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Properties

Label 650.2.t.e
Level $650$
Weight $2$
Character orbit 650.t
Analytic conductor $5.190$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [650,2,Mod(7,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([3, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.t (of order \(12\), 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: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 24x^{10} + 192x^{8} + 680x^{6} + 1104x^{4} + 672x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + \beta_{8} q^{3} + ( - \beta_{3} + 1) q^{4} + (\beta_{11} + \beta_{8} + \cdots + \beta_{5}) q^{6} + (\beta_{5} - \beta_1) q^{7} + (\beta_{6} + \beta_{4}) q^{8} + (2 \beta_{10} - 2 \beta_{9} + \beta_{8} + \cdots - 4) q^{9}+ \cdots + (\beta_{10} - 2 \beta_{8} - \beta_{7} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{4} - 24 q^{9} + 6 q^{11} + 12 q^{13} - 6 q^{16} - 12 q^{17} - 12 q^{18} + 36 q^{19} - 24 q^{21} - 6 q^{22} - 6 q^{23} + 6 q^{26} + 12 q^{27} + 6 q^{29} - 24 q^{31} + 6 q^{33} - 12 q^{34} - 24 q^{36}+ \cdots + 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 24x^{10} + 192x^{8} + 680x^{6} + 1104x^{4} + 672x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 7\nu^{10} + 136\nu^{8} + 668\nu^{6} + 702\nu^{4} + 176\nu^{2} + 380\nu + 1728 ) / 760 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{10} + 136\nu^{8} + 668\nu^{6} + 702\nu^{4} + 176\nu^{2} - 380\nu + 1728 ) / 760 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{11} + 87\nu^{9} + 576\nu^{7} + 1504\nu^{5} + 1542\nu^{3} + 456\nu + 20 ) / 40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 60 \nu^{11} + 36 \nu^{10} - 1315 \nu^{9} + 808 \nu^{8} - 8820 \nu^{7} + 5634 \nu^{6} - 23280 \nu^{5} + \cdots + 744 ) / 760 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11 \nu^{11} + 89 \nu^{10} - 173 \nu^{9} + 1892 \nu^{8} - 154 \nu^{7} + 11886 \nu^{6} + 5194 \nu^{5} + \cdots + 1016 ) / 760 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 60 \nu^{11} - 36 \nu^{10} - 1315 \nu^{9} - 808 \nu^{8} - 8820 \nu^{7} - 5634 \nu^{6} - 23280 \nu^{5} + \cdots - 744 ) / 760 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11 \nu^{11} + 82 \nu^{10} + 173 \nu^{9} + 1756 \nu^{8} + 154 \nu^{7} + 11218 \nu^{6} - 5194 \nu^{5} + \cdots - 712 ) / 760 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 36 \nu^{11} + 125 \nu^{10} - 808 \nu^{9} + 2700 \nu^{8} - 5634 \nu^{7} + 17520 \nu^{6} - 15336 \nu^{5} + \cdots + 240 ) / 760 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3 \nu^{11} - 134 \nu^{10} + 99 \nu^{9} - 2902 \nu^{8} + 1182 \nu^{7} - 18976 \nu^{6} + 6218 \nu^{5} + \cdots - 1756 ) / 760 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 105 \nu^{11} - 178 \nu^{10} - 2135 \nu^{9} - 3784 \nu^{8} - 11920 \nu^{7} - 23772 \nu^{6} + \cdots - 132 ) / 760 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -421\nu^{10} - 9048\nu^{8} - 58144\nu^{6} - 140586\nu^{4} - 112968\nu^{2} + 380\nu - 784 ) / 760 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + 2\beta_{7} - \beta_{6} + 2\beta_{5} + \beta_{4} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{11} - 4 \beta_{8} - 2 \beta_{7} + 5 \beta_{6} - 2 \beta_{5} + 5 \beta_{4} + 6 \beta_{3} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 13 \beta_{11} + 2 \beta_{10} - 4 \beta_{9} + 2 \beta_{8} - 27 \beta_{7} + 14 \beta_{6} - 29 \beta_{5} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 32 \beta_{11} - 5 \beta_{10} + 59 \beta_{8} + 24 \beta_{7} - 77 \beta_{6} + 35 \beta_{5} - 72 \beta_{4} + \cdots + 50 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 168 \beta_{11} - 24 \beta_{10} + 48 \beta_{9} - 24 \beta_{8} + 352 \beta_{7} - 174 \beta_{6} + \cdots - 292 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 422 \beta_{11} + 96 \beta_{10} - 748 \beta_{8} - 270 \beta_{7} + 1008 \beta_{6} - 478 \beta_{5} + \cdots - 680 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2122 \beta_{11} + 270 \beta_{10} - 540 \beta_{9} + 270 \beta_{8} - 4460 \beta_{7} + 2132 \beta_{6} + \cdots + 3460 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5316 \beta_{11} - 1368 \beta_{10} + 9264 \beta_{8} + 3156 \beta_{7} - 12698 \beta_{6} + 6108 \beta_{5} + \cdots + 8706 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 26474 \beta_{11} - 3156 \beta_{10} + 6312 \beta_{9} - 3156 \beta_{8} + 55718 \beta_{7} - 26196 \beta_{6} + \cdots - 42312 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 66116 \beta_{11} + 17810 \beta_{10} - 114422 \beta_{8} - 38016 \beta_{7} + 158054 \beta_{6} + \cdots - 109084 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(\beta_{4} + \beta_{6}\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
