LMFDB - Embedded newform 9464.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9464,2,Mod(1,9464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9464, 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("9464.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9464 = 2^{3} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9464.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:	$75.5704204729$
Analytic rank:	$1$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 728)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	9464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{3} -3.61803 q^{5} +1.00000 q^{7} -0.381966 q^{9} +O(q^{10})$	$q+1.61803 q^{3} -3.61803 q^{5} +1.00000 q^{7} -0.381966 q^{9} +6.09017 q^{11} -5.85410 q^{15} -1.00000 q^{17} -2.85410 q^{19} +1.61803 q^{21} +2.00000 q^{23} +8.09017 q^{25} -5.47214 q^{27} -8.09017 q^{29} +5.47214 q^{31} +9.85410 q^{33} -3.61803 q^{35} -8.94427 q^{37} +2.76393 q^{41} -7.09017 q^{43} +1.38197 q^{45} +3.00000 q^{47} +1.00000 q^{49} -1.61803 q^{51} -4.70820 q^{53} -22.0344 q^{55} -4.61803 q^{57} +9.47214 q^{59} -6.00000 q^{61} -0.381966 q^{63} -5.47214 q^{67} +3.23607 q^{69} +0.763932 q^{71} +14.9443 q^{73} +13.0902 q^{75} +6.09017 q^{77} +8.94427 q^{79} -7.70820 q^{81} -3.47214 q^{83} +3.61803 q^{85} -13.0902 q^{87} -15.3262 q^{89} +8.85410 q^{93} +10.3262 q^{95} +3.56231 q^{97} -2.32624 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 5 q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10}) $ 2 * q + q^3 - 5 * q^5 + 2 * q^7 - 3 * q^9	$ 2 q + q^{3} - 5 q^{5} + 2 q^{7} - 3 q^{9} + q^{11} - 5 q^{15} - 2 q^{17} + q^{19} + q^{21} + 4 q^{23} + 5 q^{25} - 2 q^{27} - 5 q^{29} + 2 q^{31} + 13 q^{33} - 5 q^{35} + 10 q^{41} - 3 q^{43} + 5 q^{45} + 6 q^{47} + 2 q^{49} - q^{51} + 4 q^{53} - 15 q^{55} - 7 q^{57} + 10 q^{59} - 12 q^{61} - 3 q^{63} - 2 q^{67} + 2 q^{69} + 6 q^{71} + 12 q^{73} + 15 q^{75} + q^{77} - 2 q^{81} + 2 q^{83} + 5 q^{85} - 15 q^{87} - 15 q^{89} + 11 q^{93} + 5 q^{95} - 13 q^{97} + 11 q^{99}+O(q^{100}) $ 2 * q + q^3 - 5 * q^5 + 2 * q^7 - 3 * q^9 + q^11 - 5 * q^15 - 2 * q^17 + q^19 + q^21 + 4 * q^23 + 5 * q^25 - 2 * q^27 - 5 * q^29 + 2 * q^31 + 13 * q^33 - 5 * q^35 + 10 * q^41 - 3 * q^43 + 5 * q^45 + 6 * q^47 + 2 * q^49 - q^51 + 4 * q^53 - 15 * q^55 - 7 * q^57 + 10 * q^59 - 12 * q^61 - 3 * q^63 - 2 * q^67 + 2 * q^69 + 6 * q^71 + 12 * q^73 + 15 * q^75 + q^77 - 2 * q^81 + 2 * q^83 + 5 * q^85 - 15 * q^87 - 15 * q^89 + 11 * q^93 + 5 * q^95 - 13 * q^97 + 11 * 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
$3$	1.61803	0.934172	0.467086	−	0.884212i	$-0.345304\pi$
$3$	1.61803	0.934172	0.467086	+	0.884212i	$0.345304\pi$
$4$	0	0
$5$	−3.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$5$	−3.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	6.09017	1.83626	0.918128	−	0.396285i	$-0.129701\pi$
$11$	6.09017	1.83626	0.918128	+	0.396285i	$0.129701\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−5.85410	−1.51152
$16$	0	0
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	−2.85410	−0.654776	−0.327388	−	0.944890i	$-0.606168\pi$
$19$	−2.85410	−0.654776	−0.327388	+	0.944890i	$0.606168\pi$
$20$	0	0
$21$	1.61803	0.353084
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	8.09017	1.61803
$26$	0	0
$27$	−5.47214	−1.05311
$28$	0	0
$29$	−8.09017	−1.50231	−0.751153	−	0.660128i	$-0.770502\pi$
$29$	−8.09017	−1.50231	−0.751153	+	0.660128i	$0.770502\pi$
$30$	0	0
$31$	5.47214	0.982825	0.491412	−	0.870927i	$-0.336481\pi$
$31$	5.47214	0.982825	0.491412	+	0.870927i	$0.336481\pi$
$32$	0	0
$33$	9.85410	1.71538
$34$	0	0
$35$	−3.61803	−0.611559
$36$	0	0
$37$	−8.94427	−1.47043	−0.735215	−	0.677834i	$-0.762919\pi$
$37$	−8.94427	−1.47043	−0.735215	+	0.677834i	$0.762919\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.76393	0.431654	0.215827	−	0.976432i	$-0.430755\pi$
$41$	2.76393	0.431654	0.215827	+	0.976432i	$0.430755\pi$
$42$	0	0
$43$	−7.09017	−1.08124	−0.540620	−	0.841267i	$-0.681810\pi$
$43$	−7.09017	−1.08124	−0.540620	+	0.841267i	$0.681810\pi$
$44$	0	0
$45$	1.38197	0.206011
$46$	0	0
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.61803	−0.226570
$52$	0	0
$53$	−4.70820	−0.646722	−0.323361	−	0.946276i	$-0.604813\pi$
$53$	−4.70820	−0.646722	−0.323361	+	0.946276i	$0.604813\pi$
$54$	0	0
$55$	−22.0344	−2.97112
$56$	0	0
$57$	−4.61803	−0.611674
$58$	0	0
$59$	9.47214	1.23317	0.616584	−	0.787289i	$-0.288516\pi$
$59$	9.47214	1.23317	0.616584	+	0.787289i	$0.288516\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	−0.381966	−0.0481232
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−5.47214	−0.668528	−0.334264	−	0.942480i	$-0.608488\pi$
$67$	−5.47214	−0.668528	−0.334264	+	0.942480i	$0.608488\pi$
