Properties

Label 705.2.c.a
Level $705$
Weight $2$
Character orbit 705.c
Analytic conductor $5.629$
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] = [705,2,Mod(424,705)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(705, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("705.424");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 705 = 3 \cdot 5 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 705.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: \(5.62945334250\)
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.

\(f(q)\) \(=\) \( q + 2 i q^{2} - i q^{3} - 2 q^{4} + (i + 2) q^{5} + 2 q^{6} + 4 i q^{7} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{2} - i q^{3} - 2 q^{4} + (i + 2) q^{5} + 2 q^{6} + 4 i q^{7} - q^{9} + (4 i - 2) q^{10} + 2 i q^{12} - 5 i q^{13} - 8 q^{14} + ( - 2 i + 1) q^{15} - 4 q^{16} + 6 i q^{17} - 2 i q^{18} + 2 q^{19} + ( - 2 i - 4) q^{20} + 4 q^{21} - i q^{23} + (4 i + 3) q^{25} + 10 q^{26} + i q^{27} - 8 i q^{28} + 6 q^{29} + (2 i + 4) q^{30} - 8 q^{31} - 8 i q^{32} - 12 q^{34} + (8 i - 4) q^{35} + 2 q^{36} + 2 i q^{37} + 4 i q^{38} - 5 q^{39} - 2 q^{41} + 8 i q^{42} + 5 i q^{43} + ( - i - 2) q^{45} + 2 q^{46} + i q^{47} + 4 i q^{48} - 9 q^{49} + (6 i - 8) q^{50} + 6 q^{51} + 10 i q^{52} - 12 i q^{53} - 2 q^{54} - 2 i q^{57} + 12 i q^{58} + 3 q^{59} + (4 i - 2) q^{60} - q^{61} - 16 i q^{62} - 4 i q^{63} + 8 q^{64} + ( - 10 i + 5) q^{65} + 8 i q^{67} - 12 i q^{68} - q^{69} + ( - 8 i - 16) q^{70} + q^{71} + 13 i q^{73} - 4 q^{74} + ( - 3 i + 4) q^{75} - 4 q^{76} - 10 i q^{78} + q^{79} + ( - 4 i - 8) q^{80} + q^{81} - 4 i q^{82} - 12 i q^{83} - 8 q^{84} + (12 i - 6) q^{85} - 10 q^{86} - 6 i q^{87} + 15 q^{89} + ( - 4 i + 2) q^{90} + 20 q^{91} + 2 i q^{92} + 8 i q^{93} - 2 q^{94} + (2 i + 4) q^{95} - 8 q^{96} - 8 i q^{97} - 18 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} + 4 q^{5} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} + 4 q^{5} + 4 q^{6} - 2 q^{9} - 4 q^{10} - 16 q^{14} + 2 q^{15} - 8 q^{16} + 4 q^{19} - 8 q^{20} + 8 q^{21} + 6 q^{25} + 20 q^{26} + 12 q^{29} + 8 q^{30} - 16 q^{31} - 24 q^{34} - 8 q^{35} + 4 q^{36} - 10 q^{39} - 4 q^{41} - 4 q^{45} + 4 q^{46} - 18 q^{49} - 16 q^{50} + 12 q^{51} - 4 q^{54} + 6 q^{59} - 4 q^{60} - 2 q^{61} + 16 q^{64} + 10 q^{65} - 2 q^{69} - 32 q^{70} + 2 q^{71} - 8 q^{74} + 8 q^{75} - 8 q^{76} + 2 q^{79} - 16 q^{80} + 2 q^{81} - 16 q^{84} - 12 q^{85} - 20 q^{86} + 30 q^{89} + 4 q^{90} + 40 q^{91} - 4 q^{94} + 8 q^{95} - 16 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/705\mathbb{Z}\right)^\times\).

\(n\) \(142\) \(236\) \(616\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
424.1
1.00000i
1.00000i
2.00000i 1.00000i −2.00000 2.00000 1.00000i 2.00000 4.00000i 0 −1.00000 −2.00000 4.00000i
424.2 2.00000i 1.00000i −2.00000 2.00000 + 1.00000i 2.00000 4.00000i 0 −1.00000 −2.00000 + 4.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 705.2.c.a 2
5.b even 2 1 inner 705.2.c.a 2
5.c odd 4 1 3525.2.a.a 1
5.c odd 4 1 3525.2.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.c.a 2 1.a even 1 1 trivial
705.2.c.a 2 5.b even 2 1 inner
3525.2.a.a 1 5.c odd 4 1
3525.2.a.o 1 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(705, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 25 \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 25 \) Copy content Toggle raw display
$47$ \( T^{2} + 1 \) Copy content Toggle raw display
$53$ \( T^{2} + 144 \) Copy content Toggle raw display
$59$ \( (T - 3)^{2} \) Copy content Toggle raw display
$61$ \( (T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 64 \) Copy content Toggle raw display
$71$ \( (T - 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 169 \) Copy content Toggle raw display
$79$ \( (T - 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T - 15)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 64 \) Copy content Toggle raw display
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