Properties

Label 637.2.f.k
Level $637$
Weight $2$
Character orbit 637.f
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.f (of order \(3\), degree \(2\), 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: \(x^{12} - x^{11} + 7 x^{10} - 2 x^{9} + 33 x^{8} - 11 x^{7} + 55 x^{6} + 17 x^{5} + 47 x^{4} + x^{3} + 8 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - \beta_{5} + \beta_{11} ) q^{2} + \beta_{11} q^{3} + ( \beta_{6} - \beta_{7} ) q^{4} + ( -\beta_{2} + \beta_{9} ) q^{5} + ( \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{6} + ( -1 - \beta_{5} + \beta_{6} + \beta_{10} ) q^{8} + ( \beta_{6} - \beta_{8} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} - \beta_{5} + \beta_{11} ) q^{2} + \beta_{11} q^{3} + ( \beta_{6} - \beta_{7} ) q^{4} + ( -\beta_{2} + \beta_{9} ) q^{5} + ( \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{6} + ( -1 - \beta_{5} + \beta_{6} + \beta_{10} ) q^{8} + ( \beta_{6} - \beta_{8} ) q^{9} + ( 1 + \beta_{1} + \beta_{5} - \beta_{7} + \beta_{9} ) q^{10} + ( 1 - \beta_{1} - \beta_{5} - \beta_{7} - \beta_{10} - \beta_{11} ) q^{11} + ( -1 + \beta_{2} + 2 \beta_{3} + \beta_{6} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{12} + ( -\beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{10} ) q^{13} -\beta_{10} q^{15} + ( 2 + \beta_{1} + \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} - \beta_{10} - 2 \beta_{11} ) q^{16} + ( -\beta_{3} - \beta_{6} + \beta_{7} ) q^{17} + ( -1 + \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{18} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{19} + ( -\beta_{2} + \beta_{3} - \beta_{8} ) q^{20} + ( -\beta_{1} + 2 \beta_{3} + 2 \beta_{8} ) q^{22} + ( -1 - 2 \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{8} - 3 \beta_{9} + \beta_{10} - \beta_{11} ) q^{23} + ( -2 + 2 \beta_{7} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{24} + ( -2 - \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{25} + ( 2 - 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} - 2 \beta_{10} ) q^{26} + ( -1 + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} ) q^{27} + ( 1 - 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} - \beta_{7} - 2 \beta_{8} - \beta_{10} - \beta_{11} ) q^{29} + ( \beta_{3} + \beta_{6} - \beta_{7} ) q^{30} + ( -3 + 2 \beta_{2} - 2 \beta_{5} - \beta_{6} - 2 \beta_{9} - \beta_{10} ) q^{31} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{32} + ( \beta_{2} + 2 \beta_{3} + 3 \beta_{7} + \beta_{8} ) q^{33} + ( 3 - 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{10} + 2 \beta_{11} ) q^{34} + ( -4 + 2 \beta_{4} + 4 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{36} + ( -2 + 2 \beta_{7} + 3 \beta_{9} + \beta_{10} ) q^{37} + ( 4 - \beta_{2} + 4 \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{38} + ( 