Properties

Label 20.5.d.c
Level 20
Weight 5
Character orbit 20.d
Analytic conductor 2.067
Analytic rank 0
Dimension 8
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 20.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(2.06739926168\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1816805376000000.2
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{14} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( -\beta_{2} q^{3} \) \( + ( -6 + \beta_{3} ) q^{4} \) \( + ( -5 + \beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{5} \) \( + ( \beta_{5} + \beta_{7} ) q^{6} \) \( + ( 6 \beta_{1} + \beta_{2} - 2 \beta_{4} ) q^{7} \) \( + ( -7 \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{8} \) \( + ( 39 - 3 \beta_{3} - 3 \beta_{7} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( -\beta_{2} q^{3} \) \( + ( -6 + \beta_{3} ) q^{4} \) \( + ( -5 + \beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{5} \) \( + ( \beta_{5} + \beta_{7} ) q^{6} \) \( + ( 6 \beta_{1} + \beta_{2} - 2 \beta_{4} ) q^{7} \) \( + ( -7 \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{8} \) \( + ( 39 - 3 \beta_{3} - 3 \beta_{7} ) q^{9} \) \( + ( -20 - 6 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{10} \) \( -4 \beta_{5} q^{11} \) \( + ( 3 \beta_{1} + 10 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 6 \beta_{6} + 3 \beta_{7} ) q^{12} \) \( + ( -18 \beta_{1} - \beta_{3} - 4 \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{13} \) \( + ( 80 + 8 \beta_{3} - \beta_{5} - \beta_{7} ) q^{14} \) \( + ( -18 \beta_{1} - 3 \beta_{2} - 12 \beta_{3} + 6 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} ) q^{15} \) \( + ( -104 - 12 \beta_{3} + 4 \beta_{5} - 4 \beta_{7} ) q^{16} \) \( + ( 16 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} + 4 \beta_{6} + 2 \beta_{7} ) q^{17} \) \( + ( 33 \beta_{1} + 12 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} - 12 \beta_{6} - 6 \beta_{7} ) q^{18} \) \( + ( 24 \beta_{3} + 4 \beta_{5} - 8 \beta_{7} ) q^{19} \) \( + ( -50 - 13 \beta_{1} - 22 \beta_{2} - 2 \beta_{3} - 11 \beta_{4} - 4 \beta_{5} + 6 \beta_{6} + 7 \beta_{7} ) q^{20} \) \( + ( -120 + 15 \beta_{3} + 15 \beta_{7} ) q^{21} \) \( + ( -48 \beta_{2} - 8 \beta_{3} - 8 \beta_{4} - 16 \beta_{6} - 8 \beta_{7} ) q^{22} \) \( + ( 66 \beta_{1} + 13 \beta_{2} - 22 \beta_{4} ) q^{23} \) \( + ( 240 - 4 \beta_{5} - 28 \beta_{7} ) q^{24} \) \( + ( -255 + 90 \beta_{1} + 15 \beta_{3} + 20 \beta_{4} + 10 \beta_{6} + 15 \beta_{7} ) q^{25} \) \( + ( 312 - 12 \beta_{3} - 2 \beta_{5} + 6 \beta_{7} ) q^{26} \) \( + ( -108 \beta_{1} - 6 \beta_{2} + 36 \beta_{4} ) q^{27} \) \( + ( 69 \beta_{1} - 26 \beta_{2} + 5 \beta_{3} + 27 \beta_{4} + 10 \beta_{6} + 5 \beta_{7} ) q^{28} \) \( + ( 