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Properties

Label 84.2.o.b
Level $84$
Weight $2$
Character orbit 84.o
Analytic conductor $0.671$
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] = [84,2,Mod(19,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("84.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 84.o (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: \(0.670743376979\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.562828176.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + x^{6} + 2x^{5} - 6x^{4} + 4x^{3} + 4x^{2} - 16x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + \nu^{5} + 4\nu^{4} + 2\nu^{3} + 8\nu^{2} + 4\nu - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + \nu^{5} + 4\nu^{4} - 6\nu^{3} + 4\nu - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - \nu^{5} - 4\nu^{4} - 2\nu^{3} + 8\nu^{2} - 4\nu + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - \nu^{5} + 2\nu^{3} - 4\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 4\nu^{6} - 3\nu^{5} - 8\nu^{4} + 10\nu^{3} - 12\nu + 40 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} + 2\nu^{6} + \nu^{5} - 6\nu^{4} + 10\nu^{3} + 4\nu^{2} - 12\nu + 32 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} - 2\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{7} - 2\beta_{6} - 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{6} - 2\beta_{5} - \beta_{4} + 6\beta_{3} + \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2\beta_{7} + 2\beta_{6} - 6\beta_{5} - \beta_{4} - 6\beta_{3} + 3\beta_{2} - 4\beta _1 + 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(43\) \(73\)
\(\chi(n)\) \(1\) \(-1\) \(1 - \beta_{3}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.0777157 + 1.41208i
−1.33790 − 0.458297i
1.40376 + 0.171630i
0.856419 − 1.12541i
0.0777157 − 1.41208i
−1.33790 + 0.458297i
1.40376 − 0.171630i
0.856419 + 1.12541i
−1.18404 − 0.773342i 0.500000 + 0.866025i 0.803884 + 1.83133i 0.380152 + 0.219481i 0.0777157 − 1.41208i 2.02350 + 1.70453i 0.464416 − 2.79004i −0.500000 + 0.866025i −0.280380 − 0.553861i
19.2 −0.272050 + 1.38780i 0.500000 + 0.866025i −1.85198 − 0.755103i 2.12403 + 1.22631i −1.33790 + 0.458297i −2.63169 + 0.272415i 1.55176 − 2.36475i −0.500000 + 0.866025i −2.27971 + 2.61411i
19.3 0.553244 − 1.30151i 0.500000 + 0.866025i −1.38784 − 1.44010i 0.834598 + 0.481855i 1.40376 − 0.171630i 1.20103 − 2.35744i −2.64212 + 1.00956i −0.500000 + 0.866025i 1.08887 − 0.819652i
19.4 1.40284 − 0.178976i 0.500000 + 0.866025i 1.93594 − 0.502151i −3.33878 − 1.92764i 0.856419 + 1.12541i −1.59285 + 2.11254i 2.62594 − 1.05092i −0.500000 + 0.866025i −5.02878 − 2.10662i
31.1 −1.18404 + 0.773342i 0.500000 − 0.866025i 0.803884 − 1.83133i 0.380152 − 0.219481i 0.0777157 + 1.41208i 2.02350 − 1.70453i 0.464416 + 2.79004i −0.500000 − 0.866025i −0.280380 + 0.553861i
31.2 −0.272050 − 1.38780i 0.500000 − 0.866025i −1.85198 + 0.755103i 2.12403 − 1.22631i −1.33790 − 0.458297i −2.63169 − 0.272415i 1.55176 + 2.36475i −0.500000 − 0.866025i −2.27971 − 2.61411i
31.3 0.553244 + 1.30151i 0.500000 − 0.866025i −1.38784 + 1.44010i 0.834598 − 0.481855i 1.40376 + 0.171630i 1.20103 + 2.35744i −2.64212 − 1.00956i −0.500000 − 0.866025i 1.08887 + 0.819652i
