Properties

Label 117.2.t.c
Level $117$
Weight $2$
Character orbit 117.t
Analytic conductor $0.934$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.934249703649\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \( x^{20} - 6x^{16} + 9x^{14} + 54x^{12} + 81x^{10} + 486x^{8} + 729x^{6} - 4374x^{4} + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{6} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{17} q^{2} + \beta_{10} q^{3} + (\beta_{11} - \beta_{6} + 1) q^{4} - \beta_{4} q^{5} + (\beta_{19} + \beta_{16} + \beta_{4}) q^{6} + (\beta_{17} - \beta_{15}) q^{7} + ( - \beta_{19} + \beta_{18} + \beta_{15} + \beta_{12} - \beta_{7} - \beta_{4}) q^{8} + ( - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{17} q^{2} + \beta_{10} q^{3} + (\beta_{11} - \beta_{6} + 1) q^{4} - \beta_{4} q^{5} + (\beta_{19} + \beta_{16} + \beta_{4}) q^{6} + (\beta_{17} - \beta_{15}) q^{7} + ( - \beta_{19} + \beta_{18} + \beta_{15} + \beta_{12} - \beta_{7} - \beta_{4}) q^{8} + ( - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2}) q^{9} + (\beta_{13} - \beta_{11} - 2 \beta_{10} + \beta_{5}) q^{10} + (\beta_{19} + \beta_{18} - \beta_{9}) q^{11} + (\beta_{14} - \beta_{13} + \beta_{10} + \beta_{6} - \beta_{5} - \beta_{2} - 1) q^{12} + (\beta_{14} - \beta_{12} + \beta_{4} - \beta_{2}) q^{13} + ( - \beta_{14} - \beta_{11} + 2 \beta_{6} + \beta_{2} - 2) q^{14} + ( - \beta_{17} + \beta_{15} + \beta_{8} - \beta_{7} - \beta_{4}) q^{15} + ( - \beta_{14} + \beta_{6} + \beta_{3}) q^{16} + (\beta_{13} - 2 \beta_{10} + \beta_{5} - \beta_{3}) q^{17} + ( - \beta_{19} - \beta_{18} + \beta_{17} - \beta_{16} - \beta_{15} - \beta_{12} + 2 \beta_{9} + \beta_{7} + \cdots + 2 \beta_{4}) q^{18}+ \cdots + ( - \beta_{19} - \beta_{18} + \beta_{17} - \beta_{16} + 2 \beta_{15} - \beta_{12} - \beta_{9} + \beta_{7} + \cdots + 2 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{3} + 12 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{3} + 12 q^{4} - 2 q^{9} - 16 q^{10} - 2 q^{12} - 4 q^{13} - 18 q^{14} + 4 q^{16} - 12 q^{17} - 10 q^{22} + 24 q^{23} - 12 q^{25} - 12 q^{26} - 22 q^{27} + 12 q^{29} - 54 q^{30} - 12 q^{35} + 50 q^{36} + 12 q^{38} - 8 q^{39} - 8 q^{40} + 6 q^{42} + 4 q^{43} + 38 q^{48} - 10 q^{49} - 78 q^{51} + 108 q^{53} + 20 q^{55} + 36 q^{56} - 2 q^{61} - 72 q^{62} + 8 q^{64} - 24 q^{65} + 78 q^{66} + 24 q^{68} + 72 q^{69} - 42 q^{74} - 8 q^{75} - 6 q^{77} + 66 q^{78} - 14 q^{79} + 46 q^{81} - 4 q^{82} - 54 q^{87} + 22 q^{88} + 24 q^{90} - 72 q^{91} - 84 q^{92} + 20 q^{94} + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 6x^{16} + 9x^{14} + 54x^{12} + 81x^{10} + 486x^{8} + 729x^{6} - 4374x^{4} + 59049 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{18} - 6\nu^{16} - 3\nu^{14} - 54\nu^{10} - 81\nu^{8} - 972\nu^{6} - 2916\nu^{4} - 2187\nu^{2} + 13122 ) / 6561 