# Properties

 Label 12.6.b.a Level 12 Weight 6 Character orbit 12.b Analytic conductor 1.925 Analytic rank 0 Dimension 8 CM No Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$12 = 2^{2} \cdot 3$$ Weight: $$k$$ = $$6$$ Character orbit: $$[\chi]$$ = 12.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$1.92460583776$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{18}\cdot 3^{5}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} -\beta_{1} q^{3} + ( 1 - \beta_{3} ) q^{4} + ( \beta_{2} + \beta_{5} ) q^{5} + ( 3 - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{6} + ( -\beta_{1} + 2 \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{7} + ( 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{8} + ( -3 - 2 \beta_{1} - 15 \beta_{2} + 6 \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} -\beta_{1} q^{3} + ( 1 - \beta_{3} ) q^{4} + ( \beta_{2} + \beta_{5} ) q^{5} + ( 3 - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{6} + ( -\beta_{1} + 2 \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{7} + ( 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{8} + ( -3 - 2 \beta_{1} - 15 \beta_{2} + 6 \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{9} + ( -34 + 14 \beta_{1} - 4 \beta_{3} - 4 \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{10} + ( -3 \beta_{1} - 32 \beta_{2} - \beta_{4} + 4 \beta_{7} ) q^{11} + ( -87 - 2 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 7 \beta_{7} ) q^{12} + ( 14 + 4 \beta_{1} - 12 \beta_{3} - 4 \beta_{6} - 2 \beta_{7} ) q^{13} + ( -30 \beta_{1} - 8 \beta_{2} - 10 \beta_{4} + 2 \beta_{5} - 9 \beta_{7} ) q^{14} + ( 5 \beta_{1} + 96 \beta_{2} - 6 \beta_{3} + \beta_{4} + 6 \beta_{6} - 9 \beta_{7} ) q^{15} + ( 292 - 40 \beta_{1} + 4 \beta_{3} + 16 \beta_{4} - 8 \beta_{6} - 4 \beta_{7} ) q^{16} + ( 148 \beta_{2} - 12 \beta_{5} + 20 \beta_{7} ) q^{17} + ( 462 - 10 \beta_{1} - 3 \beta_{2} + 12 \beta_{3} - 12 \beta_{4} + 8 \beta_{5} - 2 \beta_{6} + 19 \beta_{7} ) q^{18} + ( 27 \beta_{1} - 12 \beta_{3} - 13 \beta_{4} + 12 \beta_{6} + 6 \beta_{7} ) q^{19} + ( 36 \beta_{1} - 36 \beta_{2} + 12 \beta_{4} + 20 \beta_{5} - 30 \beta_{7} ) q^{20} + ( -42 + 8 \beta_{1} - 237 \beta_{2} - 24 \beta_{3} - 13 \beta_{5} - 8 \beta_{6} - 32 \beta_{7} ) q^{21} + ( -1038 + 6 \beta_{1} + 32 \beta_{3} - 4 \beta_{4} + 6 \beta_{6} + 3 \beta_{7} ) q^{22} + ( 42 \beta_{1} - 256 \beta_{2} + 14 \beta_{4} + 32 \beta_{7} ) q^{23} + ( -1404 - 2 \beta_{1} - 78 \beta_{2} - 12 \beta_{3} + 2 \beta_{4} - 18 \beta_{5} - 24 \beta_{6} + 27 \beta_{7} ) q^{24} + ( -171 - 20 \beta_{1} + 60 \beta_{3} + 20 \beta_{6} + 10 \beta_{7} ) q^{25} + ( 48 \beta_{1} + 14 \beta_{2} + 16 \beta_{4} - 16 \beta_{5} - 40 \beta_{7} ) q^{26} + ( -6 \beta_{1} + 288 \beta_{2} + 36 \beta_{3} - 15 \beta_{4} - 36 \beta_{6} - 54 \beta_{7} ) q^{27} + ( 2406 + 88 \beta_{1} + 2 \beta_{3} - 48 \beta_{4} + 56 \beta_{6} + 28 \beta_{7} ) q^{28} + ( 247 \beta_{2} + 55 \beta_{5} + 24 \beta_{7} ) q^{29} + ( 3126 + 66 \beta_{1} + 24 \beta_{2} - 96 \beta_{3} + 36 \beta_{4} - 16 \beta_{5} - 14 \beta_{6} + 25 \beta_{7} ) q^{30} + ( -185 \beta_{1} + 22 \beta_{3} + 69 \beta_{4} - 22 \beta_{6} - 11 \beta_{7} ) q^{31} + ( -168 \beta_{1} + 264 \beta_{2} - 56 \beta_{4} - 72 \beta_{5} - 4 \beta_{7} ) q^{32} + ( 528 + 10 \beta_{1} - 141 \beta_{2} - 30 \beta_{3} + 67 \beta_{5} - 10 \beta_{6} - 31 \beta_{7} ) q^{33} + ( -4552 - 168 \beta_{1} - 112 \beta_{3} + 48 \beta_{4} + 24 \beta_{6} + 12 \beta_{7} ) q^{34} + ( -246 \beta_{1} - 160 \beta_{2} - 82 \beta_{4} + 20 \beta_{7} ) q^{35} + ( -5259 + 100 \beta_{1} + 444 \beta_{2} - 21 \beta_{3} - 84 \beta_{4} + 52 \beta_{5} + 32 \beta_{6} + 2 \beta_{7} ) q^{36} + ( 806 - 4 \beta_{1} + 12 \beta_{3} + 4 \beta_{6} + 2 \beta_{7} ) q^{37} + ( 222 \beta_{1} + 48 \beta_{2} + 74 \beta_{4} + 30 \beta_{5} + 33 \beta_{7} ) q^{38} + ( -2 \beta_{1} - 288 \beta_{2} - 36 \beta_{3} + 96 \beta_{4} + 36 \beta_{6} + 54 \beta_{7} ) q^{39} + ( 6728 + 208 \beta_{1} - 24 \beta_{3} - 32 \beta_{4} - 112 \beta_{6} - 56 \beta_{7} ) q^{40} + ( -614 \beta_{2} - 102 \beta_{5} - 64 \beta_{7} ) q^{41} + ( 7386 - 86 \beta_{1} - 42 \beta_{2} + 276 \beta_{3} + 84 \beta_{4} - 32 \beta_{5} + 26 \beta_{6} - 67 \beta_{7} ) q^{42} + ( 551 \beta_{1} - 4 \beta_{3} - 185 \beta_{4} + 4 \beta_{6} + 2 \beta_{7} ) q^{43} + ( -132 \beta_{1} - 1052 \beta_{2} - 44 \beta_{4} + 76 \beta_{5} + 62 \beta_{7} ) q^{44} + ( -2208 - 68 \beta_{1} + 1029 \beta_{2} + 204 \beta_{3} - 155 \beta_{5} + 68 \beta_{6} + 182 \beta_{7} ) q^{45} + ( -8700 - 84 \beta_{1} + 256 \beta_{3} + 56 \beta_{4} - 84 \beta_{6} - 42 \beta_{7} ) q^{46} + ( 756 \beta_{1} + 1600 \beta_{2} + 252 \beta_{4} - 200 \beta_{7} ) q^{47} + ( -9372 - 336 \beta_{1} - 1464 \beta_{2} + 132 \beta_{3} + 24 \beta_{4} - 8 \beta_{5} + 56 \beta_{6} - 136 \beta_{7} ) q^{48} + ( -2693 + 164 \beta_{1} - 