LMFDB - Embedded newform 8281.2.a.bv.1.1

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:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{23})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 24x^{2} + 121 $ x^4 - 24*x^2 + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.1
Root			$2.68406$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.41421 q^{3} -1.00000 q^{4} -4.09827 q^{5} -1.41421 q^{6} -3.00000 q^{8} -1.00000 q^{9} -4.09827 q^{10} -3.79583 q^{11} +1.41421 q^{12} +5.79583 q^{15} -1.00000 q^{16} +1.26984 q^{17} -1.00000 q^{18} +2.82843 q^{19} +4.09827 q^{20} -3.79583 q^{22} -7.79583 q^{23} +4.24264 q^{24} +11.7958 q^{25} +5.65685 q^{27} +0.795832 q^{29} +5.79583 q^{30} +1.41421 q^{31} +5.00000 q^{32} +5.36812 q^{33} +1.26984 q^{34} +1.00000 q^{36} +2.79583 q^{37} +2.82843 q^{38} +12.2948 q^{40} -2.97280 q^{41} +7.79583 q^{43} +3.79583 q^{44} +4.09827 q^{45} -7.79583 q^{46} -2.82843 q^{47} +1.41421 q^{48} +11.7958 q^{50} -1.79583 q^{51} -12.5917 q^{53} +5.65685 q^{54} +15.5563 q^{55} -4.00000 q^{57} +0.795832 q^{58} +12.4392 q^{59} -5.79583 q^{60} -8.34091 q^{61} +1.41421 q^{62} +7.00000 q^{64} +5.36812 q^{66} -3.79583 q^{67} -1.26984 q^{68} +11.0250 q^{69} +6.00000 q^{71} +3.00000 q^{72} +12.5836 q^{73} +2.79583 q^{74} -16.6818 q^{75} -2.82843 q^{76} -2.20417 q^{79} +4.09827 q^{80} -5.00000 q^{81} -2.97280 q^{82} +9.89949 q^{83} -5.20417 q^{85} +7.79583 q^{86} -1.12548 q^{87} +11.3875 q^{88} +14.9789 q^{89} +4.09827 q^{90} +7.79583 q^{92} -2.00000 q^{93} -2.82843 q^{94} -11.5917 q^{95} -7.07107 q^{96} +4.24264 q^{97} +3.79583 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 4 q^{4} - 12 q^{8} - 4 q^{9} + 4 q^{11} + 4 q^{15} - 4 q^{16} - 4 q^{18} + 4 q^{22} - 12 q^{23} + 28 q^{25} - 16 q^{29} + 4 q^{30} + 20 q^{32} + 4 q^{36} - 8 q^{37} + 12 q^{43} - 4 q^{44}+ \cdots - 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 - 4 * q^4 - 12 * q^8 - 4 * q^9 + 4 * q^11 + 4 * q^15 - 4 * q^16 - 4 * q^18 + 4 * q^22 - 12 * q^23 + 28 * q^25 - 16 * q^29 + 4 * q^30 + 20 * q^32 + 4 * q^36 - 8 * q^37 + 12 * q^43 - 4 * q^44 - 12 * q^46 + 28 * q^50 + 12 * q^51 - 12 * q^53 - 16 * q^57 - 16 * q^58 - 4 * q^60 + 28 * q^64 + 4 * q^67 + 24 * q^71 + 12 * q^72 - 8 * q^74 - 28 * q^79 - 20 * q^81 - 40 * q^85 + 12 * q^86 - 12 * q^88 + 12 * q^92 - 8 * q^93 - 8 * q^95 - 4 * 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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	−1.00000	−0.500000
$5$	−4.09827	−1.83280	−0.916401	−	0.400260i	$-0.868920\pi$
$5$	−4.09827	−1.83280	−0.916401	+	0.400260i	$0.868920\pi$
$6$	−1.41421	−0.577350
$7$	0	0
$7$	0	0
$8$	−3.00000	−1.06066
$9$	−1.00000	−0.333333
$10$	−4.09827	−1.29599
$11$	−3.79583	−1.14449	−0.572243	−	0.820084i	$-0.693927\pi$
$11$	−3.79583	−1.14449	−0.572243	+	0.820084i	$0.693927\pi$
$12$	1.41421	0.408248
$13$	0	0
$13$	0	0
$14$	0	0
$15$	5.79583	1.49648
$16$	−1.00000	−0.250000
$17$	1.26984	0.307983	0.153991	−	0.988072i	$-0.450787\pi$
$17$	1.26984	0.307983	0.153991	+	0.988072i	$0.450787\pi$
$18$	−1.00000	−0.235702
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	4.09827	0.916401
$21$	0	0
$22$	−3.79583	−0.809274
$23$	−7.79583	−1.62554	−0.812772	−	0.582582i	$-0.802042\pi$
$23$	−7.79583	−1.62554	−0.812772	+	0.582582i	$0.802042\pi$
$24$	4.24264	0.866025
$25$	11.7958	2.35917
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	0.795832	0.147782	0.0738911	−	0.997266i	$-0.476458\pi$
$29$	0.795832	0.147782	0.0738911	+	0.997266i	$0.476458\pi$
$30$	5.79583	1.05817
$31$	1.41421	0.254000	0.127000	−	0.991903i	$-0.459465\pi$
$31$	1.41421	0.254000	0.127000	+	0.991903i	$0.459465\pi$
$32$	5.00000	0.883883
$33$	5.36812	0.934469
$34$	1.26984	0.217777
$35$	0	0
$36$	1.00000	0.166667
$37$	2.79583	0.459632	0.229816	−	0.973234i	$-0.426188\pi$
$37$	2.79583	0.459632	0.229816	+	0.973234i	$0.426188\pi$
$38$	2.82843	0.458831
$39$	0	0
$40$	12.2948	1.94398
$41$	−2.97280	−0.464273	−0.232136	−	0.972683i	$-0.574572\pi$
$41$	−2.97280	−0.464273	−0.232136	+	0.972683i	$0.574572\pi$
$42$	0	0
$43$	7.79583	1.18885	0.594427	−	0.804150i	$-0.297379\pi$
$43$	7.79583	1.18885	0.594427	+	0.804150i	$0.297379\pi$
$44$	3.79583	0.572243
$45$	4.09827	0.610934
$46$	−7.79583	−1.14943
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	1.41421	0.204124
$49$	0	0
$50$	11.7958	1.66818
$51$	−1.79583	−0.251467
$52$	0	0
$53$	−12.5917	−1.72960	−0.864799	−	0.502118i	$-0.832554\pi$
$53$	−12.5917	−1.72960	−0.864799	+	0.502118i	$0.832554\pi$
$54$	5.65685	0.769800
$55$	15.5563	2.09762
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0.795832	0.104498
$59$	12.4392	1.61944	0.809722	−	0.586814i	$-0.199618\pi$
$59$	12.4392	1.61944	0.809722	+	0.586814i	$0.199618\pi$
$60$	−5.79583	−0.748239
$61$	−8.34091	−1.06794	−0.533972	−	0.845502i	$-0.679301\pi$
$61$	−8.34091	−1.06794	−0.533972	+	0.845502i	$0.679301\pi$
$62$	1.41421	0.179605
$63$	0	0
$64$	7.00000	0.875000
$65$	0	0
$66$	5.36812	0.660769
$67$	−3.79583	−0.463735	−0.231867	−	0.972747i	$-0.574484\pi$
$67$	−3.79583	−0.463735	−0.231867	+	0.972747i	$0.574484\pi$
$68$	−1.26984	−0.153991
$69$	11.0250	1.32725
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	3.00000	0.353553
$73$	12.5836	1.47279	0.736397	−	0.676550i	$-0.236526\pi$
$73$	12.5836	1.47279	0.736397	+	0.676550i	$0.236526\pi$
