LMFDB - Embedded newform 8330.2.a.cn.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 = [6,-6,-2,6,6,2,0,-6,4,-6,-6,-2,0] 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:	$1$
Dimension:	$6$
Coefficient field:	6.6.12694016.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 1 $ x^6 - 8x^4 - 2x^3 + 16x^2 + 8x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.94606$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} +0.787153 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.787153 q^{6} -1.00000 q^{8} -2.38039 q^{9} -1.00000 q^{10} -2.94606 q^{11} +0.787153 q^{12} -1.11320 q^{13} +0.787153 q^{15} +1.00000 q^{16} +1.00000 q^{17} +2.38039 q^{18} +5.69821 q^{19} +1.00000 q^{20} +2.94606 q^{22} -3.99344 q^{23} -0.787153 q^{24} +1.00000 q^{25} +1.11320 q^{26} -4.23519 q^{27} +10.2331 q^{29} -0.787153 q^{30} -2.49192 q^{31} -1.00000 q^{32} -2.31900 q^{33} -1.00000 q^{34} -2.38039 q^{36} +7.19846 q^{37} -5.69821 q^{38} -0.876260 q^{39} -1.00000 q^{40} -8.70353 q^{41} -11.2218 q^{43} -2.94606 q^{44} -2.38039 q^{45} +3.99344 q^{46} -2.16665 q^{47} +0.787153 q^{48} -1.00000 q^{50} +0.787153 q^{51} -1.11320 q^{52} +6.83241 q^{53} +4.23519 q^{54} -2.94606 q^{55} +4.48536 q^{57} -10.2331 q^{58} +8.47762 q^{59} +0.787153 q^{60} -9.45164 q^{61} +2.49192 q^{62} +1.00000 q^{64} -1.11320 q^{65} +2.31900 q^{66} +6.12953 q^{67} +1.00000 q^{68} -3.14345 q^{69} +3.13998 q^{71} +2.38039 q^{72} -1.95049 q^{73} -7.19846 q^{74} +0.787153 q^{75} +5.69821 q^{76} +0.876260 q^{78} -1.03366 q^{79} +1.00000 q^{80} +3.80743 q^{81} +8.70353 q^{82} -10.8662 q^{83} +1.00000 q^{85} +11.2218 q^{86} +8.05499 q^{87} +2.94606 q^{88} -8.62613 q^{89} +2.38039 q^{90} -3.99344 q^{92} -1.96152 q^{93} +2.16665 q^{94} +5.69821 q^{95} -0.787153 q^{96} +3.16732 q^{97} +7.01278 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} + 6 q^{5} + 2 q^{6} - 6 q^{8} + 4 q^{9} - 6 q^{10} - 6 q^{11} - 2 q^{12} - 2 q^{15} + 6 q^{16} + 6 q^{17} - 4 q^{18} + 6 q^{19} + 6 q^{20} + 6 q^{22} - 2 q^{23} + 2 q^{24}+ \cdots - 8 q^{99}+O(q^{100}) $ 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 + 6 * q^5 + 2 * q^6 - 6 * q^8 + 4 * q^9 - 6 * q^10 - 6 * q^11 - 2 * q^12 - 2 * q^15 + 6 * q^16 + 6 * q^17 - 4 * q^18 + 6 * q^19 + 6 * q^20 + 6 * q^22 - 2 * q^23 + 2 * q^24 + 6 * q^25 - 20 * q^27 - 2 * q^29 + 2 * q^30 - 2 * q^31 - 6 * q^32 - 4 * q^33 - 6 * q^34 + 4 * q^36 - 12 * q^37 - 6 * q^38 - 6 * q^40 + 6 * q^41 - 26 * q^43 - 6 * q^44 + 4 * q^45 + 2 * q^46 - 30 * q^47 - 2 * q^48 - 6 * q^50 - 2 * q^51 + 14 * q^53 + 20 * q^54 - 6 * q^55 - 8 * q^57 + 2 * q^58 - 6 * q^59 - 2 * q^60 + 12 * q^61 + 2 * q^62 + 6 * q^64 + 4 * q^66 - 14 * q^67 + 6 * q^68 - 2 * q^69 + 12 * q^71 - 4 * q^72 - 12 * q^73 + 12 * q^74 - 2 * q^75 + 6 * q^76 - 36 * q^79 + 6 * q^80 - 2 * q^81 - 6 * q^82 - 32 * q^83 + 6 * q^85 + 26 * q^86 - 2 * q^87 + 6 * q^88 + 8 * q^89 - 4 * q^90 - 2 * q^92 - 12 * q^93 + 30 * q^94 + 6 * q^95 + 2 * q^96 + 16 * q^97 - 8 * 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
$3$	0.787153	0.454463	0.227231	−	0.973841i	$-0.427033\pi$
$3$	0.787153	0.454463	0.227231	+	0.973841i	$0.427033\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.787153	−0.321354
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	−2.38039	−0.793463
$10$	−1.00000	−0.316228
$11$	−2.94606	−0.888271	−0.444135	−	0.895960i	$-0.646489\pi$
$11$	−2.94606	−0.888271	−0.444135	+	0.895960i	$0.646489\pi$
$12$	0.787153	0.227231
$13$	−1.11320	−0.308747	−0.154373	−	0.988013i	$-0.549336\pi$
$13$	−1.11320	−0.308747	−0.154373	+	0.988013i	$0.549336\pi$
$14$	0	0
$15$	0.787153	0.203242
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	2.38039	0.561063
$19$	5.69821	1.30726	0.653629	−	0.756815i	$-0.273246\pi$
$19$	5.69821	1.30726	0.653629	+	0.756815i	$0.273246\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	2.94606	0.628102
$23$	−3.99344	−0.832690	−0.416345	−	0.909207i	$-0.636689\pi$
$23$	−3.99344	−0.832690	−0.416345	+	0.909207i	$0.636689\pi$
$24$	−0.787153	−0.160677
$25$	1.00000	0.200000
$26$	1.11320	0.218317
$27$	−4.23519	−0.815063
$28$	0	0
$29$	10.2331	1.90023	0.950117	−	0.311894i	$-0.100963\pi$
$29$	10.2331	1.90023	0.950117	+	0.311894i	$0.100963\pi$
$30$	−0.787153	−0.143714
$31$	−2.49192	−0.447562	−0.223781	−	0.974639i	$-0.571840\pi$
$31$	−2.49192	−0.447562	−0.223781	+	0.974639i	$0.571840\pi$
$32$	−1.00000	−0.176777
$33$	−2.31900	−0.403686
$34$	−1.00000	−0.171499
$35$	0	0
$36$	−2.38039	−0.396732
$37$	7.19846	1.18342	0.591710	−	0.806151i	$-0.298453\pi$
$37$	7.19846	1.18342	0.591710	+	0.806151i	$0.298453\pi$
$38$	−5.69821	−0.924371
$39$	−0.876260	−0.140314
$40$	−1.00000	−0.158114
$41$	−8.70353	−1.35926	−0.679632	−	0.733554i	$-0.737860\pi$
$41$	−8.70353	−1.35926	−0.679632	+	0.733554i	$0.737860\pi$
$42$	0	0
$43$	−11.2218	−1.71131	−0.855656	−	0.517545i	$-0.826846\pi$
$43$	−11.2218	−1.71131	−0.855656	+	0.517545i	$0.826846\pi$
$44$	−2.94606	−0.444135
$45$	−2.38039	−0.354848
$46$	3.99344	0.588801
$47$	−2.16665	−0.316039	−0.158019	−	0.987436i	$-0.550511\pi$
$47$	−2.16665	−0.316039	−0.158019	+	0.987436i	$0.550511\pi$
$48$	0.787153	0.113616
$49$	0	0
$50$	−1.00000	−0.141421
$51$	0.787153	0.110223
$52$	−1.11320	−0.154373
$53$	6.83241	0.938503	0.469252	−	0.883065i	$-0.344524\pi$
$53$	6.83241	0.938503	0.469252	+	0.883065i	$0.344524\pi$
$54$	4.23519	0.576336
$55$	−2.94606	−0.397247
$56$	0	0
$57$	4.48536	0.594100
$58$	−10.2331	−1.34367
$59$	8.47762	1.10369	0.551846	−	0.833946i	$-0.313924\pi$
$59$	8.47762	1.10369	0.551846	+	0.833946i	$0.313924\pi$
