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Properties

Label 448.2.t.c
Level $448$
Weight $2$
Character orbit 448.t
Analytic conductor $3.577$
Analytic rank $0$
Dimension $12$
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] = [448,2,Mod(289,448)]
 
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([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 448.t (of order \(6\), degree \(2\), 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: \(3.57729801055\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 18 x^{10} - 18 x^{9} + 11 x^{8} - 36 x^{7} + 180 x^{6} - 120 x^{5} + 31 x^{4} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{10} q^{3} + ( - \beta_{6} - \beta_{4} + \beta_{2}) q^{5} + (\beta_{10} - \beta_{5}) q^{7} + (2 \beta_{6} - \beta_{4} - \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{3} + ( - \beta_{6} - \beta_{4} + \beta_{2}) q^{5} + (\beta_{10} - \beta_{5}) q^{7} + (2 \beta_{6} - \beta_{4} - \beta_{2} + 3) q^{9} + ( - \beta_{10} - \beta_{9} + \cdots - 2 \beta_1) q^{11}+ \cdots + (2 \beta_{10} - 5 \beta_{9} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{5} + 12 q^{9} + 10 q^{17} - 2 q^{21} + 4 q^{25} + 2 q^{33} - 54 q^{37} - 16 q^{41} + 108 q^{45} - 20 q^{49} - 54 q^{53} - 4 q^{57} - 30 q^{61} + 28 q^{65} + 38 q^{73} + 66 q^{77} - 30 q^{81} + 14 q^{89} - 102 q^{93} - 112 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 18 x^{10} - 18 x^{9} + 11 x^{8} - 36 x^{7} + 180 x^{6} - 120 x^{5} + 31 x^{4} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 7740 \nu^{11} - 506995 \nu^{10} + 2434499 \nu^{9} - 5300935 \nu^{8} - 4189949 \nu^{7} + \cdots - 55851123 ) / 53101482 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 277699 \nu^{11} - 1768612 \nu^{10} + 5985242 \nu^{9} - 10273476 \nu^{8} + 18200524 \nu^{7} + \cdots + 23326206 ) / 106202964 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 201275 \nu^{11} + 586972 \nu^{10} + 244664 \nu^{9} - 8553490 \nu^{8} + 13278861 \nu^{7} + \cdots - 71902488 ) / 53101482 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 762049 \nu^{11} + 3556430 \nu^{10} - 6804492 \nu^{9} - 9663348 \nu^{8} + 23632858 \nu^{7} + \cdots + 77627910 ) / 106202964 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3281997 \nu^{11} - 22076333 \nu^{10} + 72763928 \nu^{9} - 98157007 \nu^{8} + 69886738 \nu^{7} + \cdots - 465913905 ) / 106202964 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3023 \nu^{11} - 17672 \nu^{10} + 51530 \nu^{9} - 46380 \nu^{8} + 28152 \nu^{7} - 115112 \nu^{6} + \cdots - 133206 ) / 96636 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1731304 \nu^{11} - 8906435 \nu^{10} + 22643951 \nu^{9} - 6494647 \nu^{8} - 2719650 \nu^{7} + \cdots + 42893811 ) / 53101482 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3511633 \nu^{11} + 22408838 \nu^{10} - 72050256 \nu^{9} + 91865160 \nu^{8} - 72874022 \nu^{7} + \cdots + 571705950 ) / 106202964 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1956788 \nu^{11} + 12578603 \nu^{10} - 39909583 \nu^{9} + 47983703 \nu^{8} - 29508456 \nu^{7} + \cdots + 139603275 ) / 53101482 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 4335151 \nu^{11} + 23551921 \nu^{10} - 64739774 \nu^{9} + 41686879 \nu^{8} - 24298878 \nu^{7} + \cdots - 95497173 ) / 