# Properties

 Label 33.4.e.b Level 33 Weight 4 Character orbit 33.e Analytic conductor 1.947 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 33.e (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.94706303019$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.682515625.5 Defining polynomial: $$x^{8} - 3 x^{7} + 5 x^{6} + 2 x^{5} + 19 x^{4} + 28 x^{3} + 100 x^{2} + 88 x + 121$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{2} + 3 \beta_{3} q^{3} + ( 2 - 2 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - 3 \beta_{6} - 4 \beta_{7} ) q^{4} + ( 4 - 7 \beta_{2} + 4 \beta_{3} + 3 \beta_{5} - 2 \beta_{6} ) q^{5} + ( -3 + 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{6} ) q^{6} + ( 4 - 15 \beta_{1} - 4 \beta_{2} - 6 \beta_{4} + 6 \beta_{6} ) q^{7} + ( 2 + 8 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 8 \beta_{6} - 2 \beta_{7} ) q^{8} -9 \beta_{7} q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{2} + 3 \beta_{3} q^{3} + ( 2 - 2 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - 3 \beta_{6} - 4 \beta_{7} ) q^{4} + ( 4 - 7 \beta_{2} + 4 \beta_{3} + 3 \beta_{5} - 2 \beta_{6} ) q^{5} + ( -3 + 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{6} ) q^{6} + ( 4 - 15 \beta_{1} - 4 \beta_{2} - 6 \beta_{4} + 6 \beta_{6} ) q^{7} + ( 2 + 8 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 8 \beta_{6} - 2 \beta_{7} ) q^{8} -9 \beta_{7} q^{9} + ( -3 + 11 \beta_{2} + 6 \beta_{4} + 11 \beta_{7} ) q^{10} + ( -11 + 3 \beta_{1} + 4 \beta_{2} + 11 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 14 \beta_{6} - 11 \beta_{7} ) q^{11} + ( -6 + 9 \beta_{1} + 3 \beta_{2} - 9 \beta_{4} - 9 \beta_{5} - 6 \beta_{7} ) q^{12} + ( -3 \beta_{1} + 7 \beta_{2} - 25 \beta_{3} + 15 \beta_{4} + 18 \beta_{5} + 18 \beta_{6} + 46 \beta_{7} ) q^{13} + ( -13 + 14 \beta_{1} + 2 \beta_{2} - 35 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 14 \beta_{6} + 13 \beta_{7} ) q^{14} + ( -12 + 15 \beta_{1} + 12 \beta_{2} + 9 \beta_{4} - 9 \beta_{6} ) q^{15} + ( 17 - 23 \beta_{2} + 17 \beta_{3} - 18 \beta_{5} + 3 \beta_{6} ) q^{16} + ( -11 + 26 \beta_{2} - 11 \beta_{3} + 25 \beta_{6} ) q^{17} + ( 9 + 9 \beta_{1} - 9 \beta_{2} ) q^{18} + ( 34 - 15 \beta_{1} + 27 \beta_{2} - 27 \beta_{3} - 27 \beta_{4} - 27 \beta_{5} + 15 \beta_{6} - 34 \beta_{7} ) q^{19} + ( 32 \beta_{1} + 8 \beta_{2} + 23 \beta_{3} + \beta_{4} - 31 \beta_{5} - 31 \beta_{6} - 29 \beta_{7} ) q^{20} + ( -12 - 18 \beta_{1} - 6 \beta_{2} - 27 \beta_{4} + 18 \beta_{5} + 12 \beta_{7} ) q^{21} + ( -55 - 12 \beta_{1} - 16 \beta_{2} - 11 \beta_{3} + 30 \beta_{4} + 27 \beta_{5} - 12 \beta_{6} + 11 \beta_{7} ) q^{22} + ( -21 - 39 \beta_{1} + 12 \beta_{2} - 18 \beta_{4} + 39 \beta_{5} + 51 \beta_{7} ) q^{23} + ( 18 \beta_{1} - 12 \beta_{2} + 6 \beta_{3} + 24 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} + 15 \beta_{7} ) q^{24} + ( 34 + 3 \beta_{1} - 28 \beta_{3} - 3 \beta_{6} - 34 \beta_{7} ) q^{25} + ( -83 - 76 \beta_{1} + 83 \beta_{2} + \beta_{3} + 11 \beta_{4} - 11 \beta_{6} - \beta_{7} ) q^{26} -27 \beta_{2} q^{27} + ( 