Properties

Label 15.4.e.a
Level 15
Weight 4
Character orbit 15.e
Analytic conductor 0.885
Analytic rank 0
Dimension 8
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 15.e (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.885028650086\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.28356903014400.8
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( -\beta_{3} q^{2} \) \( + ( -1 - \beta_{2} + \beta_{5} ) q^{3} \) \( + ( \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{4} \) \( + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{5} \) \( + ( -1 + 3 \beta_{1} + 3 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{6} \) \( + ( -1 + \beta_{2} + 4 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{7} \) \( + ( 3 \beta_{1} - 2 \beta_{6} + 2 \beta_{7} ) q^{8} \) \( + ( -3 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 6 \beta_{7} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{3} q^{2} \) \( + ( -1 - \beta_{2} + \beta_{5} ) q^{3} \) \( + ( \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{4} \) \( + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{5} \) \( + ( -1 + 3 \beta_{1} + 3 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{6} \) \( + ( -1 + \beta_{2} + 4 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{7} \) \( + ( 3 \beta_{1} - 2 \beta_{6} + 2 \beta_{7} ) q^{8} \) \( + ( -3 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 6 \beta_{7} ) q^{9} \) \( + ( -15 - 10 \beta_{2} - 5 \beta_{4} + 5 \beta_{5} ) q^{10} \) \( + ( -7 \beta_{1} - 7 \beta_{3} + 2 \beta_{6} ) q^{11} \) \( + ( 17 - 17 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{6} + 3 \beta_{7} ) q^{12} \) \( + ( 8 + 8 \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{13} \) \( + ( 7 \beta_{1} - 7 \beta_{3} - 8 \beta_{7} ) q^{14} \) \( + ( 10 + 9 \beta_{1} + 25 \beta_{2} - 12 \beta_{3} + 5 \beta_{5} - 6 \beta_{6} + 3 \beta_{7} ) q^{15} \) \( + ( 39 + 7 \beta_{4} - 7 \beta_{5} ) q^{16} \) \( + ( 20 \beta_{3} + 7 \beta_{6} + 7 \beta_{7} ) q^{17} \) \( + ( -30 - 21 \beta_{1} - 30 \beta_{2} - 6 \beta_{6} + 6 \beta_{7} ) q^{18} \) \( + ( -12 \beta_{2} - 6 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} ) q^{19} \) \( + ( 2 \beta_{1} + 19 \beta_{3} + 2 \beta_{6} - 6 \beta_{7} ) q^{20} \) \( + ( -62 + 6 \beta_{1} + 6 \beta_{3} - \beta_{4} + \beta_{5} - 11 \beta_{6} ) q^{21} \) \( + ( -65 + 65 \beta_{2} - 10 \beta_{4} + 5 \beta_{6} - 5 \beta_{7} ) q^{22} \) \( + ( 14 \beta_{1} + 9 \beta_{6} - 9 \beta_{7} ) q^{23} \) \( + ( -3 \beta_{1} + 39 \beta_{2} + 3 \beta_{3} + 9 \beta_{4} + 9 \beta_{5} - 9 \beta_{6} + 21 \beta_{7} ) q^{24} \) \( + ( -20 - 85 \beta_{2} + 20 \beta_{4} - 10 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} ) q^{25} \) \( + ( -4 \beta_{1} - 4 \beta_{3} + 4 \beta_{6} ) q^{26} \) \( + ( 87 - 87 \beta_{2} - 18 \beta_{3} - 3 \beta_{4} - 