$74$	2.79583	0.325009
$75$	−16.6818	−1.92625
$76$	−2.82843	−0.324443
$77$	0	0
$78$	0	0
$79$	−2.20417	−0.247988	−0.123994	−	0.992283i	$-0.539570\pi$
$79$	−2.20417	−0.247988	−0.123994	+	0.992283i	$0.539570\pi$
$80$	4.09827	0.458201
$81$	−5.00000	−0.555556
$82$	−2.97280	−0.328290
$83$	9.89949	1.08661	0.543305	−	0.839535i	$-0.317173\pi$
$83$	9.89949	1.08661	0.543305	+	0.839535i	$0.317173\pi$
$84$	0	0
$85$	−5.20417	−0.564471
$86$	7.79583	0.840646
$87$	−1.12548	−0.120664
$88$	11.3875	1.21391
$89$	14.9789	1.58776	0.793879	−	0.608076i	$-0.208058\pi$
$89$	14.9789	1.58776	0.793879	+	0.608076i	$0.208058\pi$
$90$	4.09827	0.431996
$91$	0	0
$92$	7.79583	0.812772
$93$	−2.00000	−0.207390
$94$	−2.82843	−0.291730
$95$	−11.5917	−1.18928
$96$	−7.07107	−0.721688
$97$	4.24264	0.430775	0.215387	−	0.976529i	$-0.430899\pi$
$97$	4.24264	0.430775	0.215387	+	0.976529i	$0.430899\pi$
$98$	0	0
$99$	3.79583	0.381495
$100$	−11.7958	−1.17958
$101$	9.75513	0.970671	0.485336	−	0.874328i	$-0.338697\pi$
$101$	9.75513	0.970671	0.485336	+	0.874328i	$0.338697\pi$
$102$	−1.79583	−0.177814
$103$	−5.36812	−0.528936	−0.264468	−	0.964394i	$-0.585196\pi$
$103$	−5.36812	−0.528936	−0.264468	+	0.964394i	$0.585196\pi$
$104$	0	0
$105$	0	0
$106$	−12.5917	−1.22301
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	−5.65685	−0.544331
$109$	1.59166	0.152454	0.0762268	−	0.997091i	$-0.475713\pi$
$109$	1.59166	0.152454	0.0762268	+	0.997091i	$0.475713\pi$
$110$	15.5563	1.48324
$111$	−3.95390	−0.375288
$112$	0	0
$113$	2.59166	0.243803	0.121902	−	0.992542i	$-0.461101\pi$
$113$	2.59166	0.243803	0.121902	+	0.992542i	$0.461101\pi$
$114$	−4.00000	−0.374634
$115$	31.9494	2.97930
$116$	−0.795832	−0.0738911
$117$	0	0
$118$	12.4392	1.14512
$119$	0	0
$120$	−17.3875	−1.58725
$121$	3.40834	0.309849
$122$	−8.34091	−0.755151
$123$	4.20417	0.379077
$124$	−1.41421	−0.127000
$125$	−27.8512	−2.49108
$126$	0	0
$127$	−11.5917	−1.02859	−0.514297	−	0.857612i	$-0.671947\pi$
$127$	−11.5917	−1.02859	−0.514297	+	0.857612i	$0.671947\pi$
$128$	−3.00000	−0.265165
$129$	−11.0250	−0.970695
$130$	0	0
$131$	−5.36812	−0.469015	−0.234507	−	0.972114i	$-0.575348\pi$
$131$	−5.36812	−0.469015	−0.234507	+	0.972114i	$0.575348\pi$
$132$	−5.36812	−0.467235
$133$	0	0
$134$	−3.79583	−0.327910
$135$	−23.1833	−1.99530
$136$	−3.80953	−0.326665
$137$	−18.7958	−1.60584	−0.802918	−	0.596089i	$-0.796720\pi$
$137$	−18.7958	−1.60584	−0.802918	+	0.596089i	$0.796720\pi$
$138$	11.0250	0.938508
$139$	1.70295	0.144442	0.0722212	−	0.997389i	$-0.476991\pi$
$139$	1.70295	0.144442	0.0722212	+	0.997389i	$0.476991\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	6.00000	0.503509
$143$	0	0
$144$	1.00000	0.0833333
$145$	−3.26153	−0.270856
$146$	12.5836	1.04142
$147$	0	0
$148$	−2.79583	−0.229816
$149$	−14.5917	−1.19540	−0.597698	−	0.801721i	$-0.703918\pi$
$149$	−14.5917	−1.19540	−0.597698	+	0.801721i	$0.703918\pi$
$150$	−16.6818	−1.36207
$151$	1.59166	0.129528	0.0647639	−	0.997901i	$-0.479371\pi$
$151$	1.59166	0.129528	0.0647639	+	0.997901i	$0.479371\pi$
$152$	−8.48528	−0.688247
$153$	−1.26984	−0.102661
$154$	0	0
$155$	−5.79583	−0.465532
$156$	0	0
$157$	0.144369	0.0115219	0.00576095	−	0.999983i	$-0.498166\pi$
$157$	0.144369	0.0115219	0.00576095	+	0.999983i	$0.498166\pi$
$158$	−2.20417	−0.175354
$159$	17.8073	1.41221
$160$	−20.4914	−1.61998
$161$	0	0
$162$	−5.00000	−0.392837
$163$	15.3875	1.20524	0.602621	−	0.798028i	$-0.294123\pi$
$163$	15.3875	1.20524	0.602621	+	0.798028i	$0.294123\pi$
$164$	2.97280	0.232136
$165$	−22.0000	−1.71270
$166$	9.89949	0.768350
$167$	−1.70295	−0.131778	−0.0658892	−	0.997827i	$-0.520988\pi$
$167$	−1.70295	−0.131778	−0.0658892	+	0.997827i	$0.520988\pi$
$168$	0	0
$169$	0	0
$170$	−5.20417	−0.399142
$171$	−2.82843	−0.216295
$172$	−7.79583	−0.594427
$173$	17.8073	1.35386	0.676932	−	0.736046i	$-0.263309\pi$
$173$	17.8073	1.35386	0.676932	+	0.736046i	$0.263309\pi$
$174$	−1.12548	−0.0853221
$175$	0	0
$176$	3.79583	0.286122
$177$	−17.5917	−1.32227
$178$	14.9789	1.12271
$179$	−19.5917	−1.46435	−0.732175	−	0.681117i	$-0.761495\pi$
$179$	−19.5917	−1.46435	−0.732175	+	0.681117i	$0.761495\pi$
$180$	−4.09827	−0.305467
$181$	−3.80953	−0.283160	−0.141580	−	0.989927i	$-0.545218\pi$
$181$	−3.80953	−0.283160	−0.141580	+	0.989927i	$0.545218\pi$
$182$	0	0
$183$	11.7958	0.871973
$184$	23.3875	1.72415
$185$	−11.4581	−0.842415
$186$	−2.00000	−0.146647
$187$	−4.82012	−0.352482
$188$	2.82843	0.206284
$189$	0	0
$190$	−11.5917	−0.840948
$191$	−16.2042	−1.17249	−0.586246	−	0.810133i	$-0.699395\pi$
$191$	−16.2042	−1.17249	−0.586246	+	0.810133i	$0.699395\pi$
$192$	−9.89949	−0.714435
$193$	22.5917	1.62618	0.813092	−	0.582136i	$-0.197783\pi$
$193$	22.5917	1.62618	0.813092	+	0.582136i	$0.197783\pi$
$194$	4.24264	0.304604
$195$	0	0
$196$	0	0
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	3.79583	0.269758
$199$	−5.07938	−0.360068	−0.180034	−	0.983660i	$-0.557621\pi$
$199$	−5.07938	−0.360068	−0.180034	+	0.983660i	$0.557621\pi$
$200$	−35.3875	−2.50227
$201$	5.36812	0.378638
$202$	9.75513	0.686368
$203$	0	0
$204$	1.79583	0.125733
$205$	12.1833	0.850920
$206$	−5.36812	−0.374014
$207$	7.79583	0.541848
$208$	0	0
$209$	−10.7362	−0.742641
$210$	0	0
$211$	7.79583	0.536687	0.268344	−	0.963323i	$-0.413524\pi$
$211$	7.79583	0.536687	0.268344	+	0.963323i	$0.413524\pi$
