# Properties

 Label 2450.2.a.bl Level 2450 Weight 2 Character orbit 2450.a Self dual yes Analytic conductor 19.563 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$2450 = 2 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 2450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.5633484952$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta q^{3} + q^{4} -\beta q^{6} - q^{8} + 3 q^{9} +O(q^{10})$$ $$q - q^{2} + \beta q^{3} + q^{4} -\beta q^{6} - q^{8} + 3 q^{9} -2 \beta q^{11} + \beta q^{12} + ( -2 - \beta ) q^{13} + q^{16} -2 q^{17} -3 q^{18} + ( -4 + \beta ) q^{19} + 2 \beta q^{22} + ( -2 + 2 \beta ) q^{23} -\beta q^{24} + ( 2 + \beta ) q^{26} + ( 2 + 2 \beta ) q^{29} + ( -4 - 2 \beta ) q^{31} - q^{32} -12 q^{33} + 2 q^{34} + 3 q^{36} + 2 q^{37} + ( 4 - \beta ) q^{38} + ( -6 - 2 \beta ) q^{39} + ( 6 - 2 \beta ) q^{41} + ( 4 - 2 \beta ) q^{43} -2 \beta q^{44} + ( 2 - 2 \beta ) q^{46} + ( -4 - 2 \beta ) q^{47} + \beta q^{48} -2 \beta q^{51} + ( -2 - \beta ) q^{52} + ( -6 - 2 \beta ) q^{53} + ( 6 - 4 \beta ) q^{57} + ( -2 - 2 \beta ) q^{58} + ( 4 - \beta ) q^{59} + ( -6 + \beta ) q^{61} + ( 4 + 2 \beta ) q^{62} + q^{64} + 12 q^{66} -8 q^{67} -2 q^{68} + ( 12 - 2 \beta ) q^{69} + ( -6 + 2 \beta ) q^{71} -3 q^{72} + ( 2 - 2 \beta ) q^{73} -2 q^{74} + ( -4 + \beta ) q^{76} + ( 6 + 2 \beta ) q^{78} + ( 2 + 2 \beta ) q^{79} -9 q^{81} + ( -6 + 2 \beta ) q^{82} + \beta q^{83} + ( -4 + 2 \beta ) q^{86} + ( 12 + 2 \beta ) q^{87} + 2 \beta q^{88} + 10 q^{89} + ( -2 + 2 \beta ) q^{92} + ( -12 - 4 \beta ) q^{93} + ( 4 + 2 \beta ) q^{94} -\beta q^{96} + ( -6 - 4 \beta ) q^{97} -6 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - 2q^{8} + 6q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - 2q^{8} + 6q^{9} - 4q^{13} + 2q^{16} - 4q^{17} - 6q^{18} - 8q^{19} - 4q^{23} + 4q^{26} + 4q^{29} - 8q^{31} - 2q^{32} - 24q^{33} + 4q^{34} + 6q^{36} + 4q^{37} + 8q^{38} - 12q^{39} + 12q^{41} + 8q^{43} + 4q^{46} - 8q^{47} - 4q^{52} - 12q^{53} + 12q^{57} - 4q^{58} + 8q^{59} - 12q^{61} + 8q^{62} + 2q^{64} + 24q^{66} - 16q^{67} - 4q^{68} + 24q^{69} - 12q^{71} - 6q^{72} + 4q^{73} - 4q^{74} - 8q^{76} + 12q^{78} + 4q^{79} - 18q^{81} - 12q^{82} - 8q^{86} + 24q^{87} + 20q^{89} - 4q^{92} - 24q^{93} + 8q^{94} - 12q^{97} + O(q^{100})$$

