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Properties

Label 882.2.z.b
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
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} + 2 q^{4} + 4 q^{8} + 11 q^{11} - 14 q^{13} + 2 q^{16} + 7 q^{17} - 14 q^{19} + 7 q^{20} + q^{22} + 29 q^{23} - 8 q^{25} + 7 q^{26} + 14 q^{28} - 13 q^{29} - 28 q^{31} - 2 q^{32} - 7 q^{34} + 35 q^{35} + 20 q^{37} - 35 q^{38} + 14 q^{40} - 28 q^{41} + 6 q^{43} - 3 q^{44} + 34 q^{46} - 42 q^{47} + 28 q^{49} - 16 q^{50} - 7 q^{52} + 60 q^{53} - 14 q^{55} - 7 q^{56} + 18 q^{58} - 49 q^{59} - 14 q^{61} - 4 q^{64} + 28 q^{65} + 24 q^{67} + 14 q^{68} - 28 q^{70} - 6 q^{71} - 35 q^{73} + 15 q^{74} - 49 q^{77} + 6 q^{79} + 14 q^{80} - 14 q^{82} + 77 q^{83} - 33 q^{85} + 38 q^{86} + 3 q^{88} - 21 q^{91} + 5 q^{92} - 35 q^{94} - 86 q^{95} + 98 q^{97} - 28 q^{98}+O(q^{100}) \) 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 −3.17919 − 0.980650i 0 −0.0867132 + 2.64433i 0.222521 + 0.974928i 0 3.17919 − 0.980650i
37.2 −0.826239 + 0.563320i 0 0.365341 − 0.930874i 0.427005 + 0.131714i 0 1.60505 − 2.10329i 0.222521 + 0.974928i 0 −0.427005 + 0.131714i
109.1 0.733052 − 0.680173i 0 0.0747301 − 0.997204i −0.904721 + 2.30519i 0 −1.30835 − 2.29961i −0.623490 − 0.781831i 0 0.904721 + 2.30519i
109.2 0.733052 − 0.680173i 0 0.0747301 − 0.997204i −0.578784 + 1.47472i 0 2.58946 + 0.542854i −0.623490 − 0.781831i 0 0.578784 + 1.47472i
163.1 −0.365341 − 0.930874i 0 −0.733052 + 0.680173i −1.87725 + 1.27989i 0 −2.59910 + 0.494638i 0.900969 + 0.433884i 0 1.87725 + 1.27989i
163.2 −0.365341 − 0.930874i 0 −0.733052 + 0.680173i 3.07015 − 2.09319i 0 2.50800 − 0.842585i 0.900969 + 0.433884i 0 −3.07015 − 2.09319i
235.1 0.988831 + 0.149042i 0 0.955573 + 0.294755i −0.269665 − 3.59844i 0 0.415648 − 2.61290i 0.900969 + 0.433884i 0 0.269665 − 3.59844i
235.2 0.988831 + 0.149042i 0 0.955573 + 0.294755i −0.0772188 − 1.03041i 0 1.36748 + 2.26495i 0.900969 + 0.433884i 0 0.0772188 − 1.03041i
289.1 0.988831 − 0.149042i 0 0.955573 − 0.294755i −0.269665 + 3.59844i 0 0.415648 + 2.61290i 0.900969 − 0.433884i 0 0.269665 + 3.59844i
289.2 0.988831 − 0.149042i 0 0.955573 − 0.294755i −0.0772188 + 1.03041i 0 1.36748 − 2.26495i 0.900969 − 0.433884i 0 0.0772188 + 1.03041i
415.1 −0.0747301 + 0.997204i 0 −0.988831 − 0.149042i −0.958118 − 0.889004i 0 −2.43754 + 1.02877i 0.222521 − 0.974928i 0 0.958118 − 0.889004i
415.2 −0.0747301 + 0.997204i 0 −0.988831 − 0.149042i 2.18585 + 2.02817i 0 −2.12971 − 1.56982i 0.222521 − 0.974928i 0 −2.18585 + 2.02817i
