# Properties

 Label 3800.2.a.z Level $3800$ Weight $2$ Character orbit 3800.a Self dual yes Analytic conductor $30.343$ Analytic rank $1$ 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: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.253565184.1 Defining polynomial: $$x^{6} - 2 x^{5} - 11 x^{4} + 20 x^{3} + 22 x^{2} - 32 x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 760) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( -1 - \beta_{1} - \beta_{3} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( -1 - \beta_{1} - \beta_{3} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} + ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{11} + ( -2 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{13} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{17} + q^{19} + ( 2 + 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{21} + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{23} + ( -\beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{27} + ( \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{29} + ( 2 - \beta_{2} - \beta_{4} + \beta_{5} ) q^{31} + ( -3 \beta_{2} - \beta_{4} + \beta_{5} ) q^{33} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{37} + ( -6 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{39} + ( \beta_{2} + \beta_{4} - \beta_{5} ) q^{41} + ( -1 + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{43} + ( -1 + \beta_{4} + \beta_{5} ) q^{47} + ( -1 + 3 \beta_{1} + \beta_{2} + \beta_{5} ) q^{49} + ( -4 + \beta_{1} - 4 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{51} + ( 2 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{53} -\beta_{1} q^{57} + ( \beta_{1} + \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{59} + ( 2 - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{61} + ( -5 - 4 \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{63} + ( -6 - \beta_{1} - 2 \beta_{2} ) q^{67} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{69} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{71} + ( -1 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{73} + ( -3 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{77} + ( -4 + 2 \beta_{1} ) q^{79} + ( -1 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{81} + ( -4 + 2 \beta_{2} + 2 \beta_{4} ) q^{83} + ( -8 + \beta_{1} + 2 \beta_{2} - \beta_{3} + 5 \beta_{4} + \beta_{5} ) q^{87} + ( 8 - 3 \beta_{2} + 4 \beta_{3} - \beta_{4} - \beta_{5} ) q^{89} + ( \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 5 \beta_{4} + \beta_{5} ) q^{91} + ( -2 - 2 \beta_{1} - \beta_{2} + 3 \beta_{4} + \beta_{5} ) q^{93} + ( -8 + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{97} + ( -6 + 2 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{3} - 6 q^{7} + 8 q^{9} + O(q^{10})$$ $$6 q - 2 q^{3} - 6 q^{7} + 8 q^{9} - 2 q^{11} - 14 q^{13} - 10 q^{17} + 6 q^{19} + 18 q^{21} - 2 q^{23} - 2 q^{27} - 2 q^{29} + 8 q^{31} - 8 q^{33} - 4 q^{37} - 18 q^{39} + 4 q^{41} - 4 q^{43} - 4 q^{47} + 2 q^{49} - 34 q^{51} + 14 q^{53} - 2 q^{57} - 2 q^{59} + 10 q^{61} - 44 q^{63} - 42 q^{67} + 18 q^{69} - 8 q^{71} + 2 q^{73} - 24 q^{77} - 20 q^{79} + 6 q^{81} - 16 q^{83} - 30 q^{87} + 32 q^{89} + 10 q^{91} - 12 q^{93} - 40 q^{97} - 22 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} - 11 x^{4} + 20 x^{3} + 22 x^{2} - 32 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 2 \nu^{4} + 11 \nu^{3} - 16 \nu^{2} - 26 \nu + 12$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} - 2 \nu^{4} - 11 \nu^{3} + 20 \nu^{2} + 22 \nu - 28$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} - \nu^{4} - 11 \nu^{3} + 9 \nu^{2} + 24 \nu - 10$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{5} + 4 \nu^{4} + 37 \nu^{3} - 38 \nu^{2} - 98 \nu + 48$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} + 7 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{4} + 7 \beta_{3} + 11 \beta_{2} + 9 \beta_{1} + 26$$ $$\nu^{5}$$ $$=$$ $$11 \beta_{5} + 15 \beta_{4} + 9 \beta_{3} + 2 \beta_{2} + 53 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.97875 2.43031 0.664406 0.366738 −1.59277 −2.84742
0 −2.97875 0 0 0 −4.30732 0 5.87292 0
1.2 0 −2.43031 0 0 0 −3.60737 0 2.90640 0
1.3 0 −0.664406 0 0 0 0.345799 0 −2.55856 0
1.4 0 −0.366738 0 0 0 3.08675 0 −2.86550 0
1.5 0 1.59277 0 0 0 −1.66290 0 −0.463073 0
1.6 0 2.84742 0 0 0 0.145034 0 5.10782 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.z 6
4.b odd 2 1 7600.2.a.cn 6
5.b even 2 1 3800.2.a.be 6
5.c odd 4 2 760.2.d.e 12
20.d odd 2 1 7600.2.a.cg 6
20.e even 4 2 1520.2.d.k 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.d.e 12 5.c odd 4 2
1520.2.d.k 12 20.e even 4 2
3800.2.a.z 6 1.a even 1 1 trivial
3800.2.a.be 6 5.b even 2 1
7600.2.a.cg 6 20.d odd 2 1
7600.2.a.cn 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3800))$$:

 $$T_{3}^{6} + 2 T_{3}^{5} - 11 T_{3}^{4} - 20 T_{3}^{3} + 22 T_{3}^{2} + 32 T_{3} + 8$$ $$T_{7}^{6} + 6 T_{7}^{5} - 4 T_{7}^{4} - 62 T_{7}^{3} - 49 T_{7}^{2} + 36 T_{7} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$8 + 32 T + 22 T^{2} - 20 T^{3} - 11 T^{4} + 2 T^{5} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$-4 + 36 T - 49 T^{2} - 62 T^{3} - 4 T^{4} + 6 T^{5} + T^{6}$$
$11$ $$-2584 + 456 T + 762 T^{2} - 76 T^{3} - 53 T^{4} + 2 T^{5} + T^{6}$$
$13$ $$9376 + 2080 T - 1270 T^{2} - 352 T^{3} + 27 T^{4} + 14 T^{5} + T^{6}$$
$17$ $$15864 + 6576 T - 715 T^{2} - 574 T^{3} - 34 T^{4} + 10 T^{5} + T^{6}$$
$19$ $$( -1 + T )^{6}$$
$23$ $$272 - 784 T + 608 T^{2} - 44 T^{3} - 61 T^{4} + 2 T^{5} + T^{6}$$
$29$ $$10064 + 8960 T + 1432 T^{2} - 392 T^{3} - 95 T^{4} + 2 T^{5} + T^{6}$$
$31$ $$-2304 - 3072 T + 464 T^{2} + 320 T^{3} - 38 T^{4} - 8 T^{5} + T^{6}$$
$37$ $$-12352 + 1600 T + 3640 T^{2} - 304 T^{3} - 122 T^{4} + 4 T^{5} + T^{6}$$
$41$ $$-4832 - 960 T + 1072 T^{2} + 144 T^{3} - 58 T^{4} - 4 T^{5} + T^{6}$$
$43$ $$-4632 - 5448 T + 3574 T^{2} - 148 T^{3} - 117 T^{4} + 4 T^{5} + T^{6}$$
$47$ $$-1152 + 960 T + 436 T^{2} - 124 T^{3} - 39 T^{4} + 4 T^{5} + T^{6}$$
$53$ $$2752 - 25712 T + 666 T^{2} + 1248 T^{3} - 69 T^{4} - 14 T^{5} + T^{6}$$
$59$ $$-374496 + 22752 T + 19924 T^{2} - 476 T^{3} - 265 T^{4} + 2 T^{5} + T^{6}$$
$61$ $$40256 - 44960 T + 3810 T^{2} + 1340 T^{3} - 133 T^{4} - 10 T^{5} + T^{6}$$
$67$ $$2752 + 20176 T + 16546 T^{2} + 4900 T^{3} + 665 T^{4} + 42 T^{5} + T^{6}$$
$71$ $$-64 + 128 T + 424 T^{2} - 104 T^{3} - 62 T^{4} + 8 T^{5} + T^{6}$$
$73$ $$3208 + 5192 T + 1061 T^{2} - 626 T^{3} - 162 T^{4} - 2 T^{5} + T^{6}$$
$79$ $$1024 - 768 T - 672 T^{2} + 96 T^{3} + 116 T^{4} + 20 T^{5} + T^{6}$$
$83$ $$6208 + 2560 T - 1488 T^{2} - 576 T^{3} + 12 T^{4} + 16 T^{5} + T^{6}$$
$89$ $$69504 + 81024 T - 42760 T^{2} + 4424 T^{3} + 122 T^{4} - 32 T^{5} + T^{6}$$
$97$ $$-196992 - 131328 T - 22968 T^{2} + 600 T^{3} + 478 T^{4} + 40 T^{5} + T^{6}$$