Properties

Label 637.2.e.o
Level $637$
Weight $2$
Character orbit 637.e
Analytic conductor $5.086$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 2 x^{11} + 9 x^{10} - 6 x^{9} + 34 x^{8} - 18 x^{7} + 85 x^{6} - 2 x^{5} + 92 x^{4} - 26 x^{3} + 43 x^{2} + 6 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\beta_{3} + \beta_{11} ) q^{2} + ( -\beta_{1} - \beta_{2} + \beta_{8} ) q^{3} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{4} + ( 1 - \beta_{8} + \beta_{9} ) q^{5} + ( -1 - \beta_{2} - \beta_{3} - \beta_{5} ) q^{6} + ( -\beta_{2} - \beta_{5} - \beta_{6} ) q^{8} + ( -3 \beta_{1} + \beta_{3} - \beta_{11} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{3} + \beta_{11} ) q^{2} + ( -\beta_{1} - \beta_{2} + \beta_{8} ) q^{3} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{4} + ( 1 - \beta_{8} + \beta_{9} ) q^{5} + ( -1 - \beta_{2} - \beta_{3} - \beta_{5} ) q^{6} + ( -\beta_{2} - \beta_{5} - \beta_{6} ) q^{8} + ( -3 \beta_{1} + \beta_{3} - \beta_{11} ) q^{9} + ( \beta_{1} + \beta_{2} + \beta_{8} + 2 \beta_{11} ) q^{10} + ( \beta_{4} - 2 \beta_{6} - \beta_{9} - 2 \beta_{10} ) q^{11} + ( -\beta_{1} + \beta_{3} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{12} - q^{13} + ( 2 - \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{15} + ( \beta_{3} + \beta_{7} + \beta_{9} - \beta_{11} ) q^{16} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{17} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - \beta_{9} ) q^{18} + ( \beta_{3} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{19} + ( -3 - \beta_{2} - \beta_{3} - \beta_{5} ) q^{20} + ( -3 - \beta_{2} + \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} ) q^{22} + ( \beta_{1} + \beta_{7} - \beta_{9} - 2 \beta_{10} ) q^{23} + ( -\beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{6} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{24} + ( \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{25} + ( \beta_{3} - \beta_{11} ) q^{26} + ( -2 + 4 \beta_{2} + 4 \beta_{3} + \beta_{5} ) q^{27} + ( -1 - 2 \beta_{4} - 2 \beta_{5} ) q^{29} + ( -1 + 3 \beta_{1} - 3 \beta_{3} + 2 \beta_{7} + \beta_{8} + 3 \beta_{11} ) q^{30} + ( -\beta_{1} - \beta_{2} - 3 \beta_{4} - \beta_{6} + \beta_{8} + 3 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{31} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 4 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{32} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{7} + \beta_{8} - 4 \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{33} + ( 1 + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} ) q^{34} + ( -3 + \beta_{2} + \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{36} + ( -2 \beta_{1} + 2 \beta_{3} + 2 \beta_{7} - \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{37} + ( \beta_{1} + \beta_{2} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{10} ) q^{38} + ( \beta_{1} + \beta_{2} - \beta_{8} ) q^{39} + ( 3 \beta_{1} - \beta_{3} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{40} + ( -3 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{41} + ( 1 + 4 \beta_{2} + 2 \beta_{4} + 2 \beta_{6} ) q^{43} + ( 4 \beta_{1} + 3 \beta_{3} - 2 \beta_{7} - \beta_{9} + 2 \beta_{10} - 3 \beta_{11} ) q^{44} + ( -4 \beta_{1} - 4 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} - 5 \beta_{11} ) q^{45} + ( \beta_{1} + \beta_{2} - 2 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{46} + ( 6 - 3 \beta_{3} - 3 \beta_{7} - 6 \beta_{8} + \beta_{9} + 3 \beta_{10} + 3 \beta_{11} ) q^{47} + ( 2 + \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} ) q^{48} + ( 2 + 3 \beta_{2} + \beta_{5} + \beta_{6} ) q^{50} + ( 1 + \beta_{1} - 5 \beta_{3} + 2 \beta_{7} - \beta_{8} + 4 \beta_{9} + 2 \beta_{10} + 5 \beta_{11} ) q^{51} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{52} + ( -\beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} + 2 \beta_{8} - \beta_{9} - 4 \beta_{11} ) q^{53} + ( -8 - \beta_{1} + 2 \beta_{3} + 8 \beta_{8} - 4 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{54} + ( 1 - 3 \beta_{2} + 3 \beta_{5} - 2 \beta_{6} ) q^{55} + ( \beta_{3} - 2 \beta_{4} + 2 \beta_{6} ) q^{57} + ( 2 + 3 \beta_{3} - 2 \beta_{8} + 2 \beta_{10} - 3 \beta_{11} ) q^{58} + ( \beta_{1} + \beta_{2} + 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 5 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{59} + ( \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - 3 \beta_{11} ) q^{60} + ( 2 - 3 \beta_{1} - \beta_{3} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{61} + ( -5 + 3 \beta_{3} - 3 \beta_{4} - 5 \beta_{5} + \beta_{6} ) q^{62} + ( -4 - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{5} ) q^{64} + ( -1 + \beta_{8} - \beta_{9} ) q^{65} + ( -2 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} ) q^{66} + ( -\beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} - 4 \beta_{8} + \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{67} + ( 6 - 5 \beta_{1} - 3 \beta_{3} - 4 \beta_{7} - 6 \beta_{8} + 5 \beta_{9} - \beta_{10} + 3 \beta_{11} ) q^{68} + ( 1 - 4 \beta_{2} - \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} ) q^{69} + ( 2 - \beta_{2} - 4 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} - 3 \beta_{6} ) q^{71} + ( -4 - 2 \beta_{1} + 5 \beta_{3} + 4 \beta_{8} - 5 \beta_{9} - 5 \beta_{11} ) q^{72} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} + 5 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} + 5 \beta_{10} + \beta_{11} ) q^{73} + ( \beta_{1} + \beta_{2} - 