Properties

 Label 1815.2.c.g 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: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

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

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
 0.258819 − 0.965926i −0.965926 + 0.258819i 0.965926 + 0.258819i −0.258819 − 0.965926i 0.965926 − 0.258819i −0.258819 + 0.965926i 0.258819 + 0.965926i −0.965926 − 0.258819i
1.41421i 1.00000i 0 −2.22474 0.224745i −1.41421 1.73205i 2.82843i −1.00000 −0.317837 + 3.14626i
364.2 1.41421i 1.00000i 0 0.224745 + 2.22474i −1.41421 1.73205i 2.82843i −1.00000 3.14626 0.317837i
364.3 1.41421i 1.00000i 0 −2.22474 + 0.224745i 1.41421 1.73205i 2.82843i −1.00000 0.317837 + 3.14626i
364.4 1.41421i 1.00000i 0 0.224745 2.22474i 1.41421 1.73205i 2.82843i −1.00000 −3.14626 0.317837i
364.5 1.41421i 1.00000i 0 −2.22474 0.224745i 1.41421 1.73205i 2.82843i −1.00000 0.317837 3.14626i
364.6 1.41421i 1.00000i 0 0.224745 + 2.22474i 1.41421 1.73205i 2.82843i −1.00000 −3.14626 + 0.317837i
364.7 1.41421i 1.00000i 0 −2.22474 + 0.224745i −1.41421 1.73205i 2.82843i −1.00000 −0.317837 3.14626i
364.8 1.41421i 1.00000i 0 0.224745 2.22474i −1.41421 1.73205i 2.82843i −1.00000 3.14626 + 0.317837i
 $$n$$: 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.g 8
5.b even 2 1 inner 1815.2.c.g 8
5.c odd 4 1 9075.2.a.cr 4
5.c odd 4 1 9075.2.a.cy 4
11.b odd 2 1 inner 1815.2.c.g 8
55.d odd 2 1 inner 1815.2.c.g 8
55.e even 4 1 9075.2.a.cr 4
55.e even 4 1 9075.2.a.cy 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.c.g 8 1.a even 1 1 trivial
1815.2.c.g 8 5.b even 2 1 inner
1815.2.c.g 8 11.b odd 2 1 inner
1815.2.c.g 8 55.d odd 2 1 inner
9075.2.a.cr 4 5.c odd 4 1
9075.2.a.cr 4 55.e even 4 1
9075.2.a.cy 4 5.c odd 4 1
9075.2.a.cy 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 $$S_{2}^{\mathrm{new}}(1815, [\chi])$$:

 $$T_{2}^{2} + 2$$ $$T_{19}^{2} - 3$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{4}$$
$3$ $$( 1 + T^{2} )^{4}$$
$5$ $$( 25 + 20 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$7$ $$( 3 + T^{2} )^{4}$$
$11$ $$T^{8}$$
$13$ $$( 16 + 40 T^{2} + T^{4} )^{2}$$
$17$ $$( 100 + 28 T^{2} + T^{4} )^{2}$$
$19$ $$( -3 + T^{2} )^{4}$$
$23$ $$( 4 + T^{2} )^{4}$$
$29$ $$( 2116 - 100 T^{2} + T^{4} )^{2}$$
$31$ $$( -15 + 6 T + T^{2} )^{4}$$
$37$ $$( 529 + 50 T^{2} + T^{4} )^{2}$$
$41$ $$( 400 - 88 T^{2} + T^{4} )^{2}$$
$43$ $$( 32 + T^{2} )^{4}$$
$47$ $$( 54 + T^{2} )^{4}$$
$53$ $$( 100 + T^{2} )^{4}$$
$59$ $$( -150 + T^{2} )^{4}$$
$61$ $$( 2025 - 198 T^{2} + T^{4} )^{2}$$
$67$ $$( 3249 + 210 T^{2} + T^{4} )^{2}$$
$71$ $$( 30 + 12 T + T^{2} )^{4}$$
$73$ $$( 2025 + 198 T^{2} + T^{4} )^{2}$$
$79$ $$( 4761 - 150 T^{2} + T^{4} )^{2}$$
$83$ $$( 162 + T^{2} )^{4}$$
$89$ $$( -18 - 12 T + T^{2} )^{4}$$
$97$ $$( 25 + T^{2} )^{4}$$