Properties

Label 1470.2.d.f
Level $1470$
Weight $2$
Character orbit 1470.d
Analytic conductor $11.738$
Analytic rank $0$
Dimension $8$
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.7380090971\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3317760000.3
Defining polynomial: \(x^{8} - 4 x^{6} + 7 x^{4} - 36 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( \beta_{2} + \beta_{5} ) q^{3} + q^{4} + ( -\beta_{2} - \beta_{6} ) q^{5} + ( \beta_{2} + \beta_{5} ) q^{6} + q^{8} + ( -2 - \beta_{7} ) q^{9} +O(q^{10})\) \( q + q^{2} + ( \beta_{2} + \beta_{5} ) q^{3} + q^{4} + ( -\beta_{2} - \beta_{6} ) q^{5} + ( \beta_{2} + \beta_{5} ) q^{6} + q^{8} + ( -2 - \beta_{7} ) q^{9} + ( -\beta_{2} - \beta_{6} ) q^{10} + ( \beta_{4} - \beta_{7} ) q^{11} + ( \beta_{2} + \beta_{5} ) q^{12} + ( 2 \beta_{2} + \beta_{3} ) q^{13} + ( -1 - \beta_{1} - \beta_{4} + \beta_{7} ) q^{15} + q^{16} + ( \beta_{2} + 2 \beta_{5} ) q^{17} + ( -2 - \beta_{7} ) q^{18} + ( \beta_{2} + 2 \beta_{5} - \beta_{6} ) q^{19} + ( -\beta_{2} - \beta_{6} ) q^{20} + ( \beta_{4} - \beta_{7} ) q^{22} + ( -1 - 2 \beta_{1} - \beta_{4} ) q^{23} + ( \beta_{2} + \beta_{5} ) q^{24} + ( -1 + 2 \beta_{4} ) q^{25} + ( 2 \beta_{2} + \beta_{3} ) q^{26} + ( -4 \beta_{2} - \beta_{5} ) q^{27} + ( \beta_{4} - 2 \beta_{7} ) q^{29} + ( -1 - \beta_{1} - \beta_{4} + \beta_{7} ) q^{30} + ( -\beta_{2} - 2 \beta_{5} - 2 \beta_{6} ) q^{31} + q^{32} + ( -2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{33} + ( \beta_{2} + 2 \beta_{5} ) q^{34} + ( -2 - \beta_{7} ) q^{36} + ( 2 \beta_{4} + \beta_{7} ) q^{37} + ( \beta_{2} + 2 \beta_{5} - \beta_{6} ) q^{38} + ( 2 + \beta_{1} - 2 \beta_{4} - 2 \beta_{7} ) q^{39} + ( -\beta_{2} - \beta_{6} ) q^{40} + ( -2 \beta_{2} - \beta_{3} ) q^{41} + ( \beta_{4} - 2 \beta_{7} ) q^{43} + ( \beta_{4} - \beta_{7} ) q^{44} + ( \beta_{2} - \beta_{3} - 2 \beta_{5} + 2 \beta_{6} ) q^{45} + ( -1 - 2 \beta_{1} - \beta_{4} ) q^{46} + ( 3 \beta_{2} + 6 \beta_{5} + \beta_{6} ) q^{47} + ( \beta_{2} + \beta_{5} ) q^{48} + ( -1 + 2 \beta_{4} ) q^{50} + ( -5 - \beta_{7} ) q^{51} + ( 2 \beta_{2} + \beta_{3} ) q^{52} + 5 q^{53} + ( -4 \beta_{2} - \beta_{5} ) q^{54} + ( 2 \beta_{2} - \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{55} + ( -5 - \beta_{1} - \beta_{4} - \beta_{7} ) q^{57} + ( \beta_{4} - 2 \beta_{7} ) q^{58} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{59} + ( -1 - \beta_{1} - \beta_{4} + \beta_{7} ) q^{60} -2 \beta_{6} q^{61} + ( -\beta_{2} - 2 \beta_{5} - 2 \beta_{6} ) q^{62} + q^{64} + ( -4 - 2 \beta_{1} - 3 \beta_{4} - 3 \beta_{7} ) q^{65} + ( -2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{66} + ( -4 \beta_{4} + 2 \beta_{7} ) q^{67} + ( \beta_{2} + 2 \beta_{5} ) q^{68} + ( -\beta_{2} - \beta_{3} - \beta_{5} + 5 \beta_{6} ) q^{69} + ( -\beta_{4} + 2 \beta_{7} ) q^{71} + ( -2 - \beta_{7} ) q^{72} + ( -3 \beta_{2} - 2 \beta_{3} ) q^{73} + ( 2 \beta_{4} + \beta_{7} ) q^{74} + ( -\beta_{2} + 2 \beta_{3} - \beta_{5} + 2 \beta_{6} ) q^{75} + ( \beta_{2} + 2 \beta_{5} - \beta_{6} ) q^{76} + ( 2 + \beta_{1} - 2 \beta_{4} - 2 \beta_{7} ) q^{78} + ( 6 - 2 \beta_{1} - \beta_{4} ) q^{79} + ( -\beta_{2} - \beta_{6} ) q^{80} + ( -1 + 4 \beta_{7} ) q^{81} + ( -2 \beta_{2} - \beta_{3} ) q^{82} + ( -2 \beta_{2} - 4 \beta_{5} + 6 \beta_{6} ) q^{83} + ( -2 \beta_{1} - \beta_{4} + 2 \beta_{7} ) q^{85} + ( \beta_{4} - 2 \beta_{7} ) q^{86} + ( -4 \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{87} + ( \beta_{4} - \beta_{7} ) q^{88} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{89} + ( \beta_{2} - \beta_{3} - 2 \beta_{5} + 2 \beta_{6} ) q^{90} + ( -1 - 2 \beta_{1} - \beta_{4} ) q^{92} + ( 5 - 2 \beta_{1} - 2 \beta_{4} + \beta_{7} ) q^{93} + ( 3 \beta_{2} + 6 \beta_{5} + \beta_{6} ) q^{94} + ( -3 - 2 \beta_{1} + 2 \beta_{7} ) q^{95} + ( \beta_{2} + \beta_{5} ) q^{96} + ( 3 \beta_{2} - 2 \beta_{3} ) q^{97} + ( -5 + 2 \beta_{1} - \beta_{4} + 2 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{2} + 8q^{4} + 8q^{8} - 16q^{9} + O(q^{10}) \) \( 8q + 8q^{2} + 8q^{4} + 8q^{8} - 16q^{9} - 8q^{15} + 8q^{16} - 16q^{18} - 8q^{23} - 8q^{25} - 8q^{30} + 8q^{32} - 16q^{36} + 16q^{39} - 8q^{46} - 8q^{50} - 40q^{51} + 40q^{53} - 40q^{57} - 8q^{60} + 8q^{64} - 32q^{65} - 16q^{72} + 16q^{78} + 48q^{79} - 8q^{81} - 8q^{92} + 40q^{93} - 24q^{95} - 40q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{6} + 7 x^{4} - 36 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{7} + 28 \nu^{5} - 49 \nu^{3} + 180 \nu \)\()/189\)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{7} + 7 \nu^{5} + 35 \nu^{3} + 81 \nu \)\()/189\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} + 4 \nu^{4} + 2 \nu^{2} + 18 \)\()/9\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{7} - \nu^{5} - 5 \nu^{3} + 63 \nu \)\()/27\)
\(\beta_{5}\)\(=\)\((\)\( 16 \nu^{7} - 28 \nu^{5} + 49 \nu^{3} - 324 \nu \)\()/189\)
\(\beta_{6}\)\(=\)\((\)\( -8 \nu^{6} + 14 \nu^{4} - 56 \nu^{2} + 225 \)\()/63\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{6} + 22 \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4} + \beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} - 2 \beta_{6} + \beta_{3} + 2\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{5} + 4 \beta_{2}\)
\(\nu^{4}\)\(=\)\((\)\(-4 \beta_{7} + \beta_{6} + 4 \beta_{3} + 1\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{5} - 7 \beta_{4} + 7 \beta_{2} + 5 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(-7 \beta_{7} + 22\)
\(\nu^{7}\)\(=\)\((\)\(29 \beta_{5} + 8 \beta_{4} + 8 \beta_{2} + 29 \beta_{1}\)\()/2\)

