Properties

Label 1734.2.b.c
Level $1734$
Weight $2$
Character orbit 1734.b
Analytic conductor $13.846$
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] = [1734,2,Mod(577,1734)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1734, 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("1734.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1734 = 2 \cdot 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1734.b (of order \(2\), degree \(1\), 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: \(13.8460597105\)
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.

\(f(q)\) \(=\) \( q - q^{2} - i q^{3} + q^{4} + 3 i q^{5} + i q^{6} - 4 i q^{7} - q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - i q^{3} + q^{4} + 3 i q^{5} + i q^{6} - 4 i q^{7} - q^{8} - q^{9} - 3 i q^{10} + 3 i q^{11} - i q^{12} + 2 q^{13} + 4 i q^{14} + 3 q^{15} + q^{16} + q^{18} - 8 q^{19} + 3 i q^{20} - 4 q^{21} - 3 i q^{22} + 6 i q^{23} + i q^{24} - 4 q^{25} - 2 q^{26} + i q^{27} - 4 i q^{28} - 3 i q^{29} - 3 q^{30} + 7 i q^{31} - q^{32} + 3 q^{33} + 12 q^{35} - q^{36} - 8 i q^{37} + 8 q^{38} - 2 i q^{39} - 3 i q^{40} + 6 i q^{41} + 4 q^{42} + 4 q^{43} + 3 i q^{44} - 3 i q^{45} - 6 i q^{46} - 6 q^{47} - i q^{48} - 9 q^{49} + 4 q^{50} + 2 q^{52} + 9 q^{53} - i q^{54} - 9 q^{55} + 4 i q^{56} + 8 i q^{57} + 3 i q^{58} - 15 q^{59} + 3 q^{60} + 14 i q^{61} - 7 i q^{62} + 4 i q^{63} + q^{64} + 6 i q^{65} - 3 q^{66} + 2 q^{67} + 6 q^{69} - 12 q^{70} + q^{72} + 7 i q^{73} + 8 i q^{74} + 4 i q^{75} - 8 q^{76} + 12 q^{77} + 2 i q^{78} - i q^{79} + 3 i q^{80} + q^{81} - 6 i q^{82} - 12 q^{83} - 4 q^{84} - 4 q^{86} - 3 q^{87} - 3 i q^{88} + 3 i q^{90} - 8 i q^{91} + 6 i q^{92} + 7 q^{93} + 6 q^{94} - 24 i q^{95} + i q^{96} + 13 i q^{97} + 9 q^{98} - 3 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} + 4 q^{13} + 6 q^{15} + 2 q^{16} + 2 q^{18} - 16 q^{19} - 8 q^{21} - 8 q^{25} - 4 q^{26} - 6 q^{30} - 2 q^{32} + 6 q^{33} + 24 q^{35} - 2 q^{36} + 16 q^{38} + 8 q^{42} + 8 q^{43} - 12 q^{47} - 18 q^{49} + 8 q^{50} + 4 q^{52} + 18 q^{53} - 18 q^{55} - 30 q^{59} + 6 q^{60} + 2 q^{64} - 6 q^{66} + 4 q^{67} + 12 q^{69} - 24 q^{70} + 2 q^{72} - 16 q^{76} + 24 q^{77} + 2 q^{81} - 24 q^{83} - 8 q^{84} - 8 q^{86} - 6 q^{87} + 14 q^{93} + 12 q^{94} + 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1734\mathbb{Z}\right)^\times\).

\(n\) \(1157\) \(1159\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
1.00000i
1.00000i
−1.00000 1.00000i 1.00000 3.00000i 1.00000i 4.00000i −1.00000 −1.00000 3.00000i
577.2 −1.00000 1.00000i 1.00000 3.00000i 1.00000i 4.00000i −1.00000 −1.00000 3.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1734.2.b.c 2
17.b even 2 1 inner 1734.2.b.c 2
17.c even 4 1 1734.2.a.k 1
17.c even 4 1 1734.2.a.l yes 1
17.d even 8 4 1734.2.f.c 4
51.f odd 4 1 5202.2.a.b 1
51.f odd 4 1 5202.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1734.2.a.k 1 17.c even 4 1
1734.2.a.l yes 1 17.c even 4 1
1734.2.b.c 2 1.a even 1 1 trivial
1734.2.b.c 2 17.b even 2 1 inner
1734.2.f.c 4 17.d even 8 4
5202.2.a.b 1 51.f odd 4 1
5202.2.a.e 1 51.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1734, [\chi])\):

\( T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{2} + 16 \) 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
$5$ \( T^{2} + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} + 9 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 9 \) Copy content Toggle raw display
$31$ \( T^{2} + 49 \) Copy content Toggle raw display
$37$ \( T^{2} + 64 \) Copy content Toggle raw display
$41$ \( T^{2} + 36 \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( (T - 9)^{2} \) Copy content Toggle raw display
$59$ \( (T + 15)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 196 \) Copy content Toggle raw display
$67$ \( (T - 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 49 \) Copy content Toggle raw display
$79$ \( T^{2} + 1 \) Copy content Toggle raw display
$83$ \( (T + 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 169 \) Copy content Toggle raw display
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