LMFDB - Embedded newform 5408.2.a.bk.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5408,2,Mod(1,5408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5408, 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("5408.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5408 = 2^{5} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5408.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:	$43.1830974131$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7488.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 2x + 1 $ x^4 - 2x^3 - 4x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 416)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.698857$ of defining polynomial
Character	$\chi$	$=$	5408.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.90931 q^{3} +2.00000 q^{5} +0.779548 q^{7} +5.46410 q^{9} -5.03908 q^{11} -5.81863 q^{15} -6.46410 q^{17} -0.779548 q^{19} -2.26795 q^{21} -7.16884 q^{23} -1.00000 q^{25} -7.16884 q^{27} +3.00000 q^{29} -1.55910 q^{31} +14.6603 q^{33} +1.55910 q^{35} +4.26795 q^{37} -3.19615 q^{41} +8.72794 q^{43} +10.9282 q^{45} -7.37772 q^{47} -6.39230 q^{49} +18.8061 q^{51} +9.46410 q^{53} -10.0782 q^{55} +2.26795 q^{57} +12.4168 q^{59} +3.00000 q^{61} +4.25953 q^{63} -0.779548 q^{67} +20.8564 q^{69} -15.1172 q^{71} +6.00000 q^{73} +2.90931 q^{75} -3.92820 q^{77} +10.0782 q^{79} +4.46410 q^{81} -10.0782 q^{83} -12.9282 q^{85} -8.72794 q^{87} +9.19615 q^{89} +4.53590 q^{93} -1.55910 q^{95} +10.2679 q^{97} -27.5340 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5} + 8 q^{9} - 12 q^{17} - 16 q^{21} - 4 q^{25} + 12 q^{29} + 24 q^{33} + 24 q^{37} + 8 q^{41} + 16 q^{45} + 16 q^{49} + 24 q^{53} + 16 q^{57} + 12 q^{61} + 28 q^{69} + 24 q^{73} + 12 q^{77}+ \cdots + 48 q^{97}+O(q^{100}) $ 4 * q + 8 * q^5 + 8 * q^9 - 12 * q^17 - 16 * q^21 - 4 * q^25 + 12 * q^29 + 24 * q^33 + 24 * q^37 + 8 * q^41 + 16 * q^45 + 16 * q^49 + 24 * q^53 + 16 * q^57 + 12 * q^61 + 28 * q^69 + 24 * q^73 + 12 * q^77 + 4 * q^81 - 24 * q^85 + 16 * q^89 + 32 * q^93 + 48 * q^97

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
$3$	−2.90931	−1.67969	−0.839846	−	0.542824i	$-0.817355\pi$
$3$	−2.90931	−1.67969	−0.839846	+	0.542824i	$0.817355\pi$
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	0.779548	0.294641	0.147321	−	0.989089i	$-0.452935\pi$
$7$	0.779548	0.294641	0.147321	+	0.989089i	$0.452935\pi$
$8$	0	0
$9$	5.46410	1.82137
$10$	0	0
$11$	−5.03908	−1.51934	−0.759670	−	0.650309i	$-0.774639\pi$
$11$	−5.03908	−1.51934	−0.759670	+	0.650309i	$0.774639\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−5.81863	−1.50236
$16$	0	0
$17$	−6.46410	−1.56777	−0.783887	−	0.620903i	$-0.786766\pi$
$17$	−6.46410	−1.56777	−0.783887	+	0.620903i	$0.786766\pi$
$18$	0	0
$19$	−0.779548	−0.178841	−0.0894203	−	0.995994i	$-0.528501\pi$
$19$	−0.779548	−0.178841	−0.0894203	+	0.995994i	$0.528501\pi$
$20$	0	0
$21$	−2.26795	−0.494907
$22$	0	0
$23$	−7.16884	−1.49481	−0.747404	−	0.664370i	$-0.768700\pi$
$23$	−7.16884	−1.49481	−0.747404	+	0.664370i	$0.768700\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−7.16884	−1.37964
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−1.55910	−0.280022	−0.140011	−	0.990150i	$-0.544714\pi$
$31$	−1.55910	−0.280022	−0.140011	+	0.990150i	$0.544714\pi$
$32$	0	0
$33$	14.6603	2.55202
$34$	0	0
$35$	1.55910	0.263535
$36$	0	0
$37$	4.26795	0.701647	0.350823	−	0.936442i	$-0.385902\pi$
$37$	4.26795	0.701647	0.350823	+	0.936442i	$0.385902\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.19615	−0.499155	−0.249578	−	0.968355i	$-0.580292\pi$
$41$	−3.19615	−0.499155	−0.249578	+	0.968355i	$0.580292\pi$
$42$	0	0
$43$	8.72794	1.33100	0.665499	−	0.746399i	$-0.268219\pi$
$43$	8.72794	1.33100	0.665499	+	0.746399i	$0.268219\pi$
$44$	0	0
$45$	10.9282	1.62908
$46$	0	0
$47$	−7.37772	−1.07615	−0.538076	−	0.842897i	$-0.680849\pi$
$47$	−7.37772	−1.07615	−0.538076	+	0.842897i	$0.680849\pi$
$48$	0	0
$49$	−6.39230	−0.913186
$50$	0	0
$51$	18.8061	2.63338
$52$	0	0
$53$	9.46410	1.29999	0.649997	−	0.759937i	$-0.274770\pi$
$53$	9.46410	1.29999	0.649997	+	0.759937i	$0.274770\pi$
$54$	0	0
$55$	−10.0782	−1.35894
$56$	0	0
$57$	2.26795	0.300397
$58$	0	0
$59$	12.4168	1.61653	0.808265	−	0.588819i	$-0.200407\pi$
$59$	12.4168	1.61653	0.808265	+	0.588819i	$0.200407\pi$
$60$	0	0
$61$	3.00000	0.384111	0.192055	−	0.981384i	$-0.438485\pi$
$61$	3.00000	0.384111	0.192055	+	0.981384i	$0.438485\pi$
$62$	0	0
$63$	4.25953	0.536650
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−0.779548	−0.0952370	−0.0476185	−	0.998866i	$-0.515163\pi$
