LMFDB - Embedded newform 4864.2.a.bn.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4864,2,Mod(1,4864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4864, 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("4864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4864 = 2^{8} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4864.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:	$38.8392355432$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 13x^{6} + 24x^{5} + 48x^{4} - 68x^{3} - 62x^{2} + 32x + 16 $ x^8 - 2x^7 - 13x^6 + 24x^5 + 48x^4 - 68x^3 - 62x^2 + 32*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.28103$ of defining polynomial
Character	$\chi$	$=$	4864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.13611 q^{3} -0.594041 q^{5} -3.48756 q^{7} +6.83520 q^{9} +O(q^{10})$	$q+3.13611 q^{3} -0.594041 q^{5} -3.48756 q^{7} +6.83520 q^{9} -4.83520 q^{11} -0.215597 q^{13} -1.86298 q^{15} +1.29720 q^{17} +1.00000 q^{19} -10.9374 q^{21} +4.52815 q^{23} -4.64712 q^{25} +12.0276 q^{27} -9.41093 q^{29} +1.22031 q^{31} -15.1637 q^{33} +2.07176 q^{35} -5.62653 q^{37} -0.676135 q^{39} +0.450021 q^{41} +0.794359 q^{43} -4.06039 q^{45} -12.1986 q^{47} +5.16310 q^{49} +4.06817 q^{51} -2.56409 q^{53} +2.87231 q^{55} +3.13611 q^{57} +2.75191 q^{59} -7.76665 q^{61} -23.8382 q^{63} +0.128073 q^{65} -4.11631 q^{67} +14.2008 q^{69} -7.82788 q^{71} -3.08931 q^{73} -14.5739 q^{75} +16.8631 q^{77} +10.0731 q^{79} +17.2143 q^{81} +11.6296 q^{83} -0.770591 q^{85} -29.5137 q^{87} -13.7091 q^{89} +0.751907 q^{91} +3.82703 q^{93} -0.594041 q^{95} +2.08846 q^{97} -33.0495 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5} - 4 q^{7} + 12 q^{9}+O(q^{10}) $ 8 * q - 8 * q^5 - 4 * q^7 + 12 * q^9	$ 8 q - 8 q^{5} - 4 q^{7} + 12 q^{9} + 4 q^{11} - 8 q^{13} - 4 q^{17} + 8 q^{19} - 16 q^{21} + 12 q^{25} - 28 q^{29} - 8 q^{31} - 12 q^{35} - 4 q^{37} + 4 q^{39} - 8 q^{41} - 4 q^{43} - 24 q^{45} - 12 q^{47} + 12 q^{49} + 12 q^{51} - 32 q^{53} + 8 q^{55} + 12 q^{59} - 8 q^{61} + 16 q^{63} + 8 q^{65} - 4 q^{67} - 28 q^{69} + 24 q^{71} - 24 q^{77} + 24 q^{79} - 8 q^{81} + 40 q^{83} - 24 q^{85} - 24 q^{87} + 8 q^{89} - 4 q^{91} - 32 q^{93} - 8 q^{95} + 16 q^{97} - 76 q^{99}+O(q^{100}) $ 8 * q - 8 * q^5 - 4 * q^7 + 12 * q^9 + 4 * q^11 - 8 * q^13 - 4 * q^17 + 8 * q^19 - 16 * q^21 + 12 * q^25 - 28 * q^29 - 8 * q^31 - 12 * q^35 - 4 * q^37 + 4 * q^39 - 8 * q^41 - 4 * q^43 - 24 * q^45 - 12 * q^47 + 12 * q^49 + 12 * q^51 - 32 * q^53 + 8 * q^55 + 12 * q^59 - 8 * q^61 + 16 * q^63 + 8 * q^65 - 4 * q^67 - 28 * q^69 + 24 * q^71 - 24 * q^77 + 24 * q^79 - 8 * q^81 + 40 * q^83 - 24 * q^85 - 24 * q^87 + 8 * q^89 - 4 * q^91 - 32 * q^93 - 8 * q^95 + 16 * q^97 - 76 * 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
$3$	3.13611	1.81063	0.905317	−	0.424735i	$-0.139633\pi$
$3$	3.13611	1.81063	0.905317	+	0.424735i	$0.139633\pi$
$4$	0	0
$5$	−0.594041	−0.265663	−0.132832	−	0.991139i	$-0.542407\pi$
$5$	−0.594041	−0.265663	−0.132832	+	0.991139i	$0.542407\pi$
$6$	0	0
$7$	−3.48756	−1.31818	−0.659088	−	0.752066i	$-0.729057\pi$
$7$	−3.48756	−1.31818	−0.659088	+	0.752066i	$0.729057\pi$
$8$	0	0
$9$	6.83520	2.27840
$10$	0	0
$11$	−4.83520	−1.45787	−0.728933	−	0.684585i	$-0.759984\pi$
$11$	−4.83520	−1.45787	−0.728933	+	0.684585i	$0.759984\pi$
$12$	0	0
$13$	−0.215597	−0.0597957	−0.0298979	−	0.999553i	$-0.509518\pi$
$13$	−0.215597	−0.0597957	−0.0298979	+	0.999553i	$0.509518\pi$
$14$	0	0
$15$	−1.86298	−0.481019
$16$	0	0
$17$	1.29720	0.314618	0.157309	−	0.987549i	$-0.449718\pi$
$17$	1.29720	0.314618	0.157309	+	0.987549i	$0.449718\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−10.9374	−2.38673
$22$	0	0
$23$	4.52815	0.944184	0.472092	−	0.881549i	$-0.343499\pi$
$23$	4.52815	0.944184	0.472092	+	0.881549i	$0.343499\pi$
$24$	0	0
$25$	−4.64712	−0.929423
$26$	0	0
$27$	12.0276	2.31471
$28$	0	0
$29$	−9.41093	−1.74757	−0.873783	−	0.486316i	$-0.838340\pi$
$29$	−9.41093	−1.74757	−0.873783	+	0.486316i	$0.838340\pi$
$30$	0	0
$31$	1.22031	0.219174	0.109587	−	0.993977i	$-0.465047\pi$
$31$	1.22031	0.219174	0.109587	+	0.993977i	$0.465047\pi$
$32$	0	0
$33$	−15.1637	−2.63966
$34$	0	0
$35$	2.07176	0.350191
$36$	0	0
$37$	−5.62653	−0.924995	−0.462498	−	0.886621i	$-0.653047\pi$
$37$	−5.62653	−0.924995	−0.462498	+	0.886621i	$0.653047\pi$
$38$	0	0
$39$	−0.676135	−0.108268
$40$	0	0
$41$	0.450021	0.0702815	0.0351407	−	0.999382i	$-0.488812\pi$
$41$	0.450021	0.0702815	0.0351407	+	0.999382i	$0.488812\pi$
$42$	0	0
$43$	0.794359	0.121139	0.0605693	−	0.998164i	$-0.480708\pi$
$43$	0.794359	0.121139	0.0605693	+	0.998164i	$0.480708\pi$
$44$	0	0
$45$	−4.06039	−0.605287
$46$	0	0
$47$	−12.1986	−1.77935	−0.889676	−	0.456593i	$-0.849070\pi$
$47$	−12.1986	−1.77935	−0.889676	+	0.456593i	$0.849070\pi$
$48$	0	0
$49$	5.16310	0.737586
$50$	0	0
$51$	4.06817	0.569658
$52$	0	0
