LMFDB - Embedded newform 7098.2.a.bj.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.732051 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.732051 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.732051 q^{10} +3.73205 q^{11} -1.00000 q^{12} -1.00000 q^{14} +0.732051 q^{15} +1.00000 q^{16} +0.267949 q^{17} -1.00000 q^{18} -4.46410 q^{19} -0.732051 q^{20} -1.00000 q^{21} -3.73205 q^{22} +3.46410 q^{23} +1.00000 q^{24} -4.46410 q^{25} -1.00000 q^{27} +1.00000 q^{28} -3.00000 q^{29} -0.732051 q^{30} +7.66025 q^{31} -1.00000 q^{32} -3.73205 q^{33} -0.267949 q^{34} -0.732051 q^{35} +1.00000 q^{36} -1.26795 q^{37} +4.46410 q^{38} +0.732051 q^{40} -7.00000 q^{41} +1.00000 q^{42} +0.732051 q^{43} +3.73205 q^{44} -0.732051 q^{45} -3.46410 q^{46} +4.46410 q^{47} -1.00000 q^{48} +1.00000 q^{49} +4.46410 q^{50} -0.267949 q^{51} -10.4641 q^{53} +1.00000 q^{54} -2.73205 q^{55} -1.00000 q^{56} +4.46410 q^{57} +3.00000 q^{58} +0.928203 q^{59} +0.732051 q^{60} -11.7321 q^{61} -7.66025 q^{62} +1.00000 q^{63} +1.00000 q^{64} +3.73205 q^{66} -12.9282 q^{67} +0.267949 q^{68} -3.46410 q^{69} +0.732051 q^{70} -2.19615 q^{71} -1.00000 q^{72} +6.53590 q^{73} +1.26795 q^{74} +4.46410 q^{75} -4.46410 q^{76} +3.73205 q^{77} +10.8564 q^{79} -0.732051 q^{80} +1.00000 q^{81} +7.00000 q^{82} +5.66025 q^{83} -1.00000 q^{84} -0.196152 q^{85} -0.732051 q^{86} +3.00000 q^{87} -3.73205 q^{88} -6.46410 q^{89} +0.732051 q^{90} +3.46410 q^{92} -7.66025 q^{93} -4.46410 q^{94} +3.26795 q^{95} +1.00000 q^{96} -0.732051 q^{97} -1.00000 q^{98} +3.73205 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 4 q^{11} - 2 q^{12} - 2 q^{14} - 2 q^{15} + 2 q^{16} + 4 q^{17} - 2 q^{18} - 2 q^{19} + 2 q^{20} - 2 q^{21} - 4 q^{22} + 2 q^{24} - 2 q^{25} - 2 q^{27} + 2 q^{28} - 6 q^{29} + 2 q^{30} - 2 q^{31} - 2 q^{32} - 4 q^{33} - 4 q^{34} + 2 q^{35} + 2 q^{36} - 6 q^{37} + 2 q^{38} - 2 q^{40} - 14 q^{41} + 2 q^{42} - 2 q^{43} + 4 q^{44} + 2 q^{45} + 2 q^{47} - 2 q^{48} + 2 q^{49} + 2 q^{50} - 4 q^{51} - 14 q^{53} + 2 q^{54} - 2 q^{55} - 2 q^{56} + 2 q^{57} + 6 q^{58} - 12 q^{59} - 2 q^{60} - 20 q^{61} + 2 q^{62} + 2 q^{63} + 2 q^{64} + 4 q^{66} - 12 q^{67} + 4 q^{68} - 2 q^{70} + 6 q^{71} - 2 q^{72} + 20 q^{73} + 6 q^{74} + 2 q^{75} - 2 q^{76} + 4 q^{77} - 6 q^{79} + 2 q^{80} + 2 q^{81} + 14 q^{82} - 6 q^{83} - 2 q^{84} + 10 q^{85} + 2 q^{86} + 6 q^{87} - 4 q^{88} - 6 q^{89} - 2 q^{90} + 2 q^{93} - 2 q^{94} + 10 q^{95} + 2 q^{96} + 2 q^{97} - 2 q^{98} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 + 4 * q^11 - 2 * q^12 - 2 * q^14 - 2 * q^15 + 2 * q^16 + 4 * q^17 - 2 * q^18 - 2 * q^19 + 2 * q^20 - 2 * q^21 - 4 * q^22 + 2 * q^24 - 2 * q^25 - 2 * q^27 + 2 * q^28 - 6 * q^29 + 2 * q^30 - 2 * q^31 - 2 * q^32 - 4 * q^33 - 4 * q^34 + 2 * q^35 + 2 * q^36 - 6 * q^37 + 2 * q^38 - 2 * q^40 - 14 * q^41 + 2 * q^42 - 2 * q^43 + 4 * q^44 + 2 * q^45 + 2 * q^47 - 2 * q^48 + 2 * q^49 + 2 * q^50 - 4 * q^51 - 14 * q^53 + 2 * q^54 - 2 * q^55 - 2 * q^56 + 2 * q^57 + 6 * q^58 - 12 * q^59 - 2 * q^60 - 20 * q^61 + 2 * q^62 + 2 * q^63 + 2 * q^64 + 4 * q^66 - 12 * q^67 + 4 * q^68 - 2 * q^70 + 6 * q^71 - 2 * q^72 + 20 * q^73 + 6 * q^74 + 2 * q^75 - 2 * q^76 + 4 * q^77 - 6 * q^79 + 2 * q^80 + 2 * q^81 + 14 * q^82 - 6 * q^83 - 2 * q^84 + 10 * q^85 + 2 * q^86 + 6 * q^87 - 4 * q^88 - 6 * q^89 - 2 * q^90 + 2 * q^93 - 2 * q^94 + 10 * q^95 + 2 * q^96 + 2 * q^97 - 2 * q^98 + 4 * 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
$5$	−0.732051	−0.327383	−0.163692	−	0.986512i	$-0.552340\pi$
$5$	−0.732051	−0.327383	−0.163692	+	0.986512i	$0.552340\pi$
$6$	1.00000	0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0.732051	0.231495
$11$	3.73205	1.12526	0.562628	−	0.826710i	$-0.309790\pi$
$11$	3.73205	1.12526	0.562628	+	0.826710i	$0.309790\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−1.00000	−0.267261
$15$	0.732051	0.189015
$16$	1.00000	0.250000
$17$	0.267949	0.0649872	0.0324936	−	0.999472i	$-0.489655\pi$
$17$	0.267949	0.0649872	0.0324936	+	0.999472i	$0.489655\pi$
$18$	−1.00000	−0.235702
$19$	−4.46410	−1.02414	−0.512068	−	0.858945i	$-0.671120\pi$
$19$	−4.46410	−1.02414	−0.512068	+	0.858945i	$0.671120\pi$
$20$	−0.732051	−0.163692
$21$	−1.00000	−0.218218
$22$	−3.73205	−0.795676
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	1.00000	0.204124
$25$	−4.46410	−0.892820
$26$	0	0
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	−0.732051	−0.133654
$31$	7.66025	1.37582	0.687911	−	0.725795i	$-0.258528\pi$
$31$	7.66025	1.37582	0.687911	+	0.725795i	$0.258528\pi$
$32$	−1.00000	−0.176777
$33$	−3.73205	−0.649667
$34$	−0.267949	−0.0459529
$35$	−0.732051	−0.123739
$36$	1.00000	0.166667
$37$	−1.26795	−0.208450	−0.104225	−	0.994554i	$-0.533236\pi$
$37$	−1.26795	−0.208450	−0.104225	+	0.994554i	$0.533236\pi$
$38$	4.46410	0.724173
$39$	0	0
$40$	0.732051	0.115747
$41$	−7.00000	−1.09322	−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$42$	1.00000	0.154303
$43$	0.732051	0.111637	0.0558184	−	0.998441i	$-0.482223\pi$
$43$	0.732051	0.111637	0.0558184	+	0.998441i	$0.482223\pi$
$44$	3.73205	0.562628
$45$	−0.732051	−0.109128
$46$	−3.46410	−0.510754
$47$	4.46410	0.651156	0.325578	−	0.945515i	$-0.394441\pi$
$47$	4.46410	0.651156	0.325578	+	0.945515i	$0.394441\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	4.46410	0.631319
$51$	−0.267949	−0.0375204
$52$	0	0
$53$	−10.4641	−1.43735	−0.718677	−	0.695344i	$-0.755252\pi$
$53$	−10.4641	−1.43735	−0.718677	+	0.695344i	$0.755252\pi$
$54$	1.00000	0.136083
$55$	−2.73205	−0.368390
$56$	−1.00000	−0.133631
$57$	4.46410	0.591285
$58$	3.00000	0.393919
$59$	0.928203	0.120842	0.0604209	−	0.998173i	$-0.480756\pi$
$59$	0.928203	0.120842	0.0604209	+	0.998173i	$0.480756\pi$
$60$	0.732051	0.0945074
$61$	−11.7321	−1.50214	−0.751068	−	0.660225i	$-0.770461\pi$
