Properties

Label 320.5.p.g
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
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.

\(f(q)\) \(=\) \( q + ( - i + 1) q^{3} + (20 i + 15) q^{5} + (19 i + 19) q^{7} + 79 i q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - i + 1) q^{3} + (20 i + 15) q^{5} + (19 i + 19) q^{7} + 79 i q^{9} + 202 q^{11} + ( - 99 i + 99) q^{13} + (5 i + 35) q^{15} + ( - 239 i - 239) q^{17} + 40 i q^{19} + 38 q^{21} + (541 i - 541) q^{23} + (600 i - 175) q^{25} + (160 i + 160) q^{27} + 200 i q^{29} + 758 q^{31} + ( - 202 i + 202) q^{33} + (665 i - 95) q^{35} + ( - 141 i - 141) q^{37} - 198 i q^{39} + 1042 q^{41} + (759 i - 759) q^{43} + (1185 i - 1580) q^{45} + (459 i + 459) q^{47} - 1679 i q^{49} - 478 q^{51} + ( - 1819 i + 1819) q^{53} + (4040 i + 3030) q^{55} + (40 i + 40) q^{57} + 4600 i q^{59} - 2082 q^{61} + (1501 i - 1501) q^{63} + (495 i + 3465) q^{65} + (5081 i + 5081) q^{67} + 1082 i q^{69} + 3478 q^{71} + (3479 i - 3479) q^{73} + (775 i + 425) q^{75} + (3838 i + 3838) q^{77} + 7680 i q^{79} - 6079 q^{81} + ( - 6081 i + 6081) q^{83} + ( - 8365 i + 1195) q^{85} + (200 i + 200) q^{87} - 5680 i q^{89} + 3762 q^{91} + ( - 758 i + 758) q^{93} + (600 i - 800) q^{95} + (561 i + 561) q^{97} + 15958 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 30 q^{5} + 38 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 30 q^{5} + 38 q^{7} + 404 q^{11} + 198 q^{13} + 70 q^{15} - 478 q^{17} + 76 q^{21} - 1082 q^{23} - 350 q^{25} + 320 q^{27} + 1516 q^{31} + 404 q^{33} - 190 q^{35} - 282 q^{37} + 2084 q^{41} - 1518 q^{43} - 3160 q^{45} + 918 q^{47} - 956 q^{51} + 3638 q^{53} + 6060 q^{55} + 80 q^{57} - 4164 q^{61} - 3002 q^{63} + 6930 q^{65} + 10162 q^{67} + 6956 q^{71} - 6958 q^{73} + 850 q^{75} + 7676 q^{77} - 12158 q^{81} + 12162 q^{83} + 2390 q^{85} + 400 q^{87} + 7524 q^{91} + 1516 q^{93} - 1600 q^{95} + 1122 q^{97}+O(q^{100}) \) 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 1.00000 + 1.00000i 0 15.0000 20.0000i 0 19.0000 19.0000i 0 79.0000i 0
257.1 0 1.00000 1.00000i 0 15.0000 + 20.0000i 0 19.0000 + 19.0000i 0 79.0000i 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.g 2
4.b odd 2 1 320.5.p.d 2
5.c odd 4 1 inner 320.5.p.g 2
8.b even 2 1 80.5.p.c 2
8.d odd 2 1 10.5.c.b 2
20.e even 4 1 320.5.p.d 2
24.f even 2 1 90.5.g.a 2
40.e odd 2 1 50.5.c.a 2
40.f even 2 1 400.5.p.b 2
40.i odd 4 1 80.5.p.c 2
40.i odd 4 1 400.5.p.b 2
40.k even 4 1 10.5.c.b 2
40.k even 4 1 50.5.c.a 2
120.m even 2 1 450.5.g.b 2
120.q odd 4 1 90.5.g.a 2
120.q odd 4 1 450.5.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.5.c.b 2 8.d odd 2 1
10.5.c.b 2 40.k even 4 1
50.5.c.a 2 40.e odd 2 1
50.5.c.a 2 40.k even 4 1
80.5.p.c 2 8.b even 2 1
80.5.p.c 2 40.i odd 4 1
90.5.g.a 2 24.f even 2 1
90.5.g.a 2 120.q odd 4 1
320.5.p.d 2 4.b odd 2 1
320.5.p.d 2 20.e even 4 1
320.5.p.g 2 1.a even 1 1 trivial
320.5.p.g 2 5.c odd 4 1 inner
400.5.p.b 2 40.f even 2 1
400.5.p.b 2 40.i odd 4 1
450.5.g.b 2 120.m even 2 1
450.5.g.b 2 120.q odd 4 1

Hecke kernels

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

\( T_{3}^{2} - 2T_{3} + 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 198T_{13} + 19602 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} - 30T + 625 \) Copy content Toggle raw display
$7$ \( T^{2} - 38T + 722 \) Copy content Toggle raw display
$11$ \( (T - 202)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 198T + 19602 \) Copy content Toggle raw display
$17$ \( T^{2} + 478T + 114242 \) Copy content Toggle raw display
$19$ \( T^{2} + 1600 \) Copy content Toggle raw display
$23$ \( T^{2} + 1082 T + 585362 \) Copy content Toggle raw display
$29$ \( T^{2} + 40000 \) Copy content Toggle raw display
$31$ \( (T - 758)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 282T + 39762 \) Copy content Toggle raw display
$41$ \( (T - 1042)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 1518 T + 1152162 \) Copy content Toggle raw display
$47$ \( T^{2} - 918T + 421362 \) Copy content Toggle raw display
$53$ \( T^{2} - 3638 T + 6617522 \) Copy content Toggle raw display
$59$ \( T^{2} + 21160000 \) Copy content Toggle raw display
$61$ \( (T + 2082)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 10162 T + 51633122 \) Copy content Toggle raw display
$71$ \( (T - 3478)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 6958 T + 24206882 \) Copy content Toggle raw display
$79$ \( T^{2} + 58982400 \) Copy content Toggle raw display
$83$ \( T^{2} - 12162 T + 73957122 \) Copy content Toggle raw display
$89$ \( T^{2} + 32262400 \) Copy content Toggle raw display
$97$ \( T^{2} - 1122 T + 629442 \) Copy content Toggle raw display
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