LMFDB - Embedded newform 5070.2.a.bs.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, 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("5070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.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:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.19806 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.19806 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.93900 q^{11} -1.00000 q^{12} -1.19806 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.74094 q^{17} +1.00000 q^{18} +0.911854 q^{19} -1.00000 q^{20} +1.19806 q^{21} +3.93900 q^{22} +1.24698 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -1.19806 q^{28} -3.47219 q^{29} +1.00000 q^{30} -2.15883 q^{31} +1.00000 q^{32} -3.93900 q^{33} +1.74094 q^{34} +1.19806 q^{35} +1.00000 q^{36} +4.80194 q^{37} +0.911854 q^{38} -1.00000 q^{40} +0.198062 q^{41} +1.19806 q^{42} +3.43296 q^{43} +3.93900 q^{44} -1.00000 q^{45} +1.24698 q^{46} +0.902165 q^{47} -1.00000 q^{48} -5.56465 q^{49} +1.00000 q^{50} -1.74094 q^{51} -6.91185 q^{53} -1.00000 q^{54} -3.93900 q^{55} -1.19806 q^{56} -0.911854 q^{57} -3.47219 q^{58} -10.2446 q^{59} +1.00000 q^{60} +9.92692 q^{61} -2.15883 q^{62} -1.19806 q^{63} +1.00000 q^{64} -3.93900 q^{66} -9.16852 q^{67} +1.74094 q^{68} -1.24698 q^{69} +1.19806 q^{70} +11.0707 q^{71} +1.00000 q^{72} +4.13706 q^{73} +4.80194 q^{74} -1.00000 q^{75} +0.911854 q^{76} -4.71917 q^{77} +2.73556 q^{79} -1.00000 q^{80} +1.00000 q^{81} +0.198062 q^{82} +15.5864 q^{83} +1.19806 q^{84} -1.74094 q^{85} +3.43296 q^{86} +3.47219 q^{87} +3.93900 q^{88} +8.57673 q^{89} -1.00000 q^{90} +1.24698 q^{92} +2.15883 q^{93} +0.902165 q^{94} -0.911854 q^{95} -1.00000 q^{96} +16.9976 q^{97} -5.56465 q^{98} +3.93900 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 8 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 - 8 * q^7 + 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 8 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} + 2 q^{11} - 3 q^{12} - 8 q^{14} + 3 q^{15} + 3 q^{16} - 9 q^{17} + 3 q^{18} - q^{19} - 3 q^{20} + 8 q^{21} + 2 q^{22} - q^{23} - 3 q^{24} + 3 q^{25} - 3 q^{27} - 8 q^{28} - 4 q^{29} + 3 q^{30} + 2 q^{31} + 3 q^{32} - 2 q^{33} - 9 q^{34} + 8 q^{35} + 3 q^{36} + 10 q^{37} - q^{38} - 3 q^{40} + 5 q^{41} + 8 q^{42} - 9 q^{43} + 2 q^{44} - 3 q^{45} - q^{46} + 21 q^{47} - 3 q^{48} + 5 q^{49} + 3 q^{50} + 9 q^{51} - 17 q^{53} - 3 q^{54} - 2 q^{55} - 8 q^{56} + q^{57} - 4 q^{58} + 15 q^{59} + 3 q^{60} + q^{61} + 2 q^{62} - 8 q^{63} + 3 q^{64} - 2 q^{66} + 3 q^{67} - 9 q^{68} + q^{69} + 8 q^{70} + 21 q^{71} + 3 q^{72} + 7 q^{73} + 10 q^{74} - 3 q^{75} - q^{76} - 3 q^{77} - 3 q^{79} - 3 q^{80} + 3 q^{81} + 5 q^{82} + 22 q^{83} + 8 q^{84} + 9 q^{85} - 9 q^{86} + 4 q^{87} + 2 q^{88} + 23 q^{89} - 3 q^{90} - q^{92} - 2 q^{93} + 21 q^{94} + q^{95} - 3 q^{96} + 10 q^{97} + 5 q^{98} + 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 - 8 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 + 2 * q^11 - 3 * q^12 - 8 * q^14 + 3 * q^15 + 3 * q^16 - 9 * q^17 + 3 * q^18 - q^19 - 3 * q^20 + 8 * q^21 + 2 * q^22 - q^23 - 3 * q^24 + 3 * q^25 - 3 * q^27 - 8 * q^28 - 4 * q^29 + 3 * q^30 + 2 * q^31 + 3 * q^32 - 2 * q^33 - 9 * q^34 + 8 * q^35 + 3 * q^36 + 10 * q^37 - q^38 - 3 * q^40 + 5 * q^41 + 8 * q^42 - 9 * q^43 + 2 * q^44 - 3 * q^45 - q^46 + 21 * q^47 - 3 * q^48 + 5 * q^49 + 3 * q^50 + 9 * q^51 - 17 * q^53 - 3 * q^54 - 2 * q^55 - 8 * q^56 + q^57 - 4 * q^58 + 15 * q^59 + 3 * q^60 + q^61 + 2 * q^62 - 8 * q^63 + 3 * q^64 - 2 * q^66 + 3 * q^67 - 9 * q^68 + q^69 + 8 * q^70 + 21 * q^71 + 3 * q^72 + 7 * q^73 + 10 * q^74 - 3 * q^75 - q^76 - 3 * q^77 - 3 * q^79 - 3 * q^80 + 3 * q^81 + 5 * q^82 + 22 * q^83 + 8 * q^84 + 9 * q^85 - 9 * q^86 + 4 * q^87 + 2 * q^88 + 23 * q^89 - 3 * q^90 - q^92 - 2 * q^93 + 21 * q^94 + q^95 - 3 * q^96 + 10 * q^97 + 5 * q^98 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	−1.19806	−0.452825	−0.226412	−	0.974032i	$-0.572700\pi$
$7$	−1.19806	−0.452825	−0.226412	+	0.974032i	$0.572700\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	3.93900	1.18765	0.593827	−	0.804593i	$-0.297616\pi$
$11$	3.93900	1.18765	0.593827	+	0.804593i	$0.297616\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−1.19806	−0.320196
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	1.74094	0.422240	0.211120	−	0.977460i	$-0.432289\pi$
$17$	1.74094	0.422240	0.211120	+	0.977460i	$0.432289\pi$
$18$	1.00000	0.235702
$19$	0.911854	0.209194	0.104597	−	0.994515i	$-0.466645\pi$
$19$	0.911854	0.209194	0.104597	+	0.994515i	$0.466645\pi$
$20$	−1.00000	−0.223607
$21$	1.19806	0.261439
$22$	3.93900	0.839798
$23$	1.24698	0.260013	0.130007	−	0.991513i	$-0.458500\pi$
$23$	1.24698	0.260013	0.130007	+	0.991513i	$0.458500\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−1.19806	−0.226412
$29$	−3.47219	−0.644769	−0.322385	−	0.946609i	$-0.604484\pi$
$29$	−3.47219	−0.644769	−0.322385	+	0.946609i	$0.604484\pi$
$30$	1.00000	0.182574
$31$	−2.15883	−0.387738	−0.193869	−	0.981027i	$-0.562104\pi$
$31$	−2.15883	−0.387738	−0.193869	+	0.981027i	$0.562104\pi$
$32$	1.00000	0.176777
$33$	−3.93900	−0.685692
$34$	1.74094	0.298569
$35$	1.19806	0.202509
$36$	1.00000	0.166667
$37$	4.80194	0.789434	0.394717	−	0.918803i	$-0.370843\pi$
$37$	4.80194	0.789434	0.394717	+	0.918803i	$0.370843\pi$
$38$	0.911854	0.147922
$39$	0	0
$40$	−1.00000	−0.158114
$41$	0.198062	0.0309321	0.0154661	−	0.999880i	$-0.495077\pi$
$41$	0.198062	0.0309321	0.0154661	+	0.999880i	$0.495077\pi$
$42$	1.19806	0.184865
$43$	3.43296	0.523522	0.261761	−	0.965133i	$-0.415697\pi$
$43$	3.43296	0.523522	0.261761	+	0.965133i	$0.415697\pi$
$44$	3.93900	0.593827
$45$	−1.00000	−0.149071
$46$	1.24698	0.183857
$47$	0.902165	0.131594	0.0657972	−	0.997833i	$-0.479041\pi$
$47$	0.902165	0.131594	0.0657972	+	0.997833i	$0.479041\pi$
$48$	−1.00000	−0.144338
$49$	−5.56465	−0.794950
