Properties

Label 117.4.q.f
Level $117$
Weight $4$
Character orbit 117.q
Analytic conductor $6.903$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.90322347067\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 32x^{10} + 823x^{8} - 5964x^{6} + 32913x^{4} - 47034x^{2} + 54756 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{2} + ( - \beta_{8} - 3 \beta_{3}) q^{4} + (\beta_{9} - \beta_{6} + \beta_{2}) q^{5} + ( - \beta_{11} - \beta_{5} - \beta_{3} - 2) q^{7} + ( - \beta_{10} + \beta_{4} + 3 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{2} + ( - \beta_{8} - 3 \beta_{3}) q^{4} + (\beta_{9} - \beta_{6} + \beta_{2}) q^{5} + ( - \beta_{11} - \beta_{5} - \beta_{3} - 2) q^{7} + ( - \beta_{10} + \beta_{4} + 3 \beta_{2}) q^{8} + ( - 2 \beta_{11} - 2 \beta_{8} - \beta_{7} + \beta_{5} - 10 \beta_{3} - 10) q^{10} + ( - \beta_{10} - \beta_{9} + 4 \beta_{2} + 4 \beta_1) q^{11} + ( - 4 \beta_{11} - \beta_{8} - \beta_{7} + \beta_{5} - 4 \beta_{3} - 16) q^{13} + ( - \beta_{9} - \beta_{6} + 2 \beta_{2} + 4 \beta_1) q^{14} + ( - 4 \beta_{11} - 5 \beta_{8} - 2 \beta_{7} + 3 \beta_{5} - 17 \beta_{3} + \cdots - 17) q^{16}+ \cdots + (12 \beta_{6} + 9 \beta_{4} - 289 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 16 q^{4} - 18 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 16 q^{4} - 18 q^{7} - 56 q^{10} - 154 q^{13} - 92 q^{16} - 48 q^{19} + 292 q^{22} - 92 q^{25} + 552 q^{28} - 360 q^{37} - 448 q^{40} - 810 q^{43} + 996 q^{46} - 1068 q^{49} - 700 q^{52} + 460 q^{55} + 3048 q^{58} + 1294 q^{61} - 184 q^{64} + 2658 q^{67} + 1188 q^{76} - 6380 q^{79} - 1268 q^{82} - 3768 q^{85} - 5140 q^{88} + 342 q^{91} + 1660 q^{94} + 1686 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 32x^{10} + 823x^{8} - 5964x^{6} + 32913x^{4} - 47034x^{2} + 54756 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 55141 \nu^{11} + 1684640 \nu^{9} - 43326835 \nu^{7} + 276029262 \nu^{5} - 1732704885 \nu^{3} + 480684672 \nu ) / 1995420258 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 55141 \nu^{10} - 1684640 \nu^{8} + 43326835 \nu^{6} - 276029262 \nu^{4} + 1732704885 \nu^{2} - 2476104930 ) / 1995420258 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1024\nu^{11} - 26336\nu^{9} + 677329\nu^{7} - 1053216\nu^{5} + 1505088\nu^{3} + 447354927\nu ) / 25582311 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1024\nu^{10} - 26336\nu^{8} + 677329\nu^{6} - 1053216\nu^{4} + 1505088\nu^{2} + 242696439 ) / 25582311 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23264\nu^{11} - 598321\nu^{9} + 14588621\nu^{7} - 23927751\nu^{5} + 34193718\nu^{3} + 2003386986\nu ) / 153493866 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 328789 \nu^{10} - 12986243 \nu^{8} + 344382751 \nu^{6} - 3733955064 \nu^{4} + 18916912152 \nu^{2} - 28627945398 ) / 1995420258 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 526679 \nu^{10} + 16476832 \nu^{8} - 423763523 \nu^{6} + 2954171034 \nu^{4} - 16946936613 \nu^{2} + 24217853634 ) / 1995420258 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 238924 \nu^{11} + 7921376 \nu^{9} - 203727889 \nu^{7} + 1596984255 \nu^{5} - 8147382759 \nu^{3} + 11642937462 \nu ) / 665140086 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 967807 \nu^{11} + 29953952 \nu^{9} - 770378203 \nu^{7} + 5162405130 \nu^{5} - 30808575693 \nu^{3} + 44026693074 \nu ) / 1995420258 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1076341 \nu^{10} - 32212346 \nu^{8} + 807675646 \nu^{6} - 4502835657 \nu^{4} + 20015673426 \nu^{2} - 6315279750 ) / 1995420258 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{5} + 11\beta_{3} + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{4} - 19\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} + 29\beta_{8} + 4\beta_{7} - 2\beta_{5} + 217\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 29\beta_{10} - 6\beta_{9} - 431\beta_{2} - 431\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -64\beta_{11} + 64\beta_{7} + 599\beta_{5} - 4967 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -192\beta_{6} + 727\beta_{4} - 10207\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -3292\beta_{11} - 17669\beta_{8} - 1646\beta_{7} + 16023\beta_{5} - 117901\beta_{3} - 117901 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -17669\beta_{10} + 4938\beta_{9} - 4938\beta_{6} + 17669\beta_{4} + 244439\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -40276\beta_{11} - 426067\beta_{8} - 80552\beta_{7} + 40276\beta_{5} - 2825243\beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -426067\beta_{10} + 120828\beta_{9} + 5871295\beta_{2} + 5871295\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(-\beta_{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
