Properties

Label 800.2.ba.a
Level $800$
Weight $2$
Character orbit 800.ba
Analytic conductor $6.388$
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] = [800,2,Mod(149,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.ba (of order \(8\), degree \(4\), 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: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) 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 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{8}^{3} - \zeta_{8}) q^{2} + (\zeta_{8}^{3} - 1) q^{3} + 2 q^{4} + ( - \zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8} + 1) q^{6} + ( - \zeta_{8}^{2} + 1) q^{7} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{8} + (\zeta_{8}^{3} - \zeta_{8}^{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{8}^{3} - \zeta_{8}) q^{2} + (\zeta_{8}^{3} - 1) q^{3} + 2 q^{4} + ( - \zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8} + 1) q^{6} + ( - \zeta_{8}^{2} + 1) q^{7} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{8} + (\zeta_{8}^{3} - \zeta_{8}^{2} + 1) q^{9} + ( - \zeta_{8}^{3} - \zeta_{8}^{2} + 2 \zeta_{8} - 2) q^{11} + (2 \zeta_{8}^{3} - 2) q^{12} + (\zeta_{8}^{2} + \zeta_{8}) q^{13} + 2 \zeta_{8}^{3} q^{14} + 4 q^{16} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{17} + (2 \zeta_{8}^{3} - \zeta_{8}^{2} + 1) q^{18} + ( - 2 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + 3 \zeta_{8} + 2) q^{19} + (\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} - 1) q^{21} + ( - \zeta_{8}^{3} - \zeta_{8}^{2} + 3 \zeta_{8} - 3) q^{22} + (3 \zeta_{8}^{2} - 4 \zeta_{8} + 3) q^{23} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2 \zeta_{8} + 2) q^{24} + ( - \zeta_{8}^{3} - \zeta_{8}^{2} - \zeta_{8} - 1) q^{26} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + \zeta_{8} - 1) q^{27} + ( - 2 \zeta_{8}^{2} + 2) q^{28} + (\zeta_{8}^{3} - 2 \zeta_{8}^{2} - 2 \zeta_{8} + 1) q^{29} - 4 q^{31} + (4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{32} + ( - \zeta_{8}^{3} + 2 \zeta_{8}^{2} - \zeta_{8}) q^{33} + 4 q^{34} + (2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2) q^{36} + ( - \zeta_{8}^{3} - \zeta_{8}^{2}) q^{37} + (5 \zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8} - 5) q^{38} + ( - \zeta_{8}^{2} - 2 \zeta_{8} - 1) q^{39} + (4 \zeta_{8}^{3} + 3 \zeta_{8}^{2} - 3) q^{41} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2}) q^{42} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8}^{2} + 3 \zeta_{8} + 3) q^{43} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 4 \zeta_{8} - 4) q^{44} + (4 \zeta_{8}^{2} - 6 \zeta_{8} + 4) q^{46} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8} + 6) q^{47} + (4 \zeta_{8}^{3} - 4) q^{48} + 5 \zeta_{8}^{2} q^{49} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2 \zeta_{8} + 2) q^{51} + (2 \zeta_{8}^{2} + 2 \zeta_{8}) q^{52} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + 4 \zeta_{8} + 4) q^{53} + (2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 4 \zeta_{8} - 4) q^{54} + 4 \zeta_{8}^{3} q^{56} + (4 \zeta_{8}^{3} + 5 \zeta_{8}^{2} - 5) q^{57} + (3 \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} + 3) q^{58} + (\zeta_{8}^{3} - \zeta_{8}^{2} + 4 \zeta_{8} + 4) q^{59} + (\zeta_{8}^{3} + 