Properties

Label 308.4.a.d
Level $308$
Weight $4$
Character orbit 308.a
Self dual yes
Analytic conductor $18.173$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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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: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 43x^{2} - 11x + 222 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} + \beta_{2} q^{5} - 7 q^{7} + (\beta_{2} - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} + \beta_{2} q^{5} - 7 q^{7} + (\beta_{2} - 5) q^{9} + 11 q^{11} + ( - \beta_{3} + 2 \beta_1 + 24) q^{13} + (\beta_{3} + 11 \beta_1 - 11) q^{15} + (2 \beta_{3} + \beta_{2} + 11 \beta_1 + 13) q^{17} + ( - \beta_{3} - 3 \beta_{2} + \cdots + 29) q^{19}+ \cdots + (11 \beta_{2} - 55) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{3} + q^{5} - 28 q^{7} - 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{3} + q^{5} - 28 q^{7} - 19 q^{9} + 44 q^{11} + 98 q^{13} - 33 q^{15} + 64 q^{17} + 114 q^{19} + 21 q^{21} + 231 q^{23} + 417 q^{25} + 63 q^{27} + 268 q^{29} + 33 q^{31} - 33 q^{33} - 7 q^{35} + 357 q^{37} + 140 q^{39} + 364 q^{41} + 44 q^{43} + 912 q^{45} + 720 q^{47} + 196 q^{49} + 794 q^{51} + 740 q^{53} + 11 q^{55} + 138 q^{57} + 787 q^{59} + 1020 q^{61} + 133 q^{63} + 296 q^{65} - 995 q^{67} + 1059 q^{69} - 1011 q^{71} + 1592 q^{73} - 324 q^{75} - 308 q^{77} - 178 q^{79} - 1396 q^{81} + 324 q^{83} - 94 q^{85} - 1262 q^{87} + 19 q^{89} - 686 q^{91} + 637 q^{93} - 2418 q^{95} - 555 q^{97} - 209 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 43x^{2} - 11x + 222 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu^{2} - 30\nu + 32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 21 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_{2} + 36\beta _1 + 31 \) 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
−5.25471
−2.79239
2.21425
6.83284
0 −6.25471 0 17.1213 0 −7.00000 0 12.1213 0
1.2 0 −3.79239 0 −7.61779 0 −7.00000 0 −12.6178 0
1.3 0 1.21425 0 −20.5256 0 −7.00000 0 −25.5256 0
1.4 0 5.83284 0 12.0220 0 −7.00000 0 7.02203 0
\(n\): e.g. 2-40 or 990-1000
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.d 4
4.b odd 2 1 1232.4.a.v 4
7.b odd 2 1 2156.4.a.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.4.a.d 4 1.a even 1 1 trivial
1232.4.a.v 4 4.b odd 2 1
2156.4.a.f 4 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 3T_{3}^{3} - 40T_{3}^{2} - 96T_{3} + 168 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(308))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3 T^{3} + \cdots + 168 \) Copy content Toggle raw display
$5$ \( T^{4} - T^{3} + \cdots + 32184 \) Copy content Toggle raw display
$7$ \( (T + 7)^{4} \) Copy content Toggle raw display
$11$ \( (T - 11)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 98 T^{3} + \cdots - 5715264 \) Copy content Toggle raw display
$17$ \( T^{4} - 64 T^{3} + \cdots + 80892704 \) Copy content Toggle raw display
$19$ \( T^{4} - 114 T^{3} + \cdots + 186624 \) Copy content Toggle raw display
$23$ \( T^{4} - 231 T^{3} + \cdots + 6309184 \) Copy content Toggle raw display
$29$ \( T^{4} - 268 T^{3} + \cdots - 62229648 \) Copy content Toggle raw display
$31$ \( T^{4} - 33 T^{3} + \cdots + 668400408 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 2194340904 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots - 6396705696 \) Copy content Toggle raw display
$43$ \( T^{4} - 44 T^{3} + \cdots - 221937664 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 3932145984 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 1987381584 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 4667157144 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 1375017408 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 2114400384 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 156603934656 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 8143492032 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 14872658688 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 163701302784 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 58144933944 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 5739946776 \) Copy content Toggle raw display
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