106202964 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 16071473 \nu^{11} + 92315910 \nu^{10} - 262316218 \nu^{9} + 202883076 \nu^{8} + \cdots + 141683718 ) / 106202964 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{10} - 3\beta_{9} + 3\beta_{7} - 2\beta_{6} - 2\beta_{5} + 2\beta_{3} + 2\beta_{2} - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{11} + 4 \beta_{10} - 6 \beta_{9} + \beta_{8} + 2 \beta_{7} - 6 \beta_{6} - 2 \beta_{5} + \cdots - 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{10} - 3 \beta_{9} + 4 \beta_{8} - 3 \beta_{7} + 5 \beta_{6} + 2 \beta_{5} + 6 \beta_{4} + \cdots - 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 12 \beta_{11} - 18 \beta_{10} + 25 \beta_{9} + 21 \beta_{8} - 44 \beta_{7} + 90 \beta_{6} + 42 \beta_{5} + \cdots - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 52 \beta_{11} - 134 \beta_{10} + 151 \beta_{9} + 26 \beta_{8} - 151 \beta_{7} + 294 \beta_{6} + \cdots + 250 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 125 \beta_{11} - 398 \beta_{10} + 347 \beta_{9} - 74 \beta_{8} - 263 \beta_{7} + 479 \beta_{6} + \cdots + 1183 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 276 \beta_{10} + 58 \beta_{9} - 295 \beta_{8} + 58 \beta_{7} - 487 \beta_{6} - 276 \beta_{5} + \cdots + 1311 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1250 \beta_{11} + 1240 \beta_{10} - 2620 \beta_{9} - 1870 \beta_{8} + 2895 \beta_{7} - 8943 \beta_{6} + \cdots + 50 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 5516 \beta_{11} + 10730 \beta_{10} - 11855 \beta_{9} - 2758 \beta_{8} + 11855 \beta_{7} + \cdots - 23778 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 12231 \beta_{11} + 35232 \beta_{10} - 25252 \beta_{9} + 5385 \beta_{8} + 25490 \beta_{7} + \cdots - 102905 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(-1 - \beta_{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−1.17819 + 1.17819i
0.836844 + 0.836844i
1.64901 − 1.64901i
−0.649007 − 0.649007i
0.163156 − 0.163156i
2.17819 + 2.17819i
−1.17819 − 1.17819i
0.836844 − 0.836844i
1.64901 + 1.64901i
−0.649007 + 0.649007i
0.163156 + 0.163156i
2.17819 − 2.17819i
0 −2.71294 + 1.56632i 0 2.90671 + 1.67819i 0 2.54421 − 0.725929i 0 3.40671 − 5.90059i 0
289.2 0 −1.96854 + 1.13654i 0 0.583430 + 0.336844i 0 −1.20287 − 2.35650i 0 1.08343 − 1.87656i 0
289.3 0 −0.121621 + 0.0702177i 0 −1.99014 − 1.14901i 0 −0.282982 − 2.63057i 0 −1.49014 + 2.58100i 0
289.4 0 0.121621 − 0.0702177i 0 −1.99014 − 1.14901i 0 0.282982 + 2.63057i 0 −1.49014 + 2.58100i 0
289.5 0 1.96854 − 1.13654i 0 0.583430 + 0.336844i 0 1.20287 + 2.35650i 0 1.08343 − 1.87656i 0
289.6 0 2.71294 − 1.56632i 0 2.90671 + 1.67819i 0 −2.54421 + 0.725929i 0 3.40671 − 5.90059i 0
417.1 0 −2.71294 − 1.56632i 0 2.90671 − 1.67819i 0 2.54421 + 0.725929i 0 3.40671 + 5.90059i 0
417.2 0 −1.96854 − 1.13654i 0 0.583430 − 0.336844i 0 −1.20287 + 2.35650i 0 1.08343 + 1.87656i 0
417.3 0 −0.121621 − 0.0702177i 0 −1.99014 + 1.14901i 0 −0.282982 + 2.63057i 0 −1.49014 − 2.58100i 0
417.4 0 0.121621 + 0.0702177i 0 −1.99014 + 1.14901i 0 0.282982 − 2.63057i 0 −1.49014 − 2.58100i 0
417.5 0 1.96854 + 1.13654i 0 0.583430 − 0.336844i 0 1.20287 − 2.35650i 0 1.08343 + 1.87656i 0
417.6 0 2.71294 + 1.56632i 0 2.90671 − 1.67819i 0 −2.54421 − 0.725929i 0 3.40671 + 5.90059i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
