Properties

Label 33.6.d.b
Level $33$
Weight $6$
Character orbit 33.d
Analytic conductor $5.293$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 33.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.29266605383\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{15} - 195 x^{14} - 642 x^{13} + 89670 x^{12} + 53946 x^{11} + 91115757 x^{10} - 2121785838 x^{9} + 37710373995 x^{8} - 835758339660 x^{7} + 12972600642204 x^{6} - 129499271268696 x^{5} + 2168293345395660 x^{4} - 17336133272224368 x^{3} + 169639595563975056 x^{2} - 1075523563426213440 x + 9241272870780234240\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{10} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( -3 - \beta_{2} ) q^{3} + ( 20 + \beta_{9} ) q^{4} + \beta_{8} q^{5} + ( -3 \beta_{3} + \beta_{11} ) q^{6} -\beta_{13} q^{7} + ( 14 \beta_{3} - \beta_{14} ) q^{8} + ( -15 + 2 \beta_{2} + \beta_{6} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{2} + ( -3 - \beta_{2} ) q^{3} + ( 20 + \beta_{9} ) q^{4} + \beta_{8} q^{5} + ( -3 \beta_{3} + \beta_{11} ) q^{6} -\beta_{13} q^{7} + ( 14 \beta_{3} - \beta_{14} ) q^{8} + ( -15 + 2 \beta_{2} + \beta_{6} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{9} + ( \beta_{5} + \beta_{11} + \beta_{13} ) q^{10} + ( 1 - \beta_{1} - \beta_{2} + 9 \beta_{3} + \beta_{4} - 2 \beta_{8} + 2 \beta_{10} - \beta_{14} ) q^{11} + ( -27 + 3 \beta_{1} - 18 \beta_{2} + 2 \beta_{6} + 5 \beta_{8} - 7 \beta_{9} + 5 \beta_{10} ) q^{12} + ( \beta_{7} - \beta_{11} - \beta_{13} ) q^{13} + ( -5 + 5 \beta_{1} + 2 \beta_{2} - 7 \beta_{8} - 13 \beta_{10} ) q^{14} + ( -103 + 7 \beta_{1} - 7 \beta_{2} - 9 \beta_{8} - 9 \beta_{9} + 3 \beta_{15} ) q^{15} + ( 98 - 10 \beta_{1} + 26 \beta_{2} - 2 \beta_{6} + 17 \beta_{9} - 2 \beta_{15} ) q^{16} + ( -42 \beta_{3} + 2 \beta_{4} - \beta_{6} + 3 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{17} + ( 9 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{11} + \beta_{12} - 6 \beta_{13} + \beta_{14} - \beta_{15} ) q^{18} + ( 2 \beta_{3} + 2 \beta_{5} - \beta_{7} - 3 \beta_{11} - 2 \beta_{12} ) q^{19} + ( 17 - 17 \beta_{1} - 18 \beta_{2} - 2 \beta_{6} + 49 \beta_{8} + 33 \beta_{10} + 2 \beta_{15} ) q^{20} + ( 56 \beta_{3} - 4 \beta_{4} + 2 \beta_{6} + \beta_{7} - 2 \beta_{11} + 3 \beta_{12} + 10 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{21} + ( 480 - 22 \beta_{1} + 52 \beta_{2} - 2 \beta_{3} - 3 \beta_{5} - 7 \beta_{6} - \beta_{7} + 33 \beta_{9} + 4 \beta_{11} + 2 \beta_{12} + 3 \beta_{13} - 7 \beta_{15} ) q^{22} + ( -14 + 14 \beta_{1} + 8 \beta_{2} + 3 \beta_{6} - 9 \beta_{8} - 34 \beta_{10} - 3 \beta_{15} ) q^{23} + ( -111 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 4 \beta_{11} - \beta_{12} + 12 \beta_{13} + 11 \beta_{14} - 2 \beta_{15} ) q^{24} + ( -56 + 35 \beta_{1} - 105 \beta_{2} - 60 \beta_{9} ) q^{25} + ( -23 + 23 \beta_{1} + 46 \beta_{2} - 18 \beta_{6} - 29 \beta_{8} - 23 \beta_{10} + 18 \beta_{15} ) q^{26} + ( -204 - 39 \beta_{1} + 60 \beta_{2} + 27 \beta_{9} + 54 \beta_{10} - 9 \beta_{15} ) q^{27} + ( 5 \beta_{3} - 7 \beta_{5} - 22 \beta_{11} - 5 \beta_{12} - 6 \beta_{13} ) q^{28} + ( 36 \beta_{3} + 8 \beta_{4} - 4 \beta_{6} - 9 \beta_{11} + 3 \beta_{12} + 3 \beta_{13} - 5 \beta_{14} + 4 \beta_{15} ) q^{29} + ( -323 \beta_{3} - 6 \beta_{4} - 6 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - 8 \beta_{11} - 10 \beta_{12} - 16 \beta_{13} + 15 \beta_{14} - 3 \beta_{15} ) q^{30} + ( -811 - 45 \beta_{1} + 161 \beta_{2} + 13 \beta_{6} + 2 \beta_{9} + 13 \beta_{15} ) q^{31} + ( 76 \beta_{3} + 8 \beta_{4} - 4 \beta_{6} - 30 \beta_{11} + 10 \beta_{12} + 10 \beta_{13} + 7 \beta_{14} + 4 \beta_{15} ) q^{32} + ( -338 - 28 \beta_{1} + 11 \beta_{2} - 195 \beta_{3} - 5 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - \beta_{8} - 22 \beta_{9} - 10 \beta_{10} + 3 \beta_{11} + 7 \beta_{12} - 6 \beta_{13} + 16 \beta_{14} - 19 \beta_{15} ) q^{33} + ( -2092 + 52 \beta_{1} - 160 \beta_{2} - 2 \beta_{6} - 124 \beta_{9} - 2 \beta_{15} ) q^{34} + ( 266 \beta_{3} + 10 \beta_{4} - 5 \beta_{6} + 6 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 36 \beta_{14} + 5 \beta_{15} ) q^{35} + ( 882 + 30 \beta_{1} - 88 \beta_{2} + 4 \beta_{6} - 170 \beta_{8} - 23 \beta_{9} - 62 \beta_{10} - 18 \beta_{15} ) q^{36} + ( 657 + 61 \beta_{1} - 129 \beta_{2} + 27 \beta_{6} + 38 \beta_{9} + 27 \beta_{15} ) q^{37} + ( -44 + 44 \beta_{1} + 148 \beta_{2} + 2 \beta_{6} + 124 \beta_{8} + 16 \beta_{10} - 2 \beta_{15} ) q^{38} + ( 364 \beta_{3} - 14 \beta_{4} + 18 \beta_{5} + 7 \beta_{6} - \beta_{7} + 5 \beta_{11} - 18 \beta_{12} - 25 \beta_{13} - 26 \beta_{14} - 7 \beta_{15} ) q^{39} + ( -13 \beta_{3} + 21 \beta_{5} + 4 \beta_{7} + 56 \beta_{11} + 13 \beta_{12} + 100 \beta_{13} ) q^{40} + ( -20 \beta_{3} + 16 \beta_{4} - 8 \beta_{6} + 93 \beta_{11} - 31 \beta_{12} - 31 \beta_{13} - 23 \beta_{14} + 8 \beta_{15} ) q^{41} + ( 2901 - 153 \beta_{1} + 110 \beta_{2} - 2 \beta_{6} + 67 \beta_{8} + 178 \beta_{9} + 121 \beta_{10} + 18 \beta_{15} ) q^{42} + ( -34 \beta_{3} - 10 \beta_{5} - 3 \beta_{7} + 95 \beta_{11} + 34 \beta_{12} - 54 \beta_{13} ) q^{43} + ( 51 - 51 \beta_{1} - 194 \beta_{2} + 1020 \beta_{3} - 4 \beta_{4} + 22 \beta_{6} - 69 \beta_{8} - 41 \beta_{10} - 66 \beta_{11} + 22 \beta_{12} + 22 \beta_{13} - 29 \beta_{14} - 22 \beta_{15} ) q^{44} + ( 4872 + 54 \beta_{1} + 74 \beta_{2} - 8 \beta_{6} + 43 \beta_{8} + 289 \beta_{9} - 38 \beta_{10} - 27 \beta_{15} ) q^{45} + ( 8 \beta_{3} - 15 \beta_{5} - 6 \beta_{7} - 33 \beta_{11} - 8 \beta_{12} - 91 \beta_{13} ) q^{46} + ( 21 - 21 \beta_{1} - 193 \beta_{2} + 16 \beta_{6} + 117 \beta_{8} - 130 \beta_{10} - 16 \beta_{15} ) q^{47} + ( -4923 - 21 \beta_{1} + 38 \beta_{2} + 6 \beta_{6} + 177 \beta_{8} - 237 \beta_{9} + 69 \beta_{10} + 18 \beta_{15} ) q^{48} + ( 141 + 140 \beta_{1} - 406 \beta_{2} + 7 \beta_{6} + 162 \beta_{9} + 7 \beta_{15} ) q^{49} + ( -1546 \beta_{3} + 105 \beta_{11} - 35 \beta_{12} - 35 \beta_{13} + 60 \beta_{14} ) q^{50} + ( 702 \beta_{3} - 4 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} + 7 \beta_{7} - 13 \beta_{11} + 20 \beta_{12} - 66 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{51} + ( 59 \beta_{3} + 7 \beta_{5} + 4 \beta_{7} - 174 \beta_{11} - 59 \beta_{12} - 66 \beta_{13} ) q^{52} + ( 125 - 125 \beta_{1} - 411 \beta_{2} - 53 \beta_{6} - 53 \beta_{8} - 36 \beta_{10} + 53 \beta_{15} ) q^{53} + ( 420 \beta_{3} + 18 \beta_{4} - 9 \beta_{5} - 9 \beta_{6} - 9 \beta_{7} + 12 \beta_{11} + 48 \beta_{12} + 147 \beta_{13} - 45 \beta_{14} + 9 \beta_{15} ) q^{54} + ( 7365 + 33 \beta_{1} - 67 \beta_{2} - 30 \beta_{3} + 10 \beta_{5} + 16 \beta_{6} + 7 \beta_{7} + 93 \beta_{11} + 30 \beta_{12} + 111 \beta_{13} + 16 \beta_{15} ) q^{55} + ( -169 + 169 \beta_{1} + 620 \beta_{2} - 16 \beta_{6} - 451 \beta_{8} + 113 \beta_{10} + 16 \beta_{15} ) q^{56} + ( 450 \beta_{3} + 10 \beta_{4} - 12 \beta_{5} - 5 \beta_{6} + 14 \beta_{7} - 11 \beta_{11} - 5 \beta_{12} + 39 \beta_{13} + 49 \beta_{14} + 5 \beta_{15} ) q^{57} + ( 1644 - 344 \beta_{1} + 896 \beta_{2} - 68 \beta_{6} + 198 \beta_{9} - 68 \beta_{15} ) q^{58} + ( 119 - 119 \beta_{1} - 81 \beta_{2} + 72 \beta_{6} + 214 \beta_{8} + 276 \beta_{10} - 72 \beta_{15} ) q^{59} + ( -13795 + 727 \beta_{1} - 492 \beta_{2} + 20 \beta_{6} - 409 \beta_{8} - 628 \beta_{9} - 373 \beta_{10} + 66 \beta_{15} ) q^{60} + ( 56 \beta_{3} + 16 \beta_{5} + 3 \beta_{7} - 155 \beta_{11} - 56 \beta_{12} + 33 \beta_{13} ) q^{61} + ( -875 \beta_{3} - 52 \beta_{4} + 26 \beta_{6} - 135 \beta_{11} + 45 \beta_{12} + 45 \beta_{13} + 50 \beta_{14} - 26 \beta_{15} ) q^{62} + ( -2040 \beta_{3} + 20 \beta_{4} + 12 \beta_{5} - 10 \beta_{6} - 26 \beta_{7} + 26 \beta_{11} - 76 \beta_{12} - 15 \beta_{13} - 28 \beta_{14} + 10 \beta_{15} ) q^{63} + ( -218 - 366 \beta_{1} + 1054 \beta_{2} - 22 \beta_{6} - 551 \beta_{9} - 22 \beta_{15} ) q^{64} + ( 1882 \beta_{3} - 38 \beta_{4} + 19 \beta_{6} - 246 \beta_{11} + 82 \beta_{12} + 82 \beta_{13} + 44 \beta_{14} - 19 \beta_{15} ) q^{65} + ( -10365 - 153 \beta_{1} - 226 \beta_{2} - 967 \beta_{3} + 32 \beta_{4} - 18 \beta_{5} - 9 \beta_{6} - 17 \beta_{7} + 409 \beta_{8} - 605 \beta_{9} + 31 \beta_{10} + 13 \beta_{11} + 45 \beta_{12} + 7 \beta_{13} - 10 \beta_{14} + 79 \beta_{15} ) q^{66} + ( -22713 + 137 \beta_{1} - 505 \beta_{2} - 47 \beta_{6} - 598 \beta_{9} - 47 \beta_{15} ) q^{67} + ( -3864 \beta_{3} - 56 \beta_{4} + 28 \beta_{6} + 60 \beta_{11} - 20 \beta_{12} - 20 \beta_{13} + 84 \beta_{14} - 28 \beta_{15} ) q^{68} + ( 7163 - 515 \beta_{1} + 247 \beta_{2} - 11 \beta_{6} - 5 \beta_{8} + 241 \beta_{9} + 382 \beta_{10} + 48 \beta_{15} ) q^{69} + ( 14408 - 348 \beta_{1} + 824 \beta_{2} - 110 \beta_{6} + 1366 \beta_{9} - 110 \beta_{15} ) q^{70} + ( -646 + 646 \beta_{1} + 1988 \beta_{2} - 37 \beta_{6} + 105 \beta_{8} + 50 \beta_{10} + 37 \beta_{15} ) q^{71} + ( 48 \beta_{3} + 92 \beta_{4} - 96 \beta_{5} - 46 \beta_{6} + 10 \beta_{7} - 28 \beta_{11} - 40 \beta_{12} - 132 \beta_{13} - 37 \beta_{14} + 46 \beta_{15} ) q^{72} + ( -16 \beta_{3} - 104 \beta_{5} - 36 \beta_{7} - 20 \beta_{11} + 16 \beta_{12} - 152 \beta_{13} ) q^{73} + ( 1713 \beta_{3} - 108 \beta_{4} + 