$60$	0.787153	0.101621
$61$	−9.45164	−1.21016	−0.605079	−	0.796165i	$-0.706858\pi$
$61$	−9.45164	−1.21016	−0.605079	+	0.796165i	$0.706858\pi$
$62$	2.49192	0.316474
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.11320	−0.138076
$66$	2.31900	0.285449
$67$	6.12953	0.748841	0.374420	−	0.927259i	$-0.377842\pi$
$67$	6.12953	0.748841	0.374420	+	0.927259i	$0.377842\pi$
$68$	1.00000	0.121268
$69$	−3.14345	−0.378427
$70$	0	0
$71$	3.13998	0.372647	0.186323	−	0.982488i	$-0.440343\pi$
$71$	3.13998	0.372647	0.186323	+	0.982488i	$0.440343\pi$
$72$	2.38039	0.280532
$73$	−1.95049	−0.228288	−0.114144	−	0.993464i	$-0.536413\pi$
$73$	−1.95049	−0.228288	−0.114144	+	0.993464i	$0.536413\pi$
$74$	−7.19846	−0.836804
$75$	0.787153	0.0908926
$76$	5.69821	0.653629
$77$	0	0
$78$	0.876260	0.0992169
$79$	−1.03366	−0.116296	−0.0581480	−	0.998308i	$-0.518520\pi$
$79$	−1.03366	−0.116296	−0.0581480	+	0.998308i	$0.518520\pi$
$80$	1.00000	0.111803
$81$	3.80743	0.423048
$82$	8.70353	0.961144
$83$	−10.8662	−1.19272	−0.596358	−	0.802719i	$-0.703386\pi$
$83$	−10.8662	−1.19272	−0.596358	+	0.802719i	$0.703386\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	11.2218	1.21008
$87$	8.05499	0.863586
$88$	2.94606	0.314051
$89$	−8.62613	−0.914368	−0.457184	−	0.889372i	$-0.651142\pi$
$89$	−8.62613	−0.914368	−0.457184	+	0.889372i	$0.651142\pi$
$90$	2.38039	0.250915
$91$	0	0
$92$	−3.99344	−0.416345
$93$	−1.96152	−0.203400
$94$	2.16665	0.223473
$95$	5.69821	0.584624
$96$	−0.787153	−0.0803384
$97$	3.16732	0.321592	0.160796	−	0.986988i	$-0.448594\pi$
$97$	3.16732	0.321592	0.160796	+	0.986988i	$0.448594\pi$
$98$	0	0
$99$	7.01278	0.704810
$100$	1.00000	0.100000
$101$	10.3227	1.02714	0.513572	−	0.858047i	$-0.328322\pi$
$101$	10.3227	1.02714	0.513572	+	0.858047i	$0.328322\pi$
$102$	−0.787153	−0.0779397
$103$	3.40765	0.335766	0.167883	−	0.985807i	$-0.446307\pi$
$103$	3.40765	0.335766	0.167883	+	0.985807i	$0.446307\pi$
$104$	1.11320	0.109158
$105$	0	0
$106$	−6.83241	−0.663622
$107$	−7.36100	−0.711615	−0.355808	−	0.934559i	$-0.615794\pi$
$107$	−7.36100	−0.711615	−0.355808	+	0.934559i	$0.615794\pi$
$108$	−4.23519	−0.407531
$109$	−12.4287	−1.19046	−0.595228	−	0.803557i	$-0.702938\pi$
$109$	−12.4287	−1.19046	−0.595228	+	0.803557i	$0.702938\pi$
$110$	2.94606	0.280896
$111$	5.66629	0.537820
$112$	0	0
$113$	7.82860	0.736453	0.368227	−	0.929736i	$-0.379965\pi$
$113$	7.82860	0.736453	0.368227	+	0.929736i	$0.379965\pi$
$114$	−4.48536	−0.420092
$115$	−3.99344	−0.372390
$116$	10.2331	0.950117
$117$	2.64986	0.244979
$118$	−8.47762	−0.780428
$119$	0	0
$120$	−0.787153	−0.0718569
$121$	−2.32073	−0.210975
$122$	9.45164	0.855711
$123$	−6.85101	−0.617735
$124$	−2.49192	−0.223781
$125$	1.00000	0.0894427
$126$	0	0
$127$	−20.5536	−1.82384	−0.911919	−	0.410371i	$-0.865399\pi$
$127$	−20.5536	−1.82384	−0.911919	+	0.410371i	$0.865399\pi$
$128$	−1.00000	−0.0883883
$129$	−8.83329	−0.777728
$130$	1.11320	0.0976343
$131$	6.63674	0.579855	0.289927	−	0.957049i	$-0.406369\pi$
$131$	6.63674	0.579855	0.289927	+	0.957049i	$0.406369\pi$
$132$	−2.31900	−0.201843
$133$	0	0
$134$	−6.12953	−0.529511
$135$	−4.23519	−0.364507
$136$	−1.00000	−0.0857493
$137$	12.9437	1.10585	0.552927	−	0.833230i	$-0.313511\pi$
$137$	12.9437	1.10585	0.552927	+	0.833230i	$0.313511\pi$
$138$	3.14345	0.267588
$139$	2.82900	0.239952	0.119976	−	0.992777i	$-0.461718\pi$
$139$	2.82900	0.239952	0.119976	+	0.992777i	$0.461718\pi$
$140$	0	0
$141$	−1.70549	−0.143628
$142$	−3.13998	−0.263501
$143$	3.27956	0.274251
$144$	−2.38039	−0.198366
$145$	10.2331	0.849810
$146$	1.95049	0.161424
$147$	0	0
$148$	7.19846	0.591710
$149$	−3.64947	−0.298976	−0.149488	−	0.988764i	$-0.547763\pi$
$149$	−3.64947	−0.298976	−0.149488	+	0.988764i	$0.547763\pi$
$150$	−0.787153	−0.0642708
$151$	−14.2916	−1.16304	−0.581518	−	0.813533i	$-0.697541\pi$
$151$	−14.2916	−1.16304	−0.581518	+	0.813533i	$0.697541\pi$
$152$	−5.69821	−0.462186
$153$	−2.38039	−0.192443
$154$	0	0
$155$	−2.49192	−0.200156
$156$	−0.876260	−0.0701570
$157$	−11.1535	−0.890150	−0.445075	−	0.895493i	$-0.646823\pi$
$157$	−11.1535	−0.890150	−0.445075	+	0.895493i	$0.646823\pi$
$158$	1.03366	0.0822337
$159$	5.37815	0.426515
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	−3.80743	−0.299140
$163$	−17.4228	−1.36466	−0.682331	−	0.731043i	$-0.739034\pi$
$163$	−17.4228	−1.36466	−0.682331	+	0.731043i	$0.739034\pi$
$164$	−8.70353	−0.679632
$165$	−2.31900	−0.180534
$166$	10.8662	0.843377
$167$	−24.2292	−1.87491	−0.937456	−	0.348104i	$-0.886826\pi$
$167$	−24.2292	−1.87491	−0.937456	+	0.348104i	$0.886826\pi$
$168$	0	0
$169$	−11.7608	−0.904675
$170$	−1.00000	−0.0766965
$171$	−13.5640	−1.03726
$172$	−11.2218	−0.855656
$173$	16.2943	1.23883	0.619417	−	0.785062i	$-0.287369\pi$
$173$	16.2943	1.23883	0.619417	+	0.785062i	$0.287369\pi$
$174$	−8.05499	−0.610647
$175$	0	0
$176$	−2.94606	−0.222068
$177$	6.67318	0.501587
$178$	8.62613	0.646556
$179$	12.3943	0.926397	0.463199	−	0.886255i	$-0.346702\pi$
$179$	12.3943	0.926397	0.463199	+	0.886255i	$0.346702\pi$
$180$	−2.38039	−0.177424
$181$	−15.9180	−1.18317	−0.591587	−	0.806241i	$-0.701498\pi$
$181$	−15.9180	−1.18317	−0.591587	+	0.806241i	$0.701498\pi$
$182$	0	0
$183$	−7.43988	−0.549972
$184$	3.99344	0.294400
$185$	7.19846	0.529241
$186$	1.96152	0.143826
$187$	−2.94606	−0.215437
$188$	−2.16665	−0.158019
$189$	0	0
$190$	−5.69821	−0.413391
$191$	−21.3006	−1.54126	−0.770630	−	0.637283i	$-0.780058\pi$
$191$	−21.3006	−1.54126	−0.770630	+	0.637283i	$0.780058\pi$
$192$	0.787153	0.0568079
