# Properties

 Label 138.6.a.f Level $138$ Weight $6$ Character orbit 138.a Self dual yes Analytic conductor $22.133$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$138 = 2 \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 138.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$22.1329671342$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{514})$$ Defining polynomial: $$x^{2} - 514$$ x^2 - 514 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$\beta = \sqrt{514}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 q^{2} - 9 q^{3} + 16 q^{4} + (3 \beta + 20) q^{5} - 36 q^{6} + (5 \beta - 50) q^{7} + 64 q^{8} + 81 q^{9}+O(q^{10})$$ q + 4 * q^2 - 9 * q^3 + 16 * q^4 + (3*b + 20) * q^5 - 36 * q^6 + (5*b - 50) * q^7 + 64 * q^8 + 81 * q^9 $$q + 4 q^{2} - 9 q^{3} + 16 q^{4} + (3 \beta + 20) q^{5} - 36 q^{6} + (5 \beta - 50) q^{7} + 64 q^{8} + 81 q^{9} + (12 \beta + 80) q^{10} + ( - 8 \beta + 80) q^{11} - 144 q^{12} + ( - 24 \beta + 316) q^{13} + (20 \beta - 200) q^{14} + ( - 27 \beta - 180) q^{15} + 256 q^{16} + ( - 3 \beta + 608) q^{17} + 324 q^{18} + ( - 21 \beta + 1462) q^{19} + (48 \beta + 320) q^{20} + ( - 45 \beta + 450) q^{21} + ( - 32 \beta + 320) q^{22} - 529 q^{23} - 576 q^{24} + (120 \beta + 1901) q^{25} + ( - 96 \beta + 1264) q^{26} - 729 q^{27} + (80 \beta - 800) q^{28} + ( - 102 \beta + 2650) q^{29} + ( - 108 \beta - 720) q^{30} + (182 \beta + 1256) q^{31} + 1024 q^{32} + (72 \beta - 720) q^{33} + ( - 12 \beta + 2432) q^{34} + ( - 50 \beta + 6710) q^{35} + 1296 q^{36} + (174 \beta + 10934) q^{37} + ( - 84 \beta + 5848) q^{38} + (216 \beta - 2844) q^{39} + (192 \beta + 1280) q^{40} + (128 \beta + 4766) q^{41} + ( - 180 \beta + 1800) q^{42} + ( - 259 \beta + 3306) q^{43} + ( - 128 \beta + 1280) q^{44} + (243 \beta + 1620) q^{45} - 2116 q^{46} + ( - 206 \beta + 1930) q^{47} - 2304 q^{48} + ( - 500 \beta - 1457) q^{49} + (480 \beta + 7604) q^{50} + (27 \beta - 5472) q^{51} + ( - 384 \beta + 5056) q^{52} + ( - 577 \beta + 13872) q^{53} - 2916 q^{54} + (80 \beta - 10736) q^{55} + (320 \beta - 3200) q^{56} + (189 \beta - 13158) q^{57} + ( - 408 \beta + 10600) q^{58} + ( - 50 \beta - 29570) q^{59} + ( - 432 \beta - 2880) q^{60} + ( - 46 \beta - 2862) q^{61} + (728 \beta + 5024) q^{62} + (405 \beta - 4050) q^{63} + 4096 q^{64} + (468 \beta - 30688) q^{65} + (288 \beta - 2880) q^{66} + ( - 1803 \beta - 23062) q^{67} + ( - 48 \beta + 9728) q^{68} + 4761 q^{69} + ( - 200 \beta + 26840) q^{70} + (816 \beta - 8160) q^{71} + 5184 q^{72} + (2028 \beta + 4878) q^{73} + (696 \beta + 43736) q^{74} + ( - 1080 \beta - 17109) q^{75} + ( - 336 \beta + 23392) q^{76} + (800 \beta - 