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Properties

Label 1682.2.b.j
Level $1682$
Weight $2$
Character orbit 1682.b
Analytic conductor $13.431$
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] = [1682,2,Mod(1681,1682)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1682, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1682.1681");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1682 = 2 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1682.b (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: \(13.4308376200\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{28})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 58)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} - \beta_1) q^{3} - q^{4} + ( - \beta_{10} - \beta_{8} + 1) q^{5} + ( - \beta_{9} + \beta_{7} + 2) q^{6} + ( - \beta_{10} - \beta_{9} + \beta_{7} + \cdots + 1) q^{7}+ \cdots + (2 \beta_{9} - 3 \beta_{7} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} - \beta_1) q^{3} - q^{4} + ( - \beta_{10} - \beta_{8} + 1) q^{5} + ( - \beta_{9} + \beta_{7} + 2) q^{6} + ( - \beta_{10} - \beta_{9} + \beta_{7} + \cdots + 1) q^{7}+ \cdots + ( - 2 \beta_{11} + 3 \beta_{5} + \cdots + 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4} + 12 q^{5} + 16 q^{6} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{4} + 12 q^{5} + 16 q^{6} + 4 q^{7} - 4 q^{9} + 16 q^{13} + 12 q^{16} - 12 q^{20} + 4 q^{22} + 12 q^{23} - 16 q^{24} + 8 q^{25} - 4 q^{28} + 16 q^{30} - 24 q^{33} + 24 q^{34} + 32 q^{35} + 4 q^{36} + 8 q^{38} + 24 q^{42} - 4 q^{45} - 8 q^{49} - 32 q^{51} - 16 q^{52} + 16 q^{53} - 4 q^{54} - 20 q^{57} - 48 q^{59} + 52 q^{62} - 48 q^{63} - 12 q^{64} - 12 q^{65} - 16 q^{67} - 40 q^{71} - 40 q^{74} + 12 q^{78} + 12 q^{80} + 12 q^{81} - 48 q^{82} - 56 q^{83} + 52 q^{86} - 4 q^{88} + 24 q^{91} - 12 q^{92} - 60 q^{93} - 36 q^{94} + 16 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{28}^{7} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{28}^{8} + \zeta_{28}^{6} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{28}^{9} + \zeta_{28}^{5} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{28}^{10} + \zeta_{28}^{4} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{28}^{11} + \zeta_{28}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{28}^{11} + \zeta_{28}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{28}^{10} + \zeta_{28}^{4} \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -\zeta_{28}^{9} + \zeta_{28}^{5} \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( -\zeta_{28}^{8} + \zeta_{28}^{6} \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( -\zeta_{28}^{11} + \zeta_{28}^{9} - \zeta_{28}^{7} + \zeta_{28}^{5} - \zeta_{28}^{3} + 2\zeta_{28} \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( \zeta_{28}^{10} - \zeta_{28}^{8} + \zeta_{28}^{6} - \zeta_{28}^{4} + 2\zeta_{28}^{2} - 1 \) Copy content Toggle raw display

Display \(\zeta_{28}^j\) in terms of \(\beta_i\)

\(\zeta_{28}\)\(=\) \( ( \beta_{10} + \beta_{5} - \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{28}^{2}\)\(=\) \( ( \beta_{11} - \beta_{9} + \beta_{7} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{28}^{3}\)\(=\) \( ( \beta_{6} + \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{28}^{4}\)\(=\) \( ( \beta_{7} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{28}^{5}\)\(=\) \( ( \beta_{8} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{28}^{6}\)\(=\) \( ( \beta_{9} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{28}^{7}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{28}^{8}\)\(=\) \( ( -\beta_{9} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{28}^{9}\)\(=\) \( ( -\beta_{8} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{28}^{10}\)\(=\) \( ( -\beta_{7} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{28}^{11}\)\(=\) \( ( -\beta_{6} + \beta_{5} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{28}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1682\mathbb{Z}\right)^\times\).

