LMFDB - Embedded newform 810.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,2,Mod(1,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("810.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	810.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:	$6.46788256372$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	810.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.73205 q^{7} +1.00000 q^{8} +1.00000 q^{10} -3.46410 q^{11} +3.73205 q^{13} +3.73205 q^{14} +1.00000 q^{16} +3.46410 q^{17} -7.92820 q^{19} +1.00000 q^{20} -3.46410 q^{22} +0.464102 q^{23} +1.00000 q^{25} +3.73205 q^{26} +3.73205 q^{28} -6.92820 q^{29} +5.46410 q^{31} +1.00000 q^{32} +3.46410 q^{34} +3.73205 q^{35} +8.00000 q^{37} -7.92820 q^{38} +1.00000 q^{40} +5.19615 q^{41} -1.46410 q^{43} -3.46410 q^{44} +0.464102 q^{46} -6.46410 q^{47} +6.92820 q^{49} +1.00000 q^{50} +3.73205 q^{52} -12.4641 q^{53} -3.46410 q^{55} +3.73205 q^{56} -6.92820 q^{58} +8.66025 q^{59} -8.39230 q^{61} +5.46410 q^{62} +1.00000 q^{64} +3.73205 q^{65} +8.00000 q^{67} +3.46410 q^{68} +3.73205 q^{70} -6.00000 q^{71} +6.39230 q^{73} +8.00000 q^{74} -7.92820 q^{76} -12.9282 q^{77} +6.39230 q^{79} +1.00000 q^{80} +5.19615 q^{82} -8.53590 q^{83} +3.46410 q^{85} -1.46410 q^{86} -3.46410 q^{88} +12.0000 q^{89} +13.9282 q^{91} +0.464102 q^{92} -6.46410 q^{94} -7.92820 q^{95} +1.07180 q^{97} +6.92820 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} + 2 q^{8} + 2 q^{10} + 4 q^{13} + 4 q^{14} + 2 q^{16} - 2 q^{19} + 2 q^{20} - 6 q^{23} + 2 q^{25} + 4 q^{26} + 4 q^{28} + 4 q^{31} + 2 q^{32} + 4 q^{35} + 16 q^{37}+ \cdots + 16 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 4 * q^7 + 2 * q^8 + 2 * q^10 + 4 * q^13 + 4 * q^14 + 2 * q^16 - 2 * q^19 + 2 * q^20 - 6 * q^23 + 2 * q^25 + 4 * q^26 + 4 * q^28 + 4 * q^31 + 2 * q^32 + 4 * q^35 + 16 * q^37 - 2 * q^38 + 2 * q^40 + 4 * q^43 - 6 * q^46 - 6 * q^47 + 2 * q^50 + 4 * q^52 - 18 * q^53 + 4 * q^56 + 4 * q^61 + 4 * q^62 + 2 * q^64 + 4 * q^65 + 16 * q^67 + 4 * q^70 - 12 * q^71 - 8 * q^73 + 16 * q^74 - 2 * q^76 - 12 * q^77 - 8 * q^79 + 2 * q^80 - 24 * q^83 + 4 * q^86 + 24 * q^89 + 14 * q^91 - 6 * q^92 - 6 * q^94 - 2 * q^95 + 16 * q^97

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.73205	1.41058	0.705291	−	0.708918i	$-0.250816\pi$
$7$	3.73205	1.41058	0.705291	+	0.708918i	$0.250816\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	3.73205	1.03508	0.517542	−	0.855658i	$-0.326847\pi$
$13$	3.73205	1.03508	0.517542	+	0.855658i	$0.326847\pi$
$14$	3.73205	0.997433
$15$	0	0
$16$	1.00000	0.250000
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	−7.92820	−1.81885	−0.909427	−	0.415863i	$-0.863480\pi$
$19$	−7.92820	−1.81885	−0.909427	+	0.415863i	$0.863480\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−3.46410	−0.738549
$23$	0.464102	0.0967719	0.0483859	−	0.998829i	$-0.484592\pi$
$23$	0.464102	0.0967719	0.0483859	+	0.998829i	$0.484592\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.73205	0.731915
$27$	0	0
$28$	3.73205	0.705291
$29$	−6.92820	−1.28654	−0.643268	−	0.765641i	$-0.722422\pi$
$29$	−6.92820	−1.28654	−0.643268	+	0.765641i	$0.722422\pi$
$30$	0	0
$31$	5.46410	0.981382	0.490691	−	0.871334i	$-0.336744\pi$
$31$	5.46410	0.981382	0.490691	+	0.871334i	$0.336744\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.46410	0.594089
$35$	3.73205	0.630832
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−7.92820	−1.28612
$39$	0	0
$40$	1.00000	0.158114
$41$	5.19615	0.811503	0.405751	−	0.913984i	$-0.367010\pi$
$41$	5.19615	0.811503	0.405751	+	0.913984i	$0.367010\pi$
$42$	0	0
$43$	−1.46410	−0.223273	−0.111637	−	0.993749i	$-0.535609\pi$
$43$	−1.46410	−0.223273	−0.111637	+	0.993749i	$0.535609\pi$
$44$	−3.46410	−0.522233
$45$	0	0
$46$	0.464102	0.0684280
$47$	−6.46410	−0.942886	−0.471443	−	0.881897i	$-0.656267\pi$
$47$	−6.46410	−0.942886	−0.471443	+	0.881897i	$0.656267\pi$
$48$	0	0
$49$	6.92820	0.989743
$50$	1.00000	0.141421
$51$	0	0
$52$	3.73205	0.517542
$53$	−12.4641	−1.71208	−0.856038	−	0.516913i	$-0.827081\pi$
$53$	−12.4641	−1.71208	−0.856038	+	0.516913i	$0.827081\pi$
$54$	0	0
$55$	−3.46410	−0.467099
$56$	3.73205	0.498716
$57$	0	0
$58$	−6.92820	−0.909718
$59$	8.66025	1.12747	0.563735	−	0.825956i	$-0.309364\pi$
$59$	8.66025	1.12747	0.563735	+	0.825956i	$0.309364\pi$
$60$	0	0
$61$	−8.39230	−1.07452	−0.537262	−	0.843415i	$-0.680541\pi$
$61$	−8.39230	−1.07452	−0.537262	+	0.843415i	$0.680541\pi$
$62$	5.46410	0.693942
$63$	0	0
$64$	1.00000	0.125000
$65$	3.73205	0.462904
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	3.46410	0.420084
$69$	0	0
$70$	3.73205	0.446065
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	6.39230	0.748163	0.374081	−	0.927396i	$-0.377958\pi$
$73$	6.39230	0.748163	0.374081	+	0.927396i	$0.377958\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−7.92820	−0.909427
$77$	−12.9282	−1.47331
$78$	0	0