$68$	0	0
$69$	3.23607	0.389577
$70$	0	0
$71$	0.763932	0.0906621	0.0453310	−	0.998972i	$-0.485566\pi$
$71$	0.763932	0.0906621	0.0453310	+	0.998972i	$0.485566\pi$
$72$	0	0
$73$	14.9443	1.74909	0.874547	−	0.484940i	$-0.161159\pi$
$73$	14.9443	1.74909	0.874547	+	0.484940i	$0.161159\pi$
$74$	0	0
$75$	13.0902	1.51152
$76$	0	0
$77$	6.09017	0.694039
$78$	0	0
$79$	8.94427	1.00631	0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427	1.00631	0.503155	+	0.864196i	$0.332173\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	−3.47214	−0.381116	−0.190558	−	0.981676i	$-0.561030\pi$
$83$	−3.47214	−0.381116	−0.190558	+	0.981676i	$0.561030\pi$
$84$	0	0
$85$	3.61803	0.392431
$86$	0	0
$87$	−13.0902	−1.40341
$88$	0	0
$89$	−15.3262	−1.62458	−0.812289	−	0.583255i	$-0.801779\pi$
$89$	−15.3262	−1.62458	−0.812289	+	0.583255i	$0.801779\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.85410	0.918128
$94$	0	0
$95$	10.3262	1.05945
$96$	0	0
$97$	3.56231	0.361697	0.180849	−	0.983511i	$-0.442116\pi$
$97$	3.56231	0.361697	0.180849	+	0.983511i	$0.442116\pi$
$98$	0	0
$99$	−2.32624	−0.233796
$100$	0	0
$101$	0.381966	0.0380070	0.0190035	−	0.999819i	$-0.493951\pi$
$101$	0.381966	0.0380070	0.0190035	+	0.999819i	$0.493951\pi$
$102$	0	0
$103$	−13.4721	−1.32745	−0.663724	−	0.747977i	$-0.731025\pi$
$103$	−13.4721	−1.32745	−0.663724	+	0.747977i	$0.731025\pi$
$104$	0	0
$105$	−5.85410	−0.571302
$106$	0	0
$107$	0.562306	0.0543602	0.0271801	−	0.999631i	$-0.491347\pi$
$107$	0.562306	0.0543602	0.0271801	+	0.999631i	$0.491347\pi$
$108$	0	0
$109$	11.1803	1.07088	0.535441	−	0.844573i	$-0.320145\pi$
$109$	11.1803	1.07088	0.535441	+	0.844573i	$0.320145\pi$
$110$	0	0
$111$	−14.4721	−1.37363
$112$	0	0
$113$	9.47214	0.891064	0.445532	−	0.895266i	$-0.353015\pi$
$113$	9.47214	0.891064	0.445532	+	0.895266i	$0.353015\pi$
$114$	0	0
$115$	−7.23607	−0.674767
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	26.0902	2.37183
$122$	0	0
$123$	4.47214	0.403239
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	−19.5623	−1.73587	−0.867937	−	0.496674i	$-0.834554\pi$
$127$	−19.5623	−1.73587	−0.867937	+	0.496674i	$0.834554\pi$
$128$	0	0
$129$	−11.4721	−1.01007
$130$	0	0
$131$	−13.3820	−1.16919	−0.584594	−	0.811326i	$-0.698746\pi$
$131$	−13.3820	−1.16919	−0.584594	+	0.811326i	$0.698746\pi$
$132$	0	0
$133$	−2.85410	−0.247482
$134$	0	0
$135$	19.7984	1.70397
$136$	0	0
$137$	2.85410	0.243842	0.121921	−	0.992540i	$-0.461094\pi$
$137$	2.85410	0.243842	0.121921	+	0.992540i	$0.461094\pi$
$138$	0	0
$139$	0.618034	0.0524210	0.0262105	−	0.999656i	$-0.491656\pi$
$139$	0.618034	0.0524210	0.0262105	+	0.999656i	$0.491656\pi$
$140$	0	0
$141$	4.85410	0.408789
$142$	0	0
$143$	0	0
$144$	0	0
$145$	29.2705	2.43078
$146$	0	0
$147$	1.61803	0.133453
$148$	0	0
$149$	−18.5623	−1.52068	−0.760342	−	0.649523i	$-0.774968\pi$
$149$	−18.5623	−1.52068	−0.760342	+	0.649523i	$0.774968\pi$
$150$	0	0
$151$	6.52786	0.531230	0.265615	−	0.964079i	$-0.414425\pi$
$151$	6.52786	0.531230	0.265615	+	0.964079i	$0.414425\pi$
$152$	0	0
$153$	0.381966	0.0308801
$154$	0	0
$155$	−19.7984	−1.59024
$156$	0	0
$157$	21.7426	1.73525	0.867626	−	0.497217i	$-0.165645\pi$
$157$	21.7426	1.73525	0.867626	+	0.497217i	$0.165645\pi$
$158$	0	0
$159$	−7.61803	−0.604149
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	−16.7639	−1.31305	−0.656526	−	0.754303i	$-0.727975\pi$
$163$	−16.7639	−1.31305	−0.656526	+	0.754303i	$0.727975\pi$
$164$	0	0
$165$	−35.6525	−2.77554
$166$	0	0
$167$	−11.9443	−0.924276	−0.462138	−	0.886808i	$-0.652917\pi$
$167$	−11.9443	−0.924276	−0.462138	+	0.886808i	$0.652917\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	1.09017	0.0833674
$172$	0	0
$173$	−7.00000	−0.532200	−0.266100	−	0.963945i	$-0.585735\pi$
$173$	−7.00000	−0.532200	−0.266100	+	0.963945i	$0.585735\pi$
$174$	0	0
$175$	8.09017	0.611559
$176$	0	0
$177$	15.3262	1.15199
$178$	0	0
$179$	3.76393	0.281329	0.140665	−	0.990057i	$-0.455076\pi$
$179$	3.76393	0.281329	0.140665	+	0.990057i	$0.455076\pi$
$180$	0	0
$181$	−16.6525	−1.23777	−0.618884	−	0.785482i	$-0.712415\pi$
$181$	−16.6525	−1.23777	−0.618884	+	0.785482i	$0.712415\pi$
$182$	0	0
$183$	−9.70820	−0.717651
$184$	0	0
$185$	32.3607	2.37920
$186$	0	0
$187$	−6.09017	−0.445357
$188$	0	0
$189$	−5.47214	−0.398039
$190$	0	0
$191$	−19.5066	−1.41145	−0.705723	−	0.708488i	$-0.749378\pi$
$191$	−19.5066	−1.41145	−0.705723	+	0.708488i	$0.749378\pi$
$192$	0	0
$193$	−1.05573	−0.0759930	−0.0379965	−	0.999278i	$-0.512098\pi$
$193$	−1.05573	−0.0759930	−0.0379965	+	0.999278i	$0.512098\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.7984	−1.12559	−0.562794	−	0.826597i	$-0.690273\pi$
$197$	−15.7984	−1.12559	−0.562794	+	0.826597i	$0.690273\pi$
