# Properties

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

 $$\operatorname{Tr}(f)(q) =$$ $$28q - 2q^{3} + 14q^{4} - 28q^{5} - 8q^{6} - 4q^{7} + 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q - 2q^{3} + 14q^{4} - 28q^{5} - 8q^{6} - 4q^{7} + 6q^{9} - 4q^{12} + 6q^{14} + 2q^{15} - 14q^{16} - 6q^{17} - 4q^{18} - 6q^{19} - 14q^{20} + 6q^{21} - 6q^{22} - 4q^{24} + 28q^{25} + 12q^{26} + 28q^{27} - 8q^{28} + 8q^{30} - 12q^{31} + 26q^{33} + 4q^{35} - 6q^{36} + 4q^{37} - 12q^{38} + 54q^{39} + 18q^{41} - 32q^{42} + 28q^{43} - 6q^{45} - 18q^{46} + 30q^{47} - 2q^{48} - 14q^{49} + 34q^{51} - 42q^{53} + 14q^{54} + 6q^{56} - 30q^{57} - 12q^{58} - 24q^{59} + 4q^{60} + 24q^{61} - 12q^{62} + 44q^{63} - 28q^{64} - 10q^{66} - 40q^{67} - 12q^{68} + 8q^{69} - 6q^{70} + 4q^{72} + 6q^{73} - 2q^{75} - 6q^{76} - 24q^{77} + 14q^{78} + 2q^{79} + 14q^{80} - 46q^{81} + 24q^{82} - 18q^{83} + 18q^{84} + 6q^{85} + 104q^{87} - 12q^{88} + 6q^{89} + 4q^{90} + 66q^{91} - 30q^{92} + 40q^{93} + 42q^{94} + 6q^{95} + 4q^{96} - 72q^{97} - 24q^{98} - 42q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
311.1 −0.866025 0.500000i −1.71805 0.219817i 0.500000 + 0.866025i −1.00000 1.37796 + 1.04939i −2.11261 1.59276i 1.00000i 2.90336 + 0.755312i 0.866025 + 0.500000i
311.2 −0.866025 0.500000i −1.18136 1.26664i 0.500000 + 0.866025i −1.00000 0.389766 + 1.68763i 2.05946 + 1.66090i 1.00000i −0.208774 + 2.99273i 0.866025 + 0.500000i
311.3 −0.866025 0.500000i −0.790256 + 1.54126i 0.500000 + 0.866025i −1.00000 1.45501 0.939646i −1.82075 + 1.91960i 1.00000i −1.75099 2.43599i 0.866025 + 0.500000i
311.4 −0.866025 0.500000i 0.404488 1.68416i 0.500000 + 0.866025i −1.00000 −1.19238 + 1.25628i 1.07313 + 2.41835i 1.00000i −2.67278 1.36244i 0.866025 + 0.500000i
311.5 −0.866025 0.500000i 1.16177 + 1.28464i 0.500000 + 0.866025i −1.00000 −0.363801 1.69341i −1.57438 2.12634i 1.00000i −0.300592 + 2.98490i 0.866025 + 0.500000i
311.6 −0.866025 0.500000i 1.65157 + 0.521832i 0.500000 + 0.866025i −1.00000 −1.16939 1.27771i −2.02958 + 1.69729i 1.00000i 2.45538 + 1.72369i 0.866025 + 0.500000i
311.7 −0.866025 0.500000i 1.70388 0.311090i 0.500000 + 0.866025i −1.00000 −1.63115 0.582530i 2.53870 0.744985i 1.00000i 2.80645 1.06012i 0.866025 + 0.500000i
311.8 0.866025 + 0.500000i −1.51897 + 0.832312i 0.500000 + 0.866025i −1.00000 −1.73162 0.0386792i 2.13731 + 1.55945i 1.00000i 1.61451 2.52851i −0.866025 0.500000i
311.9 0.866025 + 0.500000i −1.28599 + 1.16028i 0.500000 + 0.866025i −1.00000 −1.69383 + 0.361836i −0.555567 2.58676i 1.00000i 0.307516 2.98420i −0.866025 0.500000i
311.10 0.866025 + 0.500000i −1.22158 1.22791i 0.500000 + 0.866025i −1.00000 −0.443962 1.67419i 2.64030 + 0.169721i 1.00000i −0.0155088 + 2.99996i −0.866025 0.500000i
311.11 0.866025 + 0.500000i −0.995677 1.41726i 0.500000 + 0.866025i −1.00000 −0.153652 1.72522i −2.36047 1.19507i 1.00000i −1.01725 + 2.82227i −0.866025 0.500000i
311.12 0.866025 + 0.500000i 0.0350998 + 1.73170i 0.500000 + 0.866025i −1.00000 −0.835450 + 1.51724i −1.17997 + 2.36805i 1.00000i −2.99754 + 0.121564i −0.866025 0.500000i
311.13 0.866025 + 0.500000i 1.11018 + 1.32948i 0.500000 + 0.866025i −1.00000 0.296702 + 1.70645i −0.142082 2.64193i 1.00000i −0.535019 + 2.95191i −0.866025 0.500000i
311.14 0.866025 + 0.500000i 1.64488 0.542569i 0.500000 + 0.866025i −1.00000 1.69579 + 0.352560i −0.673503 + 2.55859i 1.00000i 2.41124 1.78492i −0.866025 0.500000i
551.1 −0.866025 + 0.500000i −1.71805 + 0.219817i 0.500000 0.866025i −1.00000 1.37796 1.04939i −2.11261 + 1.59276i 1.00000i 2.90336 0.755312i 0.866025 0.500000i
551.2 −0.866025 + 0.500000i −1.18136 + 1.26664i 0.500000 0.866025i −1.00000 0.389766 1.68763i 2.05946 1.66090i 1.00000i −0.208774 2.99273i 0.866025 0.500000i
551.3 −0.866025 + 0.500000i −0.790256 1.54126i 0.500000 0.866025i −1.00000 1.45501 + 0.939646i −1.82075 1.91960i 1.00000i −1.75099 + 2.43599i 0.866025 0.500000i
551.4 −0.866025 + 0.500000i 0.404488 + 1.68416i 0.500000 0.866025i −1.00000 −1.19238 1.25628i 1.07313 2.41835i 1.00000i −2.67278 + 1.36244i 0.866025 0.500000i
551.5 −0.866025 + 0.500000i 1.16177 1.28464i 0.500000 0.866025i −1.00000 −0.363801 + 1.69341i −1.57438 + 2.12634i 1.00000i −0.300592 2.98490i 0.866025 0.500000i
551.6 −0.866025 + 0.500000i 1.65157 0.521832i 0.500000 0.866025i −1.00000 −1.16939 + 1.27771i −2.02958 1.69729i 1.00000i 2.45538 1.72369i 0.866025 0.500000i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 551.14 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.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.b 28
3.b odd 2 1 1890.2.t.b 28
7.d odd 6 1 630.2.bk.b yes 28
9.c even 3 1 1890.2.bk.b 28
9.d odd 6 1 630.2.bk.b yes 28
21.g even 6 1 1890.2.bk.b 28
63.k odd 6 1 1890.2.t.b 28
63.s even 6 1 inner 630.2.t.b 28

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