Properties

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

Related objects

Downloads

Learn more about

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: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} -\beta_{1} q^{3} + q^{4} + ( -2 \beta_{2} - \beta_{3} ) q^{5} -\beta_{1} q^{6} + q^{8} + ( 3 \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + q^{2} -\beta_{1} q^{3} + q^{4} + ( -2 \beta_{2} - \beta_{3} ) q^{5} -\beta_{1} q^{6} + q^{8} + ( 3 \beta_{2} + \beta_{3} ) q^{9} + ( -2 \beta_{2} - \beta_{3} ) q^{10} + ( 1 + \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{11} -\beta_{1} q^{12} + 2 q^{13} + ( 3 + \beta_{1} - \beta_{3} ) q^{15} + q^{16} + ( -2 + 4 \beta_{1} - 4 \beta_{3} ) q^{17} + ( 3 \beta_{2} + \beta_{3} ) q^{18} + ( 2 - 4 \beta_{2} ) q^{19} + ( -2 \beta_{2} - \beta_{3} ) q^{20} + ( 1 + \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{22} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{23} -\beta_{1} q^{24} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{25} + 2 q^{26} + ( -3 - 2 \beta_{1} + 2 \beta_{3} ) q^{27} + ( 1 - 2 \beta_{1} + 2 \beta_{3} ) q^{29} + ( 3 + \beta_{1} - \beta_{3} ) q^{30} + ( 3 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{31} + q^{32} + ( 3 + \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{33} + ( -2 + 4 \beta_{1} - 4 \beta_{3} ) q^{34} + ( 3 \beta_{2} + \beta_{3} ) q^{36} + ( 2 - 6 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} ) q^{37} + ( 2 - 4 \beta_{2} ) q^{38} -2 \beta_{1} q^{39} + ( -2 \beta_{2} - \beta_{3} ) q^{40} + ( 4 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{41} + ( -2 - 3 \beta_{1} + 7 \beta_{2} + 3 \beta_{3} ) q^{43} + ( 1 + \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{44} + ( 3 - 4 \beta_{1} - 3 \beta_{2} ) q^{45} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{46} + ( 4 - 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{47} -\beta_{1} q^{48} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{50} + ( 12 - 2 \beta_{1} - 12 \beta_{2} ) q^{51} + 2 q^{52} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{53} + ( -3 - 2 \beta_{1} + 2 \beta_{3} ) q^{54} + ( -6 + 3 \beta_{1} + \beta_{2} ) q^{55} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{57} + ( 1 - 2 \beta_{1} + 2 \beta_{3} ) q^{58} + ( 3 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{59} + ( 3 + \beta_{1} - \beta_{3} ) q^{60} + ( 6 - 3 \beta_{1} - 9 \beta_{2} + 3 \beta_{3} ) q^{61} + ( 3 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{62} + q^{64} + ( -4 \beta_{2} - 2 \beta_{3} ) q^{65} + ( 3 + \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{66} + ( -3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{67} + ( -2 + 4 \beta_{1} - 4 \beta_{3} ) q^{68} + ( 3 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{69} + ( 4 - 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{71} + ( 3 \beta_{2} + \beta_{3} ) q^{72} + 2 q^{73} + ( 2 - 6 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} ) q^{74} + ( -9 \beta_{2} - 2 \beta_{3} ) q^{75} + ( 2 - 4 \beta_{2} ) q^{76} -2 \beta_{1} q^{78} + ( -1 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{79} + ( -2 \beta_{2} - \beta_{3} ) q^{80} + ( -6 + 5 \beta_{1} + 6 \beta_{2} ) q^{81} + ( 4 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{82} + ( -5 + 4 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{83} + ( -8 \beta_{2} + 6 \beta_{3} ) q^{85} + ( -2 - 3 \beta_{1} + 7 \beta_{2} + 3 \beta_{3} ) q^{86} + ( -6 + \beta_{1} + 6 \beta_{2} ) q^{87} + ( 1 + \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{88} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{89} + ( 3 - 4 \beta_{1} - 3 \beta_{2} ) q^{90} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{92} + ( -9 + 3 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} ) q^{93} + ( 4 - 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{94} + ( -8 + 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{95} -\beta_{1} q^{96} + ( 5 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{97} + ( 9 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - q^{3} + 4q^{4} - 3q^{5} - q^{6} + 4q^{8} + 5q^{9} + O(q^{10}) \) \( 4q + 4q^{2} - q^{3} + 4q^{4} - 3q^{5} - q^{6} + 4q^{8} + 5q^{9} - 3q^{10} - q^{12} + 8q^{13} + 14q^{15} + 4q^{16} + 5q^{18} - 3q^{20} + 6q^{23} - q^{24} + q^{25} + 8q^{26} - 16q^{27} + 14q^{30} + 4q^{32} + 10q^{33} + 5q^{36} - 2q^{39} - 3q^{40} + 18q^{41} + 2q^{45} + 6q^{46} - q^{48} + q^{50} + 22q^{51} + 8q^{52} - 6q^{53} - 16q^{54} - 19q^{55} + 6q^{57} + 18q^{59} + 14q^{60} + 4q^{64} - 6q^{65} + 10q^{66} + 15q^{69} + 5q^{72} + 8q^{73} - 16q^{75} - 2q^{78} + 2q^{79} - 3q^{80} - 7q^{81} + 18q^{82} - 22q^{85} - 11q^{87} - 6q^{89} + 2q^{90} + 6q^{92} - 12q^{93} - 18q^{95} - q^{96} + 26q^{97} + 28q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu + 3 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-2 \beta_{3} + 2 \beta_{1} + 3\)

