LMFDB - Embedded newform 6014.2.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6014,2,Mod(1,6014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6014, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6014 = 2 \cdot 31 \cdot 97 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6014.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:	$48.0220317756$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.380224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} - 2x^{2} + 5x + 2 $ x^5 - 8x^3 - 2x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.874805$ of defining polynomial
Character	$\chi$	$=$	6014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.874805 q^{3} +1.00000 q^{4} -2.68974 q^{5} +0.874805 q^{6} +1.00000 q^{8} -2.23472 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.874805 q^{3} +1.00000 q^{4} -2.68974 q^{5} +0.874805 q^{6} +1.00000 q^{8} -2.23472 q^{9} -2.68974 q^{10} +1.75652 q^{11} +0.874805 q^{12} +4.70600 q^{13} -2.35300 q^{15} +1.00000 q^{16} +3.21155 q^{17} -2.23472 q^{18} +2.76442 q^{19} -2.68974 q^{20} +1.75652 q^{22} -5.42623 q^{23} +0.874805 q^{24} +2.23472 q^{25} +4.70600 q^{26} -4.57936 q^{27} +5.19296 q^{29} -2.35300 q^{30} -1.00000 q^{31} +1.00000 q^{32} +1.53661 q^{33} +3.21155 q^{34} -2.23472 q^{36} +2.76442 q^{38} +4.11683 q^{39} -2.68974 q^{40} -6.45271 q^{41} -1.75652 q^{43} +1.75652 q^{44} +6.01081 q^{45} -5.42623 q^{46} +12.4930 q^{47} +0.874805 q^{48} -7.00000 q^{49} +2.23472 q^{50} +2.80948 q^{51} +4.70600 q^{52} +3.37558 q^{53} -4.57936 q^{54} -4.72459 q^{55} +2.41833 q^{57} +5.19296 q^{58} -5.89351 q^{59} -2.35300 q^{60} -1.25184 q^{61} -1.00000 q^{62} +1.00000 q^{64} -12.6579 q^{65} +1.53661 q^{66} +10.9534 q^{67} +3.21155 q^{68} -4.74689 q^{69} -3.29400 q^{71} -2.23472 q^{72} +11.9687 q^{73} +1.95494 q^{75} +2.76442 q^{76} +4.11683 q^{78} +9.91278 q^{79} -2.68974 q^{80} +2.69810 q^{81} -6.45271 q^{82} +7.76343 q^{83} -8.63824 q^{85} -1.75652 q^{86} +4.54283 q^{87} +1.75652 q^{88} +10.9955 q^{89} +6.01081 q^{90} -5.42623 q^{92} -0.874805 q^{93} +12.4930 q^{94} -7.43558 q^{95} +0.874805 q^{96} -1.00000 q^{97} -7.00000 q^{98} -3.92532 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{4} - 4 q^{5} + 5 q^{8} + q^{9}+O(q^{10}) $ 5 * q + 5 * q^2 + 5 * q^4 - 4 * q^5 + 5 * q^8 + q^9	$ 5 q + 5 q^{2} + 5 q^{4} - 4 q^{5} + 5 q^{8} + q^{9} - 4 q^{10} + 4 q^{11} + 5 q^{16} + 14 q^{17} + q^{18} - 10 q^{19} - 4 q^{20} + 4 q^{22} - q^{25} + 6 q^{27} + 12 q^{29} - 5 q^{31} + 5 q^{32} + 14 q^{34} + q^{36} - 10 q^{38} + 20 q^{39} - 4 q^{40} + 2 q^{41} - 4 q^{43} + 4 q^{44} + 2 q^{45} + 40 q^{47} - 35 q^{49} - q^{50} + 6 q^{51} + 30 q^{53} + 6 q^{54} - 12 q^{55} + 4 q^{57} + 12 q^{58} + 22 q^{59} - 16 q^{61} - 5 q^{62} + 5 q^{64} + 12 q^{65} + 4 q^{67} + 14 q^{68} + 34 q^{69} - 40 q^{71} + q^{72} + 18 q^{73} - 6 q^{75} - 10 q^{76} + 20 q^{78} + 20 q^{79} - 4 q^{80} + 9 q^{81} + 2 q^{82} + 38 q^{83} - 38 q^{85} - 4 q^{86} + 20 q^{87} + 4 q^{88} + 18 q^{89} + 2 q^{90} + 40 q^{94} + 8 q^{95} - 5 q^{97} - 35 q^{98} - 34 q^{99}+O(q^{100}) $ 5 * q + 5 * q^2 + 5 * q^4 - 4 * q^5 + 5 * q^8 + q^9 - 4 * q^10 + 4 * q^11 + 5 * q^16 + 14 * q^17 + q^18 - 10 * q^19 - 4 * q^20 + 4 * q^22 - q^25 + 6 * q^27 + 12 * q^29 - 5 * q^31 + 5 * q^32 + 14 * q^34 + q^36 - 10 * q^38 + 20 * q^39 - 4 * q^40 + 2 * q^41 - 4 * q^43 + 4 * q^44 + 2 * q^45 + 40 * q^47 - 35 * q^49 - q^50 + 6 * q^51 + 30 * q^53 + 6 * q^54 - 12 * q^55 + 4 * q^57 + 12 * q^58 + 22 * q^59 - 16 * q^61 - 5 * q^62 + 5 * q^64 + 12 * q^65 + 4 * q^67 + 14 * q^68 + 34 * q^69 - 40 * q^71 + q^72 + 18 * q^73 - 6 * q^75 - 10 * q^76 + 20 * q^78 + 20 * q^79 - 4 * q^80 + 9 * q^81 + 2 * q^82 + 38 * q^83 - 38 * q^85 - 4 * q^86 + 20 * q^87 + 4 * q^88 + 18 * q^89 + 2 * q^90 + 40 * q^94 + 8 * q^95 - 5 * q^97 - 35 * q^98 - 34 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.874805	0.505069	0.252535	−	0.967588i	$-0.418736\pi$
$3$	0.874805	0.505069	0.252535	+	0.967588i	$0.418736\pi$
$4$	1.00000	0.500000
$5$	−2.68974	−1.20289	−0.601445	−	0.798914i	$-0.705408\pi$
$5$	−2.68974	−1.20289	−0.601445	+	0.798914i	$0.705408\pi$
$6$	0.874805	0.357138
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	−2.23472	−0.744905
$10$	−2.68974	−0.850571
$11$	1.75652	0.529611	0.264805	−	0.964302i	$-0.414692\pi$
$11$	1.75652	0.529611	0.264805	+	0.964302i	$0.414692\pi$
$12$	0.874805	0.252535
$13$	4.70600	1.30521	0.652605	−	0.757698i	$-0.273676\pi$
$13$	4.70600	1.30521	0.652605	+	0.757698i	$0.273676\pi$
$14$	0	0
$15$	−2.35300	−0.607542
$16$	1.00000	0.250000
$17$	3.21155	0.778915	0.389457	−	0.921045i	$-0.372663\pi$
$17$	3.21155	0.778915	0.389457	+	0.921045i	$0.372663\pi$
$18$	−2.23472	−0.526728
$19$	2.76442	0.634201	0.317101	−	0.948392i	$-0.397291\pi$
$19$	2.76442	0.634201	0.317101	+	0.948392i	$0.397291\pi$
$20$	−2.68974	−0.601445
$21$	0	0
$22$	1.75652	0.374491
$23$	−5.42623	−1.13145	−0.565723	−	0.824595i	$-0.691403\pi$
$23$	−5.42623	−1.13145	−0.565723	+	0.824595i	$0.691403\pi$
$24$	0.874805	0.178569
$25$	2.23472	0.446943
$26$	4.70600	0.922923
$27$	−4.57936	−0.881298
$28$	0	0
$29$	5.19296	0.964309	0.482154	−	0.876086i	$-0.339854\pi$
$29$	5.19296	0.964309	0.482154	+	0.876086i	$0.339854\pi$
$30$	−2.35300	−0.429597
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000	0.176777
$33$	1.53661	0.267490
$34$	3.21155	0.550776
$35$	0	0
$36$	−2.23472	−0.372453
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	2.76442	0.448448
$39$	4.11683	0.659221
$40$	−2.68974	−0.425286
$41$	−6.45271	−1.00774	−0.503872	−	0.863778i	$-0.668092\pi$
$41$	−6.45271	−1.00774	−0.503872	+	0.863778i	$0.668092\pi$
$42$	0	0
$43$	−1.75652	−0.267867	−0.133933	−	0.990990i	$-0.542761\pi$
