Properties

 Label 368.6.a.e Level $368$ Weight $6$ Character orbit 368.a Self dual yes Analytic conductor $59.021$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$368 = 2^{4} \cdot 23$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 368.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$59.0212456912$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.7925.1 Defining polynomial: $$x^{3} - x^{2} - 13x + 12$$ x^3 - x^2 - 13*x + 12 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 23) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 2 \beta_1 + 7) q^{3} + (2 \beta_{2} - 3 \beta_1 - 21) q^{5} + (4 \beta_{2} - 17 \beta_1 + 87) q^{7} + (41 \beta_{2} + 25 \beta_1 + 35) q^{9}+O(q^{10})$$ q + (b2 + 2*b1 + 7) * q^3 + (2*b2 - 3*b1 - 21) * q^5 + (4*b2 - 17*b1 + 87) * q^7 + (41*b2 + 25*b1 + 35) * q^9 $$q + (\beta_{2} + 2 \beta_1 + 7) q^{3} + (2 \beta_{2} - 3 \beta_1 - 21) q^{5} + (4 \beta_{2} - 17 \beta_1 + 87) q^{7} + (41 \beta_{2} + 25 \beta_1 + 35) q^{9} + ( - 28 \beta_{2} - 3 \beta_1 - 37) q^{11} + ( - 99 \beta_{2} + 45 \beta_1 - 324) q^{13} + ( - 30 \beta_{2} - 27 \beta_1 - 249) q^{15} + ( - 146 \beta_{2} - 57 \beta_1 - 269) q^{17} + ( - 200 \beta_{2} - 40 \beta_1 - 498) q^{19} + ( - 52 \beta_{2} + 193 \beta_1 - 475) q^{21} + 529 q^{23} + ( - 88 \beta_{2} + 86 \beta_1 - 2391) q^{25} + (559 \beta_{2} - 22 \beta_1 + 3373) q^{27} + (523 \beta_{2} - 745 \beta_1 - 704) q^{29} + ( - 479 \beta_{2} + 914 \beta_1 + 1471) q^{31} + ( - 406 \beta_{2} - 329 \beta_1 - 2431) q^{33} + (148 \beta_{2} - 160 \beta_1 - 460) q^{35} + (1662 \beta_{2} - 1117 \beta_1 + 2053) q^{37} + ( - 1017 \beta_{2} - 1494 \beta_1 - 5499) q^{39} + ( - 1847 \beta_{2} + 1171 \beta_1 - 3258) q^{41} + (1150 \beta_{2} - 2050 \beta_1 + 4510) q^{43} + ( - 1392 \beta_{2} - 66 \beta_1 - 870) q^{45} + (1021 \beta_{2} + 682 \beta_1 - 7613) q^{47} + (1428 \beta_{2} - 4306 \beta_1 - 949) q^{49} + ( - 2648 \beta_{2} - 1909 \beta_1 - 16517) q^{51} + (534 \beta_{2} + 4444 \beta_1 + 7006) q^{53} + (840 \beta_{2} + 14 \beta_1 - 130) q^{55} + ( - 3338 \beta_{2} - 2836 \beta_1 - 20486) q^{57} + ( - 1980 \beta_{2} - 934 \beta_1 - 17710) q^{59} + ( - 42 \beta_{2} + 1322 \beta_1 - 23320) q^{61} + (52 \beta_{2} + 2906 \beta_1 - 12614) q^{63} + (2250 \beta_{2} + 351 \beta_1 - 351) q^{65} + (1436 \beta_{2} + 4323 \beta_1 + 21949) q^{67} + (529 \beta_{2} + 1058 \beta_1 + 3703) q^{69} + ( - 2227 \beta_{2} + 1628 \beta_1 - 31515) q^{71} + ( - 21 \beta_{2} + 3241 \beta_1 - 26170) q^{73} + ( - 2501 \beta_{2} - 5488 \beta_1 - 15929) q^{75} + ( - 876 \beta_{2} + 532 \beta_1 + 368) q^{77} + (6662 \beta_{2} + 4778 \beta_1 - 20036) q^{79} + ( - 124 \beta_{2} + 5680 \beta_1 + 51917) q^{81} + (13758 \beta_{2} - 5867 \beta_1 - 9839) q^{83} + (4476 \beta_{2} + 508 \beta_1 + 3856) q^{85} + ( - 2623 \beta_{2} + 2554 \beta_1 - 28441) q^{87} + (8666 \beta_{2} - 5964 \beta_1 + 4304) q^{89} + ( - 6984 \beta_{2} + 14229 \beta_1 - 43587) q^{91} + (5777 \beta_{2} - 455 \beta_1 + 50366) q^{93} + (5644 \beta_{2} + 894 \beta_1 + 5298) q^{95} + (618 \beta_{2} + 3351 \beta_1 - 90313) q^{97} + ( - 4118 \beta_{2} - 8116 \beta_1 - 62360) q^{99}+O(q^{100})$$ q + (b2 + 2*b1 + 7) * q^3 + (2*b2 - 3*b1 - 21) * q^5 + (4*b2 - 17*b1 + 87) * q^7 + (41*b2 + 25*b1 + 35) * q^9 + (-28*b2 - 3*b1 - 37) * q^11 + (-99*b2 + 45*b1 - 324) * q^13 + (-30*b2 - 27*b1 - 249) * q^15 + (-146*b2 - 57*b1 - 269) * q^17 + (-200*b2 - 40*b1 - 498) * q^19 + (-52*b2 + 193*b1 - 475) * q^21 + 529 * q^23 + (-88*b2 + 86*b1 - 2391) * q^25 + (559*b2 - 22*b1 + 3373) * q^27 + (523*b2 - 745*b1 - 704) * q^29 + (-479*b2 + 914*b1 + 1471) * q^31 + (-406*b2 - 329*b1 - 2431) * q^33 + (148*b2 - 160*b1 - 460) * q^35 + (1662*b2 - 1117*b1 + 2053) * q^37 + (-1017*b2 - 1494*b1 - 5499) * q^39 + (-1847*b2 + 1171*b1 - 3258) * q^41 + (1150*b2 - 2050*b1 + 4510) * q^43 + (-1392*b2 - 66*b1 - 870) * q^45 + (1021*b2 + 682*b1 - 7613) * q^47 + (1428*b2 - 4306*b1 - 949) * q^49 + (-2648*b2 - 1909*b1 - 16517) * q^51 + (534*b2 + 4444*b1 + 7006) * q^53 + (840*b2 + 14*b1 - 130) * q^55 + (-3338*b2 - 2836*b1 - 20486) * q^57 + (-1980*b2 - 934*b1 - 17710) * q^59 + (-42*b2 + 1322*b1 - 23320) * q^61 + (52*b2 + 2906*b1 - 12614) * q^63 + (2250*b2 + 351*b1 - 351) * q^65 + (1436*b2 + 4323*b1 + 21949) * q^67 + (529*b2 + 1058*b1 + 3703) * q^69 + (-2227*b2 + 1628*b1 - 31515) * q^71 + (-21*b2 + 3241*b1 - 26170) * q^73 + (-2501*b2 - 5488*b1 - 15929) * q^75 + (-876*b2 + 532*b1 + 368) * q^77 + (6662*b2 + 4778*b1 - 20036) * q^79 + (-124*b2 + 5680*b1 + 51917) * q^81 + (13758*b2 - 5867*b1 - 9839) * q^83 + (4476*b2 + 508*b1 + 3856) * q^85 + (-2623*b2 + 2554*b1 - 28441) * q^87 + (8666*b2 - 5964*b1 + 4304) * q^89 + (-6984*b2 + 14229*b1 - 43587) * q^91 + (5777*b2 - 455*b1 + 50366) * q^93 + (5644*b2 + 894*b1 + 5298) * q^95 + (618*b2 + 3351*b1 - 90313) * q^97 + (-4118*b2 - 8116*b1 - 62360) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 20 q^{3} - 58 q^{5} + 282 q^{7} + 121 q^{9}+O(q^{10})$$ 3 * q + 20 * q^3 - 58 * q^5 + 282 * q^7 + 121 * q^9 $$3 q + 20 q^{3} - 58 q^{5} + 282 q^{7} + 121 q^{9} - 136 q^{11} - 1116 q^{13} - 750 q^{15} - 896 q^{17} - 1654 q^{19} - 1670 q^{21} + 1587 q^{23} - 7347 q^{25} + 10700 q^{27} - 844 q^{29} + 3020 q^{31} - 7370 q^{33} - 1072 q^{35} + 8938 q^{37} - 16020 q^{39} - 12792 q^{41} + 16730 q^{43} - 3936 q^{45} - 22500 q^{47} + 2887 q^{49} - 50290 q^{51} + 17108 q^{53} + 436 q^{55} - 61960 q^{57} - 54176 q^{59} - 71324 q^{61} - 40696 q^{63} + 846 q^{65} + 62960 q^{67} + 10580 q^{69} - 98400 q^{71} - 81772 q^{73} - 44800 q^{75} - 304 q^{77} - 58224 q^{79} + 149947 q^{81} - 9892 q^{83} + 15536 q^{85} - 90500 q^{87} + 27542 q^{89} - 151974 q^{91} + 157330 q^{93} + 20644 q^{95} - 273672 q^{97} - 183082 q^{99}+O(q^{100})$$ 3 * q + 20 * q^3 - 58 * q^5 + 