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Properties

Label 3330.2.d.q
Level $3330$
Weight $2$
Character orbit 3330.d
Analytic conductor $26.590$
Analytic rank $0$
Dimension $14$
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 20x^{12} + 154x^{10} + 580x^{8} + 1105x^{6} + 960x^{4} + 256x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} - q^{4} + \beta_{9} q^{5} + ( - \beta_{7} + \beta_{6}) q^{7} - \beta_{6} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} - q^{4} + \beta_{9} q^{5} + ( - \beta_{7} + \beta_{6}) q^{7} - \beta_{6} q^{8} + \beta_{4} q^{10} + (\beta_{13} - \beta_{12} + \beta_{10} + \cdots + 1) q^{11}+ \cdots + ( - \beta_{12} + \beta_{11} + \cdots + \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{4} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{4} - 4 q^{5} - 6 q^{10} + 12 q^{11} - 12 q^{14} + 14 q^{16} + 4 q^{19} + 4 q^{20} + 10 q^{25} + 4 q^{26} - 16 q^{29} + 12 q^{31} - 12 q^{34} - 8 q^{35} + 6 q^{40} + 56 q^{41} - 12 q^{44} - 8 q^{46} + 18 q^{49} - 24 q^{50} + 12 q^{56} - 28 q^{59} + 12 q^{61} - 14 q^{64} + 12 q^{65} - 16 q^{70} + 48 q^{71} - 14 q^{74} - 4 q^{76} - 20 q^{79} - 4 q^{80} + 12 q^{85} + 12 q^{86} - 36 q^{89} - 32 q^{91} + 32 q^{94} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 20x^{12} + 154x^{10} + 580x^{8} + 1105x^{6} + 960x^{4} + 256x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{11} - 17\nu^{9} - 103\nu^{7} - 269\nu^{5} - 270\nu^{3} - 28\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{13} - 18 \nu^{11} - 120 \nu^{9} - 2 \nu^{8} - 374 \nu^{7} - 28 \nu^{6} - 563 \nu^{5} - 122 \nu^{4} + \cdots - 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{13} - 18 \nu^{11} - 120 \nu^{9} + 2 \nu^{8} - 374 \nu^{7} + 28 \nu^{6} - 563 \nu^{5} + 122 \nu^{4} + \cdots + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} + \nu^{10} + 17 \nu^{9} + 16 \nu^{8} + 103 \nu^{7} + 89 \nu^{6} + 271 \nu^{5} + 208 \nu^{4} + \cdots + 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{13} + 18\nu^{11} + 119\nu^{9} + 359\nu^{7} + 490\nu^{5} + 249\nu^{3} + 32\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{13} + 19\nu^{11} + 136\nu^{9} + 462\nu^{7} + 759\nu^{5} + 523\nu^{3} + 80\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{12} - 17\nu^{10} - 102\nu^{8} - 257\nu^{6} - 231\nu^{4} + 2\nu^{2} + 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{13} + 4 \nu^{12} + 18 \nu^{11} + 70 \nu^{10} + 118 \nu^{9} + 440 \nu^{8} + 344 \nu^{7} + \cdots - 24 \nu ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{13} - 4 \nu^{12} + 18 \nu^{11} - 70 \nu^{10} + 118 \nu^{9} - 440 \nu^{8} + 344 \nu^{7} + \cdots - 24 \nu ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3 \nu^{13} - 4 \nu^{12} + 50 \nu^{11} - 72 \nu^{10} + 286 \nu^{9} - 472 \nu^{8} + 624 \nu^{7} + \cdots - 72 ) / 8 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{13} - 4 \nu^{12} + 20 \nu^{11} - 72 \nu^{10} + 152 \nu^{9} - 472 \nu^{8} + 550 \nu^{7} - 1380 \nu^{6} + \cdots - 72 ) / 8 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - \nu^{13} - 6 \nu^{12} - 18 \nu^{11} - 106 \nu^{10} - 118 \nu^{9} - 676 \nu^{8} - 344 \nu^{7} + \cdots - 24 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{10} - \beta_{9} + 2\beta_{6} + \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{10} + 5\beta_{9} + 2\beta_{7} - 12\beta_{6} - 5\beta_{4} - 5\beta_{3} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{13} - \beta_{12} + \beta_{10} + 2\beta_{9} - \beta_{8} + \beta_{5} - 7\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2 \beta_{12} - 29 \beta_{10} - 27 \beta_{9} - 22 \beta_{7} + 76 \beta_{6} + 2 \beta_{5} + 27 \beta_{4} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -20\beta_{13} + 9\beta_{12} - 8\beta_{10} - 21\beta_{9} + 8\beta_{8} - 9\beta_{5} + 47\beta _1 - 88 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 24 \beta_{12} + 4 \beta_{11} + 175 \beta_{10} + 155 \beta_{9} + 182 \beta_{7} - 500 \beta_{6} + \cdots - 20 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 158 \beta_{13} - 65 \beta_{12} + 51 \beta_{10} + 172 \beta_{9} - 51 \beta_{8} + 65 \beta_{5} + \cdots + 563 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 214 \beta_{12} - 60 \beta_{11} - 1095 \beta_{10} - 941 \beta_{9} - 1378 \beta_{7} + 3372 \beta_{6} + \cdots + 154 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1164 \beta_{13} + 445 \beta_{12} - 310 \beta_{10} - 1299 \beta_{9} + 312 \beta_{8} - 445 \beta_{5} + \cdots - 3766 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1704 \beta_{12} + 608 \beta_{11} + 7069 \beta_{10} + 5973 \beta_{9} + 10058 \beta_{7} - 23084 \beta_{6} + \cdots - 1096 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 8350 \beta_{13} - 3017 \beta_{12} + 1893 \beta_{10} + 9474 \beta_{9} - 1929 \beta_{8} + 3017 \beta_{5} + \cdots + 25751 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 12842 \beta_{12} - 5240 \beta_{11} - 46765 \beta_{10} - 39163 \beta_{9} - 72118 \beta_{7} + 159436 \beta_{6} + \cdots + 7602 ) / 2 \) 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
