Properties

Label 1470.2.i.h
Level $1470$
Weight $2$
Character orbit 1470.i
Analytic conductor $11.738$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(11.7380090971\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −1.00000 0 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
961.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −1.00000 0 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
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 1470.2.i.h 2
7.b odd 2 1 1470.2.i.d 2
7.c even 3 1 1470.2.a.m 1
7.c even 3 1 inner 1470.2.i.h 2
7.d odd 6 1 210.2.a.d 1
7.d odd 6 1 1470.2.i.d 2
21.g even 6 1 630.2.a.f 1
21.h odd 6 1 4410.2.a.f 1
28.f even 6 1 1680.2.a.b 1
35.i odd 6 1 1050.2.a.a 1
35.j even 6 1 7350.2.a.bd 1
35.k even 12 2 1050.2.g.h 2
56.j odd 6 1 6720.2.a.bb 1
56.m even 6 1 6720.2.a.cc 1
84.j odd 6 1 5040.2.a.ba 1
105.p even 6 1 3150.2.a.ba 1
105.w odd 12 2 3150.2.g.o 2
140.s even 6 1 8400.2.a.cn 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.a.d 1 7.d odd 6 1
630.2.a.f 1 21.g even 6 1
1050.2.a.a 1 35.i odd 6 1
1050.2.g.h 2 35.k even 12 2
1470.2.a.m 1 7.c even 3 1
1470.2.i.d 2 7.b odd 2 1
1470.2.i.d 2 7.d odd 6 1
1470.2.i.h 2 1.a even 1 1 trivial
1470.2.i.h 2 7.c even 3 1 inner
1680.2.a.b 1 28.f even 6 1
3150.2.a.ba 1 105.p even 6 1
3150.2.g.o 2 105.w odd 12 2
4410.2.a.f 1 21.h odd 6 1
5040.2.a.ba 1 84.j odd 6 1
6720.2.a.bb 1 56.j odd 6 1
6720.2.a.cc 1 56.m even 6 1
7350.2.a.bd 1 35.j even 6 1
8400.2.a.cn 1 140.s even 6 1

Hecke kernels

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

\( T_{11} \)
\( T_{13} + 2 \)
\( T_{17}^{2} + 6 T_{17} + 36 \)
\( T_{19}^{2} + 4 T_{19} + 16 \)
\( T_{31}^{2} + 4 T_{31} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 2 + T )^{2} \)
$17$ \( 36 + 6 T + T^{2} \)
$19$ \( 16 + 4 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( 16 + 4 T + T^{2} \)
$37$ \( 4 + 2 T + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( ( -8 + T )^{2} \)
$47$ \( 144 + 12 T + T^{2} \)
$53$ \( 36 + 6 T + T^{2} \)
$59$ \( 144 + 12 T + T^{2} \)
$61$ \( 4 - 2 T + T^{2} \)
$67$ \( 64 + 8 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 196 - 14 T + T^{2} \)
$79$ \( 256 - 16 T + T^{2} \)
$83$ \( ( 12 + T )^{2} \)
$89$ \( 36 - 6 T + T^{2} \)
$97$ \( ( 14 + T )^{2} \)
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