Properties

Label 37.6.b.a
Level $37$
Weight $6$
Character orbit 37.b
Analytic conductor $5.934$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 37.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.93420133308\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 390 x^{14} + 60701 x^{12} + 4799932 x^{10} + 203487156 x^{8} + 4519465040 x^{6} + 48993644736 x^{4} + 211923220224 x^{2} + 178006118400\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -1 + \beta_{4} ) q^{3} + ( -17 + \beta_{2} ) q^{4} -\beta_{7} q^{5} + ( -3 \beta_{1} - \beta_{6} ) q^{6} + ( 12 - \beta_{2} - \beta_{4} + \beta_{5} ) q^{7} + ( -15 \beta_{1} + \beta_{3} ) q^{8} + ( 88 - 2 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{9} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -1 + \beta_{4} ) q^{3} + ( -17 + \beta_{2} ) q^{4} -\beta_{7} q^{5} + ( -3 \beta_{1} - \beta_{6} ) q^{6} + ( 12 - \beta_{2} - \beta_{4} + \beta_{5} ) q^{7} + ( -15 \beta_{1} + \beta_{3} ) q^{8} + ( 88 - 2 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{9} ) q^{9} + ( -5 + \beta_{2} + 3 \beta_{4} - \beta_{5} - \beta_{9} - \beta_{11} ) q^{10} + ( -71 + 5 \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{10} ) q^{11} + ( 86 - 4 \beta_{2} - 21 \beta_{4} + \beta_{5} + \beta_{8} + \beta_{11} ) q^{12} + ( -6 \beta_{1} + 2 \beta_{7} + \beta_{15} ) q^{13} + ( 30 \beta_{1} + 2 \beta_{6} - 3 \beta_{7} + \beta_{12} ) q^{14} + ( -26 \beta_{1} + \beta_{3} - 3 \beta_{6} + 3 \beta_{7} - \beta_{13} - \beta_{15} ) q^{15} + ( 181 - 6 \beta_{2} - 13 \beta_{4} + \beta_{8} - \beta_{10} ) q^{16} + ( -26 \beta_{1} + 3 \beta_{6} - 5 \beta_{7} + \beta_{13} - \beta_{14} ) q^{17} + ( 121 \beta_{1} - 2 \beta_{3} - 3 \beta_{6} - 15 \beta_{7} + 2 \beta_{12} + \beta_{14} - 2 \beta_{15} ) q^{18} + ( -18 \beta_{1} + 3 \beta_{3} - 3 \beta_{6} - 4 \beta_{7} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{19} + ( -24 \beta_{1} - 3 \beta_{3} - 7 \beta_{6} + 27 \beta_{7} - \beta_{12} - 2 \beta_{13} + 2 \beta_{15} ) q^{20} + ( -436 + 12 \beta_{2} + 46 \beta_{4} - \beta_{5} - 2 \beta_{8} - 3 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{21} + ( -208 \beta_{1} + 8 \beta_{3} + \beta_{6} + 7 \beta_{7} - 2 \beta_{12} - \beta_{14} - 2 \beta_{15} ) q^{22} + ( -18 \beta_{1} - 2 \beta_{3} + 7 \beta_{6} + 5 \beta_{7} - 3 \beta_{12} + \beta_{13} + 3 \beta_{15} ) q^{23} + ( 165 \beta_{1} - 9 \beta_{3} + 15 \beta_{6} - 16 \beta_{7} + \beta_{12} + 2 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{24} + ( -752 + 21 \beta_{2} + 36 \beta_{4} + 9 \beta_{5} - 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} ) q^{25} + ( 308 + \beta_{2} - 14 \beta_{4} - \beta_{5} - \beta_{8} + 11 \beta_{9} + 5 \beta_{10} + 5 \beta_{11} ) q^{26} + ( 288 - 16 \beta_{2} + 130 \beta_{4} - 6 \beta_{5} - 4 \beta_{8} + 2 \beta_{9} - 5 \beta_{10} - 4 \beta_{11} ) q^{27} + ( -1054 + 31 \beta_{2} + 69 \beta_{4} - 16 \beta_{5} - 3 \beta_{8} - 10 \beta_{9} + 5 \beta_{10} - 4 \beta_{11} ) q^{28} + ( 216 \beta_{1} - 10 \beta_{3} + 7 \beta_{6} - 17 \beta_{7} - 3 \beta_{13} + \beta_{14} - 7 \beta_{15} ) q^{29} + ( 1209 - 72 \beta_{2} - 143 \beta_{4} + 18 \beta_{5} + 6 \beta_{8} + 7 \beta_{9} - 5 \beta_{10} + 12 \beta_{11} ) q^{30} + ( 98 \beta_{1} - 16 \beta_{3} - 29 \beta_{6} + 30 \beta_{7} - 3 \beta_{12} + \beta_{13} - 2 \beta_{15} ) q^{31} + ( -69 \beta_{1} + 12 \beta_{3} + 32 \beta_{6} + 27 \beta_{7} + 2 \beta_{12} + \beta_{14} + 4 \beta_{15} ) q^{32} + ( -166 - 29 \beta_{2} - 155 \beta_{4} - 4 \beta_{5} + 8 \beta_{8} - 7 \beta_{9} - 10 \beta_{10} + 14 \beta_{11} ) q^{33} + ( 1306 - 26 \beta_{2} + 136 \beta_{4} - 4 \beta_{5} - 5 \beta_{8} + \beta_{9} - 10 \beta_{10} - 20 \beta_{11} ) q^{34} + ( 298 \beta_{1} - 13 \beta_{3} + 17 \beta_{6} + 18 \beta_{7} + 4 \beta_{12} + 3 \beta_{13} + 2 \beta_{15} ) q^{35} + ( -3276 + 169 \beta_{2} - 60 \beta_{4} - 73 \beta_{5} + 2 \beta_{8} - 34 \beta_{9} + 11 \beta_{10} - 13 \beta_{11} ) q^{36} + ( -699 - 200 \beta_{1} - 11 \beta_{2} + 16 \beta_{3} + 65 \beta_{4} - 19 \beta_{5} - 25 \beta_{6} - 32 \beta_{7} - 6 \beta_{8} - 2 \beta_{9} - 5 \beta_{10} + 4 \beta_{11} + 3 \beta_{12} - \beta_{13} + \beta_{15} ) q^{37} + ( 744 - 90 \beta_{2} - 194 \beta_{4} + 20 \beta_{5} + 7 \beta_{8} - 29 \beta_{9} - 8 \beta_{11} ) q^{38} + ( 136 \beta_{1} + 13 \beta_{3} + 44 \beta_{6} + 40 \beta_{7} + 4 \beta_{12} - 2 \beta_{13} - \beta_{14} - 10 \beta_{15} ) q^{39} + ( 1039 + 17 \beta_{2} - 259 \beta_{4} + 57 \beta_{5} + 5 \beta_{8} + 46 \beta_{9} + 10 \beta_{10} + 25 \beta_{11} ) q^{40} + ( 249 + 51 \beta_{2} + 92 \beta_{4} - 50 \beta_{5} + 2 \beta_{8} + 23 \beta_{9} + 15 \beta_{10} + 8 \beta_{11} ) q^{41} + ( -890 \beta_{1} + 22 \beta_{3} - 88 \beta_{6} + 61 \beta_{7} - 3 \beta_{12} - 4 \beta_{13} + 4 \beta_{15} ) q^{42} + ( -242 \beta_{1} + \beta_{3} + 41 \beta_{6} - 78 \beta_{7} - 5 \beta_{12} + \beta_{13} - \beta_{14} + 6 \beta_{15} ) q^{43} + ( 7909 - 328 \beta_{2} - 100 \beta_{4} + 79 \beta_{5} + 10 \beta_{8} + 22 \beta_{9} - 5 \beta_{10} - 5 \beta_{11} ) q^{44} + ( 1118 \beta_{1} - 64 \beta_{3} + 6 \beta_{6} - 41 \beta_{7} + 6 \beta_{12} - 2 \beta_{13} + 9 \beta_{15} ) q^{45} + ( 1154 - 19 \beta_{2} + 370 \beta_{4} + 109 \beta_{5} - 10 \beta_{8} + 40 \beta_{9} + \beta_{10} - 5 \beta_{11} ) q^{46} + ( 214 - \beta_{2} - 79 \beta_{4} + \beta_{5} - 8 \beta_{8} - 32 \beta_{9} + 10 \beta_{10} - 32 \beta_{11} ) q^{47} + ( -4942 + 321 \beta_{2} + 190 \beta_{4} - 53 \beta_{5} + 2 \beta_{8} - 12 \beta_{9} + 13 \beta_{10} - 13 \beta_{11} ) q^{48} + ( -2045 - 16 \beta_{2} - 84 \beta_{4} + 65 \beta_{5} + 2 \beta_{8} - 17 \beta_{9} + 5 \beta_{10} + 22 \beta_{11} ) q^{49} + ( -1662 \beta_{1} + 76 \beta_{3} - 97 \beta_{6} - 263 \beta_{7} + 5 \beta_{12} + 4 \beta_{13} - 12 \beta_{15} ) q^{50} + ( -802 \beta_{1} + 89 \beta_{3} + 123 \beta_{6} + 186 \beta_{7} - 12 \beta_{12} + \beta_{13} + 10 \beta_{15} ) q^{51} + ( 3 \beta_{1} + 39 \beta_{3} - 9 \beta_{6} - 296 \beta_{7} - \beta_{12} + 10 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{52} + ( -1562 + 224 \beta_{2} + 26 \beta_{4} - 71 \beta_{5} + 10 \beta_{8} + 7 \beta_{9} - 23 \beta_{10} - 14 \beta_{11} ) q^{53} + ( 487 \beta_{1} - 20 \beta_{3} - 242 \beta_{6} + 26 \beta_{7} + \beta_{12} - 8 \beta_{13} + 11 \beta_{14} - 2 \beta_{15} ) q^{54} + ( -804 \beta_{1} - 51 \beta_{3} - 108 \beta_{6} + 75 \beta_{7} - 16 \beta_{12} - 2 \beta_{13} - 11 \beta_{14} + 11 \beta_{15} ) q^{55} + ( -942 \beta_{1} + 59 \beta_{3} - 72 \beta_{6} + 175 \beta_{7} + 2 \beta_{12} - 8 \beta_{13} - 11 \beta_{14} + 4 \beta_{15} ) q^{56} + ( 1386 \beta_{1} - 60 \beta_{3} - 87 \beta_{6} + 107 \beta_{7} + 8 \beta_{12} + 19 \beta_{13} + \beta_{14} - 12 \beta_{15} ) q^{57} + ( -10322 + 399 \beta_{2} + 