LMFDB - Embedded newform 6027.2.a.y.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6027,2,Mod(1,6027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6027, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6027 = 3 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6027.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$48.1258372982$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 6x^{5} + 16x^{4} + 14x^{3} - 20x^{2} - 10x + 4 $ x^7 - 3x^6 - 6x^5 + 16x^4 + 14x^3 - 20x^2 - 10x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 861)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.19009$ of defining polynomial
Character	$\chi$	$=$	6027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.190092 q^{2} -1.00000 q^{3} -1.96387 q^{4} -2.28332 q^{5} +0.190092 q^{6} +0.753499 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.190092 q^{2} -1.00000 q^{3} -1.96387 q^{4} -2.28332 q^{5} +0.190092 q^{6} +0.753499 q^{8} +1.00000 q^{9} +0.434041 q^{10} +1.72995 q^{11} +1.96387 q^{12} -1.61662 q^{13} +2.28332 q^{15} +3.78450 q^{16} +3.96707 q^{17} -0.190092 q^{18} -1.10774 q^{19} +4.48414 q^{20} -0.328850 q^{22} -2.20725 q^{23} -0.753499 q^{24} +0.213562 q^{25} +0.307306 q^{26} -1.00000 q^{27} +7.83771 q^{29} -0.434041 q^{30} -3.24019 q^{31} -2.22640 q^{32} -1.72995 q^{33} -0.754109 q^{34} -1.96387 q^{36} +1.90046 q^{37} +0.210573 q^{38} +1.61662 q^{39} -1.72048 q^{40} -1.00000 q^{41} +2.91253 q^{43} -3.39739 q^{44} -2.28332 q^{45} +0.419580 q^{46} -5.86193 q^{47} -3.78450 q^{48} -0.0405965 q^{50} -3.96707 q^{51} +3.17482 q^{52} -2.97657 q^{53} +0.190092 q^{54} -3.95004 q^{55} +1.10774 q^{57} -1.48989 q^{58} -5.61410 q^{59} -4.48414 q^{60} -4.86425 q^{61} +0.615934 q^{62} -7.14577 q^{64} +3.69127 q^{65} +0.328850 q^{66} -13.1894 q^{67} -7.79080 q^{68} +2.20725 q^{69} +11.1166 q^{71} +0.753499 q^{72} +0.406178 q^{73} -0.361261 q^{74} -0.213562 q^{75} +2.17546 q^{76} -0.307306 q^{78} -2.23080 q^{79} -8.64123 q^{80} +1.00000 q^{81} +0.190092 q^{82} +1.25842 q^{83} -9.05811 q^{85} -0.553649 q^{86} -7.83771 q^{87} +1.30352 q^{88} -8.84330 q^{89} +0.434041 q^{90} +4.33474 q^{92} +3.24019 q^{93} +1.11431 q^{94} +2.52934 q^{95} +2.22640 q^{96} +18.9259 q^{97} +1.72995 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 4 q^{2} - 7 q^{3} + 8 q^{4} - q^{5} - 4 q^{6} + 12 q^{8} + 7 q^{9}+O(q^{10}) $ 7 * q + 4 * q^2 - 7 * q^3 + 8 * q^4 - q^5 - 4 * q^6 + 12 * q^8 + 7 * q^9	$ 7 q + 4 q^{2} - 7 q^{3} + 8 q^{4} - q^{5} - 4 q^{6} + 12 q^{8} + 7 q^{9} + 3 q^{10} + 11 q^{11} - 8 q^{12} + 7 q^{13} + q^{15} + 6 q^{16} - 11 q^{17} + 4 q^{18} - 4 q^{19} + 7 q^{20} + 6 q^{22} + 7 q^{23} - 12 q^{24} + 2 q^{25} + 13 q^{26} - 7 q^{27} + 4 q^{29} - 3 q^{30} + 7 q^{31} + 18 q^{32} - 11 q^{33} + 20 q^{34} + 8 q^{36} - 4 q^{38} - 7 q^{39} + 9 q^{40} - 7 q^{41} + q^{43} + 18 q^{44} - q^{45} - 17 q^{46} - 14 q^{47} - 6 q^{48} + 19 q^{50} + 11 q^{51} + 27 q^{52} + 23 q^{53} - 4 q^{54} + 30 q^{55} + 4 q^{57} - 3 q^{58} - 8 q^{59} - 7 q^{60} + 3 q^{61} + 16 q^{62} + 6 q^{64} + 15 q^{65} - 6 q^{66} + 3 q^{67} - 7 q^{69} + 7 q^{71} + 12 q^{72} + 11 q^{73} - 13 q^{74} - 2 q^{75} + 40 q^{76} - 13 q^{78} - q^{79} + 43 q^{80} + 7 q^{81} - 4 q^{82} - 10 q^{85} - 12 q^{86} - 4 q^{87} + 10 q^{88} - 32 q^{89} + 3 q^{90} - 19 q^{92} - 7 q^{93} - 21 q^{94} - 8 q^{95} - 18 q^{96} + 25 q^{97} + 11 q^{99}+O(q^{100}) $ 7 * q + 4 * q^2 - 7 * q^3 + 8 * q^4 - q^5 - 4 * q^6 + 12 * q^8 + 7 * q^9 + 3 * q^10 + 11 * q^11 - 8 * q^12 + 7 * q^13 + q^15 + 6 * q^16 - 11 * q^17 + 4 * q^18 - 4 * q^19 + 7 * q^20 + 6 * q^22 + 7 * q^23 - 12 * q^24 + 2 * q^25 + 13 * q^26 - 7 * q^27 + 4 * q^29 - 3 * q^30 + 7 * q^31 + 18 * q^32 - 11 * q^33 + 20 * q^34 + 8 * q^36 - 4 * q^38 - 7 * q^39 + 9 * q^40 - 7 * q^41 + q^43 + 18 * q^44 - q^45 - 17 * q^46 - 14 * q^47 - 6 * q^48 + 19 * q^50 + 11 * q^51 + 27 * q^52 + 23 * q^53 - 4 * q^54 + 30 * q^55 + 4 * q^57 - 3 * q^58 - 8 * q^59 - 7 * q^60 + 3 * q^61 + 16 * q^62 + 6 * q^64 + 15 * q^65 - 6 * q^66 + 3 * q^67 - 7 * q^69 + 7 * q^71 + 12 * q^72 + 11 * q^73 - 13 * q^74 - 2 * q^75 + 40 * q^76 - 13 * q^78 - q^79 + 43 * q^80 + 7 * q^81 - 4 * q^82 - 10 * q^85 - 12 * q^86 - 4 * q^87 + 10 * q^88 - 32 * q^89 + 3 * q^90 - 19 * q^92 - 7 * q^93 - 21 * q^94 - 8 * q^95 - 18 * q^96 + 25 * q^97 + 11 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−0.190092	−0.134415	−0.0672077	−	0.997739i	$-0.521409\pi$
$2$	−0.190092	−0.134415	−0.0672077	+	0.997739i	$0.521409\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.96387	−0.981933
$5$	−2.28332	−1.02113	−0.510566	−	0.859838i	$-0.670564\pi$
$5$	−2.28332	−1.02113	−0.510566	+	0.859838i	$0.670564\pi$
$6$	0.190092	0.0776047
$7$	0	0
$7$	0	0
$8$	0.753499	0.266402
$9$	1.00000	0.333333
$10$	0.434041	0.137256
$11$	1.72995	0.521600	0.260800	−	0.965393i	$-0.416014\pi$
$11$	1.72995	0.521600	0.260800	+	0.965393i	$0.416014\pi$
$12$	1.96387	0.566919
$13$	−1.61662	−0.448370	−0.224185	−	0.974547i	$-0.571972\pi$
$13$	−1.61662	−0.448370	−0.224185	+	0.974547i	$0.571972\pi$
$14$	0	0
$15$	2.28332	0.589551
$16$	3.78450	0.946124
$17$	3.96707	0.962157	0.481079	−	0.876678i	$-0.340245\pi$
$17$	3.96707	0.962157	0.481079	+	0.876678i	$0.340245\pi$
$18$	−0.190092	−0.0448051
$19$	−1.10774	−0.254134	−0.127067	−	0.991894i	$-0.540556\pi$
$19$	−1.10774	−0.254134	−0.127067	+	0.991894i	$0.540556\pi$
$20$	4.48414	1.00268
$21$	0	0
$22$	−0.328850	−0.0701111
$23$	−2.20725	−0.460243	−0.230122	−	0.973162i	$-0.573912\pi$
$23$	−2.20725	−0.460243	−0.230122	+	0.973162i	$0.573912\pi$
$24$	−0.753499	−0.153807
$25$	0.213562	0.0427124
$26$	0.307306	0.0602678
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.83771	1.45543	0.727713	−	0.685882i	$-0.240583\pi$
$29$	7.83771	1.45543	0.727713	+	0.685882i	$0.240583\pi$
$30$	−0.434041	−0.0792447
$31$	−3.24019	−0.581955	−0.290977	−	0.956730i	$-0.593980\pi$
