LMFDB - Embedded newform 6272.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6272,2,Mod(1,6272)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6272, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6272.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6272 = 2^{7} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6272.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:	$50.0821721477$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 896)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	6272.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.24914 q^{3} +3.30777 q^{5} +2.05863 q^{9} +O(q^{10})$	$q-2.24914 q^{3} +3.30777 q^{5} +2.05863 q^{9} -0.941367 q^{11} +5.19051 q^{13} -7.43965 q^{15} +6.49828 q^{17} +1.75086 q^{19} +5.55691 q^{23} +5.94137 q^{25} +2.11727 q^{27} +2.49828 q^{29} +6.61555 q^{31} +2.11727 q^{33} +4.61555 q^{37} -11.6742 q^{39} -10.4983 q^{41} -12.0552 q^{43} +6.80949 q^{45} -4.49828 q^{47} -14.6155 q^{51} +2.00000 q^{53} -3.11383 q^{55} -3.93793 q^{57} +12.6302 q^{59} +11.3078 q^{61} +17.1690 q^{65} +0.443086 q^{67} -12.4983 q^{69} +0.117266 q^{73} -13.3630 q^{75} +7.11383 q^{79} -10.9379 q^{81} -12.8647 q^{83} +21.4948 q^{85} -5.61899 q^{87} -10.9966 q^{89} -14.8793 q^{93} +5.79145 q^{95} -13.6121 q^{97} -1.93793 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 2 q^{5} + 7 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 + 2 * q^5 + 7 * q^9	$ 3 q + 2 q^{3} + 2 q^{5} + 7 q^{9} - 2 q^{11} + 6 q^{13} - 4 q^{15} + 2 q^{17} + 14 q^{19} + 17 q^{25} + 8 q^{27} - 10 q^{29} + 4 q^{31} + 8 q^{33} - 2 q^{37} - 20 q^{39} - 14 q^{41} - 2 q^{43} + 30 q^{45} + 4 q^{47} - 28 q^{51} + 6 q^{53} + 24 q^{55} + 24 q^{57} + 10 q^{59} + 26 q^{61} - 16 q^{65} + 18 q^{67} - 20 q^{69} + 2 q^{73} + 2 q^{75} - 12 q^{79} + 3 q^{81} - 14 q^{83} + 12 q^{85} - 36 q^{87} + 2 q^{89} - 8 q^{93} + 4 q^{95} + 10 q^{97} + 30 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 2 * q^5 + 7 * q^9 - 2 * q^11 + 6 * q^13 - 4 * q^15 + 2 * q^17 + 14 * q^19 + 17 * q^25 + 8 * q^27 - 10 * q^29 + 4 * q^31 + 8 * q^33 - 2 * q^37 - 20 * q^39 - 14 * q^41 - 2 * q^43 + 30 * q^45 + 4 * q^47 - 28 * q^51 + 6 * q^53 + 24 * q^55 + 24 * q^57 + 10 * q^59 + 26 * q^61 - 16 * q^65 + 18 * q^67 - 20 * q^69 + 2 * q^73 + 2 * q^75 - 12 * q^79 + 3 * q^81 - 14 * q^83 + 12 * q^85 - 36 * q^87 + 2 * q^89 - 8 * q^93 + 4 * q^95 + 10 * q^97 + 30 * 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
$3$	−2.24914	−1.29854	−0.649271	−	0.760557i	$-0.724926\pi$
$3$	−2.24914	−1.29854	−0.649271	+	0.760557i	$0.724926\pi$
$4$	0	0
$5$	3.30777	1.47928	0.739641	−	0.673002i	$-0.234995\pi$
$5$	3.30777	1.47928	0.739641	+	0.673002i	$0.234995\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.05863	0.686211
$10$	0	0
$11$	−0.941367	−0.283833	−0.141916	−	0.989879i	$-0.545326\pi$
$11$	−0.941367	−0.283833	−0.141916	+	0.989879i	$0.545326\pi$
$12$	0	0
$13$	5.19051	1.43959	0.719794	−	0.694188i	$-0.244236\pi$
$13$	5.19051	1.43959	0.719794	+	0.694188i	$0.244236\pi$
$14$	0	0
$15$	−7.43965	−1.92091
$16$	0	0
$17$	6.49828	1.57606	0.788032	−	0.615634i	$-0.211100\pi$
$17$	6.49828	1.57606	0.788032	+	0.615634i	$0.211100\pi$
$18$	0	0
$19$	1.75086	0.401675	0.200837	−	0.979625i	$-0.435634\pi$
$19$	1.75086	0.401675	0.200837	+	0.979625i	$0.435634\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.55691	1.15870	0.579348	−	0.815080i	$-0.303307\pi$
$23$	5.55691	1.15870	0.579348	+	0.815080i	$0.303307\pi$
$24$	0	0
$25$	5.94137	1.18827
$26$	0	0
$27$	2.11727	0.407468
$28$	0	0
$29$	2.49828	0.463919	0.231960	−	0.972725i	$-0.425486\pi$
$29$	2.49828	0.463919	0.231960	+	0.972725i	$0.425486\pi$
$30$	0	0
$31$	6.61555	1.18819	0.594094	−	0.804396i	$-0.297511\pi$
$31$	6.61555	1.18819	0.594094	+	0.804396i	$0.297511\pi$
$32$	0	0
$33$	2.11727	0.368569
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.61555	0.758791	0.379396	−	0.925235i	$-0.376132\pi$
$37$	4.61555	0.758791	0.379396	+	0.925235i	$0.376132\pi$
$38$	0	0
$39$	−11.6742	−1.86936
$40$	0	0
$41$	−10.4983	−1.63956	−0.819778	−	0.572681i	$-0.805903\pi$
$41$	−10.4983	−1.63956	−0.819778	+	0.572681i	$0.805903\pi$
$42$	0	0
$43$	−12.0552	−1.83840	−0.919200	−	0.393791i	$-0.871163\pi$
$43$	−12.0552	−1.83840	−0.919200	+	0.393791i	$0.871163\pi$
$44$	0	0
$45$	6.80949	1.01510
$46$	0	0
$47$	−4.49828	−0.656142	−0.328071	−	0.944653i	$-0.606398\pi$
$47$	−4.49828	−0.656142	−0.328071	+	0.944653i	$0.606398\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−14.6155	−2.04659
$52$	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
$55$	−3.11383	−0.419869
$56$	0	0
$57$	−3.93793	−0.521591
$58$	0	0
$59$	12.6302	1.64431	0.822153	−	0.569266i	$-0.192773\pi$
$59$	12.6302	1.64431	0.822153	+	0.569266i	$0.192773\pi$
$60$	0	0
$61$	11.3078	1.44781	0.723906	−	0.689899i	$-0.242345\pi$