$79$	6.39230	0.719190	0.359595	−	0.933108i	$-0.382915\pi$
$79$	6.39230	0.719190	0.359595	+	0.933108i	$0.382915\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	5.19615	0.573819
$83$	−8.53590	−0.936937	−0.468468	−	0.883480i	$-0.655194\pi$
$83$	−8.53590	−0.936937	−0.468468	+	0.883480i	$0.655194\pi$
$84$	0	0
$85$	3.46410	0.375735
$86$	−1.46410	−0.157878
$87$	0	0
$88$	−3.46410	−0.369274
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	13.9282	1.46007
$92$	0.464102	0.0483859
$93$	0	0
$94$	−6.46410	−0.666721
$95$	−7.92820	−0.813416
$96$	0	0
$97$	1.07180	0.108824	0.0544122	−	0.998519i	$-0.482671\pi$
$97$	1.07180	0.108824	0.0544122	+	0.998519i	$0.482671\pi$
$98$	6.92820	0.699854
$99$	0	0
$100$	1.00000	0.100000
$101$	−9.46410	−0.941713	−0.470857	−	0.882210i	$-0.656055\pi$
$101$	−9.46410	−0.941713	−0.470857	+	0.882210i	$0.656055\pi$
$102$	0	0
$103$	−15.1962	−1.49732	−0.748661	−	0.662953i	$-0.769303\pi$
$103$	−15.1962	−1.49732	−0.748661	+	0.662953i	$0.769303\pi$
$104$	3.73205	0.365958
$105$	0	0
$106$	−12.4641	−1.21062
$107$	−8.53590	−0.825196	−0.412598	−	0.910913i	$-0.635379\pi$
$107$	−8.53590	−0.825196	−0.412598	+	0.910913i	$0.635379\pi$
$108$	0	0
$109$	−17.8564	−1.71033	−0.855167	−	0.518353i	$-0.826545\pi$
$109$	−17.8564	−1.71033	−0.855167	+	0.518353i	$0.826545\pi$
$110$	−3.46410	−0.330289
$111$	0	0
$112$	3.73205	0.352646
$113$	−12.9282	−1.21618	−0.608092	−	0.793867i	$-0.708065\pi$
$113$	−12.9282	−1.21618	−0.608092	+	0.793867i	$0.708065\pi$
$114$	0	0
$115$	0.464102	0.0432777
$116$	−6.92820	−0.643268
$117$	0	0
$118$	8.66025	0.797241
$119$	12.9282	1.18513
$120$	0	0
$121$	1.00000	0.0909091
$122$	−8.39230	−0.759804
$123$	0	0
$124$	5.46410	0.490691
$125$	1.00000	0.0894427
$126$	0	0
$127$	−15.1962	−1.34844	−0.674220	−	0.738530i	$-0.735520\pi$
$127$	−15.1962	−1.34844	−0.674220	+	0.738530i	$0.735520\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	3.73205	0.327323
$131$	−13.7321	−1.19977	−0.599887	−	0.800084i	$-0.704788\pi$
$131$	−13.7321	−1.19977	−0.599887	+	0.800084i	$0.704788\pi$
$132$	0	0
$133$	−29.5885	−2.56564
$134$	8.00000	0.691095
$135$	0	0
$136$	3.46410	0.297044
$137$	10.3923	0.887875	0.443937	−	0.896058i	$-0.353581\pi$
$137$	10.3923	0.887875	0.443937	+	0.896058i	$0.353581\pi$
$138$	0	0
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	3.73205	0.315416
$141$	0	0
$142$	−6.00000	−0.503509
$143$	−12.9282	−1.08111
$144$	0	0
$145$	−6.92820	−0.575356
$146$	6.39230	0.529031
$147$	0	0
$148$	8.00000	0.657596
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	18.3923	1.49674	0.748372	−	0.663279i	$-0.230836\pi$
$151$	18.3923	1.49674	0.748372	+	0.663279i	$0.230836\pi$
$152$	−7.92820	−0.643062
$153$	0	0
$154$	−12.9282	−1.04178
$155$	5.46410	0.438887
$156$	0	0
$157$	14.1244	1.12725	0.563623	−	0.826032i	$-0.309407\pi$
$157$	14.1244	1.12725	0.563623	+	0.826032i	$0.309407\pi$
$158$	6.39230	0.508544
$159$	0	0
$160$	1.00000	0.0790569
$161$	1.73205	0.136505
$162$	0	0
$163$	9.85641	0.772013	0.386007	−	0.922496i	$-0.373854\pi$
$163$	9.85641	0.772013	0.386007	+	0.922496i	$0.373854\pi$
$164$	5.19615	0.405751
$165$	0	0
$166$	−8.53590	−0.662514
$167$	−6.92820	−0.536120	−0.268060	−	0.963402i	$-0.586383\pi$
$167$	−6.92820	−0.536120	−0.268060	+	0.963402i	$0.586383\pi$
$168$	0	0
$169$	0.928203	0.0714002
$170$	3.46410	0.265684
$171$	0	0
$172$	−1.46410	−0.111637
$173$	20.3205	1.54494	0.772470	−	0.635051i	$-0.219021\pi$
$173$	20.3205	1.54494	0.772470	+	0.635051i	$0.219021\pi$
$174$	0	0
$175$	3.73205	0.282117
$176$	−3.46410	−0.261116
$177$	0	0
$178$	12.0000	0.899438
$179$	−12.1244	−0.906217	−0.453108	−	0.891455i	$-0.649685\pi$
$179$	−12.1244	−0.906217	−0.453108	+	0.891455i	$0.649685\pi$
$180$	0	0
$181$	−12.5359	−0.931786	−0.465893	−	0.884841i	$-0.654267\pi$
$181$	−12.5359	−0.931786	−0.465893	+	0.884841i	$0.654267\pi$
$182$	13.9282	1.03243
$183$	0	0
$184$	0.464102	0.0342140
$185$	8.00000	0.588172
$186$	0	0
$187$	−12.0000	−0.877527
$188$	−6.46410	−0.471443
$189$	0	0
$190$	−7.92820	−0.575172
$191$	−2.53590	−0.183491	−0.0917456	−	0.995782i	$-0.529245\pi$
$191$	−2.53590	−0.183491	−0.0917456	+	0.995782i	$0.529245\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	1.07180	0.0769505
$195$	0	0
$196$	6.92820	0.494872
$197$	11.5359	0.821899	0.410949	−	0.911658i	$-0.365197\pi$
$197$	11.5359	0.821899	0.410949	+	0.911658i	$0.365197\pi$
$198$	0	0
$199$	−22.2487	−1.57717	−0.788585	−	0.614926i	$-0.789186\pi$
$199$	−22.2487	−1.57717	−0.788585	+	0.614926i	$0.789186\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−9.46410	−0.665892
$203$	−25.8564	−1.81476
$204$	0	0
$205$	5.19615	0.362915
$206$	−15.1962	−1.05877
$207$	0	0
$208$	3.73205	0.258771
$209$	27.4641	1.89973
$210$	0	0
$211$	−8.85641	−0.609700	−0.304850	−	0.952400i	$-0.598606\pi$
$211$	−8.85641	−0.609700	−0.304850	+	0.952400i	$0.598606\pi$
$212$	−12.4641	−0.856038
$213$	0	0
$214$	−8.53590	−0.583502
