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Properties

Label 102.2.i.a
Level $102$
Weight $2$
Character orbit 102.i
Analytic conductor $0.814$
Analytic rank $0$
Dimension $24$
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] = [102,2,Mod(5,102)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(102, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("102.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 102 = 2 \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 102.i (of order \(16\), degree \(8\), 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: \(0.814474100617\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(3\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{11} - 8 q^{12} + 16 q^{17} - 16 q^{18} - 48 q^{21} - 16 q^{25} + 16 q^{29} - 16 q^{31} - 16 q^{33} - 16 q^{37} + 24 q^{38} + 16 q^{39} - 32 q^{41} - 8 q^{43} + 80 q^{45} + 16 q^{46} + 16 q^{47} + 32 q^{49} - 8 q^{50} + 16 q^{52} - 64 q^{53} + 48 q^{54} + 32 q^{55} + 64 q^{57} + 16 q^{58} + 64 q^{59} + 16 q^{60} + 16 q^{61} - 16 q^{62} - 48 q^{63} + 8 q^{66} - 16 q^{69} - 16 q^{70} - 16 q^{72} - 64 q^{73} - 16 q^{74} + 32 q^{75} + 16 q^{77} + 16 q^{78} - 32 q^{79} + 16 q^{80} - 40 q^{81} - 64 q^{82} - 72 q^{83} + 16 q^{84} - 32 q^{85} + 80 q^{87} - 16 q^{88} + 48 q^{89} - 64 q^{91} - 16 q^{92} - 32 q^{93} - 32 q^{94} - 32 q^{95} - 16 q^{97} + 16 q^{98} + 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.923880 + 0.382683i −1.72898 − 0.103064i 0.707107 − 0.707107i 2.09882 + 0.417480i 1.63681 − 0.566434i 0.745978 + 3.75028i −0.382683 + 0.923880i 2.97876 + 0.356391i −2.09882 + 0.417480i
5.2 −0.923880 + 0.382683i −0.317070 − 1.70278i 0.707107 − 0.707107i −2.04899 − 0.407570i 0.944561 + 1.45183i −0.635770 − 3.19623i −0.382683 + 0.923880i −2.79893 + 1.07980i 2.04899 − 0.407570i
5.3 −0.923880 + 0.382683i 1.33894 + 1.09874i 0.707107 − 0.707107i 0.715543 + 0.142330i −1.65749 − 0.502711i −0.110207 − 0.554049i −0.382683 + 0.923880i 0.585545 + 2.94230i −0.715543 + 0.142330i
11.1 0.382683 − 0.923880i −1.48785 + 0.886741i −0.707107 − 0.707107i 2.21277 − 3.31164i 0.249867 + 1.71393i 1.38649 − 0.926424i −0.923880 + 0.382683i 1.42738 − 2.63867i −2.21277 − 3.31164i
11.2 0.382683 − 0.923880i 0.464518 − 1.66860i −0.707107 − 0.707107i −0.995238 + 1.48948i −1.36382 − 1.06770i 2.77729 − 1.85573i −0.923880 + 0.382683i −2.56845 − 1.55019i 0.995238 + 1.48948i
11.3 0.382683 − 0.923880i 1.73044 + 0.0747517i −0.707107 − 0.707107i 0.630232 − 0.943208i 0.731271 − 1.57011i −4.16378 + 2.78215i −0.923880 + 0.382683i 2.98882 + 0.258706i −0.630232 − 0.943208i
23.1 −0.382683 + 0.923880i −1.20551 − 1.24368i −0.707107 − 0.707107i −3.29353 − 2.20066i 1.61034 − 0.637806i −0.750832 − 1.12370i 0.923880 − 0.382683i −0.0935025 + 2.99854i 3.29353 − 2.20066i
23.2 −0.382683 + 0.923880i 0.573016 + 1.63452i −0.707107 − 0.707107i 0.540978 + 0.361470i −1.72938 − 0.0961059i 0.676748 + 1.01283i 0.923880 − 0.382683i −2.34331 + 1.87321i −0.540978 + 0.361470i
23.3 −0.382683 + 0.923880i 1.33960 − 1.09794i −0.707107 − 0.707107i 0.904790 + 0.604561i 0.501723 + 1.65779i 0.0740832 + 0.110873i 0.923880 − 0.382683i 0.589049 − 2.94160i −0.904790 + 0.604561i
