↑

Properties

Label 507.2.x.b
Level $507$
Weight $2$
Character orbit 507.x
Analytic conductor $4.048$
Analytic rank $0$
Dimension $2784$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,2,Mod(2,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(156))
 
chi = DirichletCharacter(H, H._module([78, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.x (of order \(156\), degree \(48\), 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: \(2784\)
Relative dimension: \(58\) over \(\Q(\zeta_{156})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{156}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 2784 q - 50 q^{3} - 92 q^{4} - 50 q^{6} - 100 q^{7} - 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 2784 q - 50 q^{3} - 92 q^{4} - 50 q^{6} - 100 q^{7} - 56 q^{9} - 116 q^{10} - 52 q^{12} - 112 q^{13} - 38 q^{15} - 204 q^{16} - 56 q^{18} - 88 q^{19} - 56 q^{21} - 48 q^{22} + 86 q^{24} - 104 q^{25} - 32 q^{27} - 124 q^{28} - 174 q^{30} - 112 q^{31} - 68 q^{33} - 68 q^{34} - 16 q^{36} - 76 q^{37} - 142 q^{39} - 96 q^{40} - 44 q^{42} - 140 q^{43} + 98 q^{45} - 58 q^{48} - 104 q^{49} - 52 q^{51} - 152 q^{52} - 98 q^{54} - 324 q^{55} - 68 q^{57} - 132 q^{58} - 96 q^{60} - 124 q^{61} - 174 q^{63} - 104 q^{64} + 58 q^{66} + 144 q^{67} + 26 q^{69} + 136 q^{70} - 64 q^{72} - 76 q^{73} - 194 q^{75} - 96 q^{76} + 28 q^{78} - 120 q^{79} - 56 q^{81} - 340 q^{82} - 56 q^{84} - 116 q^{85} - 34 q^{87} + 116 q^{88} - 52 q^{90} - 112 q^{91} + 74 q^{93} + 36 q^{94} - 406 q^{96} - 124 q^{97} - 92 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −0.0548695 + 2.72425i 0.0796955 + 1.73022i −5.42016 − 0.218425i 0.698419 − 3.81115i −4.71792 + 0.122175i 1.76691 − 2.67341i 0.563407 − 9.31422i −2.98730 + 0.275781i 10.3442 + 2.11179i
2.2 −0.0547878 + 2.72020i 1.34843 + 1.08707i −5.39809 − 0.217536i −0.502675 + 2.74301i −3.03093 + 3.60845i −0.957757 + 1.44913i 0.558940 − 9.24037i 0.636546 + 2.93169i −7.43399 − 1.51766i
2.3 −0.0543226 + 2.69710i −0.327883 − 1.70073i −5.27300 − 0.212495i 0.410508 − 2.24007i 4.60485 − 0.791944i −1.50010 + 2.26971i 0.533803 − 8.82481i −2.78499 + 1.11528i 6.01938 + 1.22887i
2.4 −0.0509749 + 2.53089i −0.290269 − 1.70755i −4.40440 − 0.177491i −0.457167 + 2.49468i 4.33642 − 0.647595i 2.49693 − 3.77795i 0.368040 − 6.08443i −2.83149 + 0.991300i −6.29045 − 1.28420i
2.5 −0.0502666 + 2.49572i −1.68648 − 0.394699i −4.22771 − 0.170371i 0.127318 − 0.694751i 1.06983 − 4.18914i 1.03440 − 1.56510i 0.336274 − 5.55927i 2.68843 + 1.33130i 1.72750 + 0.352672i
2.6 −0.0492501 + 2.44525i 1.65916 − 0.497167i −3.97844 − 0.160326i −0.00106230 + 0.00579679i 1.13398 + 4.08155i 2.10203 − 3.18046i 0.292634 − 4.83781i 2.50565 − 1.64976i −0.0141223 − 0.00288308i
2.7 −0.0488004 + 2.42292i −1.18779 + 1.26061i −3.86978 − 0.155947i −0.0407216 + 0.222211i −2.99640 − 2.93944i −1.35846 + 2.05540i 0.274050 − 4.53059i −0.178298 − 2.99470i −0.536411 − 0.109509i
2.8 −0.0447349 + 2.22107i 1.12900 − 1.31353i −2.93279 − 0.118187i −0.425593 + 2.32239i 2.86694 + 2.56635i −1.87109 + 2.83104i 0.125436 − 2.07370i −0.450722 − 2.96595i −5.13915 − 1.04916i
