LMFDB - Embedded newform 5120.2.a.t.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5120,2,Mod(1,5120)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5120.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5120 = 2^{10} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5120.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,-4,0,8,0,-4,0,8,0,-8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$40.8834058349$
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} - 12x^{6} - 8x^{5} + 21x^{4} + 12x^{3} - 10x^{2} - 4x + 1 $ x^8 - 12x^6 - 8x^5 + 21x^4 + 12x^3 - 10x^2 - 4x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 80)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$3.51762$ of defining polynomial
Character	$\chi$	$=$	5120.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.01261 q^{3} +1.00000 q^{5} -0.690576 q^{7} +1.05061 q^{9} -4.32830 q^{11} -3.30482 q^{13} +2.01261 q^{15} +5.28770 q^{17} -7.62102 q^{19} -1.38986 q^{21} +1.60841 q^{23} +1.00000 q^{25} -3.92337 q^{27} -2.40867 q^{29} +4.69807 q^{31} -8.71119 q^{33} -0.690576 q^{35} +11.1705 q^{37} -6.65131 q^{39} -5.49891 q^{41} -0.362274 q^{43} +1.05061 q^{45} -4.60743 q^{47} -6.52310 q^{49} +10.6421 q^{51} +7.06143 q^{53} -4.32830 q^{55} -15.3382 q^{57} -2.07151 q^{59} -13.1947 q^{61} -0.725523 q^{63} -3.30482 q^{65} -2.75484 q^{67} +3.23710 q^{69} -2.32246 q^{71} -1.29733 q^{73} +2.01261 q^{75} +2.98902 q^{77} -5.01968 q^{79} -11.0480 q^{81} -10.3305 q^{83} +5.28770 q^{85} -4.84772 q^{87} -1.81564 q^{89} +2.28223 q^{91} +9.45539 q^{93} -7.62102 q^{95} +5.27038 q^{97} -4.54734 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} + 8 q^{5} - 4 q^{7} + 8 q^{9} - 8 q^{11} - 4 q^{15} - 16 q^{19} - 12 q^{23} + 8 q^{25} - 16 q^{27} - 4 q^{35} - 28 q^{43} + 8 q^{45} - 20 q^{47} + 8 q^{49} - 24 q^{51} - 8 q^{55} - 16 q^{59}+ \cdots - 40 q^{99}+O(q^{100}) $ 8 * q - 4 * q^3 + 8 * q^5 - 4 * q^7 + 8 * q^9 - 8 * q^11 - 4 * q^15 - 16 * q^19 - 12 * q^23 + 8 * q^25 - 16 * q^27 - 4 * q^35 - 28 * q^43 + 8 * q^45 - 20 * q^47 + 8 * q^49 - 24 * q^51 - 8 * q^55 - 16 * q^59 - 20 * q^63 - 36 * q^67 - 16 * q^69 + 8 * q^71 - 4 * q^75 + 8 * q^79 + 8 * q^81 - 20 * q^83 - 40 * q^91 - 16 * q^95 - 40 * q^99

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
$3$	2.01261	1.16198	0.580991	−	0.813910i	$-0.302665\pi$
$3$	2.01261	1.16198	0.580991	+	0.813910i	$0.302665\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.690576	−0.261013	−0.130507	−	0.991447i	$-0.541660\pi$
$7$	−0.690576	−0.261013	−0.130507	+	0.991447i	$0.541660\pi$
$8$	0	0
$9$	1.05061	0.350202
$10$	0	0
$11$	−4.32830	−1.30503	−0.652516	−	0.757775i	$-0.726287\pi$
$11$	−4.32830	−1.30503	−0.652516	+	0.757775i	$0.726287\pi$
$12$	0	0
$13$	−3.30482	−0.916591	−0.458296	−	0.888800i	$-0.651540\pi$
$13$	−3.30482	−0.916591	−0.458296	+	0.888800i	$0.651540\pi$
$14$	0	0
$15$	2.01261	0.519654
$16$	0	0
$17$	5.28770	1.28246	0.641228	−	0.767350i	$-0.278425\pi$
$17$	5.28770	1.28246	0.641228	+	0.767350i	$0.278425\pi$
$18$	0	0
$19$	−7.62102	−1.74838	−0.874191	−	0.485583i	$-0.838607\pi$
$19$	−7.62102	−1.74838	−0.874191	+	0.485583i	$0.838607\pi$
$20$	0	0
$21$	−1.38986	−0.303293
$22$	0	0
$23$	1.60841	0.335376	0.167688	−	0.985840i	$-0.446370\pi$
$23$	1.60841	0.335376	0.167688	+	0.985840i	$0.446370\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−3.92337	−0.755053
$28$	0	0
$29$	−2.40867	−0.447279	−0.223640	−	0.974672i	$-0.571794\pi$
$29$	−2.40867	−0.447279	−0.223640	+	0.974672i	$0.571794\pi$
$30$	0	0
$31$	4.69807	0.843798	0.421899	−	0.906643i	$-0.361364\pi$
$31$	4.69807	0.843798	0.421899	+	0.906643i	$0.361364\pi$
$32$	0	0
$33$	−8.71119	−1.51642
$34$	0	0
$35$	−0.690576	−0.116729
$36$	0	0
$37$	11.1705	1.83641	0.918207	−	0.396101i	$-0.129637\pi$
$37$	11.1705	1.83641	0.918207	+	0.396101i	$0.129637\pi$
$38$	0	0
$39$	−6.65131	−1.06506
$40$	0	0
$41$	−5.49891	−0.858785	−0.429392	−	0.903118i	$-0.641272\pi$
$41$	−5.49891	−0.858785	−0.429392	+	0.903118i	$0.641272\pi$
$42$	0	0
$43$	−0.362274	−0.0552463	−0.0276231	−	0.999618i	$-0.508794\pi$
$43$	−0.362274	−0.0552463	−0.0276231	+	0.999618i	$0.508794\pi$
$44$	0	0
$45$	1.05061	0.156615
$46$	0	0
$47$	−4.60743	−0.672063	−0.336032	−	0.941851i	$-0.609085\pi$
$47$	−4.60743	−0.672063	−0.336032	+	0.941851i	$0.609085\pi$
$48$	0	0
$49$	−6.52310	−0.931872
$50$	0	0
$51$	10.6421	1.49019
$52$	0	0
$53$	7.06143	0.969961	0.484981	−	0.874525i	$-0.338827\pi$
$53$	7.06143	0.969961	0.484981	+	0.874525i	$0.338827\pi$
$54$	0	0
$55$	−4.32830	−0.583628
$56$	0	0
$57$	−15.3382	−2.03159
$58$	0	0
$59$	−2.07151	−0.269687	−0.134844	−	0.990867i	$-0.543053\pi$
$59$	−2.07151	−0.269687	−0.134844	+	0.990867i	$0.543053\pi$
$60$	0	0
$61$	−13.1947	−1.68940	−0.844702	−	0.535237i	$-0.820222\pi$
$61$	−13.1947	−1.68940	−0.844702	+	0.535237i	$0.820222\pi$
$62$	0	0
$63$	−0.725523	−0.0914074
$64$	0	0
$65$	−3.30482	−0.409912
$66$	0	0