$198$	0	0
$199$	−23.1803	−1.64321	−0.821605	−	0.570057i	$-0.806921\pi$
$199$	−23.1803	−1.64321	−0.821605	+	0.570057i	$0.806921\pi$
$200$	0	0
$201$	−8.85410	−0.624520
$202$	0	0
$203$	−8.09017	−0.567819
$204$	0	0
$205$	−10.0000	−0.698430
$206$	0	0
$207$	−0.763932	−0.0530969
$208$	0	0
$209$	−17.3820	−1.20234
$210$	0	0
$211$	−25.4721	−1.75357	−0.876787	−	0.480879i	$-0.840318\pi$
$211$	−25.4721	−1.75357	−0.876787	+	0.480879i	$0.840318\pi$
$212$	0	0
$213$	1.23607	0.0846940
$214$	0	0
$215$	25.6525	1.74948
$216$	0	0
$217$	5.47214	0.371473
$218$	0	0
$219$	24.1803	1.63396
$220$	0	0
$221$	0	0
$222$	0	0
$223$	20.1459	1.34907	0.674535	−	0.738243i	$-0.264344\pi$
$223$	20.1459	1.34907	0.674535	+	0.738243i	$0.264344\pi$
$224$	0	0
$225$	−3.09017	−0.206011
$226$	0	0
$227$	6.81966	0.452637	0.226318	−	0.974053i	$-0.427331\pi$
$227$	6.81966	0.452637	0.226318	+	0.974053i	$0.427331\pi$
$228$	0	0
$229$	−9.23607	−0.610337	−0.305168	−	0.952298i	$-0.598713\pi$
$229$	−9.23607	−0.610337	−0.305168	+	0.952298i	$0.598713\pi$
$230$	0	0
$231$	9.85410	0.648352
$232$	0	0
$233$	20.7426	1.35890	0.679448	−	0.733724i	$-0.262219\pi$
$233$	20.7426	1.35890	0.679448	+	0.733724i	$0.262219\pi$
$234$	0	0
$235$	−10.8541	−0.708044
$236$	0	0
$237$	14.4721	0.940066
$238$	0	0
$239$	−11.0000	−0.711531	−0.355765	−	0.934575i	$-0.615780\pi$
$239$	−11.0000	−0.711531	−0.355765	+	0.934575i	$0.615780\pi$
$240$	0	0
$241$	−9.67376	−0.623142	−0.311571	−	0.950223i	$-0.600855\pi$
$241$	−9.67376	−0.623142	−0.311571	+	0.950223i	$0.600855\pi$
$242$	0	0
$243$	3.94427	0.253025
$244$	0	0
$245$	−3.61803	−0.231148
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−5.61803	−0.356028
$250$	0	0
$251$	−1.81966	−0.114856	−0.0574280	−	0.998350i	$-0.518290\pi$
$251$	−1.81966	−0.114856	−0.0574280	+	0.998350i	$0.518290\pi$
$252$	0	0
$253$	12.1803	0.765771
$254$	0	0
$255$	5.85410	0.366598
$256$	0	0
$257$	8.43769	0.526329	0.263164	−	0.964751i	$-0.415234\pi$
$257$	8.43769	0.526329	0.263164	+	0.964751i	$0.415234\pi$
$258$	0	0
$259$	−8.94427	−0.555770
$260$	0	0
$261$	3.09017	0.191277
$262$	0	0
$263$	−15.1803	−0.936060	−0.468030	−	0.883713i	$-0.655036\pi$
$263$	−15.1803	−0.936060	−0.468030	+	0.883713i	$0.655036\pi$
$264$	0	0
$265$	17.0344	1.04642
$266$	0	0
$267$	−24.7984	−1.51764
$268$	0	0
$269$	2.03444	0.124042	0.0620211	−	0.998075i	$-0.480245\pi$
$269$	2.03444	0.124042	0.0620211	+	0.998075i	$0.480245\pi$
$270$	0	0
$271$	1.29180	0.0784710	0.0392355	−	0.999230i	$-0.487508\pi$
$271$	1.29180	0.0784710	0.0392355	+	0.999230i	$0.487508\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	49.2705	2.97112
$276$	0	0
$277$	−27.4721	−1.65064	−0.825320	−	0.564665i	$-0.809005\pi$
$277$	−27.4721	−1.65064	−0.825320	+	0.564665i	$0.809005\pi$
$278$	0	0
$279$	−2.09017	−0.125135
$280$	0	0
$281$	26.7639	1.59660	0.798301	−	0.602258i	$-0.205732\pi$
$281$	26.7639	1.59660	0.798301	+	0.602258i	$0.205732\pi$
$282$	0	0
$283$	23.8885	1.42003	0.710013	−	0.704188i	$-0.248689\pi$
$283$	23.8885	1.42003	0.710013	+	0.704188i	$0.248689\pi$
$284$	0	0
$285$	16.7082	0.989709
$286$	0	0
$287$	2.76393	0.163150
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	5.76393	0.337888
$292$	0	0
$293$	21.1246	1.23411	0.617056	−	0.786919i	$-0.288325\pi$
$293$	21.1246	1.23411	0.617056	+	0.786919i	$0.288325\pi$
$294$	0	0
$295$	−34.2705	−1.99531
$296$	0	0
$297$	−33.3262	−1.93378
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−7.09017	−0.408671
$302$	0	0
$303$	0.618034	0.0355051
$304$	0	0
$305$	21.7082	1.24301
$306$	0	0
$307$	−7.20163	−0.411019	−0.205509	−	0.978655i	$-0.565885\pi$
$307$	−7.20163	−0.411019	−0.205509	+	0.978655i	$0.565885\pi$
$308$	0	0
$309$	−21.7984	−1.24007
$310$	0	0
$311$	20.7984	1.17937	0.589684	−	0.807634i	$-0.299252\pi$
$311$	20.7984	1.17937	0.589684	+	0.807634i	$0.299252\pi$
$312$	0	0
$313$	−26.1803	−1.47980	−0.739900	−	0.672717i	$-0.765127\pi$
$313$	−26.1803	−1.47980	−0.739900	+	0.672717i	$0.765127\pi$
$314$	0	0
$315$	1.38197	0.0778650
$316$	0	0
$317$	−18.7082	−1.05076	−0.525379	−	0.850869i	$-0.676076\pi$
$317$	−18.7082	−1.05076	−0.525379	+	0.850869i	$0.676076\pi$
$318$	0	0
$319$	−49.2705	−2.75862
$320$	0	0
$321$	0.909830	0.0507818
$322$	0	0
$323$	2.85410	0.158806
$324$	0	0
$325$	0	0
$326$	0	0
$327$	18.0902	1.00039
$328$	0	0
$329$	3.00000	0.165395
$330$	0	0
$331$	−26.8541	−1.47603	−0.738017	−	0.674782i	$-0.764238\pi$
$331$	−26.8541	−1.47603	−0.738017	+	0.674782i	$0.764238\pi$
$332$	0	0
$333$	3.41641	0.187218
$334$	0	0
$335$	19.7984	1.08170
$336$	0	0
$337$	−6.85410	−0.373367	−0.186683	−	0.982420i	$-0.559774\pi$
$337$	−6.85410	−0.373367	−0.186683	+	0.982420i	$0.559774\pi$
$338$	0	0
$339$	15.3262	0.832407
$340$	0	0
$341$	33.3262	1.80472
