# Properties

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

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$36 = 2^{2} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 36.f (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.980928951697$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} -3 \zeta_{6} q^{3} + 4 q^{4} + ( -4 + 4 \zeta_{6} ) q^{5} -6 \zeta_{6} q^{6} + ( -4 + 2 \zeta_{6} ) q^{7} + 8 q^{8} + ( -9 + 9 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + 2 q^{2} -3 \zeta_{6} q^{3} + 4 q^{4} + ( -4 + 4 \zeta_{6} ) q^{5} -6 \zeta_{6} q^{6} + ( -4 + 2 \zeta_{6} ) q^{7} + 8 q^{8} + ( -9 + 9 \zeta_{6} ) q^{9} + ( -8 + 8 \zeta_{6} ) q^{10} + ( -14 + 7 \zeta_{6} ) q^{11} -12 \zeta_{6} q^{12} + ( 22 - 22 \zeta_{6} ) q^{13} + ( -8 + 4 \zeta_{6} ) q^{14} + 12 q^{15} + 16 q^{16} -11 q^{17} + ( -18 + 18 \zeta_{6} ) q^{18} + ( 9 - 18 \zeta_{6} ) q^{19} + ( -16 + 16 \zeta_{6} ) q^{20} + ( 6 + 6 \zeta_{6} ) q^{21} + ( -28 + 14 \zeta_{6} ) q^{22} + ( 14 + 14 \zeta_{6} ) q^{23} -24 \zeta_{6} q^{24} + 9 \zeta_{6} q^{25} + ( 44 - 44 \zeta_{6} ) q^{26} + 27 q^{27} + ( -16 + 8 \zeta_{6} ) q^{28} -34 \zeta_{6} q^{29} + 24 q^{30} + ( 4 + 4 \zeta_{6} ) q^{31} + 32 q^{32} + ( 21 + 21 \zeta_{6} ) q^{33} -22 q^{34} + ( 8 - 16 \zeta_{6} ) q^{35} + ( -36 + 36 \zeta_{6} ) q^{36} -16 q^{37} + ( 18 - 36 \zeta_{6} ) q^{38} -66 q^{39} + ( -32 + 32 \zeta_{6} ) q^{40} + ( -13 + 13 \zeta_{6} ) q^{41} + ( 12 + 12 \zeta_{6} ) q^{42} + ( -58 + 29 \zeta_{6} ) q^{43} + ( -56 + 28 \zeta_{6} ) q^{44} -36 \zeta_{6} q^{45} + ( 28 + 28 \zeta_{6} ) q^{46} + ( 4 - 2 \zeta_{6} ) q^{47} -48 \zeta_{6} q^{48} + ( -37 + 37 \zeta_{6} ) q^{49} + 18 \zeta_{6} q^{50} + 33 \zeta_{6} q^{51} + ( 88 - 88 \zeta_{6} ) q^{52} + 52 q^{53} + 54 q^{54} + ( 28 - 56 \zeta_{6} ) q^{55} + ( -32 + 16 \zeta_{6} ) q^{56} + ( -54 + 27 \zeta_{6} ) q^{57} -68 \zeta_{6} q^{58} + ( -31 - 31 \zeta_{6} ) q^{59} + 48 q^{60} + 16 \zeta_{6} q^{61} + ( 8 + 8 \zeta_{6} ) q^{62} + ( 18 - 36 \zeta_{6} ) q^{63} + 64 q^{64} + 88 \zeta_{6} q^{65} + ( 42 + 42 \zeta_{6} ) q^{66} + ( 67 + 67 \zeta_{6} ) q^{67} -44 q^{68} + ( 42 - 84 \zeta_{6} ) q^{69} + ( 16 - 32 \zeta_{6} ) q^{70} + ( -72 + 72 \zeta_{6} ) q^{72} -25 q^{73} -32 q^{74} + ( 27 - 27 \zeta_{6} ) q^{75} + ( 36 - 72 \zeta_{6} ) q^{76} + ( 42 - 42 \zeta_{6} ) q^{77} -132 q^{78} + ( 32 - 16 \zeta_{6} ) q^{79} + ( -64 + 