$67$	−2.75484	−0.336558	−0.168279	−	0.985739i	$-0.553821\pi$
$67$	−2.75484	−0.336558	−0.168279	+	0.985739i	$0.553821\pi$
$68$	0	0
$69$	3.23710	0.389701
$70$	0	0
$71$	−2.32246	−0.275625	−0.137813	−	0.990458i	$-0.544007\pi$
$71$	−2.32246	−0.275625	−0.137813	+	0.990458i	$0.544007\pi$
$72$	0	0
$73$	−1.29733	−0.151841	−0.0759206	−	0.997114i	$-0.524190\pi$
$73$	−1.29733	−0.151841	−0.0759206	+	0.997114i	$0.524190\pi$
$74$	0	0
$75$	2.01261	0.232396
$76$	0	0
$77$	2.98902	0.340631
$78$	0	0
$79$	−5.01968	−0.564758	−0.282379	−	0.959303i	$-0.591124\pi$
$79$	−5.01968	−0.564758	−0.282379	+	0.959303i	$0.591124\pi$
$80$	0	0
$81$	−11.0480	−1.22756
$82$	0	0
$83$	−10.3305	−1.13392	−0.566960	−	0.823745i	$-0.691881\pi$
$83$	−10.3305	−1.13392	−0.566960	+	0.823745i	$0.691881\pi$
$84$	0	0
$85$	5.28770	0.573532
$86$	0	0
$87$	−4.84772	−0.519730
$88$	0	0
$89$	−1.81564	−0.192458	−0.0962290	−	0.995359i	$-0.530678\pi$
$89$	−1.81564	−0.192458	−0.0962290	+	0.995359i	$0.530678\pi$
$90$	0	0
$91$	2.28223	0.239242
$92$	0	0
$93$	9.45539	0.980478
$94$	0	0
$95$	−7.62102	−0.781900
$96$	0	0
$97$	5.27038	0.535126	0.267563	−	0.963540i	$-0.413782\pi$
$97$	5.27038	0.535126	0.267563	+	0.963540i	$0.413782\pi$
$98$	0	0
$99$	−4.54734	−0.457025
$100$	0	0
$101$	−19.0214	−1.89270	−0.946351	−	0.323142i	$-0.895261\pi$
$101$	−19.0214	−1.89270	−0.946351	+	0.323142i	$0.895261\pi$
$102$	0	0
$103$	−2.64310	−0.260432	−0.130216	−	0.991486i	$-0.541567\pi$
$103$	−2.64310	−0.260432	−0.130216	+	0.991486i	$0.541567\pi$
$104$	0	0
$105$	−1.38986	−0.135637
$106$	0	0
$107$	−8.88296	−0.858748	−0.429374	−	0.903127i	$-0.641266\pi$
$107$	−8.88296	−0.858748	−0.429374	+	0.903127i	$0.641266\pi$
$108$	0	0
$109$	−9.74698	−0.933592	−0.466796	−	0.884365i	$-0.654592\pi$
$109$	−9.74698	−0.933592	−0.466796	+	0.884365i	$0.654592\pi$
$110$	0	0
$111$	22.4818	2.13388
$112$	0	0
$113$	−6.46108	−0.607807	−0.303904	−	0.952703i	$-0.598290\pi$
$113$	−6.46108	−0.607807	−0.303904	+	0.952703i	$0.598290\pi$
$114$	0	0
$115$	1.60841	0.149985
$116$	0	0
$117$	−3.47206	−0.320992
$118$	0	0
$119$	−3.65156	−0.334738
$120$	0	0
$121$	7.73420	0.703109
$122$	0	0
$123$	−11.0672	−0.997892
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	16.6123	1.47411	0.737054	−	0.675834i	$-0.236216\pi$
$127$	16.6123	1.47411	0.737054	+	0.675834i	$0.236216\pi$
$128$	0	0
$129$	−0.729117	−0.0641951
$130$	0	0
$131$	−16.6479	−1.45454	−0.727269	−	0.686353i	$-0.759211\pi$
$131$	−16.6479	−1.45454	−0.727269	+	0.686353i	$0.759211\pi$
$132$	0	0
$133$	5.26289	0.456351
$134$	0	0
$135$	−3.92337	−0.337670
$136$	0	0
$137$	−8.41495	−0.718937	−0.359469	−	0.933157i	$-0.617042\pi$
$137$	−8.41495	−0.718937	−0.359469	+	0.933157i	$0.617042\pi$
$138$	0	0
$139$	−2.14741	−0.182141	−0.0910705	−	0.995844i	$-0.529029\pi$
$139$	−2.14741	−0.182141	−0.0910705	+	0.995844i	$0.529029\pi$
$140$	0	0
$141$	−9.27297	−0.780925
$142$	0	0
$143$	14.3042	1.19618
$144$	0	0
$145$	−2.40867	−0.200029
$146$	0	0
$147$	−13.1285	−1.08282
$148$	0	0
$149$	3.69732	0.302896	0.151448	−	0.988465i	$-0.451606\pi$
$149$	3.69732	0.302896	0.151448	+	0.988465i	$0.451606\pi$
$150$	0	0
$151$	−12.7143	−1.03467	−0.517337	−	0.855782i	$-0.673077\pi$
$151$	−12.7143	−1.03467	−0.517337	+	0.855782i	$0.673077\pi$
$152$	0	0
$153$	5.55529	0.449119
$154$	0	0
$155$	4.69807	0.377358
$156$	0	0
$157$	10.1517	0.810191	0.405095	−	0.914274i	$-0.367238\pi$
$157$	10.1517	0.810191	0.405095	+	0.914274i	$0.367238\pi$
$158$	0	0
$159$	14.2119	1.12708
$160$	0	0
$161$	−1.11073	−0.0875376
$162$	0	0
$163$	−9.97694	−0.781454	−0.390727	−	0.920507i	$-0.627776\pi$
$163$	−9.97694	−0.781454	−0.390727	+	0.920507i	$0.627776\pi$
$164$	0	0
$165$	−8.71119	−0.678165
$166$	0	0
$167$	−3.90586	−0.302244	−0.151122	−	0.988515i	$-0.548289\pi$
$167$	−3.90586	−0.302244	−0.151122	+	0.988515i	$0.548289\pi$
$168$	0	0
$169$	−2.07819	−0.159861
$170$	0	0
$171$	−8.00669	−0.612287
$172$	0	0
$173$	11.5985	0.881819	0.440910	−	0.897551i	$-0.354656\pi$
$173$	11.5985	0.881819	0.440910	+	0.897551i	$0.354656\pi$
$174$	0	0
$175$	−0.690576	−0.0522026
$176$	0	0
$177$	−4.16914	−0.313372
$178$	0	0
$179$	4.38920	0.328064	0.164032	−	0.986455i	$-0.447550\pi$
$179$	4.38920	0.328064	0.164032	+	0.986455i	$0.447550\pi$
$180$	0	0
$181$	2.70172	0.200818	0.100409	−	0.994946i	$-0.467985\pi$
$181$	2.70172	0.200818	0.100409	+	0.994946i	$0.467985\pi$
$182$	0	0
$183$	−26.5557	−1.96306
$184$	0	0
$185$	11.1705	0.821269
$186$	0	0
$187$	−22.8868	−1.67365
$188$	0	0
$189$	2.70939	0.197079
$190$	0	0
$191$	−5.61041	−0.405955	−0.202977	−	0.979183i	$-0.565062\pi$
$191$	−5.61041	−0.405955	−0.202977	+	0.979183i	$0.565062\pi$
$192$	0	0
$193$	3.90696	0.281229	0.140615	−	0.990064i	$-0.455092\pi$
$193$	3.90696	0.281229	0.140615	+	0.990064i	$0.455092\pi$
$194$	0	0
$195$	−6.65131	−0.476310
$196$	0	0
$197$	0.860458	0.0613051	0.0306526	−	0.999530i	$-0.490241\pi$
$197$	0.860458	0.0613051	0.0306526	+	0.999530i	$0.490241\pi$
