# Properties

 Label 370.2.d.c Level $370$ Weight $2$ Character orbit 370.d Analytic conductor $2.954$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.399424.1 Defining polynomial: $$x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{4} + 2 \nu^{3} - \nu^{2} + 2 \nu - 2$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} - 3 \nu^{3} + 4 \nu^{2} - 2 \nu + 8$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} + 2 \nu^{4} - 3 \nu^{3} + 6 \nu^{2} - 2 \nu + 4$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} - 2 \nu^{4} + 3 \nu^{3} - 6 \nu^{2} + 10 \nu - 8$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{5} - 2 \nu^{4} + 5 \nu^{3} - 6 \nu^{2} + 2 \nu - 12$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + 2 \beta_{3} + \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_{1} + 3$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{5} + 2 \beta_{3} - 5 \beta_{2} - \beta_{1} + 4$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$4 \beta_{5} + \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 2 \beta_{1} + 5$$$$)/2$$

## Character values

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

 $$n$$ $$261$$ $$297$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
221.1
 1.40680 + 0.144584i −0.671462 + 1.24464i 0.264658 − 1.38923i 1.40680 − 0.144584i −0.671462 − 1.24464i 0.264658 + 1.38923i
1.00000i −3.10278 −1.00000 1.00000i 3.10278i 3.81361 1.00000i 6.62721 1.00000
221.2 1.00000i −1.14637 −1.00000 1.00000i 1.14637i −0.342923 1.00000i −1.68585 1.00000
221.3 1.00000i 2.24914 −1.00000 1.00000i 2.24914i 1.52932 1.00000i 2.05863 1.00000
221.4 1.00000i −3.10278 −1.00000 1.00000i 3.10278i 3.81361 1.00000i 6.62721 1.00000
221.5 1.00000i −1.14637 −1.00000 1.00000i 1.14637i −0.342923 1.00000i −1.68585 1.00000
221.6 1.00000i 2.24914 −1.00000 1.00000i 2.24914i 1.52932 1.00000i 2.05863 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 221.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.d.c 6
3.b odd 2 1 3330.2.h.n 6
4.b odd 2 1 2960.2.p.g 6
5.b even 2 1 1850.2.d.f 6
5.c odd 4 1 1850.2.c.i 6
5.c odd 4 1 1850.2.c.j 6
37.b even 2 1 inner 370.2.d.c 6
111.d odd 2 1 3330.2.h.n 6
148.b odd 2 1 2960.2.p.g 6
185.d even 2 1 1850.2.d.f 6
185.h odd 4 1 1850.2.c.i 6
185.h odd 4 1 1850.2.c.j 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.d.c 6 1.a even 1 1 trivial
370.2.d.c 6 37.b even 2 1 inner
1850.2.c.i 6 5.c odd 4 1
1850.2.c.i 6 185.h odd 4 1
1850.2.c.j 6 5.c odd 4 1
1850.2.c.j 6 185.h odd 4 1
1850.2.d.f 6 5.b even 2 1
1850.2.d.f 6 185.d even 2 1
2960.2.p.g 6 4.b odd 2 1
2960.2.p.g 6 148.b odd 2 1
3330.2.h.n 6 3.b odd 2 1
3330.2.h.n 6 111.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(370, [\chi])$$:

 $$T_{3}^{3} + 2 T_{3}^{2} - 6 T_{3} - 8$$ $$T_{7}^{3} - 5 T_{7}^{2} + 4 T_{7} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{3}$$
$3$ $$( -8 - 6 T + 2 T^{2} + T^{3} )^{2}$$
$5$ $$( 1 + T^{2} )^{3}$$
$7$ $$( 2 + 4 T - 5 T^{2} + T^{3} )^{2}$$
$11$ $$( 16 - 16 T + T^{2} + T^{3} )^{2}$$
$13$ $$256 + 320 T^{2} + 36 T^{4} + T^{6}$$
$17$ $$16384 + 2048 T^{2} + 81 T^{4} + T^{6}$$
$19$ $$16 + 260 T^{2} + 36 T^{4} + T^{6}$$
$23$ $$256 + 320 T^{2} + 36 T^{4} + T^{6}$$
$29$ $$16 + 56 T^{2} + 49 T^{4} + T^{6}$$
$31$ $$6724 + 1076 T^{2} + 57 T^{4} + T^{6}$$
$37$ $$50653 + 13690 T + 2923 T^{2} + 388 T^{3} + 79 T^{4} + 10 T^{5} + T^{6}$$
$41$ $$( 172 - 56 T - T^{2} + T^{3} )^{2}$$
$43$ $$256 + 224 T^{2} + 33 T^{4} + T^{6}$$
$47$ $$( -32 - 14 T + 2 T^{2} + T^{3} )^{2}$$
$53$ $$( 352 - 104 T - T^{2} + T^{3} )^{2}$$
$59$ $$64 + 2276 T^{2} + 96 T^{4} + T^{6}$$
$61$ $$15376 + 3800 T^{2} + 121 T^{4} + T^{6}$$
$67$ $$( -124 - 106 T - 4 T^{2} + T^{3} )^{2}$$
$71$ $$( -16 + 16 T + 10 T^{2} + T^{3} )^{2}$$
$73$ $$( -344 - 100 T + 6 T^{2} + T^{3} )^{2}$$
$79$ $$4096 + 1572 T^{2} + 80 T^{4} + T^{6}$$
$83$ $$( 8 - 62 T + 10 T^{2} + T^{3} )^{2}$$
$89$ $$262144 + 17408 T^{2} + 256 T^{4} + T^{6}$$
$97$ $$29584 + 6488 T^{2} + 289 T^{4} + T^{6}$$