Properties

Label 630.2.bk.c
Level 630
Weight 2
Character orbit 630.bk
Analytic conductor 5.031
Analytic rank 0
Dimension 32
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.bk (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32q - 2q^{3} - 32q^{4} + 16q^{5} - 2q^{6} - 2q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q - 2q^{3} - 32q^{4} + 16q^{5} - 2q^{6} - 2q^{7} - 6q^{11} + 2q^{12} + 6q^{14} + 2q^{15} + 32q^{16} - 6q^{17} + 8q^{18} - 16q^{20} + 18q^{23} + 2q^{24} - 16q^{25} - 12q^{26} - 8q^{27} + 2q^{28} + 6q^{29} - 4q^{30} + 20q^{33} + 2q^{35} + 2q^{37} + 30q^{39} + 6q^{41} + 10q^{42} - 28q^{43} + 6q^{44} + 6q^{45} + 48q^{47} - 2q^{48} + 8q^{49} + 34q^{51} + 36q^{53} - 46q^{54} - 6q^{56} + 18q^{57} - 60q^{59} - 2q^{60} + 32q^{63} - 32q^{64} - 34q^{66} - 8q^{67} + 6q^{68} - 28q^{69} + 6q^{70} - 8q^{72} - 30q^{73} - 18q^{74} + 4q^{75} - 6q^{77} - 22q^{78} - 8q^{79} + 16q^{80} + 20q^{81} - 24q^{82} - 6q^{83} + 6q^{85} + 30q^{86} - 22q^{87} + 12q^{89} + 4q^{90} - 66q^{91} - 18q^{92} - 32q^{93} - 2q^{96} - 96q^{97} + 24q^{98} - 24q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 1.00000i −1.69553 0.353784i −1.00000 0.500000 0.866025i −0.353784 + 1.69553i −1.70202 2.02561i 1.00000i 2.74967 + 1.19970i −0.866025 0.500000i
101.2 1.00000i −1.61337 0.630115i −1.00000 0.500000 0.866025i −0.630115 + 1.61337i 2.57812 + 0.594391i 1.00000i 2.20591 + 2.03321i −0.866025 0.500000i
101.3 1.00000i −0.546603 1.64354i −1.00000 0.500000 0.866025i −1.64354 + 0.546603i −0.852286 + 2.50472i 1.00000i −2.40245 + 1.79673i −0.866025 0.500000i
101.4 1.00000i −0.275993 + 1.70992i −1.00000 0.500000 0.866025i 1.70992 + 0.275993i −2.63940 + 0.183152i 1.00000i −2.84766 0.943852i −0.866025 0.500000i
101.5 1.00000i 0.271725 + 1.71060i −1.00000 0.500000 0.866025i 1.71060 0.271725i 1.13531 + 2.38979i 1.00000i −2.85233 + 0.929628i −0.866025 0.500000i
101.6 1.00000i 1.24270 1.20653i −1.00000 0.500000 0.866025i −1.20653 1.24270i −2.37260 1.17080i 1.00000i 0.0885938 2.99869i −0.866025 0.500000i
101.7 1.00000i 1.29195 + 1.15363i −1.00000 0.500000 0.866025i 1.15363 1.29195i 1.14063 2.38725i 1.00000i 0.338261 + 2.98087i −0.866025 0.500000i
101.8 1.00000i 1.69115 0.374168i −1.00000 0.500000 0.866025i −0.374168 1.69115i 1.34623 + 2.27765i 1.00000i 2.72000 1.26555i −0.866025 0.500000i
101.9 1.00000i −1.63983 + 0.557633i −1.00000 0.500000 0.866025i −0.557633 1.63983i 2.23703 1.41269i 1.00000i 2.37809 1.82885i 0.866025 + 0.500000i
101.10 1.00000i −1.61346 + 0.629881i −1.00000 0.500000 0.866025i −0.629881 1.61346i −0.166511 + 2.64051i 1.00000i 2.20650 2.03258i 0.866025 + 0.500000i
101.11 1.00000i −1.14223 1.30204i −1.00000 0.500000 0.866025i 1.30204 1.14223i 2.06989 + 1.64790i 1.00000i −0.390607 + 2.97446i 0.866025 + 0.500000i
101.12 1.00000i −0.560056 + 1.63900i −1.00000 0.500000 0.866025i −1.63900 0.560056i 0.0477786 2.64532i 1.00000i −2.37267 1.83587i 0.866025 + 0.500000i
101.13 1.00000i −0.117327 1.72807i −1.00000 0.500000 0.866025i 1.72807 0.117327i −2.63765 + 0.206895i 1.00000i −2.97247 + 0.405499i 0.866025 + 0.500000i
101.14 1.00000i 0.661104 + 1.60092i −1.00000 0.500000 0.866025i −1.60092 + 0.661104i −2.57921 + 0.589657i 1.00000i −2.12588 + 2.11675i 0.866025 + 0.500000i
101.15 1.00000i 1.51441 0.840572i −1.00000 0.500000 0.866025i 0.840572 + 1.51441i −1.15079 2.38237i 1.00000i 1.58688 2.54594i 0.866025 + 0.500000i
101.16 1.00000i 1.53137 + 0.809270i −1.00000 0.500000 0.866025i −0.809270 + 1.53137i 2.54549 + 0.721453i 1.00000i 1.69016 + 2.47858i 0.866025 + 0.500000i
131.1 1.00000i −1.63983 0.557633i −1.00000 0.500000 + 0.866025i −0.557633 + 1.63983i 2.23703 + 1.41269i 1.00000i 2.37809 + 1.82885i 0.866025 0.500000i
131.2 1.00000i −1.61346 0.629881i −1.00000 0.500000 + 0.866025i −0.629881 + 1.61346i −0.166511 2.64051i 1.00000i 2.20650 + 2.03258i 0.866025 0.500000i
131.3 1.00000i −1.14223 + 1.30204i −1.00000 0.500000 + 0.866025i 1.30204 + 1.14223i 2.06989 1.64790i 1.00000i −0.390607 2.97446i 0.866025 0.500000i
131.4 1.00000i −0.560056 1.63900i −1.00000 0.500000 + 0.866025i −1.63900 + 0.560056i 0.0477786 + 2.64532i 1.00000i −2.37267 + 1.83587i 0.866025 0.500000i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 131.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.bk.c yes 32
3.b odd 2 1 1890.2.bk.c 32
7.d odd 6 1 630.2.t.c 32
9.c even 3 1 1890.2.t.c 32
9.d odd 6 1 630.2.t.c 32
21.g even 6 1 1890.2.t.c 32
63.i even 6 1 inner 630.2.bk.c yes 32
63.t odd 6 1 1890.2.bk.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.t.c 32 7.d odd 6 1
630.2.t.c 32 9.d odd 6 1
630.2.bk.c yes 32 1.a even 1 1 trivial
630.2.bk.c yes 32 63.i even 6 1 inner
1890.2.t.c 32 9.c even 3 1
1890.2.t.c 32 21.g even 6 1
1890.2.bk.c 32 3.b odd 2 1
1890.2.bk.c 32 63.t odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{11}^{32} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database