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Properties

Label 91.2.q.a
Level $91$
Weight $2$
Character orbit 91.q
Analytic conductor $0.727$
Analytic rank $0$
Dimension $12$
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(36,91)]
 
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, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.36");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.q (of order \(6\), 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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.58891012706304.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} - 5\nu^{9} - 2\nu^{8} + 15\nu^{7} + 2\nu^{6} - 30\nu^{5} + 4\nu^{4} + 60\nu^{3} - 16\nu^{2} - 80\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3 \nu^{11} + 4 \nu^{10} + 7 \nu^{9} - 6 \nu^{8} - 13 \nu^{7} + 30 \nu^{6} - 6 \nu^{5} - 28 \nu^{4} + \cdots + 96 ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3 \nu^{11} + 2 \nu^{10} - 11 \nu^{9} - 8 \nu^{8} + 21 \nu^{7} + 4 \nu^{6} - 42 \nu^{5} + 84 \nu^{3} + \cdots - 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} + 3 \nu^{10} - 3 \nu^{9} - 13 \nu^{8} + 7 \nu^{7} + 23 \nu^{6} - 26 \nu^{5} - 38 \nu^{4} + \cdots - 64 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5 \nu^{11} + 4 \nu^{10} + 13 \nu^{9} - 10 \nu^{8} - 23 \nu^{7} + 42 \nu^{6} + 10 \nu^{5} - 68 \nu^{4} + \cdots + 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5 \nu^{11} + 2 \nu^{10} + 17 \nu^{9} - 8 \nu^{8} - 39 \nu^{7} + 44 \nu^{6} + 50 \nu^{5} - 88 \nu^{4} + \cdots - 32 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{11} + \nu^{10} + 5 \nu^{9} - 3 \nu^{8} - 13 \nu^{7} + 13 \nu^{6} + 20 \nu^{5} - 34 \nu^{4} + \cdots - 32 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{11} + 3 \nu^{10} + 5 \nu^{9} - 13 \nu^{8} - 13 \nu^{7} + 35 \nu^{6} + 12 \nu^{5} - 70 \nu^{4} + \cdots - 80 ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3 \nu^{11} + 15 \nu^{9} - 2 \nu^{8} - 37 \nu^{7} + 18 \nu^{6} + 66 \nu^{5} - 68 \nu^{4} - 92 \nu^{3} + \cdots - 64 ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - \nu^{11} + 4 \nu^{10} + 9 \nu^{9} - 18 \nu^{8} - 27 \nu^{7} + 50 \nu^{6} + 34 \nu^{5} - 116 \nu^{4} + \cdots - 160 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{9} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{9} + \beta_{7} - 2\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{11} + 4\beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \beta_{3} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 5 \beta_{11} - 3 \beta_{10} + 5 \beta_{9} + 7 \beta_{8} - \beta_{7} - 2 \beta_{6} + 3 \beta_{5} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -3\beta_{11} + 7\beta_{8} + \beta_{7} - 5\beta_{6} - \beta_{5} + 2\beta_{4} + 4\beta_{3} + 10\beta_{2} + 8\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 7 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} + 6 \beta_{8} - 4 \beta_{7} - 11 \beta_{6} + 12 \beta_{5} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 17 \beta_{11} + 4 \beta_{10} + 13 \beta_{9} + 18 \beta_{8} - 9 \beta_{7} - 8 \beta_{6} + 3 \beta_{5} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 18 \beta_{11} + 4 \beta_{10} - 6 \beta_{9} - 13 \beta_{8} - 21 \beta_{7} - 11 \beta_{6} - \beta_{5} + \cdots - 7 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(15\) \(66\)
\(\chi(n)\) \(1 - \beta_{9}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1
−1.12906 − 0.851598i
0.759479 − 1.19298i
1.34408 + 0.439820i
−1.08105 + 0.911778i
1.40744 + 0.138282i
−1.30089 + 0.554694i
−1.12906 + 0.851598i
0.759479 + 1.19298i
1.34408 − 0.439820i
−1.08105 − 0.911778i
1.40744 − 0.138282i
−1.30089 − 0.554694i
−2.34104 + 1.35160i 0.172975 + 0.299601i 2.65363 − 4.59623i − 3.25812i −0.809880 − 0.467584i 0.866025 + 0.500000i 8.94020i 1.44016 − 2.49443i 4.40367 + 7.62739i
36.2 −1.20027 + 0.692976i 1.41289 + 2.44719i −0.0395678 + 0.0685334i − 0.518957i −3.39169 − 1.95819i −0.866025 − 0.500000i − 2.88158i −2.49250 + 4.31714i 0.359625 + 0.622889i
36.3 −0.104235 + 0.0601799i 0.291146 + 0.504280i −0.992757 + 1.71951i 1.68817i −0.0606950 − 0.0350423i 0.866025 + 0.500000i − 0.479696i 1.33047 − 2.30444i −0.101594 − 0.175965i
36.4 0.713220 − 0.411778i −1.33015 − 2.30388i −0.660878 + 1.14467i − 3.16209i −1.89737 − 1.09545i 0.866025 + 0.500000i 2.73565i −2.03858 + 3.53092i −1.30208 − 2.25527i