6 \beta_{6} - 9 \beta_{7} ) q^{27} \) \( + ( 63 + 63 \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{28} \) \( + ( -3 \beta_{1} + 3 \beta_{3} + 22 \beta_{7} ) q^{29} \) \( + ( 85 - 15 \beta_{1} + 105 \beta_{2} - 10 \beta_{4} - 20 \beta_{5} + 25 \beta_{6} - 15 \beta_{7} ) q^{30} \) \( + ( 62 - 30 \beta_{4} + 30 \beta_{5} ) q^{31} \) \( + ( -43 \beta_{3} - 30 \beta_{6} - 30 \beta_{7} ) q^{32} \) \( + ( -25 + 36 \beta_{1} - 25 \beta_{2} - 20 \beta_{5} + 27 \beta_{6} - 27 \beta_{7} ) q^{33} \) \( + ( -166 \beta_{2} + 34 \beta_{4} + 34 \beta_{5} - 34 \beta_{6} + 34 \beta_{7} ) q^{34} \) \( + ( -33 \beta_{1} - 31 \beta_{3} + 17 \beta_{6} + 19 \beta_{7} ) q^{35} \) \( + ( -105 + 6 \beta_{1} + 6 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} + 24 \beta_{6} ) q^{36} \) \( + ( -146 + 146 \beta_{2} - 6 \beta_{4} + 3 \beta_{6} - 3 \beta_{7} ) q^{37} \) \( + ( -12 \beta_{1} - 12 \beta_{6} + 12 \beta_{7} ) q^{38} \) \( + ( -3 \beta_{1} + 16 \beta_{2} + 3 \beta_{3} + 7 \beta_{4} + 7 \beta_{5} - 7 \beta_{6} - 6 \beta_{7} ) q^{39} \) \( + ( -70 - 55 \beta_{2} - 15 \beta_{4} - 35 \beta_{5} + 25 \beta_{6} - 25 \beta_{7} ) q^{40} \) \( + ( 32 \beta_{1} + 32 \beta_{3} - 52 \beta_{6} ) q^{41} \) \( + ( 65 - 65 \beta_{2} + 66 \beta_{3} - 10 \beta_{4} + 7 \beta_{6} - 3 \beta_{7} ) q^{42} \) \( + ( 78 + 78 \beta_{2} - 38 \beta_{5} + 19 \beta_{6} - 19 \beta_{7} ) q^{43} \) \( + ( -29 \beta_{1} + 29 \beta_{3} + 36 \beta_{7} ) q^{44} \) \( + ( 30 + 12 \beta_{1} + 90 \beta_{2} + 69 \beta_{3} + 45 \beta_{4} + 15 \beta_{5} - 18 \beta_{6} + 24 \beta_{7} ) q^{45} \) \( + ( 108 + 32 \beta_{4} - 32 \beta_{5} ) q^{46} \) \( + ( 2 \beta_{3} + 21 \beta_{6} + 21 \beta_{7} ) q^{47} \) \( + ( -151 + 21 \beta_{1} - 151 \beta_{2} + 46 \beta_{5} - 21 \beta_{6} + 21 \beta_{7} ) q^{48} \) \( + ( 85 \beta_{2} - 16 \beta_{4} - 16 \beta_{5} + 16 \beta_{6} - 16 \beta_{7} ) q^{49} \) \( + ( 105 \beta_{1} - 40 \beta_{3} - 20 \beta_{6} - 40 \beta_{7} ) q^{50} \) \( + ( -106 - 81 \beta_{1} - 81 \beta_{3} + \beta_{4} - \beta_{5} - 19 \beta_{6} ) q^{51} \) \( + ( 24 - 24 \beta_{2} - 16 \beta_{4} + 8 \beta_{6} - 8 \beta_{7} ) q^{52} \) \( + ( -130 \beta_{1} + \beta_{6} - \beta_{7} ) q^{53} \) \( + ( 81 \beta_{1} + 147 \beta_{2} - 81 \beta_{3} - 33 \beta_{4} - 33 \beta_{5} + 33 \beta_{6} - 27 \beta_{7} ) q^{54} \) \( + ( 35 - 145 \beta_{2} - 10 \beta_{4} + 80 \beta_{5} - 35 \beta_{6} + 35 \beta_{7} ) q^{55} \) \( + ( -3 \beta_{1} - 3 \beta_{3} + 68 \beta_{6} ) q^{56} \) \( + ( 84 - 84 \beta_{2} - 18 \beta_{3} - 6 \beta_{4} + 24 \beta_{6} + 18 \beta_{7} ) q^{57} \) \( + ( -5 - 5 \beta_{2} + 50 \beta_{5} - 25 \beta_{6} + 25 \beta_{7} ) q^{58} \) \( + ( 39 \beta_{1} - 39 \beta_{3} - 136 \beta_{7} ) q^{59} \) \( + ( 55 - 69 \beta_{1} - 70 \beta_{2} - 33 \beta_{3} - 35 \beta_{4} + 15 \beta_{5} - 34 \beta_{6} - 18 \beta_{7} ) q^{60} \) \( + 