# Properties

 Label 1950.2.bc.b Level $1950$ Weight $2$ Character orbit 1950.bc 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.bc (of order $$6$$, degree $$2$$, 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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{12} - \zeta_{12}^{3} ) q^{2} -\zeta_{12}^{2} q^{3} + ( 1 - \zeta_{12}^{2} ) q^{4} -\zeta_{12} q^{6} + 2 \zeta_{12} q^{7} -\zeta_{12}^{3} q^{8} + ( -1 + \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{12} - \zeta_{12}^{3} ) q^{2} -\zeta_{12}^{2} q^{3} + ( 1 - \zeta_{12}^{2} ) q^{4} -\zeta_{12} q^{6} + 2 \zeta_{12} q^{7} -\zeta_{12}^{3} q^{8} + ( -1 + \zeta_{12}^{2} ) q^{9} + ( 4 + 3 \zeta_{12} - 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{11} - q^{12} + ( 1 - 4 \zeta_{12}^{2} ) q^{13} + 2 q^{14} -\zeta_{12}^{2} q^{16} + ( -4 + 4 \zeta_{12}^{2} ) q^{17} + \zeta_{12}^{3} q^{18} + ( 2 + 4 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{19} -2 \zeta_{12}^{3} q^{21} + ( 3 + 2 \zeta_{12} - 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{22} + ( \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{23} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{24} + ( -3 \zeta_{12} - \zeta_{12}^{3} ) q^{26} + q^{27} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{28} + ( \zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{29} + ( -1 + 2 \zeta_{12}^{2} ) q^{31} -\zeta_{12} q^{32} + ( -2 - 3 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{33} + 4 \zeta_{12}^{3} q^{34} + \zeta_{12}^{2} q^{36} + ( -6 - 4 \zeta_{12} + 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{37} + ( 4 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{38} + ( -4 + 3 \zeta_{12}^{2} ) q^{39} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{41} -2 \zeta_{12}^{2} q^{42} + ( 5 + 4 \zeta_{12} - 5 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{43} + ( 2 - 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{44} + ( 1 + 2 \zeta_{12} + \zeta_{12}^{2} ) q^{46} + ( -2 + 4 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{47} + ( -1 + \zeta_{12}^{2} ) q^{48} -3 \zeta_{12}^{2} q^{49} + 4 q^{51} + ( -3 - \zeta_{12}^{2} ) q^{52} + ( 6 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{53} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{54} + ( 2 - 2 \zeta_{12}^{2} ) q^{56} + ( 2 - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{57} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{2} ) q^{58} + ( 2 + 5 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{59} + ( 6 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{61} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{62} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{63} - q^{64} + ( -3 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{66} + ( 4 + 8 \zeta_{12} - 2 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{67} + 4 \zeta_{12}^{2} q^{68} + ( 2 + \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{69} + ( -6 - 2 \zeta_{12} - 6 \zeta_{12}^{2} ) q^{71} + \zeta_{12} q^{72} + 2 \zeta_{12}^{3} q^{73} + ( -4 - 3 \zeta_{12} + 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{74} + ( 4 + 4 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{76} + ( 6 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{77} + ( -\zeta_{12} + 4 \zeta_{12}^{3} ) q^{78} + ( -7 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{79} -\zeta_{12}^{2} q^{81} + ( 2 - 2 \zeta_{12}^{2} ) q^{82} + ( -4 + 8 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{83} -2 \zeta_{12} q^{84} + ( 4 - 8 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{86} + ( -2 + \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{87} + ( -2 \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{88} + ( 4 - 4 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{89} + ( 2 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{91} + ( 2 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{92} + ( 2 - \zeta_{12}^{2} ) q^{93} + ( 2 \zeta_{12} - 7 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{94} + \zeta_{12}^{3} q^{96} + ( 2 - 4 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{97} -3 \zeta_{12} q^{98} + ( -2 + 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} + 2q^{4} - 2q^{9} + O(q^{10})$$ $$4q - 2q^{3} + 2q^{4} - 2q^{9} + 12q^{11} - 4q^{12} - 4q^{13} + 8q^{14} - 2q^{16} - 8q^{17} + 12q^{19} + 6q^{22} + 4q^{23} + 4q^{27} - 4q^{29} - 12q^{33} + 2q^{36} - 18q^{37} + 16q^{38} - 10q^{39} - 4q^{42} + 10q^{43} + 6q^{46} - 2q^{48} - 6q^{49} + 16q^{51} - 14q^{52} + 24q^{53} + 4q^{56} + 6q^{58} + 12q^{59} - 4q^{64} - 12q^{66} + 12q^{67} + 8q^{68} + 4q^{69} - 36q^{71} - 8q^{74} + 12q^{76} + 24q^{77} - 28q^{79} - 2q^{81} + 4q^{82} - 4q^{87} - 6q^{88} + 12q^{89} + 8q^{92} + 6q^{93} - 14q^{94} + 12q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
751.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 0.866025 + 0.500000i −1.73205 1.00000i 1.00000i −0.500000 + 0.866025i 0
751.2 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0 −0.866025 0.500000i 1.73205 + 1.00000i 1.00000i −0.500000 + 0.866025i 0
901.1 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0.866025 0.500000i −1.73205 + 1.00000i 1.00000i −0.500000 0.866025i 0
901.2 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −0.866025 + 0.500000i 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
13.e 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.bc.b 4
5.b even 2 1 390.2.bb.b 4
5.c odd 4 1 1950.2.y.c 4
5.c odd 4 1 1950.2.y.f 4
13.e even 6 1 inner 1950.2.bc.b 4
15.d odd 2 1 1170.2.bs.e 4
65.l even 6 1 390.2.bb.b 4
65.l even 6 1 5070.2.b.o 4
65.n even 6 1 5070.2.b.o 4
65.r odd 12 1 1950.2.y.c 4
65.r odd 12 1 1950.2.y.f 4
65.s odd 12 1 5070.2.a.y 2
65.s odd 12 1 5070.2.a.bg 2
195.y odd 6 1 1170.2.bs.e 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 4 T_{7}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(1950, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$16 - 4 T^{2} + T^{4}$$
$11$ $$9 - 36 T + 51 T^{2} - 12 T^{3} + T^{4}$$
$13$ $$( 13 + 2 T + T^{2} )^{2}$$
$17$ $$( 16 + 4 T + T^{2} )^{2}$$
$19$ $$16 + 48 T + 44 T^{2} - 12 T^{3} + T^{4}$$
$23$ $$1 - 4 T + 15 T^{2} - 4 T^{3} + T^{4}$$
$29$ $$1 + 4 T + 15 T^{2} + 4 T^{3} + T^{4}$$
$31$ $$( 3 + T^{2} )^{2}$$
$37$ $$121 + 198 T + 119 T^{2} + 18 T^{3} + T^{4}$$
$41$ $$16 - 4 T^{2} + T^{4}$$
$43$ $$529 + 230 T + 123 T^{2} - 10 T^{3} + T^{4}$$
$47$ $$1369 + 122 T^{2} + T^{4}$$
$53$ $$( -12 - 12 T + T^{2} )^{2}$$
$59$ $$169 + 156 T + 35 T^{2} - 12 T^{3} + T^{4}$$
$61$ $$11664 + 108 T^{2} + T^{4}$$
$67$ $$2704 + 624 T - 4 T^{2} - 12 T^{3} + T^{4}$$
$71$ $$10816 + 3744 T + 536 T^{2} + 36 T^{3} + T^{4}$$
$73$ $$( 4 + T^{2} )^{2}$$
$79$ $$( 1 + 14 T + T^{2} )^{2}$$
$83$ $$1936 + 104 T^{2} + T^{4}$$
$89$ $$16 + 48 T + 44 T^{2} - 12 T^{3} + T^{4}$$
$97$ $$16 + 48 T + 44 T^{2} - 12 T^{3} + T^{4}$$