Properties

Label 2070.4.a.bj
Level $2070$
Weight $4$
Character orbit 2070.a
Self dual yes
Analytic conductor $122.134$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 68x^{2} - 111x + 342 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + 5 q^{5} + (\beta_{2} + 2 \beta_1) q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + 5 q^{5} + (\beta_{2} + 2 \beta_1) q^{7} + 8 q^{8} + 10 q^{10} + (\beta_{3} + \beta_{2} - \beta_1 + 10) q^{11} + ( - \beta_{3} + 8 \beta_1 - 5) q^{13} + (2 \beta_{2} + 4 \beta_1) q^{14} + 16 q^{16} + ( - \beta_{3} + \beta_{2} - 13 \beta_1 + 6) q^{17} + ( - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 14) q^{19} + 20 q^{20} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 20) q^{22} + 23 q^{23} + 25 q^{25} + ( - 2 \beta_{3} + 16 \beta_1 - 10) q^{26} + (4 \beta_{2} + 8 \beta_1) q^{28} + (3 \beta_{3} - 3 \beta_{2} + 14 \beta_1 - 41) q^{29} + ( - 5 \beta_{3} + 16 \beta_1 + 97) q^{31} + 32 q^{32} + ( - 2 \beta_{3} + 2 \beta_{2} - 26 \beta_1 + 12) q^{34} + (5 \beta_{2} + 10 \beta_1) q^{35} + (4 \beta_{3} - 2 \beta_{2} - 18 \beta_1 + 116) q^{37} + ( - 4 \beta_{3} + 6 \beta_{2} + 4 \beta_1 + 28) q^{38} + 40 q^{40} + (10 \beta_{3} + \beta_1 - 121) q^{41} + (6 \beta_{3} - 6 \beta_{2} + 24 \beta_1 + 222) q^{43} + (4 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 40) q^{44} + 46 q^{46} + (11 \beta_{3} + 3 \beta_{2} - 44 \beta_1 + 67) q^{47} + (9 \beta_{3} + \beta_{2} - 49 \beta_1 + 411) q^{49} + 50 q^{50} + ( - 4 \beta_{3} + 32 \beta_1 - 20) q^{52} + ( - 8 \beta_{3} - 8 \beta_{2} + 40 \beta_1 - 146) q^{53} + (5 \beta_{3} + 5 \beta_{2} - 5 \beta_1 + 50) q^{55} + (8 \beta_{2} + 16 \beta_1) q^{56} + (6 \beta_{3} - 6 \beta_{2} + 28 \beta_1 - 82) q^{58} + ( - 2 \beta_{3} - 14 \beta_{2} - 10 \beta_1 + 20) q^{59} + ( - 19 \beta_{3} + 7 \beta_{2} + 33 \beta_1 + 290) q^{61} + ( - 10 \beta_{3} + 32 \beta_1 + 194) q^{62} + 64 q^{64} + ( - 5 \beta_{3} + 40 \beta_1 - 25) q^{65} + ( - 12 \beta_{3} + 28 \beta_1 - 368) q^{67} + ( - 4 \beta_{3} + 4 \beta_{2} - 52 \beta_1 + 24) q^{68} + (10 \beta_{2} + 20 \beta_1) q^{70} + ( - 4 \beta_{3} - 28 \beta_{2} - 39 \beta_1 - 57) q^{71} + ( - 3 \beta_{3} + \beta_{2} + 44 \beta_1 + 287) q^{73} + (8 \beta_{3} - 4 \beta_{2} - 36 \beta_1 + 232) q^{74} + ( - 8 \beta_{3} + 12 \beta_{2} + 8 \beta_1 + 56) q^{76} + ( - 17 \beta_{3} + 16 \beta_{2} + 61 \beta_1 + 548) q^{77} + ( - 16 \beta_{3} - 20 \beta_{2} - 72 \beta_1 - 232) q^{79} + 80 q^{80} + (20 \beta_{3} + 2 \beta_1 - 242) q^{82} + (6 \beta_{3} - 24 \beta_{2} + 26 \beta_1 + 256) q^{83} + ( - 5 \beta_{3} + 5 \beta_{2} - 65 \beta_1 + 30) q^{85} + (12 \beta_{3} - 12 \beta_{2} + 48 \beta_1 + 444) q^{86} + (8 \beta_{3} + 8 \beta_{2} - 8 \beta_1 + 80) q^{88} + (34 \beta_{3} + 16 \beta_{2} - 30 \beta_1 + 450) q^{89} + (51 \beta_{3} - 25 \beta_{2} - 85 \beta_1 + 576) q^{91} + 92 q^{92} + (22 \beta_{3} + 6 \beta_{2} - 88 \beta_1 + 134) q^{94} + ( - 10 \beta_{3} + 15 \beta_{2} + 10 \beta_1 + 70) q^{95} + (\beta_{3} + 31 \beta_{2} - 137 \beta_1 - 504) q^{97} + (18 \beta_{3} + 2 \beta_{2} - 98 \beta_1 + 822) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8} + 40 q^{10} + 39 q^{11} - 20 q^{13} - 2 q^{14} + 64 q^{16} + 23 q^{17} + 53 q^{19} + 80 q^{20} + 78 q^{22} + 92 q^{23} + 100 q^{25} - 40 q^{26} - 4 q^{28} - 161 q^{29} + 388 q^{31} + 128 q^{32} + 46 q^{34} - 5 q^{35} + 466 q^{37} + 106 q^{38} + 160 q^{40} - 484 q^{41} + 894 q^{43} + 156 q^{44} + 184 q^{46} + 265 q^{47} + 1643 q^{49} + 200 q^{50} - 80 q^{52} - 576 q^{53} + 195 q^{55} - 8 q^{56} - 322 q^{58} + 94 q^{59} + 1153 q^{61} + 776 q^{62} + 256 q^{64} - 100 q^{65} - 1472 q^{67} + 92 