Properties

Label 531.6.a.c
Level $531$
Weight $6$
Character orbit 531.a
Self dual yes
Analytic conductor $85.164$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(85.1638083207\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 2 x^{11} - 269 x^{10} + 143 x^{9} + 25384 x^{8} + 8539 x^{7} - 1009736 x^{6} - 720516 x^{5} + 15565376 x^{4} + 6775664 x^{3} - 75006848 x^{2} + 21512960 x + 49172480\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + ( -2 + \beta_{1} ) q^{2} + ( 17 - 2 \beta_{1} + \beta_{2} ) q^{4} + ( -13 + \beta_{1} - \beta_{3} ) q^{5} + ( 35 - \beta_{1} - \beta_{11} ) q^{7} + ( -64 + 22 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{8} +O(q^{10})\) \( q + ( -2 + \beta_{1} ) q^{2} + ( 17 - 2 \beta_{1} + \beta_{2} ) q^{4} + ( -13 + \beta_{1} - \beta_{3} ) q^{5} + ( 35 - \beta_{1} - \beta_{11} ) q^{7} + ( -64 + 22 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{8} + ( 54 - 25 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{10} + ( -122 - 10 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{11} ) q^{11} + ( 40 - 2 \beta_{1} + \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{13} + ( -95 + 45 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{10} + 3 \beta_{11} ) q^{14} + ( 535 - 43 \beta_{1} + 20 \beta_{2} - 2 \beta_{4} - 4 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{10} + 3 \beta_{11} ) q^{16} + ( -126 - 44 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} - 6 \beta_{4} - 6 \beta_{5} - 5 \beta_{6} - \beta_{7} - 4 \beta_{8} - \beta_{9} - 4 \beta_{11} ) q^{17} + ( 320 + 17 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 5 \beta_{6} + \beta_{7} + \beta_{8} - 8 \beta_{9} + 11 \beta_{11} ) q^{19} + ( -687 + 32 \beta_{1} - 9 \beta_{2} - 7 \beta_{3} - 6 \beta_{4} - 6 \beta_{5} + 9 \beta_{6} - \beta_{7} + 10 \beta_{8} - 9 \beta_{9} + 5 \beta_{10} + 11 \beta_{11} ) q^{20} + ( -123 - 156 \beta_{1} - 22 \beta_{2} - 15 \beta_{3} - 11 \beta_{4} - 2 \beta_{5} + 10 \beta_{6} + 11 \beta_{7} + \beta_{8} - 2 \beta_{10} + 13 \beta_{11} ) q^{22} + ( -614 - 27 \beta_{1} + 13 \beta_{2} + 15 \beta_{3} - \beta_{4} - 13 \beta_{5} + 2 \beta_{6} - 6 \beta_{7} + 6 \beta_{8} - 4 \beta_{9} + 3 \beta_{10} + 7 \beta_{11} ) q^{23} + ( 805 - 39 \beta_{1} - 38 \beta_{2} + 14 \beta_{3} - 3 \beta_{4} - 5 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} - 5 \beta_{8} + 13 \beta_{9} - 2 \beta_{11} ) q^{25} + ( -319 + 19 \beta_{1} + 4 \beta_{2} + 38 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 16 \beta_{6} + \beta_{7} - 17 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} - 13 \beta_{11} ) q^{26} + ( 1056 - 228 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 9 \beta_{4} + 2 \beta_{5} - 17 \beta_{6} - 35 \beta_{7} - \beta_{8} - 12 \beta_{9} + \beta_{10} - 6 \beta_{11} ) q^{28} + ( -823 + 34 \beta_{1} - \beta_{2} + 16 \beta_{3} + 3 \beta_{4} + 27 \beta_{5} + 10 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} + 18 \beta_{9} - 5 \beta_{10} + 5 \beta_{11} ) q^{29} + ( 632 - 160 \beta_{1} - 46 \beta_{2} + 5 \beta_{3} + 4 \beta_{4} + 18 \beta_{5} - 22 \beta_{6} - 9 \beta_{7} + 7 \beta_{8} + 12 \beta_{9} + 5 \beta_{10} - 12 \beta_{11} ) q^{31} + ( -1063 + 440 \beta_{1} - 9 \beta_{2} + 44 \beta_{3} + 18 \beta_{4} - 14 \beta_{5} + 8 \beta_{6} + 14 \beta_{7} + 4 \beta_{9} - 12 \beta_{10} ) q^{32} + ( -1359 - 187 \beta_{1} - 102 \beta_{2} - 8 \beta_{3} + 22 \beta_{4} - 25 \beta_{5} + 64 \beta_{6} + 14 \beta_{7} + 4 \beta_{8} - 10 \beta_{9} + 2 \beta_{10} + 26 \beta_{11} ) q^{34} + ( -910 + 78 \beta_{1} + 21 \beta_{2} - 31 \beta_{3} - 16 \beta_{4} + 52 \beta_{5} + 9 \beta_{6} - 8 \beta_{7} + 13 \beta_{8} - 11 \beta_{9} - 15 \beta_{10} + 55 \beta_{11} ) q^{35} + ( 515 + 60 \beta_{1} - 81 \beta_{2} + 44 \beta_{3} - 3 \beta_{4} - 13 \beta_{5} - 12 \beta_{6} + 9 \beta_{7} - \beta_{8} - 10 \beta_{9} - 6 \beta_{10} - 10 \beta_{11} ) q^{37} + ( -250 + 27 \beta_{1} - 30 \beta_{2} + 83 \beta_{3} + 26 \beta_{4} + 4 \beta_{5} - 37 \beta_{6} - 32 \beta_{7} - 27 \beta_{8} + 22 \beta_{9} - \beta_{10} - 54 \beta_{11} ) q^{38} + ( 622 - 668 \beta_{1} - 127 \beta_{2} + 81 \beta_{3} + 9 \beta_{4} + 73 \beta_{5} - 47 \beta_{6} + 2 \beta_{7} - 62 \beta_{8} + 43 \beta_{9} - 11 \beta_{10} - 69 \beta_{11} ) q^{40} + ( -2882 - 121 \beta_{1} + 36 \beta_{2} + 6 \beta_{3} + 18 \beta_{4} - 82 \beta_{5} + 35 \beta_{6} + 18 \beta_{7} + 34 \beta_{8} + 4 \beta_{9} + 15 \beta_{10} + 28 \beta_{11} ) q^{41} + ( -636 + 336 \beta_{1} - 88 \beta_{2} + 60 \beta_{3} + 26 \beta_{4} - 6 \beta_{5} + 12 \beta_{6} - 7 \beta_{7} + 55 \beta_{8} + 16 \beta_{9} - 13 \beta_{10} + \beta_{11} ) q^{43} + ( -3298 - 789 \beta_{1} - 183 \beta_{2} + 85 \beta_{3} + 27 \beta_{4} + 18 \beta_{5} - 76 \beta_{6} - 77 \beta_{7} - 17 \beta_{8} - 4 \beta_{9} + 6 \beta_{10} - 115 \beta_{11} ) q^{44} + ( 252 - 291 \beta_{1} - 114 \beta_{2} + 23 \beta_{3} + 10 \beta_{4} + 44 \beta_{5} - 27 \beta_{6} + 46 \beta_{7} - 37 \beta_{8} + 16 \beta_{9} - 11 \beta_{10} - 26 \beta_{11} ) q^{46} + ( -5314 - 1064 \beta_{1} - 76 \beta_{2} - 54 \beta_{3} - 37 \beta_{4} + 43 \beta_{5} - 16 \beta_{6} + 93 \beta_{7} - 10 \beta_{8} + 51 \beta_{9} + 12 \beta_{10} - 39 \beta_{11} ) q^{47} + ( 2310 + 92 \beta_{1} - 145 \beta_{2} + 85 \beta_{3} - 32 \beta_{4} + 34 \beta_{5} - 38 \beta_{6} - 77 \beta_{7} + 34 \beta_{8} - 21 \beta_{9} - 15 \beta_{10} - 3 \beta_{11} ) q^{49} + ( -2590 - 277 \beta_{1} - 34 \beta_{2} - 56 \beta_{3} + 7 \beta_{4} - 73 \beta_{5} + 11 \beta_{6} - 37 \beta_{7} + 14 \beta_{8} - 80 \beta_{9} + 25 \beta_{10} - 17 \beta_{11} ) q^{50} + ( 1182 + 610 \beta_{1} + 135 \beta_{2} - 104 \beta_{3} - 16 \beta_{4} - 153 \beta_{5} + 120 \beta_{6} + 61 \beta_{7} + 57 \beta_{8} - 53 \beta_{9} + 18 \beta_{10} + 93 \beta_{11} ) q^{52} + ( -5107 - 283 \beta_{1} - 38 \beta_{2} - 107 \beta_{3} + 48 \beta_{4} + 88 \beta_{5} + 42 \beta_{6} - 2 \beta_{7} + 22 \beta_{8} - 82 \beta_{9} - 22 \beta_{10} + 60 \beta_{11} ) q^{53} + ( 4446 + 739 \beta_{1} - 112 \beta_{2} + 146 \beta_{3} - 41 \beta_{4} - 57 \beta_{5} + \beta_{6} + 62 \beta_{7} - 33 \beta_{8} + 67 \beta_{9} - 20 \beta_{10} + 16 \beta_{11} ) q^{55} + ( -9000 - 343 \beta_{1} - 283 \beta_{2} - 75 \beta_{3} - 19 \beta_{4} - 64 \beta_{5} + 124 \beta_{6} + 157 \beta_{7} + 15 \beta_{8} - 24 \beta_{9} - 18 \beta_{10} + 109 \beta_{11} ) q^{56} + ( 3234 - 100 \beta_{1} + 144 \beta_{2} - 166 \beta_{3} - 4 \beta_{4} + 6 \beta_{5} - 26 \beta_{6} - 61 \beta_{7} + 31 \beta_{8} - 115 \beta_{9} + 46 \beta_{10} - 57 \beta_{11} ) q^{58} + 3481 q^{59} + ( 5590 + 1042 \beta_{1} + 107 \beta_{2} - 104 \beta_{3} - 49 \beta_{4} + 29 \beta_{5} - 13 \beta_{6} + 28 \beta_{7} + 