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Properties

Label 224.4.i.d
Level $224$
Weight $4$
Character orbit 224.i
Analytic conductor $13.216$
Analytic rank $0$
Dimension $12$
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] = [224,4,Mod(65,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.65");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 224.i (of order \(3\), 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: \(13.2164278413\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 42 x^{10} - 74 x^{9} + 975 x^{8} - 4602 x^{7} + 21732 x^{6} - 98076 x^{5} + 355026 x^{4} + \cdots + 4977508 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2}\cdot 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + (\beta_{6} + \beta_1) q^{5} + (\beta_{9} - \beta_{3}) q^{7} + (\beta_{10} + \beta_{4} + 12 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + (\beta_{6} + \beta_1) q^{5} + (\beta_{9} - \beta_{3}) q^{7} + (\beta_{10} + \beta_{4} + 12 \beta_1) q^{9} + ( - \beta_{11} - \beta_{8} + \cdots + \beta_{2}) q^{11}+ \cdots + (22 \beta_{11} - 47 \beta_{9} + \cdots + 265 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{5} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{5} - 72 q^{9} - 144 q^{13} + 138 q^{17} + 66 q^{21} + 72 q^{25} - 144 q^{29} + 714 q^{33} + 6 q^{37} - 576 q^{41} + 120 q^{45} + 948 q^{49} + 30 q^{53} - 3420 q^{57} + 54 q^{61} - 120 q^{65} + 924 q^{69} + 2694 q^{73} + 4062 q^{77} - 3666 q^{81} - 5268 q^{85} + 3558 q^{89} - 2190 q^{93} - 768 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 42 x^{10} - 74 x^{9} + 975 x^{8} - 4602 x^{7} + 21732 x^{6} - 98076 x^{5} + 355026 x^{4} + \cdots + 4977508 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4579074758 \nu^{11} - 7952764556 \nu^{10} + 164825558269 \nu^{9} - 553564863575 \nu^{8} + \cdots - 12\!\cdots\!52 ) / 85\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 18\!\cdots\!41 \nu^{11} + \cdots + 58\!\cdots\!60 ) / 19\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 31\!\cdots\!28 \nu^{11} + \cdots - 17\!\cdots\!18 ) / 19\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 289551344732 \nu^{11} + 2625958276375 \nu^{10} - 2000604488236 \nu^{9} + \cdots + 63\!\cdots\!16 ) / 12\!\cdots\!07 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2198863635708 \nu^{11} + 16436895089611 \nu^{10} + 97923667240436 \nu^{9} + \cdots + 13\!\cdots\!24 ) / 84\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 93\!\cdots\!01 \nu^{11} + \cdots - 53\!\cdots\!76 ) / 35\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 32\!\cdots\!21 \nu^{11} + \cdots - 22\!\cdots\!94 ) / 97\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12\!\cdots\!91 \nu^{11} + \cdots - 65\!\cdots\!94 ) / 32\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 27\!\cdots\!29 \nu^{11} - 393208108185538 \nu^{10} + \cdots + 76\!\cdots\!04 ) / 36\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 19\!\cdots\!83 \nu^{11} + \cdots - 15\!\cdots\!08 ) / 12\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17\!\cdots\!39 \nu^{11} + \cdots + 48\!\cdots\!44 ) / 97\!\cdots\!90 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 6\beta_{11} - 2\beta_{9} - \beta_{8} - 3\beta_{7} + 21\beta_{6} + 18\beta_{3} - 6\beta_{2} ) / 84 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 9 \beta_{11} + 21 \beta_{10} - 53 \beta_{9} - 37 \beta_{8} - 6 \beta_{7} - 21 \beta_{6} + 21 \beta_{4} + \cdots - 756 ) / 84 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 20 \beta_{11} - 21 \beta_{10} - 2 \beta_{9} - 8 \beta_{8} + 67 \beta_{7} + 21 \beta_{6} + 63 \beta_{5} + \cdots + 896 ) / 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 60 \beta_{11} - 252 \beta_{10} + 776 \beta_{9} - 620 \beta_{8} - 474 \beta_{7} - 1911 \beta_{5} + \cdots - 6468 ) / 84 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3141 \beta_{11} + 1995 \beta_{10} - 2467 \beta_{9} + 3943 \beta_{8} - 372 \beta_{7} - 1029 \beta_{6} + \cdots + 10920 ) / 84 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 8132 \beta_{11} - 798 \beta_{10} - 32 \beta_{9} + 376 \beta_{8} + 4852 \beta_{7} + 17262 \beta_{6} + \cdots + 40684 ) / 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 8175 \beta_{11} + 2331 \beta_{10} + 817 \beta_{9} - 20287 \beta_{8} - 13872 \beta_{7} + \cdots - 328272 ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 262872 \beta_{11} - 69972 \beta_{10} - 6680 \beta_{9} + 762824 \beta_{8} + 433416 \beta_{7} + \cdots + 10541244 ) / 84 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 856041 \beta_{11} - 259497 \beta_{10} + 752361 \beta_{9} - 361035 \beta_{8} - 132918 \beta_{7} + \cdots - 3387664 ) / 28 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 14915526 \beta_{11} + 7932834 \beta_{10} - 11567338 \beta_{9} - 6397478 \beta_{8} - 12701670 \beta_{7} + \cdots - 217825020 ) / 84 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 52984935 \beta_{11} - 26142753 \beta_{10} - 11606869 \beta_{9} + 80165431 \beta_{8} + 102946398 \beta_{7} + \cdots + 1766961504 ) / 84 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−1.18461 − 3.64087i
