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Properties

Label 2160.3.e.d
Level $2160$
Weight $3$
Character orbit 2160.e
Analytic conductor $58.856$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2160,3,Mod(271,2160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2160.271");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2160.e (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: \(58.8557371018\)
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: \( x^{12} - 3 x^{11} + 17 x^{10} - 22 x^{9} + 127 x^{8} - 157 x^{7} + 552 x^{6} - 9 x^{5} + 251 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{6} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{5} - \beta_{2} q^{7} - \beta_{10} q^{11} + (\beta_{7} + \beta_{3} - 1) q^{13} + (\beta_1 - 7) q^{17} + ( - \beta_{10} - \beta_{9} + \cdots - \beta_{6}) q^{19} + ( - \beta_{11} - \beta_{8} + \cdots + \beta_{2}) q^{23}+ \cdots + ( - 5 \beta_{5} + 32 \beta_{3} + \cdots + 20) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{13} - 84 q^{17} + 60 q^{25} - 132 q^{29} + 72 q^{37} - 84 q^{41} - 204 q^{49} - 12 q^{53} + 12 q^{61} - 60 q^{65} - 84 q^{73} + 144 q^{77} - 300 q^{89} + 240 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 17 x^{10} - 22 x^{9} + 127 x^{8} - 157 x^{7} + 552 x^{6} - 9 x^{5} + 251 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 103995 \nu^{11} + 773970 \nu^{10} + 464070 \nu^{9} + 10429977 \nu^{8} + 20002182 \nu^{7} + \cdots - 82445821 ) / 14223533 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3992325 \nu^{11} - 17235734 \nu^{10} + 91074959 \nu^{9} - 192320617 \nu^{8} + 730725662 \nu^{7} + \cdots - 150738759 ) / 113788264 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 453 \nu^{11} - 692 \nu^{10} + 5725 \nu^{9} + 1223 \nu^{8} + 43732 \nu^{7} + 12439 \nu^{6} + \cdots + 20411 ) / 8104 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2276619 \nu^{11} + 3736366 \nu^{10} - 29928170 \nu^{9} - 3332617 \nu^{8} - 217019654 \nu^{7} + \cdots - 9893094 ) / 28447066 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3124347 \nu^{11} + 3705058 \nu^{10} - 38696408 \nu^{9} - 20051353 \nu^{8} - 313022954 \nu^{7} + \cdots + 153002422 ) / 28447066 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 26184127 \nu^{11} - 78376428 \nu^{10} + 445393367 \nu^{9} - 574434731 \nu^{8} + 3331647244 \nu^{7} + \cdots - 132280207 ) / 113788264 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21529392 \nu^{11} + 31520113 \nu^{10} - 273027365 \nu^{9} - 73009024 \nu^{8} + \cdots - 1565756991 ) / 56894132 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10470 \nu^{11} + 30530 \nu^{10} - 174968 \nu^{9} + 214132 \nu^{8} - 1304826 \nu^{7} + \cdots + 42557 ) / 14041 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 251724799 \nu^{11} + 733868086 \nu^{10} - 4204791829 \nu^{9} + 5169350411 \nu^{8} + \cdots + 971707293 ) / 113788264 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 261532761 \nu^{11} - 775667068 \nu^{10} + 4406374085 \nu^{9} - 5562076365 \nu^{8} + \cdots - 909225421 ) / 113788264 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 601540329 \nu^{11} + 1770509206 \nu^{10} - 10121230575 \nu^{9} + 12648064621 \nu^{8} + \cdots + 2749676847 ) / 113788264 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} - \beta_{10} + \beta_{9} + 3\beta_{8} + 7\beta_{6} - \beta_{5} - 2\beta_{3} - 4\beta_{2} - \beta _1 + 6 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{10} - 2\beta_{8} - \beta_{7} + 27\beta_{6} + \beta_{5} + \beta_{4} - 3\beta_{3} - 4\beta_{2} - 25 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{5} - \beta_{4} + 8\beta_{3} + 3\beta _1 - 30 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 7 \beta_{11} + 9 \beta_{10} - \beta_{9} + 16 \beta_{8} - 8 \beta_{7} - 214 \beta_{6} + 9 \beta_{5} + \cdots - 197 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 43 \beta_{11} + 33 \beta_{10} - 65 \beta_{9} - 193 \beta_{8} + 4 \beta_{7} - 635 \beta_{6} - 83 \beta_{5} + \cdots + 566 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 