Properties

Label 308.4.a.e
Level $308$
Weight $4$
Character orbit 308.a
Self dual yes
Analytic conductor $18.173$
Analytic rank $0$
Dimension $5$
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] = [308,4,Mod(1,308)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(308, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("308.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 308.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: \(18.1725882818\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 116x^{3} - 22x^{2} + 2859x - 2034 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + (\beta_{2} + 3) q^{5} + 7 q^{7} + (\beta_{4} + \beta_{2} - 2 \beta_1 + 20) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + (\beta_{2} + 3) q^{5} + 7 q^{7} + (\beta_{4} + \beta_{2} - 2 \beta_1 + 20) q^{9} - 11 q^{11} + ( - \beta_{4} + \beta_{3} - 4 \beta_1 + 14) q^{13} + ( - \beta_{4} - 3 \beta_{3} - 5 \beta_1 + 5) q^{15} + (2 \beta_{3} + \beta_{2} - \beta_1 + 20) q^{17} + ( - 3 \beta_{4} + \beta_{3} + \cdots - 18) q^{19}+ \cdots + ( - 11 \beta_{4} - 11 \beta_{2} + \cdots - 220) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{3} + 15 q^{5} + 35 q^{7} + 102 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{3} + 15 q^{5} + 35 q^{7} + 102 q^{9} - 55 q^{11} + 70 q^{13} + 17 q^{15} + 104 q^{17} - 94 q^{19} + 35 q^{21} + 73 q^{23} + 576 q^{25} + 365 q^{27} + 14 q^{29} + 457 q^{31} - 55 q^{33} + 105 q^{35} + 139 q^{37} + 1020 q^{39} + 724 q^{41} + 64 q^{43} + 978 q^{45} + 378 q^{47} + 245 q^{49} + 246 q^{51} + 126 q^{53} - 165 q^{55} + 290 q^{57} - 1121 q^{59} + 740 q^{61} + 714 q^{63} + 400 q^{65} + 907 q^{67} + 47 q^{69} + 1435 q^{71} + 1380 q^{73} + 842 q^{75} - 385 q^{77} - 326 q^{79} + 1057 q^{81} - 416 q^{83} + 858 q^{85} - 3798 q^{87} + 1709 q^{89} + 490 q^{91} - 1235 q^{93} - 2590 q^{95} + 723 q^{97} - 1122 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 116x^{3} - 22x^{2} + 2859x - 2034 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{4} + 21\nu^{3} + 35\nu^{2} - 1313\nu + 1194 ) / 120 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{4} - 27\nu^{3} - 485\nu^{2} + 1271\nu + 1242 ) / 120 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 21\nu^{3} + 85\nu^{2} + 1313\nu - 6714 ) / 120 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{2} + 46 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{4} + \beta_{3} + 9\beta_{2} + 66\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 77\beta_{4} + 21\beta_{3} + 104\beta_{2} + 73\beta _1 + 3056 \) 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
9.40564
5.11231
0.731352
−7.08017
−8.16912
0 −8.40564 0 16.2345 0 7.00000 0 43.6547 0
1.2 0 −4.11231 0 −17.6741 0 7.00000 0 −10.0889 0
1.3 0 0.268648 0 5.16987 0 7.00000 0 −26.9278 0
1.4 0 8.08017 0 21.9878 0 7.00000 0 38.2892 0
1.5 0 9.16912 0 −10.7180 0 7.00000 0 57.0728 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)
\(11\) \( +1 \)

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 308.4.a.e 5
4.b odd 2 1 1232.4.a.x 5
7.b odd 2 1 2156.4.a.g 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.4.a.e 5 1.a even 1 1 trivial
1232.4.a.x 5 4.b odd 2 1
2156.4.a.g 5 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - 5T_{3}^{4} - 106T_{3}^{3} + 360T_{3}^{2} + 2472T_{3} - 688 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(308))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 5 T^{4} + \cdots - 688 \) Copy content Toggle raw display
$5$ \( T^{5} - 15 T^{4} + \cdots - 349584 \) Copy content Toggle raw display
$7$ \( (T - 7)^{5} \) Copy content Toggle raw display
$11$ \( (T + 11)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} - 70 T^{4} + \cdots - 71584768 \) Copy content Toggle raw display
$17$ \( T^{5} - 104 T^{4} + \cdots - 762459840 \) Copy content Toggle raw display
$19$ \( T^{5} + 94 T^{4} + \cdots + 207309056 \) Copy content Toggle raw display
$23$ \( T^{5} + \cdots - 13952097024 \) Copy content Toggle raw display
$29$ \( T^{5} - 14 T^{4} + \cdots - 105914592 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots - 26839714544 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots + 8597743600 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots + 2108713005504 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots + 94214553600 \) Copy content Toggle raw display
$47$ \( T^{5} - 378 T^{4} + \cdots - 30164736 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots - 54102174432 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots + 6726646275120 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots - 3356972813440 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots - 19330693120 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots + 30220124785920 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots - 415472733120 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots - 41144623266816 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots + 1948582310400 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots - 227399586768 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots + 1065686219248 \) Copy content Toggle raw display
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