Properties

Label 1400.2.bh.f
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: no (minimal twist has level 56)
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.

\(f(q)\) \(=\) \( q + 3 \zeta_{12} q^{3} + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + 6 \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + 3 \zeta_{12} q^{3} + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + 6 \zeta_{12}^{2} q^{9} + ( 1 - \zeta_{12}^{2} ) q^{11} -2 \zeta_{12}^{3} q^{13} -3 \zeta_{12} q^{17} + 5 \zeta_{12}^{2} q^{19} + ( -3 + 9 \zeta_{12}^{2} ) q^{21} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{23} + 9 \zeta_{12}^{3} q^{27} + 6 q^{29} + ( 1 - \zeta_{12}^{2} ) q^{31} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{33} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{37} + ( 6 - 6 \zeta_{12}^{2} ) q^{39} -10 q^{41} + 4 \zeta_{12}^{3} q^{43} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{47} + ( -5 + 8 \zeta_{12}^{2} ) q^{49} -9 \zeta_{12}^{2} q^{51} -9 \zeta_{12} q^{53} + 15 \zeta_{12}^{3} q^{57} + ( 3 - 3 \zeta_{12}^{2} ) q^{59} -3 \zeta_{12}^{2} q^{61} + ( -6 \zeta_{12} + 18 \zeta_{12}^{3} ) q^{63} -11 \zeta_{12} q^{67} + 9 q^{69} + 16 q^{71} + 7 \zeta_{12} q^{73} + ( 3 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{77} -11 \zeta_{12}^{2} q^{79} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} + 4 \zeta_{12}^{3} q^{83} + 18 \zeta_{12} q^{87} -9 \zeta_{12}^{2} q^{89} + ( 6 - 4 \zeta_{12}^{2} ) q^{91} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{93} + 6 \zeta_{12}^{3} q^{97} + 6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} + 2q^{11} + 10q^{19} + 6q^{21} + 24q^{29} + 2q^{31} + 12q^{39} - 40q^{41} - 4q^{49} - 18q^{51} + 6q^{59} - 6q^{61} + 36q^{69} + 64q^{71} - 22q^{79} - 18q^{81} - 18q^{89} + 16q^{91} + 24q^{99} + O(q^{100}) \)