54 \beta_{6} + 183 \beta_{11} - 61 \beta_{12} - 61 \beta_{13} + 70 \beta_{14} - 54 \beta_{15} ) q^{74} + ( 20133 - 180 \beta_{1} - 519 \beta_{2} - 15 \beta_{6} - 510 \beta_{8} + 525 \beta_{9} - 510 \beta_{10} ) q^{75} + ( -24 \beta_{3} + 56 \beta_{5} + 28 \beta_{7} + 100 \beta_{11} + 24 \beta_{12} + 112 \beta_{13} ) q^{76} + ( 283 - 283 \beta_{1} - 1053 \beta_{2} + 1084 \beta_{3} + 8 \beta_{4} - 99 \beta_{6} - 687 \beta_{8} - 204 \beta_{10} - 33 \beta_{11} + 11 \beta_{12} + 11 \beta_{13} + 91 \beta_{14} + 99 \beta_{15} ) q^{77} + ( 18879 + 693 \beta_{1} + 46 \beta_{2} - 28 \beta_{6} + 965 \beta_{8} + 1250 \beta_{9} - 61 \beta_{10} + 144 \beta_{15} ) q^{78} + ( -8 \beta_{3} + 96 \beta_{5} + 4 \beta_{7} + 116 \beta_{11} + 8 \beta_{12} + 239 \beta_{13} ) q^{79} + ( 695 - 695 \beta_{1} - 1240 \beta_{2} + 28 \beta_{6} + 897 \beta_{8} + 845 \beta_{10} - 28 \beta_{15} ) q^{80} + ( -16218 + 81 \beta_{1} + 267 \beta_{2} - 15 \beta_{6} + 192 \beta_{8} - 1149 \beta_{9} - 294 \beta_{10} - 162 \beta_{15} ) q^{81} + ( 2260 + 1624 \beta_{1} - 4752 \beta_{2} + 60 \beta_{6} - 166 \beta_{9} + 60 \beta_{15} ) q^{82} + ( 712 \beta_{3} - 48 \beta_{4} + 24 \beta_{6} + 90 \beta_{11} - 30 \beta_{12} - 30 \beta_{13} - 126 \beta_{14} - 24 \beta_{15} ) q^{83} + ( 5435 \beta_{3} + 96 \beta_{4} + 87 \beta_{5} - 48 \beta_{6} - 12 \beta_{7} + 98 \beta_{11} + 37 \beta_{12} + 142 \beta_{13} - 114 \beta_{14} + 48 \beta_{15} ) q^{84} + ( 8 \beta_{3} - 80 \beta_{5} + 20 \beta_{7} - 124 \beta_{11} - 8 \beta_{12} - 144 \beta_{13} ) q^{85} + ( 1126 - 1126 \beta_{1} - 3672 \beta_{2} + 278 \beta_{6} - 270 \beta_{8} - 294 \beta_{10} - 278 \beta_{15} ) q^{86} + ( -3360 \beta_{3} - 10 \beta_{4} + 66 \beta_{5} + 5 \beta_{6} + 31 \beta_{7} - 15 \beta_{11} + 50 \beta_{12} - 30 \beta_{13} + 86 \beta_{14} - 5 \beta_{15} ) q^{87} + ( 35806 - 990 \beta_{1} + 3078 \beta_{2} - 27 \beta_{3} - 13 \beta_{5} + 54 \beta_{6} - 8 \beta_{7} + 1573 \beta_{9} + 76 \beta_{11} + 27 \beta_{12} - 196 \beta_{13} + 54 \beta_{15} ) q^{88} + ( -669 + 669 \beta_{1} + 2083 \beta_{2} + 101 \beta_{6} + 308 \beta_{8} + 76 \beta_{10} - 101 \beta_{15} ) q^{89} + ( 12510 \beta_{3} + 70 \beta_{4} + 24 \beta_{5} - 35 \beta_{6} - 19 \beta_{7} - 47 \beta_{11} - 35 \beta_{12} - 87 \beta_{13} - 359 \beta_{14} + 35 \beta_{15} ) q^{90} + ( -13782 - 328 \beta_{1} + 1642 \beta_{2} + 329 \beta_{6} - 534 \beta_{9} + 329 \beta_{15} ) q^{91} + ( -423 + 423 \beta_{1} + 798 \beta_{2} - 6 \beta_{6} - 1511 \beta_{8} - 471 \beta_{10} + 6 \beta_{15} ) q^{92} + ( -32736 + 474 \beta_{1} + 1582 \beta_{2} - 144 \beta_{6} + 72 \beta_{8} + 423 \beta_{9} + 774 \beta_{10} - 117 \beta_{15} ) q^{93} + ( -53 \beta_{3} + 85 \beta_{5} - 32 \beta_{7} + 276 \beta_{11} + 53 \beta_{12} - 122 \beta_{13} ) q^{94} + ( -7024 \beta_{3} + 120 \beta_{4} - 60 \beta_{6} + 276 \beta_{11} - 92 \beta_{12} - 92 \beta_{13} - 140 \beta_{14} + 60 \beta_{15} ) q^{95} + ( -7587 \beta_{3} + 80 \beta_{4} + 93 \beta_{5} - 40 \beta_{6} + 76 \beta_{7} + 68 \beta_{11} + 41 \beta_{12} - 48 \beta_{13} - 67 \beta_{14} + 40 \beta_{15} ) q^{96} + ( 6111 - 679 \beta_{1} + 1929 \beta_{2} - 54 \beta_{6} - 2232 \beta_{9} - 54 \beta_{15} ) q^{97} + ( 4619 \beta_{3} - 28 \beta_{4} + 14 \beta_{6} + 420 \beta_{11} - 140 \beta_{12} - 140 \beta_{13} - 134 \beta_{14} - 14 \beta_{15} ) q^{98} + ( -15783 - 951 \beta_{1} + 661 \beta_{2} - 3009 \beta_{3} - 61 \beta_{4} - 60 \beta_{5} - 41 \beta_{6} + 46 \beta_{7} - 208 \beta_{8} - 517 \beta_{9} - 100 \beta_{10} - 184 \beta_{11} - 70 \beta_{12} - 72 \beta_{13} - 181 \beta_{14} - 8 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 54q^{3} + 316q^{4} - 222q^{9} + O(q^{10}) \) \( 16q - 54q^{3} + 316q^{4} - 222q^{9} - 552q^{12} - 1674q^{15} + 1684q^{16} + 7932q^{22} - 1356q^{25} - 3240q^{27} - 11980q^{31} - 5106q^{33} - 34032q^{34} + 14016q^{36} + 9356q^{37} + 45912q^{42} + 77430q^{45} - 78012q^{48} - 1136q^{49} + 117308q^{55} + 31848q^{58} - 220548q^{60} + 5860q^{64} - 164796q^{66} - 364132q^{67} + 113790q^{69} + 231144q^{70} + 320364q^{75} + 296088q^{78} - 251334q^{81} + 4824q^{82} + 586836q^{88} - 209184q^{91} - 521046q^{93} + 119852q^{97} - 243894q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 6 x^{15} - 195 x^{14} - 642 x^{13} + 89670 x^{12} + 53946 x^{11} + 91115757 x^{10} - 2121785838 x^{9} + 37710373995 x^{8} - 835758339660 x^{7} + 12972600642204 x^{6} - 129499271268696 x^{5} + 2168293345395660 x^{4} - 17336133272224368 x^{3} + 169639595563975056 x^{2} - 1075523563426213440 x + 9241272870780234240\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(15\!