Properties

Label 91.2.r.a
Level $91$
Weight $2$
Character orbit 91.r
Analytic conductor $0.727$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.726638658394\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 11 x^{14} + 85 x^{12} - 334 x^{10} + 952 x^{8} - 1050 x^{6} + 853 x^{4} - 93 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{11} q^{2} + ( -1 + \beta_{3} + \beta_{6} ) q^{3} + ( 1 - \beta_{6} - \beta_{8} ) q^{4} -\beta_{13} q^{5} + ( -\beta_{1} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{6} + ( -\beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{7} -\beta_{12} q^{8} + ( 1 - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{9} +O(q^{10})\) \( q + \beta_{11} q^{2} + ( -1 + \beta_{3} + \beta_{6} ) q^{3} + ( 1 - \beta_{6} - \beta_{8} ) q^{4} -\beta_{13} q^{5} + ( -\beta_{1} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{6} + ( -\beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{7} -\beta_{12} q^{8} + ( 1 - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{9} + ( -2 \beta_{3} + \beta_{8} - \beta_{9} ) q^{10} + ( \beta_{1} + \beta_{5} + \beta_{10} - \beta_{14} - \beta_{15} ) q^{11} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{12} + ( -1 + \beta_{1} - \beta_{2} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{13} + ( -2 - \beta_{2} + \beta_{3} - \beta_{8} + \beta_{9} ) q^{14} + ( -2 \beta_{1} + \beta_{10} - 2 \beta_{11} + \beta_{13} ) q^{15} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{8} ) q^{16} + ( 1 - \beta_{6} + \beta_{9} ) q^{17} + ( 3 \beta_{1} - \beta_{5} - 4 \beta_{10} + \beta_{14} + \beta_{15} ) q^{18} + ( -\beta_{5} + 2 \beta_{11} + \beta_{12} ) q^{19} + ( \beta_{1} - \beta_{5} - 3 \beta_{10} + 2 \beta_{11} + \beta_{12} - 3 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{20} + ( \beta_{5} + \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{14} ) q^{21} + ( -3 - \beta_{2} - \beta_{4} - \beta_{7} ) q^{22} + ( -1 + \beta_{3} - \beta_{4} - \beta_{6} ) q^{23} + ( -3 \beta_{1} + 2 \beta_{10} - \beta_{14} - \beta_{15} ) q^{24} + ( \beta_{3} - 2 \beta_{8} + \beta_{9} ) q^{25} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{11} + \beta_{12} ) q^{26} + ( 2 + 2 \beta_{2} - \beta_{4} - \beta_{7} ) q^{27} + ( \beta_{1} + 2 \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{28} + ( -3 - 2 \beta_{2} - 3 \beta_{4} ) q^{29} + ( -2 + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 6 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{30} + ( -\beta_{1} + \beta_{14} + \beta_{15} ) q^{31} + ( -\beta_{1} + 2 \beta_{5} - \beta_{10} ) q^{32} + ( -\beta_{1} - \beta_{5} - \beta_{11} + 2 \beta_{14} - \beta_{15} ) q^{33} + ( -\beta_{1} + \beta_{5} + 3 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + 3 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{34} + ( -1 - \beta_{3} + \beta_{4} + 3 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{35} + ( -1 + 5 \beta_{4} + 2 \beta_{7} ) q^{36} + ( \beta_{1} + \beta_{5} + 3 \beta_{11} - 2 \beta_{14} + \beta_{15} ) q^{37} + ( 4 + \beta_{3} - 4 \beta_{6} - \beta_{8} ) q^{38} + ( 3 - \beta_{3} + \beta_{5} - 3 \beta_{6} - \beta_{8} - \beta_{9} + \beta_{14} + \beta_{15} ) q^{39} + ( 1 - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{40} + ( \beta_{5} - \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{41} + ( 6 - \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 6 \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{42} + ( 2 + 2 \beta_{4} - \beta_{7} ) q^{43} + ( -\beta_{1} - \beta_{5} - 2 \beta_{11} - 2 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{44} + ( 4 \beta_{1} - 2 \beta_{5} ) q^{45} + ( \beta_{1} + \beta_{5} + \beta_{10} ) q^{46} + ( \beta_{1} - 2 \beta_{5} + \beta_{11} + 3 \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{47} + ( 2 - \beta_{2} ) q^{48} + ( -2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - \beta_{7} - 2 \beta_{8} ) q^{49} + ( \beta_{5} + 4 \beta_{10} - \beta_{11} - 3 \beta_{12} + 4 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{50} + ( 1 - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - 2 \beta_{8} ) q^{51} + ( -6 - \beta_{1} + \beta_{3} - 2 \beta_{5} + 6 \beta_{6} + \beta_{8} - 2 \beta_{10} + \beta_{14} + \beta_{15} ) q^{52} + ( -3 + 3 \beta_{6} + 2 \beta_{8} ) q^{53} + ( \beta_{1} + 2 \beta_{5} + 8 \beta_{11} - \beta_{12} - 4 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{54} + ( 4 + 4 \beta_{2} + 3 \beta_{4} + \beta_{7} ) q^{55} + ( -5 + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} ) q^{56} + ( 2 \beta_{1} - \beta_{5} + 3 \beta_{11} + 4 \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{57} + ( \beta_{5} - 2 \beta_{11} - \beta_{12} - 3 \beta_{13} ) q^{58} + ( 3 \beta_{1} + \beta_{5} - 3 \beta_{10} + \beta_{14} + \beta_{15} ) q^{59} + ( -4 \beta_{1} + 3 \beta_{5} + 3 \beta_{10} - \beta_{14} - \beta_{15} ) q^{60} + ( 2 \beta_{2} - \beta_{6} + 2 \beta_{8} ) q^{61} + ( 2 + \beta_{2} - 2 \beta_{4} ) q^{62} + ( -7 \beta_{1} + 3 \beta_{10} - 3 \beta_{11} + 3 \beta_{13} - \beta_{15} ) q^{63} + ( 5 + 2 \beta_{2} - 2 \beta_{4} + \beta_{7} ) q^{64} + ( -\beta_{1} - \beta_{2} - 4 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{11} - \beta_{12} + 3 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{65} + \beta_{8} q^{66} + ( -3 \beta_{5} - 2 \beta_{10} ) q^{67} + ( -4 - 3 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} + \beta_{6} - \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{68} + ( -4 - \beta_{4} + \beta_{7} ) q^{69} + ( -4 \beta_{1} - \beta_{5} - 4 \beta_{10} - 4 \beta_{11} + 2 \beta_{12} + \beta_{14} + \beta_{15} ) q^{70} + ( -4 \beta_{1} - \beta_{10} - 4 \beta_{11} - 3 \beta_{12} - \beta_{13} ) q^{71} + ( -3 \beta_{5} - 2 \beta_{11} + 3 \beta_{12} + 3 \beta_{13} ) q^{72} + ( -2 \beta_{1} + 2 \beta_{5} + 3 \beta_{10} - 2 \beta_{14} - 2 \beta_{15} ) q^{73} + ( 6 - 6 \beta_{6} - 3 \beta_{8} ) q^{74} + ( 2 - 2 \beta_{3} + 2 \beta_{4} - \beta_{7} + \beta_{9} ) q^{75} + ( \beta_{1} + \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{76} + ( -2 + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{6} - \beta_{7} + 4 \beta_{8} ) q^{77} + ( 6 \beta_{1} + \beta_{2} - 3 \beta_{4} - \beta_{5} - 4 \beta_{10} + 7 \beta_{11} - 4 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{78} + ( -1 + \beta_{3} - \beta_{4} + 3 \beta_{6} - \beta_{7} + \beta_{9} ) q^{79} + ( 2 \beta_{1} + \beta_{5} + 2 \beta_{10} - \beta_{14} - \beta_{15} ) q^{80} + ( -5 + 2 \beta_{3} + 5 \beta_{6} + 4 \beta_{8} ) q^{81} + ( 4 - 4 \beta_{3} + 4 \beta_{4} - 4 \beta_{6} + \beta_{7} - \beta_{9} ) q^{82} + ( 6 \beta_{1} - \beta_{5} - \beta_{10} + 7 \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{83} + ( 8 \beta_{1} - 5 \beta_{10} + 4 \beta_{11} - \beta_{12} - 3 \beta_{13} + \beta_{14} ) q^{84} + ( -2 \beta_{1} - 3 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - 3 \beta_{13} ) q^{85} + ( \beta_{1} - 3 \beta_{5} + 3 \beta_{11} + 4 \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{86} + ( -8 + 8 \beta_{6} - 2 \beta_{8} + \beta_{9} ) q^{87} + ( 1 - 2 \beta_{3} - \beta_{6} + 2 \beta_{8} ) q^{88} + ( \beta_{1} + 3 \beta_{5} - 3 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{89} + ( -14 - 6 \beta_{2} + 2 \beta_{4} ) q^{90} + ( 4 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{8} + \beta_{9} - 2 \beta_{13} + \beta_{15} ) q^{91} + ( -\beta_{2} - \beta_{4} - \beta_{7} ) q^{92} + ( 2 \beta_{5} - 6 \beta_{11} - 2 \beta_{12} + 3 \beta_{13} ) q^{93} + ( -6 + 5 \beta_{3} + 6 \beta_{6} + \beta_{8} + \beta_{9} ) q^{94} + ( -1 - 3 \beta_{3} + \beta_{6} + 2 \beta_{8} - \beta_{9} ) q^{95} + ( -2 \beta_{1} + \beta_{5} - 9 \beta_{11} - 3 \beta_{12} + 4 \beta_{13} + 4 \beta_{14} - 2 \beta_{15} ) q^{96} + ( 2 \beta_{1} - \beta_{5} - \beta_{10} + 3 \beta_{11} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{97} + ( 3 \beta_{1} + 3 \beta_{10} + 6 \beta_{11} + 2 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{98} + ( 2 \beta_{1} + \beta_{5} + 2 \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 4q^{3} + 6q^{4} - 12q^{9} + O(q^{10}) \) \( 16q - 4q^{3} + 6q^{4} - 12q^{9} - 6q^{10} + 18q^{12} - 12q^{13} - 26q^{14} + 2q^{16} + 8q^{17} - 36q^{22} - 12q^{23} - 6q^{26} + 32q^{27} - 16q^{29} + 38q^{30} - 56q^{36} + 34q^{38} + 18q^{39} - 4q^{40} + 16q^{42} + 16q^{43} + 36q^{48} + 40q^{49} + 16q^{51} - 42q^{52} - 20q^{53} + 24q^{55} - 36q^{56} - 12q^{61} + 44q^{62} + 88q^{64} - 30q^{65} + 2q^{66} - 2q^{68} - 56q^{69} + 42q^{74} + 8q^{75} - 76q^{77} + 20q^{78} + 20q^{79} - 24q^{81} - 16q^{82} - 68q^{87} + 4q^{88} - 216q^{90} + 56q^{91} + 12q^{92} - 26q^{94} - 16q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 11 x^{14} + 85 x^{12} - 334 x^{10} + 952 x^{8} - 1050 x^{6} + 853 x^{4} - 93 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -24498 \nu^{14} + 246060 \nu^{12} - 1852321 \nu^{10} + 6411671 \nu^{8} - 17193085 \nu^{6} + 6321845 \nu^{4} - 690027 \nu^{2} - 40872159 \)\()/14163622\)
