# Properties

 Label 108.6.b.a Level 108 Weight 6 Character orbit 108.b Analytic conductor 17.321 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 108.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.3214525398$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{-29})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{2} ) q^{2} + ( -26 - 2 \beta_{3} ) q^{4} -17 \beta_{2} q^{5} -17 \beta_{3} q^{7} + ( 84 \beta_{1} + 20 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{2} ) q^{2} + ( -26 - 2 \beta_{3} ) q^{4} -17 \beta_{2} q^{5} -17 \beta_{3} q^{7} + ( 84 \beta_{1} + 20 \beta_{2} ) q^{8} + ( -493 - 17 \beta_{3} ) q^{10} -335 \beta_{1} q^{11} -166 q^{13} + ( 493 \beta_{1} - 51 \beta_{2} ) q^{14} + ( 328 + 104 \beta_{3} ) q^{16} + 154 \beta_{2} q^{17} -72 \beta_{3} q^{19} + ( 986 \beta_{1} + 442 \beta_{2} ) q^{20} + ( 1005 - 335 \beta_{3} ) q^{22} + 2228 \beta_{1} q^{23} -5256 q^{25} + ( 166 \beta_{1} + 166 \beta_{2} ) q^{26} + ( -2958 + 442 \beta_{3} ) q^{28} + 634 \beta_{2} q^{29} -679 \beta_{3} q^{31} + ( -3344 \beta_{1} - 16 \beta_{2} ) q^{32} + ( 4466 + 154 \beta_{3} ) q^{34} + 8381 \beta_{1} q^{35} + 15332 q^{37} + ( 2088 \beta_{1} - 216 \beta_{2} ) q^{38} + ( 9860 + 1428 \beta_{3} ) q^{40} + 1876 \beta_{2} q^{41} + 322 \beta_{3} q^{43} + ( 8710 \beta_{1} - 2010 \beta_{2} ) q^{44} + ( -6684 + 2228 \beta_{3} ) q^{46} -4082 \beta_{1} q^{47} -8336 q^{49} + ( 5256 \beta_{1} + 5256 \beta_{2} ) q^{50} + ( 4316 + 332 \beta_{3} ) q^{52} -3209 \beta_{2} q^{53} -5695 \beta_{3} q^{55} + ( -9860 \beta_{1} + 4284 \beta_{2} ) q^{56} + ( 18386 + 634 \beta_{3} ) q^{58} -16414 \beta_{1} q^{59} -53188 q^{61} + ( 19691 \beta_{1} - 2037 \beta_{2} ) q^{62} + ( 9568 - 3360 \beta_{3} ) q^{64} + 2822 \beta_{2} q^{65} -4402 \beta_{3} q^{67} + ( -8932 \beta_{1} - 4004 \beta_{2} ) q^{68} + ( -25143 + 8381 \beta_{3} ) q^{70} -15486 \beta_{1} q^{71} -30739 q^{73} + ( -15332 \beta_{1} - 15332 \beta_{2} ) q^{74} + ( -12528 + 1872 \beta_{3} ) q^{76} -17085 \beta_{2} q^{77} -7526 \beta_{3} q^{79} + ( -51272 \beta_{1} - 5576 \beta_{2} ) q^{80} + ( 54404 + 1876 \beta_{3} ) q^{82} + 6583 \beta_{1} q^{83} + 75922 q^{85} + ( -9338 \beta_{1} + 966 \beta_{2} ) q^{86} + ( -84420 + 6700 \beta_{3} ) q^{88} + 12754 \beta_{2} q^{89} + 2822 \beta_{3} q^{91} + ( -57928 \beta_{1} + 13368 \beta_{2} ) q^{92} + ( 12246 - 4082 \beta_{3} ) q^{94} + 35496 \beta_{1} q^{95} -13717 q^{97} + ( 8336 \beta_{1} + 8336 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 104q^{4} + O(q^{10})$$ $$4q - 104q^{4} - 1972q^{10} - 664q^{13} + 1312q^{16} + 4020q^{22} - 21024q^{25} - 11832q^{28} + 17864q^{34} + 61328q^{37} + 39440q^{40} - 26736q^{46} - 33344q^{49} + 17264q^{52} + 73544q^{58} - 212752q^{61} + 38272q^{64} - 100572q^{70} - 122956q^{73} - 50112q^{76} + 217616q^{82} + 