# Properties

 Label 3800.2.a.r Level $3800$ Weight $2$ Character orbit 3800.a Self dual yes Analytic conductor $30.343$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$30.3431527681$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.961.1 Defining polynomial: $$x^{3} - x^{2} - 10 x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( -2 + \beta_{1} - \beta_{2} ) q^{7} + ( 5 - \beta_{1} + 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( -2 + \beta_{1} - \beta_{2} ) q^{7} + ( 5 - \beta_{1} + 2 \beta_{2} ) q^{9} + ( -2 - \beta_{2} ) q^{11} + ( -2 + \beta_{1} ) q^{13} + ( -\beta_{1} + \beta_{2} ) q^{17} - q^{19} + ( -4 + 3 \beta_{1} ) q^{21} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{23} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{27} + ( -2 - \beta_{1} + 2 \beta_{2} ) q^{29} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{33} + 2 q^{37} + ( -8 + 3 \beta_{1} - 2 \beta_{2} ) q^{39} + ( 2 + 2 \beta_{1} ) q^{41} + ( -6 + 2 \beta_{1} + \beta_{2} ) q^{43} + ( 2 - 2 \beta_{1} + 3 \beta_{2} ) q^{47} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{49} + ( 4 - \beta_{1} ) q^{51} + ( -2 + \beta_{1} - 4 \beta_{2} ) q^{53} + \beta_{1} q^{57} + ( -8 + \beta_{1} ) q^{59} + ( 2 \beta_{1} - \beta_{2} ) q^{61} + ( -18 + 4 \beta_{1} - 3 \beta_{2} ) q^{63} + ( -4 - \beta_{1} + 2 \beta_{2} ) q^{67} + ( -16 + 3 \beta_{1} - 2 \beta_{2} ) q^{69} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( -3 \beta_{1} + \beta_{2} ) q^{73} + ( 6 + \beta_{2} ) q^{77} + ( 8 + 2 \beta_{1} ) q^{79} + ( 17 + 4 \beta_{2} ) q^{81} + ( 4 - 2 \beta_{1} + 4 \beta_{2} ) q^{83} + ( \beta_{1} - 2 \beta_{2} ) q^{87} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 8 - 5 \beta_{1} + 2 \beta_{2} ) q^{91} + ( 2 - 2 \beta_{2} ) q^{97} + ( -18 - 2 \beta_{1} - 5 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} - 4 q^{7} + 12 q^{9} + O(q^{10})$$ $$3 q - q^{3} - 4 q^{7} + 12 q^{9} - 5 q^{11} - 5 q^{13} - 2 q^{17} - 3 q^{19} - 9 q^{21} + 5 q^{23} - q^{27} - 9 q^{29} + 12 q^{33} + 6 q^{37} - 19 q^{39} + 8 q^{41} - 17 q^{43} + q^{47} + 5 q^{49} + 11 q^{51} - q^{53} + q^{57} - 23 q^{59} + 3 q^{61} - 47 q^{63} - 15 q^{67} - 43 q^{69} - 12 q^{71} - 4 q^{73} + 17 q^{77} + 26 q^{79} + 47 q^{81} + 6 q^{83} + 3 q^{87} + 18 q^{89} + 17 q^{91} + 8 q^{97} - 51 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 10 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} + \nu - 8$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2} - \beta_{1} + 8$$

## 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
 3.29707 0.786802 −3.08387
0 −3.29707 0 0 0 −1.78680 0 7.87067 0
1.2 0 −0.786802 0 0 0 2.08387 0 −2.38094 0
1.3 0 3.08387 0 0 0 −4.29707 0 6.51027 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.2.a.r 3
4.b odd 2 1 7600.2.a.bv 3
5.b even 2 1 152.2.a.c 3
5.c odd 4 2 3800.2.d.j 6
15.d odd 2 1 1368.2.a.n 3
20.d odd 2 1 304.2.a.g 3
35.c odd 2 1 7448.2.a.bf 3
40.e odd 2 1 1216.2.a.v 3
40.f even 2 1 1216.2.a.u 3
60.h even 2 1 2736.2.a.bd 3
95.d odd 2 1 2888.2.a.o 3
380.d even 2 1 5776.2.a.bp 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.a.c 3 5.b even 2 1
304.2.a.g 3 20.d odd 2 1
1216.2.a.u 3 40.f even 2 1
1216.2.a.v 3 40.e odd 2 1
1368.2.a.n 3 15.d odd 2 1
2736.2.a.bd 3 60.h even 2 1
2888.2.a.o 3 95.d odd 2 1
3800.2.a.r 3 1.a even 1 1 trivial
3800.2.d.j 6 5.c odd 4 2
5776.2.a.bp 3 380.d even 2 1
7448.2.a.bf 3 35.c odd 2 1
7600.2.a.bv 3 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3800))$$:

 $$T_{3}^{3} + T_{3}^{2} - 10 T_{3} - 8$$ $$T_{7}^{3} + 4 T_{7}^{2} - 5 T_{7} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-8 - 10 T + T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$-16 - 5 T + 4 T^{2} + T^{3}$$
$11$ $$-8 - 2 T + 5 T^{2} + T^{3}$$
$13$ $$-8 - 2 T + 5 T^{2} + T^{3}$$
$17$ $$-2 - 9 T + 2 T^{2} + T^{3}$$
$19$ $$( 1 + T )^{3}$$
$23$ $$256 - 64 T - 5 T^{2} + T^{3}$$
$29$ $$-4 - 4 T + 9 T^{2} + T^{3}$$
$31$ $$T^{3}$$
$37$ $$( -2 + T )^{3}$$
$41$ $$128 - 20 T - 8 T^{2} + T^{3}$$
$43$ $$-368 + 24 T + 17 T^{2} + T^{3}$$
$47$ $$256 - 72 T - T^{2} + T^{3}$$
$53$ $$-256 - 134 T + T^{2} + T^{3}$$
$59$ $$376 + 166 T + 23 T^{2} + T^{3}$$
$61$ $$92 - 28 T - 3 T^{2} + T^{3}$$
$67$ $$32 + 44 T + 15 T^{2} + T^{3}$$
$71$ $$-928 - 76 T + 12 T^{2} + T^{3}$$
$73$ $$-326 - 67 T + 4 T^{2} + T^{3}$$
$79$ $$-256 + 184 T - 26 T^{2} + T^{3}$$
$83$ $$736 - 112 T - 6 T^{2} + T^{3}$$
$89$ $$1024 - 16 T - 18 T^{2} + T^{3}$$
$97$ $$128 - 20 T - 8 T^{2} + T^{3}$$