Properties

Label 1400.1.m.b
Level $1400$
Weight $1$
Character orbit 1400.m
Self dual yes
Analytic conductor $0.699$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -56
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1400.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.698691017686\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.11200.2
Artin image $D_8$
Artin field Galois closure of 8.0.19208000000.2

$q$-expansion

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

\(f(q)\) \(=\) \( q - q^{2} -\beta q^{3} + q^{4} + \beta q^{6} + q^{7} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} -\beta q^{3} + q^{4} + \beta q^{6} + q^{7} - q^{8} + q^{9} -\beta q^{12} + \beta q^{13} - q^{14} + q^{16} - q^{18} -\beta q^{19} -\beta q^{21} + \beta q^{24} -\beta q^{26} + q^{28} - q^{32} + q^{36} + \beta q^{38} -2 q^{39} + \beta q^{42} -\beta q^{48} + q^{49} + \beta q^{52} - q^{56} + 2 q^{57} + \beta q^{59} + \beta q^{61} + q^{63} + q^{64} - q^{72} -\beta q^{76} + 2 q^{78} - q^{81} + \beta q^{83} -\beta q^{84} + \beta q^{91} + \beta q^{96} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} + 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} + 2q^{7} - 2q^{8} + 2q^{9} - 2q^{14} + 2q^{16} - 2q^{18} + 2q^{28} - 2q^{32} + 2q^{36} - 4q^{39} + 2q^{49} - 2q^{56} + 4q^{57} + 2q^{63} + 2q^{64} - 2q^{72} + 4q^{78} - 2q^{81} - 2q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1301.1
1.41421
−1.41421
−1.00000 −1.41421 1.00000 0 1.41421 1.00000 −1.00000 1.00000 0
1301.2 −1.00000 1.41421 1.00000 0 −1.41421 1.00000 −1.00000 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
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
7.b odd 2 1 inner
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.1.m.b 2
5.b even 2 1 1400.1.m.e 2
5.c odd 4 2 280.1.c.a 4
7.b odd 2 1 inner 1400.1.m.b 2
8.b even 2 1 inner 1400.1.m.b 2
15.e even 4 2 2520.1.h.e 4
20.e even 4 2 1120.1.c.a 4
35.c odd 2 1 1400.1.m.e 2
35.f even 4 2 280.1.c.a 4
35.k even 12 4 1960.1.bk.a 8
35.l odd 12 4 1960.1.bk.a 8
40.f even 2 1 1400.1.m.e 2
40.i odd 4 2 280.1.c.a 4
40.k even 4 2 1120.1.c.a 4
56.h odd 2 1 CM 1400.1.m.b 2
105.k odd 4 2 2520.1.h.e 4
120.w even 4 2 2520.1.h.e 4
140.j odd 4 2 1120.1.c.a 4
280.c odd 2 1 1400.1.m.e 2
280.s even 4 2 280.1.c.a 4
280.y odd 4 2 1120.1.c.a 4
280.bt odd 12 4 1960.1.bk.a 8
280.bv even 12 4 1960.1.bk.a 8
840.bp odd 4 2 2520.1.h.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.1.c.a 4 5.c odd 4 2
280.1.c.a 4 35.f even 4 2
280.1.c.a 4 40.i odd 4 2
280.1.c.a 4 280.s even 4 2
1120.1.c.a 4 20.e even 4 2
1120.1.c.a 4 40.k even 4 2
1120.1.c.a 4 140.j odd 4 2
1120.1.c.a 4 280.y odd 4 2
1400.1.m.b 2 1.a even 1 1 trivial
1400.1.m.b 2 7.b odd 2 1 inner
1400.1.m.b 2 8.b even 2 1 inner
1400.1.m.b 2 56.h odd 2 1 CM
1400.1.m.e 2 5.b even 2 1
1400.1.m.e 2 35.c odd 2 1
1400.1.m.e 2 40.f even 2 1
1400.1.m.e 2 280.c odd 2 1
1960.1.bk.a 8 35.k even 12 4
1960.1.bk.a 8 35.l odd 12 4
1960.1.bk.a 8 280.bt odd 12 4
1960.1.bk.a 8 280.bv even 12 4
2520.1.h.e 4 15.e even 4 2
2520.1.h.e 4 105.k odd 4 2
2520.1.h.e 4 120.w even 4 2
2520.1.h.e 4 840.bp odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1400, [\chi])\):

\( T_{3}^{2} - 2 \)
\( T_{11} \)
\( T_{23} \)
\( T_{113} - 2 \)

Hecke characteristic polynomials

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