14 \beta_{8} + 20 \beta_{9} - 22 \beta_{10} + 16 \beta_{11} ) q^{61} + ( -8202 - 746 \beta_{1} - 62 \beta_{2} - 120 \beta_{3} - 44 \beta_{4} - 34 \beta_{5} + 118 \beta_{6} + 59 \beta_{7} + 93 \beta_{8} - 15 \beta_{9} - 26 \beta_{10} + 137 \beta_{11} ) q^{62} + ( 5848 + 372 \beta_{1} + 437 \beta_{2} - 328 \beta_{3} - 19 \beta_{4} + 25 \beta_{5} + 10 \beta_{6} - 11 \beta_{7} - 20 \beta_{8} - 88 \beta_{9} + 34 \beta_{10} - 54 \beta_{11} ) q^{64} + ( -6091 - 2897 \beta_{1} + 75 \beta_{2} - 95 \beta_{3} + 6 \beta_{4} + 190 \beta_{5} - 97 \beta_{6} + 97 \beta_{7} - 4 \beta_{8} + 121 \beta_{9} + 52 \beta_{10} - 51 \beta_{11} ) q^{65} + ( 9382 + 639 \beta_{1} + 104 \beta_{2} - 2 \beta_{3} + 115 \beta_{4} - 33 \beta_{5} - 29 \beta_{6} - 72 \beta_{7} - 109 \beta_{8} - 29 \beta_{9} + 70 \beta_{10} - 52 \beta_{11} ) q^{67} + ( -1867 - 4863 \beta_{1} - 523 \beta_{2} + 187 \beta_{3} - 80 \beta_{4} + 221 \beta_{5} - 286 \beta_{6} - 214 \beta_{7} - 153 \beta_{8} + 110 \beta_{9} - 8 \beta_{10} - 229 \beta_{11} ) q^{68} + ( 2792 - 5 \beta_{1} + 292 \beta_{2} - 105 \beta_{3} + 145 \beta_{4} + 39 \beta_{5} - 112 \beta_{6} - 211 \beta_{7} + 113 \beta_{8} - 32 \beta_{9} + 30 \beta_{10} - 179 \beta_{11} ) q^{70} + ( -6574 - 3160 \beta_{1} + 128 \beta_{2} + 319 \beta_{3} + 19 \beta_{4} - 161 \beta_{5} - 174 \beta_{6} + 15 \beta_{7} - 118 \beta_{8} - 25 \beta_{9} - 54 \beta_{10} - 234 \beta_{11} ) q^{71} + ( 11241 + 1034 \beta_{1} + 46 \beta_{2} - 92 \beta_{3} - 149 \beta_{4} + 57 \beta_{5} - 49 \beta_{6} + 179 \beta_{7} - 66 \beta_{8} + 195 \beta_{9} + 5 \beta_{10} - 14 \beta_{11} ) q^{73} + ( 2977 - 1976 \beta_{1} + 454 \beta_{2} - 223 \beta_{3} - 57 \beta_{4} - 162 \beta_{5} + 70 \beta_{6} - 66 \beta_{7} + 68 \beta_{8} - 17 \beta_{9} + 86 \beta_{10} + 146 \beta_{11} ) q^{74} + ( -5068 - 165 \beta_{1} + 183 \beta_{2} - 496 \beta_{3} - 106 \beta_{4} - 102 \beta_{5} + 343 \beta_{6} + 322 \beta_{7} + 62 \beta_{8} + 72 \beta_{9} + 59 \beta_{10} + 155 \beta_{11} ) q^{76} + ( 932 - 3105 \beta_{1} + 545 \beta_{2} + 311 \beta_{3} + 95 \beta_{4} + 77 \beta_{5} - 122 \beta_{6} - 131 \beta_{7} - 45 \beta_{8} + 122 \beta_{9} - 22 \beta_{10} - 36 \beta_{11} ) q^{77} + ( 1631 - 804 \beta_{1} - 398 \beta_{2} + 22 \beta_{3} + 56 \beta_{4} + 484 \beta_{5} - 9 \beta_{6} - 121 \beta_{7} - 137 \beta_{8} + 146 \beta_{9} - 90 \beta_{10} + 56 \beta_{11} ) q^{79} + ( -6105 - 1635 \beta_{1} + 124 \beta_{2} - 535 \beta_{3} + 52 \beta_{4} - 226 \beta_{5} + 454 \beta_{6} + 165 \beta_{7} - 72 \beta_{8} - 93 \beta_{9} + 18 \beta_{10} + 224 \beta_{11} ) q^{80} + ( 499 - 3551 \beta_{1} - 168 \beta_{2} + 414 \beta_{3} - 3 \beta_{4} + 162 \beta_{5} - 621 \beta_{6} - 41 \beta_{7} - 222 \beta_{8} + 104 \beta_{9} - 41 \beta_{10} - 375 \beta_{11} ) q^{82} + ( -9008 - 2199 \beta_{1} + 182 \beta_{2} - 114 \beta_{3} + 74 \beta_{4} - 106 \beta_{5} + 499 \beta_{6} + 168 \beta_{7} - 106 \beta_{8} + 38 \beta_{9} - 51 \beta_{10} + 94 \beta_{11} ) q^{83} + ( -9185 + 1491 \beta_{1} + 169 \beta_{2} + 221 \beta_{3} + 141 \beta_{4} - 493 \beta_{5} + 131 \beta_{6} - 48 \beta_{7} + 238 \beta_{8} - 288 \beta_{9} - 60 \beta_{10} + 244 \beta_{11} ) q^{85} + ( 17831 - 3420 \beta_{1} + 966 \beta_{2} - 669 \beta_{3} - 311 \beta_{4} + 82 \beta_{5} - 374 \beta_{6} - 197 \beta_{7} + 143 \beta_{8} + 12 \beta_{9} + 58 \beta_{10} - \beta_{11} ) q^{86} + ( -20061 - 2923 \beta_{1} - 599 \beta_{2} - 359 \beta_{3} - 262 \beta_{4} - 119 \beta_{5} + 706 \beta_{6} + 182 \beta_{7} + 139 \beta_{8} + 52 \beta_{9} - 24 \beta_{10} + 587 \beta_{11} ) q^{88} + ( -14756 - 6178 \beta_{1} + 195 \beta_{2} + 3 \beta_{3} - 80 \beta_{4} + 242 \beta_{5} + 41 \beta_{6} - 372 \beta_{7} + 235 \beta_{8} - 247 \beta_{9} - 57 \beta_{10} + 403 \beta_{11} ) q^{89} + ( -14193 - 607 \beta_{1} - 13 \beta_{2} + 265 \beta_{3} + 165 \beta_{4} - 445 \beta_{5} + 307 \beta_{6} + 343 \beta_{7} - 253 \beta_{8} - 108 \beta_{9} + 23 \beta_{10} + 59 \beta_{11} ) q^{91} + ( 6532 - 1727 \beta_{1} + 457 \beta_{2} - 754 \beta_{3} + 50 \beta_{4} - 14 \beta_{5} + 123 \beta_{6} + 44 \beta_{7} + 14 \beta_{8} - 78 \beta_{9} + 111 \beta_{10} + 51 \beta_{11} ) q^{92} + ( -38382 - 7405 \beta_{1} - 412 \beta_{2} + 451 \beta_{3} + 367 \beta_{4} - 123 \beta_{5} - 92 \beta_{6} - 359 \beta_{7} + 159 \beta_{8} - 82 \beta_{9} + 34 \beta_{10} - 185 \beta_{11} ) q^{94} + ( -1345 - 3993 \beta_{1} + 660 \beta_{2} - 24 \beta_{3} - 124 \beta_{4} - 634 \beta_{5} - 138 \beta_{6} + 79 \beta_{7} + 127 \beta_{8} - 18 \beta_{9} + 35 \beta_{10} - 384 \beta_{11} ) q^{95} + ( -4947 - 5921 \beta_{1} + 536 \beta_{2} - 748 \beta_{3} - 163 \beta_{4} + 563 \beta_{5} - 190 \beta_{6} - 484 \beta_{7} + 209 \beta_{8} - 177 \beta_{9} + 91 \beta_{10} + 311 \beta_{11} ) q^{97} + ( 1813 - 954 \beta_{1} - 352 \beta_{2} - 900 \beta_{3} - 228 \beta_{4} + 37 \beta_{5} + 522 \beta_{6} + 34 \beta_{7} + 246 \beta_{8} + 24 \beta_{9} - 40 \beta_{10} + 358 \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 22q^{2} + 198q^{4} - 158q^{5} + 413q^{7} - 723q^{8} + O(q^{10}) \) \( 12q - 22q^{2} + 198q^{4} - 158q^{5} + 413q^{7} - 723q^{8} + 601q^{10} - 1480q^{11} + 472q^{13} - 1065q^{14} + 6370q^{16} - 1565q^{17} + 3939q^{19} - 8033q^{20} - 1738q^{22} - 7245q^{23} + 9690q^{25} - 3764q^{26} + 12154q^{28} - 10003q^{29} + 7295q^{31} - 11628q^{32} - 16344q^{34} - 11015q^{35} + 6741q^{37} - 3035q^{38} + 5572q^{40} - 34025q^{41} - 6336q^{43} - 41168q^{44} + 2345q^{46} - 66167q^{47} + 28319q^{49} - 31173q^{50} + 16440q^{52} - 62290q^{53} + 55764q^{55} - 107306q^{56} + 37952q^{58} + 41772q^{59} + 68469q^{61} - 99190q^{62} + 68525q^{64} - 80156q^{65} + 113310q^{67} - 33887q^{68} + 32034q^{70} - 84520q^{71} + 135895q^{73} + 31962q^{74} - 61848q^{76} + 3799q^{77} + 14122q^{79} - 77609q^{80} - 1501q^{82} - 114463q^{83} - 101097q^{85} + 203536q^{86} - 244967q^{88} - 189109q^{89} - 168249q^{91} + 71946q^{92} - 472284q^{94} - 21923q^{95} - 76192q^{97} + 17544q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 2 x^{11} - 269 x^{10} + 143 x^{9} + 25384 x^{8} + 8539 x^{7} - 1009736 x^{6} - 720516 x^{5} + 15565376 x^{4} + 6775664 x^{3} - 75006848 x^{2} + 21512960 x + 49172480\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 45 \)
\(\beta_{3}\)\(=\)\((\)\(350941 \nu^{11} - 1428132 \nu^{10} - 70855647 \nu^{9} + 91864075 \nu^{8} + 4536395590 \nu^{7} + 5505304837 \nu^{6} - 84243883332 \nu^{5} - 433160653132 \nu^{4} - 827674808048 \nu^{3} + 5866887507152 \nu^{2} + 17723236279616 \nu - 17699688792576\)\()/ 277202304512 \)
