# Properties

 Label 7350.2.a.du Level $7350$ Weight $2$ Character orbit 7350.a Self dual yes Analytic conductor $58.690$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.10304.1 Defining polynomial: $$x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. 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} + ( -\beta_{1} + \beta_{2} ) q^{11} + q^{12} + ( \beta_{1} - 2 \beta_{2} ) q^{13} + q^{16} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{17} + q^{18} + ( 3 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{19} + ( -\beta_{1} + \beta_{2} ) q^{22} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{23} + q^{24} + ( \beta_{1} - 2 \beta_{2} ) q^{26} + q^{27} + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{29} + ( 4 - \beta_{1} + \beta_{2} ) q^{31} + q^{32} + ( -\beta_{1} + \beta_{2} ) q^{33} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{34} + q^{36} + ( -2 - \beta_{1} - 3 \beta_{2} ) q^{37} + ( 3 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{38} + ( \beta_{1} - 2 \beta_{2} ) q^{39} + ( 3 + 2 \beta_{2} + \beta_{3} ) q^{41} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{43} + ( -\beta_{1} + \beta_{2} ) q^{44} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{46} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{47} + q^{48} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{51} + ( \beta_{1} - 2 \beta_{2} ) q^{52} + ( 5 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{53} + q^{54} + ( 3 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{57} + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{58} + ( 5 - \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{59} + ( 3 - 3 \beta_{3} ) q^{61} + ( 4 - \beta_{1} + \beta_{2} ) q^{62} + q^{64} + ( -\beta_{1} + \beta_{2} ) q^{66} + ( -1 + \beta_{2} + \beta_{3} ) q^{67} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{68} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{69} + ( -5 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{71} + q^{72} + ( -3 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{73} + ( -2 - \beta_{1} - 3 \beta_{2} ) q^{74} + ( 3 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{76} + ( \beta_{1} - 2 \beta_{2} ) q^{78} + ( -2 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{79} + q^{81} + ( 3 + 2 \beta_{2} + \beta_{3} ) q^{82} + ( 2 - 2 \beta_{2} + 2 \beta_{3} ) q^{83} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{86} + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{87} + ( -\beta_{1} + \beta_{2} ) q^{88} + ( 11 + 4 \beta_{2} + \beta_{3} ) q^{89} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{92} + ( 4 - \beta_{1} + \beta_{2} ) q^{93} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{94} + q^{96} + ( 5 - 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{97} + ( -\beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} + 4q^{3} + 4q^{4} + 4q^{6} + 4q^{8} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{2} + 4q^{3} + 4q^{4} + 4q^{6} + 4q^{8} + 4q^{9} + 4q^{12} + 4q^{16} + 4q^{17} + 4q^{18} + 12q^{19} + 4q^{23} + 4q^{24} + 4q^{27} - 8q^{29} + 16q^{31} + 4q^{32} + 4q^{34} + 4q^{36} - 8q^{37} + 12q^{38} + 12q^{41} + 4q^{43} + 4q^{46} + 12q^{47} + 4q^{48} + 4q^{51} + 20q^{53} + 4q^{54} + 12q^{57} - 8q^{58} + 20q^{59} + 12q^{61} + 16q^{62} + 4q^{64} - 4q^{67} + 4q^{68} + 4q^{69} - 20q^{71} + 4q^{72} - 12q^{73} - 8q^{74} + 12q^{76} - 8q^{79} + 4q^{81} + 12q^{82} + 8q^{83} + 4q^{86} - 8q^{87} + 44q^{89} + 4q^{92} + 16q^{93} + 12q^{94} + 4q^{96} + 20q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - \nu^{2} - 4 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - \nu - 4$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 8 \nu - 2$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 4 \beta_{2} + \beta_{1} + 9$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{3} + 4 \beta_{2} + 9 \beta_{1} + 13$$$$)/2$$

## 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
 3.16053 −0.692297 1.69230 −2.16053
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
1.3 1.00000 1.00000 1.00000 0 1.00000 0 1.00000 1.00000 0
1.4 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.du 4
5.b even 2 1 7350.2.a.dr 4
5.c odd 4 2 1470.2.g.k yes 8
7.b odd 2 1 7350.2.a.dt 4
35.c odd 2 1 7350.2.a.ds 4
35.f even 4 2 1470.2.g.j 8
35.k even 12 4 1470.2.n.l 16
35.l odd 12 4 1470.2.n.k 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.g.j 8 35.f even 4 2
1470.2.g.k yes 8 5.c odd 4 2
1470.2.n.k 16 35.l odd 12 4
1470.2.n.l 16 35.k even 12 4
7350.2.a.dr 4 5.b even 2 1
7350.2.a.ds 4 35.c odd 2 1
7350.2.a.dt 4 7.b odd 2 1
7350.2.a.du 4 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7350))$$:

 $$T_{11}^{4} - 18 T_{11}^{2} - 16 T_{11} + 32$$ $$T_{13}^{4} - 26 T_{13}^{2} + 48 T_{13} - 16$$ $$T_{17}^{4} - 4 T_{17}^{3} - 32 T_{17}^{2} + 8 T_{17} + 124$$ $$T_{19}^{4} - 12 T_{19}^{3} + 12 T_{19}^{2} + 224 T_{19} - 448$$ $$T_{23}^{4} - 4 T_{23}^{3} - 36 T_{23}^{2} + 128$$ $$T_{31}^{4} - 16 T_{31}^{3} + 78 T_{31}^{2} - 128 T_{31} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$32 - 16 T - 18 T^{2} + T^{4}$$
$13$ $$-16 + 48 T - 26 T^{2} + T^{4}$$
$17$ $$124 + 8 T - 32 T^{2} - 4 T^{3} + T^{4}$$
$19$ $$-448 + 224 T + 12 T^{2} - 12 T^{3} + T^{4}$$
$23$ $$128 - 36 T^{2} - 4 T^{3} + T^{4}$$
$29$ $$1568 - 336 T - 66 T^{2} + 8 T^{3} + T^{4}$$
$31$ $$64 - 128 T + 78 T^{2} - 16 T^{3} + T^{4}$$
$37$ $$200 - 120 T - 42 T^{2} + 8 T^{3} + T^{4}$$
$41$ $$-376 + 120 T + 26 T^{2} - 12 T^{3} + T^{4}$$
$43$ $$736 + 368 T - 110 T^{2} - 4 T^{3} + T^{4}$$
$47$ $$-1024 + 512 T - 30 T^{2} - 12 T^{3} + T^{4}$$
$53$ $$64 - 160 T + 108 T^{2} - 20 T^{3} + T^{4}$$
$59$ $$-3008 + 736 T + 52 T^{2} - 20 T^{3} + T^{4}$$
$61$ $$-648 + 1512 T - 126 T^{2} - 12 T^{3} + T^{4}$$
$67$ $$32 - 16 T - 14 T^{2} + 4 T^{3} + T^{4}$$
$71$ $$2944 - 1472 T - 36 T^{2} + 20 T^{3} + T^{4}$$
$73$ $$-248 - 536 T - 86 T^{2} + 12 T^{3} + T^{4}$$
$79$ $$-3584 - 1792 T - 176 T^{2} + 8 T^{3} + T^{4}$$
$83$ $$128 - 64 T - 88 T^{2} - 8 T^{3} + T^{4}$$
$89$ $$5048 - 3688 T + 658 T^{2} - 44 T^{3} + T^{4}$$
$97$ $$-10376 + 2184 T - 14 T^{2} - 20 T^{3} + T^{4}$$