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4 \nu^{18} + 21 \nu^{16} + 66 \nu^{14} + 99 \nu^{12} + 189 \nu^{10} + 648 \nu^{8} + 4617 \nu^{6} + 18954 \nu^{4} + 39366 \nu^{2} + 39366 ) / 19683 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5 \nu^{19} + 15 \nu^{17} - 12 \nu^{15} - 153 \nu^{13} - 27 \nu^{11} + 324 \nu^{9} + 2673 \nu^{7} + 5103 \nu^{5} - 13122 \nu^{3} - 98415 \nu ) / 59049 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4 \nu^{18} - 12 \nu^{16} + 15 \nu^{14} + 90 \nu^{12} + 135 \nu^{10} - 162 \nu^{8} - 1701 \nu^{6} - 5832 \nu^{4} + 13122 \nu^{2} + 98415 ) / 19683 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4 \nu^{18} + 3 \nu^{16} - 42 \nu^{14} - 117 \nu^{12} - 54 \nu^{10} - 81 \nu^{8} + 972 \nu^{6} + 1458 \nu^{4} - 32805 \nu^{2} - 98415 ) / 19683 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5 \nu^{19} + 6 \nu^{17} - 66 \nu^{15} - 99 \nu^{13} - 27 \nu^{11} + 81 \nu^{9} + 486 \nu^{7} + 2916 \nu^{5} - 32805 \nu^{3} - 157464 \nu ) / 59049 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4 \nu^{19} - 6 \nu^{17} - 96 \nu^{15} - 225 \nu^{13} - 297 \nu^{11} - 810 \nu^{9} - 1215 \nu^{7} - 2916 \nu^{5} - 72171 \nu^{3} - 196830 \nu ) / 59049 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2 \nu^{19} + 24 \nu^{17} + 87 \nu^{15} + 171 \nu^{13} + 378 \nu^{11} + 1053 \nu^{9} + 1944 \nu^{7} + 18225 \nu^{5} + 45927 \nu^{3} + 78732 \nu ) / 59049 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7 \nu^{18} + 12 \nu^{16} - 33 \nu^{14} - 63 \nu^{12} + 27 \nu^{10} + 405 \nu^{8} + 3159 \nu^{6} + 8019 \nu^{4} - 26244 \nu^{2} - 98415 ) / 19683 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 8 \nu^{18} - 24 \nu^{16} + 3 \nu^{14} + 18 \nu^{12} - 54 \nu^{10} - 324 \nu^{8} - 3402 \nu^{6} - 16038 \nu^{4} + 6561 \nu^{2} + 98415 ) / 19683 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10 \nu^{19} - 21 \nu^{17} + 51 \nu^{15} + 90 \nu^{13} + 216 \nu^{11} - 405 \nu^{9} - 3888 \nu^{7} - 14580 \nu^{5} + 39366 \nu^{3} + 137781 \nu ) / 59049 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 7 \nu^{18} + 3 \nu^{16} - 87 \nu^{14} - 171 \nu^{12} - 216 \nu^{10} - 324 \nu^{8} + 3159 \nu^{6} + 3645 \nu^{4} - 59049 \nu^{2} - 196830 ) / 19683 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 7 \nu^{18} - 6 \nu^{16} - 87 \nu^{14} - 198 \nu^{12} - 297 \nu^{10} - 324 \nu^{8} + 1701 \nu^{6} - 5103 \nu^{4} - 72171 \nu^{2} - 177147 ) / 19683 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 10 \nu^{19} - 3 \nu^{17} + 78 \nu^{15} + 225 \nu^{13} - 27 \nu^{11} + 81 \nu^{9} - 3888 \nu^{7} - 3645 \nu^{5} + 78732 \nu^{3} + 196830 \nu ) / 59049 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 10 \nu^{19} - 48 \nu^{17} - 30 \nu^{15} + 9 \nu^{13} - 270 \nu^{11} - 1134 \nu^{9} - 8262 \nu^{7} - 34263 \nu^{5} - 19683 \nu^{3} + 137781 \nu ) / 59049 