# Properties

 Label 1950.2.bc.a 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.

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

## 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 −2.59808 1.50000i 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 2.59808 + 1.50000i 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 −2.59808 + 1.50000i 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 2.59808 1.50000i 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.a 4
5.b even 2 1 390.2.bb.a 4
5.c odd 4 1 1950.2.y.d 4
5.c odd 4 1 1950.2.y.e 4
13.e even 6 1 inner 1950.2.bc.a 4
15.d odd 2 1 1170.2.bs.d 4
65.l even 6 1 390.2.bb.a 4
65.l even 6 1 5070.2.b.p 4
65.n even 6 1 5070.2.b.p 4
65.r odd 12 1 1950.2.y.d 4
65.r odd 12 1 1950.2.y.e 4
65.s odd 12 1 5070.2.a.ba 2
65.s odd 12 1 5070.2.a.be 2
195.y odd 6 1 1170.2.bs.d 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 9 T_{7}^{2} + 81$$ acting on $$S_{2}^{\mathrm{new}}(1950, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$81 - 9 T^{2} + T^{4}$$
$11$ $$1 - 6 T + 11 T^{2} + 6 T^{3} + T^{4}$$
$13$ $$169 + 23 T^{2} + T^{4}$$
$17$ $$( 16 - 4 T + T^{2} )^{2}$$
$19$ $$169 + 78 T - T^{2} - 6 T^{3} + T^{4}$$
$23$ $$144 + 12 T^{2} + T^{4}$$
$29$ $$64 + 32 T + 24 T^{2} - 4 T^{3} + T^{4}$$
$31$ $$1936 + 104 T^{2} + T^{4}$$
$37$ $$2209 - 1128 T + 239 T^{2} - 24 T^{3} + T^{4}$$
$41$ $$256 - 16 T^{2} + T^{4}$$
$43$ $$( 36 - 6 T + T^{2} )^{2}$$
$47$ $$9 + 42 T^{2} + T^{4}$$
$53$ $$( 1 - 4 T + T^{2} )^{2}$$
$59$ $$2704 - 624 T - 4 T^{2} + 12 T^{3} + T^{4}$$
$61$ $$16 - 32 T + 60 T^{2} - 8 T^{3} + T^{4}$$
$67$ $$64 + 96 T + 56 T^{2} + 12 T^{3} + T^{4}$$
$71$ $$144 + 288 T + 204 T^{2} + 24 T^{3} + T^{4}$$
$73$ $$( 48 + T^{2} )^{2}$$
$79$ $$( 52 - 20 T + T^{2} )^{2}$$
$83$ $$576 + 96 T^{2} + T^{4}$$
$89$ $$20449 + 6006 T + 731 T^{2} + 42 T^{3} + T^{4}$$
$97$ $$10816 + 3744 T + 536 T^{2} + 36 T^{3} + T^{4}$$