# Properties

 Label 7350.2.a.bn Level $7350$ Weight $2$ Character orbit 7350.a Self dual yes Analytic conductor $58.690$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7350.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$58.6900454856$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} + q^{9} - 5q^{11} - q^{12} - q^{13} + q^{16} + 2q^{17} + q^{18} + 7q^{19} - 5q^{22} - 3q^{23} - q^{24} - q^{26} - q^{27} - 6q^{31} + q^{32} + 5q^{33} + 2q^{34} + q^{36} + 5q^{37} + 7q^{38} + q^{39} - 9q^{41} - 10q^{43} - 5q^{44} - 3q^{46} + 13q^{47} - q^{48} - 2q^{51} - q^{52} + q^{53} - q^{54} - 7q^{57} + 4q^{59} - 2q^{61} - 6q^{62} + q^{64} + 5q^{66} - 6q^{67} + 2q^{68} + 3q^{69} - 2q^{71} + q^{72} - 4q^{73} + 5q^{74} + 7q^{76} + q^{78} - 14q^{79} + q^{81} - 9q^{82} + 10q^{83} - 10q^{86} - 5q^{88} + 10q^{89} - 3q^{92} + 6q^{93} + 13q^{94} - q^{96} - 8q^{97} - 5q^{99} + O(q^{100})$$

## 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 0 −1.00000 0 1.00000 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## 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 7350.2.a.bn 1
5.b even 2 1 7350.2.a.t 1
5.c odd 4 2 1470.2.g.f 2
7.b odd 2 1 7350.2.a.ch 1
7.c even 3 2 1050.2.i.f 2
35.c odd 2 1 7350.2.a.b 1
35.f even 4 2 1470.2.g.a 2
35.j even 6 2 1050.2.i.o 2
35.k even 12 4 1470.2.n.i 4
35.l odd 12 4 210.2.n.a 4
105.x even 12 4 630.2.u.c 4
140.w even 12 4 1680.2.di.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.n.a 4 35.l odd 12 4
630.2.u.c 4 105.x even 12 4
1050.2.i.f 2 7.c even 3 2
1050.2.i.o 2 35.j even 6 2
1470.2.g.a 2 35.f even 4 2
1470.2.g.f 2 5.c odd 4 2
1470.2.n.i 4 35.k even 12 4
1680.2.di.a 4 140.w even 12 4
7350.2.a.b 1 35.c odd 2 1
7350.2.a.t 1 5.b even 2 1
7350.2.a.bn 1 1.a even 1 1 trivial
7350.2.a.ch 1 7.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7350))$$:

 $$T_{11} + 5$$ $$T_{13} + 1$$ $$T_{17} - 2$$ $$T_{19} - 7$$ $$T_{23} + 3$$ $$T_{31} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$5 + T$$
$13$ $$1 + T$$
$17$ $$-2 + T$$
$19$ $$-7 + T$$
$23$ $$3 + T$$
$29$ $$T$$
$31$ $$6 + T$$
$37$ $$-5 + T$$
$41$ $$9 + T$$
$43$ $$10 + T$$
$47$ $$-13 + T$$
$53$ $$-1 + T$$
$59$ $$-4 + T$$
$61$ $$2 + T$$
$67$ $$6 + T$$
$71$ $$2 + T$$
$73$ $$4 + T$$
$79$ $$14 + T$$
$83$ $$-10 + T$$
$89$ $$-10 + T$$
$97$ $$8 + T$$