24560) q^{77} + (864 \beta - 11376) q^{78} + ( - 1811 \beta - 1514) q^{79} + (768 \beta + 5120) q^{80} + 6561 q^{81} + (512 \beta + 19064) q^{82} + ( - 1432 \beta + 30780) q^{83} + ( - 720 \beta + 7200) q^{84} + (1764 \beta + 7534) q^{85} + ( - 1036 \beta + 13224) q^{86} + (918 \beta - 23850) q^{87} + ( - 512 \beta + 5120) q^{88} + ( - 4671 \beta + 32796) q^{89} + (972 \beta + 6480) q^{90} + (2780 \beta - 77480) q^{91} - 8464 q^{92} + ( - 1638 \beta - 11304) q^{93} + ( - 824 \beta + 7720) q^{94} + (3966 \beta - 3142) q^{95} - 9216 q^{96} + (5306 \beta - 53362) q^{97} + ( - 2000 \beta - 5828) q^{98} + ( - 648 \beta + 6480) q^{99}+O(q^{100})$$ q + 4 * q^2 - 9 * q^3 + 16 * q^4 + (3*b + 20) * q^5 - 36 * q^6 + (5*b - 50) * q^7 + 64 * q^8 + 81 * q^9 + (12*b + 80) * q^10 + (-8*b + 80) * q^11 - 144 * q^12 + (-24*b + 316) * q^13 + (20*b - 200) * q^14 + (-27*b - 180) * q^15 + 256 * q^16 + (-3*b + 608) * q^17 + 324 * q^18 + (-21*b + 1462) * q^19 + (48*b + 320) * q^20 + (-45*b + 450) * q^21 + (-32*b + 320) * q^22 - 529 * q^23 - 576 * q^24 + (120*b + 1901) * q^25 + (-96*b + 1264) * q^26 - 729 * q^27 + (80*b - 800) * q^28 + (-102*b + 2650) * q^29 + (-108*b - 720) * q^30 + (182*b + 1256) * q^31 + 1024 * q^32 + (72*b - 720) * q^33 + (-12*b + 2432) * q^34 + (-50*b + 6710) * q^35 + 1296 * q^36 + (174*b + 10934) * q^37 + (-84*b + 5848) * q^38 + (216*b - 2844) * q^39 + (192*b + 1280) * q^40 + (128*b + 4766) * q^41 + (-180*b + 1800) * q^42 + (-259*b + 3306) * q^43 + (-128*b + 1280) * q^44 + (243*b + 1620) * q^45 - 2116 * q^46 + (-206*b + 1930) * q^47 - 2304 * q^48 + (-500*b - 1457) * q^49 + (480*b + 7604) * q^50 + (27*b - 5472) * q^51 + (-384*b + 5056) * q^52 + (-577*b + 13872) * q^53 - 2916 * q^54 + (80*b - 10736) * q^55 + (320*b - 3200) * q^56 + (189*b - 13158) * q^57 + (-408*b + 10600) * q^58 + (-50*b - 29570) * q^59 + (-432*b - 2880) * q^60 + (-46*b - 2862) * q^61 + (728*b + 5024) * q^62 + (405*b - 4050) * q^63 + 4096 * q^64 + (468*b - 30688) * q^65 + (288*b - 2880) * q^66 + (-1803*b - 23062) * q^67 + (-48*b + 9728) * q^68 + 4761 * q^69 + (-200*b + 26840) * q^70 + (816*b - 8160) * q^71 + 5184 * q^72 + (2028*b + 4878) * q^73 + (696*b + 43736) * q^74 + (-1080*b - 17109) * q^75 + (-336*b + 23392) * q^76 + (800*b - 24560) * q^77 + (864*b - 11376) * q^78 + (-1811*b - 1514) * q^79 + (768*b + 5120) * q^80 + 6561 * q^81 + (512*b + 19064) * q^82 + (-1432*b + 30780) * q^83 + (-720*b + 7200) * q^84 + (1764*b + 7534) * q^85 + (-1036*b + 13224) * q^86 + (918*b - 23850) * q^87 + (-512*b + 5120) * q^88 + (-4671*b + 32796) * q^89 + (972*b + 6480) * q^90 + (2780*b - 77480) * q^91 - 8464 * q^92 + (-1638*b - 11304) * q^93 + (-824*b + 