LMFDB - Embedded newform 5070.2.a.bo.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:	$0$
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			1.2
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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.44504 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.44504 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.24698 q^{11} +1.00000 q^{12} -3.44504 q^{14} -1.00000 q^{15} +1.00000 q^{16} +6.78986 q^{17} -1.00000 q^{18} +6.26875 q^{19} -1.00000 q^{20} +3.44504 q^{21} -4.24698 q^{22} -1.30798 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +3.44504 q^{28} -9.14675 q^{29} +1.00000 q^{30} +3.75302 q^{31} -1.00000 q^{32} +4.24698 q^{33} -6.78986 q^{34} -3.44504 q^{35} +1.00000 q^{36} +6.82908 q^{37} -6.26875 q^{38} +1.00000 q^{40} +4.26875 q^{41} -3.44504 q^{42} +3.07069 q^{43} +4.24698 q^{44} -1.00000 q^{45} +1.30798 q^{46} +7.76809 q^{47} +1.00000 q^{48} +4.86831 q^{49} -1.00000 q^{50} +6.78986 q^{51} -8.93900 q^{53} -1.00000 q^{54} -4.24698 q^{55} -3.44504 q^{56} +6.26875 q^{57} +9.14675 q^{58} -10.3327 q^{59} -1.00000 q^{60} +2.53319 q^{61} -3.75302 q^{62} +3.44504 q^{63} +1.00000 q^{64} -4.24698 q^{66} -0.0760644 q^{67} +6.78986 q^{68} -1.30798 q^{69} +3.44504 q^{70} +0.374354 q^{71} -1.00000 q^{72} -16.7114 q^{73} -6.82908 q^{74} +1.00000 q^{75} +6.26875 q^{76} +14.6310 q^{77} -1.33513 q^{79} -1.00000 q^{80} +1.00000 q^{81} -4.26875 q^{82} -0.740939 q^{83} +3.44504 q^{84} -6.78986 q^{85} -3.07069 q^{86} -9.14675 q^{87} -4.24698 q^{88} +13.3274 q^{89} +1.00000 q^{90} -1.30798 q^{92} +3.75302 q^{93} -7.76809 q^{94} -6.26875 q^{95} -1.00000 q^{96} -13.1903 q^{97} -4.86831 q^{98} +4.24698 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} + 10 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 + 10 * 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} + 10 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{10} + 8 q^{11} + 3 q^{12} - 10 q^{14} - 3 q^{15} + 3 q^{16} - 3 q^{17} - 3 q^{18} + 11 q^{19} - 3 q^{20} + 10 q^{21} - 8 q^{22} - 9 q^{23} - 3 q^{24} + 3 q^{25} + 3 q^{27} + 10 q^{28} + 3 q^{30} + 16 q^{31} - 3 q^{32} + 8 q^{33} + 3 q^{34} - 10 q^{35} + 3 q^{36} + 10 q^{37} - 11 q^{38} + 3 q^{40} + 5 q^{41} - 10 q^{42} - 3 q^{43} + 8 q^{44} - 3 q^{45} + 9 q^{46} + 3 q^{47} + 3 q^{48} + 17 q^{49} - 3 q^{50} - 3 q^{51} - 17 q^{53} - 3 q^{54} - 8 q^{55} - 10 q^{56} + 11 q^{57} + 11 q^{59} - 3 q^{60} + 11 q^{61} - 16 q^{62} + 10 q^{63} + 3 q^{64} - 8 q^{66} + 15 q^{67} - 3 q^{68} - 9 q^{69} + 10 q^{70} + 13 q^{71} - 3 q^{72} - q^{73} - 10 q^{74} + 3 q^{75} + 11 q^{76} + 29 q^{77} - 3 q^{79} - 3 q^{80} + 3 q^{81} - 5 q^{82} + 12 q^{83} + 10 q^{84} + 3 q^{85} + 3 q^{86} - 8 q^{88} + q^{89} + 3 q^{90} - 9 q^{92} + 16 q^{93} - 3 q^{94} - 11 q^{95} - 3 q^{96} - 6 q^{97} - 17 q^{98} + 8 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 + 10 * q^7 - 3 * q^8 + 3 * q^9 + 3 * q^10 + 8 * q^11 + 3 * q^12 - 10 * q^14 - 3 * q^15 + 3 * q^16 - 3 * q^17 - 3 * q^18 + 11 * q^19 - 3 * q^20 + 10 * q^21 - 8 * q^22 - 9 * q^23 - 3 * q^24 + 3 * q^25 + 3 * q^27 + 10 * q^28 + 3 * q^30 + 16 * q^31 - 3 * q^32 + 8 * q^33 + 3 * q^34 - 10 * q^35 + 3 * q^36 + 10 * q^37 - 11 * q^38 + 3 * q^40 + 5 * q^41 - 10 * q^42 - 3 * q^43 + 8 * q^44 - 3 * q^45 + 9 * q^46 + 3 * q^47 + 3 * q^48 + 17 * q^49 - 3 * q^50 - 3 * q^51 - 17 * q^53 - 3 * q^54 - 8 * q^55 - 10 * q^56 + 11 * q^57 + 11 * q^59 - 3 * q^60 + 11 * q^61 - 16 * q^62 + 10 * q^63 + 3 * q^64 - 8 * q^66 + 15 * q^67 - 3 * q^68 - 9 * q^69 + 10 * q^70 + 13 * q^71 - 3 * q^72 - q^73 - 10 * q^74 + 3 * q^75 + 11 * q^76 + 29 * q^77 - 3 * q^79 - 3 * q^80 + 3 * q^81 - 5 * q^82 + 12 * q^83 + 10 * q^84 + 3 * q^85 + 3 * q^86 - 8 * q^88 + q^89 + 3 * q^90 - 9 * q^92 + 16 * q^93 - 3 * q^94 - 11 * q^95 - 3 * q^96 - 6 * q^97 - 17 * q^98 + 8 * 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
$7$	3.44504	1.30210	0.651052	−	0.759033i	$-0.274328\pi$
$7$	3.44504	1.30210	0.651052	+	0.759033i	$0.274328\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	4.24698	1.28051	0.640256	−	0.768161i	$-0.278828\pi$
$11$	4.24698	1.28051	0.640256	+	0.768161i	$0.278828\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−3.44504	−0.920726
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	6.78986	1.64678	0.823391	−	0.567474i	$-0.192079\pi$
$17$	6.78986	1.64678	0.823391	+	0.567474i	$0.192079\pi$
$18$	−1.00000	−0.235702
$19$	6.26875	1.43815	0.719075	−	0.694933i	$-0.244566\pi$
$19$	6.26875	1.43815	0.719075	+	0.694933i	$0.244566\pi$
$20$	−1.00000	−0.223607
$21$	3.44504	0.751770
$22$	−4.24698	−0.905459
$23$	−1.30798	−0.272732	−0.136366	−	0.990658i	$-0.543542\pi$
$23$	−1.30798	−0.272732	−0.136366	+	0.990658i	$0.543542\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	3.44504	0.651052
$29$	−9.14675	−1.69851	−0.849255	−	0.527984i	$-0.822948\pi$
$29$	−9.14675	−1.69851	−0.849255	+	0.527984i	$0.822948\pi$
$30$	1.00000	0.182574
$31$	3.75302	0.674062	0.337031	−	0.941493i	$-0.390577\pi$
$31$	3.75302	0.674062	0.337031	+	0.941493i	$0.390577\pi$
$32$	−1.00000	−0.176777
$33$	4.24698	0.739304
$34$	−6.78986	−1.16445
$35$	−3.44504	−0.582318
$36$	1.00000	0.166667
$37$	6.82908	1.12269	0.561347	−	0.827580i	$-0.310283\pi$
$37$	6.82908	1.12269	0.561347	+	0.827580i	$0.310283\pi$
$38$	−6.26875	−1.01693
$39$	0	0
$40$	1.00000	0.158114
$41$	4.26875	0.666667	0.333333	−	0.942809i	$-0.391827\pi$
$41$	4.26875	0.666667	0.333333	+	0.942809i	$0.391827\pi$
$42$	−3.44504	−0.531582
$43$	3.07069	0.468275	0.234138	−	0.972203i	$-0.424773\pi$
$43$	3.07069	0.468275	0.234138	+	0.972203i	$0.424773\pi$
$44$	4.24698	0.640256
$45$	−1.00000	−0.149071
$46$	1.30798	0.192851
$47$	7.76809	1.13309	0.566546	−	0.824030i	$-0.308279\pi$
$47$	7.76809	1.13309	0.566546	+	0.824030i	$0.308279\pi$
$48$	1.00000	0.144338
$49$	4.86831	0.695473
