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Properties

Label 1860.2.z.e
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: \( x^{20} - x^{19} + 7 x^{18} + x^{17} + 148 x^{16} + 321 x^{15} + 1813 x^{14} + 3286 x^{13} + \cdots + 102400 \) 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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + q^{5} - \beta_{15} q^{7} + \beta_{2} q^{9} + ( - \beta_{7} - \beta_{3}) q^{11} + ( - \beta_{15} - \beta_{12} - \beta_{11} + \cdots - 1) q^{13} - \beta_{5} q^{15} + ( - \beta_{19} + \beta_{18} - \beta_{13} + \cdots + 1) q^{17}+ \cdots + (\beta_{6} + \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 5 q^{3} + 20 q^{5} - q^{7} - 5 q^{9} - 2 q^{11} - 10 q^{13} + 5 q^{15} + 12 q^{17} + 4 q^{19} - 4 q^{21} - 7 q^{23} + 20 q^{25} + 5 q^{27} + 3 q^{29} - 8 q^{31} - 3 q^{33} - q^{35} + 24 q^{37}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + 7 x^{18} + x^{17} + 148 x^{16} + 321 x^{15} + 1813 x^{14} + 3286 x^{13} + \cdots + 102400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 23\!\cdots\!98 \nu^{19} + \cdots + 21\!\cdots\!20 ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 28\!\cdots\!08 \nu^{19} + \cdots - 33\!\cdots\!00 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 27\!\cdots\!44 \nu^{19} + \cdots + 17\!\cdots\!00 ) / 22\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\!\cdots\!09 \nu^{19} + \cdots - 76\!\cdots\!00 ) / 78\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 79\!\cdots\!74 \nu^{19} + \cdots + 69\!\cdots\!00 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 84\!\cdots\!03 \nu^{19} + \cdots + 23\!\cdots\!00 ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!37 \nu^{19} + \cdots + 11\!\cdots\!80 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 41\!\cdots\!65 \nu^{19} + \cdots - 55\!\cdots\!40 ) / 63\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 40\!\cdots\!96 \nu^{19} + \cdots + 95\!\cdots\!00 ) / 45\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 41\!\cdots\!18 \nu^{19} + \cdots + 23\!\cdots\!00 ) / 45\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 82\!\cdots\!83 \nu^{19} + \cdots + 17\!\cdots\!80 ) / 91\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 10\!\cdots\!27 \nu^{19} + \cdots - 22\!\cdots\!20 ) / 91\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 28\!\cdots\!07 \nu^{19} + \cdots - 16\!\cdots\!20 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 14\!\cdots\!93 \nu^{19} + \cdots - 34\!\cdots\!80 ) / 91\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 30\!\cdots\!83 \nu^{19} + \cdots - 25\!\cdots\!20 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 38\!\cdots\!17 \nu^{19} + \cdots + 57\!\cdots\!20 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 62\!\cdots\!99 \nu^{19} + \cdots + 40\!\cdots\!00 ) / 24\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 30\!\cdots\!11 \nu^{19} + \cdots - 19\!