$198$	0	0
$199$	15.5282	1.10076	0.550382	−	0.834913i	$-0.314482\pi$
$199$	15.5282	1.10076	0.550382	+	0.834913i	$0.314482\pi$
$200$	0	0
$201$	−5.54443	−0.391074
$202$	0	0
$203$	1.66337	0.116746
$204$	0	0
$205$	−5.49891	−0.384060
$206$	0	0
$207$	1.68980	0.117449
$208$	0	0
$209$	32.9861	2.28169
$210$	0	0
$211$	−3.03350	−0.208835	−0.104417	−	0.994534i	$-0.533298\pi$
$211$	−3.03350	−0.208835	−0.104417	+	0.994534i	$0.533298\pi$
$212$	0	0
$213$	−4.67421	−0.320271
$214$	0	0
$215$	−0.362274	−0.0247069
$216$	0	0
$217$	−3.24437	−0.220242
$218$	0	0
$219$	−2.61103	−0.176437
$220$	0	0
$221$	−17.4749	−1.17549
$222$	0	0
$223$	2.34794	0.157230	0.0786148	−	0.996905i	$-0.474950\pi$
$223$	2.34794	0.157230	0.0786148	+	0.996905i	$0.474950\pi$
$224$	0	0
$225$	1.05061	0.0700404
$226$	0	0
$227$	18.6508	1.23790	0.618948	−	0.785432i	$-0.287559\pi$
$227$	18.6508	1.23790	0.618948	+	0.785432i	$0.287559\pi$
$228$	0	0
$229$	13.2633	0.876467	0.438233	−	0.898861i	$-0.355604\pi$
$229$	13.2633	0.876467	0.438233	+	0.898861i	$0.355604\pi$
$230$	0	0
$231$	6.01574	0.395807
$232$	0	0
$233$	16.3435	1.07070	0.535350	−	0.844630i	$-0.320180\pi$
$233$	16.3435	1.07070	0.535350	+	0.844630i	$0.320180\pi$
$234$	0	0
$235$	−4.60743	−0.300556
$236$	0	0
$237$	−10.1027	−0.656239
$238$	0	0
$239$	19.3818	1.25371	0.626854	−	0.779137i	$-0.284342\pi$
$239$	19.3818	1.25371	0.626854	+	0.779137i	$0.284342\pi$
$240$	0	0
$241$	−7.15965	−0.461193	−0.230597	−	0.973049i	$-0.574068\pi$
$241$	−7.15965	−0.461193	−0.230597	+	0.973049i	$0.574068\pi$
$242$	0	0
$243$	−10.4653	−0.671350
$244$	0	0
$245$	−6.52310	−0.416746
$246$	0	0
$247$	25.1861	1.60255
$248$	0	0
$249$	−20.7913	−1.31759
$250$	0	0
$251$	−14.7604	−0.931666	−0.465833	−	0.884873i	$-0.654245\pi$
$251$	−14.7604	−0.931666	−0.465833	+	0.884873i	$0.654245\pi$
$252$	0	0
$253$	−6.96167	−0.437676
$254$	0	0
$255$	10.6421	0.666434
$256$	0	0
$257$	5.72152	0.356899	0.178449	−	0.983949i	$-0.442892\pi$
$257$	5.72152	0.356899	0.178449	+	0.983949i	$0.442892\pi$
$258$	0	0
$259$	−7.71406	−0.479328
$260$	0	0
$261$	−2.53056	−0.156638
$262$	0	0
$263$	−27.1378	−1.67339	−0.836695	−	0.547669i	$-0.815515\pi$
$263$	−27.1378	−1.67339	−0.836695	+	0.547669i	$0.815515\pi$
$264$	0	0
$265$	7.06143	0.433780
$266$	0	0
$267$	−3.65419	−0.223633
$268$	0	0
$269$	18.4937	1.12758	0.563790	−	0.825918i	$-0.309343\pi$
$269$	18.4937	1.12758	0.563790	+	0.825918i	$0.309343\pi$
$270$	0	0
$271$	6.55264	0.398044	0.199022	−	0.979995i	$-0.436223\pi$
$271$	6.55264	0.398044	0.199022	+	0.979995i	$0.436223\pi$
$272$	0	0
$273$	4.59324	0.277995
$274$	0	0
$275$	−4.32830	−0.261006
$276$	0	0
$277$	14.5454	0.873946	0.436973	−	0.899475i	$-0.356051\pi$
$277$	14.5454	0.873946	0.436973	+	0.899475i	$0.356051\pi$
$278$	0	0
$279$	4.93582	0.295500
$280$	0	0
$281$	29.9714	1.78794	0.893971	−	0.448124i	$-0.147908\pi$
$281$	29.9714	1.78794	0.893971	+	0.448124i	$0.147908\pi$
$282$	0	0
$283$	27.0363	1.60714	0.803571	−	0.595209i	$-0.202931\pi$
$283$	27.0363	1.60714	0.803571	+	0.595209i	$0.202931\pi$
$284$	0	0
$285$	−15.3382	−0.908554
$286$	0	0
$287$	3.79741	0.224154
$288$	0	0
$289$	10.9598	0.644695
$290$	0	0
$291$	10.6072	0.621806
$292$	0	0
$293$	10.2948	0.601428	0.300714	−	0.953714i	$-0.402775\pi$
$293$	10.2948	0.601428	0.300714	+	0.953714i	$0.402775\pi$
$294$	0	0
$295$	−2.07151	−0.120608
$296$	0	0
$297$	16.9815	0.985369
$298$	0	0
$299$	−5.31549	−0.307403
$300$	0	0
$301$	0.250178	0.0144200
$302$	0	0
$303$	−38.2827	−2.19928
$304$	0	0
$305$	−13.1947	−0.755525
$306$	0	0
$307$	−9.94622	−0.567661	−0.283831	−	0.958874i	$-0.591605\pi$
$307$	−9.94622	−0.567661	−0.283831	+	0.958874i	$0.591605\pi$
$308$	0	0
$309$	−5.31953	−0.302617
$310$	0	0
$311$	14.2833	0.809929	0.404964	−	0.914332i	$-0.367284\pi$
$311$	14.2833	0.809929	0.404964	+	0.914332i	$0.367284\pi$
$312$	0	0
$313$	−18.4579	−1.04330	−0.521652	−	0.853158i	$-0.674684\pi$
$313$	−18.4579	−1.04330	−0.521652	+	0.853158i	$0.674684\pi$
$314$	0	0
$315$	−0.725523	−0.0408786
$316$	0	0
$317$	10.2079	0.573332	0.286666	−	0.958031i	$-0.407453\pi$
$317$	10.2079	0.573332	0.286666	+	0.958031i	$0.407453\pi$
$318$	0	0
$319$	10.4255	0.583714
$320$	0	0
$321$	−17.8780	−0.997850
$322$	0	0
$323$	−40.2977	−2.24222
$324$	0	0
$325$	−3.30482	−0.183318
$326$	0	0
$327$	−19.6169	−1.08482
$328$	0	0
$329$	3.18178	0.175417
$330$	0	0
$331$	−21.8987	−1.20366	−0.601830	−	0.798624i	$-0.705561\pi$
$331$	−21.8987	−1.20366	−0.601830	+	0.798624i	$0.705561\pi$
$332$	0	0
$333$	11.7358	0.643116
$334$	0	0
$335$	−2.75484	−0.150513
$336$	0	0
$337$	26.0210	1.41746	0.708728	−	0.705482i	$-0.249269\pi$
$337$	26.0210	1.41746	0.708728	+	0.705482i	$0.249269\pi$
$338$	0	0
$339$	−13.0036	−0.706261
$340$	0	0
$341$	−20.3347	−1.10118
$342$	0	0
$343$	9.33873	0.504244
$344$	0	0
$345$	3.23710	0.174280
$346$	0	0
$347$	18.1803	0.975970	0.487985	−	0.872852i	$-0.337732\pi$
$347$	18.1803	0.975970	0.487985	+	0.872852i	$0.337732\pi$
