Properties

Label 1183.2.a.q
Level $1183$
Weight $2$
Character orbit 1183.a
Self dual yes
Analytic conductor $9.446$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 15 x^{10} + 46 x^{9} + 80 x^{8} - 246 x^{7} - 199 x^{6} + 562 x^{5} + 262 x^{4} - 542 x^{3} - 157 x^{2} + 183 x + 29\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{6} ) q^{3} + ( 2 - \beta_{4} + \beta_{5} ) q^{4} + ( 1 - \beta_{4} - \beta_{7} - \beta_{11} ) q^{5} + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{6} - q^{7} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{8} + ( 3 - \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{6} ) q^{3} + ( 2 - \beta_{4} + \beta_{5} ) q^{4} + ( 1 - \beta_{4} - \beta_{7} - \beta_{11} ) q^{5} + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{6} - q^{7} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{8} + ( 3 - \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} ) q^{9} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{9} + \beta_{10} ) q^{10} + ( -1 - \beta_{2} - \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} ) q^{11} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{12} + \beta_{1} q^{14} + ( -2 + \beta_{1} - \beta_{2} - \beta_{6} - \beta_{7} + 2 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} ) q^{15} + ( 2 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} ) q^{16} + ( 3 + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{17} + ( 2 - 4 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{18} + ( 2 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} + 3 \beta_{10} ) q^{19} + ( 3 + \beta_{1} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{8} + 2 \beta_{10} ) q^{20} + ( -1 - \beta_{6} ) q^{21} + ( 2 \beta_{1} + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{11} ) q^{22} + ( 2 - \beta_{1} + \beta_{3} + \beta_{7} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{23} + ( 3 - \beta_{1} + 2 \beta_{2} + \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{11} ) q^{24} + ( 4 + \beta_{1} + 2 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{25} + ( 2 + \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{27} + ( -2 + \beta_{4} - \beta_{5} ) q^{28} + ( 2 + \beta_{1} - 2 \beta_{4} - \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} ) q^{29} + ( -3 + 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{30} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{8} + 2 \beta_{9} + \beta_{11} ) q^{31} + ( 1 - \beta_{2} - \beta_{4} + \beta_{6} + 2 \beta_{8} + 3 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{32} + ( \beta_{1} - \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{33} + ( -1 - 3 \beta_{1} - \beta_{2} + 3 \beta_{4} - \beta_{6} - \beta_{11} ) q^{34} + ( -1 + \beta_{4} + \beta_{7} + \beta_{11} ) q^{35} + ( 8 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} + \beta_{8} + 3 \beta_{9} + \beta_{10} ) q^{36} + ( 2 + \beta_{1} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{11} ) q^{37} + ( 5 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{38} + ( -6 - \beta_{3} + 6 \beta_{4} - \beta_{5} + \beta_{8} - 4 \beta_{10} + 2 \beta_{11} ) q^{40} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{41} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{42} + ( -4 + \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - 4 \beta_{10} - \beta_{11} ) q^{43} + ( -9 