Properties

Label 312.4.a.f
Level $312$
Weight $4$
Character orbit 312.a
Self dual yes
Analytic conductor $18.409$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [312,4,Mod(1,312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(312, 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("312.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 312.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: \(18.4085959218\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{43}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 43 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2\sqrt{43}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + (\beta + 6) q^{5} + (\beta + 22) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + (\beta + 6) q^{5} + (\beta + 22) q^{7} + 9 q^{9} + 26 q^{11} - 13 q^{13} + (3 \beta + 18) q^{15} + ( - 2 \beta - 10) q^{17} + ( - 5 \beta - 30) q^{19} + (3 \beta + 66) q^{21} + ( - 12 \beta - 4) q^{23} + (12 \beta + 83) q^{25} + 27 q^{27} + ( - 12 \beta + 66) q^{29} + ( - 15 \beta - 70) q^{31} + 78 q^{33} + (28 \beta + 304) q^{35} + ( - 14 \beta + 34) q^{37} - 39 q^{39} + ( - 7 \beta + 14) q^{41} + 14 \beta q^{43} + (9 \beta + 54) q^{45} + ( - 6 \beta + 18) q^{47} + (44 \beta + 313) q^{49} + ( - 6 \beta - 30) q^{51} + ( - 4 \beta + 334) q^{53} + (26 \beta + 156) q^{55} + ( - 15 \beta - 90) q^{57} + ( - 22 \beta - 254) q^{59} + ( - 8 \beta + 170) q^{61} + (9 \beta + 198) q^{63} + ( - 13 \beta - 78) q^{65} + ( - 43 \beta - 470) q^{67} + ( - 36 \beta - 12) q^{69} + (68 \beta + 150) q^{71} + ( - 6 \beta + 562) q^{73} + (36 \beta + 249) q^{75} + (26 \beta + 572) q^{77} + ( - 44 \beta - 760) q^{79} + 81 q^{81} + (42 \beta + 262) q^{83} + ( - 22 \beta - 404) q^{85} + ( - 36 \beta + 198) q^{87} + ( - \beta + 950) q^{89} + ( - 13 \beta - 286) q^{91} + ( - 45 \beta - 210) q^{93} + ( - 60 \beta - 1040) q^{95} + ( - 18 \beta - 718) q^{97} + 234 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 12 q^{5} + 44 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} + 12 q^{5} + 44 q^{7} + 18 q^{9} + 52 q^{11} - 26 q^{13} + 36 q^{15} - 20 q^{17} - 60 q^{19} + 132 q^{21} - 8 q^{23} + 166 q^{25} + 54 q^{27} + 132 q^{29} - 140 q^{31} + 156 q^{33} + 608 q^{35} + 68 q^{37} - 78 q^{39} + 28 q^{41} + 108 q^{45} + 36 q^{47} + 626 q^{49} - 60 q^{51} + 668 q^{53} + 312 q^{55} - 180 q^{57} - 508 q^{59} + 340 q^{61} + 396 q^{63} - 156 q^{65} - 940 q^{67} - 24 q^{69} + 300 q^{71} + 1124 q^{73} + 498 q^{75} + 1144 q^{77} - 1520 q^{79} + 162 q^{81} + 524 q^{83} - 808 q^{85} + 396 q^{87} + 1900 q^{89} - 572 q^{91} - 420 q^{93} - 2080 q^{95} - 1436 q^{97} + 468 q^{99}+O(q^{100}) \) 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
−6.55744
6.55744
0 3.00000 0 −7.11488 0 8.88512 0 9.00000 0
1.2 0 3.00000 0 19.1149 0 35.1149 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(13\) \( +1 \)

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 312.4.a.f 2
3.b odd 2 1 936.4.a.c 2
4.b odd 2 1 624.4.a.l 2
8.b even 2 1 2496.4.a.u 2
8.d odd 2 1 2496.4.a.bd 2
12.b even 2 1 1872.4.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.4.a.f 2 1.a even 1 1 trivial
624.4.a.l 2 4.b odd 2 1
936.4.a.c 2 3.b odd 2 1
1872.4.a.v 2 12.b even 2 1
2496.4.a.u 2 8.b even 2 1
2496.4.a.bd 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 12T_{5} - 136 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(312))\). 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
$5$ \( T^{2} - 12T - 136 \) Copy content Toggle raw display
$7$ \( T^{2} - 44T + 312 \) Copy content Toggle raw display
$11$ \( (T - 26)^{2} \) Copy content Toggle raw display
$13$ \( (T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 20T - 588 \) Copy content Toggle raw display
$19$ \( T^{2} + 60T - 3400 \) Copy content Toggle raw display
$23$ \( T^{2} + 8T - 24752 \) Copy content Toggle raw display
$29$ \( T^{2} - 132T - 20412 \) Copy content Toggle raw display
$31$ \( T^{2} + 140T - 33800 \) Copy content Toggle raw display
$37$ \( T^{2} - 68T - 32556 \) Copy content Toggle raw display
$41$ \( T^{2} - 28T - 8232 \) Copy content Toggle raw display
$43$ \( T^{2} - 33712 \) Copy content Toggle raw display
$47$ \( T^{2} - 36T - 5868 \) Copy content Toggle raw display
$53$ \( T^{2} - 668T + 108804 \) Copy content Toggle raw display
$59$ \( T^{2} + 508T - 18732 \) Copy content Toggle raw display
$61$ \( T^{2} - 340T + 17892 \) Copy content Toggle raw display
$67$ \( T^{2} + 940T - 97128 \) Copy content Toggle raw display
$71$ \( T^{2} - 300T - 772828 \) Copy content Toggle raw display
$73$ \( T^{2} - 1124 T + 309652 \) Copy content Toggle raw display
$79$ \( T^{2} + 1520 T + 244608 \) Copy content Toggle raw display
$83$ \( T^{2} - 524T - 234764 \) Copy content Toggle raw display
$89$ \( T^{2} - 1900 T + 902328 \) Copy content Toggle raw display
$97$ \( T^{2} + 1436 T + 459796 \) Copy content Toggle raw display
show more
show less