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Properties

Label 560.3.bh.e
Level $560$
Weight $3$
Character orbit 560.bh
Analytic conductor $15.259$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,3,Mod(113,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.113");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 560.bh (of order \(4\), degree \(2\), 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: \(15.2588948042\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 8 x^{10} + 8 x^{9} + 70 x^{8} - 248 x^{7} + 464 x^{6} + 432 x^{5} + 1129 x^{4} + \cdots + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + ( - \beta_{9} + \beta_{3} + \cdots - \beta_1) q^{5}+ \cdots + (\beta_{10} + 2 \beta_{8} + \cdots + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} + ( - \beta_{9} + \beta_{3} + \cdots - \beta_1) q^{5}+ \cdots + (6 \beta_{11} - 14 \beta_{10} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{3} - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{3} - 8 q^{5} + 12 q^{11} - 4 q^{13} + 64 q^{15} - 12 q^{17} + 28 q^{21} + 16 q^{23} + 64 q^{25} - 164 q^{27} + 96 q^{31} + 124 q^{33} - 104 q^{37} - 208 q^{41} - 76 q^{43} + 92 q^{45} + 164 q^{47} - 220 q^{51} - 204 q^{53} - 116 q^{55} - 236 q^{57} + 280 q^{61} - 112 q^{63} - 192 q^{65} - 324 q^{67} - 144 q^{71} - 248 q^{73} - 108 q^{75} - 56 q^{77} - 260 q^{81} + 224 q^{83} - 324 q^{85} - 244 q^{87} - 84 q^{91} + 236 q^{93} - 52 q^{95} + 564 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 8 x^{10} + 8 x^{9} + 70 x^{8} - 248 x^{7} + 464 x^{6} + 432 x^{5} + 1129 x^{4} + \cdots + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 216885938 \nu^{11} + 1060104974 \nu^{10} - 3418831332 \nu^{9} + 4776047991 \nu^{8} + \cdots + 9734414919250 ) / 2042321889134 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 650167512 \nu^{11} - 2167388474 \nu^{10} + 4094056596 \nu^{9} + 5876565360 \nu^{8} + \cdots + 412177775220 ) / 2042321889134 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 20608888761 \nu^{11} + 85686392604 \nu^{10} - 175708052458 \nu^{9} + \cdots - 8569191768820 ) / 10211609445670 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 62789472393 \nu^{11} + 265477896952 \nu^{10} - 559679834849 \nu^{9} + \cdots + 4818810061550 ) / 20423218891340 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 137676730578 \nu^{11} + 596855018097 \nu^{10} - 1287825684554 \nu^{9} + \cdots + 10093227434800 ) / 40846437782680 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 150422124598 \nu^{11} - 616051382677 \nu^{10} + 1215756736104 \nu^{9} + \cdots + 126398845752920 ) / 40846437782680 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 86748937073 \nu^{11} + 338454191902 \nu^{10} - 673792264554 \nu^{9} + \cdots - 70443542020170 ) / 20423218891340 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 360571142916 \nu^{11} + 1533377436449 \nu^{10} - 3341673499113 \nu^{9} + \cdots + 64434080701620 ) / 40846437782680 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 19958721249 \nu^{11} - 83519004130 \nu^{10} + 171613995862 \nu^{9} + 138524261748 \nu^{8} + \cdots + 8157013993600 ) / 2042321889134 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 448601700821 \nu^{11} - 1754253899729 \nu^{10} + 3449654172798 \nu^{9} + \cdots + 409254215744460 ) / 40846437782680 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + 5\beta_{4} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + 2\beta_{8} + 2\beta_{4} + 9\beta_{3} + \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{8} - 2\beta_{7} - 2\beta_{6} - 2\beta_{5} + 13\beta_{3} + 11\beta_{2} - 13\beta _1 - 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 15 \beta_{10} + 4 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - 6 \beta_{6} - 28 \beta_{5} - 38 \beta_{4} + \cdots - 42 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 8 \beta_{11} - 115 \beta_{10} + 8 \beta_{9} - 44 \beta_{8} + 32 \beta_{7} - 32 \beta_{6} + \cdots - 159 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 80 \beta_{11} - 195 \beta_{10} - 342 \beta_{8} + 136 \beta_{7} + 40 \beta_{6} - 40 \beta_{5} + \cdots + 630 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 176 \beta_{11} - 176 \beta_{9} - 526 \beta_{8} + 598 \beta_{7} + 598 \beta_{6} + 526 \beta_{5} + \cdots + 4401 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2451 \beta_{10} - 1196 \beta_{9} + 598 \beta_{8} + 598 \beta_{7} + 2178 \beta_{6} + 4112 \beta_{5} + \cdots + 8450 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2776 \beta_{11} + 14083 \beta_{10} - 2776 \beta_{9} + 9856 \beta_{8} - 5308 \beta_{7} + \cdots + 23323 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 16168 \beta_{11} + 30403 \beta_{10} + 49642 \beta_{8} - 30572 \beta_{7} - 8084 \beta_{6} + 8084 \beta_{5} + \cdots - 108230 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(1\) \(-\beta_{4}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
