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Properties

Label 62.5.b.a
Level $62$
Weight $5$
Character orbit 62.b
Analytic conductor $6.409$
Analytic rank $0$
Dimension $12$
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] = [62,5,Mod(61,62)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(62, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("62.61");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 62 = 2 \cdot 31 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 62.b (of order \(2\), degree \(1\), 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: \(6.40893771120\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 386x^{10} + 54113x^{8} + 3311904x^{6} + 82889694x^{4} + 608473296x^{2} + 747965664 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{3} q^{3} + 8 q^{4} + ( - \beta_{4} - \beta_{2} + 1) q^{5} + \beta_1 q^{6} + ( - \beta_{6} - 2 \beta_{2} + 18) q^{7} + 8 \beta_{2} q^{8} + (\beta_{6} + \beta_{5} - 5 \beta_{2} - 48) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_{3} q^{3} + 8 q^{4} + ( - \beta_{4} - \beta_{2} + 1) q^{5} + \beta_1 q^{6} + ( - \beta_{6} - 2 \beta_{2} + 18) q^{7} + 8 \beta_{2} q^{8} + (\beta_{6} + \beta_{5} - 5 \beta_{2} - 48) q^{9} + ( - \beta_{7} + \beta_{6} + \beta_{2} - 6) q^{10} + (\beta_{9} + 3 \beta_{3}) q^{11} + 8 \beta_{3} q^{12} + ( - \beta_{9} - \beta_{8} + \cdots + \beta_1) q^{13}+ \cdots + (24 \beta_{11} - 11 \beta_{10} + \cdots + 225 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 96 q^{4} + 12 q^{5} + 212 q^{7} - 572 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 96 q^{4} + 12 q^{5} + 212 q^{7} - 572 q^{9} - 64 q^{10} - 192 q^{14} + 768 q^{16} - 512 q^{18} - 212 q^{19} + 96 q^{20} + 3352 q^{25} + 1696 q^{28} - 1116 q^{31} - 3912 q^{33} - 2844 q^{35} - 4576 q^{36} + 4032 q^{38} - 5368 q^{39} - 512 q^{40} - 2748 q^{41} + 4124 q^{45} + 11328 q^{47} + 2984 q^{49} + 4992 q^{50} + 7520 q^{51} - 1536 q^{56} - 7620 q^{59} - 960 q^{62} - 35276 q^{63} + 6144 q^{64} + 4096 q^{66} + 19752 q^{67} - 7216 q^{69} - 31296 q^{70} + 17724 q^{71} - 4096 q^{72} - 1696 q^{76} - 14144 q^{78} + 768 q^{80} + 29700 q^{81} + 9344 q^{82} + 38776 q^{87} + 48832 q^{90} - 21584 q^{93} + 7616 q^{94} + 28116 q^{95} - 42732 q^{97} - 21120 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 386x^{10} + 54113x^{8} + 3311904x^{6} + 82889694x^{4} + 608473296x^{2} + 747965664 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1561 \nu^{10} + 1987450 \nu^{8} + 651523279 \nu^{6} + 61626496188 \nu^{4} + \cdots + 7737750837720 ) / 1770855095304 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1561 \nu^{11} + 1987450 \nu^{9} + 651523279 \nu^{7} + 61626496188 \nu^{5} + \cdots + 7737750837720 \nu ) / 3541710190608 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 384521477 \nu^{10} + 141534730906 \nu^{8} + 17309279862517 \nu^{6} + 777244880599968 \nu^{4} + \cdots + 30\!\cdots\!16 ) / 22\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1002711091 \nu^{10} + 358858038566 \nu^{8} + 45954089898755 \nu^{6} + \cdots + 47\!\cdots\!96 ) / 22\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 506315623 \nu^{10} - 173113896908 \nu^{8} - 20906829730355 \nu^{6} + \cdots - 65\!\cdots\!80 ) / 11\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 399402139 \nu^{10} - 145398339803 \nu^{8} - 18848157085013 \nu^{6} + \cdots - 49\!\cdots\!30 ) / 562689206532846 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1515751115 \nu^{11} - 561382598194 \nu^{9} - 73270290457363 \nu^{7} + \cdots + 12\!\cdots\!68 \nu ) / 45\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1972527073 \nu^{11} + 751516701494 \nu^{9} + 102661846375577 \nu^{7} + \cdots + 84\!\cdots\!28 \nu ) / 45\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 968428303 \nu^{11} - 358335895114 \nu^{9} - 47175027288663 \nu^{7} + \cdots - 19\!\cdots\!56 \nu ) / 15\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 638983959 \nu^{11} + 245445815770 \nu^{9} + 34292942015423 \nu^{7} + \cdots + 25\!