# Properties

 Label 1815.2.c.e Level $1815$ Weight $2$ Character orbit 1815.c Analytic conductor $14.493$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# 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: $$6$$ Coefficient field: 6.0.350464.1 Defining polynomial: $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{2} -\beta_{3} q^{3} + ( -\beta_{1} + \beta_{2} ) q^{4} + ( -\beta_{2} + \beta_{5} ) q^{5} + \beta_{2} q^{6} + ( -\beta_{3} - \beta_{4} + \beta_{5} ) q^{7} + ( \beta_{3} - \beta_{5} ) q^{8} - q^{9} +O(q^{10})$$ $$q -\beta_{4} q^{2} -\beta_{3} q^{3} + ( -\beta_{1} + \beta_{2} ) q^{4} + ( -\beta_{2} + \beta_{5} ) q^{5} + \beta_{2} q^{6} + ( -\beta_{3} - \beta_{4} + \beta_{5} ) q^{7} + ( \beta_{3} - \beta_{5} ) q^{8} - q^{9} + ( 1 + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{10} + ( \beta_{4} - \beta_{5} ) q^{12} + ( \beta_{3} - \beta_{4} + \beta_{5} ) q^{13} + ( -1 - \beta_{1} + 3 \beta_{2} ) q^{14} + ( -\beta_{1} - \beta_{4} ) q^{15} + ( -1 - 2 \beta_{1} ) q^{16} + ( -\beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{17} + \beta_{4} q^{18} + ( -2 - 2 \beta_{1} ) q^{19} + ( -1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{20} + ( -1 - \beta_{1} + \beta_{2} ) q^{21} -4 \beta_{3} q^{23} + ( 1 + \beta_{1} ) q^{24} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{25} + ( -1 - \beta_{1} + \beta_{2} ) q^{26} + \beta_{3} q^{27} + ( 3 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{28} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{29} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{30} + ( -2 + 2 \beta_{2} ) q^{31} + ( 3 \beta_{4} - 2 \beta_{5} ) q^{32} + ( 5 + 3 \beta_{1} - 3 \beta_{2} ) q^{34} + ( -2 - \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{35} + ( \beta_{1} - \beta_{2} ) q^{36} + ( 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{37} + ( -2 \beta_{3} + 4 \beta_{4} ) q^{38} + ( 1 - \beta_{1} + \beta_{2} ) q^{39} + ( 3 + \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{40} + ( -2 - 6 \beta_{1} + 2 \beta_{2} ) q^{41} + ( \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{42} + ( 5 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{43} + ( \beta_{2} - \beta_{5} ) q^{45} + 4 \beta_{2} q^{46} + ( 6 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{47} + ( \beta_{3} - 2 \beta_{5} ) q^{48} + ( 3 - 2 \beta_{1} + 4 \beta_{2} ) q^{49} + ( -4 - 2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{50} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{51} + ( 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{52} + ( -4 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{53} -\beta_{2} q^{54} + ( 3 + \beta_{1} - \beta_{2} ) q^{56} + ( 2 \beta_{3} - 2 \beta_{5} ) q^{57} + ( -6 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{58} + ( -4 - 2 \beta_{1} - 4 \beta_{2} ) q^{59} + ( 2 - \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{60} + ( 6 - 4 \beta_{2} ) q^{61} + ( 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{62} + ( \beta_{3} + \beta_{4} - \beta_{5} ) q^{63} + ( 2 - \beta_{1} - 5 \beta_{2} ) q^{64} + ( -2 + 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{65} + ( 4 \beta_{3} + 4 \beta_{5} ) q^{67} + ( -5 \beta_{3} - 5 \beta_{4} + \beta_{5} ) q^{68} -4 q^{69} + ( -5 - 3 \beta_{1} + 5 \beta_{2} + 2 \beta_{4} ) q^{70} + ( -4 - 6 \beta_{1} ) q^{71} + ( -\beta_{3} + \beta_{5} ) q^{72} + ( -5 \beta_{3} + 5 \beta_{4} - \beta_{5} ) q^{73} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{74} + ( 2 + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{75} + ( 4 - 2 \beta_{2} ) q^{76} + ( \beta_{3} + \beta_{4} - \beta_{5} ) q^{78} + ( -2 + 6 \beta_{1} + 4 \beta_{2} ) q^{79} + ( 2 + 3 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{80} + q^{81} + ( -2 \beta_{3} + 10 \beta_{4} - 2 \beta_{5} ) q^{82} + ( -3 \beta_{3} - \beta_{4} + 5 \beta_{5} ) q^{83} + ( 3 + \beta_{1} - 3 \beta_{2} ) q^{84} + ( -2 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{85} + ( -3 - 3 \beta_{1} + \beta_{2} ) q^{86} + ( -2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{87} + ( -8 - 6 \beta_{2} ) q^{89} + ( -1 - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{90} + ( -2 + 2 \beta_{2} ) q^{91} + ( 4 \beta_{4} - 4 \beta_{5} ) q^{92} + ( 2 \beta_{3} + 2 \beta_{4} ) q^{93} + ( -6 - 2 \beta_{1} - 6 \beta_{2} ) q^{94} + ( 2 + 4 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} ) q^{95} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{96} + ( 4 \beta_{3} + 4 \beta_{4} - 8 \beta_{5} ) q^{97} + ( 6 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 2q^{4} + 2q^{5} - 2q^{6} - 6q^{9} + O(q^{10})$$ $$6q - 2q^{4} + 2q^{5} - 2q^{6} - 6q^{9} + 4q^{10} - 12q^{14} - 6q^{16} - 12q^{19} - 10q^{20} - 8q^{21} + 6q^{24} - 2q^{25} - 8q^{26} + 16q^{29} - 14q^{30} - 16q^{31} + 36q^{34} - 12q^{35} + 2q^{36} + 4q^{39} + 16q^{40} - 16q^{41} - 2q^{45} - 8q^{46} + 10q^{49} - 24q^{50} + 2q^{54} + 20q^{56} - 16q^{59} + 12q^{60} + 44q^{61} + 22q^{64} - 12q^{65} - 24q^{69} - 40q^{70} - 24q^{71} - 16q^{74} + 8q^{75} + 28q^{76} - 20q^{79} + 6q^{80} + 6q^{81} + 24q^{84} + 4q^{85} - 20q^{86} - 36q^{89} - 4q^{90} - 16q^{91} - 24q^{94} + 4q^{95} + 6q^{96} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} + 8 \nu^{4} - 4 \nu^{3} - \nu^{2} + 2 \nu + 38$$$$)/23$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{5} + 17 \nu^{4} - 20 \nu^{3} - 5 \nu^{2} + 10 \nu + 29$$$$)/23$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{5} - 10 \nu^{4} + 5 \nu^{3} + 30 \nu^{2} + 32 \nu - 13$$$$)/23$$ $$\beta_{4}$$ $$=$$ $$($$$$-11 \nu^{5} + 19 \nu^{4} - 21 \nu^{3} - 11 \nu^{2} - 70 \nu + 27$$$$)/23$$ $$\beta_{5}$$ $$=$$ $$($$$$-14 \nu^{5} + 20 \nu^{4} - 10 \nu^{3} - 37 \nu^{2} - 64 \nu + 26$$$$)/23$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2 \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$-\beta_{2} + 5 \beta_{1} - 7$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{5} + 3 \beta_{4} - 9 \beta_{3} - 3 \beta_{2} + 8 \beta_{1} - 9$$

