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Properties

Label 882.2.z.c
Level $882$
Weight $2$
Character orbit 882.z
Analytic conductor $7.043$
Analytic rank $0$
Dimension $24$
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] = [882,2,Mod(37,882)] 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("882.37"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([0, 32])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.z (of order \(21\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,2,0,2,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(2\) over \(\Q(\zeta_{21})\)
Twist minimal: no (minimal twist has level 294)
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} + 2 q^{4} - q^{5} - 4 q^{8} - q^{10} - 6 q^{11} - 2 q^{13} + 2 q^{16} - 7 q^{17} + 22 q^{19} - 5 q^{20} + 5 q^{22} + 42 q^{23} + 29 q^{25} - 34 q^{26} + 14 q^{28} - 6 q^{29} + 2 q^{31}+ \cdots - 21 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 0.826239 − 0.563320i 0 0.365341 − 0.930874i −2.66126 − 0.820891i 0 0.800630 + 2.52170i −0.222521 − 0.974928i 0 −2.66126 + 0.820891i
37.2 0.826239 − 0.563320i 0 0.365341 − 0.930874i 2.26109 + 0.697454i 0 −2.27170 + 1.35623i −0.222521 − 0.974928i 0 2.26109 − 0.697454i
109.1 −0.733052 + 0.680173i 0 0.0747301 − 0.997204i −0.651816 + 1.66080i 0 1.92190 + 1.81832i 0.623490 + 0.781831i 0 −0.651816 − 1.66080i
109.2 −0.733052 + 0.680173i 0 0.0747301 − 0.997204i −0.357141 + 0.909982i 0 −1.62228 − 2.09002i 0.623490 + 0.781831i 0 −0.357141 − 0.909982i
163.1 0.365341 + 0.930874i 0 −0.733052 + 0.680173i −2.34942 + 1.60181i 0 −1.71046 − 2.01849i −0.900969 − 0.433884i 0 −2.34942 − 1.60181i
163.2 0.365341 + 0.930874i 0 −0.733052 + 0.680173i 1.94585 − 1.32666i 0 −1.66390 + 2.05705i −0.900969 − 0.433884i 0 1.94585 + 1.32666i
235.1 −0.988831 − 0.149042i 0 0.955573 + 0.294755i −0.126149 − 1.68335i 0 −1.36136 + 2.26863i −0.900969 − 0.433884i 0 −0.126149 + 1.68335i
235.2 −0.988831 − 0.149042i 0 0.955573 + 0.294755i 0.139635 + 1.86330i 0 2.53282 − 0.764745i −0.900969 − 0.433884i 0 0.139635 − 1.86330i
289.1 −0.988831 + 0.149042i 0 0.955573 − 0.294755i −0.126149 + 1.68335i 0 −1.36136 − 2.26863i −0.900969 + 0.433884i 0 −0.126149 − 1.68335i
289.2 −0.988831 + 0.149042i 0 0.955573 − 0.294755i 0.139635 − 1.86330i 0 2.53282 + 0.764745i −0.900969 + 0.433884i 0 0.139635 + 1.86330i
415.1 0.0747301 − 0.997204i 0 −0.988831 − 0.149042i 0.457838 + 0.424812i 0 −1.06155 − 2.42345i −0.222521 + 0.974928i 0 0.457838 − 0.424812i
415.2 0.0747301 − 0.997204i 0 −0.988831 − 0.149042i 2.93629 + 2.72448i 0 2.36506 + 1.18596i −0.222521 + 0.974928i 0 2.93629 − 2.72448i
487.1 0.365341 − 0.930874i 0 −0.733052 − 0.680173i −2.34942 − 1.60181i 0 −1.71046 + 2.01849i −0.900969 + 0.433884i 0 −2.34942 + 1.60181i
487.2 0.365341 − 0.930874i 0 −0.733052 − 0.680173i 1.94585 + 1.32666i 0 −1.66390 − 2.05705i −0.900969 + 0.433884i 0 1.94585 − 1.32666i
541.1 0.955573 + 0.294755i 0 0.826239 + 0.563320i −3.24239 + 0.488712i 0 −0.296027 − 2.62914i 0.623490 + 0.781831i 0 −3.24239 − 0.488712i
541.2 0.955573 + 0.294755i 0 0.826239 + 0.563320i 1.14747 − 0.172954i 0 2.36688 + 1.18233i 0.623490 + 0.781831i 0 1.14747 + 0.172954i
613.1 0.955573 − 0.294755i 0 0.826239 − 0.563320i −3.24239 − 0.488712i 0 −0.296027 + 2.62914i 0.623490 − 0.781831i 0 −3.24239 + 0.488712i
613.2 0.955573 − 0.294755i 0 0.826239 − 0.563320i 1.14747 + 0.172954i 0 2.36688 − 1.18233i 0.623490 − 0.781831i 0 1.14747 − 0.172954i
739.1 0.826239 + 0.563320i 0 0.365341 + 0.930874i −2.66126 + 0.820891i 0 0.800630 − 2.52170i −0.222521 + 0.974928i 0 −2.66126 − 0.820891i
739.2 0.826239 + 0.563320i 0 0.365341 + 0.930874i 2.26109 − 0.697454i 0 −2.27170 − 1.35623i −0.222521 + 0.974928i 0 2.26109 + 0.697454i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.2
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.z.c 24
3.b odd 2 1 294.2.m.a ✓ 24
49.g even 21 1 inner 882.2.z.c 24
147.n odd 42 1 294.2.m.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.2.m.a ✓ 24 3.b odd 2 1
294.2.m.a ✓ 24 147.n odd 42 1
882.2.z.c 24 1.a even 1 1 trivial
882.2.z.c 24 49.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + T_{5}^{23} - 19 T_{5}^{22} + 26 T_{5}^{21} + 313 T_{5}^{20} - 172 T_{5}^{19} + \cdots + 5340721 \) acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\). Copy content Toggle raw display