Properties

Label 1400.2.q.g
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
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.

\(f(q)\) \(=\) \( q + ( 3 - 3 \zeta_{6} ) q^{3} + ( -3 + 2 \zeta_{6} ) q^{7} -6 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 3 - 3 \zeta_{6} ) q^{3} + ( -3 + 2 \zeta_{6} ) q^{7} -6 \zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{11} -2 q^{13} + ( 3 - 3 \zeta_{6} ) q^{17} -5 \zeta_{6} q^{19} + ( -3 + 9 \zeta_{6} ) q^{21} -3 \zeta_{6} q^{23} -9 q^{27} -6 q^{29} + ( 1 - \zeta_{6} ) q^{31} -3 \zeta_{6} q^{33} -5 \zeta_{6} q^{37} + ( -6 + 6 \zeta_{6} ) q^{39} -10 q^{41} + 4 q^{43} + \zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} -9 \zeta_{6} q^{51} + ( -9 + 9 \zeta_{6} ) q^{53} -15 q^{57} + ( -3 + 3 \zeta_{6} ) q^{59} -3 \zeta_{6} q^{61} + ( 12 + 6 \zeta_{6} ) q^{63} + ( 11 - 11 \zeta_{6} ) q^{67} -9 q^{69} + 16 q^{71} + ( 7 - 7 \zeta_{6} ) q^{73} + ( -1 + 3 \zeta_{6} ) q^{77} + 11 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + 4 q^{83} + ( -18 + 18 \zeta_{6} ) q^{87} + 9 \zeta_{6} q^{89} + ( 6 - 4 \zeta_{6} ) q^{91} -3 \zeta_{6} q^{93} -6 q^{97} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{3} - 4q^{7} - 6q^{9} + O(q^{10}) \) \( 2q + 3q^{3} - 4q^{7} - 6q^{9} + q^{11} - 4q^{13} + 3q^{17} - 5q^{19} + 3q^{21} - 3q^{23} - 18q^{27} - 12q^{29} + q^{31} - 3q^{33} - 5q^{37} - 6q^{39} - 20q^{41} + 8q^{43} + q^{47} + 2q^{49} - 9q^{51} - 9q^{53} - 30q^{57} - 3q^{59} - 3q^{61} + 30q^{63} + 11q^{67} - 18q^{69} + 32q^{71} + 7q^{73} + q^{77} + 11q^{79} - 9q^{81} + 8q^{83} - 18q^{87} + 9q^{89} + 8q^{91} - 3q^{93} - 12q^{97} - 12q^{99} + O(q^{100}) \)

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 1.50000 + 2.59808i 0 0 0 −2.00000 1.73205i 0 −3.00000 + 5.19615i 0
1201.1 0 1.50000 2.59808i 0 0 0 −2.00000 + 1.73205i 0 −3.00000 5.19615i 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.g 2
5.b even 2 1 56.2.i.a 2
5.c odd 4 2 1400.2.bh.f 4
7.c even 3 1 inner 1400.2.q.g 2
7.c even 3 1 9800.2.a.b 1
7.d odd 6 1 9800.2.a.bp 1
15.d odd 2 1 504.2.s.e 2
20.d odd 2 1 112.2.i.c 2
35.c odd 2 1 392.2.i.f 2
35.i odd 6 1 392.2.a.a 1
35.i odd 6 1 392.2.i.f 2
35.j even 6 1 56.2.i.a 2
35.j even 6 1 392.2.a.f 1
35.l odd 12 2 1400.2.bh.f 4
40.e odd 2 1 448.2.i.a 2
40.f even 2 1 448.2.i.f 2
60.h even 2 1 1008.2.s.e 2
105.g even 2 1 3528.2.s.o 2
105.o odd 6 1 504.2.s.e 2
105.o odd 6 1 3528.2.a.r 1
105.p even 6 1 3528.2.a.k 1
105.p even 6 1 3528.2.s.o 2
140.c even 2 1 784.2.i.a 2
140.p odd 6 1 112.2.i.c 2
140.p odd 6 1 784.2.a.a 1
140.s even 6 1 784.2.a.j 1
140.s even 6 1 784.2.i.a 2
280.ba even 6 1 3136.2.a.a 1
280.bf even 6 1 448.2.i.f 2
280.bf even 6 1 3136.2.a.b 1
280.bi odd 6 1 448.2.i.a 2
280.bi odd 6 1 3136.2.a.bc 1
280.bk odd 6 1 3136.2.a.bb 1
420.ba even 6 1 1008.2.s.e 2
420.ba even 6 1 7056.2.a.bi 1
420.be odd 6 1 7056.2.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.a 2 5.b even 2 1
56.2.i.a 2 35.j even 6 1
112.2.i.c 2 20.d odd 2 1
112.2.i.c 2 140.p odd 6 1
392.2.a.a 1 35.i odd 6 1
392.2.a.f 1 35.j even 6 1
392.2.i.f 2 35.c odd 2 1
392.2.i.f 2 35.i odd 6 1
448.2.i.a 2 40.e odd 2 1
448.2.i.a 2 280.bi odd 6 1
448.2.i.f 2 40.f even 2 1
448.2.i.f 2 280.bf even 6 1
504.2.s.e 2 15.d odd 2 1
504.2.s.e 2 105.o odd 6 1
784.2.a.a 1 140.p odd 6 1
784.2.a.j 1 140.s even 6 1
784.2.i.a 2 140.c even 2 1
784.2.i.a 2 140.s even 6 1
1008.2.s.e 2 60.h even 2 1
1008.2.s.e 2 420.ba even 6 1
1400.2.q.g 2 1.a even 1 1 trivial
1400.2.q.g 2 7.c even 3 1 inner
1400.2.bh.f 4 5.c odd 4 2
1400.2.bh.f 4 35.l odd 12 2
3136.2.a.a 1 280.ba even 6 1
3136.2.a.b 1 280.bf even 6 1
3136.2.a.bb 1 280.bk odd 6 1
3136.2.a.bc 1 280.bi odd 6 1
3528.2.a.k 1 105.p even 6 1
3528.2.a.r 1 105.o odd 6 1
3528.2.s.o 2 105.g even 2 1
3528.2.s.o 2 105.p even 6 1
7056.2.a.s 1 420.be odd 6 1
7056.2.a.bi 1 420.ba even 6 1
9800.2.a.b 1 7.c even 3 1
9800.2.a.bp 1 7.d odd 6 1

Hecke kernels

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

\( T_{3}^{2} - 3 T_{3} + 9 \)
\( T_{11}^{2} - T_{11} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T^{2} ) \)
$5$ 1
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + 2 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \)
$19$ \( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \)
$23$ \( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} \)
$37$ \( 1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 10 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 4 T + 43 T^{2} )^{2} \)
$47$ \( 1 - T - 46 T^{2} - 47 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 3 T - 52 T^{2} + 183 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 16 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} ) \)
$71$ \( ( 1 - 16 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 17 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \)
$79$ \( 1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 4 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 6 T + 97 T^{2} )^{2} \)
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