\(\beta_{3}\)\(=\)\((\)\( 172099 \nu^{14} - 2170865 \nu^{12} + 17340370 \nu^{10} - 78484018 \nu^{8} + 236538400 \nu^{6} - 377649654 \nu^{4} + 218482087 \nu^{2} - 23829231 \)\()/42490866\)
\(\beta_{4}\)\(=\)\((\)\( -99072 \nu^{14} + 1000291 \nu^{12} - 7490944 \nu^{10} + 25929344 \nu^{8} - 66564370 \nu^{6} + 25566080 \nu^{4} - 2790528 \nu^{2} - 20454191 \)\()/14163622\)
\(\beta_{5}\)\(=\)\((\)\( 24498 \nu^{15} - 246060 \nu^{13} + 1852321 \nu^{11} - 6411671 \nu^{9} + 17193085 \nu^{7} - 6321845 \nu^{5} + 690027 \nu^{3} + 55035781 \nu \)\()/14163622\)
\(\beta_{6}\)\(=\)\((\)\( 539569 \nu^{14} - 5861765 \nu^{12} + 45125185 \nu^{10} - 174659083 \nu^{8} + 494434675 \nu^{6} - 514968195 \nu^{4} + 441286822 \nu^{2} - 5618970 \)\()/42490866\)
\(\beta_{7}\)\(=\)\((\)\( 102312 \nu^{14} - 1048444 \nu^{12} + 7735924 \nu^{10} - 26777324 \nu^{8} + 67022271 \nu^{6} - 26402180 \nu^{4} + 2881788 \nu^{2} + 23341327 \)\()/7081811\)
\(\beta_{8}\)\(=\)\((\)\( -515071 \nu^{14} + 5615705 \nu^{12} - 43272864 \nu^{10} + 168247412 \nu^{8} - 477241590 \nu^{6} + 508646350 \nu^{4} - 426433173 \nu^{2} + 46491129 \)\()/14163622\)
\(\beta_{9}\)\(=\)\((\)\( -3598 \nu^{14} + 40712 \nu^{12} - 317920 \nu^{10} + 1287475 \nu^{8} - 3722089 \nu^{6} + 4460568 \nu^{4} - 3361729 \nu^{2} + 366555 \)\()/95271\)
\(\beta_{10}\)\(=\)\((\)\( -123570 \nu^{15} + 1246351 \nu^{13} - 9343265 \nu^{11} + 32341015 \nu^{9} - 83757455 \nu^{7} + 31887925 \nu^{5} - 3480555 \nu^{3} - 61326350 \nu \)\()/14163622\)
\(\beta_{11}\)\(=\)\((\)\( 539569 \nu^{15} - 5861765 \nu^{13} + 45125185 \nu^{11} - 174659083 \nu^{9} + 494434675 \nu^{7} - 514968195 \nu^{5} + 441286822 \nu^{3} - 48109836 \nu \)\()/42490866\)
\(\beta_{12}\)\(=\)\((\)\( 1079138 \nu^{15} - 11723530 \nu^{13} + 90250370 \nu^{11} - 349318166 \nu^{9} + 988869350 \nu^{7} - 1029936390 \nu^{5} + 861328211 \nu^{3} - 11237940 \nu \)\()/21245433\)
\(\beta_{13}\)\(=\)\((\)\( -205171 \nu^{15} + 2261795 \nu^{13} - 17480377 \nu^{11} + 68898667 \nu^{9} - 196608895 \nu^{7} + 219868809 \nu^{5} - 176278948 \nu^{3} + 19219314 \nu \)\()/3862806\)
\(\beta_{14}\)\(=\)\((\)\( 1137374 \nu^{15} - 12639109 \nu^{13} + 98087264 \nu^{11} - 390741062 \nu^{9} + 1125399698 \nu^{7} - 1313214930 \nu^{5} + 1094093084 \nu^{3} - 167644965 \nu \)\()/11588418\)
\(\beta_{15}\)\(=\)\((\)\(-17689120 \nu^{15} + 191553668 \nu^{13} - 1470474691 \nu^{11} + 5653350919 \nu^{9} - 15847248841 \nu^{7} + 15781577445 \nu^{5} - 12180871093 \nu^{3} - 359333319 \nu\)\()/ 127472598 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} + 3 \beta_{6} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(-\beta_{12} + 4 \beta_{11} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(5 \beta_{8} + 14 \beta_{6} + \beta_{3} - 14\)
