# Properties

 Label 9310.2.a.o Level $9310$ Weight $2$ Character orbit 9310.a Self dual yes Analytic conductor $74.341$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$9310 = 2 \cdot 5 \cdot 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9310.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$74.3407242818$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} - 2q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} - 2q^{9} - q^{10} - q^{12} + q^{13} + q^{15} + q^{16} + 3q^{17} - 2q^{18} - q^{19} - q^{20} + 3q^{23} - q^{24} + q^{25} + q^{26} + 5q^{27} - 3q^{29} + q^{30} - 2q^{31} + q^{32} + 3q^{34} - 2q^{36} - 10q^{37} - q^{38} - q^{39} - q^{40} - 6q^{41} + 2q^{43} + 2q^{45} + 3q^{46} - q^{48} + q^{50} - 3q^{51} + q^{52} + 3q^{53} + 5q^{54} + q^{57} - 3q^{58} - 3q^{59} + q^{60} - 8q^{61} - 2q^{62} + q^{64} - q^{65} - 7q^{67} + 3q^{68} - 3q^{69} + 12q^{71} - 2q^{72} + 13q^{73} - 10q^{74} - q^{75} - q^{76} - q^{78} + 14q^{79} - q^{80} + q^{81} - 6q^{82} - 6q^{83} - 3q^{85} + 2q^{86} + 3q^{87} - 6q^{89} + 2q^{90} + 3q^{92} + 2q^{93} + q^{95} - q^{96} + 10q^{97} + O(q^{100})$$

## 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 0 1.00000 −2.00000 −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
$$2$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$-1$$
$$19$$ $$1$$

## 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 9310.2.a.o 1
7.b odd 2 1 190.2.a.c 1
21.c even 2 1 1710.2.a.d 1
28.d even 2 1 1520.2.a.d 1
35.c odd 2 1 950.2.a.a 1
35.f even 4 2 950.2.b.e 2
56.e even 2 1 6080.2.a.p 1
56.h odd 2 1 6080.2.a.h 1
105.g even 2 1 8550.2.a.bd 1
133.c even 2 1 3610.2.a.b 1
140.c even 2 1 7600.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.c 1 7.b odd 2 1
950.2.a.a 1 35.c odd 2 1
950.2.b.e 2 35.f even 4 2
1520.2.a.d 1 28.d even 2 1
1710.2.a.d 1 21.c even 2 1
3610.2.a.b 1 133.c even 2 1
6080.2.a.h 1 56.h odd 2 1
6080.2.a.p 1 56.e even 2 1
7600.2.a.m 1 140.c even 2 1
8550.2.a.bd 1 105.g even 2 1
9310.2.a.o 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9310))$$:

 $$T_{3} + 1$$ $$T_{11}$$ $$T_{13} - 1$$ $$T_{17} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$1 + T$$
$5$ $$1 + T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$-1 + T$$
$17$ $$-3 + T$$
$19$ $$1 + T$$
$23$ $$-3 + T$$
$29$ $$3 + T$$
$31$ $$2 + T$$
$37$ $$10 + T$$
$41$ $$6 + T$$
$43$ $$-2 + T$$
$47$ $$T$$
$53$ $$-3 + T$$
$59$ $$3 + T$$
$61$ $$8 + T$$
$67$ $$7 + T$$
$71$ $$-12 + T$$
$73$ $$-13 + T$$
$79$ $$-14 + T$$
$83$ $$6 + T$$
$89$ $$6 + T$$
$97$ $$-10 + T$$
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