31.4 1.40284 + 0.178976i 0.500000 − 0.866025i 1.93594 + 0.502151i −3.33878 + 1.92764i 0.856419 − 1.12541i −1.59285 − 2.11254i 2.62594 + 1.05092i −0.500000 − 0.866025i −5.02878 + 2.10662i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.2.o.b yes 8
3.b odd 2 1 252.2.bf.f 8
4.b odd 2 1 84.2.o.a ✓ 8
7.b odd 2 1 588.2.o.b 8
7.c even 3 1 588.2.b.a 8
7.c even 3 1 588.2.o.d 8
7.d odd 6 1 84.2.o.a ✓ 8
7.d odd 6 1 588.2.b.b 8
8.b even 2 1 1344.2.bl.i 8
8.d odd 2 1 1344.2.bl.j 8
12.b even 2 1 252.2.bf.g 8
21.g even 6 1 252.2.bf.g 8
21.g even 6 1 1764.2.b.i 8
21.h odd 6 1 1764.2.b.j 8
28.d even 2 1 588.2.o.d 8
28.f even 6 1 inner 84.2.o.b yes 8
28.f even 6 1 588.2.b.a 8
28.g odd 6 1 588.2.b.b 8
28.g odd 6 1 588.2.o.b 8
56.j odd 6 1 1344.2.bl.j 8
56.m even 6 1 1344.2.bl.i 8
84.j odd 6 1 252.2.bf.f 8
84.j odd 6 1 1764.2.b.j 8
84.n even 6 1 1764.2.b.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.o.a ✓ 8 4.b odd 2 1
84.2.o.a ✓ 8 7.d odd 6 1
84.2.o.b yes 8 1.a even 1 1 trivial
84.2.o.b yes 8 28.f even 6 1 inner
252.2.bf.f 8 3.b odd 2 1
252.2.bf.f 8 84.j odd 6 1
252.2.bf.g 8 12.b even 2 1
252.2.bf.g 8 21.g even 6 1
588.2.b.a 8 7.c even 3 1
588.2.b.a 8 28.f even 6 1
588.2.b.b 8 7.d odd 6 1
588.2.b.b 8 28.g odd 6 1
588.2.o.b 8 7.b odd 2 1
588.2.o.b 8 28.g odd 6 1
588.2.o.d 8 7.c even 3 1
588.2.o.d 8 28.d even 2 1
1344.2.bl.i 8 8.b even 2 1
1344.2.bl.i 8 56.m even 6 1
1344.2.bl.j 8 8.d odd 2 1
1344.2.bl.j 8 56.j odd 6 1
1764.2.b.i 8 21.g even 6 1
1764.2.b.i 8 84.n even 6 1
1764.2.b.j 8 21.h odd 6 1
1764.2.b.j 8 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{8} + 6T_{11}^{7} - T_{11}^{6} - 78T_{11}^{5} + 125T_{11}^{4} + 156T_{11}^{3} - 212T_{11}^{2} - 240T_{11} + 400 \) acting on \(S_{2}^{\mathrm{new}}(84, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} - 11 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 6 T^{7} + \cdots + 400 \) Copy content Toggle raw display
$13$ \( T^{8} + 38 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{8} - 28 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( T^{8} - 6 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{8} - 40 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$29$ \( (T^{4} + 8 T^{3} + \cdots - 512)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 6 T^{7} + \cdots + 4173849 \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + \cdots + 355216 \) Copy content Toggle raw display
$41$ \( T^{8} + 208 T^{6} + \cdots + 350464 \) Copy content Toggle raw display
$43$ \( T^{8} + 134 T^{6} + \cdots + 1073296 \) Copy content Toggle raw display
$47$ \( T^{8} + 4 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{8} + 4 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$59$ \( T^{8} - 14 T^{7} + \cdots + 1420864 \) Copy content Toggle raw display
$61$ \( T^{8} - 12 T^{7} + \cdots + 1048576 \) Copy content Toggle raw display
$67$ \( T^{8} + 42 T^{7} + \cdots + 4129024 \) Copy content Toggle raw display
$71$ \( T^{8} + 280 T^{6} + \cdots + 200704 \) Copy content Toggle raw display
$73$ \( T^{8} + 18 T^{7} + \cdots + 952576 \) Copy content Toggle raw display
$79$ \( T^{8} - 6 T^{7} + \cdots + 241081 \) Copy content Toggle raw display
$83$ \( (T^{4} + 2 T^{3} + \cdots + 196)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 92 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$97$ \( T^{8} + 182 T^{6} + \cdots + 246016 \) Copy content Toggle raw display
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