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Properties

Label 224.4.e.c
Level $224$
Weight $4$
Character orbit 224.e
Analytic conductor $13.216$
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] = [224,4,Mod(111,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.111");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 224.e (of order \(2\), degree \(1\), not 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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 89 x^{14} - 483 x^{12} - 119599 x^{10} + 4719476 x^{8} - 46256624 x^{6} + 215168576 x^{4} + \cdots + 8971878400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{37} \)
Twist minimal: no (minimal twist has level 56)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{8} q^{5} + ( - \beta_{4} - \beta_{2}) q^{7} + ( - 3 \beta_{6} - 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{8} q^{5} + ( - \beta_{4} - \beta_{2}) q^{7} + ( - 3 \beta_{6} - 16) q^{9} + ( - \beta_{10} + \beta_{7} - \beta_{6} + 1) q^{11} + (\beta_{12} + \beta_{8} + \cdots - \beta_{4}) q^{13}+ \cdots + (37 \beta_{10} - 31 \beta_{7} + \cdots + 311) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 280 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 280 q^{9} + 8 q^{11} + 760 q^{25} + 696 q^{35} - 1048 q^{43} - 992 q^{49} + 1680 q^{51} + 1736 q^{57} - 2184 q^{65} - 5192 q^{67} + 760 q^{81} + 5000 q^{91} + 5704 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 89 x^{14} - 483 x^{12} - 119599 x^{10} + 4719476 x^{8} - 46256624 x^{6} + 215168576 x^{4} + \cdots + 8971878400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 78\!\cdots\!27 \nu^{15} + \cdots - 64\!\cdots\!80 \nu ) / 31\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 22\!\cdots\!47 \nu^{14} + \cdots + 28\!\cdots\!40 ) / 98\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 31\!\cdots\!51 \nu^{14} + \cdots - 37\!\cdots\!80 ) / 98\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\!\cdots\!27 \nu^{15} + \cdots + 18\!\cdots\!40 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\!\cdots\!27 \nu^{15} + \cdots - 18\!\cdots\!40 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13\!\cdots\!81 \nu^{14} + \cdots - 15\!\cdots\!76 ) / 26\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21\!\cdots\!47 \nu^{14} + \cdots - 41\!\cdots\!20 ) / 38\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 29\!\cdots\!11 \nu^{15} + \cdots + 38\!\cdots\!20 \nu ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10\!\cdots\!19 \nu^{14} + \cdots + 77\!\cdots\!20 ) / 98\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 55\!\cdots\!54 \nu^{14} + \cdots - 64\!\cdots\!80 ) / 33\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 69\!\cdots\!87 \nu^{15} + \cdots - 82\!\cdots\!20 \nu ) / 15\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 38\!\cdots\!13 \nu^{15} + \cdots + 12\!\cdots\!40 \nu ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 35\!\cdots\!97 \nu^{15} + \cdots + 68\!\cdots\!60 \nu ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 14\!\cdots\!33 \nu^{15} + \cdots - 12\!\cdots\!80 \nu ) / 78\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 33\!\cdots\!21 \nu^{15} + \cdots - 16\!\cdots\!60 \nu ) / 63\!\cdots\!08 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} - \beta_{11} - \beta_{5} - \beta_{4} + 7\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{10} + 4\beta_{7} - 9\beta_{6} + 3\beta_{3} + \beta_{2} - 39 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 9 \beta_{15} + 5 \beta_{14} - 15 \beta_{13} + 29 \beta_{12} + 47 \beta_{11} - 117 \beta_{8} + \cdots - 528 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 119 \beta_{10} - 67 \beta_{9} - 143 \beta_{7} + 581 \beta_{6} - 162 \beta_{5} + 162 \beta_{4} + \cdots + 4141 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 321 \beta_{15} + 303 \beta_{14} + 1447 \beta_{13} - 2409 \beta_{12} - 3407 \beta_{11} + \cdots + 27336 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 6445 \beta_{10} + 8408 \beta_{9} + 7594 \beta_{7} - 30728 \beta_{6} + 11628 \beta_{5} - 11628 \beta_{4} + \cdots - 221628 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 18533 \beta_{15} - 1931 \beta_{14} - 128071 \beta_{13} + 208449 \beta_{12} + 145403 \beta_{11} + \cdots - 1367768 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 236659 \beta_{10} - 685347 \beta_{9} - 335797 \beta_{7} + 1191515 \beta_{6} - 1072710 \beta_{5} + \cdots + 7883195 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 451713 \beta_{15} + 130515 \beta_{14} + 10708179 \beta_{13} - 17756885 \beta_{12} + \cdots + 34400620 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 396949 \beta_{10} + 52919334 \beta_{9} - 302278 \beta_{7} - 969752 \beta_{6} + 82074928 \beta_{5} + \cdots - 16551372 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 31985483 \beta_{15} + 7148817 \beta_{14} - 774487403 \beta_{13} + 1276170261 \beta_{12} + \cdots + 2412164708 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 1273966025 \beta_{10} - 3697412443 \beta_{9} + 1623656697 \beta_{7} - 6200781443 \beta_{6} + \cdots - 43208010523 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 6783013919 \beta_{15} - 1995278065 \beta_{14} + 51275229063 \beta_{13} - 84610465209 \beta_{12} + \cdots - 516548377096 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 177896013907 \beta_{10} + 229683469880 \beta_{9} - 228203658870 \beta_{7} + 867647608712 \beta_{6} + \cdots + 6028125926132 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 777318630123 \beta_{15} + 218498294629 \beta_{14} - 2982402514071 \beta_{13} + \cdots + 59089805423528 \beta_1 ) / 16 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1
1.33401 + 8.44300i
−1.33401 + 8.44300i
−2.98428 + 0.982833i
2.98428 + 0.982833i
4.67961 − 1.69167i
−4.67961 − 1.69167i
1.29063 − 1.90734i
−1.29063 − 1.90734i
1.29063 + 1.90734i
−1.29063 + 1.90734i
4.67961 + 1.69167i
−4.67961 + 1.69167i
−2.98428 − 0.982833i
2.98428 − 0.982833i
1.33401 − 8.44300i
−1.33401 − 8.44300i
0 − 9.43218i 0 −10.6455 0 13.5311 − 12.6455i 0 −61.9661 0
111.2 0 − 9.43218i 0 10.6455 0 −13.5311 + 12.6455i 0 −61.9661 0
111.3 0 − 7.24640i 0 −15.7090 0 −6.82222 + 17.2179i 0 −25.5103 0
111.4 0 − 7.24640i 0 15.7090 0 6.82222 − 17.2179i 0 −25.5103 0
111.5 0 − 5.61621i 0 −0.0506623 0 14.0236 + 12.0971i 0 −4.54181 0
111.6 0 − 5.61621i 0 0.0506623 0 −14.0236 − 12.0971i 0 −4.54181 0
111.7 0 − 2.23200i 0 −18.1631 0 −11.6493 − 14.3977i 0 22.0182 0
111.8 0 − 2.23200i 0 18.1631 0 11.6493 + 14.3977i 0 22.0182 0
111.9 0 2.23200i 0 −18.1631 0 −11.6493 + 14.3977i 0 22.0182 0
111.10 0 2.23200i 0 18.1631 0 11.6493 − 14.3977i 0 22.0182 0
111.11 0 5.61621i 0 −0.0506623 0 14.0236 − 12.0971i 0 −4.54181 0
111.12 0 5.61621i 0 0.0506623 0 −14.0236 + 12.0971i 0 −4.54181 0
111.13 0 7.24640i 0 −15.7090 0 −6.82222 − 17.2179i 0 −25.5103 0
111.14 0 7.24640i 0 15.7090 0 6.82222 + 17.2179i 0 −25.5103 0
111.15 0 9.43218i 0 −10.6455 0 13.5311 + 12.6455i 0 −61.9661 0
111.16 0 9.43218i 0 10.6455 0 −13.5311 − 12.6455i 0 −61.9661 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 111.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
8.d odd 2 1 inner
56.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.4.e.c 16
4.b odd 2 1 56.4.e.c ✓ 16
7.b odd 2 1 inner 224.4.e.c 16
8.b even 2 1 56.4.e.c ✓ 16
8.d odd 2 1 inner 224.4.e.c 16
28.d even 2 1 56.4.e.c ✓ 16
56.e even 2 1 inner 224.4.e.c 16
56.h odd 2 1 56.4.e.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.4.e.c ✓ 16 4.b odd 2 1
56.4.e.c ✓ 16 8.b even 2 1
56.4.e.c ✓ 16 28.d even 2 1
56.4.e.c ✓ 16 56.h odd 2 1
224.4.e.c 16 1.a even 1 1 trivial
224.4.e.c 16 7.b odd 2 1 inner
224.4.e.c 16 8.d odd 2 1 inner
224.4.e.c 16 56.e even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 178T_{3}^{6} + 9996T_{3}^{4} + 192856T_{3}^{2} + 734080 \) acting on \(S_{4}^{\mathrm{new}}(224, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 178 T^{6} + \cdots + 734080)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - 690 T^{6} + \cdots + 23680)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} + \cdots + 196832)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 8883745252480)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 88952925061120)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 95217018359680)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 1078203498496)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 61344695910400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 86\!\cdots\!20)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 4893561585664)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 38\!\cdots\!80)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 262 T^{3} + \cdots - 295450400)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 15\!\cdots\!80)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 40\!\cdots\!36)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 11\!\cdots\!20)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 31\!\cdots\!20)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 1298 T^{3} + \cdots - 105907858720)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 12\!\cdots\!80)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 58\!\cdots\!80)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 55\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 13\!\cdots\!80)^{2} \) Copy content Toggle raw display
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