Properties

Label 1470.2.n.d
Level $1470$
Weight $2$
Character orbit 1470.n
Analytic conductor $11.738$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1470.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.7380090971\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{12} - \zeta_{12}^{3} ) q^{2} + \zeta_{12} q^{3} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{5} + q^{6} -\zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + ( \zeta_{12} - \zeta_{12}^{3} ) q^{2} + \zeta_{12} q^{3} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{5} + q^{6} -\zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} + ( 2 - \zeta_{12} - 2 \zeta_{12}^{2} ) q^{10} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{12} + 2 \zeta_{12}^{3} q^{13} + ( 2 - \zeta_{12}^{3} ) q^{15} -\zeta_{12}^{2} q^{16} + 2 \zeta_{12} q^{17} + \zeta_{12} q^{18} -2 \zeta_{12}^{2} q^{19} + ( -1 - 2 \zeta_{12}^{3} ) q^{20} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{23} + ( 1 - \zeta_{12}^{2} ) q^{24} + ( 3 - 4 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{25} + 2 \zeta_{12}^{2} q^{26} + \zeta_{12}^{3} q^{27} + 8 q^{29} + ( 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{30} + ( 4 - 4 \zeta_{12}^{2} ) q^{31} -\zeta_{12} q^{32} + 2 q^{34} + q^{36} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{37} -2 \zeta_{12} q^{38} + ( -2 + 2 \zeta_{12}^{2} ) q^{39} + ( -\zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{40} -10 q^{41} + 2 \zeta_{12}^{3} q^{43} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{2} ) q^{45} + ( 8 - 8 \zeta_{12}^{2} ) q^{46} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{47} -\zeta_{12}^{3} q^{48} + ( -4 - 3 \zeta_{12}^{3} ) q^{50} + 2 \zeta_{12}^{2} q^{51} + 2 \zeta_{12} q^{52} -6 \zeta_{12} q^{53} + \zeta_{12}^{2} q^{54} -2 \zeta_{12}^{3} q^{57} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{58} + ( -12 + 12 \zeta_{12}^{2} ) q^{59} + ( 2 - \zeta_{12} - 2 \zeta_{12}^{2} ) q^{60} + 2 \zeta_{12}^{2} q^{61} -4 \zeta_{12}^{3} q^{62} - q^{64} + ( 2 \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{65} -14 \zeta_{12} q^{67} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{68} + 8 q^{69} + 6 q^{71} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{72} -10 \zeta_{12} q^{73} + ( 6 - 6 \zeta_{12}^{2} ) q^{74} + ( 3 \zeta_{12} - 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{75} -2 q^{76} + 2 \zeta_{12}^{3} q^{78} + 4 \zeta_{12}^{2} q^{79} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{2} ) q^{80} + ( -1 + \zeta_{12}^{2} ) q^{81} + ( -10 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{82} -12 \zeta_{12}^{3} q^{83} + ( 4 - 2 \zeta_{12}^{3} ) q^{85} + 2 \zeta_{12}^{2} q^{86} + 8 \zeta_{12} q^{87} + 14 \zeta_{12}^{2} q^{89} + ( 2 - \zeta_{12}^{3} ) q^{90} -8 \zeta_{12}^{3} q^{92} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{93} + ( -6 + 6 \zeta_{12}^{2} ) q^{94} + ( -2 - 4 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{95} -\zeta_{12}^{2} q^{96} + 14 \zeta_{12}^{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} - 2q^{5} + 4q^{6} + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{4} - 2q^{5} + 4q^{6} + 2q^{9} + 4q^{10} + 8q^{15} - 2q^{16} - 4q^{19} - 4q^{20} + 2q^{24} + 6q^{25} + 4q^{26} + 32q^{29} - 2q^{30} + 8q^{31} + 8q^{34} + 4q^{36} - 4q^{39} - 4q^{40} - 40q^{41} + 2q^{45} + 16q^{46} - 16q^{50} + 4q^{51} + 2q^{54} - 24q^{59} + 4q^{60} + 4q^{61} - 4q^{64} + 8q^{65} + 32q^{69} + 24q^{71} + 12q^{74} - 8q^{75} - 8q^{76} + 8q^{79} - 2q^{80} - 2q^{81} + 16q^{85} + 4q^{86} + 28q^{89} + 8q^{90} - 12q^{94} - 4q^{95} - 2q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times\).

