LMFDB - Embedded newform 2576.2.a.bd.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2576,2,Mod(1,2576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2576 = 2^{4} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2576.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:	$20.5694635607$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.2147108.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 9x^{3} + 17x^{2} + 16x - 27 $ x^5 - 2x^4 - 9x^3 + 17x^2 + 16x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 161)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.69017$ of defining polynomial
Character	$\chi$	$=$	2576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.269842 q^{3} -3.51109 q^{5} -1.00000 q^{7} -2.92719 q^{9} +O(q^{10})$	$q+0.269842 q^{3} -3.51109 q^{5} -1.00000 q^{7} -2.92719 q^{9} +3.78810 q^{11} +1.24125 q^{13} -0.947441 q^{15} +5.98512 q^{17} +3.38034 q^{19} -0.269842 q^{21} +1.00000 q^{23} +7.32778 q^{25} -1.59940 q^{27} -7.02088 q^{29} -6.26984 q^{31} +1.02219 q^{33} +3.51109 q^{35} +4.84066 q^{37} +0.334942 q^{39} -1.78094 q^{41} -3.02219 q^{43} +10.2776 q^{45} -3.90322 q^{47} +1.00000 q^{49} +1.61504 q^{51} -2.47403 q^{53} -13.3004 q^{55} +0.912158 q^{57} +2.89143 q^{59} -10.3518 q^{61} +2.92719 q^{63} -4.35815 q^{65} -11.4466 q^{67} +0.269842 q^{69} -2.70157 q^{71} -14.1893 q^{73} +1.97734 q^{75} -3.78810 q^{77} -3.31407 q^{79} +8.34997 q^{81} -7.02219 q^{83} -21.0143 q^{85} -1.89453 q^{87} -1.59107 q^{89} -1.24125 q^{91} -1.69187 q^{93} -11.8687 q^{95} -11.6270 q^{97} -11.0885 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{5} - 5 q^{7} + 11 q^{9}+O(q^{10}) $ 5 * q - 4 * q^5 - 5 * q^7 + 11 * q^9	$ 5 q - 4 q^{5} - 5 q^{7} + 11 q^{9} + 4 q^{11} - 6 q^{13} - 10 q^{15} - 12 q^{17} - 6 q^{19} + 5 q^{23} + 19 q^{25} - 4 q^{29} - 30 q^{31} - 22 q^{33} + 4 q^{35} + 4 q^{37} - 16 q^{39} + 6 q^{41} + 12 q^{43} - 12 q^{45} - 10 q^{47} + 5 q^{49} + 4 q^{51} + 16 q^{53} - 18 q^{55} + 6 q^{57} - 22 q^{59} - 18 q^{61} - 11 q^{63} - 26 q^{65} + 2 q^{67} - 4 q^{71} - 2 q^{73} + 30 q^{75} - 4 q^{77} - 30 q^{79} - 3 q^{81} - 8 q^{83} - 12 q^{85} + 12 q^{87} - 20 q^{89} + 6 q^{91} - 26 q^{93} - 8 q^{95} - 12 q^{97} + 8 q^{99}+O(q^{100}) $ 5 * q - 4 * q^5 - 5 * q^7 + 11 * q^9 + 4 * q^11 - 6 * q^13 - 10 * q^15 - 12 * q^17 - 6 * q^19 + 5 * q^23 + 19 * q^25 - 4 * q^29 - 30 * q^31 - 22 * q^33 + 4 * q^35 + 4 * q^37 - 16 * q^39 + 6 * q^41 + 12 * q^43 - 12 * q^45 - 10 * q^47 + 5 * q^49 + 4 * q^51 + 16 * q^53 - 18 * q^55 + 6 * q^57 - 22 * q^59 - 18 * q^61 - 11 * q^63 - 26 * q^65 + 2 * q^67 - 4 * q^71 - 2 * q^73 + 30 * q^75 - 4 * q^77 - 30 * q^79 - 3 * q^81 - 8 * q^83 - 12 * q^85 + 12 * q^87 - 20 * q^89 + 6 * q^91 - 26 * q^93 - 8 * q^95 - 12 * q^97 + 8 * 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
$3$	0.269842	0.155793	0.0778967	−	0.996961i	$-0.475180\pi$
$3$	0.269842	0.155793	0.0778967	+	0.996961i	$0.475180\pi$
$4$	0	0
$5$	−3.51109	−1.57021	−0.785104	−	0.619363i	$-0.787391\pi$
$5$	−3.51109	−1.57021	−0.785104	+	0.619363i	$0.787391\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.92719	−0.975728
$10$	0	0
$11$	3.78810	1.14215	0.571077	−	0.820896i	$-0.306526\pi$
$11$	3.78810	1.14215	0.571077	+	0.820896i	$0.306526\pi$
$12$	0	0
$13$	1.24125	0.344261	0.172131	−	0.985074i	$-0.444935\pi$
$13$	1.24125	0.344261	0.172131	+	0.985074i	$0.444935\pi$
$14$	0	0
$15$	−0.947441	−0.244628
$16$	0	0
$17$	5.98512	1.45161	0.725803	−	0.687903i	$-0.241468\pi$
$17$	5.98512	1.45161	0.725803	+	0.687903i	$0.241468\pi$
$18$	0	0
$19$	3.38034	0.775503	0.387752	−	0.921764i	$-0.373252\pi$
$19$	3.38034	0.775503	0.387752	+	0.921764i	$0.373252\pi$
$20$	0	0
$21$	−0.269842	−0.0588844
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	7.32778	1.46556
$26$	0	0
$27$	−1.59940	−0.307805
$28$	0	0
$29$	−7.02088	−1.30374	−0.651872	−	0.758329i	$-0.726016\pi$
$29$	−7.02088	−1.30374	−0.651872	+	0.758329i	$0.726016\pi$
$30$	0	0
$31$	−6.26984	−1.12610	−0.563048	−	0.826424i	$-0.690372\pi$
$31$	−6.26984	−1.12610	−0.563048	+	0.826424i	$0.690372\pi$
$32$	0	0
$33$	1.02219	0.177940
$34$	0	0
$35$	3.51109	0.593483
$36$	0	0
$37$	4.84066	0.795799	0.397899	−	0.917429i	$-0.369739\pi$
$37$	4.84066	0.795799	0.397899	+	0.917429i	$0.369739\pi$
$38$	0	0
$39$	0.334942	0.0536336
$40$	0	0
$41$	−1.78094	−0.278135	−0.139068	−	0.990283i	$-0.544411\pi$
$41$	−1.78094	−0.278135	−0.139068	+	0.990283i	$0.544411\pi$
$42$	0	0
$43$	−3.02219	−0.460879	−0.230440	−	0.973087i	$-0.574016\pi$
$43$	−3.02219	−0.460879	−0.230440	+	0.973087i	$0.574016\pi$
$44$	0	0
$45$	10.2776	1.53210
$46$	0	0
$47$	−3.90322	−0.569343	−0.284671	−	0.958625i	$-0.591884\pi$
$47$	−3.90322	−0.569343	−0.284671	+	0.958625i	$0.591884\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.61504	0.226151
$52$	0	0
$53$	−2.47403	−0.339834	−0.169917	−	0.985458i	$-0.554350\pi$
