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Properties

Label 1512.2.bl.c
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: \( x^{16} + 26x^{14} + 521x^{12} + 3668x^{10} + 19270x^{8} + 25507x^{6} + 25166x^{4} + 8869x^{2} + 2401 \) 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.

\(f(q)\) \(=\) \( q + \beta_{14} q^{5} + (\beta_{9} - \beta_{7} + \cdots + \beta_{4}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{14} q^{5} + (\beta_{9} - \beta_{7} + \cdots + \beta_{4}) q^{7}+ \cdots + (2 \beta_{9} - 2 \beta_{7} + 4 \beta_{6} + \cdots + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{7} - 18 q^{19} - 12 q^{25} + 24 q^{31} + 16 q^{37} - 52 q^{43} + 44 q^{49} + 6 q^{61} - 4 q^{67} - 12 q^{73} + 34 q^{79} - 68 q^{85} + 102 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 26x^{14} + 521x^{12} + 3668x^{10} + 19270x^{8} + 25507x^{6} + 25166x^{4} + 8869x^{2} + 2401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2197 \nu^{14} - 58305 \nu^{12} - 1176032 \nu^{10} - 8691844 \nu^{8} - 47329087 \nu^{6} + \cdots + 38249694 ) / 134136849 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6144876 \nu^{14} + 362504276 \nu^{12} + 8364940166 \nu^{10} + 125259292438 \nu^{8} + \cdots + 1796382242508 ) / 325818406221 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4203583 \nu^{14} + 106894951 \nu^{12} + 2131367942 \nu^{10} + 14250430915 \nu^{8} + \cdots + 23431731008 ) / 46545486603 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 43522950 \nu^{14} - 1066460314 \nu^{12} - 20932312379 \nu^{10} - 124414527447 \nu^{8} + \cdots + 1081065930882 ) / 325818406221 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 51566747 \nu^{14} - 1323221352 \nu^{12} - 26432319672 \nu^{10} - 180548210584 \nu^{8} + \cdots - 431561039301 ) / 325818406221 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 128347598 \nu^{14} - 3402651537 \nu^{12} - 68559853505 \nu^{10} - 504521261566 \nu^{8} + \cdots - 1051396903438 ) / 325818406221 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 780606 \nu^{15} - 20403409 \nu^{13} - 409552671 \nu^{11} - 2920888376 \nu^{9} + \cdots - 14587104931 \nu ) / 6572705601 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 336900052 \nu^{14} - 8787513877 \nu^{12} - 176098877812 \nu^{10} - 1246366440256 \nu^{8} + \cdots - 1585943609439 ) / 325818406221 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 337932687 \nu^{15} - 7620847484 \nu^{13} - 146732505370 \nu^{11} + \cdots + 11558515964198 \nu ) / 2280728843547 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 51566747 \nu^{15} + 1323221352 \nu^{13} + 26432319672 \nu^{11} + 180548210584 \nu^{9} + \cdots + 431561039301 \nu ) / 325818406221 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 749068874 \nu^{15} + 19307170748 \nu^{13} + 386018390670 \nu^{11} + 2663143672770 \nu^{9} + \cdots + 3790865842176 \nu ) / 2280728843547 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 84985522 \nu^{15} + 2269933360 \nu^{13} + 45843545374 \nu^{11} + 343061930274 \nu^{9} + \cdots + 1287535535678 \nu ) / 207338985777 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 34629015 \nu^{15} - 860031409 \nu^{13} - 17001282474 \nu^{11} - 106235785869 \nu^{9} + \cdots + 260624410234 \nu ) / 46545486603 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2155970587 \nu^{15} - 57130072067 \nu^{13} - 1149960600162 \nu^{11} + \cdots - 20703691749286 \nu ) / 2280728843547 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{7} - 6\beta_{6} - 2\beta_{4} + \beta_{3} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{14} + \beta_{13} - 16\beta_{12} + 13\beta_{11} - \beta_{10} - 8\beta_{8} - 13\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -13\beta_{9} + 85\beta_{6} + 26\beta_{5} + 48\beta_{4} - 13\beta_{3} + 37\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 11 \beta_{15} + 35 \beta_{14} - 70 \beta_{13} + 170 \beta_{12} - 233 \beta_{11} - 35 \beta_{10} + \cdots - 35 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 187\beta_{9} - 385\beta_{7} - 385\beta_{5} - 275\beta_{4} - 11\beta_{3} - 935\beta_{2} + 1411 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 748 \beta_{15} - 286 \beta_{14} + 748 \beta_{13} + 3201 \beta_{12} + 748 \beta_{11} + 1496 \beta_{10} + \cdots + 4062 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 286\beta_{9} + 6342\beta_{7} - 24548\beta_{6} - 11787\beta_{4} + 3314\beta_{3} + 5445\beta_{2} - 24548 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 8473 \beta_{15} - 8473 \beta_{14} + 14204 \beta_{13} - 116648 \beta_{12} + 57778 \beta_{11} + \cdots - 57778 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -57778\beta_{9} + 434634\beta_{6} + 109825\beta_{5} + 311697\beta_{4} - 52047\beta_{3} + 210761\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 106667 \beta_{15} + 259650 \beta_{14} - 519300 \beta_{13} + 1050647 \beta_{12} - 1283409 \beta_{11} + \cdots - 259650 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 917092 \beta_{9} - 1940851 \beta_{7} - 1940851 \beta_{5} - 1829597 \beta_{4} - 106667 \beta_{3} + \cdots + 7746770 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 4682953 \beta_{15} - 1936264 \beta_{14} + 4682953 \beta_{13} + 18847653 \beta_{12} + \cdots + 22935229 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1936264 \beta_{9} + 34568288 \beta_{7} - 138421799 \beta_{6} - 67464800 \beta_{4} + 18252276 \beta_{3} + \cdots - 138421799 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 49212524 \beta_{15} - 49212524 \beta_{14} + 84045300 \beta_{13} - 675176976 \beta_{12} + \cdots - 326171963 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{6}\) \(1\)