$212$	12.5917	0.864799
$213$	−8.48528	−0.581402
$214$	−6.00000	−0.410152
$215$	−31.9494	−2.17893
$216$	−16.9706	−1.15470
$217$	0	0
$218$	1.59166	0.107801
$219$	−17.7958	−1.20253
$220$	−15.5563	−1.04881
$221$	0	0
$222$	−3.95390	−0.265369
$223$	5.65685	0.378811	0.189405	−	0.981899i	$-0.439344\pi$
$223$	5.65685	0.378811	0.189405	+	0.981899i	$0.439344\pi$
$224$	0	0
$225$	−11.7958	−0.786389
$226$	2.59166	0.172395
$227$	−7.35981	−0.488487	−0.244244	−	0.969714i	$-0.578540\pi$
$227$	−7.35981	−0.488487	−0.244244	+	0.969714i	$0.578540\pi$
$228$	4.00000	0.264906
$229$	−12.7279	−0.841085	−0.420542	−	0.907273i	$-0.638160\pi$
$229$	−12.7279	−0.841085	−0.420542	+	0.907273i	$0.638160\pi$
$230$	31.9494	2.10668
$231$	0	0
$232$	−2.38749	−0.156747
$233$	17.1833	1.12572	0.562859	−	0.826553i	$-0.309702\pi$
$233$	17.1833	1.12572	0.562859	+	0.826553i	$0.309702\pi$
$234$	0	0
$235$	11.5917	0.756157
$236$	−12.4392	−0.809722
$237$	3.11716	0.202482
$238$	0	0
$239$	−10.2042	−0.660053	−0.330026	−	0.943972i	$-0.607058\pi$
$239$	−10.2042	−0.660053	−0.330026	+	0.943972i	$0.607058\pi$
$240$	−5.79583	−0.374119
$241$	−11.1693	−0.719480	−0.359740	−	0.933053i	$-0.617135\pi$
$241$	−11.1693	−0.719480	−0.359740	+	0.933053i	$0.617135\pi$
$242$	3.40834	0.219096
$243$	−9.89949	−0.635053
$244$	8.34091	0.533972
$245$	0	0
$246$	4.20417	0.268048
$247$	0	0
$248$	−4.24264	−0.269408
$249$	−14.0000	−0.887214
$250$	−27.8512	−1.76146
$251$	16.6818	1.05295	0.526474	−	0.850191i	$-0.323514\pi$
$251$	16.6818	1.05295	0.526474	+	0.850191i	$0.323514\pi$
$252$	0	0
$253$	29.5917	1.86041
$254$	−11.5917	−0.727326
$255$	7.35981	0.460889
$256$	−17.0000	−1.06250
$257$	−8.62965	−0.538303	−0.269151	−	0.963098i	$-0.586743\pi$
$257$	−8.62965	−0.538303	−0.269151	+	0.963098i	$0.586743\pi$
$258$	−11.0250	−0.686385
$259$	0	0
$260$	0	0
$261$	−0.795832	−0.0492607
$262$	−5.36812	−0.331643
$263$	3.38749	0.208882	0.104441	−	0.994531i	$-0.466695\pi$
$263$	3.38749	0.208882	0.104441	+	0.994531i	$0.466695\pi$
$264$	−16.1043	−0.991154
$265$	51.6041	3.17001
$266$	0	0
$267$	−21.1833	−1.29640
$268$	3.79583	0.231867
$269$	−12.4392	−0.758430	−0.379215	−	0.925308i	$-0.623806\pi$
$269$	−12.4392	−0.758430	−0.379215	+	0.925308i	$0.623806\pi$
$270$	−23.1833	−1.41089
$271$	−17.5186	−1.06418	−0.532088	−	0.846689i	$-0.678593\pi$
$271$	−17.5186	−1.06418	−0.532088	+	0.846689i	$0.678593\pi$
$272$	−1.26984	−0.0769956
$273$	0	0
$274$	−18.7958	−1.13550
$275$	−44.7750	−2.70003
$276$	−11.0250	−0.663625
$277$	32.1833	1.93371	0.966854	−	0.255329i	$-0.0821837\pi$
$277$	32.1833	1.93371	0.966854	+	0.255329i	$0.0821837\pi$
$278$	1.70295	0.102136
$279$	−1.41421	−0.0846668
$280$	0	0
$281$	24.7958	1.47920	0.739598	−	0.673049i	$-0.235016\pi$
$281$	24.7958	1.47920	0.739598	+	0.673049i	$0.235016\pi$
$282$	4.00000	0.238197
$283$	15.8451	0.941893	0.470946	−	0.882162i	$-0.343913\pi$
$283$	15.8451	0.941893	0.470946	+	0.882162i	$0.343913\pi$
$284$	−6.00000	−0.356034
$285$	16.3931	0.971043
$286$	0	0
$287$	0	0
$288$	−5.00000	−0.294628
$289$	−15.3875	−0.905147
$290$	−3.26153	−0.191524
$291$	−6.00000	−0.351726
$292$	−12.5836	−0.736397
$293$	2.97280	0.173673	0.0868363	−	0.996223i	$-0.472324\pi$
$293$	2.97280	0.173673	0.0868363	+	0.996223i	$0.472324\pi$
$294$	0	0
$295$	−50.9792	−2.96812
$296$	−8.38749	−0.487513
$297$	−21.4725	−1.24596
$298$	−14.5917	−0.845272
$299$	0	0
$300$	16.6818	0.963126
$301$	0	0
$302$	1.59166	0.0915899
$303$	−13.7958	−0.792550
$304$	−2.82843	−0.162221
$305$	34.1833	1.95733
$306$	−1.26984	−0.0725922
$307$	20.0583	1.14478	0.572392	−	0.819980i	$-0.306015\pi$
$307$	20.0583	1.14478	0.572392	+	0.819980i	$0.306015\pi$
$308$	0	0
$309$	7.59166	0.431875
$310$	−5.79583	−0.329181
$311$	28.8323	1.63493	0.817464	−	0.575980i	$-0.195379\pi$
$311$	28.8323	1.63493	0.817464	+	0.575980i	$0.195379\pi$
$312$	0	0
$313$	−15.2676	−0.862976	−0.431488	−	0.902119i	$-0.642011\pi$
$313$	−15.2676	−0.862976	−0.431488	+	0.902119i	$0.642011\pi$
$314$	0.144369	0.00814721
$315$	0	0
$316$	2.20417	0.123994
$317$	6.59166	0.370225	0.185112	−	0.982717i	$-0.440735\pi$
$317$	6.59166	0.370225	0.185112	+	0.982717i	$0.440735\pi$
$318$	17.8073	0.998584
$319$	−3.02084	−0.169135
$320$	−28.6879	−1.60370
$321$	8.48528	0.473602
$322$	0	0
$323$	3.59166	0.199845
$324$	5.00000	0.277778
$325$	0	0
$326$	15.3875	0.852235
$327$	−2.25095	−0.124478
$328$	8.91839	0.492436
$329$	0	0
$330$	−22.0000	−1.21106
$331$	29.3875	1.61528	0.807641	−	0.589674i	$-0.200744\pi$
$331$	29.3875	1.61528	0.807641	+	0.589674i	$0.200744\pi$
$332$	−9.89949	−0.543305
$333$	−2.79583	−0.153211
$334$	−1.70295	−0.0931814
$335$	15.5563	0.849934
$336$	0	0
$337$	17.9792	0.979387	0.489694	−	0.871895i	$-0.337109\pi$
$337$	17.9792	0.979387	0.489694	+	0.871895i	$0.337109\pi$
$338$	0	0
$339$	−3.66517	−0.199064
$340$	5.20417	0.282236
$341$	−5.36812	−0.290700
$342$	−2.82843	−0.152944
$343$	0	0
$344$	−23.3875	−1.26097
$345$	−45.1833	−2.43259
$346$	17.8073	0.957326
$347$	−32.9792	−1.77041	−0.885207	−	0.465197i	$-0.845983\pi$
$347$	−32.9792	−1.77041	−0.885207	+	0.465197i	$0.845983\pi$
$348$	1.12548	0.0603318
$349$	−26.5813	−1.42287	−0.711433	−	0.702754i	$-0.751953\pi$
$349$	−26.5813	−1.42287	−0.711433	+	0.702754i	$0.751953\pi$
$350$	0	0
$351$	0	0
$352$	−18.9792	−1.01159