$43$	−1.75652	−0.267867	−0.133933	+	0.990990i	$0.542761\pi$
$44$	1.75652	0.264805
$45$	6.01081	0.896039
$46$	−5.42623	−0.800053
$47$	12.4930	1.82229	0.911146	−	0.412084i	$-0.135199\pi$
$47$	12.4930	1.82229	0.911146	+	0.412084i	$0.135199\pi$
$48$	0.874805	0.126267
$49$	−7.00000	−1.00000
$50$	2.23472	0.316037
$51$	2.80948	0.393406
$52$	4.70600	0.652605
$53$	3.37558	0.463672	0.231836	−	0.972755i	$-0.425527\pi$
$53$	3.37558	0.463672	0.231836	+	0.972755i	$0.425527\pi$
$54$	−4.57936	−0.623171
$55$	−4.72459	−0.637063
$56$	0	0
$57$	2.41833	0.320315
$58$	5.19296	0.681869
$59$	−5.89351	−0.767270	−0.383635	−	0.923485i	$-0.625328\pi$
$59$	−5.89351	−0.767270	−0.383635	+	0.923485i	$0.625328\pi$
$60$	−2.35300	−0.303771
$61$	−1.25184	−0.160282	−0.0801408	−	0.996784i	$-0.525537\pi$
$61$	−1.25184	−0.160282	−0.0801408	+	0.996784i	$0.525537\pi$
$62$	−1.00000	−0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	−12.6579	−1.57002
$66$	1.53661	0.189144
$67$	10.9534	1.33817	0.669084	−	0.743186i	$-0.266686\pi$
$67$	10.9534	1.33817	0.669084	+	0.743186i	$0.266686\pi$
$68$	3.21155	0.389457
$69$	−4.74689	−0.571459
$70$	0	0
$71$	−3.29400	−0.390926	−0.195463	−	0.980711i	$-0.562621\pi$
$71$	−3.29400	−0.390926	−0.195463	+	0.980711i	$0.562621\pi$
$72$	−2.23472	−0.263364
$73$	11.9687	1.40082	0.700412	−	0.713738i	$-0.252999\pi$
$73$	11.9687	1.40082	0.700412	+	0.713738i	$0.252999\pi$
$74$	0	0
$75$	1.95494	0.225737
$76$	2.76442	0.317101
$77$	0	0
$78$	4.11683	0.466140
$79$	9.91278	1.11528	0.557638	−	0.830085i	$-0.311708\pi$
$79$	9.91278	1.11528	0.557638	+	0.830085i	$0.311708\pi$
$80$	−2.68974	−0.300722
$81$	2.69810	0.299789
$82$	−6.45271	−0.712583
$83$	7.76343	0.852147	0.426074	−	0.904689i	$-0.359896\pi$
$83$	7.76343	0.852147	0.426074	+	0.904689i	$0.359896\pi$
$84$	0	0
$85$	−8.63824	−0.936948
$86$	−1.75652	−0.189410
$87$	4.54283	0.487042
$88$	1.75652	0.187246
$89$	10.9955	1.16552	0.582762	−	0.812643i	$-0.301972\pi$
$89$	10.9955	1.16552	0.582762	+	0.812643i	$0.301972\pi$
$90$	6.01081	0.633595
$91$	0	0
$92$	−5.42623	−0.565723
$93$	−0.874805	−0.0907131
$94$	12.4930	1.28856
$95$	−7.43558	−0.762874
$96$	0.874805	0.0892844
$97$	−1.00000	−0.101535
$97$	−1.00000	−0.101535
$98$	−7.00000	−0.707107
$99$	−3.92532	−0.394510
$100$	2.23472	0.223472
$101$	−9.02516	−0.898037	−0.449018	−	0.893523i	$-0.648226\pi$
$101$	−9.02516	−0.898037	−0.449018	+	0.893523i	$0.648226\pi$
$102$	2.80948	0.278180
$103$	6.91610	0.681463	0.340732	−	0.940161i	$-0.389325\pi$
$103$	6.91610	0.681463	0.340732	+	0.940161i	$0.389325\pi$
$104$	4.70600	0.461461
$105$	0	0
$106$	3.37558	0.327866
$107$	5.95784	0.575966	0.287983	−	0.957635i	$-0.407015\pi$
$107$	5.95784	0.575966	0.287983	+	0.957635i	$0.407015\pi$
$108$	−4.57936	−0.440649
$109$	5.02608	0.481411	0.240706	−	0.970598i	$-0.422621\pi$
$109$	5.02608	0.481411	0.240706	+	0.970598i	$0.422621\pi$
$110$	−4.72459	−0.450472
$111$	0	0
$112$	0	0
$113$	11.2224	1.05571	0.527855	−	0.849334i	$-0.322996\pi$
$113$	11.2224	1.05571	0.527855	+	0.849334i	$0.322996\pi$
$114$	2.41833	0.226497
$115$	14.5952	1.36101
$116$	5.19296	0.482154
$117$	−10.5166	−0.972258
$118$	−5.89351	−0.542542
$119$	0	0
$120$	−2.35300	−0.214799
$121$	−7.91464	−0.719512
$122$	−1.25184	−0.113336
$123$	−5.64487	−0.508980
$124$	−1.00000	−0.0898027
$125$	7.43790	0.665266
$126$	0	0
$127$	12.1494	1.07808	0.539040	−	0.842280i	$-0.318787\pi$
$127$	12.1494	1.07808	0.539040	+	0.842280i	$0.318787\pi$
$128$	1.00000	0.0883883
$129$	−1.53661	−0.135291
$130$	−12.6579	−1.11017
$131$	10.7194	0.936555	0.468277	−	0.883581i	$-0.344875\pi$
$131$	10.7194	0.936555	0.468277	+	0.883581i	$0.344875\pi$
$132$	1.53661	0.133745
$133$	0	0
$134$	10.9534	0.946228
$135$	12.3173	1.06010
$136$	3.21155	0.275388
$137$	17.3424	1.48167	0.740833	−	0.671690i	$-0.234431\pi$
$137$	17.3424	1.48167	0.740833	+	0.671690i	$0.234431\pi$
$138$	−4.74689	−0.404082
$139$	−11.4120	−0.967953	−0.483977	−	0.875081i	$-0.660808\pi$
$139$	−11.4120	−0.967953	−0.483977	+	0.875081i	$0.660808\pi$
$140$	0	0
$141$	10.9289	0.920383
$142$	−3.29400	−0.276426
$143$	8.26619	0.691253
$144$	−2.23472	−0.186226
$145$	−13.9677	−1.15996
$146$	11.9687	0.990533
$147$	−6.12364	−0.505069
$148$	0	0
$149$	−17.1853	−1.40788	−0.703938	−	0.710261i	$-0.748577\pi$
$149$	−17.1853	−1.40788	−0.703938	+	0.710261i	$0.748577\pi$
$150$	1.95494	0.159620
$151$	0.370125	0.0301203	0.0150602	−	0.999887i	$-0.495206\pi$
$151$	0.370125	0.0301203	0.0150602	+	0.999887i	$0.495206\pi$
$152$	2.76442	0.224224
$153$	−7.17690	−0.580218
$154$	0	0
$155$	2.68974	0.216045
$156$	4.11683	0.329611
$157$	0.745721	0.0595150	0.0297575	−	0.999557i	$-0.490526\pi$
$157$	0.745721	0.0595150	0.0297575	+	0.999557i	$0.490526\pi$
$158$	9.91278	0.788619
$159$	2.95298	0.234186
$160$	−2.68974	−0.212643
$161$	0	0
$162$	2.69810	0.211983
$163$	20.3917	1.59720	0.798602	−	0.601859i	$-0.205573\pi$
$163$	20.3917	1.59720	0.798602	+	0.601859i	$0.205573\pi$
$164$	−6.45271	−0.503872
$165$	−4.13309	−0.321761
$166$	7.76343	0.602559
$167$	−12.7189	−0.984218	−0.492109	−	0.870534i	$-0.663774\pi$
$167$	−12.7189	−0.984218	−0.492109	+	0.870534i	$0.663774\pi$
$168$	0	0
$169$	9.14645	0.703573
$170$	−8.63824	−0.662522
$171$	−6.17769	−0.472420
$172$	−1.75652	−0.133933
$173$	7.25720	0.551755	0.275877	−	0.961193i	$-0.411032\pi$
$173$	7.25720	0.551755	0.275877	+	0.961193i	$0.411032\pi$
$174$	4.54283	0.344391
$175$	0	0
$176$	1.75652	0.132403
$177$	−5.15568	−0.387524
$178$	10.9955	0.824151
$179$	−2.05587	−0.153663	−0.0768315	−	0.997044i	$-0.524480\pi$
$179$	−2.05587	−0.153663	−0.0768315	+	0.997044i	$0.524480\pi$
$180$	6.01081	0.448019
$181$	13.2284	0.983259	0.491630	−	0.870804i	$-0.336401\pi$
