# Properties

 Label 36.3.g.a Level $36$ Weight $3$ Character orbit 36.g Analytic conductor $0.981$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.980928951697$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - \beta_1 + 1) q^{3} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{5} + ( - 3 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 1) q^{7} + (\beta_{3} + \beta_{2} + 7 \beta_1 - 7) q^{9}+O(q^{10})$$ q + (b3 - b1 + 1) * q^3 + (b3 - 2*b2 - 2*b1 + 2) * q^5 + (-3*b3 + 3*b2 + 2*b1 + 1) * q^7 + (b3 + b2 + 7*b1 - 7) * q^9 $$q + (\beta_{3} - \beta_1 + 1) q^{3} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{5} + ( - 3 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 1) q^{7} + (\beta_{3} + \beta_{2} + 7 \beta_1 - 7) q^{9} + ( - 2 \beta_{3} - 2 \beta_{2} - 5 \beta_1 - 7) q^{11} + ( - 3 \beta_{3} + 4 \beta_1) q^{13} + (3 \beta_{3} - 12 \beta_1 - 6) q^{15} + (2 \beta_{3} - \beta_{2} + 14 \beta_1 - 8) q^{17} + (3 \beta_{2} + 2) q^{19} + ( - 3 \beta_{3} + \beta_{2} + 2 \beta_1 + 25) q^{21} + ( - \beta_{3} + 2 \beta_{2} - 16 \beta_1 + 34) q^{23} + (9 \beta_{3} - 9 \beta_{2} - 11 \beta_1 + 2) q^{25} + ( - 6 \beta_{2} + 24 \beta_1 - 15) q^{27} + (7 \beta_{3} + 7 \beta_{2} - 14 \beta_1 - 7) q^{29} + (9 \beta_{3} - 8 \beta_1) q^{31} + ( - 12 \beta_{3} + 3 \beta_{2} - 27 \beta_1 + 9) q^{33} + ( - 10 \beta_{3} + 5 \beta_{2} + 56 \beta_1 - 23) q^{35} + ( - 12 \beta_{2} - 22) q^{37} + (\beta_{3} - 4 \beta_{2} - 24 \beta_1 + 24) q^{39} + ( - 8 \beta_{3} + 16 \beta_{2} + 7 \beta_1 + 2) q^{41} + (23 \beta_1 - 23) q^{43} + ( - 15 \beta_{3} + 12 \beta_{2} + 30 \beta_1 - 30) q^{45} + ( - 3 \beta_{3} - 3 \beta_{2} - 12 \beta_1 - 15) q^{47} + (3 \beta_{3} - 27 \beta_1) q^{49} + (9 \beta_{3} - 15 \beta_{2} + 15 \beta_1 - 24) q^{51} + (16 \beta_{3} - 8 \beta_{2} - 32 \beta_1 + 8) q^{53} + (9 \beta_{2} + 27) q^{55} + ( - \beta_{3} + 3 \beta_{2} + 25 \beta_1 + 2) q^{57} + (8 \beta_{3} - 16 \beta_{2} - 25 \beta_1 + 34) q^{59} + ( - 9 \beta_{3} + 9 \beta_{2} - 16 \beta_1 + 25) q^{61} + (23 \beta_{3} - \beta_{2} - 40 \beta_1 + 49) q^{63} + ( - 7 \beta_{3} - 7 \beta_{2} + 32 \beta_1 + 25) q^{65} + ( - 18 \beta_{3} + 67 \beta_1) q^{67} + (15 \beta_{3} + 18 \beta_{2} - 24 \beta_1 + 42) q^{69} + ( - 4 \beta_{3} + 2 \beta_{2} - 28 \beta_1 + 16) q^{71} + (9 \beta_{2} + 26) q^{73} + (9 \beta_{3} + 2 \beta_{2} - 11 \beta_1 - 70) q^{75} + (17 \beta_{3} - 34 \beta_{2} + 38 \beta_1 - 110) q^{77} + ( - 15 \beta_{3} + 15 \beta_{2} - 34 \beta_1 + 49) q^{79} + (15 \beta_{3} - 30 \beta_{2} - 39 \beta_1 - 15) q^{81} + ( - 3 \beta_{3} - 3 \beta_{2} - 12 \beta_1 - 15) q^{83} + ( - 18 \beta_{3} + 18 \beta_1) q^{85} + ( - 21 \beta_{3} + 21 \beta_{2} + 126 \beta_1 - 63) q^{87} + ( - 16 \beta_{3} + 8 \beta_{2} - 112 \beta_1 + 64) q^{89} + (9 \beta_{2} - 71) q^{91} + (\beta_{3} + 8 \beta_{2} + 72 \beta_1 - 72) q^{93} + ( - 4 \beta_{3} + 8 \beta_{2} + 26 \beta_1 - 44) q^{95} + ( - 6 \beta_{3} + 6 \beta_{2} + 101 \beta_1 - 95) q^{97} + ( - 33 \beta_{3} + 30 \beta_{2} - 78 \beta_1 + 105) q^{99}+O(q^{100})$$ q + (b3 - b1 + 1) * q^3 + (b3 - 2*b2 - 2*b1 + 2) * q^5 + (-3*b3 + 3*b2 + 2*b1 + 1) * q^7 + (b3 + b2 + 7*b1 - 7) * q^9 + (-2*b3 - 2*b2 - 5*b1 - 7) * q^11 + (-3*b3 + 4*b1) * q^13 + (3*b3 - 12*b1 - 6) * q^15 + (2*b3 - b2 + 14*b1 - 8) * q^17 + (3*b2 + 2) * q^19 + (-3*b3 + b2 + 2*b1 + 25) * q^21 + (-b3 + 2*b2 - 16*b1 + 34) * q^23 + (9*b3 - 9*b2 - 11*b1 + 2) * q^25 + (-6*b2 + 24*b1 - 15) * q^27 + (7*b3 + 7*b2 - 14*b1 - 7) * q^29 + (9*b3 - 8*b1) * q^31 + (-12*b3 + 3*b2 - 27*b1 + 9) * q^33 + (-10*b3 + 5*b2 + 56*b1 - 23) * q^35 + (-12*b2 - 22) * q^37 + (b3 - 4*b2 - 24*b1 + 24) * q^39 + (-8*b3 + 16*b2 + 7*b1 + 2) * q^41 + (23*b1 - 23) * q^43 + (-15*b3 + 12*b2 + 30*b1 - 30) * q^45 + (-3*b3 - 3*b2 - 12*b1 - 15) * q^47 + (3*b3 - 27*b1) * q^49 + (9*b3 - 15*b2 + 15*b1 - 24) * q^51 + (16*b3 - 8*b2 - 32*b1 + 8) * q^53 + (9*b2 + 27) * q^55 + (-b3 + 3*b2 + 25*b1 + 2) * q^57 + (8*b3 - 16*b2 - 25*b1 + 34) * q^59 + (-9*b3 + 9*b2 - 16*b1 + 25) * q^61 + (23*b3 - b2 - 40*b1 + 49) * q^63 + (-7*b3 - 7*b2 + 32*b1 + 25) * q^65 + (-18*b3 + 67*b1) * q^67 + (15*b3 + 18*b2 - 24*b1 + 42) * q^69 + (-4*b3 + 2*b2 - 28*b1 + 16) * q^71 + (9*b2 + 26) * q^73 + (9*b3 + 2*b2 - 11*b1 - 70) * q^75 + (17*b3 - 34*b2 + 38*b1 - 110) * q^77 + (-15*b3 + 15*b2 - 34*b1 + 49) * q^79 + (15*b3 - 30*b2 - 39*b1 - 15) * q^81 + (-3*b3 - 3*b2 - 12*b1 - 15) * q^83 + (-18*b3 + 18*b1) * q^85 + (-21*b3 + 21*b2 + 126*b1 - 63) * q^87 + (-16*b3 + 8*b2 - 112*b1 + 64) * q^89 + (9*b2 - 71) * q^91 + (b3 + 8*b2 + 72*b1 - 72) * q^93 + (-4*b3 + 8*b2 + 26*b1 - 44) * q^95 + (-6*b3 + 6*b2 + 101*b1 - 95) * q^97 + (-33*b3 + 30*b2 - 78*b1 + 105) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3 q^{3} + 9 q^{5} - q^{7} - 15 q^{9}+O(q^{10})$$ 4 * q + 3 * q^3 + 9 * q^5 - q^7 - 15 * q^9 $$4 q + 3 q^{3} + 9 q^{5} - q^{7} - 15 q^{9} - 36 q^{11} + 5 q^{13} - 45 q^{15} + 2 q^{19} + 99 q^{21} + 99 q^{23} + 13 q^{25} - 63 