Properties

Label 756.2.w.a
Level $756$
Weight $2$
Character orbit 756.w
Analytic conductor $6.037$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.w (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} - 156 x^{7} + 558 x^{6} - 837 x^{5} + 1782 x^{4} - 4131 x^{3} + 3645 x^{2} - 4374 x + 6561\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{5} + ( \beta_{4} - \beta_{10} ) q^{7} +O(q^{10})\) \( q + \beta_{9} q^{5} + ( \beta_{4} - \beta_{10} ) q^{7} -\beta_{15} q^{11} + ( -\beta_{9} - \beta_{13} ) q^{13} + ( -2 \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{17} + ( -1 + \beta_{2} - 2 \beta_{7} - 2 \beta_{9} + \beta_{11} + 2 \beta_{12} - \beta_{15} ) q^{19} + ( -1 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{23} + ( 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - \beta_{10} - \beta_{14} + \beta_{15} ) q^{25} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{10} + 2 \beta_{12} + \beta_{13} ) q^{29} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{14} ) q^{31} + ( -\beta_{2} + 3 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{12} - \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{35} + ( 1 + \beta_{7} - 2 \beta_{11} + \beta_{12} ) q^{37} + ( 1 - \beta_{3} + 3 \beta_{4} - 2 \beta_{6} - \beta_{7} - \beta_{10} - \beta_{12} + \beta_{15} ) q^{41} + ( -\beta_{3} + \beta_{4} + \beta_{6} + 3 \beta_{9} - \beta_{10} - \beta_{14} + \beta_{15} ) q^{43} + ( 3 - \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{10} - 2 \beta_{11} - 2 \beta_{13} - \beta_{14} ) q^{47} + ( -2 - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - \beta_{9} + 3 \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{49} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{53} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{14} ) q^{55} + ( -1 - \beta_{2} + 4 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{11} + 4 \beta_{14} - 2 \beta_{15} ) q^{59} + ( -1 + \beta_{2} + \beta_{4} - \beta_{6} - 2 \beta_{7} - \beta_{9} + \beta_{11} ) q^{61} + ( 2 - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{9} - \beta_{11} ) q^{65} + ( 5 - 4 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{7} - 3 \beta_{8} + 3 \beta_{9} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} + 2 \beta_{15} ) q^{67} + ( 1 + \beta_{1} + 3 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{14} ) q^{71} + ( 1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{9} + \beta_{10} + \beta_{11} - 3 \beta_{12} + \beta_{13} ) q^{73} + ( -5 + 4 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{13} + 3 \beta_{14} - 2 \beta_{15} ) q^{77} + ( \beta_{2} - \beta_{7} - 3 \beta_{9} + 