Properties

Label 117.4.q.e
Level $117$
Weight $4$
Character orbit 117.q
Analytic conductor $6.903$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \( x^{10} + 70x^{8} + 1645x^{6} + 14700x^{4} + 44100x^{2} + 27648 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1) q^{2} + ( - \beta_{5} + 6 \beta_{2}) q^{4} + ( - \beta_{7} + \beta_{6} + 2 \beta_{2} - \beta_1 - 1) q^{5} + (\beta_{8} - 2 \beta_{2} + 4) q^{7} + ( - \beta_{9} + \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} + 7 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1) q^{2} + ( - \beta_{5} + 6 \beta_{2}) q^{4} + ( - \beta_{7} + \beta_{6} + 2 \beta_{2} - \beta_1 - 1) q^{5} + (\beta_{8} - 2 \beta_{2} + 4) q^{7} + ( - \beta_{9} + \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} + 7 \beta_1) q^{8} + ( - \beta_{9} - \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 3 \beta_{3} - 8 \beta_{2} + \cdots + 8) q^{10}+ \cdots + ( - 33 \beta_{8} + 21 \beta_{7} - 15 \beta_{5} + 30 \beta_{4} - 11 \beta_{3} + \cdots - 24) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 30 q^{4} + 30 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 30 q^{4} + 30 q^{7} + 40 q^{10} - 60 q^{11} + 25 q^{13} + 60 q^{14} - 250 q^{16} - 105 q^{17} + 180 q^{19} - 510 q^{20} - 290 q^{22} + 60 q^{23} - 960 q^{25} + 30 q^{26} + 150 q^{28} + 495 q^{29} - 1440 q^{32} - 60 q^{35} - 405 q^{37} + 1380 q^{38} + 2000 q^{40} - 1065 q^{41} - 370 q^{43} - 390 q^{46} + 775 q^{49} + 4320 q^{50} + 2940 q^{52} - 330 q^{53} - 260 q^{55} + 2670 q^{56} + 2040 q^{58} - 780 q^{59} - 1375 q^{61} + 780 q^{62} - 3140 q^{64} - 1605 q^{65} + 1590 q^{67} + 600 q^{68} - 1620 q^{71} - 2190 q^{74} - 5190 q^{76} + 4320 q^{77} + 1100 q^{79} - 8430 q^{80} - 2390 q^{82} + 525 q^{85} + 3170 q^{88} - 2040 q^{89} + 4770 q^{91} + 1740 q^{92} - 3230 q^{94} + 1380 q^{95} - 3750 q^{97} - 180 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 70x^{8} + 1645x^{6} + 14700x^{4} + 44100x^{2} + 27648 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 35\nu^{3} + 210\nu + 96 ) / 192 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 35\nu^{4} + 210\nu^{2} + 96\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{2} + 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 49\nu^{5} + 700\nu^{3} + 96\nu^{2} + 2940\nu + 1344 ) / 192 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{9} + 64\nu^{7} + 18\nu^{6} + 1309\nu^{5} + 918\nu^{4} + 9030\nu^{3} + 12132\nu^{2} + 15912\nu + 24192 ) / 1152 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{9} - 64\nu^{7} + 18\nu^{6} - 1309\nu^{5} + 918\nu^{4} - 9030\nu^{3} + 12132\nu^{2} - 15912\nu + 24192 ) / 1152 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{9} + 6 \nu^{8} - 70 \nu^{7} + 366 \nu^{6} - 1603 \nu^{5} + 7008 \nu^{4} - 12654 \nu^{3} + 42840 \nu^{2} - 20304 \nu + 48384 ) / 1152 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{9} - 70\nu^{7} - 1603\nu^{5} - 12654\nu^{3} - 20304\nu ) / 576 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{7} + \beta_{6} + 2\beta_{5} - \beta_{4} - 23\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} + 2\beta_{6} - 29\beta_{4} - 12\beta_{3} + 6\beta _1 + 322 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -35\beta_{9} + 35\beta_{7} - 35\beta_{6} - 70\beta_{5} + 35\beta_{4} + 192\beta_{2} + 595\beta _1 - 96 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -70\beta_{7} - 70\beta_{6} + 805\beta_{4} + 612\beta_{3} - 306\beta _1 - 8330 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1015 \beta_{9} - 1015 \beta_{7} + 1015 \beta_{6} + 2222 \beta_{5} - 1111 \beta_{4} - 9408 \beta_{2} - 15995 \beta _1 + 4704 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 96 \beta_{9} + 192 \beta_{8} + 1934 \beta_{7} + 1934 \beta_{6} - 22373 \beta_{4} - 23316 \beta_{3} + 11658 \beta _1 + 223930 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 28175 \beta_{9} + 27599 \beta_{7} - 27599 \beta_{6} - 68638 \beta_{5} + 34319 \beta_{4} + 350784 \beta_{2} + 436603 \beta _1 - 175392 \) Copy content Toggle raw display

