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Properties

Label 1400.2.bh.b
Level $1400$
Weight $2$
Character orbit 1400.bh
Analytic conductor $11.179$
Analytic rank $0$
Dimension $4$
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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{3} + ( - 2 \zeta_{12}^{3} - \zeta_{12}) q^{7} - 2 \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{3} + ( - 2 \zeta_{12}^{3} - \zeta_{12}) q^{7} - 2 \zeta_{12}^{2} q^{9} + (2 \zeta_{12}^{2} - 2) q^{11} - 4 \zeta_{12} q^{17} - 2 \zeta_{12}^{2} q^{19} + ( - 3 \zeta_{12}^{2} + 2) q^{21} + (\zeta_{12}^{3} - \zeta_{12}) q^{23} - 5 \zeta_{12}^{3} q^{27} - 9 q^{29} + (4 \zeta_{12}^{2} - 4) q^{31} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{33} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{37} + q^{41} - 9 \zeta_{12}^{3} q^{43} + (5 \zeta_{12}^{2} - 8) q^{49} - 4 \zeta_{12}^{2} q^{51} - 10 \zeta_{12} q^{53} - 2 \zeta_{12}^{3} q^{57} + (10 \zeta_{12}^{2} - 10) q^{59} - 9 \zeta_{12}^{2} q^{61} + (6 \zeta_{12}^{3} - 4 \zeta_{12}) q^{63} - 5 \zeta_{12} q^{67} - q^{69} + 14 q^{71} + 12 \zeta_{12} q^{73} + ( - 2 \zeta_{12}^{3} + 6 \zeta_{12}) q^{77} + 14 \zeta_{12}^{2} q^{79} + (\zeta_{12}^{2} - 1) q^{81} - 11 \zeta_{12}^{3} q^{83} - 9 \zeta_{12} q^{87} - 15 \zeta_{12}^{2} q^{89} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{93} - 18 \zeta_{12}^{3} q^{97} + 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} - 4 q^{11} - 4 q^{19} + 2 q^{21} - 36 q^{29} - 8 q^{31} + 4 q^{41} - 22 q^{49} - 8 q^{51} - 20 q^{59} - 18 q^{61} - 4 q^{69} + 56 q^{71} + 28 q^{79} - 2 q^{81} - 30 q^{89} + 16 q^{99}+O(q^{100}) \) 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 −0.866025 − 0.500000i 0 0 0 0.866025 + 2.50000i 0 −1.00000 − 1.73205i 0
249.2 0 0.866025 + 0.500000i 0 0 0 −0.866025 − 2.50000i 0 −1.00000 − 1.73205i 0
849.1 0 −0.866025 + 0.500000i 0 0 0 0.866025 − 2.50000i 0 −1.00000 + 1.73205i 0
849.2 0 0.866025 − 0.500000i 0 0 0 −0.866025 + 2.50000i 0 −1.00000 + 1.73205i 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.b 4
5.b even 2 1 inner 1400.2.bh.b 4
5.c odd 4 1 280.2.q.a ✓ 2
5.c odd 4 1 1400.2.q.e 2
7.c even 3 1 inner 1400.2.bh.b 4
15.e even 4 1 2520.2.bi.a 2
20.e even 4 1 560.2.q.h 2
35.f even 4 1 1960.2.q.k 2
35.j even 6 1 inner 1400.2.bh.b 4
35.k even 12 1 1960.2.a.e 1
35.k even 12 1 1960.2.q.k 2
35.k even 12 1 9800.2.a.bc 1
35.l odd 12 1 280.2.q.a ✓ 2
35.l odd 12 1 1400.2.q.e 2
35.l odd 12 1 1960.2.a.i 1
35.l odd 12 1 9800.2.a.r 1
105.x even 12 1 2520.2.bi.a 2
140.w even 12 1 560.2.q.h 2
140.w even 12 1 3920.2.a.m 1
140.x odd 12 1 3920.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.a ✓ 2 5.c odd 4 1
280.2.q.a ✓ 2 35.l odd 12 1
560.2.q.h 2 20.e even 4 1
560.2.q.h 2 140.w even 12 1
1400.2.q.e 2 5.c odd 4 1
1400.2.q.e 2 35.l odd 12 1
1400.2.bh.b 4 1.a even 1 1 trivial
1400.2.bh.b 4 5.b even 2 1 inner
1400.2.bh.b 4 7.c even 3 1 inner
1400.2.bh.b 4 35.j even 6 1 inner
1960.2.a.e 1 35.k even 12 1
1960.2.a.i 1 35.l odd 12 1
1960.2.q.k 2 35.f even 4 1
1960.2.q.k 2 35.k even 12 1
2520.2.bi.a 2 15.e even 4 1
2520.2.bi.a 2 105.x even 12 1
3920.2.a.m 1 140.w even 12 1
3920.2.a.y 1 140.x odd 12 1
9800.2.a.r 1 35.l odd 12 1
9800.2.a.bc 1 35.k even 12 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 11T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T + 9)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$41$ \( (T - 1)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$59$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$71$ \( (T - 14)^{4} \) 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
$83$ \( (T^{2} + 121)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 15 T + 225)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 324)^{2} \) Copy content Toggle raw display
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