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Properties

Label 1050.3.q.c
Level $1050$
Weight $3$
Character orbit 1050.q
Analytic conductor $28.610$
Analytic rank $0$
Dimension $16$
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(199,1050)]
 
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, 3, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.199");
 
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.q (of order \(6\), degree \(2\), 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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.11007531417600000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 210)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{13} q^{2} + (2 \beta_{8} + \beta_1) q^{3} - 2 \beta_{6} q^{4} + ( - 2 \beta_{12} - \beta_{4}) q^{6} + ( - 3 \beta_{13} + 3 \beta_{10} + \cdots - 2 \beta_{2}) q^{7}+ \cdots + ( - 3 \beta_{6} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{13} q^{2} + (2 \beta_{8} + \beta_1) q^{3} - 2 \beta_{6} q^{4} + ( - 2 \beta_{12} - \beta_{4}) q^{6} + ( - 3 \beta_{13} + 3 \beta_{10} + \cdots - 2 \beta_{2}) q^{7}+ \cdots + (12 \beta_{15} + 3 \beta_{11} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{4} - 24 q^{9} - 8 q^{11} + 80 q^{14} - 32 q^{16} - 216 q^{19} - 192 q^{26} - 144 q^{29} - 264 q^{31} - 96 q^{36} - 48 q^{39} + 16 q^{44} + 16 q^{46} - 312 q^{49} + 168 q^{51} + 32 q^{56} - 264 q^{59} + 192 q^{61} - 128 q^{64} - 144 q^{66} + 16 q^{71} + 32 q^{74} - 24 q^{79} - 72 q^{81} - 80 q^{86} - 984 q^{89} - 616 q^{91} - 960 q^{94} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{14} + 377\nu^{2} ) / 144 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{12} - 161 ) / 72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{14} + 281\nu^{2} ) / 48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{15} + 17\nu^{13} - 120\nu^{9} + 816\nu^{5} - 305\nu^{3} - 119\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{15} - 17\nu^{13} + 120\nu^{9} - 816\nu^{5} - 305\nu^{3} + 119\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{12} - 48\nu^{8} + 336\nu^{4} - 49 ) / 48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{15} + 19\nu^{13} - 132\nu^{9} + 912\nu^{5} - 341\nu^{3} - 133\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 55\nu^{14} - 384\nu^{10} + 2640\nu^{6} - 385\nu^{2} ) / 144 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{15} + 19\nu^{13} - 132\nu^{9} + 912\nu^{5} + 341\nu^{3} - 133\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -49\nu^{12} + 336\nu^{8} - 2256\nu^{4} + 7 ) / 144 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -7\nu^{14} + 48\nu^{10} - 328\nu^{6} + \nu^{2} ) / 8 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 91\nu^{15} + \nu^{13} - 624\nu^{11} + 4272\nu^{7} - 13\nu^{3} + 233\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -91\nu^{15} + \nu^{13} + 624\nu^{11} - 4272\nu^{7} + 13\nu^{3} + 233\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 199\nu^{15} + 77\nu^{13} - 1392\nu^{11} - 528\nu^{9} + 9552\nu^{7} + 3648\nu^{5} - 1393\nu^{3} - 11\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -203\nu^{15} + \nu^{13} + 1392\nu^{11} - 9552\nu^{7} + 29\nu^{3} + 521\nu ) / 144 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} - \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} - \beta_{7} + 2\beta_{5} + 2\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{10} + 7\beta_{6} + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{15} + 5\beta_{14} - 11\beta_{13} - 11\beta_{12} + 5\beta_{9} + 11\beta_{5} - 11\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{11} + 9\beta_{8} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13\beta_{15} + 13\beta_{14} + 29\beta_{13} - 29\beta_{12} - 13\beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 21\beta_{10} + 47\beta_{6} - 21\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 17\beta_{9} + 17\beta_{7} + 38\beta_{5} - 38\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 55\beta_{11} + 123\beta_{8} + 55\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -89\beta_{15} + 89\beta_{14} + 199\beta_{13} - 199\beta_{12} - 89\beta_{9} - 199\beta_{5} - 199\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -72\beta_{2} - 161 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( -233\beta_{15} - 233\beta_{14} + 521\beta_{13} + 521\beta_{12} + 233\beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 377\beta_{3} - 843\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( -305\beta_{9} + 305\beta_{7} - 682\beta_{5} - 682\beta_{4} ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(1 + \beta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
−1.56290 − 0.418778i
0.596975 + 0.159959i
−0.418778 + 1.56290i
0.159959 − 0.596975i
1.56290 + 0.418778i
−0.596975 − 0.159959i
0.418778 − 1.56290i
−0.159959 + 0.596975i
−1.56290 + 0.418778i
0.596975 − 0.159959i
