LMFDB - Embedded newform 1458.2.a.g.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1458,2,Mod(1,1458)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1458, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1458.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1458 = 2 \cdot 3^{6} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1458.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:	$11.6421886147$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.6357609.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} - x^{3} + 18x^{2} - 8 $ x^6 - 9x^4 - x^3 + 18x^2 - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 54)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.59848$ of defining polynomial
Character	$\chi$	$=$	1458.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} +3.16812 q^{5} +3.68572 q^{7} +1.00000 q^{8} +3.16812 q^{10} -2.28507 q^{11} +3.10027 q^{13} +3.68572 q^{14} +1.00000 q^{16} +1.72576 q^{17} -3.39480 q^{19} +3.16812 q^{20} -2.28507 q^{22} -3.35309 q^{23} +5.03695 q^{25} +3.10027 q^{26} +3.68572 q^{28} -0.585444 q^{29} -4.63798 q^{31} +1.00000 q^{32} +1.72576 q^{34} +11.6768 q^{35} -7.30720 q^{37} -3.39480 q^{38} +3.16812 q^{40} -7.08052 q^{41} +1.65791 q^{43} -2.28507 q^{44} -3.35309 q^{46} -3.85198 q^{47} +6.58452 q^{49} +5.03695 q^{50} +3.10027 q^{52} -2.58267 q^{53} -7.23936 q^{55} +3.68572 q^{56} -0.585444 q^{58} -9.66319 q^{59} +13.0913 q^{61} -4.63798 q^{62} +1.00000 q^{64} +9.82203 q^{65} -8.60876 q^{67} +1.72576 q^{68} +11.6768 q^{70} +1.98746 q^{71} +10.6474 q^{73} -7.30720 q^{74} -3.39480 q^{76} -8.42212 q^{77} +14.0932 q^{79} +3.16812 q^{80} -7.08052 q^{82} -3.09842 q^{83} +5.46739 q^{85} +1.65791 q^{86} -2.28507 q^{88} -17.3460 q^{89} +11.4267 q^{91} -3.35309 q^{92} -3.85198 q^{94} -10.7551 q^{95} +9.19117 q^{97} +6.58452 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 6 q^{4} + 3 q^{5} + 6 q^{7} + 6 q^{8} + 3 q^{10} + 3 q^{11} + 9 q^{13} + 6 q^{14} + 6 q^{16} + 6 q^{17} + 9 q^{19} + 3 q^{20} + 3 q^{22} - 3 q^{23} + 15 q^{25} + 9 q^{26} + 6 q^{28} + 3 q^{29}+ \cdots + 12 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 + 6 * q^4 + 3 * q^5 + 6 * q^7 + 6 * q^8 + 3 * q^10 + 3 * q^11 + 9 * q^13 + 6 * q^14 + 6 * q^16 + 6 * q^17 + 9 * q^19 + 3 * q^20 + 3 * q^22 - 3 * q^23 + 15 * q^25 + 9 * q^26 + 6 * q^28 + 3 * q^29 + 12 * q^31 + 6 * q^32 + 6 * q^34 - 3 * q^35 + 15 * q^37 + 9 * q^38 + 3 * q^40 + 3 * q^41 + 12 * q^43 + 3 * q^44 - 3 * q^46 - 9 * q^47 + 12 * q^49 + 15 * q^50 + 9 * q^52 - 6 * q^53 + 9 * q^55 + 6 * q^56 + 3 * q^58 - 3 * q^59 + 12 * q^61 + 12 * q^62 + 6 * q^64 - 3 * q^65 + 21 * q^67 + 6 * q^68 - 3 * q^70 - 12 * q^71 + 21 * q^73 + 15 * q^74 + 9 * q^76 - 3 * q^77 + 3 * q^80 + 3 * q^82 - 27 * q^83 + 12 * q^86 + 3 * q^88 - 12 * q^89 + 6 * q^91 - 3 * q^92 - 9 * q^94 - 30 * q^95 + 18 * q^97 + 12 * q^98

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
$5$	3.16812	1.41682	0.708412	−	0.705799i	$-0.249412\pi$
$5$	3.16812	1.41682	0.708412	+	0.705799i	$0.249412\pi$
$6$	0	0
$7$	3.68572	1.39307	0.696535	−	0.717522i	$-0.254724\pi$
$7$	3.68572	1.39307	0.696535	+	0.717522i	$0.254724\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.16812	1.00185
$11$	−2.28507	−0.688974	−0.344487	−	0.938791i	$-0.611947\pi$
$11$	−2.28507	−0.688974	−0.344487	+	0.938791i	$0.611947\pi$
$12$	0	0
$13$	3.10027	0.859862	0.429931	−	0.902862i	$-0.358538\pi$
$13$	3.10027	0.859862	0.429931	+	0.902862i	$0.358538\pi$
$14$	3.68572	0.985050
$15$	0	0
$16$	1.00000	0.250000
$17$	1.72576	0.418557	0.209279	−	0.977856i	$-0.432888\pi$
$17$	1.72576	0.418557	0.209279	+	0.977856i	$0.432888\pi$
$18$	0	0
$19$	−3.39480	−0.778820	−0.389410	−	0.921065i	$-0.627321\pi$
$19$	−3.39480	−0.778820	−0.389410	+	0.921065i	$0.627321\pi$
$20$	3.16812	0.708412
$21$	0	0
$22$	−2.28507	−0.487178
$23$	−3.35309	−0.699167	−0.349584	−	0.936905i	$-0.613677\pi$
$23$	−3.35309	−0.699167	−0.349584	+	0.936905i	$0.613677\pi$
$24$	0	0
$25$	5.03695	1.00739
$26$	3.10027	0.608014
$27$	0	0
$28$	3.68572	0.696535
$29$	−0.585444	−0.108714	−0.0543571	−	0.998522i	$-0.517311\pi$
$29$	−0.585444	−0.108714	−0.0543571	+	0.998522i	$0.517311\pi$
$30$	0	0
$31$	−4.63798	−0.833005	−0.416502	−	0.909135i	$-0.636744\pi$
$31$	−4.63798	−0.833005	−0.416502	+	0.909135i	$0.636744\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.72576	0.295965
$35$	11.6768	1.97374
$36$	0	0
$37$	−7.30720	−1.20130	−0.600648	−	0.799514i	$-0.705091\pi$
$37$	−7.30720	−1.20130	−0.600648	+	0.799514i	$0.705091\pi$
$38$	−3.39480	−0.550709
$39$	0	0
$40$	3.16812	0.500923
$41$	−7.08052	−1.10579	−0.552895	−	0.833251i	$-0.686477\pi$
$41$	−7.08052	−1.10579	−0.552895	+	0.833251i	$0.686477\pi$
$42$	0	0
$43$	1.65791	0.252830	0.126415	−	0.991977i	$-0.459653\pi$
$43$	1.65791	0.252830	0.126415	+	0.991977i	$0.459653\pi$
$44$	−2.28507	−0.344487
$45$	0	0
$46$	−3.35309	−0.494386
$47$	−3.85198	−0.561869	−0.280935	−	0.959727i	$-0.590644\pi$
$47$	−3.85198	−0.561869	−0.280935	+	0.959727i	$0.590644\pi$
$48$	0	0
$49$	6.58452	0.940646
$50$	5.03695	0.712333
$51$	0	0
$52$	3.10027	0.429931
$53$	−2.58267	−0.354757	−0.177379	−	0.984143i	$-0.556762\pi$
$53$	−2.58267	−0.354757	−0.177379	+	0.984143i	$0.556762\pi$
$54$	0	0
$55$	−7.23936	−0.976155
$56$	3.68572	0.492525
$57$	0	0
