LMFDB - Embedded newform 8330.2.a.cv.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8330,2,Mod(1,8330)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8330, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8330.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8330 = 2 \cdot 5 \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8330.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,10,-2,10,-10,-2,0,10,12,-10,-2,-2,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$66.5153848837$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 19x^{8} + 38x^{7} + 100x^{6} - 194x^{5} - 151x^{4} + 282x^{3} + 85x^{2} - 108x - 28 $ x^10 - 2x^9 - 19x^8 + 38x^7 + 100x^6 - 194x^5 - 151x^4 + 282x^3 + 85x^2 - 108*x - 28
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.35219$ of defining polynomial
Character	$\chi$	$=$	8330.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.35219 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.35219 q^{6} +1.00000 q^{8} -1.17159 q^{9} -1.00000 q^{10} -2.93946 q^{11} -1.35219 q^{12} +4.16153 q^{13} +1.35219 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.17159 q^{18} -2.23113 q^{19} -1.00000 q^{20} -2.93946 q^{22} +8.69719 q^{23} -1.35219 q^{24} +1.00000 q^{25} +4.16153 q^{26} +5.64077 q^{27} +2.53543 q^{29} +1.35219 q^{30} -9.15704 q^{31} +1.00000 q^{32} +3.97470 q^{33} -1.00000 q^{34} -1.17159 q^{36} -0.103526 q^{37} -2.23113 q^{38} -5.62717 q^{39} -1.00000 q^{40} -4.99088 q^{41} +0.552929 q^{43} -2.93946 q^{44} +1.17159 q^{45} +8.69719 q^{46} -9.03880 q^{47} -1.35219 q^{48} +1.00000 q^{50} +1.35219 q^{51} +4.16153 q^{52} -0.109403 q^{53} +5.64077 q^{54} +2.93946 q^{55} +3.01690 q^{57} +2.53543 q^{58} -8.41721 q^{59} +1.35219 q^{60} +6.82353 q^{61} -9.15704 q^{62} +1.00000 q^{64} -4.16153 q^{65} +3.97470 q^{66} +3.16632 q^{67} -1.00000 q^{68} -11.7602 q^{69} +1.49253 q^{71} -1.17159 q^{72} +6.19201 q^{73} -0.103526 q^{74} -1.35219 q^{75} -2.23113 q^{76} -5.62717 q^{78} -6.69724 q^{79} -1.00000 q^{80} -4.11260 q^{81} -4.99088 q^{82} -10.0600 q^{83} +1.00000 q^{85} +0.552929 q^{86} -3.42838 q^{87} -2.93946 q^{88} -1.79218 q^{89} +1.17159 q^{90} +8.69719 q^{92} +12.3820 q^{93} -9.03880 q^{94} +2.23113 q^{95} -1.35219 q^{96} +0.714370 q^{97} +3.44385 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{2} - 2 q^{3} + 10 q^{4} - 10 q^{5} - 2 q^{6} + 10 q^{8} + 12 q^{9} - 10 q^{10} - 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{15} + 10 q^{16} - 10 q^{17} + 12 q^{18} - 14 q^{19} - 10 q^{20} - 2 q^{22}+ \cdots - 12 q^{99}+O(q^{100}) $ 10 * q + 10 * q^2 - 2 * q^3 + 10 * q^4 - 10 * q^5 - 2 * q^6 + 10 * q^8 + 12 * q^9 - 10 * q^10 - 2 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^15 + 10 * q^16 - 10 * q^17 + 12 * q^18 - 14 * q^19 - 10 * q^20 - 2 * q^22 + 10 * q^23 - 2 * q^24 + 10 * q^25 - 4 * q^26 + 4 * q^27 + 10 * q^29 + 2 * q^30 - 2 * q^31 + 10 * q^32 - 4 * q^33 - 10 * q^34 + 12 * q^36 - 14 * q^38 + 48 * q^39 - 10 * q^40 + 22 * q^41 + 14 * q^43 - 2 * q^44 - 12 * q^45 + 10 * q^46 - 18 * q^47 - 2 * q^48 + 10 * q^50 + 2 * q^51 - 4 * q^52 + 22 * q^53 + 4 * q^54 + 2 * q^55 + 32 * q^57 + 10 * q^58 + 6 * q^59 + 2 * q^60 + 16 * q^61 - 2 * q^62 + 10 * q^64 + 4 * q^65 - 4 * q^66 + 14 * q^67 - 10 * q^68 - 10 * q^69 + 20 * q^71 + 12 * q^72 - 12 * q^73 - 2 * q^75 - 14 * q^76 + 48 * q^78 + 44 * q^79 - 10 * q^80 + 50 * q^81 + 22 * q^82 - 20 * q^83 + 10 * q^85 + 14 * q^86 - 6 * q^87 - 2 * q^88 - 12 * q^90 + 10 * q^92 + 4 * q^93 - 18 * q^94 + 14 * q^95 - 2 * q^96 + 40 * q^97 - 12 * 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
$2$	1.00000	0.707107
$3$	−1.35219	−0.780685	−0.390343	−	0.920670i	$-0.627643\pi$
$3$	−1.35219	−0.780685	−0.390343	+	0.920670i	$0.627643\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.35219	−0.552028
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−1.17159	−0.390530
$10$	−1.00000	−0.316228
$11$	−2.93946	−0.886280	−0.443140	−	0.896452i	$-0.646136\pi$
$11$	−2.93946	−0.886280	−0.443140	+	0.896452i	$0.646136\pi$
$12$	−1.35219	−0.390343
$13$	4.16153	1.15420	0.577101	−	0.816673i	$-0.304184\pi$
$13$	4.16153	1.15420	0.577101	+	0.816673i	$0.304184\pi$
$14$	0	0
$15$	1.35219	0.349133
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−1.17159	−0.276147
$19$	−2.23113	−0.511856	−0.255928	−	0.966696i	$-0.582381\pi$
$19$	−2.23113	−0.511856	−0.255928	+	0.966696i	$0.582381\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−2.93946	−0.626695
$23$	8.69719	1.81349	0.906745	−	0.421679i	$-0.138559\pi$
$23$	8.69719	1.81349	0.906745	+	0.421679i	$0.138559\pi$
$24$	−1.35219	−0.276014
$25$	1.00000	0.200000
$26$	4.16153	0.816143
$27$	5.64077	1.08557
$28$	0	0
$29$	2.53543	0.470818	0.235409	−	0.971896i	$-0.424357\pi$
$29$	2.53543	0.470818	0.235409	+	0.971896i	$0.424357\pi$
$30$	1.35219	0.246874
$31$	−9.15704	−1.64465	−0.822327	−	0.569016i	$-0.807324\pi$
$31$	−9.15704	−1.64465	−0.822327	+	0.569016i	$0.807324\pi$
$32$	1.00000	0.176777
$33$	3.97470	0.691906
$34$	−1.00000	−0.171499
$35$	0	0
$36$	−1.17159	−0.195265
$37$	−0.103526	−0.0170195	−0.00850975	−	0.999964i	$-0.502709\pi$
$37$	−0.103526	−0.0170195	−0.00850975	+	0.999964i	$0.502709\pi$
$38$	−2.23113	−0.361937
$39$	−5.62717	−0.901068
$40$	−1.00000	−0.158114
$41$	−4.99088	−0.779445	−0.389722	−	0.920932i	$-0.627429\pi$
$41$	−4.99088	−0.779445	−0.389722	+	0.920932i	$0.627429\pi$
$42$	0	0
$43$	0.552929	0.0843209	0.0421604	−	0.999111i	$-0.486576\pi$
$43$	0.552929	0.0843209	0.0421604	+	0.999111i	$0.486576\pi$
$44$	−2.93946	−0.443140
$45$	1.17159	0.174651
$46$	8.69719	1.28233
$47$	−9.03880	−1.31844	−0.659222	−	0.751948i	$-0.729114\pi$
