Properties

 Label 930.2.r.b 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.

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

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 −1.72090 0.196245i 1.00000 −1.99884 1.00231i −1.72090 0.196245i 1.33698 + 0.771903i 1.00000 2.92298 + 0.675436i −1.99884 1.00231i
119.2 1.00000 −1.71980 + 0.205656i 1.00000 0.949262 + 2.02457i −1.71980 + 0.205656i −1.56394 0.902941i 1.00000 2.91541 0.707375i 0.949262 + 2.02457i
119.3 1.00000 −1.67835 0.427964i 1.00000 1.21982 1.87404i −1.67835 0.427964i 3.99716 + 2.30776i 1.00000 2.63369 + 1.43654i 1.21982 1.87404i
119.4 1.00000 −1.57047 0.730502i 1.00000 2.16005 0.578104i −1.57047 0.730502i −0.447461 0.258342i 1.00000 1.93273 + 2.29446i 2.16005 0.578104i
119.5 1.00000 −1.54584 + 0.781271i 1.00000 2.22270 0.244106i −1.54584 + 0.781271i −2.13997 1.23551i 1.00000 1.77923 2.41544i 2.22270 0.244106i
119.6 1.00000 −1.41787 0.994813i 1.00000 −1.58068 + 1.58160i −1.41787 0.994813i 0.447461 + 0.258342i 1.00000 1.02069 + 2.82103i −1.58068 + 1.58160i
119.7 1.00000 −1.30243 + 1.14178i 1.00000 1.95677 + 1.08215i −1.30243 + 1.14178i 4.20156 + 2.42577i 1.00000 0.392656 2.97419i 1.95677 + 1.08215i
119.8 1.00000 −1.20980 1.23951i 1.00000 −2.23288 + 0.119376i −1.20980 1.23951i −3.99716 2.30776i 1.00000 −0.0727630 + 2.99912i −2.23288 + 0.119376i
119.9 1.00000 −1.12217 + 1.31937i 1.00000 −1.92892 + 1.13105i −1.12217 + 1.31937i −0.996061 0.575076i 1.00000 −0.481481 2.96111i −1.92892 + 1.13105i
119.10 1.00000 −1.12093 + 1.32043i 1.00000 −2.03226 0.932700i −1.12093 + 1.32043i 1.69016 + 0.975816i 1.00000 −0.487050 2.96020i −2.03226 0.932700i
119.11 1.00000 −1.03040 1.39222i 1.00000 0.131393 2.23220i −1.03040 1.39222i −1.33698 0.771903i 1.00000 −0.876543 + 2.86909i 0.131393 2.23220i
119.12 1.00000 −0.882485 + 1.49038i 1.00000 1.20301 1.88488i −0.882485 + 1.49038i −3.11576 1.79889i 1.00000 −1.44244 2.63047i 1.20301 1.88488i
119.13 1.00000 −0.681795 1.59222i 1.00000 1.27870 + 1.83437i −0.681795 1.59222i 1.56394 + 0.902941i 1.00000 −2.07031 + 2.17113i 1.27870 + 1.83437i
119.14 1.00000 −0.392310 + 1.68704i 1.00000 0.381329 + 2.20331i −0.392310 + 1.68704i −2.96938 1.71437i 1.00000 −2.69219 1.32368i 0.381329 + 2.20331i
119.15 1.00000 −0.0963182 1.72937i 1.00000 −1.32275 + 1.80286i −0.0963182 1.72937i 2.13997 + 1.23551i 1.00000 −2.98145 + 0.333140i −1.32275 + 1.80286i
119.16 1.00000 0.180653 + 1.72260i 1.00000 0.550525 2.16724i 0.180653 + 1.72260i 2.86191 + 1.65233i 1.00000 −2.93473 + 0.622388i 0.550525 2.16724i
119.17 1.00000 0.256671 + 1.71293i 1.00000 −1.42705 1.72149i 0.256671 + 1.71293i −2.76619 1.59706i 1.00000 −2.86824 + 0.879317i −1.42705 1.72149i
119.18 1.00000 0.337599 1.69883i 1.00000 −0.0412189 + 2.23569i 0.337599 1.69883i −4.20156 2.42577i 1.00000 −2.77205 1.14705i −0.0412189 + 2.23569i
119.19 1.00000 0.442238 + 1.67464i 1.00000 −1.20088 + 1.88624i 0.442238 + 1.67464i 2.75588 + 1.59111i 1.00000 −2.60885 + 1.48118i −1.20088 + 1.88624i
119.20 1.00000 0.552753 + 1.64148i 1.00000 2.23391 + 0.0983151i 0.552753 + 1.64148i 1.38670 + 0.800613i 1.00000 −2.38893 + 1.81467i 2.23391 + 0.0983151i
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.b yes 64
3.b odd 2 1 930.2.r.a 64
5.b even 2 1 930.2.r.a 64
15.d odd 2 1 inner 930.2.r.b yes 64
31.e odd 6 1 inner 930.2.r.b yes 64
93.g even 6 1 930.2.r.a 64
155.i odd 6 1 930.2.r.a 64
465.t even 6 1 inner 930.2.r.b yes 64

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