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Properties

Label 882.2.z.d
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 98)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} + 2 q^{4} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} + 2 q^{4} - 4 q^{8} + 7 q^{11} + 14 q^{13} + 2 q^{16} + 7 q^{17} + 7 q^{20} - 7 q^{22} + 21 q^{23} + 4 q^{25} + 7 q^{26} - 14 q^{28} + 11 q^{29} + 28 q^{31} + 2 q^{32} + 7 q^{34} - 21 q^{35} - 24 q^{37} + 7 q^{38} + 14 q^{40} - 28 q^{41} + 10 q^{43} - 21 q^{44} + 42 q^{46} + 70 q^{47} + 84 q^{49} - 8 q^{50} - 7 q^{52} - 26 q^{53} + 56 q^{55} - 21 q^{56} - 30 q^{58} + 7 q^{59} + 14 q^{61} - 28 q^{62} - 4 q^{64} - 36 q^{67} + 14 q^{68} + 14 q^{70} - 7 q^{73} + 11 q^{74} + 91 q^{77} - 26 q^{79} + 14 q^{82} + 7 q^{83} + 49 q^{85} + 16 q^{86} - 21 q^{88} + 56 q^{89} + 7 q^{91} - 21 q^{92} - 35 q^{94} + 14 q^{95} - 126 q^{97} + 56 q^{98}+O(q^{100}) \) 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 −1.15218 − 0.355400i 0 2.64164 + 0.147471i −0.222521 − 0.974928i 0 −1.15218 + 0.355400i
37.2 0.826239 − 0.563320i 0 0.365341 − 0.930874i 0.596780 + 0.184082i 0 −2.13962 − 1.55628i −0.222521 − 0.974928i 0 0.596780 − 0.184082i
109.1 −0.733052 + 0.680173i 0 0.0747301 − 0.997204i −0.459870 + 1.17173i 0 −1.68832 + 2.03705i 0.623490 + 0.781831i 0 −0.459870 − 1.17173i
109.2 −0.733052 + 0.680173i 0 0.0747301 − 0.997204i 1.10349 − 2.81164i 0 1.42353 − 2.23015i 0.623490 + 0.781831i 0 1.10349 + 2.81164i
163.1 0.365341 + 0.930874i 0 −0.733052 + 0.680173i −1.52046 + 1.03663i 0 2.03934 − 1.68555i −0.900969 − 0.433884i 0 −1.52046 − 1.03663i
163.2 0.365341 + 0.930874i 0 −0.733052 + 0.680173i 1.09780 − 0.748464i 0 2.60370 + 0.469838i −0.900969 − 0.433884i 0 1.09780 + 0.748464i
235.1 −0.988831 − 0.149042i 0 0.955573 + 0.294755i −0.169008 − 2.25526i 0 2.40601 − 1.10050i −0.900969 − 0.433884i 0 −0.169008 + 2.25526i
235.2 −0.988831 − 0.149042i 0 0.955573 + 0.294755i 0.0807922 + 1.07810i 0 −2.64324 − 0.115218i −0.900969 − 0.433884i 0 0.0807922 − 1.07810i
289.1 −0.988831 + 0.149042i 0 0.955573 − 0.294755i −0.169008 + 2.25526i 0 2.40601 + 1.10050i −0.900969 + 0.433884i 0 −0.169008 − 2.25526i
289.2 −0.988831 + 0.149042i 0 0.955573 − 0.294755i 0.0807922 − 1.07810i 0 −2.64324 + 0.115218i −0.900969 + 0.433884i 0 0.0807922 + 1.07810i
415.1 0.0747301 − 0.997204i 0 −0.988831 − 0.149042i −2.80532 − 2.60296i 0 −2.61464 + 0.404557i −0.222521 + 0.974928i 0 −2.80532 + 2.60296i
415.2 0.0747301 − 0.997204i 0 −0.988831 − 0.149042i 0.144243 + 0.133838i 0 2.44775 + 1.00425i −0.222521 + 0.974928i 0 0.144243 − 0.133838i