$181$	13.2284	0.983259	0.491630	+	0.870804i	$0.336401\pi$
$182$	0	0
$183$	−1.09512	−0.0809533
$184$	−5.42623	−0.400027
$185$	0	0
$186$	−0.874805	−0.0641438
$187$	5.64115	0.412522
$188$	12.4930	0.911146
$189$	0	0
$190$	−7.43558	−0.539433
$191$	6.79322	0.491540	0.245770	−	0.969328i	$-0.420959\pi$
$191$	6.79322	0.491540	0.245770	+	0.969328i	$0.420959\pi$
$192$	0.874805	0.0631336
$193$	9.22258	0.663856	0.331928	−	0.943305i	$-0.392301\pi$
$193$	9.22258	0.663856	0.331928	+	0.943305i	$0.392301\pi$
$194$	−1.00000	−0.0717958
$195$	−11.0732	−0.792970
$196$	−7.00000	−0.500000
$197$	−13.8960	−0.990046	−0.495023	−	0.868880i	$-0.664840\pi$
$197$	−13.8960	−0.990046	−0.495023	+	0.868880i	$0.664840\pi$
$198$	−3.92532	−0.278961
$199$	3.84496	0.272562	0.136281	−	0.990670i	$-0.456485\pi$
$199$	3.84496	0.272562	0.136281	+	0.990670i	$0.456485\pi$
$200$	2.23472	0.158018
$201$	9.58208	0.675868
$202$	−9.02516	−0.635008
$203$	0	0
$204$	2.80948	0.196703
$205$	17.3561	1.21220
$206$	6.91610	0.481867
$207$	12.1261	0.842821
$208$	4.70600	0.326303
$209$	4.85576	0.335880
$210$	0	0
$211$	14.8224	1.02041	0.510207	−	0.860052i	$-0.329569\pi$
$211$	14.8224	1.02041	0.510207	+	0.860052i	$0.329569\pi$
$212$	3.37558	0.231836
$213$	−2.88161	−0.197444
$214$	5.95784	0.407270
$215$	4.72459	0.322214
$216$	−4.57936	−0.311586
$217$	0	0
$218$	5.02608	0.340409
$219$	10.4702	0.707513
$220$	−4.72459	−0.318532
$221$	15.1135	1.01665
$222$	0	0
$223$	−9.63145	−0.644970	−0.322485	−	0.946575i	$-0.604518\pi$
$223$	−9.63145	−0.644970	−0.322485	+	0.946575i	$0.604518\pi$
$224$	0	0
$225$	−4.99396	−0.332930
$226$	11.2224	0.746500
$227$	−14.7681	−0.980196	−0.490098	−	0.871667i	$-0.663039\pi$
$227$	−14.7681	−0.980196	−0.490098	+	0.871667i	$0.663039\pi$
$228$	2.41833	0.160158
$229$	22.1789	1.46562	0.732811	−	0.680432i	$-0.238208\pi$
$229$	22.1789	1.46562	0.732811	+	0.680432i	$0.238208\pi$
$230$	14.5952	0.962376
$231$	0	0
$232$	5.19296	0.340935
$233$	27.6535	1.81164	0.905820	−	0.423663i	$-0.139256\pi$
$233$	27.6535	1.81164	0.905820	+	0.423663i	$0.139256\pi$
$234$	−10.5166	−0.687490
$235$	−33.6030	−2.19202
$236$	−5.89351	−0.383635
$237$	8.67175	0.563291
$238$	0	0
$239$	−17.6591	−1.14227	−0.571136	−	0.820856i	$-0.693497\pi$
$239$	−17.6591	−1.14227	−0.571136	+	0.820856i	$0.693497\pi$
$240$	−2.35300	−0.151886
$241$	−5.45115	−0.351140	−0.175570	−	0.984467i	$-0.556177\pi$
$241$	−5.45115	−0.351140	−0.175570	+	0.984467i	$0.556177\pi$
$242$	−7.91464	−0.508772
$243$	16.0984	1.03271
$244$	−1.25184	−0.0801408
$245$	18.8282	1.20289
$246$	−5.64487	−0.359903
$247$	13.0094	0.827766
$248$	−1.00000	−0.0635001
$249$	6.79149	0.430393
$250$	7.43790	0.470414
$251$	−27.4198	−1.73072	−0.865361	−	0.501148i	$-0.832911\pi$
$251$	−27.4198	−1.73072	−0.865361	+	0.501148i	$0.832911\pi$
$252$	0	0
$253$	−9.53128	−0.599226
$254$	12.1494	0.762318
$255$	−7.55677	−0.473223
$256$	1.00000	0.0625000
$257$	13.0361	0.813167	0.406583	−	0.913614i	$-0.366720\pi$
$257$	13.0361	0.813167	0.406583	+	0.913614i	$0.366720\pi$
$258$	−1.53661	−0.0956653
$259$	0	0
$260$	−12.6579	−0.785012
$261$	−11.6048	−0.718319
$262$	10.7194	0.662244
$263$	24.7726	1.52755	0.763773	−	0.645485i	$-0.223345\pi$
$263$	24.7726	1.52755	0.763773	+	0.645485i	$0.223345\pi$
$264$	1.53661	0.0945720
$265$	−9.07945	−0.557746
$266$	0	0
$267$	9.61896	0.588671
$268$	10.9534	0.669084
$269$	−27.5791	−1.68153	−0.840763	−	0.541403i	$-0.817893\pi$
$269$	−27.5791	−1.68153	−0.840763	+	0.541403i	$0.817893\pi$
$270$	12.3173	0.749606
$271$	−7.33432	−0.445528	−0.222764	−	0.974872i	$-0.571508\pi$
$271$	−7.33432	−0.445528	−0.222764	+	0.974872i	$0.571508\pi$
$272$	3.21155	0.194729
$273$	0	0
$274$	17.3424	1.04770
$275$	3.92532	0.236706
$276$	−4.74689	−0.285729
$277$	17.6449	1.06018	0.530089	−	0.847942i	$-0.322159\pi$
$277$	17.6449	1.06018	0.530089	+	0.847942i	$0.322159\pi$
$278$	−11.4120	−0.684446
$279$	2.23472	0.133789
$280$	0	0
$281$	−1.23303	−0.0735566	−0.0367783	−	0.999323i	$-0.511710\pi$
$281$	−1.23303	−0.0735566	−0.0367783	+	0.999323i	$0.511710\pi$
$282$	10.9289	0.650809
$283$	1.11330	0.0661787	0.0330894	−	0.999452i	$-0.489465\pi$
$283$	1.11330	0.0661787	0.0330894	+	0.999452i	$0.489465\pi$
$284$	−3.29400	−0.195463
$285$	−6.50468	−0.385304
$286$	8.26619	0.488790
$287$	0	0
$288$	−2.23472	−0.131682
$289$	−6.68597	−0.393292
$290$	−13.9677	−0.820213
$291$	−0.874805	−0.0512820
$292$	11.9687	0.700412
$293$	−25.1048	−1.46664	−0.733320	−	0.679884i	$-0.762030\pi$
$293$	−25.1048	−1.46664	−0.733320	+	0.679884i	$0.762030\pi$
$294$	−6.12364	−0.357138
$295$	15.8520	0.922941
$296$	0	0
$297$	−8.04373	−0.466745
$298$	−17.1853	−0.995519
$299$	−25.5358	−1.47678
$300$	1.95494	0.112869
$301$	0	0
$302$	0.370125	0.0212983
$303$	−7.89525	−0.453571
$304$	2.76442	0.158550
$305$	3.36713	0.192801
$306$	−7.17690	−0.410276
$307$	−19.2676	−1.09966	−0.549829	−	0.835277i	$-0.685307\pi$
$307$	−19.2676	−1.09966	−0.549829	+	0.835277i	$0.685307\pi$
$308$	0	0
$309$	6.05024	0.344186
$310$	2.68974	0.152767
$311$	22.4911	1.27535	0.637675	−	0.770305i	$-0.279896\pi$
$311$	22.4911	1.27535	0.637675	+	0.770305i	$0.279896\pi$
$312$	4.11683	0.233070
$313$	15.5055	0.876422	0.438211	−	0.898872i	$-0.355612\pi$
$313$	15.5055	0.876422	0.438211	+	0.898872i	$0.355612\pi$
$314$	0.745721	0.0420835
$315$	0	0
$316$	9.91278	0.557638
$317$	−19.3164	−1.08492	−0.542460	−	0.840082i	$-0.682507\pi$
$317$	−19.3164	−1.08492	−0.542460	+	0.840082i	$0.682507\pi$
$318$	2.95298	0.165595
$319$	9.12154	0.510708
$320$	−2.68974	−0.150361
$321$	5.21195	0.290903
$322$	0	0
$323$	8.87806	0.493989
$324$	2.69810	0.149895
$325$	10.5166	0.583355
$326$	20.3917	1.12939
$327$	4.39684	0.243146
$328$	−6.45271	−0.356291
$329$	0	0
$330$	−4.13309	−0.227519
