# Properties

 Label 384.10.a.f Level $384$ Weight $10$ Character orbit 384.a Self dual yes Analytic conductor $197.774$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 384.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$197.773761087$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - 2070x^{2} - 13768x + 561570$$ x^4 - 2070*x^2 - 13768*x + 561570 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{15}\cdot 3$$ 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.

 $$f(q)$$ $$=$$ $$q + 81 q^{3} + ( - \beta_{2} - 60) q^{5} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 1210) q^{7} + 6561 q^{9}+O(q^{10})$$ q + 81 * q^3 + (-b2 - 60) * q^5 + (b3 + 2*b2 + b1 + 1210) * q^7 + 6561 * q^9 $$q + 81 q^{3} + ( - \beta_{2} - 60) q^{5} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 1210) q^{7} + 6561 q^{9} + ( - 4 \beta_{3} + 13 \beta_{2} - 5 \beta_1 - 24916) q^{11} + ( - 19 \beta_{3} + 23 \beta_{2} - 5 \beta_1 - 15210) q^{13} + ( - 81 \beta_{2} - 4860) q^{15} + (10 \beta_{3} - 83 \beta_{2} - 41 \beta_1 - 108738) q^{17} + ( - 30 \beta_{3} + 29 \beta_{2} + 55 \beta_1 + 157944) q^{19} + (81 \beta_{3} + 162 \beta_{2} + 81 \beta_1 + 98010) q^{21} + (174 \beta_{3} + 216 \beta_{2} - 146 \beta_1 + 187348) q^{23} + (494 \beta_{3} - 150 \beta_{2} + 698 \beta_1 + 1497883) q^{25} + 531441 q^{27} + (202 \beta_{3} + 41 \beta_{2} - 556 \beta_1 - 1977136) q^{29} + ( - 171 \beta_{3} - 1072 \beta_{2} + 663 \beta_1 + 2837810) q^{31} + ( - 324 \beta_{3} + 1053 \beta_{2} - 405 \beta_1 - 2018196) q^{33} + ( - 1096 \beta_{3} - 6049 \beta_{2} - 1607 \beta_1 - 6391752) q^{35} + ( - 2949 \beta_{3} + 1123 \beta_{2} - 1571 \beta_1 - 3398230) q^{37} + ( - 1539 \beta_{3} + 1863 \beta_{2} - 405 \beta_1 - 1232010) q^{39} + (1018 \beta_{3} + 9263 \beta_{2} - 1171 \beta_1 - 4709722) q^{41} + ( - 30 \beta_{3} - 8681 \beta_{2} + 1509 \beta_1 + 3544480) q^{43} + ( - 6561 \beta_{2} - 393660) q^{45} + ( - 3450 \beta_{3} - 2440 \beta_{2} - 6662 \beta_1 - 9444780) q^{47} + (1774 \beta_{3} + 24828 \beta_{2} + 6296 \beta_1 + 2352333) q^{49} + (810 \beta_{3} - 6723 \beta_{2} - 3321 \beta_1 - 8807778) q^{51} + ( - 2334 \beta_{3} + 31753 \beta_{2} + 1480 \beta_1 + 28834128) q^{53} + ( - 6008 \beta_{3} + 52550 \beta_{2} - 7966 \beta_1 - 46145136) q^{55} + ( - 2430 \beta_{3} + 2349 \beta_{2} + 4455 \beta_1 + 12793464) q^{57} + (1924 \beta_{3} - 19436 \beta_{2} - 4356 \beta_1 - 28757020) q^{59} + (4043 \beta_{3} + 38587 \beta_{2} + 5041 \beta_1 + 43307162) q^{61} + (6561 \beta_{3} + 13122 \beta_{2} + 6561 \beta_1 + 7938810) q^{63} + ( - 9058 \beta_{3} + 65809 \beta_{2} - 15741 \beta_1 - 82019496) q^{65} + ( - 42716 \beta_{3} - 34324 \beta_{2} - 17388 \beta_1 - 57946276) q^{67} + (14094 \beta_{3} + 17496 \beta_{2} - 11826 \beta_1 + 15175188) q^{69} + (39230 \beta_{3} + 20804 \beta_{2} + 51562 \beta_1 + 49369052) q^{71} + (33290 \beta_{3} - 38004 \beta_{2} - 8672 \beta_1 + 11157350) q^{73} + (40014 \beta_{3} - 12150 \beta_{2} + 56538 \beta_1 + 121328523) q^{75} + ( - 8096 \beta_{3} - 22982 \beta_{2} - 14662 \beta_1 - 77029480) q^{77} + ( - 25099 \beta_{3} + 98534 \beta_{2} - 7939 \beta_1 - 88943646) q^{79} + 43046721 q^{81} + (60312 \beta_{3} - 79319 \beta_{2} + 20983 \beta_1 - 151903332) q^{83} + (39004 \beta_{3} + 235986 \beta_{2} + 69288 \beta_1 + 271837848) q^{85} + (16362 \beta_{3} + 3321 \beta_{2} - 45036 \beta_1 - 160148016) q^{87} + ( - 114996 \beta_{3} - 68962 \beta_{2} + \cdots - 289286606) q^{89}+ \cdots + ( - 26244 \beta_{3} + 85293 \beta_{2} - 32805 \beta_1 - 163473876) q^{99}+O(q^{100})$$ q + 81 * q^3 + (-b2 - 60) * q^5 + (b3 + 2*b2 + b1 + 1210) * q^7 + 6561 * q^9 + (-4*b3 + 13*b2 - 5*b1 - 24916) * q^11 + (-19*b3 + 23*b2 - 5*b1 - 15210) * q^13 + (-81*b2 - 4860) * q^15 + (10*b3 - 83*b2 - 41*b1 - 108738) * q^17 + (-30*b3 + 29*b2 + 55*b1 + 157944) * q^19 + (81*b3 + 162*b2 + 81*b1 + 98010) * q^21 + (174*b3 + 216*b2 - 146*b1 + 187348) * q^23 + (494*b3 - 150*b2 + 698*b1 + 1497883) * q^25 + 531441 * q^27 + (202*b3 + 41*b2 - 556*b1 - 1977136) * q^29 + (-171*b3 - 1072*b2 + 663*b1 + 2837810) * q^31 + (-324*b3 + 1053*b2 - 405*b1 - 2018196) * q^33 + (-1096*b3 - 6049*b2 - 1607*b1 - 6391752) * q^35 + (-2949*b3 + 1123*b2 - 1571*b1 - 3398230) * q^37 + (-1539*b3 + 1863*b2 - 405*b1 - 1232010) * q^39 + (1018*b3 + 9263*b2 - 1171*b1 - 4709722) * q^41 + (-30*b3 - 8681*b2 + 1509*b1 + 3544480) * q^43 + (-6561*b2 - 393660) * q^45 + (-3450*b3 - 2440*b2 - 6662*b1 - 9444780) * q^47 + (1774*b3 + 24828*b2 + 6296*b1 + 2352333) * q^49 + (810*b3 - 6723*b2 - 3321*b1 - 8807778) * q^51 + (-2334*b3 + 31753*b2 + 1480*b1 + 28834128) * q^53 + (-6008*b3 + 52550*b2 - 7966*b1 - 46145136) * q^55 + (-2430*b3 + 2349*b2 + 4455*b1 + 12793464) * q^57 + (1924*b3 - 19436*b2 - 4356*b1 - 28757020) * q^59 + (4043*b3 + 38587*b2 + 5041*b1 + 43307162) * q^61 + (6561*b3 + 13122*b2 + 6561*b1 + 7938810) * q^63 + (-9058*b3 + 65809*b2 - 15741*b1 - 82019496) * q^65 + (-42716*b3 - 34324*b2 - 17388*b1 - 57946276) * q^67 + (14094*b3 + 17496*b2 - 11826*b1 + 15175188) * q^69 + (39230*b3 + 20804*b2 + 51562*b1 + 49369052) * q^71 + (33290*b3 - 38004*b2 - 8672*b1 + 11157350) * q^73 + (40014*b3 - 12150*b2 + 56538*b1 + 121328523) * q^75 + (-8096*b3 - 22982*b2 - 14662*b1 - 77029480) * q^77 + (-25099*b3 + 98534*b2 - 7939*b1 - 88943646) * q^79 + 43046721 * q^81 + (60312*b3 - 79319*b2 + 20983*b1 - 151903332) * q^83 + (39004*b3 + 235986*b2 + 69288*b1 + 271837848) * q^85 + (16362*b3 + 3321*b2 - 45036*b1 - 160148016) * q^87 + (-114996*b3 - 68962*b2 - 6142*b1 - 289286606) * q^89 + (74790*b3 + 17875*b2 + 8969*b1 - 211907460) * q^91 + (-13851*b3 - 86832*b2 + 53703*b1 + 229862610) * q^93 + (-9556*b3 - 322864*b2 - 36352*b1 - 82424832) * q^95 + (36136*b3 + 41750*b2 + 49562*b1 - 399884118) * q^97 + (-26244*b3 + 85293*b2 - 32805*b1 - 163473876) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 324 q^{3} - 240 q^{5} + 4840 q^{7} + 26244 q^{9}+O(q^{10})$$ 4 * q + 324 * q^3 - 240 * q^5 + 4840 * q^7 + 26244 * q^9 $$4 q + 324 q^{3} - 240 q^{5} + 4840 q^{7} + 26244 q^{9} - 99664 q^{11} - 60840 q^{13} - 19440 q^{15} - 434952 q^{17} + 631776 q^{19} + 392040 q^{21} + 749392 q^{23} + 5991532 q^{25} + 2125764 q^{27} - 7908544 q^{29} + 11351240 q^{31} - 8072784 q^{33} - 25567008 q^{35} - 13592920 q^{37} - 4928040 q^{39} - 18838888 q^{41} + 14177920 q^{43} - 1574640 q^{45} - 37779120 q^{47} + 9409332 q^{49} - 35231112 q^{51} + 115336512 q^{53} - 184580544 q^{55} + 51173856 q^{57} - 115028080 q^{59} + 173228648 q^{61} + 31755240 q^{63} - 328077984 q^{65} - 231785104 q^{67} + 60700752 q^{69} + 197476208 q^{71} + 44629400 q^{73} + 485314092 q^{75} - 308117920 q^{77} - 355774584 q^{79} + 172186884 q^{81} - 607613328 q^{83} + 1087351392 q^{85} - 640592064 q^{87} - 1157146424 q^{89} - 847629840 q^{91} + 919450440 q^{93} - 329699328 q^{95} - 1599536472 q^{97} - 653895504 q^{99}+O(q^{100})$$ 4 * q + 324 * q^3 - 240 * q^5 + 4840 * q^7 + 26244 * q^9 - 99664 * q^11 - 60840 * q^13 - 19440 * q^15 - 434952 * q^17 + 631776 * q^19 + 392040 * q^21 + 749392 * q^23 + 5991532 * q^25 + 2125764 * q^27 - 7908544 * q^29 + 11351240 * q^31 - 8072784 * q^33 - 25567008 * q^35 - 13592920 * q^37 - 4928040 * q^39 - 18838888 * q^41 + 14177920 * q^43 - 1574640 * q^45 - 37779120 * q^47 + 9409332 * q^49 - 35231112 * q^51 + 115336512 * q^53 - 184580544 * q^55 + 51173856 * q^57 - 115028080 * q^59 + 173228648 * q^61 + 31755240 * q^63 - 328077984 * q^65 - 231785104 * q^67 + 60700752 * q^69 + 197476208 * q^71 + 44629400 * q^73 + 485314092 * q^75 - 308117920 * q^77 - 355774584 * q^79 + 172186884 * q^81 - 607613328 * q^83 + 1087351392 * q^85 - 640592064 * q^87 - 1157146424 * q^89 - 847629840 * q^91 + 919450440 * q^93 - 329699328 * q^95 - 1599536472 * q^97 - 653895504 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2070x^{2} - 13768x + 561570$$ :