$348$	0	0
$349$	28.3163	1.51574	0.757869	−	0.652406i	$-0.226240\pi$
$349$	28.3163	1.51574	0.757869	+	0.652406i	$0.226240\pi$
$350$	0	0
$351$	12.9660	0.692075
$352$	0	0
$353$	13.7062	0.729510	0.364755	−	0.931104i	$-0.381153\pi$
$353$	13.7062	0.729510	0.364755	+	0.931104i	$0.381153\pi$
$354$	0	0
$355$	−2.32246	−0.123263
$356$	0	0
$357$	−7.34918	−0.388960
$358$	0	0
$359$	−32.3506	−1.70740	−0.853700	−	0.520764i	$-0.825647\pi$
$359$	−32.3506	−1.70740	−0.853700	+	0.520764i	$0.825647\pi$
$360$	0	0
$361$	39.0799	2.05684
$362$	0	0
$363$	15.5659	0.817000
$364$	0	0
$365$	−1.29733	−0.0679055
$366$	0	0
$367$	16.3714	0.854582	0.427291	−	0.904114i	$-0.359468\pi$
$367$	16.3714	0.854582	0.427291	+	0.904114i	$0.359468\pi$
$368$	0	0
$369$	−5.77718	−0.300748
$370$	0	0
$371$	−4.87645	−0.253173
$372$	0	0
$373$	−21.9657	−1.13734	−0.568671	−	0.822565i	$-0.692542\pi$
$373$	−21.9657	−1.13734	−0.568671	+	0.822565i	$0.692542\pi$
$374$	0	0
$375$	2.01261	0.103931
$376$	0	0
$377$	7.96022	0.409972
$378$	0	0
$379$	−35.2900	−1.81273	−0.906364	−	0.422498i	$-0.861153\pi$
$379$	−35.2900	−1.81273	−0.906364	+	0.422498i	$0.861153\pi$
$380$	0	0
$381$	33.4342	1.71289
$382$	0	0
$383$	−6.24887	−0.319302	−0.159651	−	0.987174i	$-0.551037\pi$
$383$	−6.24887	−0.319302	−0.159651	+	0.987174i	$0.551037\pi$
$384$	0	0
$385$	2.98902	0.152335
$386$	0	0
$387$	−0.380607	−0.0193474
$388$	0	0
$389$	2.98119	0.151152	0.0755762	−	0.997140i	$-0.475920\pi$
$389$	2.98119	0.151152	0.0755762	+	0.997140i	$0.475920\pi$
$390$	0	0
$391$	8.50478	0.430105
$392$	0	0
$393$	−33.5058	−1.69015
$394$	0	0
$395$	−5.01968	−0.252568
$396$	0	0
$397$	33.2308	1.66780	0.833902	−	0.551912i	$-0.186102\pi$
$397$	33.2308	1.66780	0.833902	+	0.551912i	$0.186102\pi$
$398$	0	0
$399$	10.5922	0.530271
$400$	0	0
$401$	20.9893	1.04816	0.524078	−	0.851670i	$-0.324410\pi$
$401$	20.9893	1.04816	0.524078	+	0.851670i	$0.324410\pi$
$402$	0	0
$403$	−15.5263	−0.773418
$404$	0	0
$405$	−11.0480	−0.548982
$406$	0	0
$407$	−48.3492	−2.39658
$408$	0	0
$409$	18.4025	0.909944	0.454972	−	0.890506i	$-0.349649\pi$
$409$	18.4025	0.909944	0.454972	+	0.890506i	$0.349649\pi$
$410$	0	0
$411$	−16.9360	−0.835392
$412$	0	0
$413$	1.43053	0.0703919
$414$	0	0
$415$	−10.3305	−0.507104
$416$	0	0
$417$	−4.32190	−0.211644
$418$	0	0
$419$	21.1186	1.03171	0.515856	−	0.856675i	$-0.327474\pi$
$419$	21.1186	1.03171	0.515856	+	0.856675i	$0.327474\pi$
$420$	0	0
$421$	−23.0064	−1.12126	−0.560632	−	0.828065i	$-0.689442\pi$
$421$	−23.0064	−1.12126	−0.560632	+	0.828065i	$0.689442\pi$
$422$	0	0
$423$	−4.84060	−0.235358
$424$	0	0
$425$	5.28770	0.256491
$426$	0	0
$427$	9.11192	0.440957
$428$	0	0
$429$	28.7889	1.38994
$430$	0	0
$431$	7.05425	0.339791	0.169896	−	0.985462i	$-0.445657\pi$
$431$	7.05425	0.339791	0.169896	+	0.985462i	$0.445657\pi$
$432$	0	0
$433$	14.3192	0.688139	0.344069	−	0.938944i	$-0.388194\pi$
$433$	14.3192	0.688139	0.344069	+	0.938944i	$0.388194\pi$
$434$	0	0
$435$	−4.84772	−0.232430
$436$	0	0
$437$	−12.2577	−0.586365
$438$	0	0
$439$	−25.9047	−1.23637	−0.618183	−	0.786034i	$-0.712131\pi$
$439$	−25.9047	−1.23637	−0.618183	+	0.786034i	$0.712131\pi$
$440$	0	0
$441$	−6.85321	−0.326344
$442$	0	0
$443$	15.7527	0.748436	0.374218	−	0.927341i	$-0.377911\pi$
$443$	15.7527	0.748436	0.374218	+	0.927341i	$0.377911\pi$
$444$	0	0
$445$	−1.81564	−0.0860698
$446$	0	0
$447$	7.44127	0.351960
$448$	0	0
$449$	12.6659	0.597740	0.298870	−	0.954294i	$-0.403390\pi$
$449$	12.6659	0.597740	0.298870	+	0.954294i	$0.403390\pi$
$450$	0	0
$451$	23.8009	1.12074
$452$	0	0
$453$	−25.5889	−1.20227
$454$	0	0
$455$	2.28223	0.106992
$456$	0	0
$457$	−16.9442	−0.792617	−0.396308	−	0.918117i	$-0.629709\pi$
$457$	−16.9442	−0.792617	−0.396308	+	0.918117i	$0.629709\pi$
$458$	0	0
$459$	−20.7456	−0.968323
$460$	0	0
$461$	−18.6518	−0.868700	−0.434350	−	0.900744i	$-0.643022\pi$
$461$	−18.6518	−0.868700	−0.434350	+	0.900744i	$0.643022\pi$
$462$	0	0
$463$	−14.0955	−0.655074	−0.327537	−	0.944838i	$-0.606219\pi$
$463$	−14.0955	−0.655074	−0.327537	+	0.944838i	$0.606219\pi$
$464$	0	0
$465$	9.45539	0.438483
$466$	0	0
$467$	17.1003	0.791310	0.395655	−	0.918399i	$-0.370518\pi$
$467$	17.1003	0.791310	0.395655	+	0.918399i	$0.370518\pi$
$468$	0	0
$469$	1.90243	0.0878460
$470$	0	0
$471$	20.4313	0.941427
$472$	0	0
$473$	1.56803	0.0720981
$474$	0	0
$475$	−7.62102	−0.349676
$476$	0	0
$477$	7.41878	0.339682
$478$	0	0
$479$	−14.2523	−0.651202	−0.325601	−	0.945507i	$-0.605567\pi$
$479$	−14.2523	−0.651202	−0.325601	+	0.945507i	$0.605567\pi$
$480$	0	0
$481$	−36.9163	−1.68324
$482$	0	0
$483$	−2.23546	−0.101717
$484$	0	0
$485$	5.27038	0.239315
$486$	0	0
$487$	26.0424	1.18010	0.590048	−	0.807368i	$-0.299109\pi$
$487$	26.0424	1.18010	0.590048	+	0.807368i	$0.299109\pi$
$488$	0	0
$489$	−20.0797	−0.908036
$490$	0	0
$491$	4.90447	0.221336	0.110668	−	0.993857i	$-0.464701\pi$
$491$	4.90447	0.221336	0.110668	+	0.993857i	$0.464701\pi$
