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Properties

Label 592.2.w.f
Level $592$
Weight $2$
Character orbit 592.w
Analytic conductor $4.727$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [592,2,Mod(529,592)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("592.529"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(592, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.w (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,2,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.72714379966\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 28 x^{18} + 320 x^{16} + 1984 x^{14} + 7388 x^{12} + 17136 x^{10} + 24692 x^{8} + 21256 x^{6} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 296)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + (\beta_{14} + \beta_{5}) q^{3} + \beta_{4} q^{5} + ( - \beta_{10} - \beta_{2}) q^{7} + ( - \beta_{18} - \beta_{17} + \beta_{11} + \cdots - 2) q^{9} + ( - \beta_{19} - \beta_{17} - \beta_{11} + \cdots - 1) q^{11}+ \cdots + (2 \beta_{18} + 2 \beta_{17} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{3} + 6 q^{5} + 2 q^{7} - 12 q^{9} - 16 q^{11} - 6 q^{13} + 18 q^{15} - 6 q^{17} + 6 q^{19} + 2 q^{21} + 20 q^{25} + 20 q^{27} - 4 q^{33} + 6 q^{35} + 24 q^{37} - 18 q^{39} + 14 q^{41} + 8 q^{47}+ \cdots + 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 28 x^{18} + 320 x^{16} + 1984 x^{14} + 7388 x^{12} + 17136 x^{10} + 24692 x^{8} + 21256 x^{6} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 13 \nu^{19} - 328 \nu^{17} - 3264 \nu^{15} - 17056 \nu^{13} - 51708 \nu^{11} - 93824 \nu^{9} + \cdots + 64 ) / 128 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{18} - 112 \nu^{16} - 2304 \nu^{14} - 19840 \nu^{12} - 87964 \nu^{10} - 213920 \nu^{8} + \cdots - 1840 ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 29 \nu^{19} + 26 \nu^{18} - 720 \nu^{17} + 624 \nu^{16} - 6976 \nu^{15} + 5728 \nu^{14} + \cdots - 288 ) / 128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 21 \nu^{19} + 20 \nu^{18} - 536 \nu^{17} + 512 \nu^{16} - 5408 \nu^{15} + 5184 \nu^{14} + \cdots + 640 ) / 128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 35 \nu^{18} - 944 \nu^{16} - 10240 \nu^{14} - 59136 \nu^{12} - 199444 \nu^{10} - 400288 \nu^{8} + \cdots - 1936 ) / 128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 29 \nu^{19} - 26 \nu^{18} - 720 \nu^{17} - 624 \nu^{16} - 6976 \nu^{15} - 5728 \nu^{14} + \cdots + 288 ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21 \nu^{19} - 20 \nu^{18} - 536 \nu^{17} - 512 \nu^{16} - 5408 \nu^{15} - 5184 \nu^{14} + \cdots - 640 ) / 128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3 \nu^{19} - 4 \nu^{18} + 78 \nu^{17} - 100 \nu^{16} + 806 \nu^{15} - 984 \nu^{14} + 4388 \nu^{13} + \cdots - 80 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 127 \nu^{19} - 102 \nu^{18} + 3280 \nu^{17} - 2656 \nu^{16} + 33600 \nu^{15} - 27392 \nu^{14} + \cdots - 288 ) / 512 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 67 \nu^{19} + 2 \nu^{18} + 1936 \nu^{17} + 224 \nu^{16} + 22656 \nu^{15} + 4608 \nu^{14} + 140864 \nu^{13} + \cdots + 3680 ) / 512 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3 \nu^{19} + 4 \nu^{18} + 78 \nu^{17} + 100 \nu^{16} + 806 \nu^{15} + 984 \nu^{14} + 4388 \nu^{13} + \cdots + 80 ) / 16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 127 \nu^{19} - 102 \nu^{18} - 3280 \nu^{17} - 2656 \nu^{16} - 33600 \nu^{15} - 27392 \nu^{14} + \cdots - 288 ) / 512 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 25 \nu^{19} + 99 \nu^{18} - 664 \nu^{17} + 2496 \nu^{16} - 7072 \nu^{15} + 24736 \nu^{14} + \cdots + 2000 ) / 128 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 137 \nu^{19} + 70 \nu^{18} + 3760 \nu^{17} + 1888 \nu^{16} + 41536 \nu^{15} + 20480 \nu^{14} + \cdots + 3872 ) / 512 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 79 \nu^{19} + 390 \nu^{18} - 2160 \nu^{17} + 10080 \nu^{16} - 23808 \nu^{15} + 103424 \nu^{14} + \cdots + 15392 ) / 512 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 25 \nu^{19} + 99 \nu^{18} + 664 \nu^{17} + 2496 \nu^{16} + 7072 \nu^{15} + 24736 \nu^{14} + \cdots + 2000 ) / 128 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 79 \nu^{19} - 390 \nu^{18} - 2160 \nu^{17} - 10080 \nu^{16} - 23808 \nu^{15} - 103424 \nu^{14} + \cdots - 15392 ) / 512 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 129 \nu^{19} + 290 \nu^{18} + 3472 \nu^{17} + 7712 \nu^{16} + 37504 \nu^{15} + 82048 \nu^{14} + \cdots + 15200 ) / 512 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 79 \nu^{19} + 330 \nu^{18} + 2160 \nu^{17} + 9120 \nu^{16} + 23808 \nu^{15} + 101632 \nu^{14} + \cdots + 23264 ) / 512 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{19} - 2 \beta_{18} + \beta_{16} + \beta_{15} + 2 \beta_{14} - \beta_{13} - \beta_{12} - 2 \beta_{10} + \cdots - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{19} - \beta_{17} - \beta_{16} + 2 \beta_{15} - \beta_{13} - \beta_{11} + \beta_{8} + \beta_{7} + \cdots - 11 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{19} + 6 \beta_{18} - 2 \beta_{17} - 3 \beta_{16} - 5 \beta_{15} - 4 \beta_{14} + 3 \beta_{13} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 20 \beta_{19} - 2 \beta_{17} + 9 \beta_{16} - 18 \beta_{15} + 9 \beta_{13} - \beta_{12} + 4 \beta_{11} + \cdots + 56 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 48 \beta_{19} - 96 \beta_{18} + 42 \beta_{17} + 47 \beta_{16} + 90 \beta_{15} + 42 \beta_{14} + \cdots - 28 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 111 \beta_{19} + 31 \beta_{17} - 40 \beta_{16} + 80 \beta_{15} - 40 \beta_{13} + 10 \beta_{12} + \cdots - 197 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 108 \beta_{19} + 216 \beta_{18} - 98 \beta_{17} - 104 \beta_{16} - 206 \beta_{15} - 70 \beta_{14} + \cdots + 65 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1103 \beta_{19} - 373 \beta_{17} + 366 \beta_{16} - 730 \beta_{15} + 366 \beta_{13} - 121 \beta_{12} + \cdots + 1641 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2017 \beta_{19} - 4034 \beta_{18} + 1810 \beta_{17} + 1923 \beta_{16} + 3827 \beta_{15} + 1102 \beta_{14} + \cdots - 1275 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 10607 \beta_{19} + 3829 \beta_{17} - 3408 \beta_{16} + 6778 \beta_{15} - 3408 \beta_{13} + 1263 \beta_{12} + \cdots - 14729 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 19014 \beta_{19} + 38028 \beta_{18} - 16822 \beta_{17} - 18031 \beta_{16} - 35836 \beta_{15} + \cdots + 12408 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 100856 \beta_{19} - 37360 \beta_{17} + 31991 \beta_{16} - 63496 \beta_{15} + 31991 \beta_{13} + \cdots + 136264 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 179586 \beta_{19} - 359172 \beta_{18} + 157384 \beta_{17} + 169833 \beta_{16} + 336970 \beta_{15} + \cdots - 119214 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 477468 \beta_{19} + 178796 \beta_{17} - 150664 \beta_{16} + 298672 \beta_{15} - 150664 \beta_{13} + \cdots - 637942 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 848195 \beta_{19} + 1696390 \beta_{18} - 739400 \beta_{17} - 801022 \beta_{16} - 1587595 \beta_{15} + \cdots + 567891 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 4513268 \beta_{19} - 1698018 \beta_{17} + 1421105 \beta_{16} - 2815250 \beta_{15} + 1421105 \beta_{13} + \cdots + 6002184 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 8010737 \beta_{19} - 16021474 \beta_{18} + 6964098 \beta_{17} + 7560054 \beta_{16} + 14974835 \beta_{15} + \cdots - 5385629 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 42632415 \beta_{19} + 16072673 \beta_{17} - 13411781 \beta_{16} + 26559742 \beta_{15} - 13411781 \beta_{13} + \cdots - 56584629 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 75645378 \beta_{19} + 151290756 \beta_{18} - 65672732 \beta_{17} - 71365918 \beta_{16} + \cdots + 50956172 \) Copy content Toggle raw display