$58$	−0.585444	−0.0768726
$59$	−9.66319	−1.25804	−0.629020	−	0.777389i	$-0.716544\pi$
$59$	−9.66319	−1.25804	−0.629020	+	0.777389i	$0.716544\pi$
$60$	0	0
$61$	13.0913	1.67617	0.838087	−	0.545537i	$-0.183674\pi$
$61$	13.0913	1.67617	0.838087	+	0.545537i	$0.183674\pi$
$62$	−4.63798	−0.589023
$63$	0	0
$64$	1.00000	0.125000
$65$	9.82203	1.21827
$66$	0	0
$67$	−8.60876	−1.05173	−0.525864	−	0.850569i	$-0.676258\pi$
$67$	−8.60876	−1.05173	−0.525864	+	0.850569i	$0.676258\pi$
$68$	1.72576	0.209279
$69$	0	0
$70$	11.6768	1.39564
$71$	1.98746	0.235869	0.117934	−	0.993021i	$-0.462373\pi$
$71$	1.98746	0.235869	0.117934	+	0.993021i	$0.462373\pi$
$72$	0	0
$73$	10.6474	1.24619	0.623094	−	0.782147i	$-0.285876\pi$
$73$	10.6474	1.24619	0.623094	+	0.782147i	$0.285876\pi$
$74$	−7.30720	−0.849444
$75$	0	0
$76$	−3.39480	−0.389410
$77$	−8.42212	−0.959789
$78$	0	0
$79$	14.0932	1.58561	0.792804	−	0.609477i	$-0.208620\pi$
$79$	14.0932	1.58561	0.792804	+	0.609477i	$0.208620\pi$
$80$	3.16812	0.354206
$81$	0	0
$82$	−7.08052	−0.781912
$83$	−3.09842	−0.340096	−0.170048	−	0.985436i	$-0.554392\pi$
$83$	−3.09842	−0.340096	−0.170048	+	0.985436i	$0.554392\pi$
$84$	0	0
$85$	5.46739	0.593022
$86$	1.65791	0.178778
$87$	0	0
$88$	−2.28507	−0.243589
$89$	−17.3460	−1.83867	−0.919336	−	0.393472i	$-0.871274\pi$
$89$	−17.3460	−1.83867	−0.919336	+	0.393472i	$0.871274\pi$
$90$	0	0
$91$	11.4267	1.19785
$92$	−3.35309	−0.349584
$93$	0	0
$94$	−3.85198	−0.397302
$95$	−10.7551	−1.10345
$96$	0	0
$97$	9.19117	0.933221	0.466611	−	0.884463i	$-0.345475\pi$
$97$	9.19117	0.933221	0.466611	+	0.884463i	$0.345475\pi$
$98$	6.58452	0.665137
$99$	0	0
$100$	5.03695	0.503695
$101$	11.9602	1.19008	0.595041	−	0.803695i	$-0.297136\pi$
$101$	11.9602	1.19008	0.595041	+	0.803695i	$0.297136\pi$
$102$	0	0
$103$	−12.7465	−1.25595	−0.627974	−	0.778234i	$-0.716115\pi$
$103$	−12.7465	−1.25595	−0.627974	+	0.778234i	$0.716115\pi$
$104$	3.10027	0.304007
$105$	0	0
$106$	−2.58267	−0.250851
$107$	6.09894	0.589607	0.294803	−	0.955558i	$-0.404746\pi$
$107$	6.09894	0.589607	0.294803	+	0.955558i	$0.404746\pi$
$108$	0	0
$109$	11.2390	1.07650	0.538250	−	0.842785i	$-0.319086\pi$
$109$	11.2390	1.07650	0.538250	+	0.842785i	$0.319086\pi$
$110$	−7.23936	−0.690246
$111$	0	0
$112$	3.68572	0.348268
$113$	−7.03695	−0.661981	−0.330990	−	0.943634i	$-0.607383\pi$
$113$	−7.03695	−0.661981	−0.330990	+	0.943634i	$0.607383\pi$
$114$	0	0
$115$	−10.6230	−0.990597
$116$	−0.585444	−0.0543571
$117$	0	0
$118$	−9.66319	−0.889568
$119$	6.36065	0.583080
$120$	0	0
$121$	−5.77847	−0.525315
$122$	13.0913	1.18523
$123$	0	0
$124$	−4.63798	−0.416502
$125$	0.117076	0.0104716
$126$	0	0
$127$	5.98500	0.531082	0.265541	−	0.964100i	$-0.414449\pi$
$127$	5.98500	0.531082	0.265541	+	0.964100i	$0.414449\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	9.82203	0.861449
$131$	16.4150	1.43419	0.717094	−	0.696976i	$-0.245472\pi$
$131$	16.4150	1.43419	0.717094	+	0.696976i	$0.245472\pi$
$132$	0	0
$133$	−12.5123	−1.08495
$134$	−8.60876	−0.743684
$135$	0	0
$136$	1.72576	0.147982
$137$	−0.211159	−0.0180405	−0.00902025	−	0.999959i	$-0.502871\pi$
$137$	−0.211159	−0.0180405	−0.00902025	+	0.999959i	$0.502871\pi$
$138$	0	0
$139$	3.67151	0.311413	0.155707	−	0.987803i	$-0.450235\pi$
$139$	3.67151	0.311413	0.155707	+	0.987803i	$0.450235\pi$
$140$	11.6768	0.986868
$141$	0	0
$142$	1.98746	0.166784
$143$	−7.08434	−0.592422
$144$	0	0
$145$	−1.85475	−0.154029
$146$	10.6474	0.881188
$147$	0	0
$148$	−7.30720	−0.600648
$149$	15.4744	1.26771	0.633856	−	0.773451i	$-0.281471\pi$
$149$	15.4744	1.26771	0.633856	+	0.773451i	$0.281471\pi$
$150$	0	0
$151$	20.1457	1.63944	0.819718	−	0.572768i	$-0.194130\pi$
$151$	20.1457	1.63944	0.819718	+	0.572768i	$0.194130\pi$
$152$	−3.39480	−0.275354
$153$	0	0
$154$	−8.42212	−0.678673
$155$	−14.6936	−1.18022
$156$	0	0
$157$	17.9974	1.43635	0.718176	−	0.695862i	$-0.244977\pi$
$157$	17.9974	1.43635	0.718176	+	0.695862i	$0.244977\pi$
$158$	14.0932	1.12119
$159$	0	0
$160$	3.16812	0.250462
$161$	−12.3585	−0.973989
$162$	0	0
$163$	3.05289	0.239121	0.119560	−	0.992827i	$-0.461851\pi$
$163$	3.05289	0.239121	0.119560	+	0.992827i	$0.461851\pi$
$164$	−7.08052	−0.552895
$165$	0	0
$166$	−3.09842	−0.240484
$167$	−4.10119	−0.317360	−0.158680	−	0.987330i	$-0.550724\pi$
$167$	−4.10119	−0.317360	−0.158680	+	0.987330i	$0.550724\pi$
$168$	0	0
$169$	−3.38830	−0.260638
$170$	5.46739	0.419330
$171$	0	0
$172$	1.65791	0.126415
$173$	−17.0271	−1.29455	−0.647273	−	0.762258i	$-0.724091\pi$
$173$	−17.0271	−1.29455	−0.647273	+	0.762258i	$0.724091\pi$
$174$	0	0
$175$	18.5648	1.40337
$176$	−2.28507	−0.172243
$177$	0	0
$178$	−17.3460	−1.30014
$179$	−14.5560	−1.08797	−0.543985	−	0.839095i	$-0.683085\pi$
$179$	−14.5560	−1.08797	−0.543985	+	0.839095i	$0.683085\pi$
$180$	0	0
$181$	−13.0238	−0.968052	−0.484026	−	0.875054i	$-0.660826\pi$
$181$	−13.0238	−0.968052	−0.484026	+	0.875054i	$0.660826\pi$
$182$	11.4267	0.847006
$183$	0	0
$184$	−3.35309	−0.247193
$185$	−23.1500	−1.70203
$186$	0	0
$187$	−3.94347	−0.288375
$188$	−3.85198	−0.280935
$189$	0	0
$190$	−10.7551	−0.780258
$191$	3.60287	0.260695	0.130347	−	0.991468i	$-0.458391\pi$
$191$	3.60287	0.260695	0.130347	+	0.991468i	$0.458391\pi$
$192$	0	0
$193$	1.47487	0.106163	0.0530817	−	0.998590i	$-0.483096\pi$
