# Properties

 Label 330.2.m.b Level $330$ Weight $2$ Character orbit 330.m Analytic conductor $2.635$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$330 = 2 \cdot 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 330.m (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.63506326670$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/330\mathbb{Z}\right)^\times$$.

 $$n$$ $$67$$ $$211$$ $$221$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 −0.309017 − 0.951057i 0.809017 + 0.587785i −0.309017 + 0.951057i 0.809017 − 0.587785i
0.309017 0.951057i 0.809017 0.587785i −0.809017 0.587785i −0.309017 0.951057i −0.309017 0.951057i −1.00000 0.726543i −0.809017 + 0.587785i 0.309017 0.951057i −1.00000
91.1 −0.809017 + 0.587785i −0.309017 0.951057i 0.309017 0.951057i 0.809017 + 0.587785i 0.809017 + 0.587785i −1.00000 + 3.07768i 0.309017 + 0.951057i −0.809017 + 0.587785i −1.00000
181.1 0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i −0.309017 + 0.951057i −0.309017 + 0.951057i −1.00000 + 0.726543i −0.809017 0.587785i 0.309017 + 0.951057i −1.00000
301.1 −0.809017 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i 0.809017 0.587785i 0.809017 0.587785i −1.00000 3.07768i 0.309017 0.951057i −0.809017 0.587785i −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 330.2.m.b 4
3.b odd 2 1 990.2.n.e 4
11.c even 5 1 inner 330.2.m.b 4
11.c even 5 1 3630.2.a.bj 2
11.d odd 10 1 3630.2.a.bb 2
33.h odd 10 1 990.2.n.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.m.b 4 1.a even 1 1 trivial
330.2.m.b 4 11.c even 5 1 inner
990.2.n.e 4 3.b odd 2 1
990.2.n.e 4 33.h odd 10 1
3630.2.a.bb 2 11.d odd 10 1
3630.2.a.bj 2 11.c even 5 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 4 T_{7}^{3} + 16 T_{7}^{2} + 24 T_{7} + 16$$ acting on $$S_{2}^{\mathrm{new}}(330, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$3$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$5$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$7$ $$16 + 24 T + 16 T^{2} + 4 T^{3} + T^{4}$$
$11$ $$121 - 11 T - 9 T^{2} - T^{3} + T^{4}$$
$13$ $$361 + 133 T + 24 T^{2} + 2 T^{3} + T^{4}$$
$17$ $$961 - 372 T + 94 T^{2} - 13 T^{3} + T^{4}$$
$19$ $$16 - 16 T + 16 T^{2} - 6 T^{3} + T^{4}$$
$23$ $$( 1 - 7 T + T^{2} )^{2}$$
$29$ $$1 - 4 T + 6 T^{2} + T^{3} + T^{4}$$
$31$ $$25 - 25 T + 10 T^{2} + T^{4}$$
$37$ $$3025 - 275 T + 60 T^{2} - 10 T^{3} + T^{4}$$
$41$ $$16 - 16 T + 16 T^{2} - 6 T^{3} + T^{4}$$
$43$ $$( -1 - 11 T + T^{2} )^{2}$$
$47$ $$2401 + 686 T + 196 T^{2} + 21 T^{3} + T^{4}$$
$53$ $$16 + 24 T + 76 T^{2} + 14 T^{3} + T^{4}$$
$59$ $$1681 + 861 T + 166 T^{2} - 4 T^{3} + T^{4}$$
$61$ $$T^{4}$$
$67$ $$( -31 - T + T^{2} )^{2}$$
$71$ $$256 - 64 T + 96 T^{2} + 16 T^{3} + T^{4}$$
$73$ $$32400 - 5400 T + 540 T^{2} - 30 T^{3} + T^{4}$$
$79$ $$841 - 406 T + 96 T^{2} - 11 T^{3} + T^{4}$$
$83$ $$256 + 64 T + 16 T^{2} + 4 T^{3} + T^{4}$$
$89$ $$( 220 - 30 T + T^{2} )^{2}$$
$97$ $$55696 + 7552 T + 744 T^{2} + 38 T^{3} + T^{4}$$