56.k odd 6 1 inner
56.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.2.t.c yes 12
4.b odd 2 1 inner 448.2.t.c yes 12
7.c even 3 1 448.2.t.b ✓ 12
7.c even 3 1 3136.2.b.l 12
7.d odd 6 1 3136.2.b.m 12
8.b even 2 1 448.2.t.b ✓ 12
8.d odd 2 1 448.2.t.b ✓ 12
28.f even 6 1 3136.2.b.m 12
28.g odd 6 1 448.2.t.b ✓ 12
28.g odd 6 1 3136.2.b.l 12
56.j odd 6 1 3136.2.b.m 12
56.k odd 6 1 inner 448.2.t.c yes 12
56.k odd 6 1 3136.2.b.l 12
56.m even 6 1 3136.2.b.m 12
56.p even 6 1 inner 448.2.t.c yes 12
56.p even 6 1 3136.2.b.l 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.t.b ✓ 12 7.c even 3 1
448.2.t.b ✓ 12 8.b even 2 1
448.2.t.b ✓ 12 8.d odd 2 1
448.2.t.b ✓ 12 28.g odd 6 1
448.2.t.c yes 12 1.a even 1 1 trivial
448.2.t.c yes 12 4.b odd 2 1 inner
448.2.t.c yes 12 56.k odd 6 1 inner
448.2.t.c yes 12 56.p even 6 1 inner
3136.2.b.l 12 7.c even 3 1
3136.2.b.l 12 28.g odd 6 1
3136.2.b.l 12 56.k odd 6 1
3136.2.b.l 12 56.p even 6 1
3136.2.b.m 12 7.d odd 6 1
3136.2.b.m 12 28.f even 6 1
3136.2.b.m 12 56.j odd 6 1
3136.2.b.m 12 56.m even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(448, [\chi])\):

\( T_{3}^{12} - 15T_{3}^{10} + 174T_{3}^{8} - 763T_{3}^{6} + 2586T_{3}^{4} - 51T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{6} - 3T_{5}^{5} - 4T_{5}^{4} + 21T_{5}^{3} + 40T_{5}^{2} - 63T_{5} + 27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 15 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{6} - 3 T^{5} - 4 T^{4} + \cdots + 27)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + 10 T^{10} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} - 35 T^{10} + \cdots + 194481 \) Copy content Toggle raw display
$13$ \( (T^{6} + 20 T^{4} + \cdots + 48)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 5 T^{5} + \cdots + 3249)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 43 T^{10} + \cdots + 6765201 \) Copy content Toggle raw display
$23$ \( T^{12} + 69 T^{10} + \cdots + 4782969 \) Copy content Toggle raw display
$29$ \( (T^{6} + 140 T^{4} + \cdots + 3888)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 2341301769 \) Copy content Toggle raw display
$37$ \( (T^{6} + 27 T^{5} + \cdots + 149187)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 4 T^{2} - 20 T - 12)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + 60 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 113 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$53$ \( (T^{2} + 9 T + 27)^{6} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 23854493601 \) Copy content Toggle raw display
$61$ \( (T^{6} + 15 T^{5} + \cdots + 177147)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 10098039121 \) Copy content Toggle raw display
$71$ \( (T^{6} - 224 T^{4} + \cdots - 6912)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 19 T^{5} + \cdots + 159201)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 97960862169 \) Copy content Toggle raw display
$83$ \( (T^{6} + 204 T^{4} + \cdots + 46656)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 7 T^{5} + \cdots + 29241)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 28 T^{2} + \cdots - 268)^{4} \) Copy content Toggle raw display
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