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Properties

Label 1050.3.h.a
Level $1050$
Weight $3$
Character orbit 1050.h
Analytic conductor $28.610$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1050,3,Mod(349,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.349");
 
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.h (of order \(2\), degree \(1\), not 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: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{2} q^{3} - 2 q^{4} - \beta_{7} q^{6} + (\beta_{5} - 2 \beta_{3} + \cdots + 3 \beta_1) q^{7}+ \cdots + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{2} q^{3} - 2 q^{4} - \beta_{7} q^{6} + (\beta_{5} - 2 \beta_{3} + \cdots + 3 \beta_1) q^{7}+ \cdots + ( - 9 \beta_{6} - 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} + 24 q^{9} - 48 q^{11} + 48 q^{14} + 32 q^{16} + 24 q^{21} - 240 q^{29} - 48 q^{36} - 192 q^{39} + 96 q^{44} - 144 q^{46} + 40 q^{49} - 48 q^{51} - 96 q^{56} - 64 q^{64} - 240 q^{71} - 576 q^{74} - 256 q^{79} + 72 q^{81} - 48 q^{84} - 480 q^{86} - 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 4\zeta_{24}^{4} - 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{3} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{5} + 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{6} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{4} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{3} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{3} - \beta_1 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
0.965926 − 0.258819i
−0.965926 − 0.258819i
0.258819 + 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 + 0.258819i
0.258819 − 0.965926i
−0.258819 − 0.965926i
− 1.41421i −1.73205 −2.00000 0 2.44949i −6.63103 + 2.24264i 2.82843i 3.00000 0
349.2 − 1.41421i −1.73205 −2.00000 0 2.44949i 3.16693 + 6.24264i 2.82843i 3.00000 0
349.3 − 1.41421i 1.73205 −2.00000 0 − 2.44949i −3.16693 + 6.24264i 2.82843i 3.00000 0
349.4 − 1.41421i 1.73205 −2.00000 0 − 2.44949i 6.63103 + 2.24264i 2.82843i 3.00000 0
349.5 1.41421i −1.73205 −2.00000 0 − 2.44949i −6.63103 − 2.24264i − 2.82843i 3.00000 0
349.6 1.41421i −1.73205 −2.00000 0 − 2.44949i 3.16693 − 6.24264i − 2.82843i 3.00000 0
349.7 1.41421i 1.73205 −2.00000 0 2.44949i −3.16693 − 6.24264i − 2.82843i 3.00000 0
349.8 1.41421i 1.73205 −2.00000 0 2.44949i 6.63103 − 2.24264i − 2.82843i 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.h.a 8
5.b even 2 1 inner 1050.3.h.a 8
5.c odd 4 1 42.3.c.a ✓ 4
5.c odd 4 1 1050.3.f.a 4
7.b odd 2 1 inner 1050.3.h.a 8
15.e even 4 1 126.3.c.b 4
20.e even 4 1 336.3.f.c 4
35.c odd 2 1 inner 1050.3.h.a 8
35.f even 4 1 42.3.c.a ✓ 4
35.f even 4 1 1050.3.f.a 4
35.k even 12 1 294.3.g.b 4
35.k even 12 1 294.3.g.c 4
35.l odd 12 1 294.3.g.b 4
35.l odd 12 1 294.3.g.c 4
40.i odd 4 1 1344.3.f.f 4
40.k even 4 1 1344.3.f.e 4
60.l odd 4 1 1008.3.f.g 4
105.k odd 4 1 126.3.c.b 4
105.w odd 12 1 882.3.n.a 4
105.w odd 12 1 882.3.n.d 4
105.x even 12 1 882.3.n.a 4
105.x even 12 1 882.3.n.d 4
140.j odd 4 1 336.3.f.c 4
280.s even 4 1 1344.3.f.f 4
280.y odd 4 1 1344.3.f.e 4
420.w even 4 1 1008.3.f.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.c.a ✓ 4 5.c odd 4 1
42.3.c.a ✓ 4 35.f even 4 1
126.3.c.b 4 15.e even 4 1
126.3.c.b 4 105.k odd 4 1
294.3.g.b 4 35.k even 12 1
294.3.g.b 4 35.l odd 12 1
294.3.g.c 4 35.k even 12 1
294.3.g.c 4 35.l odd 12 1
336.3.f.c 4 20.e even 4 1
336.3.f.c 4 140.j odd 4 1
882.3.n.a 4 105.w odd 12 1
882.3.n.a 4 105.x even 12 1
882.3.n.d 4 105.w odd 12 1
882.3.n.d 4 105.x even 12 1
1008.3.f.g 4 60.l odd 4 1
1008.3.f.g 4 420.w even 4 1
1050.3.f.a 4 5.c odd 4 1
1050.3.f.a 4 35.f even 4 1
1050.3.h.a 8 1.a even 1 1 trivial
1050.3.h.a 8 5.b even 2 1 inner
1050.3.h.a 8 7.b odd 2 1 inner
1050.3.h.a 8 35.c odd 2 1 inner
1344.3.f.e 4 40.k even 4 1
1344.3.f.e 4 280.y odd 4 1
1344.3.f.f 4 40.i odd 4 1
1344.3.f.f 4 280.s even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} + 12T_{11} + 18 \) acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 20 T^{6} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{2} + 12 T + 18)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 432 T^{2} + 28224)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 1476 T^{2} + 509796)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 792 T^{2} + 138384)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 396 T^{2} + 15876)^{2} \) Copy content Toggle raw display
$29$ \( (T + 30)^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2592 T^{2} + 186624)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 5984 T^{2} + 4804864)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 1764 T^{2} + 86436)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 5648 T^{2} + 602176)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 4896 T^{2} + 5089536)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 6408 T^{2} + 6906384)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 9504 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 288 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 4448 T^{2} + 2715904)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 60 T - 5598)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 8208 T^{2} + 16064064)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 64 T - 9344)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 10944 T^{2} + 451584)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 22788 T^{2} + 37234404)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 12816 T^{2} + 27123264)^{2} \) Copy content Toggle raw display
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