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Properties

Label 325.2.q.a
Level $325$
Weight $2$
Character orbit 325.q
Analytic conductor $2.595$
Analytic rank $0$
Dimension $136$
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] = [325,2,Mod(116,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.116");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.q (of order \(10\), degree \(4\), 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: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(136\)
Relative dimension: \(34\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 136 q - 6 q^{3} + 28 q^{4} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 136 q - 6 q^{3} + 28 q^{4} - 40 q^{9} - 12 q^{10} - 10 q^{12} + 3 q^{13} - 2 q^{14} - 60 q^{16} - 18 q^{17} - 36 q^{22} - 16 q^{23} - 24 q^{25} - 26 q^{26} + 18 q^{27} - 20 q^{29} - 30 q^{30} + 44 q^{35} + 72 q^{36} + 10 q^{38} - 19 q^{39} + 6 q^{40} + 6 q^{42} - 44 q^{43} + 104 q^{48} - 136 q^{49} - 100 q^{51} - 10 q^{52} - 2 q^{53} + 32 q^{55} + 104 q^{56} + 2 q^{61} - 98 q^{62} + 32 q^{64} + 3 q^{65} - 34 q^{66} + 68 q^{68} - 28 q^{69} + 64 q^{74} + 96 q^{75} + 24 q^{77} + 17 q^{78} - 22 q^{79} + 60 q^{81} - 112 q^{82} + 124 q^{87} - 150 q^{88} + 104 q^{90} + 46 q^{91} - 20 q^{92} - 110 q^{94} - 76 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1 −1.58413 + 2.18037i −0.881345 − 2.71250i −1.62651 − 5.00589i −1.86845 − 1.22836i 7.31043 + 2.37530i − 3.33374i 8.36495 + 2.71794i −4.15384 + 3.01794i 5.63817 − 2.12803i
116.2 −1.56689 + 2.15664i 0.283831 + 0.873542i −1.57791 − 4.85630i 0.660442 − 2.13631i −2.32864 − 0.756622i 3.86814i 7.87512 + 2.55878i 1.74454 − 1.26748i 3.57240 + 4.77169i
116.3 −1.54680 + 2.12899i 0.0981887 + 0.302194i −1.52196 − 4.68412i 1.43289 + 1.71663i −0.795245 − 0.258391i − 1.80192i 7.32105 + 2.37875i 2.34537 − 1.70401i −5.87108 + 0.395322i
116.4 −1.41771 + 1.95131i 1.00828 + 3.10317i −1.17967 − 3.63066i −2.05015 + 0.892675i −7.48468 − 2.43192i 2.33792i 4.16916 + 1.35464i −6.18597 + 4.49437i 1.16464 − 5.26603i
116.5 −1.24478 + 1.71329i −0.727067 − 2.23768i −0.767858 − 2.36322i 2.23418 − 0.0918394i 4.73884 + 1.53974i 1.46838i 0.976515 + 0.317289i −2.05155 + 1.49054i −2.62371 + 3.94212i
116.6 −1.23128 + 1.69471i 0.271402 + 0.835290i −0.737961 − 2.27121i −1.71081 − 1.43984i −1.74975 − 0.568527i − 1.84619i 0.773175 + 0.251220i 1.80300 − 1.30996i 4.54659 − 1.12648i
116.7 −1.12062 + 1.54241i −0.598672 − 1.84252i −0.505184 − 1.55480i −0.696248 + 2.12491i 3.51280 + 1.14138i − 0.132981i −0.662164 − 0.215150i −0.609428 + 0.442775i −2.49724 − 3.45512i
116.8 −1.08425 + 1.49234i 0.673593 + 2.07311i −0.433447 − 1.33401i 0.602979 + 2.15323i −3.82412 − 1.24253i − 3.22253i −1.04794 − 0.340495i −1.41699 + 1.02951i −3.86713 − 1.43479i
