Properties

Label 3549.2.a.bh
Level $3549$
Weight $2$
Character orbit 3549.a
Self dual yes
Analytic conductor $28.339$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3389076774\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - 2 x^{14} - 27 x^{13} + 51 x^{12} + 290 x^{11} - 510 x^{10} - 1575 x^{9} + 2522 x^{8} + 4553 x^{7} - 6393 x^{6} - 6785 x^{5} + 7806 x^{4} + 4632 x^{3} - 3811 x^{2} - 1041 x + 281\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15} - 2 x^{14} - 27 x^{13} + 51 x^{12} + 290 x^{11} - 510 x^{10} - 1575 x^{9} + 2522 x^{8} + 4553 x^{7} - 6393 x^{6} - 6785 x^{5} + 7806 x^{4} + 4632 x^{3} - 3811 x^{2} - 1041 x + 281\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\(-4840 \nu^{14} - 207205 \nu^{13} + 946938 \nu^{12} + 3734994 \nu^{11} - 17154925 \nu^{10} - 24302208 \nu^{9} + 119482005 \nu^{8} + 71234831 \nu^{7} - 381510549 \nu^{6} - 106742226 \nu^{5} + 552269516 \nu^{4} + 106936663 \nu^{3} - 300569102 \nu^{2} - 53360949 \nu + 17099693\)\()/6033052\)
\(\beta_{4}\)\(=\)\((\)\(26527 \nu^{14} + 348794 \nu^{13} - 1761470 \nu^{12} - 7125065 \nu^{11} + 28878896 \nu^{10} + 56338585 \nu^{9} - 202723757 \nu^{8} - 223516985 \nu^{7} + 685631166 \nu^{6} + 489339396 \nu^{5} - 1079863813 \nu^{4} - 587154214 \nu^{3} + 627931543 \nu^{2} + 286694529 \nu - 38588056\)\()/6033052\)
\(\beta_{5}\)\(=\)\((\)\(-38567 \nu^{14} + 8311 \nu^{13} + 1175340 \nu^{12} - 610919 \nu^{11} - 13591371 \nu^{10} + 9178285 \nu^{9} + 75838990 \nu^{8} - 53203852 \nu^{7} - 212298805 \nu^{6} + 125754442 \nu^{5} + 277030943 \nu^{4} - 95988421 \nu^{3} - 128107759 \nu^{2} + 8811570 \nu + 5587509\)\()/6033052\)
\(\beta_{6}\)\(=\)\((\)\(39159 \nu^{14} + 79358 \nu^{13} - 1156542 \nu^{12} - 1536175 \nu^{11} + 12934898 \nu^{10} + 9988723 \nu^{9} - 69250381 \nu^{8} - 19347335 \nu^{7} + 183706814 \nu^{6} - 38850820 \nu^{5} - 230681231 \nu^{4} + 164127904 \nu^{3} + 140774277 \nu^{2} - 106534419 \nu - 41386850\)\()/6033052\)
\(\beta_{7}\)\(=\)\((\)\(-21755 \nu^{14} + 123497 \nu^{13} + 139769 \nu^{12} - 2498265 \nu^{11} + 3036617 \nu^{10} + 18918087 \nu^{9} - 39286903 \nu^{8} - 65837724 \nu^{7} + 171606601 \nu^{6} + 102192067 \nu^{5} - 317954439 \nu^{4} - 51129407 \nu^{3} + 212384073 \nu^{2} - 9188801 \nu - 20411763\)\()/3016526\)
\(\beta_{8}\)\(=\)\((\)\(-60853 \nu^{14} + 116866 \nu^{13} + 1435826 \nu^{12} - 2156565 \nu^{11} - 13912376 \nu^{10} + 13880105 \nu^{9} + 71541267 \nu^{8} - 33989261 \nu^{7} - 205828878 \nu^{6} + 7522680 \nu^{5} + 306145379 \nu^{4} + 77250998 \nu^{3} - 174934433 \nu^{2} - 68658319 \nu + 9987024\)\()/6033052\)
