Properties

Label 2300.4.a.b
Level $2300$
Weight $4$
Character orbit 2300.a
Self dual yes
Analytic conductor $135.704$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x + 6 \) 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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{3} + ( - 3 \beta_{2} - 5 \beta_1 + 18) q^{7} + (2 \beta_{2} - 3 \beta_1 - 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{3} + ( - 3 \beta_{2} - 5 \beta_1 + 18) q^{7} + (2 \beta_{2} - 3 \beta_1 - 10) q^{9} + ( - 11 \beta_{2} - 5 \beta_1 - 16) q^{11} + (6 \beta_{2} + 11 \beta_1 + 9) q^{13} + ( - 5 \beta_{2} - 33 \beta_1 + 42) q^{17} + ( - 16 \beta_{2} - 24 \beta_1 - 18) q^{19} + (25 \beta_{2} - \beta_1 - 10) q^{21} + 23 q^{23} + ( - 29 \beta_{2} - 12 \beta_1 + 7) q^{27} + ( - 6 \beta_{2} - \beta_1 + 105) q^{29} + (59 \beta_{2} + 8 \beta_1 - 69) q^{31} + ( - 17 \beta_{2} + 23 \beta_1 - 172) q^{33} + (3 \beta_{2} - 65 \beta_1 + 12) q^{37} + ( - 7 \beta_{2} + 4 \beta_1 + 61) q^{39} + ( - 52 \beta_{2} + 27 \beta_1 + 203) q^{41} + ( - 12 \beta_{2} - 62 \beta_1 + 184) q^{43} + ( - 47 \beta_{2} + 72 \beta_1 + 1) q^{47} + ( - 118 \beta_{2} - 102 \beta_1 + 305) q^{49} + (103 \beta_{2} - 51 \beta_1 + 94) q^{51} + ( - 42 \beta_{2} + 64 \beta_1 - 126) q^{53} + (14 \beta_{2} - 178) q^{57} + (38 \beta_{2} + 94 \beta_1 + 4) q^{59} + (96 \beta_{2} - 14 \beta_1 - 378) q^{61} + (98 \beta_{2} + 58 \beta_1 - 92) q^{63} + (61 \beta_{2} + 51 \beta_1 + 632) q^{67} + (23 \beta_{2} + 23) q^{69} + ( - 47 \beta_{2} - 58 \beta_1 + 155) q^{71} + (68 \beta_{2} - 89 \beta_1 + 91) q^{73} + ( - 240 \beta_{2} + 36 \beta_1 + 260) q^{77} + (176 \beta_{2} - 134 \beta_1 - 254) q^{79} + ( - 52 \beta_{2} + 144 \beta_1 - 139) q^{81} + (185 \beta_{2} + 97 \beta_1 + 374) q^{83} + (101 \beta_{2} + 16 \beta_1 + 13) q^{87} + (238 \beta_{2} - 16 \beta_1 + 104) q^{89} + (97 \beta_{2} - 21 \beta_1 - 534) q^{91} + ( - 26 \beta_{2} - 161 \beta_1 + 843) q^{93} + ( - 177 \beta_{2} + 23 \beta_1 - 194) q^{97} + (62 \beta_{2} + 232 \beta_1 - 104) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{3} + 46 q^{7} - 31 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 4 q^{3} + 46 q^{7} - 31 q^{9} - 64 q^{11} + 44 q^{13} + 88 q^{17} - 94 q^{19} - 6 q^{21} + 69 q^{23} - 20 q^{27} + 308 q^{29} - 140 q^{31} - 510 q^{33} - 26 q^{37} + 180 q^{39} + 584 q^{41} + 478 q^{43} + 28 q^{47} + 695 q^{49} + 334 q^{51} - 356 q^{53} - 520 q^{57} + 144 q^{59} - 1052 q^{61} - 120 q^{63} + 2008 q^{67} + 92 q^{69} + 360 q^{71} + 252 q^{73} + 576 q^{77} - 720 q^{79} - 325 q^{81} + 1404 q^{83} + 156 q^{87} + 534 q^{89} - 1526 q^{91} + 2342 q^{93} - 736 q^{97} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 7x + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + \nu + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{2} + \beta _1 + 10 ) / 2 \) 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
