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Properties

Label 1650.2.f.c
Level $1650$
Weight $2$
Character orbit 1650.f
Analytic conductor $13.175$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1650,2,Mod(1649,1650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1650.1649"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1650.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-4,-8,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.1753163335\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2051727616.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 37x^{4} + 36x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 330)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + ( - \beta_{2} - 1) q^{3} - q^{4} + \beta_1 q^{6} + ( - \beta_{7} - \beta_{6} + \cdots - \beta_{2}) q^{7} + \beta_{4} q^{8} + (\beta_{7} + \beta_{6} + \cdots + \beta_{2}) q^{9}+ \cdots + ( - 4 \beta_{7} - \beta_{6} - 7 \beta_{4} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} - 8 q^{4} - 2 q^{6} - 2 q^{9} - 6 q^{11} + 4 q^{12} + 4 q^{13} + 8 q^{16} + 8 q^{21} + 6 q^{22} + 8 q^{23} + 2 q^{24} - 10 q^{27} - 4 q^{29} - 4 q^{31} + 4 q^{33} - 4 q^{34} + 2 q^{36} - 20 q^{38}+ \cdots + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 37x^{4} + 36x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 2\nu^{6} + 9\nu^{5} + 14\nu^{4} + 27\nu^{3} + 18\nu^{2} + 30\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 13\nu^{5} + 4\nu^{4} + 47\nu^{3} + 20\nu^{2} + 34\nu + 4 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 2\nu^{6} - 9\nu^{5} - 14\nu^{4} - 19\nu^{3} - 18\nu^{2} + 2\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 9\nu^{5} + 23\nu^{3} + 14\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 2\nu^{5} + 9\nu^{4} + 14\nu^{3} + 23\nu^{2} + 22\nu + 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - 13\nu^{5} + 4\nu^{4} - 47\nu^{3} + 20\nu^{2} - 34\nu + 4 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{6} + 2\nu^{5} - 9\nu^{4} + 14\nu^{3} - 23\nu^{2} + 22\nu - 10 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} - 2\beta_{6} - 2\beta_{5} - 2\beta_{4} + 3\beta_{3} + 2\beta_{2} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{7} + 7\beta_{6} - 5\beta_{5} - 5\beta_{4} - 5\beta_{3} + 7\beta_{2} + 5\beta _1 + 28 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 19\beta_{7} + 17\beta_{6} + 19\beta_{5} + 17\beta_{4} - 31\beta_{3} - 17\beta_{2} - 31\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -13\beta_{7} - 20\beta_{6} + 13\beta_{5} + 11\beta_{4} + 11\beta_{3} - 20\beta_{2} - 11\beta _1 - 67 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -93\beta_{7} - 75\beta_{6} - 93\beta_{5} - 67\beta_{4} + 155\beta_{3} + 75\beta_{2} + 155\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1650\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(727\) \(1201\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1649.1
− 2.25619i
0.356500i
2.08963i
− 1.18994i
2.25619i
− 0.356500i
− 2.08963i
1.18994i
− 1.00000i −1.73007 − 0.0828988i −1.00000 0 −0.0828988 + 1.73007i −4.34658 1.00000i 2.98626 + 0.286841i 0
1649.2 − 1.00000i −1.19035 − 1.25820i −1.00000 0 −1.25820 + 1.19035i 3.22941 1.00000i −0.166154 + 2.99540i 0
1649.3 − 1.00000i −0.117016 + 1.72809i −1.00000 0 1.72809 + 0.117016i 0.723074 1.00000i −2.97261 − 0.404430i 0
1649.4 − 1.00000i 1.03743 − 1.38699i −1.00000 0 −1.38699 − 1.03743i 0.394100 1.00000i −0.847487 − 2.87781i 0
1649.5 1.00000i −1.73007 + 0.0828988i −1.00000 0 −0.0828988 − 1.73007i −4.34658 − 1.00000i 2.98626 − 0.286841i 0
1649.6 1.00000i −1.19035 + 1.25820i −1.00000 0 −1.25820 − 1.19035i 3.22941 − 1.00000i −0.166154 − 2.99540i 0
