Properties

Label 672.4.a.f
Level $672$
Weight $4$
Character orbit 672.a
Self dual yes
Analytic conductor $39.649$
Analytic rank $1$
Dimension $2$
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] = [672,4,Mod(1,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 672.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: \(39.6492835239\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{37}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(\beta = 2\sqrt{37}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + ( - \beta - 2) q^{5} - 7 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + ( - \beta - 2) q^{5} - 7 q^{7} + 9 q^{9} + ( - \beta + 24) q^{11} + (4 \beta - 26) q^{13} + (3 \beta + 6) q^{15} + (7 \beta - 6) q^{17} + ( - 2 \beta + 80) q^{19} + 21 q^{21} + (5 \beta + 24) q^{23} + (4 \beta + 27) q^{25} - 27 q^{27} + (6 \beta - 190) q^{29} + ( - 18 \beta + 92) q^{31} + (3 \beta - 72) q^{33} + (7 \beta + 14) q^{35} + ( - 20 \beta + 42) q^{37} + ( - 12 \beta + 78) q^{39} + ( - 3 \beta + 26) q^{41} + (16 \beta - 192) q^{43} + ( - 9 \beta - 18) q^{45} + (38 \beta + 32) q^{47} + 49 q^{49} + ( - 21 \beta + 18) q^{51} + ( - 36 \beta - 326) q^{53} + ( - 22 \beta + 100) q^{55} + (6 \beta - 240) q^{57} + ( - 22 \beta + 212) q^{59} + (6 \beta - 530) q^{61} - 63 q^{63} + (18 \beta - 540) q^{65} + (30 \beta - 656) q^{67} + ( - 15 \beta - 72) q^{69} + (51 \beta + 336) q^{71} + (2 \beta - 542) q^{73} + ( - 12 \beta - 81) q^{75} + (7 \beta - 168) q^{77} + ( - 38 \beta - 368) q^{79} + 81 q^{81} + (40 \beta - 140) q^{83} + ( - 8 \beta - 1024) q^{85} + ( - 18 \beta + 570) q^{87} + ( - 87 \beta + 474) q^{89} + ( - 28 \beta + 182) q^{91} + (54 \beta - 276) q^{93} + ( - 76 \beta + 136) q^{95} + ( - 26 \beta + 402) q^{97} + ( - 9 \beta + 216) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 4 q^{5} - 14 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 4 q^{5} - 14 q^{7} + 18 q^{9} + 48 q^{11} - 52 q^{13} + 12 q^{15} - 12 q^{17} + 160 q^{19} + 42 q^{21} + 48 q^{23} + 54 q^{25} - 54 q^{27} - 380 q^{29} + 184 q^{31} - 144 q^{33} + 28 q^{35} + 84 q^{37} + 156 q^{39} + 52 q^{41} - 384 q^{43} - 36 q^{45} + 64 q^{47} + 98 q^{49} + 36 q^{51} - 652 q^{53} + 200 q^{55} - 480 q^{57} + 424 q^{59} - 1060 q^{61} - 126 q^{63} - 1080 q^{65} - 1312 q^{67} - 144 q^{69} + 672 q^{71} - 1084 q^{73} - 162 q^{75} - 336 q^{77} - 736 q^{79} + 162 q^{81} - 280 q^{83} - 2048 q^{85} + 1140 q^{87} + 948 q^{89} + 364 q^{91} - 552 q^{93} + 272 q^{95} + 804 q^{97} + 432 q^{99}+O(q^{100}) \) 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
3.54138
−2.54138
0 −3.00000 0 −14.1655 0 −7.00000 0 9.00000 0
1.2 0 −3.00000 0 10.1655 0 −7.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)

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 672.4.a.f 2
3.b odd 2 1 2016.4.a.m 2
4.b odd 2 1 672.4.a.k yes 2
8.b even 2 1 1344.4.a.bp 2
8.d odd 2 1 1344.4.a.bh 2
12.b even 2 1 2016.4.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.a.f 2 1.a even 1 1 trivial
672.4.a.k yes 2 4.b odd 2 1
1344.4.a.bh 2 8.d odd 2 1
1344.4.a.bp 2 8.b even 2 1
2016.4.a.m 2 3.b odd 2 1
2016.4.a.n 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(672))\):

\( T_{5}^{2} + 4T_{5} - 144 \) Copy content Toggle raw display
\( T_{11}^{2} - 48T_{11} + 428 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4T - 144 \) Copy content Toggle raw display
$7$ \( (T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 48T + 428 \) Copy content Toggle raw display
$13$ \( T^{2} + 52T - 1692 \) Copy content Toggle raw display
$17$ \( T^{2} + 12T - 7216 \) Copy content Toggle raw display
$19$ \( T^{2} - 160T + 5808 \) Copy content Toggle raw display
$23$ \( T^{2} - 48T - 3124 \) Copy content Toggle raw display
$29$ \( T^{2} + 380T + 30772 \) Copy content Toggle raw display
$31$ \( T^{2} - 184T - 39488 \) Copy content Toggle raw display
$37$ \( T^{2} - 84T - 57436 \) Copy content Toggle raw display
$41$ \( T^{2} - 52T - 656 \) Copy content Toggle raw display
$43$ \( T^{2} + 384T - 1024 \) Copy content Toggle raw display
$47$ \( T^{2} - 64T - 212688 \) Copy content Toggle raw display
$53$ \( T^{2} + 652T - 85532 \) Copy content Toggle raw display
$59$ \( T^{2} - 424T - 26688 \) Copy content Toggle raw display
$61$ \( T^{2} + 1060 T + 275572 \) Copy content Toggle raw display
$67$ \( T^{2} + 1312 T + 297136 \) Copy content Toggle raw display
$71$ \( T^{2} - 672T - 272052 \) Copy content Toggle raw display
$73$ \( T^{2} + 1084 T + 293172 \) Copy content Toggle raw display
$79$ \( T^{2} + 736T - 78288 \) Copy content Toggle raw display
$83$ \( T^{2} + 280T - 217200 \) Copy content Toggle raw display
$89$ \( T^{2} - 948T - 895536 \) Copy content Toggle raw display
$97$ \( T^{2} - 804T + 61556 \) Copy content Toggle raw display
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