# Properties

 Label 930.2.r.a Level $930$ Weight $2$ Character orbit 930.r Analytic conductor $7.426$ Analytic rank $0$ Dimension $64$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.r (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$32$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64q - 64q^{2} + 64q^{4} - 2q^{5} - 64q^{8} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q - 64q^{2} + 64q^{4} - 2q^{5} - 64q^{8} - 4q^{9} + 2q^{10} + 64q^{16} - 6q^{17} + 4q^{18} + 4q^{19} - 2q^{20} + 2q^{25} + 18q^{31} - 64q^{32} + 8q^{33} + 6q^{34} - 16q^{35} - 4q^{36} - 4q^{38} - 8q^{39} + 2q^{40} + 5q^{45} + 4q^{47} + 30q^{49} - 2q^{50} - 8q^{51} + 6q^{53} - 6q^{57} - 18q^{62} - 12q^{63} + 64q^{64} - 8q^{66} - 6q^{68} - 10q^{69} + 16q^{70} + 4q^{72} + 15q^{75} + 4q^{76} + 8q^{78} - 30q^{79} - 2q^{80} - 12q^{81} + 54q^{83} + 16q^{87} - 5q^{90} + 18q^{93} - 4q^{94} + 56q^{95} - 30q^{98} - 102q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
119.1 −1.00000 −1.69794 + 0.342043i 1.00000 −2.23391 0.0983151i 1.69794 0.342043i 1.38670 + 0.800613i −1.00000 2.76601 1.16154i 2.23391 + 0.0983151i
119.2 −1.00000 −1.67140 + 0.454332i 1.00000 1.20088 1.88624i 1.67140 0.454332i 2.75588 + 1.59111i −1.00000 2.58717 1.51874i −1.20088 + 1.88624i
119.3 −1.00000 −1.66709 0.469912i 1.00000 −1.78081 1.35230i 1.66709 + 0.469912i −2.19151 1.26527i −1.00000 2.55837 + 1.56677i 1.78081 + 1.35230i
119.4 −1.00000 −1.61177 + 0.634180i 1.00000 1.42705 + 1.72149i 1.61177 0.634180i −2.76619 1.59706i −1.00000 2.19563 2.04431i −1.42705 1.72149i
119.5 −1.00000 −1.60376 0.654183i 1.00000 2.18833 + 0.459589i 1.60376 + 0.654183i −0.263981 0.152410i −1.00000 2.14409 + 2.09830i −2.18833 0.459589i
119.6 −1.00000 −1.58215 + 0.704852i 1.00000 −0.550525 + 2.16724i 1.58215 0.704852i 2.86191 + 1.65233i −1.00000 2.00637 2.23036i 0.550525 2.16724i
119.7 −1.00000 −1.36842 1.06181i 1.00000 −0.696148 + 2.12494i 1.36842 + 1.06181i 0.263981 + 0.152410i −1.00000 0.745140 + 2.90599i 0.696148 2.12494i
119.8 −1.00000 −1.26486 + 1.18327i 1.00000 −0.381329 2.20331i 1.26486 1.18327i −2.96938 1.71437i −1.00000 0.199749 2.99334i 0.381329 + 2.20331i
119.9 −1.00000 −1.24050 1.20878i 1.00000 −0.280722 2.21838i 1.24050 + 1.20878i 2.19151 + 1.26527i −1.00000 0.0776784 + 2.99899i 0.280722 + 2.21838i
119.10 −1.00000 −0.849461 + 1.50944i 1.00000 −1.20301 + 1.88488i 0.849461 1.50944i −3.11576 1.79889i −1.00000 −1.55683 2.56442i 1.20301 1.88488i
119.11 −1.00000 −0.583060 + 1.63096i 1.00000 2.03226 + 0.932700i 0.583060 1.63096i 1.69016 + 0.975816i −1.00000 −2.32008 1.90190i −2.03226 0.932700i
119.12 −1.00000 −0.581525 + 1.63151i 1.00000 1.92892 1.13105i 0.581525 1.63151i −0.996061 0.575076i −1.00000 −2.32366 1.89753i −1.92892 + 1.13105i
119.13 −1.00000 −0.552753 1.64148i 1.00000 1.03181 1.98378i 0.552753 + 1.64148i −1.38670 0.800613i −1.00000 −2.38893 + 1.81467i −1.03181 + 1.98378i
119.14 −1.00000 −0.442238 1.67464i 1.00000 −2.23397 + 0.0968727i 0.442238 + 1.67464i −2.75588 1.59111i −1.00000 −2.60885 + 1.48118i 2.23397 0.0968727i
119.15 −1.00000 −0.337599 + 1.69883i 1.00000 −1.95677 1.08215i 0.337599 1.69883i 4.20156 + 2.42577i −1.00000 −2.77205 1.14705i 1.95677 + 1.08215i
119.16 −1.00000 −0.256671 1.71293i 1.00000 0.777335 + 2.09660i 0.256671 + 1.71293i 2.76619 + 1.59706i −1.00000 −2.86824 + 0.879317i −0.777335 2.09660i
119.17 −1.00000 −0.180653 1.72260i 1.00000 2.15215 + 0.606851i 0.180653 + 1.72260i −2.86191 1.65233i −1.00000 −2.93473 + 0.622388i −2.15215 0.606851i
119.18 −1.00000 0.0963182 + 1.72937i 1.00000 −2.22270 + 0.244106i −0.0963182 1.72937i −2.13997 1.23551i −1.00000 −2.98145 + 0.333140i 2.22270 0.244106i
119.19 −1.00000 0.392310 1.68704i 1.00000 −1.71746 1.43190i −0.392310 + 1.68704i 2.96938 + 1.71437i −1.00000 −2.69219 1.32368i 1.71746 + 1.43190i
119.20 −1.00000 0.681795 + 1.59222i 1.00000 −0.949262 2.02457i −0.681795 1.59222i −1.56394 0.902941i −1.00000 −2.07031 + 2.17113i 0.949262 + 2.02457i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 719.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner
31.e odd 6 1 inner
465.t even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.r.a 64
3.b odd 2 1 930.2.r.b yes 64
5.b even 2 1 930.2.r.b yes 64
15.d odd 2 1 inner 930.2.r.a 64
31.e odd 6 1 inner 930.2.r.a 64
93.g even 6 1 930.2.r.b yes 64
155.i odd 6 1 930.2.r.b yes 64
465.t even 6 1 inner 930.2.r.a 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.r.a 64 1.a even 1 1 trivial
930.2.r.a 64 15.d odd 2 1 inner
930.2.r.a 64 31.e odd 6 1 inner
930.2.r.a 64 465.t even 6 1 inner
930.2.r.b yes 64 3.b odd 2 1
930.2.r.b yes 64 5.b even 2 1
930.2.r.b yes 64 93.g even 6 1
930.2.r.b yes 64 155.i odd 6 1