Properties

Label 2070.4.a.bm
Level $2070$
Weight $4$
Character orbit 2070.a
Self dual yes
Analytic conductor $122.134$
Analytic rank $1$
Dimension $5$
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: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 345x^{3} + 1261x^{2} + 22016x - 83268 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} - 5 q^{5} + ( - \beta_{3} - \beta_1) q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} - 5 q^{5} + ( - \beta_{3} - \beta_1) q^{7} + 8 q^{8} - 10 q^{10} + (\beta_{3} + 2 \beta_1 + 3) q^{11} + ( - \beta_{2} + \beta_1 - 1) q^{13} + ( - 2 \beta_{3} - 2 \beta_1) q^{14} + 16 q^{16} + (2 \beta_{4} + 3 \beta_{3} + \beta_1 - 10) q^{17} + ( - 3 \beta_{4} - \beta_{3} + 3 \beta_{2} - \beta_1 - 12) q^{19} - 20 q^{20} + (2 \beta_{3} + 4 \beta_1 + 6) q^{22} - 23 q^{23} + 25 q^{25} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{26} + ( - 4 \beta_{3} - 4 \beta_1) q^{28} + (2 \beta_{4} + 6 \beta_{3} - 3 \beta_1 - 5) q^{29} + ( - \beta_{4} + \beta_{3} - 6 \beta_{2} - 13 \beta_1 - 54) q^{31} + 32 q^{32} + (4 \beta_{4} + 6 \beta_{3} + 2 \beta_1 - 20) q^{34} + (5 \beta_{3} + 5 \beta_1) q^{35} + (2 \beta_{4} - \beta_{3} + 12 \beta_{2} + 9 \beta_1 + 8) q^{37} + ( - 6 \beta_{4} - 2 \beta_{3} + 6 \beta_{2} - 2 \beta_1 - 24) q^{38} - 40 q^{40} + ( - 5 \beta_{4} + 5 \beta_{3} + 6 \beta_{2} - 5 \beta_1 - 70) q^{41} + ( - 3 \beta_{4} + 3 \beta_{3} - 12 \beta_{2} - 16 \beta_1 - 89) q^{43} + (4 \beta_{3} + 8 \beta_1 + 12) q^{44} - 46 q^{46} + (3 \beta_{4} - 3 \beta_{3} + 5 \beta_{2} + 13 \beta_1 - 36) q^{47} + (7 \beta_{4} - 11 \beta_{3} - 2 \beta_{2} + 13 \beta_1 + 29) q^{49} + 50 q^{50} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{52} + ( - 4 \beta_{4} + 10 \beta_{3} - 13 \beta_{2} + 20 \beta_1 - 60) q^{53} + ( - 5 \beta_{3} - 10 \beta_1 - 15) q^{55} + ( - 8 \beta_{3} - 8 \beta_1) q^{56} + (4 \beta_{4} + 12 \beta_{3} - 6 \beta_1 - 10) q^{58} + (\beta_{4} + 7 \beta_{3} - 4 \beta_{2} + 39 \beta_1 - 190) q^{59} + (7 \beta_{4} - 10 \beta_{3} + 2 \beta_{2} + 38 \beta_1 - 8) q^{61} + ( - 2 \beta_{4} + 2 \beta_{3} - 12 \beta_{2} - 26 \beta_1 - 108) q^{62} + 64 q^{64} + (5 \beta_{2} - 5 \beta_1 + 5) q^{65} + ( - 11 \beta_{4} + \beta_{3} + 9 \beta_{2} - 22 \beta_1 - 137) q^{67} + (8 \beta_{4} + 12 \beta_{3} + 4 \beta_1 - 40) q^{68} + (10 \beta_{3} + 10 \beta_1) q^{70} + ( - 16 \beta_{3} + 4 \beta_{2} + 9 \beta_1 - 189) q^{71} + (3 \beta_{4} + 3 \beta_{3} + \beta_{2} - 7 \beta_1 + 80) q^{73} + (4 \beta_{4} - 2 \beta_{3} + 24 \beta_{2} + 18 \beta_1 + 16) q^{74} + ( - 12 \beta_{4} - 4 \beta_{3} + 12 \beta_{2} - 4 \beta_1 - 48) q^{76} + ( - 7 \beta_{4} + 3 \beta_{3} - \beta_{2} - 27 \beta_1 - 450) q^{77} + (2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 