LMFDB - Embedded newform 2576.2.a.w.1.1

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.81361$ 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)

$f(q)$	$=$	$q-3.10278 q^{3} -1.10278 q^{5} +1.00000 q^{7} +6.62721 q^{9} +O(q^{10})$	$q-3.10278 q^{3} -1.10278 q^{5} +1.00000 q^{7} +6.62721 q^{9} +5.62721 q^{11} -3.62721 q^{13} +3.42166 q^{15} +4.52444 q^{17} +0.578337 q^{19} -3.10278 q^{21} +1.00000 q^{23} -3.78389 q^{25} -11.2544 q^{27} +5.83276 q^{29} +2.52444 q^{31} -17.4600 q^{33} -1.10278 q^{35} -7.04888 q^{37} +11.2544 q^{39} +3.15667 q^{41} -7.25443 q^{43} -7.30833 q^{45} -2.52444 q^{47} +1.00000 q^{49} -14.0383 q^{51} +3.04888 q^{53} -6.20555 q^{55} -1.79445 q^{57} -9.30833 q^{59} +8.35720 q^{61} +6.62721 q^{63} +4.00000 q^{65} -5.62721 q^{67} -3.10278 q^{69} +13.0489 q^{71} -14.3033 q^{73} +11.7406 q^{75} +5.62721 q^{77} +16.9894 q^{79} +15.0383 q^{81} -13.6272 q^{83} -4.98944 q^{85} -18.0978 q^{87} -6.72999 q^{89} -3.62721 q^{91} -7.83276 q^{93} -0.637776 q^{95} +16.9355 q^{97} +37.2927 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + 4 q^{5} + 3 q^{7} + 7 q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 + 4 * q^5 + 3 * q^7 + 7 * q^9	$ 3 q - 2 q^{3} + 4 q^{5} + 3 q^{7} + 7 q^{9} + 4 q^{11} + 2 q^{13} + 12 q^{15} + 8 q^{17} - 2 q^{21} + 3 q^{23} + 5 q^{25} - 8 q^{27} - 10 q^{29} + 2 q^{31} - 12 q^{33} + 4 q^{35} - 10 q^{37} + 8 q^{39} + 6 q^{41} + 4 q^{43} - 2 q^{47} + 3 q^{49} - 2 q^{53} - 4 q^{55} - 20 q^{57} - 6 q^{59} - 8 q^{61} + 7 q^{63} + 12 q^{65} - 4 q^{67} - 2 q^{69} + 28 q^{71} - 6 q^{73} + 46 q^{75} + 4 q^{77} + 20 q^{79} + 3 q^{81} - 28 q^{83} + 16 q^{85} - 32 q^{87} + 2 q^{91} + 4 q^{93} - 20 q^{95} + 16 q^{97} + 44 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 + 4 * q^5 + 3 * q^7 + 7 * q^9 + 4 * q^11 + 2 * q^13 + 12 * q^15 + 8 * q^17 - 2 * q^21 + 3 * q^23 + 5 * q^25 - 8 * q^27 - 10 * q^29 + 2 * q^31 - 12 * q^33 + 4 * q^35 - 10 * q^37 + 8 * q^39 + 6 * q^41 + 4 * q^43 - 2 * q^47 + 3 * q^49 - 2 * q^53 - 4 * q^55 - 20 * q^57 - 6 * q^59 - 8 * q^61 + 7 * q^63 + 12 * q^65 - 4 * q^67 - 2 * q^69 + 28 * q^71 - 6 * q^73 + 46 * q^75 + 4 * q^77 + 20 * q^79 + 3 * q^81 - 28 * q^83 + 16 * q^85 - 32 * q^87 + 2 * q^91 + 4 * q^93 - 20 * q^95 + 16 * q^97 + 44 * 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
$3$	−3.10278	−1.79139	−0.895694	−	0.444671i	$-0.853321\pi$
$3$	−3.10278	−1.79139	−0.895694	+	0.444671i	$0.853321\pi$
$4$	0	0
$5$	−1.10278	−0.493176	−0.246588	−	0.969120i	$-0.579309\pi$
$5$	−1.10278	−0.493176	−0.246588	+	0.969120i	$0.579309\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	6.62721	2.20907
$10$	0	0
$11$	5.62721	1.69667	0.848334	−	0.529461i	$-0.177606\pi$
$11$	5.62721	1.69667	0.848334	+	0.529461i	$0.177606\pi$
$12$	0	0
$13$	−3.62721	−1.00601	−0.503004	−	0.864284i	$-0.667772\pi$
$13$	−3.62721	−1.00601	−0.503004	+	0.864284i	$0.667772\pi$
$14$	0	0
$15$	3.42166	0.883470
$16$	0	0
$17$	4.52444	1.09734	0.548669	−	0.836040i	$-0.315135\pi$
$17$	4.52444	1.09734	0.548669	+	0.836040i	$0.315135\pi$
$18$	0	0
$19$	0.578337	0.132680	0.0663398	−	0.997797i	$-0.478868\pi$
$19$	0.578337	0.132680	0.0663398	+	0.997797i	$0.478868\pi$
$20$	0	0
$21$	−3.10278	−0.677081
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.78389	−0.756777
$26$	0	0
$27$	−11.2544	−2.16592
$28$	0	0
$29$	5.83276	1.08312	0.541558	−	0.840663i	$-0.317834\pi$
$29$	5.83276	1.08312	0.541558	+	0.840663i	$0.317834\pi$
$30$	0	0
$31$	2.52444	0.453402	0.226701	−	0.973964i	$-0.427206\pi$
$31$	2.52444	0.453402	0.226701	+	0.973964i	$0.427206\pi$
$32$	0	0
$33$	−17.4600	−3.03939
$34$	0	0
$35$	−1.10278	−0.186403
$36$	0	0
$37$	−7.04888	−1.15883	−0.579414	−	0.815033i	$-0.696719\pi$
$37$	−7.04888	−1.15883	−0.579414	+	0.815033i	$0.696719\pi$
$38$	0	0
$39$	11.2544	1.80215
$40$	0	0
$41$	3.15667	0.492990	0.246495	−	0.969144i	$-0.420721\pi$
$41$	3.15667	0.492990	0.246495	+	0.969144i	$0.420721\pi$
$42$	0	0
$43$	−7.25443	−1.10629	−0.553145	−	0.833085i	$-0.686572\pi$
$43$	−7.25443	−1.10629	−0.553145	+	0.833085i	$0.686572\pi$
$44$	0	0
$45$	−7.30833	−1.08946
$46$	0	0
$47$	−2.52444	−0.368227	−0.184114	−	0.982905i	$-0.558941\pi$
$47$	−2.52444	−0.368227	−0.184114	+	0.982905i	$0.558941\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−14.0383	−1.96576
$52$	0	0
$53$	3.04888	0.418795	0.209398	−	0.977831i	$-0.432850\pi$
$53$	3.04888	0.418795	0.209398	+	0.977831i	$0.432850\pi$
$54$	0	0
$55$	−6.20555	−0.836756
$56$	0	0
$57$	−1.79445	−0.237681
$58$	0	0
$59$	−9.30833	−1.21184	−0.605920	−	0.795525i	$-0.707195\pi$
$59$	−9.30833	−1.21184	−0.605920	+	0.795525i	$0.707195\pi$
$60$	0	0
$61$	8.35720	1.07003	0.535015	−	0.844843i	$-0.320306\pi$
$61$	8.35720	1.07003	0.535015	+	0.844843i	$0.320306\pi$
