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Properties

Label 62.2.d.a
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: 8.0.511890625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 7x^{6} - 5x^{5} + 16x^{4} + 15x^{3} + 63x^{2} + 81x + 81 \) 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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{7} + \beta_{6} + \cdots + \beta_1) q^{3}+ \cdots + (2 \beta_{7} - \beta_{6} - 2 \beta_{4} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{7} + \beta_{6} + \cdots + \beta_1) q^{3}+ \cdots + ( - 5 \beta_{6} + 2 \beta_{5} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 4 q^{3} - 2 q^{4} - 4 q^{5} + 6 q^{6} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 4 q^{3} - 2 q^{4} - 4 q^{5} + 6 q^{6} + 2 q^{7} - 2 q^{8} + q^{10} - 2 q^{11} - 4 q^{12} - 11 q^{13} + 2 q^{14} + 21 q^{15} - 2 q^{16} - 7 q^{17} - 5 q^{18} - 14 q^{19} + q^{20} - 7 q^{21} - 2 q^{22} + 3 q^{23} + q^{24} + 14 q^{26} + 5 q^{27} + 2 q^{28} + 13 q^{29} - 14 q^{30} + 15 q^{31} + 8 q^{32} + 2 q^{33} + 3 q^{34} - 28 q^{35} + 10 q^{36} + 52 q^{37} + 21 q^{38} - 16 q^{39} + q^{40} - 11 q^{41} - 17 q^{42} - 22 q^{43} - 7 q^{44} - 19 q^{45} + 8 q^{46} - 10 q^{47} + q^{48} - 10 q^{49} - 15 q^{50} + 28 q^{51} - 11 q^{52} + 7 q^{53} + 5 q^{54} - 7 q^{55} - 8 q^{56} - 20 q^{57} - 17 q^{58} + 22 q^{59} + 21 q^{60} + 4 q^{61} + 5 q^{62} + 66 q^{63} - 2 q^{64} - 18 q^{66} - 26 q^{67} + 8 q^{68} + 4 q^{69} - 28 q^{70} - 15 q^{71} - 5 q^{72} - 29 q^{73} - 23 q^{74} - 34 q^{75} + 21 q^{76} + 34 q^{77} - q^{78} - 2 q^{79} + q^{80} - 45 q^{81} + 9 q^{82} + 38 q^{83} - 17 q^{84} + 25 q^{85} + 18 q^{86} - 20 q^{87} + 18 q^{88} + q^{89} + 46 q^{90} + 38 q^{91} - 22 q^{92} + 21 q^{93} - 20 q^{94} - 12 q^{95} - 4 q^{96} - 10 q^{97} + 60 q^{98} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 7x^{6} - 5x^{5} + 16x^{4} + 15x^{3} + 63x^{2} + 81x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 6\nu^{6} + 25\nu^{5} - 80\nu^{4} + 256\nu^{3} - 753\nu^{2} + 594\nu + 27 ) / 1728 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 30\nu^{6} - 97\nu^{5} + 176\nu^{4} - 160\nu^{3} + 273\nu^{2} + 54\nu + 621 ) / 576 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -13\nu^{7} + 66\nu^{6} - 217\nu^{5} + 416\nu^{4} - 496\nu^{3} - 51\nu^{2} - 270\nu - 459 ) / 1728 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 5\nu^{5} + 16\nu^{3} + 31\nu^{2} + 94\nu + 111 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 14\nu^{6} + 39\nu^{5} - 32\nu^{4} + 16\nu^{3} + 61\nu^{2} + 66\nu + 117 ) / 192 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -37\nu^{7} + 138\nu^{6} - 313\nu^{5} + 320\nu^{4} - 592\nu^{3} - 123\nu^{2} - 1494\nu - 459 ) / 1728 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{5} - \beta_{3} - 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{6} + 5\beta_{5} + 3\beta_{4} - 5\beta_{3} - 3\beta_{2} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} + 9\beta_{5} + 12\beta_{4} - 8\beta_{3} - 8\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 24\beta_{7} + 20\beta_{6} + 12\beta_{5} + 24\beta_{2} - 12\beta _1 - 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 60\beta_{7} + 56\beta_{6} - 60\beta_{4} + 56\beta_{3} + 96\beta_{2} - 15\beta _1 - 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 107\beta_{6} - 107\beta_{5} - 168\beta_{4} + 223\beta_{3} + 213\beta_{2} - 168 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-1 + \beta_{2} - \beta_{4} + \beta_{7}\)

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
−0.448193 + 1.37940i
0.639176 − 1.96718i
−0.892071 − 0.648127i
2.20109 + 1.59918i
−0.892071 + 0.648127i
2.20109 − 1.59918i
−0.448193 − 1.37940i
0.639176 + 1.96718i
−0.809017 − 0.587785i −2.48240 + 1.80357i 0.309017 + 0.951057i −3.34677 3.06842 0.778353 + 2.39552i 0.309017 − 0.951057i 1.98240 − 6.10121i 2.70759 + 1.96718i
