# Properties

 Label 7350.2.a.bz Level 7350 Weight 2 Character orbit 7350.a Self dual yes Analytic conductor 58.690 Analytic rank 0 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: $$0$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} + q^{9} + 2q^{11} - q^{12} - 6q^{13} + q^{16} - 4q^{17} + q^{18} + 6q^{19} + 2q^{22} - 8q^{23} - q^{24} - 6q^{26} - q^{27} + 6q^{29} + 2q^{31} + q^{32} - 2q^{33} - 4q^{34} + q^{36} + 4q^{37} + 6q^{38} + 6q^{39} - 2q^{41} + 4q^{43} + 2q^{44} - 8q^{46} - 8q^{47} - q^{48} + 4q^{51} - 6q^{52} + 6q^{53} - q^{54} - 6q^{57} + 6q^{58} + 8q^{59} + 10q^{61} + 2q^{62} + q^{64} - 2q^{66} + 8q^{67} - 4q^{68} + 8q^{69} - 6q^{71} + q^{72} + 14q^{73} + 4q^{74} + 6q^{76} + 6q^{78} - 12q^{79} + q^{81} - 2q^{82} + 8q^{83} + 4q^{86} - 6q^{87} + 2q^{88} + 10q^{89} - 8q^{92} - 2q^{93} - 8q^{94} - q^{96} + 10q^{97} + 2q^{99} + O(q^{100})$$

## 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.bz 1
5.b even 2 1 7350.2.a.bk 1
5.c odd 4 2 1470.2.g.b 2
7.b odd 2 1 1050.2.a.p 1
21.c even 2 1 3150.2.a.d 1
28.d even 2 1 8400.2.a.w 1
35.c odd 2 1 1050.2.a.d 1
35.f even 4 2 210.2.g.b 2
35.k even 12 4 1470.2.n.b 4
35.l odd 12 4 1470.2.n.f 4
105.g even 2 1 3150.2.a.bk 1
105.k odd 4 2 630.2.g.c 2
140.c even 2 1 8400.2.a.bp 1
140.j odd 4 2 1680.2.t.e 2
420.w even 4 2 5040.2.t.h 2

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

 $$T_{11} - 2$$ $$T_{13} + 6$$ $$T_{17} + 4$$ $$T_{19} - 6$$ $$T_{23} + 8$$ $$T_{31} - 2$$

## Hecke Characteristic Polynomials

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