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Properties

Label 525.2.bh.a
Level $525$
Weight $2$
Character orbit 525.bh
Analytic conductor $4.192$
Analytic rank $0$
Dimension $320$
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] = [525,2,Mod(13,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 19, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.bh (of order \(20\), degree \(8\), 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: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(40\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 320 q + 8 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q + 8 q^{7} - 24 q^{8} - 8 q^{15} + 80 q^{16} - 24 q^{22} + 36 q^{28} - 40 q^{29} - 32 q^{30} - 48 q^{32} + 28 q^{35} + 80 q^{36} - 32 q^{37} + 16 q^{42} - 144 q^{43} - 288 q^{50} + 136 q^{53} - 8 q^{57} - 32 q^{58} - 40 q^{60} - 8 q^{63} - 120 q^{64} - 40 q^{65} + 32 q^{67} - 40 q^{70} - 24 q^{72} + 24 q^{77} + 8 q^{78} + 80 q^{81} - 80 q^{84} - 48 q^{85} - 56 q^{88} + 40 q^{92} + 96 q^{93} - 88 q^{95} - 184 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −2.65979 + 0.421270i −0.891007 − 0.453990i 4.99492 − 1.62295i 0.301881 + 2.21560i 2.56115 + 0.832167i 2.34376 − 1.22751i −7.80289 + 3.97577i 0.587785 + 0.809017i −1.73630 − 5.76586i
13.2 −2.65979 + 0.421270i 0.891007 + 0.453990i 4.99492 − 1.62295i −0.301881 − 2.21560i −2.56115 − 0.832167i 1.22751 − 2.34376i −7.80289 + 3.97577i 0.587785 + 0.809017i 1.73630 + 5.76586i
13.3 −2.65246 + 0.420109i −0.891007 − 0.453990i 4.95695 − 1.61061i 1.94004 − 1.11186i 2.55409 + 0.829873i −0.972375 + 2.46059i −7.68586 + 3.91614i 0.587785 + 0.809017i −4.67879 + 3.76420i
13.4 −2.65246 + 0.420109i 0.891007 + 0.453990i 4.95695 − 1.61061i −1.94004 + 1.11186i −2.55409 − 0.829873i −2.46059 + 0.972375i −7.68586 + 3.91614i 0.587785 + 0.809017i 4.67879 − 3.76420i
13.5 −2.18124 + 0.345475i −0.891007 − 0.453990i 2.73636 − 0.889096i −1.83877 + 1.27236i 2.10034 + 0.682443i −2.14582 − 1.54772i −1.72604 + 0.879463i 0.587785 + 0.809017i 3.57124 − 3.41059i
13.6 −2.18124 + 0.345475i 0.891007 + 0.453990i 2.73636 − 0.889096i 1.83877 − 1.27236i −2.10034 − 0.682443i 1.54772 + 2.14582i −1.72604 + 0.879463i 0.587785 + 0.809017i −3.57124 + 3.41059i
13.7 −1.83690 + 0.290936i −0.891007 − 0.453990i 1.38744 − 0.450806i −0.677854 − 2.13085i 1.76877 + 0.574708i −2.16215 + 1.52483i 0.896755 − 0.456919i 0.587785 + 0.809017i 1.86509 + 3.71694i
13.8 −1.83690 + 0.290936i 0.891007 + 0.453990i 1.38744 − 0.450806i 0.677854 + 2.13085i −1.76877 − 0.574708i −1.52483 + 2.16215i 0.896755 − 0.456919i 0.587785 + 0.809017i −1.86509 − 3.71694i
13.9 −1.60091 + 0.253560i −0.891007 − 0.453990i 0.596516 − 0.193820i 2.13278 + 0.671755i 1.54154 + 0.500876i −0.369115 − 2.61988i 1.98258 − 1.01018i 0.587785 + 0.809017i −3.58472 − 0.534634i