$67$	−0.779548	−0.0952370	−0.0476185	+	0.998866i	$0.515163\pi$
$68$	0	0
$69$	20.8564	2.51082
$70$	0	0
$71$	−15.1172	−1.79409	−0.897043	−	0.441944i	$-0.854289\pi$
$71$	−15.1172	−1.79409	−0.897043	+	0.441944i	$0.854289\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	2.90931	0.335939
$76$	0	0
$77$	−3.92820	−0.447660
$78$	0	0
$79$	10.0782	1.13388	0.566941	−	0.823759i	$-0.308127\pi$
$79$	10.0782	1.13388	0.566941	+	0.823759i	$0.308127\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	−10.0782	−1.10622	−0.553111	−	0.833108i	$-0.686559\pi$
$83$	−10.0782	−1.10622	−0.553111	+	0.833108i	$0.686559\pi$
$84$	0	0
$85$	−12.9282	−1.40226
$86$	0	0
$87$	−8.72794	−0.935733
$88$	0	0
$89$	9.19615	0.974790	0.487395	−	0.873182i	$-0.337947\pi$
$89$	9.19615	0.974790	0.487395	+	0.873182i	$0.337947\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.53590	0.470351
$94$	0	0
$95$	−1.55910	−0.159960
$96$	0	0
$97$	10.2679	1.04255	0.521276	−	0.853388i	$-0.325456\pi$
$97$	10.2679	1.04255	0.521276	+	0.853388i	$0.325456\pi$
$98$	0	0
$99$	−27.5340	−2.76727
$100$	0	0
$101$	−1.39230	−0.138540	−0.0692698	−	0.997598i	$-0.522067\pi$
$101$	−1.39230	−0.138540	−0.0692698	+	0.997598i	$0.522067\pi$
$102$	0	0
$103$	20.1563	1.98606	0.993030	−	0.117860i	$-0.0376035\pi$
$103$	20.1563	1.98606	0.993030	+	0.117860i	$0.0376035\pi$
$104$	0	0
$105$	−4.53590	−0.442658
$106$	0	0
$107$	8.72794	0.843762	0.421881	−	0.906651i	$-0.361370\pi$
$107$	8.72794	0.843762	0.421881	+	0.906651i	$0.361370\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	−12.4168	−1.17855
$112$	0	0
$113$	−5.53590	−0.520774	−0.260387	−	0.965504i	$-0.583850\pi$
$113$	−5.53590	−0.520774	−0.260387	+	0.965504i	$0.583850\pi$
$114$	0	0
$115$	−14.3377	−1.33700
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.03908	−0.461932
$120$	0	0
$121$	14.3923	1.30839
$122$	0	0
$123$	9.29861	0.838427
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−8.72794	−0.774479	−0.387240	−	0.921979i	$-0.626571\pi$
$127$	−8.72794	−0.774479	−0.387240	+	0.921979i	$0.626571\pi$
$128$	0	0
$129$	−25.3923	−2.23567
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−0.607695	−0.0526939
$134$	0	0
$135$	−14.3377	−1.23399
$136$	0	0
$137$	2.80385	0.239549	0.119774	−	0.992801i	$-0.461783\pi$
$137$	2.80385	0.239549	0.119774	+	0.992801i	$0.461783\pi$
$138$	0	0
$139$	1.35022	0.114524	0.0572619	−	0.998359i	$-0.481763\pi$
$139$	1.35022	0.114524	0.0572619	+	0.998359i	$0.481763\pi$
$140$	0	0
$141$	21.4641	1.80760
$142$	0	0
$143$	0	0
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	18.5972	1.53387
$148$	0	0
$149$	−19.5885	−1.60475	−0.802374	−	0.596821i	$-0.796430\pi$
$149$	−19.5885	−1.60475	−0.802374	+	0.596821i	$0.796430\pi$
$150$	0	0
$151$	1.55910	0.126877	0.0634387	−	0.997986i	$-0.479793\pi$
$151$	1.55910	0.126877	0.0634387	+	0.997986i	$0.479793\pi$
$152$	0	0
$153$	−35.3205	−2.85549
$154$	0	0
$155$	−3.11819	−0.250459
$156$	0	0
$157$	−16.3923	−1.30825	−0.654124	−	0.756387i	$-0.726963\pi$
$157$	−16.3923	−1.30825	−0.654124	+	0.756387i	$0.726963\pi$
$158$	0	0
$159$	−27.5340	−2.18359
$160$	0	0
$161$	−5.58846	−0.440432
$162$	0	0
$163$	11.9990	0.939837	0.469919	−	0.882710i	$-0.344283\pi$
$163$	11.9990	0.939837	0.469919	+	0.882710i	$0.344283\pi$
$164$	0	0
$165$	29.3205	2.28260
$166$	0	0
$167$	17.8177	1.37877	0.689386	−	0.724394i	$-0.257880\pi$
$167$	17.8177	1.37877	0.689386	+	0.724394i	$0.257880\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−4.25953	−0.325734
$172$	0	0
$173$	3.92820	0.298656	0.149328	−	0.988788i	$-0.452289\pi$
$173$	3.92820	0.298656	0.149328	+	0.988788i	$0.452289\pi$
$174$	0	0
$175$	−0.779548	−0.0589283
$176$	0	0
$177$	−36.1244	−2.71527
$178$	0	0
$179$	−8.72794	−0.652357	−0.326178	−	0.945308i	$-0.605761\pi$
$179$	−8.72794	−0.652357	−0.326178	+	0.945308i	$0.605761\pi$
$180$	0	0
$181$	−4.39230	−0.326477	−0.163239	−	0.986587i	$-0.552194\pi$
$181$	−4.39230	−0.326477	−0.163239	+	0.986587i	$0.552194\pi$
$182$	0	0
$183$	−8.72794	−0.645188
$184$	0	0
$185$	8.53590	0.627572
$186$	0	0
$187$	32.5731	2.38198
$188$	0	0
$189$	−5.58846	−0.406500
$190$	0	0
$191$	−10.2870	−0.744344	−0.372172	−	0.928164i	$-0.621387\pi$
$191$	−10.2870	−0.744344	−0.372172	+	0.928164i	$0.621387\pi$
$192$	0	0
$193$	16.2679	1.17099	0.585496	−	0.810675i	$-0.300900\pi$
$193$	16.2679	1.17099	0.585496	+	0.810675i	$0.300900\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.58846	0.540655	0.270328	−	0.962768i	$-0.412868\pi$
$197$	7.58846	0.540655	0.270328	+	0.962768i	$0.412868\pi$
$198$	0	0
$199$	−11.4284	−0.810136	−0.405068	−	0.914287i	$-0.632752\pi$
$199$	−11.4284	−0.810136	−0.405068	+	0.914287i	$0.632752\pi$