$193$	1.47487	0.106163	0.0530817	+	0.998590i	$0.483096\pi$
$194$	9.19117	0.659887
$195$	0	0
$196$	6.58452	0.470323
$197$	2.53862	0.180869	0.0904346	−	0.995902i	$-0.471174\pi$
$197$	2.53862	0.180869	0.0904346	+	0.995902i	$0.471174\pi$
$198$	0	0
$199$	−1.85178	−0.131269	−0.0656347	−	0.997844i	$-0.520907\pi$
$199$	−1.85178	−0.131269	−0.0656347	+	0.997844i	$0.520907\pi$
$200$	5.03695	0.356166
$201$	0	0
$202$	11.9602	0.841515
$203$	−2.15778	−0.151447
$204$	0	0
$205$	−22.4319	−1.56671
$206$	−12.7465	−0.888089
$207$	0	0
$208$	3.10027	0.214965
$209$	7.75734	0.536586
$210$	0	0
$211$	−18.9883	−1.30721	−0.653604	−	0.756837i	$-0.726744\pi$
$211$	−18.9883	−1.30721	−0.653604	+	0.756837i	$0.726744\pi$
$212$	−2.58267	−0.177379
$213$	0	0
$214$	6.09894	0.416915
$215$	5.25247	0.358215
$216$	0	0
$217$	−17.0943	−1.16043
$218$	11.2390	0.761201
$219$	0	0
$220$	−7.23936	−0.488077
$221$	5.35032	0.359901
$222$	0	0
$223$	−7.25436	−0.485788	−0.242894	−	0.970053i	$-0.578097\pi$
$223$	−7.25436	−0.485788	−0.242894	+	0.970053i	$0.578097\pi$
$224$	3.68572	0.246262
$225$	0	0
$226$	−7.03695	−0.468091
$227$	−28.5272	−1.89342	−0.946708	−	0.322095i	$-0.895613\pi$
$227$	−28.5272	−1.89342	−0.946708	+	0.322095i	$0.895613\pi$
$228$	0	0
$229$	1.85538	0.122607	0.0613036	−	0.998119i	$-0.480474\pi$
$229$	1.85538	0.122607	0.0613036	+	0.998119i	$0.480474\pi$
$230$	−10.6230	−0.700458
$231$	0	0
$232$	−0.585444	−0.0384363
$233$	8.53469	0.559126	0.279563	−	0.960127i	$-0.409810\pi$
$233$	8.53469	0.559126	0.279563	+	0.960127i	$0.409810\pi$
$234$	0	0
$235$	−12.2035	−0.796070
$236$	−9.66319	−0.629020
$237$	0	0
$238$	6.36065	0.412300
$239$	−7.28796	−0.471419	−0.235710	−	0.971824i	$-0.575741\pi$
$239$	−7.28796	−0.471419	−0.235710	+	0.971824i	$0.575741\pi$
$240$	0	0
$241$	−1.33492	−0.0859897	−0.0429948	−	0.999075i	$-0.513690\pi$
$241$	−1.33492	−0.0859897	−0.0429948	+	0.999075i	$0.513690\pi$
$242$	−5.77847	−0.371454
$243$	0	0
$244$	13.0913	0.838087
$245$	20.8605	1.33273
$246$	0	0
$247$	−10.5248	−0.669677
$248$	−4.63798	−0.294512
$249$	0	0
$250$	0.117076	0.00740453
$251$	−2.87856	−0.181693	−0.0908466	−	0.995865i	$-0.528957\pi$
$251$	−2.87856	−0.181693	−0.0908466	+	0.995865i	$0.528957\pi$
$252$	0	0
$253$	7.66203	0.481708
$254$	5.98500	0.375532
$255$	0	0
$256$	1.00000	0.0625000
$257$	−25.7779	−1.60798	−0.803991	−	0.594641i	$-0.797294\pi$
$257$	−25.7779	−1.60798	−0.803991	+	0.594641i	$0.797294\pi$
$258$	0	0
$259$	−26.9323	−1.67349
$260$	9.82203	0.609136
$261$	0	0
$262$	16.4150	1.01412
$263$	−15.0697	−0.929237	−0.464619	−	0.885511i	$-0.653809\pi$
$263$	−15.0697	−0.929237	−0.464619	+	0.885511i	$0.653809\pi$
$264$	0	0
$265$	−8.18220	−0.502629
$266$	−12.5123	−0.767176
$267$	0	0
$268$	−8.60876	−0.525864
$269$	16.0615	0.979286	0.489643	−	0.871923i	$-0.337127\pi$
$269$	16.0615	0.979286	0.489643	+	0.871923i	$0.337127\pi$
$270$	0	0
$271$	9.41446	0.571888	0.285944	−	0.958246i	$-0.407693\pi$
$271$	9.41446	0.571888	0.285944	+	0.958246i	$0.407693\pi$
$272$	1.72576	0.104639
$273$	0	0
$274$	−0.211159	−0.0127566
$275$	−11.5098	−0.694066
$276$	0	0
$277$	5.57476	0.334955	0.167477	−	0.985876i	$-0.446438\pi$
$277$	5.57476	0.334955	0.167477	+	0.985876i	$0.446438\pi$
$278$	3.67151	0.220203
$279$	0	0
$280$	11.6768	0.697821
$281$	28.0048	1.67063	0.835313	−	0.549775i	$-0.185286\pi$
$281$	28.0048	1.67063	0.835313	+	0.549775i	$0.185286\pi$
$282$	0	0
$283$	6.15522	0.365890	0.182945	−	0.983123i	$-0.441437\pi$
$283$	6.15522	0.365890	0.182945	+	0.983123i	$0.441437\pi$
$284$	1.98746	0.117934
$285$	0	0
$286$	−7.08434	−0.418906
$287$	−26.0968	−1.54044
$288$	0	0
$289$	−14.0218	−0.824810
$290$	−1.85475	−0.108915
$291$	0	0
$292$	10.6474	0.623094
$293$	2.33510	0.136418	0.0682090	−	0.997671i	$-0.478272\pi$
$293$	2.33510	0.136418	0.0682090	+	0.997671i	$0.478272\pi$
$294$	0	0
$295$	−30.6141	−1.78242
$296$	−7.30720	−0.424722
$297$	0	0
$298$	15.4744	0.896408
$299$	−10.3955	−0.601187
$300$	0	0
$301$	6.11061	0.352210
$302$	20.1457	1.15926
$303$	0	0
$304$	−3.39480	−0.194705
$305$	41.4749	2.37484
$306$	0	0
$307$	6.82529	0.389540	0.194770	−	0.980849i	$-0.437604\pi$
$307$	6.82529	0.389540	0.194770	+	0.980849i	$0.437604\pi$
$308$	−8.42212	−0.479895
$309$	0	0
$310$	−14.6936	−0.834543
$311$	8.59522	0.487390	0.243695	−	0.969852i	$-0.421640\pi$
$311$	8.59522	0.487390	0.243695	+	0.969852i	$0.421640\pi$
$312$	0	0
$313$	−21.6689	−1.22480	−0.612399	−	0.790549i	$-0.709795\pi$
$313$	−21.6689	−1.22480	−0.612399	+	0.790549i	$0.709795\pi$
$314$	17.9974	1.01565
$315$	0	0
$316$	14.0932	0.792804
$317$	35.3200	1.98377	0.991883	−	0.127153i	$-0.0405838\pi$
$317$	35.3200	1.98377	0.991883	+	0.127153i	$0.0405838\pi$
$318$	0	0
$319$	1.33778	0.0749013
$320$	3.16812	0.177103
$321$	0	0
$322$	−12.3585	−0.688715
$323$	−5.85859	−0.325981
$324$	0	0
$325$	15.6159	0.866217
$326$	3.05289	0.169084
$327$	0	0
$328$	−7.08052	−0.390956
$329$	−14.1973	−0.782724
$330$	0	0
$331$	−25.8137	−1.41885	−0.709423	−	0.704783i	$-0.751045\pi$
$331$	−25.8137	−1.41885	−0.709423	+	0.704783i	$0.751045\pi$
$332$	−3.09842	−0.170048
$333$	0	0
$334$	−4.10119	−0.224407
$335$	−27.2735	−1.49011
$336$	0	0
$337$	−3.38459	−0.184370	−0.0921851	−	0.995742i	$-0.529385\pi$
$337$	−3.38459	−0.184370	−0.0921851	+	0.995742i	$0.529385\pi$
$338$	−3.38830	−0.184299
$339$	0	0
