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Properties

Label 1512.2.bl.b
Level $1512$
Weight $2$
Character orbit 1512.bl
Analytic conductor $12.073$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1512,2,Mod(593,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1512.593");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.bl (of order \(6\), degree \(2\), 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: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7x^{14} + 84x^{12} - 208x^{10} + 882x^{8} + 4424x^{6} + 10340x^{4} + 20412x^{2} + 17689 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{14} + \beta_{13} - \beta_1) q^{5} + ( - \beta_{7} - \beta_{3} - \beta_{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{14} + \beta_{13} - \beta_1) q^{5} + ( - \beta_{7} - \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{14} + \beta_{13}) q^{11} + (\beta_{10} - 2 \beta_{7} - \beta_{6} + \cdots + 1) q^{13}+ \cdots + (4 \beta_{10} - 8 \beta_{7} - 4 \beta_{6} + \cdots + 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{7} + 6 q^{19} - 6 q^{25} + 6 q^{31} - 8 q^{37} + 44 q^{43} - 34 q^{49} - 6 q^{61} + 20 q^{67} - 14 q^{79} + 148 q^{85} - 54 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7x^{14} + 84x^{12} - 208x^{10} + 882x^{8} + 4424x^{6} + 10340x^{4} + 20412x^{2} + 17689 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 473285366 \nu^{14} - 7365255737 \nu^{12} + 99059202963 \nu^{10} - 1361795236615 \nu^{8} + \cdots - 53699894554509 ) / 68560653305375 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1464256 \nu^{14} + 8200290 \nu^{12} - 96127703 \nu^{10} + 106337337 \nu^{8} + \cdots + 28717035214 ) / 49862293313 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 285040482 \nu^{14} - 1365163856 \nu^{12} + 20183383094 \nu^{10} - 19391988975 \nu^{8} + \cdots + 12025080733848 ) / 6232786664125 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5596543866 \nu^{14} - 28043200283 \nu^{12} + 388620592892 \nu^{10} - 207424548155 \nu^{8} + \cdots + 155903938433929 ) / 68560653305375 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6180636419 \nu^{14} - 68584547822 \nu^{12} + 730415203203 \nu^{10} - 3741349430170 \nu^{8} + \cdots + 5609604722561 ) / 68560653305375 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 430301 \nu^{14} - 4063683 \nu^{12} + 44665567 \nu^{10} - 186001200 \nu^{8} + 712829332 \nu^{6} + \cdots + 4869706139 ) / 3413865125 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11810126098 \nu^{15} + 65034439414 \nu^{13} - 680307081986 \nu^{11} + \cdots - 585312942065152 \nu ) / 479924573137625 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14497121862 \nu^{15} + 195428701531 \nu^{13} - 2126473529544 \nu^{11} + \cdots - 24457013066003 \nu ) / 479924573137625 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12951687442 \nu^{14} + 116767365736 \nu^{12} - 1318107889014 \nu^{10} + \cdots - 114192712088063 ) / 68560653305375 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11364082 \nu^{15} - 69298782 \nu^{13} + 897180858 \nu^{11} - 1690835135 \nu^{9} + \cdots + 419000990163 \nu ) / 349036053191 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 29563430419 \nu^{15} + 311699289027 \nu^{13} - 3379627506598 \nu^{11} + \cdots - 67096957928266 \nu ) / 479924573137625 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 99642098 \nu^{15} + 678195784 \nu^{13} - 8305587766 \nu^{11} + 19180368775 \nu^{9} + \cdots - 2207984413397 \nu ) / 1179175855375 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 430301 \nu^{15} - 4063683 \nu^{13} + 44665567 \nu^{11} - 186001200 \nu^{9} + \cdots + 1455841014 \nu ) / 3413865125 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 226631263589 \nu^{15} - 1890688796717 \nu^{13} + 21404669752508 \nu^{11} + \cdots + 24\!\cdots\!26 \nu ) / 479924573137625 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{14} - 3\beta_{13} - 4\beta_{12} - 5\beta_{11} + 2\beta_{9} + \beta_{8} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{10} + 3\beta_{7} + 5\beta_{6} + 4\beta_{5} + 2\beta_{4} + 11\beta_{3} - 3\beta_{2} - 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -11\beta_{15} + 12\beta_{14} - 15\beta_{13} - 39\beta_{12} + 17\beta_{11} + 22\beta_{9} - 5\beta_{8} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 55\beta_{10} + 135\beta_{7} - 61\beta_{6} - 9\beta_{5} + 27\beta_{4} + 54\beta_{3} + 6\beta_{2} - 154 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -54\beta_{15} + 244\beta_{14} + 41\beta_{13} + 130\beta_{12} + 221\beta_{11} + 34\beta_{9} - 2\beta_{8} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -148\beta_{10} + 490\beta_{7} - 764\beta_{6} - 190\beta_{5} - 229\beta_{4} - 220\beta_{3} + 192\beta_{2} + 268 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 220 \beta_{15} + 842 \beta_{14} + 103 \beta_{13} + 3304 \beta_{12} - 