# Properties

 Label 33.7.c.a Level 33 Weight 7 Character orbit 33.c Analytic conductor 7.592 Analytic rank 0 Dimension 12 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 33.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.59178475946$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 486 x^{10} + 82401 x^{8} + 6062364 x^{6} + 204706260 x^{4} + 2964086784 x^{2} + 15081209856$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}\cdot 3^{11}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{3} + ( -17 + \beta_{2} + \beta_{3} ) q^{4} + ( 19 - \beta_{3} + \beta_{4} ) q^{5} + ( -2 \beta_{1} - \beta_{5} ) q^{6} + ( 7 \beta_{1} - \beta_{5} + \beta_{6} ) q^{7} + ( -19 \beta_{1} - 2 \beta_{5} - \beta_{6} + \beta_{8} ) q^{8} + 243 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{3} + ( -17 + \beta_{2} + \beta_{3} ) q^{4} + ( 19 - \beta_{3} + \beta_{4} ) q^{5} + ( -2 \beta_{1} - \beta_{5} ) q^{6} + ( 7 \beta_{1} - \beta_{5} + \beta_{6} ) q^{7} + ( -19 \beta_{1} - 2 \beta_{5} - \beta_{6} + \beta_{8} ) q^{8} + 243 q^{9} + ( 59 \beta_{1} - 4 \beta_{5} + \beta_{6} - \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{10} + ( -287 - 63 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{11} + ( 162 - 17 \beta_{2} - 6 \beta_{3} - 3 \beta_{4} + 3 \beta_{7} ) q^{12} + ( 77 \beta_{1} + 4 \beta_{5} + 3 \beta_{6} + 2 \beta_{8} + 4 \beta_{10} - 3 \beta_{11} ) q^{13} + ( -557 - 84 \beta_{2} + 15 \beta_{3} - \beta_{4} + 5 \beta_{7} - 2 \beta_{9} - \beta_{10} ) q^{14} + ( 165 + 14 \beta_{2} - 9 \beta_{3} + 3 \beta_{4} - 6 \beta_{9} - 3 \beta_{10} ) q^{15} + ( 445 - 131 \beta_{2} - 23 \beta_{3} - 10 \beta_{4} + 10 \beta_{7} - 6 \beta_{9} - 3 \beta_{10} ) q^{16} + ( -126 \beta_{1} - 20 \beta_{5} - 5 \beta_{6} - 7 \beta_{8} + 11 \beta_{10} ) q^{17} + 243 \beta_{1} q^{18} + ( -28 \beta_{1} - 9 \beta_{5} + 6 \beta_{6} + 8 \beta_{8} - 4 \beta_{10} + 5 \beta_{11} ) q^{19} + ( -3675 - 122 \beta_{2} + 37 \beta_{3} - 9 \beta_{4} + 15 \beta_{7} + 8 \beta_{9} + 4 \beta_{10} ) q^{20} + ( 277 \beta_{1} - 4 \beta_{5} + 3 \beta_{8} - 12 \beta_{10} + 3 \beta_{11} ) q^{21} + ( 5037 - 679 \beta_{1} + 47 \beta_{2} - 83 \beta_{3} - 20 \beta_{4} + \beta_{5} - 9 \beta_{6} + 2 \beta_{7} + \beta_{8} - 12 \beta_{9} + 2 \beta_{10} ) q^{22} + ( -1285 + 198 \beta_{2} + 101 \beta_{3} + 21 \beta_{4} - 16 \beta_{7} + 34 \beta_{9} + 17 \beta_{10} ) q^{23} + ( 518 \beta_{1} + 16 \beta_{5} + 6 \beta_{6} - 6 \beta_{8} - 3 \beta_{10} - 9 \beta_{11} ) q^{24} + ( 7959 + 428 \beta_{2} - 24 \beta_{3} + 42 \beta_{4} - 60 \beta_{7} + 18 \beta_{9} + 9 \beta_{10} ) q^{25} + ( -6313 + 388 \beta_{2} + 39 \beta_{3} + 79 \beta_{4} - 41 \beta_{7} - 28 \beta_{9} - 14 \beta_{10} ) q^{26} + 243 \beta_{2} q^{27} + ( -915 \beta_{1} + 62 \beta_{5} + 41 \beta_{6} + 7 \beta_{8} + 3 \beta_{10} - 7 \beta_{11} ) q^{28} + ( -1026 \beta_{1} + 44 \beta_{5} - 17 \beta_{6} + 5 \beta_{8} - 88 \beta_{10} ) q^{29} + ( 557 \beta_{1} - 74 \beta_{5} - 15 \beta_{6} - 33 \beta_{8} - 3 \beta_{10} + 15 \beta_{11} ) q^{30} + ( -4936 + 236 \beta_{2} - 176 \beta_{3} - 70 \beta_{4} - 20 \beta_{7} + 66 \beta_{9} + 33 \beta_{10} ) q^{31} + ( 