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Properties

Label 660.2.y.c
Level $660$
Weight $2$
Character orbit 660.y
Analytic conductor $5.270$
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] = [660,2,Mod(181,660)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(660, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("660.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 660 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 660.y (of order \(5\), degree \(4\), 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: \(5.27012653340\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.819390625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 10x^{6} - 13x^{5} + 29x^{4} - 7x^{3} + 80x^{2} + 143x + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_{3} + \beta_{2} + 1) q^{3} - \beta_{6} q^{5} + (\beta_{7} - \beta_{5} - \beta_{3}) q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_{3} + \beta_{2} + 1) q^{3} - \beta_{6} q^{5} + (\beta_{7} - \beta_{5} - \beta_{3}) q^{7} + \beta_{2} q^{9} + (\beta_{7} + \beta_{5} + \beta_{3} + \cdots - 1) q^{11}+ \cdots + (\beta_{6} - \beta_{5} - 2 \beta_{2} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 2 q^{5} + 3 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} - 2 q^{5} + 3 q^{7} - 2 q^{9} - 7 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{17} + 10 q^{19} + 12 q^{21} + 10 q^{23} - 2 q^{25} + 2 q^{27} + 4 q^{29} - 9 q^{31} - 8 q^{33} + 3 q^{35} + 7 q^{37} - 2 q^{39} - q^{41} + 4 q^{43} + 8 q^{45} + q^{47} + 21 q^{49} + 6 q^{51} - 19 q^{53} + 3 q^{55} - 5 q^{57} + 9 q^{59} - 35 q^{61} + 3 q^{63} + 2 q^{65} + 18 q^{67} - 5 q^{69} + 25 q^{71} - 19 q^{73} + 2 q^{75} - 47 q^{77} + 18 q^{79} - 2 q^{81} - 11 q^{83} - 6 q^{85} + 6 q^{87} + 8 q^{89} - 7 q^{91} + 9 q^{93} + 10 q^{95} - 9 q^{97} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 10x^{6} - 13x^{5} + 29x^{4} - 7x^{3} + 80x^{2} + 143x + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -256\nu^{7} + 341\nu^{6} + 3310\nu^{5} - 16865\nu^{4} + 32996\nu^{3} - 59433\nu^{2} + 33270\nu - 118459 ) / 171589 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -620\nu^{7} + 2532\nu^{6} - 9045\nu^{5} + 18870\nu^{4} - 41955\nu^{3} + 81515\nu^{2} - 102225\nu - 16104 ) / 171589 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -672\nu^{7} + 2845\nu^{6} - 10810\nu^{5} + 23975\nu^{4} - 77175\nu^{3} + 52625\nu^{2} - 72556\nu - 75020 ) / 171589 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1187 \nu^{7} + 4445 \nu^{6} - 17191 \nu^{5} + 14238 \nu^{4} - 12015 \nu^{3} - 32510 \nu^{2} + \cdots - 143748 ) / 171589 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1188 \nu^{7} - 4751 \nu^{6} + 16325 \nu^{5} - 32635 \nu^{4} + 48690 \nu^{3} - 20331 \nu^{2} + \cdots + 144881 ) / 171589 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1432 \nu^{7} + 1420 \nu^{6} - 7808 \nu^{5} - 2207 \nu^{4} - 27965 \nu^{3} - 33635 \nu^{2} + \cdots - 249744 ) / 171589 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{4} + 3\beta_{3} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{6} - 5\beta_{4} + 2\beta_{3} - \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 15\beta_{6} - 8\beta_{5} - 8\beta_{4} - 5\beta_{3} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{7} - 25\beta_{6} - 28\beta_{5} + 25\beta_{2} - 2\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -55\beta_{7} + 55\beta_{4} + 22\beta_{3} + 110\beta_{2} - 45\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -165\beta_{7} + 175\beta_{6} + 165\beta_{5} + 188\beta_{4} - 80\beta_{3} - 188\beta _1 - 175 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/660\mathbb{Z}\right)^\times\).

