LMFDB - Embedded newform 5070.2.a.by.1.1

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			1.1
Root			$1.80194$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -0.801938 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -0.801938 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.04892 q^{11} +1.00000 q^{12} -0.801938 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.24698 q^{17} +1.00000 q^{18} +7.40581 q^{19} +1.00000 q^{20} -0.801938 q^{21} -3.04892 q^{22} -0.356896 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -0.801938 q^{28} +0.307979 q^{29} +1.00000 q^{30} +3.44504 q^{31} +1.00000 q^{32} -3.04892 q^{33} -1.24698 q^{34} -0.801938 q^{35} +1.00000 q^{36} +7.29590 q^{37} +7.40581 q^{38} +1.00000 q^{40} +3.40581 q^{41} -0.801938 q^{42} +6.54288 q^{43} -3.04892 q^{44} +1.00000 q^{45} -0.356896 q^{46} +0.603875 q^{47} +1.00000 q^{48} -6.35690 q^{49} +1.00000 q^{50} -1.24698 q^{51} -6.29590 q^{53} +1.00000 q^{54} -3.04892 q^{55} -0.801938 q^{56} +7.40581 q^{57} +0.307979 q^{58} +3.11529 q^{59} +1.00000 q^{60} +11.0368 q^{61} +3.44504 q^{62} -0.801938 q^{63} +1.00000 q^{64} -3.04892 q^{66} +5.96077 q^{67} -1.24698 q^{68} -0.356896 q^{69} -0.801938 q^{70} +6.67456 q^{71} +1.00000 q^{72} +10.1371 q^{73} +7.29590 q^{74} +1.00000 q^{75} +7.40581 q^{76} +2.44504 q^{77} -13.6799 q^{79} +1.00000 q^{80} +1.00000 q^{81} +3.40581 q^{82} -3.93900 q^{83} -0.801938 q^{84} -1.24698 q^{85} +6.54288 q^{86} +0.307979 q^{87} -3.04892 q^{88} -8.05861 q^{89} +1.00000 q^{90} -0.356896 q^{92} +3.44504 q^{93} +0.603875 q^{94} +7.40581 q^{95} +1.00000 q^{96} +9.73556 q^{97} -6.35690 q^{98} -3.04892 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + 2 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 + 2 * 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} + 2 q^{7} + 3 q^{8} + 3 q^{9} + 3 q^{10} + 3 q^{12} + 2 q^{14} + 3 q^{15} + 3 q^{16} + q^{17} + 3 q^{18} + 9 q^{19} + 3 q^{20} + 2 q^{21} + 3 q^{23} + 3 q^{24} + 3 q^{25} + 3 q^{27} + 2 q^{28} + 6 q^{29} + 3 q^{30} + 10 q^{31} + 3 q^{32} + q^{34} + 2 q^{35} + 3 q^{36} + 8 q^{37} + 9 q^{38} + 3 q^{40} - 3 q^{41} + 2 q^{42} + q^{43} + 3 q^{45} + 3 q^{46} - 7 q^{47} + 3 q^{48} - 15 q^{49} + 3 q^{50} + q^{51} - 5 q^{53} + 3 q^{54} + 2 q^{56} + 9 q^{57} + 6 q^{58} + 7 q^{59} + 3 q^{60} + 5 q^{61} + 10 q^{62} + 2 q^{63} + 3 q^{64} + 5 q^{67} + q^{68} + 3 q^{69} + 2 q^{70} - q^{71} + 3 q^{72} + 25 q^{73} + 8 q^{74} + 3 q^{75} + 9 q^{76} + 7 q^{77} - 17 q^{79} + 3 q^{80} + 3 q^{81} - 3 q^{82} - 2 q^{83} + 2 q^{84} + q^{85} + q^{86} + 6 q^{87} + 7 q^{89} + 3 q^{90} + 3 q^{92} + 10 q^{93} - 7 q^{94} + 9 q^{95} + 3 q^{96} + 18 q^{97} - 15 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9 + 3 * q^10 + 3 * q^12 + 2 * q^14 + 3 * q^15 + 3 * q^16 + q^17 + 3 * q^18 + 9 * q^19 + 3 * q^20 + 2 * q^21 + 3 * q^23 + 3 * q^24 + 3 * q^25 + 3 * q^27 + 2 * q^28 + 6 * q^29 + 3 * q^30 + 10 * q^31 + 3 * q^32 + q^34 + 2 * q^35 + 3 * q^36 + 8 * q^37 + 9 * q^38 + 3 * q^40 - 3 * q^41 + 2 * q^42 + q^43 + 3 * q^45 + 3 * q^46 - 7 * q^47 + 3 * q^48 - 15 * q^49 + 3 * q^50 + q^51 - 5 * q^53 + 3 * q^54 + 2 * q^56 + 9 * q^57 + 6 * q^58 + 7 * q^59 + 3 * q^60 + 5 * q^61 + 10 * q^62 + 2 * q^63 + 3 * q^64 + 5 * q^67 + q^68 + 3 * q^69 + 2 * q^70 - q^71 + 3 * q^72 + 25 * q^73 + 8 * q^74 + 3 * q^75 + 9 * q^76 + 7 * q^77 - 17 * q^79 + 3 * q^80 + 3 * q^81 - 3 * q^82 - 2 * q^83 + 2 * q^84 + q^85 + q^86 + 6 * q^87 + 7 * q^89 + 3 * q^90 + 3 * q^92 + 10 * q^93 - 7 * q^94 + 9 * q^95 + 3 * q^96 + 18 * q^97 - 15 * q^98

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
$7$	−0.801938	−0.303104	−0.151552	−	0.988449i	$-0.548427\pi$
$7$	−0.801938	−0.303104	−0.151552	+	0.988449i	$0.548427\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−3.04892	−0.919283	−0.459642	−	0.888105i	$-0.652022\pi$
$11$	−3.04892	−0.919283	−0.459642	+	0.888105i	$0.652022\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−0.801938	−0.214327
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−1.24698	−0.302437	−0.151218	−	0.988500i	$-0.548320\pi$
$17$	−1.24698	−0.302437	−0.151218	+	0.988500i	$0.548320\pi$
$18$	1.00000	0.235702
$19$	7.40581	1.69901	0.849505	−	0.527581i	$-0.176901\pi$
$19$	7.40581	1.69901	0.849505	+	0.527581i	$0.176901\pi$
$20$	1.00000	0.223607
$21$	−0.801938	−0.174997
$22$	−3.04892	−0.650031
$23$	−0.356896	−0.0744179	−0.0372090	−	0.999308i	$-0.511847\pi$
$23$	−0.356896	−0.0744179	−0.0372090	+	0.999308i	$0.511847\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	−0.801938	−0.151552
$29$	0.307979	0.0571902	0.0285951	−	0.999591i	$-0.490897\pi$
$29$	0.307979	0.0571902	0.0285951	+	0.999591i	$0.490897\pi$
$30$	1.00000	0.182574
$31$	3.44504	0.618748	0.309374	−	0.950940i	$-0.399881\pi$
$31$	3.44504	0.618748	0.309374	+	0.950940i	$0.399881\pi$
$32$	1.00000	0.176777
$33$	−3.04892	−0.530748
$34$	−1.24698	−0.213855
$35$	−0.801938	−0.135552
$36$	1.00000	0.166667
$37$	7.29590	1.19944	0.599719	−	0.800211i	$-0.295279\pi$
$37$	7.29590	1.19944	0.599719	+	0.800211i	$0.295279\pi$
$38$	7.40581	1.20138
$39$	0	0
$40$	1.00000	0.158114
$41$	3.40581	0.531899	0.265949	−	0.963987i	$-0.414315\pi$
$41$	3.40581	0.531899	0.265949	+	0.963987i	$0.414315\pi$
$42$	−0.801938	−0.123742
$43$	6.54288	0.997779	0.498890	−	0.866666i	$-0.333741\pi$
$43$	6.54288	0.997779	0.498890	+	0.866666i	$0.333741\pi$
$44$	−3.04892	−0.459642
$45$	1.00000	0.149071
$46$	−0.356896	−0.0526214
$47$	0.603875	0.0880843	0.0440421	−	0.999030i	$-0.485976\pi$
$47$	0.603875	0.0880843	0.0440421	+	0.999030i	$0.485976\pi$
$48$	1.00000	0.144338
$49$	−6.35690	−0.908128
$50$	1.00000	0.141421
$51$	−1.24698	−0.174612
$52$	0	0
$53$	−6.29590	−0.864808	−0.432404	−	0.901680i	$-0.642335\pi$
$53$	−6.29590	−0.864808	−0.432404	+	0.901680i	$0.642335\pi$
$54$	1.00000	0.136083
$55$	−3.04892	−0.411116
