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Properties

Label 1224.4.c.f
Level $1224$
Weight $4$
Character orbit 1224.c
Analytic conductor $72.218$
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] = [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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 430x^{10} + 68473x^{8} + 5023973x^{6} + 168164854x^{4} + 2001102145x^{2} + 2010724 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{19}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 408)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{5} + ( - \beta_{5} + \beta_{4}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{5} + ( - \beta_{5} + \beta_{4}) q^{7} + (\beta_{6} - \beta_{4}) q^{11} + (\beta_{2} + 2) q^{13} + ( - \beta_{8} + 2 \beta_{4} + \beta_{3} + \cdots - 1) q^{17}+ \cdots + ( - 9 \beta_{11} - 5 \beta_{10} + \cdots - 7 \beta_{4}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 22 q^{13} - 10 q^{17} - 194 q^{19} - 2 q^{25} - 964 q^{35} - 638 q^{43} + 924 q^{47} - 652 q^{49} + 1336 q^{53} - 1190 q^{55} - 64 q^{59} + 1256 q^{67} + 1524 q^{77} + 1796 q^{83} + 430 q^{85} + 2656 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 430x^{10} + 68473x^{8} + 5023973x^{6} + 168164854x^{4} + 2001102145x^{2} + 2010724 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 87620574929 \nu^{10} + 30528230724109 \nu^{8} + \cdots - 29\!\cdots\!56 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 86987829871 \nu^{10} + 33045969829491 \nu^{8} + \cdots - 13\!\cdots\!44 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1618548679 \nu^{10} + 570475220989 \nu^{8} + 66354465569311 \nu^{6} + \cdots + 49\!\cdots\!14 ) / 18\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 36021111877633 \nu^{11} + \cdots - 34\!\cdots\!88 \nu ) / 64\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 38758774069773 \nu^{11} + \cdots - 73\!\cdots\!28 \nu ) / 43\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 41835512038037 \nu^{11} + \cdots + 13\!\cdots\!32 \nu ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 189838197 \nu^{11} + 80377751457 \nu^{9} + 12554108002296 \nu^{7} + \cdots + 34\!\cdots\!44 \nu ) / 307117960494874 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10\!\cdots\!32 \nu^{11} + 820169771028225 \nu^{10} + \cdots + 67\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!32 \nu^{11} - 567151865709945 \nu^{10} + \cdots - 18\!\cdots\!20 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10\!\cdots\!32 \nu^{11} - 820169771028225 \nu^{10} + \cdots - 67\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 770028766177747 \nu^{11} + \cdots - 16\!\cdots\!92 \nu ) / 43\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{11} + 3\beta_{10} + 3\beta_{8} + 2\beta_{7} + 9\beta_{5} + 3\beta_{4} ) / 36 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 22\beta_{10} - 7\beta_{9} - 15\beta_{8} + 21\beta_{3} + 21\beta_{2} - 33\beta _1 - 2584 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -264\beta_{11} - 390\beta_{10} - 390\beta_{8} + 13\beta_{7} + 270\beta_{6} - 1512\beta_{5} - 228\beta_{4} ) / 36 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -1436\beta_{10} + 221\beta_{9} + 1215\beta_{8} - 1257\beta_{3} - 1881\beta_{2} + 1953\beta _1 + 96170 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 29349 \beta_{11} + 54837 \beta_{10} + 54837 \beta_{8} - 17564 \beta_{7} - 45738 \beta_{6} + \cdots - 3807 \beta_{4} ) / 36 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 353887 \beta_{10} - 23725 \beta_{9} - 330162 \beta_{8} + 311412 \beta_{3} + 526890 \beta_{2} + \cdots - 18794587 ) / 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3897675 \beta_{11} - 8247735 \beta_{10} - 8247735 \beta_{8} + 3828830 \beta_{7} + \cdots + 5381037 \beta_{4} ) / 36 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 37722638 \beta_{10} + 237713 \beta_{9} + 37484925 \beta_{8} - 34292097 \beta_{3} - 60033777 \beta_{2} + \cdots + 1785639146 ) / 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 574897512 \beta_{11} + 1289821674 \beta_{10} + 1289821674 \beta_{8} - 703552621 \beta_{7} + \cdots - 1396871844 \beta_{4} ) / 36 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 18077662192 \beta_{10} + 670583117 \beta_{9} - 18748245309 \beta_{8} + 17035440681 \beta_{3} + \cdots - 804460817764 ) / 36 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 89381571429 \beta_{11} - 205784699961 \beta_{10} - 205784699961 \beta_{8} + \cdots + 283655447655 \beta_{4} ) / 36 \) 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
12.7736i
5.13318i
10.4509i
− 8.51153i
− 7.66934i
− 0.0317000i
0.0317000i
7.66934i
8.51153i
− 10.4509i
− 5.13318i
− 12.7736i
0 0 0 − 18.9733i 0 − 25.3843i 0 0 0
577.2 0 0 0 − 10.8546i 0 − 4.13074i 0 0 0
577.3 0 0 0 − 10.1999i 0 27.2687i 0 0 0
577.4 0 0 0 − 8.52377i 0 − 30.5433i 0 0 0
577.5 0 0 0 − 8.20822i 0 5.68733i 0 0 0
577.6 0 0 0 − 5.39665i 0 − 3.70842i 0 0 0
577.7 0 0 0 5.39665i 0 3.70842i 0 0 0
577.8 0 0 0 8.20822i 0 − 5.68733i 0 0 0
577.9 0 0 0 8.52377i 0 30.5433i 0 0 0
577.10 0 0 0 10.1999i 0 − 27.2687i 0 0 0
577.11 0 0 0 10.8546i 0 4.13074i 0 0 0
577.12 0 0 0 18.9733i 0 25.3843i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.12
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 1224.4.c.f 12
3.b odd 2 1 408.4.c.d ✓ 12
12.b even 2 1 816.4.c.h 12
17.b even 2 1 inner 1224.4.c.f 12
51.c odd 2 1 408.4.c.d ✓ 12
204.h even 2 1 816.4.c.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
408.4.c.d ✓ 12 3.b odd 2 1
408.4.c.d ✓ 12 51.c odd 2 1
816.4.c.h 12 12.b even 2 1
816.4.c.h 12 204.h even 2 1
1224.4.c.f 12 1.a even 1 1 trivial
1224.4.c.f 12 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 \(S_{4}^{\mathrm{new}}(1224, [\chi])\):

\( T_{5}^{12} + 751T_{5}^{10} + 199519T_{5}^{8} + 25359477T_{5}^{6} + 1656029700T_{5}^{4} + 52729929216T_{5}^{2} + 629090095104 \) Copy content Toggle raw display
\( T_{47}^{6} - 462 T_{47}^{5} - 343236 T_{47}^{4} + 155845528 T_{47}^{3} + 26157484992 T_{47}^{2} + \cdots - 243688614539264 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 629090095104 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 3392669286400 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 108970883743744 \) Copy content Toggle raw display
$13$ \( (T^{6} - 11 T^{5} + \cdots - 125176136)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 14\!\cdots\!09 \) Copy content Toggle raw display
$19$ \( (T^{6} + 97 T^{5} + \cdots - 3692672)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 67\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + 319 T^{5} + \cdots + 20394020160)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 243688614539264)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 29\!\cdots\!32)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 32 T^{5} + \cdots + 584653012992)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 88\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 59\!\cdots\!04)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 95\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 29\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 510546674791424)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 18296160931136)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
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