$353$	−5.22375	−0.278032	−0.139016	−	0.990290i	$-0.544394\pi$
$353$	−5.22375	−0.278032	−0.139016	+	0.990290i	$0.544394\pi$
$354$	−17.5917	−0.934986
$355$	−24.5896	−1.30508
$356$	−14.9789	−0.793879
$357$	0	0
$358$	−19.5917	−1.03545
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	−12.2948	−0.647994
$361$	−11.0000	−0.578947
$362$	−3.80953	−0.200225
$363$	−4.82012	−0.252990
$364$	0	0
$365$	−51.5708	−2.69934
$366$	11.7958	0.616578
$367$	−21.2132	−1.10732	−0.553660	−	0.832743i	$-0.686769\pi$
$367$	−21.2132	−1.10732	−0.553660	+	0.832743i	$0.686769\pi$
$368$	7.79583	0.406386
$369$	2.97280	0.154758
$370$	−11.4581	−0.595677
$371$	0	0
$372$	2.00000	0.103695
$373$	6.59166	0.341303	0.170652	−	0.985331i	$-0.445413\pi$
$373$	6.59166	0.341303	0.170652	+	0.985331i	$0.445413\pi$
$374$	−4.82012	−0.249242
$375$	39.3875	2.03396
$376$	8.48528	0.437595
$377$	0	0
$378$	0	0
$379$	25.3875	1.30407	0.652034	−	0.758190i	$-0.273916\pi$
$379$	25.3875	1.30407	0.652034	+	0.758190i	$0.273916\pi$
$380$	11.5917	0.594640
$381$	16.3931	0.839843
$382$	−16.2042	−0.829077
$383$	−12.1504	−0.620859	−0.310429	−	0.950596i	$-0.600473\pi$
$383$	−12.1504	−0.620859	−0.310429	+	0.950596i	$0.600473\pi$
$384$	4.24264	0.216506
$385$	0	0
$386$	22.5917	1.14989
$387$	−7.79583	−0.396284
$388$	−4.24264	−0.215387
$389$	−0.387495	−0.0196468	−0.00982338	−	0.999952i	$-0.503127\pi$
$389$	−0.387495	−0.0196468	−0.00982338	+	0.999952i	$0.503127\pi$
$390$	0	0
$391$	−9.89949	−0.500639
$392$	0	0
$393$	7.59166	0.382949
$394$	−8.00000	−0.403034
$395$	9.03328	0.454514
$396$	−3.79583	−0.190748
$397$	−7.07107	−0.354887	−0.177443	−	0.984131i	$-0.556783\pi$
$397$	−7.07107	−0.354887	−0.177443	+	0.984131i	$0.556783\pi$
$398$	−5.07938	−0.254606
$399$	0	0
$400$	−11.7958	−0.589792
$401$	−4.38749	−0.219101	−0.109551	−	0.993981i	$-0.534941\pi$
$401$	−4.38749	−0.219101	−0.109551	+	0.993981i	$0.534941\pi$
$402$	5.36812	0.267737
$403$	0	0
$404$	−9.75513	−0.485336
$405$	20.4914	1.01822
$406$	0	0
$407$	−10.6125	−0.526042
$408$	5.38749	0.266721
$409$	−18.7884	−0.929027	−0.464513	−	0.885566i	$-0.653771\pi$
$409$	−18.7884	−0.929027	−0.464513	+	0.885566i	$0.653771\pi$
$410$	12.1833	0.601692
$411$	26.5813	1.31116
$412$	5.36812	0.264468
$413$	0	0
$414$	7.79583	0.383144
$415$	−40.5708	−1.99154
$416$	0	0
$417$	−2.40834	−0.117937
$418$	−10.7362	−0.525126
$419$	25.7151	1.25627	0.628133	−	0.778106i	$-0.283820\pi$
$419$	25.7151	1.25627	0.628133	+	0.778106i	$0.283820\pi$
$420$	0	0
$421$	12.5917	0.613680	0.306840	−	0.951761i	$-0.400728\pi$
$421$	12.5917	0.613680	0.306840	+	0.951761i	$0.400728\pi$
$422$	7.79583	0.379495
$423$	2.82843	0.137523
$424$	37.7750	1.83452
$425$	14.9789	0.726582
$426$	−8.48528	−0.411113
$427$	0	0
$428$	6.00000	0.290021
$429$	0	0
$430$	−31.9494	−1.54074
$431$	33.1833	1.59838	0.799192	−	0.601075i	$-0.205261\pi$
$431$	33.1833	1.59838	0.799192	+	0.601075i	$0.205261\pi$
$432$	−5.65685	−0.272166
$433$	20.4914	0.984752	0.492376	−	0.870383i	$-0.336129\pi$
$433$	20.4914	0.984752	0.492376	+	0.870383i	$0.336129\pi$
$434$	0	0
$435$	4.61251	0.221153
$436$	−1.59166	−0.0762268
$437$	−22.0499	−1.05479
$438$	−17.7958	−0.850318
$439$	20.0583	0.957328	0.478664	−	0.877998i	$-0.341121\pi$
$439$	20.0583	0.957328	0.478664	+	0.877998i	$0.341121\pi$
$440$	−46.6690	−2.22486
$441$	0	0
$442$	0	0
$443$	10.0000	0.475114	0.237557	−	0.971374i	$-0.423653\pi$
$443$	10.0000	0.475114	0.237557	+	0.971374i	$0.423653\pi$
$444$	3.95390	0.187644
$445$	−61.3875	−2.91005
$446$	5.65685	0.267860
$447$	20.6357	0.976036
$448$	0	0
$449$	23.5917	1.11336	0.556680	−	0.830727i	$-0.312075\pi$
$449$	23.5917	1.11336	0.556680	+	0.830727i	$0.312075\pi$
$450$	−11.7958	−0.556061
$451$	11.2842	0.531354
$452$	−2.59166	−0.121902
$453$	−2.25095	−0.105759
$454$	−7.35981	−0.345413
$455$	0	0
$456$	12.0000	0.561951
$457$	4.59166	0.214789	0.107394	−	0.994216i	$-0.465749\pi$
$457$	4.59166	0.214789	0.107394	+	0.994216i	$0.465749\pi$
$458$	−12.7279	−0.594737
$459$	7.18333	0.335289
$460$	−31.9494	−1.48965
$461$	−24.1860	−1.12645	−0.563227	−	0.826302i	$-0.690440\pi$
$461$	−24.1860	−1.12645	−0.563227	+	0.826302i	$0.690440\pi$
$462$	0	0
$463$	−17.3875	−0.808065	−0.404033	−	0.914745i	$-0.632392\pi$
$463$	−17.3875	−0.808065	−0.404033	+	0.914745i	$0.632392\pi$
$464$	−0.795832	−0.0369456
$465$	8.19654	0.380106
$466$	17.1833	0.796002
$467$	31.9494	1.47844	0.739222	−	0.673462i	$-0.235194\pi$
$467$	31.9494	1.47844	0.739222	+	0.673462i	$0.235194\pi$
$468$	0	0
$469$	0	0
$470$	11.5917	0.534684
$471$	−0.204168	−0.00940759
$472$	−37.3176	−1.71768
$473$	−29.5917	−1.36063
$474$	3.11716	0.143176
$475$	33.3636	1.53083
$476$	0	0
$477$	12.5917	0.576533
$478$	−10.2042	−0.466728
$479$	20.9245	0.956063	0.478032	−	0.878343i	$-0.341350\pi$
$479$	20.9245	0.956063	0.478032	+	0.878343i	$0.341350\pi$
$480$	28.9792	1.32271
$481$	0	0
$482$	−11.1693	−0.508749
$483$	0	0
$484$	−3.40834	−0.154924
$485$	−17.3875	−0.789525
$486$	−9.89949	−0.449050
$487$	0.408337	0.0185035	0.00925176	−	0.999957i	$-0.497055\pi$
$487$	0.408337	0.0185035	0.00925176	+	0.999957i	$0.497055\pi$
$488$	25.0227	1.13273
$489$	−21.7612	−0.984076
$490$	0	0
$491$	9.59166	0.432866	0.216433	−	0.976298i	$-0.430558\pi$
$491$	9.59166	0.432866	0.216433	+	0.976298i	$0.430558\pi$
$492$	−4.20417	−0.189539
$493$	1.01058	0.0455143
$494$	0	0
$495$	−15.5563	−0.699206