36.5 1.10554 − 0.638282i 0.583963 + 1.01145i −0.185192 + 0.320762i − 1.81487i 1.29118 + 0.745466i −0.866025 − 0.500000i 3.02595i 0.817975 − 1.41677i −1.15840 − 2.00641i
36.6 1.82678 − 1.05469i −1.13082 − 1.95864i 1.22476 − 2.12135i 3.60178i −4.13154 − 2.38535i −0.866025 − 0.500000i − 0.948212i −1.05753 + 1.83169i 3.79878 + 6.57967i
43.1 −2.34104 − 1.35160i 0.172975 − 0.299601i 2.65363 + 4.59623i 3.25812i −0.809880 + 0.467584i 0.866025 − 0.500000i − 8.94020i 1.44016 + 2.49443i 4.40367 − 7.62739i
43.2 −1.20027 − 0.692976i 1.41289 − 2.44719i −0.0395678 − 0.0685334i 0.518957i −3.39169 + 1.95819i −0.866025 + 0.500000i 2.88158i −2.49250 − 4.31714i 0.359625 − 0.622889i
43.3 −0.104235 − 0.0601799i 0.291146 − 0.504280i −0.992757 − 1.71951i − 1.68817i −0.0606950 + 0.0350423i 0.866025 − 0.500000i 0.479696i 1.33047 + 2.30444i −0.101594 + 0.175965i
43.4 0.713220 + 0.411778i −1.33015 + 2.30388i −0.660878 − 1.14467i 3.16209i −1.89737 + 1.09545i 0.866025 − 0.500000i − 2.73565i −2.03858 − 3.53092i −1.30208 + 2.25527i
43.5 1.10554 + 0.638282i 0.583963 − 1.01145i −0.185192 − 0.320762i 1.81487i 1.29118 − 0.745466i −0.866025 + 0.500000i − 3.02595i 0.817975 + 1.41677i −1.15840 + 2.00641i
43.6 1.82678 + 1.05469i −1.13082 + 1.95864i 1.22476 + 2.12135i − 3.60178i −4.13154 + 2.38535i −0.866025 + 0.500000i 0.948212i −1.05753 − 1.83169i 3.79878 − 6.57967i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.q.a ✓ 12
3.b odd 2 1 819.2.ct.a 12
4.b odd 2 1 1456.2.cc.c 12
7.b odd 2 1 637.2.q.h 12
7.c even 3 1 637.2.k.h 12
7.c even 3 1 637.2.u.h 12
7.d odd 6 1 637.2.k.g 12
7.d odd 6 1 637.2.u.i 12
13.c even 3 1 1183.2.c.i 12
13.e even 6 1 inner 91.2.q.a ✓ 12
13.e even 6 1 1183.2.c.i 12
13.f odd 12 1 1183.2.a.m 6
13.f odd 12 1 1183.2.a.p 6
39.h odd 6 1 819.2.ct.a 12
52.i odd 6 1 1456.2.cc.c 12
91.k even 6 1 637.2.u.h 12
91.l odd 6 1 637.2.u.i 12
91.p odd 6 1 637.2.k.g 12
91.t odd 6 1 637.2.q.h 12
91.u even 6 1 637.2.k.h 12
91.bc even 12 1 8281.2.a.by 6
91.bc even 12 1 8281.2.a.ch 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.q.a ✓ 12 1.a even 1 1 trivial
91.2.q.a ✓ 12 13.e even 6 1 inner
637.2.k.g 12 7.d odd 6 1
637.2.k.g 12 91.p odd 6 1
637.2.k.h 12 7.c even 3 1
637.2.k.h 12 91.u even 6 1
637.2.q.h 12 7.b odd 2 1
637.2.q.h 12 91.t odd 6 1
637.2.u.h 12 7.c even 3 1
637.2.u.h 12 91.k even 6 1
637.2.u.i 12 7.d odd 6 1
637.2.u.i 12 91.l odd 6 1
819.2.ct.a 12 3.b odd 2 1
819.2.ct.a 12 39.h odd 6 1
1183.2.a.m 6 13.f odd 12 1
1183.2.a.p 6 13.f odd 12 1
1183.2.c.i 12 13.c even 3 1
1183.2.c.i 12 13.e even 6 1
1456.2.cc.c 12 4.b odd 2 1
1456.2.cc.c 12 52.i odd 6 1
8281.2.a.by 6 91.bc even 12 1
8281.2.a.ch 6 91.bc even 12 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 8 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} + 11 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{12} + 40 T^{10} + \cdots + 3481 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{12} - 6 T^{11} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{12} - 4 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} + 4 T^{11} + \cdots + 241081 \) Copy content Toggle raw display
$19$ \( T^{12} - 29 T^{10} + \cdots + 55696 \) Copy content Toggle raw display
$23$ \( T^{12} + 12 T^{11} + \cdots + 38539264 \) Copy content Toggle raw display
$29$ \( T^{12} - 8 T^{11} + \cdots + 10042561 \) Copy content Toggle raw display
$31$ \( T^{12} + 136 T^{10} + \cdots + 913936 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 1755945216 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 884705536 \) Copy content Toggle raw display
$43$ \( T^{12} - 2 T^{11} + \cdots + 2408704 \) Copy content Toggle raw display
$47$ \( T^{12} + 272 T^{10} + \cdots + 9461776 \) Copy content Toggle raw display
$53$ \( (T^{6} + 22 T^{5} + \cdots - 2339)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 4571923456 \) Copy content Toggle raw display
$61$ \( T^{12} - 14 T^{11} + \cdots + 5607424 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 613651984 \) Copy content Toggle raw display
$71$ \( T^{12} + 24 T^{11} + \cdots + 46895104 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 1386221824 \) Copy content Toggle raw display
$79$ \( (T^{6} + 28 T^{5} + \cdots - 512)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 141324544 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 1834580224 \) Copy content Toggle raw display
$97$ \( T^{12} - 6 T^{11} + \cdots + 53465344 \) Copy content Toggle raw display
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