$47$	−9.03880	−1.31844	−0.659222	+	0.751948i	$0.729114\pi$
$48$	−1.35219	−0.195171
$49$	0	0
$50$	1.00000	0.141421
$51$	1.35219	0.189344
$52$	4.16153	0.577101
$53$	−0.109403	−0.0150277	−0.00751383	−	0.999972i	$-0.502392\pi$
$53$	−0.109403	−0.0150277	−0.00751383	+	0.999972i	$0.502392\pi$
$54$	5.64077	0.767612
$55$	2.93946	0.396357
$56$	0	0
$57$	3.01690	0.399598
$58$	2.53543	0.332919
$59$	−8.41721	−1.09583	−0.547913	−	0.836535i	$-0.684578\pi$
$59$	−8.41721	−1.09583	−0.547913	+	0.836535i	$0.684578\pi$
$60$	1.35219	0.174567
$61$	6.82353	0.873663	0.436832	−	0.899543i	$-0.356101\pi$
$61$	6.82353	0.873663	0.436832	+	0.899543i	$0.356101\pi$
$62$	−9.15704	−1.16295
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.16153	−0.516174
$66$	3.97470	0.489251
$67$	3.16632	0.386828	0.193414	−	0.981117i	$-0.438044\pi$
$67$	3.16632	0.386828	0.193414	+	0.981117i	$0.438044\pi$
$68$	−1.00000	−0.121268
$69$	−11.7602	−1.41576
$70$	0	0
$71$	1.49253	0.177131	0.0885656	−	0.996070i	$-0.471772\pi$
$71$	1.49253	0.177131	0.0885656	+	0.996070i	$0.471772\pi$
$72$	−1.17159	−0.138073
$73$	6.19201	0.724720	0.362360	−	0.932038i	$-0.381971\pi$
$73$	6.19201	0.724720	0.362360	+	0.932038i	$0.381971\pi$
$74$	−0.103526	−0.0120346
$75$	−1.35219	−0.156137
$76$	−2.23113	−0.255928
$77$	0	0
$78$	−5.62717	−0.637151
$79$	−6.69724	−0.753498	−0.376749	−	0.926315i	$-0.622958\pi$
$79$	−6.69724	−0.753498	−0.376749	+	0.926315i	$0.622958\pi$
$80$	−1.00000	−0.111803
$81$	−4.11260	−0.456955
$82$	−4.99088	−0.551151
$83$	−10.0600	−1.10423	−0.552116	−	0.833767i	$-0.686179\pi$
$83$	−10.0600	−1.10423	−0.552116	+	0.833767i	$0.686179\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	0.552929	0.0596239
$87$	−3.42838	−0.367561
$88$	−2.93946	−0.313347
$89$	−1.79218	−0.189971	−0.0949856	−	0.995479i	$-0.530280\pi$
$89$	−1.79218	−0.189971	−0.0949856	+	0.995479i	$0.530280\pi$
$90$	1.17159	0.123497
$91$	0	0
$92$	8.69719	0.906745
$93$	12.3820	1.28396
$94$	−9.03880	−0.932281
$95$	2.23113	0.228909
$96$	−1.35219	−0.138007
$97$	0.714370	0.0725333	0.0362667	−	0.999342i	$-0.488453\pi$
$97$	0.714370	0.0725333	0.0362667	+	0.999342i	$0.488453\pi$
$98$	0	0
$99$	3.44385	0.346119
$100$	1.00000	0.100000
$101$	3.36397	0.334728	0.167364	−	0.985895i	$-0.446475\pi$
$101$	3.36397	0.334728	0.167364	+	0.985895i	$0.446475\pi$
$102$	1.35219	0.133886
$103$	9.40703	0.926902	0.463451	−	0.886122i	$-0.346611\pi$
$103$	9.40703	0.926902	0.463451	+	0.886122i	$0.346611\pi$
$104$	4.16153	0.408072
$105$	0	0
$106$	−0.109403	−0.0106262
$107$	17.8174	1.72248	0.861238	−	0.508202i	$-0.169690\pi$
$107$	17.8174	1.72248	0.861238	+	0.508202i	$0.169690\pi$
$108$	5.64077	0.542783
$109$	6.96113	0.666755	0.333378	−	0.942793i	$-0.391812\pi$
$109$	6.96113	0.666755	0.333378	+	0.942793i	$0.391812\pi$
$110$	2.93946	0.280266
$111$	0.139986	0.0132869
$112$	0	0
$113$	7.97609	0.750327	0.375164	−	0.926959i	$-0.377586\pi$
$113$	7.97609	0.750327	0.375164	+	0.926959i	$0.377586\pi$
$114$	3.01690	0.282559
$115$	−8.69719	−0.811017
$116$	2.53543	0.235409
$117$	−4.87561	−0.450751
$118$	−8.41721	−0.774867
$119$	0	0
$120$	1.35219	0.123437
$121$	−2.35958	−0.214507
$122$	6.82353	0.617773
$123$	6.74860	0.608501
$124$	−9.15704	−0.822327
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	15.3160	1.35907	0.679536	−	0.733642i	$-0.262181\pi$
$127$	15.3160	1.35907	0.679536	+	0.733642i	$0.262181\pi$
$128$	1.00000	0.0883883
$129$	−0.747663	−0.0658281
$130$	−4.16153	−0.364990
$131$	−4.22974	−0.369555	−0.184777	−	0.982780i	$-0.559156\pi$
$131$	−4.22974	−0.369555	−0.184777	+	0.982780i	$0.559156\pi$
$132$	3.97470	0.345953
$133$	0	0
$134$	3.16632	0.273528
$135$	−5.64077	−0.485480
$136$	−1.00000	−0.0857493
$137$	−6.95264	−0.594004	−0.297002	−	0.954877i	$-0.595987\pi$
$137$	−6.95264	−0.594004	−0.297002	+	0.954877i	$0.595987\pi$
$138$	−11.7602	−1.00110
$139$	12.6339	1.07159	0.535795	−	0.844348i	$-0.320012\pi$
$139$	12.6339	1.07159	0.535795	+	0.844348i	$0.320012\pi$
$140$	0	0
$141$	12.2221	1.02929
$142$	1.49253	0.125251
$143$	−12.2327	−1.02295
$144$	−1.17159	−0.0976326
$145$	−2.53543	−0.210556
$146$	6.19201	0.512454
$147$	0	0
$148$	−0.103526	−0.00850975
$149$	4.41845	0.361974	0.180987	−	0.983486i	$-0.442071\pi$
$149$	4.41845	0.361974	0.180987	+	0.983486i	$0.442071\pi$
$150$	−1.35219	−0.110406
$151$	6.56833	0.534523	0.267261	−	0.963624i	$-0.413881\pi$
$151$	6.56833	0.534523	0.267261	+	0.963624i	$0.413881\pi$
$152$	−2.23113	−0.180968
$153$	1.17159	0.0947176
$154$	0	0
$155$	9.15704	0.735511
$156$	−5.62717	−0.450534
$157$	5.99757	0.478658	0.239329	−	0.970939i	$-0.423073\pi$
$157$	5.99757	0.478658	0.239329	+	0.970939i	$0.423073\pi$
$158$	−6.69724	−0.532804
$159$	0.147933	0.0117319
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	−4.11260	−0.323116
$163$	−0.0850324	−0.00666025	−0.00333013	−	0.999994i	$-0.501060\pi$
$163$	−0.0850324	−0.00666025	−0.00333013	+	0.999994i	$0.501060\pi$
$164$	−4.99088	−0.389722
$165$	−3.97470	−0.309430
$166$	−10.0600	−0.780810
$167$	3.86319	0.298942	0.149471	−	0.988766i	$-0.452243\pi$
$167$	3.86319	0.298942	0.149471	+	0.988766i	$0.452243\pi$
$168$	0	0
$169$	4.31834	0.332180
$170$	1.00000	0.0766965
$171$	2.61397	0.199895
$172$	0.552929	0.0421604
$173$	16.8732	1.28285	0.641424	−	0.767186i	$-0.278344\pi$
$173$	16.8732	1.28285	0.641424	+	0.767186i	$0.278344\pi$
$174$	−3.42838	−0.259905
$175$	0	0
$176$	−2.93946	−0.221570
$177$	11.3816	0.855496
$178$	−1.79218	−0.134330
$179$	−1.41488	−0.105753	−0.0528767	−	0.998601i	$-0.516839\pi$
$179$	−1.41488	−0.105753	−0.0528767	+	0.998601i	$0.516839\pi$
$180$	1.17159	0.0873253
$181$	−1.68884	−0.125530	−0.0627652	−	0.998028i	$-0.519992\pi$