$496$	−1.41421	−0.0635001
$497$	0	0
$498$	−14.0000	−0.627355
$499$	25.7958	1.15478	0.577390	−	0.816468i	$-0.304071\pi$
$499$	25.7958	1.15478	0.577390	+	0.816468i	$0.304071\pi$
$500$	27.8512	1.24554
$501$	2.40834	0.107597
$502$	16.6818	0.744546
$503$	−25.7151	−1.14658	−0.573290	−	0.819353i	$-0.694333\pi$
$503$	−25.7151	−1.14658	−0.573290	+	0.819353i	$0.694333\pi$
$504$	0	0
$505$	−39.9792	−1.77905
$506$	29.5917	1.31551
$507$	0	0
$508$	11.5917	0.514297
$509$	−6.08996	−0.269933	−0.134966	−	0.990850i	$-0.543093\pi$
$509$	−6.08996	−0.269933	−0.134966	+	0.990850i	$0.543093\pi$
$510$	7.35981	0.325898
$511$	0	0
$512$	−11.0000	−0.486136
$513$	16.0000	0.706417
$514$	−8.62965	−0.380638
$515$	22.0000	0.969436
$516$	11.0250	0.485347
$517$	10.7362	0.472179
$518$	0	0
$519$	−25.1833	−1.10543
$520$	0	0
$521$	−4.67575	−0.204848	−0.102424	−	0.994741i	$-0.532660\pi$
$521$	−4.67575	−0.204848	−0.102424	+	0.994741i	$0.532660\pi$
$522$	−0.795832	−0.0348326
$523$	−42.1377	−1.84255	−0.921276	−	0.388910i	$-0.872852\pi$
$523$	−42.1377	−1.84255	−0.921276	+	0.388910i	$0.872852\pi$
$524$	5.36812	0.234507
$525$	0	0
$526$	3.38749	0.147702
$527$	1.79583	0.0782276
$528$	−5.36812	−0.233617
$529$	37.7750	1.64239
$530$	51.6041	2.24154
$531$	−12.4392	−0.539815
$532$	0	0
$533$	0	0
$534$	−21.1833	−0.916692
$535$	24.5896	1.06310
$536$	11.3875	0.491865
$537$	27.7068	1.19564
$538$	−12.4392	−0.536291
$539$	0	0
$540$	23.1833	0.997652
$541$	6.59166	0.283398	0.141699	−	0.989910i	$-0.454744\pi$
$541$	6.59166	0.283398	0.141699	+	0.989910i	$0.454744\pi$
$542$	−17.5186	−0.752487
$543$	5.38749	0.231200
$544$	6.34922	0.272221
$545$	−6.52307	−0.279418
$546$	0	0
$547$	10.9792	0.469435	0.234717	−	0.972064i	$-0.424584\pi$
$547$	10.9792	0.469435	0.234717	+	0.972064i	$0.424584\pi$
$548$	18.7958	0.802918
$549$	8.34091	0.355981
$550$	−44.7750	−1.90921
$551$	2.25095	0.0958938
$552$	−33.0749	−1.40776
$553$	0	0
$554$	32.1833	1.36734
$555$	16.2042	0.687829
$556$	−1.70295	−0.0722212
$557$	1.40834	0.0596732	0.0298366	−	0.999555i	$-0.490501\pi$
$557$	1.40834	0.0596732	0.0298366	+	0.999555i	$0.490501\pi$
$558$	−1.41421	−0.0598684
$559$	0	0
$560$	0	0
$561$	6.81667	0.287800
$562$	24.7958	1.04595
$563$	−12.4392	−0.524249	−0.262125	−	0.965034i	$-0.584423\pi$
$563$	−12.4392	−0.524249	−0.262125	+	0.965034i	$0.584423\pi$
$564$	−4.00000	−0.168430
$565$	−10.6213	−0.446843
$566$	15.8451	0.666019
$567$	0	0
$568$	−18.0000	−0.755263
$569$	25.5917	1.07286	0.536429	−	0.843945i	$-0.319773\pi$
$569$	25.5917	1.07286	0.536429	+	0.843945i	$0.319773\pi$
$570$	16.3931	0.686631
$571$	−23.1833	−0.970192	−0.485096	−	0.874461i	$-0.661215\pi$
$571$	−23.1833	−0.970192	−0.485096	+	0.874461i	$0.661215\pi$
$572$	0	0
$573$	22.9162	0.957336
$574$	0	0
$575$	−91.9583	−3.83493
$576$	−7.00000	−0.291667
$577$	−24.7340	−1.02969	−0.514845	−	0.857283i	$-0.672151\pi$
$577$	−24.7340	−1.02969	−0.514845	+	0.857283i	$0.672151\pi$
$578$	−15.3875	−0.640035
$579$	−31.9494	−1.32777
$580$	3.26153	0.135428
$581$	0	0
$582$	−6.00000	−0.248708
$583$	47.7958	1.97950
$584$	−37.7507	−1.56213
$585$	0	0
$586$	2.97280	0.122805
$587$	−16.6818	−0.688533	−0.344266	−	0.938872i	$-0.611872\pi$
$587$	−16.6818	−0.688533	−0.344266	+	0.938872i	$0.611872\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	−50.9792	−2.09878
$591$	11.3137	0.465384
$592$	−2.79583	−0.114908
$593$	8.34091	0.342520	0.171260	−	0.985226i	$-0.445216\pi$
$593$	8.34091	0.342520	0.171260	+	0.985226i	$0.445216\pi$
$594$	−21.4725	−0.881026
$595$	0	0
$596$	14.5917	0.597698
$597$	7.18333	0.293994
$598$	0	0
$599$	−33.5917	−1.37252	−0.686259	−	0.727357i	$-0.740748\pi$
$599$	−33.5917	−1.37252	−0.686259	+	0.727357i	$0.740748\pi$
$600$	50.0455	2.04310
$601$	24.1860	0.986567	0.493284	−	0.869868i	$-0.335796\pi$
$601$	24.1860	0.986567	0.493284	+	0.869868i	$0.335796\pi$
$602$	0	0
$603$	3.79583	0.154578
$604$	−1.59166	−0.0647639
$605$	−13.9683	−0.567892
$606$	−13.7958	−0.560417
$607$	−4.79064	−0.194446	−0.0972231	−	0.995263i	$-0.530996\pi$
$607$	−4.79064	−0.194446	−0.0972231	+	0.995263i	$0.530996\pi$
$608$	14.1421	0.573539
$609$	0	0
$610$	34.1833	1.38404
$611$	0	0
$612$	1.26984	0.0513304
$613$	41.9792	1.69552	0.847761	−	0.530378i	$-0.177950\pi$
$613$	41.9792	1.69552	0.847761	+	0.530378i	$0.177950\pi$
$614$	20.0583	0.809485
$615$	−17.2298	−0.694774
$616$	0	0
$617$	4.38749	0.176634	0.0883169	−	0.996092i	$-0.471851\pi$
$617$	4.38749	0.176634	0.0883169	+	0.996092i	$0.471851\pi$
$618$	7.59166	0.305381
$619$	33.9116	1.36302	0.681512	−	0.731807i	$-0.261323\pi$
$619$	33.9116	1.36302	0.681512	+	0.731807i	$0.261323\pi$
$620$	5.79583	0.232766
$621$	−44.0999	−1.76967
$622$	28.8323	1.15607
$623$	0	0
$624$	0	0
$625$	55.1625	2.20650
$626$	−15.2676	−0.610216
$627$	15.1833	0.606364
$628$	−0.144369	−0.00576095
$629$	3.55027	0.141559
$630$	0	0
$631$	−39.5917	−1.57612	−0.788060	−	0.615599i	$-0.788914\pi$
$631$	−39.5917	−1.57612	−0.788060	+	0.615599i	$0.788914\pi$
$632$	6.61251	0.263031
$633$	−11.0250	−0.438203
$634$	6.59166	0.261788
$635$	47.5058	1.88521
$636$	−17.8073	−0.706105
$637$	0	0
$638$	−3.02084	−0.119596
$639$	−6.00000	−0.237356
$640$	12.2948	0.485995
$641$	−4.79583	−0.189424	−0.0947120	−	0.995505i	$-0.530193\pi$
$641$	−4.79583	−0.189424	−0.0947120	+	0.995505i	$0.530193\pi$
$642$	8.48528	0.334887