$53$	−2.56409	−0.352205	−0.176103	−	0.984372i	$-0.556349\pi$
$53$	−2.56409	−0.352205	−0.176103	+	0.984372i	$0.556349\pi$
$54$	0	0
$55$	2.87231	0.387302
$56$	0	0
$57$	3.13611	0.415388
$58$	0	0
$59$	2.75191	0.358268	0.179134	−	0.983825i	$-0.442670\pi$
$59$	2.75191	0.358268	0.179134	+	0.983825i	$0.442670\pi$
$60$	0	0
$61$	−7.76665	−0.994417	−0.497209	−	0.867631i	$-0.665642\pi$
$61$	−7.76665	−0.994417	−0.497209	+	0.867631i	$0.665642\pi$
$62$	0	0
$63$	−23.8382	−3.00333
$64$	0	0
$65$	0.128073	0.0158855
$66$	0	0
$67$	−4.11631	−0.502887	−0.251443	−	0.967872i	$-0.580905\pi$
$67$	−4.11631	−0.502887	−0.251443	+	0.967872i	$0.580905\pi$
$68$	0	0
$69$	14.2008	1.70957
$70$	0	0
$71$	−7.82788	−0.928999	−0.464499	−	0.885573i	$-0.653766\pi$
$71$	−7.82788	−0.928999	−0.464499	+	0.885573i	$0.653766\pi$
$72$	0	0
$73$	−3.08931	−0.361577	−0.180788	−	0.983522i	$-0.557865\pi$
$73$	−3.08931	−0.361577	−0.180788	+	0.983522i	$0.557865\pi$
$74$	0	0
$75$	−14.5739	−1.68285
$76$	0	0
$77$	16.8631	1.92172
$78$	0	0
$79$	10.0731	1.13331	0.566654	−	0.823956i	$-0.308237\pi$
$79$	10.0731	1.13331	0.566654	+	0.823956i	$0.308237\pi$
$80$	0	0
$81$	17.2143	1.91270
$82$	0	0
$83$	11.6296	1.27651	0.638255	−	0.769825i	$-0.279657\pi$
$83$	11.6296	1.27651	0.638255	+	0.769825i	$0.279657\pi$
$84$	0	0
$85$	−0.770591	−0.0835823
$86$	0	0
$87$	−29.5137	−3.16420
$88$	0	0
$89$	−13.7091	−1.45317	−0.726583	−	0.687079i	$-0.758893\pi$
$89$	−13.7091	−1.45317	−0.726583	+	0.687079i	$0.758893\pi$
$90$	0	0
$91$	0.751907	0.0788213
$92$	0	0
$93$	3.82703	0.396845
$94$	0	0
$95$	−0.594041	−0.0609473
$96$	0	0
$97$	2.08846	0.212051	0.106026	−	0.994363i	$-0.466187\pi$
$97$	2.08846	0.212051	0.106026	+	0.994363i	$0.466187\pi$
$98$	0	0
$99$	−33.0495	−3.32160
$100$	0	0
$101$	−2.77074	−0.275699	−0.137849	−	0.990453i	$-0.544019\pi$
$101$	−2.77074	−0.275699	−0.137849	+	0.990453i	$0.544019\pi$
$102$	0	0
$103$	−14.3363	−1.41260	−0.706301	−	0.707912i	$-0.749637\pi$
$103$	−14.3363	−1.41260	−0.706301	+	0.707912i	$0.749637\pi$
$104$	0	0
$105$	6.49726	0.634067
$106$	0	0
$107$	2.42388	0.234325	0.117163	−	0.993113i	$-0.462620\pi$
$107$	2.42388	0.234325	0.117163	+	0.993113i	$0.462620\pi$
$108$	0	0
$109$	−0.00123810	−0.000118589 0	−5.92945e−5	−	1.00000i	$-0.500019\pi$
$109$	−0.00123810	−0.000118589 0	−5.92945e−5		1.00000i	$0.500019\pi$
$110$	0	0
$111$	−17.6454	−1.67483
$112$	0	0
$113$	1.81614	0.170848	0.0854242	−	0.996345i	$-0.472775\pi$
$113$	1.81614	0.170848	0.0854242	+	0.996345i	$0.472775\pi$
$114$	0	0
$115$	−2.68990	−0.250835
$116$	0	0
$117$	−1.47365	−0.136239
$118$	0	0
$119$	−4.52407	−0.414721
$120$	0	0
$121$	12.3791	1.12538
$122$	0	0
$123$	1.41132	0.127254
$124$	0	0
$125$	5.73078	0.512577
$126$	0	0
$127$	16.1269	1.43103	0.715517	−	0.698596i	$-0.246191\pi$
$127$	16.1269	1.43103	0.715517	+	0.698596i	$0.246191\pi$
$128$	0	0
$129$	2.49120	0.219338
$130$	0	0
$131$	11.1477	0.973983	0.486992	−	0.873407i	$-0.338094\pi$
$131$	11.1477	0.973983	0.486992	+	0.873407i	$0.338094\pi$
$132$	0	0
$133$	−3.48756	−0.302410
$134$	0	0
$135$	−7.14489	−0.614934
$136$	0	0
$137$	−11.1666	−0.954025	−0.477013	−	0.878896i	$-0.658280\pi$
$137$	−11.1666	−0.954025	−0.477013	+	0.878896i	$0.658280\pi$
$138$	0	0
$139$	−13.0534	−1.10718	−0.553589	−	0.832790i	$-0.686742\pi$
$139$	−13.0534	−1.10718	−0.553589	+	0.832790i	$0.686742\pi$
$140$	0	0
$141$	−38.2562	−3.22176
$142$	0	0
$143$	1.04245	0.0871742
$144$	0	0
$145$	5.59048	0.464264
$146$	0	0
$147$	16.1921	1.33550
$148$	0	0
$149$	−14.3811	−1.17814	−0.589072	−	0.808080i	$-0.700507\pi$
$149$	−14.3811	−1.17814	−0.589072	+	0.808080i	$0.700507\pi$
$150$	0	0
$151$	17.3489	1.41184	0.705918	−	0.708293i	$-0.250535\pi$
$151$	17.3489	1.41184	0.705918	+	0.708293i	$0.250535\pi$
$152$	0	0
$153$	8.86663	0.716825
$154$	0	0
$155$	−0.724915	−0.0582266
$156$	0	0
$157$	9.63293	0.768791	0.384396	−	0.923168i	$-0.374410\pi$
$157$	9.63293	0.768791	0.384396	+	0.923168i	$0.374410\pi$
$158$	0	0
$159$	−8.04128	−0.637715
$160$	0	0
$161$	−15.7922	−1.24460
$162$	0	0
$163$	−22.5365	−1.76520	−0.882598	−	0.470129i	$-0.844207\pi$
$163$	−22.5365	−1.76520	−0.882598	+	0.470129i	$0.844207\pi$
$164$	0	0
$165$	9.00787	0.701262
$166$	0	0
$167$	16.5108	1.27765	0.638823	−	0.769353i	$-0.279421\pi$
$167$	16.5108	1.27765	0.638823	+	0.769353i	$0.279421\pi$
$168$	0	0
$169$	−12.9535	−0.996424
$170$	0	0
$171$	6.83520	0.522701
$172$	0	0
$173$	−5.14911	−0.391479	−0.195740	−	0.980656i	$-0.562711\pi$
$173$	−5.14911	−0.391479	−0.195740	+	0.980656i	$0.562711\pi$
$174$	0	0
$175$	16.2071	1.22514
$176$	0	0
$177$	8.63029	0.648692
$178$	0	0
$179$	−9.31580	−0.696295	−0.348148	−	0.937440i	$-0.613189\pi$
$179$	−9.31580	−0.696295	−0.348148	+	0.937440i	$0.613189\pi$
$180$	0	0
$181$	5.15517	0.383180	0.191590	−	0.981475i	$-0.438636\pi$
$181$	5.15517	0.383180	0.191590	+	0.981475i	$0.438636\pi$
$182$	0	0
$183$	−24.3571	−1.80053
$184$	0	0
$185$	3.34239	0.245737
$186$	0	0
$187$	−6.27223	−0.458671