\cdots\!00 ) / 91\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{18} + \beta_{16} - \beta_{14} + \beta_{12} - \beta_{9} + \beta_{8} + \beta_{7} + 7\beta_{5} + 2\beta_{3} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{18} - \beta_{14} - 3\beta_{11} - \beta_{10} + 2\beta_{9} + 3\beta_{8} - 9\beta_{6} + 3\beta_{5} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 17 \beta_{19} - 17 \beta_{18} - 7 \beta_{17} - 4 \beta_{15} - 2 \beta_{14} + 18 \beta_{13} - 18 \beta_{11} + \cdots - 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 34 \beta_{19} - 14 \beta_{18} - 41 \beta_{17} - 4 \beta_{16} - 18 \beta_{15} - 20 \beta_{14} + \cdots - 174 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 175 \beta_{19} - 183 \beta_{17} - 128 \beta_{16} - 211 \beta_{15} + 122 \beta_{13} - 290 \beta_{12} + \cdots - 1049 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 381 \beta_{18} - 324 \beta_{16} - 680 \beta_{15} + 324 \beta_{14} - 1061 \beta_{12} + 473 \beta_{11} + \cdots - 2021 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2533 \beta_{19} + 3873 \beta_{18} + 2534 \beta_{17} + 199 \beta_{16} + 198 \beta_{15} + \cdots + 2335 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 13114 \beta_{19} + 13114 \beta_{18} + 12611 \beta_{17} + 5541 \beta_{15} + 7301 \beta_{14} + \cdots + 33994 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 60922 \beta_{19} + 38758 \beta_{18} + 50949 \beta_{17} + 3796 \beta_{16} + 41189 \beta_{15} + \cdots + 242291 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 135310 \beta_{19} + 125436 \beta_{17} + 81229 \beta_{16} + 201587 \beta_{15} - 165952 \beta_{13} + \cdots + 914224 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 613780 \beta_{18} + 349363 \beta_{16} + 589886 \beta_{15} - 349363 \beta_{14} + 1203666 \beta_{12} + \cdots + 2271194 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2313253 \beta_{19} - 3802788 \beta_{18} - 2127986 \beta_{17} + 712249 \beta_{16} + 526982 \beta_{15} + \cdots - 2840235 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 15914923 \beta_{19} - 15914923 \beta_{18} - 14186364 \beta_{17} - 6008093 \beta_{15} - 8022359 \beta_{14} + \cdots - 37854525 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 63231158 \beta_{19} - 38771080 \beta_{18} - 57873408 \beta_{17} - 12178032 \beta_{16} - 47683279 \beta_{15} + \cdots - 251211321 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 161500497 \beta_{19} - 148545261 \beta_{17} - 87168771 \beta_{16} - 222821588 \beta_{15} + \cdots - 1030287399 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 643831807 \beta_{18} - 345782043 \beta_{16} - 656327256 \beta_{15} + 345782043 \beta_{14} + \cdots - 2558431863 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 2646032386 \beta_{19} + 4274055667 \beta_{18} + 2422177742 \beta_{17} - 811876522 \beta_{16} + \cdots + 3234054264 \beta_1 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 17257564505 \beta_{19} + 17257564505 \beta_{18} + 15725423819 \beta_{17} + 6613183230 \beta_{15} + \cdots + 42173909373 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(931\) \(1117\) \(1241\) \(1801\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(\beta_{2}\)

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
−0.130921 + 0.402935i
0.884565 − 2.72241i
0.360597 − 1.10980i
−0.924618 + 2.84568i
−0.498639 + 1.53465i
−0.735086 − 0.534071i
−1.46583 − 1.06499i
3.28269 + 2.38502i
2.01007 + 1.46040i
−2.28283 − 1.65858i
−0.735086 + 0.534071i
−1.46583 + 1.06499i
3.28269 − 2.38502i
2.01007 − 1.46040i
−2.28283 + 1.65858i
−0.130921 − 0.402935i
0.884565 + 2.72241i
0.360597 + 1.10980i
−0.924618 − 2.84568i
−0.498639 − 1.53465i
0 0.809017 + 0.587785i 0 1.00000 0 −1.51241 + 4.65471i 0 0.309017 + 0.951057i 0
481.2 0 0.809017 + 0.587785i 0 1.00000 0 −0.440479 + 1.35565i 0 0.309017 + 0.951057i 0
481.3 0 0.809017 + 0.587785i 0 1.00000 0 −0.205248 + 0.631687i 0 0.309017 + 0.951057i 0