98 - 102 \beta_{2} + 98 \beta_{3} + 39 \beta_{5} - 18 \beta_{6} ) q^{28} + ( 181 + 29 \beta_{1} - 181 \beta_{2} + 169 \beta_{3} + 26 \beta_{4} - 26 \beta_{6} - 169 \beta_{7} ) q^{29} + ( 33 + 18 \beta_{2} + 24 \beta_{3} - 18 \beta_{4} - 18 \beta_{5} - 33 \beta_{7} ) q^{30} + ( -15 \beta_{1} - 8 \beta_{2} + 20 \beta_{3} - 27 \beta_{4} - 12 \beta_{5} - 12 \beta_{6} + 49 \beta_{7} ) q^{31} + ( 4 + 41 \beta_{1} + 15 \beta_{2} - 5 \beta_{4} - 41 \beta_{5} - 26 \beta_{7} ) q^{32} + ( -54 \beta_{1} - 39 \beta_{2} - 33 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} + 12 \beta_{6} - 33 \beta_{7} ) q^{33} + ( -63 - 39 \beta_{1} - 40 \beta_{2} + 24 \beta_{4} + 39 \beta_{5} - \beta_{7} ) q^{34} + ( 21 \beta_{1} + 148 \beta_{2} - 119 \beta_{3} - 8 \beta_{4} - 29 \beta_{5} - 29 \beta_{6} + 238 \beta_{7} ) q^{35} + ( -18 - 27 \beta_{1} - 27 \beta_{2} - 36 \beta_{3} + 27 \beta_{4} + 27 \beta_{5} + 27 \beta_{6} + 18 \beta_{7} ) q^{36} + ( -161 + 117 \beta_{1} + 161 \beta_{2} + 12 \beta_{3} + 21 \beta_{4} - 21 \beta_{6} - 12 \beta_{7} ) q^{37} + ( 43 - 276 \beta_{2} + 43 \beta_{3} + 88 \beta_{5} + 85 \beta_{6} ) q^{38} + ( 75 + 108 \beta_{2} + 75 \beta_{3} - 45 \beta_{5} - 54 \beta_{6} ) q^{39} + ( 82 - 21 \beta_{1} - 82 \beta_{2} + 136 \beta_{3} - 39 \beta_{4} + 39 \beta_{6} - 136 \beta_{7} ) q^{40} + ( 99 - 97 \beta_{1} - 114 \beta_{2} - 2 \beta_{3} + 114 \beta_{4} + 114 \beta_{5} + 97 \beta_{6} - 99 \beta_{7} ) q^{41} + ( 36 \beta_{1} + 33 \beta_{2} - 39 \beta_{3} + 42 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} + 105 \beta_{7} ) q^{42} + ( -175 + 135 \beta_{1} + 156 \beta_{2} - 66 \beta_{4} - 135 \beta_{5} + 21 \beta_{7} ) q^{43} + ( -132 - 36 \beta_{1} + 282 \beta_{2} - 286 \beta_{3} - 97 \beta_{4} - 51 \beta_{5} + 8 \beta_{6} + 44 \beta_{7} ) q^{44} + ( 36 + 27 \beta_{1} - 9 \beta_{2} + 18 \beta_{4} - 27 \beta_{5} - 36 \beta_{7} ) q^{45} + ( 48 \beta_{1} + 183 \beta_{2} - 120 \beta_{3} - 15 \beta_{4} - 63 \beta_{5} - 63 \beta_{6} + 81 \beta_{7} ) q^{46} + ( -177 + 20 \beta_{1} + 87 \beta_{2} - 215 \beta_{3} - 87 \beta_{4} - 87 \beta_{5} - 20 \beta_{6} + 177 \beta_{7} ) q^{47} + ( -123 - 63 \beta_{1} + 123 \beta_{2} - 72 \beta_{3} - 54 \beta_{4} + 54 \beta_{6} + 72 \beta_{7} ) q^{48} + ( -299 - 58 \beta_{2} - 299 \beta_{3} + 45 \beta_{5} - 51 \beta_{6} ) q^{49} + ( 71 - 133 \beta_{2} + 71 \beta_{3} + 34 \beta_{5} + 68 \beta_{6} ) q^{50} + ( 78 - 75 \beta_{1} - 78 \beta_{2} + 45 \beta_{3} - 45 \beta_{7} ) q^{51} + ( 403 + 138 \beta_{1} + 135 \beta_{2} + 45 \beta_{3} - 135 \beta_{4} - 135 \beta_{5} - 138 \beta_{6} - 403 \beta_{7} ) q^{52} + ( 53 \beta_{1} - 134 \beta_{2} + 172 \beta_{3} + 15 \beta_{4} - 38 \beta_{5} - 38 \beta_{6} + 33 \beta_{7} ) q^{53} + ( -27 + 27 \beta_{2} + 27 \beta_{4} + 27 \beta_{7} ) q^{54} + ( -176 + 63 \beta_{1} + 18 \beta_{2} - 253 \beta_{3} + 123 \beta_{4} + 15 \beta_{5} + 30 \beta_{6} + 66 \beta_{7} ) q^{55} + ( 48 + 50 \beta_{1} + 316 \beta_{2} + 89 \beta_{4} - 50 \beta_{5} + 266 \beta_{7} ) q^{56} + ( -126 \beta_{1} - 183 \beta_{2} + 102 \beta_{3} - 45 \beta_{4} + 