$50$	−1.00000	−0.141421
$51$	6.78986	0.950770
$52$	0	0
$53$	−8.93900	−1.22787	−0.613933	−	0.789358i	$-0.710414\pi$
$53$	−8.93900	−1.22787	−0.613933	+	0.789358i	$0.710414\pi$
$54$	−1.00000	−0.136083
$55$	−4.24698	−0.572663
$56$	−3.44504	−0.460363
$57$	6.26875	0.830316
$58$	9.14675	1.20103
$59$	−10.3327	−1.34521	−0.672604	−	0.740003i	$-0.734824\pi$
$59$	−10.3327	−1.34521	−0.672604	+	0.740003i	$0.734824\pi$
$60$	−1.00000	−0.129099
$61$	2.53319	0.324341	0.162171	−	0.986763i	$-0.448150\pi$
$61$	2.53319	0.324341	0.162171	+	0.986763i	$0.448150\pi$
$62$	−3.75302	−0.476634
$63$	3.44504	0.434034
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.24698	−0.522767
$67$	−0.0760644	−0.00929275	−0.00464637	−	0.999989i	$-0.501479\pi$
$67$	−0.0760644	−0.00929275	−0.00464637	+	0.999989i	$0.501479\pi$
$68$	6.78986	0.823391
$69$	−1.30798	−0.157462
$70$	3.44504	0.411761
$71$	0.374354	0.0444277	0.0222138	−	0.999753i	$-0.492929\pi$
$71$	0.374354	0.0444277	0.0222138	+	0.999753i	$0.492929\pi$
$72$	−1.00000	−0.117851
$73$	−16.7114	−1.95592	−0.977961	−	0.208789i	$-0.933048\pi$
$73$	−16.7114	−1.95592	−0.977961	+	0.208789i	$0.933048\pi$
$74$	−6.82908	−0.793865
$75$	1.00000	0.115470
$76$	6.26875	0.719075
$77$	14.6310	1.66736
$78$	0	0
$79$	−1.33513	−0.150213	−0.0751067	−	0.997176i	$-0.523930\pi$
$79$	−1.33513	−0.150213	−0.0751067	+	0.997176i	$0.523930\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−4.26875	−0.471405
$83$	−0.740939	−0.0813286	−0.0406643	−	0.999173i	$-0.512947\pi$
$83$	−0.740939	−0.0813286	−0.0406643	+	0.999173i	$0.512947\pi$
$84$	3.44504	0.375885
$85$	−6.78986	−0.736463
$86$	−3.07069	−0.331121
$87$	−9.14675	−0.980635
$88$	−4.24698	−0.452730
$89$	13.3274	1.41270	0.706348	−	0.707864i	$-0.250341\pi$
$89$	13.3274	1.41270	0.706348	+	0.707864i	$0.250341\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−1.30798	−0.136366
$93$	3.75302	0.389170
$94$	−7.76809	−0.801217
$95$	−6.26875	−0.643160
$96$	−1.00000	−0.102062
$97$	−13.1903	−1.33927	−0.669636	−	0.742690i	$-0.733550\pi$
$97$	−13.1903	−1.33927	−0.669636	+	0.742690i	$0.733550\pi$
$98$	−4.86831	−0.491774
$99$	4.24698	0.426838
$100$	1.00000	0.100000
$101$	−15.0422	−1.49676	−0.748378	−	0.663272i	$-0.769167\pi$
$101$	−15.0422	−1.49676	−0.748378	+	0.663272i	$0.769167\pi$
$102$	−6.78986	−0.672296
$103$	−8.92154	−0.879066	−0.439533	−	0.898227i	$-0.644856\pi$
$103$	−8.92154	−0.879066	−0.439533	+	0.898227i	$0.644856\pi$
$104$	0	0
$105$	−3.44504	−0.336202
$106$	8.93900	0.868233
$107$	14.6746	1.41864	0.709322	−	0.704885i	$-0.249001\pi$
$107$	14.6746	1.41864	0.709322	+	0.704885i	$0.249001\pi$
$108$	1.00000	0.0962250
$109$	9.66786	0.926013	0.463006	−	0.886355i	$-0.346771\pi$
$109$	9.66786	0.926013	0.463006	+	0.886355i	$0.346771\pi$
$110$	4.24698	0.404934
$111$	6.82908	0.648188
$112$	3.44504	0.325526
$113$	−11.3448	−1.06723	−0.533615	−	0.845727i	$-0.679167\pi$
$113$	−11.3448	−1.06723	−0.533615	+	0.845727i	$0.679167\pi$
$114$	−6.26875	−0.587122
$115$	1.30798	0.121970
$116$	−9.14675	−0.849255
$117$	0	0
$118$	10.3327	0.951205
$119$	23.3913	2.14428
$120$	1.00000	0.0912871
$121$	7.03684	0.639712
$122$	−2.53319	−0.229344
$123$	4.26875	0.384900
$124$	3.75302	0.337031
$125$	−1.00000	−0.0894427
$126$	−3.44504	−0.306909
$127$	−2.54288	−0.225644	−0.112822	−	0.993615i	$-0.535989\pi$
$127$	−2.54288	−0.225644	−0.112822	+	0.993615i	$0.535989\pi$
$128$	−1.00000	−0.0883883
$129$	3.07069	0.270359
$130$	0	0
$131$	−10.8509	−0.948044	−0.474022	−	0.880513i	$-0.657198\pi$
$131$	−10.8509	−0.948044	−0.474022	+	0.880513i	$0.657198\pi$
$132$	4.24698	0.369652
$133$	21.5961	1.87262
$134$	0.0760644	0.00657096
$135$	−1.00000	−0.0860663
$136$	−6.78986	−0.582225
$137$	−18.6407	−1.59258	−0.796292	−	0.604913i	$-0.793208\pi$
$137$	−18.6407	−1.59258	−0.796292	+	0.604913i	$0.793208\pi$
$138$	1.30798	0.111343
$139$	3.53750	0.300047	0.150023	−	0.988682i	$-0.452065\pi$
$139$	3.53750	0.300047	0.150023	+	0.988682i	$0.452065\pi$
$140$	−3.44504	−0.291159
$141$	7.76809	0.654191
$142$	−0.374354	−0.0314151
$143$	0	0
$144$	1.00000	0.0833333
$145$	9.14675	0.759596
$146$	16.7114	1.38305
$147$	4.86831	0.401532
$148$	6.82908	0.561347
$149$	−16.6843	−1.36683	−0.683414	−	0.730031i	$-0.739505\pi$
$149$	−16.6843	−1.36683	−0.683414	+	0.730031i	$0.739505\pi$
$150$	−1.00000	−0.0816497
$151$	18.3327	1.49190	0.745948	−	0.666004i	$-0.231997\pi$
$151$	18.3327	1.49190	0.745948	+	0.666004i	$0.231997\pi$
$152$	−6.26875	−0.508463
$153$	6.78986	0.548927
$154$	−14.6310	−1.17900
$155$	−3.75302	−0.301450
$156$	0	0
$157$	−18.6015	−1.48456	−0.742280	−	0.670090i	$-0.766256\pi$
$157$	−18.6015	−1.48456	−0.742280	+	0.670090i	$0.766256\pi$
$158$	1.33513	0.106217
$159$	−8.93900	−0.708909
$160$	1.00000	0.0790569
$161$	−4.50604	−0.355126
$162$	−1.00000	−0.0785674
$163$	18.8944	1.47992	0.739962	−	0.672649i	$-0.234844\pi$
$163$	18.8944	1.47992	0.739962	+	0.672649i	$0.234844\pi$
$164$	4.26875	0.333333
$165$	−4.24698	−0.330627
$166$	0.740939	0.0575080
$167$	−4.24698	−0.328641	−0.164321	−	0.986407i	$-0.552543\pi$
$167$	−4.24698	−0.328641	−0.164321	+	0.986407i	$0.552543\pi$
$168$	−3.44504	−0.265791
$169$	0	0
$170$	6.78986	0.520758
$171$	6.26875	0.479383
$172$	3.07069	0.234138
$173$	1.16852	0.0888411	0.0444206	−	0.999013i	$-0.485856\pi$
$173$	1.16852	0.0888411	0.0444206	+	0.999013i	$0.485856\pi$
$174$	9.14675	0.693413
$175$	3.44504	0.260421
$176$	4.24698	0.320128
$177$	−10.3327	−0.776656
$178$	−13.3274	−0.998928
$179$	−10.1836	−0.761157	−0.380579	−	0.924749i	$-0.624275\pi$
$179$	−10.1836	−0.761157	−0.380579	+	0.924749i	$0.624275\pi$
$180$	−1.00000	−0.0745356
$181$	10.3351	0.768204	0.384102	−	0.923291i	$-0.374511\pi$
$181$	10.3351	0.768204	0.384102	+	0.923291i	$0.374511\pi$
$182$	0	0
$183$	2.53319	0.187259
$184$	1.30798	0.0964255
$185$	−6.82908	−0.502084
$186$	−3.75302	−0.275185
$187$	28.8364	2.10872
$188$	7.76809	0.566546
