Properties

Label 2394.4.a.o
Level $2394$
Weight $4$
Character orbit 2394.a
Self dual yes
Analytic conductor $141.251$
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] = [2394,4,Mod(1,2394)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2394, 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("2394.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2394.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: \(141.250572554\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.22397.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 37x + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 798)
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.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} - \beta_{2} q^{5} - 7 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} - \beta_{2} q^{5} - 7 q^{7} + 8 q^{8} - 2 \beta_{2} q^{10} + (\beta_{2} - \beta_1 - 13) q^{11} + (2 \beta_{2} - 14) q^{13} - 14 q^{14} + 16 q^{16} + (3 \beta_1 + 11) q^{17} + 19 q^{19} - 4 \beta_{2} q^{20} + (2 \beta_{2} - 2 \beta_1 - 26) q^{22} + (8 \beta_{2} + 6 \beta_1 + 14) q^{23} + ( - 8 \beta_{2} + 2 \beta_1 + 9) q^{25} + (4 \beta_{2} - 28) q^{26} - 28 q^{28} + (7 \beta_{2} - 20 \beta_1 - 32) q^{29} + (9 \beta_{2} - 7 \beta_1 + 53) q^{31} + 32 q^{32} + (6 \beta_1 + 22) q^{34} + 7 \beta_{2} q^{35} + ( - 13 \beta_{2} + 3 \beta_1 + 17) q^{37} + 38 q^{38} - 8 \beta_{2} q^{40} + (12 \beta_{2} + 30 \beta_1 - 24) q^{41} + (2 \beta_{2} + 10 \beta_1 - 14) q^{43} + (4 \beta_{2} - 4 \beta_1 - 52) q^{44} + (16 \beta_{2} + 12 \beta_1 + 28) q^{46} + ( - 4 \beta_{2} - 27 \beta_1 - 25) q^{47} + 49 q^{49} + ( - 16 \beta_{2} + 4 \beta_1 + 18) q^{50} + (8 \beta_{2} - 56) q^{52} + (25 \beta_{2} + 20 \beta_1 - 68) q^{53} + (22 \beta_{2} + 6 \beta_1 - 130) q^{55} - 56 q^{56} + (14 \beta_{2} - 40 \beta_1 - 64) q^{58} + ( - 24 \beta_{2} - 4 \beta_1 - 336) q^{59} + ( - 10 \beta_{2} - 54 \beta_1 - 168) q^{61} + (18 \beta_{2} - 14 \beta_1 + 106) q^{62} + 64 q^{64} + (30 \beta_{2} - 4 \beta_1 - 268) q^{65} + (15 \beta_{2} - 27 \beta_1 - 311) q^{67} + (12 \beta_1 + 44) q^{68} + 14 \beta_{2} q^{70} + ( - 41 \beta_{2} + 28 \beta_1 - 306) q^{71} + (50 \beta_{2} + 76 \beta_1 - 262) q^{73} + ( - 26 \beta_{2} + 6 \beta_1 + 34) q^{74} + 76 q^{76} + ( - 7 \beta_{2} + 7 \beta_1 + 91) q^{77} + ( - 20 \beta_{2} + 92 \beta_1 + 80) q^{79} - 16 \beta_{2} q^{80} + (24 \beta_{2} + 60 \beta_1 - 48) q^{82} + (3 \beta_{2} + 14 \beta_1 + 264) q^{83} + ( - 14 \beta_{2} - 24 \beta_1 - 12) q^{85} + (4 \beta_{2} + 20 \beta_1 - 28) q^{86} + (8 \beta_{2} - 8 \beta_1 - 104) q^{88} + (48 \beta_{2} - 70 \beta_1 + 52) q^{89} + ( - 14 \beta_{2} + 98) q^{91} + (32 \beta_{2} + 24 \beta_1 + 56) q^{92} + ( - 8 \beta_{2} - 54 \beta_1 - 50) q^{94} - 19 \beta_{2} q^{95} + ( - 53 \beta_{2} - 29 \beta_1 - 743) q^{97} + 98 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{2} + 12 q^{4} - 21 q^{7} + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 6 q^{2} + 12 q^{4} - 21 q^{7} + 24 q^{8} - 38 q^{11} - 42 q^{13} - 42 q^{14} + 48 q^{16} + 30 q^{17} + 57 q^{19} - 76 q^{22} + 36 q^{23} + 25 q^{25} - 84 q^{26} - 84 q^{28} - 76 q^{29} + 166 q^{31} + 96 q^{32} + 60 q^{34} + 48 q^{37} + 114 q^{38} - 102 q^{41} - 52 q^{43} - 152 q^{44} + 72 q^{46} - 48 q^{47} + 147 q^{49} + 50 q^{50} - 168 q^{52} - 224 q^{53} - 396 q^{55} - 168 q^{56} - 152 q^{58} - 1004 q^{59} - 450 q^{61} + 332 q^{62} + 192 q^{64} - 800 q^{65} - 906 q^{67} + 120 q^{68} - 946 q^{71} - 862 q^{73} + 96 q^{74} + 228 q^{76} + 266 q^{77} + 148 q^{79} - 204 q^{82} + 778 q^{83} - 12 q^{85} - 104 q^{86} - 304 q^{88} + 226 q^{89} + 294 q^{91} + 144 q^{92} - 96 q^{94} - 2200 q^{97} + 294 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 37x + 28 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{2} - 50 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{2} + 50 ) / 2 \) 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
6.22280
−5.97577
0.752971
2.00000 0 4.00000 −9.14881 0 −7.00000 8.00000 0 −18.2976
1.2 2.00000 0 4.00000 −7.13988 0 −7.00000 8.00000 0 −14.2798
1.3 2.00000 0 4.00000 16.2887 0 −7.00000 8.00000 0 32.5774
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)
\(19\) \(-1\)

