LMFDB - Embedded newform 5070.2.a.be.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.73205$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.73205 q^{11} -1.00000 q^{12} +3.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} -2.26795 q^{19} -1.00000 q^{20} -3.00000 q^{21} -3.73205 q^{22} +3.46410 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +3.00000 q^{28} -5.46410 q^{29} +1.00000 q^{30} +8.92820 q^{31} +1.00000 q^{32} +3.73205 q^{33} -4.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} -7.92820 q^{37} -2.26795 q^{38} -1.00000 q^{40} -4.00000 q^{41} -3.00000 q^{42} -6.00000 q^{43} -3.73205 q^{44} -1.00000 q^{45} +3.46410 q^{46} +0.464102 q^{47} -1.00000 q^{48} +2.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} -3.73205 q^{53} -1.00000 q^{54} +3.73205 q^{55} +3.00000 q^{56} +2.26795 q^{57} -5.46410 q^{58} -4.53590 q^{59} +1.00000 q^{60} -7.46410 q^{61} +8.92820 q^{62} +3.00000 q^{63} +1.00000 q^{64} +3.73205 q^{66} -5.46410 q^{67} -4.00000 q^{68} -3.46410 q^{69} -3.00000 q^{70} -0.928203 q^{71} +1.00000 q^{72} +6.92820 q^{73} -7.92820 q^{74} -1.00000 q^{75} -2.26795 q^{76} -11.1962 q^{77} +16.9282 q^{79} -1.00000 q^{80} +1.00000 q^{81} -4.00000 q^{82} -2.53590 q^{83} -3.00000 q^{84} +4.00000 q^{85} -6.00000 q^{86} +5.46410 q^{87} -3.73205 q^{88} -10.1244 q^{89} -1.00000 q^{90} +3.46410 q^{92} -8.92820 q^{93} +0.464102 q^{94} +2.26795 q^{95} -1.00000 q^{96} +12.3923 q^{97} +2.00000 q^{98} -3.73205 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 6 * q^7 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} - 2 q^{12} + 6 q^{14} + 2 q^{15} + 2 q^{16} - 8 q^{17} + 2 q^{18} - 8 q^{19} - 2 q^{20} - 6 q^{21} - 4 q^{22} - 2 q^{24} + 2 q^{25} - 2 q^{27} + 6 q^{28} - 4 q^{29} + 2 q^{30} + 4 q^{31} + 2 q^{32} + 4 q^{33} - 8 q^{34} - 6 q^{35} + 2 q^{36} - 2 q^{37} - 8 q^{38} - 2 q^{40} - 8 q^{41} - 6 q^{42} - 12 q^{43} - 4 q^{44} - 2 q^{45} - 6 q^{47} - 2 q^{48} + 4 q^{49} + 2 q^{50} + 8 q^{51} - 4 q^{53} - 2 q^{54} + 4 q^{55} + 6 q^{56} + 8 q^{57} - 4 q^{58} - 16 q^{59} + 2 q^{60} - 8 q^{61} + 4 q^{62} + 6 q^{63} + 2 q^{64} + 4 q^{66} - 4 q^{67} - 8 q^{68} - 6 q^{70} + 12 q^{71} + 2 q^{72} - 2 q^{74} - 2 q^{75} - 8 q^{76} - 12 q^{77} + 20 q^{79} - 2 q^{80} + 2 q^{81} - 8 q^{82} - 12 q^{83} - 6 q^{84} + 8 q^{85} - 12 q^{86} + 4 q^{87} - 4 q^{88} + 4 q^{89} - 2 q^{90} - 4 q^{93} - 6 q^{94} + 8 q^{95} - 2 q^{96} + 4 q^{97} + 4 q^{98} - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 6 * q^7 + 2 * q^8 + 2 * q^9 - 2 * q^10 - 4 * q^11 - 2 * q^12 + 6 * q^14 + 2 * q^15 + 2 * q^16 - 8 * q^17 + 2 * q^18 - 8 * q^19 - 2 * q^20 - 6 * q^21 - 4 * q^22 - 2 * q^24 + 2 * q^25 - 2 * q^27 + 6 * q^28 - 4 * q^29 + 2 * q^30 + 4 * q^31 + 2 * q^32 + 4 * q^33 - 8 * q^34 - 6 * q^35 + 2 * q^36 - 2 * q^37 - 8 * q^38 - 2 * q^40 - 8 * q^41 - 6 * q^42 - 12 * q^43 - 4 * q^44 - 2 * q^45 - 6 * q^47 - 2 * q^48 + 4 * q^49 + 2 * q^50 + 8 * q^51 - 4 * q^53 - 2 * q^54 + 4 * q^55 + 6 * q^56 + 8 * q^57 - 4 * q^58 - 16 * q^59 + 2 * q^60 - 8 * q^61 + 4 * q^62 + 6 * q^63 + 2 * q^64 + 4 * q^66 - 4 * q^67 - 8 * q^68 - 6 * q^70 + 12 * q^71 + 2 * q^72 - 2 * q^74 - 2 * q^75 - 8 * q^76 - 12 * q^77 + 20 * q^79 - 2 * q^80 + 2 * q^81 - 8 * q^82 - 12 * q^83 - 6 * q^84 + 8 * q^85 - 12 * q^86 + 4 * q^87 - 4 * q^88 + 4 * q^89 - 2 * q^90 - 4 * q^93 - 6 * q^94 + 8 * q^95 - 2 * q^96 + 4 * q^97 + 4 * q^98 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−3.73205	−1.12526	−0.562628	−	0.826710i	$-0.690210\pi$
$11$	−3.73205	−1.12526	−0.562628	+	0.826710i	$0.690210\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	3.00000	0.801784
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	1.00000	0.235702
$19$	−2.26795	−0.520303	−0.260152	−	0.965568i	$-0.583773\pi$
$19$	−2.26795	−0.520303	−0.260152	+	0.965568i	$0.583773\pi$
$20$	−1.00000	−0.223607
$21$	−3.00000	−0.654654
$22$	−3.73205	−0.795676
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	3.00000	0.566947
$29$	−5.46410	−1.01466	−0.507329	−	0.861752i	$-0.669367\pi$
$29$	−5.46410	−1.01466	−0.507329	+	0.861752i	$0.669367\pi$
$30$	1.00000	0.182574
$31$	8.92820	1.60355	0.801776	−	0.597624i	$-0.203889\pi$
$31$	8.92820	1.60355	0.801776	+	0.597624i	$0.203889\pi$
$32$	1.00000	0.176777
$33$	3.73205	0.649667
$34$	−4.00000	−0.685994
$35$	−3.00000	−0.507093
$36$	1.00000	0.166667
$37$	−7.92820	−1.30339	−0.651694	−	0.758482i	$-0.725941\pi$
$37$	−7.92820	−1.30339	−0.651694	+	0.758482i	$0.725941\pi$
$38$	−2.26795	−0.367910
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	−3.00000	−0.462910
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	−3.73205	−0.562628
$45$	−1.00000	−0.149071
$46$	3.46410	0.510754
$47$	0.464102	0.0676962	0.0338481	−	0.999427i	$-0.489224\pi$
$47$	0.464102	0.0676962	0.0338481	+	0.999427i	$0.489224\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	1.00000	0.141421
$51$	4.00000	0.560112
$52$	0	0
$53$	−3.73205	−0.512637	−0.256318	−	0.966592i	$-0.582510\pi$
$53$	−3.73205	−0.512637	−0.256318	+	0.966592i	$0.582510\pi$
$54$	−1.00000	−0.136083
$55$	3.73205	0.503230
$56$	3.00000	0.400892
$57$	2.26795	0.300397
$58$	−5.46410	−0.717472
$59$	−4.53590	−0.590524	−0.295262	−	0.955416i	$-0.595407\pi$
$59$	−4.53590	−0.590524	−0.295262	+	0.955416i	$0.595407\pi$
$60$	1.00000	0.129099
$61$	−7.46410	−0.955680	−0.477840	−	0.878447i	$-0.658580\pi$
$61$	−7.46410	−0.955680	−0.477840	+	0.878447i	$0.658580\pi$
$62$	8.92820	1.13388
$63$	3.00000	0.377964
$64$	1.00000	0.125000
$65$	0	0
$66$	3.73205	0.459384
$67$	−5.46410	−0.667546	−0.333773	−	0.942653i	$-0.608322\pi$
$67$	−5.46410	−0.667546	−0.333773	+	0.942653i	$0.608322\pi$
$68$	−4.00000	−0.485071
$69$	−3.46410	−0.417029
$70$	−3.00000	−0.358569
$71$	−0.928203	−0.110157	−0.0550787	−	0.998482i	$-0.517541\pi$