q^{29} - 7 q^{31} - 36 q^{33} - 64 q^{37} + 57 q^{39} - 18 q^{41} - 46 q^{43} - 99 q^{45} - 81 q^{47} - 51 q^{49} - 27 q^{51} + 90 q^{55} + 51 q^{57} + 126 q^{59} + 41 q^{61} + 141 q^{63} + 171 q^{65} + 116 q^{67} + 99 q^{69} + 86 q^{73} - 297 q^{75} - 279 q^{77} + 83 q^{79} - 63 q^{81} - 81 q^{83} + 18 q^{85} - 63 q^{87} - 302 q^{91} - 159 q^{93} - 144 q^{95} - 196 q^{97} + 171 q^{99}+O(q^{100})$$ 4 * q + 3 * q^3 + 9 * q^5 - q^7 - 15 * q^9 - 36 * q^11 + 5 * q^13 - 45 * q^15 + 2 * q^19 + 99 * q^21 + 99 * q^23 + 13 * q^25 - 63 * q^29 - 7 * q^31 - 36 * q^33 - 64 * q^37 + 57 * q^39 - 18 * q^41 - 46 * q^43 - 99 * q^45 - 81 * q^47 - 51 * q^49 - 27 * q^51 + 90 * q^55 + 51 * q^57 + 126 * q^59 + 41 * q^61 + 141 * q^63 + 171 * q^65 + 116 * q^67 + 99 * q^69 + 86 * q^73 - 297 * q^75 - 279 * q^77 + 83 * q^79 - 63 * q^81 - 81 * q^83 + 18 * q^85 - 63 * q^87 - 302 * q^91 - 159 * q^93 - 144 * q^95 - 196 * q^97 + 171 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6$$ (v^3 + 2*v^2 - 2*v - 3) / 6 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 5\nu ) / 3$$ (-v^3 + v^2 + 5*v) / 3 $$\beta_{3}$$ $$=$$ $$( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3$$ (2*v^3 + v^2 + 2*v - 9) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3$$ (b3 + b2 - 2*b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3$$ (-b3 + 2*b2 + 8*b1 + 1) / 3 $$\nu^{3}$$ $$=$$ $$( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3$$ (4*b3 - 2*b2 - 2*b1 + 11) / 3

## Character values

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

 $$n$$ $$19$$ $$29$$ $$\chi(n)$$ $$1$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 −1.18614 + 1.26217i 1.68614 − 0.396143i −1.18614 − 1.26217i 1.68614 + 0.396143i
0 −0.686141 + 2.92048i 0 6.55842 + 3.78651i 0 −4.55842 7.89542i 0 −8.05842 4.00772i 0
5.2 0 2.18614 2.05446i 0 −2.05842 1.18843i 0 4.05842 + 7.02939i 0 0.558422 8.98266i 0
29.1 0 −0.686141 2.92048i 0 6.55842 3.78651i 0 −4.55842 + 7.89542i 0 −8.05842 + 4.00772i 0
29.2 0 2.18614 + 2.05446i 0 −2.05842 + 1.18843i 0 4.05842 7.02939i 0 0.558422 + 8.98266i 0
 $$n$$: 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
9.d 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.g.a 4
3.b odd 2 1 108.3.g.a 4
4.b odd 2 1 144.3.q.b 4
5.b even 2 1 900.3.p.a 4
5.c odd 4 2 900.3.u.a 8
8.b even 2 1 576.3.q.d 4
8.d odd 2 1 576.3.q.g 4
9.c even 3 1 108.3.g.a 4
9.c even 3 1 324.3.c.b 4
9.d odd 6 1 inner 36.3.g.a 4
9.d odd 6 1 324.3.c.b 4
12.b even 2 1 432.3.q.b 4