338 - 4 \beta_{3} - 4 \beta_{7} ) q^{29} \) \( + ( -240 + 24 \beta_{1} + 64 \beta_{2} - 24 \beta_{3} - 48 \beta_{4} + 3 \beta_{5} + 3 \beta_{7} ) q^{30} \) \( + ( -24 \beta_{3} + 16 \beta_{5} + 8 \beta_{7} ) q^{31} \) \( + ( -104 \beta_{1} + 80 \beta_{2} - 8 \beta_{3} - 8 \beta_{4} - 16 \beta_{6} - 8 \beta_{7} ) q^{32} \) \( + ( -324 \beta_{1} + 12 \beta_{3} - 60 \beta_{4} + 24 \beta_{6} + 12 \beta_{7} ) q^{33} \) \( + ( -272 + 8 \beta_{3} + 4 \beta_{5} - 12 \beta_{7} ) q^{34} \) \( + ( -60 \beta_{1} + 35 \beta_{2} + 20 \beta_{4} - 20 \beta_{5} ) q^{35} \) \( + ( -714 + 39 \beta_{3} - 24 \beta_{5} + 24 \beta_{7} ) q^{36} \) \( + ( 162 \beta_{1} - 21 \beta_{3} + 24 \beta_{4} - 42 \beta_{6} - 21 \beta_{7} ) q^{37} \) \( + ( -48 \beta_{1} + 16 \beta_{2} + 24 \beta_{3} + 120 \beta_{4} + 48 \beta_{6} + 24 \beta_{7} ) q^{38} \) \( -24 \beta_{5} q^{39} \) \( + ( 800 - 33 \beta_{1} - 46 \beta_{2} - 17 \beta_{3} - 43 \beta_{4} + 28 \beta_{5} - 18 \beta_{6} - 5 \beta_{7} ) q^{40} \) \( + ( 182 - 79 \beta_{3} - 79 \beta_{7} ) q^{41} \) \( + ( -90 \beta_{1} - 60 \beta_{2} + 30 \beta_{3} - 30 \beta_{4} + 60 \beta_{6} + 30 \beta_{7} ) q^{42} \) \( + ( 348 \beta_{1} - 77 \beta_{2} - 116 \beta_{4} ) q^{43} \) \( + ( 240 + 24 \beta_{3} + 32 \beta_{5} + 96 \beta_{7} ) q^{44} \) \( + ( 765 + 339 \beta_{1} - 24 \beta_{3} + 60 \beta_{4} - 39 \beta_{6} - 24 \beta_{7} ) q^{45} \) \( + ( 880 + 88 \beta_{3} - 13 \beta_{5} - 13 \beta_{7} ) q^{46} \) \( + ( 102 \beta_{1} - 7 \beta_{2} - 34 \beta_{4} ) q^{47} \) \( + ( 156 \beta_{1} + 8 \beta_{2} - 36 \beta_{3} + 132 \beta_{4} - 72 \beta_{6} - 36 \beta_{7} ) q^{48} \) \( + ( -1001 + 5 \beta_{3} + 5 \beta_{7} ) q^{49} \) \( + ( -1560 - 235 \beta_{1} - 40 \beta_{2} + 80 \beta_{3} - 20 \beta_{4} + 10 \beta_{5} + 40 \beta_{6} - 10 \beta_{7} ) q^{50} \) \( + ( 24 \beta_{3} + 16 \beta_{5} - 8 \beta_{7} ) q^{51} \) \( + ( 342 \beta_{1} - 12 \beta_{2} - 10 \beta_{3} - 70 \beta_{4} - 20 \beta_{6} - 10 \beta_{7} ) q^{52} \) \( + ( -502 \beta_{1} - 59 \beta_{3} - 124 \beta_{4} - 118 \beta_{6} - 59 \beta_{7} ) q^{53} \) \( + ( -1440 - 144 \beta_{3} + 6 \beta_{5} + 6 \beta_{7} ) q^{54} \) \( + ( -468 \beta_{1} - 168 \beta_{2} + 108 \beta_{3} + 156 \beta_{4} + 4 \beta_{5} - 36 \beta_{7} ) q^{55} \) \( + ( -1840 + 32 \beta_{3} + 36 \beta_{5} - 4 \beta_{7} ) q^{56} \) \( + ( 132 \beta_{1} + 84 \beta_{3} + 60 \beta_{4} + 168 \beta_{6} + 84 \beta_{7} ) q^{57} \) \( + ( 330 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} - 16 \beta_{6} - 8 \beta_{7} ) q^{58} \) \( + ( -216 \beta_{3} + 28 \beta_{5} + 72 \beta_{7} ) q^{59} \) \( + ( 2640 - 207 \beta_{1} + 78 \beta_{2} + 57 \beta_{3} - 81 \beta_{4} - 64 \beta_{5} - 30 \beta_{6} - 79 \beta_{7} ) q^{60} \) \( + ( 