# Properties

 Label 370.2.c.a Level $370$ Weight $2$ Character orbit 370.c Analytic conductor $2.954$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.95446487479$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 19 x^{8} + 103 x^{6} + 210 x^{4} + 140 x^{2} + 16$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 19 x^{8} + 103 x^{6} + 210 x^{4} + 140 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{9} - 45 \nu^{7} - 121 \nu^{5} - 26 \nu^{3} + 12 \nu$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{9} + 6 \nu^{8} + 15 \nu^{7} + 94 \nu^{6} + 39 \nu^{5} + 302 \nu^{4} - 10 \nu^{3} + 212 \nu^{2} - 44 \nu + 16$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{9} + 2 \nu^{8} + 49 \nu^{7} + 34 \nu^{6} + 181 \nu^{5} + 142 \nu^{4} + 186 \nu^{3} + 196 \nu^{2} + 20 \nu + 64$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{9} - 6 \nu^{8} + 15 \nu^{7} - 94 \nu^{6} + 39 \nu^{5} - 302 \nu^{4} - 10 \nu^{3} - 212 \nu^{2} - 44 \nu - 16$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{9} - 2 \nu^{8} + 49 \nu^{7} - 34 \nu^{6} + 181 \nu^{5} - 142 \nu^{4} + 186 \nu^{3} - 196 \nu^{2} + 20 \nu - 64$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{8} + 49 \nu^{6} + 181 \nu^{4} + 186 \nu^{2} + 28$$$$)/4$$ $$\beta_{8}$$ $$=$$ $$($$$$3 \nu^{8} + 49 \nu^{6} + 181 \nu^{4} + 190 \nu^{2} + 44$$$$)/4$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{9} - 17 \nu^{7} - 71 \nu^{5} - 100 \nu^{3} - 46 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{8} - \beta_{7} - 4$$ $$\nu^{3}$$ $$=$$ $$-\beta_{9} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 7 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-14 \beta_{8} + 10 \beta_{7} - 3 \beta_{6} - \beta_{5} + 3 \beta_{4} + \beta_{3} + 32$$ $$\nu^{5}$$ $$=$$ $$14 \beta_{9} + 14 \beta_{6} - 17 \beta_{5} + 14 \beta_{4} - 17 \beta_{3} - 2 \beta_{2} + 68 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$170 \beta_{8} - 108 \beta_{7} + 45 \beta_{6} + 16 \beta_{5} - 45 \beta_{4} - 16 \beta_{3} - 330$$ $$\nu^{7}$$ $$=$$ $$-170 \beta_{9} - 169 \beta_{6} + 215 \beta_{5} - 169 \beta_{4} + 215 \beta_{3} + 32 \beta_{2} - 748 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-1994 \beta_{8} + 1224 \beta_{7} - 554 \beta_{6} - 201 \beta_{5} + 554 \beta_{4} + 201 \beta_{3} + 3698$$ $$\nu^{9}$$ $$=$$ $$1994 \beta_{9} + 1979 \beta_{6} - 2548 \beta_{5} + 1979 \beta_{4} - 2548 \beta_{3} - 402 \beta_{2} + 8542 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
369.1
 − 3.40359i − 1.78647i − 1.76216i − 0.987983i − 0.377861i 0.377861i 0.987983i 1.76216i 1.78647i 3.40359i
−1.00000 3.40359i 1.00000 −1.28269 1.83159i 3.40359i 2.06225i −1.00000 −8.58443 1.28269 + 1.83159i
369.2 −1.00000 1.78647i 1.00000 2.21736 + 0.288618i 1.78647i 3.14934i −1.00000 −0.191472 −2.21736 0.288618i
369.3 −1.00000 1.76216i 1.00000 −1.62868 + 1.53213i 1.76216i 1.22131i −1.00000 −0.105209 1.62868 1.53213i
369.4 −1.00000 0.987983i 1.00000 −1.85396 1.25013i 0.987983i 4.78937i −1.00000 2.02389 1.85396 + 1.25013i
369.5 −1.00000 0.377861i 1.00000 1.04797 1.97529i 0.377861i 0.631751i −1.00000 2.85722 −1.04797 + 1.97529i
369.6 −1.00000 0.377861i 1.00000 1.04797 + 1.97529i 0.377861i 0.631751i −1.00000 2.85722 −1.04797 1.97529i
369.7 −1.00000 0.987983i 1.00000 −1.85396 + 1.25013i 0.987983i 4.78937i −1.00000 2.02389 1.85396 1.25013i