81 \beta_{5} + 81 \beta_{6} + 81 \beta_{7} ) q^{57} + ( -106 - 18 \beta_{1} - 135 \beta_{2} + 73 \beta_{3} + 135 \beta_{4} + 135 \beta_{5} + 18 \beta_{6} + 106 \beta_{7} ) q^{58} + ( 130 + 74 \beta_{1} - 130 \beta_{2} + 433 \beta_{3} + 68 \beta_{4} - 68 \beta_{6} - 433 \beta_{7} ) q^{59} + ( -69 - 15 \beta_{2} - 69 \beta_{3} - 3 \beta_{5} + 93 \beta_{6} ) q^{60} + ( 238 - 12 \beta_{2} + 238 \beta_{3} + 3 \beta_{5} + 21 \beta_{6} ) q^{61} + ( -14 + 5 \beta_{1} + 14 \beta_{2} - 38 \beta_{3} - 4 \beta_{4} + 4 \beta_{6} + 38 \beta_{7} ) q^{62} + ( 36 + 54 \beta_{1} - 81 \beta_{2} + 81 \beta_{4} + 81 \beta_{5} - 54 \beta_{6} - 36 \beta_{7} ) q^{63} + ( -228 \beta_{1} - 388 \beta_{2} + 268 \beta_{3} - 108 \beta_{4} + 120 \beta_{5} + 120 \beta_{6} + 151 \beta_{7} ) q^{64} + ( 1 - 269 \beta_{1} - 239 \beta_{2} + 3 \beta_{4} + 269 \beta_{5} + 30 \beta_{7} ) q^{65} + ( 33 + 117 \beta_{1} + 90 \beta_{2} - 132 \beta_{3} - 45 \beta_{4} - 90 \beta_{5} - 81 \beta_{6} ) q^{66} + ( -151 + 3 \beta_{1} + 189 \beta_{2} + 33 \beta_{4} - 3 \beta_{5} + 186 \beta_{7} ) q^{67} + ( -58 \beta_{1} + 281 \beta_{2} - 292 \beta_{3} - 47 \beta_{4} + 11 \beta_{5} + 11 \beta_{6} - 113 \beta_{7} ) q^{68} + ( 153 + 117 \beta_{1} - 54 \beta_{2} + 90 \beta_{3} + 54 \beta_{4} + 54 \beta_{5} - 117 \beta_{6} - 153 \beta_{7} ) q^{69} + ( 124 - 222 \beta_{1} - 124 \beta_{2} + 190 \beta_{3} - 177 \beta_{4} + 177 \beta_{6} - 190 \beta_{7} ) q^{70} + ( 190 - 10 \beta_{2} + 190 \beta_{3} - 369 \beta_{5} - 161 \beta_{6} ) q^{71} + ( -18 + 135 \beta_{2} - 18 \beta_{3} - 72 \beta_{5} - 18 \beta_{6} ) q^{72} + ( 679 + 39 \beta_{1} - 679 \beta_{2} + 429 \beta_{3} - 18 \beta_{4} + 18 \beta_{6} - 429 \beta_{7} ) q^{73} + ( 128 - 19 \beta_{1} + 11 \beta_{2} + 157 \beta_{3} - 11 \beta_{4} - 11 \beta_{5} + 19 \beta_{6} - 128 \beta_{7} ) q^{74} + ( 9 \beta_{1} - 102 \beta_{2} + 102 \beta_{3} + 9 \beta_{4} + 84 \beta_{7} ) q^{75} + ( -171 - 333 \beta_{1} - 68 \beta_{2} + 318 \beta_{4} + 333 \beta_{5} + 265 \beta_{7} ) q^{76} + ( -77 + 49 \beta_{1} - 228 \beta_{2} + 231 \beta_{3} - 18 \beta_{4} + 184 \beta_{5} - 105 \beta_{6} - 660 \beta_{7} ) q^{77} + ( 249 + 33 \beta_{1} - 219 \beta_{2} - 261 \beta_{4} - 33 \beta_{5} - 252 \beta_{7} ) q^{78} + ( 144 \beta_{1} + 406 \beta_{2} - 307 \beta_{3} + 45 \beta_{4} - 99 \beta_{5} - 99 \beta_{6} + 667 \beta_{7} ) q^{79} + ( -401 + 10 \beta_{1} + 52 \beta_{2} - 310 \beta_{3} - 52 \beta_{4} - 52 \beta_{5} - 10 \beta_{6} + 401 \beta_{7} ) q^{80} + ( -81 + 81 \beta_{2} - 81 \beta_{3} + 81 \beta_{7} ) q^{81} + ( -304 + 482 \beta_{2} - 304 \beta_{3} - 129 \beta_{5} - 321 \beta_{6} ) q^{82} + ( -491 + 377 \beta_{2} - 491 \beta_{3} - 27 \beta_{5} - 167 \beta_{6} ) q^{83} + ( -189 + 171 \beta_{1} + 189 \beta_{2} + 105 \beta_{3} + 117 \beta_{4} - 117 \beta_{6} - 105 \beta_{7} ) q^{84} + ( -71 - 27 \beta_{1} - 144 \beta_{2} - 344 \beta_{3} + 144 \beta_{4} + 144 \beta_{5} + 27 \beta_{6} + 71 \beta_{7} ) q^{85} + ( -116 \beta_{1} - 523 \beta_{2} + 296 \beta_{3} + 111 \beta_{4} + 227 \beta_{5} + 227 \beta_{6} - 93 \beta_{7} ) q^{86} + ( -543 + 78 \beta_{1} + 114 \beta_{2} + 9 \beta_{4} - 78 \beta_{5} + 36 \beta_{7} ) q^{87} + ( 176 + 216 \beta_{1} - 53 \beta_{2} - 209 \beta_{3} + 21 \beta_{4} - 90 \beta_{5} + 84 \beta_{6} + 231 \beta_{7} ) q^{88} + ( -267 + 345 \beta_{1} + 840 \beta_{2} - 282 \beta_{4} - 345 \beta_{5} + 495 \beta_{7} ) q^{89} + ( -54 \beta_{1} - 153 \beta_{2} + 99 \beta_{3} + 54 \beta_{5} + 54 \beta_{6} - 72 \beta_{7} ) q^{90} + ( 339 - 501 \beta_{1} + 396 \beta_{2} - 82 \beta_{3} - 396 \beta_{4} - 396 \beta_{5} + 501 \beta_{6} - 339 \beta_{7} ) q^{91} + ( 258 + 405 \beta_{1} - 258 \beta_{2} + 519 \beta_{3} + 66 \beta_{4} - 66 \beta_{6} - 519 \beta_{7} ) q^{92} + ( -60 + 126 \beta_{2} - 60 \beta_{3} + 81 \beta_{5} + 36 \beta_{6} ) q^{93} + ( 185 - 226 \beta_{2} + 185 \beta_{3} - 3 \beta_{5} + 252 \beta_{6} ) q^{94} + ( 140 + 63 \beta_{1} - 140 \beta_{2} - 473 \beta_{3} - 22 \beta_{4} + 22 \beta_{6} + 473 \beta_{7} ) q^{95} + ( -78 - 123 \beta_{1} - 15 \beta_{2} - 66 \beta_{3} + 15 \beta_{4} + 15 \beta_{5} + 123 \beta_{6} + 78 \beta_{7} ) q^{96} + ( 171 \beta_{1} + 142 \beta_{2} - 148 \beta_{3} + 177 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} - 374 \beta_{7} ) q^{97} + ( 445 + 401 \beta_{1} + 498 \beta_{2} + \beta_{4} - 401 \beta_{5} + 97 \beta_{7} ) q^{98} + ( -99 - 18 \beta_{1} + 9 \beta_{2} - 99 \beta_{3} - 153 \beta_{4} - 9 \beta_{5} - 18 \beta_{6} + 198 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 6q^{2} - 6q^{3} - 16q^{4} + 9q^{5} - 18q^{6} + 3q^{7} + 36q^{8} - 18q^{9} + O(q^{10})$$ $$8q - 6q^{2} - 6q^{3} - 16q^{4} + 9q^{5} - 18q^{6} + 3q^{7} + 36q^{8} - 18q^{9} + 8q^{10} - 87q^{11} - 18q^{12} + 171q^{13} + 12q^{14} - 63q^{15} + 44q^{16} + 36q^{17} + 81q^{18} + 324q^{19} - 87q^{20} - 66q^{21} - 521q^{22} - 84q^{23} + 18q^{24} + 263q^{25} - 774q^{26} - 54q^{27} + 387q^{28} + 393q^{29} + 204q^{30} + 15q^{31} + 102q^{32} - 216q^{33} - 712q^{34} + 1002q^{35} - 144q^{36} - 747q^{37} - 36q^{38} + 513q^{39} + 41q^{40} + 159q^{41} + 396q^{42} - 644q^{43} + 219q^{44} + 216q^{45} + 753q^{46} - 351q^{47} - 423q^{48} - 1967q^{49} + 330q^{50} + 63q^{51} + 2871q^{52} - 531q^{53} - 162q^{54} - 716q^{55} + 1470q^{56} - 453q^{57} - 1205q^{58} - 1002q^{59} - 261q^{60} + 1449q^{61} + 99q^{62} + 27q^{63} - 1118q^{64} - 954q^{65} + 897q^{66} - 518q^{67} + 873q^{68} + 693q^{69} + 26q^{70} + 429q^{71} + 54q^{72} + 2547q^{73} + 468q^{74} - 231q^{75} - 2276q^{76} - 2697q^{77} + 1638q^{78} + 2805q^{79} - 1620q^{80} - 162q^{81} - 1631q^{82} - 2553q^{83} - 1509q^{84} - 197q^{85} - 1713q^{86} - 3906q^{87} + 2866q^{88} + 1788q^{89} - 648q^{90} + 2885q^{91} + 423q^{92} + 45q^{93} + 1159q^{94} + 3009q^{95} - 504q^{96} + 9q^{97} + 5550q^{98} + 27q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 5 x^{6} + 2 x^{5} + 19 x^{4} + 28 x^{3} + 100 x^{2} + 88 x + 121$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$528 \nu^{7} + 2098 \nu^{6} - 15725 \nu^{5} + 33439 \nu^{4} + 71401 \nu^{3} - 332708 \nu^{2} + 319181 \nu + 440220$$$$)/1168519$$ $$\beta_{3}$$ $$=$$ $$($$$$5794 \nu^{7} - 9973 \nu^{6} - 30517 \nu^{5} + 195125 \nu^{4} - 61888 \nu^{3} + 104068 \nu^{2} + 501961 \nu + 528473$$$$)/1168519$$ $$\beta_{4}$$ $$=$$ $$($$$$7409 \nu^{7} - 59487 \nu^{6} + 183537 \nu^{5} - 171974 \nu^{4} - 58164 \nu^{3} - 77439 \nu^{2} + 18601 \nu - 701074$$$$)/1168519$$ $$\beta_{5}$$ $$=$$ $$($$$$8817 \nu^{7} + 16927 \nu^{6} - 106264 \nu^{5} + 200474 \nu^{4} + 521745 \nu^{3} + 380907 \nu^{2} + 2179908 \nu + 2809884$$$$)/1168519$$ $$\beta_{6}$$ $$=$$ $$($$$$-11971 \nu^{7} + 3536 \nu^{6} + 58156 \nu^{5} - 228404 \nu^{4} - 102852 \nu^{3} - 979996 \nu^{2} - 1085964 \nu - 2305776$$$$)/1168519$$ $$\beta_{7}$$ $$=$$ $$($$$$-13790 \nu^{7} + 57068 \nu^{6} - 113608 \nu^{5} + 65418 \nu^{4} - 266949 \nu^{3} + 6060 \nu^{2} - 742824 \nu + 665808$$$$)/1168519$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{3} - 5 \beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + 6 \beta_{6} + 6 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 10 \beta_{2} - 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$12 \beta_{7} + 10 \beta_{6} + 13 \beta_{5} + 13 \beta_{4} + 14 \beta_{3} - 13 \beta_{2} - 10 \beta_{1} - 12$$ $$\nu^{5}$$ $$=$$ $$43 \beta_{7} + 25 \beta_{5} + 49 \beta_{4} + 18 \beta_{2} - 25 \beta_{1} - 62$$ $$\nu^{6}$$ $$=$$ $$97 \beta_{7} - 92 \beta_{6} + 92 \beta_{4} - 97 \beta_{3} + 221 \beta_{2} - 44 \beta_{1} - 221$$ $$\nu^{7}$$ $$=$$ $$-449 \beta_{6} - 260 \beta_{5} - 412 \beta_{3} + 896 \beta_{2} - 412$$

## Character values

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

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$-1 + \beta_{2} - \beta_{3} + \beta_{7}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.581882 + 1.79085i −0.390899 − 1.20306i 2.51217 − 1.82520i −1.20316 + 0.874145i 0.581882 − 1.79085i −0.390899 + 1.20306i 2.51217 + 1.82520i −1.20316 − 0.874145i
−2.02339 1.47008i 0.927051 2.85317i −0.539165 1.65938i −8.44146 + 6.13308i −6.07016 + 4.41023i −10.1220 31.1524i −7.53140 + 23.1793i −7.28115 5.29007i 26.0964
4.2 0.523388 + 0.380264i 0.927051 2.85317i −2.34280 7.21040i 9.01441 6.54935i 1.57016 1.14079i 8.07696 + 24.8583i 3.11499 9.58696i −7.28115 5.29007i 7.20851
16.1 −1.45957 4.49208i −2.42705 1.76336i −11.5763 + 8.41069i 1.86000 5.72450i −4.37870 + 13.4762i −8.05785 + 5.85437i 24.1083 + 17.5157i 2.78115 + 8.55951i −28.4297
16.2 −0.0404346 0.124445i −2.42705 1.76336i 6.45828 4.69222i 2.06705 6.36172i −0.121304 + 0.373335i 11.6029 8.43002i −1.69194 1.22926i 2.78115 + 8.55951i −0.875265
25.1 −2.02339 + 1.47008i 0.927051 + 2.85317i −0.539165 + 1.65938i −8.44146 6.13308i −6.07016 4.41023i −10.1220 + 31.1524i −7.53140 23.1793i −7.28115 + 5.29007i 26.0964
25.2 0.523388 0.380264i 0.927051 + 2.85317i −2.34280 + 7.21040i 9.01441 + 6.54935i 1.57016 + 1.14079i 8.07696 24.8583i 3.11499 + 9.58696i −7.28115 + 5.29007i 7.20851
31.1 −1.45957 + 4.49208i −2.42705 + 1.76336i −11.5763 8.41069i 1.86000 + 5.72450i −4.37870 13.4762i −8.05785 5.85437i 24.1083 17.5157i 2.78115 8.55951i −28.4297
31.2 −0.0404346 + 0.124445i −2.42705 + 1.76336i 6.45828 + 4.69222i 2.06705 + 6.36172i −0.121304 0.373335i 11.6029 + 8.43002i −1.69194 + 1.22926i 2.78115 8.55951i −0.875265
