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Properties

Label 588.3.p.i
Level $588$
Weight $3$
Character orbit 588.p
Analytic conductor $16.022$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [588,3,Mod(557,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.557");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 588.p (of order \(6\), degree \(2\), not 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: \(16.0218395444\)
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: \( x^{16} - 18x^{14} + 201x^{12} - 1606x^{10} + 9216x^{8} - 21516x^{6} + 38173x^{4} - 134064x^{2} + 194481 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{12} - \beta_{10}) q^{3} + \beta_{15} q^{5} + ( - \beta_{9} - \beta_{7} - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{12} - \beta_{10}) q^{3} + \beta_{15} q^{5} + ( - \beta_{9} - \beta_{7} - \beta_{2} - 1) q^{9} + 2 \beta_{8} q^{11} + ( - \beta_{15} + \beta_{13} + \cdots + \beta_1) q^{13}+ \cdots + (10 \beta_{9} - 10 \beta_{8} + \cdots - 80) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{9} - 64 q^{15} - 48 q^{25} - 104 q^{37} + 240 q^{39} + 80 q^{43} + 44 q^{51} - 440 q^{57} - 528 q^{67} - 256 q^{79} + 496 q^{81} + 1136 q^{85} - 24 q^{93} - 1312 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 18x^{14} + 201x^{12} - 1606x^{10} + 9216x^{8} - 21516x^{6} + 38173x^{4} - 134064x^{2} + 194481 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2217421 \nu^{15} - 156568074 \nu^{13} + 1654561364 \nu^{11} - 12930446174 \nu^{9} + \cdots - 744442540002 \nu ) / 2135091405069 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 376708 \nu^{14} - 5545136 \nu^{12} + 60310032 \nu^{10} - 442163312 \nu^{8} + \cdots + 11422019385 ) / 14524431327 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1419620 \nu^{14} - 20206173 \nu^{12} + 213532378 \nu^{10} - 1572569530 \nu^{8} + \cdots - 96075364329 ) / 29048862654 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1240331 \nu^{14} - 19500156 \nu^{12} + 206071416 \nu^{10} - 1539567578 \nu^{8} + \cdots - 115311987630 ) / 22593559842 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4454281 \nu^{14} - 69079847 \nu^{12} + 713119524 \nu^{10} - 5228239484 \nu^{8} + \cdots - 423832119543 ) / 29048862654 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 79879 \nu^{15} - 1376817 \nu^{13} + 14549762 \nu^{11} - 109267598 \nu^{9} + \cdots - 6546424941 \nu ) / 8489429046 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 37590360 \nu^{14} - 594571745 \nu^{12} + 6259292866 \nu^{10} - 47233355220 \nu^{8} + \cdots - 3559412714445 ) / 203342038578 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 21897609 \nu^{14} - 312112559 \nu^{12} + 3139628410 \nu^{10} - 22032451242 \nu^{8} + \cdots - 1455766112745 ) / 101671019289 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 34714576 \nu^{14} + 560562405 \nu^{12} - 5923844330 \nu^{10} + 44416446098 \nu^{8} + \cdots + 2665335850065 ) / 101671019289 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 50428425 \nu^{15} + 692768740 \nu^{13} - 6868877852 \nu^{11} + 46576214910 \nu^{9} + \cdots + 2883501401130 \nu ) / 1423394270046 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2256886 \nu^{15} - 34827224 \nu^{13} + 361321944 \nu^{11} - 2649033704 \nu^{9} + \cdots - 182289473001 \nu ) / 43573293981 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 62341021 \nu^{15} + 1001444028 \nu^{13} - 10582940408 \nu^{11} + 79480381739 \nu^{9} + \cdots + 4761619127244 \nu ) / 711697135023 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 