−4.24667 2.45182i
−2.19846 1.26928i
−1.06422 0.614429i
1.06422 + 0.614429i
2.19846 + 1.26928i
4.24667 + 2.45182i
−4.24667 + 2.45182i
−2.19846 + 1.26928i
−1.06422 + 0.614429i
1.06422 0.614429i
2.19846 1.26928i
4.24667 2.45182i
−4.24667 + 2.45182i 0 8.02279 13.8959i 5.61325i 0 2.78293 + 1.60672i 39.4526i 0 −13.7627 23.8376i
10.2 −2.19846 + 1.26928i 0 −0.777841 + 1.34726i 8.18941i 0 9.33900 + 5.39187i 24.2577i 0 10.3947 + 18.0041i
10.3 −1.06422 + 0.614429i 0 −3.24495 + 5.62043i 17.3039i 0 −16.6219 9.59667i 17.8060i 0 −10.6320 18.4152i
10.4 1.06422 0.614429i 0 −3.24495 + 5.62043i 17.3039i 0 −16.6219 9.59667i 17.8060i 0 −10.6320 18.4152i
10.5 2.19846 1.26928i 0 −0.777841 + 1.34726i 8.18941i 0 9.33900 + 5.39187i 24.2577i 0 10.3947 + 18.0041i
10.6 4.24667 2.45182i 0 8.02279 13.8959i 5.61325i 0 2.78293 + 1.60672i 39.4526i 0 −13.7627 23.8376i
82.1 −4.24667 2.45182i 0 8.02279 + 13.8959i 5.61325i 0 2.78293 1.60672i 39.4526i 0 −13.7627 + 23.8376i
82.2 −2.19846 1.26928i 0 −0.777841 1.34726i 8.18941i 0 9.33900 5.39187i 24.2577i 0 10.3947 18.0041i
82.3 −1.06422 0.614429i 0 −3.24495 5.62043i 17.3039i 0 −16.6219 + 9.59667i 17.8060i 0 −10.6320 + 18.4152i
82.4 1.06422 + 0.614429i 0 −3.24495 5.62043i 17.3039i 0 −16.6219 + 9.59667i 17.8060i 0 −10.6320 + 18.4152i
82.5 2.19846 + 1.26928i 0 −0.777841 1.34726i 8.18941i 0 9.33900 5.39187i 24.2577i 0 10.3947 18.0041i
82.6 4.24667 + 2.45182i 0 8.02279 + 13.8959i 5.61325i 0 2.78293 1.60672i 39.4526i 0 −13.7627 + 23.8376i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.e even 6 1 inner
39.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.4.q.f 12
3.b odd 2 1 inner 117.4.q.f 12
13.e even 6 1 inner 117.4.q.f 12
13.f odd 12 2 1521.4.a.bl 12
39.h odd 6 1 inner 117.4.q.f 12
39.k even 12 2 1521.4.a.bl 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.4.q.f 12 1.a even 1 1 trivial
117.4.q.f 12 3.b odd 2 1 inner
117.4.q.f 12 13.e even 6 1 inner
117.4.q.f 12 39.h odd 6 1 inner
1521.4.a.bl 12 13.f odd 12 2
1521.4.a.bl 12 39.k even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 32T_{2}^{10} + 823T_{2}^{8} - 5964T_{2}^{6} + 32913T_{2}^{4} - 47034T_{2}^{2} + 54756 \) acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 32 T^{10} + 823 T^{8} + \cdots + 54756 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 398 T^{4} + 31629 T^{2} + \cdots + 632736)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 9 T^{5} - 207 T^{4} + \cdots + 442368)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 3092 T^{10} + \cdots + 60\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{6} + 77 T^{5} + 1430 T^{4} + \cdots + 10604499373)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 11850 T^{10} + \cdots + 58\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( (T^{6} + 24 T^{5} - 4740 T^{4} + \cdots + 231475968)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 60252 T^{10} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{12} + 114954 T^{10} + \cdots + 28\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( (T^{6} + 98019 T^{4} + \cdots + 30656351232)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 180 T^{5} + \cdots + 4986821583792)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 321494 T^{10} + \cdots + 56\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{6} + 405 T^{5} + \cdots + 131524015065664)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 159044 T^{4} + \cdots + 35969118142464)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 769410 T^{4} + \cdots - 581060481251328)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 169088 T^{10} + \cdots + 62\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{6} - 647 T^{5} + \cdots + 88\!\cdots\!09)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 1329 T^{5} + \cdots + 74\!\cdots\!68)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} - 1058468 T^{10} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{6} + 112065 T^{4} + \cdots + 13909107863187)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 1595 T^{2} + 72716 T - 515213504)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 2259428 T^{4} + \cdots + 13\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} - 2817944 T^{10} + \cdots + 52\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{6} - 843 T^{5} + \cdots + 587965056519168)^{2} \) Copy content Toggle raw display
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