$340$	5.46739	0.296511
$341$	10.5981	0.573919
$342$	0	0
$343$	−1.53133	−0.0826839
$344$	1.65791	0.0893888
$345$	0	0
$346$	−17.0271	−0.915382
$347$	−16.2000	−0.869663	−0.434831	−	0.900512i	$-0.643192\pi$
$347$	−16.2000	−0.869663	−0.434831	+	0.900512i	$0.643192\pi$
$348$	0	0
$349$	4.49253	0.240480	0.120240	−	0.992745i	$-0.461634\pi$
$349$	4.49253	0.240480	0.120240	+	0.992745i	$0.461634\pi$
$350$	18.5648	0.992330
$351$	0	0
$352$	−2.28507	−0.121795
$353$	−23.2267	−1.23623	−0.618116	−	0.786087i	$-0.712104\pi$
$353$	−23.2267	−1.23623	−0.618116	+	0.786087i	$0.712104\pi$
$354$	0	0
$355$	6.29651	0.334184
$356$	−17.3460	−0.919336
$357$	0	0
$358$	−14.5560	−0.769311
$359$	−11.5539	−0.609794	−0.304897	−	0.952385i	$-0.598622\pi$
$359$	−11.5539	−0.609794	−0.304897	+	0.952385i	$0.598622\pi$
$360$	0	0
$361$	−7.47535	−0.393440
$362$	−13.0238	−0.684516
$363$	0	0
$364$	11.4267	0.598924
$365$	33.7323	1.76563
$366$	0	0
$367$	15.5697	0.812733	0.406366	−	0.913710i	$-0.366796\pi$
$367$	15.5697	0.812733	0.406366	+	0.913710i	$0.366796\pi$
$368$	−3.35309	−0.174792
$369$	0	0
$370$	−23.1500	−1.20351
$371$	−9.51900	−0.494202
$372$	0	0
$373$	−9.75726	−0.505212	−0.252606	−	0.967569i	$-0.581288\pi$
$373$	−9.75726	−0.505212	−0.252606	+	0.967569i	$0.581288\pi$
$374$	−3.94347	−0.203912
$375$	0	0
$376$	−3.85198	−0.198651
$377$	−1.81504	−0.0934792
$378$	0	0
$379$	−18.7904	−0.965197	−0.482599	−	0.875842i	$-0.660307\pi$
$379$	−18.7904	−0.965197	−0.482599	+	0.875842i	$0.660307\pi$
$380$	−10.7551	−0.551725
$381$	0	0
$382$	3.60287	0.184339
$383$	30.7680	1.57217	0.786086	−	0.618117i	$-0.212104\pi$
$383$	30.7680	1.57217	0.786086	+	0.618117i	$0.212104\pi$
$384$	0	0
$385$	−26.6822	−1.35985
$386$	1.47487	0.0750689
$387$	0	0
$388$	9.19117	0.466611
$389$	29.6007	1.50081	0.750407	−	0.660976i	$-0.229858\pi$
$389$	29.6007	1.50081	0.750407	+	0.660976i	$0.229858\pi$
$390$	0	0
$391$	−5.78661	−0.292641
$392$	6.58452	0.332569
$393$	0	0
$394$	2.53862	0.127894
$395$	44.6489	2.24653
$396$	0	0
$397$	16.1427	0.810178	0.405089	−	0.914277i	$-0.367241\pi$
$397$	16.1427	0.810178	0.405089	+	0.914277i	$0.367241\pi$
$398$	−1.85178	−0.0928215
$399$	0	0
$400$	5.03695	0.251848
$401$	25.3624	1.26654	0.633268	−	0.773932i	$-0.281713\pi$
$401$	25.3624	1.26654	0.633268	+	0.773932i	$0.281713\pi$
$402$	0	0
$403$	−14.3790	−0.716269
$404$	11.9602	0.595041
$405$	0	0
$406$	−2.15778	−0.107089
$407$	16.6974	0.827661
$408$	0	0
$409$	−22.7713	−1.12597	−0.562984	−	0.826468i	$-0.690347\pi$
$409$	−22.7713	−1.12597	−0.562984	+	0.826468i	$0.690347\pi$
$410$	−22.4319	−1.10783
$411$	0	0
$412$	−12.7465	−0.627974
$413$	−35.6158	−1.75254
$414$	0	0
$415$	−9.81616	−0.481856
$416$	3.10027	0.152003
$417$	0	0
$418$	7.75734	0.379424
$419$	−4.92276	−0.240492	−0.120246	−	0.992744i	$-0.538368\pi$
$419$	−4.92276	−0.240492	−0.120246	+	0.992744i	$0.538368\pi$
$420$	0	0
$421$	19.7176	0.960977	0.480489	−	0.877001i	$-0.340459\pi$
$421$	19.7176	0.960977	0.480489	+	0.877001i	$0.340459\pi$
$422$	−18.9883	−0.924336
$423$	0	0
$424$	−2.58267	−0.125426
$425$	8.69255	0.421651
$426$	0	0
$427$	48.2510	2.33503
$428$	6.09894	0.294803
$429$	0	0
$430$	5.25247	0.253296
$431$	−6.16323	−0.296873	−0.148436	−	0.988922i	$-0.547424\pi$
$431$	−6.16323	−0.296873	−0.148436	+	0.988922i	$0.547424\pi$
$432$	0	0
$433$	−14.7838	−0.710466	−0.355233	−	0.934778i	$-0.615599\pi$
$433$	−14.7838	−0.710466	−0.355233	+	0.934778i	$0.615599\pi$
$434$	−17.0943	−0.820551
$435$	0	0
$436$	11.2390	0.538250
$437$	11.3831	0.544525
$438$	0	0
$439$	29.7670	1.42070	0.710352	−	0.703847i	$-0.248536\pi$
$439$	29.7670	1.42070	0.710352	+	0.703847i	$0.248536\pi$
$440$	−7.23936	−0.345123
$441$	0	0
$442$	5.35032	0.254489
$443$	30.4744	1.44788	0.723940	−	0.689862i	$-0.242329\pi$
$443$	30.4744	1.44788	0.723940	+	0.689862i	$0.242329\pi$
$444$	0	0
$445$	−54.9541	−2.60508
$446$	−7.25436	−0.343504
$447$	0	0
$448$	3.68572	0.174134
$449$	11.8411	0.558816	0.279408	−	0.960172i	$-0.409862\pi$
$449$	11.8411	0.558816	0.279408	+	0.960172i	$0.409862\pi$
$450$	0	0
$451$	16.1795	0.761861
$452$	−7.03695	−0.330990
$453$	0	0
$454$	−28.5272	−1.33885
$455$	36.2012	1.69714
$456$	0	0
$457$	−0.186393	−0.00871910	−0.00435955	−	0.999990i	$-0.501388\pi$
$457$	−0.186393	−0.00871910	−0.00435955	+	0.999990i	$0.501388\pi$
$458$	1.85538	0.0866963
$459$	0	0
$460$	−10.6230	−0.495299
$461$	10.4054	0.484626	0.242313	−	0.970198i	$-0.422094\pi$
$461$	10.4054	0.484626	0.242313	+	0.970198i	$0.422094\pi$
$462$	0	0
$463$	25.7884	1.19849	0.599245	−	0.800566i	$-0.295468\pi$
$463$	25.7884	1.19849	0.599245	+	0.800566i	$0.295468\pi$
$464$	−0.585444	−0.0271786
$465$	0	0
$466$	8.53469	0.395362
$467$	21.7011	1.00421	0.502104	−	0.864807i	$-0.332559\pi$
$467$	21.7011	1.00421	0.502104	+	0.864807i	$0.332559\pi$
$468$	0	0
$469$	−31.7295	−1.46513
$470$	−12.2035	−0.562906
$471$	0	0
$472$	−9.66319	−0.444784
$473$	−3.78845	−0.174193
$474$	0	0
$475$	−17.0994	−0.784576
$476$	6.36065	0.291540
$477$	0	0
$478$	−7.28796	−0.333344
$479$	−39.1212	−1.78749	−0.893747	−	0.448571i	$-0.851933\pi$
$479$	−39.1212	−1.78749	−0.893747	+	0.448571i	$0.851933\pi$
$480$	0	0
$481$	−22.6543	−1.03295
$482$	−1.33492	−0.0608039
$483$	0	0
$484$	−5.77847	−0.262658
$485$	29.1187	1.32221
$486$	0	0
$487$	−10.4833	−0.475043	−0.237522	−	0.971382i	$-0.576335\pi$