1099 \beta_{1} + 162 \beta_{5} - 65 \beta_{6} + 17 \beta_{8} - 22 \beta_{10} - 18 \beta_{11} ) q^{32} + ( 375 - 190 \beta_{1} - 278 \beta_{2} + 6 \beta_{3} + 9 \beta_{4} + 76 \beta_{5} - 24 \beta_{6} + 42 \beta_{7} - 12 \beta_{8} - 21 \beta_{9} - 57 \beta_{10} ) q^{33} + ( 9732 - 1062 \beta_{2} + 70 \beta_{3} - 166 \beta_{4} - 14 \beta_{7} + 66 \beta_{9} + 33 \beta_{10} ) q^{34} + ( 1064 \beta_{1} - 62 \beta_{5} - 66 \beta_{6} - 66 \beta_{8} - 35 \beta_{10} + 54 \beta_{11} ) q^{35} + ( -4131 + 243 \beta_{2} + 243 \beta_{3} ) q^{36} + ( -16900 + 468 \beta_{2} + 116 \beta_{3} + 30 \beta_{4} - 156 \beta_{7} - 54 \beta_{9} - 27 \beta_{10} ) q^{37} + ( 2454 - 1112 \beta_{2} - 386 \beta_{3} - 166 \beta_{4} + 160 \beta_{7} - 66 \beta_{9} - 33 \beta_{10} ) q^{38} + ( -1231 \beta_{1} - 50 \beta_{5} - 117 \beta_{6} + 42 \beta_{8} - 87 \beta_{10} - 21 \beta_{11} ) q^{39} + ( -1725 \beta_{1} + 156 \beta_{5} + 59 \beta_{6} + 5 \beta_{8} + 211 \beta_{10} + 9 \beta_{11} ) q^{40} + ( 196 \beta_{1} - 92 \beta_{5} - 11 \beta_{6} + 5 \beta_{8} - 80 \beta_{10} - 90 \beta_{11} ) q^{41} + ( -22023 - 492 \beta_{2} + 111 \beta_{3} - 3 \beta_{4} + 93 \beta_{7} - 18 \beta_{9} - 9 \beta_{10} ) q^{42} + ( -276 \beta_{1} + 479 \beta_{5} + 230 \beta_{6} - 56 \beta_{8} + 108 \beta_{10} + 5 \beta_{11} ) q^{43} + ( 36257 + 6531 \beta_{1} - 58 \beta_{2} - 511 \beta_{3} + 31 \beta_{4} + 120 \beta_{5} + 163 \beta_{6} + 97 \beta_{7} - 67 \beta_{8} - 54 \beta_{9} + 53 \beta_{10} ) q^{44} + ( 4617 - 243 \beta_{3} + 243 \beta_{4} ) q^{45} + ( -8141 \beta_{1} - 322 \beta_{5} + 35 \beta_{6} + 237 \beta_{8} + 99 \beta_{10} - 15 \beta_{11} ) q^{46} + ( 43039 + 1888 \beta_{2} + 565 \beta_{3} - 49 \beta_{4} - 100 \beta_{7} - 36 \beta_{9} - 18 \beta_{10} ) q^{47} + ( -31356 + 565 \beta_{2} + 576 \beta_{3} + 225 \beta_{4} + 9 \beta_{7} + 18 \beta_{9} + 9 \beta_{10} ) q^{48} + ( 13189 - 2992 \beta_{2} - 514 \beta_{3} + 158 \beta_{4} + 40 \beta_{7} + 84 \beta_{9} + 42 \beta_{10} ) q^{49} + ( 8003 \beta_{1} - 1292 \beta_{5} + 96 \beta_{6} + 48 \beta_{8} + 42 \beta_{10} + 126 \beta_{11} ) q^{50} + ( 3360 \beta_{1} - 3 \beta_{5} - 174 \beta_{6} - 60 \beta_{8} + 132 \beta_{10} + 27 \beta_{11} ) q^{51} + ( -6863 \beta_{1} - 1392 \beta_{5} + 41 \beta_{6} + 55 \beta_{8} + 355 \beta_{10} + 25 \beta_{11} ) q^{52} + ( -86811 + 3112 \beta_{2} + 541 \beta_{3} - 105 \beta_{4} + 144 \beta_{7} - 76 \beta_{9} - 38 \beta_{10} ) q^{53} + ( -486 \beta_{1} - 243 \beta_{5} ) q^{54} + ( 21650 - 413 \beta_{1} + 1446 \beta_{2} + 914 \beta_{3} - 598 \beta_{4} + 702 \beta_{5} - 191 \beta_{6} + 40 \beta_{7} + 262 \beta_{8} + 156 \beta_{9} - 59 \beta_{10} - 11 \beta_{11} ) q^{55} + ( 38867 - 2098 \beta_{2} + 463 \beta_{3} + 405 \beta_{4} + 161 \beta_{7} - 280 \beta_{9} - 140 \beta_{10} ) q^{56} + ( 3342 \beta_{1} + 96 \beta_{5} + 231 \beta_{6} - 87 \beta_{8} - 165 \beta_{10} ) q^{57} + ( 86124 - 488 \beta_{2} - 1294 \beta_{3} + 968 \beta_{4} + 40 \beta_{7} - 6 \beta_{9} - 3 \beta_{10} ) q^{58} + ( -38390 - 3290 \beta_{2} - 1486 \beta_{3} + 136 \beta_{4} + 84 \beta_{7} + 138 \beta_{9} + 69 \beta_{10} ) q^{59} + ( -34881 - 3100 \beta_{2} + 1059 \beta_{3} - 489 \beta_{4} + 195 \beta_{7} - 60 \beta_{9} - 30 \beta_{10} ) q^{60} + ( 705 \beta_{1} - 476 \beta_{5} + 141 \beta_{6} - 400 \beta_{8} + 30 \beta_{10} - 35 \beta_{11} ) q^{61} + ( 8500 \beta_{1} + 604 \beta_{5} + 440 \beta_{6} + 88 \beta_{8} - 118 \beta_{10} - 162 \beta_{11} ) q^{62} + ( 1701 \beta_{1} - 243 \beta_{5} + 243 \beta_{6} ) q^{63} + ( -60149 + 4765 \beta_{2} - 1285 \beta_{3} + 532 \beta_{4} - 64 \beta_{7} - 426 \beta_{9} - 213 \beta_{10} ) q^{64} + ( -17438 \beta_{1} + 2096 \beta_{5} - 880 \beta_{6} + 82 \beta_{8} - 53 \beta_{10} + 90 \beta_{11} ) q^{65} + ( 16776 - 47 \beta_{1} + 4881 \beta_{2} + 402 \beta_{3} + 669 \beta_{4} + 626 \beta_{5} - 90 \beta_{6} - 255 \beta_{7} - 78 \beta_{8} + 144 \beta_{9} + 141 \beta_{10} - 33 \beta_{11} ) q^{66} + ( 30038 - 920 \beta_{2} - 626 \beta_{3} - 1228 \beta_{4} - 284 \beta_{7} - 438 \beta_{9} - 219 \beta_{10} ) q^{67} + ( 4248 \beta_{1} + 940 \beta_{5} - 126 \beta_{6} - 114 \beta_{8} + 310 \beta_{10} - 270 \beta_{11} ) q^{68} + ( 41619 - 892 \beta_{2} - 357 \beta_{3} - 1431 \beta_{4} + 120 \beta_{7} - 132 \beta_{9} - 66 \beta_{10} ) q^{69} + ( -86388 - 1442 \beta_{2} + 3084 \beta_{3} - 1726 \beta_{4} + 430 \beta_{7} + 768 \beta_{9} + 384 \beta_{10} ) q^{70} + ( -63465 - 10672 \beta_{2} + 2037 \beta_{3} - 77 \beta_{4} - 916 \beta_{7} + 608 \beta_{9} + 304 \beta_{10} ) q^{71} + ( -4617 \beta_{1} - 486 \beta_{5} - 243 \beta_{6} + 243 \beta_{8} ) q^{72} + ( 11738 \beta_{1} + 2060 \beta_{5} - 556 \beta_{6} - 166 \beta_{8} - 792 \beta_{10} + 58 \beta_{11} ) q^{73} + ( -26424 \beta_{1} - 3404 \beta_{5} - 332 \beta_{6} - 100 \beta_{8} - 330 \beta_{10} + 450 \beta_{11} ) q^{74} + ( 113454 + 6879 \beta_{2} - 2970 \beta_{3} - 216 \beta_{4} - 540 \beta_{7} + 54 \beta_{9} + 27 \beta_{10} ) q^{75} + ( 30330 \beta_{1} + 3398 \beta_{5} + 506 \beta_{6} - 138 \beta_{8} - 566 \beta_{10} - 34 \beta_{11} ) q^{76} + ( -8551 - 17060 \beta_{1} + 7316 \beta_{2} - 2455 \beta_{3} + 899 \beta_{4} + 1456 \beta_{5} + 217 \beta_{6} - 432 \beta_{7} - 139 \beta_{8} + 524 \beta_{9} + 129 \beta_{10} + 198 \beta_{11} ) q^{77} + ( 102051 - 8310 \beta_{2} - 4809 \beta_{3} + 1137 \beta_{4} + 69 \beta_{7} - 144 \beta_{9} - 72 \beta_{10} ) q^{78} + ( -847 \beta_{1} - 1709 \beta_{5} - 481 \beta_{6} - 436 \beta_{8} - 722 \beta_{10} + 102 \beta_{11} ) q^{79} + ( -102551 + 9242 \beta_{2} + 1717 \beta_{3} - 2425 \beta_{4} + 335 \beta_{7} + 372 \beta_{9} + 186 \beta_{10} ) q^{80} + 59049 q^{81} + ( -12186 - 11160 \beta_{2} - 364 \beta_{3} + 3642 \beta_{4} - 726 \beta_{7} - 198 \beta_{9} - 99 \beta_{10} ) q^{82} + ( -50354 \beta_{1} - 948 \beta_{5} + 90 \beta_{6} + 324 \beta_{8} + 258 \beta_{10} - 630 \beta_{11} ) q^{83} + ( -10781 \beta_{1} + 1280 \beta_{5} - 183 \beta_{6} + 231 \beta_{8} - 627 \beta_{10} + 39 \beta_{11} ) q^{84} + ( 46454 \beta_{1} - 3508 \beta_{5} - 680 \beta_{6} + 114 \beta_{8} - 1576 \beta_{10} - 170 \beta_{11} ) q^{85} + ( 20494 + 40232 \beta_{2} + 7078 \beta_{3} + 754 \beta_{4} - 1464 \beta_{7} - 2 \beta_{9} - \beta_{10} ) q^{86} + ( -806 \beta_{1} + 1235 \beta_{5} + 1086 \beta_{6} + 462 \beta_{8} + 393 \beta_{10} - 81 \beta_{11} ) q^{87} + ( -208161 + 23927 \beta_{1} + 21793 \beta_{2} + 6871 \beta_{3} - 1260 \beta_{4} + 1350 \beta_{5} - 281 \beta_{6} - 468 \beta_{7} - 663 \beta_{8} - 558 \beta_{9} + 412 \beta_{10} - 55 \beta_{11} ) q^{88} + ( -292328 - 18112 \beta_{2} + 7926 \beta_{3} + 1998 \beta_{4} - 272 \beta_{7} + 324 \beta_{9} + 162 \beta_{10} ) q^{89} + ( 14337 \beta_{1} - 972 \beta_{5} + 243 \beta_{6} - 243 \beta_{8} + 729 \beta_{10} + 243 \beta_{11} ) q^{90} + ( -56226 + 18008 \beta_{2} - 6266 \beta_{3} + 4948 \beta_{4} + 1460 \beta_{7} - 606 \beta_{9} - 303 \beta_{10} ) q^{91} + ( 575065 - 25296 \beta_{2} - 13891 \beta_{3} - 491 \beta_{4} + 1041 \beta_{7} + 180 \beta_{9} + 90 \beta_{10} ) q^{92} + ( 64710 - 3376 \beta_{2} + 3798 \beta_{3} - 1728 \beta_{4} - 1116 \beta_{7} + 342 \beta_{9} + 171 \beta_{10} ) q^{93} + ( 3975 \beta_{1} - 3932 \beta_{5} - 709 \beta_{6} + 421 \beta_{8} - 419 \beta_{10} + 223 \beta_{11} ) q^{94} + ( 20282 \beta_{1} - 270 \beta_{5} + 1306 \beta_{6} - 28 \beta_{8} + 2016 \beta_{10} + 576 \beta_{11} ) q^{95} + ( -41364 \beta_{1} - 954 \beta_{5} - 120 \beta_{6} + 264 \beta_{8} + 537 \beta_{10} - 405 \beta_{11} ) q^{96} + ( 198038 + 36512 \beta_{2} + 882 \beta_{3} + 834 \beta_{4} + 1416 \beta_{7} + 684 \beta_{9} + 342 \beta_{10} ) q^{97} + ( 49373 \beta_{1} + 4032 \beta_{5} + 850 \beta_{6} - 178 \beta_{8} + 722 \beta_{10} - 90 \beta_{11} ) q^{98} + ( -69741 - 15309 \beta_{1} + 729 \beta_{2} + 1215 \beta_{3} + 486 \beta_{4} + 243 \beta_{5} + 486 \beta_{6} + 486 \beta_{7} + 243 \beta_{8} - 243 \beta_{9} + 486 \beta_{10} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 204q^{4} + 224q^{5} + 2916q^{9} + O(q^{10})$$ $$12q - 204q^{4} + 224q^{5} + 2916q^{9} - 3464q^{11} + 1944q^{12} - 6708q^{14} + 1944q^{15} + 5316q^{16} - 44092q^{20} + 60468q^{22} - 15304q^{23} + 95652q^{25} - 76020q^{26} - 58608q^{31} + 4212q^{33} + 117768q^{34} - 49572q^{36} - 202512q^{37} + 29208q^{38} - 264708q^{42} + 434356q^{44} + 54432q^{45} + 516920q^{47} - 377136q^{48} + 157812q^{49} - 1042192q^{53} + 262656q^{55} + 463020q^{56} + 1029432q^{58} - 461008q^{59} - 417636q^{60} - 725364q^{64} + 200232q^{66} + 364752q^{67} + 504144q^{69} - 1028400q^{70} - 755176q^{71} + 1364688q^{75} - 102384q^{77} + 1219212q^{78} - 1220764q^{80} + 708588q^{81} - 158688q^{82} + 248760q^{86} - 2493252q^{88} - 3513544q^{89} - 702768q^{91} + 6899300q^{92} + 789264q^{93} + 2370192q^{97} - 841752q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 486 x^{10} + 82401 x^{8} + 6062364 x^{6} + 204706260 x^{4} + 2964086784 x^{2} + 15081209856$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$117 \nu^{10} + 56806 \nu^{8} + 9541621 \nu^{6} + 673291092 \nu^{4} + 19439761332 \nu^{2} + 162772955136$$$$)/ 269967872$$ $$\beta_{3}$$ $$=$$ $$($$$$-117 \nu^{10} - 56806 \nu^{8} - 9541621 \nu^{6} - 673291092 \nu^{4} - 19169793460 \nu^{2} - 140905557504$$$$)/ 269967872$$ $$\beta_{4}$$ $$=$$ $$($$$$40281 \nu^{10} + 18394564 \nu^{8} + 2794934961 \nu^{6} + 168462043098 \nu^{4} + 4080295276416 \nu^{2} + 30842537263488$$$$)/ 7896560256$$ $$\beta_{5}$$ $$=$$ $$($$$$-117 \nu^{11} - 56806 \nu^{9} - 9541621 \nu^{7} - 673291092 \nu^{5} - 19439761332 \nu^{3} - 163312890880 \nu$$$$)/ 269967872$$ $$\beta_{6}$$ $$=$$ $$($$$$-139765 \nu^{11} - 65082782 \nu^{9} - 10255266645 \nu^{7} - 665103431244 \nu^{5} - 18531153088740 \nu^{3} - 183148581464064 \nu$$$$)/ 252689928192$$ $$\beta_{7}$$ $$=$$ $$($$$$501165 \nu^{10} + 235863262 \nu^{8} + 37684766925 \nu^{6} + 2467300644060 \nu^{4} + 66068672738724 \nu^{2} + 534076796284416$$$$)/ 31586241024$$ $$\beta_{8}$$ $$=$$ $$($$$$-358789 \nu^{11} - 171423614 \nu^{9} - 28117181157 \nu^{7} - 1925504355468 \nu^{5} - 54669696374052 \nu^{3} - 451724893747200 \nu$$$$)/ 252689928192$$ $$\beta_{9}$$ $$=$$ $$($$$$-878169 \nu^{11} + 5125328 \nu^{10} - 409604982 \nu^{9} + 2413465184 \nu^{8} - 64509725241 \nu^{7} + 385286260944 \nu^{6} - 4130097219900 \nu^{5} + 25053952632384 \nu^{4} - 107808843253428 \nu^{3} + 653448134184768 \nu^{2} - 858826839375360 \nu + 5002935675199488$$$$)/ 505379856384$$ $$\beta_{10}$$ $$=$$ $$($$$$292723 \nu^{11} + 136534994 \nu^{9} + 21503241747 \nu^{7} + 1376699073300 \nu^{5} + 35936281084476 \nu^{3} + 286275613125120 \nu$$$$)/ 84229976064$$ $$\beta_{11}$$ $$=$$ $$($$$$-1893075 \nu^{11} - 906040978 \nu^{9} - 148746043251 \nu^{7} - 10150508591124 \nu^{5} - 283835163948732 \nu^{3} - 2347759096736256 \nu$$$$)/ 252689928192$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} - 81$$ $$\nu^{3}$$ $$=$$ $$\beta_{8} - \beta_{6} - 2 \beta_{5} - 147 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{10} - 6 \beta_{9} + 10 \beta_{7} - 10 \beta_{4} - 215 \beta_{3} - 323 \beta_{2} + 11901$$ $$\nu^{5}$$ $$=$$ $$-18 \beta_{11} - 22 \beta_{10} - 239 \beta_{8} + 191 \beta_{6} + 674 \beta_{5} + 26443 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$747 \beta_{10} + 1494 \beta_{9} - 3264 \beta_{7} + 3732 \beta_{4} + 42939 \beta_{3} + 83549 \beta_{2} - 2139957$$ $$\nu^{7}$$ $$=$$ $$7272 \beta_{11} + 7656 \beta_{10} + 48915 \beta_{8} - 36963 \beta_{6} - 177398 \beta_{5} - 5117275 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-151425 \beta_{10} - 302850 \beta_{9} + 830982 \beta_{7} - 1035030 \beta_{4} - 8632887 \beta_{3} - 19786227 \beta_{2} + 413976969$$ $$\nu^{9}$$ $$=$$ $$-2091294 \beta_{11} - 2048826 \beta_{10} - 9844287 \beta_{8} + 7421487 \beta_{6} + 42502362 \beta_{5} + 1024993047 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$29864367 \beta_{10} + 59728734 \beta_{9} - 194819004 \beta_{7} + 255722184 \beta_{4} + 1760768703 \beta_{3} + 4487920917 \beta_{2} - 82894928409$$ $$\nu^{11}$$ $$=$$ $$525902724 \beta_{11} + 496986012 \beta_{10} + 1999683639 \beta_{8} - 1521853767 \beta_{6} - 9717207990 \beta_{5} - 209471459367 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 − 14.5918i − 11.9419i − 7.49503i − 6.83773i − 3.80817i − 3.61107i 3.61107i 3.80817i 6.83773i 7.49503i 11.9419i 14.5918i
14.5918i −15.5885 −148.920 0.687766 227.463i 18.0081i 1239.14i 243.000 10.0357i
10.2 11.9419i 15.5885 −78.6078 234.218 186.155i 368.050i 174.445i 243.000 2797.00i
10.3 7.49503i 15.5885 7.82458 −189.722 116.836i 392.912i 538.327i 243.000 1421.97i
10.4 6.83773i −15.5885 17.2454 167.968 106.590i 106.676i 555.534i 243.000 1148.52i
