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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(1470, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( (T + 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 18T + 324 \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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