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Properties

Label 1050.2.u.j
Level $1050$
Weight $2$
Character orbit 1050.u
Analytic conductor $8.384$
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] = [1050,2,Mod(299,1050)]
 
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([3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.299");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.u (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: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.1523584037250322661376.8
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 50 x^{12} - 18 x^{11} - 144 x^{10} + 498 x^{9} - 1013 x^{8} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{12} q^{2} + \beta_{14} q^{3} + ( - \beta_{12} - 1) q^{4} + (\beta_{14} - \beta_{11}) q^{6} + ( - \beta_{15} - \beta_{14} + \cdots - \beta_{2}) q^{7}+ \cdots + ( - \beta_{15} - \beta_{14} + \beta_{11} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{12} q^{2} + \beta_{14} q^{3} + ( - \beta_{12} - 1) q^{4} + (\beta_{14} - \beta_{11}) q^{6} + ( - \beta_{15} - \beta_{14} + \cdots - \beta_{2}) q^{7}+ \cdots + (2 \beta_{15} + 2 \beta_{12} + \cdots + 11) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{2} - 6 q^{3} - 8 q^{4} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{2} - 6 q^{3} - 8 q^{4} - 16 q^{8} + 6 q^{12} - 8 q^{16} + 12 q^{17} + 28 q^{21} - 4 q^{23} + 6 q^{24} + 8 q^{32} - 12 q^{33} - 12 q^{39} + 14 q^{42} + 4 q^{46} - 84 q^{47} - 10 q^{51} - 28 q^{53} - 18 q^{54} - 56 q^{57} + 108 q^{61} + 16 q^{64} - 12 q^{66} - 12 q^{68} - 56 q^{77} - 24 q^{78} + 24 q^{79} + 16 q^{81} - 14 q^{84} - 48 q^{87} - 28 q^{91} + 8 q^{92} + 42 q^{93} - 84 q^{94} - 6 q^{96} + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 50 x^{12} - 18 x^{11} - 144 x^{10} + 498 x^{9} - 1013 x^{8} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 46 \nu^{15} - 36869 \nu^{14} + 118359 \nu^{13} - 195939 \nu^{12} + 320017 \nu^{11} + \cdots - 66205593 ) / 15914070 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1073 \nu^{15} + 11560 \nu^{14} - 87756 \nu^{13} + 173637 \nu^{12} - 248489 \nu^{11} + \cdots + 68897061 ) / 15914070 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7882 \nu^{15} - 22599 \nu^{14} + 121086 \nu^{13} - 219303 \nu^{12} + 282590 \nu^{11} + \cdots - 55939086 ) / 47742210 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 9951 \nu^{15} - 8101 \nu^{14} + 54501 \nu^{13} - 87084 \nu^{12} + 263373 \nu^{11} + \cdots - 60309441 ) / 15914070 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10017 \nu^{15} + 46793 \nu^{14} - 106608 \nu^{13} + 192753 \nu^{12} - 168444 \nu^{11} + \cdots + 52074657 ) / 15914070 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 32020 \nu^{15} - 17667 \nu^{14} - 66672 \nu^{13} + 102483 \nu^{12} - 493795 \nu^{11} + \cdots + 197184294 ) / 47742210 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 38444 \nu^{15} + 153579 \nu^{14} - 278262 \nu^{13} + 435420 \nu^{12} - 278224 \nu^{11} + \cdots + 12971097 ) / 47742210 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 45181 \nu^{15} - 356838 \nu^{14} + 963288 \nu^{13} - 1826640 \nu^{12} + 2407199 \nu^{11} + \cdots - 386657226 ) / 47742210 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 50377 \nu^{15} - 