↑

Properties

Label 1050.2.bc.f
Level $1050$
Weight $2$
Character orbit 1050.bc
Analytic conductor $8.384$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1050,2,Mod(157,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.157");
 
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.bc (of order \(12\), degree \(4\), 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: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 28x^{14} + 519x^{12} - 5404x^{10} + 40705x^{8} - 194544x^{6} + 672624x^{4} - 1306368x^{2} + 1679616 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{11} q^{2} + \beta_{2} q^{3} - \beta_{8} q^{4} + \beta_{9} q^{6} - \beta_{12} q^{7} + ( - \beta_{13} + \beta_{6}) q^{8} + ( - \beta_{9} + \beta_{8}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{2} + \beta_{2} q^{3} - \beta_{8} q^{4} + \beta_{9} q^{6} - \beta_{12} q^{7} + ( - \beta_{13} + \beta_{6}) q^{8} + ( - \beta_{9} + \beta_{8}) q^{9} + ( - \beta_{15} + \beta_{10} - \beta_{9} + \cdots - 1) q^{11}+ \cdots + (\beta_{15} + \beta_{10} + 2 \beta_{9} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{11} + 8 q^{14} + 8 q^{16} - 4 q^{19} - 4 q^{21} + 8 q^{24} - 16 q^{34} - 16 q^{36} - 12 q^{44} + 8 q^{46} + 96 q^{49} + 8 q^{51} + 8 q^{54} - 4 q^{56} + 40 q^{59} - 24 q^{61} + 12 q^{66} - 16 q^{69} + 104 q^{71} - 48 q^{74} - 12 q^{79} + 8 q^{81} - 4 q^{84} + 52 q^{86} + 60 q^{89} - 52 q^{91} - 4 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 28x^{14} + 519x^{12} - 5404x^{10} + 40705x^{8} - 194544x^{6} + 672624x^{4} - 1306368x^{2} + 1679616 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2107 \nu^{15} + 12398488 \nu^{13} - 340572477 \nu^{11} + 5822728240 \nu^{9} + \cdots + 1559363223744 \nu ) / 1091452828032 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2107 \nu^{14} + 12398488 \nu^{12} - 340572477 \nu^{10} + 5822728240 \nu^{8} + \cdots + 1559363223744 ) / 181908804672 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11299 \nu^{14} - 1267888 \nu^{12} + 26810709 \nu^{10} - 398280184 \nu^{8} + 2897647075 \nu^{6} + \cdots - 87800666688 ) / 12993486048 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 236299 \nu^{14} + 6070207 \nu^{12} - 109750641 \nu^{10} + 1048998769 \nu^{8} + \cdots + 100335652560 ) / 136431603504 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 90523 \nu^{15} - 908495 \nu^{13} + 32000682 \nu^{11} - 823985897 \nu^{9} + \cdots - 348009515532 \nu ) / 204647405256 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17575 \nu^{15} - 458203 \nu^{13} + 5317761 \nu^{11} - 14543173 \nu^{9} + \cdots + 94833934272 \nu ) / 38980458144 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 961463 \nu^{14} + 19357688 \nu^{12} - 307019433 \nu^{10} + 1771996688 \nu^{8} + \cdots - 667398916800 ) / 272863207008 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15973 \nu^{14} - 483532 \nu^{12} + 9009267 \nu^{10} - 98147980 \nu^{8} + 718144501 \nu^{6} + \cdots - 14125104000 ) / 3389605056 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 443203 \nu^{14} - 17467576 \nu^{12} + 339648981 \nu^{10} - 4086660592 \nu^{8} + \cdots - 242518237920 ) / 90954402336 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 198541 \nu^{15} + 5115945 \nu^{13} - 85575203 \nu^{11} + 733266583 \nu^{9} + \cdots - 63434342592 \nu ) / 90954402336 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 7603 \nu^{15} - 127492 \nu^{13} + 1857273 \nu^{11} - 4730788 \nu^{9} - 12141473 \nu^{7} + \cdots + 7949027664 \nu ) / 1917071712 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14621479 \nu^{15} + 351463840 \nu^{13} - 6005325921 \nu^{11} + 51247418824 \nu^{9} + \cdots + 1746136865088 \nu ) / 3274358484096 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 15973 \nu^{15} - 483532 \nu^{13} + 9009267 \nu^{11} - 98147980 \nu^{9} + 718144501 \nu^{7} + \cdots - 14125104000 \nu ) / 3389605056 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 14621479 \nu^{14} + 351463840 \nu^{12} - 6005325921 \nu^{10} + 51247418824 \nu^{8} + \cdots + 1746136865088 ) / 545726414016 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} + 2\beta_{9} - \beta_{8} - 8\beta_{5} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{14} + 6\beta_{13} + \beta_{12} - 8\beta_{11} - 8\beta_{7} + \beta_{6} + 6\beta_{2} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 15\beta_{15} + 15\beta_{10} + 37\beta_{9} - 9\beta_{8} - 79\beta_{5} - 16\beta_{4} + 16\beta_{3} - 63 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 22\beta_{14} + 90\beta_{13} - 22\beta_{12} - 169\beta_{11} - 79\beta_{7} - 118\beta_{6} + 6\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -28\beta_{15} + 219\beta_{10} - 132\beta_{9} + 292\beta_{8} - 191\beta_{4} + 28\beta_{3} - 693 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -351\beta_{14} - 