LMFDB - Embedded newform 5070.2.a.br.1.2

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:	$1$
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			1.2
Root			$-1.24698$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.55496 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.55496 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.24698 q^{11} -1.00000 q^{12} -2.55496 q^{14} +1.00000 q^{15} +1.00000 q^{16} +2.02177 q^{17} +1.00000 q^{18} +1.33513 q^{19} -1.00000 q^{20} +2.55496 q^{21} -2.24698 q^{22} +5.58211 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -2.55496 q^{28} +3.26875 q^{29} +1.00000 q^{30} +1.35690 q^{31} +1.00000 q^{32} +2.24698 q^{33} +2.02177 q^{34} +2.55496 q^{35} +1.00000 q^{36} -9.64071 q^{37} +1.33513 q^{38} -1.00000 q^{40} -1.06100 q^{41} +2.55496 q^{42} +0.137063 q^{43} -2.24698 q^{44} -1.00000 q^{45} +5.58211 q^{46} +12.7995 q^{47} -1.00000 q^{48} -0.472189 q^{49} +1.00000 q^{50} -2.02177 q^{51} -3.03684 q^{53} -1.00000 q^{54} +2.24698 q^{55} -2.55496 q^{56} -1.33513 q^{57} +3.26875 q^{58} -11.9608 q^{59} +1.00000 q^{60} -0.972853 q^{61} +1.35690 q^{62} -2.55496 q^{63} +1.00000 q^{64} +2.24698 q^{66} -8.51573 q^{67} +2.02177 q^{68} -5.58211 q^{69} +2.55496 q^{70} -2.69202 q^{71} +1.00000 q^{72} -1.48427 q^{73} -9.64071 q^{74} -1.00000 q^{75} +1.33513 q^{76} +5.74094 q^{77} -7.77479 q^{79} -1.00000 q^{80} +1.00000 q^{81} -1.06100 q^{82} -17.7289 q^{83} +2.55496 q^{84} -2.02177 q^{85} +0.137063 q^{86} -3.26875 q^{87} -2.24698 q^{88} -15.4601 q^{89} -1.00000 q^{90} +5.58211 q^{92} -1.35690 q^{93} +12.7995 q^{94} -1.33513 q^{95} -1.00000 q^{96} +7.14675 q^{97} -0.472189 q^{98} -2.24698 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 3 q^{17} + 3 q^{18} + 3 q^{19} - 3 q^{20} + 8 q^{21} - 2 q^{22} + 11 q^{23} - 3 q^{24} + 3 q^{25} - 3 q^{27} - 8 q^{28} + 2 q^{29} + 3 q^{30} + 3 q^{32} + 2 q^{33} + 3 q^{34} + 8 q^{35} + 3 q^{36} + 8 q^{37} + 3 q^{38} - 3 q^{40} - 13 q^{41} + 8 q^{42} - 5 q^{43} - 2 q^{44} - 3 q^{45} + 11 q^{46} - 7 q^{47} - 3 q^{48} + 5 q^{49} + 3 q^{50} - 3 q^{51} + 19 q^{53} - 3 q^{54} + 2 q^{55} - 8 q^{56} - 3 q^{57} + 2 q^{58} - 23 q^{59} + 3 q^{60} - 9 q^{61} - 8 q^{63} + 3 q^{64} + 2 q^{66} - 13 q^{67} + 3 q^{68} - 11 q^{69} + 8 q^{70} - 3 q^{71} + 3 q^{72} - 17 q^{73} + 8 q^{74} - 3 q^{75} + 3 q^{76} + 3 q^{77} - 25 q^{79} - 3 q^{80} + 3 q^{81} - 13 q^{82} - 20 q^{83} + 8 q^{84} - 3 q^{85} - 5 q^{86} - 2 q^{87} - 2 q^{88} - 21 q^{89} - 3 q^{90} + 11 q^{92} - 7 q^{94} - 3 q^{95} - 3 q^{96} - 6 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 + 3 * q^17 + 3 * q^18 + 3 * q^19 - 3 * q^20 + 8 * q^21 - 2 * q^22 + 11 * q^23 - 3 * q^24 + 3 * q^25 - 3 * q^27 - 8 * q^28 + 2 * q^29 + 3 * q^30 + 3 * q^32 + 2 * q^33 + 3 * q^34 + 8 * q^35 + 3 * q^36 + 8 * q^37 + 3 * q^38 - 3 * q^40 - 13 * q^41 + 8 * q^42 - 5 * q^43 - 2 * q^44 - 3 * q^45 + 11 * q^46 - 7 * q^47 - 3 * q^48 + 5 * q^49 + 3 * q^50 - 3 * q^51 + 19 * q^53 - 3 * q^54 + 2 * q^55 - 8 * q^56 - 3 * q^57 + 2 * q^58 - 23 * q^59 + 3 * q^60 - 9 * q^61 - 8 * q^63 + 3 * q^64 + 2 * q^66 - 13 * q^67 + 3 * q^68 - 11 * q^69 + 8 * q^70 - 3 * q^71 + 3 * q^72 - 17 * q^73 + 8 * q^74 - 3 * q^75 + 3 * q^76 + 3 * q^77 - 25 * q^79 - 3 * q^80 + 3 * q^81 - 13 * q^82 - 20 * q^83 + 8 * q^84 - 3 * q^85 - 5 * q^86 - 2 * q^87 - 2 * q^88 - 21 * q^89 - 3 * q^90 + 11 * q^92 - 7 * q^94 - 3 * q^95 - 3 * q^96 - 6 * 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
$7$	−2.55496	−0.965683	−0.482842	−	0.875708i	$-0.660395\pi$
$7$	−2.55496	−0.965683	−0.482842	+	0.875708i	$0.660395\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−2.24698	−0.677490	−0.338745	−	0.940878i	$-0.610002\pi$
$11$	−2.24698	−0.677490	−0.338745	+	0.940878i	$0.610002\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−2.55496	−0.682841
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	2.02177	0.490351	0.245176	−	0.969479i	$-0.421154\pi$
$17$	2.02177	0.490351	0.245176	+	0.969479i	$0.421154\pi$
$18$	1.00000	0.235702
$19$	1.33513	0.306299	0.153149	−	0.988203i	$-0.451058\pi$
$19$	1.33513	0.306299	0.153149	+	0.988203i	$0.451058\pi$
$20$	−1.00000	−0.223607
$21$	2.55496	0.557538
$22$	−2.24698	−0.479058
$23$	5.58211	1.16395	0.581975	−	0.813207i	$-0.302280\pi$
$23$	5.58211	1.16395	0.581975	+	0.813207i	$0.302280\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−2.55496	−0.482842
$29$	3.26875	0.606992	0.303496	−	0.952833i	$-0.401846\pi$
$29$	3.26875	0.606992	0.303496	+	0.952833i	$0.401846\pi$
$30$	1.00000	0.182574
$31$	1.35690	0.243706	0.121853	−	0.992548i	$-0.461116\pi$
$31$	1.35690	0.243706	0.121853	+	0.992548i	$0.461116\pi$
$32$	1.00000	0.176777
$33$	2.24698	0.391149
$34$	2.02177	0.346731
$35$	2.55496	0.431867
$36$	1.00000	0.166667
$37$	−9.64071	−1.58492	−0.792462	−	0.609922i	$-0.791201\pi$
$37$	−9.64071	−1.58492	−0.792462	+	0.609922i	$0.791201\pi$
$38$	1.33513	0.216586
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−1.06100	−0.165700	−0.0828501	−	0.996562i	$-0.526402\pi$
$41$	−1.06100	−0.165700	−0.0828501	+	0.996562i	$0.526402\pi$
$42$	2.55496	0.394239
$43$	0.137063	0.0209020	0.0104510	−	0.999945i	$-0.496673\pi$
$43$	0.137063	0.0209020	0.0104510	+	0.999945i	$0.496673\pi$
$44$	−2.24698	−0.338745
$45$	−1.00000	−0.149071
$46$	5.58211	0.823037
$47$	12.7995	1.86701	0.933503	−	0.358570i	$-0.116736\pi$
$47$	12.7995	1.86701	0.933503	+	0.358570i	$0.116736\pi$
$48$	−1.00000	−0.144338
$49$	−0.472189	−0.0674556
$50$	1.00000	0.141421
$51$	−2.02177	−0.283104
$52$	0	0
$53$	−3.03684	−0.417141	−0.208571	−	0.978007i	$-0.566881\pi$
$53$	−3.03684	−0.417141	−0.208571	+	0.978007i	$0.566881\pi$
$54$	−1.00000	−0.136083
$55$	2.24698	0.302983
$56$	−2.55496	−0.341421
$57$	−1.33513	−0.176842
$58$	3.26875	0.429208
$59$	−11.9608	−1.55716	−0.778580	−	0.627545i	$-0.784060\pi$
