Properties

Label 3997.1.cz.a
Level 3997
Weight 1
Character orbit 3997.cz
Analytic conductor 1.995
Analytic rank 0
Dimension 144
Projective image \(D_{285}\)
CM disc. -7
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 3997 = 7 \cdot 571 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 3997.cz (of order \(570\) and degree \(144\))

Newform invariants

Self dual: No
Analytic conductor: \(1.99476285549\)
Analytic rank: \(0\)
Dimension: \(144\)
Coefficient field: \(\Q(\zeta_{570})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{285}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{285} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -\zeta_{570}^{201} + \zeta_{570}^{244} ) q^{2} + ( -\zeta_{570}^{117} + \zeta_{570}^{160} - \zeta_{570}^{203} ) q^{4} -\zeta_{570}^{93} q^{7} + ( -\zeta_{570}^{33} + \zeta_{570}^{76} - \zeta_{570}^{119} + \zeta_{570}^{162} ) q^{8} + \zeta_{570}^{176} q^{9} +O(q^{10})\) \( q + ( -\zeta_{570}^{201} + \zeta_{570}^{244} ) q^{2} + ( -\zeta_{570}^{117} + \zeta_{570}^{160} - \zeta_{570}^{203} ) q^{4} -\zeta_{570}^{93} q^{7} + ( -\zeta_{570}^{33} + \zeta_{570}^{76} - \zeta_{570}^{119} + \zeta_{570}^{162} ) q^{8} + \zeta_{570}^{176} q^{9} + ( -\zeta_{570}^{3} + \zeta_{570}^{140} ) q^{11} + ( -\zeta_{570}^{9} + \zeta_{570}^{52} ) q^{14} + ( -\zeta_{570}^{35} + \zeta_{570}^{78} - \zeta_{570}^{121} + \zeta_{570}^{234} - \zeta_{570}^{277} ) q^{16} + ( \zeta_{570}^{92} - \zeta_{570}^{135} ) q^{18} + ( \zeta_{570}^{56} - \zeta_{570}^{99} + \zeta_{570}^{204} - \zeta_{570}^{247} ) q^{22} + ( \zeta_{570}^{106} - \zeta_{570}^{155} ) q^{23} + \zeta_{570}^{172} q^{25} + ( -\zeta_{570}^{11} + \zeta_{570}^{210} - \zeta_{570}^{253} ) q^{28} + ( \zeta_{570}^{40} - \zeta_{570}^{105} ) q^{29} + ( -\zeta_{570}^{37} + \zeta_{570}^{80} + \zeta_{570}^{150} - \zeta_{570}^{193} + \zeta_{570}^{236} - \zeta_{570}^{279} ) q^{32} + ( \zeta_{570}^{8} - \zeta_{570}^{51} + \zeta_{570}^{94} ) q^{36} + ( \zeta_{570}^{98} + \zeta_{570}^{194} ) q^{37} + ( -\zeta_{570}^{29} - \zeta_{570}^{75} ) q^{43} + ( -\zeta_{570}^{15} + \zeta_{570}^{58} + \zeta_{570}^{120} - \zeta_{570}^{163} + \zeta_{570}^{206} - \zeta_{570}^{257} ) q^{44} + ( \zeta_{570}^{22} - \zeta_{570}^{65} - \zeta_{570}^{71} + \zeta_{570}^{114} ) q^{46} + \zeta_{570}^{186} q^{49} + ( \zeta_{570}^{88} - \zeta_{570}^{131} ) q^{50} + ( \zeta_{570}^{20} + \zeta_{570}^{48} ) q^{53} + ( \zeta_{570}^{126} - \zeta_{570}^{169} + \zeta_{570}^{212} - \zeta_{570}^{255} ) q^{56} + ( -\zeta_{570}^{21} + \zeta_{570}^{64} - \zeta_{570}^{241} + \zeta_{570}^{284} ) q^{58} -\zeta_{570}^{269} q^{63} + ( -\zeta_{570}^{39} + \zeta_{570}^{66} - \zeta_{570}^{109} + \zeta_{570}^{152} - \zeta_{570}^{195} + \zeta_{570}^{238} - \zeta_{570}^{281} ) q^{64} + ( -\zeta_{570}^{19} + \zeta_{570}^{144} ) q^{67} + ( \zeta_{570}^{100} + \zeta_{570}^{166} ) q^{71} + ( \zeta_{570}^{10} - \zeta_{570}^{53} - \zeta_{570}^{209} + \zeta_{570}^{252} ) q^{72} + ( \zeta_{570}^{14} - \zeta_{570}^{57} + \zeta_{570}^{110} - \zeta_{570}^{153} ) q^{74} + ( \zeta_{570}^{96} - \zeta_{570}^{233} ) q^{77} + ( \zeta_{570}^{156} + \zeta_{570}^{170} ) q^{79} -\zeta_{570}^{67} q^{81} + ( \zeta_{570}^{34} + \zeta_{570}^{230} - \zeta_{570}^{273} + \zeta_{570}^{276} ) q^{86} + ( -\zeta_{570}^{17} + \zeta_{570}^{36} - \zeta_{570}^{79} + \zeta_{570}^{122} - \zeta_{570}^{165} - \zeta_{570}^{173} + \zeta_{570}^{216} - \zeta_{570}^{259} ) q^{88} + ( \zeta_{570}^{24} + \zeta_{570}^{30} - \zeta_{570}^{73} - \zeta_{570}^{223} + \zeta_{570}^{266} + \zeta_{570}^{272} ) q^{92} + ( \zeta_{570}^{102} - \zeta_{570}^{145} ) q^{98} + ( -\zeta_{570}^{31} - \zeta_{570}^{179} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 144q + q^{2} + 5q^{4} + 2q^{7} + 21q^{8} - q^{9} + O(q^{10}) \) \( 144q + q^{2} + 5q^{4} + 2q^{7} + 21q^{8} - q^{9} + 6q^{11} + q^{14} + 6q^{16} - 9q^{18} + 21q^{22} + 3q^{23} - q^{25} - 10q^{28} - 4q^{29} - 5q^{32} - 2q^{37} - 9q^{43} - 20q^{44} - 34q^{46} + 2q^{49} - 2q^{50} + 6q^{53} - 8q^{56} - q^{58} - q^{63} + 11q^{64} + 20q^{67} + 3q^{71} + 23q^{72} - 31q^{74} + q^{77} + 6q^{79} - q^{81} + 7q^{86} - 9q^{88} + 9q^{92} + 6q^{98} - 2q^{99} + O(q^{100}) \)

