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Properties

Label 3330.2.d.p
Level $3330$
Weight $2$
Character orbit 3330.d
Analytic conductor $26.590$
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] = [3330,2,Mod(1999,3330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3330, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3330.1999");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.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: \(26.5901838731\)
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: \( x^{10} - 2x^{9} + 2x^{8} - 4x^{7} + 51x^{6} - 124x^{5} + 154x^{4} - 46x^{3} + x^{2} + 4x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - q^{4} + ( - \beta_{9} + \beta_{5} - \beta_{4} + \cdots - 1) q^{5}+ \cdots - \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - q^{4} + ( - \beta_{9} + \beta_{5} - \beta_{4} + \cdots - 1) q^{5}+ \cdots + ( - \beta_{9} + 2 \beta_{8} + \cdots + 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{4} - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{4} - 6 q^{5} + 2 q^{10} - 6 q^{11} - 2 q^{14} + 10 q^{16} - 8 q^{19} + 6 q^{20} + 4 q^{25} + 12 q^{26} + 22 q^{29} + 46 q^{31} - 18 q^{34} - 32 q^{35} - 2 q^{40} + 14 q^{41} + 6 q^{44} + 12 q^{46} - 60 q^{49} - 8 q^{50} + 42 q^{55} + 2 q^{56} + 40 q^{59} - 18 q^{61} - 10 q^{64} - 4 q^{65} - 6 q^{70} - 12 q^{71} + 10 q^{74} + 8 q^{76} - 40 q^{79} - 6 q^{80} + 36 q^{85} + 34 q^{86} + 24 q^{89} + 32 q^{91} - 24 q^{94} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 2x^{8} - 4x^{7} + 51x^{6} - 124x^{5} + 154x^{4} - 46x^{3} + x^{2} + 4x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 8189 \nu^{9} - 27007 \nu^{8} + 63073 \nu^{7} - 48258 \nu^{6} - 261048 \nu^{5} - 1171720 \nu^{4} + \cdots + 81466 ) / 1402866 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9773 \nu^{9} + 29441 \nu^{8} + 10432 \nu^{7} - 56736 \nu^{6} - 566796 \nu^{5} + 1401884 \nu^{4} + \cdots - 972956 ) / 1402866 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11204 \nu^{9} - 16670 \nu^{8} + 30482 \nu^{7} - 71109 \nu^{6} + 540120 \nu^{5} - 1213433 \nu^{4} + \cdots - 346156 ) / 467622 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 76468 \nu^{9} + 174376 \nu^{8} - 151780 \nu^{7} + 373635 \nu^{6} - 3969330 \nu^{5} + \cdots - 912988 ) / 1402866 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 93757 \nu^{9} + 280360 \nu^{8} - 307876 \nu^{7} + 400662 \nu^{6} - 5058726 \nu^{5} + \cdots - 1174924 ) / 1402866 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 109244 \nu^{9} + 194594 \nu^{8} - 195959 \nu^{7} + 396594 \nu^{6} - 5584197 \nu^{5} + \cdots + 2825248 ) / 1402866 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 116740 \nu^{9} + 142891 \nu^{8} - 66016 \nu^{7} + 264366 \nu^{6} - 5536122 \nu^{5} + \cdots + 123896 ) / 1402866 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 150175 \nu^{9} + 250360 \nu^{8} - 304354 \nu^{7} + 635400 \nu^{6} - 7542114 \nu^{5} + \cdots - 1400668 ) / 1402866 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{6} + \beta_{3} - 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{6} + 6\beta_{4} - 6\beta_{3} + 2\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{9} + 6\beta_{7} - \beta_{5} - 9\beta_{4} + 2\beta_{3} + 2\beta _1 - 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{9} + 10\beta_{8} - 10\beta_{7} - 3\beta_{6} + 2\beta_{4} + 2\beta_{3} - 24\beta_{2} - 31\beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{9} - 39 \beta_{8} + 2 \beta_{7} + 39 \beta_{6} + 9 \beta_{5} + 27 \beta_{4} - 73 \beta_{3} + \cdots + 34 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 82 \beta_{9} + 36 \beta_{8} + 36 \beta_{7} - 82 \beta_{6} + 14 \beta_{5} - 290 \beta_{4} + 290 \beta_{3} + \cdots - 218 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 276 \beta_{9} + 29 \beta_{8} - 276 \beta_{7} + 29 \beta_{6} + 60 \beta_{5} + 576 \beta_{4} - 269 \beta_{3} + \cdots + 1132 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 329 \beta_{9} - 636 \beta_{8} + 636 \beta_{7} + 329 \beta_{6} - 298 \beta_{4} - 298 \beta_{3} + \cdots - 1810 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1999.1