$181$	−1.68884	−0.125530	−0.0627652	+	0.998028i	$0.519992\pi$
$182$	0	0
$183$	−9.22669	−0.682056
$184$	8.69719	0.641166
$185$	0.103526	0.00761135
$186$	12.3820	0.907894
$187$	2.93946	0.214955
$188$	−9.03880	−0.659222
$189$	0	0
$190$	2.23113	0.161863
$191$	−0.907269	−0.0656477	−0.0328239	−	0.999461i	$-0.510450\pi$
$191$	−0.907269	−0.0656477	−0.0328239	+	0.999461i	$0.510450\pi$
$192$	−1.35219	−0.0975857
$193$	21.3866	1.53944	0.769722	−	0.638380i	$-0.220395\pi$
$193$	21.3866	1.53944	0.769722	+	0.638380i	$0.220395\pi$
$194$	0.714370	0.0512888
$195$	5.62717	0.402970
$196$	0	0
$197$	21.0895	1.50256	0.751281	−	0.659982i	$-0.229436\pi$
$197$	21.0895	1.50256	0.751281	+	0.659982i	$0.229436\pi$
$198$	3.44385	0.244743
$199$	−14.1982	−1.00648	−0.503242	−	0.864146i	$-0.667860\pi$
$199$	−14.1982	−1.00648	−0.503242	+	0.864146i	$0.667860\pi$
$200$	1.00000	0.0707107
$201$	−4.28146	−0.301991
$202$	3.36397	0.236688
$203$	0	0
$204$	1.35219	0.0946720
$205$	4.99088	0.348578
$206$	9.40703	0.655419
$207$	−10.1896	−0.708223
$208$	4.16153	0.288550
$209$	6.55831	0.453648
$210$	0	0
$211$	26.1407	1.79960	0.899802	−	0.436299i	$-0.143711\pi$
$211$	26.1407	1.79960	0.899802	+	0.436299i	$0.143711\pi$
$212$	−0.109403	−0.00751383
$213$	−2.01818	−0.138284
$214$	17.8174	1.21797
$215$	−0.552929	−0.0377094
$216$	5.64077	0.383806
$217$	0	0
$218$	6.96113	0.471467
$219$	−8.37275	−0.565778
$220$	2.93946	0.198178
$221$	−4.16153	−0.279935
$222$	0.139986	0.00939524
$223$	16.9592	1.13567	0.567834	−	0.823143i	$-0.307781\pi$
$223$	16.9592	1.13567	0.567834	+	0.823143i	$0.307781\pi$
$224$	0	0
$225$	−1.17159	−0.0781061
$226$	7.97609	0.530562
$227$	18.8819	1.25324	0.626619	−	0.779325i	$-0.284438\pi$
$227$	18.8819	1.25324	0.626619	+	0.779325i	$0.284438\pi$
$228$	3.01690	0.199799
$229$	4.67822	0.309146	0.154573	−	0.987981i	$-0.450600\pi$
$229$	4.67822	0.309146	0.154573	+	0.987981i	$0.450600\pi$
$230$	−8.69719	−0.573476
$231$	0	0
$232$	2.53543	0.166459
$233$	17.8669	1.17050	0.585250	−	0.810853i	$-0.300996\pi$
$233$	17.8669	1.17050	0.585250	+	0.810853i	$0.300996\pi$
$234$	−4.87561	−0.318729
$235$	9.03880	0.589626
$236$	−8.41721	−0.547913
$237$	9.05592	0.588245
$238$	0	0
$239$	0.864416	0.0559144	0.0279572	−	0.999609i	$-0.491100\pi$
$239$	0.864416	0.0559144	0.0279572	+	0.999609i	$0.491100\pi$
$240$	1.35219	0.0872833
$241$	−4.74645	−0.305746	−0.152873	−	0.988246i	$-0.548852\pi$
$241$	−4.74645	−0.305746	−0.152873	+	0.988246i	$0.548852\pi$
$242$	−2.35958	−0.151680
$243$	−11.3613	−0.728828
$244$	6.82353	0.436832
$245$	0	0
$246$	6.74860	0.430275
$247$	−9.28491	−0.590785
$248$	−9.15704	−0.581473
$249$	13.6030	0.862058
$250$	−1.00000	−0.0632456
$251$	3.45527	0.218095	0.109047	−	0.994037i	$-0.465220\pi$
$251$	3.45527	0.218095	0.109047	+	0.994037i	$0.465220\pi$
$252$	0	0
$253$	−25.5650	−1.60726
$254$	15.3160	0.961010
$255$	−1.35219	−0.0846772
$256$	1.00000	0.0625000
$257$	27.2139	1.69756	0.848779	−	0.528747i	$-0.177338\pi$
$257$	27.2139	1.69756	0.848779	+	0.528747i	$0.177338\pi$
$258$	−0.747663	−0.0465475
$259$	0	0
$260$	−4.16153	−0.258087
$261$	−2.97049	−0.183869
$262$	−4.22974	−0.261314
$263$	13.1007	0.807827	0.403913	−	0.914797i	$-0.367650\pi$
$263$	13.1007	0.807827	0.403913	+	0.914797i	$0.367650\pi$
$264$	3.97470	0.244626
$265$	0.109403	0.00672058
$266$	0	0
$267$	2.42337	0.148308
$268$	3.16632	0.193414
$269$	−10.7385	−0.654740	−0.327370	−	0.944896i	$-0.606162\pi$
$269$	−10.7385	−0.654740	−0.327370	+	0.944896i	$0.606162\pi$
$270$	−5.64077	−0.343286
$271$	7.72677	0.469368	0.234684	−	0.972072i	$-0.424594\pi$
$271$	7.72677	0.469368	0.234684	+	0.972072i	$0.424594\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	−6.95264	−0.420024
$275$	−2.93946	−0.177256
$276$	−11.7602	−0.707882
$277$	−19.2239	−1.15505	−0.577525	−	0.816373i	$-0.695981\pi$
$277$	−19.2239	−1.15505	−0.577525	+	0.816373i	$0.695981\pi$
$278$	12.6339	0.757728
$279$	10.7283	0.642287
$280$	0	0
$281$	−6.45650	−0.385162	−0.192581	−	0.981281i	$-0.561686\pi$
$281$	−6.45650	−0.385162	−0.192581	+	0.981281i	$0.561686\pi$
$282$	12.2221	0.727818
$283$	−9.15294	−0.544086	−0.272043	−	0.962285i	$-0.587699\pi$
$283$	−9.15294	−0.544086	−0.272043	+	0.962285i	$0.587699\pi$
$284$	1.49253	0.0885656
$285$	−3.01690	−0.178706
$286$	−12.2327	−0.723332
$287$	0	0
$288$	−1.17159	−0.0690367
$289$	1.00000	0.0588235
$290$	−2.53543	−0.148886
$291$	−0.965962	−0.0566257
$292$	6.19201	0.362360
$293$	−4.17718	−0.244033	−0.122017	−	0.992528i	$-0.538936\pi$
$293$	−4.17718	−0.244033	−0.122017	+	0.992528i	$0.538936\pi$
$294$	0	0
$295$	8.41721	0.490069
$296$	−0.103526	−0.00601730
$297$	−16.5808	−0.962116
$298$	4.41845	0.255954
$299$	36.1936	2.09313
$300$	−1.35219	−0.0780685
$301$	0	0
$302$	6.56833	0.377965
$303$	−4.54872	−0.261317
$304$	−2.23113	−0.127964
$305$	−6.82353	−0.390714
$306$	1.17159	0.0669754
$307$	−31.3778	−1.79082	−0.895412	−	0.445239i	$-0.853119\pi$
$307$	−31.3778	−1.79082	−0.895412	+	0.445239i	$0.853119\pi$
$308$	0	0
$309$	−12.7201	−0.723619
$310$	9.15704	0.520085
$311$	−0.559867	−0.0317472	−0.0158736	−	0.999874i	$-0.505053\pi$
$311$	−0.559867	−0.0317472	−0.0158736	+	0.999874i	$0.505053\pi$
$312$	−5.62717	−0.318576
$313$	−0.824866	−0.0466242	−0.0233121	−	0.999728i	$-0.507421\pi$
$313$	−0.824866	−0.0466242	−0.0233121	+	0.999728i	$0.507421\pi$
$314$	5.99757	0.338462
$315$	0	0
$316$	−6.69724	−0.376749
$317$	−9.13508	−0.513077	−0.256539	−	0.966534i	$-0.582582\pi$
$317$	−9.13508	−0.513077	−0.256539	+	0.966534i	$0.582582\pi$
$318$	0.147933	0.00829569
$319$	−7.45280	−0.417277
$320$	−1.00000	−0.0559017
$321$	−24.0925	−1.34471
$322$	0	0
$323$	2.23113	0.124143
$324$	−4.11260	−0.228478
$325$	4.16153	0.230840
