399149	0.199574	+	0.979883i	$0.436044\pi$
$510$	0	0
$511$	−184.144	−0.360359
$512$	0	0
$513$	40.1528i	0.0782705i
$514$	0	0
$515$	26.5012	0.0514586
$516$	0	0
$517$	0	0
$518$	0	0
$519$	− 61.6832i	− 0.118850i
$520$	0	0
$521$	−349.395	−0.670623	−0.335312	−	0.942107i	$-0.608842\pi$
$521$	−349.395	−0.670623	−0.335312	+	0.942107i	$0.608842\pi$
$522$	0	0
$523$	− 792.351i	− 1.51501i	−0.652828	−	0.757506i	$-0.726418\pi$
$523$	− 792.351i	− 1.51501i	0.652828	−	0.757506i	$-0.273582\pi$
$524$	0	0
$525$	− 105.761i	− 0.201450i
$526$	0	0
$527$	− 157.248i	− 0.298383i
$528$	0	0
$529$	24.0333	0.0454316
$530$	0	0
$531$	−35.4115	−0.0666884
$532$	0	0
$533$	−27.6860	−0.0519438
$534$	0	0
$535$	48.3932i	0.0904546i
$536$	0	0
$537$	462.995	0.862188
$538$	0	0
$539$	0	0
$540$	0	0
$541$	349.588i	0.646189i	0.946367	+	0.323094i	$0.104723\pi$
$541$	349.588i	0.646189i	−0.946367	+	0.323094i	$0.895277\pi$
$542$	0	0
$543$	93.4974	0.172187
$544$	0	0
$545$	0.597304i	0.00109597i
$546$	0	0
$547$	− 228.296i	− 0.417359i	−0.977984	−	0.208680i	$-0.933083\pi$
$547$	− 228.296i	− 0.417359i	0.977984	−	0.208680i	$-0.0669166\pi$
$548$	0	0
$549$	303.115i	0.552122i
$550$	0	0
$551$	65.8564	0.119522
$552$	0	0
$553$	77.0718	0.139370
$554$	0	0
$555$	−3.47303	−0.00625771
$556$	0	0
$557$	− 441.044i	− 0.791821i	−0.918289	−	0.395910i	$-0.870429\pi$
$557$	− 441.044i	− 0.791821i	0.918289	−	0.395910i	$-0.129571\pi$
$558$	0	0
$559$	−31.7654	−0.0568254
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 373.880i	− 0.664085i	−0.943264	−	0.332043i	$-0.892262\pi$
$563$	− 373.880i	− 0.664085i	0.943264	−	0.332043i	$-0.107738\pi$
$564$	0	0
$565$	−9.09103	−0.0160903
$566$	0	0
$567$	22.0454i	0.0388808i
$568$	0	0
$569$	624.738i	1.09796i	0.835836	+	0.548979i	$0.184983\pi$
$569$	624.738i	1.09796i	−0.835836	+	0.548979i	$0.815017\pi$
$570$	0	0
$571$	504.909i	0.884253i	0.896953	+	0.442127i	$0.145776\pi$
$571$	504.909i	0.884253i	−0.896953	+	0.442127i	$0.854224\pi$
$572$	0	0
$573$	−26.7180	−0.0466282
$574$	0	0
$575$	586.228	1.01953
$576$	0	0
$577$	−783.942	−1.35865	−0.679326	−	0.733837i	$-0.737728\pi$
$577$	−783.942	−1.35865	−0.679326	+	0.733837i	$0.737728\pi$
$578$	0	0
$579$	41.0965i	0.0709784i
$580$	0	0
$581$	−237.913	−0.409488
$582$	0	0
$583$	0	0
$584$	0	0
$585$	6.01854i	0.0102881i
$586$	0	0
$587$	−65.8743	−0.112222	−0.0561110	−	0.998425i	$-0.517870\pi$
$587$	−65.8743	−0.112222	−0.0561110	+	0.998425i	$0.517870\pi$
$588$	0	0
$589$	− 65.9604i	− 0.111987i
$590$	0	0
$591$	− 98.4083i	− 0.166512i
$592$	0	0
$593$	− 441.252i	− 0.744101i	−0.928212	−	0.372050i	$-0.878655\pi$