\(\beta_{9}\)\(=\)\((\)\(73663 \nu^{14} + 73174 \nu^{13} - 2302936 \nu^{12} - 1328053 \nu^{11} + 27645810 \nu^{10} + 9206243 \nu^{9} - 163544127 \nu^{8} - 34531577 \nu^{7} + 502314938 \nu^{6} + 91388378 \nu^{5} - 757335097 \nu^{4} - 157471248 \nu^{3} + 438736369 \nu^{2} + 93954739 \nu - 27937020\)\()/6033052\)
\(\beta_{10}\)\(=\)\((\)\(-177180 \nu^{14} + 186673 \nu^{13} + 5359798 \nu^{12} - 4786396 \nu^{11} - 63240615 \nu^{10} + 44708658 \nu^{9} + 369211365 \nu^{8} - 181845153 \nu^{7} - 1110194795 \nu^{6} + 275576846 \nu^{5} + 1600220662 \nu^{4} + 22579661 \nu^{3} - 837484336 \nu^{2} - 190640833 \nu + 17466621\)\()/6033052\)
\(\beta_{11}\)\(=\)\((\)\(-121202 \nu^{14} + 43404 \nu^{13} + 3702933 \nu^{12} - 1127106 \nu^{11} - 44424222 \nu^{10} + 10040241 \nu^{9} + 265785957 \nu^{8} - 33848081 \nu^{7} - 827597654 \nu^{6} + 12720921 \nu^{5} + 1256849458 \nu^{4} + 114455160 \nu^{3} - 732374977 \nu^{2} - 101829125 \nu + 58774419\)\()/3016526\)
\(\beta_{12}\)\(=\)\((\)\(131583 \nu^{14} - 276767 \nu^{13} - 3276487 \nu^{12} + 5917076 \nu^{11} + 33124120 \nu^{10} - 47456984 \nu^{9} - 173841121 \nu^{8} + 173650765 \nu^{7} + 497057283 \nu^{6} - 271189129 \nu^{5} - 725804406 \nu^{4} + 101476522 \nu^{3} + 422840940 \nu^{2} + 49042155 \nu - 37166976\)\()/3016526\)
\(\beta_{13}\)\(=\)\((\)\(-286506 \nu^{14} + 246133 \nu^{13} + 7666606 \nu^{12} - 5103648 \nu^{11} - 82118737 \nu^{10} + 38749648 \nu^{9} + 446409499 \nu^{8} - 126820423 \nu^{7} - 1283390115 \nu^{6} + 142922354 \nu^{5} + 1825407274 \nu^{4} + 58512023 \nu^{3} - 1004383910 \nu^{2} - 112246899 \nu + 64260237\)\()/6033052\)
\(\beta_{14}\)\(=\)\((\)\(328859 \nu^{14} - 463195 \nu^{13} - 8935738 \nu^{12} + 10255723 \nu^{11} + 97315541 \nu^{10} - 84491865 \nu^{9} - 538705514 \nu^{8} + 310055960 \nu^{7} + 1581453993 \nu^{6} - 443158712 \nu^{5} - 2310023707 \nu^{4} + 24798435 \nu^{3} + 1321185415 \nu^{2} + 189938318 \nu - 101420669\)\()/6033052\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{14} - \beta_{12} + \beta_{8} + \beta_{3} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{13} + \beta_{11} + \beta_{7} - \beta_{6} - \beta_{5} + 8 \beta_{2} + 2 \beta_{1} + 24\)