−0.418778 − 1.56290i
0.159959 + 0.596975i
1.56290 − 0.418778i
−0.596975 + 0.159959i
0.418778 + 1.56290i
−0.159959 − 0.596975i
−1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 − 1.73205i 0 − 2.44949i −4.79227 + 5.10237i 2.82843i −1.50000 − 2.59808i 0
199.2 −1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 − 1.73205i 0 − 2.44949i −2.55620 − 6.51658i 2.82843i −1.50000 − 2.59808i 0
199.3 −1.22474 + 0.707107i 0.866025 − 1.50000i 1.00000 − 1.73205i 0 2.44949i −4.79227 + 5.10237i 2.82843i −1.50000 − 2.59808i 0
199.4 −1.22474 + 0.707107i 0.866025 − 1.50000i 1.00000 − 1.73205i 0 2.44949i −2.55620 − 6.51658i 2.82843i −1.50000 − 2.59808i 0
199.5 1.22474 − 0.707107i −0.866025 + 1.50000i 1.00000 − 1.73205i 0 2.44949i 2.55620 + 6.51658i − 2.82843i −1.50000 − 2.59808i 0
199.6 1.22474 − 0.707107i −0.866025 + 1.50000i 1.00000 − 1.73205i 0 2.44949i 4.79227 − 5.10237i − 2.82843i −1.50000 − 2.59808i 0
199.7 1.22474 − 0.707107i 0.866025 − 1.50000i 1.00000 − 1.73205i 0 − 2.44949i 2.55620 + 6.51658i − 2.82843i −1.50000 − 2.59808i 0
199.8 1.22474 − 0.707107i 0.866025 − 1.50000i 1.00000 − 1.73205i 0 − 2.44949i 4.79227 − 5.10237i − 2.82843i −1.50000 − 2.59808i 0
649.1 −1.22474 − 0.707107i −0.866025 − 1.50000i 1.00000 + 1.73205i 0 2.44949i −4.79227 − 5.10237i − 2.82843i −1.50000 + 2.59808i 0
649.2 −1.22474 − 0.707107i −0.866025 − 1.50000i 1.00000 + 1.73205i 0 2.44949i −2.55620 + 6.51658i − 2.82843i −1.50000 + 2.59808i 0
649.3 −1.22474 − 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 − 2.44949i −4.79227 − 5.10237i − 2.82843i −1.50000 + 2.59808i 0
649.4 −1.22474 − 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 − 2.44949i −2.55620 + 6.51658i − 2.82843i −1.50000 + 2.59808i 0
649.5 1.22474 + 0.707107i −0.866025 − 1.50000i 1.00000 + 1.73205i 0 − 2.44949i 2.55620 − 6.51658i 2.82843i −1.50000 + 2.59808i 0
649.6 1.22474 + 0.707107i −0.866025 − 1.50000i 1.00000 + 1.73205i 0 − 2.44949i 4.79227 + 5.10237i 2.82843i −1.50000 + 2.59808i 0
649.7 1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i 2.55620 − 6.51658i 2.82843i −1.50000 + 2.59808i 0
649.8 1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i 4.79227 + 5.10237i 2.82843i −1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i 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.q.c 16
5.b even 2 1 inner 1050.3.q.c 16
5.c odd 4 1 210.3.o.a ✓ 8
5.c odd 4 1 1050.3.p.b 8
7.d odd 6 1 inner 1050.3.q.c 16
15.e even 4 1 630.3.v.b 8
35.i odd 6 1 inner 1050.3.q.c 16
35.k even 12 1 210.3.o.a ✓ 8
35.k even 12 1 1050.3.p.b 8
35.k even 12 1 1470.3.f.a 8
35.l odd 12 1 1470.3.f.a 8
105.w odd 12 1 630.3.v.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.o.a ✓ 8 5.c odd 4 1
210.3.o.a ✓ 8 35.k even 12 1
630.3.v.b 8 15.e even 4 1
630.3.v.b 8 105.w odd 12 1
1050.3.p.b 8 5.c odd 4 1
1050.3.p.b 8 35.k even 12 1
1050.3.q.c 16 1.a even 1 1 trivial
1050.3.q.c 16 5.b even 2 1 inner
1050.3.q.c 16 7.d odd 6 1 inner
1050.3.q.c 16 35.i odd 6 1 inner
1470.3.f.a 8 35.k even 12 1
1470.3.f.a 8 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{8} + 4 T_{11}^{7} + 316 T_{11}^{6} + 2896 T_{11}^{5} + 93448 T_{11}^{4} + 576448 T_{11}^{3} + \cdots + 22505536 \) acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} + 3 T^{2} + 9)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 78 T^{6} + \cdots + 5764801)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 4 T^{7} + \cdots + 22505536)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 492 T^{6} + \cdots + 33189121)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 249911075147776 \) Copy content Toggle raw display
$19$ \( (T^{8} + 108 T^{7} + \cdots + 3571138081)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 1065552449536 \) Copy content Toggle raw display
$29$ \( (T^{4} + 36 T^{3} + \cdots + 20104)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 132 T^{7} + \cdots + 56085121)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 47\!\cdots\!61 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 6360766467136)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 7764 T^{6} + \cdots + 769478085601)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 31\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 19263180552256)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 96 T^{7} + \cdots + 981652934656)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 72\!\cdots\!01 \) Copy content Toggle raw display
$71$ \( (T^{4} - 4 T^{3} + \cdots - 2872184)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 27\!\cdots\!21 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 11\!\cdots\!21)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 89\!\cdots\!56)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 91315148362816)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 34\!\cdots\!56)^{2} \) Copy content Toggle raw display
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