$61$	11.3078	1.44781	0.723906	+	0.689899i	$0.242345\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	17.1690	2.12956
$66$	0	0
$67$	0.443086	0.0541315	0.0270658	−	0.999634i	$-0.491384\pi$
$67$	0.443086	0.0541315	0.0270658	+	0.999634i	$0.491384\pi$
$68$	0	0
$69$	−12.4983	−1.50462
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	0.117266	0.0137250	0.00686249	−	0.999976i	$-0.497816\pi$
$73$	0.117266	0.0137250	0.00686249	+	0.999976i	$0.497816\pi$
$74$	0	0
$75$	−13.3630	−1.54302
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.11383	0.800368	0.400184	−	0.916435i	$-0.368946\pi$
$79$	7.11383	0.800368	0.400184	+	0.916435i	$0.368946\pi$
$80$	0	0
$81$	−10.9379	−1.21533
$82$	0	0
$83$	−12.8647	−1.41208	−0.706041	−	0.708170i	$-0.749521\pi$
$83$	−12.8647	−1.41208	−0.706041	+	0.708170i	$0.749521\pi$
$84$	0	0
$85$	21.4948	2.33144
$86$	0	0
$87$	−5.61899	−0.602418
$88$	0	0
$89$	−10.9966	−1.16563	−0.582817	−	0.812604i	$-0.698049\pi$
$89$	−10.9966	−1.16563	−0.582817	+	0.812604i	$0.698049\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−14.8793	−1.54291
$94$	0	0
$95$	5.79145	0.594190
$96$	0	0
$97$	−13.6121	−1.38210	−0.691050	−	0.722807i	$-0.742852\pi$
$97$	−13.6121	−1.38210	−0.691050	+	0.722807i	$0.742852\pi$
$98$	0	0
$99$	−1.93793	−0.194769
$100$	0	0
$101$	15.8061	1.57276	0.786381	−	0.617742i	$-0.211953\pi$
$101$	15.8061	1.57276	0.786381	+	0.617742i	$0.211953\pi$
$102$	0	0
$103$	6.61555	0.651849	0.325925	−	0.945396i	$-0.394324\pi$
$103$	6.61555	0.651849	0.325925	+	0.945396i	$0.394324\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.43965	0.525871	0.262935	−	0.964813i	$-0.415309\pi$
$107$	5.43965	0.525871	0.262935	+	0.964813i	$0.415309\pi$
$108$	0	0
$109$	0.381015	0.0364946	0.0182473	−	0.999834i	$-0.494191\pi$
$109$	0.381015	0.0364946	0.0182473	+	0.999834i	$0.494191\pi$
$110$	0	0
$111$	−10.3810	−0.985322
$112$	0	0
$113$	−8.05520	−0.757769	−0.378885	−	0.925444i	$-0.623692\pi$
$113$	−8.05520	−0.757769	−0.378885	+	0.925444i	$0.623692\pi$
$114$	0	0
$115$	18.3810	1.71404
$116$	0	0
$117$	10.6854	0.987861
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.1138	−0.919439
$122$	0	0
$123$	23.6121	2.12903
$124$	0	0
$125$	3.11383	0.278509
$126$	0	0
$127$	−11.6742	−1.03592	−0.517958	−	0.855406i	$-0.673308\pi$
$127$	−11.6742	−1.03592	−0.517958	+	0.855406i	$0.673308\pi$
$128$	0	0
$129$	27.1138	2.38724
$130$	0	0
$131$	0.366407	0.0320131	0.0160066	−	0.999872i	$-0.494905\pi$
$131$	0.366407	0.0320131	0.0160066	+	0.999872i	$0.494905\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	7.00344	0.602760
$136$	0	0
$137$	−3.88273	−0.331724	−0.165862	−	0.986149i	$-0.553041\pi$
$137$	−3.88273	−0.331724	−0.165862	+	0.986149i	$0.553041\pi$
$138$	0	0
$139$	14.2491	1.20860	0.604298	−	0.796758i	$-0.293454\pi$
$139$	14.2491	1.20860	0.604298	+	0.796758i	$0.293454\pi$
$140$	0	0
$141$	10.1173	0.852028
$142$	0	0
$143$	−4.88617	−0.408602
$144$	0	0
$145$	8.26375	0.686267
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.87930	0.727420	0.363710	−	0.931512i	$-0.381510\pi$
$149$	8.87930	0.727420	0.363710	+	0.931512i	$0.381510\pi$
$150$	0	0
$151$	−22.5535	−1.83538	−0.917688	−	0.397302i	$-0.869947\pi$
$151$	−22.5535	−1.83538	−0.917688	+	0.397302i	$0.869947\pi$
$152$	0	0
$153$	13.3776	1.08151
$154$	0	0
$155$	21.8827	1.75766
$156$	0	0
$157$	−12.8026	−1.02176	−0.510880	−	0.859652i	$-0.670680\pi$
$157$	−12.8026	−1.02176	−0.510880	+	0.859652i	$0.670680\pi$
$158$	0	0
$159$	−4.49828	−0.356737
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.93793	0.778399	0.389199	−	0.921154i	$-0.372752\pi$
$163$	9.93793	0.778399	0.389199	+	0.921154i	$0.372752\pi$
$164$	0	0
$165$	7.00344	0.545217
$166$	0	0
$167$	12.4983	0.967146	0.483573	−	0.875304i	$-0.339339\pi$
$167$	12.4983	0.967146	0.483573	+	0.875304i	$0.339339\pi$
$168$	0	0
$169$	13.9414	1.07241
$170$	0	0
$171$	3.60438	0.275634
$172$	0	0
$173$	−16.0406	−1.21954	−0.609772	−	0.792577i	$-0.708739\pi$
$173$	−16.0406	−1.21954	−0.609772	+	0.792577i	$0.708739\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−28.4070	−2.13520
$178$	0	0
$179$	−12.7880	−0.955821	−0.477910	−	0.878409i	$-0.658606\pi$
$179$	−12.7880	−0.955821	−0.477910	+	0.878409i	$0.658606\pi$
$180$	0	0
$181$	9.68879	0.720162	0.360081	−	0.932921i	$-0.382749\pi$
$181$	9.68879	0.720162	0.360081	+	0.932921i	$0.382749\pi$
$182$	0	0
$183$	−25.4328	−1.88004
$184$	0	0
$185$	15.2672	1.12247
$186$	0	0
$187$	−6.11727	−0.447339
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.88273	0.715090	0.357545	−	0.933896i	$-0.383614\pi$
$191$	9.88273	0.715090	0.357545	+	0.933896i	$0.383614\pi$
$192$	0	0
$193$	8.70683	0.626732	0.313366	−	0.949632i	$-0.398543\pi$
$193$	8.70683	0.626732	0.313366	+	0.949632i	$0.398543\pi$
$194$	0	0
$195$	−38.6155	−2.76532
