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

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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:

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