LMFDB - Embedded newform 7098.2.a.cg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7098,2,Mod(1,7098)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7098, 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("7098.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7098.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:	$56.6778153547$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	7098.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.19806 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.19806 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.19806 q^{10} +1.55496 q^{11} +1.00000 q^{12} +1.00000 q^{14} +1.19806 q^{15} +1.00000 q^{16} -5.74094 q^{17} -1.00000 q^{18} +4.93900 q^{19} +1.19806 q^{20} -1.00000 q^{21} -1.55496 q^{22} -0.417895 q^{23} -1.00000 q^{24} -3.56465 q^{25} +1.00000 q^{27} -1.00000 q^{28} -2.46681 q^{29} -1.19806 q^{30} -10.2567 q^{31} -1.00000 q^{32} +1.55496 q^{33} +5.74094 q^{34} -1.19806 q^{35} +1.00000 q^{36} -2.58211 q^{37} -4.93900 q^{38} -1.19806 q^{40} +11.0761 q^{41} +1.00000 q^{42} +2.91185 q^{43} +1.55496 q^{44} +1.19806 q^{45} +0.417895 q^{46} +7.89008 q^{47} +1.00000 q^{48} +1.00000 q^{49} +3.56465 q^{50} -5.74094 q^{51} -1.62133 q^{53} -1.00000 q^{54} +1.86294 q^{55} +1.00000 q^{56} +4.93900 q^{57} +2.46681 q^{58} +6.13169 q^{59} +1.19806 q^{60} +14.7017 q^{61} +10.2567 q^{62} -1.00000 q^{63} +1.00000 q^{64} -1.55496 q^{66} +10.5646 q^{67} -5.74094 q^{68} -0.417895 q^{69} +1.19806 q^{70} +1.20775 q^{71} -1.00000 q^{72} +5.86294 q^{73} +2.58211 q^{74} -3.56465 q^{75} +4.93900 q^{76} -1.55496 q^{77} -6.67994 q^{79} +1.19806 q^{80} +1.00000 q^{81} -11.0761 q^{82} +9.06100 q^{83} -1.00000 q^{84} -6.87800 q^{85} -2.91185 q^{86} -2.46681 q^{87} -1.55496 q^{88} +3.67456 q^{89} -1.19806 q^{90} -0.417895 q^{92} -10.2567 q^{93} -7.89008 q^{94} +5.91723 q^{95} -1.00000 q^{96} +17.3937 q^{97} -1.00000 q^{98} +1.55496 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 8 q^{5} - 3 q^{6} - 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 8 * q^5 - 3 * q^6 - 3 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 8 q^{5} - 3 q^{6} - 3 q^{7} - 3 q^{8} + 3 q^{9} - 8 q^{10} + 5 q^{11} + 3 q^{12} + 3 q^{14} + 8 q^{15} + 3 q^{16} - 3 q^{17} - 3 q^{18} + 5 q^{19} + 8 q^{20} - 3 q^{21} - 5 q^{22} - 7 q^{23} - 3 q^{24} + 11 q^{25} + 3 q^{27} - 3 q^{28} - 4 q^{29} - 8 q^{30} - 4 q^{31} - 3 q^{32} + 5 q^{33} + 3 q^{34} - 8 q^{35} + 3 q^{36} - 2 q^{37} - 5 q^{38} - 8 q^{40} + 18 q^{41} + 3 q^{42} + 5 q^{43} + 5 q^{44} + 8 q^{45} + 7 q^{46} + 23 q^{47} + 3 q^{48} + 3 q^{49} - 11 q^{50} - 3 q^{51} - 12 q^{53} - 3 q^{54} + 11 q^{55} + 3 q^{56} + 5 q^{57} + 4 q^{58} + 16 q^{59} + 8 q^{60} + 17 q^{61} + 4 q^{62} - 3 q^{63} + 3 q^{64} - 5 q^{66} + 10 q^{67} - 3 q^{68} - 7 q^{69} + 8 q^{70} - 14 q^{71} - 3 q^{72} + 23 q^{73} + 2 q^{74} + 11 q^{75} + 5 q^{76} - 5 q^{77} + 4 q^{79} + 8 q^{80} + 3 q^{81} - 18 q^{82} + 37 q^{83} - 3 q^{84} - q^{85} - 5 q^{86} - 4 q^{87} - 5 q^{88} - 10 q^{89} - 8 q^{90} - 7 q^{92} - 4 q^{93} - 23 q^{94} + 11 q^{95} - 3 q^{96} + 20 q^{97} - 3 q^{98} + 5 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 8 * q^5 - 3 * q^6 - 3 * q^7 - 3 * q^8 + 3 * q^9 - 8 * q^10 + 5 * q^11 + 3 * q^12 + 3 * q^14 + 8 * q^15 + 3 * q^16 - 3 * q^17 - 3 * q^18 + 5 * q^19 + 8 * q^20 - 3 * q^21 - 5 * q^22 - 7 * q^23 - 3 * q^24 + 11 * q^25 + 3 * q^27 - 3 * q^28 - 4 * q^29 - 8 * q^30 - 4 * q^31 - 3 * q^32 + 5 * q^33 + 3 * q^34 - 8 * q^35 + 3 * q^36 - 2 * q^37 - 5 * q^38 - 8 * q^40 + 18 * q^41 + 3 * q^42 + 5 * q^43 + 5 * q^44 + 8 * q^45 + 7 * q^46 + 23 * q^47 + 3 * q^48 + 3 * q^49 - 11 * q^50 - 3 * q^51 - 12 * q^53 - 3 * q^54 + 11 * q^55 + 3 * q^56 + 5 * q^57 + 4 * q^58 + 16 * q^59 + 8 * q^60 + 17 * q^61 + 4 * q^62 - 3 * q^63 + 3 * q^64 - 5 * q^66 + 10 * q^67 - 3 * q^68 - 7 * q^69 + 8 * q^70 - 14 * q^71 - 3 * q^72 + 23 * q^73 + 2 * q^74 + 11 * q^75 + 5 * q^76 - 5 * q^77 + 4 * q^79 + 8 * q^80 + 3 * q^81 - 18 * q^82 + 37 * q^83 - 3 * q^84 - q^85 - 5 * q^86 - 4 * q^87 - 5 * q^88 - 10 * q^89 - 8 * q^90 - 7 * q^92 - 4 * q^93 - 23 * q^94 + 11 * q^95 - 3 * q^96 + 20 * q^97 - 3 * q^98 + 5 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.19806	0.535790	0.267895	−	0.963448i	$-0.413672\pi$
$5$	1.19806	0.535790	0.267895	+	0.963448i	$0.413672\pi$
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.19806	−0.378861
$11$	1.55496	0.468838	0.234419	−	0.972136i	$-0.424681\pi$
$11$	1.55496	0.468838	0.234419	+	0.972136i	$0.424681\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	1.00000	0.267261
$15$	1.19806	0.309338
$16$	1.00000	0.250000
$17$	−5.74094	−1.39238	−0.696191	−	0.717857i	$-0.745123\pi$
$17$	−5.74094	−1.39238	−0.696191	+	0.717857i	$0.745123\pi$
$18$	−1.00000	−0.235702
$19$	4.93900	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$19$	4.93900	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$20$	1.19806	0.267895
$21$	−1.00000	−0.218218
$22$	−1.55496	−0.331518
$23$	−0.417895	−0.0871371	−0.0435685	−	0.999050i	$-0.513873\pi$
$23$	−0.417895	−0.0871371	−0.0435685	+	0.999050i	$0.513873\pi$
$24$	−1.00000	−0.204124
$25$	−3.56465	−0.712929
$26$	0	0
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	−2.46681	−0.458076	−0.229038	−	0.973418i	$-0.573558\pi$
$29$	−2.46681	−0.458076	−0.229038	+	0.973418i	$0.573558\pi$
$30$	−1.19806	−0.218735
$31$	−10.2567	−1.84215	−0.921076	−	0.389383i	$-0.872688\pi$
$31$	−10.2567	−1.84215	−0.921076	+	0.389383i	$0.872688\pi$
$32$	−1.00000	−0.176777
$33$	1.55496	0.270683
$34$	5.74094	0.984563
$35$	−1.19806	−0.202509
$36$	1.00000	0.166667
$37$	−2.58211	−0.424495	−0.212248	−	0.977216i	$-0.568078\pi$
$37$	−2.58211	−0.424495	−0.212248	+	0.977216i	$0.568078\pi$
$38$	−4.93900	−0.801212
$39$	0	0
$40$	−1.19806	−0.189430
$41$	11.0761	1.72979	0.864895	−	0.501952i	$-0.167385\pi$
$41$	11.0761	1.72979	0.864895	+	0.501952i	$0.167385\pi$
$42$	1.00000	0.154303
