LMFDB - Embedded newform 7098.2.a.bj.1.2

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			1.2
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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.73205 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.73205 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.73205 q^{10} +0.267949 q^{11} -1.00000 q^{12} -1.00000 q^{14} -2.73205 q^{15} +1.00000 q^{16} +3.73205 q^{17} -1.00000 q^{18} +2.46410 q^{19} +2.73205 q^{20} -1.00000 q^{21} -0.267949 q^{22} -3.46410 q^{23} +1.00000 q^{24} +2.46410 q^{25} -1.00000 q^{27} +1.00000 q^{28} -3.00000 q^{29} +2.73205 q^{30} -9.66025 q^{31} -1.00000 q^{32} -0.267949 q^{33} -3.73205 q^{34} +2.73205 q^{35} +1.00000 q^{36} -4.73205 q^{37} -2.46410 q^{38} -2.73205 q^{40} -7.00000 q^{41} +1.00000 q^{42} -2.73205 q^{43} +0.267949 q^{44} +2.73205 q^{45} +3.46410 q^{46} -2.46410 q^{47} -1.00000 q^{48} +1.00000 q^{49} -2.46410 q^{50} -3.73205 q^{51} -3.53590 q^{53} +1.00000 q^{54} +0.732051 q^{55} -1.00000 q^{56} -2.46410 q^{57} +3.00000 q^{58} -12.9282 q^{59} -2.73205 q^{60} -8.26795 q^{61} +9.66025 q^{62} +1.00000 q^{63} +1.00000 q^{64} +0.267949 q^{66} +0.928203 q^{67} +3.73205 q^{68} +3.46410 q^{69} -2.73205 q^{70} +8.19615 q^{71} -1.00000 q^{72} +13.4641 q^{73} +4.73205 q^{74} -2.46410 q^{75} +2.46410 q^{76} +0.267949 q^{77} -16.8564 q^{79} +2.73205 q^{80} +1.00000 q^{81} +7.00000 q^{82} -11.6603 q^{83} -1.00000 q^{84} +10.1962 q^{85} +2.73205 q^{86} +3.00000 q^{87} -0.267949 q^{88} +0.464102 q^{89} -2.73205 q^{90} -3.46410 q^{92} +9.66025 q^{93} +2.46410 q^{94} +6.73205 q^{95} +1.00000 q^{96} +2.73205 q^{97} -1.00000 q^{98} +0.267949 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.73205	1.22181	0.610905	−	0.791704i	$-0.290806\pi$
$5$	2.73205	1.22181	0.610905	+	0.791704i	$0.290806\pi$
$6$	1.00000	0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.73205	−0.863950
$11$	0.267949	0.0807897	0.0403949	−	0.999184i	$-0.487138\pi$
$11$	0.267949	0.0807897	0.0403949	+	0.999184i	$0.487138\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−1.00000	−0.267261
$15$	−2.73205	−0.705412
$16$	1.00000	0.250000
$17$	3.73205	0.905155	0.452578	−	0.891725i	$-0.350505\pi$
$17$	3.73205	0.905155	0.452578	+	0.891725i	$0.350505\pi$
$18$	−1.00000	−0.235702
$19$	2.46410	0.565304	0.282652	−	0.959223i	$-0.408786\pi$
$19$	2.46410	0.565304	0.282652	+	0.959223i	$0.408786\pi$
$20$	2.73205	0.610905
$21$	−1.00000	−0.218218
$22$	−0.267949	−0.0571270
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	1.00000	0.204124
$25$	2.46410	0.492820
$26$	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	$0.589851\pi$
$30$	2.73205	0.498802
$31$	−9.66025	−1.73503	−0.867516	−	0.497409i	$-0.834285\pi$
$31$	−9.66025	−1.73503	−0.867516	+	0.497409i	$0.834285\pi$
$32$	−1.00000	−0.176777
$33$	−0.267949	−0.0466440
$34$	−3.73205	−0.640041
$35$	2.73205	0.461801
$36$	1.00000	0.166667
$37$	−4.73205	−0.777944	−0.388972	−	0.921250i	$-0.627170\pi$
$37$	−4.73205	−0.777944	−0.388972	+	0.921250i	$0.627170\pi$
$38$	−2.46410	−0.399730
$39$	0	0
$40$	−2.73205	−0.431975
$41$	−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
$43$	−2.73205	−0.416634	−0.208317	−	0.978061i	$-0.566799\pi$
$43$	−2.73205	−0.416634	−0.208317	+	0.978061i	$0.566799\pi$
$44$	0.267949	0.0403949
$45$	2.73205	0.407270
$46$	3.46410	0.510754
$47$	−2.46410	−0.359426	−0.179713	−	0.983719i	$-0.557517\pi$
$47$	−2.46410	−0.359426	−0.179713	+	0.983719i	$0.557517\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	−2.46410	−0.348477
$51$	−3.73205	−0.522592
$52$	0	0
$53$	−3.53590	−0.485693	−0.242846	−	0.970065i	$-0.578081\pi$
$53$	−3.53590	−0.485693	−0.242846	+	0.970065i	$0.578081\pi$
$54$	1.00000	0.136083
$55$	0.732051	0.0987097
$56$	−1.00000	−0.133631
$57$	−2.46410	−0.326378
$58$	3.00000	0.393919
$59$	−12.9282	−1.68311	−0.841554	−	0.540172i	$-0.818359\pi$
$59$	−12.9282	−1.68311	−0.841554	+	0.540172i	$0.818359\pi$
$60$	−2.73205	−0.352706
$61$	−8.26795	−1.05860	−0.529301	−	0.848434i	$-0.677546\pi$
$61$	−8.26795	−1.05860	−0.529301	+	0.848434i	$0.677546\pi$
$62$	9.66025	1.22685
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	0.267949	0.0329823
$67$	0.928203	0.113398	0.0566990	−	0.998391i	$-0.481942\pi$
$67$	0.928203	0.113398	0.0566990	+	0.998391i	$0.481942\pi$
$68$	3.73205	0.452578
$69$	3.46410	0.417029
$70$	−2.73205	−0.326543
$71$	8.19615	0.972704	0.486352	−	0.873763i	$-0.338327\pi$
$71$	8.19615	0.972704	0.486352	+	0.873763i	$0.338327\pi$
$72$	−1.00000	−0.117851
$73$	13.4641	1.57585	0.787927	−	0.615769i	$-0.211154\pi$
$73$	13.4641	1.57585	0.787927	+	0.615769i	$0.211154\pi$
$74$	4.73205	0.550090
$75$	−2.46410	−0.284530
$76$	2.46410	0.282652
$77$	0.267949	0.0305356
$78$	0	0
$79$	−16.8564	−1.89649	−0.948247	−	0.317534i	$-0.897145\pi$
$79$	−16.8564	−1.89649	−0.948247	+	0.317534i	$0.897145\pi$
$80$	2.73205	0.305453
$81$	1.00000	0.111111
$82$	7.00000	0.773021
$83$	−11.6603	−1.27988	−0.639940	−	0.768425i	$-0.721041\pi$
$83$	−11.6603	−1.27988	−0.639940	+	0.768425i	$0.721041\pi$
$84$	−1.00000	−0.109109
$85$	10.1962	1.10593
$86$	2.73205	0.294605
$87$	3.00000	0.321634
$88$	−0.267949	−0.0285635
$89$	0.464102	0.0491947	0.0245973	−	0.999697i	$-0.492170\pi$
$89$	0.464102	0.0491947	0.0245973	+	0.999697i	$0.492170\pi$
$90$	−2.73205	−0.287983
$91$	0	0
