LMFDB - Embedded newform 2576.2.a.s.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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
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			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.23607 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.23607 q^{5} +1.00000 q^{7} -2.00000 q^{9} -4.47214 q^{11} +0.236068 q^{13} -3.23607 q^{15} +7.23607 q^{19} +1.00000 q^{21} +1.00000 q^{23} +5.47214 q^{25} -5.00000 q^{27} -1.47214 q^{29} +9.00000 q^{31} -4.47214 q^{33} -3.23607 q^{35} -5.70820 q^{37} +0.236068 q^{39} -2.23607 q^{41} -2.47214 q^{43} +6.47214 q^{45} +3.47214 q^{47} +1.00000 q^{49} +11.2361 q^{53} +14.4721 q^{55} +7.23607 q^{57} +1.52786 q^{59} +13.4164 q^{61} -2.00000 q^{63} -0.763932 q^{65} +12.1803 q^{67} +1.00000 q^{69} +10.2361 q^{71} -6.70820 q^{73} +5.47214 q^{75} -4.47214 q^{77} +7.23607 q^{79} +1.00000 q^{81} -6.47214 q^{83} -1.47214 q^{87} +8.94427 q^{89} +0.236068 q^{91} +9.00000 q^{93} -23.4164 q^{95} +3.70820 q^{97} +8.94427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 - 4 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} - 4 q^{9} - 4 q^{13} - 2 q^{15} + 10 q^{19} + 2 q^{21} + 2 q^{23} + 2 q^{25} - 10 q^{27} + 6 q^{29} + 18 q^{31} - 2 q^{35} + 2 q^{37} - 4 q^{39} + 4 q^{43} + 4 q^{45} - 2 q^{47} + 2 q^{49} + 18 q^{53} + 20 q^{55} + 10 q^{57} + 12 q^{59} - 4 q^{63} - 6 q^{65} + 2 q^{67} + 2 q^{69} + 16 q^{71} + 2 q^{75} + 10 q^{79} + 2 q^{81} - 4 q^{83} + 6 q^{87} - 4 q^{91} + 18 q^{93} - 20 q^{95} - 6 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 - 4 * q^9 - 4 * q^13 - 2 * q^15 + 10 * q^19 + 2 * q^21 + 2 * q^23 + 2 * q^25 - 10 * q^27 + 6 * q^29 + 18 * q^31 - 2 * q^35 + 2 * q^37 - 4 * q^39 + 4 * q^43 + 4 * q^45 - 2 * q^47 + 2 * q^49 + 18 * q^53 + 20 * q^55 + 10 * q^57 + 12 * q^59 - 4 * q^63 - 6 * q^65 + 2 * q^67 + 2 * q^69 + 16 * q^71 + 2 * q^75 + 10 * q^79 + 2 * q^81 - 4 * q^83 + 6 * q^87 - 4 * q^91 + 18 * q^93 - 20 * q^95 - 6 * q^97

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
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−4.47214	−1.34840	−0.674200	−	0.738549i	$-0.735511\pi$
$11$	−4.47214	−1.34840	−0.674200	+	0.738549i	$0.735511\pi$
$12$	0	0
$13$	0.236068	0.0654735	0.0327367	−	0.999464i	$-0.489578\pi$
$13$	0.236068	0.0654735	0.0327367	+	0.999464i	$0.489578\pi$
$14$	0	0
$15$	−3.23607	−0.835549
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	7.23607	1.66007	0.830034	−	0.557713i	$-0.188321\pi$
$19$	7.23607	1.66007	0.830034	+	0.557713i	$0.188321\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	−1.47214	−0.273369	−0.136684	−	0.990615i	$-0.543645\pi$
$29$	−1.47214	−0.273369	−0.136684	+	0.990615i	$0.543645\pi$
$30$	0	0
$31$	9.00000	1.61645	0.808224	−	0.588875i	$-0.200429\pi$
$31$	9.00000	1.61645	0.808224	+	0.588875i	$0.200429\pi$
$32$	0	0
$33$	−4.47214	−0.778499
$34$	0	0
$35$	−3.23607	−0.546995
$36$	0	0
$37$	−5.70820	−0.938423	−0.469211	−	0.883086i	$-0.655462\pi$
$37$	−5.70820	−0.938423	−0.469211	+	0.883086i	$0.655462\pi$
$38$	0	0
$39$	0.236068	0.0378011
$40$	0	0
$41$	−2.23607	−0.349215	−0.174608	−	0.984638i	$-0.555866\pi$
$41$	−2.23607	−0.349215	−0.174608	+	0.984638i	$0.555866\pi$
$42$	0	0
$43$	−2.47214	−0.376997	−0.188499	−	0.982073i	$-0.560362\pi$
$43$	−2.47214	−0.376997	−0.188499	+	0.982073i	$0.560362\pi$
$44$	0	0
$45$	6.47214	0.964809
$46$	0	0
$47$	3.47214	0.506463	0.253232	−	0.967406i	$-0.418507\pi$
$47$	3.47214	0.506463	0.253232	+	0.967406i	$0.418507\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.2361	1.54339	0.771696	−	0.635991i	$-0.219409\pi$
$53$	11.2361	1.54339	0.771696	+	0.635991i	$0.219409\pi$
$54$	0	0
$55$	14.4721	1.95142
$56$	0	0
$57$	7.23607	0.958441
$58$	0	0
$59$	1.52786	0.198911	0.0994555	−	0.995042i	$-0.468290\pi$
$59$	1.52786	0.198911	0.0994555	+	0.995042i	$0.468290\pi$
$60$	0	0
$61$	13.4164	1.71780	0.858898	−	0.512148i	$-0.171150\pi$
$61$	13.4164	1.71780	0.858898	+	0.512148i	$0.171150\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	−0.763932	−0.0947541
$66$	0	0
$67$	12.1803	1.48807	0.744033	−	0.668143i	$-0.232911\pi$
$67$	12.1803	1.48807	0.744033	+	0.668143i	$0.232911\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	10.2361	1.21480	0.607399	−	0.794397i	$-0.292213\pi$
$71$	10.2361	1.21480	0.607399	+	0.794397i	$0.292213\pi$
$72$	0	0
$73$	−6.70820	−0.785136	−0.392568	−	0.919723i	$-0.628413\pi$
$73$	−6.70820	−0.785136	−0.392568	+	0.919723i	$0.628413\pi$
$74$	0	0
$75$	5.47214	0.631868
$76$	0	0
$77$	−4.47214	−0.509647
$78$	0	0