$188$	0	0
$189$	−41.9471	−3.05120
$190$	0	0
$191$	14.8805	1.07672	0.538359	−	0.842715i	$-0.319044\pi$
$191$	14.8805	1.07672	0.538359	+	0.842715i	$0.319044\pi$
$192$	0	0
$193$	−21.2754	−1.53144	−0.765719	−	0.643175i	$-0.777617\pi$
$193$	−21.2754	−1.53144	−0.765719	+	0.643175i	$0.777617\pi$
$194$	0	0
$195$	0.401652	0.0287629
$196$	0	0
$197$	3.69169	0.263022	0.131511	−	0.991315i	$-0.458017\pi$
$197$	3.69169	0.263022	0.131511	+	0.991315i	$0.458017\pi$
$198$	0	0
$199$	11.6857	0.828377	0.414188	−	0.910191i	$-0.364065\pi$
$199$	11.6857	0.828377	0.414188	+	0.910191i	$0.364065\pi$
$200$	0	0
$201$	−12.9092	−0.910545
$202$	0	0
$203$	32.8212	2.30360
$204$	0	0
$205$	−0.267331	−0.0186712
$206$	0	0
$207$	30.9508	2.15123
$208$	0	0
$209$	−4.83520	−0.334458
$210$	0	0
$211$	−7.61611	−0.524315	−0.262157	−	0.965025i	$-0.584434\pi$
$211$	−7.61611	−0.524315	−0.262157	+	0.965025i	$0.584434\pi$
$212$	0	0
$213$	−24.5491	−1.68208
$214$	0	0
$215$	−0.471882	−0.0321821
$216$	0	0
$217$	−4.25591	−0.288910
$218$	0	0
$219$	−9.68844	−0.654684
$220$	0	0
$221$	−0.279672	−0.0188128
$222$	0	0
$223$	1.95477	0.130901	0.0654505	−	0.997856i	$-0.479152\pi$
$223$	1.95477	0.130901	0.0654505	+	0.997856i	$0.479152\pi$
$224$	0	0
$225$	−31.7640	−2.11760
$226$	0	0
$227$	−13.3709	−0.887461	−0.443730	−	0.896160i	$-0.646345\pi$
$227$	−13.3709	−0.887461	−0.443730	+	0.896160i	$0.646345\pi$
$228$	0	0
$229$	11.5800	0.765225	0.382613	−	0.923909i	$-0.375024\pi$
$229$	11.5800	0.765225	0.382613	+	0.923909i	$0.375024\pi$
$230$	0	0
$231$	52.8844	3.47954
$232$	0	0
$233$	−1.58872	−0.104080	−0.0520401	−	0.998645i	$-0.516572\pi$
$233$	−1.58872	−0.104080	−0.0520401	+	0.998645i	$0.516572\pi$
$234$	0	0
$235$	7.24648	0.472708
$236$	0	0
$237$	31.5903	2.05201
$238$	0	0
$239$	−23.8219	−1.54091	−0.770455	−	0.637494i	$-0.779971\pi$
$239$	−23.8219	−1.54091	−0.770455	+	0.637494i	$0.779971\pi$
$240$	0	0
$241$	22.8554	1.47225	0.736123	−	0.676848i	$-0.236654\pi$
$241$	22.8554	1.47225	0.736123	+	0.676848i	$0.236654\pi$
$242$	0	0
$243$	17.9032	1.14849
$244$	0	0
$245$	−3.06710	−0.195950
$246$	0	0
$247$	−0.215597	−0.0137181
$248$	0	0
$249$	36.4716	2.31129
$250$	0	0
$251$	2.63524	0.166335	0.0831676	−	0.996536i	$-0.473496\pi$
$251$	2.63524	0.166335	0.0831676	+	0.996536i	$0.473496\pi$
$252$	0	0
$253$	−21.8945	−1.37649
$254$	0	0
$255$	−2.41666	−0.151337
$256$	0	0
$257$	20.0579	1.25118	0.625588	−	0.780153i	$-0.284859\pi$
$257$	20.0579	1.25118	0.625588	+	0.780153i	$0.284859\pi$
$258$	0	0
$259$	19.6229	1.21931
$260$	0	0
$261$	−64.3256	−3.98165
$262$	0	0
$263$	4.03667	0.248912	0.124456	−	0.992225i	$-0.460282\pi$
$263$	4.03667	0.248912	0.124456	+	0.992225i	$0.460282\pi$
$264$	0	0
$265$	1.52318	0.0935679
$266$	0	0
$267$	−42.9934	−2.63115
$268$	0	0
$269$	−14.3095	−0.872467	−0.436233	−	0.899834i	$-0.643688\pi$
$269$	−14.3095	−0.872467	−0.436233	+	0.899834i	$0.643688\pi$
$270$	0	0
$271$	−9.85034	−0.598366	−0.299183	−	0.954196i	$-0.596714\pi$
$271$	−9.85034	−0.598366	−0.299183	+	0.954196i	$0.596714\pi$
$272$	0	0
$273$	2.35806	0.142717
$274$	0	0
$275$	22.4697	1.35498
$276$	0	0
$277$	16.9641	1.01928	0.509638	−	0.860389i	$-0.329779\pi$
$277$	16.9641	1.01928	0.509638	+	0.860389i	$0.329779\pi$
$278$	0	0
$279$	8.34107	0.499367
$280$	0	0
$281$	10.2078	0.608944	0.304472	−	0.952521i	$-0.401520\pi$
$281$	10.2078	0.608944	0.304472	+	0.952521i	$0.401520\pi$
$282$	0	0
$283$	−18.7709	−1.11581	−0.557907	−	0.829903i	$-0.688396\pi$
$283$	−18.7709	−1.11581	−0.557907	+	0.829903i	$0.688396\pi$
$284$	0	0
$285$	−1.86298	−0.110353
$286$	0	0
$287$	−1.56948	−0.0926433
$288$	0	0
$289$	−15.3173	−0.901016
$290$	0	0
$291$	6.54966	0.383948
$292$	0	0
$293$	−19.8271	−1.15831	−0.579155	−	0.815217i	$-0.696618\pi$
$293$	−19.8271	−1.15831	−0.579155	+	0.815217i	$0.696618\pi$
$294$	0	0
$295$	−1.63475	−0.0951786
$296$	0	0
$297$	−58.1559	−3.37454
$298$	0	0
$299$	−0.976253	−0.0564582
$300$	0	0
$301$	−2.77038	−0.159682
$302$	0	0
$303$	−8.68935	−0.499190
$304$	0	0
$305$	4.61371	0.264180
$306$	0	0
$307$	−18.3935	−1.04977	−0.524886	−	0.851173i	$-0.675892\pi$
$307$	−18.3935	−1.04977	−0.524886	+	0.851173i	$0.675892\pi$
$308$	0	0
$309$	−44.9604	−2.55771
$310$	0	0
$311$	19.1747	1.08730	0.543648	−	0.839313i	$-0.317043\pi$
$311$	19.1747	1.08730	0.543648	+	0.839313i	$0.317043\pi$
$312$	0	0
$313$	−14.2274	−0.804183	−0.402091	−	0.915600i	$-0.631717\pi$
$313$	−14.2274	−0.804183	−0.402091	+	0.915600i	$0.631717\pi$
$314$	0	0
$315$	14.1609	0.797874
$316$	0	0
$317$	−0.777938	−0.0436934	−0.0218467	−	0.999761i	$-0.506955\pi$
$317$	−0.777938	−0.0436934	−0.0218467	+	0.999761i	$0.506955\pi$
$318$	0	0
$319$	45.5037	2.54772
$320$	0	0
$321$	7.60156	0.424278
$322$	0	0
$323$	1.29720	0.0721782
$324$	0	0
$325$	1.00190	0.0555755
$326$	0	0
$327$	−0.00388283	−0.000214721 0
$328$	0	0
$329$	42.5435	2.34550
$330$	0	0
$331$	16.3988	0.901359	0.450680	−	0.892686i	$-0.351182\pi$