$326$	−0.0850324	−0.00470951
$327$	−9.41275	−0.520526
$328$	−4.99088	−0.275575
$329$	0	0
$330$	−3.97470	−0.218800
$331$	−9.48144	−0.521147	−0.260573	−	0.965454i	$-0.583912\pi$
$331$	−9.48144	−0.521147	−0.260573	+	0.965454i	$0.583912\pi$
$332$	−10.0600	−0.552116
$333$	0.121290	0.00664663
$334$	3.86319	0.211384
$335$	−3.16632	−0.172995
$336$	0	0
$337$	−14.7377	−0.802811	−0.401405	−	0.915900i	$-0.631478\pi$
$337$	−14.7377	−0.802811	−0.401405	+	0.915900i	$0.631478\pi$
$338$	4.31834	0.234887
$339$	−10.7852	−0.585770
$340$	1.00000	0.0542326
$341$	26.9167	1.45762
$342$	2.61397	0.141347
$343$	0	0
$344$	0.552929	0.0298119
$345$	11.7602	0.633149
$346$	16.8732	0.907111
$347$	−19.7544	−1.06047	−0.530236	−	0.847850i	$-0.677897\pi$
$347$	−19.7544	−1.06047	−0.530236	+	0.847850i	$0.677897\pi$
$348$	−3.42838	−0.183780
$349$	25.3828	1.35871	0.679356	−	0.733808i	$-0.262259\pi$
$349$	25.3828	1.35871	0.679356	+	0.733808i	$0.262259\pi$
$350$	0	0
$351$	23.4742	1.25296
$352$	−2.93946	−0.156674
$353$	10.1367	0.539522	0.269761	−	0.962927i	$-0.413055\pi$
$353$	10.1367	0.539522	0.269761	+	0.962927i	$0.413055\pi$
$354$	11.3816	0.604927
$355$	−1.49253	−0.0792155
$356$	−1.79218	−0.0949856
$357$	0	0
$358$	−1.41488	−0.0747789
$359$	−13.8984	−0.733527	−0.366763	−	0.930314i	$-0.619534\pi$
$359$	−13.8984	−0.733527	−0.366763	+	0.930314i	$0.619534\pi$
$360$	1.17159	0.0617483
$361$	−14.0221	−0.738003
$362$	−1.68884	−0.0887633
$363$	3.19059	0.167463
$364$	0	0
$365$	−6.19201	−0.324104
$366$	−9.22669	−0.482287
$367$	9.00621	0.470121	0.235060	−	0.971981i	$-0.424471\pi$
$367$	9.00621	0.470121	0.235060	+	0.971981i	$0.424471\pi$
$368$	8.69719	0.453373
$369$	5.84727	0.304397
$370$	0.103526	0.00538204
$371$	0	0
$372$	12.3820	0.641978
$373$	−30.5019	−1.57933	−0.789665	−	0.613539i	$-0.789745\pi$
$373$	−30.5019	−1.57933	−0.789665	+	0.613539i	$0.789745\pi$
$374$	2.93946	0.151996
$375$	1.35219	0.0698266
$376$	−9.03880	−0.466141
$377$	10.5513	0.543419
$378$	0	0
$379$	29.4384	1.51215	0.756074	−	0.654487i	$-0.227115\pi$
$379$	29.4384	1.51215	0.756074	+	0.654487i	$0.227115\pi$
$380$	2.23113	0.114454
$381$	−20.7101	−1.06101
$382$	−0.907269	−0.0464199
$383$	19.6495	1.00404	0.502022	−	0.864855i	$-0.332590\pi$
$383$	19.6495	1.00404	0.502022	+	0.864855i	$0.332590\pi$
$384$	−1.35219	−0.0690035
$385$	0	0
$386$	21.3866	1.08855
$387$	−0.647807	−0.0329299
$388$	0.714370	0.0362667
$389$	−6.69989	−0.339698	−0.169849	−	0.985470i	$-0.554328\pi$
$389$	−6.69989	−0.339698	−0.169849	+	0.985470i	$0.554328\pi$
$390$	5.62717	0.284943
$391$	−8.69719	−0.439836
$392$	0	0
$393$	5.71940	0.288506
$394$	21.0895	1.06247
$395$	6.69724	0.336975
$396$	3.44385	0.173060
$397$	35.0846	1.76085	0.880423	−	0.474189i	$-0.157259\pi$
$397$	35.0846	1.76085	0.880423	+	0.474189i	$0.157259\pi$
$398$	−14.1982	−0.711691
$399$	0	0
$400$	1.00000	0.0500000
$401$	24.5806	1.22750	0.613749	−	0.789501i	$-0.289661\pi$
$401$	24.5806	1.22750	0.613749	+	0.789501i	$0.289661\pi$
$402$	−4.28146	−0.213540
$403$	−38.1073	−1.89826
$404$	3.36397	0.167364
$405$	4.11260	0.204357
$406$	0	0
$407$	0.304309	0.0150840
$408$	1.35219	0.0669432
$409$	−5.85978	−0.289748	−0.144874	−	0.989450i	$-0.546278\pi$
$409$	−5.85978	−0.289748	−0.144874	+	0.989450i	$0.546278\pi$
$410$	4.99088	0.246482
$411$	9.40127	0.463730
$412$	9.40703	0.463451
$413$	0	0
$414$	−10.1896	−0.500789
$415$	10.0600	0.493827
$416$	4.16153	0.204036
$417$	−17.0833	−0.836574
$418$	6.55831	0.320778
$419$	37.4041	1.82731	0.913656	−	0.406488i	$-0.133247\pi$
$419$	37.4041	1.82731	0.913656	+	0.406488i	$0.133247\pi$
$420$	0	0
$421$	17.3809	0.847092	0.423546	−	0.905875i	$-0.360785\pi$
$421$	17.3809	0.847092	0.423546	+	0.905875i	$0.360785\pi$
$422$	26.1407	1.27251
$423$	10.5898	0.514893
$424$	−0.109403	−0.00531308
$425$	−1.00000	−0.0485071
$426$	−2.01818	−0.0977813
$427$	0	0
$428$	17.8174	0.861238
$429$	16.5408	0.798599
$430$	−0.552929	−0.0266646
$431$	−16.4023	−0.790070	−0.395035	−	0.918666i	$-0.629268\pi$
$431$	−16.4023	−0.790070	−0.395035	+	0.918666i	$0.629268\pi$
$432$	5.64077	0.271392
$433$	22.7926	1.09534	0.547672	−	0.836693i	$-0.315514\pi$
$433$	22.7926	1.09534	0.547672	+	0.836693i	$0.315514\pi$
$434$	0	0
$435$	3.42838	0.164378
$436$	6.96113	0.333378
$437$	−19.4046	−0.928246
$438$	−8.37275	−0.400065
$439$	21.0246	1.00345	0.501724	−	0.865027i	$-0.332699\pi$
$439$	21.0246	1.00345	0.501724	+	0.865027i	$0.332699\pi$
$440$	2.93946	0.140133
$441$	0	0
$442$	−4.16153	−0.197944
$443$	−36.5497	−1.73653	−0.868264	−	0.496102i	$-0.834764\pi$
$443$	−36.5497	−1.73653	−0.868264	+	0.496102i	$0.834764\pi$
$444$	0.139986	0.00664344
$445$	1.79218	0.0849577
$446$	16.9592	0.803039
$447$	−5.97457	−0.282588
$448$	0	0
$449$	33.2693	1.57008	0.785038	−	0.619448i	$-0.212643\pi$
$449$	33.2693	1.57008	0.785038	+	0.619448i	$0.212643\pi$
$450$	−1.17159	−0.0552294
$451$	14.6705	0.690806
$452$	7.97609	0.375164
$453$	−8.88160	−0.417294
$454$	18.8819	0.886174
$455$	0	0
$456$	3.01690	0.141279
$457$	27.2381	1.27415	0.637073	−	0.770804i	$-0.280145\pi$
$457$	27.2381	1.27415	0.637073	+	0.770804i	$0.280145\pi$
$458$	4.67822	0.218599
$459$	−5.64077	−0.263289
$460$	−8.69719	−0.405509
$461$	−21.6542	−1.00853	−0.504267	−	0.863548i	$-0.668237\pi$
$461$	−21.6542	−1.00853	−0.504267	+	0.863548i	$0.668237\pi$
$462$	0	0
$463$	−32.2708	−1.49975	−0.749876	−	0.661579i	$-0.769887\pi$
$463$	−32.2708	−1.49975	−0.749876	+	0.661579i	$0.769887\pi$
$464$	2.53543	0.117705
$465$	−12.3820	−0.574203
$466$	17.8669	0.827669
$467$	−26.4552	−1.22420	−0.612101	−	0.790780i	$-0.709675\pi$
$467$	−26.4552	−1.22420	−0.612101	+	0.790780i	$0.709675\pi$
$468$	−4.87561	−0.225375
$469$	0	0
$470$	9.03880	0.416929
