LMFDB - Embedded newform 1150.2.b.j.599.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1150,2,Mod(599,1150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1150.599"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1150.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,-6,0,2,0,0,-20,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$9.18279623245$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.77580864.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 19x^{4} + 105x^{2} + 144 $ x^6 + 19x^4 + 105x^2 + 144
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.4
Root			$-2.68740i$ of defining polynomial
Character	$\chi$	$=$	1150.599
Dual form			1150.2.b.j.599.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -2.68740i q^{3} -1.00000 q^{4} +2.68740 q^{6} -4.59692i q^{7} -1.00000i q^{8} -4.22212 q^{9} +5.13163 q^{11} +2.68740i q^{12} +1.22212i q^{13} +4.59692 q^{14} +1.00000 q^{16} -4.68740i q^{17} -4.22212i q^{18} +4.59692 q^{19} -12.3537 q^{21} +5.13163i q^{22} +1.00000i q^{23} -2.68740 q^{24} -1.22212 q^{26} +3.28432i q^{27} +4.59692i q^{28} -3.37480 q^{29} -0.777884 q^{31} +1.00000i q^{32} -13.7907i q^{33} +4.68740 q^{34} +4.22212 q^{36} +5.81903i q^{37} +4.59692i q^{38} +3.28432 q^{39} -8.50643 q^{41} -12.3537i q^{42} -8.00000i q^{43} -5.13163 q^{44} -1.00000 q^{46} -6.44423i q^{47} -2.68740i q^{48} -14.1316 q^{49} -12.5969 q^{51} -1.22212i q^{52} +6.00000i q^{53} -3.28432 q^{54} -4.59692 q^{56} -12.3537i q^{57} -3.37480i q^{58} -9.37480 q^{59} +10.9507 q^{61} -0.777884i q^{62} +19.4087i q^{63} -1.00000 q^{64} +13.7907 q^{66} +15.6381i q^{67} +4.68740i q^{68} +2.68740 q^{69} +1.31260 q^{71} +4.22212i q^{72} +4.44423i q^{73} -5.81903 q^{74} -4.59692 q^{76} -23.5897i q^{77} +3.28432i q^{78} +4.88847 q^{79} -3.84008 q^{81} -8.50643i q^{82} +3.81903i q^{83} +12.3537 q^{84} +8.00000 q^{86} +9.06943i q^{87} -5.13163i q^{88} -8.93057 q^{89} +5.61797 q^{91} -1.00000i q^{92} +2.09048i q^{93} +6.44423 q^{94} +2.68740 q^{96} -18.0622i q^{97} -14.1316i q^{98} -21.6663 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} + 2 q^{6} - 20 q^{9} + 6 q^{11} - 6 q^{14} + 6 q^{16} - 6 q^{19} - 44 q^{21} - 2 q^{24} - 2 q^{26} + 8 q^{29} - 10 q^{31} + 14 q^{34} + 20 q^{36} - 28 q^{39} + 2 q^{41} - 6 q^{44} - 6 q^{46}+ \cdots - 114 q^{99}+O(q^{100}) $ 6 * q - 6 * q^4 + 2 * q^6 - 20 * q^9 + 6 * q^11 - 6 * q^14 + 6 * q^16 - 6 * q^19 - 44 * q^21 - 2 * q^24 - 2 * q^26 + 8 * q^29 - 10 * q^31 + 14 * q^34 + 20 * q^36 - 28 * q^39 + 2 * q^41 - 6 * q^44 - 6 * q^46 - 60 * q^49 - 42 * q^51 + 28 * q^54 + 6 * q^56 - 28 * q^59 + 2 * q^61 - 6 * q^64 - 18 * q^66 + 2 * q^69 + 22 * q^71 + 4 * q^74 + 6 * q^76 + 8 * q^79 + 14 * q^81 + 44 * q^84 + 48 * q^86 - 36 * q^89 + 2 * q^91 + 28 * q^94 + 2 * q^96 - 114 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	− 2.68740i	− 1.55157i	−0.630997	−	0.775785i	$-0.717354\pi$
$3$	− 2.68740i	− 1.55157i	0.630997	−	0.775785i	$-0.282646\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	2.68740	1.09713
$7$	− 4.59692i	− 1.73747i	−0.495277	−	0.868735i	$-0.664933\pi$
$7$	− 4.59692i	− 1.73747i	0.495277	−	0.868735i	$-0.335067\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−4.22212	−1.40737
$10$	0	0
$11$	5.13163	1.54725	0.773623	−	0.633647i	$-0.218443\pi$
$11$	5.13163	1.54725	0.773623	+	0.633647i	$0.218443\pi$
$12$	2.68740i	0.775785i
$13$	1.22212i	0.338954i	0.985534	+	0.169477i	$0.0542079\pi$
$13$	1.22212i	0.338954i	−0.985534	+	0.169477i	$0.945792\pi$
$14$	4.59692	1.22858
$15$	0	0
$16$	1.00000	0.250000
$17$	− 4.68740i	− 1.13686i	−0.822731	−	0.568431i	$-0.807551\pi$
$17$	− 4.68740i	− 1.13686i	0.822731	−	0.568431i	$-0.192449\pi$
$18$	− 4.22212i	− 0.995162i
$19$	4.59692	1.05460	0.527302	−	0.849678i	$-0.323204\pi$
$19$	4.59692	1.05460	0.527302	+	0.849678i	$0.323204\pi$
$20$	0	0
$21$	−12.3537	−2.69581
$22$	5.13163i	1.09407i
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	−2.68740	−0.548563
$25$	0	0
$26$	−1.22212	−0.239677
$27$	3.28432i	0.632067i
$28$	4.59692i	0.868735i
$29$	−3.37480	−0.626684	−0.313342	−	0.949640i	$-0.601449\pi$
$29$	−3.37480	−0.626684	−0.313342	+	0.949640i	$0.601449\pi$
$30$	0	0
$31$	−0.777884	−0.139712	−0.0698560	−	0.997557i	$-0.522254\pi$
$31$	−0.777884	−0.139712	−0.0698560	+	0.997557i	$0.522254\pi$
$32$	1.00000i	0.176777i
$33$	− 13.7907i	− 2.40066i
$34$	4.68740	0.803882
$35$	0	0
$36$	4.22212	0.703686
$37$	5.81903i	0.956643i	0.878185	+	0.478321i	$0.158755\pi$
$37$	5.81903i	0.956643i	−0.878185	+	0.478321i	$0.841245\pi$
$38$	4.59692i	0.745718i
$39$	3.28432	0.525911
$40$	0	0
$41$	−8.50643	−1.32848	−0.664241	−	0.747519i	$-0.731245\pi$
$41$	−8.50643	−1.32848	−0.664241	+	0.747519i	$0.731245\pi$
$42$	− 12.3537i	− 1.90622i
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	−5.13163	−0.773623
$45$	0	0
$46$	−1.00000	−0.147442
$47$	− 6.44423i	− 0.939988i	−0.882670	−	0.469994i	$-0.844256\pi$
$47$	− 6.44423i	− 0.939988i	0.882670	−	0.469994i	$-0.155744\pi$
$48$	− 2.68740i	− 0.387893i
$49$	−14.1316	−2.01880
$50$	0	0
$51$	−12.5969	−1.76392
$52$	− 1.22212i	− 0.169477i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	−3.28432	−0.446939
$55$	0	0
$56$	−4.59692	−0.614289
$57$	− 12.3537i	− 1.63629i
$58$	− 3.37480i	− 0.443133i
$59$	−9.37480	−1.22049	−0.610247	−	0.792211i	$-0.708930\pi$
$59$	−9.37480	−1.22049	−0.610247	+	0.792211i	$0.708930\pi$
$60$	0	0
$61$	10.9507	1.40209	0.701044	−	0.713118i	$-0.252717\pi$
$61$	10.9507	1.40209	0.701044	+	0.713118i	$0.252717\pi$
$62$	− 0.777884i	− 0.0987913i
