Properties

Label 2007.1.v.a
Level 2007
Weight 1
Character orbit 2007.v
Analytic conductor 1.002
Analytic rank 0
Dimension 36
Projective image \(D_{74}\)
CM disc. -3
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 2007 = 3^{2} \cdot 223 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 2007.v (of order \(74\) and degree \(36\))

Newform invariants

Self dual: No
Analytic conductor: \(1.00162348035\)
Analytic rank: \(0\)
Dimension: \(36\)
Coefficient field: \(\Q(\zeta_{74})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{74}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{74} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -\zeta_{74}^{20} q^{4} \) \( + ( \zeta_{74}^{21} + \zeta_{74}^{27} ) q^{7} \) \(+O(q^{10})\) \( q\) \( -\zeta_{74}^{20} q^{4} \) \( + ( \zeta_{74}^{21} + \zeta_{74}^{27} ) q^{7} \) \( + ( \zeta_{74}^{9} + \zeta_{74}^{32} ) q^{13} \) \( -\zeta_{74}^{3} q^{16} \) \( + ( \zeta_{74}^{24} + \zeta_{74}^{28} ) q^{19} \) \( + \zeta_{74}^{14} q^{25} \) \( + ( \zeta_{74}^{4} + \zeta_{74}^{10} ) q^{28} \) \( + ( -\zeta_{74}^{6} - \zeta_{74}^{36} ) q^{31} \) \( + ( \zeta_{74}^{8} - \zeta_{74}^{13} ) q^{37} \) \( + ( \zeta_{74}^{12} + \zeta_{74}^{34} ) q^{43} \) \( + ( -\zeta_{74}^{5} - \zeta_{74}^{11} - \zeta_{74}^{17} ) q^{49} \) \( + ( \zeta_{74}^{15} - \zeta_{74}^{29} ) q^{52} \) \( + ( -\zeta_{74}^{15} - \zeta_{74}^{26} ) q^{61} \) \( + \zeta_{74}^{23} q^{64} \) \( + ( \zeta_{74} + \zeta_{74}^{2} ) q^{67} \) \( + ( -\zeta_{74}^{2} + \zeta_{74}^{5} ) q^{73} \) \( + ( \zeta_{74}^{7} + \zeta_{74}^{11} ) q^{76} \) \( + ( \zeta_{74}^{13} + \zeta_{74}^{22} ) q^{79} \) \( + ( -\zeta_{74}^{16} - \zeta_{74}^{22} + \zeta_{74}^{30} + \zeta_{74}^{36} ) q^{91} \) \( + ( \zeta_{74}^{29} - \zeta_{74}^{31} ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(36q \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(36q \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut q^{16} \) \(\mathstrut -\mathstrut 2q^{19} \) \(\mathstrut -\mathstrut q^{25} \) \(\mathstrut -\mathstrut 2q^{28} \) \(\mathstrut +\mathstrut 2q^{31} \) \(\mathstrut -\mathstrut 2q^{37} \) \(\mathstrut -\mathstrut 2q^{43} \) \(\mathstrut -\mathstrut 3q^{49} \) \(\mathstrut +\mathstrut q^{64} \) \(\mathstrut +\mathstrut 2q^{73} \) \(\mathstrut +\mathstrut 2q^{76} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2007\mathbb{Z}\right)^\times\).

\(n\) \(226\) \(893\)
\(\chi(n)\) \(\zeta_{74}^{33}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
−0.985616 + 0.169001i
−0.524307 0.851529i
−0.0424412 0.999099i
0.721956 0.691939i
0.594633 + 0.803997i
0.292823 + 0.956167i
−0.985616 0.169001i
0.828510 + 0.559975i
−0.660675 + 0.750672i
−0.873014 0.487695i
0.594633 0.803997i
0.828510 0.559975i
0.292823 0.956167i
−0.942877 0.333140i
0.721956 + 0.691939i
−0.873014 + 0.487695i
−0.210679 0.977555i
−0.660675 0.750672i
−0.372856 0.927889i
0.996397 + 0.0848059i
0 0 0.967733 0.251978i 0 0 1.03825 1.40380i 0 0 0
118.1 0 0 −0.0424412 0.999099i 0 0 1.55047 1.25191i 0 0 0
163.1 0 0 −0.660675 + 0.750672i 0 0 0.133193 0.216319i 0 0 0
190.1 0 0 0.911228 + 0.411901i 0 0 −1.15356 0.644415i 0 0 0
208.1 0 0 −0.985616 + 0.169001i 0 0 1.71835 + 0.776745i 0 0 0
334.1 0 0 −0.942877 0.333140i 0 0 −1.02806 + 1.16810i 0 0 0
397.1 0 0 0.967733 + 0.251978i 0 0 1.03825 + 1.40380i 0 0 0
442.1 0 0 −0.778036 + 0.628220i 0 0 0.0535200 0.417946i 0 0 0
505.1 0 0 0.292823 0.956167i 0 0 0.0703259 1.65553i 0 0 0
541.1 0 0 0.721956 + 0.691939i 0 0 −0.0800337 + 0.0282777i 0 0 0
550.1 0 0 −0.985616 0.169001i 0 0 1.71835 0.776745i 0 0 0
613.1 0 0 −0.778036 0.628220i 0 0 0.0535200 + 0.417946i 0 0 0
667.1 0 0 −0.942877 + 0.333140i 0 0 −1.02806 1.16810i 0 0 0
721.1 0 0 −0.873014 0.487695i 0 0 0.307058 1.00265i 0 0 0
919.1 0 0 0.911228 0.411901i 0 0 −1.15356 + 0.644415i 0 0 0
946.1 0 0 0.721956 0.691939i 0 0 −0.0800337 0.0282777i 0 0 0
1000.1 0 0 0.450204 0.892926i 0 0 0.443426 + 1.10351i 0 0 0
1081.1 0 0 0.292823 + 0.956167i 0 0 0.0703259 + 1.65553i 0 0 0
1099.1 0 0 −0.210679 + 0.977555i 0 0 −1.76365 0.459219i 0 0 0
1108.1 0 0 0.127018 0.991900i 0 0 −0.871354 + 1.72823i 0 0 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1999.1
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 CM by \(\Q(\sqrt{-3}) \) yes
223.f Odd 1 yes
669.k Even 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(2007, [\chi])\).