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Properties

Label 875.2.s.c
Level $875$
Weight $2$
Character orbit 875.s
Analytic conductor $6.987$
Analytic rank $0$
Dimension $144$
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] = [875,2,Mod(118,875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(875, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([7, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("875.118");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 875 = 5^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 875.s (of order \(20\), degree \(8\), not 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: \(6.98691017686\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 175)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 16 q^{2} - 20 q^{4} + 14 q^{7} + 12 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 16 q^{2} - 20 q^{4} + 14 q^{7} + 12 q^{8} - 20 q^{9} - 12 q^{11} - 10 q^{14} + 12 q^{16} + 28 q^{18} - 6 q^{21} - 16 q^{22} + 8 q^{23} + 70 q^{28} + 20 q^{32} - 28 q^{36} - 4 q^{37} - 60 q^{39} + 30 q^{42} - 72 q^{43} - 20 q^{44} - 12 q^{46} - 32 q^{51} + 104 q^{53} - 22 q^{56} - 120 q^{57} + 32 q^{58} - 48 q^{63} + 40 q^{64} + 16 q^{67} - 12 q^{71} + 64 q^{72} - 74 q^{77} - 60 q^{78} - 20 q^{79} - 8 q^{81} + 190 q^{84} - 12 q^{86} - 92 q^{88} - 6 q^{91} + 20 q^{92} + 160 q^{93} - 162 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1 −1.19990 − 2.35493i −1.89936 + 0.300829i −2.93037 + 4.03331i 0 2.98746 + 4.11189i −2.60993 + 0.433889i 7.79337 + 1.23435i 0.663898 − 0.215713i 0
118.2 −1.19990 − 2.35493i 1.89936 − 0.300829i −2.93037 + 4.03331i 0 −2.98746 − 4.11189i −0.433889 + 2.60993i 7.79337 + 1.23435i 0.663898 − 0.215713i 0
118.3 −0.909735 − 1.78545i −1.06006 + 0.167897i −1.18466 + 1.63055i 0 1.26414 + 1.73994i 2.63729 − 0.211395i 0.0306105 + 0.00484822i −1.75764 + 0.571091i 0
118.4 −0.909735 − 1.78545i 1.06006 − 0.167897i −1.18466 + 1.63055i 0 −1.26414 − 1.73994i 0.211395 − 2.63729i 0.0306105 + 0.00484822i −1.75764 + 0.571091i 0
118.5 −0.489193 − 0.960095i −0.958836 + 0.151865i 0.493097 − 0.678690i 0 0.614861 + 0.846283i −1.23562 + 2.33949i −3.02137 − 0.478539i −1.95687 + 0.635824i 0
118.6 −0.489193 − 0.960095i 0.958836 − 0.151865i 0.493097 − 0.678690i 0 −0.614861 − 0.846283i −2.33949 + 1.23562i −3.02137 − 0.478539i −1.95687 + 0.635824i 0
118.7 −0.350774 − 0.688433i −2.82111 + 0.446820i 0.824673 − 1.13507i 0 1.29718 + 1.78541i 2.56878 − 0.633527i −2.59696 − 0.411318i 4.90584 − 1.59401i 0
118.8 −0.350774 − 0.688433i 2.82111 − 0.446820i 0.824673 − 1.13507i 0 −1.29718 − 1.78541i 0.633527 − 2.56878i −2.59696 − 0.411318i 4.90584 − 1.59401i 0
118.9 0.0684818 + 0.134403i −1.05542 + 0.167162i 1.16220 − 1.59963i 0 −0.0947443 − 0.130404i −0.298495 − 2.62886i 0.592558 + 0.0938519i −1.76720 + 0.574198i 0
118.10 0.0684818 + 0.134403i 1.05542 − 0.167162i 1.16220 − 1.59963i 0 0.0947443 + 0.130404i 2.62886 + 0.298495i 0.592558 + 0.0938519i −1.76720 + 0.574198i 0
