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Properties

Label 882.2.z.f
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 3 q^{2} + 3 q^{4} - 2 q^{5} - 5 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 3 q^{2} + 3 q^{4} - 2 q^{5} - 5 q^{7} - 6 q^{8} - 2 q^{10} - 9 q^{11} + 10 q^{13} + 4 q^{14} + 3 q^{16} - 7 q^{17} + 16 q^{19} - 3 q^{20} + 4 q^{22} - 42 q^{23} - 23 q^{25} + 16 q^{26} + 15 q^{28} + 12 q^{29} - 3 q^{31} + 3 q^{32} - 21 q^{34} - 26 q^{35} - 25 q^{37} + 2 q^{38} - 2 q^{40} + 28 q^{41} + 4 q^{43} + 19 q^{44} - 7 q^{46} + 4 q^{47} - 25 q^{49} + 74 q^{50} + 9 q^{52} + 37 q^{53} - 54 q^{55} + 9 q^{56} + 8 q^{58} - 23 q^{59} - 79 q^{61} + 34 q^{62} - 6 q^{64} - 55 q^{65} + 19 q^{67} + 7 q^{68} + 18 q^{70} + 34 q^{71} - 15 q^{73} + 24 q^{74} + 3 q^{76} - 129 q^{77} + 11 q^{79} + 5 q^{80} + 49 q^{82} + 44 q^{83} - 77 q^{85} - 72 q^{86} + 19 q^{88} - 39 q^{89} - 43 q^{91} - 7 q^{92} + 25 q^{94} + 36 q^{95} - 8 q^{97} + 48 q^{98}+O(q^{100}) \) 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 −3.05947 − 0.943720i 0 −1.76838 + 1.96795i −0.222521 − 0.974928i 0 −3.05947 + 0.943720i
37.2 0.826239 − 0.563320i 0 0.365341 − 0.930874i 0.203744 + 0.0628466i 0 −1.93486 − 1.80452i −0.222521 − 0.974928i 0 0.203744 − 0.0628466i
37.3 0.826239 − 0.563320i 0 0.365341 − 0.930874i 2.59837 + 0.801491i 0 2.59324 − 0.524514i −0.222521 − 0.974928i 0 2.59837 − 0.801491i
109.1 −0.733052 + 0.680173i 0 0.0747301 − 0.997204i −1.35994 + 3.46506i 0 −2.55254 + 0.696071i 0.623490 + 0.781831i 0 −1.35994 − 3.46506i
109.2 −0.733052 + 0.680173i 0 0.0747301 − 0.997204i −0.00944986 + 0.0240779i 0 2.05870 − 1.66185i 0.623490 + 0.781831i 0 −0.00944986 − 0.0240779i
109.3 −0.733052 + 0.680173i 0 0.0747301 − 0.997204i 1.05865 − 2.69740i 0 0.494639 + 2.59910i 0.623490 + 0.781831i 0 1.05865 + 2.69740i
163.1 0.365341 + 0.930874i 0 −0.733052 + 0.680173i −3.22240 + 2.19700i 0 0.294655 + 2.62929i −0.900969 − 0.433884i 0 −3.22240 − 2.19700i
163.2 0.365341 + 0.930874i 0 −0.733052 + 0.680173i 0.172293 − 0.117467i 0 −0.0217470 − 2.64566i −0.900969 − 0.433884i 0 0.172293 + 0.117467i
163.3 0.365341 + 0.930874i 0 −0.733052 + 0.680173i 1.01252 − 0.690323i 0 −2.64531 + 0.0485591i −0.900969 − 0.433884i 0 1.01252 + 0.690323i
235.1 −0.988831 − 0.149042i 0 0.955573 + 0.294755i −0.303595 − 4.05119i 0 −1.43085 − 2.22546i −0.900969 − 0.433884i 0 −0.303595 + 4.05119i
235.2 −0.988831 − 0.149042i 0 0.955573 + 0.294755i 0.119906 + 1.60003i 0 0.120308 − 2.64301i −0.900969 − 0.433884i 0 0.119906 − 1.60003i
235.3 −0.988831 − 0.149042i 0 0.955573 + 0.294755i 0.251778 + 3.35975i 0 0.0241055 + 2.64564i −0.900969 − 0.433884i 0 0.251778 − 3.35975i
289.1 −0.988831 + 0.149042i 0 0.955573 − 0.294755i −0.303595 + 4.05119i 0 −1.43085 + 2.22546i −0.900969 + 0.433884i 0 −0.303595 − 4.05119i
289.2 −0.988831 + 0.149042i 0 0.955573 − 0.294755i 0.119906 − 1.60003i 0 0.120308 + 2.64301i −0.900969 + 0.433884i 0 0.119906 + 1.60003i
289.3 −0.988831 + 0.149042i 0 0.955573 − 0.294755i 0.251778 − 3.35975i 0 0.0241055 − 2.64564i −0.900969 + 0.433884i 0 0.251778 + 3.35975i
415.1 0.0747301 − 0.997204i 0 −0.988831 − 0.149042i −0.892287 − 0.827921i 0 0.278782 + 2.63102i −0.222521 + 0.974928i 0 −0.892287 + 0.827921i
415.2 0.0747301 − 0.997204i 0 −0.988831 − 0.149042i 0.0373053 + 0.0346142i 0 0.835928 − 2.51022i −0.222521 + 0.974928i 0 0.0373053 − 0.0346142i
415.3 0.0747301 − 0.997204i 0 −0.988831 − 0.149042i 3.03776 + 2.81863i 0 −2.64177 + 0.145096i −0.222521 + 0.974928i 0 3.03776 − 2.81863i
487.1 0.365341 − 0.930874i 0 −0.733052 − 0.680173i −3.22240 − 2.19700i 0 0.294655 − 2.62929i −0.900969 + 0.433884i 0 −3.22240 + 2.19700i
487.2 0.365341 − 0.930874i 0 −0.733052 − 0.680173i 0.172293 + 0.117467i 0 −0.0217470 + 2.64566i −0.900969 + 0.433884i 0 0.172293 − 0.117467i
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.f 36
3.b odd 2 1 294.2.m.c ✓ 36
49.g even 21 1 inner 882.2.z.f 36
147.n odd 42 1 294.2.m.c ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.2.m.c ✓ 36 3.b odd 2 1
294.2.m.c ✓ 36 147.n odd 42 1
882.2.z.f 36 1.a even 1 1 trivial
882.2.z.f 36 49.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{36} + 2 T_{5}^{35} + 6 T_{5}^{34} + 44 T_{5}^{33} - 22 T_{5}^{32} + 613 T_{5}^{31} + 3398 T_{5}^{30} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\). Copy content Toggle raw display