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Properties

Label 369.3.c.a
Level $369$
Weight $3$
Character orbit 369.c
Analytic conductor $10.055$
Analytic rank $0$
Dimension $28$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [369,3,Mod(206,369)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(369, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("369.206");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 369 = 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 369.c (of order \(2\), degree \(1\), 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: \(10.0545217549\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 56 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 56 q^{4} - 8 q^{10} - 24 q^{13} + 192 q^{16} + 40 q^{19} - 104 q^{22} - 228 q^{25} + 120 q^{28} - 4 q^{31} - 176 q^{34} + 108 q^{37} + 152 q^{40} - 36 q^{43} - 24 q^{46} + 308 q^{49} - 64 q^{52} + 104 q^{55} + 56 q^{58} - 252 q^{61} - 904 q^{64} + 152 q^{67} + 752 q^{70} + 36 q^{73} - 576 q^{76} + 232 q^{79} + 16 q^{85} + 296 q^{88} - 464 q^{91} + 744 q^{94} - 144 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
206.1 − 3.90838i 0 −11.2755 − 8.78613i 0 −9.73763 28.4352i 0 −34.3396
206.2 − 3.75382i 0 −10.0912 2.64086i 0 −3.17453 22.8653i 0 9.91333
206.3 − 3.72223i 0 −9.85500 5.69700i 0 11.5124 21.7936i 0 21.2055
206.4 − 3.04162i 0 −5.25147 − 4.70569i 0 5.80931 3.80650i 0 −14.3129
206.5 − 2.86833i 0 −4.22733 2.97587i 0 −6.41898 0.652052i 0 8.53579
206.6 − 2.48154i 0 −2.15802 0.286430i 0 2.54116 − 4.57094i 0 0.710786
206.7 − 2.32717i 0 −1.41574 8.87338i 0 −7.43547 − 6.01403i 0 20.6499
206.8 − 1.94697i 0 0.209308 − 7.17454i 0 −10.8232 − 8.19540i 0 −13.9686
206.9 − 1.87789i 0 0.473512 3.17491i 0 7.32780 − 8.40078i 0 5.96216
206.10 − 1.67433i 0 1.19662 − 9.02341i 0 6.60620 − 8.70085i 0 −15.1082
206.11 − 0.851883i 0 3.27430 − 1.38107i 0 −8.21045 − 6.19685i 0 −1.17651
206.12 − 0.753779i 0 3.43182 9.10307i 0 2.07059 − 5.60195i 0 6.86170
206.13 − 0.553693i 0 3.69342 1.58943i 0 −3.10710 − 4.25979i 0 0.880056
206.14 − 0.0691318i 0 3.99522 2.69819i 0 13.0399 − 0.552724i 0 0.186530
206.15 0.0691318i 0 3.99522 − 2.69819i 0 13.0399 0.552724i 0 0.186530
206.16 0.553693i 0 3.69342 − 1.58943i 0 −3.10710 4.25979i 0 0.880056
206.17 0.753779i 0 3.43182 − 9.10307i 0 2.07059 5.60195i 0 6.86170
206.18 0.851883i 0 3.27430 1.38107i 0 −8.21045 6.19685i 0 −1.17651
206.19 1.67433i 0 1.19662 9.02341i 0 6.60620 8.70085i 0 −15.1082
206.20 1.87789i 0 0.473512 − 3.17491i 0 7.32780 8.40078i 0 5.96216
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 206.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 369.3.c.a ✓ 28
3.b odd 2 1 inner 369.3.c.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
369.3.c.a ✓ 28 1.a even 1 1 trivial
369.3.c.a ✓ 28 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(369, [\chi])\).