1649.7 1.00000i −0.117016 − 1.72809i −1.00000 0 1.72809 − 0.117016i 0.723074 − 1.00000i −2.97261 + 0.404430i 0
1649.8 1.00000i 1.03743 + 1.38699i −1.00000 0 −1.38699 + 1.03743i 0.394100 − 1.00000i −0.847487 + 2.87781i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1649.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
165.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1650.2.f.c 8
3.b odd 2 1 1650.2.f.f 8
5.b even 2 1 1650.2.f.e 8
5.c odd 4 1 330.2.d.a ✓ 8
5.c odd 4 1 1650.2.d.f 8
11.b odd 2 1 1650.2.f.d 8
15.d odd 2 1 1650.2.f.d 8
15.e even 4 1 330.2.d.b yes 8
15.e even 4 1 1650.2.d.c 8
20.e even 4 1 2640.2.f.b 8
33.d even 2 1 1650.2.f.e 8
55.d odd 2 1 1650.2.f.f 8
55.e even 4 1 330.2.d.b yes 8
55.e even 4 1 1650.2.d.c 8
60.l odd 4 1 2640.2.f.a 8
165.d even 2 1 inner 1650.2.f.c 8
165.l odd 4 1 330.2.d.a ✓ 8
165.l odd 4 1 1650.2.d.f 8
220.i odd 4 1 2640.2.f.a 8
660.q even 4 1 2640.2.f.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.d.a ✓ 8 5.c odd 4 1
330.2.d.a ✓ 8 165.l odd 4 1
330.2.d.b yes 8 15.e even 4 1
330.2.d.b yes 8 55.e even 4 1
1650.2.d.c 8 15.e even 4 1
1650.2.d.c 8 55.e even 4 1
1650.2.d.f 8 5.c odd 4 1
1650.2.d.f 8 165.l odd 4 1
1650.2.f.c 8 1.a even 1 1 trivial
1650.2.f.c 8 165.d even 2 1 inner
1650.2.f.d 8 11.b odd 2 1
1650.2.f.d 8 15.d odd 2 1
1650.2.f.e 8 5.b even 2 1
1650.2.f.e 8 33.d even 2 1
1650.2.f.f 8 3.b odd 2 1
1650.2.f.f 8 55.d odd 2 1
2640.2.f.a 8 60.l odd 4 1
2640.2.f.a 8 220.i odd 4 1
2640.2.f.b 8 20.e even 4 1
2640.2.f.b 8 660.q even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1650, [\chi])\):

\( T_{7}^{4} - 15T_{7}^{2} + 16T_{7} - 4 \) Copy content Toggle raw display
\( T_{23}^{4} - 4T_{23}^{3} - 44T_{23}^{2} - 16T_{23} + 160 \) Copy content Toggle raw display
\( T_{29}^{4} + 2T_{29}^{3} - 43T_{29}^{2} - 50T_{29} + 452 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 15 T^{2} + 16 T - 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 6 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( (T^{4} - 2 T^{3} - 20 T^{2} + \cdots + 32)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 66 T^{6} + \cdots + 4624 \) Copy content Toggle raw display
$19$ \( T^{8} + 98 T^{6} + \cdots + 73984 \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{3} + \cdots + 160)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 2 T^{3} + \cdots + 452)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2 T^{3} + \cdots + 2000)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 178 T^{6} + \cdots + 1926544 \) Copy content Toggle raw display
$41$ \( (T^{4} - 8 T^{3} - 12 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 4 T^{3} + \cdots + 512)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 20 T^{3} + \cdots - 1648)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 6 T^{3} - 11 T^{2} + \cdots + 52)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 188 T^{6} + \cdots + 640000 \) Copy content Toggle raw display
$61$ \( T^{8} + 310 T^{6} + \cdots + 712336 \) Copy content Toggle raw display
$67$ \( T^{8} + 388 T^{6} + \cdots + 14622976 \) Copy content Toggle raw display
$71$ \( T^{8} + 170 T^{6} + \cdots + 150544 \) Copy content Toggle raw display
$73$ \( (T^{4} - 6 T^{3} + \cdots + 1088)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 316 T^{6} + \cdots + 173056 \) Copy content Toggle raw display
$83$ \( T^{8} + 512 T^{6} + \cdots + 8667136 \) Copy content Toggle raw display
$89$ \( T^{8} + 262 T^{6} + \cdots + 135424 \) Copy content Toggle raw display
$97$ \( T^{8} + 448 T^{6} + \cdots + 130142464 \) Copy content Toggle raw display
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