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Properties

Label 62.2.d.b
Level $62$
Weight $2$
Character orbit 62.d
Analytic conductor $0.495$
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] = [62,2,Mod(33,62)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(62, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("62.33");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 62 = 2 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 62.d (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: \(0.495072492532\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.1903140625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 6x^{6} + x^{5} + 29x^{4} + 43x^{3} + 194x^{2} + 209x + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + \beta_{7} q^{3} + ( - \beta_{6} + \beta_{3} + \beta_{2} - 1) q^{4} + (\beta_{6} + \beta_{5} - \beta_{2}) q^{5} + \beta_{4} q^{6} + (\beta_{6} - \beta_{3}) q^{7} + \beta_{3} q^{8} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + \beta_{7} q^{3} + ( - \beta_{6} + \beta_{3} + \beta_{2} - 1) q^{4} + (\beta_{6} + \beta_{5} - \beta_{2}) q^{5} + \beta_{4} q^{6} + (\beta_{6} - \beta_{3}) q^{7} + \beta_{3} q^{8} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots - 1) q^{9}+ \cdots + (\beta_{6} - 2 \beta_{5} - \beta_{4} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} - 4 q^{7} + 2 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} - 4 q^{7} + 2 q^{8} - 12 q^{9} - 5 q^{10} + 6 q^{11} - 2 q^{12} + 7 q^{13} + 4 q^{14} - 23 q^{15} - 2 q^{16} + 5 q^{17} - 3 q^{18} - 2 q^{19} - 5 q^{20} + q^{21} - 6 q^{22} - 15 q^{23} - 3 q^{24} + 16 q^{25} + 18 q^{26} + 37 q^{27} + 6 q^{28} - 19 q^{29} - 2 q^{30} + 13 q^{31} - 8 q^{32} + 30 q^{33} + 5 q^{34} + 18 q^{36} - 40 q^{37} - 3 q^{38} + 30 q^{39} + 5 q^{40} - 7 q^{41} - q^{42} + 12 q^{43} + q^{44} - 31 q^{45} - 20 q^{46} - 12 q^{47} + 3 q^{48} + 6 q^{49} + 19 q^{50} - 22 q^{51} + 7 q^{52} + 9 q^{53} - 37 q^{54} - 13 q^{55} + 4 q^{56} + 28 q^{57} - q^{58} - 18 q^{59} - 23 q^{60} + 12 q^{61} - 3 q^{62} + 6 q^{63} - 2 q^{64} - 16 q^{65} + 10 q^{66} - 26 q^{67} - 50 q^{69} - 25 q^{71} - 3 q^{72} + 35 q^{73} - 5 q^{74} + 26 q^{75} + 3 q^{76} - 8 q^{77} + 15 q^{78} + 6 q^{79} + 5 q^{80} + 43 q^{81} - 13 q^{82} + 24 q^{83} + q^{84} - q^{85} + 8 q^{86} + 8 q^{87} + 14 q^{88} - 7 q^{89} - 4 q^{90} - 16 q^{91} - 10 q^{92} + 3 q^{93} + 12 q^{94} + 30 q^{95} + 2 q^{96} + 26 q^{97} + 44 q^{98} - 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 6x^{6} + x^{5} + 29x^{4} + 43x^{3} + 194x^{2} + 209x + 361 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16228 \nu^{7} + 164686 \nu^{6} - 1074875 \nu^{5} + 2192108 \nu^{4} + 7497629 \nu^{3} + \cdots + 55384240 ) / 205159891 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 294510 \nu^{7} + 348813 \nu^{6} + 3198495 \nu^{5} - 17987557 \nu^{4} + 10405212 \nu^{3} + \cdots - 65903001 ) / 205159891 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 28143 \nu^{7} + 261345 \nu^{6} - 931213 \nu^{5} + 997158 \nu^{4} + 325779 \nu^{3} + \cdots + 5595690 ) / 10797889 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 45075 \nu^{7} - 64855 \nu^{6} + 453785 \nu^{5} - 989293 \nu^{4} - 2983130 \nu^{3} + \cdots - 23973516 ) / 10797889 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 967254 \nu^{7} - 4292904 \nu^{6} + 9536834 \nu^{5} - 8103878 \nu^{4} + 28199801 \nu^{3} + \cdots - 119034259 ) / 205159891 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 56305 \nu^{7} + 3158 \nu^{6} - 339265 \nu^{5} + 1359136 \nu^{4} + 462427 \nu^{3} + \cdots + 23665184 ) / 10797889 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{5} - \beta_{3} - 5\beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -7\beta_{7} - \beta_{6} - 7\beta_{5} - 2\beta_{4} - 5\beta_{3} - 5\beta_{2} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -12\beta_{7} - 15\beta_{6} - 15\beta_{5} - 15\beta_{4} - 22\beta_{3} + 3\beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -63\beta_{6} - 30\beta_{5} - 64\beta_{4} + 63\beta_{2} - 87 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 127\beta_{7} - 166\beta_{6} - 127\beta_{4} + 166\beta_{3} + 350\beta_{2} - 54\beta _1 - 350 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 658\beta_{7} + 365\beta_{5} + 689\beta_{3} + 1032\beta_{2} - 365\beta _1 - 689 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/62\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
