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Properties

Label 860.1.bq.a
Level $860$
Weight $1$
Character orbit 860.bq
Analytic conductor $0.429$
Analytic rank $0$
Dimension $12$
Projective image $D_{21}$
CM discriminant -20
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [860,1,Mod(99,860)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(860, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 21, 32]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("860.99");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 860 = 2^{2} \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 860.bq (of order \(42\), degree \(12\), 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: \(0.429195910864\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{21})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{21}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{21} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{42}^{12} q^{2} + (\zeta_{42}^{18} + \zeta_{42}^{14}) q^{3} - \zeta_{42}^{3} q^{4} - \zeta_{42} q^{5} + ( - \zeta_{42}^{9} - \zeta_{42}^{5}) q^{6} + ( - \zeta_{42}^{5} + \zeta_{42}^{2}) q^{7} - \zeta_{42}^{15} q^{8} + ( - \zeta_{42}^{15} + \cdots - \zeta_{42}^{7}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{42}^{12} q^{2} + (\zeta_{42}^{18} + \zeta_{42}^{14}) q^{3} - \zeta_{42}^{3} q^{4} - \zeta_{42} q^{5} + ( - \zeta_{42}^{9} - \zeta_{42}^{5}) q^{6} + ( - \zeta_{42}^{5} + \zeta_{42}^{2}) q^{7} - \zeta_{42}^{15} q^{8} + ( - \zeta_{42}^{15} + \cdots - \zeta_{42}^{7}) q^{9} + \cdots + ( - \zeta_{42}^{19} + \cdots - \zeta_{42}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} - 8 q^{3} - 2 q^{4} + q^{5} - q^{6} + 2 q^{7} - 2 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} - 8 q^{3} - 2 q^{4} + q^{5} - q^{6} + 2 q^{7} - 2 q^{8} - 7 q^{9} + q^{10} + 13 q^{12} - 5 q^{14} - q^{15} - 2 q^{16} + q^{20} + 4 q^{21} - 5 q^{23} - q^{24} + q^{25} + 12 q^{27} + 2 q^{28} - q^{29} - q^{30} - 2 q^{32} - 4 q^{35} - 7 q^{36} + q^{40} + 2 q^{41} - 10 q^{42} + q^{43} + 2 q^{46} + 2 q^{47} - q^{48} - 4 q^{49} - 6 q^{50} - 2 q^{54} + 2 q^{56} - q^{58} - q^{60} + 2 q^{61} - 2 q^{64} - q^{67} + 12 q^{69} - 4 q^{70} + 2 q^{75} - 6 q^{80} - 8 q^{81} + 2 q^{82} + 13 q^{83} + 4 q^{84} + q^{86} - 2 q^{87} - q^{89} - 7 q^{90} + 2 q^{92} + 2 q^{94} - q^{96} + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/860\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(431\) \(517\)
\(\chi(n)\) \(-\zeta_{42}^{11}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
−0.988831 − 0.149042i
−0.988831 + 0.149042i
0.826239 − 0.563320i
0.365341 + 0.930874i
0.0747301 + 0.997204i
0.826239 + 0.563320i
0.0747301 − 0.997204i
0.955573 + 0.294755i
0.365341 − 0.930874i
0.955573 − 0.294755i
−0.733052 − 0.680173i
−0.733052 + 0.680173i
−0.222521 + 0.974928i −1.40097 + 1.29991i −0.900969 − 0.433884i −0.988831 − 0.149042i −0.955573 − 1.65510i 0.222521 − 0.385418i 0.623490 − 0.781831i 0.198220 − 2.64506i 0.365341 − 0.930874i
139.1 −0.222521 − 0.974928i −1.40097 − 1.29991i −0.900969 + 0.433884i −0.988831 + 0.149042i −0.955573 + 1.65510i 0.222521 + 0.385418i 0.623490 + 0.781831i 0.198220 + 2.64506i 0.365341 + 0.930874i
239.1 0.623490 − 0.781831i −0.722521 + 0.108903i −0.222521 − 0.974928i 0.826239 − 0.563320i −0.365341 + 0.632789i −0.623490 − 1.07992i −0.900969 − 0.433884i −0.445396 + 0.137386i 0.0747301 − 0.997204i
