# Properties

 Label 15.3.f.a Level 15 Weight 3 Character orbit 15.f Analytic conductor 0.409 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$15 = 3 \cdot 5$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 15.f (of order $$4$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.40872039654$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -1 + \beta_{1} - \beta_{2} ) q^{2}$$ $$+ \beta_{3} q^{3}$$ $$+ ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4}$$ $$+ ( -1 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{5}$$ $$+ ( -3 + \beta_{1} - \beta_{3} ) q^{6}$$ $$+ ( 1 + 2 \beta_{1} + \beta_{2} ) q^{7}$$ $$+ ( 3 - 3 \beta_{2} + \beta_{3} ) q^{8}$$ $$-3 \beta_{2} q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -1 + \beta_{1} - \beta_{2} ) q^{2}$$ $$+ \beta_{3} q^{3}$$ $$+ ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4}$$ $$+ ( -1 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{5}$$ $$+ ( -3 + \beta_{1} - \beta_{3} ) q^{6}$$ $$+ ( 1 + 2 \beta_{1} + \beta_{2} ) q^{7}$$ $$+ ( 3 - 3 \beta_{2} + \beta_{3} ) q^{8}$$ $$-3 \beta_{2} q^{9}$$ $$+ ( 1 + 2 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{10}$$ $$+ ( 4 + 3 \beta_{1} - 3 \beta_{3} ) q^{11}$$ $$+ ( 6 - \beta_{1} + 6 \beta_{2} ) q^{12}$$ $$+ ( -8 + 8 \beta_{2} - 2 \beta_{3} ) q^{13}$$ $$+ ( -\beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{14}$$ $$+ ( 6 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{15}$$ $$+ ( -5 - 4 \beta_{1} + 4 \beta_{3} ) q^{16}$$ $$+ ( -10 - 6 \beta_{1} - 10 \beta_{2} ) q^{17}$$ $$+ ( -3 + 3 \beta_{2} - 3 \beta_{3} ) q^{18}$$ $$+ ( 6 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{19}$$ $$+ ( -9 + 7 \beta_{1} + 17 \beta_{2} - 6 \beta_{3} ) q^{20}$$ $$+ ( -6 - \beta_{1} + \beta_{3} ) q^{21}$$ $$+ ( 5 - 2 \beta_{1} + 5 \beta_{2} ) q^{22}$$ $$+ ( 14 - 14 \beta_{2} + 2 \beta_{3} ) q^{23}$$ $$+ ( 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{24}$$ $$+ ( 4 - 2 \beta_{1} + 3 \beta_{2} - 14 \beta_{3} ) q^{25}$$ $$+ ( 22 - 10 \beta_{1} + 10 \beta_{3} ) q^{26}$$ $$+ 3 \beta_{1} q^{27}$$ $$+ ( 11 - 11 \beta_{2} - 2 \beta_{3} ) q^{28}$$ $$+ ( 7 \beta_{1} + 18 \beta_{2} + 7 \beta_{3} ) q^{29}$$ $$+ ( -6 + 8 \beta_{1} - 12 \beta_{2} + \beta_{3} ) q^{30}$$ $$+ ( -4 + 6 \beta_{1} - 6 \beta_{3} ) q^{31}$$ $$+ ( -19 + 7 \beta_{1} - 19 \beta_{2} ) q^{32}$$ $$+ ( -9 + 9 \beta_{2} + 4 \beta_{3} ) q^{33}$$ $$+ ( -4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{34}$$ $$+ ( -10 - 5 \beta_{1} - 10 \beta_{2} + 5 \beta_{3} ) q^{35}$$ $$+ ( 3 - 6 \beta_{1} + 6 \beta_{3} ) q^{36}$$ $$+ ( 16 - 18 \beta_{1} + 16 \beta_{2} ) q^{37}$$ $$+ ( -24 + 24 \beta_{2} - 18 \beta_{3} ) q^{38}$$ $$+ ( -8 \beta_{1} + 6 \beta_{2} - 8 \beta_{3} ) q^{39}$$ $$+ ( 12 - 6 \beta_{1} + 9 \beta_{2} + 8 \beta_{3} ) q^{40}$$ $$+ ( -14 + 6 \beta_{1} - 6 \beta_{3} ) q^{41}$$ $$+ ( 3 - 4 \beta_{1} + 3 \beta_{2} ) q^{42}$$ $$+ ( -2 + 2 \beta_{2} + 20 \beta_{3} ) q^{43}$$ $$+ ( -5 \beta_{1} - 32 \beta_{2} - 5 \beta_{3} ) q^{44}$$ $$+ ( 9 + 3 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{45}$$ $$+ ( -34 + 16 \beta_{1} - 16 \beta_{3} ) q^{46}$$ $$+ ( 32 + 10 \beta_{1} + 32 \beta_{2} ) q^{47}$$ $$+ ( 12 - 12 \beta_{2} - 5 \beta_{3} ) q^{48}$$ $$+ ( 4 \beta_{1} - 35 \beta_{2} + 4 \beta_{3} ) q^{49}$$ $$+ ( 41 - 8 \beta_{1} - 13 \beta_{2} + 19 \beta_{3} ) q^{50}$$ $$+ ( 18 + 10 \beta_{1} - 10 \beta_{3} ) q^{51}$$ $$+ ( -20 + 34 \beta_{1} - 20 \beta_{2} ) q^{52}$$ $$+ ( 14 - 14 \beta_{2} + 12 \beta_{3} ) q^{53}$$ $$+ ( -3 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} ) q^{54}$$ $$+ ( -31 - 2 \beta_{1} + 3 \beta_{2} + 16 \beta_{3} ) q^{55}$$ $$+ ( 5 \beta_{1} - 5 \beta_{3} ) q^{56}$$ $$+ ( -18 + 6 \beta_{1} - 18 \beta_{2} ) q^{57}$$ $$+ ( -3 + 3 \beta_{2} + 4 \beta_{3} ) q^{58}$$ $$+ ( -31 \beta_{1} + 36 \beta_{2} - 31 \beta_{3} ) q^{59}$$ $$+ ( -21 - 17 \beta_{1} + 18 \beta_{2} - 9 \beta_{3} ) q^{60}$$ $$+ ( 50 - 18 \beta_{1} + 18 \beta_{3} ) q^{61}$$ $$+ ( 22 - 16 \beta_{1} + 22 \beta_{2} ) q^{62}$$ $$+ ( 3 - 3 \beta_{2} - 6 \beta_{3} ) q^{63}$$ $$+ ( -10 \beta_{1} + 79 \beta_{2} - 10 \beta_{3} ) q^{64}$$ $$+ ( -28 + 14 \beta_{1} - 26 \beta_{2} - 22 \beta_{3} ) q^{65}$$ $$+ ( 6 - 5 \beta_{1} + 5 \beta_{3} ) q^{66}$$ $$+ ( -50 + 4 \beta_{1} - 50 \beta_{2} ) q^{67}$$ $$+ ( -26 + 26 \beta_{2} + 34 \beta_{3} ) q^{68}$$ $$+ ( 14 \beta_{1} - 6 \beta_{2} + 14 \beta_{3} ) q^{69}$$ $$+ ( -15 + 5 \beta_{2} - 10 \beta_{3} ) q^{70}$$ $$-68 q^{71}$$ $$+ ( -9 + 3 \beta_{1} - 9 \beta_{2} ) q^{72}$$ $$+ ( 19 - 19 \beta_{2} - 48 \beta_{3} ) q^{73}$$ $$+ ( 34 \beta_{1} - 86 \beta_{2} + 34 \beta_{3} ) q^{74}$$ $$+ ( 6 - 3 \beta_{1} + 42 \beta_{2} + 4 \beta_{3} ) q^{75}$$ $$+ ( 78 - 18 \beta_{1} + 18 \beta_{3} ) q^{76}$$ $$+ ( 22 + 14 \beta_{1} + 22 \beta_{2} ) q^{77}$$ $$+ ( 30 - 30 \beta_{2} + 22 \beta_{3} ) q^{78}$$ $$+ ( -10 \beta_{1} - 10 \beta_{3} ) q^{79}$$ $$+ ( 41 + 2 \beta_{1} - 3 \beta_{2} - 