$53$	−2.47403	−0.339834	−0.169917	+	0.985458i	$0.554350\pi$
$54$	0	0
$55$	−13.3004	−1.79342
$56$	0	0
$57$	0.912158	0.120818
$58$	0	0
$59$	2.89143	0.376433	0.188216	−	0.982128i	$-0.439729\pi$
$59$	2.89143	0.376433	0.188216	+	0.982128i	$0.439729\pi$
$60$	0	0
$61$	−10.3518	−1.32541	−0.662703	−	0.748882i	$-0.730591\pi$
$61$	−10.3518	−1.32541	−0.662703	+	0.748882i	$0.730591\pi$
$62$	0	0
$63$	2.92719	0.368791
$64$	0	0
$65$	−4.35815	−0.540562
$66$	0	0
$67$	−11.4466	−1.39843	−0.699213	−	0.714913i	$-0.746466\pi$
$67$	−11.4466	−1.39843	−0.699213	+	0.714913i	$0.746466\pi$
$68$	0	0
$69$	0.269842	0.0324852
$70$	0	0
$71$	−2.70157	−0.320617	−0.160309	−	0.987067i	$-0.551249\pi$
$71$	−2.70157	−0.320617	−0.160309	+	0.987067i	$0.551249\pi$
$72$	0	0
$73$	−14.1893	−1.66073	−0.830367	−	0.557217i	$-0.811869\pi$
$73$	−14.1893	−1.66073	−0.830367	+	0.557217i	$0.811869\pi$
$74$	0	0
$75$	1.97734	0.228324
$76$	0	0
$77$	−3.78810	−0.431694
$78$	0	0
$79$	−3.31407	−0.372862	−0.186431	−	0.982468i	$-0.559692\pi$
$79$	−3.31407	−0.372862	−0.186431	+	0.982468i	$0.559692\pi$
$80$	0	0
$81$	8.34997	0.927774
$82$	0	0
$83$	−7.02219	−0.770785	−0.385393	−	0.922753i	$-0.625934\pi$
$83$	−7.02219	−0.770785	−0.385393	+	0.922753i	$0.625934\pi$
$84$	0	0
$85$	−21.0143	−2.27932
$86$	0	0
$87$	−1.89453	−0.203115
$88$	0	0
$89$	−1.59107	−0.168653	−0.0843265	−	0.996438i	$-0.526874\pi$
$89$	−1.59107	−0.168653	−0.0843265	+	0.996438i	$0.526874\pi$
$90$	0	0
$91$	−1.24125	−0.130119
$92$	0	0
$93$	−1.69187	−0.175438
$94$	0	0
$95$	−11.8687	−1.21770
$96$	0	0
$97$	−11.6270	−1.18054	−0.590270	−	0.807206i	$-0.700979\pi$
$97$	−11.6270	−1.18054	−0.590270	+	0.807206i	$0.700979\pi$
$98$	0	0
$99$	−11.0885	−1.11443
$100$	0	0
$101$	−0.366626	−0.0364806	−0.0182403	−	0.999834i	$-0.505806\pi$
$101$	−0.366626	−0.0364806	−0.0182403	+	0.999834i	$0.505806\pi$
$102$	0	0
$103$	0.474030	0.0467076	0.0233538	−	0.999727i	$-0.492566\pi$
$103$	0.474030	0.0467076	0.0233538	+	0.999727i	$0.492566\pi$
$104$	0	0
$105$	0.947441	0.0924608
$106$	0	0
$107$	5.74697	0.555580	0.277790	−	0.960642i	$-0.410398\pi$
$107$	5.74697	0.555580	0.277790	+	0.960642i	$0.410398\pi$
$108$	0	0
$109$	15.4877	1.48346	0.741728	−	0.670700i	$-0.234006\pi$
$109$	15.4877	1.48346	0.741728	+	0.670700i	$0.234006\pi$
$110$	0	0
$111$	1.30621	0.123980
$112$	0	0
$113$	−4.48250	−0.421678	−0.210839	−	0.977521i	$-0.567620\pi$
$113$	−4.48250	−0.421678	−0.210839	+	0.977521i	$0.567620\pi$
$114$	0	0
$115$	−3.51109	−0.327411
$116$	0	0
$117$	−3.63337	−0.335906
$118$	0	0
$119$	−5.98512	−0.548655
$120$	0	0
$121$	3.34968	0.304516
$122$	0	0
$123$	−0.480572	−0.0433317
$124$	0	0
$125$	−8.17306	−0.731021
$126$	0	0
$127$	2.13061	0.189061	0.0945307	−	0.995522i	$-0.469865\pi$
$127$	2.13061	0.189061	0.0945307	+	0.995522i	$0.469865\pi$
$128$	0	0
$129$	−0.815514	−0.0718020
$130$	0	0
$131$	18.1242	1.58352	0.791760	−	0.610832i	$-0.209165\pi$
$131$	18.1242	1.58352	0.791760	+	0.610832i	$0.209165\pi$
$132$	0	0
$133$	−3.38034	−0.293113
$134$	0	0
$135$	5.61566	0.483319
$136$	0	0
$137$	−15.3062	−1.30770	−0.653849	−	0.756625i	$-0.726847\pi$
$137$	−15.3062	−1.30770	−0.653849	+	0.756625i	$0.726847\pi$
$138$	0	0
$139$	−13.5273	−1.14737	−0.573687	−	0.819074i	$-0.694488\pi$
$139$	−13.5273	−1.14737	−0.573687	+	0.819074i	$0.694488\pi$
$140$	0	0
$141$	−1.05325	−0.0886998
$142$	0	0
$143$	4.70198	0.393200
$144$	0	0
$145$	24.6510	2.04715
$146$	0	0
$147$	0.269842	0.0222562
$148$	0	0
$149$	5.75773	0.471691	0.235846	−	0.971791i	$-0.424214\pi$
$149$	5.75773	0.471691	0.235846	+	0.971791i	$0.424214\pi$
$150$	0	0
$151$	9.57813	0.779457	0.389728	−	0.920930i	$-0.372569\pi$
$151$	9.57813	0.779457	0.389728	+	0.920930i	$0.372569\pi$
$152$	0	0
$153$	−17.5196	−1.41637
$154$	0	0
$155$	22.0140	1.76821
$156$	0	0
$157$	−5.03706	−0.402001	−0.201001	−	0.979591i	$-0.564419\pi$
$157$	−5.03706	−0.402001	−0.201001	+	0.979591i	$0.564419\pi$
$158$	0	0
$159$	−0.667598	−0.0529439
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	7.35713	0.576255	0.288127	−	0.957592i	$-0.406967\pi$
$163$	7.35713	0.576255	0.288127	+	0.957592i	$0.406967\pi$
$164$	0	0
$165$	−3.58900	−0.279403
$166$	0	0
$167$	−20.9904	−1.62428	−0.812142	−	0.583460i	$-0.801698\pi$
$167$	−20.9904	−1.62428	−0.812142	+	0.583460i	$0.801698\pi$
$168$	0	0
$169$	−11.4593	−0.881484
$170$	0	0
$171$	−9.89488	−0.756681
$172$	0	0
$173$	−12.3369	−0.937955	−0.468978	−	0.883210i	$-0.655378\pi$
$173$	−12.3369	−0.937955	−0.468978	+	0.883210i	$0.655378\pi$
$174$	0	0
$175$	−7.32778	−0.553928
$176$	0	0
$177$	0.780231	0.0586457
$178$	0	0
$179$	12.9082	0.964807	0.482404	−	0.875949i	$-0.339764\pi$
$179$	12.9082	0.964807	0.482404	+	0.875949i	$0.339764\pi$
$180$	0	0
$181$	−5.86925	−0.436258	−0.218129	−	0.975920i	$-0.569995\pi$
$181$	−5.86925	−0.436258	−0.218129	+	0.975920i	$0.569995\pi$
$182$	0	0
$183$	−2.79334	−0.206489
$184$	0	0
