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Properties

Label 575.2.p.c
Level $575$
Weight $2$
Character orbit 575.p
Analytic conductor $4.591$
Analytic rank $0$
Dimension $40$
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] = [575,2,Mod(49,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.p (of order \(22\), degree \(10\), 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: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(4\) over \(\Q(\zeta_{22})\)
Twist minimal: no (minimal twist has level 115)
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 8 q^{4} - 18 q^{6} + 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 8 q^{4} - 18 q^{6} + 34 q^{9} - 16 q^{11} - 6 q^{14} - 72 q^{16} - 48 q^{19} + 50 q^{21} - 4 q^{26} - 24 q^{29} + 20 q^{31} + 14 q^{34} + 86 q^{36} + 104 q^{39} + 20 q^{41} - 12 q^{44} - 66 q^{46} - 8 q^{49} + 100 q^{51} - 46 q^{54} - 82 q^{59} - 26 q^{61} - 32 q^{64} + 112 q^{66} + 90 q^{69} - 38 q^{71} - 16 q^{74} + 8 q^{76} + 166 q^{79} + 24 q^{81} - 56 q^{84} - 146 q^{86} + 10 q^{89} - 168 q^{91} + 74 q^{94} + 28 q^{96} - 124 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 −1.89945 + 0.273100i −2.03278 − 0.928337i 1.61435 − 0.474017i 0 4.11469 + 1.20818i −0.463153 − 0.720681i 0.554206 − 0.253098i 1.30578 + 1.50695i 0
49.2 −1.89945 + 0.273100i 2.94241 + 1.34375i 1.61435 − 0.474017i 0 −5.95594 − 1.74882i 1.99362 + 3.10213i 0.554206 − 0.253098i 4.88751 + 5.64048i 0
49.3 1.89945 − 0.273100i −2.94241 − 1.34375i 1.61435 − 0.474017i 0 −5.95594 − 1.74882i −1.99362 − 3.10213i −0.554206 + 0.253098i 4.88751 + 5.64048i 0
49.4 1.89945 − 0.273100i 2.03278 + 0.928337i 1.61435 − 0.474017i 0 4.11469 + 1.20818i 0.463153 + 0.720681i −0.554206 + 0.253098i 1.30578 + 1.50695i 0
124.1 −0.474017 − 1.61435i −0.898452 + 0.778513i −0.698939 + 0.449181i 0 1.68268 + 1.08139i 2.32595 + 1.06223i −1.48666 − 1.28820i −0.225811 + 1.57055i 0
124.2 −0.474017 − 1.61435i 1.65420 − 1.43337i −0.698939 + 0.449181i 0 −3.09809 − 1.99102i −1.69805 − 0.775475i −1.48666 − 1.28820i 0.254878 − 1.77272i 0
124.3 0.474017 + 1.61435i −1.65420 + 1.43337i −0.698939 + 0.449181i 0 −3.09809 − 1.99102i 1.69805 + 0.775475i 1.48666 + 1.28820i 0.254878 − 1.77272i 0
124.4 0.474017 + 1.61435i 0.898452 − 0.778513i −0.698939 + 0.449181i 0 1.68268 + 1.08139i −2.32595 − 1.06223i 1.48666 + 1.28820i −0.225811 + 1.57055i 0
174.1 −0.215109 + 0.186393i −1.31080 + 2.03964i −0.273100 + 1.89945i 0 −0.0982103 − 0.683068i 1.43100 − 4.87354i −0.603063 − 0.938384i −1.19570 − 2.61822i 0
174.2 −0.215109 + 0.186393i 0.770157 − 1.19839i −0.273100 + 1.89945i 0 0.0577033 + 0.401335i −0.393517 + 1.34020i −0.603063 − 0.938384i 0.403254 + 0.883004i 0
174.3 0.215109 − 0.186393i −0.770157 + 1.19839i −0.273100 + 1.89945i 0 0.0577033 + 0.401335i 0.393517 − 1.34020i 0.603063 + 0.938384i 0.403254 + 0.883004i 0
174.4 0.215109 − 0.186393i 1.31080 − 2.03964i −0.273100 + 1.89945i 0 −0.0982103 − 0.683068i −1.43100 + 4.87354i 0.603063 + 0.938384i −1.19570 − 2.61822i 0
324.1 −0.449181 + 0.698939i −1.79202 + 0.257653i 0.544078 + 1.19136i 0 0.624856 − 1.36824i −1.14519 − 0.992315i −2.72183 − 0.391340i 0.266462 − 0.0782402i 0
324.2 −0.449181 + 0.698939i 2.78184 − 0.399968i 0.544078 + 1.19136i 0 −0.969995 + 2.12399i −0.151198 − 0.131014i −2.72183 − 0.391340i 4.70017 − 1.38010i 0
324.3 0.449181 − 0.698939i −2.78184 + 0.399968i 0.544078 + 1.19136i 0 −0.969995 + 2.12399i 0.151198 + 0.131014i 2.72183 + 0.391340i 4.70017 − 1.38010i 0
324.4 0.449181 − 0.698939i 1.79202 − 0.257653i 0.544078 + 1.19136i 0 0.624856 − 1.36824i 1.14519 + 0.992315i 2.72183 + 0.391340i 0.266462 − 0.0782402i 0
349.1 −1.19136 + 0.544078i −0.413883 − 1.40956i −0.186393 + 0.215109i 0 1.25999 + 1.45411i −3.78024 + 0.543517i 0.843008 − 2.87102i 0.708209 − 0.455138i 0
349.2 −1.19136 + 0.544078i 0.695616 + 2.36905i −0.186393 + 0.215109i 0 −2.11768 − 2.44393i 3.70005 − 0.531987i 0.843008 − 2.87102i −2.60476 + 1.67397i 0
349.3 1.19136 − 0.544078i −0.695616 − 2.36905i −0.186393 + 0.215109i 0 −2.11768 − 2.44393i −3.70005 + 0.531987i −0.843008 + 2.87102i −2.60476 + 1.67397i 0
349.4 1.19136 − 0.544078i 0.413883 + 1.40956i −0.186393 + 0.215109i 0 1.25999 + 1.45411i 3.78024 − 0.543517i −0.843008 + 2.87102i 0.708209 − 0.455138i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.c even 11 1 inner
115.j even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.2.p.c 40
5.b even 2 1 inner 575.2.p.c 40
5.c odd 4 1 115.2.g.b ✓ 20
5.c odd 4 1 575.2.k.c 20
23.c even 11 1 inner 575.2.p.c 40
115.j even 22 1 inner 575.2.p.c 40
115.k odd 44 1 115.2.g.b ✓ 20
115.k odd 44 1 575.2.k.c 20
115.k odd 44 1 2645.2.a.t 10
115.l even 44 1 2645.2.a.u 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.g.b ✓ 20 5.c odd 4 1
115.2.g.b ✓ 20 115.k odd 44 1
575.2.k.c 20 5.c odd 4 1
575.2.k.c 20 115.k odd 44 1
575.2.p.c 40 1.a even 1 1 trivial
575.2.p.c 40 5.b even 2 1 inner
575.2.p.c 40 23.c even 11 1 inner
575.2.p.c 40 115.j even 22 1 inner
2645.2.a.t 10 115.k odd 44 1
2645.2.a.u 10 115.l even 44 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 4 T_{2}^{18} - 6 T_{2}^{16} + 24 T_{2}^{14} + 69 T_{2}^{12} - 177 T_{2}^{10} + 224 T_{2}^{8} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(575, [\chi])\). Copy content Toggle raw display