\cdots\!79\)\( \nu^{15} - \)\(97\!\cdots\!52\)\( \nu^{14} + \)\(37\!\cdots\!11\)\( \nu^{13} - \)\(38\!\cdots\!96\)\( \nu^{12} - \)\(86\!\cdots\!60\)\( \nu^{11} - \)\(33\!\cdots\!90\)\( \nu^{10} + \)\(17\!\cdots\!09\)\( \nu^{9} - \)\(86\!\cdots\!04\)\( \nu^{8} + \)\(39\!\cdots\!97\)\( \nu^{7} - \)\(90\!\cdots\!34\)\( \nu^{6} + \)\(16\!\cdots\!94\)\( \nu^{5} - \)\(28\!\cdots\!52\)\( \nu^{4} + \)\(35\!\cdots\!56\)\( \nu^{3} - \)\(37\!\cdots\!52\)\( \nu^{2} + \)\(35\!\cdots\!80\)\( \nu - \)\(29\!\cdots\!92\)\(\)\()/ \)\(36\!\cdots\!44\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(11\!\cdots\!21\)\( \nu^{15} + \)\(41\!\cdots\!52\)\( \nu^{14} - \)\(51\!\cdots\!85\)\( \nu^{13} - \)\(87\!\cdots\!36\)\( \nu^{12} + \)\(16\!\cdots\!96\)\( \nu^{11} + \)\(69\!\cdots\!66\)\( \nu^{10} + \)\(77\!\cdots\!69\)\( \nu^{9} - \)\(13\!\cdots\!52\)\( \nu^{8} + \)\(31\!\cdots\!45\)\( \nu^{7} - \)\(19\!\cdots\!02\)\( \nu^{6} - \)\(14\!\cdots\!42\)\( \nu^{5} + \)\(12\!\cdots\!64\)\( \nu^{4} - \)\(97\!\cdots\!72\)\( \nu^{3} + \)\(13\!\cdots\!20\)\( \nu^{2} - \)\(17\!\cdots\!80\)\( \nu + \)\(16\!\cdots\!20\)\(\)\()/ \)\(11\!\cdots\!32\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(11\!\cdots\!21\)\( \nu^{15} - \)\(41\!\cdots\!52\)\( \nu^{14} + \)\(51\!\cdots\!85\)\( \nu^{13} + \)\(87\!\cdots\!36\)\( \nu^{12} - \)\(16\!\cdots\!96\)\( \nu^{11} - \)\(69\!\cdots\!66\)\( \nu^{10} - \)\(77\!\cdots\!69\)\( \nu^{9} + \)\(13\!\cdots\!52\)\( \nu^{8} - \)\(31\!\cdots\!45\)\( \nu^{7} + \)\(19\!\cdots\!02\)\( \nu^{6} + \)\(14\!\cdots\!42\)\( \nu^{5} - \)\(12\!\cdots\!64\)\( \nu^{4} + \)\(97\!\cdots\!72\)\( \nu^{3} - \)\(13\!\cdots\!20\)\( \nu^{2} + \)\(12\!\cdots\!12\)\( \nu - \)\(16\!\cdots\!20\)\(\)\()/ \)\(11\!\cdots\!32\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(32\!\cdots\!69\)\( \nu^{15} + \)\(14\!\cdots\!78\)\( \nu^{14} + \)\(41\!\cdots\!09\)\( \nu^{13} + \)\(16\!\cdots\!70\)\( \nu^{12} - \)\(20\!\cdots\!64\)\( \nu^{11} - \)\(60\!\cdots\!74\)\( \nu^{10} + \)\(39\!\cdots\!55\)\( \nu^{9} + \)\(28\!\cdots\!90\)\( \nu^{8} - \)\(13\!\cdots\!33\)\( \nu^{7} + \)\(21\!\cdots\!88\)\( \nu^{6} - \)\(12\!\cdots\!98\)\( \nu^{5} + \)\(22\!\cdots\!28\)\( \nu^{4} - \)\(26\!\cdots\!44\)\( \nu^{3} + \)\(57\!\cdots\!12\)\( \nu^{2} - \)\(21\!\cdots\!20\)\( \nu + \)\(33\!\cdots\!64\)\(\)\()/ \)\(19\!\cdots\!76\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(67\!\cdots\!86\)\( \nu^{15} - \)\(31\!\cdots\!77\)\( \nu^{14} - \)\(23\!\cdots\!28\)\( \nu^{13} + \)\(98\!\cdots\!17\)\( \nu^{12} + \)\(64\!\cdots\!74\)\( \nu^{11} - \)\(36\!\cdots\!08\)\( \nu^{10} - \)\(11\!\cdots\!72\)\( \nu^{9} - \)\(25\!\cdots\!29\)\( \nu^{8} + \)\(51\!\cdots\!00\)\( \nu^{7} - \)\(63\!\cdots\!81\)\( \nu^{6} + \)\(15\!\cdots\!04\)\( \nu^{5} - \)\(20\!\cdots\!70\)\( \nu^{4} + \)\(10\!\cdots\!24\)\( \nu^{3} - \)\(32\!\cdots\!68\)\( \nu^{2} + \)\(19\!\cdots\!36\)\( \nu - \)\(14\!\cdots\!80\)\(\)\()/ \)\(33\!\cdots\!96\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(11\!\cdots\!67\)\( \nu^{15} - \)\(24\!\cdots\!58\)\( \nu^{14} + \)\(14\!\cdots\!87\)\( \nu^{13} + \)\(53\!\cdots\!06\)\( \nu^{12} - \)\(42\!\cdots\!44\)\( \nu^{11} - \)\(84\!\cdots\!90\)\( \nu^{10} - \)\(51\!\cdots\!19\)\( \nu^{9} - \)\(10\!\cdots\!34\)\( \nu^{8} + \)\(12\!\cdots\!69\)\( \nu^{7} - \)\(22\!\cdots\!40\)\( \nu^{6} + \)\(37\!\cdots\!90\)\( \nu^{5} - \)\(73\!\cdots\!40\)\( \nu^{4} + \)\(58\!\cdots\!84\)\( \nu^{3} + \)\(10\!\cdots\!72\)\( \nu^{2} + \)\(61\!\cdots\!20\)\( \nu + \)\(30\!\cdots\!36\)\(\)\()/ \)\(49\!\cdots\!44\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(15\!\cdots\!23\)\( \nu^{15} + \)\(74\!\cdots\!62\)\( \nu^{14} - \)\(33\!\cdots\!53\)\( \nu^{13} - \)\(21\!\cdots\!68\)\( \nu^{12} - \)\(74\!\cdots\!24\)\( \nu^{11} + \)\(68\!\cdots\!56\)\( \nu^{10} - \)\(34\!\cdots\!87\)\( \nu^{9} + \)\(67\!\cdots\!82\)\( \nu^{8} - \)\(11\!\cdots\!87\)\( \nu^{7} + \)\(15\!\cdots\!18\)\( \nu^{6} - \)\(39\!\cdots\!22\)\( \nu^{5} + \)\(40\!\cdots\!94\)\( \nu^{4} - \)\(98\!\cdots\!24\)\( \nu^{3} + \)\(48\!\cdots\!16\)\( \nu^{2} + \)\(17\!\cdots\!52\)\( \nu + \)\(23\!\cdots\!20\)\(\)\()/ \)\(49\!\cdots\!44\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(17\!\cdots\!31\)\( \nu^{15} + \)\(57\!\cdots\!70\)\( \nu^{14} + \)\(23\!\cdots\!71\)\( \nu^{13} - \)\(14\!\cdots\!46\)\( \nu^{12} - \)\(17\!\cdots\!72\)\( \nu^{11} + \)\(55\!\cdots\!18\)\( \nu^{10} - \)\(14\!\cdots\!27\)\( \nu^{9} + \)\(75\!\cdots\!38\)\( \nu^{8} - \)\(14\!\cdots\!31\)\( \nu^{7} + \)\(24\!\cdots\!12\)\( \nu^{6} - \)\(43\!\cdots\!58\)\( \nu^{5} + \)\(51\!