$79$	7.23607	0.814121	0.407061	−	0.913401i	$-0.366554\pi$
$79$	7.23607	0.814121	0.407061	+	0.913401i	$0.366554\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.47214	−0.710409	−0.355205	−	0.934789i	$-0.615589\pi$
$83$	−6.47214	−0.710409	−0.355205	+	0.934789i	$0.615589\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.47214	−0.157830
$88$	0	0
$89$	8.94427	0.948091	0.474045	−	0.880500i	$-0.342793\pi$
$89$	8.94427	0.948091	0.474045	+	0.880500i	$0.342793\pi$
$90$	0	0
$91$	0.236068	0.0247466
$92$	0	0
$93$	9.00000	0.933257
$94$	0	0
$95$	−23.4164	−2.40247
$96$	0	0
$97$	3.70820	0.376511	0.188256	−	0.982120i	$-0.439717\pi$
$97$	3.70820	0.376511	0.188256	+	0.982120i	$0.439717\pi$
$98$	0	0
$99$	8.94427	0.898933
$100$	0	0
$101$	−13.4164	−1.33498	−0.667491	−	0.744618i	$-0.732632\pi$
$101$	−13.4164	−1.33498	−0.667491	+	0.744618i	$0.732632\pi$
$102$	0	0
$103$	15.7082	1.54778	0.773888	−	0.633323i	$-0.218309\pi$
$103$	15.7082	1.54778	0.773888	+	0.633323i	$0.218309\pi$
$104$	0	0
$105$	−3.23607	−0.315808
$106$	0	0
$107$	−11.2361	−1.08623	−0.543116	−	0.839658i	$-0.682756\pi$
$107$	−11.2361	−1.08623	−0.543116	+	0.839658i	$0.682756\pi$
$108$	0	0
$109$	−15.4164	−1.47662	−0.738312	−	0.674459i	$-0.764377\pi$
$109$	−15.4164	−1.47662	−0.738312	+	0.674459i	$0.764377\pi$
$110$	0	0
$111$	−5.70820	−0.541799
$112$	0	0
$113$	2.47214	0.232559	0.116279	−	0.993217i	$-0.462903\pi$
$113$	2.47214	0.232559	0.116279	+	0.993217i	$0.462903\pi$
$114$	0	0
$115$	−3.23607	−0.301765
$116$	0	0
$117$	−0.472136	−0.0436490
$118$	0	0
$119$	0	0
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	−2.23607	−0.201619
$124$	0	0
$125$	−1.52786	−0.136656
$126$	0	0
$127$	2.70820	0.240314	0.120157	−	0.992755i	$-0.461660\pi$
$127$	2.70820	0.240314	0.120157	+	0.992755i	$0.461660\pi$
$128$	0	0
$129$	−2.47214	−0.217659
$130$	0	0
$131$	−3.94427	−0.344613	−0.172306	−	0.985043i	$-0.555122\pi$
$131$	−3.94427	−0.344613	−0.172306	+	0.985043i	$0.555122\pi$
$132$	0	0
$133$	7.23607	0.627447
$134$	0	0
$135$	16.1803	1.39258
$136$	0	0
$137$	15.7082	1.34204	0.671021	−	0.741438i	$-0.265856\pi$
$137$	15.7082	1.34204	0.671021	+	0.741438i	$0.265856\pi$
$138$	0	0
$139$	−2.52786	−0.214411	−0.107205	−	0.994237i	$-0.534190\pi$
$139$	−2.52786	−0.214411	−0.107205	+	0.994237i	$0.534190\pi$
$140$	0	0
$141$	3.47214	0.292407
$142$	0	0
$143$	−1.05573	−0.0882844
$144$	0	0
$145$	4.76393	0.395623
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	15.2361	1.24819	0.624094	−	0.781350i	$-0.285468\pi$
$149$	15.2361	1.24819	0.624094	+	0.781350i	$0.285468\pi$
$150$	0	0
$151$	15.1803	1.23536	0.617679	−	0.786430i	$-0.288073\pi$
$151$	15.1803	1.23536	0.617679	+	0.786430i	$0.288073\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−29.1246	−2.33935
$156$	0	0
$157$	15.4164	1.23036	0.615182	−	0.788385i	$-0.289083\pi$
$157$	15.4164	1.23036	0.615182	+	0.788385i	$0.289083\pi$
$158$	0	0
$159$	11.2361	0.891078
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	−19.1803	−1.50232	−0.751160	−	0.660120i	$-0.770505\pi$
$163$	−19.1803	−1.50232	−0.751160	+	0.660120i	$0.770505\pi$
$164$	0	0
$165$	14.4721	1.12665
$166$	0	0
$167$	−21.8885	−1.69379	−0.846893	−	0.531763i	$-0.821530\pi$
$167$	−21.8885	−1.69379	−0.846893	+	0.531763i	$0.821530\pi$
$168$	0	0
$169$	−12.9443	−0.995713
$170$	0	0
$171$	−14.4721	−1.10671
$172$	0	0
$173$	−3.52786	−0.268219	−0.134109	−	0.990967i	$-0.542817\pi$
$173$	−3.52786	−0.268219	−0.134109	+	0.990967i	$0.542817\pi$
$174$	0	0
$175$	5.47214	0.413655
$176$	0	0
$177$	1.52786	0.114841
$178$	0	0
$179$	18.7082	1.39832	0.699158	−	0.714967i	$-0.253558\pi$
$179$	18.7082	1.39832	0.699158	+	0.714967i	$0.253558\pi$
$180$	0	0
$181$	−5.05573	−0.375789	−0.187895	−	0.982189i	$-0.560166\pi$
$181$	−5.05573	−0.375789	−0.187895	+	0.982189i	$0.560166\pi$
$182$	0	0
$183$	13.4164	0.991769
$184$	0	0
$185$	18.4721	1.35810
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−5.00000	−0.363696
$190$	0	0
$191$	24.1803	1.74963	0.874814	−	0.484459i	$-0.160984\pi$
$191$	24.1803	1.74963	0.874814	+	0.484459i	$0.160984\pi$
$192$	0	0
$193$	4.41641	0.317900	0.158950	−	0.987287i	$-0.449189\pi$
$193$	4.41641	0.317900	0.158950	+	0.987287i	$0.449189\pi$
$194$	0	0
$195$	−0.763932	−0.0547063
$196$	0	0
$197$	−21.4721	−1.52983	−0.764913	−	0.644133i	$-0.777218\pi$
$197$	−21.4721	−1.52983	−0.764913	+	0.644133i	$0.777218\pi$
$198$	0	0
$199$	−15.8885	−1.12631	−0.563155	−	0.826352i	$-0.690412\pi$
$199$	−15.8885	−1.12631	−0.563155	+	0.826352i	$0.690412\pi$
$200$	0	0
$201$	12.1803	0.859135
$202$	0	0
$203$	−1.47214	−0.103324
