Properties

Label 15.5.f.a
Level 15
Weight 5
Character orbit 15.f
Analytic conductor 1.551
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(1.55054944626\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + \beta_{5} q^{3} \) \( + ( -\beta_{1} - 12 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{4} \) \( + ( -10 + \beta_{1} + 8 \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{5} \) \( + ( 3 - 3 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{7} ) q^{6} \) \( + ( 5 + 4 \beta_{1} + 3 \beta_{3} - 5 \beta_{6} - 5 \beta_{7} ) q^{7} \) \( + ( 20 + 25 \beta_{2} + 5 \beta_{3} - 7 \beta_{4} - 2 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} ) q^{8} \) \( + 27 \beta_{2} q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + \beta_{5} q^{3} \) \( + ( -\beta_{1} - 12 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{4} \) \( + ( -10 + \beta_{1} + 8 \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{5} \) \( + ( 3 - 3 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{7} ) q^{6} \) \( + ( 5 + 4 \beta_{1} + 3 \beta_{3} - 5 \beta_{6} - 5 \beta_{7} ) q^{7} \) \( + ( 20 + 25 \beta_{2} + 5 \beta_{3} - 7 \beta_{4} - 2 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} ) q^{8} \) \( + 27 \beta_{2} q^{9} \) \( + ( 15 - 11 \beta_{1} - 43 \beta_{2} - 14 \beta_{3} + 13 \beta_{4} - 17 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} ) q^{10} \) \( + ( -38 - 9 \beta_{1} + 16 \beta_{3} - 9 \beta_{4} + 16 \beta_{5} + 4 \beta_{7} ) q^{11} \) \( + ( -45 + 9 \beta_{1} + 45 \beta_{2} + 10 \beta_{3} ) q^{12} \) \( + ( -40 - 45 \beta_{2} - 5 \beta_{3} - 8 \beta_{4} - 14 \beta_{5} + 5 \beta_{6} - 5 \beta_{7} ) q^{13} \) \( + ( -13 \beta_{1} - 10 \beta_{2} - 36 \beta_{3} + 13 \beta_{4} + 28 \beta_{5} + 8 \beta_{6} ) q^{14} \) \( + ( 15 + 9 \beta_{1} - 33 \beta_{2} - 9 \beta_{3} - 12 \beta_{4} - 7 \beta_{5} + 3 \beta_{6} + 6 \beta_{7} ) q^{15} \) \( + ( 81 + 37 \beta_{1} + 7 \beta_{3} + 37 \beta_{4} + 7 \beta_{5} - 7 \beta_{7} ) q^{16} \) \( + ( 110 - 115 \beta_{2} + 29 \beta_{3} + 5 \beta_{6} + 5 \beta_{7} ) q^{17} \) \( + 27 \beta_{4} q^{18} \) \( + ( 48 \beta_{1} + 48 \beta_{2} - 24 \beta_{3} - 48 \beta_{4} + 42 \beta_{5} - 18 \beta_{6} ) q^{19} \) \( + ( 70 + 49 \beta_{1} + 207 \beta_{2} + \beta_{3} - 42 \beta_{4} - 32 \beta_{5} - 27 \beta_{6} + \beta_{7} ) q^{20} \) \( + ( 108 - 27 \beta_{1} - 2 \beta_{3} - 27 \beta_{4} - 2 \beta_{5} - 18 \beta_{7} ) q^{21} \) \( + ( -155 - 100 \beta_{1} + 120 \beta_{2} + 63 \beta_{3} + 35 \beta_{6} + 35 \beta_{7} ) q^{22} \) \( + ( -180 - 210 \beta_{2} - 30 \beta_{3} + 94 \beta_{4} - 72 \beta_{5} + 30 \beta_{6} - 30 \beta_{7} ) q^{23} \) \( + ( -36 \beta_{1} - 216 \beta_{2} - 