$331$	10.3830	0.570702	0.285351	−	0.958423i	$-0.407890\pi$
$331$	10.3830	0.570702	0.285351	+	0.958423i	$0.407890\pi$
$332$	7.76343	0.426074
$333$	0	0
$334$	−12.7189	−0.695947
$335$	−29.4618	−1.60967
$336$	0	0
$337$	6.48483	0.353251	0.176625	−	0.984278i	$-0.443482\pi$
$337$	6.48483	0.353251	0.176625	+	0.984278i	$0.443482\pi$
$338$	9.14645	0.497501
$339$	9.81738	0.533207
$340$	−8.63824	−0.468474
$341$	−1.75652	−0.0951209
$342$	−6.17769	−0.334051
$343$	0	0
$344$	−1.75652	−0.0947052
$345$	12.7679	0.687402
$346$	7.25720	0.390150
$347$	−28.6504	−1.53803	−0.769016	−	0.639229i	$-0.779254\pi$
$347$	−28.6504	−1.53803	−0.769016	+	0.639229i	$0.779254\pi$
$348$	4.54283	0.243521
$349$	0.870796	0.0466126	0.0233063	−	0.999728i	$-0.492581\pi$
$349$	0.870796	0.0466126	0.0233063	+	0.999728i	$0.492581\pi$
$350$	0	0
$351$	−21.5505	−1.15028
$352$	1.75652	0.0936228
$353$	−13.7516	−0.731923	−0.365962	−	0.930630i	$-0.619260\pi$
$353$	−13.7516	−0.731923	−0.365962	+	0.930630i	$0.619260\pi$
$354$	−5.15568	−0.274021
$355$	8.86001	0.470240
$356$	10.9955	0.582762
$357$	0	0
$358$	−2.05587	−0.108656
$359$	0.195690	0.0103281	0.00516407	−	0.999987i	$-0.498356\pi$
$359$	0.195690	0.0103281	0.00516407	+	0.999987i	$0.498356\pi$
$360$	6.01081	0.316798
$361$	−11.3580	−0.597789
$362$	13.2284	0.695269
$363$	−6.92377	−0.363403
$364$	0	0
$365$	−32.1926	−1.68504
$366$	−1.09512	−0.0572426
$367$	6.77463	0.353633	0.176816	−	0.984244i	$-0.443420\pi$
$367$	6.77463	0.353633	0.176816	+	0.984244i	$0.443420\pi$
$368$	−5.42623	−0.282862
$369$	14.4200	0.750674
$370$	0	0
$371$	0	0
$372$	−0.874805	−0.0453565
$373$	−1.94657	−0.100790	−0.0503948	−	0.998729i	$-0.516048\pi$
$373$	−1.94657	−0.100790	−0.0503948	+	0.998729i	$0.516048\pi$
$374$	5.64115	0.291697
$375$	6.50672	0.336005
$376$	12.4930	0.644278
$377$	24.4381	1.25863
$378$	0	0
$379$	11.6151	0.596629	0.298315	−	0.954468i	$-0.403576\pi$
$379$	11.6151	0.596629	0.298315	+	0.954468i	$0.403576\pi$
$380$	−7.43558	−0.381437
$381$	10.6283	0.544505
$382$	6.79322	0.347571
$383$	−12.0688	−0.616685	−0.308343	−	0.951275i	$-0.599774\pi$
$383$	−12.0688	−0.616685	−0.308343	+	0.951275i	$0.599774\pi$
$384$	0.874805	0.0446422
$385$	0	0
$386$	9.22258	0.469417
$387$	3.92532	0.199535
$388$	−1.00000	−0.0507673
$389$	3.05288	0.154787	0.0773937	−	0.997001i	$-0.475340\pi$
$389$	3.05288	0.154787	0.0773937	+	0.997001i	$0.475340\pi$
$390$	−11.0732	−0.560715
$391$	−17.4266	−0.881300
$392$	−7.00000	−0.353553
$393$	9.37735	0.473025
$394$	−13.8960	−0.700068
$395$	−26.6628	−1.34155
$396$	−3.92532	−0.197255
$397$	6.91203	0.346905	0.173452	−	0.984842i	$-0.444508\pi$
$397$	6.91203	0.346905	0.173452	+	0.984842i	$0.444508\pi$
$398$	3.84496	0.192730
$399$	0	0
$400$	2.23472	0.111736
$401$	−25.7173	−1.28426	−0.642131	−	0.766595i	$-0.721950\pi$
$401$	−25.7173	−1.28426	−0.642131	+	0.766595i	$0.721950\pi$
$402$	9.58208	0.477910
$403$	−4.70600	−0.234423
$404$	−9.02516	−0.449018
$405$	−7.25720	−0.360613
$406$	0	0
$407$	0	0
$408$	2.80948	0.139090
$409$	4.98931	0.246706	0.123353	−	0.992363i	$-0.460635\pi$
$409$	4.98931	0.246706	0.123353	+	0.992363i	$0.460635\pi$
$410$	17.3561	0.857158
$411$	15.1713	0.748343
$412$	6.91610	0.340732
$413$	0	0
$414$	12.1261	0.595964
$415$	−20.8816	−1.02504
$416$	4.70600	0.230731
$417$	−9.98328	−0.488883
$418$	4.85576	0.237503
$419$	12.5882	0.614976	0.307488	−	0.951552i	$-0.400512\pi$
$419$	12.5882	0.614976	0.307488	+	0.951552i	$0.400512\pi$
$420$	0	0
$421$	28.4292	1.38555	0.692776	−	0.721152i	$-0.256387\pi$
$421$	28.4292	1.38555	0.692776	+	0.721152i	$0.256387\pi$
$422$	14.8224	0.721542
$423$	−27.9183	−1.35743
$424$	3.37558	0.163933
$425$	7.17690	0.348131
$426$	−2.88161	−0.139614
$427$	0	0
$428$	5.95784	0.287983
$429$	7.23130	0.349131
$430$	4.72459	0.227840
$431$	−37.8929	−1.82524	−0.912618	−	0.408813i	$-0.865943\pi$
$431$	−37.8929	−1.82524	−0.912618	+	0.408813i	$0.865943\pi$
$432$	−4.57936	−0.220324
$433$	−29.1209	−1.39946	−0.699730	−	0.714408i	$-0.746696\pi$
$433$	−29.1209	−1.39946	−0.699730	+	0.714408i	$0.746696\pi$
$434$	0	0
$435$	−12.2190	−0.585858
$436$	5.02608	0.240706
$437$	−15.0004	−0.717565
$438$	10.4702	0.500287
$439$	−24.9095	−1.18887	−0.594433	−	0.804145i	$-0.702623\pi$
$439$	−24.9095	−1.18887	−0.594433	+	0.804145i	$0.702623\pi$
$440$	−4.72459	−0.225236
$441$	15.6430	0.744905
$442$	15.1135	0.718878
$443$	−36.2476	−1.72218	−0.861088	−	0.508455i	$-0.830217\pi$
$443$	−36.2476	−1.72218	−0.861088	+	0.508455i	$0.830217\pi$
$444$	0	0
$445$	−29.5752	−1.40200
$446$	−9.63145	−0.456062
$447$	−15.0338	−0.711075
$448$	0	0
$449$	−14.0680	−0.663908	−0.331954	−	0.943296i	$-0.607708\pi$
$449$	−14.0680	−0.663908	−0.331954	+	0.943296i	$0.607708\pi$
$450$	−4.99396	−0.235417
$451$	−11.3343	−0.533712
$452$	11.2224	0.527855
$453$	0.323787	0.0152128
$454$	−14.7681	−0.693103
$455$	0	0
$456$	2.41833	0.113249
$457$	13.8844	0.649483	0.324741	−	0.945803i	$-0.394723\pi$
$457$	13.8844	0.649483	0.324741	+	0.945803i	$0.394723\pi$
$458$	22.1789	1.03635
$459$	−14.7068	−0.686456
$460$	14.5952	0.680503
$461$	3.92069	0.182605	0.0913024	−	0.995823i	$-0.470897\pi$
$461$	3.92069	0.182605	0.0913024	+	0.995823i	$0.470897\pi$
$462$	0	0
$463$	−17.8452	−0.829337	−0.414669	−	0.909972i	$-0.636102\pi$
$463$	−17.8452	−0.829337	−0.414669	+	0.909972i	$0.636102\pi$
$464$	5.19296	0.241077
$465$	2.35300	0.109118
$466$	27.6535	1.28102
$467$	−21.1077	−0.976749	−0.488374	−	0.872634i	$-0.662410\pi$
$467$	−21.1077	−0.976749	−0.488374	+	0.872634i	$0.662410\pi$
$468$	−10.5166	−0.486129
$469$	0	0
$470$	−33.6030	−1.54999
$471$	0.652360	0.0300592
$472$	−5.89351	−0.271271
$473$	−3.08536	−0.141865
$474$	8.67175	0.398307
$475$	6.17769	0.283452
$476$	0	0
$477$	−7.54347	−0.345392
