# Properties

 Label 37.4.a.a Level $37$ Weight $4$ Character orbit 37.a Self dual yes Analytic conductor $2.183$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$2.18307067021$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.21208.1 Defining polynomial: $$x^{4} - x^{3} - 8 x^{2} + 13 x - 3$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + \beta_{2} - \beta_{3} ) q^{2} + ( -2 - 2 \beta_{2} + \beta_{3} ) q^{3} + ( 4 - \beta_{1} - 3 \beta_{2} + 6 \beta_{3} ) q^{4} + ( -9 + 4 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{5} + ( -7 - \beta_{2} - 2 \beta_{3} ) q^{6} + ( -5 - 13 \beta_{1} + \beta_{2} ) q^{7} + ( -30 + 9 \beta_{1} + \beta_{2} - 20 \beta_{3} ) q^{8} + ( -5 + 3 \beta_{1} + 15 \beta_{2} - 3 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -2 + \beta_{2} - \beta_{3} ) q^{2} + ( -2 - 2 \beta_{2} + \beta_{3} ) q^{3} + ( 4 - \beta_{1} - 3 \beta_{2} + 6 \beta_{3} ) q^{4} + ( -9 + 4 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{5} + ( -7 - \beta_{2} - 2 \beta_{3} ) q^{6} + ( -5 - 13 \beta_{1} + \beta_{2} ) q^{7} + ( -30 + 9 \beta_{1} + \beta_{2} - 20 \beta_{3} ) q^{8} + ( -5 + 3 \beta_{1} + 15 \beta_{2} - 3 \beta_{3} ) q^{9} + ( 25 - 6 \beta_{2} + 13 \beta_{3} ) q^{10} + ( -4 + \beta_{1} + 9 \beta_{2} + 5 \beta_{3} ) q^{11} + ( 37 - 5 \beta_{1} + 7 \beta_{2} + 7 \beta_{3} ) q^{12} + ( -17 + 19 \beta_{1} - 13 \beta_{2} - 17 \beta_{3} ) q^{13} + ( 26 + \beta_{1} - 18 \beta_{2} + 5 \beta_{3} ) q^{14} + ( -19 \beta_{1} - 5 \beta_{2} - 8 \beta_{3} ) q^{15} + ( 122 - 31 \beta_{1} - 17 \beta_{2} + 62 \beta_{3} ) q^{16} + ( -14 - 8 \beta_{1} + 8 \beta_{2} ) q^{17} + ( 67 + 9 \beta_{1} - 5 \beta_{2} + 17 \beta_{3} ) q^{18} + ( -42 + 40 \beta_{1} - 36 \beta_{2} - 20 \beta_{3} ) q^{19} + ( -61 - 12 \beta_{1} + 22 \beta_{2} - 69 \beta_{3} ) q^{20} + ( 3 + 49 \beta_{1} + 43 \beta_{2} - 4 \beta_{3} ) q^{21} + ( 9 + 19 \beta_{1} + 2 \beta_{2} - 16 \beta_{3} ) q^{22} + ( -49 - 29 \beta_{1} - 9 \beta_{2} + 41 \beta_{3} ) q^{23} + ( -27 + 21 \beta_{1} + 47 \beta_{2} - 49 \beta_{3} ) q^{24} + ( 38 - 69 \beta_{1} + 11 \beta_{2} + 3 \beta_{3} ) q^{25} + ( 61 - 47 \beta_{1} - 15 \beta_{2} + 85 \beta_{3} ) q^{26} + ( -53 - 48 \beta_{1} - 32 \beta_{2} - 20 \beta_{3} ) q^{27} + ( -92 + 96 \beta_{1} + 24 \beta_{2} - 46 \beta_{3} ) q^{28} + ( 9 + \beta_{1} - 15 \beta_{2} + 51 \beta_{3} ) q^{29} + ( 44 - 21 \beta_{1} - 27 \beta_{2} + 32 \beta_{3} ) q^{30} + ( 35 + 10 \beta_{1} + 16 \beta_{2} + 55 \beta_{3} ) q^{31} + ( -334 + 35 \beta_{1} + 145 \beta_{2} - 210 \beta_{3} ) q^{32} + ( -35 - 46 \beta_{1} - 54 \beta_{2} + 10 \beta_{3} ) q^{33} + ( 60 + 8 \beta_{1} - 22 \beta_{2} + 14 \beta_{3} ) q^{34} + ( -152 + 126 \beta_{1} - 98 \beta_{2} + 56 \beta_{3} ) q^{35} + ( -203 + 5 \beta_{1} - 27 \beta_{2} - 111 \beta_{3} ) q^{36} + 37 q^{37} + ( 36 - 76 \beta_{1} - 22 \beta_{2} + 122 \beta_{3} ) q^{38} + ( 57 + 14 \beta_{1} + 72 \beta_{2} - 47 \beta_{3} ) q^{39} + ( 345 - 116 \beta_{1} - 94 \beta_{2} + 233 \beta_{3} ) q^{40} + ( -6 + 24 \beta_{1} - 54 \beta_{2} - 45 \beta_{3} ) q^{41} + ( 94 + 35 \beta_{1} + 48 \beta_{2} + 13 \beta_{3} ) q^{42} + ( -42 - 72 \beta_{1} + 60 \beta_{2} + 10 \beta_{3} ) q^{43} + ( 81 - 38 \beta_{1} - 60 \beta_{2} + 15 \beta_{3} ) q^{44} + ( 246 + 7 \beta_{1} + 41 \beta_{2} + 14 \beta_{3} ) q^{45} + ( -105 + 73 \beta_{1} - 37 \beta_{2} - 115 \beta_{3} ) q^{46} + ( -169 + 125 \beta_{1} + 71 \beta_{2} - 8 \beta_{3} ) q^{47} + ( 123 - 11 \beta_{1} - 111 \beta_{2} + 167 \beta_{3} ) q^{48} + ( 336 - 37 \beta_{1} + 109 \beta_{2} - 170 \beta_{3} ) q^{49} + ( 11 + 17 \beta_{1} - 28 \beta_{2} - 50 \beta_{3} ) q^{50} + ( -28 + 8 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{51} + ( -409 + 3 \beta_{1} + 203 \beta_{2} - 265 \beta_{3} ) q^{52} + ( 107 - 15 \beta_{1} - 25 \beta_{2} + 70 \beta_{3} ) q^{53} + ( 158 - 72 \beta_{1} - 121 \beta_{2} + 133 \beta_{3} ) q^{54} + ( 171 - 25 \beta_{1} + 51 \beta_{2} - 69 \beta_{3} ) q^{55} + ( 182 - 76 \beta_{1} + 102 \beta_{2} + 236 \beta_{3} ) q^{56} + ( 256 + 8 \beta_{1} + 200 \beta_{2} - 98 \beta_{3} ) q^{57} + ( -319 + 87 \beta_{1} + 61 \beta_{2} - 213 \beta_{3} ) q^{58} + ( -366 - 114 \beta_{1} - 6 \beta_{2} - 54 \beta_{3} ) q^{59} + ( -308 + 189 \beta_{1} + 95 \beta_{2} - 108 \beta_{3} ) q^{60} + ( 41 + 12 \beta_{1} - 202 \beta_{2} + 5 \beta_{3} ) q^{61} + ( -307 + 126 \beta_{1} + 100 \beta_{2} - 255 \beta_{3} ) q^{62} + ( -188 + 38 \beta_{1} - 434 \beta_{2} + 42 \beta_{3} ) q^{63} + ( 1142 - 27 \beta_{1} - 373 \beta_{2} + 678 \beta_{3} ) q^{64} + ( 246 - 231 \beta_{1} - 25 \beta_{2} + 158 \beta_{3} ) q^{65} + ( -96 - 34 \beta_{1} - 71 \beta_{2} - 5 \beta_{3} ) q^{66} + ( -73 + 128 \beta_{1} + 90 \beta_{2} - 69 \beta_{3} ) q^{67} + ( -152 + 70 \beta_{1} + 18 \beta_{2} - 116 \beta_{3} ) q^{68} + ( 325 + 20 \beta_{1} + 198 \beta_{2} - 17 \beta_{3} ) q^{69} + ( -396 + 14 \beta_{1} + 30 \beta_{2} - 72 \beta_{3} ) q^{70} + ( 139 - 267 \beta_{1} + 163 \beta_{2} + 62 \beta_{3} ) q^{71} + ( 339 - 321 \beta_{1} - 269 \beta_{2} + 511 \beta_{3} ) q^{72} + ( -188 + 3 \beta_{1} - \beta_{2} + 217 \beta_{3} ) q^{73} + ( -74 + 37 \beta_{2} - 37 \beta_{3} ) q^{74} + ( -141 + 234 \beta_{1} + 62 \beta_{2} + 52 \beta_{3} ) q^{75} + ( -336 - 98 \beta_{1} + 370 \beta_{2} - 364 \beta_{3} ) q^{76} + ( -237 + 213 \beta_{1} - 341 \beta_{2} - 26 \beta_{3} ) q^{77} + ( 323 - 22 \beta_{1} + 24 \beta_{2} + 131 \beta_{3} ) q^{78} + ( -97 + 27 \beta_{1} + 111 \beta_{2} - 267 \beta_{3} ) q^{79} + ( -1533 + 468 \beta_{1} + 286 \beta_{2} - 725 \beta_{3} ) q^{80} + ( 385 + 267 \beta_{1} + 57 \beta_{2} - 24 \beta_{3} ) q^{81} + ( 51 - 144 \beta_{1} - 27 \beta_{2} + 186 \beta_{3} ) q^{82} + ( -757 - 189 \beta_{1} - 35 \beta_{2} - 114 \beta_{3} ) q^{83} + ( -168 - 318 \beta_{1} - 202 \beta_{2} - 114 \beta_{3} ) q^{84} + ( 86 + 56 \beta_{1} - 60 \beta_{2} + 38 \beta_{3} ) q^{85} + ( 286 + 80 \beta_{1} - 104 \beta_{2} + 2 \beta_{3} ) q^{86} + ( 291 - 112 \beta_{1} + 18 \beta_{2} + 45 \beta_{3} ) q^{87} + ( -451 - 182 \beta_{1} + 42 \beta_{2} - 13 \beta_{3} ) q^{88} + ( 58 + 362 \beta_{1} - 182 \beta_{2} + 188 \beta_{3} ) q^{89} + ( -446 + 69 \beta_{1} + 267 \beta_{2} - 302 \beta_{3} ) q^{90} + ( -342 - 112 \beta_{1} + 372 \beta_{2} + 362 \beta_{3} ) q^{91} + ( 993 - 35 \beta_{1} - 75 \beta_{2} + 237 \beta_{3} ) q^{92} + ( 38 - 253 \beta_{1} - 251 \beta_{2} + 106 \beta_{3} ) q^{93} + ( 466 + 55 \beta_{1} - 52 \beta_{2} + 201 \beta_{3} ) q^{94} + ( 542 - 616 \beta_{1} - 12 \beta_{2} + 158 \beta_{3} ) q^{95} + ( -1187 + 55 \beta_{1} - 97 \beta_{2} - 399 \beta_{3} ) q^{96} + ( -604 - 122 \beta_{1} - 74 \beta_{2} - 80 \beta_{3} ) q^{97} + ( 542 - 231 \beta_{1} + 129 \beta_{2} + 344 \beta_{3} ) q^{98} + ( 596 + 289 \beta_{1} + 279 \beta_{2} - 214 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 6q^{2} - 11q^{3} + 6q^{4} - 29q^{5} - 27q^{6} - 32q^{7} - 90q^{8} + q^{9} + O(q^{10})$$ $$4q - 6q^{2} - 11q^{3} + 6q^{4} - 29q^{5} - 27q^{6} - 32q^{7} - 90q^{8} + q^{9} + 81q^{10} - 11q^{11} + 143q^{12} - 45q^{13} + 82q^{14} - 16q^{15} + 378q^{16} - 56q^{17} + 255q^{18} - 144q^{19} - 165q^{20} + 108q^{21} + 73q^{22} - 275q^{23} + 9q^{24} + 91q^{25} + 97q^{26} - 272q^{27} - 202q^{28} - 29q^{29} + 96q^{30} + 111q^{31} - 946q^{32} - 250q^{33} + 212q^{34} - 636q^{35} - 723q^{36} + 148q^{37} - 76q^{38} + 361q^{39} + 937q^{40} - 9q^{41} + 446q^{42} - 190q^{43} + 211q^{44} + 1018q^{45} - 269q^{46} - 472q^{47} + 203q^{48} + 1586q^{49} + 83q^{50} - 94q^{51} - 1165q^{52} + 318q^{53} + 306q^{54} + 779q^{55} + 518q^{56} + 1330q^{57} - 915q^{58} - 1530q^{59} - 840q^{60} - 31q^{61} - 747q^{62} - 1190q^{63} + 3490q^{64} + 570q^{65} - 484q^{66} - 5q^{67} - 404q^{68} + 1535q^{69} - 1468q^{70} + 390q^{71} + 255q^{72} - 967q^{73} - 222q^{74} - 320q^{75} - 708q^{76} - 1050q^{77} + 1163q^{78} + 17q^{79} - 4653q^{80} + 1888q^{81} - 153q^{82} - 3138q^{83} - 1078q^{84} + 302q^{85} + 1118q^{86} + 1025q^{87} - 1931q^{88} + 224q^{89} - 1146q^{90} - 1470q^{91} + 3625q^{92} - 458q^{93} + 1666q^{94} + 1382q^{95} - 4391q^{96} - 2532q^{97} + 1722q^{98} + 3166q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 8 x^{2} + 13 x - 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} + \nu^{2} - 6 \nu + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 7 \nu + 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} - \beta_{1} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 7 \beta_{1} - 5$$

