Properties

 Label 1950.2.a.bc Level $1950$ Weight $2$ Character orbit 1950.a Self dual yes Analytic conductor $15.571$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

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Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ Defining polynomial: $$x^{2} - x - 10$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{41})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} + ( -1 - \beta ) q^{7} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} + ( -1 - \beta ) q^{7} - q^{8} + q^{9} + ( 1 + \beta ) q^{11} - q^{12} + q^{13} + ( 1 + \beta ) q^{14} + q^{16} + ( -3 + \beta ) q^{17} - q^{18} + ( 2 - \beta ) q^{19} + ( 1 + \beta ) q^{21} + ( -1 - \beta ) q^{22} + q^{24} - q^{26} - q^{27} + ( -1 - \beta ) q^{28} + ( 1 - 2 \beta ) q^{29} + ( 1 - 3 \beta ) q^{31} - q^{32} + ( -1 - \beta ) q^{33} + ( 3 - \beta ) q^{34} + q^{36} + ( -2 + \beta ) q^{37} + ( -2 + \beta ) q^{38} - q^{39} -\beta q^{41} + ( -1 - \beta ) q^{42} + ( 4 + 2 \beta ) q^{43} + ( 1 + \beta ) q^{44} -7 q^{47} - q^{48} + ( 4 + 3 \beta ) q^{49} + ( 3 - \beta ) q^{51} + q^{52} + ( 5 - 2 \beta ) q^{53} + q^{54} + ( 1 + \beta ) q^{56} + ( -2 + \beta ) q^{57} + ( -1 + 2 \beta ) q^{58} + ( -1 + \beta ) q^{59} + ( 3 + 3 \beta ) q^{61} + ( -1 + 3 \beta ) q^{62} + ( -1 - \beta ) q^{63} + q^{64} + ( 1 + \beta ) q^{66} + ( -1 + 2 \beta ) q^{67} + ( -3 + \beta ) q^{68} + ( 2 - \beta ) q^{71} - q^{72} + 12 q^{73} + ( 2 - \beta ) q^{74} + ( 2 - \beta ) q^{76} + ( -11 - 3 \beta ) q^{77} + q^{78} + ( -2 - \beta ) q^{79} + q^{81} + \beta q^{82} + ( 7 + \beta ) q^{83} + ( 1 + \beta ) q^{84} + ( -4 - 2 \beta ) q^{86} + ( -1 + 2 \beta ) q^{87} + ( -1 - \beta ) q^{88} + ( 4 + 2 \beta ) q^{89} + ( -1 - \beta ) q^{91} + ( -1 + 3 \beta ) q^{93} + 7 q^{94} + q^{96} + ( 10 - 2 \beta ) q^{97} + ( -4 - 3 \beta ) q^{98} + ( 1 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 3 q^{7} - 2 q^{8} + 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 3 q^{7} - 2 q^{8} + 2 q^{9} + 3 q^{11} - 2 q^{12} + 2 q^{13} + 3 q^{14} + 2 q^{16} - 5 q^{17} - 2 q^{18} + 3 q^{19} + 3 q^{21} - 3 q^{22} + 2 q^{24} - 2 q^{26} - 2 q^{27} - 3 q^{28} - q^{31} - 2 q^{32} - 3 q^{33} + 5 q^{34} + 2 q^{36} - 3 q^{37} - 3 q^{38} - 2 q^{39} - q^{41} - 3 q^{42} + 10 q^{43} + 3 q^{44} - 14 q^{47} - 2 q^{48} + 11 q^{49} + 5 q^{51} + 2 q^{52} + 8 q^{53} + 2 q^{54} + 3 q^{56} - 3 q^{57} - q^{59} + 9 q^{61} + q^{62} - 3 q^{63} + 2 q^{64} + 3 q^{66} - 5 q^{68} + 3 q^{71} - 2 q^{72} + 24 q^{73} + 3 q^{74} + 3 q^{76} - 25 q^{77} + 2 q^{78} - 5 q^{79} + 2 q^{81} + q^{82} + 15 q^{83} + 3 q^{84} - 10 q^{86} - 3 q^{88} + 10 q^{89} - 3 q^{91} + q^{93} + 14 q^{94} + 2 q^{96} + 18 q^{97} - 11 q^{98} + 3 q^{99} + O(q^{100})$$

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
 3.70156 −2.70156
−1.00000 −1.00000 1.00000 0 1.00000 −4.70156 −1.00000 1.00000 0
1.2 −1.00000 −1.00000 1.00000 0 1.00000 1.70156 −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
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$13$$ $$-1$$

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 1950.2.a.bc 2
3.b odd 2 1 5850.2.a.cj 2
5.b even 2 1 1950.2.a.bg yes 2
5.c odd 4 2 1950.2.e.p 4
15.d odd 2 1 5850.2.a.cg 2
15.e even 4 2 5850.2.e.bi 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.a.bc 2 1.a even 1 1 trivial
1950.2.a.bg yes 2 5.b even 2 1
1950.2.e.p 4 5.c odd 4 2
5850.2.a.cg 2 15.d odd 2 1
5850.2.a.cj 2 3.b odd 2 1
5850.2.e.bi 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1950))$$:

 $$T_{7}^{2} + 3 T_{7} - 8$$ $$T_{11}^{2} - 3 T_{11} - 8$$ $$T_{17}^{2} + 5 T_{17} - 4$$ $$T_{23}$$ $$T_{31}^{2} + T_{31} - 92$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$-8 + 3 T + T^{2}$$
$11$ $$-8 - 3 T + T^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$-4 + 5 T + T^{2}$$
$19$ $$-8 - 3 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$-41 + T^{2}$$
$31$ $$-92 + T + T^{2}$$
$37$ $$-8 + 3 T + T^{2}$$
$41$ $$-10 + T + T^{2}$$
$43$ $$-16 - 10 T + T^{2}$$
$47$ $$( 7 + T )^{2}$$
$53$ $$-25 - 8 T + T^{2}$$
$59$ $$-10 + T + T^{2}$$
$61$ $$-72 - 9 T + T^{2}$$
$67$ $$-41 + T^{2}$$
$71$ $$-8 - 3 T + T^{2}$$
$73$ $$( -12 + T )^{2}$$
$79$ $$-4 + 5 T + T^{2}$$
$83$ $$46 - 15 T + T^{2}$$
$89$ $$-16 - 10 T + T^{2}$$
$97$ $$40 - 18 T + T^{2}$$
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