$62$	0	0
$63$	6.62721	0.834950
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	−5.62721	−0.687473	−0.343737	−	0.939066i	$-0.611693\pi$
$67$	−5.62721	−0.687473	−0.343737	+	0.939066i	$0.611693\pi$
$68$	0	0
$69$	−3.10278	−0.373530
$70$	0	0
$71$	13.0489	1.54862	0.774308	−	0.632809i	$-0.218098\pi$
$71$	13.0489	1.54862	0.774308	+	0.632809i	$0.218098\pi$
$72$	0	0
$73$	−14.3033	−1.67407	−0.837037	−	0.547146i	$-0.815714\pi$
$73$	−14.3033	−1.67407	−0.837037	+	0.547146i	$0.815714\pi$
$74$	0	0
$75$	11.7406	1.35568
$76$	0	0
$77$	5.62721	0.641280
$78$	0	0
$79$	16.9894	1.91146	0.955731	−	0.294243i	$-0.0950676\pi$
$79$	16.9894	1.91146	0.955731	+	0.294243i	$0.0950676\pi$
$80$	0	0
$81$	15.0383	1.67092
$82$	0	0
$83$	−13.6272	−1.49578	−0.747890	−	0.663822i	$-0.768933\pi$
$83$	−13.6272	−1.49578	−0.747890	+	0.663822i	$0.768933\pi$
$84$	0	0
$85$	−4.98944	−0.541180
$86$	0	0
$87$	−18.0978	−1.94028
$88$	0	0
$89$	−6.72999	−0.713377	−0.356689	−	0.934223i	$-0.616094\pi$
$89$	−6.72999	−0.713377	−0.356689	+	0.934223i	$0.616094\pi$
$90$	0	0
$91$	−3.62721	−0.380235
$92$	0	0
$93$	−7.83276	−0.812220
$94$	0	0
$95$	−0.637776	−0.0654344
$96$	0	0
$97$	16.9355	1.71954	0.859772	−	0.510679i	$-0.170606\pi$
$97$	16.9355	1.71954	0.859772	+	0.510679i	$0.170606\pi$
$98$	0	0
$99$	37.2927	3.74806
$100$	0	0
$101$	4.37279	0.435109	0.217554	−	0.976048i	$-0.430192\pi$
$101$	4.37279	0.435109	0.217554	+	0.976048i	$0.430192\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	3.42166	0.333920
$106$	0	0
$107$	18.5089	1.78932	0.894659	−	0.446749i	$-0.147418\pi$
$107$	18.5089	1.78932	0.894659	+	0.446749i	$0.147418\pi$
$108$	0	0
$109$	−0.843326	−0.0807760	−0.0403880	−	0.999184i	$-0.512859\pi$
$109$	−0.843326	−0.0807760	−0.0403880	+	0.999184i	$0.512859\pi$
$110$	0	0
$111$	21.8711	2.07591
$112$	0	0
$113$	3.79445	0.356952	0.178476	−	0.983944i	$-0.442883\pi$
$113$	3.79445	0.356952	0.178476	+	0.983944i	$0.442883\pi$
$114$	0	0
$115$	−1.10278	−0.102834
$116$	0	0
$117$	−24.0383	−2.22234
$118$	0	0
$119$	4.52444	0.414755
$120$	0	0
$121$	20.6655	1.87868
$122$	0	0
$123$	−9.79445	−0.883136
$124$	0	0
$125$	9.68665	0.866400
$126$	0	0
$127$	14.2056	1.26054	0.630269	−	0.776377i	$-0.282944\pi$
$127$	14.2056	1.26054	0.630269	+	0.776377i	$0.282944\pi$
$128$	0	0
$129$	22.5089	1.98179
$130$	0	0
$131$	−22.3572	−1.95336	−0.976679	−	0.214705i	$-0.931121\pi$
$131$	−22.3572	−1.95336	−0.976679	+	0.214705i	$0.931121\pi$
$132$	0	0
$133$	0.578337	0.0501482
$134$	0	0
$135$	12.4111	1.06818
$136$	0	0
$137$	−1.89220	−0.161662	−0.0808309	−	0.996728i	$-0.525757\pi$
$137$	−1.89220	−0.161662	−0.0808309	+	0.996728i	$0.525757\pi$
$138$	0	0
$139$	6.35720	0.539211	0.269605	−	0.962971i	$-0.413107\pi$
$139$	6.35720	0.539211	0.269605	+	0.962971i	$0.413107\pi$
$140$	0	0
$141$	7.83276	0.659638
$142$	0	0
$143$	−20.4111	−1.70686
$144$	0	0
$145$	−6.43223	−0.534167
$146$	0	0
$147$	−3.10278	−0.255913
$148$	0	0
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−18.6167	−1.51500	−0.757501	−	0.652834i	$-0.773580\pi$
$151$	−18.6167	−1.51500	−0.757501	+	0.652834i	$0.773580\pi$
$152$	0	0
$153$	29.9844	2.42410
$154$	0	0
$155$	−2.78389	−0.223607
$156$	0	0
$157$	6.89722	0.550458	0.275229	−	0.961379i	$-0.411246\pi$
$157$	6.89722	0.550458	0.275229	+	0.961379i	$0.411246\pi$
$158$	0	0
$159$	−9.45998	−0.750225
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	19.3622	1.51657	0.758283	−	0.651925i	$-0.226038\pi$
$163$	19.3622	1.51657	0.758283	+	0.651925i	$0.226038\pi$
$164$	0	0
$165$	19.2544	1.49896
$166$	0	0
$167$	10.5244	0.814405	0.407203	−	0.913338i	$-0.366504\pi$
$167$	10.5244	0.814405	0.407203	+	0.913338i	$0.366504\pi$
$168$	0	0
$169$	0.156674	0.0120519
$170$	0	0
$171$	3.83276	0.293099
$172$	0	0
$173$	17.4217	1.32454	0.662272	−	0.749263i	$-0.269592\pi$
$173$	17.4217	1.32454	0.662272	+	0.749263i	$0.269592\pi$
$174$	0	0
$175$	−3.78389	−0.286035
$176$	0	0
$177$	28.8816	2.17088
$178$	0	0
$179$	11.6655	0.871922	0.435961	−	0.899965i	$-0.356408\pi$
$179$	11.6655	0.871922	0.435961	+	0.899965i	$0.356408\pi$
$180$	0	0
$181$	6.25945	0.465261	0.232631	−	0.972565i	$-0.425267\pi$
$181$	6.25945	0.465261	0.232631	+	0.972565i	$0.425267\pi$
$182$	0	0
$183$	−25.9305	−1.91684
$184$	0	0
$185$	7.77332	0.571506
$186$	0	0
$187$	25.4600	1.86182
$188$	0	0
$189$	−11.2544	−0.818639
$190$	0	0
$191$	2.31335	0.167388	0.0836940	−	0.996492i	$-0.473328\pi$
$191$	2.31335	0.167388	0.0836940	+	0.996492i	$0.473328\pi$
$192$	0	0
$193$	5.58890	0.402298	0.201149	−	0.979561i	$-0.435532\pi$
$193$	5.58890	0.402298	0.201149	+	0.979561i	$0.435532\pi$
$194$	0	0
$195$	−12.4111	−0.888777
$196$	0	0
$197$	18.4111	1.31174	0.655868	−	0.754875i	$-0.272303\pi$
$197$	18.4111	1.31174	0.655868	+	0.754875i	$0.272303\pi$
