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Properties

Label 935.2.bp.b
Level $935$
Weight $2$
Character orbit 935.bp
Analytic conductor $7.466$
Analytic rank $0$
Dimension $768$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [935,2,Mod(54,935)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(935, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 8, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("935.54");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 935 = 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 935.bp (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: \(7.46601258899\)
Analytic rank: \(0\)
Dimension: \(768\)
Relative dimension: \(96\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 768 q - 32 q^{4} - 16 q^{5} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 768 q - 32 q^{4} - 16 q^{5} - 32 q^{9} - 32 q^{11} - 32 q^{14} - 16 q^{15} - 16 q^{20} + 224 q^{26} - 96 q^{31} - 32 q^{34} - 96 q^{36} - 304 q^{44} - 16 q^{45} - 96 q^{49} - 40 q^{55} - 32 q^{56} - 32 q^{59} - 16 q^{60} + 96 q^{64} + 16 q^{66} - 288 q^{70} - 32 q^{71} + 48 q^{75} + 336 q^{80} - 128 q^{81} - 32 q^{89} - 64 q^{91} - 128 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
54.1 −2.49043 + 1.03157i −0.422961 + 2.12637i 3.72390 − 3.72390i −0.122345 + 2.23272i −1.14014 − 5.73189i 1.84699 − 2.76421i −3.36951 + 8.13471i −1.57091 − 0.650694i −1.99852 − 5.68664i
54.2 −2.49043 + 1.03157i 0.422961 − 2.12637i 3.72390 − 3.72390i 2.10958 + 0.741393i 1.14014 + 5.73189i 1.84699 − 2.76421i −3.36951 + 8.13471i −1.57091 − 0.650694i −6.01857 + 0.329795i
54.3 −2.47064 + 1.02337i −0.548510 + 2.75755i 3.64257 − 3.64257i 1.90225 − 1.17535i −1.46683 − 7.37424i −1.16397 + 1.74201i −3.22503 + 7.78590i −4.53156 − 1.87703i −3.49695 + 4.85059i
54.4 −2.47064 + 1.02337i 0.548510 − 2.75755i 3.64257 − 3.64257i −1.81384 + 1.30766i 1.46683 + 7.37424i −1.16397 + 1.74201i −3.22503 + 7.78590i −4.53156 − 1.87703i 3.14313 − 5.08700i
54.5 −2.24209 + 0.928704i −0.331493 + 1.66653i 2.75027 − 2.75027i 2.09589 + 0.779250i −0.804475 − 4.04437i −0.170502 + 0.255175i −1.75475 + 4.23635i 0.104207 + 0.0431640i −5.42288 + 0.199317i
54.6 −2.24209 + 0.928704i 0.331493 − 1.66653i 2.75027 − 2.75027i −0.0821307 + 2.23456i 0.804475 + 4.04437i −0.170502 + 0.255175i −1.75475 + 4.23635i 0.104207 + 0.0431640i −1.89110 − 5.08636i
54.7 −2.22737 + 0.922605i −0.0309643 + 0.155668i 2.69575 − 2.69575i 0.317250 − 2.21345i −0.0746513 − 0.375298i −0.876068 + 1.31113i −1.67209 + 4.03679i 2.74836 + 1.13841i 1.33551 + 5.22286i
54.8 −2.22737 + 0.922605i 0.0309643 − 0.155668i 2.69575 − 2.69575i −2.16637 − 0.553949i 0.0746513 + 0.375298i −0.876068 + 1.31113i −1.67209 + 4.03679i 2.74836 + 1.13841i 5.33637 − 0.764853i
54.9 −2.18602 + 0.905480i −0.484464 + 2.43557i 2.54459 − 2.54459i −1.73480 − 1.41084i −1.14631 − 5.76288i −0.0726618 + 0.108746i −1.44749 + 3.49455i −2.92564 − 1.21184i 5.06979 + 1.51331i
54.10 −2.18602 + 0.905480i 0.484464 − 2.43557i 2.54459 − 2.54459i −0.639572 − 2.14265i 1.14631 + 5.76288i −0.0726618 + 0.108746i −1.44749 + 3.49455i −2.92564 − 1.21184i 3.33825 + 4.10476i
