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Properties

Label 1860.2.g.a
Level $1860$
Weight $2$
Character orbit 1860.g
Analytic conductor $14.852$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1860,2,Mod(1489,1860)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1860, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1860.1489");
 
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.g (of order \(2\), degree \(1\), 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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} + 20 x^{11} + 10 x^{10} - 46 x^{9} + 33 x^{8} + 664 x^{7} + 165 x^{6} - 1150 x^{5} + \cdots + 78125 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{3} - \beta_1 q^{5} + ( - \beta_{8} - \beta_{7}) q^{7} - q^{9} + (\beta_{11} + 1) q^{11} + (\beta_{10} + \beta_{9} + \cdots + \beta_{4}) q^{13} + \beta_{9} q^{15} + (\beta_{10} + \beta_{9} + \cdots - \beta_{5}) q^{17}+ \cdots + ( - \beta_{11} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{5} - 14 q^{9} + 8 q^{11} + 2 q^{15} - 20 q^{19} + 8 q^{21} + 4 q^{25} - 20 q^{29} - 14 q^{31} + 10 q^{35} - 20 q^{39} + 4 q^{41} + 2 q^{45} - 18 q^{49} + 12 q^{51} + 16 q^{55} - 28 q^{59} + 20 q^{61}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 20 x^{11} + 10 x^{10} - 46 x^{9} + 33 x^{8} + 664 x^{7} + 165 x^{6} - 1150 x^{5} + \cdots + 78125 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{13} - 2 \nu^{12} + 20 \nu^{10} + 10 \nu^{9} - 46 \nu^{8} + 33 \nu^{7} + 664 \nu^{6} + \cdots - 31250 ) / 15625 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3 \nu^{13} - \nu^{12} - 10 \nu^{11} + 60 \nu^{10} + 130 \nu^{9} - 88 \nu^{8} - 131 \nu^{7} + \cdots - 93750 ) / 31250 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 433 \nu^{13} + 5371 \nu^{12} + 7065 \nu^{11} - 10685 \nu^{10} + 29895 \nu^{9} + 243343 \nu^{8} + \cdots + 74484375 ) / 4750000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 508 \nu^{13} - 2069 \nu^{12} - 2830 \nu^{11} - 3785 \nu^{10} - 41030 \nu^{9} - 108107 \nu^{8} + \cdots - 17109375 ) / 4750000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 117 \nu^{13} - 983 \nu^{12} - 1717 \nu^{11} + 2745 \nu^{10} - 2635 \nu^{9} - 48547 \nu^{8} + \cdots - 15846875 ) / 950000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 943 \nu^{13} - 5756 \nu^{12} - 9585 \nu^{11} + 22760 \nu^{10} - 17095 \nu^{9} - 276078 \nu^{8} + \cdots - 110531250 ) / 4750000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 611 \nu^{13} - 527 \nu^{12} - 2040 \nu^{11} + 9395 \nu^{10} + 22635 \nu^{9} - 19781 \nu^{8} + \cdots - 15015625 ) / 2375000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 139 \nu^{13} + 408 \nu^{12} + 565 \nu^{11} - 3305 \nu^{10} - 1665 \nu^{9} + 19194 \nu^{8} + \cdots + 9546875 ) / 475000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 961 \nu^{13} - 1133 \nu^{12} + 2635 \nu^{11} - 9020 \nu^{10} - 56585 \nu^{9} - 68969 \nu^{8} + \cdots - 7781250 ) / 2375000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 63 \nu^{13} + 29 \nu^{12} - 160 \nu^{11} + 585 \nu^{10} + 2605 \nu^{9} + 2277 \nu^{8} + \cdots - 328125 ) / 125000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 132 \nu^{13} + 229 \nu^{12} + 470 \nu^{11} - 2065 \nu^{10} - 2270 \nu^{9} + 11847 \nu^{8} + \cdots + 6453125 ) / 250000 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 57 \nu^{13} + 6 \nu^{12} - 190 \nu^{11} + 540 \nu^{10} + 2095 \nu^{9} + 203 \nu^{8} - 6889 \nu^{7} + \cdots - 921875 ) / 62500 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} - 2 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + \beta_{6} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{13} - 4 \beta_{12} - 2 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} - 2 \beta_{7} - \beta_{6} + \cdots - 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{12} + 8 \beta_{11} + 6 \beta_{10} - 6 \beta_{9} - 4 \beta_{8} - 8 \beta_{7} + 2 \beta_{6} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 22 \beta_{13} - 32 \beta_{12} + 12 \beta_{11} - 26 \beta_{10} + 14 \beta_{9} - 40 \beta_{8} + \cdots + 35 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 24 \beta_{13} - 4 \beta_{12} + 44 \beta_{11} + 54 \beta_{10} + 74 \beta_{9} + 112 \beta_{8} + \cdots + 48 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 26 \beta_{13} + 44 \beta_{12} + 56 \beta_{11} - 62 \beta_{10} - 34 \beta_{9} - 188 \beta_{8} + \cdots + 138 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 130 \beta_{13} + 138 \beta_{12} - 342 \beta_{11} - 6 \beta_{10} + 120 \beta_{9} + 414 \beta_{8} + \cdots - 