Properties

Label 12.7.d.a
Level 12
Weight 7
Character orbit 12.d
Analytic conductor 2.761
Analytic rank 0
Dimension 6
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 12.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(2.76064900344\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.50898483.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{5} \)
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.

\(f(q)\) \(=\) \( q\) \( + ( -2 - \beta_{1} ) q^{2} \) \( -\beta_{2} q^{3} \) \( + ( 27 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{4} \) \( + ( -7 + \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{5} \) \( + ( -27 + 2 \beta_{2} - 3 \beta_{4} ) q^{6} \) \( + ( 7 + 19 \beta_{1} - 10 \beta_{2} - 3 \beta_{3} + 6 \beta_{4} - \beta_{5} ) q^{7} \) \( + ( 176 - 36 \beta_{1} + 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{8} \) \( -243 q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -2 - \beta_{1} ) q^{2} \) \( -\beta_{2} q^{3} \) \( + ( 27 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{4} \) \( + ( -7 + \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{5} \) \( + ( -27 + 2 \beta_{2} - 3 \beta_{4} ) q^{6} \) \( + ( 7 + 19 \beta_{1} - 10 \beta_{2} - 3 \beta_{3} + 6 \beta_{4} - \beta_{5} ) q^{7} \) \( + ( 176 - 36 \beta_{1} + 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{8} \) \( -243 q^{9} \) \( + ( 12 - 14 \beta_{1} + 40 \beta_{2} + 8 \beta_{5} ) q^{10} \) \( + ( -22 - 46 \beta_{1} + 8 \beta_{2} + 14 \beta_{3} + 36 \beta_{4} - 6 \beta_{5} ) q^{11} \) \( + ( -147 + 42 \beta_{1} - 31 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} ) q^{12} \) \( + ( -504 + 178 \beta_{1} - 16 \beta_{2} + 14 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{13} \) \( + ( 756 - 96 \beta_{1} + 160 \beta_{2} + 16 \beta_{3} - 4 \beta_{4} - 8 \beta_{5} ) q^{14} \) \( + ( 75 + 183 \beta_{1} - 20 \beta_{2} - 39 \beta_{3} - 18 \beta_{4} + 3 \beta_{5} ) q^{15} \) \( + ( -1628 - 48 \beta_{1} - 188 \beta_{2} - 40 \beta_{3} - 40 \beta_{4} - 4 \beta_{5} ) q^{16} \) \( + ( 1932 - 346 \beta_{1} + 32 \beta_{2} - 38 \beta_{3} - 12 \beta_{4} - 6 \beta_{5} ) q^{17} \) \( + ( 486 + 243 \beta_{1} ) q^{18} \) \( + ( -126 - 342 \beta_{1} - 80 \beta_{2} + 54 \beta_{3} - 108 \beta_{4} + 18 \beta_{5} ) q^{19} \) \( + ( 2954 + 124 \beta_{1} - 510 \beta_{2} + 34 \beta_{3} + 98 \beta_{4} - 32 \beta_{5} ) q^{20} \) \( + ( -1785 - 543 \beta_{1} + 48 \beta_{2} - 33 \beta_{3} + 30 \beta_{4} + 15 \beta_{5} ) q^{21} \) \( + ( -2236 - 64 \beta_{1} + 504 \beta_{2} - 160 \beta_{3} + 20 \beta_{4} - 48 \beta_{5} ) q^{22} \) \( + ( 46 + 70 \beta_{1} + 372 \beta_{2} - 38 \beta_{3} - 180 \beta_{4} + 30 \beta_{5} ) q^{23} \) \( + ( -1236 + 276 \beta_{1} - 196 \beta_{2} + 66 \beta_{3} - 54 \beta_{4} - 30 \beta_{5} ) q^{24} \) \( + ( 