Properties

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

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.h (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: \(x^{12} - x^{11} + 7 x^{10} - 2 x^{9} + 33 x^{8} - 11 x^{7} + 55 x^{6} + 17 x^{5} + 47 x^{4} + x^{3} + 8 x^{2} + x + 1\)
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.

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

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 \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
−0.181721 + 0.314749i
1.16700 2.02131i
0.756174 1.30973i
−0.437442 + 0.757672i
−1.02197 + 1.77010i
0.217953 0.377506i
−0.181721 0.314749i
1.16700 + 2.02131i
0.756174 + 1.30973i
−0.437442 0.757672i
−1.02197 1.77010i
0.217953 + 0.377506i
−2.38804 −1.37574 2.38285i 3.70272 0.491140 + 0.850679i 3.28532 + 5.69033i 0 −4.06616 −2.28532 + 3.95828i −1.17286 2.03145i
165.2 −1.90556 0.214224 + 0.371047i 1.63116 −0.736565 1.27577i −0.408216 0.707051i 0 0.702849 1.40822 2.43910i 1.40357 + 2.43105i
165.3 −0.851125 0.330612 + 0.572636i −1.27559 1.72074 + 2.98041i −0.281392 0.487385i 0 2.78793 1.28139 2.21944i −1.46456 2.53670i
165.4 −0.268125 −0.571504 0.989875i −1.92811 −1.28088 2.21854i 0.153235 + 0.265410i 0 1.05323 0.846765 1.46664i 0.343436 + 0.594848i
165.5 1.55469 −0.244626 0.423704i 0.417051 −0.595756 1.03188i −0.380316 0.658727i 0 −2.46099 1.38032 2.39078i −0.926214 1.60425i
165.6 1.85816 1.14703 + 1.98672i 1.45276 −0.0986811 0.170921i 2.13137 + 3.69165i 0 −1.01686 −1.13137 + 1.95960i −0.183365 0.317598i
471.1 −2.38804 −1.37574 + 2.38285i 3.70272 0.491140 0.850679i 3.28532 5.69033i 0 −4.06616 −2.28532 3.95828i −1.17286 + 2.03145i
471.2 −1.90556 0.214224 0.371047i 1.63116 −0.736565 + 1.27577i −0.408216 + 0.707051i 0 0.702849 1.40822 + 2.43910i 1.40357 2.43105i
471.3 −0.851125 0.330612 0.572636i −1.27559 1.72074 2.98041i −0.281392 + 0.487385i 0 2.78793 1.28139 + 2.21944i −1.46456 + 2.53670i
471.4 −0.268125 −0.571504 + 0.989875i −1.92811 −1.28088 + 2.21854i 0.153235 0.265410i 0 1.05323 0.846765 + 1.46664i 0.343436 0.594848i
471.5 1.55469 −0.244626 + 0.423704i 0.417051 −0.595756 + 1.03188i −0.380316 + 0.658727i 0 −2.46099 1.38032 + 2.39078i −0.926214 + 1.60425i
471.6 1.85816 1.14703 1.98672i 1.45276 −0.0986811 + 0.170921i 2.13137 3.69165i 0 −1.01686 −1.13137 1.95960i −0.183365 + 0.317598i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 471.6
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{2}^{6} + 2 T_{2}^{5} - 6 T_{2}^{4} - 11 T_{2}^{3} + 8 T_{2}^{2} + 14 T_{2} + 3 \)
\(T_{3}^{12} + \cdots\)
\(T_{5}^{12} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 3 + 14 T + 8 T^{2} - 11 T^{3} - 6 T^{4} + 2 T^{5} + T^{6} )^{2} \)
$3$ \( 1 - T + 7 T^{2} - 2 T^{3} + 33 T^{4} - 11 T^{5} + 55 T^{6} + 17 T^{7} + 47 T^{8} + T^{9} + 8 T^{10} + T^{11} + T^{12} \)
$5$ \( 9 + 51 T + 271 T^{2} + 210 T^{3} + 375 T^{4} + 269 T^{5} + 379 T^{6} + 203 T^{7} + 133 T^{8} + 25 T^{9} + 12 T^{10} + T^{11} + T^{12} \)
$7$ \( T^{12} \)
$11$ \( 6561 - 16767 T + 36288 T^{2} - 29079 T^{3} + 23994 T^{4} - 2862 T^{5} + 6811 T^{6} - 1155 T^{7} + 664 T^{8} - 68 T^{9} + 37 T^{10} - 4 T^{11} + T^{12} \)