−2.59261
2.75153
0.841083
0 −3.31427 0 0 0 35.2976 0 −16.0156 0
1.2 0 1.18060 0 0 0 −9.15411 0 −25.6062 0
1.3 0 6.13366 0 0 0 19.8565 0 10.6218 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.b 3
5.b even 2 1 92.4.a.a 3
5.c odd 4 2 2300.4.c.b 6
15.d odd 2 1 828.4.a.f 3
20.d odd 2 1 368.4.a.k 3
40.e odd 2 1 1472.4.a.p 3
40.f even 2 1 1472.4.a.w 3
115.c odd 2 1 2116.4.a.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
92.4.a.a 3 5.b even 2 1
368.4.a.k 3 20.d odd 2 1
828.4.a.f 3 15.d odd 2 1
1472.4.a.p 3 40.e odd 2 1
1472.4.a.w 3 40.f even 2 1
2116.4.a.a 3 115.c odd 2 1
2300.4.a.b 3 1.a even 1 1 trivial
2300.4.c.b 6 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 4T_{3}^{2} - 17T_{3} + 24 \) 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
$3$ \( T^{3} - 4 T^{2} - 17 T + 24 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 46 T^{2} + 196 T + 6416 \) Copy content Toggle raw display
$11$ \( T^{3} + 64 T^{2} - 1112 T - 88184 \) Copy content Toggle raw display
$13$ \( T^{3} - 44 T^{2} - 1739 T + 3334 \) Copy content Toggle raw display
$17$ \( T^{3} - 88 T^{2} - 17920 T + 1617496 \) Copy content Toggle raw display
$19$ \( T^{3} + 94 T^{2} - 9364 T - 184984 \) Copy content Toggle raw display
$23$ \( (T - 23)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 308 T^{2} + 30877 T - 1008698 \) Copy content Toggle raw display
$31$ \( T^{3} + 140 T^{2} - 66217 T - 1074768 \) Copy content Toggle raw display
$37$ \( T^{3} + 26 T^{2} - 88484 T + 4672784 \) Copy content Toggle raw display
$41$ \( T^{3} - 584 T^{2} + \cdots + 21405186 \) Copy content Toggle raw display
$43$ \( T^{3} - 478 T^{2} + \cdots + 14441984 \) Copy content Toggle raw display
$47$ \( T^{3} - 28 T^{2} - 199601 T + 25906224 \) Copy content Toggle raw display
$53$ \( T^{3} + 356 T^{2} - 116276 T - 63184 \) Copy content Toggle raw display
$59$ \( T^{3} - 144 T^{2} + \cdots - 15495744 \) Copy content Toggle raw display
$61$ \( T^{3} + 1052 T^{2} + \cdots - 55378432 \) Copy content Toggle raw display
$67$ \( T^{3} - 2008 T^{2} + \cdots - 228170232 \) Copy content Toggle raw display
$71$ \( T^{3} - 360 T^{2} - 38189 T + 7542816 \) Copy content Toggle raw display
$73$ \( T^{3} - 252 T^{2} + \cdots - 34560066 \) Copy content Toggle raw display
$79$ \( T^{3} + 720 T^{2} + \cdots - 932658192 \) Copy content Toggle raw display
$83$ \( T^{3} - 1404 T^{2} + \cdots + 464610136 \) Copy content Toggle raw display
$89$ \( T^{3} - 534 T^{2} + \cdots + 77599776 \) Copy content Toggle raw display
$97$ \( T^{3} + 736 T^{2} + \cdots - 67270584 \) Copy content Toggle raw display
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