Character values

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

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1469.1
1.01575 1.40294i
−1.72286 0.178197i
−1.72286 + 0.178197i
1.01575 + 1.40294i
1.72286 0.178197i
−1.01575 1.40294i
−1.01575 + 1.40294i
1.72286 + 0.178197i
1.00000 −0.707107 1.58114i 1.00000 1.41421 1.73205i −0.707107 1.58114i 0 1.00000 −2.00000 + 2.23607i 1.41421 1.73205i
1469.2 1.00000 −0.707107 1.58114i 1.00000 1.41421 + 1.73205i −0.707107 1.58114i 0 1.00000 −2.00000 + 2.23607i 1.41421 + 1.73205i
1469.3 1.00000 −0.707107 + 1.58114i 1.00000 1.41421 1.73205i −0.707107 + 1.58114i 0 1.00000 −2.00000 2.23607i 1.41421 1.73205i
1469.4 1.00000 −0.707107 + 1.58114i 1.00000 1.41421 + 1.73205i −0.707107 + 1.58114i 0 1.00000 −2.00000 2.23607i 1.41421 + 1.73205i
1469.5 1.00000 0.707107 1.58114i 1.00000 −1.41421 1.73205i 0.707107 1.58114i 0 1.00000 −2.00000 2.23607i −1.41421 1.73205i
1469.6 1.00000 0.707107 1.58114i 1.00000 −1.41421 + 1.73205i 0.707107 1.58114i 0 1.00000 −2.00000 2.23607i −1.41421 + 1.73205i
1469.7 1.00000 0.707107 + 1.58114i 1.00000 −1.41421 1.73205i 0.707107 + 1.58114i 0 1.00000 −2.00000 + 2.23607i −1.41421 1.73205i
1469.8 1.00000 0.707107 + 1.58114i 1.00000 −1.41421 + 1.73205i 0.707107 + 1.58114i 0 1.00000 −2.00000 + 2.23607i −1.41421 + 1.73205i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1469.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
15.d odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.d.f 8
3.b odd 2 1 1470.2.d.e 8
5.b even 2 1 1470.2.d.e 8
7.b odd 2 1 inner 1470.2.d.f 8
7.c even 3 1 210.2.t.e 8
7.d odd 6 1 210.2.t.e 8
15.d odd 2 1 inner 1470.2.d.f 8
21.c even 2 1 1470.2.d.e 8
21.g even 6 1 210.2.t.f yes 8
21.h odd 6 1 210.2.t.f yes 8
35.c odd 2 1 1470.2.d.e 8
35.i odd 6 1 210.2.t.f yes 8
35.j even 6 1 210.2.t.f yes 8
35.k even 12 2 1050.2.s.i 16
35.l odd 12 2 1050.2.s.i 16
105.g even 2 1 inner 1470.2.d.f 8
105.o odd 6 1 210.2.t.e 8
105.p even 6 1 210.2.t.e 8
105.w odd 12 2 1050.2.s.i 16
105.x even 12 2 1050.2.s.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.t.e 8 7.c even 3 1
210.2.t.e 8 7.d odd 6 1
210.2.t.e 8 105.o odd 6 1
210.2.t.e 8 105.p even 6 1
210.2.t.f yes 8 21.g even 6 1
210.2.t.f yes 8 21.h odd 6 1
210.2.t.f yes 8 35.i odd 6 1
210.2.t.f yes 8 35.j even 6 1
1050.2.s.i 16 35.k even 12 2
1050.2.s.i 16 35.l odd 12 2
1050.2.s.i 16 105.w odd 12 2
1050.2.s.i 16 105.x even 12 2
1470.2.d.e 8 3.b odd 2 1
1470.2.d.e 8 5.b even 2 1
1470.2.d.e 8 21.c even 2 1
1470.2.d.e 8 35.c odd 2 1
1470.2.d.f 8 1.a even 1 1 trivial
1470.2.d.f 8 7.b odd 2 1 inner
1470.2.d.f 8 15.d odd 2 1 inner
1470.2.d.f 8 105.g even 2 1 inner

Hecke kernels

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

\( T_{11}^{4} + 22 T_{11}^{2} + 1 \)
\( T_{13}^{4} - 46 T_{13}^{2} + 49 \)
\( T_{23}^{2} + 2 T_{23} - 29 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{8} \)
$3$ \( ( 9 + 4 T^{2} + T^{4} )^{2} \)
$5$ \( ( 25 + 2 T^{2} + T^{4} )^{2} \)
$7$ \( T^{8} \)
$11$ \( ( 1 + 22 T^{2} + T^{4} )^{2} \)
$13$ \( ( 49 - 46 T^{2} + T^{4} )^{2} \)
$17$ \( ( 10 + T^{2} )^{4} \)
$19$ \( ( 49 + 26 T^{2} + T^{4} )^{2} \)
$23$ \( ( -29 + 2 T + T^{2} )^{4} \)
$29$ \( ( 196 + 52 T^{2} + T^{4} )^{2} \)
$31$ \( ( 4 + 44 T^{2} + T^{4} )^{2} \)
$37$ \( ( 361 + 58 T^{2} + T^{4} )^{2} \)
$41$ \( ( 49 - 46 T^{2} + T^{4} )^{2} \)
$43$ \( ( 196 + 52 T^{2} + T^{4} )^{2} \)
$47$ \( ( 7569 + 186 T^{2} + T^{4} )^{2} \)
$53$ \( ( -5 + T )^{8} \)
$59$ \( ( 2704 - 136 T^{2} + T^{4} )^{2} \)
$61$ \( ( 12 + T^{2} )^{4} \)
$67$ \( ( 5776 + 232 T^{2} + T^{4} )^{2} \)
$71$ \( ( 196 + 52 T^{2} + T^{4} )^{2} \)
$73$ \( ( 1764 - 156 T^{2} + T^{4} )^{2} \)
$79$ \( ( 6 - 12 T + T^{2} )^{4} \)
$83$ \( ( 4624 + 296 T^{2} + T^{4} )^{2} \)
$89$ \( ( 2704 - 136 T^{2} + T^{4} )^{2} \)
$97$ \( ( 1764 - 156 T^{2} + T^{4} )^{2} \)
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