9919 + 1708 \beta_{1} - 160 \beta_{2} + 212 \beta_{3} + 104 \beta_{4} + 52 \beta_{5} ) q^{25} \) \( + ( -9684 + 686 \beta_{1} + 208 \beta_{2} + 128 \beta_{3} + 128 \beta_{4} + 16 \beta_{5} ) q^{26} \) \( + 243 \beta_{2} q^{27} \) \( + ( -3108 - 648 \beta_{1} - 116 \beta_{2} - 188 \beta_{3} + 388 \beta_{4} + 40 \beta_{5} ) q^{28} \) \( + ( -14485 - 1121 \beta_{1} + 96 \beta_{2} - 31 \beta_{3} + 130 \beta_{4} + 65 \beta_{5} ) q^{29} \) \( + ( 9294 - 480 \beta_{1} + 100 \beta_{2} + 336 \beta_{3} + 102 \beta_{4} + 24 \beta_{5} ) q^{30} \) \( + ( -697 - 1645 \beta_{1} + 174 \beta_{2} + 381 \beta_{3} + 390 \beta_{4} - 65 \beta_{5} ) q^{31} \) \( + ( 10616 + 1200 \beta_{1} + 376 \beta_{2} + 88 \beta_{3} - 648 \beta_{4} + 96 \beta_{5} ) q^{32} \) \( + ( 150 - 2010 \beta_{1} + 192 \beta_{2} - 294 \beta_{3} - 204 \beta_{4} - 102 \beta_{5} ) q^{33} \) \( + ( 17500 - 2506 \beta_{1} - 16 \beta_{2} - 256 \beta_{3} - 256 \beta_{4} + 48 \beta_{5} ) q^{34} \) \( + ( 1510 + 3838 \beta_{1} - 2100 \beta_{2} - 734 \beta_{3} + 252 \beta_{4} - 42 \beta_{5} ) q^{35} \) \( + ( -6561 - 486 \beta_{1} + 243 \beta_{2} + 243 \beta_{3} + 243 \beta_{4} ) q^{36} \) \( + ( 4798 + 4364 \beta_{1} - 368 \beta_{2} + 52 \beta_{3} - 632 \beta_{4} - 316 \beta_{5} ) q^{37} \) \( + ( -20628 + 1728 \beta_{1} - 2360 \beta_{2} - 288 \beta_{3} - 708 \beta_{4} + 144 \beta_{5} ) q^{38} \) \( + ( 342 + 990 \beta_{1} + 370 \beta_{2} - 126 \beta_{3} + 540 \beta_{4} - 90 \beta_{5} ) q^{39} \) \( + ( -33696 - 3512 \beta_{1} + 3520 \beta_{2} - 268 \beta_{3} - 1276 \beta_{4} - 68 \beta_{5} ) q^{40} \) \( + ( -10920 - 166 \beta_{1} + 32 \beta_{2} - 218 \beta_{3} - 372 \beta_{4} - 186 \beta_{5} ) q^{41} \) \( + ( 35664 + 1428 \beta_{1} - 984 \beta_{2} - 384 \beta_{3} - 384 \beta_{4} - 120 \beta_{5} ) q^{42} \) \( + ( -914 - 2234 \beta_{1} - 2560 \beta_{2} + 474 \beta_{3} + 204 \beta_{4} - 34 \beta_{5} ) q^{43} \) \( + ( 73876 - 920 \beta_{1} + 2404 \beta_{2} + 108 \beta_{3} + 1516 \beta_{4} + 152 \beta_{5} ) q^{44} \) \( + ( 1701 - 243 \beta_{1} + 243 \beta_{3} + 486 \beta_{4} + 243 \beta_{5} ) q^{45} \) \( + ( 13696 + 832 \beta_{1} - 3664 \beta_{2} + 544 \beta_{3} + 976 \beta_{4} + 240 \beta_{5} ) q^{46} \) \( + ( -1534 - 3862 \beta_{1} + 5524 \beta_{2} + 758 \beta_{3} - 108 \beta_{4} + 18 \beta_{5} ) q^{47} \) \( + ( -45276 + 1920 \beta_{1} + 1412 \beta_{2} - 48 \beta_{3} - 336 \beta_{4} + 228 \beta_{5} ) q^{48} \) \( + ( -20007 - 4600 \beta_{1} + 352 \beta_{2} + 376 \beta_{3} + 1456 \beta_{4} + 728 \beta_{5} ) q^{49} \) \( + ( -126614 - 6587 \beta_{1} - 800 \beta_{2} + 1280 \beta_{3} + 1280 \beta_{4} - 416 \beta_{5} ) q^{50} \) \( + ( 66 - 150 \beta_{1} - 1934 \beta_{2} - 138 \beta_{3} - 1260 \beta_{4} + 210 \beta_{5} ) q^{51} \) \( + ( -26426 + 11108 \beta_{1} - 50 \beta_{2} + 