\(\beta_{4}\)\(=\)\((\)\(-8192145 \nu^{11} - 67427634 \nu^{10} + 2640430613 \nu^{9} + 18376727045 \nu^{8} - 268943328324 \nu^{7} - 1674621922747 \nu^{6} + 10557691149516 \nu^{5} + 59174525644532 \nu^{4} - 134622184486576 \nu^{3} - 631726297910064 \nu^{2} + 320260391715264 \nu + 1009628811466240\)\()/ 3049225349632 \)
\(\beta_{5}\)\(=\)\((\)\(-299377 \nu^{11} - 524416 \nu^{10} + 88399319 \nu^{9} + 187648889 \nu^{8} - 8690291434 \nu^{7} - 19612654933 \nu^{6} + 346024848588 \nu^{5} + 775699987236 \nu^{4} - 4941116508272 \nu^{3} - 9713875467792 \nu^{2} + 13945651094144 \nu + 16317679113344\)\()/ 108900905344 \)
\(\beta_{6}\)\(=\)\((\)\(5842853 \nu^{11} + 35029081 \nu^{10} - 1903226618 \nu^{9} - 8347251655 \nu^{8} + 191946287467 \nu^{7} + 677570013916 \nu^{6} - 7467607930832 \nu^{5} - 21595833573480 \nu^{4} + 98729745120848 \nu^{3} + 206836035149408 \nu^{2} - 292465352655168 \nu - 324693088418048\)\()/ 1524612674816 \)
\(\beta_{7}\)\(=\)\((\)\(1506791 \nu^{11} + 7464662 \nu^{10} - 465055595 \nu^{9} - 2148263267 \nu^{8} + 46570135316 \nu^{7} + 202161478261 \nu^{6} - 1840580628180 \nu^{5} - 7354011389740 \nu^{4} + 25092970187984 \nu^{3} + 80769880387280 \nu^{2} - 85679977174848 \nu - 94789227549184\)\()/ 277202304512 \)
\(\beta_{8}\)\(=\)\((\)\(100767 \nu^{11} + 53452 \nu^{10} - 28964453 \nu^{9} - 48656271 \nu^{8} + 2821805338 \nu^{7} + 6891575767 \nu^{6} - 108469681492 \nu^{5} - 313727728564 \nu^{4} + 1335646794992 \nu^{3} + 3716852794288 \nu^{2} - 2979411689408 \nu - 4701282618368\)\()/ 11773070848 \)
\(\beta_{9}\)\(=\)\((\)\(14438755 \nu^{11} + 3300860 \nu^{10} - 4146949925 \nu^{9} - 4582591287 \nu^{8} + 404519679498 \nu^{7} + 624811840007 \nu^{6} - 15987697561584 \nu^{5} - 26224218394372 \nu^{4} + 222176054432320 \nu^{3} + 262460919500528 \nu^{2} - 669401779958912 \nu - 156500452428544\)\()/ 1524612674816 \)
\(\beta_{10}\)\(=\)\((\)\(24199755 \nu^{11} + 1545329 \nu^{10} - 6527695920 \nu^{9} - 8306848553 \nu^{8} + 593347027999 \nu^{7} + 1132584010846 \nu^{6} - 21677470373468 \nu^{5} - 46080636431880 \nu^{4} + 275119055854096 \nu^{3} + 458277283491552 \nu^{2} - 745180880860416 \nu - 172136514045440\)\()/ 1524612674816 \)
\(\beta_{11}\)\(=\)\((\)\(-25955403 \nu^{11} + 47327234 \nu^{10} + 6681309643 \nu^{9} + 346810155 \nu^{8} - 598161796452 \nu^{7} - 781324808245 \nu^{6} + 21825571577712 \nu^{5} + 44920811814580 \nu^{4} - 275046633380496 \nu^{3} - 576938518125872 \nu^{2} + 788942374823680 \nu + 849314392654336\)\()/ 1524612674816 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 45\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{5} + \beta_{4} + 5 \beta_{2} + 86 \beta_{1} + 86\)
\(\nu^{4}\)\(=\)\(3 \beta_{11} + 3 \beta_{10} + 2 \beta_{8} + 6 \beta_{7} - 5 \beta_{6} + 4 \beta_{5} + 6 \beta_{4} + 132 \beta_{2} + 437 \beta_{1} + 3807\)
\(\nu^{5}\)\(=\)\(30 \beta_{11} + 18 \beta_{10} + 4 \beta_{9} + 20 \beta_{8} + 162 \beta_{7} - 42 \beta_{6} + 114 \beta_{5} + 166 \beta_{4} + 44 \beta_{3} + 1063 \beta_{2} + 9386 \beta_{1} + 18767\)
\(\nu^{6}\)\(=\)\(606 \beta_{11} + 550 \beta_{10} - 40 \beta_{9} + 420 \beta_{8} + 1413 \beta_{7} - 994 \beta_{6} + 673 \beta_{5} + 1453 \beta_{4} + 200 \beta_{3} + 18249 \beta_{2} + 74944 \beta_{1} + 411640\)
\(\nu^{7}\)\(=\)\(6783 \beta_{11} + 4559 \beta_{10} + 584 \beta_{9} + 4402 \beta_{8} + 24374 \beta_{7} - 10209 \beta_{6} + 13228 \beta_{5} + 25022 \beta_{4} + 8000 \beta_{3} + 183734 \beta_{2} + 1198009 \beta_{1} + 3226133\)
