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Properties

Label 86.2.e.a
Level $86$
Weight $2$
Character orbit 86.e
Analytic conductor $0.687$
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] = [86,2,Mod(11,86)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(86, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("86.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 86 = 2 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 86.e (of order \(7\), degree \(6\), 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: \(0.686713457383\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} + 6 x^{10} + 9 x^{9} - 33 x^{8} + 36 x^{7} + 34 x^{6} - 219 x^{5} + 414 x^{4} + \cdots + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + ( - \beta_{11} - \beta_{7}) q^{3} - \beta_{3} q^{4} + (\beta_{8} + \beta_{7} + \beta_{6} + \cdots - 1) q^{5}+ \cdots + (\beta_{11} - \beta_{10} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + ( - \beta_{11} - \beta_{7}) q^{3} - \beta_{3} q^{4} + (\beta_{8} + \beta_{7} + \beta_{6} + \cdots - 1) q^{5}+ \cdots + (2 \beta_{9} - 3 \beta_{8} + 2 \beta_{7} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} - 6 q^{3} - 2 q^{4} - 3 q^{5} - 8 q^{6} - 4 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} - 6 q^{3} - 2 q^{4} - 3 q^{5} - 8 q^{6} - 4 q^{7} + 2 q^{8} + 2 q^{9} + 3 q^{10} - 6 q^{12} + 3 q^{13} + 4 q^{14} + 5 q^{15} - 2 q^{16} - 5 q^{17} + 5 q^{18} + 6 q^{19} - 3 q^{20} + 2 q^{21} + 7 q^{22} - 26 q^{23} - q^{24} + 9 q^{25} - 3 q^{26} + 3 q^{27} - 4 q^{28} - 3 q^{29} - 12 q^{30} + 13 q^{31} + 2 q^{32} - 49 q^{33} - 2 q^{34} + 8 q^{35} + 16 q^{36} + 34 q^{37} + 29 q^{38} - 40 q^{39} - 4 q^{40} - 10 q^{41} - 2 q^{42} + 33 q^{43} + 45 q^{45} - 9 q^{46} + 16 q^{47} + q^{48} - 64 q^{49} + 12 q^{50} + 48 q^{51} + 10 q^{52} - 6 q^{53} - 10 q^{54} - 7 q^{55} - 10 q^{56} + 25 q^{57} + 3 q^{58} - 40 q^{59} + 12 q^{60} - 4 q^{61} + 15 q^{62} - 3 q^{63} - 2 q^{64} - 55 q^{65} - 42 q^{66} + 27 q^{67} + 16 q^{68} + 13 q^{69} - 8 q^{70} - 14 q^{71} + 5 q^{72} + q^{74} + 20 q^{75} - 29 q^{76} - 7 q^{77} + 12 q^{78} + 12 q^{79} + 4 q^{80} + 17 q^{81} - 32 q^{82} - 36 q^{83} + 2 q^{84} + 66 q^{85} + 2 q^{86} + 54 q^{87} + 7 q^{88} + 43 q^{89} + 11 q^{90} - 22 q^{91} + 2 q^{92} - 108 q^{93} - 9 q^{94} - 19 q^{95} - q^{96} + 25 q^{97} - 20 q^{98} - 35 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5 x^{11} + 6 x^{10} + 9 x^{9} - 33 x^{8} + 36 x^{7} + 34 x^{6} - 219 x^{5} + 414 x^{4} + \cdots + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3758002732 \nu^{11} - 15345060499 \nu^{10} + 8036941801 \nu^{9} + 42568038423 \nu^{8} + \cdots - 656116970405 ) / 45309803461 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4279517737 \nu^{11} - 17584341715 \nu^{10} + 11080820926 \nu^{9} + 45137425160 \nu^{8} + \cdots - 836495555714 ) / 45309803461 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4898041125 \nu^{11} - 18616458772 \nu^{10} + 7211094441 \nu^{9} + 52462771388 \nu^{8} + \cdots - 654772555164 ) / 45309803461 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5367531297 \nu^{11} - 19684865359 \nu^{10} + 5873570405 \nu^{9} + 56260592279 \nu^{8} + \cdots - 731751474558 ) / 45309803461 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7798406703 \nu^{11} - 30038600936 \nu^{10} + 12074405592 \nu^{9} + 84163997726 \nu^{8} + \cdots - 1115717599425 ) / 45309803461 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8772564331 \nu^{11} + 34293896406 \nu^{10} - 15499723383 \nu^{9} - 95959569597 \nu^{8} + \cdots + 1359604925588 ) / 45309803461 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8812484458 \nu^{11} - 34195939950 \nu^{10} + 14619584240 \nu^{9} + 95596983867 \nu^{8} + \cdots - 1321544132805 ) / 45309803461 