282 * q^7 + 121 * q^9 - 136 * q^11 - 1116 * q^13 - 750 * q^15 - 896 * q^17 - 1654 * q^19 - 1670 * q^21 + 1587 * q^23 - 7347 * q^25 + 10700 * q^27 - 844 * q^29 + 3020 * q^31 - 7370 * q^33 - 1072 * q^35 + 8938 * q^37 - 16020 * q^39 - 12792 * q^41 + 16730 * q^43 - 3936 * q^45 - 22500 * q^47 + 2887 * q^49 - 50290 * q^51 + 17108 * q^53 + 436 * q^55 - 61960 * q^57 - 54176 * q^59 - 71324 * q^61 - 40696 * q^63 + 846 * q^65 + 62960 * q^67 + 10580 * q^69 - 98400 * q^71 - 81772 * q^73 - 44800 * q^75 - 304 * q^77 - 58224 * q^79 + 149947 * q^81 - 9892 * q^83 + 15536 * q^85 - 90500 * q^87 + 27542 * q^89 - 151974 * q^91 + 157330 * q^93 + 20644 * q^95 - 273672 * q^97 - 183082 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 13x + 12$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 9$$ v^2 + v - 9
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{2} - \beta _1 + 17 ) / 2$$ (2*b2 - b1 + 17) / 2

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
 −3.57511 0.917748 3.65736
0 −9.09413 0 3.86330 0 226.379 0 −160.297 0
1.2 0 1.43100 0 −37.9865 0 43.8366 0 −240.952 0
1.3 0 27.6631 0 −23.8768 0 11.7843 0 522.249 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$23$$ $$-1$$

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 368.6.a.e 3
4.b odd 2 1 23.6.a.a 3
12.b even 2 1 207.6.a.b 3
20.d odd 2 1 575.6.a.b 3
92.b even 2 1 529.6.a.a 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.6.a.a 3 4.b odd 2 1
207.6.a.b 3 12.b even 2 1
368.6.a.e 3 1.a even 1 1 trivial
529.6.a.a 3 92.b even 2 1
575.6.a.b 3 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} - 20T_{3}^{2} - 225T_{3} + 360$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(368))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 20 T^{2} - 225 T + 360$$
$5$ $$T^{3} + 58 T^{2} + 668 T - 3504$$
$7$ $$T^{3} - 282 T^{2} + 13108 T - 116944$$
$11$ $$T^{3} + 136 T^{2} - 43688 T - 840152$$
$13$ $$T^{3} + 1116 T^{2} + \cdots - 255615102$$
$17$ $$T^{3} + 896 T^{2} + \cdots + 220718408$$
$19$ $$T^{3} + 1654 T^{2} + \cdots - 460771768$$
$23$ $$(T - 529)^{3}$$
$29$ $$T^{3} + 844 T^{2} + \cdots - 33789223458$$
$31$ $$T^{3} - 3020 T^{2} + \cdots + 117638912880$$
$37$ $$T^{3} - 8938 T^{2} + \cdots + 1048082031344$$
$41$ $$T^{3} + 12792 T^{2} + \cdots - 1564944049486$$
$43$ $$T^{3} - 16730 T^{2} + \cdots + 95315904000$$
$47$ $$T^{3} + 22500 T^{2} + \cdots - 916008439440$$
$53$ $$T^{3} - 17108 T^{2} + \cdots + 7849670295504$$
$59$ $$T^{3} + 54176 T^{2} + \cdots + 1725012447168$$
$61$ $$T^{3} + 71324 T^{2} + \cdots + 11439907465152$$
$67$ $$T^{3} - 62960 T^{2} + \cdots + 11971711378840$$
$71$ $$T^{3} + 98400 T^{2} + \cdots + 24837760695040$$
$73$ $$T^{3} + 81772 T^{2} + \cdots + 7199078503954$$
$79$ $$T^{3} + \cdots - 235690469012368$$
$83$ $$T^{3} + \cdots + 297029282761704$$
$89$ $$T^{3} + \cdots + 125799322340896$$
$97$ $$T^{3} + \cdots + 693755159518744$$