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Properties

Label 448.6.f.d
Level $448$
Weight $6$
Character orbit 448.f
Analytic conductor $71.852$
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] = [448,6,Mod(447,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.447");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 448.f (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: \(71.8519512762\)
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: \( x^{16} - 4 x^{15} - 674 x^{14} + 3404 x^{13} + 173721 x^{12} - 919512 x^{11} - 21981508 x^{10} + \cdots + 224266997486896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{70} \)
Twist minimal: no (minimal twist has level 28)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{6} q^{5} - \beta_{8} q^{7} + ( - \beta_{5} + \beta_{4} - \beta_{3} + 101) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{6} q^{5} - \beta_{8} q^{7} + ( - \beta_{5} + \beta_{4} - \beta_{3} + 101) q^{9} + (\beta_{9} - \beta_{8} + \cdots - \beta_{2}) q^{11}+ \cdots + ( - 116 \beta_{11} + \cdots + 493 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 1616 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 1616 q^{9} + 1792 q^{21} - 9776 q^{25} - 26592 q^{29} + 26272 q^{37} + 8848 q^{49} + 41888 q^{53} - 60288 q^{57} + 66688 q^{65} - 320992 q^{77} + 56336 q^{81} - 78080 q^{85} + 335616 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 674 x^{14} + 3404 x^{13} + 173721 x^{12} - 919512 x^{11} - 21981508 x^{10} + \cdots + 224266997486896 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 93\!\cdots\!21 \nu^{15} + \cdots + 62\!\cdots\!76 ) / 35\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 25\!\cdots\!71 \nu^{15} + \cdots - 76\!\cdots\!68 ) / 71\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 28\!\cdots\!42 \nu^{15} + \cdots + 51\!\cdots\!92 ) / 77\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 71\!\cdots\!18 \nu^{15} + \cdots + 10\!\cdots\!48 ) / 15\!\cdots\!39 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 78\!\cdots\!74 \nu^{15} + \cdots - 95\!\cdots\!24 ) / 15\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 20\!\cdots\!64 \nu^{15} + \cdots - 11\!\cdots\!68 ) / 37\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 22\!\cdots\!73 \nu^{15} + \cdots + 47\!\cdots\!28 ) / 35\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 38\!\cdots\!33 \nu^{15} + \cdots - 11\!\cdots\!48 ) / 30\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 49\!\cdots\!57 \nu^{15} + \cdots + 75\!\cdots\!72 ) / 30\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11\!\cdots\!84 \nu^{15} + \cdots + 15\!\cdots\!24 ) / 46\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11\!\cdots\!99 \nu^{15} + \cdots + 27\!\cdots\!44 ) / 35\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 63\!\cdots\!88 \nu^{15} + \cdots - 93\!\cdots\!28 ) / 18\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 12\!\cdots\!84 \nu^{15} + \cdots + 86\!\cdots\!44 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 27\!\cdots\!07 \nu^{15} + \cdots + 12\!\cdots\!12 ) / 19\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 30\!\cdots\!57 \nu^{15} + \cdots + 13\!\cdots\!52 ) / 13\!\cdots\!55 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{4} + \beta_{3} - 3\beta_{2} - 64\beta _1 + 32 ) / 128 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} - 2 \beta_{12} + 2 \beta_{11} - 4 \beta_{10} + 3 \beta_{9} - \beta_{8} - 6 \beta_{7} + \cdots + 10912 ) / 128 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 16 \beta_{15} - 22 \beta_{14} - 18 \beta_{13} + 6 \beta_{12} + 412 \beta_{11} + 6 \beta_{10} + \cdots - 16480 ) / 128 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 132 \beta_{15} + 60 \beta_{14} + 196 \beta_{13} - 316 \beta_{12} - 349 \beta_{11} - 428 \beta_{10} + \cdots + 810384 ) / 64 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4424 \beta_{15} - 4754 \beta_{14} - 8350 \beta_{13} + 5010 \beta_{12} + 110325 \beta_{11} + \cdots - 10547488 ) / 128 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 62123 \beta_{15} + 37704 \beta_{14} + 181560 \beta_{13} - 187254 \beta_{12} - 798385 \beta_{11} + \cdots + 295198816 ) / 128 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1989061 \beta_{15} - 1896392 \beta_{14} - 6058696 \beta_{13} + 4583810 \beta_{12} + 60860088 \beta_{11} + \cdots - 6310027968 ) / 256 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 6847088 \beta_{15} + 4729944 \beta_{14} + 30937992 \beta_{13} - 27846616 \beta_{12} + \cdots + 28296013456 ) / 64 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 199109653 \beta_{15} - 177144814 \beta_{14} - 997109418 \beta_{13} + 818363232 \beta_{12} + \cdots - 717210847648 ) / 128 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2669361731 \beta_{15} + 1967443360 \beta_{14} + 18890797568 \beta_{13} - 16299827470 \beta_{12} + \cdots + 10196144270432 ) / 128 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 67056127663 \beta_{15} - 57459399104 \beta_{14} - 610464907712 \beta_{13} + 513975474518 \beta_{12} + \cdots - 258712258131648 ) / 256 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 199843963766 \beta_{15} + 151476472412 \beta_{14} + 2713953587572 \beta_{13} + \cdots + 719217461196432 ) / 64 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 3402142925288 \beta_{15} - 2900497585586 \beta_{14} - 87816679124486 \beta_{13} + \cdots - 14\!