10.5 3.80817i −15.5885 49.4978 −143.833 59.3636i 413.864i 432.219i 243.000 547.741i
10.6 3.61107i 15.5885 50.9602 42.6809 56.2910i 392.632i 415.129i 243.000 154.124i
10.7 3.61107i 15.5885 50.9602 42.6809 56.2910i 392.632i 415.129i 243.000 154.124i
10.8 3.80817i −15.5885 49.4978 −143.833 59.3636i 413.864i 432.219i 243.000 547.741i
10.9 6.83773i −15.5885 17.2454 167.968 106.590i 106.676i 555.534i 243.000 1148.52i
10.10 7.49503i 15.5885 7.82458 −189.722 116.836i 392.912i 538.327i 243.000 1421.97i
10.11 11.9419i 15.5885 −78.6078 234.218 186.155i 368.050i 174.445i 243.000 2797.00i
10.12 14.5918i −15.5885 −148.920 0.687766 227.463i 18.0081i 1239.14i 243.000 10.0357i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 10.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.7.c.a 12
3.b odd 2 1 99.7.c.d 12
4.b odd 2 1 528.7.j.c 12
11.b odd 2 1 inner 33.7.c.a 12
33.d even 2 1 99.7.c.d 12
44.c even 2 1 528.7.j.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.7.c.a 12 1.a even 1 1 trivial
33.7.c.a 12 11.b odd 2 1 inner
99.7.c.d 12 3.b odd 2 1
99.7.c.d 12 33.d even 2 1
528.7.j.c 12 4.b odd 2 1
528.7.j.c 12 44.c even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 282 T^{2} + 41697 T^{4} - 4219108 T^{6} + 343648212 T^{8} - 24741884928 T^{10} + 1647936307200 T^{12} - 101342760665088 T^{14} + 5765460280737792 T^{16} - 289934894052671488 T^{18} + 11736662103904223232 T^{20} -$$$$32\!\cdots\!32$$$$T^{22} +$$$$47\!\cdots\!96$$$$T^{24}$$
$3$ $$( 1 - 243 T^{2} )^{6}$$
$5$ $$( 1 - 112 T + 29234 T^{2} - 3676200 T^{3} + 608299075 T^{4} - 82098791000 T^{5} + 12395840252500 T^{6} - 1282793609375000 T^{7} + 148510516357421875 T^{8} - 14023590087890625000 T^{9} +$$$$17\!\cdots\!50$$$$T^{10} -$$$$10\!\cdots\!00$$$$T^{11} +$$$$14\!\cdots\!25$$$$T^{12} )^{2}$$
$7$ $$1 - 784800 T^{2} + 324728983446 T^{4} - 91703033397949408 T^{6} +$$$$19\!\cdots\!75$$$$T^{8} -$$$$31\!\cdots\!84$$$$T^{10} +$$$$41\!\cdots\!76$$$$T^{12} -$$$$44\!\cdots\!84$$$$T^{14} +$$$$37\!\cdots\!75$$$$T^{16} -$$$$24\!\cdots\!08$$$$T^{18} +$$$$11\!\cdots\!46$$$$T^{20} -$$$$39\!\cdots\!00$$$$T^{22} +$$$$70\!\cdots\!01$$$$T^{24}$$
$11$ $$1 + 3464 T + 8137514 T^{2} + 10918677000 T^{3} + 8619795607951 T^{4} - 1445621685494096 T^{5} - 9476672140132971092 T^{6} -$$$$25\!\cdots\!56$$$$T^{7} +$$$$27\!\cdots\!71$$$$T^{8} +$$$$60\!\cdots\!00$$$$T^{9} +$$$$80\!\cdots\!74$$$$T^{10} +$$$$60\!\cdots\!64$$$$T^{11} +$$$$30\!\cdots\!61$$$$T^{12}$$
$13$ $$1 - 11813004 T^{2} + 173649316619562 T^{4} -$$$$13\!\cdots\!40$$$$T^{6} +$$$$11\!\cdots\!27$$$$T^{8} -$$$$64\!\cdots\!96$$$$T^{10} +$$$$35\!\cdots\!44$$$$T^{12} -$$$$14\!\cdots\!76$$$$T^{14} +$$$$60\!\cdots\!47$$$$T^{16} -$$$$17\!\cdots\!40$$$$T^{18} +$$$$51\!\cdots\!02$$$$T^{20} -$$$$81\!\cdots\!04$$$$T^{22} +$$$$15\!\cdots\!81$$$$T^{24}$$
$17$ $$1 - 112969992 T^{2} + 7355558115683166 T^{4} -$$$$33\!\cdots\!68$$$$T^{6} +$$$$12\!\cdots\!35$$$$T^{8} -$$$$39\!\cdots\!52$$$$T^{10} +$$$$10\!\cdots\!48$$$$T^{12} -$$$$22\!\cdots\!72$$$$T^{14} +$$$$42\!\cdots\!35$$$$T^{16} -$$$$67\!\cdots\!08$$$$T^{18} +$$$$84\!\cdots\!06$$$$T^{20} -$$$$75\!\cdots\!92$$$$T^{22} +$$$$39\!\cdots\!61$$$$T^{24}$$