$71$	−0.928203	−0.110157	−0.0550787	+	0.998482i	$0.517541\pi$
$72$	1.00000	0.117851
$73$	6.92820	0.810885	0.405442	−	0.914121i	$-0.367117\pi$
$73$	6.92820	0.810885	0.405442	+	0.914121i	$0.367117\pi$
$74$	−7.92820	−0.921635
$75$	−1.00000	−0.115470
$76$	−2.26795	−0.260152
$77$	−11.1962	−1.27592
$78$	0	0
$79$	16.9282	1.90457	0.952286	−	0.305208i	$-0.0987259\pi$
$79$	16.9282	1.90457	0.952286	+	0.305208i	$0.0987259\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−4.00000	−0.441726
$83$	−2.53590	−0.278351	−0.139176	−	0.990268i	$-0.544445\pi$
$83$	−2.53590	−0.278351	−0.139176	+	0.990268i	$0.544445\pi$
$84$	−3.00000	−0.327327
$85$	4.00000	0.433861
$86$	−6.00000	−0.646997
$87$	5.46410	0.585813
$88$	−3.73205	−0.397838
$89$	−10.1244	−1.07318	−0.536590	−	0.843843i	$-0.680288\pi$
$89$	−10.1244	−1.07318	−0.536590	+	0.843843i	$0.680288\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	3.46410	0.361158
$93$	−8.92820	−0.925812
$94$	0.464102	0.0478684
$95$	2.26795	0.232687
$96$	−1.00000	−0.102062
$97$	12.3923	1.25825	0.629124	−	0.777305i	$-0.283414\pi$
$97$	12.3923	1.25825	0.629124	+	0.777305i	$0.283414\pi$
$98$	2.00000	0.202031
$99$	−3.73205	−0.375085
$100$	1.00000	0.100000
$101$	8.39230	0.835066	0.417533	−	0.908662i	$-0.362895\pi$
$101$	8.39230	0.835066	0.417533	+	0.908662i	$0.362895\pi$
$102$	4.00000	0.396059
$103$	−19.5885	−1.93011	−0.965054	−	0.262051i	$-0.915601\pi$
$103$	−19.5885	−1.93011	−0.965054	+	0.262051i	$0.915601\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	−3.73205	−0.362489
$107$	0.928203	0.0897328	0.0448664	−	0.998993i	$-0.485714\pi$
$107$	0.928203	0.0897328	0.0448664	+	0.998993i	$0.485714\pi$
$108$	−1.00000	−0.0962250
$109$	−10.3923	−0.995402	−0.497701	−	0.867349i	$-0.665822\pi$
$109$	−10.3923	−0.995402	−0.497701	+	0.867349i	$0.665822\pi$
$110$	3.73205	0.355837
$111$	7.92820	0.752512
$112$	3.00000	0.283473
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	2.26795	0.212413
$115$	−3.46410	−0.323029
$116$	−5.46410	−0.507329
$117$	0	0
$118$	−4.53590	−0.417563
$119$	−12.0000	−1.10004
$120$	1.00000	0.0912871
$121$	2.92820	0.266200
$122$	−7.46410	−0.675768
$123$	4.00000	0.360668
$124$	8.92820	0.801776
$125$	−1.00000	−0.0894427
$126$	3.00000	0.267261
$127$	−4.66025	−0.413531	−0.206765	−	0.978391i	$-0.566294\pi$
$127$	−4.66025	−0.413531	−0.206765	+	0.978391i	$0.566294\pi$
$128$	1.00000	0.0883883
$129$	6.00000	0.528271
$130$	0	0
$131$	−20.3205	−1.77541	−0.887706	−	0.460412i	$-0.847702\pi$
$131$	−20.3205	−1.77541	−0.887706	+	0.460412i	$0.847702\pi$
$132$	3.73205	0.324833
$133$	−6.80385	−0.589968
$134$	−5.46410	−0.472026
$135$	1.00000	0.0860663
$136$	−4.00000	−0.342997
$137$	6.53590	0.558399	0.279200	−	0.960233i	$-0.409931\pi$
$137$	6.53590	0.558399	0.279200	+	0.960233i	$0.409931\pi$
$138$	−3.46410	−0.294884
$139$	21.7846	1.84775	0.923873	−	0.382699i	$-0.125005\pi$
$139$	21.7846	1.84775	0.923873	+	0.382699i	$0.125005\pi$
$140$	−3.00000	−0.253546
$141$	−0.464102	−0.0390844
$142$	−0.928203	−0.0778931
$143$	0	0
$144$	1.00000	0.0833333
$145$	5.46410	0.453769
$146$	6.92820	0.573382
$147$	−2.00000	−0.164957
$148$	−7.92820	−0.651694
$149$	22.7846	1.86659	0.933294	−	0.359113i	$-0.116921\pi$
$149$	22.7846	1.86659	0.933294	+	0.359113i	$0.116921\pi$
$150$	−1.00000	−0.0816497
$151$	−18.7846	−1.52867	−0.764335	−	0.644819i	$-0.776933\pi$
$151$	−18.7846	−1.52867	−0.764335	+	0.644819i	$0.776933\pi$
$152$	−2.26795	−0.183955
$153$	−4.00000	−0.323381
$154$	−11.1962	−0.902212
$155$	−8.92820	−0.717131
$156$	0	0
$157$	10.8038	0.862241	0.431120	−	0.902294i	$-0.358118\pi$
$157$	10.8038	0.862241	0.431120	+	0.902294i	$0.358118\pi$
$158$	16.9282	1.34674
$159$	3.73205	0.295971
$160$	−1.00000	−0.0790569
$161$	10.3923	0.819028
$162$	1.00000	0.0785674
$163$	10.9282	0.855963	0.427981	−	0.903788i	$-0.359225\pi$
$163$	10.9282	0.855963	0.427981	+	0.903788i	$0.359225\pi$
$164$	−4.00000	−0.312348
$165$	−3.73205	−0.290540
$166$	−2.53590	−0.196824
$167$	−6.46410	−0.500207	−0.250104	−	0.968219i	$-0.580465\pi$
$167$	−6.46410	−0.500207	−0.250104	+	0.968219i	$0.580465\pi$
$168$	−3.00000	−0.231455
$169$	0	0
$170$	4.00000	0.306786
$171$	−2.26795	−0.173434
$172$	−6.00000	−0.457496
$173$	−22.1244	−1.68208	−0.841042	−	0.540970i	$-0.818057\pi$
$173$	−22.1244	−1.68208	−0.841042	+	0.540970i	$0.818057\pi$
$174$	5.46410	0.414232
$175$	3.00000	0.226779
$176$	−3.73205	−0.281314
$177$	4.53590	0.340939
$178$	−10.1244	−0.758853
$179$	−22.9282	−1.71373	−0.856867	−	0.515537i	$-0.827592\pi$
$179$	−22.9282	−1.71373	−0.856867	+	0.515537i	$0.827592\pi$
$180$	−1.00000	−0.0745356
$181$	−3.07180	−0.228325	−0.114162	−	0.993462i	$-0.536418\pi$
$181$	−3.07180	−0.228325	−0.114162	+	0.993462i	$0.536418\pi$
$182$	0	0
$183$	7.46410	0.551762
$184$	3.46410	0.255377
$185$	7.92820	0.582893
$186$	−8.92820	−0.654648
$187$	14.9282	1.09166
$188$	0.464102	0.0338481
$189$	−3.00000	−0.218218
$190$	2.26795	0.164534
$191$	17.3205	1.25327	0.626634	−	0.779314i	$-0.284432\pi$
$191$	17.3205	1.25327	0.626634	+	0.779314i	$0.284432\pi$
$192$	−1.00000	−0.0721688
$193$	−17.8564	−1.28533	−0.642666	−	0.766146i	$-0.722172\pi$
$193$	−17.8564	−1.28533	−0.642666	+	0.766146i	$0.722172\pi$
$194$	12.3923	0.889716
$195$	0	0
$196$	2.00000	0.142857
$197$	9.39230	0.669174	0.334587	−	0.942365i	$-0.391403\pi$
$197$	9.39230	0.669174	0.334587	+	0.942365i	$0.391403\pi$
$198$	−3.73205	−0.265225
$199$	11.0718	0.784859	0.392429	−	0.919782i	$-0.371635\pi$
$199$	11.0718	0.784859	0.392429	+	0.919782i	$0.371635\pi$
$200$	1.00000	0.0707107
$201$	5.46410	0.385408
$202$	8.39230	0.590481
$203$	−16.3923	−1.15051
$204$	4.00000	0.280056
$205$	4.00000	0.279372
$206$	−19.5885	−1.36479
$207$	3.46410	0.240772
$208$	0	0
$209$	8.46410	0.585474
$210$	3.00000	0.207020