$478$	−17.6591	−0.807708
$479$	28.7327	1.31283	0.656416	−	0.754399i	$-0.272071\pi$
$479$	28.7327	1.31283	0.656416	+	0.754399i	$0.272071\pi$
$480$	−2.35300	−0.107399
$481$	0	0
$482$	−5.45115	−0.248293
$483$	0	0
$484$	−7.91464	−0.359756
$485$	2.68974	0.122135
$486$	16.0984	0.730238
$487$	−38.0933	−1.72617	−0.863086	−	0.505057i	$-0.831471\pi$
$487$	−38.0933	−1.72617	−0.863086	+	0.505057i	$0.831471\pi$
$488$	−1.25184	−0.0566681
$489$	17.8388	0.806699
$490$	18.8282	0.850571
$491$	7.68293	0.346726	0.173363	−	0.984858i	$-0.444537\pi$
$491$	7.68293	0.346726	0.173363	+	0.984858i	$0.444537\pi$
$492$	−5.64487	−0.254490
$493$	16.6774	0.751114
$494$	13.0094	0.585319
$495$	10.5581	0.474552
$496$	−1.00000	−0.0449013
$497$	0	0
$498$	6.79149	0.304334
$499$	−20.5707	−0.920872	−0.460436	−	0.887693i	$-0.652307\pi$
$499$	−20.5707	−0.920872	−0.460436	+	0.887693i	$0.652307\pi$
$500$	7.43790	0.332633
$501$	−11.1266	−0.497098
$502$	−27.4198	−1.22381
$503$	0.751464	0.0335061	0.0167531	−	0.999860i	$-0.494667\pi$
$503$	0.751464	0.0335061	0.0167531	+	0.999860i	$0.494667\pi$
$504$	0	0
$505$	24.2754	1.08024
$506$	−9.53128	−0.423717
$507$	8.00136	0.355353
$508$	12.1494	0.539040
$509$	7.23320	0.320606	0.160303	−	0.987068i	$-0.448753\pi$
$509$	7.23320	0.320606	0.160303	+	0.987068i	$0.448753\pi$
$510$	−7.55677	−0.334620
$511$	0	0
$512$	1.00000	0.0441942
$513$	−12.6593	−0.558920
$514$	13.0361	0.574996
$515$	−18.6025	−0.819725
$516$	−1.53661	−0.0676456
$517$	21.9442	0.965105
$518$	0	0
$519$	6.34864	0.278674
$520$	−12.6579	−0.555087
$521$	−19.2692	−0.844199	−0.422100	−	0.906549i	$-0.638707\pi$
$521$	−19.2692	−0.844199	−0.422100	+	0.906549i	$0.638707\pi$
$522$	−11.6048	−0.507928
$523$	−28.8573	−1.26184	−0.630922	−	0.775847i	$-0.717323\pi$
$523$	−28.8573	−1.26184	−0.630922	+	0.775847i	$0.717323\pi$
$524$	10.7194	0.468277
$525$	0	0
$526$	24.7726	1.08014
$527$	−3.21155	−0.139897
$528$	1.53661	0.0668725
$529$	6.44394	0.280171
$530$	−9.07945	−0.394386
$531$	13.1703	0.571544
$532$	0	0
$533$	−30.3665	−1.31532
$534$	9.61896	0.416253
$535$	−16.0251	−0.692824
$536$	10.9534	0.473114
$537$	−1.79849	−0.0776104
$538$	−27.5791	−1.18902
$539$	−12.2956	−0.529611
$540$	12.3173	0.530052
$541$	−1.04315	−0.0448484	−0.0224242	−	0.999749i	$-0.507138\pi$
$541$	−1.04315	−0.0448484	−0.0224242	+	0.999749i	$0.507138\pi$
$542$	−7.33432	−0.315036
$543$	11.5723	0.496614
$544$	3.21155	0.137694
$545$	−13.5189	−0.579085
$546$	0	0
$547$	1.61115	0.0688880	0.0344440	−	0.999407i	$-0.489034\pi$
$547$	1.61115	0.0688880	0.0344440	+	0.999407i	$0.489034\pi$
$548$	17.3424	0.740833
$549$	2.79751	0.119395
$550$	3.92532	0.167376
$551$	14.3555	0.611566
$552$	−4.74689	−0.202041
$553$	0	0
$554$	17.6449	0.749658
$555$	0	0
$556$	−11.4120	−0.483977
$557$	25.0064	1.05955	0.529777	−	0.848137i	$-0.322276\pi$
$557$	25.0064	1.05955	0.529777	+	0.848137i	$0.322276\pi$
$558$	2.23472	0.0946031
$559$	−8.26619	−0.349622
$560$	0	0
$561$	4.93490	0.208352
$562$	−1.23303	−0.0520124
$563$	−20.9289	−0.882046	−0.441023	−	0.897496i	$-0.645384\pi$
$563$	−20.9289	−0.882046	−0.441023	+	0.897496i	$0.645384\pi$
$564$	10.9289	0.460192
$565$	−30.1853	−1.26990
$566$	1.11330	0.0467954
$567$	0	0
$568$	−3.29400	−0.138213
$569$	−14.4761	−0.606868	−0.303434	−	0.952852i	$-0.598133\pi$
$569$	−14.4761	−0.606868	−0.303434	+	0.952852i	$0.598133\pi$
$570$	−6.50468	−0.272451
$571$	−2.48888	−0.104157	−0.0520783	−	0.998643i	$-0.516585\pi$
$571$	−2.48888	−0.104157	−0.0520783	+	0.998643i	$0.516585\pi$
$572$	8.26619	0.345627
$573$	5.94274	0.248262
$574$	0	0
$575$	−12.1261	−0.505692
$576$	−2.23472	−0.0931132
$577$	13.3945	0.557619	0.278810	−	0.960346i	$-0.410060\pi$
$577$	13.3945	0.557619	0.278810	+	0.960346i	$0.410060\pi$
$578$	−6.68597	−0.278100
$579$	8.06796	0.335293
$580$	−13.9677	−0.579978
$581$	0	0
$582$	−0.874805	−0.0362618
$583$	5.92928	0.245566
$584$	11.9687	0.495266
$585$	28.2869	1.16952
$586$	−25.1048	−1.03707
$587$	30.2635	1.24911	0.624555	−	0.780981i	$-0.285280\pi$
$587$	30.2635	1.24911	0.624555	+	0.780981i	$0.285280\pi$
$588$	−6.12364	−0.252535
$589$	−2.76442	−0.113906
$590$	15.8520	0.652618
$591$	−12.1563	−0.500041
$592$	0	0
$593$	29.6607	1.21802	0.609009	−	0.793164i	$-0.291567\pi$
$593$	29.6607	1.21802	0.609009	+	0.793164i	$0.291567\pi$
$594$	−8.04373	−0.330038
$595$	0	0
$596$	−17.1853	−0.703938
$597$	3.36359	0.137663
$598$	−25.5358	−1.04424
$599$	28.4432	1.16216	0.581080	−	0.813847i	$-0.302631\pi$
$599$	28.4432	1.16216	0.581080	+	0.813847i	$0.302631\pi$
$600$	1.95494	0.0798101
$601$	20.4815	0.835459	0.417729	−	0.908572i	$-0.362826\pi$
$601$	20.4815	0.835459	0.417729	+	0.908572i	$0.362826\pi$
$602$	0	0
$603$	−24.4777	−0.996809
$604$	0.370125	0.0150602
$605$	21.2883	0.865494
$606$	−7.89525	−0.320723
$607$	45.7994	1.85894	0.929470	−	0.368897i	$-0.120264\pi$
$607$	45.7994	1.85894	0.929470	+	0.368897i	$0.120264\pi$
$608$	2.76442	0.112112
$609$	0	0
$610$	3.36713	0.136331
$611$	58.7921	2.37847
$612$	−7.17690	−0.290109
$613$	15.9625	0.644721	0.322361	−	0.946617i	$-0.395524\pi$
$613$	15.9625	0.644721	0.322361	+	0.946617i	$0.395524\pi$
$614$	−19.2676	−0.777575
$615$	15.1832	0.612247
$616$	0	0
$617$	−14.7216	−0.592668	−0.296334	−	0.955084i	$-0.595764\pi$
$617$	−14.7216	−0.592668	−0.296334	+	0.955084i	$0.595764\pi$
$618$	6.05024	0.243376
$619$	−12.5440	−0.504186	−0.252093	−	0.967703i	$-0.581119\pi$
$619$	−12.5440	−0.504186	−0.252093	+	0.967703i	$0.581119\pi$
$620$	2.68974	0.108023
$621$	24.8486	0.997141
$622$	22.4911	0.901809
$623$	0	0
$624$	4.11683	0.164805
$625$	−31.1796	−1.24718
$626$	15.5055	0.619724
$627$	4.24784	0.169642
$628$	0.745721	0.0297575
$629$	0	0
$630$	0	0
$631$	−1.22211	−0.0486516	−0.0243258	−	0.999704i	$-0.507744\pi$