$200$	0	0
$201$	2.26795	0.159969
$202$	0	0
$203$	2.33864	0.164141
$204$	0	0
$205$	−6.39230	−0.446458
$206$	0	0
$207$	−39.1713	−2.72259
$208$	0	0
$209$	3.92820	0.271719
$210$	0	0
$211$	23.4834	1.61666	0.808331	−	0.588728i	$-0.200371\pi$
$211$	23.4834	1.61666	0.808331	+	0.588728i	$0.200371\pi$
$212$	0	0
$213$	43.9808	3.01351
$214$	0	0
$215$	17.4559	1.19048
$216$	0	0
$217$	−1.21539	−0.0825061
$218$	0	0
$219$	−17.4559	−1.17956
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−0.779548	−0.0522024	−0.0261012	−	0.999659i	$-0.508309\pi$
$223$	−0.779548	−0.0522024	−0.0261012	+	0.999659i	$0.508309\pi$
$224$	0	0
$225$	−5.46410	−0.364273
$226$	0	0
$227$	2.33864	0.155221	0.0776106	−	0.996984i	$-0.475271\pi$
$227$	2.33864	0.155221	0.0776106	+	0.996984i	$0.475271\pi$
$228$	0	0
$229$	12.9282	0.854320	0.427160	−	0.904176i	$-0.359514\pi$
$229$	12.9282	0.854320	0.427160	+	0.904176i	$0.359514\pi$
$230$	0	0
$231$	11.4284	0.751932
$232$	0	0
$233$	21.4641	1.40616	0.703080	−	0.711111i	$-0.251808\pi$
$233$	21.4641	1.40616	0.703080	+	0.711111i	$0.251808\pi$
$234$	0	0
$235$	−14.7554	−0.962539
$236$	0	0
$237$	−29.3205	−1.90457
$238$	0	0
$239$	10.0782	0.651902	0.325951	−	0.945387i	$-0.394316\pi$
$239$	10.0782	0.651902	0.325951	+	0.945387i	$0.394316\pi$
$240$	0	0
$241$	9.58846	0.617647	0.308823	−	0.951119i	$-0.400065\pi$
$241$	9.58846	0.617647	0.308823	+	0.951119i	$0.400065\pi$
$242$	0	0
$243$	8.51906	0.546498
$244$	0	0
$245$	−12.7846	−0.816779
$246$	0	0
$247$	0	0
$248$	0	0
$249$	29.3205	1.85811
$250$	0	0
$251$	−19.9474	−1.25907	−0.629535	−	0.776972i	$-0.716755\pi$
$251$	−19.9474	−1.25907	−0.629535	+	0.776972i	$0.716755\pi$
$252$	0	0
$253$	36.1244	2.27112
$254$	0	0
$255$	37.6122	2.35537
$256$	0	0
$257$	−6.46410	−0.403220	−0.201610	−	0.979466i	$-0.564617\pi$
$257$	−6.46410	−0.403220	−0.201610	+	0.979466i	$0.564617\pi$
$258$	0	0
$259$	3.32707	0.206734
$260$	0	0
$261$	16.3923	1.01466
$262$	0	0
$263$	4.05065	0.249774	0.124887	−	0.992171i	$-0.460143\pi$
$263$	4.05065	0.249774	0.124887	+	0.992171i	$0.460143\pi$
$264$	0	0
$265$	18.9282	1.16275
$266$	0	0
$267$	−26.7545	−1.63735
$268$	0	0
$269$	25.3923	1.54820	0.774098	−	0.633066i	$-0.218204\pi$
$269$	25.3923	1.54820	0.774098	+	0.633066i	$0.218204\pi$
$270$	0	0
$271$	3.89774	0.236771	0.118385	−	0.992968i	$-0.462228\pi$
$271$	3.89774	0.236771	0.118385	+	0.992968i	$0.462228\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.03908	0.303868
$276$	0	0
$277$	−17.3923	−1.04500	−0.522501	−	0.852639i	$-0.675001\pi$
$277$	−17.3923	−1.04500	−0.522501	+	0.852639i	$0.675001\pi$
$278$	0	0
$279$	−8.51906	−0.510023
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	6.02751	0.358298	0.179149	−	0.983822i	$-0.442666\pi$
$283$	6.02751	0.358298	0.179149	+	0.983822i	$0.442666\pi$
$284$	0	0
$285$	4.53590	0.268683
$286$	0	0
$287$	−2.49155	−0.147072
$288$	0	0
$289$	24.7846	1.45792
$290$	0	0
$291$	−29.8727	−1.75117
$292$	0	0
$293$	14.8038	0.864850	0.432425	−	0.901670i	$-0.357658\pi$
$293$	14.8038	0.864850	0.432425	+	0.901670i	$0.357658\pi$
$294$	0	0
$295$	24.8336	1.44587
$296$	0	0
$297$	36.1244	2.09615
$298$	0	0
$299$	0	0
$300$	0	0
$301$	6.80385	0.392167
$302$	0	0
$303$	4.05065	0.232704
$304$	0	0
$305$	6.00000	0.343559
$306$	0	0
$307$	33.3527	1.90354	0.951768	−	0.306817i	$-0.0992641\pi$
$307$	33.3527	1.90354	0.951768	+	0.306817i	$0.0992641\pi$
$308$	0	0
$309$	−58.6410	−3.33597
$310$	0	0
$311$	3.11819	0.176816	0.0884082	−	0.996084i	$-0.471822\pi$
$311$	3.11819	0.176816	0.0884082	+	0.996084i	$0.471822\pi$
$312$	0	0
$313$	12.3923	0.700454	0.350227	−	0.936665i	$-0.386104\pi$
$313$	12.3923	0.700454	0.350227	+	0.936665i	$0.386104\pi$
$314$	0	0
$315$	8.51906	0.479995
$316$	0	0
$317$	4.00000	0.224662	0.112331	−	0.993671i	$-0.464168\pi$
$317$	4.00000	0.224662	0.112331	+	0.993671i	$0.464168\pi$
$318$	0	0
$319$	−15.1172	−0.846403
$320$	0	0
$321$	−25.3923	−1.41726
$322$	0	0
$323$	5.03908	0.280382
$324$	0	0
$325$	0	0
$326$	0	0
$327$	17.4559	0.965312
$328$	0	0
$329$	−5.75129	−0.317079
$330$	0	0
$331$	16.6763	0.916614	0.458307	−	0.888794i	$-0.348456\pi$
$331$	16.6763	0.916614	0.458307	+	0.888794i	$0.348456\pi$
$332$	0	0
$333$	23.3205	1.27796
$334$	0	0
$335$	−1.55910	−0.0851825
$336$	0	0
$337$	−20.3923	−1.11084	−0.555420	−	0.831570i	$-0.687442\pi$
$337$	−20.3923	−1.11084	−0.555420	+	0.831570i	$0.687442\pi$
$338$	0	0
$339$	16.1057	0.874739
$340$	0	0
$341$	7.85641	0.425448
$342$	0	0
$343$	−10.4399	−0.563704
$344$	0	0
$345$	41.7128	2.24574
$346$	0	0
$347$	−5.60975	−0.301147	−0.150573	−	0.988599i	$-0.548112\pi$
