LMFDB - Embedded newform 8281.2.a.bo.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, 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("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.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:	$66.1241179138$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.105456.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 13x^{2} + 39 $ x^4 - 13*x^2 + 39
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 637)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.16731$ of defining polynomial
Character	$\chi$	$=$	8281.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.30278 q^{2} +2.16731 q^{3} +3.30278 q^{4} -2.16731 q^{5} -4.99082 q^{6} -3.00000 q^{8} +1.69722 q^{9} +4.99082 q^{10} -4.90833 q^{11} +7.15813 q^{12} -4.69722 q^{15} +0.302776 q^{16} +7.15813 q^{17} -3.90833 q^{18} -2.16731 q^{19} -7.15813 q^{20} +11.3028 q^{22} -0.605551 q^{23} -6.50192 q^{24} -0.302776 q^{25} -2.82352 q^{27} -2.30278 q^{29} +10.8167 q^{30} +7.15813 q^{31} +5.30278 q^{32} -10.6379 q^{33} -16.4836 q^{34} +5.60555 q^{36} -8.60555 q^{37} +4.99082 q^{38} +6.50192 q^{40} +9.98165 q^{41} -12.5139 q^{43} -16.2111 q^{44} -3.67841 q^{45} +1.39445 q^{46} +1.51110 q^{47} +0.656208 q^{48} +0.697224 q^{50} +15.5139 q^{51} +2.39445 q^{53} +6.50192 q^{54} +10.6379 q^{55} -4.69722 q^{57} +5.30278 q^{58} +2.82352 q^{59} -15.5139 q^{60} +4.33462 q^{61} -16.4836 q^{62} -12.8167 q^{64} +24.4966 q^{66} -1.00000 q^{67} +23.6417 q^{68} -1.31242 q^{69} -4.00000 q^{71} -5.09167 q^{72} +4.33462 q^{73} +19.8167 q^{74} -0.656208 q^{75} -7.15813 q^{76} -6.60555 q^{79} -0.656208 q^{80} -11.2111 q^{81} -22.9855 q^{82} -2.82352 q^{83} -15.5139 q^{85} +28.8167 q^{86} -4.99082 q^{87} +14.7250 q^{88} -6.50192 q^{89} +8.47055 q^{90} -2.00000 q^{92} +15.5139 q^{93} -3.47972 q^{94} +4.69722 q^{95} +11.4927 q^{96} +13.6601 q^{97} -8.33053 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 6 q^{4} - 12 q^{8} + 14 q^{9} + 2 q^{11} - 26 q^{15} - 6 q^{16} + 6 q^{18} + 38 q^{22} + 12 q^{23} + 6 q^{25} - 2 q^{29} + 14 q^{32} + 8 q^{36} - 20 q^{37} - 14 q^{43} - 36 q^{44} + 20 q^{46}+ \cdots + 46 q^{99}+O(q^{100}) $ 4 * q - 2 * q^2 + 6 * q^4 - 12 * q^8 + 14 * q^9 + 2 * q^11 - 26 * q^15 - 6 * q^16 + 6 * q^18 + 38 * q^22 + 12 * q^23 + 6 * q^25 - 2 * q^29 + 14 * q^32 + 8 * q^36 - 20 * q^37 - 14 * q^43 - 36 * q^44 + 20 * q^46 + 10 * q^50 + 26 * q^51 + 24 * q^53 - 26 * q^57 + 14 * q^58 - 26 * q^60 - 8 * q^64 - 4 * q^67 - 16 * q^71 - 42 * q^72 + 36 * q^74 - 12 * q^79 - 16 * q^81 - 26 * q^85 + 72 * q^86 - 6 * q^88 - 8 * q^92 + 26 * q^93 + 26 * q^95 + 46 * q^99

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 $

$2$	−2.30278	−1.62831	−0.814154	−	0.580649i	$-0.802799\pi$
$2$	−2.30278	−1.62831	−0.814154	+	0.580649i	$0.802799\pi$
$3$	2.16731	1.25130	0.625648	−	0.780106i	$-0.284835\pi$
$3$	2.16731	1.25130	0.625648	+	0.780106i	$0.284835\pi$
$4$	3.30278	1.65139
$5$	−2.16731	−0.969250	−0.484625	−	0.874722i	$-0.661044\pi$
$5$	−2.16731	−0.969250	−0.484625	+	0.874722i	$0.661044\pi$
$6$	−4.99082	−2.03750
$7$	0	0
$7$	0	0
$8$	−3.00000	−1.06066
$9$	1.69722	0.565741
$10$	4.99082	1.57824
$11$	−4.90833	−1.47992	−0.739958	−	0.672653i	$-0.765155\pi$
$11$	−4.90833	−1.47992	−0.739958	+	0.672653i	$0.765155\pi$
$12$	7.15813	2.06637
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−4.69722	−1.21282
$16$	0.302776	0.0756939
$17$	7.15813	1.73610	0.868051	−	0.496475i	$-0.165373\pi$
$17$	7.15813	1.73610	0.868051	+	0.496475i	$0.165373\pi$
$18$	−3.90833	−0.921201
$19$	−2.16731	−0.497215	−0.248607	−	0.968604i	$-0.579973\pi$
$19$	−2.16731	−0.497215	−0.248607	+	0.968604i	$0.579973\pi$
$20$	−7.15813	−1.60061
$21$	0	0
$22$	11.3028	2.40976
$23$	−0.605551	−0.126266	−0.0631331	−	0.998005i	$-0.520109\pi$
$23$	−0.605551	−0.126266	−0.0631331	+	0.998005i	$0.520109\pi$
$24$	−6.50192	−1.32720
$25$	−0.302776	−0.0605551
$26$	0	0
$27$	−2.82352	−0.543386
$28$	0	0
$29$	−2.30278	−0.427615	−0.213807	−	0.976876i	$-0.568586\pi$
$29$	−2.30278	−0.427615	−0.213807	+	0.976876i	$0.568586\pi$
$30$	10.8167	1.97484
$31$	7.15813	1.28564	0.642819	−	0.766018i	$-0.277765\pi$
$31$	7.15813	1.28564	0.642819	+	0.766018i	$0.277765\pi$
$32$	5.30278	0.937407
$33$	−10.6379	−1.85181
$34$	−16.4836	−2.82691
$35$	0	0
$36$	5.60555	0.934259
$37$	−8.60555	−1.41474	−0.707372	−	0.706842i	$-0.750119\pi$
$37$	−8.60555	−1.41474	−0.707372	+	0.706842i	$0.750119\pi$
$38$	4.99082	0.809619
$39$	0	0
$40$	6.50192	1.02804
$41$	9.98165	1.55887	0.779436	−	0.626482i	$-0.215506\pi$
$41$	9.98165	1.55887	0.779436	+	0.626482i	$0.215506\pi$
$42$	0	0
$43$	−12.5139	−1.90835	−0.954174	−	0.299252i	$-0.903263\pi$
$43$	−12.5139	−1.90835	−0.954174	+	0.299252i	$0.903263\pi$
$44$	−16.2111	−2.44392
$45$	−3.67841	−0.548345
$46$	1.39445	0.205600
$47$	1.51110	0.220417	0.110208	−	0.993909i	$-0.464848\pi$
$47$	1.51110	0.220417	0.110208	+	0.993909i	$0.464848\pi$
$48$	0.656208	0.0947155
$49$	0	0
$50$	0.697224	0.0986024
$51$	15.5139	2.17238
$52$	0	0
$53$	2.39445	0.328903	0.164451	−	0.986385i	$-0.447415\pi$
$53$	2.39445	0.328903	0.164451	+	0.986385i	$0.447415\pi$
$54$	6.50192	0.884800
$55$	10.6379	1.43441
$56$	0	0
$57$	−4.69722	−0.622163
$58$	5.30278	0.696289
$59$	2.82352	0.367590	0.183795	−	0.982965i	$-0.441162\pi$
$59$	2.82352	0.367590	0.183795	+	0.982965i	$0.441162\pi$
$60$	−15.5139	−2.00283
