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 \(4\), degree \(2\), minimal)

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: \( 3^{2} \)
Twist minimal: yes
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 84q^{5} + 36q^{6} + 20q^{7} + 180q^{8} + O(q^{10}) \) \( 8q - 84q^{5} + 36q^{6} + 20q^{7} + 180q^{8} + 104q^{10} - 288q^{11} - 360q^{12} - 340q^{13} + 144q^{15} + 620q^{16} + 900q^{17} + 564q^{20} + 792q^{21} - 1100q^{22} - 1560q^{23} - 1204q^{25} - 3024q^{26} + 3580q^{28} - 2664q^{30} - 512q^{31} + 4980q^{32} + 2700q^{33} + 6600q^{35} + 2484q^{36} - 3820q^{37} - 7680q^{38} - 2952q^{40} - 2712q^{41} - 7380q^{42} - 1240q^{43} - 1944q^{45} + 13528q^{46} + 4800q^{47} + 3600q^{48} + 3744q^{50} + 6264q^{51} - 1240q^{52} + 1020q^{53} - 3644q^{55} - 30720q^{56} - 5400q^{57} + 2340q^{58} - 1044q^{60} - 4760q^{61} + 28680q^{62} + 540q^{63} - 1212q^{65} + 10008q^{66} - 8920q^{67} - 1920q^{68} + 7380q^{70} + 7536q^{71} - 4860q^{72} + 11600q^{73} - 5976q^{75} + 4344q^{76} - 360q^{77} - 4680q^{78} + 10644q^{80} - 5832q^{81} - 27200q^{82} - 32400q^{83} - 15628q^{85} + 14592q^{86} + 10620q^{87} - 14340q^{88} + 8964q^{90} + 16528q^{91} - 31800q^{92} + 14040q^{93} + 18864q^{95} - 4068q^{96} + 58640q^{97} + 46440q^{98} + O(q^{100}) \)

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.5.f.a 8
3.b odd 2 1 45.5.g.e 8
4.b odd 2 1 240.5.bg.c 8
5.b even 2 1 75.5.f.e 8
5.c odd 4 1 inner 15.5.f.a 8
5.c odd 4 1 75.5.f.e 8
15.d odd 2 1 225.5.g.m 8
15.e even 4 1 45.5.g.e 8
15.e even 4 1 225.5.g.m 8
20.e even 4 1 240.5.bg.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.5.f.a 8 1.a even 1 1 trivial
15.5.f.a 8 5.c odd 4 1 inner
45.5.g.e 8 3.b odd 2 1
45.5.g.e 8 15.e even 4 1
75.5.f.e 8 5.b even 2 1
75.5.f.e 8 5.c odd 4 1
225.5.g.m 8 15.d odd 2 1
225.5.g.m 8 15.e even 4 1
240.5.bg.c 8 4.b odd 2 1
240.5.bg.c 8 20.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(15, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 60 T^{3} - 523 T^{4} - 420 T^{5} + 1800 T^{6} + 9480 T^{7} + 180804 T^{8} + 151680 T^{9} + 460800 T^{10} - 1720320 T^{11} - 34275328 T^{12} - 62914560 T^{13} + 4294967296 T^{16} \)
$3$ \( ( 1 + 729 T^{4} )^{2} \)
$5$ \( 1 + 84 T + 4130 T^{2} + 145500 T^{3} + 4037250 T^{4} + 90937500 T^{5} + 1613281250 T^{6} + 20507812500 T^{7} + 152587890625 T^{8} \)