$61$	−11.7321	−1.50214	−0.751068	+	0.660225i	$0.770461\pi$
$62$	−7.66025	−0.972853
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	3.73205	0.459384
$67$	−12.9282	−1.57943	−0.789716	−	0.613473i	$-0.789772\pi$
$67$	−12.9282	−1.57943	−0.789716	+	0.613473i	$0.789772\pi$
$68$	0.267949	0.0324936
$69$	−3.46410	−0.417029
$70$	0.732051	0.0874968
$71$	−2.19615	−0.260635	−0.130318	−	0.991472i	$-0.541600\pi$
$71$	−2.19615	−0.260635	−0.130318	+	0.991472i	$0.541600\pi$
$72$	−1.00000	−0.117851
$73$	6.53590	0.764969	0.382485	−	0.923962i	$-0.375069\pi$
$73$	6.53590	0.764969	0.382485	+	0.923962i	$0.375069\pi$
$74$	1.26795	0.147396
$75$	4.46410	0.515470
$76$	−4.46410	−0.512068
$77$	3.73205	0.425307
$78$	0	0
$79$	10.8564	1.22144	0.610721	−	0.791846i	$-0.290880\pi$
$79$	10.8564	1.22144	0.610721	+	0.791846i	$0.290880\pi$
$80$	−0.732051	−0.0818458
$81$	1.00000	0.111111
$82$	7.00000	0.773021
$83$	5.66025	0.621294	0.310647	−	0.950525i	$-0.399454\pi$
$83$	5.66025	0.621294	0.310647	+	0.950525i	$0.399454\pi$
$84$	−1.00000	−0.109109
$85$	−0.196152	−0.0212757
$86$	−0.732051	−0.0789391
$87$	3.00000	0.321634
$88$	−3.73205	−0.397838
$89$	−6.46410	−0.685193	−0.342597	−	0.939483i	$-0.611306\pi$
$89$	−6.46410	−0.685193	−0.342597	+	0.939483i	$0.611306\pi$
$90$	0.732051	0.0771649
$91$	0	0
$92$	3.46410	0.361158
$93$	−7.66025	−0.794331
$94$	−4.46410	−0.460437
$95$	3.26795	0.335285
$96$	1.00000	0.102062
$97$	−0.732051	−0.0743285	−0.0371642	−	0.999309i	$-0.511832\pi$
$97$	−0.732051	−0.0743285	−0.0371642	+	0.999309i	$0.511832\pi$
$98$	−1.00000	−0.101015
$99$	3.73205	0.375085
$100$	−4.46410	−0.446410
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0.267949	0.0265309
$103$	−12.1962	−1.20172	−0.600861	−	0.799353i	$-0.705176\pi$
$103$	−12.1962	−1.20172	−0.600861	+	0.799353i	$0.705176\pi$
$104$	0	0
$105$	0.732051	0.0714408
$106$	10.4641	1.01636
$107$	−1.53590	−0.148481	−0.0742405	−	0.997240i	$-0.523653\pi$
$107$	−1.53590	−0.148481	−0.0742405	+	0.997240i	$0.523653\pi$
$108$	−1.00000	−0.0962250
$109$	3.66025	0.350589	0.175294	−	0.984516i	$-0.443912\pi$
$109$	3.66025	0.350589	0.175294	+	0.984516i	$0.443912\pi$
$110$	2.73205	0.260491
$111$	1.26795	0.120348
$112$	1.00000	0.0944911
$113$	−10.1962	−0.959173	−0.479587	−	0.877495i	$-0.659213\pi$
$113$	−10.1962	−0.959173	−0.479587	+	0.877495i	$0.659213\pi$
$114$	−4.46410	−0.418101
$115$	−2.53590	−0.236474
$116$	−3.00000	−0.278543
$117$	0	0
$118$	−0.928203	−0.0854480
$119$	0.267949	0.0245629
$120$	−0.732051	−0.0668268
$121$	2.92820	0.266200
$122$	11.7321	1.06217
$123$	7.00000	0.631169
$124$	7.66025	0.687911
$125$	6.92820	0.619677
$126$	−1.00000	−0.0890871
$127$	−12.5359	−1.11238	−0.556191	−	0.831055i	$-0.687738\pi$
$127$	−12.5359	−1.11238	−0.556191	+	0.831055i	$0.687738\pi$
$128$	−1.00000	−0.0883883
$129$	−0.732051	−0.0644535
$130$	0	0
$131$	18.9282	1.65376	0.826882	−	0.562375i	$-0.190112\pi$
$131$	18.9282	1.65376	0.826882	+	0.562375i	$0.190112\pi$
$132$	−3.73205	−0.324833
$133$	−4.46410	−0.387087
$134$	12.9282	1.11683
$135$	0.732051	0.0630049
$136$	−0.267949	−0.0229765
$137$	15.4641	1.32119	0.660594	−	0.750744i	$-0.270305\pi$
$137$	15.4641	1.32119	0.660594	+	0.750744i	$0.270305\pi$
$138$	3.46410	0.294884
$139$	−18.1244	−1.53729	−0.768644	−	0.639677i	$-0.779068\pi$
$139$	−18.1244	−1.53729	−0.768644	+	0.639677i	$0.779068\pi$
$140$	−0.732051	−0.0618696
$141$	−4.46410	−0.375945
$142$	2.19615	0.184297
$143$	0	0
$144$	1.00000	0.0833333
$145$	2.19615	0.182381
$146$	−6.53590	−0.540915
$147$	−1.00000	−0.0824786
$148$	−1.26795	−0.104225
$149$	18.9282	1.55066	0.775329	−	0.631557i	$-0.217584\pi$
$149$	18.9282	1.55066	0.775329	+	0.631557i	$0.217584\pi$
$150$	−4.46410	−0.364492
$151$	13.1962	1.07389	0.536944	−	0.843618i	$-0.319579\pi$
$151$	13.1962	1.07389	0.536944	+	0.843618i	$0.319579\pi$
$152$	4.46410	0.362086
$153$	0.267949	0.0216624
$154$	−3.73205	−0.300737
$155$	−5.60770	−0.450421
$156$	0	0
$157$	−15.4641	−1.23417	−0.617085	−	0.786897i	$-0.711686\pi$
$157$	−15.4641	−1.23417	−0.617085	+	0.786897i	$0.711686\pi$
$158$	−10.8564	−0.863689
$159$	10.4641	0.829857
$160$	0.732051	0.0578737
$161$	3.46410	0.273009
$162$	−1.00000	−0.0785674
$163$	−7.80385	−0.611245	−0.305622	−	0.952153i	$-0.598864\pi$
$163$	−7.80385	−0.611245	−0.305622	+	0.952153i	$0.598864\pi$
$164$	−7.00000	−0.546608
$165$	2.73205	0.212690
$166$	−5.66025	−0.439321
$167$	−5.85641	−0.453182	−0.226591	−	0.973990i	$-0.572758\pi$
$167$	−5.85641	−0.453182	−0.226591	+	0.973990i	$0.572758\pi$
$168$	1.00000	0.0771517
$169$	0	0
$170$	0.196152	0.0150442
$171$	−4.46410	−0.341378
$172$	0.732051	0.0558184
$173$	20.7321	1.57623	0.788114	−	0.615529i	$-0.211058\pi$
$173$	20.7321	1.57623	0.788114	+	0.615529i	$0.211058\pi$
$174$	−3.00000	−0.227429
$175$	−4.46410	−0.337454
$176$	3.73205	0.281314
$177$	−0.928203	−0.0697680
$178$	6.46410	0.484505
$179$	22.3923	1.67368	0.836840	−	0.547448i	$-0.184401\pi$
$179$	22.3923	1.67368	0.836840	+	0.547448i	$0.184401\pi$
$180$	−0.732051	−0.0545638
$181$	1.19615	0.0889093	0.0444547	−	0.999011i	$-0.485845\pi$
$181$	1.19615	0.0889093	0.0444547	+	0.999011i	$0.485845\pi$
$182$	0	0
$183$	11.7321	0.867258
$184$	−3.46410	−0.255377
$185$	0.928203	0.0682429
$186$	7.66025	0.561677
$187$	1.00000	0.0731272
$188$	4.46410	0.325578
$189$	−1.00000	−0.0727393
$190$	−3.26795	−0.237082
$191$	−4.19615	−0.303623	−0.151811	−	0.988409i	$-0.548511\pi$
$191$	−4.19615	−0.303623	−0.151811	+	0.988409i	$0.548511\pi$
$192$	−1.00000	−0.0721688
$193$	17.1962	1.23781	0.618903	−	0.785467i	$-0.287577\pi$
$193$	17.1962	1.23781	0.618903	+	0.785467i	$0.287577\pi$
$194$	0.732051	0.0525582
$195$	0	0
$196$	1.00000	0.0714286
$197$	2.26795	0.161585	0.0807923	−	0.996731i	$-0.474255\pi$
$197$	2.26795	0.161585	0.0807923	+	0.996731i	$0.474255\pi$
$198$	−3.73205	−0.265225