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 31.2 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.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.4.e.b 8
3.b odd 2 1 99.4.f.b 8
11.c even 5 1 inner 33.4.e.b 8
11.c even 5 1 363.4.a.p 4
11.d odd 10 1 363.4.a.t 4
33.f even 10 1 1089.4.a.z 4
33.h odd 10 1 99.4.f.b 8
33.h odd 10 1 1089.4.a.bg 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.e.b 8 1.a even 1 1 trivial
33.4.e.b 8 11.c even 5 1 inner
99.4.f.b 8 3.b odd 2 1
99.4.f.b 8 33.h odd 10 1
363.4.a.p 4 11.c even 5 1
363.4.a.t 4 11.d odd 10 1
1089.4.a.z 4 33.f even 10 1
1089.4.a.bg 4 33.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + \cdots$$ acting on $$S_{4}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 6 T + 18 T^{2} + 48 T^{3} + 153 T^{4} + 288 T^{5} + 371 T^{6} + 1419 T^{7} + 6393 T^{8} + 11352 T^{9} + 23744 T^{10} + 147456 T^{11} + 626688 T^{12} + 1572864 T^{13} + 4718592 T^{14} + 12582912 T^{15} + 16777216 T^{16}$$
$3$ $$( 1 + 3 T + 9 T^{2} + 27 T^{3} + 81 T^{4} )^{2}$$
$5$ $$1 - 9 T - 216 T^{2} - 108 T^{3} + 28704 T^{4} + 66129 T^{5} + 3870971 T^{6} - 23622552 T^{7} - 735099864 T^{8} - 2952819000 T^{9} + 60483921875 T^{10} + 129158203125 T^{11} + 7007812500000 T^{12} - 3295898437500 T^{13} - 823974609375000 T^{14} - 4291534423828125 T^{15} + 59604644775390625 T^{16}$$
$7$ $$1 - 3 T + 645 T^{2} - 8102 T^{3} + 318354 T^{4} - 5087519 T^{5} + 160735204 T^{6} - 1769175174 T^{7} + 69157603561 T^{8} - 606827084682 T^{9} + 18910336015396 T^{10} - 205299742331033 T^{11} + 4406429145587154 T^{12} - 38464743353558186 T^{13} + 1050326770652239605 T^{14} - 1675637592249852021 T^{15} +$$$$19\!\cdots\!01$$$$T^{16}$$
$11$ $$1 + 87 T + 4433 T^{2} + 229779 T^{3} + 9982500 T^{4} + 305835849 T^{5} + 7853329913 T^{6} + 205141449117 T^{7} + 3138428376721 T^{8}$$
$13$ $$1 - 171 T + 15261 T^{2} - 1179218 T^{3} + 78389016 T^{4} - 4452059033 T^{5} + 251390254120 T^{6} - 13190914668630 T^{7} + 625464463948177 T^{8} - 28980439526980110 T^{9} + 1213412741098703080 T^{10} - 47211857224007486309 T^{11} +$$$$18\!\cdots\!96$$$$T^{12} -$$$$60\!\cdots\!26$$$$T^{13} +$$$$17\!\cdots\!69$$$$T^{14} -$$$$42\!\cdots\!23$$$$T^{15} +$$$$54\!\cdots\!61$$$$T^{16}$$
$17$ $$1 - 36 T - 4704 T^{2} + 183996 T^{3} + 26983893 T^{4} - 1508244396 T^{5} - 94799670436 T^{6} + 2705638256928 T^{7} + 216503731307013 T^{8} + 13292800756287264 T^{9} - 2288233586326210084 T^{10} -$$$$17\!\cdots\!12$$$$T^{11} +$$$$15\!\cdots\!73$$$$T^{12} +$$$$52\!\cdots\!28$$$$T^{13} -$$$$66\!\cdots\!36$$$$T^{14} -$$$$24\!\cdots\!12$$$$T^{15} +$$$$33\!\cdots\!21$$$$T^{16}$$
$19$ $$1 - 324 T + 42702 T^{2} - 2758988 T^{3} + 70535517 T^{4} + 3205929520 T^{5} - 922691677142 T^{6} + 150427813848684 T^{7} - 15705053233704905 T^{8} + 1031784375188123556 T^{9} - 43408842842512952102 T^{10} +$$$$10\!\cdots\!80$$$$T^{11} +$$$$15\!\cdots\!37$$$$T^{12} -$$$$41\!\cdots\!12$$$$T^{13} +$$$$44\!\cdots\!82$$$$T^{14} -$$$$23\!\cdots\!56$$$$T^{15} +$$$$48\!\cdots\!21$$$$T^{16}$$
