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Properties

Label 417.2.d.b
Level $417$
Weight $2$
Character orbit 417.d
Analytic conductor $3.330$
Analytic rank $0$
Dimension $28$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [417,2,Mod(416,417)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(417, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("417.416");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 417 = 3 \cdot 139 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 417.d (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: \(3.32976176429\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 20 q^{4} - 6 q^{6} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 20 q^{4} - 6 q^{6} + 4 q^{7} - 4 q^{9} - 4 q^{13} + 4 q^{16} + 10 q^{24} - 76 q^{25} - 84 q^{28} + 94 q^{30} - 16 q^{31} + 80 q^{34} + 6 q^{36} - 16 q^{37} - 20 q^{42} + 36 q^{45} - 44 q^{46} + 56 q^{49} - 10 q^{51} + 52 q^{52} - 48 q^{54} - 28 q^{55} + 14 q^{57} + 14 q^{63} - 4 q^{64} - 136 q^{66} + 84 q^{67} - 86 q^{69} - 16 q^{78} - 92 q^{79} - 8 q^{81} + 164 q^{91} + 134 q^{96} + 6 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
416.1 −2.60277 −1.00408 − 1.41132i 4.77443 − 3.95502i 2.61338 + 3.67335i −4.34832 −7.22121 −0.983663 + 2.83415i 10.2940i
416.2 −2.60277 −1.00408 + 1.41132i 4.77443 3.95502i 2.61338 − 3.67335i −4.34832 −7.22121 −0.983663 − 2.83415i − 10.2940i
416.3 −2.16680 1.57651 − 0.717358i 2.69503 − 3.64560i −3.41599 + 1.55437i 1.72126 −1.50599 1.97079 − 2.26185i 7.89929i
416.4 −2.16680 1.57651 + 0.717358i 2.69503 3.64560i −3.41599 − 1.55437i 1.72126 −1.50599 1.97079 + 2.26185i − 7.89929i
416.5 −1.79526 1.41227 − 1.00275i 1.22297 1.92391i −2.53540 + 1.80019i −3.14590 1.39497 0.989005 − 2.83229i − 3.45393i
416.6 −1.79526 1.41227 + 1.00275i 1.22297 − 1.92391i −2.53540 − 1.80019i −3.14590 1.39497 0.989005 + 2.83229i 3.45393i
416.7 −1.64553 −0.272627 − 1.71046i 0.707753 − 0.686414i 0.448614 + 2.81461i 4.18919 2.12643 −2.85135 + 0.932635i 1.12951i
416.8 −1.64553 −0.272627 + 1.71046i 0.707753 0.686414i 0.448614 − 2.81461i 4.18919 2.12643 −2.85135 − 0.932635i − 1.12951i
416.9 −1.02528 −1.69533 − 0.354745i −0.948798 − 3.51235i 1.73819 + 0.363713i 1.81477 3.02335 2.74831 + 1.20282i 3.60115i
416.10 −1.02528 −1.69533 + 0.354745i −0.948798 3.51235i 1.73819 − 0.363713i 1.81477 3.02335 2.74831 − 1.20282i − 3.60115i
416.11 −0.696431 0.0484126 − 1.73137i −1.51498 − 2.63064i −0.0337160 + 1.20578i −1.86209 2.44794 −2.99531 − 0.167641i 1.83206i
416.12 −0.696431 0.0484126 + 1.73137i −1.51498 2.63064i −0.0337160 − 1.20578i −1.86209 2.44794 −2.99531 + 0.167641i − 1.83206i
416.13 −0.252180 1.24944 − 1.19954i −1.93641 − 1.27983i −0.315085 + 0.302500i 2.63109 0.992684 0.122213 − 2.99751i 0.322748i
416.14 −0.252180 1.24944 + 1.19954i −1.93641 1.27983i −0.315085 − 0.302500i 2.63109 0.992684 0.122213 + 2.99751i − 0.322748i
416.15 0.252180 −1.24944 − 1.19954i −1.93641 1.27983i −0.315085 − 0.302500i 2.63109 −0.992684 0.122213 + 2.99751i 0.322748i
416.16 0.252180 −1.24944 + 1.19954i −1.93641 − 1.27983i −0.315085 + 0.302500i 2.63109 −0.992684 0.122213 − 2.99751i − 0.322748i
416.17 0.696431 −0.0484126 − 1.73137i −1.51498 2.63064i −0.0337160 − 1.20578i −1.86209 −2.44794 −2.99531 + 0.167641i 1.83206i
416.18 0.696431 −0.0484126 + 1.73137i −1.51498 − 2.63064i −0.0337160 + 1.20578i −1.86209 −2.44794 −2.99531 − 0.167641i − 1.83206i
416.19 1.02528 1.69533 − 0.354745i −0.948798 3.51235i 1.73819 − 0.363713i 1.81477 −3.02335 2.74831 − 1.20282i 3.60115i
416.20 1.02528 1.69533 + 0.354745i −0.948798 − 3.51235i 1.73819 + 0.363713i 1.81477 −3.02335 2.74831 + 1.20282i − 3.60115i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 416.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
139.b odd 2 1 inner
417.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 417.2.d.b ✓ 28
3.b odd 2 1 inner 417.2.d.b ✓ 28
139.b odd 2 1 inner 417.2.d.b ✓ 28
417.d even 2 1 inner 417.2.d.b ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
417.2.d.b ✓ 28 1.a even 1 1 trivial
417.2.d.b ✓ 28 3.b odd 2 1 inner
417.2.d.b ✓ 28 139.b odd 2 1 inner
417.2.d.b ✓ 28 417.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 19T_{2}^{12} + 137T_{2}^{10} - 473T_{2}^{8} + 806T_{2}^{6} - 623T_{2}^{4} + 178T_{2}^{2} - 9 \) acting on \(S_{2}^{\mathrm{new}}(417, [\chi])\). Copy content Toggle raw display