$471$	−8.10983	−0.373681
$472$	−8.41721	−0.387433
$473$	−1.62531	−0.0747319
$474$	9.05592	0.415952
$475$	−2.23113	−0.102371
$476$	0	0
$477$	0.128176	0.00586876
$478$	0.864416	0.0395375
$479$	−4.15252	−0.189733	−0.0948667	−	0.995490i	$-0.530242\pi$
$479$	−4.15252	−0.189733	−0.0948667	+	0.995490i	$0.530242\pi$
$480$	1.35219	0.0617186
$481$	−0.430825	−0.0196439
$482$	−4.74645	−0.216195
$483$	0	0
$484$	−2.35958	−0.107254
$485$	−0.714370	−0.0324379
$486$	−11.3613	−0.515359
$487$	30.4531	1.37996	0.689980	−	0.723828i	$-0.257619\pi$
$487$	30.4531	1.37996	0.689980	+	0.723828i	$0.257619\pi$
$488$	6.82353	0.308887
$489$	0.114980	0.00519956
$490$	0	0
$491$	−22.0284	−0.994128	−0.497064	−	0.867714i	$-0.665589\pi$
$491$	−22.0284	−0.994128	−0.497064	+	0.867714i	$0.665589\pi$
$492$	6.74860	0.304250
$493$	−2.53543	−0.114190
$494$	−9.28491	−0.417748
$495$	−3.44385	−0.154789
$496$	−9.15704	−0.411163
$497$	0	0
$498$	13.6030	0.609567
$499$	2.45247	0.109788	0.0548939	−	0.998492i	$-0.482518\pi$
$499$	2.45247	0.109788	0.0548939	+	0.998492i	$0.482518\pi$
$500$	−1.00000	−0.0447214
$501$	−5.22375	−0.233380
$502$	3.45527	0.154216
$503$	−38.2500	−1.70548	−0.852742	−	0.522332i	$-0.825062\pi$
$503$	−38.2500	−1.70548	−0.852742	+	0.522332i	$0.825062\pi$
$504$	0	0
$505$	−3.36397	−0.149695
$506$	−25.5650	−1.13650
$507$	−5.83921	−0.259328
$508$	15.3160	0.679536
$509$	22.4482	0.995001	0.497500	−	0.867464i	$-0.334251\pi$
$509$	22.4482	0.995001	0.497500	+	0.867464i	$0.334251\pi$
$510$	−1.35219	−0.0598758
$511$	0	0
$512$	1.00000	0.0441942
$513$	−12.5853	−0.555654
$514$	27.2139	1.20036
$515$	−9.40703	−0.414523
$516$	−0.747663	−0.0329140
$517$	26.5692	1.16851
$518$	0	0
$519$	−22.8158	−1.00150
$520$	−4.16153	−0.182495
$521$	25.3855	1.11216	0.556079	−	0.831130i	$-0.312305\pi$
$521$	25.3855	1.11216	0.556079	+	0.831130i	$0.312305\pi$
$522$	−2.97049	−0.130015
$523$	43.9818	1.92319	0.961594	−	0.274474i	$-0.0885038\pi$
$523$	43.9818	1.92319	0.961594	+	0.274474i	$0.0885038\pi$
$524$	−4.22974	−0.184777
$525$	0	0
$526$	13.1007	0.571220
$527$	9.15704	0.398887
$528$	3.97470	0.172976
$529$	52.6412	2.28875
$530$	0.109403	0.00475217
$531$	9.86153	0.427954
$532$	0	0
$533$	−20.7697	−0.899636
$534$	2.42337	0.104869
$535$	−17.8174	−0.770315
$536$	3.16632	0.136764
$537$	1.91319	0.0825601
$538$	−10.7385	−0.462971
$539$	0	0
$540$	−5.64077	−0.242740
$541$	−25.7282	−1.10614	−0.553070	−	0.833135i	$-0.686544\pi$
$541$	−25.7282	−1.10614	−0.553070	+	0.833135i	$0.686544\pi$
$542$	7.72677	0.331893
$543$	2.28362	0.0979997
$544$	−1.00000	−0.0428746
$545$	−6.96113	−0.298182
$546$	0	0
$547$	11.1565	0.477019	0.238509	−	0.971140i	$-0.423341\pi$
$547$	11.1565	0.477019	0.238509	+	0.971140i	$0.423341\pi$
$548$	−6.95264	−0.297002
$549$	−7.99439	−0.341192
$550$	−2.93946	−0.125339
$551$	−5.65688	−0.240991
$552$	−11.7602	−0.500549
$553$	0	0
$554$	−19.2239	−0.816744
$555$	−0.139986	−0.00594207
$556$	12.6339	0.535795
$557$	27.5906	1.16905	0.584525	−	0.811375i	$-0.301281\pi$
$557$	27.5906	1.16905	0.584525	+	0.811375i	$0.301281\pi$
$558$	10.7283	0.454166
$559$	2.30103	0.0973233
$560$	0	0
$561$	−3.97470	−0.167812
$562$	−6.45650	−0.272351
$563$	−24.1375	−1.01728	−0.508638	−	0.860981i	$-0.669851\pi$
$563$	−24.1375	−1.01728	−0.508638	+	0.860981i	$0.669851\pi$
$564$	12.2221	0.514645
$565$	−7.97609	−0.335557
$566$	−9.15294	−0.384727
$567$	0	0
$568$	1.49253	0.0626253
$569$	−33.1126	−1.38815	−0.694076	−	0.719902i	$-0.744187\pi$
$569$	−33.1126	−1.38815	−0.694076	+	0.719902i	$0.744187\pi$
$570$	−3.01690	−0.126364
$571$	−8.53502	−0.357179	−0.178590	−	0.983924i	$-0.557153\pi$
$571$	−8.53502	−0.357179	−0.178590	+	0.983924i	$0.557153\pi$
$572$	−12.2327	−0.511473
$573$	1.22680	0.0512502
$574$	0	0
$575$	8.69719	0.362698
$576$	−1.17159	−0.0488163
$577$	14.8054	0.616355	0.308178	−	0.951329i	$-0.400281\pi$
$577$	14.8054	0.616355	0.308178	+	0.951329i	$0.400281\pi$
$578$	1.00000	0.0415945
$579$	−28.9187	−1.20182
$580$	−2.53543	−0.105278
$581$	0	0
$582$	−0.965962	−0.0400404
$583$	0.321586	0.0133187
$584$	6.19201	0.256227
$585$	4.87561	0.201582
$586$	−4.17718	−0.172558
$587$	−16.0268	−0.661495	−0.330747	−	0.943719i	$-0.607301\pi$
$587$	−16.0268	−0.661495	−0.330747	+	0.943719i	$0.607301\pi$
$588$	0	0
$589$	20.4305	0.841826
$590$	8.41721	0.346531
$591$	−28.5169	−1.17303
$592$	−0.103526	−0.00425488
$593$	8.81573	0.362019	0.181009	−	0.983481i	$-0.442064\pi$
$593$	8.81573	0.362019	0.181009	+	0.983481i	$0.442064\pi$
$594$	−16.5808	−0.680319
$595$	0	0
$596$	4.41845	0.180987
$597$	19.1986	0.785747
$598$	36.1936	1.48007
$599$	28.9323	1.18214	0.591070	−	0.806620i	$-0.298706\pi$
$599$	28.9323	1.18214	0.591070	+	0.806620i	$0.298706\pi$
$600$	−1.35219	−0.0552028
$601$	−33.6810	−1.37388	−0.686939	−	0.726715i	$-0.741046\pi$
$601$	−33.6810	−1.37388	−0.686939	+	0.726715i	$0.741046\pi$
$602$	0	0
$603$	−3.70963	−0.151068
$604$	6.56833	0.267261
$605$	2.35958	0.0959306
$606$	−4.54872	−0.184779
$607$	18.5475	0.752821	0.376411	−	0.926453i	$-0.377158\pi$
$607$	18.5475	0.752821	0.376411	+	0.926453i	$0.377158\pi$
$608$	−2.23113	−0.0904842
$609$	0	0
$610$	−6.82353	−0.276277
$611$	−37.6153	−1.52175
$612$	1.17159	0.0473588
$613$	7.12473	0.287765	0.143883	−	0.989595i	$-0.454041\pi$
$613$	7.12473	0.287765	0.143883	+	0.989595i	$0.454041\pi$
$614$	−31.3778	−1.26630
$615$	−6.74860	−0.272130
$616$	0	0
$617$	−8.50005	−0.342199	−0.171100	−	0.985254i	$-0.554732\pi$
$617$	−8.50005	−0.342199	−0.171100	+	0.985254i	$0.554732\pi$
$618$	−12.7201	−0.511676
$619$	48.5640	1.95195	0.975977	−	0.217875i	$-0.0699124\pi$
$619$	48.5640	1.95195	0.975977	+	0.217875i	$0.0699124\pi$
$620$	9.15704	0.367756
