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Properties

Label 1760.2.b.f
Level $1760$
Weight $2$
Character orbit 1760.b
Analytic conductor $14.054$
Analytic rank $0$
Dimension $14$
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] = [1760,2,Mod(1409,1760)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1760, 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("1760.1409"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1760 = 2^{5} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1760.b (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,-12,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.0536707557\)
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: \( x^{14} + 16x^{12} + 96x^{10} + 272x^{8} + 372x^{6} + 225x^{4} + 56x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{13} \)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} - \beta_{6} q^{5} + \beta_{11} q^{7} + ( - \beta_{8} - 1) q^{9} + q^{11} + \beta_{13} q^{13} + (\beta_{8} + \beta_{3} + 1) q^{15} + ( - \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{17}+ \cdots + ( - \beta_{8} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 2 q^{5} - 12 q^{9} + 14 q^{11} + 12 q^{15} - 14 q^{19} + 26 q^{21} - 4 q^{25} + 22 q^{29} - 22 q^{31} - 2 q^{35} - 8 q^{39} + 8 q^{41} - 8 q^{45} - 32 q^{49} - 14 q^{51} + 2 q^{55} - 52 q^{59} - 10 q^{61}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 16x^{12} + 96x^{10} + 272x^{8} + 372x^{6} + 225x^{4} + 56x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{13} - 46\nu^{11} - 256\nu^{9} - 628\nu^{7} - 620\nu^{5} - 115\nu^{3} + 26\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3 \nu^{13} - 12 \nu^{12} + 46 \nu^{11} - 188 \nu^{10} + 260 \nu^{9} - 1088 \nu^{8} + 680 \nu^{7} + \cdots - 116 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3 \nu^{13} - 12 \nu^{12} - 46 \nu^{11} - 188 \nu^{10} - 260 \nu^{9} - 1088 \nu^{8} - 680 \nu^{7} + \cdots - 116 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17\nu^{13} + 262\nu^{11} + 1480\nu^{9} + 3780\nu^{7} + 4208\nu^{5} + 1501\nu^{3} + 106\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17 \nu^{13} - 8 \nu^{12} + 262 \nu^{11} - 124 \nu^{10} + 1480 \nu^{9} - 704 \nu^{8} + 3780 \nu^{7} + \cdots - 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 35 \nu^{13} + 24 \nu^{12} - 542 \nu^{11} + 372 \nu^{10} - 3084 \nu^{9} + 2120 \nu^{8} - 7968 \nu^{7} + \cdots + 228 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 35 \nu^{13} - 24 \nu^{12} - 542 \nu^{11} - 372 \nu^{10} - 3084 \nu^{9} - 2120 \nu^{8} - 7968 \nu^{7} + \cdots - 228 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 11\nu^{12} + 170\nu^{10} + 965\nu^{8} + 2487\nu^{6} + 2821\nu^{4} + 1062\nu^{2} + 87 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{13} + 88 \nu^{12} + 18 \nu^{11} + 1372 \nu^{10} + 124 \nu^{9} + 7872 \nu^{8} + 408 \nu^{7} + \cdots + 740 ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{13} - 88 \nu^{12} + 18 \nu^{11} - 1372 \nu^{10} + 124 \nu^{9} - 7872 \nu^{8} + 408 \nu^{7} + \cdots - 740 ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( 10\nu^{13} + 155\nu^{11} + 883\nu^{9} + 2285\nu^{7} + 2605\nu^{5} + 989\nu^{3} + 80\nu \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( 13\nu^{12} + 202\nu^{10} + 1154\nu^{8} + 2994\nu^{6} + 3418\nu^{4} + 1299\nu^{2} + 112 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 55\nu^{13} + 854\nu^{11} + 4876\nu^{9} + 12652\nu^{7} + 14472\nu^{5} + 5539\nu^{3} + 482\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} + 2\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{12} + \beta_{10} + \beta_{9} + 2\beta_{7} + 2\beta_{5} + \beta_{3} + \beta_{2} - 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{13} + \beta_{11} - 3 \beta_{10} - 3 \beta_{9} - 7 \beta_{7} - 7 \beta_{6} - 11 \beta_{4} + \cdots - 2 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{12} - 4 \beta_{10} - 7 \beta_{9} - \beta_{8} - 10 \beta_{7} - \beta_{6} - 11 \beta_{5} + \cdots + 41 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9 \beta_{13} - 13 \beta_{11} + 12 \beta_{10} + 12 \beta_{9} + 42 \beta_{7} + 42 \beta_{6} + \cdots + 16 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 3 \beta_{12} + 16 \beta_{10} + 44 \beta_{9} + 8 \beta_{8} + 47 \beta_{7} + 13 \beta_{6} + 60 \beta_{5} + \cdots - 196 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 32 \beta_{13} + 53 \beta_{11} - 27 \beta_{10} - 27 \beta_{9} - 124 \beta_{7} - 124 \beta_{6} + \cdots - 55 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 68 \beta_{12} - 67 \beta_{10} - 268 \beta_{9} - 51 \beta_{8} - 229 \beta_{7} - 106 \beta_{6} + \cdots + 1021 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 209 \beta_{13} - 370 \beta_{11} + 132 \beta_{10} + 132 \beta_{9} + 