$61$	4.33462	0.554991	0.277495	−	0.960727i	$-0.410496\pi$
$61$	4.33462	0.554991	0.277495	+	0.960727i	$0.410496\pi$
$62$	−16.4836	−2.09342
$63$	0	0
$64$	−12.8167	−1.60208
$65$	0	0
$66$	24.4966	3.01532
$67$	−1.00000	−0.122169	−0.0610847	−	0.998133i	$-0.519456\pi$
$67$	−1.00000	−0.122169	−0.0610847	+	0.998133i	$0.519456\pi$
$68$	23.6417	2.86698
$69$	−1.31242	−0.157996
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	−5.09167	−0.600059
$73$	4.33462	0.507328	0.253664	−	0.967292i	$-0.418364\pi$
$73$	4.33462	0.507328	0.253664	+	0.967292i	$0.418364\pi$
$74$	19.8167	2.30364
$75$	−0.656208	−0.0757724
$76$	−7.15813	−0.821094
$77$	0	0
$78$	0	0
$79$	−6.60555	−0.743183	−0.371591	−	0.928396i	$-0.621188\pi$
$79$	−6.60555	−0.743183	−0.371591	+	0.928396i	$0.621188\pi$
$80$	−0.656208	−0.0733663
$81$	−11.2111	−1.24568
$82$	−22.9855	−2.53832
$83$	−2.82352	−0.309921	−0.154961	−	0.987921i	$-0.549525\pi$
$83$	−2.82352	−0.309921	−0.154961	+	0.987921i	$0.549525\pi$
$84$	0	0
$85$	−15.5139	−1.68272
$86$	28.8167	3.10738
$87$	−4.99082	−0.535073
$88$	14.7250	1.56969
$89$	−6.50192	−0.689203	−0.344601	−	0.938749i	$-0.611986\pi$
$89$	−6.50192	−0.689203	−0.344601	+	0.938749i	$0.611986\pi$
$90$	8.47055	0.892874
$91$	0	0
$92$	−2.00000	−0.208514
$93$	15.5139	1.60871
$94$	−3.47972	−0.358906
$95$	4.69722	0.481925
$96$	11.4927	1.17297
$97$	13.6601	1.38697	0.693484	−	0.720472i	$-0.256075\pi$
$97$	13.6601	1.38697	0.693484	+	0.720472i	$0.256075\pi$
$98$	0	0
$99$	−8.33053	−0.837250
$100$	−1.00000	−0.100000
$101$	−6.50192	−0.646966	−0.323483	−	0.946234i	$-0.604854\pi$
$101$	−6.50192	−0.646966	−0.323483	+	0.946234i	$0.604854\pi$
$102$	−35.7250	−3.53730
$103$	11.4927	1.13241	0.566207	−	0.824263i	$-0.308410\pi$
$103$	11.4927	1.13241	0.566207	+	0.824263i	$0.308410\pi$
$104$	0	0
$105$	0	0
$106$	−5.51388	−0.535555
$107$	5.69722	0.550771	0.275386	−	0.961334i	$-0.411194\pi$
$107$	5.69722	0.550771	0.275386	+	0.961334i	$0.411194\pi$
$108$	−9.32544	−0.897341
$109$	−8.21110	−0.786481	−0.393240	−	0.919436i	$-0.628646\pi$
$109$	−8.21110	−0.786481	−0.393240	+	0.919436i	$0.628646\pi$
$110$	−24.4966	−2.33566
$111$	−18.6509	−1.77026
$112$	0	0
$113$	6.81665	0.641257	0.320628	−	0.947205i	$-0.396106\pi$
$113$	6.81665	0.641257	0.320628	+	0.947205i	$0.396106\pi$
$114$	10.8167	1.01307
$115$	1.31242	0.122383
$116$	−7.60555	−0.706158
$117$	0	0
$118$	−6.50192	−0.598551
$119$	0	0
$120$	14.0917	1.28639
$121$	13.0917	1.19015
$122$	−9.98165	−0.903696
$123$	21.6333	1.95061
$124$	23.6417	2.12309
$125$	11.4927	1.02794
$126$	0	0
$127$	7.90833	0.701751	0.350875	−	0.936422i	$-0.385884\pi$
$127$	7.90833	0.701751	0.350875	+	0.936422i	$0.385884\pi$
$128$	18.9083	1.67128
$129$	−27.1214	−2.38791
$130$	0	0
$131$	−10.6379	−0.929434	−0.464717	−	0.885459i	$-0.653844\pi$
$131$	−10.6379	−0.929434	−0.464717	+	0.885459i	$0.653844\pi$
$132$	−35.1345	−3.05806
$133$	0	0
$134$	2.30278	0.198930
$135$	6.11943	0.526677
$136$	−21.4744	−1.84141
$137$	−6.30278	−0.538482	−0.269241	−	0.963073i	$-0.586773\pi$
$137$	−6.30278	−0.538482	−0.269241	+	0.963073i	$0.586773\pi$
$138$	3.02220	0.257267
$139$	−10.6379	−0.902291	−0.451146	−	0.892450i	$-0.648984\pi$
$139$	−10.6379	−0.902291	−0.451146	+	0.892450i	$0.648984\pi$
$140$	0	0
$141$	3.27502	0.275806
$142$	9.21110	0.772979
$143$	0	0
$144$	0.513878	0.0428232
$145$	4.99082	0.414465
$146$	−9.98165	−0.826087
$147$	0	0
$148$	−28.4222	−2.33629
$149$	17.5139	1.43479	0.717396	−	0.696665i	$-0.245334\pi$
$149$	17.5139	1.43479	0.717396	+	0.696665i	$0.245334\pi$
$150$	1.51110	0.123381
$151$	−8.21110	−0.668210	−0.334105	−	0.942536i	$-0.608434\pi$
$151$	−8.21110	−0.668210	−0.334105	+	0.942536i	$0.608434\pi$
$152$	6.50192	0.527376
$153$	12.1490	0.982185
$154$	0	0
$155$	−15.5139	−1.24610
$156$	0	0
$157$	−0.656208	−0.0523711	−0.0261856	−	0.999657i	$-0.508336\pi$
$157$	−0.656208	−0.0523711	−0.0261856	+	0.999657i	$0.508336\pi$
$158$	15.2111	1.21013
$159$	5.18951	0.411555
$160$	−11.4927	−0.908582
$161$	0	0
$162$	25.8167	2.02835
$163$	2.78890	0.218443	0.109222	−	0.994017i	$-0.465164\pi$
$163$	2.78890	0.218443	0.109222	+	0.994017i	$0.465164\pi$
$164$	32.9671	2.57430
$165$	23.0555	1.79487
$166$	6.50192	0.504647
$167$	−11.4927	−0.889336	−0.444668	−	0.895696i	$-0.646678\pi$
$167$	−11.4927	−0.889336	−0.444668	+	0.895696i	$0.646678\pi$
$168$	0	0
$169$	0	0
$170$	35.7250	2.73998
$171$	−3.67841	−0.281295
$172$	−41.3305	−3.15142
$173$	10.1803	0.773996	0.386998	−	0.922080i	$-0.373512\pi$
$173$	10.1803	0.773996	0.386998	+	0.922080i	$0.373512\pi$
$174$	11.4927	0.871263
$175$	0	0
$176$	−1.48612	−0.112021
$177$	6.11943	0.459964
$178$	14.9725	1.12223
$179$	21.6056	1.61487	0.807437	−	0.589953i	$-0.200854\pi$
$179$	21.6056	1.61487	0.807437	+	0.589953i	$0.200854\pi$
$180$	−12.1490	−0.905530
$181$	−24.2979	−1.80605	−0.903025	−	0.429588i	$-0.858659\pi$
$181$	−24.2979	−1.80605	−0.903025	+	0.429588i	$0.858659\pi$
$182$	0	0
$183$	9.39445	0.694458
$184$	1.81665	0.133925
$185$	18.6509	1.37124
$186$	−35.7250	−2.61948
$187$	−35.1345	−2.56929
$188$	4.99082	0.363993
$189$	0	0
$190$	−10.8167	−0.784723
$191$	15.6972	1.13581	0.567906	−	0.823094i	$-0.307754\pi$
$191$	15.6972	1.13581	0.567906	+	0.823094i	$0.307754\pi$
$192$	−27.7776	−2.00468
$193$	−19.8167	−1.42643	−0.713217	−	0.700943i	$-0.752763\pi$
$193$	−19.8167	−1.42643	−0.713217	+	0.700943i	$0.752763\pi$
$194$	−31.4560	−2.25841
$195$	0	0
$196$	0	0
$197$	1.09167	0.0777785	0.0388892	−	0.999244i	$-0.487618\pi$