$193$	3.07394	0.221267	0.110634	−	0.993861i	$-0.464712\pi$
$193$	3.07394	0.221267	0.110634	+	0.993861i	$0.464712\pi$
$194$	−3.16732	−0.227400
$195$	−0.876260	−0.0627503
$196$	0	0
$197$	−13.6321	−0.971250	−0.485625	−	0.874167i	$-0.661408\pi$
$197$	−13.6321	−0.971250	−0.485625	+	0.874167i	$0.661408\pi$
$198$	−7.01278	−0.498376
$199$	0.747489	0.0529881	0.0264941	−	0.999649i	$-0.491566\pi$
$199$	0.747489	0.0529881	0.0264941	+	0.999649i	$0.491566\pi$
$200$	−1.00000	−0.0707107
$201$	4.82487	0.340320
$202$	−10.3227	−0.726300
$203$	0	0
$204$	0.787153	0.0551117
$205$	−8.70353	−0.607881
$206$	−3.40765	−0.237423
$207$	9.50595	0.660709
$208$	−1.11320	−0.0771867
$209$	−16.7873	−1.16120
$210$	0	0
$211$	−3.40524	−0.234426	−0.117213	−	0.993107i	$-0.537396\pi$
$211$	−3.40524	−0.234426	−0.117213	+	0.993107i	$0.537396\pi$
$212$	6.83241	0.469252
$213$	2.47164	0.169354
$214$	7.36100	0.503188
$215$	−11.2218	−0.765322
$216$	4.23519	0.288168
$217$	0	0
$218$	12.4287	0.841779
$219$	−1.53534	−0.103748
$220$	−2.94606	−0.198623
$221$	−1.11320	−0.0748821
$222$	−5.66629	−0.380296
$223$	−26.4484	−1.77112	−0.885559	−	0.464527i	$-0.846224\pi$
$223$	−26.4484	−1.77112	−0.885559	+	0.464527i	$0.846224\pi$
$224$	0	0
$225$	−2.38039	−0.158693
$226$	−7.82860	−0.520751
$227$	−4.73814	−0.314481	−0.157241	−	0.987560i	$-0.550260\pi$
$227$	−4.73814	−0.314481	−0.157241	+	0.987560i	$0.550260\pi$
$228$	4.48536	0.297050
$229$	8.03937	0.531257	0.265628	−	0.964076i	$-0.414421\pi$
$229$	8.03937	0.531257	0.265628	+	0.964076i	$0.414421\pi$
$230$	3.99344	0.263320
$231$	0	0
$232$	−10.2331	−0.671834
$233$	4.82207	0.315904	0.157952	−	0.987447i	$-0.449511\pi$
$233$	4.82207	0.315904	0.157952	+	0.987447i	$0.449511\pi$
$234$	−2.64986	−0.173226
$235$	−2.16665	−0.141337
$236$	8.47762	0.551846
$237$	−0.813650	−0.0528522
$238$	0	0
$239$	1.02875	0.0665443	0.0332721	−	0.999446i	$-0.489407\pi$
$239$	1.02875	0.0665443	0.0332721	+	0.999446i	$0.489407\pi$
$240$	0.787153	0.0508105
$241$	15.4314	0.994024	0.497012	−	0.867744i	$-0.334430\pi$
$241$	15.4314	0.994024	0.497012	+	0.867744i	$0.334430\pi$
$242$	2.32073	0.149182
$243$	15.7026	1.00732
$244$	−9.45164	−0.605079
$245$	0	0
$246$	6.85101	0.436804
$247$	−6.34326	−0.403612
$248$	2.49192	0.158237
$249$	−8.55332	−0.542045
$250$	−1.00000	−0.0632456
$251$	−3.69171	−0.233019	−0.116509	−	0.993190i	$-0.537171\pi$
$251$	−3.69171	−0.233019	−0.116509	+	0.993190i	$0.537171\pi$
$252$	0	0
$253$	11.7649	0.739654
$254$	20.5536	1.28965
$255$	0.787153	0.0492934
$256$	1.00000	0.0625000
$257$	−20.9450	−1.30652	−0.653258	−	0.757136i	$-0.726598\pi$
$257$	−20.9450	−1.30652	−0.653258	+	0.757136i	$0.726598\pi$
$258$	8.83329	0.549936
$259$	0	0
$260$	−1.11320	−0.0690379
$261$	−24.3587	−1.50777
$262$	−6.63674	−0.410019
$263$	−16.8515	−1.03911	−0.519554	−	0.854438i	$-0.673902\pi$
$263$	−16.8515	−1.03911	−0.519554	+	0.854438i	$0.673902\pi$
$264$	2.31900	0.142725
$265$	6.83241	0.419711
$266$	0	0
$267$	−6.79008	−0.415546
$268$	6.12953	0.374420
$269$	−30.1046	−1.83551	−0.917756	−	0.397144i	$-0.870001\pi$
$269$	−30.1046	−1.83551	−0.917756	+	0.397144i	$0.870001\pi$
$270$	4.23519	0.257745
$271$	−2.74351	−0.166656	−0.0833282	−	0.996522i	$-0.526555\pi$
$271$	−2.74351	−0.166656	−0.0833282	+	0.996522i	$0.526555\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	−12.9437	−0.781957
$275$	−2.94606	−0.177654
$276$	−3.14345	−0.189213
$277$	1.84011	0.110561	0.0552807	−	0.998471i	$-0.482395\pi$
$277$	1.84011	0.110561	0.0552807	+	0.998471i	$0.482395\pi$
$278$	−2.82900	−0.169672
$279$	5.93174	0.355124
$280$	0	0
$281$	−19.5015	−1.16336	−0.581682	−	0.813416i	$-0.697605\pi$
$281$	−19.5015	−1.16336	−0.581682	+	0.813416i	$0.697605\pi$
$282$	1.70549	0.101560
$283$	7.49543	0.445557	0.222778	−	0.974869i	$-0.428487\pi$
$283$	7.49543	0.445557	0.222778	+	0.974869i	$0.428487\pi$
$284$	3.13998	0.186323
$285$	4.48536	0.265690
$286$	−3.27956	−0.193925
$287$	0	0
$288$	2.38039	0.140266
$289$	1.00000	0.0588235
$290$	−10.2331	−0.600907
$291$	2.49316	0.146152
$292$	−1.95049	−0.114144
$293$	17.5823	1.02717	0.513584	−	0.858039i	$-0.328317\pi$
$293$	17.5823	1.02717	0.513584	+	0.858039i	$0.328317\pi$
$294$	0	0
$295$	8.47762	0.493586
$296$	−7.19846	−0.418402
$297$	12.4771	0.723996
$298$	3.64947	0.211408
$299$	4.44551	0.257090
$300$	0.787153	0.0454463
$301$	0	0
$302$	14.2916	0.822391
$303$	8.12551	0.466798
$304$	5.69821	0.326815
$305$	−9.45164	−0.541199
$306$	2.38039	0.136078
$307$	1.21966	0.0696095	0.0348047	−	0.999394i	$-0.488919\pi$
$307$	1.21966	0.0696095	0.0348047	+	0.999394i	$0.488919\pi$
$308$	0	0
$309$	2.68235	0.152593
$310$	2.49192	0.141531
$311$	23.2968	1.32104	0.660519	−	0.750809i	$-0.270336\pi$
$311$	23.2968	1.32104	0.660519	+	0.750809i	$0.270336\pi$
$312$	0.876260	0.0496085
$313$	−5.26705	−0.297711	−0.148856	−	0.988859i	$-0.547559\pi$
$313$	−5.26705	−0.297711	−0.148856	+	0.988859i	$0.547559\pi$
$314$	11.1535	0.629431
$315$	0	0
$316$	−1.03366	−0.0581480
$317$	9.29454	0.522033	0.261017	−	0.965334i	$-0.415942\pi$
$317$	9.29454	0.522033	0.261017	+	0.965334i	$0.415942\pi$
$318$	−5.37815	−0.301592
$319$	−30.1473	−1.68792
$320$	1.00000	0.0559017
$321$	−5.79423	−0.323403
$322$	0	0
$323$	5.69821	0.317057
$324$	3.80743	0.211524
$325$	−1.11320	−0.0617493
$326$	17.4228	0.964962
$327$	−9.78331	−0.541018
$328$	8.70353	0.480572
$329$	0	0
$330$	2.31900	0.127657
$331$	23.2960	1.28047	0.640233	−	0.768181i	$-0.278838\pi$
$331$	23.2960	1.28047	0.640233	+	0.768181i	$0.278838\pi$
$332$	−10.8662	−0.596358
$333$	−17.1351	−0.939000
$334$	24.2292	1.32576
$335$	6.12953	0.334892