$19$ $$1 - 298936704 T^{2} + 47587972333287630 T^{4} -$$$$51\!\cdots\!12$$$$T^{6} +$$$$42\!\cdots\!31$$$$T^{8} -$$$$27\!\cdots\!04$$$$T^{10} +$$$$14\!\cdots\!72$$$$T^{12} -$$$$60\!\cdots\!44$$$$T^{14} +$$$$20\!\cdots\!51$$$$T^{16} -$$$$56\!\cdots\!72$$$$T^{18} +$$$$11\!\cdots\!30$$$$T^{20} -$$$$15\!\cdots\!04$$$$T^{22} +$$$$11\!\cdots\!61$$$$T^{24}$$
$23$ $$( 1 + 7652 T + 314675762 T^{2} + 898155402636 T^{3} + 60250870240477171 T^{4} +$$$$30\!\cdots\!60$$$$T^{5} +$$$$11\!\cdots\!60$$$$T^{6} +$$$$45\!\cdots\!40$$$$T^{7} +$$$$13\!\cdots\!91$$$$T^{8} +$$$$29\!\cdots\!84$$$$T^{9} +$$$$15\!\cdots\!42$$$$T^{10} +$$$$54\!\cdots\!48$$$$T^{11} +$$$$10\!\cdots\!61$$$$T^{12} )^{2}$$
$29$ $$1 - 3552464472 T^{2} + 7069307690910966078 T^{4} -$$$$97\!\cdots\!24$$$$T^{6} +$$$$10\!\cdots\!71$$$$T^{8} -$$$$83\!\cdots\!92$$$$T^{10} +$$$$54\!\cdots\!60$$$$T^{12} -$$$$29\!\cdots\!72$$$$T^{14} +$$$$12\!\cdots\!51$$$$T^{16} -$$$$43\!\cdots\!04$$$$T^{18} +$$$$11\!\cdots\!58$$$$T^{20} -$$$$19\!\cdots\!72$$$$T^{22} +$$$$19\!\cdots\!41$$$$T^{24}$$
$31$ $$( 1 + 29304 T + 2909337822 T^{2} + 67282988073368 T^{3} + 4222373507783705727 T^{4} +$$$$85\!\cdots\!92$$$$T^{5} +$$$$43\!\cdots\!24$$$$T^{6} +$$$$75\!\cdots\!52$$$$T^{7} +$$$$33\!\cdots\!47$$$$T^{8} +$$$$47\!\cdots\!88$$$$T^{9} +$$$$18\!\cdots\!62$$$$T^{10} +$$$$16\!\cdots\!04$$$$T^{11} +$$$$48\!\cdots\!81$$$$T^{12} )^{2}$$
$37$ $$( 1 + 101256 T + 8636966478 T^{2} + 484631270053544 T^{3} + 33956810947327452207 T^{4} +$$$$20\!\cdots\!48$$$$T^{5} +$$$$12\!\cdots\!16$$$$T^{6} +$$$$51\!\cdots\!32$$$$T^{7} +$$$$22\!\cdots\!67$$$$T^{8} +$$$$81\!\cdots\!76$$$$T^{9} +$$$$37\!\cdots\!58$$$$T^{10} +$$$$11\!\cdots\!44$$$$T^{11} +$$$$28\!\cdots\!41$$$$T^{12} )^{2}$$
$41$ $$1 - 18443226360 T^{2} +$$$$19\!\cdots\!26$$$$T^{4} -$$$$14\!\cdots\!20$$$$T^{6} +$$$$94\!\cdots\!19$$$$T^{8} -$$$$52\!\cdots\!20$$$$T^{10} +$$$$26\!\cdots\!44$$$$T^{12} -$$$$11\!\cdots\!20$$$$T^{14} +$$$$48\!\cdots\!59$$$$T^{16} -$$$$17\!\cdots\!20$$$$T^{18} +$$$$51\!\cdots\!46$$$$T^{20} -$$$$10\!\cdots\!60$$$$T^{22} +$$$$13\!\cdots\!81$$$$T^{24}$$
$43$ $$1 - 23801189952 T^{2} +$$$$39\!\cdots\!58$$$$T^{4} -$$$$47\!\cdots\!28$$$$T^{6} +$$$$45\!\cdots\!71$$$$T^{8} -$$$$36\!\cdots\!92$$$$T^{10} +$$$$24\!\cdots\!12$$$$T^{12} -$$$$14\!\cdots\!92$$$$T^{14} +$$$$72\!\cdots\!71$$$$T^{16} -$$$$30\!\cdots\!28$$$$T^{18} +$$$$10\!\cdots\!58$$$$T^{20} -$$$$24\!\cdots\!52$$$$T^{22} +$$$$40\!\cdots\!01$$$$T^{24}$$
$47$ $$( 1 - 258460 T + 80997697934 T^{2} - 13130966707228932 T^{3} +$$$$23\!\cdots\!79$$$$T^{4} -$$$$27\!\cdots\!60$$$$T^{5} +$$$$33\!\cdots\!76$$$$T^{6} -$$$$29\!\cdots\!40$$$$T^{7} +$$$$27\!\cdots\!39$$$$T^{8} -$$$$16\!\cdots\!48$$$$T^{9} +$$$$10\!\cdots\!54$$$$T^{10} -$$$$37\!\cdots\!40$$$$T^{11} +$$$$15\!\cdots\!21$$$$T^{12} )^{2}$$
$53$ $$( 1 + 521096 T + 230179608818 T^{2} + 64512279082152192 T^{3} +$$$$15\!\cdots\!39$$$$T^{4} +$$$$29\!\cdots\!76$$$$T^{5} +$$$$50\!\cdots\!52$$$$T^{6} +$$$$66\!\cdots\!04$$$$T^{7} +$$$$78\!\cdots\!99$$$$T^{8} +$$$$70\!\cdots\!88$$$$T^{9} +$$$$55\!\cdots\!58$$$$T^{10} +$$$$27\!\cdots\!04$$$$T^{11} +$$$$11\!\cdots\!21$$$$T^{12} )^{2}$$
