# 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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.76064900344$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.50898483.1 Defining polynomial: $$x^{6} + 8 x^{4} - 10 x^{3} + 64 x^{2} - 40 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{12}\cdot 3^{5}$$ Twist minimal: yes 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 10q^{2} + 156q^{4} - 44q^{5} - 162q^{6} + 1136q^{8} - 1458q^{9} + O(q^{10})$$ $$6q - 10q^{2} + 156q^{4} - 44q^{5} - 162q^{6} + 1136q^{8} - 1458q^{9} + 84q^{10} - 972q^{12} - 3348q^{13} + 4776q^{14} - 9744q^{16} + 12220q^{17} + 2430q^{18} + 17608q^{20} - 9720q^{21} - 13512q^{22} - 7776q^{24} + 56418q^{25} - 59252q^{26} - 17808q^{28} - 84860q^{29} + 57348q^{30} + 61280q^{32} + 4536q^{33} + 109404q^{34} - 37908q^{36} + 20796q^{37} - 128088q^{38} - 195552q^{40} - 65252q^{41} + 210600q^{42} + 445008q^{44} + 10692q^{45} + 81120q^{46} - 276048q^{48} - 111546q^{49} - 743118q^{50} - 179592q^{52} + 470308q^{53} + 39366q^{54} + 793728q^{56} - 204120q^{57} + 529860q^{58} - 723816q^{60} + 12924q^{61} - 513384q^{62} - 642432q^{64} + 321512q^{65} + 771768q^{66} + 690328q^{68} + 541728q^{69} + 938928q^{70} - 276048q^{72} - 1283412q^{73} - 1522916q^{74} + 67824q^{76} - 1487328q^{77} + 396252q^{78} + 1272352q^{80} + 354294q^{81} + 240444q^{82} - 143856q^{84} + 814920q^{85} - 1154568q^{86} + 489600q^{88} + 730924q^{89} - 20412q^{90} - 1338816q^{92} - 83592q^{93} - 390288q^{94} + 624672q^{96} + 3249420q^{97} + 1604918q^{98} + O(q^{100})$$

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

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 12.7.d.a 6
3.b odd 2 1 36.7.d.e 6
4.b odd 2 1 inner 12.7.d.a 6
8.b even 2 1 192.7.g.e 6
8.d odd 2 1 192.7.g.e 6
12.b even 2 1 36.7.d.e 6
16.e even 4 2 768.7.b.h 12
16.f odd 4 2 768.7.b.h 12
24.f even 2 1 576.7.g.p 6
24.h odd 2 1 576.7.g.p 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.7.d.a 6 1.a even 1 1 trivial
12.7.d.a 6 4.b odd 2 1 inner
36.7.d.e 6 3.b odd 2 1
36.7.d.e 6 12.b even 2 1
192.7.g.e 6 8.b even 2 1
192.7.g.e 6 8.d odd 2 1
576.7.g.p 6 24.f even 2 1
576.7.g.p 6 24.h odd 2 1
768.7.b.h 12 16.e even 4 2
768.7.b.h 12 16.f odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(12, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 10 T - 28 T^{2} - 992 T^{3} - 1792 T^{4} + 40960 T^{5} + 262144 T^{6}$$
$3$ $$( 1 + 243 T^{2} )^{3}$$
$5$ $$( 1 + 22 T + 9575 T^{2} + 1350100 T^{3} + 149609375 T^{4} + 5371093750 T^{5} + 3814697265625 T^{6} )^{2}$$
$7$ $$1 - 297174 T^{2} + 49120386735 T^{4} - 6352070622707828 T^{6} +$$$$67\!\cdots\!35$$$$T^{8} -$$$$56\!\cdots\!74$$$$T^{10} +$$$$26\!\cdots\!01$$$$T^{12}$$
$11$ $$1 - 2481510 T^{2} + 6024225785151 T^{4} - 16527738499614302996 T^{6} +$$$$18\!\cdots\!71$$$$T^{8} -$$$$24\!\cdots\!10$$$$T^{10} +$$$$30\!\cdots\!61$$$$T^{12}$$
$13$ $$( 1 + 1674 T + 11116791 T^{2} + 16065685420 T^{3} + 53658626849919 T^{4} + 39000994495033194 T^{5} +$$$$11\!\cdots\!29$$$$T^{6} )^{2}$$
$17$ $$( 1 - 6110 T + 66889295 T^{2} - 292605918500 T^{3} + 1614544973423855 T^{4} - 3559821869473839710 T^{5} +$$$$14\!\cdots\!09$$$$T^{6} )^{2}$$