$50$	1.00000	0.141421
$51$	−1.74094	−0.243780
$52$	0	0
$53$	−6.91185	−0.949416	−0.474708	−	0.880143i	$-0.657446\pi$
$53$	−6.91185	−0.949416	−0.474708	+	0.880143i	$0.657446\pi$
$54$	−1.00000	−0.136083
$55$	−3.93900	−0.531135
$56$	−1.19806	−0.160098
$57$	−0.911854	−0.120778
$58$	−3.47219	−0.455921
$59$	−10.2446	−1.33373	−0.666866	−	0.745178i	$-0.732365\pi$
$59$	−10.2446	−1.33373	−0.666866	+	0.745178i	$0.732365\pi$
$60$	1.00000	0.129099
$61$	9.92692	1.27101	0.635506	−	0.772096i	$-0.280792\pi$
$61$	9.92692	1.27101	0.635506	+	0.772096i	$0.280792\pi$
$62$	−2.15883	−0.274172
$63$	−1.19806	−0.150942
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.93900	−0.484858
$67$	−9.16852	−1.12011	−0.560057	−	0.828454i	$-0.689221\pi$
$67$	−9.16852	−1.12011	−0.560057	+	0.828454i	$0.689221\pi$
$68$	1.74094	0.211120
$69$	−1.24698	−0.150119
$70$	1.19806	0.143196
$71$	11.0707	1.31385	0.656924	−	0.753956i	$-0.271857\pi$
$71$	11.0707	1.31385	0.656924	+	0.753956i	$0.271857\pi$
$72$	1.00000	0.117851
$73$	4.13706	0.484207	0.242103	−	0.970250i	$-0.422163\pi$
$73$	4.13706	0.484207	0.242103	+	0.970250i	$0.422163\pi$
$74$	4.80194	0.558214
$75$	−1.00000	−0.115470
$76$	0.911854	0.104597
$77$	−4.71917	−0.537799
$78$	0	0
$79$	2.73556	0.307775	0.153887	−	0.988088i	$-0.450821\pi$
$79$	2.73556	0.307775	0.153887	+	0.988088i	$0.450821\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	0.198062	0.0218723
$83$	15.5864	1.71083	0.855416	−	0.517942i	$-0.173302\pi$
$83$	15.5864	1.71083	0.855416	+	0.517942i	$0.173302\pi$
$84$	1.19806	0.130719
$85$	−1.74094	−0.188831
$86$	3.43296	0.370186
$87$	3.47219	0.372258
$88$	3.93900	0.419899
$89$	8.57673	0.909131	0.454566	−	0.890713i	$-0.349794\pi$
$89$	8.57673	0.909131	0.454566	+	0.890713i	$0.349794\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	1.24698	0.130007
$93$	2.15883	0.223861
$94$	0.902165	0.0930512
$95$	−0.911854	−0.0935542
$96$	−1.00000	−0.102062
$97$	16.9976	1.72585	0.862923	−	0.505336i	$-0.168631\pi$
$97$	16.9976	1.72585	0.862923	+	0.505336i	$0.168631\pi$
$98$	−5.56465	−0.562114
$99$	3.93900	0.395885
$100$	1.00000	0.100000
$101$	6.67025	0.663715	0.331857	−	0.943330i	$-0.392325\pi$
$101$	6.67025	0.663715	0.331857	+	0.943330i	$0.392325\pi$
$102$	−1.74094	−0.172379
$103$	10.0315	0.988429	0.494215	−	0.869340i	$-0.335456\pi$
$103$	10.0315	0.988429	0.494215	+	0.869340i	$0.335456\pi$
$104$	0	0
$105$	−1.19806	−0.116919
$106$	−6.91185	−0.671339
$107$	−10.1032	−0.976714	−0.488357	−	0.872644i	$-0.662404\pi$
$107$	−10.1032	−0.976714	−0.488357	+	0.872644i	$0.662404\pi$
$108$	−1.00000	−0.0962250
$109$	10.5211	1.00774	0.503870	−	0.863780i	$-0.331909\pi$
$109$	10.5211	1.00774	0.503870	+	0.863780i	$0.331909\pi$
$110$	−3.93900	−0.375569
$111$	−4.80194	−0.455780
$112$	−1.19806	−0.113206
$113$	−11.2131	−1.05484	−0.527421	−	0.849604i	$-0.676841\pi$
$113$	−11.2131	−1.05484	−0.527421	+	0.849604i	$0.676841\pi$
$114$	−0.911854	−0.0854030
$115$	−1.24698	−0.116281
$116$	−3.47219	−0.322385
$117$	0	0
$118$	−10.2446	−0.943091
$119$	−2.08575	−0.191201
$120$	1.00000	0.0912871
$121$	4.51573	0.410521
$122$	9.92692	0.898741
$123$	−0.198062	−0.0178587
$124$	−2.15883	−0.193869
$125$	−1.00000	−0.0894427
$126$	−1.19806	−0.106732
$127$	−9.32736	−0.827669	−0.413834	−	0.910352i	$-0.635811\pi$
$127$	−9.32736	−0.827669	−0.413834	+	0.910352i	$0.635811\pi$
$128$	1.00000	0.0883883
$129$	−3.43296	−0.302255
$130$	0	0
$131$	4.62133	0.403768	0.201884	−	0.979409i	$-0.435294\pi$
$131$	4.62133	0.403768	0.201884	+	0.979409i	$0.435294\pi$
$132$	−3.93900	−0.342846
$133$	−1.09246	−0.0947281
$134$	−9.16852	−0.792040
$135$	1.00000	0.0860663
$136$	1.74094	0.149284
$137$	0.0760644	0.00649862	0.00324931	−	0.999995i	$-0.498966\pi$
$137$	0.0760644	0.00649862	0.00324931	+	0.999995i	$0.498966\pi$
$138$	−1.24698	−0.106150
$139$	18.7017	1.58626	0.793129	−	0.609053i	$-0.208451\pi$
$139$	18.7017	1.58626	0.793129	+	0.609053i	$0.208451\pi$
$140$	1.19806	0.101255
$141$	−0.902165	−0.0759760
$142$	11.0707	0.929031
$143$	0	0
$144$	1.00000	0.0833333
$145$	3.47219	0.288350
$146$	4.13706	0.342386
$147$	5.56465	0.458964
$148$	4.80194	0.394717
$149$	13.6799	1.12070	0.560352	−	0.828254i	$-0.310666\pi$
$149$	13.6799	1.12070	0.560352	+	0.828254i	$0.310666\pi$
$150$	−1.00000	−0.0816497
$151$	−1.15883	−0.0943045	−0.0471523	−	0.998888i	$-0.515015\pi$
$151$	−1.15883	−0.0943045	−0.0471523	+	0.998888i	$0.515015\pi$
$152$	0.911854	0.0739611
$153$	1.74094	0.140747
$154$	−4.71917	−0.380281
$155$	2.15883	0.173402
$156$	0	0
$157$	−12.2664	−0.978962	−0.489481	−	0.872014i	$-0.662814\pi$
$157$	−12.2664	−0.978962	−0.489481	+	0.872014i	$0.662814\pi$
$158$	2.73556	0.217630
$159$	6.91185	0.548146
$160$	−1.00000	−0.0790569
$161$	−1.49396	−0.117740
$162$	1.00000	0.0785674
$163$	−2.48858	−0.194921	−0.0974604	−	0.995239i	$-0.531072\pi$
$163$	−2.48858	−0.194921	−0.0974604	+	0.995239i	$0.531072\pi$
$164$	0.198062	0.0154661
$165$	3.93900	0.306651
$166$	15.5864	1.20974
$167$	7.27950	0.563305	0.281652	−	0.959516i	$-0.409117\pi$
$167$	7.27950	0.563305	0.281652	+	0.959516i	$0.409117\pi$
$168$	1.19806	0.0924325
$169$	0	0
$170$	−1.74094	−0.133524
$171$	0.911854	0.0697312
$172$	3.43296	0.261761
$173$	15.8485	1.20494	0.602468	−	0.798143i	$-0.294184\pi$
$173$	15.8485	1.20494	0.602468	+	0.798143i	$0.294184\pi$
$174$	3.47219	0.263226
$175$	−1.19806	−0.0905650
$176$	3.93900	0.296913
$177$	10.2446	0.770030
$178$	8.57673	0.642853
$179$	18.4819	1.38140	0.690700	−	0.723141i	$-0.257302\pi$
$179$	18.4819	1.38140	0.690700	+	0.723141i	$0.257302\pi$
$180$	−1.00000	−0.0745356
$181$	10.7041	0.795630	0.397815	−	0.917466i	$-0.369769\pi$
$181$	10.7041	0.795630	0.397815	+	0.917466i	$0.369769\pi$
$182$	0	0
$183$	−9.92692	−0.733819
$184$	1.24698	0.0919286
$185$	−4.80194	−0.353045
$186$	2.15883	0.158293
$187$	6.85756	0.501474
$188$	0.902165	0.0657972
$189$	1.19806	0.0871462
$190$	−0.911854	−0.0661528