\cdots\!48 \nu ) / 500168183584752 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{5} - 5\beta_{2} - 129 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} + 5\beta_{10} + 11\beta_{9} + \beta_{8} - 48\beta_{3} - 202\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 37\beta_{7} - 170\beta_{6} - 117\beta_{5} + 16\beta_{4} + 943\beta_{2} + 13659 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 101\beta_{11} - 993\beta_{10} - 1599\beta_{9} + 195\beta_{8} + 8136\beta_{3} + 23138\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -15292\beta_{7} + 47248\beta_{6} + 27779\beta_{5} - 12925\beta_{4} - 267032\beta_{2} - 3211056 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3053\beta_{11} + 158357\beta_{10} + 202919\beta_{9} - 49367\beta_{8} - 1279344\beta_{3} - 2801342\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1218527\beta_{7} - 3104838\beta_{6} - 1702490\beta_{5} + 1423229\beta_{4} + 18892509\beta_{2} + 197367909 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 960147 \beta_{11} - 23265637 \beta_{10} - 24941203 \beta_{9} + 8229547 \beta_{8} + \cdots + 349186938 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 358238496 \beta_{7} + 797397772 \beta_{6} + 427183129 \beta_{5} - 507173319 \beta_{4} + \cdots - 49724408040 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 287593681 \beta_{11} + 3285624079 \beta_{10} + 3045653509 \beta_{9} - 1225350493 \beta_{8} + \cdots - 44275529114 \beta_1 ) / 8 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
10.9760i
6.36279i
3.06836i
− 3.06836i
− 6.36279i
− 10.9760i
− 11.6360i
− 8.87141i
− 1.23636i
1.23636i
8.87141i
11.6360i
−2.82843 − 15.5224i 8.00000 27.1078 43.9040i 95.2540 −22.6274 −159.945 −76.6725
61.2 −2.82843 − 8.99834i 8.00000 −35.3666 25.4511i 1.89818 −22.6274 0.0299154 100.032
61.3 −2.82843 − 4.33931i 8.00000 16.9156 12.2734i −27.1816 −22.6274 62.1704 −47.8446
61.4 −2.82843 4.33931i 8.00000 16.9156 − 12.2734i −27.1816 −22.6274 62.1704 −47.8446
61.5 −2.82843 8.99834i 8.00000 −35.3666 − 25.4511i 1.89818 −22.6274 0.0299154 100.032
61.6 −2.82843 15.5224i 8.00000 27.1078 − 43.9040i 95.2540 −22.6274 −159.945 −76.6725
61.7 2.82843 − 16.4558i 8.00000 −42.1186 − 46.5441i 41.5993 22.6274 −189.794 −119.129
61.8 2.82843 − 12.5461i 8.00000 37.0751 − 35.4856i −49.3479 22.6274 −76.4039 104.864
61.9 2.82843 − 1.74848i 8.00000 2.38662 − 4.94545i 43.7781 22.6274 77.9428 6.75039
61.10 2.82843 1.74848i 8.00000 2.38662 4.94545i 43.7781 22.6274 77.9428 6.75039
61.11 2.82843 12.5461i 8.00000 37.0751 35.4856i −49.3479 22.6274 −76.4039 104.864
61.12 2.82843 16.4558i 8.00000 −42.1186 46.5441i 41.5993 22.6274 −189.794 −119.129
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 62.5.b.a ✓ 12
3.b odd 2 1 558.5.d.a 12
4.b odd 2 1 496.5.e.d 12
31.b odd 2 1 inner 62.5.b.a ✓ 12
93.c even 2 1 558.5.d.a 12
124.d even 2 1 496.5.e.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
62.5.b.a ✓ 12 1.a even 1 1 trivial
62.5.b.a ✓ 12 31.b odd 2 1 inner
496.5.e.d 12 4.b odd 2 1
496.5.e.d 12 124.d even 2 1
558.5.d.a 12 3.b odd 2 1
558.5.d.a 12 93.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(62, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 8)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 47869802496 \) Copy content Toggle raw display
$5$ \( (T^{6} - 6 T^{5} + \cdots + 60438672)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 106 T^{5} + \cdots + 441679072)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 215757707746032)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 45\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 62\!\cdots\!21 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 63\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 74\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 34\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 13\!\cdots\!64)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 84\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 75\!\cdots\!28)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 60\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 33\!\cdots\!04)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 18\!\cdots\!72)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 59\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 58\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 40\!\cdots\!28)^{2} \) Copy content Toggle raw display
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