# Properties

 Label 1815.2.c.f Level $1815$ Weight $2$ Character orbit 1815.c Analytic conductor $14.493$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1815.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.4928479669$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.49787136.1 Defining polynomial: $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} -\beta_{3} q^{3} + \beta_{4} q^{4} + ( 2 - \beta_{3} ) q^{5} -\beta_{7} q^{6} -2 \beta_{6} q^{7} + ( -\beta_{2} - \beta_{6} ) q^{8} - q^{9} +O(q^{10})$$ $$q + \beta_{6} q^{2} -\beta_{3} q^{3} + \beta_{4} q^{4} + ( 2 - \beta_{3} ) q^{5} -\beta_{7} q^{6} -2 \beta_{6} q^{7} + ( -\beta_{2} - \beta_{6} ) q^{8} - q^{9} + ( 2 \beta_{6} - \beta_{7} ) q^{10} + ( -\beta_{1} + \beta_{3} ) q^{12} -2 \beta_{6} q^{13} + ( 4 - 2 \beta_{4} ) q^{14} + ( -1 - 2 \beta_{3} ) q^{15} + ( 1 + \beta_{4} ) q^{16} + ( 3 \beta_{2} - \beta_{6} ) q^{17} -\beta_{6} q^{18} + ( 2 \beta_{5} + 2 \beta_{7} ) q^{19} + ( -\beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{20} + 2 \beta_{7} q^{21} + ( -2 \beta_{1} + 5 \beta_{3} ) q^{23} + ( -\beta_{5} + \beta_{7} ) q^{24} + ( 3 - 4 \beta_{3} ) q^{25} + ( 4 - 2 \beta_{4} ) q^{26} + \beta_{3} q^{27} + ( 2 \beta_{2} + 6 \beta_{6} ) q^{28} + 2 \beta_{5} q^{29} + ( -\beta_{6} - 2 \beta_{7} ) q^{30} + ( -1 + 2 \beta_{4} ) q^{31} + ( -3 \beta_{2} - 4 \beta_{6} ) q^{32} + ( 5 - \beta_{4} ) q^{34} + ( -4 \beta_{6} + 2 \beta_{7} ) q^{35} -\beta_{4} q^{36} + ( -2 \beta_{1} - 4 \beta_{3} ) q^{37} + ( 2 \beta_{1} - 8 \beta_{3} ) q^{38} + 2 \beta_{7} q^{39} + ( -2 \beta_{2} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{40} + ( -4 \beta_{5} + 4 \beta_{7} ) q^{41} + ( 2 \beta_{1} - 6 \beta_{3} ) q^{42} -2 \beta_{2} q^{43} + ( -2 + \beta_{3} ) q^{45} + ( -2 \beta_{5} + 9 \beta_{7} ) q^{46} + ( -2 \beta_{1} - \beta_{3} ) q^{47} -\beta_{1} q^{48} + ( -1 + 4 \beta_{4} ) q^{49} + ( 3 \beta_{6} - 4 \beta_{7} ) q^{50} + ( 3 \beta_{5} + \beta_{7} ) q^{51} + ( 2 \beta_{2} + 6 \beta_{6} ) q^{52} + 5 \beta_{3} q^{53} + \beta_{7} q^{54} + ( -2 + 2 \beta_{4} ) q^{56} + ( -2 \beta_{2} + 2 \beta_{6} ) q^{57} -2 \beta_{3} q^{58} + 2 \beta_{4} q^{59} + ( -2 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{60} + ( -5 \beta_{5} + 5 \beta_{7} ) q^{61} + ( -2 \beta_{2} - 7 \beta_{6} ) q^{62} + 2 \beta_{6} q^{63} + ( 7 - 2 \beta_{4} ) q^{64} + ( -4 \beta_{6} + 2 \beta_{7} ) q^{65} + ( 6 \beta_{1} - 2 \beta_{3} ) q^{67} + ( 7 \beta_{2} + 6 \beta_{6} ) q^{68} + ( 3 - 2 \beta_{4} ) q^{69} + ( 8 + 2 \beta_{1} - 6 \beta_{3} - 4 \beta_{4} ) q^{70} + ( -2 - 4 \beta_{4} ) q^{71} + ( \beta_{2} + \beta_{6} ) q^{72} + ( -8 \beta_{2} - 4 \beta_{6} ) q^{73} -2 \beta_{5} q^{74} + ( -4 - 3 \beta_{3} ) q^{75} + ( 6 \beta_{5} - 8 \beta_{7} ) q^{76} + ( 2 \beta_{1} - 6 \beta_{3} ) q^{78} + ( -\beta_{5} - \beta_{7} ) q^{79} + ( 2 - \beta_{1} + 2 \beta_{4} ) q^{80} + q^{81} + ( 4 \beta_{1} - 8 \beta_{3} ) q^{82} + ( 6 \beta_{2} + 2 \beta_{6} ) q^{83} + ( 2 \beta_{5} - 6 \beta_{7} ) q^{84} + ( 6 \beta_{2} + 3 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{85} -2 q^{86} -2 \beta_{2} q^{87} + ( -6 - 6 \beta_{4} ) q^{89} + ( -2 \beta_{6} + \beta_{7} ) q^{90} + ( -8 + 4 \beta_{4} ) q^{91} + ( 5 \beta_{1} - 15 \beta_{3} ) q^{92} + ( -2 \beta_{1} + 3 \beta_{3} ) q^{93} + ( -2 \beta_{5} + 3 \beta_{7} ) q^{94} + ( -2 \beta_{2} + 4 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{95} + ( -3 \beta_{5} + 4 \beta_{7} ) q^{96} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{97} + ( -4 \beta_{2} - 13 \beta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{4} + 16q^{5} - 8q^{9} + O(q^{10})$$ $$8q - 4q^{4} + 16q^{5} - 8q^{9} + 40q^{14} - 8q^{15} + 4q^{16} - 8q^{20} + 24q^{25} + 40q^{26} - 16q^{31} + 44q^{34} + 4q^{36} - 16q^{45} - 24q^{49} - 24q^{56} - 8q^{59} + 4q^{60} + 64q^{64} + 32q^{69} + 80q^{70} - 32q^{75} + 8q^{80} + 8q^{81} - 16q^{86} - 24q^{89} - 80q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 5 \nu^{5} + 15 \nu^{3} + 42 \nu$$$$)/20$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{6} + 5 \nu^{4} + 15 \nu^{2} + 8$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{7} - 5 \nu^{5} + 5 \nu^{3} - 16 \nu$$$$)/40$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} - 3 \nu^{4} - \nu^{2} - 8$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} - 3 \nu^{5} - 5 \nu^{3} - 4 \nu$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$7 \nu^{6} + 5 \nu^{4} + 15 \nu^{2} + 44$$$$)/20$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{5} - 3 \nu^{3} - 6 \nu$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + \beta_{4} + 2 \beta_{2} - 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} - \beta_{5} + 5 \beta_{3}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{6} - 3 \beta_{4} + \beta_{2} - 2$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{7} - 6 \beta_{5} - 6 \beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$5 \beta_{6} - 5 \beta_{2} - 9$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-10 \beta_{7} + 3 \beta_{5} - 3 \beta_{3} - 7 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1815\mathbb{Z}\right)^\times$$.