1214 \beta_{11} - 1442 \beta_{9} + \cdots + 649 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4600 \beta_{10} - 6089 \beta_{7} - 509 \beta_{6} - 413 \beta_{5} - 1888 \beta_{4} - 2427 \beta_{3} + \cdots + 7819 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2427 \beta_{15} - 8995 \beta_{14} - 6396 \beta_{13} + 5902 \beta_{12} - 17327 \beta_{11} + \cdots - 1395 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 12987 \beta_{10} - 55296 \beta_{7} + 38298 \beta_{6} + 5173 \beta_{5} + 15957 \beta_{4} + 4168 \beta_{3} + \cdots + 5270 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 4168 \beta_{15} - 84456 \beta_{14} - 23729 \beta_{13} - 156558 \beta_{12} + 37651 \beta_{11} + \cdots - 73797 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 191809 \beta_{10} + 104497 \beta_{7} + 174056 \beta_{6} + 14827 \beta_{5} + 203102 \beta_{4} + \cdots - 376941 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 89263 \beta_{15} + 91376 \beta_{14} + 450212 \beta_{13} - 803078 \beta_{12} + 1252871 \beta_{11} + \cdots - 268910 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(1\) \(-1\) \(1 - \beta_{7}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
593.1
−2.23600 − 1.01798i
−0.163224 − 1.23010i
2.10263 − 1.76088i
0.895887 − 1.04875i
−0.895887 + 1.04875i
−2.10263 + 1.76088i
0.163224 + 1.23010i
2.23600 + 1.01798i
−2.23600 + 1.01798i
−0.163224 + 1.23010i
2.10263 + 1.76088i
0.895887 + 1.04875i
−0.895887 − 1.04875i
−2.10263 − 1.76088i
0.163224 − 1.23010i
2.23600 − 1.01798i
0 0 0 −1.99959 + 3.46340i 0 0.605908 − 2.57544i 0 0 0
593.2 0 0 0 −1.14691 + 1.98651i 0 1.64363 − 2.07328i 0 0 0
593.3 0 0 0 −0.473647 + 0.820381i 0 −0.170326 + 2.64026i 0 0 0
593.4 0 0 0 −0.460302 + 0.797267i 0 −2.57921 − 0.589625i 0 0 0
593.5 0 0 0 0.460302 − 0.797267i 0 −2.57921 − 0.589625i 0 0 0
593.6 0 0 0 0.473647 − 0.820381i 0 −0.170326 + 2.64026i 0 0 0
593.7 0 0 0 1.14691 − 1.98651i 0 1.64363 − 2.07328i 0 0 0
593.8 0 0 0 1.99959 − 3.46340i 0 0.605908 − 2.57544i 0 0 0
1025.1 0 0 0 −1.99959 − 3.46340i 0 0.605908 + 2.57544i 0 0 0
1025.2 0 0 0 −1.14691 − 1.98651i 0 1.64363 + 2.07328i 0 0 0
1025.3 0 0 0 −0.473647 − 0.820381i 0 −0.170326 − 2.64026i 0 0 0
1025.4 0 0 0 −0.460302 − 0.797267i 0 −2.57921 + 0.589625i 0 0 0
1025.5 0 0 0 0.460302 + 0.797267i 0 −2.57921 + 0.589625i 0 0 0
1025.6 0 0 0 0.473647 + 0.820381i 0 −0.170326 − 2.64026i 0 0 0
1025.7 0 0 0 1.14691 + 1.98651i 0 1.64363 + 2.07328i 0 0 0
1025.8 0 0 0 1.99959 + 3.46340i 0 0.605908 + 2.57544i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 593.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.bl.b ✓ 16
3.b odd 2 1 inner 1512.2.bl.b ✓ 16
7.d odd 6 1 inner 1512.2.bl.b ✓ 16
21.g even 6 1 inner 1512.2.bl.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.bl.b ✓ 16 1.a even 1 1 trivial
1512.2.bl.b ✓ 16 3.b odd 2 1 inner
1512.2.bl.b ✓ 16 7.d odd 6 1 inner
1512.2.bl.b ✓ 16 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 23T_{5}^{14} + 407T_{5}^{12} + 2480T_{5}^{10} + 11071T_{5}^{8} + 16942T_{5}^{6} + 18761T_{5}^{4} + 10432T_{5}^{2} + 4096 \) acting on \(S_{2}^{\mathrm{new}}(1512, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 23 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$7$ \( (T^{8} + T^{7} + 9 T^{6} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 30 T^{14} + \cdots + 2401 \) Copy content Toggle raw display
$13$ \( (T^{8} + 22 T^{6} + 41 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 157351936 \) Copy content Toggle raw display
$19$ \( (T^{8} - 3 T^{7} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 94 T^{14} + \cdots + 59969536 \) Copy content Toggle raw display
$29$ \( (T^{8} + 30 T^{6} + \cdots + 49)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 3 T^{7} + \cdots + 1466521)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 4 T^{7} + \cdots + 256)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 130 T^{6} + \cdots + 5476)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 11 T^{3} + \cdots + 484)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 11412532523536 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 146999232456976 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 48354903305361 \) Copy content Toggle raw display
$61$ \( (T^{8} + 3 T^{7} + \cdots + 1016064)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 10 T^{7} + \cdots + 198916)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 29 T^{6} + \cdots + 784)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 19 T^{6} + \cdots + 169)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 7 T^{7} + \cdots + 9042049)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 298 T^{6} + \cdots + 3301489)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{8} + 551 T^{6} + \cdots + 7529536)^{2} \) Copy content Toggle raw display
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