## 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
 −0.854638 − 0.854638i 0.403032 − 0.403032i 1.45161 + 1.45161i 1.45161 − 1.45161i 0.403032 + 0.403032i −0.854638 + 0.854638i
2.17009i 1.00000i −2.70928 2.17009 0.539189i −2.17009 3.70928i 1.53919i −1.00000 −1.17009 4.70928i
364.2 1.48119i 1.00000i −0.193937 −1.48119 + 1.67513i 1.48119 1.19394i 2.67513i −1.00000 2.48119 + 2.19394i
364.3 0.311108i 1.00000i 1.90321 0.311108 + 2.21432i −0.311108 0.903212i 1.21432i −1.00000 0.688892 0.0967881i
364.4 0.311108i 1.00000i 1.90321 0.311108 2.21432i −0.311108 0.903212i 1.21432i −1.00000 0.688892 + 0.0967881i
364.5 1.48119i 1.00000i −0.193937 −1.48119 1.67513i 1.48119 1.19394i 2.67513i −1.00000 2.48119 2.19394i
364.6 2.17009i 1.00000i −2.70928 2.17009 + 0.539189i −2.17009 3.70928i 1.53919i −1.00000 −1.17009 + 4.70928i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 364.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 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.e 6
5.b even 2 1 inner 1815.2.c.e 6
5.c odd 4 1 9075.2.a.cg 3
5.c odd 4 1 9075.2.a.ch 3
11.b odd 2 1 165.2.c.b 6
33.d even 2 1 495.2.c.e 6
44.c even 2 1 2640.2.d.h 6
55.d odd 2 1 165.2.c.b 6
55.e even 4 1 825.2.a.j 3
55.e even 4 1 825.2.a.l 3
165.d even 2 1 495.2.c.e 6
165.l odd 4 1 2475.2.a.ba 3
165.l odd 4 1 2475.2.a.bc 3
220.g even 2 1 2640.2.d.h 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.c.b 6 11.b odd 2 1
165.2.c.b 6 55.d odd 2 1
495.2.c.e 6 33.d even 2 1
495.2.c.e 6 165.d even 2 1
825.2.a.j 3 55.e even 4 1
825.2.a.l 3 55.e even 4 1
1815.2.c.e 6 1.a even 1 1 trivial
1815.2.c.e 6 5.b even 2 1 inner
2475.2.a.ba 3 165.l odd 4 1
2475.2.a.bc 3 165.l odd 4 1
2640.2.d.h 6 44.c even 2 1
2640.2.d.h 6 220.g even 2 1
9075.2.a.cg 3 5.c odd 4 1
9075.2.a.ch 3 5.c odd 4 1