$331$	16.3988	0.901359	0.450680	+	0.892686i	$0.351182\pi$
$332$	0	0
$333$	−38.4584	−2.10751
$334$	0	0
$335$	2.44525	0.133599
$336$	0	0
$337$	24.6109	1.34064	0.670320	−	0.742072i	$-0.266157\pi$
$337$	24.6109	1.34064	0.670320	+	0.742072i	$0.266157\pi$
$338$	0	0
$339$	5.69563	0.309344
$340$	0	0
$341$	−5.90045	−0.319527
$342$	0	0
$343$	6.40629	0.345907
$344$	0	0
$345$	−8.43584	−0.454170
$346$	0	0
$347$	−13.4635	−0.722760	−0.361380	−	0.932419i	$-0.617694\pi$
$347$	−13.4635	−0.722760	−0.361380	+	0.932419i	$0.617694\pi$
$348$	0	0
$349$	2.83422	0.151712	0.0758560	−	0.997119i	$-0.475831\pi$
$349$	2.83422	0.151712	0.0758560	+	0.997119i	$0.475831\pi$
$350$	0	0
$351$	−2.59311	−0.138410
$352$	0	0
$353$	−5.02227	−0.267308	−0.133654	−	0.991028i	$-0.542671\pi$
$353$	−5.02227	−0.267308	−0.133654	+	0.991028i	$0.542671\pi$
$354$	0	0
$355$	4.65008	0.246801
$356$	0	0
$357$	−14.1880	−0.750909
$358$	0	0
$359$	23.6898	1.25030	0.625149	−	0.780505i	$-0.285038\pi$
$359$	23.6898	1.25030	0.625149	+	0.780505i	$0.285038\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	38.8223	2.03764
$364$	0	0
$365$	1.83518	0.0960577
$366$	0	0
$367$	16.9338	0.883938	0.441969	−	0.897030i	$-0.354280\pi$
$367$	16.9338	0.883938	0.441969	+	0.897030i	$0.354280\pi$
$368$	0	0
$369$	3.07598	0.160129
$370$	0	0
$371$	8.94244	0.464268
$372$	0	0
$373$	12.8599	0.665860	0.332930	−	0.942952i	$-0.391963\pi$
$373$	12.8599	0.665860	0.332930	+	0.942952i	$0.391963\pi$
$374$	0	0
$375$	17.9724	0.928089
$376$	0	0
$377$	2.02896	0.104497
$378$	0	0
$379$	20.7810	1.06745	0.533723	−	0.845659i	$-0.320793\pi$
$379$	20.7810	1.06745	0.533723	+	0.845659i	$0.320793\pi$
$380$	0	0
$381$	50.5758	2.59108
$382$	0	0
$383$	15.6170	0.797993	0.398996	−	0.916953i	$-0.369359\pi$
$383$	15.6170	0.797993	0.398996	+	0.916953i	$0.369359\pi$
$384$	0	0
$385$	−10.0173	−0.510531
$386$	0	0
$387$	5.42960	0.276002
$388$	0	0
$389$	10.1133	0.512764	0.256382	−	0.966576i	$-0.417470\pi$
$389$	10.1133	0.512764	0.256382	+	0.966576i	$0.417470\pi$
$390$	0	0
$391$	5.87392	0.297057
$392$	0	0
$393$	34.9606	1.76353
$394$	0	0
$395$	−5.98381	−0.301078
$396$	0	0
$397$	−4.60677	−0.231207	−0.115604	−	0.993295i	$-0.536880\pi$
$397$	−4.60677	−0.231207	−0.115604	+	0.993295i	$0.536880\pi$
$398$	0	0
$399$	−10.9374	−0.547554
$400$	0	0
$401$	−4.00112	−0.199806	−0.0999032	−	0.994997i	$-0.531853\pi$
$401$	−4.00112	−0.199806	−0.0999032	+	0.994997i	$0.531853\pi$
$402$	0	0
$403$	−0.263095	−0.0131057
$404$	0	0
$405$	−10.2260	−0.508135
$406$	0	0
$407$	27.2054	1.34852
$408$	0	0
$409$	14.5111	0.717529	0.358765	−	0.933428i	$-0.383198\pi$
$409$	14.5111	0.717529	0.358765	+	0.933428i	$0.383198\pi$
$410$	0	0
$411$	−35.0196	−1.72739
$412$	0	0
$413$	−9.59745	−0.472260
$414$	0	0
$415$	−6.90843	−0.339122
$416$	0	0
$417$	−40.9370	−2.00470
$418$	0	0
$419$	−5.02078	−0.245281	−0.122641	−	0.992451i	$-0.539136\pi$
$419$	−5.02078	−0.245281	−0.122641	+	0.992451i	$0.539136\pi$
$420$	0	0
$421$	4.10152	0.199896	0.0999479	−	0.994993i	$-0.468132\pi$
$421$	4.10152	0.199896	0.0999479	+	0.994993i	$0.468132\pi$
$422$	0	0
$423$	−83.3800	−4.05407
$424$	0	0
$425$	−6.02825	−0.292413
$426$	0	0
$427$	27.0867	1.31082
$428$	0	0
$429$	3.26925	0.157841
$430$	0	0
$431$	16.2920	0.784760	0.392380	−	0.919803i	$-0.371652\pi$
$431$	16.2920	0.784760	0.392380	+	0.919803i	$0.371652\pi$
$432$	0	0
$433$	7.44858	0.357956	0.178978	−	0.983853i	$-0.442721\pi$
$433$	7.44858	0.357956	0.178978	+	0.983853i	$0.442721\pi$
$434$	0	0
$435$	17.5324	0.840612
$436$	0	0
$437$	4.52815	0.216611
$438$	0	0
$439$	15.8898	0.758379	0.379189	−	0.925319i	$-0.376203\pi$
$439$	15.8898	0.758379	0.379189	+	0.925319i	$0.376203\pi$
$440$	0	0
$441$	35.2908	1.68052
$442$	0	0
$443$	−15.1742	−0.720949	−0.360474	−	0.932769i	$-0.617385\pi$
$443$	−15.1742	−0.720949	−0.360474	+	0.932769i	$0.617385\pi$
$444$	0	0
$445$	8.14379	0.386053
$446$	0	0
$447$	−45.1007	−2.13319
$448$	0	0
$449$	−24.3909	−1.15108	−0.575538	−	0.817775i	$-0.695207\pi$
$449$	−24.3909	−1.15108	−0.575538	+	0.817775i	$0.695207\pi$
$450$	0	0
$451$	−2.17594	−0.102461
$452$	0	0
$453$	54.4082	2.55632
$454$	0	0
$455$	−0.446664	−0.0209399
$456$	0	0
$457$	10.6860	0.499869	0.249934	−	0.968263i	$-0.419591\pi$
$457$	10.6860	0.499869	0.249934	+	0.968263i	$0.419591\pi$
$458$	0	0
$459$	15.6022	0.728250
$460$	0	0
$461$	−21.8232	−1.01641	−0.508205	−	0.861236i	$-0.669691\pi$
$461$	−21.8232	−1.01641	−0.508205	+	0.861236i	$0.669691\pi$
$462$	0	0
$463$	−0.0258442	−0.00120108	−0.000600541	−	1.00000i	$-0.500191\pi$
$463$	−0.0258442	−0.00120108	−0.000600541		1.00000i	$0.500191\pi$
$464$	0	0
$465$	−2.27341	−0.105427
$466$	0	0
$467$	4.66985	0.216095	0.108047	−	0.994146i	$-0.465540\pi$
$467$	4.66985	0.216095	0.108047	+	0.994146i	$0.465540\pi$
$468$	0	0
$469$	14.3559	0.662893
$470$	0	0
$471$	30.2099	1.39200
$472$	0	0
$473$	−3.84088	−0.176604
$474$	0	0
$475$	−4.64712	−0.213224
$476$	0	0
$477$	−17.5261	−0.802464
$478$	0	0
