Properties

Label 1470.2.n.f
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: no (minimal twist has level 210)
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} + ( -2 + 2 \zeta_{12}^{2} ) q^{11} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{12} + 6 \zeta_{12}^{3} q^{13} + ( -2 - \zeta_{12}^{3} ) q^{15} -\zeta_{12}^{2} q^{16} + 4 \zeta_{12} q^{17} + \zeta_{12} q^{18} + 6 \zeta_{12}^{2} q^{19} + ( 1 - 2 \zeta_{12}^{3} ) q^{20} + 2 \zeta_{12}^{3} q^{22} + ( 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} + 6 \zeta_{12}^{2} q^{26} -\zeta_{12}^{3} q^{27} -6 q^{29} + ( -2 \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{30} + ( -2 + 2 \zeta_{12}^{2} ) q^{31} -\zeta_{12} q^{32} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{33} + 4 q^{34} + q^{36} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{37} + 6 \zeta_{12} q^{38} + ( 6 - 6 \zeta_{12}^{2} ) q^{39} + ( \zeta_{12} - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{40} -2 q^{41} -4 \zeta_{12}^{3} q^{43} + 2 \zeta_{12}^{2} q^{44} + ( -1 + 2 \zeta_{12} + \zeta_{12}^{2} ) q^{45} + ( 8 - 8 \zeta_{12}^{2} ) q^{46} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{47} + \zeta_{12}^{3} q^{48} + ( 4 - 3 \zeta_{12}^{3} ) q^{50} -4 \zeta_{12}^{2} q^{51} + 6 \zeta_{12} q^{52} + 6 \zeta_{12} q^{53} -\zeta_{12}^{2} q^{54} + ( -2 + 4 \zeta_{12}^{3} ) q^{55} -6 \zeta_{12}^{3} q^{57} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{58} + ( 8 - 8 \zeta_{12}^{2} ) q^{59} + ( -2 - \zeta_{12} + 2 \zeta_{12}^{2} ) q^{60} -10 \zeta_{12}^{2} q^{61} + 2 \zeta_{12}^{3} q^{62} - q^{64} + ( -6 \zeta_{12} + 12 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{65} + ( 2 - 2 \zeta_{12}^{2} ) q^{66} -8 \zeta_{12} q^{67} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{68} -8 q^{69} -6 q^{71} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{72} + 14 \zeta_{12} q^{73} + ( 4 - 4 \zeta_{12}^{2} ) q^{74} + ( -3 \zeta_{12} - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{75} + 6 q^{76} -6 \zeta_{12}^{3} q^{78} -12 \zeta_{12}^{2} q^{79} + ( 1 - 2 \zeta_{12} - \zeta_{12}^{2} ) q^{80} + ( -1 + \zeta_{12}^{2} ) q^{81} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{82} -8 \zeta_{12}^{3} q^{83} + ( 8 + 4 \zeta_{12}^{3} ) q^{85} -4 \zeta_{12}^{2} q^{86} + 6 \zeta_{12} q^{87} + 2 \zeta_{12} q^{88} + 10 \zeta_{12}^{2} q^{89} + ( 2 + \zeta_{12}^{3} ) q^{90} -8 \zeta_{12}^{3} q^{92} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{93} + ( -8 + 8 \zeta_{12}^{2} ) q^{94} + ( -6 + 12 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{95} + \zeta_{12}^{2} q^{96} + 10 \zeta_{12}^{3} q^{97} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + 2q^{5} - 4q^{6} + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{4} + 2q^{5} - 4q^{6} + 2q^{9} + 4q^{10} - 4q^{11} - 8q^{15} - 2q^{16} + 12q^{19} + 4q^{20} - 2q^{24} + 6q^{25} + 12q^{26} - 24q^{29} - 2q^{30} - 4q^{31} + 16q^{34} + 4q^{36} + 12q^{39} - 4q^{40} - 8q^{41} + 4q^{44} - 2q^{45} + 16q^{46} + 16q^{50} - 8q^{51} - 2q^{54} - 8q^{55} + 16q^{59} - 4q^{60} - 20q^{61} - 4q^{64} + 24q^{65} + 4q^{66} - 32q^{69} - 24q^{71} + 8q^{74} - 8q^{75} + 24q^{76} - 24q^{79} + 2q^{80} - 2q^{81} + 32q^{85} - 8q^{86} + 20q^{89} + 8q^{90} - 16q^{94} - 12q^{95} + 2q^{96} - 8q^{99} + O(q^{100}) \)