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 2394.4.a.o 3
3.b odd 2 1 798.4.a.e 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.4.a.e 3 3.b odd 2 1
2394.4.a.o 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2394))\):

\( T_{5}^{3} - 200T_{5} - 1064 \) Copy content Toggle raw display
\( T_{11}^{3} + 38T_{11}^{2} + 136T_{11} - 3232 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 200T - 1064 \) Copy content Toggle raw display
$7$ \( (T + 7)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 38 T^{2} + 136 T - 3232 \) Copy content Toggle raw display
$13$ \( T^{3} + 42 T^{2} - 212 T + 56 \) Copy content Toggle raw display
$17$ \( T^{3} - 30 T^{2} - 1044 T + 15808 \) Copy content Toggle raw display
$19$ \( (T - 19)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 36 T^{2} - 17936 T - 116352 \) Copy content Toggle raw display
$29$ \( T^{3} + 76 T^{2} - 67048 T - 8515080 \) Copy content Toggle raw display
$31$ \( T^{3} - 166 T^{2} - 14080 T + 1115296 \) Copy content Toggle raw display
$37$ \( T^{3} - 48 T^{2} - 34220 T - 1784976 \) Copy content Toggle raw display
$41$ \( T^{3} + 102 T^{2} + \cdots - 28656504 \) Copy content Toggle raw display
$43$ \( T^{3} + 52 T^{2} - 14912 T - 637184 \) Copy content Toggle raw display
$47$ \( T^{3} + 48 T^{2} - 111728 T + 2907864 \) Copy content Toggle raw display
$53$ \( T^{3} + 224 T^{2} + \cdots - 26916888 \) Copy content Toggle raw display
$59$ \( T^{3} + 1004 T^{2} + \cdots - 14493120 \) Copy content Toggle raw display
$61$ \( T^{3} + 450 T^{2} + \cdots - 13092296 \) Copy content Toggle raw display
$67$ \( T^{3} + 906 T^{2} + \cdots - 40049632 \) Copy content Toggle raw display
$71$ \( T^{3} + 946 T^{2} + \cdots - 130358816 \) Copy content Toggle raw display
$73$ \( T^{3} + 862 T^{2} + \cdots - 983385816 \) Copy content Toggle raw display
$79$ \( T^{3} - 148 T^{2} + \cdots + 539005760 \) Copy content Toggle raw display
$83$ \( T^{3} - 778 T^{2} + \cdots - 10448544 \) Copy content Toggle raw display
$89$ \( T^{3} - 226 T^{2} + \cdots - 308767320 \) Copy content Toggle raw display
$97$ \( T^{3} + 2200 T^{2} + \cdots - 122881008 \) Copy content Toggle raw display
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