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Properties

Label 224.5.c.b
Level $224$
Weight $5$
Character orbit 224.c
Analytic conductor $23.155$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [224,5,Mod(97,224)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("224.97"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(224, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 224.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,-368] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.1548717308\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 400 x^{14} + 62280 x^{12} + 4705760 x^{10} + 180995480 x^{8} + 3374721216 x^{6} + \cdots + 520527618576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{54} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{5} q^{5} + ( - \beta_{8} - \beta_1) q^{7} + (\beta_{2} - 23) q^{9} + ( - \beta_{14} + \beta_{6}) q^{11} + (\beta_{11} - \beta_{5}) q^{13} + (\beta_{15} - \beta_{14} + \cdots + 4 \beta_{6}) q^{15}+ \cdots + ( - 36 \beta_{15} + \cdots + 15 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 368 q^{9} + 1984 q^{21} - 3824 q^{25} + 672 q^{29} - 2272 q^{37} - 6128 q^{49} - 3168 q^{53} - 17664 q^{57} + 17280 q^{65} - 11040 q^{77} + 23824 q^{81} + 16512 q^{85} - 61568 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 400 x^{14} + 62280 x^{12} + 4705760 x^{10} + 180995480 x^{8} + 3374721216 x^{6} + \cdots + 520527618576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 443133239 \nu^{14} - 167847029358 \nu^{12} - 24202613712378 \nu^{10} + \cdots - 15\!\cdots\!04 ) / 79\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 443133239 \nu^{14} - 167847029358 \nu^{12} - 24202613712378 \nu^{10} + \cdots + 18\!\cdots\!96 ) / 19\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7668772763375 \nu^{14} + \cdots - 91\!\cdots\!40 ) / 44\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 45636575700197 \nu^{14} + \cdots + 35\!\cdots\!36 ) / 44\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 29\!\cdots\!51 \nu^{15} + \cdots - 20\!\cdots\!56 \nu ) / 89\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1217614154525 \nu^{15} - 478164828567201 \nu^{13} + \cdots - 36\!\cdots\!48 \nu ) / 99\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 458269483813515 \nu^{15} + \cdots - 84\!\cdots\!84 ) / 36\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 458269483813515 \nu^{15} + \cdots - 84\!\cdots\!84 ) / 36\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13\!\cdots\!51 \nu^{15} + \cdots + 12\!\cdots\!80 ) / 89\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13\!\cdots\!51 \nu^{15} + \cdots - 51\!\cdots\!60 ) / 89\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15\!\cdots\!60 \nu^{15} + \cdots - 71\!\cdots\!00 \nu ) / 74\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!51 \nu^{15} + \cdots + 34\!\cdots\!72 \nu ) / 44\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 46\!\cdots\!31 \nu^{15} + \cdots + 28\!\cdots\!80 \nu ) / 89\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 615847577154965 \nu^{15} + \cdots + 34\!\cdots\!60 \nu ) / 90\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 56870331101551 \nu^{15} + \cdots + 20\!\cdots\!64 \nu ) / 39\!\cdots\!92 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} + 2\beta_{6} + 2\beta_{5} ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 4\beta _1 - 100 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -12\beta_{15} - 2\beta_{13} - 35\beta_{12} + 18\beta_{11} - 154\beta_{6} - 200\beta_{5} ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9 \beta_{12} - 3 \beta_{10} + 15 \beta_{9} + 40 \beta_{8} + 40 \beta_{7} + 8 \beta_{4} + 8 \beta_{3} + \cdots + 17720 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2640 \beta_{15} - 1440 \beta_{14} + 1048 \beta_{13} + 2765 \beta_{12} - 2952 \beta_{11} + \cdots + 23538 \beta_{5} ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1440 \beta_{12} + 327 \beta_{10} - 2553 \beta_{9} - 12800 \beta_{8} - 12800 \beta_{7} + \cdots - 1690640 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 225036 \beta_{15} + 181440 \beta_{14} - 77130 \beta_{13} - 113605 \beta_{12} + \cdots - 1223924 \beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 39663 \beta_{12} - 7407 \beta_{10} + 71919 \beta_{9} + 616392 \beta_{8} + 616392 \beta_{7} + \cdots + 37953860 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 34167840 \beta_{15} - 31748544 \beta_{14} + 7738480 \beta_{13} + 8694517 \beta_{12} + \cdots + 107311834 \beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3217140 \beta_{12} + 542541 \beta_{10} - 5891739 \beta_{9} - 96945760 \beta_{8} - 96945760 \beta_{7} + \cdots - 2764222760 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2396538012 \beta_{15} + 2362440960 \beta_{14} - 250692202 \beta_{13} - 226069403 \beta_{12} + \cdots - 3160706016 \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 111357243 \beta_{12} - 15016701 \beta_{10} + 207697785 \beta_{9} + 13591686488 \beta_{8} + \cdots + 74055431864 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 314734865328 \beta_{15} - 318193046496 \beta_{14} - 4734383256 \beta_{13} + \cdots - 105202039718 \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 26742599928 \beta_{12} - 5188599027 \beta_{10} + 48296600829 \beta_{9} - 1754411770944 \beta_{8} + \cdots + 26976068979376 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 19398472958076 \beta_{15} + 19820469216960 \beta_{14} + 2705883137246 \beta_{13} + \cdots + 39444444723908 \beta_{5} ) / 2 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−1.41421 − 11.2664i
1.41421 + 11.2664i
1.41421 + 7.85219i
−1.41421 − 7.85219i
1.41421 + 4.08819i
−1.41421 − 4.08819i
1.41421 + 1.64240i
−1.41421 − 1.64240i
−1.41421 + 1.64240i
1.41421 − 1.64240i
−1.41421 + 4.08819i
1.41421 − 4.08819i
−1.41421 + 7.85219i
1.41421 − 7.85219i
1.41421 − 11.2664i
−1.41421 + 11.2664i
0 − 15.9331i 0 − 36.7014i 0 0.663529 + 48.9955i 0 −172.865 0
97.2 0 − 15.9331i 0 36.7014i 0 −0.663529 + 48.9955i 0 −172.865 0
97.3 0 − 11.1047i 0 − 7.82927i 0 −25.2269 − 42.0072i 0 −42.3137 0
97.4 0 − 11.1047i 0 7.82927i 0 25.2269 − 42.0072i 0 −42.3137 0
97.5 0 − 5.78158i 0 − 7.51633i 0 45.4440 + 18.3261i 0 47.5734 0
97.6 0 − 5.78158i 0 7.51633i 0 −45.4440 + 18.3261i 0 47.5734 0
97.7 0 − 2.32270i 0 − 44.6230i 0 −36.5241 + 32.6648i 0 75.6051 0
97.8 0 − 2.32270i 0 44.6230i 0 36.5241 + 32.6648i 0 75.6051 0
97.9 0 2.32270i 0 − 44.6230i 0 36.5241 − 32.6648i 0 75.6051 0
97.10 0 2.32270i 0 44.6230i 0 −36.5241 − 32.6648i 0 75.6051 0
97.11 0 5.78158i 0 − 7.51633i 0 −45.4440 − 18.3261i 0 47.5734 0
97.12 0 5.78158i 0 7.51633i 0 45.4440 − 18.3261i 0 47.5734 0
97.13 0 11.1047i 0 − 7.82927i 0 25.2269 + 42.0072i 0 −42.3137 0
97.14 0 11.1047i 0 7.82927i 0 −25.2269 + 42.0072i 0 −42.3137 0
97.15 0 15.9331i 0 − 36.7014i 0 −0.663529 − 48.9955i 0 −172.865 0
97.16 0 15.9331i 0 36.7014i 0 0.663529 − 48.9955i 0 −172.865 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.5.c.b ✓ 16
4.b odd 2 1 inner 224.5.c.b ✓ 16
7.b odd 2 1 inner 224.5.c.b ✓ 16
8.b even 2 1 448.5.c.j 16
8.d odd 2 1 448.5.c.j 16
28.d even 2 1 inner 224.5.c.b ✓ 16
56.e even 2 1 448.5.c.j 16
56.h odd 2 1 448.5.c.j 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.5.c.b ✓ 16 1.a even 1 1 trivial
224.5.c.b ✓ 16 4.b odd 2 1 inner
224.5.c.b ✓ 16 7.b odd 2 1 inner
224.5.c.b ✓ 16 28.d even 2 1 inner
448.5.c.j 16 8.b even 2 1
448.5.c.j 16 8.d odd 2 1
448.5.c.j 16 56.e even 2 1
448.5.c.j 16 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 416T_{3}^{6} + 46128T_{3}^{4} + 1283328T_{3}^{2} + 5645376 \) acting on \(S_{5}^{\mathrm{new}}(224, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 416 T^{6} + \cdots + 5645376)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + 3456 T^{6} + \cdots + 9288333376)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 50065926729984)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 11\!\cdots\!24)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 83\!\cdots\!76)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 24\!\cdots\!56)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 168 T^{3} + \cdots + 308054618640)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 57\!\cdots\!44)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 568 T^{3} + \cdots + 426376699152)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 60\!\cdots\!04)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 38\!\cdots\!24)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 47\!\cdots\!44)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 792 T^{3} + \cdots + 553325894928)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 12\!\cdots\!64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 40\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 67\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 63\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 55\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 18\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 80\!\cdots\!76)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 86\!\cdots\!44)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 65\!\cdots\!24)^{2} \) Copy content Toggle raw display
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