$56$	−0.801938	−0.107163
$57$	7.40581	0.980924
$58$	0.307979	0.0404396
$59$	3.11529	0.405577	0.202788	−	0.979223i	$-0.435000\pi$
$59$	3.11529	0.405577	0.202788	+	0.979223i	$0.435000\pi$
$60$	1.00000	0.129099
$61$	11.0368	1.41312	0.706561	−	0.707652i	$-0.250246\pi$
$61$	11.0368	1.41312	0.706561	+	0.707652i	$0.250246\pi$
$62$	3.44504	0.437521
$63$	−0.801938	−0.101035
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.04892	−0.375296
$67$	5.96077	0.728224	0.364112	−	0.931355i	$-0.381373\pi$
$67$	5.96077	0.728224	0.364112	+	0.931355i	$0.381373\pi$
$68$	−1.24698	−0.151218
$69$	−0.356896	−0.0429652
$70$	−0.801938	−0.0958499
$71$	6.67456	0.792125	0.396062	−	0.918224i	$-0.370376\pi$
$71$	6.67456	0.792125	0.396062	+	0.918224i	$0.370376\pi$
$72$	1.00000	0.117851
$73$	10.1371	1.18645	0.593227	−	0.805035i	$-0.297854\pi$
$73$	10.1371	1.18645	0.593227	+	0.805035i	$0.297854\pi$
$74$	7.29590	0.848131
$75$	1.00000	0.115470
$76$	7.40581	0.849505
$77$	2.44504	0.278638
$78$	0	0
$79$	−13.6799	−1.53911	−0.769557	−	0.638578i	$-0.779523\pi$
$79$	−13.6799	−1.53911	−0.769557	+	0.638578i	$0.779523\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	3.40581	0.376109
$83$	−3.93900	−0.432362	−0.216181	−	0.976353i	$-0.569360\pi$
$83$	−3.93900	−0.432362	−0.216181	+	0.976353i	$0.569360\pi$
$84$	−0.801938	−0.0874986
$85$	−1.24698	−0.135254
$86$	6.54288	0.705537
$87$	0.307979	0.0330188
$88$	−3.04892	−0.325016
$89$	−8.05861	−0.854211	−0.427105	−	0.904202i	$-0.640467\pi$
$89$	−8.05861	−0.854211	−0.427105	+	0.904202i	$0.640467\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−0.356896	−0.0372090
$93$	3.44504	0.357234
$94$	0.603875	0.0622850
$95$	7.40581	0.759820
$96$	1.00000	0.102062
$97$	9.73556	0.988497	0.494248	−	0.869321i	$-0.335443\pi$
$97$	9.73556	0.988497	0.494248	+	0.869321i	$0.335443\pi$
$98$	−6.35690	−0.642143
$99$	−3.04892	−0.306428
$100$	1.00000	0.100000
$101$	7.65817	0.762016	0.381008	−	0.924572i	$-0.375577\pi$
$101$	7.65817	0.762016	0.381008	+	0.924572i	$0.375577\pi$
$102$	−1.24698	−0.123469
$103$	2.32975	0.229557	0.114778	−	0.993391i	$-0.463384\pi$
$103$	2.32975	0.229557	0.114778	+	0.993391i	$0.463384\pi$
$104$	0	0
$105$	−0.801938	−0.0782611
$106$	−6.29590	−0.611512
$107$	3.77479	0.364923	0.182461	−	0.983213i	$-0.441594\pi$
$107$	3.77479	0.364923	0.182461	+	0.983213i	$0.441594\pi$
$108$	1.00000	0.0962250
$109$	−1.05861	−0.101396	−0.0506980	−	0.998714i	$-0.516145\pi$
$109$	−1.05861	−0.101396	−0.0506980	+	0.998714i	$0.516145\pi$
$110$	−3.04892	−0.290703
$111$	7.29590	0.692496
$112$	−0.801938	−0.0757760
$113$	11.0368	1.03826	0.519129	−	0.854696i	$-0.326256\pi$
$113$	11.0368	1.03826	0.519129	+	0.854696i	$0.326256\pi$
$114$	7.40581	0.693618
$115$	−0.356896	−0.0332807
$116$	0.307979	0.0285951
$117$	0	0
$118$	3.11529	0.286786
$119$	1.00000	0.0916698
$120$	1.00000	0.0912871
$121$	−1.70410	−0.154918
$122$	11.0368	0.999228
$123$	3.40581	0.307092
$124$	3.44504	0.309374
$125$	1.00000	0.0894427
$126$	−0.801938	−0.0714423
$127$	15.9541	1.41569	0.707847	−	0.706366i	$-0.249667\pi$
$127$	15.9541	1.41569	0.707847	+	0.706366i	$0.249667\pi$
$128$	1.00000	0.0883883
$129$	6.54288	0.576068
$130$	0	0
$131$	−17.0911	−1.49326	−0.746629	−	0.665240i	$-0.768329\pi$
$131$	−17.0911	−1.49326	−0.746629	+	0.665240i	$0.768329\pi$
$132$	−3.04892	−0.265374
$133$	−5.93900	−0.514977
$134$	5.96077	0.514932
$135$	1.00000	0.0860663
$136$	−1.24698	−0.106928
$137$	−16.4373	−1.40433	−0.702165	−	0.712014i	$-0.747783\pi$
$137$	−16.4373	−1.40433	−0.702165	+	0.712014i	$0.747783\pi$
$138$	−0.356896	−0.0303810
$139$	−8.10992	−0.687874	−0.343937	−	0.938993i	$-0.611761\pi$
$139$	−8.10992	−0.687874	−0.343937	+	0.938993i	$0.611761\pi$
$140$	−0.801938	−0.0677761
$141$	0.603875	0.0508555
$142$	6.67456	0.560117
$143$	0	0
$144$	1.00000	0.0833333
$145$	0.307979	0.0255762
$146$	10.1371	0.838949
$147$	−6.35690	−0.524308
$148$	7.29590	0.599719
$149$	11.0532	0.905516	0.452758	−	0.891633i	$-0.350440\pi$
$149$	11.0532	0.905516	0.452758	+	0.891633i	$0.350440\pi$
$150$	1.00000	0.0816497
$151$	6.89546	0.561145	0.280572	−	0.959833i	$-0.409476\pi$
$151$	6.89546	0.561145	0.280572	+	0.959833i	$0.409476\pi$
$152$	7.40581	0.600691
$153$	−1.24698	−0.100812
$154$	2.44504	0.197027
$155$	3.44504	0.276712
$156$	0	0
$157$	−20.8388	−1.66312	−0.831558	−	0.555439i	$-0.812550\pi$
$157$	−20.8388	−1.66312	−0.831558	+	0.555439i	$0.812550\pi$
$158$	−13.6799	−1.08832
$159$	−6.29590	−0.499297
$160$	1.00000	0.0790569
$161$	0.286208	0.0225564
$162$	1.00000	0.0785674
$163$	−4.71917	−0.369634	−0.184817	−	0.982773i	$-0.559169\pi$
$163$	−4.71917	−0.369634	−0.184817	+	0.982773i	$0.559169\pi$
$164$	3.40581	0.265949
$165$	−3.04892	−0.237358
$166$	−3.93900	−0.305726
$167$	−14.5308	−1.12443	−0.562213	−	0.826992i	$-0.690050\pi$
$167$	−14.5308	−1.12443	−0.562213	+	0.826992i	$0.690050\pi$
$168$	−0.801938	−0.0618708
$169$	0	0
$170$	−1.24698	−0.0956390
$171$	7.40581	0.566337
$172$	6.54288	0.498890
$173$	11.4523	0.870705	0.435353	−	0.900260i	$-0.356624\pi$
$173$	11.4523	0.870705	0.435353	+	0.900260i	$0.356624\pi$
$174$	0.307979	0.0233478
$175$	−0.801938	−0.0606208
$176$	−3.04892	−0.229821
$177$	3.11529	0.234160
$178$	−8.05861	−0.604018
$179$	−4.32975	−0.323621	−0.161810	−	0.986822i	$-0.551733\pi$
$179$	−4.32975	−0.323621	−0.161810	+	0.986822i	$0.551733\pi$
$180$	1.00000	0.0745356
$181$	−24.8321	−1.84575	−0.922877	−	0.385096i	$-0.874168\pi$
$181$	−24.8321	−1.84575	−0.922877	+	0.385096i	$0.874168\pi$
$182$	0	0
$183$	11.0368	0.815866
$184$	−0.356896	−0.0263107
$185$	7.29590	0.536405
$186$	3.44504	0.252603
$187$	3.80194	0.278025
$188$	0.603875	0.0440421
$189$	−0.801938	−0.0583324
$190$	7.40581	0.537274
$191$	22.1933	1.60585	0.802925	−	0.596081i	$-0.203276\pi$
$191$	22.1933	1.60585	0.802925	+	0.596081i	$0.203276\pi$
$192$	1.00000	0.0721688
