Properties

Label 1050.2.d.g
Level $1050$
Weight $2$
Character orbit 1050.d
Analytic conductor $8.384$
Analytic rank $0$
Dimension $12$
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(1049,1050)]
 
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([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.1049");
 
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.d (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: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 4x^{8} - 30x^{6} + 36x^{4} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_{10} q^{3} + q^{4} - \beta_{10} q^{6} + (\beta_{10} + \beta_{9} + \beta_{8} + \beta_{3} - \beta_1) q^{7} - q^{8} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_{10} q^{3} + q^{4} - \beta_{10} q^{6} + (\beta_{10} + \beta_{9} + \beta_{8} + \beta_{3} - \beta_1) q^{7} - q^{8} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{9} + ( - \beta_{6} - \beta_{5}) q^{11} + \beta_{10} q^{12} + ( - \beta_{11} + \beta_{9} + \beta_{7} + \beta_{2} - \beta_1) q^{13} + ( - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{3} + \beta_1) q^{14} + q^{16} + (\beta_{11} + \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2}) q^{17} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{18} + (\beta_{9} + \beta_1) q^{19} + (\beta_{11} - \beta_{9} - \beta_{8} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 + 1) q^{21} + (\beta_{6} + \beta_{5}) q^{22} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} - 1) q^{23} - \beta_{10} q^{24} + (\beta_{11} - \beta_{9} - \beta_{7} - \beta_{2} + \beta_1) q^{26} + (\beta_{11} + \beta_{10} + 2 \beta_{9} - \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{27} + (\beta_{10} + \beta_{9} + \beta_{8} + \beta_{3} - \beta_1) q^{28} + ( - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_1) q^{29} + (\beta_{11} - \beta_{10} - \beta_{9} + 2 \beta_{8} - \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - 2 \beta_1) q^{31} - q^{32} + ( - \beta_{11} + \beta_{10} + 4 \beta_{9} - \beta_{8} + \beta_{7} + \beta_{2} - \beta_1) q^{33} + ( - \beta_{11} - \beta_{9} - \beta_{8} + \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2}) q^{34} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{36} + (\beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} - \beta_1) q^{37} + ( - \beta_{9} - \beta_1) q^{38} + ( - 3 \beta_{7} + 2 \beta_{5} + \beta_{4} + \beta_{3}) q^{39} + (\beta_{11} + 2 \beta_{9} - \beta_{7} - \beta_{2} - 2 \beta_1) q^{41} + ( - \beta_{11} + \beta_{9} + \beta_{8} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 - 1) q^{42} + (\beta_{6} + \beta_{5} - \beta_{4}) q^{43} + ( - \beta_{6} - \beta_{5}) q^{44} + (2 \beta_{7} + \beta_{6} - \beta_{5} + 1) q^{46} + (\beta_{10} - 3 \beta_{9} - \beta_{8} - 3 \beta_1) q^{47} + \beta_{10} q^{48} + (2 \beta_{10} + \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_1 - 3) q^{49} + (3 \beta_{10} + 3 \beta_{9} + 3 \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} + \cdots + 2) q^{51}+ \cdots + ( - 3 \beta_{7} + \beta_{5} - 7 \beta_{4} + 2 \beta_{3} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 12 q^{4} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 12 q^{4} - 12 q^{8} + 12 q^{16} + 14 q^{21} - 20 q^{23} - 12 q^{32} - 12 q^{39} - 14 q^{42} + 20 q^{46} - 28 q^{49} + 28 q^{51} - 20 q^{53} + 8 q^{57} - 30 q^{63} + 12 q^{64} + 44 q^{77} + 12 q^{78} - 56 q^{79} - 16 q^{81} + 14 q^{84} - 20 q^{91} - 20 q^{92} + 48 q^{93} + 28 q^{98} - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 4x^{8} - 30x^{6} + 36x^{4} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{11} + 9\nu^{8} + 50\nu^{7} + 54\nu^{6} + 111\nu^{5} - 126\nu^{4} - 306\nu^{3} + 189\nu^{2} - 324\nu - 972 ) / 972 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 3 \nu^{10} + 9 \nu^{8} - 23 \nu^{7} + 42 \nu^{6} + 51 \nu^{5} + 207 \nu^{4} - 72 \nu^{3} - 405 \nu^{2} + 891 \nu - 486 ) / 972 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{10} - 3\nu^{8} + 5\nu^{6} - 9\nu^{4} + 9\nu^{2} - 243 ) / 324 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} - 