\(\nu^{5}\)\(=\)\(10 \beta_{14} - 9 \beta_{12} + 2 \beta_{11} - \beta_{10} + 9 \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 10 \beta_{3} + \beta_{2} + 30 \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(3 \beta_{14} - 11 \beta_{13} + 13 \beta_{11} + \beta_{10} - 2 \beta_{9} + 3 \beta_{8} + 12 \beta_{7} - 10 \beta_{6} - 10 \beta_{5} + 2 \beta_{4} + \beta_{3} + 57 \beta_{2} + 25 \beta_{1} + 157\)
\(\nu^{7}\)\(=\)\(82 \beta_{14} - 3 \beta_{13} - 69 \beta_{12} + 27 \beta_{11} - 14 \beta_{10} + 78 \beta_{8} + 14 \beta_{7} + 12 \beta_{6} + 17 \beta_{5} + 13 \beta_{4} + 84 \beta_{3} + 16 \beta_{2} + 196 \beta_{1} + 16\)
\(\nu^{8}\)\(=\)\(46 \beta_{14} - 97 \beta_{13} - 5 \beta_{12} + 123 \beta_{11} + 12 \beta_{10} - 34 \beta_{9} + 52 \beta_{8} + 108 \beta_{7} - 80 \beta_{6} - 81 \beta_{5} + 31 \beta_{4} + 22 \beta_{3} + 400 \beta_{2} + 237 \beta_{1} + 1063\)
\(\nu^{9}\)\(=\)\(636 \beta_{14} - 53 \beta_{13} - 509 \beta_{12} + 267 \beta_{11} - 140 \beta_{10} - 4 \beta_{9} + 672 \beta_{8} + 140 \beta_{7} + 104 \beta_{6} + 193 \beta_{5} + 129 \beta_{4} + 670 \beta_{3} + 181 \beta_{2} + 1342 \beta_{1} + 189\)
\(\nu^{10}\)\(=\)\(503 \beta_{14} - 795 \beta_{13} - 94 \beta_{12} + 1045 \beta_{11} + 100 \beta_{10} - 386 \beta_{9} + 627 \beta_{8} + 886 \beta_{7} - 596 \beta_{6} - 609 \beta_{5} + 341 \beta_{4} + 305 \beta_{3} + 2816 \beta_{2} + 2041 \beta_{1} + 7341\)
\(\nu^{11}\)\(=\)\(4838 \beta_{14} - 634 \beta_{13} - 3711 \beta_{12} + 2366 \beta_{11} - 1232 \beta_{10} - 86 \beta_{9} + 5690 \beta_{8} + 1249 \beta_{7} + 801 \beta_{6} + 1836 \beta_{5} + 1165 \beta_{4} + 5236 \beta_{3} + 1778 \beta_{2} + 9472 \beta_{1} + 1979\)
\(\nu^{12}\)\(=\)\(4826 \beta_{14} - 6311 \beta_{13} - 1180 \beta_{12} + 8491 \beta_{11} + 705 \beta_{10} - 3702 \beta_{9} + 6497 \beta_{8} + 7001 \beta_{7} - 4309 \beta_{6} - 4400 \beta_{5} + 3276 \beta_{4} + 3450 \beta_{3} + 19981 \beta_{2} + 16828 \beta_{1} + 51420\)
\(\nu^{13}\)\(=\)\(36569 \beta_{14} - 6458 \beta_{13} - 26997 \beta_{12} + 19948 \beta_{11} - 10167 \beta_{10} - 1206 \beta_{9} + 47239 \beta_{8} + 10610 \beta_{7} + 5836 \beta_{6} + 15863 \beta_{5} + 10071 \beta_{4} + 40559 \beta_{3} + 16213 \beta_{2} + 68313 \beta_{1} + 19338\)
\(\nu^{14}\)\(=\)\(43352 \beta_{14} - 49362 \beta_{13} - 12504 \beta_{12} + 67626 \beta_{11} + 4396 \beta_{10} - 32488 \beta_{9} + 62004 \beta_{8} + 54390 \beta_{7} - 30706 \beta_{6} - 30972 \beta_{5} + 29358 \beta_{4} + 34898 \beta_{3} + 143080 \beta_{2} + 135634 \beta_{1} + 364313\)

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
−2.61511
−2.60515
−2.41695
−1.89376
−1.29981
−1.08127
−0.415563
0.182130
1.02667
1.09372
1.86754
