Properties

Label 1400.2.q.f
Level $1400$
Weight $2$
Character orbit 1400.q
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.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{3} + (3 \zeta_{6} - 2) q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{3} + (3 \zeta_{6} - 2) q^{7} - \zeta_{6} q^{9} + (4 \zeta_{6} - 4) q^{11} + 2 q^{13} + (3 \zeta_{6} - 3) q^{17} + (4 \zeta_{6} + 2) q^{21} + 3 \zeta_{6} q^{23} + 4 q^{27} - 6 q^{29} + (9 \zeta_{6} - 9) q^{31} + 8 \zeta_{6} q^{33} + ( - 4 \zeta_{6} + 4) q^{39} + 5 q^{41} + 6 q^{43} + 9 \zeta_{6} q^{47} + ( - 3 \zeta_{6} - 5) q^{49} + 6 \zeta_{6} q^{51} + (6 \zeta_{6} - 6) q^{53} + (8 \zeta_{6} - 8) q^{59} - 8 \zeta_{6} q^{61} + ( - \zeta_{6} + 3) q^{63} + ( - 14 \zeta_{6} + 14) q^{67} + 6 q^{69} + 11 q^{71} + (2 \zeta_{6} - 2) q^{73} + ( - 8 \zeta_{6} - 4) q^{77} - 9 \zeta_{6} q^{79} + ( - 11 \zeta_{6} + 11) q^{81} + 6 q^{83} + (12 \zeta_{6} - 12) q^{87} - 11 \zeta_{6} q^{89} + (6 \zeta_{6} - 4) q^{91} + 18 \zeta_{6} q^{93} + 11 q^{97} + 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - q^{7} - q^{9} - 4 q^{11} + 4 q^{13} - 3 q^{17} + 8 q^{21} + 3 q^{23} + 8 q^{27} - 12 q^{29} - 9 q^{31} + 8 q^{33} + 4 q^{39} + 10 q^{41} + 12 q^{43} + 9 q^{47} - 13 q^{49} + 6 q^{51} - 6 q^{53} - 8 q^{59} - 8 q^{61} + 5 q^{63} + 14 q^{67} + 12 q^{69} + 22 q^{71} - 2 q^{73} - 16 q^{77} - 9 q^{79} + 11 q^{81} + 12 q^{83} - 12 q^{87} - 11 q^{89} - 2 q^{91} + 18 q^{93} + 22 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.00000 + 1.73205i 0 0 0 −0.500000 2.59808i 0 −0.500000 + 0.866025i 0
1201.1 0 1.00000 1.73205i 0 0 0 −0.500000 + 2.59808i 0 −0.500000 0.866025i 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.q.f yes 2
5.b even 2 1 1400.2.q.b 2
5.c odd 4 2 1400.2.bh.d 4
7.c even 3 1 inner 1400.2.q.f yes 2
7.c even 3 1 9800.2.a.j 1
7.d odd 6 1 9800.2.a.bl 1
35.i odd 6 1 9800.2.a.k 1
35.j even 6 1 1400.2.q.b 2
35.j even 6 1 9800.2.a.bm 1
35.l odd 12 2 1400.2.bh.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.q.b 2 5.b even 2 1
1400.2.q.b 2 35.j even 6 1
1400.2.q.f yes 2 1.a even 1 1 trivial
1400.2.q.f yes 2 7.c even 3 1 inner
1400.2.bh.d 4 5.c odd 4 2
1400.2.bh.d 4 35.l odd 12 2
9800.2.a.j 1 7.c even 3 1
9800.2.a.k 1 35.i odd 6 1
9800.2.a.bl 1 7.d odd 6 1
9800.2.a.bm 1 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1400, [\chi])\):

\( T_{3}^{2} - 2T_{3} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( (T - 5)^{2} \) Copy content Toggle raw display
$43$ \( (T - 6)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$71$ \( (T - 11)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$97$ \( (T - 11)^{2} \) Copy content Toggle raw display
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