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Properties

Label 102.2.i.b
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
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 16 q^{11} - 48 q^{15} - 16 q^{17} - 16 q^{18} - 8 q^{24} - 16 q^{25} - 16 q^{29} - 16 q^{31} + 16 q^{33} - 16 q^{37} - 24 q^{38} + 16 q^{39} + 32 q^{41} + 48 q^{42} - 8 q^{43} - 16 q^{45} + 16 q^{46} - 16 q^{47} + 32 q^{49} + 8 q^{50} + 64 q^{51} + 16 q^{52} + 64 q^{53} + 8 q^{54} + 32 q^{55} + 8 q^{57} + 16 q^{58} - 64 q^{59} + 16 q^{60} + 16 q^{61} + 16 q^{62} + 48 q^{63} - 16 q^{69} - 16 q^{70} + 16 q^{72} - 64 q^{73} + 16 q^{74} - 80 q^{75} - 16 q^{77} - 16 q^{78} - 32 q^{79} - 16 q^{80} + 32 q^{81} - 64 q^{82} + 72 q^{83} - 16 q^{84} - 32 q^{85} - 32 q^{87} - 16 q^{88} - 48 q^{89} - 64 q^{91} + 16 q^{92} + 96 q^{93} - 32 q^{94} + 32 q^{95} - 16 q^{97} - 16 q^{98} + 16 q^{99}+O(q^{100}) \) 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 −1.45183 − 0.944561i 0.707107 − 0.707107i 2.04899 + 0.407570i −1.70278 − 0.317070i −0.635770 − 3.19623i 0.382683 − 0.923880i 1.21561 + 2.74268i 2.04899 − 0.407570i
5.2 0.923880 − 0.382683i 0.502711 + 1.65749i 0.707107 − 0.707107i −0.715543 − 0.142330i 1.09874 + 1.33894i −0.110207 − 0.554049i 0.382683 − 0.923880i −2.49456 + 1.66648i −0.715543 + 0.142330i
5.3 0.923880 − 0.382683i 0.566434 − 1.63681i 0.707107 − 0.707107i −2.09882 − 0.417480i −0.103064 − 1.72898i 0.745978 + 3.75028i 0.382683 − 0.923880i −2.35830 − 1.85429i −2.09882 + 0.417480i
11.1 −0.382683 + 0.923880i −1.57011 + 0.731271i −0.707107 − 0.707107i −0.630232 + 0.943208i −0.0747517 − 1.73044i −4.16378 + 2.78215i 0.923880 − 0.382683i 1.93048 − 2.29635i −0.630232 − 0.943208i
11.2 −0.382683 + 0.923880i −1.06770 − 1.36382i −0.707107 − 0.707107i 0.995238 − 1.48948i 1.66860 − 0.464518i 2.77729 − 1.85573i 0.923880 − 0.382683i −0.720018 + 2.91231i 0.995238 + 1.48948i
11.3 −0.382683 + 0.923880i 1.71393 + 0.249867i −0.707107 − 0.707107i −2.21277 + 3.31164i −0.886741 + 1.48785i 1.38649 − 0.926424i 0.923880 − 0.382683i 2.87513 + 0.856510i −2.21277 − 3.31164i
23.1 0.382683 − 0.923880i −0.637806 + 1.61034i −0.707107 − 0.707107i 3.29353 + 2.20066i 1.24368 + 1.20551i −0.750832 − 1.12370i −0.923880 + 0.382683i −2.18641 − 2.05417i 3.29353 − 2.20066i
23.2 0.382683 − 0.923880i −0.0961059 − 1.72938i −0.707107 − 0.707107i −0.540978 − 0.361470i −1.63452 − 0.573016i 0.676748 + 1.01283i −0.923880 + 0.382683i −2.98153 + 0.332408i −0.540978 + 0.361470i
23.3 0.382683 − 0.923880i 1.65779 + 0.501723i −0.707107 − 0.707107i −0.904790 − 0.604561i 1.09794 − 1.33960i 0.0740832 + 0.110873i −0.923880 + 0.382683i 2.49655 + 1.66351i −0.904790 + 0.604561i