492 \beta_{3} - 164 \beta_{6} - 82 \beta_{7} ) q^{49} + ( -240 \beta_{1} - 171 \beta_{2} - 80 \beta_{4} + 80 \beta_{5} + 200 \beta_{7} ) q^{50} + ( 20 \beta_{1} - 1632 \beta_{2} - 168 \beta_{3} - 332 \beta_{4} + 168 \beta_{6} + 288 \beta_{7} ) q^{51} + ( 10526 - 320 \beta_{1} + 34 \beta_{3} + 128 \beta_{4} - 64 \beta_{6} - 32 \beta_{7} ) q^{52} + ( -1043 \beta_{2} - 19 \beta_{5} - 128 \beta_{7} ) q^{53} + ( 9765 - 342 \beta_{1} - 144 \beta_{2} - 288 \beta_{3} - 201 \beta_{4} + 147 \beta_{5} + 87 \beta_{6} - 174 \beta_{7} ) q^{54} + ( -812 \beta_{1} - 68 \beta_{3} + 248 \beta_{4} + 68 \beta_{6} + 34 \beta_{7} ) q^{55} + ( 612 \beta_{1} + 2572 \beta_{2} + 204 \beta_{4} + 180 \beta_{5} + 250 \beta_{7} ) q^{56} + ( 5418 - 6 \beta_{1} + 1737 \beta_{2} + 18 \beta_{3} + 57 \beta_{5} + 6 \beta_{6} + 213 \beta_{7} ) q^{57} + ( -7822 + 770 \beta_{1} - 412 \beta_{3} - 220 \beta_{4} - 110 \beta_{6} - 55 \beta_{7} ) q^{58} + ( -1215 \beta_{1} + 1664 \beta_{2} - 405 \beta_{4} - 208 \beta_{7} ) q^{59} + ( -8088 + 52 \beta_{1} + 3180 \beta_{2} + 24 \beta_{3} + 316 \beta_{4} - 220 \beta_{5} - 128 \beta_{6} - 230 \beta_{7} ) q^{60} + ( 7454 - 164 \beta_{1} + 492 \beta_{3} + 164 \beta_{6} + 82 \beta_{7} ) q^{61} + ( -678 \beta_{1} - 88 \beta_{2} - 226 \beta_{4} - 326 \beta_{5} + 75 \beta_{7} ) q^{62} + ( -111 \beta_{1} - 1152 \beta_{2} + 342 \beta_{3} + 627 \beta_{4} - 342 \beta_{6} - 27 \beta_{7} ) q^{63} + ( 4816 - 672 \beta_{1} - 48 \beta_{3} + 64 \beta_{4} + 480 \beta_{6} + 240 \beta_{7} ) q^{64} + ( -2050 \beta_{2} + 318 \beta_{5} - 296 \beta_{7} ) q^{65} + ( 4170 + 1058 \beta_{1} + 528 \beta_{2} - 60 \beta_{3} - 228 \beta_{4} - 40 \beta_{5} - 134 \beta_{6} - 167 \beta_{7} ) q^{66} + ( 847 \beta_{1} + 304 \beta_{3} - 181 \beta_{4} - 304 \beta_{6} - 152 \beta_{7} ) q^{67} + ( 528 \beta_{1} - 4368 \beta_{2} + 176 \beta_{4} - 560 \beta_{5} + 200 \beta_{7} ) q^{68} + ( -11616 + 212 \beta_{1} - 138 \beta_{2} - 636 \beta_{3} + 470 \beta_{5} - 212 \beta_{6} - 182 \beta_{7} ) q^{69} + ( -3804 + 492 \beta_{1} + 160 \beta_{3} - 328 \beta_{4} + 492 \beta_{6} + 246 \beta_{7} ) q^{70} + ( 798 \beta_{1} - 2496 \beta_{2} + 266 \beta_{4} + 312 \beta_{7} ) q^{71} + ( 2568 + 1550 \beta_{1} - 5142 \beta_{2} - 600 \beta_{3} - 54 \beta_{4} + 278 \beta_{5} + 16 \beta_{6} + 19 \beta_{7} ) q^{72} + ( -16150 - 448 \beta_{1} + 1344 \beta_{3} + 448 \beta_{6} + 224 \beta_{7} ) q^{73} + ( -48 \beta_{1} + 