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 13 \nu^{19} - 30 \nu^{17} + 15 \nu^{15} + 117 \nu^{13} + 54 \nu^{11} - 891 \nu^{9} - 7533 \nu^{7} - 18954 \nu^{5} + 32805 \nu^{3} + 196830 \nu ) / 59049 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 13 \nu^{19} + 3 \nu^{17} - 231 \nu^{15} - 522 \nu^{13} - 459 \nu^{11} - 567 \nu^{9} + 1701 \nu^{7} - 7290 \nu^{5} - 183708 \nu^{3} - 492075 \nu ) / 59049 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 25 \nu^{19} - 57 \nu^{17} + 60 \nu^{15} + 252 \nu^{13} + 54 \nu^{11} - 405 \nu^{9} - 11907 \nu^{7} - 34263 \nu^{5} + 59049 \nu^{3} + 373977 \nu ) / 59049 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{17} - \beta_{15} - 2\beta_{12} - \beta_{8} - \beta_{7} + \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{18} - \beta_{9} + 2\beta_{7} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{14} + 2\beta_{13} + 3\beta_{6} + \beta_{3} + 3\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{19} - 3\beta_{18} - 3\beta_{16} - 3\beta_{15} - 3\beta_{12} + 6\beta_{8} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{14} + 3\beta_{13} + 6\beta_{11} - 6\beta_{6} + 3\beta_{3} - 6\beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3 \beta_{18} - 3 \beta_{17} - 9 \beta_{16} - 3 \beta_{15} + 3 \beta_{12} - 15 \beta_{9} - 12 \beta_{8} - 9 \beta_{7} - 9 \beta_{4} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 18 \beta_{14} - 6 \beta_{13} + 27 \beta_{11} + 54 \beta_{10} - 9 \beta_{6} - 27 \beta_{5} - 3 \beta_{3} + 18 \beta_{2} - 6 \beta _1 - 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 45 \beta_{19} - 9 \beta_{18} - 45 \beta_{17} + 9 \beta_{16} - 9 \beta_{15} - 27 \beta_{12} + 9 \beta_{9} - 9 \beta_{8} + 45 \beta_{7} + 36 \beta_{4} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -9\beta_{14} - 27\beta_{13} + 9\beta_{11} + 126\beta_{6} + 81\beta_{5} + 9\beta_{3} - 36\beta_{2} + 9\beta _1 - 117 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 27 \beta_{19} + 18 \beta_{18} + 117 \beta_{17} - 27 \beta_{16} - 99 \beta_{15} + 72 \beta_{12} + 45 \beta_{9} - 72 \beta_{8} + 54 \beta_{7} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 27 \beta_{14} + 72 \beta_{13} - 135 \beta_{11} + 81 \beta_{10} - 405 \beta_{6} + 81 \beta_{5} - 99 \beta_{3} - 63 \beta _1 - 189 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 135 \beta_{19} + 162 \beta_{18} - 162 \beta_{17} + 54 \beta_{16} + 216 \beta_{15} + 135 \beta_{12} + 27 \beta_{9} - 189 \beta_{8} + 27 \beta_{7} - 594 \beta_{4} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 432 \beta_{14} - 432 \beta_{13} - 27 \beta_{11} - 486 \beta_{10} + 432 \beta_{6} + 324 \beta_{3} - 54 \beta_{2} - 297 \beta _1 - 1107 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 81 \beta_{19} + 27 \beta_{18} - 1161 \beta_{17} + 486 \beta_{16} + 378 \beta_{15} + 999 \beta_{12} + 756 \beta_{9} + 