$492$	0	0
$493$	−12.7363	−0.573616
$494$	0	0
$495$	−4.54734	−0.204388
$496$	0	0
$497$	1.60383	0.0719418
$498$	0	0
$499$	7.49920	0.335711	0.167855	−	0.985812i	$-0.446316\pi$
$499$	7.49920	0.335711	0.167855	+	0.985812i	$0.446316\pi$
$500$	0	0
$501$	−7.86097	−0.351202
$502$	0	0
$503$	−28.8492	−1.28632	−0.643161	−	0.765731i	$-0.722378\pi$
$503$	−28.8492	−1.28632	−0.643161	+	0.765731i	$0.722378\pi$
$504$	0	0
$505$	−19.0214	−0.846442
$506$	0	0
$507$	−4.18259	−0.185755
$508$	0	0
$509$	−18.3802	−0.814689	−0.407344	−	0.913275i	$-0.633545\pi$
$509$	−18.3802	−0.814689	−0.407344	+	0.913275i	$0.633545\pi$
$510$	0	0
$511$	0.895906	0.0396326
$512$	0	0
$513$	29.9001	1.32012
$514$	0	0
$515$	−2.64310	−0.116469
$516$	0	0
$517$	19.9424	0.877064
$518$	0	0
$519$	23.3433	1.02466
$520$	0	0
$521$	13.9833	0.612618	0.306309	−	0.951932i	$-0.400906\pi$
$521$	13.9833	0.612618	0.306309	+	0.951932i	$0.400906\pi$
$522$	0	0
$523$	−8.91928	−0.390013	−0.195007	−	0.980802i	$-0.562473\pi$
$523$	−8.91928	−0.390013	−0.195007	+	0.980802i	$0.562473\pi$
$524$	0	0
$525$	−1.38986	−0.0606585
$526$	0	0
$527$	24.8420	1.08213
$528$	0	0
$529$	−20.4130	−0.887523
$530$	0	0
$531$	−2.17634	−0.0944450
$532$	0	0
$533$	18.1729	0.787154
$534$	0	0
$535$	−8.88296	−0.384044
$536$	0	0
$537$	8.83375	0.381204
$538$	0	0
$539$	28.2340	1.21612
$540$	0	0
$541$	−5.50578	−0.236712	−0.118356	−	0.992971i	$-0.537762\pi$
$541$	−5.50578	−0.236712	−0.118356	+	0.992971i	$0.537762\pi$
$542$	0	0
$543$	5.43752	0.233346
$544$	0	0
$545$	−9.74698	−0.417515
$546$	0	0
$547$	−39.3683	−1.68327	−0.841633	−	0.540050i	$-0.818405\pi$
$547$	−39.3683	−1.68327	−0.841633	+	0.540050i	$0.818405\pi$
$548$	0	0
$549$	−13.8624	−0.591633
$550$	0	0
$551$	18.3565	0.782014
$552$	0	0
$553$	3.46647	0.147409
$554$	0	0
$555$	22.4818	0.954300
$556$	0	0
$557$	2.12774	0.0901550	0.0450775	−	0.998983i	$-0.485647\pi$
$557$	2.12774	0.0901550	0.0450775	+	0.998983i	$0.485647\pi$
$558$	0	0
$559$	1.19725	0.0506382
$560$	0	0
$561$	−46.0622	−1.94475
$562$	0	0
$563$	9.42803	0.397344	0.198672	−	0.980066i	$-0.436337\pi$
$563$	9.42803	0.397344	0.198672	+	0.980066i	$0.436337\pi$
$564$	0	0
$565$	−6.46108	−0.271820
$566$	0	0
$567$	7.62952	0.320410
$568$	0	0
$569$	8.38187	0.351386	0.175693	−	0.984445i	$-0.443783\pi$
$569$	8.38187	0.351386	0.175693	+	0.984445i	$0.443783\pi$
$570$	0	0
$571$	−40.1819	−1.68156	−0.840779	−	0.541378i	$-0.817903\pi$
$571$	−40.1819	−1.68156	−0.840779	+	0.541378i	$0.817903\pi$
$572$	0	0
$573$	−11.2916	−0.471712
$574$	0	0
$575$	1.60841	0.0670752
$576$	0	0
$577$	23.2045	0.966014	0.483007	−	0.875616i	$-0.339545\pi$
$577$	23.2045	0.966014	0.483007	+	0.875616i	$0.339545\pi$
$578$	0	0
$579$	7.86320	0.326784
$580$	0	0
$581$	7.13400	0.295968
$582$	0	0
$583$	−30.5640	−1.26583
$584$	0	0
$585$	−3.47206	−0.143552
$586$	0	0
$587$	−15.5842	−0.643230	−0.321615	−	0.946871i	$-0.604226\pi$
$587$	−15.5842	−0.643230	−0.321615	+	0.946871i	$0.604226\pi$
$588$	0	0
$589$	−35.8041	−1.47528
$590$	0	0
$591$	1.73177	0.0712354
$592$	0	0
$593$	6.98847	0.286982	0.143491	−	0.989652i	$-0.454167\pi$
$593$	6.98847	0.286982	0.143491	+	0.989652i	$0.454167\pi$
$594$	0	0
$595$	−3.65156	−0.149699
$596$	0	0
$597$	31.2522	1.27907
$598$	0	0
$599$	39.9642	1.63289	0.816447	−	0.577420i	$-0.195941\pi$
$599$	39.9642	1.63289	0.816447	+	0.577420i	$0.195941\pi$
$600$	0	0
$601$	21.0830	0.859993	0.429997	−	0.902831i	$-0.358515\pi$
$601$	21.0830	0.859993	0.429997	+	0.902831i	$0.358515\pi$
$602$	0	0
$603$	−2.89425	−0.117863
$604$	0	0
$605$	7.73420	0.314440
$606$	0	0
$607$	−22.3189	−0.905897	−0.452949	−	0.891537i	$-0.649628\pi$
$607$	−22.3189	−0.905897	−0.452949	+	0.891537i	$0.649628\pi$
$608$	0	0
$609$	3.34772	0.135656
$610$	0	0
$611$	15.2267	0.616007
$612$	0	0
$613$	−14.9970	−0.605725	−0.302862	−	0.953034i	$-0.597942\pi$
$613$	−14.9970	−0.605725	−0.302862	+	0.953034i	$0.597942\pi$
$614$	0	0
$615$	−11.0672	−0.446271
$616$	0	0
$617$	−33.7636	−1.35927	−0.679635	−	0.733550i	$-0.737862\pi$
$617$	−33.7636	−1.35927	−0.679635	+	0.733550i	$0.737862\pi$
$618$	0	0
$619$	6.88406	0.276694	0.138347	−	0.990384i	$-0.455821\pi$
$619$	6.88406	0.276694	0.138347	+	0.990384i	$0.455821\pi$
$620$	0	0
$621$	−6.31038	−0.253227
$622$	0	0
$623$	1.25384	0.0502341
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	66.3882	2.65129
$628$	0	0
$629$	59.0661	2.35512
$630$	0	0
$631$	16.1348	0.642315	0.321157	−	0.947026i	$-0.395928\pi$
$631$	16.1348	0.642315	0.321157	+	0.947026i	$0.395928\pi$
$632$	0	0
$633$	−6.10526	−0.242662
$634$	0	0
$635$	16.6123	0.659241
$636$	0	0
$637$	21.5577	0.854146
$638$	0	0
$639$	−2.43999	−0.0965245
$640$	0	0
$641$	20.3125	0.802296	0.401148	−	0.916013i	$-0.368611\pi$
$641$	20.3125	0.802296	0.401148	+	0.916013i	$0.368611\pi$
$642$	0	0
$643$	11.0088	0.434146	0.217073	−	0.976155i	$-0.430349\pi$
$643$	11.0088	0.434146	0.217073	+	0.976155i	$0.430349\pi$
$644$	0	0
$645$	−0.729117	−0.0287089
$646$	0	0
