Properties

Label 605.2.g.p
Level $605$
Weight $2$
Character orbit 605.g
Analytic conductor $4.831$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.g (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - x^{11} + 6 x^{10} - 12 x^{9} + 43 x^{8} + 72 x^{7} + 155 x^{6} + 162 x^{5} + 541 x^{4} + 114 x^{3} + 24 x^{2} + 5 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} -\beta_{1} q^{3} + ( -3 + \beta_{3} + 3 \beta_{4} + \beta_{5} - 3 \beta_{7} + \beta_{8} - \beta_{9} + 3 \beta_{10} - \beta_{11} ) q^{4} + \beta_{10} q^{5} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{6} + 2 \beta_{8} + \beta_{10} ) q^{6} + ( \beta_{3} + \beta_{5} + 2 \beta_{8} - \beta_{9} - \beta_{11} ) q^{7} + ( \beta_{1} - 4 \beta_{4} - 2 \beta_{5} ) q^{8} + ( \beta_{6} - \beta_{9} ) q^{9} +O(q^{10})\) \( q + \beta_{9} q^{2} -\beta_{1} q^{3} + ( -3 + \beta_{3} + 3 \beta_{4} + \beta_{5} - 3 \beta_{7} + \beta_{8} - \beta_{9} + 3 \beta_{10} - \beta_{11} ) q^{4} + \beta_{10} q^{5} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{6} + 2 \beta_{8} + \beta_{10} ) q^{6} + ( \beta_{3} + \beta_{5} + 2 \beta_{8} - \beta_{9} - \beta_{11} ) q^{7} + ( \beta_{1} - 4 \beta_{4} - 2 \beta_{5} ) q^{8} + ( \beta_{6} - \beta_{9} ) q^{9} + \beta_{3} q^{10} + ( 2 - 2 \beta_{2} + \beta_{3} ) q^{12} + ( \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{13} + ( 3 \beta_{1} - 3 \beta_{4} - \beta_{5} ) q^{14} -\beta_{8} q^{15} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{6} - 2 \beta_{8} - 5 \beta_{10} + 4 \beta_{11} ) q^{16} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{10} ) q^{17} + ( 6 - \beta_{3} - 6 \beta_{4} - \beta_{5} + 6 \beta_{7} - 3 \beta_{8} + \beta_{9} - 6 \beta_{10} + \beta_{11} ) q^{18} + ( -\beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{19} + ( -\beta_{6} - 3 \beta_{7} - \beta_{9} ) q^{20} + ( 5 + 2 \beta_{3} ) q^{21} + ( -2 + \beta_{2} + \beta_{3} ) q^{23} + ( -\beta_{6} - 5 \beta_{7} + \beta_{9} ) q^{24} -\beta_{4} q^{25} + ( -4 - \beta_{3} + 4 \beta_{4} - \beta_{5} - 4 \beta_{7} - \beta_{8} + \beta_{9} + 4 \beta_{10} + \beta_{11} ) q^{26} + ( -4 \beta_{10} - \beta_{11} ) q^{27} + ( 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{6} - 3 \beta_{8} - 8 \beta_{10} + 2 \beta_{11} ) q^{28} + ( 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{9} - 2 \beta_{11} ) q^{29} + ( -2 \beta_{1} - \beta_{4} ) q^{30} + ( \beta_{6} + 4 \beta_{7} - \beta_{9} ) q^{31} + ( 10 - 2 \beta_{2} - 5 \beta_{3} ) q^{32} + ( -2 + 4 \beta_{2} + 2 \beta_{3} ) q^{34} + ( -2 \beta_{6} - \beta_{9} ) q^{35} + ( -3 \beta_{1} + 2 \beta_{4} + 5 \beta_{5} ) q^{36} + ( -2 + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{37} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{6} + 3 \beta_{8} + 6 \beta_{10} - 3 \beta_{11} ) q^{38} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{6} + 3 \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{39} + ( 4 - 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 4 \beta_{7} + \beta_{8} + 2 \beta_{9} - 4 \beta_{10} + 2 \beta_{11} ) q^{40} + ( 3 \beta_{1} + 3 \beta_{4} + 3 \beta_{5} ) q^{41} + ( -2 \beta_{6} - 10 \beta_{7} + 3 \beta_{9} ) q^{42} + ( -2 + 2 \beta_{2} + \beta_{3} ) q^{43} + ( \beta_{2} - \beta_{3} ) q^{45} + ( \beta_{6} - 4 \beta_{7} - 3 \beta_{9} ) q^{46} + ( -\beta_{1} + 4 \beta_{4} - 4 \beta_{5} ) q^{47} + ( -2 - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{48} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{6} + 3 \beta_{8} - 6 \beta_{10} - 3 \beta_{11} ) q^{49} + \beta_{11} q^{50} + ( -6 - 2 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} - 6 \beta_{7} + 2 \beta_{9} + 6 \beta_{10} + 2 \beta_{11} ) q^{51} + ( \beta_{1} - \beta_{5} ) q^{52} + ( -\beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{53} + ( -5 + \beta_{2} - 3 \beta_{3} ) q^{54} + ( 1 - 2 \beta_{2} - 8 \beta_{3} ) q^{56} + ( \beta_{6} + 4 \beta_{7} - \beta_{9} ) q^{57} + ( -2 \beta_{1} - 10 \beta_{4} - 2 \beta_{5} ) q^{58} + ( -2 - 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{7} + \beta_{8} + 3 \beta_{9} + 2 \beta_{10} + 3 \beta_{11} ) q^{59} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{60} + ( 5 \beta_{1} + 5 \beta_{2} - 5 \beta_{6} - 5 \beta_{8} + 3 \beta_{10} - \beta_{11} ) q^{61} + ( 6 + 3 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} + 6 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} - 6 \beta_{10} - 3 \beta_{11} ) q^{62} + ( -3 \beta_{1} - 2 \beta_{4} + 3 \beta_{5} ) q^{63} + ( -3 \beta_{6} + 13 \beta_{7} + 7 \beta_{9} ) q^{64} + ( 2 + \beta_{2} + \beta_{3} ) q^{65} + ( 6 + \beta_{2} + 2 \beta_{3} ) q^{67} + ( 2 \beta_{6} - 2 \beta_{7} - 4 \beta_{9} ) q^{68} + ( \beta_{1} + 2 \beta_{4} - \beta_{5} ) q^{69} + ( 3 - \beta_{3} - 3 \beta_{4} - \beta_{5} + 3 \beta_{7} + 3 \beta_{8} + \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{70} + ( \beta_{1} + \beta_{2} - \beta_{6} - \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{71} + ( -5 \beta_{1} - 5 \beta_{2} + 5 \beta_{6} + 5 \beta_{8} + 16 \beta_{10} - 5 \beta_{11} ) q^{72} + ( -4 - 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 4 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} + 2 \beta_{11} ) q^{73} + ( -2 \beta_{1} - 10 \beta_{4} - 4 \beta_{5} ) q^{74} + \beta_{6} q^{75} + ( -8 - \beta_{2} + 7 \beta_{3} ) q^{76} + ( -2 - 5 \beta_{2} - \beta_{3} ) q^{78} + ( -4 \beta_{6} - 2 \beta_{9} ) q^{79} + ( 2 \beta_{1} + 5 \beta_{4} + 4 \beta_{5} ) q^{80} + ( -1 + 3 \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + \beta_{10} - 3 \beta_{11} ) q^{81} + ( 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{6} - 3 \beta_{8} + 12 \beta_{10} - 6 \beta_{11} ) q^{82} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{6} + 3 \beta_{8} + 6 \beta_{10} - 3 \beta_{11} ) q^{83} + ( -7 - 3 \beta_{3} + 7 \beta_{4} - 3 \beta_{5} - 7 \beta_{7} + 7 \beta_{8} + 3 \beta_{9} + 7 \beta_{10} + 3 \beta_{11} ) q^{84} + ( 2 \beta_{1} - 2 \beta_{4} ) q^{85} + ( 3 \beta_{6} - 3 \beta_{7} - 3 \beta_{9} ) q^{86} + ( -2 + 4 \beta_{2} ) q^{87} + ( 3 - 6 \beta_{2} - 4 \beta_{3} ) q^{89} + ( 3 \beta_{6} + 6 \beta_{7} + \beta_{9} ) q^{90} + ( 7 \beta_{1} - 8 \beta_{4} + 3 \beta_{5} ) q^{91} + ( 12 - 5 \beta_{3} - 12 \beta_{4} - 5 \beta_{5} + 12 \beta_{7} - 3 \beta_{8} + 5 \beta_{9} - 12 \beta_{10} + 5 \beta_{11} ) q^{92} + ( 7 \beta_{1} + 7 \beta_{2} - 7 \beta_{6} - 7 \beta_{8} - 4 \beta_{10} - \beta_{11} ) q^{93} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{6} - 2 \beta_{8} - 19 \beta_{10} ) q^{94} + ( -2 + \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{95} + ( -2 \beta_{1} - \beta_{4} + 2 \beta_{5} ) q^{96} + ( 7 \beta_{6} - 2 \beta_{7} + 3 \beta_{9} ) q^{97} + ( -12 - 3 \beta_{2} - 3 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + q^{2} - q^{3} - 9q^{4} + 3q^{5} + 5q^{6} - q^{7} - 9q^{8} - 2q^{9} + O(q^{10}) \) \( 12q + q^{2} - q^{3} - 9q^{4} + 3q^{5} + 5q^{6} - q^{7} - 9q^{8} - 2q^{9} + 4q^{10} + 36q^{12} + 6q^{13} - 5q^{14} + q^{15} - 13q^{16} + 4q^{17} + 20q^{18} + 4q^{19} + 9q^{20} + 68q^{21} - 24q^{23} + 17q^{24} - 3q^{25} - 12q^{26} - 13q^{27} - 25q^{28} + 2q^{29} - 5q^{30} - 14q^{31} + 108q^{32} - 32q^{34} + q^{35} - 2q^{36} - 4q^{37} + 18q^{38} - 4q^{39} + 9q^{40} + 9q^{41} + 35q^{42} - 28q^{43} - 8q^{45} + 8q^{46} + 15q^{47} - 7q^{48} - 18q^{49} + q^{50} - 20q^{51} + 2q^{52} + 6q^{53} - 76q^{54} - 12q^{56} - 14q^{57} - 30q^{58} - 10q^{59} + 9q^{60} + 3q^{61} + 24q^{62} - 12q^{63} - 29q^{64} + 24q^{65} + 76q^{67} + 8q^{69} + 5q^{70} - 6q^{71} + 48q^{72} - 12q^{73} - 28q^{74} - q^{75} - 