Properties

Label 2300.4.a.a
Level $2300$
Weight $4$
Character orbit 2300.a
Self dual yes
Analytic conductor $135.704$
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] = [2300,4,Mod(1,2300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2300, 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("2300.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2300.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: \(135.704393013\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.28669.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 34x - 69 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 92)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 3) q^{3} + (\beta_1 - 14) q^{7} + (3 \beta_{2} - 3 \beta_1 + 28) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 3) q^{3} + (\beta_1 - 14) q^{7} + (3 \beta_{2} - 3 \beta_1 + 28) q^{9} + ( - 6 \beta_{2} + 3 \beta_1) q^{11} + (\beta_{2} + 3 \beta_1 - 33) q^{13} + (2 \beta_{2} - 5 \beta_1 + 10) q^{17} + ( - 6 \beta_{2} - 2 \beta_1 + 38) q^{19} + (20 \beta_{2} - 5 \beta_1 + 42) q^{21} - 23 q^{23} + ( - 19 \beta_{2} + 24 \beta_1 - 141) q^{27} + (5 \beta_{2} - 5 \beta_1 - 41) q^{29} + ( - 11 \beta_{2} + 8 \beta_1 - 29) q^{31} + (18 \beta_{2} - 33 \beta_1 + 276) q^{33} + (4 \beta_{2} + 13 \beta_1 + 72) q^{37} + (51 \beta_{2} - 12 \beta_1 + 53) q^{39} + ( - 17 \beta_{2} - 9 \beta_1 - 199) q^{41} + (52 \beta_{2} + 4 \beta_1 + 24) q^{43} + ( - 9 \beta_{2} - 16 \beta_1 + 129) q^{47} + (4 \beta_{2} - 22 \beta_1 - 55) q^{49} + ( - 40 \beta_{2} + 31 \beta_1 - 122) q^{51} + (28 \beta_{2} + 42 \beta_1 + 50) q^{53} + ( - 50 \beta_{2} - 8 \beta_1 + 162) q^{57} + (4 \beta_{2} - 22 \beta_1 + 212) q^{59} + (16 \beta_{2} + 4 \beta_1 + 118) q^{61} + ( - 72 \beta_{2} + 58 \beta_1 - 668) q^{63} + ( - 62 \beta_{2} - 53 \beta_1 - 168) q^{67} + (23 \beta_{2} + 69) q^{69} + (57 \beta_{2} + 42 \beta_1 + 315) q^{71} + (115 \beta_{2} - 37 \beta_1 - 87) q^{73} + (132 \beta_{2} - 36 \beta_1 + 276) q^{77} + ( - 46 \beta_{2} + 2 \beta_1 + 498) q^{79} + (204 \beta_{2} - 96 \beta_1 + 541) q^{81} + (64 \beta_{2} + 69 \beta_1 - 130) q^{83} + (11 \beta_{2} + 40 \beta_1 - 107) q^{87} + ( - 18 \beta_{2} - 48 \beta_1 - 276) q^{89} + ( - 8 \beta_{2} - 55 \beta_1 + 738) q^{91} + (77 \beta_{2} - 73 \beta_1 + 593) q^{93} + (54 \beta_{2} + 11 \beta_1 - 410) q^{97} + ( - 312 \beta_{2} + 138 \beta_1 - 1656) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 8 q^{3} - 42 q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 8 q^{3} - 42 q^{7} + 81 q^{9} + 6 q^{11} - 100 q^{13} + 28 q^{17} + 120 q^{19} + 106 q^{21} - 69 q^{23} - 404 q^{27} - 128 q^{29} - 76 q^{31} + 810 q^{33} + 212 q^{37} + 108 q^{39} - 580 q^{41} + 20 q^{43} + 396 q^{47} - 169 q^{49} - 326 q^{51} + 122 q^{53} + 536 q^{57} + 632 q^{59} + 338 q^{61} - 1932 q^{63} - 442 q^{67} + 184 q^{69} + 888 q^{71} - 376 q^{73} + 696 q^{77} + 1540 q^{79} + 1419 q^{81} - 454 q^{83} - 332 q^{87} - 810 q^{89} + 2222 q^{91} + 1702 q^{93} - 1284 q^{97} - 4656 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 34x - 69 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3\nu - 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} + 3\beta _1 + 46 ) / 2 \) 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