$215$	−1.46410	−0.0998509
$216$	0	0
$217$	20.3923	1.38432
$218$	−17.8564	−1.20939
$219$	0	0
$220$	−3.46410	−0.233550
$221$	12.9282	0.869645
$222$	0	0
$223$	−12.5359	−0.839466	−0.419733	−	0.907648i	$-0.637876\pi$
$223$	−12.5359	−0.839466	−0.419733	+	0.907648i	$0.637876\pi$
$224$	3.73205	0.249358
$225$	0	0
$226$	−12.9282	−0.859971
$227$	−16.3923	−1.08800	−0.543998	−	0.839087i	$-0.683090\pi$
$227$	−16.3923	−1.08800	−0.543998	+	0.839087i	$0.683090\pi$
$228$	0	0
$229$	21.8564	1.44431	0.722156	−	0.691730i	$-0.243151\pi$
$229$	21.8564	1.44431	0.722156	+	0.691730i	$0.243151\pi$
$230$	0.464102	0.0306020
$231$	0	0
$232$	−6.92820	−0.454859
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	0	0
$235$	−6.46410	−0.421671
$236$	8.66025	0.563735
$237$	0	0
$238$	12.9282	0.838011
$239$	8.53590	0.552141	0.276071	−	0.961137i	$-0.410968\pi$
$239$	8.53590	0.552141	0.276071	+	0.961137i	$0.410968\pi$
$240$	0	0
$241$	−16.9282	−1.09044	−0.545221	−	0.838293i	$-0.683554\pi$
$241$	−16.9282	−1.09044	−0.545221	+	0.838293i	$0.683554\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−8.39230	−0.537262
$245$	6.92820	0.442627
$246$	0	0
$247$	−29.5885	−1.88267
$248$	5.46410	0.346971
$249$	0	0
$250$	1.00000	0.0632456
$251$	12.1244	0.765283	0.382641	−	0.923897i	$-0.375015\pi$
$251$	12.1244	0.765283	0.382641	+	0.923897i	$0.375015\pi$
$252$	0	0
$253$	−1.60770	−0.101075
$254$	−15.1962	−0.953491
$255$	0	0
$256$	1.00000	0.0625000
$257$	31.8564	1.98715	0.993574	−	0.113184i	$-0.0361050\pi$
$257$	31.8564	1.98715	0.993574	+	0.113184i	$0.0361050\pi$
$258$	0	0
$259$	29.8564	1.85519
$260$	3.73205	0.231452
$261$	0	0
$262$	−13.7321	−0.848369
$263$	8.32051	0.513065	0.256532	−	0.966536i	$-0.417420\pi$
$263$	8.32051	0.513065	0.256532	+	0.966536i	$0.417420\pi$
$264$	0	0
$265$	−12.4641	−0.765664
$266$	−29.5885	−1.81418
$267$	0	0
$268$	8.00000	0.488678
$269$	12.9282	0.788246	0.394123	−	0.919058i	$-0.371048\pi$
$269$	12.9282	0.788246	0.394123	+	0.919058i	$0.371048\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	3.46410	0.210042
$273$	0	0
$274$	10.3923	0.627822
$275$	−3.46410	−0.208893
$276$	0	0
$277$	12.2679	0.737110	0.368555	−	0.929606i	$-0.379853\pi$
$277$	12.2679	0.737110	0.368555	+	0.929606i	$0.379853\pi$
$278$	−13.0000	−0.779688
$279$	0	0
$280$	3.73205	0.223033
$281$	−7.05256	−0.420720	−0.210360	−	0.977624i	$-0.567464\pi$
$281$	−7.05256	−0.420720	−0.210360	+	0.977624i	$0.567464\pi$
$282$	0	0
$283$	−3.32051	−0.197384	−0.0986919	−	0.995118i	$-0.531466\pi$
$283$	−3.32051	−0.197384	−0.0986919	+	0.995118i	$0.531466\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−12.9282	−0.764461
$287$	19.3923	1.14469
$288$	0	0
$289$	−5.00000	−0.294118
$290$	−6.92820	−0.406838
$291$	0	0
$292$	6.39230	0.374081
$293$	−7.39230	−0.431863	−0.215932	−	0.976409i	$-0.569279\pi$
$293$	−7.39230	−0.431863	−0.215932	+	0.976409i	$0.569279\pi$
$294$	0	0
$295$	8.66025	0.504219
$296$	8.00000	0.464991
$297$	0	0
$298$	18.0000	1.04271
$299$	1.73205	0.100167
$300$	0	0
$301$	−5.46410	−0.314946
$302$	18.3923	1.05836
$303$	0	0
$304$	−7.92820	−0.454714
$305$	−8.39230	−0.480542
$306$	0	0
$307$	12.3923	0.707266	0.353633	−	0.935384i	$-0.384946\pi$
$307$	12.3923	0.707266	0.353633	+	0.935384i	$0.384946\pi$
$308$	−12.9282	−0.736653
$309$	0	0
$310$	5.46410	0.310340
$311$	−8.53590	−0.484026	−0.242013	−	0.970273i	$-0.577808\pi$
$311$	−8.53590	−0.484026	−0.242013	+	0.970273i	$0.577808\pi$
$312$	0	0
$313$	−3.32051	−0.187686	−0.0938431	−	0.995587i	$-0.529915\pi$
$313$	−3.32051	−0.187686	−0.0938431	+	0.995587i	$0.529915\pi$
$314$	14.1244	0.797084
$315$	0	0
$316$	6.39230	0.359595
$317$	17.5359	0.984914	0.492457	−	0.870337i	$-0.336099\pi$
$317$	17.5359	0.984914	0.492457	+	0.870337i	$0.336099\pi$
$318$	0	0
$319$	24.0000	1.34374
$320$	1.00000	0.0559017
$321$	0	0
$322$	1.73205	0.0965234
$323$	−27.4641	−1.52814
$324$	0	0
$325$	3.73205	0.207017
$326$	9.85641	0.545896
$327$	0	0
$328$	5.19615	0.286910
$329$	−24.1244	−1.33002
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−8.53590	−0.468468
$333$	0	0
$334$	−6.92820	−0.379094
$335$	8.00000	0.437087
$336$	0	0
$337$	7.07180	0.385225	0.192613	−	0.981275i	$-0.438304\pi$
$337$	7.07180	0.385225	0.192613	+	0.981275i	$0.438304\pi$
$338$	0.928203	0.0504876
$339$	0	0
$340$	3.46410	0.187867
$341$	−18.9282	−1.02502
$342$	0	0
$343$	−0.267949	−0.0144679
$344$	−1.46410	−0.0789391
$345$	0	0
$346$	20.3205	1.09244
$347$	14.7846	0.793679	0.396840	−	0.917888i	$-0.370107\pi$
$347$	14.7846	0.793679	0.396840	+	0.917888i	$0.370107\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	3.73205	0.199487
$351$	0	0
$352$	−3.46410	−0.184637
$353$	−22.3923	−1.19182	−0.595911	−	0.803050i	$-0.703209\pi$
$353$	−22.3923	−1.19182	−0.595911	+	0.803050i	$0.703209\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	12.0000	0.635999
$357$	0	0
$358$	−12.1244	−0.640792
$359$	8.53590	0.450507	0.225254	−	0.974300i	$-0.427679\pi$