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(-\zeta_{12}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −2.23205 0.133975i 1.00000 0 1.00000i 0.500000 0.866025i 1.86603 + 1.23205i
79.2 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 1.23205 + 1.86603i 1.00000 0 1.00000i 0.500000 0.866025i 0.133975 + 2.23205i
949.1 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i −2.23205 + 0.133975i 1.00000 0 1.00000i 0.500000 + 0.866025i 1.86603 1.23205i
949.2 0.866025 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i 1.23205 1.86603i 1.00000 0 1.00000i 0.500000 + 0.866025i 0.133975 2.23205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.n.d 4
5.b even 2 1 inner 1470.2.n.d 4
7.b odd 2 1 1470.2.n.e 4
7.c even 3 1 1470.2.g.d yes 2
7.c even 3 1 inner 1470.2.n.d 4
7.d odd 6 1 1470.2.g.c 2
7.d odd 6 1 1470.2.n.e 4
35.c odd 2 1 1470.2.n.e 4
35.i odd 6 1 1470.2.g.c 2
35.i odd 6 1 1470.2.n.e 4
35.j even 6 1 1470.2.g.d yes 2
35.j even 6 1 inner 1470.2.n.d 4
35.k even 12 1 7350.2.a.bc 1
35.k even 12 1 7350.2.a.bx 1
35.l odd 12 1 7350.2.a.m 1
35.l odd 12 1 7350.2.a.cr 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.g.c 2 7.d odd 6 1
1470.2.g.c 2 35.i odd 6 1
1470.2.g.d yes 2 7.c even 3 1
1470.2.g.d yes 2 35.j even 6 1
1470.2.n.d 4 1.a even 1 1 trivial
1470.2.n.d 4 5.b even 2 1 inner
1470.2.n.d 4 7.c even 3 1 inner
1470.2.n.d 4 35.j even 6 1 inner
1470.2.n.e 4 7.b odd 2 1
1470.2.n.e 4 7.d odd 6 1
1470.2.n.e 4 35.c odd 2 1
1470.2.n.e 4 35.i odd 6 1
7350.2.a.m 1 35.l odd 12 1
7350.2.a.bc 1 35.k even 12 1
7350.2.a.bx 1 35.k even 12 1
7350.2.a.cr 1 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1470, [\chi])\):

\( T_{11} \)
\( T_{17}^{4} - 4 T_{17}^{2} + 16 \)
\( T_{19}^{2} + 2 T_{19} + 4 \)
\( T_{31}^{2} - 4 T_{31} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( 25 + 10 T - T^{2} + 2 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( 4 + T^{2} )^{2} \)
$17$ \( 16 - 4 T^{2} + T^{4} \)
$19$ \( ( 4 + 2 T + T^{2} )^{2} \)
$23$ \( 4096 - 64 T^{2} + T^{4} \)
$29$ \( ( -8 + T )^{4} \)
$31$ \( ( 16 - 4 T + T^{2} )^{2} \)
$37$ \( 1296 - 36 T^{2} + T^{4} \)
$41$ \( ( 10 + T )^{4} \)
$43$ \( ( 4 + T^{2} )^{2} \)
$47$ \( 1296 - 36 T^{2} + T^{4} \)
$53$ \( 1296 - 36 T^{2} + T^{4} \)
$59$ \( ( 144 + 12 T + T^{2} )^{2} \)
$61$ \( ( 4 - 2 T + T^{2} )^{2} \)
$67$ \( 38416 - 196 T^{2} + T^{4} \)
$71$ \( ( -6 + T )^{4} \)
$73$ \( 10000 - 100 T^{2} + T^{4} \)
$79$ \( ( 16 - 4 T + T^{2} )^{2} \)
$83$ \( ( 144 + T^{2} )^{2} \)
$89$ \( ( 196 - 14 T + T^{2} )^{2} \)
$97$ \( ( 196 + T^{2} )^{2} \)
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