2 q^{61} \) \( + ( 58 \beta_{3} + 60 \beta_{6} + 60 \beta_{7} ) q^{62} \) \( + ( 183 - 6 \beta_{1} + 183 \beta_{2} - 18 \beta_{5} - 48 \beta_{6} + 48 \beta_{7} ) q^{63} \) \( + ( 15 \beta_{2} - 47 \beta_{4} - 47 \beta_{5} + 47 \beta_{6} - 47 \beta_{7} ) q^{64} \) \( + ( 8 \beta_{1} + 6 \beta_{3} - 17 \beta_{6} + 31 \beta_{7} ) q^{65} \) \( + ( 290 - 15 \beta_{1} - 15 \beta_{3} + 70 \beta_{4} - 70 \beta_{5} - 40 \beta_{6} ) q^{66} \) \( + ( 84 - 84 \beta_{2} + 134 \beta_{4} - 67 \beta_{6} + 67 \beta_{7} ) q^{67} \) \( + ( 142 \beta_{1} + 12 \beta_{6} - 12 \beta_{7} ) q^{68} \) \( + ( -69 \beta_{1} - 148 \beta_{2} + 69 \beta_{3} - 13 \beta_{4} - 13 \beta_{5} + 13 \beta_{6} - 12 \beta_{7} ) q^{69} \) \( + ( -295 + 315 \beta_{2} - 30 \beta_{4} + 40 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} ) q^{70} \) \( + ( 116 \beta_{1} + 116 \beta_{3} + 74 \beta_{6} ) q^{71} \) \( + ( -210 + 210 \beta_{2} - 27 \beta_{3} + 60 \beta_{4} - 60 \beta_{6} ) q^{72} \) \( + ( -317 - 317 \beta_{2} + 12 \beta_{5} - 6 \beta_{6} + 6 \beta_{7} ) q^{73} \) \( + ( -158 \beta_{1} + 158 \beta_{3} + 12 \beta_{7} ) q^{74} \) \( + ( -385 + 45 \beta_{1} - 55 \beta_{2} + 15 \beta_{3} - 90 \beta_{4} - 5 \beta_{5} + 30 \beta_{6} + 30 \beta_{7} ) q^{75} \) \( + ( -180 + 12 \beta_{4} - 12 \beta_{5} ) q^{76} \) \( + ( -74 \beta_{3} - 78 \beta_{6} - 78 \beta_{7} ) q^{77} \) \( + ( -40 + 12 \beta_{1} - 40 \beta_{2} - 20 \beta_{5} + 24 \beta_{6} - 24 \beta_{7} ) q^{78} \) \( + ( 298 \beta_{2} - 56 \beta_{4} - 56 \beta_{5} + 56 \beta_{6} - 56 \beta_{7} ) q^{79} \) \( + ( -197 \beta_{1} + 46 \beta_{3} - 22 \beta_{6} + 46 \beta_{7} ) q^{80} \) \( + ( -9 + 72 \beta_{1} + 72 \beta_{3} - 90 \beta_{4} + 90 \beta_{5} + 108 \beta_{6} ) q^{81} \) \( + ( 340 - 340 \beta_{2} - 40 \beta_{4} + 20 \beta_{6} - 20 \beta_{7} ) q^{82} \) \( + ( 86 \beta_{1} - 7 \beta_{6} + 7 \beta_{7} ) q^{83} \) \( + ( -3 \beta_{1} - 94 \beta_{2} + 3 \beta_{3} + 62 \beta_{4} + 62 \beta_{5} - 62 \beta_{6} - 6 \beta_{7} ) q^{84} \) \( + ( 650 + 60 \beta_{2} + 30 \beta_{4} - 100 \beta_{5} + 35 \beta_{6} - 35 \beta_{7} ) q^{85} \) \( + ( -154 \beta_{1} - 154 \beta_{3} - 76 \beta_{6} ) q^{86} \) \( + ( -195 + 195 \beta_{2} - 84 \beta_{3} + 60 \beta_{4} - 3 \beta_{6} + 57 \beta_{7} ) q^{87} \) \( + ( 295 + 295 \beta_{2} + 50 \beta_{5} - 25 \beta_{6} + 25 \beta_{7} ) q^{88} \) \( + ( 276 \beta_{1} - 276 \beta_{3} + 126 \beta_{7} ) q^{89} \) \( + ( 90 + 30 \beta_{1} - 615 \beta_{2} - 90 \beta_{3} + 105 \beta_{4} + 45 \beta_{5} - 45 \beta_{6} - 15 \beta_{7} ) q^{90} \) \( + ( -144 + 38 \beta_{4} - 38 \beta_{5} ) q^{91} \) \( + ( -124 \beta_{3} + 8 \beta_{6} + 8 \beta_{7} ) q^{92} \) \( + ( 418 - 90 \beta_{1} + 418 \beta_{2} + 32 \beta_{5} + 90 \beta_{6} - 90 \beta_{7} ) q^{93} \) \( + ( 24 \beta_{2} + 44 \beta_{4} + 44 \beta_{5} - 44 \beta_{6} + 44 \beta_{7} ) q^{94} \) \( + ( -6 \beta_{1} + 78 \beta_{3} - 6 \beta_{6} - 72 \beta_{7} ) q^{95} \) \( + ( 497 + 219 \beta_{1} + 219 \beta_{3} - 47 \beta_{4} + 47 \beta_{5} - 4 \beta_{6} ) q^{96} \) \( + ( 179 - 179 \beta_{2} - 236 \beta_{4} + 118 \beta_{6} - 118 \beta_{7} ) q^{97} \) \( + ( -149 \beta_{1} - 32 \beta_{6} + 32 \beta_{7} ) q^{98} \) \( + ( -129 \beta_{1} - 540 \beta_{2} + 129 \beta_{3} - 30 \beta_{4} - 30 \beta_{5} + 30 \beta_{6} + 66 \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 12q^{6} \) \(\mathstrut -\mathstrut 16q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 12q^{6} \) \(\mathstrut -\mathstrut 16q^{7} \) \(\mathstrut -\mathstrut 100q^{10} \) \(\mathstrut +\mathstrut 132q^{12} \) \(\mathstrut +\mathstrut 68q^{13} \) \(\mathstrut +\mathstrut 90q^{15} \) \(\mathstrut +\mathstrut 284q^{16} \) \(\mathstrut -\mathstrut 240q^{18} \) \(\mathstrut -\mathstrut 492q^{21} \) \(\mathstrut -\mathstrut 500q^{22} \) \(\mathstrut -\mathstrut 220q^{25} \) \(\mathstrut +\mathstrut 702q^{27} \) \(\mathstrut +\mathstrut 508q^{28} \) \(\mathstrut +\mathstrut 660q^{30} \) \(\mathstrut +\mathstrut 616q^{31} \) \(\mathstrut -\mathstrut 240q^{33} \) \(\mathstrut -\mathstrut 804q^{36} \) \(\mathstrut -\mathstrut 1156q^{37} \) \(\mathstrut -\mathstrut 600q^{40} \) \(\mathstrut +\mathstrut 540q^{42} \) \(\mathstrut +\mathstrut 548q^{43} \) \(\mathstrut +\mathstrut 180q^{45} \) \(\mathstrut +\mathstrut 736q^{46} \) \(\mathstrut -\mathstrut 1116q^{48} \) \(\mathstrut -\mathstrut 852q^{51} \) \(\mathstrut +\mathstrut 224q^{52} \) \(\mathstrut +\mathstrut 460q^{55} \) \(\mathstrut +\mathstrut 684q^{57} \) \(\mathstrut +\mathstrut 60q^{58} \) \(\mathstrut +\mathstrut 540q^{60} \) \(\mathstrut +\mathstrut 16q^{61} \) \(\mathstrut +\mathstrut 1428q^{63} \) \(\mathstrut +\mathstrut 2040q^{66} \) \(\mathstrut +\mathstrut 404q^{67} \) \(\mathstrut -\mathstrut 2220q^{70} \) \(\mathstrut -\mathstrut 1800q^{72} \) \(\mathstrut -\mathstrut 2512q^{73} \) \(\mathstrut -\mathstrut 2910q^{75} \) \(\mathstrut -\mathstrut 1488q^{76} \) \(\mathstrut -\mathstrut 360q^{78} \) \(\mathstrut +\mathstrut 288q^{81} \) \(\mathstrut +\mathstrut 2800q^{82} \) \(\mathstrut +\mathstrut 4940q^{85} \) \(\mathstrut -\mathstrut 1680q^{87} \) \(\mathstrut +\mathstrut 2460q^{88} \) \(\mathstrut +\mathstrut 600q^{90} \) \(\mathstrut -\mathstrut 1304q^{91} \) \(\mathstrut +\mathstrut 3408q^{93} \) \(\mathstrut +\mathstrut 4164q^{96} \) \(\mathstrut +\mathstrut 1904q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut +\mathstrut \) \(209\) \(x^{4}\mathstrut +\mathstrut \) \(1600\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{6} - 249 \nu^{2} \)\()/680\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 249 \nu^{3} \)\()/680\)
\(\beta_{4}\)\(=\)\((\)\( -9 \nu^{6} - 20 \nu^{5} - 40 \nu^{4} - 1561 \nu^{2} - 3620 \nu - 4520 \)\()/1360\)
