Properties

Label 91.2.c.a
Level $91$
Weight $2$
Character orbit 91.c
Analytic conductor $0.727$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.726638658394\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_{3}) q^{2} + \beta_1 q^{3} + ( - \beta_{2} - \beta_1 - 1) q^{4} + ( - \beta_{5} - \beta_{3}) q^{5} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{6} - \beta_{3} q^{7} + ( - 2 \beta_{5} + 2 \beta_{3}) q^{8} + (\beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_{3}) q^{2} + \beta_1 q^{3} + ( - \beta_{2} - \beta_1 - 1) q^{4} + ( - \beta_{5} - \beta_{3}) q^{5} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{6} - \beta_{3} q^{7} + ( - 2 \beta_{5} + 2 \beta_{3}) q^{8} + (\beta_{2} - \beta_1) q^{9} + \beta_1 q^{10} + ( - \beta_{4} + 3 \beta_{3}) q^{11} + ( - 2 \beta_{2} - 4) q^{12} + ( - \beta_{4} + 2 \beta_{2} - \beta_1 + 2) q^{13} + ( - \beta_{2} - 1) q^{14} + (2 \beta_{5} - \beta_{4} + 3 \beta_{3}) q^{15} + (2 \beta_{2} + 2) q^{16} + ( - 2 \beta_{2} + \beta_1 - 2) q^{17} + ( - \beta_{4} - \beta_{3}) q^{18} + (4 \beta_{5} - \beta_{4}) q^{19} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3}) q^{20} + \beta_{5} q^{21} + (3 \beta_{2} + \beta_1 + 5) q^{22} + (3 \beta_{2} - \beta_1 + 2) q^{23} + ( - 2 \beta_{4} + 6 \beta_{3}) q^{24} + ( - \beta_{2} + 3 \beta_1 + 1) q^{25} + (\beta_{5} + \beta_{4} - 5 \beta_{3} + \beta_1 + 2) q^{26} + ( - 2 \beta_1 - 2) q^{27} + ( - \beta_{5} - \beta_{4} + \beta_{3}) q^{28} + ( - 2 \beta_{2} - 2 \beta_1 - 3) q^{29} + (\beta_{2} - \beta_1 + 3) q^{30} + ( - \beta_{5} - 2 \beta_{4} - 3 \beta_{3}) q^{31} + ( - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{32} + ( - 3 \beta_{5} - \beta_{4} + \beta_{3}) q^{33} + ( - \beta_{5} - \beta_{4} + 5 \beta_{3}) q^{34} + (\beta_1 - 1) q^{35} + (\beta_{2} - \beta_1 + 1) q^{36} + (3 \beta_{4} + 3 \beta_{3}) q^{37} + ( - 4 \beta_{2} - 3 \beta_1 - 2) q^{38} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1 - 1) q^{39} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{40} + (4 \beta_{4} - 2 \beta_{3}) q^{41} + ( - \beta_{2} - \beta_1 - 1) q^{42} + ( - 2 \beta_{2} + 2 \beta_1 - 5) q^{43} + (4 \beta_{5} + 4 \beta_{4} - 6 \beta_{3}) q^{44} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_{3}) q^{45} + (2 \beta_{5} + \beta_{4} - 7 \beta_{3}) q^{46} + ( - 2 \beta_{5} + \beta_{4} + 6 \beta_{3}) q^{47} + (2 \beta_{2} + 2 \beta_1 + 2) q^{48} - q^{49} + (2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3}) q^{50} + ( - \beta_{2} - 3 \beta_1 + 1) q^{51} + (\beta_{5} + \beta_{4} - 3 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 4) q^{52} + ( - \beta_{2} - \beta_1) q^{53} + ( - 2 \beta_{5} - 4 \beta_{4} + 4 \beta_{3}) q^{54} + ( - 3 \beta_1 + 2) q^{55} + (2 \beta_1 + 2) q^{56} + ( - 4 \beta_{5} + 3 \beta_{4} - 11 \beta_{3}) q^{57} + ( - 4 \beta_{5} - 5 \beta_{4} + 9 \beta_{3}) q^{58} + (3 \beta_{5} - \beta_{4} + \beta_{3}) q^{59} + (4 \beta_{5} + 2 \beta_{3}) q^{60} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{61} + ( - 2 \beta_{2} + 3 \beta_1 + 