# Properties

 Label 1950.2.y.d Level $1950$ Weight $2$ Character orbit 1950.y 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.y (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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{12}^{2} q^{2} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -1 + \zeta_{12}^{2} ) q^{4} -\zeta_{12} q^{6} + ( 3 - 3 \zeta_{12}^{2} ) q^{7} + q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q -\zeta_{12}^{2} q^{2} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -1 + \zeta_{12}^{2} ) q^{4} -\zeta_{12} q^{6} + ( 3 - 3 \zeta_{12}^{2} ) q^{7} + q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} + ( -2 - 2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{11} + \zeta_{12}^{3} q^{12} + ( 3 + \zeta_{12}^{2} ) q^{13} -3 q^{14} -\zeta_{12}^{2} q^{16} -4 \zeta_{12} q^{17} - q^{18} + ( -1 + 4 \zeta_{12} - \zeta_{12}^{2} ) q^{19} -3 \zeta_{12}^{3} q^{21} + ( 1 + 2 \zeta_{12} + \zeta_{12}^{2} ) q^{22} + ( 4 - 2 \zeta_{12}^{2} ) q^{23} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{24} + ( 1 - 4 \zeta_{12}^{2} ) q^{26} -\zeta_{12}^{3} q^{27} + 3 \zeta_{12}^{2} q^{28} + ( -2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{29} + ( 4 - 8 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{31} + ( -1 + \zeta_{12}^{2} ) q^{32} + ( -2 - \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{33} + 4 \zeta_{12}^{3} q^{34} + \zeta_{12}^{2} q^{36} + ( -4 \zeta_{12} - \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{37} + ( -1 + 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{38} + ( 4 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{39} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{41} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{42} + 6 \zeta_{12} q^{43} + ( 1 - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{44} + ( -2 - 2 \zeta_{12}^{2} ) q^{46} + ( 3 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{47} -\zeta_{12} q^{48} -2 \zeta_{12}^{2} q^{49} -4 q^{51} + ( -4 + 3 \zeta_{12}^{2} ) q^{52} + ( -1 + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{53} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{54} + ( 3 - 3 \zeta_{12}^{2} ) q^{56} + ( 4 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{57} + ( -2 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{58} + ( 2 - 8 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{59} + ( 4 + 2 \zeta_{12} - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{61} + ( -8 - 2 \zeta_{12} + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{62} -3 \zeta_{12}^{2} q^{63} + q^{64} + ( 2 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{66} + ( 2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{67} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{68} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{69} + ( -4 + 6 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{71} + ( 1 - \zeta_{12}^{2} ) q^{72} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{73} + ( -1 - 4 \zeta_{12} + \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{74} + ( 2 - 4 \zeta_{12} - \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{76} + ( -3 + 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{77} + ( -3 \zeta_{12} - \zeta_{12}^{3} ) q^{78} + ( -10 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{79} -\zeta_{12}^{2} q^{81} -4 \zeta_{12} q^{82} + ( 6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{83} + 3 \zeta_{12} q^{84} -6 \zeta_{12}^{3} q^{86} + ( -2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{87} + ( -2 - 2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{88} + ( 