$59$	−11.9608	−1.55716	−0.778580	+	0.627545i	$0.784060\pi$
$60$	1.00000	0.129099
$61$	−0.972853	−0.124561	−0.0622805	−	0.998059i	$-0.519837\pi$
$61$	−0.972853	−0.124561	−0.0622805	+	0.998059i	$0.519837\pi$
$62$	1.35690	0.172326
$63$	−2.55496	−0.321894
$64$	1.00000	0.125000
$65$	0	0
$66$	2.24698	0.276584
$67$	−8.51573	−1.04036	−0.520181	−	0.854056i	$-0.674136\pi$
$67$	−8.51573	−1.04036	−0.520181	+	0.854056i	$0.674136\pi$
$68$	2.02177	0.245176
$69$	−5.58211	−0.672006
$70$	2.55496	0.305376
$71$	−2.69202	−0.319484	−0.159742	−	0.987159i	$-0.551066\pi$
$71$	−2.69202	−0.319484	−0.159742	+	0.987159i	$0.551066\pi$
$72$	1.00000	0.117851
$73$	−1.48427	−0.173721	−0.0868604	−	0.996220i	$-0.527683\pi$
$73$	−1.48427	−0.173721	−0.0868604	+	0.996220i	$0.527683\pi$
$74$	−9.64071	−1.12071
$75$	−1.00000	−0.115470
$76$	1.33513	0.153149
$77$	5.74094	0.654241
$78$	0	0
$79$	−7.77479	−0.874732	−0.437366	−	0.899284i	$-0.644089\pi$
$79$	−7.77479	−0.874732	−0.437366	+	0.899284i	$0.644089\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−1.06100	−0.117168
$83$	−17.7289	−1.94599	−0.972997	−	0.230816i	$-0.925860\pi$
$83$	−17.7289	−1.94599	−0.972997	+	0.230816i	$0.925860\pi$
$84$	2.55496	0.278769
$85$	−2.02177	−0.219292
$86$	0.137063	0.0147799
$87$	−3.26875	−0.350447
$88$	−2.24698	−0.239529
$89$	−15.4601	−1.63877	−0.819384	−	0.573245i	$-0.805684\pi$
$89$	−15.4601	−1.63877	−0.819384	+	0.573245i	$0.805684\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	5.58211	0.581975
$93$	−1.35690	−0.140704
$94$	12.7995	1.32017
$95$	−1.33513	−0.136981
$96$	−1.00000	−0.102062
$97$	7.14675	0.725643	0.362821	−	0.931859i	$-0.381814\pi$
$97$	7.14675	0.725643	0.362821	+	0.931859i	$0.381814\pi$
$98$	−0.472189	−0.0476983
$99$	−2.24698	−0.225830
$100$	1.00000	0.100000
$101$	−4.71379	−0.469040	−0.234520	−	0.972111i	$-0.575352\pi$
$101$	−4.71379	−0.469040	−0.234520	+	0.972111i	$0.575352\pi$
$102$	−2.02177	−0.200185
$103$	12.0858	1.19084	0.595422	−	0.803413i	$-0.296985\pi$
$103$	12.0858	1.19084	0.595422	+	0.803413i	$0.296985\pi$
$104$	0	0
$105$	−2.55496	−0.249338
$106$	−3.03684	−0.294964
$107$	−8.01507	−0.774846	−0.387423	−	0.921902i	$-0.626635\pi$
$107$	−8.01507	−0.774846	−0.387423	+	0.921902i	$0.626635\pi$
$108$	−1.00000	−0.0962250
$109$	9.17390	0.878700	0.439350	−	0.898316i	$-0.355209\pi$
$109$	9.17390	0.878700	0.439350	+	0.898316i	$0.355209\pi$
$110$	2.24698	0.214241
$111$	9.64071	0.915056
$112$	−2.55496	−0.241421
$113$	5.24698	0.493594	0.246797	−	0.969067i	$-0.420622\pi$
$113$	5.24698	0.493594	0.246797	+	0.969067i	$0.420622\pi$
$114$	−1.33513	−0.125046
$115$	−5.58211	−0.520534
$116$	3.26875	0.303496
$117$	0	0
$118$	−11.9608	−1.10108
$119$	−5.16554	−0.473524
$120$	1.00000	0.0912871
$121$	−5.95108	−0.541008
$122$	−0.972853	−0.0880780
$123$	1.06100	0.0956671
$124$	1.35690	0.121853
$125$	−1.00000	−0.0894427
$126$	−2.55496	−0.227614
$127$	−12.6843	−1.12555	−0.562773	−	0.826612i	$-0.690265\pi$
$127$	−12.6843	−1.12555	−0.562773	+	0.826612i	$0.690265\pi$
$128$	1.00000	0.0883883
$129$	−0.137063	−0.0120678
$130$	0	0
$131$	−14.5332	−1.26977	−0.634885	−	0.772606i	$-0.718953\pi$
$131$	−14.5332	−1.26977	−0.634885	+	0.772606i	$0.718953\pi$
$132$	2.24698	0.195574
$133$	−3.41119	−0.295788
$134$	−8.51573	−0.735647
$135$	1.00000	0.0860663
$136$	2.02177	0.173365
$137$	−5.06100	−0.432390	−0.216195	−	0.976350i	$-0.569365\pi$
$137$	−5.06100	−0.432390	−0.216195	+	0.976350i	$0.569365\pi$
$138$	−5.58211	−0.475180
$139$	−19.9638	−1.69330	−0.846652	−	0.532147i	$-0.821385\pi$
$139$	−19.9638	−1.69330	−0.846652	+	0.532147i	$0.821385\pi$
$140$	2.55496	0.215933
$141$	−12.7995	−1.07792
$142$	−2.69202	−0.225909
$143$	0	0
$144$	1.00000	0.0833333
$145$	−3.26875	−0.271455
$146$	−1.48427	−0.122839
$147$	0.472189	0.0389455
$148$	−9.64071	−0.792462
$149$	−3.33513	−0.273224	−0.136612	−	0.990625i	$-0.543621\pi$
$149$	−3.33513	−0.273224	−0.136612	+	0.990625i	$0.543621\pi$
$150$	−1.00000	−0.0816497
$151$	15.6625	1.27459	0.637297	−	0.770618i	$-0.280052\pi$
$151$	15.6625	1.27459	0.637297	+	0.770618i	$0.280052\pi$
$152$	1.33513	0.108293
$153$	2.02177	0.163450
$154$	5.74094	0.462618
$155$	−1.35690	−0.108988
$156$	0	0
$157$	2.99761	0.239235	0.119618	−	0.992820i	$-0.461833\pi$
$157$	2.99761	0.239235	0.119618	+	0.992820i	$0.461833\pi$
$158$	−7.77479	−0.618529
$159$	3.03684	0.240837
$160$	−1.00000	−0.0790569
$161$	−14.2620	−1.12401
$162$	1.00000	0.0785674
$163$	1.46681	0.114890	0.0574448	−	0.998349i	$-0.481705\pi$
$163$	1.46681	0.114890	0.0574448	+	0.998349i	$0.481705\pi$
$164$	−1.06100	−0.0828501
$165$	−2.24698	−0.174927
$166$	−17.7289	−1.37603
$167$	−1.13706	−0.0879886	−0.0439943	−	0.999032i	$-0.514008\pi$
$167$	−1.13706	−0.0879886	−0.0439943	+	0.999032i	$0.514008\pi$
$168$	2.55496	0.197119
$169$	0	0
$170$	−2.02177	−0.155063
$171$	1.33513	0.102100
$172$	0.137063	0.0104510
$173$	0.983607	0.0747822	0.0373911	−	0.999301i	$-0.488095\pi$
$173$	0.983607	0.0747822	0.0373911	+	0.999301i	$0.488095\pi$
$174$	−3.26875	−0.247803
$175$	−2.55496	−0.193137
$176$	−2.24698	−0.169372
$177$	11.9608	0.899027
$178$	−15.4601	−1.15878
$179$	9.86592	0.737414	0.368707	−	0.929546i	$-0.379801\pi$
$179$	9.86592	0.737414	0.368707	+	0.929546i	$0.379801\pi$
$180$	−1.00000	−0.0745356
$181$	−3.66487	−0.272408	−0.136204	−	0.990681i	$-0.543490\pi$
$181$	−3.66487	−0.272408	−0.136204	+	0.990681i	$0.543490\pi$
$182$	0	0
$183$	0.972853	0.0719154
$184$	5.58211	0.411518
$185$	9.64071	0.708799
$186$	−1.35690	−0.0994924
$187$	−4.54288	−0.332208
$188$	12.7995	0.933503
$189$	2.55496	0.185846
$190$	−1.33513	−0.0968602
$191$	−12.2252	−0.884585	−0.442293	−	0.896871i	$-0.645835\pi$
$191$	−12.2252	−0.884585	−0.442293	+	0.896871i	$0.645835\pi$
$192$	−1.00000	−0.0721688
$193$	10.5579	0.759977	0.379989	−	0.924991i	$-0.375928\pi$
$193$	10.5579	0.759977	0.379989	+	0.924991i	$0.375928\pi$
$194$	7.14675	0.513107
$195$	0	0