$191$	15.3623	1.11158	0.555788	−	0.831324i	$-0.312417\pi$
$191$	15.3623	1.11158	0.555788	+	0.831324i	$0.312417\pi$
$192$	−1.00000	−0.0721688
$193$	−18.3207	−1.31875	−0.659375	−	0.751815i	$-0.729179\pi$
$193$	−18.3207	−1.31875	−0.659375	+	0.751815i	$0.729179\pi$
$194$	16.9976	1.22036
$195$	0	0
$196$	−5.56465	−0.397475
$197$	−2.31336	−0.164820	−0.0824099	−	0.996599i	$-0.526262\pi$
$197$	−2.31336	−0.164820	−0.0824099	+	0.996599i	$0.526262\pi$
$198$	3.93900	0.279933
$199$	6.36765	0.451391	0.225695	−	0.974198i	$-0.427535\pi$
$199$	6.36765	0.451391	0.225695	+	0.974198i	$0.427535\pi$
$200$	1.00000	0.0707107
$201$	9.16852	0.646698
$202$	6.67025	0.469317
$203$	4.15990	0.291968
$204$	−1.74094	−0.121890
$205$	−0.198062	−0.0138333
$206$	10.0315	0.698925
$207$	1.24698	0.0866711
$208$	0	0
$209$	3.59179	0.248450
$210$	−1.19806	−0.0826742
$211$	−11.6775	−0.803915	−0.401958	−	0.915658i	$-0.631670\pi$
$211$	−11.6775	−0.803915	−0.401958	+	0.915658i	$0.631670\pi$
$212$	−6.91185	−0.474708
$213$	−11.0707	−0.758551
$214$	−10.1032	−0.690641
$215$	−3.43296	−0.234126
$216$	−1.00000	−0.0680414
$217$	2.58642	0.175577
$218$	10.5211	0.712579
$219$	−4.13706	−0.279557
$220$	−3.93900	−0.265567
$221$	0	0
$222$	−4.80194	−0.322285
$223$	11.4004	0.763430	0.381715	−	0.924280i	$-0.375334\pi$
$223$	11.4004	0.763430	0.381715	+	0.924280i	$0.375334\pi$
$224$	−1.19806	−0.0800489
$225$	1.00000	0.0666667
$226$	−11.2131	−0.745886
$227$	8.86725	0.588540	0.294270	−	0.955722i	$-0.404924\pi$
$227$	8.86725	0.588540	0.294270	+	0.955722i	$0.404924\pi$
$228$	−0.911854	−0.0603890
$229$	12.6136	0.833528	0.416764	−	0.909015i	$-0.363164\pi$
$229$	12.6136	0.833528	0.416764	+	0.909015i	$0.363164\pi$
$230$	−1.24698	−0.0822234
$231$	4.71917	0.310498
$232$	−3.47219	−0.227960
$233$	−23.4155	−1.53400	−0.767000	−	0.641647i	$-0.778252\pi$
$233$	−23.4155	−1.53400	−0.767000	+	0.641647i	$0.778252\pi$
$234$	0	0
$235$	−0.902165	−0.0588508
$236$	−10.2446	−0.666866
$237$	−2.73556	−0.177694
$238$	−2.08575	−0.135199
$239$	8.87263	0.573922	0.286961	−	0.957942i	$-0.407355\pi$
$239$	8.87263	0.573922	0.286961	+	0.957942i	$0.407355\pi$
$240$	1.00000	0.0645497
$241$	9.72886	0.626691	0.313345	−	0.949639i	$-0.398550\pi$
$241$	9.72886	0.626691	0.313345	+	0.949639i	$0.398550\pi$
$242$	4.51573	0.290282
$243$	−1.00000	−0.0641500
$244$	9.92692	0.635506
$245$	5.56465	0.355512
$246$	−0.198062	−0.0126280
$247$	0	0
$248$	−2.15883	−0.137086
$249$	−15.5864	−0.987749
$250$	−1.00000	−0.0632456
$251$	29.8823	1.88615	0.943077	−	0.332573i	$-0.107917\pi$
$251$	29.8823	1.88615	0.943077	+	0.332573i	$0.107917\pi$
$252$	−1.19806	−0.0754708
$253$	4.91185	0.308806
$254$	−9.32736	−0.585250
$255$	1.74094	0.109022
$256$	1.00000	0.0625000
$257$	−8.52111	−0.531532	−0.265766	−	0.964038i	$-0.585625\pi$
$257$	−8.52111	−0.531532	−0.265766	+	0.964038i	$0.585625\pi$
$258$	−3.43296	−0.213727
$259$	−5.75302	−0.357475
$260$	0	0
$261$	−3.47219	−0.214923
$262$	4.62133	0.285507
$263$	−22.6450	−1.39635	−0.698176	−	0.715926i	$-0.746005\pi$
$263$	−22.6450	−1.39635	−0.698176	+	0.715926i	$0.746005\pi$
$264$	−3.93900	−0.242429
$265$	6.91185	0.424592
$266$	−1.09246	−0.0669829
$267$	−8.57673	−0.524887
$268$	−9.16852	−0.560057
$269$	−9.02608	−0.550330	−0.275165	−	0.961397i	$-0.588732\pi$
$269$	−9.02608	−0.550330	−0.275165	+	0.961397i	$0.588732\pi$
$270$	1.00000	0.0608581
$271$	−8.59611	−0.522176	−0.261088	−	0.965315i	$-0.584081\pi$
$271$	−8.59611	−0.522176	−0.261088	+	0.965315i	$0.584081\pi$
$272$	1.74094	0.105560
$273$	0	0
$274$	0.0760644	0.00459522
$275$	3.93900	0.237531
$276$	−1.24698	−0.0750594
$277$	6.31527	0.379448	0.189724	−	0.981837i	$-0.439241\pi$
$277$	6.31527	0.379448	0.189724	+	0.981837i	$0.439241\pi$
$278$	18.7017	1.12165
$279$	−2.15883	−0.129246
$280$	1.19806	0.0715979
$281$	21.7409	1.29696	0.648478	−	0.761234i	$-0.275406\pi$
$281$	21.7409	1.29696	0.648478	+	0.761234i	$0.275406\pi$
$282$	−0.902165	−0.0537232
$283$	12.9487	0.769720	0.384860	−	0.922975i	$-0.374250\pi$
$283$	12.9487	0.769720	0.384860	+	0.922975i	$0.374250\pi$
$284$	11.0707	0.656924
$285$	0.911854	0.0540136
$286$	0	0
$287$	−0.237291	−0.0140068
$288$	1.00000	0.0589256
$289$	−13.9691	−0.821714
$290$	3.47219	0.203894
$291$	−16.9976	−0.996417
$292$	4.13706	0.242103
$293$	27.3056	1.59521	0.797605	−	0.603181i	$-0.206100\pi$
$293$	27.3056	1.59521	0.797605	+	0.603181i	$0.206100\pi$
$294$	5.56465	0.324537
$295$	10.2446	0.596463
$296$	4.80194	0.279107
$297$	−3.93900	−0.228564
$298$	13.6799	0.792458
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−4.11290	−0.237064
$302$	−1.15883	−0.0666834
$303$	−6.67025	−0.383196
$304$	0.911854	0.0522984
$305$	−9.92692	−0.568414
$306$	1.74094	0.0995228
$307$	−24.9124	−1.42183	−0.710914	−	0.703279i	$-0.751719\pi$
$307$	−24.9124	−1.42183	−0.710914	+	0.703279i	$0.751719\pi$
$308$	−4.71917	−0.268900
$309$	−10.0315	−0.570670
$310$	2.15883	0.122614
$311$	21.3787	1.21227	0.606136	−	0.795361i	$-0.292719\pi$
$311$	21.3787	1.21227	0.606136	+	0.795361i	$0.292719\pi$
$312$	0	0
$313$	−23.8713	−1.34929	−0.674643	−	0.738144i	$-0.735702\pi$
$313$	−23.8713	−1.34929	−0.674643	+	0.738144i	$0.735702\pi$
$314$	−12.2664	−0.692231
$315$	1.19806	0.0675032
$316$	2.73556	0.153887
$317$	−12.6407	−0.709973	−0.354987	−	0.934871i	$-0.615515\pi$
$317$	−12.6407	−0.709973	−0.354987	+	0.934871i	$0.615515\pi$
$318$	6.91185	0.387598
$319$	−13.6770	−0.765763
$320$	−1.00000	−0.0559017
$321$	10.1032	0.563906
$322$	−1.49396	−0.0832551
$323$	1.58748	0.0883299
$324$	1.00000	0.0555556
$325$	0	0
$326$	−2.48858	−0.137830
$327$	−10.5211	−0.581819
$328$	0.198062	0.0109362
$329$	−1.08085	−0.0595892
$330$	3.93900	0.216835
$331$	20.5109	1.12738	0.563692	−	0.825985i	$-0.309381\pi$
$331$	20.5109	1.12738	0.563692	+	0.825985i	$0.309381\pi$
$332$	15.5864	0.855416
$333$	4.80194	0.263145
$334$	7.27950	0.398317
$335$	9.16852	0.500930
$336$	1.19806	0.0653597
$337$	6.82371	0.371711	0.185856	−	0.982577i	$-0.440494\pi$