$643$	−5.65685	−0.223085	−0.111542	−	0.993760i	$-0.535579\pi$
$643$	−5.65685	−0.223085	−0.111542	+	0.993760i	$0.535579\pi$
$644$	0	0
$645$	45.1833	1.77909
$646$	3.59166	0.141312
$647$	−38.7318	−1.52270	−0.761351	−	0.648339i	$-0.775464\pi$
$647$	−38.7318	−1.52270	−0.761351	+	0.648339i	$0.775464\pi$
$648$	15.0000	0.589256
$649$	−47.2170	−1.85343
$650$	0	0
$651$	0	0
$652$	−15.3875	−0.602621
$653$	−13.1833	−0.515903	−0.257952	−	0.966158i	$-0.583048\pi$
$653$	−13.1833	−0.515903	−0.257952	+	0.966158i	$0.583048\pi$
$654$	−2.25095	−0.0880192
$655$	22.0000	0.859611
$656$	2.97280	0.116068
$657$	−12.5836	−0.490931
$658$	0	0
$659$	−18.4083	−0.717087	−0.358543	−	0.933513i	$-0.616727\pi$
$659$	−18.4083	−0.717087	−0.358543	+	0.933513i	$0.616727\pi$
$660$	22.0000	0.856349
$661$	−7.50417	−0.291879	−0.145939	−	0.989294i	$-0.546620\pi$
$661$	−7.50417	−0.291879	−0.145939	+	0.989294i	$0.546620\pi$
$662$	29.3875	1.14218
$663$	0	0
$664$	−29.6985	−1.15252
$665$	0	0
$666$	−2.79583	−0.108336
$667$	−6.20417	−0.240226
$668$	1.70295	0.0658892
$669$	−8.00000	−0.309298
$670$	15.5563	0.600994
$671$	31.6607	1.22225
$672$	0	0
$673$	−23.9792	−0.924329	−0.462164	−	0.886794i	$-0.652927\pi$
$673$	−23.9792	−0.924329	−0.462164	+	0.886794i	$0.652927\pi$
$674$	17.9792	0.692531
$675$	66.7273	2.56833
$676$	0	0
$677$	−7.64854	−0.293957	−0.146979	−	0.989140i	$-0.546955\pi$
$677$	−7.64854	−0.293957	−0.146979	+	0.989140i	$0.546955\pi$
$678$	−3.66517	−0.140760
$679$	0	0
$680$	15.6125	0.598712
$681$	10.4083	0.398848
$682$	−5.36812	−0.205556
$683$	−6.77499	−0.259238	−0.129619	−	0.991564i	$-0.541375\pi$
$683$	−6.77499	−0.259238	−0.129619	+	0.991564i	$0.541375\pi$
$684$	2.82843	0.108148
$685$	77.0304	2.94318
$686$	0	0
$687$	18.0000	0.686743
$688$	−7.79583	−0.297213
$689$	0	0
$690$	−45.1833	−1.72010
$691$	2.53969	0.0966143	0.0483072	−	0.998833i	$-0.484617\pi$
$691$	2.53969	0.0966143	0.0483072	+	0.998833i	$0.484617\pi$
$692$	−17.8073	−0.676932
$693$	0	0
$694$	−32.9792	−1.25187
$695$	−6.97916	−0.264735
$696$	3.37643	0.127983
$697$	−3.77499	−0.142988
$698$	−26.5813	−1.00612
$699$	−24.3009	−0.919144
$700$	0	0
$701$	−29.5917	−1.11766	−0.558831	−	0.829282i	$-0.688750\pi$
$701$	−29.5917	−1.11766	−0.558831	+	0.829282i	$0.688750\pi$
$702$	0	0
$703$	7.90781	0.298249
$704$	−26.5708	−1.00143
$705$	−16.3931	−0.617399
$706$	−5.22375	−0.196598
$707$	0	0
$708$	17.5917	0.661135
$709$	−9.20417	−0.345670	−0.172835	−	0.984951i	$-0.555293\pi$
$709$	−9.20417	−0.345670	−0.172835	+	0.984951i	$0.555293\pi$
$710$	−24.5896	−0.922832
$711$	2.20417	0.0826628
$712$	−44.9366	−1.68407
$713$	−11.0250	−0.412888
$714$	0	0
$715$	0	0
$716$	19.5917	0.732175
$717$	14.4309	0.538931
$718$	−4.00000	−0.149279
$719$	−1.99169	−0.0742775	−0.0371387	−	0.999310i	$-0.511824\pi$
$719$	−1.99169	−0.0742775	−0.0371387	+	0.999310i	$0.511824\pi$
$720$	−4.09827	−0.152734
$721$	0	0
$722$	−11.0000	−0.409378
$723$	15.7958	0.587453
$724$	3.80953	0.141580
$725$	9.38749	0.348643
$726$	−4.82012	−0.178891
$727$	−32.4974	−1.20526	−0.602632	−	0.798020i	$-0.705881\pi$
$727$	−32.4974	−1.20526	−0.602632	+	0.798020i	$0.705881\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	−51.5708	−1.90872
$731$	9.89949	0.366146
$732$	−11.7958	−0.435986
$733$	33.2193	1.22698	0.613491	−	0.789702i	$-0.289765\pi$
$733$	33.2193	1.22698	0.613491	+	0.789702i	$0.289765\pi$
$734$	−21.2132	−0.782994
$735$	0	0
$736$	−38.9792	−1.43679
$737$	14.4083	0.530738
$738$	2.97280	0.109430
$739$	−23.1833	−0.852812	−0.426406	−	0.904532i	$-0.640221\pi$
$739$	−23.1833	−0.852812	−0.426406	+	0.904532i	$0.640221\pi$
$740$	11.4581	0.421207
$741$	0	0
$742$	0	0
$743$	−29.1833	−1.07063	−0.535316	−	0.844652i	$-0.679808\pi$
$743$	−29.1833	−1.07063	−0.535316	+	0.844652i	$0.679808\pi$
$744$	6.00000	0.219971
$745$	59.8006	2.19092
$746$	6.59166	0.241338
$747$	−9.89949	−0.362204
$748$	4.82012	0.176241
$749$	0	0
$750$	39.3875	1.43823
$751$	−7.79583	−0.284474	−0.142237	−	0.989833i	$-0.545430\pi$
$751$	−7.79583	−0.284474	−0.142237	+	0.989833i	$0.545430\pi$
$752$	2.82843	0.103142
$753$	−23.5917	−0.859728
$754$	0	0
$755$	−6.52307	−0.237399
$756$	0	0
$757$	13.1833	0.479156	0.239578	−	0.970877i	$-0.422991\pi$
$757$	13.1833	0.479156	0.239578	+	0.970877i	$0.422991\pi$
$758$	25.3875	0.922115
$759$	−41.8489	−1.51902
$760$	34.7750	1.26142
$761$	17.8073	0.645514	0.322757	−	0.946482i	$-0.395390\pi$
$761$	17.8073	0.645514	0.322757	+	0.946482i	$0.395390\pi$
$762$	16.3931	0.593859
$763$	0	0
$764$	16.2042	0.586246
$765$	5.20417	0.188157
$766$	−12.1504	−0.439013
$767$	0	0
$768$	24.0416	0.867528
$769$	−4.24264	−0.152994	−0.0764968	−	0.997070i	$-0.524373\pi$
$769$	−4.24264	−0.152994	−0.0764968	+	0.997070i	$0.524373\pi$
$770$	0	0
$771$	12.2042	0.439522
$772$	−22.5917	−0.813092
$773$	−34.7779	−1.25087	−0.625436	−	0.780275i	$-0.715079\pi$
$773$	−34.7779	−1.25087	−0.625436	+	0.780275i	$0.715079\pi$
$774$	−7.79583	−0.280215
$775$	16.6818	0.599229
$776$	−12.7279	−0.456906
$777$	0	0
$778$	−0.387495	−0.0138924
$779$	−8.40834	−0.301260
$780$	0	0
$781$	−22.7750	−0.814953
$782$	−9.89949	−0.354005
$783$	4.50190	0.160885
$784$	0	0
$785$	−0.591663	−0.0211174
$786$	7.59166	0.270786
$787$	42.3969	1.51129	0.755644	−	0.654983i	$-0.227324\pi$
$787$	42.3969	1.51129	0.755644	+	0.654983i	$0.227324\pi$
$788$	8.00000	0.284988