14 - 2 \zeta_{12} - 7 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{89} + ( 12 - 9 \zeta_{12}^{2} ) q^{91} + ( -2 + 4 \zeta_{12}^{2} ) q^{92} + ( -4 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{93} + ( 2 \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{94} + \zeta_{12}^{3} q^{96} + ( -2 - 6 \zeta_{12} + 2 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{97} + ( -2 + 2 \zeta_{12}^{2} ) q^{98} + ( -1 + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 2q^{4} + 6q^{7} + 4q^{8} + 2q^{9} + O(q^{10})$$ $$4q - 2q^{2} - 2q^{4} + 6q^{7} + 4q^{8} + 2q^{9} - 6q^{11} + 14q^{13} - 12q^{14} - 2q^{16} - 4q^{18} - 6q^{19} + 6q^{22} + 12q^{23} - 4q^{26} + 6q^{28} - 4q^{29} - 2q^{32} - 4q^{33} + 2q^{36} - 2q^{37} - 12q^{46} + 12q^{47} - 4q^{49} - 16q^{51} - 10q^{52} + 6q^{56} + 16q^{57} - 4q^{58} + 12q^{59} + 8q^{61} - 24q^{62} - 6q^{63} + 4q^{64} + 8q^{66} + 4q^{67} - 24q^{71} + 2q^{72} - 2q^{74} + 6q^{76} - 40q^{79} - 2q^{81} + 24q^{83} - 12q^{87} - 6q^{88} + 42q^{89} + 30q^{91} - 4q^{93} - 6q^{94} - 4q^{97} - 4q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.500000 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 0 0.866025 + 0.500000i 1.50000 2.59808i 1.00000 0.500000 0.866025i 0
49.2 −0.500000 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 0 −0.866025 0.500000i 1.50000 2.59808i 1.00000 0.500000 0.866025i 0
199.1 −0.500000 + 0.866025i −0.866025 0.500000i −0.500000 0.866025i 0 0.866025 0.500000i 1.50000 + 2.59808i 1.00000 0.500000 + 0.866025i 0
199.2 −0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 0 −0.866025 + 0.500000i 1.50000 + 2.59808i 1.00000 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
65.l 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.y.d 4
5.b even 2 1 1950.2.y.e 4
5.c odd 4 1 390.2.bb.a 4
5.c odd 4 1 1950.2.bc.a 4
13.e even 6 1 1950.2.y.e 4
15.e even 4 1 1170.2.bs.d 4
65.l even 6 1 inner 1950.2.y.d 4
65.o even 12 1 5070.2.a.be 2
65.q odd 12 1 5070.2.b.p 4
65.r odd 12 1 390.2.bb.a 4
65.r odd 12 1 1950.2.bc.a 4
65.r odd 12 1 5070.2.b.p 4
65.t even 12 1 5070.2.a.ba 2
195.bf even 12 1 1170.2.bs.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.a 4 5.c odd 4 1
390.2.bb.a 4 65.r odd 12 1
1170.2.bs.d 4 15.e even 4 1
1170.2.bs.d 4 195.bf even 12 1
1950.2.y.d 4 1.a even 1 1 trivial
1950.2.y.d 4 65.l even 6 1 inner
1950.2.y.e 4 5.b even 2 1
1950.2.y.e 4 13.e even 6 1
1950.2.bc.a 4 5.c odd 4 1
1950.2.bc.a 4 65.r odd 12 1
5070.2.a.ba 2 65.t even 12 1
5070.2.a.be 2 65.o even 12 1
5070.2.b.p 4 65.q odd 12 1
5070.2.b.p 4 65.r odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - 3 T_{7} + 9$$ acting on $$S_{2}^{\mathrm{new}}(1950, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 9 - 3 T + T^{2} )^{2}$$
$11$ $$1 - 6 T + 11 T^{2} + 6 T^{3} + T^{4}$$
$13$ $$( 13 - 7 T + T^{2} )^{2}$$
$17$ $$256 - 16 T^{2} + T^{4}$$
$19$ $$169 - 78 T - T^{2} + 6 T^{3} + T^{4}$$
$23$ $$( 12 - 6 T + T^{2} )^{2}$$
$29$ $$64 - 32 T + 24 T^{2} + 4 T^{3} + T^{4}$$
$31$ $$1936 + 104 T^{2} + T^{4}$$
$37$ $$2209 - 94 T + 51 T^{2} + 2 T^{3} + T^{4}$$
$41$ $$256 - 16 T^{2} + T^{4}$$
$43$ $$1296 - 36 T^{2} + T^{4}$$
$47$ $$( -3 - 6 T + T^{2} )^{2}$$
$53$ $$1 + 14 T^{2} + T^{4}$$
$59$ $$2704 + 624 T - 4 T^{2} - 12 T^{3} + T^{4}$$
$61$ $$16 - 32 T + 60 T^{2} - 8 T^{3} + T^{4}$$
$67$ $$64 + 32 T + 24 T^{2} - 4 T^{3} + T^{4}$$
$71$ $$144 + 288 T + 204 T^{2} + 24 T^{3} + T^{4}$$
$73$ $$( -48 + T^{2} )^{2}$$
$79$ $$( 52 + 20 T + T^{2} )^{2}$$
$83$ $$( 24 - 12 T + T^{2} )^{2}$$
$89$ $$20449 - 6006 T + 731 T^{2} - 42 T^{3} + T^{4}$$
$97$ $$10816 - 416 T + 120 T^{2} + 4 T^{3} + T^{4}$$