Properties

Label 2070.4.a.v
Level $2070$
Weight $4$
Character orbit 2070.a
Self dual yes
Analytic conductor $122.134$
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] = [2070,4,Mod(1,2070)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2070, 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("2070.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2070.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: \(122.133953712\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 486x - 3340 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 690)
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.

\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} + 5 q^{5} + ( - \beta_{2} + \beta_1 - 1) q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 4 q^{4} + 5 q^{5} + ( - \beta_{2} + \beta_1 - 1) q^{7} - 8 q^{8} - 10 q^{10} + ( - \beta_1 - 14) q^{11} + ( - 3 \beta_1 + 24) q^{13} + (2 \beta_{2} - 2 \beta_1 + 2) q^{14} + 16 q^{16} + ( - \beta_{2} - 49) q^{17} + (2 \beta_{2} + \beta_1 + 8) q^{19} + 20 q^{20} + (2 \beta_1 + 28) q^{22} + 23 q^{23} + 25 q^{25} + (6 \beta_1 - 48) q^{26} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{28} + (\beta_{2} + 8 \beta_1 - 71) q^{29} + (5 \beta_{2} + 5 \beta_1 + 45) q^{31} - 32 q^{32} + (2 \beta_{2} + 98) q^{34} + ( - 5 \beta_{2} + 5 \beta_1 - 5) q^{35} + (3 \beta_{2} + 7 \beta_1 + 69) q^{37} + ( - 4 \beta_{2} - 2 \beta_1 - 16) q^{38} - 40 q^{40} + (5 \beta_{2} + 8 \beta_1 - 155) q^{41} + (10 \beta_{2} - 8 \beta_1 + 138) q^{43} + ( - 4 \beta_1 - 56) q^{44} - 46 q^{46} + (6 \beta_{2} - 15 \beta_1 - 64) q^{47} + (7 \beta_{2} - 27 \beta_1 + 284) q^{49} - 50 q^{50} + ( - 12 \beta_1 + 96) q^{52} + ( - 3 \beta_{2} + 12 \beta_1 - 291) q^{53} + ( - 5 \beta_1 - 70) q^{55} + (8 \beta_{2} - 8 \beta_1 + 8) q^{56} + ( - 2 \beta_{2} - 16 \beta_1 + 142) q^{58} + (5 \beta_{2} - 12 \beta_1 - 169) q^{59} + ( - 2 \beta_{2} - 5 \beta_1 - 22) q^{61} + ( - 10 \beta_{2} - 10 \beta_1 - 90) q^{62} + 64 q^{64} + ( - 15 \beta_1 + 120) q^{65} + ( - 23 \beta_{2} + 21 \beta_1 - 187) q^{67} + ( - 4 \beta_{2} - 196) q^{68} + (10 \beta_{2} - 10 \beta_1 + 10) q^{70} + (33 \beta_{2} - 20 \beta_1 - 189) q^{71} + (36 \beta_{2} - 15 \beta_1 + 196) q^{73} + ( - 6 \beta_{2} - 14 \beta_1 - 138) q^{74} + (8 \beta_{2} + 4 \beta_1 + 32) q^{76} + ( - 5 \beta_1 - 190) q^{77} + (32 \beta_{2} + 8 \beta_1 + 80) q^{79} + 80 q^{80} + ( - 10 \beta_{2} - 16 \beta_1 + 310) q^{82} + ( - 3 \beta_{2} - 16 \beta_1 - 301) q^{83} + ( - 5 \beta_{2} - 245) q^{85} + ( - 20 \beta_{2} + 16 \beta_1 - 276) q^{86} + (8 \beta_1 + 112) q^{88} + (30 \beta_{2} - 20 \beta_1 + 8) q^{89} + ( - 66 \beta_{2} + 51 \beta_1 - 636) q^{91} + 92 q^{92} + ( - 12 \beta_{2} + 30 \beta_1 + 128) q^{94} + (10 \beta_{2} + 5 \beta_1 + 40) q^{95} + ( - 10 \beta_{2} - 34 \beta_1 - 176) q^{97} + ( - 14 \beta_{2} + 54 \beta_1 - 568) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{2} + 12 q^{4} + 15 q^{5} - 2 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 6 q^{2} + 12 q^{4} + 15 q^{5} - 2 q^{7} - 24 q^{8} - 30 q^{10} - 42 q^{11} + 72 q^{13} + 4 q^{14} + 48 q^{16} - 146 q^{17} + 22 q^{19} + 60 q^{20} + 84 q^{22} + 69 q^{23} + 75 q^{25} - 144 q^{26} - 8 q^{28} - 214 q^{29} + 130 q^{31} - 96 q^{32} + 292 q^{34} - 10 q^{35} + 204 q^{37} - 44 q^{38} - 120 q^{40} - 470 q^{41} + 404 q^{43} - 168 q^{44} - 138 q^{46} - 198 q^{47} + 845 q^{49} - 150 q^{50} + 288 q^{52} - 870 q^{53} - 210 q^{55} + 16 q^{56} + 428 q^{58} - 512 q^{59} - 64 q^{61} - 260 q^{62} + 192 q^{64} + 360 q^{65} - 538 q^{67} - 584 q^{68} + 20 q^{70} - 600 q^{71} + 552 q^{73} - 408 q^{74} + 88 q^{76} - 570 q^{77} + 208 q^{79} + 240 q^{80} + 940 q^{82} - 900 q^{83} - 730 q^{85} - 808 q^{86} + 336 q^{88} - 6 q^{89} - 1842 q^{91} + 276 q^{92} + 396 q^{94} + 110 q^{95} - 518 q^{97} - 1690 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 486x - 3340 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 8\nu - 326 ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 6\beta_{2} + 8\beta _1 + 326 \) 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
−17.0235
24.9023
−7.87875
−2.00000 0 4.00000 5.00000 0 −34.6885 −8.00000 0 −10.0000
1.2 −2.00000 0 4.00000 5.00000 0 8.08465 −8.00000 0 −10.0000
1.3 −2.00000 0 4.00000 5.00000 0 24.6038 −8.00000 0 −10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(23\) \(-1\)