5 + \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{39} + ( 1 - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{40} + ( -2 + 2 \beta_{7} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{41} + ( 2 \beta_{1} - \beta_{3} + 2 \beta_{6} - 3 \beta_{7} ) q^{43} + ( -3 - 2 \beta_{2} + \beta_{4} + 2 \beta_{5} + 2 \beta_{9} ) q^{44} + ( -2 \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} ) q^{45} + ( -2 \beta_{2} - \beta_{3} - 2 \beta_{6} + 4 \beta_{7} + 3 \beta_{8} ) q^{46} + ( -2 - 3 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{10} + 3 \beta_{11} ) q^{47} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} + 4 \beta_{7} + 2 \beta_{8} ) q^{48} + ( 2 + 3 \beta_{1} + 3 \beta_{5} - 2 \beta_{7} + \beta_{9} + \beta_{10} - 4 \beta_{11} ) q^{50} + ( 4 - \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{6} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{51} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{6} - 3 \beta_{7} + 2 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{52} + ( 3 \beta_{2} - 2 \beta_{4} - \beta_{5} - 3 \beta_{9} ) q^{53} + ( -3 - \beta_{4} + 3 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{54} + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{55} + ( 3 + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{9} + 2 \beta_{11} ) q^{57} + ( 3 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - \beta_{7} + 3 \beta_{8} ) q^{58} + ( -\beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{59} + ( -3 + 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{11} ) q^{60} + ( -6 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} ) q^{61} + ( 1 + \beta_{1} + 2 \beta_{4} + \beta_{5} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{62} + ( -2 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{64} + ( -1 + \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{65} + ( -5 - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{10} + 4 \beta_{11} ) q^{66} + ( -3 + \beta_{1} + 2 \beta_{4} + \beta_{5} + 3 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} ) q^{67} + ( 5 - \beta_{1} - \beta_{4} - \beta_{5} - 5 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} + 4 \beta_{11} ) q^{68} + ( -\beta_{1} - \beta_{2} - 5 \beta_{6} + 6 \beta_{7} + \beta_{8} ) q^{69} + ( -\beta_{2} + 3 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} ) q^{71} + ( 3 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{6} + 4 \beta_{7} + 2 \beta_{8} ) q^{72} + ( 4 - \beta_{3} - 5 \beta_{5} + \beta_{11} ) q^{73} + ( 5 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{74} + ( -\beta_{4} - \beta_{8} - 2 \beta_{9} - 4 \beta_{11} ) q^{75} + ( \beta_{1} + \beta_{5} + 3 \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{76} + ( 1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + \beta_{9} - \beta_{10} + 6 \beta_{11} ) q^{78} + ( -2 + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{9} + 3 \beta_{11} ) q^{79} + ( -2 - \beta_{1} - 2 \beta_{4} - \beta_{5} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{80} + ( -3 \beta_{4} - 3 \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{81} + ( 3 \beta_{1} + \beta_{2} - 4 \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{82} + ( -4 - \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{83} + ( \beta_{2} - \beta_{3} - \beta_{6} + \beta_{8} ) q^{85} + ( 2 + 5 \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{10} - 5 \beta_{11} ) q^{86} + ( -2 \beta_{1} + \beta_{2} + 6 \beta_{3} + 4 \beta_{7} + \beta_{8} ) q^{87} + ( \beta_{1} + 2 \beta_{4} + \beta_{5} + 2 \beta_{8} + \beta_{9} ) q^{88} + ( 4 + 8 \beta_{1} + \beta_{4} + 8 \beta_{5} - 4 \beta_{7} + \beta_{8} - 4 \beta_{10} - 5 \beta_{11} ) q^{89} + ( -1 - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{90} + ( 4 + \beta_{2} - 7 \beta_{3} - 3 \beta_{4} - \beta_{6} - \beta_{9} - \beta_{10} + 7 \beta_{11} ) q^{92} + ( -1 + \beta_{7} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{93} + ( 8 - 2 \beta_{1} + \beta_{4} - 2 \beta_{5} - 8 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} - 6 \beta_{11} ) q^{94} + ( -3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{6} + \beta_{7} - 3 \beta_{8} ) q^{95} + ( -5 + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{9} + 2 \beta_{10} + 3 \beta_{11} ) q^{96} + ( 2 \beta_{1} - 4 \beta_{3} + 3 \beta_{6} - 8 \beta_{7} - \beta_{8} ) q^{97} + ( -3 - 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 2q^{2} + q^{3} - 4q^{4} - 2q^{5} - 9q^{6} - 6q^{8} + 3q^{9} + O(q^{10}) \) \( 12q + 2q^{2} + q^{3} - 4q^{4} - 2q^{5} - 9q^{6} - 6q^{8} + 3q^{9} + 4q^{10} + 4q^{11} - 10q^{12} - 2q^{13} - 2q^{15} + 8q^{16} + 5q^{17} - 6q^{18} - q^{19} - q^{20} - 5q^{22} - q^{23} - 11q^{24} - 14q^{25} + 11q^{26} - 8q^{27} + 3q^{29} - 5q^{30} - 32q^{31} + 8q^{32} + 16q^{33} + 32q^{34} - 21q^{36} - 13q^{37} + 34q^{38} + 43q^{39} + 10q^{40} - 8q^{41} - 11q^{43} - 42q^{44} - 7q^{45} + 16q^{46} + 2q^{47} + 21q^{48} + 6q^{50} + 40q^{51} - 16q^{52} + 4q^{53} - 18q^{54} + 9q^{55} + 42q^{57} - 8q^{58} + 13q^{59} - 40q^{60} - 5q^{61} + 5q^{62} - 30q^{64} - 14q^{65} - 36q^{66} - 11q^{67} + 29q^{68} + 23q^{69} + 6q^{71} + 25q^{72} + 60q^{73} - 3q^{74} - 3q^{75} - 9q^{76} + 16q^{78} - 14q^{79} - 7q^{80} - 6q^{81} + q^{82} - 54q^{83} - q^{85} + 14q^{86} + 16q^{87} + 4q^{89} - 16q^{90} + 54q^{92} - 7q^{93} + 45q^{94} - 6q^{95} - 38q^{96} - 35q^{97} - 20q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - x^{11} + 7 x^{10} - 2 x^{9} + 33 x^{8} - 11 x^{7} + 55 x^{6} + 17 x^{5} + 47 x^{4} + x^{3} + 8 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-29696 \nu^{11} - 478424 \nu^{10} + 682506 \nu^{9} - 3846008 \nu^{8} + 2684563 \nu^{7} - 16878368 \nu^{6} + 16008568 \nu^{5} - 31119861 \nu^{4} + 8363982 \nu^{3} - 14058754 \nu^{2} + 5624108 \nu - 2119374\)\()/3318773\)