 $$n$$ $$727$$ $$1211$$ $$1696$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
364.1
 0.228425 − 1.39564i −0.228425 + 1.39564i 1.09445 + 0.895644i −1.09445 − 0.895644i −1.09445 + 0.895644i 1.09445 − 0.895644i −0.228425 − 1.39564i 0.228425 + 1.39564i
2.18890i 1.00000i −2.79129 2.00000 1.00000i −2.18890 4.37780i 1.73205i −1.00000 −2.18890 4.37780i
364.2 2.18890i 1.00000i −2.79129 2.00000 + 1.00000i 2.18890 4.37780i 1.73205i −1.00000 2.18890 4.37780i
364.3 0.456850i 1.00000i 1.79129 2.00000 1.00000i −0.456850 0.913701i 1.73205i −1.00000 −0.456850 0.913701i
364.4 0.456850i 1.00000i 1.79129 2.00000 + 1.00000i 0.456850 0.913701i 1.73205i −1.00000 0.456850 0.913701i
364.5 0.456850i 1.00000i 1.79129 2.00000 1.00000i 0.456850 0.913701i 1.73205i −1.00000 0.456850 + 0.913701i
364.6 0.456850i 1.00000i 1.79129 2.00000 + 1.00000i −0.456850 0.913701i 1.73205i −1.00000 −0.456850 + 0.913701i
364.7 2.18890i 1.00000i −2.79129 2.00000 1.00000i 2.18890 4.37780i 1.73205i −1.00000 2.18890 + 4.37780i
364.8 2.18890i 1.00000i −2.79129 2.00000 + 1.00000i −2.18890 4.37780i 1.73205i −1.00000 −2.18890 + 4.37780i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 364.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.b odd 2 1 inner
55.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1815.2.c.f 8
5.b even 2 1 inner 1815.2.c.f 8
5.c odd 4 1 9075.2.a.cs 4
5.c odd 4 1 9075.2.a.cz 4
11.b odd 2 1 inner 1815.2.c.f 8
55.d odd 2 1 inner 1815.2.c.f 8
55.e even 4 1 9075.2.a.cs 4
55.e even 4 1 9075.2.a.cz 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.c.f 8 1.a even 1 1 trivial
1815.2.c.f 8 5.b even 2 1 inner
1815.2.c.f 8 11.b odd 2 1 inner
1815.2.c.f 8 55.d odd 2 1 inner
9075.2.a.cs 4 5.c odd 4 1
9075.2.a.cs 4 55.e even 4 1
9075.2.a.cz 4 5.c odd 4 1
9075.2.a.cz 4 55.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1815, [\chi])$$:

 $$T_{2}^{4} + 5 T_{2}^{2} + 1$$ $$T_{19}^{2} - 28$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 5 T^{2} + T^{4} )^{2}$$
$3$ $$( 1 + T^{2} )^{4}$$
$5$ $$( 5 - 4 T + T^{2} )^{4}$$
$7$ $$( 16 + 20 T^{2} + T^{4} )^{2}$$
$11$ $$T^{8}$$
$13$ $$( 16 + 20 T^{2} + T^{4} )^{2}$$
$17$ $$( 625 + 62 T^{2} + T^{4} )^{2}$$
$19$ $$( -28 + T^{2} )^{4}$$
$23$ $$( 25 + 74 T^{2} + T^{4} )^{2}$$
$29$ $$( 16 - 20 T^{2} + T^{4} )^{2}$$
$31$ $$( -17 + 4 T + T^{2} )^{4}$$
$37$ $$( 16 + 92 T^{2} + T^{4} )^{2}$$
$41$ $$( -48 + T^{2} )^{4}$$
$43$ $$( 16 + 20 T^{2} + T^{4} )^{2}$$
$47$ $$( 289 + 50 T^{2} + T^{4} )^{2}$$
$53$ $$( 25 + T^{2} )^{4}$$
$59$ $$( -20 + 2 T + T^{2} )^{4}$$
$61$ $$( -75 + T^{2} )^{4}$$
$67$ $$( 35344 + 380 T^{2} + T^{4} )^{2}$$
$71$ $$( -84 + T^{2} )^{4}$$
$73$ $$( 6400 + 272 T^{2} + T^{4} )^{2}$$
$79$ $$( -7 + T^{2} )^{4}$$
$83$ $$( 400 + 152 T^{2} + T^{4} )^{2}$$
$89$ $$( -180 + 6 T + T^{2} )^{4}$$
$97$ $$( 400 + 44 T^{2} + T^{4} )^{2}$$