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Properties

Label 61.2.c.a
Level $61$
Weight $2$
Character orbit 61.c
Analytic conductor $0.487$
Analytic rank $0$
Dimension $8$
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(13,61)]
 
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([4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 61.c (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: \(0.487087452330\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.310217769.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 4x^{6} - 2x^{5} + 15x^{4} - 4x^{3} + 5x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{6} - 1) q^{3} + (\beta_{7} + \beta_{6} + \beta_{4} + \cdots - \beta_1) q^{4}+ \cdots + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_{3} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{6} - 1) q^{3} + (\beta_{7} + \beta_{6} + \beta_{4} + \cdots - \beta_1) q^{4}+ \cdots + (5 \beta_{7} + 2 \beta_{6} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{3} - 2 q^{5} - q^{6} - q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{3} - 2 q^{5} - q^{6} - q^{7} - 6 q^{8} + 2 q^{9} - 2 q^{11} - 7 q^{12} + 5 q^{13} + 4 q^{14} + 8 q^{15} + 4 q^{16} + 4 q^{17} - 10 q^{18} + q^{19} + 14 q^{20} - 2 q^{21} - q^{22} - 8 q^{23} + 10 q^{24} - 8 q^{25} + 9 q^{26} - 30 q^{27} - 4 q^{28} - 16 q^{29} + 6 q^{30} + 11 q^{31} - 4 q^{32} + 10 q^{33} - 28 q^{34} + 13 q^{35} + 15 q^{36} + 4 q^{37} + 24 q^{38} + 8 q^{39} + 15 q^{40} + 4 q^{41} - 3 q^{42} + 10 q^{43} - 14 q^{44} - 27 q^{45} - q^{46} + 12 q^{48} + 19 q^{49} - 14 q^{50} - 16 q^{51} - 8 q^{52} + 38 q^{53} + 26 q^{54} - 20 q^{55} - 8 q^{56} - 12 q^{57} - 58 q^{58} + q^{59} + 20 q^{60} + 10 q^{61} + 14 q^{62} + 18 q^{63} - 26 q^{64} - 4 q^{65} - 4 q^{66} - 17 q^{67} - 27 q^{68} - 38 q^{69} - 4 q^{70} - 3 q^{71} - 38 q^{72} - 24 q^{73} + 20 q^{74} + 29 q^{75} + 15 q^{76} + 11 q^{77} + 13 q^{78} - 16 q^{79} + 16 q^{80} + 72 q^{81} + 6 q^{82} + 16 q^{83} - 10 q^{84} + 24 q^{85} - 24 q^{86} + 23 q^{87} - 20 q^{88} + 8 q^{89} - 25 q^{90} + 33 q^{92} + 8 q^{93} + 74 q^{94} - 82 q^{95} + 26 q^{96} + 13 q^{97} - 2 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 4x^{6} - 2x^{5} + 15x^{4} - 4x^{3} + 5x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -49\nu^{7} + 64\nu^{6} - 256\nu^{5} + 338\nu^{4} - 1088\nu^{3} + 1216\nu^{2} - 1385\nu + 256 ) / 289 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 60\nu^{7} + 16\nu^{6} + 225\nu^{5} - 60\nu^{4} + 884\nu^{3} + 15\nu^{2} + 304\nu + 64 ) / 289 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 63\nu^{7} - 41\nu^{6} + 164\nu^{5} - 352\nu^{4} + 697\nu^{3} - 779\nu^{2} - 490\nu - 164 ) / 289 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -64\nu^{7} + 60\nu^{6} - 240\nu^{5} + 353\nu^{4} - 1020\nu^{3} + 1140\nu^{2} - 305\nu + 240 ) / 289 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -164\nu^{7} - 63\nu^{6} - 615\nu^{5} + 164\nu^{4} - 2108\nu^{3} - 41\nu^{2} - 41\nu + 37 ) / 289 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 180\nu^{7} + 48\nu^{6} + 675\nu^{5} - 180\nu^{4} + 2363\nu^{3} + 45\nu^{2} + 45\nu + 481 ) / 289 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{2} - \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + 3\beta_{3} - 3\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{5} - 4\beta_{4} + 4\beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5\beta_{7} + \beta_{6} + 6\beta_{5} + \beta_{4} - 11\beta_{3} - 5\beta_{2} - 5\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{7} - 15\beta_{6} + 7\beta_{3} - 7\beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -30\beta_{5} - 8\beta_{4} + 23\beta_{2} + 65\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1 + \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
0.882007 − 1.52768i
0.346911 − 0.600868i
−0.198169 + 0.343239i
−1.03075 + 1.78531i
0.882007 + 1.52768i
0.346911 + 0.600868i
−0.198169 − 0.343239i
−1.03075 − 1.78531i
−0.882007 + 1.52768i 1.30887 −0.555874 − 0.962803i −0.184881 + 0.320224i −1.15444 + 1.99954i 0.283444 − 0.490940i −1.56689 −1.28685 −0.326133 − 0.564879i
13.2 −0.346911 + 0.600868i −3.26608 0.759305 + 1.31516i −1.59438 + 2.76155i 1.13304 − 1.96248i 0.720645 − 1.24819i −2.44129 7.66727 −1.10622 − 1.91602i
13.3 0.198169 − 0.343239i −0.716159 0.921458 + 1.59601i 1.82493 − 3.16087i −0.141921 + 0.245814i −1.26155 + 2.18506i 1.52310 −2.48712 −0.723289 − 1.25277i
13.4 1.03075 − 1.78531i −0.326637 −1.12489 − 1.94836i −1.04567 + 1.81115i −0.336681 + 0.583149i −0.242542 + 0.420095i −0.514916 −2.89331 2.15564 + 3.73367i
47.1 −0.882007 − 1.52768i 1.30887 −0.555874 + 0.962803i −0.184881 − 0.320224i −1.15444 − 1.99954i 0.283444 + 0.490940i −1.56689 −1.28685 −0.326133 + 0.564879i
47.2 −0.346911 − 0.600868i −3.26608 0.759305 − 1.31516i −1.59438 − 2.76155i 1.13304 + 1.96248i 0.720645 + 1.24819i −2.44129 7.66727 −1.10622 + 1.91602i
47.3 0.198169 + 0.343239i −0.716159 0.921458 − 1.59601i 1.82493 + 3.16087i −0.141921 − 0.245814i −1.26155 − 2.18506i 1.52310 −2.48712 −0.723289 + 1.25277i
47.4 1.03075 + 1.78531i −0.326637 −1.12489 + 1.94836i −1.04567 − 1.81115i −0.336681 − 0.583149i −0.242542 − 0.420095i −0.514916 −2.89331 2.15564 − 3.73367i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.c even 3 1 inner