$197$	1.09167	0.0777785	0.0388892	+	0.999244i	$0.487618\pi$
$198$	19.1833	1.36330
$199$	−7.15813	−0.507427	−0.253713	−	0.967279i	$-0.581652\pi$
$199$	−7.15813	−0.507427	−0.253713	+	0.967279i	$0.581652\pi$
$200$	0.908327	0.0642284
$201$	−2.16731	−0.152870
$202$	14.9725	1.05346
$203$	0	0
$204$	51.2389	3.58744
$205$	−21.6333	−1.51094
$206$	−26.4652	−1.84392
$207$	−1.02776	−0.0714340
$208$	0	0
$209$	10.6379	0.735836
$210$	0	0
$211$	15.0000	1.03264	0.516321	−	0.856395i	$-0.327301\pi$
$211$	15.0000	1.03264	0.516321	+	0.856395i	$0.327301\pi$
$212$	7.90833	0.543146
$213$	−8.66923	−0.594006
$214$	−13.1194	−0.896826
$215$	27.1214	1.84967
$216$	8.47055	0.576348
$217$	0	0
$218$	18.9083	1.28063
$219$	9.39445	0.634818
$220$	35.1345	2.36876
$221$	0	0
$222$	42.9488	2.88253
$223$	17.9947	1.20501	0.602506	−	0.798114i	$-0.294169\pi$
$223$	17.9947	1.20501	0.602506	+	0.798114i	$0.294169\pi$
$224$	0	0
$225$	−0.513878	−0.0342585
$226$	−15.6972	−1.04416
$227$	18.4522	1.22472	0.612358	−	0.790581i	$-0.290221\pi$
$227$	18.4522	1.22472	0.612358	+	0.790581i	$0.290221\pi$
$228$	−15.5139	−1.02743
$229$	15.6287	1.03277	0.516386	−	0.856356i	$-0.327277\pi$
$229$	15.6287	1.03277	0.516386	+	0.856356i	$0.327277\pi$
$230$	−3.02220	−0.199278
$231$	0	0
$232$	6.90833	0.453554
$233$	−13.0917	−0.857664	−0.428832	−	0.903384i	$-0.641075\pi$
$233$	−13.0917	−0.857664	−0.428832	+	0.903384i	$0.641075\pi$
$234$	0	0
$235$	−3.27502	−0.213639
$236$	9.32544	0.607034
$237$	−14.3163	−0.929941
$238$	0	0
$239$	−4.39445	−0.284253	−0.142127	−	0.989848i	$-0.545394\pi$
$239$	−4.39445	−0.284253	−0.142127	+	0.989848i	$0.545394\pi$
$240$	−1.42221	−0.0918029
$241$	9.32544	0.600704	0.300352	−	0.953828i	$-0.402896\pi$
$241$	9.32544	0.600704	0.300352	+	0.953828i	$0.402896\pi$
$242$	−30.1472	−1.93793
$243$	−15.8274	−1.01533
$244$	14.3163	0.916505
$245$	0	0
$246$	−49.8167	−3.17619
$247$	0	0
$248$	−21.4744	−1.36363
$249$	−6.11943	−0.387803
$250$	−26.4652	−1.67381
$251$	29.9449	1.89011	0.945054	−	0.326914i	$-0.106009\pi$
$251$	29.9449	1.89011	0.945054	+	0.326914i	$0.106009\pi$
$252$	0	0
$253$	2.97224	0.186863
$254$	−18.2111	−1.14267
$255$	−33.6234	−2.10558
$256$	−17.9083	−1.11927
$257$	6.30324	0.393185	0.196593	−	0.980485i	$-0.437012\pi$
$257$	6.30324	0.393185	0.196593	+	0.980485i	$0.437012\pi$
$258$	62.4546	3.88825
$259$	0	0
$260$	0	0
$261$	−3.90833	−0.241919
$262$	24.4966	1.51340
$263$	15.4222	0.950974	0.475487	−	0.879723i	$-0.342272\pi$
$263$	15.4222	0.950974	0.475487	+	0.879723i	$0.342272\pi$
$264$	31.9136	1.96414
$265$	−5.18951	−0.318789
$266$	0	0
$267$	−14.0917	−0.862396
$268$	−3.30278	−0.201749
$269$	−0.656208	−0.0400097	−0.0200049	−	0.999800i	$-0.506368\pi$
$269$	−0.656208	−0.0400097	−0.0200049	+	0.999800i	$0.506368\pi$
$270$	−14.0917	−0.857592
$271$	−14.5149	−0.881720	−0.440860	−	0.897576i	$-0.645327\pi$
$271$	−14.5149	−0.881720	−0.440860	+	0.897576i	$0.645327\pi$
$272$	2.16731	0.131412
$273$	0	0
$274$	14.5139	0.876815
$275$	1.48612	0.0896165
$276$	−4.33462	−0.260913
$277$	−0.211103	−0.0126839	−0.00634196	−	0.999980i	$-0.502019\pi$
$277$	−0.211103	−0.0126839	−0.00634196	+	0.999980i	$0.502019\pi$
$278$	24.4966	1.46921
$279$	12.1490	0.727339
$280$	0	0
$281$	−23.8167	−1.42078	−0.710391	−	0.703807i	$-0.751482\pi$
$281$	−23.8167	−1.42078	−0.710391	+	0.703807i	$0.751482\pi$
$282$	−7.54163	−0.449098
$283$	18.6509	1.10868	0.554340	−	0.832290i	$-0.312971\pi$
$283$	18.6509	1.10868	0.554340	+	0.832290i	$0.312971\pi$
$284$	−13.2111	−0.783935
$285$	10.1803	0.603031
$286$	0	0
$287$	0	0
$288$	9.00000	0.530330
$289$	34.2389	2.01405
$290$	−11.4927	−0.674877
$291$	29.6056	1.73551
$292$	14.3163	0.837796
$293$	28.6325	1.67273	0.836365	−	0.548173i	$-0.184676\pi$
$293$	28.6325	1.67273	0.836365	+	0.548173i	$0.184676\pi$
$294$	0	0
$295$	−6.11943	−0.356287
$296$	25.8167	1.50056
$297$	13.8587	0.804166
$298$	−40.3305	−2.33628
$299$	0	0
$300$	−2.16731	−0.125130
$301$	0	0
$302$	18.9083	1.08805
$303$	−14.0917	−0.809545
$304$	−0.656208	−0.0376361
$305$	−9.39445	−0.537925
$306$	−27.9763	−1.59930
$307$	−3.67841	−0.209938	−0.104969	−	0.994476i	$-0.533474\pi$
$307$	−3.67841	−0.209938	−0.104969	+	0.994476i	$0.533474\pi$
$308$	0	0
$309$	24.9083	1.41699
$310$	35.7250	2.02904
$311$	10.6379	0.603218	0.301609	−	0.953432i	$-0.402476\pi$
$311$	10.6379	0.603218	0.301609	+	0.953432i	$0.402476\pi$
$312$	0	0
$313$	−5.64703	−0.319189	−0.159595	−	0.987183i	$-0.551019\pi$
$313$	−5.64703	−0.319189	−0.159595	+	0.987183i	$0.551019\pi$
$314$	1.51110	0.0852763
$315$	0	0
$316$	−21.8167	−1.22728
$317$	6.21110	0.348850	0.174425	−	0.984670i	$-0.444193\pi$
$317$	6.21110	0.348850	0.174425	+	0.984670i	$0.444193\pi$
$318$	−11.9503	−0.670138
$319$	11.3028	0.632834
$320$	27.7776	1.55282
$321$	12.3476	0.689178
$322$	0	0
$323$	−15.5139	−0.863215
$324$	−37.0278	−2.05710
$325$	0	0
$326$	−6.42221	−0.355693
$327$	−17.7960	−0.984120
$328$	−29.9449	−1.65343
$329$	0	0
$330$	−53.0917	−2.92260
$331$	−4.30278	−0.236502	−0.118251	−	0.992984i	$-0.537729\pi$
$331$	−4.30278	−0.236502	−0.118251	+	0.992984i	$0.537729\pi$
$332$	−9.32544	−0.511800
$333$	−14.6056	−0.800379
$334$	26.4652	1.44811
$335$	2.16731	0.118413
$336$	0	0
$337$	18.1194	0.987028	0.493514	−	0.869738i	$-0.335712\pi$
$337$	18.1194	0.987028	0.493514	+	0.869738i	$0.335712\pi$
$338$	0	0
$339$	14.7738	0.802402
$340$	−51.2389	−2.77882
$341$	−35.1345	−1.90264
$342$	8.47055	0.458035