$204$	0	0
$205$	7.23607	0.505389
$206$	0	0
$207$	−2.00000	−0.139010
$208$	0	0
$209$	−32.3607	−2.23844
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	0	0
$213$	10.2361	0.701364
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	9.00000	0.610960
$218$	0	0
$219$	−6.70820	−0.453298
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−3.41641	−0.228780	−0.114390	−	0.993436i	$-0.536491\pi$
$223$	−3.41641	−0.228780	−0.114390	+	0.993436i	$0.536491\pi$
$224$	0	0
$225$	−10.9443	−0.729618
$226$	0	0
$227$	5.88854	0.390836	0.195418	−	0.980720i	$-0.437394\pi$
$227$	5.88854	0.390836	0.195418	+	0.980720i	$0.437394\pi$
$228$	0	0
$229$	14.7639	0.975628	0.487814	−	0.872948i	$-0.337794\pi$
$229$	14.7639	0.975628	0.487814	+	0.872948i	$0.337794\pi$
$230$	0	0
$231$	−4.47214	−0.294245
$232$	0	0
$233$	−11.4721	−0.751565	−0.375782	−	0.926708i	$-0.622626\pi$
$233$	−11.4721	−0.751565	−0.375782	+	0.926708i	$0.622626\pi$
$234$	0	0
$235$	−11.2361	−0.732960
$236$	0	0
$237$	7.23607	0.470033
$238$	0	0
$239$	15.7639	1.01968	0.509842	−	0.860268i	$-0.329704\pi$
$239$	15.7639	1.01968	0.509842	+	0.860268i	$0.329704\pi$
$240$	0	0
$241$	−21.4164	−1.37955	−0.689776	−	0.724023i	$-0.742291\pi$
$241$	−21.4164	−1.37955	−0.689776	+	0.724023i	$0.742291\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	−3.23607	−0.206745
$246$	0	0
$247$	1.70820	0.108690
$248$	0	0
$249$	−6.47214	−0.410155
$250$	0	0
$251$	−8.29180	−0.523374	−0.261687	−	0.965153i	$-0.584279\pi$
$251$	−8.29180	−0.523374	−0.261687	+	0.965153i	$0.584279\pi$
$252$	0	0
$253$	−4.47214	−0.281161
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.23607	0.264239	0.132119	−	0.991234i	$-0.457822\pi$
$257$	4.23607	0.264239	0.132119	+	0.991234i	$0.457822\pi$
$258$	0	0
$259$	−5.70820	−0.354691
$260$	0	0
$261$	2.94427	0.182246
$262$	0	0
$263$	26.9443	1.66145	0.830727	−	0.556679i	$-0.187925\pi$
$263$	26.9443	1.66145	0.830727	+	0.556679i	$0.187925\pi$
$264$	0	0
$265$	−36.3607	−2.23362
$266$	0	0
$267$	8.94427	0.547381
$268$	0	0
$269$	−9.18034	−0.559735	−0.279868	−	0.960039i	$-0.590291\pi$
$269$	−9.18034	−0.559735	−0.279868	+	0.960039i	$0.590291\pi$
$270$	0	0
$271$	16.9443	1.02929	0.514646	−	0.857403i	$-0.327924\pi$
$271$	16.9443	1.02929	0.514646	+	0.857403i	$0.327924\pi$
$272$	0	0
$273$	0.236068	0.0142875
$274$	0	0
$275$	−24.4721	−1.47573
$276$	0	0
$277$	20.4164	1.22670	0.613352	−	0.789810i	$-0.289821\pi$
$277$	20.4164	1.22670	0.613352	+	0.789810i	$0.289821\pi$
$278$	0	0
$279$	−18.0000	−1.07763
$280$	0	0
$281$	3.70820	0.221213	0.110606	−	0.993864i	$-0.464721\pi$
$281$	3.70820	0.221213	0.110606	+	0.993864i	$0.464721\pi$
$282$	0	0
$283$	−18.9443	−1.12612	−0.563060	−	0.826416i	$-0.690376\pi$
$283$	−18.9443	−1.12612	−0.563060	+	0.826416i	$0.690376\pi$
$284$	0	0
$285$	−23.4164	−1.38707
$286$	0	0
$287$	−2.23607	−0.131991
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	3.70820	0.217379
$292$	0	0
$293$	−15.7082	−0.917683	−0.458842	−	0.888518i	$-0.651735\pi$
$293$	−15.7082	−0.917683	−0.458842	+	0.888518i	$0.651735\pi$
$294$	0	0
$295$	−4.94427	−0.287867
$296$	0	0
$297$	22.3607	1.29750
$298$	0	0
$299$	0.236068	0.0136522
$300$	0	0
$301$	−2.47214	−0.142492
$302$	0	0
$303$	−13.4164	−0.770752
$304$	0	0
$305$	−43.4164	−2.48602
$306$	0	0
$307$	11.4164	0.651569	0.325784	−	0.945444i	$-0.394372\pi$
$307$	11.4164	0.651569	0.325784	+	0.945444i	$0.394372\pi$
$308$	0	0
$309$	15.7082	0.893609
$310$	0	0
$311$	2.88854	0.163794	0.0818971	−	0.996641i	$-0.473902\pi$
$311$	2.88854	0.163794	0.0818971	+	0.996641i	$0.473902\pi$
$312$	0	0
$313$	2.76393	0.156227	0.0781133	−	0.996944i	$-0.475110\pi$
$313$	2.76393	0.156227	0.0781133	+	0.996944i	$0.475110\pi$
$314$	0	0
$315$	6.47214	0.364664
$316$	0	0
$317$	31.3050	1.75826	0.879131	−	0.476581i	$-0.158124\pi$
$317$	31.3050	1.75826	0.879131	+	0.476581i	$0.158124\pi$
$318$	0	0
$319$	6.58359	0.368610
$320$	0	0
$321$	−11.2361	−0.627136
$322$	0	0
$323$	0	0
$324$	0	0
$325$	1.29180	0.0716560
$326$	0	0
$327$	−15.4164	−0.852529
$328$	0	0
$329$	3.47214	0.191425
$330$	0	0
$331$	−0.708204	−0.0389264	−0.0194632	−	0.999811i	$-0.506196\pi$
$331$	−0.708204	−0.0389264	−0.0194632	+	0.999811i	$0.506196\pi$
$332$	0	0
$333$	11.4164	0.625615
$334$	0	0
$335$	−39.4164	−2.15355
$336$	0	0
$337$	−17.5967	−0.958556	−0.479278	−	0.877663i	$-0.659101\pi$
$337$	−17.5967	−0.958556	−0.479278	+	0.877663i	$0.659101\pi$
$338$	0	0
$339$	2.47214	0.134268
$340$	0	0
$341$	−40.2492	−2.17962
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−3.23607	−0.174224
$346$	0	0