$593$	− 441.252i	− 0.744101i	0.928212	−	0.372050i	$-0.121345\pi$
$594$	0	0
$595$	−12.0910	−0.0203211
$596$	0	0
$597$	−110.991	−0.185915
$598$	0	0
$599$	444.497	0.742066	0.371033	−	0.928620i	$-0.379004\pi$
$599$	444.497	0.742066	0.371033	+	0.928620i	$0.379004\pi$
$600$	0	0
$601$	465.995i	0.775366i	0.921793	+	0.387683i	$0.126724\pi$
$601$	465.995i	0.775366i	−0.921793	+	0.387683i	$0.873276\pi$
$602$	0	0
$603$	−195.258	−0.323810
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 729.957i	− 1.20257i	−0.799036	−	0.601283i	$-0.794657\pi$
$607$	− 729.957i	− 1.20257i	0.799036	−	0.601283i	$-0.205343\pi$
$608$	0	0
$609$	36.1577	0.0593722
$610$	0	0
$611$	− 183.704i	− 0.300662i
$612$	0	0
$613$	− 145.134i	− 0.236760i	−0.992968	−	0.118380i	$-0.962230\pi$
$613$	− 145.134i	− 0.236760i	0.992968	−	0.118380i	$-0.0377701\pi$
$614$	0	0
$615$	1.71615i	0.00279049i
$616$	0	0
$617$	−484.751	−0.785658	−0.392829	−	0.919611i	$-0.628504\pi$
$617$	−484.751	−0.785658	−0.392829	+	0.919611i	$0.628504\pi$
$618$	0	0
$619$	221.709	0.358173	0.179086	−	0.983833i	$-0.442686\pi$
$619$	221.709	0.358173	0.179086	+	0.983833i	$0.442686\pi$
$620$	0	0
$621$	−122.196	−0.196773
$622$	0	0
$623$	− 274.647i	− 0.440847i
$624$	0	0
$625$	619.620	0.991393
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 137.858i	− 0.219170i
$630$	0	0
$631$	840.894	1.33264	0.666318	−	0.745668i	$-0.267869\pi$
$631$	840.894	1.33264	0.666318	+	0.745668i	$0.267869\pi$
$632$	0	0
$633$	− 333.341i	− 0.526605i
$634$	0	0
$635$	17.4510i	0.0274819i
$636$	0	0
$637$	321.948i	0.505413i
$638$	0	0
$639$	237.804	0.372150
$640$	0	0
$641$	124.212	0.193778	0.0968889	−	0.995295i	$-0.469111\pi$
$641$	124.212	0.193778	0.0968889	+	0.995295i	$0.469111\pi$
$642$	0	0
$643$	−110.908	−0.172485	−0.0862423	−	0.996274i	$-0.527486\pi$
$643$	−110.908	−0.172485	−0.0862423	+	0.996274i	$0.527486\pi$
$644$	0	0
$645$	1.96902i	0.00305274i
$646$	0	0
$647$	38.8897	0.0601078	0.0300539	−	0.999548i	$-0.490432\pi$
$647$	38.8897	0.0601078	0.0300539	+	0.999548i	$0.490432\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	− 36.2147i	− 0.0556294i
$652$	0	0
$653$	50.4514	0.0772609	0.0386305	−	0.999254i	$-0.487700\pi$
$653$	50.4514	0.0772609	0.0386305	+	0.999254i	$0.487700\pi$
$654$	0	0
$655$	− 47.7189i	− 0.0728533i
$656$	0	0
$657$	225.529i	0.343271i
$658$	0	0
$659$	− 406.684i	− 0.617123i	−0.951204	−	0.308562i	$-0.900152\pi$
$659$	− 406.684i	− 0.617123i	0.951204	−	0.308562i	$-0.0998476\pi$
$660$	0	0
$661$	540.682	0.817976	0.408988	−	0.912540i	$-0.365882\pi$