$343$	0	0
$344$	37.5416	2.02411
$345$	2.84441	0.153138
$346$	−23.4430	−1.26030
$347$	15.2111	0.816575	0.408287	−	0.912853i	$-0.366126\pi$
$347$	15.2111	0.816575	0.408287	+	0.912853i	$0.366126\pi$
$348$	−16.4836	−0.883612
$349$	−10.1803	−0.544941	−0.272470	−	0.962164i	$-0.587841\pi$
$349$	−10.1803	−0.544941	−0.272470	+	0.962164i	$0.587841\pi$
$350$	0	0
$351$	0	0
$352$	−26.0278	−1.38728
$353$	26.4652	1.40860	0.704301	−	0.709902i	$-0.251261\pi$
$353$	26.4652	1.40860	0.704301	+	0.709902i	$0.251261\pi$
$354$	−14.0917	−0.748964
$355$	8.66923	0.460115
$356$	−21.4744	−1.13814
$357$	0	0
$358$	−49.7527	−2.62951
$359$	12.0917	0.638174	0.319087	−	0.947725i	$-0.396624\pi$
$359$	12.0917	0.638174	0.319087	+	0.947725i	$0.396624\pi$
$360$	11.0352	0.581607
$361$	−14.3028	−0.752778
$362$	55.9526	2.94081
$363$	28.3737	1.48923
$364$	0	0
$365$	−9.39445	−0.491728
$366$	−21.6333	−1.13079
$367$	25.6103	1.33685	0.668424	−	0.743780i	$-0.266969\pi$
$367$	25.6103	1.33685	0.668424	+	0.743780i	$0.266969\pi$
$368$	−0.183346	−0.00955758
$369$	16.9411	0.881918
$370$	−42.9488	−2.23280
$371$	0	0
$372$	51.2389	2.65661
$373$	−12.6972	−0.657437	−0.328719	−	0.944428i	$-0.606617\pi$
$373$	−12.6972	−0.657437	−0.328719	+	0.944428i	$0.606617\pi$
$374$	80.9068	4.18359
$375$	24.9083	1.28626
$376$	−4.53330	−0.233787
$377$	0	0
$378$	0	0
$379$	12.1194	0.622533	0.311267	−	0.950323i	$-0.399247\pi$
$379$	12.1194	0.622533	0.311267	+	0.950323i	$0.399247\pi$
$380$	15.5139	0.795845
$381$	17.1398	0.878098
$382$	−36.1472	−1.84945
$383$	22.3293	1.14097	0.570487	−	0.821307i	$-0.306755\pi$
$383$	22.3293	1.14097	0.570487	+	0.821307i	$0.306755\pi$
$384$	40.9802	2.09126
$385$	0	0
$386$	45.6333	2.32267
$387$	−21.2389	−1.07963
$388$	45.1161	2.29042
$389$	29.0278	1.47177	0.735883	−	0.677109i	$-0.236767\pi$
$389$	29.0278	1.47177	0.735883	+	0.677109i	$0.236767\pi$
$390$	0	0
$391$	−4.33462	−0.219211
$392$	0	0
$393$	−23.0555	−1.16300
$394$	−2.51388	−0.126647
$395$	14.3163	0.720329
$396$	−27.5139	−1.38262
$397$	−4.33462	−0.217548	−0.108774	−	0.994066i	$-0.534693\pi$
$397$	−4.33462	−0.217548	−0.108774	+	0.994066i	$0.534693\pi$
$398$	16.4836	0.826247
$399$	0	0
$400$	−0.0916731	−0.00458365
$401$	10.1194	0.505340	0.252670	−	0.967552i	$-0.418691\pi$
$401$	10.1194	0.505340	0.252670	+	0.967552i	$0.418691\pi$
$402$	4.99082	0.248920
$403$	0	0
$404$	−21.4744	−1.06839
$405$	24.2979	1.20737
$406$	0	0
$407$	42.2389	2.09370
$408$	−46.5416	−2.30415
$409$	−3.47972	−0.172061	−0.0860306	−	0.996292i	$-0.527418\pi$
$409$	−3.47972	−0.172061	−0.0860306	+	0.996292i	$0.527418\pi$
$410$	49.8167	2.46027
$411$	−13.6601	−0.673801
$412$	37.9580	1.87005
$413$	0	0
$414$	2.36669	0.116317
$415$	6.11943	0.300391
$416$	0	0
$417$	−23.0555	−1.12903
$418$	−24.4966	−1.19817
$419$	−17.1398	−0.837333	−0.418667	−	0.908140i	$-0.637502\pi$
$419$	−17.1398	−0.837333	−0.418667	+	0.908140i	$0.637502\pi$
$420$	0	0
$421$	−5.02776	−0.245038	−0.122519	−	0.992466i	$-0.539097\pi$
$421$	−5.02776	−0.245038	−0.122519	+	0.992466i	$0.539097\pi$
$422$	−34.5416	−1.68146
$423$	2.56468	0.124699
$424$	−7.18335	−0.348854
$425$	−2.16731	−0.105130
$426$	19.9633	0.967225
$427$	0	0
$428$	18.8167	0.909537
$429$	0	0
$430$	−62.4546	−3.01183
$431$	20.9361	1.00846	0.504228	−	0.863571i	$-0.331777\pi$
$431$	20.9361	1.00846	0.504228	+	0.863571i	$0.331777\pi$
$432$	−0.854892	−0.0411310
$433$	17.1398	0.823685	0.411843	−	0.911255i	$-0.364885\pi$
$433$	17.1398	0.823685	0.411843	+	0.911255i	$0.364885\pi$
$434$	0	0
$435$	10.8167	0.518619
$436$	−27.1194	−1.29879
$437$	1.31242	0.0627814
$438$	−21.6333	−1.03368
$439$	−3.67841	−0.175561	−0.0877804	−	0.996140i	$-0.527977\pi$
$439$	−3.67841	−0.175561	−0.0877804	+	0.996140i	$0.527977\pi$
$440$	−31.9136	−1.52142
$441$	0	0
$442$	0	0
$443$	−20.2389	−0.961577	−0.480789	−	0.876837i	$-0.659650\pi$
$443$	−20.2389	−0.961577	−0.480789	+	0.876837i	$0.659650\pi$
$444$	−61.5997	−2.92339
$445$	14.0917	0.668009
$446$	−41.4377	−1.96213
$447$	37.9580	1.79535
$448$	0	0
$449$	20.4222	0.963783	0.481892	−	0.876231i	$-0.339950\pi$
$449$	20.4222	0.963783	0.481892	+	0.876231i	$0.339950\pi$
$450$	1.18335	0.0557835
$451$	−48.9932	−2.30700
$452$	22.5139	1.05896
$453$	−17.7960	−0.836129
$454$	−42.4913	−1.99421
$455$	0	0
$456$	14.0917	0.659903
$457$	−20.6056	−0.963887	−0.481944	−	0.876202i	$-0.660069\pi$
$457$	−20.6056	−0.963887	−0.481944	+	0.876202i	$0.660069\pi$
$458$	−35.9893	−1.68167
$459$	−20.2111	−0.943373
$460$	4.33462	0.202103
$461$	25.1528	1.17148	0.585741	−	0.810498i	$-0.300803\pi$
$461$	25.1528	1.17148	0.585741	+	0.810498i	$0.300803\pi$
$462$	0	0
$463$	13.7889	0.640824	0.320412	−	0.947278i	$-0.396179\pi$
$463$	13.7889	0.640824	0.320412	+	0.947278i	$0.396179\pi$
$464$	−0.697224	−0.0323678
$465$	−33.6234	−1.55925
$466$	30.1472	1.39654
$467$	12.8052	0.592552	0.296276	−	0.955102i	$-0.404255\pi$
$467$	12.8052	0.592552	0.296276	+	0.955102i	$0.404255\pi$
$468$	0	0
$469$	0	0
$470$	7.54163	0.347870
$471$	−1.42221	−0.0655318
$472$	−8.47055	−0.389889
$473$	61.4222	2.82420
$474$	32.9671	1.51423
$475$	0.656208	0.0301089
$476$	0	0
$477$	4.06392	0.186074
$478$	10.1194	0.462852
$479$	16.4836	0.753154	0.376577	−	0.926385i	$-0.377101\pi$
$479$	16.4836	0.753154	0.376577	+	0.926385i	$0.377101\pi$
$480$	−24.9083	−1.13690
$481$	0	0
$482$	−21.4744	−0.978132
$483$	0	0
$484$	43.2389	1.96540
$485$	−29.6056	−1.34432
$486$	36.4469	1.65326