−4.18462
6.66032
−2.47570
0 −10.0649 0 0 0 −22.3692 0 74.3025 0
1.2 0 −4.37890 0 0 0 −0.679360 0 −7.82521 0
1.3 0 6.44381 0 0 0 −18.9514 0 14.5228 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(23\) \(1\)

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 2300.4.a.a 3
5.b even 2 1 92.4.a.b 3
5.c odd 4 2 2300.4.c.a 6
15.d odd 2 1 828.4.a.e 3
20.d odd 2 1 368.4.a.h 3
40.e odd 2 1 1472.4.a.x 3
40.f even 2 1 1472.4.a.o 3
115.c odd 2 1 2116.4.a.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
92.4.a.b 3 5.b even 2 1
368.4.a.h 3 20.d odd 2 1
828.4.a.e 3 15.d odd 2 1
1472.4.a.o 3 40.f even 2 1
1472.4.a.x 3 40.e odd 2 1
2116.4.a.b 3 115.c odd 2 1
2300.4.a.a 3 1.a even 1 1 trivial
2300.4.c.a 6 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + 8T_{3}^{2} - 49T_{3} - 284 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2300))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 8 T^{2} - 49 T - 284 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 42 T^{2} + 452 T + 288 \) Copy content Toggle raw display
$11$ \( T^{3} - 6 T^{2} - 3636 T + 89424 \) Copy content Toggle raw display
$13$ \( T^{3} + 100 T^{2} + 2021 T - 24394 \) Copy content Toggle raw display
$17$ \( T^{3} - 28 T^{2} - 3360 T + 56376 \) Copy content Toggle raw display
$19$ \( T^{3} - 120 T^{2} + 1652 T - 3984 \) Copy content Toggle raw display
$23$ \( (T + 23)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 128 T^{2} + 453 T - 231174 \) Copy content Toggle raw display
$31$ \( T^{3} + 76 T^{2} - 14761 T + 382208 \) Copy content Toggle raw display
$37$ \( T^{3} - 212 T^{2} - 9440 T - 64536 \) Copy content Toggle raw display
$41$ \( T^{3} + 580 T^{2} + 79873 T - 509682 \) Copy content Toggle raw display
$43$ \( T^{3} - 20 T^{2} - 193472 T + 25961472 \) Copy content Toggle raw display
$47$ \( T^{3} - 396 T^{2} + 10895 T + 5642304 \) Copy content Toggle raw display
$53$ \( T^{3} - 122 T^{2} + \cdots - 28384056 \) Copy content Toggle raw display
$59$ \( T^{3} - 632 T^{2} + 66720 T + 9077184 \) Copy content Toggle raw display
$61$ \( T^{3} - 338 T^{2} + 17516 T + 2020904 \) Copy content Toggle raw display
$67$ \( T^{3} + 442 T^{2} + \cdots - 105984432 \) Copy content Toggle raw display
$71$ \( T^{3} - 888 T^{2} + \cdots + 150510312 \) Copy content Toggle raw display
$73$ \( T^{3} + 376 T^{2} + \cdots - 431656494 \) Copy content Toggle raw display
$79$ \( T^{3} - 1540 T^{2} + \cdots - 66491136 \) Copy content Toggle raw display
$83$ \( T^{3} + 454 T^{2} + \cdots - 241050384 \) Copy content Toggle raw display
$89$ \( T^{3} + 810 T^{2} - 122616 T - 178848 \) Copy content Toggle raw display
$97$ \( T^{3} + 1284 T^{2} + \cdots + 22204008 \) Copy content Toggle raw display
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