\(\nu^{5}\)\(=\)\(-\beta_{13} - 6 \beta_{12} + 19 \beta_{11} + 6 \beta_{5}\)
\(\nu^{6}\)\(=\)\(\beta_{7} + 8 \beta_{4} - 24 \beta_{2} - 61\)
\(\nu^{7}\)\(=\)\(-\beta_{15} - \beta_{14} + 11 \beta_{10} + 32 \beta_{5} - 94 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-11 \beta_{9} - 115 \beta_{8} + 11 \beta_{7} - 345 \beta_{6} + 52 \beta_{4} - 52 \beta_{3} - 115 \beta_{2} + 52\)
\(\nu^{9}\)\(=\)\(-22 \beta_{15} + 11 \beta_{14} + 85 \beta_{13} + 145 \beta_{12} - 493 \beta_{11} + 85 \beta_{10} + 11 \beta_{5} - 482 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-85 \beta_{9} - 553 \beta_{8} - 1736 \beta_{6} - 315 \beta_{3} + 1736\)
\(\nu^{11}\)\(=\)\(-85 \beta_{15} + 170 \beta_{14} + 570 \beta_{13} + 698 \beta_{12} - 2544 \beta_{11} - 783 \beta_{5} - 85 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-570 \beta_{7} - 1838 \beta_{4} + 2672 \beta_{2} + 6935\)
\(\nu^{13}\)\(=\)\(570 \beta_{15} + 570 \beta_{14} - 3548 \beta_{10} - 4510 \beta_{5} + 12015 \beta_{1}\)
\(\nu^{14}\)\(=\)\(3548 \beta_{9} + 12977 \beta_{8} - 3548 \beta_{7} + 44495 \beta_{6} - 10466 \beta_{4} + 10466 \beta_{3} + 12977 \beta_{2} - 10466\)
\(\nu^{15}\)\(=\)\(7096 \beta_{15} - 3548 \beta_{14} - 21110 \beta_{13} - 16347 \beta_{12} + 68116 \beta_{11} - 21110 \beta_{10} - 3548 \beta_{5} + 64568 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(66\)
\(\chi(n)\) \(-1\) \(-\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
1.97871 + 1.14241i
1.84073 + 1.06275i
0.929293 + 0.536527i
0.287846 + 0.166188i
−0.287846 0.166188i
−0.929293 0.536527i
−1.84073 1.06275i
−1.97871 1.14241i
1.97871 1.14241i
1.84073 1.06275i
0.929293 0.536527i
0.287846 0.166188i
−0.287846 + 0.166188i
−0.929293 + 0.536527i
−1.84073 + 1.06275i
−1.97871 + 1.14241i
−1.97871 + 1.14241i −1.57521 + 2.72835i 1.61019 2.78892i −1.84030 + 1.06250i 7.19813i 2.62488 + 0.331665i 2.78832i −3.46258 5.99736i 2.42760 4.20473i
25.2 −1.84073 + 1.06275i 0.0894272 0.154892i 1.25885 2.18040i 3.12291 1.80301i 0.380153i −1.20931 + 2.35320i 1.10038i 1.48401 + 2.57037i −3.83229 + 6.63772i
25.3 −0.929293 + 0.536527i 1.21570 2.10566i −0.424277 + 0.734868i 0.541640 0.312716i 2.60903i 2.34996 1.21561i 3.05665i −1.45586 2.52163i −0.335561 + 0.581209i
25.4 −0.287846 + 0.166188i −0.729919 + 1.26426i −0.944763 + 1.63638i −1.25195 + 0.722811i 0.485214i −2.26391 1.36920i 1.29278i 0.434437 + 0.752468i 0.240245 0.416116i
25.5 0.287846 0.166188i −0.729919 + 1.26426i −0.944763 + 1.63638i 1.25195 0.722811i 0.485214i 2.26391 + 1.36920i 1.29278i 0.434437 + 0.752468i 0.240245 0.416116i
25.6 0.929293 0.536527i 1.21570 2.10566i −0.424277 + 0.734868i −0.541640 + 0.312716i 2.60903i −2.34996 + 1.21561i 3.05665i −1.45586 2.52163i −0.335561 + 0.581209i