$19$ $$1 - 146059206 T^{2} + 11584792041958815 T^{4} -$$$$63\!\cdots\!32$$$$T^{6} +$$$$25\!\cdots\!15$$$$T^{8} -$$$$71\!\cdots\!26$$$$T^{10} +$$$$10\!\cdots\!81$$$$T^{12}$$
$23$ $$1 - 607352358 T^{2} + 180562295442724527 T^{4} -$$$$33\!\cdots\!16$$$$T^{6} +$$$$39\!\cdots\!67$$$$T^{8} -$$$$29\!\cdots\!78$$$$T^{10} +$$$$10\!\cdots\!61$$$$T^{12}$$
$29$ $$( 1 + 42430 T + 2077407863 T^{2} + 50868097395076 T^{3} + 1235690644141173023 T^{4} +$$$$15\!\cdots\!30$$$$T^{5} +$$$$21\!\cdots\!61$$$$T^{6} )^{2}$$
$31$ $$1 - 3163788534 T^{2} + 5235513742747890255 T^{4} -$$$$55\!\cdots\!16$$$$T^{6} +$$$$41\!\cdots\!55$$$$T^{8} -$$$$19\!\cdots\!14$$$$T^{10} +$$$$48\!\cdots\!81$$$$T^{12}$$
$37$ $$( 1 - 10398 T + 1806728583 T^{2} - 208245913091588 T^{3} + 4635571239298248447 T^{4} -$$$$68\!\cdots\!38$$$$T^{5} +$$$$16\!\cdots\!29$$$$T^{6} )^{2}$$
$41$ $$( 1 + 32626 T + 13292008703 T^{2} + 308481662881276 T^{3} + 63138426911529209423 T^{4} +$$$$73\!\cdots\!06$$$$T^{5} +$$$$10\!\cdots\!21$$$$T^{6} )^{2}$$
$43$ $$1 - 29142871014 T^{2} +$$$$39\!\cdots\!55$$$$T^{4} -$$$$31\!\cdots\!92$$$$T^{6} +$$$$15\!\cdots\!55$$$$T^{8} -$$$$46\!\cdots\!14$$$$T^{10} +$$$$63\!\cdots\!01$$$$T^{12}$$
$47$ $$1 - 39654949638 T^{2} +$$$$75\!\cdots\!55$$$$T^{4} -$$$$94\!\cdots\!52$$$$T^{6} +$$$$87\!\cdots\!55$$$$T^{8} -$$$$53\!\cdots\!78$$$$T^{10} +$$$$15\!\cdots\!21$$$$T^{12}$$
$53$ $$( 1 - 235154 T + 80653582439 T^{2} - 10625112229848476 T^{3} +$$$$17\!\cdots\!31$$$$T^{4} -$$$$11\!\cdots\!14$$$$T^{5} +$$$$10\!\cdots\!89$$$$T^{6} )^{2}$$
$59$ $$1 - 38958254886 T^{2} +$$$$25\!\cdots\!75$$$$T^{4} -$$$$57\!\cdots\!32$$$$T^{6} +$$$$45\!\cdots\!75$$$$T^{8} -$$$$12\!\cdots\!46$$$$T^{10} +$$$$56\!\cdots\!41$$$$T^{12}$$
$61$ $$( 1 - 6462 T + 83167803063 T^{2} - 6651331196354948 T^{3} +$$$$42\!\cdots\!43$$$$T^{4} -$$$$17\!\cdots\!02$$$$T^{5} +$$$$13\!\cdots\!81$$$$T^{6} )^{2}$$
$67$ $$1 - 296399142534 T^{2} +$$$$37\!\cdots\!95$$$$T^{4} -$$$$34\!\cdots\!68$$$$T^{6} +$$$$30\!\cdots\!95$$$$T^{8} -$$$$19\!\cdots\!14$$$$T^{10} +$$$$54\!\cdots\!81$$$$T^{12}$$
$71$ $$1 - 415188323430 T^{2} +$$$$92\!\cdots\!47$$$$T^{4} -$$$$13\!\cdots\!36$$$$T^{6} +$$$$15\!\cdots\!27$$$$T^{8} -$$$$11\!\cdots\!30$$$$T^{10} +$$$$44\!\cdots\!21$$$$T^{12}$$
$73$ $$( 1 + 641706 T + 530324505759 T^{2} + 188743068109697740 T^{3} +$$$$80\!\cdots\!51$$$$T^{4} +$$$$14\!\cdots\!26$$$$T^{5} +$$$$34\!\cdots\!69$$$$T^{6} )^{2}$$
$79$ $$1 - 234724993590 T^{2} +$$$$48\!\cdots\!95$$$$T^{4} -$$$$12\!\cdots\!60$$$$T^{6} +$$$$28\!\cdots\!95$$$$T^{8} -$$$$81\!\cdots\!90$$$$T^{10} +$$$$20\!\cdots\!21$$$$T^{12}$$
$83$ $$1 - 954563652678 T^{2} +$$$$53\!\cdots\!43$$$$T^{4} -$$$$21\!\cdots\!84$$$$T^{6} +$$$$57\!\cdots\!23$$$$T^{8} -$$$$10\!\cdots\!38$$$$T^{10} +$$$$12\!\cdots\!81$$$$T^{12}$$
$89$ $$( 1 - 365462 T + 1298555750591 T^{2} - 314016079540511540 T^{3} +$$$$64\!\cdots\!51$$$$T^{4} -$$$$90\!\cdots\!02$$$$T^{5} +$$$$12\!\cdots\!81$$$$T^{6} )^{2}$$
$97$ $$( 1 - 1624710 T + 3125499884751 T^{2} - 2772177769643198996 T^{3} +$$$$26\!\cdots\!79$$$$T^{4} -$$$$11\!\cdots\!10$$$$T^{5} +$$$$57\!\cdots\!89$$$$T^{6} )^{2}$$