$189$	3.44504	0.250590
$190$	6.26875	0.454783
$191$	−9.43296	−0.682545	−0.341273	−	0.939964i	$-0.610858\pi$
$191$	−9.43296	−0.682545	−0.341273	+	0.939964i	$0.610858\pi$
$192$	1.00000	0.0721688
$193$	22.9191	1.64976	0.824878	−	0.565310i	$-0.191244\pi$
$193$	22.9191	1.64976	0.824878	+	0.565310i	$0.191244\pi$
$194$	13.1903	0.947008
$195$	0	0
$196$	4.86831	0.347737
$197$	−14.3502	−1.02241	−0.511204	−	0.859459i	$-0.670800\pi$
$197$	−14.3502	−1.02241	−0.511204	+	0.859459i	$0.670800\pi$
$198$	−4.24698	−0.301820
$199$	14.2524	1.01032	0.505161	−	0.863025i	$-0.331433\pi$
$199$	14.2524	1.01032	0.505161	+	0.863025i	$0.331433\pi$
$200$	−1.00000	−0.0707107
$201$	−0.0760644	−0.00536517
$202$	15.0422	1.05837
$203$	−31.5109	−2.21163
$204$	6.78986	0.475385
$205$	−4.26875	−0.298142
$206$	8.92154	0.621593
$207$	−1.30798	−0.0909108
$208$	0	0
$209$	26.6233	1.84157
$210$	3.44504	0.237730
$211$	0.396125	0.0272703	0.0136352	−	0.999907i	$-0.495660\pi$
$211$	0.396125	0.0272703	0.0136352	+	0.999907i	$0.495660\pi$
$212$	−8.93900	−0.613933
$213$	0.374354	0.0256503
$214$	−14.6746	−1.00313
$215$	−3.07069	−0.209419
$216$	−1.00000	−0.0680414
$217$	12.9293	0.877699
$218$	−9.66786	−0.654790
$219$	−16.7114	−1.12925
$220$	−4.24698	−0.286331
$221$	0	0
$222$	−6.82908	−0.458338
$223$	1.49635	0.100203	0.0501016	−	0.998744i	$-0.484045\pi$
$223$	1.49635	0.100203	0.0501016	+	0.998744i	$0.484045\pi$
$224$	−3.44504	−0.230182
$225$	1.00000	0.0666667
$226$	11.3448	0.754646
$227$	25.2150	1.67358	0.836791	−	0.547523i	$-0.184429\pi$
$227$	25.2150	1.67358	0.836791	+	0.547523i	$0.184429\pi$
$228$	6.26875	0.415158
$229$	−22.3666	−1.47803	−0.739013	−	0.673691i	$-0.764708\pi$
$229$	−22.3666	−1.47803	−0.739013	+	0.673691i	$0.764708\pi$
$230$	−1.30798	−0.0862456
$231$	14.6310	0.962651
$232$	9.14675	0.600514
$233$	14.0858	0.922788	0.461394	−	0.887195i	$-0.347349\pi$
$233$	14.0858	0.922788	0.461394	+	0.887195i	$0.347349\pi$
$234$	0	0
$235$	−7.76809	−0.506734
$236$	−10.3327	−0.672604
$237$	−1.33513	−0.0867257
$238$	−23.3913	−1.51624
$239$	5.38106	0.348072	0.174036	−	0.984739i	$-0.444319\pi$
$239$	5.38106	0.348072	0.174036	+	0.984739i	$0.444319\pi$
$240$	−1.00000	−0.0645497
$241$	−29.9420	−1.92873	−0.964366	−	0.264570i	$-0.914770\pi$
$241$	−29.9420	−1.92873	−0.964366	+	0.264570i	$0.914770\pi$
$242$	−7.03684	−0.452345
$243$	1.00000	0.0641500
$244$	2.53319	0.162171
$245$	−4.86831	−0.311025
$246$	−4.26875	−0.272166
$247$	0	0
$248$	−3.75302	−0.238317
$249$	−0.740939	−0.0469551
$250$	1.00000	0.0632456
$251$	−8.73556	−0.551384	−0.275692	−	0.961246i	$-0.588907\pi$
$251$	−8.73556	−0.551384	−0.275692	+	0.961246i	$0.588907\pi$
$252$	3.44504	0.217017
$253$	−5.55496	−0.349237
$254$	2.54288	0.159554
$255$	−6.78986	−0.425197
$256$	1.00000	0.0625000
$257$	21.2717	1.32689	0.663447	−	0.748223i	$-0.269093\pi$
$257$	21.2717	1.32689	0.663447	+	0.748223i	$0.269093\pi$
$258$	−3.07069	−0.191173
$259$	23.5265	1.46186
$260$	0	0
$261$	−9.14675	−0.566170
$262$	10.8509	0.670368
$263$	−22.2325	−1.37091	−0.685457	−	0.728113i	$-0.740398\pi$
$263$	−22.2325	−1.37091	−0.685457	+	0.728113i	$0.740398\pi$
$264$	−4.24698	−0.261384
$265$	8.93900	0.549118
$266$	−21.5961	−1.32414
$267$	13.3274	0.815621
$268$	−0.0760644	−0.00464637
$269$	−10.2241	−0.623377	−0.311689	−	0.950184i	$-0.600895\pi$
$269$	−10.2241	−0.623377	−0.311689	+	0.950184i	$0.600895\pi$
$270$	1.00000	0.0608581
$271$	0.716185	0.0435051	0.0217526	−	0.999763i	$-0.493075\pi$
$271$	0.716185	0.0435051	0.0217526	+	0.999763i	$0.493075\pi$
$272$	6.78986	0.411696
$273$	0	0
$274$	18.6407	1.12613
$275$	4.24698	0.256103
$276$	−1.30798	−0.0787311
$277$	−22.8713	−1.37420	−0.687102	−	0.726561i	$-0.741117\pi$
$277$	−22.8713	−1.37420	−0.687102	+	0.726561i	$0.741117\pi$
$278$	−3.53750	−0.212165
$279$	3.75302	0.224687
$280$	3.44504	0.205881
$281$	4.13036	0.246397	0.123198	−	0.992382i	$-0.460685\pi$
$281$	4.13036	0.246397	0.123198	+	0.992382i	$0.460685\pi$
$282$	−7.76809	−0.462583
$283$	16.2959	0.968691	0.484345	−	0.874877i	$-0.339058\pi$
$283$	16.2959	0.968691	0.484345	+	0.874877i	$0.339058\pi$
$284$	0.374354	0.0222138
$285$	−6.26875	−0.371329
$286$	0	0
$287$	14.7060	0.868069
$288$	−1.00000	−0.0589256
$289$	29.1021	1.71189
$290$	−9.14675	−0.537116
$291$	−13.1903	−0.773229
$292$	−16.7114	−0.977961
$293$	7.72587	0.451350	0.225675	−	0.974203i	$-0.427541\pi$
$293$	7.72587	0.451350	0.225675	+	0.974203i	$0.427541\pi$
$294$	−4.86831	−0.283926
$295$	10.3327	0.601595
$296$	−6.82908	−0.396932
$297$	4.24698	0.246435
$298$	16.6843	0.966493
$299$	0	0
$300$	1.00000	0.0577350
$301$	10.5786	0.609743
$302$	−18.3327	−1.05493
$303$	−15.0422	−0.864153
$304$	6.26875	0.359537
$305$	−2.53319	−0.145050
$306$	−6.78986	−0.388150
$307$	−5.13036	−0.292805	−0.146403	−	0.989225i	$-0.546769\pi$
$307$	−5.13036	−0.292805	−0.146403	+	0.989225i	$0.546769\pi$
$308$	14.6310	0.833680
$309$	−8.92154	−0.507529
$310$	3.75302	0.213157
$311$	33.1987	1.88252	0.941261	−	0.337679i	$-0.109642\pi$
$311$	33.1987	1.88252	0.941261	+	0.337679i	$0.109642\pi$
$312$	0	0
$313$	0.0814412	0.00460333	0.00230167	−	0.999997i	$-0.499267\pi$
$313$	0.0814412	0.00460333	0.00230167	+	0.999997i	$0.499267\pi$
$314$	18.6015	1.04974
$315$	−3.44504	−0.194106
$316$	−1.33513	−0.0751067
$317$	−19.4969	−1.09506	−0.547529	−	0.836787i	$-0.684431\pi$
$317$	−19.4969	−1.09506	−0.547529	+	0.836787i	$0.684431\pi$
$318$	8.93900	0.501274
$319$	−38.8461	−2.17496
$320$	−1.00000	−0.0559017
$321$	14.6746	0.819054
$322$	4.50604	0.251112
$323$	42.5639	2.36832
$324$	1.00000	0.0555556
$325$	0	0
$326$	−18.8944	−1.04646
$327$	9.66786	0.534634
$328$	−4.26875	−0.235702
$329$	26.7614	1.47540
$330$	4.24698	0.233789
$331$	23.8049	1.30844	0.654218	−	0.756306i	$-0.272998\pi$
$331$	23.8049	1.30844	0.654218	+	0.756306i	$0.272998\pi$
$332$	−0.740939	−0.0406643
$333$	6.82908	0.374232
$334$	4.24698	0.232384