$631$	−1.22211	−0.0486516	−0.0243258	+	0.999704i	$0.507744\pi$
$632$	9.91278	0.394309
$633$	12.9667	0.515380
$634$	−19.3164	−0.767154
$635$	−32.6786	−1.29681
$636$	2.95298	0.117093
$637$	−32.9420	−1.30521
$638$	9.12154	0.361125
$639$	7.36115	0.291203
$640$	−2.68974	−0.106321
$641$	23.0142	0.909008	0.454504	−	0.890745i	$-0.349817\pi$
$641$	23.0142	0.909008	0.454504	+	0.890745i	$0.349817\pi$
$642$	5.21195	0.205699
$643$	−26.7321	−1.05421	−0.527105	−	0.849800i	$-0.676723\pi$
$643$	−26.7321	−1.05421	−0.527105	+	0.849800i	$0.676723\pi$
$644$	0	0
$645$	4.13309	0.162740
$646$	8.87806	0.349303
$647$	30.3370	1.19267	0.596336	−	0.802735i	$-0.296623\pi$
$647$	30.3370	1.19267	0.596336	+	0.802735i	$0.296623\pi$
$648$	2.69810	0.105991
$649$	−10.3521	−0.406355
$650$	10.5166	0.412494
$651$	0	0
$652$	20.3917	0.798602
$653$	−21.8436	−0.854804	−0.427402	−	0.904062i	$-0.640571\pi$
$653$	−21.8436	−0.854804	−0.427402	+	0.904062i	$0.640571\pi$
$654$	4.39684	0.171930
$655$	−28.8323	−1.12657
$656$	−6.45271	−0.251936
$657$	−26.7465	−1.04348
$658$	0	0
$659$	−12.0387	−0.468961	−0.234480	−	0.972121i	$-0.575339\pi$
$659$	−12.0387	−0.468961	−0.234480	+	0.972121i	$0.575339\pi$
$660$	−4.13309	−0.160880
$661$	20.5624	0.799783	0.399892	−	0.916562i	$-0.369048\pi$
$661$	20.5624	0.799783	0.399892	+	0.916562i	$0.369048\pi$
$662$	10.3830	0.403548
$663$	13.2214	0.513477
$664$	7.76343	0.301280
$665$	0	0
$666$	0	0
$667$	−28.1782	−1.09106
$668$	−12.7189	−0.492109
$669$	−8.42564	−0.325754
$670$	−29.4618	−1.13821
$671$	−2.19888	−0.0848869
$672$	0	0
$673$	8.30974	0.320317	0.160158	−	0.987091i	$-0.448799\pi$
$673$	8.30974	0.320317	0.160158	+	0.987091i	$0.448799\pi$
$674$	6.48483	0.249786
$675$	−10.2336	−0.393890
$676$	9.14645	0.351787
$677$	−29.1760	−1.12132	−0.560662	−	0.828045i	$-0.689453\pi$
$677$	−29.1760	−1.12132	−0.560662	+	0.828045i	$0.689453\pi$
$678$	9.81738	0.377034
$679$	0	0
$680$	−8.63824	−0.331261
$681$	−12.9192	−0.495066
$682$	−1.75652	−0.0672606
$683$	−38.4850	−1.47259	−0.736294	−	0.676661i	$-0.763426\pi$
$683$	−38.4850	−1.47259	−0.736294	+	0.676661i	$0.763426\pi$
$684$	−6.17769	−0.236210
$685$	−46.6467	−1.78228
$686$	0	0
$687$	19.4022	0.740240
$688$	−1.75652	−0.0669667
$689$	15.8855	0.605190
$690$	12.7679	0.486066
$691$	29.0306	1.10438	0.552188	−	0.833719i	$-0.313793\pi$
$691$	29.0306	1.10438	0.552188	+	0.833719i	$0.313793\pi$
$692$	7.25720	0.275877
$693$	0	0
$694$	−28.6504	−1.08755
$695$	30.6954	1.16434
$696$	4.54283	0.172196
$697$	−20.7232	−0.784947
$698$	0.870796	0.0329601
$699$	24.1914	0.915003
$700$	0	0
$701$	6.61079	0.249686	0.124843	−	0.992177i	$-0.460157\pi$
$701$	6.61079	0.249686	0.124843	+	0.992177i	$0.460157\pi$
$702$	−21.5505	−0.813370
$703$	0	0
$704$	1.75652	0.0662013
$705$	−29.3960	−1.10712
$706$	−13.7516	−0.517548
$707$	0	0
$708$	−5.15568	−0.193762
$709$	−47.9070	−1.79919	−0.899593	−	0.436730i	$-0.856137\pi$
$709$	−47.9070	−1.79919	−0.899593	+	0.436730i	$0.856137\pi$
$710$	8.86001	0.332510
$711$	−22.1523	−0.830774
$712$	10.9955	0.412075
$713$	5.42623	0.203214
$714$	0	0
$715$	−22.2339	−0.831501
$716$	−2.05587	−0.0768315
$717$	−15.4483	−0.576926
$718$	0.195690	0.00730310
$719$	−5.63001	−0.209964	−0.104982	−	0.994474i	$-0.533478\pi$
$719$	−5.63001	−0.209964	−0.104982	+	0.994474i	$0.533478\pi$
$720$	6.01081	0.224010
$721$	0	0
$722$	−11.3580	−0.422701
$723$	−4.76870	−0.177350
$724$	13.2284	0.491630
$725$	11.6048	0.430991
$726$	−6.92377	−0.256965
$727$	46.7794	1.73495	0.867475	−	0.497481i	$-0.165741\pi$
$727$	46.7794	1.73495	0.867475	+	0.497481i	$0.165741\pi$
$728$	0	0
$729$	5.98864	0.221802
$730$	−32.1926	−1.19150
$731$	−5.64115	−0.208645
$732$	−1.09512	−0.0404767
$733$	31.8500	1.17641	0.588203	−	0.808713i	$-0.299836\pi$
$733$	31.8500	1.17641	0.588203	+	0.808713i	$0.299836\pi$
$734$	6.77463	0.250056
$735$	16.4710	0.607542
$736$	−5.42623	−0.200013
$737$	19.2398	0.708709
$738$	14.4200	0.530807
$739$	−50.0291	−1.84035	−0.920174	−	0.391509i	$-0.871953\pi$
$739$	−50.0291	−1.84035	−0.920174	+	0.391509i	$0.871953\pi$
$740$	0	0
$741$	11.3807	0.418079
$742$	0	0
$743$	5.03152	0.184589	0.0922943	−	0.995732i	$-0.470580\pi$
$743$	5.03152	0.184589	0.0922943	+	0.995732i	$0.470580\pi$
$744$	−0.874805	−0.0320719
$745$	46.2241	1.69352
$746$	−1.94657	−0.0712690
$747$	−17.3491	−0.634769
$748$	5.64115	0.206261
$749$	0	0
$750$	6.50672	0.237592
$751$	9.97979	0.364168	0.182084	−	0.983283i	$-0.441716\pi$
$751$	9.97979	0.364168	0.182084	+	0.983283i	$0.441716\pi$
$752$	12.4930	0.455573
$753$	−23.9870	−0.874134
$754$	24.4381	0.889983
$755$	−0.995541	−0.0362314
$756$	0	0
$757$	24.4243	0.887715	0.443857	−	0.896097i	$-0.353610\pi$
$757$	24.4243	0.887715	0.443857	+	0.896097i	$0.353610\pi$
$758$	11.6151	0.421881
$759$	−8.33801	−0.302651
$760$	−7.43558	−0.269717
$761$	−32.2287	−1.16829	−0.584145	−	0.811649i	$-0.698570\pi$
$761$	−32.2287	−1.16829	−0.584145	+	0.811649i	$0.698570\pi$
$762$	10.6283	0.385023
$763$	0	0
$764$	6.79322	0.245770
$765$	19.3040	0.697938
$766$	−12.0688	−0.436062
$767$	−27.7349	−1.00145
$768$	0.874805	0.0315668
$769$	−30.3289	−1.09369	−0.546845	−	0.837234i	$-0.684171\pi$
$769$	−30.3289	−1.09369	−0.546845	+	0.837234i	$0.684171\pi$
$770$	0	0
$771$	11.4040	0.410705
$772$	9.22258	0.331928
$773$	21.0586	0.757424	0.378712	−	0.925515i	$-0.376367\pi$
$773$	21.0586	0.757424	0.378712	+	0.925515i	$0.376367\pi$
$774$	3.92532	0.141093
$775$	−2.23472	−0.0802734
$776$	−1.00000	−0.0358979
$777$	0	0
$778$	3.05288	0.109451
$779$	−17.8380	−0.639113
$780$	−11.0732	−0.396485
$781$	−5.78597	−0.207038
$782$	−17.4266	−0.623173
$783$	−23.7804	−0.849843
$784$	−7.00000	−0.250000
$785$	−2.00580	−0.0715900
$786$	9.37735	0.334479
