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Properties

Label 650.2.e.h
Level $650$
Weight $2$
Character orbit 650.e
Analytic conductor $5.190$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 10x^{2} + 100 \) 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 10x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 10 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 10\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 10\beta_{3} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(27\) \(301\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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
1.58114 − 2.73861i
−1.58114 + 2.73861i
1.58114 + 2.73861i
−1.58114 − 2.73861i
0.500000 − 0.866025i −1.58114 + 2.73861i −0.500000 − 0.866025i 0 1.58114 + 2.73861i −0.500000 − 0.866025i −1.00000 −3.50000 − 6.06218i 0
451.2 0.500000 − 0.866025i 1.58114 − 2.73861i −0.500000 − 0.866025i 0 −1.58114 − 2.73861i −0.500000 − 0.866025i −1.00000 −3.50000 − 6.06218i 0
601.1 0.500000 + 0.866025i −1.58114 − 2.73861i −0.500000 + 0.866025i 0 1.58114 − 2.73861i −0.500000 + 0.866025i −1.00000 −3.50000 + 6.06218i 0
601.2 0.500000 + 0.866025i 1.58114 + 2.73861i −0.500000 + 0.866025i 0 −1.58114 + 2.73861i −0.500000 + 0.866025i −1.00000 −3.50000 + 6.06218i 0
\(n\): e.g. 2-40 or 990-1000
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.h 4
5.b even 2 1 130.2.e.c ✓ 4
5.c odd 4 2 650.2.o.g 8
13.c even 3 1 inner 650.2.e.h 4
13.c even 3 1 8450.2.a.bc 2
13.e even 6 1 8450.2.a.bj 2
15.d odd 2 1 1170.2.i.q 4
20.d odd 2 1 1040.2.q.m 4
65.d even 2 1 1690.2.e.m 4
65.g odd 4 2 1690.2.l.k 8
65.l even 6 1 1690.2.a.k 2
65.l even 6 1 1690.2.e.m 4
65.n even 6 1 130.2.e.c ✓ 4
65.n even 6 1 1690.2.a.n 2
65.q odd 12 2 650.2.o.g 8
65.s odd 12 2 1690.2.d.g 4
65.s odd 12 2 1690.2.l.k 8
195.x odd 6 1 1170.2.i.q 4
260.v odd 6 1 1040.2.q.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.e.c ✓ 4 5.b even 2 1
130.2.e.c ✓ 4 65.n even 6 1
650.2.e.h 4 1.a even 1 1 trivial
650.2.e.h 4 13.c even 3 1 inner
650.2.o.g 8 5.c odd 4 2
650.2.o.g 8 65.q odd 12 2
1040.2.q.m 4 20.d odd 2 1
1040.2.q.m 4 260.v odd 6 1
1170.2.i.q 4 15.d odd 2 1
1170.2.i.q 4 195.x odd 6 1
1690.2.a.k 2 65.l even 6 1
1690.2.a.n 2 65.n even 6 1
1690.2.d.g 4 65.s odd 12 2
1690.2.e.m 4 65.d even 2 1
1690.2.e.m 4 65.l even 6 1
1690.2.l.k 8 65.g odd 4 2
1690.2.l.k 8 65.s odd 12 2
8450.2.a.bc 2 13.c even 3 1
8450.2.a.bj 2 13.e even 6 1

Hecke kernels

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

\( T_{3}^{4} + 10T_{3}^{2} + 100 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 10T^{2} + 100 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$19$ \( T^{4} - 10 T^{3} + \cdots + 225 \) Copy content Toggle raw display
$23$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 8 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T + 6)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 6 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{4} - 8 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$47$ \( (T - 3)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 2 T - 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 8 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$61$ \( T^{4} + 4 T^{3} + \cdots + 7396 \) Copy content Toggle raw display
$67$ \( T^{4} - 8 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$71$ \( T^{4} + 4 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$73$ \( (T^{2} + 12 T + 26)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T - 74)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 90)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 2 T^{3} + \cdots + 1521 \) Copy content Toggle raw display
$97$ \( T^{4} - 4 T^{3} + \cdots + 7396 \) Copy content Toggle raw display
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