# Properties

 Label 19.4.c.a Level 19 Weight 4 Character orbit 19.c Analytic conductor 1.121 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$19$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 19.c (of order $$3$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$1.12103629011$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{55})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{2} q^{2}$$ $$+ ( \beta_{1} + \beta_{3} ) q^{3}$$ $$+ ( 7 + 7 \beta_{2} ) q^{4}$$ $$+ ( -\beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{5}$$ $$-\beta_{1} q^{6}$$ $$+ ( -7 - \beta_{3} ) q^{7}$$ $$-15 q^{8}$$ $$+ ( -28 - 28 \beta_{2} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{2} q^{2}$$ $$+ ( \beta_{1} + \beta_{3} ) q^{3}$$ $$+ ( 7 + 7 \beta_{2} ) q^{4}$$ $$+ ( -\beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{5}$$ $$-\beta_{1} q^{6}$$ $$+ ( -7 - \beta_{3} ) q^{7}$$ $$-15 q^{8}$$ $$+ ( -28 - 28 \beta_{2} ) q^{9}$$ $$+ ( 7 + \beta_{1} + 7 \beta_{2} ) q^{10}$$ $$+ ( 14 - 7 \beta_{3} ) q^{11}$$ $$+ 7 \beta_{3} q^{12}$$ $$+ ( 14 - 8 \beta_{1} + 14 \beta_{2} ) q^{13}$$ $$+ ( \beta_{1} - 7 \beta_{2} + \beta_{3} ) q^{14}$$ $$+ ( 55 + 7 \beta_{1} + 55 \beta_{2} ) q^{15}$$ $$+ 41 \beta_{2} q^{16}$$ $$+ ( 8 \beta_{1} - 56 \beta_{2} + 8 \beta_{3} ) q^{17}$$ $$+ 28 q^{18}$$ $$+ ( -70 + 7 \beta_{1} - 42 \beta_{2} + 8 \beta_{3} ) q^{19}$$ $$+ ( 49 - 7 \beta_{3} ) q^{20}$$ $$+ ( -7 \beta_{1} + 55 \beta_{2} - 7 \beta_{3} ) q^{21}$$ $$+ ( 7 \beta_{1} + 14 \beta_{2} + 7 \beta_{3} ) q^{22}$$ $$+ ( -57 - 7 \beta_{1} - 57 \beta_{2} ) q^{23}$$ $$+ ( -15 \beta_{1} - 15 \beta_{3} ) q^{24}$$ $$+ ( 21 - 14 \beta_{1} + 21 \beta_{2} ) q^{25}$$ $$+ ( -14 - 8 \beta_{3} ) q^{26}$$ $$-\beta_{3} q^{27}$$ $$+ ( -49 + 7 \beta_{1} - 49 \beta_{2} ) q^{28}$$ $$+ ( -111 + 7 \beta_{1} - 111 \beta_{2} ) q^{29}$$ $$+ ( -55 + 7 \beta_{3} ) q^{30}$$ $$+ ( 133 - 7 \beta_{3} ) q^{31}$$ $$+ ( -161 - 161 \beta_{2} ) q^{32}$$ $$+ ( 14 \beta_{1} + 385 \beta_{2} + 14 \beta_{3} ) q^{33}$$ $$+ ( 56 - 8 \beta_{1} + 56 \beta_{2} ) q^{34}$$ $$-6 \beta_{2} q^{35}$$ $$-196 \beta_{2} q^{36}$$ $$+ ( 91 - 7 \beta_{3} ) q^{37}$$ $$+ ( 42 - 8 \beta_{1} - 28 \beta_{2} - \beta_{3} ) q^{38}$$ $$+ ( 440 + 14 \beta_{3} ) q^{39}$$ $$+ ( 15 \beta_{1} + 105 \beta_{2} + 15 \beta_{3} ) q^{40}$$ $$+ ( -34 \beta_{1} + 77 \beta_{2} - 34 \beta_{3} ) q^{41}$$ $$+ ( -55 + 7 \beta_{1} - 55 \beta_{2} ) q^{42}$$ $$+ ( 42 \beta_{1} + 134 \beta_{2} + 42 \beta_{3} ) q^{43}$$ $$+ ( 98 + 49 \beta_{1} + 98 \beta_{2} ) q^{44}$$ $$+ ( -196 + 