$31$	−3.24019	−0.581955	−0.290977	+	0.956730i	$0.593980\pi$
$32$	−2.22640	−0.393576
$33$	−1.72995	−0.301146
$34$	−0.754109	−0.129329
$35$	0	0
$36$	−1.96387	−0.327311
$37$	1.90046	0.312433	0.156217	−	0.987723i	$-0.450070\pi$
$37$	1.90046	0.312433	0.156217	+	0.987723i	$0.450070\pi$
$38$	0.210573	0.0341595
$39$	1.61662	0.258866
$40$	−1.72048	−0.272032
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	2.91253	0.444157	0.222078	−	0.975029i	$-0.428716\pi$
$43$	2.91253	0.444157	0.222078	+	0.975029i	$0.428716\pi$
$44$	−3.39739	−0.512176
$45$	−2.28332	−0.340378
$46$	0.419580	0.0618637
$47$	−5.86193	−0.855050	−0.427525	−	0.904003i	$-0.640614\pi$
$47$	−5.86193	−0.855050	−0.427525	+	0.904003i	$0.640614\pi$
$48$	−3.78450	−0.546245
$49$	0	0
$50$	−0.0405965	−0.00574121
$51$	−3.96707	−0.555502
$52$	3.17482	0.440269
$53$	−2.97657	−0.408864	−0.204432	−	0.978881i	$-0.565535\pi$
$53$	−2.97657	−0.408864	−0.204432	+	0.978881i	$0.565535\pi$
$54$	0.190092	0.0258682
$55$	−3.95004	−0.532623
$56$	0	0
$57$	1.10774	0.146724
$58$	−1.48989	−0.195632
$59$	−5.61410	−0.730893	−0.365447	−	0.930832i	$-0.619084\pi$
$59$	−5.61410	−0.730893	−0.365447	+	0.930832i	$0.619084\pi$
$60$	−4.48414	−0.578900
$61$	−4.86425	−0.622803	−0.311401	−	0.950278i	$-0.600798\pi$
$61$	−4.86425	−0.622803	−0.311401	+	0.950278i	$0.600798\pi$
$62$	0.615934	0.0782237
$63$	0	0
$64$	−7.14577	−0.893221
$65$	3.69127	0.457845
$66$	0.328850	0.0404786
$67$	−13.1894	−1.61134	−0.805672	−	0.592362i	$-0.798195\pi$
$67$	−13.1894	−1.61134	−0.805672	+	0.592362i	$0.798195\pi$
$68$	−7.79080	−0.944773
$69$	2.20725	0.265722
$70$	0	0
$71$	11.1166	1.31930	0.659648	−	0.751575i	$-0.270705\pi$
$71$	11.1166	1.31930	0.659648	+	0.751575i	$0.270705\pi$
$72$	0.753499	0.0888007
$73$	0.406178	0.0475395	0.0237697	−	0.999717i	$-0.492433\pi$
$73$	0.406178	0.0475395	0.0237697	+	0.999717i	$0.492433\pi$
$74$	−0.361261	−0.0419958
$75$	−0.213562	−0.0246600
$76$	2.17546	0.249542
$77$	0	0
$78$	−0.307306	−0.0347956
$79$	−2.23080	−0.250984	−0.125492	−	0.992095i	$-0.540051\pi$
$79$	−2.23080	−0.250984	−0.125492	+	0.992095i	$0.540051\pi$
$80$	−8.64123	−0.966118
$81$	1.00000	0.111111
$82$	0.190092	0.0209921
$83$	1.25842	0.138129	0.0690646	−	0.997612i	$-0.477999\pi$
$83$	1.25842	0.138129	0.0690646	+	0.997612i	$0.477999\pi$
$84$	0	0
$85$	−9.05811	−0.982490
$86$	−0.553649	−0.0597015
$87$	−7.83771	−0.840291
$88$	1.30352	0.138955
$89$	−8.84330	−0.937388	−0.468694	−	0.883361i	$-0.655275\pi$
$89$	−8.84330	−0.937388	−0.468694	+	0.883361i	$0.655275\pi$
$90$	0.434041	0.0457520
$91$	0	0
$92$	4.33474	0.451928
$93$	3.24019	0.335992
$94$	1.11431	0.114932
$95$	2.52934	0.259504
$96$	2.22640	0.227231
$97$	18.9259	1.92163	0.960816	−	0.277187i	$-0.0894022\pi$
$97$	18.9259	1.92163	0.960816	+	0.277187i	$0.0894022\pi$
$98$	0	0
$99$	1.72995	0.173867
$100$	−0.419407	−0.0419407
$101$	−2.37750	−0.236570	−0.118285	−	0.992980i	$-0.537740\pi$
$101$	−2.37750	−0.236570	−0.118285	+	0.992980i	$0.537740\pi$
$102$	0.754109	0.0746679
$103$	−1.27186	−0.125321	−0.0626603	−	0.998035i	$-0.519958\pi$
$103$	−1.27186	−0.125321	−0.0626603	+	0.998035i	$0.519958\pi$
$104$	−1.21812	−0.119447
$105$	0	0
$106$	0.565823	0.0549576
$107$	−15.4694	−1.49548	−0.747742	−	0.663989i	$-0.768862\pi$
$107$	−15.4694	−1.49548	−0.747742	+	0.663989i	$0.768862\pi$
$108$	1.96387	0.188973
$109$	−5.96439	−0.571285	−0.285643	−	0.958336i	$-0.592207\pi$
$109$	−5.96439	−0.571285	−0.285643	+	0.958336i	$0.592207\pi$
$110$	0.750871	0.0715927
$111$	−1.90046	−0.180383
$112$	0	0
$113$	9.73825	0.916097	0.458049	−	0.888927i	$-0.348549\pi$
$113$	9.73825	0.916097	0.458049	+	0.888927i	$0.348549\pi$
$114$	−0.210573	−0.0197220
$115$	5.03986	0.469970
$116$	−15.3922	−1.42913
$117$	−1.61662	−0.149457
$118$	1.06719	0.0982432
$119$	0	0
$120$	1.72048	0.157058
$121$	−8.00727	−0.727933
$122$	0.924654	0.0837142
$123$	1.00000	0.0901670
$124$	6.36329	0.571441
$125$	10.9290	0.977518
$126$	0	0
$127$	−4.74582	−0.421124	−0.210562	−	0.977581i	$-0.567529\pi$
$127$	−4.74582	−0.421124	−0.210562	+	0.977581i	$0.567529\pi$
$128$	5.81115	0.513638
$129$	−2.91253	−0.256434
$130$	−0.701680	−0.0615414
$131$	−3.47647	−0.303740	−0.151870	−	0.988400i	$-0.548530\pi$
$131$	−3.47647	−0.303740	−0.151870	+	0.988400i	$0.548530\pi$
$132$	3.39739	0.295705
$133$	0	0
$134$	2.50720	0.216589
$135$	2.28332	0.196517
$136$	2.98919	0.256321
$137$	−8.28210	−0.707587	−0.353794	−	0.935324i	$-0.615108\pi$
$137$	−8.28210	−0.707587	−0.353794	+	0.935324i	$0.615108\pi$
$138$	−0.419580	−0.0357170
$139$	6.78978	0.575902	0.287951	−	0.957645i	$-0.407026\pi$
$139$	6.78978	0.575902	0.287951	+	0.957645i	$0.407026\pi$
$140$	0	0
$141$	5.86193	0.493663
$142$	−2.11317	−0.177334
$143$	−2.79668	−0.233870
$144$	3.78450	0.315375
$145$	−17.8960	−1.48618
$146$	−0.0772111	−0.00639004
$147$	0	0
$148$	−3.73224	−0.306788
$149$	−4.76826	−0.390631	−0.195316	−	0.980740i	$-0.562573\pi$
$149$	−4.76826	−0.390631	−0.195316	+	0.980740i	$0.562573\pi$
$150$	0.0405965	0.00331469
$151$	−7.61766	−0.619917	−0.309958	−	0.950750i	$-0.600315\pi$
$151$	−7.61766	−0.619917	−0.309958	+	0.950750i	$0.600315\pi$
$152$	−0.834683	−0.0677017
$153$	3.96707	0.320719
$154$	0	0
$155$	7.39840	0.594253
$156$	−3.17482	−0.254189
$157$	12.4677	0.995031	0.497515	−	0.867455i	$-0.334246\pi$
$157$	12.4677	0.995031	0.497515	+	0.867455i	$0.334246\pi$
$158$	0.424056	0.0337361
$159$	2.97657	0.236058
$160$	5.08359	0.401893
$161$	0	0
$162$	−0.190092	−0.0149350
$163$	22.2666	1.74406	0.872028	−	0.489457i	$-0.162805\pi$
$163$	22.2666	1.74406	0.872028	+	0.489457i	$0.162805\pi$
$164$	1.96387	0.153352
$165$	3.95004	0.307510
$166$	−0.239215	−0.0185667
$167$	−14.3144	−1.10768	−0.553842	−	0.832622i	$-0.686839\pi$
$167$	−14.3144	−1.10768	−0.553842	+	0.832622i	$0.686839\pi$
$168$	0	0
$169$	−10.3865	−0.798965
$170$	1.72187	0.132062
$171$	−1.10774	−0.0847112
$172$	−5.71982	−0.436132
$173$	8.70139	0.661554	0.330777	−	0.943709i	$-0.392689\pi$