4 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} + \beta_{8} + 4 \beta_{10} + \beta_{11} ) q^{74} + ( 2 + \beta_{1} - 3 \beta_{3} + 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + \beta_{10} + 3 \beta_{11} ) q^{75} + ( 3 - \beta_{3} + 3 \beta_{5} - \beta_{6} ) q^{76} + ( 1 + \beta_{2} + \beta_{3} + \beta_{5} ) q^{78} + ( 3 + 2 \beta_{1} - 3 \beta_{3} - 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} + 3 \beta_{11} ) q^{79} + ( \beta_{1} + \beta_{2} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - \beta_{10} - 2 \beta_{11} ) q^{80} + ( 5 \beta_{1} + 5 \beta_{2} - \beta_{4} + 5 \beta_{5} + 5 \beta_{7} - 6 \beta_{8} + \beta_{9} + 6 \beta_{11} ) q^{81} + ( -8 + 3 \beta_{1} - \beta_{3} + 3 \beta_{7} + 8 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{82} + ( -7 + 2 \beta_{2} - \beta_{4} - 2 \beta_{6} ) q^{83} + ( 6 \beta_{2} + 4 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{85} + ( 4 - 2 \beta_{1} - 5 \beta_{3} - 8 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 5 \beta_{11} ) q^{86} + ( \beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{6} - 3 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{87} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{4} + 3 \beta_{5} + 3 \beta_{7} + 8 \beta_{8} - \beta_{9} - \beta_{11} ) q^{88} + ( 4 - \beta_{1} - \beta_{3} - \beta_{7} - 4 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{89} + ( 5 - 5 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} ) q^{90} + ( 3 - 3 \beta_{2} + \beta_{3} + 4 \beta_{5} - 2 \beta_{6} ) q^{92} + ( -2 - 4 \beta_{3} + 6 \beta_{7} + 2 \beta_{8} + 5 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} ) q^{93} + ( -5 \beta_{1} - 5 \beta_{2} + 3 \beta_{5} - 6 \beta_{6} + 3 \beta_{7} - 5 \beta_{8} - 6 \beta_{10} + 4 \beta_{11} ) q^{94} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{10} - 2 \beta_{11} ) q^{95} + ( -4 + 4 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} ) q^{96} + ( 4 + 3 \beta_{2} + 3 \beta_{3} + \beta_{4} - 5 \beta_{5} - 2 \beta_{6} ) q^{97} + ( -5 \beta_{2} + 2 \beta_{3} - 7 \beta_{4} + \beta_{5} + \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 8q^{3} - 4q^{4} + 6q^{5} - 8q^{6} - 6q^{9} + O(q^{10}) \) \( 12q + 8q^{3} - 4q^{4} + 6q^{5} - 8q^{6} - 6q^{9} + 4q^{10} - 4q^{11} - 4q^{12} - 12q^{13} + 24q^{15} + 16q^{17} + 4q^{18} + 2q^{19} - 32q^{20} - 24q^{22} + 6q^{23} + 12q^{24} + 4q^{25} - 40q^{27} - 12q^{29} + 6q^{31} + 20q^{32} + 4q^{33} - 48q^{36} + 8q^{38} - 8q^{39} + 4q^{40} + 16q^{41} + 4q^{43} + 4q^{44} + 14q^{45} - 8q^{46} + 30q^{47} + 16q^{48} + 16q^{50} + 4q^{51} + 4q^{52} + 14q^{53} - 48q^{54} + 16q^{55} + 8q^{57} + 8q^{58} + 24q^{59} - 12q^{60} - 56q^{62} - 40q^{64} - 