Properties

Label 1400.2.a.s
Level $1400$
Weight $2$
Character orbit 1400.a
Self dual yes
Analytic conductor $11.179$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) 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)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9} + ( - \beta_{2} + 2) q^{11} + (2 \beta_{2} + \beta_1 + 2) q^{13} + ( - \beta_{2} - 2 \beta_1) q^{17} + (\beta_{2} - \beta_1 + 2) q^{19} - \beta_1 q^{21} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{23} + ( - \beta_{2} - 2 \beta_1 - 6) q^{27} - 3 \beta_{2} q^{29} + ( - 2 \beta_1 + 4) q^{31} + ( - \beta_{2} - 2 \beta_1 - 2) q^{33} + 6 q^{37} + (\beta_{2} - 4 \beta_1) q^{39} + (2 \beta_{2} + 2 \beta_1 + 6) q^{41} + (2 \beta_{2} + 2 \beta_1) q^{43} + (3 \beta_{2} - 2) q^{47} + q^{49} + (\beta_{2} + 4 \beta_1 + 6) q^{51} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{53} + (2 \beta_{2} + 6) q^{57} + ( - 3 \beta_{2} - 3 \beta_1 + 2) q^{59} + ( - \beta_{2} - \beta_1 + 8) q^{61} + (\beta_{2} + 2 \beta_1 + 1) q^{63} + (2 \beta_{2} + 4 \beta_1 - 4) q^{67} + (6 \beta_1 + 4) q^{69} + ( - 2 \beta_1 + 2) q^{71} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{73} + ( - \beta_{2} + 2) q^{77} + ( - \beta_{2} + 4 \beta_1 + 4) q^{79} + ( - 2 \beta_{2} + 4 \beta_1 + 3) q^{81} + ( - \beta_{2} + 5 \beta_1 - 6) q^{83} + ( - 3 \beta_{2} - 6) q^{87} + (2 \beta_{2} - 4 \beta_1 + 2) q^{89} + (2 \beta_{2} + \beta_1 + 2) q^{91} + (2 \beta_{2} + 8) q^{93} + ( - \beta_{2} + 2 \beta_1 - 4) q^{97} + (4 \beta_{2} + 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} + 3 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} + 3 q^{7} + 4 q^{9} + 7 q^{11} + 5 q^{13} - q^{17} + 4 q^{19} - q^{21} - 6 q^{23} - 19 q^{27} + 3 q^{29} + 10 q^{31} - 7 q^{33} + 18 q^{37} - 5 q^{39} + 18 q^{41} - 9 q^{47} + 3 q^{49} + 21 q^{51} + 10 q^{53} + 16 q^{57} + 6 q^{59} + 24 q^{61} + 4 q^{63} - 10 q^{67} + 18 q^{69} + 4 q^{71} - 6 q^{73} + 7 q^{77} + 17 q^{79} + 15 q^{81} - 12 q^{83} - 15 q^{87} + 5 q^{91} + 22 q^{93} - 9 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 4 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.12489
−0.363328
−1.76156
0 −3.12489 0 0 0 1.00000 0 6.76491 0
1.2 0 0.363328 0 0 0 1.00000 0 −2.86799 0
1.3 0 1.76156 0 0 0 1.00000 0 0.103084 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.a.s 3
4.b odd 2 1 2800.2.a.br 3
5.b even 2 1 1400.2.a.t 3
5.c odd 4 2 280.2.g.b 6
7.b odd 2 1 9800.2.a.cg 3
15.e even 4 2 2520.2.t.g 6
20.d odd 2 1 2800.2.a.bq 3
20.e even 4 2 560.2.g.f 6
35.c odd 2 1 9800.2.a.cd 3
35.f even 4 2 1960.2.g.c 6
40.i odd 4 2 2240.2.g.l 6
40.k even 4 2 2240.2.g.m 6
60.l odd 4 2 5040.2.t.y 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.g.b 6 5.c odd 4 2
560.2.g.f 6 20.e even 4 2
1400.2.a.s 3 1.a even 1 1 trivial
1400.2.a.t 3 5.b even 2 1
1960.2.g.c 6 35.f even 4 2
2240.2.g.l 6 40.i odd 4 2
2240.2.g.m 6 40.k even 4 2
2520.2.t.g 6 15.e even 4 2
2800.2.a.bq 3 20.d odd 2 1
2800.2.a.br 3 4.b odd 2 1
5040.2.t.y 6 60.l odd 4 2
9800.2.a.cd 3 35.c odd 2 1
9800.2.a.cg 3 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1400))\):

\( T_{3}^{3} + T_{3}^{2} - 6T_{3} + 2 \) Copy content Toggle raw display
\( T_{11}^{3} - 7T_{11}^{2} + 8T_{11} + 8 \) Copy content Toggle raw display
\( T_{13}^{3} - 5T_{13}^{2} - 22T_{13} + 106 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + T^{2} - 6T + 2 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 7 T^{2} + 8 T + 8 \) Copy content Toggle raw display
$13$ \( T^{3} - 5 T^{2} - 22 T + 106 \) Copy content Toggle raw display
$17$ \( T^{3} + T^{2} - 24 T + 20 \) Copy content Toggle raw display
$19$ \( T^{3} - 4 T^{2} - 14 T - 8 \) Copy content Toggle raw display
$23$ \( T^{3} + 6 T^{2} - 28 T - 136 \) Copy content Toggle raw display
$29$ \( T^{3} - 3 T^{2} - 72 T + 108 \) Copy content Toggle raw display
$31$ \( T^{3} - 10 T^{2} + 8 T + 80 \) Copy content Toggle raw display
$37$ \( (T - 6)^{3} \) Copy content Toggle raw display
$41$ \( T^{3} - 18 T^{2} + 68 T + 88 \) Copy content Toggle raw display
$43$ \( T^{3} - 40T + 64 \) Copy content Toggle raw display
$47$ \( T^{3} + 9 T^{2} - 48 T - 232 \) Copy content Toggle raw display
$53$ \( T^{3} - 10 T^{2} - 44 T + 472 \) Copy content Toggle raw display
$59$ \( T^{3} - 6 T^{2} - 78 T - 44 \) Copy content Toggle raw display
$61$ \( T^{3} - 24 T^{2} + 182 T - 440 \) Copy content Toggle raw display
$67$ \( T^{3} + 10 T^{2} - 64 T - 512 \) Copy content Toggle raw display
$71$ \( T^{3} - 4 T^{2} - 20 T + 64 \) Copy content Toggle raw display
$73$ \( T^{3} + 6 T^{2} - 28 T - 136 \) Copy content Toggle raw display
$79$ \( T^{3} - 17 T^{2} - 32 T + 548 \) Copy content Toggle raw display
$83$ \( T^{3} + 12 T^{2} - 142 T - 824 \) Copy content Toggle raw display
$89$ \( T^{3} - 172T - 464 \) Copy content Toggle raw display
$97$ \( T^{3} + 9 T^{2} - 16 T - 44 \) Copy content Toggle raw display
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