LMFDB - Embedded newform 8330.2.a.cw.1.7

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			1.7
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	$-0.372357\pi$
$3$	1.35219	0.780685	0.390343	+	0.920670i	$0.372357\pi$
$4$	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	$-0.695816\pi$
$13$	−4.16153	−1.15420	−0.577101	+	0.816673i	$0.695816\pi$
$14$	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	$-0.417619\pi$
$19$	2.23113	0.511856	0.255928	+	0.966696i	$0.417619\pi$
$20$	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	$-0.192676\pi$
$31$	9.15704	1.64465	0.822327	+	0.569016i	$0.192676\pi$
$32$	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	$-0.372571\pi$
$41$	4.99088	0.779445	0.389722	+	0.920932i	$0.372571\pi$
$42$	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	$-0.270886\pi$
$47$	9.03880	1.31844	0.659222	+	0.751948i	$0.270886\pi$
$48$	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	$-0.315422\pi$
$59$	8.41721	1.09583	0.547913	+	0.836535i	$0.315422\pi$
$60$	1.35219	0.174567
$61$	−6.82353	−0.873663	−0.436832	−	0.899543i	$-0.643899\pi$
$61$	−6.82353	−0.873663	−0.436832	+	0.899543i	$0.643899\pi$
$62$	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	$-0.618029\pi$
$73$	−6.19201	−0.724720	−0.362360	+	0.932038i	$0.618029\pi$
$74$	−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	$-0.313821\pi$
$83$	10.0600	1.10423	0.552116	+	0.833767i	$0.313821\pi$
$84$	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	$-0.469720\pi$
$89$	1.79218	0.189971	0.0949856	+	0.995479i	$0.469720\pi$
$90$	−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	$-0.511547\pi$
$97$	−0.714370	−0.0725333	−0.0362667	+	0.999342i	$0.511547\pi$
$98$	0	0
$99$	3.44385	0.346119
$100$	1.00000	0.100000
$101$	−3.36397	−0.334728	−0.167364	−	0.985895i	$-0.553525\pi$
$101$	−3.36397	−0.334728	−0.167364	+	0.985895i	$0.553525\pi$
$102$	1.35219	0.133886
$103$	−9.40703	−0.926902	−0.463451	−	0.886122i	$-0.653389\pi$
$103$	−9.40703	−0.926902	−0.463451	+	0.886122i	$0.653389\pi$
$104$	−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	$-0.440844\pi$
$131$	4.22974	0.369555	0.184777	+	0.982780i	$0.440844\pi$
$132$	−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	$-0.679988\pi$
$139$	−12.6339	−1.07159	−0.535795	+	0.844348i	$0.679988\pi$
$140$	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	$-0.576927\pi$
$157$	−5.99757	−0.478658	−0.239329	+	0.970939i	$0.576927\pi$
$158$	−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	$-0.547757\pi$
$167$	−3.86319	−0.298942	−0.149471	+	0.988766i	$0.547757\pi$
$168$	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	$-0.721656\pi$
$173$	−16.8732	−1.28285	−0.641424	+	0.767186i	$0.721656\pi$
$174$	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	$-0.480008\pi$
$181$	1.68884	0.125530	0.0627652	+	0.998028i	$0.480008\pi$
$182$	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	$-0.332140\pi$
$199$	14.1982	1.00648	0.503242	+	0.864146i	$0.332140\pi$
$200$	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	$-0.692219\pi$
$223$	−16.9592	−1.13567	−0.567834	+	0.823143i	$0.692219\pi$
$224$	0	0
$225$	−1.17159	−0.0781061
$226$	7.97609	0.530562
$227$	−18.8819	−1.25324	−0.626619	−	0.779325i	$-0.715562\pi$
$227$	−18.8819	−1.25324	−0.626619	+	0.779325i	$0.715562\pi$
$228$	3.01690	0.199799
$229$	−4.67822	−0.309146	−0.154573	−	0.987981i	$-0.549400\pi$
$229$	−4.67822	−0.309146	−0.154573	+	0.987981i	$0.549400\pi$
$230$	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	$-0.451148\pi$
$241$	4.74645	0.305746	0.152873	+	0.988246i	$0.451148\pi$
$242$	−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	$-0.534780\pi$
$251$	−3.45527	−0.218095	−0.109047	+	0.994037i	$0.534780\pi$
$252$	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	$-0.822662\pi$
$257$	−27.2139	−1.69756	−0.848779	+	0.528747i	$0.822662\pi$
$258$	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	$-0.393838\pi$
$269$	10.7385	0.654740	0.327370	+	0.944896i	$0.393838\pi$
$270$	−5.64077	−0.343286
$271$	−7.72677	−0.469368	−0.234684	−	0.972072i	$-0.575406\pi$