6q^{65} - 4q^{66} - 16q^{67} + 28q^{68} + 40q^{69} + 16q^{71} - 28q^{72} - 6q^{73} + 12q^{74} + 12q^{75} + 32q^{76} + 8q^{78} + 22q^{79} - 28q^{80} - 46q^{81} - 40q^{82} - 100q^{83} - 16q^{85} + 16q^{86} - 16q^{87} + 44q^{88} + 26q^{89} + 80q^{90} + 40q^{92} - 16q^{93} - 32q^{94} + 6q^{95} - 20q^{96} + 28q^{97} + 24q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 2 x^{11} + 9 x^{10} - 6 x^{9} + 34 x^{8} - 18 x^{7} + 85 x^{6} - 2 x^{5} + 92 x^{4} - 26 x^{3} + 43 x^{2} + 6 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(512378 \nu^{11} + 5336 \nu^{10} + 3065721 \nu^{9} + 4369060 \nu^{8} + 17395862 \nu^{7} + 17140164 \nu^{6} + 44372910 \nu^{5} + 58644524 \nu^{4} + 109670152 \nu^{3} + 40367568 \nu^{2} + 5740174 \nu + 9713364\)\()/62842357\)
\(\beta_{3}\)\(=\)\((\)\(517714 \nu^{11} - 1551017 \nu^{10} + 4377607 \nu^{9} - 4394050 \nu^{8} + 8967106 \nu^{7} - 16319384 \nu^{6} + 15296370 \nu^{5} + 3886852 \nu^{4} - 55980756 \nu^{3} + 6182709 \nu^{2} + 898922 \nu + 115458972\)\()/62842357\)
\(\beta_{4}\)\(=\)\((\)\(2297118 \nu^{11} - 4668361 \nu^{10} + 18068728 \nu^{9} - 6968746 \nu^{8} + 52032662 \nu^{7} - 10138403 \nu^{6} + 109881238 \nu^{5} + 95992492 \nu^{4} + 33981855 \nu^{3} + 76649413 \nu^{2} + 10958926 \nu + 103693971\)\()/62842357\)
\(\beta_{5}\)\(=\)\((\)\( -1878 \nu^{11} + 1119 \nu^{10} - 12173 \nu^{9} - 9570 \nu^{8} - 57462 \nu^{7} - 38216 \nu^{6} - 141030 \nu^{5} - 174444 \nu^{4} - 233985 \nu^{3} - 122643 \nu^{2} - 17454 \nu - 66913 \)\()/45971\)
\(\beta_{6}\)\(=\)\((\)\(6705029 \nu^{11} - 7815976 \nu^{10} + 48698198 \nu^{9} + 12543658 \nu^{8} + 184020371 \nu^{7} + 87381018 \nu^{6} + 431005791 \nu^{5} + 486894758 \nu^{4} + 510469395 \nu^{3} + 352920684 \nu^{2} + 50284591 \nu + 159636402\)\()/62842357\)
\(\beta_{7}\)\(=\)\((\)\(-9197775 \nu^{11} + 20220925 \nu^{10} - 86127174 \nu^{9} + 69981075 \nu^{8} - 318884784 \nu^{7} + 220945734 \nu^{6} - 813418056 \nu^{5} + 167410082 \nu^{4} - 854628237 \nu^{3} + 383580396 \nu^{2} - 552502129 \nu + 10820061\)\()/62842357\)
\(\beta_{8}\)\(=\)\((\)\(9713364 \nu^{11} - 19939106 \nu^{10} + 87414940 \nu^{9} - 61345905 \nu^{8} + 325885316 \nu^{7} - 192236414 \nu^{6} + 808495776 \nu^{5} - 63799638 \nu^{4} + 834984964 \nu^{3} - 362217616 \nu^{2} + 377307084 \nu + 52540010\)\()/62842357\)
\(\beta_{9}\)\(=\)\((\)\(-14791857 \nu^{11} + 31867988 \nu^{10} - 138901006 \nu^{9} + 113931690 \nu^{8} - 526956324 \nu^{7} + 359867408 \nu^{6} - 1313722872 \nu^{5} + 273803355 \nu^{4} - 1381879675 \nu^{3} + 621612052 \nu^{2} - 609066000 \nu + 17525090\)\()/62842357\)
\(\beta_{10}\)\(=\)\((\)\(15232235 \nu^{11} - 36300822 \nu^{10} + 151870530 \nu^{9} - 148661200 \nu^{8} + 577528481 \nu^{7} - 473129740 \nu^{6} + 1478901160 \nu^{5} - 492235424 \nu^{4} + 1591586825 \nu^{3} - 747084060 \nu^{2} + 943653761 \nu - 20852300\)\()/62842357\)
