Properties

Label 104.2.m.a
Level $104$
Weight $2$
Character orbit 104.m
Analytic conductor $0.830$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [104,2,Mod(83,104)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(104, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("104.83");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 104 = 2^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 104.m (of order \(4\), degree \(2\), 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: \(0.830444181021\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{26})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{2} - q^{3} + 2 \beta_{2} q^{4} - \beta_{3} q^{5} + (\beta_{2} + 1) q^{6} - \beta_1 q^{7} + ( - 2 \beta_{2} + 2) q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{2} - q^{3} + 2 \beta_{2} q^{4} - \beta_{3} q^{5} + (\beta_{2} + 1) q^{6} - \beta_1 q^{7} + ( - 2 \beta_{2} + 2) q^{8} - 2 q^{9} + (\beta_{3} - \beta_1) q^{10} + ( - \beta_{2} + 1) q^{11} - 2 \beta_{2} q^{12} - \beta_{3} q^{13} + (\beta_{3} + \beta_1) q^{14} + \beta_{3} q^{15} - 4 q^{16} + 3 \beta_{2} q^{17} + (2 \beta_{2} + 2) q^{18} + (2 \beta_{2} + 2) q^{19} + 2 \beta_1 q^{20} + \beta_1 q^{21} - 2 q^{22} + ( - \beta_{3} + \beta_1) q^{23} + (2 \beta_{2} - 2) q^{24} - 8 \beta_{2} q^{25} + (\beta_{3} - \beta_1) q^{26} + 5 q^{27} - 2 \beta_{3} q^{28} + (\beta_{3} + \beta_1) q^{29} + ( - \beta_{3} + \beta_1) q^{30} + 2 \beta_{3} q^{31} + (4 \beta_{2} + 4) q^{32} + (\beta_{2} - 1) q^{33} + ( - 3 \beta_{2} + 3) q^{34} - 13 q^{35} - 4 \beta_{2} q^{36} - \beta_1 q^{37} - 4 \beta_{2} q^{38} + \beta_{3} q^{39} + ( - 2 \beta_{3} - 2 \beta_1) q^{40} + (6 \beta_{2} + 6) q^{41} + ( - \beta_{3} - \beta_1) q^{42} + \beta_{2} q^{43} + (2 \beta_{2} + 2) q^{44} + 2 \beta_{3} q^{45} - 2 \beta_1 q^{46} - \beta_1 q^{47} + 4 q^{48} + 6 \beta_{2} q^{49} + (8 \beta_{2} - 8) q^{50} - 3 \beta_{2} q^{51} + 2 \beta_1 q^{52} + ( - \beta_{3} - \beta_1) q^{53} + ( - 5 \beta_{2} - 5) q^{54} + ( - \beta_{3} - \beta_1) q^{55} + (2 \beta_{3} - 2 \beta_1) q^{56} + ( - 2 \beta_{2} - 2) q^{57} - 2 \beta_{3} q^{58} + ( - 8 \beta_{2} + 8) q^{59} - 2 \beta_1 q^{60} + ( - 2 \beta_{3} + 2 \beta_1) q^{62} + 2 \beta_1 q^{63} - 8 \beta_{2} q^{64} - 13 \beta_{2} q^{65} + 2 q^{66} + (3 \beta_{2} + 3) q^{67} - 6 q^{68} + (\beta_{3} - \beta_1) q^{69} + (13 \beta_{2} + 13) q^{70} - 3 \beta_{3} q^{71} + (4 \beta_{2} - 4) q^{72} + (6 \beta_{2} - 6) q^{73} + (\beta_{3} + \beta_1) q^{74} + 8 \beta_{2} q^{75} + (4 \beta_{2} - 4) q^{76} + (\beta_{3} - \beta_1) q^{77} + ( - \beta_{3} + \beta_1) q^{78} + (\beta_{3} + \beta_1) q^{79} + 4 \beta_{3} q^{80} + q^{81} - 12 \beta_{2} q^{82} + (5 \beta_{2} + 5) q^{83} + 2 \beta_{3} q^{84} + 3 \beta_1 q^{85} + ( - \beta_{2} + 1) q^{86} + ( - \beta_{3} - \beta_1) q^{87} - 4 \beta_{2} q^{88} + (2 \beta_{2} - 2) q^{89} + ( - 2 \beta_{3} + 2 \beta_1) q^{90} - 13 q^{91} + (2 \beta_{3} + 2 \beta_1) q^{92} - 2 \beta_{3} q^{93} + (\beta_{3} + \beta_1) q^{94} + ( - 2 \beta_{3} + 2 \beta_1) q^{95} + ( - 4 \beta_{2} - 4) q^{96} + ( - 7 \beta_{2} - 7) q^{97} + ( - 6 \beta_{2} + 6) q^{98} + (2 \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{6} + 8 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{6} + 8 q^{8} - 8 q^{9} + 4 q^{11} - 16 q^{16} + 8 