$661$	540.682	0.817976	0.408988	+	0.912540i	$0.365882\pi$
$662$	0	0
$663$	−238.899	−0.360330
$664$	0	0
$665$	−5.07180	−0.00762676
$666$	0	0
$667$	200.419i	0.300479i
$668$	0	0
$669$	−26.0563	−0.0389482
$670$	0	0
$671$	0	0
$672$	0	0
$673$	680.797i	1.01159i	0.862655	+	0.505793i	$0.168800\pi$
$673$	680.797i	1.01159i	−0.862655	+	0.505793i	$0.831200\pi$
$674$	0	0
$675$	−129.531	−0.191897
$676$	0	0
$677$	828.271i	1.22344i	0.791073	+	0.611722i	$0.209523\pi$
$677$	828.271i	1.22344i	−0.791073	+	0.611722i	$0.790477\pi$
$678$	0	0
$679$	− 243.332i	− 0.358368i
$680$	0	0
$681$	− 456.588i	− 0.670467i
$682$	0	0
$683$	1174.40	1.71947	0.859734	−	0.510743i	$-0.170630\pi$
$683$	1174.40	1.71947	0.859734	+	0.510743i	$0.170630\pi$
$684$	0	0
$685$	57.6640	0.0841811
$686$	0	0
$687$	136.128	0.198149
$688$	0	0
$689$	− 127.282i	− 0.184734i
$690$	0	0
$691$	−586.540	−0.848827	−0.424414	−	0.905468i	$-0.639520\pi$
$691$	−586.540	−0.848827	−0.424414	+	0.905468i	$0.639520\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	45.0445i	0.0648122i
$696$	0	0
$697$	−68.1206	−0.0977340
$698$	0	0
$699$	152.049i	0.217523i
$700$	0	0
$701$	− 427.469i	− 0.609799i	−0.952385	−	0.304899i	$-0.901377\pi$
$701$	− 427.469i	− 0.609799i	0.952385	−	0.304899i	$-0.0986228\pi$
$702$	0	0
$703$	− 57.8268i	− 0.0822572i
$704$	0	0
$705$	−11.3872	−0.0161520
$706$	0	0
$707$	304.774	0.431081
$708$	0	0
$709$	388.851	0.548450	0.274225	−	0.961666i	$-0.411579\pi$
$709$	388.851	0.548450	0.274225	+	0.961666i	$0.411579\pi$
$710$	0	0
$711$	− 94.3933i	− 0.132761i
$712$	0	0
$713$	200.736	0.281537
$714$	0	0
$715$	0	0
$716$	0	0
$717$	96.6072i	0.134738i
$718$	0	0
$719$	−229.404	−0.319060	−0.159530	−	0.987193i	$-0.550998\pi$
$719$	−229.404	−0.319060	−0.159530	+	0.987193i	$0.550998\pi$
$720$	0	0
$721$	242.264i	0.336011i
$722$	0	0
$723$	313.136i	0.433106i
$724$	0	0
$725$	212.449i	0.293033i
$726$	0	0
$727$	−611.850	−0.841609	−0.420805	−	0.907151i	$-0.638252\pi$
$727$	−611.850	−0.841609	−0.420805	+	0.907151i	$0.638252\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	−78.1577	−0.106919
$732$	0	0
$733$	− 101.021i	− 0.137819i	−0.997623	−	0.0689093i	$-0.978048\pi$
$733$	− 101.021i	− 0.137819i	0.997623	−	0.0689093i	$-0.0219519\pi$
$734$	0	0
$735$	19.9564	0.0271515
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 245.881i	− 0.332721i	−0.986065	−	0.166360i	$-0.946798\pi$
$739$	− 245.881i	− 0.332721i	0.986065	−	0.166360i	$-0.0532015\pi$
$740$	0	0
$741$	−100.210	−0.135236
$742$	0	0
$743$	− 729.089i	− 0.981277i	−0.871363	−	0.490638i	$-0.836764\pi$