$621$	49.0589	1.96866
$622$	−0.559867	−0.0224486
$623$	0	0
$624$	−5.62717	−0.225267
$625$	1.00000	0.0400000
$626$	−0.824866	−0.0329683
$627$	−8.86806	−0.354156
$628$	5.99757	0.239329
$629$	0.103526	0.00412784
$630$	0	0
$631$	29.5171	1.17506	0.587529	−	0.809203i	$-0.300101\pi$
$631$	29.5171	1.17506	0.587529	+	0.809203i	$0.300101\pi$
$632$	−6.69724	−0.266402
$633$	−35.3472	−1.40492
$634$	−9.13508	−0.362800
$635$	−15.3160	−0.607796
$636$	0.147933	0.00586594
$637$	0	0
$638$	−7.45280	−0.295059
$639$	−1.74864	−0.0691751
$640$	−1.00000	−0.0395285
$641$	35.8060	1.41425	0.707126	−	0.707088i	$-0.249992\pi$
$641$	35.8060	1.41425	0.707126	+	0.707088i	$0.249992\pi$
$642$	−24.0925	−0.950855
$643$	27.2628	1.07514	0.537570	−	0.843219i	$-0.319342\pi$
$643$	27.2628	1.07514	0.537570	+	0.843219i	$0.319342\pi$
$644$	0	0
$645$	0.747663	0.0294392
$646$	2.23113	0.0877826
$647$	−28.3402	−1.11417	−0.557084	−	0.830456i	$-0.688080\pi$
$647$	−28.3402	−1.11417	−0.557084	+	0.830456i	$0.688080\pi$
$648$	−4.11260	−0.161558
$649$	24.7420	0.971210
$650$	4.16153	0.163229
$651$	0	0
$652$	−0.0850324	−0.00333013
$653$	−24.3044	−0.951103	−0.475551	−	0.879688i	$-0.657751\pi$
$653$	−24.3044	−0.951103	−0.475551	+	0.879688i	$0.657751\pi$
$654$	−9.41275	−0.368068
$655$	4.22974	0.165270
$656$	−4.99088	−0.194861
$657$	−7.25450	−0.283025
$658$	0	0
$659$	1.03631	0.0403691	0.0201845	−	0.999796i	$-0.493575\pi$
$659$	1.03631	0.0403691	0.0201845	+	0.999796i	$0.493575\pi$
$660$	−3.97470	−0.154715
$661$	−34.7225	−1.35055	−0.675274	−	0.737567i	$-0.735975\pi$
$661$	−34.7225	−1.35055	−0.675274	+	0.737567i	$0.735975\pi$
$662$	−9.48144	−0.368506
$663$	5.62717	0.218541
$664$	−10.0600	−0.390405
$665$	0	0
$666$	0.121290	0.00469988
$667$	22.0511	0.853824
$668$	3.86319	0.149471
$669$	−22.9319	−0.886600
$670$	−3.16632	−0.122326
$671$	−20.0575	−0.774311
$672$	0	0
$673$	41.7943	1.61105	0.805526	−	0.592560i	$-0.201883\pi$
$673$	41.7943	1.61105	0.805526	+	0.592560i	$0.201883\pi$
$674$	−14.7377	−0.567673
$675$	5.64077	0.217113
$676$	4.31834	0.166090
$677$	10.9366	0.420329	0.210165	−	0.977666i	$-0.432600\pi$
$677$	10.9366	0.420329	0.210165	+	0.977666i	$0.432600\pi$
$678$	−10.7852	−0.414202
$679$	0	0
$680$	1.00000	0.0383482
$681$	−25.5319	−0.978385
$682$	26.9167	1.03070
$683$	−38.1855	−1.46113	−0.730564	−	0.682844i	$-0.760743\pi$
$683$	−38.1855	−1.46113	−0.730564	+	0.682844i	$0.760743\pi$
$684$	2.61397	0.0999477
$685$	6.95264	0.265647
$686$	0	0
$687$	−6.32583	−0.241345
$688$	0.552929	0.0210802
$689$	−0.455284	−0.0173450
$690$	11.7602	0.447704
$691$	−31.0174	−1.17996	−0.589979	−	0.807419i	$-0.700864\pi$
$691$	−31.0174	−1.17996	−0.589979	+	0.807419i	$0.700864\pi$
$692$	16.8732	0.641424
$693$	0	0
$694$	−19.7544	−0.749867
$695$	−12.6339	−0.479229
$696$	−3.42838	−0.129952
$697$	4.99088	0.189043
$698$	25.3828	0.960755
$699$	−24.1594	−0.913793
$700$	0	0
$701$	−9.46262	−0.357398	−0.178699	−	0.983904i	$-0.557189\pi$
$701$	−9.46262	−0.357398	−0.178699	+	0.983904i	$0.557189\pi$
$702$	23.4742	0.885978
$703$	0.230979	0.00871153
$704$	−2.93946	−0.110785
$705$	−12.2221	−0.460313
$706$	10.1367	0.381500
$707$	0	0
$708$	11.3816	0.427748
$709$	−19.7351	−0.741168	−0.370584	−	0.928799i	$-0.620842\pi$
$709$	−19.7351	−0.741168	−0.370584	+	0.928799i	$0.620842\pi$
$710$	−1.49253	−0.0560138
$711$	7.84643	0.294264
$712$	−1.79218	−0.0671649
$713$	−79.6406	−2.98256
$714$	0	0
$715$	12.2327	0.457475
$716$	−1.41488	−0.0528767
$717$	−1.16885	−0.0436516
$718$	−13.8984	−0.518682
$719$	15.4983	0.577987	0.288994	−	0.957331i	$-0.406679\pi$
$719$	15.4983	0.577987	0.288994	+	0.957331i	$0.406679\pi$
$720$	1.17159	0.0436626
$721$	0	0
$722$	−14.0221	−0.521847
$723$	6.41809	0.238691
$724$	−1.68884	−0.0627652
$725$	2.53543	0.0941636
$726$	3.19059	0.118414
$727$	−20.3247	−0.753801	−0.376900	−	0.926254i	$-0.623010\pi$
$727$	−20.3247	−0.753801	−0.376900	+	0.926254i	$0.623010\pi$
$728$	0	0
$729$	27.7004	1.02594
$730$	−6.19201	−0.229176
$731$	−0.552929	−0.0204508
$732$	−9.22669	−0.341028
$733$	−3.61452	−0.133505	−0.0667527	−	0.997770i	$-0.521264\pi$
$733$	−3.61452	−0.133505	−0.0667527	+	0.997770i	$0.521264\pi$
$734$	9.00621	0.332425
$735$	0	0
$736$	8.69719	0.320583
$737$	−9.30727	−0.342838
$738$	5.84727	0.215241
$739$	30.2144	1.11145	0.555727	−	0.831365i	$-0.312440\pi$
$739$	30.2144	1.11145	0.555727	+	0.831365i	$0.312440\pi$
$740$	0.103526	0.00380568
$741$	12.5549	0.461217
$742$	0	0
$743$	−31.4838	−1.15503	−0.577515	−	0.816380i	$-0.695978\pi$
$743$	−31.4838	−1.15503	−0.577515	+	0.816380i	$0.695978\pi$
$744$	12.3820	0.453947
$745$	−4.41845	−0.161880
$746$	−30.5019	−1.11675
$747$	11.7862	0.431236
$748$	2.93946	0.107477
$749$	0	0
$750$	1.35219	0.0493749
$751$	−6.25998	−0.228430	−0.114215	−	0.993456i	$-0.536435\pi$
$751$	−6.25998	−0.228430	−0.114215	+	0.993456i	$0.536435\pi$
$752$	−9.03880	−0.329611
$753$	−4.67218	−0.170264
$754$	10.5513	0.384255
$755$	−6.56833	−0.239046
$756$	0	0
$757$	37.4269	1.36030	0.680151	−	0.733072i	$-0.261914\pi$
$757$	37.4269	1.36030	0.680151	+	0.733072i	$0.261914\pi$
$758$	29.4384	1.06925
$759$	34.5687	1.25476
$760$	2.23113	0.0809315
$761$	13.9313	0.505009	0.252505	−	0.967596i	$-0.418746\pi$
$761$	13.9313	0.505009	0.252505	+	0.967596i	$0.418746\pi$
$762$	−20.7101	−0.750246
$763$	0	0
$764$	−0.907269	−0.0328239
$765$	−1.17159	−0.0423590
$766$	19.6495	0.709966
$767$	−35.0285	−1.26480
$768$	−1.35219	−0.0487928
$769$	−14.8786	−0.536535	−0.268267	−	0.963344i	$-0.586451\pi$
$769$	−14.8786	−0.536535	−0.268267	+	0.963344i	$0.586451\pi$
$770$	0	0
$771$	−36.7983	−1.32526
$772$	21.3866	0.769722
$773$	−16.1961	−0.582532	−0.291266	−	0.956642i	$-0.594076\pi$