$59$ $$( 1 + 230504 T + 221620610954 T^{2} + 38998264192118184 T^{3} +$$$$20\!\cdots\!99$$$$T^{4} +$$$$28\!\cdots\!68$$$$T^{5} +$$$$11\!\cdots\!28$$$$T^{6} +$$$$12\!\cdots\!88$$$$T^{7} +$$$$37\!\cdots\!19$$$$T^{8} +$$$$29\!\cdots\!64$$$$T^{9} +$$$$70\!\cdots\!94$$$$T^{10} +$$$$30\!\cdots\!04$$$$T^{11} +$$$$56\!\cdots\!41$$$$T^{12} )^{2}$$
$61$ $$1 - 344957455356 T^{2} +$$$$58\!\cdots\!66$$$$T^{4} -$$$$63\!\cdots\!48$$$$T^{6} +$$$$51\!\cdots\!59$$$$T^{8} -$$$$32\!\cdots\!44$$$$T^{10} +$$$$17\!\cdots\!88$$$$T^{12} -$$$$86\!\cdots\!24$$$$T^{14} +$$$$35\!\cdots\!19$$$$T^{16} -$$$$11\!\cdots\!28$$$$T^{18} +$$$$28\!\cdots\!46$$$$T^{20} -$$$$45\!\cdots\!56$$$$T^{22} +$$$$34\!\cdots\!21$$$$T^{24}$$
$67$ $$( 1 - 182376 T + 317875829886 T^{2} - 34532602304668456 T^{3} +$$$$46\!\cdots\!79$$$$T^{4} -$$$$36\!\cdots\!72$$$$T^{5} +$$$$48\!\cdots\!60$$$$T^{6} -$$$$33\!\cdots\!68$$$$T^{7} +$$$$38\!\cdots\!19$$$$T^{8} -$$$$25\!\cdots\!04$$$$T^{9} +$$$$21\!\cdots\!06$$$$T^{10} -$$$$11\!\cdots\!24$$$$T^{11} +$$$$54\!\cdots\!81$$$$T^{12} )^{2}$$
$71$ $$( 1 + 377588 T + 443405411582 T^{2} + 168724078391174988 T^{3} +$$$$10\!\cdots\!11$$$$T^{4} +$$$$36\!\cdots\!64$$$$T^{5} +$$$$16\!\cdots\!84$$$$T^{6} +$$$$47\!\cdots\!44$$$$T^{7} +$$$$17\!\cdots\!51$$$$T^{8} +$$$$35\!\cdots\!68$$$$T^{9} +$$$$11\!\cdots\!42$$$$T^{10} +$$$$13\!\cdots\!88$$$$T^{11} +$$$$44\!\cdots\!21$$$$T^{12} )^{2}$$
$73$ $$1 - 1038551693052 T^{2} +$$$$52\!\cdots\!34$$$$T^{4} -$$$$17\!\cdots\!24$$$$T^{6} +$$$$44\!\cdots\!75$$$$T^{8} -$$$$89\!\cdots\!52$$$$T^{10} +$$$$14\!\cdots\!40$$$$T^{12} -$$$$20\!\cdots\!92$$$$T^{14} +$$$$23\!\cdots\!75$$$$T^{16} -$$$$20\!\cdots\!64$$$$T^{18} +$$$$14\!\cdots\!54$$$$T^{20} -$$$$65\!\cdots\!52$$$$T^{22} +$$$$14\!\cdots\!21$$$$T^{24}$$
$79$ $$1 - 2060241143040 T^{2} +$$$$20\!\cdots\!02$$$$T^{4} -$$$$12\!\cdots\!72$$$$T^{6} +$$$$58\!\cdots\!51$$$$T^{8} -$$$$20\!\cdots\!32$$$$T^{10} +$$$$55\!\cdots\!96$$$$T^{12} -$$$$11\!\cdots\!12$$$$T^{14} +$$$$20\!\cdots\!31$$$$T^{16} -$$$$26\!\cdots\!12$$$$T^{18} +$$$$24\!\cdots\!22$$$$T^{20} -$$$$14\!\cdots\!40$$$$T^{22} +$$$$42\!\cdots\!41$$$$T^{24}$$
$83$ $$1 - 1043972235372 T^{2} +$$$$84\!\cdots\!42$$$$T^{4} -$$$$43\!\cdots\!80$$$$T^{6} +$$$$20\!\cdots\!39$$$$T^{8} -$$$$72\!\cdots\!88$$$$T^{10} +$$$$25\!\cdots\!00$$$$T^{12} -$$$$77\!\cdots\!68$$$$T^{14} +$$$$23\!\cdots\!19$$$$T^{16} -$$$$53\!\cdots\!80$$$$T^{18} +$$$$11\!\cdots\!22$$$$T^{20} -$$$$14\!\cdots\!72$$$$T^{22} +$$$$14\!\cdots\!61$$$$T^{24}$$
$89$ $$( 1 + 1756772 T + 2562798883442 T^{2} + 2577409891035591252 T^{3} +$$$$24\!\cdots\!83$$$$T^{4} +$$$$19\!\cdots\!88$$$$T^{5} +$$$$14\!\cdots\!88$$$$T^{6} +$$$$96\!\cdots\!68$$$$T^{7} +$$$$61\!\cdots\!43$$$$T^{8} +$$$$31\!\cdots\!12$$$$T^{9} +$$$$15\!\cdots\!22$$$$T^{10} +$$$$53\!\cdots\!72$$$$T^{11} +$$$$15\!\cdots\!61$$$$T^{12} )^{2}$$
$97$ $$( 1 - 1185096 T + 3545530528134 T^{2} - 3370506166089168808 T^{3} +$$$$54\!\cdots\!91$$$$T^{4} -$$$$43\!\cdots\!64$$$$T^{5} +$$$$53\!\cdots\!96$$$$T^{6} -$$$$36\!\cdots\!56$$$$T^{7} +$$$$37\!\cdots\!31$$$$T^{8} -$$$$19\!\cdots\!12$$$$T^{9} +$$$$17\!\cdots\!54$$$$T^{10} -$$$$47\!\cdots\!04$$$$T^{11} +$$$$33\!\cdots\!21$$$$T^{12} )^{2}$$