# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.z (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{12} - \zeta_{12}^{3} ) q^{2} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( -1 + \zeta_{12}^{2} ) q^{6} -\zeta_{12}^{3} q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{12} - \zeta_{12}^{3} ) q^{2} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( -1 + \zeta_{12}^{2} ) q^{6} -\zeta_{12}^{3} q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} -3 \zeta_{12}^{2} q^{11} + \zeta_{12}^{3} q^{12} + ( 3 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{13} -\zeta_{12}^{2} q^{16} -6 \zeta_{12} q^{17} -\zeta_{12}^{3} q^{18} + ( -6 + 6 \zeta_{12}^{2} ) q^{19} -3 \zeta_{12} q^{22} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{23} + \zeta_{12}^{2} q^{24} + ( 3 - 4 \zeta_{12}^{2} ) q^{26} + \zeta_{12}^{3} q^{27} + 2 \zeta_{12}^{2} q^{29} -6 q^{31} -\zeta_{12} q^{32} + 3 \zeta_{12} q^{33} -6 q^{34} -\zeta_{12}^{2} q^{36} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{37} + 6 \zeta_{12}^{3} q^{38} + ( -3 + 4 \zeta_{12}^{2} ) q^{39} + 10 \zeta_{12}^{2} q^{41} -8 \zeta_{12} q^{43} -3 q^{44} + ( 5 - 5 \zeta_{12}^{2} ) q^{46} -12 \zeta_{12}^{3} q^{47} + \zeta_{12} q^{48} -7 \zeta_{12}^{2} q^{49} + 6 q^{51} + ( -\zeta_{12} - 3 \zeta_{12}^{3} ) q^{52} -12 \zeta_{12}^{3} q^{53} + \zeta_{12}^{2} q^{54} -6 \zeta_{12}^{3} q^{57} + 2 \zeta_{12} q^{58} + ( 15 - 15 \zeta_{12}^{2} ) q^{61} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{62} - q^{64} + 3 q^{66} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{67} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{68} + ( -5 + 5 \zeta_{12}^{2} ) q^{69} + ( -7 + 7 \zeta_{12}^{2} ) q^{71} -\zeta_{12} q^{72} + \zeta_{12}^{3} q^{73} + ( -9 + 9 \zeta_{12}^{2} ) q^{74} + 6 \zeta_{12}^{2} q^{76} + ( \zeta_{12} + 3 \zeta_{12}^{3} ) q^{78} -6 q^{79} -\zeta_{12}^{2} q^{81} + 10 \zeta_{12} q^{82} -5 \zeta_{12}^{3} q^{83} -8 q^{86} -2 \zeta_{12} q^{87} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{88} -6 \zeta_{12}^{2} q^{89} -5 \zeta_{12}^{3} q^{92} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{93} -12 \zeta_{12}^{2} q^{94} + q^{96} + 5 \zeta_{12} q^{97} -7 \zeta_{12} q^{98} -3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} - 2q^{6} + 2q^{9} + O(q^{10})$$ $$4q + 2q^{4} - 2q^{6} + 2q^{9} - 6q^{11} - 2q^{16} - 12q^{19} + 2q^{24} + 4q^{26} + 4q^{29} - 24q^{31} - 24q^{34} - 2q^{36} - 4q^{39} + 20q^{41} - 12q^{44} + 10q^{46} - 14q^{49} + 24q^{51} + 2q^{54} + 30q^{61} - 4q^{64} + 12q^{66} - 10q^{69} - 14q^{71} - 18q^{74} + 12q^{76} - 24q^{79} - 2q^{81} - 32q^{86} - 12q^{89} - 24q^{94} + 4q^{96} - 12q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$-1 + \zeta_{12}^{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1699.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 −0.500000 0.866025i 0 1.00000i 0.500000 + 0.866025i 0
1699.2 0.866025 + 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 0 −0.500000 0.866025i 0 1.00000i 0.500000 + 0.866025i 0
1849.1 −0.866025 + 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0 −0.500000 + 0.866025i 0 1.00000i 0.500000 0.866025i 0
1849.2 0.866025 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 0 −0.500000 + 0.866025i 0 1.00000i 0.500000 0.866025i 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
13.c even 3 1 inner
65.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.z.a 4
5.b even 2 1 inner 1950.2.z.a 4
5.c odd 4 1 1950.2.i.e 2
5.c odd 4 1 1950.2.i.v yes 2
13.c even 3 1 inner 1950.2.z.a 4
65.n even 6 1 inner 1950.2.z.a 4
65.q odd 12 1 1950.2.i.e 2
65.q odd 12 1 1950.2.i.v yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.i.e 2 5.c odd 4 1
1950.2.i.e 2 65.q odd 12 1
1950.2.i.v yes 2 5.c odd 4 1
1950.2.i.v yes 2 65.q odd 12 1
1950.2.z.a 4 1.a even 1 1 trivial
1950.2.z.a 4 5.b even 2 1 inner
1950.2.z.a 4 13.c even 3 1 inner
1950.2.z.a 4 65.n even 6 1 inner

## Hecke kernels

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

 $$T_{7}$$ $$T_{11}^{2} + 3 T_{11} + 9$$ $$T_{17}^{4} - 36 T_{17}^{2} + 1296$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 9 + 3 T + T^{2} )^{2}$$
$13$ $$169 - T^{2} + T^{4}$$
$17$ $$1296 - 36 T^{2} + T^{4}$$
$19$ $$( 36 + 6 T + T^{2} )^{2}$$
$23$ $$625 - 25 T^{2} + T^{4}$$
$29$ $$( 4 - 2 T + T^{2} )^{2}$$
$31$ $$( 6 + T )^{4}$$
$37$ $$6561 - 81 T^{2} + T^{4}$$
$41$ $$( 100 - 10 T + T^{2} )^{2}$$
$43$ $$4096 - 64 T^{2} + T^{4}$$
$47$ $$( 144 + T^{2} )^{2}$$
$53$ $$( 144 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$( 225 - 15 T + T^{2} )^{2}$$
$67$ $$256 - 16 T^{2} + T^{4}$$
$71$ $$( 49 + 7 T + T^{2} )^{2}$$
$73$ $$( 1 + T^{2} )^{2}$$
$79$ $$( 6 + T )^{4}$$
$83$ $$( 25 + T^{2} )^{2}$$
$89$ $$( 36 + 6 T + T^{2} )^{2}$$
$97$ $$625 - 25 T^{2} + T^{4}$$