64 \zeta_{6} ) q^{80} -81 \zeta_{6} q^{81} + ( -26 + 26 \zeta_{6} ) q^{82} + ( 40 - 20 \zeta_{6} ) q^{83} + ( 24 + 24 \zeta_{6} ) q^{84} + ( 44 - 44 \zeta_{6} ) q^{85} + ( -116 + 58 \zeta_{6} ) q^{86} + ( -102 + 102 \zeta_{6} ) q^{87} + ( -112 + 56 \zeta_{6} ) q^{88} -2 q^{89} -72 \zeta_{6} q^{90} + ( -44 + 88 \zeta_{6} ) q^{91} + ( 56 + 56 \zeta_{6} ) q^{92} + ( 12 - 24 \zeta_{6} ) q^{93} + ( 8 - 4 \zeta_{6} ) q^{94} + ( 36 + 36 \zeta_{6} ) q^{95} -96 \zeta_{6} q^{96} + 43 \zeta_{6} q^{97} + ( -74 + 74 \zeta_{6} ) q^{98} + ( 63 - 126 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{2} - 3q^{3} + 8q^{4} - 4q^{5} - 6q^{6} - 6q^{7} + 16q^{8} - 9q^{9} + O(q^{10})$$ $$2q + 4q^{2} - 3q^{3} + 8q^{4} - 4q^{5} - 6q^{6} - 6q^{7} + 16q^{8} - 9q^{9} - 8q^{10} - 21q^{11} - 12q^{12} + 22q^{13} - 12q^{14} + 24q^{15} + 32q^{16} - 22q^{17} - 18q^{18} - 16q^{20} + 18q^{21} - 42q^{22} + 42q^{23} - 24q^{24} + 9q^{25} + 44q^{26} + 54q^{27} - 24q^{28} - 34q^{29} + 48q^{30} + 12q^{31} + 64q^{32} + 63q^{33} - 44q^{34} - 36q^{36} - 32q^{37} - 132q^{39} - 32q^{40} - 13q^{41} + 36q^{42} - 87q^{43} - 84q^{44} - 36q^{45} + 84q^{46} + 6q^{47} - 48q^{48} - 37q^{49} + 18q^{50} + 33q^{51} + 88q^{52} + 104q^{53} + 108q^{54} - 48q^{56} - 81q^{57} - 68q^{58} - 93q^{59} + 96q^{60} + 16q^{61} + 24q^{62} + 128q^{64} + 88q^{65} + 126q^{66} + 201q^{67} - 88q^{68} - 72q^{72} - 50q^{73} - 64q^{74} + 27q^{75} + 42q^{77} - 264q^{78} + 48q^{79} - 64q^{80} - 81q^{81} - 26q^{82} + 60q^{83} + 72q^{84} + 44q^{85} - 174q^{86} - 102q^{87} - 168q^{88} - 4q^{89} - 72q^{90} + 168q^{92} + 12q^{94} + 108q^{95} - 96q^{96} + 43q^{97} - 74q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$19$$ $$29$$ $$\chi(n)$$ $$-1$$ $$-1 + \zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.5 − 0.866025i 0.5 + 0.866025i
2.00000 −1.50000 + 2.59808i 4.00000 −2.00000 3.46410i −3.00000 + 5.19615i −3.00000 1.73205i 8.00000 −4.50000 7.79423i −4.00000 6.92820i
31.1 2.00000 −1.50000 2.59808i 4.00000 −2.00000 + 3.46410i −3.00000 5.19615i −3.00000 + 1.73205i 8.00000 −4.50000 + 7.79423i −4.00000 + 6.92820i
 $$n$$: 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
36.f odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.3.f.b yes 2
3.b odd 2 1 108.3.f.a 2
4.b odd 2 1 36.3.f.a 2
8.b even 2 1 576.3.o.b 2
8.d odd 2 1 576.3.o.a 2
9.c even 3 1 36.3.f.a 2
9.c even 3 1 324.3.d.b 2
9.d odd 6 1 108.3.f.b 2