$347$	5.88854	0.316114	0.158057	−	0.987430i	$-0.449477\pi$
$347$	5.88854	0.316114	0.158057	+	0.987430i	$0.449477\pi$
$348$	0	0
$349$	1.29180	0.0691483	0.0345741	−	0.999402i	$-0.488993\pi$
$349$	1.29180	0.0691483	0.0345741	+	0.999402i	$0.488993\pi$
$350$	0	0
$351$	−1.18034	−0.0630019
$352$	0	0
$353$	−1.76393	−0.0938846	−0.0469423	−	0.998898i	$-0.514948\pi$
$353$	−1.76393	−0.0938846	−0.0469423	+	0.998898i	$0.514948\pi$
$354$	0	0
$355$	−33.1246	−1.75807
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.76393	−0.251431	−0.125715	−	0.992066i	$-0.540123\pi$
$359$	−4.76393	−0.251431	−0.125715	+	0.992066i	$0.540123\pi$
$360$	0	0
$361$	33.3607	1.75583
$362$	0	0
$363$	9.00000	0.472377
$364$	0	0
$365$	21.7082	1.13626
$366$	0	0
$367$	4.58359	0.239262	0.119631	−	0.992818i	$-0.461829\pi$
$367$	4.58359	0.239262	0.119631	+	0.992818i	$0.461829\pi$
$368$	0	0
$369$	4.47214	0.232810
$370$	0	0
$371$	11.2361	0.583348
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	−1.52786	−0.0788986
$376$	0	0
$377$	−0.347524	−0.0178984
$378$	0	0
$379$	−8.29180	−0.425921	−0.212960	−	0.977061i	$-0.568311\pi$
$379$	−8.29180	−0.425921	−0.212960	+	0.977061i	$0.568311\pi$
$380$	0	0
$381$	2.70820	0.138745
$382$	0	0
$383$	15.7082	0.802652	0.401326	−	0.915935i	$-0.368549\pi$
$383$	15.7082	0.802652	0.401326	+	0.915935i	$0.368549\pi$
$384$	0	0
$385$	14.4721	0.737568
$386$	0	0
$387$	4.94427	0.251331
$388$	0	0
$389$	−7.41641	−0.376027	−0.188013	−	0.982166i	$-0.560205\pi$
$389$	−7.41641	−0.376027	−0.188013	+	0.982166i	$0.560205\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−3.94427	−0.198962
$394$	0	0
$395$	−23.4164	−1.17821
$396$	0	0
$397$	−25.6525	−1.28746	−0.643730	−	0.765252i	$-0.722614\pi$
$397$	−25.6525	−1.28746	−0.643730	+	0.765252i	$0.722614\pi$
$398$	0	0
$399$	7.23607	0.362257
$400$	0	0
$401$	−19.8885	−0.993186	−0.496593	−	0.867983i	$-0.665416\pi$
$401$	−19.8885	−0.993186	−0.496593	+	0.867983i	$0.665416\pi$
$402$	0	0
$403$	2.12461	0.105834
$404$	0	0
$405$	−3.23607	−0.160802
$406$	0	0
$407$	25.5279	1.26537
$408$	0	0
$409$	4.12461	0.203949	0.101974	−	0.994787i	$-0.467484\pi$
$409$	4.12461	0.203949	0.101974	+	0.994787i	$0.467484\pi$
$410$	0	0
$411$	15.7082	0.774829
$412$	0	0
$413$	1.52786	0.0751813
$414$	0	0
$415$	20.9443	1.02811
$416$	0	0
$417$	−2.52786	−0.123790
$418$	0	0
$419$	−26.4721	−1.29325	−0.646624	−	0.762809i	$-0.723820\pi$
$419$	−26.4721	−1.29325	−0.646624	+	0.762809i	$0.723820\pi$
$420$	0	0
$421$	−8.58359	−0.418339	−0.209169	−	0.977879i	$-0.567076\pi$
$421$	−8.58359	−0.418339	−0.209169	+	0.977879i	$0.567076\pi$
$422$	0	0
$423$	−6.94427	−0.337642
$424$	0	0
$425$	0	0
$426$	0	0
$427$	13.4164	0.649265
$428$	0	0
$429$	−1.05573	−0.0509710
$430$	0	0
$431$	−18.1803	−0.875716	−0.437858	−	0.899044i	$-0.644263\pi$
$431$	−18.1803	−0.875716	−0.437858	+	0.899044i	$0.644263\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	4.76393	0.228413
$436$	0	0
$437$	7.23607	0.346148
$438$	0	0
$439$	−30.3050	−1.44638	−0.723188	−	0.690651i	$-0.757324\pi$
$439$	−30.3050	−1.44638	−0.723188	+	0.690651i	$0.757324\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	0	0
$443$	7.18034	0.341148	0.170574	−	0.985345i	$-0.445438\pi$
$443$	7.18034	0.341148	0.170574	+	0.985345i	$0.445438\pi$
$444$	0	0
$445$	−28.9443	−1.37209
$446$	0	0
$447$	15.2361	0.720641
$448$	0	0
$449$	32.8328	1.54948	0.774738	−	0.632282i	$-0.217882\pi$
$449$	32.8328	1.54948	0.774738	+	0.632282i	$0.217882\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	0	0
$453$	15.1803	0.713235
$454$	0	0
$455$	−0.763932	−0.0358137
$456$	0	0
$457$	14.4721	0.676978	0.338489	−	0.940970i	$-0.390084\pi$
$457$	14.4721	0.676978	0.338489	+	0.940970i	$0.390084\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−25.7639	−1.19995	−0.599973	−	0.800020i	$-0.704822\pi$
$461$	−25.7639	−1.19995	−0.599973	+	0.800020i	$0.704822\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	−29.1246	−1.35062
$466$	0	0
$467$	11.5279	0.533446	0.266723	−	0.963773i	$-0.414059\pi$
$467$	11.5279	0.533446	0.266723	+	0.963773i	$0.414059\pi$
$468$	0	0
$469$	12.1803	0.562436
$470$	0	0
$471$	15.4164	0.710351
$472$	0	0
$473$	11.0557	0.508343
$474$	0	0
$475$	39.5967	1.81682
$476$	0	0
$477$	−22.4721	−1.02893
$478$	0	0
$479$	23.1246	1.05659	0.528295	−	0.849061i	$-0.322831\pi$
$479$	23.1246	1.05659	0.528295	+	0.849061i	$0.322831\pi$
$480$	0	0
$481$	−1.34752	−0.0614418
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−12.0000	−0.544892
$486$	0	0
$487$	−20.1246	−0.911933	−0.455967	−	0.889997i	$-0.650706\pi$
$487$	−20.1246	−0.911933	−0.455967	+	0.889997i	$0.650706\pi$