$199$	−4.19615	−0.297457	−0.148729	−	0.988878i	$-0.547518\pi$
$199$	−4.19615	−0.297457	−0.148729	+	0.988878i	$0.547518\pi$
$200$	4.46410	0.315660
$201$	12.9282	0.911885
$202$	10.0000	0.703598
$203$	−3.00000	−0.210559
$204$	−0.267949	−0.0187602
$205$	5.12436	0.357901
$206$	12.1962	0.849746
$207$	3.46410	0.240772
$208$	0	0
$209$	−16.6603	−1.15241
$210$	−0.732051	−0.0505163
$211$	−15.8564	−1.09160	−0.545800	−	0.837915i	$-0.683774\pi$
$211$	−15.8564	−1.09160	−0.545800	+	0.837915i	$0.683774\pi$
$212$	−10.4641	−0.718677
$213$	2.19615	0.150478
$214$	1.53590	0.104992
$215$	−0.535898	−0.0365480
$216$	1.00000	0.0680414
$217$	7.66025	0.520012
$218$	−3.66025	−0.247904
$219$	−6.53590	−0.441655
$220$	−2.73205	−0.184195
$221$	0	0
$222$	−1.26795	−0.0850992
$223$	8.39230	0.561990	0.280995	−	0.959709i	$-0.409336\pi$
$223$	8.39230	0.561990	0.280995	+	0.959709i	$0.409336\pi$
$224$	−1.00000	−0.0668153
$225$	−4.46410	−0.297607
$226$	10.1962	0.678238
$227$	−8.53590	−0.566547	−0.283274	−	0.959039i	$-0.591420\pi$
$227$	−8.53590	−0.566547	−0.283274	+	0.959039i	$0.591420\pi$
$228$	4.46410	0.295642
$229$	−10.3205	−0.681998	−0.340999	−	0.940064i	$-0.610765\pi$
$229$	−10.3205	−0.681998	−0.340999	+	0.940064i	$0.610765\pi$
$230$	2.53590	0.167212
$231$	−3.73205	−0.245551
$232$	3.00000	0.196960
$233$	2.19615	0.143875	0.0719374	−	0.997409i	$-0.477082\pi$
$233$	2.19615	0.143875	0.0719374	+	0.997409i	$0.477082\pi$
$234$	0	0
$235$	−3.26795	−0.213177
$236$	0.928203	0.0604209
$237$	−10.8564	−0.705199
$238$	−0.267949	−0.0173686
$239$	13.8038	0.892897	0.446448	−	0.894809i	$-0.352689\pi$
$239$	13.8038	0.892897	0.446448	+	0.894809i	$0.352689\pi$
$240$	0.732051	0.0472537
$241$	−12.7846	−0.823529	−0.411765	−	0.911290i	$-0.635087\pi$
$241$	−12.7846	−0.823529	−0.411765	+	0.911290i	$0.635087\pi$
$242$	−2.92820	−0.188232
$243$	−1.00000	−0.0641500
$244$	−11.7321	−0.751068
$245$	−0.732051	−0.0467690
$246$	−7.00000	−0.446304
$247$	0	0
$248$	−7.66025	−0.486427
$249$	−5.66025	−0.358704
$250$	−6.92820	−0.438178
$251$	18.1962	1.14853	0.574265	−	0.818669i	$-0.305288\pi$
$251$	18.1962	1.14853	0.574265	+	0.818669i	$0.305288\pi$
$252$	1.00000	0.0629941
$253$	12.9282	0.812789
$254$	12.5359	0.786572
$255$	0.196152	0.0122835
$256$	1.00000	0.0625000
$257$	11.0526	0.689440	0.344720	−	0.938706i	$-0.387974\pi$
$257$	11.0526	0.689440	0.344720	+	0.938706i	$0.387974\pi$
$258$	0.732051	0.0455755
$259$	−1.26795	−0.0787865
$260$	0	0
$261$	−3.00000	−0.185695
$262$	−18.9282	−1.16939
$263$	−18.7321	−1.15507	−0.577534	−	0.816367i	$-0.695985\pi$
$263$	−18.7321	−1.15507	−0.577534	+	0.816367i	$0.695985\pi$
$264$	3.73205	0.229692
$265$	7.66025	0.470566
$266$	4.46410	0.273712
$267$	6.46410	0.395597
$268$	−12.9282	−0.789716
$269$	−23.7128	−1.44580	−0.722898	−	0.690955i	$-0.757190\pi$
$269$	−23.7128	−1.44580	−0.722898	+	0.690955i	$0.757190\pi$
$270$	−0.732051	−0.0445512
$271$	−21.5167	−1.30704	−0.653522	−	0.756908i	$-0.726709\pi$
$271$	−21.5167	−1.30704	−0.653522	+	0.756908i	$0.726709\pi$
$272$	0.267949	0.0162468
$273$	0	0
$274$	−15.4641	−0.934221
$275$	−16.6603	−1.00465
$276$	−3.46410	−0.208514
$277$	−26.3923	−1.58576	−0.792880	−	0.609378i	$-0.791419\pi$
$277$	−26.3923	−1.58576	−0.792880	+	0.609378i	$0.791419\pi$
$278$	18.1244	1.08703
$279$	7.66025	0.458607
$280$	0.732051	0.0437484
$281$	18.1962	1.08549	0.542746	−	0.839897i	$-0.317385\pi$
$281$	18.1962	1.08549	0.542746	+	0.839897i	$0.317385\pi$
$282$	4.46410	0.265833
$283$	−22.9282	−1.36294	−0.681470	−	0.731846i	$-0.738659\pi$
$283$	−22.9282	−1.36294	−0.681470	+	0.731846i	$0.738659\pi$
$284$	−2.19615	−0.130318
$285$	−3.26795	−0.193577
$286$	0	0
$287$	−7.00000	−0.413197
$288$	−1.00000	−0.0589256
$289$	−16.9282	−0.995777
$290$	−2.19615	−0.128963
$291$	0.732051	0.0429136
$292$	6.53590	0.382485
$293$	−12.7846	−0.746885	−0.373442	−	0.927653i	$-0.621823\pi$
$293$	−12.7846	−0.746885	−0.373442	+	0.927653i	$0.621823\pi$
$294$	1.00000	0.0583212
$295$	−0.679492	−0.0395615
$296$	1.26795	0.0736980
$297$	−3.73205	−0.216556
$298$	−18.9282	−1.09648
$299$	0	0
$300$	4.46410	0.257735
$301$	0.732051	0.0421947
$302$	−13.1962	−0.759353
$303$	10.0000	0.574485
$304$	−4.46410	−0.256034
$305$	8.58846	0.491774
$306$	−0.267949	−0.0153176
$307$	−33.7846	−1.92819	−0.964095	−	0.265558i	$-0.914444\pi$
$307$	−33.7846	−1.92819	−0.964095	+	0.265558i	$0.914444\pi$
$308$	3.73205	0.212653
$309$	12.1962	0.693815
$310$	5.60770	0.318496
$311$	−19.1962	−1.08851	−0.544257	−	0.838919i	$-0.683188\pi$
$311$	−19.1962	−1.08851	−0.544257	+	0.838919i	$0.683188\pi$
$312$	0	0
$313$	1.80385	0.101959	0.0509797	−	0.998700i	$-0.483766\pi$
$313$	1.80385	0.101959	0.0509797	+	0.998700i	$0.483766\pi$
$314$	15.4641	0.872690
$315$	−0.732051	−0.0412464
$316$	10.8564	0.610721
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	−10.4641	−0.586798
$319$	−11.1962	−0.626864
$320$	−0.732051	−0.0409229
$321$	1.53590	0.0857255
$322$	−3.46410	−0.193047
$323$	−1.19615	−0.0665557
$324$	1.00000	0.0555556
$325$	0	0
$326$	7.80385	0.432215
$327$	−3.66025	−0.202413
$328$	7.00000	0.386510
$329$	4.46410	0.246114
$330$	−2.73205	−0.150394
$331$	16.9282	0.930458	0.465229	−	0.885190i	$-0.345972\pi$
$331$	16.9282	0.930458	0.465229	+	0.885190i	$0.345972\pi$
$332$	5.66025	0.310647
$333$	−1.26795	−0.0694832
$334$	5.85641	0.320448
$335$	9.46410	0.517079
$336$	−1.00000	−0.0545545
$337$	27.7846	1.51352	0.756762	−	0.653690i	$-0.226780\pi$
$337$	27.7846	1.51352	0.756762	+	0.653690i	$0.226780\pi$
$338$	0	0
$339$	10.1962	0.553779
$340$	−0.196152	−0.0106379
$341$	28.5885	1.54815
$342$	4.46410	0.241391
$343$	1.00000	0.0539949
$344$	−0.732051	−0.0394695
$345$	2.53590	0.136528
$346$	−20.7321	−1.11456
$347$	22.8564	1.22700	0.613498	−	0.789696i	$-0.289762\pi$
$347$	22.8564	1.22700	0.613498	+	0.789696i	$0.289762\pi$