$43$	2.91185	0.444054	0.222027	−	0.975041i	$-0.428733\pi$
$43$	2.91185	0.444054	0.222027	+	0.975041i	$0.428733\pi$
$44$	1.55496	0.234419
$45$	1.19806	0.178597
$46$	0.417895	0.0616152
$47$	7.89008	1.15089	0.575443	−	0.817842i	$-0.304829\pi$
$47$	7.89008	1.15089	0.575443	+	0.817842i	$0.304829\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	3.56465	0.504117
$51$	−5.74094	−0.803892
$52$	0	0
$53$	−1.62133	−0.222707	−0.111354	−	0.993781i	$-0.535519\pi$
$53$	−1.62133	−0.222707	−0.111354	+	0.993781i	$0.535519\pi$
$54$	−1.00000	−0.136083
$55$	1.86294	0.251198
$56$	1.00000	0.133631
$57$	4.93900	0.654187
$58$	2.46681	0.323908
$59$	6.13169	0.798278	0.399139	−	0.916891i	$-0.369309\pi$
$59$	6.13169	0.798278	0.399139	+	0.916891i	$0.369309\pi$
$60$	1.19806	0.154669
$61$	14.7017	1.88236	0.941181	−	0.337904i	$-0.109718\pi$
$61$	14.7017	1.88236	0.941181	+	0.337904i	$0.109718\pi$
$62$	10.2567	1.30260
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.55496	−0.191402
$67$	10.5646	1.29068	0.645339	−	0.763897i	$-0.276716\pi$
$67$	10.5646	1.29068	0.645339	+	0.763897i	$0.276716\pi$
$68$	−5.74094	−0.696191
$69$	−0.417895	−0.0503086
$70$	1.19806	0.143196
$71$	1.20775	0.143334	0.0716668	−	0.997429i	$-0.477168\pi$
$71$	1.20775	0.143334	0.0716668	+	0.997429i	$0.477168\pi$
$72$	−1.00000	−0.117851
$73$	5.86294	0.686205	0.343102	−	0.939298i	$-0.388522\pi$
$73$	5.86294	0.686205	0.343102	+	0.939298i	$0.388522\pi$
$74$	2.58211	0.300164
$75$	−3.56465	−0.411610
$76$	4.93900	0.566542
$77$	−1.55496	−0.177204
$78$	0	0
$79$	−6.67994	−0.751552	−0.375776	−	0.926711i	$-0.622624\pi$
$79$	−6.67994	−0.751552	−0.375776	+	0.926711i	$0.622624\pi$
$80$	1.19806	0.133947
$81$	1.00000	0.111111
$82$	−11.0761	−1.22315
$83$	9.06100	0.994574	0.497287	−	0.867586i	$-0.334330\pi$
$83$	9.06100	0.994574	0.497287	+	0.867586i	$0.334330\pi$
$84$	−1.00000	−0.109109
$85$	−6.87800	−0.746024
$86$	−2.91185	−0.313993
$87$	−2.46681	−0.264470
$88$	−1.55496	−0.165759
$89$	3.67456	0.389503	0.194751	−	0.980853i	$-0.437610\pi$
$89$	3.67456	0.389503	0.194751	+	0.980853i	$0.437610\pi$
$90$	−1.19806	−0.126287
$91$	0	0
$92$	−0.417895	−0.0435685
$93$	−10.2567	−1.06357
$94$	−7.89008	−0.813800
$95$	5.91723	0.607095
$96$	−1.00000	−0.102062
$97$	17.3937	1.76607	0.883033	−	0.469311i	$-0.155498\pi$
$97$	17.3937	1.76607	0.883033	+	0.469311i	$0.155498\pi$
$98$	−1.00000	−0.101015
$99$	1.55496	0.156279
$100$	−3.56465	−0.356465
$101$	0.674563	0.0671215	0.0335608	−	0.999437i	$-0.489315\pi$
$101$	0.674563	0.0671215	0.0335608	+	0.999437i	$0.489315\pi$
$102$	5.74094	0.568438
$103$	−7.75063	−0.763692	−0.381846	−	0.924226i	$-0.624712\pi$
$103$	−7.75063	−0.763692	−0.381846	+	0.924226i	$0.624712\pi$
$104$	0	0
$105$	−1.19806	−0.116919
$106$	1.62133	0.157478
$107$	−6.20775	−0.600126	−0.300063	−	0.953919i	$-0.597008\pi$
$107$	−6.20775	−0.600126	−0.300063	+	0.953919i	$0.597008\pi$
$108$	1.00000	0.0962250
$109$	4.91185	0.470470	0.235235	−	0.971938i	$-0.424414\pi$
$109$	4.91185	0.470470	0.235235	+	0.971938i	$0.424414\pi$
$110$	−1.86294	−0.177624
$111$	−2.58211	−0.245083
$112$	−1.00000	−0.0944911
$113$	19.2054	1.80669	0.903344	−	0.428917i	$-0.141105\pi$
$113$	19.2054	1.80669	0.903344	+	0.428917i	$0.141105\pi$
$114$	−4.93900	−0.462580
$115$	−0.500664	−0.0466872
$116$	−2.46681	−0.229038
$117$	0	0
$118$	−6.13169	−0.564467
$119$	5.74094	0.526271
$120$	−1.19806	−0.109368
$121$	−8.58211	−0.780191
$122$	−14.7017	−1.33103
$123$	11.0761	0.998695
$124$	−10.2567	−0.921076
$125$	−10.2610	−0.917770
$126$	1.00000	0.0890871
$127$	6.72886	0.597090	0.298545	−	0.954396i	$-0.403499\pi$
$127$	6.72886	0.597090	0.298545	+	0.954396i	$0.403499\pi$
$128$	−1.00000	−0.0883883
$129$	2.91185	0.256374
$130$	0	0
$131$	3.19269	0.278946	0.139473	−	0.990226i	$-0.455459\pi$
$131$	3.19269	0.278946	0.139473	+	0.990226i	$0.455459\pi$
$132$	1.55496	0.135342
$133$	−4.93900	−0.428266
$134$	−10.5646	−0.912646
$135$	1.19806	0.103113
$136$	5.74094	0.492281
$137$	3.21014	0.274261	0.137131	−	0.990553i	$-0.456212\pi$
$137$	3.21014	0.274261	0.137131	+	0.990553i	$0.456212\pi$
$138$	0.417895	0.0355736
$139$	−10.5308	−0.893210	−0.446605	−	0.894731i	$-0.647367\pi$
$139$	−10.5308	−0.893210	−0.446605	+	0.894731i	$0.647367\pi$
$140$	−1.19806	−0.101255
$141$	7.89008	0.664465
$142$	−1.20775	−0.101352
$143$	0	0
$144$	1.00000	0.0833333
$145$	−2.95539	−0.245432
$146$	−5.86294	−0.485220
$147$	1.00000	0.0824786
$148$	−2.58211	−0.212248
$149$	2.44504	0.200306	0.100153	−	0.994972i	$-0.468067\pi$
$149$	2.44504	0.200306	0.100153	+	0.994972i	$0.468067\pi$
$150$	3.56465	0.291052
$151$	−21.8364	−1.77702	−0.888510	−	0.458858i	$-0.848259\pi$
$151$	−21.8364	−1.77702	−0.888510	+	0.458858i	$0.848259\pi$
$152$	−4.93900	−0.400606
$153$	−5.74094	−0.464127
$154$	1.55496	0.125302
$155$	−12.2881	−0.987006
$156$	0	0
$157$	15.3599	1.22585	0.612926	−	0.790140i	$-0.289992\pi$
$157$	15.3599	1.22585	0.612926	+	0.790140i	$0.289992\pi$
$158$	6.67994	0.531427
$159$	−1.62133	−0.128580
$160$	−1.19806	−0.0947151
$161$	0.417895	0.0329347
$162$	−1.00000	−0.0785674
$163$	16.0911	1.26035	0.630177	−	0.776451i	$-0.282982\pi$
$163$	16.0911	1.26035	0.630177	+	0.776451i	$0.282982\pi$
$164$	11.0761	0.864895
$165$	1.86294	0.145029
$166$	−9.06100	−0.703270
$167$	6.65519	0.514994	0.257497	−	0.966279i	$-0.417102\pi$
$167$	6.65519	0.514994	0.257497	+	0.966279i	$0.417102\pi$
$168$	1.00000	0.0771517
$169$	0	0
$170$	6.87800	0.527519
$171$	4.93900	0.377695
$172$	2.91185	0.222027
$173$	3.83446	0.291529	0.145764	−	0.989319i	$-0.453436\pi$
$173$	3.83446	0.291529	0.145764	+	0.989319i	$0.453436\pi$
$174$	2.46681	0.187009
$175$	3.56465	0.269462
$176$	1.55496	0.117209
$177$	6.13169	0.460886
$178$	−3.67456	−0.275420
$179$	−6.48188	−0.484478	−0.242239	−	0.970217i	$-0.577882\pi$
$179$	−6.48188	−0.484478	−0.242239	+	0.970217i	$0.577882\pi$
$180$	1.19806	0.0892983
$181$	10.1075	0.751286	0.375643	−	0.926764i	$-0.377422\pi$
$181$	10.1075	0.751286	0.375643	+	0.926764i	$0.377422\pi$