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 −2.59808 1.50000i 0 0 0 −1.73205 2.00000i 0 3.00000 + 5.19615i 0
249.2 0 2.59808 + 1.50000i 0 0 0 1.73205 + 2.00000i 0 3.00000 + 5.19615i 0
849.1 0 −2.59808 + 1.50000i 0 0 0 −1.73205 + 2.00000i 0 3.00000 5.19615i 0
849.2 0 2.59808 1.50000i 0 0 0 1.73205 2.00000i 0 3.00000 5.19615i 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.f 4
5.b even 2 1 inner 1400.2.bh.f 4
5.c odd 4 1 56.2.i.a 2
5.c odd 4 1 1400.2.q.g 2
7.c even 3 1 inner 1400.2.bh.f 4
15.e even 4 1 504.2.s.e 2
20.e even 4 1 112.2.i.c 2
35.f even 4 1 392.2.i.f 2
35.j even 6 1 inner 1400.2.bh.f 4
35.k even 12 1 392.2.a.a 1
35.k even 12 1 392.2.i.f 2
35.k even 12 1 9800.2.a.bp 1
35.l odd 12 1 56.2.i.a 2
35.l odd 12 1 392.2.a.f 1
35.l odd 12 1 1400.2.q.g 2
35.l odd 12 1 9800.2.a.b 1
40.i odd 4 1 448.2.i.f 2
40.k even 4 1 448.2.i.a 2
60.l odd 4 1 1008.2.s.e 2
105.k odd 4 1 3528.2.s.o 2
105.w odd 12 1 3528.2.a.k 1
105.w odd 12 1 3528.2.s.o 2
105.x even 12 1 504.2.s.e 2
105.x even 12 1 3528.2.a.r 1
140.j odd 4 1 784.2.i.a 2
140.w even 12 1 112.2.i.c 2
140.w even 12 1 784.2.a.a 1
140.x odd 12 1 784.2.a.j 1
140.x odd 12 1 784.2.i.a 2
280.bp odd 12 1 3136.2.a.a 1
280.br even 12 1 448.2.i.a 2
280.br even 12 1 3136.2.a.bc 1
280.bt odd 12 1 448.2.i.f 2
280.bt odd 12 1 3136.2.a.b 1
280.bv even 12 1 3136.2.a.bb 1
420.bp odd 12 1 1008.2.s.e 2
420.bp odd 12 1 7056.2.a.bi 1
420.br even 12 1 7056.2.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.a 2 5.c odd 4 1
56.2.i.a 2 35.l odd 12 1
112.2.i.c 2 20.e even 4 1
112.2.i.c 2 140.w even 12 1
392.2.a.a 1 35.k even 12 1
392.2.a.f 1 35.l odd 12 1
392.2.i.f 2 35.f even 4 1
392.2.i.f 2 35.k even 12 1
448.2.i.a 2 40.k even 4 1
448.2.i.a 2 280.br even 12 1
448.2.i.f 2 40.i odd 4 1
448.2.i.f 2 280.bt odd 12 1
504.2.s.e 2 15.e even 4 1
504.2.s.e 2 105.x even 12 1
784.2.a.a 1 140.w even 12 1
784.2.a.j 1 140.x odd 12 1
784.2.i.a 2 140.j odd 4 1
784.2.i.a 2 140.x odd 12 1
1008.2.s.e 2 60.l odd 4 1
1008.2.s.e 2 420.bp odd 12 1
1400.2.q.g 2 5.c odd 4 1
1400.2.q.g 2 35.l odd 12 1
1400.2.bh.f 4 1.a even 1 1 trivial
1400.2.bh.f 4 5.b even 2 1 inner
1400.2.bh.f 4 7.c even 3 1 inner
1400.2.bh.f 4 35.j even 6 1 inner
3136.2.a.a 1 280.bp odd 12 1
3136.2.a.b 1 280.bt odd 12 1
3136.2.a.bb 1 280.bv even 12 1
3136.2.a.bc 1 280.br even 12 1
3528.2.a.k 1 105.w odd 12 1
3528.2.a.r 1 105.x even 12 1
3528.2.s.o 2 105.k odd 4 1
3528.2.s.o 2 105.w odd 12 1
7056.2.a.s 1 420.br even 12 1
7056.2.a.bi 1 420.bp odd 12 1
9800.2.a.b 1 35.l odd 12 1
9800.2.a.bp 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 \(S_{2}^{\mathrm{new}}(1400, [\chi])\):

\( T_{3}^{4} - 9 T_{3}^{2} + 81 \)
\( T_{11}^{2} - T_{11} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 3 T^{2} )^{2}( 1 + 3 T^{2} + 9 T^{4} ) \)
$5$ 1
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 22 T^{2} + 169 T^{4} )^{2} \)
$17$ \( 1 + 25 T^{2} + 336 T^{4} + 7225 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} )^{2} \)
$37$ \( 1 + 49 T^{2} + 1032 T^{4} + 67081 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 + 10 T + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 70 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 + 93 T^{2} + 6440 T^{4} + 205437 T^{6} + 4879681 T^{8} \)
$53$ \( 1 + 25 T^{2} - 2184 T^{4} + 70225 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 3 T - 52 T^{2} + 183 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 109 T^{2} + 4489 T^{4} )( 1 + 122 T^{2} + 4489 T^{4} ) \)
$71$ \( ( 1 - 16 T + 71 T^{2} )^{4} \)
$73$ \( ( 1 - 46 T^{2} + 5329 T^{4} )( 1 + 143 T^{2} + 5329 T^{4} ) \)
$79$ \( ( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 150 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 9 T - 8 T^{2} + 801 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 158 T^{2} + 9409 T^{4} )^{2} \)
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