2.07185
2.49813
2.78394
2.80363
−2.61511 1.00000 4.83880 −4.09685 −2.61511 −1.00000 −7.42376 1.00000 10.7137
1.2 −2.60515 1.00000 4.78683 −1.51368 −2.60515 −1.00000 −7.26012 1.00000 3.94338
1.3 −2.41695 1.00000 3.84166 −1.98263 −2.41695 −1.00000 −4.45120 1.00000 4.79192
1.4 −1.89376 1.00000 1.58633 0.892462 −1.89376 −1.00000 0.783390 1.00000 −1.69011
1.5 −1.29981 1.00000 −0.310501 1.99349 −1.29981 −1.00000 3.00321 1.00000 −2.59115
1.6 −1.08127 1.00000 −0.830866 −1.66920 −1.08127 −1.00000 3.06092 1.00000 1.80485
1.7 −0.415563 1.00000 −1.82731 −4.32224 −0.415563 −1.00000 1.59049 1.00000 1.79616
1.8 0.182130 1.00000 −1.96683 1.12886 0.182130 −1.00000 −0.722479 1.00000 0.205599
1.9 1.02667 1.00000 −0.945953 −3.27646 1.02667 −1.00000 −3.02452 1.00000 −3.36384
1.10 1.09372 1.00000 −0.803784 3.36980 1.09372 −1.00000 −3.06655 1.00000 3.68561
1.11 1.86754 1.00000 1.48770 −4.30941 1.86754 −1.00000 −0.956744 1.00000 −8.04798
1.12 2.07185 1.00000 2.29258 3.98491 2.07185 −1.00000 0.606176 1.00000 8.25616
1.13 2.49813 1.00000 4.24068 2.35199 2.49813 −1.00000 5.59751 1.00000 5.87558
1.14 2.78394 1.00000 5.75034 1.59144 2.78394 −1.00000 10.4407 1.00000 4.43048
1.15 2.80363 1.00000 5.86034 −3.14248 2.80363 −1.00000 10.8230 1.00000 −8.81036
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(13\) \(1\)

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 3549.2.a.bh yes 15
13.b even 2 1 3549.2.a.bg 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3549.2.a.bg 15 13.b even 2 1
3549.2.a.bh yes 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3549))\):

\(T_{2}^{15} - \cdots\)
\(T_{5}^{15} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 281 - 1041 T - 3811 T^{2} + 4632 T^{3} + 7806 T^{4} - 6785 T^{5} - 6393 T^{6} + 4553 T^{7} + 2522 T^{8} - 1575 T^{9} - 510 T^{10} + 290 T^{11} + 51 T^{12} - 27 T^{13} - 2 T^{14} + T^{15} \)
$3$ \( ( -1 + T )^{15} \)
$5$ \( -397312 + 402944 T + 675648 T^{2} - 576600 T^{3} - 504860 T^{4} + 306312 T^{5} + 206759 T^{6} - 74303 T^{7} - 47961 T^{8} + 7829 T^{9} + 6081 T^{10} - 169 T^{11} - 380 T^{12} - 22 T^{13} + 9 T^{14} + T^{15} \)
$7$ \( ( 1 + T )^{15} \)
$11$ \( 695227 + 3634046 T - 12433029 T^{2} - 6623301 T^{3} + 12938774 T^{4} - 205726 T^{5} - 3221862 T^{6} + 393644 T^{7} + 349925 T^{8} - 57156 T^{9} - 18897 T^{10} + 3463 T^{11} + 495 T^{12} - 96 T^{13} - 5 T^{14} + T^{15} \)
$13$ \( T^{15} \)