$337$	6.82371	0.371711	0.185856	+	0.982577i	$0.440494\pi$
$338$	0	0
$339$	11.2131	0.609014
$340$	−1.74094	−0.0944157
$341$	−8.50365	−0.460498
$342$	0.911854	0.0493074
$343$	15.0532	0.812798
$344$	3.43296	0.185093
$345$	1.24698	0.0671351
$346$	15.8485	0.852019
$347$	−29.4741	−1.58225	−0.791127	−	0.611653i	$-0.790505\pi$
$347$	−29.4741	−1.58225	−0.791127	+	0.611653i	$0.790505\pi$
$348$	3.47219	0.186129
$349$	−34.5394	−1.84885	−0.924426	−	0.381361i	$-0.875456\pi$
$349$	−34.5394	−1.84885	−0.924426	+	0.381361i	$0.875456\pi$
$350$	−1.19806	−0.0640391
$351$	0	0
$352$	3.93900	0.209949
$353$	31.2295	1.66218	0.831090	−	0.556138i	$-0.187717\pi$
$353$	31.2295	1.66218	0.831090	+	0.556138i	$0.187717\pi$
$354$	10.2446	0.544494
$355$	−11.0707	−0.587571
$356$	8.57673	0.454566
$357$	2.08575	0.110390
$358$	18.4819	0.976798
$359$	18.2524	0.963323	0.481661	−	0.876357i	$-0.340034\pi$
$359$	18.2524	0.963323	0.481661	+	0.876357i	$0.340034\pi$
$360$	−1.00000	−0.0527046
$361$	−18.1685	−0.956238
$362$	10.7041	0.562595
$363$	−4.51573	−0.237014
$364$	0	0
$365$	−4.13706	−0.216544
$366$	−9.92692	−0.518888
$367$	−19.3773	−1.01149	−0.505744	−	0.862683i	$-0.668782\pi$
$367$	−19.3773	−1.01149	−0.505744	+	0.862683i	$0.668782\pi$
$368$	1.24698	0.0650033
$369$	0.198062	0.0103107
$370$	−4.80194	−0.249641
$371$	8.28083	0.429919
$372$	2.15883	0.111930
$373$	−13.6280	−0.705633	−0.352817	−	0.935693i	$-0.614776\pi$
$373$	−13.6280	−0.705633	−0.352817	+	0.935693i	$0.614776\pi$
$374$	6.85756	0.354596
$375$	1.00000	0.0516398
$376$	0.902165	0.0465256
$377$	0	0
$378$	1.19806	0.0616217
$379$	−1.32006	−0.0678069	−0.0339035	−	0.999425i	$-0.510794\pi$
$379$	−1.32006	−0.0678069	−0.0339035	+	0.999425i	$0.510794\pi$
$380$	−0.911854	−0.0467771
$381$	9.32736	0.477855
$382$	15.3623	0.786002
$383$	25.5308	1.30456	0.652281	−	0.757977i	$-0.273812\pi$
$383$	25.5308	1.30456	0.652281	+	0.757977i	$0.273812\pi$
$384$	−1.00000	−0.0510310
$385$	4.71917	0.240511
$386$	−18.3207	−0.932497
$387$	3.43296	0.174507
$388$	16.9976	0.862923
$389$	−6.05967	−0.307238	−0.153619	−	0.988130i	$-0.549093\pi$
$389$	−6.05967	−0.307238	−0.153619	+	0.988130i	$0.549093\pi$
$390$	0	0
$391$	2.17092	0.109788
$392$	−5.56465	−0.281057
$393$	−4.62133	−0.233115
$394$	−2.31336	−0.116545
$395$	−2.73556	−0.137641
$396$	3.93900	0.197942
$397$	5.99330	0.300795	0.150397	−	0.988626i	$-0.451945\pi$
$397$	5.99330	0.300795	0.150397	+	0.988626i	$0.451945\pi$
$398$	6.36765	0.319181
$399$	1.09246	0.0546913
$400$	1.00000	0.0500000
$401$	17.3773	0.867783	0.433891	−	0.900965i	$-0.357140\pi$
$401$	17.3773	0.867783	0.433891	+	0.900965i	$0.357140\pi$
$402$	9.16852	0.457284
$403$	0	0
$404$	6.67025	0.331857
$405$	−1.00000	−0.0496904
$406$	4.15990	0.206452
$407$	18.9148	0.937574
$408$	−1.74094	−0.0861893
$409$	29.3672	1.45211	0.726057	−	0.687635i	$-0.241351\pi$
$409$	29.3672	1.45211	0.726057	+	0.687635i	$0.241351\pi$
$410$	−0.198062	−0.00978160
$411$	−0.0760644	−0.00375198
$412$	10.0315	0.494215
$413$	12.2737	0.603947
$414$	1.24698	0.0612857
$415$	−15.5864	−0.765107
$416$	0	0
$417$	−18.7017	−0.915827
$418$	3.59179	0.175680
$419$	−26.5472	−1.29692	−0.648458	−	0.761251i	$-0.724586\pi$
$419$	−26.5472	−1.29692	−0.648458	+	0.761251i	$0.724586\pi$
$420$	−1.19806	−0.0584595
$421$	−31.0834	−1.51491	−0.757455	−	0.652887i	$-0.773558\pi$
$421$	−31.0834	−1.51491	−0.757455	+	0.652887i	$0.773558\pi$
$422$	−11.6775	−0.568454
$423$	0.902165	0.0438648
$424$	−6.91185	−0.335669
$425$	1.74094	0.0844479
$426$	−11.0707	−0.536377
$427$	−11.8931	−0.575546
$428$	−10.1032	−0.488357
$429$	0	0
$430$	−3.43296	−0.165552
$431$	36.1226	1.73996	0.869982	−	0.493084i	$-0.164130\pi$
$431$	36.1226	1.73996	0.869982	+	0.493084i	$0.164130\pi$
$432$	−1.00000	−0.0481125
$433$	−35.2868	−1.69578	−0.847888	−	0.530176i	$-0.822126\pi$
$433$	−35.2868	−1.69578	−0.847888	+	0.530176i	$0.822126\pi$
$434$	2.58642	0.124152
$435$	−3.47219	−0.166479
$436$	10.5211	0.503870
$437$	1.13706	0.0543931
$438$	−4.13706	−0.197677
$439$	−32.0170	−1.52809	−0.764044	−	0.645165i	$-0.776789\pi$
$439$	−32.0170	−1.52809	−0.764044	+	0.645165i	$0.776789\pi$
$440$	−3.93900	−0.187785
$441$	−5.56465	−0.264983
$442$	0	0
$443$	−23.5948	−1.12102	−0.560511	−	0.828147i	$-0.689395\pi$
$443$	−23.5948	−1.12102	−0.560511	+	0.828147i	$0.689395\pi$
$444$	−4.80194	−0.227890
$445$	−8.57673	−0.406576
$446$	11.4004	0.539826
$447$	−13.6799	−0.647039
$448$	−1.19806	−0.0566031
$449$	−19.8374	−0.936187	−0.468093	−	0.883679i	$-0.655059\pi$
$449$	−19.8374	−0.936187	−0.468093	+	0.883679i	$0.655059\pi$
$450$	1.00000	0.0471405
$451$	0.780167	0.0367367
$452$	−11.2131	−0.527421
$453$	1.15883	0.0544468
$454$	8.86725	0.416161
$455$	0	0
$456$	−0.911854	−0.0427015
$457$	−26.5284	−1.24095	−0.620473	−	0.784228i	$-0.713059\pi$
$457$	−26.5284	−1.24095	−0.620473	+	0.784228i	$0.713059\pi$
$458$	12.6136	0.589393
$459$	−1.74094	−0.0812601
$460$	−1.24698	−0.0581407
$461$	−12.2717	−0.571552	−0.285776	−	0.958297i	$-0.592251\pi$
$461$	−12.2717	−0.571552	−0.285776	+	0.958297i	$0.592251\pi$
$462$	4.71917	0.219556
$463$	−12.6455	−0.587686	−0.293843	−	0.955854i	$-0.594934\pi$
$463$	−12.6455	−0.587686	−0.293843	+	0.955854i	$0.594934\pi$
$464$	−3.47219	−0.161192
$465$	−2.15883	−0.100114
$466$	−23.4155	−1.08470
$467$	11.3870	0.526929	0.263464	−	0.964669i	$-0.415135\pi$
$467$	11.3870	0.526929	0.263464	+	0.964669i	$0.415135\pi$
$468$	0	0
$469$	10.9845	0.507215
$470$	−0.902165	−0.0416138
$471$	12.2664	0.565204
$472$	−10.2446	−0.471545
$473$	13.5224	0.621762
$474$	−2.73556	−0.125649
$475$	0.911854	0.0418387
$476$	−2.08575	−0.0956003
$477$	−6.91185	−0.316472
$478$	8.87263	0.405824
$479$	18.8431	0.860963	0.430481	−	0.902600i	$-0.358344\pi$
$479$	18.8431	0.860963	0.430481	+	0.902600i	$0.358344\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	9.72886	0.443137
$483$	1.49396	0.0679775
$484$	4.51573	0.205260
$485$	−16.9976	−0.771822
$486$	−1.00000	−0.0453609