\(\beta_{3}\)\(=\)\((\)\(-73788 \nu^{11} - 498559 \nu^{10} + 495146 \nu^{9} - 4188508 \nu^{8} + 1631143 \nu^{7} - 18206928 \nu^{6} + 16328192 \nu^{5} - 34289666 \nu^{4} + 8704710 \nu^{3} - 14803002 \nu^{2} + 21668998 \nu - 2229034\)\()/3318773\)
\(\beta_{4}\)\(=\)\((\)\(-109660 \nu^{11} + 153752 \nu^{10} - 747485 \nu^{9} + 406680 \nu^{8} - 3276280 \nu^{7} + 2259680 \nu^{6} - 4702740 \nu^{5} - 2183844 \nu^{4} - 1984215 \nu^{3} - 450388 \nu^{2} - 133032 \nu - 6198231\)\()/3318773\)
\(\beta_{5}\)\(=\)\((\)\(439315 \nu^{11} - 329655 \nu^{10} + 2921453 \nu^{9} - 131145 \nu^{8} + 14090715 \nu^{7} - 1556185 \nu^{6} + 21902645 \nu^{5} + 12171095 \nu^{4} + 22831649 \nu^{3} + 2423530 \nu^{2} + 646135 \nu + 572347\)\()/3318773\)
\(\beta_{6}\)\(=\)\((\)\(566698 \nu^{11} - 1732988 \nu^{10} + 5617249 \nu^{9} - 9944902 \nu^{8} + 24340355 \nu^{7} - 46353032 \nu^{6} + 58565408 \nu^{5} - 63065800 \nu^{4} + 27901335 \nu^{3} - 44235433 \nu^{2} + 12588213 \nu - 6707921\)\()/3318773\)
\(\beta_{7}\)\(=\)\((\)\(-572347 \nu^{11} + 1011662 \nu^{10} - 4336084 \nu^{9} + 4066147 \nu^{8} - 19018596 \nu^{7} + 20386532 \nu^{6} - 33035270 \nu^{5} + 12172746 \nu^{4} - 14729214 \nu^{3} + 22259302 \nu^{2} - 2155246 \nu + 3392561\)\()/3318773\)
\(\beta_{8}\)\(=\)\((\)\(-1035034 \nu^{11} + 1869572 \nu^{10} - 7924683 \nu^{9} + 7725614 \nu^{8} - 34760912 \nu^{7} + 38513384 \nu^{6} - 61367800 \nu^{5} + 26529336 \nu^{4} - 27474213 \nu^{3} + 41650219 \nu^{2} - 4177460 \nu + 6345807\)\()/3318773\)
\(\beta_{9}\)\(=\)\((\)\(1166290 \nu^{11} - 1650363 \nu^{10} + 8811506 \nu^{9} - 5639321 \nu^{8} + 40119354 \nu^{7} - 27397018 \nu^{6} + 72699666 \nu^{5} - 1266529 \nu^{4} + 44802131 \nu^{3} - 8054629 \nu^{2} + 7274619 \nu + 566698\)\()/3318773\)
\(\beta_{10}\)\(=\)\((\)\(-2686072 \nu^{11} + 3882058 \nu^{10} - 19974443 \nu^{9} + 13501144 \nu^{8} - 90433689 \nu^{7} + 66981583 \nu^{6} - 158252610 \nu^{5} + 11027874 \nu^{4} - 96392052 \nu^{3} + 33752077 \nu^{2} - 15484451 \nu - 1035561\)\()/3318773\)
\(\beta_{11}\)\(=\)\( \nu^{11} - \nu^{10} + 7 \nu^{9} - 2 \nu^{8} + 33 \nu^{7} - 11 \nu^{6} + 55 \nu^{5} + 17 \nu^{4} + 47 \nu^{3} + \nu^{2} + 8 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{8} + 2 \beta_{7} - \beta_{4} - 2\)
\(\nu^{3}\)\(=\)\(-\beta_{11} + \beta_{9} + 5 \beta_{5} + \beta_{3} - \beta_{2}\)
\(\nu^{4}\)\(=\)\(5 \beta_{8} - 8 \beta_{7} + \beta_{6} - \beta_{2} - \beta_{1}\)
\(\nu^{5}\)\(=\)\(5 \beta_{11} - \beta_{10} - 7 \beta_{9} + \beta_{8} - \beta_{7} - 24 \beta_{5} + \beta_{4} - 24 \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(\beta_{11} - 7 \beta_{10} - 9 \beta_{9} - 7 \beta_{6} - 11 \beta_{5} + 24 \beta_{4} - \beta_{3} + 9 \beta_{2} + 36\)