$335$	0.0760644	0.00415584
$336$	3.44504	0.187942
$337$	21.2064	1.15519	0.577594	−	0.816324i	$-0.303992\pi$
$337$	21.2064	1.15519	0.577594	+	0.816324i	$0.303992\pi$
$338$	0	0
$339$	−11.3448	−0.616166
$340$	−6.78986	−0.368232
$341$	15.9390	0.863145
$342$	−6.26875	−0.338975
$343$	−7.34375	−0.396525
$344$	−3.07069	−0.165560
$345$	1.30798	0.0704192
$346$	−1.16852	−0.0628201
$347$	−16.5773	−0.889917	−0.444959	−	0.895551i	$-0.646782\pi$
$347$	−16.5773	−0.889917	−0.444959	+	0.895551i	$0.646782\pi$
$348$	−9.14675	−0.490317
$349$	2.51035	0.134376	0.0671880	−	0.997740i	$-0.478597\pi$
$349$	2.51035	0.134376	0.0671880	+	0.997740i	$0.478597\pi$
$350$	−3.44504	−0.184145
$351$	0	0
$352$	−4.24698	−0.226365
$353$	−20.8412	−1.10926	−0.554632	−	0.832096i	$-0.687141\pi$
$353$	−20.8412	−1.10926	−0.554632	+	0.832096i	$0.687141\pi$
$354$	10.3327	0.549179
$355$	−0.374354	−0.0198687
$356$	13.3274	0.706348
$357$	23.3913	1.23800
$358$	10.1836	0.538219
$359$	2.42566	0.128022	0.0640108	−	0.997949i	$-0.479611\pi$
$359$	2.42566	0.128022	0.0640108	+	0.997949i	$0.479611\pi$
$360$	1.00000	0.0527046
$361$	20.2972	1.06827
$362$	−10.3351	−0.543202
$363$	7.03684	0.369338
$364$	0	0
$365$	16.7114	0.874715
$366$	−2.53319	−0.132412
$367$	2.66248	0.138980	0.0694902	−	0.997583i	$-0.477863\pi$
$367$	2.66248	0.138980	0.0694902	+	0.997583i	$0.477863\pi$
$368$	−1.30798	−0.0681831
$369$	4.26875	0.222222
$370$	6.82908	0.355027
$371$	−30.7952	−1.59881
$372$	3.75302	0.194585
$373$	5.87800	0.304351	0.152176	−	0.988353i	$-0.451372\pi$
$373$	5.87800	0.304351	0.152176	+	0.988353i	$0.451372\pi$
$374$	−28.8364	−1.49109
$375$	−1.00000	−0.0516398
$376$	−7.76809	−0.400608
$377$	0	0
$378$	−3.44504	−0.177194
$379$	36.1540	1.85711	0.928554	−	0.371196i	$-0.121052\pi$
$379$	36.1540	1.85711	0.928554	+	0.371196i	$0.121052\pi$
$380$	−6.26875	−0.321580
$381$	−2.54288	−0.130276
$382$	9.43296	0.482632
$383$	−8.98361	−0.459041	−0.229520	−	0.973304i	$-0.573716\pi$
$383$	−8.98361	−0.459041	−0.229520	+	0.973304i	$0.573716\pi$
$384$	−1.00000	−0.0510310
$385$	−14.6310	−0.745666
$386$	−22.9191	−1.16655
$387$	3.07069	0.156092
$388$	−13.1903	−0.669636
$389$	0.655186	0.0332192	0.0166096	−	0.999862i	$-0.494713\pi$
$389$	0.655186	0.0332192	0.0166096	+	0.999862i	$0.494713\pi$
$390$	0	0
$391$	−8.88099	−0.449131
$392$	−4.86831	−0.245887
$393$	−10.8509	−0.547353
$394$	14.3502	0.722952
$395$	1.33513	0.0671775
$396$	4.24698	0.213419
$397$	0.271143	0.0136083	0.00680413	−	0.999977i	$-0.497834\pi$
$397$	0.271143	0.0136083	0.00680413	+	0.999977i	$0.497834\pi$
$398$	−14.2524	−0.714406
$399$	21.5961	1.08116
$400$	1.00000	0.0500000
$401$	0.344814	0.0172192	0.00860960	−	0.999963i	$-0.497259\pi$
$401$	0.344814	0.0172192	0.00860960	+	0.999963i	$0.497259\pi$
$402$	0.0760644	0.00379375
$403$	0	0
$404$	−15.0422	−0.748378
$405$	−1.00000	−0.0496904
$406$	31.5109	1.56386
$407$	29.0030	1.43762
$408$	−6.78986	−0.336148
$409$	27.0388	1.33698	0.668490	−	0.743721i	$-0.266941\pi$
$409$	27.0388	1.33698	0.668490	+	0.743721i	$0.266941\pi$
$410$	4.26875	0.210819
$411$	−18.6407	−0.919478
$412$	−8.92154	−0.439533
$413$	−35.5967	−1.75160
$414$	1.30798	0.0642836
$415$	0.740939	0.0363713
$416$	0	0
$417$	3.53750	0.173232
$418$	−26.6233	−1.30219
$419$	5.86533	0.286540	0.143270	−	0.989684i	$-0.454238\pi$
$419$	5.86533	0.286540	0.143270	+	0.989684i	$0.454238\pi$
$420$	−3.44504	−0.168101
$421$	−28.9476	−1.41082	−0.705410	−	0.708799i	$-0.749237\pi$
$421$	−28.9476	−1.41082	−0.705410	+	0.708799i	$0.749237\pi$
$422$	−0.396125	−0.0192830
$423$	7.76809	0.377697
$424$	8.93900	0.434116
$425$	6.78986	0.329356
$426$	−0.374354	−0.0181375
$427$	8.72694	0.422326
$428$	14.6746	0.709322
$429$	0	0
$430$	3.07069	0.148082
$431$	4.87023	0.234591	0.117295	−	0.993097i	$-0.462578\pi$
$431$	4.87023	0.234591	0.117295	+	0.993097i	$0.462578\pi$
$432$	1.00000	0.0481125
$433$	−36.9788	−1.77709	−0.888544	−	0.458791i	$-0.848283\pi$
$433$	−36.9788	−1.77709	−0.888544	+	0.458791i	$0.848283\pi$
$434$	−12.9293	−0.620627
$435$	9.14675	0.438553
$436$	9.66786	0.463006
$437$	−8.19939	−0.392230
$438$	16.7114	0.798502
$439$	22.8799	1.09200	0.546000	−	0.837785i	$-0.316150\pi$
$439$	22.8799	1.09200	0.546000	+	0.837785i	$0.316150\pi$
$440$	4.24698	0.202467
$441$	4.86831	0.231824
$442$	0	0
$443$	−10.5942	−0.503345	−0.251673	−	0.967812i	$-0.580981\pi$
$443$	−10.5942	−0.503345	−0.251673	+	0.967812i	$0.580981\pi$
$444$	6.82908	0.324094
$445$	−13.3274	−0.631777
$446$	−1.49635	−0.0708543
$447$	−16.6843	−0.789138
$448$	3.44504	0.162763
$449$	19.6485	0.927269	0.463635	−	0.886027i	$-0.346545\pi$
$449$	19.6485	0.927269	0.463635	+	0.886027i	$0.346545\pi$
$450$	−1.00000	−0.0471405
$451$	18.1293	0.853675
$452$	−11.3448	−0.533615
$453$	18.3327	0.861347
$454$	−25.2150	−1.18340
$455$	0	0
$456$	−6.26875	−0.293561
$457$	−18.4239	−0.861832	−0.430916	−	0.902392i	$-0.641809\pi$
$457$	−18.4239	−0.861832	−0.430916	+	0.902392i	$0.641809\pi$
$458$	22.3666	1.04512
$459$	6.78986	0.316923
$460$	1.30798	0.0609848
$461$	−22.2446	−1.03603	−0.518017	−	0.855370i	$-0.673330\pi$
$461$	−22.2446	−1.03603	−0.518017	+	0.855370i	$0.673330\pi$
$462$	−14.6310	−0.680697
$463$	−8.68532	−0.403641	−0.201820	−	0.979423i	$-0.564686\pi$
$463$	−8.68532	−0.403641	−0.201820	+	0.979423i	$0.564686\pi$
$464$	−9.14675	−0.424627
$465$	−3.75302	−0.174042
$466$	−14.0858	−0.652510
$467$	19.6136	0.907608	0.453804	−	0.891102i	$-0.350067\pi$
$467$	19.6136	0.907608	0.453804	+	0.891102i	$0.350067\pi$
$468$	0	0
$469$	−0.262045	−0.0121001
$470$	7.76809	0.358315
$471$	−18.6015	−0.857111
$472$	10.3327	0.475603
$473$	13.0411	0.599633
$474$	1.33513	0.0613243
$475$	6.26875	0.287630
$476$	23.3913	1.07214
$477$	−8.93900	−0.409289
$478$	−5.38106	−0.246124
$479$	31.4698	1.43789	0.718946	−	0.695066i	$-0.244625\pi$
$479$	31.4698	1.43789	0.718946	+	0.695066i	$0.244625\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	29.9420	1.36382
