# Properties

 Label 3800.2.a.bb Level $3800$ Weight $2$ Character orbit 3800.a Self dual yes Analytic conductor $30.343$ Analytic rank $0$ Dimension $6$ 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: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 2 x^{5} - 12 x^{4} + 16 x^{3} + 33 x^{2} - 4 x - 7$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + \beta_{3} q^{7} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + \beta_{3} q^{7} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} + ( -\beta_{1} - \beta_{3} - \beta_{5} ) q^{11} -\beta_{4} q^{13} + ( \beta_{2} - \beta_{4} + \beta_{5} ) q^{17} + q^{19} + ( 1 - \beta_{3} - \beta_{4} ) q^{21} + ( -\beta_{2} + \beta_{3} ) q^{23} + ( -3 - 3 \beta_{1} - 2 \beta_{3} ) q^{27} + ( 2 \beta_{1} - \beta_{5} ) q^{29} + ( 2 - \beta_{1} + \beta_{3} + \beta_{4} ) q^{31} + ( 4 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{33} + ( 1 + \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} ) q^{37} + ( 3 + \beta_{1} + 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{39} + ( 2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{41} + ( -\beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{43} + ( -3 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{47} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{49} + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} ) q^{51} + ( 2 - \beta_{1} + \beta_{5} ) q^{53} -\beta_{1} q^{57} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{59} + ( 4 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{61} + ( 2 + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{63} + ( -4 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{67} + ( 4 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{69} + ( 4 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{71} + ( 4 - 2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{73} + ( -2 - 3 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{77} + ( 4 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{79} + ( 7 + 3 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{81} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{5} ) q^{83} + ( -7 - 4 \beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{87} + ( -2 - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{89} + ( 3 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{91} + ( 2 - 2 \beta_{1} + \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{93} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{97} + ( -6 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{3} - 2 q^{7} + 10 q^{9} + O(q^{10})$$ $$6 q - 2 q^{3} - 2 q^{7} + 10 q^{9} + 3 q^{11} + 3 q^{13} + 2 q^{17} + 6 q^{19} + 11 q^{21} - 4 q^{23} - 20 q^{27} + 7 q^{29} + 5 q^{31} + 16 q^{33} + 8 q^{39} + 11 q^{41} + 7 q^{43} - 20 q^{47} - 2 q^{49} + 13 q^{51} + 7 q^{53} - 2 q^{57} - 4 q^{59} + 13 q^{61} + q^{63} - 25 q^{67} + 7 q^{69} + 29 q^{71} + 19 q^{73} - 24 q^{77} + 28 q^{79} + 38 q^{81} + 15 q^{83} - 57 q^{87} - 12 q^{89} + 27 q^{93} + 13 q^{97} + 10 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} - 12 x^{4} + 16 x^{3} + 33 x^{2} - 4 x - 7$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - 9 \nu - 3$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} - \nu^{3} - 9 \nu^{2} + 6 \nu + 5$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{5} - 2 \nu^{4} - 11 \nu^{3} + 15 \nu^{2} + 24 \nu - 2$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 4$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3} + 9 \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{4} + 2 \beta_{3} + 9 \beta_{2} + 12 \beta_{1} + 34$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{5} + 4 \beta_{4} + 26 \beta_{3} + 3 \beta_{2} + 84 \beta_{1} + 43$$

## 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
 3.30105 2.70452 0.486697 −0.471016 −1.22174 −2.79951
0 −3.30105 0 0 0 1.63094 0 7.89694 0
1.2 0 −2.70452 0 0 0 −3.77934 0 4.31442 0
1.3 0 −0.486697 0 0 0 −3.63249 0 −2.76313 0
1.4 0 0.471016 0 0 0 0.567324 0 −2.77814 0
1.5 0 1.22174 0 0 0 3.08602 0 −1.50735 0
1.6 0 2.79951 0 0 0 0.127550 0 4.83726 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