2.47480 + 2.47480i
−0.109772 − 0.109772i
0.950261 + 0.950261i
−1.85871 − 1.85871i
1.90845 + 1.90845i
−1.36503 − 1.36503i
2.47480 − 2.47480i
−0.109772 + 0.109772i
0.950261 − 0.950261i
−1.85871 + 1.85871i
1.90845 − 1.90845i
−1.36503 + 1.36503i
0 −2.98009 − 2.98009i 0 −4.25027 + 2.63347i 0 −1.87083 + 1.87083i 0 8.76185i 0
113.2 0 −1.67281 − 1.67281i 0 −0.861142 − 4.92529i 0 −1.87083 + 1.87083i 0 − 3.40339i 0
113.3 0 −0.269488 − 0.269488i 0 4.00169 + 2.99774i 0 1.87083 − 1.87083i 0 − 8.85475i 0
113.4 0 −0.163806 − 0.163806i 0 −4.36579 + 2.43719i 0 1.87083 − 1.87083i 0 − 8.94634i 0
113.5 0 3.30412 + 3.30412i 0 −3.50673 − 3.56410i 0 1.87083 − 1.87083i 0 12.8345i 0
113.6 0 3.78207 + 3.78207i 0 4.98225 + 0.420986i 0 −1.87083 + 1.87083i 0 19.6082i 0
337.1 0 −2.98009 + 2.98009i 0 −4.25027 − 2.63347i 0 −1.87083 − 1.87083i 0 − 8.76185i 0
337.2 0 −1.67281 + 1.67281i 0 −0.861142 + 4.92529i 0 −1.87083 − 1.87083i 0 3.40339i 0
337.3 0 −0.269488 + 0.269488i 0 4.00169 − 2.99774i 0 1.87083 + 1.87083i 0 8.85475i 0
337.4 0 −0.163806 + 0.163806i 0 −4.36579 − 2.43719i 0 1.87083 + 1.87083i 0 8.94634i 0
337.5 0 3.30412 − 3.30412i 0 −3.50673 + 3.56410i 0 1.87083 + 1.87083i 0 − 12.8345i 0
337.6 0 3.78207 − 3.78207i 0 4.98225 − 0.420986i 0 −1.87083 − 1.87083i 0 − 19.6082i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.3.bh.e 12
4.b odd 2 1 35.3.g.a ✓ 12
5.c odd 4 1 inner 560.3.bh.e 12
12.b even 2 1 315.3.o.a 12
20.d odd 2 1 175.3.g.b 12
20.e even 4 1 35.3.g.a ✓ 12
20.e even 4 1 175.3.g.b 12
28.d even 2 1 245.3.g.a 12
28.f even 6 2 245.3.m.c 24
28.g odd 6 2 245.3.m.d 24
60.l odd 4 1 315.3.o.a 12
140.j odd 4 1 245.3.g.a 12
140.w even 12 2 245.3.m.d 24
140.x odd 12 2 245.3.m.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.g.a ✓ 12 4.b odd 2 1
35.3.g.a ✓ 12 20.e even 4 1
175.3.g.b 12 20.d odd 2 1
175.3.g.b 12 20.e even 4 1
245.3.g.a 12 28.d even 2 1
245.3.g.a 12 140.j odd 4 1
245.3.m.c 24 28.f even 6 2
245.3.m.c 24 140.x odd 12 2
245.3.m.d 24 28.g odd 6 2
245.3.m.d 24 140.w even 12 2
315.3.o.a 12 12.b even 2 1
315.3.o.a 12 60.l odd 4 1
560.3.bh.e 12 1.a even 1 1 trivial
560.3.bh.e 12 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(560, [\chi])\):

\( T_{3}^{12} - 4 T_{3}^{11} + 8 T_{3}^{10} + 68 T_{3}^{9} + 517 T_{3}^{8} - 1192 T_{3}^{7} + 2944 T_{3}^{6} + \cdots + 484 \) Copy content Toggle raw display
\( T_{11}^{6} - 6T_{11}^{5} - 407T_{11}^{4} + 2464T_{11}^{3} + 32952T_{11}^{2} - 217312T_{11} + 255536 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 4 T^{11} + \cdots + 484 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 49)^{3} \) Copy content Toggle raw display
$11$ \( (T^{6} - 6 T^{5} + \cdots + 255536)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 7014062500 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 70807081216 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 2061234490000 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 133532630919424 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 196000000 \) Copy content Toggle raw display
$31$ \( (T^{6} - 48 T^{5} + \cdots + 116022400)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 37\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{6} + 104 T^{5} + \cdots + 38452576)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 159385575040000 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 91\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} - 140 T^{5} + \cdots + 35862860384)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 72\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{6} + 72 T^{5} + \cdots + 15911820320)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 36\!\cdots\!96 \) Copy content Toggle raw display
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