# Properties

 Label 14.3.d.a Level 14 Weight 3 Character orbit 14.d Analytic conductor 0.381 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$14 = 2 \cdot 7$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 14.d (of order $$6$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.381472370104$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( -2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{3}$$ $$+ 2 \beta_{2} q^{4}$$ $$+ ( -1 - 2 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{5}$$ $$+ ( -2 - 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{6}$$ $$+ ( 3 + 4 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} ) q^{7}$$ $$+ 2 \beta_{3} q^{8}$$ $$+ 6 \beta_{1} q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( -2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{3}$$ $$+ 2 \beta_{2} q^{4}$$ $$+ ( -1 - 2 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{5}$$ $$+ ( -2 - 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{6}$$ $$+ ( 3 + 4 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} ) q^{7}$$ $$+ 2 \beta_{3} q^{8}$$ $$+ 6 \beta_{1} q^{9}$$ $$+ ( 8 - \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{10}$$ $$+ ( -3 \beta_{1} - 9 \beta_{2} - 3 \beta_{3} ) q^{11}$$ $$+ ( 2 - 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{12}$$ $$+ ( 6 - 4 \beta_{1} + 12 \beta_{2} - 2 \beta_{3} ) q^{13}$$ $$+ ( -10 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{14}$$ $$+ ( -9 + 3 \beta_{3} ) q^{15}$$ $$+ ( -4 - 4 \beta_{2} ) q^{16}$$ $$+ ( -10 - 2 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{17}$$ $$+ 12 \beta_{2} q^{18}$$ $$+ ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{19}$$ $$+ ( -2 + 8 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{20}$$ $$+ ( 8 - 8 \beta_{1} - 11 \beta_{2} - 10 \beta_{3} ) q^{21}$$ $$+ ( 6 - 9 \beta_{3} ) q^{22}$$ $$+ ( 15 - 9 \beta_{1} + 15 \beta_{2} ) q^{23}$$ $$+ ( 8 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{24}$$ $$+ ( 12 \beta_{1} - 2 \beta_{2} + 12 \beta_{3} ) q^{25}$$ $$+ ( 4 + 6 \beta_{1} - 4 \beta_{2} + 12 \beta_{3} ) q^{26}$$ $$+ ( -3 + 6 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{27}$$ $$+ ( -4 - 10 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{28}$$ $$+ ( 12 - 6 \beta_{3} ) q^{29}$$ $$+ ( -6 - 9 \beta_{1} - 6 \beta_{2} ) q^{30}$$ $$+ ( -14 + 15 \beta_{1} - 7 \beta_{2} - 15 \beta_{3} ) q^{31}$$ $$+ ( -4 \beta_{1} - 4 \beta_{3} ) q^{32}$$ $$+ ( -15 + 12 \beta_{1} + 15 \beta_{2} + 24 \beta_{3} ) q^{33}$$ $$+ ( -4 - 10 \beta_{1} - 8 \beta_{2} - 5 \beta_{3} ) q^{34}$$ $$+ ( 7 - 7 \beta_{1} + 35 \beta_{2} - 14 \beta_{3} ) q^{35}$$ $$+ 12 \beta_{3} q^{36}$$ $$+ ( -31 - 24 \beta_{1} - 31 \beta_{2} ) q^{37}$$ $$+ ( -4 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{38}$$ $$+ ( -12 \beta_{1} - 6 \beta_{2} - 12 \beta_{3} ) q^{39}$$ $$+ ( -8 - 2 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{40}$$ $$+ ( -2 + 20 \beta_{1} - 4 \beta_{2} + 10 \beta_{3} ) q^{41}$$ $$+ ( 20 + 8 \beta_{1} + 4 \beta_{2} - 11 \beta_{3} ) q^{42}$$ $$+ ( -2 - 6 \beta_{3} ) q^{43}$$ $$+ ( 18 + 6 \beta_{1} + 18 \beta_{2} ) q^{44}$$ $$+ ( 48 - 6 \beta_{1} + 24 \beta_{2} + 6 \beta_{3} ) q^{45}$$ $$+ ( 15 \beta_{1} - 18 \beta_{2} + 15 \beta_{3} ) q^{46}$$ $$+ ( 29 + \beta_{1} - 29 \beta_{2} + 2 \beta_{3} ) q^{47}$$ $$+ ( 4 + 8 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} ) q^{48}$$ $$+ ( -25 + 4 \beta_{1} - 40 \beta_{2} + 26 \beta_{3} ) q^{49}$$ $$+ ( -24 - 2 \beta_{3} ) q^{50}$$ $$+ ( 27 + 21 \beta_{1} + 27 \beta_{2} ) q^{51}$$ $$+ ( -24 + 4 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} ) q^{52}$$ $$+ ( -12 \beta_{1} + 39 \beta_{2} - 12 \beta_{3} ) q^{53}$$ $$+ ( -6 - 3 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{54}$$ $$+ ( -3 - 30 \beta_{1} - 6 \beta_{2} - 15 \beta_{3} ) q^{55}$$ $$+ ( 4 - 4 \beta_{1} - 16 \beta_{2} + 2 \beta_{3} ) q^{56}$$ $$+ 3 q^{57}$$ $$+ ( 12 + 12 \beta_{1} + 12 \beta_{2} ) q^{58}$$ $$+ ( -26 - 25 \beta_{1} - 13 \beta_{2} + 25 \beta_{3} ) q^{59}$$ $$+ ( -6 \beta_{1} - 18 \beta_{2} - 6 \beta_{3} ) q^{60}$$ $$+ ( -7 - 32 \beta_{1} + 7 \beta_{2} - 64 \beta_{3} ) q^{61}$$ $$+ ( 30 - 14 \beta_{1} + 60 \beta_{2} - 7 \beta_{3} ) q^{62}$$ $$+ ( -60 + 18 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} ) q^{63}$$ $$+ 8 q^{64}$$ $$+ ( -42 + 42 \beta_{1} - 42 \beta_{2} ) q^{65}$$ $$+ ( -48 - 15 \beta_{1} - 24 \beta_{2} + 15 \beta_{3} ) q^{66}$$ $$+ ( 45 \beta_{1} + 29 \beta_{2} + 45 \beta_{3} ) q^{67}$$ $$+ ( 10 - 4 \beta_{1} - 10 \beta_{2} - 8 \beta_{3} ) q^{68}$$ $$+ ( 3 - 12 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{69}$$ $$+ ( 28 + 7 \beta_{1} + 14 \beta_{2} + 35 \beta_{3} ) q^{70}$$ $$+ ( -6 + 30 \beta_{3} ) q^{71}$$ $$+ ( -24 - 24 \beta_{2} ) q^{72}$$ $$+ ( 106 + 16 \beta_{1} + 53 \beta_{2} - 16 \beta_{3} ) q^{73}$$ $$+ ( -31 \beta_{1} - 48 \beta_{2} - 31 \beta_{3} ) q^{74}$$ $$+ ( 22 - 10 \beta_{1} - 22 \beta_{2} - 20 \beta_{3} ) q^{75}$$ $$+ ( 2 - 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{76}$$ $$+ ( 42 + 42 \beta_{1} + 21 \beta_{2} ) q^{77}$$ $$+ ( 24 - 6 \beta_{3} ) q^{78}$$ $$+ ( 55 + 15 \beta_{1} + 55 \beta_{2} ) q^{79}$$ $$+ ( 8 - 8 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} ) q^{80}$$ $$+ ( -54 \beta_{1} - 9 \beta_{2} - 54 \beta_{3} ) q^{81}$$ $$+ ( -20 - 2 \beta_{1} + 20 \beta_{2} - 4 \beta_{3} ) q^{82}$$ $$+ ( -68 + 8 \beta_{1} - 136 \beta_{2} + 4 \beta_{3} ) q^{83}$$ $$+ ( 22 + 20 \beta_{1} + 38 \beta_{2} + 4 \beta_{3} ) q^{84}$$ $$+ ( -9 + 24 \beta_{3} ) q^{85}$$ $$+ ( 12 - 2 \beta_{1} + 12 \beta_{2} ) q^{86}$$ $$+ ( -48 - 18 \beta_{1} - 24 \beta_{2} + 18 \beta_{3} ) q^{87}$$ $$+ ( 18 \beta_{1} + 12 \beta_{2} + 18 \beta_{3} ) q^{88}$$ $$+ ( -63 + 24 \beta_{1} + 63 \beta_{2} + 48 \beta_{3} ) q^{89}$$ $$+ ( -12 + 48 \beta_{1} - 24 \beta_{2} + 24 \beta_{3} ) q^{90}$$ $$+ ( 30 - 44 \beta_{1} + 48 \beta_{2} + 8 \beta_{3} ) q^{91}$$ $$+ ( -30 - 18 \beta_{3} ) q^{92}$$ $$+ ( -69 - 24 \beta_{1} - 69 \beta_{2} ) q^{93}$$ $$+ ( -4 + 29 \beta_{1} - 2 \beta_{2} - 29 \beta_{3} ) q^{94}$$ $$+ ( -9 \beta_{1} + 15 \beta_{2} - 9 \beta_{3} ) q^{95}$$ $$+ ( -8 + 4 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{96}$$ $$+ ( 22 - 52 \beta_{1} + 44 \beta_{2} - 26 \beta_{3} ) q^{97}$$ $$+ ( -52 - 25 \beta_{1} - 44 \beta_{2} - 40 \beta_{3} ) q^{98}$$ $$+ ( 36 - 54 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 6q^{3}$$ $$\mathstrut -\mathstrut 4q^{4}$$ $$\mathstrut -\mathstrut 6q^{5}$$ $$\mathstrut +\mathstrut 8q^{7}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 6q^{3}$$ $$\mathstrut -\mathstrut 4q^{4}$$ $$\mathstrut -\mathstrut 6q^{5}$$ $$\mathstrut +\mathstrut 8q^{7}$$ $$\mathstrut +\mathstrut 24q^{10}$$ $$\mathstrut +\mathstrut 18q^{11}$$ $$\mathstrut +\mathstrut 12q^{12}$$ $$\mathstrut -\mathstrut 36q^{14}$$ $$\mathstrut -\mathstrut 36q^{15}$$ $$\mathstrut -\mathstrut 8q^{16}$$ $$\mathstrut -\mathstrut 30q^{17}$$ $$\mathstrut -\mathstrut 24q^{18}$$ $$\mathstrut +\mathstrut 6q^{19}$$ $$\mathstrut +\mathstrut 54q^{21}$$ $$\mathstrut +\mathstrut 24q^{22}$$ $$\mathstrut +\mathstrut 30q^{23}$$ $$\mathstrut +\mathstrut 24q^{24}$$ $$\mathstrut +\mathstrut 4q^{25}$$ $$\mathstrut +\mathstrut 24q^{26}$$ $$\mathstrut -\mathstrut 20q^{28}$$ $$\mathstrut +\mathstrut 48q^{29}$$ $$\mathstrut -\mathstrut 12q^{30}$$ $$\mathstrut -\mathstrut 42q^{31}$$ $$\mathstrut -\mathstrut 90q^{33}$$ $$\mathstrut -\mathstrut 42q^{35}$$ $$\mathstrut -\mathstrut 62q^{37}$$ $$\mathstrut -\mathstrut 12q^{38}$$ $$\mathstrut +\mathstrut 12q^{39}$$ $$\mathstrut -\mathstrut 48q^{40}$$ $$\mathstrut +\mathstrut 72q^{42}$$ $$\mathstrut -\mathstrut 8q^{43}$$ $$\mathstrut +\mathstrut 36q^{44}$$ $$\mathstrut +\mathstrut 144q^{45}$$ $$\mathstrut +\mathstrut 36q^{46}$$ $$\mathstrut +\mathstrut 174q^{47}$$ $$\mathstrut -\mathstrut 20q^{49}$$ $$\mathstrut -\mathstrut 