$483$	−4.50604	−0.205032
$484$	7.03684	0.319856
$485$	13.1903	0.598940
$486$	−1.00000	−0.0453609
$487$	13.2644	0.601069	0.300535	−	0.953771i	$-0.402835\pi$
$487$	13.2644	0.601069	0.300535	+	0.953771i	$0.402835\pi$
$488$	−2.53319	−0.114672
$489$	18.8944	0.854434
$490$	4.86831	0.219928
$491$	11.7259	0.529181	0.264591	−	0.964361i	$-0.414763\pi$
$491$	11.7259	0.529181	0.264591	+	0.964361i	$0.414763\pi$
$492$	4.26875	0.192450
$493$	−62.1051	−2.79707
$494$	0	0
$495$	−4.24698	−0.190888
$496$	3.75302	0.168516
$497$	1.28967	0.0578494
$498$	0.740939	0.0332023
$499$	−26.7633	−1.19809	−0.599045	−	0.800715i	$-0.704453\pi$
$499$	−26.7633	−1.19809	−0.599045	+	0.800715i	$0.704453\pi$
$500$	−1.00000	−0.0447214
$501$	−4.24698	−0.189741
$502$	8.73556	0.389887
$503$	17.2239	0.767975	0.383987	−	0.923338i	$-0.374551\pi$
$503$	17.2239	0.767975	0.383987	+	0.923338i	$0.374551\pi$
$504$	−3.44504	−0.153454
$505$	15.0422	0.669370
$506$	5.55496	0.246948
$507$	0	0
$508$	−2.54288	−0.112822
$509$	−28.2737	−1.25321	−0.626604	−	0.779338i	$-0.715556\pi$
$509$	−28.2737	−1.25321	−0.626604	+	0.779338i	$0.715556\pi$
$510$	6.78986	0.300660
$511$	−57.5715	−2.54681
$512$	−1.00000	−0.0441942
$513$	6.26875	0.276772
$514$	−21.2717	−0.938256
$515$	8.92154	0.393130
$516$	3.07069	0.135179
$517$	32.9909	1.45094
$518$	−23.5265	−1.03369
$519$	1.16852	0.0512924
$520$	0	0
$521$	1.47889	0.0647915	0.0323958	−	0.999475i	$-0.489686\pi$
$521$	1.47889	0.0647915	0.0323958	+	0.999475i	$0.489686\pi$
$522$	9.14675	0.400342
$523$	−28.2198	−1.23397	−0.616984	−	0.786976i	$-0.711646\pi$
$523$	−28.2198	−1.23397	−0.616984	+	0.786976i	$0.711646\pi$
$524$	−10.8509	−0.474022
$525$	3.44504	0.150354
$526$	22.2325	0.969383
$527$	25.4825	1.11003
$528$	4.24698	0.184826
$529$	−21.2892	−0.925617
$530$	−8.93900	−0.388285
$531$	−10.3327	−0.448402
$532$	21.5961	0.936310
$533$	0	0
$534$	−13.3274	−0.576731
$535$	−14.6746	−0.634437
$536$	0.0760644	0.00328548
$537$	−10.1836	−0.439454
$538$	10.2241	0.440794
$539$	20.6756	0.890562
$540$	−1.00000	−0.0430331
$541$	0.245915	0.0105727	0.00528635	−	0.999986i	$-0.498317\pi$
$541$	0.245915	0.0105727	0.00528635	+	0.999986i	$0.498317\pi$
$542$	−0.716185	−0.0307628
$543$	10.3351	0.443523
$544$	−6.78986	−0.291113
$545$	−9.66786	−0.414126
$546$	0	0
$547$	−13.6407	−0.583235	−0.291617	−	0.956535i	$-0.594193\pi$
$547$	−13.6407	−0.583235	−0.291617	+	0.956535i	$0.594193\pi$
$548$	−18.6407	−0.796292
$549$	2.53319	0.108114
$550$	−4.24698	−0.181092
$551$	−57.3387	−2.44271
$552$	1.30798	0.0556713
$553$	−4.59956	−0.195593
$554$	22.8713	0.971708
$555$	−6.82908	−0.289879
$556$	3.53750	0.150023
$557$	14.0248	0.594248	0.297124	−	0.954839i	$-0.403973\pi$
$557$	14.0248	0.594248	0.297124	+	0.954839i	$0.403973\pi$
$558$	−3.75302	−0.158878
$559$	0	0
$560$	−3.44504	−0.145580
$561$	28.8364	1.21747
$562$	−4.13036	−0.174229
$563$	−3.26981	−0.137806	−0.0689031	−	0.997623i	$-0.521950\pi$
$563$	−3.26981	−0.137806	−0.0689031	+	0.997623i	$0.521950\pi$
$564$	7.76809	0.327095
$565$	11.3448	0.477280
$566$	−16.2959	−0.684968
$567$	3.44504	0.144678
$568$	−0.374354	−0.0157076
$569$	4.10752	0.172196	0.0860982	−	0.996287i	$-0.472560\pi$
$569$	4.10752	0.172196	0.0860982	+	0.996287i	$0.472560\pi$
$570$	6.26875	0.262569
$571$	9.18465	0.384366	0.192183	−	0.981359i	$-0.438443\pi$
$571$	9.18465	0.384366	0.192183	+	0.981359i	$0.438443\pi$
$572$	0	0
$573$	−9.43296	−0.394068
$574$	−14.7060	−0.613817
$575$	−1.30798	−0.0545465
$576$	1.00000	0.0416667
$577$	28.0000	1.16566	0.582828	−	0.812596i	$-0.301946\pi$
$577$	28.0000	1.16566	0.582828	+	0.812596i	$0.301946\pi$
$578$	−29.1021	−1.21049
$579$	22.9191	0.952487
$580$	9.14675	0.379798
$581$	−2.55257	−0.105898
$582$	13.1903	0.546755
$583$	−37.9638	−1.57230
$584$	16.7114	0.691523
$585$	0	0
$586$	−7.72587	−0.319153
$587$	−47.6765	−1.96782	−0.983910	−	0.178668i	$-0.942821\pi$
$587$	−47.6765	−1.96782	−0.983910	+	0.178668i	$0.942821\pi$
$588$	4.86831	0.200766
$589$	23.5267	0.969403
$590$	−10.3327	−0.425392
$591$	−14.3502	−0.590288
$592$	6.82908	0.280674
$593$	−16.7681	−0.688583	−0.344291	−	0.938863i	$-0.611881\pi$
$593$	−16.7681	−0.688583	−0.344291	+	0.938863i	$0.611881\pi$
$594$	−4.24698	−0.174256
$595$	−23.3913	−0.958951
$596$	−16.6843	−0.683414
$597$	14.2524	0.583310
$598$	0	0
$599$	44.0157	1.79843	0.899215	−	0.437506i	$-0.144138\pi$
$599$	44.0157	1.79843	0.899215	+	0.437506i	$0.144138\pi$
$600$	−1.00000	−0.0408248
$601$	39.1299	1.59614	0.798071	−	0.602564i	$-0.205854\pi$
$601$	39.1299	1.59614	0.798071	+	0.602564i	$0.205854\pi$
$602$	−10.5786	−0.431153
$603$	−0.0760644	−0.00309758
$604$	18.3327	0.745948
$605$	−7.03684	−0.286088
$606$	15.0422	0.611048
$607$	6.33273	0.257038	0.128519	−	0.991707i	$-0.458978\pi$
$607$	6.33273	0.257038	0.128519	+	0.991707i	$0.458978\pi$
$608$	−6.26875	−0.254231
$609$	−31.5109	−1.27689
$610$	2.53319	0.102566
$611$	0	0
$612$	6.78986	0.274464
$613$	29.1129	1.17586	0.587929	−	0.808912i	$-0.299943\pi$
$613$	29.1129	1.17586	0.587929	+	0.808912i	$0.299943\pi$
$614$	5.13036	0.207044
$615$	−4.26875	−0.172133
$616$	−14.6310	−0.589501
$617$	−27.9933	−1.12697	−0.563484	−	0.826127i	$-0.690539\pi$
$617$	−27.9933	−1.12697	−0.563484	+	0.826127i	$0.690539\pi$
$618$	8.92154	0.358877
$619$	23.0084	0.924784	0.462392	−	0.886676i	$-0.346991\pi$
$619$	23.0084	0.924784	0.462392	+	0.886676i	$0.346991\pi$
$620$	−3.75302	−0.150725
$621$	−1.30798	−0.0524874
$622$	−33.1987	−1.33114
$623$	45.9133	1.83948
$624$	0	0
$625$	1.00000	0.0400000
$626$	−0.0814412	−0.00325505
$627$	26.6233	1.06323
$628$	−18.6015	−0.742280
$629$	46.3685	1.84883
$630$	3.44504	0.137254
$631$	−6.18731	−0.246313	−0.123156	−	0.992387i	$-0.539302\pi$
$631$	−6.18731	−0.246313	−0.123156	+	0.992387i	$0.539302\pi$
$632$	1.33513	0.0531084
$633$	0.396125	0.0157445
$634$	19.4969	0.774323
$635$	2.54288	0.100911