$173$	8.70139	0.661554	0.330777	+	0.943709i	$0.392689\pi$
$174$	1.48989	0.112948
$175$	0	0
$176$	6.54700	0.493498
$177$	5.61410	0.421981
$178$	1.68104	0.125999
$179$	−2.03988	−0.152468	−0.0762341	−	0.997090i	$-0.524290\pi$
$179$	−2.03988	−0.152468	−0.0762341	+	0.997090i	$0.524290\pi$
$180$	4.48414	0.334228
$181$	20.5069	1.52427	0.762133	−	0.647421i	$-0.224152\pi$
$181$	20.5069	1.52427	0.762133	+	0.647421i	$0.224152\pi$
$182$	0	0
$183$	4.86425	0.359575
$184$	−1.66316	−0.122610
$185$	−4.33935	−0.319036
$186$	−0.615934	−0.0451624
$187$	6.86285	0.501861
$188$	11.5120	0.839602
$189$	0	0
$190$	−0.480806	−0.0348813
$191$	10.2243	0.739803	0.369902	−	0.929071i	$-0.379391\pi$
$191$	10.2243	0.739803	0.369902	+	0.929071i	$0.379391\pi$
$192$	7.14577	0.515702
$193$	21.1438	1.52196	0.760982	−	0.648773i	$-0.224718\pi$
$193$	21.1438	1.52196	0.760982	+	0.648773i	$0.224718\pi$
$194$	−3.59766	−0.258297
$195$	−3.69127	−0.264337
$196$	0	0
$197$	−15.1685	−1.08071	−0.540355	−	0.841437i	$-0.681710\pi$
$197$	−15.1685	−1.08071	−0.540355	+	0.841437i	$0.681710\pi$
$198$	−0.328850	−0.0233704
$199$	7.99926	0.567053	0.283526	−	0.958964i	$-0.408496\pi$
$199$	7.99926	0.567053	0.283526	+	0.958964i	$0.408496\pi$
$200$	0.160919	0.0113787
$201$	13.1894	0.930310
$202$	0.451944	0.0317987
$203$	0	0
$204$	7.79080	0.545465
$205$	2.28332	0.159474
$206$	0.241771	0.0168450
$207$	−2.20725	−0.153414
$208$	−6.11809	−0.424213
$209$	−1.91634	−0.132556
$210$	0	0
$211$	−18.1670	−1.25067	−0.625335	−	0.780356i	$-0.715038\pi$
$211$	−18.1670	−1.25067	−0.625335	+	0.780356i	$0.715038\pi$
$212$	5.84559	0.401477
$213$	−11.1166	−0.761696
$214$	2.94061	0.201016
$215$	−6.65025	−0.453543
$216$	−0.753499	−0.0512691
$217$	0	0
$218$	1.13378	0.0767895
$219$	−0.406178	−0.0274469
$220$	7.75734	0.523000
$221$	−6.41325	−0.431402
$222$	0.361261	0.0242463
$223$	15.1901	1.01720	0.508601	−	0.861002i	$-0.330163\pi$
$223$	15.1901	1.01720	0.508601	+	0.861002i	$0.330163\pi$
$224$	0	0
$225$	0.213562	0.0142375
$226$	−1.85116	−0.123138
$227$	11.3294	0.751958	0.375979	−	0.926628i	$-0.377307\pi$
$227$	11.3294	0.751958	0.375979	+	0.926628i	$0.377307\pi$
$228$	−2.17546	−0.144073
$229$	16.3760	1.08216	0.541079	−	0.840972i	$-0.318016\pi$
$229$	16.3760	1.08216	0.541079	+	0.840972i	$0.318016\pi$
$230$	−0.958037	−0.0631711
$231$	0	0
$232$	5.90570	0.387729
$233$	−2.40958	−0.157857	−0.0789284	−	0.996880i	$-0.525150\pi$
$233$	−2.40958	−0.157857	−0.0789284	+	0.996880i	$0.525150\pi$
$234$	0.307306	0.0200893
$235$	13.3847	0.873120
$236$	11.0253	0.717688
$237$	2.23080	0.144906
$238$	0	0
$239$	8.04273	0.520241	0.260120	−	0.965576i	$-0.416238\pi$
$239$	8.04273	0.520241	0.260120	+	0.965576i	$0.416238\pi$
$240$	8.64123	0.557789
$241$	16.4724	1.06108	0.530541	−	0.847659i	$-0.321989\pi$
$241$	16.4724	1.06108	0.530541	+	0.847659i	$0.321989\pi$
$242$	1.52212	0.0978454
$243$	−1.00000	−0.0641500
$244$	9.55272	0.611550
$245$	0	0
$246$	−0.190092	−0.0121198
$247$	1.79080	0.113946
$248$	−2.44148	−0.155034
$249$	−1.25842	−0.0797489
$250$	−2.07751	−0.131393
$251$	12.7762	0.806430	0.403215	−	0.915105i	$-0.367893\pi$
$251$	12.7762	0.806430	0.403215	+	0.915105i	$0.367893\pi$
$252$	0	0
$253$	−3.81844	−0.240063
$254$	0.902143	0.0566055
$255$	9.05811	0.567241
$256$	13.1869	0.824181
$257$	−17.5803	−1.09663	−0.548315	−	0.836272i	$-0.684731\pi$
$257$	−17.5803	−1.09663	−0.548315	+	0.836272i	$0.684731\pi$
$258$	0.553649	0.0344687
$259$	0	0
$260$	−7.24915	−0.449573
$261$	7.83771	0.485142
$262$	0.660848	0.0408273
$263$	2.61090	0.160995	0.0804976	−	0.996755i	$-0.474349\pi$
$263$	2.61090	0.160995	0.0804976	+	0.996755i	$0.474349\pi$
$264$	−1.30352	−0.0802259
$265$	6.79648	0.417504
$266$	0	0
$267$	8.84330	0.541201
$268$	25.9022	1.58223
$269$	18.5601	1.13163	0.565816	−	0.824532i	$-0.308561\pi$
$269$	18.5601	1.13163	0.565816	+	0.824532i	$0.308561\pi$
$270$	−0.434041	−0.0264149
$271$	−21.3817	−1.29885	−0.649423	−	0.760427i	$-0.724990\pi$
$271$	−21.3817	−1.29885	−0.649423	+	0.760427i	$0.724990\pi$
$272$	15.0134	0.910320
$273$	0	0
$274$	1.57436	0.0951105
$275$	0.369452	0.0222788
$276$	−4.33474	−0.260921
$277$	14.0879	0.846462	0.423231	−	0.906022i	$-0.360896\pi$
$277$	14.0879	0.846462	0.423231	+	0.906022i	$0.360896\pi$
$278$	−1.29068	−0.0774100
$279$	−3.24019	−0.193985
$280$	0	0
$281$	−22.3895	−1.33565	−0.667823	−	0.744320i	$-0.732774\pi$
$281$	−22.3895	−1.33565	−0.667823	+	0.744320i	$0.732774\pi$
$282$	−1.11431	−0.0663559
$283$	10.5639	0.627961	0.313980	−	0.949429i	$-0.398337\pi$
$283$	10.5639	0.627961	0.313980	+	0.949429i	$0.398337\pi$
$284$	−21.8315	−1.29546
$285$	−2.52934	−0.149825
$286$	0.531625	0.0314357
$287$	0	0
$288$	−2.22640	−0.131192
$289$	−1.26232	−0.0742539
$290$	3.40189	0.199766
$291$	−18.9259	−1.10945
$292$	−0.797678	−0.0466806
$293$	8.74247	0.510740	0.255370	−	0.966843i	$-0.417803\pi$
$293$	8.74247	0.510740	0.255370	+	0.966843i	$0.417803\pi$
$294$	0	0
$295$	12.8188	0.746339
$296$	1.43199	0.0832328
$297$	−1.72995	−0.100382
$298$	0.906408	0.0525068
$299$	3.56828	0.206359
$300$	0.419407	0.0242145
$301$	0	0
$302$	1.44806	0.0833263
$303$	2.37750	0.136584
$304$	−4.19225	−0.240442
$305$	11.1066	0.635964
$306$	−0.754109	−0.0431095
$307$	−7.97557	−0.455190	−0.227595	−	0.973756i	$-0.573086\pi$
$307$	−7.97557	−0.455190	−0.227595	+	0.973756i	$0.573086\pi$
$308$	0	0
$309$	1.27186	0.0723538
$310$	−1.40638	−0.0798767
$311$	−14.1483	−0.802276	−0.401138	−	0.916018i	$-0.631385\pi$
$311$	−14.1483	−0.802276	−0.401138	+	0.916018i	$0.631385\pi$
$312$	1.21812	0.0689625
$313$	8.96795	0.506898	0.253449	−	0.967349i	$-0.418435\pi$
$313$	8.96795	0.506898	0.253449	+	0.967349i	$0.418435\pi$
$314$	−2.37001	−0.133747
$315$	0	0
$316$	4.38098	0.246449
$317$	8.04996	0.452131	0.226065	−	0.974112i	$-0.427414\pi$
$317$	8.04996	0.452131	0.226065	+	0.974112i	$0.427414\pi$
$318$	−0.565823	−0.0317298
$319$	13.5589	0.759151
$320$	16.3161	0.912098