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 5 \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{8} - 3 \beta_{10} + \beta_{11} ) q^{44} + ( -2 \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{6} - 2 \beta_{7} + 3 \beta_{9} - \beta_{10} - \beta_{11} ) q^{45} + ( 2 - \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{46} + ( 1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{47} + ( 2 - 4 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} + 3 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{48} + q^{49} + ( -3 - \beta_{1} - 2 \beta_{2} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{50} + ( -1 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{51} + ( 5 - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{53} + ( -3 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 8 \beta_{10} ) q^{54} + ( -3 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} - 3 \beta_{6} + \beta_{7} + 3 \beta_{8} + 4 \beta_{9} + \beta_{11} ) q^{55} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{56} + ( 1 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{10} + 2 \beta_{11} ) q^{57} + ( -1 - \beta_{1} + \beta_{3} - 4 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + 3 \beta_{10} ) q^{58} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{59} + ( -4 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 7 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - \beta_{7} + 3 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{60} + ( -1 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{8} ) q^{61} + ( 5 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} - 7 \beta_{4} - \beta_{6} + \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{62} + ( -3 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} ) q^{63} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} - 3 \beta_{10} - 2 \beta_{11} ) q^{64} + ( -5 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 8 \beta_{4} - 2 \beta_{5} + \beta_{6} - 4 \beta_{7} + 2 \beta_{8} + 4 \beta_{9} ) q^{66} + ( -2 - \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 3 \beta_{8} - 4 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{67} + ( 4 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{8} - \beta_{9} + 3 \beta_{10} ) q^{68} + ( -1 - 3 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{69} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{9} - \beta_{10} ) q^{70} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} + 2 \beta_{11} ) q^{71} + ( 4 - 4 \beta_{1} + \beta_{2} - 7 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} + 4 \beta_{10} + 2 \beta_{11} ) q^{72} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 3 \beta_{9} - \beta_{10} ) q^{73} + ( -8 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - 3 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{74} + ( 3 + 3 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{8} - \beta_{9} + 9 \beta_{10} + \beta_{11} ) q^{75} + ( -3 - 3 \beta_{1} - 3 \beta_{3} - \beta_{6} - 2 \beta_{7} - \beta_{8} + 3 \beta_{9} + \beta_{10} + \beta_{11} ) q^{76} + ( 1 + \beta_{2} + \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{77} + ( 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} - 4 \beta_{9} - 2 \beta_{11} ) q^{79} + ( -1 + 4 \beta_{1} + \beta_{2} - 4 \beta_{4} + \beta_{5} - 2 \beta_{6} - 3 \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{80} + ( 3 - 4 \beta_{1} - \beta_{3} + 4 \beta_{4} + \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - 6 \beta_{10} - 4 \beta_{11} ) q^{81} + ( -3 - \beta_{1} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{82} + ( 2 + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{7} + 2 \beta_{11} ) q^{83} + ( -1 + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{8} - 3 \beta_{9} + \beta_{10} - \beta_{11} ) q^{84} + ( 4 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{7} - \beta_{8} - 3 \beta_{9} - 3 \beta_{10} - 4 \beta_{11} ) q^{85} + ( -3 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} - 4 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{86} + ( 6 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + 5 \beta_{8} + 4 \beta_{9} + 6 \beta_{10} + 3 \beta_{11} ) q^{87} + ( -3 + 6 \beta_{1} - \beta_{2} + 4 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - 3 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{88} + ( 1 + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{89} + ( -5 - 4 \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - \beta_{6} - 7 \beta_{8} - 9 \beta_{9} - 3 \beta_{10} - 2 \beta_{11} ) q^{90} + ( 4 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{9} - 6 \beta_{10} - 2 \beta_{11} ) q^{92} + ( 8 + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + \beta_{6} + 3 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{93} + ( -4 + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - 3 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{94} + ( 1 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - 4 \beta_{6} - 3 \beta_{7} + 5 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} ) q^{95} + ( 15 - \beta_{1} + \beta_{2} - \beta_{3} - 5 \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} + 5 \beta_{10} - \beta_{11} ) q^{96} + ( -3 + \beta_{1} - \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{97} -\beta_{1} q^{98} + ( -8 + 2 \beta_{1} + \beta_{2} - \beta_{3} + 6 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + \beta_{7} - 6 \beta_{8} - 6 \beta_{9} - 4 \beta_{10} - 2 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} + 8 q^{3} + 15 q^{4} + 4 q^{5} - 2 q^{6} - 12 q^{7} - 12 q^{8} + 26 q^{9} + O(q^{10}) \) \( 12 q - 3 q^{2} + 8 q^{3} + 15 q^{4} + 4 q^{5} - 2 q^{6} - 12 q^{7} - 12 q^{8} + 26 q^{9} - 6 q^{10} - 12 q^{11} + 13 q^{12} + 3 q^{14} - 11 q^{15} + 13 q^{16} + 31 q^{17} + 29 q^{18} + 3 q^{19} + 18 q^{20} - 8 q^{21} - 4 q^{22} + 18 q^{23} + 6 q^{24} + 32 q^{25} + 32 q^{27} - 15 q^{28} + 15 q^{29} - 10 q^{30} + 21 q^{31} + 3 q^{32} + 29 q^{33} - 3 q^{34} - 4 q^{35} + 49 q^{36} - 5 q^{37} + 45 q^{38} - 20 q^{40} + 16 q^{41} + 2 q^{42} - 22 q^{43} - 35 q^{44} - 5 q^{45} - 2 q^{46} + 4 q^{47} + 11 q^{48} + 12 q^{49} - 13 q^{50} + 18 q^{51} + 53 q^{53} + 5 q^{54} - 26 q^{55} + 12 q^{56} + 8 q^{57} - 32 q^{58} + 26 q^{59} - 38 q^{60} + 22 q^{61} + 19 q^{62} - 26 q^{63} + 2 q^{64} - 34 q^{66} - 12 q^{67} + 34 q^{68} + 3 q^{69} + 6 q^{70} - 21 q^{71} + 4 q^{72} + 15 q^{73} - 40 q^{74} + 15 q^{75} - 43 q^{76} + 12 q^{77} + 2 q^{79} - 13 q^{80} + 36 q^{81} - 32 q^{82} + 9 q^{83} - 13 q^{84} + 39 q^{85} - 44 q^{86} + 27 q^{87} - 48 q^{88} + 22 q^{89} - 26 q^{90} + 52 q^{92} + 53 q^{93} - 44 q^{94} + 29 q^{95} + 114 q^{96} - 9 q^{97} - 3 q^{98} - 37 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} - 15 x^{10} + 46 x^{9} + 80 x^{8} - 246 x^{7} - 199 x^{6} + 562 x^{5} + 262 x^{4} - 542 x^{3} - 157 x^{2} + 183 x + 29\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 202 \nu^{11} + 1816 \nu^{10} - 5010 \nu^{9} - 29197 \nu^{8} + 39197 \nu^{7} + 161452 \nu^{6} - 110863 \nu^{5} - 346540 \nu^{4} + 77279 \nu^{3} + 193411 \nu^{2} + 16187 \nu + 25085 \)\()/14629\)