$487$	−10.4833	−0.475043	−0.237522	+	0.971382i	$0.576335\pi$
$488$	13.0913	0.592617
$489$	0	0
$490$	20.8605	0.942383
$491$	6.85702	0.309453	0.154727	−	0.987957i	$-0.450550\pi$
$491$	6.85702	0.309453	0.154727	+	0.987957i	$0.450550\pi$
$492$	0	0
$493$	−1.01033	−0.0455031
$494$	−10.5248	−0.473533
$495$	0	0
$496$	−4.63798	−0.208251
$497$	7.32523	0.328582
$498$	0	0
$499$	18.7737	0.840427	0.420213	−	0.907425i	$-0.361955\pi$
$499$	18.7737	0.840427	0.420213	+	0.907425i	$0.361955\pi$
$500$	0.117076	0.00523579
$501$	0	0
$502$	−2.87856	−0.128476
$503$	−12.4270	−0.554092	−0.277046	−	0.960857i	$-0.589356\pi$
$503$	−12.4270	−0.554092	−0.277046	+	0.960857i	$0.589356\pi$
$504$	0	0
$505$	37.8912	1.68614
$506$	7.66203	0.340619
$507$	0	0
$508$	5.98500	0.265541
$509$	32.9777	1.46171	0.730856	−	0.682532i	$-0.239121\pi$
$509$	32.9777	1.46171	0.730856	+	0.682532i	$0.239121\pi$
$510$	0	0
$511$	39.2434	1.73603
$512$	1.00000	0.0441942
$513$	0	0
$514$	−25.7779	−1.13702
$515$	−40.3823	−1.77946
$516$	0	0
$517$	8.80204	0.387113
$518$	−26.9323	−1.18334
$519$	0	0
$520$	9.82203	0.430724
$521$	14.3320	0.627895	0.313947	−	0.949440i	$-0.398348\pi$
$521$	14.3320	0.627895	0.313947	+	0.949440i	$0.398348\pi$
$522$	0	0
$523$	−5.70884	−0.249630	−0.124815	−	0.992180i	$-0.539834\pi$
$523$	−5.70884	−0.249630	−0.124815	+	0.992180i	$0.539834\pi$
$524$	16.4150	0.717094
$525$	0	0
$526$	−15.0697	−0.657070
$527$	−8.00401	−0.348660
$528$	0	0
$529$	−11.7568	−0.511165
$530$	−8.18220	−0.355412
$531$	0	0
$532$	−12.5123	−0.542476
$533$	−21.9515	−0.950827
$534$	0	0
$535$	19.3221	0.835369
$536$	−8.60876	−0.371842
$537$	0	0
$538$	16.0615	0.692459
$539$	−15.0461	−0.648081
$540$	0	0
$541$	18.0099	0.774305	0.387153	−	0.922016i	$-0.373459\pi$
$541$	18.0099	0.774305	0.387153	+	0.922016i	$0.373459\pi$
$542$	9.41446	0.404386
$543$	0	0
$544$	1.72576	0.0739911
$545$	35.6064	1.52521
$546$	0	0
$547$	−20.6277	−0.881977	−0.440989	−	0.897513i	$-0.645372\pi$
$547$	−20.6277	−0.881977	−0.440989	+	0.897513i	$0.645372\pi$
$548$	−0.211159	−0.00902025
$549$	0	0
$550$	−11.5098	−0.490779
$551$	1.98746	0.0846688
$552$	0	0
$553$	51.9435	2.20886
$554$	5.57476	0.236849
$555$	0	0
$556$	3.67151	0.155707
$557$	−3.80418	−0.161188	−0.0805942	−	0.996747i	$-0.525682\pi$
$557$	−3.80418	−0.161188	−0.0805942	+	0.996747i	$0.525682\pi$
$558$	0	0
$559$	5.13999	0.217398
$560$	11.6768	0.493434
$561$	0	0
$562$	28.0048	1.18131
$563$	−11.0714	−0.466602	−0.233301	−	0.972405i	$-0.574953\pi$
$563$	−11.0714	−0.466602	−0.233301	+	0.972405i	$0.574953\pi$
$564$	0	0
$565$	−22.2939	−0.937911
$566$	6.15522	0.258723
$567$	0	0
$568$	1.98746	0.0833921
$569$	46.7153	1.95841	0.979203	−	0.202881i	$-0.0650306\pi$
$569$	46.7153	1.95841	0.979203	+	0.202881i	$0.0650306\pi$
$570$	0	0
$571$	−31.6862	−1.32603	−0.663013	−	0.748608i	$-0.730723\pi$
$571$	−31.6862	−1.32603	−0.663013	+	0.748608i	$0.730723\pi$
$572$	−7.08434	−0.296211
$573$	0	0
$574$	−26.0968	−1.08926
$575$	−16.8894	−0.704335
$576$	0	0
$577$	44.3284	1.84541	0.922707	−	0.385502i	$-0.125972\pi$
$577$	44.3284	1.84541	0.922707	+	0.385502i	$0.125972\pi$
$578$	−14.0218	−0.583229
$579$	0	0
$580$	−1.85475	−0.0770145
$581$	−11.4199	−0.473778
$582$	0	0
$583$	5.90158	0.244418
$584$	10.6474	0.440594
$585$	0	0
$586$	2.33510	0.0964621
$587$	41.0447	1.69409	0.847047	−	0.531518i	$-0.178378\pi$
$587$	41.0447	1.69409	0.847047	+	0.531518i	$0.178378\pi$
$588$	0	0
$589$	15.7450	0.648761
$590$	−30.6141	−1.26036
$591$	0	0
$592$	−7.30720	−0.300324
$593$	−9.69265	−0.398029	−0.199015	−	0.979996i	$-0.563774\pi$
$593$	−9.69265	−0.398029	−0.199015	+	0.979996i	$0.563774\pi$
$594$	0	0
$595$	20.1513	0.826121
$596$	15.4744	0.633856
$597$	0	0
$598$	−10.3955	−0.425103
$599$	−29.8379	−1.21914	−0.609572	−	0.792731i	$-0.708659\pi$
$599$	−29.8379	−1.21914	−0.609572	+	0.792731i	$0.708659\pi$
$600$	0	0
$601$	12.5507	0.511954	0.255977	−	0.966683i	$-0.417603\pi$
$601$	12.5507	0.511954	0.255977	+	0.966683i	$0.417603\pi$
$602$	6.11061	0.249050
$603$	0	0
$604$	20.1457	0.819718
$605$	−18.3068	−0.744279
$606$	0	0
$607$	−2.78013	−0.112842	−0.0564211	−	0.998407i	$-0.517969\pi$
$607$	−2.78013	−0.112842	−0.0564211	+	0.998407i	$0.517969\pi$
$608$	−3.39480	−0.137677
$609$	0	0
$610$	41.4749	1.67927
$611$	−11.9422	−0.483130
$612$	0	0
$613$	−8.59291	−0.347064	−0.173532	−	0.984828i	$-0.555518\pi$
$613$	−8.59291	−0.347064	−0.173532	+	0.984828i	$0.555518\pi$
$614$	6.82529	0.275446
$615$	0	0
$616$	−8.42212	−0.339337
$617$	−20.0790	−0.808351	−0.404176	−	0.914681i	$-0.632442\pi$
$617$	−20.0790	−0.808351	−0.404176	+	0.914681i	$0.632442\pi$
$618$	0	0
$619$	8.39185	0.337297	0.168648	−	0.985676i	$-0.446060\pi$
$619$	8.39185	0.337297	0.168648	+	0.985676i	$0.446060\pi$
$620$	−14.6936	−0.590111
$621$	0	0
$622$	8.59522	0.344637
$623$	−63.9325	−2.56140
$624$	0	0
$625$	−24.8139	−0.992554
$626$	−21.6689	−0.866063
$627$	0	0
$628$	17.9974	0.718176
$629$	−12.6104	−0.502811
$630$	0	0
$631$	17.5631	0.699178	0.349589	−	0.936903i	$-0.386321\pi$
$631$	17.5631	0.699178	0.349589	+	0.936903i	$0.386321\pi$
$632$	14.0932	0.560597
$633$	0	0
$634$	35.3200	1.40273
$635$	18.9612	0.752450
$636$	0	0
$637$	20.4138	0.808826
$638$	1.33778	0.0529632
$639$	0	0
$640$	3.16812	0.125231
$641$	−13.4146	−0.529846	−0.264923	−	0.964270i	$-0.585347\pi$
