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Properties

Label 1224.4.c.h
Level $1224$
Weight $4$
Character orbit 1224.c
Analytic conductor $72.218$
Analytic rank $0$
Dimension $14$
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] = [1224,4,Mod(577,1224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1224, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1224.577");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1224.c (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: \(72.2183378470\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 1021 x^{12} + 377909 x^{10} + 60179667 x^{8} + 3773591630 x^{6} + 80016298092 x^{4} + \cdots + 7295353632 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{13} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} - \beta_{6} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} - \beta_{6} q^{7} + ( - \beta_{11} - \beta_1) q^{11} + (\beta_{4} - 3) q^{13} + (\beta_{8} - \beta_1 + 2) q^{17} + ( - \beta_{5} + 8) q^{19} + ( - \beta_{12} - \beta_{6} + \beta_1) q^{23} + (\beta_{2} - 21) q^{25} + (\beta_{13} + \beta_{11} + \cdots + \beta_1) q^{29}+ \cdots + ( - \beta_{13} - 6 \beta_{12} + \cdots + 16 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 46 q^{13} + 26 q^{17} + 114 q^{19} - 292 q^{25} - 724 q^{35} + 74 q^{43} - 888 q^{47} - 1322 q^{49} - 772 q^{53} + 1122 q^{55} + 420 q^{59} + 252 q^{67} + 632 q^{77} + 2892 q^{83} + 1794 q^{85} - 116 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 1021 x^{12} + 377909 x^{10} + 60179667 x^{8} + 3773591630 x^{6} + 80016298092 x^{4} + \cdots + 7295353632 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 146 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 638064476625 \nu^{12} + 653444664960491 \nu^{10} + \cdots + 32\!\cdots\!44 ) / 80\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 95966242711 \nu^{12} + 98042591630071 \nu^{10} + \cdots + 25\!\cdots\!36 ) / 75\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 516149346253 \nu^{12} + 526378478655535 \nu^{10} + \cdots + 10\!\cdots\!80 ) / 30\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12053672859619 \nu^{13} + \cdots - 12\!\cdots\!64 \nu ) / 24\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2741825428935 \nu^{12} + \cdots - 72\!\cdots\!36 ) / 80\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 18\!\cdots\!52 \nu^{13} + \cdots - 46\!\cdots\!92 ) / 24\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18\!\cdots\!52 \nu^{13} + \cdots + 46\!\cdots\!92 ) / 24\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 39\!\cdots\!29 \nu^{13} + \cdots + 10\!\cdots\!72 \nu ) / 30\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22\!\cdots\!93 \nu^{13} + \cdots + 49\!\cdots\!68 \nu ) / 15\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 92\!\cdots\!91 \nu^{13} + \cdots + 25\!\cdots\!12 \nu ) / 24\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 14\!\cdots\!95 \nu^{13} + \cdots + 35\!\cdots\!56 \nu ) / 24\!\cdots\!44 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 146 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} + 4\beta_{12} - 5\beta_{11} - 4\beta_{10} + 2\beta_{9} + 2\beta_{8} + 11\beta_{6} - 280\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -22\beta_{9} + 22\beta_{8} - 4\beta_{7} - 8\beta_{5} + 9\beta_{4} + 34\beta_{3} - 344\beta_{2} + 40989 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 435 \beta_{13} - 1628 \beta_{12} + 2571 \beta_{11} + 1228 \beta_{10} - 1100 \beta_{9} + \cdots + 84654 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8382 \beta_{9} - 8382 \beta_{8} + 832 \beta_{7} - 4756 \beta_{5} + 8689 \beta_{4} - 18534 \beta_{3} + \cdots - 12485359 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 167227 \beta_{13} + 595364 \beta_{12} - 1117159 \beta_{11} - 308548 \beta_{10} + \cdots - 26166534 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2926778 \beta_{9} + 2926778 \beta_{8} - 27524 \beta_{7} + 4352680 \beta_{5} - 6999893 \beta_{4} + \cdots + 3887134155 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 