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Properties

Label 1040.2.bg.o
Level $1040$
Weight $2$
Character orbit 1040.bg
Analytic conductor $8.304$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1040,2,Mod(577,1040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1040, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1040.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.bg (of order \(4\), degree \(2\), not minimal)

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: no
Analytic conductor: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 8 x^{10} + 6 x^{9} + 25 x^{8} - 106 x^{7} + 242 x^{6} + 268 x^{5} + 316 x^{4} + \cdots + 676 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 520)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{7} + 1) q^{5} + \beta_{10} q^{7} + (\beta_{10} - \beta_{5} - \beta_{3} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{7} + 1) q^{5} + \beta_{10} q^{7} + (\beta_{10} - \beta_{5} - \beta_{3} + \beta_1) q^{9} + (\beta_{10} + \beta_{6} + \beta_{5} + \cdots + 1) q^{11}+ \cdots + ( - \beta_{11} + \beta_{10} + 2 \beta_{9} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{3} + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{3} + 8 q^{5} + 14 q^{11} - 2 q^{13} + 12 q^{15} + 14 q^{17} + 6 q^{19} - 18 q^{21} + 2 q^{23} + 2 q^{25} + 26 q^{27} + 22 q^{31} - 4 q^{33} + 12 q^{35} - 4 q^{39} - 4 q^{41} + 8 q^{43} + 16 q^{49} - 36 q^{53} - 20 q^{55} - 32 q^{57} + 26 q^{59} + 12 q^{61} - 28 q^{63} + 12 q^{65} - 40 q^{67} - 40 q^{69} - 28 q^{71} + 60 q^{73} - 22 q^{75} - 40 q^{77} + 36 q^{81} + 10 q^{85} - 8 q^{87} + 24 q^{89} + 8 q^{91} - 34 q^{95} + 56 q^{97} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 8 x^{10} + 6 x^{9} + 25 x^{8} - 106 x^{7} + 242 x^{6} + 268 x^{5} + 316 x^{4} + \cdots + 676 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 200229143 \nu^{11} - 987855571 \nu^{10} + 3070512122 \nu^{9} - 4131767008 \nu^{8} + \cdots - 2306446672764 ) / 751256469470 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 373202047 \nu^{11} + 1319835284 \nu^{10} - 2653636663 \nu^{9} - 1822336823 \nu^{8} + \cdots - 446294621344 ) / 751256469470 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 459983417 \nu^{11} - 2988622629 \nu^{10} + 6392846413 \nu^{9} + 5677985653 \nu^{8} + \cdots - 832054875626 ) / 751256469470 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 660199144 \nu^{11} - 3013998623 \nu^{10} + 6601428436 \nu^{9} + 1307558201 \nu^{8} + \cdots + 444281997978 ) / 751256469470 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 209470774 \nu^{11} + 901610719 \nu^{10} - 2428630465 \nu^{9} + 2078555479 \nu^{8} + \cdots - 70692166662 ) / 150251293894 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1325497056 \nu^{11} - 4341084712 \nu^{10} + 6597255989 \nu^{9} + 18489030809 \nu^{8} + \cdots + 1891478600362 ) / 751256469470 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1480119086 \nu^{11} + 2122082992 \nu^{10} + 8811358056 \nu^{9} - 62716764329 \nu^{8} + \cdots - 2994907420492 ) / 751256469470 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1660682324 \nu^{11} + 7298233383 \nu^{10} - 16803272076 \nu^{9} + 2630463284 \nu^{8} + \cdots - 2032492236138 ) / 751256469470 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2267594529 \nu^{11} - 10736159208 \nu^{10} + 23752077081 \nu^{9} + 3407895981 \nu^{8} + \cdots + 1330833370568 ) / 751256469470 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3254542157 \nu^{11} + 14625556184 \nu^{10} - 33315291503 \nu^{9} - 17768135323 \nu^{8} + \cdots - 2305886175294 ) / 751256469470 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - 4\beta_{5} - \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{9} - \beta_{7} - 2\beta_{5} - 6\beta_{3} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - 2\beta_{7} - 2\beta_{6} + \beta_{4} - 9\beta_{3} - 8\beta_{2} - 9\beta _1 - 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} - 12 \beta_{10} - 2 \beta_{9} - \beta_{8} - 2 \beta_{7} - 10 \beta_{6} + 23 \beta_{5} + \cdots - 23 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4 \beta_{11} - 62 \beta_{10} - 29 \beta_{9} + 12 \beta_{7} - 8 \beta_{6} + 144 \beta_{5} + \cdots - 77 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 17 \beta_{11} - 110 \beta_{10} - 108 \beta_{9} + 17 \beta_{8} + 108 \beta_{7} + 54 \beta_{6} + \cdots + 213 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 113 \beta_{9} + 74 \beta_{8} + 326 \beta_{7} + 326 \beta_{6} - 113 \beta_{4} + 577 \beta_{3} + \cdots + 1050 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 213 \beta_{11} + 942 \beta_{10} + 474 \beta_{9} + 213 \beta_{8} + 474 \beta_{7} + 818 \beta_{6} + \cdots + 1837 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 948 \beta_{11} + 3992 \beta_{10} + 3347 \beta_{9} - 990 \beta_{7} + 42 \beta_{6} - 8144 \beta_{5} + \cdots + 5485 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2357 \beta_{11} + 7926 \beta_{10} + 9696 \beta_{9} - 2357 \beta_{8} - 9696 \beta_{7} - 7600 \beta_{6} + \cdots - 15449 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(\beta_{5}\) \(-\beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
2.10080 − 2.10080i
1.89794 − 1.89794i
1.09797 − 1.09797i
−0.413236 + 0.413236i
−1.28020 + 1.28020i
−1.40328 + 1.40328i
2.10080 + 2.10080i
1.89794 + 1.89794i
1.09797 + 1.09797i
−0.413236 − 0.413236i
−1.28020 − 1.28020i
−1.40328 − 1.40328i
0 −2.10080 + 2.10080i 0 −2.23468 − 0.0786365i 0 − 0.625152i 0 − 5.82676i 0
577.2 0 −1.89794 + 1.89794i 0 1.64127 − 1.51862i 0 0.591509i 0 − 4.20438i 0
577.3 0 −1.09797 + 1.09797i 0 1.67999 + 1.47568i 0 3.78486i 0 0.588908i 0
577.4 0 0.413236 − 0.413236i 0 0.0648426 − 2.23513i 0 2.83200i 0 2.65847i 0
577.5 0 1.28020 − 1.28020i 0 2.10570 − 0.752353i 0 − 1.83826i 0 − 0.277848i 0
577.6 0 1.40328 − 1.40328i 0 0.742882 + 2.10906i 0 − 2.74495i 0 − 0.938393i 0
593.1 0 −2.10080 − 2.10080i 0 −2.23468 + 0.0786365i 0 0.625152i 0 5.82676i 0
593.2 0 −1.89794 − 1.89794i 0 1.64127 + 1.51862i 0 − 0.591509i 0 4.20438i 0
593.3 0 −1.09797 − 1.09797i 0 1.67999 − 1.47568i 0 − 3.78486i 0 − 0.588908i 0
593.4 0 0.413236 + 0.413236i 0 0.0648426 + 2.23513i 0 − 2.83200i 0 − 2.65847i 0
593.5 0 1.28020 + 1.28020i 0 2.10570 + 0.752353i 0 1.83826i 0 0.277848i 0
593.6 0 1.40328 + 1.40328i 0 0.742882 − 2.10906i 0 2.74495i 0 0.938393i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.bg.o 12
4.b odd 2 1 520.2.w.e ✓ 12
5.c odd 4 1 1040.2.cd.o 12
13.d odd 4 1 1040.2.cd.o 12
20.e even 4 1 520.2.bh.e yes 12
52.f even 4 1 520.2.bh.e yes 12
65.k even 4 1 inner 1040.2.bg.o 12
260.s odd 4 1 520.2.w.e ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.w.e ✓ 12 4.b odd 2 1
520.2.w.e ✓ 12 260.s odd 4 1
520.2.bh.e yes 12 20.e even 4 1
520.2.bh.e yes 12 52.f even 4 1
1040.2.bg.o 12 1.a even 1 1 trivial
1040.2.bg.o 12 65.k even 4 1 inner
1040.2.cd.o 12 5.c odd 4 1
1040.2.cd.o 12 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\):