q^{68} - 10 q^{70} - 200 q^{71} + 1147 q^{73} + 932 q^{74} + 212 q^{76} + 2176 q^{77} - 908 q^{79} + 320 q^{80} - 968 q^{82} + 1048 q^{83} + 115 q^{85} + 1788 q^{86} + 312 q^{88} + 1784 q^{89} + 2329 q^{91} + 368 q^{92} + 530 q^{94} + 265 q^{95} - 2047 q^{97} + 3286 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 68x^{2} - 111x + 342 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 3\nu^{2} - 50\nu + 18 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 2\nu - 34 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2\beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} + 3\beta_{2} + 56\beta _1 + 84 \) 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
−6.57209
1.58997
8.73081
−3.74869
2.00000 0 4.00000 5.00000 0 −35.4229 8.00000 0 10.0000
1.2 2.00000 0 4.00000 5.00000 0 −18.5077 8.00000 0 10.0000
1.3 2.00000 0 4.00000 5.00000 0 23.5622 8.00000 0 10.0000
1.4 2.00000 0 4.00000 5.00000 0 29.3684 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.bj 4
3.b odd 2 1 230.4.a.h 4
12.b even 2 1 1840.4.a.m 4
15.d odd 2 1 1150.4.a.p 4
15.e even 4 2 1150.4.b.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.h 4 3.b odd 2 1
1150.4.a.p 4 15.d odd 2 1
1150.4.b.n 8 15.e even 4 2
1840.4.a.m 4 12.b even 2 1
2070.4.a.bj 4 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2070))\):

\( T_{7}^{4} + T_{7}^{3} - 1507T_{7}^{2} + 2618T_{7} + 453664 \) Copy content Toggle raw display
\( T_{11}^{4} - 39T_{11}^{3} - 1771T_{11}^{2} + 94420T_{11} - 977536 \) Copy content Toggle raw display
\( T_{17}^{4} - 23T_{17}^{3} - 14205T_{17}^{2} + 590494T_{17} + 1048184 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T - 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + T^{3} - 1507 T^{2} + \cdots + 453664 \) Copy content Toggle raw display
$11$ \( T^{4} - 39 T^{3} - 1771 T^{2} + \cdots - 977536 \) Copy content Toggle raw display
$13$ \( T^{4} + 20 T^{3} - 4948 T^{2} + \cdots + 3059002 \) Copy content Toggle raw display
$17$ \( T^{4} - 23 T^{3} - 14205 T^{2} + \cdots + 1048184 \) Copy content Toggle raw display
$19$ \( T^{4} - 53 T^{3} - 15029 T^{2} + \cdots - 78336 \) Copy content Toggle raw display
$23$ \( (T - 23)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 161 T^{3} - 27260 T^{2} + \cdots + 1597064 \) Copy content Toggle raw display
$31$ \( T^{4} - 388 T^{3} + \cdots - 397027152 \) Copy content Toggle raw display
$37$ \( T^{4} - 466 T^{3} + 37920 T^{2} + \cdots - 7921024 \) Copy content Toggle raw display
$41$ \( T^{4} + 484 T^{3} + \cdots + 1506099394 \) Copy content Toggle raw display
$43$ \( T^{4} - 894 T^{3} + \cdots - 5392023552 \) Copy content Toggle raw display
$47$ \( T^{4} - 265 T^{3} + \cdots + 1306137888 \) Copy content Toggle raw display
$53$ \( T^{4} + 576 T^{3} + \cdots - 3236491568 \) Copy content Toggle raw display
$59$ \( T^{4} - 94 T^{3} + \cdots + 11673423616 \) Copy content Toggle raw display
$61$ \( T^{4} - 1153 T^{3} + \cdots - 42772329400 \) Copy content Toggle raw display
$67$ \( T^{4} + 1472 T^{3} + \cdots - 10308616192 \) Copy content Toggle raw display
$71$ \( T^{4} + 200 T^{3} + \cdots + 274201266224 \) Copy content Toggle raw display
$73$ \( T^{4} - 1147 T^{3} + \cdots - 4754107544 \) Copy content Toggle raw display
$79$ \( T^{4} + 908 T^{3} + \cdots + 145785325568 \) Copy content Toggle raw display
$83$ \( T^{4} - 1048 T^{3} + \cdots - 178673654272 \) Copy content Toggle raw display
$89$ \( T^{4} - 1784 T^{3} + \cdots - 969417760000 \) Copy content Toggle raw display
$97$ \( T^{4} + 2047 T^{3} + \cdots - 658266694276 \) Copy content Toggle raw display
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