706 \beta_{4} + 11 \beta_{5} - 6 \beta_{8} - 60 \beta_{9} - 13 \beta_{10} - 15 \beta_{11} ) q^{58} + ( -1008 \beta_{1} + 68 \beta_{3} + 58 \beta_{6} + 126 \beta_{7} + 15 \beta_{12} - 4 \beta_{13} + 11 \beta_{14} + 2 \beta_{15} ) q^{59} + ( 2729 \beta_{1} - 61 \beta_{3} + 230 \beta_{6} - 404 \beta_{7} + 24 \beta_{12} - 8 \beta_{13} - 24 \beta_{15} ) q^{60} + ( 1798 \beta_{1} - 86 \beta_{3} + 77 \beta_{6} - 106 \beta_{7} - 8 \beta_{12} - 17 \beta_{13} + 11 \beta_{14} - 14 \beta_{15} ) q^{61} + ( -5329 + 313 \beta_{2} - 1391 \beta_{4} + 185 \beta_{5} + 17 \beta_{8} + 20 \beta_{9} - 10 \beta_{10} + 41 \beta_{11} ) q^{62} + ( 13782 - 508 \beta_{2} - 578 \beta_{4} - 64 \beta_{5} + 68 \beta_{9} + 24 \beta_{11} ) q^{63} + ( 10119 - 424 \beta_{2} + 991 \beta_{4} - 116 \beta_{5} + 7 \beta_{8} + 30 \beta_{9} - 5 \beta_{10} + 12 \beta_{11} ) q^{64} + ( 6394 - 643 \beta_{2} + 529 \beta_{4} - 73 \beta_{5} + 18 \beta_{8} - 60 \beta_{9} - 5 \beta_{10} ) q^{65} + ( 1461 \beta_{1} - 90 \beta_{3} + 408 \beta_{6} - 312 \beta_{7} - 7 \beta_{12} + 28 \beta_{13} - 11 \beta_{14} + 50 \beta_{15} ) q^{66} + ( 1589 - 332 \beta_{2} + 221 \beta_{4} + 22 \beta_{5} - 28 \beta_{8} + 2 \beta_{9} + 5 \beta_{10} - 4 \beta_{11} ) q^{67} + ( 822 \beta_{1} - 101 \beta_{3} - 262 \beta_{6} + 525 \beta_{7} + 22 \beta_{12} - 8 \beta_{13} - \beta_{14} + 8 \beta_{15} ) q^{68} + ( -2150 \beta_{1} + 58 \beta_{3} + 333 \beta_{6} + 362 \beta_{7} - 10 \beta_{12} + 3 \beta_{13} - 11 \beta_{14} - 28 \beta_{15} ) q^{69} + ( -13988 + 750 \beta_{2} + 854 \beta_{4} - 204 \beta_{5} - 39 \beta_{8} - 31 \beta_{9} + 40 \beta_{10} - 16 \beta_{11} ) q^{70} + ( -3400 + 217 \beta_{2} - 1811 \beta_{4} - \beta_{5} + 8 \beta_{8} + 32 \beta_{9} + 44 \beta_{10} + 32 \beta_{11} ) q^{71} + ( -3079 \beta_{1} + 38 \beta_{3} - 47 \beta_{6} + 825 \beta_{7} - 39 \beta_{12} - 26 \beta_{13} - 14 \beta_{15} ) q^{72} + ( -12533 + 774 \beta_{2} - 633 \beta_{4} - 177 \beta_{5} - 16 \beta_{8} - 13 \beta_{9} - 32 \beta_{10} + 8 \beta_{11} ) q^{73} + ( 8747 - 268 \beta_{1} - 476 \beta_{2} + 28 \beta_{3} - 1457 \beta_{4} - 130 \beta_{5} - 168 \beta_{6} - 301 \beta_{7} + 38 \beta_{8} - 31 \beta_{9} + 5 \beta_{10} + 8 \beta_{11} - 26 \beta_{12} + 8 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{74} + ( 13632 - 1046 \beta_{2} - 192 \beta_{4} + 196 \beta_{5} - 36 \beta_{8} + 38 \beta_{9} - 5 \beta_{10} - 12 \beta_{11} ) q^{75} + ( 3364 \beta_{1} - 47 \beta_{3} + 290 \beta_{6} + 729 \beta_{7} - 26 \beta_{12} + 16 \beta_{13} + 11 \beta_{14} + 72 \beta_{15} ) q^{76} + ( -15216 + 798 \beta_{2} - 140 \beta_{4} - 13 \beta_{5} - 26 \beta_{8} - 65 \beta_{9} - 45 \beta_{10} - 86 \beta_{11} ) q^{77} + ( -5369 - 169 \beta_{2} + 2119 \beta_{4} - 115 \beta_{5} - 24 \beta_{8} - 33 \beta_{9} - 50 \beta_{10} - 15 \beta_{11} ) q^{78} + ( -1796 \beta_{1} - 41 \beta_{3} - 166 \beta_{6} - 215 \beta_{7} + 21 \beta_{12} + 38 \beta_{13} - 11 \beta_{15} ) q^{79} + ( -780 \beta_{1} + 22 \beta_{3} + 199 \beta_{6} - 764 \beta_{7} + 41 \beta_{12} - 14 \beta_{13} + 11 \beta_{14} - 38 \beta_{15} ) q^{80} + ( 20301 - 1159 \beta_{2} + 1411 \beta_{4} + 79 \beta_{5} - 50 \beta_{8} - 16 \beta_{9} + 5 \beta_{10} - 20 \beta_{11} ) q^{81} + ( -991 \beta_{1} + 50 \beta_{3} - 125 \beta_{6} - 356 \beta_{7} - 48 \beta_{12} + 16 \beta_{13} - 72 \beta_{15} ) q^{82} + ( -13480 + 361 \beta_{2} + 