339.1 −0.222521 + 0.974928i −1.40097 − 0.432142i −0.900969 − 0.433884i 0.365341 + 0.930874i 0.733052 − 1.26968i 0.222521 + 0.385418i 0.623490 − 0.781831i 0.949729 + 0.647514i −0.988831 + 0.149042i
359.1 0.623490 − 0.781831i −0.722521 + 1.84095i −0.222521 − 0.974928i 0.0747301 + 0.997204i 0.988831 + 1.71271i −0.623490 + 1.07992i −0.900969 − 0.433884i −2.13402 − 1.98008i 0.826239 + 0.563320i
439.1 0.623490 + 0.781831i −0.722521 − 0.108903i −0.222521 + 0.974928i 0.826239 + 0.563320i −0.365341 − 0.632789i −0.623490 + 1.07992i −0.900969 + 0.433884i −0.445396 − 0.137386i 0.0747301 + 0.997204i
539.1 0.623490 + 0.781831i −0.722521 − 1.84095i −0.222521 + 0.974928i 0.0747301 − 0.997204i 0.988831 − 1.71271i −0.623490 − 1.07992i −0.900969 + 0.433884i −2.13402 + 1.98008i 0.826239 − 0.563320i
599.1 −0.900969 − 0.433884i 0.123490 − 1.64786i 0.623490 + 0.781831i 0.955573 + 0.294755i −0.826239 + 1.43109i 0.900969 + 1.56052i −0.222521 − 0.974928i −1.71135 − 0.257945i −0.733052 − 0.680173i
619.1 −0.222521 − 0.974928i −1.40097 + 0.432142i −0.900969 + 0.433884i 0.365341 − 0.930874i 0.733052 + 1.26968i 0.222521 − 0.385418i 0.623490 + 0.781831i 0.949729 − 0.647514i −0.988831 − 0.149042i
659.1 −0.900969 + 0.433884i 0.123490 + 1.64786i 0.623490 − 0.781831i 0.955573 − 0.294755i −0.826239 − 1.43109i 0.900969 − 1.56052i −0.222521 + 0.974928i −1.71135 + 0.257945i −0.733052 + 0.680173i
719.1 −0.900969 + 0.433884i 0.123490 − 0.0841939i 0.623490 − 0.781831i −0.733052 − 0.680173i −0.0747301 + 0.129436i 0.900969 + 1.56052i −0.222521 + 0.974928i −0.357180 + 0.910080i 0.955573 + 0.294755i
799.1 −0.900969 − 0.433884i 0.123490 + 0.0841939i 0.623490 + 0.781831i −0.733052 + 0.680173i −0.0747301 − 0.129436i 0.900969 − 1.56052i −0.222521 − 0.974928i −0.357180 − 0.910080i 0.955573 − 0.294755i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
43.g even 21 1 inner
860.bq odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 860.1.bq.a ✓ 12
4.b odd 2 1 860.1.bq.b yes 12
5.b even 2 1 860.1.bq.b yes 12
20.d odd 2 1 CM 860.1.bq.a ✓ 12
43.g even 21 1 inner 860.1.bq.a ✓ 12
172.o odd 42 1 860.1.bq.b yes 12
215.u even 42 1 860.1.bq.b yes 12
860.bq odd 42 1 inner 860.1.bq.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
860.1.bq.a ✓ 12 1.a even 1 1 trivial
860.1.bq.a ✓ 12 20.d odd 2 1 CM
860.1.bq.a ✓ 12 43.g even 21 1 inner
860.1.bq.a ✓ 12 860.bq odd 42 1 inner
860.1.bq.b yes 12 4.b odd 2 1
860.1.bq.b yes 12 5.b even 2 1
860.1.bq.b yes 12 172.o odd 42 1
860.1.bq.b yes 12 215.u even 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 8 T_{3}^{11} + 35 T_{3}^{10} + 104 T_{3}^{9} + 230 T_{3}^{8} + 392 T_{3}^{7} + 519 T_{3}^{6} + \cdots + 1 \) acting on \(S_{1}^{\mathrm{new}}(860, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + 8 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} - T^{11} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T^{6} - T^{5} + 3 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} + 5 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{12} - T^{11} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( (T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} - 13 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{12} \) Copy content Toggle raw display
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