Twists

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

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(61, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} + 3 T^{3} - 2 T^{2} + \cdots - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{8} + T^{7} + 5 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{4} + T^{3} - 12 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 5 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{7} + \cdots + 3721 \) Copy content Toggle raw display
$19$ \( T^{8} - T^{7} + \cdots + 67081 \) Copy content Toggle raw display
$23$ \( (T^{4} + 4 T^{3} + \cdots + 489)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 16 T^{7} + \cdots + 720801 \) Copy content Toggle raw display
$31$ \( T^{8} - 11 T^{7} + \cdots + 71289 \) Copy content Toggle raw display
$37$ \( (T^{4} - 2 T^{3} + \cdots - 513)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 2 T^{3} - 22 T^{2} + \cdots - 49)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 10 T^{7} + \cdots + 1849 \) Copy content Toggle raw display
$47$ \( T^{8} + 131 T^{6} + \cdots + 3087049 \) Copy content Toggle raw display
$53$ \( (T^{4} - 19 T^{3} + \cdots - 27)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - T^{7} + \cdots + 3493161 \) Copy content Toggle raw display
$61$ \( T^{8} - 10 T^{7} + \cdots + 13845841 \) Copy content Toggle raw display
$67$ \( T^{8} + 17 T^{7} + \cdots + 7656289 \) Copy content Toggle raw display
$71$ \( T^{8} + 3 T^{7} + \cdots + 134689 \) Copy content Toggle raw display
$73$ \( T^{8} + 24 T^{7} + \cdots + 19321 \) Copy content Toggle raw display
$79$ \( T^{8} + 16 T^{7} + \cdots + 859329 \) Copy content Toggle raw display
$83$ \( T^{8} - 16 T^{7} + \cdots + 529 \) Copy content Toggle raw display
$89$ \( (T^{4} - 4 T^{3} + \cdots - 1447)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 13 T^{7} + \cdots + 110481121 \) Copy content Toggle raw display
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