$198$	0	0
$199$	−4.41110	−0.312695	−0.156347	−	0.987702i	$-0.549972\pi$
$199$	−4.41110	−0.312695	−0.156347	+	0.987702i	$0.549972\pi$
$200$	0	0
$201$	17.4600	1.23153
$202$	0	0
$203$	5.83276	0.409380
$204$	0	0
$205$	−3.48110	−0.243131
$206$	0	0
$207$	6.62721	0.460623
$208$	0	0
$209$	3.25443	0.225113
$210$	0	0
$211$	−10.8433	−0.746485	−0.373243	−	0.927734i	$-0.621754\pi$
$211$	−10.8433	−0.746485	−0.373243	+	0.927734i	$0.621754\pi$
$212$	0	0
$213$	−40.4877	−2.77417
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	2.52444	0.171370
$218$	0	0
$219$	44.3799	2.99892
$220$	0	0
$221$	−16.4111	−1.10393
$222$	0	0
$223$	19.9844	1.33826	0.669128	−	0.743147i	$-0.266668\pi$
$223$	19.9844	1.33826	0.669128	+	0.743147i	$0.266668\pi$
$224$	0	0
$225$	−25.0766	−1.67178
$226$	0	0
$227$	−4.98944	−0.331161	−0.165580	−	0.986196i	$-0.552950\pi$
$227$	−4.98944	−0.331161	−0.165580	+	0.986196i	$0.552950\pi$
$228$	0	0
$229$	3.94610	0.260766	0.130383	−	0.991464i	$-0.458379\pi$
$229$	3.94610	0.260766	0.130383	+	0.991464i	$0.458379\pi$
$230$	0	0
$231$	−17.4600	−1.14878
$232$	0	0
$233$	−2.88164	−0.188782	−0.0943912	−	0.995535i	$-0.530090\pi$
$233$	−2.88164	−0.188782	−0.0943912	+	0.995535i	$0.530090\pi$
$234$	0	0
$235$	2.78389	0.181601
$236$	0	0
$237$	−52.7144	−3.42417
$238$	0	0
$239$	−20.7144	−1.33990	−0.669952	−	0.742405i	$-0.733685\pi$
$239$	−20.7144	−1.33990	−0.669952	+	0.742405i	$0.733685\pi$
$240$	0	0
$241$	11.0333	0.710717	0.355358	−	0.934730i	$-0.384359\pi$
$241$	11.0333	0.710717	0.355358	+	0.934730i	$0.384359\pi$
$242$	0	0
$243$	−12.8972	−0.827357
$244$	0	0
$245$	−1.10278	−0.0704537
$246$	0	0
$247$	−2.09775	−0.133477
$248$	0	0
$249$	42.2822	2.67952
$250$	0	0
$251$	−16.2439	−1.02530	−0.512652	−	0.858597i	$-0.671337\pi$
$251$	−16.2439	−1.02530	−0.512652	+	0.858597i	$0.671337\pi$
$252$	0	0
$253$	5.62721	0.353780
$254$	0	0
$255$	15.4811	0.969464
$256$	0	0
$257$	−13.6655	−0.852432	−0.426216	−	0.904621i	$-0.640154\pi$
$257$	−13.6655	−0.852432	−0.426216	+	0.904621i	$0.640154\pi$
$258$	0	0
$259$	−7.04888	−0.437996
$260$	0	0
$261$	38.6550	2.39268
$262$	0	0
$263$	−24.9894	−1.54091	−0.770457	−	0.637492i	$-0.779972\pi$
$263$	−24.9894	−1.54091	−0.770457	+	0.637492i	$0.779972\pi$
$264$	0	0
$265$	−3.36222	−0.206540
$266$	0	0
$267$	20.8816	1.27794
$268$	0	0
$269$	−17.4983	−1.06689	−0.533445	−	0.845835i	$-0.679103\pi$
$269$	−17.4983	−1.06689	−0.533445	+	0.845835i	$0.679103\pi$
$270$	0	0
$271$	18.5244	1.12528	0.562640	−	0.826702i	$-0.309786\pi$
$271$	18.5244	1.12528	0.562640	+	0.826702i	$0.309786\pi$
$272$	0	0
$273$	11.2544	0.681149
$274$	0	0
$275$	−21.2927	−1.28400
$276$	0	0
$277$	−10.9894	−0.660291	−0.330146	−	0.943930i	$-0.607098\pi$
$277$	−10.9894	−0.660291	−0.330146	+	0.943930i	$0.607098\pi$
$278$	0	0
$279$	16.7300	1.00160
$280$	0	0
$281$	−7.45998	−0.445025	−0.222512	−	0.974930i	$-0.571426\pi$
$281$	−7.45998	−0.445025	−0.222512	+	0.974930i	$0.571426\pi$
$282$	0	0
$283$	−1.51941	−0.0903198	−0.0451599	−	0.998980i	$-0.514380\pi$
$283$	−1.51941	−0.0903198	−0.0451599	+	0.998980i	$0.514380\pi$
$284$	0	0
$285$	1.97887	0.117218
$286$	0	0
$287$	3.15667	0.186333
$288$	0	0
$289$	3.47054	0.204149
$290$	0	0
$291$	−52.5472	−3.08037
$292$	0	0
$293$	−1.10278	−0.0644248	−0.0322124	−	0.999481i	$-0.510255\pi$
$293$	−1.10278	−0.0644248	−0.0322124	+	0.999481i	$0.510255\pi$
$294$	0	0
$295$	10.2650	0.597651
$296$	0	0
$297$	−63.3311	−3.67484
$298$	0	0
$299$	−3.62721	−0.209767
$300$	0	0
$301$	−7.25443	−0.418138
$302$	0	0
$303$	−13.5678	−0.779448
$304$	0	0
$305$	−9.21611	−0.527713
$306$	0	0
$307$	11.1028	0.633669	0.316834	−	0.948481i	$-0.397380\pi$
$307$	11.1028	0.633669	0.316834	+	0.948481i	$0.397380\pi$
$308$	0	0
$309$	−24.8222	−1.41209
$310$	0	0
$311$	14.2978	0.810752	0.405376	−	0.914150i	$-0.367141\pi$
$311$	14.2978	0.810752	0.405376	+	0.914150i	$0.367141\pi$
$312$	0	0
$313$	−19.4756	−1.10082	−0.550412	−	0.834893i	$-0.685529\pi$
$313$	−19.4756	−1.10082	−0.550412	+	0.834893i	$0.685529\pi$
$314$	0	0
$315$	−7.30833	−0.411777
$316$	0	0
$317$	28.3416	1.59182	0.795912	−	0.605413i	$-0.206992\pi$
$317$	28.3416	1.59182	0.795912	+	0.605413i	$0.206992\pi$
$318$	0	0
$319$	32.8222	1.83769
$320$	0	0
$321$	−57.4288	−3.20536
$322$	0	0
$323$	2.61665	0.145594
$324$	0	0
$325$	13.7250	0.761324
$326$	0	0
$327$	2.61665	0.144701
$328$	0	0
$329$	−2.52444	−0.139177
$330$	0	0
$331$	34.8122	1.91345	0.956725	−	0.290995i	$-0.0939864\pi$
$331$	34.8122	1.91345	0.956725	+	0.290995i	$0.0939864\pi$
$332$	0	0
$333$	−46.7144	−2.55993
$334$	0	0
$335$	6.20555	0.339045
$336$	0	0
$337$	−4.84333	−0.263833	−0.131916	−	0.991261i	$-0.542113\pi$
$337$	−4.84333	−0.263833	−0.131916	+	0.991261i	$0.542113\pi$
$338$	0	0
$339$	−11.7733	−0.639439