25.7 1.84073 1.06275i 0.0894272 0.154892i 1.25885 2.18040i −3.12291 + 1.80301i 0.380153i 1.20931 2.35320i 1.10038i 1.48401 + 2.57037i −3.83229 + 6.63772i
25.8 1.97871 1.14241i −1.57521 + 2.72835i 1.61019 2.78892i 1.84030 1.06250i 7.19813i −2.62488 0.331665i 2.78832i −3.46258 5.99736i 2.42760 4.20473i
51.1 −1.97871 1.14241i −1.57521 2.72835i 1.61019 + 2.78892i −1.84030 1.06250i 7.19813i 2.62488 0.331665i 2.78832i −3.46258 + 5.99736i 2.42760 + 4.20473i
51.2 −1.84073 1.06275i 0.0894272 + 0.154892i 1.25885 + 2.18040i 3.12291 + 1.80301i 0.380153i −1.20931 2.35320i 1.10038i 1.48401 2.57037i −3.83229 6.63772i
51.3 −0.929293 0.536527i 1.21570 + 2.10566i −0.424277 0.734868i 0.541640 + 0.312716i 2.60903i 2.34996 + 1.21561i 3.05665i −1.45586 + 2.52163i −0.335561 0.581209i
51.4 −0.287846 0.166188i −0.729919 1.26426i −0.944763 1.63638i −1.25195 0.722811i 0.485214i −2.26391 + 1.36920i 1.29278i 0.434437 0.752468i 0.240245 + 0.416116i
51.5 0.287846 + 0.166188i −0.729919 1.26426i −0.944763 1.63638i 1.25195 + 0.722811i 0.485214i 2.26391 1.36920i 1.29278i 0.434437 0.752468i 0.240245 + 0.416116i
51.6 0.929293 + 0.536527i 1.21570 + 2.10566i −0.424277 0.734868i −0.541640 0.312716i 2.60903i −2.34996 1.21561i 3.05665i −1.45586 + 2.52163i −0.335561 0.581209i
51.7 1.84073 + 1.06275i 0.0894272 + 0.154892i 1.25885 + 2.18040i −3.12291 1.80301i 0.380153i 1.20931 + 2.35320i 1.10038i 1.48401 2.57037i −3.83229 6.63772i
51.8 1.97871 + 1.14241i −1.57521 2.72835i 1.61019 + 2.78892i 1.84030 + 1.06250i 7.19813i −2.62488 + 0.331665i 2.78832i −3.46258 + 5.99736i 2.42760 + 4.20473i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
13.b even 2 1 inner
91.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.r.a 16
3.b odd 2 1 819.2.dl.e 16
7.b odd 2 1 637.2.r.f 16
7.c even 3 1 inner 91.2.r.a 16
7.c even 3 1 637.2.c.f 8
7.d odd 6 1 637.2.c.e 8
7.d odd 6 1 637.2.r.f 16
13.b even 2 1 inner 91.2.r.a 16
13.d odd 4 2 1183.2.e.i 16
21.h odd 6 1 819.2.dl.e 16
39.d odd 2 1 819.2.dl.e 16
91.b odd 2 1 637.2.r.f 16
91.r even 6 1 inner 91.2.r.a 16
91.r even 6 1 637.2.c.f 8
91.s odd 6 1 637.2.c.e 8
91.s odd 6 1 637.2.r.f 16
91.z odd 12 2 1183.2.e.i 16
91.z odd 12 2 8281.2.a.ck 8
91.bb even 12 2 8281.2.a.cj 8
273.w odd 6 1 819.2.dl.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.r.a 16 1.a even 1 1 trivial
91.2.r.a 16 7.c even 3 1 inner
91.2.r.a 16 13.b even 2 1 inner
91.2.r.a 16 91.r even 6 1 inner
637.2.c.e 8 7.d odd 6 1
637.2.c.e 8 91.s odd 6 1
637.2.c.f 8 7.c even 3 1
637.2.c.f 8 91.r even 6 1
637.2.r.f 16 7.b odd 2 1
637.2.r.f 16 7.d odd 6 1
637.2.r.f 16 91.b odd 2 1
637.2.r.f 16 91.s odd 6 1
819.2.dl.e 16 3.b odd 2 1
819.2.dl.e 16 21.h odd 6 1
819.2.dl.e 16 39.d odd 2 1
819.2.dl.e 16 273.w odd 6 1
1183.2.e.i 16 13.d odd 4 2
1183.2.e.i 16 91.z odd 12 2
8281.2.a.cj 8 91.bb even 12 2
8281.2.a.ck 8 91.z odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(91, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 - 93 T^{2} + 853 T^{4} - 1050 T^{6} + 952 T^{8} - 334 T^{10} + 85 T^{12} - 11 T^{14} + T^{16} \)