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10662843258 \nu^{11} + 41097088705 \nu^{10} - 17791390980 \nu^{9} - 113008785180 \nu^{8} + \cdots + 1657578123214 ) / 45309803461 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13765433193 \nu^{11} + 54120477683 \nu^{10} - 23680318705 \nu^{9} - 150936363701 \nu^{8} + \cdots + 2141813443809 ) / 45309803461 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 14686231311 \nu^{11} + 56373092259 \nu^{10} - 22999985503 \nu^{9} - 157933810003 \nu^{8} + \cdots + 2194622886391 ) / 45309803461 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 18701702638 \nu^{11} + 72570624569 \nu^{10} - 30647367604 \nu^{9} - 203131549212 \nu^{8} + \cdots + 2850532053356 ) / 45309803461 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 4 \beta_{11} - 5 \beta_{10} - 3 \beta_{9} + 2 \beta_{8} - 4 \beta_{7} - 3 \beta_{6} + \beta_{5} + \cdots + 5 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - 3 \beta_{10} + \beta_{9} + 4 \beta_{8} + 13 \beta_{7} + 8 \beta_{6} - 5 \beta_{5} + \cdots + 3 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 17 \beta_{11} - 16 \beta_{10} - 11 \beta_{9} + 5 \beta_{8} + 18 \beta_{7} - 11 \beta_{6} - \beta_{5} + \cdots + 2 ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8 \beta_{11} + 4 \beta_{10} - 13 \beta_{9} + 4 \beta_{8} + 62 \beta_{7} + 22 \beta_{6} - 54 \beta_{5} + \cdots + 3 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 43 \beta_{11} - 38 \beta_{10} - 6 \beta_{9} - 31 \beta_{8} + 6 \beta_{7} - 20 \beta_{6} + 16 \beta_{5} + \cdots + 66 ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 95 \beta_{11} - 47 \beta_{10} - 94 \beta_{9} + 65 \beta_{8} + 255 \beta_{7} + 88 \beta_{6} - 202 \beta_{5} + \cdots - 135 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 45 \beta_{11} - 121 \beta_{10} + 45 \beta_{9} - 121 \beta_{8} + 235 \beta_{7} + 122 \beta_{6} + \cdots + 198 ) / 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 72 \beta_{11} - 49 \beta_{10} - 73 \beta_{9} + 36 \beta_{8} + 117 \beta_{7} - 19 \beta_{6} - 127 \beta_{5} + \cdots - 80 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 218 \beta_{11} - 74 \beta_{10} + 510 \beta_{9} - 648 \beta_{8} + 1009 \beta_{7} + 755 \beta_{6} + \cdots + 1026 ) / 7 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2427 \beta_{11} - 2206 \beta_{10} - 1675 \beta_{9} + 531 \beta_{8} + 1605 \beta_{7} - 1675 \beta_{6} + \cdots - 1889 ) / 7 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1420 \beta_{11} + 130 \beta_{10} + 2003 \beta_{9} - 1550 \beta_{8} + 7167 \beta_{7} + 4838 \beta_{6} + \cdots + 192 ) / 7 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/86\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.238885 − 1.58047i
1.38461 + 0.798641i
2.04692 − 0.238953i
−1.94789 + 0.672836i
−0.314615 − 1.04837i
1.09209 + 0.0734422i
2.04692 + 0.238953i
−1.94789 − 0.672836i
0.238885 + 1.58047i
1.38461 − 0.798641i
−0.314615 + 1.04837i
1.09209 − 0.0734422i
−0.623490 − 0.781831i −1.36420 + 1.71065i −0.222521 + 0.974928i −2.87229 + 1.38322i 2.18801 −1.80194 0.900969 − 0.433884i −0.397728 − 1.74256i 2.87229 + 1.38322i
11.2 −0.623490 − 0.781831i 1.21021 − 1.51756i −0.222521 + 0.974928i 0.847836 − 0.408296i −1.94103 −1.80194 0.900969 − 0.433884i −0.170804 − 0.748341i −0.847836 − 0.408296i
21.1 0.900969 + 0.433884i −2.86402 + 1.37924i 0.623490 + 0.781831i 0.484834 + 2.12420i −3.17883 −0.445042 0.222521 + 0.974928i 4.42985 − 5.55486i −0.484834 + 2.12420i
21.2 0.900969 + 0.433884i 0.339565 − 0.163526i 0.623490 + 0.781831i −0.306386 − 1.34237i 0.376888 −0.445042 0.222521 + 0.974928i −1.78191 + 2.23444i 0.306386 − 1.34237i
35.1 0.222521 − 0.974928i −0.551108 − 2.41456i −0.900969 − 0.433884i −0.920681 + 1.15450i −2.47666 1.24698 −0.623490 + 0.781831i −2.82348 + 1.35972i 0.920681 + 1.15450i