$743$	− 729.089i	− 0.981277i	0.871363	−	0.490638i	$-0.163236\pi$
$744$	0	0
$745$	75.5181i	0.101367i
$746$	0	0
$747$	291.382i	0.390070i
$748$	0	0
$749$	−442.392	−0.590644
$750$	0	0
$751$	−811.055	−1.07997	−0.539983	−	0.841676i	$-0.681569\pi$
$751$	−811.055	−1.07997	−0.539983	+	0.841676i	$0.681569\pi$
$752$	0	0
$753$	38.3604	0.0509434
$754$	0	0
$755$	47.5139i	0.0629323i
$756$	0	0
$757$	−3.34352	−0.00441680	−0.00220840	−	0.999998i	$-0.500703\pi$
$757$	−3.34352	−0.00441680	−0.00220840	+	0.999998i	$0.500703\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	737.559i	0.969196i	0.874737	+	0.484598i	$0.161034\pi$
$761$	737.559i	0.969196i	−0.874737	+	0.484598i	$0.838966\pi$
$762$	0	0
$763$	−5.46033	−0.00715639
$764$	0	0
$765$	14.8084i	0.0193574i
$766$	0	0
$767$	− 88.3774i	− 0.115225i
$768$	0	0
$769$	− 328.412i	− 0.427064i	−0.976936	−	0.213532i	$-0.931503\pi$
$769$	− 328.412i	− 0.427064i	0.976936	−	0.213532i	$-0.0684967\pi$
$770$	0	0
$771$	−646.492	−0.838511
$772$	0	0
$773$	−1291.64	−1.67095	−0.835474	−	0.549529i	$-0.814807\pi$
$773$	−1291.64	−1.67095	−0.835474	+	0.549529i	$0.814807\pi$
$774$	0	0
$775$	212.785	0.274561
$776$	0	0
$777$	− 31.7491i	− 0.0408612i
$778$	0	0
$779$	−28.5744	−0.0366808
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 44.2839i	− 0.0565567i
$784$	0	0
$785$	−57.6180	−0.0733987
$786$	0	0
$787$	1062.74i	1.35036i	0.737651	+	0.675182i	$0.235935\pi$
$787$	1062.74i	1.35036i	−0.737651	+	0.675182i	$0.764065\pi$
$788$	0	0
$789$	− 430.699i	− 0.545879i
$790$	0	0
$791$	− 83.1068i	− 0.105065i
$792$	0	0
$793$	−756.491	−0.953961
$794$	0	0
$795$	−7.88973	−0.00992419
$796$	0	0
$797$	695.836	0.873069	0.436534	−	0.899688i	$-0.356206\pi$
$797$	695.836	0.873069	0.436534	+	0.899688i	$0.356206\pi$
$798$	0	0
$799$	− 451.999i	− 0.565706i
$800$	0	0
$801$	−336.373	−0.419941
$802$	0	0
$803$	0	0
$804$	0	0
$805$	− 15.4349i	− 0.0191738i
$806$	0	0
$807$	787.501	0.975838
$808$	0	0
$809$	1222.28i	1.51086i	0.655230	+	0.755429i	$0.272572\pi$
$809$	1222.28i	1.51086i	−0.655230	+	0.755429i	$0.727428\pi$
$810$	0	0
$811$	1480.11i	1.82504i	0.409028	+	0.912522i	$0.365868\pi$
$811$	1480.11i	1.82504i	−0.409028	+	0.912522i	$0.634132\pi$
$812$	0	0
$813$	629.742i	0.774590i
$814$	0	0
$815$	−66.9601	−0.0821597
$816$	0	0
$817$	−32.7846	−0.0401280
$818$	0	0
$819$	−55.0192	−0.0671786
$820$	0	0
$821$	1100.83i	1.34084i	0.741984	+	0.670418i	$0.233885\pi$
$821$	1100.83i	1.34084i	−0.741984	+	0.670418i	$0.766115\pi$
$822$	0	0
$823$	−465.212	−0.565263	−0.282632	−	0.959229i	$-0.591207\pi$
