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Properties

Label 1760.2.b.c
Level $1760$
Weight $2$
Character orbit 1760.b
Analytic conductor $14.054$
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] = [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 = [14,0,0,0,0,0,0,0,-6,0,-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: \( x^{14} + 24x^{12} + 222x^{10} + 992x^{8} + 2193x^{6} + 2184x^{4} + 784x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{11} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{5} q^{5} + \beta_{7} q^{7} + \beta_{2} q^{9} - q^{11} + (\beta_{12} + \beta_{11} + \cdots - \beta_1) q^{13} + ( - \beta_{12} - \beta_{11} + \cdots + \beta_1) q^{15} + ( - \beta_{12} - \beta_{7} + \cdots + \beta_1) q^{17}+ \cdots - \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 6 q^{9} - 14 q^{11} - 6 q^{15} - 20 q^{19} - 8 q^{25} - 12 q^{29} - 28 q^{31} + 8 q^{35} + 16 q^{39} + 12 q^{41} + 6 q^{45} + 6 q^{49} + 4 q^{51} - 28 q^{59} - 12 q^{61} - 28 q^{69} - 28 q^{71} + 22 q^{75}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 24x^{12} + 222x^{10} + 992x^{8} + 2193x^{6} + 2184x^{4} + 784x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 11\nu^{4} + 30\nu^{2} + 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} + 14\nu^{6} + 2\nu^{5} + 61\nu^{4} + 18\nu^{3} + 88\nu^{2} + 32\nu + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{8} - 14\nu^{6} + 2\nu^{5} - 61\nu^{4} + 18\nu^{3} - 88\nu^{2} + 32\nu - 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{11} - 19\nu^{9} - 127\nu^{7} - 349\nu^{5} - 336\nu^{3} - 16\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} + 19\nu^{9} + 127\nu^{7} + 349\nu^{5} + 352\nu^{3} + 112\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{10} + 19\nu^{8} + 2\nu^{7} + 127\nu^{6} + 24\nu^{5} + 349\nu^{4} + 78\nu^{3} + 344\nu^{2} + 56\nu + 72 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{12} + 21 \nu^{10} + 2 \nu^{9} + 163 \nu^{8} + 32 \nu^{7} + 571 \nu^{6} + 174 \nu^{5} + 876 \nu^{4} + \cdots + 48 ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{13} + 23 \nu^{11} + 201 \nu^{9} + 2 \nu^{8} + 829 \nu^{7} + 24 \nu^{6} + 1622 \nu^{5} + 86 \nu^{4} + \cdots + 48 ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - \nu^{13} - 23 \nu^{11} - 201 \nu^{9} + 2 \nu^{8} - 829 \nu^{7} + 24 \nu^{6} - 1622 \nu^{5} + \cdots + 48 ) / 16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{13} + 24\nu^{11} + 222\nu^{9} + 984\nu^{7} + 2089\nu^{5} + 1784\nu^{3} + 336\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{13} + 24\nu^{11} + 222\nu^{9} + 992\nu^{7} + 2185\nu^{5} + 2096\nu^{3} + 560\nu ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{5} - \beta_{4} + \beta_{3} - 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{7} - 9\beta_{6} + 2\beta_{5} + 2\beta_{4} + 38\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -11\beta_{11} - 11\beta_{10} - 11\beta_{5} + 11\beta_{4} - 9\beta_{3} + 47\beta_{2} - 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{13} - 2\beta_{12} + 69\beta_{7} + 69\beta_{6} - 24\beta_{5} - 24\beta_{4} - 250\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 93\beta_{11} + 93\beta_{10} + 89\beta_{5} - 89\beta_{4} + 65\beta_{3} - 319\beta_{2} + 543 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 28 \beta_{13} + 36 \beta_{12} + 4 \beta_{11} - 4 \beta_{10} - 505 \beta_{7} - 497 \beta_{6} + \cdots + 1678 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4 \beta_{13} + 4 \beta_{12} - 719 \beta_{11} - 719 \beta_{10} + 8 \beta_{8} - 643 \beta_{5} + \cdots - 3543 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 278 \beta_{13} - 430 \beta_{12} - 76 \beta_{11} + 76 \beta_{10} + 3637 \beta_{7} + 3469 \beta_{6} + \cdots - 11394 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 76 \beta_{13} - 92 \beta_{12} + 5337 \beta_{11} + 5353 \beta_{10} + 16 \beta_{9} - 168 \beta_{8} + \cdots + 23775 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2424 \beta_{13} + 4312 \beta_{12} + 936 \beta_{11} - 936 \beta_{10} - 26057 \beta_{7} + \cdots + 77926 \beta_1 \) 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
2.67471i
2.61008i
2.10504i
1.87427i
1.21318i
0.709735i
0.337321i
− 0.337321i
− 0.709735i
− 1.21318i
− 1.87427i
− 2.10504i
− 2.61008i
− 2.67471i
0 − 2.67471i 0 1.22842 − 1.86842i 0 − 2.55119i 0 −4.15409 0
1409.2 0 − 2.61008i 0 −2.11824 − 0.716272i 0 1.36136i 0 −3.81251 0
1409.3 0 − 2.10504i 0 −0.0936170 + 2.23411i 0 − 3.69146i 0 −1.43117 0
1409.4 0 − 1.87427i 0 1.62610 + 1.53486i 0 4.12492i 0 −0.512901 0
1409.5 0 − 1.21318i 0 −1.66535 − 1.49219i 0 2.06823i 0 1.52818 0
1409.6 0 − 0.709735i 0 −0.821838 − 2.07956i 0 0.363836i 0 2.49628 0
1409.7 0 − 0.337321i 0 1.84452 − 1.26402i 0 1.60822i 0 2.88621 0
1409.8 0 0.337321i 0 1.84452 + 1.26402i 0 − 1.60822i 0 2.88621 0
1409.9 0 0.709735i 0 −0.821838 + 2.07956i 0 − 0.363836i 0 2.49628 0
1409.10 0 1.21318i 0 −1.66535 + 1.49219i 0 − 2.06823i 0 1.52818 0
1409.11 0 1.87427i 0 1.62610 − 1.53486i 0 − 4.12492i 0 −0.512901 0
1409.12 0 2.10504i 0 −0.0936170 − 2.23411i 0 3.69146i 0 −1.43117 0
1409.13 0 2.61008i 0 −2.11824 + 0.716272i 0 − 1.36136i 0 −3.81251 0
1409.14 0 2.67471i 0 1.22842 + 1.86842i 0 2.55119i 0 −4.15409 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.c ✓ 14
4.b odd 2 1 1760.2.b.d yes 14
5.b even 2 1 inner 1760.2.b.c ✓ 14
5.c odd 4 1 8800.2.a.ce 7
5.c odd 4 1 8800.2.a.cg 7
20.d odd 2 1 1760.2.b.d yes 14
20.e even 4 1 8800.2.a.cf 7
20.e even 4 1 8800.2.a.ch 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1760.2.b.c ✓ 14 1.a even 1 1 trivial
1760.2.b.c ✓ 14 5.b even 2 1 inner
1760.2.b.d yes 14 4.b odd 2 1
1760.2.b.d yes 14 20.d odd 2 1
8800.2.a.ce 7 5.c odd 4 1
8800.2.a.cf 7 20.e even 4 1
8800.2.a.cg 7 5.c odd 4 1
8800.2.a.ch 7 20.e even 4 1