$336$	0	0
$337$	−27.4705	−1.49641	−0.748207	−	0.663466i	$-0.769085\pi$
$337$	−27.4705	−1.49641	−0.748207	+	0.663466i	$0.769085\pi$
$338$	11.7608	0.639702
$339$	6.16231	0.334691
$340$	1.00000	0.0542326
$341$	7.34134	0.397556
$342$	13.5640	0.733455
$343$	0	0
$344$	11.2218	0.605040
$345$	−3.14345	−0.169238
$346$	−16.2943	−0.875988
$347$	29.1651	1.56566	0.782832	−	0.622233i	$-0.213774\pi$
$347$	29.1651	1.56566	0.782832	+	0.622233i	$0.213774\pi$
$348$	8.05499	0.431793
$349$	22.0357	1.17954	0.589771	−	0.807571i	$-0.299218\pi$
$349$	22.0357	1.17954	0.589771	+	0.807571i	$0.299218\pi$
$350$	0	0
$351$	4.71462	0.251648
$352$	2.94606	0.157026
$353$	−20.8632	−1.11044	−0.555218	−	0.831705i	$-0.687365\pi$
$353$	−20.8632	−1.11044	−0.555218	+	0.831705i	$0.687365\pi$
$354$	−6.67318	−0.354675
$355$	3.13998	0.166653
$356$	−8.62613	−0.457184
$357$	0	0
$358$	−12.3943	−0.655062
$359$	−23.1275	−1.22062	−0.610312	−	0.792161i	$-0.708956\pi$
$359$	−23.1275	−1.22062	−0.610312	+	0.792161i	$0.708956\pi$
$360$	2.38039	0.125458
$361$	13.4696	0.708924
$362$	15.9180	0.836630
$363$	−1.82677	−0.0958803
$364$	0	0
$365$	−1.95049	−0.102093
$366$	7.43988	0.388889
$367$	−13.9632	−0.728871	−0.364435	−	0.931229i	$-0.618738\pi$
$367$	−13.9632	−0.728871	−0.364435	+	0.931229i	$0.618738\pi$
$368$	−3.99344	−0.208173
$369$	20.7178	1.07853
$370$	−7.19846	−0.374230
$371$	0	0
$372$	−1.96152	−0.101700
$373$	−17.3986	−0.900867	−0.450433	−	0.892810i	$-0.648731\pi$
$373$	−17.3986	−0.900867	−0.450433	+	0.892810i	$0.648731\pi$
$374$	2.94606	0.152337
$375$	0.787153	0.0406484
$376$	2.16665	0.111736
$377$	−11.3915	−0.586691
$378$	0	0
$379$	−22.1472	−1.13763	−0.568813	−	0.822467i	$-0.692597\pi$
$379$	−22.1472	−1.13763	−0.568813	+	0.822467i	$0.692597\pi$
$380$	5.69821	0.292312
$381$	−16.1788	−0.828867
$382$	21.3006	1.08984
$383$	32.2236	1.64655	0.823275	−	0.567643i	$-0.192145\pi$
$383$	32.2236	1.64655	0.823275	+	0.567643i	$0.192145\pi$
$384$	−0.787153	−0.0401692
$385$	0	0
$386$	−3.07394	−0.156459
$387$	26.7123	1.35786
$388$	3.16732	0.160796
$389$	−21.2771	−1.07879	−0.539395	−	0.842053i	$-0.681347\pi$
$389$	−21.2771	−1.07879	−0.539395	+	0.842053i	$0.681347\pi$
$390$	0.876260	0.0443712
$391$	−3.99344	−0.201957
$392$	0	0
$393$	5.22413	0.263522
$394$	13.6321	0.686778
$395$	−1.03366	−0.0520092
$396$	7.01278	0.352405
$397$	19.9983	1.00368	0.501842	−	0.864959i	$-0.332656\pi$
$397$	19.9983	1.00368	0.501842	+	0.864959i	$0.332656\pi$
$398$	−0.747489	−0.0374682
$399$	0	0
$400$	1.00000	0.0500000
$401$	3.05235	0.152427	0.0762136	−	0.997092i	$-0.475717\pi$
$401$	3.05235	0.152427	0.0762136	+	0.997092i	$0.475717\pi$
$402$	−4.82487	−0.240643
$403$	2.77401	0.138183
$404$	10.3227	0.513572
$405$	3.80743	0.189193
$406$	0	0
$407$	−21.2071	−1.05120
$408$	−0.787153	−0.0389699
$409$	23.1383	1.14412	0.572058	−	0.820213i	$-0.306145\pi$
$409$	23.1383	1.14412	0.572058	+	0.820213i	$0.306145\pi$
$410$	8.70353	0.429837
$411$	10.1887	0.502570
$412$	3.40765	0.167883
$413$	0	0
$414$	−9.50595	−0.467192
$415$	−10.8662	−0.533398
$416$	1.11320	0.0545792
$417$	2.22685	0.109049
$418$	16.7873	0.821092
$419$	−37.3839	−1.82632	−0.913161	−	0.407599i	$-0.866366\pi$
$419$	−37.3839	−1.82632	−0.913161	+	0.407599i	$0.866366\pi$
$420$	0	0
$421$	−23.7075	−1.15543	−0.577717	−	0.816237i	$-0.696056\pi$
$421$	−23.7075	−1.15543	−0.577717	+	0.816237i	$0.696056\pi$
$422$	3.40524	0.165765
$423$	5.15748	0.250765
$424$	−6.83241	−0.331811
$425$	1.00000	0.0485071
$426$	−2.47164	−0.119751
$427$	0	0
$428$	−7.36100	−0.355808
$429$	2.58152	0.124637
$430$	11.2218	0.541164
$431$	30.4551	1.46697	0.733485	−	0.679706i	$-0.237893\pi$
$431$	30.4551	1.46697	0.733485	+	0.679706i	$0.237893\pi$
$432$	−4.23519	−0.203766
$433$	−33.9326	−1.63070	−0.815348	−	0.578972i	$-0.803454\pi$
$433$	−33.9326	−1.63070	−0.815348	+	0.578972i	$0.803454\pi$
$434$	0	0
$435$	8.05499	0.386207
$436$	−12.4287	−0.595228
$437$	−22.7555	−1.08854
$438$	1.53534	0.0733612
$439$	−14.2887	−0.681962	−0.340981	−	0.940070i	$-0.610759\pi$
$439$	−14.2887	−0.681962	−0.340981	+	0.940070i	$0.610759\pi$
$440$	2.94606	0.140448
$441$	0	0
$442$	1.11320	0.0529496
$443$	23.8510	1.13319	0.566597	−	0.823995i	$-0.308260\pi$
$443$	23.8510	1.13319	0.566597	+	0.823995i	$0.308260\pi$
$444$	5.66629	0.268910
$445$	−8.62613	−0.408918
$446$	26.4484	1.25237
$447$	−2.87269	−0.135873
$448$	0	0
$449$	−24.6972	−1.16553	−0.582766	−	0.812640i	$-0.698029\pi$
$449$	−24.6972	−1.16553	−0.582766	+	0.812640i	$0.698029\pi$
$450$	2.38039	0.112213
$451$	25.6411	1.20739
$452$	7.82860	0.368227
$453$	−11.2497	−0.528557
$454$	4.73814	0.222372
$455$	0	0
$456$	−4.48536	−0.210046
$457$	−32.7562	−1.53227	−0.766135	−	0.642680i	$-0.777823\pi$
$457$	−32.7562	−1.53227	−0.766135	+	0.642680i	$0.777823\pi$
$458$	−8.03937	−0.375655
$459$	−4.23519	−0.197682
$460$	−3.99344	−0.186195
$461$	25.0719	1.16771	0.583857	−	0.811856i	$-0.301543\pi$
$461$	25.0719	1.16771	0.583857	+	0.811856i	$0.301543\pi$
$462$	0	0
$463$	15.6048	0.725216	0.362608	−	0.931942i	$-0.381886\pi$
$463$	15.6048	0.725216	0.362608	+	0.931942i	$0.381886\pi$
$464$	10.2331	0.475059
$465$	−1.96152	−0.0909633
$466$	−4.82207	−0.223378
$467$	9.63558	0.445882	0.222941	−	0.974832i	$-0.428434\pi$
$467$	9.63558	0.445882	0.222941	+	0.974832i	$0.428434\pi$
$468$	2.64986	0.122490
$469$	0	0
$470$	2.16665	0.0999402
$471$	−8.77954	−0.404540
$472$	−8.47762	−0.390214
$473$	33.0602	1.52011
$474$	0.813650	0.0373722
$475$	5.69821	0.261452
$476$	0	0
$477$	−16.2638	−0.744668
$478$	−1.02875	−0.0470539
$479$	−17.7902	−0.812853	−0.406426	−	0.913683i	$-0.633225\pi$