$823$	−465.212	−0.565263	−0.282632	+	0.959229i	$0.591207\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 538.409i	− 0.651039i	−0.945535	−	0.325519i	$-0.894461\pi$
$827$	− 538.409i	− 0.651039i	0.945535	−	0.325519i	$-0.105539\pi$
$828$	0	0
$829$	−652.069	−0.786573	−0.393287	−	0.919416i	$-0.628662\pi$
$829$	−652.069	−0.786573	−0.393287	+	0.919416i	$0.628662\pi$
$830$	0	0
$831$	688.465i	0.828478i
$832$	0	0
$833$	792.143i	0.950953i
$834$	0	0
$835$	− 63.1737i	− 0.0756571i
$836$	0	0
$837$	−44.3538	−0.0529914
$838$	0	0
$839$	−212.308	−0.253049	−0.126524	−	0.991964i	$-0.540382\pi$
$839$	−212.308	−0.253049	−0.126524	+	0.991964i	$0.540382\pi$
$840$	0	0
$841$	768.368	0.913636
$842$	0	0
$843$	393.510i	0.466798i
$844$	0	0
$845$	30.2628	0.0358140
$846$	0	0
$847$	0	0
$848$	0	0
$849$	− 124.126i	− 0.146203i
$850$	0	0
$851$	175.983	0.206796
$852$	0	0
$853$	− 187.375i	− 0.219666i	−0.993950	−	0.109833i	$-0.964968\pi$
$853$	− 187.375i	− 0.219666i	0.993950	−	0.109833i	$-0.0350316\pi$
$854$	0	0
$855$	6.21166i	0.00726510i
$856$	0	0
$857$	609.486i	0.711186i	0.934641	+	0.355593i	$0.115721\pi$
$857$	609.486i	0.711186i	−0.934641	+	0.355593i	$0.884279\pi$
$858$	0	0
$859$	−392.193	−0.456570	−0.228285	−	0.973594i	$-0.573312\pi$
$859$	−392.193	−0.456570	−0.228285	+	0.973594i	$0.573312\pi$
$860$	0	0
$861$	−15.6884	−0.0182212
$862$	0	0
$863$	−783.934	−0.908383	−0.454191	−	0.890904i	$-0.650072\pi$
$863$	−783.934	−0.908383	−0.454191	+	0.890904i	$0.650072\pi$
$864$	0	0
$865$	− 9.54243i	− 0.0110317i
$866$	0	0
$867$	−87.2398	−0.100623
$868$	0	0
$869$	0	0
$870$	0	0
$871$	− 487.309i	− 0.559482i
$872$	0	0
$873$	−298.019	−0.341374
$874$	0	0
$875$	− 32.7698i	− 0.0374512i
$876$	0	0
$877$	− 815.132i	− 0.929455i	−0.885454	−	0.464727i	$-0.846152\pi$
$877$	− 815.132i	− 0.929455i	0.885454	−	0.464727i	$-0.153848\pi$
$878$	0	0
$879$	− 693.515i	− 0.788982i
$880$	0	0
$881$	−1458.83	−1.65587	−0.827937	−	0.560821i	$-0.810486\pi$
$881$	−1458.83	−1.65587	−0.827937	+	0.560821i	$0.810486\pi$
$882$	0	0
$883$	−274.633	−0.311023	−0.155512	−	0.987834i	$-0.549703\pi$
$883$	−274.633	−0.311023	−0.155512	+	0.987834i	$0.549703\pi$
$884$	0	0
$885$	−5.47818	−0.00619004
$886$	0	0
$887$	693.280i	0.781601i	0.920475	+	0.390800i	$0.127802\pi$
$887$	693.280i	0.781601i	−0.920475	+	0.390800i	$0.872198\pi$
$888$	0	0
$889$	−159.531	−0.179450
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 189.599i	− 0.212317i
$894$	0	0
$895$	71.6256	0.0800285
$896$	0	0
$897$	− 304.968i	− 0.339986i
$898$	0	0
$899$	72.7467i	0.0809196i