$789$	−4.79064	−0.170551
$790$	9.03328	0.321390
$791$	0	0
$792$	−11.3875	−0.404637
$793$	0	0
$794$	−7.07107	−0.250943
$795$	−72.9792	−2.58830
$796$	5.07938	0.180034
$797$	−0.836738	−0.0296388	−0.0148194	−	0.999890i	$-0.504717\pi$
$797$	−0.836738	−0.0296388	−0.0148194	+	0.999890i	$0.504717\pi$
$798$	0	0
$799$	−3.59166	−0.127064
$800$	58.9792	2.08523
$801$	−14.9789	−0.529252
$802$	−4.38749	−0.154928
$803$	−47.7650	−1.68559
$804$	−5.36812	−0.189319
$805$	0	0
$806$	0	0
$807$	17.5917	0.619256
$808$	−29.2654	−1.02955
$809$	−37.3667	−1.31374	−0.656871	−	0.754003i	$-0.728120\pi$
$809$	−37.3667	−1.31374	−0.656871	+	0.754003i	$0.728120\pi$
$810$	20.4914	0.719993
$811$	52.0372	1.82727	0.913636	−	0.406533i	$-0.133262\pi$
$811$	52.0372	1.82727	0.913636	+	0.406533i	$0.133262\pi$
$812$	0	0
$813$	24.7750	0.868897
$814$	−10.6125	−0.371968
$815$	−63.0621	−2.20897
$816$	1.79583	0.0628667
$817$	22.0499	0.771430
$818$	−18.7884	−0.656921
$819$	0	0
$820$	−12.1833	−0.425460
$821$	36.3667	1.26920	0.634602	−	0.772839i	$-0.281164\pi$
$821$	36.3667	1.26920	0.634602	+	0.772839i	$0.281164\pi$
$822$	26.5813	0.927130
$823$	−20.7750	−0.724171	−0.362085	−	0.932145i	$-0.617935\pi$
$823$	−20.7750	−0.724171	−0.362085	+	0.932145i	$0.617935\pi$
$824$	16.1043	0.561022
$825$	63.3214	2.20457
$826$	0	0
$827$	−20.9792	−0.729517	−0.364758	−	0.931102i	$-0.618848\pi$
$827$	−20.9792	−0.729517	−0.364758	+	0.931102i	$0.618848\pi$
$828$	−7.79583	−0.270924
$829$	27.2737	0.947254	0.473627	−	0.880725i	$-0.342944\pi$
$829$	27.2737	0.947254	0.473627	+	0.880725i	$0.342944\pi$
$830$	−40.5708	−1.40823
$831$	−45.5141	−1.57887
$832$	0	0
$833$	0	0
$834$	−2.40834	−0.0833939
$835$	6.97916	0.241524
$836$	10.7362	0.371320
$837$	8.00000	0.276520
$838$	25.7151	0.888314
$839$	−5.65685	−0.195296	−0.0976481	−	0.995221i	$-0.531132\pi$
$839$	−5.65685	−0.195296	−0.0976481	+	0.995221i	$0.531132\pi$
$840$	0	0
$841$	−28.3667	−0.978160
$842$	12.5917	0.433937
$843$	−35.0666	−1.20776
$844$	−7.79583	−0.268344
$845$	0	0
$846$	2.82843	0.0972433
$847$	0	0
$848$	12.5917	0.432399
$849$	−22.4083	−0.769052
$850$	14.9789	0.513771
$851$	−21.7958	−0.747151
$852$	8.48528	0.290701
$853$	33.5080	1.14729	0.573646	−	0.819103i	$-0.305528\pi$
$853$	33.5080	1.14729	0.573646	+	0.819103i	$0.305528\pi$
$854$	0	0
$855$	11.5917	0.396427
$856$	18.0000	0.615227
$857$	17.4037	0.594498	0.297249	−	0.954800i	$-0.403931\pi$
$857$	17.4037	0.594498	0.297249	+	0.954800i	$0.403931\pi$
$858$	0	0
$859$	−10.1882	−0.347618	−0.173809	−	0.984779i	$-0.555608\pi$
$859$	−10.1882	−0.347618	−0.173809	+	0.984779i	$0.555608\pi$
$860$	31.9494	1.08947
$861$	0	0
$862$	33.1833	1.13023
$863$	8.20417	0.279273	0.139637	−	0.990203i	$-0.455407\pi$
$863$	8.20417	0.279273	0.139637	+	0.990203i	$0.455407\pi$
$864$	28.2843	0.962250
$865$	−72.9792	−2.48137
$866$	20.4914	0.696325
$867$	21.7612	0.739049
$868$	0	0
$869$	8.36665	0.283819
$870$	4.61251	0.156379
$871$	0	0
$872$	−4.77499	−0.161702
$873$	−4.24264	−0.143592
$874$	−22.0499	−0.745850
$875$	0	0
$876$	17.7958	0.601265
$877$	4.79583	0.161944	0.0809719	−	0.996716i	$-0.474198\pi$
$877$	4.79583	0.161944	0.0809719	+	0.996716i	$0.474198\pi$
$878$	20.0583	0.676933
$879$	−4.20417	−0.141803
$880$	−15.5563	−0.524404
$881$	−10.5919	−0.356849	−0.178424	−	0.983954i	$-0.557100\pi$
$881$	−10.5919	−0.356849	−0.178424	+	0.983954i	$0.557100\pi$
$882$	0	0
$883$	−45.3875	−1.52741	−0.763705	−	0.645565i	$-0.776622\pi$
$883$	−45.3875	−1.52741	−0.763705	+	0.645565i	$0.776622\pi$
$884$	0	0
$885$	72.0954	2.42346
$886$	10.0000	0.335957
$887$	42.1377	1.41484	0.707422	−	0.706791i	$-0.249858\pi$
$887$	42.1377	1.41484	0.707422	+	0.706791i	$0.249858\pi$
$888$	11.8617	0.398053
$889$	0	0
$890$	−61.3875	−2.05771
$891$	18.9792	0.635826
$892$	−5.65685	−0.189405
$893$	−8.00000	−0.267710
$894$	20.6357	0.690162
$895$	80.2920	2.68386
$896$	0	0
$897$	0	0
$898$	23.5917	0.787264
$899$	1.12548	0.0375367
$900$	11.7958	0.393194
$901$	−15.9895	−0.532686
$902$	11.2842	0.375724
$903$	0	0
$904$	−7.77499	−0.258592
$905$	15.6125	0.518977
$906$	−2.25095	−0.0747829
$907$	20.6125	0.684427	0.342214	−	0.939622i	$-0.388823\pi$
$907$	20.6125	0.684427	0.342214	+	0.939622i	$0.388823\pi$
$908$	7.35981	0.244244
$909$	−9.75513	−0.323557
$910$	0	0
$911$	−4.81667	−0.159584	−0.0797918	−	0.996812i	$-0.525426\pi$
$911$	−4.81667	−0.159584	−0.0797918	+	0.996812i	$0.525426\pi$
$912$	4.00000	0.132453
$913$	−37.5768	−1.24361
$914$	4.59166	0.151879
$915$	−48.3425	−1.59815
$916$	12.7279	0.420542
$917$	0	0
$918$	7.18333	0.237085
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	−95.8483	−3.16003
$921$	−28.3667	−0.934713
$922$	−24.1860	−0.796523
$923$	0	0
$924$	0	0
$925$	32.9792	1.08435
$926$	−17.3875	−0.571389
$927$	5.36812	0.176312
$928$	3.97916	0.130622
$929$	21.3576	0.700719	0.350360	−	0.936615i	$-0.386059\pi$
$929$	21.3576	0.700719	0.350360	+	0.936615i	$0.386059\pi$
$930$	8.19654	0.268775
$931$	0	0
$932$	−17.1833	−0.562859
$933$	−40.7750	−1.33491
$934$	31.9494	1.04542
$935$	19.7541	0.646030
$936$	0	0
$937$	13.7090	0.447854	0.223927	−	0.974606i	$-0.428112\pi$
$937$	13.7090	0.447854	0.223927	+	0.974606i	$0.428112\pi$
$938$	0	0
$939$	21.5917	0.704617
$940$	−11.5917	−0.378078
$941$	−3.66517	−0.119481	−0.0597405	−	0.998214i	$-0.519027\pi$
$941$	−3.66517	−0.119481	−0.0597405	+	0.998214i	$0.519027\pi$