333159 \nu^{14} + 996561 \nu^{13} - 1711944 \nu^{12} + 2232110 \nu^{11} + \cdots - 509592870 ) / 47742210 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 56597 \nu^{15} - 77682 \nu^{14} + 49725 \nu^{13} + 71154 \nu^{12} - 458102 \nu^{11} + \cdots + 82712340 ) / 47742210 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 19341 \nu^{15} - 85900 \nu^{14} + 201921 \nu^{13} - 355239 \nu^{12} + 393039 \nu^{11} + \cdots - 62832510 ) / 15914070 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 29722 \nu^{15} + 164562 \nu^{14} - 398880 \nu^{13} + 663750 \nu^{12} - 731018 \nu^{11} + \cdots + 127792971 ) / 23871105 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 45034 \nu^{15} - 135084 \nu^{14} + 254115 \nu^{13} - 408870 \nu^{12} + 279566 \nu^{11} + \cdots - 28280097 ) / 23871105 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 90379 \nu^{15} + 352275 \nu^{14} - 724689 \nu^{13} + 1066779 \nu^{12} - 953816 \nu^{11} + \cdots + 108171207 ) / 47742210 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 47043 \nu^{15} - 254603 \nu^{14} + 618777 \nu^{13} - 1050849 \nu^{12} + 1234395 \nu^{11} + \cdots - 212955480 ) / 15914070 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} - \beta_{14} - \beta_{13} + \beta_{11} + \beta_{10} + \beta_{8} - 2\beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{15} + \beta_{11} + \beta_{9} - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{15} + 3\beta_{14} + \beta_{13} - 5\beta_{12} + \beta_{11} + \beta_{10} - \beta_{7} + 4\beta_{5} - \beta_{3} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 4 \beta_{15} - 4 \beta_{14} - 2 \beta_{12} + 3 \beta_{11} - 2 \beta_{10} + \beta_{9} + 2 \beta_{8} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6 \beta_{15} - 6 \beta_{14} + 7 \beta_{12} - 2 \beta_{9} + 7 \beta_{8} - 23 \beta_{7} - 4 \beta_{6} + \cdots - 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{15} + 8 \beta_{14} + 4 \beta_{13} + \beta_{12} - 7 \beta_{11} + \beta_{9} - 16 \beta_{7} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7 \beta_{15} + 7 \beta_{14} + 7 \beta_{13} - 24 \beta_{12} - 7 \beta_{11} + 7 \beta_{10} - 42 \beta_{9} + \cdots - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 28 \beta_{14} + 51 \beta_{12} + 28 \beta_{11} - 24 \beta_{10} - 12 \beta_{8} - 28 \beta_{6} + \cdots + 63 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 37 \beta_{15} + 37 \beta_{14} - 35 \beta_{13} + 139 \beta_{12} - 37 \beta_{11} + 35 \beta_{10} + \cdots + 72 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 19 \beta_{15} - 19 \beta_{14} - 68 \beta_{13} + \beta_{12} + 67 \beta_{11} + 48 \beta_{9} + 21 \beta_{7} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 134 \beta_{15} - 134 \beta_{14} + 299 \beta_{12} + 268 \beta_{11} - 120 \beta_{9} - 11 \beta_{8} + \cdots - 20 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 49 \beta_{15} - 201 \beta_{14} + 320 \beta_{12} + 49 \beta_{11} - 70 \beta_{10} + 152 \beta_{9} + \cdots + 77 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 533 \beta_{15} - 533 \beta_{14} - 533 \beta_{13} - 1290 \beta_{12} - 1093 \beta_{11} + 533 \beta_{10} + \cdots - 2538 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 56 \beta_{15} - 399 \beta_{14} - 336 \beta_{13} + 63 \beta_{12} + 119 \beta_{11} - 280 \beta_{9} + \cdots + 63 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 2509 \beta_{15} + 2061 \beta_{14} + 2285 \beta_{13} + 3587 \beta_{12} - 2509 \beta_{11} + 2285 \beta_{10} + \cdots + 1456 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(-\beta_{12}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