$340$	0	0
$341$	14.2056	0.769274
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	3.42166	0.184216
$346$	0	0
$347$	23.7733	1.27622	0.638109	−	0.769946i	$-0.279717\pi$
$347$	23.7733	1.27622	0.638109	+	0.769946i	$0.279717\pi$
$348$	0	0
$349$	32.7527	1.75321	0.876606	−	0.481208i	$-0.159802\pi$
$349$	32.7527	1.75321	0.876606	+	0.481208i	$0.159802\pi$
$350$	0	0
$351$	40.8222	2.17893
$352$	0	0
$353$	13.5577	0.721605	0.360803	−	0.932642i	$-0.382503\pi$
$353$	13.5577	0.721605	0.360803	+	0.932642i	$0.382503\pi$
$354$	0	0
$355$	−14.3900	−0.763741
$356$	0	0
$357$	−14.0383	−0.742986
$358$	0	0
$359$	−6.67609	−0.352350	−0.176175	−	0.984359i	$-0.556373\pi$
$359$	−6.67609	−0.352350	−0.176175	+	0.984359i	$0.556373\pi$
$360$	0	0
$361$	−18.6655	−0.982396
$362$	0	0
$363$	−64.1205	−3.36545
$364$	0	0
$365$	15.7733	0.825614
$366$	0	0
$367$	24.3033	1.26862	0.634311	−	0.773078i	$-0.281284\pi$
$367$	24.3033	1.26862	0.634311	+	0.773078i	$0.281284\pi$
$368$	0	0
$369$	20.9200	1.08905
$370$	0	0
$371$	3.04888	0.158290
$372$	0	0
$373$	−8.50885	−0.440572	−0.220286	−	0.975435i	$-0.570699\pi$
$373$	−8.50885	−0.440572	−0.220286	+	0.975435i	$0.570699\pi$
$374$	0	0
$375$	−30.0555	−1.55206
$376$	0	0
$377$	−21.1567	−1.08962
$378$	0	0
$379$	9.21611	0.473400	0.236700	−	0.971583i	$-0.423934\pi$
$379$	9.21611	0.473400	0.236700	+	0.971583i	$0.423934\pi$
$380$	0	0
$381$	−44.0766	−2.25811
$382$	0	0
$383$	23.9688	1.22475	0.612375	−	0.790567i	$-0.290214\pi$
$383$	23.9688	1.22475	0.612375	+	0.790567i	$0.290214\pi$
$384$	0	0
$385$	−6.20555	−0.316264
$386$	0	0
$387$	−48.0766	−2.44387
$388$	0	0
$389$	−2.33447	−0.118363	−0.0591813	−	0.998247i	$-0.518849\pi$
$389$	−2.33447	−0.118363	−0.0591813	+	0.998247i	$0.518849\pi$
$390$	0	0
$391$	4.52444	0.228811
$392$	0	0
$393$	69.3694	3.49922
$394$	0	0
$395$	−18.7355	−0.942687
$396$	0	0
$397$	−21.9406	−1.10117	−0.550583	−	0.834781i	$-0.685594\pi$
$397$	−21.9406	−1.10117	−0.550583	+	0.834781i	$0.685594\pi$
$398$	0	0
$399$	−1.79445	−0.0898349
$400$	0	0
$401$	−27.3522	−1.36590	−0.682951	−	0.730464i	$-0.739304\pi$
$401$	−27.3522	−1.36590	−0.682951	+	0.730464i	$0.739304\pi$
$402$	0	0
$403$	−9.15667	−0.456126
$404$	0	0
$405$	−16.5839	−0.824059
$406$	0	0
$407$	−39.6655	−1.96615
$408$	0	0
$409$	4.95112	0.244817	0.122409	−	0.992480i	$-0.460938\pi$
$409$	4.95112	0.244817	0.122409	+	0.992480i	$0.460938\pi$
$410$	0	0
$411$	5.87108	0.289599
$412$	0	0
$413$	−9.30833	−0.458033
$414$	0	0
$415$	15.0278	0.737683
$416$	0	0
$417$	−19.7250	−0.965936
$418$	0	0
$419$	26.3416	1.28687	0.643436	−	0.765500i	$-0.277508\pi$
$419$	26.3416	1.28687	0.643436	+	0.765500i	$0.277508\pi$
$420$	0	0
$421$	−9.36222	−0.456287	−0.228143	−	0.973628i	$-0.573266\pi$
$421$	−9.36222	−0.456287	−0.228143	+	0.973628i	$0.573266\pi$
$422$	0	0
$423$	−16.7300	−0.813440
$424$	0	0
$425$	−17.1200	−0.830440
$426$	0	0
$427$	8.35720	0.404433
$428$	0	0
$429$	63.3311	3.05765
$430$	0	0
$431$	−8.82220	−0.424950	−0.212475	−	0.977166i	$-0.568152\pi$
$431$	−8.82220	−0.424950	−0.212475	+	0.977166i	$0.568152\pi$
$432$	0	0
$433$	14.6222	0.702698	0.351349	−	0.936245i	$-0.385723\pi$
$433$	14.6222	0.702698	0.351349	+	0.936245i	$0.385723\pi$
$434$	0	0
$435$	19.9577	0.956901
$436$	0	0
$437$	0.578337	0.0276656
$438$	0	0
$439$	30.2978	1.44603	0.723017	−	0.690831i	$-0.242755\pi$
$439$	30.2978	1.44603	0.723017	+	0.690831i	$0.242755\pi$
$440$	0	0
$441$	6.62721	0.315582
$442$	0	0
$443$	−17.5678	−0.834670	−0.417335	−	0.908753i	$-0.637036\pi$
$443$	−17.5678	−0.834670	−0.417335	+	0.908753i	$0.637036\pi$
$444$	0	0
$445$	7.42166	0.351821
$446$	0	0
$447$	6.20555	0.293512
$448$	0	0
$449$	11.2927	0.532937	0.266469	−	0.963844i	$-0.414143\pi$
$449$	11.2927	0.532937	0.266469	+	0.963844i	$0.414143\pi$
$450$	0	0
$451$	17.7633	0.836440
$452$	0	0
$453$	57.7633	2.71396
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	27.4600	1.28452	0.642262	−	0.766485i	$-0.277996\pi$
$457$	27.4600	1.28452	0.642262	+	0.766485i	$0.277996\pi$
$458$	0	0
$459$	−50.9200	−2.37674
$460$	0	0
$461$	29.4983	1.37387	0.686936	−	0.726718i	$-0.258955\pi$
$461$	29.4983	1.37387	0.686936	+	0.726718i	$0.258955\pi$
$462$	0	0
$463$	−14.7244	−0.684303	−0.342152	−	0.939645i	$-0.611156\pi$
$463$	−14.7244	−0.684303	−0.342152	+	0.939645i	$0.611156\pi$
$464$	0	0
$465$	8.63778	0.400567
$466$	0	0
$467$	−8.57834	−0.396958	−0.198479	−	0.980105i	$-0.563600\pi$
$467$	−8.57834	−0.396958	−0.198479	+	0.980105i	$0.563600\pi$
$468$	0	0
$469$	−5.62721	−0.259841
$470$	0	0
$471$	−21.4005	−0.986085
$472$	0	0
$473$	−40.8222	−1.87701
$474$	0	0
$475$	−2.18836	−0.100409
$476$	0	0
$477$	20.2056	0.925149
$478$	0	0
$479$	28.0766	1.28285	0.641427	−	0.767184i	$-0.278343\pi$
$479$	28.0766	1.28285	0.641427	+	0.767184i	$0.278343\pi$
