Properties

Label 91.2.e.b
Level $91$
Weight $2$
Character orbit 91.e
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.e (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: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1 + 1) q^{2} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{3} + (3 \beta_{2} + 3 \beta_1) q^{4} + (\beta_{3} - 2 \beta_1 + 1) q^{5} + ( - 3 \beta_{2} + 1) q^{6} + (2 \beta_{3} - 1) q^{7} + (4 \beta_{2} - 1) q^{8} + ( - 2 \beta_{3} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1 + 1) q^{2} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{3} + (3 \beta_{2} + 3 \beta_1) q^{4} + (\beta_{3} - 2 \beta_1 + 1) q^{5} + ( - 3 \beta_{2} + 1) q^{6} + (2 \beta_{3} - 1) q^{7} + (4 \beta_{2} - 1) q^{8} + ( - 2 \beta_{3} - 2) q^{9} + ( - \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{10} - 3 \beta_{3} q^{11} + (6 \beta_{3} + 3 \beta_1 + 6) q^{12} - q^{13} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 3) q^{14} - 5 q^{15} + ( - 5 \beta_{3} - 3 \beta_1 - 5) q^{16} + ( - 5 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{17} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{18} + ( - 3 \beta_{3} - 3) q^{19} + ( - 3 \beta_{2} + 6) q^{20} + ( - 3 \beta_{3} + 2 \beta_{2} + 6 \beta_1 - 2) q^{21} + ( - 3 \beta_{2} + 3) q^{22} + (5 \beta_{3} + 2 \beta_1 + 5) q^{23} + (7 \beta_{3} + 6 \beta_{2} + 6 \beta_1) q^{24} + ( - \beta_{3} - \beta_1 - 1) q^{26} + ( - 2 \beta_{2} - 1) q^{27} + ( - 3 \beta_{2} - 9 \beta_1) q^{28} + ( - 4 \beta_{2} - 2) q^{29} + ( - 5 \beta_{3} - 5 \beta_1 - 5) q^{30} + 5 \beta_{3} q^{31} + ( - 6 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{32} + (3 \beta_{3} - 6 \beta_1 + 3) q^{33} + (3 \beta_{2} + 1) q^{34} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 3) q^{35} - 6 \beta_{2} q^{36} + (5 \beta_{3} - 6 \beta_1 + 5) q^{37} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{38} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{39} + (7 \beta_{3} + 6 \beta_1 + 7) q^{40} + (4 \beta_{2} + 2) q^{41} + (2 \beta_{3} + 9 \beta_{2} + 6 \beta_1 - 1) q^{42} - 8 q^{43} + 9 \beta_1 q^{44} + ( - 2 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{45} + (7 \beta_{3} + 9 \beta_{2} + 9 \beta_1) q^{46} + (\beta_{3} + 4 \beta_1 + 1) q^{47} + (13 \beta_{2} - 1) q^{48} + ( - 8 \beta_{3} - 3) q^{49} + (13 \beta_{3} - 6 \beta_1 + 13) q^{51} + ( - 3 \beta_{2} - 3 \beta_1) q^{52} + ( - \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{53} + (\beta_{3} + 3 \beta_1 + 1) q^{54} + (6 \beta_{2} + 3) q^{55} + ( - 2 \beta_{3} - 12 \beta_{2} - 8 \beta_1 + 1) q^{56} + (6 \beta_{2} + 