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Properties

Label 1860.2.z.d
Level $1860$
Weight $2$
Character orbit 1860.z
Analytic conductor $14.852$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1860,2,Mod(481,1860)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1860, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1860.481");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1860.z (of order \(5\), 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: \(14.8521747760\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 7 x^{19} + 33 x^{18} - 113 x^{17} + 460 x^{16} - 1315 x^{15} + 4399 x^{14} - 12870 x^{13} + \cdots + 525625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{3} + q^{5} + ( - \beta_{11} - \beta_{3} - \beta_1) q^{7} + \beta_{6} q^{9} + (\beta_{17} + \beta_{15} + \beta_{12} + \cdots + 1) q^{11} + ( - \beta_{17} - \beta_{13}) q^{13} + \beta_{8} q^{15}+ \cdots + ( - \beta_{17} - \beta_{16} + \cdots - \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 5 q^{3} + 20 q^{5} - 5 q^{7} - 5 q^{9} + 2 q^{11} - 5 q^{15} - 6 q^{17} - 4 q^{19} - 11 q^{23} + 20 q^{25} - 5 q^{27} - 29 q^{29} - 2 q^{31} - 3 q^{33} - 5 q^{35} + 36 q^{37} - 5 q^{39} - 10 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 7 x^{19} + 33 x^{18} - 113 x^{17} + 460 x^{16} - 1315 x^{15} + 4399 x^{14} - 12870 x^{13} + \cdots + 525625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 14\!\cdots\!33 \nu^{19} + \cdots - 39\!\cdots\!00 ) / 13\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 15\!\cdots\!24 \nu^{19} + \cdots + 18\!\cdots\!75 ) / 20\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18\!\cdots\!30 \nu^{19} + \cdots + 14\!\cdots\!45 ) / 41\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 35\!\cdots\!49 \nu^{19} + \cdots + 48\!\cdots\!75 ) / 51\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 49\!\cdots\!59 \nu^{19} + \cdots - 38\!\cdots\!25 ) / 51\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10\!\cdots\!42 \nu^{19} + \cdots + 75\!\cdots\!25 ) / 10\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 72\!\cdots\!41 \nu^{19} + \cdots + 43\!\cdots\!00 ) / 51\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 21\!\cdots\!84 \nu^{19} + \cdots + 84\!\cdots\!75 ) / 13\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 21\!\cdots\!47 \nu^{19} + \cdots + 29\!\cdots\!25 ) / 67\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22\!\cdots\!76 \nu^{19} + \cdots + 16\!\cdots\!50 ) / 67\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 27\!\cdots\!62 \nu^{19} + \cdots - 31\!\cdots\!25 ) / 67\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 84\!\cdots\!49 \nu^{19} + \cdots - 34\!\cdots\!25 ) / 13\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 87\!\cdots\!39 \nu^{19} + \cdots + 23\!\cdots\!75 ) / 13\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 47\!\cdots\!56 \nu^{19} + \cdots + 17\!\cdots\!00 ) / 67\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 30\!\cdots\!68 \nu^{19} + \cdots + 10\!\cdots\!00 ) / 33\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 62\!\cdots\!51 \nu^{19} + \cdots + 57\!\cdots\!75 ) / 67\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 75\!\cdots\!58 \nu^{19} + \cdots - 22\!\cdots\!75 ) / 67\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 51\!\cdots\!31 \nu^{19} + \cdots + 18\!\cdots\!75 ) / 33\!