$487$	27.6383	1.25241	0.626206	−	0.779658i	$-0.284607\pi$
$487$	27.6383	1.25241	0.626206	+	0.779658i	$0.284607\pi$
$488$	9.92692	0.449371
$489$	2.48858	0.112538
$490$	5.56465	0.251385
$491$	−13.9866	−0.631206	−0.315603	−	0.948891i	$-0.602207\pi$
$491$	−13.9866	−0.631206	−0.315603	+	0.948891i	$0.602207\pi$
$492$	−0.198062	−0.00892934
$493$	−6.04487	−0.272247
$494$	0	0
$495$	−3.93900	−0.177045
$496$	−2.15883	−0.0969345
$497$	−13.2634	−0.594944
$498$	−15.5864	−0.698444
$499$	−8.87071	−0.397107	−0.198554	−	0.980090i	$-0.563624\pi$
$499$	−8.87071	−0.397107	−0.198554	+	0.980090i	$0.563624\pi$
$500$	−1.00000	−0.0447214
$501$	−7.27950	−0.325224
$502$	29.8823	1.33371
$503$	1.99761	0.0890689	0.0445344	−	0.999008i	$-0.485820\pi$
$503$	1.99761	0.0890689	0.0445344	+	0.999008i	$0.485820\pi$
$504$	−1.19806	−0.0533659
$505$	−6.67025	−0.296822
$506$	4.91185	0.218359
$507$	0	0
$508$	−9.32736	−0.413834
$509$	−21.7023	−0.961938	−0.480969	−	0.876738i	$-0.659715\pi$
$509$	−21.7023	−0.961938	−0.480969	+	0.876738i	$0.659715\pi$
$510$	1.74094	0.0770901
$511$	−4.95646	−0.219261
$512$	1.00000	0.0441942
$513$	−0.911854	−0.0402593
$514$	−8.52111	−0.375850
$515$	−10.0315	−0.442039
$516$	−3.43296	−0.151128
$517$	3.55363	0.156288
$518$	−5.75302	−0.252773
$519$	−15.8485	−0.695670
$520$	0	0
$521$	11.8364	0.518561	0.259281	−	0.965802i	$-0.416515\pi$
$521$	11.8364	0.518561	0.259281	+	0.965802i	$0.416515\pi$
$522$	−3.47219	−0.151974
$523$	24.6112	1.07617	0.538086	−	0.842890i	$-0.319148\pi$
$523$	24.6112	1.07617	0.538086	+	0.842890i	$0.319148\pi$
$524$	4.62133	0.201884
$525$	1.19806	0.0522877
$526$	−22.6450	−0.987370
$527$	−3.75840	−0.163718
$528$	−3.93900	−0.171423
$529$	−21.4450	−0.932393
$530$	6.91185	0.300232
$531$	−10.2446	−0.444577
$532$	−1.09246	−0.0473641
$533$	0	0
$534$	−8.57673	−0.371151
$535$	10.1032	0.436800
$536$	−9.16852	−0.396020
$537$	−18.4819	−0.797552
$538$	−9.02608	−0.389142
$539$	−21.9191	−0.944125
$540$	1.00000	0.0430331
$541$	29.1282	1.25232	0.626160	−	0.779694i	$-0.284626\pi$
$541$	29.1282	1.25232	0.626160	+	0.779694i	$0.284626\pi$
$542$	−8.59611	−0.369234
$543$	−10.7041	−0.459357
$544$	1.74094	0.0746421
$545$	−10.5211	−0.450675
$546$	0	0
$547$	−10.1752	−0.435061	−0.217531	−	0.976053i	$-0.569800\pi$
$547$	−10.1752	−0.435061	−0.217531	+	0.976053i	$0.569800\pi$
$548$	0.0760644	0.00324931
$549$	9.92692	0.423671
$550$	3.93900	0.167960
$551$	−3.16613	−0.134882
$552$	−1.24698	−0.0530750
$553$	−3.27737	−0.139368
$554$	6.31527	0.268310
$555$	4.80194	0.203831
$556$	18.7017	0.793129
$557$	13.0398	0.552515	0.276257	−	0.961084i	$-0.410906\pi$
$557$	13.0398	0.552515	0.276257	+	0.961084i	$0.410906\pi$
$558$	−2.15883	−0.0913907
$559$	0	0
$560$	1.19806	0.0506274
$561$	−6.85756	−0.289526
$562$	21.7409	0.917086
$563$	−32.9221	−1.38750	−0.693751	−	0.720215i	$-0.744043\pi$
$563$	−32.9221	−1.38750	−0.693751	+	0.720215i	$0.744043\pi$
$564$	−0.902165	−0.0379880
$565$	11.2131	0.471740
$566$	12.9487	0.544274
$567$	−1.19806	−0.0503139
$568$	11.0707	0.464516
$569$	7.53319	0.315808	0.157904	−	0.987454i	$-0.449526\pi$
$569$	7.53319	0.315808	0.157904	+	0.987454i	$0.449526\pi$
$570$	0.911854	0.0381934
$571$	36.2422	1.51669	0.758344	−	0.651854i	$-0.226009\pi$
$571$	36.2422	1.51669	0.758344	+	0.651854i	$0.226009\pi$
$572$	0	0
$573$	−15.3623	−0.641768
$574$	−0.237291	−0.00990433
$575$	1.24698	0.0520026
$576$	1.00000	0.0416667
$577$	−14.2198	−0.591979	−0.295990	−	0.955191i	$-0.595649\pi$
$577$	−14.2198	−0.591979	−0.295990	+	0.955191i	$0.595649\pi$
$578$	−13.9691	−0.581039
$579$	18.3207	0.761380
$580$	3.47219	0.144175
$581$	−18.6735	−0.774707
$582$	−16.9976	−0.704573
$583$	−27.2258	−1.12758
$584$	4.13706	0.171193
$585$	0	0
$586$	27.3056	1.12798
$587$	22.3207	0.921272	0.460636	−	0.887589i	$-0.347621\pi$
$587$	22.3207	0.921272	0.460636	+	0.887589i	$0.347621\pi$
$588$	5.56465	0.229482
$589$	−1.96854	−0.0811123
$590$	10.2446	0.421763
$591$	2.31336	0.0951587
$592$	4.80194	0.197358
$593$	−1.64742	−0.0676513	−0.0338256	−	0.999428i	$-0.510769\pi$
$593$	−1.64742	−0.0676513	−0.0338256	+	0.999428i	$0.510769\pi$
$594$	−3.93900	−0.161619
$595$	2.08575	0.0855075
$596$	13.6799	0.560352
$597$	−6.36765	−0.260611
$598$	0	0
$599$	9.41119	0.384531	0.192265	−	0.981343i	$-0.438417\pi$
$599$	9.41119	0.384531	0.192265	+	0.981343i	$0.438417\pi$
$600$	−1.00000	−0.0408248
$601$	1.62565	0.0663115	0.0331557	−	0.999450i	$-0.489444\pi$
$601$	1.62565	0.0663115	0.0331557	+	0.999450i	$0.489444\pi$
$602$	−4.11290	−0.167629
$603$	−9.16852	−0.373371
$604$	−1.15883	−0.0471523
$605$	−4.51573	−0.183591
$606$	−6.67025	−0.270960
$607$	14.9282	0.605919	0.302959	−	0.953003i	$-0.402025\pi$
$607$	14.9282	0.605919	0.302959	+	0.953003i	$0.402025\pi$
$608$	0.911854	0.0369806
$609$	−4.15990	−0.168568
$610$	−9.92692	−0.401929
$611$	0	0
$612$	1.74094	0.0703733
$613$	28.8437	1.16499	0.582493	−	0.812836i	$-0.302078\pi$
$613$	28.8437	1.16499	0.582493	+	0.812836i	$0.302078\pi$
$614$	−24.9124	−1.00538
$615$	0.198062	0.00798664
$616$	−4.71917	−0.190141
$617$	−39.7985	−1.60223	−0.801113	−	0.598513i	$-0.795759\pi$
$617$	−39.7985	−1.60223	−0.801113	+	0.598513i	$0.795759\pi$
$618$	−10.0315	−0.403524
$619$	−38.0465	−1.52922	−0.764609	−	0.644494i	$-0.777068\pi$
$619$	−38.0465	−1.52922	−0.764609	+	0.644494i	$0.777068\pi$
$620$	2.15883	0.0867008
$621$	−1.24698	−0.0500396
$622$	21.3787	0.857206
$623$	−10.2755	−0.411677
$624$	0	0
$625$	1.00000	0.0400000
$626$	−23.8713	−0.954089
$627$	−3.59179	−0.143442
$628$	−12.2664	−0.489481
$629$	8.35988	0.333330
$630$	1.19806	0.0477319
$631$	23.0901	0.919201	0.459600	−	0.888126i	$-0.347993\pi$
$631$	23.0901	0.919201	0.459600	+	0.888126i	$0.347993\pi$
$632$	2.73556	0.108815
$633$	11.6775	0.464141
$634$	−12.6407	−0.502027
$635$	9.32736	0.370145
$636$	6.91185	0.274073
$637$	0	0
$638$	−13.6770	−0.541476
$639$	11.0707	0.437950
$640$	−1.00000	−0.0395285
$641$	23.3948	0.924039	0.462019	−	0.886870i	$-0.347125\pi$
