Properties

Label 1400.2.q.j
Level $1400$
Weight $2$
Character orbit 1400.q
Analytic conductor $11.179$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.11337408.1
Defining polynomial: \(x^{6} + 18 x^{4} + 81 x^{2} + 12\)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{3} + \beta_{4} - \beta_{5} ) q^{3} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + ( -\beta_{1} - 3 \beta_{2} + \beta_{4} ) q^{9} +O(q^{10})\) \( q + ( \beta_{3} + \beta_{4} - \beta_{5} ) q^{3} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + ( -\beta_{1} - 3 \beta_{2} + \beta_{4} ) q^{9} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{4} ) q^{11} + ( -1 - \beta_{1} + 2 \beta_{3} ) q^{13} + ( 2 - 2 \beta_{2} ) q^{17} + ( -\beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{19} + ( 2 - \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{21} + ( -\beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{23} + ( 6 - 3 \beta_{3} - 3 \beta_{4} ) q^{27} + ( 4 + \beta_{1} + 2 \beta_{4} ) q^{29} + ( -4 + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{31} + ( -6 \beta_{2} + 4 \beta_{5} ) q^{33} + ( -\beta_{1} + 3 \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{37} + ( 6 - 4 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{39} + ( 3 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{41} + ( -4 - \beta_{3} - \beta_{4} ) q^{43} + ( -3 \beta_{1} - 5 \beta_{2} + 3 \beta_{4} - 2 \beta_{5} ) q^{47} + ( -1 - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{49} -2 \beta_{5} q^{51} + ( 3 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{53} + ( -6 + 2 \beta_{1} - 4 \beta_{3} ) q^{57} + ( -8 + 8 \beta_{2} ) q^{59} + ( -\beta_{1} + 2 \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{61} + ( -6 + \beta_{1} + 9 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{63} + ( 2 - 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{67} + ( \beta_{1} - 2 \beta_{3} ) q^{69} + ( 4 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{73} + ( 7 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{77} + ( -2 \beta_{1} + 6 \beta_{2} + 2 \beta_{4} - 4 \beta_{5} ) q^{79} + ( -9 + 9 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} ) q^{81} + ( 10 + \beta_{3} + \beta_{4} ) q^{83} + ( 6 - 6 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{87} + ( \beta_{1} - \beta_{4} + 2 \beta_{5} ) q^{89} + ( 9 + \beta_{1} - 10 \beta_{2} - 2 \beta_{5} ) q^{91} + ( 2 \beta_{1} + 12 \beta_{2} - 2 \beta_{4} + 4 \beta_{5} ) q^{93} + 2 q^{97} + ( 21 - \beta_{1} - 6 \beta_{3} - 8 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{7} - 9q^{9} + O(q^{10}) \) \( 6q - 6q^{7} - 9q^{9} - 3q^{11} - 6q^{13} + 6q^{17} - 3q^{19} + 3q^{23} + 36q^{27} + 24q^{29} - 12q^{31} - 18q^{33} + 9q^{37} + 18q^{39} + 18q^{41} - 24q^{43} - 15q^{47} - 12q^{49} + 9q^{53} - 36q^{57} - 24q^{59} + 6q^{61} - 9q^{63} + 6q^{67} + 39q^{77} + 18q^{79} - 27q^{81} + 60q^{83} + 18q^{87} + 24q^{91} + 36q^{93} + 12q^{97} + 126q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 18 x^{4} + 81 x^{2} + 12\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 9 \nu + 2 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{2} + \nu + 6 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} + 9 \nu^{2} - 2 \nu \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} + \nu^{4} + 15 \nu^{3} + 11 \nu^{2} + 48 \nu + 12 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{3} - \beta_{1} - 6\)
\(\nu^{3}\)\(=\)\(4 \beta_{2} - 9 \beta_{1} - 2\)
\(\nu^{4}\)\(=\)\(4 \beta_{4} - 18 \beta_{3} + 11 \beta_{1} + 54\)
\(\nu^{5}\)\(=\)\(8 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} - 60 \beta_{2} + 87 \beta_{1} + 30\)

Character values

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

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\) \(1\)