$271$	−7.72677	−0.469368	−0.234684	+	0.972072i	$0.575406\pi$
$272$	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	$-0.412301\pi$
$283$	9.15294	0.544086	0.272043	+	0.962285i	$0.412301\pi$
$284$	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	$-0.461064\pi$
$293$	4.17718	0.244033	0.122017	+	0.992528i	$0.461064\pi$
$294$	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	$-0.146881\pi$
$307$	31.3778	1.79082	0.895412	+	0.445239i	$0.146881\pi$
$308$	0	0
$309$	−12.7201	−0.723619
$310$	9.15704	0.520085
$311$	0.559867	0.0317472	0.0158736	−	0.999874i	$-0.494947\pi$
$311$	0.559867	0.0317472	0.0158736	+	0.999874i	$0.494947\pi$
$312$	−5.62717	−0.318576
$313$	0.824866	0.0466242	0.0233121	−	0.999728i	$-0.492579\pi$
$313$	0.824866	0.0466242	0.0233121	+	0.999728i	$0.492579\pi$
$314$	−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	$-0.737741\pi$
$349$	−25.3828	−1.35871	−0.679356	+	0.733808i	$0.737741\pi$
$350$	0	0
$351$	23.4742	1.25296
$352$	−2.93946	−0.156674
$353$	−10.1367	−0.539522	−0.269761	−	0.962927i	$-0.586945\pi$
$353$	−10.1367	−0.539522	−0.269761	+	0.962927i	$0.586945\pi$
$354$	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	$-0.575529\pi$
$367$	−9.00621	−0.470121	−0.235060	+	0.971981i	$0.575529\pi$
$368$	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	$-0.667410\pi$
$383$	−19.6495	−1.00404	−0.502022	+	0.864855i	$0.667410\pi$
$384$	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	$-0.842741\pi$
$397$	−35.0846	−1.76085	−0.880423	+	0.474189i	$0.842741\pi$
$398$	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	$-0.453722\pi$
$409$	5.85978	0.289748	0.144874	+	0.989450i	$0.453722\pi$
$410$	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	$-0.866753\pi$
$419$	−37.4041	−1.82731	−0.913656	+	0.406488i	$0.866753\pi$
$420$	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	$-0.684486\pi$
$433$	−22.7926	−1.09534	−0.547672	+	0.836693i	$0.684486\pi$
$434$	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	$-0.667301\pi$
$439$	−21.0246	−1.00345	−0.501724	+	0.865027i	$0.667301\pi$
$440$	−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	$-0.331763\pi$
$461$	21.6542	1.00853	0.504267	+	0.863548i	$0.331763\pi$
$462$	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	$-0.290325\pi$
$467$	26.4552	1.22420	0.612101	+	0.790780i	$0.290325\pi$
$468$	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	$-0.469758\pi$
$479$	4.15252	0.189733	0.0948667	+	0.995490i	$0.469758\pi$
$480$	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	$-0.174938\pi$
$503$	38.2500	1.70548	0.852742	+	0.522332i	$0.174938\pi$
$504$	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	$-0.665749\pi$
$509$	−22.4482	−0.995001	−0.497500	+	0.867464i	$0.665749\pi$
$510$	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	$-0.687695\pi$
$521$	−25.3855	−1.11216	−0.556079	+	0.831130i	$0.687695\pi$
$522$	−2.97049	−0.130015
$523$	−43.9818	−1.92319	−0.961594	−	0.274474i	$-0.911496\pi$
$523$	−43.9818	−1.92319	−0.961594	+	0.274474i	$0.911496\pi$
$524$	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	$-0.330149\pi$
$563$	24.1375	1.01728	0.508638	+	0.860981i	$0.330149\pi$
$564$	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	$-0.599719\pi$
$577$	−14.8054	−0.616355	−0.308178	+	0.951329i	$0.599719\pi$
$578$	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	$-0.392699\pi$
$587$	16.0268	0.661495	0.330747	+	0.943719i	$0.392699\pi$
$588$	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	$-0.557936\pi$
$593$	−8.81573	−0.362019	−0.181009	+	0.983481i	$0.557936\pi$
$594$	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	$-0.258954\pi$
$601$	33.6810	1.37388	0.686939	+	0.726715i	$0.258954\pi$
$602$	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	$-0.622842\pi$
$607$	−18.5475	−0.752821	−0.376411	+	0.926453i	$0.622842\pi$
$608$	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	$-0.930088\pi$
$619$	−48.5640	−1.95195	−0.975977	+	0.217875i	$0.930088\pi$
$620$	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	$-0.680658\pi$
