Properties

Label 1183.2.e.g
Level $1183$
Weight $2$
Character orbit 1183.e
Analytic conductor $9.446$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.44630255912\)
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.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{5} - \beta_{11} ) q^{2} -\beta_{3} q^{3} + ( \beta_{6} - \beta_{7} ) q^{4} + \beta_{9} q^{5} + ( -2 + \beta_{4} + \beta_{6} + \beta_{10} ) q^{6} + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} - \beta_{7} ) q^{7} + ( 1 + \beta_{5} - \beta_{6} - \beta_{10} ) q^{8} + ( \beta_{4} + \beta_{8} + \beta_{10} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{5} - \beta_{11} ) q^{2} -\beta_{3} q^{3} + ( \beta_{6} - \beta_{7} ) q^{4} + \beta_{9} q^{5} + ( -2 + \beta_{4} + \beta_{6} + \beta_{10} ) q^{6} + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} - \beta_{7} ) q^{7} + ( 1 + \beta_{5} - \beta_{6} - \beta_{10} ) q^{8} + ( \beta_{4} + \beta_{8} + \beta_{10} ) q^{9} + ( -\beta_{1} - \beta_{2} + \beta_{7} ) q^{10} + ( -\beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{11} + ( 1 - \beta_{7} + \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{12} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{11} ) q^{14} + ( -\beta_{6} - \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} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{6} - \beta_{7} ) q^{18} + ( 1 + 2 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{19} + ( -\beta_{2} + \beta_{3} + \beta_{4} + \beta_{9} - \beta_{11} ) q^{20} + ( 1 + \beta_{1} - 3 \beta_{3} - \beta_{7} - \beta_{10} + \beta_{11} ) q^{21} + ( -2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{11} ) q^{22} + ( -1 - 2 \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{8} - 3 \beta_{9} + \beta_{10} - \beta_{11} ) q^{23} + ( -\beta_{2} - 2 \beta_{3} - \beta_{6} + 2 \beta_{7} ) q^{24} + ( -\beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{6} + 2 \beta_{7} ) q^{25} + ( -1 + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} ) q^{27} + ( -1 + \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{10} - \beta_{11} ) q^{28} + ( -1 - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{10} + \beta_{11} ) q^{29} + ( -1 + \beta_{7} + \beta_{10} - \beta_{11} ) q^{30} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{6} - 3 \beta_{7} ) q^{31} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{32} + ( -3 + \beta_{4} + 3 \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{33} + ( -3 + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{10} - 2 \beta_{11} ) q^{34} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{35} + ( 4 - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{6} + \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{36} + ( 2 - 2 \beta_{7} - 3 \beta_{9} - \beta_{10} ) q^{37} + ( 4 \beta_{1} + \beta_{2} + \beta_{6} - 4 \beta_{7} ) q^{38} + ( -1 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{40} + ( -2 - \beta_{2} - \beta_{3} + 2 \beta_{6} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{41} + ( -7 + \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{6} + 3 \beta_{7} + \beta_{8} + 