64q^{76} - 8q^{78} + 2q^{79} + 13q^{80} - 3q^{81} + 27q^{82} + 18q^{83} - 31q^{84} - 4q^{85} + 3q^{86} - 40q^{87} + 44q^{89} - 20q^{90} - 20q^{91} + 34q^{92} - 20q^{93} - 59q^{94} - 4q^{95} - 7q^{96} + 2q^{97} - 144q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - x^{11} + 6 x^{10} - 12 x^{9} + 43 x^{8} + 72 x^{7} + 155 x^{6} + 162 x^{5} + 541 x^{4} + 114 x^{3} + 24 x^{2} + 5 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 12 \nu^{10} + 2713 \nu^{5} + 5190 \)\()/24631\)
\(\beta_{3}\)\(=\)\((\)\( 43 \nu^{10} + 7669 \nu^{5} - 67611 \)\()/24631\)
\(\beta_{4}\)\(=\)\((\)\( 43 \nu^{11} + 7669 \nu^{6} - 116873 \nu \)\()/24631\)
\(\beta_{5}\)\(=\)\((\)\( 141 \nu^{11} + 25720 \nu^{6} - 320798 \nu \)\()/24631\)
\(\beta_{6}\)\(=\)\((\)\( 43 \nu^{11} - 258 \nu^{10} + 516 \nu^{9} - 1849 \nu^{8} + 4573 \nu^{7} - 6665 \nu^{6} - 6966 \nu^{5} - 23263 \nu^{4} - 4902 \nu^{3} - 117905 \nu^{2} - 215 \nu - 43 \)\()/24631\)
\(\beta_{7}\)\(=\)\((\)\( -227 \nu^{11} + 1362 \nu^{10} - 2724 \nu^{9} + 9761 \nu^{8} - 24714 \nu^{7} + 35185 \nu^{6} + 36774 \nu^{5} + 122807 \nu^{4} + 25878 \nu^{3} + 559992 \nu^{2} + 1135 \nu + 227 \)\()/24631\)
\(\beta_{8}\)\(=\)\((\)\( 1092 \nu^{11} - 1092 \nu^{10} + 6521 \nu^{9} - 13104 \nu^{8} + 46956 \nu^{7} + 78624 \nu^{6} + 169260 \nu^{5} + 171948 \nu^{4} + 590772 \nu^{3} + 124488 \nu^{2} + 26208 \nu + 5460 \)\()/24631\)
\(\beta_{9}\)\(=\)\((\)\( -638 \nu^{11} + 3828 \nu^{10} - 7656 \nu^{9} + 27434 \nu^{8} - 69569 \nu^{7} + 98890 \nu^{6} + 103356 \nu^{5} + 345158 \nu^{4} + 72732 \nu^{3} + 1537440 \nu^{2} + 3190 \nu + 638 \)\()/24631\)
\(\beta_{10}\)\(=\)\((\)\( -5460 \nu^{11} + 6552 \nu^{10} - 33852 \nu^{9} + 72041 \nu^{8} - 247884 \nu^{7} - 346164 \nu^{6} - 767676 \nu^{5} - 715260 \nu^{4} - 2781912 \nu^{3} - 31668 \nu^{2} - 6552 \nu - 1092 \)\()/24631\)
\(\beta_{11}\)\(=\)\((\)\( 15030 \nu^{11} - 18036 \nu^{10} + 93186 \nu^{9} - 198446 \nu^{8} + 682362 \nu^{7} + 952902 \nu^{6} + 2113218 \nu^{5} + 1968930 \nu^{4} + 7637059 \nu^{3} + 87174 \nu^{2} + 18036 \nu + 3006 \)\()/24631\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{9} + 3 \beta_{7} + \beta_{6}\)
\(\nu^{3}\)\(=\)\(\beta_{11} + 4 \beta_{10} + 6 \beta_{8} + 6 \beta_{6} - 6 \beta_{2} - 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(6 \beta_{11} + 19 \beta_{10} + 6 \beta_{9} + 12 \beta_{8} - 19 \beta_{7} - 6 \beta_{5} + 19 \beta_{4} - 6 \beta_{3} - 19\)
\(\nu^{5}\)\(=\)\(-12 \beta_{3} + 43 \beta_{2} - 42\)
\(\nu^{6}\)\(=\)\(43 \beta_{5} - 141 \beta_{4} - 109 \beta_{1}\)