$182$	0	0
$183$	14.7017	1.08678
$184$	0.417895	0.0308076
$185$	−3.09352	−0.227440
$186$	10.2567	0.752055
$187$	−8.92692	−0.652801
$188$	7.89008	0.575443
$189$	−1.00000	−0.0727393
$190$	−5.91723	−0.429281
$191$	22.9681	1.66191	0.830956	−	0.556339i	$-0.187794\pi$
$191$	22.9681	1.66191	0.830956	+	0.556339i	$0.187794\pi$
$192$	1.00000	0.0721688
$193$	4.37435	0.314873	0.157436	−	0.987529i	$-0.449677\pi$
$193$	4.37435	0.314873	0.157436	+	0.987529i	$0.449677\pi$
$194$	−17.3937	−1.24880
$195$	0	0
$196$	1.00000	0.0714286
$197$	4.90515	0.349477	0.174739	−	0.984615i	$-0.444092\pi$
$197$	4.90515	0.349477	0.174739	+	0.984615i	$0.444092\pi$
$198$	−1.55496	−0.110506
$199$	−7.73125	−0.548054	−0.274027	−	0.961722i	$-0.588356\pi$
$199$	−7.73125	−0.548054	−0.274027	+	0.961722i	$0.588356\pi$
$200$	3.56465	0.252059
$201$	10.5646	0.745173
$202$	−0.674563	−0.0474621
$203$	2.46681	0.173136
$204$	−5.74094	−0.401946
$205$	13.2698	0.926804
$206$	7.75063	0.540012
$207$	−0.417895	−0.0290457
$208$	0	0
$209$	7.67994	0.531233
$210$	1.19806	0.0826742
$211$	6.35690	0.437627	0.218813	−	0.975767i	$-0.429781\pi$
$211$	6.35690	0.437627	0.218813	+	0.975767i	$0.429781\pi$
$212$	−1.62133	−0.111354
$213$	1.20775	0.0827537
$214$	6.20775	0.424353
$215$	3.48858	0.237919
$216$	−1.00000	−0.0680414
$217$	10.2567	0.696268
$218$	−4.91185	−0.332673
$219$	5.86294	0.396181
$220$	1.86294	0.125599
$221$	0	0
$222$	2.58211	0.173300
$223$	−6.49157	−0.434707	−0.217354	−	0.976093i	$-0.569743\pi$
$223$	−6.49157	−0.434707	−0.217354	+	0.976093i	$0.569743\pi$
$224$	1.00000	0.0668153
$225$	−3.56465	−0.237643
$226$	−19.2054	−1.27752
$227$	7.92931	0.526287	0.263143	−	0.964757i	$-0.415241\pi$
$227$	7.92931	0.526287	0.263143	+	0.964757i	$0.415241\pi$
$228$	4.93900	0.327093
$229$	−21.3502	−1.41086	−0.705430	−	0.708779i	$-0.749246\pi$
$229$	−21.3502	−1.41086	−0.705430	+	0.708779i	$0.749246\pi$
$230$	0.500664	0.0330128
$231$	−1.55496	−0.102309
$232$	2.46681	0.161954
$233$	10.3002	0.674789	0.337395	−	0.941363i	$-0.390454\pi$
$233$	10.3002	0.674789	0.337395	+	0.941363i	$0.390454\pi$
$234$	0	0
$235$	9.45281	0.616633
$236$	6.13169	0.399139
$237$	−6.67994	−0.433909
$238$	−5.74094	−0.372130
$239$	−19.2814	−1.24721	−0.623606	−	0.781739i	$-0.714333\pi$
$239$	−19.2814	−1.24721	−0.623606	+	0.781739i	$0.714333\pi$
$240$	1.19806	0.0773346
$241$	−10.6920	−0.688734	−0.344367	−	0.938835i	$-0.611906\pi$
$241$	−10.6920	−0.688734	−0.344367	+	0.938835i	$0.611906\pi$
$242$	8.58211	0.551679
$243$	1.00000	0.0641500
$244$	14.7017	0.941181
$245$	1.19806	0.0765414
$246$	−11.0761	−0.706184
$247$	0	0
$248$	10.2567	0.651299
$249$	9.06100	0.574217
$250$	10.2610	0.648961
$251$	−6.49827	−0.410167	−0.205084	−	0.978744i	$-0.565747\pi$
$251$	−6.49827	−0.410167	−0.205084	+	0.978744i	$0.565747\pi$
$252$	−1.00000	−0.0629941
$253$	−0.649809	−0.0408531
$254$	−6.72886	−0.422206
$255$	−6.87800	−0.430717
$256$	1.00000	0.0625000
$257$	0.907542	0.0566109	0.0283055	−	0.999599i	$-0.490989\pi$
$257$	0.907542	0.0566109	0.0283055	+	0.999599i	$0.490989\pi$
$258$	−2.91185	−0.181284
$259$	2.58211	0.160444
$260$	0	0
$261$	−2.46681	−0.152692
$262$	−3.19269	−0.197245
$263$	17.1564	1.05791	0.528956	−	0.848649i	$-0.322584\pi$
$263$	17.1564	1.05791	0.528956	+	0.848649i	$0.322584\pi$
$264$	−1.55496	−0.0957011
$265$	−1.94246	−0.119324
$266$	4.93900	0.302830
$267$	3.67456	0.224880
$268$	10.5646	0.645339
$269$	−19.7047	−1.20142	−0.600708	−	0.799468i	$-0.705115\pi$
$269$	−19.7047	−1.20142	−0.600708	+	0.799468i	$0.705115\pi$
$270$	−1.19806	−0.0729117
$271$	16.0194	0.973108	0.486554	−	0.873651i	$-0.338254\pi$
$271$	16.0194	0.973108	0.486554	+	0.873651i	$0.338254\pi$
$272$	−5.74094	−0.348096
$273$	0	0
$274$	−3.21014	−0.193932
$275$	−5.54288	−0.334248
$276$	−0.417895	−0.0251543
$277$	−10.6571	−0.640323	−0.320162	−	0.947363i	$-0.603737\pi$
$277$	−10.6571	−0.640323	−0.320162	+	0.947363i	$0.603737\pi$
$278$	10.5308	0.631595
$279$	−10.2567	−0.614051
$280$	1.19806	0.0715979
$281$	−28.6286	−1.70784	−0.853920	−	0.520404i	$-0.825782\pi$
$281$	−28.6286	−1.70784	−0.853920	+	0.520404i	$0.825782\pi$
$282$	−7.89008	−0.469848
$283$	−7.11960	−0.423217	−0.211608	−	0.977355i	$-0.567870\pi$
$283$	−7.11960	−0.423217	−0.211608	+	0.977355i	$0.567870\pi$
$284$	1.20775	0.0716668
$285$	5.91723	0.350507
$286$	0	0
$287$	−11.0761	−0.653799
$288$	−1.00000	−0.0589256
$289$	15.9584	0.938728
$290$	2.95539	0.173547
$291$	17.3937	1.01964
$292$	5.86294	0.343102
$293$	25.0978	1.46623	0.733116	−	0.680104i	$-0.238065\pi$
$293$	25.0978	1.46623	0.733116	+	0.680104i	$0.238065\pi$
$294$	−1.00000	−0.0583212
$295$	7.34614	0.427709
$296$	2.58211	0.150082
$297$	1.55496	0.0902278
$298$	−2.44504	−0.141637
$299$	0	0
$300$	−3.56465	−0.205805
$301$	−2.91185	−0.167836
$302$	21.8364	1.25654
$303$	0.674563	0.0387526
$304$	4.93900	0.283271
$305$	17.6136	1.00855
$306$	5.74094	0.328188
$307$	−8.36658	−0.477506	−0.238753	−	0.971080i	$-0.576739\pi$
$307$	−8.36658	−0.477506	−0.238753	+	0.971080i	$0.576739\pi$
$308$	−1.55496	−0.0886020
$309$	−7.75063	−0.440918
$310$	12.2881	0.697919
$311$	−21.6668	−1.22861	−0.614306	−	0.789068i	$-0.710564\pi$
$311$	−21.6668	−1.22861	−0.614306	+	0.789068i	$0.710564\pi$
$312$	0	0
$313$	14.6950	0.830611	0.415305	−	0.909682i	$-0.363675\pi$
$313$	14.6950	0.830611	0.415305	+	0.909682i	$0.363675\pi$
$314$	−15.3599	−0.866808
$315$	−1.19806	−0.0675032
$316$	−6.67994	−0.375776
$317$	27.1672	1.52586	0.762931	−	0.646480i	$-0.223760\pi$
$317$	27.1672	1.52586	0.762931	+	0.646480i	$0.223760\pi$
$318$	1.62133	0.0909199
$319$	−3.83579	−0.214763
$320$	1.19806	0.0669737
$321$	−6.20775	−0.346483
$322$	−0.417895	−0.0232884
$323$	−28.3545	−1.57769
$324$	1.00000	0.0555556
$325$	0	0
$326$	−16.0911	−0.891205
$327$	4.91185	0.271626
$328$	−11.0761	−0.611573
$329$	−7.89008	−0.434994
$330$	−1.86294	−0.102551
$331$	30.9885	1.70328	0.851641	−	0.524125i	$-0.175608\pi$