\(n\) \(843\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1681.1
0.433884 − 0.900969i
−0.433884 − 0.900969i
−0.781831 + 0.623490i
0.781831 + 0.623490i
0.974928 − 0.222521i
−0.974928 − 0.222521i
0.974928 + 0.222521i
−0.974928 + 0.222521i
−0.781831 − 0.623490i
0.781831 − 0.623490i
0.433884 + 0.900969i
−0.433884 + 0.900969i
− 1.00000i − 0.246980i −1.00000 −1.43143 −0.246980 −4.06460 1.00000i 2.93900 1.43143i
1681.2 − 1.00000i − 0.246980i −1.00000 3.43143 −0.246980 1.57064 1.00000i 2.93900 − 3.43143i
1681.3 − 1.00000i 1.44504i −1.00000 0.613807 1.44504 2.87647 1.00000i 0.911854 − 0.613807i
1681.4 − 1.00000i 1.44504i −1.00000 1.38619 1.44504 −1.98639 1.00000i 0.911854 − 1.38619i
1681.5 − 1.00000i 2.80194i −1.00000 −1.81762 2.80194 1.41574 1.00000i −4.85086 1.81762i
1681.6 − 1.00000i 2.80194i −1.00000 3.81762 2.80194 2.18813 1.00000i −4.85086 − 3.81762i
1681.7 1.00000i − 2.80194i −1.00000 −1.81762 2.80194 1.41574 − 1.00000i −4.85086 − 1.81762i
1681.8 1.00000i − 2.80194i −1.00000 3.81762 2.80194 2.18813 − 1.00000i −4.85086 3.81762i
1681.9 1.00000i − 1.44504i −1.00000 0.613807 1.44504 2.87647 − 1.00000i 0.911854 0.613807i
1681.10 1.00000i − 1.44504i −1.00000 1.38619 1.44504 −1.98639 − 1.00000i 0.911854 1.38619i
1681.11 1.00000i 0.246980i −1.00000 −1.43143 −0.246980 −4.06460 − 1.00000i 2.93900 − 1.43143i
1681.12 1.00000i 0.246980i −1.00000 3.43143 −0.246980 1.57064 − 1.00000i 2.93900 3.43143i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1681.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1682.2.b.j 12
29.b even 2 1 inner 1682.2.b.j 12
29.c odd 4 1 1682.2.a.r 6
29.c odd 4 1 1682.2.a.s 6
29.d even 7 1 58.2.e.a ✓ 12
29.e even 14 1 58.2.e.a ✓ 12
87.h odd 14 1 522.2.n.a 12
87.j odd 14 1 522.2.n.a 12
116.h odd 14 1 464.2.y.c 12
116.j odd 14 1 464.2.y.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.2.e.a ✓ 12 29.d even 7 1
58.2.e.a ✓ 12 29.e even 14 1
464.2.y.c 12 116.h odd 14 1
464.2.y.c 12 116.j odd 14 1
522.2.n.a 12 87.h odd 14 1
522.2.n.a 12 87.j odd 14 1
1682.2.a.r 6 29.c odd 4 1
1682.2.a.s 6 29.c odd 4 1
1682.2.b.j 12 1.a even 1 1 trivial
1682.2.b.j 12 29.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1682, [\chi])\):

\( T_{3}^{6} + 10T_{3}^{4} + 17T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{6} - 6T_{5}^{5} + T_{5}^{4} + 36T_{5}^{3} - 20T_{5}^{2} - 48T_{5} + 29 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$3$ \( (T^{6} + 10 T^{4} + 17 T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} - 6 T^{5} + T^{4} + \cdots + 29)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 2 T^{5} + \cdots + 113)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 66 T^{10} + \cdots + 841 \) Copy content Toggle raw display
$13$ \( (T^{6} - 8 T^{5} + \cdots + 197)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 94 T^{10} + \cdots + 12769 \) Copy content Toggle raw display
$19$ \( T^{12} + 124 T^{10} + \cdots + 175561 \) Copy content Toggle raw display
$23$ \( (T^{6} - 6 T^{5} - 34 T^{4} + \cdots - 27)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 5377435561 \) Copy content Toggle raw display
$37$ \( T^{12} + 202 T^{10} + \cdots + 3736489 \) Copy content Toggle raw display
$41$ \( T^{12} + 236 T^{10} + \cdots + 60171049 \) Copy content Toggle raw display
$43$ \( T^{12} + 234 T^{10} + \cdots + 8637721 \) Copy content Toggle raw display
$47$ \( T^{12} + 320 T^{10} + \cdots + 24571849 \) Copy content Toggle raw display
$53$ \( (T^{6} - 8 T^{5} + \cdots - 46927)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 24 T^{5} + \cdots + 35869)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 105805126729 \) Copy content Toggle raw display
$67$ \( (T^{6} + 8 T^{5} + \cdots - 47207)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 20 T^{5} + \cdots + 123089)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 164 T^{10} + \cdots + 5938969 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 4893282304 \) Copy content Toggle raw display
$83$ \( (T^{6} + 28 T^{5} + \cdots + 1421)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 378 T^{10} + \cdots + 34656769 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 10446475264 \) Copy content Toggle raw display
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