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Properties

Label 1050.3.p.a
Level $1050$
Weight $3$
Character orbit 1050.p
Analytic conductor $28.610$
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] = [1050,3,Mod(451,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.451");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.p (of order \(6\), 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: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) 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 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{2} - 2) q^{3} + 2 \beta_{2} q^{4} + (\beta_{3} + 2 \beta_1) q^{6} + (4 \beta_{3} + 5 \beta_{2} + 2 \beta_1 + 5) q^{7} - 2 \beta_{3} q^{8} + (3 \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{2} - 2) q^{3} + 2 \beta_{2} q^{4} + (\beta_{3} + 2 \beta_1) q^{6} + (4 \beta_{3} + 5 \beta_{2} + 2 \beta_1 + 5) q^{7} - 2 \beta_{3} q^{8} + (3 \beta_{2} + 3) q^{9} + 6 \beta_{2} q^{11} + ( - 2 \beta_{2} + 2) q^{12} + (8 \beta_{3} + 2 \beta_{2} + 16 \beta_1 + 1) q^{13} + ( - 5 \beta_{3} + 4 \beta_{2} + \cdots + 8) q^{14}+ \cdots - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{3} - 4 q^{4} + 10 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{3} - 4 q^{4} + 10 q^{7} + 6 q^{9} - 12 q^{11} + 12 q^{12} + 24 q^{14} - 8 q^{16} + 48 q^{17} - 42 q^{19} - 24 q^{23} + 96 q^{26} - 40 q^{28} + 102 q^{31} + 36 q^{33} - 24 q^{36} - 22 q^{37} - 24 q^{38} + 6 q^{39} - 24 q^{42} - 28 q^{43} - 24 q^{44} - 72 q^{46} + 132 q^{47} - 2 q^{49} - 48 q^{51} - 12 q^{52} - 120 q^{53} - 48 q^{56} + 84 q^{57} + 96 q^{58} - 24 q^{59} - 72 q^{61} - 30 q^{63} + 32 q^{64} - 110 q^{67} - 96 q^{68} + 312 q^{71} + 66 q^{73} - 48 q^{74} - 120 q^{77} - 192 q^{78} - 10 q^{79} - 18 q^{81} - 48 q^{82} + 60 q^{84} - 24 q^{86} - 72 q^{89} - 222 q^{91} + 96 q^{92} - 102 q^{93} - 24 q^{94} - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

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

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(1\) \(1 + \beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
0.707107 + 1.22474i
−0.707107 − 1.22474i
0.707107 − 1.22474i
−0.707107 + 1.22474i
−0.707107 − 1.22474i −1.50000 − 0.866025i −1.00000 + 1.73205i 0 2.44949i −1.74264 + 6.77962i 2.82843 1.50000 + 2.59808i 0
451.2 0.707107 + 1.22474i −1.50000 − 0.866025i −1.00000 + 1.73205i 0 − 2.44949i 6.74264 + 1.88064i −2.82843 1.50000 + 2.59808i 0
901.1 −0.707107 + 1.22474i −1.50000 + 0.866025i −1.00000 − 1.73205i 0 − 2.44949i −1.74264 − 6.77962i 2.82843 1.50000 − 2.59808i 0
901.2 0.707107 − 1.22474i −1.50000 + 0.866025i −1.00000 − 1.73205i 0 2.44949i 6.74264 − 1.88064i −2.82843 1.50000 − 2.59808i 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
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.p.a 4
5.b even 2 1 42.3.g.a ✓ 4
5.c odd 4 2 1050.3.q.a 8
7.d odd 6 1 inner 1050.3.p.a 4
15.d odd 2 1 126.3.n.a 4
20.d odd 2 1 336.3.bh.e 4
35.c odd 2 1 294.3.g.a 4
35.i odd 6 1 42.3.g.a ✓ 4
35.i odd 6 1 294.3.c.a 4
35.j even 6 1 294.3.c.a 4
35.j even 6 1 294.3.g.a 4
35.k even 12 2 1050.3.q.a 8
60.h even 2 1 1008.3.cg.h 4
105.g even 2 1 882.3.n.e 4
105.o odd 6 1 882.3.c.b 4
105.o odd 6 1 882.3.n.e 4
105.p even 6 1 126.3.n.a 4
105.p even 6 1 882.3.c.b 4
140.p odd 6 1 2352.3.f.e 4
140.s even 6 1 336.3.bh.e 4
140.s even 6 1 2352.3.f.e 4
420.be odd 6 1 1008.3.cg.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.g.a ✓ 4 5.b even 2 1
42.3.g.a ✓ 4 35.i odd 6 1
126.3.n.a 4 15.d odd 2 1
126.3.n.a 4 105.p even 6 1
294.3.c.a 4 35.i odd 6 1
294.3.c.a 4 35.j even 6 1
294.3.g.a 4 35.c odd 2 1
294.3.g.a 4 35.j even 6 1
336.3.bh.e 4 20.d odd 2 1
336.3.bh.e 4 140.s even 6 1
882.3.c.b 4 105.o odd 6 1
882.3.c.b 4 105.p even 6 1
882.3.n.e 4 105.g even 2 1
882.3.n.e 4 105.o odd 6 1
1008.3.cg.h 4 60.h even 2 1
1008.3.cg.h 4 420.be odd 6 1
1050.3.p.a 4 1.a even 1 1 trivial
1050.3.p.a 4 7.d odd 6 1 inner
1050.3.q.a 8 5.c odd 4 2
1050.3.q.a 8 35.k even 12 2
2352.3.f.e 4 140.p odd 6 1
2352.3.f.e 4 140.s even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{2} + 6T_{11} + 36 \) Copy content Toggle raw display
\( T_{17}^{4} - 48T_{17}^{3} + 936T_{17}^{2} - 8064T_{17} + 28224 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 10 T^{3} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 774 T^{2} + 145161 \) Copy content Toggle raw display
$17$ \( T^{4} - 48 T^{3} + \cdots + 28224 \) Copy content Toggle raw display
$19$ \( T^{4} + 42 T^{3} + \cdots + 15129 \) Copy content Toggle raw display
$23$ \( T^{4} + 24 T^{3} + \cdots + 254016 \) Copy content Toggle raw display
$29$ \( (T^{2} - 1152)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 102 T^{3} + \cdots + 423801 \) Copy content Toggle raw display
$37$ \( T^{4} + 22 T^{3} + \cdots + 27889 \) Copy content Toggle raw display
$41$ \( T^{4} + 4248 T^{2} + 3732624 \) Copy content Toggle raw display
$43$ \( (T^{2} + 14 T - 23)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 132 T^{3} + \cdots + 2039184 \) Copy content Toggle raw display
$53$ \( T^{4} + 120 T^{3} + \cdots + 8714304 \) Copy content Toggle raw display
$59$ \( T^{4} + 24 T^{3} + \cdots + 1272384 \) Copy content Toggle raw display
$61$ \( T^{4} + 72 T^{3} + \cdots + 2304 \) Copy content Toggle raw display
$67$ \( T^{4} + 110 T^{3} + \cdots + 253009 \) Copy content Toggle raw display
$71$ \( (T^{2} - 156 T + 2556)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 66 T^{3} + \cdots + 85322169 \) Copy content Toggle raw display
$79$ \( T^{4} + 10 T^{3} + \cdots + 75463969 \) Copy content Toggle raw display
$83$ \( T^{4} + 17928 T^{2} + 69956496 \) Copy content Toggle raw display
$89$ \( (T^{2} + 36 T + 432)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 1056 T^{2} + 112896 \) Copy content Toggle raw display
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