# Properties

 Label 7350.2.a.t 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

## 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.t 1
5.b even 2 1 7350.2.a.bn 1
5.c odd 4 2 1470.2.g.f 2
7.b odd 2 1 7350.2.a.b 1
7.c even 3 2 1050.2.i.o 2
35.c odd 2 1 7350.2.a.ch 1
35.f even 4 2 1470.2.g.a 2
35.j even 6 2 1050.2.i.f 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 35.j even 6 2
1050.2.i.o 2 7.c even 3 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 7.b odd 2 1
7350.2.a.t 1 1.a even 1 1 trivial
7350.2.a.bn 1 5.b even 2 1
7350.2.a.ch 1 35.c odd 2 1

## Atkin-Lehner signs

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

## 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

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