Properties

Label 1470.2.i.d
Level 1470
Weight 2
Character orbit 1470.i
Analytic conductor 11.738
Analytic rank 0
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
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.d 2
7.b odd 2 1 1470.2.i.h 2
7.c even 3 1 210.2.a.d 1
7.c even 3 1 inner 1470.2.i.d 2
7.d odd 6 1 1470.2.a.m 1
7.d odd 6 1 1470.2.i.h 2
21.g even 6 1 4410.2.a.f 1
21.h odd 6 1 630.2.a.f 1
28.g odd 6 1 1680.2.a.b 1
35.i odd 6 1 7350.2.a.bd 1
35.j even 6 1 1050.2.a.a 1
35.l odd 12 2 1050.2.g.h 2
56.k odd 6 1 6720.2.a.cc 1
56.p even 6 1 6720.2.a.bb 1
84.n even 6 1 5040.2.a.ba 1
105.o odd 6 1 3150.2.a.ba 1
105.x even 12 2 3150.2.g.o 2
140.p odd 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.c even 3 1
630.2.a.f 1 21.h odd 6 1
1050.2.a.a 1 35.j even 6 1
1050.2.g.h 2 35.l odd 12 2
1470.2.a.m 1 7.d odd 6 1
1470.2.i.d 2 1.a even 1 1 trivial
1470.2.i.d 2 7.c even 3 1 inner
1470.2.i.h 2 7.b odd 2 1
1470.2.i.h 2 7.d odd 6 1
1680.2.a.b 1 28.g odd 6 1
3150.2.a.ba 1 105.o odd 6 1
3150.2.g.o 2 105.x even 12 2
4410.2.a.f 1 21.g even 6 1
5040.2.a.ba 1 84.n even 6 1
6720.2.a.bb 1 56.p even 6 1
6720.2.a.cc 1 56.k odd 6 1
7350.2.a.bd 1 35.i odd 6 1
8400.2.a.cn 1 140.p odd 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

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ 1
$11$ \( 1 - 11 T^{2} + 121 T^{4} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \)
$19$ \( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \)
$23$ \( 1 - 23 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \)
$37$ \( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 12 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 14 T + 97 T^{2} )^{2} \)
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