$63$	19.4087i	2.44527i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	13.7907	1.69752
$67$	15.6381i	1.91049i	0.295810	+	0.955247i	$0.404410\pi$
$67$	15.6381i	1.91049i	−0.295810	+	0.955247i	$0.595590\pi$
$68$	4.68740i	0.568431i
$69$	2.68740	0.323525
$70$	0	0
$71$	1.31260	0.155777	0.0778885	−	0.996962i	$-0.475182\pi$
$71$	1.31260	0.155777	0.0778885	+	0.996962i	$0.475182\pi$
$72$	4.22212i	0.497581i
$73$	4.44423i	0.520158i	0.965587	+	0.260079i	$0.0837486\pi$
$73$	4.44423i	0.520158i	−0.965587	+	0.260079i	$0.916251\pi$
$74$	−5.81903	−0.676449
$75$	0	0
$76$	−4.59692	−0.527302
$77$	− 23.5897i	− 2.68829i
$78$	3.28432i	0.371875i
$79$	4.88847	0.549995	0.274998	−	0.961445i	$-0.411323\pi$
$79$	4.88847	0.549995	0.274998	+	0.961445i	$0.411323\pi$
$80$	0	0
$81$	−3.84008	−0.426676
$82$	− 8.50643i	− 0.939378i
$83$	3.81903i	0.419193i	0.977788	+	0.209597i	$0.0672151\pi$
$83$	3.81903i	0.419193i	−0.977788	+	0.209597i	$0.932785\pi$
$84$	12.3537	1.34790
$85$	0	0
$86$	8.00000	0.862662
$87$	9.06943i	0.972345i
$88$	− 5.13163i	− 0.547034i
$89$	−8.93057	−0.946638	−0.473319	−	0.880891i	$-0.656944\pi$
$89$	−8.93057	−0.946638	−0.473319	+	0.880891i	$0.656944\pi$
$90$	0	0
$91$	5.61797	0.588923
$92$	− 1.00000i	− 0.104257i
$93$	2.09048i	0.216773i
$94$	6.44423	0.664672
$95$	0	0
$96$	2.68740	0.274282
$97$	− 18.0622i	− 1.83394i	−0.398958	−	0.916969i	$-0.630628\pi$
$97$	− 18.0622i	− 1.83394i	0.398958	−	0.916969i	$-0.369372\pi$
$98$	− 14.1316i	− 1.42751i
$99$	−21.6663	−2.17755
$100$	0	0
$101$	3.37480	0.335805	0.167903	−	0.985804i	$-0.446301\pi$
$101$	3.37480	0.335805	0.167903	+	0.985804i	$0.446301\pi$
$102$	− 12.5969i	− 1.24728i
$103$	− 13.1316i	− 1.29390i	−0.762533	−	0.646949i	$-0.776045\pi$
$103$	− 13.1316i	− 1.29390i	0.762533	−	0.646949i	$-0.223955\pi$
$104$	1.22212	0.119838
$105$	0	0
$106$	−6.00000	−0.582772
$107$	12.3054i	1.18960i	0.803872	+	0.594802i	$0.202770\pi$
$107$	12.3054i	1.18960i	−0.803872	+	0.594802i	$0.797230\pi$
$108$	− 3.28432i	− 0.316033i
$109$	10.5969	1.01500	0.507500	−	0.861652i	$-0.330570\pi$
$109$	10.5969	1.01500	0.507500	+	0.861652i	$0.330570\pi$
$110$	0	0
$111$	15.6381	1.48430
$112$	− 4.59692i	− 0.434368i
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	12.3537	1.15703
$115$	0	0
$116$	3.37480	0.313342
$117$	− 5.15992i	− 0.477035i
$118$	− 9.37480i	− 0.863020i
$119$	−21.5476	−1.97526
$120$	0	0
$121$	15.3337	1.39397
$122$	10.9507i	0.991427i
$123$	22.8602i	2.06123i
$124$	0.777884	0.0698560
$125$	0	0
$126$	−19.4087	−1.72907
$127$	− 2.26326i	− 0.200832i	−0.994946	−	0.100416i	$-0.967983\pi$
$127$	− 2.26326i	− 0.200832i	0.994946	−	0.100416i	$-0.0320174\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−21.4992	−1.89290
$130$	0	0
$131$	2.93057	0.256045	0.128022	−	0.991771i	$-0.459137\pi$
$131$	2.93057	0.256045	0.128022	+	0.991771i	$0.459137\pi$
$132$	13.7907i	1.20033i
$133$	− 21.1316i	− 1.83234i
$134$	−15.6381	−1.35092
$135$	0	0
$136$	−4.68740	−0.401941
$137$	− 19.7907i	− 1.69084i	−0.534104	−	0.845419i	$-0.679351\pi$
$137$	− 19.7907i	− 1.69084i	0.534104	−	0.845419i	$-0.320649\pi$
$138$	2.68740i	0.228767i
$139$	−6.26326	−0.531243	−0.265622	−	0.964077i	$-0.585577\pi$
$139$	−6.26326	−0.531243	−0.265622	+	0.964077i	$0.585577\pi$
$140$	0	0
$141$	−17.3182	−1.45846
$142$	1.31260i	0.110151i
$143$	6.27145i	0.524445i
$144$	−4.22212	−0.351843
$145$	0	0
$146$	−4.44423	−0.367807
$147$	37.9773i	3.13232i
$148$	− 5.81903i	− 0.478321i
$149$	19.7907	1.62132	0.810661	−	0.585516i	$-0.199108\pi$
$149$	19.7907	1.62132	0.810661	+	0.585516i	$0.199108\pi$
$150$	0	0
$151$	4.06220	0.330577	0.165289	−	0.986245i	$-0.447144\pi$
$151$	4.06220	0.330577	0.165289	+	0.986245i	$0.447144\pi$
$152$	− 4.59692i	− 0.372859i
$153$	19.7907i	1.59999i
$154$	23.5897	1.90091
$155$	0	0
$156$	−3.28432	−0.262956
$157$	4.62520i	0.369131i	0.982820	+	0.184566i	$0.0590878\pi$
$157$	4.62520i	0.369131i	−0.982820	+	0.184566i	$0.940912\pi$
$158$	4.88847i	0.388905i
$159$	16.1244	1.27875
$160$	0	0
$161$	4.59692	0.362288
$162$	− 3.84008i	− 0.301705i
$163$	− 12.4159i	− 0.972492i	−0.873822	−	0.486246i	$-0.838366\pi$
$163$	− 12.4159i	− 0.972492i	0.873822	−	0.486246i	$-0.161634\pi$
$164$	8.50643	0.664241
$165$	0	0
$166$	−3.81903	−0.296414
$167$	12.8885i	0.997339i	0.866792	+	0.498670i	$0.166178\pi$
$167$	12.8885i	0.997339i	−0.866792	+	0.498670i	$0.833822\pi$
$168$	12.3537i	0.953112i
$169$	11.5064	0.885110
$170$	0	0
$171$	−19.4087	−1.48422
$172$	8.00000i	0.609994i
$173$	10.2432i	0.778774i	0.921074	+	0.389387i	$0.127313\pi$
$173$	10.2432i	0.778774i	−0.921074	+	0.389387i	$0.872687\pi$
$174$	−9.06943	−0.687552
$175$	0	0
$176$	5.13163	0.386811
$177$	25.1938i	1.89368i
$178$	− 8.93057i	− 0.669374i
$179$	13.1938	0.986153	0.493077	−	0.869986i	$-0.335872\pi$
$179$	13.1938	0.986153	0.493077	+	0.869986i	$0.335872\pi$
$180$	0	0
$181$	15.7907	1.17372	0.586858	−	0.809690i	$-0.300364\pi$
$181$	15.7907	1.17372	0.586858	+	0.809690i	$0.300364\pi$
$182$	5.61797i	0.416431i
$183$	− 29.4288i	− 2.17544i
$184$	1.00000	0.0737210
$185$	0	0
$186$	−2.09048	−0.153282
$187$	− 24.0540i	− 1.75900i
$188$	6.44423i	0.469994i
$189$	15.0977	1.09820
$190$	0	0
$191$	16.1244	1.16672	0.583360	−	0.812214i	$-0.301738\pi$
$191$	16.1244	1.16672	0.583360	+	0.812214i	$0.301738\pi$
$192$	2.68740i	0.193946i
$193$	17.9434i	1.29160i	0.763508	+	0.645798i	$0.223475\pi$
$193$	17.9434i	1.29160i	−0.763508	+	0.645798i	$0.776525\pi$
$194$	18.0622	1.29679
$195$	0	0
$196$	14.1316	1.00940
$197$	5.88123i	0.419020i	0.977806	+	0.209510i	$0.0671869\pi$
$197$	5.88123i	0.419020i	−0.977806	+	0.209510i	$0.932813\pi$