## 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
 −2.44949 2.44949
−1.00000 −2.44949 1.00000 0 2.44949 0 −1.00000 3.00000 0
1.2 −1.00000 2.44949 1.00000 0 −2.44949 0 −1.00000 3.00000 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 2450.2.a.bl 2
5.b even 2 1 2450.2.a.bq 2
5.c odd 4 2 490.2.c.e 4
7.b odd 2 1 350.2.a.g 2
21.c even 2 1 3150.2.a.bt 2
28.d even 2 1 2800.2.a.bm 2
35.c odd 2 1 350.2.a.h 2
35.f even 4 2 70.2.c.a 4
35.k even 12 4 490.2.i.c 8
35.l odd 12 4 490.2.i.f 8
105.g even 2 1 3150.2.a.bs 2
105.k odd 4 2 630.2.g.g 4
140.c even 2 1 2800.2.a.bl 2
140.j odd 4 2 560.2.g.e 4
280.s even 4 2 2240.2.g.j 4
280.y odd 4 2 2240.2.g.i 4
420.w even 4 2 5040.2.t.t 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 35.f even 4 2
350.2.a.g 2 7.b odd 2 1
350.2.a.h 2 35.c odd 2 1
490.2.c.e 4 5.c odd 4 2
490.2.i.c 8 35.k even 12 4
490.2.i.f 8 35.l odd 12 4
560.2.g.e 4 140.j odd 4 2
630.2.g.g 4 105.k odd 4 2
2240.2.g.i 4 280.y odd 4 2
2240.2.g.j 4 280.s even 4 2
2450.2.a.bl 2 1.a even 1 1 trivial
2450.2.a.bq 2 5.b even 2 1
2800.2.a.bl 2 140.c even 2 1
2800.2.a.bm 2 28.d even 2 1
3150.2.a.bs 2 105.g even 2 1
3150.2.a.bt 2 21.c even 2 1
5040.2.t.t 4 420.w even 4 2

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2450))$$:

 $$T_{3}^{2} - 6$$ $$T_{11}^{2} - 24$$ $$T_{13}^{2} + 4 T_{13} - 2$$ $$T_{17} + 2$$ $$T_{19}^{2} + 8 T_{19} + 10$$ $$T_{23}^{2} + 4 T_{23} - 20$$ $$T_{37} - 2$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$1 + 9 T^{4}$$
$5$ 1
$7$ 1
$11$ $$1 - 2 T^{2} + 121 T^{4}$$
$13$ $$1 + 4 T + 24 T^{2} + 52 T^{3} + 169 T^{4}$$
$17$ $$( 1 + 2 T + 17 T^{2} )^{2}$$
$19$ $$1 + 8 T + 48 T^{2} + 152 T^{3} + 361 T^{4}$$
$23$ $$1 + 4 T + 26 T^{2} + 92 T^{3} + 529 T^{4}$$
$29$ $$1 - 4 T + 38 T^{2} - 116 T^{3} + 841 T^{4}$$
$31$ $$1 + 8 T + 54 T^{2} + 248 T^{3} + 961 T^{4}$$
$37$ $$( 1 - 2 T + 37 T^{2} )^{2}$$
$41$ $$1 - 12 T + 94 T^{2} - 492 T^{3} + 1681 T^{4}$$
$43$ $$1 - 8 T + 78 T^{2} - 344 T^{3} + 1849 T^{4}$$
$47$ $$1 + 8 T + 86 T^{2} + 376 T^{3} + 2209 T^{4}$$
$53$ $$1 + 12 T + 118 T^{2} + 636 T^{3} + 2809 T^{4}$$
$59$ $$1 - 8 T + 128 T^{2} - 472 T^{3} + 3481 T^{4}$$
$61$ $$1 + 12 T + 152 T^{2} + 732 T^{3} + 3721 T^{4}$$
$67$ $$( 1 + 8 T + 67 T^{2} )^{2}$$
$71$ $$1 + 12 T + 154 T^{2} + 852 T^{3} + 5041 T^{4}$$
$73$ $$1 - 4 T + 126 T^{2} - 292 T^{3} + 5329 T^{4}$$
$79$ $$1 - 4 T + 138 T^{2} - 316 T^{3} + 6241 T^{4}$$
$83$ $$1 + 160 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 10 T + 89 T^{2} )^{2}$$
$97$ $$1 + 12 T + 134 T^{2} + 1164 T^{3} + 9409 T^{4}$$