# Properties

 Label 74.2.c.c Level $74$ Weight $2$ Character orbit 74.c Analytic conductor $0.591$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$74 = 2 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 74.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.590892974957$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.4406832.1 Defining polynomial: $$x^{6} - x^{5} + 6 x^{4} + 7 x^{3} + 24 x^{2} + 5 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 6 x^{4} + 7 x^{3} + 24 x^{2} + 5 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-5 \nu^{5} + 30 \nu^{4} - 31 \nu^{3} + 120 \nu^{2} + 25 \nu + 595$$$$)/149$$ $$\beta_{2}$$ $$=$$ $$($$$$7 \nu^{5} - 42 \nu^{4} + 103 \nu^{3} - 168 \nu^{2} - 35 \nu - 386$$$$)/149$$ $$\beta_{3}$$ $$=$$ $$($$$$30 \nu^{5} - 31 \nu^{4} + 186 \nu^{3} + 174 \nu^{2} + 744 \nu + 6$$$$)/149$$ $$\beta_{4}$$ $$=$$ $$($$$$88 \nu^{5} - 81 \nu^{4} + 486 \nu^{3} + 719 \nu^{2} + 1944 \nu + 405$$$$)/149$$ $$\beta_{5}$$ $$=$$ $$($$$$125 \nu^{5} - 154 \nu^{4} + 775 \nu^{3} + 725 \nu^{2} + 2951 \nu + 25$$$$)/149$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - 4 \beta_{3} + \beta_{1} - 4$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{2} + 7 \beta_{1} - 15$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-9 \beta_{5} + 3 \beta_{4} + 28 \beta_{3} + 3 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$($$$$-55 \beta_{5} + 31 \beta_{4} + 139 \beta_{3} - 55 \beta_{1} + 139$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/74\mathbb{Z}\right)^\times$$.

 $$n$$ $$39$$ $$\chi(n)$$ $$\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 −0.105378 − 0.182520i 1.43310 + 2.48220i −0.827721 − 1.43366i −0.105378 + 0.182520i 1.43310 − 2.48220i −0.827721 + 1.43366i
−0.500000 + 0.866025i −1.26704 2.19457i −0.500000 0.866025i −1.97779 3.42563i 2.53407 1.26704 + 2.19457i 1.00000 −1.71076 + 2.96312i 3.95558
47.2 −0.500000 + 0.866025i −0.258652 0.447998i −0.500000 0.866025i 2.10755 + 3.65038i 0.517304 0.258652 + 0.447998i 1.00000 1.36620 2.36632i −4.21509
47.3 −0.500000 + 0.866025i 1.52569 + 2.64257i −0.500000 0.866025i −0.629755 1.09077i −3.05137 −1.52569 2.64257i 1.00000 −3.15544 + 5.46539i 1.25951
63.1 −0.500000 0.866025i −1.26704 + 2.19457i −0.500000 + 0.866025i −1.97779 + 3.42563i 2.53407 1.26704 2.19457i 1.00000 −1.71076 2.96312i 3.95558
63.2 −0.500000 0.866025i −0.258652 + 0.447998i −0.500000 + 0.866025i 2.10755 3.65038i 0.517304 0.258652 0.447998i 1.00000 1.36620 + 2.36632i −4.21509
63.3 −0.500000 0.866025i 1.52569 2.64257i −0.500000 + 0.866025i −0.629755 + 1.09077i −3.05137 −1.52569 + 2.64257i 1.00000 −3.15544 5.46539i 1.25951
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 63.3 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.2.c.c 6
3.b odd 2 1 666.2.f.j 6
4.b odd 2 1 592.2.i.e 6
37.c even 3 1 inner 74.2.c.c 6
37.c even 3 1 2738.2.a.o 3
37.e even 6 1 2738.2.a.n 3
111.i odd 6 1 666.2.f.j 6
148.i odd 6 1 592.2.i.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.c.c 6 1.a even 1 1 trivial
74.2.c.c 6 37.c even 3 1 inner
592.2.i.e 6 4.b odd 2 1
592.2.i.e 6 148.i odd 6 1
666.2.f.j 6 3.b odd 2 1
666.2.f.j 6 111.i odd 6 1
2738.2.a.n 3 37.e even 6 1
2738.2.a.o 3 37.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 8 T_{3}^{4} + 8 T_{3}^{3} + 64 T_{3}^{2} + 32 T_{3} + 16$$ acting on $$S_{2}^{\mathrm{new}}(74, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{3}$$
$3$ $$16 + 32 T + 64 T^{2} + 8 T^{3} + 8 T^{4} + T^{6}$$
$5$ $$441 + 357 T + 310 T^{2} + 25 T^{3} + 18 T^{4} + T^{5} + T^{6}$$
$7$ $$16 - 32 T + 64 T^{2} - 8 T^{3} + 8 T^{4} + T^{6}$$
$11$ $$( 48 - 16 T - 4 T^{2} + T^{3} )^{2}$$
$13$ $$( 4 + 2 T + T^{2} )^{3}$$
$17$ $$3969 - 1827 T + 1030 T^{2} - 39 T^{3} + 38 T^{4} - 3 T^{5} + T^{6}$$
$19$ $$12544 + 896 T + 960 T^{2} + 160 T^{3} + 72 T^{4} + 8 T^{5} + T^{6}$$
$23$ $$( 12 + 4 T - 8 T^{2} + T^{3} )^{2}$$
$29$ $$( -3 - 5 T + 3 T^{2} + T^{3} )^{2}$$
$31$ $$( -16 - 24 T - 4 T^{2} + T^{3} )^{2}$$
$37$ $$50653 + 15059 T + 4958 T^{2} + 815 T^{3} + 134 T^{4} + 11 T^{5} + T^{6}$$
$41$ $$9 - 39 T + 190 T^{2} + 85 T^{3} + 62 T^{4} - 7 T^{5} + T^{6}$$
$43$ $$( 148 - 20 T - 8 T^{2} + T^{3} )^{2}$$
$47$ $$( 48 + 56 T + 16 T^{2} + T^{3} )^{2}$$
$53$ $$46656 + 30240 T + 14848 T^{2} + 2648 T^{3} + 344 T^{4} + 22 T^{5} + T^{6}$$
$59$ $$451584 + 107520 T + 28288 T^{2} + 704 T^{3} + 176 T^{4} + 4 T^{5} + T^{6}$$
$61$ $$49 + 133 T + 298 T^{2} + 157 T^{3} + 62 T^{4} + 9 T^{5} + T^{6}$$
$67$ $$256 - 1024 T + 3840 T^{2} - 992 T^{3} + 192 T^{4} - 16 T^{5} + T^{6}$$
$71$ $$1296 - 1440 T + 1168 T^{2} - 408 T^{3} + 104 T^{4} - 12 T^{5} + T^{6}$$
$73$ $$( -8 - 20 T - 2 T^{2} + T^{3} )^{2}$$
$79$ $$20736 + 10368 T + 4608 T^{2} + 576 T^{3} + 88 T^{4} - 4 T^{5} + T^{6}$$
$83$ $$17424 - 8448 T + 3568 T^{2} - 520 T^{3} + 80 T^{4} + 4 T^{5} + T^{6}$$
$89$ $$301401 + 81801 T + 27142 T^{2} - 243 T^{3} + 230 T^{4} + 9 T^{5} + T^{6}$$
$97$ $$( -7 + 27 T - 13 T^{2} + T^{3} )^{2}$$