$198$	− 21.6663i	− 1.53976i
$199$	−6.56863	−0.465638	−0.232819	−	0.972520i	$-0.574795\pi$
$199$	−6.56863	−0.465638	−0.232819	+	0.972520i	$0.574795\pi$
$200$	0	0
$201$	42.0257	2.96427
$202$	3.37480i	0.237450i
$203$	15.5137i	1.08885i
$204$	12.5969	0.881960
$205$	0	0
$206$	13.1316	0.914924
$207$	− 4.22212i	− 0.293457i
$208$	1.22212i	0.0847385i
$209$	23.5897	1.63173
$210$	0	0
$211$	−15.4571	−1.06411	−0.532055	−	0.846710i	$-0.678580\pi$
$211$	−15.4571	−1.06411	−0.532055	+	0.846710i	$0.678580\pi$
$212$	− 6.00000i	− 0.412082i
$213$	− 3.52748i	− 0.241699i
$214$	−12.3054	−0.841177
$215$	0	0
$216$	3.28432	0.223469
$217$	3.57587i	0.242746i
$218$	10.5969i	0.717714i
$219$	11.9434	0.807062
$220$	0	0
$221$	5.72855	0.385344
$222$	15.6381i	1.04956i
$223$	10.7496i	0.719846i	0.932982	+	0.359923i	$0.117197\pi$
$223$	10.7496i	0.719846i	−0.932982	+	0.359923i	$0.882803\pi$
$224$	4.59692	0.307144
$225$	0	0
$226$	−6.00000	−0.399114
$227$	− 12.3054i	− 0.816736i	−0.912817	−	0.408368i	$-0.866098\pi$
$227$	− 12.3054i	− 0.816736i	0.912817	−	0.408368i	$-0.133902\pi$
$228$	12.3537i	0.818147i
$229$	−9.63806	−0.636901	−0.318451	−	0.947939i	$-0.603162\pi$
$229$	−9.63806	−0.636901	−0.318451	+	0.947939i	$0.603162\pi$
$230$	0	0
$231$	−63.3949	−4.17108
$232$	3.37480i	0.221566i
$233$	− 13.9434i	− 0.913464i	−0.889604	−	0.456732i	$-0.849020\pi$
$233$	− 13.9434i	− 0.913464i	0.889604	−	0.456732i	$-0.150980\pi$
$234$	5.15992	0.337314
$235$	0	0
$236$	9.37480	0.610247
$237$	− 13.1373i	− 0.853357i
$238$	− 21.5476i	− 1.39672i
$239$	26.3877	1.70688	0.853438	−	0.521194i	$-0.174513\pi$
$239$	26.3877	1.70688	0.853438	+	0.521194i	$0.174513\pi$
$240$	0	0
$241$	−7.06943	−0.455382	−0.227691	−	0.973733i	$-0.573118\pi$
$241$	−7.06943	−0.455382	−0.227691	+	0.973733i	$0.573118\pi$
$242$	15.3337i	0.985684i
$243$	20.1728i	1.29408i
$244$	−10.9507	−0.701044
$245$	0	0
$246$	−22.8602	−1.45751
$247$	5.61797i	0.357463i
$248$	0.777884i	0.0493957i
$249$	10.2633	0.650408
$250$	0	0
$251$	−24.9023	−1.57182	−0.785909	−	0.618342i	$-0.787805\pi$
$251$	−24.9023	−1.57182	−0.785909	+	0.618342i	$0.787805\pi$
$252$	− 19.4087i	− 1.22263i
$253$	5.13163i	0.322623i
$254$	2.26326	0.142010
$255$	0	0
$256$	1.00000	0.0625000
$257$	0.444233i	0.0277105i	0.999904	+	0.0138552i	$0.00441040\pi$
$257$	0.444233i	0.0277105i	−0.999904	+	0.0138552i	$0.995590\pi$
$258$	− 21.4992i	− 1.33848i
$259$	26.7496	1.66214
$260$	0	0
$261$	14.2488	0.881978
$262$	2.93057i	0.181051i
$263$	23.8812i	1.47258i	0.676666	+	0.736290i	$0.263424\pi$
$263$	23.8812i	1.47258i	−0.676666	+	0.736290i	$0.736576\pi$
$264$	−13.7907	−0.848762
$265$	0	0
$266$	21.1316	1.29566
$267$	24.0000i	1.46878i
$268$	− 15.6381i	− 0.955247i
$269$	−16.2633	−0.991589	−0.495794	−	0.868440i	$-0.665123\pi$
$269$	−16.2633	−0.991589	−0.495794	+	0.868440i	$0.665123\pi$
$270$	0	0
$271$	25.6098	1.55568	0.777842	−	0.628460i	$-0.216315\pi$
$271$	25.6098	1.55568	0.777842	+	0.628460i	$0.216315\pi$
$272$	− 4.68740i	− 0.284215i
$273$	− 15.0977i	− 0.913756i
$274$	19.7907	1.19560
$275$	0	0
$276$	−2.68740	−0.161762
$277$	2.88847i	0.173551i	0.996228	+	0.0867755i	$0.0276563\pi$
$277$	2.88847i	0.173551i	−0.996228	+	0.0867755i	$0.972344\pi$
$278$	− 6.26326i	− 0.375646i
$279$	3.28432	0.196627
$280$	0	0
$281$	4.26326	0.254325	0.127163	−	0.991882i	$-0.459413\pi$
$281$	4.26326	0.254325	0.127163	+	0.991882i	$0.459413\pi$
$282$	− 17.3182i	− 1.03129i
$283$	− 30.5686i	− 1.81712i	−0.417758	−	0.908558i	$-0.637184\pi$
$283$	− 30.5686i	− 1.81712i	0.417758	−	0.908558i	$-0.362816\pi$
$284$	−1.31260	−0.0778885
$285$	0	0
$286$	−6.27145	−0.370839
$287$	39.1033i	2.30820i
$288$	− 4.22212i	− 0.248791i
$289$	−4.97171	−0.292454
$290$	0	0
$291$	−48.5403	−2.84549
$292$	− 4.44423i	− 0.260079i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	−37.9773	−2.21488
$295$	0	0
$296$	5.81903	0.338224
$297$	16.8539i	0.977962i
$298$	19.7907i	1.14645i
$299$	−1.22212	−0.0706768
$300$	0	0
$301$	−36.7753	−2.11969
$302$	4.06220i	0.233753i
$303$	− 9.06943i	− 0.521025i
$304$	4.59692	0.263651
$305$	0	0
$306$	−19.7907	−1.13136
$307$	− 8.54853i	− 0.487891i	−0.969789	−	0.243945i	$-0.921558\pi$
$307$	− 8.54853i	− 0.487891i	0.969789	−	0.243945i	$-0.0784417\pi$
$308$	23.5897i	1.34415i
$309$	−35.2899	−2.00757
$310$	0	0
$311$	−7.63806	−0.433115	−0.216557	−	0.976270i	$-0.569483\pi$
$311$	−7.63806	−0.433115	−0.216557	+	0.976270i	$0.569483\pi$
$312$	− 3.28432i	− 0.185938i
$313$	18.2350i	1.03070i	0.856979	+	0.515351i	$0.172338\pi$
$313$	18.2350i	1.03070i	−0.856979	+	0.515351i	$0.827662\pi$
$314$	−4.62520	−0.261015
$315$	0	0
$316$	−4.88847	−0.274998
$317$	− 16.8602i	− 0.946962i	−0.880804	−	0.473481i	$-0.842997\pi$
$317$	− 16.8602i	− 0.946962i	0.880804	−	0.473481i	$-0.157003\pi$
$318$	16.1244i	0.904211i
$319$	−17.3182	−0.969635
$320$	0	0
$321$	33.0694	1.84576
$322$	4.59692i	0.256176i
$323$	− 21.5476i	− 1.19894i
$324$	3.84008	0.213338
$325$	0	0
$326$	12.4159	0.687656
$327$	− 28.4781i	− 1.57485i
$328$	8.50643i	0.469689i
$329$	−29.6236	−1.63320
$330$	0	0
$331$	19.1517	1.05267	0.526337	−	0.850276i	$-0.323565\pi$
$331$	19.1517	1.05267	0.526337	+	0.850276i	$0.323565\pi$
$332$	− 3.81903i	− 0.209597i
$333$	− 24.5686i	− 1.34635i
$334$	−12.8885	−0.705225
$335$	0	0
$336$	−12.3537	−0.673952
$337$	1.70845i	0.0930652i	0.998917	+	0.0465326i	$0.0148171\pi$
$337$	1.70845i	0.0930652i	−0.998917	+	0.0465326i	$0.985183\pi$
$338$	11.5064i	0.625867i
$339$	16.1244	0.875757
$340$	0	0
$341$	−3.99181	−0.216169
$342$	− 19.4087i	− 1.04950i
$343$	32.7835i	1.77014i