$92$	−3.46410	−0.361158
$93$	9.66025	1.00172
$94$	2.46410	0.254153
$95$	6.73205	0.690694
$96$	1.00000	0.102062
$97$	2.73205	0.277398	0.138699	−	0.990335i	$-0.455708\pi$
$97$	2.73205	0.277398	0.138699	+	0.990335i	$0.455708\pi$
$98$	−1.00000	−0.101015
$99$	0.267949	0.0269299
$100$	2.46410	0.246410
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	3.73205	0.369528
$103$	−1.80385	−0.177738	−0.0888692	−	0.996043i	$-0.528325\pi$
$103$	−1.80385	−0.177738	−0.0888692	+	0.996043i	$0.528325\pi$
$104$	0	0
$105$	−2.73205	−0.266621
$106$	3.53590	0.343437
$107$	−8.46410	−0.818256	−0.409128	−	0.912477i	$-0.634167\pi$
$107$	−8.46410	−0.818256	−0.409128	+	0.912477i	$0.634167\pi$
$108$	−1.00000	−0.0962250
$109$	−13.6603	−1.30842	−0.654208	−	0.756315i	$-0.726998\pi$
$109$	−13.6603	−1.30842	−0.654208	+	0.756315i	$0.726998\pi$
$110$	−0.732051	−0.0697983
$111$	4.73205	0.449146
$112$	1.00000	0.0944911
$113$	0.196152	0.0184525	0.00922623	−	0.999957i	$-0.497063\pi$
$113$	0.196152	0.0184525	0.00922623	+	0.999957i	$0.497063\pi$
$114$	2.46410	0.230784
$115$	−9.46410	−0.882532
$116$	−3.00000	−0.278543
$117$	0	0
$118$	12.9282	1.19014
$119$	3.73205	0.342117
$120$	2.73205	0.249401
$121$	−10.9282	−0.993473
$122$	8.26795	0.748545
$123$	7.00000	0.631169
$124$	−9.66025	−0.867516
$125$	−6.92820	−0.619677
$126$	−1.00000	−0.0890871
$127$	−19.4641	−1.72716	−0.863580	−	0.504212i	$-0.831783\pi$
$127$	−19.4641	−1.72716	−0.863580	+	0.504212i	$0.831783\pi$
$128$	−1.00000	−0.0883883
$129$	2.73205	0.240544
$130$	0	0
$131$	5.07180	0.443125	0.221562	−	0.975146i	$-0.428884\pi$
$131$	5.07180	0.443125	0.221562	+	0.975146i	$0.428884\pi$
$132$	−0.267949	−0.0233220
$133$	2.46410	0.213665
$134$	−0.928203	−0.0801845
$135$	−2.73205	−0.235137
$136$	−3.73205	−0.320021
$137$	8.53590	0.729271	0.364636	−	0.931150i	$-0.381194\pi$
$137$	8.53590	0.729271	0.364636	+	0.931150i	$0.381194\pi$
$138$	−3.46410	−0.294884
$139$	6.12436	0.519461	0.259731	−	0.965681i	$-0.416366\pi$
$139$	6.12436	0.519461	0.259731	+	0.965681i	$0.416366\pi$
$140$	2.73205	0.230900
$141$	2.46410	0.207515
$142$	−8.19615	−0.687806
$143$	0	0
$144$	1.00000	0.0833333
$145$	−8.19615	−0.680653
$146$	−13.4641	−1.11430
$147$	−1.00000	−0.0824786
$148$	−4.73205	−0.388972
$149$	5.07180	0.415498	0.207749	−	0.978182i	$-0.433386\pi$
$149$	5.07180	0.415498	0.207749	+	0.978182i	$0.433386\pi$
$150$	2.46410	0.201193
$151$	2.80385	0.228174	0.114087	−	0.993471i	$-0.463606\pi$
$151$	2.80385	0.228174	0.114087	+	0.993471i	$0.463606\pi$
$152$	−2.46410	−0.199865
$153$	3.73205	0.301718
$154$	−0.267949	−0.0215920
$155$	−26.3923	−2.11988
$156$	0	0
$157$	−8.53590	−0.681239	−0.340619	−	0.940201i	$-0.610637\pi$
$157$	−8.53590	−0.681239	−0.340619	+	0.940201i	$0.610637\pi$
$158$	16.8564	1.34102
$159$	3.53590	0.280415
$160$	−2.73205	−0.215988
$161$	−3.46410	−0.273009
$162$	−1.00000	−0.0785674
$163$	−18.1962	−1.42523	−0.712616	−	0.701554i	$-0.752490\pi$
$163$	−18.1962	−1.42523	−0.712616	+	0.701554i	$0.752490\pi$
$164$	−7.00000	−0.546608
$165$	−0.732051	−0.0569901
$166$	11.6603	0.905011
$167$	21.8564	1.69130	0.845650	−	0.533738i	$-0.179213\pi$
$167$	21.8564	1.69130	0.845650	+	0.533738i	$0.179213\pi$
$168$	1.00000	0.0771517
$169$	0	0
$170$	−10.1962	−0.782009
$171$	2.46410	0.188435
$172$	−2.73205	−0.208317
$173$	17.2679	1.31286	0.656429	−	0.754388i	$-0.272066\pi$
$173$	17.2679	1.31286	0.656429	+	0.754388i	$0.272066\pi$
$174$	−3.00000	−0.227429
$175$	2.46410	0.186269
$176$	0.267949	0.0201974
$177$	12.9282	0.971743
$178$	−0.464102	−0.0347859
$179$	1.60770	0.120165	0.0600824	−	0.998193i	$-0.480864\pi$
$179$	1.60770	0.120165	0.0600824	+	0.998193i	$0.480864\pi$
$180$	2.73205	0.203635
$181$	−9.19615	−0.683545	−0.341772	−	0.939783i	$-0.611027\pi$
$181$	−9.19615	−0.683545	−0.341772	+	0.939783i	$0.611027\pi$
$182$	0	0
$183$	8.26795	0.611184
$184$	3.46410	0.255377
$185$	−12.9282	−0.950500
$186$	−9.66025	−0.708324
$187$	1.00000	0.0731272
$188$	−2.46410	−0.179713
$189$	−1.00000	−0.0727393
$190$	−6.73205	−0.488394
$191$	6.19615	0.448338	0.224169	−	0.974550i	$-0.428033\pi$
$191$	6.19615	0.448338	0.224169	+	0.974550i	$0.428033\pi$
$192$	−1.00000	−0.0721688
$193$	6.80385	0.489752	0.244876	−	0.969554i	$-0.421253\pi$
$193$	6.80385	0.489752	0.244876	+	0.969554i	$0.421253\pi$
$194$	−2.73205	−0.196150
$195$	0	0
$196$	1.00000	0.0714286
$197$	5.73205	0.408392	0.204196	−	0.978930i	$-0.434542\pi$
$197$	5.73205	0.408392	0.204196	+	0.978930i	$0.434542\pi$
$198$	−0.267949	−0.0190423
$199$	6.19615	0.439234	0.219617	−	0.975586i	$-0.429519\pi$
$199$	6.19615	0.439234	0.219617	+	0.975586i	$0.429519\pi$
$200$	−2.46410	−0.174238
$201$	−0.928203	−0.0654704
$202$	10.0000	0.703598
$203$	−3.00000	−0.210559
$204$	−3.73205	−0.261296
$205$	−19.1244	−1.33570
$206$	1.80385	0.125680
$207$	−3.46410	−0.240772
$208$	0	0
$209$	0.660254	0.0456707
$210$	2.73205	0.188529
$211$	11.8564	0.816229	0.408114	−	0.912931i	$-0.366186\pi$
$211$	11.8564	0.816229	0.408114	+	0.912931i	$0.366186\pi$
$212$	−3.53590	−0.242846
$213$	−8.19615	−0.561591
$214$	8.46410	0.578594
$215$	−7.46410	−0.509048
$216$	1.00000	0.0680414
$217$	−9.66025	−0.655781
$218$	13.6603	0.925189
$219$	−13.4641	−0.909820
$220$	0.732051	0.0493549
$221$	0	0
$222$	−4.73205	−0.317594
$223$	−12.3923	−0.829850	−0.414925	−	0.909856i	$-0.636192\pi$
$223$	−12.3923	−0.829850	−0.414925	+	0.909856i	$0.636192\pi$