\cdots\!00 ) / 128 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 15038595180927 \beta_{15} + 10511683168856 \beta_{14} + \cdots + 34\!\cdots\!44 ) / 128 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 12\!\cdots\!19 \beta_{15} + 954508584590952 \beta_{14} + \cdots + 39\!\cdots\!40 ) / 256 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
447.1
−16.2859 − 1.80141i
−16.2859 + 1.80141i
−7.16085 + 2.48366i
−7.16085 − 2.48366i
−5.20336 − 0.902498i
−5.20336 + 0.902498i
−4.07684 + 2.69664i
−4.07684 − 2.69664i
2.37022 − 2.69664i
2.37022 + 2.69664i
10.5645 + 0.902498i
10.5645 − 0.902498i
9.86747 − 2.48366i
9.86747 + 2.48366i
11.9248 + 1.80141i
11.9248 − 1.80141i
0 −28.2107 0 − 57.4402i 0 −15.0189 + 128.769i 0 552.846 0
447.2 0 −28.2107 0 57.4402i 0 −15.0189 − 128.769i 0 552.846 0
447.3 0 −17.0283 0 − 28.3847i 0 117.282 − 55.2436i 0 46.9637 0
447.4 0 −17.0283 0 28.3847i 0 117.282 + 55.2436i 0 46.9637 0
447.5 0 −15.7679 0 − 70.8301i 0 −78.9551 − 102.826i 0 5.62572 0
447.6 0 −15.7679 0 70.8301i 0 −78.9551 + 102.826i 0 5.62572 0
447.7 0 −6.44707 0 − 76.3023i 0 −120.438 + 47.9752i 0 −201.435 0
447.8 0 −6.44707 0 76.3023i 0 −120.438 − 47.9752i 0 −201.435 0
447.9 0 6.44707 0 − 76.3023i 0 120.438 − 47.9752i 0 −201.435 0
447.10 0 6.44707 0 76.3023i 0 120.438 + 47.9752i 0 −201.435 0
447.11 0 15.7679 0 − 70.8301i 0 78.9551 + 102.826i 0 5.62572 0
447.12 0 15.7679 0 70.8301i 0 78.9551 − 102.826i 0 5.62572 0
447.13 0 17.0283 0 − 28.3847i 0 −117.282 + 55.2436i 0 46.9637 0
447.14 0 17.0283 0 28.3847i 0 −117.282 − 55.2436i 0 46.9637 0
447.15 0 28.2107 0 − 57.4402i 0 15.0189 − 128.769i 0 552.846 0
447.16 0 28.2107 0 57.4402i 0 15.0189 + 128.769i 0 552.846 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 447.16
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 448.6.f.d 16
4.b odd 2 1 inner 448.6.f.d 16
7.b odd 2 1 inner 448.6.f.d 16
8.b even 2 1 28.6.d.b ✓ 16
8.d odd 2 1 28.6.d.b ✓ 16
24.f even 2 1 252.6.b.d 16
24.h odd 2 1 252.6.b.d 16
28.d even 2 1 inner 448.6.f.d 16
56.e even 2 1 28.6.d.b ✓ 16
56.h odd 2 1 28.6.d.b ✓ 16
168.e odd 2 1 252.6.b.d 16
168.i even 2 1 252.6.b.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.6.d.b ✓ 16 8.b even 2 1
28.6.d.b ✓ 16 8.d odd 2 1
28.6.d.b ✓ 16 56.e even 2 1
28.6.d.b ✓ 16 56.h odd 2 1
252.6.b.d 16 24.f even 2 1
252.6.b.d 16 24.h odd 2 1
252.6.b.d 16 168.e odd 2 1
252.6.b.d 16 168.i even 2 1
448.6.f.d 16 1.a even 1 1 trivial
448.6.f.d 16 4.b odd 2 1 inner
448.6.f.d 16 7.b odd 2 1 inner
448.6.f.d 16 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 1376T_{3}^{6} + 556192T_{3}^{4} - 78187008T_{3}^{2} + 2384750592 \) acting on \(S_{6}^{\mathrm{new}}(448, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} - 1376 T^{6} + \cdots + 2384750592)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots + 77644413167616)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 63\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 16\!\cdots\!48)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 34\!\cdots\!24)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 18\!\cdots\!36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 43\!\cdots\!08)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 27\!\cdots\!08)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 47450817355248)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 19\!\cdots\!72)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 39\!\cdots\!84)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 18\!\cdots\!96)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 22\!\cdots\!88)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 51\!\cdots\!28)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 85\!\cdots\!76)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 16\!\cdots\!08)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 75\!\cdots\!56)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 51\!\cdots\!92)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 50\!\cdots\!88)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 22\!\cdots\!04)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 63\!\cdots\!72)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 62\!\cdots\!92)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 12\!\cdots\!96)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 31\!\cdots\!84)^{2} \) Copy content Toggle raw display
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