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Properties

Label 882.2.z.h
Level $882$
Weight $2$
Character orbit 882.z
Analytic conductor $7.043$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [882,2,Mod(37,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
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

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: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(5\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 5 q^{2} + 5 q^{4} + 3 q^{5} + q^{7} - 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 5 q^{2} + 5 q^{4} + 3 q^{5} + q^{7} - 10 q^{8} + 3 q^{10} - q^{11} - 4 q^{13} + 4 q^{14} + 5 q^{16} + 6 q^{17} - 12 q^{19} - 6 q^{20} + 9 q^{22} - 2 q^{23} + 6 q^{25} + 2 q^{26} - 5 q^{28} - 6 q^{29} - 7 q^{31} + 5 q^{32} - 12 q^{34} + 40 q^{35} + 6 q^{37} - 26 q^{38} - 11 q^{40} + 12 q^{43} - 15 q^{44} - 2 q^{46} - 44 q^{47} - 43 q^{49} + 86 q^{50} + 2 q^{52} + 13 q^{53} - 45 q^{55} + 8 q^{56} - 18 q^{58} + 53 q^{59} - 32 q^{61} - 14 q^{62} - 10 q^{64} + 2 q^{65} - 4 q^{67} - 36 q^{68} - 8 q^{70} - 42 q^{71} - 40 q^{73} - 8 q^{74} - 4 q^{76} - 55 q^{77} - 3 q^{79} - 11 q^{80} + 17 q^{83} + 16 q^{85} + 64 q^{86} - 15 q^{88} + 50 q^{89} + 72 q^{91} + 4 q^{92} + 26 q^{94} - 26 q^{95} - 2 q^{97} - 53 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 0.826239 − 0.563320i 0 0.365341 − 0.930874i −2.46645 − 0.760798i 0 1.96507 − 1.77158i −0.222521 − 0.974928i 0 −2.46645 + 0.760798i
37.2 0.826239 − 0.563320i 0 0.365341 − 0.930874i −1.42188 − 0.438591i 0 −2.62709 + 0.313696i −0.222521 − 0.974928i 0 −1.42188 + 0.438591i
37.3 0.826239 − 0.563320i 0 0.365341 − 0.930874i −0.563769 − 0.173900i 0 −0.146622 + 2.64169i −0.222521 − 0.974928i 0 −0.563769 + 0.173900i
37.4 0.826239 − 0.563320i 0 0.365341 − 0.930874i 2.57241 + 0.793483i 0 1.29888 + 2.30497i −0.222521 − 0.974928i 0 2.57241 − 0.793483i
37.5 0.826239 − 0.563320i 0 0.365341 − 0.930874i 3.64801 + 1.12526i 0 0.465332 − 2.60451i −0.222521 − 0.974928i 0 3.64801 − 1.12526i
109.1 −0.733052 + 0.680173i 0 0.0747301 − 0.997204i −1.08013 + 2.75212i 0 2.52768 − 0.781556i 0.623490 + 0.781831i 0 −1.08013 − 2.75212i
109.2 −0.733052 + 0.680173i 0 0.0747301 − 0.997204i −0.455674 + 1.16104i 0 1.96995 − 1.76616i 0.623490 + 0.781831i 0 −0.455674 − 1.16104i
109.3 −0.733052 + 0.680173i 0 0.0747301 − 0.997204i −0.123680 + 0.315131i 0 −2.62269 − 0.348571i 0.623490 + 0.781831i 0 −0.123680 − 0.315131i
109.4 −0.733052 + 0.680173i 0 0.0747301 − 0.997204i 0.253636 − 0.646253i 0 −0.581112 + 2.58114i 0.623490 + 0.781831i 0 0.253636 + 0.646253i
109.5 −0.733052 + 0.680173i 0 0.0747301 − 0.997204i 1.43831 − 3.66475i 0 −0.928485 − 2.47748i 0.623490 + 0.781831i 0 1.43831 + 3.66475i
163.1 0.365341 + 0.930874i 0 −0.733052 + 0.680173i −1.77049 + 1.20710i 0 0.331365 − 2.62492i −0.900969 − 0.433884i 0 −1.77049 − 1.20710i
163.2 0.365341 + 0.930874i 0 −0.733052 + 0.680173i −0.773406 + 0.527299i 0 2.36057 + 1.19488i −0.900969 − 0.433884i 0 −0.773406 − 0.527299i
163.3 0.365341 + 0.930874i 0 −0.733052 + 0.680173i 0.194435 − 0.132563i 0 −2.59859 − 0.497323i −0.900969 − 0.433884i 0 0.194435 + 0.132563i
163.4 0.365341 + 0.930874i 0 −0.733052 + 0.680173i 2.07899 − 1.41743i 0 1.17308 − 2.37147i −0.900969 − 0.433884i 0 2.07899 + 1.41743i
163.5 0.365341 + 0.930874i 0 −0.733052 + 0.680173i 3.55697 − 2.42510i 0 −0.440185 + 2.60888i −0.900969 − 0.433884i 0 3.55697 + 2.42510i
235.1 −0.988831 − 0.149042i 0 0.955573 + 0.294755i −0.257415 − 3.43497i 0 −0.250764 + 2.63384i −0.900969 − 0.433884i 0 −0.257415 + 3.43497i
235.2 −0.988831 − 0.149042i 0 0.955573 + 0.294755i −0.163784 − 2.18555i 0 1.53324 − 2.15620i −0.900969 − 0.433884i 0 −0.163784 + 2.18555i
235.3 −0.988831 − 0.149042i 0 0.955573 + 0.294755i 0.0935118 + 1.24783i 0 −2.64522 − 0.0531479i −0.900969 − 0.433884i 0 0.0935118 − 1.24783i
235.4 −0.988831 − 0.149042i 0 0.955573 + 0.294755i 0.126871 + 1.69298i 0 −1.20000 + 2.35797i −0.900969 − 0.433884i 0 0.126871 − 1.69298i
235.5 −0.988831 − 0.149042i 0 0.955573 + 0.294755i 0.295673 + 3.94548i 0 2.63747 + 0.209149i −0.900969 − 0.433884i 0 0.295673 − 3.94548i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.5
Significant digits:
Format:

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.h yes 60
3.b odd 2 1 882.2.z.g ✓ 60
49.g even 21 1 inner 882.2.z.h yes 60
147.n odd 42 1 882.2.z.g ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.z.g ✓ 60 3.b odd 2 1
882.2.z.g ✓ 60 147.n odd 42 1
882.2.z.h yes 60 1.a even 1 1 trivial
882.2.z.h yes 60 49.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{60} - 3 T_{5}^{59} - 11 T_{5}^{58} + 88 T_{5}^{57} - 105 T_{5}^{56} - 127 T_{5}^{55} + \cdots + 40\!\cdots\!49 \) acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\). Copy content Toggle raw display