$479$	−17.7902	−0.812853	−0.406426	+	0.913683i	$0.633225\pi$
$480$	−0.787153	−0.0359284
$481$	−8.01334	−0.365377
$482$	−15.4314	−0.702881
$483$	0	0
$484$	−2.32073	−0.105488
$485$	3.16732	0.143821
$486$	−15.7026	−0.712284
$487$	−20.7253	−0.939152	−0.469576	−	0.882892i	$-0.655593\pi$
$487$	−20.7253	−0.939152	−0.469576	+	0.882892i	$0.655593\pi$
$488$	9.45164	0.427855
$489$	−13.7144	−0.620189
$490$	0	0
$491$	26.2703	1.18556	0.592782	−	0.805363i	$-0.298030\pi$
$491$	26.2703	1.18556	0.592782	+	0.805363i	$0.298030\pi$
$492$	−6.85101	−0.308867
$493$	10.2331	0.460874
$494$	6.34326	0.285397
$495$	7.01278	0.315201
$496$	−2.49192	−0.111890
$497$	0	0
$498$	8.55332	0.383283
$499$	3.31684	0.148482	0.0742411	−	0.997240i	$-0.476347\pi$
$499$	3.31684	0.148482	0.0742411	+	0.997240i	$0.476347\pi$
$500$	1.00000	0.0447214
$501$	−19.0721	−0.852078
$502$	3.69171	0.164769
$503$	−1.87937	−0.0837972	−0.0418986	−	0.999122i	$-0.513341\pi$
$503$	−1.87937	−0.0837972	−0.0418986	+	0.999122i	$0.513341\pi$
$504$	0	0
$505$	10.3227	0.459352
$506$	−11.7649	−0.523015
$507$	−9.25753	−0.411141
$508$	−20.5536	−0.911919
$509$	22.5275	0.998514	0.499257	−	0.866454i	$-0.333606\pi$
$509$	22.5275	0.998514	0.499257	+	0.866454i	$0.333606\pi$
$510$	−0.787153	−0.0348557
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−24.1330	−1.06550
$514$	20.9450	0.923846
$515$	3.40765	0.150159
$516$	−8.83329	−0.388864
$517$	6.38309	0.280728
$518$	0	0
$519$	12.8261	0.563004
$520$	1.11320	0.0488171
$521$	20.8370	0.912886	0.456443	−	0.889753i	$-0.349123\pi$
$521$	20.8370	0.912886	0.456443	+	0.889753i	$0.349123\pi$
$522$	24.3587	1.06615
$523$	−2.71013	−0.118506	−0.0592528	−	0.998243i	$-0.518872\pi$
$523$	−2.71013	−0.118506	−0.0592528	+	0.998243i	$0.518872\pi$
$524$	6.63674	0.289927
$525$	0	0
$526$	16.8515	0.734760
$527$	−2.49192	−0.108550
$528$	−2.31900	−0.100922
$529$	−7.05243	−0.306627
$530$	−6.83241	−0.296781
$531$	−20.1800	−0.875739
$532$	0	0
$533$	9.68879	0.419668
$534$	6.79008	0.293836
$535$	−7.36100	−0.318244
$536$	−6.12953	−0.264755
$537$	9.75625	0.421013
$538$	30.1046	1.29790
$539$	0	0
$540$	−4.23519	−0.182254
$541$	21.8197	0.938103	0.469052	−	0.883171i	$-0.344596\pi$
$541$	21.8197	0.938103	0.469052	+	0.883171i	$0.344596\pi$
$542$	2.74351	0.117844
$543$	−12.5299	−0.537709
$544$	−1.00000	−0.0428746
$545$	−12.4287	−0.532388
$546$	0	0
$547$	31.8855	1.36333	0.681664	−	0.731666i	$-0.261257\pi$
$547$	31.8855	1.36333	0.681664	+	0.731666i	$0.261257\pi$
$548$	12.9437	0.552927
$549$	22.4986	0.960216
$550$	2.94606	0.125620
$551$	58.3102	2.48410
$552$	3.14345	0.133794
$553$	0	0
$554$	−1.84011	−0.0781787
$555$	5.66629	0.240520
$556$	2.82900	0.119976
$557$	−3.41154	−0.144551	−0.0722757	−	0.997385i	$-0.523026\pi$
$557$	−3.41154	−0.144551	−0.0722757	+	0.997385i	$0.523026\pi$
$558$	−5.93174	−0.251110
$559$	12.4922	0.528362
$560$	0	0
$561$	−2.31900	−0.0979083
$562$	19.5015	0.822622
$563$	−1.81260	−0.0763921	−0.0381961	−	0.999270i	$-0.512161\pi$
$563$	−1.81260	−0.0763921	−0.0381961	+	0.999270i	$0.512161\pi$
$564$	−1.70549	−0.0718139
$565$	7.82860	0.329352
$566$	−7.49543	−0.315056
$567$	0	0
$568$	−3.13998	−0.131750
$569$	38.3499	1.60771	0.803855	−	0.594825i	$-0.202779\pi$
$569$	38.3499	1.60771	0.803855	+	0.594825i	$0.202779\pi$
$570$	−4.48536	−0.187871
$571$	−25.3419	−1.06052	−0.530262	−	0.847834i	$-0.677906\pi$
$571$	−25.3419	−1.06052	−0.530262	+	0.847834i	$0.677906\pi$
$572$	3.27956	0.137125
$573$	−16.7669	−0.700446
$574$	0	0
$575$	−3.99344	−0.166538
$576$	−2.38039	−0.0991829
$577$	−43.6081	−1.81543	−0.907714	−	0.419590i	$-0.862174\pi$
$577$	−43.6081	−1.81543	−0.907714	+	0.419590i	$0.862174\pi$
$578$	−1.00000	−0.0415945
$579$	2.41966	0.100558
$580$	10.2331	0.424905
$581$	0	0
$582$	−2.49316	−0.103345
$583$	−20.1287	−0.833645
$584$	1.95049	0.0807119
$585$	2.64986	0.109558
$586$	−17.5823	−0.726317
$587$	−17.1863	−0.709354	−0.354677	−	0.934989i	$-0.615409\pi$
$587$	−17.1863	−0.709354	−0.354677	+	0.934989i	$0.615409\pi$
$588$	0	0
$589$	−14.1995	−0.585079
$590$	−8.47762	−0.349018
$591$	−10.7306	−0.441397
$592$	7.19846	0.295855
$593$	−33.4269	−1.37268	−0.686340	−	0.727281i	$-0.740784\pi$
$593$	−33.4269	−1.37268	−0.686340	+	0.727281i	$0.740784\pi$
$594$	−12.4771	−0.511943
$595$	0	0
$596$	−3.64947	−0.149488
$597$	0.588388	0.0240811
$598$	−4.44551	−0.181790
$599$	−11.9139	−0.486789	−0.243395	−	0.969927i	$-0.578261\pi$
$599$	−11.9139	−0.486789	−0.243395	+	0.969927i	$0.578261\pi$
$600$	−0.787153	−0.0321354
$601$	32.7848	1.33732	0.668660	−	0.743568i	$-0.266868\pi$
$601$	32.7848	1.33732	0.668660	+	0.743568i	$0.266868\pi$
$602$	0	0
$603$	−14.5907	−0.594178
$604$	−14.2916	−0.581518
$605$	−2.32073	−0.0943509
$606$	−8.12551	−0.330076
$607$	−20.5013	−0.832123	−0.416061	−	0.909336i	$-0.636590\pi$
$607$	−20.5013	−0.832123	−0.416061	+	0.909336i	$0.636590\pi$
$608$	−5.69821	−0.231093
$609$	0	0
$610$	9.45164	0.382686
$611$	2.41192	0.0975759
$612$	−2.38039	−0.0962216
$613$	42.6746	1.72361	0.861805	−	0.507239i	$-0.169334\pi$
$613$	42.6746	1.72361	0.861805	+	0.507239i	$0.169334\pi$
$614$	−1.21966	−0.0492213
$615$	−6.85101	−0.276259
$616$	0	0
$617$	7.68208	0.309269	0.154635	−	0.987972i	$-0.450580\pi$
$617$	7.68208	0.309269	0.154635	+	0.987972i	$0.450580\pi$
$618$	−2.68235	−0.107900
$619$	27.3067	1.09755	0.548774	−	0.835970i	$-0.315095\pi$
$619$	27.3067	1.09755	0.548774	+	0.835970i	$0.315095\pi$
$620$	−2.49192	−0.100078
$621$	16.9130	0.678695
$622$	−23.2968	−0.934115
$623$	0	0
$624$	−0.876260	−0.0350785
$625$	1.00000	0.0400000
