# Properties

 Label 36.3.d.c Level $36$ Weight $3$ Character orbit 36.d Analytic conductor $0.981$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$36 = 2^{2} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 36.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.980928951697$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 12) 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 2 q^{5} + ( 4 - 8 \zeta_{6} ) q^{7} -8 q^{8} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 2 q^{5} + ( 4 - 8 \zeta_{6} ) q^{7} -8 q^{8} + 4 \zeta_{6} q^{10} + ( 4 - 8 \zeta_{6} ) q^{11} + 2 q^{13} + ( 16 - 8 \zeta_{6} ) q^{14} -16 \zeta_{6} q^{16} -10 q^{17} + ( -12 + 24 \zeta_{6} ) q^{19} + ( -8 + 8 \zeta_{6} ) q^{20} + ( 16 - 8 \zeta_{6} ) q^{22} + ( -16 + 32 \zeta_{6} ) q^{23} -21 q^{25} + 4 \zeta_{6} q^{26} + ( 16 + 16 \zeta_{6} ) q^{28} + 26 q^{29} + ( -4 + 8 \zeta_{6} ) q^{31} + ( 32 - 32 \zeta_{6} ) q^{32} -20 \zeta_{6} q^{34} + ( 8 - 16 \zeta_{6} ) q^{35} + 26 q^{37} + ( -48 + 24 \zeta_{6} ) q^{38} -16 q^{40} -58 q^{41} + ( 28 - 56 \zeta_{6} ) q^{43} + ( 16 + 16 \zeta_{6} ) q^{44} + ( -64 + 32 \zeta_{6} ) q^{46} + ( 40 - 80 \zeta_{6} ) q^{47} + q^{49} -42 \zeta_{6} q^{50} + ( -8 + 8 \zeta_{6} ) q^{52} + 74 q^{53} + ( 8 - 16 \zeta_{6} ) q^{55} + ( -32 + 64 \zeta_{6} ) q^{56} + 52 \zeta_{6} q^{58} + ( -52 + 104 \zeta_{6} ) q^{59} + 26 q^{61} + ( -16 + 8 \zeta_{6} ) q^{62} + 64 q^{64} + 4 q^{65} + ( -4 + 8 \zeta_{6} ) q^{67} + ( 40 - 40 \zeta_{6} ) q^{68} + ( 32 - 16 \zeta_{6} ) q^{70} -46 q^{73} + 52 \zeta_{6} q^{74} + ( -48 - 48 \zeta_{6} ) q^{76} -48 q^{77} + ( -68 + 136 \zeta_{6} ) q^{79} -32 \zeta_{6} q^{80} -116 \zeta_{6} q^{82} + ( 28 - 56 \zeta_{6} ) q^{83} -20 q^{85} + ( 112 - 56 \zeta_{6} ) q^{86} + ( -32 + 64 \zeta_{6} ) q^{88} -82 q^{89} + ( 8 - 16 \zeta_{6} ) q^{91} + ( -64 - 64 \zeta_{6} ) q^{92} + ( 160 - 80 \zeta_{6} ) q^{94} + ( -24 + 48 \zeta_{6} ) q^{95} + 2 q^{97} + 2 \zeta_{6} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 4q^{4} + 4q^{5} - 16q^{8} + O(q^{10})$$ $$2q + 2q^{2} - 4q^{4} + 4q^{5} - 16q^{8} + 4q^{10} + 4q^{13} + 24q^{14} - 16q^{16} - 20q^{17} - 8q^{20} + 24q^{22} - 42q^{25} + 4q^{26} + 48q^{28} + 52q^{29} + 32q^{32} - 20q^{34} + 52q^{37} - 72q^{38} - 32q^{40} - 116q^{41} + 48q^{44} - 96q^{46} + 2q^{49} - 42q^{50} - 8q^{52} + 148q^{53} + 52q^{58} + 52q^{61} - 24q^{62} + 128q^{64} + 8q^{65} + 40q^{68} + 48q^{70} - 92q^{73} + 52q^{74} - 144q^{76} - 96q^{77} - 32q^{80} - 116q^{82} - 40q^{85} + 168q^{86} - 164q^{89} - 192q^{92} + 240q^{94} + 4q^{97} + 2q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/36\mathbb{Z}\right)^\times$$.

