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Properties

Label 1428.2.d.c
Level $1428$
Weight $2$
Character orbit 1428.d
Analytic conductor $11.403$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1428,2,Mod(169,1428)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1428, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1428.169");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1428.d (of order \(2\), degree \(1\), 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: \(11.4026374086\)
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: \( x^{12} + 43x^{10} + 647x^{8} + 4049x^{6} + 10288x^{4} + 9088x^{2} + 2304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{3} + \beta_1 q^{5} - \beta_{8} q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{3} + \beta_1 q^{5} - \beta_{8} q^{7} - q^{9} + ( - \beta_{8} + \beta_{2}) q^{11} + ( - \beta_{11} - \beta_{10} + \beta_{4}) q^{13} + \beta_{4} q^{15} + (\beta_{6} - \beta_{2}) q^{17} + (\beta_{7} + 1) q^{19} + q^{21} + (\beta_{9} + \beta_{8} + \beta_{2} + \beta_1) q^{23} + (\beta_{6} + \beta_{5} + \beta_{3} - 2) q^{25} - \beta_{8} q^{27} + ( - \beta_{9} - 2 \beta_{8} + \cdots - \beta_1) q^{29}+ \cdots + (\beta_{8} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{9} - 6 q^{13} + 2 q^{15} - 2 q^{17} + 6 q^{19} + 12 q^{21} - 26 q^{25} + 10 q^{33} - 2 q^{35} - 18 q^{43} - 20 q^{47} - 12 q^{49} - 2 q^{51} + 16 q^{53} + 22 q^{55} - 12 q^{59} + 32 q^{67} - 6 q^{69} - 10 q^{77} + 12 q^{81} - 16 q^{83} + 14 q^{85} + 24 q^{87} + 4 q^{89} - 20 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 43x^{10} + 647x^{8} + 4049x^{6} + 10288x^{4} + 9088x^{2} + 2304 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -81\nu^{11} - 8775\nu^{9} - 232091\nu^{7} - 2239509\nu^{5} - 7414236\nu^{3} - 6102208\nu ) / 1574464 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -475\nu^{10} - 18657\nu^{8} - 246997\nu^{6} - 1298931\nu^{4} - 2859560\nu^{2} - 1549680 ) / 393616 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -7\nu^{10} - 309\nu^{8} - 4497\nu^{6} - 24239\nu^{4} - 43872\nu^{2} - 18240 ) / 5392 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1921 \nu^{11} + 2850 \nu^{10} - 76903 \nu^{9} + 111942 \nu^{8} - 1019003 \nu^{7} + \cdots + 25829952 ) / 4723392 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1921 \nu^{11} + 2850 \nu^{10} + 76903 \nu^{9} + 111942 \nu^{8} + 1019003 \nu^{7} + 1481982 \nu^{6} + \cdots + 25829952 ) / 4723392 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -1521\nu^{10} - 66371\nu^{8} - 1023351\nu^{6} - 6594761\nu^{4} - 16119472\nu^{2} - 8003440 ) / 393616 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 95\nu^{11} + 4001\nu^{9} + 57757\nu^{7} + 330691\nu^{5} + 686492\nu^{3} + 336896\nu ) / 64704 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -1921\nu^{11} - 76903\nu^{9} - 1019003\nu^{7} - 4814165\nu^{5} - 4963308\nu^{3} + 4260960\nu ) / 787232 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10664 \nu^{11} - 10281 \nu^{10} + 450038 \nu^{9} - 424947 \nu^{8} + 6567682 \nu^{7} + \cdots - 27736800 ) / 2361696 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10664 \nu^{11} - 10281 \nu^{10} - 450038 \nu^{9} - 424947 \nu^{8} - 6567682 \nu^{7} + \cdots - 27736800 ) / 2361696 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{3} - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - 3\beta_{6} + 3\beta_{5} - 16\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} + 2\beta_{10} - \beta_{7} - 15\beta_{6} - 15\beta_{5} - 3\beta_{4} - 23\beta_{3} + 90 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{11} - 2\beta_{10} + 24\beta_{9} + 22\beta_{8} + 55\beta_{6} - 55\beta_{5} + 8\beta_{2} + 253\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -44\beta_{11} - 44\beta_{10} + 16\beta_{7} + 229\beta_{6} + 229\beta_{5} + 82\beta_{4} + 407\beta_{3} - 1333 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 66 \beta_{11} + 66 \beta_{10} - 423 \beta_{9} - 618 \beta_{8} - 897 \beta_{6} + 897 \beta_{5} + \cdots - 3992 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 808 \beta_{11} + 808 \beta_{10} - 253 \beta_{7} - 3569 \beta_{6} - 3569 \beta_{5} - 1773 \beta_{4} + \cdots + 20568 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1520 \beta_{11} - 1520 \beta_{10} + 6934 \beta_{9} + 13028 \beta_{8} + 14325 \beta_{6} + \cdots + 63037 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 14326 \beta_{11} - 14326 \beta_{10} + 4352 \beta_{7} + 56103 \beta_{6} + 56103 \beta_{5} + \cdots - 321951 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 30852 \beta_{11} + 30852 \beta_{10} - 111177 \beta_{9} - 248860 \beta_{8} - 227735 \beta_{6} + \cdots - 996452 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1428\mathbb{Z}\right)^\times\).

