# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{1}^{\mathrm{new}}(2007, [\chi])$$.