Properties

Label 136.3.f.a
Level $136$
Weight $3$
Character orbit 136.f
Analytic conductor $3.706$
Analytic rank $0$
Dimension $32$
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] = [136,3,Mod(35,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.35");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 136.f (of order \(2\), degree \(1\), 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: \(3.70573159530\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{2} - 6 q^{4} - 6 q^{6} + 14 q^{8} + 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{2} - 6 q^{4} - 6 q^{6} + 14 q^{8} + 96 q^{9} + 26 q^{10} - 32 q^{11} + 6 q^{12} + 44 q^{14} - 30 q^{16} + 2 q^{18} - 10 q^{20} + 58 q^{22} - 54 q^{24} - 160 q^{25} + 52 q^{26} - 96 q^{27} + 16 q^{28} + 96 q^{30} - 18 q^{32} + 96 q^{35} - 14 q^{36} - 252 q^{38} + 198 q^{40} - 88 q^{42} - 32 q^{43} - 198 q^{44} - 52 q^{46} - 174 q^{48} - 256 q^{49} + 70 q^{50} + 172 q^{52} - 104 q^{54} + 76 q^{56} - 96 q^{57} - 226 q^{58} + 288 q^{59} - 40 q^{60} + 148 q^{62} - 198 q^{64} - 268 q^{66} - 128 q^{67} - 184 q^{70} + 214 q^{72} + 160 q^{73} + 246 q^{74} + 160 q^{75} + 316 q^{76} - 156 q^{78} + 94 q^{80} + 256 q^{81} - 236 q^{82} + 160 q^{83} + 24 q^{84} + 320 q^{86} + 374 q^{88} - 96 q^{89} + 158 q^{90} + 388 q^{92} + 64 q^{94} - 350 q^{96} + 256 q^{97} - 62 q^{98} - 224 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 −1.89539 0.638369i 0.544852 3.18497 + 2.41991i 9.70934i −1.03270 0.347816i 5.41501i −4.49195 6.61985i −8.70314 6.19813 18.4029i
35.2 −1.89539 + 0.638369i 0.544852 3.18497 2.41991i 9.70934i −1.03270 + 0.347816i 5.41501i −4.49195 + 6.61985i −8.70314 6.19813 + 18.4029i
35.3 −1.86718 0.716676i 4.09644 2.97275 + 2.67633i 4.29268i −7.64881 2.93582i 5.81927i −3.63261 7.12770i 7.78083 −3.07646 + 8.01523i
35.4 −1.86718 + 0.716676i 4.09644 2.97275 2.67633i 4.29268i −7.64881 + 2.93582i 5.81927i −3.63261 + 7.12770i 7.78083 −3.07646 8.01523i
35.5 −1.73617 0.992835i −5.08000 2.02856 + 3.44746i 1.34644i 8.81974 + 5.04361i 6.95342i −0.0991579 7.99939i 16.8064 1.33679 2.33764i
35.6 −1.73617 + 0.992835i −5.08000 2.02856 3.44746i 1.34644i 8.81974 5.04361i 6.95342i −0.0991579 + 7.99939i 16.8064 1.33679 + 2.33764i
35.7 −1.35773 1.46853i 0.832222 −0.313139 + 3.98772i 1.35598i −1.12993 1.22214i 13.3698i 6.28124 4.95440i −8.30741 −1.99129 + 1.84105i
35.8 −1.35773 + 1.46853i 0.832222 −0.313139 3.98772i 1.35598i −1.12993 + 1.22214i 13.3698i 6.28124 + 4.95440i −8.30741 −1.99129 1.84105i
35.9 −1.03740 1.70991i 0.111431 −1.84759 + 3.54773i 2.64262i −0.115599 0.190537i 5.52885i 7.98300 0.521210i −8.98758 −4.51864 + 2.74146i
35.10 −1.03740 + 1.70991i 0.111431 −1.84759 3.54773i 2.64262i −0.115599 + 0.190537i 5.52885i 7.98300 + 0.521210i −8.98758 −4.51864 2.74146i
35.11 −0.801563 1.83235i 5.56462 −2.71499 + 2.93748i 6.17932i −4.46039 10.1963i 1.08834i 7.55873 + 2.62023i 21.9650 11.3227 4.95311i
35.12 −0.801563 + 1.83235i 5.56462 −2.71499 2.93748i 6.17932i −4.46039 + 10.1963i 1.08834i 7.55873 2.62023i 21.9650 11.3227 + 4.95311i
35.13 −0.697886 1.87429i −3.94276 −3.02591 + 2.61608i 4.71646i 2.75160 + 7.38987i 8.04857i 7.01502 + 3.84570i 6.54538 8.84000 3.29155i
35.14 −0.697886 + 1.87429i −3.94276 −3.02591 2.61608i 4.71646i 2.75160 7.38987i 8.04857i 7.01502 3.84570i 6.54538 8.84000 + 3.29155i
35.15 0.0317538 1.99975i −4.33534 −3.99798 0.126999i 9.04730i −0.137663 + 8.66959i 6.61433i −0.380917 + 7.99093i 9.79518 −18.0923 0.287286i
35.16 0.0317538 + 1.99975i −4.33534 −3.99798 + 0.126999i 9.04730i −0.137663 8.66959i 6.61433i −0.380917 7.99093i 9.79518 −18.0923 + 0.287286i
35.17 0.336933 1.97141i 3.07030 −3.77295 1.32847i 5.04316i 1.03449 6.05284i 2.68396i −3.89020 + 6.99045i 0.426766 −9.94217 1.69921i
35.18 0.336933 + 1.97141i 3.07030 −3.77295 + 1.32847i 5.04316i 1.03449 + 6.05284i 2.68396i −3.89020 6.99045i 0.426766 −9.94217 + 1.69921i
35.19 0.376144 1.96431i −0.688578 −3.71703 1.47773i 7.39528i −0.259005 + 1.35258i 10.6434i −4.30085 + 6.74557i −8.52586 14.5266 + 2.78169i
35.20 0.376144 + 1.96431i −0.688578 −3.71703 + 1.47773i 7.39528i −0.259005 1.35258i 10.6434i −4.30085 6.74557i −8.52586 14.5266 2.78169i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.3.f.a 32
4.b odd 2 1 544.3.f.a 32
8.b even 2 1 544.3.f.a 32
8.d odd 2 1 inner 136.3.f.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.3.f.a 32 1.a even 1 1 trivial
136.3.f.a 32 8.d odd 2 1 inner
544.3.f.a 32 4.b odd 2 1
544.3.f.a 32 8.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(136, [\chi])\).