$193$	25.4306	1.83053	0.915266	−	0.402850i	$-0.131980\pi$
$193$	25.4306	1.83053	0.915266	+	0.402850i	$0.131980\pi$
$194$	9.73556	0.698973
$195$	0	0
$196$	−6.35690	−0.454064
$197$	−10.6310	−0.757429	−0.378714	−	0.925514i	$-0.623634\pi$
$197$	−10.6310	−0.757429	−0.378714	+	0.925514i	$0.623634\pi$
$198$	−3.04892	−0.216677
$199$	11.0465	0.783067	0.391534	−	0.920164i	$-0.371945\pi$
$199$	11.0465	0.783067	0.391534	+	0.920164i	$0.371945\pi$
$200$	1.00000	0.0707107
$201$	5.96077	0.420440
$202$	7.65817	0.538827
$203$	−0.246980	−0.0173346
$204$	−1.24698	−0.0873060
$205$	3.40581	0.237872
$206$	2.32975	0.162321
$207$	−0.356896	−0.0248060
$208$	0	0
$209$	−22.5797	−1.56187
$210$	−0.801938	−0.0553390
$211$	−21.3491	−1.46973	−0.734867	−	0.678211i	$-0.762756\pi$
$211$	−21.3491	−1.46973	−0.734867	+	0.678211i	$0.762756\pi$
$212$	−6.29590	−0.432404
$213$	6.67456	0.457334
$214$	3.77479	0.258039
$215$	6.54288	0.446220
$216$	1.00000	0.0680414
$217$	−2.76271	−0.187545
$218$	−1.05861	−0.0716978
$219$	10.1371	0.684999
$220$	−3.04892	−0.205558
$221$	0	0
$222$	7.29590	0.489669
$223$	−12.7603	−0.854494	−0.427247	−	0.904135i	$-0.640517\pi$
$223$	−12.7603	−0.854494	−0.427247	+	0.904135i	$0.640517\pi$
$224$	−0.801938	−0.0535817
$225$	1.00000	0.0666667
$226$	11.0368	0.734159
$227$	14.1836	0.941398	0.470699	−	0.882294i	$-0.344002\pi$
$227$	14.1836	0.941398	0.470699	+	0.882294i	$0.344002\pi$
$228$	7.40581	0.490462
$229$	−4.71140	−0.311338	−0.155669	−	0.987809i	$-0.549753\pi$
$229$	−4.71140	−0.311338	−0.155669	+	0.987809i	$0.549753\pi$
$230$	−0.356896	−0.0235330
$231$	2.44504	0.160872
$232$	0.307979	0.0202198
$233$	0.0556221	0.00364393	0.00182196	−	0.999998i	$-0.499420\pi$
$233$	0.0556221	0.00364393	0.00182196	+	0.999998i	$0.499420\pi$
$234$	0	0
$235$	0.603875	0.0393925
$236$	3.11529	0.202788
$237$	−13.6799	−0.888608
$238$	1.00000	0.0648204
$239$	21.0489	1.36154	0.680771	−	0.732497i	$-0.261645\pi$
$239$	21.0489	1.36154	0.680771	+	0.732497i	$0.261645\pi$
$240$	1.00000	0.0645497
$241$	−12.0030	−0.773180	−0.386590	−	0.922252i	$-0.626347\pi$
$241$	−12.0030	−0.773180	−0.386590	+	0.922252i	$0.626347\pi$
$242$	−1.70410	−0.109544
$243$	1.00000	0.0641500
$244$	11.0368	0.706561
$245$	−6.35690	−0.406127
$246$	3.40581	0.217147
$247$	0	0
$248$	3.44504	0.218760
$249$	−3.93900	−0.249624
$250$	1.00000	0.0632456
$251$	−2.25906	−0.142591	−0.0712953	−	0.997455i	$-0.522713\pi$
$251$	−2.25906	−0.142591	−0.0712953	+	0.997455i	$0.522713\pi$
$252$	−0.801938	−0.0505173
$253$	1.08815	0.0684112
$254$	15.9541	1.00105
$255$	−1.24698	−0.0780889
$256$	1.00000	0.0625000
$257$	−17.0043	−1.06070	−0.530350	−	0.847779i	$-0.677939\pi$
$257$	−17.0043	−1.06070	−0.530350	+	0.847779i	$0.677939\pi$
$258$	6.54288	0.407342
$259$	−5.85086	−0.363554
$260$	0	0
$261$	0.307979	0.0190634
$262$	−17.0911	−1.05589
$263$	−28.5773	−1.76215	−0.881077	−	0.472974i	$-0.843181\pi$
$263$	−28.5773	−1.76215	−0.881077	+	0.472974i	$0.843181\pi$
$264$	−3.04892	−0.187648
$265$	−6.29590	−0.386754
$266$	−5.93900	−0.364144
$267$	−8.05861	−0.493179
$268$	5.96077	0.364112
$269$	−3.28083	−0.200036	−0.100018	−	0.994986i	$-0.531890\pi$
$269$	−3.28083	−0.200036	−0.100018	+	0.994986i	$0.531890\pi$
$270$	1.00000	0.0608581
$271$	−8.70948	−0.529063	−0.264532	−	0.964377i	$-0.585217\pi$
$271$	−8.70948	−0.529063	−0.264532	+	0.964377i	$0.585217\pi$
$272$	−1.24698	−0.0756092
$273$	0	0
$274$	−16.4373	−0.993012
$275$	−3.04892	−0.183857
$276$	−0.356896	−0.0214826
$277$	−8.91185	−0.535461	−0.267731	−	0.963494i	$-0.586274\pi$
$277$	−8.91185	−0.535461	−0.267731	+	0.963494i	$0.586274\pi$
$278$	−8.10992	−0.486400
$279$	3.44504	0.206249
$280$	−0.801938	−0.0479249
$281$	−1.90648	−0.113731	−0.0568654	−	0.998382i	$-0.518111\pi$
$281$	−1.90648	−0.113731	−0.0568654	+	0.998382i	$0.518111\pi$
$282$	0.603875	0.0359603
$283$	−14.9245	−0.887171	−0.443586	−	0.896232i	$-0.646294\pi$
$283$	−14.9245	−0.887171	−0.443586	+	0.896232i	$0.646294\pi$
$284$	6.67456	0.396062
$285$	7.40581	0.438683
$286$	0	0
$287$	−2.73125	−0.161221
$288$	1.00000	0.0589256
$289$	−15.4450	−0.908532
$290$	0.307979	0.0180851
$291$	9.73556	0.570709
$292$	10.1371	0.593227
$293$	−12.8901	−0.753047	−0.376523	−	0.926407i	$-0.622881\pi$
$293$	−12.8901	−0.753047	−0.376523	+	0.926407i	$0.622881\pi$
$294$	−6.35690	−0.370742
$295$	3.11529	0.181379
$296$	7.29590	0.424065
$297$	−3.04892	−0.176916
$298$	11.0532	0.640296
$299$	0	0
$300$	1.00000	0.0577350
$301$	−5.24698	−0.302431
$302$	6.89546	0.396789
$303$	7.65817	0.439950
$304$	7.40581	0.424753
$305$	11.0368	0.631967
$306$	−1.24698	−0.0712851
$307$	19.2892	1.10089	0.550446	−	0.834871i	$-0.314458\pi$
$307$	19.2892	1.10089	0.550446	+	0.834871i	$0.314458\pi$
$308$	2.44504	0.139319
$309$	2.32975	0.132535
$310$	3.44504	0.195665
$311$	1.23729	0.0701603	0.0350802	−	0.999385i	$-0.488831\pi$
$311$	1.23729	0.0701603	0.0350802	+	0.999385i	$0.488831\pi$
$312$	0	0
$313$	8.09113	0.457338	0.228669	−	0.973504i	$-0.426563\pi$
$313$	8.09113	0.457338	0.228669	+	0.973504i	$0.426563\pi$
$314$	−20.8388	−1.17600
$315$	−0.801938	−0.0451841
$316$	−13.6799	−0.769557
$317$	−21.0368	−1.18155	−0.590773	−	0.806838i	$-0.701177\pi$
$317$	−21.0368	−1.18155	−0.590773	+	0.806838i	$0.701177\pi$
$318$	−6.29590	−0.353056
$319$	−0.939001	−0.0525740
$320$	1.00000	0.0559017
$321$	3.77479	0.210688
$322$	0.286208	0.0159498
$323$	−9.23490	−0.513843
$324$	1.00000	0.0555556
$325$	0	0
$326$	−4.71917	−0.261371
$327$	−1.05861	−0.0585410
$328$	3.40581	0.188055
$329$	−0.484271	−0.0266987
$330$	−3.04892	−0.167837
$331$	−29.4252	−1.61735	−0.808677	−	0.588252i	$-0.799816\pi$
$331$	−29.4252	−1.61735	−0.808677	+	0.588252i	$0.799816\pi$
$332$	−3.93900	−0.216181
$333$	7.29590	0.399813
$334$	−14.5308	−0.795090
$335$	5.96077	0.325672
$336$	−0.801938	−0.0437493
$337$	−17.5483	−0.955914	−0.477957	−	0.878383i	$-0.658623\pi$