$7$ \( 1 - 20 T + 200 T^{2} + 199700 T^{3} - 6227200 T^{4} + 79376300 T^{5} + 19597959000 T^{6} - 40112935980 T^{7} + 30640225075198 T^{8} - 96311159287980 T^{9} + 112978333641159000 T^{10} + 1098670165252736300 T^{11} - \)\(20\!\cdots\!00\)\( T^{12} + \)\(15\!\cdots\!00\)\( T^{13} + \)\(38\!\cdots\!00\)\( T^{14} - \)\(91\!\cdots\!20\)\( T^{15} + \)\(11\!\cdots\!01\)\( T^{16} \)
$11$ \( ( 1 + 144 T + 41822 T^{2} + 6150240 T^{3} + 823068186 T^{4} + 90045663840 T^{5} + 8964917121182 T^{6} + 451933686247824 T^{7} + 45949729863572161 T^{8} )^{2} \)
$13$ \( 1 + 340 T + 57800 T^{2} + 10577660 T^{3} + 3418178672 T^{4} + 809516994020 T^{5} + 133608496263000 T^{6} + 23843087441136780 T^{7} + 4251050341381084894 T^{8} + \)\(68\!\cdots\!80\)\( T^{9} + \)\(10\!\cdots\!00\)\( T^{10} + \)\(18\!\cdots\!20\)\( T^{11} + \)\(22\!\cdots\!52\)\( T^{12} + \)\(20\!\cdots\!60\)\( T^{13} + \)\(31\!\cdots\!00\)\( T^{14} + \)\(52\!\cdots\!40\)\( T^{15} + \)\(44\!\cdots\!81\)\( T^{16} \)
$17$ \( 1 - 900 T + 405000 T^{2} - 162344700 T^{3} + 70846529936 T^{4} - 26992676052900 T^{5} + 8778464632575000 T^{6} - 2859179891204281500 T^{7} + \)\(88\!\cdots\!86\)\( T^{8} - \)\(23\!\cdots\!00\)\( T^{9} + \)\(61\!\cdots\!00\)\( T^{10} - \)\(15\!\cdots\!00\)\( T^{11} + \)\(34\!\cdots\!16\)\( T^{12} - \)\(65\!\cdots\!00\)\( T^{13} + \)\(13\!\cdots\!00\)\( T^{14} - \)\(25\!\cdots\!00\)\( T^{15} + \)\(23\!\cdots\!61\)\( T^{16} \)
$19$ \( 1 - 475280 T^{2} + 118243003996 T^{4} - 19617731389431920 T^{6} + \)\(27\!\cdots\!66\)\( T^{8} - \)\(33\!\cdots\!20\)\( T^{10} + \)\(34\!\cdots\!76\)\( T^{12} - \)\(23\!\cdots\!80\)\( T^{14} + \)\(83\!\cdots\!61\)\( T^{16} \)
$23$ \( 1 + 1560 T + 1216800 T^{2} + 797095800 T^{3} + 382745999300 T^{4} + 92493403026600 T^{5} - 3754766037924000 T^{6} - 28706182737150138360 T^{7} - \)\(24\!\cdots\!22\)\( T^{8} - \)\(80\!\cdots\!60\)\( T^{9} - \)\(29\!\cdots\!00\)\( T^{10} + \)\(20\!\cdots\!00\)\( T^{11} + \)\(23\!\cdots\!00\)\( T^{12} + \)\(13\!\cdots\!00\)\( T^{13} + \)\(58\!\cdots\!00\)\( T^{14} + \)\(20\!\cdots\!60\)\( T^{15} + \)\(37\!\cdots\!21\)\( T^{16} \)
$29$ \( 1 - 3993236 T^{2} + 7931637401512 T^{4} - 9869586507702691580 T^{6} + \)\(83\!\cdots\!06\)\( T^{8} - \)\(49\!\cdots\!80\)\( T^{10} + \)\(19\!\cdots\!52\)\( T^{12} - \)\(49\!\cdots\!16\)\( T^{14} + \)\(62\!\cdots\!41\)\( T^{16} \)
$31$ \( ( 1 + 256 T + 1349068 T^{2} - 180990272 T^{3} + 1174676381974 T^{4} - 167148316987712 T^{5} + 1150608006098454988 T^{6} + \)\(20\!\cdots\!16\)\( T^{7} + \)\(72\!\cdots\!81\)\( T^{8} )^{2} \)