$626$	5.26705	0.210514
$627$	−13.2141	−0.527722
$628$	−11.1535	−0.445075
$629$	7.19846	0.287021
$630$	0	0
$631$	−23.9125	−0.951942	−0.475971	−	0.879461i	$-0.657903\pi$
$631$	−23.9125	−0.951942	−0.475971	+	0.879461i	$0.657903\pi$
$632$	1.03366	0.0411169
$633$	−2.68044	−0.106538
$634$	−9.29454	−0.369133
$635$	−20.5536	−0.815645
$636$	5.37815	0.213257
$637$	0	0
$638$	30.1473	1.19354
$639$	−7.47437	−0.295681
$640$	−1.00000	−0.0395285
$641$	7.09763	0.280340	0.140170	−	0.990127i	$-0.455235\pi$
$641$	7.09763	0.280340	0.140170	+	0.990127i	$0.455235\pi$
$642$	5.79423	0.228680
$643$	8.85889	0.349360	0.174680	−	0.984625i	$-0.444111\pi$
$643$	8.85889	0.349360	0.174680	+	0.984625i	$0.444111\pi$
$644$	0	0
$645$	−8.83329	−0.347810
$646$	−5.69821	−0.224193
$647$	8.62968	0.339268	0.169634	−	0.985507i	$-0.445742\pi$
$647$	8.62968	0.339268	0.169634	+	0.985507i	$0.445742\pi$
$648$	−3.80743	−0.149570
$649$	−24.9756	−0.980377
$650$	1.11320	0.0436634
$651$	0	0
$652$	−17.4228	−0.682331
$653$	2.10941	0.0825476	0.0412738	−	0.999148i	$-0.486858\pi$
$653$	2.10941	0.0825476	0.0412738	+	0.999148i	$0.486858\pi$
$654$	9.78331	0.382558
$655$	6.63674	0.259319
$656$	−8.70353	−0.339816
$657$	4.64293	0.181138
$658$	0	0
$659$	34.3393	1.33767	0.668834	−	0.743412i	$-0.266794\pi$
$659$	34.3393	1.33767	0.668834	+	0.743412i	$0.266794\pi$
$660$	−2.31900	−0.0902670
$661$	27.1956	1.05779	0.528894	−	0.848688i	$-0.322607\pi$
$661$	27.1956	1.05779	0.528894	+	0.848688i	$0.322607\pi$
$662$	−23.2960	−0.905426
$663$	−0.876260	−0.0340311
$664$	10.8662	0.421688
$665$	0	0
$666$	17.1351	0.663973
$667$	−40.8652	−1.58231
$668$	−24.2292	−0.937456
$669$	−20.8190	−0.804907
$670$	−6.12953	−0.236804
$671$	27.8451	1.07495
$672$	0	0
$673$	−10.1538	−0.391402	−0.195701	−	0.980664i	$-0.562698\pi$
$673$	−10.1538	−0.391402	−0.195701	+	0.980664i	$0.562698\pi$
$674$	27.4705	1.05812
$675$	−4.23519	−0.163013
$676$	−11.7608	−0.452338
$677$	−29.4897	−1.13338	−0.566690	−	0.823931i	$-0.691776\pi$
$677$	−29.4897	−1.13338	−0.566690	+	0.823931i	$0.691776\pi$
$678$	−6.16231	−0.236662
$679$	0	0
$680$	−1.00000	−0.0383482
$681$	−3.72964	−0.142920
$682$	−7.34134	−0.281115
$683$	−41.5991	−1.59175	−0.795874	−	0.605463i	$-0.792988\pi$
$683$	−41.5991	−1.59175	−0.795874	+	0.605463i	$0.792988\pi$
$684$	−13.5640	−0.518631
$685$	12.9437	0.494553
$686$	0	0
$687$	6.32821	0.241436
$688$	−11.2218	−0.427828
$689$	−7.60585	−0.289760
$690$	3.14345	0.119669
$691$	41.0681	1.56230	0.781152	−	0.624341i	$-0.214632\pi$
$691$	41.0681	1.56230	0.781152	+	0.624341i	$0.214632\pi$
$692$	16.2943	0.619417
$693$	0	0
$694$	−29.1651	−1.10709
$695$	2.82900	0.107310
$696$	−8.05499	−0.305324
$697$	−8.70353	−0.329670
$698$	−22.0357	−0.834062
$699$	3.79571	0.143567
$700$	0	0
$701$	−36.9938	−1.39724	−0.698618	−	0.715495i	$-0.746201\pi$
$701$	−36.9938	−1.39724	−0.698618	+	0.715495i	$0.746201\pi$
$702$	−4.71462	−0.177942
$703$	41.0183	1.54703
$704$	−2.94606	−0.111034
$705$	−1.70549	−0.0642323
$706$	20.8632	0.785197
$707$	0	0
$708$	6.67318	0.250793
$709$	−24.7679	−0.930179	−0.465090	−	0.885264i	$-0.653978\pi$
$709$	−24.7679	−0.930179	−0.465090	+	0.885264i	$0.653978\pi$
$710$	−3.13998	−0.117841
$711$	2.46052	0.0922767
$712$	8.62613	0.323278
$713$	9.95133	0.372680
$714$	0	0
$715$	3.27956	0.122649
$716$	12.3943	0.463199
$717$	0.809783	0.0302419
$718$	23.1275	0.863112
$719$	10.5252	0.392524	0.196262	−	0.980551i	$-0.437120\pi$
$719$	10.5252	0.392524	0.196262	+	0.980551i	$0.437120\pi$
$720$	−2.38039	−0.0887119
$721$	0	0
$722$	−13.4696	−0.501285
$723$	12.1469	0.451747
$724$	−15.9180	−0.591587
$725$	10.2331	0.380047
$726$	1.82677	0.0677976
$727$	−43.5027	−1.61343	−0.806714	−	0.590943i	$-0.798756\pi$
$727$	−43.5027	−1.61343	−0.806714	+	0.590943i	$0.798756\pi$
$728$	0	0
$729$	0.938054	0.0347428
$730$	1.95049	0.0721910
$731$	−11.2218	−0.415054
$732$	−7.43988	−0.274986
$733$	−0.717491	−0.0265011	−0.0132506	−	0.999912i	$-0.504218\pi$
$733$	−0.717491	−0.0265011	−0.0132506	+	0.999912i	$0.504218\pi$
$734$	13.9632	0.515389
$735$	0	0
$736$	3.99344	0.147200
$737$	−18.0580	−0.665174
$738$	−20.7178	−0.762633
$739$	−12.2008	−0.448814	−0.224407	−	0.974496i	$-0.572045\pi$
$739$	−12.2008	−0.448814	−0.224407	+	0.974496i	$0.572045\pi$
$740$	7.19846	0.264621
$741$	−4.99311	−0.183427
$742$	0	0
$743$	−24.1016	−0.884203	−0.442102	−	0.896965i	$-0.645767\pi$
$743$	−24.1016	−0.884203	−0.442102	+	0.896965i	$0.645767\pi$
$744$	1.96152	0.0719128
$745$	−3.64947	−0.133706
$746$	17.3986	0.637009
$747$	25.8657	0.946376
$748$	−2.94606	−0.107719
$749$	0	0
$750$	−0.787153	−0.0287428
$751$	−52.1531	−1.90310	−0.951548	−	0.307501i	$-0.900507\pi$
$751$	−52.1531	−1.90310	−0.951548	+	0.307501i	$0.900507\pi$
$752$	−2.16665	−0.0790096
$753$	−2.90594	−0.105898
$754$	11.3915	0.414853
$755$	−14.2916	−0.520126
$756$	0	0
$757$	17.7395	0.644751	0.322376	−	0.946612i	$-0.395519\pi$
$757$	17.7395	0.644751	0.322376	+	0.946612i	$0.395519\pi$
$758$	22.1472	0.804423
$759$	9.26079	0.336145
$760$	−5.69821	−0.206696
$761$	−11.3751	−0.412346	−0.206173	−	0.978516i	$-0.566101\pi$
$761$	−11.3751	−0.412346	−0.206173	+	0.978516i	$0.566101\pi$
$762$	16.1788	0.586097
$763$	0	0
$764$	−21.3006	−0.770630
$765$	−2.38039	−0.0860632
$766$	−32.2236	−1.16429
$767$	−9.43730	−0.340761
$768$	0.787153	0.0284039
$769$	7.67486	0.276762	0.138381	−	0.990379i	$-0.455810\pi$
$769$	7.67486	0.276762	0.138381	+	0.990379i	$0.455810\pi$
$770$	0	0
$771$	−16.4869	−0.593763
$772$	3.07394	0.110634
$773$	−11.9392	−0.429425	−0.214712	−	0.976677i	$-0.568881\pi$