$900$	0	0
$901$	− 313.173i	− 0.347584i
$902$	0	0
$903$	−18.0000	−0.0199336
$904$	0	0
$905$	14.4641	0.0159824
$906$	0	0
$907$	606.024	0.668164	0.334082	−	0.942544i	$-0.391574\pi$
$907$	606.024	0.668164	0.334082	+	0.942544i	$0.391574\pi$
$908$	0	0
$909$	− 373.271i	− 0.410639i
$910$	0	0
$911$	−347.419	−0.381360	−0.190680	−	0.981652i	$-0.561069\pi$
$911$	−347.419	−0.381360	−0.190680	+	0.981652i	$0.561069\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	46.8920i	0.0512481i
$916$	0	0
$917$	436.228	0.475712
$918$	0	0
$919$	1207.58i	1.31402i	0.753883	+	0.657009i	$0.228179\pi$
$919$	1207.58i	1.31402i	−0.753883	+	0.657009i	$0.771821\pi$
$920$	0	0
$921$	139.458i	0.151420i
$922$	0	0
$923$	593.493i	0.643004i
$924$	0	0
$925$	186.546	0.201672
$926$	0	0
$927$	296.711	0.320077
$928$	0	0
$929$	1189.26	1.28015	0.640073	−	0.768314i	$-0.278904\pi$
$929$	1189.26	1.28015	0.640073	+	0.768314i	$0.278904\pi$
$930$	0	0
$931$	332.278i	0.356905i
$932$	0	0
$933$	519.433	0.556734
$934$	0	0
$935$	0	0
$936$	0	0
$937$	844.259i	0.901023i	0.892771	+	0.450512i	$0.148758\pi$
$937$	844.259i	0.901023i	−0.892771	+	0.450512i	$0.851242\pi$
$938$	0	0
$939$	−637.501	−0.678915
$940$	0	0
$941$	1739.58i	1.84865i	0.381609	+	0.924324i	$0.375370\pi$
$941$	1739.58i	1.84865i	−0.381609	+	0.924324i	$0.624630\pi$
$942$	0	0
$943$	− 86.9598i	− 0.0922161i
$944$	0	0
$945$	3.41044i	0.00360893i
$946$	0	0
$947$	227.124	0.239836	0.119918	−	0.992784i	$-0.461737\pi$
$947$	227.124	0.239836	0.119918	+	0.992784i	$0.461737\pi$
$948$	0	0
$949$	−562.858	−0.593106
$950$	0	0
$951$	665.090	0.699358
$952$	0	0
$953$	1717.25i	1.80194i	0.433885	+	0.900968i	$0.357142\pi$
$953$	1717.25i	1.80194i	−0.433885	+	0.900968i	$0.642858\pi$
$954$	0	0
$955$	−4.13328	−0.00432805
$956$	0	0
$957$	0	0
$958$	0	0
$959$	527.143i	0.549679i
$960$	0	0
$961$	−888.138	−0.924182
$962$	0	0
$963$	541.818i	0.562635i
$964$	0	0
$965$	6.35765i	0.00658824i
$966$	0	0
$967$	− 555.142i	− 0.574087i	−0.957918	−	0.287044i	$-0.907328\pi$
$967$	− 555.142i	− 0.574087i	0.957918	−	0.287044i	$-0.0926724\pi$
$968$	0	0
$969$	−246.564	−0.254452
$970$	0	0
$971$	−411.317	−0.423601	−0.211801	−	0.977313i	$-0.567933\pi$
$971$	−411.317	−0.423601	−0.211801	+	0.977313i	$0.567933\pi$
$972$	0	0
$973$	−411.779	−0.423206
$974$	0	0
$975$	− 323.273i	− 0.331562i
$976$	0	0
$977$	−1519.62	−1.55539	−0.777697	−	0.628639i	$-0.783612\pi$
$977$	−1519.62	−1.55539	−0.777697	+	0.628639i	$0.783612\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	6.68751i	0.00681703i
$982$	0	0