$348$	3.00000	0.160817
$349$	−23.7128	−1.26932	−0.634659	−	0.772792i	$-0.718859\pi$
$349$	−23.7128	−1.26932	−0.634659	+	0.772792i	$0.718859\pi$
$350$	4.46410	0.238616
$351$	0	0
$352$	−3.73205	−0.198919
$353$	−1.46410	−0.0779263	−0.0389631	−	0.999241i	$-0.512405\pi$
$353$	−1.46410	−0.0779263	−0.0389631	+	0.999241i	$0.512405\pi$
$354$	0.928203	0.0493334
$355$	1.60770	0.0853276
$356$	−6.46410	−0.342597
$357$	−0.267949	−0.0141814
$358$	−22.3923	−1.18347
$359$	15.8038	0.834095	0.417048	−	0.908885i	$-0.363065\pi$
$359$	15.8038	0.834095	0.417048	+	0.908885i	$0.363065\pi$
$360$	0.732051	0.0385825
$361$	0.928203	0.0488528
$362$	−1.19615	−0.0628684
$363$	−2.92820	−0.153691
$364$	0	0
$365$	−4.78461	−0.250438
$366$	−11.7321	−0.613244
$367$	16.2487	0.848176	0.424088	−	0.905621i	$-0.360595\pi$
$367$	16.2487	0.848176	0.424088	+	0.905621i	$0.360595\pi$
$368$	3.46410	0.180579
$369$	−7.00000	−0.364405
$370$	−0.928203	−0.0482550
$371$	−10.4641	−0.543269
$372$	−7.66025	−0.397166
$373$	−20.5885	−1.06603	−0.533015	−	0.846106i	$-0.678941\pi$
$373$	−20.5885	−1.06603	−0.533015	+	0.846106i	$0.678941\pi$
$374$	−1.00000	−0.0517088
$375$	−6.92820	−0.357771
$376$	−4.46410	−0.230218
$377$	0	0
$378$	1.00000	0.0514344
$379$	−14.5885	−0.749359	−0.374679	−	0.927154i	$-0.622247\pi$
$379$	−14.5885	−0.749359	−0.374679	+	0.927154i	$0.622247\pi$
$380$	3.26795	0.167642
$381$	12.5359	0.642234
$382$	4.19615	0.214694
$383$	−14.6077	−0.746418	−0.373209	−	0.927747i	$-0.621743\pi$
$383$	−14.6077	−0.746418	−0.373209	+	0.927747i	$0.621743\pi$
$384$	1.00000	0.0510310
$385$	−2.73205	−0.139238
$386$	−17.1962	−0.875261
$387$	0.732051	0.0372122
$388$	−0.732051	−0.0371642
$389$	−33.1769	−1.68214	−0.841068	−	0.540929i	$-0.818073\pi$
$389$	−33.1769	−1.68214	−0.841068	+	0.540929i	$0.818073\pi$
$390$	0	0
$391$	0.928203	0.0469413
$392$	−1.00000	−0.0505076
$393$	−18.9282	−0.954802
$394$	−2.26795	−0.114258
$395$	−7.94744	−0.399879
$396$	3.73205	0.187543
$397$	−16.4641	−0.826310	−0.413155	−	0.910661i	$-0.635573\pi$
$397$	−16.4641	−0.826310	−0.413155	+	0.910661i	$0.635573\pi$
$398$	4.19615	0.210334
$399$	4.46410	0.223485
$400$	−4.46410	−0.223205
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	−12.9282	−0.644800
$403$	0	0
$404$	−10.0000	−0.497519
$405$	−0.732051	−0.0363759
$406$	3.00000	0.148888
$407$	−4.73205	−0.234559
$408$	0.267949	0.0132655
$409$	−13.2679	−0.656058	−0.328029	−	0.944668i	$-0.606384\pi$
$409$	−13.2679	−0.656058	−0.328029	+	0.944668i	$0.606384\pi$
$410$	−5.12436	−0.253074
$411$	−15.4641	−0.762788
$412$	−12.1962	−0.600861
$413$	0.928203	0.0456739
$414$	−3.46410	−0.170251
$415$	−4.14359	−0.203401
$416$	0	0
$417$	18.1244	0.887554
$418$	16.6603	0.814880
$419$	−6.19615	−0.302702	−0.151351	−	0.988480i	$-0.548362\pi$
$419$	−6.19615	−0.302702	−0.151351	+	0.988480i	$0.548362\pi$
$420$	0.732051	0.0357204
$421$	14.3923	0.701438	0.350719	−	0.936481i	$-0.385937\pi$
$421$	14.3923	0.701438	0.350719	+	0.936481i	$0.385937\pi$
$422$	15.8564	0.771878
$423$	4.46410	0.217052
$424$	10.4641	0.508182
$425$	−1.19615	−0.0580219
$426$	−2.19615	−0.106404
$427$	−11.7321	−0.567754
$428$	−1.53590	−0.0742405
$429$	0	0
$430$	0.535898	0.0258433
$431$	21.1244	1.01752	0.508762	−	0.860907i	$-0.330103\pi$
$431$	21.1244	1.01752	0.508762	+	0.860907i	$0.330103\pi$
$432$	−1.00000	−0.0481125
$433$	8.92820	0.429062	0.214531	−	0.976717i	$-0.431178\pi$
$433$	8.92820	0.429062	0.214531	+	0.976717i	$0.431178\pi$
$434$	−7.66025	−0.367704
$435$	−2.19615	−0.105297
$436$	3.66025	0.175294
$437$	−15.4641	−0.739748
$438$	6.53590	0.312297
$439$	−16.3923	−0.782362	−0.391181	−	0.920314i	$-0.627933\pi$
$439$	−16.3923	−0.782362	−0.391181	+	0.920314i	$0.627933\pi$
$440$	2.73205	0.130245
$441$	1.00000	0.0476190
$442$	0	0
$443$	−31.3923	−1.49149	−0.745747	−	0.666230i	$-0.767907\pi$
$443$	−31.3923	−1.49149	−0.745747	+	0.666230i	$0.767907\pi$
$444$	1.26795	0.0601742
$445$	4.73205	0.224321
$446$	−8.39230	−0.397387
$447$	−18.9282	−0.895273
$448$	1.00000	0.0472456
$449$	−10.7321	−0.506477	−0.253238	−	0.967404i	$-0.581496\pi$
$449$	−10.7321	−0.506477	−0.253238	+	0.967404i	$0.581496\pi$
$450$	4.46410	0.210440
$451$	−26.1244	−1.23015
$452$	−10.1962	−0.479587
$453$	−13.1962	−0.620009
$454$	8.53590	0.400610
$455$	0	0
$456$	−4.46410	−0.209051
$457$	6.78461	0.317371	0.158685	−	0.987329i	$-0.449274\pi$
$457$	6.78461	0.317371	0.158685	+	0.987329i	$0.449274\pi$
$458$	10.3205	0.482246
$459$	−0.267949	−0.0125068
$460$	−2.53590	−0.118237
$461$	27.7128	1.29071	0.645357	−	0.763881i	$-0.276709\pi$
$461$	27.7128	1.29071	0.645357	+	0.763881i	$0.276709\pi$
$462$	3.73205	0.173631
$463$	−1.19615	−0.0555899	−0.0277950	−	0.999614i	$-0.508849\pi$
$463$	−1.19615	−0.0555899	−0.0277950	+	0.999614i	$0.508849\pi$
$464$	−3.00000	−0.139272
$465$	5.60770	0.260051
$466$	−2.19615	−0.101735
$467$	−9.85641	−0.456100	−0.228050	−	0.973649i	$-0.573235\pi$
$467$	−9.85641	−0.456100	−0.228050	+	0.973649i	$0.573235\pi$
$468$	0	0
$469$	−12.9282	−0.596969
$470$	3.26795	0.150739
$471$	15.4641	0.712548
$472$	−0.928203	−0.0427240
$473$	2.73205	0.125620
$474$	10.8564	0.498651
$475$	19.9282	0.914369
$476$	0.267949	0.0122814
$477$	−10.4641	−0.479118
$478$	−13.8038	−0.631373
$479$	−31.3923	−1.43435	−0.717176	−	0.696893i	$-0.754565\pi$
$479$	−31.3923	−1.43435	−0.717176	+	0.696893i	$0.754565\pi$
$480$	−0.732051	−0.0334134
$481$	0	0
$482$	12.7846	0.582323
$483$	−3.46410	−0.157622
$484$	2.92820	0.133100
$485$	0.535898	0.0243339
$486$	1.00000	0.0453609
$487$	−21.1962	−0.960489	−0.480245	−	0.877135i	$-0.659452\pi$
$487$	−21.1962	−0.960489	−0.480245	+	0.877135i	$0.659452\pi$
$488$	11.7321	0.531085
$489$	7.80385	0.352902
$490$	0.732051	0.0330707
$491$	36.2487	1.63588	0.817941	−	0.575303i	$-0.195116\pi$
$491$	36.2487	1.63588	0.817941	+	0.575303i	$0.195116\pi$
$492$	7.00000	0.315584