12 \beta_{3} + 36 \beta_{4} + 21 \beta_{5} - 9 \beta_{6} ) q^{24} \) \( + ( -170 - 94 \beta_{1} + 58 \beta_{2} + 19 \beta_{3} - 8 \beta_{4} + 22 \beta_{5} - 23 \beta_{6} + 39 \beta_{7} ) q^{25} \) \( + ( -358 - 10 \beta_{1} - 40 \beta_{3} - 10 \beta_{4} - 40 \beta_{5} - 40 \beta_{7} ) q^{26} \) \( -27 \beta_{3} q^{27} \) \( + ( 435 + 460 \beta_{2} + 25 \beta_{3} - 164 \beta_{4} + 22 \beta_{5} - 25 \beta_{6} + 25 \beta_{7} ) q^{28} \) \( + ( \beta_{1} + 386 \beta_{2} + 72 \beta_{3} - \beta_{4} - 26 \beta_{5} - 46 \beta_{6} ) q^{29} \) \( + ( -315 + 81 \beta_{1} - 297 \beta_{2} + 19 \beta_{3} - 18 \beta_{4} + 7 \beta_{5} - 18 \beta_{6} - 36 \beta_{7} ) q^{30} \) \( + ( -88 + 102 \beta_{1} + 72 \beta_{3} + 102 \beta_{4} + 72 \beta_{5} + 48 \beta_{7} ) q^{31} \) \( + ( 630 + 49 \beta_{1} - 615 \beta_{2} - 27 \beta_{3} - 15 \beta_{6} - 15 \beta_{7} ) q^{32} \) \( + ( 345 + 330 \beta_{2} - 15 \beta_{3} + 66 \beta_{4} - 62 \beta_{5} + 15 \beta_{6} - 15 \beta_{7} ) q^{33} \) \( + ( 28 \beta_{1} - 294 \beta_{2} - 4 \beta_{3} - 28 \beta_{4} - 88 \beta_{5} + 92 \beta_{6} ) q^{34} \) \( + ( 820 - 65 \beta_{1} + 260 \beta_{2} - 110 \beta_{3} + 55 \beta_{4} + 80 \beta_{5} + 110 \beta_{6} + 10 \beta_{7} ) q^{35} \) \( + ( 297 - 27 \beta_{1} - 27 \beta_{3} - 27 \beta_{4} - 27 \beta_{5} + 27 \beta_{7} ) q^{36} \) \( + ( -440 + 144 \beta_{1} + 515 \beta_{2} - 207 \beta_{3} - 75 \beta_{6} - 75 \beta_{7} ) q^{37} \) \( + ( -990 - 930 \beta_{2} + 60 \beta_{3} - 54 \beta_{4} + 336 \beta_{5} - 60 \beta_{6} + 60 \beta_{7} ) q^{38} \) \( + ( -9 \beta_{1} - 306 \beta_{2} + 17 \beta_{3} + 9 \beta_{4} - 71 \beta_{5} + 54 \beta_{6} ) q^{39} \) \( + ( -310 + 33 \beta_{1} - 806 \beta_{2} + 92 \beta_{3} + 221 \beta_{4} + 11 \beta_{5} + 61 \beta_{6} - 118 \beta_{7} ) q^{40} \) \( + ( -398 - 168 \beta_{1} + 22 \beta_{3} - 168 \beta_{4} + 22 \beta_{5} + 118 \beta_{7} ) q^{41} \) \( + ( -915 + 102 \beta_{1} + 930 \beta_{2} - 11 \beta_{3} - 15 \beta_{6} - 15 \beta_{7} ) q^{42} \) \( + ( -150 - 160 \beta_{2} - 10 \beta_{3} + 68 \beta_{4} + 32 \beta_{5} + 10 \beta_{6} - 10 \beta_{7} ) q^{43} \) \( + ( -65 \beta_{1} + 1038 \beta_{2} + 120 \beta_{3} + 65 \beta_{4} - 180 \beta_{5} + 60 \beta_{6} ) q^{44} \) \( + ( -270 - 54 \beta_{1} - 297 \beta_{2} + 54 \beta_{3} + 27 \beta_{4} + 27 \beta_{5} + 27 \beta_{6} + 54 \beta_{7} ) q^{45} \) \( + ( 1722 - 178 \beta_{1} - 358 \beta_{3} - 178 \beta_{4} - 358 \beta_{5} - 62 \beta_{7} ) q^{46} \) \( + ( 630 + 118 \beta_{1} - 570 \beta_{2} - 168 \beta_{3} - 60 \beta_{6} - 60 \beta_{7} ) q^{47} \) \( + ( 405 + 495 \beta_{2} + 90 \beta_{3} - 243 \beta_{4} + 208 \beta_{5} - 90 \beta_{6} + 90 \beta_{7} ) q^{48} \) \( + ( -178 \beta_{1} - 