$337$	−17.5483	−0.955914	−0.477957	+	0.878383i	$0.658623\pi$
$338$	0	0
$339$	11.0368	0.599439
$340$	−1.24698	−0.0676270
$341$	−10.5036	−0.568804
$342$	7.40581	0.400461
$343$	10.7114	0.578361
$344$	6.54288	0.352768
$345$	−0.356896	−0.0192146
$346$	11.4523	0.615681
$347$	1.27844	0.0686302	0.0343151	−	0.999411i	$-0.489075\pi$
$347$	1.27844	0.0686302	0.0343151	+	0.999411i	$0.489075\pi$
$348$	0.307979	0.0165094
$349$	18.0911	0.968396	0.484198	−	0.874958i	$-0.339111\pi$
$349$	18.0911	0.968396	0.484198	+	0.874958i	$0.339111\pi$
$350$	−0.801938	−0.0428654
$351$	0	0
$352$	−3.04892	−0.162508
$353$	−18.9071	−1.00632	−0.503161	−	0.864193i	$-0.667830\pi$
$353$	−18.9071	−1.00632	−0.503161	+	0.864193i	$0.667830\pi$
$354$	3.11529	0.165576
$355$	6.67456	0.354249
$356$	−8.05861	−0.427105
$357$	1.00000	0.0529256
$358$	−4.32975	−0.228834
$359$	32.7308	1.72746	0.863732	−	0.503951i	$-0.168121\pi$
$359$	32.7308	1.72746	0.863732	+	0.503951i	$0.168121\pi$
$360$	1.00000	0.0527046
$361$	35.8461	1.88664
$362$	−24.8321	−1.30514
$363$	−1.70410	−0.0894422
$364$	0	0
$365$	10.1371	0.530598
$366$	11.0368	0.576905
$367$	−3.10321	−0.161986	−0.0809932	−	0.996715i	$-0.525809\pi$
$367$	−3.10321	−0.161986	−0.0809932	+	0.996715i	$0.525809\pi$
$368$	−0.356896	−0.0186045
$369$	3.40581	0.177300
$370$	7.29590	0.379296
$371$	5.04892	0.262127
$372$	3.44504	0.178617
$373$	13.7366	0.711256	0.355628	−	0.934628i	$-0.384267\pi$
$373$	13.7366	0.711256	0.355628	+	0.934628i	$0.384267\pi$
$374$	3.80194	0.196594
$375$	1.00000	0.0516398
$376$	0.603875	0.0311425
$377$	0	0
$378$	−0.801938	−0.0412472
$379$	34.0097	1.74696	0.873480	−	0.486860i	$-0.161858\pi$
$379$	34.0097	1.74696	0.873480	+	0.486860i	$0.161858\pi$
$380$	7.40581	0.379910
$381$	15.9541	0.817352
$382$	22.1933	1.13551
$383$	38.6902	1.97698	0.988489	−	0.151293i	$-0.0483438\pi$
$383$	38.6902	1.97698	0.988489	+	0.151293i	$0.0483438\pi$
$384$	1.00000	0.0510310
$385$	2.44504	0.124611
$386$	25.4306	1.29438
$387$	6.54288	0.332593
$388$	9.73556	0.494248
$389$	−11.2567	−0.570736	−0.285368	−	0.958418i	$-0.592116\pi$
$389$	−11.2567	−0.570736	−0.285368	+	0.958418i	$0.592116\pi$
$390$	0	0
$391$	0.445042	0.0225067
$392$	−6.35690	−0.321072
$393$	−17.0911	−0.862133
$394$	−10.6310	−0.535583
$395$	−13.6799	−0.688312
$396$	−3.04892	−0.153214
$397$	−23.4403	−1.17643	−0.588216	−	0.808704i	$-0.700170\pi$
$397$	−23.4403	−1.17643	−0.588216	+	0.808704i	$0.700170\pi$
$398$	11.0465	0.553712
$399$	−5.93900	−0.297322
$400$	1.00000	0.0500000
$401$	−37.6950	−1.88240	−0.941199	−	0.337852i	$-0.890300\pi$
$401$	−37.6950	−1.88240	−0.941199	+	0.337852i	$0.890300\pi$
$402$	5.96077	0.297296
$403$	0	0
$404$	7.65817	0.381008
$405$	1.00000	0.0496904
$406$	−0.246980	−0.0122574
$407$	−22.2446	−1.10262
$408$	−1.24698	−0.0617347
$409$	−1.19567	−0.0591220	−0.0295610	−	0.999563i	$-0.509411\pi$
$409$	−1.19567	−0.0591220	−0.0295610	+	0.999563i	$0.509411\pi$
$410$	3.40581	0.168201
$411$	−16.4373	−0.810791
$412$	2.32975	0.114778
$413$	−2.49827	−0.122932
$414$	−0.356896	−0.0175405
$415$	−3.93900	−0.193358
$416$	0	0
$417$	−8.10992	−0.397144
$418$	−22.5797	−1.10441
$419$	1.77048	0.0864935	0.0432468	−	0.999064i	$-0.486230\pi$
$419$	1.77048	0.0864935	0.0432468	+	0.999064i	$0.486230\pi$
$420$	−0.801938	−0.0391306
$421$	1.80194	0.0878211	0.0439105	−	0.999035i	$-0.486018\pi$
$421$	1.80194	0.0878211	0.0439105	+	0.999035i	$0.486018\pi$
$422$	−21.3491	−1.03926
$423$	0.603875	0.0293614
$424$	−6.29590	−0.305756
$425$	−1.24698	−0.0604874
$426$	6.67456	0.323384
$427$	−8.85086	−0.428323
$428$	3.77479	0.182461
$429$	0	0
$430$	6.54288	0.315526
$431$	−0.567040	−0.0273133	−0.0136567	−	0.999907i	$-0.504347\pi$
$431$	−0.567040	−0.0273133	−0.0136567	+	0.999907i	$0.504347\pi$
$432$	1.00000	0.0481125
$433$	17.5743	0.844569	0.422284	−	0.906463i	$-0.361228\pi$
$433$	17.5743	0.844569	0.422284	+	0.906463i	$0.361228\pi$
$434$	−2.76271	−0.132614
$435$	0.307979	0.0147664
$436$	−1.05861	−0.0506980
$437$	−2.64310	−0.126437
$438$	10.1371	0.484368
$439$	3.00969	0.143645	0.0718223	−	0.997417i	$-0.477119\pi$
$439$	3.00969	0.143645	0.0718223	+	0.997417i	$0.477119\pi$
$440$	−3.04892	−0.145351
$441$	−6.35690	−0.302709
$442$	0	0
$443$	38.4499	1.82681	0.913406	−	0.407050i	$-0.133442\pi$
$443$	38.4499	1.82681	0.913406	+	0.407050i	$0.133442\pi$
$444$	7.29590	0.346248
$445$	−8.05861	−0.382015
$446$	−12.7603	−0.604218
$447$	11.0532	0.522800
$448$	−0.801938	−0.0378880
$449$	26.9909	1.27378	0.636890	−	0.770955i	$-0.280221\pi$
$449$	26.9909	1.27378	0.636890	+	0.770955i	$0.280221\pi$
$450$	1.00000	0.0471405
$451$	−10.3840	−0.488965
$452$	11.0368	0.519129
$453$	6.89546	0.323977
$454$	14.1836	0.665669
$455$	0	0
$456$	7.40581	0.346809
$457$	19.2664	0.901242	0.450621	−	0.892715i	$-0.351203\pi$
$457$	19.2664	0.901242	0.450621	+	0.892715i	$0.351203\pi$
$458$	−4.71140	−0.220149
$459$	−1.24698	−0.0582040
$460$	−0.356896	−0.0166404
$461$	−18.5942	−0.866018	−0.433009	−	0.901390i	$-0.642548\pi$
$461$	−18.5942	−0.866018	−0.433009	+	0.901390i	$0.642548\pi$
$462$	2.44504	0.113754
$463$	5.79417	0.269278	0.134639	−	0.990895i	$-0.457013\pi$
$463$	5.79417	0.269278	0.134639	+	0.990895i	$0.457013\pi$
$464$	0.307979	0.0142975
$465$	3.44504	0.159760
$466$	0.0556221	0.00257664
$467$	11.2457	0.520387	0.260193	−	0.965557i	$-0.416214\pi$
$467$	11.2457	0.520387	0.260193	+	0.965557i	$0.416214\pi$
$468$	0	0
$469$	−4.78017	−0.220728
$470$	0.603875	0.0278547
$471$	−20.8388	−0.960200
$472$	3.11529	0.143393
$473$	−19.9487	−0.917242
$474$	−13.6799	−0.628340
$475$	7.40581	0.339802
$476$	1.00000	0.0458349
$477$	−6.29590	−0.288269
$478$	21.0489	0.962755
$479$	−35.2452	−1.61039	−0.805197	−	0.593008i	$-0.797940\pi$
$479$	−35.2452	−1.61039	−0.805197	+	0.593008i	$0.797940\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−12.0030	−0.546721
$483$	0.286208	0.0130229
$484$	−1.70410	−0.0774592