3 \beta_{10} ) q^{42} + ( 3 + \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - 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} + ( 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 - 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_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{48} + ( 1 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{49} + ( 2 - \beta_{2} + 4 \beta_{3} + 3 \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} - 4 \beta_{11} ) q^{50} + ( -4 + \beta_{4} + 4 \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{51} + ( -\beta_{1} - 3 \beta_{2} - 2 \beta_{8} ) q^{53} + ( 3 + \beta_{4} - 3 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{54} + ( -1 + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{10} - \beta_{11} ) q^{55} + ( 2 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{7} - 2 \beta_{10} + \beta_{11} ) q^{56} + ( -3 - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{9} - 2 \beta_{11} ) q^{57} + ( 1 + 3 \beta_{1} + 3 \beta_{4} + 3 \beta_{5} - \beta_{7} + 3 \beta_{8} + 2 \beta_{9} + 4 \beta_{11} ) q^{58} + ( \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{59} + ( -\beta_{1} + 2 \beta_{3} - 3 \beta_{7} - \beta_{8} ) q^{60} + ( 2 + 6 \beta_{1} + 6 \beta_{5} - 2 \beta_{7} + 2 \beta_{9} - 2 \beta_{10} - 5 \beta_{11} ) q^{61} + ( -1 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{62} + ( -4 + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + 5 \beta_{7} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{63} + ( -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} + ( -2 \beta_{1} + 4 \beta_{3} - 2 \beta_{6} + 5 \beta_{7} + 2 \beta_{8} ) q^{66} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} ) q^{67} + ( 5 - \beta_{1} - \beta_{4} - \beta_{5} - 5 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} + 4 \beta_{11} ) q^{68} + ( -6 + \beta_{2} + \beta_{4} - \beta_{5} + 5 \beta_{6} - \beta_{9} + 5 \beta_{10} ) q^{69} + ( 4 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{70} + ( 2 - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + \beta_{9} - 2 \beta_{10} - 3 \beta_{11} ) q^{71} + ( -4 + 3 \beta_{1} + 2 \beta_{4} + 3 \beta_{5} + 4 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} - 3 \beta_{11} ) q^{72} + ( 5 \beta_{1} - \beta_{3} + 4 \beta_{7} ) 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} + ( -3 \beta_{2} - \beta_{3} + \beta_{5} - 3 \beta_{6} + 3 \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{76} + ( 1 + 3 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{77} + ( 2 - \beta_{1} - \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} - 3 \beta_{11} ) q^{79} + ( -\beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} ) q^{80} + ( \beta_{2} + 2 \beta_{3} - \beta_{6} + 3 \beta_{8} ) q^{81} + ( -1 - 3 \beta_{1} + \beta_{4} - 3 \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + 4 \beta_{11} ) q^{82} + ( 4 + \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{6} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{83} + ( -2 - \beta_{1} + \beta_{2} + 4 \beta_{3} + \beta_{4} + 2 \beta_{6} - 3 \beta_{7} + \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{84} + ( \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \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} + ( -2 \beta_{1} + \beta_{2} + 6 \beta_{3} + 4 \beta_{7} + \beta_{8} ) q^{87} + ( -\beta_{1} - \beta_{2} - 2 \beta_{8} ) 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} + ( 2 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} - \beta_{6} + 8 \beta_{7} - \beta_{8} ) q^{94} + ( -3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{6} + \beta_{7} - 3 \beta_{8} ) 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 - 4 \beta_{3} + \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{10} + 4 \beta_{11} ) q^{97} + ( 9 - \beta_{1} - 3 \beta_{3} + \beta_{4} - 3 \beta_{6} - 3 \beta_{7} - \beta_{8} - 3 \beta_{10} - \beta_{11} ) q^{98} + ( 3 + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 2q^{2} + q^{3} - 4q^{4} - q^{5} - 18q^{6} + 6q^{7} + 6q^{8} + 3q^{9} + O(q^{10}) \) \( 12q - 2q^{2} + q^{3} - 4q^{4} - q^{5} - 18q^{6} + 6q^{7} + 6q^{8} + 3q^{9} + 4q^{10} - 4q^{11} + 5q^{12} - 2q^{14} - 4q^{15} + 8q^{16} + 5q^{17} - 3q^{18} + q^{19} - 2q^{20} + 9q^{21} + 10q^{22} - q^{23} + 11q^{24} + 7q^{25} - 8q^{27} - 8q^{28} - 6q^{29} - 5q^{30} - 16q^{31} - 8q^{32} - 16q^{33} - 32q^{34} - 28q^{35} + 42q^{36} + 13q^{37} - 17q^{38} - 5q^{40} - 16q^{41} - 52q^{42} + 22q^{43} - 21q^{44} + 7q^{45} - 16q^{46} + q^{47} - 42q^{48} + 6q^{49} + 12q^{50} - 20q^{51} - 2q^{53} + 18q^{54} - 18q^{55} + 9q^{56} - 42q^{57} + 8q^{58} - 13q^{59} - 20q^{60} - 5q^{61} - 10q^{62} - 8q^{63} - 30q^{64} + 18q^{66} + 11q^{67} + 29q^{68} - 46q^{69} + 39q^{70} + 12q^{71} - 25q^{72} + 30q^{73} - 3q^{74} - 3q^{75} - 18q^{76} + 11q^{77} + 7q^{79} + 7q^{80} - 6q^{81} + q^{82} + 54q^{83} - 41q^{84} - 2q^{85} + 7q^{86} + 16q^{87} - 4q^{89} - 16q^{90} + 54q^{92} + 7q^{93} + 45q^{94} - 6q^{95} - 19q^{96} - 70q^{97} + 82q^{98} + 20q^{99} + O(q^{100}) \)

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 \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(-1 + \beta_{7}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
170.1
−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
−1.19402 + 2.06810i 1.37574 + 2.38285i −1.85136 3.20665i 0.491140 0.850679i −6.57063 1.69505 2.03145i 4.06616 −2.28532 + 3.95828i 1.17286 + 2.03145i
170.2 −0.952780 + 1.65026i −0.214224 0.371047i −0.815580 1.41263i −0.736565 + 1.27577i 0.816433 −1.04402 + 2.43105i −0.702849 1.40822 2.43910i −1.40357 2.43105i
170.3 −0.425563 + 0.737096i −0.330612 0.572636i 0.637793 + 1.10469i 1.72074 2.98041i 0.562784 −0.751763 2.53670i −2.78793 1.28139 2.21944i 1.46456 + 2.53670i
170.4 −0.134063 + 0.232203i 0.571504 + 0.989875i 0.964054 + 1.66979i −1.28088 + 2.21854i −0.306470 2.57801 + 0.594848i −1.05323 0.846765 1.46664i −0.343436 0.594848i
170.5 0.777343 1.34640i 0.244626 + 0.423704i −0.208526 0.361177i −0.595756 + 1.03188i 0.760633 −2.10390 1.60425i 2.46099 1.38032 2.39078i 0.926214 + 1.60425i
170.6 0.929081 1.60921i −1.14703 1.98672i −0.726381 1.25813i −0.0986811 + 0.170921i −4.26275 2.62662 0.317598i 1.01686 −1.13137 + 1.95960i 0.183365 + 0.317598i