$347$	−5.60975	−0.301147	−0.150573	+	0.988599i	$0.548112\pi$
$348$	0	0
$349$	−21.5885	−1.15560	−0.577802	−	0.816177i	$-0.696089\pi$
$349$	−21.5885	−1.15560	−0.577802	+	0.816177i	$0.696089\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.19615	−0.0636648	−0.0318324	−	0.999493i	$-0.510134\pi$
$353$	−1.19615	−0.0636648	−0.0318324	+	0.999493i	$0.510134\pi$
$354$	0	0
$355$	−30.2345	−1.60468
$356$	0	0
$357$	14.6603	0.775903
$358$	0	0
$359$	10.0782	0.531905	0.265952	−	0.963986i	$-0.414314\pi$
$359$	10.0782	0.531905	0.265952	+	0.963986i	$0.414314\pi$
$360$	0	0
$361$	−18.3923	−0.968016
$362$	0	0
$363$	−41.8717	−2.19770
$364$	0	0
$365$	12.0000	0.628109
$366$	0	0
$367$	18.8061	0.981670	0.490835	−	0.871253i	$-0.336692\pi$
$367$	18.8061	0.981670	0.490835	+	0.871253i	$0.336692\pi$
$368$	0	0
$369$	−17.4641	−0.909145
$370$	0	0
$371$	7.37772	0.383032
$372$	0	0
$373$	−11.3923	−0.589871	−0.294936	−	0.955517i	$-0.595298\pi$
$373$	−11.3923	−0.589871	−0.294936	+	0.955517i	$0.595298\pi$
$374$	0	0
$375$	34.9118	1.80284
$376$	0	0
$377$	0	0
$378$	0	0
$379$	16.6763	0.856606	0.428303	−	0.903635i	$-0.359112\pi$
$379$	16.6763	0.856606	0.428303	+	0.903635i	$0.359112\pi$
$380$	0	0
$381$	25.3923	1.30089
$382$	0	0
$383$	27.8958	1.42541	0.712705	−	0.701464i	$-0.247470\pi$
$383$	27.8958	1.42541	0.712705	+	0.701464i	$0.247470\pi$
$384$	0	0
$385$	−7.85641	−0.400400
$386$	0	0
$387$	47.6903	2.42424
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	46.3401	2.34352
$392$	0	0
$393$	0	0
$394$	0	0
$395$	20.1563	1.01417
$396$	0	0
$397$	−13.0526	−0.655089	−0.327545	−	0.944836i	$-0.606221\pi$
$397$	−13.0526	−0.655089	−0.327545	+	0.944836i	$0.606221\pi$
$398$	0	0
$399$	1.76798	0.0885095
$400$	0	0
$401$	15.9808	0.798041	0.399021	−	0.916942i	$-0.369350\pi$
$401$	15.9808	0.798041	0.399021	+	0.916942i	$0.369350\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	8.92820	0.443646
$406$	0	0
$407$	−21.5065	−1.06604
$408$	0	0
$409$	−16.5167	−0.816696	−0.408348	−	0.912826i	$-0.633895\pi$
$409$	−16.5167	−0.816696	−0.408348	+	0.912826i	$0.633895\pi$
$410$	0	0
$411$	−8.15727	−0.402368
$412$	0	0
$413$	9.67949	0.476297
$414$	0	0
$415$	−20.1563	−0.989434
$416$	0	0
$417$	−3.92820	−0.192365
$418$	0	0
$419$	4.05065	0.197887	0.0989436	−	0.995093i	$-0.468454\pi$
$419$	4.05065	0.197887	0.0989436	+	0.995093i	$0.468454\pi$
$420$	0	0
$421$	12.9282	0.630082	0.315041	−	0.949078i	$-0.397982\pi$
$421$	12.9282	0.630082	0.315041	+	0.949078i	$0.397982\pi$
$422$	0	0
$423$	−40.3126	−1.96007
$424$	0	0
$425$	6.46410	0.313555
$426$	0	0
$427$	2.33864	0.113175
$428$	0	0
$429$	0	0
$430$	0	0
$431$	19.7945	0.953469	0.476734	−	0.879047i	$-0.341820\pi$
$431$	19.7945	0.953469	0.476734	+	0.879047i	$0.341820\pi$
$432$	0	0
$433$	7.39230	0.355251	0.177626	−	0.984098i	$-0.443158\pi$
$433$	7.39230	0.355251	0.177626	+	0.984098i	$0.443158\pi$
$434$	0	0
$435$	−17.4559	−0.836945
$436$	0	0
$437$	5.58846	0.267332
$438$	0	0
$439$	6.02751	0.287677	0.143839	−	0.989601i	$-0.454055\pi$
$439$	6.02751	0.287677	0.143839	+	0.989601i	$0.454055\pi$
$440$	0	0
$441$	−34.9282	−1.66325
$442$	0	0
$443$	25.5572	1.21426	0.607129	−	0.794603i	$-0.292321\pi$
$443$	25.5572	1.21426	0.607129	+	0.794603i	$0.292321\pi$
$444$	0	0
$445$	18.3923	0.871879
$446$	0	0
$447$	56.9890	2.69548
$448$	0	0
$449$	−6.41154	−0.302579	−0.151290	−	0.988489i	$-0.548343\pi$
$449$	−6.41154	−0.302579	−0.151290	+	0.988489i	$0.548343\pi$
$450$	0	0
$451$	16.1057	0.758386
$452$	0	0
$453$	−4.53590	−0.213115
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−0.124356	−0.00581711	−0.00290856	−	0.999996i	$-0.500926\pi$
$457$	−0.124356	−0.00581711	−0.00290856	+	0.999996i	$0.500926\pi$
$458$	0	0
$459$	46.3401	2.16297
$460$	0	0
$461$	7.19615	0.335158	0.167579	−	0.985859i	$-0.446405\pi$
$461$	7.19615	0.335158	0.167579	+	0.985859i	$0.446405\pi$
$462$	0	0
$463$	−27.1163	−1.26020	−0.630100	−	0.776514i	$-0.716986\pi$
$463$	−27.1163	−1.26020	−0.630100	+	0.776514i	$0.716986\pi$
$464$	0	0
$465$	9.07180	0.420695
$466$	0	0
$467$	1.55910	0.0721464	0.0360732	−	0.999349i	$-0.488515\pi$
$467$	1.55910	0.0721464	0.0360732	+	0.999349i	$0.488515\pi$
$468$	0	0
$469$	−0.607695	−0.0280608
$470$	0	0
$471$	47.6903	2.19746
$472$	0	0
$473$	−43.9808	−2.02224
$474$	0	0
$475$	0.779548	0.0357681
$476$	0	0
$477$	51.7128	2.36777
$478$	0	0
$479$	−25.1954	−1.15121	−0.575603	−	0.817729i	$-0.695233\pi$
$479$	−25.1954	−1.15121	−0.575603	+	0.817729i	$0.695233\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	16.2586	0.739791
$484$	0	0
$485$	20.5359	0.932487
$486$	0	0
$487$	16.6763	0.755677	0.377838	−	0.925872i	$-0.376668\pi$
