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Properties

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

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1860,2,Mod(1489,1860)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [14,0,0,0,-2,0,0,0,-14,0,8,0,0,0,2,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.8521747760\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} + 2 x^{12} + 2 x^{11} + 85 x^{10} - 158 x^{9} + 148 x^{8} + 80 x^{7} + 828 x^{6} + \cdots + 1458 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

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

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{13} q^{5} + ( - \beta_{9} + \beta_{6} - \beta_{2}) q^{7} - q^{9} + ( - \beta_{7} - \beta_{4}) q^{11} + ( - \beta_{11} + \beta_{6} + 2 \beta_1) q^{13} + \beta_{11} q^{15} + (\beta_{13} - \beta_{12} + \cdots - 3 \beta_1) q^{17}+ \cdots + (\beta_{7} + \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{5} - 14 q^{9} + 8 q^{11} + 2 q^{15} + 12 q^{19} - 8 q^{21} - 4 q^{25} + 4 q^{29} + 14 q^{31} - 22 q^{35} + 20 q^{39} + 20 q^{41} + 2 q^{45} - 2 q^{49} - 28 q^{51} - 4 q^{55} + 4 q^{59} + 4 q^{61}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 2 x^{12} + 2 x^{11} + 85 x^{10} - 158 x^{9} + 148 x^{8} + 80 x^{7} + 828 x^{6} + \cdots + 1458 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 61130558475409 \nu^{13} - 74981595984059 \nu^{12} + 24454518237833 \nu^{11} + \cdots - 57\!\cdots\!08 ) / 13\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 675304643396585 \nu^{13} - 627676955844069 \nu^{12} - 985019453540881 \nu^{11} + \cdots - 45\!\cdots\!84 ) / 13\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21\!\cdots\!87 \nu^{13} + \cdots + 11\!\cdots\!16 ) / 41\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 22\!\cdots\!13 \nu^{13} + \cdots + 38\!\cdots\!64 ) / 41\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 22\!\cdots\!01 \nu^{13} + \cdots + 15\!\cdots\!78 ) / 41\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 41\!\cdots\!09 \nu^{13} - 861656847984832 \nu^{12} + \cdots - 13\!\cdots\!66 ) / 41\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 44\!\cdots\!77 \nu^{13} + \cdots - 50\!\cdots\!40 ) / 41\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 24\!\cdots\!12 \nu^{13} - 878050401077534 \nu^{12} + \cdots - 77\!\cdots\!45 ) / 20\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 56\!\cdots\!50 \nu^{13} + \cdots - 21\!\cdots\!26 ) / 41\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 62\!\cdots\!76 \nu^{13} + \cdots + 19\!\cdots\!96 ) / 41\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 79\!\cdots\!91 \nu^{13} + \cdots - 64\!\cdots\!02 ) / 41\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 28\!\cdots\!43 \nu^{13} - 920397958263600 \nu^{12} - 994723808066341 \nu^{11} + \cdots - 37\!\cdots\!18 ) / 13\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 93\!\cdots\!01 \nu^{13} + \cdots + 13\!\cdots\!50 ) / 41\!\cdots\!74 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{13} - \beta_{12} + \beta_{10} + \beta_{9} + \beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{8} + \beta_{6} - \beta_{2} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5 \beta_{13} + 5 \beta_{12} + 2 \beta_{11} + 8 \beta_{10} + 8 \beta_{9} + \beta_{8} + \beta_{7} + \cdots - 2 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} - 2\beta_{10} + 10\beta_{7} + 11\beta_{4} + 11\beta_{3} - 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 18 \beta_{13} + 15 \beta_{12} - 9 \beta_{11} - 36 \beta_{10} - 34 \beta_{9} - 4 \beta_{8} + 4 \beta_{7} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 26 \beta_{13} - 26 \beta_{12} - 3 \beta_{11} - 106 \beta_{10} - \beta_{9} - 90 \beta_{8} + \cdots - 218 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 108 \beta_{13} - 151 \beta_{12} - 119 \beta_{11} - 286 \beta_{10} - 293 \beta_{9} - 36 \beta_{8} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 37 \beta_{13} - 37 \beta_{12} - 261 \beta_{11} + 261 \beta_{10} - 797 \beta_{7} - 15 \beta_{5} + \cdots + 952 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1329 \beta_{13} - 858 \beta_{12} + 643 \beta_{11} + 3006 \beta_{10} + 2549 \beta_{9} + 358 \beta_{8} + \cdots + 319 