1) q^{61} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8}) q^{62} + (\zeta_{8}^{3} - 2 \zeta_{8}^{2} + \zeta_{8}) q^{63} + 8 q^{64} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2 \zeta_{8}) q^{66} + ( - 3 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2 \zeta_{8} + 3) q^{67} + (4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{68} + (3 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + \zeta_{8} + 1) q^{69} + ( - 3 \zeta_{8}^{2} + 4 \zeta_{8} - 3) q^{71} + (4 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2) q^{72} + (7 \zeta_{8}^{2} + 7) q^{73} + (\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} - 1) q^{74} + ( - 4 \zeta_{8}^{3} - 6 \zeta_{8}^{2} + 6 \zeta_{8} + 4) q^{76} + ( - 3 \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} - 3) q^{77} + (2 \zeta_{8}^{2} + 2 \zeta_{8} + 2) q^{78} + 6 \zeta_{8}^{2} q^{79} + (5 \zeta_{8}^{3} + 3 \zeta_{8}^{2} + 5 \zeta_{8}) q^{81} + ( - 6 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 4) q^{82} + ( - \zeta_{8}^{3} + 4 \zeta_{8}^{2} + 4 \zeta_{8} - 1) q^{83} + (2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 2 \zeta_{8} - 2) q^{84} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} - 7 \zeta_{8} - 7) q^{86} + (\zeta_{8}^{2} + 4 \zeta_{8} + 1) q^{87} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 6 \zeta_{8} - 6) q^{88} + ( - 3 \zeta_{8}^{2} + 8 \zeta_{8} - 3) q^{89} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} + 1) q^{91} + (6 \zeta_{8}^{2} - 8 \zeta_{8} + 6) q^{92} + ( - 4 \zeta_{8}^{3} + 4) q^{93} + (6 \zeta_{8}^{3} - 6 \zeta_{8} - 8) q^{94} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 4 \zeta_{8} + 4) q^{96} + ( - 6 \zeta_{8}^{3} + 10 \zeta_{8}^{2} - 6 \zeta_{8}) q^{97} + ( - 5 \zeta_{8}^{3} - 5 \zeta_{8}) q^{98} + ( - 5 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 2 \zeta_{8} - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 8 q^{4} + 4 q^{6} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 8 q^{4} + 4 q^{6} + 4 q^{7} + 4 q^{9} - 8 q^{11} - 8 q^{12} + 16 q^{16} + 4 q^{18} + 8 q^{19} - 4 q^{21} - 12 q^{22} + 12 q^{23} + 8 q^{24} - 4 q^{26} - 4 q^{27} + 8 q^{28} + 4 q^{29} - 16 q^{31} + 16 q^{34} + 8 q^{36} - 20 q^{38} - 4 q^{39} - 12 q^{41} + 12 q^{43} - 16 q^{44} + 16 q^{46} + 24 q^{47} - 16 q^{48} + 8 q^{51} + 16 q^{53} - 16 q^{54} - 20 q^{57} + 12 q^{58} + 16 q^{59} + 4 q^{61} + 32 q^{64} + 12 q^{67} + 4 q^{69} - 12 q^{71} + 8 q^{72} + 28 q^{73} - 4 q^{74} + 16 q^{76} - 12 q^{77} + 8 q^{78} + 16 q^{82} - 4 q^{83} - 8 q^{84} - 28 q^{86} + 4 q^{87} - 24 q^{88} - 12 q^{89} + 4 q^{91} + 24 q^{92} + 16 q^{93} - 32 q^{94} + 16 q^{96} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(\zeta_{8}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
1.41421 −0.292893 0.707107i 2.00000 0 −0.414214 1.00000i 1.00000 1.00000i 2.82843 1.70711 1.70711i 0
349.1 1.41421 −0.292893 + 0.707107i 2.00000 0 −0.414214 + 1.00000i 1.00000 + 1.00000i 2.82843 1.70711 + 1.70711i 0
549.1 −1.41421 −1.70711 + 0.707107i 2.00000 0 2.41421 1.00000i 1.00000 1.00000i −2.82843 0.292893 0.292893i 0
749.1 −1.41421 −1.70711 0.707107i 2.00000 0 2.41421 + 1.00000i 1.00000 + 1.00000i −2.82843 0.292893 + 0.292893i 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