$485$	9.73556	0.442069
$486$	1.00000	0.0453609
$487$	8.49827	0.385093	0.192547	−	0.981288i	$-0.438325\pi$
$487$	8.49827	0.385093	0.192547	+	0.981288i	$0.438325\pi$
$488$	11.0368	0.499614
$489$	−4.71917	−0.213408
$490$	−6.35690	−0.287175
$491$	−25.5254	−1.15195	−0.575973	−	0.817469i	$-0.695377\pi$
$491$	−25.5254	−1.15195	−0.575973	+	0.817469i	$0.695377\pi$
$492$	3.40581	0.153546
$493$	−0.384043	−0.0172964
$494$	0	0
$495$	−3.04892	−0.137039
$496$	3.44504	0.154687
$497$	−5.35258	−0.240096
$498$	−3.93900	−0.176511
$499$	−29.4577	−1.31871	−0.659354	−	0.751832i	$-0.729170\pi$
$499$	−29.4577	−1.31871	−0.659354	+	0.751832i	$0.729170\pi$
$500$	1.00000	0.0447214
$501$	−14.5308	−0.649188
$502$	−2.25906	−0.100827
$503$	14.4480	0.644206	0.322103	−	0.946705i	$-0.395610\pi$
$503$	14.4480	0.644206	0.322103	+	0.946705i	$0.395610\pi$
$504$	−0.801938	−0.0357211
$505$	7.65817	0.340784
$506$	1.08815	0.0483740
$507$	0	0
$508$	15.9541	0.707847
$509$	4.54288	0.201359	0.100680	−	0.994919i	$-0.467898\pi$
$509$	4.54288	0.201359	0.100680	+	0.994919i	$0.467898\pi$
$510$	−1.24698	−0.0552172
$511$	−8.12929	−0.359619
$512$	1.00000	0.0441942
$513$	7.40581	0.326975
$514$	−17.0043	−0.750028
$515$	2.32975	0.102661
$516$	6.54288	0.288034
$517$	−1.84117	−0.0809744
$518$	−5.85086	−0.257072
$519$	11.4523	0.502702
$520$	0	0
$521$	−30.0911	−1.31832	−0.659158	−	0.752004i	$-0.729087\pi$
$521$	−30.0911	−1.31832	−0.659158	+	0.752004i	$0.729087\pi$
$522$	0.307979	0.0134799
$523$	−3.18359	−0.139209	−0.0696043	−	0.997575i	$-0.522174\pi$
$523$	−3.18359	−0.139209	−0.0696043	+	0.997575i	$0.522174\pi$
$524$	−17.0911	−0.746629
$525$	−0.801938	−0.0349994
$526$	−28.5773	−1.24603
$527$	−4.29590	−0.187132
$528$	−3.04892	−0.132687
$529$	−22.8726	−0.994462
$530$	−6.29590	−0.273476
$531$	3.11529	0.135192
$532$	−5.93900	−0.257488
$533$	0	0
$534$	−8.05861	−0.348730
$535$	3.77479	0.163198
$536$	5.96077	0.257466
$537$	−4.32975	−0.186842
$538$	−3.28083	−0.141447
$539$	19.3817	0.834827
$540$	1.00000	0.0430331
$541$	21.2825	0.915006	0.457503	−	0.889208i	$-0.348744\pi$
$541$	21.2825	0.915006	0.457503	+	0.889208i	$0.348744\pi$
$542$	−8.70948	−0.374104
$543$	−24.8321	−1.06565
$544$	−1.24698	−0.0534638
$545$	−1.05861	−0.0453457
$546$	0	0
$547$	27.5545	1.17814	0.589072	−	0.808080i	$-0.299493\pi$
$547$	27.5545	1.17814	0.589072	+	0.808080i	$0.299493\pi$
$548$	−16.4373	−0.702165
$549$	11.0368	0.471041
$550$	−3.04892	−0.130006
$551$	2.28083	0.0971667
$552$	−0.356896	−0.0151905
$553$	10.9705	0.466511
$554$	−8.91185	−0.378628
$555$	7.29590	0.309694
$556$	−8.10992	−0.343937
$557$	−3.88040	−0.164418	−0.0822088	−	0.996615i	$-0.526197\pi$
$557$	−3.88040	−0.164418	−0.0822088	+	0.996615i	$0.526197\pi$
$558$	3.44504	0.145840
$559$	0	0
$560$	−0.801938	−0.0338881
$561$	3.80194	0.160518
$562$	−1.90648	−0.0804199
$563$	−38.6286	−1.62800	−0.814001	−	0.580863i	$-0.802715\pi$
$563$	−38.6286	−1.62800	−0.814001	+	0.580863i	$0.802715\pi$
$564$	0.603875	0.0254277
$565$	11.0368	0.464323
$566$	−14.9245	−0.627325
$567$	−0.801938	−0.0336782
$568$	6.67456	0.280058
$569$	−27.6746	−1.16018	−0.580089	−	0.814553i	$-0.696982\pi$
$569$	−27.6746	−1.16018	−0.580089	+	0.814553i	$0.696982\pi$
$570$	7.40581	0.310195
$571$	−10.9065	−0.456422	−0.228211	−	0.973612i	$-0.573288\pi$
$571$	−10.9065	−0.456422	−0.228211	+	0.973612i	$0.573288\pi$
$572$	0	0
$573$	22.1933	0.927137
$574$	−2.73125	−0.114000
$575$	−0.356896	−0.0148836
$576$	1.00000	0.0416667
$577$	29.9758	1.24791	0.623955	−	0.781460i	$-0.285525\pi$
$577$	29.9758	1.24791	0.623955	+	0.781460i	$0.285525\pi$
$578$	−15.4450	−0.642429
$579$	25.4306	1.05686
$580$	0.307979	0.0127881
$581$	3.15883	0.131051
$582$	9.73556	0.403552
$583$	19.1957	0.795003
$584$	10.1371	0.419475
$585$	0	0
$586$	−12.8901	−0.532484
$587$	32.1551	1.32718	0.663592	−	0.748095i	$-0.269031\pi$
$587$	32.1551	1.32718	0.663592	+	0.748095i	$0.269031\pi$
$588$	−6.35690	−0.262154
$589$	25.5133	1.05126
$590$	3.11529	0.128255
$591$	−10.6310	−0.437302
$592$	7.29590	0.299860
$593$	−3.88876	−0.159692	−0.0798460	−	0.996807i	$-0.525443\pi$
$593$	−3.88876	−0.159692	−0.0798460	+	0.996807i	$0.525443\pi$
$594$	−3.04892	−0.125099
$595$	1.00000	0.0409960
$596$	11.0532	0.452758
$597$	11.0465	0.452104
$598$	0	0
$599$	−37.2301	−1.52118	−0.760591	−	0.649232i	$-0.775090\pi$
$599$	−37.2301	−1.52118	−0.760591	+	0.649232i	$0.775090\pi$
$600$	1.00000	0.0408248
$601$	−4.04593	−0.165037	−0.0825185	−	0.996590i	$-0.526296\pi$
$601$	−4.04593	−0.165037	−0.0825185	+	0.996590i	$0.526296\pi$
$602$	−5.24698	−0.213851
$603$	5.96077	0.242741
$604$	6.89546	0.280572
$605$	−1.70410	−0.0692816
$606$	7.65817	0.311092
$607$	−6.36658	−0.258412	−0.129206	−	0.991618i	$-0.541243\pi$
$607$	−6.36658	−0.258412	−0.129206	+	0.991618i	$0.541243\pi$
$608$	7.40581	0.300345
$609$	−0.246980	−0.0100081
$610$	11.0368	0.446868
$611$	0	0
$612$	−1.24698	−0.0504062
$613$	−42.2519	−1.70654	−0.853269	−	0.521471i	$-0.825383\pi$
$613$	−42.2519	−1.70654	−0.853269	+	0.521471i	$0.825383\pi$
$614$	19.2892	0.778448
$615$	3.40581	0.137336
$616$	2.44504	0.0985135
$617$	0.978230	0.0393820	0.0196910	−	0.999806i	$-0.493732\pi$
$617$	0.978230	0.0393820	0.0196910	+	0.999806i	$0.493732\pi$
$618$	2.32975	0.0937162
$619$	19.0392	0.765251	0.382626	−	0.923903i	$-0.375020\pi$
$619$	19.0392	0.765251	0.382626	+	0.923903i	$0.375020\pi$
$620$	3.44504	0.138356
$621$	−0.356896	−0.0143217
$622$	1.23729	0.0496108
$623$	6.46250	0.258915
$624$	0	0
$625$	1.00000	0.0400000
$626$	8.09113	0.323387
$627$	−22.5797	−0.901747
$628$	−20.8388	−0.831558
$629$	−9.09783	−0.362754
$630$	−0.801938	−0.0319500
$631$	37.2664	1.48355	0.741775	−	0.670649i	$-0.233984\pi$
$631$	37.2664	1.48355	0.741775	+	0.670649i	$0.233984\pi$
$632$	−13.6799	−0.544159
$633$	−21.3491	−0.848552
$634$	−21.0368	−0.835479
$635$	15.9541	0.633118
$636$	−6.29590	−0.249649