$37$ \( 1 + 3820 T + 7296200 T^{2} + 10974405380 T^{3} + 13446367756400 T^{4} + 10277980840235420 T^{5} + 1373285107740096600 T^{6} - \)\(20\!\cdots\!20\)\( T^{7} - \)\(46\!\cdots\!82\)\( T^{8} - \)\(38\!\cdots\!20\)\( T^{9} + \)\(48\!\cdots\!00\)\( T^{10} + \)\(67\!\cdots\!20\)\( T^{11} + \)\(16\!\cdots\!00\)\( T^{12} + \)\(25\!\cdots\!80\)\( T^{13} + \)\(31\!\cdots\!00\)\( T^{14} + \)\(31\!\cdots\!20\)\( T^{15} + \)\(15\!\cdots\!81\)\( T^{16} \)
$41$ \( ( 1 + 1356 T + 8325968 T^{2} + 6593365668 T^{3} + 29888933841054 T^{4} + 18631275563373348 T^{5} + 66482231940054114128 T^{6} + \)\(30\!\cdots\!36\)\( T^{7} + \)\(63\!\cdots\!41\)\( T^{8} )^{2} \)
$43$ \( 1 + 1240 T + 768800 T^{2} + 4155585560 T^{3} + 45689035751396 T^{4} + 43012988909739080 T^{5} + 26844821235643471200 T^{6} + \)\(14\!\cdots\!20\)\( T^{7} + \)\(79\!\cdots\!06\)\( T^{8} + \)\(49\!\cdots\!20\)\( T^{9} + \)\(31\!\cdots\!00\)\( T^{10} + \)\(17\!\cdots\!80\)\( T^{11} + \)\(62\!\cdots\!96\)\( T^{12} + \)\(19\!\cdots\!60\)\( T^{13} + \)\(12\!\cdots\!00\)\( T^{14} + \)\(67\!\cdots\!40\)\( T^{15} + \)\(18\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 - 4800 T + 11520000 T^{2} - 34026427200 T^{3} + 139196198193956 T^{4} - 374777003457297600 T^{5} + \)\(77\!\cdots\!00\)\( T^{6} - \)\(20\!\cdots\!00\)\( T^{7} + \)\(52\!\cdots\!26\)\( T^{8} - \)\(98\!\cdots\!00\)\( T^{9} + \)\(18\!\cdots\!00\)\( T^{10} - \)\(43\!\cdots\!00\)\( T^{11} + \)\(78\!\cdots\!76\)\( T^{12} - \)\(94\!\cdots\!00\)\( T^{13} + \)\(15\!\cdots\!00\)\( T^{14} - \)\(31\!\cdots\!00\)\( T^{15} + \)\(32\!\cdots\!41\)\( T^{16} \)
$53$ \( 1 - 1020 T + 520200 T^{2} - 3711679620 T^{3} + 27373065594800 T^{4} - 31066466654514780 T^{5} + 24336610065951787800 T^{6} - \)\(12\!\cdots\!80\)\( T^{7} - \)\(51\!\cdots\!42\)\( T^{8} - \)\(95\!\cdots\!80\)\( T^{9} + \)\(15\!\cdots\!00\)\( T^{10} - \)\(15\!\cdots\!80\)\( T^{11} + \)\(10\!\cdots\!00\)\( T^{12} - \)\(11\!\cdots\!20\)\( T^{13} + \)\(12\!\cdots\!00\)\( T^{14} - \)\(19\!\cdots\!20\)\( T^{15} + \)\(15\!\cdots\!41\)\( T^{16} \)
$59$ \( 1 - 68266460 T^{2} + 2308748573454136 T^{4} - \)\(49\!\cdots\!40\)\( T^{6} + \)\(71\!\cdots\!06\)\( T^{8} - \)\(72\!\cdots\!40\)\( T^{10} + \)\(49\!\cdots\!76\)\( T^{12} - \)\(21\!\cdots\!60\)\( T^{14} + \)\(46\!\cdots\!81\)\( T^{16} \)
$61$ \( ( 1 + 2380 T + 35645776 T^{2} + 28234432420 T^{3} + 552631006355806 T^{4} + 390929462012565220 T^{5} + \)\(68\!\cdots\!56\)\( T^{6} + \)\(63\!\cdots\!80\)\( T^{7} + \)\(36\!\cdots\!61\)\( T^{8} )^{2} \)