733 \beta_{7} + 733 \beta_{6} + \cdots + 358 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 144 \beta_{12} + 74 \beta_{10} + 404 \beta_{9} + 76 \beta_{8} + 293 \beta_{7} + 185 \beta_{6} + \cdots - 1405 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2624 \beta_{13} + 4820 \beta_{11} - 1375 \beta_{10} - 1375 \beta_{9} - 8697 \beta_{7} + \cdots - 4520 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 4031 \beta_{12} - 1385 \beta_{10} - 9713 \beta_{9} - 1776 \beta_{8} - 6278 \beta_{7} - 4820 \beta_{6} + \cdots + 32018 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 16141 \beta_{13} - 30301 \beta_{11} + 7503 \beta_{10} + 7503 \beta_{9} + 51767 \beta_{7} + \cdots + 28002 \beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1760\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(991\) \(1057\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1409.1
1.71085i
0.700339i
− 1.51372i
2.44914i
− 0.340436i
0.687115i
1.92480i
− 1.92480i
− 0.687115i
0.340436i
− 2.44914i
1.51372i
− 0.700339i
− 1.71085i
0 − 3.19254i 0 −0.395001 + 2.20090i 0 4.81353i 0 −7.19231 0
1409.2 0 − 2.54571i 0 0.743143 + 2.10897i 0 − 0.823764i 0 −3.48064 0
1409.3 0 − 2.13376i 0 2.12532 − 0.694983i 0 1.20437i 0 −1.55295 0
1409.4 0 − 1.64481i 0 −0.124172 − 2.23262i 0 − 1.42015i 0 0.294593 0
1409.5 0 − 1.45210i 0 −2.14731 − 0.623754i 0 1.03231i 0 0.891413 0
1409.6 0 − 0.874674i 0 2.17805 + 0.506082i 0 − 4.39214i 0 2.23495 0
1409.7 0 − 0.441641i 0 −1.38003 − 1.75941i 0 4.16263i 0 2.80495 0
1409.8 0 0.441641i 0 −1.38003 + 1.75941i 0 − 4.16263i 0 2.80495 0
1409.9 0 0.874674i 0 2.17805 − 0.506082i 0 4.39214i 0 2.23495 0
1409.10 0 1.45210i 0 −2.14731 + 0.623754i 0 − 1.03231i 0 0.891413 0
1409.11 0 1.64481i 0 −0.124172 + 2.23262i 0 1.42015i 0 0.294593 0
1409.12 0 2.13376i 0 2.12532 + 0.694983i 0 − 1.20437i 0 −1.55295 0
1409.13 0 2.54571i 0 0.743143 − 2.10897i 0 0.823764i 0 −3.48064 0
1409.14 0 3.19254i 0 −0.395001 − 2.20090i 0 − 4.81353i 0 −7.19231 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1409.14
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 1760.2.b.f yes 14
4.b odd 2 1 1760.2.b.e ✓ 14
5.b even 2 1 inner 1760.2.b.f yes 14
5.c odd 4 1 8800.2.a.cd 7
5.c odd 4 1 8800.2.a.cj 7
20.d odd 2 1 1760.2.b.e ✓ 14
20.e even 4 1 8800.2.a.cc 7
20.e even 4 1 8800.2.a.ci 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1760.2.b.e ✓ 14 4.b odd 2 1
1760.2.b.e ✓ 14 20.d odd 2 1
1760.2.b.f yes 14 1.a even 1 1 trivial
1760.2.b.f yes 14 5.b even 2 1 inner
8800.2.a.cc 7 20.e even 4 1
8800.2.a.cd 7 5.c odd 4 1
8800.2.a.ci 7 20.e even 4 1
8800.2.a.cj 7 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1760, [\chi])\):

\( T_{3}^{14} + 27T_{3}^{12} + 275T_{3}^{10} + 1349T_{3}^{8} + 3356T_{3}^{6} + 4048T_{3}^{4} + 1984T_{3}^{2} + 256 \) Copy content Toggle raw display
\( T_{19}^{7} + 7T_{19}^{6} - 52T_{19}^{5} - 340T_{19}^{4} + 608T_{19}^{3} + 3584T_{19}^{2} + 3584T_{19} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + 27 T^{12} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{14} - 2 T^{13} + \cdots + 78125 \) Copy content Toggle raw display
$7$ \( T^{14} + 65 T^{12} + \cdots + 16384 \) Copy content Toggle raw display
$11$ \( (T - 1)^{14} \) Copy content Toggle raw display
$13$ \( T^{14} + 100 T^{12} + \cdots + 1048576 \) Copy content Toggle raw display
$17$ \( T^{14} + 77 T^{12} + \cdots + 16384 \) Copy content Toggle raw display
$19$ \( (T^{7} + 7 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 116985856 \) Copy content Toggle raw display
$29$ \( (T^{7} - 11 T^{6} + \cdots - 256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} + 11 T^{6} + \cdots + 7424)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + 215 T^{12} + \cdots + 65536 \) Copy content Toggle raw display
$41$ \( (T^{7} - 4 T^{6} + \cdots - 652544)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 435306496 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 1593492176896 \) Copy content Toggle raw display
$53$ \( T^{14} + 433 T^{12} + \cdots + 67108864 \) Copy content Toggle raw display
$59$ \( (T^{7} + 26 T^{6} + \cdots - 708608)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} + 5 T^{6} + \cdots + 20224)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + 250 T^{12} + \cdots + 28217344 \) Copy content Toggle raw display
$71$ \( (T^{7} - 3 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 35063072702464 \) Copy content Toggle raw display
$79$ \( (T^{7} - 32 T^{6} + \cdots - 483328)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 220463104 \) Copy content Toggle raw display
$89$ \( (T^{7} + 11 T^{6} + \cdots + 859696)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 710230933504 \) Copy content Toggle raw display
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