$773$	−16.1961	−0.582532	−0.291266	+	0.956642i	$0.594076\pi$
$774$	−0.647807	−0.0232849
$775$	−9.15704	−0.328931
$776$	0.714370	0.0256444
$777$	0	0
$778$	−6.69989	−0.240203
$779$	11.1353	0.398963
$780$	5.62717	0.201485
$781$	−4.38724	−0.156988
$782$	−8.69719	−0.311011
$783$	14.3018	0.511104
$784$	0	0
$785$	−5.99757	−0.214062
$786$	5.71940	0.204004
$787$	−29.6891	−1.05830	−0.529151	−	0.848528i	$-0.677490\pi$
$787$	−29.6891	−1.05830	−0.529151	+	0.848528i	$0.677490\pi$
$788$	21.0895	0.751281
$789$	−17.7147	−0.630658
$790$	6.69724	0.238277
$791$	0	0
$792$	3.44385	0.122372
$793$	28.3963	1.00838
$794$	35.0846	1.24511
$795$	−0.147933	−0.00524666
$796$	−14.1982	−0.503242
$797$	28.7915	1.01985	0.509924	−	0.860219i	$-0.329673\pi$
$797$	28.7915	1.01985	0.509924	+	0.860219i	$0.329673\pi$
$798$	0	0
$799$	9.03880	0.319770
$800$	1.00000	0.0353553
$801$	2.09971	0.0741895
$802$	24.5806	0.867972
$803$	−18.2012	−0.642305
$804$	−4.28146	−0.150995
$805$	0	0
$806$	−38.1073	−1.34227
$807$	14.5205	0.511146
$808$	3.36397	0.118344
$809$	44.9627	1.58080	0.790402	−	0.612589i	$-0.209872\pi$
$809$	44.9627	1.58080	0.790402	+	0.612589i	$0.209872\pi$
$810$	4.11260	0.144502
$811$	25.5372	0.896733	0.448367	−	0.893850i	$-0.352006\pi$
$811$	25.5372	0.896733	0.448367	+	0.893850i	$0.352006\pi$
$812$	0	0
$813$	−10.4480	−0.366429
$814$	0.304309	0.0106660
$815$	0.0850324	0.00297855
$816$	1.35219	0.0473360
$817$	−1.23366	−0.0431601
$818$	−5.85978	−0.204882
$819$	0	0
$820$	4.99088	0.174289
$821$	−39.4365	−1.37634	−0.688172	−	0.725548i	$-0.741586\pi$
$821$	−39.4365	−1.37634	−0.688172	+	0.725548i	$0.741586\pi$
$822$	9.40127	0.327907
$823$	−29.9492	−1.04396	−0.521981	−	0.852957i	$-0.674807\pi$
$823$	−29.9492	−1.04396	−0.521981	+	0.852957i	$0.674807\pi$
$824$	9.40703	0.327710
$825$	3.97470	0.138381
$826$	0	0
$827$	41.9013	1.45705	0.728525	−	0.685019i	$-0.240206\pi$
$827$	41.9013	1.45705	0.728525	+	0.685019i	$0.240206\pi$
$828$	−10.1896	−0.354112
$829$	33.4094	1.16036	0.580178	−	0.814490i	$-0.302983\pi$
$829$	33.4094	1.16036	0.580178	+	0.814490i	$0.302983\pi$
$830$	10.0600	0.349189
$831$	25.9942	0.901730
$832$	4.16153	0.144275
$833$	0	0
$834$	−17.0833	−0.591547
$835$	−3.86319	−0.133691
$836$	6.55831	0.226824
$837$	−51.6528	−1.78538
$838$	37.4041	1.29210
$839$	−13.4764	−0.465255	−0.232628	−	0.972566i	$-0.574732\pi$
$839$	−13.4764	−0.465255	−0.232628	+	0.972566i	$0.574732\pi$
$840$	0	0
$841$	−22.5716	−0.778330
$842$	17.3809	0.598984
$843$	8.73039	0.300690
$844$	26.1407	0.899802
$845$	−4.31834	−0.148556
$846$	10.5898	0.364084
$847$	0	0
$848$	−0.109403	−0.00375692
$849$	12.3765	0.424760
$850$	−1.00000	−0.0342997
$851$	−0.900382	−0.0308647
$852$	−2.01818	−0.0691418
$853$	−20.4939	−0.701698	−0.350849	−	0.936432i	$-0.614107\pi$
$853$	−20.4939	−0.701698	−0.350849	+	0.936432i	$0.614107\pi$
$854$	0	0
$855$	−2.61397	−0.0893959
$856$	17.8174	0.608987
$857$	−11.4675	−0.391722	−0.195861	−	0.980632i	$-0.562750\pi$
$857$	−11.4675	−0.391722	−0.195861	+	0.980632i	$0.562750\pi$
$858$	16.5408	0.564695
$859$	39.9145	1.36187	0.680933	−	0.732346i	$-0.261574\pi$
$859$	39.9145	1.36187	0.680933	+	0.732346i	$0.261574\pi$
$860$	−0.552929	−0.0188547
$861$	0	0
$862$	−16.4023	−0.558664
$863$	9.03946	0.307707	0.153853	−	0.988094i	$-0.450832\pi$
$863$	9.03946	0.307707	0.153853	+	0.988094i	$0.450832\pi$
$864$	5.64077	0.191903
$865$	−16.8732	−0.573707
$866$	22.7926	0.774525
$867$	−1.35219	−0.0459227
$868$	0	0
$869$	19.6863	0.667811
$870$	3.42838	0.116233
$871$	13.1767	0.446477
$872$	6.96113	0.235734
$873$	−0.836950	−0.0283265
$874$	−19.4046	−0.656369
$875$	0	0
$876$	−8.37275	−0.282889
$877$	16.5623	0.559269	0.279634	−	0.960107i	$-0.409787\pi$
$877$	16.5623	0.559269	0.279634	+	0.960107i	$0.409787\pi$
$878$	21.0246	0.709546
$879$	5.64833	0.190513
$880$	2.93946	0.0990891
$881$	−23.7542	−0.800298	−0.400149	−	0.916450i	$-0.631042\pi$
$881$	−23.7542	−0.800298	−0.400149	+	0.916450i	$0.631042\pi$
$882$	0	0
$883$	−35.2191	−1.18522	−0.592609	−	0.805490i	$-0.701902\pi$
$883$	−35.2191	−1.18522	−0.592609	+	0.805490i	$0.701902\pi$
$884$	−4.16153	−0.139967
$885$	−11.3816	−0.382589
$886$	−36.5497	−1.22791
$887$	−38.4622	−1.29143	−0.645717	−	0.763576i	$-0.723442\pi$
$887$	−38.4622	−1.29143	−0.645717	+	0.763576i	$0.723442\pi$
$888$	0.139986	0.00469762
$889$	0	0
$890$	1.79218	0.0600741
$891$	12.0888	0.404991
$892$	16.9592	0.567834
$893$	20.1667	0.674854
$894$	−5.97457	−0.199820
$895$	1.41488	0.0472944
$896$	0	0
$897$	−48.9406	−1.63408
$898$	33.2693	1.11021
$899$	−23.2171	−0.774332
$900$	−1.17159	−0.0390530
$901$	0.109403	0.00364474
$902$	14.6705	0.488474
$903$	0	0
$904$	7.97609	0.265281
$905$	1.68884	0.0561389
$906$	−8.88160	−0.295072
$907$	−28.3655	−0.941861	−0.470930	−	0.882170i	$-0.656082\pi$
$907$	−28.3655	−0.941861	−0.470930	+	0.882170i	$0.656082\pi$
$908$	18.8819	0.626619
$909$	−3.94120	−0.130721
$910$	0	0
$911$	12.4552	0.412659	0.206329	−	0.978483i	$-0.433848\pi$
$911$	12.4552	0.412659	0.206329	+	0.978483i	$0.433848\pi$
$912$	3.01690	0.0998996
$913$	29.5710	0.978659
$914$	27.2381	0.900957
$915$	9.22669	0.305025
$916$	4.67822	0.154573
$917$	0	0
$918$	−5.64077	−0.186173
$919$	3.91129	0.129022	0.0645108	−	0.997917i	$-0.479451\pi$
$919$	3.91129	0.129022	0.0645108	+	0.997917i	$0.479451\pi$
$920$	−8.69719	−0.286738
$921$	42.4286	1.39807
$922$	−21.6542	−0.713142
$923$	6.21122	0.204445
$924$	0	0
$925$	−0.103526	−0.00340390
$926$	−32.2708	−1.06048
$927$	−11.0212	−0.361984
$928$	2.53543	0.0832297
$929$	55.0856	1.80730	0.903650	−	0.428272i	$-0.140877\pi$
$929$	55.0856	1.80730	0.903650	+	0.428272i	$0.140877\pi$
$930$	−12.3820	−0.406023
$931$	0	0
$932$	17.8669	0.585250
$933$	0.757045	0.0247845
