Properties

Label 1400.2.a.t
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: \(x^{3} - x^{2} - 6 x - 2\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 4\)

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
−1.76156
−0.363328
3.12489
0 −1.76156 0 0 0 −1.00000 0 0.103084 0
1.2 0 −0.363328 0 0 0 −1.00000 0 −2.86799 0
1.3 0 3.12489 0 0 0 −1.00000 0 6.76491 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.t 3
4.b odd 2 1 2800.2.a.bq 3
5.b even 2 1 1400.2.a.s 3
5.c odd 4 2 280.2.g.b 6
7.b odd 2 1 9800.2.a.cd 3
15.e even 4 2 2520.2.t.g 6
20.d odd 2 1 2800.2.a.br 3
20.e even 4 2 560.2.g.f 6
35.c odd 2 1 9800.2.a.cg 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 5.b even 2 1
1400.2.a.t 3 1.a even 1 1 trivial
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 4.b odd 2 1
2800.2.a.br 3 20.d odd 2 1
5040.2.t.y 6 60.l odd 4 2
9800.2.a.cd 3 7.b odd 2 1
9800.2.a.cg 3 35.c odd 2 1

Hecke kernels

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

\( T_{3}^{3} - T_{3}^{2} - 6 T_{3} - 2 \)
\( T_{11}^{3} - 7 T_{11}^{2} + 8 T_{11} + 8 \)
\( T_{13}^{3} + 5 T_{13}^{2} - 22 T_{13} - 106 \)

Hecke characteristic polynomials

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