881 \beta_{4} + 145 \beta_{5} + 60 \beta_{8} - 42 \beta_{9} + 8 \beta_{10} + 60 \beta_{11} ) q^{83} + ( 27324 - 1241 \beta_{2} - 3531 \beta_{4} + 270 \beta_{5} + 49 \beta_{8} + 74 \beta_{9} - 45 \beta_{10} + 130 \beta_{11} ) q^{84} + ( -21700 + 1186 \beta_{2} + 1690 \beta_{4} + 46 \beta_{5} + 4 \beta_{8} + 72 \beta_{9} - 86 \beta_{10} + 48 \beta_{11} ) q^{85} + ( 12696 - 240 \beta_{2} + 2378 \beta_{4} + 82 \beta_{5} - 43 \beta_{8} + 17 \beta_{9} - 6 \beta_{10} - 118 \beta_{11} ) q^{86} + ( -5866 \beta_{1} - 124 \beta_{3} - 501 \beta_{6} - 468 \beta_{7} - 40 \beta_{12} - 37 \beta_{13} + \beta_{14} + 48 \beta_{15} ) q^{87} + ( 10120 \beta_{1} - 155 \beta_{3} + 305 \beta_{6} + 233 \beta_{7} + 57 \beta_{12} - 10 \beta_{13} - 78 \beta_{15} ) q^{88} + ( 3082 \beta_{1} - 191 \beta_{6} - 373 \beta_{7} + 8 \beta_{12} - 37 \beta_{13} - 11 \beta_{14} + 52 \beta_{15} ) q^{89} + ( -54161 + 2870 \beta_{2} + 1177 \beta_{4} - 314 \beta_{5} - 83 \beta_{8} + 24 \beta_{9} + 141 \beta_{10} + 4 \beta_{11} ) q^{90} + ( 134 \beta_{1} + 89 \beta_{3} - 101 \beta_{6} - 180 \beta_{7} - 4 \beta_{12} + 17 \beta_{13} - 44 \beta_{14} ) q^{91} + ( -1793 \beta_{1} + 70 \beta_{3} - 377 \beta_{6} - 733 \beta_{7} + 47 \beta_{12} + 22 \beta_{13} + 44 \beta_{14} - 6 \beta_{15} ) q^{92} + ( 8922 \beta_{1} + 12 \beta_{3} - 84 \beta_{6} - 259 \beta_{7} + 28 \beta_{12} + 20 \beta_{13} + 44 \beta_{14} - 43 \beta_{15} ) q^{93} + ( 282 \beta_{1} + 24 \beta_{3} - 168 \beta_{6} + 1581 \beta_{7} - 17 \beta_{12} - 64 \beta_{13} - 10 \beta_{14} + 28 \beta_{15} ) q^{94} + ( -8382 + 1290 \beta_{2} + 1264 \beta_{4} + 278 \beta_{5} + 112 \beta_{8} + 256 \beta_{9} + 96 \beta_{10} + 160 \beta_{11} ) q^{95} + ( -8929 \beta_{1} - 28 \beta_{3} + 233 \beta_{6} + 477 \beta_{7} - 31 \beta_{12} + 38 \beta_{13} - 44 \beta_{14} + 66 \beta_{15} ) q^{96} + ( 736 \beta_{1} + 184 \beta_{3} + 260 \beta_{6} - 318 \beta_{7} - 60 \beta_{12} + 20 \beta_{13} + 68 \beta_{15} ) q^{97} + ( -1931 \beta_{1} + 154 \beta_{3} + 370 \beta_{6} - 877 \beta_{7} + 23 \beta_{12} + 44 \beta_{13} - 44 \beta_{14} + 28 \beta_{15} ) q^{98} + ( -37160 + 2741 \beta_{2} - 3449 \beta_{4} - 289 \beta_{5} - 216 \beta_{9} + 55 \beta_{10} - 216 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 18q^{3} - 268q^{4} + 190q^{7} + 1394q^{9} + O(q^{10}) \) \( 16q - 18q^{3} - 268q^{4} + 190q^{7} + 1394q^{9} - 74q^{10} - 1110q^{11} + 1402q^{12} + 2900q^{16} - 7010q^{21} - 12052q^{25} + 4902q^{26} + 4266q^{27} - 16824q^{28} + 19280q^{30} - 2478q^{33} + 20556q^{34} - 51402q^{36} - 11400q^{37} + 12108q^{38} + 16966q^{40} + 3918q^{41} + 125394q^{44} + 17470q^{46} + 3822q^{47} - 78034q^{48} - 32618q^{49} - 24126q^{53} - 164718q^{58} - 81426q^{62} + 219268q^{63} + 158076q^{64} + 98976q^{65} + 23560q^{67} - 222404q^{70} - 50046q^{71} - 196274q^{73} + 141216q^{74} + 214054q^{75} - 239574q^{77} - 90822q^{78} + 317312q^{81} - 215814q^{83} + 438572q^{84} - 346472q^{85} + 197640q^{86} - 857612q^{90} - 132504q^{95} - 574860q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 390 x^{14} + 60701 x^{12} + 4799932 x^{10} + 203487156 x^{8} + 4519465040 x^{6} + 48993644736 x^{4} + 211923220224 x^{2} + 178006118400\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 49 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 79 \nu \)
