# Properties

 Label 1950.2.i.z 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: $$\Q(\sqrt{-3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - x^{3} + 5 x^{2} + 4 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{2} ) q^{2} + ( -1 + \beta_{2} ) q^{3} -\beta_{2} q^{4} -\beta_{2} q^{6} + ( \beta_{1} + \beta_{2} ) q^{7} + q^{8} -\beta_{2} q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{2} ) q^{2} + ( -1 + \beta_{2} ) q^{3} -\beta_{2} q^{4} -\beta_{2} q^{6} + ( \beta_{1} + \beta_{2} ) q^{7} + q^{8} -\beta_{2} q^{9} + ( -2 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{11} + q^{12} + ( -1 + \beta_{1} - \beta_{3} ) q^{13} + ( -1 + \beta_{3} ) q^{14} + ( -1 + \beta_{2} ) q^{16} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{17} + q^{18} + ( -\beta_{1} + 5 \beta_{2} ) q^{19} + ( -1 + \beta_{3} ) q^{21} + ( \beta_{1} - 2 \beta_{2} ) q^{22} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{23} + ( -1 + \beta_{2} ) q^{24} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{26} + q^{27} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{28} + ( 4 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{29} + 4 q^{31} -\beta_{2} q^{32} + ( \beta_{1} - 2 \beta_{2} ) q^{33} + ( 2 + 2 \beta_{3} ) q^{34} + ( -1 + \beta_{2} ) q^{36} + ( 2 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{37} + ( -5 - \beta_{3} ) q^{38} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{39} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{41} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{42} + ( \beta_{1} - 3 \beta_{2} ) q^{43} + ( 2 + \beta_{3} ) q^{44} + 3 \beta_{1} q^{46} + 4 q^{47} -\beta_{2} q^{48} + ( 2 + 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{49} + ( 2 + 2 \beta_{3} ) q^{51} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{52} + ( -6 - 4 \beta_{3} ) q^{53} + ( -1 + \beta_{2} ) q^{54} + ( \beta_{1} + \beta_{2} ) q^{56} + ( -5 - \beta_{3} ) q^{57} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{58} + 4 \beta_{1} q^{59} + ( 3 \beta_{1} - 6 \beta_{2} ) q^{61} + ( -4 + 4 \beta_{2} ) q^{62} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{63} + q^{64} + ( 2 + \beta_{3} ) q^{66} + ( 1 + 5 \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{67} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{68} + 3 \beta_{1} q^{69} + ( -\beta_{1} - 10 \beta_{2} ) q^{71} -\beta_{2} q^{72} + 9 q^{73} + ( -\beta_{1} + 2 \beta_{2} ) q^{74} + ( 5 + \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{76} + 2 q^{77} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{78} + ( -3 + 5 \beta_{3} ) q^{79} + ( -1 + \beta_{2} ) q^{81} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{82} + ( 4 - \beta_{3} ) q^{83} + ( \beta_{1} + \beta_{2} ) q^{84} + ( 3 + \beta_{3} ) q^{86} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{87} + ( -2 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{88} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{89} + ( -4 + 3 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{91} + 3 \beta_{3} q^{92} + ( -4 + 4 \beta_{2} ) q^{93} + ( -4 + 4 \beta_{2} ) q^{94} + q^{96} + ( 5 \beta_{1} - 10 \beta_{2} ) q^{97} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{98} + ( 2 + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 2q^{3} - 2q^{4} - 2q^{6} + 3q^{7} + 4q^{8} - 2q^{9} + O(q^{10})$$ $$4q - 2q^{2} - 2q^{3} - 2q^{4} - 2q^{6} + 3q^{7} + 4q^{8} - 2q^{9} - 3q^{11} + 4q^{12} - q^{13} - 6q^{14} - 2q^{16} - 2q^{17} + 4q^{18} + 9q^{19} - 6q^{21} - 3q^{22} + 3q^{23} - 2q^{24} - q^{26} + 4q^{27} + 3q^{28} + 6q^{29} + 16q^{31} - 2q^{32} - 3q^{33} + 4q^{34} - 2q^{36} + 3q^{37} - 18q^{38} - q^{39} - 2q^{41} + 3q^{42} - 5q^{43} + 6q^{44} + 3q^{46} + 16q^{47} - 2q^{48} + q^{49} + 4q^{51} + 2q^{52} - 16q^{53} - 2q^{54} + 3q^{56} - 18q^{57} + 6q^{58} + 4q^{59} - 9q^{61} - 8q^{62} + 3q^{63} + 4q^{64} + 6q^{66} - 3q^{67} - 2q^{68} + 3q^{69} - 21q^{71} - 2q^{72} + 36q^{73} + 3q^{74} + 9q^{76} + 8q^{77} + 2q^{78} - 22q^{79} - 2q^{81} - 2q^{82} + 18q^{83} + 3q^{84} + 10q^{86} + 6q^{87} - 3q^{88} - 4q^{89} - 3q^{91} - 6q^{92} - 8q^{93} - 8q^{94} + 4q^{96} - 15q^{97} + q^{98} + 6q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5 x^{2} + 4 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 5 \nu^{2} - 5 \nu + 16$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 4$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4 \beta_{2} + \beta_{1} - 4$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3} - 4$$