$211$	−21.9282	−1.50960	−0.754800	−	0.655955i	$-0.772266\pi$
$211$	−21.9282	−1.50960	−0.754800	+	0.655955i	$0.772266\pi$
$212$	−3.73205	−0.256318
$213$	0.928203	0.0635994
$214$	0.928203	0.0634507
$215$	6.00000	0.409197
$216$	−1.00000	−0.0680414
$217$	26.7846	1.81826
$218$	−10.3923	−0.703856
$219$	−6.92820	−0.468165
$220$	3.73205	0.251615
$221$	0	0
$222$	7.92820	0.532106
$223$	8.85641	0.593069	0.296534	−	0.955022i	$-0.404169\pi$
$223$	8.85641	0.593069	0.296534	+	0.955022i	$0.404169\pi$
$224$	3.00000	0.200446
$225$	1.00000	0.0666667
$226$	−12.0000	−0.798228
$227$	−13.4641	−0.893644	−0.446822	−	0.894623i	$-0.647444\pi$
$227$	−13.4641	−0.893644	−0.446822	+	0.894623i	$0.647444\pi$
$228$	2.26795	0.150199
$229$	11.4641	0.757569	0.378785	−	0.925485i	$-0.376342\pi$
$229$	11.4641	0.757569	0.378785	+	0.925485i	$0.376342\pi$
$230$	−3.46410	−0.228416
$231$	11.1962	0.736653
$232$	−5.46410	−0.358736
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	−0.464102	−0.0302747
$236$	−4.53590	−0.295262
$237$	−16.9282	−1.09960
$238$	−12.0000	−0.777844
$239$	−3.46410	−0.224074	−0.112037	−	0.993704i	$-0.535738\pi$
$239$	−3.46410	−0.224074	−0.112037	+	0.993704i	$0.535738\pi$
$240$	1.00000	0.0645497
$241$	−14.8038	−0.953600	−0.476800	−	0.879012i	$-0.658203\pi$
$241$	−14.8038	−0.953600	−0.476800	+	0.879012i	$0.658203\pi$
$242$	2.92820	0.188232
$243$	−1.00000	−0.0641500
$244$	−7.46410	−0.477840
$245$	−2.00000	−0.127775
$246$	4.00000	0.255031
$247$	0	0
$248$	8.92820	0.566941
$249$	2.53590	0.160706
$250$	−1.00000	−0.0632456
$251$	−26.4641	−1.67040	−0.835200	−	0.549947i	$-0.814648\pi$
$251$	−26.4641	−1.67040	−0.835200	+	0.549947i	$0.814648\pi$
$252$	3.00000	0.188982
$253$	−12.9282	−0.812789
$254$	−4.66025	−0.292410
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	−21.4641	−1.33889	−0.669447	−	0.742860i	$-0.733469\pi$
$257$	−21.4641	−1.33889	−0.669447	+	0.742860i	$0.733469\pi$
$258$	6.00000	0.373544
$259$	−23.7846	−1.47790
$260$	0	0
$261$	−5.46410	−0.338219
$262$	−20.3205	−1.25541
$263$	20.5167	1.26511	0.632556	−	0.774515i	$-0.282006\pi$
$263$	20.5167	1.26511	0.632556	+	0.774515i	$0.282006\pi$
$264$	3.73205	0.229692
$265$	3.73205	0.229258
$266$	−6.80385	−0.417171
$267$	10.1244	0.619601
$268$	−5.46410	−0.333773
$269$	−30.9282	−1.88573	−0.942863	−	0.333181i	$-0.891878\pi$
$269$	−30.9282	−1.88573	−0.942863	+	0.333181i	$0.891878\pi$
$270$	1.00000	0.0608581
$271$	−3.60770	−0.219152	−0.109576	−	0.993978i	$-0.534949\pi$
$271$	−3.60770	−0.219152	−0.109576	+	0.993978i	$0.534949\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	6.53590	0.394848
$275$	−3.73205	−0.225051
$276$	−3.46410	−0.208514
$277$	19.5885	1.17696	0.588478	−	0.808513i	$-0.299727\pi$
$277$	19.5885	1.17696	0.588478	+	0.808513i	$0.299727\pi$
$278$	21.7846	1.30655
$279$	8.92820	0.534518
$280$	−3.00000	−0.179284
$281$	6.92820	0.413302	0.206651	−	0.978415i	$-0.433744\pi$
$281$	6.92820	0.413302	0.206651	+	0.978415i	$0.433744\pi$
$282$	−0.464102	−0.0276368
$283$	−14.3923	−0.855534	−0.427767	−	0.903889i	$-0.640700\pi$
$283$	−14.3923	−0.855534	−0.427767	+	0.903889i	$0.640700\pi$
$284$	−0.928203	−0.0550787
$285$	−2.26795	−0.134342
$286$	0	0
$287$	−12.0000	−0.708338
$288$	1.00000	0.0589256
$289$	−1.00000	−0.0588235
$290$	5.46410	0.320863
$291$	−12.3923	−0.726450
$292$	6.92820	0.405442
$293$	−19.2487	−1.12452	−0.562261	−	0.826960i	$-0.690068\pi$
$293$	−19.2487	−1.12452	−0.562261	+	0.826960i	$0.690068\pi$
$294$	−2.00000	−0.116642
$295$	4.53590	0.264090
$296$	−7.92820	−0.460817
$297$	3.73205	0.216556
$298$	22.7846	1.31988
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−18.0000	−1.03750
$302$	−18.7846	−1.08093
$303$	−8.39230	−0.482125
$304$	−2.26795	−0.130076
$305$	7.46410	0.427393
$306$	−4.00000	−0.228665
$307$	−20.2487	−1.15565	−0.577827	−	0.816159i	$-0.696099\pi$
$307$	−20.2487	−1.15565	−0.577827	+	0.816159i	$0.696099\pi$
$308$	−11.1962	−0.637960
$309$	19.5885	1.11435
$310$	−8.92820	−0.507088
$311$	−5.07180	−0.287595	−0.143798	−	0.989607i	$-0.545931\pi$
$311$	−5.07180	−0.287595	−0.143798	+	0.989607i	$0.545931\pi$
$312$	0	0
$313$	1.32051	0.0746395	0.0373198	−	0.999303i	$-0.488118\pi$
$313$	1.32051	0.0746395	0.0373198	+	0.999303i	$0.488118\pi$
$314$	10.8038	0.609696
$315$	−3.00000	−0.169031
$316$	16.9282	0.952286
$317$	23.5359	1.32191	0.660954	−	0.750427i	$-0.270152\pi$
$317$	23.5359	1.32191	0.660954	+	0.750427i	$0.270152\pi$
$318$	3.73205	0.209283
$319$	20.3923	1.14175
$320$	−1.00000	−0.0559017
$321$	−0.928203	−0.0518073
$322$	10.3923	0.579141
$323$	9.07180	0.504768
$324$	1.00000	0.0555556
$325$	0	0
$326$	10.9282	0.605257
$327$	10.3923	0.574696
$328$	−4.00000	−0.220863
$329$	1.39230	0.0767603
$330$	−3.73205	−0.205443
$331$	6.39230	0.351353	0.175676	−	0.984448i	$-0.443789\pi$
$331$	6.39230	0.351353	0.175676	+	0.984448i	$0.443789\pi$
$332$	−2.53590	−0.139176
$333$	−7.92820	−0.434463
$334$	−6.46410	−0.353700
$335$	5.46410	0.298536
$336$	−3.00000	−0.163663
$337$	5.60770	0.305471	0.152735	−	0.988267i	$-0.451192\pi$
$337$	5.60770	0.305471	0.152735	+	0.988267i	$0.451192\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	4.00000	0.216930
$341$	−33.3205	−1.80441
$342$	−2.26795	−0.122637
$343$	−15.0000	−0.809924
$344$	−6.00000	−0.323498
$345$	3.46410	0.186501
$346$	−22.1244	−1.18941
$347$	−16.3923	−0.879985	−0.439993	−	0.898001i	$-0.645019\pi$
$347$	−16.3923	−0.879985	−0.439993	+	0.898001i	$0.645019\pi$
$348$	5.46410	0.292907
$349$	17.4641	0.934832	0.467416	−	0.884038i	$-0.345185\pi$
$349$	17.4641	0.934832	0.467416	+	0.884038i	$0.345185\pi$
$350$	3.00000	0.160357
$351$	0	0
$352$	−3.73205	−0.198919
$353$	28.3923	1.51117	0.755585	−	0.655051i	$-0.227353\pi$
$353$	28.3923	1.51117	0.755585	+	0.655051i	$0.227353\pi$
$354$	4.53590	0.241080
