Properties

Label 448.4.p.f
Level $448$
Weight $4$
Character orbit 448.p
Analytic conductor $26.433$
Analytic rank $0$
Dimension $6$
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] = [448,4,Mod(255,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("448.255");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 448.p (of order \(6\), degree \(2\), 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: \(26.4328556826\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.12258833328.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 29x^{4} - 20x^{3} + 808x^{2} - 672x + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + 2 \beta_{3}) q^{3} + (\beta_{2} + \beta_1) q^{5} + ( - \beta_{4} + 9 \beta_{3} - 4) q^{7} + ( - 2 \beta_{4} - 25 \beta_{3} + \beta_{2} - \beta_1 - 25) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + 2 \beta_{3}) q^{3} + (\beta_{2} + \beta_1) q^{5} + ( - \beta_{4} + 9 \beta_{3} - 4) q^{7} + ( - 2 \beta_{4} - 25 \beta_{3} + \beta_{2} - \beta_1 - 25) q^{9} + (\beta_{5} - \beta_{4} + 11 \beta_{3} + \beta_1 + 22) q^{11} + ( - \beta_{4} + 46 \beta_{3} + \beta_{2} + 23) q^{13} + (2 \beta_{5} + 7 \beta_{4} + 38 \beta_{3} - 7 \beta_{2} + \beta_1 + 19) q^{15} + (8 \beta_{5} - 5 \beta_{4} + 11 \beta_1) q^{17} + ( - \beta_{5} - 2 \beta_{4} + 49 \beta_{3} + \beta_{2} - 2 \beta_1 + 49) q^{19} + ( - 16 \beta_{5} + \beta_{4} + 3 \beta_{3} + 5 \beta_{2} - 2 \beta_1 + 1) q^{21} + (\beta_{5} + 6 \beta_{2} + 5 \beta_1) q^{23} + ( - 24 \beta_{5} - 94 \beta_{3}) q^{25} + (7 \beta_{4} + 7 \beta_{2} + 15 \beta_1 + 53) q^{27} + ( - \beta_{4} - \beta_{2} + 22 \beta_1 + 51) q^{29} + ( - 23 \beta_{5} - \beta_{4} + 77 \beta_{3} + 2 \beta_{2} + \beta_1) q^{31} + (8 \beta_{5} + 51 \beta_{3} + 4 \beta_{2} - 4 \beta_1 - 51) q^{33} + ( - \beta_{5} + 9 \beta_{4} - 19 \beta_{3} - 8 \beta_{2} - 22 \beta_1 - 200) q^{35} + (24 \beta_{5} + 4 \beta_{4} + 73 \beta_{3} - 2 \beta_{2} + 26 \beta_1 + 73) q^{37} + ( - 22 \beta_{5} + 7 \beta_{4} - 27 \beta_{3} - 37 \beta_1 - 54) q^{39} + ( - 16 \beta_{5} + 9 \beta_{4} + 282 \beta_{3} - 9 \beta_{2} - 8 \beta_1 + 141) q^{41} + ( - 44 \beta_{5} + 16 \beta_{4} + 40 \beta_{3} - 16 \beta_{2} - 22 \beta_1 + 20) q^{43} + ( - 24 \beta_{5} - 25 \beta_{4} - 219 \beta_{3} - 73 \beta_1 - 438) q^{45} + (21 \beta_{5} + 2 \beta_{4} - 33 \beta_{3} - \beta_{2} + 22 \beta_1 - 33) q^{47} + (40 \beta_{5} - \beta_{4} + 28 \beta_{3} + 19 \beta_{2} + 26 \beta_1 + 126) q^{49} + ( - 21 \beta_{5} + 289 \beta_{3} + 11 \beta_{2} + 32 \beta_1 - 289) q^{51} + (48 \beta_{5} + 6 \beta_{4} - 237 \beta_{3} - 12 \beta_{2} - 6 \beta_1) q^{53} + (16 \beta_{4} + 16 \beta_{2} + 11 \beta_1 - 162) q^{55} + (8 \beta_{4} + 8 \beta_{2} - 32 \beta_1 + 7) q^{57} + ( - 3 \beta_{5} + 2 \beta_{4} - 108 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{59} + ( - 8 \beta_{5} - 189 \beta_{3} - 6 \beta_{2} + 2 \beta_1 + 189) q^{61} + (20 \beta_{5} + 32 \beta_{4} + 481 \beta_{3} - 15 \beta_{2} + 13 \beta_1 + 487) q^{63} + ( - 24 \beta_{5} + 46 \beta_{4} - 219 \beta_{3} - 23 \beta_{2} + \cdots - 219) q^{65}+ \cdots + ( - 8 \beta_{5} - 23 \beta_{4} - 226 \beta_{3} + 23 \beta_{2} - 4 \beta_1 - 113) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 7 q^{3} + 3 q^{5} - 52 q^{7} - 78 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 7 q^{3} + 3 q^{5} - 52 q^{7} - 78 q^{9} + 99 q^{11} + 9 q^{17} + 143 q^{19} + 15 q^{21} + 15 q^{23} + 306 q^{25} + 362 q^{27} + 348 q^{29} - 205 q^{31} - 471 q^{33} - 1185 q^{35} + 249 q^{37} - 288 q^{39} - 2118 q^{45} - 75 q^{47} + 702 q^{49} - 2505 q^{51} + 645 q^{53} - 918 q^{55} - 6 q^{57} + 321 q^{59} + 1707 q^{61} + 1502 q^{63} - 612 q^{65} + 447 q^{67} + 705 q^{73} + 4138 q^{75} - 555 q^{77} + 3447 q^{79} + 225 q^{81} + 24 q^{83} - 3786 q^{85} - 3642 q^{87} - 2607 q^{89} - 2448 q^{91} + 2991 q^{93} - 2085 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 29x^{4} - 20x^{3} + 808x^{2} - 672x + 576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 29\nu^{5} - 841\nu^{4} + 1629\nu^{3} - 23432\nu^{2} + 19488\nu - 428592 ) / 45520 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 64\nu^{5} + 989\nu^{4} + 5459\nu^{3} + 30793\nu^{2} - 82172\nu + 596328 ) / 68280 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -203\nu^{5} + 197\nu^{4} - 5713\nu^{3} - 986\nu^{2} - 159176\nu - 4176 ) / 136560 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -133\nu^{5} + 1012\nu^{4} + 4792\nu^{3} + 24959\nu^{2} + 35804\nu + 543504 ) / 68280 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 581\nu^{5} + 221\nu^{4} + 16351\nu^{3} + 2822\nu^{2} + 481472\nu + 11952 ) / 45520 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - 3\beta_{3} - 2\beta_{2} - \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -12\beta_{5} + 2\beta_{4} - 111\beta_{3} - \beta_{2} - 11\beta _1 - 111 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 29\beta_{4} + 29\beta_{2} + 46\beta _1 - 51 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 116\beta_{5} - 11\beta_{4} + 1029\beta_{3} + 22\beta_{2} + 11\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 396\beta_{5} - 1642\beta_{4} + 1851\beta_{3} + 821\beta_{2} - 425\beta _1 + 1851 ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(1 + \beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
255.1
−2.61524 + 4.52973i
2.68858 4.65676i
0.426664 0.739004i
−2.61524 4.52973i
2.68858 + 4.65676i
0.426664 + 0.739004i
0 −4.64712 + 8.04905i 0 17.1915 9.92549i 0 −18.3972 2.13127i 0 −29.6915 51.4271i 0
255.2 0 −2.38417 + 4.12950i 0 −14.6315 + 8.44749i 0 8.89981 + 16.2417i 0 2.13148 + 3.69182i 0
255.3 0 3.53129 6.11637i 0 −1.05999 + 0.611983i 0 −16.5026 + 8.40621i 0 −11.4400 19.8147i 0
383.1 0 −4.64712 8.04905i 0 17.1915 + 9.92549i 0 −18.3972 + 2.13127i 0 −29.6915 + 51.4271i 0
383.2 0 −2.38417 4.12950i 0 −14.6315 8.44749i 0 8.89981 16.2417i 0 2.13148 3.69182i 0
383.3 0 3.53129 + 6.11637i 0 −1.05999 0.611983i 0 −16.5026 8.40621i 0 −11.4400 + 19.8147i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 255.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.4.p.f 6