$641$	−13.4146	−0.529846	−0.264923	+	0.964270i	$0.585347\pi$
$642$	0	0
$643$	−10.4351	−0.411520	−0.205760	−	0.978602i	$-0.565967\pi$
$643$	−10.4351	−0.411520	−0.205760	+	0.978602i	$0.565967\pi$
$644$	−12.3585	−0.486995
$645$	0	0
$646$	−5.85859	−0.230503
$647$	−37.9585	−1.49230	−0.746152	−	0.665775i	$-0.768101\pi$
$647$	−37.9585	−1.49230	−0.746152	+	0.665775i	$0.768101\pi$
$648$	0	0
$649$	22.0810	0.866756
$650$	15.6159	0.612508
$651$	0	0
$652$	3.05289	0.119560
$653$	−27.3063	−1.06858	−0.534289	−	0.845302i	$-0.679421\pi$
$653$	−27.3063	−1.06858	−0.534289	+	0.845302i	$0.679421\pi$
$654$	0	0
$655$	52.0047	2.03199
$656$	−7.08052	−0.276448
$657$	0	0
$658$	−14.1973	−0.553469
$659$	−34.3549	−1.33828	−0.669139	−	0.743138i	$-0.733337\pi$
$659$	−34.3549	−1.33828	−0.669139	+	0.743138i	$0.733337\pi$
$660$	0	0
$661$	−24.5777	−0.955962	−0.477981	−	0.878370i	$-0.658631\pi$
$661$	−24.5777	−0.955962	−0.477981	+	0.878370i	$0.658631\pi$
$662$	−25.8137	−1.00328
$663$	0	0
$664$	−3.09842	−0.120242
$665$	−39.6403	−1.53719
$666$	0	0
$667$	1.96305	0.0760094
$668$	−4.10119	−0.158680
$669$	0	0
$670$	−27.2735	−1.05367
$671$	−29.9146	−1.15484
$672$	0	0
$673$	42.8754	1.65273	0.826363	−	0.563137i	$-0.190406\pi$
$673$	42.8754	1.65273	0.826363	+	0.563137i	$0.190406\pi$
$674$	−3.38459	−0.130369
$675$	0	0
$676$	−3.38830	−0.130319
$677$	7.48743	0.287765	0.143883	−	0.989595i	$-0.454041\pi$
$677$	7.48743	0.287765	0.143883	+	0.989595i	$0.454041\pi$
$678$	0	0
$679$	33.8761	1.30004
$680$	5.46739	0.209665
$681$	0	0
$682$	10.5981	0.405822
$683$	16.7545	0.641093	0.320546	−	0.947233i	$-0.396134\pi$
$683$	16.7545	0.641093	0.320546	+	0.947233i	$0.396134\pi$
$684$	0	0
$685$	−0.668975	−0.0255602
$686$	−1.53133	−0.0584664
$687$	0	0
$688$	1.65791	0.0632074
$689$	−8.00699	−0.305042
$690$	0	0
$691$	−14.9534	−0.568856	−0.284428	−	0.958697i	$-0.591804\pi$
$691$	−14.9534	−0.568856	−0.284428	+	0.958697i	$0.591804\pi$
$692$	−17.0271	−0.647273
$693$	0	0
$694$	−16.2000	−0.614945
$695$	11.6318	0.441218
$696$	0	0
$697$	−12.2192	−0.462837
$698$	4.49253	0.170045
$699$	0	0
$700$	18.5648	0.701683
$701$	−42.1025	−1.59019	−0.795094	−	0.606486i	$-0.792579\pi$
$701$	−42.1025	−1.59019	−0.795094	+	0.606486i	$0.792579\pi$
$702$	0	0
$703$	24.8065	0.935593
$704$	−2.28507	−0.0861217
$705$	0	0
$706$	−23.2267	−0.874147
$707$	44.0819	1.65787
$708$	0	0
$709$	28.4856	1.06980	0.534899	−	0.844916i	$-0.320350\pi$
$709$	28.4856	1.06980	0.534899	+	0.844916i	$0.320350\pi$
$710$	6.29651	0.236304
$711$	0	0
$712$	−17.3460	−0.650069
$713$	15.5515	0.582410
$714$	0	0
$715$	−22.4440	−0.839358
$716$	−14.5560	−0.543985
$717$	0	0
$718$	−11.5539	−0.431189
$719$	2.53487	0.0945348	0.0472674	−	0.998882i	$-0.484949\pi$
$719$	2.53487	0.0945348	0.0472674	+	0.998882i	$0.484949\pi$
$720$	0	0
$721$	−46.9799	−1.74962
$722$	−7.47535	−0.278204
$723$	0	0
$724$	−13.0238	−0.484026
$725$	−2.94885	−0.109518
$726$	0	0
$727$	19.8834	0.737434	0.368717	−	0.929542i	$-0.379797\pi$
$727$	19.8834	0.737434	0.368717	+	0.929542i	$0.379797\pi$
$728$	11.4267	0.423503
$729$	0	0
$730$	33.7323	1.24849
$731$	2.86116	0.105824
$732$	0	0
$733$	−10.0111	−0.369767	−0.184884	−	0.982760i	$-0.559191\pi$
$733$	−10.0111	−0.369767	−0.184884	+	0.982760i	$0.559191\pi$
$734$	15.5697	0.574689
$735$	0	0
$736$	−3.35309	−0.123596
$737$	19.6716	0.724613
$738$	0	0
$739$	−40.3914	−1.48582	−0.742911	−	0.669390i	$-0.766556\pi$
$739$	−40.3914	−1.48582	−0.742911	+	0.669390i	$0.766556\pi$
$740$	−23.1500	−0.851013
$741$	0	0
$742$	−9.51900	−0.349453
$743$	35.6502	1.30788	0.653939	−	0.756547i	$-0.273115\pi$
$743$	35.6502	1.30788	0.653939	+	0.756547i	$0.273115\pi$
$744$	0	0
$745$	49.0247	1.79612
$746$	−9.75726	−0.357239
$747$	0	0
$748$	−3.94347	−0.144187
$749$	22.4790	0.821364
$750$	0	0
$751$	−9.46873	−0.345519	−0.172759	−	0.984964i	$-0.555268\pi$
$751$	−9.46873	−0.345519	−0.172759	+	0.984964i	$0.555268\pi$
$752$	−3.85198	−0.140467
$753$	0	0
$754$	−1.81504	−0.0660998
$755$	63.8240	2.32279
$756$	0	0
$757$	−37.9651	−1.37987	−0.689933	−	0.723873i	$-0.742360\pi$
$757$	−37.9651	−1.37987	−0.689933	+	0.723873i	$0.742360\pi$
$758$	−18.7904	−0.682498
$759$	0	0
$760$	−10.7551	−0.390129
$761$	44.8858	1.62711	0.813554	−	0.581490i	$-0.197530\pi$
$761$	44.8858	1.62711	0.813554	+	0.581490i	$0.197530\pi$
$762$	0	0
$763$	41.4238	1.49964
$764$	3.60287	0.130347
$765$	0	0
$766$	30.7680	1.11169
$767$	−29.9585	−1.08174
$768$	0	0
$769$	45.7368	1.64931	0.824656	−	0.565634i	$-0.191369\pi$
$769$	45.7368	1.64931	0.824656	+	0.565634i	$0.191369\pi$
$770$	−26.6822	−0.961561
$771$	0	0
$772$	1.47487	0.0530817
$773$	−20.5707	−0.739876	−0.369938	−	0.929056i	$-0.620621\pi$
$773$	−20.5707	−0.739876	−0.369938	+	0.929056i	$0.620621\pi$
$774$	0	0
$775$	−23.3613	−0.839162
$776$	9.19117	0.329944
$777$	0	0
$778$	29.6007	1.06124
$779$	24.0369	0.861212
$780$	0	0
$781$	−4.54149	−0.162507
$782$	−5.78661	−0.206929
$783$	0	0
$784$	6.58452	0.235162
$785$	57.0180	2.03506
$786$	0	0
$787$	44.5635	1.58852	0.794259	−	0.607579i	$-0.207859\pi$
$787$	44.5635	1.58852	0.794259	+	0.607579i	$0.207859\pi$
$788$	2.53862	0.0904346
$789$	0	0
$790$	44.6489	1.58853
$791$	−25.9362	−0.922186
$792$	0	0
$793$	40.5867	1.44128
$794$	16.1427	0.572882
$795$	0	0
$796$	−1.85178	−0.0656347
$797$	43.8601	1.55361	0.776803	−	0.629744i	$-0.216840\pi$