## Character values

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

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$ $$1$$

## 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
 −0.780776 − 1.35234i 1.28078 + 2.21837i −0.780776 + 1.35234i 1.28078 − 2.21837i
−0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0 −0.500000 0.866025i −0.280776 0.486319i 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 1.78078 + 3.08440i 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 −0.280776 + 0.486319i 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 1.78078 3.08440i 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.z 4
5.b even 2 1 1950.2.i.bg yes 4
5.c odd 4 2 1950.2.z.m 8
13.c even 3 1 inner 1950.2.i.z 4
65.n even 6 1 1950.2.i.bg yes 4
65.q odd 12 2 1950.2.z.m 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.i.z 4 1.a even 1 1 trivial
1950.2.i.z 4 13.c even 3 1 inner
1950.2.i.bg yes 4 5.b even 2 1
1950.2.i.bg yes 4 65.n even 6 1
1950.2.z.m 8 5.c odd 4 2
1950.2.z.m 8 65.q odd 12 2

## Hecke kernels

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

 $$T_{7}^{4} - 3 T_{7}^{3} + 11 T_{7}^{2} + 6 T_{7} + 4$$ $$T_{11}^{4} + 3 T_{11}^{3} + 11 T_{11}^{2} - 6 T_{11} + 4$$ $$T_{17}^{4} + 2 T_{17}^{3} + 20 T_{17}^{2} - 32 T_{17} + 256$$ $$T_{19}^{4} - 9 T_{19}^{3} + 65 T_{19}^{2} - 144 T_{19} + 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$4 + 6 T + 11 T^{2} - 3 T^{3} + T^{4}$$
$11$ $$4 - 6 T + 11 T^{2} + 3 T^{3} + T^{4}$$
$13$ $$169 + 13 T - 12 T^{2} + T^{3} + T^{4}$$
$17$ $$256 - 32 T + 20 T^{2} + 2 T^{3} + T^{4}$$
$19$ $$256 - 144 T + 65 T^{2} - 9 T^{3} + T^{4}$$
$23$ $$1296 + 108 T + 45 T^{2} - 3 T^{3} + T^{4}$$
$29$ $$64 + 48 T + 44 T^{2} - 6 T^{3} + T^{4}$$
$31$ $$( -4 + T )^{4}$$
$37$ $$4 + 6 T + 11 T^{2} - 3 T^{3} + T^{4}$$
$41$ $$256 - 32 T + 20 T^{2} + 2 T^{3} + T^{4}$$
$43$ $$4 + 10 T + 23 T^{2} + 5 T^{3} + T^{4}$$
$47$ $$( -4 + T )^{4}$$
$53$ $$( -52 + 8 T + T^{2} )^{2}$$
$59$ $$4096 + 256 T + 80 T^{2} - 4 T^{3} + T^{4}$$
$61$ $$324 - 162 T + 99 T^{2} + 9 T^{3} + T^{4}$$
$67$ $$10816 - 312 T + 113 T^{2} + 3 T^{3} + T^{4}$$
$71$ $$11236 + 2226 T + 335 T^{2} + 21 T^{3} + T^{4}$$
$73$ $$( -9 + T )^{4}$$
$79$ $$( -76 + 11 T + T^{2} )^{2}$$
$83$ $$( 16 - 9 T + T^{2} )^{2}$$
$89$ $$4096 - 256 T + 80 T^{2} + 4 T^{3} + T^{4}$$
$97$ $$2500 - 750 T + 275 T^{2} + 15 T^{3} + T^{4}$$