\(\nu^{7}\)\(=\)\(109 \beta_{9} - 370 \beta_{7} - 336 \beta_{6}\)
\(\nu^{8}\)\(=\)\(-336 \beta_{11} - 1117 \beta_{10} - 924 \beta_{8} - 924 \beta_{6} + 924 \beta_{2} + 924 \beta_{1}\)
\(\nu^{9}\)\(=\)\(-924 \beta_{11} - 3108 \beta_{10} - 924 \beta_{9} - 2713 \beta_{8} + 3108 \beta_{7} + 924 \beta_{5} - 3108 \beta_{4} + 924 \beta_{3} + 3108\)
\(\nu^{10}\)\(=\)\(2713 \beta_{3} - 7669 \beta_{2} + 9063\)
\(\nu^{11}\)\(=\)\(-7669 \beta_{5} + 25720 \beta_{4} + 22158 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(1\) \(-1 + \beta_{4} - \beta_{7} + \beta_{10}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
0.0651271 0.200441i
0.511560 1.57442i
−0.885704 + 2.72592i
2.31880 1.68471i
−1.33928 + 0.973045i
−0.170505 + 0.123879i
0.0651271 + 0.200441i
0.511560 + 1.57442i
−0.885704 2.72592i
2.31880 + 1.68471i
−1.33928 0.973045i
−0.170505 0.123879i
−2.22061 1.61337i −0.0651271 + 0.200441i 1.71012 + 5.26321i 0.809017 0.587785i 0.468007 0.340027i 0.717944 + 2.20960i 2.99759 9.22564i 2.39112 + 1.73725i −2.74483
81.2 1.12933 + 0.820508i −0.511560 + 1.57442i −0.0158755 0.0488598i 0.809017 0.587785i −1.86955 + 1.35830i −1.45449 4.47645i 0.884894 2.72343i 0.209948 + 0.152536i 1.39593
81.3 1.90030 + 1.38065i 0.885704 2.72592i 1.08691 + 3.34515i 0.809017 0.587785i 5.44662 3.95720i 1.04556 + 3.21790i −1.10133 + 3.38955i −4.21910 3.06535i 2.34889
251.1 −0.725848 + 2.23393i −2.31880 + 1.68471i −2.84556 2.06742i −0.309017 0.951057i −2.08042 6.40289i −2.73731 1.98877i 2.88333 2.09486i 1.61155 4.95985i 2.34889
251.2 −0.431367 + 1.32761i 1.33928 0.973045i 0.0415626 + 0.0301970i −0.309017 0.951057i 0.714103 + 2.19778i 3.80789 + 2.76660i −2.31668 + 1.68317i −0.0801932 + 0.246809i 1.39593
251.3 0.848198 2.61048i 0.170505 0.123879i −4.47716 3.25284i −0.309017 0.951057i −0.178763 0.550175i −1.87960 1.36561i −7.84779 + 5.70176i −0.913325 + 2.81093i −2.74483
366.1 −2.22061 + 1.61337i −0.0651271 0.200441i 1.71012 5.26321i 0.809017 + 0.587785i 0.468007 + 0.340027i 0.717944 2.20960i 2.99759 + 9.22564i 2.39112 1.73725i −2.74483
366.2 1.12933 0.820508i −0.511560 1.57442i −0.0158755 + 0.0488598i 0.809017 + 0.587785i −1.86955 1.35830i −1.45449 + 4.47645i 0.884894 + 2.72343i 0.209948 0.152536i 1.39593
366.3 1.90030 1.38065i 0.885704 + 2.72592i 1.08691 3.34515i 0.809017 + 0.587785i 5.44662 + 3.95720i 1.04556 3.21790i −1.10133 3.38955i −4.21910 + 3.06535i 2.34889
511.1 −0.725848 2.23393i −2.31880 1.68471i −2.84556 + 2.06742i −0.309017 + 0.951057i −2.08042 + 6.40289i −2.73731 + 1.98877i 2.88333 + 2.09486i 1.61155 + 4.95985i 2.34889