$934$	−26.4552	−0.865642
$935$	−2.93946	−0.0961306
$936$	−4.87561	−0.159364
$937$	−7.00856	−0.228960	−0.114480	−	0.993426i	$-0.536520\pi$
$937$	−7.00856	−0.228960	−0.114480	+	0.993426i	$0.536520\pi$
$938$	0	0
$939$	1.11537	0.0363988
$940$	9.03880	0.294813
$941$	37.4110	1.21957	0.609783	−	0.792569i	$-0.291257\pi$
$941$	37.4110	1.21957	0.609783	+	0.792569i	$0.291257\pi$
$942$	−8.10983	−0.264233
$943$	−43.4067	−1.41352
$944$	−8.41721	−0.273957
$945$	0	0
$946$	−1.62531	−0.0528435
$947$	−7.49182	−0.243452	−0.121726	−	0.992564i	$-0.538843\pi$
$947$	−7.49182	−0.243452	−0.121726	+	0.992564i	$0.538843\pi$
$948$	9.05592	0.294122
$949$	25.7682	0.836472
$950$	−2.23113	−0.0723874
$951$	12.3523	0.400552
$952$	0	0
$953$	4.19859	0.136006	0.0680028	−	0.997685i	$-0.478337\pi$
$953$	4.19859	0.136006	0.0680028	+	0.997685i	$0.478337\pi$
$954$	0.128176	0.00414984
$955$	0.907269	0.0293585
$956$	0.864416	0.0279572
$957$	10.0776	0.325762
$958$	−4.15252	−0.134162
$959$	0	0
$960$	1.35219	0.0436416
$961$	52.8514	1.70488
$962$	−0.430825	−0.0138904
$963$	−20.8747	−0.672679
$964$	−4.74645	−0.152873
$965$	−21.3866	−0.688460
$966$	0	0
$967$	−28.3977	−0.913208	−0.456604	−	0.889670i	$-0.650934\pi$
$967$	−28.3977	−0.913208	−0.456604	+	0.889670i	$0.650934\pi$
$968$	−2.35958	−0.0758398
$969$	−3.01690	−0.0969169
$970$	−0.714370	−0.0229370
$971$	−25.1233	−0.806245	−0.403123	−	0.915146i	$-0.632075\pi$
$971$	−25.1233	−0.806245	−0.403123	+	0.915146i	$0.632075\pi$
$972$	−11.3613	−0.364414
$973$	0	0
$974$	30.4531	0.975779
$975$	−5.62717	−0.180214
$976$	6.82353	0.218416
$977$	−28.8029	−0.921485	−0.460743	−	0.887534i	$-0.652417\pi$
$977$	−28.8029	−0.921485	−0.460743	+	0.887534i	$0.652417\pi$
$978$	0.114980	0.00367664
$979$	5.26805	0.168368
$980$	0	0
$981$	−8.15560	−0.260388
$982$	−22.0284	−0.702955
$983$	−47.1719	−1.50455	−0.752275	−	0.658850i	$-0.771043\pi$
$983$	−47.1719	−1.50455	−0.752275	+	0.658850i	$0.771043\pi$
$984$	6.74860	0.215138
$985$	−21.0895	−0.671967
$986$	−2.53543	−0.0807446
$987$	0	0
$988$	−9.28491	−0.295392
$989$	4.80893	0.152915
$990$	−3.44385	−0.109453
$991$	8.38756	0.266440	0.133220	−	0.991087i	$-0.457468\pi$
$991$	8.38756	0.266440	0.133220	+	0.991087i	$0.457468\pi$
$992$	−9.15704	−0.290736
$993$	12.8207	0.406852
$994$	0	0
$995$	14.1982	0.450113
$996$	13.6030	0.431029
$997$	−20.6155	−0.652901	−0.326450	−	0.945214i	$-0.605853\pi$
$997$	−20.6155	−0.652901	−0.326450	+	0.945214i	$0.605853\pi$
$998$	2.45247	0.0776317
$999$	−0.583964	−0.0184758

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8330.2.a.cv.1.4	✓	10
7.6	odd	2		8330.2.a.cw.1.7	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8330.2.a.cv.1.4	✓	10	1.1	even	1	trivial
8330.2.a.cw.1.7	yes	10	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8330 = 2 \cdot 5 \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8330.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.35219 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.35219 q^{6} +1.00000 q^{8} -1.17159 q^{9} -1.00000 q^{10} -2.93946 q^{11} -1.35219 q^{12} +4.16153 q^{13} +1.35219 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.17159 q^{18} -2.23113 q^{19} -1.00000 q^{20} -2.93946 q^{22} +8.69719 q^{23} -1.35219 q^{24} +1.00000 q^{25} +4.16153 q^{26} +5.64077 q^{27} +2.53543 q^{29} +1.35219 q^{30} -9.15704 q^{31} +1.00000 q^{32} +3.97470 q^{33} -1.00000 q^{34} -1.17159 q^{36} -0.103526 q^{37} -2.23113 q^{38} -5.62717 q^{39} -1.00000 q^{40} -4.99088 q^{41} +0.552929 q^{43} -2.93946 q^{44} +1.17159 q^{45} +8.69719 q^{46} -9.03880 q^{47} -1.35219 q^{48} +1.00000 q^{50} +1.35219 q^{51} +4.16153 q^{52} -0.109403 q^{53} +5.64077 q^{54} +2.93946 q^{55} +3.01690 q^{57} +2.53543 q^{58} -8.41721 q^{59} +1.35219 q^{60} +6.82353 q^{61} -9.15704 q^{62} +1.00000 q^{64} -4.16153 q^{65} +3.97470 q^{66} +3.16632 q^{67} -1.00000 q^{68} -11.7602 q^{69} +1.49253 q^{71} -1.17159 q^{72} +6.19201 q^{73} -0.103526 q^{74} -1.35219 q^{75} -2.23113 q^{76} -5.62717 q^{78} -6.69724 q^{79} -1.00000 q^{80} -4.11260 q^{81} -4.99088 q^{82} -10.0600 q^{83} +1.00000 q^{85} +0.552929 q^{86} -3.42838 q^{87} -2.93946 q^{88} -1.79218 q^{89} +1.17159 q^{90} +8.69719 q^{92} +12.3820 q^{93} -9.03880 q^{94} +2.23113 q^{95} -1.35219 q^{96} +0.714370 q^{97} +3.44385 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{2} - 2 q^{3} + 10 q^{4} - 10 q^{5} - 2 q^{6} + 10 q^{8} + 12 q^{9} - 10 q^{10} - 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{15} + 10 q^{16} - 10 q^{17} + 12 q^{18} - 14 q^{19} - 10 q^{20} - 2 q^{22}+ \cdots - 12 q^{99}+O(q^{100}) \) 10 * q + 10 * q^2 - 2 * q^3 + 10 * q^4 - 10 * q^5 - 2 * q^6 + 10 * q^8 + 12 * q^9 - 10 * q^10 - 2 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^15 + 10 * q^16 - 10 * q^17 + 12 * q^18 - 14 * q^19 - 10 * q^20 - 2 * q^22 + 10 * q^23 - 2 * q^24 + 10 * q^25 - 4 * q^26 + 4 * q^27 + 10 * q^29 + 2 * q^30 - 2 * q^31 + 10 * q^32 - 4 * q^33 - 10 * q^34 + 12 * q^36 - 14 * q^38 + 48 * q^39 - 10 * q^40 + 22 * q^41 + 14 * q^43 - 2 * q^44 - 12 * q^45 + 10 * q^46 - 18 * q^47 - 2 * q^48 + 10 * q^50 + 2 * q^51 - 4 * q^52 + 22 * q^53 + 4 * q^54 + 2 * q^55 + 32 * q^57 + 10 * q^58 + 6 * q^59 + 2 * q^60 + 16 * q^61 - 2 * q^62 + 10 * q^64 + 4 * q^65 - 4 * q^66 + 14 * q^67 - 10 * q^68 - 10 * q^69 + 20 * q^71 + 12 * q^72 - 12 * q^73 - 2 * q^75 - 14 * q^76 + 48 * q^78 + 44 * q^79 - 10 * q^80 + 50 * q^81 + 22 * q^82 - 20 * q^83 + 10 * q^85 + 14 * q^86 - 6 * q^87 - 2 * q^88 - 12 * q^90 + 10 * q^92 + 4 * q^93 - 18 * q^94 + 14 * q^95 - 2 * q^96 + 40 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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