$487$	16.6763	0.755677	0.377838	+	0.925872i	$0.376668\pi$
$488$	0	0
$489$	−34.9090	−1.57864
$490$	0	0
$491$	−10.2870	−0.464247	−0.232124	−	0.972686i	$-0.574567\pi$
$491$	−10.2870	−0.464247	−0.232124	+	0.972686i	$0.574567\pi$
$492$	0	0
$493$	−19.3923	−0.873385
$494$	0	0
$495$	−55.0681	−2.47513
$496$	0	0
$497$	−11.7846	−0.528612
$498$	0	0
$499$	−1.55910	−0.0697947	−0.0348974	−	0.999391i	$-0.511110\pi$
$499$	−1.55910	−0.0697947	−0.0348974	+	0.999391i	$0.511110\pi$
$500$	0	0
$501$	−51.8372	−2.31591
$502$	0	0
$503$	−13.4052	−0.597710	−0.298855	−	0.954299i	$-0.596605\pi$
$503$	−13.4052	−0.597710	−0.298855	+	0.954299i	$0.596605\pi$
$504$	0	0
$505$	−2.78461	−0.123914
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16.8038	0.744817	0.372409	−	0.928069i	$-0.378532\pi$
$509$	16.8038	0.744817	0.372409	+	0.928069i	$0.378532\pi$
$510$	0	0
$511$	4.67729	0.206911
$512$	0	0
$513$	5.58846	0.246736
$514$	0	0
$515$	40.3126	1.77639
$516$	0	0
$517$	37.1769	1.63504
$518$	0	0
$519$	−11.4284	−0.501650
$520$	0	0
$521$	−42.2487	−1.85095	−0.925475	−	0.378809i	$-0.876334\pi$
$521$	−42.2487	−1.85095	−0.925475	+	0.378809i	$0.876334\pi$
$522$	0	0
$523$	−36.2620	−1.58563	−0.792813	−	0.609465i	$-0.791384\pi$
$523$	−36.2620	−1.58563	−0.792813	+	0.609465i	$0.791384\pi$
$524$	0	0
$525$	2.26795	0.0989814
$526$	0	0
$527$	10.0782	0.439011
$528$	0	0
$529$	28.3923	1.23445
$530$	0	0
$531$	67.8467	2.94429
$532$	0	0
$533$	0	0
$534$	0	0
$535$	17.4559	0.754683
$536$	0	0
$537$	25.3923	1.09576
$538$	0	0
$539$	32.2113	1.38744
$540$	0	0
$541$	6.92820	0.297867	0.148933	−	0.988847i	$-0.452416\pi$
$541$	6.92820	0.297867	0.148933	+	0.988847i	$0.452416\pi$
$542$	0	0
$543$	12.7786	0.548382
$544$	0	0
$545$	−12.0000	−0.514024
$546$	0	0
$547$	−20.1563	−0.861822	−0.430911	−	0.902395i	$-0.641808\pi$
$547$	−20.1563	−0.861822	−0.430911	+	0.902395i	$0.641808\pi$
$548$	0	0
$549$	16.3923	0.699607
$550$	0	0
$551$	−2.33864	−0.0996296
$552$	0	0
$553$	7.85641	0.334088
$554$	0	0
$555$	−24.8336	−1.05413
$556$	0	0
$557$	33.1962	1.40657	0.703283	−	0.710910i	$-0.251717\pi$
$557$	33.1962	1.40657	0.703283	+	0.710910i	$0.251717\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−94.7654	−4.00100
$562$	0	0
$563$	−30.8611	−1.30064	−0.650320	−	0.759660i	$-0.725365\pi$
$563$	−30.8611	−1.30064	−0.650320	+	0.759660i	$0.725365\pi$
$564$	0	0
$565$	−11.0718	−0.465794
$566$	0	0
$567$	3.47998	0.146145
$568$	0	0
$569$	3.92820	0.164679	0.0823394	−	0.996604i	$-0.473761\pi$
$569$	3.92820	0.164679	0.0823394	+	0.996604i	$0.473761\pi$
$570$	0	0
$571$	−14.7554	−0.617496	−0.308748	−	0.951144i	$-0.599910\pi$
$571$	−14.7554	−0.617496	−0.308748	+	0.951144i	$0.599910\pi$
$572$	0	0
$573$	29.9282	1.25027
$574$	0	0
$575$	7.16884	0.298961
$576$	0	0
$577$	16.1436	0.672067	0.336033	−	0.941850i	$-0.390915\pi$
$577$	16.1436	0.672067	0.336033	+	0.941850i	$0.390915\pi$
$578$	0	0
$579$	−47.3286	−1.96691
$580$	0	0
$581$	−7.85641	−0.325939
$582$	0	0
$583$	−47.6903	−1.97513
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−17.8177	−0.735414	−0.367707	−	0.929942i	$-0.619857\pi$
$587$	−17.8177	−0.735414	−0.367707	+	0.929942i	$0.619857\pi$
$588$	0	0
$589$	1.21539	0.0500793
$590$	0	0
$591$	−22.0772	−0.908135
$592$	0	0
$593$	18.7846	0.771391	0.385696	−	0.922626i	$-0.373961\pi$
$593$	18.7846	0.771391	0.385696	+	0.922626i	$0.373961\pi$
$594$	0	0
$595$	−10.0782	−0.413164
$596$	0	0
$597$	33.2487	1.36078
$598$	0	0
$599$	30.2345	1.23535	0.617673	−	0.786435i	$-0.288075\pi$
$599$	30.2345	1.23535	0.617673	+	0.786435i	$0.288075\pi$
$600$	0	0
$601$	−15.3923	−0.627865	−0.313933	−	0.949445i	$-0.601647\pi$
$601$	−15.3923	−0.627865	−0.313933	+	0.949445i	$0.601647\pi$
$602$	0	0
$603$	−4.25953	−0.173461
$604$	0	0
$605$	28.7846	1.17026
$606$	0	0
$607$	−28.8842	−1.17238	−0.586188	−	0.810175i	$-0.699372\pi$
$607$	−28.8842	−1.17238	−0.586188	+	0.810175i	$0.699372\pi$
$608$	0	0
$609$	−6.80385	−0.275706
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−11.1962	−0.452208	−0.226104	−	0.974103i	$-0.572599\pi$
$613$	−11.1962	−0.452208	−0.226104	+	0.974103i	$0.572599\pi$
$614$	0	0
$615$	18.5972	0.749912
$616$	0	0
$617$	−19.5885	−0.788602	−0.394301	−	0.918981i	$-0.629013\pi$
$617$	−19.5885	−0.788602	−0.394301	+	0.918981i	$0.629013\pi$
$618$	0	0
$619$	1.55910	0.0626654	0.0313327	−	0.999509i	$-0.490025\pi$
$619$	1.55910	0.0626654	0.0313327	+	0.999509i	$0.490025\pi$
$620$	0	0
$621$	51.3923	2.06230
$622$	0	0
$623$	7.16884	0.287214
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−11.4284	−0.456405
$628$	0	0
$629$	−27.5885	−1.10002
$630$	0	0
$631$	43.7926	1.74336	0.871678	−	0.490079i	$-0.163032\pi$