33.2 −0.809017 − 0.587785i 0.364368 − 0.264729i 0.309017 + 0.951057i 2.34677 −0.450384 −1.39639 − 4.29764i 0.309017 − 0.951057i −0.864368 + 2.66025i −1.89858 − 1.37940i
35.1 0.309017 + 0.951057i −0.531724 + 1.63648i −0.809017 + 0.587785i −1.68148 −1.72069 3.90218 − 2.83510i −0.809017 − 0.587785i 0.0317236 + 0.0230486i −0.519606 − 1.59918i
35.2 0.309017 + 0.951057i 0.649758 − 1.99975i −0.809017 + 0.587785i 0.681481 2.10266 −2.28414 + 1.65953i −0.809017 − 0.587785i −1.14976 − 0.835348i 0.210589 + 0.648127i
39.1 0.309017 − 0.951057i −0.531724 − 1.63648i −0.809017 − 0.587785i −1.68148 −1.72069 3.90218 + 2.83510i −0.809017 + 0.587785i 0.0317236 − 0.0230486i −0.519606 + 1.59918i
39.2 0.309017 − 0.951057i 0.649758 + 1.99975i −0.809017 − 0.587785i 0.681481 2.10266 −2.28414 − 1.65953i −0.809017 + 0.587785i −1.14976 + 0.835348i 0.210589 − 0.648127i
47.1 −0.809017 + 0.587785i −2.48240 − 1.80357i 0.309017 − 0.951057i −3.34677 3.06842 0.778353 − 2.39552i 0.309017 + 0.951057i 1.98240 + 6.10121i 2.70759 − 1.96718i
47.2 −0.809017 + 0.587785i 0.364368 + 0.264729i 0.309017 − 0.951057i 2.34677 −0.450384 −1.39639 + 4.29764i 0.309017 + 0.951057i −0.864368 − 2.66025i −1.89858 + 1.37940i
\(n\): 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.a ✓ 8
3.b odd 2 1 558.2.i.i 8
4.b odd 2 1 496.2.n.e 8
31.d even 5 1 inner 62.2.d.a ✓ 8
31.d even 5 1 1922.2.a.r 4
31.f odd 10 1 1922.2.a.n 4
93.l odd 10 1 558.2.i.i 8
124.l odd 10 1 496.2.n.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
62.2.d.a ✓ 8 1.a even 1 1 trivial
62.2.d.a ✓ 8 31.d even 5 1 inner
496.2.n.e 8 4.b odd 2 1
496.2.n.e 8 124.l odd 10 1
558.2.i.i 8 3.b odd 2 1
558.2.i.i 8 93.l odd 10 1
1922.2.a.n 4 31.f odd 10 1
1922.2.a.r 4 31.d even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 4T_{3}^{7} + 11T_{3}^{6} + 19T_{3}^{5} + 56T_{3}^{4} + 25T_{3}^{3} + 85T_{3}^{2} - 75T_{3} + 25 \) 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
$3$ \( T^{8} + 4 T^{7} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( (T^{4} + 2 T^{3} - 8 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 2 T^{7} + \cdots + 24025 \) Copy content Toggle raw display
$11$ \( T^{8} + 2 T^{7} + \cdots + 2025 \) Copy content Toggle raw display
$13$ \( T^{8} + 11 T^{7} + \cdots + 10201 \) Copy content Toggle raw display
$17$ \( T^{8} + 7 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{8} + 14 T^{7} + \cdots + 2025 \) Copy content Toggle raw display
$23$ \( T^{8} - 3 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( T^{8} - 13 T^{7} + \cdots + 2025 \) Copy content Toggle raw display
$31$ \( T^{8} - 15 T^{7} + \cdots + 923521 \) Copy content Toggle raw display
$37$ \( (T^{4} - 26 T^{3} + \cdots + 1145)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 11 T^{7} + \cdots + 826281 \) Copy content Toggle raw display
$43$ \( T^{8} + 22 T^{7} + \cdots + 436921 \) Copy content Toggle raw display
$47$ \( T^{8} + 10 T^{7} + \cdots + 826281 \) Copy content Toggle raw display
$53$ \( T^{8} - 7 T^{7} + \cdots + 29241 \) Copy content Toggle raw display
$59$ \( T^{8} - 22 T^{7} + \cdots + 731025 \) Copy content Toggle raw display
$61$ \( (T^{4} - 2 T^{3} + \cdots + 279)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 13 T^{3} + \cdots - 139)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 15 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$73$ \( T^{8} + 29 T^{7} + \cdots + 4818025 \) Copy content Toggle raw display
$79$ \( T^{8} + 2 T^{7} + \cdots + 10272025 \) Copy content Toggle raw display
$83$ \( T^{8} - 38 T^{7} + \cdots + 3606201 \) Copy content Toggle raw display
$89$ \( T^{8} - T^{7} + \cdots + 2025 \) Copy content Toggle raw display
$97$ \( T^{8} + 10 T^{7} + \cdots + 72361 \) Copy content Toggle raw display
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