$185$	−16.9960	−1.24957
$186$	0	0
$187$	22.6722	1.65796
$188$	0	0
$189$	1.59940	0.116340
$190$	0	0
$191$	−11.9343	−0.863539	−0.431769	−	0.901984i	$-0.642111\pi$
$191$	−11.9343	−0.863539	−0.431769	+	0.901984i	$0.642111\pi$
$192$	0	0
$193$	0.752900	0.0541949	0.0270975	−	0.999633i	$-0.491374\pi$
$193$	0.752900	0.0541949	0.0270975	+	0.999633i	$0.491374\pi$
$194$	0	0
$195$	−1.17601	−0.0842160
$196$	0	0
$197$	25.4074	1.81020	0.905100	−	0.425200i	$-0.139796\pi$
$197$	25.4074	1.81020	0.905100	+	0.425200i	$0.139796\pi$
$198$	0	0
$199$	−15.1988	−1.07741	−0.538707	−	0.842493i	$-0.681087\pi$
$199$	−15.1988	−1.07741	−0.538707	+	0.842493i	$0.681087\pi$
$200$	0	0
$201$	−3.08878	−0.217866
$202$	0	0
$203$	7.02088	0.492769
$204$	0	0
$205$	6.25303	0.436731
$206$	0	0
$207$	−2.92719	−0.203453
$208$	0	0
$209$	12.8051	0.885744
$210$	0	0
$211$	24.2797	1.67148	0.835742	−	0.549123i	$-0.185038\pi$
$211$	24.2797	1.67148	0.835742	+	0.549123i	$0.185038\pi$
$212$	0	0
$213$	−0.728997	−0.0499500
$214$	0	0
$215$	10.6112	0.723677
$216$	0	0
$217$	6.26984	0.425625
$218$	0	0
$219$	−3.82887	−0.258731
$220$	0	0
$221$	7.42905	0.499732
$222$	0	0
$223$	−17.7344	−1.18758	−0.593791	−	0.804619i	$-0.702369\pi$
$223$	−17.7344	−1.18758	−0.593791	+	0.804619i	$0.702369\pi$
$224$	0	0
$225$	−21.4498	−1.42998
$226$	0	0
$227$	15.7087	1.04263	0.521313	−	0.853366i	$-0.325442\pi$
$227$	15.7087	1.04263	0.521313	+	0.853366i	$0.325442\pi$
$228$	0	0
$229$	−4.51633	−0.298448	−0.149224	−	0.988803i	$-0.547678\pi$
$229$	−4.51633	−0.298448	−0.149224	+	0.988803i	$0.547678\pi$
$230$	0	0
$231$	−1.02219	−0.0672550
$232$	0	0
$233$	−16.2641	−1.06549	−0.532747	−	0.846275i	$-0.678840\pi$
$233$	−16.2641	−1.06549	−0.532747	+	0.846275i	$0.678840\pi$
$234$	0	0
$235$	13.7046	0.893987
$236$	0	0
$237$	−0.894275	−0.0580894
$238$	0	0
$239$	−2.14756	−0.138914	−0.0694571	−	0.997585i	$-0.522127\pi$
$239$	−2.14756	−0.138914	−0.0694571	+	0.997585i	$0.522127\pi$
$240$	0	0
$241$	12.4676	0.803111	0.401555	−	0.915835i	$-0.368470\pi$
$241$	12.4676	0.803111	0.401555	+	0.915835i	$0.368470\pi$
$242$	0	0
$243$	7.05139	0.452347
$244$	0	0
$245$	−3.51109	−0.224316
$246$	0	0
$247$	4.19585	0.266976
$248$	0	0
$249$	−1.89488	−0.120083
$250$	0	0
$251$	−4.91478	−0.310218	−0.155109	−	0.987897i	$-0.549573\pi$
$251$	−4.91478	−0.310218	−0.155109	+	0.987897i	$0.549573\pi$
$252$	0	0
$253$	3.78810	0.238156
$254$	0	0
$255$	−5.67055	−0.355104
$256$	0	0
$257$	28.4782	1.77642	0.888212	−	0.459433i	$-0.151948\pi$
$257$	28.4782	1.77642	0.888212	+	0.459433i	$0.151948\pi$
$258$	0	0
$259$	−4.84066	−0.300784
$260$	0	0
$261$	20.5514	1.27210
$262$	0	0
$263$	−1.89322	−0.116741	−0.0583703	−	0.998295i	$-0.518590\pi$
$263$	−1.89322	−0.116741	−0.0583703	+	0.998295i	$0.518590\pi$
$264$	0	0
$265$	8.68655	0.533611
$266$	0	0
$267$	−0.429338	−0.0262750
$268$	0	0
$269$	−2.51156	−0.153133	−0.0765663	−	0.997064i	$-0.524396\pi$
$269$	−2.51156	−0.153133	−0.0765663	+	0.997064i	$0.524396\pi$
$270$	0	0
$271$	−25.1098	−1.52531	−0.762656	−	0.646804i	$-0.776105\pi$
$271$	−25.1098	−1.52531	−0.762656	+	0.646804i	$0.776105\pi$
$272$	0	0
$273$	−0.334942	−0.0202716
$274$	0	0
$275$	27.7583	1.67389
$276$	0	0
$277$	9.48643	0.569984	0.284992	−	0.958530i	$-0.408009\pi$
$277$	9.48643	0.569984	0.284992	+	0.958530i	$0.408009\pi$
$278$	0	0
$279$	18.3530	1.09876
$280$	0	0
$281$	8.71197	0.519713	0.259856	−	0.965647i	$-0.416325\pi$
$281$	8.71197	0.519713	0.259856	+	0.965647i	$0.416325\pi$
$282$	0	0
$283$	−30.5543	−1.81627	−0.908133	−	0.418683i	$-0.862492\pi$
$283$	−30.5543	−1.81627	−0.908133	+	0.418683i	$0.862492\pi$
$284$	0	0
$285$	−3.20267	−0.189710
$286$	0	0
$287$	1.78094	0.105125
$288$	0	0
$289$	18.8217	1.10716
$290$	0	0
$291$	−3.13745	−0.183920
$292$	0	0
$293$	12.7063	0.742312	0.371156	−	0.928570i	$-0.378961\pi$
$293$	12.7063	0.742312	0.371156	+	0.928570i	$0.378961\pi$
$294$	0	0
$295$	−10.1521	−0.591078
$296$	0	0
$297$	−6.05870	−0.351561
$298$	0	0
$299$	1.24125	0.0717835
$300$	0	0
$301$	3.02219	0.174196
$302$	0	0
$303$	−0.0989310	−0.00568344
$304$	0	0
$305$	36.3460	2.08116
$306$	0	0
$307$	−22.1153	−1.26219	−0.631094	−	0.775706i	$-0.717394\pi$
$307$	−22.1153	−1.26219	−0.631094	+	0.775706i	$0.717394\pi$
$308$	0	0
$309$	0.127913	0.00727673
$310$	0	0
$311$	−21.3945	−1.21317	−0.606586	−	0.795018i	$-0.707461\pi$
$311$	−21.3945	−1.21317	−0.606586	+	0.795018i	$0.707461\pi$
$312$	0	0
$313$	−29.9051	−1.69034	−0.845169	−	0.534498i	$-0.820501\pi$
$313$	−29.9051	−1.69034	−0.845169	+	0.534498i	$0.820501\pi$
$314$	0	0
$315$	−10.2776	−0.579078
$316$	0	0
$317$	7.06103	0.396587	0.198294	−	0.980143i	$-0.436460\pi$
$317$	7.06103	0.396587	0.198294	+	0.980143i	$0.436460\pi$
$318$	0	0
$319$	−26.5958	−1.48908
$320$	0	0
$321$	1.55077	0.0865557
$322$	0	0
$323$	20.2318	1.12573
$324$	0	0
$325$	9.09562	0.504534
$326$	0	0
$327$	4.17925	0.231113
$328$	0	0
$329$	3.90322	0.215191