822 + 131 \beta_{3} + 131 \beta_{7} ) q^{61} \) \( + ( 48 \beta_{1} + 224 \beta_{2} + 16 \beta_{3} - 80 \beta_{4} + 32 \beta_{6} + 16 \beta_{7} ) q^{62} \) \( + ( 54 \beta_{1} + 279 \beta_{2} - 18 \beta_{4} ) q^{63} \) \( + ( 864 - 80 \beta_{3} - 96 \beta_{5} - 32 \beta_{7} ) q^{64} \) \( + ( -240 + 126 \beta_{1} - 41 \beta_{3} + 40 \beta_{4} + 74 \beta_{6} - 41 \beta_{7} ) q^{65} \) \( + ( 5760 - 288 \beta_{3} + 24 \beta_{5} - 72 \beta_{7} ) q^{66} \) \( + ( -300 \beta_{1} + 139 \beta_{2} + 100 \beta_{4} ) q^{67} \) \( + ( -316 \beta_{1} + 56 \beta_{2} + 4 \beta_{3} + 92 \beta_{4} + 8 \beta_{6} + 4 \beta_{7} ) q^{68} \) \( + ( -1560 + 171 \beta_{3} + 171 \beta_{7} ) q^{69} \) \( + ( -800 - 240 \beta_{2} - 120 \beta_{3} - 40 \beta_{4} - 35 \beta_{5} - 80 \beta_{6} - 75 \beta_{7} ) q^{70} \) \( + ( 216 \beta_{3} - 120 \beta_{5} - 72 \beta_{7} ) q^{71} \) \( + ( -681 \beta_{1} - 414 \beta_{2} + 15 \beta_{3} - 51 \beta_{4} + 30 \beta_{6} + 15 \beta_{7} ) q^{72} \) \( + ( -108 \beta_{1} + 144 \beta_{3} + 36 \beta_{4} + 288 \beta_{6} + 144 \beta_{7} ) q^{73} \) \( + ( -2952 + 180 \beta_{3} - 42 \beta_{5} + 126 \beta_{7} ) q^{74} \) \( + ( 360 \beta_{1} + 415 \beta_{2} - 120 \beta_{4} + 120 \beta_{5} ) q^{75} \) \( + ( -6000 - 216 \beta_{3} + 32 \beta_{5} - 160 \beta_{7} ) q^{76} \) \( + ( 180 \beta_{1} - 140 \beta_{3} - 20 \beta_{4} - 280 \beta_{6} - 140 \beta_{7} ) q^{77} \) \( + ( -288 \beta_{2} - 48 \beta_{3} - 48 \beta_{4} - 96 \beta_{6} - 48 \beta_{7} ) q^{78} \) \( + ( -24 \beta_{3} + 96 \beta_{5} + 8 \beta_{7} ) q^{79} \) \( + ( 2440 + 820 \beta_{1} + 344 \beta_{2} + 80 \beta_{3} + 12 \beta_{4} + 28 \beta_{5} + 104 \beta_{6} + 152 \beta_{7} ) q^{80} \) \( + ( -2439 + 9 \beta_{3} + 9 \beta_{7} ) q^{81} \) \( + ( 24 \beta_{1} + 316 \beta_{2} - 158 \beta_{3} + 158 \beta_{4} - 316 \beta_{6} - 158 \beta_{7} ) q^{82} \) \( + ( -204 \beta_{1} - 693 \beta_{2} + 68 \beta_{4} ) q^{83} \) \( + ( 3120 - 120 \beta_{3} + 120 \beta_{5} - 120 \beta_{7} ) q^{84} \) \( + ( 1120 - 216 \beta_{1} + 26 \beta_{3} - 60 \beta_{4} - 84 \beta_{6} + 26 \beta_{7} ) q^{85} \) \( + ( 4640 + 464 \beta_{3} + 77 \beta_{5} + 77 \beta_{7} ) q^{86} \) \( + ( -144 \beta_{1} - 402 \beta_{2} + 48 \beta_{4} ) q^{87} \) \( + ( 504 \beta_{1} + 144 \beta_{2} + 184 \beta_{3} - 344 \beta_{4} + 368 \beta_{6} + 184 \beta_{7} ) q^{88} \) \( + ( 3458 - 340 \beta_{3} - 340 \beta_{7} ) q^{89} \) \( + ( -6060 + 756 \beta_{1} + 18 \beta_{2} + 309 \beta_{3} + 9 \beta_{4} - 39 \beta_{5} - 18 \beta_{6} + 108 \beta_{7} ) q^{90} \) \( + ( -216 \beta_{3} - 8 \beta_{5} + 72 \beta_{7} ) q^{91} \) \( + ( 753 \beta_{1} - 306 \beta_{2} + 49 \beta_{3} + 303 \beta_{4} + 98 \beta_{6} + 49 \beta_{7} ) q^{92} \) \( + ( 1488 \beta_{1} - 