# Properties

 Label 7350.2.a.do Level 7350 Weight 2 Character orbit 7350.a Self dual yes Analytic conductor 58.690 Analytic rank 0 Dimension 3 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: $$3$$ Coefficient field: 3.3.2700.1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 15 x - 20$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 10$$

## 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
 −1.61323 −2.80560 4.41883
−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
 $$n$$: 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.do 3
5.b even 2 1 7350.2.a.dp 3
5.c odd 4 2 1470.2.g.h 6
7.b odd 2 1 7350.2.a.dn 3
7.d odd 6 2 1050.2.i.v 6
35.c odd 2 1 7350.2.a.dq 3
35.f even 4 2 1470.2.g.i 6
35.i odd 6 2 1050.2.i.u 6
35.k even 12 4 210.2.n.b 12
35.l odd 12 4 1470.2.n.j 12
105.w odd 12 4 630.2.u.f 12
140.x odd 12 4 1680.2.di.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.n.b 12 35.k even 12 4
630.2.u.f 12 105.w odd 12 4
1050.2.i.u 6 35.i odd 6 2
1050.2.i.v 6 7.d odd 6 2
1470.2.g.h 6 5.c odd 4 2
1470.2.g.i 6 35.f even 4 2
1470.2.n.j 12 35.l odd 12 4
1680.2.di.c 12 140.x odd 12 4
7350.2.a.dn 3 7.b odd 2 1
7350.2.a.do 3 1.a even 1 1 trivial
7350.2.a.dp 3 5.b even 2 1
7350.2.a.dq 3 35.c odd 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7350))$$:

 $$T_{11}^{3} - 3 T_{11}^{2} - 27 T_{11} + 49$$ $$T_{13}^{3} - 3 T_{13}^{2} - 12 T_{13} + 24$$ $$T_{17}^{3} - 6 T_{17}^{2} - 48 T_{17} + 272$$ $$T_{19}^{3} + 3 T_{19}^{2} - 12 T_{19} - 24$$ $$T_{23}^{3} + 9 T_{23}^{2} + 12 T_{23} - 8$$ $$T_{31}^{3} - 15 T_{31} + 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$( 1 - T )^{3}$$
$5$ 1
$7$ 1
$11$ $$1 - 3 T + 6 T^{2} - 17 T^{3} + 66 T^{4} - 363 T^{5} + 1331 T^{6}$$
$13$ $$1 - 3 T + 27 T^{2} - 54 T^{3} + 351 T^{4} - 507 T^{5} + 2197 T^{6}$$
$17$ $$1 - 6 T + 3 T^{2} + 68 T^{3} + 51 T^{4} - 1734 T^{5} + 4913 T^{6}$$
$19$ $$1 + 3 T + 45 T^{2} + 90 T^{3} + 855 T^{4} + 1083 T^{5} + 6859 T^{6}$$
$23$ $$1 + 9 T + 81 T^{2} + 406 T^{3} + 1863 T^{4} + 4761 T^{5} + 12167 T^{6}$$
$29$ $$1 - 12 T + 120 T^{2} - 720 T^{3} + 3480 T^{4} - 10092 T^{5} + 24389 T^{6}$$
$31$ $$1 + 78 T^{2} + 10 T^{3} + 2418 T^{4} + 29791 T^{6}$$
$37$ $$1 + 9 T + 123 T^{2} + 658 T^{3} + 4551 T^{4} + 12321 T^{5} + 50653 T^{6}$$
$41$ $$1 - 9 T + 75 T^{2} - 610 T^{3} + 3075 T^{4} - 15129 T^{5} + 68921 T^{6}$$
$43$ $$( 1 - 2 T + 43 T^{2} )^{3}$$
$47$ $$1 + 3 T + 9 T^{2} - 122 T^{3} + 423 T^{4} + 6627 T^{5} + 103823 T^{6}$$
$53$ $$1 - 9 T + 66 T^{2} - 141 T^{3} + 3498 T^{4} - 25281 T^{5} + 148877 T^{6}$$
$59$ $$1 - 12 T + 150 T^{2} - 980 T^{3} + 8850 T^{4} - 41772 T^{5} + 205379 T^{6}$$
$61$ $$1 + 6 T + 75 T^{2} + 20 T^{3} + 4575 T^{4} + 22326 T^{5} + 226981 T^{6}$$
$67$ $$1 - 6 T + 93 T^{2} - 412 T^{3} + 6231 T^{4} - 26934 T^{5} + 300763 T^{6}$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{3}$$
$73$ $$( 1 - 4 T + 73 T^{2} )^{3}$$
$79$ $$1 - 24 T + 414 T^{2} - 4194 T^{3} + 32706 T^{4} - 149784 T^{5} + 493039 T^{6}$$
$83$ $$1 - 12 T + 222 T^{2} - 1906 T^{3} + 18426 T^{4} - 82668 T^{5} + 571787 T^{6}$$
$89$ $$1 - 6 T + 159 T^{2} - 676 T^{3} + 14151 T^{4} - 47526 T^{5} + 704969 T^{6}$$
$97$ $$1 + 24 T + 408 T^{2} + 4768 T^{3} + 39576 T^{4} + 225816 T^{5} + 912673 T^{6}$$