Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.726638658394\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \(x^{4} - x^{3} + 2 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 2 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 1 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu^{2} - 2 \nu - 1 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(2 \beta_{2} - 1\)

Character values

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

\(n\) \(15\) \(66\)
\(\chi(n)\) \(\beta_{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
0.809017 1.40126i
−0.309017 + 0.535233i
0.809017 + 1.40126i
−0.309017 0.535233i
−1.30902 + 2.26728i 1.30902 2.26728i −2.42705 4.20378i 2.61803 3.42705 + 5.93583i −0.500000 0.866025i 7.47214 −1.92705 3.33775i −3.42705 + 5.93583i
22.2 −0.190983 + 0.330792i 0.190983 0.330792i 0.927051 + 1.60570i 0.381966 0.0729490 + 0.126351i −0.500000 0.866025i −1.47214 1.42705 + 2.47172i −0.0729490 + 0.126351i
29.1 −1.30902 2.26728i 1.30902 + 2.26728i −2.42705 + 4.20378i 2.61803 3.42705 5.93583i −0.500000 + 0.866025i 7.47214 −1.92705 + 3.33775i −3.42705 5.93583i
29.2 −0.190983 0.330792i 0.190983 + 0.330792i 0.927051 1.60570i 0.381966 0.0729490 0.126351i −0.500000 + 0.866025i −1.47214 1.42705 2.47172i −0.0729490 0.126351i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.f.a 4
3.b odd 2 1 819.2.o.c 4
4.b odd 2 1 1456.2.s.h 4
7.b odd 2 1 637.2.f.c 4
7.c even 3 1 637.2.g.b 4
7.c even 3 1 637.2.h.g 4
7.d odd 6 1 637.2.g.c 4
7.d odd 6 1 637.2.h.f 4
13.c even 3 1 inner 91.2.f.a 4
13.c even 3 1 1183.2.a.g 2
13.e even 6 1 1183.2.a.c 2
13.f odd 12 2 1183.2.c.c 4
39.i odd 6 1 819.2.o.c 4
52.j odd 6 1 1456.2.s.h 4
91.g even 3 1 637.2.h.g 4
91.h even 3 1 637.2.g.b 4
91.m odd 6 1 637.2.h.f 4
91.n odd 6 1 637.2.f.c 4
91.n odd 6 1 8281.2.a.bb 2
91.t odd 6 1 8281.2.a.n 2
91.v odd 6 1 637.2.g.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.a 4 1.a even 1 1 trivial
91.2.f.a 4 13.c even 3 1 inner
637.2.f.c 4 7.b odd 2 1
637.2.f.c 4 91.n odd 6 1
637.2.g.b 4 7.c even 3 1
637.2.g.b 4 91.h even 3 1
637.2.g.c 4 7.d odd 6 1
637.2.g.c 4 91.v odd 6 1
637.2.h.f 4 7.d odd 6 1
637.2.h.f 4 91.m odd 6 1
637.2.h.g 4 7.c even 3 1
637.2.h.g 4 91.g even 3 1
819.2.o.c 4 3.b odd 2 1
819.2.o.c 4 39.i odd 6 1
1183.2.a.c 2 13.e even 6 1
1183.2.a.g 2 13.c even 3 1
1183.2.c.c 4 13.f odd 12 2
1456.2.s.h 4 4.b odd 2 1
1456.2.s.h 4 52.j odd 6 1
8281.2.a.n 2 91.t odd 6 1
8281.2.a.bb 2 91.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3 T_{2}^{3} + 8 T_{2}^{2} + 3 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(91, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 8 T^{2} + 3 T^{3} + T^{4} \)
$3$ \( 1 - 3 T + 8 T^{2} - 3 T^{3} + T^{4} \)
$5$ \( ( 1 - 3 T + T^{2} )^{2} \)
$7$ \( ( 1 + T + T^{2} )^{2} \)
$11$ \( 81 + 27 T + 18 T^{2} - 3 T^{3} + T^{4} \)
$13$ \( ( 13 + 5 T + T^{2} )^{2} \)
$17$ \( 121 - 66 T + 47 T^{2} + 6 T^{3} + T^{4} \)
$19$ \( 81 - 27 T + 18 T^{2} + 3 T^{3} + T^{4} \)
$23$ \( 400 + 20 T^{2} + T^{4} \)
$29$ \( 841 - 87 T + 38 T^{2} + 3 T^{3} + T^{4} \)
$31$ \( ( -41 - 4 T + T^{2} )^{2} \)
$37$ \( ( 16 + 4 T + T^{2} )^{2} \)
$41$ \( 16 + 24 T + 32 T^{2} + 6 T^{3} + T^{4} \)
$43$ \( 9025 - 475 T + 120 T^{2} + 5 T^{3} + T^{4} \)
$47$ \( ( -5 + T^{2} )^{2} \)
$53$ \( ( 31 - 12 T + T^{2} )^{2} \)
$59$ \( 25 + 5 T^{2} + T^{4} \)
$61$ \( ( 36 - 6 T + T^{2} )^{2} \)
$67$ \( 81 + 108 T + 153 T^{2} - 12 T^{3} + T^{4} \)
$71$ \( 13456 + 696 T + 152 T^{2} - 6 T^{3} + T^{4} \)
$73$ \( ( 2 + T )^{4} \)
$79$ \( ( -4 + T )^{4} \)
$83$ \( ( -45 + T^{2} )^{2} \)
$89$ \( 6241 + 1659 T + 362 T^{2} + 21 T^{3} + T^{4} \)
$97$ \( 52441 + 7099 T + 732 T^{2} + 31 T^{3} + T^{4} \)
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