# Properties

 Label 1950.2.i.o Level $1950$ Weight $2$ Character orbit 1950.i Analytic conductor $15.571$ Analytic rank $0$ Dimension $2$ 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.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{2} + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + \zeta_{6} q^{6} -3 \zeta_{6} q^{7} - q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{2} + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + \zeta_{6} q^{6} -3 \zeta_{6} q^{7} - q^{8} -\zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{11} + q^{12} + ( -4 + 3 \zeta_{6} ) q^{13} -3 q^{14} + ( -1 + \zeta_{6} ) q^{16} - q^{18} + 5 \zeta_{6} q^{19} + 3 q^{21} + \zeta_{6} q^{22} + ( -4 + 4 \zeta_{6} ) q^{23} + ( 1 - \zeta_{6} ) q^{24} + ( -1 + 4 \zeta_{6} ) q^{26} + q^{27} + ( -3 + 3 \zeta_{6} ) q^{28} + 10 q^{31} + \zeta_{6} q^{32} -\zeta_{6} q^{33} + ( -1 + \zeta_{6} ) q^{36} + ( -1 + \zeta_{6} ) q^{37} + 5 q^{38} + ( 1 - 4 \zeta_{6} ) q^{39} + ( -6 + 6 \zeta_{6} ) q^{41} + ( 3 - 3 \zeta_{6} ) q^{42} -2 \zeta_{6} q^{43} + q^{44} + 4 \zeta_{6} q^{46} + 9 q^{47} -\zeta_{6} q^{48} + ( -2 + 2 \zeta_{6} ) q^{49} + ( 3 + \zeta_{6} ) q^{52} + 13 q^{53} + ( 1 - \zeta_{6} ) q^{54} + 3 \zeta_{6} q^{56} -5 q^{57} -4 \zeta_{6} q^{59} + 2 \zeta_{6} q^{61} + ( 10 - 10 \zeta_{6} ) q^{62} + ( -3 + 3 \zeta_{6} ) q^{63} + q^{64} - q^{66} + ( -12 + 12 \zeta_{6} ) q^{67} -4 \zeta_{6} q^{69} + 2 \zeta_{6} q^{71} + \zeta_{6} q^{72} + 16 q^{73} + \zeta_{6} q^{74} + ( 5 - 5 \zeta_{6} ) q^{76} + 3 q^{77} + ( -3 - \zeta_{6} ) q^{78} -10 q^{79} + ( -1 + \zeta_{6} ) q^{81} + 6 \zeta_{6} q^{82} -12 q^{83} -3 \zeta_{6} q^{84} -2 q^{86} + ( 1 - \zeta_{6} ) q^{88} + ( -1 + \zeta_{6} ) q^{89} + ( 9 + 3 \zeta_{6} ) q^{91} + 4 q^{92} + ( -10 + 10 \zeta_{6} ) q^{93} + ( 9 - 9 \zeta_{6} ) q^{94} - q^{96} + 12 \zeta_{6} q^{97} + 2 \zeta_{6} q^{98} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{3} - q^{4} + q^{6} - 3q^{7} - 2q^{8} - q^{9} + O(q^{10})$$ $$2q + q^{2} - q^{3} - q^{4} + q^{6} - 3q^{7} - 2q^{8} - q^{9} - q^{11} + 2q^{12} - 5q^{13} - 6q^{14} - q^{16} - 2q^{18} + 5q^{19} + 6q^{21} + q^{22} - 4q^{23} + q^{24} + 2q^{26} + 2q^{27} - 3q^{28} + 20q^{31} + q^{32} - q^{33} - q^{36} - q^{37} + 10q^{38} - 2q^{39} - 6q^{41} + 3q^{42} - 2q^{43} + 2q^{44} + 4q^{46} + 18q^{47} - q^{48} - 2q^{49} + 7q^{52} + 26q^{53} + q^{54} + 3q^{56} - 10q^{57} - 4q^{59} + 2q^{61} + 10q^{62} - 3q^{63} + 2q^{64} - 2q^{66} - 12q^{67} - 4q^{69} + 2q^{71} + q^{72} + 32q^{73} + q^{74} + 5q^{76} + 6q^{77} - 7q^{78} - 20q^{79} - q^{81} + 6q^{82} - 24q^{83} - 3q^{84} - 4q^{86} + q^{88} - q^{89} + 21q^{91} + 8q^{92} - 10q^{93} + 9q^{94} - 2q^{96} + 12q^{97} + 2q^{98} + 2q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0 0.500000 + 0.866025i −1.50000 2.59808i −1.00000 −0.500000 0.866025i 0
601.1 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0 0.500000 0.866025i −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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.i.o 2
5.b even 2 1 390.2.i.b 2
5.c odd 4 2 1950.2.z.i 4
13.c even 3 1 inner 1950.2.i.o 2
15.d odd 2 1 1170.2.i.j 2
65.l even 6 1 5070.2.a.c 1
65.n even 6 1 390.2.i.b 2
65.n even 6 1 5070.2.a.q 1
65.q odd 12 2 1950.2.z.i 4
65.s odd 12 2 5070.2.b.a 2
195.x odd 6 1 1170.2.i.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.b 2 5.b even 2 1
390.2.i.b 2 65.n even 6 1
1170.2.i.j 2 15.d odd 2 1
1170.2.i.j 2 195.x odd 6 1
1950.2.i.o 2 1.a even 1 1 trivial
1950.2.i.o 2 13.c even 3 1 inner
1950.2.z.i 4 5.c odd 4 2
1950.2.z.i 4 65.q odd 12 2
5070.2.a.c 1 65.l even 6 1
5070.2.a.q 1 65.n even 6 1
5070.2.b.a 2 65.s odd 12 2

## Hecke kernels

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

 $$T_{7}^{2} + 3 T_{7} + 9$$ $$T_{11}^{2} + T_{11} + 1$$ $$T_{17}$$ $$T_{19}^{2} - 5 T_{19} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$9 + 3 T + T^{2}$$
$11$ $$1 + T + T^{2}$$
$13$ $$13 + 5 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$25 - 5 T + T^{2}$$
$23$ $$16 + 4 T + T^{2}$$
$29$ $$T^{2}$$
$31$ $$( -10 + T )^{2}$$
$37$ $$1 + T + T^{2}$$
$41$ $$36 + 6 T + T^{2}$$
$43$ $$4 + 2 T + T^{2}$$
$47$ $$( -9 + T )^{2}$$
$53$ $$( -13 + T )^{2}$$
$59$ $$16 + 4 T + T^{2}$$
$61$ $$4 - 2 T + T^{2}$$
$67$ $$144 + 12 T + T^{2}$$
$71$ $$4 - 2 T + T^{2}$$
$73$ $$( -16 + T )^{2}$$
$79$ $$( 10 + T )^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$1 + T + T^{2}$$
$97$ $$144 - 12 T + T^{2}$$