$479$	30.5428	1.39554	0.697768	−	0.716324i	$-0.254177\pi$
$479$	30.5428	1.39554	0.697768	+	0.716324i	$0.254177\pi$
$480$	0	0
$481$	1.21306	0.0553108
$482$	0	0
$483$	−49.5261	−2.25352
$484$	0	0
$485$	−1.24063	−0.0563342
$486$	0	0
$487$	−2.37133	−0.107455	−0.0537277	−	0.998556i	$-0.517110\pi$
$487$	−2.37133	−0.107455	−0.0537277	+	0.998556i	$0.517110\pi$
$488$	0	0
$489$	−70.6770	−3.19613
$490$	0	0
$491$	41.3206	1.86477	0.932385	−	0.361466i	$-0.117724\pi$
$491$	41.3206	1.86477	0.932385	+	0.361466i	$0.117724\pi$
$492$	0	0
$493$	−12.2079	−0.549815
$494$	0	0
$495$	19.6328	0.882427
$496$	0	0
$497$	27.3002	1.22458
$498$	0	0
$499$	15.8610	0.710035	0.355018	−	0.934860i	$-0.384475\pi$
$499$	15.8610	0.710035	0.355018	+	0.934860i	$0.384475\pi$
$500$	0	0
$501$	51.7798	2.31335
$502$	0	0
$503$	−18.6879	−0.833254	−0.416627	−	0.909078i	$-0.636788\pi$
$503$	−18.6879	−0.833254	−0.416627	+	0.909078i	$0.636788\pi$
$504$	0	0
$505$	1.64593	0.0732431
$506$	0	0
$507$	−40.6237	−1.80416
$508$	0	0
$509$	−33.2172	−1.47233	−0.736163	−	0.676804i	$-0.763364\pi$
$509$	−33.2172	−1.47233	−0.736163	+	0.676804i	$0.763364\pi$
$510$	0	0
$511$	10.7742	0.476622
$512$	0	0
$513$	12.0276	0.531032
$514$	0	0
$515$	8.51637	0.375276
$516$	0	0
$517$	58.9827	2.59406
$518$	0	0
$519$	−16.1482	−0.708826
$520$	0	0
$521$	10.9831	0.481178	0.240589	−	0.970627i	$-0.422659\pi$
$521$	10.9831	0.481178	0.240589	+	0.970627i	$0.422659\pi$
$522$	0	0
$523$	8.00433	0.350005	0.175002	−	0.984568i	$-0.444007\pi$
$523$	8.00433	0.350005	0.175002	+	0.984568i	$0.444007\pi$
$524$	0	0
$525$	50.8273	2.21829
$526$	0	0
$527$	1.58299	0.0689561
$528$	0	0
$529$	−2.49589	−0.108517
$530$	0	0
$531$	18.8098	0.816277
$532$	0	0
$533$	−0.0970230	−0.00420253
$534$	0	0
$535$	−1.43988	−0.0622516
$536$	0	0
$537$	−29.2154	−1.26074
$538$	0	0
$539$	−24.9646	−1.07530
$540$	0	0
$541$	24.4359	1.05058	0.525291	−	0.850923i	$-0.323956\pi$
$541$	24.4359	1.05058	0.525291	+	0.850923i	$0.323956\pi$
$542$	0	0
$543$	16.1672	0.693800
$544$	0	0
$545$	0.000735485 0	3.15047e−5 0
$546$	0	0
$547$	5.25284	0.224595	0.112298	−	0.993675i	$-0.464179\pi$
$547$	5.25284	0.224595	0.112298	+	0.993675i	$0.464179\pi$
$548$	0	0
$549$	−53.0866	−2.26568
$550$	0	0
$551$	−9.41093	−0.400919
$552$	0	0
$553$	−35.1305	−1.49390
$554$	0	0
$555$	10.4821	0.444940
$556$	0	0
$557$	3.56791	0.151177	0.0755886	−	0.997139i	$-0.475916\pi$
$557$	3.56791	0.151177	0.0755886	+	0.997139i	$0.475916\pi$
$558$	0	0
$559$	−0.171261	−0.00724357
$560$	0	0
$561$	−19.6704	−0.830485
$562$	0	0
$563$	17.0555	0.718805	0.359403	−	0.933183i	$-0.382980\pi$
$563$	17.0555	0.718805	0.359403	+	0.933183i	$0.382980\pi$
$564$	0	0
$565$	−1.07886	−0.0453881
$566$	0	0
$567$	−60.0361	−2.52128
$568$	0	0
$569$	0.858816	0.0360034	0.0180017	−	0.999838i	$-0.494270\pi$
$569$	0.858816	0.0360034	0.0180017	+	0.999838i	$0.494270\pi$
$570$	0	0
$571$	40.5440	1.69671	0.848356	−	0.529426i	$-0.177593\pi$
$571$	40.5440	1.69671	0.848356	+	0.529426i	$0.177593\pi$
$572$	0	0
$573$	46.6671	1.94954
$574$	0	0
$575$	−21.0428	−0.877546
$576$	0	0
$577$	36.5405	1.52120	0.760600	−	0.649220i	$-0.224905\pi$
$577$	36.5405	1.52120	0.760600	+	0.649220i	$0.224905\pi$
$578$	0	0
$579$	−66.7221	−2.77288
$580$	0	0
$581$	−40.5588	−1.68266
$582$	0	0
$583$	12.3979	0.513468
$584$	0	0
$585$	0.875406	0.0361936
$586$	0	0
$587$	30.4022	1.25483	0.627416	−	0.778684i	$-0.284113\pi$
$587$	30.4022	1.25483	0.627416	+	0.778684i	$0.284113\pi$
$588$	0	0
$589$	1.22031	0.0502821
$590$	0	0
$591$	11.5776	0.476237
$592$	0	0
$593$	−16.3814	−0.672703	−0.336351	−	0.941737i	$-0.609193\pi$
$593$	−16.3814	−0.672703	−0.336351	+	0.941737i	$0.609193\pi$
$594$	0	0
$595$	2.68749	0.110176
$596$	0	0
$597$	36.6476	1.49989
$598$	0	0
$599$	−2.24749	−0.0918298	−0.0459149	−	0.998945i	$-0.514620\pi$
$599$	−2.24749	−0.0918298	−0.0459149	+	0.998945i	$0.514620\pi$
$600$	0	0
$601$	33.1805	1.35346	0.676730	−	0.736231i	$-0.263396\pi$
$601$	33.1805	1.35346	0.676730	+	0.736231i	$0.263396\pi$
$602$	0	0
$603$	−28.1358	−1.14578
$604$	0	0
$605$	−7.35371	−0.298971
$606$	0	0
$607$	−35.7847	−1.45246	−0.726228	−	0.687454i	$-0.758728\pi$
$607$	−35.7847	−1.45246	−0.726228	+	0.687454i	$0.758728\pi$
$608$	0	0
$609$	102.931	4.17098
$610$	0	0
$611$	2.62998	0.106398
$612$	0	0
$613$	−42.2856	−1.70790	−0.853950	−	0.520355i	$-0.825800\pi$
$613$	−42.2856	−1.70790	−0.853950	+	0.520355i	$0.825800\pi$
$614$	0	0
$615$	−0.838380	−0.0338067
$616$	0	0
$617$	26.2079	1.05509	0.527545	−	0.849527i	$-0.323113\pi$
$617$	26.2079	1.05509	0.527545	+	0.849527i	$0.323113\pi$
$618$	0	0
$619$	−40.7494	−1.63786	−0.818929	−	0.573896i	$-0.805432\pi$
$619$	−40.7494	−1.63786	−0.818929	+	0.573896i	$0.805432\pi$
$620$	0	0
$621$	54.4628	2.18552
$622$	0	0
$623$	47.8115	1.91553
$624$	0	0
$625$	19.8313	0.793250
$626$	0	0
$627$	−15.1637	−0.605581
$628$	0	0
$629$	−7.29874	−0.291020
$630$	0	0
$631$	42.3804	1.68714	0.843569	−	0.537021i	$-0.180450\pi$
