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: \(x^{2} + 1\)
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.

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

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 \(S_{2}^{\mathrm{new}}(1400, [\chi])\):

\( T_{3}^{2} + 1 \)
\( T_{11} + 5 \)

Hecke characteristic polynomials

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