# Properties

 Label 1950.2.i.bh Level $1950$ Weight $2$ Character orbit 1950.i 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.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{12}^{2} ) q^{2} + ( 1 - \zeta_{12}^{2} ) q^{3} -\zeta_{12}^{2} q^{4} -\zeta_{12}^{2} q^{6} + ( -\zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{7} - q^{8} -\zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{12}^{2} ) q^{2} + ( 1 - \zeta_{12}^{2} ) q^{3} -\zeta_{12}^{2} q^{4} -\zeta_{12}^{2} q^{6} + ( -\zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{7} - q^{8} -\zeta_{12}^{2} q^{9} + ( 1 + \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{11} - q^{12} + ( 3 + 2 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{13} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + ( -2 \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{17} - q^{18} + ( -\zeta_{12} - 3 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{19} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{21} + ( -\zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{22} + ( 1 - 3 \zeta_{12} - \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{23} + ( -1 + \zeta_{12}^{2} ) q^{24} + ( 2 \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{26} - q^{27} + ( 1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{28} + ( 3 - 2 \zeta_{12} - 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{29} + 4 q^{31} + \zeta_{12}^{2} q^{32} + ( -\zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{33} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{34} + ( -1 + \zeta_{12}^{2} ) q^{36} + ( 1 - 4 \zeta_{12} - \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{37} + ( -3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{38} + ( 2 \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{39} + ( 4 - 3 \zeta_{12} - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{41} + ( 1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{42} + ( -\zeta_{12} - 5 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{43} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{44} + ( 3 \zeta_{12} - \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{46} + ( 1 + 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{47} + \zeta_{12}^{2} q^{48} + ( 3 + 2 \zeta_{12} - 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{49} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{51} + ( -3 - 2 \zeta_{12}^{3} ) q^{52} + ( -6 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{53} + ( -1 + \zeta_{12}^{2} ) q^{54} + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{56} + ( -3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{57} + ( 2 \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{58} + ( -6 \zeta_{12} + 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{59} + ( -7 \zeta_{12} - 2 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{61} + ( 4 - 4 \zeta_{12}^{2} ) q^{62} + ( 1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{63} + q^{64} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{66} + ( -1 - 5 \zeta_{12} + \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{67} + ( 1 - 2 \zeta_{12} - \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{68} + ( 3 \zeta_{12} - \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{69} + ( -\zeta_{12} - 3 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{71} + \zeta_{12}^{2} q^{72} + ( -4 - 10 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{73} + ( 4 \zeta_{12} - \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{74} + ( -3 - \zeta_{12} + 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{76} -2 q^{77} + ( -3 - 2 \zeta_{12}^{3} ) q^{78} + 12 q^{79} + ( -1 + \zeta_{12}^{2} ) q^{81} + ( 3 \zeta_{12} - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{82} + ( -7 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{83} + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{84} + ( -5 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{86} + ( 2 \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{87} + ( -1 - \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{88} + ( -2 - 4 \zeta_{12} + 2 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{89} + ( 5 - 6 \zeta_{12} - 4 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{91} + ( -1 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{92} + ( 4 - 4 \zeta_{12}^{2} ) q^{93} + ( 1 + 5 \zeta_{12} - \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{94} + q^{96} -10 \zeta_{12}^{2} q^{97} + ( -2 \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{98} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{6} + 2 q^{7} - 4 q^{8} - 2 q^{9} + O(q^{10})$$ $$4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{6} + 2 q^{7} - 4 q^{8} - 2 q^{9} + 2 q^{11} - 4 q^{12} + 6 q^{13} + 4 q^{14} - 2 q^{16} + 2 q^{17} - 4 q^{18} - 6 q^{19} + 4 q^{21} - 2 q^{22} + 2 q^{23} - 2 q^{24} - 6 q^{26} - 4 q^{27} + 2 q^{28} + 6 q^{29} + 16 q^{31} + 2 q^{32} - 2 q^{33} + 4 q^{34} - 2 q^{36} + 2 q^{37} - 12 q^{38} - 6 q^{39} + 8 q^{41} + 2 q^{42} - 10 q^{43} - 4 q^{44} - 2 q^{46} + 4 q^{47} + 2 q^{48} + 6 q^{49} + 4 q^{51} - 12 q^{52} - 24 q^{53} - 2 q^{54} - 2 q^{56} - 12 q^{57} - 6 q^{58} + 4 q^{59} - 4 q^{61} + 8 q^{62} + 2 q^{63} + 4 q^{64} - 4 q^{66} - 2 q^{67} + 2 q^{68} - 2 q^{69} - 6 q^{71} + 2 q^{72} - 16 q^{73} - 2 q^{74} - 6 q^{76} - 8 q^{77} - 12 q^{78} + 48 q^{79} - 2 q^{81} - 8 q^{82} - 28 q^{83} - 2 q^{84} - 20 q^{86} - 6 q^{87} - 2 q^{88} - 4 q^{89} + 12 q^{91} - 4 q^{92} + 8 q^{93} + 2 q^{94} + 4 q^{96} - 20 q^{97} - 6 q^{98} - 4 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$-\zeta_{12}^{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i 0 −0.500000 0.866025i −0.366025 0.633975i −1.00000 −0.500000 0.866025i 0
451.2 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i 0 −0.500000 0.866025i 1.36603 + 2.36603i −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 −0.366025 + 0.633975i −1.00000 −0.500000 + 0.866025i 0
601.2 0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.36603 2.36603i −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.bh 4
5.b even 2 1 1950.2.i.y 4
5.c odd 4 1 390.2.y.b 4
5.c odd 4 1 390.2.y.c yes 4
13.c even 3 1 inner 1950.2.i.bh 4
15.e even 4 1 1170.2.bp.d 4
15.e even 4 1 1170.2.bp.e 4
65.n even 6 1 1950.2.i.y 4
65.q odd 12 1 390.2.y.b 4
65.q odd 12 1 390.2.y.c yes 4
195.bl even 12 1 1170.2.bp.d 4
195.bl even 12 1 1170.2.bp.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.y.b 4 5.c odd 4 1
390.2.y.b 4 65.q odd 12 1
390.2.y.c yes 4 5.c odd 4 1
390.2.y.c yes 4 65.q odd 12 1
1170.2.bp.d 4 15.e even 4 1
1170.2.bp.d 4 195.bl even 12 1
1170.2.bp.e 4 15.e even 4 1
1170.2.bp.e 4 195.bl even 12 1
1950.2.i.y 4 5.b even 2 1
1950.2.i.y 4 65.n even 6 1
1950.2.i.bh 4 1.a even 1 1 trivial
1950.2.i.bh 4 13.c even 3 1 inner