481.4 0 0.809017 + 0.587785i 0 1.00000 0 −0.0467593 + 0.143910i 0 0.309017 + 0.951057i 0
481.5 0 0.809017 + 0.587785i 0 1.00000 0 1.39587 − 4.29606i 0 0.309017 + 0.951057i 0
721.1 0 −0.309017 − 0.951057i 0 1.00000 0 −2.57440 − 1.87041i 0 −0.809017 + 0.587785i 0
721.2 0 −0.309017 − 0.951057i 0 1.00000 0 −1.44773 − 1.05183i 0 −0.809017 + 0.587785i 0
721.3 0 −0.309017 − 0.951057i 0 1.00000 0 −0.945262 − 0.686773i 0 −0.809017 + 0.587785i 0
721.4 0 −0.309017 − 0.951057i 0 1.00000 0 1.46802 + 1.06658i 0 −0.809017 + 0.587785i 0
721.5 0 −0.309017 − 0.951057i 0 1.00000 0 3.80838 + 2.76695i 0 −0.809017 + 0.587785i 0
841.1 0 −0.309017 + 0.951057i 0 1.00000 0 −2.57440 + 1.87041i 0 −0.809017 − 0.587785i 0
841.2 0 −0.309017 + 0.951057i 0 1.00000 0 −1.44773 + 1.05183i 0 −0.809017 − 0.587785i 0
841.3 0 −0.309017 + 0.951057i 0 1.00000 0 −0.945262 + 0.686773i 0 −0.809017 − 0.587785i 0
841.4 0 −0.309017 + 0.951057i 0 1.00000 0 1.46802 − 1.06658i 0 −0.809017 − 0.587785i 0
841.5 0 −0.309017 + 0.951057i 0 1.00000 0 3.80838 − 2.76695i 0 −0.809017 − 0.587785i 0
901.1 0 0.809017 − 0.587785i 0 1.00000 0 −1.51241 − 4.65471i 0 0.309017 − 0.951057i 0
901.2 0 0.809017 − 0.587785i 0 1.00000 0 −0.440479 − 1.35565i 0 0.309017 − 0.951057i 0
901.3 0 0.809017 − 0.587785i 0 1.00000 0 −0.205248 − 0.631687i 0 0.309017 − 0.951057i 0
901.4 0 0.809017 − 0.587785i 0 1.00000 0 −0.0467593 − 0.143910i 0 0.309017 − 0.951057i 0
901.5 0 0.809017 − 0.587785i 0 1.00000 0 1.39587 + 4.29606i 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.e ✓ 20
31.d even 5 1 inner 1860.2.z.e ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1860.2.z.e ✓ 20 1.a even 1 1 trivial
1860.2.z.e ✓ 20 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{20} + T_{7}^{19} + 26 T_{7}^{18} + 25 T_{7}^{17} + 433 T_{7}^{16} + 1575 T_{7}^{15} + \cdots + 32400 \) 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
$7$ \( T^{20} + T^{19} + \cdots + 32400 \) Copy content Toggle raw display
$11$ \( T^{20} + 2 T^{19} + \cdots + 102400 \) Copy content Toggle raw display
$13$ \( T^{20} + 10 T^{19} + \cdots + 1860496 \) Copy content Toggle raw display
$17$ \( T^{20} - 12 T^{19} + \cdots + 34857216 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 375390625 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 43149260176 \) Copy content Toggle raw display
$29$ \( T^{20} - 3 T^{19} + \cdots + 91853056 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 819628286980801 \) Copy content Toggle raw display
$37$ \( (T^{10} - 12 T^{9} + \cdots + 30351116)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 24395854694656 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 870136487701776 \) Copy content Toggle raw display
$47$ \( T^{20} - 2 T^{19} + \cdots + 495616 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 64890429696 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 403723183722496 \) Copy content Toggle raw display
$61$ \( (T^{10} + 4 T^{9} + \cdots - 222345)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} - 9 T^{9} + \cdots + 21296484)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 71\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 262471782400 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 50\!\cdots\!21 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 51\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 89267802970896 \) Copy content Toggle raw display
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