Properties

 Label 930.2.a.h Level $930$ Weight $2$ Character orbit 930.a Self dual yes Analytic conductor $7.426$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$7.42608738798$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 2q^{7} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 2q^{7} - q^{8} + q^{9} - q^{10} - 4q^{11} + q^{12} - 4q^{13} + 2q^{14} + q^{15} + q^{16} + 2q^{17} - q^{18} - 8q^{19} + q^{20} - 2q^{21} + 4q^{22} - 8q^{23} - q^{24} + q^{25} + 4q^{26} + q^{27} - 2q^{28} + 4q^{29} - q^{30} - q^{31} - q^{32} - 4q^{33} - 2q^{34} - 2q^{35} + q^{36} - 12q^{37} + 8q^{38} - 4q^{39} - q^{40} + 10q^{41} + 2q^{42} + 8q^{43} - 4q^{44} + q^{45} + 8q^{46} - 4q^{47} + q^{48} - 3q^{49} - q^{50} + 2q^{51} - 4q^{52} + 6q^{53} - q^{54} - 4q^{55} + 2q^{56} - 8q^{57} - 4q^{58} + 2q^{59} + q^{60} + 10q^{61} + q^{62} - 2q^{63} + q^{64} - 4q^{65} + 4q^{66} - 6q^{67} + 2q^{68} - 8q^{69} + 2q^{70} + 6q^{71} - q^{72} - 4q^{73} + 12q^{74} + q^{75} - 8q^{76} + 8q^{77} + 4q^{78} - 8q^{79} + q^{80} + q^{81} - 10q^{82} + 4q^{83} - 2q^{84} + 2q^{85} - 8q^{86} + 4q^{87} + 4q^{88} - q^{90} + 8q^{91} - 8q^{92} - q^{93} + 4q^{94} - 8q^{95} - q^{96} - 18q^{97} + 3q^{98} - 4q^{99} + O(q^{100})$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 1.00000 1.00000 1.00000 −1.00000 −2.00000 −1.00000 1.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$31$$ $$1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.a.h 1
3.b odd 2 1 2790.2.a.p 1
4.b odd 2 1 7440.2.a.m 1
5.b even 2 1 4650.2.a.bg 1
5.c odd 4 2 4650.2.d.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.a.h 1 1.a even 1 1 trivial
2790.2.a.p 1 3.b odd 2 1
4650.2.a.bg 1 5.b even 2 1
4650.2.d.b 2 5.c odd 4 2
7440.2.a.m 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(930))$$:

 $$T_{7} + 2$$ $$T_{11} + 4$$ $$T_{19} + 8$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$-1 + T$$
$5$ $$-1 + T$$
$7$ $$2 + T$$
$11$ $$4 + T$$
$13$ $$4 + T$$
$17$ $$-2 + T$$
$19$ $$8 + T$$
$23$ $$8 + T$$
$29$ $$-4 + T$$
$31$ $$1 + T$$
$37$ $$12 + T$$
$41$ $$-10 + T$$
$43$ $$-8 + T$$
$47$ $$4 + T$$
$53$ $$-6 + T$$
$59$ $$-2 + T$$
$61$ $$-10 + T$$
$67$ $$6 + T$$
$71$ $$-6 + T$$
$73$ $$4 + T$$
$79$ $$8 + T$$
$83$ $$-4 + T$$
$89$ $$T$$
$97$ $$18 + T$$