$321$	15.4694	0.863419
$322$	0	0
$323$	−4.39450	−0.244517
$324$	−1.96387	−0.109104
$325$	−0.345249	−0.0191510
$326$	−4.23270	−0.234428
$327$	5.96439	0.329832
$328$	−0.753499	−0.0416050
$329$	0	0
$330$	−0.750871	−0.0413341
$331$	17.3497	0.953628	0.476814	−	0.879004i	$-0.341792\pi$
$331$	17.3497	0.953628	0.476814	+	0.879004i	$0.341792\pi$
$332$	−2.47136	−0.135634
$333$	1.90046	0.104144
$334$	2.72106	0.148890
$335$	30.1157	1.64540
$336$	0	0
$337$	−29.8239	−1.62461	−0.812307	−	0.583230i	$-0.801788\pi$
$337$	−29.8239	−1.62461	−0.812307	+	0.583230i	$0.801788\pi$
$338$	1.97440	0.107393
$339$	−9.73825	−0.528909
$340$	17.7889	0.964739
$341$	−5.60537	−0.303548
$342$	0.210573	0.0113865
$343$	0	0
$344$	2.19459	0.118324
$345$	−5.03986	−0.271337
$346$	−1.65406	−0.0889230
$347$	18.3160	0.983255	0.491628	−	0.870805i	$-0.336402\pi$
$347$	18.3160	0.983255	0.491628	+	0.870805i	$0.336402\pi$
$348$	15.3922	0.825109
$349$	6.08069	0.325492	0.162746	−	0.986668i	$-0.447965\pi$
$349$	6.08069	0.325492	0.162746	+	0.986668i	$0.447965\pi$
$350$	0	0
$351$	1.61662	0.0862888
$352$	−3.85157	−0.205289
$353$	6.85088	0.364635	0.182318	−	0.983240i	$-0.441640\pi$
$353$	6.85088	0.364635	0.182318	+	0.983240i	$0.441640\pi$
$354$	−1.06719	−0.0567207
$355$	−25.3827	−1.34718
$356$	17.3670	0.920452
$357$	0	0
$358$	0.387766	0.0204940
$359$	0.354000	0.0186834	0.00934171	−	0.999956i	$-0.497026\pi$
$359$	0.354000	0.0186834	0.00934171	+	0.999956i	$0.497026\pi$
$360$	−1.72048	−0.0906773
$361$	−17.7729	−0.935416
$362$	−3.89820	−0.204885
$363$	8.00727	0.420272
$364$	0	0
$365$	−0.927435	−0.0485441
$366$	−0.924654	−0.0483324
$367$	20.8456	1.08813	0.544065	−	0.839043i	$-0.316884\pi$
$367$	20.8456	1.08813	0.544065	+	0.839043i	$0.316884\pi$
$368$	−8.35333	−0.435447
$369$	−1.00000	−0.0520579
$370$	0.824876	0.0428833
$371$	0	0
$372$	−6.36329	−0.329921
$373$	35.6438	1.84557	0.922783	−	0.385320i	$-0.125909\pi$
$373$	35.6438	1.84557	0.922783	+	0.385320i	$0.125909\pi$
$374$	−1.30457	−0.0674578
$375$	−10.9290	−0.564370
$376$	−4.41696	−0.227787
$377$	−12.6706	−0.652569
$378$	0	0
$379$	15.9661	0.820125	0.410063	−	0.912057i	$-0.365507\pi$
$379$	15.9661	0.820125	0.410063	+	0.912057i	$0.365507\pi$
$380$	−4.96727	−0.254816
$381$	4.74582	0.243136
$382$	−1.94355	−0.0994408
$383$	14.1502	0.723043	0.361521	−	0.932364i	$-0.382257\pi$
$383$	14.1502	0.723043	0.361521	+	0.932364i	$0.382257\pi$
$384$	−5.81115	−0.296549
$385$	0	0
$386$	−4.01926	−0.204575
$387$	2.91253	0.148052
$388$	−37.1679	−1.88691
$389$	9.66026	0.489795	0.244897	−	0.969549i	$-0.421246\pi$
$389$	9.66026	0.489795	0.244897	+	0.969549i	$0.421246\pi$
$390$	0.701680	0.0355309
$391$	−8.75632	−0.442826
$392$	0	0
$393$	3.47647	0.175365
$394$	2.88340	0.145264
$395$	5.09363	0.256288
$396$	−3.39739	−0.170725
$397$	−9.54771	−0.479186	−0.239593	−	0.970873i	$-0.577014\pi$
$397$	−9.54771	−0.479186	−0.239593	+	0.970873i	$0.577014\pi$
$398$	−1.52060	−0.0762206
$399$	0	0
$400$	0.808225	0.0404113
$401$	34.5897	1.72733	0.863663	−	0.504070i	$-0.168165\pi$
$401$	34.5897	1.72733	0.863663	+	0.504070i	$0.168165\pi$
$402$	−2.50720	−0.125048
$403$	5.23815	0.260931
$404$	4.66909	0.232296
$405$	−2.28332	−0.113459
$406$	0	0
$407$	3.28770	0.162965
$408$	−2.98919	−0.147987
$409$	−27.9927	−1.38415	−0.692074	−	0.721826i	$-0.743303\pi$
$409$	−27.9927	−1.38415	−0.692074	+	0.721826i	$0.743303\pi$
$410$	−0.434041	−0.0214358
$411$	8.28210	0.408526
$412$	2.49777	0.123056
$413$	0	0
$414$	0.419580	0.0206212
$415$	−2.87337	−0.141048
$416$	3.59924	0.176467
$417$	−6.78978	−0.332497
$418$	0.364281	0.0178176
$419$	−20.1883	−0.986263	−0.493131	−	0.869955i	$-0.664148\pi$
$419$	−20.1883	−0.986263	−0.493131	+	0.869955i	$0.664148\pi$
$420$	0	0
$421$	7.88294	0.384191	0.192096	−	0.981376i	$-0.438472\pi$
$421$	7.88294	0.384191	0.192096	+	0.981376i	$0.438472\pi$
$422$	3.45341	0.168109
$423$	−5.86193	−0.285017
$424$	−2.24285	−0.108922
$425$	0.847217	0.0410961
$426$	2.11317	0.102384
$427$	0	0
$428$	30.3798	1.46847
$429$	2.79668	0.135025
$430$	1.26416	0.0609631
$431$	−4.43615	−0.213682	−0.106841	−	0.994276i	$-0.534074\pi$
$431$	−4.43615	−0.213682	−0.106841	+	0.994276i	$0.534074\pi$
$432$	−3.78450	−0.182082
$433$	7.00079	0.336436	0.168218	−	0.985750i	$-0.446199\pi$
$433$	7.00079	0.336436	0.168218	+	0.985750i	$0.446199\pi$
$434$	0	0
$435$	17.8960	0.858048
$436$	11.7133	0.560964
$437$	2.44507	0.116963
$438$	0.0772111	0.00368929
$439$	11.0416	0.526988	0.263494	−	0.964661i	$-0.415125\pi$
$439$	11.0416	0.526988	0.263494	+	0.964661i	$0.415125\pi$
$440$	−2.97635	−0.141892
$441$	0	0
$442$	1.21911	0.0579870
$443$	−6.88937	−0.327324	−0.163662	−	0.986516i	$-0.552331\pi$
$443$	−6.88937	−0.327324	−0.163662	+	0.986516i	$0.552331\pi$
$444$	3.73224	0.177124
$445$	20.1921	0.957197
$446$	−2.88751	−0.136728
$447$	4.76826	0.225531
$448$	0	0
$449$	27.0247	1.27538	0.637688	−	0.770295i	$-0.279891\pi$
$449$	27.0247	1.27538	0.637688	+	0.770295i	$0.279891\pi$
$450$	−0.0405965	−0.00191374
$451$	−1.72995	−0.0814603
$452$	−19.1246	−0.899546
$453$	7.61766	0.357909
$454$	−2.15362	−0.101075
$455$	0	0
$456$	0.834683	0.0390876
$457$	2.39212	0.111899	0.0559494	−	0.998434i	$-0.482181\pi$
$457$	2.39212	0.111899	0.0559494	+	0.998434i	$0.482181\pi$
$458$	−3.11295	−0.145459
$459$	−3.96707	−0.185167
$460$	−9.89761	−0.461478
$461$	−15.9987	−0.745136	−0.372568	−	0.928005i	$-0.621523\pi$
$461$	−15.9987	−0.745136	−0.372568	+	0.928005i	$0.621523\pi$
$462$	0	0
$463$	17.3408	0.805893	0.402947	−	0.915224i	$-0.367986\pi$
$463$	17.3408	0.805893	0.402947	+	0.915224i	$0.367986\pi$
$464$	29.6618	1.37701
$465$	−7.39840	−0.343092
$466$	0.458042	0.0212184
$467$	31.7666	1.46998	0.734990	−	0.678078i	$-0.237187\pi$
$467$	31.7666	1.46998	0.734990	+	0.678078i	$0.237187\pi$
$468$	3.17482	0.146756
$469$	0	0
$470$	−2.54432	−0.117361
$471$	−12.4677	−0.574481
$472$	−4.23022	−0.194711
$473$	5.03854	0.231672
$474$	−0.424056	−0.0194776