## 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
 0.280359 1.54469 −3.07664 2.25158
−5.64104 2.22256 23.8213 −12.1011 −12.5375 −9.22618 −89.2487 −22.0602 68.2629
1.2 −2.06923 0.265551 −3.71830 −5.08678 −0.549486 −27.2773 24.2478 −26.9295 10.5257
1.3 0.389055 −4.19207 −7.84864 −19.1145 −1.63095 34.7993 −6.16599 −9.42651 −7.43658
1.4 1.32121 −9.29603 −6.25440 7.30237 −12.2820 −30.2958 −18.8331 59.4162 9.64798
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$37$$ $$-1$$

## 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 37.4.a.a 4
3.b odd 2 1 333.4.a.e 4
4.b odd 2 1 592.4.a.f 4
5.b even 2 1 925.4.a.a 4
7.b odd 2 1 1813.4.a.b 4
8.b even 2 1 2368.4.a.l 4
8.d odd 2 1 2368.4.a.g 4
37.b even 2 1 1369.4.a.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.4.a.a 4 1.a even 1 1 trivial
333.4.a.e 4 3.b odd 2 1
592.4.a.f 4 4.b odd 2 1
925.4.a.a 4 5.b even 2 1
1369.4.a.c 4 37.b even 2 1
1813.4.a.b 4 7.b odd 2 1
2368.4.a.g 4 8.d odd 2 1
2368.4.a.l 4 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 6 T_{2}^{3} - T_{2}^{2} - 16 T_{2} + 6$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(37))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$6 - 16 T - T^{2} + 6 T^{3} + T^{4}$$
$3$ $$23 - 89 T + 6 T^{2} + 11 T^{3} + T^{4}$$
$5$ $$-8592 - 1672 T + 125 T^{2} + 29 T^{3} + T^{4}$$
$7$ $$-265324 - 39618 T - 967 T^{2} + 32 T^{3} + T^{4}$$
$11$ $$169317 - 18301 T - 1432 T^{2} + 11 T^{3} + T^{4}$$
$13$ $$-4630592 - 307600 T - 4635 T^{2} + 45 T^{3} + T^{4}$$
$17$ $$-1776 - 8224 T + 344 T^{2} + 56 T^{3} + T^{4}$$
$19$ $$-115380208 - 2961344 T - 13640 T^{2} + 144 T^{3} + T^{4}$$
$23$ $$-93600684 - 1528864 T + 12461 T^{2} + 275 T^{3} + T^{4}$$
$29$ $$-20921868 - 1763168 T - 24515 T^{2} + 29 T^{3} + T^{4}$$
$31$ $$443578864 + 2426408 T - 40665 T^{2} - 111 T^{3} + T^{4}$$
$37$ $$( -37 + T )^{4}$$
$41$ $$239475447 - 2562219 T - 55962 T^{2} + 9 T^{3} + T^{4}$$
$43$ $$118152128 - 1558016 T - 41956 T^{2} + 190 T^{3} + T^{4}$$
$47$ $$-4884453612 - 63380002 T - 140511 T^{2} + 472 T^{3} + T^{4}$$
$53$ $$1515396 + 133348 T - 4891 T^{2} - 318 T^{3} + T^{4}$$
$59$ $$-19839940416 + 45543168 T + 687780 T^{2} + 1530 T^{3} + T^{4}$$
$61$ $$-1974966496 + 76291616 T - 404631 T^{2} + 31 T^{3} + T^{4}$$
$67$ $$3584880464 - 31714408 T - 234035 T^{2} + 5 T^{3} + T^{4}$$
$71$ $$-21095220732 + 225705452 T - 548443 T^{2} - 390 T^{3} + T^{4}$$
$73$ $$-10092864749 - 240493869 T - 141322 T^{2} + 967 T^{3} + T^{4}$$
$79$ $$-29390715884 + 266725624 T - 653187 T^{2} - 17 T^{3} + T^{4}$$
$83$ $$-146330370096 + 725722108 T + 3026753 T^{2} + 3138 T^{3} + T^{4}$$
$89$ $$-212866753728 + 1237975760 T - 1689756 T^{2} - 224 T^{3} + T^{4}$$
$97$ $$3402009472 + 487325120 T + 1996596 T^{2} + 2532 T^{3} + T^{4}$$