Properties

Label 1470.2.g.e
Level $1470$
Weight $2$
Character orbit 1470.g
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.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.7380090971\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} -i q^{3} - q^{4} + ( 1 - 2 i ) q^{5} + q^{6} -i q^{8} - q^{9} +O(q^{10})\) \( q + i q^{2} -i q^{3} - q^{4} + ( 1 - 2 i ) q^{5} + q^{6} -i q^{8} - q^{9} + ( 2 + i ) q^{10} -2 q^{11} + i q^{12} + 2 i q^{13} + ( -2 - i ) q^{15} + q^{16} -8 i q^{17} -i q^{18} -2 q^{19} + ( -1 + 2 i ) q^{20} -2 i q^{22} - q^{24} + ( -3 - 4 i ) q^{25} -2 q^{26} + i q^{27} + 6 q^{29} + ( 1 - 2 i ) q^{30} -6 q^{31} + i q^{32} + 2 i q^{33} + 8 q^{34} + q^{36} + 8 i q^{37} -2 i q^{38} + 2 q^{39} + ( -2 - i ) q^{40} -6 q^{41} -8 i q^{43} + 2 q^{44} + ( -1 + 2 i ) q^{45} -4 i q^{47} -i q^{48} + ( 4 - 3 i ) q^{50} -8 q^{51} -2 i q^{52} + 2 i q^{53} - q^{54} + ( -2 + 4 i ) q^{55} + 2 i q^{57} + 6 i q^{58} -8 q^{59} + ( 2 + i ) q^{60} -10 q^{61} -6 i q^{62} - q^{64} + ( 4 + 2 i ) q^{65} -2 q^{66} -12 i q^{67} + 8 i q^{68} -14 q^{71} + i q^{72} -10 i q^{73} -8 q^{74} + ( -4 + 3 i ) q^{75} + 2 q^{76} + 2 i q^{78} -4 q^{79} + ( 1 - 2 i ) q^{80} + q^{81} -6 i q^{82} + 16 i q^{83} + ( -16 - 8 i ) q^{85} + 8 q^{86} -6 i q^{87} + 2 i q^{88} + 10 q^{89} + ( -2 - i ) q^{90} + 6 i q^{93} + 4 q^{94} + ( -2 + 4 i ) q^{95} + q^{96} -10 i q^{97} + 2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{5} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{5} + 2q^{6} - 2q^{9} + 4q^{10} - 4q^{11} - 4q^{15} + 2q^{16} - 4q^{19} - 2q^{20} - 2q^{24} - 6q^{25} - 4q^{26} + 12q^{29} + 2q^{30} - 12q^{31} + 16q^{34} + 2q^{36} + 4q^{39} - 4q^{40} - 12q^{41} + 4q^{44} - 2q^{45} + 8q^{50} - 16q^{51} - 2q^{54} - 4q^{55} - 16q^{59} + 4q^{60} - 20q^{61} - 2q^{64} + 8q^{65} - 4q^{66} - 28q^{71} - 16q^{74} - 8q^{75} + 4q^{76} - 8q^{79} + 2q^{80} + 2q^{81} - 32q^{85} + 16q^{86} + 20q^{89} - 4q^{90} + 8q^{94} - 4q^{95} + 2q^{96} + 4q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
589.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 1.00000 + 2.00000i 1.00000 0 1.00000i −1.00000 2.00000 1.00000i
589.2 1.00000i 1.00000i −1.00000 1.00000 2.00000i 1.00000 0 1.00000i −1.00000 2.00000 + 1.00000i
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.g.e 2
5.b even 2 1 inner 1470.2.g.e 2
5.c odd 4 1 7350.2.a.g 1
5.c odd 4 1 7350.2.a.co 1
7.b odd 2 1 210.2.g.a 2
7.c even 3 2 1470.2.n.c 4
7.d odd 6 2 1470.2.n.g 4
21.c even 2 1 630.2.g.d 2
28.d even 2 1 1680.2.t.d 2
35.c odd 2 1 210.2.g.a 2
35.f even 4 1 1050.2.a.g 1
35.f even 4 1 1050.2.a.m 1
35.i odd 6 2 1470.2.n.g 4
35.j even 6 2 1470.2.n.c 4
84.h odd 2 1 5040.2.t.k 2
105.g even 2 1 630.2.g.d 2
105.k odd 4 1 3150.2.a.q 1
105.k odd 4 1 3150.2.a.be 1
140.c even 2 1 1680.2.t.d 2
140.j odd 4 1 8400.2.a.bd 1
140.j odd 4 1 8400.2.a.ca 1
420.o odd 2 1 5040.2.t.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.g.a 2 7.b odd 2 1
210.2.g.a 2 35.c odd 2 1
630.2.g.d 2 21.c even 2 1
630.2.g.d 2 105.g even 2 1
1050.2.a.g 1 35.f even 4 1
1050.2.a.m 1 35.f even 4 1
1470.2.g.e 2 1.a even 1 1 trivial
1470.2.g.e 2 5.b even 2 1 inner
1470.2.n.c 4 7.c even 3 2
1470.2.n.c 4 35.j even 6 2
1470.2.n.g 4 7.d odd 6 2
1470.2.n.g 4 35.i odd 6 2
1680.2.t.d 2 28.d even 2 1
1680.2.t.d 2 140.c even 2 1
3150.2.a.q 1 105.k odd 4 1
3150.2.a.be 1 105.k odd 4 1
5040.2.t.k 2 84.h odd 2 1
5040.2.t.k 2 420.o odd 2 1
7350.2.a.g 1 5.c odd 4 1
7350.2.a.co 1 5.c odd 4 1
8400.2.a.bd 1 140.j odd 4 1
8400.2.a.ca 1 140.j odd 4 1

Hecke kernels

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

\( T_{11} + 2 \)
\( T_{17}^{2} + 64 \)
\( T_{19} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 5 - 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 2 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 64 + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( ( 6 + T )^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( 16 + T^{2} \)
$53$ \( 4 + T^{2} \)
$59$ \( ( 8 + T )^{2} \)
$61$ \( ( 10 + T )^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( ( 14 + T )^{2} \)
$73$ \( 100 + T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( 256 + T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( 100 + T^{2} \)
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