$224$	−1.00000	−0.0668153
$225$	2.46410	0.164273
$226$	−0.196152	−0.0130479
$227$	−15.4641	−1.02639	−0.513194	−	0.858272i	$-0.671538\pi$
$227$	−15.4641	−1.02639	−0.513194	+	0.858272i	$0.671538\pi$
$228$	−2.46410	−0.163189
$229$	24.3205	1.60714	0.803572	−	0.595207i	$-0.202930\pi$
$229$	24.3205	1.60714	0.803572	+	0.595207i	$0.202930\pi$
$230$	9.46410	0.624044
$231$	−0.267949	−0.0176298
$232$	3.00000	0.196960
$233$	−8.19615	−0.536948	−0.268474	−	0.963287i	$-0.586519\pi$
$233$	−8.19615	−0.536948	−0.268474	+	0.963287i	$0.586519\pi$
$234$	0	0
$235$	−6.73205	−0.439151
$236$	−12.9282	−0.841554
$237$	16.8564	1.09494
$238$	−3.73205	−0.241913
$239$	24.1962	1.56512	0.782559	−	0.622576i	$-0.213914\pi$
$239$	24.1962	1.56512	0.782559	+	0.622576i	$0.213914\pi$
$240$	−2.73205	−0.176353
$241$	28.7846	1.85418	0.927090	−	0.374839i	$-0.122302\pi$
$241$	28.7846	1.85418	0.927090	+	0.374839i	$0.122302\pi$
$242$	10.9282	0.702492
$243$	−1.00000	−0.0641500
$244$	−8.26795	−0.529301
$245$	2.73205	0.174544
$246$	−7.00000	−0.446304
$247$	0	0
$248$	9.66025	0.613427
$249$	11.6603	0.738939
$250$	6.92820	0.438178
$251$	7.80385	0.492574	0.246287	−	0.969197i	$-0.420789\pi$
$251$	7.80385	0.492574	0.246287	+	0.969197i	$0.420789\pi$
$252$	1.00000	0.0629941
$253$	−0.928203	−0.0583556
$254$	19.4641	1.22129
$255$	−10.1962	−0.638508
$256$	1.00000	0.0625000
$257$	−27.0526	−1.68749	−0.843746	−	0.536742i	$-0.819655\pi$
$257$	−27.0526	−1.68749	−0.843746	+	0.536742i	$0.819655\pi$
$258$	−2.73205	−0.170090
$259$	−4.73205	−0.294035
$260$	0	0
$261$	−3.00000	−0.185695
$262$	−5.07180	−0.313337
$263$	−15.2679	−0.941462	−0.470731	−	0.882277i	$-0.656010\pi$
$263$	−15.2679	−0.941462	−0.470731	+	0.882277i	$0.656010\pi$
$264$	0.267949	0.0164911
$265$	−9.66025	−0.593425
$266$	−2.46410	−0.151084
$267$	−0.464102	−0.0284026
$268$	0.928203	0.0566990
$269$	31.7128	1.93356	0.966782	−	0.255602i	$-0.0822736\pi$
$269$	31.7128	1.93356	0.966782	+	0.255602i	$0.0822736\pi$
$270$	2.73205	0.166267
$271$	23.5167	1.42854	0.714268	−	0.699873i	$-0.246760\pi$
$271$	23.5167	1.42854	0.714268	+	0.699873i	$0.246760\pi$
$272$	3.73205	0.226289
$273$	0	0
$274$	−8.53590	−0.515672
$275$	0.660254	0.0398148
$276$	3.46410	0.208514
$277$	−5.60770	−0.336934	−0.168467	−	0.985707i	$-0.553882\pi$
$277$	−5.60770	−0.336934	−0.168467	+	0.985707i	$0.553882\pi$
$278$	−6.12436	−0.367314
$279$	−9.66025	−0.578344
$280$	−2.73205	−0.163271
$281$	7.80385	0.465539	0.232769	−	0.972532i	$-0.425221\pi$
$281$	7.80385	0.465539	0.232769	+	0.972532i	$0.425221\pi$
$282$	−2.46410	−0.146735
$283$	−9.07180	−0.539262	−0.269631	−	0.962964i	$-0.586902\pi$
$283$	−9.07180	−0.539262	−0.269631	+	0.962964i	$0.586902\pi$
$284$	8.19615	0.486352
$285$	−6.73205	−0.398772
$286$	0	0
$287$	−7.00000	−0.413197
$288$	−1.00000	−0.0589256
$289$	−3.07180	−0.180694
$290$	8.19615	0.481295
$291$	−2.73205	−0.160156
$292$	13.4641	0.787927
$293$	28.7846	1.68161	0.840807	−	0.541334i	$-0.182081\pi$
$293$	28.7846	1.68161	0.840807	+	0.541334i	$0.182081\pi$
$294$	1.00000	0.0583212
$295$	−35.3205	−2.05644
$296$	4.73205	0.275045
$297$	−0.267949	−0.0155480
$298$	−5.07180	−0.293801
$299$	0	0
$300$	−2.46410	−0.142265
$301$	−2.73205	−0.157473
$302$	−2.80385	−0.161343
$303$	10.0000	0.574485
$304$	2.46410	0.141326
$305$	−22.5885	−1.29341
$306$	−3.73205	−0.213347
$307$	7.78461	0.444291	0.222146	−	0.975014i	$-0.428694\pi$
$307$	7.78461	0.444291	0.222146	+	0.975014i	$0.428694\pi$
$308$	0.267949	0.0152678
$309$	1.80385	0.102617
$310$	26.3923	1.49898
$311$	−8.80385	−0.499220	−0.249610	−	0.968346i	$-0.580302\pi$
$311$	−8.80385	−0.499220	−0.249610	+	0.968346i	$0.580302\pi$
$312$	0	0
$313$	12.1962	0.689367	0.344684	−	0.938719i	$-0.387986\pi$
$313$	12.1962	0.689367	0.344684	+	0.938719i	$0.387986\pi$
$314$	8.53590	0.481709
$315$	2.73205	0.153934
$316$	−16.8564	−0.948247
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	−3.53590	−0.198283
$319$	−0.803848	−0.0450068
$320$	2.73205	0.152726
$321$	8.46410	0.472420
$322$	3.46410	0.193047
$323$	9.19615	0.511688
$324$	1.00000	0.0555556
$325$	0	0
$326$	18.1962	1.00779
$327$	13.6603	0.755414
$328$	7.00000	0.386510
$329$	−2.46410	−0.135850
$330$	0.732051	0.0402981
$331$	3.07180	0.168841	0.0844206	−	0.996430i	$-0.473096\pi$
$331$	3.07180	0.168841	0.0844206	+	0.996430i	$0.473096\pi$
$332$	−11.6603	−0.639940
$333$	−4.73205	−0.259315
$334$	−21.8564	−1.19593
$335$	2.53590	0.138551
$336$	−1.00000	−0.0545545
$337$	−13.7846	−0.750896	−0.375448	−	0.926844i	$-0.622511\pi$
$337$	−13.7846	−0.750896	−0.375448	+	0.926844i	$0.622511\pi$
$338$	0	0
$339$	−0.196152	−0.0106535
$340$	10.1962	0.552964
$341$	−2.58846	−0.140173
$342$	−2.46410	−0.133243
$343$	1.00000	0.0539949
$344$	2.73205	0.147302
$345$	9.46410	0.509530
$346$	−17.2679	−0.928331
$347$	−4.85641	−0.260706	−0.130353	−	0.991468i	$-0.541611\pi$
$347$	−4.85641	−0.260706	−0.130353	+	0.991468i	$0.541611\pi$
$348$	3.00000	0.160817
$349$	31.7128	1.69755	0.848774	−	0.528756i	$-0.177341\pi$
$349$	31.7128	1.69755	0.848774	+	0.528756i	$0.177341\pi$
$350$	−2.46410	−0.131712
$351$	0	0
$352$	−0.267949	−0.0142817
$353$	5.46410	0.290825	0.145412	−	0.989371i	$-0.453549\pi$
$353$	5.46410	0.290825	0.145412	+	0.989371i	$0.453549\pi$
$354$	−12.9282	−0.687126
$355$	22.3923	1.18846
$356$	0.464102	0.0245973
$357$	−3.73205	−0.197521