$330$	0	0
$331$	−16.2063	−0.890777	−0.445388	−	0.895338i	$-0.646934\pi$
$331$	−16.2063	−0.890777	−0.445388	+	0.895338i	$0.646934\pi$
$332$	0	0
$333$	−14.1695	−0.776484
$334$	0	0
$335$	40.1901	2.19582
$336$	0	0
$337$	5.21804	0.284245	0.142122	−	0.989849i	$-0.454607\pi$
$337$	5.21804	0.284245	0.142122	+	0.989849i	$0.454607\pi$
$338$	0	0
$339$	−1.20957	−0.0656947
$340$	0	0
$341$	−23.7508	−1.28618
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−0.947441	−0.0510085
$346$	0	0
$347$	−4.19094	−0.224982	−0.112491	−	0.993653i	$-0.535883\pi$
$347$	−4.19094	−0.224982	−0.112491	+	0.993653i	$0.535883\pi$
$348$	0	0
$349$	−18.6611	−0.998903	−0.499452	−	0.866342i	$-0.666465\pi$
$349$	−18.6611	−0.998903	−0.499452	+	0.866342i	$0.666465\pi$
$350$	0	0
$351$	−1.98526	−0.105966
$352$	0	0
$353$	−11.5781	−0.616241	−0.308121	−	0.951347i	$-0.599700\pi$
$353$	−11.5781	−0.616241	−0.308121	+	0.951347i	$0.599700\pi$
$354$	0	0
$355$	9.48546	0.503436
$356$	0	0
$357$	−1.61504	−0.0854769
$358$	0	0
$359$	−29.4910	−1.55647	−0.778237	−	0.627970i	$-0.783886\pi$
$359$	−29.4910	−1.55647	−0.778237	+	0.627970i	$0.783886\pi$
$360$	0	0
$361$	−7.57330	−0.398595
$362$	0	0
$363$	0.903884	0.0474416
$364$	0	0
$365$	49.8200	2.60770
$366$	0	0
$367$	24.7858	1.29381	0.646903	−	0.762572i	$-0.276064\pi$
$367$	24.7858	1.29381	0.646903	+	0.762572i	$0.276064\pi$
$368$	0	0
$369$	5.21313	0.271385
$370$	0	0
$371$	2.47403	0.128445
$372$	0	0
$373$	17.1524	0.888117	0.444058	−	0.895998i	$-0.353538\pi$
$373$	17.1524	0.888117	0.444058	+	0.895998i	$0.353538\pi$
$374$	0	0
$375$	−2.20544	−0.113888
$376$	0	0
$377$	−8.71467	−0.448829
$378$	0	0
$379$	−32.8211	−1.68591	−0.842953	−	0.537986i	$-0.819185\pi$
$379$	−32.8211	−1.68591	−0.842953	+	0.537986i	$0.819185\pi$
$380$	0	0
$381$	0.574930	0.0294545
$382$	0	0
$383$	8.42609	0.430553	0.215277	−	0.976553i	$-0.430935\pi$
$383$	8.42609	0.430553	0.215277	+	0.976553i	$0.430935\pi$
$384$	0	0
$385$	13.3004	0.677849
$386$	0	0
$387$	8.84650	0.449693
$388$	0	0
$389$	−25.3004	−1.28278	−0.641390	−	0.767215i	$-0.721642\pi$
$389$	−25.3004	−1.28278	−0.641390	+	0.767215i	$0.721642\pi$
$390$	0	0
$391$	5.98512	0.302681
$392$	0	0
$393$	4.89068	0.246702
$394$	0	0
$395$	11.6360	0.585471
$396$	0	0
$397$	20.3650	1.02209	0.511045	−	0.859554i	$-0.329259\pi$
$397$	20.3650	1.02209	0.511045	+	0.859554i	$0.329259\pi$
$398$	0	0
$399$	−0.912158	−0.0456650
$400$	0	0
$401$	4.10277	0.204883	0.102441	−	0.994739i	$-0.467335\pi$
$401$	4.10277	0.204883	0.102441	+	0.994739i	$0.467335\pi$
$402$	0	0
$403$	−7.78245	−0.387672
$404$	0	0
$405$	−29.3175	−1.45680
$406$	0	0
$407$	18.3369	0.908925
$408$	0	0
$409$	−26.8449	−1.32739	−0.663697	−	0.748002i	$-0.731013\pi$
$409$	−26.8449	−1.32739	−0.663697	+	0.748002i	$0.731013\pi$
$410$	0	0
$411$	−4.13026	−0.203731
$412$	0	0
$413$	−2.89143	−0.142278
$414$	0	0
$415$	24.6556	1.21029
$416$	0	0
$417$	−3.65025	−0.178753
$418$	0	0
$419$	−0.149494	−0.00730327	−0.00365163	−	0.999993i	$-0.501162\pi$
$419$	−0.149494	−0.00730327	−0.00365163	+	0.999993i	$0.501162\pi$
$420$	0	0
$421$	−10.5269	−0.513049	−0.256524	−	0.966538i	$-0.582577\pi$
$421$	−10.5269	−0.513049	−0.256524	+	0.966538i	$0.582577\pi$
$422$	0	0
$423$	11.4254	0.555524
$424$	0	0
$425$	43.8577	2.12741
$426$	0	0
$427$	10.3518	0.500956
$428$	0	0
$429$	1.26879	0.0612579
$430$	0	0
$431$	2.28665	0.110144	0.0550720	−	0.998482i	$-0.482461\pi$
$431$	2.28665	0.110144	0.0550720	+	0.998482i	$0.482461\pi$
$432$	0	0
$433$	−20.4569	−0.983094	−0.491547	−	0.870851i	$-0.663568\pi$
$433$	−20.4569	−0.983094	−0.491547	+	0.870851i	$0.663568\pi$
$434$	0	0
$435$	6.65186	0.318933
$436$	0	0
$437$	3.38034	0.161704
$438$	0	0
$439$	−8.93821	−0.426597	−0.213299	−	0.976987i	$-0.568421\pi$
$439$	−8.93821	−0.426597	−0.213299	+	0.976987i	$0.568421\pi$
$440$	0	0
$441$	−2.92719	−0.139390
$442$	0	0
$443$	6.71589	0.319082	0.159541	−	0.987191i	$-0.448999\pi$
$443$	6.71589	0.319082	0.159541	+	0.987191i	$0.448999\pi$
$444$	0	0
$445$	5.58640	0.264821
$446$	0	0
$447$	1.55368	0.0734864
$448$	0	0
$449$	39.7178	1.87440	0.937200	−	0.348792i	$-0.113408\pi$
$449$	39.7178	1.87440	0.937200	+	0.348792i	$0.113408\pi$
$450$	0	0
$451$	−6.74636	−0.317674
$452$	0	0
$453$	2.58458	0.121434
$454$	0	0
$455$	4.35815	0.204313
$456$	0	0
$457$	−35.7388	−1.67179	−0.835895	−	0.548889i	$-0.815051\pi$
$457$	−35.7388	−1.67179	−0.835895	+	0.548889i	$0.815051\pi$
$458$	0	0
$459$	−9.57263	−0.446812
$460$	0	0
$461$	−18.3516	−0.854720	−0.427360	−	0.904082i	$-0.640556\pi$
$461$	−18.3516	−0.854720	−0.427360	+	0.904082i	$0.640556\pi$
$462$	0	0
$463$	−11.1821	−0.519678	−0.259839	−	0.965652i	$-0.583670\pi$
$463$	−11.1821	−0.519678	−0.259839	+	0.965652i	$0.583670\pi$
$464$	0	0
$465$	5.94031	0.275475
$466$	0	0
$467$	−31.7231	−1.46797	−0.733984	−	0.679167i	$-0.762341\pi$
$467$	−31.7231	−1.46797	−0.733984	+	0.679167i	$0.762341\pi$
$468$	0	0
