Properties

Label 80.5.b.b
Level $80$
Weight $5$
Character orbit 80.b
Analytic conductor $8.270$
Analytic rank $0$
Dimension $4$
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] = [80,5,Mod(31,80)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(80, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("80.31");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 80.b (of order \(2\), degree \(1\), 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: \(8.26959704671\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - 5 \beta_1 q^{5} + (7 \beta_{3} + 2 \beta_{2}) q^{7} + (18 \beta_1 + 39) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - 5 \beta_1 q^{5} + (7 \beta_{3} + 2 \beta_{2}) q^{7} + (18 \beta_1 + 39) q^{9} + (30 \beta_{3} + 4 \beta_{2}) q^{11} + (54 \beta_1 - 68) q^{13} + (5 \beta_{3} + 10 \beta_{2}) q^{15} + (120 \beta_1 - 54) q^{17} + (74 \beta_{3} + 34 \beta_{2}) q^{19} + (78 \beta_1 - 126) q^{21} + (111 \beta_{3} - 32 \beta_{2}) q^{23} + 125 q^{25} + ( - 18 \beta_{3} + 84 \beta_{2}) q^{27} + ( - 276 \beta_1 + 918) q^{29} + ( - 62 \beta_{3} - 28 \beta_{2}) q^{31} + (252 \beta_1 - 348) q^{33} + ( - 60 \beta_{3} + 55 \beta_{2}) q^{35} + ( - 594 \beta_1 - 940) q^{37} + ( - 54 \beta_{3} - 176 \beta_{2}) q^{39} + (246 \beta_1 - 108) q^{41} + ( - 326 \beta_{3} - 241 \beta_{2}) q^{43} + ( - 195 \beta_1 - 450) q^{45} + ( - 597 \beta_{3} + 194 \beta_{2}) q^{47} + ( - 54 \beta_1 + 1183) q^{49} + ( - 120 \beta_{3} - 294 \beta_{2}) q^{51} + ( - 1122 \beta_1 + 756) q^{53} + ( - 280 \beta_{3} + 190 \beta_{2}) q^{55} + (1056 \beta_1 - 1872) q^{57} + (570 \beta_{3} - 626 \beta_{2}) q^{59} + ( - 1998 \beta_1 + 1376) q^{61} + (489 \beta_{3} - 120 \beta_{2}) q^{63} + (340 \beta_1 - 1350) q^{65} + ( - 1214 \beta_{3} + 1133 \beta_{2}) q^{67} + (90 \beta_1 + 678) q^{69} + (606 \beta_{3} + 664 \beta_{2}) q^{71} + (1296 \beta_1 + 5506) q^{73} + 125 \beta_{2} q^{75} + ( - 588 \beta_1 - 4644) q^{77} + (1320 \beta_{3} - 1500 \beta_{2}) q^{79} + (2862 \beta_1 - 261) q^{81} + (1962 \beta_{3} + 87 \beta_{2}) q^{83} + (270 \beta_1 - 3000) q^{85} + (276 \beta_{3} + 1470 \beta_{2}) q^{87} + (5112 \beta_1 + 2106) q^{89} + (172 \beta_{3} - 730 \beta_{2}) q^{91} + ( - 876 \beta_1 + 1548) q^{93} + ( - 570 \beta_{3} + 710 \beta_{2}) q^{95} + ( - 540 \beta_1 - 8882) q^{97} + (2178 \beta_{3} - 528 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 156 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 156 q^{9} - 272 q^{13} - 216 q^{17} - 504 q^{21} + 500 q^{25} + 3672 q^{29} - 1392 q^{33} - 3760 q^{37} - 432 q^{41} - 1800 q^{45} + 4732 q^{49} + 3024 q^{53} - 7488 q^{57} + 5504 q^{61} - 5400 q^{65} + 2712 q^{69} + 22024 q^{73} - 18576 q^{77} - 1044 q^{81} - 12000 q^{85} + 8424 q^{89} + 6192 q^{93} - 35528 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} + 4\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\nu^{3} - 4\nu^{2} + 8\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 - 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta _1 - 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.809017 1.40126i
−0.309017 + 0.535233i
−0.309017 0.535233i
0.809017 + 1.40126i
0 9.06914i 0 11.1803 0 33.1248i 0 −1.24922 0
31.2 0 1.32317i 0 −11.1803 0 36.5889i 0 79.2492 0
31.3 0 1.32317i 0 −11.1803 0 36.5889i 0 79.2492 0
31.4 0 9.06914i 0 11.1803 0 33.1248i 0 −1.24922 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.5.b.b 4
3.b odd 2 1 720.5.e.b 4
4.b odd 2 1 inner 80.5.b.b 4
5.b even 2 1 400.5.b.h 4
5.c odd 4 2 400.5.h.c 8
8.b even 2 1 320.5.b.b 4
8.d odd 2 1 320.5.b.b 4
12.b even 2 1 720.5.e.b 4
20.d odd 2 1 400.5.b.h 4
20.e even 4 2 400.5.h.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.5.b.b 4 1.a even 1 1 trivial
80.5.b.b 4 4.b odd 2 1 inner
320.5.b.b 4 8.b even 2 1
320.5.b.b 4 8.d odd 2 1
400.5.b.h 4 5.b even 2 1
400.5.b.h 4 20.d odd 2 1
400.5.h.c 8 5.c odd 4 2
400.5.h.c 8 20.e even 4 2
720.5.e.b 4 3.b odd 2 1
720.5.e.b 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 84T_{3}^{2} + 144 \) acting on \(S_{5}^{\mathrm{new}}(80, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 84T^{2} + 144 \) Copy content Toggle raw display
$5$ \( (T^{2} - 125)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 2436 T^{2} + \cdots + 1468944 \) Copy content Toggle raw display
$11$ \( T^{4} + 36624 T^{2} + \cdots + 267911424 \) Copy content Toggle raw display
$13$ \( (T^{2} + 136 T - 9956)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 108 T - 69084)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 354624 T^{2} + \cdots + 29793521664 \) Copy content Toggle raw display
$23$ \( T^{4} + 444324 T^{2} + \cdots + 1220244624 \) Copy content Toggle raw display
$29$ \( (T^{2} - 1836 T + 461844)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 245904 T^{2} + \cdots + 14411522304 \) Copy content Toggle raw display
$37$ \( (T^{2} + 1880 T - 880580)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 216 T - 290916)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 10590324 T^{2} + \cdots + 18918219842064 \) Copy content Toggle raw display
$47$ \( T^{4} + 13212516 T^{2} + \cdots + 3006714384144 \) Copy content Toggle raw display
$53$ \( (T^{2} - 1512 T - 5722884)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 321523343069184 \) Copy content Toggle raw display
$61$ \( (T^{2} - 2752 T - 18066644)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 127875444 T^{2} + \cdots + 40\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 339683772582144 \) Copy content Toggle raw display
$73$ \( (T^{2} - 11012 T + 21917956)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 204206400 T^{2} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + 143312436 T^{2} + \cdots + 29\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{2} - 4212 T - 126227484)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 17764 T + 77431924)^{2} \) Copy content Toggle raw display
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