↑

Properties

Label 875.2.s.b
Level $875$
Weight $2$
Character orbit 875.s
Analytic conductor $6.987$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 4 q^{2} + 20 q^{4} - 14 q^{7} - 52 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 4 q^{2} + 20 q^{4} - 14 q^{7} - 52 q^{8} + 20 q^{9} - 12 q^{11} + 10 q^{14} + 12 q^{16} - 28 q^{18} - 6 q^{21} - 4 q^{22} + 72 q^{23} + 40 q^{28} - 20 q^{32} - 28 q^{36} + 24 q^{37} + 60 q^{39} + 20 q^{42} + 72 q^{43} + 20 q^{44} - 12 q^{46} - 32 q^{51} + 16 q^{53} - 22 q^{56} + 120 q^{57} + 108 q^{58} + 28 q^{63} - 40 q^{64} + 124 q^{67} - 12 q^{71} - 204 q^{72} - 66 q^{77} + 20 q^{78} + 20 q^{79} - 8 q^{81} - 190 q^{84} - 12 q^{86} + 92 q^{88} - 6 q^{91} - 300 q^{92} - 160 q^{93} - 68 q^{98}+O(q^{100}) \) 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 −0.995551 − 1.95388i −1.71532 + 0.271680i −1.65095 + 2.27234i 0 2.23852 + 3.08106i −0.759807 + 2.53430i 1.75170 + 0.277442i 0.0153479 − 0.00498685i 0
118.2 −0.995551 − 1.95388i 1.71532 − 0.271680i −1.65095 + 2.27234i 0 −2.23852 − 3.08106i −2.53430 + 0.759807i 1.75170 + 0.277442i 0.0153479 − 0.00498685i 0
118.3 −0.810703 − 1.59109i −2.87561 + 0.455453i −0.698771 + 0.961776i 0 3.05594 + 4.20614i −0.970101 − 2.46148i −1.43071 − 0.226602i 5.20855 − 1.69236i 0
118.4 −0.810703 − 1.59109i 2.87561 − 0.455453i −0.698771 + 0.961776i 0 −3.05594 − 4.20614i 2.46148 + 0.970101i −1.43071 − 0.226602i 5.20855 − 1.69236i 0
118.5 −0.527094 − 1.03448i −1.04877 + 0.166108i 0.383250 − 0.527498i 0 0.724634 + 0.997373i 0.833689 + 2.51097i −3.04115 − 0.481672i −1.78085 + 0.578633i 0
118.6 −0.527094 − 1.03448i 1.04877 − 0.166108i 0.383250 − 0.527498i 0 −0.724634 − 0.997373i −2.51097 − 0.833689i −3.04115 − 0.481672i −1.78085 + 0.578633i 0
118.7 −0.167491 − 0.328720i −0.0205527 + 0.00325523i 1.09557 − 1.50792i 0 0.00451245 + 0.00621086i 0.878617 − 2.49560i −1.40796 − 0.222999i −2.85276 + 0.926917i 0
118.8 −0.167491 − 0.328720i 0.0205527 − 0.00325523i 1.09557 − 1.50792i 0 −0.00451245 − 0.00621086i 2.49560 − 0.878617i −1.40796 − 0.222999i −2.85276 + 0.926917i 0
118.9 0.00287269 + 0.00563796i −2.69153 + 0.426296i 1.17555 − 1.61800i 0 −0.0101354 − 0.0139501i −2.62833 + 0.303137i 0.0249987 + 0.00395940i 4.20942 − 1.36772i 0
118.10 0.00287269 + 0.00563796i 2.69153 − 0.426296i 1.17555 − 1.61800i 0 0.0101354 + 0.0139501i −0.303137 + 2.62833i 0.0249987 + 0.00395940i 4.20942 − 1.36772i 0
118.11 0.324189 + 0.636257i −2.34302 + 0.371098i 0.875846 − 1.20550i 0 −0.995696 − 1.37046i 2.28336 + 1.33651i 2.46154 + 0.389869i 2.49888 − 0.811934i 0