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−10.2432	−0.550676
$347$	10.6874i	0.573730i	0.957971	+	0.286865i	$0.0926131\pi$
$347$	10.6874i	0.573730i	−0.957971	+	0.286865i	$0.907387\pi$
$348$	− 9.06943i	− 0.486173i
$349$	12.3877	0.663096	0.331548	−	0.943438i	$-0.392429\pi$
$349$	12.3877	0.663096	0.331548	+	0.943438i	$0.392429\pi$
$350$	0	0
$351$	−4.01382	−0.214242
$352$	5.13163i	0.273517i
$353$	− 5.45710i	− 0.290452i	−0.989399	−	0.145226i	$-0.953609\pi$
$353$	− 5.45710i	− 0.290452i	0.989399	−	0.145226i	$-0.0463909\pi$
$354$	−25.1938	−1.33904
$355$	0	0
$356$	8.93057	0.473319
$357$	57.9070i	3.06476i
$358$	13.1938i	0.697316i
$359$	−20.1810	−1.06511	−0.532555	−	0.846395i	$-0.678768\pi$
$359$	−20.1810	−1.06511	−0.532555	+	0.846395i	$0.678768\pi$
$360$	0	0
$361$	2.13163	0.112191
$362$	15.7907i	0.829943i
$363$	− 41.2076i	− 2.16284i
$364$	−5.61797	−0.294461
$365$	0	0
$366$	29.4288	1.53827
$367$	2.74960i	0.143528i	0.997422	+	0.0717639i	$0.0228628\pi$
$367$	2.74960i	0.143528i	−0.997422	+	0.0717639i	$0.977137\pi$
$368$	1.00000i	0.0521286i
$369$	35.9151	1.86967
$370$	0	0
$371$	27.5815	1.43196
$372$	− 2.09048i	− 0.108387i
$373$	12.0823i	0.625598i	0.949819	+	0.312799i	$0.101267\pi$
$373$	12.0823i	0.625598i	−0.949819	+	0.312799i	$0.898733\pi$
$374$	24.0540	1.24380
$375$	0	0
$376$	−6.44423	−0.332336
$377$	− 4.12440i	− 0.212417i
$378$	15.0977i	0.776543i
$379$	−5.25603	−0.269984	−0.134992	−	0.990847i	$-0.543101\pi$
$379$	−5.25603	−0.269984	−0.134992	+	0.990847i	$0.543101\pi$
$380$	0	0
$381$	−6.08230	−0.311605
$382$	16.1244i	0.824996i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−2.68740	−0.137141
$385$	0	0
$386$	−17.9434	−0.913297
$387$	33.7769i	1.71698i
$388$	18.0622i	0.916969i
$389$	−0.325463	−0.0165017	−0.00825083	−	0.999966i	$-0.502626\pi$
$389$	−0.325463	−0.0165017	−0.00825083	+	0.999966i	$0.502626\pi$
$390$	0	0
$391$	4.68740	0.237052
$392$	14.1316i	0.713755i
$393$	− 7.87560i	− 0.397272i
$394$	−5.88123	−0.296292
$395$	0	0
$396$	21.6663	1.08878
$397$	− 17.7568i	− 0.891190i	−0.895235	−	0.445595i	$-0.852992\pi$
$397$	− 17.7568i	− 0.891190i	0.895235	−	0.445595i	$-0.147008\pi$
$398$	− 6.56863i	− 0.329256i
$399$	−56.7891	−2.84301
$400$	0	0
$401$	3.91770	0.195641	0.0978204	−	0.995204i	$-0.468813\pi$
$401$	3.91770	0.195641	0.0978204	+	0.995204i	$0.468813\pi$
$402$	42.0257i	2.09605i
$403$	− 0.950664i	− 0.0473560i
$404$	−3.37480	−0.167903
$405$	0	0
$406$	−15.5137	−0.769930
$407$	29.8611i	1.48016i
$408$	12.5969i	0.623640i
$409$	−9.58405	−0.473901	−0.236950	−	0.971522i	$-0.576148\pi$
$409$	−9.58405	−0.473901	−0.236950	+	0.971522i	$0.576148\pi$
$410$	0	0
$411$	−53.1856	−2.62345
$412$	13.1316i	0.646949i
$413$	43.0952i	2.12057i
$414$	4.22212	0.207506
$415$	0	0
$416$	−1.22212	−0.0599192
$417$	16.8319i	0.824261i
$418$	23.5897i	1.15381i
$419$	20.5265	1.00279	0.501393	−	0.865219i	$-0.332821\pi$
$419$	20.5265	1.00279	0.501393	+	0.865219i	$0.332821\pi$
$420$	0	0
$421$	2.29155	0.111683	0.0558417	−	0.998440i	$-0.482216\pi$
$421$	2.29155	0.111683	0.0558417	+	0.998440i	$0.482216\pi$
$422$	− 15.4571i	− 0.752440i
$423$	27.2083i	1.32291i
$424$	6.00000	0.291386
$425$	0	0
$426$	3.52748	0.170907
$427$	− 50.3393i	− 2.43609i
$428$	− 12.3054i	− 0.594802i
$429$	16.8539	0.813714
$430$	0	0
$431$	−8.83189	−0.425417	−0.212709	−	0.977116i	$-0.568229\pi$
$431$	−8.83189	−0.425417	−0.212709	+	0.977116i	$0.568229\pi$
$432$	3.28432i	0.158017i
$433$	− 33.3465i	− 1.60253i	−0.598309	−	0.801266i	$-0.704160\pi$
$433$	− 33.3465i	− 1.60253i	0.598309	−	0.801266i	$-0.295840\pi$
$434$	−3.57587	−0.171647
$435$	0	0
$436$	−10.5969	−0.507500
$437$	4.59692i	0.219900i
$438$	11.9434i	0.570679i
$439$	6.02829	0.287714	0.143857	−	0.989598i	$-0.454049\pi$
$439$	6.02829	0.287714	0.143857	+	0.989598i	$0.454049\pi$
$440$	0	0
$441$	59.6654	2.84121
$442$	5.72855i	0.272479i
$443$	15.5275i	0.737733i	0.929482	+	0.368866i	$0.120254\pi$
$443$	15.5275i	0.737733i	−0.929482	+	0.368866i	$0.879746\pi$
$444$	−15.6381	−0.742150
$445$	0	0
$446$	−10.7496	−0.509008
$447$	− 53.1856i	− 2.51559i
$448$	4.59692i	0.217184i
$449$	18.3594	0.866433	0.433216	−	0.901290i	$-0.357379\pi$
$449$	18.3594	0.866433	0.433216	+	0.901290i	$0.357379\pi$
$450$	0	0
$451$	−43.6519	−2.05549
$452$	− 6.00000i	− 0.282216i
$453$	− 10.9168i	− 0.512914i
$454$	12.3054	0.577519
$455$	0	0
$456$	−12.3537	−0.578517
$457$	− 11.4992i	− 0.537910i	−0.963153	−	0.268955i	$-0.913322\pi$
$457$	− 11.4992i	− 0.537910i	0.963153	−	0.268955i	$-0.0866782\pi$
$458$	− 9.63806i	− 0.450357i
$459$	15.3949	0.718572
$460$	0	0
$461$	1.33270	0.0620700	0.0310350	−	0.999518i	$-0.490120\pi$
$461$	1.33270	0.0620700	0.0310350	+	0.999518i	$0.490120\pi$
$462$	− 63.3949i	− 2.94940i
$463$	− 35.8190i	− 1.66465i	−0.554287	−	0.832326i	$-0.687009\pi$
$463$	− 35.8190i	− 1.66465i	0.554287	−	0.832326i	$-0.312991\pi$
$464$	−3.37480	−0.156671
$465$	0	0
$466$	13.9434	0.645917
$467$	− 23.7625i	− 1.09960i	−0.835298	−	0.549798i	$-0.814705\pi$
$467$	− 23.7625i	− 1.09960i	0.835298	−	0.549798i	$-0.185295\pi$
$468$	5.15992i	0.238517i
$469$	71.8869	3.31943
$470$	0	0
$471$	12.4298	0.572733
$472$	9.37480i	0.431510i
$473$	− 41.0531i	− 1.88762i
$474$	13.1373	0.603414
$475$	0	0
$476$	21.5476	0.987632
$477$	− 25.3327i	− 1.15990i
$478$	26.3877i	1.20694i
$479$	2.04210	0.0933060	0.0466530	−	0.998911i	$-0.485145\pi$
$479$	2.04210	0.0933060	0.0466530	+	0.998911i	$0.485145\pi$
$480$	0	0
$481$	−7.11153	−0.324258
$482$	− 7.06943i	− 0.322004i
$483$	− 12.3537i	− 0.562115i
$484$	−15.3337	−0.696984
$485$	0	0
$486$	−20.1728	−0.915056
