# Properties

 Label 108.4.e.a Level 108 Weight 4 Character orbit 108.e Analytic conductor 6.372 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 108.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.37220628062$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.6831243.2 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}\cdot 3^{5}$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \beta_{1} + \beta_{5} ) q^{5} + ( -2 \beta_{1} - \beta_{2} - \beta_{4} ) q^{7} +O(q^{10})$$ $$q + ( -2 + 2 \beta_{1} + \beta_{5} ) q^{5} + ( -2 \beta_{1} - \beta_{2} - \beta_{4} ) q^{7} + ( -17 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} ) q^{11} + ( 4 - 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 5 \beta_{5} ) q^{13} + ( 37 + 6 \beta_{3} - \beta_{4} + \beta_{5} ) q^{17} + ( 5 - 2 \beta_{3} - 7 \beta_{4} + 7 \beta_{5} ) q^{19} + ( -70 + 70 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 5 \beta_{5} ) q^{23} + ( -\beta_{1} - 6 \beta_{2} + 3 \beta_{4} ) q^{25} + ( -152 \beta_{1} + 5 \beta_{4} ) q^{29} + ( 16 - 16 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 15 \beta_{5} ) q^{31} + ( 184 - 15 \beta_{3} - \beta_{4} + \beta_{5} ) q^{35} + ( -16 + 8 \beta_{3} + 10 \beta_{4} - 10 \beta_{5} ) q^{37} + ( -299 + 299 \beta_{1} + 12 \beta_{2} + 12 \beta_{3} - 14 \beta_{5} ) q^{41} + ( 43 \beta_{1} + 27 \beta_{2} ) q^{43} + ( -174 \beta_{1} + 21 \beta_{2} - 39 \beta_{4} ) q^{47} + ( -75 + 75 \beta_{1} + 22 \beta_{2} + 22 \beta_{3} + 13 \beta_{5} ) q^{49} + ( 368 + 22 \beta_{4} - 22 \beta_{5} ) q^{53} + ( -36 - 15 \beta_{3} + 33 \beta_{4} - 33 \beta_{5} ) q^{55} + ( -151 + 151 \beta_{1} - 15 \beta_{2} - 15 \beta_{3} - 34 \beta_{5} ) q^{59} + ( -134 \beta_{1} - 12 \beta_{2} - 3 \beta_{4} ) q^{61} + ( -370 \beta_{1} + 6 \beta_{2} + 37 \beta_{4} ) q^{65} + ( -71 + 71 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 24 \beta_{5} ) q^{67} + ( -20 - 12 \beta_{3} - 52 \beta_{4} + 52 \beta_{5} ) q^{71} + ( 125 + 30 \beta_{3} - 21 \beta_{4} + 21 \beta_{5} ) q^{73} + ( -376 + 376 \beta_{1} + 12 \beta_{2} + 12 \beta_{3} + 65 \beta_{5} ) q^{77} + ( 184 \beta_{1} - 5 \beta_{2} - 23 \beta_{4} ) q^{79} + ( 204 \beta_{1} - 33 \beta_{2} + 15 \beta_{4} ) q^{83} + ( 396 - 396 \beta_{1} - 60 \beta_{2} - 60 \beta_{3} + 30 \beta_{5} ) q^{85} + ( 154 + 60 \beta_{3} + 44 \beta_{4} - 44 \beta_{5} ) q^{89} + ( -44 - 33 \beta_{3} - 75 \beta_{4} + 75 \beta_{5} ) q^{91} + ( 728 - 728 \beta_{1} - 24 \beta_{2} - 24 \beta_{3} + 44 \beta_{5} ) q^{95} + ( 31 \beta_{1} - 56 \beta_{2} + 70 \beta_{4} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{5} - 6q^{7} + O(q^{10})$$ $$6q - 6q^{5} - 6q^{7} - 51q^{11} + 12q^{13} + 222q^{17} + 30q^{19} - 