2 \beta_{11} + 2 \beta_{12} ) q^{79} + ( -\beta_{1} - 2 \beta_{3} + \beta_{4} - 2 \beta_{10} + \beta_{13} - \beta_{14} ) q^{83} + ( -3 - 2 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} - 4 \beta_{9} - 2 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{85} + ( -5 + 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{14} - \beta_{15} ) q^{89} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + 3 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{91} + ( 3 + \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + 3 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} + 2 \beta_{14} ) q^{95} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - 4 \beta_{4} + 2 \beta_{6} + 3 \beta_{10} - 3 \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - q^{7} + O(q^{10}) \) \( 16q - q^{7} + 6q^{11} - 3q^{13} - 9q^{17} - 21q^{23} - 8q^{25} - 6q^{29} + 15q^{35} + q^{37} + 6q^{41} - 2q^{43} + 36q^{47} - 5q^{49} + 30q^{59} + 14q^{67} - 3q^{77} + 2q^{79} + 6q^{85} - 21q^{89} + 9q^{91} - 3q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} - 156 x^{7} + 558 x^{6} - 837 x^{5} + 1782 x^{4} - 4131 x^{3} + 3645 x^{2} - 4374 x + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(19 \nu^{15} + 139 \nu^{14} + 1928 \nu^{13} + 8221 \nu^{12} + 10009 \nu^{11} + 14762 \nu^{10} - 23272 \nu^{9} - 19426 \nu^{8} - 26486 \nu^{7} - 17106 \nu^{6} - 123732 \nu^{5} - 231723 \nu^{4} - 63747 \nu^{3} + 2064528 \nu^{2} + 2597427 \nu + 4788801\)\()/103518\)
\(\beta_{2}\)\(=\)\((\)\(1307 \nu^{15} - 5068 \nu^{14} - 824 \nu^{13} - 49267 \nu^{12} - 2716 \nu^{11} - 77018 \nu^{10} + 113602 \nu^{9} + 7210 \nu^{8} + 181946 \nu^{7} - 84090 \nu^{6} + 1174032 \nu^{5} - 900801 \nu^{4} + 2054484 \nu^{3} - 11094408 \nu^{2} - 4573017 \nu - 19166868\)\()/621108\)
\(\beta_{3}\)\(=\)\((\)\(-3695 \nu^{15} + 20725 \nu^{14} - 51544 \nu^{13} + 99223 \nu^{12} - 215537 \nu^{11} + 360098 \nu^{10} - 187876 \nu^{9} + 298928 \nu^{8} - 711356 \nu^{7} + 844320 \nu^{6} - 2586978 \nu^{5} + 9488205 \nu^{4} - 18766647 \nu^{3} + 20620980 \nu^{2} - 33241671 \nu + 46300977\)\()/1242216\)
\(\beta_{4}\)\(=\)\((\)\(-6677 \nu^{15} + 21577 \nu^{14} - 16612 \nu^{13} + 91129 \nu^{12} - 145673 \nu^{11} + 11630 \nu^{10} - 101824 \nu^{9} + 277628 \nu^{8} - 214640 \nu^{7} + 202764 \nu^{6} - 3624714 \nu^{5} + 7713063 \nu^{4} - 3514995 \nu^{3} + 15997176 \nu^{2} - 16161201 \nu - 5872095\)\()/1242216\)
\(\beta_{5}\)\(=\)\((\)\(1669 \nu^{15} + 1930 \nu^{14} + 23477 \nu^{13} + 5248 \nu^{12} + 44857 \nu^{11} - 82192 \nu^{10} - 18340 \nu^{9} - 86836 \nu^{8} + 54754 \nu^{7} - 419040 \nu^{6} + 118692 \nu^{5} - 910737 \nu^{4} + 6259194 \nu^{3} + 2423439 \nu^{2} + 11432178 \nu - 4533651\)\()/310554\)