$797$	43.8601	1.55361	0.776803	+	0.629744i	$0.216840\pi$
$798$	0	0
$799$	−6.64758	−0.235174
$800$	5.03695	0.178083
$801$	0	0
$802$	25.3624	0.895577
$803$	−24.3301	−0.858590
$804$	0	0
$805$	−39.1533	−1.37997
$806$	−14.3790	−0.506479
$807$	0	0
$808$	11.9602	0.420758
$809$	−15.5821	−0.547836	−0.273918	−	0.961753i	$-0.588320\pi$
$809$	−15.5821	−0.547836	−0.273918	+	0.961753i	$0.588320\pi$
$810$	0	0
$811$	−30.4691	−1.06992	−0.534958	−	0.844879i	$-0.679673\pi$
$811$	−30.4691	−1.06992	−0.534958	+	0.844879i	$0.679673\pi$
$812$	−2.15778	−0.0757233
$813$	0	0
$814$	16.6974	0.585245
$815$	9.67191	0.338792
$816$	0	0
$817$	−5.62828	−0.196909
$818$	−22.7713	−0.796180
$819$	0	0
$820$	−22.4319	−0.783356
$821$	−1.90102	−0.0663460	−0.0331730	−	0.999450i	$-0.510561\pi$
$821$	−1.90102	−0.0663460	−0.0331730	+	0.999450i	$0.510561\pi$
$822$	0	0
$823$	−22.0410	−0.768302	−0.384151	−	0.923270i	$-0.625506\pi$
$823$	−22.0410	−0.768302	−0.384151	+	0.923270i	$0.625506\pi$
$824$	−12.7465	−0.444045
$825$	0	0
$826$	−35.6158	−1.23923
$827$	48.4976	1.68643	0.843213	−	0.537579i	$-0.180661\pi$
$827$	48.4976	1.68643	0.843213	+	0.537579i	$0.180661\pi$
$828$	0	0
$829$	−20.3186	−0.705693	−0.352846	−	0.935681i	$-0.614786\pi$
$829$	−20.3186	−0.705693	−0.352846	+	0.935681i	$0.614786\pi$
$830$	−9.81616	−0.340724
$831$	0	0
$832$	3.10027	0.107483
$833$	11.3633	0.393714
$834$	0	0
$835$	−12.9931	−0.449643
$836$	7.75734	0.268293
$837$	0	0
$838$	−4.92276	−0.170054
$839$	−5.06635	−0.174910	−0.0874549	−	0.996168i	$-0.527873\pi$
$839$	−5.06635	−0.174910	−0.0874549	+	0.996168i	$0.527873\pi$
$840$	0	0
$841$	−28.6573	−0.988181
$842$	19.7176	0.679514
$843$	0	0
$844$	−18.9883	−0.653604
$845$	−10.7345	−0.369278
$846$	0	0
$847$	−21.2978	−0.731801
$848$	−2.58267	−0.0886893
$849$	0	0
$850$	8.69255	0.298152
$851$	24.5017	0.839907
$852$	0	0
$853$	26.4679	0.906242	0.453121	−	0.891449i	$-0.350311\pi$
$853$	26.4679	0.906242	0.453121	+	0.891449i	$0.350311\pi$
$854$	48.2510	1.65112
$855$	0	0
$856$	6.09894	0.208457
$857$	−18.3116	−0.625514	−0.312757	−	0.949833i	$-0.601252\pi$
$857$	−18.3116	−0.625514	−0.312757	+	0.949833i	$0.601252\pi$
$858$	0	0
$859$	−9.43937	−0.322067	−0.161034	−	0.986949i	$-0.551483\pi$
$859$	−9.43937	−0.322067	−0.161034	+	0.986949i	$0.551483\pi$
$860$	5.25247	0.179108
$861$	0	0
$862$	−6.16323	−0.209921
$863$	−14.2154	−0.483898	−0.241949	−	0.970289i	$-0.577787\pi$
$863$	−14.2154	−0.483898	−0.241949	+	0.970289i	$0.577787\pi$
$864$	0	0
$865$	−53.9438	−1.83414
$866$	−14.7838	−0.502375
$867$	0	0
$868$	−17.0943	−0.580217
$869$	−32.2039	−1.09244
$870$	0	0
$871$	−26.6895	−0.904340
$872$	11.2390	0.380600
$873$	0	0
$874$	11.3831	0.385038
$875$	0.431509	0.0145877
$876$	0	0
$877$	39.1054	1.32050	0.660248	−	0.751047i	$-0.270451\pi$
$877$	39.1054	1.32050	0.660248	+	0.751047i	$0.270451\pi$
$878$	29.7670	1.00459
$879$	0	0
$880$	−7.23936	−0.244039
$881$	−22.8938	−0.771313	−0.385657	−	0.922642i	$-0.626025\pi$
$881$	−22.8938	−0.771313	−0.385657	+	0.922642i	$0.626025\pi$
$882$	0	0
$883$	−17.1509	−0.577174	−0.288587	−	0.957454i	$-0.593185\pi$
$883$	−17.1509	−0.577174	−0.288587	+	0.957454i	$0.593185\pi$
$884$	5.35032	0.179951
$885$	0	0
$886$	30.4744	1.02381
$887$	18.2570	0.613011	0.306506	−	0.951869i	$-0.400840\pi$
$887$	18.2570	0.613011	0.306506	+	0.951869i	$0.400840\pi$
$888$	0	0
$889$	22.0590	0.739835
$890$	−54.9541	−1.84207
$891$	0	0
$892$	−7.25436	−0.242894
$893$	13.0767	0.437595
$894$	0	0
$895$	−46.1152	−1.54146
$896$	3.68572	0.123131
$897$	0	0
$898$	11.8411	0.395143
$899$	2.71527	0.0905595
$900$	0	0
$901$	−4.45706	−0.148486
$902$	16.1795	0.538717
$903$	0	0
$904$	−7.03695	−0.234046
$905$	−41.2609	−1.37156
$906$	0	0
$907$	29.1916	0.969293	0.484646	−	0.874710i	$-0.338948\pi$
$907$	29.1916	0.969293	0.484646	+	0.874710i	$0.338948\pi$
$908$	−28.5272	−0.946708
$909$	0	0
$910$	36.2012	1.20006
$911$	54.6359	1.81017	0.905084	−	0.425233i	$-0.139808\pi$
$911$	54.6359	1.81017	0.905084	+	0.425233i	$0.139808\pi$
$912$	0	0
$913$	7.08010	0.234317
$914$	−0.186393	−0.00616533
$915$	0	0
$916$	1.85538	0.0613036
$917$	60.5012	1.99793
$918$	0	0
$919$	41.0995	1.35575	0.677873	−	0.735179i	$-0.262902\pi$
$919$	41.0995	1.35575	0.677873	+	0.735179i	$0.262902\pi$
$920$	−10.6230	−0.350229
$921$	0	0
$922$	10.4054	0.342682
$923$	6.16168	0.202814
$924$	0	0
$925$	−36.8060	−1.21017
$926$	25.7884	0.847460
$927$	0	0
$928$	−0.585444	−0.0192181
$929$	14.5961	0.478882	0.239441	−	0.970911i	$-0.423036\pi$
$929$	14.5961	0.478882	0.239441	+	0.970911i	$0.423036\pi$
$930$	0	0
$931$	−22.3531	−0.732594
$932$	8.53469	0.279563
$933$	0	0
$934$	21.7011	0.710083
$935$	−12.4934	−0.408576
$936$	0	0
$937$	24.9282	0.814370	0.407185	−	0.913346i	$-0.366511\pi$
$937$	24.9282	0.814370	0.407185	+	0.913346i	$0.366511\pi$
$938$	−31.7295	−1.03600
$939$	0	0
$940$	−12.2035	−0.398035
$941$	20.8443	0.679506	0.339753	−	0.940515i	$-0.389657\pi$
$941$	20.8443	0.679506	0.339753	+	0.940515i	$0.389657\pi$
$942$	0	0
$943$	23.7416	0.773133
$944$	−9.66319	−0.314510
$945$	0	0
$946$	−3.78845	−0.123173
$947$	35.9130	1.16702	0.583508	−	0.812107i	$-0.301680\pi$
$947$	35.9130	1.16702	0.583508	+	0.812107i	$0.301680\pi$
$948$	0	0
$949$	33.0100	1.07155
$950$	−17.0994	−0.554779
$951$	0	0
$952$	6.36065	0.206150