$17$ \( 13071808 + 41168832 T - 210545120 T^{2} + 49389008 T^{3} + 94448292 T^{4} - 35728680 T^{5} - 10153087 T^{6} + 5111978 T^{7} + 420567 T^{8} - 318075 T^{9} - 5919 T^{10} + 10052 T^{11} - 32 T^{12} - 159 T^{13} + T^{14} + T^{15} \)
$19$ \( 5898752 + 37750272 T - 106280800 T^{2} - 14034616 T^{3} + 120043624 T^{4} - 15238642 T^{5} - 23063773 T^{6} + 4204448 T^{7} + 1566129 T^{8} - 331519 T^{9} - 46895 T^{10} + 11224 T^{11} + 628 T^{12} - 173 T^{13} - 3 T^{14} + T^{15} \)
$23$ \( 900147443 - 90200601 T - 989900104 T^{2} + 126312863 T^{3} + 353769690 T^{4} - 48656958 T^{5} - 51126170 T^{6} + 8378968 T^{7} + 3138163 T^{8} - 586297 T^{9} - 82362 T^{10} + 17077 T^{11} + 951 T^{12} - 217 T^{13} - 4 T^{14} + T^{15} \)
$29$ \( 2666274624 - 2816766144 T - 1661737920 T^{2} + 1292983600 T^{3} + 455601772 T^{4} - 203314456 T^{5} - 56760423 T^{6} + 15501492 T^{7} + 3495671 T^{8} - 650665 T^{9} - 109329 T^{10} + 15514 T^{11} + 1632 T^{12} - 197 T^{13} - 9 T^{14} + T^{15} \)
$31$ \( 59072 + 672928 T - 193616 T^{2} - 8666984 T^{3} + 13286428 T^{4} - 2255866 T^{5} - 5140679 T^{6} + 1723046 T^{7} + 730693 T^{8} - 247915 T^{9} - 47493 T^{10} + 10942 T^{11} + 1106 T^{12} - 187 T^{13} - 7 T^{14} + T^{15} \)
$37$ \( -1666363 - 4254613 T + 92767491 T^{2} - 257613799 T^{3} + 175201482 T^{4} + 67360486 T^{5} - 38059964 T^{6} - 5686348 T^{7} + 3106135 T^{8} + 179277 T^{9} - 121895 T^{10} - 485 T^{11} + 2311 T^{12} - 75 T^{13} - 17 T^{14} + T^{15} \)
$41$ \( 85163678208 - 9886810752 T - 74867710176 T^{2} - 21203355032 T^{3} + 7868604372 T^{4} + 3588799324 T^{5} - 71445603 T^{6} - 187174169 T^{7} - 15263316 T^{8} + 3587741 T^{9} + 530073 T^{10} - 17731 T^{11} - 5973 T^{12} - 128 T^{13} + 22 T^{14} + T^{15} \)
$43$ \( 2841571328 + 8418838528 T - 1799891200 T^{2} - 12982199616 T^{3} - 4357405248 T^{4} + 1692405264 T^{5} + 490038864 T^{6} - 135091420 T^{7} - 14048236 T^{8} + 5115507 T^{9} - 37300 T^{10} - 75354 T^{11} + 4996 T^{12} + 255 T^{13} - 36 T^{14} + T^{15} \)
$47$ \( 21639614464 + 31766960128 T + 2924525568 T^{2} - 11884556288 T^{3} - 3373343232 T^{4} + 1314918656 T^{5} + 512860608 T^{6} - 34201408 T^{7} - 25031744 T^{8} - 558872 T^{9} + 487236 T^{10} + 27406 T^{11} - 4059 T^{12} - 297 T^{13} + 12 T^{14} + T^{15} \)
$53$ \( 156125119424 - 143175123840 T - 23977900880 T^{2} + 42138630768 T^{3} - 785075472 T^{4} - 4498179640 T^{5} + 249033707 T^{6} + 227581235 T^{7} - 12547255 T^{8} - 5826509 T^{9} + 240971 T^{10} + 74367 T^{11} - 1868 T^{12} - 446 T^{13} + 5 T^{14} + T^{15} \)