\(n\) \(221\) \(331\) \(397\) \(541\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
−0.575405 − 1.77091i
0.766388 + 2.35870i
2.06426 + 1.49977i
−0.755243 − 0.548716i
−0.575405 + 1.77091i
0.766388 − 2.35870i
2.06426 − 1.49977i
−0.755243 + 0.548716i
0 0.809017 + 0.587785i 0 0.309017 − 0.951057i 0 −1.62844 + 1.18313i 0 0.309017 + 0.951057i 0
181.2 0 0.809017 + 0.587785i 0 0.309017 − 0.951057i 0 4.05549 − 2.94648i 0 0.309017 + 0.951057i 0
301.1 0 −0.309017 + 0.951057i 0 −0.809017 + 0.587785i 0 −0.796323 − 2.45083i 0 −0.809017 − 0.587785i 0
301.2 0 −0.309017 + 0.951057i 0 −0.809017 + 0.587785i 0 −0.130728 − 0.402341i 0 −0.809017 − 0.587785i 0
361.1 0 0.809017 − 0.587785i 0 0.309017 + 0.951057i 0 −1.62844 − 1.18313i 0 0.309017 − 0.951057i 0
361.2 0 0.809017 − 0.587785i 0 0.309017 + 0.951057i 0 4.05549 + 2.94648i 0 0.309017 − 0.951057i 0
421.1 0 −0.309017 − 0.951057i 0 −0.809017 − 0.587785i 0 −0.796323 + 2.45083i 0 −0.809017 + 0.587785i 0
421.2 0 −0.309017 − 0.951057i 0 −0.809017 − 0.587785i 0 −0.130728 + 0.402341i 0 −0.809017 + 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 660.2.y.c ✓ 8
3.b odd 2 1 1980.2.z.f 8
11.c even 5 1 inner 660.2.y.c ✓ 8
11.c even 5 1 7260.2.a.bc 4
11.d odd 10 1 7260.2.a.bd 4
33.h odd 10 1 1980.2.z.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
660.2.y.c ✓ 8 1.a even 1 1 trivial
660.2.y.c ✓ 8 11.c even 5 1 inner
1980.2.z.f 8 3.b odd 2 1
1980.2.z.f 8 33.h odd 10 1
7260.2.a.bc 4 11.c even 5 1
7260.2.a.bd 4 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 3T_{7}^{7} + T_{7}^{6} + 21T_{7}^{5} + 204T_{7}^{4} + 543T_{7}^{3} + 839T_{7}^{2} + 264T_{7} + 121 \) acting on \(S_{2}^{\mathrm{new}}(660, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 3 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( T^{8} + 7 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} - 2 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{7} + \cdots + 36481 \) Copy content Toggle raw display
$19$ \( T^{8} - 10 T^{7} + \cdots + 9801 \) Copy content Toggle raw display
$23$ \( (T^{4} - 5 T^{3} + \cdots + 275)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 4 T^{7} + \cdots + 109561 \) Copy content Toggle raw display
$31$ \( T^{8} + 9 T^{7} + \cdots + 841 \) Copy content Toggle raw display
$37$ \( T^{8} - 7 T^{7} + \cdots + 923521 \) Copy content Toggle raw display
$41$ \( T^{8} + T^{7} + \cdots + 57121 \) Copy content Toggle raw display
$43$ \( (T^{4} - 2 T^{3} + \cdots + 1301)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - T^{7} + \cdots + 121 \) Copy content Toggle raw display
$53$ \( T^{8} + 19 T^{7} + \cdots + 1874161 \) Copy content Toggle raw display
$59$ \( T^{8} - 9 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{8} + 35 T^{7} + \cdots + 26409321 \) Copy content Toggle raw display
$67$ \( (T^{4} - 9 T^{3} + \cdots - 5679)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 25 T^{7} + \cdots + 2758921 \) Copy content Toggle raw display
$73$ \( T^{8} + 19 T^{7} + \cdots + 72361 \) Copy content Toggle raw display
$79$ \( T^{8} - 18 T^{7} + \cdots + 192721 \) Copy content Toggle raw display
$83$ \( T^{8} + 11 T^{7} + \cdots + 48288601 \) Copy content Toggle raw display
$89$ \( (T^{4} - 4 T^{3} - 239 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 9 T^{7} + \cdots + 3463321 \) Copy content Toggle raw display
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