Properties

Label 1407.2.i.e
Level $1407$
Weight $2$
Character orbit 1407.i
Analytic conductor $11.235$
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] = [1407,2,Mod(403,1407)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1407, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1407.403");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1407 = 3 \cdot 7 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1407.i (of order \(3\), 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: \(11.2349515644\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{2} + 1) q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{5} - q^{6} + ( - \beta_{3} - 2 \beta_1 + 1) q^{7} - 3 q^{8} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{2} + 1) q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{5} - q^{6} + ( - \beta_{3} - 2 \beta_1 + 1) q^{7} - 3 q^{8} + \beta_{2} q^{9} + ( - \beta_{2} + \beta_1 - 1) q^{10} - 3 \beta_1 q^{11} + \beta_{2} q^{12} + \beta_{3} q^{13} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{14} + ( - \beta_{3} - 1) q^{15} - \beta_{2} q^{16} + ( - 2 \beta_{2} - 3 \beta_1 - 2) q^{17} + ( - \beta_{2} - 1) q^{18} + (\beta_{3} - 4 \beta_{2} + \beta_1) q^{19} + ( - \beta_{3} - 1) q^{20} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{21} - 3 \beta_{3} q^{22} + ( - 6 \beta_{3} + \beta_{2} - 6 \beta_1) q^{23} + ( - 3 \beta_{2} - 3) q^{24} + (2 \beta_{2} + 2 \beta_1 + 2) q^{25} + ( - \beta_{3} - \beta_1) q^{26} - q^{27} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{28} + (2 \beta_{3} + 6) q^{29} + (\beta_{3} - \beta_{2} + \beta_1) q^{30} + (\beta_{2} + \beta_1 + 1) q^{31} + ( - 5 \beta_{2} - 5) q^{32} + ( - 3 \beta_{3} - 3 \beta_1) q^{33} + ( - 3 \beta_{3} + 2) q^{34} + ( - 2 \beta_{3} - \beta_{2} - 4) q^{35} - q^{36} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{37} + (4 \beta_{2} - \beta_1 + 4) q^{38} - \beta_1 q^{39} + (3 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{40} + 10 q^{41} + (\beta_{3} + 2 \beta_1 - 1) q^{42} + (2 \beta_{3} - 8) q^{43} + ( - 3 \beta_{3} - 3 \beta_1) q^{44} + ( - \beta_{2} + \beta_1 - 1) q^{45} + ( - \beta_{2} + 6 \beta_1 - 1) q^{46} - \beta_{2} q^{47} + q^{48} + ( - 2 \beta_{3} - 4 \beta_1 - 5) q^{49} + (2 \beta_{3} - 2) q^{50} + ( - 3 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{51} - \beta_1 q^{52} + ( - 7 \beta_{2} + \beta_1 - 7) q^{53} - \beta_{2} q^{54} + ( - 3 \beta_{3} - 6) q^{55} + (3 \beta_{3} + 6 \beta_1 - 3) q^{56} + (\beta_{3} + 4) q^{57} + ( - 2 \beta_{3} + 6 \beta_{2} - 2 \beta_1) q^{58} + ( - 6 \beta_{2} + 2 \beta_1 - 6) q^{59} + ( - \beta_{2} + \beta_1 - 1) q^{60} + (4 \beta_{3} - 8 \beta_{2} + 4 \beta_1) q^{61} + (\beta_{3} - 1) q^{62} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{63} + 7 q^{64} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{65} + 3 \beta_1 q^{66} + (\beta_{2} + 1) q^{67} + ( - 3 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{68} + ( - 6 \beta_{3} - 1) q^{69} + (2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 1) q^{70} - q^{71} - 3 \beta_{2} q^{72} + (\beta_{2} - 4 \beta_1 + 1) q^{73} + (4 \beta_{2} - 2 \beta_1 + 4) q^{74} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{75} + (\beta_{3} + 4) q^{76} + (6 \beta_{2} - 3 \beta_1 - 6) q^{77} - \beta_{3} q^{78} + (5 \beta_{3} - 9 \beta_{2} + 5 \beta_1) q^{79} + (\beta_{2} - \beta_1 + 1) q^{80} + ( - \beta_{2} - 1) q^{81} + 10 \beta_{2} q^{82} + (8 \beta_{3} + 3) q^{83} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{84} + ( - \beta_{3} - 4) q^{85} + ( - 2 \beta_{3} - 8 \beta_{2} - 2 \beta_1) q^{86} + (6 \beta_{2} - 2 \beta_1 + 6) q^{87} + 9 \beta_1 q^{88} + (\beta_{3} + 12 \beta_{2} + \beta_1) q^{89} + (\beta_{3} + 1) q^{90} + (\beta_{3} + 4 \beta_{2} + 2) q^{91} + ( - 6 \beta_{3} - 1) q^{92} + (\beta_{3} + \beta_{2} + \beta_1) q^{93} + (\beta_{2} + 1) q^{94} + (6 \beta_{2} - 5 \beta_1 + 6) q^{95} - 5 \beta_{2} q^{96} + ( - 5 \beta_{3} + 4) q^{97} + ( - 2 \beta_{3} - 5 \beta_{2} + 2 \beta_1) q^{98} - 3 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{7} - 12 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{7} - 12 q^{8} - 2 q^{9} - 2 q^{10} - 2 q^{12} - 2 q^{14} - 4 q^{15} + 2 q^{16} - 4 q^{17} - 2 q^{18} + 8 q^{19} - 4 q^{20} + 2 q^{21} - 2 q^{23} - 6 q^{24} + 4 q^{25} - 4 q^{27} + 2 q^{28} + 24 q^{29} + 2 q^{30} + 2 q^{31} - 10 q^{32} + 8 q^{34} - 14 q^{35} - 4 q^{36} + 8 q^{37} + 8 q^{38} + 6 q^{40} + 40 q^{41} - 4 q^{42} - 32 q^{43} - 2 q^{45} - 2 q^{46} + 2 q^{47} + 4 q^{48} - 20 q^{49} - 8 q^{50} + 4 q^{51} - 14 q^{53} + 2 q^{54} - 24 q^{55} - 12 q^{56} + 16 q^{57} - 12 q^{58} - 12 q^{59} - 2 q^{60} + 16 q^{61} - 4 q^{62} - 2 q^{63} + 28 q^{64} - 4 q^{65} + 2 q^{67} + 4 q^{68} - 4 q^{69} + 10 q^{70} - 4 q^{71} + 6 q^{72} + 2 q^{73} + 8 q^{74} - 4 q^{75} + 16 q^{76} - 36 q^{77} + 18 q^{79} + 2 q^{80} - 2 q^{81} - 20 q^{82} + 12 q^{83} - 2 q^{84} - 16 q^{85} + 16 q^{86} + 12 q^{87} - 24 q^{89} + 4 q^{90} - 4 q^{92} - 2 q^{93} + 2 q^{94} + 12 q^{95} + 10 q^{96} + 16 q^{97} + 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1407\mathbb{Z}\right)^\times\).

