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Properties

Label 575.2.k.g
Level $575$
Weight $2$
Character orbit 575.k
Analytic conductor $4.591$
Analytic rank $0$
Dimension $100$
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(26,575)]
 
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([0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.k (of order \(11\), degree \(10\), 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: \(100\)
Relative dimension: \(10\) over \(\Q(\zeta_{11})\)
Twist minimal: no (minimal twist has level 115)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 100 q + 14 q^{4} - 18 q^{6} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 100 q + 14 q^{4} - 18 q^{6} + 12 q^{9} - 26 q^{11} + 26 q^{14} - 18 q^{16} + 14 q^{19} - 22 q^{21} + 68 q^{24} - 42 q^{26} + 24 q^{29} - 12 q^{31} - 8 q^{34} - 10 q^{36} - 14 q^{39} + 8 q^{41} - 166 q^{44} - 18 q^{46} - 32 q^{49} - 22 q^{51} - 116 q^{54} - 116 q^{56} - 50 q^{59} - 38 q^{61} - 10 q^{64} - 28 q^{66} - 80 q^{69} - 110 q^{71} - 22 q^{74} + 4 q^{76} - 42 q^{79} + 204 q^{81} - 56 q^{84} + 132 q^{86} + 66 q^{89} + 76 q^{91} + 70 q^{94} + 236 q^{96} + 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −0.336428 − 2.33991i 1.10646 − 2.42282i −3.44299 + 1.01095i 0 −6.04141 − 1.77392i 2.96780 − 1.90729i 1.55980 + 3.41548i −2.68120 − 3.09427i 0
26.2 −0.317048 − 2.20512i 0.194898 − 0.426766i −2.84303 + 0.834790i 0 −1.00286 − 0.294467i −2.53951 + 1.63204i 0.891272 + 1.95161i 1.82044 + 2.10090i 0
26.3 −0.179828 − 1.25073i −0.104253 + 0.228282i 0.387001 − 0.113634i 0 0.304267 + 0.0893407i 0.933848 − 0.600148i −1.26155 − 2.76240i 1.92334 + 2.21965i 0
26.4 −0.152174 − 1.05840i −1.14653 + 2.51056i 0.821942 − 0.241344i 0 2.83164 + 0.831443i −2.32389 + 1.49347i −1.26891 − 2.77851i −3.02378 − 3.48962i 0
26.5 −0.0448489 − 0.311931i 0.875852 − 1.91785i 1.82370 − 0.535486i 0 −0.637517 − 0.187192i −1.15522 + 0.742416i −0.510652 − 1.11817i −0.946445 − 1.09226i 0
26.6 0.0448489 + 0.311931i −0.875852 + 1.91785i 1.82370 − 0.535486i 0 −0.637517 − 0.187192i 1.15522 − 0.742416i 0.510652 + 1.11817i −0.946445 − 1.09226i 0
26.7 0.152174 + 1.05840i 1.14653 − 2.51056i 0.821942 − 0.241344i 0 2.83164 + 0.831443i 2.32389 − 1.49347i 1.26891 + 2.77851i −3.02378 − 3.48962i 0
26.8 0.179828 + 1.25073i 0.104253 − 0.228282i 0.387001 − 0.113634i 0 0.304267 + 0.0893407i −0.933848 + 0.600148i 1.26155 + 2.76240i 1.92334 + 2.21965i 0
26.9 0.317048 + 2.20512i −0.194898 + 0.426766i −2.84303 + 0.834790i 0 −1.00286 − 0.294467i 2.53951 − 1.63204i −0.891272 − 1.95161i 1.82044 + 2.10090i 0
26.10 0.336428 + 2.33991i −1.10646 + 2.42282i −3.44299 + 1.01095i 0 −6.04141 − 1.77392i −2.96780 + 1.90729i −1.55980 − 3.41548i −2.68120 − 3.09427i 0
101.1 −2.53313 + 0.743795i −1.16633 − 1.34601i 4.18102 − 2.68698i 0 3.95561 + 2.54212i 0.855264 − 1.87277i −5.13475 + 5.92582i −0.0244873 + 0.170313i 0
101.2 −2.21797 + 0.651256i 0.794936 + 0.917405i 2.81277 − 1.80765i 0 −2.36061 − 1.51707i −0.779189 + 1.70619i −2.03383 + 2.34716i 0.217236 − 1.51091i 0
101.3 −1.34433 + 0.394732i −1.70867 − 1.97191i −0.0310904 + 0.0199806i 0 3.07540 + 1.97644i 1.22473 − 2.68179i 1.86894 − 2.15687i −0.541936 + 3.76925i 0
101.4 −1.09283 + 0.320885i 1.07866 + 1.24484i −0.591189 + 0.379934i 0 −1.57825 − 1.01428i 0.643697 − 1.40950i 2.01589 − 2.32646i 0.0408222 − 0.283925i 0
101.5 −0.601743 + 0.176688i 0.488459 + 0.563711i −1.35163 + 0.868641i 0 −0.393527 − 0.252905i 0.283387 − 0.620531i 1.48124 − 1.70945i 0.347766 − 2.41877i 0
101.6 0.601743 − 0.176688i −0.488459 − 0.563711i −1.35163 + 0.868641i 0 −0.393527 − 0.252905i −0.283387 + 0.620531i −1.48124 + 1.70945i 0.347766 − 2.41877i 0
101.7 1.09283 − 0.320885i −1.07866 − 1.24484i −0.591189 + 0.379934i 0 −1.57825 − 1.01428i −0.643697 + 1.40950i −2.01589 + 2.32646i 0.0408222 − 0.283925i 0
101.8 1.34433 − 0.394732i 1.70867 + 1.97191i −0.0310904 + 0.0199806i 0 3.07540 + 1.97644i −1.22473 + 2.68179i −1.86894 + 2.15687i −0.541936 + 3.76925i 0
101.9 2.21797 − 0.651256i −0.794936 − 0.917405i 2.81277 − 1.80765i 0 −2.36061 − 1.51707i 0.779189 − 1.70619i 2.03383 − 2.34716i 0.217236 − 1.51091i 0
101.10 2.53313 − 0.743795i 1.16633 + 1.34601i 4.18102 − 2.68698i 0 3.95561 + 2.54212i −0.855264 + 1.87277i 5.13475 − 5.92582i −0.0244873 + 0.170313i 0
See all 100 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.10
Significant digits:
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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.k.g 100
5.b even 2 1 inner 575.2.k.g 100
5.c odd 4 2 115.2.j.a ✓ 100
23.c even 11 1 inner 575.2.k.g 100
115.j even 22 1 inner 575.2.k.g 100
115.k odd 44 2 115.2.j.a ✓ 100
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.j.a ✓ 100 5.c odd 4 2
115.2.j.a ✓ 100 115.k odd 44 2
575.2.k.g 100 1.a even 1 1 trivial
575.2.k.g 100 5.b even 2 1 inner
575.2.k.g 100 23.c even 11 1 inner
575.2.k.g 100 115.j even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{100} + 3 T_{2}^{98} + 41 T_{2}^{96} + 429 T_{2}^{94} + 2160 T_{2}^{92} + 20830 T_{2}^{90} + \cdots + 62742241 \) acting on \(S_{2}^{\mathrm{new}}(575, [\chi])\). Copy content Toggle raw display