118.11 0.533640 + 1.04733i −1.75333 + 0.277700i 0.363447 − 0.500242i 0 −1.22649 − 1.68812i −1.38762 − 2.25267i 3.03981 + 0.481459i 0.143880 − 0.0467495i 0
118.12 0.533640 + 1.04733i 1.75333 − 0.277700i 0.363447 − 0.500242i 0 1.22649 + 1.68812i 2.25267 + 1.38762i 3.03981 + 0.481459i 0.143880 − 0.0467495i 0
118.13 0.640960 + 1.25795i −1.84312 + 0.291921i 0.00395121 − 0.00543838i 0 −1.54859 − 2.13145i −1.60411 + 2.10400i 2.79828 + 0.443204i 0.458693 − 0.149038i 0
118.14 0.640960 + 1.25795i 1.84312 − 0.291921i 0.00395121 − 0.00543838i 0 1.54859 + 2.13145i −2.10400 + 1.60411i 2.79828 + 0.443204i 0.458693 − 0.149038i 0
118.15 0.944119 + 1.85294i −3.15320 + 0.499417i −1.36645 + 1.88075i 0 −3.90238 − 5.37117i 2.60645 + 0.454304i −0.667005 − 0.105643i 6.84006 − 2.22247i 0
118.16 0.944119 + 1.85294i 3.15320 − 0.499417i −1.36645 + 1.88075i 0 3.90238 + 5.37117i −0.454304 − 2.60645i −0.667005 − 0.105643i 6.84006 − 2.22247i 0
118.17 1.17461 + 2.30531i −0.418653 + 0.0663081i −2.75916 + 3.79765i 0 −0.644615 − 0.887237i 1.37268 + 2.26180i −6.88479 − 1.09044i −2.68230 + 0.871531i 0
118.18 1.17461 + 2.30531i 0.418653 − 0.0663081i −2.75916 + 3.79765i 0 0.644615 + 0.887237i −2.26180 − 1.37268i −6.88479 − 1.09044i −2.68230 + 0.871531i 0
132.1 −1.95388 + 0.995551i −0.271680 − 1.71532i 1.65095 − 2.27234i 0 2.23852 + 3.08106i 2.53430 + 0.759807i −0.277442 + 1.75170i −0.0153479 + 0.00498685i 0
132.2 −1.95388 + 0.995551i 0.271680 + 1.71532i 1.65095 − 2.27234i 0 −2.23852 − 3.08106i 0.759807 + 2.53430i −0.277442 + 1.75170i −0.0153479 + 0.00498685i 0
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 118.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
25.f odd 20 1 inner
175.s even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 875.2.s.c 144
5.b even 2 1 175.2.s.a ✓ 144
5.c odd 4 1 875.2.s.a 144
5.c odd 4 1 875.2.s.b 144
7.b odd 2 1 inner 875.2.s.c 144
25.d even 5 1 875.2.s.a 144
25.e even 10 1 875.2.s.b 144
25.f odd 20 1 175.2.s.a ✓ 144
25.f odd 20 1 inner 875.2.s.c 144
35.c odd 2 1 175.2.s.a ✓ 144
35.f even 4 1 875.2.s.a 144
35.f even 4 1 875.2.s.b 144
175.l odd 10 1 875.2.s.a 144
175.m odd 10 1 875.2.s.b 144
175.s even 20 1 175.2.s.a ✓ 144
175.s even 20 1 inner 875.2.s.c 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.s.a ✓ 144 5.b even 2 1
175.2.s.a ✓ 144 25.f odd 20 1
175.2.s.a ✓ 144 35.c odd 2 1
175.2.s.a ✓ 144 175.s even 20 1
875.2.s.a 144 5.c odd 4 1
875.2.s.a 144 25.d even 5 1
875.2.s.a 144 35.f even 4 1
875.2.s.a 144 175.l odd 10 1
875.2.s.b 144 5.c odd 4 1
875.2.s.b 144 25.e even 10 1
875.2.s.b 144 35.f even 4 1
875.2.s.b 144 175.m odd 10 1
875.2.s.c 144 1.a even 1 1 trivial
875.2.s.c 144 7.b odd 2 1 inner
875.2.s.c 144 25.f odd 20 1 inner
875.2.s.c 144 175.s even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} - 8 T_{2}^{71} + 37 T_{2}^{70} - 130 T_{2}^{69} + 319 T_{2}^{68} - 484 T_{2}^{67} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(875, [\chi])\). Copy content Toggle raw display