9.d odd 6 1 324.3.d.c 2
12.b even 2 1 108.3.f.b 2
24.f even 2 1 1728.3.o.b 2
24.h odd 2 1 1728.3.o.a 2
36.f odd 6 1 inner 36.3.f.b yes 2
36.f odd 6 1 324.3.d.b 2
36.h even 6 1 108.3.f.a 2
36.h even 6 1 324.3.d.c 2
72.j odd 6 1 1728.3.o.b 2
72.l even 6 1 1728.3.o.a 2
72.n even 6 1 576.3.o.a 2
72.p odd 6 1 576.3.o.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.f.a 2 4.b odd 2 1
36.3.f.a 2 9.c even 3 1
36.3.f.b yes 2 1.a even 1 1 trivial
36.3.f.b yes 2 36.f odd 6 1 inner
108.3.f.a 2 3.b odd 2 1
108.3.f.a 2 36.h even 6 1
108.3.f.b 2 9.d odd 6 1
108.3.f.b 2 12.b even 2 1
324.3.d.b 2 9.c even 3 1
324.3.d.b 2 36.f odd 6 1
324.3.d.c 2 9.d odd 6 1
324.3.d.c 2 36.h even 6 1
576.3.o.a 2 8.d odd 2 1
576.3.o.a 2 72.n even 6 1
576.3.o.b 2 8.b even 2 1
576.3.o.b 2 72.p odd 6 1
1728.3.o.a 2 24.h odd 2 1
1728.3.o.a 2 72.l even 6 1
1728.3.o.b 2 24.f even 2 1
1728.3.o.b 2 72.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(36, [\chi])$$:

 $$T_{5}^{2} + 4 T_{5} + 16$$ $$T_{7}^{2} + 6 T_{7} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T )^{2}$$
$3$ $$1 + 3 T + 9 T^{2}$$
$5$ $$1 + 4 T - 9 T^{2} + 100 T^{3} + 625 T^{4}$$
$7$ $$1 + 6 T + 61 T^{2} + 294 T^{3} + 2401 T^{4}$$
$11$ $$1 + 21 T + 268 T^{2} + 2541 T^{3} + 14641 T^{4}$$
$13$ $$( 1 - 23 T + 169 T^{2} )( 1 + T + 169 T^{2} )$$
$17$ $$( 1 + 11 T + 289 T^{2} )^{2}$$
$19$ $$1 - 479 T^{2} + 130321 T^{4}$$
$23$ $$1 - 42 T + 1117 T^{2} - 22218 T^{3} + 279841 T^{4}$$
$29$ $$1 + 34 T + 315 T^{2} + 28594 T^{3} + 707281 T^{4}$$
$31$ $$1 - 12 T + 1009 T^{2} - 11532 T^{3} + 923521 T^{4}$$
$37$ $$( 1 + 16 T + 1369 T^{2} )^{2}$$
$41$ $$1 + 13 T - 1512 T^{2} + 21853 T^{3} + 2825761 T^{4}$$
$43$ $$1 + 87 T + 4372 T^{2} + 160863 T^{3} + 3418801 T^{4}$$
$47$ $$1 - 6 T + 2221 T^{2} - 13254 T^{3} + 4879681 T^{4}$$
$53$ $$( 1 - 52 T + 2809 T^{2} )^{2}$$
$59$ $$1 + 93 T + 6364 T^{2} + 323733 T^{3} + 12117361 T^{4}$$
$61$ $$1 - 16 T - 3465 T^{2} - 59536 T^{3} + 13845841 T^{4}$$
$67$ $$( 1 - 67 T )^{2}( 1 - 67 T + 4489 T^{2} )$$
$71$ $$( 1 - 71 T )^{2}( 1 + 71 T )^{2}$$
$73$ $$( 1 + 25 T + 5329 T^{2} )^{2}$$
$79$ $$1 - 48 T + 7009 T^{2} - 299568 T^{3} + 38950081 T^{4}$$
$83$ $$1 - 60 T + 8089 T^{2} - 413340 T^{3} + 47458321 T^{4}$$
$89$ $$( 1 + 2 T + 7921 T^{2} )^{2}$$
$97$ $$1 - 43 T - 7560 T^{2} - 404587 T^{3} + 88529281 T^{4}$$
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