299.1
−1.72729 + 0.128387i
1.29384 − 1.15151i
1.56007 − 0.752456i
0.128387 + 1.72729i
1.15151 + 1.29384i
−0.752456 − 1.56007i
−0.350317 + 1.69625i
1.69625 + 0.350317i
−1.72729 − 0.128387i
1.29384 + 1.15151i
1.56007 + 0.752456i
0.128387 − 1.72729i
1.15151 − 1.29384i
−0.752456 + 1.56007i
−0.350317 − 1.69625i
1.69625 − 0.350317i
0.500000 − 0.866025i −1.69625 + 0.350317i −0.500000 − 0.866025i 0 −0.544743 + 1.64416i −1.53439 − 2.15538i −1.00000 2.75456 − 1.18845i 0
299.2 0.500000 − 0.866025i −1.56007 − 0.752456i −0.500000 − 0.866025i 0 −1.43168 + 0.974830i −2.15538 + 1.53439i −1.00000 1.86762 + 2.34776i 0
299.3 0.500000 − 0.866025i −1.29384 − 1.15151i −0.500000 − 0.866025i 0 −1.64416 + 0.544743i 1.53439 + 2.15538i −1.00000 0.348046 + 2.97974i 0
299.4 0.500000 − 0.866025i −1.15151 + 1.29384i −0.500000 − 0.866025i 0 0.544743 + 1.64416i 1.53439 + 2.15538i −1.00000 −0.348046 − 2.97974i 0
299.5 0.500000 − 0.866025i −0.128387 + 1.72729i −0.500000 − 0.866025i 0 1.43168 + 0.974830i 2.15538 − 1.53439i −1.00000 −2.96703 − 0.443523i 0
299.6 0.500000 − 0.866025i 0.350317 + 1.69625i −0.500000 − 0.866025i 0 1.64416 + 0.544743i −1.53439 − 2.15538i −1.00000 −2.75456 + 1.18845i 0
299.7 0.500000 − 0.866025i 0.752456 − 1.56007i −0.500000 − 0.866025i 0 −0.974830 − 1.43168i −2.15538 + 1.53439i −1.00000 −1.86762 − 2.34776i 0
299.8 0.500000 − 0.866025i 1.72729 + 0.128387i −0.500000 − 0.866025i 0 0.974830 − 1.43168i 2.15538 − 1.53439i −1.00000 2.96703 + 0.443523i 0
899.1 0.500000 + 0.866025i −1.69625 − 0.350317i −0.500000 + 0.866025i 0 −0.544743 − 1.64416i −1.53439 + 2.15538i −1.00000 2.75456 + 1.18845i 0
899.2 0.500000 + 0.866025i −1.56007 + 0.752456i −0.500000 + 0.866025i 0 −1.43168 − 0.974830i −2.15538 − 1.53439i −1.00000 1.86762 − 2.34776i 0
899.3 0.500000 + 0.866025i −1.29384 + 1.15151i −0.500000 + 0.866025i 0 −1.64416 − 0.544743i 1.53439 − 2.15538i −1.00000 0.348046 − 2.97974i 0
899.4 0.500000 + 0.866025i −1.15151 − 1.29384i −0.500000 + 0.866025i 0 0.544743 − 1.64416i 1.53439 − 2.15538i −1.00000 −0.348046 + 2.97974i 0
899.5 0.500000 + 0.866025i −0.128387 − 1.72729i −0.500000 + 0.866025i 0 1.43168 − 0.974830i 2.15538 + 1.53439i −1.00000 −2.96703 + 0.443523i 0
899.6 0.500000 + 0.866025i 0.350317 − 1.69625i −0.500000 + 0.866025i 0 1.64416 − 0.544743i −1.53439 + 2.15538i −1.00000 −2.75456 − 1.18845i 0
899.7 0.500000 + 0.866025i 0.752456 + 1.56007i −0.500000 + 0.866025i 0 −0.974830 + 1.43168i −2.15538 − 1.53439i −1.00000 −1.86762 + 2.34776i 0
899.8 0.500000 + 0.866025i 1.72729 − 0.128387i −0.500000 + 0.866025i 0 0.974830 + 1.43168i 2.15538 + 1.53439i −1.00000 2.96703 − 0.443523i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 299.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
15.d odd 2 1 inner
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.u.j 16
3.b odd 2 1 1050.2.u.i 16
5.b even 2 1 1050.2.u.i 16
5.c odd 4 1 1050.2.s.h ✓ 16
5.c odd 4 1 1050.2.s.j yes 16
7.d odd 6 1 inner 1050.2.u.j 16