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−11.7082	−0.630349
$346$	0	0
$347$	−21.7082	−1.16536	−0.582679	−	0.812703i	$-0.697995\pi$
$347$	−21.7082	−1.16536	−0.582679	+	0.812703i	$0.697995\pi$
$348$	0	0
$349$	6.70820	0.359082	0.179541	−	0.983750i	$-0.442539\pi$
$349$	6.70820	0.359082	0.179541	+	0.983750i	$0.442539\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7.32624	0.389936	0.194968	−	0.980810i	$-0.437540\pi$
$353$	7.32624	0.389936	0.194968	+	0.980810i	$0.437540\pi$
$354$	0	0
$355$	−2.76393	−0.146694
$356$	0	0
$357$	−1.61803	−0.0856354
$358$	0	0
$359$	−8.61803	−0.454842	−0.227421	−	0.973796i	$-0.573029\pi$
$359$	−8.61803	−0.454842	−0.227421	+	0.973796i	$0.573029\pi$
$360$	0	0
$361$	−10.8541	−0.571269
$362$	0	0
$363$	42.2148	2.21570
$364$	0	0
$365$	−54.0689	−2.83009
$366$	0	0
$367$	−35.8885	−1.87337	−0.936683	−	0.350177i	$-0.886121\pi$
$367$	−35.8885	−1.87337	−0.936683	+	0.350177i	$0.886121\pi$
$368$	0	0
$369$	−1.05573	−0.0549590
$370$	0	0
$371$	−4.70820	−0.244438
$372$	0	0
$373$	4.50658	0.233342	0.116671	−	0.993171i	$-0.462778\pi$
$373$	4.50658	0.233342	0.116671	+	0.993171i	$0.462778\pi$
$374$	0	0
$375$	−18.0902	−0.934172
$376$	0	0
$377$	0	0
$378$	0	0
$379$	23.6180	1.21318	0.606588	−	0.795016i	$-0.292538\pi$
$379$	23.6180	1.21318	0.606588	+	0.795016i	$0.292538\pi$
$380$	0	0
$381$	−31.6525	−1.62161
$382$	0	0
$383$	1.67376	0.0855252	0.0427626	−	0.999085i	$-0.486384\pi$
$383$	1.67376	0.0855252	0.0427626	+	0.999085i	$0.486384\pi$
$384$	0	0
$385$	−22.0344	−1.12298
$386$	0	0
$387$	2.70820	0.137666
$388$	0	0
$389$	5.05573	0.256336	0.128168	−	0.991752i	$-0.459090\pi$
$389$	5.05573	0.256336	0.128168	+	0.991752i	$0.459090\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	0	0
$393$	−21.6525	−1.09222
$394$	0	0
$395$	−32.3607	−1.62824
$396$	0	0
$397$	7.52786	0.377813	0.188906	−	0.981995i	$-0.439506\pi$
$397$	7.52786	0.377813	0.188906	+	0.981995i	$0.439506\pi$
$398$	0	0
$399$	−4.61803	−0.231191
$400$	0	0
$401$	−15.3262	−0.765356	−0.382678	−	0.923882i	$-0.624998\pi$
$401$	−15.3262	−0.765356	−0.382678	+	0.923882i	$0.624998\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	27.8885	1.38579
$406$	0	0
$407$	−54.4721	−2.70008
$408$	0	0
$409$	−22.7426	−1.12455	−0.562276	−	0.826950i	$-0.690074\pi$
$409$	−22.7426	−1.12455	−0.562276	+	0.826950i	$0.690074\pi$
$410$	0	0
$411$	4.61803	0.227791
$412$	0	0
$413$	9.47214	0.466093
$414$	0	0
$415$	12.5623	0.616659
$416$	0	0
$417$	1.00000	0.0489702
$418$	0	0
$419$	14.2361	0.695477	0.347739	−	0.937591i	$-0.386950\pi$
$419$	14.2361	0.695477	0.347739	+	0.937591i	$0.386950\pi$
$420$	0	0
$421$	−20.4721	−0.997751	−0.498875	−	0.866674i	$-0.666253\pi$
$421$	−20.4721	−0.997751	−0.498875	+	0.866674i	$0.666253\pi$
$422$	0	0
$423$	−1.14590	−0.0557155
$424$	0	0
$425$	−8.09017	−0.392431
$426$	0	0
$427$	−6.00000	−0.290360
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.97871	−0.432489	−0.216245	−	0.976339i	$-0.569381\pi$
$431$	−8.97871	−0.432489	−0.216245	+	0.976339i	$0.569381\pi$
$432$	0	0
$433$	16.5279	0.794278	0.397139	−	0.917758i	$-0.370003\pi$
$433$	16.5279	0.794278	0.397139	+	0.917758i	$0.370003\pi$
$434$	0	0
$435$	47.3607	2.27077
$436$	0	0
$437$	−5.70820	−0.273060
$438$	0	0
$439$	3.96556	0.189266	0.0946329	−	0.995512i	$-0.469832\pi$
$439$	3.96556	0.189266	0.0946329	+	0.995512i	$0.469832\pi$
$440$	0	0
$441$	−0.381966	−0.0181889
$442$	0	0
$443$	−14.6525	−0.696160	−0.348080	−	0.937465i	$-0.613166\pi$
$443$	−14.6525	−0.696160	−0.348080	+	0.937465i	$0.613166\pi$
$444$	0	0
$445$	55.4508	2.62862
$446$	0	0
$447$	−30.0344	−1.42058
$448$	0	0
$449$	−21.4164	−1.01070	−0.505351	−	0.862914i	$-0.668637\pi$
$449$	−21.4164	−1.01070	−0.505351	+	0.862914i	$0.668637\pi$
$450$	0	0
$451$	16.8328	0.792626
$452$	0	0
$453$	10.5623	0.496260
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−37.3050	−1.74505	−0.872526	−	0.488568i	$-0.837520\pi$
$457$	−37.3050	−1.74505	−0.872526	+	0.488568i	$0.837520\pi$
$458$	0	0
$459$	5.47214	0.255417
$460$	0	0
$461$	−27.6180	−1.28630	−0.643150	−	0.765740i	$-0.722373\pi$
$461$	−27.6180	−1.28630	−0.643150	+	0.765740i	$0.722373\pi$
$462$	0	0
$463$	10.5279	0.489271	0.244636	−	0.969615i	$-0.421332\pi$
$463$	10.5279	0.489271	0.244636	+	0.969615i	$0.421332\pi$
$464$	0	0
$465$	−32.0344	−1.48556
$466$	0	0
$467$	25.9443	1.20056	0.600279	−	0.799791i	$-0.295056\pi$
$467$	25.9443	1.20056	0.600279	+	0.799791i	$0.295056\pi$
$468$	0	0
$469$	−5.47214	−0.252680
$470$	0	0
$471$	35.1803	1.62102
$472$	0	0
$473$	−43.1803	−1.98543
$474$	0	0
$475$	−23.0902	−1.05945
$476$	0	0
$477$	1.79837	0.0823419
$478$	0	0
$479$	−27.2148	−1.24348	−0.621738	−	0.783226i	$-0.713573\pi$
$479$	−27.2148	−1.24348	−0.621738	+	0.783226i	$0.713573\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	3.23607	0.147246