$59$ \( 144963371008 + 111649898496 T - 102475229184 T^{2} - 58177561088 T^{3} + 10461927680 T^{4} + 7394302592 T^{5} - 58903168 T^{6} - 347727296 T^{7} - 20930544 T^{8} + 6835608 T^{9} + 710452 T^{10} - 47368 T^{11} - 8013 T^{12} - 37 T^{13} + 29 T^{14} + T^{15} \)
$61$ \( 28606050304 - 5631082496 T - 38925092864 T^{2} - 1413225984 T^{3} + 8454666496 T^{4} + 131870080 T^{5} - 769735040 T^{6} + 15783456 T^{7} + 33244480 T^{8} - 1701600 T^{9} - 641052 T^{10} + 44074 T^{11} + 4927 T^{12} - 383 T^{13} - 12 T^{14} + T^{15} \)
$67$ \( 1364176809472 - 1034311669504 T - 319321224896 T^{2} + 401212562848 T^{3} - 47856863792 T^{4} - 24621596600 T^{5} + 4009123989 T^{6} + 735747573 T^{7} - 108060794 T^{8} - 13051891 T^{9} + 1235841 T^{10} + 123887 T^{11} - 6329 T^{12} - 570 T^{13} + 12 T^{14} + T^{15} \)
$71$ \( -140599367987 + 432355900691 T + 67296284692 T^{2} - 117571288913 T^{3} - 3430271780 T^{4} + 10806134468 T^{5} - 277134104 T^{6} - 448882344 T^{7} + 26415227 T^{8} + 8933669 T^{9} - 758610 T^{10} - 75721 T^{11} + 8927 T^{12} + 107 T^{13} - 36 T^{14} + T^{15} \)
$73$ \( -15834251264 - 1238663168 T + 38905536512 T^{2} - 25199712256 T^{3} - 1788084736 T^{4} + 3687088896 T^{5} + 19835328 T^{6} - 236451968 T^{7} - 9571040 T^{8} + 7051520 T^{9} + 673148 T^{10} - 55372 T^{11} - 8869 T^{12} - 68 T^{13} + 29 T^{14} + T^{15} \)
$79$ \( 150736832 + 2158731200 T + 8936142480 T^{2} + 6743831680 T^{3} - 14786267116 T^{4} + 4934152132 T^{5} + 456161015 T^{6} - 398023508 T^{7} + 21099149 T^{8} + 9679355 T^{9} - 921999 T^{10} - 77194 T^{11} + 10636 T^{12} + 19 T^{13} - 35 T^{14} + T^{15} \)
$83$ \( -10802692096 - 52337180672 T - 52417691648 T^{2} + 40509210624 T^{3} + 44366738432 T^{4} - 20035565568 T^{5} - 1528933184 T^{6} + 1263777792 T^{7} - 38013312 T^{8} - 22672960 T^{9} + 902860 T^{10} + 182868 T^{11} - 5393 T^{12} - 693 T^{13} + 10 T^{14} + T^{15} \)
$89$ \( 150573032217152 + 5975396152000 T - 17764174982448 T^{2} - 584290368576 T^{3} + 849343547880 T^{4} + 18419262068 T^{5} - 21720889889 T^{6} - 180853069 T^{7} + 323342801 T^{8} - 1806539 T^{9} - 2804477 T^{10} + 52757 T^{11} + 13066 T^{12} - 398 T^{13} - 25 T^{14} + T^{15} \)
$97$ \( -19521302528 - 22540277760 T + 137434228736 T^{2} - 80266006528 T^{3} - 21065460992 T^{4} + 14899556736 T^{5} + 2363230464 T^{6} - 754667488 T^{7} - 125537552 T^{8} + 10161296 T^{9} + 2253496 T^{10} - 3146 T^{11} - 13583 T^{12} - 351 T^{13} + 26 T^{14} + T^{15} \)
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