Properties

Label 2312.4.a.d
Level $2312$
Weight $4$
Character orbit 2312.a
Self dual yes
Analytic conductor $136.412$
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] = [2312,4,Mod(1,2312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2312, 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("2312.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2312 = 2^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2312.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: \(136.412415933\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.8396.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 20x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 136)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{3} + (2 \beta_{2} - \beta_1 + 3) q^{5} + (\beta_{2} + 3 \beta_1 + 2) q^{7} + ( - 6 \beta_{2} - \beta_1 - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{3} + (2 \beta_{2} - \beta_1 + 3) q^{5} + (\beta_{2} + 3 \beta_1 + 2) q^{7} + ( - 6 \beta_{2} - \beta_1 - 4) q^{9} + ( - \beta_{2} - 2 \beta_1 + 27) q^{11} + (2 \beta_{2} + 3 \beta_1 - 15) q^{13} + (6 \beta_{2} + 4 \beta_1 - 46) q^{15} + (6 \beta_{2} + 2 \beta_1 - 72) q^{19} + (6 \beta_{2} - 5 \beta_1 - 5) q^{21} + ( - \beta_{2} + \beta_1 + 78) q^{23} + (8 \beta_{2} - 22 \beta_1 + 49) q^{25} + ( - 4 \beta_1 + 96) q^{27} + ( - 18 \beta_{2} - 15 \beta_1 + 13) q^{29} + ( - 3 \beta_{2} + 25 \beta_1 + 8) q^{31} + ( - 34 \beta_{2} + 3 \beta_1 + 39) q^{33} + ( - 30 \beta_{2} + 20 \beta_1 - 146) q^{35} + ( - 38 \beta_{2} - 27 \beta_1 - 55) q^{37} + (28 \beta_{2} - 4 \beta_1 - 44) q^{39} + (40 \beta_{2} - 14 \beta_1 - 64) q^{41} + ( - 78 \beta_{2} - 26 \beta_1 - 20) q^{43} + (26 \beta_{2} + 25 \beta_1 - 239) q^{45} + (52 \beta_{2} + 30 \beta_1 + 22) q^{47} + (66 \beta_{2} + 29 \beta_1 + 166) q^{49} + (12 \beta_{2} - 48 \beta_1 + 142) q^{53} + (78 \beta_{2} - 40 \beta_1 + 166) q^{55} + (104 \beta_{2} + 2 \beta_1 - 194) q^{57} + (86 \beta_{2} - 42 \beta_1 - 8) q^{59} + ( - 114 \beta_{2} + 25 \beta_1 - 183) q^{61} + (3 \beta_{2} - 65 \beta_1 - 216) q^{63} + ( - 68 \beta_{2} + 32 \beta_1 - 148) q^{65} + (60 \beta_{2} + 102 \beta_1 - 282) q^{67} + ( - 82 \beta_{2} - 3 \beta_1 + 105) q^{69} + (59 \beta_{2} + 51 \beta_1 - 136) q^{71} + (96 \beta_{2} - 56 \beta_1 - 266) q^{73} + ( - 31 \beta_{2} + 52 \beta_1 - 237) q^{75} + ( - 14 \beta_{2} + 63 \beta_1 - 285) q^{77} + (43 \beta_{2} - 23 \beta_1 - 406) q^{79} + (62 \beta_{2} + 35 \beta_1 + 184) q^{81} + ( - 102 \beta_{2} - 22 \beta_1 - 424) q^{83} + ( - 118 \beta_{2} + 12 \beta_1 + 334) q^{87} + ( - 18 \beta_{2} + 43 \beta_1 - 827) q^{89} + (44 \beta_{2} - 14 \beta_1 + 482) q^{91} + (2 \beta_{2} - 53 \beta_1 + 199) q^{93} + ( - 188 \beta_{2} + 60 \beta_1 - 56) q^{95} + (92 \beta_{2} + 152 \beta_1 - 618) q^{97} + ( - 179 \beta_{2} + 14 \beta_1 + 73) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{3} + 8 q^{5} + 2 q^{7} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 4 q^{3} + 8 q^{5} + 2 q^{7} - 5 q^{9} + 84 q^{11} - 50 q^{13} - 148 q^{15} - 224 q^{19} - 16 q^{21} + 234 q^{23} + 161 q^{25} + 292 q^{27} + 72 q^{29} + 2 q^{31} + 148 q^{33} - 428 q^{35} - 100 q^{37} - 156 q^{39} - 218 q^{41} + 44 q^{43} - 768 q^{45} - 16 q^{47} + 403 q^{49} + 462 q^{53} + 460 q^{55} - 688 q^{57} - 68 q^{59} - 460 q^{61} - 586 q^{63} - 408 q^{65} - 1008 q^{67} + 400 q^{69} - 518 q^{71} - 838 q^{73} - 732 q^{75} - 904 q^{77} - 1238 q^{79} + 455 q^{81} - 1148 q^{83} + 1108 q^{87} - 2506 q^{89} + 1416 q^{91} + 648 q^{93} - 40 q^{95} - 2098 q^{97} + 384 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 20x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - \nu - 14 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{2} + \beta _1 + 29 ) / 2 \) 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