270 \beta_{3} + 1422 \beta_{4} - 320 \beta_{5} ) q^{52} \) \( + ( 78379 + 15 \beta_{1} - 32 \beta_{2} + 369 \beta_{3} + 674 \beta_{4} + 337 \beta_{5} ) q^{53} \) \( + ( 6561 - 486 \beta_{2} + 729 \beta_{4} ) q^{54} \) \( + ( 4760 + 11192 \beta_{1} + 8920 \beta_{2} - 2616 \beta_{3} - 2832 \beta_{4} + 472 \beta_{5} ) q^{55} \) \( + ( 131792 - 784 \beta_{1} + 1936 \beta_{2} - 232 \beta_{3} - 648 \beta_{4} - 936 \beta_{5} ) q^{56} \) \( + ( -31050 + 9774 \beta_{1} - 864 \beta_{2} + 594 \beta_{3} - 540 \beta_{4} - 270 \beta_{5} ) q^{57} \) \( + ( 93228 + 14506 \beta_{1} - 3368 \beta_{2} - 768 \beta_{3} - 768 \beta_{4} - 520 \beta_{5} ) q^{58} \) \( + ( -6776 - 17048 \beta_{1} - 7132 \beta_{2} + 3352 \beta_{3} - 432 \beta_{4} + 72 \beta_{5} ) q^{59} \) \( + ( -121818 - 4692 \beta_{1} - 2690 \beta_{2} - 1446 \beta_{3} - 102 \beta_{4} - 300 \beta_{5} ) q^{60} \) \( + ( -4374 - 21584 \beta_{1} + 2000 \beta_{2} - 2416 \beta_{3} - 832 \beta_{4} - 416 \beta_{5} ) q^{61} \) \( + ( -83556 + 2976 \beta_{1} + 2992 \beta_{2} - 3568 \beta_{3} - 668 \beta_{4} - 520 \beta_{5} ) q^{62} \) \( + ( -1701 - 4617 \beta_{1} + 2430 \beta_{2} + 729 \beta_{3} - 1458 \beta_{4} + 243 \beta_{5} ) q^{63} \) \( + ( -108992 - 4512 \beta_{1} - 10816 \beta_{2} + 2160 \beta_{3} + 1584 \beta_{4} + 912 \beta_{5} ) q^{64} \) \( + ( 56406 + 9262 \beta_{1} - 800 \beta_{2} + 338 \beta_{3} - 924 \beta_{4} - 462 \beta_{5} ) q^{65} \) \( + ( 127752 - 4980 \beta_{1} + 2544 \beta_{2} - 1536 \beta_{3} - 1536 \beta_{4} + 816 \beta_{5} ) q^{66} \) \( + ( 5192 + 13352 \beta_{1} - 16340 \beta_{2} - 2472 \beta_{3} + 1488 \beta_{4} - 248 \beta_{5} ) q^{67} \) \( + ( 109230 - 18988 \beta_{1} - 5994 \beta_{2} - 1194 \beta_{3} - 2858 \beta_{4} + 320 \beta_{5} ) q^{68} \) \( + ( 93702 + 11298 \beta_{1} - 1056 \beta_{2} + 1374 \beta_{3} + 636 \beta_{4} + 318 \beta_{5} ) q^{69} \) \( + ( 149944 - 13760 \beta_{1} + 16160 \beta_{2} + 5536 \beta_{3} - 2168 \beta_{4} - 336 \beta_{5} ) q^{70} \) \( + ( 6246 + 16542 \beta_{1} + 11188 \beta_{2} - 2814 \beta_{3} + 3708 \beta_{4} - 618 \beta_{5} ) q^{71} \) \( + ( -42768 + 8748 \beta_{1} - 1458 \beta_{3} + 486 \beta_{4} - 486 \beta_{5} ) q^{72} \) \( + ( -219350 - 18040 \beta_{1} + 1696 \beta_{2} - 2312 \beta_{3} - 1232 \beta_{4} - 616 \beta_{5} ) q^{73} \) \( + ( -256052 - 6282 \beta_{1} + 15584 \beta_{2} + 2944 \beta_{3} + 2944 \beta_{4} + 2528 \beta_{5} ) q^{74} \) \( + ( -1980 - 3276 \beta_{1} - 9315 \beta_{2} + 1548 \beta_{3} + 6696 \beta_{4} - 1116 \beta_{5} ) q^{75} \) \( + ( 17724 + 22584 \beta_{1} - 5972 \beta_{2} + 4164 \beta_{3} - 6204 \beta_{4} + 840 \beta_{5} ) q^{76} \) \( + ( -238348 + 31436 \beta_{1} - 2816 \beta_{2} + 2356 \beta_{3} - 920 \beta_{4} - 460 \beta_{5} ) q^{77} \) \( + ( 63594 - 6336 \beta_{1} + 10420 \beta_{2} + 288 \beta_{3} + 