806 \beta_{2} - 16 \beta_{4} + 16 \beta_{5} + 40 \beta_{7} ) q^{74} + ( 111 \beta_{1} + 1440 \beta_{2} + 180 \beta_{3} - 480 \beta_{4} - 180 \beta_{6} - 270 \beta_{7} ) q^{75} + ( -10782 - 24 \beta_{1} - 138 \beta_{3} + 176 \beta_{4} - 504 \beta_{6} - 252 \beta_{7} ) q^{76} + ( 1074 \beta_{2} - 206 \beta_{5} + 160 \beta_{7} ) q^{77} + ( -10470 - 144 \beta_{1} + 144 \beta_{2} + 288 \beta_{3} + 274 \beta_{4} - 398 \beta_{5} - 322 \beta_{6} + 182 \beta_{7} ) q^{78} + ( -1537 \beta_{1} - 442 \beta_{3} + 365 \beta_{4} + 442 \beta_{6} + 221 \beta_{7} ) q^{79} + ( -336 \beta_{1} + 6416 \beta_{2} - 112 \beta_{4} + 368 \beta_{5} - 904 \beta_{7} ) q^{80} + ( 25065 + 24 \beta_{1} - 3222 \beta_{2} - 72 \beta_{3} - 822 \beta_{5} - 24 \beta_{6} - 312 \beta_{7} ) q^{81} + ( 19340 - 1428 \beta_{1} + 920 \beta_{3} + 408 \beta_{4} + 204 \beta_{6} + 102 \beta_{7} ) q^{82} + ( -165 \beta_{1} - 800 \beta_{2} - 55 \beta_{4} + 100 \beta_{7} ) q^{83} + ( 20982 - 2452 \beta_{1} + 7188 \beta_{2} + 138 \beta_{3} - 348 \beta_{4} + 188 \beta_{5} - 128 \beta_{6} + 550 \beta_{7} ) q^{84} + ( 28672 + 720 \beta_{1} - 2160 \beta_{3} - 720 \beta_{6} - 360 \beta_{7} ) q^{85} + ( 1158 \beta_{1} + 16 \beta_{2} + 386 \beta_{4} + 1094 \beta_{5} - 531 \beta_{7} ) q^{86} + ( 371 \beta_{1} + 4704 \beta_{2} - 618 \beta_{3} - 329 \beta_{4} + 618 \beta_{6} - 279 \beta_{7} ) q^{87} + ( -18792 + 1328 \beta_{1} + 824 \beta_{3} - 480 \beta_{4} + 112 \beta_{6} + 56 \beta_{7} ) q^{88} + ( 11830 \beta_{2} - 234 \beta_{5} + 1508 \beta_{7} ) q^{89} + ( -31434 - 2986 \beta_{1} - 2208 \beta_{2} - 564 \beta_{3} + 348 \beta_{4} + 272 \beta_{5} + 310 \beta_{6} + 835 \beta_{7} ) q^{90} + ( 1642 \beta_{1} - 668 \beta_{3} - 770 \beta_{4} + 668 \beta_{6} + 334 \beta_{7} ) q^{91} + ( -2376 \beta_{1} - 9208 \beta_{2} - 792 \beta_{4} + 344 \beta_{5} - 164 \beta_{7} ) q^{92} + ( -43266 - 260 \beta_{1} - 5217 \beta_{2} + 780 \beta_{3} + 31 \beta_{5} + 260 \beta_{6} - 526 \beta_{7} ) q^{93} + ( 48264 - 1512 \beta_{1} - 1600 \beta_{3} + 1008 \beta_{4} - 1512 \beta_{6} - 756 \beta_{7} ) q^{94} + ( 1182 \beta_{1} - 3072 \beta_{2} + 394 \beta_{4} + 384 \beta_{7} ) q^{95} + ( 41424 - 712 \beta_{1} - 9336 \beta_{2} + 1488 \beta_{3} - 376 \beta_{4} - 200 \beta_{5} + 224 \beta_{6} + 812 \beta_{7} ) q^{96} + ( -49006 + 260 \beta_{1} - 780 \beta_{3} - 260 \beta_{6} - 130 \beta_{7} ) q^{97} + ( 1968 \beta_{1} - 2693 \beta_{2} + 656 \beta_{4} - 656 \beta_{5} - 1640 \beta_{7} ) q^{98} + ( -267 \beta_{1} + 6336 \beta_{2} - 180 \beta_{3} + 723 \beta_{4} + 180 \beta_{6} - 702 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{4} + 24q^{6} - 24q^{9} + O(q^{10})$$ $$8q + 8q^{4} + 24q^{6} - 24q^{9} - 272q^{10} - 696q^{12} + 112q^{13} + 2336q^{16} + 3696q^{18} - 336q^{21} - 8304q^{22} - 11232q^{24} - 1368q^{25} + 19248q^{28} + 25008q^{30} + 4224q^{33} - 36416q^{34} - 42072q^{36} + 6448q^{37} + 53824q^{40} + 59088q^{42} - 17664q^{45} - 69600q^{46} - 74976q^{48} - 21544q^{49} + 84208q^{52} + 78120q^{54} + 43344q^{57} - 62576q^{58} - 64704q^{60} + 59632q^{61} + 38528q^{64} + 33360q^{66} - 92928q^{69} - 30432q^{70} + 20544q^{72} - 129200q^{73} - 86256q^{76} - 83760q^{78} + 200520q^{81} + 154720q^{82} + 167856q^{84} + 229376q^{85} - 150336q^{88} - 251472q^{90} - 346128q^{93} + 386112q^{94} + 331392q^{96} - 392048q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 7 x^{6} + 6 x^{5} - 11 x^{4} - 73 x^{3} + 223 x^{2} - 768 x + 912$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-9436 \nu^{7} + 30607 \nu^{6} - 62704 \nu^{5} - 231 \nu^{4} - 423169 \nu^{3} - 610094 \nu^{2} - 8744833 \nu + 4238706$$$$)/953786$$ $$\beta_{2}$$ $$=$$ $$($$$$-5694 \nu^{7} - 2252 \nu^{6} - 73822 \nu^{5} - 139528 \nu^{4} - 251008 \nu^{3} - 81692 \nu^{2} - 673868 \nu + 5096288$$$$)/476893$$ $$\beta_{3}$$ $$=$$ $$($$$$-8084 \nu^{7} + 72516 \nu^{6} + 129032 \nu^{5} + 488016 \nu^{4} + 1228252 \nu^{3} + 3220092 \nu^{2} - 3343360 \nu - 1507299$$$$)/476893$$ $$\beta_{4}$$ $$=$$ $$($$$$98436 \nu^{7} - 85191 \nu^{6} + 753588 \nu^{5} + 1046763 \nu^{4} - 436119 \nu^{3} - 10020102 \nu^{2} + 30261501 \nu - 78787206$$$$)/953786$$ $$\beta_{5}$$ $$=$$ $$($$$$-70008 \nu^{7} + 87389 \nu^{6} - 396078 \nu^{5} - 1183333 \nu^{4} + 723461 \nu^{3} + 1647902 \nu^{2} - 24231757 \nu + 43615544$$$$)/476893$$ $$\beta_{6}$$ $$=$$ $$($$$$-164992 \nu^{7} - 286197 \nu^{6} - 1331716 \nu^{5} - 2951719 \nu^{4} - 3169901 \nu^{3} + 4344198 \nu^{2} + 2922499 \nu + 53217788$$$$)/953786$$ $$\beta_{7}$$ $$=$$ $$($$$$-83652 \nu^{7} - 63236 \nu^{6} - 485532 \nu^{5} - 1647824 \nu^{4} - 1820252 \nu^{3} + 5776852 \nu^{2} - 14368388 \nu + 26548384$$$$)/476893$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{4} + 3 \beta_{3} + 18 \beta_{2} - 6 \beta_{1} + 9$$$$)/72$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 5 \beta_{4} + 3 \beta_{2} - 12 \beta_{1} - 117$$$$)/72$$ $$\nu^{3}$$ $$=$$ $$($$$$-21 \beta_{7} + 12 \beta_{6} + 18 \beta_{5} - 22 \beta_{4} - 24 \beta_{3} - 198 \beta_{2} - 54 \beta_{1} - 684$$$$)/144$$ $$\nu^{4}$$ $$=$$ $$($$$$-7 \beta_{7} + 9 \beta_{6} - 7 \beta_{5} - 5 \beta_{4} + 6 \beta_{3} - 33 \beta_{2} + 48 \beta_{1} + 273$$$$)/24$$ $$\nu^{5}$$ $$=$$ $$($$$$72 \beta_{7} - 12 \beta_{6} + 24 \beta_{5} + 130 \beta_{4} + 69 \beta_{3} - 198 \beta_{2} + 54 \beta_{1} + 7299$$$$)/72$$ $$\nu^{6}$$ $$=$$ $$($$$$111 \beta_{7} - 402 \beta_{6} + 120 \beta_{5} + 496 \beta_{4} + 846 \beta_{3} + 6744 \beta_{2} - 1134 \beta_{1} - 18108$$$$)/144$$ $$\nu^{7}$$ $$=$$ $$($$$$-21 \beta_{7} - 648 \beta_{6} - 174 \beta_{5} - 1096 \beta_{4} - 1329 \beta_{3} - 180 \beta_{2} - 1932 \beta_{1} - 30987$$$$)/72$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −2.29661 + 1.35416i −2.29661 − 1.35416i 0.713157 − 2.93555i 0.713157 + 2.93555i 1.67284 + 0.274906i 1.67284 − 0.274906i 0.410606 − 2.17330i 0.410606 + 2.17330i
−5.41443 1.63829i 4.17995 + 15.0176i 26.6320 + 17.7408i 73.1202i 1.97116 88.1596i 51.8209i −115.132 139.687i −208.056 + 125.545i 119.792 395.904i
11.2 −5.41443 + 1.63829i 4.17995 15.0176i 26.6320 17.7408i 73.1202i 1.97116 + 88.1596i 51.8209i −115.132 + 139.687i −208.056 125.545i 119.792 + 395.904i
11.3 −1.91937 5.32128i −14.9174 4.52459i −24.6320 + 20.4270i 35.2908i 4.55538 + 88.0639i 190.564i 155.976 + 91.8667i 202.056 + 134.990i −187.792 + 67.7362i
11.4 −1.91937 + 5.32128i −14.9174 + 4.52459i −24.6320 20.4270i 35.2908i 4.55538 88.0639i 190.564i 155.976 91.8667i 202.056 134.990i −187.792 67.7362i
11.5 1.91937 5.32128i 14.9174 + 4.52459i −24.6320 20.4270i 35.2908i 52.7086 70.6951i 190.564i −155.976 + 91.8667i 202.056 + 134.990i −187.792 67.7362i
11.6 1.91937 + 5.32128i 14.9174 4.52459i −24.6320 + 20.4270i 35.2908i 52.7086 + 70.6951i 190.564i −155.976 91.8667i 202.056 134.990i −187.792 + 67.7362i
11.7 5.41443 1.63829i −4.17995 15.0176i 26.6320 17.7408i 73.1202i −47.2352 74.4637i 51.8209i 115.132 139.687i −208.056 + 125.545i 119.792 + 395.904i
11.8 5.41443 + 1.63829i −4.17995 + 15.0176i 26.6320 + 17.7408i 73.1202i −47.2352 + 74.4637i 51.8209i 115.132 + 139.687i −208.056 125.545i 119.792 395.904i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.8 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
3.b Odd 1 yes
4.b Odd 1 yes
12.b Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{6}^{\mathrm{new}}(12, [\chi])$$.