168\beta_{13} - 702\beta_{12} - 1314\beta_{11} - 1848\beta_{6} - 1146\beta_{2} - 884\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2900 \beta_{15} + 519 \beta_{10} - 9493 \beta_{9} + 5006 \beta_{8} + 10585 \beta_{5} + 519 \beta_{4} + \cdots - 519 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 10012 \beta_{14} - 17400 \beta_{13} - 5006 \beta_{12} + 7471 \beta_{11} + 10585 \beta_{7} + \cdots - 10585 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 29877 \beta_{15} - 29877 \beta_{10} - 97910 \beta_{9} - 159 \beta_{8} + 131384 \beta_{5} + \cdots + 93387 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 68033 \beta_{14} - 179262 \beta_{13} + 68033 \beta_{12} + 310646 \beta_{11} + 131384 \beta_{7} + \cdots - 48720 \beta_{2} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 116753 \beta_{15} - 495432 \beta_{10} + 408198 \beta_{9} - 933149 \beta_{8} + 378679 \beta_{4} + \cdots + 1283736 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 903630 \beta_{14} + 700518 \beta_{13} + 1807260 \beta_{12} + 2972592 \beta_{11} + 4079334 \beta_{6} + \cdots + 1662415 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 6442267 \beta_{15} - 1604148 \beta_{10} + 22123946 \beta_{9} - 11864047 \beta_{8} - 21254876 \beta_{5} + \cdots + 1604148 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 23728094 \beta_{14} + 38653602 \beta_{13} + 11864047 \beta_{12} - 11629988 \beta_{11} + \cdots + 21254876 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(\beta_{9}\) \(1 + \beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
−1.90899 + 1.10215i
2.35727 − 1.36097i
−2.35727 + 1.36097i
1.90899 − 1.10215i
1.44378 − 0.833568i
−3.11681 + 1.79949i
3.11681 − 1.79949i
−1.44378 + 0.833568i
1.44378 + 0.833568i
−3.11681 − 1.79949i
3.11681 + 1.79949i
−1.44378 − 0.833568i
−1.90899 − 1.10215i
2.35727 + 1.36097i
−2.35727 − 1.36097i
1.90899 + 1.10215i
−0.258819 + 0.965926i 0.965926 − 0.258819i −0.866025 − 0.500000i 0 1.00000i −2.46313 − 0.965926i 0.707107 − 0.707107i 0.866025 − 0.500000i 0
157.2 −0.258819 + 0.965926i 0.965926 − 0.258819i −0.866025 − 0.500000i 0 1.00000i 2.46313 − 0.965926i 0.707107 − 0.707107i 0.866025 − 0.500000i 0
157.3 0.258819 − 0.965926i −0.965926 + 0.258819i −0.866025 − 0.500000i 0 1.00000i −2.46313 + 0.965926i −0.707107 + 0.707107i 0.866025 − 0.500000i 0
157.4 0.258819 − 0.965926i −0.965926 + 0.258819i −0.866025 − 0.500000i 0 1.00000i 2.46313 + 0.965926i −0.707107 + 0.707107i 0.866025 − 0.500000i 0
493.1 −0.965926 − 0.258819i 0.258819 + 0.965926i 0.866025 + 0.500000i 0 − 1.00000i −2.63306 + 0.258819i −0.707107 − 0.707107i −0.866025 + 0.500000i 0
493.2 −0.965926 − 0.258819i 0.258819 + 0.965926i 0.866025 + 0.500000i 0 − 1.00000i 2.63306 + 0.258819i −0.707107 − 0.707107i −0.866025 + 0.500000i 0
493.3 0.965926 + 0.258819i −0.258819 − 0.965926i 0.866025 + 0.500000i 0 − 1.00000i −2.63306 − 0.258819i 0.707107 + 0.707107i −0.866025 + 0.500000i 0
493.4 0.965926 + 0.258819i −0.258819 − 0.965926i 0.866025 + 0.500000i 0 − 1.00000i 2.63306 − 0.258819i 0.707107 + 0.707107i −0.866025 + 0.500000i 0
607.1 −0.965926 + 0.258819i 0.258819 − 0.965926i 0.866025 − 0.500000i 0 1.00000i −2.63306 − 0.258819i −0.707107 + 0.707107i −0.866025 − 0.500000i 0
607.2 −0.965926 + 0.258819i 0.258819 − 0.965926i 0.866025 − 0.500000i 0 1.00000i 2.63306 − 0.258819i −0.707107 + 0.707107i −0.866025 − 0.500000i 0
607.3 0.965926 − 0.258819i −0.258819 + 0.965926i 0.866025 − 0.500000i 0 1.00000i −2.63306 + 0.258819i 0.707107 − 0.707107i −0.866025 − 0.500000i 0
607.4 0.965926 − 0.258819i −0.258819 + 0.965926i 0.866025 − 0.500000i 0 1.00000i 2.63306 + 0.258819i 0.707107 − 0.707107i −0.866025 − 0.500000i 0
943.1 −0.258819 − 0.965926i 0.965926 + 0.258819i −0.866025 + 0.500000i 0 − 1.00000i −2.46313 + 0.965926i 0.707107 + 0.707107i 0.866025 + 0.500000i 0
943.2 −0.258819 − 0.965926i 0.965926 + 0.258819i −0.866025 + 0.500000i 0 − 1.00000i 2.46313 + 0.965926i 0.707107 + 0.707107i 0.866025 + 0.500000i 0
943.3 0.258819 + 0.965926i −0.965926 − 0.258819i −0.866025 + 0.500000i 0 − 1.00000i −2.46313 − 0.965926i −0.707107 − 0.707107i 0.866025 + 0.500000i 0
943.4 0.258819 + 0.965926i −0.965926 − 0.258819i −0.866025 + 0.500000i 0 − 1.00000i 2.46313 − 0.965926i −0.707107 − 0.707107i 0.866025 + 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
35.k even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.bc.f yes 16
5.b even 2 1 inner 1050.2.bc.f yes 16
5.c odd 4 2 1050.2.bc.e ✓ 16
7.d odd 6 1 1050.2.bc.e ✓ 16
35.i odd 6 1 1050.2.bc.e ✓ 16
35.k even 12 2 inner 1050.2.bc.f yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.bc.e ✓ 16 5.c odd 4 2
1050.2.bc.e ✓ 16 7.d odd 6 1
1050.2.bc.e ✓ 16 35.i odd 6 1
1050.2.bc.f yes 16 1.a even 1 1 trivial
1050.2.bc.f yes 16 5.b even 2 1 inner
1050.2.bc.f yes 16 35.k even 12 2 inner