3) q^{57} + (2 \beta_{3} + 6 \beta_1 + 2) q^{58} + (5 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{59} + ( - 15 \beta_{2} - 15 \beta_1) q^{60} + ( - 3 \beta_{3} - 3) q^{61} + (5 \beta_{2} - 5) q^{62} + (2 \beta_{3} + 6) q^{63} + ( - 6 \beta_{2} - 1) q^{64} + ( - \beta_{3} + 2 \beta_1 - 1) q^{65} + ( - 3 \beta_{3} - 9 \beta_{2} - 9 \beta_1) q^{66} - 3 \beta_{3} q^{67} + ( - 12 \beta_{3} + 3 \beta_1 - 12) q^{68} + ( - 12 \beta_{2} - 1) q^{69} + (3 \beta_{3} + 3 \beta_{2} + 9 \beta_1 + 2) q^{70} + (8 \beta_{2} + 4) q^{71} + (2 \beta_{3} + 8 \beta_1 + 2) q^{72} + (7 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{73} + ( - \beta_{3} - 7 \beta_{2} - 7 \beta_1) q^{74} - 9 \beta_{2} q^{76} + (9 \beta_{3} + 6) q^{77} + (3 \beta_{2} - 1) q^{78} + ( - 7 \beta_{3} + 6 \beta_1 - 7) q^{79} + (\beta_{3} + 13 \beta_{2} + 13 \beta_1) q^{80} - 11 \beta_{3} q^{81} + ( - 2 \beta_{3} - 6 \beta_1 - 2) q^{82} + ( - 6 \beta_{3} + 6 \beta_{2} - 3 \beta_1 - 18) q^{84} + (6 \beta_{2} + 13) q^{85} + ( - 8 \beta_{3} - 8 \beta_1 - 8) q^{86} - 10 \beta_{3} q^{87} + (3 \beta_{3} + 12 \beta_{2} + 12 \beta_1) q^{88} + (\beta_{3} - 2 \beta_1 + 1) q^{89} + (6 \beta_{2} - 2) q^{90} + ( - 2 \beta_{3} + 1) q^{91} + (21 \beta_{2} - 6) q^{92} + ( - 5 \beta_{3} + 10 \beta_1 - 5) q^{93} + (5 \beta_{3} + 9 \beta_{2} + 9 \beta_1) q^{94} + ( - 3 \beta_{3} + 6 \beta_{2} + 6 \beta_1) q^{95} - 15 \beta_1 q^{96} + ( - 12 \beta_{2} - 10) q^{97} + ( - 3 \beta_{3} - 8 \beta_{2} - 3 \beta_1 + 5) q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} - 3 q^{4} + 10 q^{6} - 8 q^{7} - 12 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} - 3 q^{4} + 10 q^{6} - 8 q^{7} - 12 q^{8} - 4 q^{9} + 5 q^{10} + 6 q^{11} + 15 q^{12} - 4 q^{13} - 15 q^{14} - 20 q^{15} - 13 q^{16} + 6 q^{17} + 6 q^{18} - 6 q^{19} + 30 q^{20} + 18 q^{22} + 12 q^{23} - 20 q^{24} - 3 q^{26} - 3 q^{28} - 15 q^{30} - 10 q^{31} + 15 q^{32} - 2 q^{34} + 12 q^{36} + 4 q^{37} + 9 q^{38} + 20 q^{40} - 20 q^{42} - 32 q^{43} + 9 q^{44} - 23 q^{46} + 6 q^{47} - 30 q^{48} + 4 q^{49} + 20 q^{51} + 3 q^{52} + 6 q^{53} + 5 q^{54} + 24 q^{56} + 10 q^{58} - 6 q^{59} + 15 q^{60} - 6 q^{61} - 30 q^{62} + 20 q^{63} + 8 q^{64} + 15 q^{66} + 6 q^{67} - 21 q^{68} + 20 q^{69} + 5 q^{70} + 12 q^{72} - 8 q^{73} + 9 q^{74} + 18 q^{76} + 6 q^{77} - 10 q^{78} - 8 q^{79} - 15 q^{80} + 22 q^{81} - 10 q^{82} - 75 q^{84} + 40 q^{85} - 24 q^{86} + 20 q^{87} - 18 q^{88} - 20 q^{90} + 8 q^{91} - 66 q^{92} - 19 q^{94} - 15 q^{96} - 16 q^{97} + 39 q^{98} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 1 \) Copy content Toggle raw display

