↑

Properties

Label 448.3.s.h
Level $448$
Weight $3$
Character orbit 448.s
Analytic conductor $12.207$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,3,Mod(129,448)]
 
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, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.129");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 448.s (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: \(12.2071158433\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 26 x^{14} - 16 x^{13} + 469 x^{12} + 144 x^{11} - 4526 x^{10} + 4440 x^{9} + 32608 x^{8} + \cdots + 208849 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{20}\cdot 7 \)
Twist minimal: no (minimal twist has level 224)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + \beta_{5} q^{5} + ( - \beta_{12} - \beta_{2}) q^{7} + (\beta_{13} - \beta_{11} + \cdots - 5 \beta_{7}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + \beta_{5} q^{5} + ( - \beta_{12} - \beta_{2}) q^{7} + (\beta_{13} - \beta_{11} + \cdots - 5 \beta_{7}) q^{9}+ \cdots + ( - 4 \beta_{15} + 5 \beta_{14} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 40 q^{9} - 48 q^{17} + 136 q^{21} + 80 q^{25} + 16 q^{29} - 264 q^{33} - 72 q^{37} - 312 q^{45} + 128 q^{49} - 40 q^{53} + 368 q^{57} - 216 q^{61} - 168 q^{65} - 312 q^{73} - 64 q^{77} - 384 q^{81} + 1072 q^{85} + 24 q^{89} + 168 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 26 x^{14} - 16 x^{13} + 469 x^{12} + 144 x^{11} - 4526 x^{10} + 4440 x^{9} + 32608 x^{8} + \cdots + 208849 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 29\!\cdots\!02 \nu^{15} + \cdots + 19\!\cdots\!09 ) / 77\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 35\!\cdots\!52 \nu^{15} + \cdots - 32\!\cdots\!44 ) / 77\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 81\!\cdots\!73 \nu^{15} + \cdots + 73\!\cdots\!63 ) / 38\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 46\!\cdots\!90 \nu^{15} + \cdots - 25\!\cdots\!23 ) / 11\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 33\!\cdots\!38 \nu^{15} + \cdots + 15\!\cdots\!90 ) / 77\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 87\!\cdots\!58 \nu^{15} + \cdots + 36\!\cdots\!77 ) / 17\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 24\!\cdots\!00 \nu^{15} + \cdots + 11\!\cdots\!90 ) / 48\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 351675045716698 \nu^{15} + 84134512548244 \nu^{14} + \cdots + 24\!\cdots\!04 ) / 45\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 63\!\cdots\!74 \nu^{15} + \cdots - 14\!\cdots\!10 ) / 77\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 70\!\cdots\!40 \nu^{15} + \cdots - 58\!\cdots\!40 ) / 77\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 60\!\cdots\!76 \nu^{15} + \cdots - 10\!\cdots\!08 ) / 38\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 12\!\cdots\!84 \nu^{15} + \cdots + 27\!\cdots\!13 ) / 77\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 16\!\cdots\!52 \nu^{15} + \cdots + 84\!\cdots\!70 ) / 77\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 29\!\cdots\!76 \nu^{15} + \cdots - 29\!\cdots\!49 ) / 11\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 15\!\cdots\!26 \nu^{15} + \cdots + 10\!\cdots\!62 ) / 55\!