# Properties

 Label 1900.2.a.k Level $1900$ Weight $2$ Character orbit 1900.a Self dual yes 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.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.1715763840$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.56310016.1 Defining polynomial: $$x^{6} - 9 x^{4} + 14 x^{2} - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 380) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} -\beta_{4} q^{7} + ( 2 - \beta_{1} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} -\beta_{4} q^{7} + ( 2 - \beta_{1} + \beta_{5} ) q^{9} + ( 3 - \beta_{1} ) q^{11} + \beta_{2} q^{13} + ( -\beta_{2} + \beta_{3} - \beta_{4} ) q^{17} + q^{19} -2 \beta_{5} q^{21} + ( \beta_{2} - \beta_{3} ) q^{23} + ( -\beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{27} + ( 2 \beta_{1} + 2 \beta_{5} ) q^{29} + ( 2 + 2 \beta_{1} + 2 \beta_{5} ) q^{31} + ( -\beta_{2} + 5 \beta_{3} ) q^{33} + \beta_{3} q^{37} + ( -1 + \beta_{1} - \beta_{5} ) q^{39} + ( 2 \beta_{1} - 2 \beta_{5} ) q^{41} + ( -2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{43} + ( 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{47} + ( 1 - 2 \beta_{1} - 3 \beta_{5} ) q^{49} + ( 6 - 2 \beta_{1} ) q^{51} + ( 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{53} + \beta_{3} q^{57} + ( 4 - 2 \beta_{5} ) q^{59} + ( 5 - \beta_{1} ) q^{61} + ( -4 \beta_{3} - \beta_{4} ) q^{63} + ( \beta_{2} + 2 \beta_{3} + 4 \beta_{4} ) q^{67} + ( -6 + 2 \beta_{1} - 2 \beta_{5} ) q^{69} + ( 8 + 2 \beta_{5} ) q^{71} + ( -3 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{73} + ( -\beta_{2} - \beta_{3} - 5 \beta_{4} ) q^{77} + 4 q^{79} + ( 10 - \beta_{1} + 5 \beta_{5} ) q^{81} + ( -\beta_{2} + \beta_{3} ) q^{83} + ( 2 \beta_{2} + 4 \beta_{4} ) q^{87} + ( 2 + 2 \beta_{5} ) q^{89} + 4 q^{91} + ( 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} ) q^{93} + ( 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{97} + ( 17 - 3 \beta_{1} + 6 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 10q^{9} + O(q^{10})$$ $$6q + 10q^{9} + 18q^{11} + 6q^{19} + 4q^{21} - 4q^{29} + 8q^{31} - 4q^{39} + 4q^{41} + 12q^{49} + 36q^{51} + 28q^{59} + 30q^{61} - 32q^{69} + 44q^{71} + 24q^{79} + 50q^{81} + 8q^{89} + 24q^{91} + 90q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 9 x^{4} + 14 x^{2} - 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 9 \nu^{3} - 10 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} - 9 \nu^{3} + 14 \nu$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$\nu^{5} - 8 \nu^{3} + 6 \nu$$ $$\beta_{5}$$ $$=$$ $$\nu^{4} - 8 \nu^{2} + 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} + 2 \beta_{3} + 4 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$\beta_{5} + 8 \beta_{1} + 18$$ $$\nu^{5}$$ $$=$$ $$9 \beta_{4} + 13 \beta_{3} + 29 \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.608430 −1.23277 −2.66648 2.66648 1.23277 0.608430
0 −3.28715 0 0 0 1.93210 0 7.80536 0
1.2 0 −1.62236 0 0 0 −4.74397 0 −0.367938 0
1.3 0 −0.750054 0 0 0 −0.872810 0 −2.43742 0
1.4 0 0.750054 0 0 0 0.872810 0 −2.43742 0
1.5 0 1.62236 0 0 0 4.74397 0 −0.367938 0
1.6 0 3.28715 0 0 0 −1.93210 0 7.80536 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$19$$ $$-1$$

## 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.a.k 6
4.b odd 2 1 7600.2.a.cj 6
5.b even 2 1 inner 1900.2.a.k 6
5.c odd 4 2 380.2.c.b 6
15.e even 4 2 3420.2.f.c 6
20.d odd 2 1 7600.2.a.cj 6
20.e even 4 2 1520.2.d.i 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.c.b 6 5.c odd 4 2
1520.2.d.i 6 20.e even 4 2
1900.2.a.k 6 1.a even 1 1 trivial
1900.2.a.k 6 5.b even 2 1 inner
3420.2.f.c 6 15.e even 4 2
7600.2.a.cj 6 4.b odd 2 1
7600.2.a.cj 6 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1900))$$:

 $$T_{3}^{6} - 14 T_{3}^{4} + 36 T_{3}^{2} - 16$$ $$T_{7}^{6} - 27 T_{7}^{4} + 104 T_{7}^{2} - 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$-16 + 36 T^{2} - 14 T^{4} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$-64 + 104 T^{2} - 27 T^{4} + T^{6}$$
$11$ $$( 28 + 14 T - 9 T^{2} + T^{3} )^{2}$$
$13$ $$-64 + 108 T^{2} - 26 T^{4} + T^{6}$$
$17$ $$-3136 + 728 T^{2} - 51 T^{4} + T^{6}$$
$19$ $$( -1 + T )^{6}$$
$23$ $$-256 + 448 T^{2} - 44 T^{4} + T^{6}$$
$29$ $$( 88 - 84 T + 2 T^{2} + T^{3} )^{2}$$
$31$ $$( 256 - 80 T - 4 T^{2} + T^{3} )^{2}$$
$37$ $$-16 + 36 T^{2} - 14 T^{4} + T^{6}$$
$41$ $$( 488 - 116 T - 2 T^{2} + T^{3} )^{2}$$
$43$ $$-3136 + 12520 T^{2} - 227 T^{4} + T^{6}$$
$47$ $$-118336 + 14024 T^{2} - 243 T^{4} + T^{6}$$
$53$ $$-23104 + 3500 T^{2} - 114 T^{4} + T^{6}$$
$59$ $$( 128 + 16 T - 14 T^{2} + T^{3} )^{2}$$
$61$ $$( -44 + 62 T - 15 T^{2} + T^{3} )^{2}$$
$67$ $$-222784 + 35660 T^{2} - 378 T^{4} + T^{6}$$
$71$ $$( -32 + 112 T - 22 T^{2} + T^{3} )^{2}$$
$73$ $$-118336 + 16344 T^{2} - 251 T^{4} + T^{6}$$
$79$ $$( -4 + T )^{6}$$
$83$ $$-256 + 448 T^{2} - 44 T^{4} + T^{6}$$
$89$ $$( 64 - 44 T - 4 T^{2} + T^{3} )^{2}$$
$97$ $$-23104 + 3500 T^{2} - 114 T^{4} + T^{6}$$