2730 \beta_{4} - 720 \beta_{5} ) q^{78} \) \( + ( -9077 - 20441 \beta_{1} + 30694 \beta_{2} + 5289 \beta_{3} + 9006 \beta_{4} - 1501 \beta_{5} ) q^{79} \) \( + ( 225336 + 35168 \beta_{1} - 17160 \beta_{2} - 1840 \beta_{3} + 5840 \beta_{4} + 2824 \beta_{5} ) q^{80} \) \( + 59049 q^{81} \) \( + ( 42844 + 6566 \beta_{1} + 7184 \beta_{2} - 256 \beta_{3} - 256 \beta_{4} + 1488 \beta_{5} ) q^{82} \) \( + ( 8378 + 19874 \beta_{1} - 34096 \beta_{2} - 4546 \beta_{3} - 4284 \beta_{4} + 714 \beta_{5} ) q^{83} \) \( + ( -38028 - 41160 \beta_{1} + 8100 \beta_{2} + 2244 \beta_{3} - 1788 \beta_{4} + 1248 \beta_{5} ) q^{84} \) \( + ( 135670 - 450 \beta_{1} + 450 \beta_{3} + 900 \beta_{4} + 450 \beta_{5} ) q^{85} \) \( + ( -189180 + 5952 \beta_{1} + 4120 \beta_{2} - 4064 \beta_{3} - 9676 \beta_{4} - 272 \beta_{5} ) q^{86} \) \( + ( -6027 - 15639 \beta_{1} + 16720 \beta_{2} + 2823 \beta_{3} - 2286 \beta_{4} + 381 \beta_{5} ) q^{87} \) \( + ( 55216 - 77360 \beta_{1} - 528 \beta_{2} - 1848 \beta_{3} + 7656 \beta_{4} - 3640 \beta_{5} ) q^{88} \) \( + ( 117390 - 14380 \beta_{1} + 1088 \beta_{2} + 1324 \beta_{3} + 4824 \beta_{4} + 2412 \beta_{5} ) q^{89} \) \( + ( -2916 + 3402 \beta_{1} - 9720 \beta_{2} - 1944 \beta_{5} ) q^{90} \) \( + ( 7238 + 16190 \beta_{1} - 35524 \beta_{2} - 4254 \beta_{3} - 7620 \beta_{4} + 1270 \beta_{5} ) q^{91} \) \( + ( -226288 - 6880 \beta_{1} + 1744 \beta_{2} - 336 \beta_{3} - 9424 \beta_{4} - 2912 \beta_{5} ) q^{92} \) \( + ( -17745 - 12831 \beta_{1} + 1392 \beta_{2} - 3873 \beta_{3} - 4962 \beta_{4} - 2481 \beta_{5} ) q^{93} \) \( + ( -58696 + 12992 \beta_{1} - 20608 \beta_{2} - 5920 \beta_{3} + 12584 \beta_{4} + 144 \beta_{5} ) q^{94} \) \( + ( -7680 - 21504 \beta_{1} + 32600 \beta_{2} + 3072 \beta_{3} - 9216 \beta_{4} + 1536 \beta_{5} ) q^{95} \) \( + ( 119496 + 50160 \beta_{1} - 15544 \beta_{2} + 3768 \beta_{3} + 5592 \beta_{4} - 240 \beta_{5} ) q^{96} \) \( + ( 551566 + 32804 \beta_{1} - 2816 \beta_{2} + 988 \beta_{3} - 3656 \beta_{4} - 1828 \beta_{5} ) q^{97} \) \( + ( 276606 + 30367 \beta_{1} - 31936 \beta_{2} - 2816 \beta_{3} - 2816 \beta_{4} - 5824 \beta_{5} ) q^{98} \) \( + ( 5346 + 11178 \beta_{1} - 1944 \beta_{2} - 3402 \beta_{3} - 8748 \beta_{4} + 1458 \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 10q^{2} \) \(\mathstrut +\mathstrut 156q^{4} \) \(\mathstrut -\mathstrut 44q^{5} \) \(\mathstrut -\mathstrut 162q^{6} \) \(\mathstrut +\mathstrut 1136q^{8} \) \(\mathstrut -\mathstrut 1458q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 10q^{2} \) \(\mathstrut +\mathstrut 156q^{4} \) \(\mathstrut -\mathstrut 44q^{5} \) \(\mathstrut -\mathstrut 162q^{6} \) \(\mathstrut +\mathstrut 1136q^{8} \) \(\mathstrut -\mathstrut 1458q^{9} \) \(\mathstrut +\mathstrut 84q^{10} \) \(\mathstrut -\mathstrut 