Properties

Label 1400.2.a.r
Level $1400$
Weight $2$
Character orbit 1400.a
Self dual yes
Analytic conductor $11.179$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + q^{7} + ( 5 + \beta ) q^{9} +O(q^{10})\) \( q + \beta q^{3} + q^{7} + ( 5 + \beta ) q^{9} + ( 4 - \beta ) q^{11} + ( -2 + \beta ) q^{13} + ( -2 - \beta ) q^{17} + 2 \beta q^{19} + \beta q^{21} -2 \beta q^{23} + ( 8 + 3 \beta ) q^{27} + ( -2 + \beta ) q^{29} -8 q^{31} + ( -8 + 3 \beta ) q^{33} + 2 q^{37} + ( 8 - \beta ) q^{39} + ( 2 - 2 \beta ) q^{41} + ( 4 - 2 \beta ) q^{43} -3 \beta q^{47} + q^{49} + ( -8 - 3 \beta ) q^{51} + ( -6 + 2 \beta ) q^{53} + ( 16 + 2 \beta ) q^{57} + 8 q^{59} + ( 2 + 2 \beta ) q^{61} + ( 5 + \beta ) q^{63} + 4 q^{67} + ( -16 - 2 \beta ) q^{69} + 8 q^{71} + 6 q^{73} + ( 4 - \beta ) q^{77} + ( 8 - 3 \beta ) q^{79} + ( 9 + 8 \beta ) q^{81} -4 \beta q^{83} + ( 8 - \beta ) q^{87} + ( 10 - 2 \beta ) q^{89} + ( -2 + \beta ) q^{91} -8 \beta q^{93} + ( -2 - 5 \beta ) q^{97} + ( 12 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 2q^{7} + 11q^{9} + O(q^{10}) \) \( 2q + q^{3} + 2q^{7} + 11q^{9} + 7q^{11} - 3q^{13} - 5q^{17} + 2q^{19} + q^{21} - 2q^{23} + 19q^{27} - 3q^{29} - 16q^{31} - 13q^{33} + 4q^{37} + 15q^{39} + 2q^{41} + 6q^{43} - 3q^{47} + 2q^{49} - 19q^{51} - 10q^{53} + 34q^{57} + 16q^{59} + 6q^{61} + 11q^{63} + 8q^{67} - 34q^{69} + 16q^{71} + 12q^{73} + 7q^{77} + 13q^{79} + 26q^{81} - 4q^{83} + 15q^{87} + 18q^{89} - 3q^{91} - 8q^{93} - 9q^{97} + 22q^{99} + O(q^{100}) \)

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
−2.37228
3.37228
0 −2.37228 0 0 0 1.00000 0 2.62772 0
1.2 0 3.37228 0 0 0 1.00000 0 8.37228 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.r 2
4.b odd 2 1 2800.2.a.bk 2
5.b even 2 1 280.2.a.c 2
5.c odd 4 2 1400.2.g.i 4
7.b odd 2 1 9800.2.a.bu 2
15.d odd 2 1 2520.2.a.x 2
20.d odd 2 1 560.2.a.h 2
20.e even 4 2 2800.2.g.r 4
35.c odd 2 1 1960.2.a.s 2
35.i odd 6 2 1960.2.q.r 4
35.j even 6 2 1960.2.q.t 4
40.e odd 2 1 2240.2.a.bg 2
40.f even 2 1 2240.2.a.bk 2
60.h even 2 1 5040.2.a.by 2
140.c even 2 1 3920.2.a.bt 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.c 2 5.b even 2 1
560.2.a.h 2 20.d odd 2 1
1400.2.a.r 2 1.a even 1 1 trivial
1400.2.g.i 4 5.c odd 4 2
1960.2.a.s 2 35.c odd 2 1
1960.2.q.r 4 35.i odd 6 2
1960.2.q.t 4 35.j even 6 2
2240.2.a.bg 2 40.e odd 2 1
2240.2.a.bk 2 40.f even 2 1
2520.2.a.x 2 15.d odd 2 1
2800.2.a.bk 2 4.b odd 2 1
2800.2.g.r 4 20.e even 4 2
3920.2.a.bt 2 140.c even 2 1
5040.2.a.by 2 60.h even 2 1
9800.2.a.bu 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1400))\):

\( T_{3}^{2} - T_{3} - 8 \)
\( T_{11}^{2} - 7 T_{11} + 4 \)
\( T_{13}^{2} + 3 T_{13} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -8 - T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( 4 - 7 T + T^{2} \)
$13$ \( -6 + 3 T + T^{2} \)
$17$ \( -2 + 5 T + T^{2} \)
$19$ \( -32 - 2 T + T^{2} \)
$23$ \( -32 + 2 T + T^{2} \)
$29$ \( -6 + 3 T + T^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( -32 - 2 T + T^{2} \)
$43$ \( -24 - 6 T + T^{2} \)
$47$ \( -72 + 3 T + T^{2} \)
$53$ \( -8 + 10 T + T^{2} \)
$59$ \( ( -8 + T )^{2} \)
$61$ \( -24 - 6 T + T^{2} \)
$67$ \( ( -4 + T )^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( ( -6 + T )^{2} \)
$79$ \( -32 - 13 T + T^{2} \)
$83$ \( -128 + 4 T + T^{2} \)
$89$ \( 48 - 18 T + T^{2} \)
$97$ \( -186 + 9 T + T^{2} \)
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