$13$ \( 4826809 - 742586 T - 456976 T^{2} - 6591 T^{3} + 102583 T^{4} - 5629 T^{5} - 5615 T^{6} - 433 T^{7} + 607 T^{8} - 3 T^{9} - 16 T^{10} - 2 T^{11} + T^{12} \)
$17$ \( ( -9 - 8 T + 20 T^{2} + 14 T^{3} - 12 T^{4} - 5 T^{5} + T^{6} )^{2} \)
$19$ \( 762129 - 1346166 T + 1828647 T^{2} - 1163724 T^{3} + 622675 T^{4} - 128430 T^{5} + 52781 T^{6} - 7388 T^{7} + 3578 T^{8} - 158 T^{9} + 65 T^{10} - T^{11} + T^{12} \)
$23$ \( ( -24387 - 440 T + 3031 T^{2} + 63 T^{3} - 106 T^{4} - T^{5} + T^{6} )^{2} \)
$29$ \( 40401 + 225924 T + 1362670 T^{2} - 457168 T^{3} + 502614 T^{4} - 54205 T^{5} + 94294 T^{6} - 17192 T^{7} + 6322 T^{8} - 254 T^{9} + 87 T^{10} - 3 T^{11} + T^{12} \)
$31$ \( 6135529 + 9908000 T + 10941966 T^{2} + 6706570 T^{3} + 3113614 T^{4} + 962758 T^{5} + 248171 T^{6} + 46594 T^{7} + 9262 T^{8} + 1390 T^{9} + 206 T^{10} + 16 T^{11} + T^{12} \)
$37$ \( ( -13477 + 17436 T - 7753 T^{2} + 1351 T^{3} - 38 T^{4} - 13 T^{5} + T^{6} )^{2} \)
$41$ \( 4173849 - 2939877 T + 2648890 T^{2} - 728671 T^{3} + 523034 T^{4} - 155456 T^{5} + 63915 T^{6} - 11805 T^{7} + 2948 T^{8} - 388 T^{9} + 85 T^{10} - 8 T^{11} + T^{12} \)
$43$ \( 1369 + 59940 T + 2634945 T^{2} - 442016 T^{3} + 512108 T^{4} + 72977 T^{5} + 53295 T^{6} + 7624 T^{7} + 3212 T^{8} + 543 T^{9} + 120 T^{10} + 11 T^{11} + T^{12} \)
$47$ \( 318515409 - 112846581 T + 141422677 T^{2} + 39081004 T^{3} + 28592513 T^{4} + 2756381 T^{5} + 984441 T^{6} + 2115 T^{7} + 25733 T^{8} + T^{9} + 178 T^{10} - T^{11} + T^{12} \)
$53$ \( 4761 - 23046 T + 187801 T^{2} + 343402 T^{3} + 1276249 T^{4} - 138868 T^{5} + 144290 T^{6} + 14514 T^{7} + 9267 T^{8} + 172 T^{9} + 104 T^{10} + 2 T^{11} + T^{12} \)
$59$ \( ( 9123 - 18461 T + 666 T^{2} + 996 T^{3} - 59 T^{4} - 13 T^{5} + T^{6} )^{2} \)
$61$ \( 1055015361 - 1195333281 T + 1062894069 T^{2} - 390333384 T^{3} + 121103191 T^{4} - 6648335 T^{5} + 2541805 T^{6} - 133207 T^{7} + 36059 T^{8} - 847 T^{9} + 226 T^{10} - 5 T^{11} + T^{12} \)
$67$ \( 276324129 - 183966741 T + 160212699 T^{2} - 4433604 T^{3} + 13229425 T^{4} + 145321 T^{5} + 875958 T^{6} + 55361 T^{7} + 18745 T^{8} + 612 T^{9} + 227 T^{10} + 11 T^{11} + T^{12} \)
$71$ \( 530979849 - 315527799 T + 189871678 T^{2} - 50943317 T^{3} + 18814920 T^{4} - 4116692 T^{5} + 1239901 T^{6} - 175105 T^{7} + 26800 T^{8} - 1426 T^{9} + 177 T^{10} - 6 T^{11} + T^{12} \)
$73$ \( 196812841 - 343233514 T + 480685440 T^{2} - 198569706 T^{3} + 67825152 T^{4} - 13334350 T^{5} + 2769075 T^{6} - 420036 T^{7} + 72578 T^{8} - 7642 T^{9} + 662 T^{10} - 30 T^{11} + T^{12} \)
$79$ \( 110859841 + 199598253 T + 283043128 T^{2} + 130891313 T^{3} + 48229623 T^{4} + 7784759 T^{5} + 1322709 T^{6} + 74563 T^{7} + 16825 T^{8} + 416 T^{9} + 197 T^{10} - 7 T^{11} + T^{12} \)
$83$ \( ( 2673 - 1188 T - 1797 T^{2} + 403 T^{3} + 158 T^{4} - 27 T^{5} + T^{6} )^{2} \)
$89$ \( ( -304479 + 79486 T + 32872 T^{2} + 132 T^{3} - 367 T^{4} - 4 T^{5} + T^{6} )^{2} \)
$97$ \( 15202201 - 33180490 T + 68189685 T^{2} - 18481778 T^{3} + 12693220 T^{4} - 4789025 T^{5} + 2092673 T^{6} - 500330 T^{7} + 92800 T^{8} - 10403 T^{9} + 860 T^{10} - 35 T^{11} + T^{12} \)
show more
show less