LMFDB - Embedded newform 1232.2.a.s.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,2,Mod(1,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1232.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1232.a (trivial)

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:	yes
Analytic conductor:	$9.83756952902$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.11348.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + x + 2 $ x^4 - x^3 - 5*x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.723742$ of defining polynomial
Character	$\chi$	$=$	1232.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.76342 q^{3} +3.31593 q^{5} +1.00000 q^{7} +4.63646 q^{9} +O(q^{10})$	$q+2.76342 q^{3} +3.31593 q^{5} +1.00000 q^{7} +4.63646 q^{9} -1.00000 q^{11} -4.39988 q^{13} +9.16329 q^{15} +3.44748 q^{17} -7.84736 q^{19} +2.76342 q^{21} +9.16329 q^{23} +5.99540 q^{25} +4.52223 q^{27} +0.894968 q^{29} -7.71581 q^{31} -2.76342 q^{33} +3.31593 q^{35} +5.63646 q^{37} -12.1587 q^{39} +3.44748 q^{41} -8.15869 q^{43} +15.3742 q^{45} -9.60618 q^{47} +1.00000 q^{49} +9.52683 q^{51} +2.00000 q^{53} -3.31593 q^{55} -21.6855 q^{57} -5.65838 q^{59} +1.12695 q^{61} +4.63646 q^{63} -14.5897 q^{65} -8.52223 q^{67} +25.3220 q^{69} +1.89037 q^{71} -9.35227 q^{73} +16.5678 q^{75} -1.00000 q^{77} +17.5268 q^{79} -1.41260 q^{81} +2.47923 q^{83} +11.4316 q^{85} +2.47317 q^{87} +8.26833 q^{89} -4.39988 q^{91} -21.3220 q^{93} -26.0213 q^{95} -10.5222 q^{97} -4.63646 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 5 q^{5} + 4 q^{7} + 9 q^{9}+O(q^{10}) $ 4 * q - q^3 + 5 * q^5 + 4 * q^7 + 9 * q^9	$ 4 q - q^{3} + 5 q^{5} + 4 q^{7} + 9 q^{9} - 4 q^{11} + 4 q^{13} + 3 q^{15} + 10 q^{17} - 6 q^{19} - q^{21} + 3 q^{23} + 17 q^{25} - 13 q^{27} - 4 q^{29} - q^{31} + q^{33} + 5 q^{35} + 13 q^{37} - 8 q^{39} + 10 q^{41} + 8 q^{43} + 12 q^{45} + 6 q^{47} + 4 q^{49} + 14 q^{51} + 8 q^{53} - 5 q^{55} - 22 q^{57} - 3 q^{59} + 2 q^{61} + 9 q^{63} - 3 q^{67} + 27 q^{69} - 7 q^{71} + 2 q^{73} + 18 q^{75} - 4 q^{77} + 46 q^{79} + 40 q^{81} - 32 q^{83} - 14 q^{85} + 34 q^{87} + 7 q^{89} + 4 q^{91} - 11 q^{93} + 14 q^{95} - 11 q^{97} - 9 q^{99}+O(q^{100}) $ 4 * q - q^3 + 5 * q^5 + 4 * q^7 + 9 * q^9 - 4 * q^11 + 4 * q^13 + 3 * q^15 + 10 * q^17 - 6 * q^19 - q^21 + 3 * q^23 + 17 * q^25 - 13 * q^27 - 4 * q^29 - q^31 + q^33 + 5 * q^35 + 13 * q^37 - 8 * q^39 + 10 * q^41 + 8 * q^43 + 12 * q^45 + 6 * q^47 + 4 * q^49 + 14 * q^51 + 8 * q^53 - 5 * q^55 - 22 * q^57 - 3 * q^59 + 2 * q^61 + 9 * q^63 - 3 * q^67 + 27 * q^69 - 7 * q^71 + 2 * q^73 + 18 * q^75 - 4 * q^77 + 46 * q^79 + 40 * q^81 - 32 * q^83 - 14 * q^85 + 34 * q^87 + 7 * q^89 + 4 * q^91 - 11 * q^93 + 14 * q^95 - 11 * q^97 - 9 * q^99

Display 100 coefficients 10 coefficients

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
$3$	2.76342	1.59546	0.797729	−	0.603016i	$-0.206034\pi$
$3$	2.76342	1.59546	0.797729	+	0.603016i	$0.206034\pi$
$4$	0	0
$5$	3.31593	1.48293	0.741465	−	0.670992i	$-0.234131\pi$
$5$	3.31593	1.48293	0.741465	+	0.670992i	$0.234131\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	4.63646	1.54549
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−4.39988	−1.22031	−0.610153	−	0.792283i	$-0.708892\pi$
$13$	−4.39988	−1.22031	−0.610153	+	0.792283i	$0.708892\pi$
$14$	0	0
$15$	9.16329	2.36595
$16$	0	0
$17$	3.44748	0.836138	0.418069	−	0.908415i	$-0.362707\pi$
$17$	3.44748	0.836138	0.418069	+	0.908415i	$0.362707\pi$
$18$	0	0
$19$	−7.84736	−1.80031	−0.900154	−	0.435571i	$-0.856546\pi$
$19$	−7.84736	−1.80031	−0.900154	+	0.435571i	$0.856546\pi$
$20$	0	0
$21$	2.76342	0.603027
$22$	0	0
$23$	9.16329	1.91068	0.955339	−	0.295511i	$-0.0954898\pi$
$23$	9.16329	1.91068	0.955339	+	0.295511i	$0.0954898\pi$
$24$	0	0
$25$	5.99540	1.19908
$26$	0	0
$27$	4.52223	0.870303
$28$	0	0
$29$	0.894968	0.166191	0.0830957	−	0.996542i	$-0.473519\pi$
$29$	0.894968	0.166191	0.0830957	+	0.996542i	$0.473519\pi$
$30$	0	0
$31$	−7.71581	−1.38580	−0.692900	−	0.721034i	$-0.743667\pi$
$31$	−7.71581	−1.38580	−0.692900	+	0.721034i	$0.743667\pi$
$32$	0	0
$33$	−2.76342	−0.481049
$34$	0	0
$35$	3.31593	0.560495
$36$	0	0
$37$	5.63646	0.926629	0.463314	−	0.886194i	$-0.346660\pi$
$37$	5.63646	0.926629	0.463314	+	0.886194i	$0.346660\pi$
$38$	0	0
$39$	−12.1587	−1.94695
$40$	0	0
$41$	3.44748	0.538407	0.269203	−	0.963083i	$-0.413240\pi$
$41$	3.44748	0.538407	0.269203	+	0.963083i	$0.413240\pi$
$42$	0	0
$43$	−8.15869	−1.24419	−0.622094	−	0.782942i	$-0.713718\pi$
$43$	−8.15869	−1.24419	−0.622094	+	0.782942i	$0.713718\pi$
$44$	0	0
$45$	15.3742	2.29185
$46$	0	0
$47$	−9.60618	−1.40120	−0.700602	−	0.713552i	$-0.747085\pi$
$47$	−9.60618	−1.40120	−0.700602	+	0.713552i	$0.747085\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	9.52683	1.33402
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	−3.31593	−0.447120