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.bb 6
4.b odd 2 1 7600.2.a.cm 6
5.b even 2 1 3800.2.a.bd yes 6
5.c odd 4 2 3800.2.d.p 12
20.d odd 2 1 7600.2.a.ci 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.2.a.bb 6 1.a even 1 1 trivial
3800.2.a.bd yes 6 5.b even 2 1
3800.2.d.p 12 5.c odd 4 2
7600.2.a.ci 6 20.d odd 2 1
7600.2.a.cm 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3800))$$:

 $$T_{3}^{6} + 2 T_{3}^{5} - 12 T_{3}^{4} - 16 T_{3}^{3} + 33 T_{3}^{2} + 4 T_{3} - 7$$ $$T_{7}^{6} + 2 T_{7}^{5} - 18 T_{7}^{4} - 16 T_{7}^{3} + 87 T_{7}^{2} - 50 T_{7} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$-7 + 4 T + 33 T^{2} - 16 T^{3} - 12 T^{4} + 2 T^{5} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$5 - 50 T + 87 T^{2} - 16 T^{3} - 18 T^{4} + 2 T^{5} + T^{6}$$
$11$ $$-5400 - 1080 T + 906 T^{2} + 115 T^{3} - 52 T^{4} - 3 T^{5} + T^{6}$$
$13$ $$37 + 303 T + 491 T^{2} + 61 T^{3} - 43 T^{4} - 3 T^{5} + T^{6}$$
$17$ $$-745 - 624 T + 303 T^{2} + 182 T^{3} - 62 T^{4} - 2 T^{5} + T^{6}$$
$19$ $$( -1 + T )^{6}$$
$23$ $$-587 + 714 T + 469 T^{2} - 114 T^{3} - 42 T^{4} + 4 T^{5} + T^{6}$$
$29$ $$-14717 - 3169 T + 1957 T^{2} + 335 T^{3} - 73 T^{4} - 7 T^{5} + T^{6}$$
$31$ $$-296 + 1228 T - 1528 T^{2} + 609 T^{3} - 70 T^{4} - 5 T^{5} + T^{6}$$
$37$ $$-111157 + 19221 T + 11924 T^{2} - 189 T^{3} - 208 T^{4} + T^{6}$$
$41$ $$-3584 - 960 T + 3136 T^{2} + 639 T^{3} - 84 T^{4} - 11 T^{5} + T^{6}$$
$43$ $$-79576 - 18112 T + 12686 T^{2} + 975 T^{3} - 215 T^{4} - 7 T^{5} + T^{6}$$
$47$ $$-90017 - 72555 T - 21020 T^{2} - 2337 T^{3} + 12 T^{4} + 20 T^{5} + T^{6}$$
$53$ $$-1187 - 714 T + 358 T^{2} + 181 T^{3} - 32 T^{4} - 7 T^{5} + T^{6}$$
$59$ $$-11944 + 4496 T + 1926 T^{2} - 355 T^{3} - 93 T^{4} + 4 T^{5} + T^{6}$$
$61$ $$-3880 - 35240 T + 3438 T^{2} + 1547 T^{3} - 120 T^{4} - 13 T^{5} + T^{6}$$
$67$ $$14207 - 26469 T - 14043 T^{2} - 1603 T^{3} + 103 T^{4} + 25 T^{5} + T^{6}$$
$71$ $$-39960 + 71064 T - 22974 T^{2} + 1769 T^{3} + 177 T^{4} - 29 T^{5} + T^{6}$$
$73$ $$1723 - 2797 T - 1207 T^{2} + 377 T^{3} + 67 T^{4} - 19 T^{5} + T^{6}$$
$79$ $$15040 - 13216 T - 6260 T^{2} + 689 T^{3} + 185 T^{4} - 28 T^{5} + T^{6}$$
$83$ $$-200 + 1292 T - 2100 T^{2} + 627 T^{3} - 2 T^{4} - 15 T^{5} + T^{6}$$
$89$ $$-274808 + 88364 T + 14514 T^{2} - 2179 T^{3} - 217 T^{4} + 12 T^{5} + T^{6}$$
$97$ $$1784 - 3556 T - 8524 T^{2} + 2517 T^{3} - 134 T^{4} - 13 T^{5} + T^{6}$$