$359$	8.53590	0.450507	0.225254	+	0.974300i	$0.427679\pi$
$360$	0	0
$361$	43.8564	2.30823
$362$	−12.5359	−0.658872
$363$	0	0
$364$	13.9282	0.730036
$365$	6.39230	0.334589
$366$	0	0
$367$	6.39230	0.333676	0.166838	−	0.985984i	$-0.446644\pi$
$367$	6.39230	0.333676	0.166838	+	0.985984i	$0.446644\pi$
$368$	0.464102	0.0241930
$369$	0	0
$370$	8.00000	0.415900
$371$	−46.5167	−2.41502
$372$	0	0
$373$	−10.9282	−0.565841	−0.282920	−	0.959143i	$-0.591303\pi$
$373$	−10.9282	−0.565841	−0.282920	+	0.959143i	$0.591303\pi$
$374$	−12.0000	−0.620505
$375$	0	0
$376$	−6.46410	−0.333361
$377$	−25.8564	−1.33167
$378$	0	0
$379$	36.8564	1.89319	0.946593	−	0.322430i	$-0.104500\pi$
$379$	36.8564	1.89319	0.946593	+	0.322430i	$0.104500\pi$
$380$	−7.92820	−0.406708
$381$	0	0
$382$	−2.53590	−0.129748
$383$	6.46410	0.330300	0.165150	−	0.986268i	$-0.447189\pi$
$383$	6.46410	0.330300	0.165150	+	0.986268i	$0.447189\pi$
$384$	0	0
$385$	−12.9282	−0.658882
$386$	−16.0000	−0.814379
$387$	0	0
$388$	1.07180	0.0544122
$389$	−23.3205	−1.18240	−0.591198	−	0.806526i	$-0.701345\pi$
$389$	−23.3205	−1.18240	−0.591198	+	0.806526i	$0.701345\pi$
$390$	0	0
$391$	1.60770	0.0813046
$392$	6.92820	0.349927
$393$	0	0
$394$	11.5359	0.581170
$395$	6.39230	0.321632
$396$	0	0
$397$	9.85641	0.494679	0.247339	−	0.968929i	$-0.420444\pi$
$397$	9.85641	0.494679	0.247339	+	0.968929i	$0.420444\pi$
$398$	−22.2487	−1.11523
$399$	0	0
$400$	1.00000	0.0500000
$401$	39.5885	1.97695	0.988477	−	0.151374i	$-0.0483697\pi$
$401$	39.5885	1.97695	0.988477	+	0.151374i	$0.0483697\pi$
$402$	0	0
$403$	20.3923	1.01581
$404$	−9.46410	−0.470857
$405$	0	0
$406$	−25.8564	−1.28323
$407$	−27.7128	−1.37367
$408$	0	0
$409$	−13.9282	−0.688705	−0.344353	−	0.938840i	$-0.611902\pi$
$409$	−13.9282	−0.688705	−0.344353	+	0.938840i	$0.611902\pi$
$410$	5.19615	0.256620
$411$	0	0
$412$	−15.1962	−0.748661
$413$	32.3205	1.59039
$414$	0	0
$415$	−8.53590	−0.419011
$416$	3.73205	0.182979
$417$	0	0
$418$	27.4641	1.34331
$419$	3.46410	0.169232	0.0846162	−	0.996414i	$-0.473034\pi$
$419$	3.46410	0.169232	0.0846162	+	0.996414i	$0.473034\pi$
$420$	0	0
$421$	9.60770	0.468250	0.234125	−	0.972206i	$-0.424777\pi$
$421$	9.60770	0.468250	0.234125	+	0.972206i	$0.424777\pi$
$422$	−8.85641	−0.431123
$423$	0	0
$424$	−12.4641	−0.605310
$425$	3.46410	0.168034
$426$	0	0
$427$	−31.3205	−1.51571
$428$	−8.53590	−0.412598
$429$	0	0
$430$	−1.46410	−0.0706052
$431$	28.3923	1.36761	0.683805	−	0.729665i	$-0.260324\pi$
$431$	28.3923	1.36761	0.683805	+	0.729665i	$0.260324\pi$
$432$	0	0
$433$	2.92820	0.140720	0.0703602	−	0.997522i	$-0.477585\pi$
$433$	2.92820	0.140720	0.0703602	+	0.997522i	$0.477585\pi$
$434$	20.3923	0.978862
$435$	0	0
$436$	−17.8564	−0.855167
$437$	−3.67949	−0.176014
$438$	0	0
$439$	18.3923	0.877817	0.438908	−	0.898532i	$-0.355365\pi$
$439$	18.3923	0.877817	0.438908	+	0.898532i	$0.355365\pi$
$440$	−3.46410	−0.165145
$441$	0	0
$442$	12.9282	0.614932
$443$	31.8564	1.51354	0.756772	−	0.653679i	$-0.226775\pi$
$443$	31.8564	1.51354	0.756772	+	0.653679i	$0.226775\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	−12.5359	−0.593592
$447$	0	0
$448$	3.73205	0.176323
$449$	32.9090	1.55307	0.776535	−	0.630074i	$-0.216975\pi$
$449$	32.9090	1.55307	0.776535	+	0.630074i	$0.216975\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	−12.9282	−0.608092
$453$	0	0
$454$	−16.3923	−0.769329
$455$	13.9282	0.652964
$456$	0	0
$457$	−20.3923	−0.953912	−0.476956	−	0.878927i	$-0.658260\pi$
$457$	−20.3923	−0.953912	−0.476956	+	0.878927i	$0.658260\pi$
$458$	21.8564	1.02128
$459$	0	0
$460$	0.464102	0.0216388
$461$	−8.78461	−0.409140	−0.204570	−	0.978852i	$-0.565580\pi$
$461$	−8.78461	−0.409140	−0.204570	+	0.978852i	$0.565580\pi$
$462$	0	0
$463$	−4.80385	−0.223254	−0.111627	−	0.993750i	$-0.535606\pi$
$463$	−4.80385	−0.223254	−0.111627	+	0.993750i	$0.535606\pi$
$464$	−6.92820	−0.321634
$465$	0	0
$466$	12.0000	0.555889
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	0	0
$469$	29.8564	1.37864
$470$	−6.46410	−0.298167
$471$	0	0
$472$	8.66025	0.398621
$473$	5.07180	0.233201
$474$	0	0
$475$	−7.92820	−0.363771
$476$	12.9282	0.592563
$477$	0	0
$478$	8.53590	0.390423
$479$	−12.9282	−0.590705	−0.295352	−	0.955388i	$-0.595437\pi$
$479$	−12.9282	−0.590705	−0.295352	+	0.955388i	$0.595437\pi$
$480$	0	0
$481$	29.8564	1.36133
$482$	−16.9282	−0.771059
$483$	0	0
$484$	1.00000	0.0454545
$485$	1.07180	0.0486678
$486$	0	0
$487$	−27.4449	−1.24365	−0.621823	−	0.783158i	$-0.713608\pi$
$487$	−27.4449	−1.24365	−0.621823	+	0.783158i	$0.713608\pi$
$488$	−8.39230	−0.379902
$489$	0	0
$490$	6.92820	0.312984
$491$	12.1244	0.547165	0.273582	−	0.961849i	$-0.411791\pi$
$491$	12.1244	0.547165	0.273582	+	0.961849i	$0.411791\pi$
$492$	0	0
$493$	−24.0000	−1.08091
$494$	−29.5885	−1.33125
$495$	0	0
$496$	5.46410	0.245345
$497$	−22.3923	−1.00443
$498$	0	0
$499$	−3.78461	−0.169422	−0.0847112	−	0.996406i	$-0.526997\pi$