4.b odd 2 1 448.4.p.g 6
7.d odd 6 1 448.4.p.g 6
8.b even 2 1 112.4.p.g yes 6
8.d odd 2 1 112.4.p.f 6
28.f even 6 1 inner 448.4.p.f 6
56.j odd 6 1 112.4.p.f 6
56.j odd 6 1 784.4.f.h 6
56.k odd 6 1 784.4.f.h 6
56.m even 6 1 112.4.p.g yes 6
56.m even 6 1 784.4.f.g 6
56.p even 6 1 784.4.f.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.4.p.f 6 8.d odd 2 1
112.4.p.f 6 56.j odd 6 1
112.4.p.g yes 6 8.b even 2 1
112.4.p.g yes 6 56.m even 6 1
448.4.p.f 6 1.a even 1 1 trivial
448.4.p.f 6 28.f even 6 1 inner
448.4.p.g 6 4.b odd 2 1
448.4.p.g 6 7.d odd 6 1
784.4.f.g 6 56.m even 6 1
784.4.f.g 6 56.p even 6 1
784.4.f.h 6 56.j odd 6 1
784.4.f.h 6 56.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 7T_{3}^{5} + 104T_{3}^{4} + 241T_{3}^{3} + 5216T_{3}^{2} + 17215T_{3} + 97969 \) acting on \(S_{4}^{\mathrm{new}}(448, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 7 T^{5} + 104 T^{4} + \cdots + 97969 \) Copy content Toggle raw display
$5$ \( T^{6} - 3 T^{5} - 336 T^{4} + \cdots + 168507 \) Copy content Toggle raw display
$7$ \( T^{6} + 52 T^{5} + 1001 T^{4} + \cdots + 40353607 \) Copy content Toggle raw display
$11$ \( T^{6} - 99 T^{5} + 3990 T^{4} + \cdots + 4876875 \) Copy content Toggle raw display
$13$ \( T^{6} + 5304 T^{4} + \cdots + 2167603200 \) Copy content Toggle raw display
$17$ \( T^{6} - 9 T^{5} - 14640 T^{4} + \cdots + 710833347 \) Copy content Toggle raw display
$19$ \( T^{6} - 143 T^{5} + \cdots + 1106959441 \) Copy content Toggle raw display
$23$ \( T^{6} - 15 T^{5} + \cdots + 8915001507 \) Copy content Toggle raw display
$29$ \( (T^{3} - 174 T^{2} - 27504 T + 4622400)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 205 T^{5} + \cdots + 2349411462841 \) Copy content Toggle raw display
$37$ \( T^{6} - 249 T^{5} + \cdots + 14539922265625 \) Copy content Toggle raw display
$41$ \( T^{6} + 235176 T^{4} + \cdots + 38591270836992 \) Copy content Toggle raw display
$43$ \( T^{6} + 251796 T^{4} + \cdots + 83474849412288 \) Copy content Toggle raw display
$47$ \( T^{6} + 75 T^{5} + \cdots + 20686781241 \) Copy content Toggle raw display
$53$ \( T^{6} - 645 T^{5} + \cdots + 37\!\cdots\!21 \) Copy content Toggle raw display
$59$ \( T^{6} - 321 T^{5} + \cdots + 825904716849 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 912934776095523 \) Copy content Toggle raw display
$67$ \( T^{6} - 447 T^{5} + \cdots + 34\!\cdots\!75 \) Copy content Toggle raw display
$71$ \( T^{6} + 1416948 T^{4} + \cdots + 10\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{6} - 705 T^{5} + \cdots + 48\!\cdots\!03 \) Copy content Toggle raw display
$79$ \( T^{6} - 3447 T^{5} + \cdots + 65\!\cdots\!23 \) Copy content Toggle raw display
$83$ \( (T^{3} - 12 T^{2} - 856512 T + 308212992)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 2607 T^{5} + \cdots + 60\!\cdots\!03 \) Copy content Toggle raw display
$97$ \( T^{6} + 1038696 T^{4} + \cdots + 19\!\cdots\!08 \) Copy content Toggle raw display
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