$487$	31.3028	1.41846	0.709232	−	0.704975i	$-0.249042\pi$
$487$	31.3028	1.41846	0.709232	+	0.704975i	$0.249042\pi$
$488$	−13.0038	−0.588657
$489$	6.04440	0.273337
$490$	0	0
$491$	19.7250	0.890176	0.445088	−	0.895487i	$-0.353172\pi$
$491$	19.7250	0.890176	0.445088	+	0.895487i	$0.353172\pi$
$492$	71.4500	3.22121
$493$	−16.4836	−0.742383
$494$	0	0
$495$	18.0548	0.811504
$496$	2.16731	0.0973150
$497$	0	0
$498$	14.0917	0.631463
$499$	8.33053	0.372926	0.186463	−	0.982462i	$-0.440298\pi$
$499$	8.33053	0.372926	0.186463	+	0.982462i	$0.440298\pi$
$500$	37.9580	1.69753
$501$	−24.9083	−1.11282
$502$	−68.9565	−3.07768
$503$	−19.5058	−0.869719	−0.434860	−	0.900498i	$-0.643202\pi$
$503$	−19.5058	−0.869719	−0.434860	+	0.900498i	$0.643202\pi$
$504$	0	0
$505$	14.0917	0.627071
$506$	−6.84441	−0.304271
$507$	0	0
$508$	26.1194	1.15886
$509$	−21.4744	−0.951836	−0.475918	−	0.879490i	$-0.657884\pi$
$509$	−21.4744	−0.951836	−0.475918	+	0.879490i	$0.657884\pi$
$510$	77.4270	3.42853
$511$	0	0
$512$	3.42221	0.151242
$513$	6.11943	0.270179
$514$	−14.5149	−0.640227
$515$	−24.9083	−1.09759
$516$	−89.5760	−3.94336
$517$	−7.41697	−0.326198
$518$	0	0
$519$	22.0639	0.968498
$520$	0	0
$521$	−21.6731	−0.949515	−0.474757	−	0.880117i	$-0.657464\pi$
$521$	−21.6731	−0.949515	−0.474757	+	0.880117i	$0.657464\pi$
$522$	9.00000	0.393919
$523$	24.2979	1.06247	0.531237	−	0.847223i	$-0.321727\pi$
$523$	24.2979	1.06247	0.531237	+	0.847223i	$0.321727\pi$
$524$	−35.1345	−1.53486
$525$	0	0
$526$	−35.5139	−1.54848
$527$	51.2389	2.23200
$528$	−3.22088	−0.140171
$529$	−22.6333	−0.984057
$530$	11.9503	0.519087
$531$	4.79214	0.207961
$532$	0	0
$533$	0	0
$534$	32.4500	1.40425
$535$	−12.3476	−0.533835
$536$	3.00000	0.129580
$537$	46.8259	2.02069
$538$	1.51110	0.0651481
$539$	0	0
$540$	20.2111	0.869747
$541$	−27.9361	−1.20107	−0.600533	−	0.799600i	$-0.705045\pi$
$541$	−27.9361	−1.20107	−0.600533	+	0.799600i	$0.705045\pi$
$542$	33.4247	1.43571
$543$	−52.6611	−2.25990
$544$	37.9580	1.62743
$545$	17.7960	0.762296
$546$	0	0
$547$	29.0000	1.23995	0.619975	−	0.784621i	$-0.287143\pi$
$547$	29.0000	1.23995	0.619975	+	0.784621i	$0.287143\pi$
$548$	−20.8167	−0.889243
$549$	7.35682	0.313981
$550$	−3.42221	−0.145923
$551$	4.99082	0.212616
$552$	3.93725	0.167580
$553$	0	0
$554$	0.486122	0.0206533
$555$	40.4222	1.71583
$556$	−35.1345	−1.49003
$557$	−6.09167	−0.258112	−0.129056	−	0.991637i	$-0.541195\pi$
$557$	−6.09167	−0.258112	−0.129056	+	0.991637i	$0.541195\pi$
$558$	−27.9763	−1.18433
$559$	0	0
$560$	0	0
$561$	−76.1472	−3.21494
$562$	54.8444	2.31347
$563$	1.31242	0.0553117	0.0276559	−	0.999618i	$-0.491196\pi$
$563$	1.31242	0.0553117	0.0276559	+	0.999618i	$0.491196\pi$
$564$	10.8167	0.455463
$565$	−14.7738	−0.621538
$566$	−42.9488	−1.80527
$567$	0	0
$568$	12.0000	0.503509
$569$	−34.6056	−1.45074	−0.725370	−	0.688359i	$-0.758331\pi$
$569$	−34.6056	−1.45074	−0.725370	+	0.688359i	$0.758331\pi$
$570$	−23.4430	−0.981920
$571$	−10.7250	−0.448826	−0.224413	−	0.974494i	$-0.572047\pi$
$571$	−10.7250	−0.448826	−0.224413	+	0.974494i	$0.572047\pi$
$572$	0	0
$573$	34.0207	1.42124
$574$	0	0
$575$	0.183346	0.00764606
$576$	−21.7527	−0.906364
$577$	−33.1658	−1.38071	−0.690356	−	0.723470i	$-0.742546\pi$
$577$	−33.1658	−1.38071	−0.690356	+	0.723470i	$0.742546\pi$
$578$	−78.8444	−3.27950
$579$	−42.9488	−1.78489
$580$	16.4836	0.684443
$581$	0	0
$582$	−68.1749	−2.82594
$583$	−11.7527	−0.486749
$584$	−13.0038	−0.538103
$585$	0	0
$586$	−65.9343	−2.72372
$587$	−1.96862	−0.0812538	−0.0406269	−	0.999174i	$-0.512936\pi$
$587$	−1.96862	−0.0812538	−0.0406269	+	0.999174i	$0.512936\pi$
$588$	0	0
$589$	−15.5139	−0.639238
$590$	14.0917	0.580145
$591$	2.36599	0.0973239
$592$	−2.60555	−0.107087
$593$	−6.30324	−0.258843	−0.129422	−	0.991590i	$-0.541312\pi$
$593$	−6.30324	−0.258843	−0.129422	+	0.991590i	$0.541312\pi$
$594$	−31.9136	−1.30943
$595$	0	0
$596$	57.8444	2.36940
$597$	−15.5139	−0.634941
$598$	0	0
$599$	−10.5139	−0.429585	−0.214793	−	0.976660i	$-0.568908\pi$
$599$	−10.5139	−0.429585	−0.214793	+	0.976660i	$0.568908\pi$
$600$	1.96862	0.0803687
$601$	9.12676	0.372288	0.186144	−	0.982522i	$-0.440401\pi$
$601$	9.12676	0.372288	0.186144	+	0.982522i	$0.440401\pi$
$602$	0	0
$603$	−1.69722	−0.0691163
$604$	−27.1194	−1.10347
$605$	−28.3737	−1.15355
$606$	32.4500	1.31819
$607$	38.8129	1.57537	0.787683	−	0.616081i	$-0.211281\pi$
$607$	38.8129	1.57537	0.787683	+	0.616081i	$0.211281\pi$
$608$	−11.4927	−0.466093
$609$	0	0
$610$	21.6333	0.875907
$611$	0	0
$612$	40.1253	1.62197
$613$	−31.9083	−1.28877	−0.644383	−	0.764703i	$-0.722886\pi$
$613$	−31.9083	−1.28877	−0.644383	+	0.764703i	$0.722886\pi$
$614$	8.47055	0.341844
$615$	−46.8860	−1.89063
$616$	0	0
$617$	−15.8444	−0.637872	−0.318936	−	0.947776i	$-0.603325\pi$
$617$	−15.8444	−0.637872	−0.318936	+	0.947776i	$0.603325\pi$
$618$	−57.3583	−2.30729
$619$	29.9449	1.20359	0.601794	−	0.798651i	$-0.294453\pi$
$619$	29.9449	1.20359	0.601794	+	0.798651i	$0.294453\pi$
$620$	−51.2389	−2.05780
$621$	1.70978	0.0686113
$622$	−24.4966	−0.982224
$623$	0	0
$624$	0	0
$625$	−23.3944	−0.935778
$626$	13.0038	0.519738
$627$	23.0555	0.920748
$628$	−2.16731	−0.0864850
$629$	−61.5997	−2.45614
$630$	0	0
$631$	−22.9083	−0.911966	−0.455983	−	0.889988i	$-0.650712\pi$
$631$	−22.9083	−0.911966	−0.455983	+	0.889988i	$0.650712\pi$
$632$	19.8167	0.788264
$633$	32.5096	1.29214
$634$	−14.3028	−0.568036
$635$	−17.1398	−0.680171