$488$	0	0
$489$	−19.1803	−0.867365
$490$	0	0
$491$	−11.7639	−0.530899	−0.265449	−	0.964125i	$-0.585520\pi$
$491$	−11.7639	−0.530899	−0.265449	+	0.964125i	$0.585520\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−28.9443	−1.30095
$496$	0	0
$497$	10.2361	0.459150
$498$	0	0
$499$	36.7082	1.64328	0.821642	−	0.570003i	$-0.193058\pi$
$499$	36.7082	1.64328	0.821642	+	0.570003i	$0.193058\pi$
$500$	0	0
$501$	−21.8885	−0.977908
$502$	0	0
$503$	15.5967	0.695425	0.347712	−	0.937601i	$-0.386959\pi$
$503$	15.5967	0.695425	0.347712	+	0.937601i	$0.386959\pi$
$504$	0	0
$505$	43.4164	1.93200
$506$	0	0
$507$	−12.9443	−0.574875
$508$	0	0
$509$	−12.8197	−0.568221	−0.284111	−	0.958791i	$-0.591698\pi$
$509$	−12.8197	−0.568221	−0.284111	+	0.958791i	$0.591698\pi$
$510$	0	0
$511$	−6.70820	−0.296753
$512$	0	0
$513$	−36.1803	−1.59740
$514$	0	0
$515$	−50.8328	−2.23996
$516$	0	0
$517$	−15.5279	−0.682915
$518$	0	0
$519$	−3.52786	−0.154856
$520$	0	0
$521$	21.3050	0.933387	0.466693	−	0.884419i	$-0.345445\pi$
$521$	21.3050	0.933387	0.466693	+	0.884419i	$0.345445\pi$
$522$	0	0
$523$	7.70820	0.337056	0.168528	−	0.985697i	$-0.446099\pi$
$523$	7.70820	0.337056	0.168528	+	0.985697i	$0.446099\pi$
$524$	0	0
$525$	5.47214	0.238824
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−3.05573	−0.132607
$532$	0	0
$533$	−0.527864	−0.0228643
$534$	0	0
$535$	36.3607	1.57201
$536$	0	0
$537$	18.7082	0.807319
$538$	0	0
$539$	−4.47214	−0.192629
$540$	0	0
$541$	21.0000	0.902861	0.451430	−	0.892306i	$-0.350914\pi$
$541$	21.0000	0.902861	0.451430	+	0.892306i	$0.350914\pi$
$542$	0	0
$543$	−5.05573	−0.216962
$544$	0	0
$545$	49.8885	2.13699
$546$	0	0
$547$	−43.1803	−1.84626	−0.923129	−	0.384490i	$-0.874377\pi$
$547$	−43.1803	−1.84626	−0.923129	+	0.384490i	$0.874377\pi$
$548$	0	0
$549$	−26.8328	−1.14520
$550$	0	0
$551$	−10.6525	−0.453811
$552$	0	0
$553$	7.23607	0.307709
$554$	0	0
$555$	18.4721	0.784099
$556$	0	0
$557$	−26.7639	−1.13402	−0.567012	−	0.823709i	$-0.691901\pi$
$557$	−26.7639	−1.13402	−0.567012	+	0.823709i	$0.691901\pi$
$558$	0	0
$559$	−0.583592	−0.0246833
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.59675	0.404455	0.202227	−	0.979339i	$-0.435182\pi$
$563$	9.59675	0.404455	0.202227	+	0.979339i	$0.435182\pi$
$564$	0	0
$565$	−8.00000	−0.336563
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−8.18034	−0.342938	−0.171469	−	0.985190i	$-0.554851\pi$
$569$	−8.18034	−0.342938	−0.171469	+	0.985190i	$0.554851\pi$
$570$	0	0
$571$	−16.2918	−0.681790	−0.340895	−	0.940101i	$-0.610730\pi$
$571$	−16.2918	−0.681790	−0.340895	+	0.940101i	$0.610730\pi$
$572$	0	0
$573$	24.1803	1.01015
$574$	0	0
$575$	5.47214	0.228204
$576$	0	0
$577$	−15.2918	−0.636606	−0.318303	−	0.947989i	$-0.603113\pi$
$577$	−15.2918	−0.636606	−0.318303	+	0.947989i	$0.603113\pi$
$578$	0	0
$579$	4.41641	0.183540
$580$	0	0
$581$	−6.47214	−0.268509
$582$	0	0
$583$	−50.2492	−2.08111
$584$	0	0
$585$	1.52786	0.0631694
$586$	0	0
$587$	−20.8885	−0.862162	−0.431081	−	0.902313i	$-0.641868\pi$
$587$	−20.8885	−0.862162	−0.431081	+	0.902313i	$0.641868\pi$
$588$	0	0
$589$	65.1246	2.68341
$590$	0	0
$591$	−21.4721	−0.883246
$592$	0	0
$593$	34.3607	1.41102	0.705512	−	0.708698i	$-0.250717\pi$
$593$	34.3607	1.41102	0.705512	+	0.708698i	$0.250717\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−15.8885	−0.650275
$598$	0	0
$599$	2.47214	0.101009	0.0505044	−	0.998724i	$-0.483917\pi$
$599$	2.47214	0.101009	0.0505044	+	0.998724i	$0.483917\pi$
$600$	0	0
$601$	22.2361	0.907028	0.453514	−	0.891249i	$-0.350170\pi$
$601$	22.2361	0.907028	0.453514	+	0.891249i	$0.350170\pi$
$602$	0	0
$603$	−24.3607	−0.992044
$604$	0	0
$605$	−29.1246	−1.18408
$606$	0	0
$607$	33.3050	1.35181	0.675903	−	0.736990i	$-0.263754\pi$
$607$	33.3050	1.35181	0.675903	+	0.736990i	$0.263754\pi$
$608$	0	0
$609$	−1.47214	−0.0596540
$610$	0	0
$611$	0.819660	0.0331599
$612$	0	0
$613$	−22.1803	−0.895855	−0.447928	−	0.894070i	$-0.647838\pi$
$613$	−22.1803	−0.895855	−0.447928	+	0.894070i	$0.647838\pi$
$614$	0	0
$615$	7.23607	0.291786
$616$	0	0
$617$	35.2361	1.41855	0.709275	−	0.704932i	$-0.249022\pi$
$617$	35.2361	1.41855	0.709275	+	0.704932i	$0.249022\pi$
$618$	0	0
$619$	47.4853	1.90860	0.954298	−	0.298858i	$-0.0966058\pi$
$619$	47.4853	1.90860	0.954298	+	0.298858i	$0.0966058\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	0	0
$623$	8.94427	0.358345
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	−32.3607	−1.29236
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−3.41641	−0.136005	−0.0680025	−	0.997685i	$-0.521663\pi$