$487$	− 18.6252i	− 0.843988i	−0.906599	−	0.421994i	$-0.861330\pi$
$487$	− 18.6252i	− 0.843988i	0.906599	−	0.421994i	$-0.138670\pi$
$488$	− 10.9507i	− 0.495713i
$489$	−33.3666	−1.50889
$490$	0	0
$491$	33.7204	1.52178	0.760889	−	0.648882i	$-0.224763\pi$
$491$	33.7204	1.52178	0.760889	+	0.648882i	$0.224763\pi$
$492$	− 22.8602i	− 1.03062i
$493$	15.8190i	0.712453i
$494$	−5.61797	−0.252764
$495$	0	0
$496$	−0.777884	−0.0349280
$497$	− 6.03391i	− 0.270658i
$498$	10.2633i	0.459908i
$499$	16.8885	0.756032	0.378016	−	0.925799i	$-0.376607\pi$
$499$	16.8885	0.756032	0.378016	+	0.925799i	$0.376607\pi$
$500$	0	0
$501$	34.6365	1.54744
$502$	− 24.9023i	− 1.11144i
$503$	− 11.4031i	− 0.508438i	−0.967147	−	0.254219i	$-0.918182\pi$
$503$	− 11.4031i	− 0.508438i	0.967147	−	0.254219i	$-0.0818185\pi$
$504$	19.4087	0.864533
$505$	0	0
$506$	−5.13163	−0.228129
$507$	− 30.9224i	− 1.37331i
$508$	2.26326i	0.100416i
$509$	1.87560	0.0831346	0.0415673	−	0.999136i	$-0.486765\pi$
$509$	1.87560	0.0831346	0.0415673	+	0.999136i	$0.486765\pi$
$510$	0	0
$511$	20.4298	0.903759
$512$	1.00000i	0.0441942i
$513$	15.0977i	0.666581i
$514$	−0.444233	−0.0195943
$515$	0	0
$516$	21.4992	0.946449
$517$	− 33.0694i	− 1.45439i
$518$	26.7496i	1.17531i
$519$	27.5275	1.20832
$520$	0	0
$521$	5.11153	0.223940	0.111970	−	0.993712i	$-0.464284\pi$
$521$	5.11153	0.223940	0.111970	+	0.993712i	$0.464284\pi$
$522$	14.2488i	0.623653i
$523$	− 19.4571i	− 0.850799i	−0.905006	−	0.425400i	$-0.860134\pi$
$523$	− 19.4571i	− 0.850799i	0.905006	−	0.425400i	$-0.139866\pi$
$524$	−2.93057	−0.128022
$525$	0	0
$526$	−23.8812	−1.04127
$527$	3.64625i	0.158833i
$528$	− 13.7907i	− 0.600165i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	39.5815	1.71769
$532$	21.1316i	0.916172i
$533$	− 10.3958i	− 0.450294i
$534$	−24.0000	−1.03858
$535$	0	0
$536$	15.6381	0.675461
$537$	− 35.4571i	− 1.53009i
$538$	− 16.2633i	− 0.701159i
$539$	−72.5183	−3.12359
$540$	0	0
$541$	−2.70750	−0.116404	−0.0582022	−	0.998305i	$-0.518537\pi$
$541$	−2.70750	−0.116404	−0.0582022	+	0.998305i	$0.518537\pi$
$542$	25.6098i	1.10003i
$543$	− 42.4360i	− 1.82111i
$544$	4.68740	0.200971
$545$	0	0
$546$	15.0977	0.646123
$547$	− 1.66635i	− 0.0712479i	−0.999365	−	0.0356240i	$-0.988658\pi$
$547$	− 1.66635i	− 0.0712479i	0.999365	−	0.0356240i	$-0.0113419\pi$
$548$	19.7907i	0.845419i
$549$	−46.2350	−1.97326
$550$	0	0
$551$	−15.5137	−0.660904
$552$	− 2.68740i	− 0.114383i
$553$	− 22.4719i	− 0.955601i
$554$	−2.88847	−0.122719
$555$	0	0
$556$	6.26326	0.265622
$557$	− 20.9306i	− 0.886857i	−0.896310	−	0.443428i	$-0.853762\pi$
$557$	− 20.9306i	− 0.886857i	0.896310	−	0.443428i	$-0.146238\pi$
$558$	3.28432i	0.139036i
$559$	9.77693	0.413520
$560$	0	0
$561$	−64.6427	−2.72922
$562$	4.26326i	0.179835i
$563$	15.2761i	0.643812i	0.946772	+	0.321906i	$0.104324\pi$
$563$	15.2761i	0.643812i	−0.946772	+	0.321906i	$0.895676\pi$
$564$	17.3182	0.729229
$565$	0	0
$566$	30.5686	1.28490
$567$	17.6525i	0.741337i
$568$	− 1.31260i	− 0.0550755i
$569$	−20.3877	−0.854695	−0.427348	−	0.904087i	$-0.640552\pi$
$569$	−20.3877	−0.854695	−0.427348	+	0.904087i	$0.640552\pi$
$570$	0	0
$571$	5.49357	0.229899	0.114949	−	0.993371i	$-0.463329\pi$
$571$	5.49357	0.229899	0.114949	+	0.993371i	$0.463329\pi$
$572$	− 6.27145i	− 0.262223i
$573$	− 43.3327i	− 1.81025i
$574$	−39.1033	−1.63214
$575$	0	0
$576$	4.22212	0.175922
$577$	17.2761i	0.719215i	0.933104	+	0.359607i	$0.117089\pi$
$577$	17.2761i	0.719215i	−0.933104	+	0.359607i	$0.882911\pi$
$578$	− 4.97171i	− 0.206796i
$579$	48.2212	2.00400
$580$	0	0
$581$	17.5558	0.728336
$582$	− 48.5403i	− 2.01206i
$583$	30.7898i	1.27518i
$584$	4.44423	0.183904
$585$	0	0
$586$	−6.00000	−0.247858
$587$	31.8247i	1.31354i	0.754089	+	0.656772i	$0.228079\pi$
$587$	31.8247i	1.31354i	−0.754089	+	0.656772i	$0.771921\pi$
$588$	− 37.9773i	− 1.56616i
$589$	−3.57587	−0.147341
$590$	0	0
$591$	15.8052	0.650140
$592$	5.81903i	0.239161i
$593$	12.4442i	0.511023i	0.966806	+	0.255512i	$0.0822439\pi$
$593$	12.4442i	0.511023i	−0.966806	+	0.255512i	$0.917756\pi$
$594$	−16.8539	−0.691524
$595$	0	0
$596$	−19.7907	−0.810661
$597$	17.6525i	0.722470i
$598$	− 1.22212i	− 0.0499761i
$599$	−17.4370	−0.712456	−0.356228	−	0.934399i	$-0.615937\pi$
$599$	−17.4370	−0.712456	−0.356228	+	0.934399i	$0.615937\pi$
$600$	0	0
$601$	−0.916751	−0.0373951	−0.0186975	−	0.999825i	$-0.505952\pi$
$601$	−0.916751	−0.0373951	−0.0186975	+	0.999825i	$0.505952\pi$
$602$	− 36.7753i	− 1.49885i
$603$	− 66.0257i	− 2.68878i
$604$	−4.06220	−0.165289
$605$	0	0
$606$	9.06943	0.368421
$607$	36.7753i	1.49266i	0.665574	+	0.746332i	$0.268187\pi$
$607$	36.7753i	1.49266i	−0.665574	+	0.746332i	$0.731813\pi$
$608$	4.59692i	0.186430i
$609$	41.6914	1.68942
$610$	0	0
$611$	7.87560	0.318613
$612$	− 19.7907i	− 0.799994i
$613$	− 4.38766i	− 0.177216i	−0.996067	−	0.0886080i	$-0.971758\pi$
$613$	− 4.38766i	− 0.177216i	0.996067	−	0.0886080i	$-0.0282418\pi$
$614$	8.54853	0.344991
$615$	0	0
$616$	−23.5897	−0.950455
$617$	40.4499i	1.62845i	0.580549	+	0.814225i	$0.302838\pi$
$617$	40.4499i	1.62845i	−0.580549	+	0.814225i	$0.697162\pi$
$618$	− 35.2899i	− 1.41957i
$619$	39.8165	1.60036	0.800180	−	0.599760i	$-0.204737\pi$
$619$	39.8165	1.60036	0.800180	+	0.599760i	$0.204737\pi$
$620$	0	0
$621$	−3.28432	−0.131795
$622$	− 7.63806i	− 0.306258i
$623$	41.0531i	1.64476i
$624$	3.28432	0.131478
$625$	0	0
$626$	−18.2350	−0.728816
$627$	− 63.3949i	− 2.53175i
$628$	− 4.62520i	− 0.184566i
$629$	27.2761	1.08757
$630$	0	0
$631$	25.9013	1.03112	0.515558	−	0.856855i	$-0.327585\pi$