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
2.78499i
3.17656i
0.391571i
2.78499i
3.17656i
0.391571i
0 −1.64497 2.84918i 0 0 0 −2.64497 0.0641892i 0 −3.91187 + 6.77556i 0
401.2 0 0.352860 + 0.611171i 0 0 0 −0.647140 2.56539i 0 1.25098 2.16676i 0
401.3 0 1.29211 + 2.23800i 0 0 0 0.292113 + 2.62958i 0 −1.83911 + 3.18543i 0
1201.1 0 −1.64497 + 2.84918i 0 0 0 −2.64497 + 0.0641892i 0 −3.91187 6.77556i 0
1201.2 0 0.352860 0.611171i 0 0 0 −0.647140 + 2.56539i 0 1.25098 + 2.16676i 0
1201.3 0 1.29211 2.23800i 0 0 0 0.292113 2.62958i 0 −1.83911 3.18543i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1201.3
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.j 6
5.b even 2 1 280.2.q.e 6
5.c odd 4 2 1400.2.bh.i 12
7.c even 3 1 inner 1400.2.q.j 6
7.c even 3 1 9800.2.a.ce 3
7.d odd 6 1 9800.2.a.cf 3
15.d odd 2 1 2520.2.bi.q 6
20.d odd 2 1 560.2.q.l 6
35.c odd 2 1 1960.2.q.w 6
35.i odd 6 1 1960.2.a.v 3
35.i odd 6 1 1960.2.q.w 6
35.j even 6 1 280.2.q.e 6
35.j even 6 1 1960.2.a.w 3
35.l odd 12 2 1400.2.bh.i 12
105.o odd 6 1 2520.2.bi.q 6
140.p odd 6 1 560.2.q.l 6
140.p odd 6 1 3920.2.a.cc 3
140.s even 6 1 3920.2.a.cb 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.e 6 5.b even 2 1
280.2.q.e 6 35.j even 6 1
560.2.q.l 6 20.d odd 2 1
560.2.q.l 6 140.p odd 6 1
1400.2.q.j 6 1.a even 1 1 trivial
1400.2.q.j 6 7.c even 3 1 inner
1400.2.bh.i 12 5.c odd 4 2
1400.2.bh.i 12 35.l odd 12 2
1960.2.a.v 3 35.i odd 6 1
1960.2.a.w 3 35.j even 6 1
1960.2.q.w 6 35.c odd 2 1
1960.2.q.w 6 35.i odd 6 1
2520.2.bi.q 6 15.d odd 2 1
2520.2.bi.q 6 105.o odd 6 1
3920.2.a.cb 3 140.s even 6 1
3920.2.a.cc 3 140.p odd 6 1
9800.2.a.ce 3 7.c even 3 1
9800.2.a.cf 3 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 \(S_{2}^{\mathrm{new}}(1400, [\chi])\):

\( T_{3}^{6} + 9 T_{3}^{4} - 12 T_{3}^{3} + 81 T_{3}^{2} - 54 T_{3} + 36 \)
\( T_{11}^{6} + 3 T_{11}^{5} + 33 T_{11}^{4} + 16 T_{11}^{3} + 708 T_{11}^{2} + 1056 T_{11} + 1936 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 36 - 54 T + 81 T^{2} - 12 T^{3} + 9 T^{4} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 343 + 294 T + 168 T^{2} + 80 T^{3} + 24 T^{4} + 6 T^{5} + T^{6} \)
$11$ \( 1936 + 1056 T + 708 T^{2} + 16 T^{3} + 33 T^{4} + 3 T^{5} + T^{6} \)
$13$ \( ( -68 - 24 T + 3 T^{2} + T^{3} )^{2} \)
$17$ \( ( 4 - 2 T + T^{2} )^{3} \)
$19$ \( 256 - 384 T + 528 T^{2} - 104 T^{3} + 33 T^{4} + 3 T^{5} + T^{6} \)
$23$ \( 49 + 105 T + 204 T^{2} + 59 T^{3} + 24 T^{4} - 3 T^{5} + T^{6} \)
$29$ \( ( 26 + 21 T - 12 T^{2} + T^{3} )^{2} \)
$31$ \( 16384 - 1536 T + 1680 T^{2} + 400 T^{3} + 132 T^{4} + 12 T^{5} + T^{6} \)
$37$ \( 9216 + 864 T^{2} - 192 T^{3} + 81 T^{4} - 9 T^{5} + T^{6} \)
$41$ \( ( 381 - 45 T - 9 T^{2} + T^{3} )^{2} \)
$43$ \( ( 22 + 39 T + 12 T^{2} + T^{3} )^{2} \)
$47$ \( 2521744 + 152448 T + 33036 T^{2} + 1736 T^{3} + 321 T^{4} + 15 T^{5} + T^{6} \)
$53$ \( 389376 - 44928 T + 10800 T^{2} - 600 T^{3} + 153 T^{4} - 9 T^{5} + T^{6} \)
$59$ \( ( 64 + 8 T + T^{2} )^{3} \)
$61$ \( 295936 - 47328 T + 10833 T^{2} - 566 T^{3} + 123 T^{4} - 6 T^{5} + T^{6} \)
$67$ \( 64 + 552 T + 4713 T^{2} + 430 T^{3} + 105 T^{4} - 6 T^{5} + T^{6} \)
$71$ \( T^{6} \)
$73$ \( 112896 - 36288 T + 11664 T^{2} - 672 T^{3} + 108 T^{4} + T^{6} \)
$79$ \( 589824 + 13824 T^{2} - 1536 T^{3} + 324 T^{4} - 18 T^{5} + T^{6} \)
$83$ \( ( -904 + 291 T - 30 T^{2} + T^{3} )^{2} \)
$89$ \( 1764 + 1134 T + 729 T^{2} + 84 T^{3} + 27 T^{4} + T^{6} \)
$97$ \( ( -2 + T )^{6} \)
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