$355$	0.928203	0.0492639
$356$	−10.1244	−0.536590
$357$	12.0000	0.635107
$358$	−22.9282	−1.21179
$359$	12.9282	0.682324	0.341162	−	0.940004i	$-0.389179\pi$
$359$	12.9282	0.682324	0.341162	+	0.940004i	$0.389179\pi$
$360$	−1.00000	−0.0527046
$361$	−13.8564	−0.729285
$362$	−3.07180	−0.161450
$363$	−2.92820	−0.153691
$364$	0	0
$365$	−6.92820	−0.362639
$366$	7.46410	0.390155
$367$	−13.3205	−0.695325	−0.347662	−	0.937620i	$-0.613024\pi$
$367$	−13.3205	−0.695325	−0.347662	+	0.937620i	$0.613024\pi$
$368$	3.46410	0.180579
$369$	−4.00000	−0.208232
$370$	7.92820	0.412168
$371$	−11.1962	−0.581275
$372$	−8.92820	−0.462906
$373$	22.9282	1.18718	0.593589	−	0.804769i	$-0.297711\pi$
$373$	22.9282	1.18718	0.593589	+	0.804769i	$0.297711\pi$
$374$	14.9282	0.771919
$375$	1.00000	0.0516398
$376$	0.464102	0.0239342
$377$	0	0
$378$	−3.00000	−0.154303
$379$	−6.26795	−0.321963	−0.160981	−	0.986957i	$-0.551466\pi$
$379$	−6.26795	−0.321963	−0.160981	+	0.986957i	$0.551466\pi$
$380$	2.26795	0.116343
$381$	4.66025	0.238752
$382$	17.3205	0.886194
$383$	36.7846	1.87961	0.939803	−	0.341717i	$-0.111008\pi$
$383$	36.7846	1.87961	0.939803	+	0.341717i	$0.111008\pi$
$384$	−1.00000	−0.0510310
$385$	11.1962	0.570609
$386$	−17.8564	−0.908867
$387$	−6.00000	−0.304997
$388$	12.3923	0.629124
$389$	9.85641	0.499740	0.249870	−	0.968279i	$-0.419612\pi$
$389$	9.85641	0.499740	0.249870	+	0.968279i	$0.419612\pi$
$390$	0	0
$391$	−13.8564	−0.700749
$392$	2.00000	0.101015
$393$	20.3205	1.02503
$394$	9.39230	0.473177
$395$	−16.9282	−0.851750
$396$	−3.73205	−0.187543
$397$	7.92820	0.397905	0.198953	−	0.980009i	$-0.436246\pi$
$397$	7.92820	0.397905	0.198953	+	0.980009i	$0.436246\pi$
$398$	11.0718	0.554979
$399$	6.80385	0.340618
$400$	1.00000	0.0500000
$401$	−2.12436	−0.106085	−0.0530426	−	0.998592i	$-0.516892\pi$
$401$	−2.12436	−0.106085	−0.0530426	+	0.998592i	$0.516892\pi$
$402$	5.46410	0.272525
$403$	0	0
$404$	8.39230	0.417533
$405$	−1.00000	−0.0496904
$406$	−16.3923	−0.813536
$407$	29.5885	1.46665
$408$	4.00000	0.198030
$409$	0.947441	0.0468479	0.0234240	−	0.999726i	$-0.492543\pi$
$409$	0.947441	0.0468479	0.0234240	+	0.999726i	$0.492543\pi$
$410$	4.00000	0.197546
$411$	−6.53590	−0.322392
$412$	−19.5885	−0.965054
$413$	−13.6077	−0.669591
$414$	3.46410	0.170251
$415$	2.53590	0.124482
$416$	0	0
$417$	−21.7846	−1.06680
$418$	8.46410	0.413993
$419$	−17.8564	−0.872343	−0.436171	−	0.899864i	$-0.643666\pi$
$419$	−17.8564	−0.872343	−0.436171	+	0.899864i	$0.643666\pi$
$420$	3.00000	0.146385
$421$	−5.85641	−0.285424	−0.142712	−	0.989764i	$-0.545582\pi$
$421$	−5.85641	−0.285424	−0.142712	+	0.989764i	$0.545582\pi$
$422$	−21.9282	−1.06745
$423$	0.464102	0.0225654
$424$	−3.73205	−0.181244
$425$	−4.00000	−0.194029
$426$	0.928203	0.0449716
$427$	−22.3923	−1.08364
$428$	0.928203	0.0448664
$429$	0	0
$430$	6.00000	0.289346
$431$	13.8564	0.667440	0.333720	−	0.942672i	$-0.391696\pi$
$431$	13.8564	0.667440	0.333720	+	0.942672i	$0.391696\pi$
$432$	−1.00000	−0.0481125
$433$	−32.7846	−1.57553	−0.787764	−	0.615977i	$-0.788761\pi$
$433$	−32.7846	−1.57553	−0.787764	+	0.615977i	$0.788761\pi$
$434$	26.7846	1.28570
$435$	−5.46410	−0.261984
$436$	−10.3923	−0.497701
$437$	−7.85641	−0.375823
$438$	−6.92820	−0.331042
$439$	−21.3205	−1.01757	−0.508786	−	0.860893i	$-0.669906\pi$
$439$	−21.3205	−1.01757	−0.508786	+	0.860893i	$0.669906\pi$
$440$	3.73205	0.177919
$441$	2.00000	0.0952381
$442$	0	0
$443$	7.85641	0.373269	0.186635	−	0.982429i	$-0.440242\pi$
$443$	7.85641	0.373269	0.186635	+	0.982429i	$0.440242\pi$
$444$	7.92820	0.376256
$445$	10.1244	0.479940
$446$	8.85641	0.419363
$447$	−22.7846	−1.07768
$448$	3.00000	0.141737
$449$	18.1244	0.855341	0.427671	−	0.903935i	$-0.359334\pi$
$449$	18.1244	0.855341	0.427671	+	0.903935i	$0.359334\pi$
$450$	1.00000	0.0471405
$451$	14.9282	0.702942
$452$	−12.0000	−0.564433
$453$	18.7846	0.882578
$454$	−13.4641	−0.631902
$455$	0	0
$456$	2.26795	0.106206
$457$	−0.535898	−0.0250683	−0.0125341	−	0.999921i	$-0.503990\pi$
$457$	−0.535898	−0.0250683	−0.0125341	+	0.999921i	$0.503990\pi$
$458$	11.4641	0.535682
$459$	4.00000	0.186704
$460$	−3.46410	−0.161515
$461$	32.3923	1.50866	0.754330	−	0.656495i	$-0.227962\pi$
$461$	32.3923	1.50866	0.754330	+	0.656495i	$0.227962\pi$
$462$	11.1962	0.520892
$463$	−0.784610	−0.0364639	−0.0182320	−	0.999834i	$-0.505804\pi$
$463$	−0.784610	−0.0364639	−0.0182320	+	0.999834i	$0.505804\pi$
$464$	−5.46410	−0.253665
$465$	8.92820	0.414036
$466$	−18.0000	−0.833834
$467$	−3.60770	−0.166944	−0.0834721	−	0.996510i	$-0.526601\pi$
$467$	−3.60770	−0.166944	−0.0834721	+	0.996510i	$0.526601\pi$
$468$	0	0
$469$	−16.3923	−0.756926
$470$	−0.464102	−0.0214074
$471$	−10.8038	−0.497815
$472$	−4.53590	−0.208782
$473$	22.3923	1.02960
$474$	−16.9282	−0.777538
$475$	−2.26795	−0.104061
$476$	−12.0000	−0.550019
$477$	−3.73205	−0.170879
$478$	−3.46410	−0.158444
$479$	−26.2487	−1.19933	−0.599667	−	0.800250i	$-0.704700\pi$
$479$	−26.2487	−1.19933	−0.599667	+	0.800250i	$0.704700\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−14.8038	−0.674297
$483$	−10.3923	−0.472866
$484$	2.92820	0.133100
$485$	−12.3923	−0.562706
$486$	−1.00000	−0.0453609
$487$	−21.0000	−0.951601	−0.475800	−	0.879553i	$-0.657842\pi$
$487$	−21.0000	−0.951601	−0.475800	+	0.879553i	$0.657842\pi$
$488$	−7.46410	−0.337884
$489$	−10.9282	−0.494190
$490$	−2.00000	−0.0903508
$491$	15.3923	0.694645	0.347322	−	0.937746i	$-0.387091\pi$
$491$	15.3923	0.694645	0.347322	+	0.937746i	$0.387091\pi$
$492$	4.00000	0.180334
$493$	21.8564	0.984363
$494$	0	0
$495$	3.73205	0.167743
$496$	8.92820	0.400888
$497$	−2.78461	−0.124907
$498$	2.53590	0.113636
$499$	−1.32051	−0.0591141	−0.0295570	−	0.999563i	$-0.509410\pi$
$499$	−1.32051	−0.0591141	−0.0295570	+	0.999563i	$0.509410\pi$
