Properties

Label 308.4.a.c
Level $308$
Weight $4$
Character orbit 308.a
Self dual yes
Analytic conductor $18.173$
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] = [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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.5925.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 18x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{3} + (3 \beta_1 + 2) q^{5} - 7 q^{7} + (\beta_{2} - 4 \beta_1 - 11) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 2) q^{3} + (3 \beta_1 + 2) q^{5} - 7 q^{7} + (\beta_{2} - 4 \beta_1 - 11) q^{9} - 11 q^{11} + ( - 5 \beta_{2} - 5 \beta_1 - 28) q^{13} + ( - 3 \beta_{2} + 4 \beta_1 - 32) q^{15} + (6 \beta_{2} + 4 \beta_1 - 24) q^{17} + (7 \beta_{2} + 13 \beta_1 + 4) q^{19} + (7 \beta_1 - 14) q^{21} + ( - \beta_{2} - 6 \beta_1 - 44) q^{23} + (9 \beta_{2} + 12 \beta_1 - 13) q^{25} + (5 \beta_{2} + 24 \beta_1 - 28) q^{27} + ( - 22 \beta_{2} + 6 \beta_1 - 82) q^{29} + ( - 19 \beta_{2} + 24 \beta_1 - 58) q^{31} + (11 \beta_1 - 22) q^{33} + ( - 21 \beta_1 - 14) q^{35} + (11 \beta_{2} - 70 \beta_1 - 126) q^{37} + (48 \beta_1 + 4) q^{39} + ( - 2 \beta_{2} - 104 \beta_1) q^{41} + (42 \beta_{2} + 14 \beta_1 - 52) q^{43} + ( - 7 \beta_{2} - 23 \beta_1 - 166) q^{45} + ( - 32 \beta_{2} + 2 \beta_1 - 14) q^{47} + 49 q^{49} + (2 \beta_{2} - 4 \beta_1 - 96) q^{51} + (38 \beta_{2} - 74 \beta_1 - 254) q^{53} + ( - 33 \beta_1 - 22) q^{55} + ( - 6 \beta_{2} - 20 \beta_1 - 148) q^{57} + (48 \beta_{2} + 5 \beta_1 - 326) q^{59} + ( - 17 \beta_{2} + 17 \beta_1 - 8) q^{61} + ( - 7 \beta_{2} + 28 \beta_1 + 77) q^{63} + ( - 40 \beta_{2} - 184 \beta_1 - 236) q^{65} + ( - 93 \beta_{2} - 56 \beta_1 - 40) q^{67} + (5 \beta_{2} + 38 \beta_1 - 16) q^{69} + (3 \beta_{2} + 144 \beta_1 - 88) q^{71} + (20 \beta_{2} + 246 \beta_1 - 92) q^{73} + ( - 3 \beta_{2} - 17 \beta_1 - 170) q^{75} + 77 q^{77} + (58 \beta_{2} - 64 \beta_1 + 88) q^{79} + ( - 46 \beta_{2} + 154 \beta_1 - 47) q^{81} + ( - 3 \beta_{2} - 3 \beta_1 + 364) q^{83} + (42 \beta_{2} + 44 \beta_1 + 96) q^{85} + ( - 28 \beta_{2} + 226 \beta_1 - 236) q^{87} + (133 \beta_{2} - 142 \beta_1 + 226) q^{89} + (35 \beta_{2} + 35 \beta_1 + 196) q^{91} + ( - 43 \beta_{2} + 220 \beta_1 - 404) q^{93} + (74 \beta_{2} + 164 \beta_1 + 476) q^{95} + ( - 7 \beta_{2} - 208 \beta_1 - 138) q^{97} + ( - 11 \beta_{2} + 44 \beta_1 + 121) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 5 q^{3} + 9 q^{5} - 21 q^{7} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 5 q^{3} + 9 q^{5} - 21 q^{7} - 36 q^{9} - 33 q^{11} - 94 q^{13} - 95 q^{15} - 62 q^{17} + 32 q^{19} - 35 q^{21} - 139 q^{23} - 18 q^{25} - 55 q^{27} - 262 q^{29} - 169 q^{31} - 55 q^{33} - 63 q^{35} - 437 q^{37} + 60 q^{39} - 106 q^{41} - 100 q^{43} - 528 q^{45} - 72 q^{47} + 147 q^{49} - 290 q^{51} - 798 q^{53} - 99 q^{55} - 470 q^{57} - 925 q^{59} - 24 q^{61} + 252 q^{63} - 932 q^{65} - 269 q^{67} - 5 q^{69} - 117 q^{71} - 10 q^{73} - 530 q^{75} + 231 q^{77} + 258 q^{79} - 33 q^{81} + 1086 q^{83} + 374 q^{85} - 510 q^{87} + 669 q^{89} + 658 q^{91} - 1035 q^{93} + 1666 q^{95} - 629 q^{97} + 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 18x + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) 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