\(\beta_{6}\)\(=\)\((\)\(2739 \nu^{15} - 9229 \nu^{14} + 1724 \nu^{13} - 88319 \nu^{12} - 8827 \nu^{11} - 147394 \nu^{10} + 195292 \nu^{9} + 28864 \nu^{8} + 332068 \nu^{7} - 278440 \nu^{6} + 2194674 \nu^{5} - 1523241 \nu^{4} + 4418847 \nu^{3} - 19275408 \nu^{2} - 9423297 \nu - 35170605\)\()/414072\)
\(\beta_{7}\)\(=\)\((\)\(-8405 \nu^{15} - 16535 \nu^{14} - 52096 \nu^{13} - 27863 \nu^{12} - 20561 \nu^{11} + 66614 \nu^{10} + 87524 \nu^{9} + 178940 \nu^{8} + 318940 \nu^{7} + 1092156 \nu^{6} - 981702 \nu^{5} - 2209221 \nu^{4} - 11460447 \nu^{3} - 13165740 \nu^{2} - 15432201 \nu - 11234619\)\()/621108\)
\(\beta_{8}\)\(=\)\((\)\(-4934 \nu^{15} + 6616 \nu^{14} - 21253 \nu^{13} + 86401 \nu^{12} - 24581 \nu^{11} + 193856 \nu^{10} - 186610 \nu^{9} + 38834 \nu^{8} - 305642 \nu^{7} + 605064 \nu^{6} - 2581722 \nu^{5} + 1955178 \nu^{4} - 9141984 \nu^{3} + 17226513 \nu^{2} + 614547 \nu + 34596153\)\()/310554\)
\(\beta_{9}\)\(=\)\((\)\(13862 \nu^{15} - 1333 \nu^{14} + 45760 \nu^{13} - 164782 \nu^{12} - 15775 \nu^{11} - 343040 \nu^{10} + 361210 \nu^{9} - 100070 \nu^{8} + 284726 \nu^{7} - 1370658 \nu^{6} + 5698206 \nu^{5} - 63018 \nu^{4} + 18020475 \nu^{3} - 29851092 \nu^{2} - 7676370 \nu - 61443765\)\()/621108\)
\(\beta_{10}\)\(=\)\((\)\(27959 \nu^{15} + 45263 \nu^{14} + 129088 \nu^{13} - 58111 \nu^{12} - 95755 \nu^{11} - 442826 \nu^{10} + 157996 \nu^{9} - 326936 \nu^{8} - 426532 \nu^{7} - 3257256 \nu^{6} + 5542434 \nu^{5} + 10347075 \nu^{4} + 36196875 \nu^{3} + 5383908 \nu^{2} + 7598367 \nu - 47062053\)\()/1242216\)
\(\beta_{11}\)\(=\)\((\)\(-16952 \nu^{15} + 9175 \nu^{14} - 73804 \nu^{13} + 123904 \nu^{12} - 128807 \nu^{11} + 247964 \nu^{10} - 166462 \nu^{9} + 398066 \nu^{8} - 282422 \nu^{7} + 1677798 \nu^{6} - 5939262 \nu^{5} + 5124384 \nu^{4} - 20396205 \nu^{3} + 16524972 \nu^{2} - 21030192 \nu + 23648031\)\()/621108\)
\(\beta_{12}\)\(=\)\((\)\(-19793 \nu^{15} - 1430 \nu^{14} - 108688 \nu^{13} + 147133 \nu^{12} - 121322 \nu^{11} + 529214 \nu^{10} - 286834 \nu^{9} + 407870 \nu^{8} - 442766 \nu^{7} + 2453682 \nu^{6} - 6290388 \nu^{5} + 3336147 \nu^{4} - 34358094 \nu^{3} + 21512304 \nu^{2} - 26451765 \nu + 66410442\)\()/621108\)
\(\beta_{13}\)\(=\)\((\)\(3656 \nu^{15} - 865 \nu^{14} + 15076 \nu^{13} - 31234 \nu^{12} + 13505 \nu^{11} - 75428 \nu^{10} + 55828 \nu^{9} - 61412 \nu^{8} + 79862 \nu^{7} - 379710 \nu^{6} + 1284126 \nu^{5} - 523818 \nu^{4} + 4880493 \nu^{3} - 4856112 \nu^{2} + 1887138 \nu - 10890531\)\()/103518\)
