# Properties

 Label 74.4.a.c Level $74$ Weight $4$ Character orbit 74.a Self dual yes Analytic conductor $4.366$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$74 = 2 \cdot 37$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 74.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.36614134042$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.15629.1 Defining polynomial: $$x^{3} - 26 x - 45$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + 4 q^{4} + ( 2 + 3 \beta_{1} + \beta_{2} ) q^{5} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{6} + ( 2 - 4 \beta_{1} + \beta_{2} ) q^{7} -8 q^{8} + ( 15 - \beta_{1} - 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q -2 q^{2} + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + 4 q^{4} + ( 2 + 3 \beta_{1} + \beta_{2} ) q^{5} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{6} + ( 2 - 4 \beta_{1} + \beta_{2} ) q^{7} -8 q^{8} + ( 15 - \beta_{1} - 2 \beta_{2} ) q^{9} + ( -4 - 6 \beta_{1} - 2 \beta_{2} ) q^{10} + ( 26 + 5 \beta_{1} - 4 \beta_{2} ) q^{11} + ( 4 + 4 \beta_{1} + 4 \beta_{2} ) q^{12} + ( 35 - 9 \beta_{1} + 2 \beta_{2} ) q^{13} + ( -4 + 8 \beta_{1} - 2 \beta_{2} ) q^{14} + ( 65 + 8 \beta_{1} - 5 \beta_{2} ) q^{15} + 16 q^{16} + ( 6 - 8 \beta_{1} + 6 \beta_{2} ) q^{17} + ( -30 + 2 \beta_{1} + 4 \beta_{2} ) q^{18} + ( 78 + 8 \beta_{1} - 6 \beta_{2} ) q^{19} + ( 8 + 12 \beta_{1} + 4 \beta_{2} ) q^{20} + ( -12 - 20 \beta_{1} + 9 \beta_{2} ) q^{21} + ( -52 - 10 \beta_{1} + 8 \beta_{2} ) q^{22} + ( -53 - 9 \beta_{1} + 10 \beta_{2} ) q^{23} + ( -8 - 8 \beta_{1} - 8 \beta_{2} ) q^{24} + ( 32 + 33 \beta_{1} - 4 \beta_{2} ) q^{25} + ( -70 + 18 \beta_{1} - 4 \beta_{2} ) q^{26} + ( -83 - 4 \beta_{1} - 8 \beta_{2} ) q^{27} + ( 8 - 16 \beta_{1} + 4 \beta_{2} ) q^{28} + ( -105 - 5 \beta_{1} - 10 \beta_{2} ) q^{29} + ( -130 - 16 \beta_{1} + 10 \beta_{2} ) q^{30} + ( 16 + \beta_{1} - 25 \beta_{2} ) q^{31} -32 q^{32} + ( -39 + 70 \beta_{1} + 20 \beta_{2} ) q^{33} + ( -12 + 16 \beta_{1} - 12 \beta_{2} ) q^{34} + ( -158 - 44 \beta_{1} - 4 \beta_{2} ) q^{35} + ( 60 - 4 \beta_{1} - 8 \beta_{2} ) q^{36} -37 q^{37} + ( -156 - 16 \beta_{1} + 12 \beta_{2} ) q^{38} + ( -4 - 13 \beta_{1} + 51 \beta_{2} ) q^{39} + ( -16 - 24 \beta_{1} - 8 \beta_{2} ) q^{40} + ( -143 - 55 \beta_{1} - 35 \beta_{2} ) q^{41} + ( 24 + 40 \beta_{1} - 18 \beta_{2} ) q^{42} + ( 204 - 82 \beta_{1} - 24 \beta_{2} ) q^{43} + ( 104 + 20 \beta_{1} - 16 \beta_{2} ) q^{44} + ( -51 + 46 \beta_{1} + 27 \beta_{2} ) q^{45} + ( 106 + 18 \beta_{1} - 20 \beta_{2} ) q^{46} + ( 64 + 32 \beta_{1} - 37 \beta_{2} ) q^{47} + ( 16 + 16 \beta_{1} + 16 \beta_{2} ) q^{48} + ( 17 + 26 \beta_{1} + 45 \beta_{2} ) q^{49} + ( -64 - 66 \beta_{1} + 8 \beta_{2} ) q^{50} + ( 98 - 62 \beta_{1} + 16 \beta_{2} ) q^{51} + ( 140 - 36 \beta_{1} + 8 \beta_{2} ) q^{52} + ( -202 + 38 \beta_{1} + 27 \beta_{2} ) q^{53} + ( 166 + 8 \beta_{1} + 16 \beta_{2} ) q^{54} + ( 205 + 157 \beta_{1} + 50 \beta_{2} ) q^{55} + ( -16 + 32 \beta_{1} - 8 \beta_{2} ) q^{56} + ( -14 + 146 \beta_{1} + 68 \beta_{2} ) q^{57} + ( 210 + 10 \beta_{1} + 20 \beta_{2} ) q^{58} + ( 446 + 34 \beta_{1} - 80 \beta_{2} ) q^{59} + ( 260 + 32 \beta_{1} - 20 \beta_{2} ) q^{60} + ( 122 - 115 \beta_{1} - 75 \beta_{2} ) q^{61} + ( -32 - 2 \beta_{1} + 50 \beta_{2} ) q^{62} + ( -16 - 38 \beta_{1} - 8 \beta_{2} ) q^{63} + 64 q^{64} + ( -299 - 6 \beta_{1} + 23 \beta_{2} ) q^{65} + ( 78 - 140 \beta_{1} - 40 \beta_{2} ) q^{66} + ( -54 - 77 \beta_{1} + 35 \beta_{2} ) q^{67} + ( 24 - 32 \beta_{1} + 24 \beta_{2} ) q^{68} + ( 148 - 149 \beta_{1} - 45 \beta_{2} ) q^{69} + ( 316 + 88 \beta_{1} + 8 \beta_{2} ) q^{70} + ( -54 - 170 \beta_{1} - 39 \beta_{2} ) q^{71} + ( -120 + 8 \beta_{1} + 16 \beta_{2} ) q^{72} + ( 60 - 29 \beta_{1} - 138 \beta_{2} ) q^{73} + 74 q^{74} + ( 275 + 188 \beta_{1} - 30 \beta_{2} ) q^{75} + ( 312 + 32 \beta_{1} - 24 \beta_{2} ) q^{76} + ( -558 - 130 \beta_{1} - 69 \beta_{2} ) q^{77} + ( 8 + 26 \beta_{1} - 102 \beta_{2} ) q^{78} + ( 111 + 159 \beta_{1} + 80 \beta_{2} ) q^{79} + ( 32 + 48 \beta_{1} + 16 \beta_{2} ) q^{80} + ( -772 - 24 \beta_{1} - 13 \beta_{2} ) q^{81} + ( 286 + 110 \beta_{1} + 70 \beta_{2} ) q^{82} + ( -364 + 22 \beta_{1} + 55 \beta_{2} ) q^{83} + ( -48 - 80 \beta_{1} + 36 \beta_{2} ) q^{84} + ( -240 - 106 \beta_{1} - 30 \beta_{2} ) q^{85} + ( -408 + 164 \beta_{1} + 48 \beta_{2} ) q^{86} + ( -460 - 65 \beta_{1} - 85 \beta_{2} ) q^{87} + ( -208 - 40 \beta_{1} + 32 \beta_{2} ) q^{88} + ( 156 + 92 \beta_{1} + 192 \beta_{2} ) q^{89} + ( 102 - 92 \beta_{1} - 54 \beta_{2} ) q^{90} + ( 856 - 62 \beta_{1} + 128 \beta_{2} ) q^{91} + ( -212 - 36 \beta_{1} + 40 \beta_{2} ) q^{92} + ( -723 + 170 \beta_{1} + 39 \beta_{2} ) q^{93} + ( -128 - 64 \beta_{1} + 74 \beta_{2} ) q^{94} + ( 408 + 358 \beta_{1} + 114 \beta_{2} ) q^{95} + ( -32 - 32 \beta_{1} - 32 \beta_{2} ) q^{96} + ( 442 - 56 \beta_{1} + 58 \beta_{2} ) q^{97} + ( -34 - 52 \beta_{1} - 90 \beta_{2} ) q^{98} + ( 629 - 14 \beta_{1} - 91 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 6 q^{2} + 4 q^{3} + 12 q^{4} + 7 q^{5} - 8 q^{6} + 7 q^{7} - 24 q^{8} + 43 q^{9} + O(q^{10})$$ $$3 q - 6 q^{2} + 4 q^{3} + 12 q^{4} + 7 q^{5} - 8 q^{6} + 7 q^{7} - 24 q^{8} + 43 q^{9} - 14 q^{10} + 74 q^{11} + 16 q^{12} + 107 q^{13} - 14 q^{14} + 190 q^{15} + 48 q^{16} + 24 q^{17} - 86 q^{18} + 228 q^{19} + 28 q^{20} - 27 q^{21} - 148 q^{22} - 149 q^{23} - 32 q^{24} + 92 q^{25} - 214 q^{26} - 257 q^{27} + 28 q^{28} - 325 q^{29} - 380 q^{30} + 23 q^{31} - 96 q^{32} - 97 q^{33} - 48 q^{34} - 478 q^{35} + 172 q^{36} - 111 q^{37} - 456 q^{38} + 39 q^{39} - 56 q^{40} - 464 q^{41} + 54 q^{42} + 588 q^{43} + 296 q^{44} - 126 q^{45} + 298 q^{46} + 155 q^{47} + 64 q^{48} + 96 q^{49} - 184 q^{50} + 310 q^{51} + 428 q^{52} - 579 