$196$	0	0
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	−23.6121	−1.67382	−0.836909	−	0.547342i	$-0.815640\pi$
$199$	−23.6121	−1.67382	−0.836909	+	0.547342i	$0.815640\pi$
$200$	0	0
$201$	−0.996562	−0.0702921
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−34.7259	−2.42536
$206$	0	0
$207$	11.4396	0.795110
$208$	0	0
$209$	−1.64820	−0.114008
$210$	0	0
$211$	−2.32582	−0.160116	−0.0800580	−	0.996790i	$-0.525511\pi$
$211$	−2.32582	−0.160116	−0.0800580	+	0.996790i	$0.525511\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−39.8759	−2.71951
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−0.263748	−0.0178225
$220$	0	0
$221$	33.7294	2.26888
$222$	0	0
$223$	6.87930	0.460672	0.230336	−	0.973111i	$-0.426018\pi$
$223$	6.87930	0.460672	0.230336	+	0.973111i	$0.426018\pi$
$224$	0	0
$225$	12.2311	0.815406
$226$	0	0
$227$	−14.2491	−0.945749	−0.472874	−	0.881130i	$-0.656784\pi$
$227$	−14.2491	−0.945749	−0.472874	+	0.881130i	$0.656784\pi$
$228$	0	0
$229$	−3.80605	−0.251511	−0.125756	−	0.992061i	$-0.540136\pi$
$229$	−3.80605	−0.251511	−0.125756	+	0.992061i	$0.540136\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−24.8793	−1.62990	−0.814948	−	0.579533i	$-0.803235\pi$
$233$	−24.8793	−1.62990	−0.814948	+	0.579533i	$0.803235\pi$
$234$	0	0
$235$	−14.8793	−0.970618
$236$	0	0
$237$	−16.0000	−1.03931
$238$	0	0
$239$	−17.5569	−1.13566	−0.567831	−	0.823145i	$-0.692217\pi$
$239$	−17.5569	−1.13566	−0.567831	+	0.823145i	$0.692217\pi$
$240$	0	0
$241$	−0.850080	−0.0547585	−0.0273792	−	0.999625i	$-0.508716\pi$
$241$	−0.850080	−0.0547585	−0.0273792	+	0.999625i	$0.508716\pi$
$242$	0	0
$243$	18.2491	1.17068
$244$	0	0
$245$	0	0
$246$	0	0
$247$	9.08785	0.578246
$248$	0	0
$249$	28.9345	1.83365
$250$	0	0
$251$	−8.36641	−0.528083	−0.264041	−	0.964511i	$-0.585056\pi$
$251$	−8.36641	−0.528083	−0.264041	+	0.964511i	$0.585056\pi$
$252$	0	0
$253$	−5.23109	−0.328876
$254$	0	0
$255$	−48.3449	−3.02748
$256$	0	0
$257$	14.9966	0.935460	0.467730	−	0.883871i	$-0.345072\pi$
$257$	14.9966	0.935460	0.467730	+	0.883871i	$0.345072\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	5.14304	0.318346
$262$	0	0
$263$	14.2277	0.877315	0.438657	−	0.898654i	$-0.355454\pi$
$263$	14.2277	0.877315	0.438657	+	0.898654i	$0.355454\pi$
$264$	0	0
$265$	6.61555	0.406390
$266$	0	0
$267$	24.7328	1.51362
$268$	0	0
$269$	4.19395	0.255709	0.127855	−	0.991793i	$-0.459191\pi$
$269$	4.19395	0.255709	0.127855	+	0.991793i	$0.459191\pi$
$270$	0	0
$271$	1.64820	0.100121	0.0500605	−	0.998746i	$-0.484059\pi$
$271$	1.64820	0.100121	0.0500605	+	0.998746i	$0.484059\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.59301	−0.337271
$276$	0	0
$277$	−8.11727	−0.487719	−0.243860	−	0.969811i	$-0.578414\pi$
$277$	−8.11727	−0.487719	−0.243860	+	0.969811i	$0.578414\pi$
$278$	0	0
$279$	13.6190	0.815347
$280$	0	0
$281$	−14.2345	−0.849161	−0.424581	−	0.905390i	$-0.639578\pi$
$281$	−14.2345	−0.849161	−0.424581	+	0.905390i	$0.639578\pi$
$282$	0	0
$283$	31.4802	1.87131	0.935653	−	0.352922i	$-0.114812\pi$
$283$	31.4802	1.87131	0.935653	+	0.352922i	$0.114812\pi$
$284$	0	0
$285$	−13.0258	−0.771580
$286$	0	0
$287$	0	0
$288$	0	0
$289$	25.2277	1.48398
$290$	0	0
$291$	30.6155	1.79472
$292$	0	0
$293$	−1.19051	−0.0695502	−0.0347751	−	0.999395i	$-0.511071\pi$
$293$	−1.19051	−0.0695502	−0.0347751	+	0.999395i	$0.511071\pi$
$294$	0	0
$295$	41.7777	2.43239
$296$	0	0
$297$	−1.99312	−0.115653
$298$	0	0
$299$	28.8432	1.66805
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−35.5500	−2.04230
$304$	0	0
$305$	37.4036	2.14172
$306$	0	0
$307$	4.60094	0.262589	0.131295	−	0.991343i	$-0.458087\pi$
$307$	4.60094	0.262589	0.131295	+	0.991343i	$0.458087\pi$
$308$	0	0
$309$	−14.8793	−0.846454
$310$	0	0
$311$	13.2311	0.750267	0.375133	−	0.926971i	$-0.377597\pi$
$311$	13.2311	0.750267	0.375133	+	0.926971i	$0.377597\pi$
$312$	0	0
$313$	−10.4983	−0.593398	−0.296699	−	0.954971i	$-0.595886\pi$
$313$	−10.4983	−0.593398	−0.296699	+	0.954971i	$0.595886\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.00344	0.505683	0.252842	−	0.967508i	$-0.418635\pi$
$317$	9.00344	0.505683	0.252842	+	0.967508i	$0.418635\pi$
$318$	0	0
$319$	−2.35180	−0.131675
$320$	0	0
$321$	−12.2345	−0.682865
$322$	0	0
$323$	11.3776	0.633065
$324$	0	0
$325$	30.8387	1.71062
$326$	0	0
$327$	−0.856956	−0.0473898
$328$	0	0
$329$	0	0
$330$	0	0
$331$	10.1725	0.559129	0.279565	−	0.960127i	$-0.409810\pi$
$331$	10.1725	0.559129	0.279565	+	0.960127i	$0.409810\pi$
$332$	0	0
$333$	9.50172	0.520691
$334$	0	0
$335$	1.46563	0.0800758
$336$	0	0
$337$	23.8207	1.29759	0.648797	−	0.760961i	$-0.275272\pi$
$337$	23.8207	1.29759	0.648797	+	0.760961i	$0.275272\pi$
$338$	0	0
$339$	18.1173	0.983995
$340$	0	0
$341$	−6.22766	−0.337247
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−41.3415	−2.22575