\cdots\!75 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{19} - \beta_{18} - \beta_{17} - \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} + 6\beta_{6} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{17} + \beta_{13} - \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + 10 \beta_{7} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{19} + 12 \beta_{18} + 12 \beta_{16} - 13 \beta_{15} - 2 \beta_{14} - 13 \beta_{11} - 49 \beta_{8} + \cdots - 75 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 29 \beta_{18} - 28 \beta_{17} - 28 \beta_{16} - 29 \beta_{15} + 4 \beta_{14} - 4 \beta_{13} + \cdots - 70 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 39 \beta_{19} - 149 \beta_{18} + 126 \beta_{17} + 135 \beta_{15} - 161 \beta_{14} + 210 \beta_{13} + \cdots - 23 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 11 \beta_{19} + 137 \beta_{18} - 11 \beta_{17} + 363 \beta_{16} - 340 \beta_{14} + \cdots + 340 \beta_{2} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1268 \beta_{19} - 631 \beta_{17} - 1268 \beta_{16} - 1541 \beta_{15} + 2046 \beta_{13} - 1541 \beta_{12} + \cdots + 387 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4690 \beta_{19} - 2305 \beta_{18} + 378 \beta_{16} + 4544 \beta_{15} + 4406 \beta_{14} + 2371 \beta_{11} + \cdots + 15521 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 19160 \beta_{18} + 9595 \beta_{17} + 9595 \beta_{16} + 19160 \beta_{15} + 26704 \beta_{14} + \cdots + 120933 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 60848 \beta_{19} + 55017 \beta_{18} + 8801 \beta_{17} - 34479 \beta_{15} + 46974 \beta_{14} + \cdots - 42554 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 108264 \beta_{19} - 213204 \beta_{18} - 108264 \beta_{17} - 140954 \beta_{16} + \cdots - 205442 \beta_{2} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 173045 \beta_{19} + 793320 \beta_{17} - 173045 \beta_{16} + 485833 \beta_{15} - 797992 \beta_{13} + \cdots + 417998 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 2023929 \beta_{19} + 2570797 \beta_{18} + 818352 \beta_{16} - 2495562 \beta_{15} - 2969234 \beta_{14} + \cdots - 17779974 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 7786097 \beta_{18} - 10391952 \beta_{17} - 10391952 \beta_{16} - 7786097 \beta_{15} - 12597609 \beta_{14} + \cdots - 40522293 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 28599328 \beta_{19} - 28515691 \beta_{18} + 3637459 \beta_{17} + 31389322 \beta_{15} + \cdots - 19168049 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 52006848 \beta_{19} + 88092091 \beta_{18} - 52006848 \beta_{17} + 136721592 \beta_{16} + \cdots + 153050633 \beta_{2} \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 36842605 \beta_{19} - 399491290 \beta_{17} + 36842605 \beta_{16} - 387097160 \beta_{15} + \cdots + 284920729 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 1805919288 \beta_{19} - 1157283960 \beta_{18} + 838186099 \beta_{16} + 1059593388 \beta_{15} + \cdots + 7298995689 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1860\mathbb{Z}\right)^\times\).

\(n\) \(931\) \(1117\) \(1241\) \(1801\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
481.1
−2.63196 − 1.91223i
3.00149 + 2.18071i
−0.785116 − 0.570420i
1.08483 + 0.788178i
2.75780 + 2.00366i
0.932859 + 2.87105i
0.239105 + 0.735888i
−0.406012 − 1.24958i
0.389291 + 1.19812i
−1.08229 − 3.33096i
0.932859 − 2.87105i
0.239105 − 0.735888i
−0.406012 + 1.24958i
0.389291 − 1.19812i
−1.08229 + 3.33096i
−2.63196 + 1.91223i
3.00149 − 2.18071i
−0.785116 + 0.570420i
1.08483 − 0.788178i
2.75780 − 2.00366i
0 −0.809017 − 0.587785i 0 1.00000 0 −1.24936 + 3.84513i 0 0.309017 + 0.951057i 0
481.2 0 −0.809017 − 0.587785i 0 1.00000 0 −0.566036 + 1.74208i 0 0.309017 + 0.951057i 0
481.3 0 −0.809017 − 0.587785i 0 1.00000 0 0.219103 − 0.674328i 0 0.309017 + 0.951057i 0