$773$	−11.9392	−0.429425	−0.214712	+	0.976677i	$0.568881\pi$
$774$	−26.7123	−0.960154
$775$	−2.49192	−0.0895123
$776$	−3.16732	−0.113700
$777$	0	0
$778$	21.2771	0.762820
$779$	−49.5945	−1.77691
$780$	−0.876260	−0.0313751
$781$	−9.25056	−0.331011
$782$	3.99344	0.142805
$783$	−43.3390	−1.54881
$784$	0	0
$785$	−11.1535	−0.398087
$786$	−5.22413	−0.186338
$787$	−3.51065	−0.125141	−0.0625705	−	0.998041i	$-0.519930\pi$
$787$	−3.51065	−0.125141	−0.0625705	+	0.998041i	$0.519930\pi$
$788$	−13.6321	−0.485625
$789$	−13.2647	−0.472236
$790$	1.03366	0.0367760
$791$	0	0
$792$	−7.01278	−0.249188
$793$	10.5216	0.373632
$794$	−19.9983	−0.709712
$795$	5.37815	0.190743
$796$	0.747489	0.0264941
$797$	1.17713	0.0416962	0.0208481	−	0.999783i	$-0.493363\pi$
$797$	1.17713	0.0416962	0.0208481	+	0.999783i	$0.493363\pi$
$798$	0	0
$799$	−2.16665	−0.0766506
$800$	−1.00000	−0.0353553
$801$	20.5336	0.725517
$802$	−3.05235	−0.107782
$803$	5.74627	0.202781
$804$	4.82487	0.170160
$805$	0	0
$806$	−2.77401	−0.0977103
$807$	−23.6970	−0.834172
$808$	−10.3227	−0.363150
$809$	−33.7525	−1.18667	−0.593337	−	0.804954i	$-0.702190\pi$
$809$	−33.7525	−1.18667	−0.593337	+	0.804954i	$0.702190\pi$
$810$	−3.80743	−0.133779
$811$	8.13532	0.285670	0.142835	−	0.989747i	$-0.454378\pi$
$811$	8.13532	0.285670	0.142835	+	0.989747i	$0.454378\pi$
$812$	0	0
$813$	−2.15956	−0.0757391
$814$	21.2071	0.743308
$815$	−17.4228	−0.610296
$816$	0.787153	0.0275559
$817$	−63.9443	−2.23713
$818$	−23.1383	−0.809013
$819$	0	0
$820$	−8.70353	−0.303940
$821$	14.2609	0.497710	0.248855	−	0.968541i	$-0.419946\pi$
$821$	14.2609	0.497710	0.248855	+	0.968541i	$0.419946\pi$
$822$	−10.1887	−0.355371
$823$	−47.8267	−1.66713	−0.833567	−	0.552418i	$-0.813705\pi$
$823$	−47.8267	−1.66713	−0.833567	+	0.552418i	$0.813705\pi$
$824$	−3.40765	−0.118711
$825$	−2.31900	−0.0807372
$826$	0	0
$827$	−45.0419	−1.56626	−0.783130	−	0.621858i	$-0.786378\pi$
$827$	−45.0419	−1.56626	−0.783130	+	0.621858i	$0.786378\pi$
$828$	9.50595	0.330355
$829$	8.95431	0.310996	0.155498	−	0.987836i	$-0.450302\pi$
$829$	8.95431	0.310996	0.155498	+	0.987836i	$0.450302\pi$
$830$	10.8662	0.377170
$831$	1.44845	0.0502461
$832$	−1.11320	−0.0385933
$833$	0	0
$834$	−2.22685	−0.0771096
$835$	−24.2292	−0.838486
$836$	−16.7873	−0.580600
$837$	10.5537	0.364791
$838$	37.3839	1.29140
$839$	46.0842	1.59100	0.795502	−	0.605951i	$-0.207207\pi$
$839$	46.0842	1.59100	0.795502	+	0.605951i	$0.207207\pi$
$840$	0	0
$841$	75.7158	2.61089
$842$	23.7075	0.817015
$843$	−15.3507	−0.528706
$844$	−3.40524	−0.117213
$845$	−11.7608	−0.404583
$846$	−5.15748	−0.177318
$847$	0	0
$848$	6.83241	0.234626
$849$	5.90005	0.202489
$850$	−1.00000	−0.0342997
$851$	−28.7466	−0.985421
$852$	2.47164	0.0846770
$853$	22.7434	0.778719	0.389360	−	0.921086i	$-0.372696\pi$
$853$	22.7434	0.778719	0.389360	+	0.921086i	$0.372696\pi$
$854$	0	0
$855$	−13.5640	−0.463878
$856$	7.36100	0.251594
$857$	53.3998	1.82410	0.912051	−	0.410076i	$-0.134498\pi$
$857$	53.3998	1.82410	0.912051	+	0.410076i	$0.134498\pi$
$858$	−2.58152	−0.0881315
$859$	44.6070	1.52197	0.760986	−	0.648769i	$-0.224716\pi$
$859$	44.6070	1.52197	0.760986	+	0.648769i	$0.224716\pi$
$860$	−11.2218	−0.382661
$861$	0	0
$862$	−30.4551	−1.03730
$863$	−30.9061	−1.05206	−0.526028	−	0.850467i	$-0.676319\pi$
$863$	−30.9061	−1.05206	−0.526028	+	0.850467i	$0.676319\pi$
$864$	4.23519	0.144084
$865$	16.2943	0.554024
$866$	33.9326	1.15308
$867$	0.787153	0.0267331
$868$	0	0
$869$	3.04523	0.103302
$870$	−8.05499	−0.273090
$871$	−6.82340	−0.231202
$872$	12.4287	0.420890
$873$	−7.53945	−0.255172
$874$	22.7555	0.769715
$875$	0	0
$876$	−1.53534	−0.0518742
$877$	−33.8726	−1.14380	−0.571898	−	0.820325i	$-0.693793\pi$
$877$	−33.8726	−1.14380	−0.571898	+	0.820325i	$0.693793\pi$
$878$	14.2887	0.482220
$879$	13.8399	0.466810
$880$	−2.94606	−0.0993117
$881$	55.1938	1.85953	0.929763	−	0.368158i	$-0.120011\pi$
$881$	55.1938	1.85953	0.929763	+	0.368158i	$0.120011\pi$
$882$	0	0
$883$	−22.9974	−0.773923	−0.386961	−	0.922096i	$-0.626475\pi$
$883$	−22.9974	−0.773923	−0.386961	+	0.922096i	$0.626475\pi$
$884$	−1.11320	−0.0374410
$885$	6.67318	0.224316
$886$	−23.8510	−0.801289
$887$	−36.7976	−1.23554	−0.617771	−	0.786358i	$-0.711964\pi$
$887$	−36.7976	−1.23554	−0.617771	+	0.786358i	$0.711964\pi$
$888$	−5.66629	−0.190148
$889$	0	0
$890$	8.62613	0.289148
$891$	−11.2169	−0.375781
$892$	−26.4484	−0.885559
$893$	−12.3460	−0.413144
$894$	2.87269	0.0960770
$895$	12.3943	0.414297
$896$	0	0
$897$	3.49929	0.116838
$898$	24.6972	0.824155
$899$	−25.5000	−0.850472
$900$	−2.38039	−0.0793463
$901$	6.83241	0.227620
$902$	−25.6411	−0.853756
$903$	0	0
$904$	−7.82860	−0.260375
$905$	−15.9180	−0.529131
$906$	11.2497	0.373746
$907$	5.93095	0.196934	0.0984670	−	0.995140i	$-0.468606\pi$
$907$	5.93095	0.196934	0.0984670	+	0.995140i	$0.468606\pi$
$908$	−4.73814	−0.157241
$909$	−24.5720	−0.815001
$910$	0	0
$911$	45.8793	1.52005	0.760024	−	0.649895i	$-0.225187\pi$
$911$	45.8793	1.52005	0.760024	+	0.649895i	$0.225187\pi$
$912$	4.48536	0.148525
$913$	32.0123	1.05945
$914$	32.7562	1.08348
$915$	−7.43988	−0.245955
$916$	8.03937	0.265628
$917$	0	0
$918$	4.23519	0.139782
$919$	−0.0236088	−0.000778784 0	−0.000389392	−	1.00000i	$-0.500124\pi$
$919$	−0.0236088	−0.000778784 0	−0.000389392		1.00000i	$0.500124\pi$
$920$	3.99344	0.131660
$921$	0.960056	0.0316349
$922$	−25.0719	−0.825699
$923$	−3.49543	−0.115053
$924$	0	0
$925$	7.19846	0.236684
$926$	−15.6048	−0.512805
$927$	−8.11155	−0.266418
$928$	−10.2331	−0.335917