487.1 −0.365341 + 0.930874i 0 −0.733052 − 0.680173i −1.87725 − 1.27989i 0 −2.59910 − 0.494638i 0.900969 − 0.433884i 0 1.87725 − 1.27989i
487.2 −0.365341 + 0.930874i 0 −0.733052 − 0.680173i 3.07015 + 2.09319i 0 2.50800 + 0.842585i 0.900969 − 0.433884i 0 −3.07015 + 2.09319i
541.1 −0.955573 − 0.294755i 0 0.826239 + 0.563320i 0.0821245 − 0.0123783i 0 −2.45519 − 0.985929i −0.623490 − 0.781831i 0 −0.0821245 − 0.0123783i
541.2 −0.955573 − 0.294755i 0 0.826239 + 0.563320i 2.07983 − 0.313484i 0 2.53097 − 0.770831i −0.623490 − 0.781831i 0 −2.07983 − 0.313484i
613.1 −0.955573 + 0.294755i 0 0.826239 − 0.563320i 0.0821245 + 0.0123783i 0 −2.45519 + 0.985929i −0.623490 + 0.781831i 0 −0.0821245 + 0.0123783i
613.2 −0.955573 + 0.294755i 0 0.826239 − 0.563320i 2.07983 + 0.313484i 0 2.53097 + 0.770831i −0.623490 + 0.781831i 0 −2.07983 + 0.313484i
739.1 −0.826239 − 0.563320i 0 0.365341 + 0.930874i −3.17919 + 0.980650i 0 −0.0867132 − 2.64433i 0.222521 − 0.974928i 0 3.17919 + 0.980650i
739.2 −0.826239 − 0.563320i 0 0.365341 + 0.930874i 0.427005 − 0.131714i 0 1.60505 + 2.10329i 0.222521 − 0.974928i 0 −0.427005 − 0.131714i
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.b 24
3.b odd 2 1 98.2.g.b ✓ 24
12.b even 2 1 784.2.bg.b 24
21.c even 2 1 686.2.g.e 24
21.g even 6 1 686.2.e.d 24
21.g even 6 1 686.2.g.d 24
21.h odd 6 1 686.2.e.c 24
21.h odd 6 1 686.2.g.f 24
49.g even 21 1 inner 882.2.z.b 24
147.k even 14 1 686.2.g.d 24
147.l odd 14 1 686.2.g.f 24
147.n odd 42 1 98.2.g.b ✓ 24
147.n odd 42 1 686.2.e.c 24
147.n odd 42 1 4802.2.a.o 12
147.o even 42 1 686.2.e.d 24
147.o even 42 1 686.2.g.e 24
147.o even 42 1 4802.2.a.l 12
588.bb even 42 1 784.2.bg.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.g.b ✓ 24 3.b odd 2 1
98.2.g.b ✓ 24 147.n odd 42 1
686.2.e.c 24 21.h odd 6 1
686.2.e.c 24 147.n odd 42 1
686.2.e.d 24 21.g even 6 1
686.2.e.d 24 147.o even 42 1
686.2.g.d 24 21.g even 6 1
686.2.g.d 24 147.k even 14 1
686.2.g.e 24 21.c even 2 1
686.2.g.e 24 147.o even 42 1
686.2.g.f 24 21.h odd 6 1
686.2.g.f 24 147.l odd 14 1
784.2.bg.b 24 12.b even 2 1
784.2.bg.b 24 588.bb even 42 1
882.2.z.b 24 1.a even 1 1 trivial
882.2.z.b 24 49.g even 21 1 inner
4802.2.a.l 12 147.o even 42 1
4802.2.a.o 12 147.n odd 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - T_{5}^{22} + 7 T_{5}^{21} + 7 T_{5}^{20} + 455 T_{5}^{19} + 1597 T_{5}^{18} - 1253 T_{5}^{17} + \cdots + 15625 \) acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\). Copy content Toggle raw display