 $$n$$ $$19$$ $$29$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.00000 1.73205i 0 −2.00000 3.46410i 2.00000 0 6.92820i −8.00000 0 2.00000 3.46410i
19.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 2.00000 0 6.92820i −8.00000 0 2.00000 + 3.46410i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.3.d.c 2
3.b odd 2 1 12.3.d.a 2
4.b odd 2 1 inner 36.3.d.c 2
5.b even 2 1 900.3.c.e 2
5.c odd 4 2 900.3.f.c 4
8.b even 2 1 576.3.g.e 2
8.d odd 2 1 576.3.g.e 2
9.c even 3 1 324.3.f.a 2
9.c even 3 1 324.3.f.g 2
9.d odd 6 1 324.3.f.d 2
9.d odd 6 1 324.3.f.j 2
12.b even 2 1 12.3.d.a 2
15.d odd 2 1 300.3.c.b 2
15.e even 4 2 300.3.f.a 4
16.e even 4 2 2304.3.b.l 4
16.f odd 4 2 2304.3.b.l 4
20.d odd 2 1 900.3.c.e 2
20.e even 4 2 900.3.f.c 4
21.c even 2 1 588.3.g.b 2
24.f even 2 1 192.3.g.b 2
24.h odd 2 1 192.3.g.b 2
36.f odd 6 1 324.3.f.a 2
36.f odd 6 1 324.3.f.g 2
36.h even 6 1 324.3.f.d 2
36.h even 6 1 324.3.f.j 2
48.i odd 4 2 768.3.b.c 4
48.k even 4 2 768.3.b.c 4
60.h even 2 1 300.3.c.b 2
60.l odd 4 2 300.3.f.a 4
84.h odd 2 1 588.3.g.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.d.a 2 3.b odd 2 1
12.3.d.a 2 12.b even 2 1
36.3.d.c 2 1.a even 1 1 trivial
36.3.d.c 2 4.b odd 2 1 inner
192.3.g.b 2 24.f even 2 1
192.3.g.b 2 24.h odd 2 1
300.3.c.b 2 15.d odd 2 1
300.3.c.b 2 60.h even 2 1
300.3.f.a 4 15.e even 4 2
300.3.f.a 4 60.l odd 4 2
324.3.f.a 2 9.c even 3 1
324.3.f.a 2 36.f odd 6 1
324.3.f.d 2 9.d odd 6 1
324.3.f.d 2 36.h even 6 1
324.3.f.g 2 9.c even 3 1
324.3.f.g 2 36.f odd 6 1
324.3.f.j 2 9.d odd 6 1
324.3.f.j 2 36.h even 6 1
576.3.g.e 2 8.b even 2 1
576.3.g.e 2 8.d odd 2 1
588.3.g.b 2 21.c even 2 1
588.3.g.b 2 84.h odd 2 1
768.3.b.c 4 48.i odd 4 2
768.3.b.c 4 48.k even 4 2
900.3.c.e 2 5.b even 2 1
900.3.c.e 2 20.d odd 2 1
900.3.f.c 4 5.c odd 4 2
900.3.f.c 4 20.e even 4 2
2304.3.b.l 4 16.e even 4 2
2304.3.b.l 4 16.f odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 2$$ acting on $$S_{3}^{\mathrm{new}}(36, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -2 + T )^{2}$$
$7$ $$48 + T^{2}$$
$11$ $$48 + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$( 10 + T )^{2}$$
$19$ $$432 + T^{2}$$
$23$ $$768 + T^{2}$$
$29$ $$( -26 + T )^{2}$$
$31$ $$48 + T^{2}$$
$37$ $$( -26 + T )^{2}$$
$41$ $$( 58 + T )^{2}$$
$43$ $$2352 + T^{2}$$
$47$ $$4800 + T^{2}$$
$53$ $$( -74 + T )^{2}$$
$59$ $$8112 + T^{2}$$
$61$ $$( -26 + T )^{2}$$
$67$ $$48 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 46 + T )^{2}$$
$79$ $$13872 + T^{2}$$
$83$ $$2352 + T^{2}$$
$89$ $$( 82 + T )^{2}$$
$97$ $$( -2 + T )^{2}$$