2) q^{62} + ( - \beta_{5} + \beta_{4}) q^{63} + 4 \beta_{2} q^{64} + ( - 4 \beta_{5} + \beta_{4} - 3 \beta_{3} - 1) q^{65} + (4 \beta_{2} + 4 \beta_1 + 6) q^{66} + ( - 2 \beta_{5} + 4 \beta_{3}) q^{67} + (2 \beta_{2} + 4 \beta_1 + 4) q^{68} + (2 \beta_{2} + 3 \beta_1) q^{69} + \beta_{5} q^{70} + ( - 2 \beta_{5} - 5 \beta_{4} - 3 \beta_{3}) q^{71} + ( - 2 \beta_{4} - 4 \beta_{3}) q^{72} + (4 \beta_{5} + 3 \beta_{4} - 4 \beta_{3}) q^{73} + (3 \beta_{2} - 3 \beta_1 - 3) q^{74} + (2 \beta_{2} - 2 \beta_1 + 8) q^{75} + (\beta_{5} - 7 \beta_{4} + 13 \beta_{3}) q^{76} + (\beta_{2} + 3) q^{77} + (4 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{78} + (2 \beta_{2} + 4 \beta_1 + 5) q^{79} - 2 \beta_{5} q^{80} + ( - 5 \beta_{2} + 3 \beta_1 - 6) q^{81} + ( - 2 \beta_{2} - 4 \beta_1 - 10) q^{82} + (5 \beta_{5} - 4 \beta_{4} + \beta_{3}) q^{83} + ( - 2 \beta_{4} + 4 \beta_{3}) q^{84} + (4 \beta_{5} - \beta_{4} + 3 \beta_{3}) q^{85} + ( - 3 \beta_{4} + 7 \beta_{3}) q^{86} + ( - 4 \beta_{2} - \beta_1 - 8) q^{87} + ( - 4 \beta_{2} - 6 \beta_1 - 8) q^{88} + (2 \beta_{5} + 3 \beta_{4}) q^{89} + (\beta_1 - 2) q^{90} + ( - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \beta_{2}) q^{91} + ( - 3 \beta_{2} - 5 \beta_1 - 7) q^{92} + (4 \beta_{5} - 3 \beta_{4} + 5 \beta_{3}) q^{93} + (8 \beta_{2} + \beta_1 + 6) q^{94} + (4 \beta_{2} - 8 \beta_1 + 11) q^{95} + (4 \beta_{5} + 4 \beta_{3}) q^{96} + ( - 2 \beta_{5} + 9 \beta_{4} - 8 \beta_{3}) q^{97} + ( - \beta_{4} + \beta_{3}) q^{98} + (2 \beta_{5} - \beta_{4} + \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{4} - 2 q^{9} - 20 q^{12} + 8 q^{13} - 4 q^{14} + 8 q^{16} - 8 q^{17} + 24 q^{22} + 6 q^{23} + 8 q^{25} + 12 q^{26} - 12 q^{27} - 14 q^{29} + 16 q^{30} - 6 q^{35} + 4 q^{36} - 4 q^{38} - 8 q^{39} - 20 q^{40} - 4 q^{42} - 26 q^{43} + 8 q^{48} - 6 q^{49} + 8 q^{51} - 20 q^{52} + 2 q^{53} + 12 q^{55} + 12 q^{56} + 28 q^{61} + 16 q^{62} - 8 q^{64} - 6 q^{65} + 28 q^{66} + 20 q^{68} - 4 q^{69} - 24 q^{74} + 44 q^{75} + 16 q^{77} + 16 q^{78} + 26 q^{79} - 26 q^{81} - 56 q^{82} - 40 q^{87} - 40 q^{88} - 12 q^{90} - 2 q^{91} - 36 q^{92} + 20 q^{94} + 58 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{2} + 5\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \) Copy content Toggle raw display

Character values

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

\(n\) \(15\) \(66\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
0.403032 + 0.403032i
−0.854638 + 0.854638i
1.45161 + 1.45161i
1.45161 1.45161i
−0.854638 0.854638i
0.403032 0.403032i
2.48119i 1.67513 −4.15633 0.675131i 4.15633i 1.00000i 5.35026i −0.193937 1.67513
64.2 1.17009i 0.539189 0.630898 0.460811i 0.630898i 1.00000i 3.07838i −2.70928 0.539189
64.3 0.688892i −2.21432 1.52543 3.21432i 1.52543i 1.00000i 2.42864i 1.90321 −2.21432
64.4 0.688892i −2.21432 1.52543 3.21432i 1.52543i 1.00000i 2.42864i 1.90321 −2.21432
64.5 1.17009i 0.539189 0.630898 0.460811i 0.630898i 1.00000i 3.07838i −2.70928 0.539189
64.6 2.48119i 1.67513 −4.15633 0.675131i 4.15633i 1.00000i 5.35026i −0.193937 1.67513