$480$	0	0
$481$	25.5678	1.16579
$482$	0	0
$483$	−3.10278	−0.141181
$484$	0	0
$485$	−18.6761	−0.848038
$486$	0	0
$487$	−22.7244	−1.02974	−0.514872	−	0.857267i	$-0.672160\pi$
$487$	−22.7244	−1.02974	−0.514872	+	0.857267i	$0.672160\pi$
$488$	0	0
$489$	−60.0766	−2.71676
$490$	0	0
$491$	18.7244	0.845023	0.422511	−	0.906358i	$-0.361149\pi$
$491$	18.7244	0.845023	0.422511	+	0.906358i	$0.361149\pi$
$492$	0	0
$493$	26.3900	1.18854
$494$	0	0
$495$	−41.1255	−1.84845
$496$	0	0
$497$	13.0489	0.585322
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	−32.6550	−1.45892
$502$	0	0
$503$	−13.5678	−0.604957	−0.302479	−	0.953156i	$-0.597814\pi$
$503$	−13.5678	−0.604957	−0.302479	+	0.953156i	$0.597814\pi$
$504$	0	0
$505$	−4.82220	−0.214585
$506$	0	0
$507$	−0.486125	−0.0215896
$508$	0	0
$509$	−4.14611	−0.183773	−0.0918866	−	0.995769i	$-0.529290\pi$
$509$	−4.14611	−0.183773	−0.0918866	+	0.995769i	$0.529290\pi$
$510$	0	0
$511$	−14.3033	−0.632741
$512$	0	0
$513$	−6.50885	−0.287373
$514$	0	0
$515$	−8.82220	−0.388753
$516$	0	0
$517$	−14.2056	−0.624759
$518$	0	0
$519$	−54.0555	−2.37277
$520$	0	0
$521$	−28.2978	−1.23975	−0.619874	−	0.784702i	$-0.712816\pi$
$521$	−28.2978	−1.23975	−0.619874	+	0.784702i	$0.712816\pi$
$522$	0	0
$523$	4.98944	0.218173	0.109086	−	0.994032i	$-0.465207\pi$
$523$	4.98944	0.218173	0.109086	+	0.994032i	$0.465207\pi$
$524$	0	0
$525$	11.7406	0.512400
$526$	0	0
$527$	11.4217	0.497535
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−61.6883	−2.67704
$532$	0	0
$533$	−11.4499	−0.495952
$534$	0	0
$535$	−20.4111	−0.882449
$536$	0	0
$537$	−36.1955	−1.56195
$538$	0	0
$539$	5.62721	0.242381
$540$	0	0
$541$	28.3416	1.21850	0.609251	−	0.792978i	$-0.291470\pi$
$541$	28.3416	1.21850	0.609251	+	0.792978i	$0.291470\pi$
$542$	0	0
$543$	−19.4217	−0.833463
$544$	0	0
$545$	0.929999	0.0398368
$546$	0	0
$547$	−1.26447	−0.0540649	−0.0270325	−	0.999635i	$-0.508606\pi$
$547$	−1.26447	−0.0540649	−0.0270325	+	0.999635i	$0.508606\pi$
$548$	0	0
$549$	55.3850	2.36377
$550$	0	0
$551$	3.37330	0.143708
$552$	0	0
$553$	16.9894	0.722464
$554$	0	0
$555$	−24.1189	−1.02379
$556$	0	0
$557$	−2.21560	−0.0938778	−0.0469389	−	0.998898i	$-0.514947\pi$
$557$	−2.21560	−0.0938778	−0.0469389	+	0.998898i	$0.514947\pi$
$558$	0	0
$559$	26.3133	1.11294
$560$	0	0
$561$	−78.9966	−3.33524
$562$	0	0
$563$	20.7738	0.875513	0.437757	−	0.899094i	$-0.355773\pi$
$563$	20.7738	0.875513	0.437757	+	0.899094i	$0.355773\pi$
$564$	0	0
$565$	−4.18442	−0.176040
$566$	0	0
$567$	15.0383	0.631550
$568$	0	0
$569$	−35.0177	−1.46802	−0.734009	−	0.679139i	$-0.762353\pi$
$569$	−35.0177	−1.46802	−0.734009	+	0.679139i	$0.762353\pi$
$570$	0	0
$571$	−35.6655	−1.49256	−0.746278	−	0.665634i	$-0.768161\pi$
$571$	−35.6655	−1.49256	−0.746278	+	0.665634i	$0.768161\pi$
$572$	0	0
$573$	−7.17780	−0.299857
$574$	0	0
$575$	−3.78389	−0.157799
$576$	0	0
$577$	12.6167	0.525238	0.262619	−	0.964900i	$-0.415414\pi$
$577$	12.6167	0.525238	0.262619	+	0.964900i	$0.415414\pi$
$578$	0	0
$579$	−17.3411	−0.720671
$580$	0	0
$581$	−13.6272	−0.565352
$582$	0	0
$583$	17.1567	0.710557
$584$	0	0
$585$	26.5089	1.09601
$586$	0	0
$587$	21.0816	0.870133	0.435066	−	0.900398i	$-0.356725\pi$
$587$	21.0816	0.870133	0.435066	+	0.900398i	$0.356725\pi$
$588$	0	0
$589$	1.45998	0.0601573
$590$	0	0
$591$	−57.1255	−2.34983
$592$	0	0
$593$	−1.25443	−0.0515131	−0.0257566	−	0.999668i	$-0.508199\pi$
$593$	−1.25443	−0.0515131	−0.0257566	+	0.999668i	$0.508199\pi$
$594$	0	0
$595$	−4.98944	−0.204547
$596$	0	0
$597$	13.6867	0.560157
$598$	0	0
$599$	38.3900	1.56857	0.784286	−	0.620400i	$-0.213030\pi$
$599$	38.3900	1.56857	0.784286	+	0.620400i	$0.213030\pi$
$600$	0	0
$601$	14.7144	0.600213	0.300106	−	0.953906i	$-0.402978\pi$
$601$	14.7144	0.600213	0.300106	+	0.953906i	$0.402978\pi$
$602$	0	0
$603$	−37.2927	−1.51868
$604$	0	0
$605$	−22.7894	−0.926522
$606$	0	0
$607$	−29.5633	−1.19994	−0.599968	−	0.800024i	$-0.704820\pi$
$607$	−29.5633	−1.19994	−0.599968	+	0.800024i	$0.704820\pi$
$608$	0	0
$609$	−18.0978	−0.733358
$610$	0	0
$611$	9.15667	0.370439
$612$	0	0
$613$	−15.3522	−0.620069	−0.310034	−	0.950725i	$-0.600341\pi$
$613$	−15.3522	−0.620069	−0.310034	+	0.950725i	$0.600341\pi$
$614$	0	0
$615$	10.8011	0.435541
$616$	0	0
$617$	41.3311	1.66393	0.831963	−	0.554831i	$-0.187217\pi$
$617$	41.3311	1.66393	0.831963	+	0.554831i	$0.187217\pi$
$618$	0	0
$619$	5.84281	0.234842	0.117421	−	0.993082i	$-0.462537\pi$
$619$	5.84281	0.234842	0.117421	+	0.993082i	$0.462537\pi$
$620$	0	0
$621$	−11.2544	−0.451625
$622$	0	0
$623$	−6.72999	−0.269631
$624$	0	0
$625$	8.23724	0.329490
$626$	0	0
$627$	−10.0978	−0.403265
$628$	0	0
$629$	−31.8922	−1.27163
$630$	0	0
$631$	−6.34162	−0.252456	−0.126228	−	0.992001i	$-0.540287\pi$