$500$	−1.00000	−0.0447214
$501$	6.46410	0.288795
$502$	−26.4641	−1.18115
$503$	0.267949	0.0119473	0.00597363	−	0.999982i	$-0.498099\pi$
$503$	0.267949	0.0119473	0.00597363	+	0.999982i	$0.498099\pi$
$504$	3.00000	0.133631
$505$	−8.39230	−0.373453
$506$	−12.9282	−0.574729
$507$	0	0
$508$	−4.66025	−0.206765
$509$	−5.32051	−0.235827	−0.117914	−	0.993024i	$-0.537621\pi$
$509$	−5.32051	−0.235827	−0.117914	+	0.993024i	$0.537621\pi$
$510$	−4.00000	−0.177123
$511$	20.7846	0.919457
$512$	1.00000	0.0441942
$513$	2.26795	0.100132
$514$	−21.4641	−0.946741
$515$	19.5885	0.863171
$516$	6.00000	0.264135
$517$	−1.73205	−0.0761755
$518$	−23.7846	−1.04504
$519$	22.1244	0.971151
$520$	0	0
$521$	−27.3923	−1.20008	−0.600039	−	0.799970i	$-0.704848\pi$
$521$	−27.3923	−1.20008	−0.600039	+	0.799970i	$0.704848\pi$
$522$	−5.46410	−0.239157
$523$	13.7128	0.599619	0.299810	−	0.953999i	$-0.403077\pi$
$523$	13.7128	0.599619	0.299810	+	0.953999i	$0.403077\pi$
$524$	−20.3205	−0.887706
$525$	−3.00000	−0.130931
$526$	20.5167	0.894569
$527$	−35.7128	−1.55567
$528$	3.73205	0.162417
$529$	−11.0000	−0.478261
$530$	3.73205	0.162110
$531$	−4.53590	−0.196841
$532$	−6.80385	−0.294984
$533$	0	0
$534$	10.1244	0.438124
$535$	−0.928203	−0.0401297
$536$	−5.46410	−0.236013
$537$	22.9282	0.989425
$538$	−30.9282	−1.33341
$539$	−7.46410	−0.321502
$540$	1.00000	0.0430331
$541$	26.7846	1.15156	0.575780	−	0.817605i	$-0.304698\pi$
$541$	26.7846	1.15156	0.575780	+	0.817605i	$0.304698\pi$
$542$	−3.60770	−0.154964
$543$	3.07180	0.131823
$544$	−4.00000	−0.171499
$545$	10.3923	0.445157
$546$	0	0
$547$	29.3205	1.25365	0.626827	−	0.779158i	$-0.284353\pi$
$547$	29.3205	1.25365	0.626827	+	0.779158i	$0.284353\pi$
$548$	6.53590	0.279200
$549$	−7.46410	−0.318560
$550$	−3.73205	−0.159135
$551$	12.3923	0.527930
$552$	−3.46410	−0.147442
$553$	50.7846	2.15958
$554$	19.5885	0.832234
$555$	−7.92820	−0.336533
$556$	21.7846	0.923873
$557$	43.3923	1.83859	0.919295	−	0.393568i	$-0.128759\pi$
$557$	43.3923	1.83859	0.919295	+	0.393568i	$0.128759\pi$
$558$	8.92820	0.377961
$559$	0	0
$560$	−3.00000	−0.126773
$561$	−14.9282	−0.630269
$562$	6.92820	0.292249
$563$	−19.3205	−0.814262	−0.407131	−	0.913370i	$-0.633471\pi$
$563$	−19.3205	−0.814262	−0.407131	+	0.913370i	$0.633471\pi$
$564$	−0.464102	−0.0195422
$565$	12.0000	0.504844
$566$	−14.3923	−0.604954
$567$	3.00000	0.125988
$568$	−0.928203	−0.0389465
$569$	−18.3205	−0.768036	−0.384018	−	0.923326i	$-0.625460\pi$
$569$	−18.3205	−0.768036	−0.384018	+	0.923326i	$0.625460\pi$
$570$	−2.26795	−0.0949939
$571$	16.8564	0.705419	0.352709	−	0.935733i	$-0.385260\pi$
$571$	16.8564	0.705419	0.352709	+	0.935733i	$0.385260\pi$
$572$	0	0
$573$	−17.3205	−0.723575
$574$	−12.0000	−0.500870
$575$	3.46410	0.144463
$576$	1.00000	0.0416667
$577$	25.3205	1.05411	0.527053	−	0.849832i	$-0.323297\pi$
$577$	25.3205	1.05411	0.527053	+	0.849832i	$0.323297\pi$
$578$	−1.00000	−0.0415945
$579$	17.8564	0.742087
$580$	5.46410	0.226884
$581$	−7.60770	−0.315620
$582$	−12.3923	−0.513678
$583$	13.9282	0.576847
$584$	6.92820	0.286691
$585$	0	0
$586$	−19.2487	−0.795157
$587$	4.92820	0.203409	0.101704	−	0.994815i	$-0.467570\pi$
$587$	4.92820	0.203409	0.101704	+	0.994815i	$0.467570\pi$
$588$	−2.00000	−0.0824786
$589$	−20.2487	−0.834334
$590$	4.53590	0.186740
$591$	−9.39230	−0.386348
$592$	−7.92820	−0.325847
$593$	3.21539	0.132040	0.0660201	−	0.997818i	$-0.478970\pi$
$593$	3.21539	0.132040	0.0660201	+	0.997818i	$0.478970\pi$
$594$	3.73205	0.153128
$595$	12.0000	0.491952
$596$	22.7846	0.933294
$597$	−11.0718	−0.453138
$598$	0	0
$599$	15.0718	0.615817	0.307908	−	0.951416i	$-0.400371\pi$
$599$	15.0718	0.615817	0.307908	+	0.951416i	$0.400371\pi$
$600$	−1.00000	−0.0408248
$601$	−24.7128	−1.00806	−0.504028	−	0.863687i	$-0.668149\pi$
$601$	−24.7128	−1.00806	−0.504028	+	0.863687i	$0.668149\pi$
$602$	−18.0000	−0.733625
$603$	−5.46410	−0.222515
$604$	−18.7846	−0.764335
$605$	−2.92820	−0.119048
$606$	−8.39230	−0.340914
$607$	37.4449	1.51984	0.759920	−	0.650017i	$-0.225238\pi$
$607$	37.4449	1.51984	0.759920	+	0.650017i	$0.225238\pi$
$608$	−2.26795	−0.0919775
$609$	16.3923	0.664250
$610$	7.46410	0.302213
$611$	0	0
$612$	−4.00000	−0.161690
$613$	45.0000	1.81753	0.908766	−	0.417305i	$-0.137025\pi$
$613$	45.0000	1.81753	0.908766	+	0.417305i	$0.137025\pi$
$614$	−20.2487	−0.817171
$615$	−4.00000	−0.161296
$616$	−11.1962	−0.451106
$617$	28.5359	1.14881	0.574406	−	0.818571i	$-0.305233\pi$
$617$	28.5359	1.14881	0.574406	+	0.818571i	$0.305233\pi$
$618$	19.5885	0.787963
$619$	42.5167	1.70889	0.854444	−	0.519543i	$-0.173898\pi$
$619$	42.5167	1.70889	0.854444	+	0.519543i	$0.173898\pi$
$620$	−8.92820	−0.358565
$621$	−3.46410	−0.139010
$622$	−5.07180	−0.203361
$623$	−30.3731	−1.21687
$624$	0	0
$625$	1.00000	0.0400000
$626$	1.32051	0.0527781
$627$	−8.46410	−0.338024
$628$	10.8038	0.431120
$629$	31.7128	1.26447
$630$	−3.00000	−0.119523
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	16.9282	0.673368
$633$	21.9282	0.871568
$634$	23.5359	0.934730
$635$	4.66025	0.184937
$636$	3.73205	0.147985
$637$	0	0
$638$	20.3923	0.807339
$639$	−0.928203	−0.0367192
$640$	−1.00000	−0.0395285
$641$	−7.53590	−0.297650	−0.148825	−	0.988864i	$-0.547549\pi$
$641$	−7.53590	−0.297650	−0.148825	+	0.988864i	$0.547549\pi$
$642$	−0.928203	−0.0366333
$643$	28.7846	1.13515	0.567577	−	0.823320i	$-0.307881\pi$
$643$	28.7846	1.13515	0.567577	+	0.823320i	$0.307881\pi$
$644$	10.3923	0.409514
$645$	−6.00000	−0.236250
$646$	9.07180	0.356925
$647$	32.2679	1.26858	0.634292	−	0.773094i	$-0.281292\pi$
$647$	32.2679	1.26858	0.634292	+	0.773094i	$0.281292\pi$
$648$	1.00000	0.0392837
$649$	16.9282	0.664490
$650$	0	0
$651$	−26.7846	−1.04977
$652$	10.9282	0.427981
$653$	29.5885	1.15789	0.578943	−	0.815368i	$-0.303465\pi$
