Properties

Label 630.2.k.i
Level 630
Weight 2
Character orbit 630.k
Analytic conductor 5.031
Analytic rank 0
Dimension 4
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: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/7\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(7 \beta_{2}\)
\(\nu^{3}\)\(=\)\(7 \beta_{3}\)

Character values

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

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{2}\)

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
1.32288 + 2.29129i
−1.32288 2.29129i
1.32288 2.29129i
−1.32288 + 2.29129i
−0.500000 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −2.64575 1.00000 0 0.500000 0.866025i
361.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 2.64575 1.00000 0 0.500000 0.866025i
541.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 −2.64575 1.00000 0 0.500000 + 0.866025i
541.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 2.64575 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.i 4
3.b odd 2 1 630.2.k.j yes 4
7.c even 3 1 inner 630.2.k.i 4
7.c even 3 1 4410.2.a.bv 2
7.d odd 6 1 4410.2.a.ca 2
21.g even 6 1 4410.2.a.bo 2
21.h odd 6 1 630.2.k.j yes 4
21.h odd 6 1 4410.2.a.bq 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.k.i 4 1.a even 1 1 trivial
630.2.k.i 4 7.c even 3 1 inner
630.2.k.j yes 4 3.b odd 2 1
630.2.k.j yes 4 21.h odd 6 1
4410.2.a.bo 2 21.g even 6 1
4410.2.a.bq 2 21.h odd 6 1
4410.2.a.bv 2 7.c even 3 1
4410.2.a.ca 2 7.d odd 6 1

Hecke kernels

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

\( T_{11}^{4} + 4 T_{11}^{3} + 19 T_{11}^{2} - 12 T_{11} + 9 \)
\( T_{13}^{2} + 4 T_{13} - 3 \)
\( T_{17}^{4} - 4 T_{17}^{3} + 40 T_{17}^{2} + 96 T_{17} + 576 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ 1
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( 1 + 4 T - 3 T^{2} - 12 T^{3} + 64 T^{4} - 132 T^{5} - 363 T^{6} + 5324 T^{7} + 14641 T^{8} \)
$13$ \( ( 1 + 4 T + 23 T^{2} + 52 T^{3} + 169 T^{4} )^{2} \)
$17$ \( 1 - 4 T + 6 T^{2} + 96 T^{3} - 461 T^{4} + 1632 T^{5} + 1734 T^{6} - 19652 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 2 T - 7 T^{2} - 54 T^{3} - 316 T^{4} - 1026 T^{5} - 2527 T^{6} + 13718 T^{7} + 130321 T^{8} \)
$23$ \( ( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 29 T^{2} )^{4} \)
$31$ \( ( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} )^{2} \)
$37$ \( 1 + 4 T + T^{2} - 236 T^{3} - 1736 T^{4} - 8732 T^{5} + 1369 T^{6} + 202612 T^{7} + 1874161 T^{8} \)
$41$ \( ( 1 - 8 T + 91 T^{2} - 328 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 12 T + 94 T^{2} - 516 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( 1 + 2 T + 21 T^{2} - 222 T^{3} - 2108 T^{4} - 10434 T^{5} + 46389 T^{6} + 207646 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 10 T - 3 T^{2} - 30 T^{3} + 2500 T^{4} - 1590 T^{5} - 8427 T^{6} + 1488770 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 8 T - 42 T^{2} + 96 T^{3} + 3979 T^{4} + 5664 T^{5} - 146202 T^{6} - 1643032 T^{7} + 12117361 T^{8} \)
$61$ \( 1 + 12 T + 14 T^{2} + 96 T^{3} + 4395 T^{4} + 5856 T^{5} + 52094 T^{6} + 2723772 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 20 T + 194 T^{2} + 1440 T^{3} + 11147 T^{4} + 96480 T^{5} + 870866 T^{6} + 6015260 T^{7} + 20151121 T^{8} \)
$71$ \( ( 1 + 16 T + 178 T^{2} + 1136 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( 1 + 12 T - 22 T^{2} + 96 T^{3} + 9939 T^{4} + 7584 T^{5} - 137302 T^{6} + 5916468 T^{7} + 38950081 T^{8} \)
$83$ \( ( 1 - 20 T + 238 T^{2} - 1660 T^{3} + 6889 T^{4} )^{2} \)
$89$ \( 1 - 8 T - 18 T^{2} + 768 T^{3} - 6893 T^{4} + 68352 T^{5} - 142578 T^{6} - 5639752 T^{7} + 62742241 T^{8} \)
$97$ \( ( 1 + 4 T + 97 T^{2} )^{4} \)
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