$475$	−0.236572	−0.0108547
$476$	0	0
$477$	−2.97657	−0.136288
$478$	−1.52886	−0.0699283
$479$	−12.5261	−0.572331	−0.286166	−	0.958180i	$-0.592381\pi$
$479$	−12.5261	−0.572331	−0.286166	+	0.958180i	$0.592381\pi$
$480$	−5.08359	−0.232033
$481$	−3.07232	−0.140086
$482$	−3.13128	−0.142626
$483$	0	0
$484$	15.7252	0.714781
$485$	−43.2139	−1.96224
$486$	0.190092	0.00862275
$487$	18.9271	0.857670	0.428835	−	0.903383i	$-0.358924\pi$
$487$	18.9271	0.857670	0.428835	+	0.903383i	$0.358924\pi$
$488$	−3.66520	−0.165916
$489$	−22.2666	−1.00693
$490$	0	0
$491$	39.3059	1.77385	0.886926	−	0.461912i	$-0.152837\pi$
$491$	39.3059	1.77385	0.886926	+	0.461912i	$0.152837\pi$
$492$	−1.96387	−0.0885379
$493$	31.0928	1.40035
$494$	−0.340417	−0.0153161
$495$	−3.95004	−0.177541
$496$	−12.2625	−0.550602
$497$	0	0
$498$	0.239215	0.0107195
$499$	−0.0965158	−0.00432064	−0.00216032	−	0.999998i	$-0.500688\pi$
$499$	−0.0965158	−0.00432064	−0.00216032	+	0.999998i	$0.500688\pi$
$500$	−21.4630	−0.959857
$501$	14.3144	0.639521
$502$	−2.42866	−0.108396
$503$	−35.7044	−1.59198	−0.795990	−	0.605310i	$-0.793049\pi$
$503$	−35.7044	−1.59198	−0.795990	+	0.605310i	$0.793049\pi$
$504$	0	0
$505$	5.42860	0.241570
$506$	0.725854	0.0322681
$507$	10.3865	0.461282
$508$	9.32016	0.413515
$509$	−6.76202	−0.299721	−0.149861	−	0.988707i	$-0.547883\pi$
$509$	−6.76202	−0.299721	−0.149861	+	0.988707i	$0.547883\pi$
$510$	−1.72187	−0.0762459
$511$	0	0
$512$	−14.1290	−0.624421
$513$	1.10774	0.0489081
$514$	3.34188	0.147404
$515$	2.90408	0.127969
$516$	5.71982	0.251801
$517$	−10.1409	−0.445994
$518$	0	0
$519$	−8.70139	−0.381949
$520$	2.78136	0.121971
$521$	10.8582	0.475708	0.237854	−	0.971301i	$-0.423556\pi$
$521$	10.8582	0.475708	0.237854	+	0.971301i	$0.423556\pi$
$522$	−1.48989	−0.0652105
$523$	8.65423	0.378423	0.189212	−	0.981936i	$-0.439407\pi$
$523$	8.65423	0.378423	0.189212	+	0.981936i	$0.439407\pi$
$524$	6.82731	0.298252
$525$	0	0
$526$	−0.496312	−0.0216402
$527$	−12.8541	−0.559932
$528$	−6.54700	−0.284921
$529$	−18.1281	−0.788176
$530$	−1.29196	−0.0561190
$531$	−5.61410	−0.243631
$532$	0	0
$533$	1.61662	0.0700236
$534$	−1.68104	−0.0727457
$535$	35.3217	1.52709
$536$	−9.93821	−0.429265
$537$	2.03988	0.0880275
$538$	−3.52813	−0.152109
$539$	0	0
$540$	−4.48414	−0.192967
$541$	34.5488	1.48537	0.742684	−	0.669642i	$-0.233552\pi$
$541$	34.5488	1.48537	0.742684	+	0.669642i	$0.233552\pi$
$542$	4.06449	0.174585
$543$	−20.5069	−0.880035
$544$	−8.83229	−0.378682
$545$	13.6186	0.583358
$546$	0	0
$547$	6.28734	0.268827	0.134414	−	0.990925i	$-0.457085\pi$
$547$	6.28734	0.268827	0.134414	+	0.990925i	$0.457085\pi$
$548$	16.2649	0.694803
$549$	−4.86425	−0.207601
$550$	−0.0702299	−0.00299461
$551$	−8.68217	−0.369873
$552$	1.66316	0.0707888
$553$	0	0
$554$	−2.67800	−0.113777
$555$	4.33935	0.184195
$556$	−13.3342	−0.565497
$557$	−23.1197	−0.979612	−0.489806	−	0.871832i	$-0.662932\pi$
$557$	−23.1197	−0.979612	−0.489806	+	0.871832i	$0.662932\pi$
$558$	0.615934	0.0260746
$559$	−4.70846	−0.199146
$560$	0	0
$561$	−6.86285	−0.289750
$562$	4.25606	0.179531
$563$	−43.4261	−1.83019	−0.915095	−	0.403238i	$-0.867885\pi$
$563$	−43.4261	−1.83019	−0.915095	+	0.403238i	$0.867885\pi$
$564$	−11.5120	−0.484744
$565$	−22.2356	−0.935457
$566$	−2.00812	−0.0844076
$567$	0	0
$568$	8.37633	0.351463
$569$	0.0739827	0.00310152	0.00155076	−	0.999999i	$-0.499506\pi$
$569$	0.0739827	0.00310152	0.00155076	+	0.999999i	$0.499506\pi$
$570$	0.480806	0.0201388
$571$	−5.95013	−0.249005	−0.124503	−	0.992219i	$-0.539734\pi$
$571$	−5.95013	−0.249005	−0.124503	+	0.992219i	$0.539734\pi$
$572$	5.49229	0.229644
$573$	−10.2243	−0.427125
$574$	0	0
$575$	−0.471385	−0.0196581
$576$	−7.14577	−0.297740
$577$	29.6618	1.23484	0.617418	−	0.786636i	$-0.288179\pi$
$577$	29.6618	1.23484	0.617418	+	0.786636i	$0.288179\pi$
$578$	0.239956	0.00998086
$579$	−21.1438	−0.878706
$580$	35.1454	1.45933
$581$	0	0
$582$	3.59766	0.149128
$583$	−5.14933	−0.213264
$584$	0.306054	0.0126646
$585$	3.69127	0.152615
$586$	−1.66187	−0.0686513
$587$	1.71541	0.0708025	0.0354012	−	0.999373i	$-0.488729\pi$
$587$	1.71541	0.0708025	0.0354012	+	0.999373i	$0.488729\pi$
$588$	0	0
$589$	3.58930	0.147894
$590$	−2.43675	−0.100319
$591$	15.1685	0.623948
$592$	7.19227	0.295600
$593$	−3.69486	−0.151730	−0.0758650	−	0.997118i	$-0.524172\pi$
$593$	−3.69486	−0.151730	−0.0758650	+	0.997118i	$0.524172\pi$
$594$	0.328850	0.0134929
$595$	0	0
$596$	9.36422	0.383573
$597$	−7.99926	−0.327388
$598$	−0.678302	−0.0277378
$599$	−23.5445	−0.962002	−0.481001	−	0.876720i	$-0.659727\pi$
$599$	−23.5445	−0.962002	−0.481001	+	0.876720i	$0.659727\pi$
$600$	−0.160919	−0.00656949
$601$	9.90791	0.404152	0.202076	−	0.979370i	$-0.435231\pi$
$601$	9.90791	0.404152	0.202076	+	0.979370i	$0.435231\pi$
$602$	0	0
$603$	−13.1894	−0.537115
$604$	14.9601	0.608716
$605$	18.2832	0.743317
$606$	−0.451944	−0.0183590
$607$	27.9949	1.13628	0.568139	−	0.822933i	$-0.307664\pi$
$607$	27.9949	1.13628	0.568139	+	0.822933i	$0.307664\pi$
$608$	2.46628	0.100021
$609$	0	0
$610$	−2.11128	−0.0854833
$611$	9.47651	0.383379
$612$	−7.79080	−0.314924
$613$	−12.2122	−0.493245	−0.246622	−	0.969112i	$-0.579321\pi$
$613$	−12.2122	−0.493245	−0.246622	+	0.969112i	$0.579321\pi$
$614$	1.51609	0.0611845
$615$	−2.28332	−0.0920725
$616$	0	0
$617$	3.44647	0.138750	0.0693748	−	0.997591i	$-0.477900\pi$
$617$	3.44647	0.138750	0.0693748	+	0.997591i	$0.477900\pi$
$618$	−0.241771	−0.00972546
$619$	19.3249	0.776734	0.388367	−	0.921505i	$-0.373039\pi$
$619$	19.3249	0.776734	0.388367	+	0.921505i	$0.373039\pi$
$620$	−14.5294	−0.583517
$621$	2.20725	0.0885739
$622$	2.68948	0.107838
$623$	0	0
$624$	6.11809	0.244920
$625$	−26.0222	−1.04089
$626$	−1.70473	−0.0681349
$627$	1.91634	0.0765314
$628$	−24.4849	−0.977053
$629$	7.53925	0.300610
$630$	0	0
$631$	11.9587	0.476068	0.238034	−	0.971257i	$-0.423497\pi$