144 \beta_{3} + 240 \beta_{4} - 288 \beta_{6} - 144 \beta_{7} ) q^{93} \) \( + ( 1360 + 136 \beta_{3} + 7 \beta_{5} + 7 \beta_{7} ) q^{94} \) \( + ( 1044 \beta_{1} - 536 \beta_{2} - 324 \beta_{3} - 348 \beta_{4} - 132 \beta_{5} + 108 \beta_{7} ) q^{95} \) \( + ( -9600 + 96 \beta_{3} - 80 \beta_{5} + 208 \beta_{7} ) q^{96} \) \( + ( -1224 \beta_{1} + 22 \beta_{3} - 236 \beta_{4} + 44 \beta_{6} + 22 \beta_{7} ) q^{97} \) \( + ( -991 \beta_{1} - 20 \beta_{2} + 10 \beta_{3} - 10 \beta_{4} + 20 \beta_{6} + 10 \beta_{7} ) q^{98} \) \( + ( 648 \beta_{3} - 252 \beta_{5} - 216 \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 48q^{4} \) \(\mathstrut -\mathstrut 40q^{5} \) \(\mathstrut +\mathstrut 312q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 48q^{4} \) \(\mathstrut -\mathstrut 40q^{5} \) \(\mathstrut +\mathstrut 312q^{9} \) \(\mathstrut -\mathstrut 160q^{10} \) \(\mathstrut +\mathstrut 640q^{14} \) \(\mathstrut -\mathstrut 832q^{16} \) \(\mathstrut -\mathstrut 400q^{20} \) \(\mathstrut -\mathstrut 960q^{21} \) \(\mathstrut +\mathstrut 1920q^{24} \) \(\mathstrut -\mathstrut 2040q^{25} \) \(\mathstrut +\mathstrut 2496q^{26} \) \(\mathstrut +\mathstrut 2704q^{29} \) \(\mathstrut -\mathstrut 1920q^{30} \) \(\mathstrut -\mathstrut 2176q^{34} \) \(\mathstrut -\mathstrut 5712q^{36} \) \(\mathstrut +\mathstrut 6400q^{40} \) \(\mathstrut +\mathstrut 1456q^{41} \) \(\mathstrut +\mathstrut 1920q^{44} \) \(\mathstrut +\mathstrut 6120q^{45} \) \(\mathstrut +\mathstrut 7040q^{46} \) \(\mathstrut -\mathstrut 8008q^{49} \) \(\mathstrut -\mathstrut 12480q^{50} \) \(\mathstrut -\mathstrut 11520q^{54} \) \(\mathstrut -\mathstrut 14720q^{56} \) \(\mathstrut +\mathstrut 21120q^{60} \) \(\mathstrut +\mathstrut 6576q^{61} \) \(\mathstrut +\mathstrut 6912q^{64} \) \(\mathstrut -\mathstrut 1920q^{65} \) \(\mathstrut +\mathstrut 46080q^{66} \) \(\mathstrut -\mathstrut 12480q^{69} \) \(\mathstrut -\mathstrut 6400q^{70} \) \(\mathstrut -\mathstrut 23616q^{74} \) \(\mathstrut -\mathstrut 48000q^{76} \) \(\mathstrut +\mathstrut 19520q^{80} \) \(\mathstrut -\mathstrut 19512q^{81} \) \(\mathstrut +\mathstrut 24960q^{84} \) \(\mathstrut +\mathstrut 8960q^{85} \) \(\mathstrut +\mathstrut 37120q^{86} \) \(\mathstrut +\mathstrut 27664q^{89} \) \(\mathstrut -\mathstrut 48480q^{90} \) \(\mathstrut +\mathstrut 10880q^{94} \) \(\mathstrut -\mathstrut 76800q^{96} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut +\mathstrut \) \(6\) \(x^{6}\mathstrut +\mathstrut \) \(31\) \(x^{4}\mathstrut +\mathstrut \) \(96\) \(x^{2}\mathstrut +\mathstrut \) \(256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + 10 \nu^{5} + \nu^{3} + 80 \nu \)\()/32\)
