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Properties

Label 1428.2.j.b
Level $1428$
Weight $2$
Character orbit 1428.j
Analytic conductor $11.403$
Analytic rank $0$
Dimension $22$
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] = [1428,2,Mod(545,1428)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1428, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1428.545");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1428.j (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: \(11.4026374086\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 4 q^{7} + 4 q^{9} + 4 q^{15} + 22 q^{17} + 12 q^{21} + 30 q^{25} + 6 q^{27} - 4 q^{33} - 22 q^{35} - 8 q^{37} - 6 q^{39} - 16 q^{41} + 4 q^{43} - 26 q^{45} + 16 q^{47} + 14 q^{49} - 22 q^{57} + 34 q^{63} + 20 q^{67} + 14 q^{69} - 44 q^{75} + 6 q^{77} + 28 q^{79} - 12 q^{81} - 16 q^{83} - 8 q^{87} + 36 q^{89} - 14 q^{91} + 4 q^{93} + 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
545.1 0 −1.72378 − 0.169097i 0 2.20743 0 0.199241 − 2.63824i 0 2.94281 + 0.582971i 0
545.2 0 −1.72378 + 0.169097i 0 2.20743 0 0.199241 + 2.63824i 0 2.94281 − 0.582971i 0
545.3 0 −1.64884 − 0.530393i 0 −3.86225 0 −1.03251 − 2.43596i 0 2.43737 + 1.74907i 0
545.4 0 −1.64884 + 0.530393i 0 −3.86225 0 −1.03251 + 2.43596i 0 2.43737 − 1.74907i 0
545.5 0 −1.24278 − 1.20644i 0 −0.899779 0 −2.46934 + 0.949916i 0 0.0889847 + 2.99868i 0
545.6 0 −1.24278 + 1.20644i 0 −0.899779 0 −2.46934 − 0.949916i 0 0.0889847 − 2.99868i 0
545.7 0 −0.900103 − 1.47980i 0 −2.46431 0 2.11638 − 1.58775i 0 −1.37963 + 2.66395i 0
545.8 0 −0.900103 + 1.47980i 0 −2.46431 0 2.11638 + 1.58775i 0 −1.37963 − 2.66395i 0
545.9 0 −0.484970 − 1.66277i 0 4.34156 0 −2.52605 − 0.786802i 0 −2.52961 + 1.61279i 0
545.10 0 −0.484970 + 1.66277i 0 4.34156 0 −2.52605 + 0.786802i 0 −2.52961 − 1.61279i 0
545.11 0 −0.457256 − 1.67060i 0 1.65479 0 1.26399 + 2.32429i 0 −2.58183 + 1.52779i 0
545.12 0 −0.457256 + 1.67060i 0 1.65479 0 1.26399 − 2.32429i 0 −2.58183 − 1.52779i 0
545.13 0 0.658136 − 1.60214i 0 0.893430 0 −1.71290 − 2.01642i 0 −2.13371 − 2.10885i 0
545.14 0 0.658136 + 1.60214i 0 0.893430 0 −1.71290 + 2.01642i 0 −2.13371 + 2.10885i 0
545.15 0 1.17866 − 1.26916i 0 −3.63424 0 2.51056 + 0.834931i 0 −0.221528 − 2.99181i 0
545.16 0 1.17866 + 1.26916i 0 −3.63424 0 2.51056 − 0.834931i 0 −0.221528 + 2.99181i 0
545.17 0 1.34982 − 1.08535i 0 1.75820 0 −0.376360 + 2.61885i 0 0.644009 − 2.93006i 0
545.18 0 1.34982 + 1.08535i 0 1.75820 0 −0.376360 − 2.61885i 0 0.644009 + 2.93006i 0
545.19 0 1.54479 − 0.783333i 0 −1.52109 0 −2.53207 + 0.767202i 0 1.77278 − 2.42018i 0
545.20 0 1.54479 + 0.783333i 0 −1.52109 0 −2.53207 − 0.767202i 0 1.77278 + 2.42018i 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 545.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1428.2.j.b yes 22
3.b odd 2 1 1428.2.j.a ✓ 22
7.b odd 2 1 1428.2.j.a ✓ 22
21.c even 2 1 inner 1428.2.j.b yes 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1428.2.j.a ✓ 22 3.b odd 2 1
1428.2.j.a ✓ 22 7.b odd 2 1
1428.2.j.b yes 22 1.a even 1 1 trivial
1428.2.j.b yes 22 21.c even 2 1 inner