# Properties

 Label 7350.2.a.dk 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

# 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} +O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} + ( -2 + 3 \beta ) q^{11} + q^{12} + ( -3 - 2 \beta ) q^{13} + q^{16} + ( -1 - 4 \beta ) q^{17} + q^{18} + ( -4 - \beta ) q^{19} + ( -2 + 3 \beta ) q^{22} + ( 3 + \beta ) q^{23} + q^{24} + ( -3 - 2 \beta ) q^{26} + q^{27} + ( -5 + 4 \beta ) q^{29} + ( -3 + 3 \beta ) q^{31} + q^{32} + ( -2 + 3 \beta ) q^{33} + ( -1 - 4 \beta ) q^{34} + q^{36} + ( -4 - 3 \beta ) q^{37} + ( -4 - \beta ) q^{38} + ( -3 - 2 \beta ) q^{39} -7 q^{41} + ( -5 - \beta ) q^{43} + ( -2 + 3 \beta ) q^{44} + ( 3 + \beta ) q^{46} -8 q^{47} + q^{48} + ( -1 - 4 \beta ) q^{51} + ( -3 - 2 \beta ) q^{52} + ( -3 - 4 \beta ) q^{53} + q^{54} + ( -4 - \beta ) q^{57} + ( -5 + 4 \beta ) q^{58} + ( 3 + \beta ) q^{59} + ( 3 + 4 \beta ) q^{61} + ( -3 + 3 \beta ) q^{62} + q^{64} + ( -2 + 3 \beta ) q^{66} + ( 4 - 4 \beta ) q^{67} + ( -1 - 4 \beta ) q^{68} + ( 3 + \beta ) q^{69} -2 q^{71} + q^{72} + ( -4 + 3 \beta ) q^{73} + ( -4 - 3 \beta ) q^{74} + ( -4 - \beta ) q^{76} + ( -3 - 2 \beta ) q^{78} + ( 2 - 3 \beta ) q^{79} + q^{81} -7 q^{82} + ( 7 + \beta ) q^{83} + ( -5 - \beta ) q^{86} + ( -5 + 4 \beta ) q^{87} + ( -2 + 3 \beta ) q^{88} + ( -4 + 8 \beta ) q^{89} + ( 3 + \beta ) q^{92} + ( -3 + 3 \beta ) q^{93} -8 q^{94} + q^{96} -14 q^{97} + ( -2 + 3 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 2q^{8} + 2q^{9} - 4q^{11} + 2q^{12} - 6q^{13} + 2q^{16} - 2q^{17} + 2q^{18} - 8q^{19} - 4q^{22} + 6q^{23} + 2q^{24} - 6q^{26} + 2q^{27} - 10q^{29} - 6q^{31} + 2q^{32} - 4q^{33} - 2q^{34} + 2q^{36} - 8q^{37} - 8q^{38} - 6q^{39} - 14q^{41} - 10q^{43} - 4q^{44} + 6q^{46} - 16q^{47} + 2q^{48} - 2q^{51} - 6q^{52} - 6q^{53} + 2q^{54} - 8q^{57} - 10q^{58} + 6q^{59} + 6q^{61} - 6q^{62} + 2q^{64} - 4q^{66} + 8q^{67} - 2q^{68} + 6q^{69} - 4q^{71} + 2q^{72} - 8q^{73} - 8q^{74} - 8q^{76} - 6q^{78} + 4q^{79} + 2q^{81} - 14q^{82} + 14q^{83} - 10q^{86} - 10q^{87} - 4q^{88} - 8q^{89} + 6q^{92} - 6q^{93} - 16q^{94} + 2q^{96} - 28q^{97} - 4q^{99} + O(q^{100})$$

## 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.dk yes 2
5.b even 2 1 7350.2.a.db 2
7.b odd 2 1 7350.2.a.di yes 2
35.c odd 2 1 7350.2.a.de yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7350.2.a.db 2 5.b even 2 1
7350.2.a.de yes 2 35.c odd 2 1
7350.2.a.di yes 2 7.b odd 2 1
7350.2.a.dk yes 2 1.a even 1 1 trivial

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7350))$$:

 $$T_{11}^{2} + 4 T_{11} - 14$$ $$T_{13}^{2} + 6 T_{13} + 1$$ $$T_{17}^{2} + 2 T_{17} - 31$$ $$T_{19}^{2} + 8 T_{19} + 14$$ $$T_{23}^{2} - 6 T_{23} + 7$$ $$T_{31}^{2} + 6 T_{31} - 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{2}$$
$3$ $$( 1 - T )^{2}$$
$5$ 1
$7$ 1
$11$ $$1 + 4 T + 8 T^{2} + 44 T^{3} + 121 T^{4}$$
$13$ $$1 + 6 T + 27 T^{2} + 78 T^{3} + 169 T^{4}$$
$17$ $$1 + 2 T + 3 T^{2} + 34 T^{3} + 289 T^{4}$$
$19$ $$1 + 8 T + 52 T^{2} + 152 T^{3} + 361 T^{4}$$
$23$ $$1 - 6 T + 53 T^{2} - 138 T^{3} + 529 T^{4}$$
$29$ $$1 + 10 T + 51 T^{2} + 290 T^{3} + 841 T^{4}$$
$31$ $$1 + 6 T + 53 T^{2} + 186 T^{3} + 961 T^{4}$$
$37$ $$1 + 8 T + 72 T^{2} + 296 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 7 T + 41 T^{2} )^{2}$$
$43$ $$1 + 10 T + 109 T^{2} + 430 T^{3} + 1849 T^{4}$$
$47$ $$( 1 + 8 T + 47 T^{2} )^{2}$$
$53$ $$1 + 6 T + 83 T^{2} + 318 T^{3} + 2809 T^{4}$$
$59$ $$1 - 6 T + 125 T^{2} - 354 T^{3} + 3481 T^{4}$$
$61$ $$1 - 6 T + 99 T^{2} - 366 T^{3} + 3721 T^{4}$$
$67$ $$1 - 8 T + 118 T^{2} - 536 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 2 T + 71 T^{2} )^{2}$$
$73$ $$1 + 8 T + 144 T^{2} + 584 T^{3} + 5329 T^{4}$$
$79$ $$1 - 4 T + 144 T^{2} - 316 T^{3} + 6241 T^{4}$$
$83$ $$1 - 14 T + 213 T^{2} - 1162 T^{3} + 6889 T^{4}$$
$89$ $$1 + 8 T + 66 T^{2} + 712 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 14 T + 97 T^{2} )^{2}$$
show more
show less