$196$	−0.472189	−0.0337278
$197$	−5.46011	−0.389017	−0.194508	−	0.980901i	$-0.562311\pi$
$197$	−5.46011	−0.389017	−0.194508	+	0.980901i	$0.562311\pi$
$198$	−2.24698	−0.159686
$199$	5.00969	0.355127	0.177564	−	0.984109i	$-0.443178\pi$
$199$	5.00969	0.355127	0.177564	+	0.984109i	$0.443178\pi$
$200$	1.00000	0.0707107
$201$	8.51573	0.600653
$202$	−4.71379	−0.331661
$203$	−8.35152	−0.586162
$204$	−2.02177	−0.141552
$205$	1.06100	0.0741034
$206$	12.0858	0.842054
$207$	5.58211	0.387983
$208$	0	0
$209$	−3.00000	−0.207514
$210$	−2.55496	−0.176309
$211$	−9.38404	−0.646024	−0.323012	−	0.946395i	$-0.604695\pi$
$211$	−9.38404	−0.646024	−0.323012	+	0.946395i	$0.604695\pi$
$212$	−3.03684	−0.208571
$213$	2.69202	0.184454
$214$	−8.01507	−0.547899
$215$	−0.137063	−0.00934764
$216$	−1.00000	−0.0680414
$217$	−3.46681	−0.235343
$218$	9.17390	0.621335
$219$	1.48427	0.100298
$220$	2.24698	0.151491
$221$	0	0
$222$	9.64071	0.647042
$223$	−26.5773	−1.77975	−0.889874	−	0.456205i	$-0.849208\pi$
$223$	−26.5773	−1.77975	−0.889874	+	0.456205i	$0.849208\pi$
$224$	−2.55496	−0.170710
$225$	1.00000	0.0666667
$226$	5.24698	0.349024
$227$	−22.2271	−1.47527	−0.737633	−	0.675202i	$-0.764057\pi$
$227$	−22.2271	−1.47527	−0.737633	+	0.675202i	$0.764057\pi$
$228$	−1.33513	−0.0884209
$229$	19.1782	1.26733	0.633666	−	0.773607i	$-0.281549\pi$
$229$	19.1782	1.26733	0.633666	+	0.773607i	$0.281549\pi$
$230$	−5.58211	−0.368073
$231$	−5.74094	−0.377726
$232$	3.26875	0.214604
$233$	19.5483	1.28065	0.640324	−	0.768105i	$-0.278800\pi$
$233$	19.5483	1.28065	0.640324	+	0.768105i	$0.278800\pi$
$234$	0	0
$235$	−12.7995	−0.834950
$236$	−11.9608	−0.778580
$237$	7.77479	0.505027
$238$	−5.16554	−0.334832
$239$	−12.8194	−0.829218	−0.414609	−	0.910000i	$-0.636082\pi$
$239$	−12.8194	−0.829218	−0.414609	+	0.910000i	$0.636082\pi$
$240$	1.00000	0.0645497
$241$	8.95407	0.576782	0.288391	−	0.957513i	$-0.406880\pi$
$241$	8.95407	0.576782	0.288391	+	0.957513i	$0.406880\pi$
$242$	−5.95108	−0.382550
$243$	−1.00000	−0.0641500
$244$	−0.972853	−0.0622805
$245$	0.472189	0.0301670
$246$	1.06100	0.0676468
$247$	0	0
$248$	1.35690	0.0861630
$249$	17.7289	1.12352
$250$	−1.00000	−0.0632456
$251$	−0.276520	−0.0174538	−0.00872688	−	0.999962i	$-0.502778\pi$
$251$	−0.276520	−0.0174538	−0.00872688	+	0.999962i	$0.502778\pi$
$252$	−2.55496	−0.160947
$253$	−12.5429	−0.788564
$254$	−12.6843	−0.795881
$255$	2.02177	0.126608
$256$	1.00000	0.0625000
$257$	9.77048	0.609466	0.304733	−	0.952438i	$-0.401433\pi$
$257$	9.77048	0.609466	0.304733	+	0.952438i	$0.401433\pi$
$258$	−0.137063	−0.00853319
$259$	24.6316	1.53053
$260$	0	0
$261$	3.26875	0.202331
$262$	−14.5332	−0.897863
$263$	−12.7071	−0.783553	−0.391776	−	0.920061i	$-0.628139\pi$
$263$	−12.7071	−0.783553	−0.391776	+	0.920061i	$0.628139\pi$
$264$	2.24698	0.138292
$265$	3.03684	0.186551
$266$	−3.41119	−0.209153
$267$	15.4601	0.946143
$268$	−8.51573	−0.520181
$269$	9.47757	0.577857	0.288929	−	0.957351i	$-0.406701\pi$
$269$	9.47757	0.577857	0.288929	+	0.957351i	$0.406701\pi$
$270$	1.00000	0.0608581
$271$	−20.7657	−1.26143	−0.630713	−	0.776016i	$-0.717237\pi$
$271$	−20.7657	−1.26143	−0.630713	+	0.776016i	$0.717237\pi$
$272$	2.02177	0.122588
$273$	0	0
$274$	−5.06100	−0.305746
$275$	−2.24698	−0.135498
$276$	−5.58211	−0.336003
$277$	15.0911	0.906738	0.453369	−	0.891323i	$-0.350222\pi$
$277$	15.0911	0.906738	0.453369	+	0.891323i	$0.350222\pi$
$278$	−19.9638	−1.19735
$279$	1.35690	0.0812352
$280$	2.55496	0.152688
$281$	17.6015	1.05002	0.525008	−	0.851097i	$-0.324062\pi$
$281$	17.6015	1.05002	0.525008	+	0.851097i	$0.324062\pi$
$282$	−12.7995	−0.762202
$283$	−2.07606	−0.123409	−0.0617046	−	0.998094i	$-0.519654\pi$
$283$	−2.07606	−0.123409	−0.0617046	+	0.998094i	$0.519654\pi$
$284$	−2.69202	−0.159742
$285$	1.33513	0.0790860
$286$	0	0
$287$	2.71081	0.160014
$288$	1.00000	0.0589256
$289$	−12.9124	−0.759556
$290$	−3.26875	−0.191948
$291$	−7.14675	−0.418950
$292$	−1.48427	−0.0868604
$293$	26.0930	1.52437	0.762186	−	0.647358i	$-0.224126\pi$
$293$	26.0930	1.52437	0.762186	+	0.647358i	$0.224126\pi$
$294$	0.472189	0.0275386
$295$	11.9608	0.696383
$296$	−9.64071	−0.560355
$297$	2.24698	0.130383
$298$	−3.33513	−0.193199
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−0.350191	−0.0201847
$302$	15.6625	0.901275
$303$	4.71379	0.270800
$304$	1.33513	0.0765747
$305$	0.972853	0.0557054
$306$	2.02177	0.115577
$307$	15.8092	0.902281	0.451140	−	0.892453i	$-0.351017\pi$
$307$	15.8092	0.902281	0.451140	+	0.892453i	$0.351017\pi$
$308$	5.74094	0.327120
$309$	−12.0858	−0.687534
$310$	−1.35690	−0.0770665
$311$	−23.2121	−1.31624	−0.658118	−	0.752915i	$-0.728647\pi$
$311$	−23.2121	−1.31624	−0.658118	+	0.752915i	$0.728647\pi$
$312$	0	0
$313$	28.5297	1.61260	0.806298	−	0.591510i	$-0.201468\pi$
$313$	28.5297	1.61260	0.806298	+	0.591510i	$0.201468\pi$
$314$	2.99761	0.169165
$315$	2.55496	0.143956
$316$	−7.77479	−0.437366
$317$	−0.709480	−0.0398484	−0.0199242	−	0.999801i	$-0.506342\pi$
$317$	−0.709480	−0.0398484	−0.0199242	+	0.999801i	$0.506342\pi$
$318$	3.03684	0.170297
$319$	−7.34481	−0.411231
$320$	−1.00000	−0.0559017
$321$	8.01507	0.447357
$322$	−14.2620	−0.794793
$323$	2.69932	0.150194
$324$	1.00000	0.0555556
$325$	0	0
$326$	1.46681	0.0812392
$327$	−9.17390	−0.507318
$328$	−1.06100	−0.0585839
$329$	−32.7023	−1.80294
$330$	−2.24698	−0.123692
$331$	−22.8605	−1.25653	−0.628265	−	0.778000i	$-0.716234\pi$
$331$	−22.8605	−1.25653	−0.628265	+	0.778000i	$0.716234\pi$
$332$	−17.7289	−0.972997
$333$	−9.64071	−0.528308
$334$	−1.13706	−0.0622173
$335$	8.51573	0.465264
$336$	2.55496	0.139384
$337$	33.0629	1.80105	0.900526	−	0.434802i	$-0.143182\pi$
$337$	33.0629	1.80105	0.900526	+	0.434802i	$0.143182\pi$
$338$	0	0
$339$	−5.24698	−0.284977
$340$	−2.02177	−0.109646
$341$	−3.04892	−0.165108
$342$	1.33513	0.0721953
$343$	19.0911	1.03082