511.2 −0.431367 1.32761i 1.33928 + 0.973045i 0.0415626 0.0301970i −0.309017 + 0.951057i 0.714103 2.19778i 3.80789 2.76660i −2.31668 1.68317i −0.0801932 0.246809i 1.39593
511.3 0.848198 + 2.61048i 0.170505 + 0.123879i −4.47716 + 3.25284i −0.309017 + 0.951057i −0.178763 + 0.550175i −1.87960 + 1.36561i −7.84779 5.70176i −0.913325 2.81093i −2.74483
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 511.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.g.p 12
11.b odd 2 1 605.2.g.o 12
11.c even 5 1 605.2.a.g 3
11.c even 5 3 inner 605.2.g.p 12
11.d odd 10 1 605.2.a.h yes 3
11.d odd 10 3 605.2.g.o 12
33.f even 10 1 5445.2.a.bb 3
33.h odd 10 1 5445.2.a.bd 3
44.g even 10 1 9680.2.a.cb 3
44.h odd 10 1 9680.2.a.bz 3
55.h odd 10 1 3025.2.a.p 3
55.j even 10 1 3025.2.a.u 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.g 3 11.c even 5 1
605.2.a.h yes 3 11.d odd 10 1
605.2.g.o 12 11.b odd 2 1
605.2.g.o 12 11.d odd 10 3
605.2.g.p 12 1.a even 1 1 trivial
605.2.g.p 12 11.c even 5 3 inner
3025.2.a.p 3 55.h odd 10 1
3025.2.a.u 3 55.j even 10 1
5445.2.a.bb 3 33.f even 10 1
5445.2.a.bd 3 33.h odd 10 1
9680.2.a.bz 3 44.h odd 10 1
9680.2.a.cb 3 44.g even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 6561 - 5103 T + 4698 T^{2} - 3492 T^{3} + 2671 T^{4} - 656 T^{5} + 419 T^{6} - 102 T^{7} + 53 T^{8} - 6 T^{9} + 8 T^{10} - T^{11} + T^{12} \)
$3$ \( 1 - 5 T + 24 T^{2} - 114 T^{3} + 541 T^{4} - 162 T^{5} + 155 T^{6} - 72 T^{7} + 43 T^{8} + 12 T^{9} + 6 T^{10} + T^{11} + T^{12} \)
$5$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{3} \)
$7$ \( 1874161 + 962407 T + 544862 T^{2} + 255152 T^{3} + 119739 T^{4} + 22556 T^{5} + 7923 T^{6} + 1442 T^{7} + 345 T^{8} + 2 T^{9} + 20 T^{10} + T^{11} + T^{12} \)
$11$ \( T^{12} \)
$13$ \( 256 + 256 T + 640 T^{2} + 1088 T^{3} + 2112 T^{4} - 1952 T^{5} + 1488 T^{6} - 912 T^{7} + 832 T^{8} - 164 T^{9} + 32 T^{10} - 6 T^{11} + T^{12} \)
$17$ \( 5308416 - 1769472 T + 1032192 T^{2} - 380928 T^{3} + 176128 T^{4} - 11264 T^{5} + 10496 T^{6} - 768 T^{7} + 896 T^{8} - 144 T^{9} + 32 T^{10} - 4 T^{11} + T^{12} \)
$19$ \( 1679616 - 559872 T + 373248 T^{2} - 139968 T^{3} + 72576 T^{4} - 5184 T^{5} + 5904 T^{6} - 528 T^{7} + 688 T^{8} - 124 T^{9} + 28 T^{10} - 4 T^{11} + T^{12} \)
$23$ \( ( -12 + 4 T + 6 T^{2} + T^{3} )^{4} \)
$29$ \( 26873856 - 10450944 T + 4810752 T^{2} - 1787904 T^{3} + 683776 T^{4} - 83968 T^{5} + 26816 T^{6} - 3264 T^{7} + 848 T^{8} - 48 T^{9} + 32 T^{10} - 2 T^{11} + T^{12} \)