\(\beta_{3}\)\(=\)\((\)\( 202 \nu^{11} + 1816 \nu^{10} - 5010 \nu^{9} - 29197 \nu^{8} + 39197 \nu^{7} + 161452 \nu^{6} - 110863 \nu^{5} - 346540 \nu^{4} + 91908 \nu^{3} + 193411 \nu^{2} - 56958 \nu + 10456 \)\()/14629\)
\(\beta_{4}\)\(=\)\((\)\( 266 \nu^{11} + 1812 \nu^{10} - 9639 \nu^{9} - 26136 \nu^{8} + 98110 \nu^{7} + 125845 \nu^{6} - 384542 \nu^{5} - 239507 \nu^{4} + 573947 \nu^{3} + 165757 \nu^{2} - 263443 \nu - 14765 \)\()/14629\)
\(\beta_{5}\)\(=\)\((\)\( 266 \nu^{11} + 1812 \nu^{10} - 9639 \nu^{9} - 26136 \nu^{8} + 98110 \nu^{7} + 125845 \nu^{6} - 384542 \nu^{5} - 239507 \nu^{4} + 573947 \nu^{3} + 180386 \nu^{2} - 263443 \nu - 73281 \)\()/14629\)
\(\beta_{6}\)\(=\)\((\)\( 721 \nu^{11} - 2788 \nu^{10} - 8033 \nu^{9} + 41570 \nu^{8} + 13387 \nu^{7} - 212101 \nu^{6} + 95671 \nu^{5} + 447600 \nu^{4} - 310269 \nu^{3} - 374170 \nu^{2} + 204863 \nu + 63537 \)\()/14629\)
\(\beta_{7}\)\(=\)\((\)\( 1025 \nu^{11} - 2807 \nu^{10} - 19049 \nu^{9} + 45138 \nu^{8} + 135962 \nu^{7} - 245916 \nu^{6} - 469197 \nu^{5} + 506165 \nu^{4} + 761552 \nu^{3} - 264148 \nu^{2} - 384615 \nu - 16033 \)\()/14629\)
\(\beta_{8}\)\(=\)\((\)\( 1181 \nu^{11} - 6474 \nu^{10} - 9303 \nu^{9} + 90086 \nu^{8} - 23075 \nu^{7} - 408596 \nu^{6} + 301924 \nu^{5} + 705799 \nu^{4} - 620810 \nu^{3} - 446758 \nu^{2} + 361943 \nu + 60552 \)\()/14629\)
\(\beta_{9}\)\(=\)\((\)\( 1429 \nu^{11} + 825 \nu^{10} - 29069 \nu^{9} - 13256 \nu^{8} + 214356 \nu^{7} + 76988 \nu^{6} - 690923 \nu^{5} - 201544 \nu^{4} + 930739 \nu^{3} + 225077 \nu^{2} - 425386 \nu - 68266 \)\()/14629\)
\(\beta_{10}\)\(=\)\((\)\( 2354 \nu^{11} - 5633 \nu^{10} - 34485 \nu^{9} + 79215 \nu^{8} + 175064 \nu^{7} - 364728 \nu^{6} - 391458 \nu^{5} + 632025 \nu^{4} + 415204 \nu^{3} - 345129 \nu^{2} - 144501 \nu + 5396 \)\()/14629\)
\(\beta_{11}\)\(=\)\((\)\( -4628 \nu^{11} + 11261 \nu^{10} + 66841 \nu^{9} - 158261 \nu^{8} - 324031 \nu^{7} + 725177 \nu^{6} + 627337 \nu^{5} - 1236319 \nu^{4} - 443637 \nu^{3} + 648376 \nu^{2} + 34553 \nu + 5226 \)\()/14629\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{4} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} - \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{9} + \beta_{7} + 7 \beta_{5} - 7 \beta_{4} + \beta_{3} + \beta_{2} + 22\)
\(\nu^{5}\)\(=\)\(-2 \beta_{11} - \beta_{10} - 3 \beta_{9} - 2 \beta_{8} - \beta_{6} + \beta_{4} + 8 \beta_{3} - 7 \beta_{2} + 28 \beta_{1} + 7\)
\(\nu^{6}\)\(=\)\(-2 \beta_{11} - 3 \beta_{10} - 10 \beta_{9} - \beta_{8} + 11 \beta_{7} - 2 \beta_{6} + 45 \beta_{5} - 48 \beta_{4} + 10 \beta_{3} + 11 \beta_{2} - \beta_{1} + 131\)
\(\nu^{7}\)\(=\)\(-23 \beta_{11} - 9 \beta_{10} - 37 \beta_{9} - 25 \beta_{8} + \beta_{7} - 13 \beta_{6} + 10 \beta_{4} + 57 \beta_{3} - 43 \beta_{2} + 166 \beta_{1} + 43\)
\(\nu^{8}\)\(=\)\(-27 \beta_{11} - 36 \beta_{10} - 84 \beta_{9} - 19 \beta_{8} + 93 \beta_{7} - 24 \beta_{6} + 289 \beta_{5} - 330 \beta_{4} + 80 \beta_{3} + 94 \beta_{2} - 14 \beta_{1} + 812\)