$631$	42.3804	1.68714	0.843569	+	0.537021i	$0.180450\pi$
$632$	0	0
$633$	−23.8850	−0.949343
$634$	0	0
$635$	−9.58005	−0.380173
$636$	0	0
$637$	−1.11315	−0.0441045
$638$	0	0
$639$	−53.5051	−2.11663
$640$	0	0
$641$	−40.0356	−1.58131	−0.790656	−	0.612261i	$-0.790260\pi$
$641$	−40.0356	−1.58131	−0.790656	+	0.612261i	$0.790260\pi$
$642$	0	0
$643$	2.25369	0.0888767	0.0444384	−	0.999012i	$-0.485850\pi$
$643$	2.25369	0.0888767	0.0444384	+	0.999012i	$0.485850\pi$
$644$	0	0
$645$	−1.47987	−0.0582700
$646$	0	0
$647$	−6.89272	−0.270981	−0.135490	−	0.990779i	$-0.543261\pi$
$647$	−6.89272	−0.270981	−0.135490	+	0.990779i	$0.543261\pi$
$648$	0	0
$649$	−13.3060	−0.522307
$650$	0	0
$651$	−13.3470	−0.523111
$652$	0	0
$653$	5.75398	0.225170	0.112585	−	0.993642i	$-0.464087\pi$
$653$	5.75398	0.225170	0.112585	+	0.993642i	$0.464087\pi$
$654$	0	0
$655$	−6.62222	−0.258751
$656$	0	0
$657$	−21.1161	−0.823817
$658$	0	0
$659$	−34.1024	−1.32844	−0.664220	−	0.747537i	$-0.731236\pi$
$659$	−34.1024	−1.32844	−0.664220	+	0.747537i	$0.731236\pi$
$660$	0	0
$661$	−46.3767	−1.80384	−0.901922	−	0.431900i	$-0.857843\pi$
$661$	−46.3767	−1.80384	−0.901922	+	0.431900i	$0.857843\pi$
$662$	0	0
$663$	−0.877084	−0.0340631
$664$	0	0
$665$	2.07176	0.0803392
$666$	0	0
$667$	−42.6141	−1.65002
$668$	0	0
$669$	6.13038	0.237014
$670$	0	0
$671$	37.5533	1.44973
$672$	0	0
$673$	−29.1579	−1.12395	−0.561977	−	0.827153i	$-0.689959\pi$
$673$	−29.1579	−1.12395	−0.561977	+	0.827153i	$0.689959\pi$
$674$	0	0
$675$	−55.8937	−2.15135
$676$	0	0
$677$	10.0436	0.386006	0.193003	−	0.981198i	$-0.438177\pi$
$677$	10.0436	0.386006	0.193003	+	0.981198i	$0.438177\pi$
$678$	0	0
$679$	−7.28365	−0.279521
$680$	0	0
$681$	−41.9328	−1.60687
$682$	0	0
$683$	−25.5866	−0.979042	−0.489521	−	0.871991i	$-0.662828\pi$
$683$	−25.5866	−0.979042	−0.489521	+	0.871991i	$0.662828\pi$
$684$	0	0
$685$	6.63341	0.253449
$686$	0	0
$687$	36.3161	1.38554
$688$	0	0
$689$	0.552809	0.0210604
$690$	0	0
$691$	−42.2386	−1.60683	−0.803417	−	0.595417i	$-0.796987\pi$
$691$	−42.2386	−1.60683	−0.803417	+	0.595417i	$0.796987\pi$
$692$	0	0
$693$	115.262	4.37845
$694$	0	0
$695$	7.75428	0.294136
$696$	0	0
$697$	0.583768	0.0221118
$698$	0	0
$699$	−4.98239	−0.188451
$700$	0	0
$701$	−43.9811	−1.66114	−0.830572	−	0.556911i	$-0.811986\pi$
$701$	−43.9811	−1.66114	−0.830572	+	0.556911i	$0.811986\pi$
$702$	0	0
$703$	−5.62653	−0.212208
$704$	0	0
$705$	22.7258	0.855902
$706$	0	0
$707$	9.66314	0.363420
$708$	0	0
$709$	11.1307	0.418023	0.209012	−	0.977913i	$-0.432975\pi$
$709$	11.1307	0.418023	0.209012	+	0.977913i	$0.432975\pi$
$710$	0	0
$711$	68.8514	2.58213
$712$	0	0
$713$	5.52575	0.206941
$714$	0	0
$715$	−0.619259	−0.0231590
$716$	0	0
$717$	−74.7081	−2.79003
$718$	0	0
$719$	−17.8961	−0.667413	−0.333706	−	0.942677i	$-0.608299\pi$
$719$	−17.8961	−0.667413	−0.333706	+	0.942677i	$0.608299\pi$
$720$	0	0
$721$	49.9989	1.86206
$722$	0	0
$723$	71.6771	2.66570
$724$	0	0
$725$	43.7337	1.62423
$726$	0	0
$727$	−35.8772	−1.33061	−0.665306	−	0.746571i	$-0.731699\pi$
$727$	−35.8772	−1.33061	−0.665306	+	0.746571i	$0.731699\pi$
$728$	0	0
$729$	4.50357	0.166799
$730$	0	0
$731$	1.03044	0.0381123
$732$	0	0
$733$	24.4618	0.903516	0.451758	−	0.892141i	$-0.350797\pi$
$733$	24.4618	0.903516	0.451758	+	0.892141i	$0.350797\pi$
$734$	0	0
$735$	−9.61875	−0.354793
$736$	0	0
$737$	19.9032	0.733142
$738$	0	0
$739$	−26.7820	−0.985193	−0.492596	−	0.870258i	$-0.663952\pi$
$739$	−26.7820	−0.985193	−0.492596	+	0.870258i	$0.663952\pi$
$740$	0	0
$741$	−0.676135	−0.0248384
$742$	0	0
$743$	23.7835	0.872534	0.436267	−	0.899817i	$-0.356300\pi$
$743$	23.7835	0.872534	0.436267	+	0.899817i	$0.356300\pi$
$744$	0	0
$745$	8.54295	0.312990
$746$	0	0
$747$	79.4903	2.90840
$748$	0	0
$749$	−8.45344	−0.308882
$750$	0	0
$751$	−22.4303	−0.818494	−0.409247	−	0.912424i	$-0.634209\pi$
$751$	−22.4303	−0.818494	−0.409247	+	0.912424i	$0.634209\pi$
$752$	0	0
$753$	8.26442	0.301172
$754$	0	0
$755$	−10.3060	−0.375073
$756$	0	0
$757$	−32.2421	−1.17186	−0.585930	−	0.810362i	$-0.699271\pi$
$757$	−32.2421	−1.17186	−0.585930	+	0.810362i	$0.699271\pi$
$758$	0	0
$759$	−68.6635	−2.49233
$760$	0	0
$761$	−16.3559	−0.592902	−0.296451	−	0.955048i	$-0.595803\pi$
$761$	−16.3559	−0.592902	−0.296451	+	0.955048i	$0.595803\pi$
$762$	0	0
$763$	0.00431797	0.000156321 0
$764$	0	0
$765$	−5.26714	−0.190434
$766$	0	0
$767$	−0.593302	−0.0214229
$768$	0	0
$769$	−19.6832	−0.709794	−0.354897	−	0.934905i	$-0.615484\pi$
$769$	−19.6832	−0.709794	−0.354897	+	0.934905i	$0.615484\pi$
$770$	0	0
$771$	62.9038	2.26542
$772$	0	0
$773$	−33.0819	−1.18987	−0.594936	−	0.803773i	$-0.702823\pi$
$773$	−33.0819	−1.18987	−0.594936	+	0.803773i	$0.702823\pi$
$774$	0	0
$775$	−5.67093	−0.203706
$776$	0	0
$777$	61.5395	2.20772
$778$	0	0
$779$	0.450021	0.0161237
$780$	0	0
$781$	37.8494	1.35436
$782$	0	0
$783$	−113.191	−4.04512
$784$	0	0