$637$	0	0
$638$	−0.939001	−0.0371754
$639$	6.67456	0.264042
$640$	1.00000	0.0395285
$641$	−29.7017	−1.17315	−0.586574	−	0.809896i	$-0.699523\pi$
$641$	−29.7017	−1.17315	−0.586574	+	0.809896i	$0.699523\pi$
$642$	3.77479	0.148979
$643$	−21.5907	−0.851455	−0.425728	−	0.904851i	$-0.639982\pi$
$643$	−21.5907	−0.851455	−0.425728	+	0.904851i	$0.639982\pi$
$644$	0.286208	0.0112782
$645$	6.54288	0.257626
$646$	−9.23490	−0.363342
$647$	−29.3623	−1.15435	−0.577175	−	0.816620i	$-0.695845\pi$
$647$	−29.3623	−1.15435	−0.577175	+	0.816620i	$0.695845\pi$
$648$	1.00000	0.0392837
$649$	−9.49827	−0.372840
$650$	0	0
$651$	−2.76271	−0.108279
$652$	−4.71917	−0.184817
$653$	33.4470	1.30888	0.654440	−	0.756114i	$-0.272904\pi$
$653$	33.4470	1.30888	0.654440	+	0.756114i	$0.272904\pi$
$654$	−1.05861	−0.0413948
$655$	−17.0911	−0.667806
$656$	3.40581	0.132975
$657$	10.1371	0.395485
$658$	−0.484271	−0.0188788
$659$	39.3715	1.53370	0.766848	−	0.641829i	$-0.221824\pi$
$659$	39.3715	1.53370	0.766848	+	0.641829i	$0.221824\pi$
$660$	−3.04892	−0.118679
$661$	−10.1274	−0.393909	−0.196955	−	0.980413i	$-0.563105\pi$
$661$	−10.1274	−0.393909	−0.196955	+	0.980413i	$0.563105\pi$
$662$	−29.4252	−1.14364
$663$	0	0
$664$	−3.93900	−0.152863
$665$	−5.93900	−0.230305
$666$	7.29590	0.282710
$667$	−0.109916	−0.00425597
$668$	−14.5308	−0.562213
$669$	−12.7603	−0.493342
$670$	5.96077	0.230285
$671$	−33.6504	−1.29906
$672$	−0.801938	−0.0309354
$673$	−10.1400	−0.390870	−0.195435	−	0.980717i	$-0.562612\pi$
$673$	−10.1400	−0.390870	−0.195435	+	0.980717i	$0.562612\pi$
$674$	−17.5483	−0.675933
$675$	1.00000	0.0384900
$676$	0	0
$677$	22.9952	0.883778	0.441889	−	0.897070i	$-0.354309\pi$
$677$	22.9952	0.883778	0.441889	+	0.897070i	$0.354309\pi$
$678$	11.0368	0.423867
$679$	−7.80731	−0.299617
$680$	−1.24698	−0.0478195
$681$	14.1836	0.543516
$682$	−10.5036	−0.402205
$683$	−4.10800	−0.157188	−0.0785941	−	0.996907i	$-0.525043\pi$
$683$	−4.10800	−0.157188	−0.0785941	+	0.996907i	$0.525043\pi$
$684$	7.40581	0.283168
$685$	−16.4373	−0.628036
$686$	10.7114	0.408963
$687$	−4.71140	−0.179751
$688$	6.54288	0.249445
$689$	0	0
$690$	−0.356896	−0.0135868
$691$	30.1618	1.14741	0.573705	−	0.819062i	$-0.305506\pi$
$691$	30.1618	1.14741	0.573705	+	0.819062i	$0.305506\pi$
$692$	11.4523	0.435353
$693$	2.44504	0.0928795
$694$	1.27844	0.0485289
$695$	−8.10992	−0.307627
$696$	0.307979	0.0116739
$697$	−4.24698	−0.160866
$698$	18.0911	0.684759
$699$	0.0556221	0.00210382
$700$	−0.801938	−0.0303104
$701$	−37.5066	−1.41661	−0.708303	−	0.705909i	$-0.750539\pi$
$701$	−37.5066	−1.41661	−0.708303	+	0.705909i	$0.750539\pi$
$702$	0	0
$703$	54.0320	2.03786
$704$	−3.04892	−0.114910
$705$	0.603875	0.0227433
$706$	−18.9071	−0.711577
$707$	−6.14138	−0.230970
$708$	3.11529	0.117080
$709$	−4.20211	−0.157814	−0.0789068	−	0.996882i	$-0.525143\pi$
$709$	−4.20211	−0.157814	−0.0789068	+	0.996882i	$0.525143\pi$
$710$	6.67456	0.250492
$711$	−13.6799	−0.513038
$712$	−8.05861	−0.302009
$713$	−1.22952	−0.0460459
$714$	1.00000	0.0374241
$715$	0	0
$716$	−4.32975	−0.161810
$717$	21.0489	0.786086
$718$	32.7308	1.22150
$719$	−28.3532	−1.05740	−0.528698	−	0.848810i	$-0.677319\pi$
$719$	−28.3532	−1.05740	−0.528698	+	0.848810i	$0.677319\pi$
$720$	1.00000	0.0372678
$721$	−1.86831	−0.0695796
$722$	35.8461	1.33405
$723$	−12.0030	−0.446396
$724$	−24.8321	−0.922877
$725$	0.307979	0.0114380
$726$	−1.70410	−0.0632452
$727$	10.2892	0.381605	0.190803	−	0.981628i	$-0.438891\pi$
$727$	10.2892	0.381605	0.190803	+	0.981628i	$0.438891\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	10.1371	0.375190
$731$	−8.15883	−0.301765
$732$	11.0368	0.407933
$733$	−48.8963	−1.80603	−0.903013	−	0.429613i	$-0.858650\pi$
$733$	−48.8963	−1.80603	−0.903013	+	0.429613i	$0.858650\pi$
$734$	−3.10321	−0.114542
$735$	−6.35690	−0.234478
$736$	−0.356896	−0.0131554
$737$	−18.1739	−0.669444
$738$	3.40581	0.125370
$739$	−23.7496	−0.873642	−0.436821	−	0.899549i	$-0.643896\pi$
$739$	−23.7496	−0.873642	−0.436821	+	0.899549i	$0.643896\pi$
$740$	7.29590	0.268203
$741$	0	0
$742$	5.04892	0.185352
$743$	−51.5153	−1.88991	−0.944956	−	0.327198i	$-0.893896\pi$
$743$	−51.5153	−1.88991	−0.944956	+	0.327198i	$0.893896\pi$
$744$	3.44504	0.126301
$745$	11.0532	0.404959
$746$	13.7366	0.502934
$747$	−3.93900	−0.144121
$748$	3.80194	0.139013
$749$	−3.02715	−0.110610
$750$	1.00000	0.0365148
$751$	43.8939	1.60171	0.800856	−	0.598857i	$-0.204378\pi$
$751$	43.8939	1.60171	0.800856	+	0.598857i	$0.204378\pi$
$752$	0.603875	0.0220211
$753$	−2.25906	−0.0823248
$754$	0	0
$755$	6.89546	0.250952
$756$	−0.801938	−0.0291662
$757$	−18.6136	−0.676521	−0.338261	−	0.941052i	$-0.609839\pi$
$757$	−18.6136	−0.676521	−0.338261	+	0.941052i	$0.609839\pi$
$758$	34.0097	1.23529
$759$	1.08815	0.0394972
$760$	7.40581	0.268637
$761$	−44.8364	−1.62532	−0.812659	−	0.582740i	$-0.801981\pi$
$761$	−44.8364	−1.62532	−0.812659	+	0.582740i	$0.801981\pi$
$762$	15.9541	0.577955
$763$	0.848936	0.0307336
$764$	22.1933	0.802925
$765$	−1.24698	−0.0450846
$766$	38.6902	1.39793
$767$	0	0
$768$	1.00000	0.0360844
$769$	39.5303	1.42550	0.712750	−	0.701418i	$-0.247449\pi$
$769$	39.5303	1.42550	0.712750	+	0.701418i	$0.247449\pi$
$770$	2.44504	0.0881132
$771$	−17.0043	−0.612395
$772$	25.4306	0.915266
$773$	49.8665	1.79357	0.896787	−	0.442463i	$-0.145895\pi$
$773$	49.8665	1.79357	0.896787	+	0.442463i	$0.145895\pi$
$774$	6.54288	0.235179
$775$	3.44504	0.123750
$776$	9.73556	0.349486
$777$	−5.85086	−0.209898
$778$	−11.2567	−0.403571
$779$	25.2228	0.903701
$780$	0	0
$781$	−20.3502	−0.728187
$782$	0.445042	0.0159147
$783$	0.307979	0.0110063
$784$	−6.35690	−0.227032
$785$	−20.8388	−0.743768
$786$	−17.0911	−0.609620
$787$	−26.0398	−0.928219	−0.464110	−	0.885778i	$-0.653626\pi$
$787$	−26.0398	−0.928219	−0.464110	+	0.885778i	$0.653626\pi$
$788$	−10.6310	−0.378714
