Properties

Label 1400.2.a.q
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: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + q^{7} + (\beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + q^{7} + (\beta + 1) q^{9} + (2 \beta - 3) q^{11} + 2 q^{13} - \beta q^{17} + ( - \beta + 2) q^{19} + \beta q^{21} + (\beta + 3) q^{23} + ( - \beta + 4) q^{27} + (\beta + 5) q^{29} + (2 \beta - 6) q^{31} + ( - \beta + 8) q^{33} + ( - 5 \beta + 1) q^{37} + 2 \beta q^{39} + ( - \beta - 4) q^{41} + ( - \beta + 5) q^{43} + (4 \beta - 2) q^{47} + q^{49} + ( - \beta - 4) q^{51} + (2 \beta + 2) q^{53} + (\beta - 4) q^{57} + ( - 6 \beta + 2) q^{59} + ( - 2 \beta + 8) q^{61} + (\beta + 1) q^{63} + (2 \beta + 11) q^{67} + (4 \beta + 4) q^{69} + ( - 3 \beta - 3) q^{71} + (\beta + 8) q^{73} + (2 \beta - 3) q^{77} + (3 \beta - 1) q^{79} - 7 q^{81} + ( - 3 \beta - 6) q^{83} + (6 \beta + 4) q^{87} + ( - 5 \beta + 2) q^{89} + 2 q^{91} + ( - 4 \beta + 8) q^{93} + 6 q^{97} + (\beta + 5) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 2 q^{7} + 3 q^{9} - 4 q^{11} + 4 q^{13} - q^{17} + 3 q^{19} + q^{21} + 7 q^{23} + 7 q^{27} + 11 q^{29} - 10 q^{31} + 15 q^{33} - 3 q^{37} + 2 q^{39} - 9 q^{41} + 9 q^{43} + 2 q^{49} - 9 q^{51} + 6 q^{53} - 7 q^{57} - 2 q^{59} + 14 q^{61} + 3 q^{63} + 24 q^{67} + 12 q^{69} - 9 q^{71} + 17 q^{73} - 4 q^{77} + q^{79} - 14 q^{81} - 15 q^{83} + 14 q^{87} - q^{89} + 4 q^{91} + 12 q^{93} + 12 q^{97} + 11 q^{99}+O(q^{100}) \) 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
−1.56155
2.56155
0 −1.56155 0 0 0 1.00000 0 −0.561553 0
1.2 0 2.56155 0 0 0 1.00000 0 3.56155 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.q yes 2
4.b odd 2 1 2800.2.a.bj 2
5.b even 2 1 1400.2.a.o 2
5.c odd 4 2 1400.2.g.j 4
7.b odd 2 1 9800.2.a.bt 2
20.d odd 2 1 2800.2.a.bo 2
20.e even 4 2 2800.2.g.v 4
35.c odd 2 1 9800.2.a.bx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.a.o 2 5.b even 2 1
1400.2.a.q yes 2 1.a even 1 1 trivial
1400.2.g.j 4 5.c odd 4 2
2800.2.a.bj 2 4.b odd 2 1
2800.2.a.bo 2 20.d odd 2 1
2800.2.g.v 4 20.e even 4 2
9800.2.a.bt 2 7.b odd 2 1
9800.2.a.bx 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1400))\):

\( T_{3}^{2} - T_{3} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 13 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 13 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$23$ \( T^{2} - 7T + 8 \) Copy content Toggle raw display
$29$ \( T^{2} - 11T + 26 \) Copy content Toggle raw display
$31$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$37$ \( T^{2} + 3T - 104 \) Copy content Toggle raw display
$41$ \( T^{2} + 9T + 16 \) Copy content Toggle raw display
$43$ \( T^{2} - 9T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 68 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$59$ \( T^{2} + 2T - 152 \) Copy content Toggle raw display
$61$ \( T^{2} - 14T + 32 \) Copy content Toggle raw display
$67$ \( T^{2} - 24T + 127 \) Copy content Toggle raw display
$71$ \( T^{2} + 9T - 18 \) Copy content Toggle raw display
$73$ \( T^{2} - 17T + 68 \) Copy content Toggle raw display
$79$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$83$ \( T^{2} + 15T + 18 \) Copy content Toggle raw display
$89$ \( T^{2} + T - 106 \) Copy content Toggle raw display
$97$ \( (T - 6)^{2} \) Copy content Toggle raw display
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