Properties

Label 1400.2.q.a
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: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
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.

\(f(q)\) \(=\) \( q + ( -2 + 2 \zeta_{6} ) q^{3} + ( -3 + 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( -2 + 2 \zeta_{6} ) q^{3} + ( -3 + 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{11} + 3 q^{13} + ( -2 + 2 \zeta_{6} ) q^{17} + 5 \zeta_{6} q^{19} + ( 2 - 6 \zeta_{6} ) q^{21} + 7 \zeta_{6} q^{23} -4 q^{27} -6 q^{29} + ( -4 + 4 \zeta_{6} ) q^{31} + 2 \zeta_{6} q^{33} -5 \zeta_{6} q^{37} + ( -6 + 6 \zeta_{6} ) q^{39} -5 q^{41} -6 q^{43} -9 \zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} -4 \zeta_{6} q^{51} + ( 11 - 11 \zeta_{6} ) q^{53} -10 q^{57} + ( -8 + 8 \zeta_{6} ) q^{59} + 12 \zeta_{6} q^{61} + ( 2 + \zeta_{6} ) q^{63} + ( -4 + 4 \zeta_{6} ) q^{67} -14 q^{69} -4 q^{71} + ( 12 - 12 \zeta_{6} ) q^{73} + ( -1 + 3 \zeta_{6} ) q^{77} -14 \zeta_{6} q^{79} + ( 11 - 11 \zeta_{6} ) q^{81} + 4 q^{83} + ( 12 - 12 \zeta_{6} ) q^{87} -6 \zeta_{6} q^{89} + ( -9 + 6 \zeta_{6} ) q^{91} -8 \zeta_{6} q^{93} -6 q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 4q^{7} - q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 4q^{7} - q^{9} + q^{11} + 6q^{13} - 2q^{17} + 5q^{19} - 2q^{21} + 7q^{23} - 8q^{27} - 12q^{29} - 4q^{31} + 2q^{33} - 5q^{37} - 6q^{39} - 10q^{41} - 12q^{43} - 9q^{47} + 2q^{49} - 4q^{51} + 11q^{53} - 20q^{57} - 8q^{59} + 12q^{61} + 5q^{63} - 4q^{67} - 28q^{69} - 8q^{71} + 12q^{73} + q^{77} - 14q^{79} + 11q^{81} + 8q^{83} + 12q^{87} - 6q^{89} - 12q^{91} - 8q^{93} - 12q^{97} - 2q^{99} + O(q^{100}) \)

Character values

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

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

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 −2.00000 1.73205i 0 −0.500000 + 0.866025i 0
1201.1 0 −1.00000 + 1.73205i 0 0 0 −2.00000 + 1.73205i 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.a 2
5.b even 2 1 280.2.q.c 2
5.c odd 4 2 1400.2.bh.e 4
7.c even 3 1 inner 1400.2.q.a 2
7.c even 3 1 9800.2.a.bi 1
7.d odd 6 1 9800.2.a.g 1
15.d odd 2 1 2520.2.bi.e 2
20.d odd 2 1 560.2.q.c 2
35.c odd 2 1 1960.2.q.c 2
35.i odd 6 1 1960.2.a.m 1
35.i odd 6 1 1960.2.q.c 2
35.j even 6 1 280.2.q.c 2
35.j even 6 1 1960.2.a.a 1
35.l odd 12 2 1400.2.bh.e 4
105.o odd 6 1 2520.2.bi.e 2
140.p odd 6 1 560.2.q.c 2
140.p odd 6 1 3920.2.a.bf 1
140.s even 6 1 3920.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.c 2 5.b even 2 1
280.2.q.c 2 35.j even 6 1
560.2.q.c 2 20.d odd 2 1
560.2.q.c 2 140.p odd 6 1
1400.2.q.a 2 1.a even 1 1 trivial
1400.2.q.a 2 7.c even 3 1 inner
1400.2.bh.e 4 5.c odd 4 2
1400.2.bh.e 4 35.l odd 12 2
1960.2.a.a 1 35.j even 6 1
1960.2.a.m 1 35.i odd 6 1
1960.2.q.c 2 35.c odd 2 1
1960.2.q.c 2 35.i odd 6 1
2520.2.bi.e 2 15.d odd 2 1
2520.2.bi.e 2 105.o odd 6 1
3920.2.a.i 1 140.s even 6 1
3920.2.a.bf 1 140.p odd 6 1
9800.2.a.g 1 7.d odd 6 1
9800.2.a.bi 1 7.c even 3 1

Hecke kernels

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

\( T_{3}^{2} + 2 T_{3} + 4 \)
\( T_{11}^{2} - T_{11} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 + 2 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 + 4 T + T^{2} \)
$11$ \( 1 - T + T^{2} \)
$13$ \( ( -3 + T )^{2} \)
$17$ \( 4 + 2 T + T^{2} \)
$19$ \( 25 - 5 T + T^{2} \)
$23$ \( 49 - 7 T + T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( 16 + 4 T + T^{2} \)
$37$ \( 25 + 5 T + T^{2} \)
$41$ \( ( 5 + T )^{2} \)
$43$ \( ( 6 + T )^{2} \)
$47$ \( 81 + 9 T + T^{2} \)
$53$ \( 121 - 11 T + T^{2} \)
$59$ \( 64 + 8 T + T^{2} \)
$61$ \( 144 - 12 T + T^{2} \)
$67$ \( 16 + 4 T + T^{2} \)
$71$ \( ( 4 + T )^{2} \)
$73$ \( 144 - 12 T + T^{2} \)
$79$ \( 196 + 14 T + T^{2} \)
$83$ \( ( -4 + T )^{2} \)
$89$ \( 36 + 6 T + T^{2} \)
$97$ \( ( 6 + T )^{2} \)
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