972q^{12} \) \(\mathstrut -\mathstrut 3348q^{13} \) \(\mathstrut +\mathstrut 4776q^{14} \) \(\mathstrut -\mathstrut 9744q^{16} \) \(\mathstrut +\mathstrut 12220q^{17} \) \(\mathstrut +\mathstrut 2430q^{18} \) \(\mathstrut +\mathstrut 17608q^{20} \) \(\mathstrut -\mathstrut 9720q^{21} \) \(\mathstrut -\mathstrut 13512q^{22} \) \(\mathstrut -\mathstrut 7776q^{24} \) \(\mathstrut +\mathstrut 56418q^{25} \) \(\mathstrut -\mathstrut 59252q^{26} \) \(\mathstrut -\mathstrut 17808q^{28} \) \(\mathstrut -\mathstrut 84860q^{29} \) \(\mathstrut +\mathstrut 57348q^{30} \) \(\mathstrut +\mathstrut 61280q^{32} \) \(\mathstrut +\mathstrut 4536q^{33} \) \(\mathstrut +\mathstrut 109404q^{34} \) \(\mathstrut -\mathstrut 37908q^{36} \) \(\mathstrut +\mathstrut 20796q^{37} \) \(\mathstrut -\mathstrut 128088q^{38} \) \(\mathstrut -\mathstrut 195552q^{40} \) \(\mathstrut -\mathstrut 65252q^{41} \) \(\mathstrut +\mathstrut 210600q^{42} \) \(\mathstrut +\mathstrut 445008q^{44} \) \(\mathstrut +\mathstrut 10692q^{45} \) \(\mathstrut +\mathstrut 81120q^{46} \) \(\mathstrut -\mathstrut 276048q^{48} \) \(\mathstrut -\mathstrut 111546q^{49} \) \(\mathstrut -\mathstrut 743118q^{50} \) \(\mathstrut -\mathstrut 179592q^{52} \) \(\mathstrut +\mathstrut 470308q^{53} \) \(\mathstrut +\mathstrut 39366q^{54} \) \(\mathstrut +\mathstrut 793728q^{56} \) \(\mathstrut -\mathstrut 204120q^{57} \) \(\mathstrut +\mathstrut 529860q^{58} \) \(\mathstrut -\mathstrut 723816q^{60} \) \(\mathstrut +\mathstrut 12924q^{61} \) \(\mathstrut -\mathstrut 513384q^{62} \) \(\mathstrut -\mathstrut 642432q^{64} \) \(\mathstrut +\mathstrut 321512q^{65} \) \(\mathstrut +\mathstrut 771768q^{66} \) \(\mathstrut +\mathstrut 690328q^{68} \) \(\mathstrut +\mathstrut 541728q^{69} \) \(\mathstrut +\mathstrut 938928q^{70} \) \(\mathstrut -\mathstrut 276048q^{72} \) \(\mathstrut -\mathstrut 1283412q^{73} \) \(\mathstrut -\mathstrut 1522916q^{74} \) \(\mathstrut +\mathstrut 67824q^{76} \) \(\mathstrut -\mathstrut 1487328q^{77} \) \(\mathstrut +\mathstrut 396252q^{78} \) \(\mathstrut +\mathstrut 1272352q^{80} \) \(\mathstrut +\mathstrut 354294q^{81} \) \(\mathstrut +\mathstrut 240444q^{82} \) \(\mathstrut -\mathstrut 143856q^{84} \) \(\mathstrut +\mathstrut 814920q^{85} \) \(\mathstrut -\mathstrut 1154568q^{86} \) \(\mathstrut +\mathstrut 489600q^{88} \) \(\mathstrut +\mathstrut 730924q^{89} \) \(\mathstrut -\mathstrut 20412q^{90} \) \(\mathstrut -\mathstrut 1338816q^{92} \) \(\mathstrut -\mathstrut 83592q^{93} \) \(\mathstrut -\mathstrut 390288q^{94} \) \(\mathstrut +\mathstrut 624672q^{96} \) \(\mathstrut +\mathstrut 3249420q^{97} \) \(\mathstrut +\mathstrut 1604918q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut +\mathstrut \) \(8\) \(x^{4}\mathstrut -\mathstrut \) \(10\) \(x^{3}\mathstrut +\mathstrut \) \(64\) \(x^{2}\mathstrut -\mathstrut \) \(40\) \(x\mathstrut +\mathstrut \) \(25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{5} - 5 \nu^{4} - 13 \nu^{3} - 35 \nu^{2} - 79 \nu - 175 \)\()/20\)
