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Properties

Label 91.2.f.c
Level $91$
Weight $2$
Character orbit 91.f
Analytic conductor $0.727$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [91,2,Mod(22,91)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(91, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.22");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.f (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.726638658394\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 7x^{6} + 38x^{4} - 16x^{3} + 15x^{2} + 3x + 1 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{5} q^{3} + ( - \beta_{7} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + ( - \beta_{3} - 2) q^{5} + (\beta_{4} - \beta_{3} - \beta_1 + 1) q^{6} + (\beta_{4} + 1) q^{7} + (\beta_{6} + 3 \beta_{3} + \beta_{2} - 1) q^{8} + (\beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{5} q^{3} + ( - \beta_{7} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + ( - \beta_{3} - 2) q^{5} + (\beta_{4} - \beta_{3} - \beta_1 + 1) q^{6} + (\beta_{4} + 1) q^{7} + (\beta_{6} + 3 \beta_{3} + \beta_{2} - 1) q^{8} + (\beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{2} - 2) q^{9} + ( - \beta_{7} - 3 \beta_{4} - \beta_1) q^{10} + \beta_{5} q^{11} + ( - 2 \beta_{6} - 2 \beta_{3} - \beta_{2} + 3) q^{12} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 1) q^{13} - \beta_{3} q^{14} + (2 \beta_{5} + \beta_{4} - \beta_1) q^{15} + (\beta_{7} + \beta_{5} + 4 \beta_{4} - 4 \beta_1) q^{16} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{17} + ( - \beta_{6} - 2 \beta_{3} - 1) q^{18} + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{3} + \beta_{2} + 3 \beta_1) q^{19} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{20} - \beta_{6} q^{21} + ( - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{22} + (\beta_{7} + \beta_{5} + \beta_1) q^{23} + ( - 2 \beta_{7} - \beta_{5} + 4 \beta_1) q^{24} + (3 \beta_{3} - \beta_{2} + 2) q^{25} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - \beta_{2} - 3 \beta_1) q^{26} + (3 \beta_{6} + \beta_{3} - 7) q^{27} + ( - \beta_{7} - \beta_{4} + \beta_1) q^{28} + ( - \beta_{7} - \beta_{5}) q^{29} + (\beta_{7} + \beta_{4} - \beta_{2} + 1) q^{30} + (\beta_{2} - 1) q^{31} + (2 \beta_{7} + \beta_{6} - \beta_{5} + 7 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 7) q^{32} + ( - \beta_{7} - \beta_{6} + \beta_{5} + 5 \beta_{4} + \beta_{2} + 5) q^{33} + ( - \beta_{6} - 4 \beta_{3} - \beta_{2}) q^{34} + ( - 2 \beta_{4} - \beta_{3} - \beta_1 - 2) q^{35} + ( - 2 \beta_{5} - 9 \beta_{4}) q^{36} + (\beta_{5} - 2 \beta_{4} + \beta_1) q^{37} + (\beta_{6} + 4 \beta_{3} + 3 \beta_{2} - 5) q^{38} + ( - \beta_{7} + \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{39} + ( - \beta_{6} + \beta_{2} - 4) q^{40} + (\beta_{7} - 6 \beta_{4} - 2 \beta_1) q^{41} + (\beta_{4} - \beta_1) q^{42} + ( - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{43} + (2 \beta_{6} + 2 \beta_{3} + \beta_{2} - 3) q^{44} + ( - 2 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots + 3) q^{45}+ \cdots + ( - 6 \beta_{6} - \beta_{3} + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - q^{3} - 5 q^{4} - 14 q^{5} + 5 q^{6} + 4 q^{7} - 12 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} - q^{3} - 5 q^{4} - 14 q^{5} + 5 q^{6} + 4 q^{7} - 12 q^{8} - 7 q^{9} + 11 q^{10} + q^{11} + 24 q^{12} + 4 q^{13} + 2 q^{14} - 3 q^{15} - 19 q^{16} + 4 q^{17} - 6 q^{18} - q^{19} + 2 q^{20} - 2 q^{21} - 5 q^{22} + 2 q^{23} + 3 q^{24} + 10 q^{25} + 12 q^{26} - 52 q^{27} + 5 q^{28} - q^{29} + 4 q^{30} - 8 q^{31} + 33 q^{32} + 19 q^{33} + 6 q^{34} - 7 q^{35} + 34 q^{36} + 10 q^{37} - 46 q^{38} + 20 q^{39} - 34 q^{40} + 22 q^{41} - 5 