# Properties

 Label 1900.2.c.g Level $1900$ Weight $2$ Character orbit 1900.c Analytic conductor $15.172$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1900.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.1715763840$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.4227136.2 Defining polynomial: $$x^{6} + 9 x^{4} + 22 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$5^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + \beta_{1} q^{7} + \beta_{5} q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + \beta_{1} q^{7} + \beta_{5} q^{9} -\beta_{2} q^{11} + ( \beta_{1} - \beta_{3} ) q^{13} + \beta_{1} q^{17} + q^{19} + ( 3 - \beta_{5} ) q^{21} + ( \beta_{1} + \beta_{4} ) q^{23} + ( 2 \beta_{1} - \beta_{3} - \beta_{4} ) q^{27} + ( 2 + \beta_{2} ) q^{29} + ( 3 + \beta_{2} - \beta_{5} ) q^{31} + \beta_{4} q^{33} + ( -2 \beta_{1} + 3 \beta_{3} + \beta_{4} ) q^{37} + ( 2 - \beta_{5} ) q^{39} + ( 4 - \beta_{5} ) q^{41} + ( \beta_{1} - \beta_{3} + \beta_{4} ) q^{43} + ( \beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{47} + ( 4 + \beta_{5} ) q^{49} + ( 3 - \beta_{5} ) q^{51} + ( \beta_{1} - \beta_{3} + \beta_{4} ) q^{53} -\beta_{1} q^{57} + ( 2 - 2 \beta_{2} - 2 \beta_{5} ) q^{59} + ( 3 - \beta_{2} + \beta_{5} ) q^{61} + ( -5 \beta_{1} + \beta_{3} + \beta_{4} ) q^{63} + ( 2 \beta_{1} + 3 \beta_{3} + \beta_{4} ) q^{67} + ( 4 + \beta_{2} ) q^{69} + ( -1 + 2 \beta_{5} ) q^{71} -3 \beta_{3} q^{73} -\beta_{4} q^{77} + ( 5 + \beta_{2} - 2 \beta_{5} ) q^{79} + ( 4 - \beta_{2} ) q^{81} + ( 6 \beta_{1} - 5 \beta_{3} ) q^{83} + ( -2 \beta_{1} - \beta_{4} ) q^{87} + ( 1 + 3 \beta_{5} ) q^{89} + ( -2 + \beta_{5} ) q^{91} + ( -8 \beta_{1} + \beta_{3} ) q^{93} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{97} + ( 1 - 2 \beta_{2} + \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 2q^{9} + O(q^{10})$$ $$6q - 2q^{9} - 2q^{11} + 6q^{19} + 20q^{21} + 14q^{29} + 22q^{31} + 14q^{39} + 26q^{41} + 22q^{49} + 20q^{51} + 12q^{59} + 14q^{61} + 26q^{69} - 10q^{71} + 36q^{79} + 22q^{81} - 14q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 9 x^{4} + 22 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 6 \nu^{3} + 4 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$\nu^{4} + 3 \nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 9 \nu^{3} + 19 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$-\nu^{5} - 5 \nu^{3} - 4 \nu$$ $$\beta_{5}$$ $$=$$ $$2 \nu^{4} + 11 \nu^{2} + 8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{4} + \beta_{3} - 4 \beta_{1}$$$$)/5$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} - 2 \beta_{2} - 14$$$$)/5$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} + 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$($$$$-3 \beta_{5} + 11 \beta_{2} + 57$$$$)/5$$ $$\nu^{5}$$ $$=$$ $$($$$$-26 \beta_{4} - 4 \beta_{3} - 59 \beta_{1}$$$$)/5$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1749.1
 − 1.91223i − 2.19869i 0.713538i − 0.713538i 2.19869i 1.91223i
0 2.91223i 0 0 0 2.91223i 0 −5.48108 0
1749.2 0 1.19869i 0 0 0 1.19869i 0 1.56314 0
1749.3 0 0.286462i 0 0 0 0.286462i 0 2.91794 0
1749.4 0 0.286462i 0 0 0 0.286462i 0 2.91794 0
1749.5 0 1.19869i 0 0 0 1.19869i 0 1.56314 0
1749.6 0 2.91223i 0 0 0 2.91223i 0 −5.48108 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1749.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.2.c.g 6
5.b even 2 1 inner 1900.2.c.g 6
5.c odd 4 1 1900.2.a.f 3
5.c odd 4 1 1900.2.a.h yes 3
20.e even 4 1 7600.2.a.bj 3
20.e even 4 1 7600.2.a.by 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.2.a.f 3 5.c odd 4 1
1900.2.a.h yes 3 5.c odd 4 1
1900.2.c.g 6 1.a even 1 1 trivial
1900.2.c.g 6 5.b even 2 1 inner
7600.2.a.bj 3 20.e even 4 1
7600.2.a.by 3 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1900, [\chi])$$:

 $$T_{3}^{6} + 10 T_{3}^{4} + 13 T_{3}^{2} + 1$$ $$T_{7}^{6} + 10 T_{7}^{4} + 13 T_{7}^{2} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$1 + 13 T^{2} + 10 T^{4} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$1 + 13 T^{2} + 10 T^{4} + T^{6}$$
$11$ $$( 15 - 26 T + T^{2} + T^{3} )^{2}$$
$13$ $$9 + 70 T^{2} + 17 T^{4} + T^{6}$$
$17$ $$1 + 13 T^{2} + 10 T^{4} + T^{6}$$
$19$ $$( -1 + T )^{6}$$
$23$ $$81 + 153 T^{2} + 70 T^{4} + T^{6}$$
$29$ $$( 25 - 10 T - 7 T^{2} + T^{3} )^{2}$$
$31$ $$( 241 - 6 T - 11 T^{2} + T^{3} )^{2}$$
$37$ $$164025 + 9234 T^{2} + 169 T^{4} + T^{6}$$
$41$ $$( -25 + 36 T - 13 T^{2} + T^{3} )^{2}$$
$43$ $$729 + 1123 T^{2} + 75 T^{4} + T^{6}$$
$47$ $$3969 + 7018 T^{2} + 225 T^{4} + T^{6}$$
$53$ $$729 + 1123 T^{2} + 75 T^{4} + T^{6}$$
$59$ $$( 856 - 176 T - 6 T^{2} + T^{3} )^{2}$$
$61$ $$( -25 - 30 T - 7 T^{2} + T^{3} )^{2}$$
$67$ $$358801 + 19978 T^{2} + 265 T^{4} + T^{6}$$
$71$ $$( 123 - 73 T + 5 T^{2} + T^{3} )^{2}$$
$73$ $$729 + 810 T^{2} + 117 T^{4} + T^{6}$$
$79$ $$( 607 + T - 18 T^{2} + T^{3} )^{2}$$
$83$ $$561001 + 65998 T^{2} + 505 T^{4} + T^{6}$$
$89$ $$( 857 - 183 T + T^{3} )^{2}$$
$97$ $$6561 + 5670 T^{2} + 153 T^{4} + T^{6}$$