Properties

Label 1400.2.bh.d
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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Newspace parameters

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

Self dual: no
Analytic conductor: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + 2 \zeta_{12} q^{3} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + 2 \zeta_{12} q^{3} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{2} q^{9} + ( -4 + 4 \zeta_{12}^{2} ) q^{11} + 2 \zeta_{12}^{3} q^{13} + 3 \zeta_{12} q^{17} + ( 2 + 4 \zeta_{12}^{2} ) q^{21} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{23} -4 \zeta_{12}^{3} q^{27} + 6 q^{29} + ( -9 + 9 \zeta_{12}^{2} ) q^{31} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{33} + ( -4 + 4 \zeta_{12}^{2} ) q^{39} + 5 q^{41} + 6 \zeta_{12}^{3} q^{43} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{47} + ( 5 + 3 \zeta_{12}^{2} ) q^{49} + 6 \zeta_{12}^{2} q^{51} -6 \zeta_{12} q^{53} + ( 8 - 8 \zeta_{12}^{2} ) q^{59} -8 \zeta_{12}^{2} q^{61} + ( \zeta_{12} + 2 \zeta_{12}^{3} ) q^{63} -14 \zeta_{12} q^{67} -6 q^{69} + 11 q^{71} -2 \zeta_{12} q^{73} + ( -8 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{77} + 9 \zeta_{12}^{2} q^{79} + ( 11 - 11 \zeta_{12}^{2} ) q^{81} + 6 \zeta_{12}^{3} q^{83} + 12 \zeta_{12} q^{87} + 11 \zeta_{12}^{2} q^{89} + ( -4 + 6 \zeta_{12}^{2} ) q^{91} + ( -18 \zeta_{12} + 18 \zeta_{12}^{3} ) q^{93} -11 \zeta_{12}^{3} q^{97} -4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{9} - 8q^{11} + 16q^{21} + 24q^{29} - 18q^{31} - 8q^{39} + 20q^{41} + 26q^{49} + 12q^{51} + 16q^{59} - 16q^{61} - 24q^{69} + 44q^{71} + 18q^{79} + 22q^{81} + 22q^{89} - 4q^{91} - 16q^{99} + O(q^{100}) \)

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.

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 −2.59808 0.500000i 0 0.500000 + 0.866025i 0
249.2 0 1.73205 + 1.00000i 0 0 0 2.59808 + 0.500000i 0 0.500000 + 0.866025i 0
849.1 0 −1.73205 + 1.00000i 0 0 0 −2.59808 + 0.500000i 0 0.500000 0.866025i 0
849.2 0 1.73205 1.00000i 0 0 0 2.59808 0.500000i 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.d 4
5.b even 2 1 inner 1400.2.bh.d 4
5.c odd 4 1 1400.2.q.b 2
5.c odd 4 1 1400.2.q.f yes 2
7.c even 3 1 inner 1400.2.bh.d 4
35.j even 6 1 inner 1400.2.bh.d 4
35.k even 12 1 9800.2.a.k 1
35.k even 12 1 9800.2.a.bl 1
35.l odd 12 1 1400.2.q.b 2
35.l odd 12 1 1400.2.q.f yes 2
35.l odd 12 1 9800.2.a.j 1
35.l odd 12 1 9800.2.a.bm 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.q.b 2 5.c odd 4 1
1400.2.q.b 2 35.l odd 12 1
1400.2.q.f yes 2 5.c odd 4 1
1400.2.q.f yes 2 35.l odd 12 1
1400.2.bh.d 4 1.a even 1 1 trivial
1400.2.bh.d 4 5.b even 2 1 inner
1400.2.bh.d 4 7.c even 3 1 inner
1400.2.bh.d 4 35.j even 6 1 inner
9800.2.a.j 1 35.l odd 12 1
9800.2.a.k 1 35.k even 12 1
9800.2.a.bl 1 35.k even 12 1
9800.2.a.bm 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 \(S_{2}^{\mathrm{new}}(1400, [\chi])\):

\( T_{3}^{4} - 4 T_{3}^{2} + 16 \)
\( T_{11}^{2} + 4 T_{11} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 16 - 4 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 49 - 13 T^{2} + T^{4} \)
$11$ \( ( 16 + 4 T + T^{2} )^{2} \)
$13$ \( ( 4 + T^{2} )^{2} \)
$17$ \( 81 - 9 T^{2} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( 81 - 9 T^{2} + T^{4} \)
$29$ \( ( -6 + T )^{4} \)
$31$ \( ( 81 + 9 T + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( ( -5 + T )^{4} \)
$43$ \( ( 36 + T^{2} )^{2} \)
$47$ \( 6561 - 81 T^{2} + T^{4} \)
$53$ \( 1296 - 36 T^{2} + T^{4} \)
$59$ \( ( 64 - 8 T + T^{2} )^{2} \)
$61$ \( ( 64 + 8 T + T^{2} )^{2} \)
$67$ \( 38416 - 196 T^{2} + T^{4} \)
$71$ \( ( -11 + T )^{4} \)
$73$ \( 16 - 4 T^{2} + T^{4} \)
$79$ \( ( 81 - 9 T + T^{2} )^{2} \)
$83$ \( ( 36 + T^{2} )^{2} \)
$89$ \( ( 121 - 11 T + T^{2} )^{2} \)
$97$ \( ( 121 + T^{2} )^{2} \)
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