$499$	−3.78461	−0.169422	−0.0847112	+	0.996406i	$0.526997\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	12.1244	0.541136
$503$	5.07180	0.226140	0.113070	−	0.993587i	$-0.463932\pi$
$503$	5.07180	0.226140	0.113070	+	0.993587i	$0.463932\pi$
$504$	0	0
$505$	−9.46410	−0.421147
$506$	−1.60770	−0.0714708
$507$	0	0
$508$	−15.1962	−0.674220
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	23.8564	1.05535
$512$	1.00000	0.0441942
$513$	0	0
$514$	31.8564	1.40513
$515$	−15.1962	−0.669622
$516$	0	0
$517$	22.3923	0.984812
$518$	29.8564	1.31182
$519$	0	0
$520$	3.73205	0.163661
$521$	17.1962	0.753377	0.376689	−	0.926340i	$-0.377063\pi$
$521$	17.1962	0.753377	0.376689	+	0.926340i	$0.377063\pi$
$522$	0	0
$523$	2.00000	0.0874539	0.0437269	−	0.999044i	$-0.486077\pi$
$523$	2.00000	0.0874539	0.0437269	+	0.999044i	$0.486077\pi$
$524$	−13.7321	−0.599887
$525$	0	0
$526$	8.32051	0.362791
$527$	18.9282	0.824525
$528$	0	0
$529$	−22.7846	−0.990635
$530$	−12.4641	−0.541406
$531$	0	0
$532$	−29.5885	−1.28282
$533$	19.3923	0.839974
$534$	0	0
$535$	−8.53590	−0.369039
$536$	8.00000	0.345547
$537$	0	0
$538$	12.9282	0.557374
$539$	−24.0000	−1.03375
$540$	0	0
$541$	26.9282	1.15773	0.578867	−	0.815422i	$-0.303495\pi$
$541$	26.9282	1.15773	0.578867	+	0.815422i	$0.303495\pi$
$542$	14.0000	0.601351
$543$	0	0
$544$	3.46410	0.148522
$545$	−17.8564	−0.764884
$546$	0	0
$547$	12.3923	0.529857	0.264928	−	0.964268i	$-0.414652\pi$
$547$	12.3923	0.529857	0.264928	+	0.964268i	$0.414652\pi$
$548$	10.3923	0.443937
$549$	0	0
$550$	−3.46410	−0.147710
$551$	54.9282	2.34002
$552$	0	0
$553$	23.8564	1.01448
$554$	12.2679	0.521215
$555$	0	0
$556$	−13.0000	−0.551323
$557$	−22.1769	−0.939666	−0.469833	−	0.882755i	$-0.655686\pi$
$557$	−22.1769	−0.939666	−0.469833	+	0.882755i	$0.655686\pi$
$558$	0	0
$559$	−5.46410	−0.231107
$560$	3.73205	0.157708
$561$	0	0
$562$	−7.05256	−0.297494
$563$	−13.8564	−0.583978	−0.291989	−	0.956422i	$-0.594317\pi$
$563$	−13.8564	−0.583978	−0.291989	+	0.956422i	$0.594317\pi$
$564$	0	0
$565$	−12.9282	−0.543894
$566$	−3.32051	−0.139571
$567$	0	0
$568$	−6.00000	−0.251754
$569$	12.1244	0.508279	0.254140	−	0.967168i	$-0.418208\pi$
$569$	12.1244	0.508279	0.254140	+	0.967168i	$0.418208\pi$
$570$	0	0
$571$	14.9282	0.624726	0.312363	−	0.949963i	$-0.398880\pi$
$571$	14.9282	0.624726	0.312363	+	0.949963i	$0.398880\pi$
$572$	−12.9282	−0.540555
$573$	0	0
$574$	19.3923	0.809419
$575$	0.464102	0.0193544
$576$	0	0
$577$	−24.7846	−1.03180	−0.515898	−	0.856650i	$-0.672542\pi$
$577$	−24.7846	−1.03180	−0.515898	+	0.856650i	$0.672542\pi$
$578$	−5.00000	−0.207973
$579$	0	0
$580$	−6.92820	−0.287678
$581$	−31.8564	−1.32163
$582$	0	0
$583$	43.1769	1.78821
$584$	6.39230	0.264515
$585$	0	0
$586$	−7.39230	−0.305373
$587$	−22.3923	−0.924229	−0.462115	−	0.886820i	$-0.652909\pi$
$587$	−22.3923	−0.924229	−0.462115	+	0.886820i	$0.652909\pi$
$588$	0	0
$589$	−43.3205	−1.78499
$590$	8.66025	0.356537
$591$	0	0
$592$	8.00000	0.328798
$593$	−3.46410	−0.142254	−0.0711268	−	0.997467i	$-0.522659\pi$
$593$	−3.46410	−0.142254	−0.0711268	+	0.997467i	$0.522659\pi$
$594$	0	0
$595$	12.9282	0.530005
$596$	18.0000	0.737309
$597$	0	0
$598$	1.73205	0.0708288
$599$	−18.0000	−0.735460	−0.367730	−	0.929933i	$-0.619865\pi$
$599$	−18.0000	−0.735460	−0.367730	+	0.929933i	$0.619865\pi$
$600$	0	0
$601$	5.92820	0.241816	0.120908	−	0.992664i	$-0.461419\pi$
$601$	5.92820	0.241816	0.120908	+	0.992664i	$0.461419\pi$
$602$	−5.46410	−0.222700
$603$	0	0
$604$	18.3923	0.748372
$605$	1.00000	0.0406558
$606$	0	0
$607$	27.1769	1.10308	0.551538	−	0.834149i	$-0.314041\pi$
$607$	27.1769	1.10308	0.551538	+	0.834149i	$0.314041\pi$
$608$	−7.92820	−0.321531
$609$	0	0
$610$	−8.39230	−0.339794
$611$	−24.1244	−0.975967
$612$	0	0
$613$	−11.9808	−0.483898	−0.241949	−	0.970289i	$-0.577787\pi$
$613$	−11.9808	−0.483898	−0.241949	+	0.970289i	$0.577787\pi$
$614$	12.3923	0.500113
$615$	0	0
$616$	−12.9282	−0.520892
$617$	22.3923	0.901480	0.450740	−	0.892655i	$-0.351160\pi$
$617$	22.3923	0.901480	0.450740	+	0.892655i	$0.351160\pi$
$618$	0	0
$619$	−15.7846	−0.634437	−0.317219	−	0.948352i	$-0.602749\pi$
$619$	−15.7846	−0.634437	−0.317219	+	0.948352i	$0.602749\pi$
$620$	5.46410	0.219444
$621$	0	0
$622$	−8.53590	−0.342258
$623$	44.7846	1.79426
$624$	0	0
$625$	1.00000	0.0400000
$626$	−3.32051	−0.132714
$627$	0	0
$628$	14.1244	0.563623
$629$	27.7128	1.10498
$630$	0	0
$631$	−19.7128	−0.784755	−0.392377	−	0.919804i	$-0.628347\pi$
$631$	−19.7128	−0.784755	−0.392377	+	0.919804i	$0.628347\pi$
$632$	6.39230	0.254272
$633$	0	0
$634$	17.5359	0.696439
$635$	−15.1962	−0.603041
$636$	0	0
$637$	25.8564	1.02447
$638$	24.0000	0.950169
$639$	0	0
$640$	1.00000	0.0395285
$641$	−3.21539	−0.127000	−0.0635001	−	0.997982i	$-0.520226\pi$
$641$	−3.21539	−0.127000	−0.0635001	+	0.997982i	$0.520226\pi$
$642$	0	0
$643$	0.392305	0.0154710	0.00773550	−	0.999970i	$-0.497538\pi$