160.z even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.ba.a 4
5.b even 2 1 800.2.ba.b 4
5.c odd 4 1 32.2.g.a 4
5.c odd 4 1 800.2.y.a 4
15.e even 4 1 288.2.v.a 4
20.e even 4 1 128.2.g.a 4
32.g even 8 1 800.2.ba.b 4
40.i odd 4 1 256.2.g.b 4
40.k even 4 1 256.2.g.a 4
60.l odd 4 1 1152.2.v.a 4
80.i odd 4 1 512.2.g.d 4
80.j even 4 1 512.2.g.c 4
80.s even 4 1 512.2.g.b 4
80.t odd 4 1 512.2.g.a 4
160.u even 8 1 128.2.g.a 4
160.u even 8 1 256.2.g.a 4
160.v odd 8 1 512.2.g.a 4
160.v odd 8 1 512.2.g.d 4
160.v odd 8 1 800.2.y.a 4
160.z even 8 1 inner 800.2.ba.a 4
160.ba even 8 1 512.2.g.b 4
160.ba even 8 1 512.2.g.c 4
160.bb odd 8 1 32.2.g.a 4
160.bb odd 8 1 256.2.g.b 4
320.bc odd 16 2 4096.2.a.e 4
320.bd even 16 2 4096.2.a.f 4
480.bq odd 8 1 1152.2.v.a 4
480.cb even 8 1 288.2.v.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.g.a 4 5.c odd 4 1
32.2.g.a 4 160.bb odd 8 1
128.2.g.a 4 20.e even 4 1
128.2.g.a 4 160.u even 8 1
256.2.g.a 4 40.k even 4 1
256.2.g.a 4 160.u even 8 1
256.2.g.b 4 40.i odd 4 1
256.2.g.b 4 160.bb odd 8 1
288.2.v.a 4 15.e even 4 1
288.2.v.a 4 480.cb even 8 1
512.2.g.a 4 80.t odd 4 1
512.2.g.a 4 160.v odd 8 1
512.2.g.b 4 80.s even 4 1
512.2.g.b 4 160.ba even 8 1
512.2.g.c 4 80.j even 4 1
512.2.g.c 4 160.ba even 8 1
512.2.g.d 4 80.i odd 4 1
512.2.g.d 4 160.v odd 8 1
800.2.y.a 4 5.c odd 4 1
800.2.y.a 4 160.v odd 8 1
800.2.ba.a 4 1.a even 1 1 trivial
800.2.ba.a 4 160.z even 8 1 inner
800.2.ba.b 4 5.b even 2 1
800.2.ba.b 4 32.g even 8 1
1152.2.v.a 4 60.l odd 4 1
1152.2.v.a 4 480.bq odd 8 1
4096.2.a.e 4 320.bc odd 16 2
4096.2.a.f 4 320.bd even 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 4T_{3}^{3} + 6T_{3}^{2} + 4T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 8 T^{3} + 18 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{2} + 4 T + 2 \) Copy content Toggle raw display
$17$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 8 T^{3} + 18 T^{2} - 68 T + 578 \) Copy content Toggle raw display
$23$ \( T^{4} - 12 T^{3} + 72 T^{2} - 24 T + 4 \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + 6 T^{2} - 28 T + 98 \) Copy content Toggle raw display
$31$ \( (T + 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{2} + 4 T + 2 \) Copy content Toggle raw display
$41$ \( T^{4} + 12 T^{3} + 72 T^{2} + 24 T + 4 \) Copy content Toggle raw display
$43$ \( T^{4} - 12 T^{3} + 38 T^{2} + \cdots + 1922 \) Copy content Toggle raw display
$47$ \( (T^{2} - 12 T + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 16 T^{3} + 82 T^{2} - 84 T + 98 \) Copy content Toggle raw display
$59$ \( T^{4} - 16 T^{3} + 114 T^{2} + \cdots + 1058 \) Copy content Toggle raw display
$61$ \( T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$67$ \( T^{4} - 12 T^{3} + 86 T^{2} + \cdots + 578 \) Copy content Toggle raw display
$71$ \( T^{4} + 12 T^{3} + 72 T^{2} + 24 T + 4 \) Copy content Toggle raw display
$73$ \( (T^{2} - 14 T + 98)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 4 T^{3} + 22 T^{2} + \cdots + 1058 \) Copy content Toggle raw display
$89$ \( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$97$ \( T^{4} + 344T^{2} + 784 \) Copy content Toggle raw display
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