# Properties

 Label 1950.2.z.h 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: no (minimal twist has level 390) 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.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{2} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( 1 - \zeta_{12}^{2} ) q^{6} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$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} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} -\zeta_{12}^{2} q^{11} + \zeta_{12}^{3} q^{12} + ( -4 \zeta_{12} + \zeta_{12}^{3} ) q^{13} -2 q^{14} -\zeta_{12}^{2} q^{16} + 2 \zeta_{12} q^{17} + \zeta_{12}^{3} q^{18} + ( 6 - 6 \zeta_{12}^{2} ) q^{19} -2 q^{21} + \zeta_{12} q^{22} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{23} -\zeta_{12}^{2} q^{24} + ( 4 - \zeta_{12}^{2} ) q^{26} + \zeta_{12}^{3} q^{27} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{28} -\zeta_{12}^{2} q^{29} -3 q^{31} + \zeta_{12} q^{32} + \zeta_{12} q^{33} -2 q^{34} -\zeta_{12}^{2} q^{36} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{37} + 6 \zeta_{12}^{3} q^{38} + ( 4 - \zeta_{12}^{2} ) q^{39} -10 \zeta_{12}^{2} q^{41} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{42} -5 \zeta_{12} q^{43} - q^{44} + ( 3 - 3 \zeta_{12}^{2} ) q^{46} -3 \zeta_{12}^{3} q^{47} + \zeta_{12} q^{48} -3 \zeta_{12}^{2} q^{49} -2 q^{51} + ( -3 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{52} + 14 \zeta_{12}^{3} q^{53} -\zeta_{12}^{2} q^{54} + ( -2 + 2 \zeta_{12}^{2} ) q^{56} + 6 \zeta_{12}^{3} q^{57} + \zeta_{12} q^{58} + ( -5 + 5 \zeta_{12}^{2} ) q^{59} + ( 10 - 10 \zeta_{12}^{2} ) q^{61} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{62} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{63} - q^{64} - q^{66} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{68} + ( 3 - 3 \zeta_{12}^{2} ) q^{69} + ( -4 + 4 \zeta_{12}^{2} ) q^{71} + \zeta_{12} q^{72} -2 \zeta_{12}^{3} q^{73} + ( -5 + 5 \zeta_{12}^{2} ) q^{74} -6 \zeta_{12}^{2} q^{76} -2 \zeta_{12}^{3} q^{77} + ( -3 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{78} -5 q^{79} -\zeta_{12}^{2} q^{81} + 10 \zeta_{12} q^{82} -6 \zeta_{12}^{3} q^{83} + ( -2 + 2 \zeta_{12}^{2} ) q^{84} + 5 q^{86} + \zeta_{12} q^{87} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{88} + 10 \zeta_{12}^{2} q^{89} + ( -2 - 6 \zeta_{12}^{2} ) q^{91} + 3 \zeta_{12}^{3} q^{92} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{93} + 3 \zeta_{12}^{2} q^{94} - q^{96} -10 \zeta_{12} q^{97} + 3 \zeta_{12} q^{98} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + 2q^{6} + 2q^{9} + O(q^{10})$$ $$4q + 2q^{4} + 2q^{6} + 2q^{9} - 2q^{11} - 8q^{14} - 2q^{16} + 12q^{19} - 8q^{21} - 2q^{24} + 14q^{26} - 2q^{29} - 12q^{31} - 8q^{34} - 2q^{36} + 14q^{39} - 20q^{41} - 4q^{44} + 6q^{46} - 6q^{49} - 8q^{51} - 2q^{54} - 4q^{56} - 10q^{59} + 20q^{61} - 4q^{64} - 4q^{66} + 6q^{69} - 8q^{71} - 10q^{74} - 12q^{76} - 20q^{79} - 2q^{81} - 4q^{84} + 20q^{86} + 20q^{89} - 20q^{91} + 6q^{94} - 4q^{96} - 4q^{99} + O(q^{100})$$

## 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 1.73205 1.00000i 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 −1.73205 + 1.00000i 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 1.73205 + 1.00000i 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 −1.73205 1.00000i 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.h 4
5.b even 2 1 inner 1950.2.z.h 4
5.c odd 4 1 390.2.i.a 2
5.c odd 4 1 1950.2.i.s 2
13.c even 3 1 inner 1950.2.z.h 4
15.e even 4 1 1170.2.i.k 2
65.n even 6 1 inner 1950.2.z.h 4
65.o even 12 1 5070.2.b.g 2
65.q odd 12 1 390.2.i.a 2
65.q odd 12 1 1950.2.i.s 2
65.q odd 12 1 5070.2.a.o 1
65.r odd 12 1 5070.2.a.f 1
65.t even 12 1 5070.2.b.g 2
195.bl even 12 1 1170.2.i.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.a 2 5.c odd 4 1
390.2.i.a 2 65.q odd 12 1
1170.2.i.k 2 15.e even 4 1
1170.2.i.k 2 195.bl even 12 1
1950.2.i.s 2 5.c odd 4 1
1950.2.i.s 2 65.q odd 12 1
1950.2.z.h 4 1.a even 1 1 trivial
1950.2.z.h 4 5.b even 2 1 inner
1950.2.z.h 4 13.c even 3 1 inner
1950.2.z.h 4 65.n even 6 1 inner
5070.2.a.f 1 65.r odd 12 1
5070.2.a.o 1 65.q odd 12 1
5070.2.b.g 2 65.o even 12 1
5070.2.b.g 2 65.t even 12 1

## Hecke kernels

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

 $$T_{7}^{4} - 4 T_{7}^{2} + 16$$ $$T_{11}^{2} + T_{11} + 1$$ $$T_{17}^{4} - 4 T_{17}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$16 - 4 T^{2} + T^{4}$$
$11$ $$( 1 + T + T^{2} )^{2}$$
$13$ $$169 - 22 T^{2} + T^{4}$$
$17$ $$16 - 4 T^{2} + T^{4}$$
$19$ $$( 36 - 6 T + T^{2} )^{2}$$
$23$ $$81 - 9 T^{2} + T^{4}$$
$29$ $$( 1 + T + T^{2} )^{2}$$
$31$ $$( 3 + T )^{4}$$
$37$ $$625 - 25 T^{2} + T^{4}$$
$41$ $$( 100 + 10 T + T^{2} )^{2}$$
$43$ $$625 - 25 T^{2} + T^{4}$$
$47$ $$( 9 + T^{2} )^{2}$$
$53$ $$( 196 + T^{2} )^{2}$$
$59$ $$( 25 + 5 T + T^{2} )^{2}$$
$61$ $$( 100 - 10 T + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$( 16 + 4 T + T^{2} )^{2}$$
$73$ $$( 4 + T^{2} )^{2}$$
$79$ $$( 5 + T )^{4}$$
$83$ $$( 36 + T^{2} )^{2}$$
$89$ $$( 100 - 10 T + T^{2} )^{2}$$
$97$ $$10000 - 100 T^{2} + T^{4}$$