↑

Properties

Label 507.2.t.a
Level $507$
Weight $2$
Character orbit 507.t
Analytic conductor $4.048$
Analytic rank $0$
Dimension $336$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,2,Mod(4,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(78))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.t (of order \(78\), degree \(24\), minimal)

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: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(14\) over \(\Q(\zeta_{78})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{78}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 336 q + 14 q^{3} - 12 q^{4} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 336 q + 14 q^{3} - 12 q^{4} + 14 q^{9} + 4 q^{10} + 24 q^{12} + 26 q^{13} + 4 q^{14} + 8 q^{16} + 4 q^{17} + 72 q^{22} - 48 q^{23} + 68 q^{25} + 39 q^{26} - 28 q^{27} - 45 q^{29} + 4 q^{30} + 26 q^{31} + 65 q^{32} - 13 q^{33} + 65 q^{34} - 82 q^{35} - 12 q^{36} - 73 q^{38} + 24 q^{40} - 132 q^{42} - 72 q^{43} + 39 q^{44} + 8 q^{48} + 68 q^{49} + 52 q^{50} - 8 q^{51} - 65 q^{52} + 37 q^{53} - 53 q^{55} + 14 q^{56} - 26 q^{57} + 26 q^{58} - 208 q^{59} + 78 q^{60} - 12 q^{61} - 49 q^{62} + 14 q^{64} + 52 q^{65} + 64 q^{66} - 26 q^{67} - 33 q^{68} + 4 q^{69} - 78 q^{71} + 52 q^{73} + 205 q^{74} - 8 q^{75} - 26 q^{76} - 114 q^{77} - 65 q^{78} + 28 q^{79} - 468 q^{80} + 14 q^{81} - 45 q^{82} - 78 q^{83} + 13 q^{85} + 13 q^{86} + 46 q^{87} + 26 q^{88} - 8 q^{90} - 260 q^{91} + 8 q^{92} - 25 q^{94} - 90 q^{95} - 65 q^{96} - 26 q^{97} + 104 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.42393 − 0.0976810i 0.278217 − 0.960518i 3.87238 + 0.312611i −2.76576 − 1.04892i −0.768204 + 2.30105i −0.673721 − 1.58128i −4.53942 − 0.551185i −0.845190 − 0.534466i 6.60156 + 2.81266i
4.2 −2.28165 − 0.0919475i 0.278217 − 0.960518i 3.20398 + 0.258652i −1.50502 − 0.570779i −0.723113 + 2.16599i 1.60130 + 3.75840i −2.75287 − 0.334259i −0.845190 − 0.534466i 3.38146 + 1.44070i
4.3 −2.22635 − 0.0897189i 0.278217 − 0.960518i 2.95507 + 0.238558i 1.89329 + 0.718032i −0.705586 + 2.11349i −0.669684 − 1.57181i −2.13381 − 0.259091i −0.845190 − 0.534466i −4.15072 − 1.76846i
4.4 −1.11129 − 0.0447836i 0.278217 − 0.960518i −0.760548 − 0.0613978i 3.40515 + 1.29140i −0.352197 + 1.05496i 1.25015 + 2.93422i 3.05061 + 0.370412i −0.845190 − 0.534466i −3.72629 − 1.58762i
4.5 −1.05386 − 0.0424689i 0.278217 − 0.960518i −0.884707 − 0.0714209i −0.307541 − 0.116635i −0.333993 + 1.00043i 0.927898 + 2.17786i 3.02336 + 0.367103i −0.845190 − 0.534466i 0.319151 + 0.135977i
4.6 −1.04226 − 0.0420016i 0.278217 − 0.960518i −0.908975 − 0.0733801i 0.954453 + 0.361976i −0.330318 + 0.989423i −1.88928 − 4.43431i 3.01530 + 0.366124i −0.845190 − 0.534466i −0.979583 − 0.417361i
4.7 −0.0608806 − 0.00245341i 0.278217 − 0.960518i −1.98981 − 0.160634i 0.970084 + 0.367904i −0.0192946 + 0.0577944i 0.630770 + 1.48047i 0.241719 + 0.0293500i −0.845190 − 0.534466i −0.0581567 − 0.0247783i