28 \beta_{3} ) q^{45}$$ $$+ ( 57 - 7 \beta_{3} ) q^{46}$$ $$+ ( -63 - 15 \beta_{1} - 63 \beta_{2} ) q^{47}$$ $$-41 \beta_{1} q^{48}$$ $$+ ( -239 + 14 \beta_{3} ) q^{49}$$ $$+ ( -21 - 14 \beta_{3} ) q^{50}$$ $$+ ( -440 + 56 \beta_{1} - 440 \beta_{2} ) q^{51}$$ $$+ ( -56 \beta_{1} + 98 \beta_{2} - 56 \beta_{3} ) q^{52}$$ $$+ ( 442 - 28 \beta_{1} + 442 \beta_{2} ) q^{53}$$ $$+ ( \beta_{1} + \beta_{3} ) q^{54}$$ $$+ ( -63 \beta_{1} - 483 \beta_{2} - 63 \beta_{3} ) q^{55}$$ $$+ ( 105 + 15 \beta_{3} ) q^{56}$$ $$+ ( -385 - 28 \beta_{1} - 440 \beta_{2} - 70 \beta_{3} ) q^{57}$$ $$+ ( 111 + 7 \beta_{3} ) q^{58}$$ $$+ ( 13 \beta_{1} + 56 \beta_{2} + 13 \beta_{3} ) q^{59}$$ $$+ ( 49 \beta_{1} + 385 \beta_{2} + 49 \beta_{3} ) q^{60}$$ $$+ ( -273 + 29 \beta_{1} - 273 \beta_{2} ) q^{61}$$ $$+ ( 7 \beta_{1} + 133 \beta_{2} + 7 \beta_{3} ) q^{62}$$ $$+ ( 196 - 28 \beta_{1} + 196 \beta_{2} ) q^{63}$$ $$-167 q^{64}$$ $$+ ( -342 + 42 \beta_{3} ) q^{65}$$ $$+ ( -385 - 14 \beta_{1} - 385 \beta_{2} ) q^{66}$$ $$+ ( 370 + 63 \beta_{1} + 370 \beta_{2} ) q^{67}$$ $$+ ( 392 + 56 \beta_{3} ) q^{68}$$ $$+ ( 385 - 57 \beta_{3} ) q^{69}$$ $$+ ( 6 + 6 \beta_{2} ) q^{70}$$ $$+ ( 42 \beta_{1} + 216 \beta_{2} + 42 \beta_{3} ) q^{71}$$ $$+ ( 420 + 420 \beta_{2} ) q^{72}$$ $$+ ( 6 \beta_{1} + 175 \beta_{2} + 6 \beta_{3} ) q^{73}$$ $$+ ( 7 \beta_{1} + 91 \beta_{2} + 7 \beta_{3} ) q^{74}$$ $$+ ( 770 + 21 \beta_{3} ) q^{75}$$ $$+ ( -196 - 7 \beta_{1} - 490 \beta_{2} + 49 \beta_{3} ) q^{76}$$ $$+ ( 287 + 35 \beta_{3} ) q^{77}$$ $$+ ( -14 \beta_{1} + 440 \beta_{2} - 14 \beta_{3} ) q^{78}$$ $$+ ( -56 \beta_{1} - 76 \beta_{2} - 56 \beta_{3} ) q^{79}$$ $$+ ( 287 + 41 \beta_{1} + 287 \beta_{2} ) q^{80}$$ $$-701 \beta_{2} q^{81}$$ $$+ ( -77 + 34 \beta_{1} - 77 \beta_{2} ) q^{82}$$ $$+ ( -952 - 35 \beta_{3} ) q^{83}$$ $$+ ( -385 - 49 \beta_{3} ) q^{84}$$ $$+ ( 48 + 48 \beta_{2} ) q^{85}$$ $$+ ( -134 - 42 \beta_{1} - 134 \beta_{2} ) q^{86}$$ $$+ ( -385 - 111 \beta_{3} ) q^{87}$$ $$+ ( -210 + 105 \beta_{3} ) q^{88}$$ $$+ ( -56 - 104 \beta_{1} - 56 \beta_{2} ) q^{89}$$ $$+ ( -28 \beta_{1} - 196 \beta_{2} - 28 \beta_{3} ) q^{90}$$ $$+ ( -538 + 70 \beta_{1} - 538 \beta_{2} ) q^{91}$$ $$+ ( -49 \beta_{1} - 399 \beta_{2} - 49 \beta_{3} ) q^{92}$$ $$+ ( 133 \beta_{1} + 385 \beta_{2} + 133 \beta_{3} ) q^{93}$$ $$+ ( 63 - 15 \beta_{3} ) q^{94}$$ $$+ ( 91 + 84 \beta_{1} + 636 \beta_{2} + 77 \beta_{3} ) q^{95}$$ $$-161 \beta_{3} q^{96}$$ $$+ ( -62 \beta_{1} - 273 \beta_{2} - 62 \beta_{3} ) q^{97}$$ $$+ ( -14 \beta_{1} - 239 \beta_{2} - 