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Properties

Label 975.2.bh.b
Level $975$
Weight $2$
Character orbit 975.bh
Analytic conductor $7.785$
Analytic rank $0$
Dimension $128$
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] = [975,2,Mod(181,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bh (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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 32 q^{3} + 20 q^{4} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 32 q^{3} + 20 q^{4} - 32 q^{9} + 34 q^{10} + 30 q^{12} + 6 q^{13} + 40 q^{14} - 32 q^{16} - 12 q^{17} - 24 q^{23} + 6 q^{25} + 70 q^{26} - 32 q^{27} + 44 q^{29} - 16 q^{30} - 8 q^{35} + 30 q^{36} + 18 q^{38} + 6 q^{39} - 32 q^{40} + 30 q^{42} + 24 q^{43} - 32 q^{48} - 84 q^{49} + 68 q^{51} + 14 q^{52} - 28 q^{53} + 36 q^{55} - 62 q^{56} + 6 q^{61} + 156 q^{62} + 68 q^{64} + 7 q^{65} + 10 q^{66} - 176 q^{68} + 6 q^{69} + 68 q^{74} + 46 q^{75} - 36 q^{77} + 20 q^{78} + 8 q^{79} - 32 q^{81} + 156 q^{82} - 46 q^{87} - 82 q^{88} - 6 q^{90} + 3 q^{91} - 60 q^{92} - 54 q^{94} - 44 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1 −2.51953 + 0.818646i −0.809017 + 0.587785i 4.05984 − 2.94965i 2.23554 − 0.0486999i 1.55716 − 2.14324i 4.14054i −4.69987 + 6.46882i 0.309017 − 0.951057i −5.59265 + 1.95282i
181.2 −2.48207 + 0.806475i −0.809017 + 0.587785i 3.89225 − 2.82789i −0.860659 + 2.06380i 1.53401 − 2.11138i 0.951254i −4.31223 + 5.93527i 0.309017 − 0.951057i 0.471818 − 5.81660i
181.3 −2.34803 + 0.762921i −0.809017 + 0.587785i 3.31316 − 2.40715i −2.23606 − 0.00424791i 1.45116 − 1.99735i − 2.12193i −3.04061 + 4.18504i 0.309017 − 0.951057i 5.25359 − 1.69597i
181.4 −2.14102 + 0.695659i −0.809017 + 0.587785i 2.48199 − 1.80327i 0.612095 − 2.15066i 1.32322 − 1.82126i − 0.511787i −1.41308 + 1.94493i 0.309017 − 0.951057i 0.185621 + 5.03041i
181.5 −1.95739 + 0.635994i −0.809017 + 0.587785i 1.80885 − 1.31421i −1.06125 − 1.96818i 1.20973 − 1.66505i 4.50709i −0.285328 + 0.392720i 0.309017 − 0.951057i 3.32904 + 3.17755i
181.6 −1.77481 + 0.576670i −0.809017 + 0.587785i 1.19936 − 0.871385i 0.358216 + 2.20719i 1.09689 − 1.50974i 1.62866i 0.567653 − 0.781308i 0.309017 − 0.951057i −1.90858 − 3.71076i
181.7 −1.60957 + 0.522982i −0.809017 + 0.587785i 0.699184 − 0.507987i −1.79781 − 1.32961i 0.994772 − 1.36919i − 2.45263i 1.12982 − 1.55507i 0.309017 − 0.951057i 3.58908 + 1.19988i
181.8 −1.50843 + 0.490118i −0.809017 + 0.587785i 0.417104 − 0.303044i 1.48702 + 1.66996i 0.932260 − 1.28315i 0.00306347i 1.38387 − 1.90474i 0.309017 − 0.951057i −3.06154 − 1.79020i
181.9 −1.07535 + 0.349401i −0.809017 + 0.587785i −0.583744 + 0.424115i −2.19858 + 0.407722i 0.664601 − 0.914745i 0.378674i 1.80874 − 2.48952i 0.309017 − 0.951057i 2.22178 − 1.20663i