 $$\beta_{1}$$ $$=$$ $$( -88\nu^{3} - 100\nu^{2} + 76928\nu + 1012188 ) / 483$$ (-88*v^3 - 100*v^2 + 76928*v + 1012188) / 483 $$\beta_{2}$$ $$=$$ $$( -8\nu^{3} + 1396\nu^{2} - 15488\nu - 1362252 ) / 483$$ (-8*v^3 + 1396*v^2 - 15488*v - 1362252) / 483 $$\beta_{3}$$ $$=$$ $$( -160\nu^{3} + 4736\nu^{2} + 246656\nu - 3249600 ) / 483$$ (-160*v^3 + 4736*v^2 + 246656*v - 3249600) / 483
 $$\nu$$ $$=$$ $$( 2\beta_{3} - 7\beta_{2} - 3\beta_1 ) / 768$$ (2*b3 - 7*b2 - 3*b1) / 768 $$\nu^{2}$$ $$=$$ $$( 4\beta_{3} + 19\beta_{2} - 9\beta _1 + 99360 ) / 96$$ (4*b3 + 19*b2 - 9*b1 + 99360) / 96 $$\nu^{3}$$ $$=$$ $$( 428\beta_{3} - 1573\beta_{2} - 1689\beta _1 + 1982592 ) / 192$$ (428*b3 - 1573*b2 - 1689*b1 + 1982592) / 192

