# Properties

 Label 108.3.d.d Level 108 Weight 3 Character orbit 108.d Analytic conductor 2.943 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 108.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.94278685509$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.207360000.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}\cdot 3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( -1 + \beta_{5} ) q^{4} + ( -\beta_{1} - \beta_{2} ) q^{5} + ( -1 - \beta_{3} + \beta_{5} ) q^{7} + ( \beta_{2} + \beta_{7} ) q^{8} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( -1 + \beta_{5} ) q^{4} + ( -\beta_{1} - \beta_{2} ) q^{5} + ( -1 - \beta_{3} + \beta_{5} ) q^{7} + ( \beta_{2} + \beta_{7} ) q^{8} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{10} + ( \beta_{1} - \beta_{6} - 2 \beta_{7} ) q^{11} + ( -2 + \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{13} + ( 2 \beta_{1} + \beta_{2} + \beta_{6} + \beta_{7} ) q^{14} + ( 7 - \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{16} + ( -3 \beta_{1} + 3 \beta_{2} + 2 \beta_{6} ) q^{17} + \beta_{4} q^{19} + 4 \beta_{1} q^{20} + ( 1 + 3 \beta_{3} - 5 \beta_{4} + \beta_{5} ) q^{22} + ( \beta_{1} + 10 \beta_{2} - 3 \beta_{6} - 2 \beta_{7} ) q^{23} + ( 5 - 2 \beta_{3} + 2 \beta_{4} - 6 \beta_{5} ) q^{25} + ( 3 \beta_{2} - 2 \beta_{6} + 2 \beta_{7} ) q^{26} + ( -12 + \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{28} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{29} + ( 4 + 4 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} ) q^{31} + ( -4 \beta_{1} - 10 \beta_{2} + 4 \beta_{6} + 2 \beta_{7} ) q^{32} + ( -21 - 3 \beta_{3} - 3 \beta_{4} - 5 \beta_{5} ) q^{34} + ( \beta_{1} - 20 \beta_{2} + 3 \beta_{6} - 2 \beta_{7} ) q^{35} + ( -6 - \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{37} + ( -2 \beta_{1} - \beta_{2} + \beta_{6} + \beta_{7} ) q^{38} + ( -8 + 4 \beta_{3} + 4 \beta_{4} ) q^{40} + ( -12 \beta_{2} - 4 \beta_{6} ) q^{41} + ( 2 + 2 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} ) q^{43} + ( 4 \beta_{1} + 4 \beta_{2} - 8 \beta_{6} - 4 \beta_{7} ) q^{44} + ( 41 + 3 \beta_{3} - 5 \beta_{4} - 7 \beta_{5} ) q^{46} + ( -3 \beta_{1} + 10 \beta_{2} + \beta_{6} + 6 \beta_{7} ) q^{47} + ( -16 + 2 \beta_{3} - 2 \beta_{4} + 6 \beta_{5} ) q^{49} + ( -7 \beta_{2} + 4 \beta_{6} - 4 \beta_{7} ) q^{50} + ( 45 - 2 \beta_{3} + 6 \beta_{4} - \beta_{5} ) q^{52} + ( 10 \beta_{1} + 22 \beta_{2} + 4 \beta_{6} ) q^{53} + ( -2 - 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} ) q^{55} + ( -12 \beta_{1} + 7 \beta_{2} + 4 \beta_{6} + 3 \beta_{7} ) q^{56} + ( 4 - 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{58} + ( -3 \beta_{1} + 20 \beta_{2} - \beta_{6} + 6 \beta_{7} ) q^{59} + ( 4 + \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{61} + ( -8 \beta_{6} - 8 \beta_{7} ) q^{62} + ( -50 - 6 \beta_{3} + 2 \beta_{4} + 6 \beta_{5} ) q^{64} + ( 15 \beta_{1} + 9 \beta_{2} - 2 \beta_{6} ) q^{65} + ( -8 - 8 \beta_{3} + 9 \beta_{4} + 8 \beta_{5} ) q^{67} + ( 12 \beta_{1} + 24 \beta_{2} - 8 \beta_{7} ) q^{68} + ( -79 + 3 \beta_{3} - 5 \beta_{4} + 17 \beta_{5} ) q^{70} + ( -2 \beta_{1} - 40 \beta_{2} + 10 \beta_{6} + 4 \beta_{7} ) q^{71} + ( 27 + 2 \beta_{3} - 2 \beta_{4} + 6 \beta_{5} ) q^{73} + ( 5 \beta_{2} + 2 \beta_{6} - 2 \beta_{7} ) q^{74} + ( -6 - 3 \beta_{3} + \beta_{4} ) q^{76} + ( 11 \beta_{1} - 13 \beta_{2} - 8 \beta_{6} ) q^{77} + ( -7 - 7 \beta_{3} - 16 \beta_{4} + 7 \beta_{5} ) q^{79} + ( -16 \beta_{1} + 4 \beta_{2} + 4 \beta_{7} ) q^{80} + ( 48 + 16 \beta_{5} ) q^{82} + ( -2 \beta_{1} + 2 \beta_{6} + 4 \beta_{7} ) q^{83} + ( 58 - 6 \beta_{3} + 6 \beta_{4} - 18 \beta_{5} ) q^{85} + ( -16 \beta_{1} - 8 \beta_{2} + 4 \beta_{6} + 4 \beta_{7} ) q^{86} + ( 92 + 8 \beta_{3} - 8 \beta_{4} + 4 \beta_{5} ) q^{88} + ( -3 \beta_{1} + 3 \beta_{2} + 2 \beta_{6} ) q^{89} + ( 4 + 4 \beta_{3} + 19 \beta_{4} - 4 \beta_{5} ) q^{91} + ( 4 \beta_{1} - 36 \beta_{2} - 8 \beta_{6} - 12 \beta_{7} ) q^{92} + ( 37 - 9 \beta_{3} + 15 \beta_{4} - 11 \beta_{5} ) q^{94} + ( -\beta_{1} + 10 \beta_{2} - \beta_{6} + 2 \beta_{7} ) q^{95} + ( -39 + 8 \beta_{3} - 8 \beta_{4} + 24 \beta_{5} ) q^{97} + ( 18 \beta_{2} - 4 \beta_{6} + 4 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{4} + O(q^{10})$$ $$8q - 4q^{4} + 16q^{10} - 8q^{13} + 56q^{16} + 24q^{25} - 108q^{28} - 176q^{34} - 56q^{37} - 80q^{40} + 288q^{46} - 112q^{49} + 364q^{52} + 64q^{58} + 40q^{61} - 352q^{64} - 576q^{70} + 232q^{73} - 36q^{76} + 448q^{82} + 416q^{85} + 720q^{88} + 288q^{94} - 248q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 6 x^{6} + 32 x^{4} + 24 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$3 \nu^{7} + 16 \nu^{5} + 128 \nu^{3} + 72 \nu$$$$)/64$$ $$\beta_{2}$$ $$=$$ $$($$$$-9 \nu^{7} - 48 \nu^{5} - 256 \nu^{3} - 88 \nu$$$$)/64$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{6} + 24 \nu^{4} + 96 \nu^{2} + 88$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$-9 \nu^{6} - 48 \nu^{4} - 288 \nu^{2} - 120$$$$)/32$$ $$\beta_{5}$$ $$=$$ $$($$$$-7 \nu^{6} - 32 \nu^{4} - 160 \nu^{2} + 56$$$$)/32$$ $$\beta_{6}$$ $$=$$ $$($$$$-33 \nu^{7} - 176 \nu^{5} - 896 \nu^{3} + 104 \nu$$$$)/64$$ $$\beta_{7}$$ $$=$$ $$($$$$33 \nu^{7} + 208 \nu^{5} + 1088 \nu^{3} + 792 \nu$$$$)/64$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} - 4 \beta_{2} - \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{5} - 3 \beta_{4} - \beta_{3} - 11$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{6} + 7 \beta_{2} + 10 \beta_{1}$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$($$$$4 \beta_{4} + 6 \beta_{3} - 18$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$6 \beta_{7} - 5 \beta_{6} + 35 \beta_{2} - 16 \beta_{1}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$-48 \beta_{5} + 16 \beta_{4} - 16 \beta_{3} + 232$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-32 \beta_{7} + 36 \beta_{6} - 288 \beta_{2} - 52 \beta_{1}$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$55$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.437016 − 0.756934i 0.437016 + 0.756934i −1.14412 − 1.98168i −1.14412 + 1.98168i 1.14412 − 1.98168i 1.14412 + 1.98168i −0.437016 − 0.756934i −0.437016 + 0.756934i
