Properties

Label 630.2.g.b
Level 630
Weight 2
Character orbit 630.g
Analytic conductor 5.031
Analytic rank 1
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 630.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.03057532734\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} + ( -1 + 2 i ) q^{5} -i q^{7} -i q^{8} +O(q^{10})\) \( q + i q^{2} - q^{4} + ( -1 + 2 i ) q^{5} -i q^{7} -i q^{8} + ( -2 - i ) q^{10} -6 q^{11} -2 i q^{13} + q^{14} + q^{16} -2 i q^{17} -4 q^{19} + ( 1 - 2 i ) q^{20} -6 i q^{22} + 4 i q^{23} + ( -3 - 4 i ) q^{25} + 2 q^{26} + i q^{28} + 2 q^{29} -2 q^{31} + i q^{32} + 2 q^{34} + ( 2 + i ) q^{35} -10 i q^{37} -4 i q^{38} + ( 2 + i ) q^{40} -6 q^{41} + 2 i q^{43} + 6 q^{44} -4 q^{46} + 2 i q^{47} - q^{49} + ( 4 - 3 i ) q^{50} + 2 i q^{52} -6 i q^{53} + ( 6 - 12 i ) q^{55} - q^{56} + 2 i q^{58} -4 q^{59} -12 q^{61} -2 i q^{62} - q^{64} + ( 4 + 2 i ) q^{65} + 10 i q^{67} + 2 i q^{68} + ( -1 + 2 i ) q^{70} -12 q^{71} + 2 i q^{73} + 10 q^{74} + 4 q^{76} + 6 i q^{77} + 16 q^{79} + ( -1 + 2 i ) q^{80} -6 i q^{82} + 12 i q^{83} + ( 4 + 2 i ) q^{85} -2 q^{86} + 6 i q^{88} + 14 q^{89} -2 q^{91} -4 i q^{92} -2 q^{94} + ( 4 - 8 i ) q^{95} + 18 i q^{97} -i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{5} - 4q^{10} - 12q^{11} + 2q^{14} + 2q^{16} - 8q^{19} + 2q^{20} - 6q^{25} + 4q^{26} + 4q^{29} - 4q^{31} + 4q^{34} + 4q^{35} + 4q^{40} - 12q^{41} + 12q^{44} - 8q^{46} - 2q^{49} + 8q^{50} + 12q^{55} - 2q^{56} - 8q^{59} - 24q^{61} - 2q^{64} + 8q^{65} - 2q^{70} - 24q^{71} + 20q^{74} + 8q^{76} + 32q^{79} - 2q^{80} + 8q^{85} - 4q^{86} + 28q^{89} - 4q^{91} - 4q^{94} + 8q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
1.00000i
1.00000i
1.00000i 0 −1.00000 −1.00000 2.00000i 0 1.00000i 1.00000i 0 −2.00000 + 1.00000i
379.2 1.00000i 0 −1.00000 −1.00000 + 2.00000i 0 1.00000i 1.00000i 0 −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 630.2.g.b 2
3.b odd 2 1 630.2.g.e yes 2
4.b odd 2 1 5040.2.t.i 2
5.b even 2 1 inner 630.2.g.b 2
5.c odd 4 1 3150.2.a.l 1
5.c odd 4 1 3150.2.a.v 1
12.b even 2 1 5040.2.t.j 2
15.d odd 2 1 630.2.g.e yes 2
15.e even 4 1 3150.2.a.k 1
15.e even 4 1 3150.2.a.br 1
20.d odd 2 1 5040.2.t.i 2
60.h even 2 1 5040.2.t.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.g.b 2 1.a even 1 1 trivial
630.2.g.b 2 5.b even 2 1 inner
630.2.g.e yes 2 3.b odd 2 1
630.2.g.e yes 2 15.d odd 2 1
3150.2.a.k 1 15.e even 4 1
3150.2.a.l 1 5.c odd 4 1
3150.2.a.v 1 5.c odd 4 1
3150.2.a.br 1 15.e even 4 1
5040.2.t.i 2 4.b odd 2 1
5040.2.t.i 2 20.d odd 2 1
5040.2.t.j 2 12.b even 2 1
5040.2.t.j 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\):

\( T_{11} + 6 \)
\( T_{29} - 2 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ 1
$5$ \( 1 + 2 T + 5 T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 + 6 T + 11 T^{2} )^{2} \)
$13$ \( 1 - 22 T^{2} + 169 T^{4} \)
$17$ \( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \)
$19$ \( ( 1 + 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 30 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 2 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 2 T + 31 T^{2} )^{2} \)
$37$ \( 1 + 26 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 82 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 90 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 4 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 12 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 34 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 142 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 16 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 22 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 14 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 8 T + 97 T^{2} )( 1 + 8 T + 97 T^{2} ) \)
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