# Properties

 Label 7350.2.a.dc Level 7350 Weight 2 Character orbit 7350.a Self dual yes Analytic conductor 58.690 Analytic rank 1 Dimension 2 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} + ( 2 + \beta ) q^{11} - q^{12} -3 q^{13} + q^{16} + ( -3 - 2 \beta ) q^{17} - q^{18} -\beta q^{19} + ( -2 - \beta ) q^{22} + ( 1 - \beta ) q^{23} + q^{24} + 3 q^{26} - q^{27} + q^{29} + ( 1 + \beta ) q^{31} - q^{32} + ( -2 - \beta ) q^{33} + ( 3 + 2 \beta ) q^{34} + q^{36} + 5 \beta q^{37} + \beta q^{38} + 3 q^{39} + ( 3 + 2 \beta ) q^{41} + ( 3 - \beta ) q^{43} + ( 2 + \beta ) q^{44} + ( -1 + \beta ) q^{46} + ( -4 + 4 \beta ) q^{47} - q^{48} + ( 3 + 2 \beta ) q^{51} -3 q^{52} + ( 7 + 4 \beta ) q^{53} + q^{54} + \beta q^{57} - q^{58} + ( -3 - 3 \beta ) q^{59} + ( 3 - 2 \beta ) q^{61} + ( -1 - \beta ) q^{62} + q^{64} + ( 2 + \beta ) q^{66} + ( -4 - 8 \beta ) q^{67} + ( -3 - 2 \beta ) q^{68} + ( -1 + \beta ) q^{69} + ( 2 - 8 \beta ) q^{71} - q^{72} + ( -8 - 5 \beta ) q^{73} -5 \beta q^{74} -\beta q^{76} -3 q^{78} + ( -6 + 5 \beta ) q^{79} + q^{81} + ( -3 - 2 \beta ) q^{82} + ( -7 + 5 \beta ) q^{83} + ( -3 + \beta ) q^{86} - q^{87} + ( -2 - \beta ) q^{88} -4 q^{89} + ( 1 - \beta ) q^{92} + ( -1 - \beta ) q^{93} + ( 4 - 4 \beta ) q^{94} + q^{96} + ( 6 - 4 \beta ) q^{97} + ( 2 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 2q^{8} + 2q^{9} + 4q^{11} - 2q^{12} - 6q^{13} + 2q^{16} - 6q^{17} - 2q^{18} - 4q^{22} + 2q^{23} + 2q^{24} + 6q^{26} - 2q^{27} + 2q^{29} + 2q^{31} - 2q^{32} - 4q^{33} + 6q^{34} + 2q^{36} + 6q^{39} + 6q^{41} + 6q^{43} + 4q^{44} - 2q^{46} - 8q^{47} - 2q^{48} + 6q^{51} - 6q^{52} + 14q^{53} + 2q^{54} - 2q^{58} - 6q^{59} + 6q^{61} - 2q^{62} + 2q^{64} + 4q^{66} - 8q^{67} - 6q^{68} - 2q^{69} + 4q^{71} - 2q^{72} - 16q^{73} - 6q^{78} - 12q^{79} + 2q^{81} - 6q^{82} - 14q^{83} - 6q^{86} - 2q^{87} - 4q^{88} - 8q^{89} + 2q^{92} - 2q^{93} + 8q^{94} + 2q^{96} + 12q^{97} + 4q^{99} + O(q^{100})$$

## 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
 −1.41421 1.41421
−1.00000 −1.00000 1.00000 0 1.00000 0 −1.00000 1.00000 0
1.2 −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.dc 2
5.b even 2 1 7350.2.a.dm yes 2
7.b odd 2 1 7350.2.a.dg yes 2
35.c odd 2 1 7350.2.a.dj yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7350.2.a.dc 2 1.a even 1 1 trivial
7350.2.a.dg yes 2 7.b odd 2 1
7350.2.a.dj yes 2 35.c odd 2 1
7350.2.a.dm yes 2 5.b 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7350))$$:

 $$T_{11}^{2} - 4 T_{11} + 2$$ $$T_{13} + 3$$ $$T_{17}^{2} + 6 T_{17} + 1$$ $$T_{19}^{2} - 2$$ $$T_{23}^{2} - 2 T_{23} - 1$$ $$T_{31}^{2} - 2 T_{31} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ 1
$7$ 1
$11$ $$1 - 4 T + 24 T^{2} - 44 T^{3} + 121 T^{4}$$
$13$ $$( 1 + 3 T + 13 T^{2} )^{2}$$
$17$ $$1 + 6 T + 35 T^{2} + 102 T^{3} + 289 T^{4}$$
$19$ $$1 + 36 T^{2} + 361 T^{4}$$
$23$ $$1 - 2 T + 45 T^{2} - 46 T^{3} + 529 T^{4}$$
$29$ $$( 1 - T + 29 T^{2} )^{2}$$
$31$ $$1 - 2 T + 61 T^{2} - 62 T^{3} + 961 T^{4}$$
$37$ $$1 + 24 T^{2} + 1369 T^{4}$$
$41$ $$1 - 6 T + 83 T^{2} - 246 T^{3} + 1681 T^{4}$$
$43$ $$1 - 6 T + 93 T^{2} - 258 T^{3} + 1849 T^{4}$$
$47$ $$1 + 8 T + 78 T^{2} + 376 T^{3} + 2209 T^{4}$$
$53$ $$1 - 14 T + 123 T^{2} - 742 T^{3} + 2809 T^{4}$$
$59$ $$1 + 6 T + 109 T^{2} + 354 T^{3} + 3481 T^{4}$$
$61$ $$1 - 6 T + 123 T^{2} - 366 T^{3} + 3721 T^{4}$$
$67$ $$1 + 8 T + 22 T^{2} + 536 T^{3} + 4489 T^{4}$$
$71$ $$1 - 4 T + 18 T^{2} - 284 T^{3} + 5041 T^{4}$$
$73$ $$1 + 16 T + 160 T^{2} + 1168 T^{3} + 5329 T^{4}$$
$79$ $$1 + 12 T + 144 T^{2} + 948 T^{3} + 6241 T^{4}$$
$83$ $$1 + 14 T + 165 T^{2} + 1162 T^{3} + 6889 T^{4}$$
$89$ $$( 1 + 4 T + 89 T^{2} )^{2}$$
$97$ $$1 - 12 T + 198 T^{2} - 1164 T^{3} + 9409 T^{4}$$