\(\nu^{9}\)\(=\)\(-198 \beta_{11} - 61 \beta_{10} - 331 \beta_{9} - 231 \beta_{8} + 14 \beta_{7} - 120 \beta_{6} - \beta_{5} + 67 \beta_{4} + 396 \beta_{3} - 259 \beta_{2} + 1021 \beta_{1} + 265\)
\(\nu^{10}\)\(=\)\(-257 \beta_{11} - 306 \beta_{10} - 658 \beta_{9} - 224 \beta_{8} + 710 \beta_{7} - 212 \beta_{6} + 1874 \beta_{5} - 2277 \beta_{4} + 593 \beta_{3} + 734 \beta_{2} - 132 \beta_{1} + 5167\)
\(\nu^{11}\)\(=\)\(-1539 \beta_{11} - 370 \beta_{10} - 2625 \beta_{9} - 1909 \beta_{8} + 136 \beta_{7} - 967 \beta_{6} - 16 \beta_{5} + 356 \beta_{4} + 2724 \beta_{3} - 1555 \beta_{2} + 6448 \beta_{1} + 1686\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.62803
2.47725
2.23724
2.06743
0.983820
0.842530
−0.149660
−0.961590
−1.10989
−1.35819
−2.07140
−2.58557
−2.62803 −1.76762 4.90656 2.94291 4.64535 −1.00000 −7.63852 0.124466 −7.73406
1.2 −2.47725 0.982981 4.13677 −1.35413 −2.43509 −1.00000 −5.29330 −2.03375 3.35452
1.3 −2.23724 3.02592 3.00523 3.28547 −6.76971 −1.00000 −2.24893 6.15622 −7.35037
1.4 −2.06743 2.11889 2.27425 −2.43928 −4.38065 −1.00000 −0.566992 1.48970 5.04303
1.5 −0.983820 −1.57171 −1.03210 −0.398447 1.54628 −1.00000 2.98304 −0.529731 0.392000
1.6 −0.842530 0.161973 −1.29014 3.72786 −0.136467 −1.00000 2.77204 −2.97376 −3.14083
1.7 0.149660 2.76031 −1.97760 −4.13443 0.413107 −1.00000 −0.595288 4.61930 −0.618759
1.8 0.961590 −1.98737 −1.07534 −3.39320 −1.91103 −1.00000 −2.95722 0.949635 −3.26287
1.9 1.10989 0.955760 −0.768150 3.55862 1.06079 −1.00000 −3.07233 −2.08652 3.94966
1.10 1.35819 3.39737 −0.155322 0.772491 4.61428 −1.00000 −2.92733 8.54215 1.04919
1.11 2.07140 −3.01646 2.29068 2.69245 −6.24828 −1.00000 0.602118 6.09903 5.57713
1.12 2.58557 2.93994 4.68518 −1.26031 7.60143 −1.00000 6.94272 5.64327 −3.25863
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.a.q 12
7.b odd 2 1 8281.2.a.cn 12
13.b even 2 1 1183.2.a.r yes 12
13.d odd 4 2 1183.2.c.j 24
91.b odd 2 1 8281.2.a.cq 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1183.2.a.q 12 1.a even 1 1 trivial
1183.2.a.r yes 12 13.b even 2 1
1183.2.c.j 24 13.d odd 4 2
8281.2.a.cn 12 7.b odd 2 1
8281.2.a.cq 12 91.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1183))\):

\(T_{2}^{12} + \cdots\)
\(T_{11}^{12} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 29 - 183 T - 157 T^{2} + 542 T^{3} + 262 T^{4} - 562 T^{5} - 199 T^{6} + 246 T^{7} + 80 T^{8} - 46 T^{9} - 15 T^{10} + 3 T^{11} + T^{12} \)
$3$ \( 448 - 3584 T + 4928 T^{2} + 1616 T^{3} - 5492 T^{4} + 920 T^{5} + 1973 T^{6} - 691 T^{7} - 249 T^{8} + 136 T^{9} + T^{10} - 8 T^{11} + T^{12} \)
$5$ \( 6208 + 13216 T - 16848 T^{2} - 25848 T^{3} + 8916 T^{4} + 11870 T^{5} - 2897 T^{6} - 2122 T^{7} + 500 T^{8} + 157 T^{9} - 38 T^{10} - 4 T^{11} + T^{12} \)
$7$ \( ( 1 + T )^{12} \)
$11$ \( -2701133 - 900676 T + 1617506 T^{2} + 505685 T^{3} - 339321 T^{4} - 112394 T^{5} + 30684 T^{6} + 12005 T^{7} - 994 T^{8} - 613 T^{9} - 13 T^{10} + 12 T^{11} + T^{12} \)
$13$ \( T^{12} \)
$17$ \( 110272 - 472544 T + 675872 T^{2} - 210520 T^{3} - 402728 T^{4} + 483052 T^{5} - 228079 T^{6} + 48450 T^{7} - 861 T^{8} - 1749 T^{9} + 364 T^{10} - 31 T^{11} + T^{12} \)
$19$ \( 1395008 - 523872 T - 3138960 T^{2} - 181112 T^{3} + 1033252 T^{4} + 81510 T^{5} - 123297 T^{6} - 8681 T^{7} + 6082 T^{8} + 296 T^{9} - 131 T^{10} - 3 T^{11} + T^{12} \)