$631$	25.9013	1.03112	0.515558	+	0.856855i	$0.327585\pi$
$632$	− 4.88847i	− 0.194453i
$633$	41.5394i	1.65104i
$634$	16.8602	0.669603
$635$	0	0
$636$	−16.1244	−0.639374
$637$	− 17.2705i	− 0.684282i
$638$	− 17.3182i	− 0.685635i
$639$	−5.54195	−0.219236
$640$	0	0
$641$	−43.8448	−1.73176	−0.865882	−	0.500248i	$-0.833242\pi$
$641$	−43.8448	−1.73176	−0.865882	+	0.500248i	$0.833242\pi$
$642$	33.0694i	1.30515i
$643$	− 3.94343i	− 0.155514i	−0.996972	−	0.0777568i	$-0.975224\pi$
$643$	− 3.94343i	− 0.155514i	0.996972	−	0.0777568i	$-0.0247758\pi$
$644$	−4.59692	−0.181144
$645$	0	0
$646$	21.5476	0.847778
$647$	24.3456i	0.957123i	0.878054	+	0.478561i	$0.158842\pi$
$647$	24.3456i	0.957123i	−0.878054	+	0.478561i	$0.841158\pi$
$648$	3.84008i	0.150853i
$649$	−48.1080	−1.88841
$650$	0	0
$651$	9.60978	0.376637
$652$	12.4159i	0.486246i
$653$	− 37.6921i	− 1.47500i	−0.675344	−	0.737502i	$-0.736005\pi$
$653$	− 37.6921i	− 1.47500i	0.675344	−	0.737502i	$-0.263995\pi$
$654$	28.4781	1.11358
$655$	0	0
$656$	−8.50643	−0.332120
$657$	− 18.7641i	− 0.732056i
$658$	− 29.6236i	− 1.15485i
$659$	−24.6107	−0.958698	−0.479349	−	0.877624i	$-0.659127\pi$
$659$	−24.6107	−0.958698	−0.479349	+	0.877624i	$0.659127\pi$
$660$	0	0
$661$	27.3126	1.06234	0.531169	−	0.847266i	$-0.321753\pi$
$661$	27.3126	1.06234	0.531169	+	0.847266i	$0.321753\pi$
$662$	19.1517i	0.744353i
$663$	− 15.3949i	− 0.597888i
$664$	3.81903	0.148207
$665$	0	0
$666$	24.5686	0.952015
$667$	− 3.37480i	− 0.130673i
$668$	− 12.8885i	− 0.498670i
$669$	28.8885	1.11689
$670$	0	0
$671$	56.1948	2.16938
$672$	− 12.3537i	− 0.476556i
$673$	− 22.5265i	− 0.868334i	−0.900832	−	0.434167i	$-0.857043\pi$
$673$	− 22.5265i	− 0.868334i	0.900832	−	0.434167i	$-0.142957\pi$
$674$	−1.70845	−0.0658070
$675$	0	0
$676$	−11.5064	−0.442555
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	16.1244i	0.619254i
$679$	−83.0304	−3.18641
$680$	0	0
$681$	−33.0694	−1.26722
$682$	− 3.99181i	− 0.152854i
$683$	31.9974i	1.22435i	0.790723	+	0.612174i	$0.209705\pi$
$683$	31.9974i	1.22435i	−0.790723	+	0.612174i	$0.790295\pi$
$684$	19.4087	0.742111
$685$	0	0
$686$	−32.7835	−1.25168
$687$	25.9013i	0.988197i
$688$	− 8.00000i	− 0.304997i
$689$	−7.33270	−0.279354
$690$	0	0
$691$	−2.80617	−0.106752	−0.0533758	−	0.998574i	$-0.516998\pi$
$691$	−2.80617	−0.106752	−0.0533758	+	0.998574i	$0.516998\pi$
$692$	− 10.2432i	− 0.389387i
$693$	99.5984i	3.78343i
$694$	−10.6874	−0.405688
$695$	0	0
$696$	9.06943	0.343776
$697$	39.8730i	1.51030i
$698$	12.3877i	0.468880i
$699$	−37.4716	−1.41730
$700$	0	0
$701$	8.37385	0.316276	0.158138	−	0.987417i	$-0.449451\pi$
$701$	8.37385	0.316276	0.158138	+	0.987417i	$0.449451\pi$
$702$	− 4.01382i	− 0.151492i
$703$	26.7496i	1.00888i
$704$	−5.13163	−0.193406
$705$	0	0
$706$	5.45710	0.205381
$707$	− 15.5137i	− 0.583451i
$708$	− 25.1938i	− 0.946842i
$709$	−21.2139	−0.796706	−0.398353	−	0.917232i	$-0.630418\pi$
$709$	−21.2139	−0.796706	−0.398353	+	0.917232i	$0.630418\pi$
$710$	0	0
$711$	−20.6397	−0.774048
$712$	8.93057i	0.334687i
$713$	− 0.777884i	− 0.0291320i
$714$	−57.9070	−2.16711
$715$	0	0
$716$	−13.1938	−0.493077
$717$	− 70.9142i	− 2.64834i
$718$	− 20.1810i	− 0.753147i
$719$	28.1106	1.04835	0.524174	−	0.851611i	$-0.324374\pi$
$719$	28.1106	1.04835	0.524174	+	0.851611i	$0.324374\pi$
$720$	0	0
$721$	−60.3650	−2.24811
$722$	2.13163i	0.0793311i
$723$	18.9984i	0.706558i
$724$	−15.7907	−0.586858
$725$	0	0
$726$	41.2076	1.52936
$727$	− 39.0330i	− 1.44765i	−0.689982	−	0.723826i	$-0.742382\pi$
$727$	− 39.0330i	− 1.44765i	0.689982	−	0.723826i	$-0.257618\pi$
$728$	− 5.61797i	− 0.208216i
$729$	42.6921	1.58119
$730$	0	0
$731$	−37.4992	−1.38696
$732$	29.4288i	1.08772i
$733$	− 47.1373i	− 1.74105i	−0.492120	−	0.870527i	$-0.663778\pi$
$733$	− 47.1373i	− 1.74105i	0.492120	−	0.870527i	$-0.336222\pi$
$734$	−2.74960	−0.101490
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	80.2488i	2.95600i
$738$	35.9151i	1.32205i
$739$	24.5265	0.902223	0.451111	−	0.892468i	$-0.351028\pi$
$739$	24.5265	0.902223	0.451111	+	0.892468i	$0.351028\pi$
$740$	0	0
$741$	15.0977	0.554629
$742$	27.5815i	1.01255i
$743$	7.34651i	0.269517i	0.990878	+	0.134759i	$0.0430259\pi$
$743$	7.34651i	0.269517i	−0.990878	+	0.134759i	$0.956974\pi$
$744$	2.09048	0.0766409
$745$	0	0
$746$	−12.0823	−0.442364
$747$	− 16.1244i	− 0.589961i
$748$	24.0540i	0.879502i
$749$	56.5667	2.06690
$750$	0	0
$751$	11.2359	0.410005	0.205002	−	0.978761i	$-0.434280\pi$
$751$	11.2359	0.410005	0.205002	+	0.978761i	$0.434280\pi$
$752$	− 6.44423i	− 0.234997i
$753$	66.9224i	2.43879i
$754$	4.12440	0.150202
$755$	0	0
$756$	−15.0977	−0.549099
$757$	49.1794i	1.78745i	0.448611	+	0.893727i	$0.351919\pi$
$757$	49.1794i	1.78745i	−0.448611	+	0.893727i	$0.648081\pi$
$758$	− 5.25603i	− 0.190908i
$759$	13.7907	0.500572
$760$	0	0
$761$	43.2478	1.56773	0.783867	−	0.620929i	$-0.213245\pi$
$761$	43.2478	1.56773	0.783867	+	0.620929i	$0.213245\pi$
$762$	− 6.08230i	− 0.220338i
$763$	− 48.7131i	− 1.76353i
$764$	−16.1244	−0.583360
$765$	0	0
$766$	0	0
$767$	− 11.4571i	− 0.413692i
$768$	− 2.68740i	− 0.0969732i
$769$	39.6638	1.43031	0.715156	−	0.698964i	$-0.246355\pi$
$769$	39.6638	1.43031	0.715156	+	0.698964i	$0.246355\pi$
$770$	0	0
$771$	1.19383	0.0429948
$772$	− 17.9434i	− 0.645798i
$773$	2.18097i	0.0784440i	0.999231	+	0.0392220i	$0.0124879\pi$
$773$	2.18097i	0.0784440i	−0.999231	+	0.0392220i	$0.987512\pi$
$774$	−33.7769	−1.21409
$775$	0	0
$776$	−18.0622	−0.648395
$777$	− 71.8869i	− 2.57893i
$778$	− 0.325463i	− 0.0116684i
$779$	−39.1033	−1.40102
$780$	0	0