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
1.68614 + 0.396143i
1.68614 0.396143i
−1.18614 + 1.26217i
−1.18614 1.26217i
1.00000 −1.68614 0.396143i 1.00000 −2.18614 0.469882i −1.68614 0.396143i 0 1.00000 2.68614 + 1.33591i −2.18614 0.469882i
1469.2 1.00000 −1.68614 + 0.396143i 1.00000 −2.18614 + 0.469882i −1.68614 + 0.396143i 0 1.00000 2.68614 1.33591i −2.18614 + 0.469882i
1469.3 1.00000 1.18614 1.26217i 1.00000 0.686141 + 2.12819i 1.18614 1.26217i 0 1.00000 −0.186141 2.99422i 0.686141 + 2.12819i
1469.4 1.00000 1.18614 + 1.26217i 1.00000 0.686141 2.12819i 1.18614 + 1.26217i 0 1.00000 −0.186141 + 2.99422i 0.686141 2.12819i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.c 4
3.b odd 2 1 1470.2.d.a 4
5.b even 2 1 1470.2.d.b 4
7.b odd 2 1 1470.2.d.d 4
7.c even 3 1 210.2.t.b yes 4
7.d odd 6 1 210.2.t.a 4
15.d odd 2 1 1470.2.d.d 4
21.c even 2 1 1470.2.d.b 4
21.g even 6 1 210.2.t.c yes 4
21.h odd 6 1 210.2.t.d yes 4
35.c odd 2 1 1470.2.d.a 4
35.i odd 6 1 210.2.t.d yes 4
35.j even 6 1 210.2.t.c yes 4
35.k even 12 2 1050.2.s.e 8
35.l odd 12 2 1050.2.s.d 8
105.g even 2 1 inner 1470.2.d.c 4
105.o odd 6 1 210.2.t.a 4
105.p even 6 1 210.2.t.b yes 4
105.w odd 12 2 1050.2.s.d 8
105.x even 12 2 1050.2.s.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.t.a 4 7.d odd 6 1
210.2.t.a 4 105.o odd 6 1
210.2.t.b yes 4 7.c even 3 1
210.2.t.b yes 4 105.p even 6 1
210.2.t.c yes 4 21.g even 6 1
210.2.t.c yes 4 35.j even 6 1
210.2.t.d yes 4 21.h odd 6 1
210.2.t.d yes 4 35.i odd 6 1
1050.2.s.d 8 35.l odd 12 2
1050.2.s.d 8 105.w odd 12 2
1050.2.s.e 8 35.k even 12 2
1050.2.s.e 8 105.x even 12 2
1470.2.d.a 4 3.b odd 2 1
1470.2.d.a 4 35.c odd 2 1
1470.2.d.b 4 5.b even 2 1
1470.2.d.b 4 21.c even 2 1
1470.2.d.c 4 1.a even 1 1 trivial
1470.2.d.c 4 105.g even 2 1 inner
1470.2.d.d 4 7.b odd 2 1
1470.2.d.d 4 15.d odd 2 1

Hecke kernels

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

\( T_{11}^{4} + 19 T_{11}^{2} + 16 \)
\( T_{13} - 2 \)
\( T_{23}^{2} - 3 T_{23} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( 9 + 3 T - 2 T^{2} + T^{3} + T^{4} \)
$5$ \( 25 + 15 T + 4 T^{2} + 3 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 16 + 19 T^{2} + T^{4} \)
$13$ \( ( -2 + T )^{4} \)
$17$ \( ( 44 + T^{2} )^{2} \)
$19$ \( ( 12 + T^{2} )^{2} \)
$23$ \( ( -6 - 3 T + T^{2} )^{2} \)
$29$ \( ( 11 + T^{2} )^{2} \)
$31$ \( 324 + 63 T^{2} + T^{4} \)
$37$ \( 9216 + 204 T^{2} + T^{4} \)
$41$ \( ( 12 - 9 T + T^{2} )^{2} \)
$43$ \( 144 + 123 T^{2} + T^{4} \)
$47$ \( 256 + 76 T^{2} + T^{4} \)
$53$ \( ( -6 + 3 T + T^{2} )^{2} \)
$59$ \( ( -54 - 9 T + T^{2} )^{2} \)
$61$ \( 1296 + 171 T^{2} + T^{4} \)
$67$ \( 324 + 63 T^{2} + T^{4} \)
$71$ \( 256 + 76 T^{2} + T^{4} \)
$73$ \( ( -2 + T )^{4} \)
$79$ \( ( -74 - T + T^{2} )^{2} \)
$83$ \( 289 + 142 T^{2} + T^{4} \)
$89$ \( ( -6 + 3 T + T^{2} )^{2} \)
$97$ \( ( -32 - 13 T + T^{2} )^{2} \)
show more
show less