$346$	0	0
$347$	−17.2863	−0.927977	−0.463988	−	0.885841i	$-0.653582\pi$
$347$	−17.2863	−0.927977	−0.463988	+	0.885841i	$0.653582\pi$
$348$	0	0
$349$	16.6922	0.893514	0.446757	−	0.894655i	$-0.352579\pi$
$349$	16.6922	0.893514	0.446757	+	0.894655i	$0.352579\pi$
$350$	0	0
$351$	10.9897	0.586586
$352$	0	0
$353$	14.3449	0.763503	0.381752	−	0.924265i	$-0.375321\pi$
$353$	14.3449	0.763503	0.381752	+	0.924265i	$0.375321\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.43965	−0.181538	−0.0907688	−	0.995872i	$-0.528932\pi$
$359$	−3.43965	−0.181538	−0.0907688	+	0.995872i	$0.528932\pi$
$360$	0	0
$361$	−15.9345	−0.838657
$362$	0	0
$363$	22.7474	1.19393
$364$	0	0
$365$	0.387890	0.0203031
$366$	0	0
$367$	−10.1173	−0.528117	−0.264059	−	0.964507i	$-0.585061\pi$
$367$	−10.1173	−0.528117	−0.264059	+	0.964507i	$0.585061\pi$
$368$	0	0
$369$	−21.6121	−1.12508
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−20.8793	−1.08109	−0.540544	−	0.841315i	$-0.681782\pi$
$373$	−20.8793	−1.08109	−0.540544	+	0.841315i	$0.681782\pi$
$374$	0	0
$375$	−7.00344	−0.361656
$376$	0	0
$377$	12.9673	0.667852
$378$	0	0
$379$	13.9379	0.715943	0.357972	−	0.933732i	$-0.383468\pi$
$379$	13.9379	0.715943	0.357972	+	0.933732i	$0.383468\pi$
$380$	0	0
$381$	26.2569	1.34518
$382$	0	0
$383$	22.4914	1.14926	0.574629	−	0.818414i	$-0.305147\pi$
$383$	22.4914	1.14926	0.574629	+	0.818414i	$0.305147\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−24.8172	−1.26153
$388$	0	0
$389$	17.8466	0.904861	0.452430	−	0.891800i	$-0.350557\pi$
$389$	17.8466	0.904861	0.452430	+	0.891800i	$0.350557\pi$
$390$	0	0
$391$	36.1104	1.82618
$392$	0	0
$393$	−0.824101	−0.0415704
$394$	0	0
$395$	23.5309	1.18397
$396$	0	0
$397$	0.692226	0.0347418	0.0173709	−	0.999849i	$-0.494470\pi$
$397$	0.692226	0.0347418	0.0173709	+	0.999849i	$0.494470\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.40699	−0.120200	−0.0600998	−	0.998192i	$-0.519142\pi$
$401$	−2.40699	−0.120200	−0.0600998	+	0.998192i	$0.519142\pi$
$402$	0	0
$403$	34.3380	1.71050
$404$	0	0
$405$	−36.1802	−1.79781
$406$	0	0
$407$	−4.34492	−0.215370
$408$	0	0
$409$	−21.6121	−1.06865	−0.534325	−	0.845279i	$-0.679434\pi$
$409$	−21.6121	−1.06865	−0.534325	+	0.845279i	$0.679434\pi$
$410$	0	0
$411$	8.73281	0.430758
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−42.5535	−2.08887
$416$	0	0
$417$	−32.0483	−1.56941
$418$	0	0
$419$	−9.25258	−0.452018	−0.226009	−	0.974125i	$-0.572568\pi$
$419$	−9.25258	−0.452018	−0.226009	+	0.974125i	$0.572568\pi$
$420$	0	0
$421$	−14.3449	−0.699129	−0.349564	−	0.936912i	$-0.613670\pi$
$421$	−14.3449	−0.699129	−0.349564	+	0.936912i	$0.613670\pi$
$422$	0	0
$423$	−9.26031	−0.450252
$424$	0	0
$425$	38.6087	1.87280
$426$	0	0
$427$	0	0
$428$	0	0
$429$	10.9897	0.530587
$430$	0	0
$431$	−24.4362	−1.17705	−0.588525	−	0.808479i	$-0.700291\pi$
$431$	−24.4362	−1.17705	−0.588525	+	0.808479i	$0.700291\pi$
$432$	0	0
$433$	16.4914	0.792526	0.396263	−	0.918137i	$-0.370307\pi$
$433$	16.4914	0.792526	0.396263	+	0.918137i	$0.370307\pi$
$434$	0	0
$435$	−18.5863	−0.891146
$436$	0	0
$437$	9.72938	0.465419
$438$	0	0
$439$	38.8793	1.85561	0.927804	−	0.373069i	$-0.121694\pi$
$439$	38.8793	1.85561	0.927804	+	0.373069i	$0.121694\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	15.0878	0.716845	0.358423	−	0.933559i	$-0.383315\pi$
$443$	15.0878	0.716845	0.358423	+	0.933559i	$0.383315\pi$
$444$	0	0
$445$	−36.3741	−1.72430
$446$	0	0
$447$	−19.9708	−0.944586
$448$	0	0
$449$	28.8793	1.36290	0.681449	−	0.731865i	$-0.261350\pi$
$449$	28.8793	1.36290	0.681449	+	0.731865i	$0.261350\pi$
$450$	0	0
$451$	9.88273	0.465360
$452$	0	0
$453$	50.7259	2.38331
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.2897	1.13623	0.568113	−	0.822951i	$-0.307674\pi$
$457$	24.2897	1.13623	0.568113	+	0.822951i	$0.307674\pi$
$458$	0	0
$459$	13.7586	0.642196
$460$	0	0
$461$	−10.0767	−0.469318	−0.234659	−	0.972078i	$-0.575397\pi$
$461$	−10.0767	−0.469318	−0.234659	+	0.972078i	$0.575397\pi$
$462$	0	0
$463$	−34.8793	−1.62098	−0.810489	−	0.585754i	$-0.800799\pi$
$463$	−34.8793	−1.62098	−0.810489	+	0.585754i	$0.800799\pi$
$464$	0	0
$465$	−49.2173	−2.28240
$466$	0	0
$467$	26.7474	1.23772	0.618862	−	0.785500i	$-0.287594\pi$
$467$	26.7474	1.23772	0.618862	+	0.785500i	$0.287594\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	28.7949	1.32680
$472$	0	0
$473$	11.3484	0.521798
$474$	0	0
$475$	10.4025	0.477299
$476$	0	0
$477$	4.11727	0.188517
$478$	0	0
$479$	1.38445	0.0632573	0.0316286	−	0.999500i	$-0.489931\pi$
$479$	1.38445	0.0632573	0.0316286	+	0.999500i	$0.489931\pi$
$480$	0	0
$481$	23.9570	1.09235
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−45.0258	−2.04452
$486$	0	0
$487$	−6.20855	−0.281336	−0.140668	−	0.990057i	$-0.544925\pi$
