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Properties

Label 61.2.f.b
Level $61$
Weight $2$
Character orbit 61.f
Analytic conductor $0.487$
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] = [61,2,Mod(14,61)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(61, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.14");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 61.f (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: \(0.487087452330\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.542936601.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 2x^{6} - 4x^{5} + x^{4} - 8x^{3} + 8x^{2} - 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
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.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_1) q^{2} + (\beta_{3} - \beta_1) q^{3} + (2 \beta_{7} - \beta_{6} + \cdots - \beta_{2}) q^{4}+ \cdots + (\beta_{7} - \beta_{6} + \cdots + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_1) q^{2} + (\beta_{3} - \beta_1) q^{3} + (2 \beta_{7} - \beta_{6} + \cdots - \beta_{2}) q^{4}+ \cdots + (\beta_{7} - \beta_{6} - 4 \beta_{4} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 8 q^{4} + 6 q^{5} - 21 q^{6} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} + 8 q^{4} + 6 q^{5} - 21 q^{6} - 3 q^{7} + 2 q^{9} - 6 q^{10} + q^{12} - 3 q^{13} - 6 q^{14} - 6 q^{15} - 20 q^{16} + 6 q^{17} + 12 q^{18} + 3 q^{19} + 30 q^{20} + 6 q^{21} + 5 q^{22} - 4 q^{25} - 15 q^{26} - 14 q^{27} - 42 q^{30} - 3 q^{31} + 48 q^{32} + 8 q^{34} + 3 q^{35} + 35 q^{36} + 18 q^{39} + 21 q^{40} - 24 q^{41} + 27 q^{42} - 18 q^{44} + 9 q^{45} - 17 q^{46} + 12 q^{47} - 22 q^{48} - 13 q^{49} - 12 q^{51} - 12 q^{52} - 18 q^{54} - 6 q^{55} - 6 q^{56} + 22 q^{58} - 3 q^{59} - 12 q^{60} - 8 q^{61} + 66 q^{62} - 36 q^{63} - 98 q^{64} + 24 q^{65} + 10 q^{66} - 15 q^{67} - 57 q^{68} - 36 q^{70} + 15 q^{71} - 16 q^{73} + 6 q^{74} + 13 q^{75} + 21 q^{76} + 3 q^{77} + 33 q^{78} + 42 q^{79} + 42 q^{80} + 8 q^{81} + 60 q^{82} - 42 q^{83} - 42 q^{86} + 15 q^{87} + 84 q^{88} + 63 q^{90} - 69 q^{92} - 48 q^{93} + 54 q^{95} + 102 q^{96} + 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - \nu^{6} + 7\nu^{3} + 6\nu^{2} + 8\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - \nu^{6} + 2\nu^{5} - 4\nu^{4} + \nu^{3} - 8\nu^{2} + 8\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 3\nu^{5} + \nu^{3} - 7\nu^{2} - 2\nu - 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + \nu^{6} + 4\nu^{5} + \nu^{3} - 10\nu^{2} - 8\nu - 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} + \nu^{6} + 6\nu^{5} + 3\nu^{3} - 20\nu^{2} - 4\nu - 32 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 2\nu^{5} - \nu^{4} + \nu^{3} - 5\nu^{2} + \nu - 8 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} - \beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{4} + \beta_{2} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{3} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} - 3\beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{3} + 2\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{7} + 3\beta_{5} - 6\beta_{4} - 2\beta_{3} + 4\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4\beta_{7} + \beta_{6} - 3\beta_{5} - \beta_{4} - 4\beta_{3} - \beta_{2} - 3\beta _1 - 1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/61\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(1 + \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1
−1.06969 + 0.925071i
0.295677 + 1.38296i
−0.139771 − 1.40729i
1.41379 − 0.0347146i
−1.06969 − 0.925071i
0.295677 − 1.38296i
−0.139771 + 1.40729i
1.41379 + 0.0347146i
−2.40567 + 1.38892i 2.13939 2.85818 − 4.95052i 0.766288 + 1.32725i −5.14667 + 2.97143i −1.02220 + 0.590167i 10.3234i 1.57698 −3.68688 − 2.12862i
