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Properties

Label 175.5.g.d
Level $175$
Weight $5$
Character orbit 175.g
Analytic conductor $18.090$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,5,Mod(43,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.43");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 175.g (of order \(4\), degree \(2\), 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: \(18.0897435397\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 248 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 248 q^{6} - 1008 q^{11} - 1732 q^{16} - 392 q^{21} - 4296 q^{26} + 9064 q^{31} - 14388 q^{36} + 1440 q^{41} + 26076 q^{46} + 11576 q^{51} + 12936 q^{56} - 6664 q^{61} + 20664 q^{66} - 11496 q^{71} - 78624 q^{76} - 24112 q^{81} - 17412 q^{86} + 25480 q^{91} + 144 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −5.21263 + 5.21263i −3.06857 − 3.06857i − 38.3430i 0 31.9907 13.0958 − 13.0958i 116.466 + 116.466i − 62.1677i 0
43.2 −4.62285 + 4.62285i 6.85998 + 6.85998i − 26.7415i 0 −63.4253 13.0958 − 13.0958i 49.6561 + 49.6561i 13.1187i 0
43.3 −4.13842 + 4.13842i −3.82134 − 3.82134i − 18.2531i 0 31.6286 −13.0958 + 13.0958i 9.32435 + 9.32435i − 51.7948i 0
43.4 −3.03869 + 3.03869i −7.54494 − 7.54494i − 2.46726i 0 45.8535 −13.0958 + 13.0958i −41.1218 − 41.1218i 32.8523i 0
43.5 −2.13708 + 2.13708i −11.3544 − 11.3544i 6.86575i 0 48.5306 13.0958 − 13.0958i −48.8660 − 48.8660i 176.845i 0
43.6 −1.31243 + 1.31243i 10.0771 + 10.0771i 12.5551i 0 −26.4509 −13.0958 + 13.0958i −37.4765 − 37.4765i 122.095i 0
43.7 −1.20735 + 1.20735i 2.55450 + 2.55450i 13.0846i 0 −6.16838 13.0958 − 13.0958i −35.1154 − 35.1154i − 67.9490i 0
43.8 −0.922118 + 0.922118i −0.0223741 − 0.0223741i 14.2994i 0 0.0412631 13.0958 − 13.0958i −27.9396 − 27.9396i − 80.9990i 0
43.9 0.922118 − 0.922118i 0.0223741 + 0.0223741i 14.2994i 0 0.0412631 −13.0958 + 13.0958i 27.9396 + 27.9396i − 80.9990i 0
43.10 1.20735 − 1.20735i −2.55450 − 2.55450i 13.0846i 0 −6.16838 −13.0958 + 13.0958i 35.1154 + 35.1154i − 67.9490i 0
43.11 1.31243 − 1.31243i −10.0771 − 10.0771i 12.5551i 0 −26.4509 13.0958 − 13.0958i 37.4765 + 37.4765i 122.095i 0
43.12 2.13708 − 2.13708i 11.3544 + 11.3544i 6.86575i 0 48.5306 −13.0958 + 13.0958i 48.8660 + 48.8660i 176.845i 0
43.13 3.03869 − 3.03869i 7.54494 + 7.54494i − 2.46726i 0 45.8535 13.0958 − 13.0958i 41.1218 + 41.1218i 32.8523i 0
43.14 4.13842 − 4.13842i 3.82134 + 3.82134i − 18.2531i 0 31.6286 13.0958 − 13.0958i −9.32435 − 9.32435i − 51.7948i 0
43.15 4.62285 − 4.62285i −6.85998 − 6.85998i − 26.7415i 0 −63.4253 −13.0958 + 13.0958i −49.6561 − 49.6561i 13.1187i 0
43.16 5.21263 − 5.21263i 3.06857 + 3.06857i − 38.3430i 0 31.9907 −13.0958 + 13.0958i −116.466 − 116.466i − 62.1677i 0
57.1 −5.21263 − 5.21263i −3.06857 + 3.06857i 38.3430i 0 31.9907 13.0958 + 13.0958i 116.466 − 116.466i 62.1677i 0
57.2 −4.62285 − 4.62285i 6.85998 − 6.85998i 26.7415i 0 −63.4253 13.0958 + 13.0958i 49.6561 − 49.6561i − 13.1187i 0
57.3 −4.13842 − 4.13842i −3.82134 + 3.82134i 18.2531i 0 31.6286 −13.0958 − 13.0958i 9.32435 − 9.32435i 51.7948i 0
57.4 −3.03869 − 3.03869i −7.54494 + 7.54494i 2.46726i 0 45.8535 −13.0958 − 13.0958i −41.1218 + 41.1218i − 32.8523i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.5.g.d ✓ 32
5.b even 2 1 inner 175.5.g.d ✓ 32
5.c odd 4 2 inner 175.5.g.d ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.5.g.d ✓ 32 1.a even 1 1 trivial
175.5.g.d ✓ 32 5.b even 2 1 inner
175.5.g.d ✓ 32 5.c odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 6401 T_{2}^{28} + 13707136 T_{2}^{24} + 11486077025 T_{2}^{20} + 3261864859825 T_{2}^{16} + \cdots + 52\!\cdots\!00 \) acting on \(S_{5}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display