# Properties

 Label 1900.3.e.d Level $1900$ Weight $3$ Character orbit 1900.e Self dual yes Analytic conductor $51.771$ Analytic rank $0$ Dimension $4$ CM discriminant -19 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1900.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$51.7712502285$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{19})$$ Defining polynomial: $$x^{4} - 11 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{3}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 380) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{7} + 9 q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{7} + 9 q^{9} + ( 1 - \beta_{3} ) q^{11} + ( \beta_{1} + \beta_{2} ) q^{17} + 19 q^{19} + ( -4 \beta_{1} - \beta_{2} ) q^{23} + ( 5 \beta_{1} - 3 \beta_{2} ) q^{43} + ( -7 \beta_{1} + 2 \beta_{2} ) q^{47} + ( 14 + 3 \beta_{3} ) q^{49} + ( 53 + 3 \beta_{3} ) q^{61} -9 \beta_{1} q^{63} + ( 7 \beta_{1} + 3 \beta_{2} ) q^{73} + ( 15 \beta_{1} - 3 \beta_{2} ) q^{77} + 81 q^{81} + ( 16 \beta_{1} + 4 \beta_{2} ) q^{83} + ( 9 - 9 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 36 q^{9} + O(q^{10})$$ $$4 q + 36 q^{9} + 6 q^{11} + 76 q^{19} + 50 q^{49} + 206 q^{61} + 324 q^{81} + 54 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 11 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$-3 \nu^{3} + 25 \nu$$ $$\beta_{3}$$ $$=$$ $$5 \nu^{2} - 28$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 12 \beta_{1}$$$$)/40$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 28$$$$)/5$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{2} + 20 \beta_{1}$$$$)/8$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1101.1
 3.04547 1.31342 −1.31342 −3.04547
0 0 0 0 0 −10.8685 0 9.00000 0
1101.2 0 0 0 0 0 −2.20822 0 9.00000 0
1101.3 0 0 0 0 0 2.20822 0 9.00000 0
1101.4 0 0 0 0 0 10.8685 0 9.00000 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
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
5.b even 2 1 inner
95.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.3.e.d 4
5.b even 2 1 inner 1900.3.e.d 4
5.c odd 4 2 380.3.g.b 4
15.e even 4 2 3420.3.h.a 4
19.b odd 2 1 CM 1900.3.e.d 4
95.d odd 2 1 inner 1900.3.e.d 4
95.g even 4 2 380.3.g.b 4
285.j odd 4 2 3420.3.h.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.3.g.b 4 5.c odd 4 2
380.3.g.b 4 95.g even 4 2
1900.3.e.d 4 1.a even 1 1 trivial
1900.3.e.d 4 5.b even 2 1 inner
1900.3.e.d 4 19.b odd 2 1 CM
1900.3.e.d 4 95.d odd 2 1 inner
3420.3.h.a 4 15.e even 4 2
3420.3.h.a 4 285.j odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1900, [\chi])$$:

 $$T_{3}$$ $$T_{7}^{4} - 123 T_{7}^{2} + 576$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$576 - 123 T^{2} + T^{4}$$
$11$ $$( -354 - 3 T + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$4096 - 803 T^{2} + T^{4}$$
$19$ $$( -19 + T )^{4}$$
$23$ $$( -1216 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$28901376 - 10923 T^{2} + T^{4}$$
$47$ $$11669056 - 10043 T^{2} + T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( -554 - 103 T + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$22127616 - 11283 T^{2} + T^{4}$$
$79$ $$T^{4}$$
$83$ $$( -19456 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$