Properties

Label 448.7.c.e
Level $448$
Weight $7$
Character orbit 448.c
Analytic conductor $103.064$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,7,Mod(321,448)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.321");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 448.c (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: \(103.064229462\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.211968.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 30x^{2} + 207 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
Twist minimal: no (minimal twist has level 14)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + 5 \beta_{2} q^{5} + (7 \beta_{3} + 7 \beta_{2} - 14 \beta_1 - 77) q^{7} + (26 \beta_{3} + 273) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + 5 \beta_{2} q^{5} + (7 \beta_{3} + 7 \beta_{2} - 14 \beta_1 - 77) q^{7} + (26 \beta_{3} + 273) q^{9} + ( - 4 \beta_{3} - 1110) q^{11} + ( - 43 \beta_{2} - 94 \beta_1) q^{13} + ( - 110 \beta_{3} + 1080) q^{15} + ( - 134 \beta_{2} + 150 \beta_1) q^{17} + (50 \beta_{2} - 283 \beta_1) q^{19} + (210 \beta_{3} + 21 \beta_{2} + 182 \beta_1 - 4872) q^{21} + (88 \beta_{3} - 10146) q^{23} + ( - 950 \beta_{3} - 10175) q^{25} + (78 \beta_{2} - 612 \beta_1) q^{27} + ( - 2188 \beta_{3} + 4566) q^{29} + ( - 1038 \beta_{2} - 54 \beta_1) q^{31} + ( - 12 \beta_{2} + 1050 \beta_1) q^{33} + ( - 2870 \beta_{3} + 140 \beta_{2} - 735 \beta_1 - 21000) q^{35} + ( - 2556 \beta_{3} + 5798) q^{37} + (3390 \beta_{3} - 52152) q^{39} + ( - 2098 \beta_{2} + 1980 \beta_1) q^{41} + (6080 \beta_{3} - 11174) q^{43} + (3315 \beta_{2} - 2730 \beta_1) q^{45} + (986 \beta_{2} + 4014 \beta_1) q^{47} + ( - 2156 \beta_{3} + 980 \beta_{2} + 3038 \beta_1 - 77567) q^{49} + ( - 952 \beta_{3} + 39456) q^{51} + (3656 \beta_{3} - 62154) q^{53} + ( - 5850 \beta_{2} + 420 \beta_1) q^{55} + (6258 \beta_{3} - 118248) q^{57} + (6710 \beta_{2} - 13485 \beta_1) q^{59} + ( - 2945 \beta_{2} - 9368 \beta_1) q^{61} + ( - 91 \beta_{3} + 5733 \beta_{2} - 2184 \beta_1 + 31395) q^{63} + ( - 2170 \beta_{3} + 323400) q^{65} + ( - 136 \beta_{3} - 108694) q^{67} + (264 \beta_{2} + 11466 \beta_1) q^{69} + (19382 \beta_{3} + 112902) q^{71} + ( - 11008 \beta_{2} + 3494 \beta_1) q^{73} + ( - 2850 \beta_{2} - 4075 \beta_1) q^{75} + ( - 7462 \beta_{3} - 8358 \beta_{2} + 15288 \beta_1 + 77406) q^{77} + ( - 4230 \beta_{3} - 523226) q^{79} + (33150 \beta_{3} - 63207) q^{81} + ( - 14380 \beta_{2} + 16983 \beta_1) q^{83} + (41960 \beta_{3} + 529440) q^{85} + ( - 6564 \beta_{2} - 37386 \beta_1) q^{87} + (8076 \beta_{2} + 8430 \beta_1) q^{89} + (44422 \beta_{3} + 770 \beta_{2} + 23429 \beta_1 - 277368) q^{91} + (24240 \beta_{3} - 248832) q^{93} + ( - 40630 \beta_{3} + 47640) q^{95} + (3562 \beta_{2} - 6254 \beta_1) q^{97} + ( - 29952 \beta_{3} - 332982) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 308 q^{7} + 1092 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 308 q^{7} + 1092 q^{9} - 4440 q^{11} + 4320 q^{15} - 19488 q^{21} - 40584 q^{23} - 40700 q^{25} + 18264 q^{29} - 84000 q^{35} + 23192 q^{37} - 208608 q^{39} - 44696 q^{43} - 310268 q^{49} + 157824 q^{51} - 248616 q^{53} - 472992 q^{57} + 125580 q^{63} + 1293600 q^{65} - 434776 q^{67} + 451608 q^{71} + 309624 q^{77} - 2092904 q^{79} - 252828 q^{81} + 2117760 q^{85} - 1109472 q^{91} - 995328 q^{93} + 190560 q^{95} - 1331928 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 30x^{2} + 207 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} + 18\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 34\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} + 60 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 3\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 60 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{2} + 51\beta_1 ) / 16 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1