$493$	−0.803848	−0.0362035
$494$	0	0
$495$	−2.73205	−0.122797
$496$	7.66025	0.343956
$497$	−2.19615	−0.0985109
$498$	5.66025	0.253642
$499$	12.9808	0.581099	0.290549	−	0.956860i	$-0.406162\pi$
$499$	12.9808	0.581099	0.290549	+	0.956860i	$0.406162\pi$
$500$	6.92820	0.309839
$501$	5.85641	0.261645
$502$	−18.1962	−0.812134
$503$	−23.7128	−1.05730	−0.528651	−	0.848839i	$-0.677302\pi$
$503$	−23.7128	−1.05730	−0.528651	+	0.848839i	$0.677302\pi$
$504$	−1.00000	−0.0445435
$505$	7.32051	0.325758
$506$	−12.9282	−0.574729
$507$	0	0
$508$	−12.5359	−0.556191
$509$	−26.0526	−1.15476	−0.577380	−	0.816476i	$-0.695925\pi$
$509$	−26.0526	−1.15476	−0.577380	+	0.816476i	$0.695925\pi$
$510$	−0.196152	−0.00868578
$511$	6.53590	0.289131
$512$	−1.00000	−0.0441942
$513$	4.46410	0.197095
$514$	−11.0526	−0.487507
$515$	8.92820	0.393424
$516$	−0.732051	−0.0322267
$517$	16.6603	0.732717
$518$	1.26795	0.0557105
$519$	−20.7321	−0.910036
$520$	0	0
$521$	−2.26795	−0.0993607	−0.0496803	−	0.998765i	$-0.515820\pi$
$521$	−2.26795	−0.0993607	−0.0496803	+	0.998765i	$0.515820\pi$
$522$	3.00000	0.131306
$523$	−18.8038	−0.822235	−0.411117	−	0.911582i	$-0.634861\pi$
$523$	−18.8038	−0.822235	−0.411117	+	0.911582i	$0.634861\pi$
$524$	18.9282	0.826882
$525$	4.46410	0.194829
$526$	18.7321	0.816756
$527$	2.05256	0.0894109
$528$	−3.73205	−0.162417
$529$	−11.0000	−0.478261
$530$	−7.66025	−0.332740
$531$	0.928203	0.0402806
$532$	−4.46410	−0.193543
$533$	0	0
$534$	−6.46410	−0.279729
$535$	1.12436	0.0486101
$536$	12.9282	0.558413
$537$	−22.3923	−0.966299
$538$	23.7128	1.02233
$539$	3.73205	0.160751
$540$	0.732051	0.0315025
$541$	30.1962	1.29823	0.649117	−	0.760689i	$-0.275139\pi$
$541$	30.1962	1.29823	0.649117	+	0.760689i	$0.275139\pi$
$542$	21.5167	0.924220
$543$	−1.19615	−0.0513318
$544$	−0.267949	−0.0114882
$545$	−2.67949	−0.114777
$546$	0	0
$547$	−12.8756	−0.550523	−0.275261	−	0.961369i	$-0.588764\pi$
$547$	−12.8756	−0.550523	−0.275261	+	0.961369i	$0.588764\pi$
$548$	15.4641	0.660594
$549$	−11.7321	−0.500712
$550$	16.6603	0.710396
$551$	13.3923	0.570531
$552$	3.46410	0.147442
$553$	10.8564	0.461661
$554$	26.3923	1.12130
$555$	−0.928203	−0.0394000
$556$	−18.1244	−0.768644
$557$	−36.6603	−1.55334	−0.776672	−	0.629905i	$-0.783094\pi$
$557$	−36.6603	−1.55334	−0.776672	+	0.629905i	$0.783094\pi$
$558$	−7.66025	−0.324284
$559$	0	0
$560$	−0.732051	−0.0309348
$561$	−1.00000	−0.0422200
$562$	−18.1962	−0.767558
$563$	−33.2679	−1.40208	−0.701038	−	0.713123i	$-0.747280\pi$
$563$	−33.2679	−1.40208	−0.701038	+	0.713123i	$0.747280\pi$
$564$	−4.46410	−0.187973
$565$	7.46410	0.314017
$566$	22.9282	0.963744
$567$	1.00000	0.0419961
$568$	2.19615	0.0921485
$569$	22.7321	0.952977	0.476489	−	0.879181i	$-0.341909\pi$
$569$	22.7321	0.952977	0.476489	+	0.879181i	$0.341909\pi$
$570$	3.26795	0.136879
$571$	19.8038	0.828765	0.414383	−	0.910103i	$-0.363998\pi$
$571$	19.8038	0.828765	0.414383	+	0.910103i	$0.363998\pi$
$572$	0	0
$573$	4.19615	0.175297
$574$	7.00000	0.292174
$575$	−15.4641	−0.644898
$576$	1.00000	0.0416667
$577$	13.2679	0.552352	0.276176	−	0.961107i	$-0.410933\pi$
$577$	13.2679	0.552352	0.276176	+	0.961107i	$0.410933\pi$
$578$	16.9282	0.704120
$579$	−17.1962	−0.714648
$580$	2.19615	0.0911903
$581$	5.66025	0.234827
$582$	−0.732051	−0.0303445
$583$	−39.0526	−1.61739
$584$	−6.53590	−0.270457
$585$	0	0
$586$	12.7846	0.528127
$587$	−31.3731	−1.29491	−0.647453	−	0.762106i	$-0.724166\pi$
$587$	−31.3731	−1.29491	−0.647453	+	0.762106i	$0.724166\pi$
$588$	−1.00000	−0.0412393
$589$	−34.1962	−1.40903
$590$	0.679492	0.0279742
$591$	−2.26795	−0.0932910
$592$	−1.26795	−0.0521124
$593$	13.6795	0.561749	0.280875	−	0.959744i	$-0.409375\pi$
$593$	13.6795	0.561749	0.280875	+	0.959744i	$0.409375\pi$
$594$	3.73205	0.153128
$595$	−0.196152	−0.00804147
$596$	18.9282	0.775329
$597$	4.19615	0.171737
$598$	0	0
$599$	−21.7128	−0.887161	−0.443581	−	0.896234i	$-0.646292\pi$
$599$	−21.7128	−0.887161	−0.443581	+	0.896234i	$0.646292\pi$
$600$	−4.46410	−0.182246
$601$	−12.8756	−0.525208	−0.262604	−	0.964904i	$-0.584581\pi$
$601$	−12.8756	−0.525208	−0.262604	+	0.964904i	$0.584581\pi$
$602$	−0.732051	−0.0298362
$603$	−12.9282	−0.526477
$604$	13.1962	0.536944
$605$	−2.14359	−0.0871495
$606$	−10.0000	−0.406222
$607$	39.9090	1.61985	0.809927	−	0.586530i	$-0.199506\pi$
$607$	39.9090	1.61985	0.809927	+	0.586530i	$0.199506\pi$
$608$	4.46410	0.181043
$609$	3.00000	0.121566
$610$	−8.58846	−0.347736
$611$	0	0
$612$	0.267949	0.0108312
$613$	41.8564	1.69056	0.845282	−	0.534320i	$-0.179432\pi$
$613$	41.8564	1.69056	0.845282	+	0.534320i	$0.179432\pi$
$614$	33.7846	1.36344
$615$	−5.12436	−0.206634
$616$	−3.73205	−0.150369
$617$	−41.5167	−1.67140	−0.835699	−	0.549188i	$-0.814937\pi$
$617$	−41.5167	−1.67140	−0.835699	+	0.549188i	$0.814937\pi$
$618$	−12.1962	−0.490601
$619$	2.32051	0.0932691	0.0466345	−	0.998912i	$-0.485150\pi$
$619$	2.32051	0.0932691	0.0466345	+	0.998912i	$0.485150\pi$
$620$	−5.60770	−0.225210
$621$	−3.46410	−0.139010
$622$	19.1962	0.769696
$623$	−6.46410	−0.258979
$624$	0	0
$625$	17.2487	0.689948
$626$	−1.80385	−0.0720962
$627$	16.6603	0.665346
$628$	−15.4641	−0.617085
$629$	−0.339746	−0.0135466
$630$	0.732051	0.0291656
$631$	−15.9808	−0.636184	−0.318092	−	0.948060i	$-0.603042\pi$
$631$	−15.9808	−0.636184	−0.318092	+	0.948060i	$0.603042\pi$
$632$	−10.8564	−0.431845
$633$	15.8564	0.630236
$634$	−18.0000	−0.714871
$635$	9.17691	0.364175
$636$	10.4641	0.414929
$637$	0	0
$638$	11.1962	0.443260
$639$	−2.19615	−0.0868784
$640$	0.732051	0.0289368
$641$	−16.2487	−0.641786	−0.320893	−	0.947116i	$-0.603983\pi$
$641$	−16.2487	−0.641786	−0.320893	+	0.947116i	$0.603983\pi$
$642$	−1.53590	−0.0606171
$643$	−1.67949	−0.0662327	−0.0331163	−	0.999452i	$-0.510543\pi$
$643$	−1.67949	−0.0662327	−0.0331163	+	0.999452i	$0.510543\pi$