\(\beta_{4}\)\(=\)\((\)\(-674098535 \nu^{14} - 245321490958 \nu^{12} - 34551894466035 \nu^{10} - 2345419900098968 \nu^{8} - 77024862646765324 \nu^{6} - 1048010799769888512 \nu^{4} - 3811004402134104384 \nu^{2} + 2806943637099303936\)\()/ 281490730186825728 \)
\(\beta_{5}\)\(=\)\((\)\(-4384882567 \nu^{14} - 1545295917902 \nu^{12} - 210316644411027 \nu^{10} - 13749061517048728 \nu^{8} - 433442391758304140 \nu^{6} - 5742848407425817344 \nu^{4} - 25543812247940921664 \nu^{2} - 30122943449991736320\)\()/ 281490730186825728 \)
\(\beta_{6}\)\(=\)\((\)\(674098535 \nu^{15} + 245321490958 \nu^{13} + 34551894466035 \nu^{11} + 2345419900098968 \nu^{9} + 77024862646765324 \nu^{7} + 1048010799769888512 \nu^{5} + 3811004402134104384 \nu^{3} - 3369925097472955392 \nu\)\()/ 281490730186825728 \)
\(\beta_{7}\)\(=\)\((\)\(7252561847 \nu^{15} + 2718118676590 \nu^{13} + 401348460870147 \nu^{11} + 29424453299549144 \nu^{9} + 1107364244835972172 \nu^{7} + 20072925997007639040 \nu^{5} + 155570009624238610752 \nu^{3} + 380234029172397493248 \nu\)\()/ 2814907301868257280 \)
\(\beta_{8}\)\(=\)\((\)\(10784407991 \nu^{14} + 4028491445614 \nu^{12} + 594960727700739 \nu^{10} + 43918856194199000 \nu^{8} + 1683896535217190476 \nu^{6} + 31746127134088266240 \nu^{4} + 263317078101435830592 \nu^{2} + 614266126630313978880\)\()/ 281490730186825728 \)
\(\beta_{9}\)\(=\)\((\)\(15874352675 \nu^{14} + 6039075706054 \nu^{12} + 893613182617311 \nu^{10} + 64200426182579768 \nu^{8} + 2266108332782959804 \nu^{6} + 34955184490513996800 \nu^{4} + 191965017593430952512 \nu^{2} + 215254558030479694848\)\()/ 281490730186825728 \)
\(\beta_{10}\)\(=\)\((\)\(9773844473 \nu^{14} + 3608835414034 \nu^{12} + 522067677879597 \nu^{10} + 37204657447742792 \nu^{8} + 1342609874812569844 \nu^{6} + 22544388400454995584 \nu^{4} + 142074040425061481664 \nu^{2} + 128860449562801087488\)\()/ 140745365093412864 \)
\(\beta_{11}\)\(=\)\((\)\(-8183270029 \nu^{14} - 3039557947162 \nu^{12} - 441895753596305 \nu^{10} - 31443839426081480 \nu^{8} - 1111405967019873540 \nu^{6} - 17444103791917527552 \nu^{4} - 97749317017820629440 \nu^{2} - 97808409941670061056\)\()/ 93830243395608576 \)
\(\beta_{12}\)\(=\)\((\)\(-9610708493 \nu^{15} - 3250606019610 \nu^{13} - 414880002053393 \nu^{11} - 24223818090943176 \nu^{9} - 594193269847592708 \nu^{7} - 2563271360311380480 \nu^{5} + 56782318356699104832 \nu^{3} + 230067643123522649088 \nu\)\()/ 938302433956085760 \)
\(\beta_{13}\)\(=\)\((\)\(647921873 \nu^{15} + 259554609880 \nu^{13} + 41451572192793 \nu^{11} + 3360687888477566 \nu^{9} + 145930357949679748 \nu^{7} + 3301879613634737880 \nu^{5} + 35316625895261654208 \nu^{3} + 128356180397222723712 \nu\)\()/ 43982926591691520 \)
\(\beta_{14}\)\(=\)\((\)\(5239827795 \nu^{15} + 1952240948390 \nu^{13} + 281406349626831 \nu^{11} + 19480262373745208 \nu^{9} + 646753974914742396 \nu^{7} + 8888361500104240640 \nu^{5} + 45166675995135581760 \nu^{3} + 176878509073032723456 \nu\)\()/ 187660486791217152 \)
\(\beta_{15}\)\(=\)\((\)\(-108115966759 \nu^{15} - 40418014870670 \nu^{13} - 5906207476176819 \nu^{11} - 422963908545003928 \nu^{9} - 15170735165542933004 \nu^{7} - 248115836220849411840 \nu^{5} - 1527894117896966601024 \nu^{3} - 1762108634575731280896 \nu\)\()/ 2814907301868257280 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 49\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 79 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{10} + \beta_{8} - 13 \beta_{4} - 102 \beta_{2} + 3861\)