$787$	1.65077	0.0588438	0.0294219	−	0.999567i	$-0.490633\pi$
$787$	1.65077	0.0588438	0.0294219	+	0.999567i	$0.490633\pi$
$788$	−13.8960	−0.495023
$789$	21.6712	0.771516
$790$	−26.6628	−0.948621
$791$	0	0
$792$	−3.92532	−0.139480
$793$	−5.89116	−0.209201
$794$	6.91203	0.245299
$795$	−7.94275	−0.281700
$796$	3.84496	0.136281
$797$	−31.7766	−1.12558	−0.562792	−	0.826599i	$-0.690273\pi$
$797$	−31.7766	−1.12558	−0.562792	+	0.826599i	$0.690273\pi$
$798$	0	0
$799$	40.1219	1.41941
$800$	2.23472	0.0790091
$801$	−24.5719	−0.868206
$802$	−25.7173	−0.908111
$803$	21.0232	0.741892
$804$	9.58208	0.337934
$805$	0	0
$806$	−4.70600	−0.165762
$807$	−24.1263	−0.849287
$808$	−9.02516	−0.317504
$809$	−26.2995	−0.924642	−0.462321	−	0.886713i	$-0.652983\pi$
$809$	−26.2995	−0.924642	−0.462321	+	0.886713i	$0.652983\pi$
$810$	−7.25720	−0.254992
$811$	11.7066	0.411075	0.205537	−	0.978649i	$-0.434106\pi$
$811$	11.7066	0.411075	0.205537	+	0.978649i	$0.434106\pi$
$812$	0	0
$813$	−6.41610	−0.225022
$814$	0	0
$815$	−54.8486	−1.92126
$816$	2.80948	0.0983514
$817$	−4.85576	−0.169881
$818$	4.98931	0.174447
$819$	0	0
$820$	17.3561	0.606102
$821$	6.78232	0.236705	0.118352	−	0.992972i	$-0.462239\pi$
$821$	6.78232	0.236705	0.118352	+	0.992972i	$0.462239\pi$
$822$	15.1713	0.529159
$823$	−12.9693	−0.452080	−0.226040	−	0.974118i	$-0.572578\pi$
$823$	−12.9693	−0.452080	−0.226040	+	0.974118i	$0.572578\pi$
$824$	6.91610	0.240934
$825$	3.43389	0.119553
$826$	0	0
$827$	47.1982	1.64124	0.820622	−	0.571472i	$-0.193627\pi$
$827$	47.1982	1.64124	0.820622	+	0.571472i	$0.193627\pi$
$828$	12.1261	0.421410
$829$	48.1572	1.67257	0.836285	−	0.548295i	$-0.184723\pi$
$829$	48.1572	1.67257	0.836285	+	0.548295i	$0.184723\pi$
$830$	−20.8816	−0.724812
$831$	15.4358	0.535463
$832$	4.70600	0.163151
$833$	−22.4808	−0.778915
$834$	−9.98328	−0.345693
$835$	34.2106	1.18391
$836$	4.85576	0.167940
$837$	4.57936	0.158286
$838$	12.5882	0.434854
$839$	−4.36932	−0.150846	−0.0754228	−	0.997152i	$-0.524031\pi$
$839$	−4.36932	−0.150846	−0.0754228	+	0.997152i	$0.524031\pi$
$840$	0	0
$841$	−2.03315	−0.0701087
$842$	28.4292	0.979734
$843$	−1.07866	−0.0371512
$844$	14.8224	0.510207
$845$	−24.6016	−0.846321
$846$	−27.9183	−0.959851
$847$	0	0
$848$	3.37558	0.115918
$849$	0.973919	0.0334248
$850$	7.17690	0.246165
$851$	0	0
$852$	−2.88161	−0.0987222
$853$	39.1314	1.33983	0.669916	−	0.742437i	$-0.266330\pi$
$853$	39.1314	1.33983	0.669916	+	0.742437i	$0.266330\pi$
$854$	0	0
$855$	16.6164	0.568269
$856$	5.95784	0.203635
$857$	−27.2747	−0.931687	−0.465844	−	0.884867i	$-0.654249\pi$
$857$	−27.2747	−0.931687	−0.465844	+	0.884867i	$0.654249\pi$
$858$	7.23130	0.246873
$859$	−17.4100	−0.594022	−0.297011	−	0.954874i	$-0.595990\pi$
$859$	−17.4100	−0.594022	−0.297011	+	0.954874i	$0.595990\pi$
$860$	4.72459	0.161107
$861$	0	0
$862$	−37.8929	−1.29064
$863$	17.5672	0.597993	0.298996	−	0.954254i	$-0.403348\pi$
$863$	17.5672	0.597993	0.298996	+	0.954254i	$0.403348\pi$
$864$	−4.57936	−0.155793
$865$	−19.5200	−0.663700
$866$	−29.1209	−0.989567
$867$	−5.84892	−0.198640
$868$	0	0
$869$	17.4120	0.590662
$870$	−12.2190	−0.414264
$871$	51.5466	1.74659
$872$	5.02608	0.170205
$873$	2.23472	0.0756337
$874$	−15.0004	−0.507395
$875$	0	0
$876$	10.4702	0.353757
$877$	−26.5511	−0.896566	−0.448283	−	0.893892i	$-0.647964\pi$
$877$	−26.5511	−0.896566	−0.448283	+	0.893892i	$0.647964\pi$
$878$	−24.9095	−0.840655
$879$	−21.9618	−0.740754
$880$	−4.72459	−0.159266
$881$	−31.6673	−1.06690	−0.533449	−	0.845832i	$-0.679104\pi$
$881$	−31.6673	−1.06690	−0.533449	+	0.845832i	$0.679104\pi$
$882$	15.6430	0.526728
$883$	−31.9845	−1.07636	−0.538182	−	0.842829i	$-0.680889\pi$
$883$	−31.9845	−1.07636	−0.538182	+	0.842829i	$0.680889\pi$
$884$	15.1135	0.508324
$885$	13.8674	0.466149
$886$	−36.2476	−1.21776
$887$	20.7730	0.697490	0.348745	−	0.937218i	$-0.386608\pi$
$887$	20.7730	0.697490	0.348745	+	0.937218i	$0.386608\pi$
$888$	0	0
$889$	0	0
$890$	−29.5752	−0.991362
$891$	4.73927	0.158772
$892$	−9.63145	−0.322485
$893$	34.5359	1.15570
$894$	−15.0338	−0.502806
$895$	5.52976	0.184840
$896$	0	0
$897$	−22.3389	−0.745873
$898$	−14.0680	−0.469454
$899$	−5.19296	−0.173195
$900$	−4.99396	−0.166465
$901$	10.8408	0.361161
$902$	−11.3343	−0.377392
$903$	0	0
$904$	11.2224	0.373250
$905$	−35.5810	−1.18275
$906$	0.323787	0.0107571
$907$	23.0713	0.766070	0.383035	−	0.923734i	$-0.374879\pi$
$907$	23.0713	0.766070	0.383035	+	0.923734i	$0.374879\pi$
$908$	−14.7681	−0.490098
$909$	20.1687	0.668952
$910$	0	0
$911$	−36.7959	−1.21910	−0.609551	−	0.792747i	$-0.708650\pi$
$911$	−36.7959	−1.21910	−0.609551	+	0.792747i	$0.708650\pi$
$912$	2.41833	0.0800788
$913$	13.6366	0.451306
$914$	13.8844	0.459254
$915$	2.94558	0.0973779
$916$	22.1789	0.732811
$917$	0	0
$918$	−14.7068	−0.485397
$919$	38.0598	1.25548	0.627739	−	0.778424i	$-0.283980\pi$
$919$	38.0598	1.25548	0.627739	+	0.778424i	$0.283980\pi$
$920$	14.5952	0.481188
$921$	−16.8554	−0.555403
$922$	3.92069	0.129121
$923$	−15.5016	−0.510240
$924$	0	0
$925$	0	0
$926$	−17.8452	−0.586430
$927$	−15.4555	−0.507626
$928$	5.19296	0.170467
$929$	−36.8027	−1.20746	−0.603728	−	0.797190i	$-0.706319\pi$
$929$	−36.8027	−1.20746	−0.603728	+	0.797190i	$0.706319\pi$
$930$	2.35300	0.0771579
$931$	−19.3509	−0.634201
$932$	27.6535	0.905820
$933$	19.6753	0.644140
$934$	−21.1077	−0.690666
$935$	−15.1732	−0.496218
$936$	−10.5166	−0.343745
$937$	24.9426	0.814839	0.407420	−	0.913241i	$-0.366429\pi$
$937$	24.9426	0.814839	0.407420	+	0.913241i	$0.366429\pi$
$938$	0	0
$939$	13.5643	0.442654
$940$	−33.6030	−1.09601
$941$	−40.1644	−1.30932	−0.654660	−	0.755923i	$-0.727188\pi$
$941$	−40.1644	−1.30932	−0.654660	+	0.755923i	$0.727188\pi$