$31$ \( 1679616 + 2239488 T + 2332800 T^{2} + 2286144 T^{3} + 2203200 T^{4} + 759456 T^{5} + 219312 T^{6} + 58272 T^{7} + 13504 T^{8} + 1436 T^{9} + 148 T^{10} + 14 T^{11} + T^{12} \)
$37$ \( 65536 - 98304 T + 131072 T^{2} - 167936 T^{3} + 212992 T^{4} - 58368 T^{5} + 23808 T^{6} - 7808 T^{7} + 2112 T^{8} + 272 T^{9} + 40 T^{10} + 4 T^{11} + T^{12} \)
$41$ \( 7780827681 - 1178913285 T + 414405882 T^{2} - 72315342 T^{3} + 19545219 T^{4} - 599238 T^{5} + 439587 T^{6} - 18954 T^{7} + 14175 T^{8} - 1242 T^{9} + 126 T^{10} - 9 T^{11} + T^{12} \)
$43$ \( ( -63 - 3 T + 7 T^{2} + T^{3} )^{4} \)
$47$ \( 411651843201 - 14903749629 T + 8248422456 T^{2} - 63806058 T^{3} + 138168493 T^{4} - 19400442 T^{5} + 3210155 T^{6} - 307032 T^{7} + 47011 T^{8} - 3444 T^{9} + 254 T^{10} - 15 T^{11} + T^{12} \)
$53$ \( 20736 + 6912 T + 12672 T^{2} + 9408 T^{3} + 10048 T^{4} - 6816 T^{5} + 3536 T^{6} - 1392 T^{7} + 736 T^{8} - 156 T^{9} + 32 T^{10} - 6 T^{11} + T^{12} \)
$59$ \( 14666178816 + 2191497984 T + 748907136 T^{2} + 132735552 T^{3} + 35056960 T^{4} + 489824 T^{5} + 699152 T^{6} + 17808 T^{7} + 21344 T^{8} + 1692 T^{9} + 152 T^{10} + 10 T^{11} + T^{12} \)
$61$ \( 713283282721 - 124960400999 T + 24220320358 T^{2} - 3874939606 T^{3} + 621943239 T^{4} - 40060742 T^{5} + 4832067 T^{6} - 300546 T^{7} + 24835 T^{8} - 74 T^{9} + 170 T^{10} - 3 T^{11} + T^{12} \)
$67$ \( ( -59 + 95 T - 19 T^{2} + T^{3} )^{4} \)
$71$ \( 20736 - 6912 T + 12672 T^{2} - 9408 T^{3} + 10048 T^{4} + 6816 T^{5} + 3536 T^{6} + 1392 T^{7} + 736 T^{8} + 156 T^{9} + 32 T^{10} + 6 T^{11} + T^{12} \)
$73$ \( 1048576 - 524288 T + 655360 T^{2} - 557056 T^{3} + 540672 T^{4} + 249856 T^{5} + 95232 T^{6} + 29184 T^{7} + 13312 T^{8} + 1312 T^{9} + 128 T^{10} + 12 T^{11} + T^{12} \)
$79$ \( 7676563456 - 1971009536 T + 557938688 T^{2} - 130637824 T^{3} + 30653184 T^{4} - 2887168 T^{5} + 507072 T^{6} - 46144 T^{7} + 5520 T^{8} - 16 T^{9} + 80 T^{10} - 2 T^{11} + T^{12} \)
$83$ \( 11019960576 + 1224440064 T + 748268928 T^{2} + 185177664 T^{3} + 65924928 T^{4} - 14906592 T^{5} + 2577744 T^{6} - 338256 T^{7} + 59616 T^{8} - 4212 T^{9} + 288 T^{10} - 18 T^{11} + T^{12} \)
$89$ \( ( 1719 - 157 T - 11 T^{2} + T^{3} )^{4} \)
$97$ \( 754507653376 + 184579125504 T + 43535434880 T^{2} + 9444649408 T^{3} + 2019023680 T^{4} + 113691168 T^{5} + 13346448 T^{6} + 854704 T^{7} + 58464 T^{8} - 1852 T^{9} + 232 T^{10} - 2 T^{11} + T^{12} \)
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