\(\nu^{5}\)\(=\)\(4 \beta_{15} + \beta_{14} + 2 \beta_{12} + 27 \beta_{7} + 32 \beta_{6} - 116 \beta_{3} + 6971 \beta_{1}\)
\(\nu^{6}\)\(=\)\(12 \beta_{11} + 155 \beta_{10} + 30 \beta_{9} - 153 \beta_{8} - 116 \beta_{5} + 3071 \beta_{4} + 9752 \beta_{2} - 339353\)
\(\nu^{7}\)\(=\)\(-676 \beta_{15} - 137 \beta_{14} + 24 \beta_{13} - 406 \beta_{12} - 4663 \beta_{7} - 6112 \beta_{6} + 11804 \beta_{3} - 640347 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-1612 \beta_{11} - 18471 \beta_{10} - 5614 \beta_{9} + 18837 \beta_{8} + 22500 \beta_{5} - 454123 \beta_{4} - 932740 \beta_{2} + 31058425\)
\(\nu^{9}\)\(=\)\(85844 \beta_{15} + 14469 \beta_{14} - 3224 \beta_{13} + 55806 \beta_{12} + 580859 \beta_{7} + 841696 \beta_{6} - 1173328 \beta_{3} + 60132987 \beta_{1}\)
\(\nu^{10}\)\(=\)\(124924 \beta_{11} + 2015023 \beta_{10} + 729814 \beta_{9} - 2138981 \beta_{8} - 3077108 \beta_{5} + 56865011 \beta_{4} + 89921244 \beta_{2} - 2907542169\)
\(\nu^{11}\)\(=\)\(-9767636 \beta_{15} - 1410133 \beta_{14} + 249848 \beta_{13} - 6626222 \beta_{12} - 63271419 \beta_{7} - 102022656 \beta_{6} + 116063920 \beta_{3} - 5733574315 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-3656924 \beta_{11} - 210974255 \beta_{10} - 81252086 \beta_{9} + 232820453 \beta_{8} + 369064788 \beta_{5} - 6574085779 \beta_{4} - 8735801996 \beta_{2} + 276525368009\)
\(\nu^{13}\)\(=\)\(1050093588 \beta_{15} + 133379093 \beta_{14} - 7313848 \beta_{13} + 735264334 \beta_{12} + 6402272347 \beta_{7} + 11588553888 \beta_{6} - 11472214368 \beta_{3} + 552923669851 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-834774532 \beta_{11} + 21606729663 \beta_{10} + 8267141558 \beta_{9} - 24705433237 \beta_{8} - 41621032244 \beta_{5} + 726717324611 \beta_{4} + 854084680204 \beta_{2} - 26612991785209\)
\(\nu^{15}\)\(=\)\(-109158608916 \beta_{15} - 12504813573 \beta_{14} - 1669549064 \beta_{13} - 78831279054 \beta_{12} - 617054034251 \beta_{7} - 1267551660224 \beta_{6} + 1134338659296 \beta_{3} - 53781326022347 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1
10.0606i
9.87320i
8.69480i
8.11553i
4.68483i
4.32146i
2.85171i
1.04262i
1.04262i
2.85171i
4.32146i
4.68483i
8.11553i
8.69480i
9.87320i
10.0606i
10.0606i −25.7580 −69.2166 77.4051i 259.142i 168.370 374.424i 420.475 −778.746
36.2 9.87320i −2.75305 −65.4800 100.817i 27.1814i −37.8971 330.555i −235.421 995.382
36.3 8.69480i 28.4744 −43.5996 14.1700i 247.579i 47.5239 100.856i 567.789 −123.205
36.4 8.11553i 2.07043 −33.8618 43.8828i 16.8026i −152.059 15.1093i −238.713 −356.132
36.5 4.68483i 5.81584 10.0523 4.49750i 27.2462i 210.637 197.008i −209.176 −21.0700
36.6 4.32146i −24.1711 13.3250 61.4962i 104.454i 41.8425 195.870i 341.241 265.753
36.7 2.85171i −12.3080 23.8677 38.2713i 35.0990i −96.3094 159.319i −91.5127 −109.139
36.8 1.04262i 19.6295 30.9129 86.4713i 20.4662i −87.1072 65.5944i 142.318 90.1568
36.9 1.04262i 19.6295 30.9129 86.4713i 20.4662i −87.1072 65.5944i 142.318 90.1568
36.10 2.85171i −12.3080 23.8677 38.2713i 35.0990i −96.3094 159.319i −91.5127 −109.139
36.11 4.32146i −24.1711 13.3250 61.4962i 104.454i 41.8425 195.870i 341.241 265.753
36.12 4.68483i 5.81584 10.0523 4.49750i 27.2462i 210.637 197.008i −209.176 −21.0700
36.13 8.11553i 2.07043 −33.8618 43.8828i 16.8026i −152.059 15.1093i −238.713 −356.132
