Properties

Label 8.5.d.b
Level $8$
Weight $5$
Character orbit 8.d
Analytic conductor $0.827$
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] = [8,5,Mod(3,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.3");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 8.d (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: \(0.826959704671\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-15}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{-15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + 6 q^{3} + (2 \beta - 14) q^{4} + 8 \beta q^{5} + ( - 6 \beta - 6) q^{6} - 16 \beta q^{7} + (12 \beta + 44) q^{8} - 45 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} + 6 q^{3} + (2 \beta - 14) q^{4} + 8 \beta q^{5} + ( - 6 \beta - 6) q^{6} - 16 \beta q^{7} + (12 \beta + 44) q^{8} - 45 q^{9} + ( - 8 \beta + 120) q^{10} - 26 q^{11} + (12 \beta - 84) q^{12} - 8 \beta q^{13} + (16 \beta - 240) q^{14} + 48 \beta q^{15} + ( - 56 \beta + 136) q^{16} + 226 q^{17} + (45 \beta + 45) q^{18} + 134 q^{19} + ( - 112 \beta - 240) q^{20} - 96 \beta q^{21} + (26 \beta + 26) q^{22} + 80 \beta q^{23} + (72 \beta + 264) q^{24} - 335 q^{25} + (8 \beta - 120) q^{26} - 756 q^{27} + (224 \beta + 480) q^{28} + 88 \beta q^{29} + ( - 48 \beta + 720) q^{30} - 320 \beta q^{31} + ( - 80 \beta - 976) q^{32} - 156 q^{33} + ( - 226 \beta - 226) q^{34} + 1920 q^{35} + ( - 90 \beta + 630) q^{36} + 456 \beta q^{37} + ( - 134 \beta - 134) q^{38} - 48 \beta q^{39} + (352 \beta - 1440) q^{40} + 994 q^{41} + (96 \beta - 1440) q^{42} - 1882 q^{43} + ( - 52 \beta + 364) q^{44} - 360 \beta q^{45} + ( - 80 \beta + 1200) q^{46} + 544 \beta q^{47} + ( - 336 \beta + 816) q^{48} - 1439 q^{49} + (335 \beta + 335) q^{50} + 1356 q^{51} + (112 \beta + 240) q^{52} - 984 \beta q^{53} + (756 \beta + 756) q^{54} - 208 \beta q^{55} + ( - 704 \beta + 2880) q^{56} + 804 q^{57} + ( - 88 \beta + 1320) q^{58} - 5018 q^{59} + ( - 672 \beta - 1440) q^{60} + 536 \beta q^{61} + (320 \beta - 4800) q^{62} + 720 \beta q^{63} + (1056 \beta - 224) q^{64} + 960 q^{65} + (156 \beta + 156) q^{66} + 8006 q^{67} + (452 \beta - 3164) q^{68} + 480 \beta q^{69} + ( - 1920 \beta - 1920) q^{70} - 144 \beta q^{71} + ( - 540 \beta - 1980) q^{72} + 386 q^{73} + ( - 456 \beta + 6840) q^{74} - 2010 q^{75} + (268 \beta - 1876) q^{76} + 416 \beta q^{77} + (48 \beta - 720) q^{78} - 2848 \beta q^{79} + (1088 \beta + 6720) q^{80} - 891 q^{81} + ( - 994 \beta - 994) q^{82} - 2234 q^{83} + (1344 \beta + 2880) q^{84} + 1808 \beta q^{85} + (1882 \beta + 1882) q^{86} + 528 \beta q^{87} + ( - 312 \beta - 1144) q^{88} - 10046 q^{89} + (360 \beta - 5400) q^{90} - 1920 q^{91} + ( - 1120 \beta - 2400) q^{92} - 1920 \beta q^{93} + ( - 544 \beta + 8160) q^{94} + 1072 \beta q^{95} + ( - 480 \beta - 5856) q^{96} + 8738 q^{97} + (1439 \beta + 1439) q^{98} + 1170 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 12 q^{3} - 28 q^{4} - 12 q^{6} + 88 q^{8} - 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 12 q^{3} - 28 q^{4} - 12 q^{6} + 88 q^{8} - 90 q^{9} + 240 q^{10} - 52 q^{11} - 168 q^{12} - 480 q^{14} + 272 q^{16} + 452 q^{17} + 90 q^{18} + 268 q^{19} - 480 q^{20} + 52 q^{22} + 528 q^{24} - 670 q^{25} - 240 q^{26} - 1512 q^{27} + 960 q^{28} + 1440 q^{30} - 1952 q^{32} - 312 q^{33} - 452 q^{34} + 3840 q^{35} + 1260 q^{36} - 268 q^{38} - 2880 q^{40} + 1988 q^{41} - 2880 q^{42} - 3764 q^{43} + 728 q^{44} + 2400 q^{46} + 1632 q^{48} - 2878 q^{49} + 670 q^{50} + 2712 