## Hecke kernels

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

 $$T_{7}^{4} - 2 T_{7}^{3} + 6 T_{7}^{2} + 4 T_{7} + 4$$ $$T_{11}^{4} - 2 T_{11}^{3} + 6 T_{11}^{2} + 4 T_{11} + 4$$ $$T_{17}^{4} - 2 T_{17}^{3} + 15 T_{17}^{2} + 22 T_{17} + 121$$ $$T_{19}^{4} + 6 T_{19}^{3} + 30 T_{19}^{2} + 36 T_{19} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$( 1 - T + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$4 + 4 T + 6 T^{2} - 2 T^{3} + T^{4}$$
$11$ $$4 + 4 T + 6 T^{2} - 2 T^{3} + T^{4}$$
$13$ $$169 - 78 T + 23 T^{2} - 6 T^{3} + T^{4}$$
$17$ $$121 + 22 T + 15 T^{2} - 2 T^{3} + T^{4}$$
$19$ $$36 + 36 T + 30 T^{2} + 6 T^{3} + T^{4}$$
$23$ $$676 + 52 T + 30 T^{2} - 2 T^{3} + T^{4}$$
$29$ $$9 + 18 T + 39 T^{2} - 6 T^{3} + T^{4}$$
$31$ $$( -4 + T )^{4}$$
$37$ $$2209 + 94 T + 51 T^{2} - 2 T^{3} + T^{4}$$
$41$ $$121 + 88 T + 75 T^{2} - 8 T^{3} + T^{4}$$
$43$ $$484 + 220 T + 78 T^{2} + 10 T^{3} + T^{4}$$
$47$ $$( -74 - 2 T + T^{2} )^{2}$$
$53$ $$( 33 + 12 T + T^{2} )^{2}$$
$59$ $$10816 + 416 T + 120 T^{2} - 4 T^{3} + T^{4}$$
$61$ $$20449 - 572 T + 159 T^{2} + 4 T^{3} + T^{4}$$
$67$ $$5476 - 148 T + 78 T^{2} + 2 T^{3} + T^{4}$$
$71$ $$36 + 36 T + 30 T^{2} + 6 T^{3} + T^{4}$$
$73$ $$( -59 + 8 T + T^{2} )^{2}$$
$79$ $$( -12 + T )^{4}$$
$83$ $$( 46 + 14 T + T^{2} )^{2}$$
$89$ $$1936 - 176 T + 60 T^{2} + 4 T^{3} + T^{4}$$
$97$ $$( 100 + 10 T + T^{2} )^{2}$$