1026 \beta_{8} - 810 \beta_{7} - 81 \beta_{4} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 243 \beta_{14} - 216 \beta_{13} - 648 \beta_{11} - 243 \beta_{10} - 1863 \beta_{6} - 972 \beta_{5} - 1242 \beta_{3} - 1620 \beta_{2} - 378 \beta _1 + 3321 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 2268 \beta_{19} + 3240 \beta_{18} + 3564 \beta_{17} + 81 \beta_{16} + 2916 \beta_{15} - 81 \beta_{12} + 1215 \beta_{9} - 3402 \beta_{8} - 2754 \beta_{7} + 324 \beta_{4} \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 5022 \beta_{14} - 4212 \beta_{13} - 4536 \beta_{11} - 1458 \beta_{10} + 891 \beta_{6} + 3645 \beta_{5} + 405 \beta_{3} + 891 \beta_{2} + 3888 \beta _1 + 2106 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 3645 \beta_{19} - 5184 \beta_{18} - 162 \beta_{17} + 9720 \beta_{16} - 3078 \beta_{15} - 2754 \beta_{12} + 4050 \beta_{9} + 810 \beta_{8} + 8019 \beta_{7} + 4131 \beta_{4} \) Copy content Toggle raw display

Character values

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

\(n\) \(28\) \(92\)
\(\chi(n)\) \(-1\) \(-1 + \beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
1.66095 + 0.491165i
−1.23798 + 1.21137i
0.651881 + 1.60470i
1.65391 0.514376i
0.219737 1.71806i
−0.219737 + 1.71806i
−1.65391 + 0.514376i
−0.651881 1.60470i
1.23798 1.21137i
−1.66095 0.491165i
1.66095 0.491165i
−1.23798 1.21137i
0.651881 1.60470i
1.65391 + 0.514376i
0.219737 + 1.71806i
−0.219737 1.71806i
−1.65391 0.514376i
−0.651881 + 1.60470i
1.23798 + 1.21137i
−1.66095 + 0.491165i
−2.14539 1.23864i −1.72976 + 0.0890572i 2.06847 + 3.58269i −0.771397 + 0.445366i 3.82132 + 1.95149i 0.850723 + 0.491165i 5.29379i 2.98414 0.308095i 2.20660
25.2 −1.97712 1.14149i 0.833228 1.51846i 1.60600 + 2.78168i 2.78501 1.60793i −3.38070 + 2.05106i 2.09815 + 1.21137i 2.76698i −1.61146 2.53045i −7.34174
25.3 −1.41717 0.818205i 0.471101 + 1.66675i 0.338918 + 0.587023i −0.950358 + 0.548689i 0.696113 2.74753i 2.77942 + 1.60470i 2.16360i −2.55613 + 1.57042i 1.79576
25.4 −0.929969 0.536918i −0.744247 1.56400i −0.423439 0.733417i −1.10543 + 0.638222i −0.147613 + 1.85407i −0.890926 0.514376i 3.05708i −1.89219 + 2.32800i 1.37069
25.5 −0.784270 0.452798i 1.66968 + 0.460628i −0.589947 1.02182i 1.94254 1.12153i −1.10091 1.11728i −2.97576 1.71806i 2.87970i 2.57564 + 1.53820i −2.03130
25.6 0.784270 + 0.452798i 1.66968 + 0.460628i −0.589947 1.02182i −1.94254 + 1.12153i 1.10091 + 1.11728i 2.97576 + 1.71806i 2.87970i 2.57564 + 1.53820i −2.03130
25.7 0.929969 + 0.536918i −0.744247 1.56400i −0.423439 0.733417i 1.10543 0.638222i 0.147613 1.85407i 0.890926 + 0.514376i 3.05708i −1.89219 + 2.32800i 1.37069
25.8 1.41717 + 0.818205i 0.471101 + 1.66675i 0.338918 + 0.587023i 0.950358 0.548689i −0.696113 + 2.74753i −2.77942 1.60470i 2.16360i −2.55613 + 1.57042i 1.79576