14.2 −0.754161 + 0.435415i −0.591355 −0.620827 + 1.07530i 1.84552 + 3.19653i 0.445977 − 0.257485i 2.17067 − 1.25323i − 2.82293i −2.65030 −2.78363 − 1.60713i
14.3 1.00909 − 0.582599i 0.279543 −0.321156 + 0.556259i −0.788634 − 1.36595i 0.282084 − 0.162861i −0.327706 + 0.189201i 3.07882i −2.92186 −1.59161 − 0.918915i
14.4 2.15074 − 1.24173i −2.82757 2.08380 − 3.60925i 1.17683 + 2.03833i −6.08139 + 3.51109i −2.32076 + 1.33989i − 5.38317i 4.99518 5.06212 + 2.92262i
48.1 −2.40567 − 1.38892i 2.13939 2.85818 + 4.95052i 0.766288 − 1.32725i −5.14667 − 2.97143i −1.02220 − 0.590167i − 10.3234i 1.57698 −3.68688 + 2.12862i
48.2 −0.754161 − 0.435415i −0.591355 −0.620827 − 1.07530i 1.84552 − 3.19653i 0.445977 + 0.257485i 2.17067 + 1.25323i 2.82293i −2.65030 −2.78363 + 1.60713i
48.3 1.00909 + 0.582599i 0.279543 −0.321156 − 0.556259i −0.788634 + 1.36595i 0.282084 + 0.162861i −0.327706 − 0.189201i − 3.07882i −2.92186 −1.59161 + 0.918915i
48.4 2.15074 + 1.24173i −2.82757 2.08380 + 3.60925i 1.17683 − 2.03833i −6.08139 − 3.51109i −2.32076 − 1.33989i 5.38317i 4.99518 5.06212 − 2.92262i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 61.2.f.b ✓ 8
3.b odd 2 1 549.2.s.j 8
4.b odd 2 1 976.2.ba.c 8
61.f even 6 1 inner 61.2.f.b ✓ 8
61.h odd 12 2 3721.2.a.i 8
183.i odd 6 1 549.2.s.j 8
244.i odd 6 1 976.2.ba.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.2.f.b ✓ 8 1.a even 1 1 trivial
61.2.f.b ✓ 8 61.f even 6 1 inner
549.2.s.j 8 3.b odd 2 1
549.2.s.j 8 183.i odd 6 1
976.2.ba.c 8 4.b odd 2 1
976.2.ba.c 8 244.i odd 6 1
3721.2.a.i 8 61.h odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 8T_{2}^{6} + 57T_{2}^{4} - 24T_{2}^{3} - 53T_{2}^{2} + 21T_{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(61, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 8 T^{6} + \cdots + 49 \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} - 6 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 6 T^{7} + \cdots + 441 \) Copy content Toggle raw display
$7$ \( T^{8} + 3 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( T^{8} + 19 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{8} + 3 T^{7} + \cdots + 3969 \) Copy content Toggle raw display
$17$ \( T^{8} - 6 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$19$ \( T^{8} - 3 T^{7} + \cdots + 441 \) Copy content Toggle raw display
$23$ \( T^{8} + 94 T^{6} + \cdots + 43681 \) Copy content Toggle raw display
$29$ \( T^{8} - 50 T^{6} + \cdots + 38809 \) Copy content Toggle raw display
$31$ \( T^{8} + 3 T^{7} + \cdots + 47961 \) Copy content Toggle raw display
$37$ \( T^{8} + 120 T^{6} + \cdots + 16641 \) Copy content Toggle raw display
$41$ \( (T^{4} + 12 T^{3} + \cdots - 147)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 90 T^{6} + \cdots + 4761 \) Copy content Toggle raw display
$47$ \( T^{8} - 12 T^{7} + \cdots + 5625 \) Copy content Toggle raw display
$53$ \( T^{8} + 223 T^{6} + \cdots + 1907161 \) Copy content Toggle raw display
$59$ \( T^{8} + 3 T^{7} + \cdots + 1274641 \) Copy content Toggle raw display
$61$ \( T^{8} + 8 T^{7} + \cdots + 13845841 \) Copy content Toggle raw display
$67$ \( T^{8} + 15 T^{7} + \cdots + 1447209 \) Copy content Toggle raw display
$71$ \( T^{8} - 15 T^{7} + \cdots + 3721 \) Copy content Toggle raw display
$73$ \( T^{8} + 16 T^{7} + \cdots + 49 \) Copy content Toggle raw display
$79$ \( T^{8} - 42 T^{7} + \cdots + 2350089 \) Copy content Toggle raw display
$83$ \( T^{8} + 42 T^{7} + \cdots + 126900225 \) Copy content Toggle raw display
$89$ \( T^{8} + 220 T^{6} + \cdots + 2455489 \) Copy content Toggle raw display
$97$ \( T^{8} - 3 T^{7} + \cdots + 1199025 \) Copy content Toggle raw display
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