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Properties

Label 3360.2.d.a
Level $3360$
Weight $2$
Character orbit 3360.d
Analytic conductor $26.830$
Analytic rank $0$
Dimension $16$
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] = [3360,2,Mod(2911,3360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3360.2911");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3360.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: \(26.8297350792\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 24x^{14} + 238x^{12} + 1262x^{10} + 3861x^{8} + 6834x^{6} + 6589x^{4} + 2916x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta_{5} q^{5} - \beta_{10} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + \beta_{5} q^{5} - \beta_{10} q^{7} + q^{9} + ( - \beta_{15} - \beta_{5}) q^{11} + ( - \beta_{15} - \beta_{6}) q^{13} - \beta_{5} q^{15} + (\beta_{14} - \beta_{12} + \cdots - \beta_{3}) q^{17}+ \cdots + ( - \beta_{15} - \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{3} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{3} + 16 q^{9} + 16 q^{19} - 16 q^{25} - 16 q^{27} + 8 q^{29} - 8 q^{31} - 8 q^{37} + 16 q^{47} - 16 q^{49} - 48 q^{53} + 8 q^{55} - 16 q^{57} + 16 q^{59} - 8 q^{65} + 16 q^{75} + 16 q^{77} + 16 q^{81} - 24 q^{83} + 8 q^{85} - 8 q^{87} + 24 q^{91} + 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{14} + 238x^{12} + 1262x^{10} + 3861x^{8} + 6834x^{6} + 6589x^{4} + 2916x^{2} + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{14} + 39\nu^{12} + 296\nu^{10} + 1111\nu^{8} + 2187\nu^{6} + 2274\nu^{4} + 1226\nu^{2} + 18\nu + 216 ) / 18 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{15} - 84\nu^{13} - 488\nu^{11} - 982\nu^{9} + 639\nu^{7} + 4332\nu^{5} + 3613\nu^{3} + 90\nu ) / 54 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{14} + 24\nu^{12} + 238\nu^{10} + 1244\nu^{8} + 3591\nu^{6} + 5412\nu^{4} + 3493\nu^{2} + 486 ) / 18 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 25 \nu^{15} - 510 \nu^{13} - 4114 \nu^{11} - 16718 \nu^{9} - 35973 \nu^{7} - 39036 \nu^{5} + \cdots - 1422 \nu ) / 216 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 23 \nu^{15} - 498 \nu^{13} - 4286 \nu^{11} - 18658 \nu^{9} - 43227 \nu^{7} - 51396 \nu^{5} + \cdots - 3618 \nu ) / 216 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{15} - 102\nu^{13} - 830\nu^{11} - 3466\nu^{9} - 7965\nu^{7} - 9996\nu^{5} - 6197\nu^{3} - 1278\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -13\nu^{14} - 258\nu^{12} - 2014\nu^{10} - 7892\nu^{8} - 16425\nu^{6} - 17598\nu^{4} - 8347\nu^{2} - 936 ) / 18 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{14} - 21\nu^{12} - 176\nu^{10} - 753\nu^{8} - 1740\nu^{6} - 2092\nu^{4} - 1109\nu^{2} - 2\nu - 136 ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 17 \nu^{15} - 90 \nu^{14} - 390 \nu^{13} - 1836 \nu^{12} - 3614 \nu^{11} - 14832 \nu^{10} + \cdots - 8100 ) / 216 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{14} - 21\nu^{12} - 176\nu^{10} - 753\nu^{8} - 1740\nu^{6} - 2092\nu^{4} - 1109\nu^{2} + 2\nu - 136 ) / 2 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 17 \nu^{15} + 90 \nu^{14} - 390 \nu^{13} + 1836 \nu^{12} - 3614 \nu^{11} + 14832 \nu^{10} + \cdots + 8100 ) / 216 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 19\nu^{14} + 402\nu^{12} + 3406\nu^{10} + 14798\nu^{8} + 34911\nu^{6} + 43032\nu^{4} + 23203\nu^{2} + 2610 ) / 36 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 31 \nu^{15} + 45 \nu^{14} - 618 \nu^{13} + 918 \nu^{12} - 4840 \nu^{11} + 7416 \nu^{10} + \cdots + 4050 ) / 108 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 113 \nu^{15} + 2406 \nu^{13} + 20522 \nu^{11} + 89758 \nu^{9} + 213093 \nu^{7} + 264588 \nu^{5} + \cdots + 18918 \nu ) / 216 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{14} - \beta_{12} - 4\beta_{11} + 4\beta_{9} - \beta_{5} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{13} - \beta_{11} - \beta_{9} + \beta_{4} - 7\beta _1 + 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2 \beta_{15} - 7 \beta_{14} + 9 \beta_{12} + 18 \beta_{11} + 2 \beta_{10} - 18 \beta_{9} + \cdots + 7 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 18 \beta_{13} + \beta_{12} + 8 \beta_{11} - \beta_{10} + 11 \beta_{9} + \beta_{8} - 8 \beta_{4} + \cdots - 108 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 22 \beta_{15} + 43 \beta_{14} - 63 \beta_{12} - 87 \beta_{11} - 20 \beta_{10} + 87 \beta_{9} + \cdots - 41 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 126 \beta_{13} - 16 \beta_{12} - 53 \beta_{11} + 16 \beta_{10} - 89 \beta_{9} - 14 \beta_{8} + \cdots + 524 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 178 \beta_{15} - 258 \beta_{14} + 406 \beta_{12} + 442 \beta_{11} + 148 \beta_{10} + \cdots + 226 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 812 \beta_{13} + 163 \beta_{12} + 337 \beta_{11} - 163 \beta_{10} + 640 \beta_{9} + 131 \beta_{8} + \cdots - 2686 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1280 \beta_{15} + 1542 \beta_{14} - 2526 \beta_{12} - 2331 \beta_{11} - 984 \beta_{10} + \cdots - 1212 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 5052 \beta_{13} - 1367 \beta_{12} - 2119 \beta_{11} + 1367 \beta_{10} - 4340 \beta_{9} - 1033 \beta_{8} + \cdots + 14360 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 8680 \beta_{15} - 9221 \beta_{14} + 15461 \beta_{12} + 12655 \beta_{11} + 6240 \beta_{10} + \cdots + 6417 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 30922 \beta_{13} + 10327 \beta_{12} + 13266 \beta_{11} - 10327 \beta_{10} + 28467 \beta_{9} + \cdots - 79302 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 56934 \beta_{15} + 55238 \beta_{14} - 93922 \beta_{12} - 70303 \beta_{11} - 38684 \beta_{10} + \cdots - 33820 \beta_{3} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times\).