210q^{23} - 3q^{25} - 456q^{29} + 48q^{31} + 1104q^{35} - 96q^{37} - 897q^{41} + 129q^{43} - 522q^{47} - 225q^{49} + 2208q^{53} - 216q^{55} - 453q^{59} - 402q^{61} - 1110q^{65} - 213q^{67} - 120q^{71} + 750q^{73} - 1128q^{77} + 552q^{79} + 612q^{83} + 1188q^{85} + 924q^{89} - 264q^{91} + 2184q^{95} + 93q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 13 x^{4} + 49 x^{2} + 48$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 9 \nu^{3} + 17 \nu + 4$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 21 \nu^{3} + 12 \nu^{2} + 77 \nu + 52$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$-6 \nu^{2} - 26$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{5} + 6 \nu^{4} - 3 \nu^{3} + 54 \nu^{2} + 31 \nu + 92$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{5} - 6 \nu^{4} - 3 \nu^{3} - 54 \nu^{2} + 31 \nu - 92$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 12 \beta_{1} - 6$$$$)/18$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} - 26$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{5} - 5 \beta_{4} + 4 \beta_{3} + 8 \beta_{2} - 36 \beta_{1} + 18$$$$)/9$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{5} + 2 \beta_{4} + 9 \beta_{3} + 142$$$$)/6$$ $$\nu^{5}$$ $$=$$ $$($$$$56 \beta_{5} + 56 \beta_{4} - 55 \beta_{3} - 110 \beta_{2} + 588 \beta_{1} - 294$$$$)/18$$

## Character values

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

 $$n$$ $$29$$ $$55$$ $$\chi(n)$$ $$-1 + \beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 1.23396i 2.63162i − 2.13353i − 1.23396i − 2.63162i 2.13353i
0 0 0 −6.92194 + 11.9892i 0 −15.3540 26.5939i 0 0 0
37.2 0 0 0 −2.44901 + 4.24182i 0 5.32725 + 9.22708i 0 0 0
37.3 0 0 0 6.37096 11.0348i 0 7.02674 + 12.1707i 0 0 0
73.1 0 0 0 −6.92194 11.9892i 0 −15.3540 + 26.5939i 0 0 0
73.2 0 0 0 −2.44901 4.24182i 0 5.32725 9.22708i 0 0 0
73.3 0 0 0 6.37096 + 11.0348i 0 7.02674 12.1707i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 73.3 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.4.e.a 6
3.b odd 2 1 36.4.e.a 6
4.b odd 2 1 432.4.i.d 6
9.c even 3 1 inner 108.4.e.a 6
9.c even 3 1 324.4.a.d 3
9.d odd 6 1 36.4.e.a 6
9.d odd 6 1 324.4.a.c 3
12.b even 2 1 144.4.i.d 6
36.f odd 6 1 432.4.i.d 6
36.f odd 6 1 1296.4.a.w 3
36.h even 6 1 144.4.i.d 6
36.h even 6 1 1296.4.a.v 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.4.e.a 6 3.b odd 2 1
36.4.e.a 6 9.d odd 6 1
108.4.e.a 6 1.a even 1 1 trivial
108.4.e.a 6 9.c even 3 1 inner
144.4.i.d 6 12.b even 2 1
144.4.i.d 6 36.h even 6 1
324.4.a.c 3 9.d odd 6 1
324.4.a.d 3 9.c even 3 1
432.4.i.d 6 4.b odd 2 1
432.4.i.d 6 36.f odd 6 1
1296.4.a.v 3 36.h even 6 1
