Properties

Label 475.2.bb.c
Level $475$
Weight $2$
Character orbit 475.bb
Analytic conductor $3.793$
Analytic rank $0$
Dimension $168$
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] = [475,2,Mod(32,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([9, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.32");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.bb (of order \(36\), degree \(12\), 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: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(14\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 168 q - 36 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 168 q - 36 q^{6} - 72 q^{16} - 60 q^{21} + 72 q^{26} - 72 q^{31} - 252 q^{36} - 36 q^{41} - 180 q^{46} + 108 q^{61} + 144 q^{66} + 168 q^{71} - 24 q^{76} + 372 q^{81} + 480 q^{86} + 36 q^{91} + 696 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −0.219903 + 2.51350i −0.420327 0.901394i −4.29970 0.758153i 0 2.35808 0.858272i 0.630014 0.168812i 1.54508 5.76632i 1.29253 1.54037i 0
32.2 −0.205646 + 2.35054i 0.0966746 + 0.207319i −3.51313 0.619460i 0 −0.507193 + 0.184603i −4.84313 + 1.29771i 0.957149 3.57213i 1.89473 2.25805i 0
32.3 −0.158607 + 1.81289i 1.18655 + 2.54457i −1.29180 0.227779i 0 −4.80122 + 1.74750i 4.44244 1.19035i −0.324178 + 1.20985i −3.13855 + 3.74038i 0
32.4 −0.122983 + 1.40570i 0.364684 + 0.782068i 0.00874799 + 0.00154251i 0 −1.14420 + 0.416456i −1.95768 + 0.524559i −0.733667 + 2.73808i 1.44973 1.72772i 0
32.5 −0.0696587 + 0.796203i −1.25628 2.69410i 1.34053 + 0.236371i 0 2.23256 0.812585i 4.61801 1.23739i −0.695298 + 2.59489i −3.75157 + 4.47094i 0
32.6 −0.0525703 + 0.600882i 0.0661813 + 0.141926i 1.61132 + 0.284119i 0 −0.0887601 + 0.0323060i 0.465075 0.124617i −0.567657 + 2.11852i 1.91260 2.27935i 0
32.7 −0.00427917 + 0.0489112i 1.29767 + 2.78287i 1.96724 + 0.346878i 0 −0.141666 + 0.0515624i 1.74113 0.466534i −0.0507994 + 0.189586i −4.13205 + 4.92438i 0
32.8 0.00427917 0.0489112i −1.29767 2.78287i 1.96724 + 0.346878i 0 −0.141666 + 0.0515624i −1.74113 + 0.466534i 0.0507994 0.189586i −4.13205 + 4.92438i 0
32.9 0.0525703 0.600882i −0.0661813 0.141926i 1.61132 + 0.284119i 0 −0.0887601 + 0.0323060i −0.465075 + 0.124617i 0.567657 2.11852i 1.91260 2.27935i 0
32.10 0.0696587 0.796203i 1.25628 + 2.69410i 1.34053 + 0.236371i 0 2.23256 0.812585i −4.61801 + 1.23739i 0.695298 2.59489i −3.75157 + 4.47094i 0
32.11 0.122983 1.40570i −0.364684 0.782068i 0.00874799 + 0.00154251i 0 −1.14420 + 0.416456i 1.95768 0.524559i 0.733667 2.73808i 1.44973 1.72772i 0
32.12 0.158607 1.81289i −1.18655 2.54457i −1.29180 0.227779i 0 −4.80122 + 1.74750i −4.44244 + 1.19035i 0.324178 1.20985i −3.13855 + 3.74038i 0
32.13 0.205646 2.35054i −0.0966746 0.207319i −3.51313 0.619460i 0 −0.507193 + 0.184603i 4.84313 1.29771i −0.957149 + 3.57213i 1.89473 2.25805i 0
32.14 0.219903 2.51350i 0.420327 + 0.901394i −4.29970 0.758153i 0 2.35808 0.858272i −0.630014 + 0.168812i −1.54508 + 5.76632i 1.29253 1.54037i 0
143.1 −1.56389 + 2.23346i 0.0995607 1.13798i −1.85858 5.10639i 0 2.38595 + 2.00205i 0.344190 + 1.28454i 9.04425 + 2.42340i 1.66933 + 0.294348i 0
143.2 −1.51810 + 2.16807i −0.234015 + 2.67481i −1.71186 4.70330i 0 −5.44391 4.56798i −1.16345 4.34206i 7.68278 + 2.05859i −4.14540 0.730946i 0
143.3 −1.14859 + 1.64036i 0.200764 2.29474i −0.687469 1.88881i 0 3.53360 + 2.96504i −0.402975 1.50392i 0.0193927 + 0.00519625i −2.27111 0.400458i 0
143.4 −1.12765 + 1.61045i −0.263219 + 3.00861i −0.637911 1.75265i 0 −4.54838 3.81655i 0.583268 + 2.17679i −0.256124 0.0686282i −6.02800 1.06290i 0
143.5 −0.751242 + 1.07289i −0.0801886 + 0.916560i 0.0973225 + 0.267391i 0 −0.923122 0.774592i 0.629559 + 2.34955i −2.89024 0.774437i 2.12077 + 0.373949i 0
143.6 −0.558716 + 0.797930i 0.285949 3.26841i 0.359512 + 0.987752i 0 2.44820 + 2.05428i −0.476978 1.78010i −2.87082 0.769235i −7.64630 1.34825i 0
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.14
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
19.f odd 18 1 inner
95.o odd 18 1 inner
95.r even 36 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.bb.c 168
5.b even 2 1 inner 475.2.bb.c 168
5.c odd 4 2 inner 475.2.bb.c 168
19.f odd 18 1 inner 475.2.bb.c 168
95.o odd 18 1 inner 475.2.bb.c 168
95.r even 36 2 inner 475.2.bb.c 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.bb.c 168 1.a even 1 1 trivial
475.2.bb.c 168 5.b even 2 1 inner
475.2.bb.c 168 5.c odd 4 2 inner
475.2.bb.c 168 19.f odd 18 1 inner
475.2.bb.c 168 95.o odd 18 1 inner
475.2.bb.c 168 95.r even 36 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{168} + 90 T_{2}^{164} + 4437 T_{2}^{160} - 19482 T_{2}^{156} - 6597594 T_{2}^{152} - 468213210 T_{2}^{148} + 529796541 T_{2}^{144} + 462484115838 T_{2}^{140} + 37018558161000 T_{2}^{136} + \cdots + 282429536481 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display