$942$	0.652360	0.0212551
$943$	35.0139	1.14021
$944$	−5.89351	−0.191818
$945$	0	0
$946$	−3.08536	−0.100314
$947$	43.3492	1.40866	0.704330	−	0.709873i	$-0.251248\pi$
$947$	43.3492	1.40866	0.704330	+	0.709873i	$0.251248\pi$
$948$	8.67175	0.281645
$949$	56.3245	1.82837
$950$	6.17769	0.200431
$951$	−16.8981	−0.547959
$952$	0	0
$953$	−8.09931	−0.262362	−0.131181	−	0.991358i	$-0.541877\pi$
$953$	−8.09931	−0.262362	−0.131181	+	0.991358i	$0.541877\pi$
$954$	−7.54347	−0.244229
$955$	−18.2720	−0.591268
$956$	−17.6591	−0.571136
$957$	7.97957	0.257943
$958$	28.7327	0.928313
$959$	0	0
$960$	−2.35300	−0.0759428
$961$	1.00000	0.0322581
$962$	0	0
$963$	−13.3141	−0.429040
$964$	−5.45115	−0.175570
$965$	−24.8064	−0.798545
$966$	0	0
$967$	6.81311	0.219095	0.109547	−	0.993982i	$-0.465060\pi$
$967$	6.81311	0.219095	0.109547	+	0.993982i	$0.465060\pi$
$968$	−7.91464	−0.254386
$969$	7.76657	0.249498
$970$	2.68974	0.0863624
$971$	55.4745	1.78026	0.890131	−	0.455705i	$-0.150613\pi$
$971$	55.4745	1.78026	0.890131	+	0.455705i	$0.150613\pi$
$972$	16.0984	0.516356
$973$	0	0
$974$	−38.0933	−1.22059
$975$	9.19996	0.294634
$976$	−1.25184	−0.0400704
$977$	−3.80370	−0.121691	−0.0608455	−	0.998147i	$-0.519380\pi$
$977$	−3.80370	−0.121691	−0.0608455	+	0.998147i	$0.519380\pi$
$978$	17.8388	0.570422
$979$	19.3139	0.617275
$980$	18.8282	0.601445
$981$	−11.2319	−0.358606
$982$	7.68293	0.245172
$983$	−39.7310	−1.26722	−0.633611	−	0.773651i	$-0.718428\pi$
$983$	−39.7310	−1.26722	−0.633611	+	0.773651i	$0.718428\pi$
$984$	−5.64487	−0.179952
$985$	37.3765	1.19092
$986$	16.6774	0.531118
$987$	0	0
$988$	13.0094	0.413883
$989$	9.53128	0.303077
$990$	10.5581	0.335559
$991$	48.8277	1.55106	0.775531	−	0.631309i	$-0.217482\pi$
$991$	48.8277	1.55106	0.775531	+	0.631309i	$0.217482\pi$
$992$	−1.00000	−0.0317500
$993$	9.08312	0.288244
$994$	0	0
$995$	−10.3419	−0.327862
$996$	6.79149	0.215197
$997$	−24.1694	−0.765454	−0.382727	−	0.923862i	$-0.625015\pi$
$997$	−24.1694	−0.765454	−0.382727	+	0.923862i	$0.625015\pi$
$998$	−20.5707	−0.651155
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6014.2.a.d.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6014.2.a.d.1.4	✓	5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6014 = 2 \cdot 31 \cdot 97 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6014.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.874805 q^{3} +1.00000 q^{4} -2.68974 q^{5} +0.874805 q^{6} +1.00000 q^{8} -2.23472 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.874805 q^{3} +1.00000 q^{4} -2.68974 q^{5} +0.874805 q^{6} +1.00000 q^{8} -2.23472 q^{9} -2.68974 q^{10} +1.75652 q^{11} +0.874805 q^{12} +4.70600 q^{13} -2.35300 q^{15} +1.00000 q^{16} +3.21155 q^{17} -2.23472 q^{18} +2.76442 q^{19} -2.68974 q^{20} +1.75652 q^{22} -5.42623 q^{23} +0.874805 q^{24} +2.23472 q^{25} +4.70600 q^{26} -4.57936 q^{27} +5.19296 q^{29} -2.35300 q^{30} -1.00000 q^{31} +1.00000 q^{32} +1.53661 q^{33} +3.21155 q^{34} -2.23472 q^{36} +2.76442 q^{38} +4.11683 q^{39} -2.68974 q^{40} -6.45271 q^{41} -1.75652 q^{43} +1.75652 q^{44} +6.01081 q^{45} -5.42623 q^{46} +12.4930 q^{47} +0.874805 q^{48} -7.00000 q^{49} +2.23472 q^{50} +2.80948 q^{51} +4.70600 q^{52} +3.37558 q^{53} -4.57936 q^{54} -4.72459 q^{55} +2.41833 q^{57} +5.19296 q^{58} -5.89351 q^{59} -2.35300 q^{60} -1.25184 q^{61} -1.00000 q^{62} +1.00000 q^{64} -12.6579 q^{65} +1.53661 q^{66} +10.9534 q^{67} +3.21155 q^{68} -4.74689 q^{69} -3.29400 q^{71} -2.23472 q^{72} +11.9687 q^{73} +1.95494 q^{75} +2.76442 q^{76} +4.11683 q^{78} +9.91278 q^{79} -2.68974 q^{80} +2.69810 q^{81} -6.45271 q^{82} +7.76343 q^{83} -8.63824 q^{85} -1.75652 q^{86} +4.54283 q^{87} +1.75652 q^{88} +10.9955 q^{89} +6.01081 q^{90} -5.42623 q^{92} -0.874805 q^{93} +12.4930 q^{94} -7.43558 q^{95} +0.874805 q^{96} -1.00000 q^{97} -7.00000 q^{98} -3.92532 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} - 4 q^{5} + 5 q^{8} + q^{9}+O(q^{10}) \) 5 * q + 5 * q^2 + 5 * q^4 - 4 * q^5 + 5 * q^8 + q^9	\( 5 q + 5 q^{2} + 5 q^{4} - 4 q^{5} + 5 q^{8} + q^{9} - 4 q^{10} + 4 q^{11} + 5 q^{16} + 14 q^{17} + q^{18} - 10 q^{19} - 4 q^{20} + 4 q^{22} - q^{25} + 6 q^{27} + 12 q^{29} - 5 q^{31} + 5 q^{32} + 14 q^{34} + q^{36} - 10 q^{38} + 20 q^{39} - 4 q^{40} + 2 q^{41} - 4 q^{43} + 4 q^{44} + 2 q^{45} + 40 q^{47} - 35 q^{49} - q^{50} + 6 q^{51} + 30 q^{53} + 6 q^{54} - 12 q^{55} + 4 q^{57} + 12 q^{58} + 22 q^{59} - 16 q^{61} - 5 q^{62} + 5 q^{64} + 12 q^{65} + 4 q^{67} + 14 q^{68} + 34 q^{69} - 40 q^{71} + q^{72} + 18 q^{73} - 6 q^{75} - 10 q^{76} + 20 q^{78} + 20 q^{79} - 4 q^{80} + 9 q^{81} + 2 q^{82} + 38 q^{83} - 38 q^{85} - 4 q^{86} + 20 q^{87} + 4 q^{88} + 18 q^{89} + 2 q^{90} + 40 q^{94} + 8 q^{95} - 5 q^{97} - 35 q^{98} - 34 q^{99}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^4 - 4 * q^5 + 5 * q^8 + q^9 - 4 * q^10 + 4 * q^11 + 5 * q^16 + 14 * q^17 + q^18 - 10 * q^19 - 4 * q^20 + 4 * q^22 - q^25 + 6 * q^27 + 12 * q^29 - 5 * q^31 + 5 * q^32 + 14 * q^34 + q^36 - 10 * q^38 + 20 * q^39 - 4 * q^40 + 2 * q^41 - 4 * q^43 + 4 * q^44 + 2 * q^45 + 40 * q^47 - 35 * q^49 - q^50 + 6 * q^51 + 30 * q^53 + 6 * q^54 - 12 * q^55 + 4 * q^57 + 12 * q^58 + 22 * q^59 - 16 * q^61 - 5 * q^62 + 5 * q^64 + 12 * q^65 + 4 * q^67 + 14 * q^68 + 34 * q^69 - 40 * q^71 + q^72 + 18 * q^73 - 6 * q^75 - 10 * q^76 + 20 * q^78 + 20 * q^79 - 4 * q^80 + 9 * q^81 + 2 * q^82 + 38 * q^83 - 38 * q^85 - 4 * q^86 + 20 * q^87 + 4 * q^88 + 18 * q^89 + 2 * q^90 + 40 * q^94 + 8 * q^95 - 5 * q^97 - 35 * q^98 - 34 * q^99

Introduction

L-functions

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Groups

Database

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