$331$	30.9885	1.70328	0.851641	+	0.524125i	$0.175608\pi$
$332$	9.06100	0.497287
$333$	−2.58211	−0.141498
$334$	−6.65519	−0.364156
$335$	12.6571	0.691532
$336$	−1.00000	−0.0545545
$337$	−1.95539	−0.106517	−0.0532586	−	0.998581i	$-0.516961\pi$
$337$	−1.95539	−0.106517	−0.0532586	+	0.998581i	$0.516961\pi$
$338$	0	0
$339$	19.2054	1.04309
$340$	−6.87800	−0.373012
$341$	−15.9487	−0.863670
$342$	−4.93900	−0.267071
$343$	−1.00000	−0.0539949
$344$	−2.91185	−0.156997
$345$	−0.500664	−0.0269548
$346$	−3.83446	−0.206142
$347$	22.9259	1.23072	0.615362	−	0.788244i	$-0.289010\pi$
$347$	22.9259	1.23072	0.615362	+	0.788244i	$0.289010\pi$
$348$	−2.46681	−0.132235
$349$	3.67994	0.196983	0.0984913	−	0.995138i	$-0.468598\pi$
$349$	3.67994	0.196983	0.0984913	+	0.995138i	$0.468598\pi$
$350$	−3.56465	−0.190538
$351$	0	0
$352$	−1.55496	−0.0828795
$353$	26.6353	1.41766	0.708828	−	0.705381i	$-0.249224\pi$
$353$	26.6353	1.41766	0.708828	+	0.705381i	$0.249224\pi$
$354$	−6.13169	−0.325895
$355$	1.44696	0.0767967
$356$	3.67456	0.194751
$357$	5.74094	0.303843
$358$	6.48188	0.342578
$359$	−3.05967	−0.161483	−0.0807416	−	0.996735i	$-0.525729\pi$
$359$	−3.05967	−0.161483	−0.0807416	+	0.996735i	$0.525729\pi$
$360$	−1.19806	−0.0631434
$361$	5.39373	0.283881
$362$	−10.1075	−0.531240
$363$	−8.58211	−0.450444
$364$	0	0
$365$	7.02416	0.367662
$366$	−14.7017	−0.768471
$367$	−25.6582	−1.33935	−0.669673	−	0.742656i	$-0.733566\pi$
$367$	−25.6582	−1.33935	−0.669673	+	0.742656i	$0.733566\pi$
$368$	−0.417895	−0.0217843
$369$	11.0761	0.576597
$370$	3.09352	0.160825
$371$	1.62133	0.0841755
$372$	−10.2567	−0.531783
$373$	−21.6612	−1.12157	−0.560786	−	0.827961i	$-0.689501\pi$
$373$	−21.6612	−1.12157	−0.560786	+	0.827961i	$0.689501\pi$
$374$	8.92692	0.461600
$375$	−10.2610	−0.529875
$376$	−7.89008	−0.406900
$377$	0	0
$378$	1.00000	0.0514344
$379$	−6.93602	−0.356279	−0.178140	−	0.984005i	$-0.557008\pi$
$379$	−6.93602	−0.356279	−0.178140	+	0.984005i	$0.557008\pi$
$380$	5.91723	0.303548
$381$	6.72886	0.344730
$382$	−22.9681	−1.17515
$383$	19.9734	1.02060	0.510298	−	0.859998i	$-0.329535\pi$
$383$	19.9734	1.02060	0.510298	+	0.859998i	$0.329535\pi$
$384$	−1.00000	−0.0510310
$385$	−1.86294	−0.0949440
$386$	−4.37435	−0.222649
$387$	2.91185	0.148018
$388$	17.3937	0.883033
$389$	−30.7289	−1.55801	−0.779007	−	0.627015i	$-0.784277\pi$
$389$	−30.7289	−1.55801	−0.779007	+	0.627015i	$0.784277\pi$
$390$	0	0
$391$	2.39911	0.121328
$392$	−1.00000	−0.0505076
$393$	3.19269	0.161050
$394$	−4.90515	−0.247118
$395$	−8.00298	−0.402674
$396$	1.55496	0.0781396
$397$	−12.3405	−0.619352	−0.309676	−	0.950842i	$-0.600221\pi$
$397$	−12.3405	−0.619352	−0.309676	+	0.950842i	$0.600221\pi$
$398$	7.73125	0.387533
$399$	−4.93900	−0.247259
$400$	−3.56465	−0.178232
$401$	10.2241	0.510569	0.255285	−	0.966866i	$-0.417831\pi$
$401$	10.2241	0.510569	0.255285	+	0.966866i	$0.417831\pi$
$402$	−10.5646	−0.526917
$403$	0	0
$404$	0.674563	0.0335608
$405$	1.19806	0.0595322
$406$	−2.46681	−0.122426
$407$	−4.01507	−0.199019
$408$	5.74094	0.284219
$409$	14.0127	0.692882	0.346441	−	0.938072i	$-0.387390\pi$
$409$	14.0127	0.692882	0.346441	+	0.938072i	$0.387390\pi$
$410$	−13.2698	−0.655349
$411$	3.21014	0.158345
$412$	−7.75063	−0.381846
$413$	−6.13169	−0.301721
$414$	0.417895	0.0205384
$415$	10.8556	0.532882
$416$	0	0
$417$	−10.5308	−0.515695
$418$	−7.67994	−0.375638
$419$	1.10859	0.0541581	0.0270790	−	0.999633i	$-0.491379\pi$
$419$	1.10859	0.0541581	0.0270790	+	0.999633i	$0.491379\pi$
$420$	−1.19806	−0.0584595
$421$	−5.63342	−0.274556	−0.137278	−	0.990533i	$-0.543835\pi$
$421$	−5.63342	−0.274556	−0.137278	+	0.990533i	$0.543835\pi$
$422$	−6.35690	−0.309449
$423$	7.89008	0.383629
$424$	1.62133	0.0787389
$425$	20.4644	0.992670
$426$	−1.20775	−0.0585157
$427$	−14.7017	−0.711466
$428$	−6.20775	−0.300063
$429$	0	0
$430$	−3.48858	−0.168234
$431$	18.6907	0.900299	0.450150	−	0.892953i	$-0.351371\pi$
$431$	18.6907	0.900299	0.450150	+	0.892953i	$0.351371\pi$
$432$	1.00000	0.0481125
$433$	−26.2640	−1.26217	−0.631083	−	0.775715i	$-0.717389\pi$
$433$	−26.2640	−1.26217	−0.631083	+	0.775715i	$0.717389\pi$
$434$	−10.2567	−0.492336
$435$	−2.95539	−0.141700
$436$	4.91185	0.235235
$437$	−2.06398	−0.0987337
$438$	−5.86294	−0.280142
$439$	11.8092	0.563624	0.281812	−	0.959470i	$-0.409065\pi$
$439$	11.8092	0.563624	0.281812	+	0.959470i	$0.409065\pi$
$440$	−1.86294	−0.0888120
$441$	1.00000	0.0476190
$442$	0	0
$443$	7.16660	0.340496	0.170248	−	0.985401i	$-0.445543\pi$
$443$	7.16660	0.340496	0.170248	+	0.985401i	$0.445543\pi$
$444$	−2.58211	−0.122541
$445$	4.40236	0.208692
$446$	6.49157	0.307385
$447$	2.44504	0.115647
$448$	−1.00000	−0.0472456
$449$	31.3357	1.47882	0.739412	−	0.673253i	$-0.235104\pi$
$449$	31.3357	1.47882	0.739412	+	0.673253i	$0.235104\pi$
$450$	3.56465	0.168039
$451$	17.2228	0.810991
$452$	19.2054	0.903344
$453$	−21.8364	−1.02596
$454$	−7.92931	−0.372141
$455$	0	0
$456$	−4.93900	−0.231290
$457$	−25.4058	−1.18843	−0.594217	−	0.804305i	$-0.702538\pi$
$457$	−25.4058	−1.18843	−0.594217	+	0.804305i	$0.702538\pi$
$458$	21.3502	0.997629
$459$	−5.74094	−0.267964
$460$	−0.500664	−0.0233436
$461$	−37.0417	−1.72521	−0.862603	−	0.505882i	$-0.831167\pi$
$461$	−37.0417	−1.72521	−0.862603	+	0.505882i	$0.831167\pi$
$462$	1.55496	0.0723432
$463$	−10.2929	−0.478352	−0.239176	−	0.970976i	$-0.576877\pi$
$463$	−10.2929	−0.478352	−0.239176	+	0.970976i	$0.576877\pi$
$464$	−2.46681	−0.114519
$465$	−12.2881	−0.569848
$466$	−10.3002	−0.477148
$467$	−12.1890	−0.564038	−0.282019	−	0.959409i	$-0.591004\pi$
$467$	−12.1890	−0.564038	−0.282019	+	0.959409i	$0.591004\pi$
$468$	0	0
$469$	−10.5646	−0.487830
$470$	−9.45281	−0.436026
$471$	15.3599	0.707746
$472$	−6.13169	−0.282234
$473$	4.52781	0.208189
$474$	6.67994	0.306820
$475$	−17.6058	−0.807809
$476$	5.74094	0.263135
$477$	−1.62133	−0.0742358
$478$	19.2814	0.881912
$479$	42.4413	1.93919	0.969597	−	0.244708i	$-0.0786922\pi$