$484$	0	0
$485$	−12.8885	−0.585239
$486$	0	0
$487$	−23.8541	−1.08093	−0.540466	−	0.841366i	$-0.681752\pi$
$487$	−23.8541	−1.08093	−0.540466	+	0.841366i	$0.681752\pi$
$488$	0	0
$489$	−27.1246	−1.22662
$490$	0	0
$491$	0.618034	0.0278915	0.0139457	−	0.999903i	$-0.495561\pi$
$491$	0.618034	0.0278915	0.0139457	+	0.999903i	$0.495561\pi$
$492$	0	0
$493$	8.09017	0.364363
$494$	0	0
$495$	8.41641	0.378289
$496$	0	0
$497$	0.763932	0.0342670
$498$	0	0
$499$	−6.50658	−0.291274	−0.145637	−	0.989338i	$-0.546523\pi$
$499$	−6.50658	−0.291274	−0.145637	+	0.989338i	$0.546523\pi$
$500$	0	0
$501$	−19.3262	−0.863433
$502$	0	0
$503$	36.6312	1.63330	0.816652	−	0.577130i	$-0.195828\pi$
$503$	36.6312	1.63330	0.816652	+	0.577130i	$0.195828\pi$
$504$	0	0
$505$	−1.38197	−0.0614967
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−12.5967	−0.558341	−0.279171	−	0.960242i	$-0.590059\pi$
$509$	−12.5967	−0.558341	−0.279171	+	0.960242i	$0.590059\pi$
$510$	0	0
$511$	14.9443	0.661096
$512$	0	0
$513$	15.6180	0.689553
$514$	0	0
$515$	48.7426	2.14786
$516$	0	0
$517$	18.2705	0.803536
$518$	0	0
$519$	−11.3262	−0.497167
$520$	0	0
$521$	−13.8197	−0.605450	−0.302725	−	0.953078i	$-0.597896\pi$
$521$	−13.8197	−0.605450	−0.302725	+	0.953078i	$0.597896\pi$
$522$	0	0
$523$	21.5967	0.944360	0.472180	−	0.881502i	$-0.343467\pi$
$523$	21.5967	0.944360	0.472180	+	0.881502i	$0.343467\pi$
$524$	0	0
$525$	13.0902	0.571302
$526$	0	0
$527$	−5.47214	−0.238370
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−3.61803	−0.157009
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−2.03444	−0.0879566
$536$	0	0
$537$	6.09017	0.262810
$538$	0	0
$539$	6.09017	0.262322
$540$	0	0
$541$	−23.0344	−0.990328	−0.495164	−	0.868800i	$-0.664892\pi$
$541$	−23.0344	−0.990328	−0.495164	+	0.868800i	$0.664892\pi$
$542$	0	0
$543$	−26.9443	−1.15629
$544$	0	0
$545$	−40.4508	−1.73272
$546$	0	0
$547$	17.1803	0.734578	0.367289	−	0.930107i	$-0.380286\pi$
$547$	17.1803	0.734578	0.367289	+	0.930107i	$0.380286\pi$
$548$	0	0
$549$	2.29180	0.0978115
$550$	0	0
$551$	23.0902	0.983674
$552$	0	0
$553$	8.94427	0.380349
$554$	0	0
$555$	52.3607	2.22259
$556$	0	0
$557$	42.6869	1.80870	0.904351	−	0.426789i	$-0.140355\pi$
$557$	42.6869	1.80870	0.904351	+	0.426789i	$0.140355\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−9.85410	−0.416041
$562$	0	0
$563$	−24.4721	−1.03138	−0.515689	−	0.856776i	$-0.672464\pi$
$563$	−24.4721	−1.03138	−0.515689	+	0.856776i	$0.672464\pi$
$564$	0	0
$565$	−34.2705	−1.44177
$566$	0	0
$567$	−7.70820	−0.323714
$568$	0	0
$569$	26.9443	1.12956	0.564781	−	0.825241i	$-0.308961\pi$
$569$	26.9443	1.12956	0.564781	+	0.825241i	$0.308961\pi$
$570$	0	0
$571$	−20.2016	−0.845412	−0.422706	−	0.906267i	$-0.638920\pi$
$571$	−20.2016	−0.845412	−0.422706	+	0.906267i	$0.638920\pi$
$572$	0	0
$573$	−31.5623	−1.31853
$574$	0	0
$575$	16.1803	0.674767
$576$	0	0
$577$	−27.4721	−1.14368	−0.571840	−	0.820365i	$-0.693770\pi$
$577$	−27.4721	−1.14368	−0.571840	+	0.820365i	$0.693770\pi$
$578$	0	0
$579$	−1.70820	−0.0709905
$580$	0	0
$581$	−3.47214	−0.144048
$582$	0	0
$583$	−28.6738	−1.18755
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9.45085	−0.390078	−0.195039	−	0.980795i	$-0.562483\pi$
$587$	−9.45085	−0.390078	−0.195039	+	0.980795i	$0.562483\pi$
$588$	0	0
$589$	−15.6180	−0.643530
$590$	0	0
$591$	−25.5623	−1.05149
$592$	0	0
$593$	28.5066	1.17062	0.585312	−	0.810808i	$-0.300972\pi$
$593$	28.5066	1.17062	0.585312	+	0.810808i	$0.300972\pi$
$594$	0	0
$595$	3.61803	0.148325
$596$	0	0
$597$	−37.5066	−1.53504
$598$	0	0
$599$	24.3262	0.993943	0.496972	−	0.867767i	$-0.334445\pi$
$599$	24.3262	0.993943	0.496972	+	0.867767i	$0.334445\pi$
$600$	0	0
$601$	−27.3262	−1.11466	−0.557330	−	0.830291i	$-0.688174\pi$
$601$	−27.3262	−1.11466	−0.557330	+	0.830291i	$0.688174\pi$
$602$	0	0
$603$	2.09017	0.0851183
$604$	0	0
$605$	−94.3951	−3.83771
$606$	0	0
$607$	−23.0689	−0.936337	−0.468169	−	0.883639i	$-0.655086\pi$
$607$	−23.0689	−0.936337	−0.468169	+	0.883639i	$0.655086\pi$
$608$	0	0
$609$	−13.0902	−0.530440
$610$	0	0
$611$	0	0
$612$	0	0
$613$	25.9230	1.04702	0.523510	−	0.852020i	$-0.324622\pi$
$613$	25.9230	1.04702	0.523510	+	0.852020i	$0.324622\pi$
$614$	0	0
$615$	−16.1803	−0.652454
$616$	0	0
$617$	32.8885	1.32404	0.662021	−	0.749485i	$-0.269699\pi$
$617$	32.8885	1.32404	0.662021	+	0.749485i	$0.269699\pi$
$618$	0	0
$619$	39.8885	1.60326	0.801628	−	0.597823i	$-0.203968\pi$
$619$	39.8885	1.60326	0.801628	+	0.597823i	$0.203968\pi$
$620$	0	0
$621$	−10.9443	−0.439179
$622$	0	0
$623$	−15.3262	−0.614033
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−28.1246	−1.12319
$628$	0	0
$629$	8.94427	0.356631
$630$	0	0
