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Properties

Label 976.2.bw.c
Level $976$
Weight $2$
Character orbit 976.bw
Analytic conductor $7.793$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [976,2,Mod(225,976)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("976.225"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(976, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([0, 0, 28])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 976 = 2^{4} \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 976.bw (of order \(15\), degree \(8\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.79339923728\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 61)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{3} + 2 q^{5} - q^{7} - 2 q^{9} + 18 q^{11} - 2 q^{15} - 24 q^{17} - 9 q^{19} - 3 q^{21} + 2 q^{23} + 28 q^{25} - 35 q^{27} - 4 q^{29} + 11 q^{31} - 35 q^{33} + 58 q^{35} - 14 q^{37} - 17 q^{39}+ \cdots + 31 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
225.1 0 −0.508502 + 1.56501i 0 −0.0637708 + 0.606739i 0 0.455761 − 0.506174i 0 0.236374 + 0.171736i 0
225.2 0 −0.379526 + 1.16806i 0 0.174208 − 1.65748i 0 −1.99411 + 2.21469i 0 1.20672 + 0.876735i 0
225.3 0 0.507067 − 1.56059i 0 −0.0538027 + 0.511898i 0 2.89428 − 3.21442i 0 0.248718 + 0.180704i 0
225.4 0 0.880961 − 2.71132i 0 0.308047 − 2.93087i 0 −2.02506 + 2.24905i 0 −4.14811 − 3.01378i 0
321.1 0 −0.508502 − 1.56501i 0 −0.0637708 − 0.606739i 0 0.455761 + 0.506174i 0 0.236374 − 0.171736i 0
321.2 0 −0.379526 − 1.16806i 0 0.174208 + 1.65748i 0 −1.99411 − 2.21469i 0 1.20672 − 0.876735i 0
321.3 0 0.507067 + 1.56059i 0 −0.0538027 − 0.511898i 0 2.89428 + 3.21442i 0 0.248718 − 0.180704i 0
321.4 0 0.880961 + 2.71132i 0 0.308047 + 2.93087i 0 −2.02506 − 2.24905i 0 −4.14811 + 3.01378i 0
449.1 0 −0.520127 + 1.60079i 0 −3.28207 + 1.46127i 0 −3.82105 − 0.812190i 0 0.135065 + 0.0981308i 0
449.2 0 −0.0959745 + 0.295379i 0 −1.01852 + 0.453476i 0 4.83747 + 1.02824i 0 2.34901 + 1.70666i 0
449.3 0 0.289378 − 0.890614i 0 2.60246 − 1.15869i 0 −0.0927128 − 0.0197067i 0 1.71760 + 1.24791i 0
449.4 0 0.826723 − 2.54439i 0 −2.63869 + 1.17482i 0 0.0544423 + 0.0115721i 0 −3.36341 − 2.44366i 0
513.1 0 −0.520127 − 1.60079i 0 −3.28207 − 1.46127i 0 −3.82105 + 0.812190i 0 0.135065 − 0.0981308i 0
513.2 0 −0.0959745 − 0.295379i 0 −1.01852 − 0.453476i 0 4.83747 − 1.02824i 0 2.34901 − 1.70666i 0
513.3 0 0.289378 + 0.890614i 0 2.60246 + 1.15869i 0 −0.0927128 + 0.0197067i 0 1.71760 − 1.24791i 0
513.4 0 0.826723 + 2.54439i 0 −2.63869 − 1.17482i 0 0.0544423 − 0.0115721i 0 −3.36341 + 2.44366i 0
545.1 0 −1.09846 − 0.798080i 0 2.01896 − 2.24228i 0 2.42543 + 1.07987i 0 −0.357362 − 1.09985i 0
545.2 0 −0.768086 − 0.558047i 0 −0.799461 + 0.887892i 0 −3.76417 − 1.67592i 0 −0.648512 − 1.99591i 0
545.3 0 0.246407 + 0.179025i 0 −0.519162 + 0.576588i 0 0.0513167 + 0.0228477i 0 −0.898385 − 2.76494i 0
545.4 0 2.12014 + 1.54037i 0 −0.176149 + 0.195633i 0 0.373885 + 0.166464i 0 1.19520 + 3.67845i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 225.4
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.i even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 976.2.bw.c 32
4.b odd 2 1 61.2.i.a ✓ 32
12.b even 2 1 549.2.bl.b 32
61.i even 15 1 inner 976.2.bw.c 32
244.u odd 30 1 61.2.i.a ✓ 32
244.u odd 30 1 3721.2.a.j 16
244.v odd 30 1 3721.2.a.l 16
732.bl even 30 1 549.2.bl.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.2.i.a ✓ 32 4.b odd 2 1
61.2.i.a ✓ 32 244.u odd 30 1
549.2.bl.b 32 12.b even 2 1
549.2.bl.b 32 732.bl even 30 1
976.2.bw.c 32 1.a even 1 1 trivial
976.2.bw.c 32 61.i even 15 1 inner
3721.2.a.j 16 244.u odd 30 1
3721.2.a.l 16 244.v odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} - 4 T_{3}^{31} + 21 T_{3}^{30} - 39 T_{3}^{29} + 133 T_{3}^{28} - 171 T_{3}^{27} + \cdots + 32761 \) acting on \(S_{2}^{\mathrm{new}}(976, [\chi])\). Copy content Toggle raw display