$643$	0.392305	0.0154710	0.00773550	+	0.999970i	$0.497538\pi$
$644$	1.73205	0.0682524
$645$	0	0
$646$	−27.4641	−1.08056
$647$	22.6410	0.890110	0.445055	−	0.895503i	$-0.353184\pi$
$647$	22.6410	0.890110	0.445055	+	0.895503i	$0.353184\pi$
$648$	0	0
$649$	−30.0000	−1.17760
$650$	3.73205	0.146383
$651$	0	0
$652$	9.85641	0.386007
$653$	−12.9282	−0.505920	−0.252960	−	0.967477i	$-0.581404\pi$
$653$	−12.9282	−0.505920	−0.252960	+	0.967477i	$0.581404\pi$
$654$	0	0
$655$	−13.7321	−0.536556
$656$	5.19615	0.202876
$657$	0	0
$658$	−24.1244	−0.940465
$659$	−4.94744	−0.192725	−0.0963625	−	0.995346i	$-0.530721\pi$
$659$	−4.94744	−0.192725	−0.0963625	+	0.995346i	$0.530721\pi$
$660$	0	0
$661$	−11.6077	−0.451487	−0.225744	−	0.974187i	$-0.572481\pi$
$661$	−11.6077	−0.451487	−0.225744	+	0.974187i	$0.572481\pi$
$662$	8.00000	0.310929
$663$	0	0
$664$	−8.53590	−0.331257
$665$	−29.5885	−1.14739
$666$	0	0
$667$	−3.21539	−0.124500
$668$	−6.92820	−0.268060
$669$	0	0
$670$	8.00000	0.309067
$671$	29.0718	1.12230
$672$	0	0
$673$	39.1769	1.51016	0.755080	−	0.655633i	$-0.227598\pi$
$673$	39.1769	1.51016	0.755080	+	0.655633i	$0.227598\pi$
$674$	7.07180	0.272395
$675$	0	0
$676$	0.928203	0.0357001
$677$	15.2487	0.586056	0.293028	−	0.956104i	$-0.405337\pi$
$677$	15.2487	0.586056	0.293028	+	0.956104i	$0.405337\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	3.46410	0.132842
$681$	0	0
$682$	−18.9282	−0.724798
$683$	−21.7128	−0.830818	−0.415409	−	0.909635i	$-0.636361\pi$
$683$	−21.7128	−0.830818	−0.415409	+	0.909635i	$0.636361\pi$
$684$	0	0
$685$	10.3923	0.397070
$686$	−0.267949	−0.0102303
$687$	0	0
$688$	−1.46410	−0.0558184
$689$	−46.5167	−1.77214
$690$	0	0
$691$	−39.7846	−1.51348	−0.756739	−	0.653717i	$-0.773209\pi$
$691$	−39.7846	−1.51348	−0.756739	+	0.653717i	$0.773209\pi$
$692$	20.3205	0.772470
$693$	0	0
$694$	14.7846	0.561216
$695$	−13.0000	−0.493118
$696$	0	0
$697$	18.0000	0.681799
$698$	2.00000	0.0757011
$699$	0	0
$700$	3.73205	0.141058
$701$	41.3205	1.56065	0.780327	−	0.625372i	$-0.215053\pi$
$701$	41.3205	1.56065	0.780327	+	0.625372i	$0.215053\pi$
$702$	0	0
$703$	−63.4256	−2.39214
$704$	−3.46410	−0.130558
$705$	0	0
$706$	−22.3923	−0.842746
$707$	−35.3205	−1.32836
$708$	0	0
$709$	−21.3205	−0.800708	−0.400354	−	0.916360i	$-0.631113\pi$
$709$	−21.3205	−0.800708	−0.400354	+	0.916360i	$0.631113\pi$
$710$	−6.00000	−0.225176
$711$	0	0
$712$	12.0000	0.449719
$713$	2.53590	0.0949701
$714$	0	0
$715$	−12.9282	−0.483487
$716$	−12.1244	−0.453108
$717$	0	0
$718$	8.53590	0.318557
$719$	−35.5692	−1.32651	−0.663254	−	0.748394i	$-0.730825\pi$
$719$	−35.5692	−1.32651	−0.663254	+	0.748394i	$0.730825\pi$
$720$	0	0
$721$	−56.7128	−2.11210
$722$	43.8564	1.63217
$723$	0	0
$724$	−12.5359	−0.465893
$725$	−6.92820	−0.257307
$726$	0	0
$727$	−28.8038	−1.06828	−0.534138	−	0.845397i	$-0.679364\pi$
$727$	−28.8038	−1.06828	−0.534138	+	0.845397i	$0.679364\pi$
$728$	13.9282	0.516214
$729$	0	0
$730$	6.39230	0.236590
$731$	−5.07180	−0.187587
$732$	0	0
$733$	−16.0000	−0.590973	−0.295487	−	0.955347i	$-0.595482\pi$
$733$	−16.0000	−0.590973	−0.295487	+	0.955347i	$0.595482\pi$
$734$	6.39230	0.235944
$735$	0	0
$736$	0.464102	0.0171070
$737$	−27.7128	−1.02081
$738$	0	0
$739$	37.5692	1.38201	0.691003	−	0.722852i	$-0.257169\pi$
$739$	37.5692	1.38201	0.691003	+	0.722852i	$0.257169\pi$
$740$	8.00000	0.294086
$741$	0	0
$742$	−46.5167	−1.70768
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	−10.9282	−0.400110
$747$	0	0
$748$	−12.0000	−0.438763
$749$	−31.8564	−1.16401
$750$	0	0
$751$	22.7846	0.831422	0.415711	−	0.909497i	$-0.363533\pi$
$751$	22.7846	0.831422	0.415711	+	0.909497i	$0.363533\pi$
$752$	−6.46410	−0.235722
$753$	0	0
$754$	−25.8564	−0.941635
$755$	18.3923	0.669365
$756$	0	0
$757$	8.80385	0.319981	0.159991	−	0.987119i	$-0.448854\pi$
$757$	8.80385	0.319981	0.159991	+	0.987119i	$0.448854\pi$
$758$	36.8564	1.33868
$759$	0	0
$760$	−7.92820	−0.287586
$761$	−13.7321	−0.497786	−0.248893	−	0.968531i	$-0.580067\pi$
$761$	−13.7321	−0.497786	−0.248893	+	0.968531i	$0.580067\pi$
$762$	0	0
$763$	−66.6410	−2.41257
$764$	−2.53590	−0.0917456
$765$	0	0
$766$	6.46410	0.233557
$767$	32.3205	1.16703
$768$	0	0
$769$	−27.0718	−0.976234	−0.488117	−	0.872778i	$-0.662316\pi$
$769$	−27.0718	−0.976234	−0.488117	+	0.872778i	$0.662316\pi$
$770$	−12.9282	−0.465900
$771$	0	0
$772$	−16.0000	−0.575853
$773$	−24.9282	−0.896605	−0.448303	−	0.893882i	$-0.647971\pi$
$773$	−24.9282	−0.896605	−0.448303	+	0.893882i	$0.647971\pi$
$774$	0	0
$775$	5.46410	0.196276
$776$	1.07180	0.0384753
$777$	0	0
$778$	−23.3205	−0.836081
$779$	−41.1962	−1.47601
$780$	0	0
$781$	20.7846	0.743732
$782$	1.60770	0.0574911
$783$	0	0
$784$	6.92820	0.247436
$785$	14.1244	0.504120
$786$	0	0
$787$	−34.0000	−1.21197	−0.605985	−	0.795476i	$-0.707221\pi$
$787$	−34.0000	−1.21197	−0.605985	+	0.795476i	$0.707221\pi$
$788$	11.5359	0.410949
$789$	0	0