$23$ \( 7017331 - 11388083 T + 3342704 T^{2} + 2962993 T^{3} - 1682702 T^{4} - 103895 T^{5} + 199072 T^{6} - 20950 T^{7} - 7201 T^{8} + 1447 T^{9} + 12 T^{10} - 18 T^{11} + T^{12} \)
$29$ \( 177673 - 867913 T + 828198 T^{2} + 217418 T^{3} - 481608 T^{4} + 59784 T^{5} + 77202 T^{6} - 18103 T^{7} - 3607 T^{8} + 1221 T^{9} - 24 T^{10} - 15 T^{11} + T^{12} \)
$31$ \( -15261184 - 271744 T + 13600640 T^{2} - 1249464 T^{3} - 3652072 T^{4} + 539022 T^{5} + 346335 T^{6} - 67095 T^{7} - 8438 T^{8} + 2180 T^{9} + 7 T^{10} - 21 T^{11} + T^{12} \)
$37$ \( 545930729 - 4008911 T - 216159072 T^{2} + 11928582 T^{3} + 23149628 T^{4} - 1293842 T^{5} - 1076680 T^{6} + 51131 T^{7} + 24107 T^{8} - 863 T^{9} - 252 T^{10} + 5 T^{11} + T^{12} \)
$41$ \( -101827648 - 127486208 T + 53137520 T^{2} + 51226496 T^{3} - 27517248 T^{4} + 1654660 T^{5} + 1222195 T^{6} - 214272 T^{7} - 6045 T^{8} + 3532 T^{9} - 135 T^{10} - 16 T^{11} + T^{12} \)
$43$ \( -9002449 - 10108897 T + 53854685 T^{2} - 513456 T^{3} - 11811616 T^{4} - 474599 T^{5} + 880317 T^{6} + 89963 T^{7} - 21642 T^{8} - 3728 T^{9} - 28 T^{10} + 22 T^{11} + T^{12} \)
$47$ \( 17156608 - 5974528 T - 28506016 T^{2} + 6563096 T^{3} + 8846240 T^{4} - 510390 T^{5} - 760339 T^{6} + 1170 T^{7} + 23426 T^{8} + 457 T^{9} - 262 T^{10} - 4 T^{11} + T^{12} \)
$53$ \( 62267981 + 177543116 T - 37586257 T^{2} - 92629676 T^{3} + 43097846 T^{4} - 1557851 T^{5} - 2436927 T^{6} + 473888 T^{7} - 1896 T^{8} - 8533 T^{9} + 1054 T^{10} - 53 T^{11} + T^{12} \)
$59$ \( 913870784 - 980465920 T + 130935936 T^{2} + 118535488 T^{3} - 31898324 T^{4} - 3347084 T^{5} + 1779215 T^{6} - 69801 T^{7} - 32990 T^{8} + 3699 T^{9} + 65 T^{10} - 26 T^{11} + T^{12} \)
$61$ \( -69982144 - 18517536 T + 49214640 T^{2} + 6247144 T^{3} - 12613036 T^{4} + 309150 T^{5} + 1268015 T^{6} - 196445 T^{7} - 19825 T^{8} + 5344 T^{9} - 129 T^{10} - 22 T^{11} + T^{12} \)
$67$ \( 1207037 + 1648589 T - 3785554 T^{2} - 685362 T^{3} + 2433927 T^{4} - 370595 T^{5} - 308286 T^{6} + 63359 T^{7} + 11336 T^{8} - 2004 T^{9} - 199 T^{10} + 12 T^{11} + T^{12} \)
$71$ \( -22571863 - 45796935 T + 10711179 T^{2} + 29155208 T^{3} - 2962125 T^{4} - 4813988 T^{5} + 281595 T^{6} + 258995 T^{7} + 109 T^{8} - 4211 T^{9} - 125 T^{10} + 21 T^{11} + T^{12} \)
$73$ \( -4111861312 + 4922870400 T + 430370544 T^{2} - 923841136 T^{3} + 61231368 T^{4} + 41485060 T^{5} - 3463011 T^{6} - 748304 T^{7} + 58607 T^{8} + 5707 T^{9} - 406 T^{10} - 15 T^{11} + T^{12} \)
$79$ \( -33746539 + 393597671 T - 336730518 T^{2} - 15641243 T^{3} + 46453724 T^{4} - 238149 T^{5} - 2240776 T^{6} - 7420 T^{7} + 44193 T^{8} + 343 T^{9} - 364 T^{10} - 2 T^{11} + T^{12} \)
$83$ \( -478100288 + 299570240 T + 91803616 T^{2} - 78899128 T^{3} - 1911072 T^{4} + 6691902 T^{5} - 420269 T^{6} - 204661 T^{7} + 19015 T^{8} + 2362 T^{9} - 248 T^{10} - 9 T^{11} + T^{12} \)
$89$ \( -639815168 + 2900248960 T - 3531001216 T^{2} + 1826781592 T^{3} - 406843464 T^{4} + 12278850 T^{5} + 9911493 T^{6} - 1335627 T^{7} - 15214 T^{8} + 11148 T^{9} - 360 T^{10} - 22 T^{11} + T^{12} \)
$97$ \( -6873362944 - 12657080576 T + 2910438944 T^{2} + 1985269000 T^{3} - 4751296 T^{4} - 74671606 T^{5} - 4261095 T^{6} + 974036 T^{7} + 78756 T^{8} - 5062 T^{9} - 488 T^{10} + 9 T^{11} + T^{12} \)
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