\(\beta_{5}\)\(=\)\((\)\( -9 \nu^{6} - 20 \nu^{5} + 40 \nu^{4} - 1561 \nu^{2} - 3620 \nu + 4520 \)\()/1360\)
\(\beta_{6}\)\(=\)\((\)\( -13 \nu^{7} - 40 \nu^{5} - 2557 \nu^{3} - 7240 \nu \)\()/2720\)
\(\beta_{7}\)\(=\)\((\)\( -13 \nu^{7} + 40 \nu^{5} - 2557 \nu^{3} + 7240 \nu \)\()/2720\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(9\) \(\beta_{2}\)
\(\nu^{3}\)\(=\)\(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(13\) \(\beta_{3}\)
\(\nu^{4}\)\(=\)\(17\) \(\beta_{5}\mathstrut -\mathstrut \) \(17\) \(\beta_{4}\mathstrut -\mathstrut \) \(113\)
\(\nu^{5}\)\(=\)\(34\) \(\beta_{7}\mathstrut -\mathstrut \) \(34\) \(\beta_{6}\mathstrut -\mathstrut \) \(181\) \(\beta_{1}\)
\(\nu^{6}\)\(=\)\(-\)\(249\) \(\beta_{7}\mathstrut +\mathstrut \) \(249\) \(\beta_{6}\mathstrut -\mathstrut \) \(249\) \(\beta_{5}\mathstrut -\mathstrut \) \(249\) \(\beta_{4}\mathstrut +\mathstrut \) \(1561\) \(\beta_{2}\)
\(\nu^{7}\)\(=\)\(-\)\(498\) \(\beta_{7}\mathstrut -\mathstrut \) \(498\) \(\beta_{6}\mathstrut -\mathstrut \) \(2557\) \(\beta_{3}\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-\beta_{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
−2.66260 2.66260i
−1.18766 1.18766i
1.18766 + 1.18766i
2.66260 + 2.66260i
−2.66260 + 2.66260i
−1.18766 + 1.18766i
1.18766 1.18766i
2.66260 2.66260i
−2.66260 + 2.66260i −4.37420 + 2.80471i 6.17891i 9.55729 + 5.80157i 4.17891 19.1146i 9.35782 + 9.35782i −4.84884 4.84884i 11.2672 24.5367i −40.8945 + 10.0000i
2.2 −1.18766 + 1.18766i 5.11173 + 0.932827i 5.17891i −2.48157 10.9015i −7.17891 + 4.96314i −13.3578 13.3578i −15.6521 15.6521i 25.2597 + 9.53673i 15.8945 + 10.0000i
2.3 1.18766 1.18766i −0.932827 5.11173i 5.17891i 2.48157 + 10.9015i −7.17891 4.96314i −13.3578 13.3578i 15.6521 + 15.6521i −25.2597 + 9.53673i 15.8945 + 10.0000i
2.4 2.66260 2.66260i −2.80471 + 4.37420i 6.17891i −9.55729 5.80157i 4.17891 + 19.1146i 9.35782 + 9.35782i 4.84884 + 4.84884i −11.2672 24.5367i −40.8945 + 10.0000i
8.1 −2.66260 2.66260i −4.37420 2.80471i 6.17891i 9.55729 5.80157i 4.17891 + 19.1146i 9.35782 9.35782i −4.84884 + 4.84884i 11.2672 + 24.5367i −40.8945 10.0000i
8.2 −1.18766 1.18766i 5.11173 0.932827i 5.17891i −2.48157 + 10.9015i −7.17891 4.96314i −13.3578 + 13.3578i −15.6521 + 15.6521i 25.2597 9.53673i 15.8945 10.0000i
8.3 1.18766 + 1.18766i −0.932827 + 5.11173i 5.17891i 2.48157 10.9015i −7.17891 + 4.96314i −13.3578 + 13.3578i 15.6521 15.6521i −25.2597 9.53673i 15.8945 10.0000i
8.4 2.66260 + 2.66260i −2.80471 4.37420i 6.17891i −9.55729 + 5.80157i 4.17891 19.1146i 9.35782 9.35782i 4.84884 4.84884i −11.2672 + 24.5367i −40.8945 10.0000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.4
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
5.c Odd 1 yes
15.e Even 1 yes

Hecke kernels

There are no other newforms in \(S_{4}^{\mathrm{new}}(15, [\chi])\).