$641$	23.3948	0.924039	0.462019	+	0.886870i	$0.347125\pi$
$642$	10.1032	0.398742
$643$	−6.00836	−0.236947	−0.118473	−	0.992957i	$-0.537800\pi$
$643$	−6.00836	−0.236947	−0.118473	+	0.992957i	$0.537800\pi$
$644$	−1.49396	−0.0588702
$645$	3.43296	0.135173
$646$	1.58748	0.0624586
$647$	−4.38511	−0.172396	−0.0861982	−	0.996278i	$-0.527472\pi$
$647$	−4.38511	−0.172396	−0.0861982	+	0.996278i	$0.527472\pi$
$648$	1.00000	0.0392837
$649$	−40.3534	−1.58401
$650$	0	0
$651$	−2.58642	−0.101370
$652$	−2.48858	−0.0974604
$653$	−20.6160	−0.806765	−0.403382	−	0.915032i	$-0.632166\pi$
$653$	−20.6160	−0.806765	−0.403382	+	0.915032i	$0.632166\pi$
$654$	−10.5211	−0.411408
$655$	−4.62133	−0.180570
$656$	0.198062	0.00773303
$657$	4.13706	0.161402
$658$	−1.08085	−0.0421359
$659$	18.4910	0.720306	0.360153	−	0.932893i	$-0.382724\pi$
$659$	18.4910	0.720306	0.360153	+	0.932893i	$0.382724\pi$
$660$	3.93900	0.153325
$661$	17.6420	0.686196	0.343098	−	0.939300i	$-0.388524\pi$
$661$	17.6420	0.686196	0.343098	+	0.939300i	$0.388524\pi$
$662$	20.5109	0.797180
$663$	0	0
$664$	15.5864	0.604870
$665$	1.09246	0.0423637
$666$	4.80194	0.186071
$667$	−4.32975	−0.167649
$668$	7.27950	0.281652
$669$	−11.4004	−0.440766
$670$	9.16852	0.354211
$671$	39.1021	1.50952
$672$	1.19806	0.0462163
$673$	17.1473	0.660981	0.330491	−	0.943809i	$-0.392786\pi$
$673$	17.1473	0.660981	0.330491	+	0.943809i	$0.392786\pi$
$674$	6.82371	0.262839
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	31.7982	1.22210	0.611052	−	0.791590i	$-0.290746\pi$
$677$	31.7982	1.22210	0.611052	+	0.791590i	$0.290746\pi$
$678$	11.2131	0.430638
$679$	−20.3642	−0.781506
$680$	−1.74094	−0.0667620
$681$	−8.86725	−0.339794
$682$	−8.50365	−0.325622
$683$	6.32783	0.242128	0.121064	−	0.992645i	$-0.461369\pi$
$683$	6.32783	0.242128	0.121064	+	0.992645i	$0.461369\pi$
$684$	0.911854	0.0348656
$685$	−0.0760644	−0.00290627
$686$	15.0532	0.574735
$687$	−12.6136	−0.481237
$688$	3.43296	0.130880
$689$	0	0
$690$	1.24698	0.0474717
$691$	−15.1825	−0.577570	−0.288785	−	0.957394i	$-0.593251\pi$
$691$	−15.1825	−0.577570	−0.288785	+	0.957394i	$0.593251\pi$
$692$	15.8485	0.602468
$693$	−4.71917	−0.179266
$694$	−29.4741	−1.11882
$695$	−18.7017	−0.709396
$696$	3.47219	0.131613
$697$	0.344814	0.0130608
$698$	−34.5394	−1.30734
$699$	23.4155	0.885656
$700$	−1.19806	−0.0452825
$701$	−23.6528	−0.893354	−0.446677	−	0.894695i	$-0.647393\pi$
$701$	−23.6528	−0.893354	−0.446677	+	0.894695i	$0.647393\pi$
$702$	0	0
$703$	4.37867	0.165145
$704$	3.93900	0.148457
$705$	0.902165	0.0339775
$706$	31.2295	1.17534
$707$	−7.99138	−0.300547
$708$	10.2446	0.385015
$709$	−40.9353	−1.53736	−0.768678	−	0.639636i	$-0.779085\pi$
$709$	−40.9353	−1.53736	−0.768678	+	0.639636i	$0.779085\pi$
$710$	−11.0707	−0.415476
$711$	2.73556	0.102592
$712$	8.57673	0.321426
$713$	−2.69202	−0.100817
$714$	2.08575	0.0780573
$715$	0	0
$716$	18.4819	0.690700
$717$	−8.87263	−0.331354
$718$	18.2524	0.681172
$719$	4.55363	0.169822	0.0849109	−	0.996389i	$-0.472939\pi$
$719$	4.55363	0.169822	0.0849109	+	0.996389i	$0.472939\pi$
$720$	−1.00000	−0.0372678
$721$	−12.0183	−0.447585
$722$	−18.1685	−0.676162
$723$	−9.72886	−0.361820
$724$	10.7041	0.397815
$725$	−3.47219	−0.128954
$726$	−4.51573	−0.167594
$727$	18.1612	0.673563	0.336781	−	0.941583i	$-0.390662\pi$
$727$	18.1612	0.673563	0.336781	+	0.941583i	$0.390662\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−4.13706	−0.153120
$731$	5.97657	0.221052
$732$	−9.92692	−0.366910
$733$	9.89115	0.365338	0.182669	−	0.983174i	$-0.441526\pi$
$733$	9.89115	0.365338	0.182669	+	0.983174i	$0.441526\pi$
$734$	−19.3773	−0.715231
$735$	−5.56465	−0.205255
$736$	1.24698	0.0459643
$737$	−36.1148	−1.33031
$738$	0.198062	0.00729077
$739$	−23.9608	−0.881411	−0.440706	−	0.897652i	$-0.645272\pi$
$739$	−23.9608	−0.881411	−0.440706	+	0.897652i	$0.645272\pi$
$740$	−4.80194	−0.176523
$741$	0	0
$742$	8.28083	0.303999
$743$	46.6708	1.71219	0.856094	−	0.516821i	$-0.172885\pi$
$743$	46.6708	1.71219	0.856094	+	0.516821i	$0.172885\pi$
$744$	2.15883	0.0791467
$745$	−13.6799	−0.501194
$746$	−13.6280	−0.498958
$747$	15.5864	0.570277
$748$	6.85756	0.250737
$749$	12.1043	0.442281
$750$	1.00000	0.0365148
$751$	−30.9168	−1.12817	−0.564084	−	0.825717i	$-0.690771\pi$
$751$	−30.9168	−1.12817	−0.564084	+	0.825717i	$0.690771\pi$
$752$	0.902165	0.0328986
$753$	−29.8823	−1.08897
$754$	0	0
$755$	1.15883	0.0421743
$756$	1.19806	0.0435731
$757$	24.4566	0.888892	0.444446	−	0.895806i	$-0.353401\pi$
$757$	24.4566	0.888892	0.444446	+	0.895806i	$0.353401\pi$
$758$	−1.32006	−0.0479467
$759$	−4.91185	−0.178289
$760$	−0.911854	−0.0330764
$761$	−20.5536	−0.745069	−0.372534	−	0.928018i	$-0.621511\pi$
$761$	−20.5536	−0.745069	−0.372534	+	0.928018i	$0.621511\pi$
$762$	9.32736	0.337894
$763$	−12.6049	−0.456329
$764$	15.3623	0.555788
$765$	−1.74094	−0.0629438
$766$	25.5308	0.922465
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−21.0084	−0.757581	−0.378790	−	0.925482i	$-0.623660\pi$
$769$	−21.0084	−0.757581	−0.378790	+	0.925482i	$0.623660\pi$
$770$	4.71917	0.170067
$771$	8.52111	0.306880
$772$	−18.3207	−0.659375
$773$	−36.2626	−1.30428	−0.652138	−	0.758100i	$-0.726128\pi$
$773$	−36.2626	−1.30428	−0.652138	+	0.758100i	$0.726128\pi$
$774$	3.43296	0.123395
$775$	−2.15883	−0.0775476
$776$	16.9976	0.610179
$777$	5.75302	0.206388
$778$	−6.05967	−0.217250
$779$	0.180604	0.00647081
$780$	0	0
$781$	43.6075	1.56040
$782$	2.17092	0.0776318
$783$	3.47219	0.124086
$784$	−5.56465	−0.198737
$785$	12.2664	0.437805
$786$	−4.62133	−0.164838
$787$	−1.55927	−0.0555820	−0.0277910	−	0.999614i	$-0.508847\pi$
$787$	−1.55927	−0.0555820	−0.0277910	+	0.999614i	$0.508847\pi$
$788$	−2.31336	−0.0824099
$789$	22.6450	0.806184
$790$	−2.73556	−0.0973269
$791$	13.4340	0.477659
$792$	3.93900	0.139966
$793$	0	0
$794$	5.99330	0.212694
$795$	−6.91185	−0.245138
$796$	6.36765	0.225695
$797$	5.16793	0.183058	0.0915288	−	0.995802i	$-0.470825\pi$