$56$	0	0
$57$	−21.6855	−2.87232
$58$	0	0
$59$	−5.65838	−0.736659	−0.368329	−	0.929695i	$-0.620070\pi$
$59$	−5.65838	−0.736659	−0.368329	+	0.929695i	$0.620070\pi$
$60$	0	0
$61$	1.12695	0.144291	0.0721457	−	0.997394i	$-0.477015\pi$
$61$	1.12695	0.144291	0.0721457	+	0.997394i	$0.477015\pi$
$62$	0	0
$63$	4.63646	0.584140
$64$	0	0
$65$	−14.5897	−1.80963
$66$	0	0
$67$	−8.52223	−1.04116	−0.520578	−	0.853814i	$-0.674283\pi$
$67$	−8.52223	−1.04116	−0.520578	+	0.853814i	$0.674283\pi$
$68$	0	0
$69$	25.3220	3.04841
$70$	0	0
$71$	1.89037	0.224345	0.112173	−	0.993689i	$-0.464219\pi$
$71$	1.89037	0.224345	0.112173	+	0.993689i	$0.464219\pi$
$72$	0	0
$73$	−9.35227	−1.09460	−0.547300	−	0.836936i	$-0.684344\pi$
$73$	−9.35227	−1.09460	−0.547300	+	0.836936i	$0.684344\pi$
$74$	0	0
$75$	16.5678	1.91308
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	17.5268	1.97192	0.985961	−	0.166974i	$-0.0533997\pi$
$79$	17.5268	1.97192	0.985961	+	0.166974i	$0.0533997\pi$
$80$	0	0
$81$	−1.41260	−0.156955
$82$	0	0
$83$	2.47923	0.272130	0.136065	−	0.990700i	$-0.456554\pi$
$83$	2.47923	0.272130	0.136065	+	0.990700i	$0.456554\pi$
$84$	0	0
$85$	11.4316	1.23993
$86$	0	0
$87$	2.47317	0.265152
$88$	0	0
$89$	8.26833	0.876441	0.438220	−	0.898868i	$-0.355609\pi$
$89$	8.26833	0.876441	0.438220	+	0.898868i	$0.355609\pi$
$90$	0	0
$91$	−4.39988	−0.461233
$92$	0	0
$93$	−21.3220	−2.21099
$94$	0	0
$95$	−26.0213	−2.66973
$96$	0	0
$97$	−10.5222	−1.06837	−0.534185	−	0.845367i	$-0.679382\pi$
$97$	−10.5222	−1.06837	−0.534185	+	0.845367i	$0.679382\pi$
$98$	0	0
$99$	−4.63646	−0.465982
$100$	0	0
$101$	5.50491	0.547759	0.273880	−	0.961764i	$-0.411693\pi$
$101$	5.50491	0.547759	0.273880	+	0.961764i	$0.411693\pi$
$102$	0	0
$103$	3.65755	0.360389	0.180194	−	0.983631i	$-0.442327\pi$
$103$	3.65755	0.360389	0.180194	+	0.983631i	$0.442327\pi$
$104$	0	0
$105$	9.16329	0.894246
$106$	0	0
$107$	−4.42180	−0.427471	−0.213736	−	0.976892i	$-0.568563\pi$
$107$	−4.42180	−0.427471	−0.213736	+	0.976892i	$0.568563\pi$
$108$	0	0
$109$	−12.7906	−1.22511	−0.612556	−	0.790427i	$-0.709859\pi$
$109$	−12.7906	−1.22511	−0.612556	+	0.790427i	$0.709859\pi$
$110$	0	0
$111$	15.5759	1.47840
$112$	0	0
$113$	−4.99540	−0.469928	−0.234964	−	0.972004i	$-0.575497\pi$
$113$	−4.99540	−0.469928	−0.234964	+	0.972004i	$0.575497\pi$
$114$	0	0
$115$	30.3849	2.83340
$116$	0	0
$117$	−20.3999	−1.88597
$118$	0	0
$119$	3.44748	0.316030
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	9.52683	0.859005
$124$	0	0
$125$	3.30067	0.295221
$126$	0	0
$127$	16.4218	1.45720	0.728600	−	0.684940i	$-0.240172\pi$
$127$	16.4218	1.45720	0.728600	+	0.684940i	$0.240172\pi$
$128$	0	0
$129$	−22.5459	−1.98505
$130$	0	0
$131$	0.311329	0.0272010	0.0136005	−	0.999908i	$-0.495671\pi$
$131$	0.311329	0.0272010	0.0136005	+	0.999908i	$0.495671\pi$
$132$	0	0
$133$	−7.84736	−0.680453
$134$	0	0
$135$	14.9954	1.29060
$136$	0	0
$137$	7.62726	0.651641	0.325820	−	0.945432i	$-0.394360\pi$
$137$	7.62726	0.651641	0.325820	+	0.945432i	$0.394360\pi$
$138$	0	0
$139$	−3.20630	−0.271955	−0.135977	−	0.990712i	$-0.543417\pi$
$139$	−3.20630	−0.271955	−0.135977	+	0.990712i	$0.543417\pi$
$140$	0	0
$141$	−26.5459	−2.23556
$142$	0	0
$143$	4.39988	0.367936
$144$	0	0
$145$	2.96765	0.246450
$146$	0	0
$147$	2.76342	0.227923
$148$	0	0
$149$	3.78994	0.310484	0.155242	−	0.987876i	$-0.450384\pi$
$149$	3.78994	0.310484	0.155242	+	0.987876i	$0.450384\pi$
$150$	0	0
$151$	8.37796	0.681789	0.340894	−	0.940102i	$-0.389270\pi$
$151$	8.37796	0.681789	0.340894	+	0.940102i	$0.389270\pi$
$152$	0	0
$153$	15.9841	1.29224
$154$	0	0
$155$	−25.5851	−2.05504
$156$	0	0
$157$	24.2744	1.93731	0.968653	−	0.248417i	$-0.0799102\pi$
$157$	24.2744	1.93731	0.968653	+	0.248417i	$0.0799102\pi$
$158$	0	0
$159$	5.52683	0.438306
$160$	0	0
$161$	9.16329	0.722169
$162$	0	0
$163$	−19.8534	−1.55504	−0.777520	−	0.628858i	$-0.783523\pi$
$163$	−19.8534	−1.55504	−0.777520	+	0.628858i	$0.783523\pi$
$164$	0	0
$165$	−9.16329	−0.713361
$166$	0	0
$167$	−10.6319	−0.822718	−0.411359	−	0.911473i	$-0.634946\pi$
$167$	−10.6319	−0.822718	−0.411359	+	0.911473i	$0.634946\pi$
$168$	0	0
$169$	6.35893	0.489149
$170$	0	0
$171$	−36.3840	−2.78236
$172$	0	0
$173$	2.14598	0.163156	0.0815778	−	0.996667i	$-0.474004\pi$
$173$	2.14598	0.163156	0.0815778	+	0.996667i	$0.474004\pi$
$174$	0	0
$175$	5.99540	0.453210
$176$	0	0
$177$	−15.6365	−1.17531
$178$	0	0
$179$	20.5949	1.53934	0.769668	−	0.638444i	$-0.220422\pi$
$179$	20.5949	1.53934	0.769668	+	0.638444i	$0.220422\pi$
$180$	0	0
$181$	3.57904	0.266028	0.133014	−	0.991114i	$-0.457535\pi$
$181$	3.57904	0.266028	0.133014	+	0.991114i	$0.457535\pi$
$182$	0	0
$183$	3.11423	0.230211
$184$	0	0
$185$	18.6901	1.37413
$186$	0	0
$187$	−3.44748	−0.252105
$188$	0	0
$189$	4.52223	0.328944
$190$	0	0
$191$	−8.47839	−0.613475	−0.306737	−	0.951794i	$-0.599237\pi$