$487$	−6.20855	−0.281336	−0.140668	+	0.990057i	$0.544925\pi$
$488$	0	0
$489$	−22.3518	−1.01078
$490$	0	0
$491$	30.7811	1.38913	0.694567	−	0.719428i	$-0.255596\pi$
$491$	30.7811	1.38913	0.694567	+	0.719428i	$0.255596\pi$
$492$	0	0
$493$	16.2345	0.731167
$494$	0	0
$495$	−6.41023	−0.288118
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−33.6742	−1.50746	−0.753732	−	0.657182i	$-0.771748\pi$
$499$	−33.6742	−1.50746	−0.753732	+	0.657182i	$0.771748\pi$
$500$	0	0
$501$	−28.1104	−1.25588
$502$	0	0
$503$	29.8827	1.33241	0.666203	−	0.745771i	$-0.267919\pi$
$503$	29.8827	1.33241	0.666203	+	0.745771i	$0.267919\pi$
$504$	0	0
$505$	52.2829	2.32656
$506$	0	0
$507$	−31.3561	−1.39257
$508$	0	0
$509$	−3.54231	−0.157010	−0.0785050	−	0.996914i	$-0.525015\pi$
$509$	−3.54231	−0.157010	−0.0785050	+	0.996914i	$0.525015\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3.70704	0.163670
$514$	0	0
$515$	21.8827	0.964268
$516$	0	0
$517$	4.23453	0.186235
$518$	0	0
$519$	36.0775	1.58363
$520$	0	0
$521$	38.8432	1.70175	0.850876	−	0.525367i	$-0.176072\pi$
$521$	38.8432	1.70175	0.850876	+	0.525367i	$0.176072\pi$
$522$	0	0
$523$	3.63359	0.158886	0.0794430	−	0.996839i	$-0.474686\pi$
$523$	3.63359	0.158886	0.0794430	+	0.996839i	$0.474686\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	42.9897	1.87266
$528$	0	0
$529$	7.87930	0.342578
$530$	0	0
$531$	26.0009	1.12834
$532$	0	0
$533$	−54.4914	−2.36028
$534$	0	0
$535$	17.9931	0.777911
$536$	0	0
$537$	28.7620	1.24117
$538$	0	0
$539$	0	0
$540$	0	0
$541$	22.1104	0.950600	0.475300	−	0.879824i	$-0.342340\pi$
$541$	22.1104	0.950600	0.475300	+	0.879824i	$0.342340\pi$
$542$	0	0
$543$	−21.7914	−0.935160
$544$	0	0
$545$	1.26031	0.0539858
$546$	0	0
$547$	39.1690	1.67475	0.837373	−	0.546632i	$-0.184090\pi$
$547$	39.1690	1.67475	0.837373	+	0.546632i	$0.184090\pi$
$548$	0	0
$549$	23.2786	0.993505
$550$	0	0
$551$	4.37414	0.186345
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−34.3380	−1.45757
$556$	0	0
$557$	13.7655	0.583262	0.291631	−	0.956531i	$-0.405802\pi$
$557$	13.7655	0.583262	0.291631	+	0.956531i	$0.405802\pi$
$558$	0	0
$559$	−62.5726	−2.64654
$560$	0	0
$561$	13.7586	0.580888
$562$	0	0
$563$	−21.6267	−0.911457	−0.455729	−	0.890119i	$-0.650621\pi$
$563$	−21.6267	−0.911457	−0.455729	+	0.890119i	$0.650621\pi$
$564$	0	0
$565$	−26.6448	−1.12095
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.16902	0.300541	0.150271	−	0.988645i	$-0.451986\pi$
$569$	7.16902	0.300541	0.150271	+	0.988645i	$0.451986\pi$
$570$	0	0
$571$	−36.0552	−1.50886	−0.754431	−	0.656379i	$-0.772087\pi$
$571$	−36.0552	−1.50886	−0.754431	+	0.656379i	$0.772087\pi$
$572$	0	0
$573$	−22.2277	−0.928574
$574$	0	0
$575$	33.0157	1.37685
$576$	0	0
$577$	34.1104	1.42003	0.710017	−	0.704184i	$-0.248687\pi$
$577$	34.1104	1.42003	0.710017	+	0.704184i	$0.248687\pi$
$578$	0	0
$579$	−19.5829	−0.813837
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−1.88273	−0.0779749
$584$	0	0
$585$	35.3447	1.46132
$586$	0	0
$587$	−2.74742	−0.113398	−0.0566991	−	0.998391i	$-0.518058\pi$
$587$	−2.74742	−0.113398	−0.0566991	+	0.998391i	$0.518058\pi$
$588$	0	0
$589$	11.5829	0.477265
$590$	0	0
$591$	−22.4914	−0.925173
$592$	0	0
$593$	−32.2277	−1.32343	−0.661716	−	0.749755i	$-0.730171\pi$
$593$	−32.2277	−1.32343	−0.661716	+	0.749755i	$0.730171\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	53.1070	2.17352
$598$	0	0
$599$	0.469065	0.0191655	0.00958274	−	0.999954i	$-0.496950\pi$
$599$	0.469065	0.0191655	0.00958274	+	0.999954i	$0.496950\pi$
$600$	0	0
$601$	−37.4588	−1.52797	−0.763987	−	0.645231i	$-0.776761\pi$
$601$	−37.4588	−1.52797	−0.763987	+	0.645231i	$0.776761\pi$
$602$	0	0
$603$	0.912151	0.0371457
$604$	0	0
$605$	−33.4543	−1.36011
$606$	0	0
$607$	25.4656	1.03362	0.516809	−	0.856101i	$-0.327120\pi$
$607$	25.4656	1.03362	0.516809	+	0.856101i	$0.327120\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−23.3484	−0.944574
$612$	0	0
$613$	1.03265	0.0417085	0.0208542	−	0.999783i	$-0.493361\pi$
$613$	1.03265	0.0417085	0.0208542	+	0.999783i	$0.493361\pi$
$614$	0	0
$615$	78.1035	3.14944
$616$	0	0
$617$	6.93449	0.279172	0.139586	−	0.990210i	$-0.455423\pi$
$617$	6.93449	0.279172	0.139586	+	0.990210i	$0.455423\pi$
$618$	0	0
$619$	−2.98195	−0.119855	−0.0599274	−	0.998203i	$-0.519087\pi$
$619$	−2.98195	−0.119855	−0.0599274	+	0.998203i	$0.519087\pi$
$620$	0	0
$621$	11.7655	0.472132
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19.4070	−0.776280
$626$	0	0
$627$	3.70704	0.148045
$628$	0	0
$629$	29.9931	1.19590
$630$	0	0
$631$	−38.7552	−1.54282	−0.771409	−	0.636339i	$-0.780448\pi$
$631$	−38.7552	−1.54282	−0.771409	+	0.636339i	$0.780448\pi$
$632$	0	0
$633$	5.23109	0.207917
$634$	0	0
$635$	−38.6155	−1.53241
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24.0552	0.950123	0.475062	−	0.879953i	$-0.342426\pi$