0.671745 + 2.06742i
−0.480762 − 1.47963i
2.68070 − 1.94764i
−1.37168 + 0.996583i
2.68070 + 1.94764i
−1.37168 − 0.996583i
0.671745 − 2.06742i
−0.480762 + 1.47963i
0.809017 + 0.587785i −1.75865 + 1.27773i 0.309017 + 0.951057i 1.89927 −2.17381 −0.500000 − 1.53884i −0.309017 + 0.951057i 0.533195 − 1.64101i 1.53654 + 1.11636i
33.2 0.809017 + 0.587785i 1.25865 − 0.914463i 0.309017 + 0.951057i −4.13533 1.55578 −0.500000 − 1.53884i −0.309017 + 0.951057i −0.179093 + 0.551192i −3.34556 − 2.43069i
35.1 −0.309017 − 0.951057i −1.02393 + 3.15135i −0.809017 + 0.587785i 2.66590 3.31352 −0.500000 + 0.363271i 0.809017 + 0.587785i −6.45549 − 4.69019i −0.823809 − 2.53542i
35.2 −0.309017 − 0.951057i 0.523934 − 1.61250i −0.809017 + 0.587785i −0.429835 −1.69549 −0.500000 + 0.363271i 0.809017 + 0.587785i 0.101388 + 0.0736626i 0.132826 + 0.408797i
39.1 −0.309017 + 0.951057i −1.02393 − 3.15135i −0.809017 − 0.587785i 2.66590 3.31352 −0.500000 − 0.363271i 0.809017 − 0.587785i −6.45549 + 4.69019i −0.823809 + 2.53542i
39.2 −0.309017 + 0.951057i 0.523934 + 1.61250i −0.809017 − 0.587785i −0.429835 −1.69549 −0.500000 − 0.363271i 0.809017 − 0.587785i 0.101388 − 0.0736626i 0.132826 − 0.408797i
47.1 0.809017 − 0.587785i −1.75865 − 1.27773i 0.309017 − 0.951057i 1.89927 −2.17381 −0.500000 + 1.53884i −0.309017 − 0.951057i 0.533195 + 1.64101i 1.53654 − 1.11636i
47.2 0.809017 − 0.587785i 1.25865 + 0.914463i 0.309017 − 0.951057i −4.13533 1.55578 −0.500000 + 1.53884i −0.309017 − 0.951057i −0.179093 − 0.551192i −3.34556 + 2.43069i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 62.2.d.b ✓ 8
3.b odd 2 1 558.2.i.g 8
4.b odd 2 1 496.2.n.d 8
31.d even 5 1 inner 62.2.d.b ✓ 8
31.d even 5 1 1922.2.a.i 4
31.f odd 10 1 1922.2.a.l 4
93.l odd 10 1 558.2.i.g 8
124.l odd 10 1 496.2.n.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
62.2.d.b ✓ 8 1.a even 1 1 trivial
62.2.d.b ✓ 8 31.d even 5 1 inner
496.2.n.d 8 4.b odd 2 1
496.2.n.d 8 124.l odd 10 1
558.2.i.g 8 3.b odd 2 1
558.2.i.g 8 93.l odd 10 1
1922.2.a.i 4 31.d even 5 1
1922.2.a.l 4 31.f odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 2T_{3}^{7} + 11T_{3}^{6} + T_{3}^{5} + 14T_{3}^{4} + 13T_{3}^{3} + 99T_{3}^{2} - 171T_{3} + 361 \) acting on \(S_{2}^{\mathrm{new}}(62, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + \cdots + 361 \) Copy content Toggle raw display
$5$ \( (T^{4} - 14 T^{2} + 15 T + 9)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 6 T^{7} + \cdots + 9801 \) Copy content Toggle raw display
$13$ \( T^{8} - 7 T^{7} + \cdots + 7921 \) Copy content Toggle raw display
$17$ \( T^{8} - 5 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{8} + 2 T^{7} + \cdots + 281961 \) Copy content Toggle raw display
$23$ \( T^{8} + 15 T^{7} + \cdots + 50625 \) Copy content Toggle raw display
$29$ \( T^{8} + 19 T^{7} + \cdots + 9801 \) Copy content Toggle raw display
$31$ \( T^{8} - 13 T^{7} + \cdots + 923521 \) Copy content Toggle raw display
$37$ \( (T^{4} + 20 T^{3} + \cdots + 209)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 7 T^{7} + \cdots + 9801 \) Copy content Toggle raw display
$43$ \( T^{8} - 12 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$47$ \( (T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 9 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$59$ \( T^{8} + 18 T^{7} + \cdots + 531441 \) Copy content Toggle raw display
$61$ \( (T^{4} - 6 T^{3} + \cdots + 251)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 13 T^{3} + \cdots + 171)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 25 T^{7} + \cdots + 2954961 \) Copy content Toggle raw display
$73$ \( T^{8} - 35 T^{7} + \cdots + 600625 \) Copy content Toggle raw display
$79$ \( T^{8} - 6 T^{7} + \cdots + 9801 \) Copy content Toggle raw display
$83$ \( (T^{4} - 12 T^{3} + \cdots + 81)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 7 T^{7} + \cdots + 160807761 \) Copy content Toggle raw display
$97$ \( T^{8} - 26 T^{7} + \cdots + 83521 \) Copy content Toggle raw display
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