$983$	445.577	0.453283	0.226641	−	0.973978i	$-0.427225\pi$
$983$	445.577	0.453283	0.226641	+	0.973978i	$0.427225\pi$
$984$	0	0
$985$	− 15.2238i	− 0.0154557i
$986$	0	0
$987$	− 104.097i	− 0.105468i
$988$	0	0
$989$	− 99.7727i	− 0.100882i
$990$	0	0
$991$	−1044.25	−1.05373	−0.526866	−	0.849948i	$-0.676633\pi$
$991$	−1044.25	−1.05373	−0.526866	+	0.849948i	$0.676633\pi$
$992$	0	0
$993$	405.118	0.407974
$994$	0	0
$995$	−17.1704	−0.0172567
$996$	0	0
$997$	− 170.660i	− 0.171174i	−0.996331	−	0.0855868i	$-0.972724\pi$
$997$	− 170.660i	− 0.171174i	0.996331	−	0.0855868i	$-0.0272765\pi$
$998$	0	0
$999$	−38.8846	−0.0389235

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1452.3.f.a.241.4	yes	4
3.2	odd	2		4356.3.f.d.1693.4		4
11.10	odd	2	inner	1452.3.f.a.241.3	✓	4
33.32	even	2		4356.3.f.d.1693.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1452.3.f.a.241.3	✓	4	11.10	odd	2	inner
1452.3.f.a.241.4	yes	4	1.1	even	1	trivial
4356.3.f.d.1693.3		4	33.32	even	2
4356.3.f.d.1693.4		4	3.2	odd	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:	\( N \)	\(=\)	\( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1452.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +0.267949 q^{5} +2.44949i q^{7} +3.00000 q^{9} +7.48717i q^{13} +0.464102 q^{15} +18.4219i q^{17} +7.72741i q^{19} +4.24264i q^{21} -23.5167 q^{23} -24.9282 q^{25} +5.19615 q^{27} -8.52245i q^{29} -8.53590 q^{31} +0.656339i q^{35} -7.48334 q^{37} +12.9682i q^{39} +3.69780i q^{41} +4.24264i q^{43} +0.803848 q^{45} -24.5359 q^{47} +43.0000 q^{49} +31.9077i q^{51} -17.0000 q^{53} +13.3843i q^{57} -11.8038 q^{59} +101.038i q^{61} +7.34847i q^{63} +2.00618i q^{65} -65.0859 q^{67} -40.7321 q^{69} +79.2679 q^{71} +75.1763i q^{73} -43.1769 q^{75} -31.4644i q^{79} +9.00000 q^{81} +97.1275i q^{83} +4.93614i q^{85} -14.7613i q^{87} -112.124 q^{89} -18.3397 q^{91} -14.7846 q^{93} +2.07055i q^{95} -99.3397 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5} + 12 q^{9} - 12 q^{15} - 4 q^{23} - 72 q^{25} - 48 q^{31} - 120 q^{37} + 24 q^{45} - 112 q^{47} + 172 q^{49} - 68 q^{53} - 68 q^{59} - 4 q^{67} - 156 q^{69} + 324 q^{71} - 48 q^{75} + 36 q^{81}+ \cdots - 432 q^{97}+O(q^{100}) \) 4 * q + 8 * q^5 + 12 * q^9 - 12 * q^15 - 4 * q^23 - 72 * q^25 - 48 * q^31 - 120 * q^37 + 24 * q^45 - 112 * q^47 + 172 * q^49 - 68 * q^53 - 68 * q^59 - 4 * q^67 - 156 * q^69 + 324 * q^71 - 48 * q^75 + 36 * q^81 - 400 * q^89 - 108 * q^91 + 24 * q^93 - 432 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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