Properties

Label 1400.2.g.e
Level $1400$
Weight $2$
Character orbit 1400.g
Analytic conductor $11.179$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) 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)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + i q^{3} - i q^{7} + 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{3} - i q^{7} + 2 q^{9} - 5 q^{11} - i q^{13} + 3 i q^{17} + 6 q^{19} + q^{21} + 6 i q^{23} + 5 i q^{27} + 9 q^{29} - 5 i q^{33} + 6 i q^{37} + q^{39} + 8 q^{41} - 6 i q^{43} + 3 i q^{47} - q^{49} - 3 q^{51} + 12 i q^{53} + 6 i q^{57} - 8 q^{59} - 4 q^{61} - 2 i q^{63} - 4 i q^{67} - 6 q^{69} + 8 q^{71} - 10 i q^{73} + 5 i q^{77} + 3 q^{79} + q^{81} + 12 i q^{83} + 9 i q^{87} + 16 q^{89} - q^{91} + 7 i q^{97} - 10 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{9} - 10 q^{11} + 12 q^{19} + 2 q^{21} + 18 q^{29} + 2 q^{39} + 16 q^{41} - 2 q^{49} - 6 q^{51} - 16 q^{59} - 8 q^{61} - 12 q^{69} + 16 q^{71} + 6 q^{79} + 2 q^{81} + 32 q^{89} - 2 q^{91} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.00000i
1.00000i
0 1.00000i 0 0 0 1.00000i 0 2.00000 0
449.2 0 1.00000i 0 0 0 1.00000i 0 2.00000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.g.e 2
4.b odd 2 1 2800.2.g.m 2
5.b even 2 1 inner 1400.2.g.e 2
5.c odd 4 1 280.2.a.b 1
5.c odd 4 1 1400.2.a.k 1
15.e even 4 1 2520.2.a.p 1
20.d odd 2 1 2800.2.g.m 2
20.e even 4 1 560.2.a.e 1
20.e even 4 1 2800.2.a.i 1
35.f even 4 1 1960.2.a.k 1
35.f even 4 1 9800.2.a.n 1
35.k even 12 2 1960.2.q.e 2
35.l odd 12 2 1960.2.q.m 2
40.i odd 4 1 2240.2.a.v 1
40.k even 4 1 2240.2.a.j 1
60.l odd 4 1 5040.2.a.be 1
140.j odd 4 1 3920.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.b 1 5.c odd 4 1
560.2.a.e 1 20.e even 4 1
1400.2.a.k 1 5.c odd 4 1
1400.2.g.e 2 1.a even 1 1 trivial
1400.2.g.e 2 5.b even 2 1 inner
1960.2.a.k 1 35.f even 4 1
1960.2.q.e 2 35.k even 12 2
1960.2.q.m 2 35.l odd 12 2
2240.2.a.j 1 40.k even 4 1
2240.2.a.v 1 40.i odd 4 1
2520.2.a.p 1 15.e even 4 1
2800.2.a.i 1 20.e even 4 1
2800.2.g.m 2 4.b odd 2 1
2800.2.g.m 2 20.d odd 2 1
3920.2.a.r 1 140.j odd 4 1
5040.2.a.be 1 60.l odd 4 1
9800.2.a.n 1 35.f even 4 1

Hecke kernels

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

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{11} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T + 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 9 \) Copy content Toggle raw display
$19$ \( (T - 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T - 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T - 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 36 \) Copy content Toggle raw display
$47$ \( T^{2} + 9 \) Copy content Toggle raw display
$53$ \( T^{2} + 144 \) Copy content Toggle raw display
$59$ \( (T + 8)^{2} \) Copy content Toggle raw display
$61$ \( (T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( (T - 3)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T - 16)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 49 \) Copy content Toggle raw display
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