21 \beta_{3} ) q^{80}$$ $$-9 q^{81}$$ $$+ ( 32 - 26 \beta_{1} + 32 \beta_{2} ) q^{82}$$ $$+ ( -4 + 4 \beta_{2} - 14 \beta_{3} ) q^{83}$$ $$+ ( 11 \beta_{1} + 6 \beta_{2} + 11 \beta_{3} ) q^{84}$$ $$+ ( 58 + 36 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} ) q^{85}$$ $$+ ( -56 + 18 \beta_{1} - 18 \beta_{3} ) q^{86}$$ $$+ ( -21 - 18 \beta_{1} - 21 \beta_{2} ) q^{87}$$ $$+ ( 3 - 3 \beta_{2} - 14 \beta_{3} ) q^{88}$$ $$+ ( 36 \beta_{1} - 6 \beta_{2} + 36 \beta_{3} ) q^{89}$$ $$+ ( -24 + 12 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} ) q^{90}$$ $$+ ( -4 - 14 \beta_{1} + 14 \beta_{3} ) q^{91}$$ $$+ ( 26 - 58 \beta_{1} + 26 \beta_{2} ) q^{92}$$ $$+ ( -18 + 18 \beta_{2} - 4 \beta_{3} ) q^{93}$$ $$+ ( 22 \beta_{1} - 34 \beta_{2} + 22 \beta_{3} ) q^{94}$$ $$+ ( 36 - 18 \beta_{1} - 48 \beta_{2} + 24 \beta_{3} ) q^{95}$$ $$+ ( -21 + 19 \beta_{1} - 19 \beta_{3} ) q^{96}$$ $$+ ( -5 - 16 \beta_{1} - 5 \beta_{2} ) q^{97}$$ $$+ ( -47 + 47 \beta_{2} - 43 \beta_{3} ) q^{98}$$ $$+ ( -9 \beta_{1} - 12 \beta_{2} - 9 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 4q^{2}$$ $$\mathstrut -\mathstrut 4q^{5}$$ $$\mathstrut -\mathstrut 12q^{6}$$ $$\mathstrut +\mathstrut 4q^{7}$$ $$\mathstrut +\mathstrut 12q^{8}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 4q^{2}$$ $$\mathstrut -\mathstrut 4q^{5}$$ $$\mathstrut -\mathstrut 12q^{6}$$ $$\mathstrut +\mathstrut 4q^{7}$$ $$\mathstrut +\mathstrut 12q^{8}$$ $$\mathstrut +\mathstrut 4q^{10}$$ $$\mathstrut +\mathstrut 16q^{11}$$ $$\mathstrut +\mathstrut 24q^{12}$$ $$\mathstrut -\mathstrut 32q^{13}$$ $$\mathstrut +\mathstrut 24q^{15}$$ $$\mathstrut -\mathstrut 20q^{16}$$ $$\mathstrut -\mathstrut 40q^{17}$$ $$\mathstrut -\mathstrut 12q^{18}$$ $$\mathstrut -\mathstrut 36q^{20}$$ $$\mathstrut -\mathstrut 24q^{21}$$ $$\mathstrut +\mathstrut 20q^{22}$$ $$\mathstrut +\mathstrut 56q^{23}$$ $$\mathstrut +\mathstrut 16q^{25}$$ $$\mathstrut +\mathstrut 88q^{26}$$ $$\mathstrut +\mathstrut 44q^{28}$$ $$\mathstrut -\mathstrut 24q^{30}$$ $$\mathstrut -\mathstrut 16q^{31}$$ $$\mathstrut -\mathstrut 76q^{32}$$ $$\mathstrut -\mathstrut 36q^{33}$$ $$\mathstrut -\mathstrut 40q^{35}$$ $$\mathstrut +\mathstrut 12q^{36}$$ $$\mathstrut +\mathstrut 64q^{37}$$ $$\mathstrut -\mathstrut 96q^{38}$$ $$\mathstrut +\mathstrut 48q^{40}$$ $$\mathstrut -\mathstrut 56q^{41}$$ $$\mathstrut +\mathstrut 12q^{42}$$ $$\mathstrut -\mathstrut 8q^{43}$$ $$\mathstrut +\mathstrut 36q^{45}$$ $$\mathstrut -\mathstrut 136q^{46}$$ $$\mathstrut +\mathstrut 128q^{47}$$ $$\mathstrut +\mathstrut 48q^{48}$$ $$\mathstrut +\mathstrut 164q^{50}$$ $$\mathstrut +\mathstrut 72q^{51}$$ $$\mathstrut -\mathstrut 80q^{52}$$ $$\mathstrut +\mathstrut 56q^{53}$$ $$\mathstrut -\mathstrut 124q^{55}$$ $$\mathstrut -\mathstrut 72q^{57}$$ $$\mathstrut -\mathstrut 12q^{58}$$ $$\mathstrut -\mathstrut 84q^{60}$$ $$\mathstrut +\mathstrut 200q^{61}$$ $$\mathstrut +\mathstrut 88q^{62}$$ $$\mathstrut +\mathstrut 12q^{63}$$ $$\mathstrut -\mathstrut 112q^{65}$$ $$\mathstrut +\mathstrut 24q^{66}$$ $$\mathstrut -\mathstrut 200q^{67}$$ $$\mathstrut -\mathstrut 104q^{68}$$ $$\mathstrut -\mathstrut 60q^{70}$$ $$\mathstrut -\mathstrut 272q^{71}$$ $$\mathstrut -\mathstrut 36q^{72}$$ $$\mathstrut +\mathstrut 76q^{73}$$ $$\mathstrut +\mathstrut 24q^{75}$$ $$\mathstrut +\mathstrut 312q^{76}$$ $$\mathstrut +\mathstrut 88q^{77}$$ $$\mathstrut +\mathstrut 120q^{78}$$ $$\mathstrut +\mathstrut 164q^{80}$$ $$\mathstrut -\mathstrut 36q^{81}$$ $$\mathstrut +\mathstrut 128q^{82}$$ $$\mathstrut -\mathstrut 16q^{83}$$ $$\mathstrut +\mathstrut 232q^{85}$$ $$\mathstrut -\mathstrut 224q^{86}$$ $$\mathstrut -\mathstrut 84q^{87}$$ $$\mathstrut +\mathstrut 12q^{88}$$ $$\mathstrut -\mathstrut 96q^{90}$$ $$\mathstrut -\mathstrut 16q^{91}$$ $$\mathstrut +\mathstrut 104q^{92}$$ $$\mathstrut -\mathstrut 72q^{93}$$ $$\mathstrut +\mathstrut 144q^{95}$$ $$\mathstrut -\mathstrut 84q^{96}$$ $$\mathstrut -\mathstrut 20q^{97}$$ $$\mathstrut -\mathstrut 188q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut +\mathstrut$$ $$9$$:

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

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
−2.22474 2.22474i 1.22474 1.22474i 5.89898i 2.67423 + 4.22474i −5.44949 −1.44949 1.44949i 4.22474 4.22474i 3.00000i 3.44949 15.3485i
7.2 0.224745 + 0.224745i −1.22474 + 1.22474i 3.89898i −4.67423 + 1.77526i −0.550510 3.44949 + 3.44949i 1.77526 1.77526i 3.00000i −1.44949 0.651531i
13.1 −2.22474 + 2.22474i 1.22474 + 1.22474i 5.89898i 2.67423 4.22474i −5.44949 −1.44949 + 1.44949i 4.22474 + 4.22474i 3.00000i 3.44949 + 15.3485i
13.2 0.224745 0.224745i −1.22474 1.22474i 3.89898i −4.67423 1.77526i −0.550510 3.44949 3.44949i 1.77526 + 1.77526i 3.00000i −1.44949 + 0.651531i
 $$n$$: 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
5.c Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{3}^{\mathrm{new}}(15, [\chi])$$.