\(\nu^{7}\)\(=\)\(-11 \beta_{8} + 12 \beta_{7} - 9 \beta_{6} - 24 \beta_{3} + 40 \beta_{2} + 117 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-11 \beta_{11} + 40 \beta_{10} + 60 \beta_{9} - 117 \beta_{8} + 170 \beta_{7} + 85 \beta_{5} - 117 \beta_{4} + 85 \beta_{1} - 170\)
\(\nu^{9}\)\(=\)\(-117 \beta_{11} + 60 \beta_{10} + 217 \beta_{9} + 60 \beta_{6} + 581 \beta_{5} - 85 \beta_{4} + 117 \beta_{3} - 217 \beta_{2} - 99\)
\(\nu^{10}\)\(=\)\(581 \beta_{8} - 828 \beta_{7} + 217 \beta_{6} + 85 \beta_{3} - 362 \beta_{2} - 571 \beta_{1}\)
\(\nu^{11}\)\(=\)\(581 \beta_{11} - 362 \beta_{10} - 1160 \beta_{9} + 571 \beta_{8} - 695 \beta_{7} - 2933 \beta_{5} + 571 \beta_{4} - 2933 \beta_{1} + 695\)

Character values

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

\(n\) \(197\) \(248\)
\(\chi(n)\) \(-\beta_{7}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
295.1
0.217953 + 0.377506i
−1.02197 1.77010i
−0.437442 0.757672i
0.756174 + 1.30973i
1.16700 + 2.02131i
−0.181721 0.314749i
0.217953 0.377506i
−1.02197 + 1.77010i
−0.437442 + 0.757672i
0.756174 1.30973i
1.16700 2.02131i
−0.181721 + 0.314749i
−0.929081 + 1.60921i −1.14703 + 1.98672i −0.726381 1.25813i −0.197362 −2.13137 3.69165i 0 −1.01686 −1.13137 1.95960i 0.183365 0.317598i
295.2 −0.777343 + 1.34640i 0.244626 0.423704i −0.208526 0.361177i −1.19151 0.380316 + 0.658727i 0 −2.46099 1.38032 + 2.39078i 0.926214 1.60425i
295.3 0.134063 0.232203i 0.571504 0.989875i 0.964054 + 1.66979i −2.56175 −0.153235 0.265410i 0 1.05323 0.846765 + 1.46664i −0.343436 + 0.594848i
295.4 0.425563 0.737096i −0.330612 + 0.572636i 0.637793 + 1.10469i 3.44148 0.281392 + 0.487385i 0 2.78793 1.28139 + 2.21944i 1.46456 2.53670i
295.5 0.952780 1.65026i −0.214224 + 0.371047i −0.815580 1.41263i −1.47313 0.408216 + 0.707051i 0 0.702849 1.40822 + 2.43910i −1.40357 + 2.43105i
295.6 1.19402 2.06810i 1.37574 2.38285i −1.85136 3.20665i 0.982280 −3.28532 5.69033i 0 −4.06616 −2.28532 3.95828i 1.17286 2.03145i
393.1 −0.929081 1.60921i −1.14703 1.98672i −0.726381 + 1.25813i −0.197362 −2.13137 + 3.69165i 0 −1.01686 −1.13137 + 1.95960i 0.183365 + 0.317598i
393.2 −0.777343 1.34640i 0.244626 + 0.423704i −0.208526 + 0.361177i −1.19151 0.380316 0.658727i 0 −2.46099 1.38032 2.39078i 0.926214 + 1.60425i
393.3 0.134063 + 0.232203i 0.571504 + 0.989875i 0.964054 1.66979i −2.56175 −0.153235 + 0.265410i 0 1.05323 0.846765 1.46664i −0.343436 0.594848i
393.4 0.425563 + 0.737096i −0.330612 0.572636i 0.637793 1.10469i 3.44148 0.281392 0.487385i 0 2.78793 1.28139 2.21944i 1.46456 + 2.53670i
393.5 0.952780 + 1.65026i −0.214224 0.371047i −0.815580 + 1.41263i −1.47313 0.408216 0.707051i 0 0.702849 1.40822 2.43910i −1.40357 2.43105i