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 −1.23205 1.86603i −1.00000 0 1.00000i 0.500000 0.866025i 0.133975 + 2.23205i
79.2 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
949.1 −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
949.2 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
\(n\): 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.f 4
5.b even 2 1 inner 1470.2.n.f 4
7.b odd 2 1 1470.2.n.b 4
7.c even 3 1 1470.2.g.b 2
7.c even 3 1 inner 1470.2.n.f 4
7.d odd 6 1 210.2.g.b 2
7.d odd 6 1 1470.2.n.b 4
21.g even 6 1 630.2.g.c 2
28.f even 6 1 1680.2.t.e 2
35.c odd 2 1 1470.2.n.b 4
35.i odd 6 1 210.2.g.b 2
35.i odd 6 1 1470.2.n.b 4
35.j even 6 1 1470.2.g.b 2
35.j even 6 1 inner 1470.2.n.f 4
35.k even 12 1 1050.2.a.d 1
35.k even 12 1 1050.2.a.p 1
35.l odd 12 1 7350.2.a.bk 1
35.l odd 12 1 7350.2.a.bz 1
84.j odd 6 1 5040.2.t.h 2
105.p even 6 1 630.2.g.c 2
105.w odd 12 1 3150.2.a.d 1
105.w odd 12 1 3150.2.a.bk 1
140.s even 6 1 1680.2.t.e 2
140.x odd 12 1 8400.2.a.w 1
140.x odd 12 1 8400.2.a.bp 1
420.be odd 6 1 5040.2.t.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.g.b 2 7.d odd 6 1
210.2.g.b 2 35.i odd 6 1
630.2.g.c 2 21.g even 6 1
630.2.g.c 2 105.p even 6 1
1050.2.a.d 1 35.k even 12 1
1050.2.a.p 1 35.k even 12 1
1470.2.g.b 2 7.c even 3 1
1470.2.g.b 2 35.j even 6 1
1470.2.n.b 4 7.b odd 2 1
1470.2.n.b 4 7.d odd 6 1
1470.2.n.b 4 35.c odd 2 1
1470.2.n.b 4 35.i odd 6 1
1470.2.n.f 4 1.a even 1 1 trivial
1470.2.n.f 4 5.b even 2 1 inner
1470.2.n.f 4 7.c even 3 1 inner
1470.2.n.f 4 35.j even 6 1 inner
1680.2.t.e 2 28.f even 6 1
1680.2.t.e 2 140.s even 6 1
3150.2.a.d 1 105.w odd 12 1
3150.2.a.bk 1 105.w odd 12 1
5040.2.t.h 2 84.j odd 6 1
5040.2.t.h 2 420.be odd 6 1
7350.2.a.bk 1 35.l odd 12 1
7350.2.a.bz 1 35.l odd 12 1
8400.2.a.w 1 140.x odd 12 1
8400.2.a.bp 1 140.x odd 12 1

Hecke kernels

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

\( T_{11}^{2} + 2 T_{11} + 4 \)
\( T_{17}^{4} - 16 T_{17}^{2} + 256 \)
\( T_{19}^{2} - 6 T_{19} + 36 \)
\( T_{31}^{2} + 2 T_{31} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( 25 - 10 T - T^{2} - 2 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 4 + 2 T + T^{2} )^{2} \)
$13$ \( ( 36 + T^{2} )^{2} \)
$17$ \( 256 - 16 T^{2} + T^{4} \)
$19$ \( ( 36 - 6 T + T^{2} )^{2} \)
$23$ \( 4096 - 64 T^{2} + T^{4} \)
$29$ \( ( 6 + T )^{4} \)
$31$ \( ( 4 + 2 T + T^{2} )^{2} \)
$37$ \( 256 - 16 T^{2} + T^{4} \)
$41$ \( ( 2 + T )^{4} \)
$43$ \( ( 16 + T^{2} )^{2} \)
$47$ \( 4096 - 64 T^{2} + T^{4} \)
$53$ \( 1296 - 36 T^{2} + T^{4} \)
$59$ \( ( 64 - 8 T + T^{2} )^{2} \)
$61$ \( ( 100 + 10 T + T^{2} )^{2} \)
$67$ \( 4096 - 64 T^{2} + T^{4} \)
$71$ \( ( 6 + T )^{4} \)
$73$ \( 38416 - 196 T^{2} + T^{4} \)
$79$ \( ( 144 + 12 T + T^{2} )^{2} \)
$83$ \( ( 64 + T^{2} )^{2} \)
$89$ \( ( 100 - 10 T + T^{2} )^{2} \)
$97$ \( ( 100 + T^{2} )^{2} \)
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