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Properties

Label 224.3.h.d
Level $224$
Weight $3$
Character orbit 224.h
Analytic conductor $6.104$
Analytic rank $0$
Dimension $8$
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] = [224,3,Mod(209,224)]
 
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([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.209");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.h (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: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.976966189056.51
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 10x^{6} + 123x^{4} - 300x^{3} + 86x^{2} + 300x + 526 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + (\beta_{3} - \beta_{2}) q^{5} + ( - \beta_{4} + \beta_1 + 2) q^{7} + (2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + (\beta_{3} - \beta_{2}) q^{5} + ( - \beta_{4} + \beta_1 + 2) q^{7} + (2 \beta_1 + 1) q^{9} + ( - \beta_{6} - \beta_{5}) q^{11} + ( - \beta_{3} - 3 \beta_{2}) q^{13} + (\beta_1 - 8) q^{15} + (\beta_{7} - \beta_{4} + \beta_1) q^{17} + ( - 3 \beta_{3} - 4 \beta_{2}) q^{19} + ( - 2 \beta_{6} - 3 \beta_{5} + \cdots - \beta_{2}) q^{21}+ \cdots + ( - 9 \beta_{6} - 5 \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{7} + 8 q^{9} - 64 q^{15} + 32 q^{23} - 24 q^{25} + 128 q^{39} - 280 q^{49} + 304 q^{57} + 112 q^{63} + 272 q^{65} - 544 q^{71} + 32 q^{79} - 328 q^{81} + 256 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 10x^{6} + 123x^{4} - 300x^{3} + 86x^{2} + 300x + 526 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 868 \nu^{7} - 9000 \nu^{6} - 26304 \nu^{5} + 91850 \nu^{4} + 320592 \nu^{3} - 425950 \nu^{2} + \cdots + 4767050 ) / 1582309 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2679764 \nu^{7} + 208186 \nu^{6} - 31814108 \nu^{5} - 10053780 \nu^{4} + 411714584 \nu^{3} + \cdots + 982876622 ) / 685139797 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3872412 \nu^{7} + 6558661 \nu^{6} - 34456727 \nu^{5} - 55216902 \nu^{4} + 394683447 \nu^{3} + \cdots + 1714881404 ) / 685139797 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12992 \nu^{7} + 18417 \nu^{6} + 167753 \nu^{5} - 3353 \nu^{4} - 1377571 \nu^{3} + 4853780 \nu^{2} + \cdots + 556407 ) / 1582309 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5735372 \nu^{7} + 4313372 \nu^{6} - 52238584 \nu^{5} - 59878610 \nu^{4} + 684612832 \nu^{3} + \cdots - 98379406 ) / 685139797 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8071492 \nu^{7} - 164500 \nu^{6} - 46895176 \nu^{5} + 109451638 \nu^{4} + 845538352 \nu^{3} + \cdots + 1079942072 ) / 685139797 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 19240 \nu^{7} + 143219 \nu^{6} + 372167 \nu^{5} - 1048471 \nu^{4} - 3401197 \nu^{3} + \cdots - 15107251 ) / 1582309 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - 2\beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} - 3\beta_{4} - 4\beta_{3} + 2\beta_{2} + 2\beta _1 + 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} - 4\beta_{6} + 28\beta_{5} + 21\beta_{4} - 16\beta_{3} + 10\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8\beta_{7} + 21\beta_{6} - 33\beta_{5} - 24\beta_{4} + 16\beta_{3} - 32\beta_{2} + 66\beta _1 - 146 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -25\beta_{7} + 6\beta_{6} + 112\beta_{5} + 175\beta_{4} - 376\beta_{3} + 502\beta_{2} - 522\beta _1 + 1500 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -16\beta_{7} + 24\beta_{6} + 93\beta_{5} + 148\beta_{4} + 804\beta_{3} - 1068\beta_{2} + 743\beta _1 - 2948 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 693 \beta_{7} + 1306 \beta_{6} - 2864 \beta_{5} - 2751 \beta_{4} - 3876 \beta_{3} + 5654 \beta_{2} + \cdots + 18900 ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