$479$	42.4413	1.93919	0.969597	+	0.244708i	$0.0786922\pi$
$480$	−1.19806	−0.0546838
$481$	0	0
$482$	10.6920	0.487008
$483$	0.417895	0.0190149
$484$	−8.58211	−0.390096
$485$	20.8388	0.946240
$486$	−1.00000	−0.0453609
$487$	−7.26742	−0.329318	−0.164659	−	0.986351i	$-0.552652\pi$
$487$	−7.26742	−0.329318	−0.164659	+	0.986351i	$0.552652\pi$
$488$	−14.7017	−0.665515
$489$	16.0911	0.727666
$490$	−1.19806	−0.0541229
$491$	14.3763	0.648792	0.324396	−	0.945921i	$-0.394839\pi$
$491$	14.3763	0.648792	0.324396	+	0.945921i	$0.394839\pi$
$492$	11.0761	0.499348
$493$	14.1618	0.637816
$494$	0	0
$495$	1.86294	0.0837328
$496$	−10.2567	−0.460538
$497$	−1.20775	−0.0541750
$498$	−9.06100	−0.406033
$499$	−5.55602	−0.248722	−0.124361	−	0.992237i	$-0.539688\pi$
$499$	−5.55602	−0.248722	−0.124361	+	0.992237i	$0.539688\pi$
$500$	−10.2610	−0.458885
$501$	6.65519	0.297332
$502$	6.49827	0.290032
$503$	38.2747	1.70659	0.853293	−	0.521432i	$-0.174602\pi$
$503$	38.2747	1.70659	0.853293	+	0.521432i	$0.174602\pi$
$504$	1.00000	0.0445435
$505$	0.808169	0.0359630
$506$	0.649809	0.0288875
$507$	0	0
$508$	6.72886	0.298545
$509$	−33.3685	−1.47903	−0.739516	−	0.673138i	$-0.764946\pi$
$509$	−33.3685	−1.47903	−0.739516	+	0.673138i	$0.764946\pi$
$510$	6.87800	0.304563
$511$	−5.86294	−0.259361
$512$	−1.00000	−0.0441942
$513$	4.93900	0.218062
$514$	−0.907542	−0.0400300
$515$	−9.28573	−0.409178
$516$	2.91185	0.128187
$517$	12.2687	0.539579
$518$	−2.58211	−0.113451
$519$	3.83446	0.168314
$520$	0	0
$521$	14.8291	0.649674	0.324837	−	0.945770i	$-0.394691\pi$
$521$	14.8291	0.649674	0.324837	+	0.945770i	$0.394691\pi$
$522$	2.46681	0.107969
$523$	−27.8625	−1.21834	−0.609170	−	0.793040i	$-0.708497\pi$
$523$	−27.8625	−1.21834	−0.609170	+	0.793040i	$0.708497\pi$
$524$	3.19269	0.139473
$525$	3.56465	0.155574
$526$	−17.1564	−0.748056
$527$	58.8829	2.56498
$528$	1.55496	0.0676709
$529$	−22.8254	−0.992407
$530$	1.94246	0.0843750
$531$	6.13169	0.266093
$532$	−4.93900	−0.214133
$533$	0	0
$534$	−3.67456	−0.159014
$535$	−7.43727	−0.321541
$536$	−10.5646	−0.456323
$537$	−6.48188	−0.279714
$538$	19.7047	0.849530
$539$	1.55496	0.0669768
$540$	1.19806	0.0515564
$541$	−27.8019	−1.19530	−0.597649	−	0.801758i	$-0.703898\pi$
$541$	−27.8019	−1.19530	−0.597649	+	0.801758i	$0.703898\pi$
$542$	−16.0194	−0.688091
$543$	10.1075	0.433755
$544$	5.74094	0.246141
$545$	5.88471	0.252073
$546$	0	0
$547$	−11.2500	−0.481014	−0.240507	−	0.970647i	$-0.577314\pi$
$547$	−11.2500	−0.481014	−0.240507	+	0.970647i	$0.577314\pi$
$548$	3.21014	0.137131
$549$	14.7017	0.627454
$550$	5.54288	0.236349
$551$	−12.1836	−0.519038
$552$	0.417895	0.0177868
$553$	6.67994	0.284060
$554$	10.6571	0.452777
$555$	−3.09352	−0.131313
$556$	−10.5308	−0.446605
$557$	30.0062	1.27140	0.635702	−	0.771934i	$-0.280711\pi$
$557$	30.0062	1.27140	0.635702	+	0.771934i	$0.280711\pi$
$558$	10.2567	0.434199
$559$	0	0
$560$	−1.19806	−0.0506274
$561$	−8.92692	−0.376895
$562$	28.6286	1.20763
$563$	39.3491	1.65837	0.829184	−	0.558976i	$-0.188806\pi$
$563$	39.3491	1.65837	0.829184	+	0.558976i	$0.188806\pi$
$564$	7.89008	0.332232
$565$	23.0092	0.968005
$566$	7.11960	0.299259
$567$	−1.00000	−0.0419961
$568$	−1.20775	−0.0506761
$569$	9.36360	0.392543	0.196271	−	0.980550i	$-0.437117\pi$
$569$	9.36360	0.392543	0.196271	+	0.980550i	$0.437117\pi$
$570$	−5.91723	−0.247846
$571$	24.1540	1.01082	0.505408	−	0.862881i	$-0.331342\pi$
$571$	24.1540	1.01082	0.505408	+	0.862881i	$0.331342\pi$
$572$	0	0
$573$	22.9681	0.959505
$574$	11.0761	0.462306
$575$	1.48965	0.0621226
$576$	1.00000	0.0416667
$577$	−28.7885	−1.19848	−0.599241	−	0.800569i	$-0.704531\pi$
$577$	−28.7885	−1.19848	−0.599241	+	0.800569i	$0.704531\pi$
$578$	−15.9584	−0.663781
$579$	4.37435	0.181792
$580$	−2.95539	−0.122716
$581$	−9.06100	−0.375914
$582$	−17.3937	−0.720993
$583$	−2.52111	−0.104414
$584$	−5.86294	−0.242610
$585$	0	0
$586$	−25.0978	−1.03678
$587$	6.58211	0.271673	0.135836	−	0.990731i	$-0.456628\pi$
$587$	6.58211	0.271673	0.135836	+	0.990731i	$0.456628\pi$
$588$	1.00000	0.0412393
$589$	−50.6577	−2.08731
$590$	−7.34614	−0.302436
$591$	4.90515	0.201771
$592$	−2.58211	−0.106124
$593$	1.66355	0.0683137	0.0341568	−	0.999416i	$-0.489125\pi$
$593$	1.66355	0.0683137	0.0341568	+	0.999416i	$0.489125\pi$
$594$	−1.55496	−0.0638007
$595$	6.87800	0.281971
$596$	2.44504	0.100153
$597$	−7.73125	−0.316419
$598$	0	0
$599$	32.9788	1.34748	0.673739	−	0.738969i	$-0.264687\pi$
$599$	32.9788	1.34748	0.673739	+	0.738969i	$0.264687\pi$
$600$	3.56465	0.145526
$601$	−30.2107	−1.23232	−0.616161	−	0.787620i	$-0.711313\pi$
$601$	−30.2107	−1.23232	−0.616161	+	0.787620i	$0.711313\pi$
$602$	2.91185	0.118678
$603$	10.5646	0.430226
$604$	−21.8364	−0.888510
$605$	−10.2819	−0.418019
$606$	−0.674563	−0.0274023
$607$	7.50471	0.304607	0.152303	−	0.988334i	$-0.451331\pi$
$607$	7.50471	0.304607	0.152303	+	0.988334i	$0.451331\pi$
$608$	−4.93900	−0.200303
$609$	2.46681	0.0999603
$610$	−17.6136	−0.713152
$611$	0	0
$612$	−5.74094	−0.232064
$613$	18.5187	0.747964	0.373982	−	0.927436i	$-0.377992\pi$
$613$	18.5187	0.747964	0.373982	+	0.927436i	$0.377992\pi$
$614$	8.36658	0.337648
$615$	13.2698	0.535091
$616$	1.55496	0.0626510
$617$	−31.8998	−1.28424	−0.642118	−	0.766606i	$-0.721944\pi$
$617$	−31.8998	−1.28424	−0.642118	+	0.766606i	$0.721944\pi$
$618$	7.75063	0.311776
$619$	−48.4566	−1.94764	−0.973819	−	0.227327i	$-0.927002\pi$
$619$	−48.4566	−1.94764	−0.973819	+	0.227327i	$0.927002\pi$
$620$	−12.2881	−0.493503
$621$	−0.417895	−0.0167695
$622$	21.6668	0.868759
$623$	−3.67456	−0.147218
$624$	0	0
$625$	5.52994	0.221198
$626$	−14.6950	−0.587331
$627$	7.67994	0.306707
$628$	15.3599	0.612926
$629$	14.8237	0.591060
$630$	1.19806	0.0477319
$631$	3.20477	0.127580	0.0637899	−	0.997963i	$-0.479681\pi$
$631$	3.20477	0.127580	0.0637899	+	0.997963i	$0.479681\pi$
$632$	6.67994	0.265714
$633$	6.35690	0.252664
$634$	−27.1672	−1.07895
$635$	8.06159	0.319914
$636$	−1.62133	−0.0642901
$637$	0	0