$789$	−28.5773	−1.01738
$790$	−13.6799	−0.486710
$791$	−8.85086	−0.314700
$792$	−3.04892	−0.108339
$793$	0	0
$794$	−23.4403	−0.831864
$795$	−6.29590	−0.223292
$796$	11.0465	0.391534
$797$	−23.9855	−0.849611	−0.424805	−	0.905285i	$-0.639657\pi$
$797$	−23.9855	−0.849611	−0.424805	+	0.905285i	$0.639657\pi$
$798$	−5.93900	−0.210238
$799$	−0.753020	−0.0266399
$800$	1.00000	0.0353553
$801$	−8.05861	−0.284737
$802$	−37.6950	−1.33106
$803$	−30.9071	−1.09069
$804$	5.96077	0.210220
$805$	0.286208	0.0100875
$806$	0	0
$807$	−3.28083	−0.115491
$808$	7.65817	0.269413
$809$	52.4825	1.84519	0.922593	−	0.385776i	$-0.126066\pi$
$809$	52.4825	1.84519	0.922593	+	0.385776i	$0.126066\pi$
$810$	1.00000	0.0351364
$811$	3.63666	0.127700	0.0638502	−	0.997959i	$-0.479662\pi$
$811$	3.63666	0.127700	0.0638502	+	0.997959i	$0.479662\pi$
$812$	−0.246980	−0.00866728
$813$	−8.70948	−0.305455
$814$	−22.2446	−0.779672
$815$	−4.71917	−0.165305
$816$	−1.24698	−0.0436530
$817$	48.4553	1.69524
$818$	−1.19567	−0.0418056
$819$	0	0
$820$	3.40581	0.118936
$821$	7.98984	0.278847	0.139424	−	0.990233i	$-0.455475\pi$
$821$	7.98984	0.278847	0.139424	+	0.990233i	$0.455475\pi$
$822$	−16.4373	−0.573316
$823$	28.1395	0.980880	0.490440	−	0.871475i	$-0.336836\pi$
$823$	28.1395	0.980880	0.490440	+	0.871475i	$0.336836\pi$
$824$	2.32975	0.0811606
$825$	−3.04892	−0.106150
$826$	−2.49827	−0.0869260
$827$	−30.9256	−1.07539	−0.537694	−	0.843140i	$-0.680704\pi$
$827$	−30.9256	−1.07539	−0.537694	+	0.843140i	$0.680704\pi$
$828$	−0.356896	−0.0124030
$829$	−31.9172	−1.10853	−0.554266	−	0.832340i	$-0.687001\pi$
$829$	−31.9172	−1.10853	−0.554266	+	0.832340i	$0.687001\pi$
$830$	−3.93900	−0.136725
$831$	−8.91185	−0.309149
$832$	0	0
$833$	7.92692	0.274651
$834$	−8.10992	−0.280823
$835$	−14.5308	−0.502859
$836$	−22.5797	−0.780936
$837$	3.44504	0.119078
$838$	1.77048	0.0611602
$839$	24.9866	0.862633	0.431316	−	0.902201i	$-0.358049\pi$
$839$	24.9866	0.862633	0.431316	+	0.902201i	$0.358049\pi$
$840$	−0.801938	−0.0276695
$841$	−28.9051	−0.996729
$842$	1.80194	0.0620989
$843$	−1.90648	−0.0656626
$844$	−21.3491	−0.734867
$845$	0	0
$846$	0.603875	0.0207617
$847$	1.36658	0.0469564
$848$	−6.29590	−0.216202
$849$	−14.9245	−0.512208
$850$	−1.24698	−0.0427710
$851$	−2.60388	−0.0892597
$852$	6.67456	0.228667
$853$	35.4416	1.21350	0.606748	−	0.794894i	$-0.292474\pi$
$853$	35.4416	1.21350	0.606748	+	0.794894i	$0.292474\pi$
$854$	−8.85086	−0.302870
$855$	7.40581	0.253273
$856$	3.77479	0.129020
$857$	−14.6541	−0.500575	−0.250288	−	0.968172i	$-0.580525\pi$
$857$	−14.6541	−0.500575	−0.250288	+	0.968172i	$0.580525\pi$
$858$	0	0
$859$	46.3153	1.58026	0.790128	−	0.612941i	$-0.210014\pi$
$859$	46.3153	1.58026	0.790128	+	0.612941i	$0.210014\pi$
$860$	6.54288	0.223110
$861$	−2.73125	−0.0930808
$862$	−0.567040	−0.0193134
$863$	−40.3642	−1.37401	−0.687007	−	0.726651i	$-0.741076\pi$
$863$	−40.3642	−1.37401	−0.687007	+	0.726651i	$0.741076\pi$
$864$	1.00000	0.0340207
$865$	11.4523	0.389391
$866$	17.5743	0.597200
$867$	−15.4450	−0.524541
$868$	−2.76271	−0.0937725
$869$	41.7090	1.41488
$870$	0.307979	0.0104414
$871$	0	0
$872$	−1.05861	−0.0358489
$873$	9.73556	0.329499
$874$	−2.64310	−0.0894043
$875$	−0.801938	−0.0271104
$876$	10.1371	0.342500
$877$	−35.0103	−1.18221	−0.591107	−	0.806593i	$-0.701309\pi$
$877$	−35.0103	−1.18221	−0.591107	+	0.806593i	$0.701309\pi$
$878$	3.00969	0.101572
$879$	−12.8901	−0.434772
$880$	−3.04892	−0.102779
$881$	23.8689	0.804164	0.402082	−	0.915604i	$-0.368287\pi$
$881$	23.8689	0.804164	0.402082	+	0.915604i	$0.368287\pi$
$882$	−6.35690	−0.214048
$883$	26.1806	0.881047	0.440524	−	0.897741i	$-0.354793\pi$
$883$	26.1806	0.881047	0.440524	+	0.897741i	$0.354793\pi$
$884$	0	0
$885$	3.11529	0.104719
$886$	38.4499	1.29175
$887$	25.8625	0.868377	0.434188	−	0.900822i	$-0.357035\pi$
$887$	25.8625	0.868377	0.434188	+	0.900822i	$0.357035\pi$
$888$	7.29590	0.244834
$889$	−12.7942	−0.429103
$890$	−8.05861	−0.270125
$891$	−3.04892	−0.102143
$892$	−12.7603	−0.427247
$893$	4.47219	0.149656
$894$	11.0532	0.369675
$895$	−4.32975	−0.144728
$896$	−0.801938	−0.0267909
$897$	0	0
$898$	26.9909	0.900698
$899$	1.06100	0.0353863
$900$	1.00000	0.0333333
$901$	7.85086	0.261550
$902$	−10.3840	−0.345751
$903$	−5.24698	−0.174609
$904$	11.0368	0.367080
$905$	−24.8321	−0.825446
$906$	6.89546	0.229086
$907$	12.3980	0.411670	0.205835	−	0.978587i	$-0.434009\pi$
$907$	12.3980	0.411670	0.205835	+	0.978587i	$0.434009\pi$
$908$	14.1836	0.470699
$909$	7.65817	0.254005
$910$	0	0
$911$	5.34183	0.176983	0.0884914	−	0.996077i	$-0.471795\pi$
$911$	5.34183	0.176983	0.0884914	+	0.996077i	$0.471795\pi$
$912$	7.40581	0.245231
$913$	12.0097	0.397463
$914$	19.2664	0.637274
$915$	11.0368	0.364867
$916$	−4.71140	−0.155669
$917$	13.7060	0.452613
$918$	−1.24698	−0.0411565
$919$	32.3327	1.06656	0.533279	−	0.845939i	$-0.320960\pi$
$919$	32.3327	1.06656	0.533279	+	0.845939i	$0.320960\pi$
$920$	−0.356896	−0.0117665
$921$	19.2892	0.635600
$922$	−18.5942	−0.612367
$923$	0	0
$924$	2.44504	0.0804360
$925$	7.29590	0.239888
$926$	5.79417	0.190408
$927$	2.32975	0.0765190
$928$	0.307979	0.0101099
$929$	19.3381	0.634463	0.317231	−	0.948348i	$-0.397247\pi$
$929$	19.3381	0.634463	0.317231	+	0.948348i	$0.397247\pi$
$930$	3.44504	0.112967
$931$	−47.0780	−1.54292
$932$	0.0556221	0.00182196
$933$	1.23729	0.0405071
$934$	11.2457	0.367969
$935$	3.80194	0.124337
$936$	0	0
$937$	−40.4271	−1.32070	−0.660348	−	0.750960i	$-0.729591\pi$
$937$	−40.4271	−1.32070	−0.660348	+	0.750960i	$0.729591\pi$
$938$	−4.78017	−0.156078
$939$	8.09113	0.264044
$940$	0.603875	0.0196962
$941$	−9.38537	−0.305954	−0.152977	−	0.988230i	$-0.548886\pi$
$941$	−9.38537	−0.305954	−0.152977	+	0.988230i	$0.548886\pi$
$942$	−20.8388	−0.678964
$943$	−1.21552	−0.0395828
$944$	3.11529	0.101394
$945$	−0.801938	−0.0260870