$653$	29.5885	1.15789	0.578943	+	0.815368i	$0.303465\pi$
$654$	10.3923	0.406371
$655$	20.3205	0.793988
$656$	−4.00000	−0.156174
$657$	6.92820	0.270295
$658$	1.39230	0.0542777
$659$	15.7128	0.612084	0.306042	−	0.952018i	$-0.400995\pi$
$659$	15.7128	0.612084	0.306042	+	0.952018i	$0.400995\pi$
$660$	−3.73205	−0.145270
$661$	42.3923	1.64887	0.824435	−	0.565957i	$-0.191493\pi$
$661$	42.3923	1.64887	0.824435	+	0.565957i	$0.191493\pi$
$662$	6.39230	0.248444
$663$	0	0
$664$	−2.53590	−0.0984119
$665$	6.80385	0.263842
$666$	−7.92820	−0.307212
$667$	−18.9282	−0.732903
$668$	−6.46410	−0.250104
$669$	−8.85641	−0.342408
$670$	5.46410	0.211097
$671$	27.8564	1.07538
$672$	−3.00000	−0.115728
$673$	−24.7846	−0.955376	−0.477688	−	0.878529i	$-0.658525\pi$
$673$	−24.7846	−0.955376	−0.477688	+	0.878529i	$0.658525\pi$
$674$	5.60770	0.216000
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−2.92820	−0.112540	−0.0562700	−	0.998416i	$-0.517921\pi$
$677$	−2.92820	−0.112540	−0.0562700	+	0.998416i	$0.517921\pi$
$678$	12.0000	0.460857
$679$	37.1769	1.42672
$680$	4.00000	0.153393
$681$	13.4641	0.515945
$682$	−33.3205	−1.27591
$683$	6.67949	0.255584	0.127792	−	0.991801i	$-0.459211\pi$
$683$	6.67949	0.255584	0.127792	+	0.991801i	$0.459211\pi$
$684$	−2.26795	−0.0867172
$685$	−6.53590	−0.249724
$686$	−15.0000	−0.572703
$687$	−11.4641	−0.437383
$688$	−6.00000	−0.228748
$689$	0	0
$690$	3.46410	0.131876
$691$	−23.0526	−0.876961	−0.438480	−	0.898741i	$-0.644483\pi$
$691$	−23.0526	−0.876961	−0.438480	+	0.898741i	$0.644483\pi$
$692$	−22.1244	−0.841042
$693$	−11.1962	−0.425307
$694$	−16.3923	−0.622243
$695$	−21.7846	−0.826337
$696$	5.46410	0.207116
$697$	16.0000	0.606043
$698$	17.4641	0.661026
$699$	18.0000	0.680823
$700$	3.00000	0.113389
$701$	−16.3923	−0.619129	−0.309564	−	0.950878i	$-0.600183\pi$
$701$	−16.3923	−0.619129	−0.309564	+	0.950878i	$0.600183\pi$
$702$	0	0
$703$	17.9808	0.678157
$704$	−3.73205	−0.140657
$705$	0.464102	0.0174791
$706$	28.3923	1.06856
$707$	25.1769	0.946875
$708$	4.53590	0.170470
$709$	7.32051	0.274927	0.137464	−	0.990507i	$-0.456105\pi$
$709$	7.32051	0.274927	0.137464	+	0.990507i	$0.456105\pi$
$710$	0.928203	0.0348348
$711$	16.9282	0.634857
$712$	−10.1244	−0.379426
$713$	30.9282	1.15827
$714$	12.0000	0.449089
$715$	0	0
$716$	−22.9282	−0.856867
$717$	3.46410	0.129369
$718$	12.9282	0.482476
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	−1.00000	−0.0372678
$721$	−58.7654	−2.18854
$722$	−13.8564	−0.515682
$723$	14.8038	0.550561
$724$	−3.07180	−0.114162
$725$	−5.46410	−0.202932
$726$	−2.92820	−0.108676
$727$	−12.6603	−0.469543	−0.234771	−	0.972051i	$-0.575434\pi$
$727$	−12.6603	−0.469543	−0.234771	+	0.972051i	$0.575434\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−6.92820	−0.256424
$731$	24.0000	0.887672
$732$	7.46410	0.275881
$733$	−6.85641	−0.253247	−0.126624	−	0.991951i	$-0.540414\pi$
$733$	−6.85641	−0.253247	−0.126624	+	0.991951i	$0.540414\pi$
$734$	−13.3205	−0.491669
$735$	2.00000	0.0737711
$736$	3.46410	0.127688
$737$	20.3923	0.751160
$738$	−4.00000	−0.147242
$739$	43.8372	1.61258	0.806288	−	0.591523i	$-0.201473\pi$
$739$	43.8372	1.61258	0.806288	+	0.591523i	$0.201473\pi$
$740$	7.92820	0.291447
$741$	0	0
$742$	−11.1962	−0.411024
$743$	2.92820	0.107425	0.0537127	−	0.998556i	$-0.482894\pi$
$743$	2.92820	0.107425	0.0537127	+	0.998556i	$0.482894\pi$
$744$	−8.92820	−0.327324
$745$	−22.7846	−0.834764
$746$	22.9282	0.839461
$747$	−2.53590	−0.0927837
$748$	14.9282	0.545829
$749$	2.78461	0.101747
$750$	1.00000	0.0365148
$751$	−41.1769	−1.50257	−0.751283	−	0.659980i	$-0.770565\pi$
$751$	−41.1769	−1.50257	−0.751283	+	0.659980i	$0.770565\pi$
$752$	0.464102	0.0169240
$753$	26.4641	0.964405
$754$	0	0
$755$	18.7846	0.683642
$756$	−3.00000	−0.109109
$757$	18.2679	0.663960	0.331980	−	0.943286i	$-0.392283\pi$
$757$	18.2679	0.663960	0.331980	+	0.943286i	$0.392283\pi$
$758$	−6.26795	−0.227662
$759$	12.9282	0.469264
$760$	2.26795	0.0822672
$761$	−43.9808	−1.59430	−0.797151	−	0.603780i	$-0.793660\pi$
$761$	−43.9808	−1.59430	−0.797151	+	0.603780i	$0.793660\pi$
$762$	4.66025	0.168823
$763$	−31.1769	−1.12868
$764$	17.3205	0.626634
$765$	4.00000	0.144620
$766$	36.7846	1.32908
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−39.7128	−1.43208	−0.716040	−	0.698059i	$-0.754047\pi$
$769$	−39.7128	−1.43208	−0.716040	+	0.698059i	$0.754047\pi$
$770$	11.1962	0.403481
$771$	21.4641	0.773011
$772$	−17.8564	−0.642666
$773$	53.3923	1.92039	0.960194	−	0.279334i	$-0.0901138\pi$
$773$	53.3923	1.92039	0.960194	+	0.279334i	$0.0901138\pi$
$774$	−6.00000	−0.215666
$775$	8.92820	0.320711
$776$	12.3923	0.444858
$777$	23.7846	0.853268
$778$	9.85641	0.353369
$779$	9.07180	0.325031
$780$	0	0
$781$	3.46410	0.123955
$782$	−13.8564	−0.495504
$783$	5.46410	0.195271
$784$	2.00000	0.0714286
$785$	−10.8038	−0.385606
$786$	20.3205	0.724809
$787$	−5.46410	−0.194774	−0.0973871	−	0.995247i	$-0.531048\pi$
$787$	−5.46410	−0.194774	−0.0973871	+	0.995247i	$0.531048\pi$
$788$	9.39230	0.334587
$789$	−20.5167	−0.730412
$790$	−16.9282	−0.602278
$791$	−36.0000	−1.28001
$792$	−3.73205	−0.132613
$793$	0	0
$794$	7.92820	0.281361
$795$	−3.73205	−0.132362
$796$	11.0718	0.392429
$797$	37.8564	1.34094	0.670471	−	0.741935i	$-0.266092\pi$
$797$	37.8564	1.34094	0.670471	+	0.741935i	$0.266092\pi$
$798$	6.80385	0.240854
$799$	−1.85641	−0.0656749
$800$	1.00000	0.0353553
$801$	−10.1244	−0.357727
$802$	−2.12436	−0.0750136
$803$	−25.8564	−0.912453
$804$	5.46410	0.192704
$805$	−10.3923	−0.366281
$806$	0	0
$807$	30.9282	1.08872
$808$	8.39230	0.295240
$809$	34.7846	1.22296	0.611481	−	0.791259i	$-0.290574\pi$
$809$	34.7846	1.22296	0.611481	+	0.791259i	$0.290574\pi$
$810$	−1.00000	−0.0351364