$638$	3.83579	0.151860
$639$	1.20775	0.0477779
$640$	−1.19806	−0.0473576
$641$	−32.5515	−1.28571	−0.642853	−	0.765989i	$-0.722250\pi$
$641$	−32.5515	−1.28571	−0.642853	+	0.765989i	$0.722250\pi$
$642$	6.20775	0.245000
$643$	−9.77107	−0.385333	−0.192667	−	0.981264i	$-0.561714\pi$
$643$	−9.77107	−0.385333	−0.192667	+	0.981264i	$0.561714\pi$
$644$	0.417895	0.0164674
$645$	3.48858	0.137363
$646$	28.3545	1.11559
$647$	−12.4829	−0.490755	−0.245378	−	0.969428i	$-0.578912\pi$
$647$	−12.4829	−0.490755	−0.245378	+	0.969428i	$0.578912\pi$
$648$	−1.00000	−0.0392837
$649$	9.53452	0.374262
$650$	0	0
$651$	10.2567	0.401991
$652$	16.0911	0.630177
$653$	42.5488	1.66506	0.832532	−	0.553976i	$-0.186890\pi$
$653$	42.5488	1.66506	0.832532	+	0.553976i	$0.186890\pi$
$654$	−4.91185	−0.192069
$655$	3.82504	0.149457
$656$	11.0761	0.432448
$657$	5.86294	0.228735
$658$	7.89008	0.307587
$659$	−40.8068	−1.58961	−0.794804	−	0.606866i	$-0.792427\pi$
$659$	−40.8068	−1.58961	−0.794804	+	0.606866i	$0.792427\pi$
$660$	1.86294	0.0725147
$661$	−39.2597	−1.52702	−0.763512	−	0.645794i	$-0.776527\pi$
$661$	−39.2597	−1.52702	−0.763512	+	0.645794i	$0.776527\pi$
$662$	−30.9885	−1.20440
$663$	0	0
$664$	−9.06100	−0.351635
$665$	−5.91723	−0.229460
$666$	2.58211	0.100055
$667$	1.03087	0.0399154
$668$	6.65519	0.257497
$669$	−6.49157	−0.250978
$670$	−12.6571	−0.488987
$671$	22.8605	0.882522
$672$	1.00000	0.0385758
$673$	40.8353	1.57409	0.787043	−	0.616898i	$-0.211611\pi$
$673$	40.8353	1.57409	0.787043	+	0.616898i	$0.211611\pi$
$674$	1.95539	0.0753190
$675$	−3.56465	−0.137203
$676$	0	0
$677$	−22.3817	−0.860197	−0.430098	−	0.902782i	$-0.641521\pi$
$677$	−22.3817	−0.860197	−0.430098	+	0.902782i	$0.641521\pi$
$678$	−19.2054	−0.737577
$679$	−17.3937	−0.667510
$680$	6.87800	0.263759
$681$	7.92931	0.303852
$682$	15.9487	0.610707
$683$	−9.96482	−0.381293	−0.190647	−	0.981659i	$-0.561058\pi$
$683$	−9.96482	−0.381293	−0.190647	+	0.981659i	$0.561058\pi$
$684$	4.93900	0.188847
$685$	3.84595	0.146946
$686$	1.00000	0.0381802
$687$	−21.3502	−0.814561
$688$	2.91185	0.111013
$689$	0	0
$690$	0.500664	0.0190600
$691$	24.9366	0.948633	0.474317	−	0.880354i	$-0.342695\pi$
$691$	24.9366	0.948633	0.474317	+	0.880354i	$0.342695\pi$
$692$	3.83446	0.145764
$693$	−1.55496	−0.0590680
$694$	−22.9259	−0.870254
$695$	−12.6165	−0.478573
$696$	2.46681	0.0935043
$697$	−63.5870	−2.40853
$698$	−3.67994	−0.139288
$699$	10.3002	0.389590
$700$	3.56465	0.134731
$701$	12.2567	0.462928	0.231464	−	0.972843i	$-0.425648\pi$
$701$	12.2567	0.462928	0.231464	+	0.972843i	$0.425648\pi$
$702$	0	0
$703$	−12.7530	−0.480989
$704$	1.55496	0.0586047
$705$	9.45281	0.356013
$706$	−26.6353	−1.00243
$707$	−0.674563	−0.0253696
$708$	6.13169	0.230443
$709$	−27.1825	−1.02086	−0.510431	−	0.859919i	$-0.670514\pi$
$709$	−27.1825	−1.02086	−0.510431	+	0.859919i	$0.670514\pi$
$710$	−1.44696	−0.0543035
$711$	−6.67994	−0.250517
$712$	−3.67456	−0.137710
$713$	4.28621	0.160520
$714$	−5.74094	−0.214849
$715$	0	0
$716$	−6.48188	−0.242239
$717$	−19.2814	−0.720078
$718$	3.05967	0.114186
$719$	24.6625	0.919755	0.459878	−	0.887982i	$-0.347893\pi$
$719$	24.6625	0.919755	0.459878	+	0.887982i	$0.347893\pi$
$720$	1.19806	0.0446491
$721$	7.75063	0.288648
$722$	−5.39373	−0.200734
$723$	−10.6920	−0.397641
$724$	10.1075	0.375643
$725$	8.79331	0.326575
$726$	8.58211	0.318512
$727$	38.9657	1.44516	0.722578	−	0.691289i	$-0.242957\pi$
$727$	38.9657	1.44516	0.722578	+	0.691289i	$0.242957\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−7.02416	−0.259976
$731$	−16.7168	−0.618292
$732$	14.7017	0.543391
$733$	−8.73855	−0.322766	−0.161383	−	0.986892i	$-0.551595\pi$
$733$	−8.73855	−0.322766	−0.161383	+	0.986892i	$0.551595\pi$
$734$	25.6582	0.947060
$735$	1.19806	0.0441912
$736$	0.417895	0.0154038
$737$	16.4276	0.605118
$738$	−11.0761	−0.407716
$739$	21.8200	0.802661	0.401331	−	0.915933i	$-0.368548\pi$
$739$	21.8200	0.802661	0.401331	+	0.915933i	$0.368548\pi$
$740$	−3.09352	−0.113720
$741$	0	0
$742$	−1.62133	−0.0595210
$743$	25.6437	0.940776	0.470388	−	0.882460i	$-0.344114\pi$
$743$	25.6437	0.940776	0.470388	+	0.882460i	$0.344114\pi$
$744$	10.2567	0.376028
$745$	2.92931	0.107322
$746$	21.6612	0.793071
$747$	9.06100	0.331525
$748$	−8.92692	−0.326401
$749$	6.20775	0.226826
$750$	10.2610	0.374678
$751$	43.4034	1.58381	0.791907	−	0.610642i	$-0.209089\pi$
$751$	43.4034	1.58381	0.791907	+	0.610642i	$0.209089\pi$
$752$	7.89008	0.287722
$753$	−6.49827	−0.236810
$754$	0	0
$755$	−26.1613	−0.952109
$756$	−1.00000	−0.0363696
$757$	39.9111	1.45059	0.725297	−	0.688436i	$-0.241702\pi$
$757$	39.9111	1.45059	0.725297	+	0.688436i	$0.241702\pi$
$758$	6.93602	0.251927
$759$	−0.649809	−0.0235866
$760$	−5.91723	−0.214641
$761$	3.46011	0.125429	0.0627144	−	0.998032i	$-0.480024\pi$
$761$	3.46011	0.125429	0.0627144	+	0.998032i	$0.480024\pi$
$762$	−6.72886	−0.243761
$763$	−4.91185	−0.177821
$764$	22.9681	0.830956
$765$	−6.87800	−0.248675
$766$	−19.9734	−0.721670
$767$	0	0
$768$	1.00000	0.0360844
$769$	−31.8146	−1.14726	−0.573632	−	0.819113i	$-0.694466\pi$
$769$	−31.8146	−1.14726	−0.573632	+	0.819113i	$0.694466\pi$
$770$	1.86294	0.0671356
$771$	0.907542	0.0326843
$772$	4.37435	0.157436
$773$	−1.18300	−0.0425494	−0.0212747	−	0.999774i	$-0.506772\pi$
$773$	−1.18300	−0.0425494	−0.0212747	+	0.999774i	$0.506772\pi$
$774$	−2.91185	−0.104664
$775$	36.5614	1.31332
$776$	−17.3937	−0.624399
$777$	2.58211	0.0926325
$778$	30.7289	1.10168
$779$	54.7047	1.96000
$780$	0	0
$781$	1.87800	0.0672002
$782$	−2.39911	−0.0857919
$783$	−2.46681	−0.0881567
$784$	1.00000	0.0357143
$785$	18.4021	0.656799
$786$	−3.19269	−0.113879
$787$	15.2553	0.543794	0.271897	−	0.962326i	$-0.412349\pi$
$787$	15.2553	0.543794	0.271897	+	0.962326i	$0.412349\pi$
$788$	4.90515	0.174739
$789$	17.1564	0.610785
$790$	8.00298	0.284733
$791$	−19.2054	−0.682864
$792$	−1.55496	−0.0552530
$793$	0	0
$794$	12.3405	0.437948
$795$	−1.94246	−0.0688919