15.d odd 2 1 2700.3.p.b 4
15.e even 4 2 2700.3.u.b 8
24.f even 2 1 1728.3.q.h 4
24.h odd 2 1 1728.3.q.g 4
36.f odd 6 1 432.3.q.b 4
36.f odd 6 1 1296.3.e.e 4
36.h even 6 1 144.3.q.b 4
36.h even 6 1 1296.3.e.e 4
45.h odd 6 1 900.3.p.a 4
45.j even 6 1 2700.3.p.b 4
45.k odd 12 2 2700.3.u.b 8
45.l even 12 2 900.3.u.a 8
72.j odd 6 1 576.3.q.d 4
72.l even 6 1 576.3.q.g 4
72.n even 6 1 1728.3.q.g 4
72.p odd 6 1 1728.3.q.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.g.a 4 1.a even 1 1 trivial
36.3.g.a 4 9.d odd 6 1 inner
108.3.g.a 4 3.b odd 2 1
108.3.g.a 4 9.c even 3 1
144.3.q.b 4 4.b odd 2 1
144.3.q.b 4 36.h even 6 1
324.3.c.b 4 9.c even 3 1
324.3.c.b 4 9.d odd 6 1
432.3.q.b 4 12.b even 2 1
432.3.q.b 4 36.f odd 6 1
576.3.q.d 4 8.b even 2 1
576.3.q.d 4 72.j odd 6 1
576.3.q.g 4 8.d odd 2 1
576.3.q.g 4 72.l even 6 1
900.3.p.a 4 5.b even 2 1
900.3.p.a 4 45.h odd 6 1
900.3.u.a 8 5.c odd 4 2
900.3.u.a 8 45.l even 12 2
1296.3.e.e 4 36.f odd 6 1
1296.3.e.e 4 36.h even 6 1
1728.3.q.g 4 24.h odd 2 1
1728.3.q.g 4 72.n even 6 1
1728.3.q.h 4 24.f even 2 1
1728.3.q.h 4 72.p odd 6 1
2700.3.p.b 4 15.d odd 2 1
2700.3.p.b 4 45.j even 6 1
2700.3.u.b 8 15.e even 4 2
2700.3.u.b 8 45.k odd 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(36, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 3 T^{3} + 12 T^{2} - 27 T + 81$$
$5$ $$T^{4} - 9 T^{3} + 9 T^{2} + 162 T + 324$$
$7$ $$T^{4} + T^{3} + 75 T^{2} - 74 T + 5476$$
$11$ $$T^{4} + 36 T^{3} + 441 T^{2} + \cdots + 81$$
$13$ $$T^{4} - 5 T^{3} + 93 T^{2} + \cdots + 4624$$
$17$ $$T^{4} + 387 T^{2} + 20736$$
$19$ $$(T^{2} - T - 74)^{2}$$
$23$ $$T^{4} - 99 T^{3} + 4059 T^{2} + \cdots + 627264$$
$29$ $$T^{4} + 63 T^{3} + 441 T^{2} + \cdots + 777924$$
$31$ $$T^{4} + 7 T^{3} + 705 T^{2} + \cdots + 430336$$
$37$ $$(T^{2} + 32 T - 932)^{2}$$
$41$ $$T^{4} + 18 T^{3} - 1449 T^{2} + \cdots + 2424249$$
$43$ $$(T^{2} + 23 T + 529)^{2}$$
$47$ $$T^{4} + 81 T^{3} + 2511 T^{2} + \cdots + 104976$$
$53$ $$T^{4} + 4032 T^{2} + \cdots + 1327104$$
$59$ $$T^{4} - 126 T^{3} + 5031 T^{2} + \cdots + 68121$$
$61$ $$T^{4} - 41 T^{3} + 1929 T^{2} + \cdots + 61504$$
$67$ $$T^{4} - 116 T^{3} + 12765 T^{2} + \cdots + 477481$$
$71$ $$T^{4} + 1548 T^{2} + 331776$$
$73$ $$(T^{2} - 43 T - 206)^{2}$$
$79$ $$T^{4} - 83 T^{3} + 7023 T^{2} + \cdots + 17956$$
$83$ $$T^{4} + 81 T^{3} + 2511 T^{2} + \cdots + 104976$$
$89$ $$T^{4} + 24768 T^{2} + \cdots + 84934656$$
$97$ $$T^{4} + 196 T^{3} + \cdots + 86620249$$