9840451 \nu^{15} + 150973769 \nu^{13} - 1575432204 \nu^{11} + 11550289364 \nu^{9} + \cdots + 414669778461 \nu ) / 87146587962 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 5394370 \nu^{15} - 83720891 \nu^{13} + 863625480 \nu^{11} - 6331674680 \nu^{9} + \cdots - 448065709101 \nu ) / 43573293981 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 316546710 \nu^{15} - 5101754408 \nu^{13} + 54403963123 \nu^{11} - 412942041972 \nu^{9} + \cdots - 31685326689996 \nu ) / 2135091405069 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} - 2\beta_{13} + 2\beta_{12} - 2\beta_{11} + 2\beta_{10} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} + 8\beta_{4} - 2\beta_{3} + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} - \beta_{13} + \beta_{12} - 19\beta_{6} + 5\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{8} + 6\beta_{7} - 20\beta_{5} + 26\beta_{4} - 20\beta_{3} - 20 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -34\beta_{15} - 73\beta_{14} + 434\beta_{11} + 124\beta_{10} - 434\beta_{6} + 73\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -20\beta_{9} + 20\beta_{8} - 64\beta_{5} - 51\beta_{2} - 83 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -325\beta_{14} - 502\beta_{13} + 1870\beta_{12} + 4124\beta_{11} + 1870\beta_{10} ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -163\beta_{9} - 1098\beta_{7} + 2800\beta_{4} + 452\beta_{3} - 1098\beta_{2} + 1702 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2564\beta_{15} - 2564\beta_{13} + 9341\beta_{12} + 15760\beta_{6} + 265\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 299\beta_{8} - 9042\beta_{7} - 2410\beta_{5} + 38152\beta_{4} - 2410\beta_{3} - 2410 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 41776\beta_{15} - 31733\beta_{14} - 175286\beta_{11} - 144844\beta_{10} + 175286\beta_{6} + 31733\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -8179\beta_{9} + 8179\beta_{8} - 34442\beta_{5} + 28032\beta_{2} - 193039 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( -439397\beta_{14} + 258838\beta_{13} - 823102\beta_{12} - 340490\beta_{11} - 823102\beta_{10} ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -225755\beta_{9} + 185796\beta_{7} - 2848924\beta_{4} + 862522\beta_{3} + 185796\beta_{2} - 2663128 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( -429083\beta_{15} + 429083\beta_{13} - 927341\beta_{12} + 3833981\beta_{6} - 2155216\beta_1 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(-1\) \(1\) \(-1 - \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1
2.10226 + 2.03024i
−0.858455 − 1.31213i
2.80937 + 0.805494i
−1.56556 − 0.0873809i
1.56556 + 0.0873809i
−2.80937 − 0.805494i
0.858455 + 1.31213i
−2.10226 − 2.03024i
2.10226 − 2.03024i
−0.858455 + 1.31213i
2.80937 − 0.805494i
−1.56556 + 0.0873809i
1.56556 − 0.0873809i
−2.80937 + 0.805494i
0.858455 − 1.31213i
−2.10226 + 2.03024i
0 −2.97890 + 0.355159i 0 −0.750856 + 0.433507i 0 0 0 8.74772 − 2.11597i 0
557.2 0 −2.68280 + 1.34261i 0 5.28547 − 3.05157i 0 0 0 5.39478 − 7.20391i 0
557.3 0 −1.18187 − 2.75738i 0 −0.750856 + 0.433507i 0 0 0 −6.20634 + 6.51777i 0
557.4 0 −0.178660 − 2.99468i 0 5.28547 − 3.05157i 0 0 0 −8.93616 + 1.07006i 0
557.5 0 0.178660 + 2.99468i 0 −5.28547 + 3.05157i 0 0 0 −8.93616 + 1.07006i 0
557.6 0 1.18187 + 2.75738i 0 0.750856 − 0.433507i 0 0 0 −6.20634 + 6.51777i 0
557.7 0 2.68280 − 1.34261i 0 −5.28547 + 3.05157i 0 0 0 5.39478 − 7.20391i 0
557.8 0 2.97890 − 0.355159i 0 0.750856 − 0.433507i 0 0 0 8.74772 − 2.11597i 0