32\beta_{7} - 40\beta_{5} - 30\beta_{4} + 96\beta_{3} - 7\beta _1 + 803 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 311 \beta_{11} - 287 \beta_{10} + 533 \beta_{9} + 1413 \beta_{8} + 66 \beta_{7} + 5783 \beta_{6} + \cdots + 5186 ) / 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 358 \beta_{11} - 606 \beta_{10} + 226 \beta_{9} - 781 \beta_{8} - 516 \beta_{7} + 14335 \beta_{6} + \cdots - 13181 ) / 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -400\beta_{7} + 2853\beta_{5} - 281\beta_{4} + 2296\beta_{3} + 1525\beta _1 - 23394 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2565 \beta_{11} + 4975 \beta_{10} - 2480 \beta_{9} + 5494 \beta_{8} - 4195 \beta_{7} - 118241 \beta_{6} + \cdots - 108705 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 16269 \beta_{11} + 21455 \beta_{10} - 36841 \beta_{9} - 74659 \beta_{8} + 8586 \beta_{7} + \cdots + 417434 ) / 24 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(1297\) \(1621\) \(2081\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
1.46938 − 2.54504i
−1.34180 − 2.32406i
0.0633998 + 0.109812i
0.0633998 − 0.109812i
−1.34180 + 2.32406i
1.46938 + 2.54504i
0.232776 + 0.403180i
1.44728 − 2.50676i
−0.371039 − 0.642658i
−0.371039 + 0.642658i
1.44728 + 2.50676i
0.232776 − 0.403180i
0 0 0 −2.23607 0 − 8.95014i 0 0 0
271.2 0 0 0 −2.23607 0 − 4.03609i 0 0 0
271.3 0 0 0 −2.23607 0 − 2.77311i 0 0 0
271.4 0 0 0 −2.23607 0 2.77311i 0 0 0
271.5 0 0 0 −2.23607 0 4.03609i 0 0 0
271.6 0 0 0 −2.23607 0 8.95014i 0 0 0
271.7 0 0 0 2.23607 0 − 12.4278i 0 0 0
271.8 0 0 0 2.23607 0 − 10.9675i 0 0 0
271.9 0 0 0 2.23607 0 − 4.14472i 0 0 0
271.10 0 0 0 2.23607 0 4.14472i 0 0 0
271.11 0 0 0 2.23607 0 10.9675i 0 0 0
271.12 0 0 0 2.23607 0 12.4278i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.3.e.d ✓ 12
3.b odd 2 1 2160.3.e.e yes 12
4.b odd 2 1 inner 2160.3.e.d ✓ 12
12.b even 2 1 2160.3.e.e yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2160.3.e.d ✓ 12 1.a even 1 1 trivial
2160.3.e.d ✓ 12 4.b odd 2 1 inner
2160.3.e.e yes 12 3.b odd 2 1
2160.3.e.e yes 12 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2160, [\chi])\):

\( T_{7}^{12} + 396T_{7}^{10} + 55728T_{7}^{8} + 3351456T_{7}^{6} + 83820096T_{7}^{4} + 886837248T_{7}^{2} + 3202654464 \) Copy content Toggle raw display
\( T_{17}^{6} + 42T_{17}^{5} - 9T_{17}^{4} - 17388T_{17}^{3} - 141993T_{17}^{2} + 1008450T_{17} + 9920961 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 3202654464 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 33077424384 \) Copy content Toggle raw display
$13$ \( (T^{6} + 6 T^{5} + \cdots + 165616)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 42 T^{5} + \cdots + 9920961)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 2538595330209 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 231876360335169 \) Copy content Toggle raw display
$29$ \( (T^{6} + 66 T^{5} + \cdots + 1761264)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 12\!\cdots\!89 \) Copy content Toggle raw display
$37$ \( (T^{6} - 36 T^{5} + \cdots - 10282541504)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 42 T^{5} + \cdots + 683265456)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 32\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{6} + 6 T^{5} + \cdots + 28701795069)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 45\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{6} - 6 T^{5} + \cdots + 82334401)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 24\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( (T^{6} + 42 T^{5} + \cdots - 112495289744)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 16\!\cdots\!89 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 14\!\cdots\!69 \) Copy content Toggle raw display
$89$ \( (T^{6} + 150 T^{5} + \cdots + 254818404144)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 120 T^{5} + \cdots + 140420018176)^{2} \) Copy content Toggle raw display
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