393.6 1.19402 + 2.06810i 1.37574 + 2.38285i −1.85136 + 3.20665i 0.982280 −3.28532 + 5.69033i 0 −4.06616 −2.28532 + 3.95828i 1.17286 + 2.03145i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 393.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.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.f.k 12
7.b odd 2 1 637.2.f.j 12
7.c even 3 1 91.2.g.b 12
7.c even 3 1 91.2.h.b yes 12
7.d odd 6 1 637.2.g.l 12
7.d odd 6 1 637.2.h.l 12
13.c even 3 1 inner 637.2.f.k 12
13.c even 3 1 8281.2.a.bz 6
13.e even 6 1 8281.2.a.ce 6
21.h odd 6 1 819.2.n.d 12
21.h odd 6 1 819.2.s.d 12
91.g even 3 1 91.2.h.b yes 12
91.g even 3 1 1183.2.e.h 12
91.h even 3 1 91.2.g.b 12
91.h even 3 1 1183.2.e.h 12
91.k even 6 1 1183.2.e.g 12
91.m odd 6 1 637.2.h.l 12
91.n odd 6 1 637.2.f.j 12
91.n odd 6 1 8281.2.a.ca 6
91.t odd 6 1 8281.2.a.cf 6
91.u even 6 1 1183.2.e.g 12
91.v odd 6 1 637.2.g.l 12
273.s odd 6 1 819.2.n.d 12
273.bm odd 6 1 819.2.s.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.b 12 7.c even 3 1
91.2.g.b 12 91.h even 3 1
91.2.h.b yes 12 7.c even 3 1
91.2.h.b yes 12 91.g even 3 1
637.2.f.j 12 7.b odd 2 1
637.2.f.j 12 91.n odd 6 1
637.2.f.k 12 1.a even 1 1 trivial
637.2.f.k 12 13.c even 3 1 inner
637.2.g.l 12 7.d odd 6 1
637.2.g.l 12 91.v odd 6 1
637.2.h.l 12 7.d odd 6 1
637.2.h.l 12 91.m odd 6 1
819.2.n.d 12 21.h odd 6 1
819.2.n.d 12 273.s odd 6 1
819.2.s.d 12 21.h odd 6 1
819.2.s.d 12 273.bm odd 6 1
1183.2.e.g 12 91.k even 6 1
1183.2.e.g 12 91.u even 6 1
1183.2.e.h 12 91.g even 3 1
1183.2.e.h 12 91.h even 3 1
8281.2.a.bz 6 13.c even 3 1
8281.2.a.ca 6 91.n odd 6 1
8281.2.a.ce 6 13.e even 6 1
8281.2.a.cf 6 91.t odd 6 1

Hecke kernels

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

\(T_{2}^{12} - \cdots\)
\(T_{3}^{12} - \cdots\)
\( T_{5}^{6} + T_{5}^{5} - 11 T_{5}^{4} - 18 T_{5}^{3} + 6 T_{5}^{2} + 17 T_{5} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 - 42 T + 172 T^{2} - 178 T^{3} + 236 T^{4} - 86 T^{5} + 147 T^{6} - 48 T^{7} + 50 T^{8} - 10 T^{9} + 10 T^{10} - 2 T^{11} + T^{12} \)
$3$ \( 1 + T + 7 T^{2} + 2 T^{3} + 33 T^{4} + 11 T^{5} + 55 T^{6} - 17 T^{7} + 47 T^{8} - T^{9} + 8 T^{10} - T^{11} + T^{12} \)
$5$ \( ( 3 + 17 T + 6 T^{2} - 18 T^{3} - 11 T^{4} + T^{5} + T^{6} )^{2} \)
$7$ \( T^{12} \)
$11$ \( 6561 - 16767 T + 36288 T^{2} - 29079 T^{3} + 23994 T^{4} - 2862 T^{5} + 6811 T^{6} - 1155 T^{7} + 664 T^{8} - 68 T^{9} + 37 T^{10} - 4 T^{11} + T^{12} \)
$13$ \( 4826809 + 742586 T - 456976 T^{2} + 6591 T^{3} + 102583 T^{4} + 5629 T^{5} - 5615 T^{6} + 433 T^{7} + 607 T^{8} + 3 T^{9} - 16 T^{10} + 2 T^{11} + T^{12} \)
$17$ \( 81 + 72 T + 244 T^{2} + 92 T^{3} + 404 T^{4} + 133 T^{5} + 378 T^{6} + 24 T^{7} + 194 T^{8} + 32 T^{9} + 37 T^{10} - 5 T^{11} + T^{12} \)