29.1 −0.923880 + 0.382683i −1.72545 + 0.151023i 0.707107 − 0.707107i 0.456232 − 2.29363i 1.53632 − 0.799829i 3.60046 − 0.716176i −0.382683 + 0.923880i 2.95438 − 0.521165i 0.456232 + 2.29363i
29.2 −0.923880 + 0.382683i 0.600082 + 1.62478i 0.707107 − 0.707107i −0.284770 + 1.43164i −1.17618 − 1.27146i 0.146842 − 0.0292088i −0.382683 + 0.923880i −2.27980 + 1.95000i −0.284770 − 1.43164i
29.3 −0.923880 + 0.382683i 1.50806 − 0.851920i 0.707107 − 0.707107i 0.593905 − 2.98576i −1.06725 + 1.36418i −3.74730 + 0.745385i −0.382683 + 0.923880i 1.54846 − 2.56949i 0.593905 + 2.98576i
41.1 0.923880 + 0.382683i −1.45183 + 0.944561i 0.707107 + 0.707107i 2.04899 − 0.407570i −1.70278 + 0.317070i −0.635770 + 3.19623i 0.382683 + 0.923880i 1.21561 − 2.74268i 2.04899 + 0.407570i
41.2 0.923880 + 0.382683i 0.502711 − 1.65749i 0.707107 + 0.707107i −0.715543 + 0.142330i 1.09874 − 1.33894i −0.110207 + 0.554049i 0.382683 + 0.923880i −2.49456 − 1.66648i −0.715543 − 0.142330i
41.3 0.923880 + 0.382683i 0.566434 + 1.63681i 0.707107 + 0.707107i −2.09882 + 0.417480i −0.103064 + 1.72898i 0.745978 − 3.75028i 0.382683 + 0.923880i −2.35830 + 1.85429i −2.09882 − 0.417480i
65.1 −0.382683 − 0.923880i −1.57011 − 0.731271i −0.707107 + 0.707107i −0.630232 − 0.943208i −0.0747517 + 1.73044i −4.16378 − 2.78215i 0.923880 + 0.382683i 1.93048 + 2.29635i −0.630232 + 0.943208i
65.2 −0.382683 − 0.923880i −1.06770 + 1.36382i −0.707107 + 0.707107i 0.995238 + 1.48948i 1.66860 + 0.464518i 2.77729 + 1.85573i 0.923880 + 0.382683i −0.720018 − 2.91231i 0.995238 − 1.48948i
65.3 −0.382683 − 0.923880i 1.71393 − 0.249867i −0.707107 + 0.707107i −2.21277 − 3.31164i −0.886741 − 1.48785i 1.38649 + 0.926424i 0.923880 + 0.382683i 2.87513 − 0.856510i −2.21277 + 3.31164i
71.1 0.382683 + 0.923880i −0.637806 − 1.61034i −0.707107 + 0.707107i 3.29353 − 2.20066i 1.24368 − 1.20551i −0.750832 + 1.12370i −0.923880 − 0.382683i −2.18641 + 2.05417i 3.29353 + 2.20066i
71.2 0.382683 + 0.923880i −0.0961059 + 1.72938i −0.707107 + 0.707107i −0.540978 + 0.361470i −1.63452 + 0.573016i 0.676748 − 1.01283i −0.923880 − 0.382683i −2.98153 − 0.332408i −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.b yes 24
3.b odd 2 1 102.2.i.a ✓ 24
4.b odd 2 1 816.2.cj.a 24
12.b even 2 1 816.2.cj.b 24
17.e odd 16 1 102.2.i.a ✓ 24
51.i even 16 1 inner 102.2.i.b yes 24
68.i even 16 1 816.2.cj.b 24
204.t odd 16 1 816.2.cj.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.i.a ✓ 24 3.b odd 2 1
102.2.i.a ✓ 24 17.e odd 16 1
102.2.i.b yes 24 1.a even 1 1 trivial
102.2.i.b yes 24 51.i even 16 1 inner
816.2.cj.a 24 4.b odd 2 1
816.2.cj.a 24 204.t odd 16 1
816.2.cj.b 24 12.b even 2 1
816.2.cj.b 24 68.i even 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