$23$ $$( 1 + 42 T + 28427 T^{2} - 413280 T^{3} + 349199196 T^{4} - 5028377760 T^{5} + 4208216216603 T^{6} + 75648411781446 T^{7} + 21914624432020321 T^{8} )^{2}$$
$29$ $$1 - 393 T + 51762 T^{2} + 2821128 T^{3} - 2373469998 T^{4} + 460592296485 T^{5} - 25589160757657 T^{6} - 7422933439897836 T^{7} + 1994632640290085280 T^{8} -$$$$18\!\cdots\!04$$$$T^{9} -$$$$15\!\cdots\!97$$$$T^{10} +$$$$66\!\cdots\!65$$$$T^{11} -$$$$83\!\cdots\!18$$$$T^{12} +$$$$24\!\cdots\!72$$$$T^{13} +$$$$10\!\cdots\!82$$$$T^{14} -$$$$20\!\cdots\!97$$$$T^{15} +$$$$12\!\cdots\!81$$$$T^{16}$$
$31$ $$1 - 15 T - 48960 T^{2} + 4414990 T^{3} + 1983425964 T^{4} - 36049577855 T^{5} - 67992515413865 T^{6} + 334846364616300 T^{7} + 2537520588739268776 T^{8} + 9975408048284193300 T^{9} -$$$$60\!\cdots\!65$$$$T^{10} -$$$$95\!\cdots\!05$$$$T^{11} +$$$$15\!\cdots\!04$$$$T^{12} +$$$$10\!\cdots\!90$$$$T^{13} -$$$$34\!\cdots\!60$$$$T^{14} -$$$$31\!\cdots\!65$$$$T^{15} +$$$$62\!\cdots\!21$$$$T^{16}$$
$37$ $$1 + 747 T + 208089 T^{2} + 18431284 T^{3} - 1384403574 T^{4} + 152950360711 T^{5} - 32557198551620 T^{6} - 149480116479179160 T^{7} - 55527587568079442063 T^{8} -$$$$75\!\cdots\!80$$$$T^{9} -$$$$83\!\cdots\!80$$$$T^{10} +$$$$19\!\cdots\!47$$$$T^{11} -$$$$91\!\cdots\!94$$$$T^{12} +$$$$61\!\cdots\!12$$$$T^{13} +$$$$35\!\cdots\!81$$$$T^{14} +$$$$63\!\cdots\!39$$$$T^{15} +$$$$43\!\cdots\!61$$$$T^{16}$$
$41$ $$1 - 159 T - 208197 T^{2} + 22172226 T^{3} + 14361268182 T^{4} - 959553734445 T^{5} + 186051111389942 T^{6} + 9057919749889008 T^{7} - 68887839310262606715 T^{8} +$$$$62\!\cdots\!68$$$$T^{9} +$$$$88\!\cdots\!22$$$$T^{10} -$$$$31\!\cdots\!45$$$$T^{11} +$$$$32\!\cdots\!42$$$$T^{12} +$$$$34\!\cdots\!26$$$$T^{13} -$$$$22\!\cdots\!37$$$$T^{14} -$$$$11\!\cdots\!19$$$$T^{15} +$$$$50\!\cdots\!61$$$$T^{16}$$
$43$ $$( 1 + 322 T + 181243 T^{2} + 48993496 T^{3} + 21697488148 T^{4} + 3895325886472 T^{5} + 1145702803089907 T^{6} + 161834821043663446 T^{7} + 39959630797262576401 T^{8} )^{2}$$
$47$ $$1 + 351 T - 13296 T^{2} + 44932770 T^{3} + 25775818116 T^{4} + 3450345166263 T^{5} + 2013116151255215 T^{6} + 899717908387248252 T^{7} +$$$$20\!\cdots\!32$$$$T^{8} +$$$$93\!\cdots\!96$$$$T^{9} +$$$$21\!\cdots\!35$$$$T^{10} +$$$$38\!\cdots\!21$$$$T^{11} +$$$$29\!\cdots\!56$$$$T^{12} +$$$$54\!\cdots\!10$$$$T^{13} -$$$$16\!\cdots\!44$$$$T^{14} +$$$$45\!\cdots\!97$$$$T^{15} +$$$$13\!\cdots\!81$$$$T^{16}$$
$53$ $$1 + 531 T - 23214 T^{2} + 55947168 T^{3} + 86011920246 T^{4} + 24095935682613 T^{5} + 567789066963755 T^{6} + 3215216614335806880 T^{7} +$$$$27\!\cdots\!92$$$$T^{8} +$$$$47\!\cdots\!60$$$$T^{9} +$$$$12\!\cdots\!95$$$$T^{10} +$$$$79\!\cdots\!29$$$$T^{11} +$$$$42\!\cdots\!86$$$$T^{12} +$$$$40\!\cdots\!76$$$$T^{13} -$$$$25\!\cdots\!46$$$$T^{14} +$$$$86\!\cdots\!43$$$$T^{15} +$$$$24\!\cdots\!81$$$$T^{16}$$
$59$ $$1 + 1002 T + 688980 T^{2} + 485993670 T^{3} + 355241156445 T^{4} + 202661193907746 T^{5} + 100593726036552992 T^{6} + 53249849322765499920 T^{7} +$$$$26\!\cdots\!05$$$$T^{8} +$$$$10\!\cdots\!80$$$$T^{9} +$$$$42\!\cdots\!72$$$$T^{10} +$$$$17\!\cdots\!94$$$$T^{11} +$$$$63\!\cdots\!45$$$$T^{12} +$$$$17\!\cdots\!30$$$$T^{13} +$$$$51\!\cdots\!80$$$$T^{14} +$$$$15\!\cdots\!18$$$$T^{15} +$$$$31\!\cdots\!61$$$$T^{16}$$
