Properties

Label 448.7.c.c
Level $448$
Weight $7$
Character orbit 448.c
Analytic conductor $103.064$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-510}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 510 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 7)
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 \(\beta = 2\sqrt{-510}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + \beta q^{5} + ( - 7 \beta - 133) q^{7} - 1311 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + \beta q^{5} + ( - 7 \beta - 133) q^{7} - 1311 q^{9} + 874 q^{11} - 49 \beta q^{13} - 2040 q^{15} - 132 \beta q^{17} + 69 \beta q^{19} + ( - 133 \beta + 14280) q^{21} - 4738 q^{23} + 13585 q^{25} - 582 \beta q^{27} - 11146 q^{29} + 608 \beta q^{31} + 874 \beta q^{33} + ( - 133 \beta + 14280) q^{35} - 3002 q^{37} + 99960 q^{39} + 1274 \beta q^{41} + 31418 q^{43} - 1311 \beta q^{45} + 1604 \beta q^{47} + (1862 \beta - 82271) q^{49} + 269280 q^{51} + 76406 q^{53} + 874 \beta q^{55} - 140760 q^{57} + 2507 \beta q^{59} + 6091 \beta q^{61} + (9177 \beta + 174363) q^{63} + 99960 q^{65} + 495242 q^{67} - 4738 \beta q^{69} + 184406 q^{71} - 1350 \beta q^{73} + 13585 \beta q^{75} + ( - 6118 \beta - 116242) q^{77} + 534934 q^{79} + 231561 q^{81} - 15827 \beta q^{83} + 269280 q^{85} - 11146 \beta q^{87} - 13938 \beta q^{89} + (6517 \beta - 699720) q^{91} - 1240320 q^{93} - 140760 q^{95} + 18032 \beta q^{97} - 1145814 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 266 q^{7} - 2622 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 266 q^{7} - 2622 q^{9} + 1748 q^{11} - 4080 q^{15} + 28560 q^{21} - 9476 q^{23} + 27170 q^{25} - 22292 q^{29} + 28560 q^{35} - 6004 q^{37} + 199920 q^{39} + 62836 q^{43} - 164542 q^{49} + 538560 q^{51} + 152812 q^{53} - 281520 q^{57} + 348726 q^{63} + 199920 q^{65} + 990484 q^{67} + 368812 q^{71} - 232484 q^{77} + 1069868 q^{79} + 463122 q^{81} + 538560 q^{85} - 1399440 q^{91} - 2480640 q^{93} - 281520 q^{95} - 2291628 q^{99}+O(q^{100}) \) 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
22.5832i
22.5832i
0 45.1664i 0 45.1664i 0 −133.000 + 316.165i 0 −1311.00 0
321.2 0 45.1664i 0 45.1664i 0 −133.000 316.165i 0 −1311.00 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.c 2
4.b odd 2 1 448.7.c.d 2
7.b odd 2 1 inner 448.7.c.c 2
8.b even 2 1 112.7.c.b 2
8.d odd 2 1 7.7.b.b 2
24.f even 2 1 63.7.d.d 2
28.d even 2 1 448.7.c.d 2
40.e odd 2 1 175.7.d.e 2
40.k even 4 2 175.7.c.c 4
56.e even 2 1 7.7.b.b 2
56.h odd 2 1 112.7.c.b 2
56.k odd 6 2 49.7.d.d 4
56.m even 6 2 49.7.d.d 4
168.e odd 2 1 63.7.d.d 2
280.n even 2 1 175.7.d.e 2
280.y odd 4 2 175.7.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.7.b.b 2 8.d odd 2 1
7.7.b.b 2 56.e even 2 1
49.7.d.d 4 56.k odd 6 2
49.7.d.d 4 56.m even 6 2
63.7.d.d 2 24.f even 2 1
63.7.d.d 2 168.e odd 2 1
112.7.c.b 2 8.b even 2 1
112.7.c.b 2 56.h odd 2 1
175.7.c.c 4 40.k even 4 2
175.7.c.c 4 280.y odd 4 2
175.7.d.e 2 40.e odd 2 1
175.7.d.e 2 280.n even 2 1
448.7.c.c 2 1.a even 1 1 trivial
448.7.c.c 2 7.b odd 2 1 inner
448.7.c.d 2 4.b odd 2 1
448.7.c.d 2 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 \(S_{7}^{\mathrm{new}}(448, [\chi])\):

\( T_{3}^{2} + 2040 \) Copy content Toggle raw display
\( T_{11} - 874 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2040 \) Copy content Toggle raw display
$5$ \( T^{2} + 2040 \) Copy content Toggle raw display
$7$ \( T^{2} + 266T + 117649 \) Copy content Toggle raw display
$11$ \( (T - 874)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4898040 \) Copy content Toggle raw display
$17$ \( T^{2} + 35544960 \) Copy content Toggle raw display
$19$ \( T^{2} + 9712440 \) Copy content Toggle raw display
$23$ \( (T + 4738)^{2} \) Copy content Toggle raw display
$29$ \( (T + 11146)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 754114560 \) Copy content Toggle raw display
$37$ \( (T + 3002)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 3311075040 \) Copy content Toggle raw display
$43$ \( (T - 31418)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 5248544640 \) Copy content Toggle raw display
$53$ \( (T - 76406)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 12821499960 \) Copy content Toggle raw display
$61$ \( T^{2} + 75684573240 \) Copy content Toggle raw display
$67$ \( (T - 495242)^{2} \) Copy content Toggle raw display
$71$ \( (T - 184406)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 3717900000 \) Copy content Toggle raw display
$79$ \( (T - 534934)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 511007615160 \) Copy content Toggle raw display
$89$ \( T^{2} + 396306401760 \) Copy content Toggle raw display
$97$ \( T^{2} + 663312168960 \) Copy content Toggle raw display
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