$469$	11.4466	0.528556
$470$	0	0
$471$	−1.35921	−0.0626292
$472$	0	0
$473$	−11.4483	−0.526395
$474$	0	0
$475$	24.7704	1.13654
$476$	0	0
$477$	7.24194	0.331586
$478$	0	0
$479$	9.81725	0.448562	0.224281	−	0.974525i	$-0.427997\pi$
$479$	9.81725	0.448562	0.224281	+	0.974525i	$0.427997\pi$
$480$	0	0
$481$	6.00847	0.273963
$482$	0	0
$483$	−0.269842	−0.0122782
$484$	0	0
$485$	40.8234	1.85369
$486$	0	0
$487$	36.2778	1.64390	0.821952	−	0.569557i	$-0.192885\pi$
$487$	36.2778	1.64390	0.821952	+	0.569557i	$0.192885\pi$
$488$	0	0
$489$	1.98526	0.0897767
$490$	0	0
$491$	17.6053	0.794514	0.397257	−	0.917707i	$-0.369962\pi$
$491$	17.6053	0.794514	0.397257	+	0.917707i	$0.369962\pi$
$492$	0	0
$493$	−42.0208	−1.89252
$494$	0	0
$495$	38.9326	1.74989
$496$	0	0
$497$	2.70157	0.121182
$498$	0	0
$499$	−12.1142	−0.542308	−0.271154	−	0.962536i	$-0.587405\pi$
$499$	−12.1142	−0.542308	−0.271154	+	0.962536i	$0.587405\pi$
$500$	0	0
$501$	−5.66408	−0.253053
$502$	0	0
$503$	−10.2723	−0.458020	−0.229010	−	0.973424i	$-0.573549\pi$
$503$	−10.2723	−0.458020	−0.229010	+	0.973424i	$0.573549\pi$
$504$	0	0
$505$	1.28726	0.0572822
$506$	0	0
$507$	−3.09220	−0.137329
$508$	0	0
$509$	−6.45082	−0.285928	−0.142964	−	0.989728i	$-0.545663\pi$
$509$	−6.45082	−0.285928	−0.142964	+	0.989728i	$0.545663\pi$
$510$	0	0
$511$	14.1893	0.627698
$512$	0	0
$513$	−5.40653	−0.238704
$514$	0	0
$515$	−1.66436	−0.0733407
$516$	0	0
$517$	−14.7858	−0.650277
$518$	0	0
$519$	−3.32901	−0.146127
$520$	0	0
$521$	34.4858	1.51085	0.755425	−	0.655235i	$-0.227430\pi$
$521$	34.4858	1.51085	0.755425	+	0.655235i	$0.227430\pi$
$522$	0	0
$523$	25.7852	1.12751	0.563753	−	0.825943i	$-0.309357\pi$
$523$	25.7852	1.12751	0.563753	+	0.825943i	$0.309357\pi$
$524$	0	0
$525$	−1.97734	−0.0862984
$526$	0	0
$527$	−37.5258	−1.63465
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−8.46376	−0.367296
$532$	0	0
$533$	−2.21059	−0.0957513
$534$	0	0
$535$	−20.1781	−0.872377
$536$	0	0
$537$	3.48319	0.150311
$538$	0	0
$539$	3.78810	0.163165
$540$	0	0
$541$	−21.1475	−0.909203	−0.454601	−	0.890695i	$-0.650218\pi$
$541$	−21.1475	−0.909203	−0.454601	+	0.890695i	$0.650218\pi$
$542$	0	0
$543$	−1.58377	−0.0679661
$544$	0	0
$545$	−54.3789	−2.32934
$546$	0	0
$547$	3.97188	0.169825	0.0849126	−	0.996388i	$-0.472939\pi$
$547$	3.97188	0.169825	0.0849126	+	0.996388i	$0.472939\pi$
$548$	0	0
$549$	30.3015	1.29324
$550$	0	0
$551$	−23.7329	−1.01106
$552$	0	0
$553$	3.31407	0.140928
$554$	0	0
$555$	−4.58624	−0.194675
$556$	0	0
$557$	−13.5585	−0.574491	−0.287246	−	0.957857i	$-0.592740\pi$
$557$	−13.5585	−0.574491	−0.287246	+	0.957857i	$0.592740\pi$
$558$	0	0
$559$	−3.75130	−0.158663
$560$	0	0
$561$	6.11792	0.258299
$562$	0	0
$563$	32.4740	1.36862	0.684309	−	0.729193i	$-0.260104\pi$
$563$	32.4740	1.36862	0.684309	+	0.729193i	$0.260104\pi$
$564$	0	0
$565$	15.7385	0.662123
$566$	0	0
$567$	−8.34997	−0.350666
$568$	0	0
$569$	−26.7707	−1.12228	−0.561142	−	0.827719i	$-0.689638\pi$
$569$	−26.7707	−1.12228	−0.561142	+	0.827719i	$0.689638\pi$
$570$	0	0
$571$	−5.30884	−0.222168	−0.111084	−	0.993811i	$-0.535432\pi$
$571$	−5.30884	−0.222168	−0.111084	+	0.993811i	$0.535432\pi$
$572$	0	0
$573$	−3.22039	−0.134534
$574$	0	0
$575$	7.32778	0.305590
$576$	0	0
$577$	43.9737	1.83065	0.915325	−	0.402717i	$-0.131934\pi$
$577$	43.9737	1.83065	0.915325	+	0.402717i	$0.131934\pi$
$578$	0	0
$579$	0.203164	0.00844321
$580$	0	0
$581$	7.02219	0.291329
$582$	0	0
$583$	−9.37187	−0.388143
$584$	0	0
$585$	12.7571	0.527442
$586$	0	0
$587$	34.5314	1.42526	0.712631	−	0.701539i	$-0.247503\pi$
$587$	34.5314	1.42526	0.712631	+	0.701539i	$0.247503\pi$
$588$	0	0
$589$	−21.1942	−0.873292
$590$	0	0
$591$	6.85597	0.282017
$592$	0	0
$593$	2.06074	0.0846245	0.0423123	−	0.999104i	$-0.486528\pi$
$593$	2.06074	0.0846245	0.0423123	+	0.999104i	$0.486528\pi$
$594$	0	0
$595$	21.0143	0.861504
$596$	0	0
$597$	−4.10128	−0.167854
$598$	0	0
$599$	28.2627	1.15478	0.577392	−	0.816467i	$-0.304070\pi$
$599$	28.2627	1.15478	0.577392	+	0.816467i	$0.304070\pi$
$600$	0	0
$601$	−25.0036	−1.01992	−0.509959	−	0.860199i	$-0.670340\pi$
$601$	−25.0036	−1.01992	−0.509959	+	0.860199i	$0.670340\pi$
$602$	0	0
$603$	33.5064	1.36448
$604$	0	0
$605$	−11.7610	−0.478154
$606$	0	0
$607$	29.0255	1.17811	0.589054	−	0.808093i	$-0.299500\pi$
$607$	29.0255	1.17811	0.589054	+	0.808093i	$0.299500\pi$
$608$	0	0
$609$	1.89453	0.0767701
$610$	0	0
$611$	−4.84487	−0.196003
$612$	0	0
$613$	−25.2180	−1.01855	−0.509274	−	0.860605i	$-0.670086\pi$
$613$	−25.2180	−1.01855	−0.509274	+	0.860605i	$0.670086\pi$
$614$	0	0
$615$	1.68733	0.0680398
$616$	0	0
$617$	41.7462	1.68064	0.840319	−	0.542093i	$-0.182368\pi$
$617$	41.7462	1.68064	0.840319	+	0.542093i	$0.182368\pi$
$618$	0	0
$619$	0.503782	0.0202487	0.0101243	−	0.999949i	$-0.496777\pi$
$619$	0.503782	0.0202487	0.0101243	+	0.999949i	$0.496777\pi$
$620$	0	0