$344$	0.137063	0.00738996
$345$	5.58211	0.300530
$346$	0.983607	0.0528790
$347$	23.4808	1.26052	0.630258	−	0.776386i	$-0.282949\pi$
$347$	23.4808	1.26052	0.630258	+	0.776386i	$0.282949\pi$
$348$	−3.26875	−0.175223
$349$	−35.9004	−1.92170	−0.960851	−	0.277065i	$-0.910638\pi$
$349$	−35.9004	−1.92170	−0.960851	+	0.277065i	$0.910638\pi$
$350$	−2.55496	−0.136568
$351$	0	0
$352$	−2.24698	−0.119764
$353$	−31.0562	−1.65296	−0.826478	−	0.562969i	$-0.809659\pi$
$353$	−31.0562	−1.65296	−0.826478	+	0.562969i	$0.809659\pi$
$354$	11.9608	0.635708
$355$	2.69202	0.142878
$356$	−15.4601	−0.819384
$357$	5.16554	0.273389
$358$	9.86592	0.521430
$359$	6.14675	0.324413	0.162207	−	0.986757i	$-0.448139\pi$
$359$	6.14675	0.324413	0.162207	+	0.986757i	$0.448139\pi$
$360$	−1.00000	−0.0527046
$361$	−17.2174	−0.906181
$362$	−3.66487	−0.192622
$363$	5.95108	0.312351
$364$	0	0
$365$	1.48427	0.0776903
$366$	0.972853	0.0508518
$367$	−26.5163	−1.38414	−0.692070	−	0.721830i	$-0.743301\pi$
$367$	−26.5163	−1.38414	−0.692070	+	0.721830i	$0.743301\pi$
$368$	5.58211	0.290987
$369$	−1.06100	−0.0552334
$370$	9.64071	0.501197
$371$	7.75899	0.402827
$372$	−1.35690	−0.0703518
$373$	−27.0616	−1.40120	−0.700598	−	0.713556i	$-0.747083\pi$
$373$	−27.0616	−1.40120	−0.700598	+	0.713556i	$0.747083\pi$
$374$	−4.54288	−0.234907
$375$	1.00000	0.0516398
$376$	12.7995	0.660086
$377$	0	0
$378$	2.55496	0.131413
$379$	16.7313	0.859427	0.429713	−	0.902965i	$-0.358615\pi$
$379$	16.7313	0.859427	0.429713	+	0.902965i	$0.358615\pi$
$380$	−1.33513	−0.0684905
$381$	12.6843	0.649834
$382$	−12.2252	−0.625496
$383$	−34.3569	−1.75556	−0.877778	−	0.479068i	$-0.840975\pi$
$383$	−34.3569	−1.75556	−0.877778	+	0.479068i	$0.840975\pi$
$384$	−1.00000	−0.0510310
$385$	−5.74094	−0.292585
$386$	10.5579	0.537385
$387$	0.137063	0.00696732
$388$	7.14675	0.362821
$389$	11.6732	0.591857	0.295928	−	0.955210i	$-0.404371\pi$
$389$	11.6732	0.591857	0.295928	+	0.955210i	$0.404371\pi$
$390$	0	0
$391$	11.2857	0.570744
$392$	−0.472189	−0.0238491
$393$	14.5332	0.733102
$394$	−5.46011	−0.275076
$395$	7.77479	0.391192
$396$	−2.24698	−0.112915
$397$	−12.6431	−0.634539	−0.317270	−	0.948335i	$-0.602766\pi$
$397$	−12.6431	−0.634539	−0.317270	+	0.948335i	$0.602766\pi$
$398$	5.00969	0.251113
$399$	3.41119	0.170773
$400$	1.00000	0.0500000
$401$	−29.7724	−1.48676	−0.743381	−	0.668868i	$-0.766779\pi$
$401$	−29.7724	−1.48676	−0.743381	+	0.668868i	$0.766779\pi$
$402$	8.51573	0.424726
$403$	0	0
$404$	−4.71379	−0.234520
$405$	−1.00000	−0.0496904
$406$	−8.35152	−0.414479
$407$	21.6625	1.07377
$408$	−2.02177	−0.100093
$409$	−5.61117	−0.277455	−0.138727	−	0.990331i	$-0.544301\pi$
$409$	−5.61117	−0.277455	−0.138727	+	0.990331i	$0.544301\pi$
$410$	1.06100	0.0523990
$411$	5.06100	0.249641
$412$	12.0858	0.595422
$413$	30.5593	1.50372
$414$	5.58211	0.274346
$415$	17.7289	0.870275
$416$	0	0
$417$	19.9638	0.977629
$418$	−3.00000	−0.146735
$419$	−1.24937	−0.0610358	−0.0305179	−	0.999534i	$-0.509716\pi$
$419$	−1.24937	−0.0610358	−0.0305179	+	0.999534i	$0.509716\pi$
$420$	−2.55496	−0.124669
$421$	−2.76749	−0.134879	−0.0674397	−	0.997723i	$-0.521483\pi$
$421$	−2.76749	−0.134879	−0.0674397	+	0.997723i	$0.521483\pi$
$422$	−9.38404	−0.456808
$423$	12.7995	0.622335
$424$	−3.03684	−0.147482
$425$	2.02177	0.0980703
$426$	2.69202	0.130429
$427$	2.48560	0.120287
$428$	−8.01507	−0.387423
$429$	0	0
$430$	−0.137063	−0.00660978
$431$	−19.1793	−0.923833	−0.461917	−	0.886923i	$-0.652838\pi$
$431$	−19.1793	−0.923833	−0.461917	+	0.886923i	$0.652838\pi$
$432$	−1.00000	−0.0481125
$433$	36.9051	1.77355	0.886774	−	0.462203i	$-0.152941\pi$
$433$	36.9051	1.77355	0.886774	+	0.462203i	$0.152941\pi$
$434$	−3.46681	−0.166412
$435$	3.26875	0.156725
$436$	9.17390	0.439350
$437$	7.45281	0.356516
$438$	1.48427	0.0709212
$439$	14.6213	0.697838	0.348919	−	0.937153i	$-0.386549\pi$
$439$	14.6213	0.697838	0.348919	+	0.937153i	$0.386549\pi$
$440$	2.24698	0.107121
$441$	−0.472189	−0.0224852
$442$	0	0
$443$	−35.1377	−1.66944	−0.834720	−	0.550674i	$-0.814371\pi$
$443$	−35.1377	−1.66944	−0.834720	+	0.550674i	$0.814371\pi$
$444$	9.64071	0.457528
$445$	15.4601	0.732879
$446$	−26.5773	−1.25847
$447$	3.33513	0.157746
$448$	−2.55496	−0.120710
$449$	6.67994	0.315246	0.157623	−	0.987499i	$-0.449617\pi$
$449$	6.67994	0.315246	0.157623	+	0.987499i	$0.449617\pi$
$450$	1.00000	0.0471405
$451$	2.38404	0.112260
$452$	5.24698	0.246797
$453$	−15.6625	−0.735888
$454$	−22.2271	−1.04317
$455$	0	0
$456$	−1.33513	−0.0625230
$457$	22.9963	1.07572	0.537860	−	0.843034i	$-0.319233\pi$
$457$	22.9963	1.07572	0.537860	+	0.843034i	$0.319233\pi$
$458$	19.1782	0.896139
$459$	−2.02177	−0.0943682
$460$	−5.58211	−0.260267
$461$	−33.9705	−1.58216	−0.791081	−	0.611711i	$-0.790481\pi$
$461$	−33.9705	−1.58216	−0.791081	+	0.611711i	$0.790481\pi$
$462$	−5.74094	−0.267093
$463$	−26.9594	−1.25291	−0.626456	−	0.779457i	$-0.715495\pi$
$463$	−26.9594	−1.25291	−0.626456	+	0.779457i	$0.715495\pi$
$464$	3.26875	0.151748
$465$	1.35690	0.0629245
$466$	19.5483	0.905555
$467$	−25.0411	−1.15877	−0.579383	−	0.815055i	$-0.696706\pi$
$467$	−25.0411	−1.15877	−0.579383	+	0.815055i	$0.696706\pi$
$468$	0	0
$469$	21.7573	1.00466
$470$	−12.7995	−0.590399
$471$	−2.99761	−0.138122
$472$	−11.9608	−0.550539
$473$	−0.307979	−0.0141609
$474$	7.77479	0.357108
$475$	1.33513	0.0612598
$476$	−5.16554	−0.236762
$477$	−3.03684	−0.139047
$478$	−12.8194	−0.586346
$479$	−7.21983	−0.329883	−0.164941	−	0.986303i	$-0.552743\pi$
$479$	−7.21983	−0.329883	−0.164941	+	0.986303i	$0.552743\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	8.95407	0.407847
$483$	14.2620	0.648946
$484$	−5.95108	−0.270504
$485$	−7.14675	−0.324517
$486$	−1.00000	−0.0453609
$487$	24.1497	1.09433	0.547164	−	0.837025i	$-0.315707\pi$
$487$	24.1497	1.09433	0.547164	+	0.837025i	$0.315707\pi$