116.9 −0.904762 + 1.24530i −0.700549 − 2.15607i −0.114139 − 0.351284i −1.60277 − 1.55920i 3.31878 + 1.07834i 4.50178i −2.38715 − 0.775632i −1.73081 + 1.25750i 3.39180 − 0.585223i
116.10 −0.878044 + 1.20852i 0.0468143 + 0.144080i −0.0715345 − 0.220160i 1.13784 − 1.92492i −0.215229 − 0.0699320i − 1.27944i −2.51253 − 0.816370i 2.40848 − 1.74987i 1.32723 + 3.06527i
116.11 −0.801184 + 1.10274i 0.884381 + 2.72185i 0.0439045 + 0.135124i 2.01087 − 0.977958i −3.71003 − 1.20546i 0.692490i −2.77687 − 0.902259i −4.19926 + 3.05094i −0.532648 + 3.00098i
116.12 −0.584688 + 0.804754i 0.335715 + 1.03323i 0.312265 + 0.961053i −0.0936828 + 2.23410i −1.02778 − 0.333946i 4.80957i −2.84808 − 0.925397i 1.47220 − 1.06962i −1.74313 − 1.38165i
116.13 −0.471981 + 0.649626i −0.406107 − 1.24987i 0.418786 + 1.28889i −1.71003 + 1.44077i 1.00362 + 0.326097i − 4.79790i −2.56232 − 0.832548i 1.02980 − 0.748194i −0.128861 − 1.79089i
116.14 −0.355291 + 0.489016i −0.363025 − 1.11728i 0.505129 + 1.55463i 1.77881 + 1.35493i 0.675345 + 0.219433i 0.311570i −2.08945 − 0.678903i 1.31053 − 0.952157i −1.29458 + 0.388471i
116.15 −0.150977 + 0.207802i −1.03155 − 3.17477i 0.597646 + 1.83937i 1.51190 − 1.64747i 0.815463 + 0.264960i − 2.87266i −0.961026 − 0.312256i −6.58803 + 4.78649i 0.114087 + 0.562905i
116.16 −0.136859 + 0.188370i 0.759716 + 2.33816i 0.601281 + 1.85055i −0.891637 − 2.05061i −0.544413 − 0.176891i 2.54954i −0.873763 − 0.283903i −2.46279 + 1.78932i 0.508301 + 0.112686i
116.17 −0.0758175 + 0.104354i 0.155406 + 0.478289i 0.612893 + 1.88629i −2.22811 + 0.188508i −0.0616937 − 0.0200455i 1.36391i −0.488660 − 0.158775i 2.22244 − 1.61470i 0.149258 − 0.246804i
116.18 0.0758175 − 0.104354i 0.155406 + 0.478289i 0.612893 + 1.88629i 2.22811 − 0.188508i 0.0616937 + 0.0200455i − 1.36391i 0.488660 + 0.158775i 2.22244 − 1.61470i 0.149258 − 0.246804i
116.19 0.136859 − 0.188370i 0.759716 + 2.33816i 0.601281 + 1.85055i 0.891637 + 2.05061i 0.544413 + 0.176891i − 2.54954i 0.873763 + 0.283903i −2.46279 + 1.78932i 0.508301 + 0.112686i
116.20 0.150977 − 0.207802i −1.03155 − 3.17477i 0.597646 + 1.83937i −1.51190 + 1.64747i −0.815463 − 0.264960i 2.87266i 0.961026 + 0.312256i −6.58803 + 4.78649i 0.114087 + 0.562905i
See next 80 embeddings (of 136 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
25.d even 5 1 inner
325.q even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.q.a ✓ 136
13.b even 2 1 inner 325.2.q.a ✓ 136
25.d even 5 1 inner 325.2.q.a ✓ 136
325.q even 10 1 inner 325.2.q.a ✓ 136
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.q.a ✓ 136 1.a even 1 1 trivial
325.2.q.a ✓ 136 13.b even 2 1 inner
325.2.q.a ✓ 136 25.d even 5 1 inner
325.2.q.a ✓ 136 325.q even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(325, [\chi])\).