Properties

Label 630.2.t.c
Level 630
Weight 2
Character orbit 630.t
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.t (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.

\(\operatorname{Tr}(f)(q) = \) \( 32q + 2q^{3} + 16q^{4} + 32q^{5} + 2q^{6} - 2q^{7} + 6q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q + 2q^{3} + 16q^{4} + 32q^{5} + 2q^{6} - 2q^{7} + 6q^{9} + 4q^{12} + 2q^{15} - 16q^{16} + 6q^{17} - 4q^{18} + 16q^{20} + 16q^{21} + 4q^{24} + 32q^{25} + 12q^{26} + 8q^{27} + 2q^{28} + 6q^{29} + 2q^{30} + 18q^{31} + 16q^{33} - 2q^{35} + 2q^{37} - 18q^{39} - 6q^{41} + 6q^{42} - 28q^{43} - 6q^{44} + 6q^{45} + 24q^{47} + 2q^{48} + 32q^{49} - 26q^{51} - 36q^{53} - 32q^{54} - 6q^{56} + 18q^{57} - 30q^{59} + 4q^{60} + 54q^{61} - 94q^{63} - 32q^{64} - 44q^{66} + 4q^{67} + 12q^{68} + 28q^{69} + 4q^{72} - 30q^{73} + 2q^{75} - 6q^{77} - 22q^{78} + 4q^{79} - 16q^{80} + 26q^{81} - 24q^{82} + 6q^{83} - 4q^{84} + 6q^{85} - 8q^{87} - 12q^{89} - 4q^{90} - 66q^{91} - 18q^{92} - 32q^{93} - 42q^{94} + 2q^{96} + 96q^{97} - 24q^{98} - 24q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
311.1 −0.866025 0.500000i −1.69665 + 0.348398i 0.500000 + 0.866025i 1.00000 1.64354 + 0.546603i −1.74301 1.99046i 1.00000i 2.75724 1.18222i −0.866025 0.500000i
311.2 −0.866025 0.500000i −1.35238 1.08216i 0.500000 + 0.866025i 1.00000 0.630115 + 1.61337i −1.80382 + 1.93552i 1.00000i 0.657860 + 2.92698i −0.866025 0.500000i
311.3 −0.866025 0.500000i −1.15415 1.29148i 0.500000 + 0.866025i 1.00000 0.353784 + 1.69553i 2.60525 0.461188i 1.00000i −0.335862 + 2.98114i −0.866025 0.500000i
311.4 −0.866025 0.500000i −0.423533 + 1.67947i 0.500000 + 0.866025i 1.00000 1.20653 1.24270i 2.20024 1.46933i 1.00000i −2.64124 1.42262i −0.866025 0.500000i
311.5 −0.866025 0.500000i 0.521538 + 1.65167i 0.500000 + 0.866025i 1.00000 0.374168 1.69115i −2.64561 + 0.0270445i 1.00000i −2.45600 + 1.72281i −0.866025 0.500000i
311.6 −0.866025 0.500000i 1.34284 1.09398i 0.500000 + 0.866025i 1.00000 −1.70992 + 0.275993i 1.16109 2.37737i 1.00000i 0.606428 2.93807i −0.866025 0.500000i
311.7 −0.866025 0.500000i 1.61729 0.619981i 0.500000 + 0.866025i 1.00000 −1.71060 0.271725i −2.63727 0.211686i 1.00000i 2.23125 2.00538i −0.866025 0.500000i
311.8 −0.866025 0.500000i 1.64505 + 0.542044i 0.500000 + 0.866025i 1.00000 −1.15363 1.29195i 1.49710 + 2.18144i 1.00000i 2.41238 + 1.78338i −0.866025 0.500000i
311.9 0.866025 + 0.500000i −1.69871 0.338184i 0.500000 + 0.866025i 1.00000 −1.30204 1.14223i −2.46207 + 0.968625i 1.00000i 2.77126 + 1.14896i 0.866025 + 0.500000i
311.10 0.866025 + 0.500000i −1.55522 + 0.762428i 0.500000 + 0.866025i 1.00000 −1.72807 0.117327i 1.13965 2.38772i 1.00000i 1.83741 2.37148i 0.866025 + 0.500000i
311.11 0.866025 + 0.500000i −0.336991 1.69895i 0.500000 + 0.866025i 1.00000 0.557633 1.63983i 0.104916 + 2.64367i 1.00000i −2.77287 + 1.14506i 0.866025 + 0.500000i
311.12 0.866025 + 0.500000i −0.261236 1.71224i 0.500000 + 0.866025i 1.00000 0.629881 1.61346i −2.20349 1.46446i 1.00000i −2.86351 + 0.894597i 0.866025 + 0.500000i
311.13 0.866025 + 0.500000i 0.0292487 + 1.73180i 0.500000 + 0.866025i 1.00000 −0.840572 + 1.51441i 2.63859 + 0.194573i 1.00000i −2.99829 + 0.101306i 0.866025 + 0.500000i
311.14 0.866025 + 0.500000i 1.13939 1.30453i 0.500000 + 0.866025i 1.00000 1.63900 0.560056i 2.26702 + 1.36404i 1.00000i −0.403573 2.97273i 0.866025 + 0.500000i
311.15 0.866025 + 0.500000i 1.46653 + 0.921567i 0.500000 + 0.866025i 1.00000 0.809270 + 1.53137i −1.89754 + 1.84373i 1.00000i 1.30143 + 2.70301i 0.866025 + 0.500000i
311.16 0.866025 + 0.500000i 1.71699 0.227927i 0.500000 + 0.866025i 1.00000 1.60092 + 0.661104i 0.778946 2.52849i 1.00000i 2.89610 0.782695i 0.866025 + 0.500000i
551.1 −0.866025 + 0.500000i −1.69665 0.348398i 0.500000 0.866025i 1.00000 1.64354 0.546603i −1.74301 + 1.99046i 1.00000i 2.75724 + 1.18222i −0.866025 + 0.500000i
551.2 −0.866025 + 0.500000i −1.35238 + 1.08216i 0.500000 0.866025i 1.00000 0.630115 1.61337i −1.80382 1.93552i 1.00000i 0.657860 2.92698i −0.866025 + 0.500000i
551.3 −0.866025 + 0.500000i −1.15415 + 1.29148i 0.500000 0.866025i 1.00000 0.353784 1.69553i 2.60525 + 0.461188i 1.00000i −0.335862 2.98114i −0.866025 + 0.500000i
551.4 −0.866025 + 0.500000i −0.423533 1.67947i 0.500000 0.866025i 1.00000 1.20653 + 1.24270i 2.20024 + 1.46933i 1.00000i −2.64124 + 1.42262i −0.866025 + 0.500000i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 551.16
Significant digits:
Format:

Inner twists

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