$358$	−1.60770	−0.0849693
$359$	26.1962	1.38258	0.691290	−	0.722577i	$-0.257043\pi$
$359$	26.1962	1.38258	0.691290	+	0.722577i	$0.257043\pi$
$360$	−2.73205	−0.143992
$361$	−12.9282	−0.680432
$362$	9.19615	0.483339
$363$	10.9282	0.573582
$364$	0	0
$365$	36.7846	1.92539
$366$	−8.26795	−0.432173
$367$	−32.2487	−1.68337	−0.841685	−	0.539970i	$-0.818436\pi$
$367$	−32.2487	−1.68337	−0.841685	+	0.539970i	$0.818436\pi$
$368$	−3.46410	−0.180579
$369$	−7.00000	−0.364405
$370$	12.9282	0.672105
$371$	−3.53590	−0.183575
$372$	9.66025	0.500861
$373$	10.5885	0.548250	0.274125	−	0.961694i	$-0.411612\pi$
$373$	10.5885	0.548250	0.274125	+	0.961694i	$0.411612\pi$
$374$	−1.00000	−0.0517088
$375$	6.92820	0.357771
$376$	2.46410	0.127076
$377$	0	0
$378$	1.00000	0.0514344
$379$	16.5885	0.852092	0.426046	−	0.904702i	$-0.359906\pi$
$379$	16.5885	0.852092	0.426046	+	0.904702i	$0.359906\pi$
$380$	6.73205	0.345347
$381$	19.4641	0.997176
$382$	−6.19615	−0.317023
$383$	−35.3923	−1.80846	−0.904231	−	0.427043i	$-0.859555\pi$
$383$	−35.3923	−1.80846	−0.904231	+	0.427043i	$0.859555\pi$
$384$	1.00000	0.0510310
$385$	0.732051	0.0373088
$386$	−6.80385	−0.346307
$387$	−2.73205	−0.138878
$388$	2.73205	0.138699
$389$	29.1769	1.47933	0.739664	−	0.672976i	$-0.234984\pi$
$389$	29.1769	1.47933	0.739664	+	0.672976i	$0.234984\pi$
$390$	0	0
$391$	−12.9282	−0.653807
$392$	−1.00000	−0.0505076
$393$	−5.07180	−0.255838
$394$	−5.73205	−0.288777
$395$	−46.0526	−2.31716
$396$	0.267949	0.0134650
$397$	−9.53590	−0.478593	−0.239297	−	0.970947i	$-0.576917\pi$
$397$	−9.53590	−0.478593	−0.239297	+	0.970947i	$0.576917\pi$
$398$	−6.19615	−0.310585
$399$	−2.46410	−0.123359
$400$	2.46410	0.123205
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0.928203	0.0462946
$403$	0	0
$404$	−10.0000	−0.497519
$405$	2.73205	0.135757
$406$	3.00000	0.148888
$407$	−1.26795	−0.0628499
$408$	3.73205	0.184764
$409$	−16.7321	−0.827347	−0.413673	−	0.910425i	$-0.635754\pi$
$409$	−16.7321	−0.827347	−0.413673	+	0.910425i	$0.635754\pi$
$410$	19.1244	0.944485
$411$	−8.53590	−0.421045
$412$	−1.80385	−0.0888692
$413$	−12.9282	−0.636155
$414$	3.46410	0.170251
$415$	−31.8564	−1.56377
$416$	0	0
$417$	−6.12436	−0.299911
$418$	−0.660254	−0.0322941
$419$	4.19615	0.204995	0.102498	−	0.994733i	$-0.467317\pi$
$419$	4.19615	0.204995	0.102498	+	0.994733i	$0.467317\pi$
$420$	−2.73205	−0.133310
$421$	−6.39230	−0.311542	−0.155771	−	0.987793i	$-0.549786\pi$
$421$	−6.39230	−0.311542	−0.155771	+	0.987793i	$0.549786\pi$
$422$	−11.8564	−0.577161
$423$	−2.46410	−0.119809
$424$	3.53590	0.171718
$425$	9.19615	0.446079
$426$	8.19615	0.397105
$427$	−8.26795	−0.400114
$428$	−8.46410	−0.409128
$429$	0	0
$430$	7.46410	0.359951
$431$	−3.12436	−0.150495	−0.0752475	−	0.997165i	$-0.523975\pi$
$431$	−3.12436	−0.150495	−0.0752475	+	0.997165i	$0.523975\pi$
$432$	−1.00000	−0.0481125
$433$	−4.92820	−0.236834	−0.118417	−	0.992964i	$-0.537782\pi$
$433$	−4.92820	−0.236834	−0.118417	+	0.992964i	$0.537782\pi$
$434$	9.66025	0.463707
$435$	8.19615	0.392975
$436$	−13.6603	−0.654208
$437$	−8.53590	−0.408327
$438$	13.4641	0.643340
$439$	4.39230	0.209633	0.104817	−	0.994492i	$-0.466574\pi$
$439$	4.39230	0.209633	0.104817	+	0.994492i	$0.466574\pi$
$440$	−0.732051	−0.0348992
$441$	1.00000	0.0476190
$442$	0	0
$443$	−10.6077	−0.503987	−0.251993	−	0.967729i	$-0.581086\pi$
$443$	−10.6077	−0.503987	−0.251993	+	0.967729i	$0.581086\pi$
$444$	4.73205	0.224573
$445$	1.26795	0.0601066
$446$	12.3923	0.586793
$447$	−5.07180	−0.239888
$448$	1.00000	0.0472456
$449$	−7.26795	−0.342996	−0.171498	−	0.985184i	$-0.554861\pi$
$449$	−7.26795	−0.342996	−0.171498	+	0.985184i	$0.554861\pi$
$450$	−2.46410	−0.116159
$451$	−1.87564	−0.0883206
$452$	0.196152	0.00922623
$453$	−2.80385	−0.131736
$454$	15.4641	0.725766
$455$	0	0
$456$	2.46410	0.115392
$457$	−34.7846	−1.62716	−0.813578	−	0.581456i	$-0.802483\pi$
$457$	−34.7846	−1.62716	−0.813578	+	0.581456i	$0.802483\pi$
$458$	−24.3205	−1.13642
$459$	−3.73205	−0.174197
$460$	−9.46410	−0.441266
$461$	−27.7128	−1.29071	−0.645357	−	0.763881i	$-0.723291\pi$
$461$	−27.7128	−1.29071	−0.645357	+	0.763881i	$0.723291\pi$
$462$	0.267949	0.0124661
$463$	9.19615	0.427381	0.213691	−	0.976901i	$-0.431452\pi$
$463$	9.19615	0.427381	0.213691	+	0.976901i	$0.431452\pi$
$464$	−3.00000	−0.139272
$465$	26.3923	1.22391
$466$	8.19615	0.379679
$467$	17.8564	0.826296	0.413148	−	0.910664i	$-0.364429\pi$
$467$	17.8564	0.826296	0.413148	+	0.910664i	$0.364429\pi$
$468$	0	0
$469$	0.928203	0.0428604
$470$	6.73205	0.310526
$471$	8.53590	0.393313
$472$	12.9282	0.595069
$473$	−0.732051	−0.0336597
$474$	−16.8564	−0.774240
$475$	6.07180	0.278593
$476$	3.73205	0.171058
$477$	−3.53590	−0.161898
$478$	−24.1962	−1.10671
$479$	−10.6077	−0.484678	−0.242339	−	0.970192i	$-0.577915\pi$
$479$	−10.6077	−0.484678	−0.242339	+	0.970192i	$0.577915\pi$
$480$	2.73205	0.124700
$481$	0	0
$482$	−28.7846	−1.31110
$483$	3.46410	0.157622
$484$	−10.9282	−0.496737
$485$	7.46410	0.338927
$486$	1.00000	0.0453609
$487$	−10.8038	−0.489569	−0.244785	−	0.969578i	$-0.578717\pi$
$487$	−10.8038	−0.489569	−0.244785	+	0.969578i	$0.578717\pi$
$488$	8.26795	0.374272
$489$	18.1962	0.822858
$490$	−2.73205	−0.123421
$491$	−12.2487	−0.552777	−0.276388	−	0.961046i	$-0.589138\pi$
$491$	−12.2487	−0.552777	−0.276388	+	0.961046i	$0.589138\pi$