$797$	5.16793	0.183058	0.0915288	+	0.995802i	$0.470825\pi$
$798$	1.09246	0.0386726
$799$	1.57061	0.0555644
$800$	1.00000	0.0353553
$801$	8.57673	0.303044
$802$	17.3773	0.613615
$803$	16.2959	0.575070
$804$	9.16852	0.323349
$805$	1.49396	0.0526551
$806$	0	0
$807$	9.02608	0.317733
$808$	6.67025	0.234659
$809$	−31.8605	−1.12016	−0.560079	−	0.828440i	$-0.689229\pi$
$809$	−31.8605	−1.12016	−0.560079	+	0.828440i	$0.689229\pi$
$810$	−1.00000	−0.0351364
$811$	−15.3056	−0.537452	−0.268726	−	0.963217i	$-0.586603\pi$
$811$	−15.3056	−0.537452	−0.268726	+	0.963217i	$0.586603\pi$
$812$	4.15990	0.145984
$813$	8.59611	0.301479
$814$	18.9148	0.662965
$815$	2.48858	0.0871712
$816$	−1.74094	−0.0609450
$817$	3.13036	0.109517
$818$	29.3672	1.02680
$819$	0	0
$820$	−0.198062	−0.00691663
$821$	32.0248	1.11767	0.558836	−	0.829278i	$-0.311248\pi$
$821$	32.0248	1.11767	0.558836	+	0.829278i	$0.311248\pi$
$822$	−0.0760644	−0.00265305
$823$	−36.8571	−1.28476	−0.642379	−	0.766387i	$-0.722052\pi$
$823$	−36.8571	−1.28476	−0.642379	+	0.766387i	$0.722052\pi$
$824$	10.0315	0.349462
$825$	−3.93900	−0.137138
$826$	12.2737	0.427055
$827$	−37.9269	−1.31885	−0.659424	−	0.751771i	$-0.729200\pi$
$827$	−37.9269	−1.31885	−0.659424	+	0.751771i	$0.729200\pi$
$828$	1.24698	0.0433355
$829$	−34.2349	−1.18903	−0.594514	−	0.804086i	$-0.702655\pi$
$829$	−34.2349	−1.18903	−0.594514	+	0.804086i	$0.702655\pi$
$830$	−15.5864	−0.541012
$831$	−6.31527	−0.219074
$832$	0	0
$833$	−9.68771	−0.335659
$834$	−18.7017	−0.647587
$835$	−7.27950	−0.251918
$836$	3.59179	0.124225
$837$	2.15883	0.0746202
$838$	−26.5472	−0.917057
$839$	18.7318	0.646695	0.323347	−	0.946280i	$-0.395192\pi$
$839$	18.7318	0.646695	0.323347	+	0.946280i	$0.395192\pi$
$840$	−1.19806	−0.0413371
$841$	−16.9439	−0.584273
$842$	−31.0834	−1.07120
$843$	−21.7409	−0.748798
$844$	−11.6775	−0.401958
$845$	0	0
$846$	0.902165	0.0310171
$847$	−5.41013	−0.185894
$848$	−6.91185	−0.237354
$849$	−12.9487	−0.444398
$850$	1.74094	0.0597137
$851$	5.98792	0.205263
$852$	−11.0707	−0.379276
$853$	−4.64071	−0.158895	−0.0794475	−	0.996839i	$-0.525316\pi$
$853$	−4.64071	−0.158895	−0.0794475	+	0.996839i	$0.525316\pi$
$854$	−11.8931	−0.406972
$855$	−0.911854	−0.0311847
$856$	−10.1032	−0.345321
$857$	11.1347	0.380353	0.190177	−	0.981750i	$-0.439094\pi$
$857$	11.1347	0.380353	0.190177	+	0.981750i	$0.439094\pi$
$858$	0	0
$859$	28.9661	0.988312	0.494156	−	0.869373i	$-0.335477\pi$
$859$	28.9661	0.988312	0.494156	+	0.869373i	$0.335477\pi$
$860$	−3.43296	−0.117063
$861$	0.237291	0.00808685
$862$	36.1226	1.23034
$863$	33.1371	1.12800	0.563999	−	0.825775i	$-0.309262\pi$
$863$	33.1371	1.12800	0.563999	+	0.825775i	$0.309262\pi$
$864$	−1.00000	−0.0340207
$865$	−15.8485	−0.538864
$866$	−35.2868	−1.19909
$867$	13.9691	0.474417
$868$	2.58642	0.0877887
$869$	10.7754	0.365530
$870$	−3.47219	−0.117718
$871$	0	0
$872$	10.5211	0.356290
$873$	16.9976	0.575282
$874$	1.13706	0.0384617
$875$	1.19806	0.0405019
$876$	−4.13706	−0.139778
$877$	19.1371	0.646213	0.323106	−	0.946363i	$-0.395273\pi$
$877$	19.1371	0.646213	0.323106	+	0.946363i	$0.395273\pi$
$878$	−32.0170	−1.08052
$879$	−27.3056	−0.920995
$880$	−3.93900	−0.132784
$881$	−40.3387	−1.35905	−0.679523	−	0.733655i	$-0.737813\pi$
$881$	−40.3387	−1.35905	−0.679523	+	0.733655i	$0.737813\pi$
$882$	−5.56465	−0.187371
$883$	−4.31336	−0.145156	−0.0725780	−	0.997363i	$-0.523123\pi$
$883$	−4.31336	−0.145156	−0.0725780	+	0.997363i	$0.523123\pi$
$884$	0	0
$885$	−10.2446	−0.344368
$886$	−23.5948	−0.792682
$887$	−9.38404	−0.315085	−0.157543	−	0.987512i	$-0.550357\pi$
$887$	−9.38404	−0.315085	−0.157543	+	0.987512i	$0.550357\pi$
$888$	−4.80194	−0.161142
$889$	11.1748	0.374789
$890$	−8.57673	−0.287493
$891$	3.93900	0.131962
$892$	11.4004	0.381715
$893$	0.822643	0.0275287
$894$	−13.6799	−0.457526
$895$	−18.4819	−0.617781
$896$	−1.19806	−0.0400245
$897$	0	0
$898$	−19.8374	−0.661984
$899$	7.49588	0.250002
$900$	1.00000	0.0333333
$901$	−12.0331	−0.400881
$902$	0.780167	0.0259767
$903$	4.11290	0.136869
$904$	−11.2131	−0.372943
$905$	−10.7041	−0.355816
$906$	1.15883	0.0384997
$907$	14.2131	0.471939	0.235970	−	0.971760i	$-0.424173\pi$
$907$	14.2131	0.471939	0.235970	+	0.971760i	$0.424173\pi$
$908$	8.86725	0.294270
$909$	6.67025	0.221238
$910$	0	0
$911$	40.2707	1.33423	0.667113	−	0.744956i	$-0.267530\pi$
$911$	40.2707	1.33423	0.667113	+	0.744956i	$0.267530\pi$
$912$	−0.911854	−0.0301945
$913$	61.3949	2.03188
$914$	−26.5284	−0.877482
$915$	9.92692	0.328174
$916$	12.6136	0.416764
$917$	−5.53665	−0.182836
$918$	−1.74094	−0.0574595
$919$	7.03577	0.232089	0.116044	−	0.993244i	$-0.462979\pi$
$919$	7.03577	0.232089	0.116044	+	0.993244i	$0.462979\pi$
$920$	−1.24698	−0.0411117
$921$	24.9124	0.820893
$922$	−12.2717	−0.404148
$923$	0	0
$924$	4.71917	0.155249
$925$	4.80194	0.157887
$926$	−12.6455	−0.415557
$927$	10.0315	0.329476
$928$	−3.47219	−0.113980
$929$	−21.3575	−0.700716	−0.350358	−	0.936616i	$-0.613940\pi$
$929$	−21.3575	−0.700716	−0.350358	+	0.936616i	$0.613940\pi$
$930$	−2.15883	−0.0707909
$931$	−5.07415	−0.166298
$932$	−23.4155	−0.767000
$933$	−21.3787	−0.699906
$934$	11.3870	0.372595
$935$	−6.85756	−0.224266
$936$	0	0
$937$	41.7396	1.36357	0.681787	−	0.731551i	$-0.261203\pi$
$937$	41.7396	1.36357	0.681787	+	0.731551i	$0.261203\pi$
$938$	10.9845	0.358655
$939$	23.8713	0.779010
$940$	−0.902165	−0.0294254
$941$	−26.2078	−0.854348	−0.427174	−	0.904169i	$-0.640491\pi$
$941$	−26.2078	−0.854348	−0.427174	+	0.904169i	$0.640491\pi$
$942$	12.2664	0.399660
$943$	0.246980	0.00804276
$944$	−10.2446	−0.333433
$945$	−1.19806	−0.0389730
$946$	13.5224	0.439652
$947$	−3.15239	−0.102439	−0.0512195	−	0.998687i	$-0.516311\pi$
$947$	−3.15239	−0.102439	−0.0512195	+	0.998687i	$0.516311\pi$
$948$	−2.73556	−0.0888469
$949$	0	0
$950$	0.911854	0.0295845
$951$	12.6407	0.409903
$952$	−2.08575	−0.0675996
$953$	−3.56060	−0.115339	−0.0576695	−	0.998336i	$-0.518367\pi$