36.14 8.69480i 28.4744 −43.5996 14.1700i 247.579i 47.5239 100.856i 567.789 −123.205
36.15 9.87320i −2.75305 −65.4800 100.817i 27.1814i −37.8971 330.555i −235.421 995.382
36.16 10.0606i −25.7580 −69.2166 77.4051i 259.142i 168.370 374.424i 420.475 −778.746
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.6.b.a 16
3.b odd 2 1 333.6.c.d 16
4.b odd 2 1 592.6.g.c 16
37.b even 2 1 inner 37.6.b.a 16
111.d odd 2 1 333.6.c.d 16
148.b odd 2 1 592.6.g.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.6.b.a 16 1.a even 1 1 trivial
37.6.b.a 16 37.b even 2 1 inner
333.6.c.d 16 3.b odd 2 1
333.6.c.d 16 111.d odd 2 1
592.6.g.c 16 4.b odd 2 1
592.6.g.c 16 148.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(37, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 178006118400 + 211923220224 T^{2} + 48993644736 T^{4} + 4519465040 T^{6} + 203487156 T^{8} + 4799932 T^{10} + 60701 T^{12} + 390 T^{14} + T^{16} \)
$3$ \( ( 141986196 - 30714957 T - 25673805 T^{2} + 2751084 T^{3} + 422488 T^{4} - 11016 T^{5} - 1280 T^{6} + 9 T^{7} + T^{8} )^{2} \)
$5$ \( \)\(19\!\cdots\!44\)\( + \)\(11\!\cdots\!84\)\( T^{2} + \)\(68\!\cdots\!56\)\( T^{4} + 10475224449621004436 T^{6} + 6834316986168825 T^{8} + 2220056867434 T^{10} + 373650779 T^{12} + 31026 T^{14} + T^{16} \)
$7$ \( ( 3409346511153792 - 2731120576272 T - 3494999585616 T^{2} + 534484632 T^{3} + 907038560 T^{4} + 2410919 T^{5} - 54561 T^{6} - 95 T^{7} + T^{8} )^{2} \)
$11$ \( ( 64484644334705602164 - 2413591630240011543 T - 317735542999485 T^{2} + 80459289952272 T^{3} + 117553655916 T^{4} - 411615756 T^{5} - 697116 T^{6} + 555 T^{7} + T^{8} )^{2} \)
$13$ \( \)\(85\!\cdots\!00\)\( + \)\(11\!\cdots\!56\)\( T^{2} + \)\(42\!\cdots\!24\)\( T^{4} + \)\(58\!\cdots\!84\)\( T^{6} + \)\(38\!\cdots\!57\)\( T^{8} + 1350468104982414570 T^{10} + 2576662431099 T^{12} + 2530866 T^{14} + T^{16} \)
$17$ \( \)\(13\!\cdots\!96\)\( + \)\(69\!\cdots\!60\)\( T^{2} + \)\(14\!\cdots\!60\)\( T^{4} + \)\(16\!\cdots\!08\)\( T^{6} + \)\(11\!\cdots\!32\)\( T^{8} + \)\(48\!\cdots\!64\)\( T^{10} + 123283110665216 T^{12} + 17116188 T^{14} + T^{16} \)
$19$ \( \)\(17\!\cdots\!36\)\( + \)\(23\!\cdots\!92\)\( T^{2} + \)\(80\!\cdots\!60\)\( T^{4} + \)\(10\!\cdots\!76\)\( T^{6} + \)\(66\!\cdots\!44\)\( T^{8} + \)\(20\!\cdots\!56\)\( T^{10} + 346874082231120 T^{12} + 29435112 T^{14} + T^{16} \)
$23$ \( \)\(14\!\cdots\!76\)\( + \)\(74\!\cdots\!40\)\( T^{2} + \)\(95\!\cdots\!80\)\( T^{4} + \)\(47\!\cdots\!64\)\( T^{6} + \)\(12\!\cdots\!17\)\( T^{8} + \)\(17\!\cdots\!14\)\( T^{10} + 1388084473811267 T^{12} + 58780098 T^{14} + T^{16} \)
$29$ \( \)\(13\!\cdots\!04\)\( + \)\(49\!\cdots\!68\)\( T^{2} + \)\(37\!\cdots\!72\)\( T^{4} + \)\(97\!\cdots\!24\)\( T^{6} + \)\(11\!\cdots\!41\)\( T^{8} + \)\(57\!\cdots\!38\)\( T^{10} + 15212305941055715 T^{12} + 196771278 T^{14} + T^{16} \)
$31$ \( \)\(28\!\cdots\!00\)\( + \)\(93\!\cdots\!44\)\( T^{2} + \)\(13\!\cdots\!72\)\( T^{4} + \)\(10\!\cdots\!76\)\( T^{6} + \)\(47\!\cdots\!73\)\( T^{8} + \)\(13\!\cdots\!90\)\( T^{10} + 24270609343777899 T^{12} + 238357770 T^{14} + T^{16} \)
$37$ \( \)\(53\!\cdots\!01\)\( + \)\(87\!\cdots\!00\)\( T + \)\(30\!\cdots\!88\)\( T^{2} - \)\(10\!\cdots\!24\)\( T^{3} - \)\(21\!\cdots\!92\)\( T^{4} - \)\(10\!\cdots\!40\)\( T^{5} + \)\(20\!\cdots\!28\)\( T^{6} + \)\(18\!\cdots\!84\)\( T^{7} + \)\(23\!\cdots\!62\)\( T^{8} + \)\(26\!\cdots\!12\)\( T^{9} + \)\(41\!\cdots\!72\)\( T^{10} - 3210736144030661880 T^{11} - 9199071723822692 T^{12} - 648232575432 T^{13} + 27632512 T^{14} + 11400 T^{15} + T^{16} \)