$781$	6.73578	0.241025
$782$	4.68740i	0.167621i
$783$	− 11.0839i	− 0.396106i
$784$	−14.1316	−0.504701
$785$	0	0
$786$	7.87560	0.280913
$787$	4.76407i	0.169821i	0.996389	+	0.0849103i	$0.0270604\pi$
$787$	4.76407i	0.169821i	−0.996389	+	0.0849103i	$0.972940\pi$
$788$	− 5.88123i	− 0.209510i
$789$	64.1784	2.28481
$790$	0	0
$791$	27.5815	0.980685
$792$	21.6663i	0.769880i
$793$	13.3830i	0.475244i
$794$	17.7568	0.630166
$795$	0	0
$796$	6.56863	0.232819
$797$	39.7204i	1.40697i	0.710711	+	0.703484i	$0.248373\pi$
$797$	39.7204i	1.40697i	−0.710711	+	0.703484i	$0.751627\pi$
$798$	− 56.7891i	− 2.01031i
$799$	−30.2067	−1.06864
$800$	0	0
$801$	37.7059	1.33227
$802$	3.91770i	0.138339i
$803$	22.8062i	0.804812i
$804$	−42.0257	−1.48213
$805$	0	0
$806$	0.950664	0.0334857
$807$	43.7059i	1.53852i
$808$	− 3.37480i	− 0.118725i
$809$	−46.8941	−1.64871	−0.824354	−	0.566074i	$-0.808462\pi$
$809$	−46.8941	−1.64871	−0.824354	+	0.566074i	$0.808462\pi$
$810$	0	0
$811$	−36.8319	−1.29334	−0.646671	−	0.762769i	$-0.723839\pi$
$811$	−36.8319	−1.29334	−0.646671	+	0.762769i	$0.723839\pi$
$812$	− 15.5137i	− 0.544423i
$813$	− 68.8237i	− 2.41375i
$814$	−29.8611	−1.04663
$815$	0	0
$816$	−12.5969	−0.440980
$817$	− 36.7753i	− 1.28661i
$818$	− 9.58405i	− 0.335099i
$819$	−23.7197	−0.828834
$820$	0	0
$821$	−18.6107	−0.649519	−0.324759	−	0.945797i	$-0.605283\pi$
$821$	−18.6107	−0.649519	−0.324759	+	0.945797i	$0.605283\pi$
$822$	− 53.1856i	− 1.85506i
$823$	9.31823i	0.324813i	0.986724	+	0.162407i	$0.0519256\pi$
$823$	9.31823i	0.324813i	−0.986724	+	0.162407i	$0.948074\pi$
$824$	−13.1316	−0.457462
$825$	0	0
$826$	−43.0952	−1.49947
$827$	53.0129i	1.84344i	0.387858	+	0.921719i	$0.373215\pi$
$827$	53.0129i	1.84344i	−0.387858	+	0.921719i	$0.626785\pi$
$828$	4.22212i	0.146729i
$829$	25.2761	0.877876	0.438938	−	0.898517i	$-0.355355\pi$
$829$	25.2761	0.877876	0.438938	+	0.898517i	$0.355355\pi$
$830$	0	0
$831$	7.76246	0.269277
$832$	− 1.22212i	− 0.0423693i
$833$	66.2406i	2.29510i
$834$	−16.8319	−0.582841
$835$	0	0
$836$	−23.5897	−0.815866
$837$	− 2.55481i	− 0.0883073i
$838$	20.5265i	0.709077i
$839$	−11.6946	−0.403744	−0.201872	−	0.979412i	$-0.564702\pi$
$839$	−11.6946	−0.403744	−0.201872	+	0.979412i	$0.564702\pi$
$840$	0	0
$841$	−17.6107	−0.607267
$842$	2.29155i	0.0789720i
$843$	− 11.4571i	− 0.394603i
$844$	15.4571	0.532055
$845$	0	0
$846$	−27.2083	−0.935441
$847$	− 70.4875i	− 2.42198i
$848$	6.00000i	0.206041i
$849$	−82.1501	−2.81938
$850$	0	0
$851$	−5.81903	−0.199474
$852$	3.52748i	0.120850i
$853$	− 38.6371i	− 1.32291i	−0.749985	−	0.661455i	$-0.769939\pi$
$853$	− 38.6371i	− 1.32291i	0.749985	−	0.661455i	$-0.230061\pi$
$854$	50.3393	1.72257
$855$	0	0
$856$	12.3054	0.420589
$857$	26.2488i	0.896642i	0.893873	+	0.448321i	$0.147978\pi$
$857$	26.2488i	0.896642i	−0.893873	+	0.448321i	$0.852022\pi$
$858$	16.8539i	0.575383i
$859$	−37.5558	−1.28139	−0.640693	−	0.767797i	$-0.721353\pi$
$859$	−37.5558	−1.28139	−0.640693	+	0.767797i	$0.721353\pi$
$860$	0	0
$861$	105.086	3.58133
$862$	− 8.83189i	− 0.300816i
$863$	21.9855i	0.748396i	0.927349	+	0.374198i	$0.122082\pi$
$863$	21.9855i	0.748396i	−0.927349	+	0.374198i	$0.877918\pi$
$864$	−3.28432	−0.111735
$865$	0	0
$866$	33.3465	1.13316
$867$	13.3610i	0.453763i
$868$	− 3.57587i	− 0.121373i
$869$	25.0858	0.850978
$870$	0	0
$871$	−19.1115	−0.647570
$872$	− 10.5969i	− 0.358857i
$873$	76.2607i	2.58103i
$874$	−4.59692	−0.155493
$875$	0	0
$876$	−11.9434	−0.403531
$877$	− 36.5064i	− 1.23273i	−0.787459	−	0.616367i	$-0.788604\pi$
$877$	− 36.5064i	− 1.23273i	0.787459	−	0.616367i	$-0.211396\pi$
$878$	6.02829i	0.203445i
$879$	16.1244	0.543862
$880$	0	0
$881$	−17.4571	−0.588145	−0.294072	−	0.955783i	$-0.595011\pi$
$881$	−17.4571	−0.588145	−0.294072	+	0.955783i	$0.595011\pi$
$882$	59.6654i	2.00904i
$883$	− 22.9444i	− 0.772140i	−0.922470	−	0.386070i	$-0.873832\pi$
$883$	− 22.9444i	− 0.772140i	0.922470	−	0.386070i	$-0.126168\pi$
$884$	−5.72855	−0.192672
$885$	0	0
$886$	−15.5275	−0.521656
$887$	− 11.7223i	− 0.393595i	−0.980444	−	0.196798i	$-0.936946\pi$
$887$	− 11.7223i	− 0.393595i	0.980444	−	0.196798i	$-0.0630542\pi$
$888$	− 15.6381i	− 0.524779i
$889$	−10.4040	−0.348940
$890$	0	0
$891$	−19.7059	−0.660172
$892$	− 10.7496i	− 0.359923i
$893$	− 29.6236i	− 0.991316i
$894$	53.1856	1.77879
$895$	0	0
$896$	−4.59692	−0.153572
$897$	3.28432i	0.109660i
$898$	18.3594i	0.612660i
$899$	2.62520	0.0875554
$900$	0	0
$901$	28.1244	0.936960
$902$	− 43.6519i	− 1.45345i
$903$	98.8300i	3.28886i
$904$	6.00000	0.199557
$905$	0	0
$906$	10.9168	0.362685
$907$	− 1.91770i	− 0.0636763i	−0.999493	−	0.0318381i	$-0.989864\pi$
$907$	− 1.91770i	− 0.0636763i	0.999493	−	0.0318381i	$-0.0101361\pi$
$908$	12.3054i	0.408368i
$909$	−14.2488	−0.472603
$910$	0	0
$911$	−56.8319	−1.88292	−0.941462	−	0.337118i	$-0.890548\pi$
$911$	−56.8319	−1.88292	−0.941462	+	0.337118i	$0.890548\pi$
$912$	− 12.3537i	− 0.409073i
$913$	19.5979i	0.648595i
$914$	11.4992	0.380360
$915$	0	0
$916$	9.63806	0.318451
$917$	− 13.4716i	− 0.444870i
$918$	15.3949i	0.508107i
$919$	26.0257	0.858509	0.429255	−	0.903183i	$-0.358776\pi$
$919$	26.0257	0.858509	0.429255	+	0.903183i	$0.358776\pi$
$920$	0	0
$921$	−22.9733	−0.756997
$922$	1.33270i	0.0438901i
$923$	1.60415i	0.0528013i
$924$	63.3949	2.08554
$925$	0	0
$926$	35.8190	1.17709
$927$	55.4433i	1.82100i
$928$	− 3.37480i	− 0.110783i
$929$	17.1115	0.561411	0.280706	−	0.959794i	$-0.409432\pi$
$929$	17.1115	0.561411	0.280706	+	0.959794i	$0.409432\pi$
$930$	0	0
$931$	−64.9619	−2.12904
$932$	13.9434i	0.456732i