$644$	3.46410	0.136505
$645$	0.535898	0.0211010
$646$	1.19615	0.0470620
$647$	1.73205	0.0680939	0.0340470	−	0.999420i	$-0.489160\pi$
$647$	1.73205	0.0680939	0.0340470	+	0.999420i	$0.489160\pi$
$648$	−1.00000	−0.0392837
$649$	3.46410	0.135978
$650$	0	0
$651$	−7.66025	−0.300229
$652$	−7.80385	−0.305622
$653$	11.6795	0.457054	0.228527	−	0.973538i	$-0.426609\pi$
$653$	11.6795	0.457054	0.228527	+	0.973538i	$0.426609\pi$
$654$	3.66025	0.143127
$655$	−13.8564	−0.541415
$656$	−7.00000	−0.273304
$657$	6.53590	0.254990
$658$	−4.46410	−0.174029
$659$	33.9282	1.32166	0.660828	−	0.750538i	$-0.270205\pi$
$659$	33.9282	1.32166	0.660828	+	0.750538i	$0.270205\pi$
$660$	2.73205	0.106345
$661$	−27.6077	−1.07381	−0.536907	−	0.843641i	$-0.680408\pi$
$661$	−27.6077	−1.07381	−0.536907	+	0.843641i	$0.680408\pi$
$662$	−16.9282	−0.657933
$663$	0	0
$664$	−5.66025	−0.219660
$665$	3.26795	0.126726
$666$	1.26795	0.0491320
$667$	−10.3923	−0.402392
$668$	−5.85641	−0.226591
$669$	−8.39230	−0.324465
$670$	−9.46410	−0.365630
$671$	−43.7846	−1.69029
$672$	1.00000	0.0385758
$673$	−47.9282	−1.84750	−0.923748	−	0.383000i	$-0.874891\pi$
$673$	−47.9282	−1.84750	−0.923748	+	0.383000i	$0.874891\pi$
$674$	−27.7846	−1.07022
$675$	4.46410	0.171823
$676$	0	0
$677$	−47.7654	−1.83577	−0.917886	−	0.396844i	$-0.870105\pi$
$677$	−47.7654	−1.83577	−0.917886	+	0.396844i	$0.870105\pi$
$678$	−10.1962	−0.391581
$679$	−0.732051	−0.0280935
$680$	0.196152	0.00752210
$681$	8.53590	0.327096
$682$	−28.5885	−1.09471
$683$	−26.3923	−1.00987	−0.504937	−	0.863156i	$-0.668484\pi$
$683$	−26.3923	−1.00987	−0.504937	+	0.863156i	$0.668484\pi$
$684$	−4.46410	−0.170689
$685$	−11.3205	−0.432534
$686$	−1.00000	−0.0381802
$687$	10.3205	0.393752
$688$	0.732051	0.0279092
$689$	0	0
$690$	−2.53590	−0.0965400
$691$	40.7846	1.55152	0.775760	−	0.631028i	$-0.217367\pi$
$691$	40.7846	1.55152	0.775760	+	0.631028i	$0.217367\pi$
$692$	20.7321	0.788114
$693$	3.73205	0.141769
$694$	−22.8564	−0.867617
$695$	13.2679	0.503282
$696$	−3.00000	−0.113715
$697$	−1.87564	−0.0710451
$698$	23.7128	0.897543
$699$	−2.19615	−0.0830661
$700$	−4.46410	−0.168727
$701$	14.3205	0.540878	0.270439	−	0.962737i	$-0.412831\pi$
$701$	14.3205	0.540878	0.270439	+	0.962737i	$0.412831\pi$
$702$	0	0
$703$	5.66025	0.213481
$704$	3.73205	0.140657
$705$	3.26795	0.123078
$706$	1.46410	0.0551022
$707$	−10.0000	−0.376089
$708$	−0.928203	−0.0348840
$709$	24.0526	0.903313	0.451656	−	0.892192i	$-0.350833\pi$
$709$	24.0526	0.903313	0.451656	+	0.892192i	$0.350833\pi$
$710$	−1.60770	−0.0603357
$711$	10.8564	0.407147
$712$	6.46410	0.242252
$713$	26.5359	0.993777
$714$	0.267949	0.0100277
$715$	0	0
$716$	22.3923	0.836840
$717$	−13.8038	−0.515514
$718$	−15.8038	−0.589794
$719$	3.05256	0.113841	0.0569206	−	0.998379i	$-0.481872\pi$
$719$	3.05256	0.113841	0.0569206	+	0.998379i	$0.481872\pi$
$720$	−0.732051	−0.0272819
$721$	−12.1962	−0.454208
$722$	−0.928203	−0.0345441
$723$	12.7846	0.475465
$724$	1.19615	0.0444547
$725$	13.3923	0.497378
$726$	2.92820	0.108676
$727$	1.46410	0.0543005	0.0271503	−	0.999631i	$-0.491357\pi$
$727$	1.46410	0.0543005	0.0271503	+	0.999631i	$0.491357\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	4.78461	0.177086
$731$	0.196152	0.00725496
$732$	11.7321	0.433629
$733$	6.85641	0.253247	0.126624	−	0.991951i	$-0.459586\pi$
$733$	6.85641	0.253247	0.126624	+	0.991951i	$0.459586\pi$
$734$	−16.2487	−0.599751
$735$	0.732051	0.0270021
$736$	−3.46410	−0.127688
$737$	−48.2487	−1.77726
$738$	7.00000	0.257674
$739$	−6.39230	−0.235145	−0.117572	−	0.993064i	$-0.537511\pi$
$739$	−6.39230	−0.235145	−0.117572	+	0.993064i	$0.537511\pi$
$740$	0.928203	0.0341214
$741$	0	0
$742$	10.4641	0.384149
$743$	−19.5167	−0.715997	−0.357998	−	0.933722i	$-0.616541\pi$
$743$	−19.5167	−0.715997	−0.357998	+	0.933722i	$0.616541\pi$
$744$	7.66025	0.280839
$745$	−13.8564	−0.507659
$746$	20.5885	0.753797
$747$	5.66025	0.207098
$748$	1.00000	0.0365636
$749$	−1.53590	−0.0561205
$750$	6.92820	0.252982
$751$	−6.85641	−0.250194	−0.125097	−	0.992145i	$-0.539924\pi$
$751$	−6.85641	−0.250194	−0.125097	+	0.992145i	$0.539924\pi$
$752$	4.46410	0.162789
$753$	−18.1962	−0.663105
$754$	0	0
$755$	−9.66025	−0.351573
$756$	−1.00000	−0.0363696
$757$	−38.0526	−1.38304	−0.691522	−	0.722356i	$-0.743059\pi$
$757$	−38.0526	−1.38304	−0.691522	+	0.722356i	$0.743059\pi$
$758$	14.5885	0.529877
$759$	−12.9282	−0.469264
$760$	−3.26795	−0.118541
$761$	−0.679492	−0.0246316	−0.0123158	−	0.999924i	$-0.503920\pi$
$761$	−0.679492	−0.0246316	−0.0123158	+	0.999924i	$0.503920\pi$
$762$	−12.5359	−0.454128
$763$	3.66025	0.132510
$764$	−4.19615	−0.151811
$765$	−0.196152	−0.00709191
$766$	14.6077	0.527797
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	24.4449	0.881504	0.440752	−	0.897629i	$-0.354712\pi$
$769$	24.4449	0.881504	0.440752	+	0.897629i	$0.354712\pi$
$770$	2.73205	0.0984563
$771$	−11.0526	−0.398048
$772$	17.1962	0.618903
$773$	−42.3923	−1.52475	−0.762373	−	0.647138i	$-0.775966\pi$
$773$	−42.3923	−1.52475	−0.762373	+	0.647138i	$0.775966\pi$
$774$	−0.732051	−0.0263130
$775$	−34.1962	−1.22836
$776$	0.732051	0.0262791
$777$	1.26795	0.0454874
$778$	33.1769	1.18945
$779$	31.2487	1.11960
$780$	0	0
$781$	−8.19615	−0.293281
$782$	−0.928203	−0.0331925
$783$	3.00000	0.107211
$784$	1.00000	0.0357143
$785$	11.3205	0.404046
$786$	18.9282	0.675147
$787$	−55.7846	−1.98851	−0.994253	−	0.107053i	$-0.965858\pi$
$787$	−55.7846	−1.98851	−0.994253	+	0.107053i	$0.965858\pi$
$788$	2.26795	0.0807923
$789$	18.7321	0.666879
$790$	7.94744	0.282757
$791$	−10.1962	−0.362533
$792$	−3.73205	−0.132613
$793$	0	0
$794$	16.4641	0.584289
$795$	−7.66025	−0.271681
$796$	−4.19615	−0.148729
$797$	−0.732051	−0.0259306	−0.0129653	−	0.999916i	$-0.504127\pi$
$797$	−0.732051	−0.0259306	−0.0129653	+	0.999916i	$0.504127\pi$