$191$	−8.47839	−0.613475	−0.306737	+	0.951794i	$0.599237\pi$
$192$	0	0
$193$	6.64107	0.478034	0.239017	−	0.971015i	$-0.423175\pi$
$193$	6.64107	0.478034	0.239017	+	0.971015i	$0.423175\pi$
$194$	0	0
$195$	−40.3174	−2.88719
$196$	0	0
$197$	−0.0952106	−0.00678348	−0.00339174	−	0.999994i	$-0.501080\pi$
$197$	−0.0952106	−0.00678348	−0.00339174	+	0.999994i	$0.501080\pi$
$198$	0	0
$199$	−17.6062	−1.24807	−0.624034	−	0.781397i	$-0.714507\pi$
$199$	−17.6062	−1.24807	−0.624034	+	0.781397i	$0.714507\pi$
$200$	0	0
$201$	−23.5505	−1.66112
$202$	0	0
$203$	0.894968	0.0628145
$204$	0	0
$205$	11.4316	0.798419
$206$	0	0
$207$	42.4853	2.95293
$208$	0	0
$209$	7.84736	0.542813
$210$	0	0
$211$	23.4755	1.61612	0.808059	−	0.589102i	$-0.200518\pi$
$211$	23.4755	1.61612	0.808059	+	0.589102i	$0.200518\pi$
$212$	0	0
$213$	5.22387	0.357934
$214$	0	0
$215$	−27.0537	−1.84504
$216$	0	0
$217$	−7.71581	−0.523783
$218$	0	0
$219$	−25.8442	−1.74639
$220$	0	0
$221$	−15.1685	−1.02034
$222$	0	0
$223$	18.3477	1.22865	0.614325	−	0.789053i	$-0.289428\pi$
$223$	18.3477	1.22865	0.614325	+	0.789053i	$0.289428\pi$
$224$	0	0
$225$	27.7974	1.85316
$226$	0	0
$227$	24.4279	1.62133	0.810667	−	0.585508i	$-0.199105\pi$
$227$	24.4279	1.62133	0.810667	+	0.585508i	$0.199105\pi$
$228$	0	0
$229$	−23.4746	−1.55125	−0.775623	−	0.631196i	$-0.782564\pi$
$229$	−23.4746	−1.55125	−0.775623	+	0.631196i	$0.782564\pi$
$230$	0	0
$231$	−2.76342	−0.181819
$232$	0	0
$233$	−3.10503	−0.203417	−0.101709	−	0.994814i	$-0.532431\pi$
$233$	−3.10503	−0.203417	−0.101709	+	0.994814i	$0.532431\pi$
$234$	0	0
$235$	−31.8534	−2.07789
$236$	0	0
$237$	48.4339	3.14612
$238$	0	0
$239$	4.95845	0.320735	0.160368	−	0.987057i	$-0.448732\pi$
$239$	4.95845	0.320735	0.160368	+	0.987057i	$0.448732\pi$
$240$	0	0
$241$	−2.54331	−0.163829	−0.0819146	−	0.996639i	$-0.526103\pi$
$241$	−2.54331	−0.163829	−0.0819146	+	0.996639i	$0.526103\pi$
$242$	0	0
$243$	−17.4703	−1.12072
$244$	0	0
$245$	3.31593	0.211847
$246$	0	0
$247$	34.5274	2.19693
$248$	0	0
$249$	6.85113	0.434173
$250$	0	0
$251$	−6.12235	−0.386439	−0.193220	−	0.981156i	$-0.561893\pi$
$251$	−6.12235	−0.386439	−0.193220	+	0.981156i	$0.561893\pi$
$252$	0	0
$253$	−9.16329	−0.576091
$254$	0	0
$255$	31.5903	1.97826
$256$	0	0
$257$	−15.5995	−0.973071	−0.486536	−	0.873661i	$-0.661740\pi$
$257$	−15.5995	−0.973071	−0.486536	+	0.873661i	$0.661740\pi$
$258$	0	0
$259$	5.63646	0.350233
$260$	0	0
$261$	4.14949	0.256847
$262$	0	0
$263$	−3.78073	−0.233130	−0.116565	−	0.993183i	$-0.537188\pi$
$263$	−3.78073	−0.233130	−0.116565	+	0.993183i	$0.537188\pi$
$264$	0	0
$265$	6.63186	0.407392
$266$	0	0
$267$	22.8488	1.39832
$268$	0	0
$269$	30.3840	1.85255	0.926273	−	0.376853i	$-0.122994\pi$
$269$	30.3840	1.85255	0.926273	+	0.376853i	$0.122994\pi$
$270$	0	0
$271$	17.4408	1.05945	0.529727	−	0.848168i	$-0.322294\pi$
$271$	17.4408	1.05945	0.529727	+	0.848168i	$0.322294\pi$
$272$	0	0
$273$	−12.1587	−0.735878
$274$	0	0
$275$	−5.99540	−0.361536
$276$	0	0
$277$	−17.2729	−1.03783	−0.518915	−	0.854826i	$-0.673664\pi$
$277$	−17.2729	−1.03783	−0.518915	+	0.854826i	$0.673664\pi$
$278$	0	0
$279$	−35.7741	−2.14174
$280$	0	0
$281$	−24.4853	−1.46067	−0.730335	−	0.683089i	$-0.760636\pi$
$281$	−24.4853	−1.46067	−0.730335	+	0.683089i	$0.760636\pi$
$282$	0	0
$283$	−20.2253	−1.20227	−0.601135	−	0.799147i	$-0.705285\pi$
$283$	−20.2253	−1.20227	−0.601135	+	0.799147i	$0.705285\pi$
$284$	0	0
$285$	−71.9077	−4.25944
$286$	0	0
$287$	3.44748	0.203499
$288$	0	0
$289$	−5.11485	−0.300874
$290$	0	0
$291$	−29.0773	−1.70454
$292$	0	0
$293$	−12.9804	−0.758321	−0.379161	−	0.925331i	$-0.623787\pi$
$293$	−12.9804	−0.758321	−0.379161	+	0.925331i	$0.623787\pi$
$294$	0	0
$295$	−18.7628	−1.09241
$296$	0	0
$297$	−4.52223	−0.262406
$298$	0	0
$299$	−40.3174	−2.33161
$300$	0	0
$301$	−8.15869	−0.470259
$302$	0	0
$303$	15.2124	0.873927
$304$	0	0
$305$	3.73689	0.213974
$306$	0	0
$307$	−2.62581	−0.149863	−0.0749313	−	0.997189i	$-0.523874\pi$
$307$	−2.62581	−0.149863	−0.0749313	+	0.997189i	$0.523874\pi$
$308$	0	0
$309$	10.1073	0.574985
$310$	0	0
$311$	23.8601	1.35298	0.676490	−	0.736451i	$-0.263500\pi$
$311$	23.8601	1.35298	0.676490	+	0.736451i	$0.263500\pi$
$312$	0	0
$313$	−18.4801	−1.04455	−0.522277	−	0.852776i	$-0.674917\pi$
$313$	−18.4801	−1.04455	−0.522277	+	0.852776i	$0.674917\pi$
$314$	0	0
$315$	15.3742	0.866238
$316$	0	0
$317$	−12.0491	−0.676743	−0.338371	−	0.941013i	$-0.609876\pi$
$317$	−12.0491	−0.676743	−0.338371	+	0.941013i	$0.609876\pi$
$318$	0	0
$319$	−0.894968	−0.0501086
$320$	0	0
$321$	−12.2193	−0.682013
$322$	0	0
$323$	−27.0537	−1.50531
$324$	0	0
$325$	−26.3790	−1.46325
$326$	0	0
$327$	−35.3456	−1.95462
$328$	0	0
$329$	−9.60618	−0.529606
$330$	0	0
$331$	7.53205	0.413999	0.206999	−	0.978341i	$-0.433630\pi$
$331$	7.53205	0.413999	0.206999	+	0.978341i	$0.433630\pi$