29.1 0.923880 − 0.382683i −1.27146 − 1.17618i 0.707107 − 0.707107i 0.284770 − 1.43164i −1.62478 − 0.600082i 0.146842 − 0.0292088i 0.382683 − 0.923880i 0.233206 + 2.99092i −0.284770 − 1.43164i
29.2 0.923880 − 0.382683i −0.799829 + 1.53632i 0.707107 − 0.707107i −0.456232 + 2.29363i −0.151023 + 1.72545i 3.60046 − 0.716176i 0.382683 − 0.923880i −1.72055 − 2.45758i 0.456232 + 2.29363i
29.3 0.923880 − 0.382683i 1.36418 − 1.06725i 0.707107 − 0.707107i −0.593905 + 2.98576i 0.851920 − 1.50806i −3.74730 + 0.745385i 0.382683 − 0.923880i 0.721973 − 2.91183i 0.593905 + 2.98576i
41.1 −0.923880 − 0.382683i −1.72898 + 0.103064i 0.707107 + 0.707107i 2.09882 − 0.417480i 1.63681 + 0.566434i 0.745978 − 3.75028i −0.382683 − 0.923880i 2.97876 − 0.356391i −2.09882 − 0.417480i
41.2 −0.923880 − 0.382683i −0.317070 + 1.70278i 0.707107 + 0.707107i −2.04899 + 0.407570i 0.944561 − 1.45183i −0.635770 + 3.19623i −0.382683 − 0.923880i −2.79893 − 1.07980i 2.04899 + 0.407570i
41.3 −0.923880 − 0.382683i 1.33894 − 1.09874i 0.707107 + 0.707107i 0.715543 − 0.142330i −1.65749 + 0.502711i −0.110207 + 0.554049i −0.382683 − 0.923880i 0.585545 − 2.94230i −0.715543 − 0.142330i
65.1 0.382683 + 0.923880i −1.48785 − 0.886741i −0.707107 + 0.707107i 2.21277 + 3.31164i 0.249867 − 1.71393i 1.38649 + 0.926424i −0.923880 − 0.382683i 1.42738 + 2.63867i −2.21277 + 3.31164i
65.2 0.382683 + 0.923880i 0.464518 + 1.66860i −0.707107 + 0.707107i −0.995238 − 1.48948i −1.36382 + 1.06770i 2.77729 + 1.85573i −0.923880 − 0.382683i −2.56845 + 1.55019i 0.995238 − 1.48948i
65.3 0.382683 + 0.923880i 1.73044 − 0.0747517i −0.707107 + 0.707107i 0.630232 + 0.943208i 0.731271 + 1.57011i −4.16378 − 2.78215i −0.923880 − 0.382683i 2.98882 − 0.258706i −0.630232 + 0.943208i
71.1 −0.382683 − 0.923880i −1.20551 + 1.24368i −0.707107 + 0.707107i −3.29353 + 2.20066i 1.61034 + 0.637806i −0.750832 + 1.12370i 0.923880 + 0.382683i −0.0935025 − 2.99854i 3.29353 + 2.20066i
71.2 −0.382683 − 0.923880i 0.573016 − 1.63452i −0.707107 + 0.707107i 0.540978 − 0.361470i −1.72938 + 0.0961059i 0.676748 − 1.01283i 0.923880 + 0.382683i −2.34331 − 1.87321i −0.540978 − 0.361470i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
51.i even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 102.2.i.a ✓ 24
3.b odd 2 1 102.2.i.b yes 24
4.b odd 2 1 816.2.cj.b 24
12.b even 2 1 816.2.cj.a 24
17.e odd 16 1 102.2.i.b yes 24
51.i even 16 1 inner 102.2.i.a ✓ 24
68.i even 16 1 816.2.cj.a 24
204.t odd 16 1 816.2.cj.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.i.a ✓ 24 1.a even 1 1 trivial
102.2.i.a ✓ 24 51.i even 16 1 inner
102.2.i.b yes 24 3.b odd 2 1
102.2.i.b yes 24 17.e odd 16 1
816.2.cj.a 24 12.b even 2 1
816.2.cj.a 24 68.i even 16 1
816.2.cj.b 24 4.b odd 2 1
816.2.cj.b 24 204.t odd 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + 8 T_{5}^{22} + 16 T_{5}^{21} + 156 T_{5}^{20} - 224 T_{5}^{19} + 1256 T_{5}^{18} + \cdots + 591872 \) acting on \(S_{2}^{\mathrm{new}}(102, [\chi])\). Copy content Toggle raw display