14 \beta_{3} ) q^{98}$$ $$+ ( -392 - 196 \beta_{1} - 392 \beta_{2} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 2q^{2}$$ $$\mathstrut +\mathstrut 14q^{4}$$ $$\mathstrut +\mathstrut 14q^{5}$$ $$\mathstrut -\mathstrut 28q^{7}$$ $$\mathstrut -\mathstrut 60q^{8}$$ $$\mathstrut -\mathstrut 56q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 2q^{2}$$ $$\mathstrut +\mathstrut 14q^{4}$$ $$\mathstrut +\mathstrut 14q^{5}$$ $$\mathstrut -\mathstrut 28q^{7}$$ $$\mathstrut -\mathstrut 60q^{8}$$ $$\mathstrut -\mathstrut 56q^{9}$$ $$\mathstrut +\mathstrut 14q^{10}$$ $$\mathstrut +\mathstrut 56q^{11}$$ $$\mathstrut +\mathstrut 28q^{13}$$ $$\mathstrut +\mathstrut 14q^{14}$$ $$\mathstrut +\mathstrut 110q^{15}$$ $$\mathstrut -\mathstrut 82q^{16}$$ $$\mathstrut +\mathstrut 112q^{17}$$ $$\mathstrut +\mathstrut 112q^{18}$$ $$\mathstrut -\mathstrut 196q^{19}$$ $$\mathstrut +\mathstrut 196q^{20}$$ $$\mathstrut -\mathstrut 110q^{21}$$ $$\mathstrut -\mathstrut 28q^{22}$$ $$\mathstrut -\mathstrut 114q^{23}$$ $$\mathstrut +\mathstrut 42q^{25}$$ $$\mathstrut -\mathstrut 56q^{26}$$ $$\mathstrut -\mathstrut 98q^{28}$$ $$\mathstrut -\mathstrut 222q^{29}$$ $$\mathstrut -\mathstrut 220q^{30}$$ $$\mathstrut +\mathstrut 532q^{31}$$ $$\mathstrut -\mathstrut 322q^{32}$$ $$\mathstrut -\mathstrut 770q^{33}$$ $$\mathstrut +\mathstrut 112q^{34}$$ $$\mathstrut +\mathstrut 12q^{35}$$ $$\mathstrut +\mathstrut 392q^{36}$$ $$\mathstrut +\mathstrut 364q^{37}$$ $$\mathstrut +\mathstrut 224q^{38}$$ $$\mathstrut +\mathstrut 1760q^{39}$$ $$\mathstrut -\mathstrut 210q^{40}$$ $$\mathstrut -\mathstrut 154q^{41}$$ $$\mathstrut -\mathstrut 110q^{42}$$ $$\mathstrut -\mathstrut 268q^{43}$$ $$\mathstrut +\mathstrut 196q^{44}$$ $$\mathstrut -\mathstrut 784q^{45}$$ $$\mathstrut +\mathstrut 228q^{46}$$ $$\mathstrut -\mathstrut 126q^{47}$$ $$\mathstrut -\mathstrut 956q^{49}$$ $$\mathstrut -\mathstrut 84q^{50}$$ $$\mathstrut -\mathstrut 880q^{51}$$ $$\mathstrut -\mathstrut 196q^{52}$$ $$\mathstrut +\mathstrut 884q^{53}$$ $$\mathstrut +\mathstrut 966q^{55}$$ $$\mathstrut +\mathstrut 420q^{56}$$ $$\mathstrut -\mathstrut 660q^{57}$$ $$\mathstrut +\mathstrut 444q^{58}$$ $$\mathstrut -\mathstrut 112q^{59}$$ $$\mathstrut -\mathstrut 770q^{60}$$ $$\mathstrut -\mathstrut 546q^{61}$$ $$\mathstrut -\mathstrut 266q^{62}$$ $$\mathstrut +\mathstrut 392q^{63}$$ $$\mathstrut -\mathstrut 668q^{64}$$ $$\mathstrut -\mathstrut 1368q^{65}$$ $$\mathstrut -\mathstrut 770q^{66}$$ $$\mathstrut +\mathstrut 740q^{67}$$ $$\mathstrut +\mathstrut 