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Properties

Label 592.2.be.e
Level $592$
Weight $2$
Character orbit 592.be
Analytic conductor $4.727$
Analytic rank $0$
Dimension $20$
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] = [592,2,Mod(319,592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(592, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("592.319");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.be (of order \(12\), degree \(4\), 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: \(4.72714379966\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 46 x^{18} + 885 x^{16} + 9292 x^{14} + 58264 x^{12} + 224256 x^{10} + 523884 x^{8} + 706272 x^{6} + \cdots + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{13} - \beta_{10}) q^{3} + (\beta_{16} + \beta_{14} + \beta_{11} + \cdots - 1) q^{5}+ \cdots + (\beta_{10} - 2 \beta_{9} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{13} - \beta_{10}) q^{3} + (\beta_{16} + \beta_{14} + \beta_{11} + \cdots - 1) q^{5}+ \cdots + ( - \beta_{19} - \beta_{18} + 2 \beta_{17} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 4 q^{3} - 4 q^{5} - 6 q^{7} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 4 q^{3} - 4 q^{5} - 6 q^{7} - 16 q^{9} + 4 q^{11} + 4 q^{13} + 8 q^{15} - 4 q^{17} + 10 q^{19} - 18 q^{21} + 8 q^{23} + 42 q^{25} + 68 q^{27} - 8 q^{29} + 28 q^{31} - 20 q^{33} - 10 q^{35} - 24 q^{37} - 14 q^{39} - 6 q^{41} + 32 q^{43} + 8 q^{45} - 12 q^{49} - 58 q^{51} + 6 q^{53} + 26 q^{55} - 2 q^{57} - 56 q^{59} - 8 q^{61} + 6 q^{65} + 20 q^{67} + 26 q^{69} - 30 q^{71} + 60 q^{77} + 50 q^{79} - 22 q^{81} + 36 q^{83} - 32 q^{87} - 20 q^{89} - 50 q^{91} - 50 q^{93} - 72 q^{95} + 32 q^{97} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 46 x^{18} + 885 x^{16} + 9292 x^{14} + 58264 x^{12} + 224256 x^{10} + 523884 x^{8} + 706272 x^{6} + \cdots + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 818479 \nu^{19} - 496071 \nu^{18} + 32692722 \nu^{17} - 18837444 \nu^{16} + 522746193 \nu^{15} + \cdots - 1921192272 ) / 5618795616 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 145 \nu^{18} - 6654 \nu^{16} - 124749 \nu^{14} - 1230679 \nu^{12} - 6864483 \nu^{10} + \cdots - 34704 ) / 824112 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 145 \nu^{18} - 6654 \nu^{16} - 124749 \nu^{14} - 1230679 \nu^{12} - 6864483 \nu^{10} + \cdots - 34704 ) / 824112 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 165357 \nu^{18} + 6279148 \nu^{16} + 95698361 \nu^{14} + 769692390 \nu^{12} + 3706418548 \nu^{10} + \cdots + 640397424 ) / 936465936 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 662794 \nu^{19} + 582235 \nu^{18} + 37342902 \nu^{17} + 24737616 \nu^{16} + 873447462 \nu^{15} + \cdots - 8195859936 ) / 5618795616 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 582235 \nu^{18} + 24737616 \nu^{16} + 421318635 \nu^{14} + 3644473348 \nu^{12} + \cdots - 8195859936 ) / 2809397808 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 241 \nu^{19} + 10941 \nu^{17} + 206631 \nu^{15} + 2114623 \nu^{13} + 12810945 \nu^{11} + \cdots - 412056 ) / 824112 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1639979 \nu^{19} + 817268 \nu^{18} - 73543460 \nu^{17} + 37650024 \nu^{16} + \cdots - 4726766592 ) / 5618795616 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1639979 \nu^{19} + 817268 \nu^{18} + 73543460 \nu^{17} + 37650024 \nu^{16} + 1370156331 \nu^{15} + \cdots - 4726766592 ) / 5618795616 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 408634 \nu^{19} + 947787 \nu^{18} + 18825012 \nu^{17} + 40612542 \nu^{16} + 360980349 \nu^{15} + \cdots + 118078488 ) / 2809397808 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 33080 \nu^{19} - 1493500 \nu^{18} + 1378507 \nu^{17} - 60243573 \nu^{16} + 20486261 \nu^{15} + \cdots - 2694632832 ) / 2809397808 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1320730 \nu^{19} - 3469099 \nu^{18} + 60740460 \nu^{17} - 145095390 \nu^{16} + \cdots + 14073909264 ) / 5618795616 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3257 \nu^{18} + 139562 \nu^{16} + 2457688 \nu^{14} + 