\(\beta_{11}\)\(=\)\((\)\(18914350 \nu^{11} - 39883548 \nu^{10} + 171764159 \nu^{9} - 127060870 \nu^{8} + 634374770 \nu^{7} - 401612992 \nu^{6} + 1572618642 \nu^{5} - 186243800 \nu^{4} + 1560299776 \nu^{3} - 701960443 \nu^{2} + 686031637 \nu + 95366656\)\()/62842357\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{11} - 2 \beta_{8} + \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{3} + 4 \beta_{2} - 1\)
\(\nu^{4}\)\(=\)\(-5 \beta_{11} - \beta_{9} + 7 \beta_{8} - \beta_{7} + 5 \beta_{3} - 6 \beta_{1} - 7\)
\(\nu^{5}\)\(=\)\(-8 \beta_{11} - \beta_{10} - 2 \beta_{9} + 8 \beta_{8} - 6 \beta_{7} - \beta_{6} - 6 \beta_{5} + 2 \beta_{4} - 18 \beta_{2} - 18 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-2 \beta_{6} - 9 \beta_{5} + 9 \beta_{4} - 26 \beta_{3} - 33 \beta_{2} + 30\)
\(\nu^{7}\)\(=\)\(51 \beta_{11} + 9 \beta_{10} + 20 \beta_{9} - 49 \beta_{8} + 33 \beta_{7} - 51 \beta_{3} + 87 \beta_{1} + 49\)
\(\nu^{8}\)\(=\)\(140 \beta_{11} + 20 \beta_{10} + 62 \beta_{9} - 143 \beta_{8} + 62 \beta_{7} + 20 \beta_{6} + 62 \beta_{5} - 62 \beta_{4} + 178 \beta_{2} + 178 \beta_{1}\)
\(\nu^{9}\)\(=\)\(62 \beta_{6} + 182 \beta_{5} - 144 \beta_{4} + 302 \beta_{3} + 441 \beta_{2} - 278\)
\(\nu^{10}\)\(=\)\(-767 \beta_{11} - 144 \beta_{10} - 388 \beta_{9} + 724 \beta_{8} - 384 \beta_{7} + 767 \beta_{3} - 959 \beta_{1} - 724\)
\(\nu^{11}\)\(=\)\(-1731 \beta_{11} - 388 \beta_{10} - 916 \beta_{9} + 1539 \beta_{8} - 1011 \beta_{7} - 388 \beta_{6} - 1011 \beta_{5} + 916 \beta_{4} - 2306 \beta_{2} - 2306 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(248\)
\(\chi(n)\) \(1\) \(-1 + \beta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.379209 0.656810i
−0.0731214 + 0.126650i
0.954516 1.65327i
−0.602377 + 1.04335i
1.17550 2.03602i
−0.833726 + 1.44406i
0.379209 + 0.656810i
−0.0731214 0.126650i
0.954516 + 1.65327i
−0.602377 1.04335i
1.17550 + 2.03602i
−0.833726 1.44406i
−1.09161 + 1.89072i 0.879209 + 1.52284i −1.38322 2.39581i 1.05533 1.82788i −3.83901 0 1.67333 −0.0460183 + 0.0797060i 2.30401 + 3.99066i
79.2 −0.916185 + 1.58688i 0.426879 + 0.739375i −0.678791 1.17570i 1.31278 2.27379i −1.56440 0 −1.17715 1.13555 1.96683i 2.40549 + 4.16643i
79.3 −0.132313 + 0.229173i 1.45452 + 2.51930i 0.964986 + 1.67141i −0.717577 + 1.24288i −0.769807 0 −1.03998 −2.73124 + 4.73064i −0.189890 0.328899i
79.4 0.328092 0.568272i −0.102377 0.177322i 0.784711 + 1.35916i −0.679981 + 1.17776i −0.134356 0 2.34220 1.47904 2.56177i 0.446193 + 0.772828i
79.5 0.588093 1.01861i 1.67550 + 2.90205i 0.308293 + 0.533979i 1.57431 2.72679i 3.94140 0 3.07759 −4.11459 + 7.12667i −1.85168 3.20721i