$790$	6.39230	0.227428
$791$	−48.2487	−1.71553
$792$	0	0
$793$	−31.3205	−1.11222
$794$	9.85641	0.349791
$795$	0	0
$796$	−22.2487	−0.788585
$797$	16.1436	0.571835	0.285918	−	0.958254i	$-0.407702\pi$
$797$	16.1436	0.571835	0.285918	+	0.958254i	$0.407702\pi$
$798$	0	0
$799$	−22.3923	−0.792183
$800$	1.00000	0.0353553
$801$	0	0
$802$	39.5885	1.39792
$803$	−22.1436	−0.781430
$804$	0	0
$805$	1.73205	0.0610468
$806$	20.3923	0.718288
$807$	0	0
$808$	−9.46410	−0.332946
$809$	−51.5885	−1.81375	−0.906877	−	0.421396i	$-0.861540\pi$
$809$	−51.5885	−1.81375	−0.906877	+	0.421396i	$0.861540\pi$
$810$	0	0
$811$	−2.14359	−0.0752717	−0.0376359	−	0.999292i	$-0.511983\pi$
$811$	−2.14359	−0.0752717	−0.0376359	+	0.999292i	$0.511983\pi$
$812$	−25.8564	−0.907382
$813$	0	0
$814$	−27.7128	−0.971334
$815$	9.85641	0.345255
$816$	0	0
$817$	11.6077	0.406102
$818$	−13.9282	−0.486988
$819$	0	0
$820$	5.19615	0.181458
$821$	41.3205	1.44210	0.721048	−	0.692885i	$-0.243661\pi$
$821$	41.3205	1.44210	0.721048	+	0.692885i	$0.243661\pi$
$822$	0	0
$823$	25.3205	0.882617	0.441309	−	0.897355i	$-0.354514\pi$
$823$	25.3205	0.882617	0.441309	+	0.897355i	$0.354514\pi$
$824$	−15.1962	−0.529383
$825$	0	0
$826$	32.3205	1.12457
$827$	56.1051	1.95097	0.975483	−	0.220075i	$-0.0706301\pi$
$827$	56.1051	1.95097	0.975483	+	0.220075i	$0.0706301\pi$
$828$	0	0
$829$	37.5692	1.30483	0.652416	−	0.757861i	$-0.273755\pi$
$829$	37.5692	1.30483	0.652416	+	0.757861i	$0.273755\pi$
$830$	−8.53590	−0.296285
$831$	0	0
$832$	3.73205	0.129386
$833$	24.0000	0.831551
$834$	0	0
$835$	−6.92820	−0.239760
$836$	27.4641	0.949866
$837$	0	0
$838$	3.46410	0.119665
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	19.0000	0.655172
$842$	9.60770	0.331103
$843$	0	0
$844$	−8.85641	−0.304850
$845$	0.928203	0.0319312
$846$	0	0
$847$	3.73205	0.128235
$848$	−12.4641	−0.428019
$849$	0	0
$850$	3.46410	0.118818
$851$	3.71281	0.127274
$852$	0	0
$853$	−7.21539	−0.247050	−0.123525	−	0.992341i	$-0.539420\pi$
$853$	−7.21539	−0.247050	−0.123525	+	0.992341i	$0.539420\pi$
$854$	−31.3205	−1.07177
$855$	0	0
$856$	−8.53590	−0.291751
$857$	45.7128	1.56152	0.780760	−	0.624831i	$-0.214832\pi$
$857$	45.7128	1.56152	0.780760	+	0.624831i	$0.214832\pi$
$858$	0	0
$859$	52.7846	1.80099	0.900494	−	0.434869i	$-0.143205\pi$
$859$	52.7846	1.80099	0.900494	+	0.434869i	$0.143205\pi$
$860$	−1.46410	−0.0499255
$861$	0	0
$862$	28.3923	0.967046
$863$	9.67949	0.329494	0.164747	−	0.986336i	$-0.447319\pi$
$863$	9.67949	0.329494	0.164747	+	0.986336i	$0.447319\pi$
$864$	0	0
$865$	20.3205	0.690918
$866$	2.92820	0.0995044
$867$	0	0
$868$	20.3923	0.692160
$869$	−22.1436	−0.751170
$870$	0	0
$871$	29.8564	1.01165
$872$	−17.8564	−0.604694
$873$	0	0
$874$	−3.67949	−0.124461
$875$	3.73205	0.126166
$876$	0	0
$877$	−27.4449	−0.926747	−0.463374	−	0.886163i	$-0.653361\pi$
$877$	−27.4449	−0.926747	−0.463374	+	0.886163i	$0.653361\pi$
$878$	18.3923	0.620710
$879$	0	0
$880$	−3.46410	−0.116775
$881$	29.5692	0.996212	0.498106	−	0.867116i	$-0.334029\pi$
$881$	29.5692	0.996212	0.498106	+	0.867116i	$0.334029\pi$
$882$	0	0
$883$	23.4641	0.789630	0.394815	−	0.918761i	$-0.370809\pi$
$883$	23.4641	0.789630	0.394815	+	0.918761i	$0.370809\pi$
$884$	12.9282	0.434823
$885$	0	0
$886$	31.8564	1.07024
$887$	−53.1051	−1.78310	−0.891548	−	0.452927i	$-0.850380\pi$
$887$	−53.1051	−1.78310	−0.891548	+	0.452927i	$0.850380\pi$
$888$	0	0
$889$	−56.7128	−1.90209
$890$	12.0000	0.402241
$891$	0	0
$892$	−12.5359	−0.419733
$893$	51.2487	1.71497
$894$	0	0
$895$	−12.1244	−0.405273
$896$	3.73205	0.124679
$897$	0	0
$898$	32.9090	1.09819
$899$	−37.8564	−1.26258
$900$	0	0
$901$	−43.1769	−1.43843
$902$	−18.0000	−0.599334
$903$	0	0
$904$	−12.9282	−0.429986
$905$	−12.5359	−0.416707
$906$	0	0
$907$	−27.5692	−0.915421	−0.457710	−	0.889101i	$-0.651330\pi$
$907$	−27.5692	−0.915421	−0.457710	+	0.889101i	$0.651330\pi$
$908$	−16.3923	−0.543998
$909$	0	0
$910$	13.9282	0.461715
$911$	−46.3923	−1.53705	−0.768523	−	0.639822i	$-0.779008\pi$
$911$	−46.3923	−1.53705	−0.768523	+	0.639822i	$0.779008\pi$
$912$	0	0
$913$	29.5692	0.978598
$914$	−20.3923	−0.674517
$915$	0	0
$916$	21.8564	0.722156
$917$	−51.2487	−1.69238
$918$	0	0
$919$	16.7846	0.553673	0.276837	−	0.960917i	$-0.410714\pi$
$919$	16.7846	0.553673	0.276837	+	0.960917i	$0.410714\pi$
$920$	0.464102	0.0153010
$921$	0	0
$922$	−8.78461	−0.289306
$923$	−22.3923	−0.737052
$924$	0	0
$925$	8.00000	0.263038
$926$	−4.80385	−0.157864
$927$	0	0
$928$	−6.92820	−0.227429
$929$	−30.9282	−1.01472	−0.507361	−	0.861734i	$-0.669379\pi$
$929$	−30.9282	−1.01472	−0.507361	+	0.861734i	$0.669379\pi$
$930$	0	0
$931$	−54.9282	−1.80020
$932$	12.0000	0.393073
$933$	0	0
$934$	−6.00000	−0.196326
$935$	−12.0000	−0.392442
$936$	0	0
$937$	6.14359	0.200702	0.100351	−	0.994952i	$-0.468003\pi$
$937$	6.14359	0.200702	0.100351	+	0.994952i	$0.468003\pi$
$938$	29.8564	0.974846
$939$	0	0
$940$	−6.46410	−0.210836