\( T_{3}^{12} + 4 T_{3}^{11} + 8 T_{3}^{10} - 6 T_{3}^{9} + 25 T_{3}^{8} + 106 T_{3}^{7} + 242 T_{3}^{6} + \cdots + 676 \) Copy content Toggle raw display
\( T_{7}^{12} + 34T_{7}^{10} + 409T_{7}^{8} + 2112T_{7}^{6} + 4328T_{7}^{4} + 2416T_{7}^{2} + 400 \) Copy content Toggle raw display
\( T_{11}^{12} - 14 T_{11}^{11} + 98 T_{11}^{10} - 312 T_{11}^{9} + 648 T_{11}^{8} - 2228 T_{11}^{7} + \cdots + 1140624 \) Copy content Toggle raw display
\( T_{19}^{12} - 6 T_{19}^{11} + 18 T_{19}^{10} - 156 T_{19}^{9} + 3860 T_{19}^{8} - 28220 T_{19}^{7} + \cdots + 67897600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 4 T^{11} + \cdots + 676 \) Copy content Toggle raw display
$5$ \( T^{12} - 8 T^{11} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} + 34 T^{10} + \cdots + 400 \) Copy content Toggle raw display
$11$ \( T^{12} - 14 T^{11} + \cdots + 1140624 \) Copy content Toggle raw display
$13$ \( T^{12} + 2 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} - 14 T^{11} + \cdots + 3600 \) Copy content Toggle raw display
$19$ \( T^{12} - 6 T^{11} + \cdots + 67897600 \) Copy content Toggle raw display
$23$ \( T^{12} - 2 T^{11} + \cdots + 400 \) Copy content Toggle raw display
$29$ \( T^{12} + 164 T^{10} + \cdots + 77862976 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 11580342544 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 1576090000 \) Copy content Toggle raw display
$41$ \( T^{12} + 4 T^{11} + \cdots + 6150400 \) Copy content Toggle raw display
$43$ \( T^{12} - 8 T^{11} + \cdots + 2196324 \) Copy content Toggle raw display
$47$ \( T^{12} + 166 T^{10} + \cdots + 32490000 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 578691136 \) Copy content Toggle raw display
$59$ \( T^{12} - 26 T^{11} + \cdots + 1600 \) Copy content Toggle raw display
$61$ \( (T^{6} - 6 T^{5} + \cdots - 78768)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 20 T^{5} + \cdots - 520)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 282672988900 \) Copy content Toggle raw display
$73$ \( (T^{6} - 30 T^{5} + \cdots + 61696)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 347449600 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 4361281600 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 63463686400 \) Copy content Toggle raw display
$97$ \( (T^{6} - 28 T^{5} + \cdots - 78000)^{2} \) Copy content Toggle raw display
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