$631$	−3.41641	−0.136005	−0.0680025	+	0.997685i	$0.521663\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	0	0
$635$	−8.76393	−0.347786
$636$	0	0
$637$	0.236068	0.00935335
$638$	0	0
$639$	−20.4721	−0.809865
$640$	0	0
$641$	7.05573	0.278685	0.139342	−	0.990244i	$-0.455501\pi$
$641$	7.05573	0.278685	0.139342	+	0.990244i	$0.455501\pi$
$642$	0	0
$643$	17.7082	0.698343	0.349172	−	0.937059i	$-0.386463\pi$
$643$	17.7082	0.698343	0.349172	+	0.937059i	$0.386463\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	18.5279	0.728405	0.364203	−	0.931320i	$-0.381342\pi$
$647$	18.5279	0.728405	0.364203	+	0.931320i	$0.381342\pi$
$648$	0	0
$649$	−6.83282	−0.268211
$650$	0	0
$651$	9.00000	0.352738
$652$	0	0
$653$	24.5279	0.959849	0.479925	−	0.877310i	$-0.340664\pi$
$653$	24.5279	0.959849	0.479925	+	0.877310i	$0.340664\pi$
$654$	0	0
$655$	12.7639	0.498728
$656$	0	0
$657$	13.4164	0.523424
$658$	0	0
$659$	−6.76393	−0.263485	−0.131743	−	0.991284i	$-0.542057\pi$
$659$	−6.76393	−0.263485	−0.131743	+	0.991284i	$0.542057\pi$
$660$	0	0
$661$	−41.7082	−1.62226	−0.811131	−	0.584865i	$-0.801147\pi$
$661$	−41.7082	−1.62226	−0.811131	+	0.584865i	$0.801147\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−23.4164	−0.908049
$666$	0	0
$667$	−1.47214	−0.0570013
$668$	0	0
$669$	−3.41641	−0.132086
$670$	0	0
$671$	−60.0000	−2.31627
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	0	0
$675$	−27.3607	−1.05311
$676$	0	0
$677$	23.2361	0.893035	0.446517	−	0.894775i	$-0.352664\pi$
$677$	23.2361	0.893035	0.446517	+	0.894775i	$0.352664\pi$
$678$	0	0
$679$	3.70820	0.142308
$680$	0	0
$681$	5.88854	0.225649
$682$	0	0
$683$	5.18034	0.198220	0.0991101	−	0.995076i	$-0.468400\pi$
$683$	5.18034	0.198220	0.0991101	+	0.995076i	$0.468400\pi$
$684$	0	0
$685$	−50.8328	−1.94222
$686$	0	0
$687$	14.7639	0.563279
$688$	0	0
$689$	2.65248	0.101051
$690$	0	0
$691$	−42.8328	−1.62944	−0.814719	−	0.579857i	$-0.803109\pi$
$691$	−42.8328	−1.62944	−0.814719	+	0.579857i	$0.803109\pi$
$692$	0	0
$693$	8.94427	0.339765
$694$	0	0
$695$	8.18034	0.310298
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−11.4721	−0.433916
$700$	0	0
$701$	−22.7639	−0.859782	−0.429891	−	0.902881i	$-0.641448\pi$
$701$	−22.7639	−0.859782	−0.429891	+	0.902881i	$0.641448\pi$
$702$	0	0
$703$	−41.3050	−1.55785
$704$	0	0
$705$	−11.2361	−0.423175
$706$	0	0
$707$	−13.4164	−0.504576
$708$	0	0
$709$	34.2492	1.28626	0.643128	−	0.765758i	$-0.277636\pi$
$709$	34.2492	1.28626	0.643128	+	0.765758i	$0.277636\pi$
$710$	0	0
$711$	−14.4721	−0.542748
$712$	0	0
$713$	9.00000	0.337053
$714$	0	0
$715$	3.41641	0.127766
$716$	0	0
$717$	15.7639	0.588715
$718$	0	0
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	15.7082	0.585004
$722$	0	0
$723$	−21.4164	−0.796485
$724$	0	0
$725$	−8.05573	−0.299182
$726$	0	0
$727$	−3.23607	−0.120019	−0.0600096	−	0.998198i	$-0.519113\pi$
$727$	−3.23607	−0.120019	−0.0600096	+	0.998198i	$0.519113\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	0	0
$732$	0	0
$733$	12.9443	0.478108	0.239054	−	0.971006i	$-0.423163\pi$
$733$	12.9443	0.478108	0.239054	+	0.971006i	$0.423163\pi$
$734$	0	0
$735$	−3.23607	−0.119364
$736$	0	0
$737$	−54.4721	−2.00651
$738$	0	0
$739$	−8.70820	−0.320336	−0.160168	−	0.987090i	$-0.551204\pi$
$739$	−8.70820	−0.320336	−0.160168	+	0.987090i	$0.551204\pi$
$740$	0	0
$741$	1.70820	0.0627524
$742$	0	0
$743$	25.5279	0.936527	0.468263	−	0.883589i	$-0.344880\pi$
$743$	25.5279	0.936527	0.468263	+	0.883589i	$0.344880\pi$
$744$	0	0
$745$	−49.3050	−1.80639
$746$	0	0
$747$	12.9443	0.473606
$748$	0	0
$749$	−11.2361	−0.410557
$750$	0	0
$751$	8.58359	0.313220	0.156610	−	0.987661i	$-0.449943\pi$
$751$	8.58359	0.313220	0.156610	+	0.987661i	$0.449943\pi$
$752$	0	0
$753$	−8.29180	−0.302170
$754$	0	0
$755$	−49.1246	−1.78783
$756$	0	0
$757$	−28.8328	−1.04795	−0.523973	−	0.851735i	$-0.675551\pi$
$757$	−28.8328	−1.04795	−0.523973	+	0.851735i	$0.675551\pi$
$758$	0	0
$759$	−4.47214	−0.162328
$760$	0	0
$761$	−31.6525	−1.14740	−0.573701	−	0.819065i	$-0.694493\pi$
$761$	−31.6525	−1.14740	−0.573701	+	0.819065i	$0.694493\pi$
$762$	0	0
$763$	−15.4164	−0.558111
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.360680	0.0130234
$768$	0	0
$769$	−43.2361	−1.55913	−0.779566	−	0.626320i	$-0.784560\pi$
$769$	−43.2361	−1.55913	−0.779566	+	0.626320i	$0.784560\pi$
$770$	0	0
$771$	4.23607	0.152558
$772$	0	0
$773$	40.6525	1.46217	0.731084	−	0.682288i	$-0.239015\pi$
$773$	40.6525	1.46217	0.731084	+	0.682288i	$0.239015\pi$
$774$	0	0
$775$	49.2492	1.76908
