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Properties

Label 882.2.z.e
Level $882$
Weight $2$
Character orbit 882.z
Analytic conductor $7.043$
Analytic rank $0$
Dimension $36$
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: \(36\)
Relative dimension: \(3\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 3 q^{2} + 3 q^{4} - q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 3 q^{2} + 3 q^{4} - q^{7} + 6 q^{8} - 13 q^{11} + 2 q^{13} + 4 q^{14} + 3 q^{16} - q^{17} - 2 q^{19} - 7 q^{20} - 12 q^{22} - 42 q^{23} + 33 q^{25} - 34 q^{26} - 9 q^{28} + 4 q^{29} - q^{31} - 3 q^{32} + 5 q^{34} + 4 q^{35} + 45 q^{37} + 30 q^{38} + 58 q^{41} + q^{44} - 7 q^{46} + 68 q^{47} - 29 q^{49} - 74 q^{50} - 15 q^{52} + 19 q^{53} + 48 q^{55} + 15 q^{56} - 40 q^{58} + 123 q^{59} - 13 q^{61} - 2 q^{62} - 6 q^{64} - 9 q^{65} - 7 q^{67} - q^{68} - 16 q^{70} + 38 q^{71} + 65 q^{73} - 38 q^{74} - 17 q^{76} - 25 q^{77} - 71 q^{79} - 7 q^{80} + 29 q^{82} - 8 q^{83} + 73 q^{85} + 14 q^{86} - q^{88} - 45 q^{89} - 35 q^{91} + 7 q^{92} + 9 q^{94} + 144 q^{95} + 56 q^{97} - 24 q^{98}+O(q^{100}) \) 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 −3.02490 − 0.933059i 0 −2.44171 − 1.01884i 0.222521 + 0.974928i 0 3.02490 − 0.933059i
37.2 −0.826239 + 0.563320i 0 0.365341 − 0.930874i 0.409482 + 0.126309i 0 2.34610 + 1.22304i 0.222521 + 0.974928i 0 −0.409482 + 0.126309i
37.3 −0.826239 + 0.563320i 0 0.365341 − 0.930874i 4.26921 + 1.31688i 0 −1.15057 − 2.38248i 0.222521 + 0.974928i 0 −4.26921 + 1.31688i
109.1 0.733052 − 0.680173i 0 0.0747301 − 0.997204i −1.26635 + 3.22660i 0 −1.11625 + 2.39875i −0.623490 − 0.781831i 0 1.26635 + 3.22660i
109.2 0.733052 − 0.680173i 0 0.0747301 − 0.997204i 0.513031 − 1.30718i 0 −0.0547842 − 2.64518i −0.623490 − 0.781831i 0 −0.513031 − 1.30718i
109.3 0.733052 − 0.680173i 0 0.0747301 − 0.997204i 1.17326 − 2.98942i 0 1.68654 + 2.03853i −0.623490 − 0.781831i 0 −1.17326 − 2.98942i
163.1 −0.365341 − 0.930874i 0 −0.733052 + 0.680173i −2.54357 + 1.73418i 0 −0.146091 − 2.64171i 0.900969 + 0.433884i 0 2.54357 + 1.73418i
163.2 −0.365341 − 0.930874i 0 −0.733052 + 0.680173i −0.374506 + 0.255334i 0 1.59832 + 2.10840i 0.900969 + 0.433884i 0 0.374506 + 0.255334i
163.3 −0.365341 − 0.930874i 0 −0.733052 + 0.680173i 2.53296 − 1.72694i 0 −2.53425 + 0.759993i 0.900969 + 0.433884i 0 −2.53296 − 1.72694i
235.1 0.988831 + 0.149042i 0 0.955573 + 0.294755i −0.0491854 − 0.656334i 0 2.17925 + 1.50029i 0.900969 + 0.433884i 0 0.0491854 − 0.656334i
235.2 0.988831 + 0.149042i 0 0.955573 + 0.294755i −0.0260923 − 0.348178i 0 −0.245125 − 2.63437i 0.900969 + 0.433884i 0 0.0260923 − 0.348178i
235.3 0.988831 + 0.149042i 0 0.955573 + 0.294755i 0.292828 + 3.90752i 0 −2.59908 + 0.494735i 0.900969 + 0.433884i 0 −0.292828 + 3.90752i
289.1 0.988831 − 0.149042i 0 0.955573 − 0.294755i −0.0491854 + 0.656334i 0 2.17925 − 1.50029i 0.900969 − 0.433884i 0 0.0491854 + 0.656334i
289.2 0.988831 − 0.149042i 0 0.955573 − 0.294755i −0.0260923 + 0.348178i 0 −0.245125 + 2.63437i 0.900969 − 0.433884i 0 0.0260923 + 0.348178i
289.3 0.988831 − 0.149042i 0 0.955573 − 0.294755i 0.292828 − 3.90752i 0 −2.59908 − 0.494735i 0.900969 − 0.433884i 0 −0.292828 − 3.90752i
415.1 −0.0747301 + 0.997204i 0 −0.988831 − 0.149042i −0.794619 − 0.737299i 0 2.24297 − 1.40324i 0.222521 − 0.974928i 0 0.794619 − 0.737299i
415.2 −0.0747301 + 0.997204i 0 −0.988831 − 0.149042i −0.738585 − 0.685307i 0 −1.26941 + 2.32133i 0.222521 − 0.974928i 0 0.738585 − 0.685307i
415.3 −0.0747301 + 0.997204i 0 −0.988831 − 0.149042i 2.24988 + 2.08758i 0 1.57456 + 2.12621i 0.222521 − 0.974928i 0 −2.24988 + 2.08758i
487.1 −0.365341 + 0.930874i 0 −0.733052 − 0.680173i −2.54357 − 1.73418i 0 −0.146091 + 2.64171i 0.900969 − 0.433884i 0 2.54357 − 1.73418i
487.2 −0.365341 + 0.930874i 0 −0.733052 − 0.680173i −0.374506 − 0.255334i 0 1.59832 − 2.10840i 0.900969 − 0.433884i 0 0.374506 − 0.255334i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.3
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.e 36
3.b odd 2 1 294.2.m.d ✓ 36
49.g even 21 1 inner 882.2.z.e 36
147.n odd 42 1 294.2.m.d ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.2.m.d ✓ 36 3.b odd 2 1
294.2.m.d ✓ 36 147.n odd 42 1
882.2.z.e 36 1.a even 1 1 trivial
882.2.z.e 36 49.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{36} - 24 T_{5}^{34} + 18 T_{5}^{33} + 56 T_{5}^{32} - 235 T_{5}^{31} + 4260 T_{5}^{30} + \cdots + 439279681 \) acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\). Copy content Toggle raw display