$631$	−34.5623	−1.37590	−0.687952	−	0.725756i	$-0.741490\pi$
$631$	−34.5623	−1.37590	−0.687952	+	0.725756i	$0.741490\pi$
$632$	0	0
$633$	−41.2148	−1.63814
$634$	0	0
$635$	70.7771	2.80870
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−0.291796	−0.0115433
$640$	0	0
$641$	46.5066	1.83690	0.918450	−	0.395538i	$-0.129442\pi$
$641$	46.5066	1.83690	0.918450	+	0.395538i	$0.129442\pi$
$642$	0	0
$643$	−19.1803	−0.756399	−0.378199	−	0.925724i	$-0.623457\pi$
$643$	−19.1803	−0.756399	−0.378199	+	0.925724i	$0.623457\pi$
$644$	0	0
$645$	41.5066	1.63432
$646$	0	0
$647$	−3.70820	−0.145785	−0.0728923	−	0.997340i	$-0.523223\pi$
$647$	−3.70820	−0.145785	−0.0728923	+	0.997340i	$0.523223\pi$
$648$	0	0
$649$	57.6869	2.26441
$650$	0	0
$651$	8.85410	0.347020
$652$	0	0
$653$	−36.5066	−1.42861	−0.714306	−	0.699833i	$-0.753258\pi$
$653$	−36.5066	−1.42861	−0.714306	+	0.699833i	$0.753258\pi$
$654$	0	0
$655$	48.4164	1.89179
$656$	0	0
$657$	−5.70820	−0.222698
$658$	0	0
$659$	−21.4164	−0.834265	−0.417132	−	0.908846i	$-0.636965\pi$
$659$	−21.4164	−0.834265	−0.417132	+	0.908846i	$0.636965\pi$
$660$	0	0
$661$	−37.5967	−1.46234	−0.731172	−	0.682193i	$-0.761026\pi$
$661$	−37.5967	−1.46234	−0.731172	+	0.682193i	$0.761026\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	10.3262	0.400434
$666$	0	0
$667$	−16.1803	−0.626505
$668$	0	0
$669$	32.5967	1.26026
$670$	0	0
$671$	−36.5410	−1.41065
$672$	0	0
$673$	19.3607	0.746299	0.373150	−	0.927771i	$-0.378278\pi$
$673$	19.3607	0.746299	0.373150	+	0.927771i	$0.378278\pi$
$674$	0	0
$675$	−44.2705	−1.70397
$676$	0	0
$677$	−35.1459	−1.35077	−0.675383	−	0.737467i	$-0.736022\pi$
$677$	−35.1459	−1.35077	−0.675383	+	0.737467i	$0.736022\pi$
$678$	0	0
$679$	3.56231	0.136709
$680$	0	0
$681$	11.0344	0.422841
$682$	0	0
$683$	25.5410	0.977300	0.488650	−	0.872480i	$-0.337490\pi$
$683$	25.5410	0.977300	0.488650	+	0.872480i	$0.337490\pi$
$684$	0	0
$685$	−10.3262	−0.394545
$686$	0	0
$687$	−14.9443	−0.570160
$688$	0	0
$689$	0	0
$690$	0	0
$691$	27.0344	1.02844	0.514219	−	0.857659i	$-0.328082\pi$
$691$	27.0344	1.02844	0.514219	+	0.857659i	$0.328082\pi$
$692$	0	0
$693$	−2.32624	−0.0883665
$694$	0	0
$695$	−2.23607	−0.0848189
$696$	0	0
$697$	−2.76393	−0.104691
$698$	0	0
$699$	33.5623	1.26944
$700$	0	0
$701$	−1.81966	−0.0687276	−0.0343638	−	0.999409i	$-0.510940\pi$
$701$	−1.81966	−0.0687276	−0.0343638	+	0.999409i	$0.510940\pi$
$702$	0	0
$703$	25.5279	0.962802
$704$	0	0
$705$	−17.5623	−0.661435
$706$	0	0
$707$	0.381966	0.0143653
$708$	0	0
$709$	38.4508	1.44405	0.722026	−	0.691866i	$-0.243211\pi$
$709$	38.4508	1.44405	0.722026	+	0.691866i	$0.243211\pi$
$710$	0	0
$711$	−3.41641	−0.128125
$712$	0	0
$713$	10.9443	0.409866
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−17.7984	−0.664692
$718$	0	0
$719$	32.8885	1.22654	0.613268	−	0.789875i	$-0.289855\pi$
$719$	32.8885	1.22654	0.613268	+	0.789875i	$0.289855\pi$
$720$	0	0
$721$	−13.4721	−0.501729
$722$	0	0
$723$	−15.6525	−0.582122
$724$	0	0
$725$	−65.4508	−2.43078
$726$	0	0
$727$	−19.8328	−0.735558	−0.367779	−	0.929913i	$-0.619882\pi$
$727$	−19.8328	−0.735558	−0.367779	+	0.929913i	$0.619882\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	7.09017	0.262239
$732$	0	0
$733$	−16.6180	−0.613801	−0.306901	−	0.951742i	$-0.599292\pi$
$733$	−16.6180	−0.613801	−0.306901	+	0.951742i	$0.599292\pi$
$734$	0	0
$735$	−5.85410	−0.215932
$736$	0	0
$737$	−33.3262	−1.22759
$738$	0	0
$739$	−28.2918	−1.04073	−0.520365	−	0.853944i	$-0.674204\pi$
$739$	−28.2918	−1.04073	−0.520365	+	0.853944i	$0.674204\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	22.6869	0.832302	0.416151	−	0.909295i	$-0.363379\pi$
$743$	22.6869	0.832302	0.416151	+	0.909295i	$0.363379\pi$
$744$	0	0
$745$	67.1591	2.46052
$746$	0	0
$747$	1.32624	0.0485245
$748$	0	0
$749$	0.562306	0.0205462
$750$	0	0
$751$	3.58359	0.130767	0.0653836	−	0.997860i	$-0.479173\pi$
$751$	3.58359	0.130767	0.0653836	+	0.997860i	$0.479173\pi$
$752$	0	0
$753$	−2.94427	−0.107295
$754$	0	0
$755$	−23.6180	−0.859548
$756$	0	0
$757$	34.8328	1.26602	0.633010	−	0.774144i	$-0.281819\pi$
$757$	34.8328	1.26602	0.633010	+	0.774144i	$0.281819\pi$
$758$	0	0
$759$	19.7082	0.715362
$760$	0	0
$761$	−4.03444	−0.146248	−0.0731242	−	0.997323i	$-0.523297\pi$
$761$	−4.03444	−0.146248	−0.0731242	+	0.997323i	$0.523297\pi$
$762$	0	0
$763$	11.1803	0.404755
$764$	0	0
$765$	−1.38197	−0.0499651
$766$	0	0
$767$	0	0
$768$	0	0
$769$	41.7214	1.50451	0.752255	−	0.658872i	$-0.228966\pi$
$769$	41.7214	1.50451	0.752255	+	0.658872i	$0.228966\pi$
$770$	0	0
$771$	13.6525	0.491682
$772$	0	0
$773$	27.2492	0.980086	0.490043	−	0.871698i	$-0.336981\pi$
$773$	27.2492	0.980086	0.490043	+	0.871698i	$0.336981\pi$
$774$	0	0
$775$	44.2705	1.59024
$776$	0	0
$777$	−14.4721	−0.519185