1296.4.a.w 3 36.f odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(108, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 6 T - 168 T^{2} - 48 T^{3} + 12300 T^{4} - 82506 T^{5} - 1453754 T^{6} - 10313250 T^{7} + 192187500 T^{8} - 93750000 T^{9} - 41015625000 T^{10} + 183105468750 T^{11} + 3814697265625 T^{12}$$
$7$ $$1 + 6 T - 384 T^{2} - 14908 T^{3} - 19944 T^{4} + 2637054 T^{5} + 91900446 T^{6} + 904509522 T^{7} - 2346391656 T^{8} - 601591573156 T^{9} - 5315054285184 T^{10} + 28485369059658 T^{11} + 1628413597910449 T^{12}$$
$11$ $$1 + 51 T - 195 T^{2} + 3534 T^{3} + 550059 T^{4} - 94832265 T^{5} - 5462621714 T^{6} - 126221744715 T^{7} + 974463072099 T^{8} + 8332987139994 T^{9} - 611993533460595 T^{10} + 213039656640198201 T^{11} + 5559917313492231481 T^{12}$$
$13$ $$1 - 12 T - 1176 T^{2} - 39316 T^{3} - 1016784 T^{4} + 27173412 T^{5} + 16823666094 T^{6} + 59699986164 T^{7} - 4907822162256 T^{8} - 416926497348868 T^{9} - 27398548104037656 T^{10} - 614230716169089084 T^{11} +$$$$11\!\cdots\!29$$$$T^{12}$$
$17$ $$( 1 - 111 T + 8115 T^{2} - 513210 T^{3} + 39868995 T^{4} - 2679270159 T^{5} + 118587876497 T^{6} )^{2}$$
$19$ $$( 1 - 15 T + 13065 T^{2} - 422138 T^{3} + 89612835 T^{4} - 705688215 T^{5} + 322687697779 T^{6} )^{2}$$
$23$ $$1 + 210 T - 2760 T^{2} - 384324 T^{3} + 552096960 T^{4} + 31196345610 T^{5} - 3505833433730 T^{6} + 379565937036870 T^{7} + 81730164287797440 T^{8} - 692226195464106012 T^{9} - 60484363432376085960 T^{10} +$$$$55\!\cdots\!70$$$$T^{11} +$$$$32\!\cdots\!69$$$$T^{12}$$
$29$ $$1 + 456 T + 70032 T^{2} + 12639372 T^{3} + 4198346304 T^{4} + 668355791208 T^{5} + 72647268176734 T^{6} + 16300529391771912 T^{7} + 2497274291253355584 T^{8} +$$$$18\!\cdots\!68$$$$T^{9} +$$$$24\!\cdots\!12$$$$T^{10} +$$$$39\!\cdots\!44$$$$T^{11} +$$$$21\!\cdots\!61$$$$T^{12}$$
$31$ $$1 - 48 T - 52116 T^{2} - 3001864 T^{3} + 1349663076 T^{4} + 123641386272 T^{5} - 37354727750178 T^{6} + 3683400538429152 T^{7} + 1197830948059782756 T^{8} - 79368149937720490744 T^{9} -$$$$41\!\cdots\!76$$$$T^{10} -$$$$11\!\cdots\!48$$$$T^{11} +$$$$69\!\cdots\!41$$$$T^{12}$$
$37$ $$( 1 + 48 T + 131451 T^{2} + 4180336 T^{3} + 6658387503 T^{4} + 123154867632 T^{5} + 129961739795077 T^{6} )^{2}$$
$41$ $$1 + 897 T + 374295 T^{2} + 115110024 T^{3} + 34022054553 T^{4} + 9337061108679 T^{5} + 2414891032993726 T^{6} + 643519588671265359 T^{7} +$$$$16\!\cdots\!73$$$$T^{8} +$$$$37\!\cdots\!64$$$$T^{9} +$$$$84\!\cdots\!95$$$$T^{10} +$$$$13\!\cdots\!97$$$$T^{11} +$$$$10\!\cdots\!21$$$$T^{12}$$
$43$ $$1 - 129 T - 34971 T^{2} + 31517366 T^{3} - 3902024493 T^{4} - 741901043133 T^{5} + 1052539440287118 T^{6} - 58986326236375431 T^{7} - 24666113446343159157 T^{8} +$$$$15\!\cdots\!38$$$$T^{9} -$$$$13\!\cdots\!71$$$$T^{10} -$$$$40\!\cdots\!03$$$$T^{11} +$$$$25\!\cdots\!49$$$$T^{12}$$