Character Values

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

\(n\) \(1716\) \(2285\)
\(\chi(n)\) \(\zeta_{570}^{176}\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
−0.999757 0.0220445i
−0.618553 + 0.785743i
−0.815447 0.578832i
0.868768 + 0.495219i
−0.224056 + 0.974576i
0.999939 + 0.0110229i
−0.775409 0.631460i
−0.999028 + 0.0440782i
−0.840168 0.542326i
0.782322 + 0.622874i
0.997024 + 0.0770854i
0.170028 0.985439i
−0.988116 + 0.153712i
−0.761300 0.648400i
−0.0935596 + 0.995614i
0.319482 0.947592i
−0.959210 + 0.282694i
0.0605901 + 0.998163i
−0.884667 + 0.466224i
−0.731980 + 0.681326i
0.341150 1.74650i 0 −2.00737 0.815325i 0 0 −0.461341 + 0.887223i −1.13548 + 1.73798i −0.739446 0.673216i 0
34.1 1.66575 0.0919017i 0 1.77233 0.196161i 0 0 −0.724425 0.689353i 1.28870 0.215046i −0.421786 0.906696i 0
83.1 0.967953 1.17532i 0 −0.252734 1.29386i 0 0 0.652586 + 0.757715i −0.426247 0.230673i −0.256156 + 0.966635i 0
97.1 1.62671 + 1.12794i 0 1.02331 + 2.73313i 0 0 0.490424 + 0.871484i −0.932234 + 3.68131i −0.997024 0.0770854i 0
139.1 1.15042 1.11915i 0 0.0434243 1.57535i 0 0 0.828009 + 0.560715i −0.626073 0.680097i −0.480787 + 0.876837i 0
174.1 −0.298517 0.362469i 0 0.149439 0.765045i 0 0 −0.518970 0.854793i −0.734890 + 0.397702i −0.360939 + 0.932589i 0
202.1 −0.315447 1.00854i 0 −0.0958675 + 0.0664730i 0 0 0.746821 + 0.665025i −0.736619 0.573334i 0.618553 + 0.785743i 0
223.1 −1.08088 + 0.439016i 0 0.258785 0.251750i 0 0 −0.574329 + 0.818625i 0.299439 0.682651i 0.0935596 0.995614i 0
237.1 −0.579560 + 1.85296i 0 −2.27578 1.57799i 0 0 −0.995083 + 0.0990455i 2.71079 2.10989i 0.938430 + 0.345471i 0
244.1 1.75911 + 0.714490i 0 1.86720 + 1.81644i 0 0 −0.956036 + 0.293250i 1.22410 + 2.79067i 0.509516 0.860461i 0
279.1 1.98014 0.219162i 0 2.89711 0.649259i 0 0 −0.627176 0.778877i 3.71011 1.27368i 0.528360 + 0.849020i 0
293.1 −0.874198 1.72746i 0 −1.62767 + 2.21452i 0 0 0.180881 0.983505i 3.33875 + 0.557139i 0.224056 0.974576i 0
321.1 1.92132 + 0.430577i 0 2.60171 + 1.22778i 0 0 −0.213300 0.976987i 2.91626 + 2.26982i −0.441671 0.897177i 0
342.1 −1.70487 + 0.188695i 0 1.89518 0.424719i 0 0 −0.934564 + 0.355794i −1.52855 + 0.524751i 0.0715891 0.997434i 0
482.1 −0.660728 1.76473i 0 −1.92359 + 1.67525i 0 0 0.652586 + 0.757715i 2.57009 + 1.39086i −0.709053 0.705155i 0
531.1 −1.26766 1.53924i 0 −0.570575 + 2.92103i 0 0 0.922290 + 0.386499i 3.46574 1.87557i 0.775409 + 0.631460i 0
538.1 1.17577 1.59970i 0 −0.878074 2.80736i 0 0 0.0495838 0.998770i −3.64560 1.25154i 0.984487 0.175457i 0
552.1 −0.238657 1.72107i 0 −1.94285 + 0.549385i 0 0 0.601081 0.799188i 0.711243 + 1.62147i −0.319482 + 0.947592i 0
559.1 −0.485567 + 0.959507i 0 −0.0926427 0.126045i 0 0 0.431754 0.901991i −0.894782 + 0.149313i −0.857640 + 0.514250i 0
580.1 1.76667 0.499567i 0 2.01965 1.24147i 0 0 0.828009 0.560715i 1.70440 1.85147i 0.999757 + 0.0220445i 0
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3884.1
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.b Odd 1 CM by \(\Q(\sqrt{-7}) \) yes
571.o Even 1 yes
3997.cz Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(3997, [\chi])\).