$332$	0	0
$333$	26.1333	1.43209
$334$	0	0
$335$	−28.2591	−1.54396
$336$	0	0
$337$	12.0635	0.657140	0.328570	−	0.944480i	$-0.393433\pi$
$337$	12.0635	0.657140	0.328570	+	0.944480i	$0.393433\pi$
$338$	0	0
$339$	−13.8044	−0.749750
$340$	0	0
$341$	7.71581	0.417835
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	83.9660	4.52058
$346$	0	0
$347$	13.2216	0.709770	0.354885	−	0.934910i	$-0.384520\pi$
$347$	13.2216	0.709770	0.354885	+	0.934910i	$0.384520\pi$
$348$	0	0
$349$	−24.3393	−1.30285	−0.651427	−	0.758712i	$-0.725829\pi$
$349$	−24.3393	−1.30285	−0.651427	+	0.758712i	$0.725829\pi$
$350$	0	0
$351$	−19.8973	−1.06204
$352$	0	0
$353$	6.98619	0.371838	0.185919	−	0.982565i	$-0.440474\pi$
$353$	6.98619	0.371838	0.185919	+	0.982565i	$0.440474\pi$
$354$	0	0
$355$	6.26833	0.332688
$356$	0	0
$357$	9.52683	0.504213
$358$	0	0
$359$	−23.2124	−1.22510	−0.612551	−	0.790431i	$-0.709856\pi$
$359$	−23.2124	−1.22510	−0.612551	+	0.790431i	$0.709856\pi$
$360$	0	0
$361$	42.5811	2.24111
$362$	0	0
$363$	2.76342	0.145042
$364$	0	0
$365$	−31.0115	−1.62322
$366$	0	0
$367$	13.0839	0.682977	0.341488	−	0.939886i	$-0.389069\pi$
$367$	13.0839	0.682977	0.341488	+	0.939886i	$0.389069\pi$
$368$	0	0
$369$	15.9841	0.832101
$370$	0	0
$371$	2.00000	0.103835
$372$	0	0
$373$	−10.4218	−0.539620	−0.269810	−	0.962914i	$-0.586961\pi$
$373$	−10.4218	−0.539620	−0.269810	+	0.962914i	$0.586961\pi$
$374$	0	0
$375$	9.12113	0.471013
$376$	0	0
$377$	−3.93775	−0.202805
$378$	0	0
$379$	−3.98558	−0.204725	−0.102363	−	0.994747i	$-0.532640\pi$
$379$	−3.98558	−0.204725	−0.102363	+	0.994747i	$0.532640\pi$
$380$	0	0
$381$	45.3802	2.32490
$382$	0	0
$383$	8.60157	0.439520	0.219760	−	0.975554i	$-0.429473\pi$
$383$	8.60157	0.439520	0.219760	+	0.975554i	$0.429473\pi$
$384$	0	0
$385$	−3.31593	−0.168995
$386$	0	0
$387$	−37.8275	−1.92288
$388$	0	0
$389$	21.4172	1.08590	0.542948	−	0.839767i	$-0.317308\pi$
$389$	21.4172	1.08590	0.542948	+	0.839767i	$0.317308\pi$
$390$	0	0
$391$	31.5903	1.59759
$392$	0	0
$393$	0.860332	0.0433980
$394$	0	0
$395$	58.1178	2.92422
$396$	0	0
$397$	−18.0921	−0.908015	−0.454007	−	0.890998i	$-0.650006\pi$
$397$	−18.0921	−0.908015	−0.454007	+	0.890998i	$0.650006\pi$
$398$	0	0
$399$	−21.6855	−1.08563
$400$	0	0
$401$	25.3895	1.26789	0.633944	−	0.773379i	$-0.281435\pi$
$401$	25.3895	1.26789	0.633944	+	0.773379i	$0.281435\pi$
$402$	0	0
$403$	33.9486	1.69110
$404$	0	0
$405$	−4.68407	−0.232753
$406$	0	0
$407$	−5.63646	−0.279389
$408$	0	0
$409$	−7.44748	−0.368254	−0.184127	−	0.982902i	$-0.558946\pi$
$409$	−7.44748	−0.368254	−0.184127	+	0.982902i	$0.558946\pi$
$410$	0	0
$411$	21.0773	1.03967
$412$	0	0
$413$	−5.65838	−0.278431
$414$	0	0
$415$	8.22094	0.403550
$416$	0	0
$417$	−8.86033	−0.433892
$418$	0	0
$419$	−18.0219	−0.880428	−0.440214	−	0.897893i	$-0.645097\pi$
$419$	−18.0219	−0.880428	−0.440214	+	0.897893i	$0.645097\pi$
$420$	0	0
$421$	−6.84360	−0.333537	−0.166768	−	0.985996i	$-0.553333\pi$
$421$	−6.84360	−0.333537	−0.166768	+	0.985996i	$0.553333\pi$
$422$	0	0
$423$	−44.5387	−2.16555
$424$	0	0
$425$	20.6690	1.00260
$426$	0	0
$427$	1.12695	0.0545370
$428$	0	0
$429$	12.1587	0.587027
$430$	0	0
$431$	−20.9146	−1.00742	−0.503711	−	0.863872i	$-0.668032\pi$
$431$	−20.9146	−1.00742	−0.503711	+	0.863872i	$0.668032\pi$
$432$	0	0
$433$	−9.06808	−0.435784	−0.217892	−	0.975973i	$-0.569918\pi$
$433$	−9.06808	−0.435784	−0.217892	+	0.975973i	$0.569918\pi$
$434$	0	0
$435$	8.20086	0.393201
$436$	0	0
$437$	−71.9077	−3.43981
$438$	0	0
$439$	4.90417	0.234063	0.117032	−	0.993128i	$-0.462662\pi$
$439$	4.90417	0.234063	0.117032	+	0.993128i	$0.462662\pi$
$440$	0	0
$441$	4.63646	0.220784
$442$	0	0
$443$	−30.9215	−1.46912	−0.734562	−	0.678541i	$-0.762613\pi$
$443$	−30.9215	−1.46912	−0.734562	+	0.678541i	$0.762613\pi$
$444$	0	0
$445$	27.4172	1.29970
$446$	0	0
$447$	10.4732	0.495364
$448$	0	0
$449$	12.6901	0.598884	0.299442	−	0.954115i	$-0.403199\pi$
$449$	12.6901	0.598884	0.299442	+	0.954115i	$0.403199\pi$
$450$	0	0
$451$	−3.44748	−0.162336
$452$	0	0
$453$	23.1518	1.08777
$454$	0	0
$455$	−14.5897	−0.683975
$456$	0	0
$457$	37.4755	1.75303	0.876514	−	0.481376i	$-0.159863\pi$
$457$	37.4755	1.75303	0.876514	+	0.481376i	$0.159863\pi$
$458$	0	0
$459$	15.5903	0.727693
$460$	0	0
$461$	5.24180	0.244135	0.122068	−	0.992522i	$-0.461048\pi$
$461$	5.24180	0.244135	0.122068	+	0.992522i	$0.461048\pi$
$462$	0	0
$463$	−14.9532	−0.694936	−0.347468	−	0.937692i	$-0.612958\pi$
$463$	−14.9532	−0.694936	−0.347468	+	0.937692i	$0.612958\pi$
$464$	0	0
$465$	−70.7022	−3.27874
$466$	0	0
$467$	−4.97348	−0.230145	−0.115073	−	0.993357i	$-0.536710\pi$
$467$	−4.97348	−0.230145	−0.115073	+	0.993357i	$0.536710\pi$
$468$	0	0
$469$	−8.52223	−0.393520
$470$	0	0
$471$	67.0802	3.09089
$472$	0	0
$473$	8.15869	0.375137
$474$	0	0
$475$	−47.0481	−2.15871
$476$	0	0