$636$	−8.93900	−0.354454
$637$	0	0
$638$	38.8461	1.53793
$639$	0.374354	0.0148092
$640$	1.00000	0.0395285
$641$	39.1159	1.54498	0.772492	−	0.635024i	$-0.219010\pi$
$641$	39.1159	1.54498	0.772492	+	0.635024i	$0.219010\pi$
$642$	−14.6746	−0.579159
$643$	8.78209	0.346332	0.173166	−	0.984893i	$-0.444600\pi$
$643$	8.78209	0.346332	0.173166	+	0.984893i	$0.444600\pi$
$644$	−4.50604	−0.177563
$645$	−3.07069	−0.120908
$646$	−42.5639	−1.67465
$647$	−30.1333	−1.18466	−0.592332	−	0.805694i	$-0.701793\pi$
$647$	−30.1333	−1.18466	−0.592332	+	0.805694i	$0.701793\pi$
$648$	−1.00000	−0.0392837
$649$	−43.8829	−1.72255
$650$	0	0
$651$	12.9293	0.506740
$652$	18.8944	0.739962
$653$	−27.0858	−1.05995	−0.529974	−	0.848014i	$-0.677798\pi$
$653$	−27.0858	−1.05995	−0.529974	+	0.848014i	$0.677798\pi$
$654$	−9.66786	−0.378043
$655$	10.8509	0.423978
$656$	4.26875	0.166667
$657$	−16.7114	−0.651974
$658$	−26.7614	−1.04327
$659$	5.69681	0.221916	0.110958	−	0.993825i	$-0.464608\pi$
$659$	5.69681	0.221916	0.110958	+	0.993825i	$0.464608\pi$
$660$	−4.24698	−0.165313
$661$	−17.1008	−0.665145	−0.332572	−	0.943078i	$-0.607917\pi$
$661$	−17.1008	−0.665145	−0.332572	+	0.943078i	$0.607917\pi$
$662$	−23.8049	−0.925205
$663$	0	0
$664$	0.740939	0.0287540
$665$	−21.5961	−0.837461
$666$	−6.82908	−0.264622
$667$	11.9638	0.463238
$668$	−4.24698	−0.164321
$669$	1.49635	0.0578523
$670$	−0.0760644	−0.00293862
$671$	10.7584	0.415323
$672$	−3.44504	−0.132895
$673$	−43.1957	−1.66507	−0.832535	−	0.553972i	$-0.813111\pi$
$673$	−43.1957	−1.66507	−0.832535	+	0.553972i	$0.813111\pi$
$674$	−21.2064	−0.816841
$675$	1.00000	0.0384900
$676$	0	0
$677$	−7.78746	−0.299297	−0.149648	−	0.988739i	$-0.547814\pi$
$677$	−7.78746	−0.299297	−0.149648	+	0.988739i	$0.547814\pi$
$678$	11.3448	0.435695
$679$	−45.4411	−1.74387
$680$	6.78986	0.260379
$681$	25.2150	0.966243
$682$	−15.9390	−0.610336
$683$	30.9138	1.18288	0.591441	−	0.806348i	$-0.298559\pi$
$683$	30.9138	1.18288	0.591441	+	0.806348i	$0.298559\pi$
$684$	6.26875	0.239692
$685$	18.6407	0.712225
$686$	7.34375	0.280386
$687$	−22.3666	−0.853338
$688$	3.07069	0.117069
$689$	0	0
$690$	−1.30798	−0.0497939
$691$	−6.20237	−0.235949	−0.117975	−	0.993017i	$-0.537640\pi$
$691$	−6.20237	−0.235949	−0.117975	+	0.993017i	$0.537640\pi$
$692$	1.16852	0.0444206
$693$	14.6310	0.555787
$694$	16.5773	0.629266
$695$	−3.53750	−0.134185
$696$	9.14675	0.346707
$697$	28.9842	1.09785
$698$	−2.51035	−0.0950182
$699$	14.0858	0.532772
$700$	3.44504	0.130210
$701$	26.3220	0.994167	0.497084	−	0.867703i	$-0.334404\pi$
$701$	26.3220	0.994167	0.497084	+	0.867703i	$0.334404\pi$
$702$	0	0
$703$	42.8098	1.61460
$704$	4.24698	0.160064
$705$	−7.76809	−0.292563
$706$	20.8412	0.784368
$707$	−51.8211	−1.94893
$708$	−10.3327	−0.388328
$709$	33.7429	1.26724	0.633620	−	0.773645i	$-0.281568\pi$
$709$	33.7429	1.26724	0.633620	+	0.773645i	$0.281568\pi$
$710$	0.374354	0.0140493
$711$	−1.33513	−0.0500711
$712$	−13.3274	−0.499464
$713$	−4.90887	−0.183839
$714$	−23.3913	−0.875399
$715$	0	0
$716$	−10.1836	−0.380579
$717$	5.38106	0.200959
$718$	−2.42566	−0.0905250
$719$	−9.50173	−0.354355	−0.177177	−	0.984179i	$-0.556697\pi$
$719$	−9.50173	−0.354355	−0.177177	+	0.984179i	$0.556697\pi$
$720$	−1.00000	−0.0372678
$721$	−30.7351	−1.14463
$722$	−20.2972	−0.755384
$723$	−29.9420	−1.11355
$724$	10.3351	0.384102
$725$	−9.14675	−0.339702
$726$	−7.03684	−0.261161
$727$	−12.2631	−0.454814	−0.227407	−	0.973800i	$-0.573025\pi$
$727$	−12.2631	−0.454814	−0.227407	+	0.973800i	$0.573025\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−16.7114	−0.618517
$731$	20.8495	0.771148
$732$	2.53319	0.0936293
$733$	1.52217	0.0562227	0.0281113	−	0.999605i	$-0.491051\pi$
$733$	1.52217	0.0562227	0.0281113	+	0.999605i	$0.491051\pi$
$734$	−2.66248	−0.0982740
$735$	−4.86831	−0.179570
$736$	1.30798	0.0482127
$737$	−0.323044	−0.0118995
$738$	−4.26875	−0.157135
$739$	51.1075	1.88002	0.940010	−	0.341146i	$-0.110815\pi$
$739$	51.1075	1.88002	0.940010	+	0.341146i	$0.110815\pi$
$740$	−6.82908	−0.251042
$741$	0	0
$742$	30.7952	1.13053
$743$	−9.08144	−0.333166	−0.166583	−	0.986027i	$-0.553273\pi$
$743$	−9.08144	−0.333166	−0.166583	+	0.986027i	$0.553273\pi$
$744$	−3.75302	−0.137592
$745$	16.6843	0.611264
$746$	−5.87800	−0.215209
$747$	−0.740939	−0.0271095
$748$	28.8364	1.05436
$749$	50.5545	1.84722
$750$	1.00000	0.0365148
$751$	31.7090	1.15708	0.578539	−	0.815655i	$-0.303623\pi$
$751$	31.7090	1.15708	0.578539	+	0.815655i	$0.303623\pi$
$752$	7.76809	0.283273
$753$	−8.73556	−0.318342
$754$	0	0
$755$	−18.3327	−0.667196
$756$	3.44504	0.125295
$757$	−8.49934	−0.308914	−0.154457	−	0.988000i	$-0.549363\pi$
$757$	−8.49934	−0.308914	−0.154457	+	0.988000i	$0.549363\pi$
$758$	−36.1540	−1.31317
$759$	−5.55496	−0.201632
$760$	6.26875	0.227391
$761$	−16.4198	−0.595218	−0.297609	−	0.954688i	$-0.596189\pi$
$761$	−16.4198	−0.595218	−0.297609	+	0.954688i	$0.596189\pi$
$762$	2.54288	0.0921187
$763$	33.3062	1.20576
$764$	−9.43296	−0.341273
$765$	−6.78986	−0.245488
$766$	8.98361	0.324591
$767$	0	0
$768$	1.00000	0.0360844
$769$	−27.1011	−0.977290	−0.488645	−	0.872483i	$-0.662509\pi$
$769$	−27.1011	−0.977290	−0.488645	+	0.872483i	$0.662509\pi$
$770$	14.6310	0.527265
$771$	21.2717	0.766083
$772$	22.9191	0.824878
$773$	12.6297	0.454259	0.227129	−	0.973865i	$-0.427066\pi$
$773$	12.6297	0.454259	0.227129	+	0.973865i	$0.427066\pi$
$774$	−3.07069	−0.110374
$775$	3.75302	0.134812
$776$	13.1903	0.473504
$777$	23.5265	0.844008
$778$	−0.655186	−0.0234895
$779$	26.7597	0.958767
$780$	0	0
$781$	1.58987	0.0568902
$782$	8.88099	0.317583
$783$	−9.14675	−0.326878
$784$	4.86831	0.173868
$785$	18.6015	0.663915
$786$	10.8509	0.387037
$787$	3.02954	0.107991	0.0539957	−	0.998541i	$-0.482804\pi$
$787$	3.02954	0.107991	0.0539957	+	0.998541i	$0.482804\pi$
$788$	−14.3502	−0.511204
$789$	−22.2325	−0.791498
