# Properties

 Label 33.2.e.a Level 33 Weight 2 Character orbit 33.e Analytic conductor 0.264 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$33 = 3 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 33.e (of order $$5$$ and degree $$4$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.26350632667$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$\zeta_{10}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.809017 + 0.587785i −0.309017 − 0.951057i 0.809017 − 0.587785i −0.309017 + 0.951057i
−1.30902 0.951057i 0.309017 0.951057i 0.190983 + 0.587785i 0.309017 0.224514i −1.30902 + 0.951057i 0.927051 + 2.85317i −0.690983 + 2.12663i −0.809017 0.587785i −0.618034
16.1 −0.190983 0.587785i −0.809017 0.587785i 1.30902 0.951057i −0.809017 + 2.48990i −0.190983 + 0.587785i −2.42705 + 1.76336i −1.80902 1.31433i 0.309017 + 0.951057i 1.61803
25.1 −1.30902 + 0.951057i 0.309017 + 0.951057i 0.190983 0.587785i 0.309017 + 0.224514i −1.30902 0.951057i 0.927051 2.85317i −0.690983 2.12663i −0.809017 + 0.587785i −0.618034
31.1 −0.190983 + 0.587785i −0.809017 + 0.587785i 1.30902 + 0.951057i −0.809017 2.48990i −0.190983 0.587785i −2.42705 1.76336i −1.80902 + 1.31433i 0.309017 0.951057i 1.61803
 $$n$$: 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
11.c Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{4}$$ $$\mathstrut +\mathstrut 3 T_{2}^{3}$$ $$\mathstrut +\mathstrut 4 T_{2}^{2}$$ $$\mathstrut +\mathstrut 2 T_{2}$$ $$\mathstrut +\mathstrut 1$$ acting on $$S_{2}^{\mathrm{new}}(33, [\chi])$$.