Properties

Label 3997.1.ce.a
Level $3997$
Weight $1$
Character orbit 3997.ce
Analytic conductor $1.995$
Analytic rank $0$
Dimension $36$
Projective image $D_{57}$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3997,1,Mod(258,3997)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3997, base_ring=CyclotomicField(114))
 
chi = DirichletCharacter(H, H._module([57, 86]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3997.258");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3997 = 7 \cdot 571 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3997.ce (of order \(114\), degree \(36\), 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: \(1.99476285549\)
Analytic rank: \(0\)
Dimension: \(36\)
Coefficient field: \(\Q(\zeta_{57})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{36} - x^{35} + x^{33} - x^{32} + x^{30} - x^{29} + x^{27} - x^{26} + x^{24} - x^{23} + x^{21} - x^{20} + x^{18} - x^{16} + x^{15} - x^{13} + x^{12} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{57}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{57} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{114}^{47} + \zeta_{114}^{24}) q^{2} + (\zeta_{114}^{48} - \zeta_{114}^{37} + \zeta_{114}^{14}) q^{4} + \zeta_{114}^{6} q^{7} + (\zeta_{114}^{38} - \zeta_{114}^{27} - \zeta_{114}^{15} - \zeta_{114}^{4}) q^{8} + \zeta_{114}^{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{114}^{47} + \zeta_{114}^{24}) q^{2} + (\zeta_{114}^{48} - \zeta_{114}^{37} + \zeta_{114}^{14}) q^{4} + \zeta_{114}^{6} q^{7} + (\zeta_{114}^{38} - \zeta_{114}^{27} - \zeta_{114}^{15} - \zeta_{114}^{4}) q^{8} + \zeta_{114}^{4} q^{9} + ( - \zeta_{114}^{55} + \zeta_{114}^{48}) q^{11} + ( - \zeta_{114}^{53} + \zeta_{114}^{30}) q^{14} + ( - \zeta_{114}^{51} - \zeta_{114}^{39} + \zeta_{114}^{28} - \zeta_{114}^{17} + \zeta_{114}^{5}) q^{16} + ( - \zeta_{114}^{51} + \zeta_{114}^{28}) q^{18} + ( - \zeta_{114}^{45} + \zeta_{114}^{38} + \zeta_{114}^{22} - \zeta_{114}^{15}) q^{22} + (\zeta_{114}^{10} - \zeta_{114}^{5}) q^{23} - \zeta_{114}^{35} q^{25} + (\zeta_{114}^{54} - \zeta_{114}^{43} + \zeta_{114}^{20}) q^{28} + (\zeta_{114}^{32} - \zeta_{114}^{27}) q^{29} + (\zeta_{114}^{52} - \zeta_{114}^{41} - \zeta_{114}^{29} + \zeta_{114}^{18} - \zeta_{114}^{7} + \zeta_{114}^{6}) q^{32} + (\zeta_{114}^{52} - \zeta_{114}^{41} + \zeta_{114}^{18}) q^{36} + (\zeta_{114}^{10} - \zeta_{114}^{7}) q^{37} + (\zeta_{114}^{46} - \zeta_{114}^{3}) q^{43} + (\zeta_{114}^{46} - \zeta_{114}^{39} - \zeta_{114}^{35} + \zeta_{114}^{28} + \zeta_{114}^{12} - \zeta_{114}^{5}) q^{44} + (\zeta_{114}^{52} + \zeta_{114}^{34} - \zeta_{114}^{29} + 1) q^{46} + \zeta_{114}^{12} q^{49} + ( - \zeta_{114}^{25} + \zeta_{114}^{2}) q^{50} + ( - \zeta_{114}^{27} + \zeta_{114}^{16}) q^{53} + (\zeta_{114}^{44} - \zeta_{114}^{33} - \zeta_{114}^{21} + \zeta_{114}^{10}) q^{56} + (\zeta_{114}^{56} - \zeta_{114}^{51} + \zeta_{114}^{22} - \zeta_{114}^{17}) q^{58} + \zeta_{114}^{10} q^{63} + (\zeta_{114}^{54} - \zeta_{114}^{53} + \zeta_{114}^{42} - \zeta_{114}^{31} + \zeta_{114}^{30} - \zeta_{114}^{19} - \zeta_{114}^{8}) q^{64} + (\zeta_{114}^{38} + \zeta_{114}^{24}) q^{67} + ( - \zeta_{114}^{53} - \zeta_{114}^{23}) q^{71} + (\zeta_{114}^{42} - \zeta_{114}^{31} - \zeta_{114}^{19} + \zeta_{114}^{8}) q^{72} + (\zeta_{114}^{54} + \zeta_{114}^{34} - \zeta_{114}^{31} + 