# Properties

 Label 69.2.c.a Level $69$ Weight $2$ Character orbit 69.c Analytic conductor $0.551$ Analytic rank $0$ Dimension $6$ CM discriminant -23 Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$69 = 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 69.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.550967773947$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.8869743.1 Defining polynomial: $$x^{6} - 3 x^{3} + 8$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} -\beta_{1} q^{3} + ( -2 + \beta_{1} + \beta_{3} ) q^{4} + ( 1 + \beta_{2} + \beta_{4} ) q^{6} + ( -1 - 2 \beta_{2} - \beta_{4} + \beta_{5} ) q^{8} + ( -\beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} -\beta_{1} q^{3} + ( -2 + \beta_{1} + \beta_{3} ) q^{4} + ( 1 + \beta_{2} + \beta_{4} ) q^{6} + ( -1 - 2 \beta_{2} - \beta_{4} + \beta_{5} ) q^{8} + ( -\beta_{2} - \beta_{3} ) q^{9} + ( -2 + 2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{12} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{13} + ( 4 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{16} + ( 4 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} ) q^{18} + ( 1 + \beta_{4} - \beta_{5} ) q^{23} + ( -2 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{24} -5 q^{25} + ( -1 - 3 \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{26} + ( 1 - \beta_{4} + \beta_{5} ) q^{27} + ( 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{29} + ( \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{31} + ( 2 + 3 \beta_{1} + 6 \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{32} + ( -5 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{5} ) q^{36} + ( -5 + 3 \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{39} + ( -3 \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{41} + ( 4 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{46} + ( -3 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{47} + ( 11 - 4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + \beta_{5} ) q^{48} + 7 q^{49} -5 \beta_{2} q^{50} + ( -1 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{52} + ( -4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{54} + ( 5 + \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{58} + ( 2 + 2 \beta_{4} - 2 \beta_{5} ) q^{59} + ( -1 + 3 \beta_{1} - 4 \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{62} + ( -15 + 8 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{64} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{69} + ( 3 \beta_{1} + 5 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{71} + ( -8 + 2 \beta_{1} - 7 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{72} + ( -5 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{73} + 5 \beta_{1} q^{75} + ( -8 + \beta_{1} - 4 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{78} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{81} + ( 11 - 5 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{82} + ( 7 + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{87} + ( -2 - 3 \beta_{1} - 6 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{92} + ( 1 - 3 \beta_{2} + \beta_{3} - 4 \beta_{5} ) q^{93} + ( -13 + \beta_{1} + 2 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{94} + ( 14 - 2 \beta_{1} + 12 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{96} + 7 \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 12q^{4} + 3q^{6} + O(q^{10})$$ $$6q - 12q^{4} + 3q^{6} - 15q^{12} + 24q^{16} + 21q^{18} - 6q^{24} - 30q^{25} + 12q^{27} - 33q^{36} - 24q^{39} + 69q^{48} + 42q^{49} - 6q^{52} + 30q^{58} - 90q^{64} - 42q^{72} - 51q^{78} + 66q^{82} + 48q^{87} - 6q^{93} - 78q^{94} + 69q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{3} + 8$$:

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$$.

 $$n$$ $$28$$ $$47$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
68.1
 −0.261988 − 1.38973i −1.07255 − 0.921756i 1.33454 − 0.467979i 1.33454 + 0.467979i −1.07255 + 0.921756i −0.261988 + 1.38973i
2.77947i 1.60074 + 0.661546i −5.72545 0 1.83875 4.44920i 0 10.3548i 2.12471 + 2.11792i 0
68.2 1.84351i −1.37328 1.05550i −1.39853 0 −1.94584 + 2.53166i 0 1.10881i 0.771819 + 2.89902i 0
68.3 0.935958i −0.227452 + 1.71705i 1.12398 0 1.60709 + 0.212885i 0 2.92392i −2.89653 0.781094i 0
68.4 0.935958i −0.227452 1.71705i 1.12398 0 1.60709 0.212885i 0 2.92392i −2.89653 + 0.781094i 0
68.5 1.84351i −1.37328 + 1.05550i −1.39853 0 −1.94584 2.53166i 0 1.10881i 0.771819 2.89902i 0
68.6 2.77947i 1.60074 0.661546i −5.72545 0 1.83875 + 4.44920i 0 10.3548i 2.12471 2.11792i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 68.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by $$\Q(\sqrt{-23})$$
3.b odd 2 1 inner
69.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.2.c.a 6
3.b odd 2 1 inner 69.2.c.a 6
4.b odd 2 1 1104.2.m.a 6
12.b even 2 1 1104.2.m.a 6
23.b odd 2 1 CM 69.2.c.a 6
69.c even 2 1 inner 69.2.c.a 6
92.b even 2 1 1104.2.m.a 6
276.h odd 2 1 1104.2.m.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.2.c.a 6 1.a even 1 1 trivial
69.2.c.a 6 3.b odd 2 1 inner
69.2.c.a 6 23.b odd 2 1 CM
69.2.c.a 6 69.c even 2 1 inner
1104.2.m.a 6 4.b odd 2 1
1104.2.m.a 6 12.b even 2 1
1104.2.m.a 6 92.b even 2 1
1104.2.m.a 6 276.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(69, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$23 + 36 T^{2} + 12 T^{4} + T^{6}$$
$3$ $$27 - 4 T^{3} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$T^{6}$$
$11$ $$T^{6}$$
$13$ $$( -74 - 39 T + T^{3} )^{2}$$
$17$ $$T^{6}$$
$19$ $$T^{6}$$
$23$ $$( 23 + T^{2} )^{3}$$
$29$ $$18032 + 7569 T^{2} + 174 T^{4} + T^{6}$$
$31$ $$( -344 - 93 T + T^{3} )^{2}$$
$37$ $$T^{6}$$
$41$ $$94208 + 15129 T^{2} + 246 T^{4} + T^{6}$$
$43$ $$T^{6}$$
$47$ $$412988 + 19881 T^{2} + 282 T^{4} + T^{6}$$
$53$ $$T^{6}$$
$59$ $$( 92 + T^{2} )^{3}$$
$61$ $$T^{6}$$
$67$ $$T^{6}$$
$71$ $$48668 + 45369 T^{2} + 426 T^{4} + T^{6}$$
$73$ $$( -1226 - 219 T + T^{3} )^{2}$$
$79$ $$T^{6}$$
$83$ $$T^{6}$$
$89$ $$T^{6}$$
$97$ $$T^{6}$$
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