508.1 −1.19402 2.06810i 1.37574 2.38285i −1.85136 + 3.20665i 0.491140 + 0.850679i −6.57063 1.69505 + 2.03145i 4.06616 −2.28532 3.95828i 1.17286 2.03145i
508.2 −0.952780 1.65026i −0.214224 + 0.371047i −0.815580 + 1.41263i −0.736565 1.27577i 0.816433 −1.04402 2.43105i −0.702849 1.40822 + 2.43910i −1.40357 + 2.43105i
508.3 −0.425563 0.737096i −0.330612 + 0.572636i 0.637793 1.10469i 1.72074 + 2.98041i 0.562784 −0.751763 + 2.53670i −2.78793 1.28139 + 2.21944i 1.46456 2.53670i
508.4 −0.134063 0.232203i 0.571504 0.989875i 0.964054 1.66979i −1.28088 2.21854i −0.306470 2.57801 0.594848i −1.05323 0.846765 + 1.46664i −0.343436 + 0.594848i
508.5 0.777343 + 1.34640i 0.244626 0.423704i −0.208526 + 0.361177i −0.595756 1.03188i 0.760633 −2.10390 + 1.60425i 2.46099 1.38032 + 2.39078i 0.926214 1.60425i
508.6 0.929081 + 1.60921i −1.14703 + 1.98672i −0.726381 + 1.25813i −0.0986811 0.170921i −4.26275 2.62662 + 0.317598i 1.01686 −1.13137 1.95960i 0.183365 0.317598i
\(n\): 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 1183.2.e.g 12
7.c even 3 1 inner 1183.2.e.g 12
7.c even 3 1 8281.2.a.ce 6
7.d odd 6 1 8281.2.a.cf 6
13.b even 2 1 1183.2.e.h 12
13.e even 6 1 91.2.g.b 12
13.e even 6 1 91.2.h.b yes 12
39.h odd 6 1 819.2.n.d 12
39.h odd 6 1 819.2.s.d 12
91.k even 6 1 91.2.g.b 12
91.k even 6 1 637.2.f.k 12
91.l odd 6 1 637.2.f.j 12
91.l odd 6 1 637.2.g.l 12
91.p odd 6 1 637.2.f.j 12
91.p odd 6 1 637.2.h.l 12
91.r even 6 1 1183.2.e.h 12
91.r even 6 1 8281.2.a.bz 6
91.s odd 6 1 8281.2.a.ca 6
91.t odd 6 1 637.2.g.l 12
91.t odd 6 1 637.2.h.l 12
91.u even 6 1 91.2.h.b yes 12
91.u even 6 1 637.2.f.k 12
273.x odd 6 1 819.2.s.d 12
273.bp odd 6 1 819.2.n.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.b 12 13.e even 6 1
91.2.g.b 12 91.k even 6 1
91.2.h.b yes 12 13.e even 6 1
91.2.h.b yes 12 91.u even 6 1
637.2.f.j 12 91.l odd 6 1
637.2.f.j 12 91.p odd 6 1
637.2.f.k 12 91.k even 6 1
637.2.f.k 12 91.u even 6 1
637.2.g.l 12 91.l odd 6 1
637.2.g.l 12 91.t odd 6 1
637.2.h.l 12 91.p odd 6 1
637.2.h.l 12 91.t odd 6 1
819.2.n.d 12 39.h odd 6 1
819.2.n.d 12 273.bp odd 6 1
819.2.s.d 12 39.h odd 6 1
819.2.s.d 12 273.x odd 6 1
1183.2.e.g 12 1.a even 1 1 trivial
1183.2.e.g 12 7.c even 3 1 inner
1183.2.e.h 12 13.b even 2 1
1183.2.e.h 12 91.r even 6 1
8281.2.a.bz 6 91.r even 6 1
8281.2.a.ca 6 91.s odd 6 1
8281.2.a.ce 6 7.c even 3 1
8281.2.a.cf 6 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1183, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + 42 T + 172 T^{2} + 178 T^{3} + 236 T^{4} + 86 T^{5} + 147 T^{6} + 48 T^{7} + 50 T^{8} + 10 T^{9} + 10 T^{10} + 2 T^{11} + T^{12} \)
$3$ \( 1 + T + 7 T^{2} + 2 T^{3} + 33 T^{4} + 11 T^{5} + 55 T^{6} - 17 T^{7} + 47 T^{8} - T^{9} + 8 T^{10} - T^{11} + T^{12} \)
$5$ \( 9 + 51 T + 271 T^{2} + 210 T^{3} + 375 T^{4} + 269 T^{5} + 379 T^{6} + 203 T^{7} + 133 T^{8} + 25 T^{9} + 12 T^{10} + T^{11} + T^{12} \)
$7$ \( 117649 - 100842 T + 36015 T^{2} - 18522 T^{3} + 10731 T^{4} - 3612 T^{5} + 1069 T^{6} - 516 T^{7} + 219 T^{8} - 54 T^{9} + 15 T^{10} - 6 T^{11} + T^{12} \)