\(n\) \(409\) \(715\) \(953\) \(1261\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1
− 4.02527i
− 1.81912i
− 0.655072i
0.981755i
2.60512i
3.91259i
− 3.91259i
− 2.60512i
− 0.981755i
0.655072i
1.81912i
4.02527i
0 − 1.00000i 0 − 4.02527i 0 1.00000i 0 −1.00000 0
169.2 0 − 1.00000i 0 − 1.81912i 0 1.00000i 0 −1.00000 0
169.3 0 − 1.00000i 0 − 0.655072i 0 1.00000i 0 −1.00000 0
169.4 0 − 1.00000i 0 0.981755i 0 1.00000i 0 −1.00000 0
169.5 0 − 1.00000i 0 2.60512i 0 1.00000i 0 −1.00000 0
169.6 0 − 1.00000i 0 3.91259i 0 1.00000i 0 −1.00000 0
169.7 0 1.00000i 0 − 3.91259i 0 − 1.00000i 0 −1.00000 0
169.8 0 1.00000i 0 − 2.60512i 0 − 1.00000i 0 −1.00000 0
169.9 0 1.00000i 0 − 0.981755i 0 − 1.00000i 0 −1.00000 0
169.10 0 1.00000i 0 0.655072i 0 − 1.00000i 0 −1.00000 0
169.11 0 1.00000i 0 1.81912i 0 − 1.00000i 0 −1.00000 0
169.12 0 1.00000i 0 4.02527i 0 − 1.00000i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 169.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1428.2.d.c ✓ 12
3.b odd 2 1 4284.2.d.f 12
17.b even 2 1 inner 1428.2.d.c ✓ 12
51.c odd 2 1 4284.2.d.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1428.2.d.c ✓ 12 1.a even 1 1 trivial
1428.2.d.c ✓ 12 17.b even 2 1 inner
4284.2.d.f 12 3.b odd 2 1
4284.2.d.f 12 51.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 43T_{5}^{10} + 647T_{5}^{8} + 4049T_{5}^{6} + 10288T_{5}^{4} + 9088T_{5}^{2} + 2304 \) acting on \(S_{2}^{\mathrm{new}}(1428, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + 43 T^{10} + \cdots + 2304 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + 67 T^{10} + \cdots + 2304 \) Copy content Toggle raw display
$13$ \( (T^{6} + 3 T^{5} + \cdots - 5976)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 2 T^{11} + \cdots + 24137569 \) Copy content Toggle raw display
$19$ \( (T^{6} - 3 T^{5} + \cdots + 3240)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 104530176 \) Copy content Toggle raw display
$29$ \( T^{12} + 232 T^{10} + \cdots + 6718464 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 3681091584 \) Copy content Toggle raw display
$37$ \( T^{12} + 348 T^{10} + \cdots + 10240000 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 389193984 \) Copy content Toggle raw display
$43$ \( (T^{6} + 9 T^{5} + \cdots + 5760)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 10 T^{5} + \cdots - 60672)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 8 T^{5} + \cdots + 19728)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 6 T^{5} + \cdots - 144000)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 1081094400 \) Copy content Toggle raw display
$67$ \( (T^{6} - 16 T^{5} + \cdots + 178944)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 516198400 \) Copy content Toggle raw display
$73$ \( (T^{6} + 76 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 24908414976 \) Copy content Toggle raw display
$83$ \( (T^{6} + 8 T^{5} + \cdots + 165888)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 2 T^{5} + \cdots + 388224)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 268 T^{4} + \cdots + 14400)^{2} \) Copy content Toggle raw display
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