$492$	7.00000	0.315584
$493$	−11.1962	−0.504249
$494$	0	0
$495$	0.732051	0.0329032
$496$	−9.66025	−0.433758
$497$	8.19615	0.367648
$498$	−11.6603	−0.522508
$499$	−38.9808	−1.74502	−0.872509	−	0.488598i	$-0.837509\pi$
$499$	−38.9808	−1.74502	−0.872509	+	0.488598i	$0.837509\pi$
$500$	−6.92820	−0.309839
$501$	−21.8564	−0.976472
$502$	−7.80385	−0.348303
$503$	31.7128	1.41400	0.707002	−	0.707211i	$-0.250047\pi$
$503$	31.7128	1.41400	0.707002	+	0.707211i	$0.250047\pi$
$504$	−1.00000	−0.0445435
$505$	−27.3205	−1.21575
$506$	0.928203	0.0412637
$507$	0	0
$508$	−19.4641	−0.863580
$509$	12.0526	0.534220	0.267110	−	0.963666i	$-0.413931\pi$
$509$	12.0526	0.534220	0.267110	+	0.963666i	$0.413931\pi$
$510$	10.1962	0.451493
$511$	13.4641	0.595617
$512$	−1.00000	−0.0441942
$513$	−2.46410	−0.108793
$514$	27.0526	1.19324
$515$	−4.92820	−0.217163
$516$	2.73205	0.120272
$517$	−0.660254	−0.0290379
$518$	4.73205	0.207914
$519$	−17.2679	−0.757979
$520$	0	0
$521$	−5.73205	−0.251126	−0.125563	−	0.992086i	$-0.540074\pi$
$521$	−5.73205	−0.251126	−0.125563	+	0.992086i	$0.540074\pi$
$522$	3.00000	0.131306
$523$	−29.1962	−1.27666	−0.638329	−	0.769763i	$-0.720374\pi$
$523$	−29.1962	−1.27666	−0.638329	+	0.769763i	$0.720374\pi$
$524$	5.07180	0.221562
$525$	−2.46410	−0.107542
$526$	15.2679	0.665714
$527$	−36.0526	−1.57047
$528$	−0.267949	−0.0116610
$529$	−11.0000	−0.478261
$530$	9.66025	0.419615
$531$	−12.9282	−0.561036
$532$	2.46410	0.106832
$533$	0	0
$534$	0.464102	0.0200836
$535$	−23.1244	−0.999753
$536$	−0.928203	−0.0400923
$537$	−1.60770	−0.0693772
$538$	−31.7128	−1.36724
$539$	0.267949	0.0115414
$540$	−2.73205	−0.117569
$541$	19.8038	0.851434	0.425717	−	0.904856i	$-0.360022\pi$
$541$	19.8038	0.851434	0.425717	+	0.904856i	$0.360022\pi$
$542$	−23.5167	−1.01013
$543$	9.19615	0.394645
$544$	−3.73205	−0.160010
$545$	−37.3205	−1.59863
$546$	0	0
$547$	−37.1244	−1.58732	−0.793661	−	0.608360i	$-0.791828\pi$
$547$	−37.1244	−1.58732	−0.793661	+	0.608360i	$0.791828\pi$
$548$	8.53590	0.364636
$549$	−8.26795	−0.352867
$550$	−0.660254	−0.0281533
$551$	−7.39230	−0.314923
$552$	−3.46410	−0.147442
$553$	−16.8564	−0.716807
$554$	5.60770	0.238248
$555$	12.9282	0.548772
$556$	6.12436	0.259731
$557$	−19.3397	−0.819451	−0.409726	−	0.912209i	$-0.634376\pi$
$557$	−19.3397	−0.819451	−0.409726	+	0.912209i	$0.634376\pi$
$558$	9.66025	0.408951
$559$	0	0
$560$	2.73205	0.115450
$561$	−1.00000	−0.0422200
$562$	−7.80385	−0.329185
$563$	−36.7321	−1.54807	−0.774036	−	0.633142i	$-0.781765\pi$
$563$	−36.7321	−1.54807	−0.774036	+	0.633142i	$0.781765\pi$
$564$	2.46410	0.103757
$565$	0.535898	0.0225454
$566$	9.07180	0.381316
$567$	1.00000	0.0419961
$568$	−8.19615	−0.343903
$569$	19.2679	0.807754	0.403877	−	0.914813i	$-0.367662\pi$
$569$	19.2679	0.807754	0.403877	+	0.914813i	$0.367662\pi$
$570$	6.73205	0.281975
$571$	30.1962	1.26367	0.631835	−	0.775103i	$-0.282302\pi$
$571$	30.1962	1.26367	0.631835	+	0.775103i	$0.282302\pi$
$572$	0	0
$573$	−6.19615	−0.258848
$574$	7.00000	0.292174
$575$	−8.53590	−0.355972
$576$	1.00000	0.0416667
$577$	16.7321	0.696564	0.348282	−	0.937390i	$-0.386765\pi$
$577$	16.7321	0.696564	0.348282	+	0.937390i	$0.386765\pi$
$578$	3.07180	0.127770
$579$	−6.80385	−0.282758
$580$	−8.19615	−0.340327
$581$	−11.6603	−0.483749
$582$	2.73205	0.113247
$583$	−0.947441	−0.0392390
$584$	−13.4641	−0.557148
$585$	0	0
$586$	−28.7846	−1.18908
$587$	41.3731	1.70765	0.853825	−	0.520561i	$-0.174277\pi$
$587$	41.3731	1.70765	0.853825	+	0.520561i	$0.174277\pi$
$588$	−1.00000	−0.0412393
$589$	−23.8038	−0.980820
$590$	35.3205	1.45412
$591$	−5.73205	−0.235785
$592$	−4.73205	−0.194486
$593$	48.3205	1.98429	0.992143	−	0.125111i	$-0.0399288\pi$
$593$	48.3205	1.98429	0.992143	+	0.125111i	$0.0399288\pi$
$594$	0.267949	0.0109941
$595$	10.1962	0.418001
$596$	5.07180	0.207749
$597$	−6.19615	−0.253592
$598$	0	0
$599$	33.7128	1.37747	0.688734	−	0.725014i	$-0.258167\pi$
$599$	33.7128	1.37747	0.688734	+	0.725014i	$0.258167\pi$
$600$	2.46410	0.100597
$601$	−37.1244	−1.51433	−0.757167	−	0.653221i	$-0.773417\pi$
$601$	−37.1244	−1.51433	−0.757167	+	0.653221i	$0.773417\pi$
$602$	2.73205	0.111350
$603$	0.928203	0.0377994
$604$	2.80385	0.114087
$605$	−29.8564	−1.21384
$606$	−10.0000	−0.406222
$607$	−25.9090	−1.05161	−0.525806	−	0.850604i	$-0.676236\pi$
$607$	−25.9090	−1.05161	−0.525806	+	0.850604i	$0.676236\pi$
$608$	−2.46410	−0.0999325
$609$	3.00000	0.121566
$610$	22.5885	0.914580
$611$	0	0
$612$	3.73205	0.150859
$613$	14.1436	0.571254	0.285627	−	0.958341i	$-0.407798\pi$
$613$	14.1436	0.571254	0.285627	+	0.958341i	$0.407798\pi$
$614$	−7.78461	−0.314161
$615$	19.1244	0.771168
$616$	−0.267949	−0.0107960
$617$	3.51666	0.141575	0.0707877	−	0.997491i	$-0.477449\pi$
$617$	3.51666	0.141575	0.0707877	+	0.997491i	$0.477449\pi$
$618$	−1.80385	−0.0725614
$619$	−32.3205	−1.29907	−0.649535	−	0.760331i	$-0.725037\pi$
$619$	−32.3205	−1.29907	−0.649535	+	0.760331i	$0.725037\pi$
$620$	−26.3923	−1.05994
$621$	3.46410	0.139010
$622$	8.80385	0.353002
$623$	0.464102	0.0185938
$624$	0	0
$625$	−31.2487	−1.24995
$626$	−12.1962	−0.487456
$627$	−0.660254	−0.0263680
$628$	−8.53590	−0.340619
$629$	−17.6603	−0.704160
$630$	−2.73205	−0.108848
$631$	35.9808	1.43237	0.716186	−	0.697910i	$-0.245886\pi$
$631$	35.9808	1.43237	0.716186	+	0.697910i	$0.245886\pi$
$632$	16.8564	0.670512