$785$	−5.72235	−0.204240
$786$	0	0
$787$	35.1617	1.25338	0.626690	−	0.779269i	$-0.284409\pi$
$787$	35.1617	1.25338	0.626690	+	0.779269i	$0.284409\pi$
$788$	0	0
$789$	12.6594	0.450688
$790$	0	0
$791$	−6.33392	−0.225208
$792$	0	0
$793$	1.67446	0.0594619
$794$	0	0
$795$	4.77685	0.169417
$796$	0	0
$797$	−3.75101	−0.132868	−0.0664338	−	0.997791i	$-0.521162\pi$
$797$	−3.75101	−0.132868	−0.0664338	+	0.997791i	$0.521162\pi$
$798$	0	0
$799$	−15.8241	−0.559815
$800$	0	0
$801$	−93.7046	−3.31089
$802$	0	0
$803$	14.9374	0.527131
$804$	0	0
$805$	9.38121	0.330644
$806$	0	0
$807$	−44.8763	−1.57972
$808$	0	0
$809$	−23.9158	−0.840836	−0.420418	−	0.907331i	$-0.638117\pi$
$809$	−23.9158	−0.840836	−0.420418	+	0.907331i	$0.638117\pi$
$810$	0	0
$811$	−52.0538	−1.82785	−0.913927	−	0.405878i	$-0.866966\pi$
$811$	−52.0538	−1.82785	−0.913927	+	0.405878i	$0.866966\pi$
$812$	0	0
$813$	−30.8918	−1.08342
$814$	0	0
$815$	13.3876	0.468947
$816$	0	0
$817$	0.794359	0.0277911
$818$	0	0
$819$	5.13943	0.179586
$820$	0	0
$821$	0.703590	0.0245555	0.0122777	−	0.999925i	$-0.496092\pi$
$821$	0.703590	0.0245555	0.0122777	+	0.999925i	$0.496092\pi$
$822$	0	0
$823$	23.4559	0.817622	0.408811	−	0.912619i	$-0.365943\pi$
$823$	23.4559	0.817622	0.408811	+	0.912619i	$0.365943\pi$
$824$	0	0
$825$	70.4676	2.45337
$826$	0	0
$827$	29.5282	1.02680	0.513399	−	0.858150i	$-0.328386\pi$
$827$	29.5282	1.02680	0.513399	+	0.858150i	$0.328386\pi$
$828$	0	0
$829$	4.42946	0.153841	0.0769207	−	0.997037i	$-0.475491\pi$
$829$	4.42946	0.153841	0.0769207	+	0.997037i	$0.475491\pi$
$830$	0	0
$831$	53.2014	1.84554
$832$	0	0
$833$	6.69759	0.232058
$834$	0	0
$835$	−9.80811	−0.339424
$836$	0	0
$837$	14.6774	0.507326
$838$	0	0
$839$	−36.4506	−1.25841	−0.629207	−	0.777237i	$-0.716620\pi$
$839$	−36.4506	−1.25841	−0.629207	+	0.777237i	$0.716620\pi$
$840$	0	0
$841$	59.5656	2.05399
$842$	0	0
$843$	32.0127	1.10257
$844$	0	0
$845$	7.69492	0.264713
$846$	0	0
$847$	−43.1730	−1.48344
$848$	0	0
$849$	−58.8677	−2.02033
$850$	0	0
$851$	−25.4777	−0.873366
$852$	0	0
$853$	−52.7043	−1.80456	−0.902281	−	0.431149i	$-0.858108\pi$
$853$	−52.7043	−1.80456	−0.902281	+	0.431149i	$0.858108\pi$
$854$	0	0
$855$	−4.06039	−0.138862
$856$	0	0
$857$	−13.7199	−0.468662	−0.234331	−	0.972157i	$-0.575290\pi$
$857$	−13.7199	−0.468662	−0.234331	+	0.972157i	$0.575290\pi$
$858$	0	0
$859$	55.0757	1.87916	0.939580	−	0.342330i	$-0.111216\pi$
$859$	55.0757	1.87916	0.939580	+	0.342330i	$0.111216\pi$
$860$	0	0
$861$	−4.92206	−0.167743
$862$	0	0
$863$	−38.3384	−1.30505	−0.652527	−	0.757766i	$-0.726291\pi$
$863$	−38.3384	−1.30505	−0.652527	+	0.757766i	$0.726291\pi$
$864$	0	0
$865$	3.05878	0.104002
$866$	0	0
$867$	−48.0367	−1.63141
$868$	0	0
$869$	−48.7053	−1.65221
$870$	0	0
$871$	0.887462	0.0300705
$872$	0	0
$873$	14.2751	0.483138
$874$	0	0
$875$	−19.9865	−0.675666
$876$	0	0
$877$	−21.4788	−0.725288	−0.362644	−	0.931928i	$-0.618126\pi$
$877$	−21.4788	−0.725288	−0.362644	+	0.931928i	$0.618126\pi$
$878$	0	0
$879$	−62.1800	−2.09728
$880$	0	0
$881$	−11.2383	−0.378629	−0.189314	−	0.981917i	$-0.560627\pi$
$881$	−11.2383	−0.378629	−0.189314	+	0.981917i	$0.560627\pi$
$882$	0	0
$883$	15.7137	0.528809	0.264405	−	0.964412i	$-0.414825\pi$
$883$	15.7137	0.528809	0.264405	+	0.964412i	$0.414825\pi$
$884$	0	0
$885$	−5.12674	−0.172334
$886$	0	0
$887$	4.48008	0.150426	0.0752131	−	0.997167i	$-0.476036\pi$
$887$	4.48008	0.150426	0.0752131	+	0.997167i	$0.476036\pi$
$888$	0	0
$889$	−56.2437	−1.88635
$890$	0	0
$891$	−83.2347	−2.78847
$892$	0	0
$893$	−12.1986	−0.408211
$894$	0	0
$895$	5.53396	0.184980
$896$	0	0
$897$	−3.06164	−0.102225
$898$	0	0
$899$	−11.4843	−0.383022
$900$	0	0
$901$	−3.32614	−0.110810
$902$	0	0
$903$	−8.68822	−0.289126
$904$	0	0
$905$	−3.06238	−0.101797
$906$	0	0
$907$	6.37021	0.211519	0.105760	−	0.994392i	$-0.466273\pi$
$907$	6.37021	0.211519	0.105760	+	0.994392i	$0.466273\pi$
$908$	0	0
$909$	−18.9386	−0.628152
$910$	0	0
$911$	30.5275	1.01142	0.505711	−	0.862703i	$-0.331230\pi$
$911$	30.5275	1.01142	0.505711	+	0.862703i	$0.331230\pi$
$912$	0	0
$913$	−56.2312	−1.86098
$914$	0	0
$915$	14.4691	0.478334
$916$	0	0
$917$	−38.8785	−1.28388
$918$	0	0
$919$	−48.0650	−1.58552	−0.792758	−	0.609536i	$-0.791356\pi$
$919$	−48.0650	−1.58552	−0.792758	+	0.609536i	$0.791356\pi$
$920$	0	0
$921$	−57.6840	−1.90075
$922$	0	0
$923$	1.68767	0.0555502
$924$	0	0
$925$	26.1471	0.859712
$926$	0	0
$927$	−97.9917	−3.21847
$928$	0	0
$929$	47.2953	1.55171	0.775854	−	0.630912i	$-0.217319\pi$
$929$	47.2953	1.55171	0.775854	+	0.630912i	$0.217319\pi$
$930$	0	0
$931$	5.16310	0.169214
$932$	0	0
$933$	60.1339	1.96870
$934$	0	0
$935$	3.72596	0.121852
$936$	0	0
$937$	8.30237	0.271227	0.135613	−	0.990762i	$-0.456699\pi$
$937$	8.30237	0.271227	0.135613	+	0.990762i	$0.456699\pi$
$938$	0	0
$939$	−44.6189	−1.45608
$940$	0	0
$941$	31.3955	1.02347	0.511733	−	0.859145i	$-0.329004\pi$