3.13776 − 1.28557i
−3.78890 + 3.84640i
0.674870 + 1.54726i
0.215668 − 4.52275i
0.945207 + 4.05553i
−1.18461 + 3.64087i
3.13776 + 1.28557i
−3.78890 − 3.84640i
0.674870 − 1.54726i
0.215668 + 4.52275i
0.945207 − 4.05553i
0 −4.83694 − 8.37782i 0 −0.978804 + 1.69534i 0 16.9872 + 7.37790i 0 −33.2919 + 57.6633i 0
65.2 0 −2.14765 − 3.71985i 0 6.20686 − 10.7506i 0 −15.5092 − 10.1224i 0 4.27516 − 7.40479i 0
65.3 0 −1.11427 − 1.92998i 0 −6.72806 + 11.6533i 0 −10.1930 + 15.4630i 0 11.0168 − 19.0816i 0
65.4 0 1.11427 + 1.92998i 0 −6.72806 + 11.6533i 0 10.1930 − 15.4630i 0 11.0168 − 19.0816i 0
65.5 0 2.14765 + 3.71985i 0 6.20686 − 10.7506i 0 15.5092 + 10.1224i 0 4.27516 − 7.40479i 0
65.6 0 4.83694 + 8.37782i 0 −0.978804 + 1.69534i 0 −16.9872 − 7.37790i 0 −33.2919 + 57.6633i 0
193.1 0 −4.83694 + 8.37782i 0 −0.978804 − 1.69534i 0 16.9872 − 7.37790i 0 −33.2919 − 57.6633i 0
193.2 0 −2.14765 + 3.71985i 0 6.20686 + 10.7506i 0 −15.5092 + 10.1224i 0 4.27516 + 7.40479i 0
193.3 0 −1.11427 + 1.92998i 0 −6.72806 − 11.6533i 0 −10.1930 − 15.4630i 0 11.0168 + 19.0816i 0
193.4 0 1.11427 − 1.92998i 0 −6.72806 − 11.6533i 0 10.1930 + 15.4630i 0 11.0168 + 19.0816i 0
193.5 0 2.14765 − 3.71985i 0 6.20686 + 10.7506i 0 15.5092 − 10.1224i 0 4.27516 + 7.40479i 0
193.6 0 4.83694 − 8.37782i 0 −0.978804 − 1.69534i 0 −16.9872 + 7.37790i 0 −33.2919 − 57.6633i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.4.i.d ✓ 12
4.b odd 2 1 inner 224.4.i.d ✓ 12
7.c even 3 1 inner 224.4.i.d ✓ 12
7.c even 3 1 1568.4.a.bh 6
7.d odd 6 1 1568.4.a.bg 6
8.b even 2 1 448.4.i.p 12
8.d odd 2 1 448.4.i.p 12
28.f even 6 1 1568.4.a.bg 6
28.g odd 6 1 inner 224.4.i.d ✓ 12
28.g odd 6 1 1568.4.a.bh 6
56.k odd 6 1 448.4.i.p 12
56.p even 6 1 448.4.i.p 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.4.i.d ✓ 12 1.a even 1 1 trivial
224.4.i.d ✓ 12 4.b odd 2 1 inner
224.4.i.d ✓ 12 7.c even 3 1 inner
224.4.i.d ✓ 12 28.g odd 6 1 inner
448.4.i.p 12 8.b even 2 1
448.4.i.p 12 8.d odd 2 1
448.4.i.p 12 56.k odd 6 1
448.4.i.p 12 56.p even 6 1
1568.4.a.bg 6 7.d odd 6 1
1568.4.a.bg 6 28.f even 6 1
1568.4.a.bh 6 7.c even 3 1
1568.4.a.bh 6 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 117T_{3}^{10} + 11406T_{3}^{8} + 249961T_{3}^{6} + 4208814T_{3}^{4} + 19576725T_{3}^{2} + 73530625 \) acting on \(S_{4}^{\mathrm{new}}(224, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 117 T^{10} + \cdots + 73530625 \) Copy content Toggle raw display
$5$ \( (T^{6} + 3 T^{5} + \cdots + 106929)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 16\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 20\!\cdots\!49 \) Copy content Toggle raw display
$13$ \( (T^{3} + 36 T^{2} + \cdots + 12544)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} - 69 T^{5} + \cdots + 26151094369)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 41\!\cdots\!49 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 22\!\cdots\!69 \) Copy content Toggle raw display
$29$ \( (T^{3} + 36 T^{2} + \cdots + 4555456)^{4} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 27157095295009)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 144 T^{2} + \cdots - 10003232)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 378376502898688)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 83\!\cdots\!09 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 110460163060009)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 20\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 32\!\cdots\!21)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 33\!\cdots\!29 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 831235451977728)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 45\!\cdots\!49)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 19\!\cdots\!69 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 66\!\cdots\!48)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 20\!\cdots\!09)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 192 T^{2} + \cdots - 376824544)^{4} \) Copy content Toggle raw display
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