79.6 1.22392 2.11990i −0.333726 0.578030i −1.99598 3.45713i 0.455143 0.788331i −1.63382 0 −4.87599 1.27725 2.21227i −1.11412 1.92971i
508.1 −1.09161 1.89072i 0.879209 1.52284i −1.38322 + 2.39581i 1.05533 + 1.82788i −3.83901 0 1.67333 −0.0460183 0.0797060i 2.30401 3.99066i
508.2 −0.916185 1.58688i 0.426879 0.739375i −0.678791 + 1.17570i 1.31278 + 2.27379i −1.56440 0 −1.17715 1.13555 + 1.96683i 2.40549 4.16643i
508.3 −0.132313 0.229173i 1.45452 2.51930i 0.964986 1.67141i −0.717577 1.24288i −0.769807 0 −1.03998 −2.73124 4.73064i −0.189890 + 0.328899i
508.4 0.328092 + 0.568272i −0.102377 + 0.177322i 0.784711 1.35916i −0.679981 1.17776i −0.134356 0 2.34220 1.47904 + 2.56177i 0.446193 0.772828i
508.5 0.588093 + 1.01861i 1.67550 2.90205i 0.308293 0.533979i 1.57431 + 2.72679i 3.94140 0 3.07759 −4.11459 7.12667i −1.85168 + 3.20721i
508.6 1.22392 + 2.11990i −0.333726 + 0.578030i −1.99598 + 3.45713i 0.455143 + 0.788331i −1.63382 0 −4.87599 1.27725 + 2.21227i −1.11412 + 1.92971i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 508.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.e.o 12
7.b odd 2 1 637.2.e.n 12
7.c even 3 1 637.2.a.m 6
7.c even 3 1 inner 637.2.e.o 12
7.d odd 6 1 637.2.a.n yes 6
7.d odd 6 1 637.2.e.n 12
21.g even 6 1 5733.2.a.br 6
21.h odd 6 1 5733.2.a.bu 6
91.r even 6 1 8281.2.a.cc 6
91.s odd 6 1 8281.2.a.cd 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.a.m 6 7.c even 3 1
637.2.a.n yes 6 7.d odd 6 1
637.2.e.n 12 7.b odd 2 1
637.2.e.n 12 7.d odd 6 1
637.2.e.o 12 1.a even 1 1 trivial
637.2.e.o 12 7.c even 3 1 inner
5733.2.a.br 6 21.g even 6 1
5733.2.a.bu 6 21.h odd 6 1
8281.2.a.cc 6 91.r even 6 1
8281.2.a.cd 6 91.s odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):

\(T_{2}^{12} + \cdots\)
\(T_{3}^{12} - \cdots\)
\(T_{5}^{12} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 8 T + 44 T^{2} - 56 T^{3} + 180 T^{4} - 64 T^{5} + 108 T^{6} - 4 T^{7} + 50 T^{8} + 8 T^{10} + T^{12} \)
$3$ \( 4 + 16 T + 88 T^{2} - 48 T^{3} + 200 T^{4} - 192 T^{5} + 452 T^{6} - 424 T^{7} + 316 T^{8} - 136 T^{9} + 44 T^{10} - 8 T^{11} + T^{12} \)
$5$ \( 961 - 806 T + 1637 T^{2} - 682 T^{3} + 1430 T^{4} - 670 T^{5} + 637 T^{6} - 278 T^{7} + 200 T^{8} - 78 T^{9} + 31 T^{10} - 6 T^{11} + T^{12} \)
$7$ \( T^{12} \)
$11$ \( 315844 - 388904 T + 583396 T^{2} - 46632 T^{3} + 121192 T^{4} + 21328 T^{5} + 27512 T^{6} + 5132 T^{7} + 1882 T^{8} + 160 T^{9} + 54 T^{10} + 4 T^{11} + T^{12} \)
$13$ \( ( 1 + T )^{12} \)
$17$ \( 28026436 - 35956848 T + 31350416 T^{2} - 15532752 T^{3} + 5891120 T^{4} - 1580608 T^{5} + 359412 T^{6} - 64408 T^{7} + 11112 T^{8} - 1544 T^{9} + 200 T^{10} - 16 T^{11} + T^{12} \)
$19$ \( 5329 + 438 T + 6095 T^{2} + 1838 T^{3} + 5744 T^{4} + 1270 T^{5} + 1509 T^{6} + 54 T^{7} + 238 T^{8} + 2 T^{9} + 21 T^{10} - 2 T^{11} + T^{12} \)