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
−0.314872 + 0.545374i
0.508571 − 0.880870i
−2.11513 + 3.66351i
−1.29168 + 2.23726i
1.29168 − 2.23726i
2.11513 − 3.66351i
−0.508571 + 0.880870i
0.314872 − 0.545374i
−0.314872 − 0.545374i
0.508571 + 0.880870i
−2.11513 − 3.66351i
−1.29168 − 2.23726i
1.29168 + 2.23726i
2.11513 + 3.66351i
−0.508571 − 0.880870i
0.314872 + 0.545374i
0 0 0 −1.69652 + 2.93845i 0 −2.25634 + 1.38164i 0 0 0
593.2 0 0 0 −1.59716 + 2.76636i 0 −1.60184 − 2.10573i 0 0 0
593.3 0 0 0 −0.779738 + 1.35055i 0 2.28402 + 1.33539i 0 0 0
593.4 0 0 0 −0.680382 + 1.17846i 0 2.57416 − 0.611303i 0 0 0
593.5 0 0 0 0.680382 − 1.17846i 0 2.57416 − 0.611303i 0 0 0
593.6 0 0 0 0.779738 − 1.35055i 0 2.28402 + 1.33539i 0 0 0
593.7 0 0 0 1.59716 − 2.76636i 0 −1.60184 − 2.10573i 0 0 0
593.8 0 0 0 1.69652 − 2.93845i 0 −2.25634 + 1.38164i 0 0 0
1025.1 0 0 0 −1.69652 − 2.93845i 0 −2.25634 − 1.38164i 0 0 0
1025.2 0 0 0 −1.59716 − 2.76636i 0 −1.60184 + 2.10573i 0 0 0
1025.3 0 0 0 −0.779738 − 1.35055i 0 2.28402 − 1.33539i 0 0 0
1025.4 0 0 0 −0.680382 − 1.17846i 0 2.57416 + 0.611303i 0 0 0
1025.5 0 0 0 0.680382 + 1.17846i 0 2.57416 + 0.611303i 0 0 0
1025.6 0 0 0 0.779738 + 1.35055i 0 2.28402 − 1.33539i 0 0 0
1025.7 0 0 0 1.59716 + 2.76636i 0 −1.60184 + 2.10573i 0 0 0
1025.8 0 0 0 1.69652 + 2.93845i 0 −2.25634 − 1.38164i 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.c ✓ 16
3.b odd 2 1 inner 1512.2.bl.c ✓ 16
7.d odd 6 1 inner 1512.2.bl.c ✓ 16
21.g even 6 1 inner 1512.2.bl.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.bl.c ✓ 16 1.a even 1 1 trivial
1512.2.bl.c ✓ 16 3.b odd 2 1 inner
1512.2.bl.c ✓ 16 7.d odd 6 1 inner
1512.2.bl.c ✓ 16 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 26 T_{5}^{14} + 461 T_{5}^{12} + 4388 T_{5}^{10} + 30070 T_{5}^{8} + 101707 T_{5}^{6} + \cdots + 279841 \) 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
$5$ \( T^{16} + 26 T^{14} + \cdots + 279841 \) Copy content Toggle raw display
$7$ \( (T^{8} - 2 T^{7} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 69 T^{14} + \cdots + 40960000 \) Copy content Toggle raw display
$13$ \( (T^{8} + 106 T^{6} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 34 T^{14} + \cdots + 2085136 \) Copy content Toggle raw display
$19$ \( (T^{8} + 9 T^{7} + \cdots + 44944)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 82 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( (T^{8} + 78 T^{6} + \cdots + 117649)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 12 T^{7} + \cdots + 58564)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 8 T^{7} + \cdots + 784)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 106 T^{6} + \cdots + 56644)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 13 T^{3} + \cdots - 2576)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 201511210000 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 21808162146241 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 169647525064161 \) Copy content Toggle raw display
$61$ \( (T^{8} - 3 T^{7} + \cdots + 8100)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 2 T^{7} + \cdots + 28900)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 497 T^{6} + \cdots + 41344900)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 6 T^{7} + \cdots + 1311025)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 17 T^{7} + \cdots + 11730625)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 235 T^{6} + \cdots + 5522500)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 5636405776 \) Copy content Toggle raw display
$97$ \( (T^{8} + 524 T^{6} + \cdots + 10445824)^{2} \) Copy content Toggle raw display
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