$11$ \( 6561 + 16767 T + 36288 T^{2} + 29079 T^{3} + 23994 T^{4} + 2862 T^{5} + 6811 T^{6} + 1155 T^{7} + 664 T^{8} + 68 T^{9} + 37 T^{10} + 4 T^{11} + T^{12} \)
$13$ \( T^{12} \)
$17$ \( 81 + 72 T + 244 T^{2} + 92 T^{3} + 404 T^{4} + 133 T^{5} + 378 T^{6} + 24 T^{7} + 194 T^{8} + 32 T^{9} + 37 T^{10} - 5 T^{11} + T^{12} \)
$19$ \( 762129 - 1346166 T + 1828647 T^{2} - 1163724 T^{3} + 622675 T^{4} - 128430 T^{5} + 52781 T^{6} - 7388 T^{7} + 3578 T^{8} - 158 T^{9} + 65 T^{10} - T^{11} + T^{12} \)
$23$ \( 594725769 - 10730280 T + 74110597 T^{2} - 1739122 T^{3} + 6629659 T^{4} - 122060 T^{5} + 276041 T^{6} + 1056 T^{7} + 8268 T^{8} + 20 T^{9} + 107 T^{10} + T^{11} + T^{12} \)
$29$ \( ( -201 + 1124 T + 494 T^{2} - 244 T^{3} - 78 T^{4} + 3 T^{5} + T^{6} )^{2} \)
$31$ \( 6135529 + 9908000 T + 10941966 T^{2} + 6706570 T^{3} + 3113614 T^{4} + 962758 T^{5} + 248171 T^{6} + 46594 T^{7} + 9262 T^{8} + 1390 T^{9} + 206 T^{10} + 16 T^{11} + T^{12} \)
$37$ \( 181629529 - 234984972 T + 199526915 T^{2} - 98766454 T^{3} + 36040847 T^{4} - 8973966 T^{5} + 1730301 T^{6} - 235480 T^{7} + 26760 T^{8} - 2208 T^{9} + 207 T^{10} - 13 T^{11} + T^{12} \)
$41$ \( ( 2043 + 1439 T - 283 T^{2} - 278 T^{3} - 21 T^{4} + 8 T^{5} + T^{6} )^{2} \)
$43$ \( ( 37 - 1620 T - 285 T^{2} + 266 T^{3} + T^{4} - 11 T^{5} + T^{6} )^{2} \)
$47$ \( 318515409 - 112846581 T + 141422677 T^{2} + 39081004 T^{3} + 28592513 T^{4} + 2756381 T^{5} + 984441 T^{6} + 2115 T^{7} + 25733 T^{8} + T^{9} + 178 T^{10} - T^{11} + T^{12} \)
$53$ \( 4761 - 23046 T + 187801 T^{2} + 343402 T^{3} + 1276249 T^{4} - 138868 T^{5} + 144290 T^{6} + 14514 T^{7} + 9267 T^{8} + 172 T^{9} + 104 T^{10} + 2 T^{11} + T^{12} \)
$59$ \( 83229129 + 168419703 T + 334732603 T^{2} + 30468042 T^{3} + 19368969 T^{4} + 1633661 T^{5} + 809563 T^{6} + 59909 T^{7} + 15763 T^{8} + 1225 T^{9} + 228 T^{10} + 13 T^{11} + T^{12} \)
$61$ \( 1055015361 + 1195333281 T + 1062894069 T^{2} + 390333384 T^{3} + 121103191 T^{4} + 6648335 T^{5} + 2541805 T^{6} + 133207 T^{7} + 36059 T^{8} + 847 T^{9} + 226 T^{10} + 5 T^{11} + T^{12} \)
$67$ \( 276324129 + 183966741 T + 160212699 T^{2} + 4433604 T^{3} + 13229425 T^{4} - 145321 T^{5} + 875958 T^{6} - 55361 T^{7} + 18745 T^{8} - 612 T^{9} + 227 T^{10} - 11 T^{11} + T^{12} \)
$71$ \( ( 23043 - 13693 T - 103 T^{2} + 1136 T^{3} - 141 T^{4} - 6 T^{5} + T^{6} )^{2} \)
$73$ \( 196812841 - 343233514 T + 480685440 T^{2} - 198569706 T^{3} + 67825152 T^{4} - 13334350 T^{5} + 2769075 T^{6} - 420036 T^{7} + 72578 T^{8} - 7642 T^{9} + 662 T^{10} - 30 T^{11} + T^{12} \)
$79$ \( 110859841 + 199598253 T + 283043128 T^{2} + 130891313 T^{3} + 48229623 T^{4} + 7784759 T^{5} + 1322709 T^{6} + 74563 T^{7} + 16825 T^{8} + 416 T^{9} + 197 T^{10} - 7 T^{11} + T^{12} \)
$83$ \( ( 2673 - 1188 T - 1797 T^{2} + 403 T^{3} + 158 T^{4} - 27 T^{5} + T^{6} )^{2} \)
$89$ \( 92707461441 + 24201817794 T + 16326857884 T^{2} - 2693246248 T^{3} + 958332439 T^{4} - 63899744 T^{5} + 11790434 T^{6} - 294018 T^{7} + 102345 T^{8} - 1204 T^{9} + 383 T^{10} + 4 T^{11} + T^{12} \)
$97$ \( ( -3899 - 8510 T - 1085 T^{2} + 1186 T^{3} + 365 T^{4} + 35 T^{5} + T^{6} )^{2} \)
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