# Properties

 Label 370.2.h.c Level $370$ Weight $2$ Character orbit 370.h Analytic conductor $2.954$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.h (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/370\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$297$$ $$\chi(n)$$ $$-\zeta_{8}^{2}$$ $$\zeta_{8}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
117.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
1.00000 −1.41421 1.41421i 1.00000 −0.707107 + 2.12132i −1.41421 1.41421i 2.70711 + 2.70711i 1.00000 1.00000i −0.707107 + 2.12132i
117.2 1.00000 1.41421 + 1.41421i 1.00000 0.707107 2.12132i 1.41421 + 1.41421i 1.29289 + 1.29289i 1.00000 1.00000i 0.707107 2.12132i
253.1 1.00000 −1.41421 + 1.41421i 1.00000 −0.707107 2.12132i −1.41421 + 1.41421i 2.70711 2.70711i 1.00000 1.00000i −0.707107 2.12132i
253.2 1.00000 1.41421 1.41421i 1.00000 0.707107 + 2.12132i 1.41421 1.41421i 1.29289 1.29289i 1.00000 1.00000i 0.707107 + 2.12132i
 $$n$$: 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
185.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.h.c yes 4
5.c odd 4 1 370.2.g.c 4
37.d odd 4 1 370.2.g.c 4
185.k even 4 1 inner 370.2.h.c yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.g.c 4 5.c odd 4 1
370.2.g.c 4 37.d odd 4 1
370.2.h.c yes 4 1.a even 1 1 trivial
370.2.h.c yes 4 185.k even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 16$$ acting on $$S_{2}^{\mathrm{new}}(370, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{4}$$
$3$ $$16 + T^{4}$$
$5$ $$25 + 8 T^{2} + T^{4}$$
$7$ $$49 - 56 T + 32 T^{2} - 8 T^{3} + T^{4}$$
$11$ $$49 + 18 T^{2} + T^{4}$$
$13$ $$( -14 + 4 T + T^{2} )^{2}$$
$17$ $$49 + 18 T^{2} + T^{4}$$
$19$ $$4 - 24 T + 72 T^{2} - 12 T^{3} + T^{4}$$
$23$ $$( 14 + 8 T + T^{2} )^{2}$$
$29$ $$5041 - 1704 T + 288 T^{2} - 24 T^{3} + T^{4}$$
$31$ $$1 - 8 T + 32 T^{2} + 8 T^{3} + T^{4}$$
$37$ $$1369 + 24 T^{2} + T^{4}$$
$41$ $$( 49 + T^{2} )^{2}$$
$43$ $$( 7 + T )^{4}$$
$47$ $$16 - 64 T + 128 T^{2} + 16 T^{3} + T^{4}$$
$53$ $$81 + T^{4}$$
$59$ $$256 - 256 T + 128 T^{2} - 16 T^{3} + T^{4}$$
$61$ $$39601 - 7960 T + 800 T^{2} - 40 T^{3} + T^{4}$$
$67$ $$324 + 216 T + 72 T^{2} - 12 T^{3} + T^{4}$$
$71$ $$( 4 + 12 T + T^{2} )^{2}$$
$73$ $$( 72 + 12 T + T^{2} )^{2}$$
$79$ $$1024 - 512 T + 128 T^{2} + 16 T^{3} + T^{4}$$
$83$ $$64 + 64 T + 32 T^{2} - 8 T^{3} + T^{4}$$
$89$ $$4624 + 1632 T + 288 T^{2} + 24 T^{3} + T^{4}$$
$97$ $$( 49 + T^{2} )^{2}$$