62588335 \beta_{13} - 213563388 \beta_{12} + 449441215 \beta_{11} + 69394012 \beta_{10} + \cdots + 8176024450 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1024145390 \beta_{9} - 1024145390 \beta_{8} - 67105528 \beta_{7} - 2311452108 \beta_{5} + \cdots - 1223308066635 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 23145465531 \beta_{13} + 75972274452 \beta_{12} - 172687804879 \beta_{11} + \cdots - 2577121013126 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 361030563986 \beta_{9} + 361030563986 \beta_{8} + 43931694580 \beta_{7} + 1031448388592 \beta_{5} + \cdots + 388343977023315 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 8480989496407 \beta_{13} - 26865614674604 \beta_{12} + 64394788665935 \beta_{11} + \cdots + 819046540486842 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(613\) \(649\) \(919\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
− 18.5213i
− 17.0074i
− 16.7972i
− 7.65234i
− 6.88320i
− 0.613755i
− 0.499341i
0.499341i
0.613755i
6.88320i
7.65234i
16.7972i
17.0074i
18.5213i
0 0 0 − 18.5213i 0 − 35.0744i 0 0 0
577.2 0 0 0 − 17.0074i 0 6.49642i 0 0 0
577.3 0 0 0 − 16.7972i 0 10.0103i 0 0 0
577.4 0 0 0 − 7.65234i 0 10.8265i 0 0 0
577.5 0 0 0 − 6.88320i 0 − 6.22891i 0 0 0
577.6 0 0 0 − 0.613755i 0 − 30.2998i 0 0 0
577.7 0 0 0 − 0.499341i 0 − 24.8050i 0 0 0
577.8 0 0 0 0.499341i 0 24.8050i 0 0 0
577.9 0 0 0 0.613755i 0 30.2998i 0 0 0
577.10 0 0 0 6.88320i 0 6.22891i 0 0 0
577.11 0 0 0 7.65234i 0 − 10.8265i 0 0 0
577.12 0 0 0 16.7972i 0 − 10.0103i 0 0 0
577.13 0 0 0 17.0074i 0 − 6.49642i 0 0 0
577.14 0 0 0 18.5213i 0 35.0744i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.14
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 1224.4.c.h yes 14
3.b odd 2 1 1224.4.c.g ✓ 14
17.b even 2 1 inner 1224.4.c.h yes 14
51.c odd 2 1 1224.4.c.g ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1224.4.c.g ✓ 14 3.b odd 2 1
1224.4.c.g ✓ 14 51.c odd 2 1
1224.4.c.h yes 14 1.a even 1 1 trivial
1224.4.c.h yes 14 17.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1224, [\chi])\):

\( T_{5}^{14} + 1021 T_{5}^{12} + 377909 T_{5}^{10} + 60179667 T_{5}^{8} + 3773591630 T_{5}^{6} + \cdots + 7295353632 \) Copy content Toggle raw display
\( T_{47}^{7} + 444 T_{47}^{6} - 363536 T_{47}^{5} - 120238592 T_{47}^{4} + 44832357632 T_{47}^{3} + \cdots - 24\!\cdots\!72 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 7295353632 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 13\!\cdots\!48 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 17\!\cdots\!28 \) Copy content Toggle raw display
$13$ \( (T^{7} + 23 T^{6} + \cdots - 2344224656)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 69\!\cdots\!17 \) Copy content Toggle raw display
$19$ \( (T^{7} - 57 T^{6} + \cdots + 219759460096)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 52\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 86\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 92\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 12\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 20\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots + 13\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots - 24\!\cdots\!72)^{2} \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots - 32\!\cdots\!64)^{2} \) Copy content Toggle raw display
$59$ \( (T^{7} + \cdots + 46\!\cdots\!24)^{2} \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 44\!\cdots\!88 \) Copy content Toggle raw display
$67$ \( (T^{7} + \cdots + 75\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 42\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 20\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 54\!\cdots\!92 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots + 17\!\cdots\!76)^{2} \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots - 32\!\cdots\!36)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 14\!\cdots\!48 \) Copy content Toggle raw display
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