$811$	−7.58846	−0.266467	−0.133233	−	0.991085i	$-0.542536\pi$
$811$	−7.58846	−0.266467	−0.133233	+	0.991085i	$0.542536\pi$
$812$	−16.3923	−0.575257
$813$	3.60770	0.126527
$814$	29.5885	1.03707
$815$	−10.9282	−0.382798
$816$	4.00000	0.140028
$817$	13.6077	0.476073
$818$	0.947441	0.0331265
$819$	0	0
$820$	4.00000	0.139686
$821$	−49.3205	−1.72130	−0.860649	−	0.509199i	$-0.829942\pi$
$821$	−49.3205	−1.72130	−0.860649	+	0.509199i	$0.829942\pi$
$822$	−6.53590	−0.227966
$823$	39.5885	1.37997	0.689983	−	0.723825i	$-0.257618\pi$
$823$	39.5885	1.37997	0.689983	+	0.723825i	$0.257618\pi$
$824$	−19.5885	−0.682396
$825$	3.73205	0.129933
$826$	−13.6077	−0.473472
$827$	50.4974	1.75597	0.877984	−	0.478690i	$-0.158888\pi$
$827$	50.4974	1.75597	0.877984	+	0.478690i	$0.158888\pi$
$828$	3.46410	0.120386
$829$	39.3205	1.36566	0.682829	−	0.730578i	$-0.260749\pi$
$829$	39.3205	1.36566	0.682829	+	0.730578i	$0.260749\pi$
$830$	2.53590	0.0880223
$831$	−19.5885	−0.679516
$832$	0	0
$833$	−8.00000	−0.277184
$834$	−21.7846	−0.754339
$835$	6.46410	0.223699
$836$	8.46410	0.292737
$837$	−8.92820	−0.308604
$838$	−17.8564	−0.616839
$839$	−43.1769	−1.49063	−0.745316	−	0.666711i	$-0.767702\pi$
$839$	−43.1769	−1.49063	−0.745316	+	0.666711i	$0.767702\pi$
$840$	3.00000	0.103510
$841$	0.856406	0.0295313
$842$	−5.85641	−0.201825
$843$	−6.92820	−0.238620
$844$	−21.9282	−0.754800
$845$	0	0
$846$	0.464102	0.0159561
$847$	8.78461	0.301843
$848$	−3.73205	−0.128159
$849$	14.3923	0.493943
$850$	−4.00000	−0.137199
$851$	−27.4641	−0.941457
$852$	0.928203	0.0317997
$853$	−44.6410	−1.52848	−0.764240	−	0.644932i	$-0.776886\pi$
$853$	−44.6410	−1.52848	−0.764240	+	0.644932i	$0.776886\pi$
$854$	−22.3923	−0.766249
$855$	2.26795	0.0775622
$856$	0.928203	0.0317253
$857$	15.0718	0.514843	0.257421	−	0.966299i	$-0.417127\pi$
$857$	15.0718	0.514843	0.257421	+	0.966299i	$0.417127\pi$
$858$	0	0
$859$	−19.9282	−0.679942	−0.339971	−	0.940436i	$-0.610417\pi$
$859$	−19.9282	−0.679942	−0.339971	+	0.940436i	$0.610417\pi$
$860$	6.00000	0.204598
$861$	12.0000	0.408959
$862$	13.8564	0.471951
$863$	−10.9282	−0.372000	−0.186000	−	0.982550i	$-0.559553\pi$
$863$	−10.9282	−0.372000	−0.186000	+	0.982550i	$0.559553\pi$
$864$	−1.00000	−0.0340207
$865$	22.1244	0.752251
$866$	−32.7846	−1.11407
$867$	1.00000	0.0339618
$868$	26.7846	0.909129
$869$	−63.1769	−2.14313
$870$	−5.46410	−0.185250
$871$	0	0
$872$	−10.3923	−0.351928
$873$	12.3923	0.419416
$874$	−7.85641	−0.265747
$875$	−3.00000	−0.101419
$876$	−6.92820	−0.234082
$877$	33.7128	1.13840	0.569200	−	0.822199i	$-0.307253\pi$
$877$	33.7128	1.13840	0.569200	+	0.822199i	$0.307253\pi$
$878$	−21.3205	−0.719532
$879$	19.2487	0.649243
$880$	3.73205	0.125807
$881$	37.1051	1.25010	0.625052	−	0.780583i	$-0.285078\pi$
$881$	37.1051	1.25010	0.625052	+	0.780583i	$0.285078\pi$
$882$	2.00000	0.0673435
$883$	−21.3205	−0.717492	−0.358746	−	0.933435i	$-0.616796\pi$
$883$	−21.3205	−0.717492	−0.358746	+	0.933435i	$0.616796\pi$
$884$	0	0
$885$	−4.53590	−0.152473
$886$	7.85641	0.263941
$887$	52.2679	1.75499	0.877493	−	0.479589i	$-0.159214\pi$
$887$	52.2679	1.75499	0.877493	+	0.479589i	$0.159214\pi$
$888$	7.92820	0.266053
$889$	−13.9808	−0.468900
$890$	10.1244	0.339369
$891$	−3.73205	−0.125028
$892$	8.85641	0.296534
$893$	−1.05256	−0.0352225
$894$	−22.7846	−0.762031
$895$	22.9282	0.766405
$896$	3.00000	0.100223
$897$	0	0
$898$	18.1244	0.604818
$899$	−48.7846	−1.62706
$900$	1.00000	0.0333333
$901$	14.9282	0.497331
$902$	14.9282	0.497055
$903$	18.0000	0.599002
$904$	−12.0000	−0.399114
$905$	3.07180	0.102110
$906$	18.7846	0.624077
$907$	−7.07180	−0.234815	−0.117408	−	0.993084i	$-0.537458\pi$
$907$	−7.07180	−0.234815	−0.117408	+	0.993084i	$0.537458\pi$
$908$	−13.4641	−0.446822
$909$	8.39230	0.278355
$910$	0	0
$911$	−28.7846	−0.953677	−0.476838	−	0.878991i	$-0.658217\pi$
$911$	−28.7846	−0.953677	−0.476838	+	0.878991i	$0.658217\pi$
$912$	2.26795	0.0750993
$913$	9.46410	0.313216
$914$	−0.535898	−0.0177259
$915$	−7.46410	−0.246756
$916$	11.4641	0.378785
$917$	−60.9615	−2.01313
$918$	4.00000	0.132020
$919$	55.9615	1.84600	0.923000	−	0.384800i	$-0.125729\pi$
$919$	55.9615	1.84600	0.923000	+	0.384800i	$0.125729\pi$
$920$	−3.46410	−0.114208
$921$	20.2487	0.667218
$922$	32.3923	1.06678
$923$	0	0
$924$	11.1962	0.368326
$925$	−7.92820	−0.260678
$926$	−0.784610	−0.0257839
$927$	−19.5885	−0.643369
$928$	−5.46410	−0.179368
$929$	−30.1436	−0.988979	−0.494490	−	0.869184i	$-0.664645\pi$
$929$	−30.1436	−0.988979	−0.494490	+	0.869184i	$0.664645\pi$
$930$	8.92820	0.292767
$931$	−4.53590	−0.148658
$932$	−18.0000	−0.589610
$933$	5.07180	0.166043
$934$	−3.60770	−0.118047
$935$	−14.9282	−0.488204
$936$	0	0
$937$	33.7128	1.10135	0.550675	−	0.834720i	$-0.314370\pi$
$937$	33.7128	1.10135	0.550675	+	0.834720i	$0.314370\pi$
$938$	−16.3923	−0.535228
$939$	−1.32051	−0.0430932
$940$	−0.464102	−0.0151373
$941$	36.4974	1.18978	0.594891	−	0.803806i	$-0.297195\pi$
$941$	36.4974	1.18978	0.594891	+	0.803806i	$0.297195\pi$
$942$	−10.8038	−0.352008
$943$	−13.8564	−0.451227
$944$	−4.53590	−0.147631
$945$	3.00000	0.0975900
$946$	22.3923	0.728037
$947$	−13.6077	−0.442191	−0.221095	−	0.975252i	$-0.570963\pi$
$947$	−13.6077	−0.442191	−0.221095	+	0.975252i	$0.570963\pi$
$948$	−16.9282	−0.549802
$949$	0	0
$950$	−2.26795	−0.0735820
$951$	−23.5359	−0.763204
$952$	−12.0000	−0.388922
$953$	18.6795	0.605088	0.302544	−	0.953135i	$-0.402164\pi$
$953$	18.6795	0.605088	0.302544	+	0.953135i	$0.402164\pi$
$954$	−3.73205	−0.120830
$955$	−17.3205	−0.560478
$956$	−3.46410	−0.112037
$957$	−20.3923	−0.659190
$958$	−26.2487	−0.848057
$959$	19.6077	0.633165
$960$	1.00000	0.0322749
$961$	48.7128	1.57138
$962$	0	0
$963$	0.928203	0.0299109
$964$	−14.8038	−0.476800
$965$	17.8564	0.574818
$966$	−10.3923	−0.334367