1059 \beta_{2} + 364 \beta_{3} + 178 \beta_{4} - 182 \beta_{5} - 182 \beta_{6} ) q^{49} \) \( + ( 470 + 34 \beta_{1} + 2512 \beta_{2} + 416 \beta_{3} - 37 \beta_{4} + 58 \beta_{5} - 122 \beta_{6} - 4 \beta_{7} ) q^{50} \) \( + ( 768 + 15 \beta_{1} + 125 \beta_{3} + 15 \beta_{4} + 125 \beta_{5} + 30 \beta_{7} ) q^{51} \) \( + ( -150 - 130 \beta_{1} + 160 \beta_{2} - 66 \beta_{3} - 10 \beta_{6} - 10 \beta_{7} ) q^{52} \) \( + ( 140 + 115 \beta_{2} - 25 \beta_{3} - 186 \beta_{4} - 362 \beta_{5} + 25 \beta_{6} - 25 \beta_{7} ) q^{53} \) \( + ( 81 \beta_{1} + 162 \beta_{2} + 27 \beta_{3} - 81 \beta_{4} + 54 \beta_{5} - 81 \beta_{6} ) q^{54} \) \( + ( -475 + 146 \beta_{1} - 1292 \beta_{2} - 221 \beta_{3} - 398 \beta_{4} + 262 \beta_{5} - 23 \beta_{6} + 39 \beta_{7} ) q^{55} \) \( + ( -3830 + 425 \beta_{1} - 140 \beta_{3} + 425 \beta_{4} - 140 \beta_{5} - 20 \beta_{7} ) q^{56} \) \( + ( -720 - 342 \beta_{1} + 630 \beta_{2} + 18 \beta_{3} + 90 \beta_{6} + 90 \beta_{7} ) q^{57} \) \( + ( 355 + 230 \beta_{2} - 125 \beta_{3} + 682 \beta_{4} + 82 \beta_{5} + 125 \beta_{6} - 125 \beta_{7} ) q^{58} \) \( + ( 257 \beta_{1} + 382 \beta_{2} - 96 \beta_{3} - 257 \beta_{4} - 52 \beta_{5} + 148 \beta_{6} ) q^{59} \) \( + ( -165 - 354 \beta_{1} - 1452 \beta_{2} - 121 \beta_{3} - 18 \beta_{4} + 87 \beta_{5} + 42 \beta_{6} + 69 \beta_{7} ) q^{60} \) \( + ( -478 - 36 \beta_{1} + 234 \beta_{3} - 36 \beta_{4} + 234 \beta_{5} - 234 \beta_{7} ) q^{61} \) \( + ( 3450 - 532 \beta_{1} - 3720 \beta_{2} + 6 \beta_{3} + 270 \beta_{6} + 270 \beta_{7} ) q^{62} \) \( + ( 135 - 135 \beta_{3} + 108 \beta_{4} - 54 \beta_{5} + 135 \beta_{6} - 135 \beta_{7} ) q^{63} \) \( + ( 55 \beta_{1} + 334 \beta_{2} - 370 \beta_{3} - 55 \beta_{4} + 305 \beta_{5} + 65 \beta_{6} ) q^{64} \) \( + ( -130 + 38 \beta_{1} + 1229 \beta_{2} - 163 \beta_{3} + 76 \beta_{4} + 306 \beta_{5} - 229 \beta_{6} - 43 \beta_{7} ) q^{65} \) \( + ( 1296 + 405 \beta_{1} - 250 \beta_{3} + 405 \beta_{4} - 250 \beta_{5} - 90 \beta_{7} ) q^{66} \) \( + ( -1230 + 380 \beta_{1} + 1000 \beta_{2} + 234 \beta_{3} + 230 \beta_{6} + 230 \beta_{7} ) q^{67} \) \( + ( -180 - 300 \beta_{2} - 120 \beta_{3} + 14 \beta_{4} - 120 \beta_{5} + 120 \beta_{6} - 120 \beta_{7} ) q^{68} \) \( + ( 372 \beta_{1} - 660 \beta_{2} + 184 \beta_{3} - 372 \beta_{4} - 82 \beta_{5} - 102 \beta_{6} ) q^{69} \) \( + ( 725 + 630 \beta_{1} + 740 \beta_{2} + 395 \beta_{3} - 430 \beta_{4} - 400 \beta_{5} - 185 \beta_{6} + 395 \beta_{7} ) q^{70} \) \( + ( 1132 + 240 \beta_{1} + 280 \beta_{3} + 240 \beta_{4} + 280 \beta_{5} - 380 \beta_{7} ) q^{71} \) \( + ( -540 + 189 \beta_{1} + 675 \beta_{2} + 189 \beta_{3} - 135 \beta_{6} - 135 \beta_{7} ) q^{72} \) \( + ( 