$636$	17.1398	0.679637
$637$	0	0
$638$	−26.0278	−1.03045
$639$	−6.78890	−0.268565
$640$	−40.9802	−1.61988
$641$	−14.5139	−0.573264	−0.286632	−	0.958041i	$-0.592536\pi$
$641$	−14.5139	−0.573264	−0.286632	+	0.958041i	$0.592536\pi$
$642$	−28.4338	−1.12219
$643$	−17.1398	−0.675927	−0.337963	−	0.941159i	$-0.609738\pi$
$643$	−17.1398	−0.675927	−0.337963	+	0.941159i	$0.609738\pi$
$644$	0	0
$645$	58.7805	2.31448
$646$	35.7250	1.40558
$647$	−32.9671	−1.29607	−0.648036	−	0.761610i	$-0.724409\pi$
$647$	−32.9671	−1.29607	−0.648036	+	0.761610i	$0.724409\pi$
$648$	33.6333	1.32124
$649$	−13.8587	−0.544003
$650$	0	0
$651$	0	0
$652$	9.21110	0.360735
$653$	−46.7527	−1.82958	−0.914788	−	0.403934i	$-0.867642\pi$
$653$	−46.7527	−1.82958	−0.914788	+	0.403934i	$0.867642\pi$
$654$	40.9802	1.60245
$655$	23.0555	0.900853
$656$	3.02220	0.117997
$657$	7.35682	0.287017
$658$	0	0
$659$	−19.6333	−0.764805	−0.382403	−	0.923996i	$-0.624903\pi$
$659$	−19.6333	−0.764805	−0.382403	+	0.923996i	$0.624903\pi$
$660$	76.1472	2.96403
$661$	22.9855	0.894032	0.447016	−	0.894526i	$-0.352487\pi$
$661$	22.9855	0.894032	0.447016	+	0.894526i	$0.352487\pi$
$662$	9.90833	0.385098
$663$	0	0
$664$	8.47055	0.328721
$665$	0	0
$666$	33.6333	1.30326
$667$	1.39445	0.0539933
$668$	−37.9580	−1.46864
$669$	39.0000	1.50783
$670$	−4.99082	−0.192812
$671$	−21.2757	−0.821340
$672$	0	0
$673$	−2.21110	−0.0852317	−0.0426159	−	0.999092i	$-0.513569\pi$
$673$	−2.21110	−0.0852317	−0.0426159	+	0.999092i	$0.513569\pi$
$674$	−41.7250	−1.60719
$675$	0.854892	0.0329048
$676$	0	0
$677$	−26.4652	−1.01714	−0.508571	−	0.861020i	$-0.669826\pi$
$677$	−26.4652	−1.01714	−0.508571	+	0.861020i	$0.669826\pi$
$678$	−34.0207	−1.30656
$679$	0	0
$680$	46.5416	1.78479
$681$	39.9916	1.53248
$682$	80.9068	3.09808
$683$	3.60555	0.137963	0.0689813	−	0.997618i	$-0.478025\pi$
$683$	3.60555	0.137963	0.0689813	+	0.997618i	$0.478025\pi$
$684$	−12.1490	−0.464527
$685$	13.6601	0.521924
$686$	0	0
$687$	33.8722	1.29230
$688$	−3.78890	−0.144450
$689$	0	0
$690$	−6.55004	−0.249356
$691$	37.7593	1.43643	0.718215	−	0.695821i	$-0.244959\pi$
$691$	37.7593	1.43643	0.718215	+	0.695821i	$0.244959\pi$
$692$	33.6234	1.27817
$693$	0	0
$694$	−35.0278	−1.32964
$695$	23.0555	0.874545
$696$	14.9725	0.567530
$697$	71.4500	2.70636
$698$	23.4430	0.887331
$699$	−28.3737	−1.07319
$700$	0	0
$701$	9.02776	0.340974	0.170487	−	0.985360i	$-0.445466\pi$
$701$	9.02776	0.340974	0.170487	+	0.985360i	$0.445466\pi$
$702$	0	0
$703$	18.6509	0.703431
$704$	62.9083	2.37095
$705$	−7.09798	−0.267325
$706$	−60.9435	−2.29364
$707$	0	0
$708$	20.2111	0.759580
$709$	−32.7250	−1.22901	−0.614506	−	0.788912i	$-0.710645\pi$
$709$	−32.7250	−1.22901	−0.614506	+	0.788912i	$0.710645\pi$
$710$	−19.9633	−0.749209
$711$	−11.2111	−0.420449
$712$	19.5058	0.731010
$713$	−4.33462	−0.162333
$714$	0	0
$715$	0	0
$716$	71.3583	2.66678
$717$	−9.52412	−0.355685
$718$	−27.8444	−1.03914
$719$	−22.7868	−0.849805	−0.424902	−	0.905239i	$-0.639692\pi$
$719$	−22.7868	−0.849805	−0.424902	+	0.905239i	$0.639692\pi$
$720$	−1.11373	−0.0415064
$721$	0	0
$722$	32.9361	1.22575
$723$	20.2111	0.751659
$724$	−80.2506	−2.98249
$725$	0.697224	0.0258943
$726$	−65.3382	−2.42493
$727$	37.1031	1.37608	0.688038	−	0.725674i	$-0.258472\pi$
$727$	37.1031	1.37608	0.688038	+	0.725674i	$0.258472\pi$
$728$	0	0
$729$	−0.669468	−0.0247951
$730$	21.6333	0.800685
$731$	−89.5760	−3.31309
$732$	31.0278	1.14682
$733$	6.70061	0.247493	0.123746	−	0.992314i	$-0.460509\pi$
$733$	6.70061	0.247493	0.123746	+	0.992314i	$0.460509\pi$
$734$	−58.9748	−2.17680
$735$	0	0
$736$	−3.21110	−0.118363
$737$	4.90833	0.180801
$738$	−39.0115	−1.43603
$739$	33.2111	1.22169	0.610845	−	0.791750i	$-0.290830\pi$
$739$	33.2111	1.22169	0.610845	+	0.791750i	$0.290830\pi$
$740$	61.5997	2.26445
$741$	0	0
$742$	0	0
$743$	41.3028	1.51525	0.757626	−	0.652689i	$-0.226359\pi$
$743$	41.3028	1.51525	0.757626	+	0.652689i	$0.226359\pi$
$744$	−46.5416	−1.70630
$745$	−37.9580	−1.39067
$746$	29.2389	1.07051
$747$	−4.79214	−0.175335
$748$	−116.041	−4.24289
$749$	0	0
$750$	−57.3583	−2.09443
$751$	−11.6056	−0.423493	−0.211746	−	0.977325i	$-0.567915\pi$
$751$	−11.6056	−0.423493	−0.211746	+	0.977325i	$0.567915\pi$
$752$	0.457524	0.0166842
$753$	64.8999	2.36508
$754$	0	0
$755$	17.7960	0.647662
$756$	0	0
$757$	6.23886	0.226755	0.113378	−	0.993552i	$-0.463833\pi$
$757$	6.23886	0.226755	0.113378	+	0.993552i	$0.463833\pi$
$758$	−27.9083	−1.01368
$759$	6.44177	0.233821
$760$	−14.0917	−0.511159
$761$	42.2926	1.53311	0.766553	−	0.642182i	$-0.221971\pi$
$761$	42.2926	1.53311	0.766553	+	0.642182i	$0.221971\pi$
$762$	−39.4691	−1.42981
$763$	0	0
$764$	51.8444	1.87566
$765$	−26.3305	−0.951982
$766$	−51.4193	−1.85786
$767$	0	0
$768$	−38.8129	−1.40054
$769$	−47.0847	−1.69792	−0.848959	−	0.528458i	$-0.822770\pi$
$769$	−47.0847	−1.69792	−0.848959	+	0.528458i	$0.822770\pi$
$770$	0	0
$771$	13.6611	0.491991
$772$	−65.4500	−2.35560
$773$	−28.4338	−1.02269	−0.511347	−	0.859374i	$-0.670853\pi$
$773$	−28.4338	−1.02269	−0.511347	+	0.859374i	$0.670853\pi$
$774$	48.9083	1.75797
$775$	−2.16731	−0.0778520
$776$	−40.9802	−1.47110
$777$	0	0
$778$	−66.8444	−2.39649
$779$	−21.6333	−0.775094
$780$	0	0
$781$	19.6333	0.702535
$782$	9.98165	0.356943
$783$	6.50192	0.232360
$784$	0	0
$785$	1.42221	0.0507607
$786$	53.0917	1.89372
$787$	31.2574	1.11420	0.557102	−	0.830444i	$-0.311913\pi$
