Properties

Label 448.6.f.b
Level $448$
Weight $6$
Character orbit 448.f
Analytic conductor $71.852$
Analytic rank $0$
Dimension $8$
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] = [448,6,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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 208x^{6} + 15517x^{4} + 287808x^{2} + 1830609 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{20}\cdot 7^{5} \)
Twist minimal: no (minimal twist has level 112)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + \beta_{3} q^{5} + ( - \beta_{5} - 2 \beta_{4}) q^{7} + ( - \beta_1 + 149) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + \beta_{3} q^{5} + ( - \beta_{5} - 2 \beta_{4}) q^{7} + ( - \beta_1 + 149) q^{9} - \beta_{2} q^{11} + (\beta_{6} + 5 \beta_{3}) q^{13} + (\beta_{7} + 7 \beta_{5} - 3 \beta_{2}) q^{15} + (\beta_{6} - 10 \beta_{3}) q^{17} + ( - \beta_{7} + 7 \beta_{5} - 5 \beta_{4}) q^{19} + ( - \beta_{6} - 23 \beta_{3} + 5 \beta_1 - 784) q^{21} + (2 \beta_{7} + 14 \beta_{5} + 5 \beta_{2}) q^{23} + (15 \beta_1 - 2131) q^{25} + ( - 3 \beta_{7} + 21 \beta_{5} + 172 \beta_{4}) q^{27} + ( - 6 \beta_1 + 1218) q^{29} + (5 \beta_{7} - 35 \beta_{5} - 62 \beta_{4}) q^{31} + ( - \beta_{6} + 124 \beta_{3}) q^{33} + ( - 7 \beta_{7} - 13 \beta_{5} + 303 \beta_{4} + 7 \beta_{2}) q^{35} + (18 \beta_1 - 2422) q^{37} + (11 \beta_{7} + 77 \beta_{5} - 9 \beta_{2}) q^{39} + (10 \beta_{6} - 54 \beta_{3}) q^{41} + ( - 2 \beta_{7} - 14 \beta_{5} - 27 \beta_{2}) q^{43} + (11 \beta_{6} + 451 \beta_{3}) q^{45} + (9 \beta_{7} - 63 \beta_{5} + 1014 \beta_{4}) q^{47} + ( - 11 \beta_{6} + 188 \beta_{3} + 6 \beta_1 + 1225) q^{49} + ( - 4 \beta_{7} - 28 \beta_{5} + 36 \beta_{2}) q^{51} + (60 \beta_1 - 12774) q^{53} + (3 \beta_{7} - 21 \beta_{5} - 1668 \beta_{4}) q^{55} + ( - 37 \beta_1 - 1960) q^{57} + (9 \beta_{7} - 63 \beta_{5} - 759 \beta_{4}) q^{59} + ( - 12 \beta_{6} - 249 \beta_{3}) q^{61} + ( - 14 \beta_{7} - 65 \beta_{5} - 1628 \beta_{4} + 63 \beta_{2}) q^{63} + (141 \beta_1 - 24168) q^{65} + (42 \beta_{7} + 294 \beta_{5} - 27 \beta_{2}) q^{67} + (33 \beta_{6} + 24 \beta_{3}) q^{69} + (5 \beta_{7} + 35 \beta_{5} - 92 \beta_{2}) q^{71} + (45 \beta_{6} + 144 \beta_{3}) q^{73} + (45 \beta_{7} - 315 \beta_{5} - 6121 \beta_{4}) q^{75} + ( - 44 \beta_{6} - 326 \beta_{3} - 123 \beta_1 + 3234) q^{77} + ( - 29 \beta_{7} - 203 \beta_{5} + 54 \beta_{2}) q^{79} + ( - 55 \beta_1 + 31217) q^{81} + (54 \beta_{7} - 378 \beta_{5} + 1953 \beta_{4}) q^{83} + ( - 84 \beta_1 + 54672) q^{85} + ( - 18 \beta_{7} + 126 \beta_{5} + 2814 \beta_{4}) q^{87} + ( - 55 \beta_{6} - 172 \beta_{3}) q^{89} + (56 \beta_{7} - 130 \beta_{5} + 3135 \beta_{4} + 189 \beta_{2}) q^{91} + (272 \beta_1 - 24304) q^{93} + (29 \beta_{7} + 203 \beta_{5} + \beta_{2}) q^{95} + (11 \beta_{6} - 458 \beta_{3}) q^{97} + (118 \beta_{7} + 826 \beta_{5} - 135 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 1192 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 1192 q^{9} - 6272 q^{21} - 17048 q^{25} + 9744 q^{29} - 19376 q^{37} + 9800 q^{49} - 102192 q^{53} - 15680 q^{57} - 193344 q^{65} + 25872 q^{77} + 249736 q^{81} + 437376 q^{85} - 194432 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 208x^{6} + 15517x^{4} + 287808x^{2} + 1830609 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 196\nu^{6} + 27552\nu^{4} + 567504\nu^{2} - 45923360 ) / 154009 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4040\nu^{6} - 831924\nu^{4} - 56404144\nu^{2} - 570933138 ) / 462027 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1222\nu^{7} + 303786\nu^{5} + 27739628\nu^{3} + 809257932\nu ) / 18943107 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -93146\nu^{7} - 18153962\nu^{5} - 1214256788\nu^{3} - 11248398780\nu ) / 625122531 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 40870 \nu^{7} + 406802 \nu^{6} - 8341306 \nu^{5} + 77029898 \nu^{4} - 552201520 \nu^{3} + 5186588396 \nu^{2} - 5108756376 \nu + 52462110921 ) / 208374177 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 77668\nu^{7} + 12942020\nu^{5} + 713838720\nu^{3} + 17813518128\nu ) / 208374177 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 286090 \nu^{7} + 2847614 \nu^{6} + 58389142 \nu^{5} + 539209286 \nu^{4} + 3865410640 \nu^{3} + 36306118772 \nu^{2} + 35761294632 \nu + 367234776447 ) / 208374177 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 7\beta_{6} - 7\beta_{5} + 42\beta_{4} + 14\beta_{3} ) / 784 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 9\beta_{7} + 63\beta_{5} + 21\beta_{2} - 49\beta _1 - 20384 ) / 392 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -317\beta_{7} - 525\beta_{6} + 2219\beta_{5} - 6846\beta_{4} + 714\beta_{3} ) / 784 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -762\beta_{7} - 5334\beta_{5} - 2121\beta_{2} + 1792\beta _1 + 599270 ) / 196 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 23483\beta_{7} + 11949\beta_{6} - 164381\beta_{5} + 453264\beta_{4} - 21672\beta_{3} ) / 392 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 94086\beta_{7} + 658602\beta_{5} + 267750\beta_{2} - 26957\beta _1 - 8806672 ) / 196 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -367279\beta_{7} + 95781\beta_{6} + 2570953\beta_{5} - 6983508\beta_{4} - 182196\beta_{3} ) / 56 \) Copy content Toggle raw display