$19$ \( 762129 + 1346166 T + 1828647 T^{2} + 1163724 T^{3} + 622675 T^{4} + 128430 T^{5} + 52781 T^{6} + 7388 T^{7} + 3578 T^{8} + 158 T^{9} + 65 T^{10} + T^{11} + T^{12} \)
$23$ \( 594725769 - 10730280 T + 74110597 T^{2} - 1739122 T^{3} + 6629659 T^{4} - 122060 T^{5} + 276041 T^{6} + 1056 T^{7} + 8268 T^{8} + 20 T^{9} + 107 T^{10} + T^{11} + T^{12} \)
$29$ \( 40401 + 225924 T + 1362670 T^{2} - 457168 T^{3} + 502614 T^{4} - 54205 T^{5} + 94294 T^{6} - 17192 T^{7} + 6322 T^{8} - 254 T^{9} + 87 T^{10} - 3 T^{11} + T^{12} \)
$31$ \( ( -2477 - 4000 T - 2042 T^{2} - 295 T^{3} + 50 T^{4} + 16 T^{5} + T^{6} )^{2} \)
$37$ \( 181629529 + 234984972 T + 199526915 T^{2} + 98766454 T^{3} + 36040847 T^{4} + 8973966 T^{5} + 1730301 T^{6} + 235480 T^{7} + 26760 T^{8} + 2208 T^{9} + 207 T^{10} + 13 T^{11} + T^{12} \)
$41$ \( 4173849 + 2939877 T + 2648890 T^{2} + 728671 T^{3} + 523034 T^{4} + 155456 T^{5} + 63915 T^{6} + 11805 T^{7} + 2948 T^{8} + 388 T^{9} + 85 T^{10} + 8 T^{11} + T^{12} \)
$43$ \( 1369 + 59940 T + 2634945 T^{2} - 442016 T^{3} + 512108 T^{4} + 72977 T^{5} + 53295 T^{6} + 7624 T^{7} + 3212 T^{8} + 543 T^{9} + 120 T^{10} + 11 T^{11} + T^{12} \)
$47$ \( ( -17847 + 6323 T + 5684 T^{2} + 88 T^{3} - 177 T^{4} - T^{5} + T^{6} )^{2} \)
$53$ \( ( -69 - 334 T + 1105 T^{2} + 186 T^{3} - 100 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$59$ \( 83229129 - 168419703 T + 334732603 T^{2} - 30468042 T^{3} + 19368969 T^{4} - 1633661 T^{5} + 809563 T^{6} - 59909 T^{7} + 15763 T^{8} - 1225 T^{9} + 228 T^{10} - 13 T^{11} + T^{12} \)
$61$ \( 1055015361 + 1195333281 T + 1062894069 T^{2} + 390333384 T^{3} + 121103191 T^{4} + 6648335 T^{5} + 2541805 T^{6} + 133207 T^{7} + 36059 T^{8} + 847 T^{9} + 226 T^{10} + 5 T^{11} + T^{12} \)
$67$ \( 276324129 - 183966741 T + 160212699 T^{2} - 4433604 T^{3} + 13229425 T^{4} + 145321 T^{5} + 875958 T^{6} + 55361 T^{7} + 18745 T^{8} + 612 T^{9} + 227 T^{10} + 11 T^{11} + T^{12} \)
$71$ \( 530979849 - 315527799 T + 189871678 T^{2} - 50943317 T^{3} + 18814920 T^{4} - 4116692 T^{5} + 1239901 T^{6} - 175105 T^{7} + 26800 T^{8} - 1426 T^{9} + 177 T^{10} - 6 T^{11} + T^{12} \)
$73$ \( ( -14029 + 24466 T - 8404 T^{2} + 251 T^{3} + 238 T^{4} - 30 T^{5} + T^{6} )^{2} \)
$79$ \( ( 10529 - 18957 T + 7249 T^{2} - 310 T^{3} - 148 T^{4} + 7 T^{5} + T^{6} )^{2} \)
$83$ \( ( 2673 + 1188 T - 1797 T^{2} - 403 T^{3} + 158 T^{4} + 27 T^{5} + T^{6} )^{2} \)
$89$ \( 92707461441 - 24201817794 T + 16326857884 T^{2} + 2693246248 T^{3} + 958332439 T^{4} + 63899744 T^{5} + 11790434 T^{6} + 294018 T^{7} + 102345 T^{8} + 1204 T^{9} + 383 T^{10} - 4 T^{11} + T^{12} \)
$97$ \( 15202201 + 33180490 T + 68189685 T^{2} + 18481778 T^{3} + 12693220 T^{4} + 4789025 T^{5} + 2092673 T^{6} + 500330 T^{7} + 92800 T^{8} + 10403 T^{9} + 860 T^{10} + 35 T^{11} + T^{12} \)
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