$787$	31.2574	1.11420	0.557102	+	0.830444i	$0.311913\pi$
$788$	3.60555	0.128442
$789$	33.4247	1.18995
$790$	−32.9671	−1.17292
$791$	0	0
$792$	24.9916	0.888038
$793$	0	0
$794$	9.98165	0.354235
$795$	−11.2473	−0.398899
$796$	−23.6417	−0.837958
$797$	26.0077	0.921240	0.460620	−	0.887597i	$-0.347627\pi$
$797$	26.0077	0.921240	0.460620	+	0.887597i	$0.347627\pi$
$798$	0	0
$799$	10.8167	0.382666
$800$	−1.60555	−0.0567648
$801$	−11.0352	−0.389910
$802$	−23.3028	−0.822850
$803$	−21.2757	−0.750804
$804$	−7.15813	−0.252448
$805$	0	0
$806$	0	0
$807$	−1.42221	−0.0500640
$808$	19.5058	0.686211
$809$	20.0278	0.704138	0.352069	−	0.935974i	$-0.385478\pi$
$809$	20.0278	0.704138	0.352069	+	0.935974i	$0.385478\pi$
$810$	−55.9526	−1.96598
$811$	−0.854892	−0.0300193	−0.0150097	−	0.999887i	$-0.504778\pi$
$811$	−0.854892	−0.0300193	−0.0150097	+	0.999887i	$0.504778\pi$
$812$	0	0
$813$	−31.4584	−1.10329
$814$	−97.2666	−3.40919
$815$	−6.04440	−0.211726
$816$	4.69722	0.164436
$817$	27.1214	0.948859
$818$	8.01302	0.280169
$819$	0	0
$820$	−71.4500	−2.49514
$821$	55.8444	1.94898	0.974492	−	0.224424i	$-0.0720499\pi$
$821$	55.8444	1.94898	0.974492	+	0.224424i	$0.0720499\pi$
$822$	31.4560	1.09716
$823$	42.2666	1.47332	0.736661	−	0.676262i	$-0.236401\pi$
$823$	42.2666	1.47332	0.736661	+	0.676262i	$0.236401\pi$
$824$	−34.4782	−1.20111
$825$	3.22088	0.112137
$826$	0	0
$827$	16.7527	0.582550	0.291275	−	0.956639i	$-0.405921\pi$
$827$	16.7527	0.582550	0.291275	+	0.956639i	$0.405921\pi$
$828$	−3.39445	−0.117965
$829$	50.7631	1.76308	0.881538	−	0.472113i	$-0.156508\pi$
$829$	50.7631	1.76308	0.881538	+	0.472113i	$0.156508\pi$
$830$	−14.0917	−0.489129
$831$	−0.457524	−0.0158713
$832$	0	0
$833$	0	0
$834$	53.0917	1.83841
$835$	24.9083	0.861988
$836$	35.1345	1.21515
$837$	−20.2111	−0.698598
$838$	39.4691	1.36344
$839$	−13.4614	−0.464738	−0.232369	−	0.972628i	$-0.574648\pi$
$839$	−13.4614	−0.464738	−0.232369	+	0.972628i	$0.574648\pi$
$840$	0	0
$841$	−23.6972	−0.817146
$842$	11.5778	0.398997
$843$	−51.6180	−1.77782
$844$	49.5416	1.70529
$845$	0	0
$846$	−5.90587	−0.203048
$847$	0	0
$848$	0.724981	0.0248959
$849$	40.4222	1.38729
$850$	4.99082	0.171184
$851$	5.21110	0.178634
$852$	−28.6325	−0.980934
$853$	−12.8052	−0.438440	−0.219220	−	0.975675i	$-0.570351\pi$
$853$	−12.8052	−0.438440	−0.219220	+	0.975675i	$0.570351\pi$
$854$	0	0
$855$	7.97224	0.272645
$856$	−17.0917	−0.584181
$857$	3.28104	0.112078	0.0560391	−	0.998429i	$-0.482153\pi$
$857$	3.28104	0.112078	0.0560391	+	0.998429i	$0.482153\pi$
$858$	0	0
$859$	3.02220	0.103116	0.0515581	−	0.998670i	$-0.483581\pi$
$859$	3.02220	0.103116	0.0515581	+	0.998670i	$0.483581\pi$
$860$	89.5760	3.05452
$861$	0	0
$862$	−48.2111	−1.64208
$863$	−9.81665	−0.334163	−0.167081	−	0.985943i	$-0.553434\pi$
$863$	−9.81665	−0.334163	−0.167081	+	0.985943i	$0.553434\pi$
$864$	−14.9725	−0.509374
$865$	−22.0639	−0.750196
$866$	−39.4691	−1.34121
$867$	74.2062	2.52017
$868$	0	0
$869$	32.4222	1.09985
$870$	−24.9083	−0.844471
$871$	0	0
$872$	24.6333	0.834189
$873$	23.1842	0.784666
$874$	−3.02220	−0.102227
$875$	0	0
$876$	31.0278	1.04833
$877$	45.6056	1.53999	0.769995	−	0.638050i	$-0.220259\pi$
$877$	45.6056	1.53999	0.769995	+	0.638050i	$0.220259\pi$
$878$	8.47055	0.285867
$879$	62.0555	2.09308
$880$	3.22088	0.108576
$881$	18.8496	0.635058	0.317529	−	0.948249i	$-0.397147\pi$
$881$	18.8496	0.635058	0.317529	+	0.948249i	$0.397147\pi$
$882$	0	0
$883$	−24.3944	−0.820939	−0.410469	−	0.911874i	$-0.634635\pi$
$883$	−24.3944	−0.820939	−0.410469	+	0.911874i	$0.634635\pi$
$884$	0	0
$885$	−13.2627	−0.445820
$886$	46.6056	1.56574
$887$	34.6769	1.16434	0.582169	−	0.813068i	$-0.302204\pi$
$887$	34.6769	1.16434	0.582169	+	0.813068i	$0.302204\pi$
$888$	55.9526	1.87765
$889$	0	0
$890$	−32.4500	−1.08773
$891$	55.0278	1.84350
$892$	59.4324	1.98994
$893$	−3.27502	−0.109594
$894$	−87.4087	−2.92338
$895$	−46.8259	−1.56522
$896$	0	0
$897$	0	0
$898$	−47.0278	−1.56934
$899$	−16.4836	−0.549758
$900$	−1.69722	−0.0565741
$901$	17.1398	0.571009
$902$	112.820	3.75651
$903$	0	0
$904$	−20.4500	−0.680156
$905$	52.6611	1.75051
$906$	40.9802	1.36147
$907$	−38.8444	−1.28981	−0.644904	−	0.764264i	$-0.723103\pi$
$907$	−38.8444	−1.28981	−0.644904	+	0.764264i	$0.723103\pi$
$908$	60.9435	2.02248
$909$	−11.0352	−0.366015
$910$	0	0
$911$	35.9361	1.19062	0.595308	−	0.803498i	$-0.297030\pi$
$911$	35.9361	1.19062	0.595308	+	0.803498i	$0.297030\pi$
$912$	−1.42221	−0.0470939
$913$	13.8587	0.458657
$914$	47.4500	1.56951
$915$	−20.3607	−0.673103
$916$	51.6180	1.70551
$917$	0	0
$918$	46.5416	1.53610
$919$	53.4500	1.76315	0.881576	−	0.472043i	$-0.156483\pi$
$919$	53.4500	1.76315	0.881576	+	0.472043i	$0.156483\pi$
$920$	−3.93725	−0.129807
$921$	−7.97224	−0.262694
$922$	−57.9213	−1.90754
$923$	0	0
$924$	0	0
$925$	2.60555	0.0856700
$926$	−31.7527	−1.04346
$927$	19.5058	0.640654
$928$	−12.2111	−0.400849
$929$	17.1398	0.562338	0.281169	−	0.959658i	$-0.409278\pi$
$929$	17.1398	0.562338	0.281169	+	0.959658i	$0.409278\pi$
$930$	77.4270	2.53893
$931$	0	0
$932$	−43.2389	−1.41634
$933$	23.0555	0.754804
$934$	−29.4874	−0.964858
$935$	76.1472	2.49028
$936$	0	0
$937$	−19.7646	−0.645682	−0.322841	−	0.946453i	$-0.604638\pi$
$937$	−19.7646	−0.645682	−0.322841	+	0.946453i	$0.604638\pi$
$938$	0	0
$939$	−12.2389	−0.399400
$940$	−10.8167	−0.352800
$941$	5.44835	0.177611	0.0888055	−	0.996049i	$-0.471695\pi$