Character values

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

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

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
−0.785404 3.50041i
−0.785404 + 3.50041i
−2.52649 + 9.93716i
−2.52649 9.93716i
2.52649 + 9.93716i
2.52649 9.93716i
0.785404 3.50041i
0.785404 + 3.50041i
0 −26.7378 0 100.497i 0 89.7065 93.5935i 0 471.912 0
447.2 0 −26.7378 0 100.497i 0 89.7065 + 93.5935i 0 471.912 0
447.3 0 −8.31193 0 20.3058i 0 −99.9236 + 82.5970i 0 −173.912 0
447.4 0 −8.31193 0 20.3058i 0 −99.9236 82.5970i 0 −173.912 0
447.5 0 8.31193 0 20.3058i 0 99.9236 82.5970i 0 −173.912 0
447.6 0 8.31193 0 20.3058i 0 99.9236 + 82.5970i 0 −173.912 0
447.7 0 26.7378 0 100.497i 0 −89.7065 + 93.5935i 0 471.912 0
447.8 0 26.7378 0 100.497i 0 −89.7065 93.5935i 0 471.912 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 447.8
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.b 8
4.b odd 2 1 inner 448.6.f.b 8
7.b odd 2 1 inner 448.6.f.b 8
8.b even 2 1 112.6.f.a 8
8.d odd 2 1 112.6.f.a 8
24.f even 2 1 1008.6.b.g 8
24.h odd 2 1 1008.6.b.g 8
28.d even 2 1 inner 448.6.f.b 8
56.e even 2 1 112.6.f.a 8
56.h odd 2 1 112.6.f.a 8
168.e odd 2 1 1008.6.b.g 8
168.i even 2 1 1008.6.b.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.6.f.a 8 8.b even 2 1
112.6.f.a 8 8.d odd 2 1
112.6.f.a 8 56.e even 2 1
112.6.f.a 8 56.h odd 2 1
448.6.f.b 8 1.a even 1 1 trivial
448.6.f.b 8 4.b odd 2 1 inner
448.6.f.b 8 7.b odd 2 1 inner
448.6.f.b 8 28.d even 2 1 inner
1008.6.b.g 8 24.f even 2 1
1008.6.b.g 8 24.h odd 2 1
1008.6.b.g 8 168.e odd 2 1
1008.6.b.g 8 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 784T_{3}^{2} + 49392 \) acting on \(S_{6}^{\mathrm{new}}(448, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 784 T^{2} + 49392)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 10512 T^{2} + 4164336)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 4900 T^{6} + \cdots + 79\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{4} + 405720 T^{2} + \cdots + 41092118928)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 1587792 T^{2} + \cdots + 532363517424)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 2502912 T^{2} + \cdots + 1219231678464)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 2939216 T^{2} + \cdots + 390652133872)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 20548248 T^{2} + \cdots + 2172546584208)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 2436 T - 2270268)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 76004096 T^{2} + \cdots + 10\!\cdots\!72)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4844 T - 27918044)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 171938112 T^{2} + \cdots + 19\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 298206552 T^{2} + \cdots + 19\!\cdots\!28)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 1042594560 T^{2} + \cdots + 27\!\cdots\!72)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 25548 T - 212204124)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 688136400 T^{2} + \cdots + 43\!\cdots\!68)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 823393296 T^{2} + \cdots + 57\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 5888524824 T^{2} + \cdots + 61\!\cdots\!72)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 3593673720 T^{2} + \cdots + 28\!\cdots\!28)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 2931878592 T^{2} + \cdots + 12\!\cdots\!16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 4035162936 T^{2} + \cdots + 36\!\cdots\!72)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 11503940112 T^{2} + \cdots + 67\!\cdots\!28)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 4366945728 T^{2} + \cdots + 27\!\cdots\!96)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 2413035264 T^{2} + \cdots + 93\!\cdots\!24)^{2} \) Copy content Toggle raw display
show more
show less