$933$	20.5265i	0.672008i
$934$	23.7625	0.777531
$935$	0	0
$936$	−5.15992	−0.168657
$937$	16.3738i	0.534910i	0.963570	+	0.267455i	$0.0861827\pi$
$937$	16.3738i	0.534910i	−0.963570	+	0.267455i	$0.913817\pi$
$938$	71.8869i	2.34719i
$939$	49.0047	1.59921
$940$	0	0
$941$	−16.4097	−0.534940	−0.267470	−	0.963566i	$-0.586188\pi$
$941$	−16.4097	−0.534940	−0.267470	+	0.963566i	$0.586188\pi$
$942$	12.4298i	0.404984i
$943$	− 8.50643i	− 0.277008i
$944$	−9.37480	−0.305124
$945$	0	0
$946$	41.0531	1.33475
$947$	− 21.3886i	− 0.695037i	−0.937673	−	0.347518i	$-0.887024\pi$
$947$	− 21.3886i	− 0.695037i	0.937673	−	0.347518i	$-0.112976\pi$
$948$	13.1373i	0.426678i
$949$	−5.43137	−0.176310
$950$	0	0
$951$	−45.3100	−1.46928
$952$	21.5476i	0.698361i
$953$	16.7276i	0.541860i	0.962599	+	0.270930i	$0.0873312\pi$
$953$	16.7276i	0.541860i	−0.962599	+	0.270930i	$0.912669\pi$
$954$	25.3327	0.820176
$955$	0	0
$956$	−26.3877	−0.853438
$957$	46.5410i	1.50446i
$958$	2.04210i	0.0659773i
$959$	−90.9764	−2.93778
$960$	0	0
$961$	−30.3949	−0.980481
$962$	− 7.11153i	− 0.229285i
$963$	− 51.9547i	− 1.67422i
$964$	7.06943	0.227691
$965$	0	0
$966$	12.3537	0.397475
$967$	45.2617i	1.45552i	0.685834	+	0.727758i	$0.259438\pi$
$967$	45.2617i	1.45552i	−0.685834	+	0.727758i	$0.740562\pi$
$968$	− 15.3337i	− 0.492842i
$969$	−57.9070	−1.86024
$970$	0	0
$971$	45.2560	1.45234	0.726168	−	0.687518i	$-0.241300\pi$
$971$	45.2560	1.45234	0.726168	+	0.687518i	$0.241300\pi$
$972$	− 20.1728i	− 0.647042i
$973$	28.7917i	0.923019i
$974$	18.6252	0.596790
$975$	0	0
$976$	10.9507	0.350522
$977$	− 13.3465i	− 0.426993i	−0.976944	−	0.213496i	$-0.931515\pi$
$977$	− 13.3465i	− 0.426993i	0.976944	−	0.213496i	$-0.0684852\pi$
$978$	− 33.3666i	− 1.06695i
$979$	−45.8284	−1.46468
$980$	0	0
$981$	−44.7414	−1.42848
$982$	33.7204i	1.07606i
$983$	− 5.13163i	− 0.163674i	−0.996646	−	0.0818368i	$-0.973921\pi$
$983$	− 5.13163i	− 0.163674i	0.996646	−	0.0818368i	$-0.0260786\pi$
$984$	22.8602	0.728756
$985$	0	0
$986$	−15.8190	−0.503781
$987$	79.6104i	2.53403i
$988$	− 5.61797i	− 0.178731i
$989$	8.00000	0.254385
$990$	0	0
$991$	17.7989	0.565402	0.282701	−	0.959208i	$-0.408770\pi$
$991$	17.7989	0.565402	0.282701	+	0.959208i	$0.408770\pi$
$992$	− 0.777884i	− 0.0246978i
$993$	− 51.4684i	− 1.63330i
$994$	6.03391	0.191384
$995$	0	0
$996$	−10.2633	−0.325204
$997$	40.3877i	1.27909i	0.768754	+	0.639545i	$0.220877\pi$
$997$	40.3877i	1.27909i	−0.768754	+	0.639545i	$0.779123\pi$
$998$	16.8885i	0.534595i
$999$	−19.1115	−0.604662

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.2.b.j.599.4		6
5.2	odd	4		1150.2.a.q.1.1		3
5.3	odd	4		230.2.a.d.1.3	✓	3
5.4	even	2	inner	1150.2.b.j.599.3		6
15.8	even	4		2070.2.a.z.1.1		3
20.3	even	4		1840.2.a.r.1.1		3
20.7	even	4		9200.2.a.cf.1.3		3
40.3	even	4		7360.2.a.ce.1.3		3
40.13	odd	4		7360.2.a.bz.1.1		3
115.68	even	4		5290.2.a.r.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.a.d.1.3	✓	3	5.3	odd	4
1150.2.a.q.1.1		3	5.2	odd	4
1150.2.b.j.599.3		6	5.4	even	2	inner
1150.2.b.j.599.4		6	1.1	even	1	trivial
1840.2.a.r.1.1		3	20.3	even	4
2070.2.a.z.1.1		3	15.8	even	4
5290.2.a.r.1.3		3	115.68	even	4
7360.2.a.bz.1.1		3	40.13	odd	4
7360.2.a.ce.1.3		3	40.3	even	4
9200.2.a.cf.1.3		3	20.7	even	4

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:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1150.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -2.68740i q^{3} -1.00000 q^{4} +2.68740 q^{6} -4.59692i q^{7} -1.00000i q^{8} -4.22212 q^{9} +5.13163 q^{11} +2.68740i q^{12} +1.22212i q^{13} +4.59692 q^{14} +1.00000 q^{16} -4.68740i q^{17} -4.22212i q^{18} +4.59692 q^{19} -12.3537 q^{21} +5.13163i q^{22} +1.00000i q^{23} -2.68740 q^{24} -1.22212 q^{26} +3.28432i q^{27} +4.59692i q^{28} -3.37480 q^{29} -0.777884 q^{31} +1.00000i q^{32} -13.7907i q^{33} +4.68740 q^{34} +4.22212 q^{36} +5.81903i q^{37} +4.59692i q^{38} +3.28432 q^{39} -8.50643 q^{41} -12.3537i q^{42} -8.00000i q^{43} -5.13163 q^{44} -1.00000 q^{46} -6.44423i q^{47} -2.68740i q^{48} -14.1316 q^{49} -12.5969 q^{51} -1.22212i q^{52} +6.00000i q^{53} -3.28432 q^{54} -4.59692 q^{56} -12.3537i q^{57} -3.37480i q^{58} -9.37480 q^{59} +10.9507 q^{61} -0.777884i q^{62} +19.4087i q^{63} -1.00000 q^{64} +13.7907 q^{66} +15.6381i q^{67} +4.68740i q^{68} +2.68740 q^{69} +1.31260 q^{71} +4.22212i q^{72} +4.44423i q^{73} -5.81903 q^{74} -4.59692 q^{76} -23.5897i q^{77} +3.28432i q^{78} +4.88847 q^{79} -3.84008 q^{81} -8.50643i q^{82} +3.81903i q^{83} +12.3537 q^{84} +8.00000 q^{86} +9.06943i q^{87} -5.13163i q^{88} -8.93057 q^{89} +5.61797 q^{91} -1.00000i q^{92} +2.09048i q^{93} +6.44423 q^{94} +2.68740 q^{96} -18.0622i q^{97} -14.1316i q^{98} -21.6663 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} + 2 q^{6} - 20 q^{9} + 6 q^{11} - 6 q^{14} + 6 q^{16} - 6 q^{19} - 44 q^{21} - 2 q^{24} - 2 q^{26} + 8 q^{29} - 10 q^{31} + 14 q^{34} + 20 q^{36} - 28 q^{39} + 2 q^{41} - 6 q^{44} - 6 q^{46}+ \cdots - 114 q^{99}+O(q^{100}) \) 6 * q - 6 * q^4 + 2 * q^6 - 20 * q^9 + 6 * q^11 - 6 * q^14 + 6 * q^16 - 6 * q^19 - 44 * q^21 - 2 * q^24 - 2 * q^26 + 8 * q^29 - 10 * q^31 + 14 * q^34 + 20 * q^36 - 28 * q^39 + 2 * q^41 - 6 * q^44 - 6 * q^46 - 60 * q^49 - 42 * q^51 + 28 * q^54 + 6 * q^56 - 28 * q^59 + 2 * q^61 - 6 * q^64 - 18 * q^66 + 2 * q^69 + 22 * q^71 + 4 * q^74 + 6 * q^76 + 8 * q^79 + 14 * q^81 + 44 * q^84 + 48 * q^86 - 36 * q^89 + 2 * q^91 + 28 * q^94 + 2 * q^96 - 114 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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