$631$	43.7926	1.74336	0.871678	+	0.490079i	$0.163032\pi$
$632$	0	0
$633$	−68.3205	−2.71550
$634$	0	0
$635$	−17.4559	−0.692715
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−82.6021	−3.26769
$640$	0	0
$641$	−1.39230	−0.0549927	−0.0274964	−	0.999622i	$-0.508753\pi$
$641$	−1.39230	−0.0549927	−0.0274964	+	0.999622i	$0.508753\pi$
$642$	0	0
$643$	−21.3536	−0.842104	−0.421052	−	0.907036i	$-0.638339\pi$
$643$	−21.3536	−0.842104	−0.421052	+	0.907036i	$0.638339\pi$
$644$	0	0
$645$	−50.7846	−1.99964
$646$	0	0
$647$	26.1838	1.02939	0.514696	−	0.857373i	$-0.327905\pi$
$647$	26.1838	1.02939	0.514696	+	0.857373i	$0.327905\pi$
$648$	0	0
$649$	−62.5692	−2.45606
$650$	0	0
$651$	3.53595	0.138585
$652$	0	0
$653$	−33.2487	−1.30112	−0.650561	−	0.759454i	$-0.725466\pi$
$653$	−33.2487	−1.30112	−0.650561	+	0.759454i	$0.725466\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	32.7846	1.27905
$658$	0	0
$659$	23.0656	0.898509	0.449255	−	0.893404i	$-0.351690\pi$
$659$	23.0656	0.898509	0.449255	+	0.893404i	$0.351690\pi$
$660$	0	0
$661$	25.0526	0.974432	0.487216	−	0.873282i	$-0.338013\pi$
$661$	25.0526	0.974432	0.487216	+	0.873282i	$0.338013\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.21539	−0.0471308
$666$	0	0
$667$	−21.5065	−0.832736
$668$	0	0
$669$	2.26795	0.0876840
$670$	0	0
$671$	−15.1172	−0.583594
$672$	0	0
$673$	17.7846	0.685546	0.342773	−	0.939418i	$-0.388634\pi$
$673$	17.7846	0.685546	0.342773	+	0.939418i	$0.388634\pi$
$674$	0	0
$675$	7.16884	0.275929
$676$	0	0
$677$	2.53590	0.0974625	0.0487312	−	0.998812i	$-0.484482\pi$
$677$	2.53590	0.0974625	0.0487312	+	0.998812i	$0.484482\pi$
$678$	0	0
$679$	8.00436	0.307179
$680$	0	0
$681$	−6.80385	−0.260724
$682$	0	0
$683$	12.4168	0.475116	0.237558	−	0.971373i	$-0.423653\pi$
$683$	12.4168	0.475116	0.237558	+	0.971373i	$0.423653\pi$
$684$	0	0
$685$	5.60770	0.214259
$686$	0	0
$687$	−37.6122	−1.43499
$688$	0	0
$689$	0	0
$690$	0	0
$691$	22.9127	0.871641	0.435820	−	0.900034i	$-0.356458\pi$
$691$	22.9127	0.871641	0.435820	+	0.900034i	$0.356458\pi$
$692$	0	0
$693$	−21.4641	−0.815354
$694$	0	0
$695$	2.70043	0.102433
$696$	0	0
$697$	20.6603	0.782563
$698$	0	0
$699$	−62.4458	−2.36192
$700$	0	0
$701$	44.1051	1.66583	0.832914	−	0.553403i	$-0.186671\pi$
$701$	44.1051	1.66583	0.832914	+	0.553403i	$0.186671\pi$
$702$	0	0
$703$	−3.32707	−0.125483
$704$	0	0
$705$	42.9282	1.61677
$706$	0	0
$707$	−1.08537	−0.0408195
$708$	0	0
$709$	−38.9090	−1.46126	−0.730628	−	0.682775i	$-0.760773\pi$
$709$	−38.9090	−1.46126	−0.730628	+	0.682775i	$0.760773\pi$
$710$	0	0
$711$	55.0681	2.06521
$712$	0	0
$713$	11.1769	0.418579
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−29.3205	−1.09499
$718$	0	0
$719$	10.2870	0.383642	0.191821	−	0.981430i	$-0.438561\pi$
$719$	10.2870	0.383642	0.191821	+	0.981430i	$0.438561\pi$
$720$	0	0
$721$	15.7128	0.585176
$722$	0	0
$723$	−27.8958	−1.03746
$724$	0	0
$725$	−3.00000	−0.111417
$726$	0	0
$727$	40.3126	1.49511	0.747556	−	0.664199i	$-0.231227\pi$
$727$	40.3126	1.49511	0.747556	+	0.664199i	$0.231227\pi$
$728$	0	0
$729$	−38.1769	−1.41396
$730$	0	0
$731$	−56.4183	−2.08671
$732$	0	0
$733$	39.7128	1.46683	0.733413	−	0.679783i	$-0.237926\pi$
$733$	39.7128	1.46683	0.733413	+	0.679783i	$0.237926\pi$
$734$	0	0
$735$	37.1944	1.37194
$736$	0	0
$737$	3.92820	0.144697
$738$	0	0
$739$	26.3367	0.968812	0.484406	−	0.874843i	$-0.339036\pi$
$739$	26.3367	0.968812	0.484406	+	0.874843i	$0.339036\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	30.5963	1.12247	0.561234	−	0.827657i	$-0.310327\pi$
$743$	30.5963	1.12247	0.561234	+	0.827657i	$0.310327\pi$
$744$	0	0
$745$	−39.1769	−1.43533
$746$	0	0
$747$	−55.0681	−2.01484
$748$	0	0
$749$	6.80385	0.248607
$750$	0	0
$751$	−18.8061	−0.686244	−0.343122	−	0.939291i	$-0.611484\pi$
$751$	−18.8061	−0.686244	−0.343122	+	0.939291i	$0.611484\pi$
$752$	0	0
$753$	58.0333	2.11485
$754$	0	0
$755$	3.11819	0.113483
$756$	0	0
$757$	−43.3923	−1.57712	−0.788560	−	0.614958i	$-0.789173\pi$
$757$	−43.3923	−1.57712	−0.788560	+	0.614958i	$0.789173\pi$
$758$	0	0
$759$	−105.097	−3.81478
$760$	0	0
$761$	−7.19615	−0.260860	−0.130430	−	0.991457i	$-0.541636\pi$
$761$	−7.19615	−0.260860	−0.130430	+	0.991457i	$0.541636\pi$
$762$	0	0
$763$	−4.67729	−0.169329
$764$	0	0
$765$	−70.6410	−2.55403
$766$	0	0
$767$	0	0
$768$	0	0
$769$	17.8756	0.644612	0.322306	−	0.946636i	$-0.395542\pi$
$769$	17.8756	0.644612	0.322306	+	0.946636i	$0.395542\pi$
$770$	0	0
$771$	18.8061	0.677285
$772$	0	0
$773$	27.1962	0.978178	0.489089	−	0.872234i	$-0.337329\pi$
$773$	27.1962	0.978178	0.489089	+	0.872234i	$0.337329\pi$