$778$	0	0
$779$	−7.88854	−0.282636
$780$	0	0
$781$	4.65248	0.166479
$782$	0	0
$783$	44.2705	1.58210
$784$	0	0
$785$	−78.6656	−2.80770
$786$	0	0
$787$	42.9443	1.53080	0.765399	−	0.643556i	$-0.222542\pi$
$787$	42.9443	1.53080	0.765399	+	0.643556i	$0.222542\pi$
$788$	0	0
$789$	−24.5623	−0.874441
$790$	0	0
$791$	9.47214	0.336790
$792$	0	0
$793$	0	0
$794$	0	0
$795$	27.5623	0.977534
$796$	0	0
$797$	40.0689	1.41931	0.709656	−	0.704548i	$-0.248850\pi$
$797$	40.0689	1.41931	0.709656	+	0.704548i	$0.248850\pi$
$798$	0	0
$799$	−3.00000	−0.106132
$800$	0	0
$801$	5.85410	0.206845
$802$	0	0
$803$	91.0132	3.21178
$804$	0	0
$805$	−7.23607	−0.255038
$806$	0	0
$807$	3.29180	0.115877
$808$	0	0
$809$	11.3607	0.399420	0.199710	−	0.979855i	$-0.436000\pi$
$809$	11.3607	0.399420	0.199710	+	0.979855i	$0.436000\pi$
$810$	0	0
$811$	6.38197	0.224101	0.112051	−	0.993703i	$-0.464258\pi$
$811$	6.38197	0.224101	0.112051	+	0.993703i	$0.464258\pi$
$812$	0	0
$813$	2.09017	0.0733055
$814$	0	0
$815$	60.6525	2.12456
$816$	0	0
$817$	20.2361	0.707970
$818$	0	0
$819$	0	0
$820$	0	0
$821$	38.7771	1.35333	0.676665	−	0.736291i	$-0.263425\pi$
$821$	38.7771	1.35333	0.676665	+	0.736291i	$0.263425\pi$
$822$	0	0
$823$	7.29180	0.254176	0.127088	−	0.991891i	$-0.459437\pi$
$823$	7.29180	0.254176	0.127088	+	0.991891i	$0.459437\pi$
$824$	0	0
$825$	79.7214	2.77554
$826$	0	0
$827$	33.5623	1.16708	0.583538	−	0.812086i	$-0.301668\pi$
$827$	33.5623	1.16708	0.583538	+	0.812086i	$0.301668\pi$
$828$	0	0
$829$	−24.9656	−0.867090	−0.433545	−	0.901132i	$-0.642737\pi$
$829$	−24.9656	−0.867090	−0.433545	+	0.901132i	$0.642737\pi$
$830$	0	0
$831$	−44.4508	−1.54198
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	43.2148	1.49551
$836$	0	0
$837$	−29.9443	−1.03503
$838$	0	0
$839$	22.3820	0.772711	0.386356	−	0.922350i	$-0.373734\pi$
$839$	22.3820	0.772711	0.386356	+	0.922350i	$0.373734\pi$
$840$	0	0
$841$	36.4508	1.25693
$842$	0	0
$843$	43.3050	1.49150
$844$	0	0
$845$	0	0
$846$	0	0
$847$	26.0902	0.896469
$848$	0	0
$849$	38.6525	1.32655
$850$	0	0
$851$	−17.8885	−0.613211
$852$	0	0
$853$	−0.819660	−0.0280646	−0.0140323	−	0.999902i	$-0.504467\pi$
$853$	−0.819660	−0.0280646	−0.0140323	+	0.999902i	$0.504467\pi$
$854$	0	0
$855$	−3.94427	−0.134891
$856$	0	0
$857$	−25.0902	−0.857064	−0.428532	−	0.903527i	$-0.640969\pi$
$857$	−25.0902	−0.857064	−0.428532	+	0.903527i	$0.640969\pi$
$858$	0	0
$859$	−17.0557	−0.581934	−0.290967	−	0.956733i	$-0.593977\pi$
$859$	−17.0557	−0.581934	−0.290967	+	0.956733i	$0.593977\pi$
$860$	0	0
$861$	4.47214	0.152410
$862$	0	0
$863$	−4.47214	−0.152233	−0.0761166	−	0.997099i	$-0.524252\pi$
$863$	−4.47214	−0.152233	−0.0761166	+	0.997099i	$0.524252\pi$
$864$	0	0
$865$	25.3262	0.861118
$866$	0	0
$867$	−25.8885	−0.879221
$868$	0	0
$869$	54.4721	1.84784
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−1.36068	−0.0460520
$874$	0	0
$875$	−11.1803	−0.377964
$876$	0	0
$877$	2.81966	0.0952132	0.0476066	−	0.998866i	$-0.484841\pi$
$877$	2.81966	0.0952132	0.0476066	+	0.998866i	$0.484841\pi$
$878$	0	0
$879$	34.1803	1.15287
$880$	0	0
$881$	42.2361	1.42297	0.711485	−	0.702702i	$-0.248023\pi$
$881$	42.2361	1.42297	0.711485	+	0.702702i	$0.248023\pi$
$882$	0	0
$883$	8.23607	0.277166	0.138583	−	0.990351i	$-0.455745\pi$
$883$	8.23607	0.277166	0.138583	+	0.990351i	$0.455745\pi$
$884$	0	0
$885$	−55.4508	−1.86396
$886$	0	0
$887$	−8.87539	−0.298006	−0.149003	−	0.988837i	$-0.547606\pi$
$887$	−8.87539	−0.298006	−0.149003	+	0.988837i	$0.547606\pi$
$888$	0	0
$889$	−19.5623	−0.656099
$890$	0	0
$891$	−46.9443	−1.57269
$892$	0	0
$893$	−8.56231	−0.286527
$894$	0	0
$895$	−13.6180	−0.455201
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−44.2705	−1.47650
$900$	0	0
$901$	4.70820	0.156853
$902$	0	0
$903$	−11.4721	−0.381769
$904$	0	0
$905$	60.2492	2.00275
$906$	0	0
$907$	−25.8885	−0.859615	−0.429807	−	0.902921i	$-0.641419\pi$
$907$	−25.8885	−0.859615	−0.429807	+	0.902921i	$0.641419\pi$
$908$	0	0
$909$	−0.145898	−0.00483913
$910$	0	0
$911$	27.8673	0.923283	0.461642	−	0.887066i	$-0.347261\pi$
$911$	27.8673	0.923283	0.461642	+	0.887066i	$0.347261\pi$
$912$	0	0
$913$	−21.1459	−0.699827
$914$	0	0
$915$	35.1246	1.16118
$916$	0	0
$917$	−13.3820	−0.441911
$918$	0	0
$919$	−56.4721	−1.86284	−0.931422	−	0.363941i	$-0.881431\pi$
$919$	−56.4721	−1.86284	−0.931422	+	0.363941i	$0.881431\pi$
$920$	0	0
$921$	−11.6525	−0.383962
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−72.3607	−2.37920
$926$	0	0
$927$	5.14590	0.169013
$928$	0	0
$929$	15.7639	0.517198	0.258599	−	0.965985i	$-0.416739\pi$
$929$	15.7639	0.517198	0.258599	+	0.965985i	$0.416739\pi$
$930$	0	0
$931$	−2.85410	−0.0935394
$932$	0	0
$933$	33.6525	1.10173
$934$	0	0
$935$	22.0344	0.720603