\(n\) \(421\) \(1121\) \(1471\) \(1921\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2911.1
− 2.47382i
− 1.95785i
− 1.62047i
− 0.400519i
1.10688i
1.22519i
1.91057i
2.21001i
2.47382i
1.95785i
1.62047i
0.400519i
− 1.10688i
− 1.22519i
− 1.91057i
− 2.21001i
0 −1.00000 0 − 1.00000i 0 −2.47382 + 0.938205i 0 1.00000 0
2911.2 0 −1.00000 0 − 1.00000i 0 −1.95785 − 1.77956i 0 1.00000 0
2911.3 0 −1.00000 0 − 1.00000i 0 −1.62047 − 2.09143i 0 1.00000 0
2911.4 0 −1.00000 0 − 1.00000i 0 −0.400519 + 2.61526i 0 1.00000 0
2911.5 0 −1.00000 0 − 1.00000i 0 1.10688 − 2.40308i 0 1.00000 0
2911.6 0 −1.00000 0 − 1.00000i 0 1.22519 + 2.34498i 0 1.00000 0
2911.7 0 −1.00000 0 − 1.00000i 0 1.91057 + 1.83022i 0 1.00000 0
2911.8 0 −1.00000 0 − 1.00000i 0 2.21001 − 1.45459i 0 1.00000 0
2911.9 0 −1.00000 0 1.00000i 0 −2.47382 − 0.938205i 0 1.00000 0
2911.10 0 −1.00000 0 1.00000i 0 −1.95785 + 1.77956i 0 1.00000 0
2911.11 0 −1.00000 0 1.00000i 0 −1.62047 + 2.09143i 0 1.00000 0
2911.12 0 −1.00000 0 1.00000i 0 −0.400519 − 2.61526i 0 1.00000 0
2911.13 0 −1.00000 0 1.00000i 0 1.10688 + 2.40308i 0 1.00000 0
2911.14 0 −1.00000 0 1.00000i 0 1.22519 − 2.34498i 0 1.00000 0
2911.15 0 −1.00000 0 1.00000i 0 1.91057 − 1.83022i 0 1.00000 0
2911.16 0 −1.00000 0 1.00000i 0 2.21001 + 1.45459i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2911.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3360.2.d.a ✓ 16
4.b odd 2 1 3360.2.d.d yes 16
7.b odd 2 1 3360.2.d.d yes 16
28.d even 2 1 inner 3360.2.d.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.2.d.a ✓ 16 1.a even 1 1 trivial
3360.2.d.a ✓ 16 28.d even 2 1 inner
3360.2.d.d yes 16 4.b odd 2 1
3360.2.d.d yes 16 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3360, [\chi])\):

\( T_{11}^{16} + 92 T_{11}^{14} + 3300 T_{11}^{12} + 59712 T_{11}^{10} + 579968 T_{11}^{8} + 2933248 T_{11}^{6} + \cdots + 16384 \) Copy content Toggle raw display
\( T_{19}^{8} - 8T_{19}^{7} - 82T_{19}^{6} + 816T_{19}^{5} + 752T_{19}^{4} - 21088T_{19}^{3} + 41504T_{19}^{2} + 36480T_{19} - 91008 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T + 1)^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + 8 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} + 92 T^{14} + \cdots + 16384 \) Copy content Toggle raw display
$13$ \( T^{16} + 124 T^{14} + \cdots + 16384 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 5174788096 \) Copy content Toggle raw display
$19$ \( (T^{8} - 8 T^{7} + \cdots - 91008)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 112 T^{14} + \cdots + 5308416 \) Copy content Toggle raw display
$29$ \( (T^{8} - 4 T^{7} + \cdots - 35072)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 4 T^{7} + \cdots + 128)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 4 T^{7} + \cdots + 1711872)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 296 T^{14} + \cdots + 21233664 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 339738624 \) Copy content Toggle raw display
$47$ \( (T^{8} - 8 T^{7} + \cdots + 116992)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 24 T^{7} + \cdots - 325504)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 8 T^{7} + \cdots - 5053952)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 257446641664 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 4111875506176 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 2847399755776 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 5623803559936 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 168612069376 \) Copy content Toggle raw display
$83$ \( (T^{8} + 12 T^{7} + \cdots + 131072)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 311718516424704 \) Copy content Toggle raw display
$97$ \( T^{16} + 684 T^{14} + \cdots + 82591744 \) Copy content Toggle raw display
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