181.10 −1.04898 + 0.340834i −0.809017 + 0.587785i −0.633845 + 0.460515i 0.789777 − 2.09195i 0.648305 − 0.892315i − 4.13696i 1.80454 − 2.48374i 0.309017 − 0.951057i −0.115453 + 2.46359i
181.11 −1.01963 + 0.331299i −0.809017 + 0.587785i −0.688140 + 0.499963i 1.75245 − 1.38885i 0.630168 − 0.867352i 1.06183i 1.79635 − 2.47246i 0.309017 − 0.951057i −1.32673 + 1.99671i
181.12 −0.899442 + 0.292246i −0.809017 + 0.587785i −0.894446 + 0.649853i −1.76232 + 1.37631i 0.555886 − 0.765111i 1.95813i 1.72636 − 2.37613i 0.309017 − 0.951057i 1.18288 − 1.75295i
181.13 −0.716254 + 0.232725i −0.809017 + 0.587785i −1.15917 + 0.842190i −1.23688 − 1.86283i 0.442669 − 0.609282i 4.05496i 1.51960 − 2.09156i 0.309017 − 0.951057i 1.31945 + 1.04640i
181.14 −0.400724 + 0.130203i −0.809017 + 0.587785i −1.47441 + 1.07122i 1.83073 + 1.28391i 0.247661 − 0.340876i − 0.666399i 0.946677 − 1.30299i 0.309017 − 0.951057i −0.900787 − 0.276128i
181.15 −0.0598336 + 0.0194411i −0.809017 + 0.587785i −1.61483 + 1.17324i −1.55791 + 1.60403i 0.0369792 − 0.0508975i − 3.07747i 0.147771 − 0.203389i 0.309017 − 0.951057i 0.0620315 − 0.126262i
181.16 −0.0587760 + 0.0190975i −0.809017 + 0.587785i −1.61494 + 1.17333i −1.05147 − 1.97343i 0.0363256 − 0.0499978i − 4.58361i 0.145163 − 0.199800i 0.309017 − 0.951057i 0.0994885 + 0.0959098i
181.17 0.0587760 − 0.0190975i −0.809017 + 0.587785i −1.61494 + 1.17333i 1.05147 + 1.97343i −0.0363256 + 0.0499978i 4.58361i −0.145163 + 0.199800i 0.309017 − 0.951057i 0.0994885 + 0.0959098i
181.18 0.0598336 − 0.0194411i −0.809017 + 0.587785i −1.61483 + 1.17324i 1.55791 − 1.60403i −0.0369792 + 0.0508975i 3.07747i −0.147771 + 0.203389i 0.309017 − 0.951057i 0.0620315 − 0.126262i
181.19 0.400724 − 0.130203i −0.809017 + 0.587785i −1.47441 + 1.07122i −1.83073 − 1.28391i −0.247661 + 0.340876i 0.666399i −0.946677 + 1.30299i 0.309017 − 0.951057i −0.900787 − 0.276128i
181.20 0.716254 − 0.232725i −0.809017 + 0.587785i −1.15917 + 0.842190i 1.23688 + 1.86283i −0.442669 + 0.609282i − 4.05496i −1.51960 + 2.09156i 0.309017 − 0.951057i 1.31945 + 1.04640i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.32
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 975.2.bh.b ✓ 128
13.b even 2 1 inner 975.2.bh.b ✓ 128
25.d even 5 1 inner 975.2.bh.b ✓ 128
325.q even 10 1 inner 975.2.bh.b ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.bh.b ✓ 128 1.a even 1 1 trivial
975.2.bh.b ✓ 128 13.b even 2 1 inner
975.2.bh.b ✓ 128 25.d even 5 1 inner
975.2.bh.b ✓ 128 325.q even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{128} - 42 T_{2}^{126} + 999 T_{2}^{124} - 17720 T_{2}^{122} + 260578 T_{2}^{120} - 3308178 T_{2}^{118} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\). Copy content Toggle raw display