$41$ \( ( \)\(14\!\cdots\!82\)\( - \)\(16\!\cdots\!91\)\( T - \)\(19\!\cdots\!83\)\( T^{2} + 7727486445914500188 T^{3} + 51041738859331086 T^{4} + 399403138806 T^{5} - 424066746 T^{6} - 1959 T^{7} + T^{8} )^{2} \)
$43$ \( \)\(44\!\cdots\!76\)\( + \)\(17\!\cdots\!64\)\( T^{2} + \)\(16\!\cdots\!76\)\( T^{4} + \)\(70\!\cdots\!52\)\( T^{6} + \)\(16\!\cdots\!28\)\( T^{8} + \)\(22\!\cdots\!52\)\( T^{10} + 173382315537048576 T^{12} + 682918464 T^{14} + T^{16} \)
$47$ \( ( -\)\(24\!\cdots\!68\)\( - \)\(12\!\cdots\!60\)\( T - \)\(14\!\cdots\!56\)\( T^{2} + \)\(14\!\cdots\!36\)\( T^{3} + 256231707047151876 T^{4} - 1806635864565 T^{5} - 964309845 T^{6} - 1911 T^{7} + T^{8} )^{2} \)
$53$ \( ( -\)\(39\!\cdots\!16\)\( - \)\(35\!\cdots\!76\)\( T - \)\(29\!\cdots\!68\)\( T^{2} + \)\(59\!\cdots\!84\)\( T^{3} + 490870446314379822 T^{4} - 16364993513751 T^{5} - 1355283219 T^{6} + 12063 T^{7} + T^{8} )^{2} \)
$59$ \( \)\(18\!\cdots\!96\)\( + \)\(27\!\cdots\!12\)\( T^{2} + \)\(12\!\cdots\!16\)\( T^{4} + \)\(21\!\cdots\!12\)\( T^{6} + \)\(15\!\cdots\!80\)\( T^{8} + \)\(50\!\cdots\!72\)\( T^{10} + 7649086741655474192 T^{12} + 4831145364 T^{14} + T^{16} \)
$61$ \( \)\(84\!\cdots\!00\)\( + \)\(17\!\cdots\!44\)\( T^{2} + \)\(71\!\cdots\!04\)\( T^{4} + \)\(68\!\cdots\!20\)\( T^{6} + \)\(25\!\cdots\!73\)\( T^{8} + \)\(39\!\cdots\!46\)\( T^{10} + 27143242905303496611 T^{12} + 8542615374 T^{14} + T^{16} \)
$67$ \( ( -\)\(47\!\cdots\!72\)\( - \)\(38\!\cdots\!48\)\( T - \)\(46\!\cdots\!32\)\( T^{2} - \)\(12\!\cdots\!60\)\( T^{3} + 127652522570507585 T^{4} + 35487139320284 T^{5} - 1938067731 T^{6} - 11780 T^{7} + T^{8} )^{2} \)
$71$ \( ( -\)\(18\!\cdots\!72\)\( - \)\(44\!\cdots\!08\)\( T - \)\(22\!\cdots\!56\)\( T^{2} + \)\(13\!\cdots\!12\)\( T^{3} + 8460152638657159680 T^{4} - 103882701967335 T^{5} - 5909294721 T^{6} + 25023 T^{7} + T^{8} )^{2} \)
$73$ \( ( \)\(51\!\cdots\!46\)\( + \)\(14\!\cdots\!69\)\( T - \)\(24\!\cdots\!07\)\( T^{2} - \)\(30\!\cdots\!20\)\( T^{3} - 22073237184393354442 T^{4} - 539068556569978 T^{5} - 1909438950 T^{6} + 98137 T^{7} + T^{8} )^{2} \)
$79$ \( \)\(21\!\cdots\!64\)\( + \)\(54\!\cdots\!60\)\( T^{2} + \)\(44\!\cdots\!16\)\( T^{4} + \)\(13\!\cdots\!80\)\( T^{6} + \)\(13\!\cdots\!73\)\( T^{8} + \)\(68\!\cdots\!98\)\( T^{10} + \)\(17\!\cdots\!11\)\( T^{12} + 21090741462 T^{14} + T^{16} \)
$83$ \( ( -\)\(15\!\cdots\!40\)\( + \)\(94\!\cdots\!64\)\( T + \)\(86\!\cdots\!36\)\( T^{2} + \)\(19\!\cdots\!04\)\( T^{3} - 17247312935287483284 T^{4} - 1033010175019203 T^{5} - 5935745181 T^{6} + 107907 T^{7} + T^{8} )^{2} \)
$89$ \( \)\(37\!\cdots\!00\)\( + \)\(50\!\cdots\!24\)\( T^{2} + \)\(19\!\cdots\!68\)\( T^{4} + \)\(22\!\cdots\!88\)\( T^{6} + \)\(12\!\cdots\!08\)\( T^{8} + \)\(33\!\cdots\!84\)\( T^{10} + \)\(50\!\cdots\!00\)\( T^{12} + 36586542516 T^{14} + T^{16} \)
$97$ \( \)\(81\!\cdots\!96\)\( + \)\(32\!\cdots\!72\)\( T^{2} + \)\(31\!\cdots\!00\)\( T^{4} + \)\(13\!\cdots\!48\)\( T^{6} + \)\(26\!\cdots\!32\)\( T^{8} + \)\(22\!\cdots\!36\)\( T^{10} + \)\(48\!\cdots\!76\)\( T^{12} + 38390541864 T^{14} + T^{16} \)
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