Properties

Label 912.2.u.a
Level $912$
Weight $2$
Character orbit 912.u
Analytic conductor $7.282$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 4 q^{4} + 8 q^{11} - 8 q^{12} - 12 q^{14} - 4 q^{16} - 4 q^{18} - 44 q^{20} - 52 q^{26} + 52 q^{28} + 16 q^{29} + 8 q^{30} - 64 q^{31} + 40 q^{34} + 16 q^{37} + 4 q^{40} - 8 q^{42} + 8 q^{43} + 68 q^{44} + 104 q^{47} + 16 q^{48} - 72 q^{49} + 116 q^{50} - 8 q^{52} - 16 q^{53} - 28 q^{56} - 92 q^{58} - 32 q^{59} + 32 q^{60} + 24 q^{62} - 40 q^{63} + 28 q^{64} + 16 q^{65} + 28 q^{66} - 8 q^{67} - 100 q^{68} - 20 q^{70} + 4 q^{72} - 124 q^{74} - 16 q^{75} + 8 q^{76} - 16 q^{77} - 12 q^{78} + 48 q^{79} + 44 q^{80} - 72 q^{81} + 52 q^{82} - 8 q^{84} - 64 q^{85} + 48 q^{86} - 32 q^{88} + 8 q^{90} + 8 q^{91} + 96 q^{92} + 12 q^{94} - 20 q^{96} + 208 q^{98} + 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 229.36
Significant digits:
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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 \(S_{2}^{\mathrm{new}}(912, [\chi])\). Copy content Toggle raw display