$477$	9.27293	0.424578
$478$	0	0
$479$	−23.5360	−1.07539	−0.537694	−	0.843140i	$-0.680705\pi$
$479$	−23.5360	−1.07539	−0.537694	+	0.843140i	$0.680705\pi$
$480$	0	0
$481$	−24.7998	−1.13077
$482$	0	0
$483$	25.3220	1.15219
$484$	0	0
$485$	−34.8910	−1.58432
$486$	0	0
$487$	−2.64628	−0.119915	−0.0599573	−	0.998201i	$-0.519096\pi$
$487$	−2.64628	−0.119915	−0.0599573	+	0.998201i	$0.519096\pi$
$488$	0	0
$489$	−54.8632	−2.48100
$490$	0	0
$491$	26.7906	1.20904	0.604520	−	0.796590i	$-0.293365\pi$
$491$	26.7906	1.20904	0.604520	+	0.796590i	$0.293365\pi$
$492$	0	0
$493$	3.08539	0.138959
$494$	0	0
$495$	−15.3742	−0.691019
$496$	0	0
$497$	1.89037	0.0847945
$498$	0	0
$499$	11.5360	0.516424	0.258212	−	0.966088i	$-0.416867\pi$
$499$	11.5360	0.516424	0.258212	+	0.966088i	$0.416867\pi$
$500$	0	0
$501$	−29.3802	−1.31261
$502$	0	0
$503$	−19.6509	−0.876190	−0.438095	−	0.898929i	$-0.644347\pi$
$503$	−19.6509	−0.876190	−0.438095	+	0.898929i	$0.644347\pi$
$504$	0	0
$505$	18.2539	0.812288
$506$	0	0
$507$	17.5724	0.780417
$508$	0	0
$509$	16.8428	0.746542	0.373271	−	0.927722i	$-0.378236\pi$
$509$	16.8428	0.746542	0.373271	+	0.927722i	$0.378236\pi$
$510$	0	0
$511$	−9.35227	−0.413720
$512$	0	0
$513$	−35.4876	−1.56681
$514$	0	0
$515$	12.1282	0.534431
$516$	0	0
$517$	9.60618	0.422479
$518$	0	0
$519$	5.93022	0.260308
$520$	0	0
$521$	−13.5759	−0.594771	−0.297385	−	0.954758i	$-0.596115\pi$
$521$	−13.5759	−0.594771	−0.297385	+	0.954758i	$0.596115\pi$
$522$	0	0
$523$	−7.88953	−0.344985	−0.172493	−	0.985011i	$-0.555182\pi$
$523$	−7.88953	−0.344985	−0.172493	+	0.985011i	$0.555182\pi$
$524$	0	0
$525$	16.5678	0.723077
$526$	0	0
$527$	−26.6001	−1.15872
$528$	0	0
$529$	60.9660	2.65069
$530$	0	0
$531$	−26.2349	−1.13850
$532$	0	0
$533$	−15.1685	−0.657021
$534$	0	0
$535$	−14.6624	−0.633910
$536$	0	0
$537$	56.9123	2.45595
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	20.9677	0.901470	0.450735	−	0.892658i	$-0.351162\pi$
$541$	20.9677	0.901470	0.450735	+	0.892658i	$0.351162\pi$
$542$	0	0
$543$	9.89037	0.424436
$544$	0	0
$545$	−42.4126	−1.81676
$546$	0	0
$547$	22.1257	0.946028	0.473014	−	0.881055i	$-0.343166\pi$
$547$	22.1257	0.946028	0.473014	+	0.881055i	$0.343166\pi$
$548$	0	0
$549$	5.22507	0.223001
$550$	0	0
$551$	−7.02314	−0.299196
$552$	0	0
$553$	17.5268	0.745317
$554$	0	0
$555$	51.6486	2.19236
$556$	0	0
$557$	−13.1869	−0.558748	−0.279374	−	0.960182i	$-0.590127\pi$
$557$	−13.1869	−0.558748	−0.279374	+	0.960182i	$0.590127\pi$
$558$	0	0
$559$	35.8973	1.51829
$560$	0	0
$561$	−9.52683	−0.402223
$562$	0	0
$563$	34.1740	1.44026	0.720130	−	0.693839i	$-0.244082\pi$
$563$	34.1740	1.44026	0.720130	+	0.693839i	$0.244082\pi$
$564$	0	0
$565$	−16.5644	−0.696869
$566$	0	0
$567$	−1.41260	−0.0593234
$568$	0	0
$569$	−16.0635	−0.673416	−0.336708	−	0.941609i	$-0.609314\pi$
$569$	−16.0635	−0.673416	−0.336708	+	0.941609i	$0.609314\pi$
$570$	0	0
$571$	7.57987	0.317208	0.158604	−	0.987342i	$-0.449301\pi$
$571$	7.57987	0.317208	0.158604	+	0.987342i	$0.449301\pi$
$572$	0	0
$573$	−23.4293	−0.978773
$574$	0	0
$575$	54.9376	2.29106
$576$	0	0
$577$	39.0894	1.62731	0.813656	−	0.581346i	$-0.197474\pi$
$577$	39.0894	1.62731	0.813656	+	0.581346i	$0.197474\pi$
$578$	0	0
$579$	18.3520	0.762684
$580$	0	0
$581$	2.47923	0.102856
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	0	0
$585$	−67.6446	−2.79676
$586$	0	0
$587$	0.117131	0.00483450	0.00241725	−	0.999997i	$-0.499231\pi$
$587$	0.117131	0.00483450	0.00241725	+	0.999997i	$0.499231\pi$
$588$	0	0
$589$	60.5488	2.49487
$590$	0	0
$591$	−0.263107	−0.0108228
$592$	0	0
$593$	16.2472	0.667194	0.333597	−	0.942716i	$-0.391737\pi$
$593$	16.2472	0.667194	0.333597	+	0.942716i	$0.391737\pi$
$594$	0	0
$595$	11.4316	0.468651
$596$	0	0
$597$	−48.6532	−1.99124
$598$	0	0
$599$	−1.61284	−0.0658988	−0.0329494	−	0.999457i	$-0.510490\pi$
$599$	−1.61284	−0.0658988	−0.0329494	+	0.999457i	$0.510490\pi$
$600$	0	0
$601$	−1.92065	−0.0783451	−0.0391726	−	0.999232i	$-0.512472\pi$
$601$	−1.92065	−0.0783451	−0.0391726	+	0.999232i	$0.512472\pi$
$602$	0	0
$603$	−39.5130	−1.60909
$604$	0	0
$605$	3.31593	0.134812
$606$	0	0
$607$	36.5367	1.48298	0.741488	−	0.670966i	$-0.234120\pi$
$607$	36.5367	1.48298	0.741488	+	0.670966i	$0.234120\pi$
$608$	0	0
$609$	2.47317	0.100218
$610$	0	0
$611$	42.2660	1.70990
$612$	0	0
$613$	32.4853	1.31207	0.656034	−	0.754731i	$-0.272233\pi$
$613$	32.4853	1.31207	0.656034	+	0.754731i	$0.272233\pi$
$614$	0	0
$615$	31.5903	1.27384
$616$	0	0
$617$	14.0121	0.564106	0.282053	−	0.959399i	$-0.408985\pi$
$617$	14.0121	0.564106	0.282053	+	0.959399i	$0.408985\pi$
$618$	0	0
$619$	1.54353	0.0620398	0.0310199	−	0.999519i	$-0.490124\pi$
$619$	1.54353	0.0620398	0.0310199	+	0.999519i	$0.490124\pi$
$620$	0	0
$621$	41.4385	1.66287
$622$	0	0
$623$	8.26833	0.331263
$624$	0	0
$625$	−19.0322	−0.761288
$626$	0	0