$633$	−11.8564	−0.471250
$634$	−18.0000	−0.714871
$635$	−53.1769	−2.11026
$636$	3.53590	0.140207
$637$	0	0
$638$	0.803848	0.0318246
$639$	8.19615	0.324235
$640$	−2.73205	−0.107994
$641$	32.2487	1.27375	0.636874	−	0.770968i	$-0.280227\pi$
$641$	32.2487	1.27375	0.636874	+	0.770968i	$0.280227\pi$
$642$	−8.46410	−0.334051
$643$	−36.3205	−1.43234	−0.716171	−	0.697925i	$-0.754107\pi$
$643$	−36.3205	−1.43234	−0.716171	+	0.697925i	$0.754107\pi$
$644$	−3.46410	−0.136505
$645$	7.46410	0.293899
$646$	−9.19615	−0.361818
$647$	−1.73205	−0.0680939	−0.0340470	−	0.999420i	$-0.510840\pi$
$647$	−1.73205	−0.0680939	−0.0340470	+	0.999420i	$0.510840\pi$
$648$	−1.00000	−0.0392837
$649$	−3.46410	−0.135978
$650$	0	0
$651$	9.66025	0.378615
$652$	−18.1962	−0.712616
$653$	46.3205	1.81266	0.906331	−	0.422569i	$-0.138872\pi$
$653$	46.3205	1.81266	0.906331	+	0.422569i	$0.138872\pi$
$654$	−13.6603	−0.534158
$655$	13.8564	0.541415
$656$	−7.00000	−0.273304
$657$	13.4641	0.525285
$658$	2.46410	0.0960607
$659$	20.0718	0.781886	0.390943	−	0.920415i	$-0.372149\pi$
$659$	20.0718	0.781886	0.390943	+	0.920415i	$0.372149\pi$
$660$	−0.732051	−0.0284950
$661$	−48.3923	−1.88224	−0.941121	−	0.338069i	$-0.890226\pi$
$661$	−48.3923	−1.88224	−0.941121	+	0.338069i	$0.890226\pi$
$662$	−3.07180	−0.119389
$663$	0	0
$664$	11.6603	0.452506
$665$	6.73205	0.261058
$666$	4.73205	0.183363
$667$	10.3923	0.402392
$668$	21.8564	0.845650
$669$	12.3923	0.479114
$670$	−2.53590	−0.0979703
$671$	−2.21539	−0.0855242
$672$	1.00000	0.0385758
$673$	−34.0718	−1.31337	−0.656686	−	0.754164i	$-0.728042\pi$
$673$	−34.0718	−1.31337	−0.656686	+	0.754164i	$0.728042\pi$
$674$	13.7846	0.530963
$675$	−2.46410	−0.0948433
$676$	0	0
$677$	45.7654	1.75891	0.879453	−	0.475986i	$-0.157909\pi$
$677$	45.7654	1.75891	0.879453	+	0.475986i	$0.157909\pi$
$678$	0.196152	0.00753319
$679$	2.73205	0.104846
$680$	−10.1962	−0.391005
$681$	15.4641	0.592586
$682$	2.58846	0.0991172
$683$	−5.60770	−0.214572	−0.107286	−	0.994228i	$-0.534216\pi$
$683$	−5.60770	−0.214572	−0.107286	+	0.994228i	$0.534216\pi$
$684$	2.46410	0.0942173
$685$	23.3205	0.891031
$686$	−1.00000	−0.0381802
$687$	−24.3205	−0.927885
$688$	−2.73205	−0.104158
$689$	0	0
$690$	−9.46410	−0.360292
$691$	−0.784610	−0.0298480	−0.0149240	−	0.999889i	$-0.504751\pi$
$691$	−0.784610	−0.0298480	−0.0149240	+	0.999889i	$0.504751\pi$
$692$	17.2679	0.656429
$693$	0.267949	0.0101785
$694$	4.85641	0.184347
$695$	16.7321	0.634683
$696$	−3.00000	−0.113715
$697$	−26.1244	−0.989531
$698$	−31.7128	−1.20035
$699$	8.19615	0.310007
$700$	2.46410	0.0931343
$701$	−20.3205	−0.767495	−0.383747	−	0.923438i	$-0.625367\pi$
$701$	−20.3205	−0.767495	−0.383747	+	0.923438i	$0.625367\pi$
$702$	0	0
$703$	−11.6603	−0.439775
$704$	0.267949	0.0100987
$705$	6.73205	0.253544
$706$	−5.46410	−0.205644
$707$	−10.0000	−0.376089
$708$	12.9282	0.485872
$709$	−14.0526	−0.527755	−0.263877	−	0.964556i	$-0.585001\pi$
$709$	−14.0526	−0.527755	−0.263877	+	0.964556i	$0.585001\pi$
$710$	−22.3923	−0.840368
$711$	−16.8564	−0.632165
$712$	−0.464102	−0.0173929
$713$	33.4641	1.25324
$714$	3.73205	0.139668
$715$	0	0
$716$	1.60770	0.0600824
$717$	−24.1962	−0.903622
$718$	−26.1962	−0.977632
$719$	−35.0526	−1.30724	−0.653620	−	0.756823i	$-0.726750\pi$
$719$	−35.0526	−1.30724	−0.653620	+	0.756823i	$0.726750\pi$
$720$	2.73205	0.101818
$721$	−1.80385	−0.0671788
$722$	12.9282	0.481138
$723$	−28.7846	−1.07051
$724$	−9.19615	−0.341772
$725$	−7.39230	−0.274543
$726$	−10.9282	−0.405584
$727$	−5.46410	−0.202652	−0.101326	−	0.994853i	$-0.532309\pi$
$727$	−5.46410	−0.202652	−0.101326	+	0.994853i	$0.532309\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−36.7846	−1.36146
$731$	−10.1962	−0.377118
$732$	8.26795	0.305592
$733$	−20.8564	−0.770349	−0.385174	−	0.922844i	$-0.625859\pi$
$733$	−20.8564	−0.770349	−0.385174	+	0.922844i	$0.625859\pi$
$734$	32.2487	1.19032
$735$	−2.73205	−0.100773
$736$	3.46410	0.127688
$737$	0.248711	0.00916140
$738$	7.00000	0.257674
$739$	14.3923	0.529429	0.264715	−	0.964327i	$-0.414722\pi$
$739$	14.3923	0.529429	0.264715	+	0.964327i	$0.414722\pi$
$740$	−12.9282	−0.475250
$741$	0	0
$742$	3.53590	0.129807
$743$	25.5167	0.936115	0.468058	−	0.883698i	$-0.344954\pi$
$743$	25.5167	0.936115	0.468058	+	0.883698i	$0.344954\pi$
$744$	−9.66025	−0.354162
$745$	13.8564	0.507659
$746$	−10.5885	−0.387671
$747$	−11.6603	−0.426626
$748$	1.00000	0.0365636
$749$	−8.46410	−0.309272
$750$	−6.92820	−0.252982
$751$	20.8564	0.761061	0.380531	−	0.924768i	$-0.375741\pi$
$751$	20.8564	0.761061	0.380531	+	0.924768i	$0.375741\pi$
$752$	−2.46410	−0.0898565
$753$	−7.80385	−0.284388
$754$	0	0
$755$	7.66025	0.278785
$756$	−1.00000	−0.0363696
$757$	0.0525589	0.00191029	0.000955143	−	1.00000i	$-0.499696\pi$
$757$	0.0525589	0.00191029	0.000955143		1.00000i	$0.499696\pi$
$758$	−16.5885	−0.602520
$759$	0.928203	0.0336916
$760$	−6.73205	−0.244197
$761$	−35.3205	−1.28037	−0.640184	−	0.768222i	$-0.721142\pi$
$761$	−35.3205	−1.28037	−0.640184	+	0.768222i	$0.721142\pi$
$762$	−19.4641	−0.705110
$763$	−13.6603	−0.494534
$764$	6.19615	0.224169
$765$	10.1962	0.368643
$766$	35.3923	1.27878
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−34.4449	−1.24211	−0.621057	−	0.783766i	$-0.713296\pi$
$769$	−34.4449	−1.24211	−0.621057	+	0.783766i	$0.713296\pi$
$770$	−0.732051	−0.0263813