$47$ $$1 + 522 T + 122448 T^{2} - 9346068 T^{3} - 15946914792 T^{4} - 4059391014414 T^{5} - 1114759766751410 T^{6} - 421458153289504722 T^{7} -$$$$17\!\cdots\!68$$$$T^{8} -$$$$10\!\cdots\!56$$$$T^{9} +$$$$14\!\cdots\!68$$$$T^{10} +$$$$62\!\cdots\!46$$$$T^{11} +$$$$12\!\cdots\!89$$$$T^{12}$$
$53$ $$( 1 - 1104 T + 764331 T^{2} - 340574064 T^{3} + 113791306287 T^{4} - 24469454686416 T^{5} + 3299763591802133 T^{6} )^{2}$$
$59$ $$1 + 453 T - 119643 T^{2} - 31203366 T^{3} - 1587796437 T^{4} - 12368009864103 T^{5} - 5050047108050786 T^{6} - 2540129497879610037 T^{7} - 66974101025938437117 T^{8} -$$$$27\!\cdots\!74$$$$T^{9} -$$$$21\!\cdots\!83$$$$T^{10} +$$$$16\!\cdots\!47$$$$T^{11} +$$$$75\!\cdots\!21$$$$T^{12}$$
$61$ $$1 + 402 T - 527280 T^{2} - 85634608 T^{3} + 245321825076 T^{4} + 19877822718498 T^{5} - 59790452550726954 T^{6} + 4511888078467394538 T^{7} +$$$$12\!\cdots\!36$$$$T^{8} -$$$$10\!\cdots\!28$$$$T^{9} -$$$$13\!\cdots\!80$$$$T^{10} +$$$$24\!\cdots\!02$$$$T^{11} +$$$$13\!\cdots\!81$$$$T^{12}$$
$67$ $$1 + 213 T - 776787 T^{2} - 87320974 T^{3} + 400716501795 T^{4} + 21812260017825 T^{5} - 135770111734985634 T^{6} + 6560320759741100475 T^{7} +$$$$36\!\cdots\!55$$$$T^{8} -$$$$23\!\cdots\!78$$$$T^{9} -$$$$63\!\cdots\!07$$$$T^{10} +$$$$52\!\cdots\!59$$$$T^{11} +$$$$74\!\cdots\!09$$$$T^{12}$$
$71$ $$( 1 + 60 T + 650661 T^{2} - 70262328 T^{3} + 232878729171 T^{4} + 7686017035260 T^{5} + 45848500718449031 T^{6} )^{2}$$
$73$ $$( 1 - 375 T + 786003 T^{2} - 133393466 T^{3} + 305768529051 T^{4} - 56750334858375 T^{5} + 58871586708267913 T^{6} )^{2}$$
$79$ $$1 - 552 T - 1152564 T^{2} + 248520632 T^{3} + 1114530677604 T^{4} - 108173831172696 T^{5} - 604232203209226626 T^{6} - 53333917547554863144 T^{7} +$$$$27\!\cdots\!84$$$$T^{8} +$$$$29\!\cdots\!08$$$$T^{9} -$$$$68\!\cdots\!24$$$$T^{10} -$$$$16\!\cdots\!48$$$$T^{11} +$$$$14\!\cdots\!61$$$$T^{12}$$
$83$ $$1 - 612 T - 1223124 T^{2} + 415004184 T^{3} + 1212827483508 T^{4} - 194529560235444 T^{5} - 723127027932774218 T^{6} -$$$$11\!\cdots\!28$$$$T^{7} +$$$$39\!\cdots\!52$$$$T^{8} +$$$$77\!\cdots\!52$$$$T^{9} -$$$$13\!\cdots\!64$$$$T^{10} -$$$$37\!\cdots\!84$$$$T^{11} +$$$$34\!\cdots\!09$$$$T^{12}$$
$89$ $$( 1 - 462 T + 1340727 T^{2} - 821513604 T^{3} + 945170972463 T^{4} - 229605356423982 T^{5} + 350356403707485209 T^{6} )^{2}$$
$97$ $$1 - 93 T - 1689729 T^{2} + 354366248 T^{3} + 1310601420297 T^{4} - 224207143910091 T^{5} - 1053719651856288018 T^{6} -$$$$20\!\cdots\!43$$$$T^{7} +$$$$10\!\cdots\!13$$$$T^{8} +$$$$26\!\cdots\!16$$$$T^{9} -$$$$11\!\cdots\!89$$$$T^{10} -$$$$58\!\cdots\!49$$$$T^{11} +$$$$57\!\cdots\!89$$$$T^{12}$$