$61$ $$1 - 1449 T + 559458 T^{2} - 1297814 T^{3} + 56591949732 T^{4} - 46750854974495 T^{5} - 2095488968582993 T^{6} - 15938076497384457402 T^{7} +$$$$18\!\cdots\!00$$$$T^{8} -$$$$36\!\cdots\!62$$$$T^{9} -$$$$10\!\cdots\!73$$$$T^{10} -$$$$54\!\cdots\!95$$$$T^{11} +$$$$15\!\cdots\!72$$$$T^{12} -$$$$78\!\cdots\!14$$$$T^{13} +$$$$76\!\cdots\!98$$$$T^{14} -$$$$44\!\cdots\!89$$$$T^{15} +$$$$70\!\cdots\!41$$$$T^{16}$$
$67$ $$( 1 + 259 T + 1117027 T^{2} + 221089363 T^{3} + 492802725568 T^{4} + 66495500083969 T^{5} + 101044455259091563 T^{6} + 7046492408640391273 T^{7} +$$$$81\!\cdots\!61$$$$T^{8} )^{2}$$
$71$ $$1 - 429 T + 427302 T^{2} + 117188664 T^{3} - 140939575254 T^{4} + 134614525047165 T^{5} + 258031718426513 T^{6} - 49916639131846946880 T^{7} +$$$$41\!\cdots\!52$$$$T^{8} -$$$$17\!\cdots\!80$$$$T^{9} +$$$$33\!\cdots\!73$$$$T^{10} +$$$$61\!\cdots\!15$$$$T^{11} -$$$$23\!\cdots\!14$$$$T^{12} +$$$$68\!\cdots\!64$$$$T^{13} +$$$$89\!\cdots\!22$$$$T^{14} -$$$$32\!\cdots\!59$$$$T^{15} +$$$$26\!\cdots\!81$$$$T^{16}$$
$73$ $$1 - 2547 T + 2079912 T^{2} + 411582106 T^{3} - 2098827853302 T^{4} + 1679415683125579 T^{5} - 313174419721114481 T^{6} -$$$$56\!\cdots\!82$$$$T^{7} +$$$$58\!\cdots\!08$$$$T^{8} -$$$$22\!\cdots\!94$$$$T^{9} -$$$$47\!\cdots\!09$$$$T^{10} +$$$$98\!\cdots\!27$$$$T^{11} -$$$$48\!\cdots\!42$$$$T^{12} +$$$$36\!\cdots\!42$$$$T^{13} +$$$$72\!\cdots\!28$$$$T^{14} -$$$$34\!\cdots\!31$$$$T^{15} +$$$$52\!\cdots\!41$$$$T^{16}$$
$79$ $$1 - 2805 T + 2454909 T^{2} + 477147760 T^{3} - 2482607356860 T^{4} + 1889298883532485 T^{5} - 272522469312909914 T^{6} -$$$$74\!\cdots\!60$$$$T^{7} +$$$$81\!\cdots\!89$$$$T^{8} -$$$$36\!\cdots\!40$$$$T^{9} -$$$$66\!\cdots\!94$$$$T^{10} +$$$$22\!\cdots\!15$$$$T^{11} -$$$$14\!\cdots\!60$$$$T^{12} +$$$$13\!\cdots\!40$$$$T^{13} +$$$$35\!\cdots\!49$$$$T^{14} -$$$$19\!\cdots\!95$$$$T^{15} +$$$$34\!\cdots\!81$$$$T^{16}$$
$83$ $$1 + 2553 T + 3276861 T^{2} + 3557232930 T^{3} + 3857530610346 T^{4} + 3425756556730389 T^{5} + 2523855999787432520 T^{6} +$$$$20\!\cdots\!94$$$$T^{7} +$$$$16\!\cdots\!37$$$$T^{8} +$$$$11\!\cdots\!78$$$$T^{9} +$$$$82\!\cdots\!80$$$$T^{10} +$$$$64\!\cdots\!67$$$$T^{11} +$$$$41\!\cdots\!06$$$$T^{12} +$$$$21\!\cdots\!10$$$$T^{13} +$$$$11\!\cdots\!49$$$$T^{14} +$$$$51\!\cdots\!99$$$$T^{15} +$$$$11\!\cdots\!21$$$$T^{16}$$
$89$ $$( 1 - 894 T + 1324211 T^{2} - 258930180 T^{3} + 627382282200 T^{4} - 182537750064420 T^{5} + 658108092284756771 T^{6} -$$$$31\!\cdots\!46$$$$T^{7} +$$$$24\!\cdots\!21$$$$T^{8} )^{2}$$
$97$ $$1 - 9 T - 1418094 T^{2} - 1082718536 T^{3} + 1552923605058 T^{4} + 351454828977301 T^{5} - 769417621790276381 T^{6} -$$$$34\!\cdots\!08$$$$T^{7} +$$$$13\!\cdots\!08$$$$T^{8} -$$$$31\!\cdots\!84$$$$T^{9} -$$$$64\!\cdots\!49$$$$T^{10} +$$$$26\!\cdots\!17$$$$T^{11} +$$$$10\!\cdots\!78$$$$T^{12} -$$$$68\!\cdots\!48$$$$T^{13} -$$$$81\!\cdots\!66$$$$T^{14} -$$$$47\!\cdots\!73$$$$T^{15} +$$$$48\!\cdots\!81$$$$T^{16}$$