$627$	21.6855	0.866036
$628$	0	0
$629$	19.4316	0.774789
$630$	0	0
$631$	42.1656	1.67859	0.839293	−	0.543680i	$-0.182969\pi$
$631$	42.1656	1.67859	0.839293	+	0.543680i	$0.182969\pi$
$632$	0	0
$633$	64.8724	2.57845
$634$	0	0
$635$	54.4536	2.16092
$636$	0	0
$637$	−4.39988	−0.174330
$638$	0	0
$639$	8.76462	0.346723
$640$	0	0
$641$	29.0075	1.14573	0.572864	−	0.819651i	$-0.305832\pi$
$641$	29.0075	1.14573	0.572864	+	0.819651i	$0.305832\pi$
$642$	0	0
$643$	24.9129	0.982469	0.491235	−	0.871027i	$-0.336546\pi$
$643$	24.9129	0.982469	0.491235	+	0.871027i	$0.336546\pi$
$644$	0	0
$645$	−74.7605	−2.94369
$646$	0	0
$647$	6.62919	0.260620	0.130310	−	0.991473i	$-0.458403\pi$
$647$	6.62919	0.260620	0.130310	+	0.991473i	$0.458403\pi$
$648$	0	0
$649$	5.65838	0.222111
$650$	0	0
$651$	−21.3220	−0.835675
$652$	0	0
$653$	0.354333	0.0138661	0.00693306	−	0.999976i	$-0.497793\pi$
$653$	0.354333	0.0138661	0.00693306	+	0.999976i	$0.497793\pi$
$654$	0	0
$655$	1.03235	0.0403371
$656$	0	0
$657$	−43.3615	−1.69169
$658$	0	0
$659$	−20.3780	−0.793813	−0.396906	−	0.917859i	$-0.629916\pi$
$659$	−20.3780	−0.793813	−0.396906	+	0.917859i	$0.629916\pi$
$660$	0	0
$661$	16.5797	0.644874	0.322437	−	0.946591i	$-0.395498\pi$
$661$	16.5797	0.644874	0.322437	+	0.946591i	$0.395498\pi$
$662$	0	0
$663$	−41.9169	−1.62792
$664$	0	0
$665$	−26.0213	−1.00906
$666$	0	0
$667$	8.20086	0.317539
$668$	0	0
$669$	50.7022	1.96026
$670$	0	0
$671$	−1.12695	−0.0435055
$672$	0	0
$673$	−10.3780	−0.400041	−0.200020	−	0.979792i	$-0.564101\pi$
$673$	−10.3780	−0.400041	−0.200020	+	0.979792i	$0.564101\pi$
$674$	0	0
$675$	27.1126	1.04356
$676$	0	0
$677$	5.08311	0.195360	0.0976799	−	0.995218i	$-0.468858\pi$
$677$	5.08311	0.195360	0.0976799	+	0.995218i	$0.468858\pi$
$678$	0	0
$679$	−10.5222	−0.403806
$680$	0	0
$681$	67.5043	2.58677
$682$	0	0
$683$	−27.7068	−1.06017	−0.530086	−	0.847944i	$-0.677840\pi$
$683$	−27.7068	−1.06017	−0.530086	+	0.847944i	$0.677840\pi$
$684$	0	0
$685$	25.2915	0.966337
$686$	0	0
$687$	−64.8701	−2.47495
$688$	0	0
$689$	−8.79976	−0.335244
$690$	0	0
$691$	39.9227	1.51873	0.759366	−	0.650664i	$-0.225509\pi$
$691$	39.9227	1.51873	0.759366	+	0.650664i	$0.225509\pi$
$692$	0	0
$693$	−4.63646	−0.176125
$694$	0	0
$695$	−10.6319	−0.403290
$696$	0	0
$697$	11.8851	0.450182
$698$	0	0
$699$	−8.58049	−0.324544
$700$	0	0
$701$	−7.00062	−0.264410	−0.132205	−	0.991222i	$-0.542206\pi$
$701$	−7.00062	−0.264410	−0.132205	+	0.991222i	$0.542206\pi$
$702$	0	0
$703$	−44.2314	−1.66822
$704$	0	0
$705$	−88.0242	−3.31518
$706$	0	0
$707$	5.50491	0.207033
$708$	0	0
$709$	1.63646	0.0614587	0.0307293	−	0.999528i	$-0.490217\pi$
$709$	1.63646	0.0614587	0.0307293	+	0.999528i	$0.490217\pi$
$710$	0	0
$711$	81.2625	3.04758
$712$	0	0
$713$	−70.7022	−2.64782
$714$	0	0
$715$	14.5897	0.545624
$716$	0	0
$717$	13.7023	0.511720
$718$	0	0
$719$	−40.5594	−1.51261	−0.756305	−	0.654219i	$-0.772998\pi$
$719$	−40.5594	−1.51261	−0.756305	+	0.654219i	$0.772998\pi$
$720$	0	0
$721$	3.65755	0.136214
$722$	0	0
$723$	−7.02823	−0.261383
$724$	0	0
$725$	5.36569	0.199277
$726$	0	0
$727$	37.5479	1.39258	0.696288	−	0.717763i	$-0.254834\pi$
$727$	37.5479	1.39258	0.696288	+	0.717763i	$0.254834\pi$
$728$	0	0
$729$	−44.0398	−1.63110
$730$	0	0
$731$	−28.1270	−1.04031
$732$	0	0
$733$	15.1177	0.558387	0.279193	−	0.960235i	$-0.409933\pi$
$733$	15.1177	0.558387	0.279193	+	0.960235i	$0.409933\pi$
$734$	0	0
$735$	9.16329	0.337993
$736$	0	0
$737$	8.52223	0.313920
$738$	0	0
$739$	26.6624	0.980792	0.490396	−	0.871500i	$-0.336852\pi$
$739$	26.6624	0.980792	0.490396	+	0.871500i	$0.336852\pi$
$740$	0	0
$741$	95.4137	3.50511
$742$	0	0
$743$	51.0658	1.87342	0.936711	−	0.350104i	$-0.113854\pi$
$743$	51.0658	1.87342	0.936711	+	0.350104i	$0.113854\pi$
$744$	0	0
$745$	12.5672	0.460425
$746$	0	0
$747$	11.4948	0.420574
$748$	0	0
$749$	−4.42180	−0.161569
$750$	0	0
$751$	−9.68951	−0.353575	−0.176788	−	0.984249i	$-0.556571\pi$
$751$	−9.68951	−0.353575	−0.176788	+	0.984249i	$0.556571\pi$
$752$	0	0
$753$	−16.9186	−0.616548
$754$	0	0
$755$	27.7807	1.01104
$756$	0	0
$757$	−7.88638	−0.286635	−0.143318	−	0.989677i	$-0.545777\pi$
$757$	−7.88638	−0.286635	−0.143318	+	0.989677i	$0.545777\pi$
$758$	0	0
$759$	−25.3220	−0.919130
$760$	0	0
$761$	45.1768	1.63766	0.818830	−	0.574036i	$-0.194623\pi$
$761$	45.1768	1.63766	0.818830	+	0.574036i	$0.194623\pi$
$762$	0	0
$763$	−12.7906	−0.463049
$764$	0	0
$765$	53.0023	1.91630
$766$	0	0
$767$	24.8962	0.898950
$768$	0	0
$769$	−26.6177	−0.959858	−0.479929	−	0.877307i	$-0.659338\pi$
$769$	−26.6177	−0.959858	−0.479929	+	0.877307i	$0.659338\pi$
$770$	0	0
$771$	−43.1079	−1.55249
$772$	0	0
$773$	8.98703	0.323241	0.161621	−	0.986853i	$-0.448328\pi$
$773$	8.98703	0.323241	0.161621	+	0.986853i	$0.448328\pi$
$774$	0	0
$775$	−46.2594	−1.66169
$776$	0	0
$777$	15.5759	0.558782
$778$	0	0