35.2 0.222521 − 0.974928i 0.229556 + 1.00575i −0.900969 − 0.433884i 1.26669 − 1.58838i 1.03162 1.24698 −0.623490 + 0.781831i 1.74407 − 0.839899i −1.26669 − 1.58838i
41.1 0.900969 − 0.433884i −2.86402 − 1.37924i 0.623490 − 0.781831i 0.484834 − 2.12420i −3.17883 −0.445042 0.222521 − 0.974928i 4.42985 + 5.55486i −0.484834 − 2.12420i
41.2 0.900969 − 0.433884i 0.339565 + 0.163526i 0.623490 − 0.781831i −0.306386 + 1.34237i 0.376888 −0.445042 0.222521 − 0.974928i −1.78191 − 2.23444i 0.306386 + 1.34237i
47.1 −0.623490 + 0.781831i −1.36420 − 1.71065i −0.222521 − 0.974928i −2.87229 − 1.38322i 2.18801 −1.80194 0.900969 + 0.433884i −0.397728 + 1.74256i 2.87229 − 1.38322i
47.2 −0.623490 + 0.781831i 1.21021 + 1.51756i −0.222521 − 0.974928i 0.847836 + 0.408296i −1.94103 −1.80194 0.900969 + 0.433884i −0.170804 + 0.748341i −0.847836 + 0.408296i
59.1 0.222521 + 0.974928i −0.551108 + 2.41456i −0.900969 + 0.433884i −0.920681 − 1.15450i −2.47666 1.24698 −0.623490 − 0.781831i −2.82348 − 1.35972i 0.920681 − 1.15450i
59.2 0.222521 + 0.974928i 0.229556 − 1.00575i −0.900969 + 0.433884i 1.26669 + 1.58838i 1.03162 1.24698 −0.623490 − 0.781831i 1.74407 + 0.839899i −1.26669 + 1.58838i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 86.2.e.a ✓ 12
3.b odd 2 1 774.2.u.f 12
4.b odd 2 1 688.2.u.d 12
43.e even 7 1 inner 86.2.e.a ✓ 12
43.e even 7 1 3698.2.a.l 6
43.f odd 14 1 3698.2.a.m 6
129.l odd 14 1 774.2.u.f 12
172.k odd 14 1 688.2.u.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
86.2.e.a ✓ 12 1.a even 1 1 trivial
86.2.e.a ✓ 12 43.e even 7 1 inner
688.2.u.d 12 4.b odd 2 1
688.2.u.d 12 172.k odd 14 1
774.2.u.f 12 3.b odd 2 1
774.2.u.f 12 129.l odd 14 1
3698.2.a.l 6 43.e even 7 1
3698.2.a.m 6 43.f odd 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 6 T_{3}^{11} + 20 T_{3}^{10} + 45 T_{3}^{9} + 90 T_{3}^{8} + 135 T_{3}^{7} + 385 T_{3}^{6} + \cdots + 169 \) acting on \(S_{2}^{\mathrm{new}}(86, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + 6 T^{11} + \cdots + 169 \) Copy content Toggle raw display
$5$ \( T^{12} + 3 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$7$ \( (T^{3} + T^{2} - 2 T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{12} - 14 T^{10} + \cdots + 1750329 \) Copy content Toggle raw display
$13$ \( T^{12} - 3 T^{11} + \cdots + 32761 \) Copy content Toggle raw display
$17$ \( T^{12} + 5 T^{11} + \cdots + 3674889 \) Copy content Toggle raw display
$19$ \( T^{12} - 6 T^{11} + \cdots + 1849 \) Copy content Toggle raw display
$23$ \( T^{12} + 26 T^{11} + \cdots + 123201 \) Copy content Toggle raw display
$29$ \( T^{12} + 3 T^{11} + \cdots + 9308601 \) Copy content Toggle raw display
$31$ \( T^{12} - 13 T^{11} + \cdots + 75359761 \) Copy content Toggle raw display
$37$ \( (T^{6} - 17 T^{5} + \cdots - 293)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 10 T^{11} + \cdots + 613089 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 6321363049 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 103612041 \) Copy content Toggle raw display
$53$ \( T^{12} + 6 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$59$ \( T^{12} + 40 T^{11} + \cdots + 88792929 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 12136767889 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 493595089 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 250283080089 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 7111380241 \) Copy content Toggle raw display
$79$ \( (T^{6} - 6 T^{5} + \cdots + 22303)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 36 T^{11} + \cdots + 1225449 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 893352321 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 1047752161 \) Copy content Toggle raw display
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