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 2070.4.a.v 3
3.b odd 2 1 690.4.a.r 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.4.a.r 3 3.b odd 2 1
2070.4.a.v 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2070))\):

\( T_{7}^{3} + 2T_{7}^{2} - 935T_{7} + 6900 \) Copy content Toggle raw display
\( T_{11}^{3} + 42T_{11}^{2} + 102T_{11} - 720 \) Copy content Toggle raw display
\( T_{17}^{3} + 146T_{17}^{2} + 6281T_{17} + 66046 \) 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
$5$ \( (T - 5)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 2 T^{2} - 935 T + 6900 \) Copy content Toggle raw display
$11$ \( T^{3} + 42 T^{2} + 102 T - 720 \) Copy content Toggle raw display
$13$ \( T^{3} - 72 T^{2} - 2646 T + 181332 \) Copy content Toggle raw display
$17$ \( T^{3} + 146 T^{2} + 6281 T + 66046 \) Copy content Toggle raw display
$19$ \( T^{3} - 22 T^{2} - 4370 T + 104856 \) Copy content Toggle raw display
$23$ \( (T - 23)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 214 T^{2} - 19655 T - 4596918 \) Copy content Toggle raw display
$31$ \( T^{3} - 130 T^{2} - 36475 T + 1738000 \) Copy content Toggle raw display
$37$ \( T^{3} - 204 T^{2} - 25215 T - 4282 \) Copy content Toggle raw display
$41$ \( T^{3} + 470 T^{2} + 6961 T - 9879270 \) Copy content Toggle raw display
$43$ \( T^{3} - 404 T^{2} - 29212 T + 5718800 \) Copy content Toggle raw display
$47$ \( T^{3} + 198 T^{2} + \cdots - 14645360 \) Copy content Toggle raw display
$53$ \( T^{3} + 870 T^{2} + 188361 T + 6160050 \) Copy content Toggle raw display
$59$ \( T^{3} + 512 T^{2} + \cdots - 11151872 \) Copy content Toggle raw display
$61$ \( T^{3} + 64 T^{2} - 17822 T + 447660 \) Copy content Toggle raw display
$67$ \( T^{3} + 538 T^{2} + \cdots - 10794444 \) Copy content Toggle raw display
$71$ \( T^{3} + 600 T^{2} + \cdots - 131564200 \) Copy content Toggle raw display
$73$ \( T^{3} - 552 T^{2} + \cdots + 367172180 \) Copy content Toggle raw display
$79$ \( T^{3} - 208 T^{2} + \cdots + 395108352 \) Copy content Toggle raw display
$83$ \( T^{3} + 900 T^{2} + 120213 T + 4374116 \) Copy content Toggle raw display
$89$ \( T^{3} + 6 T^{2} - 711888 T - 11043792 \) Copy content Toggle raw display
$97$ \( T^{3} + 518 T^{2} + \cdots + 118990560 \) Copy content Toggle raw display
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