q^{42} - 3 q^{43} - 24 q^{44} + 11 q^{45} - 24 q^{46} + 4 q^{47} - 11 q^{48} - 4 q^{49} - 43 q^{50} + 14 q^{51} + 65 q^{52} + 4 q^{53} - 5 q^{54} + 3 q^{55} - 6 q^{56} - 34 q^{57} + 11 q^{58} + 8 q^{59} - 22 q^{60} - 8 q^{61} + 5 q^{62} + 7 q^{63} + 28 q^{64} + 7 q^{65} + 12 q^{66} + 6 q^{67} + 33 q^{68} + 18 q^{69} + 22 q^{70} + 14 q^{71} - 5 q^{72} - 16 q^{73} - 20 q^{74} + 7 q^{75} - 32 q^{76} + 2 q^{77} - q^{78} - 52 q^{79} - 7 q^{80} - 24 q^{81} + 14 q^{82} + 12 q^{84} - 5 q^{85} + 24 q^{86} - 13 q^{87} - 3 q^{88} + q^{89} + 52 q^{90} - 4 q^{91} + 24 q^{92} + 7 q^{93} - 33 q^{94} - 21 q^{95} - 116 q^{96} - 3 q^{97} + q^{98} + 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 7x^{6} + 38x^{4} - 16x^{3} + 15x^{2} + 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -12\nu^{7} - 7\nu^{6} - 76\nu^{5} - 44\nu^{4} - 602\nu^{3} - 36\nu^{2} - 8\nu + 1249 ) / 458 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 57\nu^{7} - 24\nu^{6} + 361\nu^{5} + 209\nu^{4} + 2287\nu^{3} + 171\nu^{2} + 38\nu + 193 ) / 916 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 193\nu^{7} - 250\nu^{6} + 1375\nu^{5} - 361\nu^{4} + 7125\nu^{3} - 5375\nu^{2} + 2724\nu - 375 ) / 916 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 174\nu^{7} - 242\nu^{6} + 1331\nu^{5} - 507\nu^{4} + 6897\nu^{3} - 5203\nu^{2} + 5383\nu - 363 ) / 458 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -375\nu^{7} + 182\nu^{6} - 2375\nu^{5} - 1375\nu^{4} - 13889\nu^{3} - 1125\nu^{2} - 250\nu - 2933 ) / 916 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -273\nu^{7} + 356\nu^{6} - 1958\nu^{5} + 602\nu^{4} - 10146\nu^{3} + 7654\nu^{2} - 3617\nu + 534 ) / 458 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - 3\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + 7\beta_{3} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{7} + \beta_{5} + 18\beta_{4} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{7} - 7\beta_{6} + 7\beta_{5} + 15\beta_{4} - 48\beta_{3} - 10\beta_{2} - 48\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{6} - 86\beta_{3} - 48\beta_{2} + 117 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -86\beta_{7} - 48\beta_{5} - 152\beta_{4} + 337\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(15\) \(66\)
\(\chi(n)\) \(-1 - \beta_{4}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
−1.11000 + 1.92258i
−0.115680 + 0.200364i
0.355143 − 0.615126i
1.37054 − 2.37385i
−1.11000 − 1.92258i
−0.115680 − 0.200364i
0.355143 + 0.615126i
1.37054 + 2.37385i
−1.11000 + 1.92258i −0.274776 + 0.475925i −1.46422 − 2.53610i −4.22001 −0.610004 − 1.05656i 0.500000 + 0.866025i 2.06113 1.34900 + 2.33653i 4.68423 − 8.11332i
22.2 −0.115680 + 0.200364i 1.66113 − 2.87716i 0.973236 + 1.68569i −2.23136 0.384320 + 0.665661i 0.500000 + 0.866025i −0.913059 −4.01868 − 6.96056i 0.258125 − 0.447085i
22.3 0.355143 − 0.615126i −1.20394 + 2.08529i 0.747746 + 1.29513i −1.28971 0.855143 + 1.48115i 0.500000 + 0.866025i 2.48280 −1.39895 − 2.42305i −0.458033 + 0.793337i
22.4 1.37054 − 2.37385i −0.682410 + 1.18197i −2.75677 − 4.77486i 0.741082 1.87054 + 3.23987i 0.500000 + 0.866025i −9.63087 0.568634 + 0.984903i 1.01568 − 1.75921i
29.1 −1.11000 − 1.92258i −0.274776 − 0.475925i −1.46422 + 2.53610i −4.22001 −0.610004 + 1.05656i 0.500000 − 0.866025i 2.06113 1.34900 − 2.33653i 4.68423 + 8.11332i
29.2 −0.115680 − 0.200364i 1.66113 + 2.87716i 0.973236 − 1.68569i −2.23136 0.384320 − 0.665661i 0.500000 − 0.866025i −0.913059 −4.01868 + 6.96056i 0.258125 + 0.447085i
29.3 0.355143 + 0.615126i −1.20394 − 2.08529i 0.747746 − 1.29513i −1.28971 0.855143 − 1.48115i 0.500000 − 0.866025i 2.48280 −1.39895 + 2.42305i −0.458033 − 0.793337i
