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

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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:

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