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.c.a 6
3.b odd 2 1 819.2.c.b 6
4.b odd 2 1 1456.2.k.c 6
7.b odd 2 1 637.2.c.d 6
7.c even 3 2 637.2.r.e 12
7.d odd 6 2 637.2.r.d 12
13.b even 2 1 inner 91.2.c.a 6
13.d odd 4 1 1183.2.a.h 3
13.d odd 4 1 1183.2.a.j 3
39.d odd 2 1 819.2.c.b 6
52.b odd 2 1 1456.2.k.c 6
91.b odd 2 1 637.2.c.d 6
91.i even 4 1 8281.2.a.be 3
91.i even 4 1 8281.2.a.bi 3
91.r even 6 2 637.2.r.e 12
91.s odd 6 2 637.2.r.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.c.a 6 1.a even 1 1 trivial
91.2.c.a 6 13.b even 2 1 inner
637.2.c.d 6 7.b odd 2 1
637.2.c.d 6 91.b odd 2 1
637.2.r.d 12 7.d odd 6 2
637.2.r.d 12 91.s odd 6 2
637.2.r.e 12 7.c even 3 2
637.2.r.e 12 91.r even 6 2
819.2.c.b 6 3.b odd 2 1
819.2.c.b 6 39.d odd 2 1
1183.2.a.h 3 13.d odd 4 1
1183.2.a.j 3 13.d odd 4 1
1456.2.k.c 6 4.b odd 2 1
1456.2.k.c 6 52.b odd 2 1
8281.2.a.be 3 91.i even 4 1
8281.2.a.bi 3 91.i even 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 8 T^{4} + 12 T^{2} + 4 \) Copy content Toggle raw display
$3$ \( (T^{3} - 4 T + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} + 11 T^{4} + 7 T^{2} + 1 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} + 28 T^{4} + 164 T^{2} + \cdots + 100 \) Copy content Toggle raw display
$13$ \( T^{6} - 8 T^{5} + 7 T^{4} + 64 T^{3} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( (T^{3} + 4 T^{2} - 8 T - 34)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 119 T^{4} + 3867 T^{2} + \cdots + 37249 \) Copy content Toggle raw display
$23$ \( (T^{3} - 3 T^{2} - 25 T + 79)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 7 T^{2} - 21 T + 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 83 T^{4} + 1791 T^{2} + \cdots + 4225 \) Copy content Toggle raw display
$37$ \( T^{6} + 108 T^{4} + 1620 T^{2} + \cdots + 2916 \) Copy content Toggle raw display
$41$ \( T^{6} + 108 T^{4} + 2864 T^{2} + \cdots + 1600 \) Copy content Toggle raw display
$43$ \( (T^{3} + 13 T^{2} + 35 T - 17)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 151 T^{4} + 5819 T^{2} + \cdots + 18769 \) Copy content Toggle raw display
$53$ \( (T^{3} - T^{2} - 9 T + 13)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 68 T^{4} + 816 T^{2} + \cdots + 2704 \) Copy content Toggle raw display
$61$ \( (T^{3} - 14 T^{2} + 28 T + 152)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 80 T^{4} + 1408 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$71$ \( T^{6} + 304 T^{4} + 27836 T^{2} + \cdots + 792100 \) Copy content Toggle raw display
$73$ \( T^{6} + 263 T^{4} + 7723 T^{2} + \cdots + 961 \) Copy content Toggle raw display
$79$ \( (T^{3} - 13 T^{2} - 37 T + 185)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 227 T^{4} + 13095 T^{2} + \cdots + 26569 \) Copy content Toggle raw display
$89$ \( T^{6} + 119 T^{4} + 4387 T^{2} + \cdots + 51529 \) Copy content Toggle raw display
$97$ \( T^{6} + 575 T^{4} + 96115 T^{2} + \cdots + 4765489 \) Copy content Toggle raw display
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