29.4 1.37054 + 2.37385i −0.682410 − 1.18197i −2.75677 + 4.77486i 0.741082 1.87054 − 3.23987i 0.500000 − 0.866025i −9.63087 0.568634 − 0.984903i 1.01568 + 1.75921i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.4
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.c ✓ 8
3.b odd 2 1 819.2.o.h 8
4.b odd 2 1 1456.2.s.q 8
7.b odd 2 1 637.2.f.i 8
7.c even 3 1 637.2.g.k 8
7.c even 3 1 637.2.h.h 8
7.d odd 6 1 637.2.g.j 8
7.d odd 6 1 637.2.h.i 8
13.c even 3 1 inner 91.2.f.c ✓ 8
13.c even 3 1 1183.2.a.k 4
13.e even 6 1 1183.2.a.l 4
13.f odd 12 2 1183.2.c.g 8
39.i odd 6 1 819.2.o.h 8
52.j odd 6 1 1456.2.s.q 8
91.g even 3 1 637.2.h.h 8
91.h even 3 1 637.2.g.k 8
91.m odd 6 1 637.2.h.i 8
91.n odd 6 1 637.2.f.i 8
91.n odd 6 1 8281.2.a.bp 4
91.t odd 6 1 8281.2.a.bt 4
91.v odd 6 1 637.2.g.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.c ✓ 8 1.a even 1 1 trivial
91.2.f.c ✓ 8 13.c even 3 1 inner
637.2.f.i 8 7.b odd 2 1
637.2.f.i 8 91.n odd 6 1
637.2.g.j 8 7.d odd 6 1
637.2.g.j 8 91.v odd 6 1
637.2.g.k 8 7.c even 3 1
637.2.g.k 8 91.h even 3 1
637.2.h.h 8 7.c even 3 1
637.2.h.h 8 91.g even 3 1
637.2.h.i 8 7.d odd 6 1
637.2.h.i 8 91.m odd 6 1
819.2.o.h 8 3.b odd 2 1
819.2.o.h 8 39.i odd 6 1
1183.2.a.k 4 13.c even 3 1
1183.2.a.l 4 13.e even 6 1
1183.2.c.g 8 13.f odd 12 2
1456.2.s.q 8 4.b odd 2 1
1456.2.s.q 8 52.j odd 6 1
8281.2.a.bp 4 91.n odd 6 1
8281.2.a.bt 4 91.t odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - T_{2}^{7} + 7T_{2}^{6} + 38T_{2}^{4} - 16T_{2}^{3} + 15T_{2}^{2} + 3T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(91, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + 7 T^{6} + 38 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + T^{7} + 10 T^{6} + 23 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$5$ \( (T^{4} + 7 T^{3} + 12 T^{2} - T - 9)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} - T^{7} + 10 T^{6} - 23 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$13$ \( T^{8} - 4 T^{7} + 16 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{7} + 28 T^{6} + \cdots + 2809 \) Copy content Toggle raw display
$19$ \( T^{8} + T^{7} + 56 T^{6} + \cdots + 250000 \) Copy content Toggle raw display
$23$ \( T^{8} - 2 T^{7} + 42 T^{6} + \cdots + 5184 \) Copy content Toggle raw display
$29$ \( T^{8} + T^{7} + 23 T^{6} - 64 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$31$ \( (T^{4} + 4 T^{3} - 13 T^{2} - 26 T + 54)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 10 T^{7} + 83 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( T^{8} - 22 T^{7} + 335 T^{6} + \cdots + 318096 \) Copy content Toggle raw display
$43$ \( T^{8} + 3 T^{7} + 12 T^{6} + 7 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$47$ \( (T^{4} - 2 T^{3} - 51 T^{2} - 12 T + 100)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 2 T^{3} - 140 T^{2} + 890 T - 1389)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 8 T^{7} + 141 T^{6} + \cdots + 498436 \) Copy content Toggle raw display
$61$ \( T^{8} + 8 T^{7} + 79 T^{6} + \cdots + 10000 \) Copy content Toggle raw display
$67$ \( T^{8} - 6 T^{7} + 247 T^{6} + \cdots + 121220100 \) Copy content Toggle raw display
$71$ \( T^{8} - 14 T^{7} + 212 T^{6} + \cdots + 4129024 \) Copy content Toggle raw display
$73$ \( (T^{4} + 8 T^{3} - 119 T^{2} - 88 T + 1772)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 26 T^{3} + 134 T^{2} - 1024 T - 7680)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 97 T^{2} - 442 T - 426)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - T^{7} + 72 T^{6} - 297 T^{5} + \cdots + 11664 \) Copy content Toggle raw display
$97$ \( T^{8} + 3 T^{7} + 314 T^{6} + \cdots + 246238864 \) Copy content Toggle raw display
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