$941$	31.3955	1.02347	0.511733	+	0.859145i	$0.329004\pi$
$942$	0	0
$943$	2.03776	0.0663586
$944$	0	0
$945$	24.9183	0.810591
$946$	0	0
$947$	2.14805	0.0698024	0.0349012	−	0.999391i	$-0.488888\pi$
$947$	2.14805	0.0698024	0.0349012	+	0.999391i	$0.488888\pi$
$948$	0	0
$949$	0.666046	0.0216208
$950$	0	0
$951$	−2.43970	−0.0791127
$952$	0	0
$953$	−25.3661	−0.821689	−0.410844	−	0.911705i	$-0.634766\pi$
$953$	−25.3661	−0.821689	−0.410844	+	0.911705i	$0.634766\pi$
$954$	0	0
$955$	−8.83965	−0.286044
$956$	0	0
$957$	142.705	4.61299
$958$	0	0
$959$	38.9442	1.25757
$960$	0	0
$961$	−29.5108	−0.951963
$962$	0	0
$963$	16.5677	0.533887
$964$	0	0
$965$	12.6385	0.406847
$966$	0	0
$967$	−12.8395	−0.412892	−0.206446	−	0.978458i	$-0.566190\pi$
$967$	−12.8395	−0.412892	−0.206446	+	0.978458i	$0.566190\pi$
$968$	0	0
$969$	4.06817	0.130688
$970$	0	0
$971$	−8.77909	−0.281734	−0.140867	−	0.990028i	$-0.544989\pi$
$971$	−8.77909	−0.281734	−0.140867	+	0.990028i	$0.544989\pi$
$972$	0	0
$973$	45.5247	1.45945
$974$	0	0
$975$	3.14208	0.100627
$976$	0	0
$977$	−45.9701	−1.47071	−0.735357	−	0.677680i	$-0.762986\pi$
$977$	−45.9701	−1.47071	−0.735357	+	0.677680i	$0.762986\pi$
$978$	0	0
$979$	66.2864	2.11852
$980$	0	0
$981$	−0.00846269	−0.000270193 0
$982$	0	0
$983$	−51.4992	−1.64257	−0.821284	−	0.570519i	$-0.806742\pi$
$983$	−51.4992	−1.64257	−0.821284	+	0.570519i	$0.806742\pi$
$984$	0	0
$985$	−2.19302	−0.0698753
$986$	0	0
$987$	133.421	4.24684
$988$	0	0
$989$	3.59697	0.114377
$990$	0	0
$991$	29.4762	0.936343	0.468171	−	0.883638i	$-0.344913\pi$
$991$	29.4762	0.936343	0.468171	+	0.883638i	$0.344913\pi$
$992$	0	0
$993$	51.4284	1.63203
$994$	0	0
$995$	−6.94178	−0.220069
$996$	0	0
$997$	54.3591	1.72157	0.860786	−	0.508968i	$-0.169973\pi$
$997$	54.3591	1.72157	0.860786	+	0.508968i	$0.169973\pi$
$998$	0	0
$999$	−67.6737	−2.14110

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.bn.1.8		8
4.3	odd	2		4864.2.a.bo.1.1		8
8.3	odd	2		4864.2.a.bq.1.8		8
8.5	even	2		4864.2.a.bp.1.1		8
16.3	odd	4		152.2.c.b.77.6	yes	16
16.5	even	4		608.2.c.b.305.1		16
16.11	odd	4		152.2.c.b.77.5	✓	16
16.13	even	4		608.2.c.b.305.16		16
48.5	odd	4		5472.2.g.b.2737.9		16
48.11	even	4		1368.2.g.b.685.12		16
48.29	odd	4		5472.2.g.b.2737.8		16
48.35	even	4		1368.2.g.b.685.11		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.c.b.77.5	✓	16	16.11	odd	4
152.2.c.b.77.6	yes	16	16.3	odd	4
608.2.c.b.305.1		16	16.5	even	4
608.2.c.b.305.16		16	16.13	even	4
1368.2.g.b.685.11		16	48.35	even	4
1368.2.g.b.685.12		16	48.11	even	4
4864.2.a.bn.1.8		8	1.1	even	1	trivial
4864.2.a.bo.1.1		8	4.3	odd	2
4864.2.a.bp.1.1		8	8.5	even	2
4864.2.a.bq.1.8		8	8.3	odd	2
5472.2.g.b.2737.8		16	48.29	odd	4
5472.2.g.b.2737.9		16	48.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4864 = 2^{8} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4864.a (trivial)

\(f(q)\)	\(=\)	\(q+3.13611 q^{3} -0.594041 q^{5} -3.48756 q^{7} +6.83520 q^{9} +O(q^{10})\)	\(q+3.13611 q^{3} -0.594041 q^{5} -3.48756 q^{7} +6.83520 q^{9} -4.83520 q^{11} -0.215597 q^{13} -1.86298 q^{15} +1.29720 q^{17} +1.00000 q^{19} -10.9374 q^{21} +4.52815 q^{23} -4.64712 q^{25} +12.0276 q^{27} -9.41093 q^{29} +1.22031 q^{31} -15.1637 q^{33} +2.07176 q^{35} -5.62653 q^{37} -0.676135 q^{39} +0.450021 q^{41} +0.794359 q^{43} -4.06039 q^{45} -12.1986 q^{47} +5.16310 q^{49} +4.06817 q^{51} -2.56409 q^{53} +2.87231 q^{55} +3.13611 q^{57} +2.75191 q^{59} -7.76665 q^{61} -23.8382 q^{63} +0.128073 q^{65} -4.11631 q^{67} +14.2008 q^{69} -7.82788 q^{71} -3.08931 q^{73} -14.5739 q^{75} +16.8631 q^{77} +10.0731 q^{79} +17.2143 q^{81} +11.6296 q^{83} -0.770591 q^{85} -29.5137 q^{87} -13.7091 q^{89} +0.751907 q^{91} +3.82703 q^{93} -0.594041 q^{95} +2.08846 q^{97} -33.0495 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5} - 4 q^{7} + 12 q^{9}+O(q^{10}) \) 8 * q - 8 * q^5 - 4 * q^7 + 12 * q^9	\( 8 q - 8 q^{5} - 4 q^{7} + 12 q^{9} + 4 q^{11} - 8 q^{13} - 4 q^{17} + 8 q^{19} - 16 q^{21} + 12 q^{25} - 28 q^{29} - 8 q^{31} - 12 q^{35} - 4 q^{37} + 4 q^{39} - 8 q^{41} - 4 q^{43} - 24 q^{45} - 12 q^{47} + 12 q^{49} + 12 q^{51} - 32 q^{53} + 8 q^{55} + 12 q^{59} - 8 q^{61} + 16 q^{63} + 8 q^{65} - 4 q^{67} - 28 q^{69} + 24 q^{71} - 24 q^{77} + 24 q^{79} - 8 q^{81} + 40 q^{83} - 24 q^{85} - 24 q^{87} + 8 q^{89} - 4 q^{91} - 32 q^{93} - 8 q^{95} + 16 q^{97} - 76 q^{99}+O(q^{100}) \) 8 * q - 8 * q^5 - 4 * q^7 + 12 * q^9 + 4 * q^11 - 8 * q^13 - 4 * q^17 + 8 * q^19 - 16 * q^21 + 12 * q^25 - 28 * q^29 - 8 * q^31 - 12 * q^35 - 4 * q^37 + 4 * q^39 - 8 * q^41 - 4 * q^43 - 24 * q^45 - 12 * q^47 + 12 * q^49 + 12 * q^51 - 32 * q^53 + 8 * q^55 + 12 * q^59 - 8 * q^61 + 16 * q^63 + 8 * q^65 - 4 * q^67 - 28 * q^69 + 24 * q^71 - 24 * q^77 + 24 * q^79 - 8 * q^81 + 40 * q^83 - 24 * q^85 - 24 * q^87 + 8 * q^89 - 4 * q^91 - 32 * q^93 - 8 * q^95 + 16 * q^97 - 76 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more