$796$	−7.73125	−0.274027
$797$	−20.1575	−0.714015	−0.357008	−	0.934101i	$-0.616203\pi$
$797$	−20.1575	−0.714015	−0.357008	+	0.934101i	$0.616203\pi$
$798$	4.93900	0.174839
$799$	−45.2965	−1.60247
$800$	3.56465	0.126029
$801$	3.67456	0.129834
$802$	−10.2241	−0.361027
$803$	9.11662	0.321719
$804$	10.5646	0.372586
$805$	0.500664	0.0176461
$806$	0	0
$807$	−19.7047	−0.693638
$808$	−0.674563	−0.0237310
$809$	−27.9842	−0.983872	−0.491936	−	0.870631i	$-0.663711\pi$
$809$	−27.9842	−0.983872	−0.491936	+	0.870631i	$0.663711\pi$
$810$	−1.19806	−0.0420956
$811$	43.1196	1.51413	0.757067	−	0.653337i	$-0.226632\pi$
$811$	43.1196	1.51413	0.757067	+	0.653337i	$0.226632\pi$
$812$	2.46681	0.0865681
$813$	16.0194	0.561824
$814$	4.01507	0.140728
$815$	19.2782	0.675285
$816$	−5.74094	−0.200973
$817$	14.3817	0.503150
$818$	−14.0127	−0.489942
$819$	0	0
$820$	13.2698	0.463402
$821$	−46.3075	−1.61614	−0.808072	−	0.589084i	$-0.799489\pi$
$821$	−46.3075	−1.61614	−0.808072	+	0.589084i	$0.799489\pi$
$822$	−3.21014	−0.111967
$823$	0.214456	0.00747546	0.00373773	−	0.999993i	$-0.498810\pi$
$823$	0.214456	0.00747546	0.00373773	+	0.999993i	$0.498810\pi$
$824$	7.75063	0.270006
$825$	−5.54288	−0.192978
$826$	6.13169	0.213349
$827$	5.83100	0.202764	0.101382	−	0.994848i	$-0.467674\pi$
$827$	5.83100	0.202764	0.101382	+	0.994848i	$0.467674\pi$
$828$	−0.417895	−0.0145228
$829$	−11.0718	−0.384538	−0.192269	−	0.981342i	$-0.561585\pi$
$829$	−11.0718	−0.384538	−0.192269	+	0.981342i	$0.561585\pi$
$830$	−10.8556	−0.376805
$831$	−10.6571	−0.369691
$832$	0	0
$833$	−5.74094	−0.198912
$834$	10.5308	0.364652
$835$	7.97333	0.275928
$836$	7.67994	0.265616
$837$	−10.2567	−0.354522
$838$	−1.10859	−0.0382955
$839$	49.6601	1.71446	0.857228	−	0.514936i	$-0.172185\pi$
$839$	49.6601	1.71446	0.857228	+	0.514936i	$0.172185\pi$
$840$	1.19806	0.0413371
$841$	−22.9148	−0.790167
$842$	5.63342	0.194140
$843$	−28.6286	−0.986022
$844$	6.35690	0.218813
$845$	0	0
$846$	−7.89008	−0.271267
$847$	8.58211	0.294885
$848$	−1.62133	−0.0556768
$849$	−7.11960	−0.244344
$850$	−20.4644	−0.701924
$851$	1.07905	0.0369893
$852$	1.20775	0.0413769
$853$	45.7120	1.56515	0.782574	−	0.622557i	$-0.213906\pi$
$853$	45.7120	1.56515	0.782574	+	0.622557i	$0.213906\pi$
$854$	14.7017	0.503082
$855$	5.91723	0.202365
$856$	6.20775	0.212177
$857$	27.3913	0.935670	0.467835	−	0.883816i	$-0.345034\pi$
$857$	27.3913	0.935670	0.467835	+	0.883816i	$0.345034\pi$
$858$	0	0
$859$	−35.4325	−1.20894	−0.604470	−	0.796628i	$-0.706615\pi$
$859$	−35.4325	−1.20894	−0.604470	+	0.796628i	$0.706615\pi$
$860$	3.48858	0.118960
$861$	−11.0761	−0.377471
$862$	−18.6907	−0.636608
$863$	10.0489	0.342069	0.171035	−	0.985265i	$-0.445289\pi$
$863$	10.0489	0.342069	0.171035	+	0.985265i	$0.445289\pi$
$864$	−1.00000	−0.0340207
$865$	4.59392	0.156198
$866$	26.2640	0.892486
$867$	15.9584	0.541975
$868$	10.2567	0.348134
$869$	−10.3870	−0.352356
$870$	2.95539	0.100197
$871$	0	0
$872$	−4.91185	−0.166336
$873$	17.3937	0.588689
$874$	2.06398	0.0698153
$875$	10.2610	0.346884
$876$	5.86294	0.198090
$877$	−43.7676	−1.47793	−0.738964	−	0.673745i	$-0.764684\pi$
$877$	−43.7676	−1.47793	−0.738964	+	0.673745i	$0.764684\pi$
$878$	−11.8092	−0.398542
$879$	25.0978	0.846529
$880$	1.86294	0.0627996
$881$	15.3612	0.517532	0.258766	−	0.965940i	$-0.416684\pi$
$881$	15.3612	0.517532	0.258766	+	0.965940i	$0.416684\pi$
$882$	−1.00000	−0.0336718
$883$	−6.10752	−0.205535	−0.102767	−	0.994705i	$-0.532770\pi$
$883$	−6.10752	−0.205535	−0.102767	+	0.994705i	$0.532770\pi$
$884$	0	0
$885$	7.34614	0.246938
$886$	−7.16660	−0.240767
$887$	−7.64801	−0.256795	−0.128397	−	0.991723i	$-0.540983\pi$
$887$	−7.64801	−0.256795	−0.128397	+	0.991723i	$0.540983\pi$
$888$	2.58211	0.0866498
$889$	−6.72886	−0.225679
$890$	−4.40236	−0.147567
$891$	1.55496	0.0520931
$892$	−6.49157	−0.217354
$893$	38.9691	1.30405
$894$	−2.44504	−0.0817744
$895$	−7.76569	−0.259578
$896$	1.00000	0.0334077
$897$	0	0
$898$	−31.3357	−1.04569
$899$	25.3013	0.843845
$900$	−3.56465	−0.118822
$901$	9.30798	0.310094
$902$	−17.2228	−0.573457
$903$	−2.91185	−0.0969004
$904$	−19.2054	−0.638761
$905$	12.1094	0.402531
$906$	21.8364	0.725465
$907$	33.3038	1.10583	0.552917	−	0.833236i	$-0.313515\pi$
$907$	33.3038	1.10583	0.552917	+	0.833236i	$0.313515\pi$
$908$	7.92931	0.263143
$909$	0.674563	0.0223738
$910$	0	0
$911$	20.0780	0.665213	0.332607	−	0.943066i	$-0.392072\pi$
$911$	20.0780	0.665213	0.332607	+	0.943066i	$0.392072\pi$
$912$	4.93900	0.163547
$913$	14.0895	0.466294
$914$	25.4058	0.840350
$915$	17.6136	0.582286
$916$	−21.3502	−0.705430
$917$	−3.19269	−0.105432
$918$	5.74094	0.189479
$919$	46.9541	1.54887	0.774436	−	0.632652i	$-0.218034\pi$
$919$	46.9541	1.54887	0.774436	+	0.632652i	$0.218034\pi$
$920$	0.500664	0.0165064
$921$	−8.36658	−0.275688
$922$	37.0417	1.21990
$923$	0	0
$924$	−1.55496	−0.0511544
$925$	9.20429	0.302635
$926$	10.2929	0.338246
$927$	−7.75063	−0.254564
$928$	2.46681	0.0809771
$929$	−3.66727	−0.120319	−0.0601596	−	0.998189i	$-0.519161\pi$
$929$	−3.66727	−0.120319	−0.0601596	+	0.998189i	$0.519161\pi$
$930$	12.2881	0.402944
$931$	4.93900	0.161869
$932$	10.3002	0.337395
$933$	−21.6668	−0.709339
$934$	12.1890	0.398835
$935$	−10.6950	−0.349764
$936$	0	0
$937$	−3.10992	−0.101597	−0.0507983	−	0.998709i	$-0.516177\pi$
$937$	−3.10992	−0.101597	−0.0507983	+	0.998709i	$0.516177\pi$
$938$	10.5646	0.344948
$939$	14.6950	0.479553
$940$	9.45281	0.308317
$941$	−7.25475	−0.236498	−0.118249	−	0.992984i	$-0.537728\pi$
$941$	−7.25475	−0.236498	−0.118249	+	0.992984i	$0.537728\pi$
$942$	−15.3599	−0.500452
$943$	−4.62863	−0.150729
$944$	6.13169	0.199569
$945$	−1.19806	−0.0389730
$946$	−4.52781	−0.147212
$947$	−9.91782	−0.322286	−0.161143	−	0.986931i	$-0.551518\pi$
$947$	−9.91782	−0.322286	−0.161143	+	0.986931i	$0.551518\pi$
$948$	−6.67994	−0.216954
$949$	0	0
$950$	17.6058	0.571207
$951$	27.1672	0.880957
$952$	−5.74094	−0.186065