q^{51} + 480 q^{52} + 1512 q^{54} + 5760 q^{56} + 1608 q^{57} + 2640 q^{58} - 10036 q^{59} - 2880 q^{60} - 9600 q^{62} - 448 q^{64} + 1920 q^{65} + 312 q^{66} + 16012 q^{67} - 6328 q^{68} - 3840 q^{70} - 3960 q^{72} + 772 q^{73} + 13680 q^{74} - 4020 q^{75} - 3752 q^{76} - 1440 q^{78} + 13440 q^{80} - 1782 q^{81} - 1988 q^{82} - 4468 q^{83} + 5760 q^{84} + 3764 q^{86} - 2288 q^{88} - 20092 q^{89} - 10800 q^{90} - 3840 q^{91} - 4800 q^{92} + 16320 q^{94} - 11712 q^{96} + 17476 q^{97} + 2878 q^{98} + 2340 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.500000 + 1.93649i
0.500000 1.93649i
−1.00000 3.87298i 6.00000 −14.0000 + 7.74597i 30.9839i −6.00000 23.2379i 61.9677i 44.0000 + 46.4758i −45.0000 120.000 30.9839i
3.2 −1.00000 + 3.87298i 6.00000 −14.0000 7.74597i 30.9839i −6.00000 + 23.2379i 61.9677i 44.0000 46.4758i −45.0000 120.000 + 30.9839i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.5.d.b 2
3.b odd 2 1 72.5.b.b 2
4.b odd 2 1 32.5.d.b 2
5.b even 2 1 200.5.g.d 2
5.c odd 4 2 200.5.e.c 4
8.b even 2 1 32.5.d.b 2
8.d odd 2 1 inner 8.5.d.b 2
12.b even 2 1 288.5.b.b 2
16.e even 4 2 256.5.c.i 4
16.f odd 4 2 256.5.c.i 4
20.d odd 2 1 800.5.g.d 2
20.e even 4 2 800.5.e.c 4
24.f even 2 1 72.5.b.b 2
24.h odd 2 1 288.5.b.b 2
40.e odd 2 1 200.5.g.d 2
40.f even 2 1 800.5.g.d 2
40.i odd 4 2 800.5.e.c 4
40.k even 4 2 200.5.e.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.5.d.b 2 1.a even 1 1 trivial
8.5.d.b 2 8.d odd 2 1 inner
32.5.d.b 2 4.b odd 2 1
32.5.d.b 2 8.b even 2 1
72.5.b.b 2 3.b odd 2 1
72.5.b.b 2 24.f even 2 1
200.5.e.c 4 5.c odd 4 2
200.5.e.c 4 40.k even 4 2
200.5.g.d 2 5.b even 2 1
200.5.g.d 2 40.e odd 2 1
256.5.c.i 4 16.e even 4 2
256.5.c.i 4 16.f odd 4 2
288.5.b.b 2 12.b even 2 1
288.5.b.b 2 24.h odd 2 1
800.5.e.c 4 20.e even 4 2
800.5.e.c 4 40.i odd 4 2
800.5.g.d 2 20.d odd 2 1
800.5.g.d 2 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 6 \) acting on \(S_{5}^{\mathrm{new}}(8, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 16 \) Copy content Toggle raw display
$3$ \( (T - 6)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 960 \) Copy content Toggle raw display
$7$ \( T^{2} + 3840 \) Copy content Toggle raw display
$11$ \( (T + 26)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 960 \) Copy content Toggle raw display
$17$ \( (T - 226)^{2} \) Copy content Toggle raw display
$19$ \( (T - 134)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 96000 \) Copy content Toggle raw display
$29$ \( T^{2} + 116160 \) Copy content Toggle raw display
$31$ \( T^{2} + 1536000 \) Copy content Toggle raw display
$37$ \( T^{2} + 3119040 \) Copy content Toggle raw display
$41$ \( (T - 994)^{2} \) Copy content Toggle raw display
$43$ \( (T + 1882)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 4439040 \) Copy content Toggle raw display
$53$ \( T^{2} + 14523840 \) Copy content Toggle raw display
$59$ \( (T + 5018)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 4309440 \) Copy content Toggle raw display
$67$ \( (T - 8006)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 311040 \) Copy content Toggle raw display
$73$ \( (T - 386)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 121666560 \) Copy content Toggle raw display
$83$ \( (T + 2234)^{2} \) Copy content Toggle raw display
$89$ \( (T + 10046)^{2} \) Copy content Toggle raw display
$97$ \( (T - 8738)^{2} \) Copy content Toggle raw display
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