## Hecke kernels

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

 $$T_{2}^{6} + 7 T_{2}^{4} + 11 T_{2}^{2} + 1$$ $$T_{19}^{3} + 6 T_{19}^{2} - 4 T_{19} - 40$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 11 T^{2} + 7 T^{4} + T^{6}$$
$3$ $$( 1 + T^{2} )^{3}$$
$5$ $$125 - 50 T + 15 T^{2} - 12 T^{3} + 3 T^{4} - 2 T^{5} + T^{6}$$
$7$ $$16 + 32 T^{2} + 16 T^{4} + T^{6}$$
$11$ $$T^{6}$$
$13$ $$16 + 32 T^{2} + 12 T^{4} + T^{6}$$
$17$ $$2704 + 784 T^{2} + 56 T^{4} + T^{6}$$
$19$ $$( -40 - 4 T + 6 T^{2} + T^{3} )^{2}$$
$23$ $$( 16 + T^{2} )^{3}$$
$29$ $$( 160 - 16 T - 8 T^{2} + T^{3} )^{2}$$
$31$ $$( -16 + 8 T + 8 T^{2} + T^{3} )^{2}$$
$37$ $$1024 + 512 T^{2} + 48 T^{4} + T^{6}$$
$41$ $$( -928 - 112 T + 8 T^{2} + T^{3} )^{2}$$
$43$ $$21904 + 3552 T^{2} + 144 T^{4} + T^{6}$$
$47$ $$1024 + 1280 T^{2} + 160 T^{4} + T^{6}$$
$53$ $$256 + 512 T^{2} + 192 T^{4} + T^{6}$$
$59$ $$( 80 - 64 T + 8 T^{2} + T^{3} )^{2}$$
$61$ $$( -8 + 108 T - 22 T^{2} + T^{3} )^{2}$$
$67$ $$4096 + 1792 T^{2} + 176 T^{4} + T^{6}$$
$71$ $$( -944 - 96 T + 12 T^{2} + T^{3} )^{2}$$
$73$ $$150544 + 9696 T^{2} + 188 T^{4} + T^{6}$$
$79$ $$( -1720 - 212 T + 10 T^{2} + T^{3} )^{2}$$
$83$ $$364816 + 15680 T^{2} + 220 T^{4} + T^{6}$$
$89$ $$( -520 - 12 T + 18 T^{2} + T^{3} )^{2}$$
$97$ $$5914624 + 103424 T^{2} + 576 T^{4} + T^{6}$$