Properties

Label 320.5.p.h
Level $320$
Weight $5$
Character orbit 320.p
Analytic conductor $33.078$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,5,Mod(193,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.193");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 320.p (of order \(4\), 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: \(33.0783881868\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 6 i + 6) q^{3} + ( - 15 i - 20) q^{5} + ( - 26 i - 26) q^{7} + 9 i q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 6 i + 6) q^{3} + ( - 15 i - 20) q^{5} + ( - 26 i - 26) q^{7} + 9 i q^{9} + 8 q^{11} + (139 i - 139) q^{13} + (30 i - 210) q^{15} + ( - i - 1) q^{17} - 180 i q^{19} - 312 q^{21} + (166 i - 166) q^{23} + (600 i + 175) q^{25} + (540 i + 540) q^{27} + 480 i q^{29} + 572 q^{31} + ( - 48 i + 48) q^{33} + (910 i + 130) q^{35} + (251 i + 251) q^{37} + 1668 i q^{39} - 1688 q^{41} + (1474 i - 1474) q^{43} + ( - 180 i + 135) q^{45} + (2474 i + 2474) q^{47} - 1049 i q^{49} - 12 q^{51} + ( - 3331 i + 3331) q^{53} + ( - 120 i - 160) q^{55} + ( - 1080 i - 1080) q^{57} + 3660 i q^{59} - 1592 q^{61} + ( - 234 i + 234) q^{63} + ( - 695 i + 4865) q^{65} + ( - 874 i - 874) q^{67} + 1992 i q^{69} - 6068 q^{71} + (791 i - 791) q^{73} + (2550 i + 4650) q^{75} + ( - 208 i - 208) q^{77} + 9120 i q^{79} + 5751 q^{81} + (5654 i - 5654) q^{83} + (35 i + 5) q^{85} + (2880 i + 2880) q^{87} + 2160 i q^{89} + 7228 q^{91} + ( - 3432 i + 3432) q^{93} + (3600 i - 2700) q^{95} + ( - 6551 i - 6551) q^{97} + 72 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 12 q^{3} - 40 q^{5} - 52 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 12 q^{3} - 40 q^{5} - 52 q^{7} + 16 q^{11} - 278 q^{13} - 420 q^{15} - 2 q^{17} - 624 q^{21} - 332 q^{23} + 350 q^{25} + 1080 q^{27} + 1144 q^{31} + 96 q^{33} + 260 q^{35} + 502 q^{37} - 3376 q^{41} - 2948 q^{43} + 270 q^{45} + 4948 q^{47} - 24 q^{51} + 6662 q^{53} - 320 q^{55} - 2160 q^{57} - 3184 q^{61} + 468 q^{63} + 9730 q^{65} - 1748 q^{67} - 12136 q^{71} - 1582 q^{73} + 9300 q^{75} - 416 q^{77} + 11502 q^{81} - 11308 q^{83} + 10 q^{85} + 5760 q^{87} + 14456 q^{91} + 6864 q^{93} - 5400 q^{95} - 13102 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(i\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
1.00000i
1.00000i
0 6.00000 + 6.00000i 0 −20.0000 + 15.0000i 0 −26.0000 + 26.0000i 0 9.00000i 0
257.1 0 6.00000 6.00000i 0 −20.0000 15.0000i 0 −26.0000 26.0000i 0 9.00000i 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
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.5.p.h 2
4.b odd 2 1 320.5.p.c 2
5.c odd 4 1 inner 320.5.p.h 2
8.b even 2 1 5.5.c.a 2
8.d odd 2 1 80.5.p.d 2
20.e even 4 1 320.5.p.c 2
24.h odd 2 1 45.5.g.b 2
40.e odd 2 1 400.5.p.a 2
40.f even 2 1 25.5.c.a 2
40.i odd 4 1 5.5.c.a 2
40.i odd 4 1 25.5.c.a 2
40.k even 4 1 80.5.p.d 2
40.k even 4 1 400.5.p.a 2
120.i odd 2 1 225.5.g.b 2
120.w even 4 1 45.5.g.b 2
120.w even 4 1 225.5.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.5.c.a 2 8.b even 2 1
5.5.c.a 2 40.i odd 4 1
25.5.c.a 2 40.f even 2 1
25.5.c.a 2 40.i odd 4 1
45.5.g.b 2 24.h odd 2 1
45.5.g.b 2 120.w even 4 1
80.5.p.d 2 8.d odd 2 1
80.5.p.d 2 40.k even 4 1
225.5.g.b 2 120.i odd 2 1
225.5.g.b 2 120.w even 4 1
320.5.p.c 2 4.b odd 2 1
320.5.p.c 2 20.e even 4 1
320.5.p.h 2 1.a even 1 1 trivial
320.5.p.h 2 5.c odd 4 1 inner
400.5.p.a 2 40.e odd 2 1
400.5.p.a 2 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{2} - 12T_{3} + 72 \) Copy content Toggle raw display
\( T_{13}^{2} + 278T_{13} + 38642 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 12T + 72 \) Copy content Toggle raw display
$5$ \( T^{2} + 40T + 625 \) Copy content Toggle raw display
$7$ \( T^{2} + 52T + 1352 \) Copy content Toggle raw display
$11$ \( (T - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 278T + 38642 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 32400 \) Copy content Toggle raw display
$23$ \( T^{2} + 332T + 55112 \) Copy content Toggle raw display
$29$ \( T^{2} + 230400 \) Copy content Toggle raw display
$31$ \( (T - 572)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 502T + 126002 \) Copy content Toggle raw display
$41$ \( (T + 1688)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 2948 T + 4345352 \) Copy content Toggle raw display
$47$ \( T^{2} - 4948 T + 12241352 \) Copy content Toggle raw display
$53$ \( T^{2} - 6662 T + 22191122 \) Copy content Toggle raw display
$59$ \( T^{2} + 13395600 \) Copy content Toggle raw display
$61$ \( (T + 1592)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1748 T + 1527752 \) Copy content Toggle raw display
$71$ \( (T + 6068)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1582 T + 1251362 \) Copy content Toggle raw display
$79$ \( T^{2} + 83174400 \) Copy content Toggle raw display
$83$ \( T^{2} + 11308 T + 63935432 \) Copy content Toggle raw display
$89$ \( T^{2} + 4665600 \) Copy content Toggle raw display
$97$ \( T^{2} + 13102 T + 85831202 \) Copy content Toggle raw display
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