$790$	−1.33513	−0.0475016
$791$	−39.0834	−1.38964
$792$	−4.24698	−0.150910
$793$	0	0
$794$	−0.271143	−0.00962250
$795$	8.93900	0.317034
$796$	14.2524	0.505161
$797$	40.2218	1.42473	0.712364	−	0.701810i	$-0.247625\pi$
$797$	40.2218	1.42473	0.712364	+	0.701810i	$0.247625\pi$
$798$	−21.5961	−0.764494
$799$	52.7442	1.86596
$800$	−1.00000	−0.0353553
$801$	13.3274	0.470899
$802$	−0.344814	−0.0121758
$803$	−70.9730	−2.50458
$804$	−0.0760644	−0.00268259
$805$	4.50604	0.158817
$806$	0	0
$807$	−10.2241	−0.359907
$808$	15.0422	0.529183
$809$	33.7918	1.18806	0.594028	−	0.804445i	$-0.297537\pi$
$809$	33.7918	1.18806	0.594028	+	0.804445i	$0.297537\pi$
$810$	1.00000	0.0351364
$811$	33.6534	1.18173	0.590865	−	0.806770i	$-0.298787\pi$
$811$	33.6534	1.18173	0.590865	+	0.806770i	$0.298787\pi$
$812$	−31.5109	−1.10582
$813$	0.716185	0.0251177
$814$	−29.0030	−1.01655
$815$	−18.8944	−0.661842
$816$	6.78986	0.237693
$817$	19.2494	0.673450
$818$	−27.0388	−0.945388
$819$	0	0
$820$	−4.26875	−0.149071
$821$	12.6939	0.443022	0.221511	−	0.975158i	$-0.428901\pi$
$821$	12.6939	0.443022	0.221511	+	0.975158i	$0.428901\pi$
$822$	18.6407	0.650169
$823$	−8.79523	−0.306583	−0.153291	−	0.988181i	$-0.548987\pi$
$823$	−8.79523	−0.306583	−0.153291	+	0.988181i	$0.548987\pi$
$824$	8.92154	0.310797
$825$	4.24698	0.147861
$826$	35.5967	1.23857
$827$	32.1758	1.11886	0.559431	−	0.828877i	$-0.311020\pi$
$827$	32.1758	1.11886	0.559431	+	0.828877i	$0.311020\pi$
$828$	−1.30798	−0.0454554
$829$	21.1812	0.735653	0.367827	−	0.929894i	$-0.380102\pi$
$829$	21.1812	0.735653	0.367827	+	0.929894i	$0.380102\pi$
$830$	−0.740939	−0.0257184
$831$	−22.8713	−0.793397
$832$	0	0
$833$	33.0551	1.14529
$834$	−3.53750	−0.122494
$835$	4.24698	0.146973
$836$	26.6233	0.920784
$837$	3.75302	0.129723
$838$	−5.86533	−0.202614
$839$	26.5931	0.918097	0.459048	−	0.888411i	$-0.348190\pi$
$839$	26.5931	0.918097	0.459048	+	0.888411i	$0.348190\pi$
$840$	3.44504	0.118865
$841$	54.6631	1.88493
$842$	28.9476	0.997601
$843$	4.13036	0.142257
$844$	0.396125	0.0136352
$845$	0	0
$846$	−7.76809	−0.267072
$847$	24.2422	0.832972
$848$	−8.93900	−0.306967
$849$	16.2959	0.559274
$850$	−6.78986	−0.232890
$851$	−8.93230	−0.306195
$852$	0.374354	0.0128252
$853$	−32.2457	−1.10407	−0.552035	−	0.833821i	$-0.686149\pi$
$853$	−32.2457	−1.10407	−0.552035	+	0.833821i	$0.686149\pi$
$854$	−8.72694	−0.298630
$855$	−6.26875	−0.214387
$856$	−14.6746	−0.501566
$857$	−43.0176	−1.46945	−0.734726	−	0.678364i	$-0.762689\pi$
$857$	−43.0176	−1.46945	−0.734726	+	0.678364i	$0.762689\pi$
$858$	0	0
$859$	−31.1997	−1.06452	−0.532260	−	0.846581i	$-0.678657\pi$
$859$	−31.1997	−1.06452	−0.532260	+	0.846581i	$0.678657\pi$
$860$	−3.07069	−0.104710
$861$	14.7060	0.501180
$862$	−4.87023	−0.165881
$863$	−34.5424	−1.17584	−0.587919	−	0.808920i	$-0.700052\pi$
$863$	−34.5424	−1.17584	−0.587919	+	0.808920i	$0.700052\pi$
$864$	−1.00000	−0.0340207
$865$	−1.16852	−0.0397309
$866$	36.9788	1.25659
$867$	29.1021	0.988361
$868$	12.9293	0.438849
$869$	−5.67025	−0.192350
$870$	−9.14675	−0.310104
$871$	0	0
$872$	−9.66786	−0.327395
$873$	−13.1903	−0.446424
$874$	8.19939	0.277349
$875$	−3.44504	−0.116464
$876$	−16.7114	−0.564626
$877$	2.12093	0.0716188	0.0358094	−	0.999359i	$-0.488599\pi$
$877$	2.12093	0.0716188	0.0358094	+	0.999359i	$0.488599\pi$
$878$	−22.8799	−0.772160
$879$	7.72587	0.260587
$880$	−4.24698	−0.143166
$881$	27.3489	0.921407	0.460703	−	0.887554i	$-0.347597\pi$
$881$	27.3489	0.921407	0.460703	+	0.887554i	$0.347597\pi$
$882$	−4.86831	−0.163925
$883$	−48.2959	−1.62529	−0.812643	−	0.582762i	$-0.801972\pi$
$883$	−48.2959	−1.62529	−0.812643	+	0.582762i	$0.801972\pi$
$884$	0	0
$885$	10.3327	0.347331
$886$	10.5942	0.355919
$887$	25.1943	0.845943	0.422972	−	0.906143i	$-0.360987\pi$
$887$	25.1943	0.845943	0.422972	+	0.906143i	$0.360987\pi$
$888$	−6.82908	−0.229169
$889$	−8.76032	−0.293812
$890$	13.3274	0.446734
$891$	4.24698	0.142279
$892$	1.49635	0.0501016
$893$	48.6962	1.62956
$894$	16.6843	0.558005
$895$	10.1836	0.340400
$896$	−3.44504	−0.115091
$897$	0	0
$898$	−19.6485	−0.655678
$899$	−34.3279	−1.14490
$900$	1.00000	0.0333333
$901$	−60.6945	−2.02203
$902$	−18.1293	−0.603639
$903$	10.5786	0.352035
$904$	11.3448	0.377323
$905$	−10.3351	−0.343551
$906$	−18.3327	−0.609064
$907$	−20.0949	−0.667239	−0.333619	−	0.942708i	$-0.608270\pi$
$907$	−20.0949	−0.667239	−0.333619	+	0.942708i	$0.608270\pi$
$908$	25.2150	0.836791
$909$	−15.0422	−0.498919
$910$	0	0
$911$	−7.44563	−0.246685	−0.123342	−	0.992364i	$-0.539361\pi$
$911$	−7.44563	−0.246685	−0.123342	+	0.992364i	$0.539361\pi$
$912$	6.26875	0.207579
$913$	−3.14675	−0.104142
$914$	18.4239	0.609407
$915$	−2.53319	−0.0837446
$916$	−22.3666	−0.739013
$917$	−37.3817	−1.23445
$918$	−6.78986	−0.224099
$919$	−14.2765	−0.470939	−0.235469	−	0.971882i	$-0.575663\pi$
$919$	−14.2765	−0.470939	−0.235469	+	0.971882i	$0.575663\pi$
$920$	−1.30798	−0.0431228
$921$	−5.13036	−0.169051
$922$	22.2446	0.732586
$923$	0	0
$924$	14.6310	0.481325
$925$	6.82908	0.224539
$926$	8.68532	0.285417
$927$	−8.92154	−0.293022
$928$	9.14675	0.300257
$929$	30.9148	1.01428	0.507141	−	0.861863i	$-0.330702\pi$
$929$	30.9148	1.01428	0.507141	+	0.861863i	$0.330702\pi$
$930$	3.75302	0.123066
$931$	30.5182	1.00019
$932$	14.0858	0.461394
$933$	33.1987	1.08688
$934$	−19.6136	−0.641775
$935$	−28.8364	−0.943050
$936$	0	0
$937$	−16.7187	−0.546176	−0.273088	−	0.961989i	$-0.588045\pi$
$937$	−16.7187	−0.546176	−0.273088	+	0.961989i	$0.588045\pi$
$938$	0.262045	0.00855608
$939$	0.0814412	0.00265773
$940$	−7.76809	−0.253367
$941$	36.9033	1.20301	0.601507	−	0.798867i	$-0.294567\pi$
$941$	36.9033	1.20301	0.601507	+	0.798867i	$0.294567\pi$
$942$	18.6015	0.606069
$943$	−5.58343	−0.181822
$944$	−10.3327	−0.336302
$945$	−3.44504	−0.112067
$946$	−13.0411	−0.424004
$947$	−39.1661	−1.27273	−0.636364	−	0.771389i	$-0.719562\pi$