$488$	−0.972853	−0.0440390
$489$	−1.46681	−0.0663315
$490$	0.472189	0.0213313
$491$	−19.7259	−0.890216	−0.445108	−	0.895477i	$-0.646835\pi$
$491$	−19.7259	−0.890216	−0.445108	+	0.895477i	$0.646835\pi$
$492$	1.06100	0.0478335
$493$	6.60866	0.297639
$494$	0	0
$495$	2.24698	0.100994
$496$	1.35690	0.0609264
$497$	6.87800	0.308521
$498$	17.7289	0.794449
$499$	35.3599	1.58293	0.791463	−	0.611217i	$-0.209320\pi$
$499$	35.3599	1.58293	0.791463	+	0.611217i	$0.209320\pi$
$500$	−1.00000	−0.0447214
$501$	1.13706	0.0508002
$502$	−0.276520	−0.0123417
$503$	14.3558	0.640095	0.320047	−	0.947402i	$-0.396301\pi$
$503$	14.3558	0.640095	0.320047	+	0.947402i	$0.396301\pi$
$504$	−2.55496	−0.113807
$505$	4.71379	0.209761
$506$	−12.5429	−0.557599
$507$	0	0
$508$	−12.6843	−0.562773
$509$	−21.4426	−0.950429	−0.475214	−	0.879870i	$-0.657629\pi$
$509$	−21.4426	−0.950429	−0.475214	+	0.879870i	$0.657629\pi$
$510$	2.02177	0.0895255
$511$	3.79225	0.167759
$512$	1.00000	0.0441942
$513$	−1.33513	−0.0589472
$514$	9.77048	0.430957
$515$	−12.0858	−0.532562
$516$	−0.137063	−0.00603388
$517$	−28.7603	−1.26488
$518$	24.6316	1.08225
$519$	−0.983607	−0.0431755
$520$	0	0
$521$	9.58748	0.420035	0.210018	−	0.977698i	$-0.432648\pi$
$521$	9.58748	0.420035	0.210018	+	0.977698i	$0.432648\pi$
$522$	3.26875	0.143069
$523$	32.5241	1.42218	0.711090	−	0.703101i	$-0.248202\pi$
$523$	32.5241	1.42218	0.711090	+	0.703101i	$0.248202\pi$
$524$	−14.5332	−0.634885
$525$	2.55496	0.111508
$526$	−12.7071	−0.554055
$527$	2.74333	0.119501
$528$	2.24698	0.0977872
$529$	8.15990	0.354778
$530$	3.03684	0.131912
$531$	−11.9608	−0.519053
$532$	−3.41119	−0.147894
$533$	0	0
$534$	15.4601	0.669024
$535$	8.01507	0.346521
$536$	−8.51573	−0.367823
$537$	−9.86592	−0.425746
$538$	9.47757	0.408607
$539$	1.06100	0.0457005
$540$	1.00000	0.0430331
$541$	−27.5381	−1.18395	−0.591977	−	0.805954i	$-0.701653\pi$
$541$	−27.5381	−1.18395	−0.591977	+	0.805954i	$0.701653\pi$
$542$	−20.7657	−0.891963
$543$	3.66487	0.157275
$544$	2.02177	0.0866827
$545$	−9.17390	−0.392967
$546$	0	0
$547$	−18.0670	−0.772488	−0.386244	−	0.922397i	$-0.626228\pi$
$547$	−18.0670	−0.772488	−0.386244	+	0.922397i	$0.626228\pi$
$548$	−5.06100	−0.216195
$549$	−0.972853	−0.0415204
$550$	−2.24698	−0.0958115
$551$	4.36419	0.185921
$552$	−5.58211	−0.237590
$553$	19.8643	0.844714
$554$	15.0911	0.641161
$555$	−9.64071	−0.409225
$556$	−19.9638	−0.846652
$557$	33.5609	1.42202	0.711011	−	0.703181i	$-0.248238\pi$
$557$	33.5609	1.42202	0.711011	+	0.703181i	$0.248238\pi$
$558$	1.35690	0.0574420
$559$	0	0
$560$	2.55496	0.107967
$561$	4.54288	0.191800
$562$	17.6015	0.742474
$563$	12.6974	0.535132	0.267566	−	0.963540i	$-0.413781\pi$
$563$	12.6974	0.535132	0.267566	+	0.963540i	$0.413781\pi$
$564$	−12.7995	−0.538958
$565$	−5.24698	−0.220742
$566$	−2.07606	−0.0872635
$567$	−2.55496	−0.107298
$568$	−2.69202	−0.112955
$569$	30.1317	1.26319	0.631593	−	0.775300i	$-0.282401\pi$
$569$	30.1317	1.26319	0.631593	+	0.775300i	$0.282401\pi$
$570$	1.33513	0.0559223
$571$	0.693349	0.0290158	0.0145079	−	0.999895i	$-0.495382\pi$
$571$	0.693349	0.0290158	0.0145079	+	0.999895i	$0.495382\pi$
$572$	0	0
$573$	12.2252	0.510715
$574$	2.71081	0.113147
$575$	5.58211	0.232790
$576$	1.00000	0.0416667
$577$	−34.6983	−1.44451	−0.722254	−	0.691628i	$-0.756894\pi$
$577$	−34.6983	−1.44451	−0.722254	+	0.691628i	$0.756894\pi$
$578$	−12.9124	−0.537087
$579$	−10.5579	−0.438773
$580$	−3.26875	−0.135727
$581$	45.2965	1.87921
$582$	−7.14675	−0.296242
$583$	6.82371	0.282609
$584$	−1.48427	−0.0614196
$585$	0	0
$586$	26.0930	1.07789
$587$	−0.891149	−0.0367816	−0.0183908	−	0.999831i	$-0.505854\pi$
$587$	−0.891149	−0.0367816	−0.0183908	+	0.999831i	$0.505854\pi$
$588$	0.472189	0.0194727
$589$	1.81163	0.0746468
$590$	11.9608	0.492417
$591$	5.46011	0.224599
$592$	−9.64071	−0.396231
$593$	−34.7439	−1.42676	−0.713381	−	0.700776i	$-0.752837\pi$
$593$	−34.7439	−1.42676	−0.713381	+	0.700776i	$0.752837\pi$
$594$	2.24698	0.0921947
$595$	5.16554	0.211766
$596$	−3.33513	−0.136612
$597$	−5.00969	−0.205033
$598$	0	0
$599$	10.4843	0.428376	0.214188	−	0.976792i	$-0.431290\pi$
$599$	10.4843	0.428376	0.214188	+	0.976792i	$0.431290\pi$
$600$	−1.00000	−0.0408248
$601$	−2.28813	−0.0933347	−0.0466673	−	0.998910i	$-0.514860\pi$
$601$	−2.28813	−0.0933347	−0.0466673	+	0.998910i	$0.514860\pi$
$602$	−0.350191	−0.0142727
$603$	−8.51573	−0.346787
$604$	15.6625	0.637297
$605$	5.95108	0.241946
$606$	4.71379	0.191485
$607$	0.0499823	0.00202872	0.00101436	−	0.999999i	$-0.499677\pi$
$607$	0.0499823	0.00202872	0.00101436	+	0.999999i	$0.499677\pi$
$608$	1.33513	0.0541465
$609$	8.35152	0.338421
$610$	0.972853	0.0393897
$611$	0	0
$612$	2.02177	0.0817252
$613$	41.2737	1.66703	0.833514	−	0.552499i	$-0.186326\pi$
$613$	41.2737	1.66703	0.833514	+	0.552499i	$0.186326\pi$
$614$	15.8092	0.638009
$615$	−1.06100	−0.0427836
$616$	5.74094	0.231309
$617$	−2.35450	−0.0947887	−0.0473944	−	0.998876i	$-0.515092\pi$
$617$	−2.35450	−0.0947887	−0.0473944	+	0.998876i	$0.515092\pi$
$618$	−12.0858	−0.486160
$619$	30.0097	1.20619	0.603096	−	0.797669i	$-0.293934\pi$
$619$	30.0097	1.20619	0.603096	+	0.797669i	$0.293934\pi$
$620$	−1.35690	−0.0544942
$621$	−5.58211	−0.224002
$622$	−23.2121	−0.930719
$623$	39.4999	1.58253
$624$	0	0
$625$	1.00000	0.0400000
$626$	28.5297	1.14028
$627$	3.00000	0.119808
$628$	2.99761	0.119618
$629$	−19.4913	−0.777169
$630$	2.55496	0.101792
$631$	−35.6209	−1.41804	−0.709022	−	0.705186i	$-0.750863\pi$
$631$	−35.6209	−1.41804	−0.709022	+	0.705186i	$0.750863\pi$
$632$	−7.77479	−0.309265
$633$	9.38404	0.372982
$634$	−0.709480	−0.0281770
$635$	12.6843	0.503359
$636$	3.03684	0.120418
$637$	0	0
$638$	−7.34481	−0.290784
$639$	−2.69202	−0.106495
$640$	−1.00000	−0.0395285
$641$	−1.93230	−0.0763211	−0.0381606	−	0.999272i	$-0.512150\pi$
$641$	−1.93230	−0.0763211	−0.0381606	+	0.999272i	$0.512150\pi$