66) q^{79} - 80 q^{80} + ( - 10 \beta_{4} + 10 \beta_{3} + 12 \beta_{2} - 10 \beta_1 - 140) q^{82} + ( - 18 \beta_{4} + \beta_{3} + 4 \beta_{2} + 13 \beta_1 - 462) q^{83} + ( - 10 \beta_{4} - 15 \beta_{3} - 5 \beta_1 + 50) q^{85} + ( - 6 \beta_{4} + 6 \beta_{3} - 24 \beta_{2} - 32 \beta_1 - 178) q^{86} + (8 \beta_{3} + 16 \beta_1 + 24) q^{88} + (29 \beta_{4} + 36 \beta_{3} - 25 \beta_{2} - 21 \beta_1 - 297) q^{89} + (11 \beta_{4} - 11 \beta_{3} - 9 \beta_{2} - 5 \beta_1 - 2) q^{91} - 92 q^{92} + (6 \beta_{4} - 6 \beta_{3} + 10 \beta_{2} + 26 \beta_1 - 72) q^{94} + (15 \beta_{4} + 5 \beta_{3} - 15 \beta_{2} + 5 \beta_1 + 60) q^{95} + (5 \beta_{4} + 25 \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 225) q^{97} + (14 \beta_{4} - 22 \beta_{3} - 4 \beta_{2} + 26 \beta_1 + 58) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 10 q^{2} + 20 q^{4} - 25 q^{5} + 40 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 10 q^{2} + 20 q^{4} - 25 q^{5} + 40 q^{8} - 50 q^{10} + 16 q^{11} - 6 q^{13} + 80 q^{16} - 56 q^{17} - 48 q^{19} - 100 q^{20} + 32 q^{22} - 115 q^{23} + 125 q^{25} - 12 q^{26} - 38 q^{29} - 294 q^{31} + 160 q^{32} - 112 q^{34} + 70 q^{37} - 96 q^{38} - 200 q^{40} - 338 q^{41} - 482 q^{43} + 64 q^{44} - 230 q^{46} - 160 q^{47} + 151 q^{49} + 250 q^{50} - 24 q^{52} - 308 q^{53} - 80 q^{55} - 76 q^{58} - 928 q^{59} - 2 q^{61} - 588 q^{62} + 320 q^{64} + 30 q^{65} - 668 q^{67} - 224 q^{68} - 912 q^{71} + 386 q^{73} + 140 q^{74} - 192 q^{76} - 2268 q^{77} - 332 q^{79} - 400 q^{80} - 676 q^{82} - 2254 q^{83} + 280 q^{85} - 964 q^{86} + 128 q^{88} - 1650 q^{89} - 44 q^{91} - 460 q^{92} - 320 q^{94} + 240 q^{95} + 1100 q^{97} + 302 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 345x^{3} + 1261x^{2} + 22016x - 83268 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{4} + 131\nu^{3} - 653\nu^{2} - 22667\nu + 21672 ) / 3114 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{4} - \nu^{3} + 1768\nu^{2} - 1277\nu - 128196 ) / 3114 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\nu^{4} + 89\nu^{3} - 2344\nu^{2} - 9869\nu + 81750 ) / 1038 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + 5\beta_{3} - 2\beta_{2} - 3\beta _1 + 141 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -12\beta_{4} - 30\beta_{3} + 48\beta_{2} + 223\beta _1 - 624 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 445\beta_{4} + 1439\beta_{3} - 896\beta_{2} - 1701\beta _1 + 30429 \) 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
12.9390
10.8375
−17.6988
−8.86315
3.78546
2.00000 0 4.00000 −5.00000 0 −24.8190 8.00000 0 −10.0000
1.2 2.00000 0 4.00000 −5.00000 0 −13.7810 8.00000 0 −10.0000
1.3 2.00000 0 4.00000 −5.00000 0 −1.97872 8.00000 0 −10.0000