$942$	−0.204168	−0.00665217
$943$	23.1754	0.754695
$944$	−12.4392	−0.404861
$945$	0	0
$946$	−29.5917	−0.962108
$947$	37.7958	1.22820	0.614100	−	0.789228i	$-0.289519\pi$
$947$	37.7958	1.22820	0.614100	+	0.789228i	$0.289519\pi$
$948$	−3.11716	−0.101241
$949$	0	0
$950$	33.3636	1.08246
$951$	−9.32202	−0.302287
$952$	0	0
$953$	−43.1833	−1.39885	−0.699423	−	0.714708i	$-0.746559\pi$
$953$	−43.1833	−1.39885	−0.699423	+	0.714708i	$0.746559\pi$
$954$	12.5917	0.407670
$955$	66.4091	2.14895
$956$	10.2042	0.330026
$957$	4.27212	0.138098
$958$	20.9245	0.676039
$959$	0	0
$960$	40.5708	1.30942
$961$	−29.0000	−0.935484
$962$	0	0
$963$	6.00000	0.193347
$964$	11.1693	0.359740
$965$	−92.5868	−2.98047
$966$	0	0
$967$	−17.3875	−0.559144	−0.279572	−	0.960125i	$-0.590193\pi$
$967$	−17.3875	−0.559144	−0.279572	+	0.960125i	$0.590193\pi$
$968$	−10.2250	−0.328644
$969$	−5.07938	−0.163173
$970$	−17.3875	−0.558279
$971$	22.6274	0.726148	0.363074	−	0.931760i	$-0.381727\pi$
$971$	22.6274	0.726148	0.363074	+	0.931760i	$0.381727\pi$
$972$	9.89949	0.317526
$973$	0	0
$974$	0.408337	0.0130840
$975$	0	0
$976$	8.34091	0.266986
$977$	−28.3875	−0.908196	−0.454098	−	0.890952i	$-0.650038\pi$
$977$	−28.3875	−0.908196	−0.454098	+	0.890952i	$0.650038\pi$
$978$	−21.7612	−0.695847
$979$	−56.8573	−1.81717
$980$	0	0
$981$	−1.59166	−0.0508179
$982$	9.59166	0.306082
$983$	−0.866213	−0.0276279	−0.0138140	−	0.999905i	$-0.504397\pi$
$983$	−0.866213	−0.0276279	−0.0138140	+	0.999905i	$0.504397\pi$
$984$	−12.6125	−0.402072
$985$	32.7862	1.04465
$986$	1.01058	0.0321835
$987$	0	0
$988$	0	0
$989$	−60.7750	−1.93253
$990$	−15.5563	−0.494413
$991$	27.7958	0.882964	0.441482	−	0.897270i	$-0.354453\pi$
$991$	27.7958	0.882964	0.441482	+	0.897270i	$0.354453\pi$
$992$	7.07107	0.224507
$993$	−41.5602	−1.31887
$994$	0	0
$995$	20.8167	0.659933
$996$	14.0000	0.443607
$997$	−1.26984	−0.0402164	−0.0201082	−	0.999798i	$-0.506401\pi$
$997$	−1.26984	−0.0402164	−0.0201082	+	0.999798i	$0.506401\pi$
$998$	25.7958	0.816553
$999$	15.8156	0.500384

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.bv.1.1		4
7.6	odd	2	inner	8281.2.a.bv.1.4		4
13.3	even	3		637.2.f.g.295.3	yes	8
13.9	even	3		637.2.f.g.393.3	yes	8
13.12	even	2		8281.2.a.bn.1.2		4
91.3	odd	6		637.2.g.h.373.3		8
91.9	even	3		637.2.g.h.263.2		8
91.16	even	3		637.2.h.k.165.4		8
91.48	odd	6		637.2.f.g.393.2	yes	8
91.55	odd	6		637.2.f.g.295.2	✓	8
91.61	odd	6		637.2.g.h.263.3		8
91.68	odd	6		637.2.h.k.165.1		8
91.74	even	3		637.2.h.k.471.4		8
91.81	even	3		637.2.g.h.373.2		8
91.87	odd	6		637.2.h.k.471.1		8
91.90	odd	2		8281.2.a.bn.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.f.g.295.2	✓	8	91.55	odd	6
637.2.f.g.295.3	yes	8	13.3	even	3
637.2.f.g.393.2	yes	8	91.48	odd	6
637.2.f.g.393.3	yes	8	13.9	even	3
637.2.g.h.263.2		8	91.9	even	3
637.2.g.h.263.3		8	91.61	odd	6
637.2.g.h.373.2		8	91.81	even	3
637.2.g.h.373.3		8	91.3	odd	6
637.2.h.k.165.1		8	91.68	odd	6
637.2.h.k.165.4		8	91.16	even	3
637.2.h.k.471.1		8	91.87	odd	6
637.2.h.k.471.4		8	91.74	even	3
8281.2.a.bn.1.2		4	13.12	even	2
8281.2.a.bn.1.3		4	91.90	odd	2
8281.2.a.bv.1.1		4	1.1	even	1	trivial
8281.2.a.bv.1.4		4	7.6	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.41421 q^{3} -1.00000 q^{4} -4.09827 q^{5} -1.41421 q^{6} -3.00000 q^{8} -1.00000 q^{9} -4.09827 q^{10} -3.79583 q^{11} +1.41421 q^{12} +5.79583 q^{15} -1.00000 q^{16} +1.26984 q^{17} -1.00000 q^{18} +2.82843 q^{19} +4.09827 q^{20} -3.79583 q^{22} -7.79583 q^{23} +4.24264 q^{24} +11.7958 q^{25} +5.65685 q^{27} +0.795832 q^{29} +5.79583 q^{30} +1.41421 q^{31} +5.00000 q^{32} +5.36812 q^{33} +1.26984 q^{34} +1.00000 q^{36} +2.79583 q^{37} +2.82843 q^{38} +12.2948 q^{40} -2.97280 q^{41} +7.79583 q^{43} +3.79583 q^{44} +4.09827 q^{45} -7.79583 q^{46} -2.82843 q^{47} +1.41421 q^{48} +11.7958 q^{50} -1.79583 q^{51} -12.5917 q^{53} +5.65685 q^{54} +15.5563 q^{55} -4.00000 q^{57} +0.795832 q^{58} +12.4392 q^{59} -5.79583 q^{60} -8.34091 q^{61} +1.41421 q^{62} +7.00000 q^{64} +5.36812 q^{66} -3.79583 q^{67} -1.26984 q^{68} +11.0250 q^{69} +6.00000 q^{71} +3.00000 q^{72} +12.5836 q^{73} +2.79583 q^{74} -16.6818 q^{75} -2.82843 q^{76} -2.20417 q^{79} +4.09827 q^{80} -5.00000 q^{81} -2.97280 q^{82} +9.89949 q^{83} -5.20417 q^{85} +7.79583 q^{86} -1.12548 q^{87} +11.3875 q^{88} +14.9789 q^{89} +4.09827 q^{90} +7.79583 q^{92} -2.00000 q^{93} -2.82843 q^{94} -11.5917 q^{95} -7.07107 q^{96} +4.24264 q^{97} +3.79583 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 4 q^{4} - 12 q^{8} - 4 q^{9} + 4 q^{11} + 4 q^{15} - 4 q^{16} - 4 q^{18} + 4 q^{22} - 12 q^{23} + 28 q^{25} - 16 q^{29} + 4 q^{30} + 20 q^{32} + 4 q^{36} - 8 q^{37} + 12 q^{43} - 4 q^{44}+ \cdots - 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 4 * q^4 - 12 * q^8 - 4 * q^9 + 4 * q^11 + 4 * q^15 - 4 * q^16 - 4 * q^18 + 4 * q^22 - 12 * q^23 + 28 * q^25 - 16 * q^29 + 4 * q^30 + 20 * q^32 + 4 * q^36 - 8 * q^37 + 12 * q^43 - 4 * q^44 - 12 * q^46 + 28 * q^50 + 12 * q^51 - 12 * q^53 - 16 * q^57 - 16 * q^58 - 4 * q^60 + 28 * q^64 + 4 * q^67 + 24 * q^71 + 12 * q^72 - 8 * q^74 - 28 * q^79 - 20 * q^81 - 40 * q^85 + 12 * q^86 - 12 * q^88 + 12 * q^92 - 8 * q^93 - 8 * q^95 - 4 * q^99

Introduction

L-functions

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