4.8 0.0547950 + 0.00220816i 0.278217 − 0.960518i −1.99052 − 0.160691i −2.83821 − 1.07639i 0.0173659 − 0.0520173i 0.272711 + 0.640076i −0.217595 − 0.0264208i −0.845190 − 0.534466i −0.153143 − 0.0652481i
4.9 0.756241 + 0.0304755i 0.278217 − 0.960518i −1.42254 − 0.114840i −1.80244 − 0.683575i 0.239672 − 0.717904i 0.988668 + 2.32049i −2.57496 − 0.312656i −0.845190 − 0.534466i −1.34225 − 0.571877i
4.10 0.819680 + 0.0330320i 0.278217 − 0.960518i −1.32273 − 0.106782i 1.33494 + 0.506274i 0.259777 − 0.778127i −1.47285 − 3.45690i −2.70942 − 0.328983i −0.845190 − 0.534466i 1.07750 + 0.459078i
4.11 1.51468 + 0.0610396i 0.278217 − 0.960518i 0.297021 + 0.0239780i −3.09424 − 1.17349i 0.480041 − 1.43790i −0.201466 − 0.472857i −2.56129 − 0.310997i −0.845190 − 0.534466i −4.61516 − 1.96633i
4.12 1.90487 + 0.0767637i 0.278217 − 0.960518i 1.62913 + 0.131517i 2.99420 + 1.13555i 0.603702 − 1.80831i −0.0821119 − 0.192724i −0.691848 − 0.0840056i −0.845190 − 0.534466i 5.61640 + 2.39293i
4.13 2.45059 + 0.0987553i 0.278217 − 0.960518i 4.00211 + 0.323084i 0.599077 + 0.227200i 0.776653 − 2.32636i 0.599988 + 1.40822i 4.90623 + 0.595725i −0.845190 − 0.534466i 1.44565 + 0.615935i
4.14 2.69937 + 0.108781i 0.278217 − 0.960518i 5.28123 + 0.426345i −2.66969 − 1.01248i 0.855497 − 2.56253i −1.46999 − 3.45019i 8.84587 + 1.07408i −0.845190 − 0.534466i −7.09634 − 3.02347i
10.1 −2.70648 − 0.552531i 0.987050 − 0.160411i 5.17976 + 2.20689i −0.330566 + 1.34116i −2.76006 − 0.111227i −2.19257 + 1.04039i −8.25286 − 5.69654i 0.948536 − 0.316668i 1.63570 − 3.44716i
10.2 −2.16315 − 0.441610i 0.987050 − 0.160411i 2.64424 + 1.12661i −0.634667 + 2.57494i −2.20598 − 0.0888979i 4.23396 − 2.00904i −1.58846 − 1.09644i 0.948536 − 0.316668i 2.51000 − 5.28972i
10.3 −1.86580 − 0.380906i 0.987050 − 0.160411i 1.49616 + 0.637456i 0.0804692 − 0.326476i −1.90274 − 0.0766778i 2.39714 − 1.13746i 0.585658 + 0.404250i 0.948536 − 0.316668i −0.274496 + 0.578489i
10.4 −1.78276 − 0.363953i 0.987050 − 0.160411i 1.20581 + 0.513748i 0.718046 − 2.91323i −1.81806 − 0.0732651i −2.45740 + 1.16605i 1.03220 + 0.712476i 0.948536 − 0.316668i −2.34038 + 4.93225i
10.5 −0.886049 − 0.180888i 0.987050 − 0.160411i −1.08760 − 0.463382i 0.108486 − 0.440146i −0.903591 − 0.0364135i −1.78657 + 0.847738i 2.36833 + 1.63474i 0.948536 − 0.316668i −0.175741 + 0.370367i
10.6 −0.561687 − 0.114669i 0.987050 − 0.160411i −1.53762 − 0.655117i −0.327138 + 1.32725i −0.572808 − 0.0230834i −1.99021 + 0.944367i 1.73212 + 1.19560i 0.948536 − 0.316668i 0.335944 − 0.707987i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.k even 78 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.t.a ✓ 336
169.k even 78 1 inner 507.2.t.a ✓ 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.t.a ✓ 336 1.a even 1 1 trivial
507.2.t.a ✓ 336 169.k even 78 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{336} + 20 T_{2}^{334} + 152 T_{2}^{332} - 13 T_{2}^{331} - 73 T_{2}^{330} - 390 T_{2}^{329} + \cdots + 1446003845001 \) acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\). Copy content Toggle raw display