23089872 \nu^{12} + 125624345 \nu^{10} + \cdots + 405768 ) / 4827144 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 243635 \nu^{19} - 4198374 \nu^{18} + 12208098 \nu^{17} - 184783456 \nu^{16} + 265530825 \nu^{15} + \cdots - 8640892800 ) / 5618795616 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1320730 \nu^{19} - 3469099 \nu^{18} - 60740460 \nu^{17} - 145095390 \nu^{16} + \cdots + 14073909264 ) / 5618795616 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 3158188 \nu^{19} + 2562987 \nu^{18} + 146878162 \nu^{17} + 107779476 \nu^{16} + \cdots + 12739523328 ) / 5618795616 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 60495 \nu^{19} + 4578589 \nu^{18} - 4069098 \nu^{17} + 195216572 \nu^{16} + \cdots - 2868847776 ) / 5618795616 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 60495 \nu^{19} - 4578589 \nu^{18} - 4069098 \nu^{17} - 195216572 \nu^{16} + \cdots - 2749947840 ) / 5618795616 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 3523593 \nu^{19} - 4198374 \nu^{18} - 159295018 \nu^{17} - 184783456 \nu^{16} + \cdots - 8640892800 ) / 5618795616 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{4} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{19} + \beta_{18} + \beta_{17} + 2 \beta_{15} + \beta_{14} - 2 \beta_{12} + 2 \beta_{9} + \cdots - 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{19} + \beta_{18} - \beta_{17} - 3 \beta_{16} + \beta_{15} - 2 \beta_{14} - 13 \beta_{13} + \cdots + 33 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14 \beta_{19} - 14 \beta_{18} - 14 \beta_{17} + 3 \beta_{16} - 32 \beta_{15} - 14 \beta_{14} + 32 \beta_{12} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 27 \beta_{19} - 6 \beta_{18} + 6 \beta_{17} + 46 \beta_{16} - 20 \beta_{15} + 27 \beta_{14} + 150 \beta_{13} + \cdots - 312 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 168 \beta_{19} + 156 \beta_{18} + 156 \beta_{17} - 75 \beta_{16} + 398 \beta_{15} + 168 \beta_{14} + \cdots + 59 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 279 \beta_{19} - 18 \beta_{18} + 18 \beta_{17} - 566 \beta_{16} + 308 \beta_{15} - 279 \beta_{14} + \cdots + 3186 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1936 \beta_{19} - 1618 \beta_{18} - 1618 \beta_{17} + 1239 \beta_{16} - 4604 \beta_{15} - 1936 \beta_{14} + \cdots - 807 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2541 \beta_{19} + 1116 \beta_{18} - 1116 \beta_{17} + 6540 \beta_{16} - 4226 \beta_{15} + 2541 \beta_{14} + \cdots - 34116 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 21932 \beta_{19} + 16334 \beta_{18} + 16334 \beta_{17} - 17451 \beta_{16} + 51944 \beta_{15} + \cdots + 9619 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 20745 \beta_{19} - 21684 \beta_{18} + 21684 \beta_{17} - 73876 \beta_{16} + 54482 \beta_{15} + \cdots + 375720 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 246384 \beta_{19} - 163182 \beta_{18} - 163182 \beta_{17} + 227559 \beta_{16} - 581060 \beta_{15} + \cdots - 107327 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 144825 \beta_{19} + 334212 \beta_{18} - 334212 \beta_{17} + 827444 \beta_{16} - 677138 \beta_{15} + \cdots - 4204824 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 2757616 \beta_{19} + 1624846 \beta_{18} + 1624846 \beta_{17} - 2844663 \beta_{16} + 6486404 \beta_{15} + \cdots + 1156335 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 689097 \beta_{19} - 4653900 \beta_{18} + 4653900 \beta_{17} - 9244020 \beta_{16} + 8225930 \beta_{15} + \cdots + 47486184 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 30834752 \beta_{19} - 16181222 \beta_{18} - 16181222 \beta_{17} + 34674687 \beta_{16} - 72454100 \beta_{15} + \cdots - 12224455 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 1931391 \beta_{19} + 61250340 \beta_{18} - 61250340 \beta_{17} + 103288852 \beta_{16} + \cdots - 539017944 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 345006744 \beta_{19} + 161467830 \beta_{18} + 161467830 \beta_{17} - 415902351 \beta_{16} + \cdots + 127994039 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/592\mathbb{Z}\right)^\times\).