1.58005i
− 0.299503i
2.13761i
− 1.79842i
0.546514i
− 1.52076i
− 2.64550i
− 1.58005i
0.299503i
− 2.13761i
1.79842i
− 0.546514i
1.52076i
2.64550i
− 1.00000i 0 −1.00000 −2.20427 − 0.375782i 0 − 0.496550i 1.00000i 0 −0.375782 + 2.20427i
1999.2 − 1.00000i 0 −1.00000 −1.89211 + 1.19161i 0 1.91030i 1.00000i 0 1.19161 + 1.89211i
1999.3 − 1.00000i 0 −1.00000 −1.37172 − 1.76589i 0 − 2.56940i 1.00000i 0 −1.76589 + 1.37172i
1999.4 − 1.00000i 0 −1.00000 −1.13070 + 1.92912i 0 − 1.23431i 1.00000i 0 1.92912 + 1.13070i
1999.5 − 1.00000i 0 −1.00000 0.605899 − 2.15241i 0 1.70132i 1.00000i 0 −2.15241 − 0.605899i
1999.6 − 1.00000i 0 −1.00000 1.81993 − 1.29917i 0 − 0.312701i 1.00000i 0 −1.29917 − 1.81993i
1999.7 − 1.00000i 0 −1.00000 2.17297 − 0.527467i 0 − 4.99866i 1.00000i 0 −0.527467 − 2.17297i
1999.8 1.00000i 0 −1.00000 −2.20427 + 0.375782i 0 0.496550i − 1.00000i 0 −0.375782 − 2.20427i
1999.9 1.00000i 0 −1.00000 −1.89211 − 1.19161i 0 − 1.91030i − 1.00000i 0 1.19161 − 1.89211i
1999.10 1.00000i 0 −1.00000 −1.37172 + 1.76589i 0 2.56940i − 1.00000i 0 −1.76589 − 1.37172i
1999.11 1.00000i 0 −1.00000 −1.13070 − 1.92912i 0 1.23431i − 1.00000i 0 1.92912 − 1.13070i
1999.12 1.00000i 0 −1.00000 0.605899 + 2.15241i 0 − 1.70132i − 1.00000i 0 −2.15241 + 0.605899i
1999.13 1.00000i 0 −1.00000 1.81993 + 1.29917i 0 0.312701i − 1.00000i 0 −1.29917 + 1.81993i
1999.14 1.00000i 0 −1.00000 2.17297 + 0.527467i 0 4.99866i − 1.00000i 0 −0.527467 + 2.17297i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1999.14
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.q ✓ 14
3.b odd 2 1 3330.2.d.r yes 14
5.b even 2 1 inner 3330.2.d.q ✓ 14
15.d odd 2 1 3330.2.d.r yes 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3330.2.d.q ✓ 14 1.a even 1 1 trivial
3330.2.d.q ✓ 14 5.b even 2 1 inner
3330.2.d.r yes 14 3.b odd 2 1
3330.2.d.r yes 14 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 \(S_{2}^{\mathrm{new}}(3330, [\chi])\):

\( T_{7}^{14} + 40T_{7}^{12} + 454T_{7}^{10} + 2148T_{7}^{8} + 4593T_{7}^{6} + 4044T_{7}^{4} + 1008T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{7} - 6T_{11}^{6} - 26T_{11}^{5} + 152T_{11}^{4} + 169T_{11}^{3} - 1150T_{11}^{2} - 60T_{11} + 2148 \) Copy content Toggle raw display
\( T_{17}^{14} + 128 T_{17}^{12} + 5662 T_{17}^{10} + 103404 T_{17}^{8} + 824313 T_{17}^{6} + \cdots + 1065024 \) Copy content Toggle raw display
\( T_{29}^{7} + 8T_{29}^{6} - 96T_{29}^{5} - 1016T_{29}^{4} + 356T_{29}^{3} + 27056T_{29}^{2} + 76320T_{29} + 37824 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 4 T^{13} + \cdots + 78125 \) Copy content Toggle raw display
$7$ \( T^{14} + 40 T^{12} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T^{7} - 6 T^{6} + \cdots + 2148)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + 72 T^{12} + \cdots + 141376 \) Copy content Toggle raw display
$17$ \( T^{14} + 128 T^{12} + \cdots + 1065024 \) Copy content Toggle raw display
$19$ \( (T^{7} - 2 T^{6} - 34 T^{5} + \cdots - 60)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + 148 T^{12} + \cdots + 82944 \) Copy content Toggle raw display
$29$ \( (T^{7} + 8 T^{6} + \cdots + 37824)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} - 6 T^{6} + \cdots - 8192)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$41$ \( (T^{7} - 28 T^{6} + \cdots + 30592)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 112869376 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 101929216 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 6004852230400 \) Copy content Toggle raw display
$59$ \( (T^{7} + 14 T^{6} + \cdots - 2976)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} - 6 T^{6} + \cdots + 960)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 219953496064 \) Copy content Toggle raw display
$71$ \( (T^{7} - 24 T^{6} + \cdots - 160896)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + 396 T^{12} + \cdots + 82591744 \) Copy content Toggle raw display
$79$ \( (T^{7} + 10 T^{6} + \cdots + 6560)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 357663744 \) Copy content Toggle raw display
$89$ \( (T^{7} + 18 T^{6} + \cdots + 187752)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 86590915819776 \) Copy content Toggle raw display
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