# Properties

 Label 7350.2.a.cd 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

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

## 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.cd 1
5.b even 2 1 1470.2.a.f 1
7.b odd 2 1 7350.2.a.cx 1
7.c even 3 2 1050.2.i.i 2
15.d odd 2 1 4410.2.a.bh 1
35.c odd 2 1 1470.2.a.e 1
35.i odd 6 2 1470.2.i.p 2
35.j even 6 2 210.2.i.c 2
35.l odd 12 4 1050.2.o.c 4
105.g even 2 1 4410.2.a.w 1
105.o odd 6 2 630.2.k.a 2
140.p odd 6 2 1680.2.bg.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.c 2 35.j even 6 2
630.2.k.a 2 105.o odd 6 2
1050.2.i.i 2 7.c even 3 2
1050.2.o.c 4 35.l odd 12 4
1470.2.a.e 1 35.c odd 2 1
1470.2.a.f 1 5.b even 2 1
1470.2.i.p 2 35.i odd 6 2
1680.2.bg.n 2 140.p odd 6 2
4410.2.a.w 1 105.g even 2 1
4410.2.a.bh 1 15.d odd 2 1
7350.2.a.cd 1 1.a even 1 1 trivial
7350.2.a.cx 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7350))$$:

 $$T_{11} - 3$$ $$T_{13} + 5$$ $$T_{17}$$ $$T_{19} - 5$$ $$T_{23} - 9$$ $$T_{31} + 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$-3 + T$$
$13$ $$5 + T$$
$17$ $$T$$
$19$ $$-5 + T$$
$23$ $$-9 + T$$
$29$ $$T$$
$31$ $$10 + T$$
$37$ $$-1 + T$$
$41$ $$-9 + T$$
$43$ $$8 + T$$
$47$ $$3 + T$$
$53$ $$-3 + T$$
$59$ $$-12 + T$$
$61$ $$-8 + T$$
$67$ $$8 + T$$
$71$ $$6 + T$$
$73$ $$2 + T$$
$79$ $$-8 + T$$
$83$ $$T$$
$89$ $$-6 + T$$
$97$ $$8 + T$$