$779$	−27.0537	−0.969298
$780$	0	0
$781$	−1.89037	−0.0676426
$782$	0	0
$783$	4.04725	0.144637
$784$	0	0
$785$	80.4922	2.87289
$786$	0	0
$787$	15.0159	0.535258	0.267629	−	0.963522i	$-0.413760\pi$
$787$	15.0159	0.535258	0.267629	+	0.963522i	$0.413760\pi$
$788$	0	0
$789$	−10.4477	−0.371949
$790$	0	0
$791$	−4.99540	−0.177616
$792$	0	0
$793$	−4.95845	−0.176080
$794$	0	0
$795$	18.3266	0.649977
$796$	0	0
$797$	−39.7498	−1.40801	−0.704006	−	0.710194i	$-0.748607\pi$
$797$	−39.7498	−1.40801	−0.704006	+	0.710194i	$0.748607\pi$
$798$	0	0
$799$	−33.1171	−1.17160
$800$	0	0
$801$	38.3358	1.35453
$802$	0	0
$803$	9.35227	0.330035
$804$	0	0
$805$	30.3849	1.07093
$806$	0	0
$807$	83.9636	2.95566
$808$	0	0
$809$	−53.6947	−1.88781	−0.943903	−	0.330223i	$-0.892876\pi$
$809$	−53.6947	−1.88781	−0.943903	+	0.330223i	$0.892876\pi$
$810$	0	0
$811$	8.13931	0.285810	0.142905	−	0.989736i	$-0.454356\pi$
$811$	8.13931	0.285810	0.142905	+	0.989736i	$0.454356\pi$
$812$	0	0
$813$	48.1962	1.69032
$814$	0	0
$815$	−65.8326	−2.30601
$816$	0	0
$817$	64.0242	2.23992
$818$	0	0
$819$	−20.3999	−0.712829
$820$	0	0
$821$	14.8436	0.518045	0.259023	−	0.965871i	$-0.416600\pi$
$821$	14.8436	0.518045	0.259023	+	0.965871i	$0.416600\pi$
$822$	0	0
$823$	23.0242	0.802575	0.401287	−	0.915952i	$-0.368563\pi$
$823$	23.0242	0.802575	0.401287	+	0.915952i	$0.368563\pi$
$824$	0	0
$825$	−16.5678	−0.576816
$826$	0	0
$827$	−3.02823	−0.105302	−0.0526509	−	0.998613i	$-0.516767\pi$
$827$	−3.02823	−0.105302	−0.0526509	+	0.998613i	$0.516767\pi$
$828$	0	0
$829$	−51.5473	−1.79031	−0.895156	−	0.445753i	$-0.852936\pi$
$829$	−51.5473	−1.79031	−0.895156	+	0.445753i	$0.852936\pi$
$830$	0	0
$831$	−47.7323	−1.65581
$832$	0	0
$833$	3.44748	0.119448
$834$	0	0
$835$	−35.2545	−1.22003
$836$	0	0
$837$	−34.8927	−1.20607
$838$	0	0
$839$	25.3575	0.875438	0.437719	−	0.899112i	$-0.355786\pi$
$839$	25.3575	0.875438	0.437719	+	0.899112i	$0.355786\pi$
$840$	0	0
$841$	−28.1990	−0.972380
$842$	0	0
$843$	−67.6630	−2.33044
$844$	0	0
$845$	21.0858	0.725373
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−55.8910	−1.91817
$850$	0	0
$851$	51.6486	1.77049
$852$	0	0
$853$	−21.9705	−0.752257	−0.376129	−	0.926567i	$-0.622745\pi$
$853$	−21.9705	−0.752257	−0.376129	+	0.926567i	$0.622745\pi$
$854$	0	0
$855$	−120.647	−4.12604
$856$	0	0
$857$	49.9144	1.70504	0.852521	−	0.522693i	$-0.175073\pi$
$857$	49.9144	1.70504	0.852521	+	0.522693i	$0.175073\pi$
$858$	0	0
$859$	−55.2833	−1.88624	−0.943121	−	0.332448i	$-0.892125\pi$
$859$	−55.2833	−1.88624	−0.943121	+	0.332448i	$0.892125\pi$
$860$	0	0
$861$	9.52683	0.324674
$862$	0	0
$863$	22.5459	0.767470	0.383735	−	0.923443i	$-0.374638\pi$
$863$	22.5459	0.767470	0.383735	+	0.923443i	$0.374638\pi$
$864$	0	0
$865$	7.11591	0.241948
$866$	0	0
$867$	−14.1345	−0.480031
$868$	0	0
$869$	−17.5268	−0.594557
$870$	0	0
$871$	37.4968	1.27053
$872$	0	0
$873$	−48.7859	−1.65115
$874$	0	0
$875$	3.30067	0.111583
$876$	0	0
$877$	−54.1892	−1.82984	−0.914920	−	0.403636i	$-0.867746\pi$
$877$	−54.1892	−1.82984	−0.914920	+	0.403636i	$0.867746\pi$
$878$	0	0
$879$	−35.8702	−1.20987
$880$	0	0
$881$	22.2458	0.749480	0.374740	−	0.927130i	$-0.377732\pi$
$881$	22.2458	0.749480	0.374740	+	0.927130i	$0.377732\pi$
$882$	0	0
$883$	−51.3895	−1.72939	−0.864696	−	0.502295i	$-0.832489\pi$
$883$	−51.3895	−1.72939	−0.864696	+	0.502295i	$0.832489\pi$
$884$	0	0
$885$	−51.8494	−1.74290
$886$	0	0
$887$	−34.3888	−1.15466	−0.577332	−	0.816509i	$-0.695906\pi$
$887$	−34.3888	−1.15466	−0.577332	+	0.816509i	$0.695906\pi$
$888$	0	0
$889$	16.4218	0.550769
$890$	0	0
$891$	1.41260	0.0473237
$892$	0	0
$893$	75.3832	2.52260
$894$	0	0
$895$	68.2913	2.28273
$896$	0	0
$897$	−111.414	−3.71999
$898$	0	0
$899$	−6.90541	−0.230308
$900$	0	0
$901$	6.89497	0.229705
$902$	0	0
$903$	−22.5459	−0.750279
$904$	0	0
$905$	11.8678	0.394500
$906$	0	0
$907$	23.5176	0.780890	0.390445	−	0.920626i	$-0.372321\pi$
$907$	23.5176	0.780890	0.390445	+	0.920626i	$0.372321\pi$
$908$	0	0
$909$	25.5233	0.846555
$910$	0	0
$911$	0.684905	0.0226919	0.0113460	−	0.999936i	$-0.496388\pi$
$911$	0.684905	0.0226919	0.0113460	+	0.999936i	$0.496388\pi$
$912$	0	0
$913$	−2.47923	−0.0820504
$914$	0	0
$915$	10.3266	0.341386
$916$	0	0
$917$	0.311329	0.0102810
$918$	0	0
$919$	−20.3358	−0.670816	−0.335408	−	0.942073i	$-0.608874\pi$
$919$	−20.3358	−0.670816	−0.335408	+	0.942073i	$0.608874\pi$
$920$	0	0
$921$	−7.25619	−0.239100
$922$	0	0
$923$	−8.31738	−0.273770
$924$	0	0
$925$	33.7928	1.11110
$926$	0	0
$927$	16.9581	0.556977
$928$	0	0
$929$	−49.7507	−1.63227	−0.816133	−	0.577864i	$-0.803887\pi$
$929$	−49.7507	−1.63227	−0.816133	+	0.577864i	$0.803887\pi$
$930$	0	0
$931$	−7.84736	−0.257187
$932$	0	0
$933$	65.9353	2.15863
$934$	0	0
$935$	−11.4316	−0.373854
$936$	0	0
$937$	14.9940	0.489831	0.244916	−	0.969544i	$-0.421240\pi$