$776$	0	0
$777$	−5.70820	−0.204781
$778$	0	0
$779$	−16.1803	−0.579721
$780$	0	0
$781$	−45.7771	−1.63803
$782$	0	0
$783$	7.36068	0.263049
$784$	0	0
$785$	−49.8885	−1.78060
$786$	0	0
$787$	0.360680	0.0128568	0.00642842	−	0.999979i	$-0.497954\pi$
$787$	0.360680	0.0128568	0.00642842	+	0.999979i	$0.497954\pi$
$788$	0	0
$789$	26.9443	0.959241
$790$	0	0
$791$	2.47214	0.0878990
$792$	0	0
$793$	3.16718	0.112470
$794$	0	0
$795$	−36.3607	−1.28958
$796$	0	0
$797$	−36.7639	−1.30225	−0.651123	−	0.758973i	$-0.725702\pi$
$797$	−36.7639	−1.30225	−0.651123	+	0.758973i	$0.725702\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−17.8885	−0.632061
$802$	0	0
$803$	30.0000	1.05868
$804$	0	0
$805$	−3.23607	−0.114056
$806$	0	0
$807$	−9.18034	−0.323163
$808$	0	0
$809$	−32.8328	−1.15434	−0.577170	−	0.816624i	$-0.695843\pi$
$809$	−32.8328	−1.15434	−0.577170	+	0.816624i	$0.695843\pi$
$810$	0	0
$811$	45.3607	1.59283	0.796414	−	0.604751i	$-0.206727\pi$
$811$	45.3607	1.59283	0.796414	+	0.604751i	$0.206727\pi$
$812$	0	0
$813$	16.9443	0.594262
$814$	0	0
$815$	62.0689	2.17418
$816$	0	0
$817$	−17.8885	−0.625841
$818$	0	0
$819$	−0.472136	−0.0164978
$820$	0	0
$821$	22.5836	0.788173	0.394086	−	0.919073i	$-0.371061\pi$
$821$	22.5836	0.788173	0.394086	+	0.919073i	$0.371061\pi$
$822$	0	0
$823$	−25.5410	−0.890304	−0.445152	−	0.895455i	$-0.646850\pi$
$823$	−25.5410	−0.890304	−0.445152	+	0.895455i	$0.646850\pi$
$824$	0	0
$825$	−24.4721	−0.852010
$826$	0	0
$827$	17.3050	0.601752	0.300876	−	0.953663i	$-0.402721\pi$
$827$	17.3050	0.601752	0.300876	+	0.953663i	$0.402721\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	20.4164	0.708237
$832$	0	0
$833$	0	0
$834$	0	0
$835$	70.8328	2.45127
$836$	0	0
$837$	−45.0000	−1.55543
$838$	0	0
$839$	−20.0689	−0.692855	−0.346427	−	0.938077i	$-0.612605\pi$
$839$	−20.0689	−0.692855	−0.346427	+	0.938077i	$0.612605\pi$
$840$	0	0
$841$	−26.8328	−0.925270
$842$	0	0
$843$	3.70820	0.127717
$844$	0	0
$845$	41.8885	1.44101
$846$	0	0
$847$	9.00000	0.309244
$848$	0	0
$849$	−18.9443	−0.650166
$850$	0	0
$851$	−5.70820	−0.195675
$852$	0	0
$853$	44.8328	1.53505	0.767523	−	0.641021i	$-0.221489\pi$
$853$	44.8328	1.53505	0.767523	+	0.641021i	$0.221489\pi$
$854$	0	0
$855$	46.8328	1.60165
$856$	0	0
$857$	9.54102	0.325915	0.162958	−	0.986633i	$-0.447897\pi$
$857$	9.54102	0.325915	0.162958	+	0.986633i	$0.447897\pi$
$858$	0	0
$859$	−11.5836	−0.395227	−0.197614	−	0.980280i	$-0.563319\pi$
$859$	−11.5836	−0.395227	−0.197614	+	0.980280i	$0.563319\pi$
$860$	0	0
$861$	−2.23607	−0.0762050
$862$	0	0
$863$	−2.23607	−0.0761166	−0.0380583	−	0.999276i	$-0.512117\pi$
$863$	−2.23607	−0.0761166	−0.0380583	+	0.999276i	$0.512117\pi$
$864$	0	0
$865$	11.4164	0.388170
$866$	0	0
$867$	−17.0000	−0.577350
$868$	0	0
$869$	−32.3607	−1.09776
$870$	0	0
$871$	2.87539	0.0974288
$872$	0	0
$873$	−7.41641	−0.251007
$874$	0	0
$875$	−1.52786	−0.0516512
$876$	0	0
$877$	−42.0000	−1.41824	−0.709120	−	0.705088i	$-0.750907\pi$
$877$	−42.0000	−1.41824	−0.709120	+	0.705088i	$0.750907\pi$
$878$	0	0
$879$	−15.7082	−0.529825
$880$	0	0
$881$	25.5967	0.862376	0.431188	−	0.902262i	$-0.358095\pi$
$881$	25.5967	0.862376	0.431188	+	0.902262i	$0.358095\pi$
$882$	0	0
$883$	−9.88854	−0.332776	−0.166388	−	0.986060i	$-0.553210\pi$
$883$	−9.88854	−0.332776	−0.166388	+	0.986060i	$0.553210\pi$
$884$	0	0
$885$	−4.94427	−0.166200
$886$	0	0
$887$	−16.8885	−0.567062	−0.283531	−	0.958963i	$-0.591506\pi$
$887$	−16.8885	−0.567062	−0.283531	+	0.958963i	$0.591506\pi$
$888$	0	0
$889$	2.70820	0.0908302
$890$	0	0
$891$	−4.47214	−0.149822
$892$	0	0
$893$	25.1246	0.840763
$894$	0	0
$895$	−60.5410	−2.02366
$896$	0	0
$897$	0.236068	0.00788208
$898$	0	0
$899$	−13.2492	−0.441886
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−2.47214	−0.0822675
$904$	0	0
$905$	16.3607	0.543847
$906$	0	0
$907$	−12.5410	−0.416418	−0.208209	−	0.978084i	$-0.566763\pi$
$907$	−12.5410	−0.416418	−0.208209	+	0.978084i	$0.566763\pi$
$908$	0	0
$909$	26.8328	0.889988
$910$	0	0
$911$	21.7771	0.721507	0.360754	−	0.932661i	$-0.382520\pi$
$911$	21.7771	0.721507	0.360754	+	0.932661i	$0.382520\pi$
$912$	0	0
$913$	28.9443	0.957916
$914$	0	0
$915$	−43.4164	−1.43530
$916$	0	0
$917$	−3.94427	−0.130251
$918$	0	0
$919$	−3.70820	−0.122322	−0.0611612	−	0.998128i	$-0.519480\pi$
$919$	−3.70820	−0.122322	−0.0611612	+	0.998128i	$0.519480\pi$
$920$	0	0
$921$	11.4164	0.376183
$922$	0	0
$923$	2.41641	0.0795370
$924$	0	0
$925$	−31.2361	−1.02704
$926$	0	0
$927$	−31.4164	−1.03185
$928$	0	0
$929$	51.6525	1.69466	0.847331	−	0.531065i	$-0.178208\pi$