$929$	−14.9056	−0.489036	−0.244518	−	0.969645i	$-0.578630\pi$
$929$	−14.9056	−0.489036	−0.244518	+	0.969645i	$0.578630\pi$
$930$	1.96152	0.0643208
$931$	0	0
$932$	4.82207	0.157952
$933$	18.3381	0.600363
$934$	−9.63558	−0.315286
$935$	−2.94606	−0.0963465
$936$	−2.64986	−0.0866132
$937$	−7.62327	−0.249041	−0.124521	−	0.992217i	$-0.539739\pi$
$937$	−7.62327	−0.249041	−0.124521	+	0.992217i	$0.539739\pi$
$938$	0	0
$939$	−4.14597	−0.135299
$940$	−2.16665	−0.0706684
$941$	54.6942	1.78298	0.891491	−	0.453039i	$-0.149660\pi$
$941$	54.6942	1.78298	0.891491	+	0.453039i	$0.149660\pi$
$942$	8.77954	0.286053
$943$	34.7570	1.13184
$944$	8.47762	0.275923
$945$	0	0
$946$	−33.0602	−1.07488
$947$	17.4023	0.565499	0.282750	−	0.959194i	$-0.408753\pi$
$947$	17.4023	0.565499	0.282750	+	0.959194i	$0.408753\pi$
$948$	−0.813650	−0.0264261
$949$	2.17129	0.0704831
$950$	−5.69821	−0.184874
$951$	7.31622	0.237245
$952$	0	0
$953$	−58.1506	−1.88368	−0.941841	−	0.336059i	$-0.890906\pi$
$953$	−58.1506	−1.88368	−0.941841	+	0.336059i	$0.890906\pi$
$954$	16.2638	0.526560
$955$	−21.3006	−0.689273
$956$	1.02875	0.0332721
$957$	−23.7305	−0.767098
$958$	17.7902	0.574774
$959$	0	0
$960$	0.787153	0.0254052
$961$	−24.7903	−0.799689
$962$	8.01334	0.258360
$963$	17.5221	0.564641
$964$	15.4314	0.497012
$965$	3.07394	0.0989536
$966$	0	0
$967$	−1.56273	−0.0502540	−0.0251270	−	0.999684i	$-0.507999\pi$
$967$	−1.56273	−0.0502540	−0.0251270	+	0.999684i	$0.507999\pi$
$968$	2.32073	0.0745909
$969$	4.48536	0.144091
$970$	−3.16732	−0.101696
$971$	−21.8647	−0.701670	−0.350835	−	0.936437i	$-0.614102\pi$
$971$	−21.8647	−0.701670	−0.350835	+	0.936437i	$0.614102\pi$
$972$	15.7026	0.503661
$973$	0	0
$974$	20.7253	0.664081
$975$	−0.876260	−0.0280628
$976$	−9.45164	−0.302539
$977$	−24.5694	−0.786046	−0.393023	−	0.919529i	$-0.628571\pi$
$977$	−24.5694	−0.786046	−0.393023	+	0.919529i	$0.628571\pi$
$978$	13.7144	0.438540
$979$	25.4131	0.812206
$980$	0	0
$981$	29.5852	0.944583
$982$	−26.2703	−0.838320
$983$	13.0013	0.414676	0.207338	−	0.978269i	$-0.433520\pi$
$983$	13.0013	0.414676	0.207338	+	0.978269i	$0.433520\pi$
$984$	6.85101	0.218402
$985$	−13.6321	−0.434356
$986$	−10.2331	−0.325887
$987$	0	0
$988$	−6.34326	−0.201806
$989$	44.8137	1.42499
$990$	−7.01278	−0.222881
$991$	44.3146	1.40770	0.703850	−	0.710349i	$-0.251463\pi$
$991$	44.3146	1.40770	0.703850	+	0.710349i	$0.251463\pi$
$992$	2.49192	0.0791185
$993$	18.3375	0.581924
$994$	0	0
$995$	0.747489	0.0236970
$996$	−8.55332	−0.271022
$997$	27.8035	0.880545	0.440272	−	0.897864i	$-0.354882\pi$
$997$	27.8035	0.880545	0.440272	+	0.897864i	$0.354882\pi$
$998$	−3.31684	−0.104993
$999$	−30.4868	−0.964561

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8330.2.a.cn.1.4	✓	6
7.6	odd	2		8330.2.a.cq.1.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8330.2.a.cn.1.4	✓	6	1.1	even	1	trivial
8330.2.a.cq.1.3	yes	6	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.787153 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.787153 q^{6} -1.00000 q^{8} -2.38039 q^{9} -1.00000 q^{10} -2.94606 q^{11} +0.787153 q^{12} -1.11320 q^{13} +0.787153 q^{15} +1.00000 q^{16} +1.00000 q^{17} +2.38039 q^{18} +5.69821 q^{19} +1.00000 q^{20} +2.94606 q^{22} -3.99344 q^{23} -0.787153 q^{24} +1.00000 q^{25} +1.11320 q^{26} -4.23519 q^{27} +10.2331 q^{29} -0.787153 q^{30} -2.49192 q^{31} -1.00000 q^{32} -2.31900 q^{33} -1.00000 q^{34} -2.38039 q^{36} +7.19846 q^{37} -5.69821 q^{38} -0.876260 q^{39} -1.00000 q^{40} -8.70353 q^{41} -11.2218 q^{43} -2.94606 q^{44} -2.38039 q^{45} +3.99344 q^{46} -2.16665 q^{47} +0.787153 q^{48} -1.00000 q^{50} +0.787153 q^{51} -1.11320 q^{52} +6.83241 q^{53} +4.23519 q^{54} -2.94606 q^{55} +4.48536 q^{57} -10.2331 q^{58} +8.47762 q^{59} +0.787153 q^{60} -9.45164 q^{61} +2.49192 q^{62} +1.00000 q^{64} -1.11320 q^{65} +2.31900 q^{66} +6.12953 q^{67} +1.00000 q^{68} -3.14345 q^{69} +3.13998 q^{71} +2.38039 q^{72} -1.95049 q^{73} -7.19846 q^{74} +0.787153 q^{75} +5.69821 q^{76} +0.876260 q^{78} -1.03366 q^{79} +1.00000 q^{80} +3.80743 q^{81} +8.70353 q^{82} -10.8662 q^{83} +1.00000 q^{85} +11.2218 q^{86} +8.05499 q^{87} +2.94606 q^{88} -8.62613 q^{89} +2.38039 q^{90} -3.99344 q^{92} -1.96152 q^{93} +2.16665 q^{94} +5.69821 q^{95} -0.787153 q^{96} +3.16732 q^{97} +7.01278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} + 6 q^{5} + 2 q^{6} - 6 q^{8} + 4 q^{9} - 6 q^{10} - 6 q^{11} - 2 q^{12} - 2 q^{15} + 6 q^{16} + 6 q^{17} - 4 q^{18} + 6 q^{19} + 6 q^{20} + 6 q^{22} - 2 q^{23} + 2 q^{24}+ \cdots - 8 q^{99}+O(q^{100}) \) 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 + 6 * q^5 + 2 * q^6 - 6 * q^8 + 4 * q^9 - 6 * q^10 - 6 * q^11 - 2 * q^12 - 2 * q^15 + 6 * q^16 + 6 * q^17 - 4 * q^18 + 6 * q^19 + 6 * q^20 + 6 * q^22 - 2 * q^23 + 2 * q^24 + 6 * q^25 - 20 * q^27 - 2 * q^29 + 2 * q^30 - 2 * q^31 - 6 * q^32 - 4 * q^33 - 6 * q^34 + 4 * q^36 - 12 * q^37 - 6 * q^38 - 6 * q^40 + 6 * q^41 - 26 * q^43 - 6 * q^44 + 4 * q^45 + 2 * q^46 - 30 * q^47 - 2 * q^48 - 6 * q^50 - 2 * q^51 + 14 * q^53 + 20 * q^54 - 6 * q^55 - 8 * q^57 + 2 * q^58 - 6 * q^59 - 2 * q^60 + 12 * q^61 + 2 * q^62 + 6 * q^64 + 4 * q^66 - 14 * q^67 + 6 * q^68 - 2 * q^69 + 12 * q^71 - 4 * q^72 - 12 * q^73 + 12 * q^74 - 2 * q^75 + 6 * q^76 - 36 * q^79 + 6 * q^80 - 2 * q^81 - 6 * q^82 - 32 * q^83 + 6 * q^85 + 26 * q^86 - 2 * q^87 + 6 * q^88 + 8 * q^89 - 4 * q^90 - 2 * q^92 - 12 * q^93 + 30 * q^94 + 6 * q^95 + 2 * q^96 + 16 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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