6\nu^{10} - 23\nu^{7} + 57\nu^{6} + 51\nu^{5} + 180\nu^{4} - 72\nu^{3} + 594\nu^{2} + 891\nu - 1215 ) / 972 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{11} + 6\nu^{10} + 23\nu^{7} + 105\nu^{6} - 51\nu^{5} - 180\nu^{4} + 72\nu^{3} + 54\nu^{2} - 891\nu - 1215 ) / 972 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} + 3\nu^{10} - 27\nu^{8} - 23\nu^{7} + 12\nu^{6} + 51\nu^{5} + 45\nu^{4} - 72\nu^{3} + 675\nu^{2} + 891\nu ) / 972 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{11} + 3\nu^{9} + 5\nu^{7} + 15\nu^{5} - 9\nu^{3} - 27\nu ) / 324 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} + 4\nu^{7} - 30\nu^{5} + 36\nu^{3} ) / 243 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{11} - 3\nu^{9} + 5\nu^{7} - 9\nu^{5} + 9\nu^{3} - 243\nu ) / 324 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4 \nu^{11} + 3 \nu^{10} - 18 \nu^{9} - 18 \nu^{8} + 43 \nu^{7} + 66 \nu^{6} - 30 \nu^{5} - 81 \nu^{4} + 792 \nu^{3} + 864 \nu^{2} - 1053 \nu - 972 ) / 972 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{10} + 3\beta_{9} + 3\beta_{8} + 2\beta_{7} - \beta_{6} - 3\beta_{4} + 3\beta_{3} - 3\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{11} - 4\beta_{10} - 5\beta_{9} - 2\beta_{6} - 2\beta_{5} + 2\beta_{3} + 2\beta_{2} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6\beta_{6} + 2\beta_{5} + 4\beta_{4} + 4\beta_{3} + 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{11} + 10 \beta_{10} + 16 \beta_{9} + 14 \beta_{8} + 4 \beta_{7} - 6 \beta_{6} - 6 \beta_{5} + 6 \beta_{3} + 10 \beta_{2} + 5 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 12 \beta_{10} - 12 \beta_{9} - 12 \beta_{8} - 26 \beta_{7} + 4 \beta_{6} + 21 \beta_{5} - 45 \beta_{4} - 15 \beta_{3} + 12 \beta _1 - 10 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 5 \beta_{11} - 41 \beta_{10} + 17 \beta_{9} + 57 \beta_{8} - 3 \beta_{7} + 8 \beta_{6} + 8 \beta_{5} - 8 \beta_{3} - 11 \beta_{2} - 25 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 9 \beta_{10} + 9 \beta_{9} + 9 \beta_{8} + 60 \beta_{7} + 27 \beta_{6} - 44 \beta_{5} - 151 \beta_{4} + 29 \beta_{3} - 9 \beta _1 - 120 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 16 \beta_{11} - 124 \beta_{10} + 65 \beta_{9} - 92 \beta_{8} + 20 \beta_{7} - 36 \beta_{6} - 36 \beta_{5} + 36 \beta_{3} + 56 \beta_{2} - 116 \beta_1 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1049.1
0.721683 1.57454i
0.721683 + 1.57454i
−1.06864 + 1.36309i
−1.06864 1.36309i
−1.68439 + 0.403509i
−1.68439 0.403509i
1.68439 + 0.403509i
1.68439 0.403509i
1.06864 + 1.36309i
1.06864 1.36309i
−0.721683 1.57454i
−0.721683 + 1.57454i
−1.00000 −1.57454 0.721683i 1.00000 0 1.57454 + 0.721683i −2.29622 + 1.31429i −1.00000 1.95835 + 2.27264i 0
1049.2 −1.00000 −1.57454 + 0.721683i 1.00000 0 1.57454 0.721683i −2.29622 1.31429i −1.00000 1.95835 2.27264i 0
1049.3 −1.00000 −1.36309 1.06864i 1.00000 0 1.36309 + 1.06864i −0.294447 + 2.62932i −1.00000 0.716015 + 2.91330i 0
1049.4 −1.00000 −1.36309 + 1.06864i 1.00000 0 1.36309 1.06864i −0.294447 2.62932i −1.00000 0.716015 2.91330i 0
1049.5 −1.00000 −0.403509 1.68439i 1.00000 0 0.403509 + 1.68439i 1.28088 2.31502i −1.00000 −2.67436 + 1.35934i 0
1049.6 −1.00000 −0.403509 + 1.68439i 1.00000 0 0.403509 1.68439i 1.28088 + 2.31502i −1.00000 −2.67436 1.35934i 0
1049.7 −1.00000 0.403509 1.68439i 1.00000 0 −0.403509 + 1.68439i −1.28088 + 2.31502i −1.00000 −2.67436 1.35934i 0
1049.8 −1.00000 0.403509 + 1.68439i 1.00000 0 −0.403509 1.68439i −1.28088 2.31502i −1.00000 −2.67436 + 1.35934i 0
1049.9 −1.00000 1.36309 1.06864i 1.00000 0 −1.36309 + 1.06864i 0.294447 2.62932i −1.00000 0.716015 2.91330i 0
1049.10 −1.00000 1.36309 + 1.06864i 1.00000 0 −1.36309 1.06864i 0.294447 + 2.62932i −1.00000 0.716015 + 2.91330i 0
1049.11 −1.00000 1.57454 0.721683i 1.00000 0 −1.57454 + 0.721683i 2.29622 1.31429i −1.00000 1.95835 2.27264i 0
1049.12 −1.00000 1.57454 + 0.721683i 1.00000 0 −1.57454 0.721683i 2.29622 + 1.31429i −1.00000 1.95835 + 2.27264i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1049.12
Significant digits:
Format:

Inner twists

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

Twists

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

Hecke kernels

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

\( T_{11}^{6} + 46T_{11}^{4} + 529T_{11}^{2} + 900 \) Copy content Toggle raw display
\( T_{13}^{6} - 63T_{13}^{4} + 1300T_{13}^{2} - 8748 \) Copy content Toggle raw display
\( T_{23}^{3} + 5T_{23}^{2} - 20T_{23} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 4 T^{8} + 30 T^{6} + 36 T^{4} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 14 T^{10} + 99 T^{8} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( (T^{6} + 46 T^{4} + 529 T^{2} + 900)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 63 T^{4} + 1300 T^{2} + \cdots - 8748)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 73 T^{4} + 1123 T^{2} + \cdots + 4563)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 18 T^{4} + 85 T^{2} + 48)^{2} \) Copy content Toggle raw display
$23$ \( (T^{3} + 5 T^{2} - 20 T - 6)^{4} \) Copy content Toggle raw display
$29$ \( (T^{6} + 65 T^{4} + 460 T^{2} + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 135 T^{4} + 4948 T^{2} + \cdots + 50700)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 152 T^{4} + 4868 T^{2} + \cdots + 23104)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 249 T^{4} + 18787 T^{2} + \cdots - 408483)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 49 T^{4} + 352 T^{2} + 64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 204 T^{4} + 12532 T^{2} + \cdots + 228528)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 5 T^{2} - 64 T + 18)^{4} \) Copy content Toggle raw display
$59$ \( (T^{6} - 175 T^{4} + 7672 T^{2} + \cdots - 97200)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 183 T^{4} + 1288 T^{2} + \cdots + 1200)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 142 T^{4} + 4401 T^{2} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 216 T^{4} + 7396 T^{2} + \cdots + 57600)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 106 T^{4} + 3373 T^{2} + \cdots - 33708)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 14 T^{2} - 44 T - 40)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 189 T^{4} + 3751 T^{2} + \cdots + 7803)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 186 T^{4} + 5437 T^{2} + \cdots - 43200)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 424 T^{4} + 44752 T^{2} + \cdots - 12288)^{2} \) Copy content Toggle raw display
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