\(\beta_{3}\)\(=\)\( 4 \nu^{2} + 6 \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + 6 \nu^{5} + 31 \nu^{3} + 80 \nu \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} + 2 \nu^{4} + \nu^{2} + 28 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( -7 \nu^{7} + 8 \nu^{6} - 26 \nu^{5} + 48 \nu^{4} - 57 \nu^{3} + 120 \nu^{2} - 176 \nu + 384 \)\()/32\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{6} - 6 \nu^{4} - 23 \nu^{2} - 60 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(6\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut -\mathstrut \) \(7\) \(\beta_{1}\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(26\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{2}\mathstrut -\mathstrut \) \(13\) \(\beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(6\) \(\beta_{5}\mathstrut -\mathstrut \) \(5\) \(\beta_{3}\mathstrut +\mathstrut \) \(54\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(19\) \(\beta_{7}\mathstrut -\mathstrut \) \(38\) \(\beta_{6}\mathstrut -\mathstrut \) \(17\) \(\beta_{4}\mathstrut -\mathstrut \) \(19\) \(\beta_{3}\mathstrut -\mathstrut \) \(58\) \(\beta_{2}\mathstrut +\mathstrut \) \(53\) \(\beta_{1}\)\()/8\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−1.42849 1.39980i
−1.42849 + 1.39980i
−0.677813 1.88164i
−0.677813 + 1.88164i
0.677813 1.88164i
0.677813 + 1.88164i
1.42849 1.39980i
1.42849 + 1.39980i
−2.85697 2.79959i −6.64118 0.324555 + 15.9967i −17.6491 + 17.7062i 18.9737 + 18.5926i −39.0703 43.8570 46.6107i −36.8947 99.9931 1.17570i
19.2 −2.85697 + 2.79959i −6.64118 0.324555 15.9967i −17.6491 17.7062i 18.9737 18.5926i −39.0703 43.8570 + 46.6107i −36.8947 99.9931 + 1.17570i
19.3 −1.35563 3.76328i 13.9962 −12.3246 + 10.2032i 7.64911 23.8011i −18.9737 52.6718i −35.6863 55.1050 + 32.5490i 114.895 −99.9394 + 3.47961i
19.4 −1.35563 + 3.76328i 13.9962 −12.3246 10.2032i 7.64911 + 23.8011i −18.9737 + 52.6718i −35.6863 55.1050 32.5490i 114.895 −99.9394 3.47961i
19.5 1.35563 3.76328i −13.9962 −12.3246 10.2032i 7.64911 23.8011i −18.9737 + 52.6718i 35.6863 −55.1050 + 32.5490i 114.895 −79.2008 61.0511i
19.6 1.35563 + 3.76328i −13.9962 −12.3246 + 10.2032i 7.64911 + 23.8011i −18.9737 52.6718i 35.6863 −55.1050 32.5490i 114.895 −79.2008 + 61.0511i
19.7 2.85697 2.79959i 6.64118 0.324555 15.9967i −17.6491 + 17.7062i 18.9737 18.5926i 39.0703 −43.8570 46.6107i −36.8947 −0.852871 + 99.9964i
19.8 2.85697 + 2.79959i 6.64118 0.324555 + 15.9967i −17.6491 17.7062i 18.9737 + 18.5926i 39.0703 −43.8570 + 46.6107i −36.8947 −0.852871 99.9964i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 yes
5.b Even 1 yes
20.d Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{4} \) \(\mathstrut -\mathstrut 240 T_{3}^{2} \) \(\mathstrut +\mathstrut 8640 \) acting on \(S_{5}^{\mathrm{new}}(20, [\chi])\).