\cdots\!24\)\( \nu^{4} - \)\(39\!\cdots\!52\)\( \nu^{3} + \)\(58\!\cdots\!48\)\( \nu^{2} - \)\(33\!\cdots\!40\)\( \nu + \)\(16\!\cdots\!28\)\(\)\()/ \)\(29\!\cdots\!64\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(10\!\cdots\!42\)\( \nu^{15} + \)\(11\!\cdots\!91\)\( \nu^{14} - \)\(37\!\cdots\!54\)\( \nu^{13} - \)\(18\!\cdots\!57\)\( \nu^{12} + \)\(14\!\cdots\!54\)\( \nu^{11} + \)\(80\!\cdots\!77\)\( \nu^{10} + \)\(73\!\cdots\!06\)\( \nu^{9} - \)\(16\!\cdots\!92\)\( \nu^{8} + \)\(16\!\cdots\!16\)\( \nu^{7} - \)\(41\!\cdots\!12\)\( \nu^{6} + \)\(63\!\cdots\!56\)\( \nu^{5} - \)\(10\!\cdots\!04\)\( \nu^{4} + \)\(77\!\cdots\!44\)\( \nu^{3} - \)\(48\!\cdots\!88\)\( \nu^{2} - \)\(71\!\cdots\!80\)\( \nu + \)\(15\!\cdots\!72\)\(\)\()/ \)\(16\!\cdots\!99\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(10\!\cdots\!55\)\( \nu^{15} + \)\(53\!\cdots\!89\)\( \nu^{14} + \)\(26\!\cdots\!19\)\( \nu^{13} - \)\(11\!\cdots\!47\)\( \nu^{12} - \)\(10\!\cdots\!72\)\( \nu^{11} + \)\(57\!\cdots\!52\)\( \nu^{10} - \)\(94\!\cdots\!49\)\( \nu^{9} + \)\(18\!\cdots\!09\)\( \nu^{8} - \)\(34\!\cdots\!79\)\( \nu^{7} + \)\(65\!\cdots\!89\)\( \nu^{6} - \)\(87\!\cdots\!32\)\( \nu^{5} + \)\(68\!\cdots\!32\)\( \nu^{4} - \)\(14\!\cdots\!52\)\( \nu^{3} + \)\(33\!\cdots\!16\)\( \nu^{2} - \)\(12\!\cdots\!00\)\( \nu + \)\(87\!\cdots\!24\)\(\)\()/ \)\(14\!\cdots\!32\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(89\!\cdots\!17\)\( \nu^{15} - \)\(14\!\cdots\!01\)\( \nu^{14} - \)\(13\!\cdots\!13\)\( \nu^{13} + \)\(25\!\cdots\!25\)\( \nu^{12} + \)\(79\!\cdots\!36\)\( \nu^{11} - \)\(10\!\cdots\!46\)\( \nu^{10} + \)\(78\!\cdots\!27\)\( \nu^{9} - \)\(25\!\cdots\!53\)\( \nu^{8} + \)\(52\!\cdots\!73\)\( \nu^{7} - \)\(95\!\cdots\!71\)\( \nu^{6} + \)\(15\!\cdots\!44\)\( \nu^{5} - \)\(17\!\cdots\!02\)\( \nu^{4} + \)\(19\!\cdots\!20\)\( \nu^{3} - \)\(17\!\cdots\!36\)\( \nu^{2} + \)\(14\!\cdots\!60\)\( \nu - \)\(19\!\cdots\!40\)\(\)\()/ \)\(55\!\cdots\!16\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(17\!\cdots\!65\)\( \nu^{15} - \)\(70\!\cdots\!97\)\( \nu^{14} - \)\(15\!\cdots\!91\)\( \nu^{13} + \)\(19\!\cdots\!81\)\( \nu^{12} + \)\(12\!\cdots\!42\)\( \nu^{11} - \)\(70\!\cdots\!98\)\( \nu^{10} + \)\(16\!\cdots\!03\)\( \nu^{9} - \)\(83\!\cdots\!93\)\( \nu^{8} + \)\(17\!\cdots\!43\)\( \nu^{7} - \)\(27\!\cdots\!35\)\( \nu^{6} + \)\(45\!\cdots\!42\)\( \nu^{5} - \)\(51\!\cdots\!18\)\( \nu^{4} + \)\(38\!\cdots\!68\)\( \nu^{3} - \)\(44\!\cdots\!36\)\( \nu^{2} + \)\(29\!\cdots\!72\)\( \nu - \)\(30\!\cdots\!60\)\(\)\()/ \)\(99\!\cdots\!88\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(49\!\cdots\!76\)\( \nu^{15} - \)\(29\!\cdots\!33\)\( \nu^{14} - \)\(93\!\cdots\!54\)\( \nu^{13} + \)\(11\!\cdots\!81\)\( \nu^{12} + \)\(37\!\cdots\!54\)\( \nu^{11} - \)\(14\!\cdots\!08\)\( \nu^{10} + \)\(46\!\cdots\!26\)\( \nu^{9} - \)\(99\!\cdots\!97\)\( \nu^{8} + \)\(18\!\cdots\!06\)\( \nu^{7} - \)\(38\!\cdots\!25\)\( \nu^{6} + \)\(54\!\cdots\!04\)\( \nu^{5} - \)\(54\!\cdots\!98\)\( \nu^{4} + \)\(79\!\cdots\!20\)\( \nu^{3} - \)\(45\!\cdots\!40\)\( \nu^{2} + \)\(64\!\cdots\!16\)\( \nu - \)\(18\!\cdots\!60\)\(\)\()/ \)\(23\!\cdots\!56\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(54\!\cdots\!65\)\( \nu^{15} - \)\(69\!\cdots\!76\)\( \nu^{14} - \)\(19\!\cdots\!65\)\( \nu^{13} - \)\(10\!\cdots\!20\)\( \nu^{12} + \)\(67\!\cdots\!64\)\( \nu^{11} + \)\(32\!\cdots\!50\)\( \nu^{10} + \)\(41\!\cdots\!81\)\( \nu^{9} - \)\(92\!\cdots\!56\)\( \nu^{8} + \)\(85\!\cdots\!41\)\( \nu^{7} - \)\(26\!\cdots\!66\)\( \nu^{6} + \)\(35\!\cdots\!58\)\( \nu^{5} - \)\(45\!\cdots\!04\)\( \nu^{4} + \)\(48\!\cdots\!32\)\( \nu^{3} - \)\(30\!\cdots\!64\)\( \nu^{2} - \)\(82\!\cdots\!44\)\( \nu - \)\(32\!\cdots\!20\)\(\)\()/ \)\(18\!\cdots\!72\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(73\!\cdots\!99\)\( \nu^{15} + \)\(33\!\cdots\!31\)\( \nu^{14} + \)\(16\!\cdots\!11\)\( \nu^{13} + \)\(22\!\cdots\!23\)\( \nu^{12} - \)\(57\!\cdots\!80\)\( \nu^{11} - \)\(15\!\cdots\!08\)\( \nu^{10} - \)\(70\!\cdots\!01\)\( \nu^{9} + \)\(14\!\cdots\!55\)\( \nu^{8} - \)\(23\!\cdots\!47\)\( \nu^{7} + \)\(51\!\cdots\!55\)\( \nu^{6} - \)\(69\!\cdots\!00\)\( \nu^{5} + \)\(39\!\cdots\!20\)\( \nu^{4} - \)\(87\!\cdots\!80\)\( \nu^{3} + \)\(28\!\cdots\!48\)\( \nu^{2} - \)\(16\!\cdots\!60\)\( \nu + \)\(73\!\cdots\!52\)\(\)\()/ \)\(24\!\cdots\!72\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{3} + \beta_{2}\)
\(\nu^{2}\)\(=\)\(-2 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + \beta_{6} - 4 \beta_{2} + 28\)
