Properties

Label 1470.2.i.q
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: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
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.

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

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.q 2
7.b odd 2 1 1470.2.i.o 2
7.c even 3 1 1470.2.a.d 1
7.c even 3 1 inner 1470.2.i.q 2
7.d odd 6 1 30.2.a.a 1
7.d odd 6 1 1470.2.i.o 2
21.g even 6 1 90.2.a.c 1
21.h odd 6 1 4410.2.a.z 1
28.f even 6 1 240.2.a.b 1
35.i odd 6 1 150.2.a.b 1
35.j even 6 1 7350.2.a.ct 1
35.k even 12 2 150.2.c.a 2
56.j odd 6 1 960.2.a.e 1
56.m even 6 1 960.2.a.p 1
63.i even 6 1 810.2.e.b 2
63.k odd 6 1 810.2.e.l 2
63.s even 6 1 810.2.e.b 2
63.t odd 6 1 810.2.e.l 2
77.i even 6 1 3630.2.a.w 1
84.j odd 6 1 720.2.a.j 1
91.s odd 6 1 5070.2.a.w 1
91.bb even 12 2 5070.2.b.k 2
105.p even 6 1 450.2.a.d 1
105.w odd 12 2 450.2.c.b 2
112.v even 12 2 3840.2.k.f 2
112.x odd 12 2 3840.2.k.y 2
119.h odd 6 1 8670.2.a.g 1
140.s even 6 1 1200.2.a.k 1
140.x odd 12 2 1200.2.f.e 2
168.ba even 6 1 2880.2.a.a 1
168.be odd 6 1 2880.2.a.q 1
280.ba even 6 1 4800.2.a.d 1
280.bk odd 6 1 4800.2.a.cq 1
280.bp odd 12 2 4800.2.f.w 2
280.bv even 12 2 4800.2.f.p 2
420.be odd 6 1 3600.2.a.f 1
420.br even 12 2 3600.2.f.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.a.a 1 7.d odd 6 1
90.2.a.c 1 21.g even 6 1
150.2.a.b 1 35.i odd 6 1
150.2.c.a 2 35.k even 12 2
240.2.a.b 1 28.f even 6 1
450.2.a.d 1 105.p even 6 1
450.2.c.b 2 105.w odd 12 2
720.2.a.j 1 84.j odd 6 1
810.2.e.b 2 63.i even 6 1
810.2.e.b 2 63.s even 6 1
810.2.e.l 2 63.k odd 6 1
810.2.e.l 2 63.t odd 6 1
960.2.a.e 1 56.j odd 6 1
960.2.a.p 1 56.m even 6 1
1200.2.a.k 1 140.s even 6 1
1200.2.f.e 2 140.x odd 12 2
1470.2.a.d 1 7.c even 3 1
1470.2.i.o 2 7.b odd 2 1
1470.2.i.o 2 7.d odd 6 1
1470.2.i.q 2 1.a even 1 1 trivial
1470.2.i.q 2 7.c even 3 1 inner
2880.2.a.a 1 168.ba even 6 1
2880.2.a.q 1 168.be odd 6 1
3600.2.a.f 1 420.be odd 6 1
3600.2.f.i 2 420.br even 12 2
3630.2.a.w 1 77.i even 6 1
3840.2.k.f 2 112.v even 12 2
3840.2.k.y 2 112.x odd 12 2
4410.2.a.z 1 21.h odd 6 1
4800.2.a.d 1 280.ba even 6 1
4800.2.a.cq 1 280.bk odd 6 1
4800.2.f.p 2 280.bv even 12 2
4800.2.f.w 2 280.bp odd 12 2
5070.2.a.w 1 91.s odd 6 1
5070.2.b.k 2 91.bb even 12 2
7350.2.a.ct 1 35.j even 6 1
8670.2.a.g 1 119.h odd 6 1

Hecke kernels

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

\( T_{11} \)
\( T_{13} + 2 \)
\( T_{17}^{2} - 6 T_{17} + 36 \)
\( T_{19}^{2} + 4 T_{19} + 16 \)
\( T_{31}^{2} - 8 T_{31} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ 1
$11$ \( 1 - 11 T^{2} + 121 T^{4} \)
$13$ \( ( 1 + 2 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \)
$19$ \( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \)
$23$ \( 1 - 23 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} \)
$37$ \( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 4 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 12 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 18 T + 235 T^{2} - 1602 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 2 T + 97 T^{2} )^{2} \)
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