$23$ \( 279841 - 462346 T + 853277 T^{2} - 140070 T^{3} + 285862 T^{4} - 113818 T^{5} + 63545 T^{6} - 12966 T^{7} + 3170 T^{8} - 322 T^{9} + 73 T^{10} - 6 T^{11} + T^{12} \)
$29$ \( ( 529 + 230 T - 401 T^{2} - 268 T^{3} - 33 T^{4} + 6 T^{5} + T^{6} )^{2} \)
$31$ \( 1957974001 + 444525454 T + 285927185 T^{2} + 2954658 T^{3} + 17495494 T^{4} + 78862 T^{5} + 590105 T^{6} - 18294 T^{7} + 12092 T^{8} - 326 T^{9} + 151 T^{10} - 6 T^{11} + T^{12} \)
$37$ \( 64516 + 181864 T + 416644 T^{2} + 390536 T^{3} + 332688 T^{4} + 28216 T^{5} + 87200 T^{6} + 20068 T^{7} + 6346 T^{8} + 472 T^{9} + 82 T^{10} + T^{12} \)
$41$ \( ( 28784 - 19744 T + 1720 T^{2} + 968 T^{3} - 120 T^{4} - 8 T^{5} + T^{6} )^{2} \)
$43$ \( ( 35153 + 29214 T + 6575 T^{2} - 188 T^{3} - 161 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$47$ \( 18391970689 - 14558484950 T + 8049379343 T^{2} - 2348988030 T^{3} + 526715296 T^{4} - 80011070 T^{5} + 10648181 T^{6} - 1111710 T^{7} + 116246 T^{8} - 9410 T^{9} + 685 T^{10} - 30 T^{11} + T^{12} \)
$53$ \( 1739761 + 4466134 T + 13046477 T^{2} - 2265974 T^{3} + 3691278 T^{4} + 583222 T^{5} + 456721 T^{6} + 5026 T^{7} + 9690 T^{8} - 842 T^{9} + 233 T^{10} - 14 T^{11} + T^{12} \)
$59$ \( 2347024 + 1164320 T + 2471152 T^{2} - 1074176 T^{3} + 1270584 T^{4} - 313072 T^{5} + 203696 T^{6} - 52144 T^{7} + 23608 T^{8} - 3592 T^{9} + 430 T^{10} - 24 T^{11} + T^{12} \)
$61$ \( 46908629056 + 4498016512 T + 3951232992 T^{2} - 289006720 T^{3} + 213173856 T^{4} - 8397632 T^{5} + 3577368 T^{6} - 48320 T^{7} + 44264 T^{8} - 224 T^{9} + 246 T^{10} + T^{12} \)
$67$ \( 37356544 + 22687744 T + 25905152 T^{2} + 2414592 T^{3} + 6905856 T^{4} + 1684992 T^{5} + 592832 T^{6} + 67200 T^{7} + 14784 T^{8} + 1600 T^{9} + 256 T^{10} + 16 T^{11} + T^{12} \)
$71$ \( ( -1206162 - 104028 T + 33274 T^{2} + 1856 T^{3} - 316 T^{4} - 8 T^{5} + T^{6} )^{2} \)
$73$ \( 20351019649 - 3171835738 T + 2207803983 T^{2} + 36518862 T^{3} + 127848856 T^{4} + 155958 T^{5} + 3128797 T^{6} + 72830 T^{7} + 50918 T^{8} + 170 T^{9} + 277 T^{10} + 6 T^{11} + T^{12} \)
$79$ \( 62615569 - 75062718 T + 178047973 T^{2} + 55369622 T^{3} + 154458578 T^{4} - 36708454 T^{5} + 9145333 T^{6} - 705342 T^{7} + 85138 T^{8} - 4914 T^{9} + 549 T^{10} - 22 T^{11} + T^{12} \)
$83$ \( ( -167041 - 13034 T + 26145 T^{2} + 7992 T^{3} + 941 T^{4} + 50 T^{5} + T^{6} )^{2} \)
$89$ \( 99181681 - 258934 T + 90159503 T^{2} - 23427206 T^{3} + 80683068 T^{4} - 11007086 T^{5} + 2616529 T^{6} - 315154 T^{7} + 57102 T^{8} - 5782 T^{9} + 545 T^{10} - 26 T^{11} + T^{12} \)
$97$ \( ( 217287 - 193902 T + 13171 T^{2} + 3620 T^{3} - 261 T^{4} - 14 T^{5} + T^{6} )^{2} \)
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