## 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
 −34.7879 45.8607 −25.0977 14.0249
0 81.0000 0 −2550.24 0 11484.8 0 6561.00 0
1.2 0 81.0000 0 −250.262 0 −1655.62 0 6561.00 0
1.3 0 81.0000 0 −126.806 0 −5939.66 0 6561.00 0
1.4 0 81.0000 0 2687.31 0 950.516 0 6561.00 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
$$2$$ $$-1$$
$$3$$ $$-1$$

## 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 384.10.a.f yes 4
4.b odd 2 1 384.10.a.b 4
8.b even 2 1 384.10.a.c yes 4
8.d odd 2 1 384.10.a.g yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.10.a.b 4 4.b odd 2 1
384.10.a.c yes 4 8.b even 2 1
384.10.a.f yes 4 1.a even 1 1 trivial
384.10.a.g yes 4 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 $$S_{10}^{\mathrm{new}}(\Gamma_0(384))$$:

 $$T_{5}^{4} + 240T_{5}^{3} - 6873216T_{5}^{2} - 2588500480T_{5} - 217486924800$$ T5^4 + 240*T5^3 - 6873216*T5^2 - 2588500480*T5 - 217486924800 $$T_{7}^{4} - 4840T_{7}^{3} - 73699080T_{7}^{2} - 39372619552T_{7} + 107350205672080$$ T7^4 - 4840*T7^3 - 73699080*T7^2 - 39372619552*T7 + 107350205672080

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T - 81)^{4}$$
$5$ $$T^{4} + 240 T^{3} + \cdots - 217486924800$$
$7$ $$T^{4} + \cdots + 107350205672080$$
$11$ $$T^{4} + 99664 T^{3} + \cdots - 54\!\cdots\!24$$
$13$ $$T^{4} + 60840 T^{3} + \cdots - 34\!\cdots\!24$$
$17$ $$T^{4} + 434952 T^{3} + \cdots - 64\!\cdots\!80$$
$19$ $$T^{4} - 631776 T^{3} + \cdots - 21\!\cdots\!60$$
$23$ $$T^{4} - 749392 T^{3} + \cdots + 85\!\cdots\!00$$
$29$ $$T^{4} + 7908544 T^{3} + \cdots - 23\!\cdots\!00$$
$31$ $$T^{4} - 11351240 T^{3} + \cdots - 39\!\cdots\!00$$
$37$ $$T^{4} + 13592920 T^{3} + \cdots - 18\!\cdots\!52$$
$41$ $$T^{4} + 18838888 T^{3} + \cdots + 83\!\cdots\!80$$
$43$ $$T^{4} - 14177920 T^{3} + \cdots + 20\!\cdots\!36$$
$47$ $$T^{4} + 37779120 T^{3} + \cdots + 56\!\cdots\!60$$
$53$ $$T^{4} - 115336512 T^{3} + \cdots - 30\!\cdots\!88$$
$59$ $$T^{4} + 115028080 T^{3} + \cdots + 76\!\cdots\!28$$
$61$ $$T^{4} - 173228648 T^{3} + \cdots - 31\!\cdots\!40$$
$67$ $$T^{4} + 231785104 T^{3} + \cdots - 96\!\cdots\!32$$
$71$ $$T^{4} - 197476208 T^{3} + \cdots + 35\!\cdots\!52$$
$73$ $$T^{4} - 44629400 T^{3} + \cdots + 67\!\cdots\!08$$
$79$ $$T^{4} + 355774584 T^{3} + \cdots + 61\!\cdots\!44$$
$83$ $$T^{4} + 607613328 T^{3} + \cdots - 66\!\cdots\!96$$
$89$ $$T^{4} + 1157146424 T^{3} + \cdots - 10\!\cdots\!80$$
$97$ $$T^{4} + 1599536472 T^{3} + \cdots + 57\!\cdots\!00$$