\(\beta_{14}\)\(=\)\((\)\(-27554 \nu^{15} + 18943 \nu^{14} - 80794 \nu^{13} + 318436 \nu^{12} - 60809 \nu^{11} + 523952 \nu^{10} - 614170 \nu^{9} + 383294 \nu^{8} - 708026 \nu^{7} + 2355330 \nu^{6} - 11690478 \nu^{5} + 5227686 \nu^{4} - 29717685 \nu^{3} + 56735154 \nu^{2} + 4219452 \nu + 92446677\)\()/621108\)
\(\beta_{15}\)\(=\)\((\)\(-23045 \nu^{15} + 19169 \nu^{14} - 116460 \nu^{13} + 283569 \nu^{12} - 185713 \nu^{11} + 740670 \nu^{10} - 571592 \nu^{9} + 458764 \nu^{8} - 995208 \nu^{7} + 2760316 \nu^{6} - 9787698 \nu^{5} + 8671095 \nu^{4} - 41432283 \nu^{3} + 51803064 \nu^{2} - 24427089 \nu + 111152817\)\()/414072\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{15} - \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{7} - \beta_{6} + 2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{15} - \beta_{13} - 2 \beta_{11} + \beta_{10} - \beta_{8} - \beta_{5} + \beta_{4} - 3 \beta_{3} - \beta_{1}\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-4 \beta_{15} + 4 \beta_{14} + \beta_{13} + \beta_{12} + \beta_{10} - 4 \beta_{9} + \beta_{8} - 3 \beta_{7} + \beta_{6} - 5 \beta_{5} - 3 \beta_{4} + 5 \beta_{3} + 6 \beta_{2} - \beta_{1} - 4\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(-2 \beta_{15} + 3 \beta_{14} + 4 \beta_{13} + 4 \beta_{12} - \beta_{11} + 5 \beta_{9} + 2 \beta_{8} + 4 \beta_{7} - 3 \beta_{6} - \beta_{5} + 3 \beta_{4} - 2 \beta_{2} + 3 \beta_{1} + 10\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(6 \beta_{15} - 14 \beta_{14} - 2 \beta_{13} - \beta_{12} + \beta_{11} - 10 \beta_{10} - 5 \beta_{8} - 7 \beta_{7} - 11 \beta_{6} - 2 \beta_{5} + 24 \beta_{4} - 17 \beta_{3} - 4 \beta_{2} + \beta_{1} - 3\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(-5 \beta_{15} + 9 \beta_{14} - 10 \beta_{13} - 13 \beta_{12} - 13 \beta_{11} + 3 \beta_{10} - 6 \beta_{9} + 9 \beta_{7} - 12 \beta_{5} - 10 \beta_{4} + 14 \beta_{3} - 24 \beta_{2} - 21 \beta_{1} + 5\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(12 \beta_{15} - 9 \beta_{14} + 2 \beta_{13} + 10 \beta_{12} - 15 \beta_{11} + 2 \beta_{10} - 15 \beta_{9} - 26 \beta_{8} - 6 \beta_{7} - 34 \beta_{6} + 7 \beta_{5} + 26 \beta_{4} - 16 \beta_{3} + 87 \beta_{2} - 5 \beta_{1} + 60\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(61 \beta_{15} - 25 \beta_{14} + 25 \beta_{13} - 41 \beta_{12} + 6 \beta_{11} - 46 \beta_{10} + 54 \beta_{9} - 56 \beta_{8} + 19 \beta_{7} - 61 \beta_{6} + \beta_{5} + 64 \beta_{4} - 64 \beta_{3} + \beta_{2} + 40 \beta_{1} + 15\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(-56 \beta_{15} + 13 \beta_{14} - 67 \beta_{13} + 14 \beta_{12} + 7 \beta_{11} + 118 \beta_{10} - 136 \beta_{9} - 47 \beta_{8} + 9 \beta_{7} + 40 \beta_{6} - 62 \beta_{5} - 110 \beta_{4} + 54 \beta_{3} - 66 \beta_{2} - 28 \beta_{1} - 81\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(-185 \beta_{15} + 207 \beta_{14} + 14 \beta_{13} + 99 \beta_{12} - 61 \beta_{11} + 112 \beta_{10} - 169 \beta_{9} + 19 \beta_{8} - 68 \beta_{7} + 25 \beta_{6} - 125 \beta_{5} - 197 \beta_{4} + 301 \beta_{3} + 412 \beta_{2} - 49 \beta_{1} - 632\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(-7 \beta_{15} - 106 \beta_{14} + 234 \beta_{13} + 286 \beta_{12} + 274 \beta_{11} + 48 \beta_{10} - 81 \beta_{9} - 117 \beta_{8} - 125 \beta_{7} - 136 \beta_{6} + 78 \beta_{5} - 46 \beta_{4} - 193 \beta_{3} + 28 \beta_{2} + 72 \beta_{1} - 63\)\()/3\)
\(\nu^{12}\)\(=\)\((\)\(-144 \beta_{15} - 439 \beta_{14} + 36 \beta_{13} - 147 \beta_{12} + 673 \beta_{11} - 220 \beta_{10} - 419 \beta_{9} - 28 \beta_{8} - 150 \beta_{7} - 40 \beta_{6} - 13 \beta_{5} + 52 \beta_{4} + 290 \beta_{3} - 858 \beta_{2} + 181 \beta_{1} - 502\)\()/3\)
\(\nu^{13}\)\(=\)\((\)\(-373 \beta_{15} + 399 \beta_{14} - 651 \beta_{13} - 44 \beta_{12} - 632 \beta_{11} + 544 \beta_{10} - 32 \beta_{9} + 210 \beta_{8} + 794 \beta_{7} + 475 \beta_{6} + 426 \beta_{5} - 875 \beta_{4} + 1117 \beta_{3} - 1357 \beta_{2} - 673 \beta_{1} - 517\)\()/3\)
\(\nu^{14}\)\(=\)\((\)\(1073 \beta_{15} - 1398 \beta_{14} - 11 \beta_{13} + 363 \beta_{12} + 530 \beta_{11} - 367 \beta_{10} - 807 \beta_{9} - 1190 \beta_{8} - 1674 \beta_{7} - 1065 \beta_{6} + 1111 \beta_{5} + 617 \beta_{4} - 1560 \beta_{3} + 4110 \beta_{2} - 365 \beta_{1} - 1014\)\()/3\)
\(\nu^{15}\)\(=\)\((\)\(1627 \beta_{15} - 544 \beta_{14} + 1472 \beta_{13} - 2368 \beta_{12} + 2547 \beta_{11} - 361 \beta_{10} + 886 \beta_{9} - 838 \beta_{8} + 1875 \beta_{7} + 188 \beta_{6} + 998 \beta_{5} - 2247 \beta_{4} + 157 \beta_{3} - 3141 \beta_{2} + 1555 \beta_{1} + 1741\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(1 - \beta_{2}\) \(1 - \beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
341.1
−0.213160 1.71888i
−0.811340 + 1.53027i
−1.61108 + 0.635951i
1.68124 0.416458i
1.08696 + 1.34852i
−0.268067 1.71118i
1.68042 + 0.419752i
−0.544978 + 1.64408i
−0.213160 + 1.71888i
−0.811340 1.53027i
−1.61108 0.635951i
1.68124 + 0.416458i
1.08696 1.34852i
−0.268067 + 1.71118i
1.68042 0.419752i
−0.544978 1.64408i
0 0 0 −1.43402 2.48379i 0 2.56899 0.632668i 0 0 0
341.2 0 0 0 −1.37166 2.37578i 0 −2.60476 + 0.463945i 0 0 0
341.3 0 0 0 −1.09150 1.89054i 0 −1.25859 + 2.32722i 0 0 0
341.4 0 0 0 −0.349828 0.605920i 0 2.48683 + 0.903137i 0 0 0
341.5 0 0 0 −0.0382122 0.0661855i 0 0.232935 2.63548i 0 0 0
341.6 0 0 0 0.842869 + 1.45989i 0 −2.27938 1.34329i 0 0 0
341.7 0 0 0 1.48494 + 2.57199i 0 −0.200279 + 2.63816i 0 0 0