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2433 \beta_{13} + 2433 \beta_{12} + 648 \beta_{11} + 9200 \beta_{10} + 656 \beta_{9} + \cdots + 16082 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 7123 \beta_{13} + 11862 \beta_{12} + 10202 \beta_{11} + 22683 \beta_{10} + 22314 \beta_{9} + 3648 \beta_{8} + \cdots - 3433 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5881 \beta_{13} + 5881 \beta_{12} + 22114 \beta_{11} - 22114 \beta_{10} + 62372 \beta_{7} + 1306 \beta_{5} + \cdots - 71672 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 106345 \beta_{13} + 60270 \beta_{12} - 46548 \beta_{11} - 249914 \beta_{10} - 196111 \beta_{9} + \cdots - 35867 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1489.1
2.13698 + 2.13698i
0.936042 + 0.936042i
−0.814859 − 0.814859i
−1.19291 − 1.19291i
0.617380 + 0.617380i
1.36982 + 1.36982i
−2.05245 − 2.05245i
2.13698 − 2.13698i
0.936042 − 0.936042i
−0.814859 + 0.814859i
−1.19291 + 1.19291i
0.617380 − 0.617380i
1.36982 − 1.36982i
−2.05245 + 2.05245i
0 − 1.00000i 0 −2.23582 − 0.0330592i 0 − 0.846429i 0 −1.00000 0
1489.2 0 − 1.00000i 0 −2.15115 + 0.610357i 0 1.73400i 0 −1.00000 0
1489.3 0 − 1.00000i 0 −0.542183 − 2.16934i 0 − 3.92312i 0 −1.00000 0
1489.4 0 − 1.00000i 0 −0.110468 + 2.23334i 0 0.0132572i 0 −1.00000 0
1489.5 0 − 1.00000i 0 0.648570 − 2.13994i 0 − 2.05452i 0 −1.00000 0
1489.6 0 − 1.00000i 0 1.24989 + 1.85412i 0 − 3.07303i 0 −1.00000 0
1489.7 0 − 1.00000i 0 2.14116 + 0.644527i 0 4.14985i 0 −1.00000 0
1489.8 0 1.00000i 0 −2.23582 + 0.0330592i 0 0.846429i 0 −1.00000 0
1489.9 0 1.00000i 0 −2.15115 − 0.610357i 0 − 1.73400i 0 −1.00000 0
1489.10 0 1.00000i 0 −0.542183 + 2.16934i 0 3.92312i 0 −1.00000 0
1489.11 0 1.00000i 0 −0.110468 − 2.23334i 0 − 0.0132572i 0 −1.00000 0
1489.12 0 1.00000i 0 0.648570 + 2.13994i 0 2.05452i 0 −1.00000 0
1489.13 0 1.00000i 0 1.24989 − 1.85412i 0 3.07303i 0 −1.00000 0
1489.14 0 1.00000i 0 2.14116 − 0.644527i 0 − 4.14985i 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.b ✓ 14
3.b odd 2 1 5580.2.g.e 14
5.b even 2 1 inner 1860.2.g.b ✓ 14
5.c odd 4 1 9300.2.a.bd 7
5.c odd 4 1 9300.2.a.be 7
15.d odd 2 1 5580.2.g.e 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1860.2.g.b ✓ 14 1.a even 1 1 trivial
1860.2.g.b ✓ 14 5.b even 2 1 inner
5580.2.g.e 14 3.b odd 2 1
5580.2.g.e 14 15.d odd 2 1
9300.2.a.bd 7 5.c odd 4 1
9300.2.a.be 7 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{14} + 50T_{7}^{12} + 925T_{7}^{10} + 7816T_{7}^{8} + 30508T_{7}^{6} + 49944T_{7}^{4} + 22768T_{7}^{2} + 4 \) 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
$7$ \( T^{14} + 50 T^{12} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( (T^{7} - 4 T^{6} - 24 T^{5} + \cdots + 32)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + 88 T^{12} + \cdots + 50176 \) Copy content Toggle raw display
$17$ \( T^{14} + 144 T^{12} + \cdots + 7225344 \) Copy content Toggle raw display
$19$ \( (T^{7} - 6 T^{6} + \cdots + 1008)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + 164 T^{12} + \cdots + 26873856 \) Copy content Toggle raw display
$29$ \( (T^{7} - 2 T^{6} + \cdots + 8632)^{2} \) Copy content Toggle raw display
$31$ \( (T - 1)^{14} \) Copy content Toggle raw display
$37$ \( T^{14} + 244 T^{12} + \cdots + 34012224 \) Copy content Toggle raw display
$41$ \( (T^{7} - 10 T^{6} + \cdots + 62184)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 136060650496 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 287099136 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 11505565696 \) Copy content Toggle raw display
$59$ \( (T^{7} - 2 T^{6} + \cdots + 562)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} - 2 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + 264 T^{12} + \cdots + 14807104 \) Copy content Toggle raw display
$71$ \( (T^{7} - 6 T^{6} + \cdots + 1213926)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 156224601032256 \) Copy content Toggle raw display
$79$ \( (T^{7} - 24 T^{6} + \cdots + 1587088)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 3371424788736 \) Copy content Toggle raw display
$89$ \( (T^{7} - 12 T^{6} + \cdots + 8865336)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 342056540736 \) Copy content Toggle raw display
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