$631$	−6.34162	−0.252456	−0.126228	+	0.992001i	$0.540287\pi$
$632$	0	0
$633$	33.6444	1.33724
$634$	0	0
$635$	−15.6655	−0.621667
$636$	0	0
$637$	−3.62721	−0.143715
$638$	0	0
$639$	86.4777	3.42100
$640$	0	0
$641$	−32.3133	−1.27630	−0.638150	−	0.769912i	$-0.720300\pi$
$641$	−32.3133	−1.27630	−0.638150	+	0.769912i	$0.720300\pi$
$642$	0	0
$643$	−28.1672	−1.11081	−0.555404	−	0.831581i	$-0.687436\pi$
$643$	−28.1672	−1.11081	−0.555404	+	0.831581i	$0.687436\pi$
$644$	0	0
$645$	−24.8222	−0.977373
$646$	0	0
$647$	−30.2978	−1.19113	−0.595564	−	0.803308i	$-0.703071\pi$
$647$	−30.2978	−1.19113	−0.595564	+	0.803308i	$0.703071\pi$
$648$	0	0
$649$	−52.3799	−2.05609
$650$	0	0
$651$	−7.83276	−0.306990
$652$	0	0
$653$	−4.64782	−0.181883	−0.0909417	−	0.995856i	$-0.528988\pi$
$653$	−4.64782	−0.181883	−0.0909417	+	0.995856i	$0.528988\pi$
$654$	0	0
$655$	24.6550	0.963349
$656$	0	0
$657$	−94.7910	−3.69815
$658$	0	0
$659$	11.8811	0.462823	0.231411	−	0.972856i	$-0.425666\pi$
$659$	11.8811	0.462823	0.231411	+	0.972856i	$0.425666\pi$
$660$	0	0
$661$	3.94610	0.153486	0.0767428	−	0.997051i	$-0.475548\pi$
$661$	3.94610	0.153486	0.0767428	+	0.997051i	$0.475548\pi$
$662$	0	0
$663$	50.9200	1.97757
$664$	0	0
$665$	−0.637776	−0.0247319
$666$	0	0
$667$	5.83276	0.225845
$668$	0	0
$669$	−62.0071	−2.39733
$670$	0	0
$671$	47.0278	1.81549
$672$	0	0
$673$	24.0383	0.926609	0.463304	−	0.886199i	$-0.346664\pi$
$673$	24.0383	0.926609	0.463304	+	0.886199i	$0.346664\pi$
$674$	0	0
$675$	42.5855	1.63912
$676$	0	0
$677$	39.2005	1.50660	0.753299	−	0.657678i	$-0.228461\pi$
$677$	39.2005	1.50660	0.753299	+	0.657678i	$0.228461\pi$
$678$	0	0
$679$	16.9355	0.649926
$680$	0	0
$681$	15.4811	0.593237
$682$	0	0
$683$	0.745574	0.0285286	0.0142643	−	0.999898i	$-0.495459\pi$
$683$	0.745574	0.0285286	0.0142643	+	0.999898i	$0.495459\pi$
$684$	0	0
$685$	2.08667	0.0797277
$686$	0	0
$687$	−12.2439	−0.467133
$688$	0	0
$689$	−11.0589	−0.421311
$690$	0	0
$691$	11.7094	0.445446	0.222723	−	0.974882i	$-0.428505\pi$
$691$	11.7094	0.445446	0.222723	+	0.974882i	$0.428505\pi$
$692$	0	0
$693$	37.2927	1.41663
$694$	0	0
$695$	−7.01056	−0.265926
$696$	0	0
$697$	14.2822	0.540976
$698$	0	0
$699$	8.94108	0.338183
$700$	0	0
$701$	7.45998	0.281759	0.140880	−	0.990027i	$-0.455007\pi$
$701$	7.45998	0.281759	0.140880	+	0.990027i	$0.455007\pi$
$702$	0	0
$703$	−4.07663	−0.153753
$704$	0	0
$705$	−8.63778	−0.325317
$706$	0	0
$707$	4.37279	0.164456
$708$	0	0
$709$	5.69670	0.213944	0.106972	−	0.994262i	$-0.465884\pi$
$709$	5.69670	0.213944	0.106972	+	0.994262i	$0.465884\pi$
$710$	0	0
$711$	112.593	4.22255
$712$	0	0
$713$	2.52444	0.0945409
$714$	0	0
$715$	22.5089	0.841783
$716$	0	0
$717$	64.2721	2.40029
$718$	0	0
$719$	−1.67107	−0.0623202	−0.0311601	−	0.999514i	$-0.509920\pi$
$719$	−1.67107	−0.0623202	−0.0311601	+	0.999514i	$0.509920\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	−34.2338	−1.27317
$724$	0	0
$725$	−22.0705	−0.819678
$726$	0	0
$727$	−36.0766	−1.33801	−0.669004	−	0.743259i	$-0.733279\pi$
$727$	−36.0766	−1.33801	−0.669004	+	0.743259i	$0.733279\pi$
$728$	0	0
$729$	−5.09775	−0.188806
$730$	0	0
$731$	−32.8222	−1.21397
$732$	0	0
$733$	−12.0227	−0.444070	−0.222035	−	0.975039i	$-0.571270\pi$
$733$	−12.0227	−0.444070	−0.222035	+	0.975039i	$0.571270\pi$
$734$	0	0
$735$	3.42166	0.126210
$736$	0	0
$737$	−31.6655	−1.16641
$738$	0	0
$739$	41.5366	1.52795	0.763974	−	0.645247i	$-0.223245\pi$
$739$	41.5366	1.52795	0.763974	+	0.645247i	$0.223245\pi$
$740$	0	0
$741$	6.50885	0.239109
$742$	0	0
$743$	13.1849	0.483709	0.241854	−	0.970313i	$-0.422244\pi$
$743$	13.1849	0.483709	0.241854	+	0.970313i	$0.422244\pi$
$744$	0	0
$745$	2.20555	0.0808051
$746$	0	0
$747$	−90.3104	−3.30429
$748$	0	0
$749$	18.5089	0.676299
$750$	0	0
$751$	7.83276	0.285822	0.142911	−	0.989736i	$-0.454354\pi$
$751$	7.83276	0.285822	0.142911	+	0.989736i	$0.454354\pi$
$752$	0	0
$753$	50.4011	1.83672
$754$	0	0
$755$	20.5300	0.747162
$756$	0	0
$757$	36.5089	1.32694	0.663468	−	0.748204i	$-0.269084\pi$
$757$	36.5089	1.32694	0.663468	+	0.748204i	$0.269084\pi$
$758$	0	0
$759$	−17.4600	−0.633757
$760$	0	0
$761$	35.2444	1.27761	0.638804	−	0.769370i	$-0.279430\pi$
$761$	35.2444	1.27761	0.638804	+	0.769370i	$0.279430\pi$
$762$	0	0
$763$	−0.843326	−0.0305304
$764$	0	0
$765$	−33.0661	−1.19551
$766$	0	0
$767$	33.7633	1.21912
$768$	0	0
$769$	−24.4933	−0.883250	−0.441625	−	0.897200i	$-0.645598\pi$
$769$	−24.4933	−0.883250	−0.441625	+	0.897200i	$0.645598\pi$
$770$	0	0
$771$	42.4011	1.52704
$772$	0	0
$773$	−7.94610	−0.285801	−0.142901	−	0.989737i	$-0.545643\pi$
$773$	−7.94610	−0.285801	−0.142901	+	0.989737i	$0.545643\pi$
$774$	0	0
$775$	−9.55219	−0.343125
$776$	0	0
$777$	21.8711	0.784620
