Properties

Label 574.2.c.a
Level $574$
Weight $2$
Character orbit 574.c
Analytic conductor $4.583$
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] = [574,2,Mod(491,574)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(574, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("574.491");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 574 = 2 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 574.c (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: \(4.58341307602\)
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: 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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + i q^{3} + q^{4} + q^{5} + i q^{6} + i q^{7} + q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + i q^{3} + q^{4} + q^{5} + i q^{6} + i q^{7} + q^{8} + 2 q^{9} + q^{10} + i q^{12} - 4 i q^{13} + i q^{14} + i q^{15} + q^{16} + 3 i q^{17} + 2 q^{18} + 4 i q^{19} + q^{20} - q^{21} + 4 q^{23} + i q^{24} - 4 q^{25} - 4 i q^{26} + 5 i q^{27} + i q^{28} - i q^{29} + i q^{30} - 3 q^{31} + q^{32} + 3 i q^{34} + i q^{35} + 2 q^{36} - 2 q^{37} + 4 i q^{38} + 4 q^{39} + q^{40} + ( - 5 i - 4) q^{41} - q^{42} + 9 q^{43} + 2 q^{45} + 4 q^{46} - 2 i q^{47} + i q^{48} - q^{49} - 4 q^{50} - 3 q^{51} - 4 i q^{52} - 9 i q^{53} + 5 i q^{54} + i q^{56} - 4 q^{57} - i q^{58} - 10 q^{59} + i q^{60} + 7 q^{61} - 3 q^{62} + 2 i q^{63} + q^{64} - 4 i q^{65} - 2 i q^{67} + 3 i q^{68} + 4 i q^{69} + i q^{70} + 5 i q^{71} + 2 q^{72} - 16 q^{73} - 2 q^{74} - 4 i q^{75} + 4 i q^{76} + 4 q^{78} - 11 i q^{79} + q^{80} + q^{81} + ( - 5 i - 4) q^{82} - 6 q^{83} - q^{84} + 3 i q^{85} + 9 q^{86} + q^{87} - i q^{89} + 2 q^{90} + 4 q^{91} + 4 q^{92} - 3 i q^{93} - 2 i q^{94} + 4 i q^{95} + i q^{96} - 7 i q^{97} - q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + 4 q^{9} + 2 q^{10} + 2 q^{16} + 4 q^{18} + 2 q^{20} - 2 q^{21} + 8 q^{23} - 8 q^{25} - 6 q^{31} + 2 q^{32} + 4 q^{36} - 4 q^{37} + 8 q^{39} + 2 q^{40} - 8 q^{41} - 2 q^{42} + 18 q^{43} + 4 q^{45} + 8 q^{46} - 2 q^{49} - 8 q^{50} - 6 q^{51} - 8 q^{57} - 20 q^{59} + 14 q^{61} - 6 q^{62} + 2 q^{64} + 4 q^{72} - 32 q^{73} - 4 q^{74} + 8 q^{78} + 2 q^{80} + 2 q^{81} - 8 q^{82} - 12 q^{83} - 2 q^{84} + 18 q^{86} + 2 q^{87} + 4 q^{90} + 8 q^{91} + 8 q^{92} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/574\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(493\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
491.1
1.00000i
1.00000i
1.00000 1.00000i 1.00000 1.00000 1.00000i 1.00000i 1.00000 2.00000 1.00000
491.2 1.00000 1.00000i 1.00000 1.00000 1.00000i 1.00000i 1.00000 2.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 574.2.c.a 2
41.b even 2 1 inner 574.2.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.c.a 2 1.a even 1 1 trivial
574.2.c.a 2 41.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 1 \) Copy content Toggle raw display
$31$ \( (T + 3)^{2} \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 8T + 41 \) Copy content Toggle raw display
$43$ \( (T - 9)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 81 \) Copy content Toggle raw display
$59$ \( (T + 10)^{2} \) Copy content Toggle raw display
$61$ \( (T - 7)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 25 \) Copy content Toggle raw display
$73$ \( (T + 16)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 121 \) Copy content Toggle raw display
$83$ \( (T + 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 1 \) Copy content Toggle raw display
$97$ \( T^{2} + 49 \) Copy content Toggle raw display
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