1375 + 1525 \beta_{2} + 150 \beta_{3} - 156 \beta_{4} - 372 \beta_{5} - 150 \beta_{6} + 150 \beta_{7} ) q^{73} \) \( + ( -38 \beta_{1} - 990 \beta_{2} - 456 \beta_{3} + 38 \beta_{4} + 1008 \beta_{5} - 552 \beta_{6} ) q^{74} \) \( + ( -630 + 189 \beta_{1} - 198 \beta_{2} - 114 \beta_{3} + 423 \beta_{4} - 257 \beta_{5} - 162 \beta_{6} - 234 \beta_{7} ) q^{75} \) \( + ( 270 - 936 \beta_{1} + 294 \beta_{3} - 936 \beta_{4} + 294 \beta_{5} + 546 \beta_{7} ) q^{76} \) \( + ( -622 \beta_{1} + 90 \beta_{2} + 366 \beta_{3} - 90 \beta_{6} - 90 \beta_{7} ) q^{77} \) \( + ( -510 - 660 \beta_{2} - 150 \beta_{3} - 60 \beta_{4} - 518 \beta_{5} + 150 \beta_{6} - 150 \beta_{7} ) q^{78} \) \( + ( -650 \beta_{1} + 2000 \beta_{2} - 340 \beta_{3} + 650 \beta_{4} + 80 \beta_{5} + 260 \beta_{6} ) q^{79} \) \( + ( 1330 - 856 \beta_{1} - 193 \beta_{2} - 419 \beta_{3} + 33 \beta_{4} - 1132 \beta_{5} + 533 \beta_{6} + \beta_{7} ) q^{80} \) \( -729 q^{81} \) \( + ( -3290 + 278 \beta_{1} + 3510 \beta_{2} + 1116 \beta_{3} - 220 \beta_{6} - 220 \beta_{7} ) q^{82} \) \( + ( -4250 - 3850 \beta_{2} + 400 \beta_{3} - 334 \beta_{4} + 140 \beta_{5} - 400 \beta_{6} + 400 \beta_{7} ) q^{83} \) \( + ( -567 \beta_{1} - 990 \beta_{2} - 524 \beta_{3} + 567 \beta_{4} + 182 \beta_{5} + 342 \beta_{6} ) q^{84} \) \( + ( -1720 - 18 \beta_{1} + 1081 \beta_{2} - 707 \beta_{3} - 176 \beta_{4} - 416 \beta_{5} + 209 \beta_{6} - 467 \beta_{7} ) q^{85} \) \( + ( 1732 - 354 \beta_{1} - 44 \beta_{3} - 354 \beta_{4} - 44 \beta_{5} + 184 \beta_{7} ) q^{86} \) \( + ( 1395 - 144 \beta_{1} - 1260 \beta_{2} - 339 \beta_{3} - 135 \beta_{6} - 135 \beta_{7} ) q^{87} \) \( + ( -1865 - 1720 \beta_{2} + 145 \beta_{3} + 208 \beta_{4} + 478 \beta_{5} - 145 \beta_{6} + 145 \beta_{7} ) q^{88} \) \( + ( -162 \beta_{1} + 2388 \beta_{2} - 24 \beta_{3} + 162 \beta_{4} - 558 \beta_{5} + 582 \beta_{6} ) q^{89} \) \( + ( 1080 - 351 \beta_{1} + 297 \beta_{2} + 351 \beta_{3} - 297 \beta_{4} - 297 \beta_{5} + 108 \beta_{6} + 81 \beta_{7} ) q^{90} \) \( + ( 1872 + 122 \beta_{1} + 632 \beta_{3} + 122 \beta_{4} + 632 \beta_{5} + 388 \beta_{7} ) q^{91} \) \( + ( -3620 + 2474 \beta_{1} + 4330 \beta_{2} + 538 \beta_{3} - 710 \beta_{6} - 710 \beta_{7} ) q^{92} \) \( + ( 1530 + 1980 \beta_{2} + 450 \beta_{3} - 468 \beta_{4} + 464 \beta_{5} - 450 \beta_{6} + 450 \beta_{7} ) q^{93} \) \( + ( 956 \beta_{1} - 856 \beta_{2} - 368 \beta_{3} - 956 \beta_{4} + 814 \beta_{5} - 446 \beta_{6} ) q^{94} \) \( + ( 2370 + 774 \beta_{1} - 4218 \beta_{2} - 924 \beta_{3} + 1128 \beta_{4} - 552 \beta_{5} + 108 \beta_{6} - 24 \beta_{7} ) q^{95} \) \( + ( -537 - 192 \beta_{1} + 683 \beta_{3} - 192 \beta_{4} + 683 \beta_{5} + 57 \beta_{7} ) q^{96} \) \( + ( 7295 - 140 \beta_{1} - 7365 \beta_{2} - 1146 \beta_{3} + 70 \beta_{6} + 70 \beta_{7} ) q^{97} \) \( + ( 6080 + 5530 \beta_{2} - 550 \beta_{3} + 223 \beta_{4} - 356 \beta_{5} + 550 \beta_{6} - 550 \beta_{7} ) q^{98} \) \( + ( 243 \beta_{1} - 918 \beta_{2} - 324 \beta_{3} - 243 \beta_{4} + 432 \beta_{5} - 108 \beta_{6} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 84q^{5} \) \(\mathstrut +\mathstrut 36q^{6} \) \(\mathstrut +\mathstrut 20q^{7} \) \(\mathstrut +\mathstrut 180q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 84q^{5} \) \(\mathstrut +\mathstrut 36q^{6} \) \(\mathstrut +\mathstrut 20q^{7} \) \(\mathstrut +\mathstrut 180q^{8} \) \(\mathstrut +\mathstrut 104q^{10} \) \(\mathstrut -\mathstrut 288q^{11} \) \(\mathstrut -\mathstrut 360q^{12} \) \(\mathstrut -\mathstrut 340q^{13} \) \(\mathstrut +\mathstrut 144q^{15} \) \(\mathstrut +\mathstrut 620q^{16} \) \(\mathstrut +\mathstrut 900q^{17} \) \(\mathstrut +\mathstrut 564q^{20} \) \(\mathstrut +\mathstrut 792q^{21} \) \(\mathstrut -\mathstrut 1100q^{22} \) \(\mathstrut -\mathstrut 1560q^{23} \) \(\mathstrut -\mathstrut 1204q^{25} \) \(\mathstrut -\mathstrut 3024q^{26} \) \(\mathstrut +\mathstrut 3580q^{28} \) \(\mathstrut -\mathstrut 2664q^{30} \) \(\mathstrut -\mathstrut 512q^{31} \) \(\mathstrut +\mathstrut 4980q^{32} \) \(\mathstrut +\mathstrut 2700q^{33} \) \(\mathstrut +\mathstrut 6600q^{35} \) \(\mathstrut +\mathstrut 2484q^{36} \) \(\mathstrut -\mathstrut 3820q^{37} \) \(\mathstrut -\mathstrut 7680q^{38} \) \(\mathstrut -\mathstrut 2952q^{40} \) \(\mathstrut -\mathstrut 2712q^{41} \) \(\mathstrut -\mathstrut 7380q^{42} \) \(\mathstrut -\mathstrut 1240q^{43} \) \(\mathstrut -\mathstrut 1944q^{45} \) \(\mathstrut +\mathstrut 13528q^{46} \) \(\mathstrut +\mathstrut 4800q^{47} \) \(\mathstrut +\mathstrut 3600q^{48} \) \(\mathstrut +\mathstrut 3744q^{50} \) \(\mathstrut +\mathstrut 6264q^{51} \) \(\mathstrut -\mathstrut 1240q^{52} \) \(\mathstrut +\mathstrut 1020q^{53} \) \(\mathstrut -\mathstrut 3644q^{55} \) \(\mathstrut -\mathstrut 30720q^{56} \) \(\mathstrut -\mathstrut 5400q^{57} \) \(\mathstrut +\mathstrut 2340q^{58} \) \(\mathstrut -\mathstrut 1044q^{60} \) \(\mathstrut -\mathstrut 4760q^{61} \) \(\mathstrut +\mathstrut 28680q^{62} \) \(\mathstrut +\mathstrut 540q^{63} \) \(\mathstrut -\mathstrut 1212q^{65} \) \(\mathstrut +\mathstrut 10008q^{66} \) \(\mathstrut -\mathstrut 8920q^{67} \) \(\mathstrut -\mathstrut 1920q^{68} \) \(\mathstrut +\mathstrut 7380q^{70} \) \(\mathstrut +\mathstrut 7536q^{71} \) \(\mathstrut -\mathstrut 4860q^{72} \) \(\mathstrut +\mathstrut 11600q^{73} \) \(\mathstrut -\mathstrut 5976q^{75} \) \(\mathstrut +\mathstrut 