$941$	5.44835	0.177611	0.0888055	+	0.996049i	$0.471695\pi$
$942$	3.27502	0.106706
$943$	−6.04440	−0.196833
$944$	0.854892	0.0278244
$945$	0	0
$946$	−141.442	−4.59866
$947$	−27.8806	−0.905997	−0.452998	−	0.891511i	$-0.649646\pi$
$947$	−27.8806	−0.905997	−0.452998	+	0.891511i	$0.649646\pi$
$948$	−47.2834	−1.53569
$949$	0	0
$950$	−1.51110	−0.0490266
$951$	13.4614	0.436515
$952$	0	0
$953$	34.3944	1.11415	0.557073	−	0.830464i	$-0.311924\pi$
$953$	34.3944	1.11415	0.557073	+	0.830464i	$0.311924\pi$
$954$	−9.35829	−0.302986
$955$	−34.0207	−1.10088
$956$	−14.5139	−0.469412
$957$	24.4966	0.791863
$958$	−37.9580	−1.22637
$959$	0	0
$960$	60.2027	1.94303
$961$	20.2389	0.652866
$962$	0	0
$963$	9.66947	0.311594
$964$	30.7998	0.991996
$965$	42.9488	1.38257
$966$	0	0
$967$	42.4500	1.36510	0.682549	−	0.730839i	$-0.260871\pi$
$967$	42.4500	1.36510	0.682549	+	0.730839i	$0.260871\pi$
$968$	−39.2750	−1.26235
$969$	−33.6234	−1.08014
$970$	68.1749	2.18897
$971$	−31.4560	−1.00947	−0.504736	−	0.863274i	$-0.668410\pi$
$971$	−31.4560	−1.00947	−0.504736	+	0.863274i	$0.668410\pi$
$972$	−52.2742	−1.67670
$973$	0	0
$974$	−72.0833	−2.30970
$975$	0	0
$976$	1.31242	0.0420094
$977$	3.97224	0.127083	0.0635417	−	0.997979i	$-0.479760\pi$
$977$	3.97224	0.127083	0.0635417	+	0.997979i	$0.479760\pi$
$978$	−13.9189	−0.445077
$979$	31.9136	1.01996
$980$	0	0
$981$	−13.9361	−0.444945
$982$	−45.4222	−1.44948
$983$	−29.2887	−0.934166	−0.467083	−	0.884214i	$-0.654695\pi$
$983$	−29.2887	−0.934166	−0.467083	+	0.884214i	$0.654695\pi$
$984$	−64.8999	−2.06893
$985$	−2.36599	−0.0753868
$986$	37.9580	1.20883
$987$	0	0
$988$	0	0
$989$	7.57779	0.240960
$990$	−41.5762	−1.32138
$991$	−6.66947	−0.211863	−0.105931	−	0.994373i	$-0.533782\pi$
$991$	−6.66947	−0.211863	−0.105931	+	0.994373i	$0.533782\pi$
$992$	37.9580	1.20517
$993$	−9.32544	−0.295934
$994$	0	0
$995$	15.5139	0.491823
$996$	−20.2111	−0.640413
$997$	52.7318	1.67003	0.835016	−	0.550226i	$-0.185458\pi$
$997$	52.7318	1.67003	0.835016	+	0.550226i	$0.185458\pi$
$998$	−19.1833	−0.607238
$999$	24.2979	0.768752

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.bo.1.2		4
7.6	odd	2	inner	8281.2.a.bo.1.1		4
13.4	even	6		637.2.f.h.393.1	yes	8
13.10	even	6		637.2.f.h.295.1	✓	8
13.12	even	2		8281.2.a.bu.1.4		4
91.4	even	6		637.2.h.j.471.3		8
91.10	odd	6		637.2.g.i.373.1		8
91.17	odd	6		637.2.h.j.471.4		8
91.23	even	6		637.2.h.j.165.3		8
91.30	even	6		637.2.g.i.263.2		8
91.62	odd	6		637.2.f.h.295.2	yes	8
91.69	odd	6		637.2.f.h.393.2	yes	8
91.75	odd	6		637.2.h.j.165.4		8
91.82	odd	6		637.2.g.i.263.1		8
91.88	even	6		637.2.g.i.373.2		8
91.90	odd	2		8281.2.a.bu.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.f.h.295.1	✓	8	13.10	even	6
637.2.f.h.295.2	yes	8	91.62	odd	6
637.2.f.h.393.1	yes	8	13.4	even	6
637.2.f.h.393.2	yes	8	91.69	odd	6
637.2.g.i.263.1		8	91.82	odd	6
637.2.g.i.263.2		8	91.30	even	6
637.2.g.i.373.1		8	91.10	odd	6
637.2.g.i.373.2		8	91.88	even	6
637.2.h.j.165.3		8	91.23	even	6
637.2.h.j.165.4		8	91.75	odd	6
637.2.h.j.471.3		8	91.4	even	6
637.2.h.j.471.4		8	91.17	odd	6
8281.2.a.bo.1.1		4	7.6	odd	2	inner
8281.2.a.bo.1.2		4	1.1	even	1	trivial
8281.2.a.bu.1.3		4	91.90	odd	2
8281.2.a.bu.1.4		4	13.12	even	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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30278 q^{2} +2.16731 q^{3} +3.30278 q^{4} -2.16731 q^{5} -4.99082 q^{6} -3.00000 q^{8} +1.69722 q^{9} +4.99082 q^{10} -4.90833 q^{11} +7.15813 q^{12} -4.69722 q^{15} +0.302776 q^{16} +7.15813 q^{17} -3.90833 q^{18} -2.16731 q^{19} -7.15813 q^{20} +11.3028 q^{22} -0.605551 q^{23} -6.50192 q^{24} -0.302776 q^{25} -2.82352 q^{27} -2.30278 q^{29} +10.8167 q^{30} +7.15813 q^{31} +5.30278 q^{32} -10.6379 q^{33} -16.4836 q^{34} +5.60555 q^{36} -8.60555 q^{37} +4.99082 q^{38} +6.50192 q^{40} +9.98165 q^{41} -12.5139 q^{43} -16.2111 q^{44} -3.67841 q^{45} +1.39445 q^{46} +1.51110 q^{47} +0.656208 q^{48} +0.697224 q^{50} +15.5139 q^{51} +2.39445 q^{53} +6.50192 q^{54} +10.6379 q^{55} -4.69722 q^{57} +5.30278 q^{58} +2.82352 q^{59} -15.5139 q^{60} +4.33462 q^{61} -16.4836 q^{62} -12.8167 q^{64} +24.4966 q^{66} -1.00000 q^{67} +23.6417 q^{68} -1.31242 q^{69} -4.00000 q^{71} -5.09167 q^{72} +4.33462 q^{73} +19.8167 q^{74} -0.656208 q^{75} -7.15813 q^{76} -6.60555 q^{79} -0.656208 q^{80} -11.2111 q^{81} -22.9855 q^{82} -2.82352 q^{83} -15.5139 q^{85} +28.8167 q^{86} -4.99082 q^{87} +14.7250 q^{88} -6.50192 q^{89} +8.47055 q^{90} -2.00000 q^{92} +15.5139 q^{93} -3.47972 q^{94} +4.69722 q^{95} +11.4927 q^{96} +13.6601 q^{97} -8.33053 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 6 q^{4} - 12 q^{8} + 14 q^{9} + 2 q^{11} - 26 q^{15} - 6 q^{16} + 6 q^{18} + 38 q^{22} + 12 q^{23} + 6 q^{25} - 2 q^{29} + 14 q^{32} + 8 q^{36} - 20 q^{37} - 14 q^{43} - 36 q^{44} + 20 q^{46}+ \cdots + 46 q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 + 6 * q^4 - 12 * q^8 + 14 * q^9 + 2 * q^11 - 26 * q^15 - 6 * q^16 + 6 * q^18 + 38 * q^22 + 12 * q^23 + 6 * q^25 - 2 * q^29 + 14 * q^32 + 8 * q^36 - 20 * q^37 - 14 * q^43 - 36 * q^44 + 20 * q^46 + 10 * q^50 + 26 * q^51 + 24 * q^53 - 26 * q^57 + 14 * q^58 - 26 * q^60 - 8 * q^64 - 4 * q^67 - 16 * q^71 - 42 * q^72 + 36 * q^74 - 12 * q^79 - 16 * q^81 - 26 * q^85 + 72 * q^86 - 6 * q^88 - 8 * q^92 + 26 * q^93 + 26 * q^95 + 46 * q^99

Introduction

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