$647$	21.7693	0.855840	0.427920	−	0.903817i	$-0.359246\pi$
$647$	21.7693	0.855840	0.427920	+	0.903817i	$0.359246\pi$
$648$	0	0
$649$	8.96611	0.351951
$650$	0	0
$651$	−6.52966	−0.255918
$652$	0	0
$653$	−37.2106	−1.45616	−0.728081	−	0.685491i	$-0.759587\pi$
$653$	−37.2106	−1.45616	−0.728081	+	0.685491i	$0.759587\pi$
$654$	0	0
$655$	−16.6479	−0.650489
$656$	0	0
$657$	−1.36298	−0.0531751
$658$	0	0
$659$	28.6222	1.11496	0.557481	−	0.830190i	$-0.311768\pi$
$659$	28.6222	1.11496	0.557481	+	0.830190i	$0.311768\pi$
$660$	0	0
$661$	−9.64359	−0.375092	−0.187546	−	0.982256i	$-0.560053\pi$
$661$	−9.64359	−0.375092	−0.187546	+	0.982256i	$0.560053\pi$
$662$	0	0
$663$	−35.1702	−1.36590
$664$	0	0
$665$	5.26289	0.204086
$666$	0	0
$667$	−3.87412	−0.150007
$668$	0	0
$669$	4.72549	0.182698
$670$	0	0
$671$	57.1105	2.20473
$672$	0	0
$673$	−8.19512	−0.315899	−0.157949	−	0.987447i	$-0.550488\pi$
$673$	−8.19512	−0.315899	−0.157949	+	0.987447i	$0.550488\pi$
$674$	0	0
$675$	−3.92337	−0.151011
$676$	0	0
$677$	18.2199	0.700249	0.350125	−	0.936703i	$-0.386139\pi$
$677$	18.2199	0.700249	0.350125	+	0.936703i	$0.386139\pi$
$678$	0	0
$679$	−3.63960	−0.139675
$680$	0	0
$681$	37.5368	1.43841
$682$	0	0
$683$	21.3084	0.815344	0.407672	−	0.913128i	$-0.366341\pi$
$683$	21.3084	0.815344	0.407672	+	0.913128i	$0.366341\pi$
$684$	0	0
$685$	−8.41495	−0.321519
$686$	0	0
$687$	26.6940	1.01844
$688$	0	0
$689$	−23.3367	−0.889058
$690$	0	0
$691$	7.40670	0.281764	0.140882	−	0.990026i	$-0.455006\pi$
$691$	7.40670	0.281764	0.140882	+	0.990026i	$0.455006\pi$
$692$	0	0
$693$	3.14028	0.119290
$694$	0	0
$695$	−2.14741	−0.0814559
$696$	0	0
$697$	−29.0766	−1.10135
$698$	0	0
$699$	32.8932	1.24414
$700$	0	0
$701$	−30.7824	−1.16263	−0.581317	−	0.813677i	$-0.697462\pi$
$701$	−30.7824	−1.16263	−0.581317	+	0.813677i	$0.697462\pi$
$702$	0	0
$703$	−85.1304	−3.21075
$704$	0	0
$705$	−9.27297	−0.349240
$706$	0	0
$707$	13.1357	0.494020
$708$	0	0
$709$	−33.8906	−1.27279	−0.636394	−	0.771364i	$-0.719575\pi$
$709$	−33.8906	−1.27279	−0.636394	+	0.771364i	$0.719575\pi$
$710$	0	0
$711$	−5.27371	−0.197779
$712$	0	0
$713$	7.55640	0.282990
$714$	0	0
$715$	14.3042	0.534948
$716$	0	0
$717$	39.0081	1.45679
$718$	0	0
$719$	−44.4408	−1.65736	−0.828681	−	0.559721i	$-0.810908\pi$
$719$	−44.4408	−1.65736	−0.828681	+	0.559721i	$0.810908\pi$
$720$	0	0
$721$	1.82526	0.0679762
$722$	0	0
$723$	−14.4096	−0.535898
$724$	0	0
$725$	−2.40867	−0.0894558
$726$	0	0
$727$	46.6543	1.73031	0.865155	−	0.501504i	$-0.167220\pi$
$727$	46.6543	1.73031	0.865155	+	0.501504i	$0.167220\pi$
$728$	0	0
$729$	12.0815	0.447464
$730$	0	0
$731$	−1.91560	−0.0708509
$732$	0	0
$733$	−27.4644	−1.01442	−0.507210	−	0.861823i	$-0.669323\pi$
$733$	−27.4644	−1.01442	−0.507210	+	0.861823i	$0.669323\pi$
$734$	0	0
$735$	−13.1285	−0.484251
$736$	0	0
$737$	11.9238	0.439219
$738$	0	0
$739$	−29.0257	−1.06773	−0.533865	−	0.845570i	$-0.679261\pi$
$739$	−29.0257	−1.06773	−0.533865	+	0.845570i	$0.679261\pi$
$740$	0	0
$741$	50.6898	1.86214
$742$	0	0
$743$	12.9245	0.474154	0.237077	−	0.971491i	$-0.423811\pi$
$743$	12.9245	0.474154	0.237077	+	0.971491i	$0.423811\pi$
$744$	0	0
$745$	3.69732	0.135459
$746$	0	0
$747$	−10.8533	−0.397101
$748$	0	0
$749$	6.13436	0.224145
$750$	0	0
$751$	−52.2694	−1.90734	−0.953668	−	0.300861i	$-0.902726\pi$
$751$	−52.2694	−1.90734	−0.953668	+	0.300861i	$0.902726\pi$
$752$	0	0
$753$	−29.7069	−1.08258
$754$	0	0
$755$	−12.7143	−0.462720
$756$	0	0
$757$	48.7216	1.77082	0.885409	−	0.464812i	$-0.153878\pi$
$757$	48.7216	1.77082	0.885409	+	0.464812i	$0.153878\pi$
$758$	0	0
$759$	−14.0111	−0.508572
$760$	0	0
$761$	−47.7467	−1.73082	−0.865408	−	0.501067i	$-0.832941\pi$
$761$	−47.7467	−1.73082	−0.865408	+	0.501067i	$0.832941\pi$
$762$	0	0
$763$	6.73103	0.243680
$764$	0	0
$765$	5.55529	0.200852
$766$	0	0
$767$	6.84595	0.247193
$768$	0	0
$769$	−17.9108	−0.645882	−0.322941	−	0.946419i	$-0.604671\pi$
$769$	−17.9108	−0.645882	−0.322941	+	0.946419i	$0.604671\pi$
$770$	0	0
$771$	11.5152	0.414710
$772$	0	0
$773$	5.27742	0.189815	0.0949077	−	0.995486i	$-0.469744\pi$
$773$	5.27742	0.189815	0.0949077	+	0.995486i	$0.469744\pi$
$774$	0	0
$775$	4.69807	0.168760
$776$	0	0
$777$	−15.5254	−0.556971
$778$	0	0
$779$	41.9073	1.50148
$780$	0	0
$781$	10.0523	0.359700
$782$	0	0
$783$	9.45012	0.337720
$784$	0	0
$785$	10.1517	0.362328
$786$	0	0
$787$	−3.39637	−0.121067	−0.0605337	−	0.998166i	$-0.519280\pi$
$787$	−3.39637	−0.121067	−0.0605337	+	0.998166i	$0.519280\pi$
$788$	0	0
$789$	−54.6179	−1.94445
$790$	0	0
$791$	4.46187	0.158646
$792$	0	0
$793$	43.6060	1.54849
$794$	0	0
$795$	14.2119	0.504044
$796$	0	0
$797$	−50.2007	−1.77820	−0.889099	−	0.457714i	$-0.848668\pi$
$797$	−50.2007	−1.77820	−0.889099	+	0.457714i	$0.848668\pi$
$798$	0	0
$799$	−24.3627	−0.861892
$800$	0	0
$801$	−1.90753	−0.0673992
$802$	0	0
$803$	5.61524	0.198158
$804$	0	0
$805$	−1.11073	−0.0391480
$806$	0	0