$937$	14.9940	0.489831	0.244916	+	0.969544i	$0.421240\pi$
$938$	0	0
$939$	−51.0681	−1.66654
$940$	0	0
$941$	−8.47256	−0.276198	−0.138099	−	0.990418i	$-0.544099\pi$
$941$	−8.47256	−0.276198	−0.138099	+	0.990418i	$0.544099\pi$
$942$	0	0
$943$	31.5903	1.02872
$944$	0	0
$945$	14.9954	0.487800
$946$	0	0
$947$	−11.5136	−0.374143	−0.187072	−	0.982346i	$-0.559900\pi$
$947$	−11.5136	−0.374143	−0.187072	+	0.982346i	$0.559900\pi$
$948$	0	0
$949$	41.1489	1.33575
$950$	0	0
$951$	−33.2966	−1.07971
$952$	0	0
$953$	47.3802	1.53480	0.767398	−	0.641171i	$-0.221551\pi$
$953$	47.3802	1.53480	0.767398	+	0.641171i	$0.221551\pi$
$954$	0	0
$955$	−28.1138	−0.909740
$956$	0	0
$957$	−2.47317	−0.0799462
$958$	0	0
$959$	7.62726	0.246297
$960$	0	0
$961$	28.5337	0.920443
$962$	0	0
$963$	−20.5015	−0.660652
$964$	0	0
$965$	22.0213	0.708891
$966$	0	0
$967$	5.64168	0.181424	0.0907121	−	0.995877i	$-0.471086\pi$
$967$	5.64168	0.181424	0.0907121	+	0.995877i	$0.471086\pi$
$968$	0	0
$969$	−74.7605	−2.40165
$970$	0	0
$971$	−30.6895	−0.984874	−0.492437	−	0.870348i	$-0.663894\pi$
$971$	−30.6895	−0.984874	−0.492437	+	0.870348i	$0.663894\pi$
$972$	0	0
$973$	−3.20630	−0.102789
$974$	0	0
$975$	−72.8962	−2.33455
$976$	0	0
$977$	−39.7317	−1.27113	−0.635564	−	0.772048i	$-0.719233\pi$
$977$	−39.7317	−1.27113	−0.635564	+	0.772048i	$0.719233\pi$
$978$	0	0
$979$	−8.26833	−0.264257
$980$	0	0
$981$	−59.3029	−1.89340
$982$	0	0
$983$	50.0424	1.59610	0.798052	−	0.602588i	$-0.205864\pi$
$983$	50.0424	1.59610	0.798052	+	0.602588i	$0.205864\pi$
$984$	0	0
$985$	−0.315712	−0.0100594
$986$	0	0
$987$	−26.5459	−0.844964
$988$	0	0
$989$	−74.7605	−2.37725
$990$	0	0
$991$	−58.6018	−1.86155	−0.930774	−	0.365595i	$-0.880865\pi$
$991$	−58.6018	−1.86155	−0.930774	+	0.365595i	$0.880865\pi$
$992$	0	0
$993$	20.8142	0.660518
$994$	0	0
$995$	−58.3809	−1.85080
$996$	0	0
$997$	19.6122	0.621126	0.310563	−	0.950553i	$-0.399482\pi$
$997$	19.6122	0.621126	0.310563	+	0.950553i	$0.399482\pi$
$998$	0	0
$999$	25.4894	0.806448

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.2.a.s.1.4		4
4.3	odd	2		616.2.a.h.1.1	✓	4
7.6	odd	2		8624.2.a.cy.1.1		4
8.3	odd	2		4928.2.a.cc.1.4		4
8.5	even	2		4928.2.a.ch.1.1		4
12.11	even	2		5544.2.a.bm.1.2		4
28.27	even	2		4312.2.a.z.1.4		4
44.43	even	2		6776.2.a.bb.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.2.a.h.1.1	✓	4	4.3	odd	2
1232.2.a.s.1.4		4	1.1	even	1	trivial
4312.2.a.z.1.4		4	28.27	even	2
4928.2.a.cc.1.4		4	8.3	odd	2
4928.2.a.ch.1.1		4	8.5	even	2
5544.2.a.bm.1.2		4	12.11	even	2
6776.2.a.bb.1.1		4	44.43	even	2
8624.2.a.cy.1.1		4	7.6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1232.a (trivial)

\(f(q)\)	\(=\)	\(q+2.76342 q^{3} +3.31593 q^{5} +1.00000 q^{7} +4.63646 q^{9} +O(q^{10})\)	\(q+2.76342 q^{3} +3.31593 q^{5} +1.00000 q^{7} +4.63646 q^{9} -1.00000 q^{11} -4.39988 q^{13} +9.16329 q^{15} +3.44748 q^{17} -7.84736 q^{19} +2.76342 q^{21} +9.16329 q^{23} +5.99540 q^{25} +4.52223 q^{27} +0.894968 q^{29} -7.71581 q^{31} -2.76342 q^{33} +3.31593 q^{35} +5.63646 q^{37} -12.1587 q^{39} +3.44748 q^{41} -8.15869 q^{43} +15.3742 q^{45} -9.60618 q^{47} +1.00000 q^{49} +9.52683 q^{51} +2.00000 q^{53} -3.31593 q^{55} -21.6855 q^{57} -5.65838 q^{59} +1.12695 q^{61} +4.63646 q^{63} -14.5897 q^{65} -8.52223 q^{67} +25.3220 q^{69} +1.89037 q^{71} -9.35227 q^{73} +16.5678 q^{75} -1.00000 q^{77} +17.5268 q^{79} -1.41260 q^{81} +2.47923 q^{83} +11.4316 q^{85} +2.47317 q^{87} +8.26833 q^{89} -4.39988 q^{91} -21.3220 q^{93} -26.0213 q^{95} -10.5222 q^{97} -4.63646 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 5 q^{5} + 4 q^{7} + 9 q^{9}+O(q^{10}) \) 4 * q - q^3 + 5 * q^5 + 4 * q^7 + 9 * q^9	\( 4 q - q^{3} + 5 q^{5} + 4 q^{7} + 9 q^{9} - 4 q^{11} + 4 q^{13} + 3 q^{15} + 10 q^{17} - 6 q^{19} - q^{21} + 3 q^{23} + 17 q^{25} - 13 q^{27} - 4 q^{29} - q^{31} + q^{33} + 5 q^{35} + 13 q^{37} - 8 q^{39} + 10 q^{41} + 8 q^{43} + 12 q^{45} + 6 q^{47} + 4 q^{49} + 14 q^{51} + 8 q^{53} - 5 q^{55} - 22 q^{57} - 3 q^{59} + 2 q^{61} + 9 q^{63} - 3 q^{67} + 27 q^{69} - 7 q^{71} + 2 q^{73} + 18 q^{75} - 4 q^{77} + 46 q^{79} + 40 q^{81} - 32 q^{83} - 14 q^{85} + 34 q^{87} + 7 q^{89} + 4 q^{91} - 11 q^{93} + 14 q^{95} - 11 q^{97} - 9 q^{99}+O(q^{100}) \) 4 * q - q^3 + 5 * q^5 + 4 * q^7 + 9 * q^9 - 4 * q^11 + 4 * q^13 + 3 * q^15 + 10 * q^17 - 6 * q^19 - q^21 + 3 * q^23 + 17 * q^25 - 13 * q^27 - 4 * q^29 - q^31 + q^33 + 5 * q^35 + 13 * q^37 - 8 * q^39 + 10 * q^41 + 8 * q^43 + 12 * q^45 + 6 * q^47 + 4 * q^49 + 14 * q^51 + 8 * q^53 - 5 * q^55 - 22 * q^57 - 3 * q^59 + 2 * q^61 + 9 * q^63 - 3 * q^67 + 27 * q^69 - 7 * q^71 + 2 * q^73 + 18 * q^75 - 4 * q^77 + 46 * q^79 + 40 * q^81 - 32 * q^83 - 14 * q^85 + 34 * q^87 + 7 * q^89 + 4 * q^91 - 11 * q^93 + 14 * q^95 - 11 * q^97 - 9 * q^99

Introduction

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