# Properties

 Label 7350.2.a.cs 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

# Learn more about

## 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

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

## 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.cs 1
5.b even 2 1 1470.2.a.b 1
7.b odd 2 1 1050.2.a.k 1
15.d odd 2 1 4410.2.a.bi 1
21.c even 2 1 3150.2.a.f 1
28.d even 2 1 8400.2.a.cm 1
35.c odd 2 1 210.2.a.b 1
35.f even 4 2 1050.2.g.c 2
35.i odd 6 2 1470.2.i.l 2
35.j even 6 2 1470.2.i.s 2
105.g even 2 1 630.2.a.h 1
105.k odd 4 2 3150.2.g.i 2
140.c even 2 1 1680.2.a.g 1
280.c odd 2 1 6720.2.a.n 1
280.n even 2 1 6720.2.a.bi 1
420.o odd 2 1 5040.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.a.b 1 35.c odd 2 1
630.2.a.h 1 105.g even 2 1
1050.2.a.k 1 7.b odd 2 1
1050.2.g.c 2 35.f even 4 2
1470.2.a.b 1 5.b even 2 1
1470.2.i.l 2 35.i odd 6 2
1470.2.i.s 2 35.j even 6 2
1680.2.a.g 1 140.c even 2 1
3150.2.a.f 1 21.c even 2 1
3150.2.g.i 2 105.k odd 4 2
4410.2.a.bi 1 15.d odd 2 1
5040.2.a.g 1 420.o odd 2 1
6720.2.a.n 1 280.c odd 2 1
6720.2.a.bi 1 280.n even 2 1
7350.2.a.cs 1 1.a even 1 1 trivial
8400.2.a.cm 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7350))$$:

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

## Hecke characteristic polynomials

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