\(\nu^{8}\)\(=\)\(100910 \beta_{11} + 85242 \beta_{10} - 6060 \beta_{9} + 69276 \beta_{8} + 253564 \beta_{7} - 162306 \beta_{6} + 106068 \beta_{5} + 263088 \beta_{4} + 56452 \beta_{3} + 2644521 \beta_{2} + 12019798 \beta_{1} + 52195939\)
\(\nu^{9}\)\(=\)\(1187260 \beta_{11} + 850124 \beta_{10} + 55096 \beta_{9} + 768448 \beta_{8} + 3661489 \beta_{7} - 1839132 \beta_{6} + 1696449 \beta_{5} + 3763833 \beta_{4} + 1216848 \beta_{3} + 29635057 \beta_{2} + 167636034 \beta_{1} + 518319894\)
\(\nu^{10}\)\(=\)\(15917723 \beta_{11} + 12861355 \beta_{10} - 684976 \beta_{9} + 10742610 \beta_{8} + 41567962 \beta_{7} - 25302541 \beta_{6} + 16617000 \beta_{5} + 43157930 \beta_{4} + 11082352 \beta_{3} + 393717500 \beta_{2} + 1879589677 \beta_{1} + 7274855475\)
\(\nu^{11}\)\(=\)\(191789770 \beta_{11} + 142131150 \beta_{10} + 3961508 \beta_{9} + 124512460 \beta_{8} + 553464350 \beta_{7} - 300500302 \beta_{6} + 235486966 \beta_{5} + 570061442 \beta_{4} + 180809340 \beta_{3} + 4645221699 \beta_{2} + 24598348998 \beta_{1} + 81136464075\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−8.57072
−8.49197
−7.23155
−5.43261
−3.21799
−0.689340
1.38446
1.70991
4.10256
7.96013
8.10567
12.3715
−10.5707 0 79.7401 45.7686 0 −1.90644 −504.647 0 −483.807
1.2 −10.4920 0 78.0815 −46.0665 0 183.145 −483.486 0 483.328
1.3 −9.23155 0 53.2215 −89.5822 0 −121.386 −195.908 0 826.983
1.4 −7.43261 0 23.2437 −48.3022 0 120.542 65.0821 0 359.012
1.5 −5.21799 0 −4.77260 21.6600 0 150.168 191.879 0 −113.022
1.6 −2.68934 0 −24.7675 83.5049 0 −48.3401 152.667 0 −224.573
1.7 −0.615542 0 −31.6211 −61.9372 0 −209.534 39.1615 0 38.1250
1.8 −0.290087 0 −31.9158 −87.1048 0 167.610 18.5412 0 25.2680
1.9 2.10256 0 −27.5792 81.1594 0 −97.6213 −125.269 0 170.642
1.10 5.96013 0 3.52316 −65.6425 0 119.026 −169.726 0 −391.238
1.11 6.10567 0 5.27923 41.8018 0 208.248 −163.148 0 255.228
1.12 10.3715 0 75.5670 −33.2592 0 −56.9512 451.853 0 −344.946
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(59\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 531.6.a.c 12
3.b odd 2 1 177.6.a.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.6.a.c 12 3.b odd 2 1
531.6.a.c 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{12} + \cdots\) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(531))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 15131776 + 78147904 T + 88035904 T^{2} - 12144672 T^{3} - 24399488 T^{4} - 2586728 T^{5} + 1174642 T^{6} + 191819 T^{7} - 15182 T^{8} - 3917 T^{9} - 49 T^{10} + 22 T^{11} + T^{12} \)
$3$ \( T^{12} \)
$5$ \( -\)\(65\!\cdots\!40\)\( - 6767956329512233920 T + 1953100084230636496 T^{2} + 35490493399256368 T^{3} - 1661090387328638 T^{4} - 39664756772442 T^{5} + 484029707037 T^{6} + 16727756618 T^{7} - 11859943 T^{8} - 2786782 T^{9} - 11113 T^{10} + 158 T^{11} + T^{12} \)
$7$ \( \)\(17\!\cdots\!12\)\( + \)\(98\!\cdots\!48\)\( T + \)\(19\!\cdots\!56\)\( T^{2} - 35283152612462006888 T^{3} - 622725300436496463 T^{4} + 6882289430733117 T^{5} + 61742243285267 T^{6} - 701221763620 T^{7} - 1729721320 T^{8} + 30924452 T^{9} - 29717 T^{10} - 413 T^{11} + T^{12} \)
$11$ \( -\)\(33\!\cdots\!20\)\( - \)\(25\!\cdots\!20\)\( T - \)\(14\!\cdots\!12\)\( T^{2} - \)\(16\!\cdots\!68\)\( T^{3} - \)\(11\!\cdots\!35\)\( T^{4} + 5664578792566664692 T^{5} + 23561665253073632 T^{6} - 16851341306852 T^{7} - 271814348806 T^{8} - 429186564 T^{9} + 378788 T^{10} + 1480 T^{11} + T^{12} \)
