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Properties

Label 1050.2.j.e
Level $1050$
Weight $2$
Character orbit 1050.j
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(407,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.407");
 
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.j (of order \(4\), degree \(2\), 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 \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 12x^{14} + 103x^{12} - 396x^{10} + 1089x^{8} - 1584x^{6} + 1648x^{4} - 768x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{11} q^{2} + ( - \beta_{13} + \beta_{11}) q^{3} + \beta_{5} q^{4} + (\beta_{8} - \beta_{5}) q^{6} - \beta_{14} q^{7} - \beta_{14} q^{8} + ( - \beta_{8} + \beta_{6} + \beta_{5} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{11} q^{2} + ( - \beta_{13} + \beta_{11}) q^{3} + \beta_{5} q^{4} + (\beta_{8} - \beta_{5}) q^{6} - \beta_{14} q^{7} - \beta_{14} q^{8} + ( - \beta_{8} + \beta_{6} + \beta_{5} + \cdots - 1) q^{9}+ \cdots + (2 \beta_{9} - 5 \beta_{8} + \cdots - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{6} - 16 q^{9} - 16 q^{14} - 16 q^{16} + 8 q^{21} + 8 q^{24} - 12 q^{36} - 48 q^{39} - 16 q^{44} - 8 q^{46} + 44 q^{51} - 16 q^{54} - 40 q^{59} + 32 q^{61} - 20 q^{66} + 48 q^{69} + 16 q^{74} + 24 q^{76} - 4 q^{81} + 4 q^{84} + 32 q^{89} - 56 q^{91} + 4 q^{96} + 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 12x^{14} + 103x^{12} - 396x^{10} + 1089x^{8} - 1584x^{6} + 1648x^{4} - 768x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 49 \nu^{14} - 1296 \nu^{12} + 13739 \nu^{10} - 91168 \nu^{8} + 318901 \nu^{6} - 655028 \nu^{4} + \cdots - 195760 ) / 209352 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 623 \nu^{14} + 11040 \nu^{12} - 99913 \nu^{10} + 531080 \nu^{8} - 1407791 \nu^{6} + \cdots + 2081120 ) / 837408 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4903 \nu^{14} - 65592 \nu^{12} + 570801 \nu^{10} - 2443056 \nu^{8} + 6424935 \nu^{6} + \cdots - 176640 ) / 2512224 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 645 \nu^{14} - 7932 \nu^{12} + 68147 \nu^{10} - 268860 \nu^{8} + 724309 \nu^{6} - 1050096 \nu^{4} + \cdots - 292224 ) / 152256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2871 \nu^{14} - 32935 \nu^{12} + 279477 \nu^{10} - 1001385 \nu^{8} + 2701611 \nu^{6} + \cdots - 1245352 ) / 628056 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2871 \nu^{15} + 32935 \nu^{13} - 279477 \nu^{11} + 1001385 \nu^{9} - 2701611 \nu^{7} + \cdots + 1873408 \nu ) / 1256112 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12765 \nu^{14} + 149308 \nu^{12} - 1252779 \nu^{10} + 4504764 \nu^{8} - 11203005 \nu^{6} + \cdots + 3851200 ) / 2512224 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 41\nu^{14} - 444\nu^{12} + 3663\nu^{10} - 11484\nu^{8} + 27225\nu^{6} - 19008\nu^{4} + 8960\nu^{2} + 9600 ) / 6336 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2835 \nu^{15} - 37372 \nu^{13} + 334053 \nu^{11} - 1477836 \nu^{9} + 4466259 \nu^{7} + \cdots - 3388864 \nu ) / 913536 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12235 \nu^{15} + 146036 \nu^{13} - 1239469 \nu^{11} + 4625236 \nu^{9} - 11865227 \nu^{7} + \cdots - 1218624 \nu ) / 3349632 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 645 \nu^{15} - 7932 \nu^{13} + 68147 \nu^{11} - 268860 \nu^{9} + 724309 \nu^{7} - 1050096 \nu^{5} + \cdots - 292224 \nu ) / 152256 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 26899 \nu^{15} - 306924 \nu^{13} + 2576805 \nu^{11} - 8989596 \nu^{9} + 22745811 \nu^{7} + \cdots - 1015296 \nu ) / 5024448 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 19433 \nu^{15} + 230704 \nu^{13} - 1957439 \nu^{11} + 7295816 \nu^{9} - 19038217 \nu^{7} + \cdots + 8279808 \nu ) / 3349632 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 5641 \nu^{15} + 59088 \nu^{13} - 484479 \nu^{11} + 1431720 \nu^{9} - 3446217 \nu^{7} + \cdots - 1580928 \nu ) / 913536 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{6} - \beta_{3} - \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} - 4\beta_{7} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + 7\beta_{8} + 4\beta_{6} + 3\beta_{5} + 7\beta_{4} + \beta_{3} - 8\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} + 15\beta_{14} - 10\beta_{13} + 20\beta_{12} - 6\beta_{11} - 8\beta_{10} - 23\beta_{7} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -20\beta_{9} - 12\beta_{8} + 59\beta_{4} + 59\beta_{3} - 12\beta_{2} - 105 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -59\beta_{15} + 35\beta_{14} - 158\beta_{13} + 79\beta_{12} - 