Properties

Label 1400.1.c.a
Level $1400$
Weight $1$
Character orbit 1400.c
Analytic conductor $0.699$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -7, -56, 8
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.698691017686\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 $D_4:C_2$
Artin field Galois closure of 8.0.2401000000.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
1.00000i
1.00000i
1.00000i 0 −1.00000 0 0 1.00000i 1.00000i 1.00000 0
349.2 1.00000i 0 −1.00000 0 0 1.00000i 1.00000i 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 \(\Q(\sqrt{-14}) \)
5.b even 2 1 inner
35.c odd 2 1 inner
40.f even 2 1 inner
280.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.1.c.a 2
5.b even 2 1 inner 1400.1.c.a 2
5.c odd 4 1 56.1.h.a 1
5.c odd 4 1 1400.1.m.a 1
7.b odd 2 1 CM 1400.1.c.a 2
8.b even 2 1 RM 1400.1.c.a 2
15.e even 4 1 504.1.l.a 1
20.e even 4 1 224.1.h.a 1
35.c odd 2 1 inner 1400.1.c.a 2
35.f even 4 1 56.1.h.a 1
35.f even 4 1 1400.1.m.a 1
35.k even 12 2 392.1.j.a 2
35.l odd 12 2 392.1.j.a 2
40.f even 2 1 inner 1400.1.c.a 2
40.i odd 4 1 56.1.h.a 1
40.i odd 4 1 1400.1.m.a 1
40.k even 4 1 224.1.h.a 1
56.h odd 2 1 CM 1400.1.c.a 2
60.l odd 4 1 2016.1.l.a 1
80.i odd 4 1 1792.1.c.b 1
80.j even 4 1 1792.1.c.a 1
80.s even 4 1 1792.1.c.a 1
80.t odd 4 1 1792.1.c.b 1
105.k odd 4 1 504.1.l.a 1
105.w odd 12 2 3528.1.bw.a 2
105.x even 12 2 3528.1.bw.a 2
120.q odd 4 1 2016.1.l.a 1
120.w even 4 1 504.1.l.a 1
140.j odd 4 1 224.1.h.a 1
140.w even 12 2 1568.1.n.a 2
140.x odd 12 2 1568.1.n.a 2
280.c odd 2 1 inner 1400.1.c.a 2
280.s even 4 1 56.1.h.a 1
280.s even 4 1 1400.1.m.a 1
280.y odd 4 1 224.1.h.a 1
280.bp odd 12 2 1568.1.n.a 2
280.br even 12 2 1568.1.n.a 2
280.bt odd 12 2 392.1.j.a 2
280.bv even 12 2 392.1.j.a 2
420.w even 4 1 2016.1.l.a 1
560.r even 4 1 1792.1.c.b 1
560.u odd 4 1 1792.1.c.a 1
560.bm odd 4 1 1792.1.c.a 1
560.bn even 4 1 1792.1.c.b 1
840.bm even 4 1 2016.1.l.a 1
840.bp odd 4 1 504.1.l.a 1
840.dc even 12 2 3528.1.bw.a 2
840.dh odd 12 2 3528.1.bw.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.1.h.a 1 5.c odd 4 1
56.1.h.a 1 35.f even 4 1
56.1.h.a 1 40.i odd 4 1
56.1.h.a 1 280.s even 4 1
224.1.h.a 1 20.e even 4 1
224.1.h.a 1 40.k even 4 1
224.1.h.a 1 140.j odd 4 1
224.1.h.a 1 280.y odd 4 1
392.1.j.a 2 35.k even 12 2
392.1.j.a 2 35.l odd 12 2
392.1.j.a 2 280.bt odd 12 2
392.1.j.a 2 280.bv even 12 2
504.1.l.a 1 15.e even 4 1
504.1.l.a 1 105.k odd 4 1
504.1.l.a 1 120.w even 4 1
504.1.l.a 1 840.bp odd 4 1
1400.1.c.a 2 1.a even 1 1 trivial
1400.1.c.a 2 5.b even 2 1 inner
1400.1.c.a 2 7.b odd 2 1 CM
1400.1.c.a 2 8.b even 2 1 RM
1400.1.c.a 2 35.c odd 2 1 inner
1400.1.c.a 2 40.f even 2 1 inner
1400.1.c.a 2 56.h odd 2 1 CM
1400.1.c.a 2 280.c odd 2 1 inner
1400.1.m.a 1 5.c odd 4 1
1400.1.m.a 1 35.f even 4 1
1400.1.m.a 1 40.i odd 4 1
1400.1.m.a 1 280.s even 4 1
1568.1.n.a 2 140.w even 12 2
1568.1.n.a 2 140.x odd 12 2
1568.1.n.a 2 280.bp odd 12 2
1568.1.n.a 2 280.br even 12 2
1792.1.c.a 1 80.j even 4 1
1792.1.c.a 1 80.s even 4 1
1792.1.c.a 1 560.u odd 4 1
1792.1.c.a 1 560.bm odd 4 1
1792.1.c.b 1 80.i odd 4 1
1792.1.c.b 1 80.t odd 4 1
1792.1.c.b 1 560.r even 4 1
1792.1.c.b 1 560.bn even 4 1
2016.1.l.a 1 60.l odd 4 1
2016.1.l.a 1 120.q odd 4 1
2016.1.l.a 1 420.w even 4 1
2016.1.l.a 1 840.bm even 4 1
3528.1.bw.a 2 105.w odd 12 2
3528.1.bw.a 2 105.x even 12 2
3528.1.bw.a 2 840.dc even 12 2
3528.1.bw.a 2 840.dh odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} \) acting on \(S_{1}^{\mathrm{new}}(1400, [\chi])\).