$967$	−28.8564	−0.927959	−0.463980	−	0.885846i	$-0.653579\pi$
$967$	−28.8564	−0.927959	−0.463980	+	0.885846i	$0.653579\pi$
$968$	2.92820	0.0941160
$969$	−9.07180	−0.291428
$970$	−12.3923	−0.397893
$971$	−20.6077	−0.661332	−0.330666	−	0.943748i	$-0.607273\pi$
$971$	−20.6077	−0.661332	−0.330666	+	0.943748i	$0.607273\pi$
$972$	−1.00000	−0.0320750
$973$	65.3538	2.09515
$974$	−21.0000	−0.672883
$975$	0	0
$976$	−7.46410	−0.238920
$977$	1.46410	0.0468408	0.0234204	−	0.999726i	$-0.492544\pi$
$977$	1.46410	0.0468408	0.0234204	+	0.999726i	$0.492544\pi$
$978$	−10.9282	−0.349445
$979$	37.7846	1.20760
$980$	−2.00000	−0.0638877
$981$	−10.3923	−0.331801
$982$	15.3923	0.491188
$983$	16.1769	0.515963	0.257982	−	0.966150i	$-0.416943\pi$
$983$	16.1769	0.515963	0.257982	+	0.966150i	$0.416943\pi$
$984$	4.00000	0.127515
$985$	−9.39230	−0.299264
$986$	21.8564	0.696050
$987$	−1.39230	−0.0443176
$988$	0	0
$989$	−20.7846	−0.660912
$990$	3.73205	0.118612
$991$	−23.1769	−0.736239	−0.368119	−	0.929778i	$-0.619998\pi$
$991$	−23.1769	−0.736239	−0.368119	+	0.929778i	$0.619998\pi$
$992$	8.92820	0.283471
$993$	−6.39230	−0.202854
$994$	−2.78461	−0.0883225
$995$	−11.0718	−0.351000
$996$	2.53590	0.0803530
$997$	10.0192	0.317312	0.158656	−	0.987334i	$-0.449284\pi$
$997$	10.0192	0.317312	0.158656	+	0.987334i	$0.449284\pi$
$998$	−1.32051	−0.0418000
$999$	7.92820	0.250837

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.be.1.1		2
13.5	odd	4		5070.2.b.p.1351.1		4
13.6	odd	12		390.2.bb.a.361.1	yes	4
13.8	odd	4		5070.2.b.p.1351.4		4
13.11	odd	12		390.2.bb.a.121.1	✓	4
13.12	even	2		5070.2.a.ba.1.2		2
39.11	even	12		1170.2.bs.d.901.2		4
39.32	even	12		1170.2.bs.d.361.2		4
65.19	odd	12		1950.2.bc.a.751.2		4
65.24	odd	12		1950.2.bc.a.901.2		4
65.32	even	12		1950.2.y.d.49.2		4
65.37	even	12		1950.2.y.e.199.1		4
65.58	even	12		1950.2.y.e.49.1		4
65.63	even	12		1950.2.y.d.199.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.bb.a.121.1	✓	4	13.11	odd	12
390.2.bb.a.361.1	yes	4	13.6	odd	12
1170.2.bs.d.361.2		4	39.32	even	12
1170.2.bs.d.901.2		4	39.11	even	12
1950.2.y.d.49.2		4	65.32	even	12
1950.2.y.d.199.2		4	65.63	even	12
1950.2.y.e.49.1		4	65.58	even	12
1950.2.y.e.199.1		4	65.37	even	12
1950.2.bc.a.751.2		4	65.19	odd	12
1950.2.bc.a.901.2		4	65.24	odd	12
5070.2.a.ba.1.2		2	13.12	even	2
5070.2.a.be.1.1		2	1.1	even	1	trivial
5070.2.b.p.1351.1		4	13.5	odd	4
5070.2.b.p.1351.4		4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.73205 q^{11} -1.00000 q^{12} +3.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} -2.26795 q^{19} -1.00000 q^{20} -3.00000 q^{21} -3.73205 q^{22} +3.46410 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +3.00000 q^{28} -5.46410 q^{29} +1.00000 q^{30} +8.92820 q^{31} +1.00000 q^{32} +3.73205 q^{33} -4.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} -7.92820 q^{37} -2.26795 q^{38} -1.00000 q^{40} -4.00000 q^{41} -3.00000 q^{42} -6.00000 q^{43} -3.73205 q^{44} -1.00000 q^{45} +3.46410 q^{46} +0.464102 q^{47} -1.00000 q^{48} +2.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} -3.73205 q^{53} -1.00000 q^{54} +3.73205 q^{55} +3.00000 q^{56} +2.26795 q^{57} -5.46410 q^{58} -4.53590 q^{59} +1.00000 q^{60} -7.46410 q^{61} +8.92820 q^{62} +3.00000 q^{63} +1.00000 q^{64} +3.73205 q^{66} -5.46410 q^{67} -4.00000 q^{68} -3.46410 q^{69} -3.00000 q^{70} -0.928203 q^{71} +1.00000 q^{72} +6.92820 q^{73} -7.92820 q^{74} -1.00000 q^{75} -2.26795 q^{76} -11.1962 q^{77} +16.9282 q^{79} -1.00000 q^{80} +1.00000 q^{81} -4.00000 q^{82} -2.53590 q^{83} -3.00000 q^{84} +4.00000 q^{85} -6.00000 q^{86} +5.46410 q^{87} -3.73205 q^{88} -10.1244 q^{89} -1.00000 q^{90} +3.46410 q^{92} -8.92820 q^{93} +0.464102 q^{94} +2.26795 q^{95} -1.00000 q^{96} +12.3923 q^{97} +2.00000 q^{98} -3.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 6 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} - 2 q^{12} + 6 q^{14} + 2 q^{15} + 2 q^{16} - 8 q^{17} + 2 q^{18} - 8 q^{19} - 2 q^{20} - 6 q^{21} - 4 q^{22} - 2 q^{24} + 2 q^{25} - 2 q^{27} + 6 q^{28} - 4 q^{29} + 2 q^{30} + 4 q^{31} + 2 q^{32} + 4 q^{33} - 8 q^{34} - 6 q^{35} + 2 q^{36} - 2 q^{37} - 8 q^{38} - 2 q^{40} - 8 q^{41} - 6 q^{42} - 12 q^{43} - 4 q^{44} - 2 q^{45} - 6 q^{47} - 2 q^{48} + 4 q^{49} + 2 q^{50} + 8 q^{51} - 4 q^{53} - 2 q^{54} + 4 q^{55} + 6 q^{56} + 8 q^{57} - 4 q^{58} - 16 q^{59} + 2 q^{60} - 8 q^{61} + 4 q^{62} + 6 q^{63} + 2 q^{64} + 4 q^{66} - 4 q^{67} - 8 q^{68} - 6 q^{70} + 12 q^{71} + 2 q^{72} - 2 q^{74} - 2 q^{75} - 8 q^{76} - 12 q^{77} + 20 q^{79} - 2 q^{80} + 2 q^{81} - 8 q^{82} - 12 q^{83} - 6 q^{84} + 8 q^{85} - 12 q^{86} + 4 q^{87} - 4 q^{88} + 4 q^{89} - 2 q^{90} - 4 q^{93} - 6 q^{94} + 8 q^{95} - 2 q^{96} + 4 q^{97} + 4 q^{98} - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 6 * q^7 + 2 * q^8 + 2 * q^9 - 2 * q^10 - 4 * q^11 - 2 * q^12 + 6 * q^14 + 2 * q^15 + 2 * q^16 - 8 * q^17 + 2 * q^18 - 8 * q^19 - 2 * q^20 - 6 * q^21 - 4 * q^22 - 2 * q^24 + 2 * q^25 - 2 * q^27 + 6 * q^28 - 4 * q^29 + 2 * q^30 + 4 * q^31 + 2 * q^32 + 4 * q^33 - 8 * q^34 - 6 * q^35 + 2 * q^36 - 2 * q^37 - 8 * q^38 - 2 * q^40 - 8 * q^41 - 6 * q^42 - 12 * q^43 - 4 * q^44 - 2 * q^45 - 6 * q^47 - 2 * q^48 + 4 * q^49 + 2 * q^50 + 8 * q^51 - 4 * q^53 - 2 * q^54 + 4 * q^55 + 6 * q^56 + 8 * q^57 - 4 * q^58 - 16 * q^59 + 2 * q^60 - 8 * q^61 + 4 * q^62 + 6 * q^63 + 2 * q^64 + 4 * q^66 - 4 * q^67 - 8 * q^68 - 6 * q^70 + 12 * q^71 + 2 * q^72 - 2 * q^74 - 2 * q^75 - 8 * q^76 - 12 * q^77 + 20 * q^79 - 2 * q^80 + 2 * q^81 - 8 * q^82 - 12 * q^83 - 6 * q^84 + 8 * q^85 - 12 * q^86 + 4 * q^87 - 4 * q^88 + 4 * q^89 - 2 * q^90 - 4 * q^93 - 6 * q^94 + 8 * q^95 - 2 * q^96 + 4 * q^97 + 4 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more