106\beta_{11} - 12\beta_{10} - 152\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -79\beta_{9} - 428\beta_{8} - 152\beta_{6} - 237\beta_{5} + 103\beta_{4} + 325\beta_{3} + 325\beta_{2} - 477 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 325 \beta_{15} - 531 \beta_{14} - 586 \beta_{13} - 586 \beta_{12} - 531 \beta_{11} + \cdots - 1057 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 586\beta_{9} - 2293\beta_{8} - 1057\beta_{6} - 1758\beta_{5} - 2293\beta_{4} - 792\beta_{3} + 3085\beta_{2} + 1849 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 792\beta_{15} - 5378\beta_{14} + 4257\beta_{13} - 8514\beta_{12} + 1501\beta_{11} + 3085\beta_{10} + 7492\beta_{7} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 8514\beta_{9} + 5841\beta_{8} - 22176\beta_{4} - 22176\beta_{3} + 5841\beta_{2} + 37160 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 22176 \beta_{15} - 10494 \beta_{14} + 61380 \beta_{13} - 30690 \beta_{12} + 38511 \beta_{11} + \cdots + 53495 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 30690 \beta_{9} + 159227 \beta_{8} + 53495 \beta_{6} + 92070 \beta_{5} - 42372 \beta_{4} - 116855 \beta_{3} + \cdots + 170350 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 116855 \beta_{15} + 201599 \beta_{14} + 220607 \beta_{13} + 220607 \beta_{12} + 201599 \beta_{11} + \cdots + 383072 \beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
407.1
2.31967 − 1.33926i
−1.02094 − 0.589442i
−0.646642 + 0.373339i
1.46923 + 0.848261i
−1.46923 − 0.848261i
0.646642 − 0.373339i
1.02094 + 0.589442i
−2.31967 + 1.33926i
2.31967 + 1.33926i
−1.02094 + 0.589442i
−0.646642 − 0.373339i
1.46923 − 0.848261i
−1.46923 + 0.848261i
0.646642 + 0.373339i
1.02094 − 0.589442i
−2.31967 − 1.33926i
−0.707107 + 0.707107i −0.632158 + 1.61257i − 1.00000i 0 −0.693255 − 1.58726i 0.707107 + 0.707107i 0.707107 + 0.707107i −2.20075 − 2.03880i 0
407.2 −0.707107 + 0.707107i 0.117665 − 1.72805i − 1.00000i 0 1.13871 + 1.30512i 0.707107 + 0.707107i 0.707107 + 0.707107i −2.97231 − 0.406663i 0
407.3 −0.707107 + 0.707107i 1.08045 − 1.35375i − 1.00000i 0 0.193255 + 1.72124i 0.707107 + 0.707107i 0.707107 + 0.707107i −0.665273 − 2.92531i 0
407.4 −0.707107 + 0.707107i 1.55537 + 0.762124i − 1.00000i 0 −1.63871 + 0.560908i 0.707107 + 0.707107i 0.707107 + 0.707107i 1.83834 + 2.37076i 0
407.5 0.707107 − 0.707107i −1.55537 − 0.762124i − 1.00000i 0 −1.63871 + 0.560908i −0.707107 − 0.707107i −0.707107 − 0.707107i 1.83834 + 2.37076i 0
407.6 0.707107 − 0.707107i −1.08045 + 1.35375i − 1.00000i 0 0.193255 + 1.72124i −0.707107 − 0.707107i −0.707107 − 0.707107i −0.665273 − 2.92531i 0
407.7 0.707107 − 0.707107i −0.117665 + 1.72805i − 1.00000i 0 1.13871 + 1.30512i −0.707107 − 0.707107i −0.707107 − 0.707107i −2.97231 − 0.406663i 0
407.8 0.707107 − 0.707107i 0.632158 − 1.61257i − 1.00000i 0 −0.693255 − 1.58726i −0.707107 − 0.707107i −0.707107 − 0.707107i −2.20075 − 2.03880i 0
743.1 −0.707107 − 0.707107i −0.632158 − 1.61257i 1.00000i 0 −0.693255 + 1.58726i 0.707107 − 0.707107i 0.707107 − 0.707107i −2.20075 + 2.03880i 0
743.2 −0.707107 − 0.707107i 0.117665 + 1.72805i 1.00000i 0 1.13871 − 1.30512i 0.707107 − 0.707107i 0.707107 − 0.707107i −2.97231 + 0.406663i 0
743.3 −0.707107 − 0.707107i 1.08045 + 1.35375i 1.00000i 0 0.193255 − 1.72124i 0.707107 − 0.707107i 0.707107 − 0.707107i −0.665273 + 2.92531i 0
743.4 −0.707107 − 0.707107i 1.55537 − 0.762124i 1.00000i 0 −1.63871 − 0.560908i 0.707107 − 0.707107i 0.707107 − 0.707107i 1.83834 − 2.37076i 0
743.5 0.707107 + 0.707107i −1.55537 + 0.762124i 1.00000i 0 −1.63871 − 0.560908i −0.707107 + 0.707107i −0.707107 + 0.707107i 1.83834 − 2.37076i 0
743.6 0.707107 + 0.707107i −1.08045 − 1.35375i 1.00000i 0 0.193255 − 1.72124i −0.707107 + 0.707107i −0.707107 + 0.707107i −0.665273 + 2.92531i 0
743.7 0.707107 + 0.707107i −0.117665 − 1.72805i 1.00000i 0 1.13871 − 1.30512i −0.707107 + 0.707107i −0.707107 + 0.707107i −2.97231 + 0.406663i 0
743.8 0.707107 + 0.707107i 0.632158 + 1.61257i 1.00000i 0 −0.693255 + 1.58726i −0.707107 + 0.707107i −0.707107 + 0.707107i −2.20075 + 2.03880i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 407.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.j.e ✓ 16
3.b odd 2 1 1050.2.j.f yes 16
5.b even 2 1 inner 1050.2.j.e ✓ 16
5.c odd 4 2 1050.2.j.f yes 16
15.d odd 2 1 1050.2.j.f yes 16
15.e even 4 2 inner 1050.2.j.e ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.j.e ✓ 16 1.a even 1 1 trivial
1050.2.j.e ✓ 16 5.b even 2 1 inner
1050.2.j.e ✓ 16 15.e even 4 2 inner
1050.2.j.f yes 16 3.b odd 2 1
1050.2.j.f yes 16 5.c odd 4 2
1050.2.j.f yes 16 15.d odd 2 1

