Properties

Label 1400.2.x.c
Level $1400$
Weight $2$
Character orbit 1400.x
Analytic conductor $11.179$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.x (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32q + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q - 16q^{11} - 40q^{21} + 32q^{51} + 128q^{71} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
657.1 0 −2.09284 2.09284i 0 0 0 0.510946 2.59595i 0 5.75996i 0
657.2 0 −2.09284 2.09284i 0 0 0 2.59595 0.510946i 0 5.75996i 0
657.3 0 −1.26512 1.26512i 0 0 0 −1.75168 1.98283i 0 0.201074i 0
657.4 0 −1.26512 1.26512i 0 0 0 1.98283 + 1.75168i 0 0.201074i 0
657.5 0 −0.923076 0.923076i 0 0 0 −2.48246 + 0.915096i 0 1.29586i 0
657.6 0 −0.923076 0.923076i 0 0 0 −0.915096 + 2.48246i 0 1.29586i 0
657.7 0 −0.409160 0.409160i 0 0 0 0.738062 2.54072i 0 2.66518i 0
657.8 0 −0.409160 0.409160i 0 0 0 2.54072 0.738062i 0 2.66518i 0
657.9 0 0.409160 + 0.409160i 0 0 0 −2.54072 + 0.738062i 0 2.66518i 0
657.10 0 0.409160 + 0.409160i 0 0 0 −0.738062 + 2.54072i 0 2.66518i 0
657.11 0 0.923076 + 0.923076i 0 0 0 0.915096 2.48246i 0 1.29586i 0
657.12 0 0.923076 + 0.923076i 0 0 0 2.48246 0.915096i 0 1.29586i 0
657.13 0 1.26512 + 1.26512i 0 0 0 −1.98283 1.75168i 0 0.201074i 0
657.14 0 1.26512 + 1.26512i 0 0 0 1.75168 + 1.98283i 0 0.201074i 0
657.15 0 2.09284 + 2.09284i 0 0 0 −2.59595 + 0.510946i 0 5.75996i 0
657.16 0 2.09284 + 2.09284i 0 0 0 −0.510946 + 2.59595i 0 5.75996i 0
993.1 0 −2.09284 + 2.09284i 0 0 0 0.510946 + 2.59595i 0 5.75996i 0
993.2 0 −2.09284 + 2.09284i 0 0 0 2.59595 + 0.510946i 0 5.75996i 0
993.3 0 −1.26512 + 1.26512i 0 0 0 −1.75168 + 1.98283i 0 0.201074i 0
993.4 0 −1.26512 + 1.26512i 0 0 0 1.98283 1.75168i 0 0.201074i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 993.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
7.b odd 2 1 inner
35.c odd 2 1 inner
35.f even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.x.c 32
5.b even 2 1 inner 1400.2.x.c 32
5.c odd 4 2 inner 1400.2.x.c 32
7.b odd 2 1 inner 1400.2.x.c 32
35.c odd 2 1 inner 1400.2.x.c 32
35.f even 4 2 inner 1400.2.x.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.x.c 32 1.a even 1 1 trivial
1400.2.x.c 32 5.b even 2 1 inner
1400.2.x.c 32 5.c odd 4 2 inner
1400.2.x.c 32 7.b odd 2 1 inner
1400.2.x.c 32 35.c odd 2 1 inner
1400.2.x.c 32 35.f even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 90 T_{3}^{12} + 1049 T_{3}^{8} + 2400 T_{3}^{4} + 256 \) acting on \(S_{2}^{\mathrm{new}}(1400, [\chi])\).