q^{53} + 514 q^{54} + 665 q^{55} - 56 q^{56} + 26 q^{57} + 650 q^{58} + 1258 q^{59} + 760 q^{60} + 291 q^{61} - 46 q^{62} - 56 q^{63} + 192 q^{64} - 874 q^{65} + 194 q^{66} - 127 q^{67} + 96 q^{68} + 399 q^{69} + 956 q^{70} - 201 q^{71} - 344 q^{72} + 42 q^{73} + 222 q^{74} + 795 q^{75} + 912 q^{76} - 1743 q^{77} - 78 q^{78} + 413 q^{79} + 112 q^{80} - 2329 q^{81} + 928 q^{82} - 1037 q^{83} - 108 q^{84} - 750 q^{85} - 1176 q^{86} - 1465 q^{87} - 592 q^{88} + 660 q^{89} + 252 q^{90} + 2696 q^{91} - 596 q^{92} - 2130 q^{93} - 310 q^{94} + 1338 q^{95} - 128 q^{96} + 1384 q^{97} - 192 q^{98} + 1796 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 26 x - 45$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3 \nu - 17$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3 \beta_{1} + 17$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.07379 −3.73537 5.80916
−2.00000 −7.55181 4.00000 −10.6994 15.1036 3.81714 −8.00000 30.0298 21.3988
1.2 −2.00000 5.42375 4.00000 −1.04699 −10.8475 25.1006 −8.00000 2.41712 2.09398
1.3 −2.00000 6.12805 4.00000 18.7464 −12.2561 −21.9178 −8.00000 10.5531 −37.4928
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$37$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.4.a.c 3
3.b odd 2 1 666.4.a.n 3
4.b odd 2 1 592.4.a.c 3
5.b even 2 1 1850.4.a.i 3
8.b even 2 1 2368.4.a.e 3
8.d odd 2 1 2368.4.a.f 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.4.a.c 3 1.a even 1 1 trivial
592.4.a.c 3 4.b odd 2 1
666.4.a.n 3 3.b odd 2 1
1850.4.a.i 3 5.b even 2 1
2368.4.a.e 3 8.b even 2 1
2368.4.a.f 3 8.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(74))$$:

 $$T_{3}^{3} - 4 T_{3}^{2} - 54 T_{3} + 251$$ $$T_{5}^{3} - 7 T_{5}^{2} - 209 T_{5} - 210$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T )^{3}$$
$3$ $$251 - 54 T - 4 T^{2} + T^{3}$$
$5$ $$-210 - 209 T - 7 T^{2} + T^{3}$$
$7$ $$2100 - 538 T - 7 T^{2} + T^{3}$$
$11$ $$60751 - 114 T - 74 T^{2} + T^{3}$$
$13$ $$64466 + 1115 T - 107 T^{2} + T^{3}$$
$17$ $$-61536 - 4436 T - 24 T^{2} + T^{3}$$
$19$ $$10800 + 12700 T - 228 T^{2} + T^{3}$$
$23$ $$-691166 - 2029 T + 149 T^{2} + T^{3}$$
$29$ $$637750 + 30175 T + 325 T^{2} + T^{3}$$
$31$ $$1309500 - 34333 T - 23 T^{2} + T^{3}$$
$37$ $$( 37 + T )^{3}$$
$41$ $$-19358953 - 33018 T + 464 T^{2} + T^{3}$$
$43$ $$42631272 - 49544 T - 588 T^{2} + T^{3}$$
$47$ $$23334804 - 117862 T - 155 T^{2} + T^{3}$$
$53$ $$-20188 + 56140 T + 579 T^{2} + T^{3}$$
$59$ $$208191120 + 92612 T - 1258 T^{2} + T^{3}$$
$61$ $$-25288348 - 440123 T - 291 T^{2} + T^{3}$$
$67$ $$-33004112 - 271931 T + 127 T^{2} + T^{3}$$
$71$ $$147028116 - 681344 T + 201 T^{2} + T^{3}$$
$73$ $$-14055069 - 971960 T - 42 T^{2} + T^{3}$$
$79$ $$122660820 - 681063 T - 413 T^{2} + T^{3}$$
$83$ $$-539472 + 206924 T + 1037 T^{2} + T^{3}$$
$89$ $$986707200 - 1706864 T - 660 T^{2} + T^{3}$$
$97$ $$-15830176 + 305964 T - 1384 T^{2} + T^{3}$$