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
Defining polynomial: \(x^{6} + 13 x^{4} + 49 x^{2} + 48\)
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

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( T^{6} \)
$5$ \( 746496 + 147744 T + 34425 T^{2} + 702 T^{3} + 207 T^{4} + 6 T^{5} + T^{6} \)
$7$ \( 21141604 - 2800182 T + 343293 T^{2} - 12850 T^{3} + 645 T^{4} + 6 T^{5} + T^{6} \)
$11$ \( 4386545361 + 79278507 T + 4810590 T^{2} + 71415 T^{3} + 3798 T^{4} + 51 T^{5} + T^{6} \)
$13$ \( 4155865156 - 339800286 T + 28557033 T^{2} - 65680 T^{3} + 5415 T^{4} - 12 T^{5} + T^{6} \)
$17$ \( ( 577476 - 6624 T - 111 T^{2} + T^{3} )^{2} \)
$19$ \( ( -216368 - 7512 T - 15 T^{2} + T^{3} )^{2} \)
$23$ \( 5391684 + 24053598 T + 106821261 T^{2} + 2170746 T^{3} + 33741 T^{4} + 210 T^{5} + T^{6} \)
$29$ \( 8292430196964 + 186420419946 T + 2877755121 T^{2} + 23760756 T^{3} + 143199 T^{4} + 456 T^{5} + T^{6} \)
$31$ \( 9331729724944 - 106774004964 T + 1368342033 T^{2} - 4431832 T^{3} + 37257 T^{4} - 48 T^{5} + T^{6} \)
$37$ \( ( -682352 - 20508 T + 48 T^{2} + T^{3} )^{2} \)
$41$ \( 139158426750849 + 2637128984193 T + 39393550530 T^{2} + 176932161 T^{3} + 581058 T^{4} + 897 T^{5} + T^{6} \)
$43$ \( 2031049672201 - 266373174441 T + 35118818502 T^{2} + 21260963 T^{3} + 203550 T^{4} - 129 T^{5} + T^{6} \)
$47$ \( 4167836333323524 + 10421917854606 T + 59760297693 T^{2} + 44849538 T^{3} + 433917 T^{4} + 522 T^{5} + T^{6} \)
$53$ \( ( -11853648 + 317700 T - 1104 T^{2} + T^{3} )^{2} \)
$59$ \( 9388197832500369 + 28223393906205 T + 128739350214 T^{2} + 61833321 T^{3} + 496494 T^{4} + 453 T^{5} + T^{6} \)
$61$ \( 1463461189696 - 9606513576 T + 549373353 T^{2} + 5611754 T^{3} + 153663 T^{4} + 402 T^{5} + T^{6} \)
$67$ \( 9579414973969 - 248016683379 T + 5762049270 T^{2} - 23258455 T^{3} + 125502 T^{4} + 213 T^{5} + T^{6} \)
$71$ \( ( -113211648 - 423072 T + 60 T^{2} + T^{3} )^{2} \)
$73$ \( ( 158369284 - 381048 T - 375 T^{2} + T^{3} )^{2} \)
$79$ \( 318578661907984 - 389977819428 T + 10329900945 T^{2} - 23636896 T^{3} + 326553 T^{4} - 552 T^{5} + T^{6} \)
$83$ \( 12101965948944 - 409428996084 T + 15980660505 T^{2} + 65070540 T^{3} + 492237 T^{4} - 612 T^{5} + T^{6} \)
$89$ \( ( -170122248 - 774180 T - 462 T^{2} + T^{3} )^{2} \)
$97$ \( 7465050374673529 + 89825526132243 T + 1072818160242 T^{2} + 269487659 T^{3} + 1048290 T^{4} - 93 T^{5} + T^{6} \)
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