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Properties

Label 1400.2.bh.e
Level $1400$
Weight $2$
Character orbit 1400.bh
Analytic conductor $11.179$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1400,2,Mod(249,1400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1400, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1400.249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.bh (of order \(6\), degree \(2\), not 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: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{12} q^{3} + ( - \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} + \zeta_{12}^{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{12} q^{3} + ( - \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} + \zeta_{12}^{2} q^{9} + ( - \zeta_{12}^{2} + 1) q^{11} - 3 \zeta_{12}^{3} q^{13} - 2 \zeta_{12} q^{17} - 5 \zeta_{12}^{2} q^{19} + ( - 6 \zeta_{12}^{2} + 2) q^{21} + ( - 7 \zeta_{12}^{3} + 7 \zeta_{12}) q^{23} - 4 \zeta_{12}^{3} q^{27} + 6 q^{29} + (4 \zeta_{12}^{2} - 4) q^{31} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{33} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{37} + ( - 6 \zeta_{12}^{2} + 6) q^{39} - 5 q^{41} + 6 \zeta_{12}^{3} q^{43} + ( - 9 \zeta_{12}^{3} + 9 \zeta_{12}) q^{47} + (8 \zeta_{12}^{2} - 5) q^{49} - 4 \zeta_{12}^{2} q^{51} - 11 \zeta_{12} q^{53} - 10 \zeta_{12}^{3} q^{57} + ( - 8 \zeta_{12}^{2} + 8) q^{59} + 12 \zeta_{12}^{2} q^{61} + ( - 3 \zeta_{12}^{3} + \zeta_{12}) q^{63} - 4 \zeta_{12} q^{67} + 14 q^{69} - 4 q^{71} - 12 \zeta_{12} q^{73} + (2 \zeta_{12}^{3} - 3 \zeta_{12}) q^{77} + 14 \zeta_{12}^{2} q^{79} + ( - 11 \zeta_{12}^{2} + 11) q^{81} - 4 \zeta_{12}^{3} q^{83} + 12 \zeta_{12} q^{87} + 6 \zeta_{12}^{2} q^{89} + (6 \zeta_{12}^{2} - 9) q^{91} + (8 \zeta_{12}^{3} - 8 \zeta_{12}) q^{93} - 6 \zeta_{12}^{3} q^{97} + q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{9} + 2 q^{11} - 10 q^{19} - 4 q^{21} + 24 q^{29} - 8 q^{31} + 12 q^{39} - 20 q^{41} - 4 q^{49} - 8 q^{51} + 16 q^{59} + 24 q^{61} + 56 q^{69} - 16 q^{71} + 28 q^{79} + 22 q^{81} + 12 q^{89} - 24 q^{91} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{12}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
249.1
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
0 −1.73205 − 1.00000i 0 0 0 1.73205 + 2.00000i 0 0.500000 + 0.866025i 0
249.2 0 1.73205 + 1.00000i 0 0 0 −1.73205 − 2.00000i 0 0.500000 + 0.866025i 0
849.1 0 −1.73205 + 1.00000i 0 0 0 1.73205 − 2.00000i 0 0.500000 − 0.866025i 0
849.2 0 1.73205 − 1.00000i 0 0 0 −1.73205 + 2.00000i 0 0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.bh.e 4
5.b even 2 1 inner 1400.2.bh.e 4
5.c odd 4 1 280.2.q.c ✓ 2
5.c odd 4 1 1400.2.q.a 2
7.c even 3 1 inner 1400.2.bh.e 4
15.e even 4 1 2520.2.bi.e 2
20.e even 4 1 560.2.q.c 2
35.f even 4 1 1960.2.q.c 2
35.j even 6 1 inner 1400.2.bh.e 4
35.k even 12 1 1960.2.a.m 1
35.k even 12 1 1960.2.q.c 2
35.k even 12 1 9800.2.a.g 1
35.l odd 12 1 280.2.q.c ✓ 2
35.l odd 12 1 1400.2.q.a 2
35.l odd 12 1 1960.2.a.a 1
35.l odd 12 1 9800.2.a.bi 1
105.x even 12 1 2520.2.bi.e 2
140.w even 12 1 560.2.q.c 2
140.w even 12 1 3920.2.a.bf 1
140.x odd 12 1 3920.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.c ✓ 2 5.c odd 4 1
280.2.q.c ✓ 2 35.l odd 12 1
560.2.q.c 2 20.e even 4 1
560.2.q.c 2 140.w even 12 1
1400.2.q.a 2 5.c odd 4 1
1400.2.q.a 2 35.l odd 12 1
1400.2.bh.e 4 1.a even 1 1 trivial
1400.2.bh.e 4 5.b even 2 1 inner
1400.2.bh.e 4 7.c even 3 1 inner
1400.2.bh.e 4 35.j even 6 1 inner
1960.2.a.a 1 35.l odd 12 1
1960.2.a.m 1 35.k even 12 1
1960.2.q.c 2 35.f even 4 1
1960.2.q.c 2 35.k even 12 1
2520.2.bi.e 2 15.e even 4 1
2520.2.bi.e 2 105.x even 12 1
3920.2.a.i 1 140.x odd 12 1
3920.2.a.bf 1 140.w even 12 1
9800.2.a.g 1 35.k even 12 1
9800.2.a.bi 1 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1400, [\chi])\):

\( T_{3}^{4} - 4T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} - T_{11} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$29$ \( (T - 6)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$41$ \( (T + 5)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$53$ \( T^{4} - 121 T^{2} + 14641 \) Copy content Toggle raw display
$59$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 12 T + 144)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$71$ \( (T + 4)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 144 T^{2} + 20736 \) Copy content Toggle raw display
$79$ \( (T^{2} - 14 T + 196)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
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