54.11 −2.10026 + 0.869957i −0.0233556 + 0.117417i 2.24006 − 2.24006i 2.05050 − 0.891883i −0.0530945 − 0.266924i 2.60224 − 3.89452i −1.01605 + 2.45296i 2.75840 + 1.14257i −3.53068 + 3.65704i
54.12 −2.10026 + 0.869957i 0.0233556 − 0.117417i 2.24006 − 2.24006i −1.60868 + 1.55310i 0.0530945 + 0.266924i 2.60224 − 3.89452i −1.01605 + 2.45296i 2.75840 + 1.14257i 2.02753 − 4.66141i
54.13 −2.01175 + 0.833295i −0.491147 + 2.46916i 1.93855 − 1.93855i −2.22452 + 0.226930i −1.06948 − 5.37662i 0.662531 − 0.991548i −0.617911 + 1.49177i −3.08391 − 1.27740i 4.28609 − 2.31021i
54.14 −2.01175 + 0.833295i 0.491147 − 2.46916i 1.93855 − 1.93855i 1.06094 − 1.96835i 1.06948 + 5.37662i 0.662531 − 0.991548i −0.617911 + 1.49177i −3.08391 − 1.27740i −0.494141 + 4.84391i
54.15 −1.81652 + 0.752428i −0.433116 + 2.17742i 1.31939 − 1.31939i 0.772136 − 2.09852i −0.851589 − 4.28123i −2.11245 + 3.16150i 0.100900 − 0.243593i −1.78194 − 0.738105i 0.176386 + 4.39299i
54.16 −1.81652 + 0.752428i 0.433116 − 2.17742i 1.31939 − 1.31939i −2.23427 − 0.0897096i 0.851589 + 4.28123i −2.11245 + 3.16150i 0.100900 − 0.243593i −1.78194 − 0.738105i 4.12610 − 1.51817i
54.17 −1.78333 + 0.738680i −0.640663 + 3.22083i 1.22041 − 1.22041i −0.224961 + 2.22472i −1.23665 − 6.21705i −0.940104 + 1.40696i 0.202460 − 0.488783i −7.19167 − 2.97889i −1.24218 − 4.13359i
54.18 −1.78333 + 0.738680i 0.640663 − 3.22083i 1.22041 − 1.22041i 2.14146 + 0.643528i 1.23665 + 6.21705i −0.940104 + 1.40696i 0.202460 − 0.488783i −7.19167 − 2.97889i −4.29430 + 0.434233i
54.19 −1.62214 + 0.671911i −0.233038 + 1.17156i 0.765649 − 0.765649i 2.01601 − 0.967313i −0.409165 − 2.05701i 0.841222 − 1.25898i 0.616282 − 1.48784i 1.45339 + 0.602015i −2.62030 + 2.92370i
54.20 −1.62214 + 0.671911i 0.233038 − 1.17156i 0.765649 − 0.765649i −1.66518 + 1.49238i 0.409165 + 2.05701i 0.841222 − 1.25898i 0.616282 − 1.48784i 1.45339 + 0.602015i 1.69840 − 3.53969i
See next 80 embeddings (of 768 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 54.96
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.b odd 2 1 inner
17.e odd 16 1 inner
55.d odd 2 1 inner
85.p odd 16 1 inner
187.m even 16 1 inner
935.bp even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 935.2.bp.b ✓ 768
5.b even 2 1 inner 935.2.bp.b ✓ 768
11.b odd 2 1 inner 935.2.bp.b ✓ 768
17.e odd 16 1 inner 935.2.bp.b ✓ 768
55.d odd 2 1 inner 935.2.bp.b ✓ 768
85.p odd 16 1 inner 935.2.bp.b ✓ 768
187.m even 16 1 inner 935.2.bp.b ✓ 768
935.bp even 16 1 inner 935.2.bp.b ✓ 768
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
935.2.bp.b ✓ 768 1.a even 1 1 trivial
935.2.bp.b ✓ 768 5.b even 2 1 inner
935.2.bp.b ✓ 768 11.b odd 2 1 inner
935.2.bp.b ✓ 768 17.e odd 16 1 inner
935.2.bp.b ✓ 768 55.d odd 2 1 inner
935.2.bp.b ✓ 768 85.p odd 16 1 inner
935.2.bp.b ✓ 768 187.m even 16 1 inner
935.2.bp.b ✓ 768 935.bp even 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{384} + 8 T_{2}^{382} + 32 T_{2}^{380} - 72 T_{2}^{378} + 21742 T_{2}^{376} + \cdots + 10\!\cdots\!24 \) acting on \(S_{2}^{\mathrm{new}}(935, [\chi])\). Copy content Toggle raw display