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Properties

Label 960.4.d.b
Level $960$
Weight $4$
Character orbit 960.d
Analytic conductor $56.642$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,4,Mod(289,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.289");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.d (of order \(2\), degree \(1\), 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: \(56.6418336055\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 18 x^{10} + 300 x^{9} + 14598 x^{8} - 60928 x^{7} + 147804 x^{6} + 1450180 x^{5} + \cdots + 2737382400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

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

\(f(q)\) \(=\) \( q + 3 q^{3} + \beta_{4} q^{5} + \beta_{6} q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + \beta_{4} q^{5} + \beta_{6} q^{7} + 9 q^{9} + (\beta_{11} + 2 \beta_{3}) q^{11} + (\beta_{10} + \beta_{5}) q^{13} + 3 \beta_{4} q^{15} + (\beta_{11} - \beta_{9} + 5 \beta_{3}) q^{17} + (2 \beta_{11} + \beta_{9} - 2 \beta_{3}) q^{19} + 3 \beta_{6} q^{21} + (2 \beta_{8} + 5 \beta_{7} + \cdots - 5 \beta_{4}) q^{23}+ \cdots + (9 \beta_{11} + 18 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 36 q^{3} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 36 q^{3} + 108 q^{9} - 44 q^{25} + 324 q^{27} + 296 q^{35} + 792 q^{41} - 1472 q^{43} - 1484 q^{49} + 952 q^{65} + 1152 q^{67} - 132 q^{75} + 972 q^{81} - 1216 q^{83} - 3112 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 18 x^{10} + 300 x^{9} + 14598 x^{8} - 60928 x^{7} + 147804 x^{6} + 1450180 x^{5} + \cdots + 2737382400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 28606200764657 \nu^{11} - 107678553130919 \nu^{10} + \cdots + 85\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5249469155 \nu^{11} - 145077666139 \nu^{10} + 3094091359655 \nu^{9} - 52134431764465 \nu^{8} + \cdots - 45\!\cdots\!00 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6915331370263 \nu^{11} - 80794468014023 \nu^{10} + 323442252748465 \nu^{9} + \cdots + 20\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 40\!\cdots\!51 \nu^{11} + \cdots + 30\!\cdots\!00 ) / 70\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11\!\cdots\!19 \nu^{11} + \cdots - 78\!\cdots\!00 ) / 64\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\!\cdots\!93 \nu^{11} + \cdots + 72\!\cdots\!00 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 16\!\cdots\!19 \nu^{11} + \cdots - 66\!\cdots\!00 ) / 70\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 215567487415101 \nu^{11} + \cdots + 64\!\cdots\!00 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!61 \nu^{11} + \cdots - 25\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 169860038210899 \nu^{11} + 661777164565559 \nu^{10} + \cdots - 99\!\cdots\!00 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 26\!\cdots\!45 \nu^{11} + \cdots + 64\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} - \beta_{8} - 4\beta_{7} - \beta_{3} + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{11} + 2\beta_{9} - \beta_{8} - 2\beta_{7} + 2\beta_{4} - 58\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 12 \beta_{11} - 90 \beta_{10} + 6 \beta_{9} - 73 \beta_{8} - 34 \beta_{7} - 90 \beta_{6} + 90 \beta_{5} + \cdots - 671 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -178\beta_{10} + 288\beta_{7} + 180\beta_{5} + 288\beta_{4} + 159\beta_{3} - 111\beta_{2} + 318\beta _1 - 22043 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2660 \beta_{11} - 10764 \beta_{10} - 1570 \beta_{9} + 7499 \beta_{8} + 36526 \beta_{7} + \cdots - 107981 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 68416 \beta_{11} - 42608 \beta_{9} + 21779 \beta_{8} + 62808 \beta_{7} + 39690 \beta_{6} + \cdots + 687052 \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 460348 \beta_{11} + 1362080 \beta_{10} - 287294 \beta_{9} + 897887 \beta_{8} + 928386 \beta_{7} + \cdots + 17617219 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5156872 \beta_{10} - 6779232 \beta_{7} - 6787440 \beta_{5} - 6779232 \beta_{4} - 2775691 \beta_{3} + \cdots + 331530127 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 73347180 \beta_{11} + 175719976 \beta_{10} + 46541190 \beta_{9} - 113872741 \beta_{8} + \cdots + 2774167879 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1136905264 \beta_{11} + 724446632 \beta_{9} - 519123541 \beta_{8} - 1604238032 \beta_{7} + \cdots - 11242050388 \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 11181281892 \beta_{11} - 22961104200 \beta_{10} + 7128161106 \beta_{9} - 14804378353 \beta_{8} + \cdots - 421812793541 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
4.41710 + 4.41710i
4.41710 − 4.41710i
8.31198 − 8.31198i
8.31198 + 8.31198i
−2.21647 − 2.21647i
−2.21647 + 2.21647i
3.21647 − 3.21647i
3.21647 + 3.21647i
−7.31198 − 7.31198i
−7.31198 + 7.31198i
−3.41710 + 3.41710i
−3.41710 − 3.41710i
0 3.00000 0 −10.7841 − 2.94995i 0 11.9816i 0 9.00000 0
289.2 0 3.00000 0 −10.7841 + 2.94995i 0 − 11.9816i 0 9.00000 0
289.3 0 3.00000 0 −6.59835 − 9.02562i 0 28.2854i 0 9.00000 0
289.4 0 3.00000 0 −6.59835 + 9.02562i 0 − 28.2854i 0 9.00000 0
289.5 0 3.00000 0 −4.70786 − 10.1408i 0 − 21.3630i 0 9.00000 0
289.6 0 3.00000 0 −4.70786 + 10.1408i 0 21.3630i 0 9.00000 0
289.7 0 3.00000 0 4.70786 − 10.1408i 0 − 21.3630i 0 9.00000 0
289.8 0 3.00000 0 4.70786 + 10.1408i 0 21.3630i 0 9.00000 0
289.9 0 3.00000 0 6.59835 − 9.02562i 0 28.2854i 0 9.00000 0
289.10 0 3.00000 0 6.59835 + 9.02562i 0 − 28.2854i 0 9.00000 0
289.11 0 3.00000 0 10.7841 − 2.94995i 0 11.9816i 0 9.00000 0
289.12 0 3.00000 0 10.7841 + 2.94995i 0 − 11.9816i 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
20.d odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.4.d.b yes 12
4.b odd 2 1 960.4.d.a ✓ 12
5.b even 2 1 960.4.d.a ✓ 12
8.b even 2 1 960.4.d.a ✓ 12
8.d odd 2 1 inner 960.4.d.b yes 12
20.d odd 2 1 inner 960.4.d.b yes 12
40.e odd 2 1 960.4.d.a ✓ 12
40.f even 2 1 inner 960.4.d.b yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.4.d.a ✓ 12 4.b odd 2 1
960.4.d.a ✓ 12 5.b even 2 1
960.4.d.a ✓ 12 8.b even 2 1
960.4.d.a ✓ 12 40.e odd 2 1
960.4.d.b yes 12 1.a even 1 1 trivial
960.4.d.b yes 12 8.d odd 2 1 inner
960.4.d.b yes 12 20.d odd 2 1 inner
960.4.d.b yes 12 40.f even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(960, [\chi])\):