487.1 0.365341 − 0.930874i 0 −0.733052 − 0.680173i −1.52046 − 1.03663i 0 2.03934 + 1.68555i −0.900969 + 0.433884i 0 −1.52046 + 1.03663i
487.2 0.365341 − 0.930874i 0 −0.733052 − 0.680173i 1.09780 + 0.748464i 0 2.60370 − 0.469838i −0.900969 + 0.433884i 0 1.09780 − 0.748464i
541.1 0.955573 + 0.294755i 0 0.826239 + 0.563320i −0.926902 + 0.139708i 0 −2.29875 + 1.30987i 0.623490 + 0.781831i 0 −0.926902 − 0.139708i
541.2 0.955573 + 0.294755i 0 0.826239 + 0.563320i 4.01065 − 0.604508i 0 −2.17740 − 1.50297i 0.623490 + 0.781831i 0 4.01065 + 0.604508i
613.1 0.955573 − 0.294755i 0 0.826239 − 0.563320i −0.926902 − 0.139708i 0 −2.29875 − 1.30987i 0.623490 − 0.781831i 0 −0.926902 + 0.139708i
613.2 0.955573 − 0.294755i 0 0.826239 − 0.563320i 4.01065 + 0.604508i 0 −2.17740 + 1.50297i 0.623490 − 0.781831i 0 4.01065 − 0.604508i
739.1 0.826239 + 0.563320i 0 0.365341 + 0.930874i −1.15218 + 0.355400i 0 2.64164 − 0.147471i −0.222521 + 0.974928i 0 −1.15218 − 0.355400i
739.2 0.826239 + 0.563320i 0 0.365341 + 0.930874i 0.596780 − 0.184082i 0 −2.13962 + 1.55628i −0.222521 + 0.974928i 0 0.596780 + 0.184082i
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.d 24
3.b odd 2 1 98.2.g.a ✓ 24
12.b even 2 1 784.2.bg.a 24
21.c even 2 1 686.2.g.b 24
21.g even 6 1 686.2.e.e 24
21.g even 6 1 686.2.g.c 24
21.h odd 6 1 686.2.e.f 24
21.h odd 6 1 686.2.g.a 24
49.g even 21 1 inner 882.2.z.d 24
147.k even 14 1 686.2.g.c 24
147.l odd 14 1 686.2.g.a 24
147.n odd 42 1 98.2.g.a ✓ 24
147.n odd 42 1 686.2.e.f 24
147.n odd 42 1 4802.2.a.i 12
147.o even 42 1 686.2.e.e 24
147.o even 42 1 686.2.g.b 24
147.o even 42 1 4802.2.a.k 12
588.bb even 42 1 784.2.bg.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.g.a ✓ 24 3.b odd 2 1
98.2.g.a ✓ 24 147.n odd 42 1
686.2.e.e 24 21.g even 6 1
686.2.e.e 24 147.o even 42 1
686.2.e.f 24 21.h odd 6 1
686.2.e.f 24 147.n odd 42 1
686.2.g.a 24 21.h odd 6 1
686.2.g.a 24 147.l odd 14 1
686.2.g.b 24 21.c even 2 1
686.2.g.b 24 147.o even 42 1
686.2.g.c 24 21.g even 6 1
686.2.g.c 24 147.k even 14 1
784.2.bg.a 24 12.b even 2 1
784.2.bg.a 24 588.bb even 42 1
882.2.z.d 24 1.a even 1 1 trivial
882.2.z.d 24 49.g even 21 1 inner
4802.2.a.i 12 147.n odd 42 1
4802.2.a.k 12 147.o even 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 7 T_{5}^{22} - 49 T_{5}^{21} + 7 T_{5}^{20} - 343 T_{5}^{19} + 469 T_{5}^{18} + 3871 T_{5}^{17} + \cdots + 2401 \) acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\). Copy content Toggle raw display