341.8 0 0 0 1.95741 + 3.39033i 0 0.554241 2.58705i 0 0 0
521.1 0 0 0 −1.43402 + 2.48379i 0 2.56899 + 0.632668i 0 0 0
521.2 0 0 0 −1.37166 + 2.37578i 0 −2.60476 0.463945i 0 0 0
521.3 0 0 0 −1.09150 + 1.89054i 0 −1.25859 2.32722i 0 0 0
521.4 0 0 0 −0.349828 + 0.605920i 0 2.48683 0.903137i 0 0 0
521.5 0 0 0 −0.0382122 + 0.0661855i 0 0.232935 + 2.63548i 0 0 0
521.6 0 0 0 0.842869 1.45989i 0 −2.27938 + 1.34329i 0 0 0
521.7 0 0 0 1.48494 2.57199i 0 −0.200279 2.63816i 0 0 0
521.8 0 0 0 1.95741 3.39033i 0 0.554241 + 2.58705i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 521.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.w.a 16
3.b odd 2 1 252.2.w.a 16
4.b odd 2 1 3024.2.ca.d 16
7.b odd 2 1 5292.2.w.b 16
7.c even 3 1 5292.2.x.a 16
7.c even 3 1 5292.2.bm.a 16
7.d odd 6 1 756.2.bm.a 16
7.d odd 6 1 5292.2.x.b 16
9.c even 3 1 252.2.bm.a yes 16
9.c even 3 1 2268.2.t.a 16
9.d odd 6 1 756.2.bm.a 16
9.d odd 6 1 2268.2.t.b 16
12.b even 2 1 1008.2.ca.d 16
21.c even 2 1 1764.2.w.b 16
21.g even 6 1 252.2.bm.a yes 16
21.g even 6 1 1764.2.x.b 16
21.h odd 6 1 1764.2.x.a 16
21.h odd 6 1 1764.2.bm.a 16
28.f even 6 1 3024.2.df.d 16
36.f odd 6 1 1008.2.df.d 16
36.h even 6 1 3024.2.df.d 16
63.g even 3 1 1764.2.x.b 16
63.h even 3 1 1764.2.w.b 16
63.i even 6 1 inner 756.2.w.a 16
63.j odd 6 1 5292.2.w.b 16
63.k odd 6 1 1764.2.x.a 16
63.k odd 6 1 2268.2.t.b 16
63.l odd 6 1 1764.2.bm.a 16
63.n odd 6 1 5292.2.x.b 16
63.o even 6 1 5292.2.bm.a 16
63.s even 6 1 2268.2.t.a 16
63.s even 6 1 5292.2.x.a 16
63.t odd 6 1 252.2.w.a 16
84.j odd 6 1 1008.2.df.d 16
252.r odd 6 1 3024.2.ca.d 16
252.bj even 6 1 1008.2.ca.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.w.a 16 3.b odd 2 1
252.2.w.a 16 63.t odd 6 1
252.2.bm.a yes 16 9.c even 3 1
252.2.bm.a yes 16 21.g even 6 1
756.2.w.a 16 1.a even 1 1 trivial
756.2.w.a 16 63.i even 6 1 inner
756.2.bm.a 16 7.d odd 6 1
756.2.bm.a 16 9.d odd 6 1
1008.2.ca.d 16 12.b even 2 1
1008.2.ca.d 16 252.bj even 6 1
1008.2.df.d 16 36.f odd 6 1
1008.2.df.d 16 84.j odd 6 1
1764.2.w.b 16 21.c even 2 1
1764.2.w.b 16 63.h even 3 1
1764.2.x.a 16 21.h odd 6 1
1764.2.x.a 16 63.k odd 6 1
1764.2.x.b 16 21.g even 6 1
1764.2.x.b 16 63.g even 3 1
1764.2.bm.a 16 21.h odd 6 1
1764.2.bm.a 16 63.l odd 6 1
2268.2.t.a 16 9.c even 3 1
2268.2.t.a 16 63.s even 6 1
2268.2.t.b 16 9.d odd 6 1
2268.2.t.b 16 63.k odd 6 1
3024.2.ca.d 16 4.b odd 2 1
3024.2.ca.d 16 252.r odd 6 1
3024.2.df.d 16 28.f even 6 1
3024.2.df.d 16 36.h even 6 1
5292.2.w.b 16 7.b odd 2 1
5292.2.w.b 16 63.j odd 6 1
5292.2.x.a 16 7.c even 3 1
5292.2.x.a 16 63.s even 6 1
5292.2.x.b 16 7.d odd 6 1
5292.2.x.b 16 63.n odd 6 1
5292.2.bm.a 16 7.c even 3 1