$641$	24.0552	0.950123	0.475062	+	0.879953i	$0.342426\pi$
$642$	0	0
$643$	22.8286	0.900272	0.450136	−	0.892960i	$-0.351376\pi$
$643$	22.8286	0.900272	0.450136	+	0.892960i	$0.351376\pi$
$644$	0	0
$645$	89.6864	3.53140
$646$	0	0
$647$	−5.14992	−0.202464	−0.101232	−	0.994863i	$-0.532278\pi$
$647$	−5.14992	−0.202464	−0.101232	+	0.994863i	$0.532278\pi$
$648$	0	0
$649$	−11.8896	−0.466708
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.73281	0.263475	0.131738	−	0.991285i	$-0.457944\pi$
$653$	6.73281	0.263475	0.131738	+	0.991285i	$0.457944\pi$
$654$	0	0
$655$	1.21199	0.0473564
$656$	0	0
$657$	0.241408	0.00941824
$658$	0	0
$659$	−36.8172	−1.43420	−0.717098	−	0.696973i	$-0.754530\pi$
$659$	−36.8172	−1.43420	−0.717098	+	0.696973i	$0.754530\pi$
$660$	0	0
$661$	−21.4250	−0.833337	−0.416669	−	0.909058i	$-0.636802\pi$
$661$	−21.4250	−0.833337	−0.416669	+	0.909058i	$0.636802\pi$
$662$	0	0
$663$	−75.8621	−2.94624
$664$	0	0
$665$	0	0
$666$	0	0
$667$	13.8827	0.537542
$668$	0	0
$669$	−15.4725	−0.598202
$670$	0	0
$671$	−10.6448	−0.410937
$672$	0	0
$673$	−6.76203	−0.260657	−0.130329	−	0.991471i	$-0.541603\pi$
$673$	−6.76203	−0.260657	−0.130329	+	0.991471i	$0.541603\pi$
$674$	0	0
$675$	12.5795	0.484183
$676$	0	0
$677$	−42.2975	−1.62562	−0.812812	−	0.582526i	$-0.802064\pi$
$677$	−42.2975	−1.62562	−0.812812	+	0.582526i	$0.802064\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	32.0483	1.22809
$682$	0	0
$683$	−32.7880	−1.25460	−0.627299	−	0.778778i	$-0.715840\pi$
$683$	−32.7880	−1.25460	−0.627299	+	0.778778i	$0.715840\pi$
$684$	0	0
$685$	−12.8432	−0.490714
$686$	0	0
$687$	8.56035	0.326598
$688$	0	0
$689$	10.3810	0.395485
$690$	0	0
$691$	22.1250	0.841675	0.420837	−	0.907136i	$-0.361736\pi$
$691$	22.1250	0.841675	0.420837	+	0.907136i	$0.361736\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	47.1329	1.78785
$696$	0	0
$697$	−68.2208	−2.58405
$698$	0	0
$699$	55.9570	2.11649
$700$	0	0
$701$	−36.7259	−1.38712	−0.693560	−	0.720399i	$-0.743959\pi$
$701$	−36.7259	−1.38712	−0.693560	+	0.720399i	$0.743959\pi$
$702$	0	0
$703$	8.08117	0.304787
$704$	0	0
$705$	33.4656	1.26039
$706$	0	0
$707$	0	0
$708$	0	0
$709$	6.38789	0.239902	0.119951	−	0.992780i	$-0.461726\pi$
$709$	6.38789	0.239902	0.119951	+	0.992780i	$0.461726\pi$
$710$	0	0
$711$	14.6448	0.549222
$712$	0	0
$713$	36.7620	1.37675
$714$	0	0
$715$	−16.1623	−0.604438
$716$	0	0
$717$	39.4880	1.47471
$718$	0	0
$719$	45.4948	1.69667	0.848336	−	0.529459i	$-0.177605\pi$
$719$	45.4948	1.69667	0.848336	+	0.529459i	$0.177605\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1.91195	0.0711062
$724$	0	0
$725$	14.8432	0.551263
$726$	0	0
$727$	−50.2569	−1.86392	−0.931962	−	0.362556i	$-0.881904\pi$
$727$	−50.2569	−1.86392	−0.931962	+	0.362556i	$0.881904\pi$
$728$	0	0
$729$	−8.23109	−0.304855
$730$	0	0
$731$	−78.3380	−2.89744
$732$	0	0
$733$	13.3009	0.491280	0.245640	−	0.969361i	$-0.421002\pi$
$733$	13.3009	0.491280	0.245640	+	0.969361i	$0.421002\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.417106	−0.0153643
$738$	0	0
$739$	2.40699	0.0885427	0.0442714	−	0.999020i	$-0.485903\pi$
$739$	2.40699	0.0885427	0.0442714	+	0.999020i	$0.485903\pi$
$740$	0	0
$741$	−20.4398	−0.750877
$742$	0	0
$743$	14.0844	0.516707	0.258353	−	0.966050i	$-0.416820\pi$
$743$	14.0844	0.516707	0.258353	+	0.966050i	$0.416820\pi$
$744$	0	0
$745$	29.3707	1.07606
$746$	0	0
$747$	−26.4837	−0.968987
$748$	0	0
$749$	0	0
$750$	0	0
$751$	7.78457	0.284063	0.142032	−	0.989862i	$-0.454637\pi$
$751$	7.78457	0.284063	0.142032	+	0.989862i	$0.454637\pi$
$752$	0	0
$753$	18.8172	0.685738
$754$	0	0
$755$	−74.6018	−2.71504
$756$	0	0
$757$	−24.4914	−0.890155	−0.445078	−	0.895492i	$-0.646824\pi$
$757$	−24.4914	−0.890155	−0.445078	+	0.895492i	$0.646824\pi$
$758$	0	0
$759$	11.7655	0.427059
$760$	0	0
$761$	−10.9673	−0.397566	−0.198783	−	0.980044i	$-0.563699\pi$
$761$	−10.9673	−0.397566	−0.198783	+	0.980044i	$0.563699\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	44.2500	1.59986
$766$	0	0
$767$	65.5569	2.36712
$768$	0	0
$769$	6.62242	0.238811	0.119405	−	0.992846i	$-0.461901\pi$
$769$	6.62242	0.238811	0.119405	+	0.992846i	$0.461901\pi$
$770$	0	0
$771$	−33.7294	−1.21473
$772$	0	0
$773$	−3.91645	−0.140865	−0.0704324	−	0.997517i	$-0.522438\pi$
$773$	−3.91645	−0.140865	−0.0704324	+	0.997517i	$0.522438\pi$
$774$	0	0
$775$	39.3054	1.41189
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−18.3810	−0.658568
$780$	0	0
$781$	0	0
$782$	0	0
$783$	5.28953	0.189032
$784$	0	0
$785$	−42.3482	−1.51147
$786$	0	0
$787$	−9.01805	−0.321459	−0.160729	−	0.986999i	$-0.551385\pi$
$787$	−9.01805	−0.321459	−0.160729	+	0.986999i	$0.551385\pi$
$788$	0	0
$789$	−32.0000	−1.13923
$790$	0	0
$791$	0	0
$792$	0	0
$793$	58.6931	2.08425
$794$	0	0
$795$	−14.8793	−0.527714