−0.639946 − 0.719687i
−0.639946 + 0.719687i
1.52814 − 1.99551i
1.52814 + 1.99551i
−3.26020 + 1.99551i
−3.26020 − 1.99551i
2.37200 + 0.719687i
2.37200 − 0.719687i
0 −4.11439 0 1.10245 0 3.73205 − 5.92214i 0 7.92820 0
209.2 0 −4.11439 0 1.10245 0 3.73205 + 5.92214i 0 7.92820 0
209.3 0 −1.75265 0 6.54099 0 0.267949 − 6.99487i 0 −5.92820 0
209.4 0 −1.75265 0 6.54099 0 0.267949 + 6.99487i 0 −5.92820 0
209.5 0 1.75265 0 −6.54099 0 0.267949 − 6.99487i 0 −5.92820 0
209.6 0 1.75265 0 −6.54099 0 0.267949 + 6.99487i 0 −5.92820 0
209.7 0 4.11439 0 −1.10245 0 3.73205 − 5.92214i 0 7.92820 0
209.8 0 4.11439 0 −1.10245 0 3.73205 + 5.92214i 0 7.92820 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.h.d 8
3.b odd 2 1 2016.3.l.f 8
4.b odd 2 1 56.3.h.d ✓ 8
7.b odd 2 1 inner 224.3.h.d 8
8.b even 2 1 inner 224.3.h.d 8
8.d odd 2 1 56.3.h.d ✓ 8
12.b even 2 1 504.3.l.f 8
21.c even 2 1 2016.3.l.f 8
24.f even 2 1 504.3.l.f 8
24.h odd 2 1 2016.3.l.f 8
28.d even 2 1 56.3.h.d ✓ 8
28.f even 6 2 392.3.j.d 16
28.g odd 6 2 392.3.j.d 16
56.e even 2 1 56.3.h.d ✓ 8
56.h odd 2 1 inner 224.3.h.d 8
56.k odd 6 2 392.3.j.d 16
56.m even 6 2 392.3.j.d 16
84.h odd 2 1 504.3.l.f 8
168.e odd 2 1 504.3.l.f 8
168.i even 2 1 2016.3.l.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.h.d ✓ 8 4.b odd 2 1
56.3.h.d ✓ 8 8.d odd 2 1
56.3.h.d ✓ 8 28.d even 2 1
56.3.h.d ✓ 8 56.e even 2 1
224.3.h.d 8 1.a even 1 1 trivial
224.3.h.d 8 7.b odd 2 1 inner
224.3.h.d 8 8.b even 2 1 inner
224.3.h.d 8 56.h odd 2 1 inner
392.3.j.d 16 28.f even 6 2
392.3.j.d 16 28.g odd 6 2
392.3.j.d 16 56.k odd 6 2
392.3.j.d 16 56.m even 6 2
504.3.l.f 8 12.b even 2 1
504.3.l.f 8 24.f even 2 1
504.3.l.f 8 84.h odd 2 1
504.3.l.f 8 168.e odd 2 1
2016.3.l.f 8 3.b odd 2 1
2016.3.l.f 8 21.c even 2 1
2016.3.l.f 8 24.h odd 2 1
2016.3.l.f 8 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 20T_{3}^{2} + 52 \) acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 20 T^{2} + 52)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 44 T^{2} + 52)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 8 T^{3} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 120 T^{2} + 528)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 332 T^{2} + 27508)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 1008 T^{2} + 247104)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 788 T^{2} + 114868)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 8 T - 416)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 1920 T^{2} + 33792)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 3984 T^{2} + 3322176)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 864 T^{2} + 76032)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 1344 T^{2} + 439296)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 6072 T^{2} + 7730448)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 3984 T^{2} + 3322176)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 3456 T^{2} + 2737152)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 4724 T^{2} + 5029492)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 44 T^{2} + 52)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 6072 T^{2} + 7730448)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 136 T + 2596)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 11952 T^{2} + 29899584)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 8 T - 956)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 20948 T^{2} + 114868)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 14256 T^{2} + 29899584)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 24048 T^{2} + 102163776)^{2} \) Copy content Toggle raw display
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