1) q^{74} + (\zeta_{114}^{54} + \zeta_{114}^{4}) q^{77} + ( - \zeta_{114}^{45} + \zeta_{114}^{22}) q^{79} + \zeta_{114}^{8} q^{81} + (\zeta_{114}^{50} + \zeta_{114}^{36} - \zeta_{114}^{27} - \zeta_{114}^{13}) q^{86} + (\zeta_{114}^{52} + \zeta_{114}^{36} - \zeta_{114}^{29} - \zeta_{114}^{25} + \zeta_{114}^{18} - \zeta_{114}^{13} + \zeta_{114}^{6} + \zeta_{114}^{2}) q^{88} + ( - \zeta_{114}^{53} - \zeta_{114}^{47} + \zeta_{114}^{42} + \zeta_{114}^{24} - \zeta_{114}^{19} - \zeta_{114}) q^{92} + (\zeta_{114}^{36} + \zeta_{114}^{2}) q^{98} + (\zeta_{114}^{52} + \zeta_{114}^{2}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q - q^{2} - 2 q^{7} - 21 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 36 q - q^{2} - 2 q^{7} - 21 q^{8} + q^{9} - q^{11} - q^{14} - q^{16} - q^{18} - 21 q^{22} + 2 q^{23} + q^{25} - q^{29} + 2 q^{37} - q^{43} + 39 q^{46} - 2 q^{49} + 2 q^{50} - q^{53} - 2 q^{56} + q^{58} + q^{63} - 21 q^{64} - 20 q^{67} + 2 q^{71} - 18 q^{72} + 36 q^{74} - q^{77} - q^{79} + q^{81} - 2 q^{86} - q^{88} - 19 q^{92} - q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3997\mathbb{Z}\right)^\times\).

\(n\) \(1716\) \(2285\)
\(\chi(n)\) \(\zeta_{114}^{4}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
258.1
−0.926494 + 0.376309i
0.716783 0.697297i
0.0275543 + 0.999620i
0.716783 + 0.697297i
−0.962268 + 0.272103i
0.993931 0.110008i
−0.926494 0.376309i
−0.298515 + 0.954405i
−0.962268 0.272103i
−0.298515 0.954405i
−0.754107 0.656752i
−0.592235 0.805765i
0.993931 + 0.110008i
0.0275543 0.999620i
−0.754107 + 0.656752i
0.350638 + 0.936511i
0.975796 0.218681i
−0.592235 + 0.805765i
0.350638 0.936511i
0.635724 + 0.771917i
−1.74047 0.821347i 0 1.71889 + 2.08714i 0 0 −0.677282 0.735724i −0.804971 3.17876i 0.0275543 0.999620i 0
328.1 1.08317 + 1.31522i 0 −0.364838 + 1.86777i 0 0 −0.0825793 + 0.996584i −1.35323 + 0.732331i −0.998482 0.0550878i 0
496.1 −0.173127 0.886316i 0 0.170911 0.0694180i 0 0 −0.986361 + 0.164595i −0.585046 0.895480i 0.993931 0.110008i 0
524.1 1.08317 1.31522i 0 −0.364838 1.86777i 0 0 −0.0825793 0.996584i −1.35323 0.732331i −0.998482 + 0.0550878i 0
587.1 0.0193232 + 0.0516099i 0 0.751816 0.654757i 0 0 −0.0825793 0.996584i 0.0967861 + 0.0523780i 0.451533 0.892254i 0
944.1 −0.427940 + 0.416307i 0 −0.0177328 + 0.643311i 0 0 0.789141 0.614213i −0.664583 0.721930i 0.904357 0.426776i 0
1007.1 −1.74047 + 0.821347i 0 1.71889 2.08714i 0 0 −0.677282 + 0.735724i −0.804971 + 3.17876i 0.0275543 + 0.999620i 0
1014.1 1.54088 + 0.947175i 0 1.02563 + 2.02671i 0 0 0.245485 0.969400i −0.189907 + 2.29183i 0.350638 + 0.936511i 0
1035.1 0.0193232 0.0516099i 0 0.751816 + 0.654757i 0 0 −0.0825793 + 0.996584i 0.0967861 0.0523780i 0.451533 + 0.892254i 0
1084.1 1.54088 0.947175i 0 1.02563 2.02671i 0 0 0.245485 + 0.969400i −0.189907 2.29183i 0.350638 0.936511i 0
1140.1 0.553144 1.76850i 0 −1.99985 1.38666i 0 0 −0.401695 0.915773i −2.09626 + 1.63158i −0.962268 + 0.272103i 0
1224.1 −1.87796 0.531035i 0 2.39280 + 1.47085i 0 0 0.789141 0.614213i −2.39072 2.59701i −0.821778 0.569808i 0
1266.1 −0.427940 0.416307i 0 −0.0177328 0.643311i 0 0 0.789141 + 0.614213i −0.664583 + 0.721930i 0.904357 + 0.426776i 0