Hecke kernels

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

\( T_{11}^{8} + 54T_{11}^{6} + 953T_{11}^{4} + 6048T_{11}^{2} + 10816 \) Copy content Toggle raw display
\( T_{17}^{16} + 3988T_{17}^{12} + 2361462T_{17}^{8} + 30432820T_{17}^{4} + 13845841 \) Copy content Toggle raw display
\( T_{29}^{4} - 79T_{29}^{2} + 96T_{29} + 676 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + 8 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} + 54 T^{6} + \cdots + 10816)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 1970 T^{12} + \cdots + 7311616 \) Copy content Toggle raw display
$17$ \( T^{16} + 3988 T^{12} + \cdots + 13845841 \) Copy content Toggle raw display
$19$ \( (T^{8} + 146 T^{6} + \cdots + 59536)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 122825015296 \) Copy content Toggle raw display
$29$ \( (T^{4} - 79 T^{2} + \cdots + 676)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 67 T^{2} + \cdots + 976)^{4} \) Copy content Toggle raw display
$37$ \( T^{16} + 2336 T^{12} + \cdots + 16777216 \) Copy content Toggle raw display
$41$ \( (T^{8} + 72 T^{6} + \cdots + 1369)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 73901944197376 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 1871773696 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{4} + 10 T^{3} + \cdots - 524)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} - 8 T^{3} + \cdots - 908)^{4} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 340701020416 \) Copy content Toggle raw display
$71$ \( (T^{4} + 152 T^{2} + 2704)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + 3330 T^{12} + \cdots + 26873856 \) Copy content Toggle raw display
$79$ \( (T^{8} + 168 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 371752372136161 \) Copy content Toggle raw display
$89$ \( (T^{4} - 8 T^{3} + \cdots - 428)^{4} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 68719476736 \) Copy content Toggle raw display
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