$3$ \( ( 4 - 20 T + 114 T^{2} + 62 T^{3} + 67 T^{4} + 6 T^{5} + 11 T^{6} + 2 T^{7} + T^{8} )^{2} \)
$5$ \( 2304 - 7680 T^{2} + 20656 T^{4} - 14560 T^{6} + 7361 T^{8} - 1740 T^{10} + 297 T^{12} - 20 T^{14} + T^{16} \)
$7$ \( 5764801 - 2352980 T^{2} + 521017 T^{4} - 78988 T^{6} + 10652 T^{8} - 1612 T^{10} + 217 T^{12} - 20 T^{14} + T^{16} \)
$11$ \( 729 - 9180 T^{2} + 99508 T^{4} - 199832 T^{6} + 337509 T^{8} - 30312 T^{10} + 2108 T^{12} - 52 T^{14} + T^{16} \)
$13$ \( ( 28561 + 13182 T + 4732 T^{2} + 1690 T^{3} + 598 T^{4} + 130 T^{5} + 28 T^{6} + 6 T^{7} + T^{8} )^{2} \)
$17$ \( ( 15129 + 6396 T + 5164 T^{2} - 56 T^{3} + 485 T^{4} - 24 T^{5} + 36 T^{6} - 4 T^{7} + T^{8} )^{2} \)
$19$ \( 10673289 - 7618644 T^{2} + 3674044 T^{4} - 971784 T^{6} + 185725 T^{8} - 19096 T^{10} + 1396 T^{12} - 44 T^{14} + T^{16} \)
$23$ \( ( 36 + 60 T + 130 T^{2} + 22 T^{3} + 91 T^{4} + 50 T^{5} + 31 T^{6} + 6 T^{7} + T^{8} )^{2} \)
$29$ \( ( 624 - 208 T - 63 T^{2} + 4 T^{3} + T^{4} )^{4} \)
$31$ \( 1136229264 - 657036336 T^{2} + 309454636 T^{4} - 35364492 T^{6} + 2779213 T^{8} - 128296 T^{10} + 4309 T^{12} - 80 T^{14} + T^{16} \)
$37$ \( 76527504 - 433655856 T^{2} + 2419355628 T^{4} - 213389964 T^{6} + 12939021 T^{8} - 422496 T^{10} + 10053 T^{12} - 120 T^{14} + T^{16} \)
$41$ \( ( 292032 + 88192 T^{2} + 5732 T^{4} + 132 T^{6} + T^{8} )^{2} \)
$43$ \( ( -104 - 156 T - 66 T^{2} - 4 T^{3} + T^{4} )^{4} \)
$47$ \( 57728231289 - 58599199164 T^{2} + 56549167060 T^{4} - 2884224440 T^{6} + 101089845 T^{8} - 1905768 T^{10} + 26204 T^{12} - 196 T^{14} + T^{16} \)
$53$ \( ( 7569 - 11310 T + 16900 T^{2} - 1740 T^{3} + 1213 T^{4} + 260 T^{5} + 100 T^{6} + 10 T^{7} + T^{8} )^{2} \)
$59$ \( 12487392009 - 18073959780 T^{2} + 24989166028 T^{4} - 1652371368 T^{6} + 79227709 T^{8} - 1646008 T^{10} + 24868 T^{12} - 188 T^{14} + T^{16} \)
$61$ \( ( 49729 - 20962 T + 14188 T^{2} - 420 T^{3} + 917 T^{4} + 44 T^{5} + 60 T^{6} + 6 T^{7} + T^{8} )^{2} \)
$67$ \( 66330457209 - 73439011956 T^{2} + 75732974260 T^{4} - 6027737800 T^{6} + 387569525 T^{8} - 5578872 T^{10} + 59004 T^{12} - 284 T^{14} + T^{16} \)
$71$ \( ( 397488 + 253084 T^{2} + 19829 T^{4} + 292 T^{6} + T^{8} )^{2} \)
$73$ \( 8437677133824 - 1371747640320 T^{2} + 164987876800 T^{4} - 7922514640 T^{6} + 273313457 T^{8} - 4249020 T^{10} + 47625 T^{12} - 260 T^{14} + T^{16} \)
$79$ \( ( 64 - 480 T + 3672 T^{2} + 380 T^{3} + 689 T^{4} - 210 T^{5} + 91 T^{6} - 10 T^{7} + T^{8} )^{2} \)
$83$ \( ( 5483712 + 959920 T^{2} + 28692 T^{4} + 296 T^{6} + T^{8} )^{2} \)
$89$ \( 58102628210064 - 22401057000432 T^{2} + 8157120819724 T^{4} - 178140025756 T^{6} + 2655587933 T^{8} - 21797952 T^{10} + 130701 T^{12} - 440 T^{14} + T^{16} \)
$97$ \( ( 192 + 4816 T^{2} + 2740 T^{4} + 104 T^{6} + T^{8} )^{2} \)
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