$946$	−19.9487	−0.648588
$947$	−24.4252	−0.793712	−0.396856	−	0.917881i	$-0.629899\pi$
$947$	−24.4252	−0.793712	−0.396856	+	0.917881i	$0.629899\pi$
$948$	−13.6799	−0.444304
$949$	0	0
$950$	7.40581	0.240276
$951$	−21.0368	−0.682166
$952$	1.00000	0.0324102
$953$	57.1657	1.85178	0.925889	−	0.377797i	$-0.123318\pi$
$953$	57.1657	1.85178	0.925889	+	0.377797i	$0.123318\pi$
$954$	−6.29590	−0.203837
$955$	22.1933	0.718158
$956$	21.0489	0.680771
$957$	−0.939001	−0.0303536
$958$	−35.2452	−1.13872
$959$	13.1817	0.425658
$960$	1.00000	0.0322749
$961$	−19.1317	−0.617151
$962$	0	0
$963$	3.77479	0.121641
$964$	−12.0030	−0.386590
$965$	25.4306	0.818639
$966$	0.286208	0.00920860
$967$	−5.76271	−0.185316	−0.0926581	−	0.995698i	$-0.529536\pi$
$967$	−5.76271	−0.185316	−0.0926581	+	0.995698i	$0.529536\pi$
$968$	−1.70410	−0.0547719
$969$	−9.23490	−0.296668
$970$	9.73556	0.312590
$971$	−1.72694	−0.0554201	−0.0277100	−	0.999616i	$-0.508822\pi$
$971$	−1.72694	−0.0554201	−0.0277100	+	0.999616i	$0.508822\pi$
$972$	1.00000	0.0320750
$973$	6.50365	0.208497
$974$	8.49827	0.272302
$975$	0	0
$976$	11.0368	0.353281
$977$	26.4747	0.847001	0.423500	−	0.905896i	$-0.360801\pi$
$977$	26.4747	0.847001	0.423500	+	0.905896i	$0.360801\pi$
$978$	−4.71917	−0.150902
$979$	24.5700	0.785261
$980$	−6.35690	−0.203064
$981$	−1.05861	−0.0337987
$982$	−25.5254	−0.814549
$983$	6.81727	0.217437	0.108719	−	0.994073i	$-0.465325\pi$
$983$	6.81727	0.217437	0.108719	+	0.994073i	$0.465325\pi$
$984$	3.40581	0.108573
$985$	−10.6310	−0.338733
$986$	−0.384043	−0.0122304
$987$	−0.484271	−0.0154145
$988$	0	0
$989$	−2.33513	−0.0742527
$990$	−3.04892	−0.0969010
$991$	8.31203	0.264040	0.132020	−	0.991247i	$-0.457854\pi$
$991$	8.31203	0.264040	0.132020	+	0.991247i	$0.457854\pi$
$992$	3.44504	0.109380
$993$	−29.4252	−0.933780
$994$	−5.35258	−0.169774
$995$	11.0465	0.350198
$996$	−3.93900	−0.124812
$997$	49.6002	1.57085	0.785426	−	0.618955i	$-0.212444\pi$
$997$	49.6002	1.57085	0.785426	+	0.618955i	$0.212444\pi$
$998$	−29.4577	−0.932468
$999$	7.29590	0.230832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.by.1.1	yes	3
13.5	odd	4		5070.2.b.x.1351.1		6
13.8	odd	4		5070.2.b.x.1351.6		6
13.12	even	2		5070.2.a.bn.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bn.1.3	✓	3	13.12	even	2
5070.2.a.by.1.1	yes	3	1.1	even	1	trivial
5070.2.b.x.1351.1		6	13.5	odd	4
5070.2.b.x.1351.6		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -0.801938 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -0.801938 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.04892 q^{11} +1.00000 q^{12} -0.801938 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.24698 q^{17} +1.00000 q^{18} +7.40581 q^{19} +1.00000 q^{20} -0.801938 q^{21} -3.04892 q^{22} -0.356896 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -0.801938 q^{28} +0.307979 q^{29} +1.00000 q^{30} +3.44504 q^{31} +1.00000 q^{32} -3.04892 q^{33} -1.24698 q^{34} -0.801938 q^{35} +1.00000 q^{36} +7.29590 q^{37} +7.40581 q^{38} +1.00000 q^{40} +3.40581 q^{41} -0.801938 q^{42} +6.54288 q^{43} -3.04892 q^{44} +1.00000 q^{45} -0.356896 q^{46} +0.603875 q^{47} +1.00000 q^{48} -6.35690 q^{49} +1.00000 q^{50} -1.24698 q^{51} -6.29590 q^{53} +1.00000 q^{54} -3.04892 q^{55} -0.801938 q^{56} +7.40581 q^{57} +0.307979 q^{58} +3.11529 q^{59} +1.00000 q^{60} +11.0368 q^{61} +3.44504 q^{62} -0.801938 q^{63} +1.00000 q^{64} -3.04892 q^{66} +5.96077 q^{67} -1.24698 q^{68} -0.356896 q^{69} -0.801938 q^{70} +6.67456 q^{71} +1.00000 q^{72} +10.1371 q^{73} +7.29590 q^{74} +1.00000 q^{75} +7.40581 q^{76} +2.44504 q^{77} -13.6799 q^{79} +1.00000 q^{80} +1.00000 q^{81} +3.40581 q^{82} -3.93900 q^{83} -0.801938 q^{84} -1.24698 q^{85} +6.54288 q^{86} +0.307979 q^{87} -3.04892 q^{88} -8.05861 q^{89} +1.00000 q^{90} -0.356896 q^{92} +3.44504 q^{93} +0.603875 q^{94} +7.40581 q^{95} +1.00000 q^{96} +9.73556 q^{97} -6.35690 q^{98} -3.04892 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9} + 3 q^{10} + 3 q^{12} + 2 q^{14} + 3 q^{15} + 3 q^{16} + q^{17} + 3 q^{18} + 9 q^{19} + 3 q^{20} + 2 q^{21} + 3 q^{23} + 3 q^{24} + 3 q^{25} + 3 q^{27} + 2 q^{28} + 6 q^{29} + 3 q^{30} + 10 q^{31} + 3 q^{32} + q^{34} + 2 q^{35} + 3 q^{36} + 8 q^{37} + 9 q^{38} + 3 q^{40} - 3 q^{41} + 2 q^{42} + q^{43} + 3 q^{45} + 3 q^{46} - 7 q^{47} + 3 q^{48} - 15 q^{49} + 3 q^{50} + q^{51} - 5 q^{53} + 3 q^{54} + 2 q^{56} + 9 q^{57} + 6 q^{58} + 7 q^{59} + 3 q^{60} + 5 q^{61} + 10 q^{62} + 2 q^{63} + 3 q^{64} + 5 q^{67} + q^{68} + 3 q^{69} + 2 q^{70} - q^{71} + 3 q^{72} + 25 q^{73} + 8 q^{74} + 3 q^{75} + 9 q^{76} + 7 q^{77} - 17 q^{79} + 3 q^{80} + 3 q^{81} - 3 q^{82} - 2 q^{83} + 2 q^{84} + q^{85} + q^{86} + 6 q^{87} + 7 q^{89} + 3 q^{90} + 3 q^{92} + 10 q^{93} - 7 q^{94} + 9 q^{95} + 3 q^{96} + 18 q^{97} - 15 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9 + 3 * q^10 + 3 * q^12 + 2 * q^14 + 3 * q^15 + 3 * q^16 + q^17 + 3 * q^18 + 9 * q^19 + 3 * q^20 + 2 * q^21 + 3 * q^23 + 3 * q^24 + 3 * q^25 + 3 * q^27 + 2 * q^28 + 6 * q^29 + 3 * q^30 + 10 * q^31 + 3 * q^32 + q^34 + 2 * q^35 + 3 * q^36 + 8 * q^37 + 9 * q^38 + 3 * q^40 - 3 * q^41 + 2 * q^42 + q^43 + 3 * q^45 + 3 * q^46 - 7 * q^47 + 3 * q^48 - 15 * q^49 + 3 * q^50 + q^51 - 5 * q^53 + 3 * q^54 + 2 * q^56 + 9 * q^57 + 6 * q^58 + 7 * q^59 + 3 * q^60 + 5 * q^61 + 10 * q^62 + 2 * q^63 + 3 * q^64 + 5 * q^67 + q^68 + 3 * q^69 + 2 * q^70 - q^71 + 3 * q^72 + 25 * q^73 + 8 * q^74 + 3 * q^75 + 9 * q^76 + 7 * q^77 - 17 * q^79 + 3 * q^80 + 3 * q^81 - 3 * q^82 - 2 * q^83 + 2 * q^84 + q^85 + q^86 + 6 * q^87 + 7 * q^89 + 3 * q^90 + 3 * q^92 + 10 * q^93 - 7 * q^94 + 9 * q^95 + 3 * q^96 + 18 * q^97 - 15 * q^98

Introduction

L-functions

Modular forms

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Database

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