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(\beta_{9}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
− 3.11417i
− 1.55981i
− 0.873351i
0.0328919i
1.78239i
3.11417i
1.55981i
0.873351i
− 0.0328919i
− 1.78239i
− 3.38161i
− 2.56563i
1.04018i
2.13532i
2.50378i
3.38161i
2.56563i
− 1.04018i
− 2.13532i
− 2.50378i
0 −1.55708 − 2.69695i 0 −0.386109 + 1.44098i 0 −1.53894 + 0.888509i 0 −3.34902 + 5.80068i 0
319.2 0 −0.779906 − 1.35084i 0 0.765103 − 2.85540i 0 1.56078 − 0.901118i 0 0.283492 − 0.491022i 0
319.3 0 −0.436676 − 0.756344i 0 −0.192218 + 0.717368i 0 −1.90253 + 1.09843i 0 1.11863 − 1.93752i 0
319.4 0 0.0164460 + 0.0284852i 0 −0.664639 + 2.48047i 0 2.50293 − 1.44507i 0 1.49946 − 2.59714i 0
319.5 0 0.891195 + 1.54359i 0 0.343889 − 1.28341i 0 −3.85429 + 2.22528i 0 −0.0884563 + 0.153211i 0
399.1 0 −1.55708 + 2.69695i 0 −0.386109 − 1.44098i 0 −1.53894 − 0.888509i 0 −3.34902 − 5.80068i 0
399.2 0 −0.779906 + 1.35084i 0 0.765103 + 2.85540i 0 1.56078 + 0.901118i 0 0.283492 + 0.491022i 0
399.3 0 −0.436676 + 0.756344i 0 −0.192218 − 0.717368i 0 −1.90253 − 1.09843i 0 1.11863 + 1.93752i 0
399.4 0 0.0164460 − 0.0284852i 0 −0.664639 − 2.48047i 0 2.50293 + 1.44507i 0 1.49946 + 2.59714i 0
399.5 0 0.891195 − 1.54359i 0 0.343889 + 1.28341i 0 −3.85429 − 2.22528i 0 −0.0884563 − 0.153211i 0
415.1 0 −1.69080 + 2.92856i 0 −3.56248 + 0.954562i 0 1.92706 + 1.11259i 0 −4.21763 − 7.30516i 0
415.2 0 −1.28281 + 2.22190i 0 3.31311 − 0.887746i 0 −0.921902 − 0.532260i 0 −1.79122 − 3.10248i 0
415.3 0 0.520091 − 0.900825i 0 −3.71558 + 0.995586i 0 −0.391848 − 0.226234i 0 0.959010 + 1.66105i 0
415.4 0 1.06766 − 1.84924i 0 1.91328 − 0.512663i 0 1.95018 + 1.12593i 0 −0.779794 − 1.35064i 0
415.5 0 1.25189 − 2.16834i 0 0.185630 − 0.0497393i 0 −2.33143 − 1.34605i 0 −1.63446 − 2.83097i 0
495.1 0 −1.69080 − 2.92856i 0 −3.56248 − 0.954562i 0 1.92706 − 1.11259i 0 −4.21763 + 7.30516i 0
495.2 0 −1.28281 − 2.22190i 0 3.31311 + 0.887746i 0 −0.921902 + 0.532260i 0 −1.79122 + 3.10248i 0
495.3 0 0.520091 + 0.900825i 0 −3.71558 − 0.995586i 0 −0.391848 + 0.226234i 0 0.959010 − 1.66105i 0
495.4 0 1.06766 + 1.84924i 0 1.91328 + 0.512663i 0 1.95018 − 1.12593i 0 −0.779794 + 1.35064i 0
495.5 0 1.25189 + 2.16834i 0 0.185630 + 0.0497393i 0 −2.33143 + 1.34605i 0 −1.63446 + 2.83097i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 319.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
148.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 592.2.be.e ✓ 20
4.b odd 2 1 592.2.be.f yes 20
37.g odd 12 1 592.2.be.f yes 20
148.l even 12 1 inner 592.2.be.e ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
592.2.be.e ✓ 20 1.a even 1 1 trivial
592.2.be.e ✓ 20 148.l even 12 1 inner
592.2.be.f yes 20 4.b odd 2 1
592.2.be.f yes 20 37.g odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 4 T_{3}^{19} + 31 T_{3}^{18} + 64 T_{3}^{17} + 391 T_{3}^{16} + 598 T_{3}^{15} + 3382 T_{3}^{14} + \cdots + 144 \) acting on \(S_{2}^{\mathrm{new}}(592, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 4 T^{19} + \cdots + 144 \) Copy content Toggle raw display
$5$ \( T^{20} + 4 T^{19} + \cdots + 42849 \) Copy content Toggle raw display
$7$ \( T^{20} + 6 T^{19} + \cdots + 345744 \) Copy content Toggle raw display
$11$ \( (T^{10} - 2 T^{9} + \cdots - 18432)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 107827456 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 309795201 \) Copy content Toggle raw display
$19$ \( T^{20} - 10 T^{19} + \cdots + 31945104 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 10218149269056 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 5627700324 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 97\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 48\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 10595202489 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 1199375424 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 144157867243776 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 927331125260544 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 294966610366464 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 206213167449 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 14883145505424 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 141997919698944 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 34\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 73\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 83841180929289 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
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