369.8 −1.00000 1.76216i 1.00000 −1.62868 1.53213i 1.76216i 1.22131i −1.00000 −0.105209 1.62868 + 1.53213i
369.9 −1.00000 1.78647i 1.00000 2.21736 0.288618i 1.78647i 3.14934i −1.00000 −0.191472 −2.21736 + 0.288618i
369.10 −1.00000 3.40359i 1.00000 −1.28269 + 1.83159i 3.40359i 2.06225i −1.00000 −8.58443 1.28269 1.83159i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 369.10 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.d 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.c.a 10
3.b odd 2 1 3330.2.e.d 10
5.b even 2 1 370.2.c.b yes 10
5.c odd 4 2 1850.2.d.i 20
15.d odd 2 1 3330.2.e.c 10
37.b even 2 1 370.2.c.b yes 10
111.d odd 2 1 3330.2.e.c 10
185.d even 2 1 inner 370.2.c.a 10
185.h odd 4 2 1850.2.d.i 20
555.b odd 2 1 3330.2.e.d 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.c.a 10 1.a even 1 1 trivial
370.2.c.a 10 185.d even 2 1 inner
370.2.c.b yes 10 5.b even 2 1
370.2.c.b yes 10 37.b even 2 1
1850.2.d.i 20 5.c odd 4 2
1850.2.d.i 20 185.h odd 4 2
3330.2.e.c 10 15.d odd 2 1
3330.2.e.c 10 111.d odd 2 1
3330.2.e.d 10 3.b odd 2 1
3330.2.e.d 10 555.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{5} + T_{13}^{4} - 39 T_{13}^{3} - 100 T_{13}^{2} + 160 T_{13} + 488$$ acting on $$S_{2}^{\mathrm{new}}(370, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{10}$$
$3$ $$16 + 140 T^{2} + 210 T^{4} + 103 T^{6} + 19 T^{8} + T^{10}$$
$5$ $$3125 + 1875 T + 250 T^{2} - 400 T^{3} - 95 T^{4} - 22 T^{5} - 19 T^{6} - 16 T^{7} + 2 T^{8} + 3 T^{9} + T^{10}$$
$7$ $$576 + 2048 T^{2} + 1684 T^{4} + 438 T^{6} + 39 T^{8} + T^{10}$$
$11$ $$( -48 + 16 T + 51 T^{2} - 28 T^{3} + T^{5} )^{2}$$
$13$ $$( 488 + 160 T - 100 T^{2} - 39 T^{3} + T^{4} + T^{5} )^{2}$$
$17$ $$( -144 - 112 T + 108 T^{2} + 4 T^{3} - 9 T^{4} + T^{5} )^{2}$$
$19$ $$9216 + 374336 T^{2} + 72240 T^{4} + 4668 T^{6} + 118 T^{8} + T^{10}$$
$23$ $$( -768 + 256 T + 308 T^{2} - 63 T^{3} - 5 T^{4} + T^{5} )^{2}$$
$29$ $$2262016 + 1251872 T^{2} + 194729 T^{4} + 10258 T^{6} + 182 T^{8} + T^{10}$$
$31$ $$60516 + 346130 T^{2} + 68053 T^{4} + 4502 T^{6} + 116 T^{8} + T^{10}$$
$37$ $$69343957 + 14993288 T + 4913341 T^{2} + 788544 T^{3} + 195434 T^{4} + 25584 T^{5} + 5282 T^{6} + 576 T^{7} + 97 T^{8} + 8 T^{9} + T^{10}$$
$41$ $$( -36 - 8 T + 89 T^{2} - 44 T^{3} + 2 T^{4} + T^{5} )^{2}$$
$43$ $$( -2624 + 1856 T - 76 T^{2} - 88 T^{3} + 5 T^{4} + T^{5} )^{2}$$
$47$ $$4596736 + 10494016 T^{2} + 992816 T^{4} + 29148 T^{6} + 310 T^{8} + T^{10}$$
$53$ $$39337984 + 10468352 T^{2} + 826000 T^{4} + 22964 T^{6} + 257 T^{8} + T^{10}$$
$59$ $$21827584 + 7068992 T^{2} + 590448 T^{4} + 18844 T^{6} + 238 T^{8} + T^{10}$$
$61$ $$82944 + 89792 T^{2} + 28169 T^{4} + 3402 T^{6} + 150 T^{8} + T^{10}$$
$67$ $$559417104 + 70672748 T^{2} + 2839394 T^{4} + 48727 T^{6} + 367 T^{8} + T^{10}$$
$71$ $$( 4608 - 2816 T - 1896 T^{2} - 164 T^{3} + 10 T^{4} + T^{5} )^{2}$$
$73$ $$589824 + 785408 T^{2} + 189152 T^{4} + 10965 T^{6} + 189 T^{8} + T^{10}$$
$79$ $$186486336 + 29601332 T^{2} + 1472598 T^{4} + 31019 T^{6} + 291 T^{8} + T^{10}$$
$83$ $$1024 + 23736128 T^{2} + 2459520 T^{4} + 62212 T^{6} + 466 T^{8} + T^{10}$$
$89$ $$1230045184 + 117901312 T^{2} + 3949632 T^{4} + 59888 T^{6} + 412 T^{8} + T^{10}$$
$97$ $$( 1168 + 2824 T - 248 T^{2} - 178 T^{3} - T^{4} + T^{5} )^{2}$$