1.24331 + 1.24331i
−1.95884 − 1.95884i
−0.282359 − 0.282359i
0.478560 + 0.478560i
1.51933 + 1.51933i
1.24331 − 1.24331i
−1.95884 + 1.95884i
−0.282359 + 0.282359i
0.478560 − 0.478560i
1.51933 − 1.51933i
− 1.00000i 0 −1.00000 −2.07757 + 0.826871i 0 4.67211i 1.00000i 0 0.826871 + 2.07757i
1999.2 − 1.00000i 0 −1.00000 −1.74265 + 1.40113i 0 − 3.20984i 1.00000i 0 1.40113 + 1.74265i
1999.3 − 1.00000i 0 −1.00000 −1.70518 − 1.44650i 0 − 1.83227i 1.00000i 0 −1.44650 + 1.70518i
1999.4 − 1.00000i 0 −1.00000 1.10390 + 1.94458i 0 3.51336i 1.00000i 0 1.94458 − 1.10390i
1999.5 − 1.00000i 0 −1.00000 1.42149 − 1.72608i 0 − 4.14336i 1.00000i 0 −1.72608 − 1.42149i
1999.6 1.00000i 0 −1.00000 −2.07757 − 0.826871i 0 − 4.67211i − 1.00000i 0 0.826871 − 2.07757i
1999.7 1.00000i 0 −1.00000 −1.74265 − 1.40113i 0 3.20984i − 1.00000i 0 1.40113 − 1.74265i
1999.8 1.00000i 0 −1.00000 −1.70518 + 1.44650i 0 1.83227i − 1.00000i 0 −1.44650 − 1.70518i
1999.9 1.00000i 0 −1.00000 1.10390 − 1.94458i 0 − 3.51336i − 1.00000i 0 1.94458 + 1.10390i
1999.10 1.00000i 0 −1.00000 1.42149 + 1.72608i 0 4.14336i − 1.00000i 0 −1.72608 + 1.42149i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1999.10
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 3330.2.d.p 10
3.b odd 2 1 370.2.b.d ✓ 10
5.b even 2 1 inner 3330.2.d.p 10
15.d odd 2 1 370.2.b.d ✓ 10
15.e even 4 1 1850.2.a.bd 5
15.e even 4 1 1850.2.a.be 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.b.d ✓ 10 3.b odd 2 1
370.2.b.d ✓ 10 15.d odd 2 1
1850.2.a.bd 5 15.e even 4 1
1850.2.a.be 5 15.e even 4 1
3330.2.d.p 10 1.a even 1 1 trivial
3330.2.d.p 10 5.b 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 \(S_{2}^{\mathrm{new}}(3330, [\chi])\):

\( T_{7}^{10} + 65T_{7}^{8} + 1592T_{7}^{6} + 18096T_{7}^{4} + 92800T_{7}^{2} + 160000 \) Copy content Toggle raw display
\( T_{11}^{5} + 3T_{11}^{4} - 23T_{11}^{3} - 71T_{11}^{2} + 19T_{11} + 1 \) Copy content Toggle raw display
\( T_{17}^{10} + 89T_{17}^{8} + 2752T_{17}^{6} + 35424T_{17}^{4} + 172800T_{17}^{2} + 135424 \) Copy content Toggle raw display
\( T_{29}^{5} - 11T_{29}^{4} - 9T_{29}^{3} + 251T_{29}^{2} + 5T_{29} - 335 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 6 T^{9} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( T^{10} + 65 T^{8} + \cdots + 160000 \) Copy content Toggle raw display
$11$ \( (T^{5} + 3 T^{4} - 23 T^{3} + \cdots + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 94 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{10} + 89 T^{8} + \cdots + 135424 \) Copy content Toggle raw display
$19$ \( (T^{5} + 4 T^{4} + \cdots + 640)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 86 T^{8} + \cdots + 82944 \) Copy content Toggle raw display
$29$ \( (T^{5} - 11 T^{4} + \cdots - 335)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 23 T^{4} + \cdots - 10511)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$41$ \( (T^{5} - 7 T^{4} + \cdots - 5821)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 185 T^{8} + \cdots + 9339136 \) Copy content Toggle raw display
$47$ \( T^{10} + 232 T^{8} + \cdots + 64000000 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 1076889856 \) Copy content Toggle raw display
$59$ \( (T^{5} - 20 T^{4} + \cdots - 3136)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 9 T^{4} + \cdots - 2323)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 202 T^{8} + \cdots + 153664 \) Copy content Toggle raw display
$71$ \( (T^{5} + 6 T^{4} + \cdots + 13792)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 216855076 \) Copy content Toggle raw display
$79$ \( (T^{5} + 20 T^{4} + \cdots + 8168)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 384 T^{8} + \cdots + 7573504 \) Copy content Toggle raw display
$89$ \( (T^{5} - 12 T^{4} + \cdots - 55552)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 497 T^{8} + \cdots + 1016064 \) Copy content Toggle raw display
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