Properties

Label 91.2.f.b
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: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(15\) \(66\)
\(\chi(n)\) \(-1 + \zeta_{12}\) \(1\)

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
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 1.50000i −1.36603 + 2.36603i −0.500000 0.866025i 1.73205 −2.36603 4.09808i −0.500000 0.866025i −1.73205 −2.23205 3.86603i −1.50000 + 2.59808i
22.2 0.866025 1.50000i 0.366025 0.633975i −0.500000 0.866025i −1.73205 −0.633975 1.09808i −0.500000 0.866025i 1.73205 1.23205 + 2.13397i −1.50000 + 2.59808i
29.1 −0.866025 1.50000i −1.36603 2.36603i −0.500000 + 0.866025i 1.73205 −2.36603 + 4.09808i −0.500000 + 0.866025i −1.73205 −2.23205 + 3.86603i −1.50000 2.59808i
29.2 0.866025 + 1.50000i 0.366025 + 0.633975i −0.500000 + 0.866025i −1.73205 −0.633975 + 1.09808i −0.500000 + 0.866025i 1.73205 1.23205 2.13397i −1.50000 2.59808i
\(n\): 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.b 4
3.b odd 2 1 819.2.o.b 4
4.b odd 2 1 1456.2.s.o 4
7.b odd 2 1 637.2.f.d 4
7.c even 3 1 637.2.g.e 4
7.c even 3 1 637.2.h.d 4
7.d odd 6 1 637.2.g.d 4
7.d odd 6 1 637.2.h.e 4
13.c even 3 1 inner 91.2.f.b 4
13.c even 3 1 1183.2.a.f 2
13.e even 6 1 1183.2.a.e 2
13.f odd 12 2 1183.2.c.e 4
39.i odd 6 1 819.2.o.b 4
52.j odd 6 1 1456.2.s.o 4
91.g even 3 1 637.2.h.d 4
91.h even 3 1 637.2.g.e 4
91.m odd 6 1 637.2.h.e 4
91.n odd 6 1 637.2.f.d 4
91.n odd 6 1 8281.2.a.r 2
91.t odd 6 1 8281.2.a.t 2
91.v odd 6 1 637.2.g.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.b 4 1.a even 1 1 trivial
91.2.f.b 4 13.c even 3 1 inner
637.2.f.d 4 7.b odd 2 1
637.2.f.d 4 91.n odd 6 1
637.2.g.d 4 7.d odd 6 1
637.2.g.d 4 91.v odd 6 1
637.2.g.e 4 7.c even 3 1
637.2.g.e 4 91.h even 3 1
637.2.h.d 4 7.c even 3 1
637.2.h.d 4 91.g even 3 1
637.2.h.e 4 7.d odd 6 1
637.2.h.e 4 91.m odd 6 1
819.2.o.b 4 3.b odd 2 1
819.2.o.b 4 39.i odd 6 1
1183.2.a.e 2 13.e even 6 1
1183.2.a.f 2 13.c even 3 1
1183.2.c.e 4 13.f odd 12 2
1456.2.s.o 4 4.b odd 2 1
1456.2.s.o 4 52.j odd 6 1
8281.2.a.r 2 91.n odd 6 1
8281.2.a.t 2 91.t odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3 T_{2}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(91, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + 3 T^{2} + T^{4} \)
$3$ \( 4 - 4 T + 6 T^{2} + 2 T^{3} + T^{4} \)
$5$ \( ( -3 + T^{2} )^{2} \)
$7$ \( ( 1 + T + T^{2} )^{2} \)
$11$ \( 36 + 36 T + 30 T^{2} + 6 T^{3} + T^{4} \)
$13$ \( 169 - 52 T + 3 T^{2} - 4 T^{3} + T^{4} \)
$17$ \( 1089 - 396 T + 111 T^{2} - 12 T^{3} + T^{4} \)
$19$ \( ( 4 + 2 T + T^{2} )^{2} \)
$23$ \( 36 + 36 T + 30 T^{2} + 6 T^{3} + T^{4} \)
$29$ \( ( 9 - 3 T + T^{2} )^{2} \)
$31$ \( ( -26 + 2 T + T^{2} )^{2} \)
$37$ \( ( 49 - 7 T + T^{2} )^{2} \)
$41$ \( 729 + 27 T^{2} + T^{4} \)
$43$ \( 4 - 20 T + 102 T^{2} + 10 T^{3} + T^{4} \)
$47$ \( ( -12 - 12 T + T^{2} )^{2} \)
$53$ \( ( -39 + 6 T + T^{2} )^{2} \)
$59$ \( 6084 + 1404 T + 246 T^{2} + 18 T^{3} + T^{4} \)
$61$ \( 5329 - 1460 T + 327 T^{2} - 20 T^{3} + T^{4} \)
$67$ \( 676 + 52 T + 30 T^{2} - 2 T^{3} + T^{4} \)
$71$ \( ( 36 + 6 T + T^{2} )^{2} \)
$73$ \( ( -23 - 4 T + T^{2} )^{2} \)
$79$ \( ( 94 - 22 T + T^{2} )^{2} \)
$83$ \( ( -18 - 6 T + T^{2} )^{2} \)
$89$ \( 144 - 144 T + 156 T^{2} + 12 T^{3} + T^{4} \)
$97$ \( 8464 + 736 T + 156 T^{2} - 8 T^{3} + T^{4} \)
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