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Properties

Label 1620.4.d.a
Level $1620$
Weight $4$
Character orbit 1620.d
Analytic conductor $95.583$
Analytic rank $0$
Dimension $10$
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] = [1620,4,Mod(649,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1620.649");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1620.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: \(95.5830942093\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} + 74 x^{8} + 552 x^{7} - 11155 x^{6} + 179300 x^{5} - 1394375 x^{4} + \cdots + 30517578125 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{5} - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{5} - \beta_1 q^{7} + (\beta_{5} + 6) q^{11} + \beta_{2} q^{13} + (\beta_{9} + \beta_{4}) q^{17} + (\beta_{6} - 4) q^{19} + (\beta_{8} + \beta_{7} + \cdots - \beta_{2}) q^{23}+ \cdots + ( - 3 \beta_{9} - \beta_{8} + \cdots - 10) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{5} + 56 q^{11} - 42 q^{19} - 144 q^{25} + 96 q^{29} - 44 q^{31} + 110 q^{35} - 298 q^{41} - 636 q^{49} - 88 q^{55} - 370 q^{59} - 232 q^{61} + 420 q^{65} + 672 q^{71} + 168 q^{79} + 1010 q^{85} - 2036 q^{89} + 6 q^{91} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} + 74 x^{8} + 552 x^{7} - 11155 x^{6} + 179300 x^{5} - 1394375 x^{4} + \cdots + 30517578125 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 771 \nu^{9} - 12917 \nu^{8} - 1248321 \nu^{7} + 10443217 \nu^{6} - 22145130 \nu^{5} + \cdots - 15283447265625 ) / 1380859375000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1683 \nu^{9} - 34634 \nu^{8} + 873333 \nu^{7} - 2459766 \nu^{6} + 147657240 \nu^{5} + \cdots + 6802734375000 ) / 690429687500 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{9} + 2 \nu^{8} - 74 \nu^{7} - 552 \nu^{6} + 11155 \nu^{5} - 179300 \nu^{4} + \cdots + 244140625 ) / 244140625 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{9} + 2 \nu^{8} - 74 \nu^{7} - 552 \nu^{6} + 11155 \nu^{5} - 179300 \nu^{4} + \cdots + 488281250 ) / 244140625 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 852 \nu^{9} + 29079 \nu^{8} + 38452 \nu^{7} - 1491429 \nu^{6} + 41646310 \nu^{5} + \cdots + 3929443359375 ) / 197265625000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1228 \nu^{9} + 23169 \nu^{8} - 304128 \nu^{7} - 6504019 \nu^{6} - 5459590 \nu^{5} + \cdots + 3240478515625 ) / 197265625000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10849 \nu^{9} + 201948 \nu^{8} - 1397701 \nu^{7} - 3118898 \nu^{6} + \cdots + 35697265625000 ) / 1380859375000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 18047 \nu^{9} + 467906 \nu^{8} - 2094397 \nu^{7} + 79445444 \nu^{6} - 49590660 \nu^{5} + \cdots + 33157714843750 ) / 1380859375000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17972 \nu^{9} + 96819 \nu^{8} - 639178 \nu^{7} + 9756081 \nu^{6} - 130013090 \nu^{5} + \cdots - 33859130859375 ) / 690429687500 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{3} + 1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{7} - 5\beta_{5} - 7\beta_{4} + 2\beta_{3} - 5\beta _1 - 73 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5 \beta_{9} + 30 \beta_{8} - 10 \beta_{7} + 35 \beta_{6} - 85 \beta_{5} + 191 \beta_{4} - 16 \beta_{3} + \cdots - 1101 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 360 \beta_{9} - 115 \beta_{8} - 140 \beta_{7} + 320 \beta_{6} + 325 \beta_{5} - 2318 \beta_{4} + \cdots + 25743 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4400 \beta_{9} - 200 \beta_{8} - 50 \beta_{7} - 1950 \beta_{6} + 9700 \beta_{5} + 6689 \beta_{4} + \cdots - 269714 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5550 \beta_{9} + 22800 \beta_{8} + 49680 \beta_{7} - 78150 \beta_{6} - 29130 \beta_{5} - 91182 \beta_{4} + \cdots + 1394727 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 271880 \beta_{9} + 146030 \beta_{8} - 358960 \beta_{7} - 345090 \beta_{6} + 164740 \beta_{5} + \cdots - 18088351 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5681160 \beta_{9} + 1819060 \beta_{8} + 2836535 \beta_{7} + 3463420 \beta_{6} + 18790325 \beta_{5} + \cdots - 192518157 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 118478525 \beta_{9} + 40466050 \beta_{8} - 27711750 \beta_{7} + 45277175 \beta_{6} + \cdots - 6049474089 ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
10.3462 − 4.23743i
10.3462 + 4.23743i
5.47180 − 9.74984i
5.47180 + 9.74984i
0.265737 − 11.1772i
0.265737 + 11.1772i
−3.95824 − 10.4562i
−3.95824 + 10.4562i
−11.1255 − 1.10587i
−11.1255 + 1.10587i
0 0 0 −10.3462 − 4.23743i 0 6.04534i 0 0 0
649.2 0 0 0 −10.3462 + 4.23743i 0 − 6.04534i 0 0 0
649.3 0 0 0 −5.47180 − 9.74984i 0 − 27.9960i 0 0 0
649.4 0 0 0 −5.47180 + 9.74984i 0 27.9960i 0 0 0
649.5 0 0 0 −0.265737 − 11.1772i 0 28.9801i 0 0 0
649.6 0 0 0 −0.265737 + 11.1772i 0 − 28.9801i 0 0 0
649.7 0 0 0 3.95824 − 10.4562i 0 − 0.0213561i 0 0 0
649.8 0 0 0 3.95824 + 10.4562i 0 0.0213561i 0 0 0
649.9 0 0 0 11.1255 − 1.10587i 0 − 19.3089i 0 0 0
649.10 0 0 0 11.1255 + 1.10587i 0 19.3089i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.4.d.a ✓ 10
3.b odd 2 1 1620.4.d.b yes 10
5.b even 2 1 inner 1620.4.d.a ✓ 10
15.d odd 2 1 1620.4.d.b yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.4.d.a ✓ 10 1.a even 1 1 trivial
1620.4.d.a ✓ 10 5.b even 2 1 inner
1620.4.d.b yes 10 3.b odd 2 1
1620.4.d.b yes 10 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{10} + 2033T_{7}^{8} + 1336552T_{7}^{6} + 291596608T_{7}^{4} + 8969155712T_{7}^{2} + 4090624 \) Copy content Toggle raw display
\( T_{11}^{5} - 28T_{11}^{4} - 3581T_{11}^{3} + 44540T_{11}^{2} + 2236036T_{11} - 10659856 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 30517578125 \) Copy content Toggle raw display
$7$ \( T^{10} + 2033 T^{8} + \cdots + 4090624 \) Copy content Toggle raw display
$11$ \( (T^{5} - 28 T^{4} + \cdots - 10659856)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 168818204160000 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{5} + 21 T^{4} + \cdots - 1461871260)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 31\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( (T^{5} - 48 T^{4} + \cdots + 88589666304)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 22 T^{4} + \cdots + 1585436000)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{5} + 149 T^{4} + \cdots - 908061766852)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 88\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots - 1699063165936)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 116 T^{4} + \cdots + 556567333000)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 86\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 6521788165680)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 158848600780800)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 22696295877890)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
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