$947$	−39.1661	−1.27273	−0.636364	+	0.771389i	$0.719562\pi$
$948$	−1.33513	−0.0433629
$949$	0	0
$950$	−6.26875	−0.203385
$951$	−19.4969	−0.632232
$952$	−23.3913	−0.758118
$953$	−19.6200	−0.635554	−0.317777	−	0.948165i	$-0.602936\pi$
$953$	−19.6200	−0.635554	−0.317777	+	0.948165i	$0.602936\pi$
$954$	8.93900	0.289411
$955$	9.43296	0.305243
$956$	5.38106	0.174036
$957$	−38.8461	−1.25572
$958$	−31.4698	−1.01674
$959$	−64.2180	−2.07371
$960$	−1.00000	−0.0322749
$961$	−16.9148	−0.545640
$962$	0	0
$963$	14.6746	0.472881
$964$	−29.9420	−0.964366
$965$	−22.9191	−0.737794
$966$	4.50604	0.144979
$967$	−43.1564	−1.38782	−0.693909	−	0.720063i	$-0.744113\pi$
$967$	−43.1564	−1.38782	−0.693909	+	0.720063i	$0.744113\pi$
$968$	−7.03684	−0.226172
$969$	42.5639	1.36735
$970$	−13.1903	−0.423515
$971$	39.5032	1.26772	0.633859	−	0.773449i	$-0.281470\pi$
$971$	39.5032	1.26772	0.633859	+	0.773449i	$0.281470\pi$
$972$	1.00000	0.0320750
$973$	12.1868	0.390692
$974$	−13.2644	−0.425020
$975$	0	0
$976$	2.53319	0.0810854
$977$	−3.84117	−0.122890	−0.0614449	−	0.998110i	$-0.519571\pi$
$977$	−3.84117	−0.122890	−0.0614449	+	0.998110i	$0.519571\pi$
$978$	−18.8944	−0.604176
$979$	56.6010	1.80898
$980$	−4.86831	−0.155513
$981$	9.66786	0.308671
$982$	−11.7259	−0.374188
$983$	−32.4300	−1.03436	−0.517178	−	0.855878i	$-0.673017\pi$
$983$	−32.4300	−1.03436	−0.517178	+	0.855878i	$0.673017\pi$
$984$	−4.26875	−0.136083
$985$	14.3502	0.457235
$986$	62.1051	1.97783
$987$	26.7614	0.851824
$988$	0	0
$989$	−4.01639	−0.127714
$990$	4.24698	0.134978
$991$	−61.9807	−1.96888	−0.984442	−	0.175712i	$-0.943777\pi$
$991$	−61.9807	−1.96888	−0.984442	+	0.175712i	$0.943777\pi$
$992$	−3.75302	−0.119159
$993$	23.8049	0.755426
$994$	−1.28967	−0.0409057
$995$	−14.2524	−0.451830
$996$	−0.740939	−0.0234775
$997$	28.9162	0.915784	0.457892	−	0.889008i	$-0.348605\pi$
$997$	28.9162	0.915784	0.457892	+	0.889008i	$0.348605\pi$
$998$	26.7633	0.847177
$999$	6.82908	0.216063

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bo.1.2	✓	3	1.1	even	1	trivial
5070.2.a.bx.1.2	yes	3	13.12	even	2
5070.2.b.w.1351.2		6	13.8	odd	4
5070.2.b.w.1351.5		6	13.5	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.44504 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.44504 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.24698 q^{11} +1.00000 q^{12} -3.44504 q^{14} -1.00000 q^{15} +1.00000 q^{16} +6.78986 q^{17} -1.00000 q^{18} +6.26875 q^{19} -1.00000 q^{20} +3.44504 q^{21} -4.24698 q^{22} -1.30798 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +3.44504 q^{28} -9.14675 q^{29} +1.00000 q^{30} +3.75302 q^{31} -1.00000 q^{32} +4.24698 q^{33} -6.78986 q^{34} -3.44504 q^{35} +1.00000 q^{36} +6.82908 q^{37} -6.26875 q^{38} +1.00000 q^{40} +4.26875 q^{41} -3.44504 q^{42} +3.07069 q^{43} +4.24698 q^{44} -1.00000 q^{45} +1.30798 q^{46} +7.76809 q^{47} +1.00000 q^{48} +4.86831 q^{49} -1.00000 q^{50} +6.78986 q^{51} -8.93900 q^{53} -1.00000 q^{54} -4.24698 q^{55} -3.44504 q^{56} +6.26875 q^{57} +9.14675 q^{58} -10.3327 q^{59} -1.00000 q^{60} +2.53319 q^{61} -3.75302 q^{62} +3.44504 q^{63} +1.00000 q^{64} -4.24698 q^{66} -0.0760644 q^{67} +6.78986 q^{68} -1.30798 q^{69} +3.44504 q^{70} +0.374354 q^{71} -1.00000 q^{72} -16.7114 q^{73} -6.82908 q^{74} +1.00000 q^{75} +6.26875 q^{76} +14.6310 q^{77} -1.33513 q^{79} -1.00000 q^{80} +1.00000 q^{81} -4.26875 q^{82} -0.740939 q^{83} +3.44504 q^{84} -6.78986 q^{85} -3.07069 q^{86} -9.14675 q^{87} -4.24698 q^{88} +13.3274 q^{89} +1.00000 q^{90} -1.30798 q^{92} +3.75302 q^{93} -7.76809 q^{94} -6.26875 q^{95} -1.00000 q^{96} -13.1903 q^{97} -4.86831 q^{98} +4.24698 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} + 10 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 + 10 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} + 10 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{10} + 8 q^{11} + 3 q^{12} - 10 q^{14} - 3 q^{15} + 3 q^{16} - 3 q^{17} - 3 q^{18} + 11 q^{19} - 3 q^{20} + 10 q^{21} - 8 q^{22} - 9 q^{23} - 3 q^{24} + 3 q^{25} + 3 q^{27} + 10 q^{28} + 3 q^{30} + 16 q^{31} - 3 q^{32} + 8 q^{33} + 3 q^{34} - 10 q^{35} + 3 q^{36} + 10 q^{37} - 11 q^{38} + 3 q^{40} + 5 q^{41} - 10 q^{42} - 3 q^{43} + 8 q^{44} - 3 q^{45} + 9 q^{46} + 3 q^{47} + 3 q^{48} + 17 q^{49} - 3 q^{50} - 3 q^{51} - 17 q^{53} - 3 q^{54} - 8 q^{55} - 10 q^{56} + 11 q^{57} + 11 q^{59} - 3 q^{60} + 11 q^{61} - 16 q^{62} + 10 q^{63} + 3 q^{64} - 8 q^{66} + 15 q^{67} - 3 q^{68} - 9 q^{69} + 10 q^{70} + 13 q^{71} - 3 q^{72} - q^{73} - 10 q^{74} + 3 q^{75} + 11 q^{76} + 29 q^{77} - 3 q^{79} - 3 q^{80} + 3 q^{81} - 5 q^{82} + 12 q^{83} + 10 q^{84} + 3 q^{85} + 3 q^{86} - 8 q^{88} + q^{89} + 3 q^{90} - 9 q^{92} + 16 q^{93} - 3 q^{94} - 11 q^{95} - 3 q^{96} - 6 q^{97} - 17 q^{98} + 8 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^5 - 3 * q^6 + 10 * q^7 - 3 * q^8 + 3 * q^9 + 3 * q^10 + 8 * q^11 + 3 * q^12 - 10 * q^14 - 3 * q^15 + 3 * q^16 - 3 * q^17 - 3 * q^18 + 11 * q^19 - 3 * q^20 + 10 * q^21 - 8 * q^22 - 9 * q^23 - 3 * q^24 + 3 * q^25 + 3 * q^27 + 10 * q^28 + 3 * q^30 + 16 * q^31 - 3 * q^32 + 8 * q^33 + 3 * q^34 - 10 * q^35 + 3 * q^36 + 10 * q^37 - 11 * q^38 + 3 * q^40 + 5 * q^41 - 10 * q^42 - 3 * q^43 + 8 * q^44 - 3 * q^45 + 9 * q^46 + 3 * q^47 + 3 * q^48 + 17 * q^49 - 3 * q^50 - 3 * q^51 - 17 * q^53 - 3 * q^54 - 8 * q^55 - 10 * q^56 + 11 * q^57 + 11 * q^59 - 3 * q^60 + 11 * q^61 - 16 * q^62 + 10 * q^63 + 3 * q^64 - 8 * q^66 + 15 * q^67 - 3 * q^68 - 9 * q^69 + 10 * q^70 + 13 * q^71 - 3 * q^72 - q^73 - 10 * q^74 + 3 * q^75 + 11 * q^76 + 29 * q^77 - 3 * q^79 - 3 * q^80 + 3 * q^81 - 5 * q^82 + 12 * q^83 + 10 * q^84 + 3 * q^85 + 3 * q^86 - 8 * q^88 + q^89 + 3 * q^90 - 9 * q^92 + 16 * q^93 - 3 * q^94 - 11 * q^95 - 3 * q^96 - 6 * q^97 - 17 * q^98 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more