239 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 448 \beta_{13} + 664 \beta_{12} - 488 \beta_{11} + 260 \beta_{10} - 644 \beta_{9} + 1000 \beta_{8} + \cdots + 48 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 392 \beta_{13} - 40 \beta_{12} - 668 \beta_{11} - 96 \beta_{10} - 1504 \beta_{9} - 644 \beta_{8} + \cdots - 2104 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 696 \beta_{13} - 2104 \beta_{12} - 1040 \beta_{11} + 236 \beta_{10} - 772 \beta_{9} - 1520 \beta_{8} + \cdots + 4417 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 536 \beta_{13} - 3224 \beta_{12} + 5436 \beta_{11} + 4672 \beta_{10} + 2624 \beta_{9} - 5204 \beta_{8} + \cdots + 12008 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(931\) \(1117\) \(1241\) \(1801\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1489.1
1.83034 + 1.28447i
1.64227 − 1.51755i
1.34603 − 1.78556i
1.22189 + 1.87269i
−1.06240 − 1.96756i
−1.75818 + 1.38160i
−2.21994 − 0.268087i
1.83034 − 1.28447i
1.64227 + 1.51755i
1.34603 + 1.78556i
1.22189 − 1.87269i
−1.06240 + 1.96756i
−1.75818 − 1.38160i
−2.21994 + 0.268087i
0 − 1.00000i 0 −1.83034 − 1.28447i 0 4.13530i 0 −1.00000 0
1489.2 0 − 1.00000i 0 −1.64227 + 1.51755i 0 − 4.29083i 0 −1.00000 0
1489.3 0 − 1.00000i 0 −1.34603 + 1.78556i 0 4.07725i 0 −1.00000 0
1489.4 0 − 1.00000i 0 −1.22189 − 1.87269i 0 − 0.386084i 0 −1.00000 0
1489.5 0 − 1.00000i 0 1.06240 + 1.96756i 0 0.690103i 0 −1.00000 0
1489.6 0 − 1.00000i 0 1.75818 − 1.38160i 0 1.50165i 0 −1.00000 0
1489.7 0 − 1.00000i 0 2.21994 + 0.268087i 0 − 1.72739i 0 −1.00000 0
1489.8 0 1.00000i 0 −1.83034 + 1.28447i 0 − 4.13530i 0 −1.00000 0
1489.9 0 1.00000i 0 −1.64227 − 1.51755i 0 4.29083i 0 −1.00000 0
1489.10 0 1.00000i 0 −1.34603 − 1.78556i 0 − 4.07725i 0 −1.00000 0
1489.11 0 1.00000i 0 −1.22189 + 1.87269i 0 0.386084i 0 −1.00000 0
1489.12 0 1.00000i 0 1.06240 − 1.96756i 0 − 0.690103i 0 −1.00000 0
1489.13 0 1.00000i 0 1.75818 + 1.38160i 0 − 1.50165i 0 −1.00000 0
1489.14 0 1.00000i 0 2.21994 − 0.268087i 0 1.72739i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1489.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1860.2.g.a ✓ 14
3.b odd 2 1 5580.2.g.d 14
5.b even 2 1 inner 1860.2.g.a ✓ 14
5.c odd 4 1 9300.2.a.bc 7
5.c odd 4 1 9300.2.a.bf 7
15.d odd 2 1 5580.2.g.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1860.2.g.a ✓ 14 1.a even 1 1 trivial
1860.2.g.a ✓ 14 5.b even 2 1 inner
5580.2.g.d 14 3.b odd 2 1
5580.2.g.d 14 15.d odd 2 1
9300.2.a.bc 7 5.c odd 4 1
9300.2.a.bf 7 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{14} + 58T_{7}^{12} + 1221T_{7}^{10} + 11072T_{7}^{8} + 40052T_{7}^{6} + 56904T_{7}^{4} + 24400T_{7}^{2} + 2500 \) acting on \(S_{2}^{\mathrm{new}}(1860, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{14} + 2 T^{13} + \cdots + 78125 \) Copy content Toggle raw display
$7$ \( T^{14} + 58 T^{12} + \cdots + 2500 \) Copy content Toggle raw display
$11$ \( (T^{7} - 4 T^{6} + \cdots + 384)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + 104 T^{12} + \cdots + 3240000 \) Copy content Toggle raw display
$17$ \( T^{14} + 96 T^{12} + \cdots + 11943936 \) Copy content Toggle raw display
$19$ \( (T^{7} + 10 T^{6} + \cdots - 144)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + 108 T^{12} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{7} + 10 T^{6} + \cdots + 18624)^{2} \) Copy content Toggle raw display
$31$ \( (T + 1)^{14} \) Copy content Toggle raw display
$37$ \( T^{14} + 180 T^{12} + \cdots + 1638400 \) Copy content Toggle raw display
$41$ \( (T^{7} - 2 T^{6} + \cdots + 9736)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 11286937600 \) Copy content Toggle raw display
$47$ \( T^{14} + 228 T^{12} + \cdots + 22278400 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 212926464 \) Copy content Toggle raw display
$59$ \( (T^{7} + 14 T^{6} + \cdots + 90)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} - 10 T^{6} + \cdots + 344576)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 1998378049600 \) Copy content Toggle raw display
$71$ \( (T^{7} + 2 T^{6} + \cdots + 4014)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + 344 T^{12} + \cdots + 55115776 \) Copy content Toggle raw display
$79$ \( (T^{7} + 24 T^{6} + \cdots + 116208)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 36943915264 \) Copy content Toggle raw display
$89$ \( (T^{7} + 12 T^{6} + \cdots + 193096)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 106879371709504 \) Copy content Toggle raw display
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