# 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 $$2$$ 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{7}^{\mathrm{new}}(12, [\chi])$$.