$796$	0	0
$797$	−23.5715	−0.834946	−0.417473	−	0.908689i	$-0.637084\pi$
$797$	−23.5715	−0.834946	−0.417473	+	0.908689i	$0.637084\pi$
$798$	0	0
$799$	−29.2311	−1.03412
$800$	0	0
$801$	−22.6379	−0.799870
$802$	0	0
$803$	−0.110391	−0.00389560
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−9.43277	−0.332049
$808$	0	0
$809$	−20.1656	−0.708984	−0.354492	−	0.935059i	$-0.615346\pi$
$809$	−20.1656	−0.708984	−0.354492	+	0.935059i	$0.615346\pi$
$810$	0	0
$811$	−21.9716	−0.771529	−0.385764	−	0.922597i	$-0.626062\pi$
$811$	−21.9716	−0.771529	−0.385764	+	0.922597i	$0.626062\pi$
$812$	0	0
$813$	−3.70704	−0.130011
$814$	0	0
$815$	32.8724	1.15147
$816$	0	0
$817$	−21.1070	−0.738439
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.10695	−0.108433	−0.0542167	−	0.998529i	$-0.517266\pi$
$821$	−3.10695	−0.108433	−0.0542167	+	0.998529i	$0.517266\pi$
$822$	0	0
$823$	12.7620	0.444856	0.222428	−	0.974949i	$-0.428602\pi$
$823$	12.7620	0.444856	0.222428	+	0.974949i	$0.428602\pi$
$824$	0	0
$825$	12.5795	0.437960
$826$	0	0
$827$	4.91215	0.170812	0.0854061	−	0.996346i	$-0.472781\pi$
$827$	4.91215	0.170812	0.0854061	+	0.996346i	$0.472781\pi$
$828$	0	0
$829$	33.5354	1.16473	0.582367	−	0.812926i	$-0.302127\pi$
$829$	33.5354	1.16473	0.582367	+	0.812926i	$0.302127\pi$
$830$	0	0
$831$	18.2569	0.633324
$832$	0	0
$833$	0	0
$834$	0	0
$835$	41.3415	1.43068
$836$	0	0
$837$	14.0069	0.484148
$838$	0	0
$839$	−6.61555	−0.228394	−0.114197	−	0.993458i	$-0.536430\pi$
$839$	−6.61555	−0.228394	−0.114197	+	0.993458i	$0.536430\pi$
$840$	0	0
$841$	−22.7586	−0.784779
$842$	0	0
$843$	32.0155	1.10267
$844$	0	0
$845$	46.1149	1.58640
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−70.8035	−2.42997
$850$	0	0
$851$	25.6482	0.879209
$852$	0	0
$853$	30.1871	1.03359	0.516793	−	0.856111i	$-0.327126\pi$
$853$	30.1871	1.03359	0.516793	+	0.856111i	$0.327126\pi$
$854$	0	0
$855$	11.9225	0.407740
$856$	0	0
$857$	−16.3810	−0.559565	−0.279782	−	0.960063i	$-0.590262\pi$
$857$	−16.3810	−0.559565	−0.279782	+	0.960063i	$0.590262\pi$
$858$	0	0
$859$	−0.366407	−0.0125016	−0.00625082	−	0.999980i	$-0.501990\pi$
$859$	−0.366407	−0.0125016	−0.00625082	+	0.999980i	$0.501990\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−13.6482	−0.464590	−0.232295	−	0.972645i	$-0.574624\pi$
$863$	−13.6482	−0.464590	−0.232295	+	0.972645i	$0.574624\pi$
$864$	0	0
$865$	−53.0586	−1.80405
$866$	0	0
$867$	−56.7405	−1.92701
$868$	0	0
$869$	−6.69672	−0.227171
$870$	0	0
$871$	2.29984	0.0779271
$872$	0	0
$873$	−28.0223	−0.948413
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15.8535	0.535335	0.267668	−	0.963511i	$-0.413747\pi$
$877$	15.8535	0.535335	0.267668	+	0.963511i	$0.413747\pi$
$878$	0	0
$879$	2.67762	0.0903138
$880$	0	0
$881$	−7.35524	−0.247804	−0.123902	−	0.992294i	$-0.539541\pi$
$881$	−7.35524	−0.247804	−0.123902	+	0.992294i	$0.539541\pi$
$882$	0	0
$883$	25.2571	0.849968	0.424984	−	0.905201i	$-0.360280\pi$
$883$	25.2571	0.849968	0.424984	+	0.905201i	$0.360280\pi$
$884$	0	0
$885$	−93.9639	−3.15856
$886$	0	0
$887$	−43.1950	−1.45035	−0.725173	−	0.688567i	$-0.758240\pi$
$887$	−43.1950	−1.45035	−0.725173	+	0.688567i	$0.758240\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	10.2966	0.344949
$892$	0	0
$893$	−7.87586	−0.263556
$894$	0	0
$895$	−42.2998	−1.41393
$896$	0	0
$897$	−64.8724	−2.16603
$898$	0	0
$899$	16.5275	0.551223
$900$	0	0
$901$	12.9966	0.432978
$902$	0	0
$903$	0	0
$904$	0	0
$905$	32.0483	1.06532
$906$	0	0
$907$	41.0810	1.36407	0.682036	−	0.731319i	$-0.261095\pi$
$907$	41.0810	1.36407	0.682036	+	0.731319i	$0.261095\pi$
$908$	0	0
$909$	32.5389	1.07925
$910$	0	0
$911$	21.4465	0.710555	0.355278	−	0.934761i	$-0.384386\pi$
$911$	21.4465	0.710555	0.355278	+	0.934761i	$0.384386\pi$
$912$	0	0
$913$	12.1104	0.400795
$914$	0	0
$915$	−84.1259	−2.78111
$916$	0	0
$917$	0	0
$918$	0	0
$919$	4.76203	0.157085	0.0785424	−	0.996911i	$-0.474973\pi$
$919$	4.76203	0.157085	0.0785424	+	0.996911i	$0.474973\pi$
$920$	0	0
$921$	−10.3482	−0.340983
$922$	0	0
$923$	0	0
$924$	0	0
$925$	27.4227	0.901652
$926$	0	0
$927$	13.6190	0.447306
$928$	0	0
$929$	−20.9605	−0.687691	−0.343845	−	0.939026i	$-0.611730\pi$
$929$	−20.9605	−0.687691	−0.343845	+	0.939026i	$0.611730\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−29.7586	−0.974253
$934$	0	0
$935$	−20.2345	−0.661740
$936$	0	0
$937$	−7.10695	−0.232174	−0.116087	−	0.993239i	$-0.537035\pi$
$937$	−7.10695	−0.232174	−0.116087	+	0.993239i	$0.537035\pi$
$938$	0	0
$939$	23.6121	0.770552
$940$	0	0
$941$	3.57152	0.116428	0.0582141	−	0.998304i	$-0.481459\pi$
$941$	3.57152	0.116428	0.0582141	+	0.998304i	$0.481459\pi$
$942$	0	0
$943$	−58.3380	−1.89975
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25.4104	0.825728	0.412864	−	0.910793i	$-0.364528\pi$
