# Properties

 Label 7350.2.a.co 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} - 2q^{11} + q^{12} - 2q^{13} + q^{16} - 8q^{17} + q^{18} + 2q^{19} - 2q^{22} + q^{24} - 2q^{26} + q^{27} - 6q^{29} - 6q^{31} + q^{32} - 2q^{33} - 8q^{34} + q^{36} + 8q^{37} + 2q^{38} - 2q^{39} - 6q^{41} + 8q^{43} - 2q^{44} - 4q^{47} + q^{48} - 8q^{51} - 2q^{52} - 2q^{53} + q^{54} + 2q^{57} - 6q^{58} + 8q^{59} - 10q^{61} - 6q^{62} + q^{64} - 2q^{66} - 12q^{67} - 8q^{68} - 14q^{71} + q^{72} + 10q^{73} + 8q^{74} + 2q^{76} - 2q^{78} + 4q^{79} + q^{81} - 6q^{82} - 16q^{83} + 8q^{86} - 6q^{87} - 2q^{88} - 10q^{89} - 6q^{93} - 4q^{94} + q^{96} - 10q^{97} - 2q^{99} + O(q^{100})$$

## 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.co 1
5.b even 2 1 7350.2.a.g 1
5.c odd 4 2 1470.2.g.e 2
7.b odd 2 1 1050.2.a.m 1
21.c even 2 1 3150.2.a.q 1
28.d even 2 1 8400.2.a.ca 1
35.c odd 2 1 1050.2.a.g 1
35.f even 4 2 210.2.g.a 2
35.k even 12 4 1470.2.n.g 4
35.l odd 12 4 1470.2.n.c 4
105.g even 2 1 3150.2.a.be 1
105.k odd 4 2 630.2.g.d 2
140.c even 2 1 8400.2.a.bd 1
140.j odd 4 2 1680.2.t.d 2
420.w even 4 2 5040.2.t.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.g.a 2 35.f even 4 2
630.2.g.d 2 105.k odd 4 2
1050.2.a.g 1 35.c odd 2 1
1050.2.a.m 1 7.b odd 2 1
1470.2.g.e 2 5.c odd 4 2
1470.2.n.c 4 35.l odd 12 4
1470.2.n.g 4 35.k even 12 4
1680.2.t.d 2 140.j odd 4 2
3150.2.a.q 1 21.c even 2 1
3150.2.a.be 1 105.g even 2 1
5040.2.t.k 2 420.w even 4 2
7350.2.a.g 1 5.b even 2 1
7350.2.a.co 1 1.a even 1 1 trivial
8400.2.a.bd 1 140.c even 2 1
8400.2.a.ca 1 28.d even 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7350))$$:

 $$T_{11} + 2$$ $$T_{13} + 2$$ $$T_{17} + 8$$ $$T_{19} - 2$$ $$T_{23}$$ $$T_{31} + 6$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T$$
$3$ $$1 - T$$
$5$ 1
$7$ 1
$11$ $$1 + 2 T + 11 T^{2}$$
$13$ $$1 + 2 T + 13 T^{2}$$
$17$ $$1 + 8 T + 17 T^{2}$$
$19$ $$1 - 2 T + 19 T^{2}$$
$23$ $$1 + 23 T^{2}$$
$29$ $$1 + 6 T + 29 T^{2}$$
$31$ $$1 + 6 T + 31 T^{2}$$
$37$ $$1 - 8 T + 37 T^{2}$$
$41$ $$1 + 6 T + 41 T^{2}$$
$43$ $$1 - 8 T + 43 T^{2}$$
$47$ $$1 + 4 T + 47 T^{2}$$
$53$ $$1 + 2 T + 53 T^{2}$$
$59$ $$1 - 8 T + 59 T^{2}$$
$61$ $$1 + 10 T + 61 T^{2}$$
$67$ $$1 + 12 T + 67 T^{2}$$
$71$ $$1 + 14 T + 71 T^{2}$$
$73$ $$1 - 10 T + 73 T^{2}$$
$79$ $$1 - 4 T + 79 T^{2}$$
$83$ $$1 + 16 T + 83 T^{2}$$
$89$ $$1 + 10 T + 89 T^{2}$$
$97$ $$1 + 10 T + 97 T^{2}$$