118.12 0.324189 + 0.636257i 2.34302 − 0.371098i 0.875846 − 1.20550i 0 0.995696 + 1.37046i −1.33651 − 2.28336i 2.46154 + 0.389869i 2.49888 − 0.811934i 0
118.13 0.778176 + 1.52726i −0.388524 + 0.0615362i −0.551382 + 0.758912i 0 −0.396322 − 0.545490i −2.42681 − 1.05384i 1.79783 + 0.284748i −2.70601 + 0.879234i 0
118.14 0.778176 + 1.52726i 0.388524 − 0.0615362i −0.551382 + 0.758912i 0 0.396322 + 0.545490i 1.05384 + 2.42681i 1.79783 + 0.284748i −2.70601 + 0.879234i 0
118.15 0.998824 + 1.96030i −1.37363 + 0.217562i −1.66957 + 2.29796i 0 −1.79850 − 2.47543i 0.119740 − 2.64304i −1.82629 − 0.289255i −1.01364 + 0.329351i 0
118.16 0.998824 + 1.96030i 1.37363 − 0.217562i −1.66957 + 2.29796i 0 1.79850 + 2.47543i 2.64304 − 0.119740i −1.82629 − 0.289255i −1.01364 + 0.329351i 0
118.17 1.20579 + 2.36651i −2.81777 + 0.446290i −2.97084 + 4.08900i 0 −4.45380 − 6.13013i −1.75636 + 1.97869i −8.01229 − 1.26902i 4.88747 − 1.58803i 0
118.18 1.20579 + 2.36651i 2.81777 − 0.446290i −2.97084 + 4.08900i 0 4.45380 + 6.13013i −1.97869 + 1.75636i −8.01229 − 1.26902i 4.88747 − 1.58803i 0
132.1 −2.35493 + 1.19990i −0.300829 − 1.89936i 2.93037 − 4.03331i 0 2.98746 + 4.11189i 0.433889 + 2.60993i −1.23435 + 7.79337i −0.663898 + 0.215713i 0
132.2 −2.35493 + 1.19990i 0.300829 + 1.89936i 2.93037 − 4.03331i 0 −2.98746 − 4.11189i 2.60993 + 0.433889i −1.23435 + 7.79337i −0.663898 + 0.215713i 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.b 144
5.b even 2 1 875.2.s.a 144
5.c odd 4 1 175.2.s.a ✓ 144
5.c odd 4 1 875.2.s.c 144
7.b odd 2 1 inner 875.2.s.b 144
25.d even 5 1 175.2.s.a ✓ 144
25.e even 10 1 875.2.s.c 144
25.f odd 20 1 875.2.s.a 144
25.f odd 20 1 inner 875.2.s.b 144
35.c odd 2 1 875.2.s.a 144
35.f even 4 1 175.2.s.a ✓ 144
35.f even 4 1 875.2.s.c 144
175.l odd 10 1 175.2.s.a ✓ 144
175.m odd 10 1 875.2.s.c 144
175.s even 20 1 875.2.s.a 144
175.s even 20 1 inner 875.2.s.b 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.s.a ✓ 144 5.c odd 4 1
175.2.s.a ✓ 144 25.d even 5 1
175.2.s.a ✓ 144 35.f even 4 1
175.2.s.a ✓ 144 175.l odd 10 1
875.2.s.a 144 5.b even 2 1
875.2.s.a 144 25.f odd 20 1
875.2.s.a 144 35.c odd 2 1
875.2.s.a 144 175.s even 20 1
875.2.s.b 144 1.a even 1 1 trivial
875.2.s.b 144 7.b odd 2 1 inner
875.2.s.b 144 25.f odd 20 1 inner
875.2.s.b 144 175.s even 20 1 inner
875.2.s.c 144 5.c odd 4 1
875.2.s.c 144 25.e even 10 1
875.2.s.c 144 35.f even 4 1
875.2.s.c 144 175.m odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} - 2 T_{2}^{71} - 3 T_{2}^{70} + 20 T_{2}^{69} - 81 T_{2}^{68} + 74 T_{2}^{67} + 344 T_{2}^{66} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(875, [\chi])\). Copy content Toggle raw display