569.1 0 −2.97890 − 0.355159i 0 −0.750856 − 0.433507i 0 0 0 8.74772 + 2.11597i 0
569.2 0 −2.68280 − 1.34261i 0 5.28547 + 3.05157i 0 0 0 5.39478 + 7.20391i 0
569.3 0 −1.18187 + 2.75738i 0 −0.750856 − 0.433507i 0 0 0 −6.20634 − 6.51777i 0
569.4 0 −0.178660 + 2.99468i 0 5.28547 + 3.05157i 0 0 0 −8.93616 − 1.07006i 0
569.5 0 0.178660 − 2.99468i 0 −5.28547 − 3.05157i 0 0 0 −8.93616 − 1.07006i 0
569.6 0 1.18187 − 2.75738i 0 0.750856 + 0.433507i 0 0 0 −6.20634 − 6.51777i 0
569.7 0 2.68280 + 1.34261i 0 −5.28547 − 3.05157i 0 0 0 5.39478 + 7.20391i 0
569.8 0 2.97890 + 0.355159i 0 0.750856 + 0.433507i 0 0 0 8.74772 + 2.11597i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 557.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.3.p.i 16
3.b odd 2 1 inner 588.3.p.i 16
7.b odd 2 1 inner 588.3.p.i 16
7.c even 3 1 588.3.c.j ✓ 8
7.c even 3 1 inner 588.3.p.i 16
7.d odd 6 1 588.3.c.j ✓ 8
7.d odd 6 1 inner 588.3.p.i 16
21.c even 2 1 inner 588.3.p.i 16
21.g even 6 1 588.3.c.j ✓ 8
21.g even 6 1 inner 588.3.p.i 16
21.h odd 6 1 588.3.c.j ✓ 8
21.h odd 6 1 inner 588.3.p.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.3.c.j ✓ 8 7.c even 3 1
588.3.c.j ✓ 8 7.d odd 6 1
588.3.c.j ✓ 8 21.g even 6 1
588.3.c.j ✓ 8 21.h odd 6 1
588.3.p.i 16 1.a even 1 1 trivial
588.3.p.i 16 3.b odd 2 1 inner
588.3.p.i 16 7.b odd 2 1 inner
588.3.p.i 16 7.c even 3 1 inner
588.3.p.i 16 7.d odd 6 1 inner
588.3.p.i 16 21.c even 2 1 inner
588.3.p.i 16 21.g even 6 1 inner
588.3.p.i 16 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(588, [\chi])\):

\( T_{5}^{8} - 38T_{5}^{6} + 1416T_{5}^{4} - 1064T_{5}^{2} + 784 \) Copy content Toggle raw display
\( T_{13}^{4} - 414T_{13}^{2} + 15876 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 2 T^{14} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( (T^{8} - 38 T^{6} + \cdots + 784)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 304 T^{6} + \cdots + 3211264)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 414 T^{2} + 15876)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} - 776 T^{6} + \cdots + 22264220944)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 334 T^{6} + \cdots + 759333136)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 2140 T^{6} + \cdots + 440959746304)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 532 T^{2} + 5488)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 72 T^{2} + 5184)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 26 T^{3} + \cdots + 7997584)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 8168 T^{2} + 15708028)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 10 T - 2972)^{8} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 104343119663104)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 18893001105664)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 226324704104464)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 1350439223056)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 66 T + 4356)^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + 11700 T^{2} + 25483248)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 9172 T^{6} + \cdots + 303120718096)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 64 T^{3} + \cdots + 18524416)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 3146 T^{2} + 1392412)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} - 22616 T^{6} + \cdots + 476337389584)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 42964 T^{2} + 260951716)^{4} \) Copy content Toggle raw display
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