$953$	5.80207	0.187947	0.0939737	−	0.995575i	$-0.470043\pi$
$953$	5.80207	0.187947	0.0939737	+	0.995575i	$0.470043\pi$
$954$	0	0
$955$	11.4143	0.369359
$956$	−7.28796	−0.235710
$957$	0	0
$958$	−39.1212	−1.26395
$959$	−0.778271	−0.0251317
$960$	0	0
$961$	−9.48919	−0.306103
$962$	−22.6543	−0.730405
$963$	0	0
$964$	−1.33492	−0.0429948
$965$	4.67256	0.150415
$966$	0	0
$967$	29.6589	0.953766	0.476883	−	0.878967i	$-0.341767\pi$
$967$	29.6589	0.953766	0.476883	+	0.878967i	$0.341767\pi$
$968$	−5.77847	−0.185727
$969$	0	0
$970$	29.1187	0.934944
$971$	47.1522	1.51318	0.756592	−	0.653887i	$-0.226863\pi$
$971$	47.1522	1.51318	0.756592	+	0.653887i	$0.226863\pi$
$972$	0	0
$973$	13.5322	0.433821
$974$	−10.4833	−0.335906
$975$	0	0
$976$	13.0913	0.419044
$977$	−23.8029	−0.761524	−0.380762	−	0.924673i	$-0.624338\pi$
$977$	−23.8029	−0.761524	−0.380762	+	0.924673i	$0.624338\pi$
$978$	0	0
$979$	39.6368	1.26680
$980$	20.8605	0.666365
$981$	0	0
$982$	6.85702	0.218816
$983$	11.0823	0.353469	0.176734	−	0.984259i	$-0.443447\pi$
$983$	11.0823	0.353469	0.176734	+	0.984259i	$0.443447\pi$
$984$	0	0
$985$	8.04264	0.256260
$986$	−1.01033	−0.0321756
$987$	0	0
$988$	−10.5248	−0.334839
$989$	−5.55913	−0.176770
$990$	0	0
$991$	11.4369	0.363306	0.181653	−	0.983363i	$-0.441855\pi$
$991$	11.4369	0.363306	0.181653	+	0.983363i	$0.441855\pi$
$992$	−4.63798	−0.147256
$993$	0	0
$994$	7.32523	0.232342
$995$	−5.86666	−0.185986
$996$	0	0
$997$	−27.8945	−0.883428	−0.441714	−	0.897156i	$-0.645629\pi$
$997$	−27.8945	−0.883428	−0.441714	+	0.897156i	$0.645629\pi$
$998$	18.7737	0.594271
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1458.2.a.g.1.5		6
3.2	odd	2		1458.2.a.f.1.2		6
9.2	odd	6		1458.2.c.g.973.5		12
9.4	even	3		1458.2.c.f.487.2		12
9.5	odd	6		1458.2.c.g.487.5		12
9.7	even	3		1458.2.c.f.973.2		12
27.2	odd	18		486.2.e.g.433.1		12
27.4	even	9		486.2.e.h.379.1		12
27.5	odd	18		162.2.e.b.73.1		12
27.7	even	9		486.2.e.h.109.1		12
27.11	odd	18		162.2.e.b.91.1		12
27.13	even	9		486.2.e.f.55.2		12
27.14	odd	18		486.2.e.g.55.1		12
27.16	even	9		54.2.e.b.13.1	✓	12
27.20	odd	18		486.2.e.e.109.2		12
27.22	even	9		54.2.e.b.25.1	yes	12
27.23	odd	18		486.2.e.e.379.2		12
27.25	even	9		486.2.e.f.433.2		12
108.43	odd	18		432.2.u.b.337.2		12
108.103	odd	18		432.2.u.b.241.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.2.e.b.13.1	✓	12	27.16	even	9
54.2.e.b.25.1	yes	12	27.22	even	9
162.2.e.b.73.1		12	27.5	odd	18
162.2.e.b.91.1		12	27.11	odd	18
432.2.u.b.241.2		12	108.103	odd	18
432.2.u.b.337.2		12	108.43	odd	18
486.2.e.e.109.2		12	27.20	odd	18
486.2.e.e.379.2		12	27.23	odd	18
486.2.e.f.55.2		12	27.13	even	9
486.2.e.f.433.2		12	27.25	even	9
486.2.e.g.55.1		12	27.14	odd	18
486.2.e.g.433.1		12	27.2	odd	18
486.2.e.h.109.1		12	27.7	even	9
486.2.e.h.379.1		12	27.4	even	9
1458.2.a.f.1.2		6	3.2	odd	2
1458.2.a.g.1.5		6	1.1	even	1	trivial
1458.2.c.f.487.2		12	9.4	even	3
1458.2.c.f.973.2		12	9.7	even	3
1458.2.c.g.487.5		12	9.5	odd	6
1458.2.c.g.973.5		12	9.2	odd	6

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:	\( N \)	\(=\)	\( 1458 = 2 \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1458.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.16812 q^{5} +3.68572 q^{7} +1.00000 q^{8} +3.16812 q^{10} -2.28507 q^{11} +3.10027 q^{13} +3.68572 q^{14} +1.00000 q^{16} +1.72576 q^{17} -3.39480 q^{19} +3.16812 q^{20} -2.28507 q^{22} -3.35309 q^{23} +5.03695 q^{25} +3.10027 q^{26} +3.68572 q^{28} -0.585444 q^{29} -4.63798 q^{31} +1.00000 q^{32} +1.72576 q^{34} +11.6768 q^{35} -7.30720 q^{37} -3.39480 q^{38} +3.16812 q^{40} -7.08052 q^{41} +1.65791 q^{43} -2.28507 q^{44} -3.35309 q^{46} -3.85198 q^{47} +6.58452 q^{49} +5.03695 q^{50} +3.10027 q^{52} -2.58267 q^{53} -7.23936 q^{55} +3.68572 q^{56} -0.585444 q^{58} -9.66319 q^{59} +13.0913 q^{61} -4.63798 q^{62} +1.00000 q^{64} +9.82203 q^{65} -8.60876 q^{67} +1.72576 q^{68} +11.6768 q^{70} +1.98746 q^{71} +10.6474 q^{73} -7.30720 q^{74} -3.39480 q^{76} -8.42212 q^{77} +14.0932 q^{79} +3.16812 q^{80} -7.08052 q^{82} -3.09842 q^{83} +5.46739 q^{85} +1.65791 q^{86} -2.28507 q^{88} -17.3460 q^{89} +11.4267 q^{91} -3.35309 q^{92} -3.85198 q^{94} -10.7551 q^{95} +9.19117 q^{97} +6.58452 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 3 q^{5} + 6 q^{7} + 6 q^{8} + 3 q^{10} + 3 q^{11} + 9 q^{13} + 6 q^{14} + 6 q^{16} + 6 q^{17} + 9 q^{19} + 3 q^{20} + 3 q^{22} - 3 q^{23} + 15 q^{25} + 9 q^{26} + 6 q^{28} + 3 q^{29}+ \cdots + 12 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 + 6 * q^4 + 3 * q^5 + 6 * q^7 + 6 * q^8 + 3 * q^10 + 3 * q^11 + 9 * q^13 + 6 * q^14 + 6 * q^16 + 6 * q^17 + 9 * q^19 + 3 * q^20 + 3 * q^22 - 3 * q^23 + 15 * q^25 + 9 * q^26 + 6 * q^28 + 3 * q^29 + 12 * q^31 + 6 * q^32 + 6 * q^34 - 3 * q^35 + 15 * q^37 + 9 * q^38 + 3 * q^40 + 3 * q^41 + 12 * q^43 + 3 * q^44 - 3 * q^46 - 9 * q^47 + 12 * q^49 + 15 * q^50 + 9 * q^52 - 6 * q^53 + 9 * q^55 + 6 * q^56 + 3 * q^58 - 3 * q^59 + 12 * q^61 + 12 * q^62 + 6 * q^64 - 3 * q^65 + 21 * q^67 + 6 * q^68 - 3 * q^70 - 12 * q^71 + 21 * q^73 + 15 * q^74 + 9 * q^76 - 3 * q^77 + 3 * q^80 + 3 * q^82 - 27 * q^83 + 12 * q^86 + 3 * q^88 - 12 * q^89 + 6 * q^91 - 3 * q^92 - 9 * q^94 - 30 * q^95 + 18 * q^97 + 12 * q^98

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