$953$	−3.56060	−0.115339	−0.0576695	+	0.998336i	$0.518367\pi$
$954$	−6.91185	−0.223780
$955$	−15.3623	−0.497111
$956$	8.87263	0.286961
$957$	13.6770	0.442113
$958$	18.8431	0.608792
$959$	−0.0911299	−0.00294274
$960$	1.00000	0.0322749
$961$	−26.3394	−0.849659
$962$	0	0
$963$	−10.1032	−0.325571
$964$	9.72886	0.313345
$965$	18.3207	0.589763
$966$	1.49396	0.0480673
$967$	−7.26875	−0.233747	−0.116874	−	0.993147i	$-0.537287\pi$
$967$	−7.26875	−0.233747	−0.116874	+	0.993147i	$0.537287\pi$
$968$	4.51573	0.145141
$969$	−1.58748	−0.0509973
$970$	−16.9976	−0.545760
$971$	−9.53511	−0.305996	−0.152998	−	0.988226i	$-0.548893\pi$
$971$	−9.53511	−0.305996	−0.152998	+	0.988226i	$0.548893\pi$
$972$	−1.00000	−0.0320750
$973$	−22.4058	−0.718297
$974$	27.6383	0.885589
$975$	0	0
$976$	9.92692	0.317753
$977$	−17.5700	−0.562115	−0.281057	−	0.959691i	$-0.590685\pi$
$977$	−17.5700	−0.562115	−0.281057	+	0.959691i	$0.590685\pi$
$978$	2.48858	0.0795761
$979$	33.7837	1.07973
$980$	5.56465	0.177756
$981$	10.5211	0.335913
$982$	−13.9866	−0.446330
$983$	−43.4935	−1.38723	−0.693613	−	0.720347i	$-0.743982\pi$
$983$	−43.4935	−1.38723	−0.693613	+	0.720347i	$0.743982\pi$
$984$	−0.198062	−0.00631399
$985$	2.31336	0.0737096
$986$	−6.04487	−0.192508
$987$	1.08085	0.0344038
$988$	0	0
$989$	4.28083	0.136123
$990$	−3.93900	−0.125190
$991$	38.8232	1.23326	0.616630	−	0.787253i	$-0.288497\pi$
$991$	38.8232	1.23326	0.616630	+	0.787253i	$0.288497\pi$
$992$	−2.15883	−0.0685430
$993$	−20.5109	−0.650895
$994$	−13.2634	−0.420689
$995$	−6.36765	−0.201868
$996$	−15.5864	−0.493875
$997$	−21.9807	−0.696137	−0.348069	−	0.937469i	$-0.613162\pi$
$997$	−21.9807	−0.696137	−0.348069	+	0.937469i	$0.613162\pi$
$998$	−8.87071	−0.280797
$999$	−4.80194	−0.151927

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.bs.1.3	yes	3
13.5	odd	4		5070.2.b.v.1351.3		6
13.8	odd	4		5070.2.b.v.1351.4		6
13.12	even	2		5070.2.a.bl.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bl.1.1	✓	3	13.12	even	2
5070.2.a.bs.1.3	yes	3	1.1	even	1	trivial
5070.2.b.v.1351.3		6	13.5	odd	4
5070.2.b.v.1351.4		6	13.8	odd	4

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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.19806 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.19806 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.93900 q^{11} -1.00000 q^{12} -1.19806 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.74094 q^{17} +1.00000 q^{18} +0.911854 q^{19} -1.00000 q^{20} +1.19806 q^{21} +3.93900 q^{22} +1.24698 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -1.19806 q^{28} -3.47219 q^{29} +1.00000 q^{30} -2.15883 q^{31} +1.00000 q^{32} -3.93900 q^{33} +1.74094 q^{34} +1.19806 q^{35} +1.00000 q^{36} +4.80194 q^{37} +0.911854 q^{38} -1.00000 q^{40} +0.198062 q^{41} +1.19806 q^{42} +3.43296 q^{43} +3.93900 q^{44} -1.00000 q^{45} +1.24698 q^{46} +0.902165 q^{47} -1.00000 q^{48} -5.56465 q^{49} +1.00000 q^{50} -1.74094 q^{51} -6.91185 q^{53} -1.00000 q^{54} -3.93900 q^{55} -1.19806 q^{56} -0.911854 q^{57} -3.47219 q^{58} -10.2446 q^{59} +1.00000 q^{60} +9.92692 q^{61} -2.15883 q^{62} -1.19806 q^{63} +1.00000 q^{64} -3.93900 q^{66} -9.16852 q^{67} +1.74094 q^{68} -1.24698 q^{69} +1.19806 q^{70} +11.0707 q^{71} +1.00000 q^{72} +4.13706 q^{73} +4.80194 q^{74} -1.00000 q^{75} +0.911854 q^{76} -4.71917 q^{77} +2.73556 q^{79} -1.00000 q^{80} +1.00000 q^{81} +0.198062 q^{82} +15.5864 q^{83} +1.19806 q^{84} -1.74094 q^{85} +3.43296 q^{86} +3.47219 q^{87} +3.93900 q^{88} +8.57673 q^{89} -1.00000 q^{90} +1.24698 q^{92} +2.15883 q^{93} +0.902165 q^{94} -0.911854 q^{95} -1.00000 q^{96} +16.9976 q^{97} -5.56465 q^{98} +3.93900 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 8 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 - 8 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 8 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} + 2 q^{11} - 3 q^{12} - 8 q^{14} + 3 q^{15} + 3 q^{16} - 9 q^{17} + 3 q^{18} - q^{19} - 3 q^{20} + 8 q^{21} + 2 q^{22} - q^{23} - 3 q^{24} + 3 q^{25} - 3 q^{27} - 8 q^{28} - 4 q^{29} + 3 q^{30} + 2 q^{31} + 3 q^{32} - 2 q^{33} - 9 q^{34} + 8 q^{35} + 3 q^{36} + 10 q^{37} - q^{38} - 3 q^{40} + 5 q^{41} + 8 q^{42} - 9 q^{43} + 2 q^{44} - 3 q^{45} - q^{46} + 21 q^{47} - 3 q^{48} + 5 q^{49} + 3 q^{50} + 9 q^{51} - 17 q^{53} - 3 q^{54} - 2 q^{55} - 8 q^{56} + q^{57} - 4 q^{58} + 15 q^{59} + 3 q^{60} + q^{61} + 2 q^{62} - 8 q^{63} + 3 q^{64} - 2 q^{66} + 3 q^{67} - 9 q^{68} + q^{69} + 8 q^{70} + 21 q^{71} + 3 q^{72} + 7 q^{73} + 10 q^{74} - 3 q^{75} - q^{76} - 3 q^{77} - 3 q^{79} - 3 q^{80} + 3 q^{81} + 5 q^{82} + 22 q^{83} + 8 q^{84} + 9 q^{85} - 9 q^{86} + 4 q^{87} + 2 q^{88} + 23 q^{89} - 3 q^{90} - q^{92} - 2 q^{93} + 21 q^{94} + q^{95} - 3 q^{96} + 10 q^{97} + 5 q^{98} + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 - 8 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 + 2 * q^11 - 3 * q^12 - 8 * q^14 + 3 * q^15 + 3 * q^16 - 9 * q^17 + 3 * q^18 - q^19 - 3 * q^20 + 8 * q^21 + 2 * q^22 - q^23 - 3 * q^24 + 3 * q^25 - 3 * q^27 - 8 * q^28 - 4 * q^29 + 3 * q^30 + 2 * q^31 + 3 * q^32 - 2 * q^33 - 9 * q^34 + 8 * q^35 + 3 * q^36 + 10 * q^37 - q^38 - 3 * q^40 + 5 * q^41 + 8 * q^42 - 9 * q^43 + 2 * q^44 - 3 * q^45 - q^46 + 21 * q^47 - 3 * q^48 + 5 * q^49 + 3 * q^50 + 9 * q^51 - 17 * q^53 - 3 * q^54 - 2 * q^55 - 8 * q^56 + q^57 - 4 * q^58 + 15 * q^59 + 3 * q^60 + q^61 + 2 * q^62 - 8 * q^63 + 3 * q^64 - 2 * q^66 + 3 * q^67 - 9 * q^68 + q^69 + 8 * q^70 + 21 * q^71 + 3 * q^72 + 7 * q^73 + 10 * q^74 - 3 * q^75 - q^76 - 3 * q^77 - 3 * q^79 - 3 * q^80 + 3 * q^81 + 5 * q^82 + 22 * q^83 + 8 * q^84 + 9 * q^85 - 9 * q^86 + 4 * q^87 + 2 * q^88 + 23 * q^89 - 3 * q^90 - q^92 - 2 * q^93 + 21 * q^94 + q^95 - 3 * q^96 + 10 * q^97 + 5 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

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