Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [912,2,Mod(229,912)]
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, 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("912.229");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 912.u (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: | \(72\) |
Relative dimension: | \(36\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
229.1 | −1.40358 | + | 0.173100i | 0.707107 | − | 0.707107i | 1.94007 | − | 0.485920i | 1.49138 | + | 1.49138i | −0.870080 | + | 1.11488i | − | 4.92462i | −2.63893 | + | 1.01785i | − | 1.00000i | −2.35143 | − | 1.83511i | ||
229.2 | −1.39928 | − | 0.204975i | −0.707107 | + | 0.707107i | 1.91597 | + | 0.573636i | −2.50960 | − | 2.50960i | 1.13438 | − | 0.844501i | − | 3.66014i | −2.56340 | − | 1.19540i | − | 1.00000i | 2.99723 | + | 4.02605i | ||
229.3 | −1.39560 | + | 0.228670i | −0.707107 | + | 0.707107i | 1.89542 | − | 0.638264i | −1.62266 | − | 1.62266i | 0.825147 | − | 1.14853i | 4.46507i | −2.49930 | + | 1.32419i | − | 1.00000i | 2.63564 | + | 1.89354i | |||
229.4 | −1.33099 | − | 0.477978i | −0.707107 | + | 0.707107i | 1.54307 | + | 1.27237i | 2.04814 | + | 2.04814i | 1.27913 | − | 0.603171i | − | 4.27873i | −1.44565 | − | 2.43107i | − | 1.00000i | −1.74709 | − | 3.70502i | ||
229.5 | −1.31725 | − | 0.514642i | 0.707107 | − | 0.707107i | 1.47029 | + | 1.35582i | −0.472067 | − | 0.472067i | −1.29534 | + | 0.567528i | 2.70142i | −1.23897 | − | 2.54263i | − | 1.00000i | 0.378884 | + | 0.864774i | |||
229.6 | −1.29742 | + | 0.562767i | 0.707107 | − | 0.707107i | 1.36659 | − | 1.46029i | −0.872136 | − | 0.872136i | −0.519477 | + | 1.31535i | 1.66350i | −0.951231 | + | 2.66367i | − | 1.00000i | 1.62233 | + | 0.640715i | |||
229.7 | −1.23568 | − | 0.687811i | 0.707107 | − | 0.707107i | 1.05383 | + | 1.69983i | −2.71335 | − | 2.71335i | −1.36012 | + | 0.387405i | − | 4.93074i | −0.133038 | − | 2.82530i | − | 1.00000i | 1.48657 | + | 5.21912i | ||
229.8 | −1.19214 | + | 0.760786i | −0.707107 | + | 0.707107i | 0.842409 | − | 1.81393i | 0.545592 | + | 0.545592i | 0.305015 | − | 1.38093i | − | 4.08514i | 0.375743 | + | 2.80336i | − | 1.00000i | −1.06550 | − | 0.235345i | ||
229.9 | −0.967261 | + | 1.03170i | −0.707107 | + | 0.707107i | −0.128812 | − | 1.99585i | 0.195347 | + | 0.195347i | −0.0455655 | − | 1.41348i | 2.13602i | 2.18371 | + | 1.79761i | − | 1.00000i | −0.390491 | + | 0.0125880i | |||
229.10 | −0.860657 | + | 1.12217i | 0.707107 | − | 0.707107i | −0.518540 | − | 1.93161i | 2.13495 | + | 2.13495i | 0.184919 | + | 1.40207i | 1.27583i | 2.61388 | + | 1.08056i | − | 1.00000i | −4.23324 | + | 0.558322i | |||
229.11 | −0.848555 | − | 1.13135i | −0.707107 | + | 0.707107i | −0.559910 | + | 1.92003i | −2.78534 | − | 2.78534i | 1.40000 | + | 0.199967i | 1.10989i | 2.64734 | − | 0.995792i | − | 1.00000i | −0.787683 | + | 5.51470i | |||
229.12 | −0.797241 | − | 1.16808i | 0.707107 | − | 0.707107i | −0.728813 | + | 1.86248i | 1.90354 | + | 1.90354i | −1.38969 | − | 0.262221i | 2.09292i | 2.75656 | − | 0.633536i | − | 1.00000i | 0.705903 | − | 3.74107i | |||
229.13 | −0.755025 | + | 1.19580i | 0.707107 | − | 0.707107i | −0.859875 | − | 1.80572i | −2.26608 | − | 2.26608i | 0.311675 | + | 1.37944i | 3.94241i | 2.80850 | + | 0.335123i | − | 1.00000i | 4.42073 | − | 0.998833i | |||
229.14 | −0.751656 | − | 1.19792i | −0.707107 | + | 0.707107i | −0.870028 | + | 1.80085i | 2.95011 | + | 2.95011i | 1.37856 | + | 0.315557i | − | 1.65157i | 2.81123 | − | 0.311393i | − | 1.00000i | 1.31653 | − | 5.75146i | ||
229.15 | −0.393309 | − | 1.35842i | −0.707107 | + | 0.707107i | −1.69062 | + | 1.06856i | 1.09899 | + | 1.09899i | 1.23866 | + | 0.682437i | − | 0.506031i | 2.11649 | + | 1.87629i | − | 1.00000i | 1.06064 | − | 1.92513i | ||
229.16 | −0.340084 | + | 1.37271i | −0.707107 | + | 0.707107i | −1.76869 | − | 0.933675i | 1.74488 | + | 1.74488i | −0.730180 | − | 1.21113i | 2.10527i | 1.88317 | − | 2.11037i | − | 1.00000i | −2.98862 | + | 1.80181i | |||
229.17 | −0.168934 | + | 1.40409i | 0.707107 | − | 0.707107i | −1.94292 | − | 0.474396i | −0.928065 | − | 0.928065i | 0.873386 | + | 1.11229i | − | 3.69407i | 0.994318 | − | 2.64789i | − | 1.00000i | 1.45987 | − | 1.14630i | ||
229.18 | −0.0883174 | − | 1.41145i | 0.707107 | − | 0.707107i | −1.98440 | + | 0.249312i | 0.529240 | + | 0.529240i | −1.06050 | − | 0.935598i | 2.76825i | 0.527149 | + | 2.77887i | − | 1.00000i | 0.700257 | − | 0.793739i | |||
229.19 | −0.0417641 | − | 1.41360i | 0.707107 | − | 0.707107i | −1.99651 | + | 0.118075i | 1.69566 | + | 1.69566i | −1.02910 | − | 0.970032i | − | 4.01648i | 0.250293 | + | 2.81733i | − | 1.00000i | 2.32616 | − | 2.46780i | ||
229.20 | 0.303797 | − | 1.38120i | −0.707107 | + | 0.707107i | −1.81541 | − | 0.839207i | 0.459913 | + | 0.459913i | 0.761837 | + | 1.19147i | − | 0.276216i | −1.71063 | + | 2.25250i | − | 1.00000i | 0.774951 | − | 0.495510i | ||
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.e | 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.u.a | ✓ | 72 |
16.e | even | 4 | 1 | inner | 912.2.u.a | ✓ | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
912.2.u.a | ✓ | 72 | 1.a | even | 1 | 1 | trivial |
912.2.u.a | ✓ | 72 | 16.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{72} + 1192 T_{5}^{68} - 96 T_{5}^{67} + 1488 T_{5}^{65} + 603908 T_{5}^{64} - 97456 T_{5}^{63} + 4608 T_{5}^{62} + 1618032 T_{5}^{61} + 170543032 T_{5}^{60} - 38602288 T_{5}^{59} + \cdots + 24199473135616 \)
acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\).