$774$	0	0
$775$	1.55910	0.0560044
$776$	0	0
$777$	−9.67949	−0.347250
$778$	0	0
$779$	2.49155	0.0892692
$780$	0	0
$781$	76.1769	2.72582
$782$	0	0
$783$	−21.5065	−0.768581
$784$	0	0
$785$	−32.7846	−1.17013
$786$	0	0
$787$	53.1472	1.89449	0.947246	−	0.320507i	$-0.103853\pi$
$787$	53.1472	1.89449	0.947246	+	0.320507i	$0.103853\pi$
$788$	0	0
$789$	−11.7846	−0.419543
$790$	0	0
$791$	−4.31550	−0.153441
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−55.0681	−1.95306
$796$	0	0
$797$	24.7128	0.875373	0.437686	−	0.899128i	$-0.355798\pi$
$797$	24.7128	0.875373	0.437686	+	0.899128i	$0.355798\pi$
$798$	0	0
$799$	47.6903	1.68716
$800$	0	0
$801$	50.2487	1.77545
$802$	0	0
$803$	−30.2345	−1.06695
$804$	0	0
$805$	−11.1769	−0.393934
$806$	0	0
$807$	−73.8742	−2.60049
$808$	0	0
$809$	−24.4641	−0.860112	−0.430056	−	0.902802i	$-0.641506\pi$
$809$	−24.4641	−0.860112	−0.430056	+	0.902802i	$0.641506\pi$
$810$	0	0
$811$	−19.0150	−0.667706	−0.333853	−	0.942625i	$-0.608349\pi$
$811$	−19.0150	−0.667706	−0.333853	+	0.942625i	$0.608349\pi$
$812$	0	0
$813$	−11.3397	−0.397702
$814$	0	0
$815$	23.9981	0.840616
$816$	0	0
$817$	−6.80385	−0.238036
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28.3731	0.990227	0.495113	−	0.868828i	$-0.335126\pi$
$821$	28.3731	0.990227	0.495113	+	0.868828i	$0.335126\pi$
$822$	0	0
$823$	−20.7829	−0.724448	−0.362224	−	0.932091i	$-0.617982\pi$
$823$	−20.7829	−0.724448	−0.362224	+	0.932091i	$0.617982\pi$
$824$	0	0
$825$	−14.6603	−0.510405
$826$	0	0
$827$	4.67729	0.162645	0.0813226	−	0.996688i	$-0.474086\pi$
$827$	4.67729	0.162645	0.0813226	+	0.996688i	$0.474086\pi$
$828$	0	0
$829$	38.5692	1.33956	0.669782	−	0.742558i	$-0.266387\pi$
$829$	38.5692	1.33956	0.669782	+	0.742558i	$0.266387\pi$
$830$	0	0
$831$	50.5997	1.75528
$832$	0	0
$833$	41.3205	1.43167
$834$	0	0
$835$	35.6353	1.23321
$836$	0	0
$837$	11.1769	0.386331
$838$	0	0
$839$	−19.7945	−0.683383	−0.341691	−	0.939812i	$-0.611000\pi$
$839$	−19.7945	−0.683383	−0.341691	+	0.939812i	$0.611000\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	29.0931	1.00202
$844$	0	0
$845$	0	0
$846$	0	0
$847$	11.2195	0.385506
$848$	0	0
$849$	−17.5359	−0.601830
$850$	0	0
$851$	−30.5963	−1.04883
$852$	0	0
$853$	48.9282	1.67527	0.837635	−	0.546231i	$-0.183938\pi$
$853$	48.9282	1.67527	0.837635	+	0.546231i	$0.183938\pi$
$854$	0	0
$855$	−8.51906	−0.291346
$856$	0	0
$857$	51.0333	1.74327	0.871633	−	0.490160i	$-0.163062\pi$
$857$	51.0333	1.74327	0.871633	+	0.490160i	$0.163062\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	7.24871	0.247035
$862$	0	0
$863$	30.2345	1.02919	0.514597	−	0.857432i	$-0.327942\pi$
$863$	30.2345	1.02919	0.514597	+	0.857432i	$0.327942\pi$
$864$	0	0
$865$	7.85641	0.267126
$866$	0	0
$867$	−72.1062	−2.44885
$868$	0	0
$869$	−50.7846	−1.72275
$870$	0	0
$871$	0	0
$872$	0	0
$873$	56.1051	1.89887
$874$	0	0
$875$	−9.35458	−0.316242
$876$	0	0
$877$	−51.5885	−1.74202	−0.871009	−	0.491267i	$-0.836534\pi$
$877$	−51.5885	−1.74202	−0.871009	+	0.491267i	$0.836534\pi$
$878$	0	0
$879$	−43.0690	−1.45268
$880$	0	0
$881$	−44.3205	−1.49320	−0.746598	−	0.665276i	$-0.768314\pi$
$881$	−44.3205	−1.49320	−0.746598	+	0.665276i	$0.768314\pi$
$882$	0	0
$883$	25.5572	0.860068	0.430034	−	0.902813i	$-0.358502\pi$
$883$	25.5572	0.860068	0.430034	+	0.902813i	$0.358502\pi$
$884$	0	0
$885$	−72.2487	−2.42861
$886$	0	0
$887$	14.9643	0.502453	0.251226	−	0.967928i	$-0.419166\pi$
$887$	14.9643	0.502453	0.251226	+	0.967928i	$0.419166\pi$
$888$	0	0
$889$	−6.80385	−0.228194
$890$	0	0
$891$	−22.4950	−0.753609
$892$	0	0
$893$	5.75129	0.192460
$894$	0	0
$895$	−17.4559	−0.583486
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4.67729	−0.155996
$900$	0	0
$901$	−61.1769	−2.03810
$902$	0	0
$903$	−19.7945	−0.658720
$904$	0	0
$905$	−8.78461	−0.292010
$906$	0	0
$907$	−3.32707	−0.110474	−0.0552368	−	0.998473i	$-0.517591\pi$
$907$	−3.32707	−0.110474	−0.0552368	+	0.998473i	$0.517591\pi$
$908$	0	0
$909$	−7.60770	−0.252331
$910$	0	0
$911$	−6.23638	−0.206621	−0.103310	−	0.994649i	$-0.532943\pi$
$911$	−6.23638	−0.206621	−0.103310	+	0.994649i	$0.532943\pi$
$912$	0	0
$913$	50.7846	1.68073
$914$	0	0
$915$	−17.4559	−0.577074
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−18.8061	−0.620356	−0.310178	−	0.950679i	$-0.600389\pi$
$919$	−18.8061	−0.620356	−0.310178	+	0.950679i	$0.600389\pi$
$920$	0	0
$921$	−97.0333	−3.19736
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−4.26795	−0.140329
$926$	0	0
$927$	110.136	3.61735
$928$	0	0
$929$	−37.5885	−1.23324	−0.616619	−	0.787262i	$-0.711498\pi$
$929$	−37.5885	−1.23324	−0.616619	+	0.787262i	$0.711498\pi$
