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Properties

Label 882.2.z.a
Level $882$
Weight $2$
Character orbit 882.z
Analytic conductor $7.043$
Analytic rank $0$
Dimension $24$
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: \(24\)
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} + 2 q^{4} - q^{5} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} + 2 q^{4} - q^{5} + 4 q^{8} + q^{10} - 8 q^{11} - 2 q^{13} + 2 q^{16} - 9 q^{17} - 4 q^{19} - 5 q^{20} + 5 q^{22} + 38 q^{23} - 19 q^{25} + 20 q^{26} - 14 q^{28} + 14 q^{29} + 4 q^{31} - 2 q^{32} + 17 q^{34} - 42 q^{35} + 7 q^{37} + 18 q^{38} - 13 q^{40} - 2 q^{41} + 4 q^{43} + 6 q^{44} + 11 q^{46} + 8 q^{47} - 42 q^{49} + 4 q^{50} + 15 q^{52} + 10 q^{53} + 33 q^{55} + 7 q^{56} - 28 q^{58} - 14 q^{59} + 7 q^{61} + 8 q^{62} - 4 q^{64} - 23 q^{65} + 23 q^{67} - 9 q^{68} + 28 q^{70} - 36 q^{71} + 15 q^{73} + 14 q^{74} + q^{76} + 28 q^{77} + 38 q^{79} - 8 q^{80} - q^{82} - 27 q^{83} - 47 q^{85} - 26 q^{86} - 6 q^{88} + 121 q^{89} + 21 q^{91} + q^{92} - q^{94} - 44 q^{95} + 14 q^{97} + 21 q^{98}+O(q^{100}) \) 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 −1.32658 − 0.409195i 0 0.662652 − 2.56142i 0.222521 + 0.974928i 0 1.32658 − 0.409195i
37.2 −0.826239 + 0.563320i 0 0.365341 − 0.930874i 3.12319 + 0.963376i 0 −2.56883 + 0.633346i 0.222521 + 0.974928i 0 −3.12319 + 0.963376i
109.1 0.733052 − 0.680173i 0 0.0747301 − 0.997204i −0.380567 + 0.969668i 0 −2.32432 + 1.26394i −0.623490 − 0.781831i 0 0.380567 + 0.969668i
109.2 0.733052 − 0.680173i 0 0.0747301 − 0.997204i 1.49873 − 3.81871i 0 −0.683647 − 2.55590i −0.623490 − 0.781831i 0 −1.49873 − 3.81871i
163.1 −0.365341 − 0.930874i 0 −0.733052 + 0.680173i −2.56909 + 1.75158i 0 1.92409 − 1.81601i 0.900969 + 0.433884i 0 2.56909 + 1.75158i
163.2 −0.365341 − 0.930874i 0 −0.733052 + 0.680173i 0.549960 − 0.374956i 0 −0.0527428 + 2.64523i 0.900969 + 0.433884i 0 −0.549960 − 0.374956i
235.1 0.988831 + 0.149042i 0 0.955573 + 0.294755i 0.0829648 + 1.10709i 0 −2.15873 − 1.52966i 0.900969 + 0.433884i 0 −0.0829648 + 1.10709i
235.2 0.988831 + 0.149042i 0 0.955573 + 0.294755i 0.189189 + 2.52456i 0 2.49030 + 0.893546i 0.900969 + 0.433884i 0 −0.189189 + 2.52456i
289.1 0.988831 − 0.149042i 0 0.955573 − 0.294755i 0.0829648 − 1.10709i 0 −2.15873 + 1.52966i 0.900969 − 0.433884i 0 −0.0829648 − 1.10709i
289.2 0.988831 − 0.149042i 0 0.955573 − 0.294755i 0.189189 − 2.52456i 0 2.49030 − 0.893546i 0.900969 − 0.433884i 0 −0.189189 − 2.52456i
415.1 −0.0747301 + 0.997204i 0 −0.988831 − 0.149042i −1.71019 − 1.58683i 0 1.81916 + 1.92111i 0.222521 − 0.974928i 0 1.71019 − 1.58683i
415.2 −0.0747301 + 0.997204i 0 −0.988831 − 0.149042i 1.21552 + 1.12784i 0 0.254582 − 2.63347i 0.222521 − 0.974928i 0 −1.21552 + 1.12784i
487.1 −0.365341 + 0.930874i 0 −0.733052 − 0.680173i −2.56909 − 1.75158i 0 1.92409 + 1.81601i 0.900969 − 0.433884i 0 2.56909 − 1.75158i
487.2 −0.365341 + 0.930874i 0 −0.733052 − 0.680173i 0.549960 + 0.374956i 0 −0.0527428 − 2.64523i 0.900969 − 0.433884i 0 −0.549960 + 0.374956i
541.1 −0.955573 − 0.294755i 0 0.826239 + 0.563320i −1.22261 + 0.184279i 0 0.792174 + 2.52437i −0.623490 − 0.781831i 0 1.22261 + 0.184279i
541.2 −0.955573 − 0.294755i 0 0.826239 + 0.563320i 0.0494904 − 0.00745948i 0 −0.154679 − 2.64123i −0.623490 − 0.781831i 0 −0.0494904 − 0.00745948i
613.1 −0.955573 + 0.294755i 0 0.826239 − 0.563320i −1.22261 − 0.184279i 0 0.792174 − 2.52437i −0.623490 + 0.781831i 0 1.22261 − 0.184279i
613.2 −0.955573 + 0.294755i 0 0.826239 − 0.563320i 0.0494904 + 0.00745948i 0 −0.154679 + 2.64123i −0.623490 + 0.781831i 0 −0.0494904 + 0.00745948i
739.1 −0.826239 − 0.563320i 0 0.365341 + 0.930874i −1.32658 + 0.409195i 0 0.662652 + 2.56142i 0.222521 − 0.974928i 0 1.32658 + 0.409195i
739.2 −0.826239 − 0.563320i 0 0.365341 + 0.930874i 3.12319 − 0.963376i 0 −2.56883 − 0.633346i 0.222521 − 0.974928i 0 −3.12319 − 0.963376i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.2
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.a 24
3.b odd 2 1 294.2.m.b ✓ 24
49.g even 21 1 inner 882.2.z.a 24
147.n odd 42 1 294.2.m.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.2.m.b ✓ 24 3.b odd 2 1
294.2.m.b ✓ 24 147.n odd 42 1
882.2.z.a 24 1.a even 1 1 trivial
882.2.z.a 24 49.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + T_{5}^{23} + 5 T_{5}^{22} + 28 T_{5}^{21} + 35 T_{5}^{20} - 476 T_{5}^{19} - 741 T_{5}^{18} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\). Copy content Toggle raw display