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Properties

Label 912.2.r.a
Level $912$
Weight $2$
Character orbit 912.r
Analytic conductor $7.282$
Analytic rank $0$
Dimension $312$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,2,Mod(341,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.341");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.r (of order \(4\), degree \(2\), 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: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(312\)
Relative dimension: \(156\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 312 q - 8 q^{4} - 4 q^{6} - 12 q^{19} - 20 q^{24} + 24 q^{28} - 40 q^{30} + 12 q^{36} + 60 q^{42} - 8 q^{43} - 24 q^{45} - 280 q^{49} + 40 q^{54} - 8 q^{58} - 40 q^{61} + 48 q^{63} + 40 q^{64} - 68 q^{66}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
341.1 −1.41379 − 0.0345843i −0.685383 − 1.59068i 1.99761 + 0.0977898i 0.255976 − 0.255976i 0.913976 + 2.27259i 2.27025i −2.82082 − 0.207340i −2.06050 + 2.18044i −0.370749 + 0.353044i
341.2 −1.41379 − 0.0345843i 1.59068 + 0.685383i 1.99761 + 0.0977898i −0.255976 + 0.255976i −2.22518 − 1.02400i 2.27025i −2.82082 − 0.207340i 2.06050 + 2.18044i 0.370749 − 0.353044i
341.3 −1.41256 + 0.0683968i 0.703921 − 1.58256i 1.99064 − 0.193229i −3.02347 + 3.02347i −0.886087 + 2.28360i 2.60060i −2.79868 + 0.409101i −2.00899 − 2.22799i 4.06403 − 4.47762i
341.4 −1.41256 + 0.0683968i 1.58256 − 0.703921i 1.99064 − 0.193229i 3.02347 − 3.02347i −2.18731 + 1.10257i 2.60060i −2.79868 + 0.409101i 2.00899 − 2.22799i −4.06403 + 4.47762i
341.5 −1.39341 − 0.241666i 1.11976 − 1.32141i 1.88319 + 0.673482i −0.229660 + 0.229660i −1.87963 + 1.57066i − 3.31284i −2.46131 − 1.39354i −0.492263 − 2.95934i 0.375512 − 0.264510i
341.6 −1.39341 − 0.241666i 1.32141 − 1.11976i 1.88319 + 0.673482i 0.229660 − 0.229660i −2.11188 + 1.24095i − 3.31284i −2.46131 − 1.39354i 0.492263 − 2.95934i −0.375512 + 0.264510i
341.7 −1.39263 + 0.246117i −1.55590 + 0.761034i 1.87885 − 0.685500i 0.0738937 − 0.0738937i 1.97950 − 1.44277i − 1.09781i −2.44784 + 1.41707i 1.84165 − 2.36819i −0.0847204 + 0.121093i
341.8 −1.39263 + 0.246117i −0.761034 + 1.55590i 1.87885 − 0.685500i −0.0738937 + 0.0738937i 0.676908 − 2.35410i − 1.09781i −2.44784 + 1.41707i −1.84165 − 2.36819i 0.0847204 − 0.121093i
341.9 −1.39224 + 0.248305i −1.53787 − 0.796835i 1.87669 − 0.691401i −1.06379 + 1.06379i 2.33895 + 0.727528i − 4.45201i −2.44113 + 1.42859i 1.73011 + 2.45086i 1.21691 − 1.74520i
341.10 −1.39224 + 0.248305i 0.796835 + 1.53787i 1.87669 − 0.691401i 1.06379 − 1.06379i −1.49125 − 1.94324i − 4.45201i −2.44113 + 1.42859i −1.73011 + 2.45086i −1.21691 + 1.74520i