13.10 −1.60091 + 0.253560i 0.891007 + 0.453990i 0.596516 − 0.193820i −2.13278 − 0.671755i −1.54154 − 0.500876i 2.61988 + 0.369115i 1.98258 − 1.01018i 0.587785 + 0.809017i 3.58472 + 0.534634i
13.11 −1.56905 + 0.248513i −0.891007 − 0.453990i 0.498045 − 0.161825i −1.85206 + 1.25295i 1.51086 + 0.490907i 0.803473 + 2.52080i 2.08968 − 1.06474i 0.587785 + 0.809017i 2.59460 − 2.42620i
13.12 −1.56905 + 0.248513i 0.891007 + 0.453990i 0.498045 − 0.161825i 1.85206 − 1.25295i −1.51086 − 0.490907i −2.52080 − 0.803473i 2.08968 − 1.06474i 0.587785 + 0.809017i −2.59460 + 2.42620i
13.13 −1.44172 + 0.228346i −0.891007 − 0.453990i 0.124296 − 0.0403862i 1.71338 + 1.43678i 1.38825 + 0.451069i 1.34100 + 2.28073i 2.43121 − 1.23876i 0.587785 + 0.809017i −2.79829 − 1.68019i
13.14 −1.44172 + 0.228346i 0.891007 + 0.453990i 0.124296 − 0.0403862i −1.71338 − 1.43678i −1.38825 − 0.451069i −2.28073 − 1.34100i 2.43121 − 1.23876i 0.587785 + 0.809017i 2.79829 + 1.68019i
13.15 −0.809024 + 0.128137i −0.891007 − 0.453990i −1.26401 + 0.410703i −1.82130 − 1.29725i 0.779018 + 0.253118i 2.49866 − 0.869897i 2.42965 − 1.23797i 0.587785 + 0.809017i 1.63970 + 0.816128i
13.16 −0.809024 + 0.128137i 0.891007 + 0.453990i −1.26401 + 0.410703i 1.82130 + 1.29725i −0.779018 − 0.253118i 0.869897 − 2.49866i 2.42965 − 1.23797i 0.587785 + 0.809017i −1.63970 − 0.816128i
13.17 −0.112696 + 0.0178493i −0.891007 − 0.453990i −1.88973 + 0.614011i −2.18359 + 0.481591i 0.108517 + 0.0352592i −1.28646 − 2.31193i 0.405335 − 0.206529i 0.587785 + 0.809017i 0.237486 − 0.0932491i
13.18 −0.112696 + 0.0178493i 0.891007 + 0.453990i −1.88973 + 0.614011i 2.18359 − 0.481591i −0.108517 − 0.0352592i 2.31193 + 1.28646i 0.405335 − 0.206529i 0.587785 + 0.809017i −0.237486 + 0.0932491i
13.19 −0.0439745 + 0.00696487i −0.891007 − 0.453990i −1.90023 + 0.617421i 1.50515 − 1.65364i 0.0423435 + 0.0137582i 2.54086 + 0.737568i 0.158601 − 0.0808113i 0.587785 + 0.809017i −0.0546709 + 0.0832010i
13.20 −0.0439745 + 0.00696487i 0.891007 + 0.453990i −1.90023 + 0.617421i −1.50515 + 1.65364i −0.0423435 − 0.0137582i −0.737568 − 2.54086i 0.158601 − 0.0808113i 0.587785 + 0.809017i 0.0546709 − 0.0832010i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.40
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 525.2.bh.a ✓ 320
7.b odd 2 1 inner 525.2.bh.a ✓ 320
25.f odd 20 1 inner 525.2.bh.a ✓ 320
175.s even 20 1 inner 525.2.bh.a ✓ 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bh.a ✓ 320 1.a even 1 1 trivial
525.2.bh.a ✓ 320 7.b odd 2 1 inner
525.2.bh.a ✓ 320 25.f odd 20 1 inner
525.2.bh.a ✓ 320 175.s even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(525, [\chi])\).