# Properties

 Label 1950.2.e.p Level $1950$ Weight $2$ Character orbit 1950.e Analytic conductor $15.571$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{41})$$ Defining polynomial: $$x^{4} + 21 x^{2} + 100$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + \beta_{2} q^{3} - q^{4} + q^{6} + ( \beta_{1} - \beta_{2} ) q^{7} + \beta_{2} q^{8} - q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + \beta_{2} q^{3} - q^{4} + q^{6} + ( \beta_{1} - \beta_{2} ) q^{7} + \beta_{2} q^{8} - q^{9} + ( 2 - \beta_{3} ) q^{11} -\beta_{2} q^{12} -\beta_{2} q^{13} + ( -2 + \beta_{3} ) q^{14} + q^{16} + ( -\beta_{1} - 3 \beta_{2} ) q^{17} + \beta_{2} q^{18} + ( -1 - \beta_{3} ) q^{19} + ( 2 - \beta_{3} ) q^{21} + ( \beta_{1} - \beta_{2} ) q^{22} - q^{24} - q^{26} -\beta_{2} q^{27} + ( -\beta_{1} + \beta_{2} ) q^{28} + ( 1 - 2 \beta_{3} ) q^{29} + ( -2 + 3 \beta_{3} ) q^{31} -\beta_{2} q^{32} + ( -\beta_{1} + \beta_{2} ) q^{33} + ( -2 - \beta_{3} ) q^{34} + q^{36} + ( -\beta_{1} - 2 \beta_{2} ) q^{37} + ( \beta_{1} + 2 \beta_{2} ) q^{38} + q^{39} + ( -1 + \beta_{3} ) q^{41} + ( \beta_{1} - \beta_{2} ) q^{42} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{43} + ( -2 + \beta_{3} ) q^{44} -7 \beta_{2} q^{47} + \beta_{2} q^{48} + ( -7 + 3 \beta_{3} ) q^{49} + ( 2 + \beta_{3} ) q^{51} + \beta_{2} q^{52} + ( -2 \beta_{1} - 5 \beta_{2} ) q^{53} - q^{54} + ( 2 - \beta_{3} ) q^{56} + ( -\beta_{1} - 2 \beta_{2} ) q^{57} + ( 2 \beta_{1} + \beta_{2} ) q^{58} + \beta_{3} q^{59} + ( 6 - 3 \beta_{3} ) q^{61} + ( -3 \beta_{1} - \beta_{2} ) q^{62} + ( -\beta_{1} + \beta_{2} ) q^{63} - q^{64} + ( 2 - \beta_{3} ) q^{66} + ( -2 \beta_{1} - \beta_{2} ) q^{67} + ( \beta_{1} + 3 \beta_{2} ) q^{68} + ( 1 + \beta_{3} ) q^{71} -\beta_{2} q^{72} -12 \beta_{2} q^{73} + ( -1 - \beta_{3} ) q^{74} + ( 1 + \beta_{3} ) q^{76} + ( 3 \beta_{1} - 11 \beta_{2} ) q^{77} -\beta_{2} q^{78} + ( 3 - \beta_{3} ) q^{79} + q^{81} -\beta_{1} q^{82} + ( \beta_{1} - 7 \beta_{2} ) q^{83} + ( -2 + \beta_{3} ) q^{84} + ( -6 + 2 \beta_{3} ) q^{86} + ( -2 \beta_{1} - \beta_{2} ) q^{87} + ( -\beta_{1} + \beta_{2} ) q^{88} + ( -6 + 2 \beta_{3} ) q^{89} + ( -2 + \beta_{3} ) q^{91} + ( 3 \beta_{1} + \beta_{2} ) q^{93} -7 q^{94} + q^{96} + ( 2 \beta_{1} + 10 \beta_{2} ) q^{97} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{98} + ( -2 + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 6 q^{11} - 6 q^{14} + 4 q^{16} - 6 q^{19} + 6 q^{21} - 4 q^{24} - 4 q^{26} - 2 q^{31} - 10 q^{34} + 4 q^{36} + 4 q^{39} - 2 q^{41} - 6 q^{44} - 22 q^{49} + 10 q^{51} - 4 q^{54} + 6 q^{56} + 2 q^{59} + 18 q^{61} - 4 q^{64} + 6 q^{66} + 6 q^{71} - 6 q^{74} + 6 q^{76} + 10 q^{79} + 4 q^{81} - 6 q^{84} - 20 q^{86} - 20 q^{89} - 6 q^{91} - 28 q^{94} + 4 q^{96} - 6 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 21 x^{2} + 100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 11 \nu$$$$)/10$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 11$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 11$$ $$\nu^{3}$$ $$=$$ $$10 \beta_{2} - 11 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1249.1
 − 3.70156i 2.70156i − 2.70156i 3.70156i
1.00000i 1.00000i −1.00000 0 1.00000 4.70156i 1.00000i −1.00000 0
1249.2 1.00000i 1.00000i −1.00000 0 1.00000 1.70156i 1.00000i −1.00000 0
1249.3 1.00000i 1.00000i −1.00000 0 1.00000 1.70156i 1.00000i −1.00000 0
1249.4 1.00000i 1.00000i −1.00000 0 1.00000 4.70156i 1.00000i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

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

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

## Hecke kernels

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

 $$T_{7}^{4} + 25 T_{7}^{2} + 64$$ $$T_{11}^{2} - 3 T_{11} - 8$$ $$T_{17}^{4} + 33 T_{17}^{2} + 16$$ $$T_{31}^{2} + T_{31} - 92$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$64 + 25 T^{2} + T^{4}$$
$11$ $$( -8 - 3 T + T^{2} )^{2}$$
$13$ $$( 1 + T^{2} )^{2}$$
$17$ $$16 + 33 T^{2} + T^{4}$$
$19$ $$( -8 + 3 T + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$( -41 + T^{2} )^{2}$$
$31$ $$( -92 + T + T^{2} )^{2}$$
$37$ $$64 + 25 T^{2} + T^{4}$$
$41$ $$( -10 + T + T^{2} )^{2}$$
$43$ $$256 + 132 T^{2} + T^{4}$$
$47$ $$( 49 + T^{2} )^{2}$$
$53$ $$625 + 114 T^{2} + T^{4}$$
$59$ $$( -10 - T + T^{2} )^{2}$$
$61$ $$( -72 - 9 T + T^{2} )^{2}$$
$67$ $$( 41 + T^{2} )^{2}$$
$71$ $$( -8 - 3 T + T^{2} )^{2}$$
$73$ $$( 144 + T^{2} )^{2}$$
$79$ $$( -4 - 5 T + T^{2} )^{2}$$
$83$ $$2116 + 133 T^{2} + T^{4}$$
$89$ $$( -16 + 10 T + T^{2} )^{2}$$
$97$ $$1600 + 244 T^{2} + T^{4}$$