−1.85123 0.756934i 0 2.85410 + 2.80252i −1.08036 0 6.01392i −3.16228 7.34847i 0 2.00000 + 0.817763i
55.2 −1.85123 + 0.756934i 0 2.85410 2.80252i −1.08036 0 6.01392i −3.16228 + 7.34847i 0 2.00000 0.817763i
55.3 −0.270091 1.98168i 0 −3.85410 + 1.07047i −7.40492 0 9.47802i 3.16228 + 7.34847i 0 2.00000 + 14.6742i
55.4 −0.270091 + 1.98168i 0 −3.85410 1.07047i −7.40492 0 9.47802i 3.16228 7.34847i 0 2.00000 14.6742i
55.5 0.270091 1.98168i 0 −3.85410 1.07047i 7.40492 0 9.47802i −3.16228 + 7.34847i 0 2.00000 14.6742i
55.6 0.270091 + 1.98168i 0 −3.85410 + 1.07047i 7.40492 0 9.47802i −3.16228 7.34847i 0 2.00000 + 14.6742i
55.7 1.85123 0.756934i 0 2.85410 2.80252i 1.08036 0 6.01392i 3.16228 7.34847i 0 2.00000 0.817763i
55.8 1.85123 + 0.756934i 0 2.85410 + 2.80252i 1.08036 0 6.01392i 3.16228 + 7.34847i 0 2.00000 + 0.817763i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 55.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 56 T_{5}^{2} + 64$$ acting on $$S_{3}^{\mathrm{new}}(108, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T^{2} - 12 T^{4} + 32 T^{6} + 256 T^{8}$$
$3$ 1
$5$ $$( 1 + 44 T^{2} + 1014 T^{4} + 27500 T^{6} + 390625 T^{8} )^{2}$$
$7$ $$( 1 - 70 T^{2} + 5307 T^{4} - 168070 T^{6} + 5764801 T^{8} )^{2}$$
$11$ $$( 1 - 124 T^{2} + 15126 T^{4} - 1815484 T^{6} + 214358881 T^{8} )^{2}$$
$13$ $$( 1 + 2 T + 159 T^{2} + 338 T^{3} + 28561 T^{4} )^{4}$$
$17$ $$( 1 + 140 T^{2} + 136662 T^{4} + 11692940 T^{6} + 6975757441 T^{8} )^{2}$$
$19$ $$( 1 - 695 T^{2} + 130321 T^{4} )^{4}$$
$23$ $$( 1 - 604 T^{2} + 632886 T^{4} - 169023964 T^{6} + 78310985281 T^{8} )^{2}$$
$29$ $$( 1 + 2468 T^{2} + 2752998 T^{4} + 1745569508 T^{6} + 500246412961 T^{8} )^{2}$$
$31$ $$( 1 - 1540 T^{2} + 1702662 T^{4} - 1422222340 T^{6} + 852891037441 T^{8} )^{2}$$
$37$ $$( 1 + 14 T + 2607 T^{2} + 19166 T^{3} + 1874161 T^{4} )^{4}$$
$41$ $$( 1 + 3140 T^{2} + 5167302 T^{4} + 8872889540 T^{6} + 7984925229121 T^{8} )^{2}$$
$43$ $$( 1 - 4516 T^{2} + 10784166 T^{4} - 15439305316 T^{6} + 11688200277601 T^{8} )^{2}$$
$47$ $$( 1 - 4444 T^{2} + 14436726 T^{4} - 21685302364 T^{6} + 23811286661761 T^{8} )^{2}$$
$53$ $$( 1 - 508 T^{2} + 14912358 T^{4} - 4008364348 T^{6} + 62259690411361 T^{8} )^{2}$$
$59$ $$( 1 - 6076 T^{2} + 23942166 T^{4} - 73625085436 T^{6} + 146830437604321 T^{8} )^{2}$$
$61$ $$( 1 - 10 T + 7287 T^{2} - 37210 T^{3} + 13845841 T^{4} )^{4}$$
$67$ $$( 1 - 8110 T^{2} + 40110387 T^{4} - 163425591310 T^{6} + 406067677556641 T^{8} )^{2}$$
$71$ $$( 1 - 292 T^{2} + 42446598 T^{4} - 7420210852 T^{6} + 645753531245761 T^{8} )^{2}$$
$73$ $$( 1 - 58 T + 10779 T^{2} - 309082 T^{3} + 28398241 T^{4} )^{4}$$
$79$ $$( 1 - 934 T^{2} - 28603749 T^{4} - 36379375654 T^{6} + 1517108809906561 T^{8} )^{2}$$
$83$ $$( 1 - 26116 T^{2} + 265140006 T^{4} - 1239421511236 T^{6} + 2252292232139041 T^{8} )^{2}$$
$89$ $$( 1 + 30668 T^{2} + 360580758 T^{4} + 1924179046988 T^{6} + 3936588805702081 T^{8} )^{2}$$
$97$ $$( 1 + 62 T + 8259 T^{2} + 583358 T^{3} + 88529281 T^{4} )^{4}$$