# Properties

 Label 1950.2.f.p Level $1950$ Weight $2$ Character orbit 1950.f 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.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9 x^{2} + 16$$:

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

## Character values

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

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 1.56155i − 2.56155i − 1.56155i 2.56155i
1.00000 1.00000i 1.00000 0 1.00000i 0.438447 1.00000 −1.00000 0
649.2 1.00000 1.00000i 1.00000 0 1.00000i 4.56155 1.00000 −1.00000 0
649.3 1.00000 1.00000i 1.00000 0 1.00000i 0.438447 1.00000 −1.00000 0
649.4 1.00000 1.00000i 1.00000 0 1.00000i 4.56155 1.00000 −1.00000 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
65.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.f.p 4
5.b even 2 1 1950.2.f.k 4
5.c odd 4 1 1950.2.b.i 4
5.c odd 4 1 1950.2.b.j yes 4
13.b even 2 1 1950.2.f.k 4
65.d even 2 1 inner 1950.2.f.p 4
65.h odd 4 1 1950.2.b.i 4
65.h odd 4 1 1950.2.b.j yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.b.i 4 5.c odd 4 1
1950.2.b.i 4 65.h odd 4 1
1950.2.b.j yes 4 5.c odd 4 1
1950.2.b.j yes 4 65.h odd 4 1
1950.2.f.k 4 5.b even 2 1
1950.2.f.k 4 13.b even 2 1
1950.2.f.p 4 1.a even 1 1 trivial
1950.2.f.p 4 65.d even 2 1 inner

## Hecke kernels

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

 $$T_{7}^{2} - 5 T_{7} + 2$$ $$T_{11}^{4} + 9 T_{11}^{2} + 16$$ $$T_{19}^{4} + 81 T_{19}^{2} + 1296$$ $$T_{37}^{2} - 5 T_{37} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{4}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 2 - 5 T + T^{2} )^{2}$$
$11$ $$16 + 9 T^{2} + T^{4}$$
$13$ $$169 + 78 T + 18 T^{2} + 6 T^{3} + T^{4}$$
$17$ $$1444 + 77 T^{2} + T^{4}$$
$19$ $$1296 + 81 T^{2} + T^{4}$$
$23$ $$256 + 36 T^{2} + T^{4}$$
$29$ $$( -1 + T )^{4}$$
$31$ $$64 + 33 T^{2} + T^{4}$$
$37$ $$( 2 - 5 T + T^{2} )^{2}$$
$41$ $$4 + 21 T^{2} + T^{4}$$
$43$ $$1024 + 132 T^{2} + T^{4}$$
$47$ $$( -1 - 8 T + T^{2} )^{2}$$
$53$ $$( 25 + T^{2} )^{2}$$
$59$ $$2704 + 121 T^{2} + T^{4}$$
$61$ $$( -8 - 11 T + T^{2} )^{2}$$
$67$ $$( -17 + T^{2} )^{2}$$
$71$ $$43264 + 417 T^{2} + T^{4}$$
$73$ $$( -8 - 6 T + T^{2} )^{2}$$
$79$ $$( 34 - 17 T + T^{2} )^{2}$$
$83$ $$( -208 - T + T^{2} )^{2}$$
$89$ $$256 + 36 T^{2} + T^{4}$$
$97$ $$( 64 + 18 T + T^{2} )^{2}$$