$67$ \( 1 + 8920 T + 39783200 T^{2} + 240429760280 T^{3} + 743487179691236 T^{4} - 1082655630131333560 T^{5} - \)\(10\!\cdots\!00\)\( T^{6} - \)\(87\!\cdots\!40\)\( T^{7} - \)\(61\!\cdots\!14\)\( T^{8} - \)\(17\!\cdots\!40\)\( T^{9} - \)\(41\!\cdots\!00\)\( T^{10} - \)\(88\!\cdots\!60\)\( T^{11} + \)\(12\!\cdots\!16\)\( T^{12} + \)\(79\!\cdots\!80\)\( T^{13} + \)\(26\!\cdots\!00\)\( T^{14} + \)\(12\!\cdots\!20\)\( T^{15} + \)\(27\!\cdots\!61\)\( T^{16} \)
$71$ \( ( 1 - 3768 T + 59990708 T^{2} - 219866356776 T^{3} + 1753029620362470 T^{4} - 5587173721023900456 T^{5} + \)\(38\!\cdots\!88\)\( T^{6} - \)\(61\!\cdots\!88\)\( T^{7} + \)\(41\!\cdots\!21\)\( T^{8} )^{2} \)
$73$ \( 1 - 11600 T + 67280000 T^{2} - 460834857520 T^{3} + 3747670062415100 T^{4} - 21684328079009172880 T^{5} + \)\(10\!\cdots\!00\)\( T^{6} - \)\(63\!\cdots\!00\)\( T^{7} + \)\(38\!\cdots\!78\)\( T^{8} - \)\(18\!\cdots\!00\)\( T^{9} + \)\(85\!\cdots\!00\)\( T^{10} - \)\(49\!\cdots\!80\)\( T^{11} + \)\(24\!\cdots\!00\)\( T^{12} - \)\(85\!\cdots\!20\)\( T^{13} + \)\(35\!\cdots\!00\)\( T^{14} - \)\(17\!\cdots\!00\)\( T^{15} + \)\(42\!\cdots\!21\)\( T^{16} \)
$79$ \( 1 - 192979448 T^{2} + 19833375036960508 T^{4} - \)\(13\!\cdots\!96\)\( T^{6} + \)\(60\!\cdots\!70\)\( T^{8} - \)\(19\!\cdots\!56\)\( T^{10} + \)\(45\!\cdots\!68\)\( T^{12} - \)\(67\!\cdots\!88\)\( T^{14} + \)\(52\!\cdots\!41\)\( T^{16} \)
$83$ \( 1 + 32400 T + 524880000 T^{2} + 6014129561040 T^{3} + 60088627758116132 T^{4} + \)\(56\!\cdots\!80\)\( T^{5} + \)\(48\!\cdots\!00\)\( T^{6} + \)\(37\!\cdots\!80\)\( T^{7} + \)\(27\!\cdots\!14\)\( T^{8} + \)\(18\!\cdots\!80\)\( T^{9} + \)\(11\!\cdots\!00\)\( T^{10} + \)\(60\!\cdots\!80\)\( T^{11} + \)\(30\!\cdots\!92\)\( T^{12} + \)\(14\!\cdots\!40\)\( T^{13} + \)\(59\!\cdots\!00\)\( T^{14} + \)\(17\!\cdots\!00\)\( T^{15} + \)\(25\!\cdots\!61\)\( T^{16} \)
$89$ \( 1 - 372973760 T^{2} + 67192056713379196 T^{4} - \)\(75\!\cdots\!40\)\( T^{6} + \)\(56\!\cdots\!26\)\( T^{8} - \)\(29\!\cdots\!40\)\( T^{10} + \)\(10\!\cdots\!56\)\( T^{12} - \)\(22\!\cdots\!60\)\( T^{14} + \)\(24\!\cdots\!21\)\( T^{16} \)
$97$ \( 1 - 58640 T + 1719324800 T^{2} - 34474607566960 T^{3} + 547427197512783356 T^{4} - \)\(74\!\cdots\!80\)\( T^{5} + \)\(91\!\cdots\!00\)\( T^{6} - \)\(10\!\cdots\!20\)\( T^{7} + \)\(10\!\cdots\!26\)\( T^{8} - \)\(89\!\cdots\!20\)\( T^{9} + \)\(71\!\cdots\!00\)\( T^{10} - \)\(51\!\cdots\!80\)\( T^{11} + \)\(33\!\cdots\!76\)\( T^{12} - \)\(18\!\cdots\!60\)\( T^{13} + \)\(82\!\cdots\!00\)\( T^{14} - \)\(24\!\cdots\!40\)\( T^{15} + \)\(37\!\cdots\!41\)\( T^{16} \)
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