$953$	−0.0204423	−0.000662191 0	−0.000331096	−	1.00000i	$-0.500105\pi$
$953$	−0.0204423	−0.000662191 0	−0.000331096		1.00000i	$0.500105\pi$
$954$	1.62133	0.0524926
$955$	27.5172	0.890435
$956$	−19.2814	−0.623606
$957$	−3.83579	−0.123993
$958$	−42.4413	−1.37122
$959$	−3.21014	−0.103661
$960$	1.19806	0.0386673
$961$	74.1992	2.39352
$962$	0	0
$963$	−6.20775	−0.200042
$964$	−10.6920	−0.344367
$965$	5.24075	0.168706
$966$	−0.417895	−0.0134455
$967$	−13.2731	−0.426833	−0.213416	−	0.976961i	$-0.568459\pi$
$967$	−13.2731	−0.426833	−0.213416	+	0.976961i	$0.568459\pi$
$968$	8.58211	0.275839
$969$	−28.3545	−0.910878
$970$	−20.8388	−0.669093
$971$	55.2409	1.77276	0.886382	−	0.462955i	$-0.153211\pi$
$971$	55.2409	1.77276	0.886382	+	0.462955i	$0.153211\pi$
$972$	1.00000	0.0320750
$973$	10.5308	0.337602
$974$	7.26742	0.232863
$975$	0	0
$976$	14.7017	0.470590
$977$	−35.9235	−1.14929	−0.574647	−	0.818402i	$-0.694861\pi$
$977$	−35.9235	−1.14929	−0.574647	+	0.818402i	$0.694861\pi$
$978$	−16.0911	−0.514538
$979$	5.71379	0.182614
$980$	1.19806	0.0382707
$981$	4.91185	0.156823
$982$	−14.3763	−0.458765
$983$	31.1360	0.993084	0.496542	−	0.868013i	$-0.334603\pi$
$983$	31.1360	0.993084	0.496542	+	0.868013i	$0.334603\pi$
$984$	−11.0761	−0.353092
$985$	5.87667	0.187246
$986$	−14.1618	−0.451004
$987$	−7.89008	−0.251144
$988$	0	0
$989$	−1.21685	−0.0386935
$990$	−1.86294	−0.0592080
$991$	−51.4161	−1.63329	−0.816643	−	0.577143i	$-0.804167\pi$
$991$	−51.4161	−1.63329	−0.816643	+	0.577143i	$0.804167\pi$
$992$	10.2567	0.325650
$993$	30.9885	0.983391
$994$	1.20775	0.0383075
$995$	−9.26252	−0.293642
$996$	9.06100	0.287109
$997$	−32.4993	−1.02926	−0.514632	−	0.857411i	$-0.672072\pi$
$997$	−32.4993	−1.02926	−0.514632	+	0.857411i	$0.672072\pi$
$998$	5.55602	0.175873
$999$	−2.58211	−0.0816942

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7098.2.a.cg.1.1	✓	3
13.12	even	2		7098.2.a.cl.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7098.2.a.cg.1.1	✓	3	1.1	even	1	trivial
7098.2.a.cl.1.3	yes	3	13.12	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7098.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.19806 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.19806 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.19806 q^{10} +1.55496 q^{11} +1.00000 q^{12} +1.00000 q^{14} +1.19806 q^{15} +1.00000 q^{16} -5.74094 q^{17} -1.00000 q^{18} +4.93900 q^{19} +1.19806 q^{20} -1.00000 q^{21} -1.55496 q^{22} -0.417895 q^{23} -1.00000 q^{24} -3.56465 q^{25} +1.00000 q^{27} -1.00000 q^{28} -2.46681 q^{29} -1.19806 q^{30} -10.2567 q^{31} -1.00000 q^{32} +1.55496 q^{33} +5.74094 q^{34} -1.19806 q^{35} +1.00000 q^{36} -2.58211 q^{37} -4.93900 q^{38} -1.19806 q^{40} +11.0761 q^{41} +1.00000 q^{42} +2.91185 q^{43} +1.55496 q^{44} +1.19806 q^{45} +0.417895 q^{46} +7.89008 q^{47} +1.00000 q^{48} +1.00000 q^{49} +3.56465 q^{50} -5.74094 q^{51} -1.62133 q^{53} -1.00000 q^{54} +1.86294 q^{55} +1.00000 q^{56} +4.93900 q^{57} +2.46681 q^{58} +6.13169 q^{59} +1.19806 q^{60} +14.7017 q^{61} +10.2567 q^{62} -1.00000 q^{63} +1.00000 q^{64} -1.55496 q^{66} +10.5646 q^{67} -5.74094 q^{68} -0.417895 q^{69} +1.19806 q^{70} +1.20775 q^{71} -1.00000 q^{72} +5.86294 q^{73} +2.58211 q^{74} -3.56465 q^{75} +4.93900 q^{76} -1.55496 q^{77} -6.67994 q^{79} +1.19806 q^{80} +1.00000 q^{81} -11.0761 q^{82} +9.06100 q^{83} -1.00000 q^{84} -6.87800 q^{85} -2.91185 q^{86} -2.46681 q^{87} -1.55496 q^{88} +3.67456 q^{89} -1.19806 q^{90} -0.417895 q^{92} -10.2567 q^{93} -7.89008 q^{94} +5.91723 q^{95} -1.00000 q^{96} +17.3937 q^{97} -1.00000 q^{98} +1.55496 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 8 q^{5} - 3 q^{6} - 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 8 * q^5 - 3 * q^6 - 3 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 8 q^{5} - 3 q^{6} - 3 q^{7} - 3 q^{8} + 3 q^{9} - 8 q^{10} + 5 q^{11} + 3 q^{12} + 3 q^{14} + 8 q^{15} + 3 q^{16} - 3 q^{17} - 3 q^{18} + 5 q^{19} + 8 q^{20} - 3 q^{21} - 5 q^{22} - 7 q^{23} - 3 q^{24} + 11 q^{25} + 3 q^{27} - 3 q^{28} - 4 q^{29} - 8 q^{30} - 4 q^{31} - 3 q^{32} + 5 q^{33} + 3 q^{34} - 8 q^{35} + 3 q^{36} - 2 q^{37} - 5 q^{38} - 8 q^{40} + 18 q^{41} + 3 q^{42} + 5 q^{43} + 5 q^{44} + 8 q^{45} + 7 q^{46} + 23 q^{47} + 3 q^{48} + 3 q^{49} - 11 q^{50} - 3 q^{51} - 12 q^{53} - 3 q^{54} + 11 q^{55} + 3 q^{56} + 5 q^{57} + 4 q^{58} + 16 q^{59} + 8 q^{60} + 17 q^{61} + 4 q^{62} - 3 q^{63} + 3 q^{64} - 5 q^{66} + 10 q^{67} - 3 q^{68} - 7 q^{69} + 8 q^{70} - 14 q^{71} - 3 q^{72} + 23 q^{73} + 2 q^{74} + 11 q^{75} + 5 q^{76} - 5 q^{77} + 4 q^{79} + 8 q^{80} + 3 q^{81} - 18 q^{82} + 37 q^{83} - 3 q^{84} - q^{85} - 5 q^{86} - 4 q^{87} - 5 q^{88} - 10 q^{89} - 8 q^{90} - 7 q^{92} - 4 q^{93} - 23 q^{94} + 11 q^{95} - 3 q^{96} + 20 q^{97} - 3 q^{98} + 5 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 8 * q^5 - 3 * q^6 - 3 * q^7 - 3 * q^8 + 3 * q^9 - 8 * q^10 + 5 * q^11 + 3 * q^12 + 3 * q^14 + 8 * q^15 + 3 * q^16 - 3 * q^17 - 3 * q^18 + 5 * q^19 + 8 * q^20 - 3 * q^21 - 5 * q^22 - 7 * q^23 - 3 * q^24 + 11 * q^25 + 3 * q^27 - 3 * q^28 - 4 * q^29 - 8 * q^30 - 4 * q^31 - 3 * q^32 + 5 * q^33 + 3 * q^34 - 8 * q^35 + 3 * q^36 - 2 * q^37 - 5 * q^38 - 8 * q^40 + 18 * q^41 + 3 * q^42 + 5 * q^43 + 5 * q^44 + 8 * q^45 + 7 * q^46 + 23 * q^47 + 3 * q^48 + 3 * q^49 - 11 * q^50 - 3 * q^51 - 12 * q^53 - 3 * q^54 + 11 * q^55 + 3 * q^56 + 5 * q^57 + 4 * q^58 + 16 * q^59 + 8 * q^60 + 17 * q^61 + 4 * q^62 - 3 * q^63 + 3 * q^64 - 5 * q^66 + 10 * q^67 - 3 * q^68 - 7 * q^69 + 8 * q^70 - 14 * q^71 - 3 * q^72 + 23 * q^73 + 2 * q^74 + 11 * q^75 + 5 * q^76 - 5 * q^77 + 4 * q^79 + 8 * q^80 + 3 * q^81 - 18 * q^82 + 37 * q^83 - 3 * q^84 - q^85 - 5 * q^86 - 4 * q^87 - 5 * q^88 - 10 * q^89 - 8 * q^90 - 7 * q^92 - 4 * q^93 - 23 * q^94 + 11 * q^95 - 3 * q^96 + 20 * q^97 - 3 * q^98 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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