$778$	0	0
$779$	1.82562	0.0654097
$780$	0	0
$781$	73.4288	2.62749
$782$	0	0
$783$	−65.6444	−2.34594
$784$	0	0
$785$	−7.60609	−0.271473
$786$	0	0
$787$	−19.5295	−0.696150	−0.348075	−	0.937467i	$-0.613165\pi$
$787$	−19.5295	−0.696150	−0.348075	+	0.937467i	$0.613165\pi$
$788$	0	0
$789$	77.5366	2.76038
$790$	0	0
$791$	3.79445	0.134915
$792$	0	0
$793$	−30.3133	−1.07646
$794$	0	0
$795$	10.4322	0.369993
$796$	0	0
$797$	−17.7406	−0.628403	−0.314201	−	0.949356i	$-0.601737\pi$
$797$	−17.7406	−0.628403	−0.314201	+	0.949356i	$0.601737\pi$
$798$	0	0
$799$	−11.4217	−0.404069
$800$	0	0
$801$	−44.6011	−1.57590
$802$	0	0
$803$	−80.4877	−2.84035
$804$	0	0
$805$	−1.10278	−0.0388677
$806$	0	0
$807$	54.2933	1.91121
$808$	0	0
$809$	−18.4111	−0.647300	−0.323650	−	0.946177i	$-0.604910\pi$
$809$	−18.4111	−0.647300	−0.323650	+	0.946177i	$0.604910\pi$
$810$	0	0
$811$	−20.2594	−0.711405	−0.355703	−	0.934599i	$-0.615758\pi$
$811$	−20.2594	−0.711405	−0.355703	+	0.934599i	$0.615758\pi$
$812$	0	0
$813$	−57.4772	−2.01581
$814$	0	0
$815$	−21.3522	−0.747934
$816$	0	0
$817$	−4.19550	−0.146782
$818$	0	0
$819$	−24.0383	−0.839967
$820$	0	0
$821$	31.8116	1.11023	0.555117	−	0.831772i	$-0.312674\pi$
$821$	31.8116	1.11023	0.555117	+	0.831772i	$0.312674\pi$
$822$	0	0
$823$	−32.9099	−1.14717	−0.573584	−	0.819147i	$-0.694447\pi$
$823$	−32.9099	−1.14717	−0.573584	+	0.819147i	$0.694447\pi$
$824$	0	0
$825$	66.0666	2.30014
$826$	0	0
$827$	37.4288	1.30153	0.650764	−	0.759280i	$-0.274449\pi$
$827$	37.4288	1.30153	0.650764	+	0.759280i	$0.274449\pi$
$828$	0	0
$829$	28.5572	0.991833	0.495916	−	0.868370i	$-0.334832\pi$
$829$	28.5572	0.991833	0.495916	+	0.868370i	$0.334832\pi$
$830$	0	0
$831$	34.0978	1.18284
$832$	0	0
$833$	4.52444	0.156762
$834$	0	0
$835$	−11.6061	−0.401645
$836$	0	0
$837$	−28.4111	−0.982031
$838$	0	0
$839$	−13.3833	−0.462045	−0.231022	−	0.972948i	$-0.574207\pi$
$839$	−13.3833	−0.462045	−0.231022	+	0.972948i	$0.574207\pi$
$840$	0	0
$841$	5.02113	0.173142
$842$	0	0
$843$	23.1466	0.797212
$844$	0	0
$845$	−0.172776	−0.00594369
$846$	0	0
$847$	20.6655	0.710076
$848$	0	0
$849$	4.71440	0.161798
$850$	0	0
$851$	−7.04888	−0.241632
$852$	0	0
$853$	−31.4882	−1.07814	−0.539068	−	0.842262i	$-0.681224\pi$
$853$	−31.4882	−1.07814	−0.539068	+	0.842262i	$0.681224\pi$
$854$	0	0
$855$	−4.22668	−0.144549
$856$	0	0
$857$	31.2333	1.06691	0.533455	−	0.845829i	$-0.320894\pi$
$857$	31.2333	1.06691	0.533455	+	0.845829i	$0.320894\pi$
$858$	0	0
$859$	53.0816	1.81112	0.905561	−	0.424216i	$-0.139450\pi$
$859$	53.0816	1.81112	0.905561	+	0.424216i	$0.139450\pi$
$860$	0	0
$861$	−9.79445	−0.333794
$862$	0	0
$863$	−5.04888	−0.171866	−0.0859329	−	0.996301i	$-0.527387\pi$
$863$	−5.04888	−0.171866	−0.0859329	+	0.996301i	$0.527387\pi$
$864$	0	0
$865$	−19.2122	−0.653234
$866$	0	0
$867$	−10.7683	−0.365711
$868$	0	0
$869$	95.6032	3.24312
$870$	0	0
$871$	20.4111	0.691604
$872$	0	0
$873$	112.235	3.79859
$874$	0	0
$875$	9.68665	0.327469
$876$	0	0
$877$	11.7350	0.396263	0.198132	−	0.980175i	$-0.436513\pi$
$877$	11.7350	0.396263	0.198132	+	0.980175i	$0.436513\pi$
$878$	0	0
$879$	3.42166	0.115410
$880$	0	0
$881$	38.9255	1.31143	0.655717	−	0.755007i	$-0.272367\pi$
$881$	38.9255	1.31143	0.655717	+	0.755007i	$0.272367\pi$
$882$	0	0
$883$	−17.3522	−0.583947	−0.291974	−	0.956426i	$-0.594312\pi$
$883$	−17.3522	−0.583947	−0.291974	+	0.956426i	$0.594312\pi$
$884$	0	0
$885$	−31.8500	−1.07062
$886$	0	0
$887$	1.55219	0.0521174	0.0260587	−	0.999660i	$-0.491704\pi$
$887$	1.55219	0.0521174	0.0260587	+	0.999660i	$0.491704\pi$
$888$	0	0
$889$	14.2056	0.476439
$890$	0	0
$891$	84.6238	2.83500
$892$	0	0
$893$	−1.45998	−0.0488562
$894$	0	0
$895$	−12.8645	−0.430011
$896$	0	0
$897$	11.2544	0.375774
$898$	0	0
$899$	14.7244	0.491088
$900$	0	0
$901$	13.7944	0.459560
$902$	0	0
$903$	22.5089	0.749048
$904$	0	0
$905$	−6.90276	−0.229456
$906$	0	0
$907$	−43.5466	−1.44594	−0.722971	−	0.690878i	$-0.757224\pi$
$907$	−43.5466	−1.44594	−0.722971	+	0.690878i	$0.757224\pi$
$908$	0	0
$909$	28.9794	0.961186
$910$	0	0
$911$	25.8116	0.855178	0.427589	−	0.903973i	$-0.359363\pi$
$911$	25.8116	0.855178	0.427589	+	0.903973i	$0.359363\pi$
$912$	0	0
$913$	−76.6832	−2.53784
$914$	0	0
$915$	28.5955	0.945339
$916$	0	0
$917$	−22.3572	−0.738300
$918$	0	0
$919$	17.1083	0.564351	0.282176	−	0.959363i	$-0.408944\pi$
$919$	17.1083	0.564351	0.282176	+	0.959363i	$0.408944\pi$
$920$	0	0
$921$	−34.4494	−1.13515
$922$	0	0
$923$	−47.3311	−1.55792
$924$	0	0
$925$	26.6722	0.876975
$926$	0	0
$927$	53.0177	1.74133
$928$	0	0
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	0.578337	0.0189542
$932$	0	0
$933$	−44.3627	−1.45237
$934$	0	0
$935$	−28.0766	−0.918204
$936$	0	0