25.9 1.97712 + 1.14149i 0.833228 1.51846i 1.60600 + 2.78168i −2.78501 + 1.60793i 3.38070 2.05106i −2.09815 1.21137i 2.76698i −1.61146 2.53045i −7.34174
25.10 2.14539 + 1.23864i −1.72976 + 0.0890572i 2.06847 + 3.58269i 0.771397 0.445366i −3.82132 1.95149i −0.850723 0.491165i 5.29379i 2.98414 0.308095i 2.20660
103.1 −2.14539 + 1.23864i −1.72976 0.0890572i 2.06847 3.58269i −0.771397 0.445366i 3.82132 1.95149i 0.850723 0.491165i 5.29379i 2.98414 + 0.308095i 2.20660
103.2 −1.97712 + 1.14149i 0.833228 + 1.51846i 1.60600 2.78168i 2.78501 + 1.60793i −3.38070 2.05106i 2.09815 1.21137i 2.76698i −1.61146 + 2.53045i −7.34174
103.3 −1.41717 + 0.818205i 0.471101 1.66675i 0.338918 0.587023i −0.950358 0.548689i 0.696113 + 2.74753i 2.77942 1.60470i 2.16360i −2.55613 1.57042i 1.79576
103.4 −0.929969 + 0.536918i −0.744247 + 1.56400i −0.423439 + 0.733417i −1.10543 0.638222i −0.147613 1.85407i −0.890926 + 0.514376i 3.05708i −1.89219 2.32800i 1.37069
103.5 −0.784270 + 0.452798i 1.66968 0.460628i −0.589947 + 1.02182i 1.94254 + 1.12153i −1.10091 + 1.11728i −2.97576 + 1.71806i 2.87970i 2.57564 1.53820i −2.03130
103.6 0.784270 0.452798i 1.66968 0.460628i −0.589947 + 1.02182i −1.94254 1.12153i 1.10091 1.11728i 2.97576 1.71806i 2.87970i 2.57564 1.53820i −2.03130
103.7 0.929969 0.536918i −0.744247 + 1.56400i −0.423439 + 0.733417i 1.10543 + 0.638222i 0.147613 + 1.85407i 0.890926 0.514376i 3.05708i −1.89219 2.32800i 1.37069
103.8 1.41717 0.818205i 0.471101 1.66675i 0.338918 0.587023i 0.950358 + 0.548689i −0.696113 2.74753i −2.77942 + 1.60470i 2.16360i −2.55613 1.57042i 1.79576
103.9 1.97712 1.14149i 0.833228 + 1.51846i 1.60600 2.78168i −2.78501 1.60793i 3.38070 + 2.05106i −2.09815 + 1.21137i 2.76698i −1.61146 + 2.53045i −7.34174
103.10 2.14539 1.23864i −1.72976 0.0890572i 2.06847 3.58269i 0.771397 + 0.445366i −3.82132 + 1.95149i −0.850723 + 0.491165i 5.29379i 2.98414 + 0.308095i 2.20660
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 103.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
13.b even 2 1 inner
117.t even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.t.c 20
3.b odd 2 1 351.2.t.c 20
9.c even 3 1 inner 117.2.t.c 20
9.c even 3 1 1053.2.b.j 10
9.d odd 6 1 351.2.t.c 20
9.d odd 6 1 1053.2.b.i 10
13.b even 2 1 inner 117.2.t.c 20
39.d odd 2 1 351.2.t.c 20
117.n odd 6 1 351.2.t.c 20
117.n odd 6 1 1053.2.b.i 10
117.t even 6 1 inner 117.2.t.c 20
117.t even 6 1 1053.2.b.j 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.t.c 20 1.a even 1 1 trivial
117.2.t.c 20 9.c even 3 1 inner
117.2.t.c 20 13.b even 2 1 inner
117.2.t.c 20 117.t even 6 1 inner
351.2.t.c 20 3.b odd 2 1
351.2.t.c 20 9.d odd 6 1
351.2.t.c 20 39.d odd 2 1
351.2.t.c 20 117.n odd 6 1
1053.2.b.i 10 9.d odd 6 1
1053.2.b.i 10 117.n odd 6 1
1053.2.b.j 10 9.c even 3 1
1053.2.b.j 10 117.t even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(117, [\chi])\):