$631$	11.9587	0.476068	0.238034	+	0.971257i	$0.423497\pi$
$632$	−1.68090	−0.0668627
$633$	18.1670	0.722075
$634$	−1.53023	−0.0607733
$635$	10.8362	0.430023
$636$	−5.84559	−0.231793
$637$	0	0
$638$	−2.57743	−0.102041
$639$	11.1166	0.439765
$640$	−13.2687	−0.524493
$641$	−46.6531	−1.84269	−0.921343	−	0.388751i	$-0.872907\pi$
$641$	−46.6531	−1.84269	−0.921343	+	0.388751i	$0.872907\pi$
$642$	−2.94061	−0.116057
$643$	7.35668	0.290119	0.145059	−	0.989423i	$-0.453663\pi$
$643$	7.35668	0.290119	0.145059	+	0.989423i	$0.453663\pi$
$644$	0	0
$645$	6.65025	0.261853
$646$	0.835359	0.0328668
$647$	−2.57068	−0.101064	−0.0505320	−	0.998722i	$-0.516092\pi$
$647$	−2.57068	−0.101064	−0.0505320	+	0.998722i	$0.516092\pi$
$648$	0.753499	0.0296002
$649$	−9.71212	−0.381234
$650$	0.0656290	0.00257418
$651$	0	0
$652$	−43.7286	−1.71254
$653$	−31.3911	−1.22843	−0.614215	−	0.789139i	$-0.710527\pi$
$653$	−31.3911	−1.22843	−0.614215	+	0.789139i	$0.710527\pi$
$654$	−1.13378	−0.0443344
$655$	7.93789	0.310159
$656$	−3.78450	−0.147760
$657$	0.406178	0.0158465
$658$	0	0
$659$	2.55964	0.0997094	0.0498547	−	0.998756i	$-0.484124\pi$
$659$	2.55964	0.0997094	0.0498547	+	0.998756i	$0.484124\pi$
$660$	−7.75734	−0.301954
$661$	19.6281	0.763445	0.381723	−	0.924277i	$-0.375331\pi$
$661$	19.6281	0.763445	0.381723	+	0.924277i	$0.375331\pi$
$662$	−3.29805	−0.128182
$663$	6.41325	0.249070
$664$	0.948216	0.0367979
$665$	0	0
$666$	−0.361261	−0.0139986
$667$	−17.2998	−0.669850
$668$	28.1116	1.08767
$669$	−15.1901	−0.587282
$670$	−5.72475	−0.221166
$671$	−8.41491	−0.324854
$672$	0	0
$673$	17.9831	0.693198	0.346599	−	0.938013i	$-0.387337\pi$
$673$	17.9831	0.693198	0.346599	+	0.938013i	$0.387337\pi$
$674$	5.66929	0.218373
$675$	−0.213562	−0.00822001
$676$	20.3978	0.784529
$677$	49.9993	1.92163	0.960815	−	0.277189i	$-0.0894029\pi$
$677$	49.9993	1.92163	0.960815	+	0.277189i	$0.0894029\pi$
$678$	1.85116	0.0710935
$679$	0	0
$680$	−6.82528	−0.261737
$681$	−11.3294	−0.434143
$682$	1.06554	0.0408015
$683$	−0.152907	−0.00585081	−0.00292541	−	0.999996i	$-0.500931\pi$
$683$	−0.152907	−0.00585081	−0.00292541	+	0.999996i	$0.500931\pi$
$684$	2.17546	0.0831807
$685$	18.9107	0.722541
$686$	0	0
$687$	−16.3760	−0.624784
$688$	11.0225	0.420227
$689$	4.81199	0.183322
$690$	0.958037	0.0364719
$691$	5.74600	0.218588	0.109294	−	0.994009i	$-0.465141\pi$
$691$	5.74600	0.218588	0.109294	+	0.994009i	$0.465141\pi$
$692$	−17.0884	−0.649602
$693$	0	0
$694$	−3.48173	−0.132165
$695$	−15.5033	−0.588072
$696$	−5.90570	−0.223855
$697$	−3.96707	−0.150264
$698$	−1.15589	−0.0437511
$699$	2.40958	0.0911387
$700$	0	0
$701$	−34.6654	−1.30929	−0.654647	−	0.755934i	$-0.727183\pi$
$701$	−34.6654	−1.30929	−0.654647	+	0.755934i	$0.727183\pi$
$702$	−0.307306	−0.0115985
$703$	−2.10522	−0.0793998
$704$	−12.3618	−0.465904
$705$	−13.3847	−0.504096
$706$	−1.30230	−0.0490126
$707$	0	0
$708$	−11.0253	−0.414357
$709$	−12.1602	−0.456684	−0.228342	−	0.973581i	$-0.573331\pi$
$709$	−12.1602	−0.456684	−0.228342	+	0.973581i	$0.573331\pi$
$710$	4.82506	0.181081
$711$	−2.23080	−0.0836614
$712$	−6.66341	−0.249722
$713$	7.15190	0.267841
$714$	0	0
$715$	6.38571	0.238812
$716$	4.00606	0.149713
$717$	−8.04273	−0.300361
$718$	−0.0672926	−0.00251134
$719$	−15.6701	−0.584396	−0.292198	−	0.956358i	$-0.594387\pi$
$719$	−15.6701	−0.584396	−0.292198	+	0.956358i	$0.594387\pi$
$720$	−8.64123	−0.322039
$721$	0	0
$722$	3.37849	0.125734
$723$	−16.4724	−0.612616
$724$	−40.2728	−1.49673
$725$	1.67384	0.0621648
$726$	−1.52212	−0.0564910
$727$	−5.01300	−0.185922	−0.0929609	−	0.995670i	$-0.529633\pi$
$727$	−5.01300	−0.185922	−0.0929609	+	0.995670i	$0.529633\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0.176298	0.00652508
$731$	11.5542	0.427349
$732$	−9.55272	−0.353079
$733$	−8.01404	−0.296005	−0.148003	−	0.988987i	$-0.547284\pi$
$733$	−8.01404	−0.296005	−0.148003	+	0.988987i	$0.547284\pi$
$734$	−3.96258	−0.146261
$735$	0	0
$736$	4.91422	0.181141
$737$	−22.8171	−0.840477
$738$	0.190092	0.00699738
$739$	0.161151	0.00592802	0.00296401	−	0.999996i	$-0.499057\pi$
$739$	0.161151	0.00592802	0.00296401	+	0.999996i	$0.499057\pi$
$740$	8.52191	0.313272
$741$	−1.79080	−0.0657867
$742$	0	0
$743$	−10.1913	−0.373884	−0.186942	−	0.982371i	$-0.559858\pi$
$743$	−10.1913	−0.373884	−0.186942	+	0.982371i	$0.559858\pi$
$744$	2.44148	0.0895089
$745$	10.8875	0.398886
$746$	−6.77560	−0.248072
$747$	1.25842	0.0460431
$748$	−13.4777	−0.492794
$749$	0	0
$750$	2.07751	0.0758600
$751$	−5.99474	−0.218751	−0.109376	−	0.994000i	$-0.534885\pi$
$751$	−5.99474	−0.218751	−0.109376	+	0.994000i	$0.534885\pi$
$752$	−22.1844	−0.808983
$753$	−12.7762	−0.465592
$754$	2.40858	0.0877153
$755$	17.3936	0.633017
$756$	0	0
$757$	−45.7296	−1.66207	−0.831035	−	0.556220i	$-0.812251\pi$
$757$	−45.7296	−1.66207	−0.831035	+	0.556220i	$0.812251\pi$
$758$	−3.03503	−0.110237
$759$	3.81844	0.138600
$760$	1.90585	0.0691325
$761$	−5.82408	−0.211123	−0.105561	−	0.994413i	$-0.533664\pi$
$761$	−5.82408	−0.211123	−0.105561	+	0.994413i	$0.533664\pi$
$762$	−0.902143	−0.0326812
$763$	0	0
$764$	−20.0791	−0.726437
$765$	−9.05811	−0.327497
$766$	−2.68984	−0.0971880
$767$	9.07586	0.327710
$768$	−13.1869	−0.475841
$769$	−18.0668	−0.651506	−0.325753	−	0.945455i	$-0.605618\pi$
$769$	−18.0668	−0.651506	−0.325753	+	0.945455i	$0.605618\pi$
$770$	0	0
$771$	17.5803	0.633140
$772$	−41.5235	−1.49447
$773$	−6.14546	−0.221037	−0.110518	−	0.993874i	$-0.535251\pi$
$773$	−6.14546	−0.221037	−0.110518	+	0.993874i	$0.535251\pi$
$774$	−0.553649	−0.0199005
$775$	−0.691982	−0.0248567
$776$	14.2606	0.511927
$777$	0	0
$778$	−1.83634	−0.0658359
$779$	1.10774	0.0396890
$780$	7.24915	0.259561
$781$	19.2312	0.688145
$782$	1.66451	0.0595226
$783$	−7.83771	−0.280097
$784$	0	0
$785$	−28.4678	−1.01606
$786$	−0.660848	−0.0235717
$787$	−42.5487	−1.51670	−0.758349	−	0.651849i	$-0.773993\pi$
$787$	−42.5487	−1.51670	−0.758349	+	0.651849i	$0.773993\pi$