$807$	37.2206	1.31023
$808$	0	0
$809$	11.9182	0.419021	0.209510	−	0.977806i	$-0.432813\pi$
$809$	11.9182	0.419021	0.209510	+	0.977806i	$0.432813\pi$
$810$	0	0
$811$	−31.3240	−1.09994	−0.549968	−	0.835186i	$-0.685360\pi$
$811$	−31.3240	−1.09994	−0.549968	+	0.835186i	$0.685360\pi$
$812$	0	0
$813$	13.1879	0.462520
$814$	0	0
$815$	−9.97694	−0.349477
$816$	0	0
$817$	2.76090	0.0965915
$818$	0	0
$819$	2.39772	0.0837832
$820$	0	0
$821$	−18.9375	−0.660924	−0.330462	−	0.943819i	$-0.607205\pi$
$821$	−18.9375	−0.660924	−0.330462	+	0.943819i	$0.607205\pi$
$822$	0	0
$823$	43.9496	1.53199	0.765994	−	0.642848i	$-0.222247\pi$
$823$	43.9496	1.53199	0.765994	+	0.642848i	$0.222247\pi$
$824$	0	0
$825$	−8.71119	−0.303285
$826$	0	0
$827$	−2.53280	−0.0880740	−0.0440370	−	0.999030i	$-0.514022\pi$
$827$	−2.53280	−0.0880740	−0.0440370	+	0.999030i	$0.514022\pi$
$828$	0	0
$829$	−19.0889	−0.662984	−0.331492	−	0.943458i	$-0.607552\pi$
$829$	−19.0889	−0.662984	−0.331492	+	0.943458i	$0.607552\pi$
$830$	0	0
$831$	29.2742	1.01551
$832$	0	0
$833$	−34.4923	−1.19509
$834$	0	0
$835$	−3.90586	−0.135168
$836$	0	0
$837$	−18.4323	−0.637113
$838$	0	0
$839$	14.5332	0.501741	0.250870	−	0.968021i	$-0.419283\pi$
$839$	14.5332	0.501741	0.250870	+	0.968021i	$0.419283\pi$
$840$	0	0
$841$	−23.1983	−0.799941
$842$	0	0
$843$	60.3208	2.07756
$844$	0	0
$845$	−2.07819	−0.0714919
$846$	0	0
$847$	−5.34105	−0.183521
$848$	0	0
$849$	54.4136	1.86747
$850$	0	0
$851$	17.9667	0.615889
$852$	0	0
$853$	−16.3817	−0.560898	−0.280449	−	0.959869i	$-0.590483\pi$
$853$	−16.3817	−0.560898	−0.280449	+	0.959869i	$0.590483\pi$
$854$	0	0
$855$	−8.00669	−0.273823
$856$	0	0
$857$	15.6443	0.534399	0.267200	−	0.963641i	$-0.413902\pi$
$857$	15.6443	0.534399	0.267200	+	0.963641i	$0.413902\pi$
$858$	0	0
$859$	−17.1044	−0.583596	−0.291798	−	0.956480i	$-0.594254\pi$
$859$	−17.1044	−0.583596	−0.291798	+	0.956480i	$0.594254\pi$
$860$	0	0
$861$	7.64272	0.260463
$862$	0	0
$863$	9.28120	0.315936	0.157968	−	0.987444i	$-0.449506\pi$
$863$	9.28120	0.315936	0.157968	+	0.987444i	$0.449506\pi$
$864$	0	0
$865$	11.5985	0.394362
$866$	0	0
$867$	22.0579	0.749124
$868$	0	0
$869$	21.7267	0.737028
$870$	0	0
$871$	9.10425	0.308486
$872$	0	0
$873$	5.53709	0.187402
$874$	0	0
$875$	−0.690576	−0.0233457
$876$	0	0
$877$	−4.20884	−0.142122	−0.0710612	−	0.997472i	$-0.522639\pi$
$877$	−4.20884	−0.142122	−0.0710612	+	0.997472i	$0.522639\pi$
$878$	0	0
$879$	20.7194	0.698849
$880$	0	0
$881$	−29.3318	−0.988214	−0.494107	−	0.869401i	$-0.664505\pi$
$881$	−29.3318	−0.988214	−0.494107	+	0.869401i	$0.664505\pi$
$882$	0	0
$883$	−50.2890	−1.69236	−0.846180	−	0.532897i	$-0.821103\pi$
$883$	−50.2890	−1.69236	−0.846180	+	0.532897i	$0.821103\pi$
$884$	0	0
$885$	−4.16914	−0.140144
$886$	0	0
$887$	4.51671	0.151656	0.0758282	−	0.997121i	$-0.475840\pi$
$887$	4.51671	0.151656	0.0758282	+	0.997121i	$0.475840\pi$
$888$	0	0
$889$	−11.4721	−0.384762
$890$	0	0
$891$	47.8193	1.60201
$892$	0	0
$893$	35.1133	1.17502
$894$	0	0
$895$	4.38920	0.146715
$896$	0	0
$897$	−10.6980	−0.357196
$898$	0	0
$899$	−11.3161	−0.377413
$900$	0	0
$901$	37.3387	1.24393
$902$	0	0
$903$	0.503511	0.0167558
$904$	0	0
$905$	2.70172	0.0898083
$906$	0	0
$907$	−7.16679	−0.237969	−0.118985	−	0.992896i	$-0.537964\pi$
$907$	−7.16679	−0.237969	−0.118985	+	0.992896i	$0.537964\pi$
$908$	0	0
$909$	−19.9840	−0.662828
$910$	0	0
$911$	36.7140	1.21639	0.608194	−	0.793788i	$-0.291894\pi$
$911$	36.7140	1.21639	0.608194	+	0.793788i	$0.291894\pi$
$912$	0	0
$913$	44.7135	1.47980
$914$	0	0
$915$	−26.5557	−0.877906
$916$	0	0
$917$	11.4967	0.379653
$918$	0	0
$919$	−21.5651	−0.711365	−0.355683	−	0.934607i	$-0.615752\pi$
$919$	−21.5651	−0.711365	−0.355683	+	0.934607i	$0.615752\pi$
$920$	0	0
$921$	−20.0179	−0.659612
$922$	0	0
$923$	7.67530	0.252635
$924$	0	0
$925$	11.1705	0.367283
$926$	0	0
$927$	−2.77685	−0.0912038
$928$	0	0
$929$	−45.6603	−1.49807	−0.749033	−	0.662532i	$-0.769482\pi$
$929$	−45.6603	−1.49807	−0.749033	+	0.662532i	$0.769482\pi$
$930$	0	0
$931$	49.7127	1.62927
$932$	0	0
$933$	28.7466	0.941123
$934$	0	0
$935$	−22.8868	−0.748478
$936$	0	0
$937$	−2.29807	−0.0750746	−0.0375373	−	0.999295i	$-0.511951\pi$
$937$	−2.29807	−0.0750746	−0.0375373	+	0.999295i	$0.511951\pi$
$938$	0	0
$939$	−37.1486	−1.21230
$940$	0	0
$941$	−34.2238	−1.11566	−0.557832	−	0.829954i	$-0.688367\pi$
$941$	−34.2238	−1.11566	−0.557832	+	0.829954i	$0.688367\pi$
$942$	0	0
$943$	−8.84448	−0.288016
$944$	0	0
$945$	2.70939	0.0881364
$946$	0	0
$947$	−34.6739	−1.12675	−0.563375	−	0.826201i	$-0.690498\pi$
$947$	−34.6739	−1.12675	−0.563375	+	0.826201i	$0.690498\pi$
$948$	0	0
$949$	4.28744	0.139176
$950$	0	0
$951$	20.5445	0.666202
$952$	0	0
$953$	−32.3462	−1.04780	−0.523898	−	0.851781i	$-0.675523\pi$
$953$	−32.3462	−1.04780	−0.523898	+	0.851781i	$0.675523\pi$
$954$	0	0
$955$	−5.61041	−0.181548
$956$	0	0
$957$	20.9824	0.678265
$958$	0	0
$959$	5.81116	0.187652