$798$	−4.46410	−0.158027
$799$	1.19615	0.0423168
$800$	4.46410	0.157830
$801$	−6.46410	−0.228398
$802$	−10.0000	−0.353112
$803$	24.3923	0.860786
$804$	12.9282	0.455943
$805$	−2.53590	−0.0893787
$806$	0	0
$807$	23.7128	0.834731
$808$	10.0000	0.351799
$809$	−0.196152	−0.00689635	−0.00344818	−	0.999994i	$-0.501098\pi$
$809$	−0.196152	−0.00689635	−0.00344818	+	0.999994i	$0.501098\pi$
$810$	0.732051	0.0257216
$811$	15.4641	0.543018	0.271509	−	0.962436i	$-0.412477\pi$
$811$	15.4641	0.543018	0.271509	+	0.962436i	$0.412477\pi$
$812$	−3.00000	−0.105279
$813$	21.5167	0.754622
$814$	4.73205	0.165858
$815$	5.71281	0.200111
$816$	−0.267949	−0.00938010
$817$	−3.26795	−0.114331
$818$	13.2679	0.463903
$819$	0	0
$820$	5.12436	0.178950
$821$	−45.5885	−1.59105	−0.795524	−	0.605922i	$-0.792804\pi$
$821$	−45.5885	−1.59105	−0.795524	+	0.605922i	$0.792804\pi$
$822$	15.4641	0.539372
$823$	7.07180	0.246507	0.123254	−	0.992375i	$-0.460667\pi$
$823$	7.07180	0.246507	0.123254	+	0.992375i	$0.460667\pi$
$824$	12.1962	0.424873
$825$	16.6603	0.580036
$826$	−0.928203	−0.0322963
$827$	−19.6077	−0.681826	−0.340913	−	0.940095i	$-0.610736\pi$
$827$	−19.6077	−0.681826	−0.340913	+	0.940095i	$0.610736\pi$
$828$	3.46410	0.120386
$829$	16.6603	0.578635	0.289317	−	0.957233i	$-0.406572\pi$
$829$	16.6603	0.578635	0.289317	+	0.957233i	$0.406572\pi$
$830$	4.14359	0.143826
$831$	26.3923	0.915539
$832$	0	0
$833$	0.267949	0.00928389
$834$	−18.1244	−0.627595
$835$	4.28719	0.148364
$836$	−16.6603	−0.576207
$837$	−7.66025	−0.264777
$838$	6.19615	0.214043
$839$	6.39230	0.220687	0.110343	−	0.993894i	$-0.464805\pi$
$839$	6.39230	0.220687	0.110343	+	0.993894i	$0.464805\pi$
$840$	−0.732051	−0.0252582
$841$	−20.0000	−0.689655
$842$	−14.3923	−0.495992
$843$	−18.1962	−0.626709
$844$	−15.8564	−0.545800
$845$	0	0
$846$	−4.46410	−0.153479
$847$	2.92820	0.100614
$848$	−10.4641	−0.359339
$849$	22.9282	0.786894
$850$	1.19615	0.0410277
$851$	−4.39230	−0.150566
$852$	2.19615	0.0752389
$853$	12.6077	0.431679	0.215840	−	0.976429i	$-0.430751\pi$
$853$	12.6077	0.431679	0.215840	+	0.976429i	$0.430751\pi$
$854$	11.7321	0.401463
$855$	3.26795	0.111762
$856$	1.53590	0.0524959
$857$	−42.3923	−1.44809	−0.724047	−	0.689751i	$-0.757720\pi$
$857$	−42.3923	−1.44809	−0.724047	+	0.689751i	$0.757720\pi$
$858$	0	0
$859$	−6.94744	−0.237044	−0.118522	−	0.992951i	$-0.537816\pi$
$859$	−6.94744	−0.237044	−0.118522	+	0.992951i	$0.537816\pi$
$860$	−0.535898	−0.0182740
$861$	7.00000	0.238559
$862$	−21.1244	−0.719498
$863$	−40.6410	−1.38344	−0.691718	−	0.722168i	$-0.743146\pi$
$863$	−40.6410	−1.38344	−0.691718	+	0.722168i	$0.743146\pi$
$864$	1.00000	0.0340207
$865$	−15.1769	−0.516031
$866$	−8.92820	−0.303393
$867$	16.9282	0.574912
$868$	7.66025	0.260006
$869$	40.5167	1.37443
$870$	2.19615	0.0744565
$871$	0	0
$872$	−3.66025	−0.123952
$873$	−0.732051	−0.0247762
$874$	15.4641	0.523081
$875$	6.92820	0.234216
$876$	−6.53590	−0.220828
$877$	−50.3013	−1.69855	−0.849277	−	0.527948i	$-0.822962\pi$
$877$	−50.3013	−1.69855	−0.849277	+	0.527948i	$0.822962\pi$
$878$	16.3923	0.553213
$879$	12.7846	0.431214
$880$	−2.73205	−0.0920974
$881$	14.9282	0.502944	0.251472	−	0.967865i	$-0.419085\pi$
$881$	14.9282	0.502944	0.251472	+	0.967865i	$0.419085\pi$
$882$	−1.00000	−0.0336718
$883$	−24.4449	−0.822635	−0.411318	−	0.911492i	$-0.634931\pi$
$883$	−24.4449	−0.822635	−0.411318	+	0.911492i	$0.634931\pi$
$884$	0	0
$885$	0.679492	0.0228409
$886$	31.3923	1.05465
$887$	6.94744	0.233272	0.116636	−	0.993175i	$-0.462789\pi$
$887$	6.94744	0.233272	0.116636	+	0.993175i	$0.462789\pi$
$888$	−1.26795	−0.0425496
$889$	−12.5359	−0.420441
$890$	−4.73205	−0.158619
$891$	3.73205	0.125028
$892$	8.39230	0.280995
$893$	−19.9282	−0.666872
$894$	18.9282	0.633054
$895$	−16.3923	−0.547934
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	10.7321	0.358133
$899$	−22.9808	−0.766451
$900$	−4.46410	−0.148803
$901$	−2.80385	−0.0934097
$902$	26.1244	0.869846
$903$	−0.732051	−0.0243611
$904$	10.1962	0.339119
$905$	−0.875644	−0.0291074
$906$	13.1962	0.438413
$907$	−20.5359	−0.681883	−0.340942	−	0.940084i	$-0.610746\pi$
$907$	−20.5359	−0.681883	−0.340942	+	0.940084i	$0.610746\pi$
$908$	−8.53590	−0.283274
$909$	−10.0000	−0.331679
$910$	0	0
$911$	38.0526	1.26074	0.630369	−	0.776296i	$-0.282904\pi$
$911$	38.0526	1.26074	0.630369	+	0.776296i	$0.282904\pi$
$912$	4.46410	0.147821
$913$	21.1244	0.699114
$914$	−6.78461	−0.224415
$915$	−8.58846	−0.283926
$916$	−10.3205	−0.340999
$917$	18.9282	0.625064
$918$	0.267949	0.00884364
$919$	−27.6410	−0.911793	−0.455896	−	0.890033i	$-0.650681\pi$
$919$	−27.6410	−0.911793	−0.455896	+	0.890033i	$0.650681\pi$
$920$	2.53590	0.0836061
$921$	33.7846	1.11324
$922$	−27.7128	−0.912673
$923$	0	0
$924$	−3.73205	−0.122775
$925$	5.66025	0.186108
$926$	1.19615	0.0393080
$927$	−12.1962	−0.400574
$928$	3.00000	0.0984798
$929$	−50.4641	−1.65567	−0.827837	−	0.560969i	$-0.810429\pi$
$929$	−50.4641	−1.65567	−0.827837	+	0.560969i	$0.810429\pi$
$930$	−5.60770	−0.183884
$931$	−4.46410	−0.146305
$932$	2.19615	0.0719374
$933$	19.1962	0.628454
$934$	9.85641	0.322511
$935$	−0.732051	−0.0239406
$936$	0	0
$937$	−37.5692	−1.22733	−0.613666	−	0.789565i	$-0.710306\pi$
$937$	−37.5692	−1.22733	−0.613666	+	0.789565i	$0.710306\pi$
$938$	12.9282	0.422121
$939$	−1.80385	−0.0588663
$940$	−3.26795	−0.106589
$941$	−13.1769	−0.429555	−0.214778	−	0.976663i	$-0.568903\pi$
$941$	−13.1769	−0.429555	−0.214778	+	0.976663i	$0.568903\pi$
$942$	−15.4641	−0.503848
$943$	−24.2487	−0.789647
$944$	0.928203	0.0302104
$945$	0.732051	0.0238136
$946$	−2.73205	−0.0888266
$947$	51.0526	1.65899	0.829493	−	0.558518i	$-0.188630\pi$
$947$	51.0526	1.65899	0.829493	+	0.558518i	$0.188630\pi$
$948$	−10.8564	−0.352600
$949$	0	0
$950$	−19.9282	−0.646556
$951$	−18.0000	−0.583690