481.4 0 −0.809017 − 0.587785i 0 1.00000 0 0.898158 − 2.76425i 0 0.309017 + 0.951057i 0
481.5 0 −0.809017 − 0.587785i 0 1.00000 0 1.12519 − 3.46297i 0 0.309017 + 0.951057i 0
721.1 0 0.309017 + 0.951057i 0 1.00000 0 −4.13441 − 3.00382i 0 −0.809017 + 0.587785i 0
721.2 0 0.309017 + 0.951057i 0 1.00000 0 −2.28346 − 1.65903i 0 −0.809017 + 0.587785i 0
721.3 0 0.309017 + 0.951057i 0 1.00000 0 −1.13511 − 0.824707i 0 −0.809017 + 0.587785i 0
721.4 0 0.309017 + 0.951057i 0 1.00000 0 1.83739 + 1.33494i 0 −0.809017 + 0.587785i 0
721.5 0 0.309017 + 0.951057i 0 1.00000 0 2.78854 + 2.02599i 0 −0.809017 + 0.587785i 0
841.1 0 0.309017 − 0.951057i 0 1.00000 0 −4.13441 + 3.00382i 0 −0.809017 − 0.587785i 0
841.2 0 0.309017 − 0.951057i 0 1.00000 0 −2.28346 + 1.65903i 0 −0.809017 − 0.587785i 0
841.3 0 0.309017 − 0.951057i 0 1.00000 0 −1.13511 + 0.824707i 0 −0.809017 − 0.587785i 0
841.4 0 0.309017 − 0.951057i 0 1.00000 0 1.83739 − 1.33494i 0 −0.809017 − 0.587785i 0
841.5 0 0.309017 − 0.951057i 0 1.00000 0 2.78854 − 2.02599i 0 −0.809017 − 0.587785i 0
901.1 0 −0.809017 + 0.587785i 0 1.00000 0 −1.24936 − 3.84513i 0 0.309017 − 0.951057i 0
901.2 0 −0.809017 + 0.587785i 0 1.00000 0 −0.566036 − 1.74208i 0 0.309017 − 0.951057i 0
901.3 0 −0.809017 + 0.587785i 0 1.00000 0 0.219103 + 0.674328i 0 0.309017 − 0.951057i 0
901.4 0 −0.809017 + 0.587785i 0 1.00000 0 0.898158 + 2.76425i 0 0.309017 − 0.951057i 0
901.5 0 −0.809017 + 0.587785i 0 1.00000 0 1.12519 + 3.46297i 0 0.309017 − 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 481.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1860.2.z.d ✓ 20
31.d even 5 1 inner 1860.2.z.d ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1860.2.z.d ✓ 20 1.a even 1 1 trivial
1860.2.z.d ✓ 20 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{20} + 5 T_{7}^{19} + 30 T_{7}^{18} + 87 T_{7}^{17} + 581 T_{7}^{16} + 817 T_{7}^{15} + \cdots + 77510416 \) acting on \(S_{2}^{\mathrm{new}}(1860, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$5$ \( (T - 1)^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + 5 T^{19} + \cdots + 77510416 \) Copy content Toggle raw display
$11$ \( T^{20} - 2 T^{19} + \cdots + 2027776 \) Copy content Toggle raw display
$13$ \( T^{20} + 17 T^{18} + \cdots + 8202496 \) Copy content Toggle raw display
$17$ \( T^{20} + 6 T^{19} + \cdots + 6400 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 18066537725625 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 828288400 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 144384256 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 819628286980801 \) Copy content Toggle raw display
$37$ \( (T^{10} - 18 T^{9} + \cdots + 2284)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 57753702400 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 17920659558400 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 198488070400 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 6140880486400 \) Copy content Toggle raw display
$61$ \( (T^{10} - 194 T^{8} + \cdots - 1333489)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} - T^{9} + \cdots - 2079076)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 11963109376 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 68792824692736 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 109690128025 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 118057873317136 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 328446753610000 \) Copy content Toggle raw display
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