2.9 −0.0419920 + 2.08489i −1.62925 − 0.587818i −2.34662 − 0.0945657i 0.137218 − 0.748772i 1.29395 − 3.37213i −1.58590 + 2.39953i 0.0438824 − 0.725462i 2.30894 + 1.91541i 1.55534 + 0.317526i
2.10 −0.0409543 + 2.03336i 0.256039 + 1.71302i −2.13452 − 0.0860180i −0.267381 + 1.45905i −3.49368 + 0.450464i 0.531912 − 0.804804i 0.0167307 − 0.276592i −2.86889 + 0.877200i −2.95583 − 0.603437i
2.11 −0.0408434 + 2.02786i 0.845552 − 1.51164i −2.11217 − 0.0851176i 0.624365 − 3.40705i 3.03085 + 1.77640i −0.0807157 + 0.122126i 0.0139470 − 0.230572i −1.57008 − 2.55633i 6.88352 + 1.40528i
2.12 −0.0377660 + 1.87507i 1.72833 + 0.113543i −1.51607 − 0.0610955i 0.388121 − 2.11791i −0.278173 + 3.23644i 0.693572 − 1.04940i −0.0546594 + 0.903628i 2.97422 + 0.392479i 3.95656 + 0.807737i
2.13 −0.0374341 + 1.85859i 1.06249 + 1.36789i −1.45458 − 0.0586174i 0.440958 − 2.40623i −2.58212 + 1.92352i −2.25911 + 3.41812i −0.0610868 + 1.00988i −0.742237 + 2.90673i 4.45569 + 0.909636i
2.14 −0.0362418 + 1.79939i −0.541421 + 1.64525i −1.23812 − 0.0498947i −0.615732 + 3.35994i −2.94084 − 1.03386i 0.961160 − 1.45427i −0.0826814 + 1.36689i −2.41373 − 1.78155i −6.02354 − 1.22971i
2.15 −0.0337489 + 1.67562i −1.44454 + 0.955666i −0.808183 − 0.0325687i 0.693384 − 3.78367i −1.55258 − 2.45276i −0.672208 + 1.01708i −0.120536 + 1.99269i 1.17340 − 2.76100i 6.31660 + 1.28954i
2.16 −0.0292596 + 1.45273i 1.72672 − 0.135794i −0.111191 − 0.00448085i −0.589576 + 3.21721i 0.146749 + 2.51243i −0.0810588 + 0.122645i −0.165700 + 2.73935i 2.96312 − 0.468956i −4.65649 − 0.950629i
2.17 −0.0287183 + 1.42585i −0.617987 − 1.61805i −0.0338548 − 0.00136430i −0.204323 + 1.11496i 2.32485 − 0.834692i 0.633950 − 0.959192i −0.169299 + 2.79885i −2.23618 + 1.99987i −1.58390 − 0.323354i
2.18 −0.0282490 + 1.40255i −1.72318 − 0.175026i 0.0320220 + 0.00129044i −0.476786 + 2.60174i 0.294162 − 2.41191i 1.45655 − 2.20382i −0.172117 + 2.84543i 2.93873 + 0.603205i −3.63560 − 0.742214i
2.19 −0.0230681 + 1.14532i −0.546057 − 1.64372i 0.687151 + 0.0276912i −0.0695333 + 0.379431i 1.89518 − 0.587493i −1.90590 + 2.88371i −0.185900 + 3.07330i −2.40364 + 1.79513i −0.432966 − 0.0883907i
2.20 −0.0219978 + 1.09218i 1.01159 + 1.40595i 0.805998 + 0.0324806i 0.345605 − 1.88591i −1.55781 + 1.07391i 2.55979 − 3.87306i −0.185120 + 3.06041i −0.953385 + 2.84448i 2.05215 + 0.418950i
See next 80 embeddings (of 2784 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
169.l odd 156 1 inner
507.x even 156 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.x.b ✓ 2784
3.b odd 2 1 inner 507.2.x.b ✓ 2784
169.l odd 156 1 inner 507.2.x.b ✓ 2784
507.x even 156 1 inner 507.2.x.b ✓ 2784
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.x.b ✓ 2784 1.a even 1 1 trivial
507.2.x.b ✓ 2784 3.b odd 2 1 inner
507.2.x.b ✓ 2784 169.l odd 156 1 inner
507.2.x.b ✓ 2784 507.x even 156 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2784} + 46 T_{2}^{2782} + 1521 T_{2}^{2780} + 37538 T_{2}^{2778} + 776642 T_{2}^{2776} + \cdots + 26\!\cdots\!61 \) acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\). Copy content Toggle raw display