$621$	−1.59940	−0.0641819
$622$	0	0
$623$	1.59107	0.0637449
$624$	0	0
$625$	−7.94253	−0.317701
$626$	0	0
$627$	3.45534	0.137993
$628$	0	0
$629$	28.9719	1.15519
$630$	0	0
$631$	17.6868	0.704102	0.352051	−	0.935981i	$-0.385484\pi$
$631$	17.6868	0.704102	0.352051	+	0.935981i	$0.385484\pi$
$632$	0	0
$633$	6.55168	0.260406
$634$	0	0
$635$	−7.48079	−0.296866
$636$	0	0
$637$	1.24125	0.0491802
$638$	0	0
$639$	7.90799	0.312835
$640$	0	0
$641$	−37.4714	−1.48003	−0.740015	−	0.672590i	$-0.765182\pi$
$641$	−37.4714	−1.48003	−0.740015	+	0.672590i	$0.765182\pi$
$642$	0	0
$643$	−0.178907	−0.00705539	−0.00352770	−	0.999994i	$-0.501123\pi$
$643$	−0.178907	−0.00705539	−0.00352770	+	0.999994i	$0.501123\pi$
$644$	0	0
$645$	2.86334	0.112744
$646$	0	0
$647$	−2.94235	−0.115676	−0.0578379	−	0.998326i	$-0.518421\pi$
$647$	−2.94235	−0.115676	−0.0578379	+	0.998326i	$0.518421\pi$
$648$	0	0
$649$	10.9530	0.429944
$650$	0	0
$651$	1.69187	0.0663095
$652$	0	0
$653$	44.1719	1.72858	0.864290	−	0.502994i	$-0.167768\pi$
$653$	44.1719	1.72858	0.864290	+	0.502994i	$0.167768\pi$
$654$	0	0
$655$	−63.6358	−2.48646
$656$	0	0
$657$	41.5347	1.62042
$658$	0	0
$659$	−13.9282	−0.542564	−0.271282	−	0.962500i	$-0.587448\pi$
$659$	−13.9282	−0.542564	−0.271282	+	0.962500i	$0.587448\pi$
$660$	0	0
$661$	−2.15327	−0.0837525	−0.0418763	−	0.999123i	$-0.513334\pi$
$661$	−2.15327	−0.0837525	−0.0418763	+	0.999123i	$0.513334\pi$
$662$	0	0
$663$	2.00467	0.0778549
$664$	0	0
$665$	11.8687	0.460248
$666$	0	0
$667$	−7.02088	−0.271849
$668$	0	0
$669$	−4.78548	−0.185017
$670$	0	0
$671$	−39.2134	−1.51382
$672$	0	0
$673$	−2.96276	−0.114206	−0.0571030	−	0.998368i	$-0.518186\pi$
$673$	−2.96276	−0.114206	−0.0571030	+	0.998368i	$0.518186\pi$
$674$	0	0
$675$	−11.7201	−0.451106
$676$	0	0
$677$	9.59927	0.368930	0.184465	−	0.982839i	$-0.440945\pi$
$677$	9.59927	0.368930	0.184465	+	0.982839i	$0.440945\pi$
$678$	0	0
$679$	11.6270	0.446202
$680$	0	0
$681$	4.23888	0.162434
$682$	0	0
$683$	24.1908	0.925636	0.462818	−	0.886453i	$-0.346838\pi$
$683$	24.1908	0.925636	0.462818	+	0.886453i	$0.346838\pi$
$684$	0	0
$685$	53.7416	2.05336
$686$	0	0
$687$	−1.21870	−0.0464962
$688$	0	0
$689$	−3.07089	−0.116992
$690$	0	0
$691$	−30.4784	−1.15945	−0.579726	−	0.814811i	$-0.696841\pi$
$691$	−30.4784	−1.15945	−0.579726	+	0.814811i	$0.696841\pi$
$692$	0	0
$693$	11.0885	0.421216
$694$	0	0
$695$	47.4958	1.80162
$696$	0	0
$697$	−10.6591	−0.403743
$698$	0	0
$699$	−4.38873	−0.165997
$700$	0	0
$701$	17.4101	0.657570	0.328785	−	0.944405i	$-0.393361\pi$
$701$	17.4101	0.657570	0.328785	+	0.944405i	$0.393361\pi$
$702$	0	0
$703$	16.3631	0.617145
$704$	0	0
$705$	3.69807	0.139277
$706$	0	0
$707$	0.366626	0.0137884
$708$	0	0
$709$	13.6465	0.512504	0.256252	−	0.966610i	$-0.417512\pi$
$709$	13.6465	0.512504	0.256252	+	0.966610i	$0.417512\pi$
$710$	0	0
$711$	9.70089	0.363812
$712$	0	0
$713$	−6.26984	−0.234807
$714$	0	0
$715$	−16.5091	−0.617405
$716$	0	0
$717$	−0.579503	−0.0216419
$718$	0	0
$719$	29.8264	1.11234	0.556169	−	0.831069i	$-0.312271\pi$
$719$	29.8264	1.11234	0.556169	+	0.831069i	$0.312271\pi$
$720$	0	0
$721$	−0.474030	−0.0176538
$722$	0	0
$723$	3.36429	0.125119
$724$	0	0
$725$	−51.4474	−1.91071
$726$	0	0
$727$	−6.92789	−0.256941	−0.128471	−	0.991713i	$-0.541007\pi$
$727$	−6.92789	−0.256941	−0.128471	+	0.991713i	$0.541007\pi$
$728$	0	0
$729$	−23.1471	−0.857302
$730$	0	0
$731$	−18.0882	−0.669015
$732$	0	0
$733$	−10.8092	−0.399246	−0.199623	−	0.979873i	$-0.563972\pi$
$733$	−10.8092	−0.399246	−0.199623	+	0.979873i	$0.563972\pi$
$734$	0	0
$735$	−0.947441	−0.0349469
$736$	0	0
$737$	−43.3609	−1.59722
$738$	0	0
$739$	−34.5455	−1.27078	−0.635388	−	0.772193i	$-0.719160\pi$
$739$	−34.5455	−1.27078	−0.635388	+	0.772193i	$0.719160\pi$
$740$	0	0
$741$	1.13222	0.0415931
$742$	0	0
$743$	−19.7750	−0.725475	−0.362737	−	0.931891i	$-0.618158\pi$
$743$	−19.7750	−0.725475	−0.362737	+	0.931891i	$0.618158\pi$
$744$	0	0
$745$	−20.2159	−0.740654
$746$	0	0
$747$	20.5552	0.752077
$748$	0	0
$749$	−5.74697	−0.209990
$750$	0	0
$751$	31.0704	1.13378	0.566888	−	0.823795i	$-0.308147\pi$
$751$	31.0704	1.13378	0.566888	+	0.823795i	$0.308147\pi$
$752$	0	0
$753$	−1.32622	−0.0483300
$754$	0	0
$755$	−33.6297	−1.22391
$756$	0	0
$757$	8.32104	0.302433	0.151217	−	0.988501i	$-0.451681\pi$
$757$	8.32104	0.302433	0.151217	+	0.988501i	$0.451681\pi$
$758$	0	0
$759$	1.02219	0.0371031
$760$	0	0
$761$	−3.82034	−0.138487	−0.0692436	−	0.997600i	$-0.522059\pi$
$761$	−3.82034	−0.138487	−0.0692436	+	0.997600i	$0.522059\pi$
$762$	0	0
$763$	−15.4877	−0.560694
$764$	0	0
$765$	61.5128	2.22400
$766$	0	0
$767$	3.58900	0.129591
$768$	0	0
$769$	8.05891	0.290612	0.145306	−	0.989387i	$-0.453583\pi$
$769$	8.05891	0.290612	0.145306	+	0.989387i	$0.453583\pi$
$770$	0	0
$771$	7.68463	0.276755
$772$	0	0
$773$	−20.3254	−0.731054	−0.365527	−	0.930801i	$-0.619111\pi$
