Properties

Label 1400.2.g.b
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
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.

\(f(q)\) \(=\) \( q + 2 i q^{3} -i q^{7} - q^{9} +O(q^{10})\) \( q + 2 i q^{3} -i q^{7} - q^{9} + 2 i q^{17} + 2 q^{19} + 2 q^{21} + 8 i q^{23} + 4 i q^{27} -2 q^{29} + 4 q^{31} + 6 i q^{37} -2 q^{41} + 8 i q^{43} + 4 i q^{47} - q^{49} -4 q^{51} -10 i q^{53} + 4 i q^{57} -6 q^{59} + 4 q^{61} + i q^{63} + 12 i q^{67} -16 q^{69} -14 i q^{73} + 8 q^{79} -11 q^{81} + 6 i q^{83} -4 i q^{87} -10 q^{89} + 8 i q^{93} + 2 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} + 4q^{19} + 4q^{21} - 4q^{29} + 8q^{31} - 4q^{41} - 2q^{49} - 8q^{51} - 12q^{59} + 8q^{61} - 32q^{69} + 16q^{79} - 22q^{81} - 20q^{89} + O(q^{100}) \)

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 2.00000i 0 0 0 1.00000i 0 −1.00000 0
449.2 0 2.00000i 0 0 0 1.00000i 0 −1.00000 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.b 2
4.b odd 2 1 2800.2.g.g 2
5.b even 2 1 inner 1400.2.g.b 2
5.c odd 4 1 56.2.a.b 1
5.c odd 4 1 1400.2.a.a 1
15.e even 4 1 504.2.a.h 1
20.d odd 2 1 2800.2.g.g 2
20.e even 4 1 112.2.a.a 1
20.e even 4 1 2800.2.a.bd 1
35.f even 4 1 392.2.a.b 1
35.f even 4 1 9800.2.a.bj 1
35.k even 12 2 392.2.i.e 2
35.l odd 12 2 392.2.i.a 2
40.i odd 4 1 448.2.a.c 1
40.k even 4 1 448.2.a.h 1
55.e even 4 1 6776.2.a.h 1
60.l odd 4 1 1008.2.a.m 1
65.h odd 4 1 9464.2.a.h 1
80.i odd 4 1 1792.2.b.a 2
80.j even 4 1 1792.2.b.h 2
80.s even 4 1 1792.2.b.h 2
80.t odd 4 1 1792.2.b.a 2
105.k odd 4 1 3528.2.a.b 1
105.w odd 12 2 3528.2.s.ba 2
105.x even 12 2 3528.2.s.a 2
120.q odd 4 1 4032.2.a.a 1
120.w even 4 1 4032.2.a.d 1
140.j odd 4 1 784.2.a.i 1
140.w even 12 2 784.2.i.j 2
140.x odd 12 2 784.2.i.b 2
280.s even 4 1 3136.2.a.w 1
280.y odd 4 1 3136.2.a.c 1
420.w even 4 1 7056.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.b 1 5.c odd 4 1
112.2.a.a 1 20.e even 4 1
392.2.a.b 1 35.f even 4 1
392.2.i.a 2 35.l odd 12 2
392.2.i.e 2 35.k even 12 2
448.2.a.c 1 40.i odd 4 1
448.2.a.h 1 40.k even 4 1
504.2.a.h 1 15.e even 4 1
784.2.a.i 1 140.j odd 4 1
784.2.i.b 2 140.x odd 12 2
784.2.i.j 2 140.w even 12 2
1008.2.a.m 1 60.l odd 4 1
1400.2.a.a 1 5.c odd 4 1
1400.2.g.b 2 1.a even 1 1 trivial
1400.2.g.b 2 5.b even 2 1 inner
1792.2.b.a 2 80.i odd 4 1
1792.2.b.a 2 80.t odd 4 1
1792.2.b.h 2 80.j even 4 1
1792.2.b.h 2 80.s even 4 1
2800.2.a.bd 1 20.e even 4 1
2800.2.g.g 2 4.b odd 2 1
2800.2.g.g 2 20.d odd 2 1
3136.2.a.c 1 280.y odd 4 1
3136.2.a.w 1 280.s even 4 1
3528.2.a.b 1 105.k odd 4 1
3528.2.s.a 2 105.x even 12 2
3528.2.s.ba 2 105.w odd 12 2
4032.2.a.a 1 120.q odd 4 1
4032.2.a.d 1 120.w even 4 1
6776.2.a.h 1 55.e even 4 1
7056.2.a.c 1 420.w even 4 1
9464.2.a.h 1 65.h odd 4 1
9800.2.a.bj 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 \(S_{2}^{\mathrm{new}}(1400, [\chi])\):

\( T_{3}^{2} + 4 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( 64 + T^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( 16 + T^{2} \)
$53$ \( 100 + T^{2} \)
$59$ \( ( 6 + T )^{2} \)
$61$ \( ( -4 + T )^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 196 + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( 4 + T^{2} \)
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