$941$	−5.32051	−0.173444	−0.0867218	−	0.996233i	$-0.527639\pi$
$941$	−5.32051	−0.173444	−0.0867218	+	0.996233i	$0.527639\pi$
$942$	0	0
$943$	2.41154	0.0785306
$944$	8.66025	0.281867
$945$	0	0
$946$	5.07180	0.164898
$947$	14.5359	0.472353	0.236177	−	0.971710i	$-0.424106\pi$
$947$	14.5359	0.472353	0.236177	+	0.971710i	$0.424106\pi$
$948$	0	0
$949$	23.8564	0.774412
$950$	−7.92820	−0.257225
$951$	0	0
$952$	12.9282	0.419005
$953$	18.2487	0.591134	0.295567	−	0.955322i	$-0.404491\pi$
$953$	18.2487	0.591134	0.295567	+	0.955322i	$0.404491\pi$
$954$	0	0
$955$	−2.53590	−0.0820597
$956$	8.53590	0.276071
$957$	0	0
$958$	−12.9282	−0.417691
$959$	38.7846	1.25242
$960$	0	0
$961$	−1.14359	−0.0368901
$962$	29.8564	0.962609
$963$	0	0
$964$	−16.9282	−0.545221
$965$	−16.0000	−0.515058
$966$	0	0
$967$	6.39230	0.205563	0.102781	−	0.994704i	$-0.467226\pi$
$967$	6.39230	0.205563	0.102781	+	0.994704i	$0.467226\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	1.07180	0.0344133
$971$	−43.3013	−1.38960	−0.694802	−	0.719201i	$-0.744508\pi$
$971$	−43.3013	−1.38960	−0.694802	+	0.719201i	$0.744508\pi$
$972$	0	0
$973$	−48.5167	−1.55537
$974$	−27.4449	−0.879390
$975$	0	0
$976$	−8.39230	−0.268631
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	−41.5692	−1.32856
$980$	6.92820	0.221313
$981$	0	0
$982$	12.1244	0.386904
$983$	−36.4974	−1.16409	−0.582043	−	0.813158i	$-0.697747\pi$
$983$	−36.4974	−1.16409	−0.582043	+	0.813158i	$0.697747\pi$
$984$	0	0
$985$	11.5359	0.367564
$986$	−24.0000	−0.764316
$987$	0	0
$988$	−29.5885	−0.941334
$989$	−0.679492	−0.0216066
$990$	0	0
$991$	11.2154	0.356269	0.178134	−	0.984006i	$-0.442994\pi$
$991$	11.2154	0.356269	0.178134	+	0.984006i	$0.442994\pi$
$992$	5.46410	0.173485
$993$	0	0
$994$	−22.3923	−0.710241
$995$	−22.2487	−0.705332
$996$	0	0
$997$	−3.19615	−0.101223	−0.0506116	−	0.998718i	$-0.516117\pi$
$997$	−3.19615	−0.101223	−0.0506116	+	0.998718i	$0.516117\pi$
$998$	−3.78461	−0.119800
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.2.a.l.1.2	yes	2
3.2	odd	2		810.2.a.j.1.2	✓	2
4.3	odd	2		6480.2.a.bj.1.1		2
5.2	odd	4		4050.2.c.x.649.4		4
5.3	odd	4		4050.2.c.x.649.1		4
5.4	even	2		4050.2.a.bk.1.1		2
9.2	odd	6		810.2.e.n.271.1		4
9.4	even	3		810.2.e.m.541.1		4
9.5	odd	6		810.2.e.n.541.1		4
9.7	even	3		810.2.e.m.271.1		4
12.11	even	2		6480.2.a.bb.1.1		2
15.2	even	4		4050.2.c.z.649.2		4
15.8	even	4		4050.2.c.z.649.3		4
15.14	odd	2		4050.2.a.bt.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
810.2.a.j.1.2	✓	2	3.2	odd	2
810.2.a.l.1.2	yes	2	1.1	even	1	trivial
810.2.e.m.271.1		4	9.7	even	3
810.2.e.m.541.1		4	9.4	even	3
810.2.e.n.271.1		4	9.2	odd	6
810.2.e.n.541.1		4	9.5	odd	6
4050.2.a.bk.1.1		2	5.4	even	2
4050.2.a.bt.1.1		2	15.14	odd	2
4050.2.c.x.649.1		4	5.3	odd	4
4050.2.c.x.649.4		4	5.2	odd	4
4050.2.c.z.649.2		4	15.2	even	4
4050.2.c.z.649.3		4	15.8	even	4
6480.2.a.bb.1.1		2	12.11	even	2
6480.2.a.bj.1.1		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	810.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.73205 q^{7} +1.00000 q^{8} +1.00000 q^{10} -3.46410 q^{11} +3.73205 q^{13} +3.73205 q^{14} +1.00000 q^{16} +3.46410 q^{17} -7.92820 q^{19} +1.00000 q^{20} -3.46410 q^{22} +0.464102 q^{23} +1.00000 q^{25} +3.73205 q^{26} +3.73205 q^{28} -6.92820 q^{29} +5.46410 q^{31} +1.00000 q^{32} +3.46410 q^{34} +3.73205 q^{35} +8.00000 q^{37} -7.92820 q^{38} +1.00000 q^{40} +5.19615 q^{41} -1.46410 q^{43} -3.46410 q^{44} +0.464102 q^{46} -6.46410 q^{47} +6.92820 q^{49} +1.00000 q^{50} +3.73205 q^{52} -12.4641 q^{53} -3.46410 q^{55} +3.73205 q^{56} -6.92820 q^{58} +8.66025 q^{59} -8.39230 q^{61} +5.46410 q^{62} +1.00000 q^{64} +3.73205 q^{65} +8.00000 q^{67} +3.46410 q^{68} +3.73205 q^{70} -6.00000 q^{71} +6.39230 q^{73} +8.00000 q^{74} -7.92820 q^{76} -12.9282 q^{77} +6.39230 q^{79} +1.00000 q^{80} +5.19615 q^{82} -8.53590 q^{83} +3.46410 q^{85} -1.46410 q^{86} -3.46410 q^{88} +12.0000 q^{89} +13.9282 q^{91} +0.464102 q^{92} -6.46410 q^{94} -7.92820 q^{95} +1.07180 q^{97} +6.92820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} + 2 q^{8} + 2 q^{10} + 4 q^{13} + 4 q^{14} + 2 q^{16} - 2 q^{19} + 2 q^{20} - 6 q^{23} + 2 q^{25} + 4 q^{26} + 4 q^{28} + 4 q^{31} + 2 q^{32} + 4 q^{35} + 16 q^{37}+ \cdots + 16 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 4 * q^7 + 2 * q^8 + 2 * q^10 + 4 * q^13 + 4 * q^14 + 2 * q^16 - 2 * q^19 + 2 * q^20 - 6 * q^23 + 2 * q^25 + 4 * q^26 + 4 * q^28 + 4 * q^31 + 2 * q^32 + 4 * q^35 + 16 * q^37 - 2 * q^38 + 2 * q^40 + 4 * q^43 - 6 * q^46 - 6 * q^47 + 2 * q^50 + 4 * q^52 - 18 * q^53 + 4 * q^56 + 4 * q^61 + 4 * q^62 + 2 * q^64 + 4 * q^65 + 16 * q^67 + 4 * q^70 - 12 * q^71 - 8 * q^73 + 16 * q^74 - 2 * q^76 - 12 * q^77 - 8 * q^79 + 2 * q^80 - 24 * q^83 + 4 * q^86 + 24 * q^89 + 14 * q^91 - 6 * q^92 - 6 * q^94 - 2 * q^95 + 16 * q^97

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