Hecke kernels

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

\( T_{11}^{8} + 2T_{11}^{7} + 30T_{11}^{6} - 4T_{11}^{5} + 580T_{11}^{4} + 48T_{11}^{3} + 4320T_{11}^{2} - 3456T_{11} + 20736 \) Copy content Toggle raw display
\( T_{13}^{16} + 1714T_{13}^{12} + 283569T_{13}^{8} + 11618176T_{13}^{4} + 4096 \) Copy content Toggle raw display
\( T_{17}^{16} - 96 T_{17}^{14} + 3383 T_{17}^{12} - 29856 T_{17}^{10} + 12481 T_{17}^{8} + 806112 T_{17}^{6} + \cdots + 1679616 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 24 T^{6} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 2 T^{7} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 1714 T^{12} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( T^{16} - 96 T^{14} + \cdots + 1679616 \) Copy content Toggle raw display
$19$ \( (T^{8} + 2 T^{7} + \cdots + 4356)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 96 T^{14} + \cdots + 1679616 \) Copy content Toggle raw display
$29$ \( (T^{8} + 168 T^{6} + \cdots + 1679616)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 25 T^{6} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 403540761128976 \) Copy content Toggle raw display
$41$ \( (T^{8} + 170 T^{6} + \cdots + 46656)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 393460125696 \) Copy content Toggle raw display
$47$ \( T^{16} - 72 T^{14} + \cdots + 1679616 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 2176782336 \) Copy content Toggle raw display
$59$ \( (T^{8} - 20 T^{7} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 12 T^{7} + \cdots + 181476)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 251928510529536 \) Copy content Toggle raw display
$71$ \( (T^{4} - 26 T^{3} + \cdots + 792)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 49763656400896 \) Copy content Toggle raw display
$79$ \( (T^{8} + 6 T^{7} + \cdots + 3200521)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 2176782336 \) Copy content Toggle raw display
$89$ \( (T^{8} - 30 T^{7} + \cdots + 382124304)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 103682919315361 \) Copy content Toggle raw display
show more
show less