15.d odd 2 1 inner 1050.2.u.j 16
15.e even 4 1 1050.2.s.h ✓ 16
15.e even 4 1 1050.2.s.j yes 16
21.g even 6 1 1050.2.u.i 16
35.i odd 6 1 1050.2.u.i 16
35.k even 12 1 1050.2.s.h ✓ 16
35.k even 12 1 1050.2.s.j yes 16
105.p even 6 1 inner 1050.2.u.j 16
105.w odd 12 1 1050.2.s.h ✓ 16
105.w odd 12 1 1050.2.s.j yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.s.h ✓ 16 5.c odd 4 1
1050.2.s.h ✓ 16 15.e even 4 1
1050.2.s.h ✓ 16 35.k even 12 1
1050.2.s.h ✓ 16 105.w odd 12 1
1050.2.s.j yes 16 5.c odd 4 1
1050.2.s.j yes 16 15.e even 4 1
1050.2.s.j yes 16 35.k even 12 1
1050.2.s.j yes 16 105.w odd 12 1
1050.2.u.i 16 3.b odd 2 1
1050.2.u.i 16 5.b even 2 1
1050.2.u.i 16 21.g even 6 1
1050.2.u.i 16 35.i odd 6 1
1050.2.u.j 16 1.a even 1 1 trivial
1050.2.u.j 16 7.d odd 6 1 inner
1050.2.u.j 16 15.d odd 2 1 inner
1050.2.u.j 16 105.p even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{16} - 44 T_{11}^{14} + 1336 T_{11}^{12} - 20288 T_{11}^{10} + 220912 T_{11}^{8} + \cdots + 21381376 \) Copy content Toggle raw display
\( T_{13}^{8} - 48T_{13}^{6} + 332T_{13}^{4} - 528T_{13}^{2} + 16 \) Copy content Toggle raw display
\( T_{17}^{8} - 6T_{17}^{7} - 22T_{17}^{6} + 204T_{17}^{5} + 723T_{17}^{4} - 5916T_{17}^{3} + 7202T_{17}^{2} + 14790T_{17} + 7225 \) Copy content Toggle raw display
\( T_{23}^{8} + 2T_{23}^{7} + 76T_{23}^{6} + 380T_{23}^{5} + 5557T_{23}^{4} + 18260T_{23}^{3} + 79516T_{23}^{2} - 39562T_{23} + 22801 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + 6 T^{15} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 77 T^{4} + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 44 T^{14} + \cdots + 21381376 \) Copy content Toggle raw display
$13$ \( (T^{8} - 48 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 6 T^{7} + \cdots + 7225)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 42 T^{6} + \cdots + 784)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 2 T^{7} + \cdots + 22801)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 172 T^{6} + \cdots + 2226064)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 84 T^{6} + \cdots + 49)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} - 92 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( (T^{8} - 48 T^{6} + \cdots + 3481)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 228 T^{6} + \cdots + 1336336)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 42 T^{7} + \cdots + 693889)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 14 T^{7} + \cdots + 490000)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 479785216 \) Copy content Toggle raw display
$61$ \( (T^{8} - 54 T^{7} + \cdots + 10000)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 65421179248896 \) Copy content Toggle raw display
$71$ \( (T^{8} + 396 T^{6} + \cdots + 2679769)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 112 T^{6} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 12 T^{7} + \cdots + 72361)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 420 T^{6} + \cdots + 490000)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 214503236273041 \) Copy content Toggle raw display
$97$ \( (T^{8} - 308 T^{6} + \cdots + 5368489)^{2} \) Copy content Toggle raw display
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