4.38664i
3.27984i
3.27984i
4.38664i
0 29.9539i 0 98.3767i 0 −195.794 281.627i 0 −168.235 0
321.2 0 3.84257i 0 204.749i 0 41.7939 340.444i 0 714.235 0
321.3 0 3.84257i 0 204.749i 0 41.7939 + 340.444i 0 714.235 0
321.4 0 29.9539i 0 98.3767i 0 −195.794 + 281.627i 0 −168.235 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.7.c.e 4
4.b odd 2 1 448.7.c.h 4
7.b odd 2 1 inner 448.7.c.e 4
8.b even 2 1 112.7.c.c 4
8.d odd 2 1 14.7.b.a 4
24.f even 2 1 126.7.c.a 4
28.d even 2 1 448.7.c.h 4
40.e odd 2 1 350.7.b.a 4
40.k even 4 2 350.7.d.a 8
56.e even 2 1 14.7.b.a 4
56.h odd 2 1 112.7.c.c 4
56.k odd 6 2 98.7.d.b 8
56.m even 6 2 98.7.d.b 8
168.e odd 2 1 126.7.c.a 4
280.n even 2 1 350.7.b.a 4
280.y odd 4 2 350.7.d.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.7.b.a 4 8.d odd 2 1
14.7.b.a 4 56.e even 2 1
98.7.d.b 8 56.k odd 6 2
98.7.d.b 8 56.m even 6 2
112.7.c.c 4 8.b even 2 1
112.7.c.c 4 56.h odd 2 1
126.7.c.a 4 24.f even 2 1
126.7.c.a 4 168.e odd 2 1
350.7.b.a 4 40.e odd 2 1
350.7.b.a 4 280.n even 2 1
350.7.d.a 8 40.k even 4 2
350.7.d.a 8 280.y odd 4 2
448.7.c.e 4 1.a even 1 1 trivial
448.7.c.e 4 7.b odd 2 1 inner
448.7.c.h 4 4.b odd 2 1
448.7.c.h 4 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(448, [\chi])\):

\( T_{3}^{4} + 912T_{3}^{2} + 13248 \) Copy content Toggle raw display
\( T_{11}^{2} + 2220T_{11} + 1227492 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 912 T^{2} + 13248 \) Copy content Toggle raw display
$5$ \( T^{4} + 51600 T^{2} + \cdots + 405720000 \) Copy content Toggle raw display
$7$ \( T^{4} + 308 T^{3} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( (T^{2} + 2220 T + 1227492)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 15367056 T^{2} + \cdots + 26266196601792 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 126735731638272 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 551809934313408 \) Copy content Toggle raw display
$23$ \( (T^{2} + 20292 T + 100711044)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 9132 T - 1357906716)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2274932736 T^{2} + \cdots + 86\!\cdots\!12 \) Copy content Toggle raw display
$37$ \( (T^{2} - 11596 T - 1847926364)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 9071224896 T^{2} + \cdots + 28\!\cdots\!32 \) Copy content Toggle raw display
$43$ \( (T^{2} + 22348 T - 10521464924)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 20120477952 T^{2} + \cdots + 12\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( (T^{2} + 124308 T + 13614948)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 180594109200 T^{2} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + 121774406928 T^{2} + \cdots + 82\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( (T^{2} + 217388 T + 11809058788)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 225804 T - 95443772508)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 228009998400 T^{2} + \cdots + 26\!\cdots\!72 \) Copy content Toggle raw display
$79$ \( (T^{2} + 1046452 T + 268612291876)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 478841902608 T^{2} + \cdots + 21\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{4} + 258250641984 T^{2} + \cdots + 15\!\cdots\!52 \) Copy content Toggle raw display
$97$ \( T^{4} + 42611214336 T^{2} + \cdots + 39\!\cdots\!12 \) Copy content Toggle raw display
show more
show less