$947$	25.4104	0.825728	0.412864	+	0.910793i	$0.364528\pi$
$948$	0	0
$949$	0.608672	0.0197583
$950$	0	0
$951$	−20.2500	−0.656651
$952$	0	0
$953$	−28.2277	−0.914383	−0.457192	−	0.889368i	$-0.651145\pi$
$953$	−28.2277	−0.914383	−0.457192	+	0.889368i	$0.651145\pi$
$954$	0	0
$955$	32.6898	1.05782
$956$	0	0
$957$	5.28953	0.170986
$958$	0	0
$959$	0	0
$960$	0	0
$961$	12.7655	0.411789
$962$	0	0
$963$	11.1982	0.360858
$964$	0	0
$965$	28.8002	0.927112
$966$	0	0
$967$	40.0191	1.28693	0.643464	−	0.765477i	$-0.277497\pi$
$967$	40.0191	1.28693	0.643464	+	0.765477i	$0.277497\pi$
$968$	0	0
$969$	−25.5898	−0.822062
$970$	0	0
$971$	16.8647	0.541214	0.270607	−	0.962690i	$-0.412776\pi$
$971$	16.8647	0.541214	0.270607	+	0.962690i	$0.412776\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−69.3606	−2.22132
$976$	0	0
$977$	−12.1173	−0.387666	−0.193833	−	0.981035i	$-0.562092\pi$
$977$	−12.1173	−0.387666	−0.193833	+	0.981035i	$0.562092\pi$
$978$	0	0
$979$	10.3518	0.330845
$980$	0	0
$981$	0.784370	0.0250430
$982$	0	0
$983$	49.6052	1.58216	0.791081	−	0.611712i	$-0.209519\pi$
$983$	49.6052	1.58216	0.791081	+	0.611712i	$0.209519\pi$
$984$	0	0
$985$	33.0777	1.05394
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−66.9897	−2.13015
$990$	0	0
$991$	−35.8759	−1.13963	−0.569817	−	0.821772i	$-0.692986\pi$
$991$	−35.8759	−1.13963	−0.569817	+	0.821772i	$0.692986\pi$
$992$	0	0
$993$	−22.8793	−0.726053
$994$	0	0
$995$	−78.1035	−2.47605
$996$	0	0
$997$	−9.45426	−0.299419	−0.149710	−	0.988730i	$-0.547834\pi$
$997$	−9.45426	−0.299419	−0.149710	+	0.988730i	$0.547834\pi$
$998$	0	0
$999$	9.77234	0.309183

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6272.2.a.x.1.1		3
4.3	odd	2		6272.2.a.v.1.3		3
7.6	odd	2		896.2.a.i.1.3	✓	3
8.3	odd	2		6272.2.a.w.1.1		3
8.5	even	2		6272.2.a.u.1.3		3
21.20	even	2		8064.2.a.ce.1.3		3
28.27	even	2		896.2.a.k.1.1	yes	3
56.13	odd	2		896.2.a.l.1.1	yes	3
56.27	even	2		896.2.a.j.1.3	yes	3
84.83	odd	2		8064.2.a.ch.1.3		3
112.13	odd	4		1792.2.b.p.897.5		6
112.27	even	4		1792.2.b.o.897.5		6
112.69	odd	4		1792.2.b.p.897.2		6
112.83	even	4		1792.2.b.o.897.2		6
168.83	odd	2		8064.2.a.cb.1.1		3
168.125	even	2		8064.2.a.bu.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.a.i.1.3	✓	3	7.6	odd	2
896.2.a.j.1.3	yes	3	56.27	even	2
896.2.a.k.1.1	yes	3	28.27	even	2
896.2.a.l.1.1	yes	3	56.13	odd	2
1792.2.b.o.897.2		6	112.83	even	4
1792.2.b.o.897.5		6	112.27	even	4
1792.2.b.p.897.2		6	112.69	odd	4
1792.2.b.p.897.5		6	112.13	odd	4
6272.2.a.u.1.3		3	8.5	even	2
6272.2.a.v.1.3		3	4.3	odd	2
6272.2.a.w.1.1		3	8.3	odd	2
6272.2.a.x.1.1		3	1.1	even	1	trivial
8064.2.a.bu.1.1		3	168.125	even	2
8064.2.a.cb.1.1		3	168.83	odd	2
8064.2.a.ce.1.3		3	21.20	even	2
8064.2.a.ch.1.3		3	84.83	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:	\( N \)	\(=\)	\( 6272 = 2^{7} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6272.a (trivial)

\(f(q)\)	\(=\)	\(q-2.24914 q^{3} +3.30777 q^{5} +2.05863 q^{9} +O(q^{10})\)	\(q-2.24914 q^{3} +3.30777 q^{5} +2.05863 q^{9} -0.941367 q^{11} +5.19051 q^{13} -7.43965 q^{15} +6.49828 q^{17} +1.75086 q^{19} +5.55691 q^{23} +5.94137 q^{25} +2.11727 q^{27} +2.49828 q^{29} +6.61555 q^{31} +2.11727 q^{33} +4.61555 q^{37} -11.6742 q^{39} -10.4983 q^{41} -12.0552 q^{43} +6.80949 q^{45} -4.49828 q^{47} -14.6155 q^{51} +2.00000 q^{53} -3.11383 q^{55} -3.93793 q^{57} +12.6302 q^{59} +11.3078 q^{61} +17.1690 q^{65} +0.443086 q^{67} -12.4983 q^{69} +0.117266 q^{73} -13.3630 q^{75} +7.11383 q^{79} -10.9379 q^{81} -12.8647 q^{83} +21.4948 q^{85} -5.61899 q^{87} -10.9966 q^{89} -14.8793 q^{93} +5.79145 q^{95} -13.6121 q^{97} -1.93793 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 2 q^{5} + 7 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 + 2 * q^5 + 7 * q^9	\( 3 q + 2 q^{3} + 2 q^{5} + 7 q^{9} - 2 q^{11} + 6 q^{13} - 4 q^{15} + 2 q^{17} + 14 q^{19} + 17 q^{25} + 8 q^{27} - 10 q^{29} + 4 q^{31} + 8 q^{33} - 2 q^{37} - 20 q^{39} - 14 q^{41} - 2 q^{43} + 30 q^{45} + 4 q^{47} - 28 q^{51} + 6 q^{53} + 24 q^{55} + 24 q^{57} + 10 q^{59} + 26 q^{61} - 16 q^{65} + 18 q^{67} - 20 q^{69} + 2 q^{73} + 2 q^{75} - 12 q^{79} + 3 q^{81} - 14 q^{83} + 12 q^{85} - 36 q^{87} + 2 q^{89} - 8 q^{93} + 4 q^{95} + 10 q^{97} + 30 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 2 * q^5 + 7 * q^9 - 2 * q^11 + 6 * q^13 - 4 * q^15 + 2 * q^17 + 14 * q^19 + 17 * q^25 + 8 * q^27 - 10 * q^29 + 4 * q^31 + 8 * q^33 - 2 * q^37 - 20 * q^39 - 14 * q^41 - 2 * q^43 + 30 * q^45 + 4 * q^47 - 28 * q^51 + 6 * q^53 + 24 * q^55 + 24 * q^57 + 10 * q^59 + 26 * q^61 - 16 * q^65 + 18 * q^67 - 20 * q^69 + 2 * q^73 + 2 * q^75 - 12 * q^79 + 3 * q^81 - 14 * q^83 + 12 * q^85 - 36 * q^87 + 2 * q^89 - 8 * q^93 + 4 * q^95 + 10 * q^97 + 30 * q^99

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