\( T_{2}^{20} - 16 T_{2}^{18} + 165 T_{2}^{16} - 1012 T_{2}^{14} + 4501 T_{2}^{12} - 12987 T_{2}^{10} + 27240 T_{2}^{8} - 35874 T_{2}^{6} + 34002 T_{2}^{4} - 18468 T_{2}^{2} + 6561 \) Copy content Toggle raw display
\( T_{5}^{20} - 19 T_{5}^{18} + 249 T_{5}^{16} - 1618 T_{5}^{14} + 7456 T_{5}^{12} - 19407 T_{5}^{10} + 36270 T_{5}^{8} - 43821 T_{5}^{6} + 38394 T_{5}^{4} - 19683 T_{5}^{2} + 6561 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 16 T^{18} + 165 T^{16} + \cdots + 6561 \) Copy content Toggle raw display
$3$ \( (T^{10} - T^{9} + T^{8} + 3 T^{7} - 9 T^{6} + \cdots + 243)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} - 19 T^{18} + 249 T^{16} + \cdots + 6561 \) Copy content Toggle raw display
$7$ \( T^{20} - 30 T^{18} + 591 T^{16} + \cdots + 531441 \) Copy content Toggle raw display
$11$ \( T^{20} - 49 T^{18} + \cdots + 276922881 \) Copy content Toggle raw display
$13$ \( T^{20} + 4 T^{19} + \cdots + 137858491849 \) Copy content Toggle raw display
$17$ \( (T^{5} + 3 T^{4} - 33 T^{3} - 90 T^{2} + \cdots + 81)^{4} \) Copy content Toggle raw display
$19$ \( (T^{10} + 129 T^{8} + 5727 T^{6} + \cdots + 700569)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} - 12 T^{9} + 144 T^{8} + \cdots + 531441)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} - 6 T^{9} + 69 T^{8} + 198 T^{7} + \cdots + 6561)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 138251528157681 \) Copy content Toggle raw display
$37$ \( (T^{10} + 231 T^{8} + 16365 T^{6} + \cdots + 41641209)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} - 118 T^{18} + \cdots + 273245607441 \) Copy content Toggle raw display
$43$ \( (T^{10} - 2 T^{9} + 57 T^{8} + 234 T^{7} + \cdots + 32041)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} - 205 T^{18} + \cdots + 49241109874401 \) Copy content Toggle raw display
$53$ \( (T^{5} - 27 T^{4} + 246 T^{3} - 855 T^{2} + \cdots - 243)^{4} \) Copy content Toggle raw display
$59$ \( T^{20} - 145 T^{18} + \cdots + 75017234240001 \) Copy content Toggle raw display
$61$ \( (T^{10} + T^{9} + 180 T^{8} + \cdots + 118091689)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} - 270 T^{18} + \cdots + 282429536481 \) Copy content Toggle raw display
$71$ \( (T^{10} + 97 T^{8} + 3046 T^{6} + \cdots + 178929)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 369 T^{8} + 46521 T^{6} + \cdots + 56746089)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + 7 T^{9} + 186 T^{8} + \cdots + 1481089)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} - 580 T^{18} + \cdots + 20\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{10} + 637 T^{8} + 134917 T^{6} + \cdots + 5851791009)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} - 612 T^{18} + \cdots + 47048089623921 \) Copy content Toggle raw display
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