$929$	51.6525	1.69466	0.847331	+	0.531065i	$0.178208\pi$
$930$	0	0
$931$	7.23607	0.237153
$932$	0	0
$933$	2.88854	0.0945667
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−57.1246	−1.86618	−0.933090	−	0.359643i	$-0.882898\pi$
$937$	−57.1246	−1.86618	−0.933090	+	0.359643i	$0.882898\pi$
$938$	0	0
$939$	2.76393	0.0901975
$940$	0	0
$941$	53.0132	1.72818	0.864090	−	0.503338i	$-0.167895\pi$
$941$	53.0132	1.72818	0.864090	+	0.503338i	$0.167895\pi$
$942$	0	0
$943$	−2.23607	−0.0728164
$944$	0	0
$945$	16.1803	0.526346
$946$	0	0
$947$	22.5967	0.734296	0.367148	−	0.930163i	$-0.380334\pi$
$947$	22.5967	0.734296	0.367148	+	0.930163i	$0.380334\pi$
$948$	0	0
$949$	−1.58359	−0.0514056
$950$	0	0
$951$	31.3050	1.01513
$952$	0	0
$953$	12.1115	0.392329	0.196164	−	0.980571i	$-0.437151\pi$
$953$	12.1115	0.392329	0.196164	+	0.980571i	$0.437151\pi$
$954$	0	0
$955$	−78.2492	−2.53209
$956$	0	0
$957$	6.58359	0.212817
$958$	0	0
$959$	15.7082	0.507244
$960$	0	0
$961$	50.0000	1.61290
$962$	0	0
$963$	22.4721	0.724154
$964$	0	0
$965$	−14.2918	−0.460069
$966$	0	0
$967$	31.0689	0.999108	0.499554	−	0.866283i	$-0.333497\pi$
$967$	31.0689	0.999108	0.499554	+	0.866283i	$0.333497\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−33.7082	−1.08175	−0.540874	−	0.841104i	$-0.681906\pi$
$971$	−33.7082	−1.08175	−0.540874	+	0.841104i	$0.681906\pi$
$972$	0	0
$973$	−2.52786	−0.0810396
$974$	0	0
$975$	1.29180	0.0413706
$976$	0	0
$977$	−18.6525	−0.596746	−0.298373	−	0.954449i	$-0.596444\pi$
$977$	−18.6525	−0.596746	−0.298373	+	0.954449i	$0.596444\pi$
$978$	0	0
$979$	−40.0000	−1.27841
$980$	0	0
$981$	30.8328	0.984416
$982$	0	0
$983$	4.18034	0.133332	0.0666661	−	0.997775i	$-0.478764\pi$
$983$	4.18034	0.133332	0.0666661	+	0.997775i	$0.478764\pi$
$984$	0	0
$985$	69.4853	2.21399
$986$	0	0
$987$	3.47214	0.110519
$988$	0	0
$989$	−2.47214	−0.0786094
$990$	0	0
$991$	48.9443	1.55477	0.777383	−	0.629028i	$-0.216547\pi$
$991$	48.9443	1.55477	0.777383	+	0.629028i	$0.216547\pi$
$992$	0	0
$993$	−0.708204	−0.0224742
$994$	0	0
$995$	51.4164	1.63001
$996$	0	0
$997$	−27.3050	−0.864756	−0.432378	−	0.901692i	$-0.642325\pi$
$997$	−27.3050	−0.864756	−0.432378	+	0.901692i	$0.642325\pi$
$998$	0	0
$999$	28.5410	0.902998

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2576.2.a.s.1.1		2
4.3	odd	2		161.2.a.b.1.2	✓	2
12.11	even	2		1449.2.a.i.1.1		2
20.19	odd	2		4025.2.a.i.1.1		2
28.27	even	2		1127.2.a.d.1.2		2
92.91	even	2		3703.2.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.a.b.1.2	✓	2	4.3	odd	2
1127.2.a.d.1.2		2	28.27	even	2
1449.2.a.i.1.1		2	12.11	even	2
2576.2.a.s.1.1		2	1.1	even	1	trivial
3703.2.a.b.1.2		2	92.91	even	2
4025.2.a.i.1.1		2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.23607 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.23607 q^{5} +1.00000 q^{7} -2.00000 q^{9} -4.47214 q^{11} +0.236068 q^{13} -3.23607 q^{15} +7.23607 q^{19} +1.00000 q^{21} +1.00000 q^{23} +5.47214 q^{25} -5.00000 q^{27} -1.47214 q^{29} +9.00000 q^{31} -4.47214 q^{33} -3.23607 q^{35} -5.70820 q^{37} +0.236068 q^{39} -2.23607 q^{41} -2.47214 q^{43} +6.47214 q^{45} +3.47214 q^{47} +1.00000 q^{49} +11.2361 q^{53} +14.4721 q^{55} +7.23607 q^{57} +1.52786 q^{59} +13.4164 q^{61} -2.00000 q^{63} -0.763932 q^{65} +12.1803 q^{67} +1.00000 q^{69} +10.2361 q^{71} -6.70820 q^{73} +5.47214 q^{75} -4.47214 q^{77} +7.23607 q^{79} +1.00000 q^{81} -6.47214 q^{83} -1.47214 q^{87} +8.94427 q^{89} +0.236068 q^{91} +9.00000 q^{93} -23.4164 q^{95} +3.70820 q^{97} +8.94427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 - 4 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} - 4 q^{9} - 4 q^{13} - 2 q^{15} + 10 q^{19} + 2 q^{21} + 2 q^{23} + 2 q^{25} - 10 q^{27} + 6 q^{29} + 18 q^{31} - 2 q^{35} + 2 q^{37} - 4 q^{39} + 4 q^{43} + 4 q^{45} - 2 q^{47} + 2 q^{49} + 18 q^{53} + 20 q^{55} + 10 q^{57} + 12 q^{59} - 4 q^{63} - 6 q^{65} + 2 q^{67} + 2 q^{69} + 16 q^{71} + 2 q^{75} + 10 q^{79} + 2 q^{81} - 4 q^{83} + 6 q^{87} - 4 q^{91} + 18 q^{93} - 20 q^{95} - 6 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 - 4 * q^9 - 4 * q^13 - 2 * q^15 + 10 * q^19 + 2 * q^21 + 2 * q^23 + 2 * q^25 - 10 * q^27 + 6 * q^29 + 18 * q^31 - 2 * q^35 + 2 * q^37 - 4 * q^39 + 4 * q^43 + 4 * q^45 - 2 * q^47 + 2 * q^49 + 18 * q^53 + 20 * q^55 + 10 * q^57 + 12 * q^59 - 4 * q^63 - 6 * q^65 + 2 * q^67 + 2 * q^69 + 16 * q^71 + 2 * q^75 + 10 * q^79 + 2 * q^81 - 4 * q^83 + 6 * q^87 - 4 * q^91 + 18 * q^93 - 20 * q^95 - 6 * q^97

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