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

Character values

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

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1
2.07608i
− 0.720268i
1.02686i
3.07265i
− 1.66557i
1.76294i
0.196178i
1.05984i
− 1.81095i
− 1.53366i
− 2.07608i
0.720268i
− 1.02686i
− 3.07265i
1.66557i
− 1.76294i
− 0.196178i
− 1.05984i
1.81095i
1.53366i
0 −1.61426 + 2.79599i 0 0.539298 + 0.311364i 0 0.278439 − 0.482271i 0 −3.71170 − 6.42885i 0
529.2 0 −1.23226 + 2.13434i 0 −3.58978 − 2.07256i 0 0.348255 − 0.603195i 0 −1.53694 − 2.66206i 0
529.3 0 −0.879285 + 1.52297i 0 2.15955 + 1.24682i 0 −0.765920 + 1.32661i 0 −0.0462834 − 0.0801651i 0
529.4 0 −0.215732 + 0.373658i 0 3.10678 + 1.79370i 0 1.85805 − 3.21823i 0 1.40692 + 2.43686i 0
529.5 0 0.0774953 − 0.134226i 0 −0.658229 − 0.380029i 0 0.575138 − 0.996168i 0 1.48799 + 2.57727i 0
529.6 0 0.446681 − 0.773674i 0 −1.20245 − 0.694237i 0 −1.61095 + 2.79025i 0 1.10095 + 1.90690i 0
529.7 0 0.626614 − 1.08533i 0 −0.986981 − 0.569834i 0 1.16967 − 2.02592i 0 0.714711 + 1.23792i 0
529.8 0 0.960833 − 1.66421i 0 2.42947 + 1.40265i 0 −2.46150 + 4.26344i 0 −0.346401 − 0.599984i 0
529.9 0 1.29167 − 2.23723i 0 −2.18526 − 1.26166i 0 0.249875 − 0.432796i 0 −1.83681 − 3.18145i 0
529.10 0 1.53825 − 2.66433i 0 3.38761 + 1.95584i 0 1.35895 − 2.35377i 0 −3.23244 − 5.59875i 0
545.1 0 −1.61426 − 2.79599i 0 0.539298 − 0.311364i 0 0.278439 + 0.482271i 0 −3.71170 + 6.42885i 0
545.2 0 −1.23226 − 2.13434i 0 −3.58978 + 2.07256i 0 0.348255 + 0.603195i 0 −1.53694 + 2.66206i 0
545.3 0 −0.879285 − 1.52297i 0 2.15955 − 1.24682i 0 −0.765920 − 1.32661i 0 −0.0462834 + 0.0801651i 0
545.4 0 −0.215732 − 0.373658i 0 3.10678 − 1.79370i 0 1.85805 + 3.21823i 0 1.40692 − 2.43686i 0
545.5 0 0.0774953 + 0.134226i 0 −0.658229 + 0.380029i 0 0.575138 + 0.996168i 0 1.48799 − 2.57727i 0
545.6 0 0.446681 + 0.773674i 0 −1.20245 + 0.694237i 0 −1.61095 − 2.79025i 0 1.10095 − 1.90690i 0
545.7 0 0.626614 + 1.08533i 0 −0.986981 + 0.569834i 0 1.16967 + 2.02592i 0 0.714711 − 1.23792i 0
545.8 0 0.960833 + 1.66421i 0 2.42947 − 1.40265i 0 −2.46150 − 4.26344i 0 −0.346401 + 0.599984i 0
545.9 0 1.29167 + 2.23723i 0 −2.18526 + 1.26166i 0 0.249875 + 0.432796i 0 −1.83681 + 3.18145i 0
545.10 0 1.53825 + 2.66433i 0 3.38761 − 1.95584i 0 1.35895 + 2.35377i 0 −3.23244 + 5.59875i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 592.2.w.f 20
4.b odd 2 1 296.2.o.a ✓ 20
12.b even 2 1 2664.2.cq.c 20
37.e even 6 1 inner 592.2.w.f 20
148.j odd 6 1 296.2.o.a ✓ 20
444.p even 6 1 2664.2.cq.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
296.2.o.a ✓ 20 4.b odd 2 1
296.2.o.a ✓ 20 148.j odd 6 1
592.2.w.f 20 1.a even 1 1 trivial
592.2.w.f 20 37.e even 6 1 inner
2664.2.cq.c 20 12.b even 2 1
2664.2.cq.c 20 444.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(592, [\chi])\):

\( T_{3}^{20} - 2 T_{3}^{19} + 23 T_{3}^{18} - 42 T_{3}^{17} + 337 T_{3}^{16} - 584 T_{3}^{15} + 2824 T_{3}^{14} + \cdots + 256 \) Copy content Toggle raw display
\( T_{5}^{20} - 6 T_{5}^{19} - 17 T_{5}^{18} + 174 T_{5}^{17} + 247 T_{5}^{16} - 3588 T_{5}^{15} + \cdots + 591361 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} - 2 T^{19} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{20} - 6 T^{19} + \cdots + 591361 \) Copy content Toggle raw display
$7$ \( T^{20} - 2 T^{19} + \cdots + 16384 \) Copy content Toggle raw display
$11$ \( (T^{10} + 8 T^{9} + \cdots - 512)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + 6 T^{19} + \cdots + 65536 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 576048001 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 34334607616 \) Copy content Toggle raw display
$23$ \( T^{20} + 264 T^{18} + \cdots + 21827584 \) Copy content Toggle raw display
$29$ \( T^{20} + 168 T^{18} + \cdots + 3748096 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 86031929344 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 48\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 73948115277481 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 107039384928256 \) Copy content Toggle raw display
$47$ \( (T^{10} - 4 T^{9} + \cdots - 32768)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{20} - 18 T^{19} + \cdots + 16777216 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 87\!\cdots\!61 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 340043631433984 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 1292241485824 \) Copy content Toggle raw display
$73$ \( (T^{10} + 8 T^{9} + \cdots + 48357376)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 3360289024 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 353330381056 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 194616785062144 \) Copy content Toggle raw display
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