$937$	39.1411	1.27868	0.639342	−	0.768923i	$-0.279207\pi$
$937$	39.1411	1.27868	0.639342	+	0.768923i	$0.279207\pi$
$938$	0	0
$939$	60.4283	1.97200
$940$	0	0
$941$	−17.4061	−0.567422	−0.283711	−	0.958910i	$-0.591566\pi$
$941$	−17.4061	−0.567422	−0.283711	+	0.958910i	$0.591566\pi$
$942$	0	0
$943$	3.15667	0.102795
$944$	0	0
$945$	12.4111	0.403733
$946$	0	0
$947$	7.89220	0.256462	0.128231	−	0.991744i	$-0.459070\pi$
$947$	7.89220	0.256462	0.128231	+	0.991744i	$0.459070\pi$
$948$	0	0
$949$	51.8811	1.68413
$950$	0	0
$951$	−87.9377	−2.85157
$952$	0	0
$953$	22.7456	0.736801	0.368401	−	0.929667i	$-0.379905\pi$
$953$	22.7456	0.736801	0.368401	+	0.929667i	$0.379905\pi$
$954$	0	0
$955$	−2.55110	−0.0825518
$956$	0	0
$957$	−101.840	−3.29202
$958$	0	0
$959$	−1.89220	−0.0611024
$960$	0	0
$961$	−24.6272	−0.794426
$962$	0	0
$963$	122.662	3.95273
$964$	0	0
$965$	−6.16330	−0.198404
$966$	0	0
$967$	−39.6655	−1.27556	−0.637779	−	0.770220i	$-0.720147\pi$
$967$	−39.6655	−1.27556	−0.637779	+	0.770220i	$0.720147\pi$
$968$	0	0
$969$	−8.11888	−0.260816
$970$	0	0
$971$	−32.7628	−1.05141	−0.525704	−	0.850668i	$-0.676198\pi$
$971$	−32.7628	−1.05141	−0.525704	+	0.850668i	$0.676198\pi$
$972$	0	0
$973$	6.35720	0.203803
$974$	0	0
$975$	−42.5855	−1.36383
$976$	0	0
$977$	13.5577	0.433750	0.216875	−	0.976199i	$-0.430414\pi$
$977$	13.5577	0.433750	0.216875	+	0.976199i	$0.430414\pi$
$978$	0	0
$979$	−37.8711	−1.21036
$980$	0	0
$981$	−5.58890	−0.178440
$982$	0	0
$983$	47.0278	1.49995	0.749976	−	0.661465i	$-0.230065\pi$
$983$	47.0278	1.49995	0.749976	+	0.661465i	$0.230065\pi$
$984$	0	0
$985$	−20.3033	−0.646917
$986$	0	0
$987$	7.83276	0.249320
$988$	0	0
$989$	−7.25443	−0.230677
$990$	0	0
$991$	−42.8888	−1.36241	−0.681204	−	0.732094i	$-0.738543\pi$
$991$	−42.8888	−1.36241	−0.681204	+	0.732094i	$0.738543\pi$
$992$	0	0
$993$	−108.014	−3.42773
$994$	0	0
$995$	4.86445	0.154213
$996$	0	0
$997$	7.10831	0.225123	0.112561	−	0.993645i	$-0.464095\pi$
$997$	7.10831	0.225123	0.112561	+	0.993645i	$0.464095\pi$
$998$	0	0
$999$	79.3311	2.50992

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2576.2.a.w.1.1		3
4.3	odd	2		322.2.a.g.1.3	✓	3
12.11	even	2		2898.2.a.be.1.3		3
20.19	odd	2		8050.2.a.bh.1.1		3
28.27	even	2		2254.2.a.p.1.1		3
92.91	even	2		7406.2.a.x.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.a.g.1.3	✓	3	4.3	odd	2
2254.2.a.p.1.1		3	28.27	even	2
2576.2.a.w.1.1		3	1.1	even	1	trivial
2898.2.a.be.1.3		3	12.11	even	2
7406.2.a.x.1.3		3	92.91	even	2
8050.2.a.bh.1.1		3	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)

\(f(q)\)	\(=\)	\(q-3.10278 q^{3} -1.10278 q^{5} +1.00000 q^{7} +6.62721 q^{9} +O(q^{10})\)	\(q-3.10278 q^{3} -1.10278 q^{5} +1.00000 q^{7} +6.62721 q^{9} +5.62721 q^{11} -3.62721 q^{13} +3.42166 q^{15} +4.52444 q^{17} +0.578337 q^{19} -3.10278 q^{21} +1.00000 q^{23} -3.78389 q^{25} -11.2544 q^{27} +5.83276 q^{29} +2.52444 q^{31} -17.4600 q^{33} -1.10278 q^{35} -7.04888 q^{37} +11.2544 q^{39} +3.15667 q^{41} -7.25443 q^{43} -7.30833 q^{45} -2.52444 q^{47} +1.00000 q^{49} -14.0383 q^{51} +3.04888 q^{53} -6.20555 q^{55} -1.79445 q^{57} -9.30833 q^{59} +8.35720 q^{61} +6.62721 q^{63} +4.00000 q^{65} -5.62721 q^{67} -3.10278 q^{69} +13.0489 q^{71} -14.3033 q^{73} +11.7406 q^{75} +5.62721 q^{77} +16.9894 q^{79} +15.0383 q^{81} -13.6272 q^{83} -4.98944 q^{85} -18.0978 q^{87} -6.72999 q^{89} -3.62721 q^{91} -7.83276 q^{93} -0.637776 q^{95} +16.9355 q^{97} +37.2927 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + 4 q^{5} + 3 q^{7} + 7 q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 + 4 * q^5 + 3 * q^7 + 7 * q^9	\( 3 q - 2 q^{3} + 4 q^{5} + 3 q^{7} + 7 q^{9} + 4 q^{11} + 2 q^{13} + 12 q^{15} + 8 q^{17} - 2 q^{21} + 3 q^{23} + 5 q^{25} - 8 q^{27} - 10 q^{29} + 2 q^{31} - 12 q^{33} + 4 q^{35} - 10 q^{37} + 8 q^{39} + 6 q^{41} + 4 q^{43} - 2 q^{47} + 3 q^{49} - 2 q^{53} - 4 q^{55} - 20 q^{57} - 6 q^{59} - 8 q^{61} + 7 q^{63} + 12 q^{65} - 4 q^{67} - 2 q^{69} + 28 q^{71} - 6 q^{73} + 46 q^{75} + 4 q^{77} + 20 q^{79} + 3 q^{81} - 28 q^{83} + 16 q^{85} - 32 q^{87} + 2 q^{91} + 4 q^{93} - 20 q^{95} + 16 q^{97} + 44 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 + 4 * q^5 + 3 * q^7 + 7 * q^9 + 4 * q^11 + 2 * q^13 + 12 * q^15 + 8 * q^17 - 2 * q^21 + 3 * q^23 + 5 * q^25 - 8 * q^27 - 10 * q^29 + 2 * q^31 - 12 * q^33 + 4 * q^35 - 10 * q^37 + 8 * q^39 + 6 * q^41 + 4 * q^43 - 2 * q^47 + 3 * q^49 - 2 * q^53 - 4 * q^55 - 20 * q^57 - 6 * q^59 - 8 * q^61 + 7 * q^63 + 12 * q^65 - 4 * q^67 - 2 * q^69 + 28 * q^71 - 6 * q^73 + 46 * q^75 + 4 * q^77 + 20 * q^79 + 3 * q^81 - 28 * q^83 + 16 * q^85 - 32 * q^87 + 2 * q^91 + 4 * q^93 - 20 * q^95 + 16 * q^97 + 44 * q^99

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