−4.20657
4.81129
0.395276
0 −2.95089 0 20.3149 0 −22.2885 0 −18.2922 0
1.2 0 −1.16862 0 −1.28534 0 30.0364 0 −25.6343 0
1.3 0 8.11952 0 −11.0296 0 −5.74786 0 38.9265 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(1\)

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 2312.4.a.d 3
17.b even 2 1 136.4.a.b 3
51.c odd 2 1 1224.4.a.i 3
68.d odd 2 1 272.4.a.j 3
136.e odd 2 1 1088.4.a.u 3
136.h even 2 1 1088.4.a.y 3
204.h even 2 1 2448.4.a.bj 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.a.b 3 17.b even 2 1
272.4.a.j 3 68.d odd 2 1
1088.4.a.u 3 136.e odd 2 1
1088.4.a.y 3 136.h even 2 1
1224.4.a.i 3 51.c odd 2 1
2312.4.a.d 3 1.a even 1 1 trivial
2448.4.a.bj 3 204.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 4T_{3}^{2} - 30T_{3} - 28 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2312))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 4 T^{2} - 30 T - 28 \) Copy content Toggle raw display
$5$ \( T^{3} - 8 T^{2} - 236 T - 288 \) Copy content Toggle raw display
$7$ \( T^{3} - 2 T^{2} - 714 T - 3848 \) Copy content Toggle raw display
$11$ \( T^{3} - 84 T^{2} + 2026 T - 10972 \) Copy content Toggle raw display
$13$ \( T^{3} + 50 T^{2} + 64 T - 16048 \) Copy content Toggle raw display
$17$ \( T^{3} \) Copy content Toggle raw display
$19$ \( T^{3} + 224 T^{2} + 15336 T + 322592 \) Copy content Toggle raw display
$23$ \( T^{3} - 234 T^{2} + 18118 T - 463504 \) Copy content Toggle raw display
$29$ \( T^{3} - 72 T^{2} - 23340 T + 1862624 \) Copy content Toggle raw display
$31$ \( T^{3} - 2 T^{2} - 52450 T + 1252344 \) Copy content Toggle raw display
$37$ \( T^{3} + 100 T^{2} - 89196 T + 4014352 \) Copy content Toggle raw display
$41$ \( T^{3} + 218 T^{2} - 66340 T - 7651496 \) Copy content Toggle raw display
$43$ \( T^{3} - 44 T^{2} - 234152 T - 18647936 \) Copy content Toggle raw display
$47$ \( T^{3} + 16 T^{2} - 141616 T - 7663872 \) Copy content Toggle raw display
$53$ \( T^{3} - 462 T^{2} + \cdots + 10502936 \) Copy content Toggle raw display
$59$ \( T^{3} + 68 T^{2} - 465864 T - 81654208 \) Copy content Toggle raw display
$61$ \( T^{3} + 460 T^{2} + \cdots - 116249648 \) Copy content Toggle raw display
$67$ \( T^{3} + 1008 T^{2} + \cdots - 534256128 \) Copy content Toggle raw display
$71$ \( T^{3} + 518 T^{2} + \cdots - 93696432 \) Copy content Toggle raw display
$73$ \( T^{3} + 838 T^{2} + \cdots - 324704504 \) Copy content Toggle raw display
$79$ \( T^{3} + 1238 T^{2} + \cdots + 7088608 \) Copy content Toggle raw display
$83$ \( T^{3} + 1148 T^{2} + \cdots - 158784832 \) Copy content Toggle raw display
$89$ \( T^{3} + 2506 T^{2} + \cdots + 456757648 \) Copy content Toggle raw display
$97$ \( T^{3} + 2098 T^{2} + \cdots - 1961875944 \) Copy content Toggle raw display
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