4344q^{76} \) \(\mathstrut -\mathstrut 360q^{77} \) \(\mathstrut -\mathstrut 4680q^{78} \) \(\mathstrut +\mathstrut 10644q^{80} \) \(\mathstrut -\mathstrut 5832q^{81} \) \(\mathstrut -\mathstrut 27200q^{82} \) \(\mathstrut -\mathstrut 32400q^{83} \) \(\mathstrut -\mathstrut 15628q^{85} \) \(\mathstrut +\mathstrut 14592q^{86} \) \(\mathstrut +\mathstrut 10620q^{87} \) \(\mathstrut -\mathstrut 14340q^{88} \) \(\mathstrut +\mathstrut 8964q^{90} \) \(\mathstrut +\mathstrut 16528q^{91} \) \(\mathstrut -\mathstrut 31800q^{92} \) \(\mathstrut +\mathstrut 14040q^{93} \) \(\mathstrut +\mathstrut 18864q^{95} \) \(\mathstrut -\mathstrut 4068q^{96} \) \(\mathstrut +\mathstrut 58640q^{97} \) \(\mathstrut +\mathstrut 46440q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(60\) \(x^{5}\mathstrut +\mathstrut \) \(1973\) \(x^{4}\mathstrut -\mathstrut \) \(3300\) \(x^{3}\mathstrut +\mathstrut \) \(1800\) \(x^{2}\mathstrut +\mathstrut \) \(31560\) \(x\mathstrut +\mathstrut \) \(276676\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 19857 \nu^{7} - 350053 \nu^{6} - 190938 \nu^{5} - 1295568 \nu^{4} + 60124233 \nu^{3} - 934582407 \nu^{2} + 716887878 \nu + 368125308 \)\()/\)\(10440376750\)
\(\beta_{3}\)\(=\)\((\)\( -1224717 \nu^{7} + 40833643 \nu^{6} + 212097928 \nu^{5} + 1138297958 \nu^{4} - 4285577073 \nu^{3} + 47520018367 \nu^{2} + 89682965932 \nu + 621224634602 \)\()/\)\(83523014000\)
\(\beta_{4}\)\(=\)\((\)\( -1331 \nu^{7} - 726 \nu^{6} - 396 \nu^{5} + 79644 \nu^{4} - 3304389 \nu^{3} + 2589906 \nu^{2} - 983124 \nu - 20889564 \)\()/39697250\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(6913292\) \(\nu^{7}\mathstrut -\mathstrut \) \(45272557\) \(\nu^{6}\mathstrut -\mathstrut \) \(24694122\) \(\nu^{5}\mathstrut +\mathstrut \) \(1350453158\) \(\nu^{4}\mathstrut -\mathstrut \) \(5477763248\) \(\nu^{3}\mathstrut -\mathstrut \) \(37379548933\) \(\nu^{2}\mathstrut +\mathstrut \) \(47175232482\) \(\nu\mathstrut +\mathstrut \) \(599396712902\)\()/\)\(83523014000\)
\(\beta_{6}\)\(=\)\((\)\( 468490 \nu^{7} + 2002219 \nu^{6} + 16278122 \nu^{5} + 18734582 \nu^{4} + 693074870 \nu^{3} + 45879131 \nu^{2} + 15134603638 \nu + 30778570718 \)\()/\)\(3340920560\)
\(\beta_{7}\)\(=\)\((\)\( -47399 \nu^{7} - 25854 \nu^{6} + 707666 \nu^{5} + 4673476 \nu^{4} - 77977231 \nu^{3} + 70577574 \nu^{2} + 999020754 \nu - 1154530656 \)\()/\)\(317578000\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(28\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\)
\(\nu^{3}\)\(=\)\(5\) \(\beta_{7}\mathstrut -\mathstrut \) \(5\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(39\) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut +\mathstrut \) \(25\) \(\beta_{2}\mathstrut +\mathstrut \) \(20\)
