Properties

Label 1400.1.m.a
Level $1400$
Weight $1$
Character orbit 1400.m
Self dual yes
Analytic conductor $0.699$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -7, -56, 8
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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{2}, \sqrt{-7})\)
Artin image $D_4$
Artin field Galois closure of 4.0.9800.2

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{7} + q^{8} - q^{9} + O(q^{10}) \) \( q + q^{2} + q^{4} + q^{7} + q^{8} - q^{9} + q^{14} + q^{16} - q^{18} - 2q^{23} + q^{28} + q^{32} - q^{36} - 2q^{46} + q^{49} + q^{56} - q^{63} + q^{64} - 2q^{71} - q^{72} - 2q^{79} + q^{81} - 2q^{92} + q^{98} + O(q^{100}) \)

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
0
1.00000 0 1.00000 0 0 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
8.b even 2 1 RM by \(\Q(\sqrt{2}) \)
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.1.m.a 1
5.b even 2 1 56.1.h.a 1
5.c odd 4 2 1400.1.c.a 2
7.b odd 2 1 CM 1400.1.m.a 1
8.b even 2 1 RM 1400.1.m.a 1
15.d odd 2 1 504.1.l.a 1
20.d odd 2 1 224.1.h.a 1
35.c odd 2 1 56.1.h.a 1
35.f even 4 2 1400.1.c.a 2
35.i odd 6 2 392.1.j.a 2
35.j even 6 2 392.1.j.a 2
40.e odd 2 1 224.1.h.a 1
40.f even 2 1 56.1.h.a 1
40.i odd 4 2 1400.1.c.a 2
56.h odd 2 1 CM 1400.1.m.a 1
60.h even 2 1 2016.1.l.a 1
80.k odd 4 2 1792.1.c.a 1
80.q even 4 2 1792.1.c.b 1
105.g even 2 1 504.1.l.a 1
105.o odd 6 2 3528.1.bw.a 2
105.p even 6 2 3528.1.bw.a 2
120.i odd 2 1 504.1.l.a 1
120.m even 2 1 2016.1.l.a 1
140.c even 2 1 224.1.h.a 1
140.p odd 6 2 1568.1.n.a 2
140.s even 6 2 1568.1.n.a 2
280.c odd 2 1 56.1.h.a 1
280.n even 2 1 224.1.h.a 1
280.s even 4 2 1400.1.c.a 2
280.ba even 6 2 1568.1.n.a 2
280.bf even 6 2 392.1.j.a 2
280.bi odd 6 2 1568.1.n.a 2
280.bk odd 6 2 392.1.j.a 2
420.o odd 2 1 2016.1.l.a 1
560.be even 4 2 1792.1.c.a 1
560.bf odd 4 2 1792.1.c.b 1
840.b odd 2 1 2016.1.l.a 1
840.u even 2 1 504.1.l.a 1
840.cb even 6 2 3528.1.bw.a 2
840.cg odd 6 2 3528.1.bw.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.1.h.a 1 5.b even 2 1
56.1.h.a 1 35.c odd 2 1
56.1.h.a 1 40.f even 2 1
56.1.h.a 1 280.c odd 2 1
224.1.h.a 1 20.d odd 2 1
224.1.h.a 1 40.e odd 2 1
224.1.h.a 1 140.c even 2 1
224.1.h.a 1 280.n even 2 1
392.1.j.a 2 35.i odd 6 2
392.1.j.a 2 35.j even 6 2
392.1.j.a 2 280.bf even 6 2
392.1.j.a 2 280.bk odd 6 2
504.1.l.a 1 15.d odd 2 1
504.1.l.a 1 105.g even 2 1
504.1.l.a 1 120.i odd 2 1
504.1.l.a 1 840.u even 2 1
1400.1.c.a 2 5.c odd 4 2
1400.1.c.a 2 35.f even 4 2
1400.1.c.a 2 40.i odd 4 2
1400.1.c.a 2 280.s even 4 2
1400.1.m.a 1 1.a even 1 1 trivial
1400.1.m.a 1 7.b odd 2 1 CM
1400.1.m.a 1 8.b even 2 1 RM
1400.1.m.a 1 56.h odd 2 1 CM
1568.1.n.a 2 140.p odd 6 2
1568.1.n.a 2 140.s even 6 2
1568.1.n.a 2 280.ba even 6 2
1568.1.n.a 2 280.bi odd 6 2
1792.1.c.a 1 80.k odd 4 2
1792.1.c.a 1 560.be even 4 2
1792.1.c.b 1 80.q even 4 2
1792.1.c.b 1 560.bf odd 4 2
2016.1.l.a 1 60.h even 2 1
2016.1.l.a 1 120.m even 2 1
2016.1.l.a 1 420.o odd 2 1
2016.1.l.a 1 840.b odd 2 1
3528.1.bw.a 2 105.o odd 6 2
3528.1.bw.a 2 105.p even 6 2
3528.1.bw.a 2 840.cb even 6 2
3528.1.bw.a 2 840.cg odd 6 2

Hecke kernels

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

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