$643$	−27.2628	−1.07514	−0.537570	+	0.843219i	$0.680658\pi$
$644$	0	0
$645$	0.747663	0.0294392
$646$	2.23113	0.0877826
$647$	28.3402	1.11417	0.557084	−	0.830456i	$-0.311920\pi$
$647$	28.3402	1.11417	0.557084	+	0.830456i	$0.311920\pi$
$648$	−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	$-0.264025\pi$
$661$	34.7225	1.35055	0.675274	+	0.737567i	$0.264025\pi$
$662$	−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	$-0.567400\pi$
$677$	−10.9366	−0.420329	−0.210165	+	0.977666i	$0.567400\pi$
$678$	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	$-0.299136\pi$
$691$	31.0174	1.17996	0.589979	+	0.807419i	$0.299136\pi$
$692$	−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	$-0.593321\pi$
$719$	−15.4983	−0.577987	−0.288994	+	0.957331i	$0.593321\pi$
$720$	−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	$-0.376990\pi$
$727$	20.3247	0.753801	0.376900	+	0.926254i	$0.376990\pi$
$728$	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	$-0.478736\pi$
$733$	3.61452	0.133505	0.0667527	+	0.997770i	$0.478736\pi$
$734$	−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	$-0.581254\pi$
$761$	−13.9313	−0.505009	−0.252505	+	0.967596i	$0.581254\pi$
$762$	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	$-0.413549\pi$
$769$	14.8786	0.536535	0.268267	+	0.963344i	$0.413549\pi$
$770$	0	0
$771$	−36.7983	−1.32526
$772$	21.3866	0.769722
$773$	16.1961	0.582532	0.291266	−	0.956642i	$-0.405924\pi$
$773$	16.1961	0.582532	0.291266	+	0.956642i	$0.405924\pi$
$774$	−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	$-0.322510\pi$
$787$	29.6891	1.05830	0.529151	+	0.848528i	$0.322510\pi$
$788$	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	$-0.670327\pi$
$797$	−28.7915	−1.01985	−0.509924	+	0.860219i	$0.670327\pi$
$798$	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	$-0.647994\pi$
$811$	−25.5372	−0.896733	−0.448367	+	0.893850i	$0.647994\pi$
$812$	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	$-0.697017\pi$
$829$	−33.4094	−1.16036	−0.580178	+	0.814490i	$0.697017\pi$
$830$	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	$-0.425268\pi$
$839$	13.4764	0.465255	0.232628	+	0.972566i	$0.425268\pi$
$840$	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	$-0.385893\pi$
$853$	20.4939	0.701698	0.350849	+	0.936432i	$0.385893\pi$
$854$	0	0
$855$	−2.61397	−0.0893959
$856$	17.8174	0.608987
$857$	11.4675	0.391722	0.195861	−	0.980632i	$-0.437250\pi$
$857$	11.4675	0.391722	0.195861	+	0.980632i	$0.437250\pi$
$858$	16.5408	0.564695
$859$	−39.9145	−1.36187	−0.680933	−	0.732346i	$-0.738426\pi$
$859$	−39.9145	−1.36187	−0.680933	+	0.732346i	$0.738426\pi$
$860$	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	$-0.368958\pi$
$881$	23.7542	0.800298	0.400149	+	0.916450i	$0.368958\pi$
$882$	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	$-0.276558\pi$
$887$	38.4622	1.29143	0.645717	+	0.763576i	$0.276558\pi$
$888$	−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	$-0.859123\pi$
$929$	−55.0856	−1.80730	−0.903650	+	0.428272i	$0.859123\pi$
$930$	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	$-0.463480\pi$
$937$	7.00856	0.228960	0.114480	+	0.993426i	$0.463480\pi$
$938$	0	0
$939$	1.11537	0.0363988
$940$	9.03880	0.294813
$941$	−37.4110	−1.21957	−0.609783	−	0.792569i	$-0.708743\pi$
$941$	−37.4110	−1.21957	−0.609783	+	0.792569i	$0.708743\pi$
$942$	−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	$-0.367925\pi$
$971$	25.1233	0.806245	0.403123	+	0.915146i	$0.367925\pi$
$972$	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	$-0.228957\pi$
$983$	47.1719	1.50455	0.752275	+	0.658850i	$0.228957\pi$
$984$	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	$-0.394147\pi$
$997$	20.6155	0.652901	0.326450	+	0.945214i	$0.394147\pi$
$998$	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.cw.1.7	yes	10
7.6	odd	2		8330.2.a.cv.1.4	✓	10

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

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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