96q^{50}$$ $$\mathstrut +\mathstrut 54q^{51}$$ $$\mathstrut -\mathstrut 72q^{52}$$ $$\mathstrut -\mathstrut 78q^{53}$$ $$\mathstrut -\mathstrut 36q^{54}$$ $$\mathstrut +\mathstrut 48q^{56}$$ $$\mathstrut +\mathstrut 12q^{57}$$ $$\mathstrut +\mathstrut 24q^{58}$$ $$\mathstrut -\mathstrut 78q^{59}$$ $$\mathstrut +\mathstrut 36q^{60}$$ $$\mathstrut -\mathstrut 42q^{61}$$ $$\mathstrut -\mathstrut 216q^{63}$$ $$\mathstrut +\mathstrut 32q^{64}$$ $$\mathstrut -\mathstrut 84q^{65}$$ $$\mathstrut -\mathstrut 144q^{66}$$ $$\mathstrut -\mathstrut 58q^{67}$$ $$\mathstrut +\mathstrut 60q^{68}$$ $$\mathstrut +\mathstrut 84q^{70}$$ $$\mathstrut -\mathstrut 24q^{71}$$ $$\mathstrut -\mathstrut 48q^{72}$$ $$\mathstrut +\mathstrut 318q^{73}$$ $$\mathstrut +\mathstrut 96q^{74}$$ $$\mathstrut +\mathstrut 132q^{75}$$ $$\mathstrut +\mathstrut 126q^{77}$$ $$\mathstrut +\mathstrut 96q^{78}$$ $$\mathstrut +\mathstrut 110q^{79}$$ $$\mathstrut +\mathstrut 24q^{80}$$ $$\mathstrut +\mathstrut 18q^{81}$$ $$\mathstrut -\mathstrut 120q^{82}$$ $$\mathstrut +\mathstrut 12q^{84}$$ $$\mathstrut -\mathstrut 36q^{85}$$ $$\mathstrut +\mathstrut 24q^{86}$$ $$\mathstrut -\mathstrut 144q^{87}$$ $$\mathstrut -\mathstrut 24q^{88}$$ $$\mathstrut -\mathstrut 378q^{89}$$ $$\mathstrut +\mathstrut 24q^{91}$$ $$\mathstrut -\mathstrut 120q^{92}$$ $$\mathstrut -\mathstrut 138q^{93}$$ $$\mathstrut -\mathstrut 12q^{94}$$ $$\mathstrut -\mathstrut 30q^{95}$$ $$\mathstrut -\mathstrut 48q^{96}$$ $$\mathstrut -\mathstrut 120q^{98}$$ $$\mathstrut +\mathstrut 144q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut +\mathstrut$$ $$2$$ $$x^{2}\mathstrut +\mathstrut$$ $$4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2$$ $$\beta_{3}$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/14\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−0.707107 1.22474i 0.621320 + 0.358719i −1.00000 + 1.73205i −5.74264 + 3.31552i 1.01461i 6.24264 3.16693i 2.82843 −4.24264 7.34847i 8.12132 + 4.68885i
3.2 0.707107 + 1.22474i −3.62132 2.09077i −1.00000 + 1.73205i 2.74264 1.58346i 5.91359i −2.24264 + 6.63103i −2.82843 4.24264 + 7.34847i 3.87868 + 2.23936i
5.1 −0.707107 + 1.22474i 0.621320 0.358719i −1.00000 1.73205i −5.74264 3.31552i 1.01461i 6.24264 + 3.16693i 2.82843 −4.24264 + 7.34847i 8.12132 4.68885i
5.2 0.707107 1.22474i −3.62132 + 2.09077i −1.00000 1.73205i 2.74264 + 1.58346i 5.91359i −2.24264 6.63103i −2.82843 4.24264 7.34847i 3.87868 2.23936i
 $$n$$: e.g. 2-40 or 990-1000 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
7.d Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{3}^{\mathrm{new}}(14, [\chi])$$.