341.11 −1.38174 − 0.301340i −1.49274 + 0.878474i 1.81839 + 0.832743i −1.56026 + 1.56026i 2.32730 − 0.763997i 3.98722i −2.26159 − 1.69858i 1.45657 − 2.62267i 2.62603 − 1.68570i
341.12 −1.38174 − 0.301340i −0.878474 + 1.49274i 1.81839 + 0.832743i 1.56026 − 1.56026i 1.66364 − 1.79786i 3.98722i −2.26159 − 1.69858i −1.45657 − 2.62267i −2.62603 + 1.68570i
341.13 −1.36478 − 0.370630i −1.67084 − 0.456392i 1.72527 + 1.01166i 2.14913 − 2.14913i 2.11118 + 1.24214i − 1.36586i −1.97966 − 2.02013i 2.58341 + 1.52512i −3.72963 + 2.13657i
341.14 −1.36478 − 0.370630i 0.456392 + 1.67084i 1.72527 + 1.01166i −2.14913 + 2.14913i −0.00361275 − 2.44949i − 1.36586i −1.97966 − 2.02013i −2.58341 + 1.52512i 3.72963 − 2.13657i
341.15 −1.36132 − 0.383166i −1.72932 − 0.0971903i 1.70637 + 1.04322i −2.68033 + 2.68033i 2.31692 + 0.794924i − 1.41487i −1.92318 − 2.07398i 2.98111 + 0.336147i 4.67578 − 2.62176i
341.16 −1.36132 − 0.383166i 0.0971903 + 1.72932i 1.70637 + 1.04322i 2.68033 − 2.68033i 0.530310 − 2.39140i − 1.41487i −1.92318 − 2.07398i −2.98111 + 0.336147i −4.67578 + 2.62176i
341.17 −1.36035 + 0.386597i −0.0554358 − 1.73116i 1.70109 − 1.05181i 2.23244 − 2.23244i 0.744674 + 2.33355i − 1.87702i −1.90744 + 2.08846i −2.99385 + 0.191937i −2.17384 + 3.89996i
341.18 −1.36035 + 0.386597i 1.73116 + 0.0554358i 1.70109 − 1.05181i −2.23244 + 2.23244i −2.37641 + 0.593851i − 1.87702i −1.90744 + 2.08846i 2.99385 + 0.191937i 2.17384 − 3.89996i
341.19 −1.31465 + 0.521235i −1.21642 − 1.23301i 1.45663 − 1.37049i −1.28422 + 1.28422i 2.24186 + 0.986945i 2.11010i −1.20062 + 2.56096i −0.0406447 + 2.99972i 1.01892 − 2.35769i
341.20 −1.31465 + 0.521235i 1.23301 + 1.21642i 1.45663 − 1.37049i 1.28422 − 1.28422i −2.25503 − 0.956481i 2.11010i −1.20062 + 2.56096i 0.0406447 + 2.99972i −1.01892 + 2.35769i
See next 80 embeddings (of 312 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 341.156
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
19.b odd 2 1 inner
48.i odd 4 1 inner
57.d even 2 1 inner
304.j odd 4 1 inner
912.r even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.r.a ✓ 312
3.b odd 2 1 inner 912.2.r.a ✓ 312
16.e even 4 1 inner 912.2.r.a ✓ 312
19.b odd 2 1 inner 912.2.r.a ✓ 312
48.i odd 4 1 inner 912.2.r.a ✓ 312
57.d even 2 1 inner 912.2.r.a ✓ 312
304.j odd 4 1 inner 912.2.r.a ✓ 312
912.r even 4 1 inner 912.2.r.a ✓ 312
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.r.a ✓ 312 1.a even 1 1 trivial
912.2.r.a ✓ 312 3.b odd 2 1 inner
912.2.r.a ✓ 312 16.e even 4 1 inner
912.2.r.a ✓ 312 19.b odd 2 1 inner
912.2.r.a ✓ 312 48.i odd 4 1 inner
912.2.r.a ✓ 312 57.d even 2 1 inner
912.2.r.a ✓ 312 304.j odd 4 1 inner
912.2.r.a ✓ 312 912.r even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(912, [\chi])\).