\cdots\!19 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{14} - \beta_{12} + \beta_{11} + \beta_{6} - 2\beta_{2} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{13} + \beta_{11} - \beta_{10} + 13\beta_{7} - 2\beta_{2} + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{15} - 6 \beta_{13} - 17 \beta_{12} + 6 \beta_{10} - 11 \beta_{9} + 19 \beta_{8} + 33 \beta_{6} + \cdots + 24 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{15} - 13 \beta_{13} - 8 \beta_{12} + 5 \beta_{11} - 26 \beta_{10} - 13 \beta_{8} + \cdots + 4 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5 \beta_{15} + 35 \beta_{14} - 170 \beta_{13} - 5 \beta_{12} - 22 \beta_{11} - 85 \beta_{10} + \cdots + 680 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 80 \beta_{15} + 37 \beta_{14} + 135 \beta_{13} - 177 \beta_{12} - 135 \beta_{10} - 122 \beta_{9} + \cdots - 1261 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 368 \beta_{15} + 397 \beta_{14} - 3780 \beta_{13} + 1469 \beta_{12} + \beta_{11} - 7560 \beta_{10} + \cdots - 933 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 340 \beta_{15} + 1600 \beta_{14} + 1558 \beta_{13} + 340 \beta_{12} + 55 \beta_{11} + 779 \beta_{10} + \cdots - 6799 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1506 \beta_{15} - 1016 \beta_{14} + 70938 \beta_{13} - 931 \beta_{12} - 70938 \beta_{10} - 369 \beta_{9} + \cdots - 660288 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 17857 \beta_{15} + 20318 \beta_{14} - 12464 \beta_{13} + 72554 \beta_{12} + 4297 \beta_{11} + \cdots - 46436 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 4253 \beta_{15} + 83314 \beta_{14} + 1157706 \beta_{13} - 4253 \beta_{12} - 74343 \beta_{11} + \cdots - 5445440 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 154836 \beta_{15} - 57704 \beta_{14} + 308517 \beta_{13} + 531899 \beta_{12} - 308517 \beta_{10} + \cdots - 2949878 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 3245192 \beta_{15} + 4023903 \beta_{14} + 16342196 \beta_{13} + 12940335 \beta_{12} + \cdots - 8482119 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 386420 \beta_{15} - 5675214 \beta_{14} + 32304774 \beta_{13} - 386420 \beta_{12} - 2503213 \beta_{11} + \cdots - 153106199 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 82151478 \beta_{15} - 29923152 \beta_{14} - 186984966 \beta_{13} + 291303441 \beta_{12} + \cdots + 1769317544 ) / 8 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(-\beta_{7}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
−3.16141 + 2.64174i
2.08703 − 2.02145i
1.20279 − 1.51093i
−0.162551 + 0.910345i
0.869658 + 0.314400i
−1.90990 + 0.286185i
−2.79414 + 0.796701i
3.86852 − 1.41699i
−3.16141 − 2.64174i
2.08703 + 2.02145i
1.20279 + 1.51093i
−0.162551 − 0.910345i
0.869658 − 0.314400i
−1.90990 − 0.286185i
−2.79414 − 0.796701i
3.86852 + 1.41699i
0 −4.97091 − 2.86995i 0 −5.45949 + 3.15204i 0 −5.64993 + 4.13259i 0 11.9733 + 20.7383i 0
129.2 0 −3.45151 − 1.99273i 0 7.80961 − 4.50888i 0 −5.54917 − 4.26693i 0 3.44195 + 5.96164i 0
129.3 0 −2.20101 − 1.27075i 0 −3.56697 + 2.05939i 0 6.98814 + 0.407289i 0 −1.27038 − 2.20036i 0
129.4 0 −0.729881 − 0.421397i 0 1.21685 − 0.702550i 0 −1.56553 − 6.82269i 0 −4.14485 − 7.17909i 0
129.5 0 0.729881 + 0.421397i 0 1.21685 − 0.702550i 0 1.56553 + 6.82269i 0 −4.14485 − 7.17909i 0
129.6 0 2.20101 + 1.27075i 0 −3.56697 + 2.05939i 0 −6.98814 − 0.407289i 0 −1.27038 − 2.20036i 0
129.7 0 3.45151 + 1.99273i 0 7.80961 − 4.50888i 0 5.54917 + 4.26693i 0 3.44195 + 5.96164i 0
129.8 0 4.97091 + 2.86995i 0 −5.45949 + 3.15204i 0 5.64993 − 4.13259i 0 11.9733 + 20.7383i 0
