# Properties

 Label 7350.2.a.df Level 7350 Weight 2 Character orbit 7350.a Self dual yes Analytic conductor 58.690 Analytic rank 0 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1470) 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.

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

## 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.df 2
5.b even 2 1 1470.2.a.u 2
7.b odd 2 1 7350.2.a.dd 2
15.d odd 2 1 4410.2.a.br 2
35.c odd 2 1 1470.2.a.v yes 2
35.i odd 6 2 1470.2.i.u 4
35.j even 6 2 1470.2.i.v 4
105.g even 2 1 4410.2.a.bn 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.u 2 5.b even 2 1
1470.2.a.v yes 2 35.c odd 2 1
1470.2.i.u 4 35.i odd 6 2
1470.2.i.v 4 35.j even 6 2
4410.2.a.bn 2 105.g even 2 1
4410.2.a.br 2 15.d odd 2 1
7350.2.a.dd 2 7.b odd 2 1
7350.2.a.df 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7350))$$:

 $$T_{11}^{2} - 4 T_{11} + 2$$ $$T_{13}$$ $$T_{17}^{2} - 2$$ $$T_{19}^{2} - 8$$ $$T_{23}^{2} + 4 T_{23} - 4$$ $$T_{31}^{2} - 4 T_{31} - 46$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( 1 - T )^{2}$$
$5$ 1
$7$ 1
$11$ $$1 - 4 T + 24 T^{2} - 44 T^{3} + 121 T^{4}$$
$13$ $$( 1 + 13 T^{2} )^{2}$$
$17$ $$1 + 32 T^{2} + 289 T^{4}$$
$19$ $$1 + 30 T^{2} + 361 T^{4}$$
$23$ $$1 + 4 T + 42 T^{2} + 92 T^{3} + 529 T^{4}$$
$29$ $$1 - 8 T + 56 T^{2} - 232 T^{3} + 841 T^{4}$$
$31$ $$1 - 4 T + 16 T^{2} - 124 T^{3} + 961 T^{4}$$
$37$ $$1 + 72 T^{2} + 1369 T^{4}$$
$41$ $$1 - 12 T + 110 T^{2} - 492 T^{3} + 1681 T^{4}$$
$43$ $$1 + 12 T + 120 T^{2} + 516 T^{3} + 1849 T^{4}$$
$47$ $$1 + 4 T + 48 T^{2} + 188 T^{3} + 2209 T^{4}$$
$53$ $$1 + 4 T - 18 T^{2} + 212 T^{3} + 2809 T^{4}$$
$59$ $$1 - 12 T + 82 T^{2} - 708 T^{3} + 3481 T^{4}$$
$61$ $$1 - 12 T + 126 T^{2} - 732 T^{3} + 3721 T^{4}$$
$67$ $$1 - 4 T + 40 T^{2} - 268 T^{3} + 4489 T^{4}$$
$71$ $$1 - 16 T + 198 T^{2} - 1136 T^{3} + 5041 T^{4}$$
$73$ $$1 - 4 T + 118 T^{2} - 292 T^{3} + 5329 T^{4}$$
$79$ $$1 + 30 T^{2} + 6241 T^{4}$$
$83$ $$1 + 16 T + 222 T^{2} + 1328 T^{3} + 6889 T^{4}$$
$89$ $$1 + 4 T + 110 T^{2} + 356 T^{3} + 7921 T^{4}$$
$97$ $$1 - 12 T + 222 T^{2} - 1164 T^{3} + 9409 T^{4}$$