\(\nu^{3}\)\(=\)\(6 \beta_{15} + 2 \beta_{14} - 18 \beta_{13} + 3 \beta_{12} + 12 \beta_{11} - 51 \beta_{10} - 24 \beta_{9} + 3 \beta_{8} - 3 \beta_{7} - 12 \beta_{6} - 9 \beta_{5} - 6 \beta_{4} + 78 \beta_{3} + 99 \beta_{2} + 30 \beta_{1} + 294\)
\(\nu^{4}\)\(=\)\(-372 \beta_{15} + 112 \beta_{14} - 420 \beta_{13} - 224 \beta_{12} - 356 \beta_{11} - 330 \beta_{10} - 802 \beta_{9} - 1140 \beta_{8} + 80 \beta_{7} + 317 \beta_{6} + 132 \beta_{5} + 16 \beta_{4} - 1512 \beta_{3} - 931 \beta_{2} - 109 \beta_{1} - 14861\)
\(\nu^{5}\)\(=\)\(10546 \beta_{15} + 2264 \beta_{14} + 385 \beta_{13} + 3670 \beta_{12} + 2685 \beta_{11} - 8331 \beta_{10} - 6777 \beta_{9} + 11703 \beta_{8} - 2205 \beta_{7} - 3157 \beta_{6} - 3825 \beta_{5} + 1718 \beta_{4} - 201017 \beta_{3} - 12798 \beta_{2} + 48801 \beta_{1} - 46695\)
\(\nu^{6}\)\(=\)\(-216177 \beta_{15} + 28482 \beta_{14} - 103968 \beta_{13} - 175044 \beta_{12} + 159864 \beta_{11} - 16848 \beta_{10} - 723825 \beta_{9} - 181116 \beta_{8} + 19110 \beta_{7} + 82761 \beta_{6} + 66456 \beta_{5} - 65820 \beta_{4} + 1954422 \beta_{3} + 410349 \beta_{2} - 805513 \beta_{1} - 40028783\)
\(\nu^{7}\)\(=\)\(3944952 \beta_{15} + 377721 \beta_{14} + 2513083 \beta_{13} + 3029053 \beta_{12} - 3622161 \beta_{11} + 2633169 \beta_{10} + 3119583 \beta_{9} + 7272363 \beta_{8} - 128142 \beta_{7} + 1485453 \beta_{6} + 61614 \beta_{5} + 1080936 \beta_{4} - 176608291 \beta_{3} - 64789293 \beta_{2} + 25966488 \beta_{1} + 633552018\)
\(\nu^{8}\)\(=\)\(-41284029 \beta_{15} + 5620056 \beta_{14} - 29124024 \beta_{13} - 45453096 \beta_{12} + 285952776 \beta_{11} + 73430730 \beta_{10} - 266860902 \beta_{9} + 218626308 \beta_{8} - 8146824 \beta_{7} - 80221050 \beta_{6} - 2403072 \beta_{5} - 20758800 \beta_{4} + 3005998944 \beta_{3} + 1504750398 \beta_{2} - 492086154 \beta_{1} - 23981003268\)
\(\nu^{9}\)\(=\)\(-596148018 \beta_{15} - 293960496 \beta_{14} + 2183715858 \beta_{13} + 310073349 \beta_{12} - 5993803422 \beta_{11} + 2392866063 \beta_{10} + 5813350506 \beta_{9} - 1758363201 \beta_{8} + 445788117 \beta_{7} + 2842254150 \beta_{6} + 1337519385 \beta_{5} + 651650334 \beta_{4} - 63427666272 \beta_{3} - 47927087181 \beta_{2} + 4092536340 \beta_{1} + 459497229300\)
\(\nu^{10}\)\(=\)\(44698815648 \beta_{15} - 9499750164 \beta_{14} + 1636782426 \beta_{13} + 25378702308 \beta_{12} + 160693447338 \beta_{11} + 18933574878 \beta_{10} - 5286503178 \beta_{9} + 220737216744 \beta_{8} - 16916490684 \beta_{7} - 92426402205 \beta_{6} - 33353480214 \beta_{5} - 8052235344 \beta_{4} + 1240630541100 \beta_{3} + 1027904065821 \beta_{2} - 20764834941 \beta_{1} - 5057485863405\)
\(\nu^{11}\)\(=\)\(-2073828990276 \beta_{15} - 235257555618 \beta_{14} + 48686983167 \beta_{13} - 963707954640 \beta_{12} - 3008734006437 \beta_{11} + 807152400009 \beta_{10} + 2205546013143 \beta_{9} - 4646100346209 \beta_{8} + 500104823427 \beta_{7} + 1977792296985 \beta_{6} + 1097399827755 \beta_{5} + 96087488742 \beta_{4} + 7682089413381 \beta_{3} - 14725734810084 \beta_{2} - 5813813008557 \beta_{1} + 62609026750263\)
\(\nu^{12}\)\(=\)\(59813051150205 \beta_{15} - 5027460218316 \beta_{14} + 14118283742760 \beta_{13} + 37146417204600 \beta_{12} + 27891054717336 \beta_{11} - 18955491880680 \beta_{10} + 73907117637039 \beta_{9} + 104904101066040 \beta_{8} - 11544881827140 \beta_{7} - 45279327964779 \beta_{6} - 26315618626008 \beta_{5} + 5654337370632 \beta_{4} - 353727165393684 \beta_{3} + 169379297370747 \beta_{2} + 194428963129653 \beta_{1} + 4167859515901695\)
\(\nu^{13}\)\(=\)\(-1556967664353204 \beta_{15} - 53168082834627 \beta_{14} - 664617308228793 \beta_{13} - 983911535888265 \beta_{12} + 48723963283467 \beta_{11} - 14444084522151 \beta_{10} - 1310512854032343 \beta_{9} - 2708039151040545 \beta_{8} + 223316627083992 \beta_{7} + 510089327298603 \beta_{6} + 429143141602656 \beta_{5} - 221435167890564 \beta_{4} + 28898083059034257 \beta_{3} + 5720529909880899 \beta_{2} - 6516808123196700 \beta_{1} - 145416069337449366\)
\(\nu^{14}\)\(=\)\(31637433098282679 \beta_{15} - 688692625062714 \beta_{14} + 13791150478491750 \beta_{13} + 21947450374777914 \beta_{12} - 37204422624941850 \beta_{11} - 10678110838130826 \beta_{10} + 63535946115887424 \beta_{9} + 13304557659416832 \beta_{8} - 2631242503600074 \beta_{7} + 557081961396030 \beta_{6} - 7252437236043456 \beta_{5} + 7682515014987036 \beta_{4} - 841552705567223676 \beta_{3} - 282483489401752752 \beta_{2} + 171776955973216896 \beta_{1} + 5430739871867089626\)
