Properties

Label 15.5.c.a
Level 15
Weight 5
Character orbit 15.c
Analytic conductor 1.551
Analytic rank 0
Dimension 6
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 15.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.55054944626\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3\cdot 5^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 1 - \beta_{2} ) q^{3} \) \( + ( -8 + \beta_{2} + \beta_{4} ) q^{4} \) \( + \beta_{3} q^{5} \) \( + ( 3 \beta_{1} - 3 \beta_{3} + \beta_{5} ) q^{6} \) \( + ( 13 + 2 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{7} \) \( + ( -1 - 9 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - \beta_{5} ) q^{8} \) \( + ( 19 - 9 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 1 - \beta_{2} ) q^{3} \) \( + ( -8 + \beta_{2} + \beta_{4} ) q^{4} \) \( + \beta_{3} q^{5} \) \( + ( 3 \beta_{1} - 3 \beta_{3} + \beta_{5} ) q^{6} \) \( + ( 13 + 2 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{7} \) \( + ( -1 - 9 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - \beta_{5} ) q^{8} \) \( + ( 19 - 9 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{9} \) \( + ( 7 - 5 \beta_{2} - \beta_{4} + \beta_{5} ) q^{10} \) \( + ( 4 - 2 \beta_{1} + 8 \beta_{2} + 4 \beta_{5} ) q^{11} \) \( + ( -73 + 3 \beta_{1} + 10 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} - 3 \beta_{5} ) q^{12} \) \( + ( -70 + 2 \beta_{2} + 2 \beta_{4} ) q^{13} \) \( + ( -2 + 36 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} - 2 \beta_{5} ) q^{14} \) \( + ( 9 - 15 \beta_{1} + 3 \beta_{4} + 2 \beta_{5} ) q^{15} \) \( + ( 126 - 31 \beta_{2} + \beta_{4} + 8 \beta_{5} ) q^{16} \) \( + ( -8 - 20 \beta_{1} - 16 \beta_{2} + 6 \beta_{3} - 8 \beta_{5} ) q^{17} \) \( + ( 194 + 63 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} ) q^{18} \) \( + ( -34 + 26 \beta_{2} + 2 \beta_{4} - 6 \beta_{5} ) q^{19} \) \( + ( 5 + 35 \beta_{1} + 10 \beta_{2} - 8 \beta_{3} + 5 \beta_{5} ) q^{20} \) \( + ( -150 + 21 \beta_{1} - 21 \beta_{2} + 21 \beta_{3} - 3 \beta_{4} + 9 \beta_{5} ) q^{21} \) \( + ( 64 + 30 \beta_{2} - 2 \beta_{4} - 8 \beta_{5} ) q^{22} \) \( + ( 7 - 72 \beta_{1} + 14 \beta_{2} - 48 \beta_{3} + 7 \beta_{5} ) q^{23} \) \( + ( -132 - 144 \beta_{1} - 3 \beta_{2} + 18 \beta_{3} + 9 \beta_{4} ) q^{24} \) \( -125 q^{25} \) \( + ( -2 - 104 \beta_{1} - 4 \beta_{2} + 12 \beta_{3} - 2 \beta_{5} ) q^{26} \) \( + ( -70 + 18 \beta_{1} - 29 \beta_{2} + 48 \beta_{3} - 24 \beta_{4} - 5 \beta_{5} ) q^{27} \) \( + ( -622 + 22 \beta_{2} - 2 \beta_{4} - 6 \beta_{5} ) q^{28} \) \( + ( -2 + 146 \beta_{1} - 4 \beta_{2} - 48 \beta_{3} - 2 \beta_{5} ) q^{29} \) \( + ( 365 - 30 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - 15 \beta_{4} ) q^{30} \) \( + ( 636 + 30 \beta_{2} - 2 \beta_{4} - 8 \beta_{5} ) q^{31} \) \( + ( 15 + 53 \beta_{1} + 30 \beta_{2} - 42 \beta_{3} + 15 \beta_{5} ) q^{32} \) \( + ( 744 + 102 \beta_{1} + 12 \beta_{2} + 42 \beta_{3} + 36 \beta_{4} + 10 \beta_{5} ) q^{33} \) \( + ( 490 - 114 \beta_{2} - 26 \beta_{4} + 22 \beta_{5} ) q^{34} \) \( + ( -15 + 20 \beta_{1} - 30 \beta_{2} + 20 \beta_{3} - 15 \beta_{5} ) q^{35} \) \( + ( -1252 + 126 \beta_{1} + 73 \beta_{2} - 42 \beta_{3} + 21 \beta_{4} - 26 \beta_{5} ) q^{36} \) \( + ( 306 - 58 \beta_{2} + 54 \beta_{4} + 28 \beta_{5} ) q^{37} \) \( + ( -26 - 134 \beta_{1} - 52 \beta_{2} + 120 \beta_{3} - 26 \beta_{5} ) q^{38} \) \( + ( -200 + 6 \beta_{1} + 74 \beta_{2} - 12 \beta_{3} + 12 \beta_{4} - 6 \beta_{5} ) q^{39} \) \( + ( -764 + 35 \beta_{2} + 27 \beta_{4} - 2 \beta_{5} ) q^{40} \) \( + ( -14 + 182 \beta_{1} - 28 \beta_{2} - 90 \beta_{3} - 14 \beta_{5} ) q^{41} \) \( + ( -318 - 36 \beta_{1} - 6 \beta_{2} - 126 \beta_{3} + 42 \beta_{5} ) q^{42} \) \( + ( -1277 - 168 \beta_{2} - 52 \beta_{4} + 29 \beta_{5} ) q^{43} \) \( + ( 34 - 22 \beta_{1} + 68 \beta_{2} + 132 \beta_{3} + 34 \beta_{5} ) q^{44} \) \( + ( 310 - 45 \beta_{1} + 5 \beta_{2} + 24 \beta_{3} + 15 \beta_{4} - 25 \beta_{5} ) q^{45} \) \( + ( 1420 + 224 \beta_{2} - 24 \beta_{4} - 62 \beta_{5} ) q^{46} \) \( + ( 31 - 212 \beta_{1} + 62 \beta_{2} + 72 \beta_{3} + 31 \beta_{5} ) q^{47} \) \( + ( 2405 - 213 \beta_{1} - 92 \beta_{2} - 78 \beta_{3} - 66 \beta_{4} - 27 \beta_{5} ) q^{48} \) \( + ( -187 + 126 \beta_{2} + 14 \beta_{4} - 28 \beta_{5} ) q^{49} \) \( -125 \beta_{1} q^{50} \) \( + ( -1434 - 366 \beta_{1} - 24 \beta_{2} - 12 \beta_{3} - 54 \beta_{4} - 32 \beta_{5} ) q^{51} \) \( + ( 1452 - 148 \beta_{2} - 84 \beta_{4} + 16 \beta_{5} ) q^{52} \) \( + ( 48 + 340 \beta_{1} + 96 \beta_{2} + 6 \beta_{3} + 48 \beta_{5} ) q^{53} \) \( + ( -92 + 333 \beta_{1} - 214 \beta_{2} - 129 \beta_{3} - 30 \beta_{4} + 77 \beta_{5} ) q^{54} \) \( + ( -210 + 50 \beta_{2} - 70 \beta_{4} - 30 \beta_{5} ) q^{55} \) \( + ( -54 - 78 \beta_{1} - 108 \beta_{2} + 192 \beta_{3} - 54 \beta_{5} ) q^{56} \) \( + ( -1922 + 168 \beta_{1} + 14 \beta_{2} + 42 \beta_{3} + 66 \beta_{4} + 12 \beta_{5} ) q^{57} \) \( + ( -3848 + 370 \beta_{2} + 194 \beta_{4} - 44 \beta_{5} ) q^{58} \) \( + ( -54 - 758 \beta_{1} - 108 \beta_{2} - 96 \beta_{3} - 54 \beta_{5} ) q^{59} \) \( + ( 858 + 360 \beta_{1} + 15 \beta_{2} - 60 \beta_{3} + 21 \beta_{4} + 34 \beta_{5} ) q^{60} \) \( + ( 1048 - 150 \beta_{2} + 122 \beta_{4} + 68 \beta_{5} ) q^{61} \) \( + ( -30 + 582 \beta_{1} - 60 \beta_{2} + 132 \beta_{3} - 30 \beta_{5} ) q^{62} \) \( + ( 2547 - 342 \beta_{1} + 180 \beta_{2} - 72 \beta_{3} + 18 \beta_{4} + 63 \beta_{5} ) q^{63} \) \( + ( 510 - 113 \beta_{2} + 111 \beta_{4} + 56 \beta_{5} ) q^{64} \) \( + ( 10 + 70 \beta_{1} + 20 \beta_{2} - 70 \beta_{3} + 10 \beta_{5} ) q^{65} \) \( + ( -2150 + 210 \beta_{1} - 100 \beta_{2} + 84 \beta_{3} + 60 \beta_{4} + 30 \beta_{5} ) q^{66} \) \( + ( 2299 + 32 \beta_{2} - 140 \beta_{4} - 43 \beta_{5} ) q^{67} \) \( + ( -14 + 854 \beta_{1} - 28 \beta_{2} - 456 \beta_{3} - 14 \beta_{5} ) q^{68} \) \( + ( 870 + 693 \beta_{1} + 21 \beta_{2} + 279 \beta_{3} - 81 \beta_{4} - 147 \beta_{5} ) q^{69} \) \( + ( -400 - 200 \beta_{2} + 50 \beta_{5} ) q^{70} \) \( + ( 78 + 236 \beta_{1} + 156 \beta_{2} + 120 \beta_{3} + 78 \beta_{5} ) q^{71} \) \( + ( -339 - 783 \beta_{1} + 150 \beta_{2} + 342 \beta_{3} + 72 \beta_{4} - 147 \beta_{5} ) q^{72} \) \( + ( 6 - 308 \beta_{2} - 196 \beta_{4} + 28 \beta_{5} ) q^{73} \) \( + ( 58 - 304 \beta_{1} + 116 \beta_{2} - 180 \beta_{3} + 58 \beta_{5} ) q^{74} \) \( + ( -125 + 125 \beta_{2} ) q^{75} \) \( + ( 3408 - 526 \beta_{2} - 222 \beta_{4} + 76 \beta_{5} ) q^{76} \) \( + ( -68 - 576 \beta_{1} - 136 \beta_{2} - 516 \beta_{3} - 68 \beta_{5} ) q^{77} \) \( + ( -264 - 546 \beta_{1} - 6 \beta_{2} + 294 \beta_{3} + 18 \beta_{4} - 86 \beta_{5} ) q^{78} \) \( + ( -2594 + 386 \beta_{2} - 38 \beta_{4} - 106 \beta_{5} ) q^{79} \) \( + ( 45 - 685 \beta_{1} + 90 \beta_{2} + 70 \beta_{3} + 45 \beta_{5} ) q^{80} \) \( + ( 907 - 828 \beta_{1} + 2 \beta_{2} + 60 \beta_{3} - 30 \beta_{4} + 176 \beta_{5} ) q^{81} \) \( + ( -5054 + 520 \beta_{2} + 272 \beta_{4} - 62 \beta_{5} ) q^{82} \) \( + ( -21 - 288 \beta_{1} - 42 \beta_{2} + 408 \beta_{3} - 21 \beta_{5} ) q^{83} \) \( + ( -2250 + 156 \beta_{1} + 594 \beta_{2} + 66 \beta_{3} + 42 \beta_{4} + 24 \beta_{5} ) q^{84} \) \( + ( -498 + 20 \beta_{2} + 164 \beta_{4} + 36 \beta_{5} ) q^{85} \) \( + ( 168 - 74 \beta_{1} + 336 \beta_{2} - 834 \beta_{3} + 168 \beta_{5} ) q^{86} \) \( + ( -804 + 1104 \beta_{1} - 6 \beta_{2} - 456 \beta_{3} - 162 \beta_{4} + 44 \beta_{5} ) q^{87} \) \( + ( 2612 + 70 \beta_{2} - 186 \beta_{4} - 64 \beta_{5} ) q^{88} \) \( + ( -156 - 12 \beta_{1} - 312 \beta_{2} + 396 \beta_{3} - 156 \beta_{5} ) q^{89} \) \( + ( 1133 - 395 \beta_{2} + 210 \beta_{3} - 69 \beta_{4} + 19 \beta_{5} ) q^{90} \) \( + ( -1946 - 64 \beta_{2} + 104 \beta_{4} + 42 \beta_{5} ) q^{91} \) \( + ( -112 - 6 \beta_{1} - 224 \beta_{2} + 204 \beta_{3} - 112 \beta_{5} ) q^{92} \) \( + ( -1578 + 210 \beta_{1} - 672 \beta_{2} + 84 \beta_{3} + 60 \beta_{4} + 30 \beta_{5} ) q^{93} \) \( + ( 5716 - 324 \beta_{2} - 284 \beta_{4} + 10 \beta_{5} ) q^{94} \) \( + ( -20 + 610 \beta_{1} - 40 \beta_{2} + 8 \beta_{3} - 20 \beta_{5} ) q^{95} \) \( + ( 2412 + 1194 \beta_{1} + 45 \beta_{2} - 24 \beta_{3} + 9 \beta_{4} + 14 \beta_{5} ) q^{96} \) \( + ( 1822 + 712 \beta_{2} + 216 \beta_{4} - 124 \beta_{5} ) q^{97} \) \( + ( -126 - 733 \beta_{1} - 252 \beta_{2} + 588 \beta_{3} - 126 \beta_{5} ) q^{98} \) \( + ( 1424 - 342 \beta_{1} - 632 \beta_{2} - 564 \beta_{3} + 300 \beta_{4} + 64 \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut +\mathstrut 8q^{3} \) \(\mathstrut -\mathstrut 50q^{4} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut 76q^{7} \) \(\mathstrut +\mathstrut 118q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut +\mathstrut 8q^{3} \) \(\mathstrut -\mathstrut 50q^{4} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut 76q^{7} \) \(\mathstrut +\mathstrut 118q^{9} \) \(\mathstrut +\mathstrut 50q^{10} \) \(\mathstrut -\mathstrut 452q^{12} \) \(\mathstrut -\mathstrut 424q^{13} \) \(\mathstrut +\mathstrut 50q^{15} \) \(\mathstrut +\mathstrut 802q^{16} \) \(\mathstrut +\mathstrut 1160q^{18} \) \(\mathstrut -\mathstrut 244q^{19} \) \(\mathstrut -\mathstrut 876q^{21} \) \(\mathstrut +\mathstrut 340q^{22} \) \(\mathstrut -\mathstrut 786q^{24} \) \(\mathstrut -\mathstrut 750q^{25} \) \(\mathstrut -\mathstrut 352q^{27} \) \(\mathstrut -\mathstrut 3764q^{28} \) \(\mathstrut +\mathstrut 2200q^{30} \) \(\mathstrut +\mathstrut 3772q^{31} \) \(\mathstrut +\mathstrut 4420q^{33} \) \(\mathstrut +\mathstrut 3124q^{34} \) \(\mathstrut -\mathstrut 7606q^{36} \) \(\mathstrut +\mathstrut 1896q^{37} \) \(\mathstrut -\mathstrut 1336q^{39} \) \(\mathstrut -\mathstrut 4650q^{40} \) \(\mathstrut -\mathstrut 1980q^{42} \) \(\mathstrut -\mathstrut 7384q^{43} \) \(\mathstrut +\mathstrut 1900q^{45} \) \(\mathstrut +\mathstrut 8196q^{46} \) \(\mathstrut +\mathstrut 14668q^{48} \) \(\mathstrut -\mathstrut 1318q^{49} \) \(\mathstrut -\mathstrut 8492q^{51} \) \(\mathstrut +\mathstrut 8976q^{52} \) \(\mathstrut -\mathstrut 278q^{54} \) \(\mathstrut -\mathstrut 1300q^{55} \) \(\mathstrut -\mathstrut 11584q^{57} \) \(\mathstrut -\mathstrut 23740q^{58} \) \(\mathstrut +\mathstrut 5050q^{60} \) \(\mathstrut +\mathstrut 6452q^{61} \) \(\mathstrut +\mathstrut 14796q^{63} \) \(\mathstrut +\mathstrut 3174q^{64} \) \(\mathstrut -\mathstrut 12760q^{66} \) \(\mathstrut +\mathstrut 13816q^{67} \) \(\mathstrut +\mathstrut 5472q^{69} \) \(\mathstrut -\mathstrut 2100q^{70} \) \(\mathstrut -\mathstrut 2040q^{72} \) \(\mathstrut +\mathstrut 596q^{73} \) \(\mathstrut -\mathstrut 1000q^{75} \) \(\mathstrut +\mathstrut 21348q^{76} \) \(\mathstrut -\mathstrut 1400q^{78} \) \(\mathstrut -\mathstrut 16124q^{79} \) \(\mathstrut +\mathstrut 5086q^{81} \) \(\mathstrut -\mathstrut 31240q^{82} \) \(\mathstrut -\mathstrut 14736q^{84} \) \(\mathstrut -\mathstrut 3100q^{85} \) \(\mathstrut -\mathstrut 4900q^{87} \) \(\mathstrut +\mathstrut 15660q^{88} \) \(\mathstrut +\mathstrut 7550q^{90} \) \(\mathstrut -\mathstrut 11632q^{91} \) \(\mathstrut -\mathstrut 8184q^{93} \) \(\mathstrut +\mathstrut 34924q^{94} \) \(\mathstrut +\mathstrut 14354q^{96} \) \(\mathstrut +\mathstrut 9756q^{97} \) \(\mathstrut +\mathstrut 9680q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut +\mathstrut \) \(73\) \(x^{4}\mathstrut +\mathstrut \) \(1096\) \(x^{2}\mathstrut +\mathstrut \) \(180\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - \nu^{4} + 71 \nu^{3} - 47 \nu^{2} + 1002 \nu - 114 \)\()/48\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + 79 \nu^{3} + 1330 \nu \)\()/48\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 71 \nu^{3} + 95 \nu^{2} - 1002 \nu + 1266 \)\()/48\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} + \nu^{4} + 142 \nu^{3} + 47 \nu^{2} + 2004 \nu + 90 \)\()/24\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(24\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{5}\mathstrut +\mathstrut \) \(6\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut -\mathstrut \) \(41\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(8\) \(\beta_{5}\mathstrut -\mathstrut \) \(47\) \(\beta_{4}\mathstrut -\mathstrut \) \(79\) \(\beta_{2}\mathstrut +\mathstrut \) \(1022\)
\(\nu^{5}\)\(=\)\(79\) \(\beta_{5}\mathstrut -\mathstrut \) \(426\) \(\beta_{3}\mathstrut +\mathstrut \) \(158\) \(\beta_{2}\mathstrut +\mathstrut \) \(1909\) \(\beta_{1}\mathstrut +\mathstrut \) \(79\)

Character Values

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

\(n\) \(7\) \(11\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
7.20990i
4.56632i
0.407512i
0.407512i
4.56632i
7.20990i
7.20990i 8.77108 + 2.01697i −35.9827 11.1803i 14.5422 63.2386i 23.3388 144.073i 72.8637 + 35.3820i 80.6091
11.2 4.56632i −7.98405 4.15391i −4.85128 11.1803i −18.9681 + 36.4577i 61.6068 50.9086i 46.4900 + 66.3301i −51.0530
11.3 0.407512i 3.21297 + 8.40695i 15.8339 11.1803i 3.42594 1.30932i −46.9457 12.9727i −60.3537 + 54.0225i −4.55613
11.4 0.407512i 3.21297 8.40695i 15.8339 11.1803i 3.42594 + 1.30932i −46.9457 12.9727i −60.3537 54.0225i −4.55613
11.5 4.56632i −7.98405 + 4.15391i −4.85128 11.1803i −18.9681 36.4577i 61.6068 50.9086i 46.4900 66.3301i −51.0530
11.6 7.20990i 8.77108 2.01697i −35.9827 11.1803i 14.5422 + 63.2386i 23.3388 144.073i 72.8637 35.3820i 80.6091
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.6
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{5}^{\mathrm{new}}(15, [\chi])\).