$13$ \( -\)\(17\!\cdots\!60\)\( + \)\(37\!\cdots\!76\)\( T - \)\(22\!\cdots\!94\)\( T^{2} - \)\(16\!\cdots\!22\)\( T^{3} + \)\(20\!\cdots\!15\)\( T^{4} + \)\(18\!\cdots\!54\)\( T^{5} - 318127682424420370 T^{6} - 697398554727854 T^{7} + 1319432358072 T^{8} + 984558370 T^{9} - 2008336 T^{10} - 472 T^{11} + T^{12} \)
$17$ \( \)\(68\!\cdots\!28\)\( - \)\(26\!\cdots\!92\)\( T - \)\(65\!\cdots\!28\)\( T^{2} + \)\(19\!\cdots\!12\)\( T^{3} + \)\(25\!\cdots\!19\)\( T^{4} - \)\(39\!\cdots\!83\)\( T^{5} - 39724671046095865535 T^{6} + 33946994996183446 T^{7} + 27740048302598 T^{8} - 12435602130 T^{9} - 8804725 T^{10} + 1565 T^{11} + T^{12} \)
$19$ \( \)\(37\!\cdots\!00\)\( + \)\(24\!\cdots\!40\)\( T - \)\(26\!\cdots\!48\)\( T^{2} - \)\(27\!\cdots\!00\)\( T^{3} - \)\(27\!\cdots\!28\)\( T^{4} + \)\(73\!\cdots\!48\)\( T^{5} + 5989482340746602519 T^{6} - 72007006950508063 T^{7} + 3134068890308 T^{8} + 28830582141 T^{9} - 4707528 T^{10} - 3939 T^{11} + T^{12} \)
$23$ \( -\)\(83\!\cdots\!52\)\( - \)\(44\!\cdots\!40\)\( T + \)\(58\!\cdots\!32\)\( T^{2} + \)\(41\!\cdots\!44\)\( T^{3} - \)\(89\!\cdots\!64\)\( T^{4} - \)\(10\!\cdots\!24\)\( T^{5} + \)\(15\!\cdots\!21\)\( T^{6} + 114973957775336745 T^{7} - 70465096827146 T^{8} - 49427738255 T^{9} + 6217238 T^{10} + 7245 T^{11} + T^{12} \)
$29$ \( -\)\(34\!\cdots\!80\)\( - \)\(77\!\cdots\!20\)\( T + \)\(40\!\cdots\!28\)\( T^{2} + \)\(74\!\cdots\!20\)\( T^{3} - \)\(41\!\cdots\!50\)\( T^{4} - \)\(46\!\cdots\!36\)\( T^{5} + \)\(10\!\cdots\!25\)\( T^{6} + 9793958586444167867 T^{7} - 417659362983198 T^{8} - 742411469525 T^{9} - 54457480 T^{10} + 10003 T^{11} + T^{12} \)
$31$ \( -\)\(26\!\cdots\!24\)\( - \)\(78\!\cdots\!00\)\( T - \)\(29\!\cdots\!00\)\( T^{2} + \)\(68\!\cdots\!16\)\( T^{3} + \)\(59\!\cdots\!14\)\( T^{4} + \)\(59\!\cdots\!20\)\( T^{5} - \)\(71\!\cdots\!53\)\( T^{6} - 14035132982507777759 T^{7} + 3561677478358512 T^{8} + 641056156181 T^{9} - 100851746 T^{10} - 7295 T^{11} + T^{12} \)
$37$ \( \)\(35\!\cdots\!20\)\( + \)\(47\!\cdots\!76\)\( T - \)\(16\!\cdots\!82\)\( T^{2} - \)\(32\!\cdots\!08\)\( T^{3} + \)\(40\!\cdots\!41\)\( T^{4} + \)\(69\!\cdots\!25\)\( T^{5} - \)\(81\!\cdots\!85\)\( T^{6} - 53725710742019855456 T^{7} + 7272908831275654 T^{8} + 1285385157840 T^{9} - 183896193 T^{10} - 6741 T^{11} + T^{12} \)
$41$ \( -\)\(20\!\cdots\!00\)\( - \)\(15\!\cdots\!20\)\( T + \)\(46\!\cdots\!96\)\( T^{2} + \)\(66\!\cdots\!12\)\( T^{3} - \)\(14\!\cdots\!05\)\( T^{4} - \)\(23\!\cdots\!83\)\( T^{5} + \)\(15\!\cdots\!01\)\( T^{6} + \)\(30\!\cdots\!74\)\( T^{7} - 43041460629711582 T^{8} - 16817134584398 T^{9} - 219775449 T^{10} + 34025 T^{11} + T^{12} \)
$43$ \( \)\(25\!\cdots\!96\)\( - \)\(23\!\cdots\!72\)\( T - \)\(16\!\cdots\!44\)\( T^{2} + \)\(40\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!33\)\( T^{4} - \)\(15\!\cdots\!52\)\( T^{5} - \)\(35\!\cdots\!60\)\( T^{6} + \)\(14\!\cdots\!68\)\( T^{7} + 292793572892837014 T^{8} - 5084337789976 T^{9} - 933348484 T^{10} + 6336 T^{11} + T^{12} \)
$47$ \( -\)\(14\!\cdots\!80\)\( + \)\(89\!\cdots\!60\)\( T + \)\(57\!\cdots\!12\)\( T^{2} - \)\(54\!\cdots\!96\)\( T^{3} - \)\(76\!\cdots\!00\)\( T^{4} - \)\(14\!\cdots\!24\)\( T^{5} + \)\(43\!\cdots\!91\)\( T^{6} + \)\(16\!\cdots\!75\)\( T^{7} - 1009768677266494104 T^{8} - 57972079582133 T^{9} + 379262884 T^{10} + 66167 T^{11} + T^{12} \)