$642$	8.01507	0.316329
$643$	31.8812	1.25727	0.628637	−	0.777699i	$-0.283613\pi$
$643$	31.8812	1.25727	0.628637	+	0.777699i	$0.283613\pi$
$644$	−14.2620	−0.562003
$645$	0.137063	0.00539686
$646$	2.69932	0.106203
$647$	34.1527	1.34268	0.671341	−	0.741149i	$-0.265719\pi$
$647$	34.1527	1.34268	0.671341	+	0.741149i	$0.265719\pi$
$648$	1.00000	0.0392837
$649$	26.8756	1.05496
$650$	0	0
$651$	3.46681	0.135875
$652$	1.46681	0.0574448
$653$	11.7909	0.461414	0.230707	−	0.973023i	$-0.425896\pi$
$653$	11.7909	0.461414	0.230707	+	0.973023i	$0.425896\pi$
$654$	−9.17390	−0.358728
$655$	14.5332	0.567859
$656$	−1.06100	−0.0414250
$657$	−1.48427	−0.0579069
$658$	−32.7023	−1.27487
$659$	−1.19806	−0.0466699	−0.0233349	−	0.999728i	$-0.507428\pi$
$659$	−1.19806	−0.0466699	−0.0233349	+	0.999728i	$0.507428\pi$
$660$	−2.24698	−0.0874636
$661$	−2.95348	−0.114877	−0.0574384	−	0.998349i	$-0.518293\pi$
$661$	−2.95348	−0.114877	−0.0574384	+	0.998349i	$0.518293\pi$
$662$	−22.8605	−0.888500
$663$	0	0
$664$	−17.7289	−0.688013
$665$	3.41119	0.132280
$666$	−9.64071	−0.373570
$667$	18.2465	0.706508
$668$	−1.13706	−0.0439943
$669$	26.5773	1.02754
$670$	8.51573	0.328991
$671$	2.18598	0.0843888
$672$	2.55496	0.0985596
$673$	32.4771	1.25190	0.625950	−	0.779863i	$-0.284711\pi$
$673$	32.4771	1.25190	0.625950	+	0.779863i	$0.284711\pi$
$674$	33.0629	1.27354
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	11.8418	0.455116	0.227558	−	0.973765i	$-0.426926\pi$
$677$	11.8418	0.455116	0.227558	+	0.973765i	$0.426926\pi$
$678$	−5.24698	−0.201509
$679$	−18.2597	−0.700741
$680$	−2.02177	−0.0775314
$681$	22.2271	0.851745
$682$	−3.04892	−0.116749
$683$	−7.34481	−0.281042	−0.140521	−	0.990078i	$-0.544878\pi$
$683$	−7.34481	−0.281042	−0.140521	+	0.990078i	$0.544878\pi$
$684$	1.33513	0.0510498
$685$	5.06100	0.193371
$686$	19.0911	0.728903
$687$	−19.1782	−0.731694
$688$	0.137063	0.00522549
$689$	0	0
$690$	5.58211	0.212507
$691$	32.8678	1.25035	0.625176	−	0.780484i	$-0.285027\pi$
$691$	32.8678	1.25035	0.625176	+	0.780484i	$0.285027\pi$
$692$	0.983607	0.0373911
$693$	5.74094	0.218080
$694$	23.4808	0.891319
$695$	19.9638	0.757268
$696$	−3.26875	−0.123902
$697$	−2.14510	−0.0812513
$698$	−35.9004	−1.35885
$699$	−19.5483	−0.739383
$700$	−2.55496	−0.0965683
$701$	−48.7133	−1.83988	−0.919938	−	0.392063i	$-0.871761\pi$
$701$	−48.7133	−1.83988	−0.919938	+	0.392063i	$0.871761\pi$
$702$	0	0
$703$	−12.8716	−0.485460
$704$	−2.24698	−0.0846862
$705$	12.7995	0.482059
$706$	−31.0562	−1.16882
$707$	12.0435	0.452944
$708$	11.9608	0.449513
$709$	29.6209	1.11243	0.556217	−	0.831037i	$-0.312252\pi$
$709$	29.6209	1.11243	0.556217	+	0.831037i	$0.312252\pi$
$710$	2.69202	0.101030
$711$	−7.77479	−0.291577
$712$	−15.4601	−0.579392
$713$	7.57434	0.283661
$714$	5.16554	0.193315
$715$	0	0
$716$	9.86592	0.368707
$717$	12.8194	0.478749
$718$	6.14675	0.229395
$719$	−33.5599	−1.25157	−0.625786	−	0.779995i	$-0.715222\pi$
$719$	−33.5599	−1.25157	−0.625786	+	0.779995i	$0.715222\pi$
$720$	−1.00000	−0.0372678
$721$	−30.8786	−1.14998
$722$	−17.2174	−0.640767
$723$	−8.95407	−0.333005
$724$	−3.66487	−0.136204
$725$	3.26875	0.121398
$726$	5.95108	0.220865
$727$	29.8950	1.10874	0.554372	−	0.832269i	$-0.312959\pi$
$727$	29.8950	1.10874	0.554372	+	0.832269i	$0.312959\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	1.48427	0.0549353
$731$	0.277111	0.0102493
$732$	0.972853	0.0359577
$733$	−19.8812	−0.734331	−0.367165	−	0.930156i	$-0.619672\pi$
$733$	−19.8812	−0.734331	−0.367165	+	0.930156i	$0.619672\pi$
$734$	−26.5163	−0.978735
$735$	−0.472189	−0.0174170
$736$	5.58211	0.205759
$737$	19.1347	0.704835
$738$	−1.06100	−0.0390559
$739$	35.4144	1.30274	0.651371	−	0.758760i	$-0.274194\pi$
$739$	35.4144	1.30274	0.651371	+	0.758760i	$0.274194\pi$
$740$	9.64071	0.354400
$741$	0	0
$742$	7.75899	0.284841
$743$	−14.9815	−0.549617	−0.274809	−	0.961499i	$-0.588614\pi$
$743$	−14.9815	−0.549617	−0.274809	+	0.961499i	$0.588614\pi$
$744$	−1.35690	−0.0497462
$745$	3.33513	0.122190
$746$	−27.0616	−0.990795
$747$	−17.7289	−0.648665
$748$	−4.54288	−0.166104
$749$	20.4782	0.748256
$750$	1.00000	0.0365148
$751$	−34.3599	−1.25381	−0.626905	−	0.779096i	$-0.715679\pi$
$751$	−34.3599	−1.25381	−0.626905	+	0.779096i	$0.715679\pi$
$752$	12.7995	0.466751
$753$	0.276520	0.0100769
$754$	0	0
$755$	−15.6625	−0.570016
$756$	2.55496	0.0929229
$757$	−20.8653	−0.758363	−0.379182	−	0.925322i	$-0.623794\pi$
$757$	−20.8653	−0.758363	−0.379182	+	0.925322i	$0.623794\pi$
$758$	16.7313	0.607706
$759$	12.5429	0.455278
$760$	−1.33513	−0.0484301
$761$	−48.0974	−1.74353	−0.871764	−	0.489926i	$-0.837024\pi$
$761$	−48.0974	−1.74353	−0.871764	+	0.489926i	$0.837024\pi$
$762$	12.6843	0.459502
$763$	−23.4389	−0.848546
$764$	−12.2252	−0.442293
$765$	−2.02177	−0.0730973
$766$	−34.3569	−1.24137
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	15.5235	0.559792	0.279896	−	0.960030i	$-0.409700\pi$
$769$	15.5235	0.559792	0.279896	+	0.960030i	$0.409700\pi$
$770$	−5.74094	−0.206889
$771$	−9.77048	−0.351875
$772$	10.5579	0.379989
$773$	−15.0043	−0.539668	−0.269834	−	0.962907i	$-0.586969\pi$
$773$	−15.0043	−0.539668	−0.269834	+	0.962907i	$0.586969\pi$
$774$	0.137063	0.00492664
$775$	1.35690	0.0487411
$776$	7.14675	0.256553
$777$	−24.6316	−0.883654
$778$	11.6732	0.418506
$779$	−1.41657	−0.0507538
$780$	0	0
$781$	6.04892	0.216447
$782$	11.2857	0.403577
$783$	−3.26875	−0.116816
$784$	−0.472189	−0.0168639
$785$	−2.99761	−0.106989
$786$	14.5332	0.518382
$787$	−18.2024	−0.648845	−0.324422	−	0.945912i	$-0.605170\pi$
$787$	−18.2024	−0.648845	−0.324422	+	0.945912i	$0.605170\pi$
$788$	−5.46011	−0.194508
$789$	12.7071	0.452384
$790$	7.77479	0.276615
$791$	−13.4058	−0.476656
$792$	−2.24698	−0.0798429
$793$	0	0
$794$	−12.6431	−0.448687
$795$	−3.03684	−0.107705
$796$	5.00969	0.177564