1378.1 −0.173127 + 0.886316i 0 0.170911 + 0.0694180i 0 0 −0.986361 0.164595i −0.585046 + 0.895480i 0.993931 + 0.110008i 0
1427.1 0.553144 + 1.76850i 0 −1.99985 + 1.38666i 0 0 −0.401695 + 0.915773i −2.09626 1.63158i −0.962268 0.272103i 0
1504.1 0.227076 0.308947i 0 0.254630 + 0.814096i 0 0 0.546948 + 0.837166i 0.671980 + 0.230691i 0.137354 0.990522i 0
1518.1 −0.0452871 + 1.64293i 0 −1.69869 0.0937194i 0 0 0.245485 0.969400i 0.0951797 1.14865i 0.635724 0.771917i 0
1525.1 −1.87796 + 0.531035i 0 2.39280 1.47085i 0 0 0.789141 + 0.614213i −2.39072 + 2.59701i −0.821778 + 0.569808i 0
1560.1 0.227076 + 0.308947i 0 0.254630 0.814096i 0 0 0.546948 0.837166i 0.671980 0.230691i 0.137354 + 0.990522i 0
1595.1 −1.49906 + 0.165916i 0 1.24385 0.278754i 0 0 0.546948 0.837166i −0.391868 + 0.134528i −0.926494 0.376309i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 258.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
571.k even 57 1 inner
3997.ce odd 114 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3997.1.ce.a 36
7.b odd 2 1 CM 3997.1.ce.a 36
571.k even 57 1 inner 3997.1.ce.a 36
3997.ce odd 114 1 inner 3997.1.ce.a 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3997.1.ce.a 36 1.a even 1 1 trivial
3997.1.ce.a 36 7.b odd 2 1 CM
3997.1.ce.a 36 571.k even 57 1 inner
3997.1.ce.a 36 3997.ce odd 114 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3997, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{36} + T^{35} + 18 T^{33} + 18 T^{32} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{36} \) Copy content Toggle raw display
$5$ \( T^{36} \) Copy content Toggle raw display
$7$ \( (T^{18} + T^{17} + T^{16} + T^{15} + T^{14} + T^{13} + T^{12} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{36} + T^{35} - T^{33} - T^{32} + 19 T^{31} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{36} \) Copy content Toggle raw display
$17$ \( T^{36} \) Copy content Toggle raw display
$19$ \( T^{36} \) Copy content Toggle raw display
$23$ \( T^{36} - 2 T^{35} + 3 T^{34} - 4 T^{33} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{36} + T^{35} - T^{33} - T^{32} + T^{30} + 96 T^{29} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{36} \) Copy content Toggle raw display
$37$ \( T^{36} - 2 T^{35} + 8 T^{33} - 16 T^{32} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{36} \) Copy content Toggle raw display
$43$ \( T^{36} + T^{35} + 18 T^{33} + 18 T^{32} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{36} \) Copy content Toggle raw display
$53$ \( T^{36} + T^{35} - T^{33} - T^{32} + T^{30} + T^{29} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{36} \) Copy content Toggle raw display
$61$ \( T^{36} \) Copy content Toggle raw display
$67$ \( T^{36} + 20 T^{35} + 209 T^{34} + 1500 T^{33} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( (T^{18} - T^{17} + 9 T^{16} - 6 T^{15} + 50 T^{14} + \cdots + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{36} \) Copy content Toggle raw display
$79$ \( T^{36} + T^{35} - T^{33} - T^{32} + 19 T^{31} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{36} \) Copy content Toggle raw display
$89$ \( T^{36} \) Copy content Toggle raw display
$97$ \( T^{36} \) Copy content Toggle raw display
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