$952$	−0.267949	−0.00868428
$953$	51.3731	1.66414	0.832068	−	0.554673i	$-0.187157\pi$
$953$	51.3731	1.66414	0.832068	+	0.554673i	$0.187157\pi$
$954$	10.4641	0.338788
$955$	3.07180	0.0994010
$956$	13.8038	0.446448
$957$	11.1962	0.361920
$958$	31.3923	1.01424
$959$	15.4641	0.499362
$960$	0.732051	0.0236268
$961$	27.6795	0.892887
$962$	0	0
$963$	−1.53590	−0.0494936
$964$	−12.7846	−0.411765
$965$	−12.5885	−0.405237
$966$	3.46410	0.111456
$967$	−36.0000	−1.15768	−0.578841	−	0.815440i	$-0.696495\pi$
$967$	−36.0000	−1.15768	−0.578841	+	0.815440i	$0.696495\pi$
$968$	−2.92820	−0.0941160
$969$	1.19615	0.0384260
$970$	−0.535898	−0.0172067
$971$	44.2487	1.42001	0.710004	−	0.704197i	$-0.248693\pi$
$971$	44.2487	1.42001	0.710004	+	0.704197i	$0.248693\pi$
$972$	−1.00000	−0.0320750
$973$	−18.1244	−0.581040
$974$	21.1962	0.679169
$975$	0	0
$976$	−11.7321	−0.375534
$977$	18.2487	0.583828	0.291914	−	0.956445i	$-0.405708\pi$
$977$	18.2487	0.583828	0.291914	+	0.956445i	$0.405708\pi$
$978$	−7.80385	−0.249540
$979$	−24.1244	−0.771018
$980$	−0.732051	−0.0233845
$981$	3.66025	0.116863
$982$	−36.2487	−1.15674
$983$	−28.1436	−0.897641	−0.448821	−	0.893622i	$-0.648156\pi$
$983$	−28.1436	−0.897641	−0.448821	+	0.893622i	$0.648156\pi$
$984$	−7.00000	−0.223152
$985$	−1.66025	−0.0529001
$986$	0.803848	0.0255997
$987$	−4.46410	−0.142094
$988$	0	0
$989$	2.53590	0.0806369
$990$	2.73205	0.0868303
$991$	−4.60770	−0.146368	−0.0731841	−	0.997318i	$-0.523316\pi$
$991$	−4.60770	−0.146368	−0.0731841	+	0.997318i	$0.523316\pi$
$992$	−7.66025	−0.243213
$993$	−16.9282	−0.537200
$994$	2.19615	0.0696577
$995$	3.07180	0.0973825
$996$	−5.66025	−0.179352
$997$	42.2295	1.33742	0.668710	−	0.743523i	$-0.266847\pi$
$997$	42.2295	1.33742	0.668710	+	0.743523i	$0.266847\pi$
$998$	−12.9808	−0.410899
$999$	1.26795	0.0401161

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7098.2.a.bj.1.1		2
13.2	odd	12		546.2.s.d.43.1	✓	4
13.7	odd	12		546.2.s.d.127.1	yes	4
13.12	even	2		7098.2.a.bs.1.2		2
39.2	even	12		1638.2.bj.d.1135.2		4
39.20	even	12		1638.2.bj.d.127.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.s.d.43.1	✓	4	13.2	odd	12
546.2.s.d.127.1	yes	4	13.7	odd	12
1638.2.bj.d.127.2		4	39.20	even	12
1638.2.bj.d.1135.2		4	39.2	even	12
7098.2.a.bj.1.1		2	1.1	even	1	trivial
7098.2.a.bs.1.2		2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.732051 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.732051 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.732051 q^{10} +3.73205 q^{11} -1.00000 q^{12} -1.00000 q^{14} +0.732051 q^{15} +1.00000 q^{16} +0.267949 q^{17} -1.00000 q^{18} -4.46410 q^{19} -0.732051 q^{20} -1.00000 q^{21} -3.73205 q^{22} +3.46410 q^{23} +1.00000 q^{24} -4.46410 q^{25} -1.00000 q^{27} +1.00000 q^{28} -3.00000 q^{29} -0.732051 q^{30} +7.66025 q^{31} -1.00000 q^{32} -3.73205 q^{33} -0.267949 q^{34} -0.732051 q^{35} +1.00000 q^{36} -1.26795 q^{37} +4.46410 q^{38} +0.732051 q^{40} -7.00000 q^{41} +1.00000 q^{42} +0.732051 q^{43} +3.73205 q^{44} -0.732051 q^{45} -3.46410 q^{46} +4.46410 q^{47} -1.00000 q^{48} +1.00000 q^{49} +4.46410 q^{50} -0.267949 q^{51} -10.4641 q^{53} +1.00000 q^{54} -2.73205 q^{55} -1.00000 q^{56} +4.46410 q^{57} +3.00000 q^{58} +0.928203 q^{59} +0.732051 q^{60} -11.7321 q^{61} -7.66025 q^{62} +1.00000 q^{63} +1.00000 q^{64} +3.73205 q^{66} -12.9282 q^{67} +0.267949 q^{68} -3.46410 q^{69} +0.732051 q^{70} -2.19615 q^{71} -1.00000 q^{72} +6.53590 q^{73} +1.26795 q^{74} +4.46410 q^{75} -4.46410 q^{76} +3.73205 q^{77} +10.8564 q^{79} -0.732051 q^{80} +1.00000 q^{81} +7.00000 q^{82} +5.66025 q^{83} -1.00000 q^{84} -0.196152 q^{85} -0.732051 q^{86} +3.00000 q^{87} -3.73205 q^{88} -6.46410 q^{89} +0.732051 q^{90} +3.46410 q^{92} -7.66025 q^{93} -4.46410 q^{94} +3.26795 q^{95} +1.00000 q^{96} -0.732051 q^{97} -1.00000 q^{98} +3.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 4 q^{11} - 2 q^{12} - 2 q^{14} - 2 q^{15} + 2 q^{16} + 4 q^{17} - 2 q^{18} - 2 q^{19} + 2 q^{20} - 2 q^{21} - 4 q^{22} + 2 q^{24} - 2 q^{25} - 2 q^{27} + 2 q^{28} - 6 q^{29} + 2 q^{30} - 2 q^{31} - 2 q^{32} - 4 q^{33} - 4 q^{34} + 2 q^{35} + 2 q^{36} - 6 q^{37} + 2 q^{38} - 2 q^{40} - 14 q^{41} + 2 q^{42} - 2 q^{43} + 4 q^{44} + 2 q^{45} + 2 q^{47} - 2 q^{48} + 2 q^{49} + 2 q^{50} - 4 q^{51} - 14 q^{53} + 2 q^{54} - 2 q^{55} - 2 q^{56} + 2 q^{57} + 6 q^{58} - 12 q^{59} - 2 q^{60} - 20 q^{61} + 2 q^{62} + 2 q^{63} + 2 q^{64} + 4 q^{66} - 12 q^{67} + 4 q^{68} - 2 q^{70} + 6 q^{71} - 2 q^{72} + 20 q^{73} + 6 q^{74} + 2 q^{75} - 2 q^{76} + 4 q^{77} - 6 q^{79} + 2 q^{80} + 2 q^{81} + 14 q^{82} - 6 q^{83} - 2 q^{84} + 10 q^{85} + 2 q^{86} + 6 q^{87} - 4 q^{88} - 6 q^{89} - 2 q^{90} + 2 q^{93} - 2 q^{94} + 10 q^{95} + 2 q^{96} + 2 q^{97} - 2 q^{98} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 + 4 * q^11 - 2 * q^12 - 2 * q^14 - 2 * q^15 + 2 * q^16 + 4 * q^17 - 2 * q^18 - 2 * q^19 + 2 * q^20 - 2 * q^21 - 4 * q^22 + 2 * q^24 - 2 * q^25 - 2 * q^27 + 2 * q^28 - 6 * q^29 + 2 * q^30 - 2 * q^31 - 2 * q^32 - 4 * q^33 - 4 * q^34 + 2 * q^35 + 2 * q^36 - 6 * q^37 + 2 * q^38 - 2 * q^40 - 14 * q^41 + 2 * q^42 - 2 * q^43 + 4 * q^44 + 2 * q^45 + 2 * q^47 - 2 * q^48 + 2 * q^49 + 2 * q^50 - 4 * q^51 - 14 * q^53 + 2 * q^54 - 2 * q^55 - 2 * q^56 + 2 * q^57 + 6 * q^58 - 12 * q^59 - 2 * q^60 - 20 * q^61 + 2 * q^62 + 2 * q^63 + 2 * q^64 + 4 * q^66 - 12 * q^67 + 4 * q^68 - 2 * q^70 + 6 * q^71 - 2 * q^72 + 20 * q^73 + 6 * q^74 + 2 * q^75 - 2 * q^76 + 4 * q^77 - 6 * q^79 + 2 * q^80 + 2 * q^81 + 14 * q^82 - 6 * q^83 - 2 * q^84 + 10 * q^85 + 2 * q^86 + 6 * q^87 - 4 * q^88 - 6 * q^89 - 2 * q^90 + 2 * q^93 - 2 * q^94 + 10 * q^95 + 2 * q^96 + 2 * q^97 - 2 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

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