# Properties

 Label 108.6.a.b Level 108 Weight 6 Character orbit 108.a Self dual yes Analytic conductor 17.321 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 108.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$17.3214525398$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 9\sqrt{41}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{5} + ( -16 + 3 \beta ) q^{7} +O(q^{10})$$ $$q -\beta q^{5} + ( -16 + 3 \beta ) q^{7} + ( -243 + 8 \beta ) q^{11} + ( 104 - 12 \beta ) q^{13} + ( -972 - 20 \beta ) q^{17} + ( -316 + 6 \beta ) q^{19} + ( -3402 + 4 \beta ) q^{23} + 196 q^{25} + ( -5832 + 38 \beta ) q^{29} + ( 1664 + 81 \beta ) q^{31} + ( -9963 + 16 \beta ) q^{35} + ( -4978 - 60 \beta ) q^{37} + ( -6804 - 58 \beta ) q^{41} + ( 2480 - 162 \beta ) q^{43} + ( -9234 - 264 \beta ) q^{47} + ( 13338 - 96 \beta ) q^{49} + ( 5832 + 341 \beta ) q^{53} + ( -26568 + 243 \beta ) q^{55} + ( -972 + 376 \beta ) q^{59} + ( 4088 + 720 \beta ) q^{61} + ( 39852 - 104 \beta ) q^{65} + ( 45032 - 450 \beta ) q^{67} + ( 22356 - 788 \beta ) q^{71} + ( -60607 - 504 \beta ) q^{73} + ( 83592 - 857 \beta ) q^{77} + ( 14384 - 840 \beta ) q^{79} + ( 7533 + 1752 \beta ) q^{83} + ( 66420 + 972 \beta ) q^{85} + ( 89424 + 358 \beta ) q^{89} + ( -121220 + 504 \beta ) q^{91} + ( -19926 + 316 \beta ) q^{95} + ( 44471 - 1272 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 32q^{7} + O(q^{10})$$ $$2q - 32q^{7} - 486q^{11} + 208q^{13} - 1944q^{17} - 632q^{19} - 6804q^{23} + 392q^{25} - 11664q^{29} + 3328q^{31} - 19926q^{35} - 9956q^{37} - 13608q^{41} + 4960q^{43} - 18468q^{47} + 26676q^{49} + 11664q^{53} - 53136q^{55} - 1944q^{59} + 8176q^{61} + 79704q^{65} + 90064q^{67} + 44712q^{71} - 121214q^{73} + 167184q^{77} + 28768q^{79} + 15066q^{83} + 132840q^{85} + 178848q^{89} - 242440q^{91} - 39852q^{95} + 88942q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 3.70156 −2.70156
0 0 0 −57.6281 0 156.884 0 0 0
1.2 0 0 0 57.6281 0 −188.884 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.6.a.b 2
3.b odd 2 1 108.6.a.c yes 2
4.b odd 2 1 432.6.a.r 2
9.c even 3 2 324.6.e.g 4
9.d odd 6 2 324.6.e.f 4
12.b even 2 1 432.6.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.6.a.b 2 1.a even 1 1 trivial
108.6.a.c yes 2 3.b odd 2 1
324.6.e.f 4 9.d odd 6 2
324.6.e.g 4 9.c even 3 2
432.6.a.q 2 12.b even 2 1
432.6.a.r 2 4.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(108))$$:

 $$T_{5}^{2} - 3321$$ $$T_{11}^{2} + 486 T_{11} - 153495$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 2929 T^{2} + 9765625 T^{4}$$
$7$ $$1 + 32 T + 3981 T^{2} + 537824 T^{3} + 282475249 T^{4}$$
$11$ $$1 + 486 T + 168607 T^{2} + 78270786 T^{3} + 25937424601 T^{4}$$
$13$ $$1 - 208 T + 275178 T^{2} - 77228944 T^{3} + 137858491849 T^{4}$$
$17$ $$1 + 1944 T + 2456098 T^{2} + 2760202008 T^{3} + 2015993900449 T^{4}$$
$19$ $$1 + 632 T + 4932498 T^{2} + 1564894568 T^{3} + 6131066257801 T^{4}$$
$23$ $$1 + 6804 T + 24393154 T^{2} + 43792877772 T^{3} + 41426511213649 T^{4}$$
$29$ $$1 + 11664 T + 70238998 T^{2} + 239242041936 T^{3} + 420707233300201 T^{4}$$
$31$ $$1 - 3328 T + 38238117 T^{2} - 95277814528 T^{3} + 819628286980801 T^{4}$$
$37$ $$1 + 9956 T + 151512798 T^{2} + 690388435892 T^{3} + 4808584372417849 T^{4}$$
$41$ $$1 + 13608 T + 266834974 T^{2} + 1576571183208 T^{3} + 13422659310152401 T^{4}$$
$43$ $$1 - 4960 T + 213010962 T^{2} - 729161877280 T^{3} + 21611482313284249 T^{4}$$
$47$ $$1 + 18468 T + 312496354 T^{2} + 4235543589276 T^{3} + 52599132235830049 T^{4}$$
$53$ $$1 - 11664 T + 484234009 T^{2} - 4877832230352 T^{3} + 174887470365513049 T^{4}$$
$59$ $$1 + 1944 T + 961283686 T^{2} + 1389812837256 T^{3} + 511116753300641401 T^{4}$$
$61$ $$1 - 8176 T - 15702054 T^{2} - 6905419356976 T^{3} + 713342911662882601 T^{4}$$
$67$ $$1 - 90064 T + 4055628738 T^{2} - 121597667636848 T^{3} + 1822837804551761449 T^{4}$$
$71$ $$1 - 44712 T + 2046094414 T^{2} - 80670702741912 T^{3} + 3255243551009881201 T^{4}$$
$73$ $$1 + 121214 T + 6975764499 T^{2} + 251285300073902 T^{3} + 4297625829703557649 T^{4}$$
$79$ $$1 - 28768 T + 4017714654 T^{2} - 88520758486432 T^{3} + 9468276082626847201 T^{4}$$
$83$ $$1 - 15066 T - 2258995409 T^{2} - 59345586327438 T^{3} + 15516041187205853449 T^{4}$$
$89$ $$1 - 178848 T + 18739138030 T^{2} - 998697864334752 T^{3} + 31181719929966183601 T^{4}$$
$97$ $$1 - 88942 T + 13779025491 T^{2} - 763775217138094 T^{3} + 73742412689492826049 T^{4}$$