\(\nu^{15}\)\(=\)\(-453282050778304968 \beta_{15} + 33423554431424268 \beta_{14} - 503851590016638660 \beta_{13} - 419697395738549595 \beta_{12} + 1346803538397806916 \beta_{11} - 188582413777670637 \beta_{10} - 1769632123135709736 \beta_{9} - 118294972797777849 \beta_{8} - 15053375028933837 \beta_{7} - 389447080581470598 \beta_{6} - 60555859459657245 \beta_{5} - 208377343245368214 \beta_{4} + 22115742150613485402 \beta_{3} + 10736398536305877945 \beta_{2} - 3362465158130546442 \beta_{1} - 148995909354352328370\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
−2.15316 + 11.0840i
−2.15316 11.0840i
−23.0756 + 10.4175i
−23.0756 10.4175i
6.80653 + 6.27606i
6.80653 6.27606i
−6.72395 + 15.5879i
−6.72395 15.5879i
0.456001 + 15.5879i
0.456001 15.5879i
15.7319 + 6.27606i
15.7319 6.27606i
−6.11708 + 10.4175i
−6.11708 10.4175i
18.0754 + 11.0840i
18.0754 11.0840i
−10.1143 −10.9611 11.0840i 70.2982 88.5196i 110.863 + 112.106i 126.220i −387.358 −2.70883 + 242.985i 895.310i
32.2 −10.1143 −10.9611 + 11.0840i 70.2982 88.5196i 110.863 112.106i 126.220i −387.358 −2.70883 242.985i 895.310i
32.3 −8.47928 11.5964 10.4175i 39.8981 35.7023i −98.3287 + 88.3330i 11.5666i −66.9703 25.9508 241.610i 302.729i
32.4 −8.47928 11.5964 + 10.4175i 39.8981 35.7023i −98.3287 88.3330i 11.5666i −66.9703 25.9508 + 241.610i 302.729i
32.5 −4.46270 −14.2692 6.27606i −12.0843 59.8956i 63.6794 + 28.0082i 169.425i 196.735 164.222 + 179.109i 267.296i
32.6 −4.46270 −14.2692 + 6.27606i −12.0843 59.8956i 63.6794 28.0082i 169.425i 196.735 164.222 179.109i 267.296i
32.7 −3.58998 0.133977 15.5879i −19.1121 11.8803i −0.480974 + 55.9601i 150.804i 183.491 −242.964 4.17684i 42.6500i
32.8 −3.58998 0.133977 + 15.5879i −19.1121 11.8803i −0.480974 55.9601i 150.804i 183.491 −242.964 + 4.17684i 42.6500i
32.9 3.58998 0.133977 15.5879i −19.1121 11.8803i 0.480974 55.9601i 150.804i −183.491 −242.964 4.17684i 42.6500i
32.10 3.58998 0.133977 + 15.5879i −19.1121 11.8803i 0.480974 + 55.9601i 150.804i −183.491 −242.964 + 4.17684i 42.6500i
32.11 4.46270 −14.2692 6.27606i −12.0843 59.8956i −63.6794 28.0082i 169.425i −196.735 164.222 + 179.109i 267.296i
32.12 4.46270 −14.2692 + 6.27606i −12.0843 59.8956i −63.6794 + 28.0082i 169.425i −196.735 164.222 179.109i 267.296i
32.13 8.47928 11.5964 10.4175i 39.8981 35.7023i 98.3287 88.3330i 11.5666i 66.9703 25.9508 241.610i 302.729i
32.14 8.47928 11.5964 + 10.4175i 39.8981 35.7023i 98.3287 + 88.3330i 11.5666i 66.9703 25.9508 + 241.610i 302.729i
32.15 10.1143 −10.9611 11.0840i 70.2982 88.5196i −110.863 112.106i 126.220i 387.358 −2.70883 + 242.985i 895.310i
32.16 10.1143 −10.9611 + 11.0840i 70.2982 88.5196i −110.863 + 112.106i 126.220i 387.358 −2.70883 242.985i 895.310i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.6.d.b 16
3.b odd 2 1 inner 33.6.d.b 16
11.b odd 2 1 inner 33.6.d.b 16
33.d even 2 1 inner 33.6.d.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.d.b 16 1.a even 1 1 trivial
33.6.d.b 16 3.b odd 2 1 inner
33.6.d.b 16 11.b odd 2 1 inner
33.6.d.b 16 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 207 T_{2}^{6} + 13326 T_{2}^{4} - 285984 T_{2}^{2} + 1887840 \) acting on \(S_{6}^{\mathrm{new}}(33, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 49 T^{2} + 2254 T^{4} + 75232 T^{6} + 3200608 T^{8} + 77037568 T^{10} + 2363490304 T^{12} + 52613349376 T^{14} + 1099511627776 T^{16} )^{2} \)
$3$ \( ( 1 + 27 T + 420 T^{2} + 5319 T^{3} + 89910 T^{4} + 1292517 T^{5} + 24800580 T^{6} + 387420489 T^{7} + 3486784401 T^{8} )^{2} \)
$5$ \( ( 1 - 12161 T^{2} + 77169586 T^{4} - 342209368091 T^{6} + 1188160998850786 T^{8} - 3341888360263671875 T^{10} + \)\(73\!\cdots\!50\)\( T^{12} - \)\(11\!\cdots\!25\)\( T^{14} + \)\(90\!\cdots\!25\)\( T^{16} )^{2} \)
$7$ \( ( 1 - 66944 T^{2} + 2582701168 T^{4} - 68803513304192 T^{6} + 1331111347651038046 T^{8} - \)\(19\!\cdots\!08\)\( T^{10} + \)\(20\!\cdots\!68\)\( T^{12} - \)\(15\!\cdots\!56\)\( T^{14} + \)\(63\!\cdots\!01\)\( T^{16} )^{2} \)
$11$ \( 1 - 523232 T^{2} + 145162030684 T^{4} - 27820940688325664 T^{6} + \)\(45\!\cdots\!98\)\( T^{8} - \)\(72\!\cdots\!64\)\( T^{10} + \)\(97\!\cdots\!84\)\( T^{12} - \)\(91\!\cdots\!32\)\( T^{14} + \)\(45\!\cdots\!01\)\( T^{16} \)
$13$ \( ( 1 - 516428 T^{2} + 556537683592 T^{4} - 204833489102832932 T^{6} + \)\(11\!\cdots\!86\)\( T^{8} - \)\(28\!\cdots\!68\)\( T^{10} + \)\(10\!\cdots\!92\)\( T^{12} - \)\(13\!\cdots\!72\)\( T^{14} + \)\(36\!\cdots\!01\)\( T^{16} )^{2} \)
$17$ \( ( 1 + 8439940 T^{2} + 33982555335940 T^{4} + 85217776986692055292 T^{6} + \)\(14\!\cdots\!66\)\( T^{8} + \)\(17\!\cdots\!08\)\( T^{10} + \)\(13\!\cdots\!40\)\( T^{12} + \)\(69\!\cdots\!60\)\( T^{14} + \)\(16\!\cdots\!01\)\( T^{16} )^{2} \)