1.4 2.00000 0 4.00000 −5.00000 0 9.49877 8.00000 0 −10.0000
1.5 2.00000 0 4.00000 −5.00000 0 31.0799 8.00000 0 −10.0000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.bm yes 5
3.b odd 2 1 2070.4.a.bl 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2070.4.a.bl 5 3.b odd 2 1
2070.4.a.bm yes 5 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2070))\):

\( T_{7}^{5} - 933T_{7}^{3} - 4322T_{7}^{2} + 96060T_{7} + 199800 \) Copy content Toggle raw display
\( T_{11}^{5} - 16T_{11}^{4} - 1578T_{11}^{3} + 3696T_{11}^{2} + 663336T_{11} + 5343840 \) Copy content Toggle raw display
\( T_{17}^{5} + 56T_{17}^{4} - 13101T_{17}^{3} - 827150T_{17}^{2} + 27038204T_{17} + 1744652040 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{5} \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( (T + 5)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 933 T^{3} - 4322 T^{2} + \cdots + 199800 \) Copy content Toggle raw display
$11$ \( T^{5} - 16 T^{4} - 1578 T^{3} + \cdots + 5343840 \) Copy content Toggle raw display
$13$ \( T^{5} + 6 T^{4} - 1858 T^{3} + \cdots + 811296 \) Copy content Toggle raw display
$17$ \( T^{5} + 56 T^{4} + \cdots + 1744652040 \) Copy content Toggle raw display
$19$ \( T^{5} + 48 T^{4} + \cdots + 4338192384 \) Copy content Toggle raw display
$23$ \( (T + 23)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} + 38 T^{4} + \cdots + 4403148336 \) Copy content Toggle raw display
$31$ \( T^{5} + 294 T^{4} + \cdots - 5598781824 \) Copy content Toggle raw display
$37$ \( T^{5} - 70 T^{4} + \cdots + 843390184064 \) Copy content Toggle raw display
$41$ \( T^{5} + 338 T^{4} + \cdots - 1892211840 \) Copy content Toggle raw display
$43$ \( T^{5} + 482 T^{4} + \cdots + 379092659328 \) Copy content Toggle raw display
$47$ \( T^{5} + 160 T^{4} + \cdots + 12506112000 \) Copy content Toggle raw display
$53$ \( T^{5} + 308 T^{4} + \cdots + 43534649448 \) Copy content Toggle raw display
$59$ \( T^{5} + 928 T^{4} + \cdots + 9991424770200 \) Copy content Toggle raw display
$61$ \( T^{5} + 2 T^{4} + \cdots - 38169789552 \) Copy content Toggle raw display
$67$ \( T^{5} + 668 T^{4} + \cdots + 980998631856 \) Copy content Toggle raw display
$71$ \( T^{5} + 912 T^{4} + \cdots - 1168952858064 \) Copy content Toggle raw display
$73$ \( T^{5} - 386 T^{4} + \cdots - 16564651440 \) Copy content Toggle raw display
$79$ \( T^{5} + 332 T^{4} + \cdots - 1940078592 \) Copy content Toggle raw display
$83$ \( T^{5} + 2254 T^{4} + \cdots - 47036465266272 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots + 656771742201600 \) Copy content Toggle raw display
$97$ \( T^{5} - 1100 T^{4} + \cdots + 3891896438784 \) Copy content Toggle raw display
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