303688q^{85} - 337680q^{88} + 48984q^{94} - 54868q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 13 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 21 \nu$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 13$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 13$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{2} + 21 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 −0.866025 + 2.69258i −0.866025 − 2.69258i 0.866025 + 2.69258i 0.866025 − 2.69258i
−1.73205 5.38516i 0 −26.0000 + 18.6548i 91.5478i 0 158.565i 145.492 + 107.703i 0 −493.000 + 158.565i
107.2 −1.73205 + 5.38516i 0 −26.0000 18.6548i 91.5478i 0 158.565i 145.492 107.703i 0 −493.000 158.565i
107.3 1.73205 5.38516i 0 −26.0000 18.6548i 91.5478i 0 158.565i −145.492 + 107.703i 0 −493.000 158.565i
107.4 1.73205 + 5.38516i 0 −26.0000 + 18.6548i 91.5478i 0 158.565i −145.492 107.703i 0 −493.000 + 158.565i
 $$n$$: 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
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 108.6.b.a 4
3.b odd 2 1 inner 108.6.b.a 4
4.b odd 2 1 inner 108.6.b.a 4
12.b even 2 1 inner 108.6.b.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.6.b.a 4 1.a even 1 1 trivial
108.6.b.a 4 3.b odd 2 1 inner
108.6.b.a 4 4.b odd 2 1 inner
108.6.b.a 4 12.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 8381$$ acting on $$S_{6}^{\mathrm{new}}(108, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 52 T^{2} + 1024 T^{4}$$
$3$ 1
$5$ $$( 1 + 2131 T^{2} + 9765625 T^{4} )^{2}$$
$7$ $$( 1 - 8471 T^{2} + 282475249 T^{4} )^{2}$$
$11$ $$( 1 - 14573 T^{2} + 25937424601 T^{4} )^{2}$$
$13$ $$( 1 + 166 T + 371293 T^{2} )^{4}$$
$17$ $$( 1 - 2151950 T^{2} + 2015993900449 T^{4} )^{2}$$
$19$ $$( 1 - 4501190 T^{2} + 6131066257801 T^{4} )^{2}$$
$23$ $$( 1 - 2019266 T^{2} + 41426511213649 T^{4} )^{2}$$
$29$ $$( 1 - 29365574 T^{2} + 420707233300201 T^{4} )^{2}$$
$31$ $$( 1 - 17147735 T^{2} + 819628286980801 T^{4} )^{2}$$
$37$ $$( 1 - 15332 T + 69343957 T^{2} )^{4}$$
$41$ $$( 1 - 129650498 T^{2} + 13422659310152401 T^{4} )^{2}$$
$43$ $$( 1 - 284996378 T^{2} + 21611482313284249 T^{4} )^{2}$$
$47$ $$( 1 + 408701842 T^{2} + 52599132235830049 T^{4} )^{2}$$
$53$ $$( 1 - 537758237 T^{2} + 174887470365513049 T^{4} )^{2}$$
$59$ $$( 1 + 621590410 T^{2} + 511116753300641401 T^{4} )^{2}$$
$61$ $$( 1 + 53188 T + 844596301 T^{2} )^{4}$$
$67$ $$( 1 - 1014398666 T^{2} + 1822837804551761449 T^{4} )^{2}$$
$71$ $$( 1 + 2889010114 T^{2} + 3255243551009881201 T^{4} )^{2}$$
$73$ $$( 1 + 30739 T + 2073071593 T^{2} )^{4}$$
$79$ $$( 1 - 1226373986 T^{2} + 9468276082626847201 T^{4} )^{2}$$
$83$ $$( 1 + 7748073619 T^{2} + 15516041187205853449 T^{4} )^{2}$$
$89$ $$( 1 - 6450847934 T^{2} + 31181719929966183601 T^{4} )^{2}$$
$97$ $$( 1 + 13717 T + 8587340257 T^{2} )^{4}$$