# Properties

 Label 630.2.bk.b Level 630 Weight 2 Character orbit 630.bk Analytic conductor 5.031 Analytic rank 0 Dimension 28 CM no Inner twists 2

# Related objects

## 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: $$28$$ Relative dimension: $$14$$ over $$\Q(\zeta_{6})$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28q + 2q^{3} - 28q^{4} - 14q^{5} + 8q^{6} + 8q^{7} + 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q + 2q^{3} - 28q^{4} - 14q^{5} + 8q^{6} + 8q^{7} + 6q^{9} - 2q^{12} + 2q^{15} + 28q^{16} + 6q^{17} + 8q^{18} - 6q^{19} + 14q^{20} + 16q^{21} - 6q^{22} + 30q^{23} - 8q^{24} - 14q^{25} - 12q^{26} - 28q^{27} - 8q^{28} - 4q^{30} - 20q^{33} - 4q^{35} - 6q^{36} + 4q^{37} - 6q^{38} - 42q^{39} - 18q^{41} + 28q^{43} - 12q^{45} - 18q^{46} + 60q^{47} + 2q^{48} - 20q^{49} - 74q^{51} + 42q^{53} + 10q^{54} - 30q^{57} + 6q^{58} - 48q^{59} - 2q^{60} + 12q^{62} - 28q^{63} - 28q^{64} - 26q^{66} + 80q^{67} - 6q^{68} - 8q^{69} + 6q^{70} - 8q^{72} + 6q^{73} - 4q^{75} + 6q^{76} - 18q^{77} + 14q^{78} - 4q^{79} - 14q^{80} + 38q^{81} + 24q^{82} + 18q^{83} - 16q^{84} + 6q^{85} - 96q^{86} + 52q^{87} + 6q^{88} - 6q^{89} - 4q^{90} + 66q^{91} - 30q^{92} + 22q^{93} + 8q^{96} + 72q^{97} + 24q^{98} - 42q^{99} + O(q^{100})$$

## 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 −1.68763 0.389766i −1.00000 −0.500000 + 0.866025i −0.389766 + 1.68763i 0.408654 2.61400i 1.00000i 2.69616 + 1.31556i 0.866025 + 0.500000i
101.2 1.00000i −1.25628 + 1.19238i −1.00000 −0.500000 + 0.866025i 1.19238 + 1.25628i 1.55779 2.13853i 1.00000i 0.156478 2.99592i 0.866025 + 0.500000i
101.3 1.00000i −1.04939 1.37796i −1.00000 −0.500000 + 0.866025i −1.37796 + 1.04939i −0.323069 + 2.62595i 1.00000i −0.797561 + 2.89204i 0.866025 + 0.500000i
101.4 1.00000i 0.582530 + 1.63115i −1.00000 −0.500000 + 0.866025i 1.63115 0.582530i −1.91453 1.82609i 1.00000i −2.32132 + 1.90039i 0.866025 + 0.500000i
101.5 1.00000i 0.939646 1.45501i −1.00000 −0.500000 + 0.866025i −1.45501 0.939646i 2.57280 + 0.617016i 1.00000i −1.23413 2.73440i 0.866025 + 0.500000i
101.6 1.00000i 1.27771 + 1.16939i −1.00000 −0.500000 + 0.866025i 1.16939 1.27771i 2.48469 + 0.909025i 1.00000i 0.265066 + 2.98827i 0.866025 + 0.500000i
101.7 1.00000i 1.69341 + 0.363801i −1.00000 −0.500000 + 0.866025i 0.363801 1.69341i −1.05428 + 2.42662i 1.00000i 2.73530 + 1.23213i 0.866025 + 0.500000i
101.8 1.00000i −1.72522 0.153652i −1.00000 −0.500000 + 0.866025i 0.153652 1.72522i 0.145275 + 2.64176i 1.00000i 2.95278 + 0.530167i −0.866025 0.500000i
101.9 1.00000i −1.67419 0.443962i −1.00000 −0.500000 + 0.866025i 0.443962 1.67419i −1.17317 2.37143i 1.00000i 2.60580 + 1.48655i −0.866025 0.500000i
101.10 1.00000i −0.0386792 1.73162i −1.00000 −0.500000 + 0.866025i 1.73162 0.0386792i 0.281867 2.63069i 1.00000i −2.99701 + 0.133955i −0.866025 0.500000i
101.11 1.00000i 0.352560 + 1.69579i −1.00000 −0.500000 + 0.866025i −1.69579 + 0.352560i 2.55256 0.696025i 1.00000i −2.75140 + 1.19573i −0.866025 0.500000i
101.12 1.00000i 0.361836 1.69383i −1.00000 −0.500000 + 0.866025i 1.69383 + 0.361836i −1.96242 + 1.77452i 1.00000i −2.73815 1.22578i −0.866025 0.500000i
101.13 1.00000i 1.51724 0.835450i −1.00000 −0.500000 + 0.866025i 0.835450 + 1.51724i 2.64078 0.162142i 1.00000i 1.60405 2.53516i −0.866025 0.500000i
101.14 1.00000i 1.70645 + 0.296702i −1.00000 −0.500000 + 0.866025i −0.296702 + 1.70645i −2.21694 + 1.44401i 1.00000i 2.82394 + 1.01261i −0.866025 0.500000i
131.1 1.00000i −1.72522 + 0.153652i −1.00000 −0.500000 0.866025i 0.153652 + 1.72522i 0.145275 2.64176i 1.00000i 2.95278 0.530167i −0.866025 + 0.500000i
131.2 1.00000i −1.67419 + 0.443962i −1.00000 −0.500000 0.866025i 0.443962 + 1.67419i −1.17317 + 2.37143i 1.00000i 2.60580 1.48655i −0.866025 + 0.500000i
131.3 1.00000i −0.0386792 + 1.73162i −1.00000 −0.500000 0.866025i 1.73162 + 0.0386792i 0.281867 + 2.63069i 1.00000i −2.99701 0.133955i −0.866025 + 0.500000i
131.4 1.00000i 0.352560 1.69579i −1.00000 −0.500000 0.866025i −1.69579 0.352560i 2.55256 + 0.696025i 1.00000i −2.75140 1.19573i −0.866025 + 0.500000i
131.5 1.00000i 0.361836 + 1.69383i −1.00000 −0.500000 0.866025i 1.69383 0.361836i −1.96242 1.77452i 1.00000i −2.73815 + 1.22578i −0.866025 + 0.500000i
131.6 1.00000i 1.51724 + 0.835450i −1.00000 −0.500000 0.866025i 0.835450 1.51724i 2.64078 + 0.162142i 1.00000i 1.60405 + 2.53516i −0.866025 + 0.500000i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 131.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.b yes 28
3.b odd 2 1 1890.2.bk.b 28
7.d odd 6 1 630.2.t.b 28
9.c even 3 1 1890.2.t.b 28
9.d odd 6 1 630.2.t.b 28
21.g even 6 1 1890.2.t.b 28
63.i even 6 1 inner 630.2.bk.b yes 28
63.t odd 6 1 1890.2.bk.b 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.t.b 28 7.d odd 6 1
630.2.t.b 28 9.d odd 6 1
630.2.bk.b yes 28 1.a even 1 1 trivial
630.2.bk.b yes 28 63.i even 6 1 inner
1890.2.t.b 28 9.c even 3 1
1890.2.t.b 28 21.g even 6 1
1890.2.bk.b 28 3.b odd 2 1
1890.2.bk.b 28 63.t odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{28} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(630, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database