$773$	−20.3254	−0.731054	−0.365527	+	0.930801i	$0.619111\pi$
$774$	0	0
$775$	−45.9440	−1.65036
$776$	0	0
$777$	−1.30621	−0.0468601
$778$	0	0
$779$	−6.02017	−0.215695
$780$	0	0
$781$	−10.2338	−0.366194
$782$	0	0
$783$	11.2292	0.401299
$784$	0	0
$785$	17.6856	0.631226
$786$	0	0
$787$	47.7951	1.70371	0.851856	−	0.523775i	$-0.175477\pi$
$787$	47.7951	1.70371	0.851856	+	0.523775i	$0.175477\pi$
$788$	0	0
$789$	−0.510869	−0.0181874
$790$	0	0
$791$	4.48250	0.159379
$792$	0	0
$793$	−12.8491	−0.456286
$794$	0	0
$795$	2.34400	0.0831330
$796$	0	0
$797$	−7.57090	−0.268175	−0.134088	−	0.990969i	$-0.542810\pi$
$797$	−7.57090	−0.268175	−0.134088	+	0.990969i	$0.542810\pi$
$798$	0	0
$799$	−23.3612	−0.826461
$800$	0	0
$801$	4.65736	0.164560
$802$	0	0
$803$	−53.7505	−1.89681
$804$	0	0
$805$	3.51109	0.123750
$806$	0	0
$807$	−0.677725	−0.0238571
$808$	0	0
$809$	8.55279	0.300700	0.150350	−	0.988633i	$-0.451960\pi$
$809$	8.55279	0.300700	0.150350	+	0.988633i	$0.451960\pi$
$810$	0	0
$811$	45.0023	1.58024	0.790122	−	0.612950i	$-0.210017\pi$
$811$	45.0023	1.58024	0.790122	+	0.612950i	$0.210017\pi$
$812$	0	0
$813$	−6.77568	−0.237634
$814$	0	0
$815$	−25.8316	−0.904841
$816$	0	0
$817$	−10.2160	−0.357413
$818$	0	0
$819$	3.63337	0.126960
$820$	0	0
$821$	41.4358	1.44612	0.723060	−	0.690785i	$-0.242735\pi$
$821$	41.4358	1.44612	0.723060	+	0.690785i	$0.242735\pi$
$822$	0	0
$823$	19.1508	0.667553	0.333777	−	0.942652i	$-0.391677\pi$
$823$	19.1508	0.667553	0.333777	+	0.942652i	$0.391677\pi$
$824$	0	0
$825$	7.49037	0.260781
$826$	0	0
$827$	32.2820	1.12256	0.561278	−	0.827627i	$-0.310310\pi$
$827$	32.2820	1.12256	0.561278	+	0.827627i	$0.310310\pi$
$828$	0	0
$829$	9.85700	0.342348	0.171174	−	0.985241i	$-0.445244\pi$
$829$	9.85700	0.342348	0.171174	+	0.985241i	$0.445244\pi$
$830$	0	0
$831$	2.55984	0.0887998
$832$	0	0
$833$	5.98512	0.207372
$834$	0	0
$835$	73.6991	2.55046
$836$	0	0
$837$	10.0280	0.346619
$838$	0	0
$839$	−11.7098	−0.404267	−0.202133	−	0.979358i	$-0.564787\pi$
$839$	−11.7098	−0.404267	−0.202133	+	0.979358i	$0.564787\pi$
$840$	0	0
$841$	20.2927	0.699748
$842$	0	0
$843$	2.35086	0.0809678
$844$	0	0
$845$	40.2347	1.38411
$846$	0	0
$847$	−3.34968	−0.115096
$848$	0	0
$849$	−8.24484	−0.282962
$850$	0	0
$851$	4.84066	0.165936
$852$	0	0
$853$	−29.2441	−1.00130	−0.500650	−	0.865650i	$-0.666906\pi$
$853$	−29.2441	−1.00130	−0.500650	+	0.865650i	$0.666906\pi$
$854$	0	0
$855$	34.7419	1.18815
$856$	0	0
$857$	38.6873	1.32153	0.660766	−	0.750592i	$-0.270232\pi$
$857$	38.6873	1.32153	0.660766	+	0.750592i	$0.270232\pi$
$858$	0	0
$859$	49.3913	1.68521	0.842604	−	0.538534i	$-0.181021\pi$
$859$	49.3913	1.68521	0.842604	+	0.538534i	$0.181021\pi$
$860$	0	0
$861$	0.480572	0.0163778
$862$	0	0
$863$	2.32156	0.0790267	0.0395134	−	0.999219i	$-0.487419\pi$
$863$	2.32156	0.0790267	0.0395134	+	0.999219i	$0.487419\pi$
$864$	0	0
$865$	43.3159	1.47279
$866$	0	0
$867$	5.07889	0.172488
$868$	0	0
$869$	−12.5540	−0.425865
$870$	0	0
$871$	−14.2081	−0.481424
$872$	0	0
$873$	34.0343	1.15189
$874$	0	0
$875$	8.17306	0.276300
$876$	0	0
$877$	36.2775	1.22500	0.612502	−	0.790469i	$-0.290163\pi$
$877$	36.2775	1.22500	0.612502	+	0.790469i	$0.290163\pi$
$878$	0	0
$879$	3.42871	0.115647
$880$	0	0
$881$	30.1252	1.01494	0.507472	−	0.861668i	$-0.330580\pi$
$881$	30.1252	1.01494	0.507472	+	0.861668i	$0.330580\pi$
$882$	0	0
$883$	−1.94158	−0.0653394	−0.0326697	−	0.999466i	$-0.510401\pi$
$883$	−1.94158	−0.0653394	−0.0326697	+	0.999466i	$0.510401\pi$
$884$	0	0
$885$	−2.73946	−0.0920860
$886$	0	0
$887$	−52.3596	−1.75806	−0.879031	−	0.476765i	$-0.841809\pi$
$887$	−52.3596	−1.75806	−0.879031	+	0.476765i	$0.841809\pi$
$888$	0	0
$889$	−2.13061	−0.0714585
$890$	0	0
$891$	31.6305	1.05966
$892$	0	0
$893$	−13.1942	−0.441527
$894$	0	0
$895$	−45.3221	−1.51495
$896$	0	0
$897$	0.334942	0.0111834
$898$	0	0
$899$	44.0198	1.46814
$900$	0	0
$901$	−14.8074	−0.493305
$902$	0	0
$903$	0.815514	0.0271386
$904$	0	0
$905$	20.6075	0.685016
$906$	0	0
$907$	−9.62846	−0.319708	−0.159854	−	0.987141i	$-0.551102\pi$
$907$	−9.62846	−0.319708	−0.159854	+	0.987141i	$0.551102\pi$
$908$	0	0
$909$	1.07318	0.0355952
$910$	0	0
$911$	24.6036	0.815154	0.407577	−	0.913171i	$-0.366374\pi$
$911$	24.6036	0.815154	0.407577	+	0.913171i	$0.366374\pi$
$912$	0	0
$913$	−26.6007	−0.880356
$914$	0	0
$915$	9.80767	0.324232
$916$	0	0
$917$	−18.1242	−0.598514
$918$	0	0
$919$	−18.4685	−0.609220	−0.304610	−	0.952477i	$-0.598526\pi$
$919$	−18.4685	−0.609220	−0.304610	+	0.952477i	$0.598526\pi$
$920$	0	0
$921$	−5.96765	−0.196641
$922$	0	0
$923$	−3.35333	−0.110376
$924$	0	0
$925$	35.4713	1.16629
$926$	0	0
$927$	−1.38757	−0.0455739
$928$	0	0
$929$	21.6032	0.708778	0.354389	−	0.935098i	$-0.384689\pi$
$929$	21.6032	0.708778	0.354389	+	0.935098i	$0.384689\pi$
$930$	0	0
$931$	3.38034	0.110786
$932$	0	0
$933$	−5.77314	−0.189004
