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Properties

Label 560.5.p.h
Level $560$
Weight $5$
Character orbit 560.p
Analytic conductor $57.887$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,5,Mod(209,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.209");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 560.p (of order \(2\), degree \(1\), not 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: \(57.8871793270\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 1342 x^{10} + 1715866 x^{8} + 1068594118 x^{6} + 652472238169 x^{4} + 172600636071304 x^{2} + 37\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 140)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + (\beta_{7} + \beta_1) q^{5} + (\beta_{10} + \beta_{8} + \beta_{5} + \beta_1) q^{7} + ( - \beta_{3} + 25) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} + (\beta_{7} + \beta_1) q^{5} + (\beta_{10} + \beta_{8} + \beta_{5} + \beta_1) q^{7} + ( - \beta_{3} + 25) q^{9} + (\beta_{3} - \beta_{2} + 1) q^{11} + ( - 5 \beta_{10} - \beta_{7}) q^{13} + ( - \beta_{10} - \beta_{9} - \beta_{8} + \cdots + 55) q^{15}+ \cdots + (82 \beta_{3} - 81 \beta_{2} - 5695) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 296 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 296 q^{9} + 20 q^{11} + 672 q^{15} - 1352 q^{21} - 2608 q^{25} - 1796 q^{29} + 2788 q^{35} + 332 q^{39} - 11792 q^{49} - 16844 q^{51} + 6928 q^{65} - 27448 q^{71} + 7948 q^{79} - 14636 q^{81} + 1748 q^{85} - 948 q^{91} + 29940 q^{95} - 67688 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 1342 x^{10} + 1715866 x^{8} + 1068594118 x^{6} + 652472238169 x^{4} + 172600636071304 x^{2} + 37\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3048135891721 \nu^{10} + 74116474532460 \nu^{8} + \cdots - 80\!\cdots\!75 ) / 54\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10911111877826 \nu^{10} + \cdots + 27\!\cdots\!25 ) / 16\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 28\!\cdots\!56 \nu^{10} + \cdots + 24\!\cdots\!00 ) / 24\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 93\!\cdots\!61 \nu^{11} + \cdots - 27\!\cdots\!44 \nu ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11\!\cdots\!87 \nu^{11} + \cdots + 43\!\cdots\!52 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 35\!\cdots\!01 \nu^{11} + \cdots - 58\!\cdots\!04 \nu ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 23\!\cdots\!37 \nu^{11} + \cdots + 11\!\cdots\!00 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 16\!\cdots\!59 \nu^{11} + \cdots - 28\!\cdots\!60 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 23\!\cdots\!37 \nu^{11} + \cdots + 64\!\cdots\!92 \nu ) / 90\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11\!\cdots\!21 \nu^{11} + \cdots + 30\!\cdots\!00 ) / 42\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{10} + \beta_{9} + 5\beta_{8} + \beta_{4} + 2\beta_{3} - 224 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} + 150\beta_{10} + 602\beta_{7} - 13\beta_{6} - 493\beta_{5} + \beta_{2} - 352\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3252\beta_{10} - 1024\beta_{9} - 5480\beta_{8} - 184\beta_{4} - 779\beta_{3} - 2873\beta_{2} - 272834 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7916 \beta_{11} - 327084 \beta_{10} - 634242 \beta_{7} + 4811 \beta_{6} + 105628 \beta_{5} + 3958 \beta_{2} + \cdots + 3958 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1272857 \beta_{10} + 88151 \beta_{9} + 2457563 \beta_{8} - 391797 \beta_{4} - 2461450 \beta_{3} + \cdots + 215703711 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 9781688 \beta_{11} + 85381212 \beta_{10} - 66973774 \beta_{7} + 10820887 \beta_{6} + 90028161 \beta_{5} + \cdots - 4890844 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1421716856 \beta_{10} + 612113376 \beta_{9} + 2231320336 \beta_{8} + 119759920 \beta_{4} + \cdots - 17696205279 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2528807422 \beta_{11} + 228475182354 \beta_{10} + 594370136430 \beta_{7} - 7715128733 \beta_{6} + \cdots + 1264403711 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2086055648921 \beta_{10} - 606408916599 \beta_{9} - 3565702381243 \beta_{8} + 166779629909 \beta_{4} + \cdots - 52622089557724 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4722309278058 \beta_{11} - 194130480871098 \beta_{10} - 420693496698958 \beta_{7} - 4318235165457 \beta_{6} + \cdots + 2361154639029 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
13.5031 − 24.9836i
13.5031 + 24.9836i
−9.11517 − 15.9519i
−9.11517 + 15.9519i
18.0854 − 19.6177i
18.0854 + 19.6177i
−18.0854 − 19.6177i
−18.0854 + 19.6177i
9.11517 − 15.9519i
9.11517 + 15.9519i
−13.5031 − 24.9836i
−13.5031 + 24.9836i
0 −14.4082 0 −0.905133 − 24.9836i 0 17.8756 − 45.6231i 0 126.597 0
209.2 0 −14.4082 0 −0.905133 + 24.9836i 0 17.8756 + 45.6231i 0 126.597 0
209.3 0 −10.1342 0 −19.2493 − 15.9519i 0 −2.89857 + 48.9142i 0 21.7012 0
209.4 0 −10.1342 0 −19.2493 + 15.9519i 0 −2.89857 − 48.9142i 0 21.7012 0
209.5 0 −2.58877 0 15.4966 − 19.6177i 0 42.4212 + 24.5242i 0 −74.2983 0
209.6 0 −2.58877 0 15.4966 + 19.6177i 0 42.4212 − 24.5242i 0 −74.2983 0
209.7 0 2.58877 0 −15.4966 − 19.6177i 0 −42.4212 − 24.5242i 0 −74.2983 0
209.8 0 2.58877 0 −15.4966 + 19.6177i 0 −42.4212 + 24.5242i 0 −74.2983 0
209.9 0 10.1342 0 19.2493 − 15.9519i 0 2.89857 − 48.9142i 0 21.7012 0
209.10 0 10.1342 0 19.2493 + 15.9519i 0 2.89857 + 48.9142i 0 21.7012 0
209.11 0 14.4082 0 0.905133 − 24.9836i 0 −17.8756 + 45.6231i 0 126.597 0
209.12 0 14.4082 0 0.905133 + 24.9836i 0 −17.8756 − 45.6231i 0 126.597 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.5.p.h 12
4.b odd 2 1 140.5.h.c ✓ 12
5.b even 2 1 inner 560.5.p.h 12
7.b odd 2 1 inner 560.5.p.h 12
12.b even 2 1 1260.5.p.c 12
20.d odd 2 1 140.5.h.c ✓ 12
20.e even 4 2 700.5.d.e 12
28.d even 2 1 140.5.h.c ✓ 12
35.c odd 2 1 inner 560.5.p.h 12
60.h even 2 1 1260.5.p.c 12
84.h odd 2 1 1260.5.p.c 12
140.c even 2 1 140.5.h.c ✓ 12
140.j odd 4 2 700.5.d.e 12
420.o odd 2 1 1260.5.p.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.5.h.c ✓ 12 4.b odd 2 1
140.5.h.c ✓ 12 20.d odd 2 1
140.5.h.c ✓ 12 28.d even 2 1
140.5.h.c ✓ 12 140.c even 2 1
560.5.p.h 12 1.a even 1 1 trivial
560.5.p.h 12 5.b even 2 1 inner
560.5.p.h 12 7.b odd 2 1 inner
560.5.p.h 12 35.c odd 2 1 inner
700.5.d.e 12 20.e even 4 2
700.5.d.e 12 140.j odd 4 2
1260.5.p.c 12 12.b even 2 1
1260.5.p.c 12 60.h even 2 1
1260.5.p.c 12 84.h odd 2 1
1260.5.p.c 12 420.o odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 317T_{3}^{4} + 23400T_{3}^{2} - 142884 \) acting on \(S_{5}^{\mathrm{new}}(560, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 317 T^{4} + \cdots - 142884)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{3} - 5 T^{2} + \cdots + 458696)^{4} \) Copy content Toggle raw display
$13$ \( (T^{6} - 35267 T^{4} + \cdots - 822518397184)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 63548957497600)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 94\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 660841654982400)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 449 T^{2} + \cdots - 1406019524)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 494155 T^{4} + \cdots - 109145658384)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + 6862 T^{2} + \cdots - 52002684160)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 29\!\cdots\!76)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 1987 T^{2} + \cdots - 215103740544)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 21\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 13\!\cdots\!04)^{2} \) Copy content Toggle raw display
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