Properties

Label 630.2.k.b
Level $630$
Weight $2$
Character orbit 630.k
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.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.03057532734\)
Analytic rank: \(1\)
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 70)
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.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 1) q^{7} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 1) q^{7} + q^{8} + (\zeta_{6} - 1) q^{10} + (6 \zeta_{6} - 6) q^{11} - 4 q^{13} + (2 \zeta_{6} - 3) q^{14} - \zeta_{6} q^{16} - 2 \zeta_{6} q^{19} + q^{20} + 6 q^{22} - 3 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 4 \zeta_{6} q^{26} + (\zeta_{6} + 2) q^{28} + 3 q^{29} + (8 \zeta_{6} - 8) q^{31} + (\zeta_{6} - 1) q^{32} + (2 \zeta_{6} - 3) q^{35} + 4 \zeta_{6} q^{37} + (2 \zeta_{6} - 2) q^{38} - \zeta_{6} q^{40} - 9 q^{41} - 7 q^{43} - 6 \zeta_{6} q^{44} + (3 \zeta_{6} - 3) q^{46} + (3 \zeta_{6} - 8) q^{49} + q^{50} + ( - 4 \zeta_{6} + 4) q^{52} + (6 \zeta_{6} - 6) q^{53} + 6 q^{55} + ( - 3 \zeta_{6} + 1) q^{56} - 3 \zeta_{6} q^{58} + (6 \zeta_{6} - 6) q^{59} - 5 \zeta_{6} q^{61} + 8 q^{62} + q^{64} + 4 \zeta_{6} q^{65} + (5 \zeta_{6} - 5) q^{67} + (\zeta_{6} + 2) q^{70} + 6 q^{71} + ( - 16 \zeta_{6} + 16) q^{73} + ( - 4 \zeta_{6} + 4) q^{74} + 2 q^{76} + (6 \zeta_{6} + 12) q^{77} - 2 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{80} + 9 \zeta_{6} q^{82} - 3 q^{83} + 7 \zeta_{6} q^{86} + (6 \zeta_{6} - 6) q^{88} - 15 \zeta_{6} q^{89} + (12 \zeta_{6} - 4) q^{91} + 3 q^{92} + (2 \zeta_{6} - 2) q^{95} + 14 q^{97} + (5 \zeta_{6} + 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - q^{5} - q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} - q^{5} - q^{7} + 2 q^{8} - q^{10} - 6 q^{11} - 8 q^{13} - 4 q^{14} - q^{16} - 2 q^{19} + 2 q^{20} + 12 q^{22} - 3 q^{23} - q^{25} + 4 q^{26} + 5 q^{28} + 6 q^{29} - 8 q^{31} - q^{32} - 4 q^{35} + 4 q^{37} - 2 q^{38} - q^{40} - 18 q^{41} - 14 q^{43} - 6 q^{44} - 3 q^{46} - 13 q^{49} + 2 q^{50} + 4 q^{52} - 6 q^{53} + 12 q^{55} - q^{56} - 3 q^{58} - 6 q^{59} - 5 q^{61} + 16 q^{62} + 2 q^{64} + 4 q^{65} - 5 q^{67} + 5 q^{70} + 12 q^{71} + 16 q^{73} + 4 q^{74} + 4 q^{76} + 30 q^{77} - 2 q^{79} - q^{80} + 9 q^{82} - 6 q^{83} + 7 q^{86} - 6 q^{88} - 15 q^{89} + 4 q^{91} + 6 q^{92} - 2 q^{95} + 28 q^{97} + 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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 −0.500000 + 0.866025i −0.500000 0.866025i 0 −0.500000 2.59808i 1.00000 0 −0.500000 + 0.866025i
541.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 −0.500000 + 2.59808i 1.00000 0 −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 630.2.k.b 2
3.b odd 2 1 70.2.e.c 2
7.c even 3 1 inner 630.2.k.b 2
7.c even 3 1 4410.2.a.bm 1
7.d odd 6 1 4410.2.a.bd 1
12.b even 2 1 560.2.q.g 2
15.d odd 2 1 350.2.e.e 2
15.e even 4 2 350.2.j.b 4
21.c even 2 1 490.2.e.h 2
21.g even 6 1 490.2.a.b 1
21.g even 6 1 490.2.e.h 2
21.h odd 6 1 70.2.e.c 2
21.h odd 6 1 490.2.a.c 1
84.j odd 6 1 3920.2.a.bc 1
84.n even 6 1 560.2.q.g 2
84.n even 6 1 3920.2.a.p 1
105.o odd 6 1 350.2.e.e 2
105.o odd 6 1 2450.2.a.w 1
105.p even 6 1 2450.2.a.bc 1
105.w odd 12 2 2450.2.c.l 2
105.x even 12 2 350.2.j.b 4
105.x even 12 2 2450.2.c.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.c 2 3.b odd 2 1
70.2.e.c 2 21.h odd 6 1
350.2.e.e 2 15.d odd 2 1
350.2.e.e 2 105.o odd 6 1
350.2.j.b 4 15.e even 4 2
350.2.j.b 4 105.x even 12 2
490.2.a.b 1 21.g even 6 1
490.2.a.c 1 21.h odd 6 1
490.2.e.h 2 21.c even 2 1
490.2.e.h 2 21.g even 6 1
560.2.q.g 2 12.b even 2 1
560.2.q.g 2 84.n even 6 1
630.2.k.b 2 1.a even 1 1 trivial
630.2.k.b 2 7.c even 3 1 inner
2450.2.a.w 1 105.o odd 6 1
2450.2.a.bc 1 105.p even 6 1
2450.2.c.g 2 105.x even 12 2
2450.2.c.l 2 105.w odd 12 2
3920.2.a.p 1 84.n even 6 1
3920.2.a.bc 1 84.j odd 6 1
4410.2.a.bd 1 7.d odd 6 1
4410.2.a.bm 1 7.c even 3 1

Hecke kernels

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

\( T_{11}^{2} + 6T_{11} + 36 \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$13$ \( (T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$41$ \( (T + 9)^{2} \) Copy content Toggle raw display
$43$ \( (T + 7)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$67$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$79$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$83$ \( (T + 3)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$97$ \( (T - 14)^{2} \) Copy content Toggle raw display
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