$936$	0	0
$937$	40.1246	1.31081	0.655407	−	0.755276i	$-0.272497\pi$
$937$	40.1246	1.31081	0.655407	+	0.755276i	$0.272497\pi$
$938$	0	0
$939$	−42.3607	−1.38239
$940$	0	0
$941$	−37.6525	−1.22744	−0.613718	−	0.789525i	$-0.710327\pi$
$941$	−37.6525	−1.22744	−0.613718	+	0.789525i	$0.710327\pi$
$942$	0	0
$943$	5.52786	0.180012
$944$	0	0
$945$	19.7984	0.644041
$946$	0	0
$947$	−14.3820	−0.467351	−0.233676	−	0.972315i	$-0.575075\pi$
$947$	−14.3820	−0.467351	−0.233676	+	0.972315i	$0.575075\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−30.2705	−0.981589
$952$	0	0
$953$	−23.8328	−0.772021	−0.386010	−	0.922494i	$-0.626147\pi$
$953$	−23.8328	−0.772021	−0.386010	+	0.922494i	$0.626147\pi$
$954$	0	0
$955$	70.5755	2.28377
$956$	0	0
$957$	−79.7214	−2.57703
$958$	0	0
$959$	2.85410	0.0921638
$960$	0	0
$961$	−1.05573	−0.0340557
$962$	0	0
$963$	−0.214782	−0.00692124
$964$	0	0
$965$	3.81966	0.122959
$966$	0	0
$967$	2.23607	0.0719071	0.0359535	−	0.999353i	$-0.488553\pi$
$967$	2.23607	0.0719071	0.0359535	+	0.999353i	$0.488553\pi$
$968$	0	0
$969$	4.61803	0.148353
$970$	0	0
$971$	44.2361	1.41960	0.709801	−	0.704402i	$-0.248785\pi$
$971$	44.2361	1.41960	0.709801	+	0.704402i	$0.248785\pi$
$972$	0	0
$973$	0.618034	0.0198133
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−21.4721	−0.686954	−0.343477	−	0.939161i	$-0.611605\pi$
$977$	−21.4721	−0.686954	−0.343477	+	0.939161i	$0.611605\pi$
$978$	0	0
$979$	−93.3394	−2.98314
$980$	0	0
$981$	−4.27051	−0.136347
$982$	0	0
$983$	−24.2148	−0.772332	−0.386166	−	0.922429i	$-0.626201\pi$
$983$	−24.2148	−0.772332	−0.386166	+	0.922429i	$0.626201\pi$
$984$	0	0
$985$	57.1591	1.82124
$986$	0	0
$987$	4.85410	0.154508
$988$	0	0
$989$	−14.1803	−0.450909
$990$	0	0
$991$	6.74265	0.214187	0.107094	−	0.994249i	$-0.465846\pi$
$991$	6.74265	0.214187	0.107094	+	0.994249i	$0.465846\pi$
$992$	0	0
$993$	−43.4508	−1.37887
$994$	0	0
$995$	83.8673	2.65877
$996$	0	0
$997$	22.7771	0.721358	0.360679	−	0.932690i	$-0.382545\pi$
$997$	22.7771	0.721358	0.360679	+	0.932690i	$0.382545\pi$
$998$	0	0
$999$	48.9443	1.54853

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9464.2.a.q.1.2		2
13.4	even	6		728.2.s.b.393.1	yes	4
13.10	even	6		728.2.s.b.113.1	✓	4
13.12	even	2		9464.2.a.s.1.2		2
52.23	odd	6		1456.2.s.n.113.2		4
52.43	odd	6		1456.2.s.n.1121.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.s.b.113.1	✓	4	13.10	even	6
728.2.s.b.393.1	yes	4	13.4	even	6
1456.2.s.n.113.2		4	52.23	odd	6
1456.2.s.n.1121.2		4	52.43	odd	6
9464.2.a.q.1.2		2	1.1	even	1	trivial
9464.2.a.s.1.2		2	13.12	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 9464 = 2^{3} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9464.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{3} -3.61803 q^{5} +1.00000 q^{7} -0.381966 q^{9} +O(q^{10})\)	\(q+1.61803 q^{3} -3.61803 q^{5} +1.00000 q^{7} -0.381966 q^{9} +6.09017 q^{11} -5.85410 q^{15} -1.00000 q^{17} -2.85410 q^{19} +1.61803 q^{21} +2.00000 q^{23} +8.09017 q^{25} -5.47214 q^{27} -8.09017 q^{29} +5.47214 q^{31} +9.85410 q^{33} -3.61803 q^{35} -8.94427 q^{37} +2.76393 q^{41} -7.09017 q^{43} +1.38197 q^{45} +3.00000 q^{47} +1.00000 q^{49} -1.61803 q^{51} -4.70820 q^{53} -22.0344 q^{55} -4.61803 q^{57} +9.47214 q^{59} -6.00000 q^{61} -0.381966 q^{63} -5.47214 q^{67} +3.23607 q^{69} +0.763932 q^{71} +14.9443 q^{73} +13.0902 q^{75} +6.09017 q^{77} +8.94427 q^{79} -7.70820 q^{81} -3.47214 q^{83} +3.61803 q^{85} -13.0902 q^{87} -15.3262 q^{89} +8.85410 q^{93} +10.3262 q^{95} +3.56231 q^{97} -2.32624 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 5 q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10}) \) 2 * q + q^3 - 5 * q^5 + 2 * q^7 - 3 * q^9	\( 2 q + q^{3} - 5 q^{5} + 2 q^{7} - 3 q^{9} + q^{11} - 5 q^{15} - 2 q^{17} + q^{19} + q^{21} + 4 q^{23} + 5 q^{25} - 2 q^{27} - 5 q^{29} + 2 q^{31} + 13 q^{33} - 5 q^{35} + 10 q^{41} - 3 q^{43} + 5 q^{45} + 6 q^{47} + 2 q^{49} - q^{51} + 4 q^{53} - 15 q^{55} - 7 q^{57} + 10 q^{59} - 12 q^{61} - 3 q^{63} - 2 q^{67} + 2 q^{69} + 6 q^{71} + 12 q^{73} + 15 q^{75} + q^{77} - 2 q^{81} + 2 q^{83} + 5 q^{85} - 15 q^{87} - 15 q^{89} + 11 q^{93} + 5 q^{95} - 13 q^{97} + 11 q^{99}+O(q^{100}) \) 2 * q + q^3 - 5 * q^5 + 2 * q^7 - 3 * q^9 + q^11 - 5 * q^15 - 2 * q^17 + q^19 + q^21 + 4 * q^23 + 5 * q^25 - 2 * q^27 - 5 * q^29 + 2 * q^31 + 13 * q^33 - 5 * q^35 + 10 * q^41 - 3 * q^43 + 5 * q^45 + 6 * q^47 + 2 * q^49 - q^51 + 4 * q^53 - 15 * q^55 - 7 * q^57 + 10 * q^59 - 12 * q^61 - 3 * q^63 - 2 * q^67 + 2 * q^69 + 6 * q^71 + 12 * q^73 + 15 * q^75 + q^77 - 2 * q^81 + 2 * q^83 + 5 * q^85 - 15 * q^87 - 15 * q^89 + 11 * q^93 + 5 * q^95 - 13 * q^97 + 11 * q^99

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