$930$	0	0
$931$	4.98311	0.163315
$932$	0	0
$933$	−9.07180	−0.296997
$934$	0	0
$935$	65.1462	2.13051
$936$	0	0
$937$	25.1769	0.822494	0.411247	−	0.911524i	$-0.365093\pi$
$937$	25.1769	0.822494	0.411247	+	0.911524i	$0.365093\pi$
$938$	0	0
$939$	−36.0531	−1.17655
$940$	0	0
$941$	4.78461	0.155974	0.0779869	−	0.996954i	$-0.475151\pi$
$941$	4.78461	0.155974	0.0779869	+	0.996954i	$0.475151\pi$
$942$	0	0
$943$	22.9127	0.746141
$944$	0	0
$945$	−11.1769	−0.363585
$946$	0	0
$947$	−10.4399	−0.339253	−0.169626	−	0.985508i	$-0.554256\pi$
$947$	−10.4399	−0.339253	−0.169626	+	0.985508i	$0.554256\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−11.6373	−0.377364
$952$	0	0
$953$	−54.7128	−1.77232	−0.886161	−	0.463378i	$-0.846637\pi$
$953$	−54.7128	−1.77232	−0.886161	+	0.463378i	$0.846637\pi$
$954$	0	0
$955$	−20.5741	−0.665761
$956$	0	0
$957$	43.9808	1.42170
$958$	0	0
$959$	2.18573	0.0705810
$960$	0	0
$961$	−28.5692	−0.921588
$962$	0	0
$963$	47.6903	1.53680
$964$	0	0
$965$	32.5359	1.04737
$966$	0	0
$967$	62.0280	1.99469	0.997343	−	0.0728422i	$-0.0232070\pi$
$967$	62.0280	1.99469	0.997343	+	0.0728422i	$0.0232070\pi$
$968$	0	0
$969$	−14.6603	−0.470955
$970$	0	0
$971$	−2.49155	−0.0799578	−0.0399789	−	0.999201i	$-0.512729\pi$
$971$	−2.49155	−0.0799578	−0.0399789	+	0.999201i	$0.512729\pi$
$972$	0	0
$973$	1.05256	0.0337435
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9.19615	−0.294211	−0.147105	−	0.989121i	$-0.546996\pi$
$977$	−9.19615	−0.294211	−0.147105	+	0.989121i	$0.546996\pi$
$978$	0	0
$979$	−46.3401	−1.48104
$980$	0	0
$981$	−32.7846	−1.04673
$982$	0	0
$983$	−44.9899	−1.43496	−0.717478	−	0.696582i	$-0.754703\pi$
$983$	−44.9899	−1.43496	−0.717478	+	0.696582i	$0.754703\pi$
$984$	0	0
$985$	15.1769	0.483577
$986$	0	0
$987$	16.7323	0.532595
$988$	0	0
$989$	−62.5692	−1.98959
$990$	0	0
$991$	−40.9393	−1.30048	−0.650239	−	0.759730i	$-0.725331\pi$
$991$	−40.9393	−1.30048	−0.650239	+	0.759730i	$0.725331\pi$
$992$	0	0
$993$	−48.5167	−1.53963
$994$	0	0
$995$	−22.8567	−0.724608
$996$	0	0
$997$	−29.0000	−0.918439	−0.459220	−	0.888323i	$-0.651871\pi$
$997$	−29.0000	−0.918439	−0.459220	+	0.888323i	$0.651871\pi$
$998$	0	0
$999$	−30.5963	−0.968023

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5408.2.a.bk.1.1		4
4.3	odd	2	inner	5408.2.a.bk.1.4		4
13.2	odd	12		416.2.w.c.225.4	yes	8
13.7	odd	12		416.2.w.c.257.4	yes	8
13.12	even	2		5408.2.a.bg.1.1		4
52.7	even	12		416.2.w.c.257.1	yes	8
52.15	even	12		416.2.w.c.225.1	✓	8
52.51	odd	2		5408.2.a.bg.1.4		4
104.59	even	12		832.2.w.h.257.4		8
104.67	even	12		832.2.w.h.641.4		8
104.85	odd	12		832.2.w.h.257.1		8
104.93	odd	12		832.2.w.h.641.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.w.c.225.1	✓	8	52.15	even	12
416.2.w.c.225.4	yes	8	13.2	odd	12
416.2.w.c.257.1	yes	8	52.7	even	12
416.2.w.c.257.4	yes	8	13.7	odd	12
832.2.w.h.257.1		8	104.85	odd	12
832.2.w.h.257.4		8	104.59	even	12
832.2.w.h.641.1		8	104.93	odd	12
832.2.w.h.641.4		8	104.67	even	12
5408.2.a.bg.1.1		4	13.12	even	2
5408.2.a.bg.1.4		4	52.51	odd	2
5408.2.a.bk.1.1		4	1.1	even	1	trivial
5408.2.a.bk.1.4		4	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 5408 = 2^{5} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5408.a (trivial)

\(f(q)\)	\(=\)	\(q-2.90931 q^{3} +2.00000 q^{5} +0.779548 q^{7} +5.46410 q^{9} -5.03908 q^{11} -5.81863 q^{15} -6.46410 q^{17} -0.779548 q^{19} -2.26795 q^{21} -7.16884 q^{23} -1.00000 q^{25} -7.16884 q^{27} +3.00000 q^{29} -1.55910 q^{31} +14.6603 q^{33} +1.55910 q^{35} +4.26795 q^{37} -3.19615 q^{41} +8.72794 q^{43} +10.9282 q^{45} -7.37772 q^{47} -6.39230 q^{49} +18.8061 q^{51} +9.46410 q^{53} -10.0782 q^{55} +2.26795 q^{57} +12.4168 q^{59} +3.00000 q^{61} +4.25953 q^{63} -0.779548 q^{67} +20.8564 q^{69} -15.1172 q^{71} +6.00000 q^{73} +2.90931 q^{75} -3.92820 q^{77} +10.0782 q^{79} +4.46410 q^{81} -10.0782 q^{83} -12.9282 q^{85} -8.72794 q^{87} +9.19615 q^{89} +4.53590 q^{93} -1.55910 q^{95} +10.2679 q^{97} -27.5340 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5} + 8 q^{9} - 12 q^{17} - 16 q^{21} - 4 q^{25} + 12 q^{29} + 24 q^{33} + 24 q^{37} + 8 q^{41} + 16 q^{45} + 16 q^{49} + 24 q^{53} + 16 q^{57} + 12 q^{61} + 28 q^{69} + 24 q^{73} + 12 q^{77}+ \cdots + 48 q^{97}+O(q^{100}) \) 4 * q + 8 * q^5 + 8 * q^9 - 12 * q^17 - 16 * q^21 - 4 * q^25 + 12 * q^29 + 24 * q^33 + 24 * q^37 + 8 * q^41 + 16 * q^45 + 16 * q^49 + 24 * q^53 + 16 * q^57 + 12 * q^61 + 28 * q^69 + 24 * q^73 + 12 * q^77 + 4 * q^81 - 24 * q^85 + 16 * q^89 + 32 * q^93 + 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more