4.44326
0.658440
−4.10170
0 −2.44326 0 15.3298 0 −7.00000 0 −21.0305 0
1.2 0 1.34156 0 3.97532 0 −7.00000 0 −25.2002 0
1.3 0 6.10170 0 −10.3051 0 −7.00000 0 10.2307 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.c 3
4.b odd 2 1 1232.4.a.r 3
7.b odd 2 1 2156.4.a.e 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.4.a.c 3 1.a even 1 1 trivial
1232.4.a.r 3 4.b odd 2 1
2156.4.a.e 3 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 5T_{3}^{2} - 10T_{3} + 20 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(308))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 5 T^{2} + \cdots + 20 \) Copy content Toggle raw display
$5$ \( T^{3} - 9 T^{2} + \cdots + 628 \) Copy content Toggle raw display
$7$ \( (T + 7)^{3} \) Copy content Toggle raw display
$11$ \( (T + 11)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 94 T^{2} + \cdots - 74608 \) Copy content Toggle raw display
$17$ \( T^{3} + 62 T^{2} + \cdots - 41856 \) Copy content Toggle raw display
$19$ \( T^{3} - 32 T^{2} + \cdots - 123384 \) Copy content Toggle raw display
$23$ \( T^{3} + 139 T^{2} + \cdots + 69072 \) Copy content Toggle raw display
$29$ \( T^{3} + 262 T^{2} + \cdots - 8469496 \) Copy content Toggle raw display
$31$ \( T^{3} + 169 T^{2} + \cdots - 4337908 \) Copy content Toggle raw display
$37$ \( T^{3} + 437 T^{2} + \cdots - 22557516 \) Copy content Toggle raw display
$41$ \( T^{3} + 106 T^{2} + \cdots - 9028992 \) Copy content Toggle raw display
$43$ \( T^{3} + 100 T^{2} + \cdots + 16519040 \) Copy content Toggle raw display
$47$ \( T^{3} + 72 T^{2} + \cdots - 15959776 \) Copy content Toggle raw display
$53$ \( T^{3} + 798 T^{2} + \cdots - 49873624 \) Copy content Toggle raw display
$59$ \( T^{3} + 925 T^{2} + \cdots - 6848460 \) Copy content Toggle raw display
$61$ \( T^{3} + 24 T^{2} + \cdots - 2045608 \) Copy content Toggle raw display
$67$ \( T^{3} + 269 T^{2} + \cdots - 260917008 \) Copy content Toggle raw display
$71$ \( T^{3} + 117 T^{2} + \cdots - 10649936 \) Copy content Toggle raw display
$73$ \( T^{3} + 10 T^{2} + \cdots - 187353120 \) Copy content Toggle raw display
$79$ \( T^{3} - 258 T^{2} + \cdots + 99545984 \) Copy content Toggle raw display
$83$ \( T^{3} - 1086 T^{2} + \cdots - 47003728 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 1273682108 \) Copy content Toggle raw display
$97$ \( T^{3} + 629 T^{2} + \cdots - 147567588 \) Copy content Toggle raw display
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