5292.2.bm.a 16 63.o even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(756, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( 324 + 4698 T + 62289 T^{2} + 89424 T^{3} + 143289 T^{4} + 45738 T^{5} + 71361 T^{6} + 21573 T^{7} + 23103 T^{8} + 4167 T^{9} + 3600 T^{10} + 423 T^{11} + 405 T^{12} + 24 T^{13} + 24 T^{14} + T^{16} \)
$7$ \( 5764801 + 823543 T + 352947 T^{2} - 117649 T^{3} - 196882 T^{4} - 48363 T^{5} - 980 T^{6} + 7777 T^{7} + 5328 T^{8} + 1111 T^{9} - 20 T^{10} - 141 T^{11} - 82 T^{12} - 7 T^{13} + 3 T^{14} + T^{15} + T^{16} \)
$11$ \( 26244 + 13122 T - 269001 T^{2} - 135594 T^{3} + 2920374 T^{4} - 4745790 T^{5} + 3147093 T^{6} - 725355 T^{7} - 137538 T^{8} + 80433 T^{9} + 7047 T^{10} - 8019 T^{11} + 711 T^{12} + 234 T^{13} - 27 T^{14} - 6 T^{15} + T^{16} \)
$13$ \( 3337929 - 17462466 T + 38097783 T^{2} - 40000230 T^{3} + 18330786 T^{4} + 831303 T^{5} - 3499173 T^{6} + 321543 T^{7} + 494064 T^{8} - 66258 T^{9} - 35379 T^{10} + 4608 T^{11} + 1980 T^{12} - 162 T^{13} - 51 T^{14} + 3 T^{15} + T^{16} \)
$17$ \( 13549761 - 35746191 T + 94137876 T^{2} - 25350003 T^{3} + 32433372 T^{4} + 1254096 T^{5} + 8181405 T^{6} + 690741 T^{7} + 912402 T^{8} + 115920 T^{9} + 73566 T^{10} + 8910 T^{11} + 3321 T^{12} + 420 T^{13} + 105 T^{14} + 9 T^{15} + T^{16} \)
$19$ \( 2099601 + 29655234 T + 166640004 T^{2} + 381649968 T^{3} + 320108679 T^{4} - 85622103 T^{5} - 40684761 T^{6} + 9517095 T^{7} + 4276089 T^{8} - 316089 T^{9} - 188478 T^{10} + 7974 T^{11} + 6156 T^{12} - 93 T^{14} + T^{16} \)
$23$ \( 15198451524 - 14629258530 T + 2044217331 T^{2} + 2550348180 T^{3} - 604378908 T^{4} - 340227216 T^{5} + 112333068 T^{6} + 20560716 T^{7} - 7612947 T^{8} - 1006263 T^{9} + 350730 T^{10} + 37422 T^{11} - 7767 T^{12} - 1008 T^{13} + 99 T^{14} + 21 T^{15} + T^{16} \)
$29$ \( 15752961 + 9001692 T - 182284263 T^{2} - 105142212 T^{3} + 2135730888 T^{4} - 464916834 T^{5} - 210471048 T^{6} + 53675541 T^{7} + 20028303 T^{8} - 2348109 T^{9} - 577287 T^{10} + 58239 T^{11} + 12753 T^{12} - 828 T^{13} - 126 T^{14} + 6 T^{15} + T^{16} \)
$31$ \( 3910251024 + 10103222664 T^{2} + 4911390369 T^{4} + 724329054 T^{6} + 47767077 T^{8} + 1632609 T^{10} + 29898 T^{12} + 276 T^{14} + T^{16} \)
$37$ \( 52765696 + 3719168 T + 115382016 T^{2} + 5193472 T^{3} + 232814336 T^{4} + 12859776 T^{5} + 40260736 T^{6} + 1907864 T^{7} + 5129568 T^{8} + 175166 T^{9} + 263287 T^{10} - 10395 T^{11} + 8996 T^{12} - 167 T^{13} + 108 T^{14} - T^{15} + T^{16} \)
$41$ \( 91647269289 + 3013404282 T + 36627754095 T^{2} + 8471239848 T^{3} + 12508926552 T^{4} + 1968463728 T^{5} + 1038978252 T^{6} + 28504467 T^{7} + 44578395 T^{8} + 430317 T^{9} + 1135899 T^{10} - 20637 T^{11} + 18891 T^{12} - 330 T^{13} + 186 T^{14} - 6 T^{15} + T^{16} \)