1.62980i
− 0.0775341i
− 1.55227i
− 1.62980i
0.0775341i
1.55227i
2.19540i
1.32083i
− 3.51623i
− 2.19540i
− 1.32083i
3.51623i
0.866025 + 0.500000i −0.814901 − 3.04125i 0.500000 + 0.866025i 0 0.814901 − 3.04125i −0.402397 − 0.696972i 1.00000i −5.98707 + 3.45663i 0
7.2 0.866025 + 0.500000i 0.0387670 + 0.144681i 0.500000 + 0.866025i 0 −0.0387670 + 0.144681i −1.10259 − 1.90974i 1.00000i 2.57865 − 1.48878i 0
7.3 0.866025 + 0.500000i 0.776134 + 2.89657i 0.500000 + 0.866025i 0 −0.776134 + 2.89657i 0.638961 + 1.10671i 1.00000i −5.18966 + 2.99625i 0
93.1 0.866025 − 0.500000i −0.814901 + 3.04125i 0.500000 − 0.866025i 0 0.814901 + 3.04125i −0.402397 + 0.696972i − 1.00000i −5.98707 − 3.45663i 0
93.2 0.866025 − 0.500000i 0.0387670 − 0.144681i 0.500000 − 0.866025i 0 −0.0387670 − 0.144681i −1.10259 + 1.90974i − 1.00000i 2.57865 + 1.48878i 0
93.3 0.866025 − 0.500000i 0.776134 − 2.89657i 0.500000 − 0.866025i 0 −0.776134 − 2.89657i 0.638961 − 1.10671i − 1.00000i −5.18966 − 2.99625i 0
557.1 −0.866025 + 0.500000i −1.09770 − 0.294128i 0.500000 − 0.866025i 0 1.09770 − 0.294128i 2.54283 − 4.40431i 1.00000i −1.47964 − 0.854273i 0
557.2 −0.866025 + 0.500000i −0.660414 − 0.176957i 0.500000 − 0.866025i 0 0.660414 − 0.176957i −0.189447 + 0.328132i 1.00000i −2.19324 − 1.26627i 0
557.3 −0.866025 + 0.500000i 1.75811 + 0.471085i 0.500000 − 0.866025i 0 −1.75811 + 0.471085i −1.48736 + 2.57618i 1.00000i 0.270964 + 0.156441i 0
643.1 −0.866025 − 0.500000i −1.09770 + 0.294128i 0.500000 + 0.866025i 0 1.09770 + 0.294128i 2.54283 + 4.40431i − 1.00000i −1.47964 + 0.854273i 0
643.2 −0.866025 − 0.500000i −0.660414 + 0.176957i 0.500000 + 0.866025i 0 0.660414 + 0.176957i −0.189447 − 0.328132i − 1.00000i −2.19324 + 1.26627i 0
643.3 −0.866025 − 0.500000i 1.75811 − 0.471085i 0.500000 + 0.866025i 0 −1.75811 − 0.471085i −1.48736 − 2.57618i − 1.00000i 0.270964 − 0.156441i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.t even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.t.e 12
5.b even 2 1 130.2.p.a ✓ 12
5.c odd 4 1 130.2.s.a yes 12
5.c odd 4 1 650.2.w.e 12
13.f odd 12 1 650.2.w.e 12
65.o even 12 1 130.2.p.a ✓ 12
65.s odd 12 1 130.2.s.a yes 12
65.t even 12 1 inner 650.2.t.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.p.a ✓ 12 5.b even 2 1
130.2.p.a ✓ 12 65.o even 12 1
130.2.s.a yes 12 5.c odd 4 1
130.2.s.a yes 12 65.s odd 12 1
650.2.t.e 12 1.a even 1 1 trivial
650.2.t.e 12 65.t even 12 1 inner
650.2.w.e 12 5.c odd 4 1
650.2.w.e 12 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 12 T_{3}^{10} - 4 T_{3}^{9} + 24 T_{3}^{8} - 24 T_{3}^{7} - 280 T_{3}^{6} - 48 T_{3}^{5} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} + 12 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 21 T^{10} + \cdots + 169 \) Copy content Toggle raw display
$11$ \( T^{12} - 6 T^{11} + \cdots + 6889 \) Copy content Toggle raw display
$13$ \( T^{12} - 12 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} + 12 T^{11} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{12} - 36 T^{11} + \cdots + 221841 \) Copy content Toggle raw display
$23$ \( T^{12} + 6 T^{11} + \cdots + 23078416 \) Copy content Toggle raw display
$29$ \( T^{12} - 6 T^{11} + \cdots + 8248384 \) Copy content Toggle raw display
$31$ \( T^{12} + 24 T^{11} + \cdots + 26152996 \) Copy content Toggle raw display
$37$ \( T^{12} + 93 T^{10} + \cdots + 769129 \) Copy content Toggle raw display
$41$ \( T^{12} - 18 T^{11} + \cdots + 1690000 \) Copy content Toggle raw display
$43$ \( T^{12} - 36 T^{10} + \cdots + 22886656 \) Copy content Toggle raw display
$47$ \( (T^{6} + 6 T^{5} + \cdots - 6047)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 18 T^{11} + \cdots + 36881329 \) Copy content Toggle raw display
$59$ \( T^{12} - 18 T^{11} + \cdots + 576 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 307721764 \) Copy content Toggle raw display
$67$ \( T^{12} + 12 T^{11} + \cdots + 2304 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 880427584 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 704902500 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 1427177284 \) Copy content Toggle raw display
$83$ \( (T^{6} - 24 T^{5} + \cdots - 31050)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} - 6 T^{11} + \cdots + 39828721 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 553217613796 \) Copy content Toggle raw display
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