$771$	27.0526	0.974274
$772$	6.80385	0.244876
$773$	−21.6077	−0.777175	−0.388587	−	0.921412i	$-0.627037\pi$
$773$	−21.6077	−0.777175	−0.388587	+	0.921412i	$0.627037\pi$
$774$	2.73205	0.0982015
$775$	−23.8038	−0.855059
$776$	−2.73205	−0.0980749
$777$	4.73205	0.169761
$778$	−29.1769	−1.04604
$779$	−17.2487	−0.617999
$780$	0	0
$781$	2.19615	0.0785845
$782$	12.9282	0.462312
$783$	3.00000	0.107211
$784$	1.00000	0.0357143
$785$	−23.3205	−0.832345
$786$	5.07180	0.180905
$787$	−14.2154	−0.506724	−0.253362	−	0.967372i	$-0.581536\pi$
$787$	−14.2154	−0.506724	−0.253362	+	0.967372i	$0.581536\pi$
$788$	5.73205	0.204196
$789$	15.2679	0.543553
$790$	46.0526	1.63848
$791$	0.196152	0.00697438
$792$	−0.267949	−0.00952116
$793$	0	0
$794$	9.53590	0.338416
$795$	9.66025	0.342614
$796$	6.19615	0.219617
$797$	2.73205	0.0967742	0.0483871	−	0.998829i	$-0.484592\pi$
$797$	2.73205	0.0967742	0.0483871	+	0.998829i	$0.484592\pi$
$798$	2.46410	0.0872283
$799$	−9.19615	−0.325336
$800$	−2.46410	−0.0871191
$801$	0.464102	0.0163982
$802$	−10.0000	−0.353112
$803$	3.60770	0.127313
$804$	−0.928203	−0.0327352
$805$	−9.46410	−0.333566
$806$	0	0
$807$	−31.7128	−1.11634
$808$	10.0000	0.351799
$809$	10.1962	0.358478	0.179239	−	0.983806i	$-0.442637\pi$
$809$	10.1962	0.358478	0.179239	+	0.983806i	$0.442637\pi$
$810$	−2.73205	−0.0959945
$811$	8.53590	0.299736	0.149868	−	0.988706i	$-0.452115\pi$
$811$	8.53590	0.299736	0.149868	+	0.988706i	$0.452115\pi$
$812$	−3.00000	−0.105279
$813$	−23.5167	−0.824765
$814$	1.26795	0.0444416
$815$	−49.7128	−1.74136
$816$	−3.73205	−0.130648
$817$	−6.73205	−0.235525
$818$	16.7321	0.585022
$819$	0	0
$820$	−19.1244	−0.667851
$821$	−14.4115	−0.502966	−0.251483	−	0.967862i	$-0.580918\pi$
$821$	−14.4115	−0.502966	−0.251483	+	0.967862i	$0.580918\pi$
$822$	8.53590	0.297724
$823$	20.9282	0.729511	0.364756	−	0.931103i	$-0.381153\pi$
$823$	20.9282	0.729511	0.364756	+	0.931103i	$0.381153\pi$
$824$	1.80385	0.0628400
$825$	−0.660254	−0.0229871
$826$	12.9282	0.449830
$827$	−40.3923	−1.40458	−0.702289	−	0.711892i	$-0.747839\pi$
$827$	−40.3923	−1.40458	−0.702289	+	0.711892i	$0.747839\pi$
$828$	−3.46410	−0.120386
$829$	−0.660254	−0.0229316	−0.0114658	−	0.999934i	$-0.503650\pi$
$829$	−0.660254	−0.0229316	−0.0114658	+	0.999934i	$0.503650\pi$
$830$	31.8564	1.10575
$831$	5.60770	0.194529
$832$	0	0
$833$	3.73205	0.129308
$834$	6.12436	0.212069
$835$	59.7128	2.06645
$836$	0.660254	0.0228354
$837$	9.66025	0.333907
$838$	−4.19615	−0.144954
$839$	−14.3923	−0.496878	−0.248439	−	0.968648i	$-0.579917\pi$
$839$	−14.3923	−0.496878	−0.248439	+	0.968648i	$0.579917\pi$
$840$	2.73205	0.0942647
$841$	−20.0000	−0.689655
$842$	6.39230	0.220293
$843$	−7.80385	−0.268779
$844$	11.8564	0.408114
$845$	0	0
$846$	2.46410	0.0847176
$847$	−10.9282	−0.375498
$848$	−3.53590	−0.121423
$849$	9.07180	0.311343
$850$	−9.19615	−0.315425
$851$	16.3923	0.561921
$852$	−8.19615	−0.280796
$853$	33.3923	1.14333	0.571665	−	0.820487i	$-0.306298\pi$
$853$	33.3923	1.14333	0.571665	+	0.820487i	$0.306298\pi$
$854$	8.26795	0.282923
$855$	6.73205	0.230231
$856$	8.46410	0.289297
$857$	−21.6077	−0.738105	−0.369052	−	0.929409i	$-0.620318\pi$
$857$	−21.6077	−0.738105	−0.369052	+	0.929409i	$0.620318\pi$
$858$	0	0
$859$	−45.0526	−1.53717	−0.768587	−	0.639746i	$-0.779040\pi$
$859$	−45.0526	−1.53717	−0.768587	+	0.639746i	$0.779040\pi$
$860$	−7.46410	−0.254524
$861$	7.00000	0.238559
$862$	3.12436	0.106416
$863$	28.6410	0.974952	0.487476	−	0.873137i	$-0.337918\pi$
$863$	28.6410	0.974952	0.487476	+	0.873137i	$0.337918\pi$
$864$	1.00000	0.0340207
$865$	47.1769	1.60406
$866$	4.92820	0.167467
$867$	3.07180	0.104324
$868$	−9.66025	−0.327890
$869$	−4.51666	−0.153217
$870$	−8.19615	−0.277876
$871$	0	0
$872$	13.6603	0.462595
$873$	2.73205	0.0924659
$874$	8.53590	0.288731
$875$	−6.92820	−0.234216
$876$	−13.4641	−0.454910
$877$	36.3013	1.22581	0.612903	−	0.790158i	$-0.290001\pi$
$877$	36.3013	1.22581	0.612903	+	0.790158i	$0.290001\pi$
$878$	−4.39230	−0.148233
$879$	−28.7846	−0.970881
$880$	0.732051	0.0246774
$881$	1.07180	0.0361098	0.0180549	−	0.999837i	$-0.494253\pi$
$881$	1.07180	0.0361098	0.0180549	+	0.999837i	$0.494253\pi$
$882$	−1.00000	−0.0336718
$883$	34.4449	1.15916	0.579581	−	0.814915i	$-0.303216\pi$
$883$	34.4449	1.15916	0.579581	+	0.814915i	$0.303216\pi$
$884$	0	0
$885$	35.3205	1.18729
$886$	10.6077	0.356372
$887$	45.0526	1.51272	0.756358	−	0.654157i	$-0.226977\pi$
$887$	45.0526	1.51272	0.756358	+	0.654157i	$0.226977\pi$
$888$	−4.73205	−0.158797
$889$	−19.4641	−0.652805
$890$	−1.26795	−0.0425018
$891$	0.267949	0.00897664
$892$	−12.3923	−0.414925
$893$	−6.07180	−0.203185
$894$	5.07180	0.169626
$895$	4.39230	0.146819
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	7.26795	0.242535
$899$	28.9808	0.966563
$900$	2.46410	0.0821367
$901$	−13.1962	−0.439628
$902$	1.87564	0.0624521
$903$	2.73205	0.0909170
$904$	−0.196152	−0.00652393
$905$	−25.1244	−0.835162
$906$	2.80385	0.0931516
$907$	−27.4641	−0.911931	−0.455965	−	0.889998i	$-0.650706\pi$
$907$	−27.4641	−0.911931	−0.455965	+	0.889998i	$0.650706\pi$
$908$	−15.4641	−0.513194
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−0.0525589	−0.00174135	−0.000870677	−	1.00000i	$-0.500277\pi$
$911$	−0.0525589	−0.00174135	−0.000870677		1.00000i	$0.500277\pi$
$912$	−2.46410	−0.0815946