$53$ \( -\)\(11\!\cdots\!68\)\( - \)\(20\!\cdots\!04\)\( T - \)\(83\!\cdots\!92\)\( T^{2} - \)\(13\!\cdots\!32\)\( T^{3} - \)\(72\!\cdots\!46\)\( T^{4} + \)\(52\!\cdots\!54\)\( T^{5} + \)\(78\!\cdots\!01\)\( T^{6} + \)\(18\!\cdots\!66\)\( T^{7} - 1509063873576592231 T^{8} - 85907870747366 T^{9} - 240337733 T^{10} + 62290 T^{11} + T^{12} \)
$59$ \( ( -3481 + T )^{12} \)
$61$ \( -\)\(74\!\cdots\!72\)\( + \)\(48\!\cdots\!08\)\( T + \)\(56\!\cdots\!56\)\( T^{2} - \)\(41\!\cdots\!32\)\( T^{3} - \)\(45\!\cdots\!14\)\( T^{4} + \)\(12\!\cdots\!52\)\( T^{5} + \)\(88\!\cdots\!39\)\( T^{6} - \)\(31\!\cdots\!05\)\( T^{7} - 572213248643671512 T^{8} + 28665455565979 T^{9} + 843694894 T^{10} - 68469 T^{11} + T^{12} \)
$67$ \( \)\(14\!\cdots\!56\)\( - \)\(39\!\cdots\!36\)\( T + \)\(29\!\cdots\!48\)\( T^{2} + \)\(68\!\cdots\!44\)\( T^{3} - \)\(16\!\cdots\!64\)\( T^{4} + \)\(35\!\cdots\!96\)\( T^{5} + \)\(24\!\cdots\!85\)\( T^{6} - \)\(89\!\cdots\!22\)\( T^{7} - 9251741319212442121 T^{8} + 603214443311248 T^{9} - 2200806425 T^{10} - 113310 T^{11} + T^{12} \)
$71$ \( -\)\(44\!\cdots\!60\)\( - \)\(37\!\cdots\!56\)\( T + \)\(25\!\cdots\!90\)\( T^{2} + \)\(24\!\cdots\!42\)\( T^{3} - \)\(31\!\cdots\!25\)\( T^{4} - \)\(47\!\cdots\!78\)\( T^{5} - \)\(44\!\cdots\!98\)\( T^{6} + \)\(34\!\cdots\!70\)\( T^{7} + 24281846713175067204 T^{8} - 996634678671726 T^{9} - 10134997760 T^{10} + 84520 T^{11} + T^{12} \)
$73$ \( \)\(51\!\cdots\!20\)\( - \)\(81\!\cdots\!80\)\( T - \)\(12\!\cdots\!08\)\( T^{2} + \)\(32\!\cdots\!92\)\( T^{3} - \)\(16\!\cdots\!76\)\( T^{4} - \)\(34\!\cdots\!68\)\( T^{5} + \)\(46\!\cdots\!09\)\( T^{6} - \)\(63\!\cdots\!19\)\( T^{7} - 28108702555938854658 T^{8} + 786960487271757 T^{9} - 734641042 T^{10} - 135895 T^{11} + T^{12} \)
$79$ \( \)\(18\!\cdots\!76\)\( + \)\(41\!\cdots\!12\)\( T - \)\(34\!\cdots\!08\)\( T^{2} - \)\(65\!\cdots\!96\)\( T^{3} + \)\(58\!\cdots\!03\)\( T^{4} + \)\(42\!\cdots\!86\)\( T^{5} - \)\(39\!\cdots\!14\)\( T^{6} - \)\(14\!\cdots\!58\)\( T^{7} + \)\(12\!\cdots\!60\)\( T^{8} + 240034389357310 T^{9} - 18158812650 T^{10} - 14122 T^{11} + T^{12} \)
$83$ \( \)\(28\!\cdots\!32\)\( + \)\(90\!\cdots\!84\)\( T + \)\(82\!\cdots\!96\)\( T^{2} + \)\(22\!\cdots\!12\)\( T^{3} - \)\(25\!\cdots\!79\)\( T^{4} - \)\(16\!\cdots\!01\)\( T^{5} - \)\(41\!\cdots\!19\)\( T^{6} + \)\(36\!\cdots\!22\)\( T^{7} + \)\(22\!\cdots\!90\)\( T^{8} - 3396406944922658 T^{9} - 26706961559 T^{10} + 114463 T^{11} + T^{12} \)
$89$ \( \)\(19\!\cdots\!44\)\( + \)\(19\!\cdots\!56\)\( T - \)\(55\!\cdots\!64\)\( T^{2} + \)\(25\!\cdots\!80\)\( T^{3} + \)\(39\!\cdots\!00\)\( T^{4} - \)\(17\!\cdots\!36\)\( T^{5} + \)\(29\!\cdots\!81\)\( T^{6} + \)\(47\!\cdots\!13\)\( T^{7} + 43485356375723600330 T^{8} - 5599723928934319 T^{9} - 21725874894 T^{10} + 189109 T^{11} + T^{12} \)
$97$ \( -\)\(10\!\cdots\!92\)\( - \)\(21\!\cdots\!24\)\( T + \)\(84\!\cdots\!20\)\( T^{2} + \)\(13\!\cdots\!88\)\( T^{3} + \)\(73\!\cdots\!80\)\( T^{4} - \)\(27\!\cdots\!52\)\( T^{5} - \)\(20\!\cdots\!05\)\( T^{6} + \)\(20\!\cdots\!24\)\( T^{7} + \)\(19\!\cdots\!67\)\( T^{8} - 6778612636727092 T^{9} - 73673351755 T^{10} + 76192 T^{11} + T^{12} \)
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