Character values

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

\(n\) \(28\) \(92\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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
5.04537i
3.27897i
0.917374i
2.04224i
5.36472i
5.04537i
3.27897i
0.917374i
2.04224i
5.36472i
−4.36942 + 2.52268i 0 8.72787 15.1171i 20.1174i 0 −13.3609 7.71395i 47.7076i 0 50.7498 + 87.9013i
10.2 −2.83967 + 1.63949i 0 1.37583 2.38302i 17.5414i 0 23.1228 + 13.3499i 17.2091i 0 −28.7589 49.8119i
10.3 0.794469 0.458687i 0 −3.57921 + 6.19938i 15.4704i 0 17.8257 + 10.2917i 13.9059i 0 −7.09608 12.2908i
10.4 1.76863 1.02112i 0 −1.91462 + 3.31622i 12.0825i 0 −25.7533 14.8686i 24.1582i 0 12.3377 + 21.3694i
10.5 4.64599 2.68236i 0 10.3901 17.9962i 2.69631i 0 13.1657 + 7.60123i 68.5626i 0 −7.23249 12.5270i
82.1 −4.36942 2.52268i 0 8.72787 + 15.1171i 20.1174i 0 −13.3609 + 7.71395i 47.7076i 0 50.7498 87.9013i
82.2 −2.83967 1.63949i 0 1.37583 + 2.38302i 17.5414i 0 23.1228 13.3499i 17.2091i 0 −28.7589 + 49.8119i
82.3 0.794469 + 0.458687i 0 −3.57921 6.19938i 15.4704i 0 17.8257 10.2917i 13.9059i 0 −7.09608 + 12.2908i
82.4 1.76863 + 1.02112i 0 −1.91462 3.31622i 12.0825i 0 −25.7533 + 14.8686i 24.1582i 0 12.3377 21.3694i
82.5 4.64599 + 2.68236i 0 10.3901 + 17.9962i 2.69631i 0 13.1657 7.60123i 68.5626i 0 −7.23249 + 12.5270i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 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.e 10
3.b odd 2 1 39.4.j.c 10
12.b even 2 1 624.4.bv.h 10
13.e even 6 1 inner 117.4.q.e 10
13.f odd 12 2 1521.4.a.bk 10
39.h odd 6 1 39.4.j.c 10
39.h odd 6 1 507.4.b.i 10
39.i odd 6 1 507.4.b.i 10
39.k even 12 2 507.4.a.r 10
156.r even 6 1 624.4.bv.h 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.c 10 3.b odd 2 1
39.4.j.c 10 39.h odd 6 1
117.4.q.e 10 1.a even 1 1 trivial
117.4.q.e 10 13.e even 6 1 inner
507.4.a.r 10 39.k even 12 2
507.4.b.i 10 39.h odd 6 1
507.4.b.i 10 39.i odd 6 1
624.4.bv.h 10 12.b even 2 1
624.4.bv.h 10 156.r even 6 1
1521.4.a.bk 10 13.f odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 35T_{2}^{8} + 1015T_{2}^{6} + 288T_{2}^{5} - 7350T_{2}^{4} + 44100T_{2}^{2} - 60480T_{2} + 27648 \) acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 35 T^{8} + 1015 T^{6} + \cdots + 27648 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 1105 T^{8} + \cdots + 31632011568 \) Copy content Toggle raw display
$7$ \( T^{10} - 30 T^{9} + \cdots + 14692478786352 \) Copy content Toggle raw display
$11$ \( T^{10} + 60 T^{9} + \cdots + 50\!\cdots\!32 \) Copy content Toggle raw display
$13$ \( T^{10} - 25 T^{9} + \cdots + 51\!\cdots\!57 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 332127005819904 \) Copy content Toggle raw display
$19$ \( T^{10} - 180 T^{9} + \cdots + 19\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( T^{10} - 60 T^{9} + \cdots + 66\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{10} - 495 T^{9} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( T^{10} + 116790 T^{8} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} + 405 T^{9} + \cdots + 48\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( T^{10} + 1065 T^{9} + \cdots + 36\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{10} + 370 T^{9} + \cdots + 51\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{10} + 181660 T^{8} + \cdots + 21\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( (T^{5} + 165 T^{4} + \cdots + 46733997168)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 780 T^{9} + \cdots + 41\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( T^{10} + 1375 T^{9} + \cdots + 87\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{10} - 1590 T^{9} + \cdots + 13\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{10} + 1620 T^{9} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{10} + 600615 T^{8} + \cdots + 20\!\cdots\!75 \) Copy content Toggle raw display
$79$ \( (T^{5} - 550 T^{4} + \cdots - 920208867136)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 3406900 T^{8} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{10} + 2040 T^{9} + \cdots + 28\!\cdots\!28 \) Copy content Toggle raw display
$97$ \( T^{10} + 3750 T^{9} + \cdots + 15\!\cdots\!68 \) Copy content Toggle raw display
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