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Properties

Label 650.2.e.k
Level $650$
Weight $2$
Character orbit 650.e
Analytic conductor $5.190$
Analytic rank $0$
Dimension $6$
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] = [650,2,Mod(451,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.451");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.e (of order \(3\), degree \(2\), 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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.591408.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 4x^{4} + x^{3} + 10x^{2} - 3x + 1 \) 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: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 4x^{4} + x^{3} + 10x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - 16\nu^{3} + 10\nu^{2} - 3\nu + 12 ) / 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{5} + 16\nu^{4} - 27\nu^{3} + 40\nu^{2} - 12\nu + 85 ) / 37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\nu^{5} - 11\nu^{4} + 44\nu^{3} + 28\nu^{2} + 110\nu + 4 ) / 37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -25\nu^{5} + 26\nu^{4} - 104\nu^{3} - 9\nu^{2} - 260\nu + 78 ) / 37 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{4} - \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{5} - 7\beta_{4} + 4\beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{5} - 8\beta_{4} + 17\beta_{2} - 17\beta _1 + 8 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(27\) \(301\)
\(\chi(n)\) \(1\) \(-1 + \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
−0.740597 + 1.28275i
1.08504 − 1.87935i
0.155554 − 0.269427i
−0.740597 − 1.28275i
1.08504 + 1.87935i
0.155554 + 0.269427i
0.500000 − 0.866025i −0.837565 + 1.45071i −0.500000 − 0.866025i 0 0.837565 + 1.45071i −0.903032 − 1.56410i −1.00000 0.0969683 + 0.167954i 0
451.2 0.500000 − 0.866025i −0.269594 + 0.466951i −0.500000 − 0.866025i 0 0.269594 + 0.466951i 0.354638 + 0.614250i −1.00000 1.35464 + 2.34630i 0
451.3 0.500000 − 0.866025i 1.10716 − 1.91766i −0.500000 − 0.866025i 0 −1.10716 − 1.91766i −1.95161 − 3.38028i −1.00000 −0.951606 − 1.64823i 0
601.1 0.500000 + 0.866025i −0.837565 − 1.45071i −0.500000 + 0.866025i 0 0.837565 − 1.45071i −0.903032 + 1.56410i −1.00000 0.0969683 − 0.167954i 0
601.2 0.500000 + 0.866025i −0.269594 − 0.466951i −0.500000 + 0.866025i 0 0.269594 − 0.466951i 0.354638 − 0.614250i −1.00000 1.35464 − 2.34630i 0
601.3 0.500000 + 0.866025i 1.10716 + 1.91766i −0.500000 + 0.866025i 0 −1.10716 + 1.91766i −1.95161 + 3.38028i −1.00000 −0.951606 + 1.64823i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 451.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.e.k 6
5.b even 2 1 650.2.e.j 6
5.c odd 4 2 130.2.n.a ✓ 12
13.c even 3 1 inner 650.2.e.k 6
13.c even 3 1 8450.2.a.bu 3
13.e even 6 1 8450.2.a.ca 3
15.e even 4 2 1170.2.bp.h 12
20.e even 4 2 1040.2.dh.b 12
65.l even 6 1 8450.2.a.bt 3
65.n even 6 1 650.2.e.j 6
65.n even 6 1 8450.2.a.cb 3
65.o even 12 2 1690.2.c.b 6
65.q odd 12 2 130.2.n.a ✓ 12
65.q odd 12 2 1690.2.b.c 6
65.r odd 12 2 1690.2.b.b 6
65.t even 12 2 1690.2.c.c 6
195.bl even 12 2 1170.2.bp.h 12
260.bj even 12 2 1040.2.dh.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.n.a ✓ 12 5.c odd 4 2
130.2.n.a ✓ 12 65.q odd 12 2
650.2.e.j 6 5.b even 2 1
650.2.e.j 6 65.n even 6 1
650.2.e.k 6 1.a even 1 1 trivial
650.2.e.k 6 13.c even 3 1 inner
1040.2.dh.b 12 20.e even 4 2
1040.2.dh.b 12 260.bj even 12 2
1170.2.bp.h 12 15.e even 4 2
1170.2.bp.h 12 195.bl even 12 2
1690.2.b.b 6 65.r odd 12 2
1690.2.b.c 6 65.q odd 12 2
1690.2.c.b 6 65.o even 12 2
1690.2.c.c 6 65.t even 12 2
8450.2.a.bt 3 65.l even 6 1
8450.2.a.bu 3 13.c even 3 1
8450.2.a.ca 3 13.e even 6 1
8450.2.a.cb 3 65.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\):

\( T_{3}^{6} + 4T_{3}^{4} + 4T_{3}^{3} + 16T_{3}^{2} + 8T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{6} + 5T_{7}^{5} + 22T_{7}^{4} + 25T_{7}^{3} + 34T_{7}^{2} - 15T_{7} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 4 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 5 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{5} + \cdots + 961 \) Copy content Toggle raw display
$13$ \( T^{6} - 3 T^{5} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( T^{6} - 4 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{6} - 13 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$23$ \( T^{6} + 40 T^{4} + \cdots + 5776 \) Copy content Toggle raw display
$29$ \( T^{6} - 14 T^{5} + \cdots + 23104 \) Copy content Toggle raw display
$31$ \( (T^{3} - 6 T^{2} + \cdots + 218)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 5 T^{5} + \cdots + 11449 \) Copy content Toggle raw display
$41$ \( T^{6} + 2 T^{5} + \cdots + 400 \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T + 4)^{3} \) Copy content Toggle raw display
$47$ \( (T^{3} - 7 T^{2} + 11 T - 1)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 9 T^{2} - 7 T - 13)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 8 T^{5} + \cdots + 73984 \) Copy content Toggle raw display
$61$ \( T^{6} + 4 T^{5} + \cdots + 372100 \) Copy content Toggle raw display
$67$ \( T^{6} + 12 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$71$ \( T^{6} - 20 T^{5} + \cdots + 215296 \) Copy content Toggle raw display
$73$ \( (T^{3} - 6 T^{2} + \cdots - 270)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 14 T^{2} + \cdots - 158)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 4 T^{2} - 128 T + 46)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 7 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$97$ \( T^{6} + 26 T^{5} + \cdots + 217156 \) Copy content Toggle raw display
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