$797$	−36.0079	−1.27546	−0.637732	−	0.770258i	$-0.720127\pi$
$797$	−36.0079	−1.27546	−0.637732	+	0.770258i	$0.720127\pi$
$798$	3.41119	0.120755
$799$	25.8777	0.915489
$800$	1.00000	0.0353553
$801$	−15.4601	−0.546256
$802$	−29.7724	−1.05130
$803$	3.33513	0.117694
$804$	8.51573	0.300327
$805$	14.2620	0.502671
$806$	0	0
$807$	−9.47757	−0.333626
$808$	−4.71379	−0.165831
$809$	21.6832	0.762340	0.381170	−	0.924505i	$-0.375521\pi$
$809$	21.6832	0.762340	0.381170	+	0.924505i	$0.375521\pi$
$810$	−1.00000	−0.0351364
$811$	16.0301	0.562894	0.281447	−	0.959577i	$-0.409186\pi$
$811$	16.0301	0.562894	0.281447	+	0.959577i	$0.409186\pi$
$812$	−8.35152	−0.293081
$813$	20.7657	0.728285
$814$	21.6625	0.759270
$815$	−1.46681	−0.0513802
$816$	−2.02177	−0.0707761
$817$	0.182997	0.00640225
$818$	−5.61117	−0.196190
$819$	0	0
$820$	1.06100	0.0370517
$821$	10.2543	0.357877	0.178938	−	0.983860i	$-0.442734\pi$
$821$	10.2543	0.357877	0.178938	+	0.983860i	$0.442734\pi$
$822$	5.06100	0.176523
$823$	3.81807	0.133089	0.0665447	−	0.997783i	$-0.478802\pi$
$823$	3.81807	0.133089	0.0665447	+	0.997783i	$0.478802\pi$
$824$	12.0858	0.421027
$825$	2.24698	0.0782298
$826$	30.5593	1.06329
$827$	29.9172	1.04032	0.520162	−	0.854068i	$-0.325872\pi$
$827$	29.9172	1.04032	0.520162	+	0.854068i	$0.325872\pi$
$828$	5.58211	0.193992
$829$	48.3105	1.67789	0.838946	−	0.544214i	$-0.183172\pi$
$829$	48.3105	1.67789	0.838946	+	0.544214i	$0.183172\pi$
$830$	17.7289	0.615378
$831$	−15.0911	−0.523505
$832$	0	0
$833$	−0.954658	−0.0330769
$834$	19.9638	0.691288
$835$	1.13706	0.0393497
$836$	−3.00000	−0.103757
$837$	−1.35690	−0.0469012
$838$	−1.24937	−0.0431589
$839$	−28.9275	−0.998689	−0.499344	−	0.866404i	$-0.666426\pi$
$839$	−28.9275	−0.998689	−0.499344	+	0.866404i	$0.666426\pi$
$840$	−2.55496	−0.0881544
$841$	−18.3153	−0.631561
$842$	−2.76749	−0.0953742
$843$	−17.6015	−0.606227
$844$	−9.38404	−0.323012
$845$	0	0
$846$	12.7995	0.440057
$847$	15.2048	0.522442
$848$	−3.03684	−0.104285
$849$	2.07606	0.0712503
$850$	2.02177	0.0693461
$851$	−53.8155	−1.84477
$852$	2.69202	0.0922271
$853$	−54.7193	−1.87355	−0.936776	−	0.349929i	$-0.886206\pi$
$853$	−54.7193	−1.87355	−0.936776	+	0.349929i	$0.886206\pi$
$854$	2.48560	0.0850554
$855$	−1.33513	−0.0456603
$856$	−8.01507	−0.273949
$857$	−57.7904	−1.97408	−0.987042	−	0.160462i	$-0.948702\pi$
$857$	−57.7904	−1.97408	−0.987042	+	0.160462i	$0.948702\pi$
$858$	0	0
$859$	−11.0261	−0.376205	−0.188103	−	0.982149i	$-0.560234\pi$
$859$	−11.0261	−0.376205	−0.188103	+	0.982149i	$0.560234\pi$
$860$	−0.137063	−0.00467382
$861$	−2.71081	−0.0923841
$862$	−19.1793	−0.653249
$863$	47.8501	1.62884	0.814418	−	0.580278i	$-0.197056\pi$
$863$	47.8501	1.62884	0.814418	+	0.580278i	$0.197056\pi$
$864$	−1.00000	−0.0340207
$865$	−0.983607	−0.0334436
$866$	36.9051	1.25409
$867$	12.9124	0.438530
$868$	−3.46681	−0.117671
$869$	17.4698	0.592622
$870$	3.26875	0.110821
$871$	0	0
$872$	9.17390	0.310667
$873$	7.14675	0.241881
$874$	7.45281	0.252095
$875$	2.55496	0.0863733
$876$	1.48427	0.0501489
$877$	−1.76569	−0.0596232	−0.0298116	−	0.999556i	$-0.509491\pi$
$877$	−1.76569	−0.0596232	−0.0298116	+	0.999556i	$0.509491\pi$
$878$	14.6213	0.493446
$879$	−26.0930	−0.880097
$880$	2.24698	0.0757457
$881$	46.8394	1.57806	0.789029	−	0.614356i	$-0.210584\pi$
$881$	46.8394	1.57806	0.789029	+	0.614356i	$0.210584\pi$
$882$	−0.472189	−0.0158994
$883$	−11.3730	−0.382733	−0.191366	−	0.981519i	$-0.561292\pi$
$883$	−11.3730	−0.382733	−0.191366	+	0.981519i	$0.561292\pi$
$884$	0	0
$885$	−11.9608	−0.402057
$886$	−35.1377	−1.18047
$887$	−53.1245	−1.78375	−0.891873	−	0.452286i	$-0.850609\pi$
$887$	−53.1245	−1.78375	−0.891873	+	0.452286i	$0.850609\pi$
$888$	9.64071	0.323521
$889$	32.4077	1.08692
$890$	15.4601	0.518224
$891$	−2.24698	−0.0752766
$892$	−26.5773	−0.889874
$893$	17.0890	0.571862
$894$	3.33513	0.111543
$895$	−9.86592	−0.329781
$896$	−2.55496	−0.0853552
$897$	0	0
$898$	6.67994	0.222912
$899$	4.43535	0.147927
$900$	1.00000	0.0333333
$901$	−6.13978	−0.204546
$902$	2.38404	0.0793799
$903$	0.350191	0.0116536
$904$	5.24698	0.174512
$905$	3.66487	0.121825
$906$	−15.6625	−0.520351
$907$	−0.871560	−0.0289397	−0.0144698	−	0.999895i	$-0.504606\pi$
$907$	−0.871560	−0.0289397	−0.0144698	+	0.999895i	$0.504606\pi$
$908$	−22.2271	−0.737633
$909$	−4.71379	−0.156347
$910$	0	0
$911$	−34.8431	−1.15440	−0.577201	−	0.816602i	$-0.695855\pi$
$911$	−34.8431	−1.15440	−0.577201	+	0.816602i	$0.695855\pi$
$912$	−1.33513	−0.0442104
$913$	39.8364	1.31839
$914$	22.9963	0.760649
$915$	−0.972853	−0.0321615
$916$	19.1782	0.633666
$917$	37.1317	1.22620
$918$	−2.02177	−0.0667284
$919$	36.6765	1.20985	0.604923	−	0.796284i	$-0.293204\pi$
$919$	36.6765	1.20985	0.604923	+	0.796284i	$0.293204\pi$
$920$	−5.58211	−0.184037
$921$	−15.8092	−0.520932
$922$	−33.9705	−1.11876
$923$	0	0
$924$	−5.74094	−0.188863
$925$	−9.64071	−0.316985
$926$	−26.9594	−0.885942
$927$	12.0858	0.396948
$928$	3.26875	0.107302
$929$	20.2446	0.664203	0.332102	−	0.943244i	$-0.392242\pi$
$929$	20.2446	0.664203	0.332102	+	0.943244i	$0.392242\pi$
$930$	1.35690	0.0444944
$931$	−0.630432	−0.0206616
$932$	19.5483	0.640324
$933$	23.2121	0.759929
$934$	−25.0411	−0.819371
$935$	4.54288	0.148568
$936$	0	0
$937$	7.66189	0.250303	0.125152	−	0.992138i	$-0.460058\pi$
$937$	7.66189	0.250303	0.125152	+	0.992138i	$0.460058\pi$
$938$	21.7573	0.710402
$939$	−28.5297	−0.931033
$940$	−12.7995	−0.417475
$941$	21.4107	0.697969	0.348985	−	0.937128i	$-0.386527\pi$
$941$	21.4107	0.697969	0.348985	+	0.937128i	$0.386527\pi$
$942$	−2.99761	−0.0976673
$943$	−5.92261	−0.192867
$944$	−11.9608	−0.389290
$945$	−2.55496	−0.0831128
$946$	−0.307979	−0.0100132
$947$	36.3631	1.18164	0.590821	−	0.806802i	$-0.298804\pi$
$947$	36.3631	1.18164	0.590821	+	0.806802i	$0.298804\pi$
$948$	7.77479	0.252513
$949$	0	0
$950$	1.33513	0.0433172