$934$	0	0
$935$	−79.6043	−2.60334
$936$	0	0
$937$	4.95446	0.161855	0.0809276	−	0.996720i	$-0.474212\pi$
$937$	4.95446	0.161855	0.0809276	+	0.996720i	$0.474212\pi$
$938$	0	0
$939$	−8.06967	−0.263344
$940$	0	0
$941$	−37.1398	−1.21072	−0.605361	−	0.795951i	$-0.706971\pi$
$941$	−37.1398	−1.21072	−0.605361	+	0.795951i	$0.706971\pi$
$942$	0	0
$943$	−1.78094	−0.0579953
$944$	0	0
$945$	−5.61566	−0.182677
$946$	0	0
$947$	11.0644	0.359543	0.179772	−	0.983708i	$-0.442464\pi$
$947$	11.0644	0.359543	0.179772	+	0.983708i	$0.442464\pi$
$948$	0	0
$949$	−17.6125	−0.571726
$950$	0	0
$951$	1.90536	0.0617857
$952$	0	0
$953$	19.3888	0.628065	0.314033	−	0.949412i	$-0.398320\pi$
$953$	19.3888	0.628065	0.314033	+	0.949412i	$0.398320\pi$
$954$	0	0
$955$	41.9026	1.35594
$956$	0	0
$957$	−7.17665	−0.231988
$958$	0	0
$959$	15.3062	0.494263
$960$	0	0
$961$	8.31092	0.268094
$962$	0	0
$963$	−16.8224	−0.542095
$964$	0	0
$965$	−2.64350	−0.0850974
$966$	0	0
$967$	32.5154	1.04563	0.522813	−	0.852448i	$-0.324883\pi$
$967$	32.5154	1.04563	0.522813	+	0.852448i	$0.324883\pi$
$968$	0	0
$969$	5.45938	0.175381
$970$	0	0
$971$	−1.32316	−0.0424622	−0.0212311	−	0.999775i	$-0.506759\pi$
$971$	−1.32316	−0.0424622	−0.0212311	+	0.999775i	$0.506759\pi$
$972$	0	0
$973$	13.5273	0.433667
$974$	0	0
$975$	2.45438	0.0786031
$976$	0	0
$977$	−31.6195	−1.01160	−0.505799	−	0.862651i	$-0.668802\pi$
$977$	−31.6195	−1.01160	−0.505799	+	0.862651i	$0.668802\pi$
$978$	0	0
$979$	−6.02713	−0.192628
$980$	0	0
$981$	−45.3355	−1.44745
$982$	0	0
$983$	12.1696	0.388149	0.194074	−	0.980987i	$-0.437830\pi$
$983$	12.1696	0.388149	0.194074	+	0.980987i	$0.437830\pi$
$984$	0	0
$985$	−89.2076	−2.84239
$986$	0	0
$987$	1.05325	0.0335254
$988$	0	0
$989$	−3.02219	−0.0961000
$990$	0	0
$991$	−2.18738	−0.0694844	−0.0347422	−	0.999396i	$-0.511061\pi$
$991$	−2.18738	−0.0694844	−0.0347422	+	0.999396i	$0.511061\pi$
$992$	0	0
$993$	−4.37313	−0.138777
$994$	0	0
$995$	53.3644	1.69177
$996$	0	0
$997$	−3.78052	−0.119730	−0.0598652	−	0.998206i	$-0.519067\pi$
$997$	−3.78052	−0.119730	−0.0598652	+	0.998206i	$0.519067\pi$
$998$	0	0
$999$	−7.74216	−0.244951

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2576.2.a.bd.1.3		5
4.3	odd	2		161.2.a.d.1.5	✓	5
12.11	even	2		1449.2.a.r.1.1		5
20.19	odd	2		4025.2.a.p.1.1		5
28.27	even	2		1127.2.a.h.1.5		5
92.91	even	2		3703.2.a.j.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.a.d.1.5	✓	5	4.3	odd	2
1127.2.a.h.1.5		5	28.27	even	2
1449.2.a.r.1.1		5	12.11	even	2
2576.2.a.bd.1.3		5	1.1	even	1	trivial
3703.2.a.j.1.5		5	92.91	even	2
4025.2.a.p.1.1		5	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2576 = 2^{4} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2576.a (trivial)

\(f(q)\)	\(=\)	\(q+0.269842 q^{3} -3.51109 q^{5} -1.00000 q^{7} -2.92719 q^{9} +O(q^{10})\)	\(q+0.269842 q^{3} -3.51109 q^{5} -1.00000 q^{7} -2.92719 q^{9} +3.78810 q^{11} +1.24125 q^{13} -0.947441 q^{15} +5.98512 q^{17} +3.38034 q^{19} -0.269842 q^{21} +1.00000 q^{23} +7.32778 q^{25} -1.59940 q^{27} -7.02088 q^{29} -6.26984 q^{31} +1.02219 q^{33} +3.51109 q^{35} +4.84066 q^{37} +0.334942 q^{39} -1.78094 q^{41} -3.02219 q^{43} +10.2776 q^{45} -3.90322 q^{47} +1.00000 q^{49} +1.61504 q^{51} -2.47403 q^{53} -13.3004 q^{55} +0.912158 q^{57} +2.89143 q^{59} -10.3518 q^{61} +2.92719 q^{63} -4.35815 q^{65} -11.4466 q^{67} +0.269842 q^{69} -2.70157 q^{71} -14.1893 q^{73} +1.97734 q^{75} -3.78810 q^{77} -3.31407 q^{79} +8.34997 q^{81} -7.02219 q^{83} -21.0143 q^{85} -1.89453 q^{87} -1.59107 q^{89} -1.24125 q^{91} -1.69187 q^{93} -11.8687 q^{95} -11.6270 q^{97} -11.0885 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{5} - 5 q^{7} + 11 q^{9}+O(q^{10}) \) 5 * q - 4 * q^5 - 5 * q^7 + 11 * q^9	\( 5 q - 4 q^{5} - 5 q^{7} + 11 q^{9} + 4 q^{11} - 6 q^{13} - 10 q^{15} - 12 q^{17} - 6 q^{19} + 5 q^{23} + 19 q^{25} - 4 q^{29} - 30 q^{31} - 22 q^{33} + 4 q^{35} + 4 q^{37} - 16 q^{39} + 6 q^{41} + 12 q^{43} - 12 q^{45} - 10 q^{47} + 5 q^{49} + 4 q^{51} + 16 q^{53} - 18 q^{55} + 6 q^{57} - 22 q^{59} - 18 q^{61} - 11 q^{63} - 26 q^{65} + 2 q^{67} - 4 q^{71} - 2 q^{73} + 30 q^{75} - 4 q^{77} - 30 q^{79} - 3 q^{81} - 8 q^{83} - 12 q^{85} + 12 q^{87} - 20 q^{89} + 6 q^{91} - 26 q^{93} - 8 q^{95} - 12 q^{97} + 8 q^{99}+O(q^{100}) \) 5 * q - 4 * q^5 - 5 * q^7 + 11 * q^9 + 4 * q^11 - 6 * q^13 - 10 * q^15 - 12 * q^17 - 6 * q^19 + 5 * q^23 + 19 * q^25 - 4 * q^29 - 30 * q^31 - 22 * q^33 + 4 * q^35 + 4 * q^37 - 16 * q^39 + 6 * q^41 + 12 * q^43 - 12 * q^45 - 10 * q^47 + 5 * q^49 + 4 * q^51 + 16 * q^53 - 18 * q^55 + 6 * q^57 - 22 * q^59 - 18 * q^61 - 11 * q^63 - 26 * q^65 + 2 * q^67 - 4 * q^71 - 2 * q^73 + 30 * q^75 - 4 * q^77 - 30 * q^79 - 3 * q^81 - 8 * q^83 - 12 * q^85 + 12 * q^87 - 20 * q^89 + 6 * q^91 - 26 * q^93 - 8 * q^95 - 12 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more