q^{18} + 8 q^{19} - 8 q^{22} - 8 q^{24} + 20 q^{27} + 16 q^{32} - 4 q^{33} + 12 q^{34} - 52 q^{35} + 24 q^{41} + 8 q^{44} + 16 q^{48} - 32 q^{50} - 20 q^{54} - 8 q^{57} + 32 q^{59} + 8 q^{66} + 12 q^{67} - 24 q^{68} + 52 q^{70} - 16 q^{72} - 24 q^{73} - 16 q^{76} + 4 q^{81} + 20 q^{83} + 4 q^{86} - 8 q^{89} - 52 q^{91} - 16 q^{96} - 28 q^{97} + 24 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 13\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 13\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/104\mathbb{Z}\right)^\times\).

\(n\) \(41\) \(53\) \(79\)
\(\chi(n)\) \(\beta_{2}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1
−2.54951 + 2.54951i
2.54951 2.54951i
−2.54951 2.54951i
2.54951 + 2.54951i
−1.00000 + 1.00000i −1.00000 2.00000i −2.54951 2.54951i 1.00000 1.00000i 2.54951 2.54951i 2.00000 + 2.00000i −2.00000 5.09902
83.2 −1.00000 + 1.00000i −1.00000 2.00000i 2.54951 + 2.54951i 1.00000 1.00000i −2.54951 + 2.54951i 2.00000 + 2.00000i −2.00000 −5.09902
99.1 −1.00000 1.00000i −1.00000 2.00000i −2.54951 + 2.54951i 1.00000 + 1.00000i 2.54951 + 2.54951i 2.00000 2.00000i −2.00000 5.09902
99.2 −1.00000 1.00000i −1.00000 2.00000i 2.54951 2.54951i 1.00000 + 1.00000i −2.54951 2.54951i 2.00000 2.00000i −2.00000 −5.09902
\(n\): 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
13.d odd 4 1 inner
104.m even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 104.2.m.a 4
3.b odd 2 1 936.2.w.g 4
4.b odd 2 1 416.2.u.a 4
8.b even 2 1 416.2.u.a 4
8.d odd 2 1 inner 104.2.m.a 4
13.d odd 4 1 inner 104.2.m.a 4
24.f even 2 1 936.2.w.g 4
39.f even 4 1 936.2.w.g 4
52.f even 4 1 416.2.u.a 4
104.j odd 4 1 416.2.u.a 4
104.m even 4 1 inner 104.2.m.a 4
312.w odd 4 1 936.2.w.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.m.a 4 1.a even 1 1 trivial
104.2.m.a 4 8.d odd 2 1 inner
104.2.m.a 4 13.d odd 4 1 inner
104.2.m.a 4 104.m even 4 1 inner
416.2.u.a 4 4.b odd 2 1
416.2.u.a 4 8.b even 2 1
416.2.u.a 4 52.f even 4 1
416.2.u.a 4 104.j odd 4 1
936.2.w.g 4 3.b odd 2 1
936.2.w.g 4 24.f even 2 1
936.2.w.g 4 39.f even 4 1
936.2.w.g 4 312.w odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(104, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 169 \) Copy content Toggle raw display
$7$ \( T^{4} + 169 \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 169 \) Copy content Toggle raw display
$17$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T + 8)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 26)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 26)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2704 \) Copy content Toggle raw display
$37$ \( T^{4} + 169 \) Copy content Toggle raw display
$41$ \( (T^{2} - 12 T + 72)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 169 \) Copy content Toggle raw display
$53$ \( (T^{2} + 26)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 16 T + 128)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 13689 \) Copy content Toggle raw display
$73$ \( (T^{2} + 12 T + 72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 26)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 10 T + 50)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 14 T + 98)^{2} \) Copy content Toggle raw display
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