$43$ \( 28009034881 + 1271593682 T + 16892706132 T^{2} + 953140042 T^{3} + 8330008163 T^{4} + 324910827 T^{5} + 1058091937 T^{6} - 87622111 T^{7} + 97936317 T^{8} - 3925843 T^{9} + 2048842 T^{10} - 18162 T^{11} + 30590 T^{12} - 104 T^{13} + 207 T^{14} + 2 T^{15} + T^{16} \)
$47$ \( ( 1404144 - 1925352 T + 887031 T^{2} - 134271 T^{3} - 8811 T^{4} + 3468 T^{5} - 114 T^{6} - 18 T^{7} + T^{8} )^{2} \)
$53$ \( 531441 - 18068994 T + 191318760 T^{2} + 457747848 T^{3} + 292233501 T^{4} - 102922407 T^{5} - 60026589 T^{6} + 16579647 T^{7} + 9705177 T^{8} - 1451439 T^{9} - 466074 T^{10} + 56376 T^{11} + 19764 T^{12} - 153 T^{14} + T^{16} \)
$59$ \( ( -406908 + 656136 T + 132849 T^{2} - 92961 T^{3} - 2979 T^{4} + 3201 T^{5} - 168 T^{6} - 15 T^{7} + T^{8} )^{2} \)
$61$ \( 1475481744 + 4244295672 T^{2} + 2010944961 T^{4} + 341040105 T^{6} + 25911891 T^{8} + 1015731 T^{10} + 21420 T^{12} + 231 T^{14} + T^{16} \)
$67$ \( ( -1454288 + 1471592 T - 106757 T^{2} - 93061 T^{3} + 10750 T^{4} + 1703 T^{5} - 221 T^{6} - 7 T^{7} + T^{8} )^{2} \)
$71$ \( 780959242139904 + 134744717006208 T^{2} + 7956570857364 T^{4} + 234856231407 T^{6} + 3949834995 T^{8} + 39561129 T^{10} + 233316 T^{12} + 747 T^{14} + T^{16} \)
$73$ \( 7523023152969 + 2752955922474 T - 1191205799235 T^{2} - 558789791238 T^{3} + 238887968904 T^{4} + 213060248478 T^{5} + 62170270956 T^{6} + 8564672457 T^{7} + 317278665 T^{8} - 59680611 T^{9} - 5521941 T^{10} + 432909 T^{11} + 71235 T^{12} - 306 T^{14} + T^{16} \)
$79$ \( ( -3248 + 4616 T + 2863 T^{2} - 6523 T^{3} + 1996 T^{4} + 461 T^{5} - 143 T^{6} - T^{7} + T^{8} )^{2} \)
$83$ \( 669184533369 + 2061188196012 T + 5773577517018 T^{2} + 1903485074616 T^{3} + 720461504880 T^{4} + 85919425596 T^{5} + 25248534300 T^{6} + 2208157416 T^{7} + 614588409 T^{8} + 32727636 T^{9} + 8300268 T^{10} + 290448 T^{11} + 80712 T^{12} + 1272 T^{13} + 330 T^{14} + T^{16} \)
$89$ \( 7161826993281 + 1051545832029 T + 1294382305422 T^{2} + 236676836007 T^{3} + 180123305796 T^{4} + 30425509008 T^{5} + 9647366571 T^{6} + 1422741915 T^{7} + 335911536 T^{8} + 43169922 T^{9} + 6642648 T^{10} + 611550 T^{11} + 69741 T^{12} + 5130 T^{13} + 459 T^{14} + 21 T^{15} + T^{16} \)
$97$ \( 22864161681 - 104576900445 T - 8224102287 T^{2} + 766864072890 T^{3} + 1208668941423 T^{4} - 71751605004 T^{5} - 41675283648 T^{6} + 2243025486 T^{7} + 1044816804 T^{8} - 41968233 T^{9} - 12357171 T^{10} + 394884 T^{11} + 109872 T^{12} - 1161 T^{13} - 384 T^{14} + 3 T^{15} + T^{16} \)
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