\(\beta_{2}\)\(=\)\((\)\( -9 \nu^{5} - 72 \nu^{3} + 45 \nu^{2} - 576 \nu + 180 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{5} - 25 \nu^{4} + 31 \nu^{3} - 210 \nu^{2} + 693 \nu - 1175 \)\()/20\)
\(\beta_{4}\)\(=\)\((\)\( -33 \nu^{5} - 15 \nu^{4} - 279 \nu^{3} + 285 \nu^{2} - 1917 \nu + 735 \)\()/20\)
\(\beta_{5}\)\(=\)\((\)\( 63 \nu^{5} + 10 \nu^{4} + 314 \nu^{3} - 715 \nu^{2} + 3262 \nu - 1170 \)\()/20\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{2}\mathstrut -\mathstrut \) \(29\) \(\beta_{1}\mathstrut -\mathstrut \) \(11\)\()/144\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(\beta_{5}\mathstrut +\mathstrut \) \(10\) \(\beta_{4}\mathstrut -\mathstrut \) \(5\) \(\beta_{3}\mathstrut -\mathstrut \) \(44\) \(\beta_{2}\mathstrut -\mathstrut \) \(7\) \(\beta_{1}\mathstrut -\mathstrut \) \(385\)\()/144\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(4\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(94\)\()/18\)
\(\nu^{4}\)\(=\)\((\)\(21\) \(\beta_{5}\mathstrut -\mathstrut \) \(54\) \(\beta_{4}\mathstrut +\mathstrut \) \(33\) \(\beta_{3}\mathstrut +\mathstrut \) \(412\) \(\beta_{2}\mathstrut -\mathstrut \) \(537\) \(\beta_{1}\mathstrut -\mathstrut \) \(3255\)\()/144\)
\(\nu^{5}\)\(=\)\((\)\(59\) \(\beta_{5}\mathstrut +\mathstrut \) \(178\) \(\beta_{4}\mathstrut -\mathstrut \) \(281\) \(\beta_{3}\mathstrut -\mathstrut \) \(732\) \(\beta_{2}\mathstrut +\mathstrut \) \(989\) \(\beta_{1}\mathstrut -\mathstrut \) \(4357\)\()/144\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−1.55022 2.68505i
−1.55022 + 2.68505i
1.21966 2.11251i
1.21966 + 2.11251i
0.330560 0.572547i
0.330560 + 0.572547i
−7.12493 3.63805i 15.5885i 37.5291 + 51.8417i 172.232 56.7117 111.067i 545.623i −78.7890 505.901i −243.000 −1227.14 626.588i
7.2 −7.12493 + 3.63805i 15.5885i 37.5291 51.8417i 172.232 56.7117 + 111.067i 545.623i −78.7890 + 505.901i −243.000 −1227.14 + 626.588i
7.3 −5.33979 5.95706i 15.5885i −6.97319 + 63.6190i −212.349 −92.8614 + 83.2392i 87.6116i 416.218 298.173i −243.000 1133.90 + 1264.98i
7.4 −5.33979 + 5.95706i 15.5885i −6.97319 63.6190i −212.349 −92.8614 83.2392i 87.6116i 416.218 + 298.173i −243.000 1133.90 1264.98i
7.5 7.46472 2.87714i 15.5885i 47.4441 42.9542i 18.1171 −44.8502 116.363i 321.465i 230.571 457.144i −243.000 135.239 52.1255i
7.6 7.46472 + 2.87714i 15.5885i 47.4441 + 42.9542i 18.1171 −44.8502 + 116.363i 321.465i 230.571 + 457.144i −243.000 135.239 + 52.1255i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.6
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{7}^{\mathrm{new}}(12, [\chi])\).