\(\nu^{4}\)\(=\)\(-\)\(55\) \(\beta_{7}\mathstrut +\mathstrut \) \(55\) \(\beta_{5}\mathstrut +\mathstrut \) \(85\) \(\beta_{4}\mathstrut +\mathstrut \) \(55\) \(\beta_{3}\mathstrut +\mathstrut \) \(85\) \(\beta_{1}\mathstrut -\mathstrut \) \(959\)
\(\nu^{5}\)\(=\)\(305\) \(\beta_{7}\mathstrut +\mathstrut \) \(305\) \(\beta_{6}\mathstrut -\mathstrut \) \(475\) \(\beta_{3}\mathstrut -\mathstrut \) \(2215\) \(\beta_{2}\mathstrut -\mathstrut \) \(1679\) \(\beta_{1}\mathstrut +\mathstrut \) \(1910\)
\(\nu^{6}\)\(=\)\(-\)\(2799\) \(\beta_{6}\mathstrut -\mathstrut \) \(2559\) \(\beta_{5}\mathstrut -\mathstrut \) \(5319\) \(\beta_{4}\mathstrut +\mathstrut \) \(5358\) \(\beta_{3}\mathstrut +\mathstrut \) \(42542\) \(\beta_{2}\mathstrut +\mathstrut \) \(5319\) \(\beta_{1}\)
\(\nu^{7}\)\(=\)\(-\)\(15795\) \(\beta_{7}\mathstrut +\mathstrut \) \(15795\) \(\beta_{6}\mathstrut +\mathstrut \) \(11598\) \(\beta_{5}\mathstrut +\mathstrut \) \(76931\) \(\beta_{4}\mathstrut -\mathstrut \) \(15795\) \(\beta_{3}\mathstrut -\mathstrut \) \(139095\) \(\beta_{2}\mathstrut -\mathstrut \) \(123300\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−5.02811 5.02811i
−2.08045 2.08045i
3.30519 + 3.30519i
3.80336 + 3.80336i
−5.02811 + 5.02811i
−2.08045 + 2.08045i
3.30519 3.30519i
3.80336 3.80336i
−5.02811 5.02811i −3.67423 + 3.67423i 34.5637i −23.8949 7.35070i 36.9489 −38.5593 38.5593i 93.3405 93.3405i 27.0000i 83.1861 + 157.106i
7.2 −2.08045 2.08045i 3.67423 3.67423i 7.34348i −8.43390 23.5344i −15.2881 65.1093 + 65.1093i −48.5649 + 48.5649i 27.0000i −31.4158 + 66.5084i
7.3 3.30519 + 3.30519i 3.67423 3.67423i 5.84858i −16.2403 + 19.0066i 24.2881 −33.1649 33.1649i 33.5524 33.5524i 27.0000i −116.498 + 9.14312i
7.4 3.80336 + 3.80336i −3.67423 + 3.67423i 12.9311i 6.56915 24.1215i −27.9489 16.6149 + 16.6149i 11.6720 11.6720i 27.0000i 116.728 66.7579i
13.1 −5.02811 + 5.02811i −3.67423 3.67423i 34.5637i −23.8949 + 7.35070i 36.9489 −38.5593 + 38.5593i 93.3405 + 93.3405i 27.0000i 83.1861 157.106i
13.2 −2.08045 + 2.08045i 3.67423 + 3.67423i 7.34348i −8.43390 + 23.5344i −15.2881 65.1093 65.1093i −48.5649 48.5649i 27.0000i −31.4158 66.5084i
13.3 3.30519 3.30519i 3.67423 + 3.67423i 5.84858i −16.2403 19.0066i 24.2881 −33.1649 + 33.1649i 33.5524 + 33.5524i 27.0000i −116.498 9.14312i
13.4 3.80336 3.80336i −3.67423 3.67423i 12.9311i 6.56915 + 24.1215i −27.9489 16.6149 16.6149i 11.6720 + 11.6720i 27.0000i 116.728 + 66.7579i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.4
Significant digits:
Format:

Inner twists

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

Hecke kernels

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