$913$	−3.12436	−0.103401
$914$	34.7846	1.15057
$915$	22.5885	0.746751
$916$	24.3205	0.803572
$917$	5.07180	0.167485
$918$	3.73205	0.123176
$919$	41.6410	1.37361	0.686805	−	0.726842i	$-0.259013\pi$
$919$	41.6410	1.37361	0.686805	+	0.726842i	$0.259013\pi$
$920$	9.46410	0.312022
$921$	−7.78461	−0.256512
$922$	27.7128	0.912673
$923$	0	0
$924$	−0.267949	−0.00881488
$925$	−11.6603	−0.383387
$926$	−9.19615	−0.302204
$927$	−1.80385	−0.0592461
$928$	3.00000	0.0984798
$929$	−43.5359	−1.42837	−0.714183	−	0.699959i	$-0.753202\pi$
$929$	−43.5359	−1.42837	−0.714183	+	0.699959i	$0.753202\pi$
$930$	−26.3923	−0.865438
$931$	2.46410	0.0807577
$932$	−8.19615	−0.268474
$933$	8.80385	0.288225
$934$	−17.8564	−0.584279
$935$	2.73205	0.0893476
$936$	0	0
$937$	45.5692	1.48868	0.744341	−	0.667800i	$-0.232764\pi$
$937$	45.5692	1.48868	0.744341	+	0.667800i	$0.232764\pi$
$938$	−0.928203	−0.0303069
$939$	−12.1962	−0.398006
$940$	−6.73205	−0.219575
$941$	49.1769	1.60312	0.801561	−	0.597913i	$-0.204003\pi$
$941$	49.1769	1.60312	0.801561	+	0.597913i	$0.204003\pi$
$942$	−8.53590	−0.278115
$943$	24.2487	0.789647
$944$	−12.9282	−0.420777
$945$	−2.73205	−0.0888736
$946$	0.732051	0.0238010
$947$	12.9474	0.420735	0.210368	−	0.977622i	$-0.432534\pi$
$947$	12.9474	0.420735	0.210368	+	0.977622i	$0.432534\pi$
$948$	16.8564	0.547471
$949$	0	0
$950$	−6.07180	−0.196995
$951$	−18.0000	−0.583690
$952$	−3.73205	−0.120956
$953$	−21.3731	−0.692342	−0.346171	−	0.938172i	$-0.612518\pi$
$953$	−21.3731	−0.692342	−0.346171	+	0.938172i	$0.612518\pi$
$954$	3.53590	0.114479
$955$	16.9282	0.547784
$956$	24.1962	0.782559
$957$	0.803848	0.0259847
$958$	10.6077	0.342719
$959$	8.53590	0.275639
$960$	−2.73205	−0.0881766
$961$	62.3205	2.01034
$962$	0	0
$963$	−8.46410	−0.272752
$964$	28.7846	0.927090
$965$	18.5885	0.598384
$966$	−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	$0.696495\pi$
$968$	10.9282	0.351246
$969$	−9.19615	−0.295423
$970$	−7.46410	−0.239658
$971$	−4.24871	−0.136348	−0.0681738	−	0.997673i	$-0.521717\pi$
$971$	−4.24871	−0.136348	−0.0681738	+	0.997673i	$0.521717\pi$
$972$	−1.00000	−0.0320750
$973$	6.12436	0.196338
$974$	10.8038	0.346178
$975$	0	0
$976$	−8.26795	−0.264651
$977$	−30.2487	−0.967742	−0.483871	−	0.875139i	$-0.660770\pi$
$977$	−30.2487	−0.967742	−0.483871	+	0.875139i	$0.660770\pi$
$978$	−18.1962	−0.581849
$979$	0.124356	0.00397442
$980$	2.73205	0.0872722
$981$	−13.6603	−0.436138
$982$	12.2487	0.390872
$983$	−55.8564	−1.78154	−0.890771	−	0.454452i	$-0.849835\pi$
$983$	−55.8564	−1.78154	−0.890771	+	0.454452i	$0.849835\pi$
$984$	−7.00000	−0.223152
$985$	15.6603	0.498977
$986$	11.1962	0.356558
$987$	2.46410	0.0784332
$988$	0	0
$989$	9.46410	0.300941
$990$	−0.732051	−0.0232661
$991$	−25.3923	−0.806613	−0.403307	−	0.915065i	$-0.632139\pi$
$991$	−25.3923	−0.806613	−0.403307	+	0.915065i	$0.632139\pi$
$992$	9.66025	0.306713
$993$	−3.07180	−0.0974805
$994$	−8.19615	−0.259966
$995$	16.9282	0.536660
$996$	11.6603	0.369469
$997$	−58.2295	−1.84415	−0.922073	−	0.387016i	$-0.873506\pi$
$997$	−58.2295	−1.84415	−0.922073	+	0.387016i	$0.873506\pi$
$998$	38.9808	1.23391
$999$	4.73205	0.149715

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7098.2.a.bj.1.2		2
13.6	odd	12		546.2.s.d.127.2	yes	4
13.11	odd	12		546.2.s.d.43.2	✓	4
13.12	even	2		7098.2.a.bs.1.1		2
39.11	even	12		1638.2.bj.d.1135.1		4
39.32	even	12		1638.2.bj.d.127.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.s.d.43.2	✓	4	13.11	odd	12
546.2.s.d.127.2	yes	4	13.6	odd	12
1638.2.bj.d.127.1		4	39.32	even	12
1638.2.bj.d.1135.1		4	39.11	even	12
7098.2.a.bj.1.2		2	1.1	even	1	trivial
7098.2.a.bs.1.1		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.73205 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.73205 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.73205 q^{10} +0.267949 q^{11} -1.00000 q^{12} -1.00000 q^{14} -2.73205 q^{15} +1.00000 q^{16} +3.73205 q^{17} -1.00000 q^{18} +2.46410 q^{19} +2.73205 q^{20} -1.00000 q^{21} -0.267949 q^{22} -3.46410 q^{23} +1.00000 q^{24} +2.46410 q^{25} -1.00000 q^{27} +1.00000 q^{28} -3.00000 q^{29} +2.73205 q^{30} -9.66025 q^{31} -1.00000 q^{32} -0.267949 q^{33} -3.73205 q^{34} +2.73205 q^{35} +1.00000 q^{36} -4.73205 q^{37} -2.46410 q^{38} -2.73205 q^{40} -7.00000 q^{41} +1.00000 q^{42} -2.73205 q^{43} +0.267949 q^{44} +2.73205 q^{45} +3.46410 q^{46} -2.46410 q^{47} -1.00000 q^{48} +1.00000 q^{49} -2.46410 q^{50} -3.73205 q^{51} -3.53590 q^{53} +1.00000 q^{54} +0.732051 q^{55} -1.00000 q^{56} -2.46410 q^{57} +3.00000 q^{58} -12.9282 q^{59} -2.73205 q^{60} -8.26795 q^{61} +9.66025 q^{62} +1.00000 q^{63} +1.00000 q^{64} +0.267949 q^{66} +0.928203 q^{67} +3.73205 q^{68} +3.46410 q^{69} -2.73205 q^{70} +8.19615 q^{71} -1.00000 q^{72} +13.4641 q^{73} +4.73205 q^{74} -2.46410 q^{75} +2.46410 q^{76} +0.267949 q^{77} -16.8564 q^{79} +2.73205 q^{80} +1.00000 q^{81} +7.00000 q^{82} -11.6603 q^{83} -1.00000 q^{84} +10.1962 q^{85} +2.73205 q^{86} +3.00000 q^{87} -0.267949 q^{88} +0.464102 q^{89} -2.73205 q^{90} -3.46410 q^{92} +9.66025 q^{93} +2.46410 q^{94} +6.73205 q^{95} +1.00000 q^{96} +2.73205 q^{97} -1.00000 q^{98} +0.267949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 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

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