Character values

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

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

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
−0.309017 0.535233i
0.809017 + 1.40126i
−0.309017 + 0.535233i
0.809017 1.40126i
0.190983 + 0.330792i −1.11803 + 1.93649i 0.927051 1.60570i 1.11803 + 1.93649i −0.854102 −2.00000 + 1.73205i 1.47214 −1.00000 1.73205i −0.427051 + 0.739674i
53.2 1.30902 + 2.26728i 1.11803 1.93649i −2.42705 + 4.20378i −1.11803 1.93649i 5.85410 −2.00000 + 1.73205i −7.47214 −1.00000 1.73205i 2.92705 5.06980i
79.1 0.190983 0.330792i −1.11803 1.93649i 0.927051 + 1.60570i 1.11803 1.93649i −0.854102 −2.00000 1.73205i 1.47214 −1.00000 + 1.73205i −0.427051 0.739674i
79.2 1.30902 2.26728i 1.11803 + 1.93649i −2.42705 4.20378i −1.11803 + 1.93649i 5.85410 −2.00000 1.73205i −7.47214 −1.00000 + 1.73205i 2.92705 + 5.06980i
\(n\): 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.b 4
3.b odd 2 1 819.2.j.c 4
4.b odd 2 1 1456.2.r.j 4
7.b odd 2 1 637.2.e.h 4
7.c even 3 1 inner 91.2.e.b 4
7.c even 3 1 637.2.a.f 2
7.d odd 6 1 637.2.a.e 2
7.d odd 6 1 637.2.e.h 4
13.b even 2 1 1183.2.e.d 4
21.g even 6 1 5733.2.a.w 2
21.h odd 6 1 819.2.j.c 4
21.h odd 6 1 5733.2.a.v 2
28.g odd 6 1 1456.2.r.j 4
91.r even 6 1 1183.2.e.d 4
91.r even 6 1 8281.2.a.z 2
91.s odd 6 1 8281.2.a.ba 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.b 4 1.a even 1 1 trivial
91.2.e.b 4 7.c even 3 1 inner
637.2.a.e 2 7.d odd 6 1
637.2.a.f 2 7.c even 3 1
637.2.e.h 4 7.b odd 2 1
637.2.e.h 4 7.d odd 6 1
819.2.j.c 4 3.b odd 2 1
819.2.j.c 4 21.h odd 6 1
1183.2.e.d 4 13.b even 2 1
1183.2.e.d 4 91.r even 6 1
1456.2.r.j 4 4.b odd 2 1
1456.2.r.j 4 28.g odd 6 1
5733.2.a.v 2 21.h odd 6 1
5733.2.a.w 2 21.g even 6 1
8281.2.a.z 2 91.r even 6 1
8281.2.a.ba 2 91.s odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 3 T^{3} + 8 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 5T^{2} + 25 \) Copy content Toggle raw display
$5$ \( T^{4} + 5T^{2} + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$13$ \( (T + 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + 47 T^{2} + 66 T + 121 \) Copy content Toggle raw display
$19$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 12 T^{3} + 113 T^{2} + \cdots + 961 \) Copy content Toggle raw display
$29$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 4 T^{3} + 57 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$41$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$43$ \( (T + 8)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 6 T^{3} + 47 T^{2} + 66 T + 121 \) Copy content Toggle raw display
$53$ \( T^{4} - 6 T^{3} + 47 T^{2} + 66 T + 121 \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} + 47 T^{2} - 66 T + 121 \) Copy content Toggle raw display
$61$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 8 T^{3} + 93 T^{2} - 232 T + 841 \) Copy content Toggle raw display
$79$ \( T^{4} + 8 T^{3} + 93 T^{2} - 232 T + 841 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} + 5T^{2} + 25 \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T - 164)^{2} \) Copy content Toggle raw display
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