Hecke kernels

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

\( T_{3}^{14} + 24T_{3}^{12} + 222T_{3}^{10} + 992T_{3}^{8} + 2193T_{3}^{6} + 2184T_{3}^{4} + 784T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{19}^{7} + 10T_{19}^{6} - 15T_{19}^{5} - 404T_{19}^{4} - 1052T_{19}^{3} + 304T_{19}^{2} + 3264T_{19} + 2304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + 24 T^{12} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{14} + 4 T^{12} + \cdots + 78125 \) Copy content Toggle raw display
$7$ \( T^{14} + 46 T^{12} + \cdots + 4096 \) Copy content Toggle raw display
$11$ \( (T + 1)^{14} \) Copy content Toggle raw display
$13$ \( T^{14} + 124 T^{12} + \cdots + 21233664 \) Copy content Toggle raw display
$17$ \( T^{14} + 142 T^{12} + \cdots + 331776 \) Copy content Toggle raw display
$19$ \( (T^{7} + 10 T^{6} + \cdots + 2304)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + 166 T^{12} + \cdots + 16384 \) Copy content Toggle raw display
$29$ \( (T^{7} + 6 T^{6} + \cdots - 19584)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} + 14 T^{6} + \cdots + 27072)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 613453824 \) Copy content Toggle raw display
$41$ \( (T^{7} - 6 T^{6} + \cdots + 8192)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 490356736 \) Copy content Toggle raw display
$47$ \( T^{14} + 96 T^{12} + \cdots + 4194304 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 326488817664 \) Copy content Toggle raw display
$59$ \( (T^{7} + 14 T^{6} + \cdots + 156672)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} + 6 T^{6} + \cdots - 19584)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 1044385890304 \) Copy content Toggle raw display
$71$ \( (T^{7} + 14 T^{6} + \cdots - 1562688)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + 404 T^{12} + \cdots + 1327104 \) Copy content Toggle raw display
$79$ \( (T^{7} - 20 T^{6} + \cdots + 36864)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 2599472594944 \) Copy content Toggle raw display
$89$ \( (T^{7} - 10 T^{6} + \cdots + 4536072)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 11439763881984 \) Copy content Toggle raw display
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