\(n\) \(337\) \(470\) \(1207\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
403.1
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
−0.500000 0.866025i 0.500000 0.866025i 0.500000 0.866025i −1.20711 2.09077i −1.00000 1.00000 2.44949i −3.00000 −0.500000 0.866025i −1.20711 + 2.09077i
403.2 −0.500000 0.866025i 0.500000 0.866025i 0.500000 0.866025i 0.207107 + 0.358719i −1.00000 1.00000 + 2.44949i −3.00000 −0.500000 0.866025i 0.207107 0.358719i
604.1 −0.500000 + 0.866025i 0.500000 + 0.866025i 0.500000 + 0.866025i −1.20711 + 2.09077i −1.00000 1.00000 + 2.44949i −3.00000 −0.500000 + 0.866025i −1.20711 2.09077i
604.2 −0.500000 + 0.866025i 0.500000 + 0.866025i 0.500000 + 0.866025i 0.207107 0.358719i −1.00000 1.00000 2.44949i −3.00000 −0.500000 + 0.866025i 0.207107 + 0.358719i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1407.2.i.e 4
7.c even 3 1 inner 1407.2.i.e 4
7.c even 3 1 9849.2.a.w 2
7.d odd 6 1 9849.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1407.2.i.e 4 1.a even 1 1 trivial
1407.2.i.e 4 7.c even 3 1 inner
9849.2.a.w 2 7.c even 3 1
9849.2.a.x 2 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1407, [\chi])\):

\( T_{2}^{2} + T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{4} + 2T_{5}^{3} + 5T_{5}^{2} - 2T_{5} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} + 18T_{11}^{2} + 324 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$13$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + 30 T^{2} - 56 T + 196 \) Copy content Toggle raw display
$19$ \( T^{4} - 8 T^{3} + 50 T^{2} - 112 T + 196 \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} + 75 T^{2} + \cdots + 5041 \) Copy content Toggle raw display
$29$ \( (T^{2} - 12 T + 28)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$37$ \( T^{4} - 8 T^{3} + 56 T^{2} - 64 T + 64 \) Copy content Toggle raw display
$41$ \( (T - 10)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 16 T + 56)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 14 T^{3} + 149 T^{2} + \cdots + 2209 \) Copy content Toggle raw display
$59$ \( T^{4} + 12 T^{3} + 116 T^{2} + \cdots + 784 \) Copy content Toggle raw display
$61$ \( T^{4} - 16 T^{3} + 224 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$67$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$71$ \( (T + 1)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 2 T^{3} + 35 T^{2} + 62 T + 961 \) Copy content Toggle raw display
$79$ \( T^{4} - 18 T^{3} + 293 T^{2} + \cdots + 961 \) Copy content Toggle raw display
$83$ \( (T^{2} - 6 T - 119)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 24 T^{3} + 434 T^{2} + \cdots + 20164 \) Copy content Toggle raw display
$97$ \( (T^{2} - 8 T - 34)^{2} \) Copy content Toggle raw display
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