$788$	29.7888	1.06118
$789$	−2.61090	−0.0929506
$790$	−0.968257	−0.0344491
$791$	0	0
$792$	1.30352	0.0463185
$793$	7.86364	0.279246
$794$	1.81494	0.0644099
$795$	−6.79648	−0.241046
$796$	−15.7095	−0.556808
$797$	−40.4705	−1.43354	−0.716770	−	0.697310i	$-0.754380\pi$
$797$	−40.4705	−1.43354	−0.716770	+	0.697310i	$0.754380\pi$
$798$	0	0
$799$	−23.2547	−0.822692
$800$	−0.475475	−0.0168106
$801$	−8.84330	−0.312463
$802$	−6.57522	−0.232179
$803$	0.702668	0.0247966
$804$	−25.9022	−0.913501
$805$	0	0
$806$	−0.995731	−0.0350731
$807$	−18.5601	−0.653348
$808$	−1.79144	−0.0630228
$809$	52.1135	1.83221	0.916107	−	0.400934i	$-0.131314\pi$
$809$	52.1135	1.83221	0.916107	+	0.400934i	$0.131314\pi$
$810$	0.434041	0.0152507
$811$	−15.9764	−0.561006	−0.280503	−	0.959853i	$-0.590501\pi$
$811$	−15.9764	−0.561006	−0.280503	+	0.959853i	$0.590501\pi$
$812$	0	0
$813$	21.3817	0.749889
$814$	−0.624965	−0.0219050
$815$	−50.8418	−1.78091
$816$	−15.0134	−0.525573
$817$	−3.22634	−0.112875
$818$	5.32118	0.186051
$819$	0	0
$820$	−4.48414	−0.156593
$821$	23.4163	0.817235	0.408617	−	0.912706i	$-0.366011\pi$
$821$	23.4163	0.817235	0.408617	+	0.912706i	$0.366011\pi$
$822$	−1.57436	−0.0549121
$823$	50.2024	1.74994	0.874972	−	0.484173i	$-0.160880\pi$
$823$	50.2024	1.74994	0.874972	+	0.484173i	$0.160880\pi$
$824$	−0.958348	−0.0333856
$825$	−0.369452	−0.0128627
$826$	0	0
$827$	−4.03629	−0.140355	−0.0701777	−	0.997535i	$-0.522357\pi$
$827$	−4.03629	−0.140355	−0.0701777	+	0.997535i	$0.522357\pi$
$828$	4.33474	0.150643
$829$	49.7726	1.72867	0.864337	−	0.502913i	$-0.167738\pi$
$829$	49.7726	1.72867	0.864337	+	0.502913i	$0.167738\pi$
$830$	0.546205	0.0189590
$831$	−14.0879	−0.488705
$832$	11.5520	0.400493
$833$	0	0
$834$	1.29068	0.0446927
$835$	32.6845	1.13109
$836$	3.76344	0.130161
$837$	3.24019	0.111997
$838$	3.83763	0.132569
$839$	7.36072	0.254120	0.127060	−	0.991895i	$-0.459446\pi$
$839$	7.36072	0.254120	0.127060	+	0.991895i	$0.459446\pi$
$840$	0	0
$841$	32.4297	1.11827
$842$	−1.49848	−0.0516412
$843$	22.3895	0.771135
$844$	35.6776	1.22807
$845$	23.7158	0.815849
$846$	1.11431	0.0383106
$847$	0	0
$848$	−11.2648	−0.386836
$849$	−10.5639	−0.362553
$850$	−0.161049	−0.00552394
$851$	−4.19478	−0.143795
$852$	21.8315	0.747934
$853$	−12.6755	−0.434001	−0.217001	−	0.976171i	$-0.569627\pi$
$853$	−12.6755	−0.434001	−0.217001	+	0.976171i	$0.569627\pi$
$854$	0	0
$855$	2.52934	0.0865014
$856$	−11.6562	−0.398400
$857$	−14.3118	−0.488883	−0.244442	−	0.969664i	$-0.578605\pi$
$857$	−14.3118	−0.488883	−0.244442	+	0.969664i	$0.578605\pi$
$858$	−0.531625	−0.0181494
$859$	−12.6064	−0.430126	−0.215063	−	0.976600i	$-0.568996\pi$
$859$	−12.6064	−0.430126	−0.215063	+	0.976600i	$0.568996\pi$
$860$	13.0602	0.445349
$861$	0	0
$862$	0.843276	0.0287221
$863$	−2.85848	−0.0973037	−0.0486518	−	0.998816i	$-0.515492\pi$
$863$	−2.85848	−0.0973037	−0.0486518	+	0.998816i	$0.515492\pi$
$864$	2.22640	0.0757437
$865$	−19.8681	−0.675535
$866$	−1.33079	−0.0452222
$867$	1.26232	0.0428705
$868$	0	0
$869$	−3.85917	−0.130913
$870$	−3.40189	−0.115335
$871$	21.3223	0.722478
$872$	−4.49416	−0.152192
$873$	18.9259	0.640544
$874$	−0.464787	−0.0157217
$875$	0	0
$876$	0.797678	0.0269510
$877$	−18.9915	−0.641297	−0.320648	−	0.947198i	$-0.603901\pi$
$877$	−18.9915	−0.641297	−0.320648	+	0.947198i	$0.603901\pi$
$878$	−2.09893	−0.0708353
$879$	−8.74247	−0.294876
$880$	−14.9489	−0.503928
$881$	4.63709	0.156228	0.0781138	−	0.996944i	$-0.475110\pi$
$881$	4.63709	0.156228	0.0781138	+	0.996944i	$0.475110\pi$
$882$	0	0
$883$	44.5951	1.50074	0.750372	−	0.661016i	$-0.229874\pi$
$883$	44.5951	1.50074	0.750372	+	0.661016i	$0.229874\pi$
$884$	12.5948	0.423608
$885$	−12.8188	−0.430899
$886$	1.30961	0.0439973
$887$	56.2119	1.88741	0.943706	−	0.330785i	$-0.107313\pi$
$887$	56.2119	1.88741	0.943706	+	0.330785i	$0.107313\pi$
$888$	−1.43199	−0.0480545
$889$	0	0
$890$	−3.83836	−0.128662
$891$	1.72995	0.0579556
$892$	−29.8312	−0.998824
$893$	6.49351	0.217297
$894$	−0.906408	−0.0303148
$895$	4.65771	0.155690
$896$	0	0
$897$	−3.56828	−0.119142
$898$	−5.13718	−0.171430
$899$	−25.3957	−0.846992
$900$	−0.419407	−0.0139802
$901$	−11.8083	−0.393391
$902$	0.328850	0.0109495
$903$	0	0
$904$	7.33776	0.244050
$905$	−46.8239	−1.55648
$906$	−1.44806	−0.0481085
$907$	−21.4550	−0.712403	−0.356201	−	0.934409i	$-0.615928\pi$
$907$	−21.4550	−0.712403	−0.356201	+	0.934409i	$0.615928\pi$
$908$	−22.2494	−0.738372
$909$	−2.37750	−0.0788567
$910$	0	0
$911$	21.3499	0.707354	0.353677	−	0.935368i	$-0.384931\pi$
$911$	21.3499	0.707354	0.353677	+	0.935368i	$0.384931\pi$
$912$	4.19225	0.138819
$913$	2.17700	0.0720482
$914$	−0.454723	−0.0150409
$915$	−11.1066	−0.367174
$916$	−32.1603	−1.06261
$917$	0	0
$918$	0.754109	0.0248893
$919$	−12.4136	−0.409487	−0.204744	−	0.978816i	$-0.565636\pi$
$919$	−12.4136	−0.409487	−0.204744	+	0.978816i	$0.565636\pi$
$920$	3.79753	0.125201
$921$	7.97557	0.262804
$922$	3.04123	0.100158
$923$	−17.9713	−0.591532
$924$	0	0
$925$	0.405866	0.0133448
$926$	−3.29634	−0.108324
$927$	−1.27186	−0.0417735
$928$	−17.4499	−0.572820
$929$	−12.7962	−0.419829	−0.209914	−	0.977720i	$-0.567319\pi$
$929$	−12.7962	−0.419829	−0.209914	+	0.977720i	$0.567319\pi$
$930$	1.40638	0.0461169
$931$	0	0
$932$	4.73209	0.155005
$933$	14.1483	0.463194
$934$	−6.03857	−0.197588
$935$	−15.6701	−0.512467
$936$	−1.21812	−0.0398155
$937$	40.7436	1.33104	0.665518	−	0.746382i	$-0.268211\pi$
$937$	40.7436	1.33104	0.665518	+	0.746382i	$0.268211\pi$
$938$	0	0
$939$	−8.96795	−0.292658
$940$	−26.2857	−0.857345
$941$	37.3259	1.21679	0.608395	−	0.793634i	$-0.291814\pi$
$941$	37.3259	1.21679	0.608395	+	0.793634i	$0.291814\pi$
$942$	2.37001	0.0772191
$943$	2.20725	0.0718779
$944$	−21.2465	−0.691516
$945$	0	0
$946$	−0.957786	−0.0311403
$947$	−3.95826	−0.128626	−0.0643131	−	0.997930i	$-0.520486\pi$
$947$	−3.95826	−0.128626	−0.0643131	+	0.997930i	$0.520486\pi$
