Properties

Label 447.2.a.a
Level $447$
Weight $2$
Character orbit 447.a
Self dual yes
Analytic conductor $3.569$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [447,2,Mod(1,447)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(447, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("447.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 447 = 3 \cdot 149 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 447.a (trivial)

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: yes
Analytic conductor: \(3.56931297035\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.53209
−0.347296
1.87939
−2.53209 1.00000 4.41147 −2.00000 −2.53209 0.532089 −6.10607 1.00000 5.06418
1.2 −1.34730 1.00000 −0.184793 −2.00000 −1.34730 −0.652704 2.94356 1.00000 2.69459
1.3 0.879385 1.00000 −1.22668 −2.00000 0.879385 −2.87939 −2.83750 1.00000 −1.75877
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(149\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 447.2.a.a 3
3.b odd 2 1 1341.2.a.c 3
4.b odd 2 1 7152.2.a.l 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
447.2.a.a 3 1.a even 1 1 trivial
1341.2.a.c 3 3.b odd 2 1
7152.2.a.l 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 3T_{2}^{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(447))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 3T^{2} - 3 \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( (T + 2)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 3T^{2} - 1 \) Copy content Toggle raw display
$11$ \( T^{3} + 9 T^{2} + 18 T + 9 \) Copy content Toggle raw display
$13$ \( T^{3} - 12T + 8 \) Copy content Toggle raw display
$17$ \( T^{3} + 6T^{2} - 8 \) Copy content Toggle raw display
$19$ \( T^{3} + 3T^{2} - 1 \) Copy content Toggle raw display
$23$ \( T^{3} + 3 T^{2} - 60 T - 71 \) Copy content Toggle raw display
$29$ \( T^{3} + 6 T^{2} - 36 T - 152 \) Copy content Toggle raw display
$31$ \( T^{3} - 3 T^{2} - 24 T + 53 \) Copy content Toggle raw display
$37$ \( T^{3} + 9 T^{2} + 18 T - 9 \) Copy content Toggle raw display
$41$ \( T^{3} + 15 T^{2} + 36 T - 51 \) Copy content Toggle raw display
$43$ \( T^{3} - 6 T^{2} - 24 T + 136 \) Copy content Toggle raw display
$47$ \( T^{3} + 12 T^{2} + 36 T + 24 \) Copy content Toggle raw display
$53$ \( T^{3} - 156T - 152 \) Copy content Toggle raw display
$59$ \( T^{3} + 27 T^{2} + 222 T + 557 \) Copy content Toggle raw display
$61$ \( T^{3} - 9 T^{2} - 90 T + 477 \) Copy content Toggle raw display
$67$ \( T^{3} + 3 T^{2} - 216 T - 489 \) Copy content Toggle raw display
$71$ \( T^{3} + 3 T^{2} - 126 T + 321 \) Copy content Toggle raw display
$73$ \( T^{3} + 21 T^{2} + 54 T - 597 \) Copy content Toggle raw display
$79$ \( T^{3} - 6T^{2} + 8 \) Copy content Toggle raw display
$83$ \( T^{3} + 9 T^{2} - 36 T - 333 \) Copy content Toggle raw display
$89$ \( T^{3} - 15 T^{2} - 6 T + 37 \) Copy content Toggle raw display
$97$ \( T^{3} + 6 T^{2} - 96 T - 424 \) Copy content Toggle raw display
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