$960$	0	0
$961$	−8.92816	−0.288005
$962$	0	0
$963$	−9.33249	−0.300735
$964$	0	0
$965$	3.90696	0.125770
$966$	0	0
$967$	−14.6983	−0.472665	−0.236333	−	0.971672i	$-0.575946\pi$
$967$	−14.6983	−0.472665	−0.236333	+	0.971672i	$0.575946\pi$
$968$	0	0
$969$	−81.1036	−2.60542
$970$	0	0
$971$	41.1628	1.32098	0.660489	−	0.750836i	$-0.270349\pi$
$971$	41.1628	1.32098	0.660489	+	0.750836i	$0.270349\pi$
$972$	0	0
$973$	1.48295	0.0475412
$974$	0	0
$975$	−6.65131	−0.213012
$976$	0	0
$977$	−17.3533	−0.555180	−0.277590	−	0.960700i	$-0.589536\pi$
$977$	−17.3533	−0.555180	−0.277590	+	0.960700i	$0.589536\pi$
$978$	0	0
$979$	7.85866	0.251164
$980$	0	0
$981$	−10.2402	−0.326946
$982$	0	0
$983$	−27.5174	−0.877668	−0.438834	−	0.898568i	$-0.644608\pi$
$983$	−27.5174	−0.877668	−0.438834	+	0.898568i	$0.644608\pi$
$984$	0	0
$985$	0.860458	0.0274165
$986$	0	0
$987$	6.40369	0.203832
$988$	0	0
$989$	−0.582684	−0.0185283
$990$	0	0
$991$	−6.96363	−0.221207	−0.110604	−	0.993865i	$-0.535278\pi$
$991$	−6.96363	−0.221207	−0.110604	+	0.993865i	$0.535278\pi$
$992$	0	0
$993$	−44.0735	−1.39863
$994$	0	0
$995$	15.5282	0.492277
$996$	0	0
$997$	−22.2103	−0.703408	−0.351704	−	0.936111i	$-0.614398\pi$
$997$	−22.2103	−0.703408	−0.351704	+	0.936111i	$0.614398\pi$
$998$	0	0
$999$	−43.8259	−1.38659

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5120.2.a.t.1.7		8
4.3	odd	2		5120.2.a.v.1.2		8
8.3	odd	2		5120.2.a.s.1.7		8
8.5	even	2		5120.2.a.u.1.2		8
32.3	odd	8		640.2.l.b.161.2		16
32.5	even	8		320.2.l.a.241.2		16
32.11	odd	8		640.2.l.b.481.2		16
32.13	even	8		320.2.l.a.81.2		16
32.19	odd	8		80.2.l.a.61.5	yes	16
32.21	even	8		640.2.l.a.481.7		16
32.27	odd	8		80.2.l.a.21.5	✓	16
32.29	even	8		640.2.l.a.161.7		16
96.5	odd	8		2880.2.t.c.2161.2		16
96.59	even	8		720.2.t.c.181.4		16
96.77	odd	8		2880.2.t.c.721.3		16
96.83	even	8		720.2.t.c.541.4		16
160.13	odd	8		1600.2.q.h.849.7		16
160.19	odd	8		400.2.l.h.301.4		16
160.27	even	8		400.2.q.g.149.1		16
160.37	odd	8		1600.2.q.h.49.7		16
160.59	odd	8		400.2.l.h.101.4		16
160.69	even	8		1600.2.l.i.1201.7		16
160.77	odd	8		1600.2.q.g.849.2		16
160.83	even	8		400.2.q.g.349.1		16
160.109	even	8		1600.2.l.i.401.7		16
160.123	even	8		400.2.q.h.149.8		16
160.133	odd	8		1600.2.q.g.49.2		16
160.147	even	8		400.2.q.h.349.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.2.l.a.21.5	✓	16	32.27	odd	8
80.2.l.a.61.5	yes	16	32.19	odd	8
320.2.l.a.81.2		16	32.13	even	8
320.2.l.a.241.2		16	32.5	even	8
400.2.l.h.101.4		16	160.59	odd	8
400.2.l.h.301.4		16	160.19	odd	8
400.2.q.g.149.1		16	160.27	even	8
400.2.q.g.349.1		16	160.83	even	8
400.2.q.h.149.8		16	160.123	even	8
400.2.q.h.349.8		16	160.147	even	8
640.2.l.a.161.7		16	32.29	even	8
640.2.l.a.481.7		16	32.21	even	8
640.2.l.b.161.2		16	32.3	odd	8
640.2.l.b.481.2		16	32.11	odd	8
720.2.t.c.181.4		16	96.59	even	8
720.2.t.c.541.4		16	96.83	even	8
1600.2.l.i.401.7		16	160.109	even	8
1600.2.l.i.1201.7		16	160.69	even	8
1600.2.q.g.49.2		16	160.133	odd	8
1600.2.q.g.849.2		16	160.77	odd	8
1600.2.q.h.49.7		16	160.37	odd	8
1600.2.q.h.849.7		16	160.13	odd	8
2880.2.t.c.721.3		16	96.77	odd	8
2880.2.t.c.2161.2		16	96.5	odd	8
5120.2.a.s.1.7		8	8.3	odd	2
5120.2.a.t.1.7		8	1.1	even	1	trivial
5120.2.a.u.1.2		8	8.5	even	2
5120.2.a.v.1.2		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5120 = 2^{10} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5120.a (trivial)

\(f(q)\)	\(=\)	\(q+2.01261 q^{3} +1.00000 q^{5} -0.690576 q^{7} +1.05061 q^{9} -4.32830 q^{11} -3.30482 q^{13} +2.01261 q^{15} +5.28770 q^{17} -7.62102 q^{19} -1.38986 q^{21} +1.60841 q^{23} +1.00000 q^{25} -3.92337 q^{27} -2.40867 q^{29} +4.69807 q^{31} -8.71119 q^{33} -0.690576 q^{35} +11.1705 q^{37} -6.65131 q^{39} -5.49891 q^{41} -0.362274 q^{43} +1.05061 q^{45} -4.60743 q^{47} -6.52310 q^{49} +10.6421 q^{51} +7.06143 q^{53} -4.32830 q^{55} -15.3382 q^{57} -2.07151 q^{59} -13.1947 q^{61} -0.725523 q^{63} -3.30482 q^{65} -2.75484 q^{67} +3.23710 q^{69} -2.32246 q^{71} -1.29733 q^{73} +2.01261 q^{75} +2.98902 q^{77} -5.01968 q^{79} -11.0480 q^{81} -10.3305 q^{83} +5.28770 q^{85} -4.84772 q^{87} -1.81564 q^{89} +2.28223 q^{91} +9.45539 q^{93} -7.62102 q^{95} +5.27038 q^{97} -4.54734 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} + 8 q^{5} - 4 q^{7} + 8 q^{9} - 8 q^{11} - 4 q^{15} - 16 q^{19} - 12 q^{23} + 8 q^{25} - 16 q^{27} - 4 q^{35} - 28 q^{43} + 8 q^{45} - 20 q^{47} + 8 q^{49} - 24 q^{51} - 8 q^{55} - 16 q^{59}+ \cdots - 40 q^{99}+O(q^{100}) \) 8 * q - 4 * q^3 + 8 * q^5 - 4 * q^7 + 8 * q^9 - 8 * q^11 - 4 * q^15 - 16 * q^19 - 12 * q^23 + 8 * q^25 - 16 * q^27 - 4 * q^35 - 28 * q^43 + 8 * q^45 - 20 * q^47 + 8 * q^49 - 24 * q^51 - 8 * q^55 - 16 * q^59 - 20 * q^63 - 36 * q^67 - 16 * q^69 + 8 * q^71 - 4 * q^75 + 8 * q^79 + 8 * q^81 - 20 * q^83 - 40 * q^91 - 16 * q^95 - 40 * q^99

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