Properties

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

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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 98) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta q^{3} + q^{4} -\beta q^{6} - q^{8} - q^{9} +O(q^{10})$$ $$q - q^{2} + \beta q^{3} + q^{4} -\beta q^{6} - q^{8} - q^{9} -2 q^{11} + \beta q^{12} + q^{16} + \beta q^{17} + q^{18} -5 \beta q^{19} + 2 q^{22} + 4 q^{23} -\beta q^{24} -4 \beta q^{27} + 2 q^{29} + 6 \beta q^{31} - q^{32} -2 \beta q^{33} -\beta q^{34} - q^{36} -10 q^{37} + 5 \beta q^{38} -7 \beta q^{41} -2 q^{43} -2 q^{44} -4 q^{46} -2 \beta q^{47} + \beta q^{48} + 2 q^{51} + 2 q^{53} + 4 \beta q^{54} -10 q^{57} -2 q^{58} -\beta q^{59} + 2 \beta q^{61} -6 \beta q^{62} + q^{64} + 2 \beta q^{66} -12 q^{67} + \beta q^{68} + 4 \beta q^{69} -12 q^{71} + q^{72} + \beta q^{73} + 10 q^{74} -5 \beta q^{76} -4 q^{79} -5 q^{81} + 7 \beta q^{82} -7 \beta q^{83} + 2 q^{86} + 2 \beta q^{87} + 2 q^{88} -5 \beta q^{89} + 4 q^{92} + 12 q^{93} + 2 \beta q^{94} -\beta q^{96} -7 \beta q^{97} + 2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - 2q^{8} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - 2q^{8} - 2q^{9} - 4q^{11} + 2q^{16} + 2q^{18} + 4q^{22} + 8q^{23} + 4q^{29} - 2q^{32} - 2q^{36} - 20q^{37} - 4q^{43} - 4q^{44} - 8q^{46} + 4q^{51} + 4q^{53} - 20q^{57} - 4q^{58} + 2q^{64} - 24q^{67} - 24q^{71} + 2q^{72} + 20q^{74} - 8q^{79} - 10q^{81} + 4q^{86} + 4q^{88} + 8q^{92} + 24q^{93} + 4q^{99} + O(q^{100})$$

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 −1.41421 1.00000 0 1.41421 0 −1.00000 −1.00000 0
1.2 −1.00000 1.41421 1.00000 0 −1.41421 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
$$2$$ $$1$$
$$5$$ $$1$$
$$7$$ $$1$$

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2450.2.a.bj 2
5.b even 2 1 98.2.a.b 2
5.c odd 4 2 2450.2.c.v 4
7.b odd 2 1 inner 2450.2.a.bj 2
15.d odd 2 1 882.2.a.n 2
20.d odd 2 1 784.2.a.l 2
35.c odd 2 1 98.2.a.b 2
35.f even 4 2 2450.2.c.v 4
35.i odd 6 2 98.2.c.c 4
35.j even 6 2 98.2.c.c 4
40.e odd 2 1 3136.2.a.bm 2
40.f even 2 1 3136.2.a.bn 2
60.h even 2 1 7056.2.a.cl 2
105.g even 2 1 882.2.a.n 2
105.o odd 6 2 882.2.g.l 4
105.p even 6 2 882.2.g.l 4
140.c even 2 1 784.2.a.l 2
140.p odd 6 2 784.2.i.m 4
140.s even 6 2 784.2.i.m 4
280.c odd 2 1 3136.2.a.bn 2
280.n even 2 1 3136.2.a.bm 2
420.o odd 2 1 7056.2.a.cl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.a.b 2 5.b even 2 1
98.2.a.b 2 35.c odd 2 1
98.2.c.c 4 35.i odd 6 2
98.2.c.c 4 35.j even 6 2
784.2.a.l 2 20.d odd 2 1
784.2.a.l 2 140.c even 2 1
784.2.i.m 4 140.p odd 6 2
784.2.i.m 4 140.s even 6 2
882.2.a.n 2 15.d odd 2 1
882.2.a.n 2 105.g even 2 1
882.2.g.l 4 105.o odd 6 2
882.2.g.l 4 105.p even 6 2
2450.2.a.bj 2 1.a even 1 1 trivial
2450.2.a.bj 2 7.b odd 2 1 inner
2450.2.c.v 4 5.c odd 4 2
2450.2.c.v 4 35.f even 4 2
3136.2.a.bm 2 40.e odd 2 1
3136.2.a.bm 2 280.n even 2 1
3136.2.a.bn 2 40.f even 2 1
3136.2.a.bn 2 280.c odd 2 1
7056.2.a.cl 2 60.h even 2 1
7056.2.a.cl 2 420.o odd 2 1

Hecke kernels

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

 $$T_{3}^{2} - 2$$ $$T_{11} + 2$$ $$T_{13}$$ $$T_{17}^{2} - 2$$ $$T_{19}^{2} - 50$$ $$T_{23} - 4$$ $$T_{37} + 10$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$-2 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$-2 + T^{2}$$
$19$ $$-50 + T^{2}$$
$23$ $$( -4 + T )^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$-72 + T^{2}$$
$37$ $$( 10 + T )^{2}$$
$41$ $$-98 + T^{2}$$
$43$ $$( 2 + T )^{2}$$
$47$ $$-8 + T^{2}$$
$53$ $$( -2 + T )^{2}$$
$59$ $$-2 + T^{2}$$
$61$ $$-8 + T^{2}$$
$67$ $$( 12 + T )^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$-2 + T^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$-98 + T^{2}$$
$89$ $$-50 + T^{2}$$
$97$ $$-98 + T^{2}$$