Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.726638658394\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{4} + ( 2 - 3 \zeta_{6} ) q^{7} -3 q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{4} + ( 2 - 3 \zeta_{6} ) q^{7} -3 q^{8} + 3 \zeta_{6} q^{9} + ( 3 - 3 \zeta_{6} ) q^{11} - q^{13} + ( -3 + \zeta_{6} ) q^{14} + \zeta_{6} q^{16} + ( -7 + 7 \zeta_{6} ) q^{17} + ( 3 - 3 \zeta_{6} ) q^{18} + 7 \zeta_{6} q^{19} -3 q^{22} + 6 \zeta_{6} q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} + \zeta_{6} q^{26} + ( -1 - 2 \zeta_{6} ) q^{28} -5 q^{29} + ( -5 + 5 \zeta_{6} ) q^{32} + 7 q^{34} + 3 q^{36} -8 \zeta_{6} q^{37} + ( 7 - 7 \zeta_{6} ) q^{38} + 2 q^{43} -3 \zeta_{6} q^{44} + ( 6 - 6 \zeta_{6} ) q^{46} -7 \zeta_{6} q^{47} + ( -5 - 3 \zeta_{6} ) q^{49} -5 q^{50} + ( -1 + \zeta_{6} ) q^{52} + ( 3 - 3 \zeta_{6} ) q^{53} + ( -6 + 9 \zeta_{6} ) q^{56} + 5 \zeta_{6} q^{58} + ( 7 - 7 \zeta_{6} ) q^{59} + 7 \zeta_{6} q^{61} + ( 9 - 3 \zeta_{6} ) q^{63} + 7 q^{64} + ( 3 - 3 \zeta_{6} ) q^{67} + 7 \zeta_{6} q^{68} -5 q^{71} -9 \zeta_{6} q^{72} + ( -14 + 14 \zeta_{6} ) q^{73} + ( -8 + 8 \zeta_{6} ) q^{74} + 7 q^{76} + ( -3 - 6 \zeta_{6} ) q^{77} + 6 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} -2 \zeta_{6} q^{86} + ( -9 + 9 \zeta_{6} ) q^{88} + ( -2 + 3 \zeta_{6} ) q^{91} + 6 q^{92} + ( -7 + 7 \zeta_{6} ) q^{94} -14 q^{97} + ( -3 + 8 \zeta_{6} ) q^{98} + 9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + q^{4} + q^{7} - 6q^{8} + 3q^{9} + O(q^{10}) \) \( 2q - q^{2} + q^{4} + q^{7} - 6q^{8} + 3q^{9} + 3q^{11} - 2q^{13} - 5q^{14} + q^{16} - 7q^{17} + 3q^{18} + 7q^{19} - 6q^{22} + 6q^{23} + 5q^{25} + q^{26} - 4q^{28} - 10q^{29} - 5q^{32} + 14q^{34} + 6q^{36} - 8q^{37} + 7q^{38} + 4q^{43} - 3q^{44} + 6q^{46} - 7q^{47} - 13q^{49} - 10q^{50} - q^{52} + 3q^{53} - 3q^{56} + 5q^{58} + 7q^{59} + 7q^{61} + 15q^{63} + 14q^{64} + 3q^{67} + 7q^{68} - 10q^{71} - 9q^{72} - 14q^{73} - 8q^{74} + 14q^{76} - 12q^{77} + 6q^{79} - 9q^{81} - 2q^{86} - 9q^{88} - q^{91} + 12q^{92} - 7q^{94} - 28q^{97} + 2q^{98} + 18q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 0 0.500000 0.866025i 0 0 0.500000 2.59808i −3.00000 1.50000 + 2.59808i 0
79.1 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 0 0.500000 + 2.59808i −3.00000 1.50000 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.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.e.a 2
3.b odd 2 1 819.2.j.b 2
4.b odd 2 1 1456.2.r.g 2
7.b odd 2 1 637.2.e.a 2
7.c even 3 1 inner 91.2.e.a 2
7.c even 3 1 637.2.a.c 1
7.d odd 6 1 637.2.a.d 1
7.d odd 6 1 637.2.e.a 2
13.b even 2 1 1183.2.e.b 2
21.g even 6 1 5733.2.a.d 1
21.h odd 6 1 819.2.j.b 2
21.h odd 6 1 5733.2.a.c 1
28.g odd 6 1 1456.2.r.g 2
91.r even 6 1 1183.2.e.b 2
91.r even 6 1 8281.2.a.f 1
91.s odd 6 1 8281.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.a 2 1.a even 1 1 trivial
91.2.e.a 2 7.c even 3 1 inner
637.2.a.c 1 7.c even 3 1
637.2.a.d 1 7.d odd 6 1
637.2.e.a 2 7.b odd 2 1
637.2.e.a 2 7.d odd 6 1
819.2.j.b 2 3.b odd 2 1
819.2.j.b 2 21.h odd 6 1
1183.2.e.b 2 13.b even 2 1
1183.2.e.b 2 91.r even 6 1
1456.2.r.g 2 4.b odd 2 1
1456.2.r.g 2 28.g odd 6 1
5733.2.a.c 1 21.h odd 6 1
5733.2.a.d 1 21.g even 6 1
8281.2.a.e 1 91.s odd 6 1
8281.2.a.f 1 91.r even 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 - T + T^{2} \)
$11$ \( 9 - 3 T + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( 49 + 7 T + T^{2} \)
$19$ \( 49 - 7 T + T^{2} \)
$23$ \( 36 - 6 T + T^{2} \)
$29$ \( ( 5 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 64 + 8 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -2 + T )^{2} \)
$47$ \( 49 + 7 T + T^{2} \)
$53$ \( 9 - 3 T + T^{2} \)
$59$ \( 49 - 7 T + T^{2} \)
$61$ \( 49 - 7 T + T^{2} \)
$67$ \( 9 - 3 T + T^{2} \)
$71$ \( ( 5 + T )^{2} \)
$73$ \( 196 + 14 T + T^{2} \)
$79$ \( 36 - 6 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( 14 + T )^{2} \)
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