257.1 0 −4.97091 + 2.86995i 0 −5.45949 − 3.15204i 0 −5.64993 − 4.13259i 0 11.9733 − 20.7383i 0
257.2 0 −3.45151 + 1.99273i 0 7.80961 + 4.50888i 0 −5.54917 + 4.26693i 0 3.44195 − 5.96164i 0
257.3 0 −2.20101 + 1.27075i 0 −3.56697 − 2.05939i 0 6.98814 − 0.407289i 0 −1.27038 + 2.20036i 0
257.4 0 −0.729881 + 0.421397i 0 1.21685 + 0.702550i 0 −1.56553 + 6.82269i 0 −4.14485 + 7.17909i 0
257.5 0 0.729881 − 0.421397i 0 1.21685 + 0.702550i 0 1.56553 − 6.82269i 0 −4.14485 + 7.17909i 0
257.6 0 2.20101 − 1.27075i 0 −3.56697 − 2.05939i 0 −6.98814 + 0.407289i 0 −1.27038 + 2.20036i 0
257.7 0 3.45151 − 1.99273i 0 7.80961 + 4.50888i 0 5.54917 − 4.26693i 0 3.44195 − 5.96164i 0
257.8 0 4.97091 − 2.86995i 0 −5.45949 − 3.15204i 0 5.64993 + 4.13259i 0 11.9733 − 20.7383i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.d odd 6 1 inner
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 448.3.s.h 16
4.b odd 2 1 inner 448.3.s.h 16
7.d odd 6 1 inner 448.3.s.h 16
8.b even 2 1 224.3.s.b ✓ 16
8.d odd 2 1 224.3.s.b ✓ 16
28.f even 6 1 inner 448.3.s.h 16
56.j odd 6 1 224.3.s.b ✓ 16
56.j odd 6 1 1568.3.c.g 16
56.k odd 6 1 1568.3.c.g 16
56.m even 6 1 224.3.s.b ✓ 16
56.m even 6 1 1568.3.c.g 16
56.p even 6 1 1568.3.c.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.s.b ✓ 16 8.b even 2 1
224.3.s.b ✓ 16 8.d odd 2 1
224.3.s.b ✓ 16 56.j odd 6 1
224.3.s.b ✓ 16 56.m even 6 1
448.3.s.h 16 1.a even 1 1 trivial
448.3.s.h 16 4.b odd 2 1 inner
448.3.s.h 16 7.d odd 6 1 inner
448.3.s.h 16 28.f even 6 1 inner
1568.3.c.g 16 56.j odd 6 1
1568.3.c.g 16 56.k odd 6 1
1568.3.c.g 16 56.m even 6 1
1568.3.c.g 16 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 56 T_{3}^{14} + 2258 T_{3}^{12} - 41216 T_{3}^{10} + 545827 T_{3}^{8} - 3222016 T_{3}^{6} + \cdots + 5764801 \) acting on \(S_{3}^{\mathrm{new}}(448, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 56 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$5$ \( (T^{8} - 70 T^{6} + \cdots + 108241)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 361696533659041 \) Copy content Toggle raw display
$13$ \( (T^{8} + 296 T^{6} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 24 T^{7} + \cdots + 717007729)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 40\!\cdots\!21 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$29$ \( (T^{4} - 4 T^{3} + \cdots + 5008)^{4} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 13\!\cdots\!41 \) Copy content Toggle raw display
$37$ \( (T^{8} + 36 T^{7} + \cdots + 628956241)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 2632 T^{6} + \cdots + 31899904)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 12876384903424)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 69\!\cdots\!21 \) Copy content Toggle raw display
$53$ \( (T^{8} + 20 T^{7} + \cdots + 607989789169)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 34\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 11484080659489)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 12\!\cdots\!21 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 159740883210496)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 46497574501569)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 72\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 5740355977216)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 32\!\cdots\!25)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 16556565688576)^{2} \) Copy content Toggle raw display
show more
show less