$951$	0.709480	0.0230065
$952$	−5.16554	−0.167416
$953$	48.8256	1.58162	0.790809	−	0.612064i	$-0.209660\pi$
$953$	48.8256	1.58162	0.790809	+	0.612064i	$0.209660\pi$
$954$	−3.03684	−0.0983212
$955$	12.2252	0.395598
$956$	−12.8194	−0.414609
$957$	7.34481	0.237424
$958$	−7.21983	−0.233262
$959$	12.9306	0.417552
$960$	1.00000	0.0322749
$961$	−29.1588	−0.940608
$962$	0	0
$963$	−8.01507	−0.258282
$964$	8.95407	0.288391
$965$	−10.5579	−0.339872
$966$	14.2620	0.458874
$967$	4.23623	0.136228	0.0681139	−	0.997678i	$-0.478302\pi$
$967$	4.23623	0.136228	0.0681139	+	0.997678i	$0.478302\pi$
$968$	−5.95108	−0.191275
$969$	−2.69932	−0.0867146
$970$	−7.14675	−0.229468
$971$	−25.7735	−0.827110	−0.413555	−	0.910479i	$-0.635713\pi$
$971$	−25.7735	−0.827110	−0.413555	+	0.910479i	$0.635713\pi$
$972$	−1.00000	−0.0320750
$973$	51.0066	1.63520
$974$	24.1497	0.773807
$975$	0	0
$976$	−0.972853	−0.0311403
$977$	33.2881	1.06498	0.532491	−	0.846436i	$-0.321256\pi$
$977$	33.2881	1.06498	0.532491	+	0.846436i	$0.321256\pi$
$978$	−1.46681	−0.0469035
$979$	34.7385	1.11025
$980$	0.472189	0.0150835
$981$	9.17390	0.292900
$982$	−19.7259	−0.629478
$983$	−45.1605	−1.44040	−0.720198	−	0.693769i	$-0.755949\pi$
$983$	−45.1605	−1.44040	−0.720198	+	0.693769i	$0.755949\pi$
$984$	1.06100	0.0338234
$985$	5.46011	0.173973
$986$	6.60866	0.210463
$987$	32.7023	1.04093
$988$	0	0
$989$	0.765102	0.0243288
$990$	2.24698	0.0714137
$991$	−24.5364	−0.779426	−0.389713	−	0.920936i	$-0.627426\pi$
$991$	−24.5364	−0.779426	−0.389713	+	0.920936i	$0.627426\pi$
$992$	1.35690	0.0430815
$993$	22.8605	0.725457
$994$	6.87800	0.218157
$995$	−5.00969	−0.158818
$996$	17.7289	0.561760
$997$	20.6286	0.653315	0.326658	−	0.945143i	$-0.394078\pi$
$997$	20.6286	0.653315	0.326658	+	0.945143i	$0.394078\pi$
$998$	35.3599	1.11930
$999$	9.64071	0.305019

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.br.1.2	yes	3
13.5	odd	4		5070.2.b.u.1351.2		6
13.8	odd	4		5070.2.b.u.1351.5		6
13.12	even	2		5070.2.a.bm.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bm.1.2	✓	3	13.12	even	2
5070.2.a.br.1.2	yes	3	1.1	even	1	trivial
5070.2.b.u.1351.2		6	13.5	odd	4
5070.2.b.u.1351.5		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.55496 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.55496 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.24698 q^{11} -1.00000 q^{12} -2.55496 q^{14} +1.00000 q^{15} +1.00000 q^{16} +2.02177 q^{17} +1.00000 q^{18} +1.33513 q^{19} -1.00000 q^{20} +2.55496 q^{21} -2.24698 q^{22} +5.58211 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -2.55496 q^{28} +3.26875 q^{29} +1.00000 q^{30} +1.35690 q^{31} +1.00000 q^{32} +2.24698 q^{33} +2.02177 q^{34} +2.55496 q^{35} +1.00000 q^{36} -9.64071 q^{37} +1.33513 q^{38} -1.00000 q^{40} -1.06100 q^{41} +2.55496 q^{42} +0.137063 q^{43} -2.24698 q^{44} -1.00000 q^{45} +5.58211 q^{46} +12.7995 q^{47} -1.00000 q^{48} -0.472189 q^{49} +1.00000 q^{50} -2.02177 q^{51} -3.03684 q^{53} -1.00000 q^{54} +2.24698 q^{55} -2.55496 q^{56} -1.33513 q^{57} +3.26875 q^{58} -11.9608 q^{59} +1.00000 q^{60} -0.972853 q^{61} +1.35690 q^{62} -2.55496 q^{63} +1.00000 q^{64} +2.24698 q^{66} -8.51573 q^{67} +2.02177 q^{68} -5.58211 q^{69} +2.55496 q^{70} -2.69202 q^{71} +1.00000 q^{72} -1.48427 q^{73} -9.64071 q^{74} -1.00000 q^{75} +1.33513 q^{76} +5.74094 q^{77} -7.77479 q^{79} -1.00000 q^{80} +1.00000 q^{81} -1.06100 q^{82} -17.7289 q^{83} +2.55496 q^{84} -2.02177 q^{85} +0.137063 q^{86} -3.26875 q^{87} -2.24698 q^{88} -15.4601 q^{89} -1.00000 q^{90} +5.58211 q^{92} -1.35690 q^{93} +12.7995 q^{94} -1.33513 q^{95} -1.00000 q^{96} +7.14675 q^{97} -0.472189 q^{98} -2.24698 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 8 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 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	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 8 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} - 2 q^{11} - 3 q^{12} - 8 q^{14} + 3 q^{15} + 3 q^{16} + 3 q^{17} + 3 q^{18} + 3 q^{19} - 3 q^{20} + 8 q^{21} - 2 q^{22} + 11 q^{23} - 3 q^{24} + 3 q^{25} - 3 q^{27} - 8 q^{28} + 2 q^{29} + 3 q^{30} + 3 q^{32} + 2 q^{33} + 3 q^{34} + 8 q^{35} + 3 q^{36} + 8 q^{37} + 3 q^{38} - 3 q^{40} - 13 q^{41} + 8 q^{42} - 5 q^{43} - 2 q^{44} - 3 q^{45} + 11 q^{46} - 7 q^{47} - 3 q^{48} + 5 q^{49} + 3 q^{50} - 3 q^{51} + 19 q^{53} - 3 q^{54} + 2 q^{55} - 8 q^{56} - 3 q^{57} + 2 q^{58} - 23 q^{59} + 3 q^{60} - 9 q^{61} - 8 q^{63} + 3 q^{64} + 2 q^{66} - 13 q^{67} + 3 q^{68} - 11 q^{69} + 8 q^{70} - 3 q^{71} + 3 q^{72} - 17 q^{73} + 8 q^{74} - 3 q^{75} + 3 q^{76} + 3 q^{77} - 25 q^{79} - 3 q^{80} + 3 q^{81} - 13 q^{82} - 20 q^{83} + 8 q^{84} - 3 q^{85} - 5 q^{86} - 2 q^{87} - 2 q^{88} - 21 q^{89} - 3 q^{90} + 11 q^{92} - 7 q^{94} - 3 q^{95} - 3 q^{96} - 6 q^{97} + 5 q^{98} - 2 q^{99}+O(q^{100}) \) 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 + 3 * q^17 + 3 * q^18 + 3 * q^19 - 3 * q^20 + 8 * q^21 - 2 * q^22 + 11 * q^23 - 3 * q^24 + 3 * q^25 - 3 * q^27 - 8 * q^28 + 2 * q^29 + 3 * q^30 + 3 * q^32 + 2 * q^33 + 3 * q^34 + 8 * q^35 + 3 * q^36 + 8 * q^37 + 3 * q^38 - 3 * q^40 - 13 * q^41 + 8 * q^42 - 5 * q^43 - 2 * q^44 - 3 * q^45 + 11 * q^46 - 7 * q^47 - 3 * q^48 + 5 * q^49 + 3 * q^50 - 3 * q^51 + 19 * q^53 - 3 * q^54 + 2 * q^55 - 8 * q^56 - 3 * q^57 + 2 * q^58 - 23 * q^59 + 3 * q^60 - 9 * q^61 - 8 * q^63 + 3 * q^64 + 2 * q^66 - 13 * q^67 + 3 * q^68 - 11 * q^69 + 8 * q^70 - 3 * q^71 + 3 * q^72 - 17 * q^73 + 8 * q^74 - 3 * q^75 + 3 * q^76 + 3 * q^77 - 25 * q^79 - 3 * q^80 + 3 * q^81 - 13 * q^82 - 20 * q^83 + 8 * q^84 - 3 * q^85 - 5 * q^86 - 2 * q^87 - 2 * q^88 - 21 * q^89 - 3 * q^90 + 11 * q^92 - 7 * q^94 - 3 * q^95 - 3 * q^96 - 6 * q^97 + 5 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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