\( T_{7}^{6} + 1400T_{7}^{4} + 545504T_{7}^{2} + 52417600 \) Copy content Toggle raw display
\( T_{43}^{3} + 368T_{43}^{2} - 62688T_{43} - 10272960 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T - 3)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 3814697265625 \) Copy content Toggle raw display
$7$ \( (T^{6} + 1400 T^{4} + \cdots + 52417600)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 4560 T^{4} + \cdots + 257281600)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 3896 T^{4} + \cdots - 7744)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 20764 T^{4} + \cdots + 236196000000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 26884 T^{4} + \cdots + 234937967616)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 28324 T^{4} + \cdots + 475636432896)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 18868250937600)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 31288 T^{4} + \cdots - 436868121600)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 42248 T^{4} + \cdots - 351649000000)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 198 T^{2} + \cdots + 4440408)^{4} \) Copy content Toggle raw display
$43$ \( (T^{3} + 368 T^{2} + \cdots - 10272960)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + 112068 T^{4} + \cdots + 666639590400)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 20194957454400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 33\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 288 T^{2} + \cdots - 139828032)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 83244456345600)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} + 304 T^{2} + \cdots - 450726336)^{4} \) Copy content Toggle raw display
$89$ \( (T^{3} + 778 T^{2} + \cdots - 773822440)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 35747006054400)^{2} \) Copy content Toggle raw display
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