1568q^{68}$$ $$\mathstrut +\mathstrut 1540q^{69}$$ $$\mathstrut +\mathstrut 12q^{70}$$ $$\mathstrut -\mathstrut 432q^{71}$$ $$\mathstrut +\mathstrut 840q^{72}$$ $$\mathstrut -\mathstrut 350q^{73}$$ $$\mathstrut -\mathstrut 182q^{74}$$ $$\mathstrut +\mathstrut 3080q^{75}$$ $$\mathstrut +\mathstrut 196q^{76}$$ $$\mathstrut +\mathstrut 1148q^{77}$$ $$\mathstrut -\mathstrut 880q^{78}$$ $$\mathstrut +\mathstrut 152q^{79}$$ $$\mathstrut +\mathstrut 574q^{80}$$ $$\mathstrut +\mathstrut 1402q^{81}$$ $$\mathstrut -\mathstrut 154q^{82}$$ $$\mathstrut -\mathstrut 3808q^{83}$$ $$\mathstrut -\mathstrut 1540q^{84}$$ $$\mathstrut +\mathstrut 96q^{85}$$ $$\mathstrut -\mathstrut 268q^{86}$$ $$\mathstrut -\mathstrut 1540q^{87}$$ $$\mathstrut -\mathstrut 840q^{88}$$ $$\mathstrut -\mathstrut 112q^{89}$$ $$\mathstrut +\mathstrut 392q^{90}$$ $$\mathstrut -\mathstrut 1076q^{91}$$ $$\mathstrut +\mathstrut 798q^{92}$$ $$\mathstrut -\mathstrut 770q^{93}$$ $$\mathstrut +\mathstrut 252q^{94}$$ $$\mathstrut -\mathstrut 908q^{95}$$ $$\mathstrut +\mathstrut 546q^{97}$$ $$\mathstrut +\mathstrut 478q^{98}$$ $$\mathstrut -\mathstrut 784q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut +\mathstrut$$ $$55$$ $$x^{2}\mathstrut +\mathstrut$$ $$3025$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/55$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/55$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$55$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$55$$ $$\beta_{3}$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/19\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 3.70810 − 6.42262i −3.70810 + 6.42262i 3.70810 + 6.42262i −3.70810 − 6.42262i
−0.500000 0.866025i −3.70810 6.42262i 3.50000 6.06218i 7.20810 + 12.4848i −3.70810 + 6.42262i 0.416198 −15.0000 −14.0000 + 24.2487i 7.20810 12.4848i
7.2 −0.500000 0.866025i 3.70810 + 6.42262i 3.50000 6.06218i −0.208099 0.360438i 3.70810 6.42262i −14.4162 −15.0000 −14.0000 + 24.2487i −0.208099 + 0.360438i
11.1 −0.500000 + 0.866025i −3.70810 + 6.42262i 3.50000 + 6.06218i 7.20810 12.4848i −3.70810 6.42262i 0.416198 −15.0000 −14.0000 24.2487i 7.20810 + 12.4848i
11.2 −0.500000 + 0.866025i 3.70810 6.42262i 3.50000 + 6.06218i −0.208099 + 0.360438i 3.70810 + 6.42262i −14.4162 −15.0000 −14.0000 24.2487i −0.208099 0.360438i
 $$n$$: 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
19.c Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{2}$$ $$\mathstrut +\mathstrut T_{2}$$ $$\mathstrut +\mathstrut 1$$ acting on $$S_{4}^{\mathrm{new}}(19, [\chi])$$.