Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1470.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.7380090971\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(-\zeta_{12}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −0.133975 + 2.23205i −1.00000 0 1.00000i 0.500000 0.866025i 1.23205 1.86603i
79.2 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −1.86603 + 1.23205i −1.00000 0 1.00000i 0.500000 0.866025i −2.23205 + 0.133975i
949.1 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i −0.133975 2.23205i −1.00000 0 1.00000i 0.500000 + 0.866025i 1.23205 + 1.86603i
949.2 0.866025 0.500000i −0.866025 0.500000i 0.500000 0.866025i −1.86603 1.23205i −1.00000 0 1.00000i 0.500000 + 0.866025i −2.23205 0.133975i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.n.a 4
5.b even 2 1 inner 1470.2.n.a 4
7.b odd 2 1 1470.2.n.h 4
7.c even 3 1 1470.2.g.g 2
7.c even 3 1 inner 1470.2.n.a 4
7.d odd 6 1 30.2.c.a 2
7.d odd 6 1 1470.2.n.h 4
21.g even 6 1 90.2.c.a 2
28.f even 6 1 240.2.f.a 2
35.c odd 2 1 1470.2.n.h 4
35.i odd 6 1 30.2.c.a 2
35.i odd 6 1 1470.2.n.h 4
35.j even 6 1 1470.2.g.g 2
35.j even 6 1 inner 1470.2.n.a 4
35.k even 12 1 150.2.a.a 1
35.k even 12 1 150.2.a.c 1
35.l odd 12 1 7350.2.a.bg 1
35.l odd 12 1 7350.2.a.cc 1
56.j odd 6 1 960.2.f.h 2
56.m even 6 1 960.2.f.i 2
63.i even 6 1 810.2.i.b 4
63.k odd 6 1 810.2.i.e 4
63.s even 6 1 810.2.i.b 4
63.t odd 6 1 810.2.i.e 4
84.j odd 6 1 720.2.f.f 2
105.p even 6 1 90.2.c.a 2
105.w odd 12 1 450.2.a.b 1
105.w odd 12 1 450.2.a.f 1
112.v even 12 1 3840.2.d.j 2
112.v even 12 1 3840.2.d.x 2
112.x odd 12 1 3840.2.d.g 2
112.x odd 12 1 3840.2.d.y 2
140.s even 6 1 240.2.f.a 2
140.x odd 12 1 1200.2.a.g 1
140.x odd 12 1 1200.2.a.m 1
168.ba even 6 1 2880.2.f.e 2
168.be odd 6 1 2880.2.f.c 2
280.ba even 6 1 960.2.f.i 2
280.bk odd 6 1 960.2.f.h 2
280.bp odd 12 1 4800.2.a.m 1
280.bp odd 12 1 4800.2.a.cj 1
280.bv even 12 1 4800.2.a.l 1
280.bv even 12 1 4800.2.a.cg 1
315.q odd 6 1 810.2.i.e 4
315.u even 6 1 810.2.i.b 4
315.bn odd 6 1 810.2.i.e 4
315.bq even 6 1 810.2.i.b 4
420.be odd 6 1 720.2.f.f 2
420.br even 12 1 3600.2.a.o 1
420.br even 12 1 3600.2.a.bg 1
560.cn odd 12 1 3840.2.d.g 2
560.cn odd 12 1 3840.2.d.y 2
560.co even 12 1 3840.2.d.j 2
560.co even 12 1 3840.2.d.x 2
840.cb even 6 1 2880.2.f.e 2
840.ct odd 6 1 2880.2.f.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.c.a 2 7.d odd 6 1
30.2.c.a 2 35.i odd 6 1
90.2.c.a 2 21.g even 6 1
90.2.c.a 2 105.p even 6 1
150.2.a.a 1 35.k even 12 1
150.2.a.c 1 35.k even 12 1
240.2.f.a 2 28.f even 6 1
240.2.f.a 2 140.s even 6 1
450.2.a.b 1 105.w odd 12 1
450.2.a.f 1 105.w odd 12 1
720.2.f.f 2 84.j odd 6 1
720.2.f.f 2 420.be odd 6 1
810.2.i.b 4 63.i even 6 1
810.2.i.b 4 63.s even 6 1
810.2.i.b 4 315.u even 6 1
810.2.i.b 4 315.bq even 6 1
810.2.i.e 4 63.k odd 6 1
810.2.i.e 4 63.t odd 6 1
810.2.i.e 4 315.q odd 6 1
810.2.i.e 4 315.bn odd 6 1
960.2.f.h 2 56.j odd 6 1
960.2.f.h 2 280.bk odd 6 1
960.2.f.i 2 56.m even 6 1
960.2.f.i 2 280.ba even 6 1
1200.2.a.g 1 140.x odd 12 1
1200.2.a.m 1 140.x odd 12 1
1470.2.g.g 2 7.c even 3 1
1470.2.g.g 2 35.j even 6 1
1470.2.n.a 4 1.a even 1 1 trivial
1470.2.n.a 4 5.b even 2 1 inner
1470.2.n.a 4 7.c even 3 1 inner
1470.2.n.a 4 35.j even 6 1 inner
1470.2.n.h 4 7.b odd 2 1
1470.2.n.h 4 7.d odd 6 1
1470.2.n.h 4 35.c odd 2 1
1470.2.n.h 4 35.i odd 6 1
2880.2.f.c 2 168.be odd 6 1
2880.2.f.c 2 840.ct odd 6 1
2880.2.f.e 2 168.ba even 6 1
2880.2.f.e 2 840.cb even 6 1
3600.2.a.o 1 420.br even 12 1
3600.2.a.bg 1 420.br even 12 1
3840.2.d.g 2 112.x odd 12 1
3840.2.d.g 2 560.cn odd 12 1
3840.2.d.j 2 112.v even 12 1
3840.2.d.j 2 560.co even 12 1
3840.2.d.x 2 112.v even 12 1
3840.2.d.x 2 560.co even 12 1
3840.2.d.y 2 112.x odd 12 1
3840.2.d.y 2 560.cn odd 12 1
4800.2.a.l 1 280.bv even 12 1
4800.2.a.m 1 280.bp odd 12 1
4800.2.a.cg 1 280.bv even 12 1
4800.2.a.cj 1 280.bp odd 12 1
7350.2.a.bg 1 35.l odd 12 1
7350.2.a.cc 1 35.l odd 12 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + 11 T^{2} + 20 T + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T + 2)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$53$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$59$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$71$ \( (T - 12)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
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