$948$	−4.38098	−0.142288
$949$	−0.656635	−0.0213153
$950$	0.0449704	0.00145903
$951$	−8.04996	−0.261038
$952$	0	0
$953$	−12.5625	−0.406940	−0.203470	−	0.979081i	$-0.565222\pi$
$953$	−12.5625	−0.406940	−0.203470	+	0.979081i	$0.565222\pi$
$954$	0.565823	0.0183192
$955$	−23.3453	−0.755437
$956$	−15.7948	−0.510841
$957$	−13.5589	−0.438296
$958$	2.38111	0.0769301
$959$	0	0
$960$	−16.3161	−0.526600
$961$	−20.5012	−0.661328
$962$	0.584023	0.0188296
$963$	−15.4694	−0.498495
$964$	−32.3496	−1.04191
$965$	−48.2781	−1.55413
$966$	0	0
$967$	35.4422	1.13974	0.569871	−	0.821734i	$-0.306993\pi$
$967$	35.4422	1.13974	0.569871	+	0.821734i	$0.306993\pi$
$968$	−6.03346	−0.193923
$969$	4.39450	0.141172
$970$	8.21461	0.263755
$971$	−28.0134	−0.898994	−0.449497	−	0.893282i	$-0.648397\pi$
$971$	−28.0134	−0.898994	−0.449497	+	0.893282i	$0.648397\pi$
$972$	1.96387	0.0629910
$973$	0	0
$974$	−3.59790	−0.115284
$975$	0.345249	0.0110568
$976$	−18.4087	−0.589249
$977$	6.84262	0.218915	0.109457	−	0.993991i	$-0.465089\pi$
$977$	6.84262	0.218915	0.109457	+	0.993991i	$0.465089\pi$
$978$	4.23270	0.135347
$979$	−15.2985	−0.488942
$980$	0	0
$981$	−5.96439	−0.190428
$982$	−7.47174	−0.238433
$983$	35.9377	1.14623	0.573117	−	0.819474i	$-0.305734\pi$
$983$	35.9377	1.14623	0.573117	+	0.819474i	$0.305734\pi$
$984$	0.753499	0.0240207
$985$	34.6345	1.10355
$986$	−5.91049	−0.188228
$987$	0	0
$988$	−3.51689	−0.111887
$989$	−6.42868	−0.204420
$990$	0.750871	0.0238642
$991$	49.4628	1.57124	0.785619	−	0.618710i	$-0.212345\pi$
$991$	49.4628	1.57124	0.785619	+	0.618710i	$0.212345\pi$
$992$	7.21395	0.229043
$993$	−17.3497	−0.550577
$994$	0	0
$995$	−18.2649	−0.579036
$996$	2.47136	0.0783081
$997$	47.1017	1.49173	0.745864	−	0.666099i	$-0.232037\pi$
$997$	47.1017	1.49173	0.745864	+	0.666099i	$0.232037\pi$
$998$	0.0183469	0.000580760 0
$999$	−1.90046	−0.0601278

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.y.1.3		7
7.6	odd	2		861.2.a.m.1.3	✓	7
21.20	even	2		2583.2.a.u.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
861.2.a.m.1.3	✓	7	7.6	odd	2
2583.2.a.u.1.5		7	21.20	even	2
6027.2.a.y.1.3		7	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

\(f(q)\)	\(=\)	\(q-0.190092 q^{2} -1.00000 q^{3} -1.96387 q^{4} -2.28332 q^{5} +0.190092 q^{6} +0.753499 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.190092 q^{2} -1.00000 q^{3} -1.96387 q^{4} -2.28332 q^{5} +0.190092 q^{6} +0.753499 q^{8} +1.00000 q^{9} +0.434041 q^{10} +1.72995 q^{11} +1.96387 q^{12} -1.61662 q^{13} +2.28332 q^{15} +3.78450 q^{16} +3.96707 q^{17} -0.190092 q^{18} -1.10774 q^{19} +4.48414 q^{20} -0.328850 q^{22} -2.20725 q^{23} -0.753499 q^{24} +0.213562 q^{25} +0.307306 q^{26} -1.00000 q^{27} +7.83771 q^{29} -0.434041 q^{30} -3.24019 q^{31} -2.22640 q^{32} -1.72995 q^{33} -0.754109 q^{34} -1.96387 q^{36} +1.90046 q^{37} +0.210573 q^{38} +1.61662 q^{39} -1.72048 q^{40} -1.00000 q^{41} +2.91253 q^{43} -3.39739 q^{44} -2.28332 q^{45} +0.419580 q^{46} -5.86193 q^{47} -3.78450 q^{48} -0.0405965 q^{50} -3.96707 q^{51} +3.17482 q^{52} -2.97657 q^{53} +0.190092 q^{54} -3.95004 q^{55} +1.10774 q^{57} -1.48989 q^{58} -5.61410 q^{59} -4.48414 q^{60} -4.86425 q^{61} +0.615934 q^{62} -7.14577 q^{64} +3.69127 q^{65} +0.328850 q^{66} -13.1894 q^{67} -7.79080 q^{68} +2.20725 q^{69} +11.1166 q^{71} +0.753499 q^{72} +0.406178 q^{73} -0.361261 q^{74} -0.213562 q^{75} +2.17546 q^{76} -0.307306 q^{78} -2.23080 q^{79} -8.64123 q^{80} +1.00000 q^{81} +0.190092 q^{82} +1.25842 q^{83} -9.05811 q^{85} -0.553649 q^{86} -7.83771 q^{87} +1.30352 q^{88} -8.84330 q^{89} +0.434041 q^{90} +4.33474 q^{92} +3.24019 q^{93} +1.11431 q^{94} +2.52934 q^{95} +2.22640 q^{96} +18.9259 q^{97} +1.72995 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 4 q^{2} - 7 q^{3} + 8 q^{4} - q^{5} - 4 q^{6} + 12 q^{8} + 7 q^{9}+O(q^{10}) \) 7 * q + 4 * q^2 - 7 * q^3 + 8 * q^4 - q^5 - 4 * q^6 + 12 * q^8 + 7 * q^9	\( 7 q + 4 q^{2} - 7 q^{3} + 8 q^{4} - q^{5} - 4 q^{6} + 12 q^{8} + 7 q^{9} + 3 q^{10} + 11 q^{11} - 8 q^{12} + 7 q^{13} + q^{15} + 6 q^{16} - 11 q^{17} + 4 q^{18} - 4 q^{19} + 7 q^{20} + 6 q^{22} + 7 q^{23} - 12 q^{24} + 2 q^{25} + 13 q^{26} - 7 q^{27} + 4 q^{29} - 3 q^{30} + 7 q^{31} + 18 q^{32} - 11 q^{33} + 20 q^{34} + 8 q^{36} - 4 q^{38} - 7 q^{39} + 9 q^{40} - 7 q^{41} + q^{43} + 18 q^{44} - q^{45} - 17 q^{46} - 14 q^{47} - 6 q^{48} + 19 q^{50} + 11 q^{51} + 27 q^{52} + 23 q^{53} - 4 q^{54} + 30 q^{55} + 4 q^{57} - 3 q^{58} - 8 q^{59} - 7 q^{60} + 3 q^{61} + 16 q^{62} + 6 q^{64} + 15 q^{65} - 6 q^{66} + 3 q^{67} - 7 q^{69} + 7 q^{71} + 12 q^{72} + 11 q^{73} - 13 q^{74} - 2 q^{75} + 40 q^{76} - 13 q^{78} - q^{79} + 43 q^{80} + 7 q^{81} - 4 q^{82} - 10 q^{85} - 12 q^{86} - 4 q^{87} + 10 q^{88} - 32 q^{89} + 3 q^{90} - 19 q^{92} - 7 q^{93} - 21 q^{94} - 8 q^{95} - 18 q^{96} + 25 q^{97} + 11 q^{99}+O(q^{100}) \) 7 * q + 4 * q^2 - 7 * q^3 + 8 * q^4 - q^5 - 4 * q^6 + 12 * q^8 + 7 * q^9 + 3 * q^10 + 11 * q^11 - 8 * q^12 + 7 * q^13 + q^15 + 6 * q^16 - 11 * q^17 + 4 * q^18 - 4 * q^19 + 7 * q^20 + 6 * q^22 + 7 * q^23 - 12 * q^24 + 2 * q^25 + 13 * q^26 - 7 * q^27 + 4 * q^29 - 3 * q^30 + 7 * q^31 + 18 * q^32 - 11 * q^33 + 20 * q^34 + 8 * q^36 - 4 * q^38 - 7 * q^39 + 9 * q^40 - 7 * q^41 + q^43 + 18 * q^44 - q^45 - 17 * q^46 - 14 * q^47 - 6 * q^48 + 19 * q^50 + 11 * q^51 + 27 * q^52 + 23 * q^53 - 4 * q^54 + 30 * q^55 + 4 * q^57 - 3 * q^58 - 8 * q^59 - 7 * q^60 + 3 * q^61 + 16 * q^62 + 6 * q^64 + 15 * q^65 - 6 * q^66 + 3 * q^67 - 7 * q^69 + 7 * q^71 + 12 * q^72 + 11 * q^73 - 13 * q^74 - 2 * q^75 + 40 * q^76 - 13 * q^78 - q^79 + 43 * q^80 + 7 * q^81 - 4 * q^82 - 10 * q^85 - 12 * q^86 - 4 * q^87 + 10 * q^88 - 32 * q^89 + 3 * q^90 - 19 * q^92 - 7 * q^93 - 21 * q^94 - 8 * q^95 - 18 * q^96 + 25 * q^97 + 11 * q^99

Introduction

L-functions

Modular forms

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