# Properties

 Label 385.2.n.b Level $385$ Weight $2$ Character orbit 385.n Analytic conductor $3.074$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$385 = 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 385.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.07424047782$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/385\mathbb{Z}\right)^\times$$.

 $$n$$ $$211$$ $$232$$ $$276$$ $$\chi(n)$$ $$-\zeta_{10}^{3}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
36.1
 0.809017 + 0.587785i −0.309017 + 0.951057i −0.309017 − 0.951057i 0.809017 − 0.587785i
1.80902 1.31433i 0.190983 + 0.587785i 0.927051 2.85317i 0.809017 + 0.587785i 1.11803 + 0.812299i −0.309017 + 0.951057i −0.690983 2.12663i 2.11803 1.53884i 2.23607
71.1 0.690983 + 2.12663i 1.30902 + 0.951057i −2.42705 + 1.76336i −0.309017 + 0.951057i −1.11803 + 3.44095i 0.809017 0.587785i −1.80902 1.31433i −0.118034 0.363271i −2.23607
141.1 0.690983 2.12663i 1.30902 0.951057i −2.42705 1.76336i −0.309017 0.951057i −1.11803 3.44095i 0.809017 + 0.587785i −1.80902 + 1.31433i −0.118034 + 0.363271i −2.23607
246.1 1.80902 + 1.31433i 0.190983 0.587785i 0.927051 + 2.85317i 0.809017 0.587785i 1.11803 0.812299i −0.309017 0.951057i −0.690983 + 2.12663i 2.11803 + 1.53884i 2.23607
 $$n$$: 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 385.2.n.b 4
11.c even 5 1 inner 385.2.n.b 4
11.c even 5 1 4235.2.a.j 2
11.d odd 10 1 4235.2.a.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.n.b 4 1.a even 1 1 trivial
385.2.n.b 4 11.c even 5 1 inner
4235.2.a.j 2 11.c even 5 1
4235.2.a.k 2 11.d odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 5 T_{2}^{3} + 15 T_{2}^{2} - 25 T_{2} + 25$$ acting on $$S_{2}^{\mathrm{new}}(385, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$25 - 25 T + 15 T^{2} - 5 T^{3} + T^{4}$$
$3$ $$1 - 2 T + 4 T^{2} - 3 T^{3} + T^{4}$$
$5$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$7$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$11$ $$121 - 11 T - 9 T^{2} - T^{3} + T^{4}$$
$13$ $$121 + 11 T + 16 T^{2} + 6 T^{3} + T^{4}$$
$17$ $$361 + 38 T + 24 T^{2} + 7 T^{3} + T^{4}$$
$19$ $$16 + 24 T + 16 T^{2} + 4 T^{3} + T^{4}$$
$23$ $$( -20 + T^{2} )^{2}$$
$29$ $$25 - 25 T + 40 T^{2} - 10 T^{3} + T^{4}$$
$31$ $$16 + 16 T + 16 T^{2} + 6 T^{3} + T^{4}$$
$37$ $$400 - 200 T + 40 T^{2} + T^{4}$$
$41$ $$256 - 128 T + 64 T^{2} - 12 T^{3} + T^{4}$$
$43$ $$( 4 + 6 T + T^{2} )^{2}$$
$47$ $$81 + 27 T + 54 T^{2} - 12 T^{3} + T^{4}$$
$53$ $$256 - 256 T + 96 T^{2} + 4 T^{3} + T^{4}$$
$59$ $$1936 - 528 T + 64 T^{2} - 2 T^{3} + T^{4}$$
$61$ $$1936 - 528 T + 64 T^{2} - 2 T^{3} + T^{4}$$
$67$ $$( -116 + 6 T + T^{2} )^{2}$$
$71$ $$121 - 99 T + 256 T^{2} + 26 T^{3} + T^{4}$$
$73$ $$3025 + 275 T + 60 T^{2} + 10 T^{3} + T^{4}$$
$79$ $$22201 + 4321 T + 456 T^{2} + 26 T^{3} + T^{4}$$
$83$ $$10201 + 2323 T + 304 T^{2} + 22 T^{3} + T^{4}$$
$89$ $$( 220 - 30 T + T^{2} )^{2}$$
$97$ $$25 - 50 T + 240 T^{2} - 25 T^{3} + T^{4}$$