$19$ \( ( 1 - 13543964 T^{2} + 90505658469700 T^{4} - \)\(38\!\cdots\!32\)\( T^{6} + \)\(11\!\cdots\!66\)\( T^{8} - \)\(23\!\cdots\!32\)\( T^{10} + \)\(34\!\cdots\!00\)\( T^{12} - \)\(31\!\cdots\!64\)\( T^{14} + \)\(14\!\cdots\!01\)\( T^{16} )^{2} \)
$23$ \( ( 1 - 33097349 T^{2} + 537570467487082 T^{4} - \)\(56\!\cdots\!55\)\( T^{6} + \)\(42\!\cdots\!34\)\( T^{8} - \)\(23\!\cdots\!95\)\( T^{10} + \)\(92\!\cdots\!82\)\( T^{12} - \)\(23\!\cdots\!01\)\( T^{14} + \)\(29\!\cdots\!01\)\( T^{16} )^{2} \)
$29$ \( ( 1 + 123527836 T^{2} + 7350528812010628 T^{4} + \)\(27\!\cdots\!72\)\( T^{6} + \)\(66\!\cdots\!86\)\( T^{8} + \)\(11\!\cdots\!72\)\( T^{10} + \)\(13\!\cdots\!28\)\( T^{12} + \)\(91\!\cdots\!36\)\( T^{14} + \)\(31\!\cdots\!01\)\( T^{16} )^{2} \)
$31$ \( ( 1 + 2995 T + 70788940 T^{2} + 114310683799 T^{3} + 2385311558674918 T^{4} + 3272617827394824649 T^{5} + \)\(58\!\cdots\!40\)\( T^{6} + \)\(70\!\cdots\!45\)\( T^{7} + \)\(67\!\cdots\!01\)\( T^{8} )^{4} \)
$37$ \( ( 1 - 2339 T + 164657482 T^{2} - 340451052785 T^{3} + 14946695959559434 T^{4} - 23608223164927770245 T^{5} + \)\(79\!\cdots\!18\)\( T^{6} - \)\(77\!\cdots\!27\)\( T^{7} + \)\(23\!\cdots\!01\)\( T^{8} )^{4} \)
$41$ \( ( 1 + 188324764 T^{2} + 32097757212980164 T^{4} + \)\(48\!\cdots\!32\)\( T^{6} + \)\(71\!\cdots\!02\)\( T^{8} + \)\(65\!\cdots\!32\)\( T^{10} + \)\(57\!\cdots\!64\)\( T^{12} + \)\(45\!\cdots\!64\)\( T^{14} + \)\(32\!\cdots\!01\)\( T^{16} )^{2} \)
$43$ \( ( 1 - 126398156 T^{2} - 4682972857463132 T^{4} - \)\(14\!\cdots\!28\)\( T^{6} + \)\(88\!\cdots\!86\)\( T^{8} - \)\(31\!\cdots\!72\)\( T^{10} - \)\(21\!\cdots\!32\)\( T^{12} - \)\(12\!\cdots\!44\)\( T^{14} + \)\(21\!\cdots\!01\)\( T^{16} )^{2} \)
$47$ \( ( 1 - 1205991188 T^{2} + 710569059038402200 T^{4} - \)\(26\!\cdots\!04\)\( T^{6} + \)\(72\!\cdots\!18\)\( T^{8} - \)\(14\!\cdots\!96\)\( T^{10} + \)\(19\!\cdots\!00\)\( T^{12} - \)\(17\!\cdots\!12\)\( T^{14} + \)\(76\!\cdots\!01\)\( T^{16} )^{2} \)
$53$ \( ( 1 - 2266729916 T^{2} + 2533327026351638056 T^{4} - \)\(18\!\cdots\!28\)\( T^{6} + \)\(89\!\cdots\!66\)\( T^{8} - \)\(31\!\cdots\!72\)\( T^{10} + \)\(77\!\cdots\!56\)\( T^{12} - \)\(12\!\cdots\!84\)\( T^{14} + \)\(93\!\cdots\!01\)\( T^{16} )^{2} \)
$59$ \( ( 1 - 2925095969 T^{2} + 4804505895375021442 T^{4} - \)\(54\!\cdots\!07\)\( T^{6} + \)\(44\!\cdots\!26\)\( T^{8} - \)\(27\!\cdots\!07\)\( T^{10} + \)\(12\!\cdots\!42\)\( T^{12} - \)\(39\!\cdots\!69\)\( T^{14} + \)\(68\!\cdots\!01\)\( T^{16} )^{2} \)
$61$ \( ( 1 - 4448729132 T^{2} + 9755875885631452168 T^{4} - \)\(13\!\cdots\!76\)\( T^{6} + \)\(13\!\cdots\!78\)\( T^{8} - \)\(98\!\cdots\!76\)\( T^{10} + \)\(49\!\cdots\!68\)\( T^{12} - \)\(16\!\cdots\!32\)\( T^{14} + \)\(25\!\cdots\!01\)\( T^{16} )^{2} \)
$67$ \( ( 1 + 91033 T + 7075541680 T^{2} + 333861885426913 T^{3} + 14553622094902605454 T^{4} + \)\(45\!\cdots\!91\)\( T^{5} + \)\(12\!\cdots\!20\)\( T^{6} + \)\(22\!\cdots\!19\)\( T^{7} + \)\(33\!\cdots\!01\)\( T^{8} )^{4} \)
$71$ \( ( 1 - 6293085533 T^{2} + 24199856209130825914 T^{4} - \)\(65\!\cdots\!11\)\( T^{6} + \)\(13\!\cdots\!58\)\( T^{8} - \)\(21\!\cdots\!11\)\( T^{10} + \)\(25\!\cdots\!14\)\( T^{12} - \)\(21\!\cdots\!33\)\( T^{14} + \)\(11\!\cdots\!01\)\( T^{16} )^{2} \)
$73$ \( ( 1 - 5123388680 T^{2} + 12857066040387608668 T^{4} - \)\(37\!\cdots\!32\)\( T^{6} + \)\(10\!\cdots\!46\)\( T^{8} - \)\(16\!\cdots\!68\)\( T^{10} + \)\(23\!\cdots\!68\)\( T^{12} - \)\(40\!\cdots\!20\)\( T^{14} + \)\(34\!\cdots\!01\)\( T^{16} )^{2} \)
$79$ \( ( 1 - 12358523072 T^{2} + 48649054043753229424 T^{4} - \)\(10\!\cdots\!64\)\( T^{6} - \)\(31\!\cdots\!82\)\( T^{8} - \)\(10\!\cdots\!64\)\( T^{10} + \)\(43\!\cdots\!24\)\( T^{12} - \)\(10\!\cdots\!72\)\( T^{14} + \)\(80\!\cdots\!01\)\( T^{16} )^{2} \)
$83$ \( ( 1 + 26359778920 T^{2} + \)\(32\!\cdots\!08\)\( T^{4} + \)\(23\!\cdots\!64\)\( T^{6} + \)\(11\!\cdots\!38\)\( T^{8} + \)\(36\!\cdots\!36\)\( T^{10} + \)\(77\!\cdots\!08\)\( T^{12} + \)\(98\!\cdots\!80\)\( T^{14} + \)\(57\!\cdots\!01\)\( T^{16} )^{2} \)
$89$ \( ( 1 - 33267956981 T^{2} + \)\(52\!\cdots\!78\)\( T^{4} - \)\(51\!\cdots\!87\)\( T^{6} + \)\(34\!\cdots\!06\)\( T^{8} - \)\(15\!\cdots\!87\)\( T^{10} + \)\(50\!\cdots\!78\)\( T^{12} - \)\(10\!\cdots\!81\)\( T^{14} + \)\(94\!\cdots\!01\)\( T^{16} )^{2} \)
$97$ \( ( 1 - 29963 T + 16625594254 T^{2} - 436974456024497 T^{3} + \)\(21\!\cdots\!50\)\( T^{4} - \)\(37\!\cdots\!29\)\( T^{5} + \)\(12\!\cdots\!46\)\( T^{6} - \)\(18\!\cdots\!59\)\( T^{7} + \)\(54\!\cdots\!01\)\( T^{8} )^{4} \)
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