7720) * q^94 + (3966*b - 3142) * q^95 - 9216 * q^96 + (5306*b - 53362) * q^97 + (-2000*b - 5828) * q^98 + (-648*b + 6480) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{2} - 18 q^{3} + 32 q^{4} + 40 q^{5} - 72 q^{6} - 100 q^{7} + 128 q^{8} + 162 q^{9}+O(q^{10})$$ 2 * q + 8 * q^2 - 18 * q^3 + 32 * q^4 + 40 * q^5 - 72 * q^6 - 100 * q^7 + 128 * q^8 + 162 * q^9 $$2 q + 8 q^{2} - 18 q^{3} + 32 q^{4} + 40 q^{5} - 72 q^{6} - 100 q^{7} + 128 q^{8} + 162 q^{9} + 160 q^{10} + 160 q^{11} - 288 q^{12} + 632 q^{13} - 400 q^{14} - 360 q^{15} + 512 q^{16} + 1216 q^{17} + 648 q^{18} + 2924 q^{19} + 640 q^{20} + 900 q^{21} + 640 q^{22} - 1058 q^{23} - 1152 q^{24} + 3802 q^{25} + 2528 q^{26} - 1458 q^{27} - 1600 q^{28} + 5300 q^{29} - 1440 q^{30} + 2512 q^{31} + 2048 q^{32} - 1440 q^{33} + 4864 q^{34} + 13420 q^{35} + 2592 q^{36} + 21868 q^{37} + 11696 q^{38} - 5688 q^{39} + 2560 q^{40} + 9532 q^{41} + 3600 q^{42} + 6612 q^{43} + 2560 q^{44} + 3240 q^{45} - 4232 q^{46} + 3860 q^{47} - 4608 q^{48} - 2914 q^{49} + 15208 q^{50} - 10944 q^{51} + 10112 q^{52} + 27744 q^{53} - 5832 q^{54} - 21472 q^{55} - 6400 q^{56} - 26316 q^{57} + 21200 q^{58} - 59140 q^{59} - 5760 q^{60} - 5724 q^{61} + 10048 q^{62} - 8100 q^{63} + 8192 q^{64} - 61376 q^{65} - 5760 q^{66} - 46124 q^{67} + 19456 q^{68} + 9522 q^{69} + 53680 q^{70} - 16320 q^{71} + 10368 q^{72} + 9756 q^{73} + 87472 q^{74} - 34218 q^{75} + 46784 q^{76} - 49120 q^{77} - 22752 q^{78} - 3028 q^{79} + 10240 q^{80} + 13122 q^{81} + 38128 q^{82} + 61560 q^{83} + 14400 q^{84} + 15068 q^{85} + 26448 q^{86} - 47700 q^{87} + 10240 q^{88} + 65592 q^{89} + 12960 q^{90} - 154960 q^{91} - 16928 q^{92} - 22608 q^{93} + 15440 q^{94} - 6284 q^{95} - 18432 q^{96} - 106724 q^{97} - 11656 q^{98} + 12960 q^{99}+O(q^{100})$$ 2 * q + 8 * q^2 - 18 * q^3 + 32 * q^4 + 40 * q^5 - 72 * q^6 - 100 * q^7 + 128 * q^8 + 162 * q^9 + 160 * q^10 + 160 * q^11 - 288 * q^12 + 632 * q^13 - 400 * q^14 - 360 * q^15 + 512 * q^16 + 1216 * q^17 + 648 * q^18 + 2924 * q^19 + 640 * q^20 + 900 * q^21 + 640 * q^22 - 1058 * q^23 - 1152 * q^24 + 3802 * q^25 + 2528 * q^26 - 1458 * q^27 - 1600 * q^28 + 5300 * q^29 - 1440 * q^30 + 2512 * q^31 + 2048 * q^32 - 1440 * q^33 + 4864 * q^34 + 13420 * q^35 + 2592 * q^36 + 21868 * q^37 + 11696 * q^38 - 5688 * q^39 + 2560 * q^40 + 9532 * q^41 + 3600 * q^42 + 6612 * q^43 + 2560 * q^44 + 3240 * q^45 - 4232 * q^46 + 3860 * q^47 - 4608 * q^48 - 2914 * q^49 + 15208 * q^50 - 10944 * q^51 + 10112 * q^52 + 27744 * q^53 - 5832 * q^54 - 21472 * q^55 - 6400 * q^56 - 26316 * q^57 + 21200 * q^58 - 59140 * q^59 - 5760 * q^60 - 5724 * q^61 + 10048 * q^62 - 8100 * q^63 + 8192 * q^64 - 61376 * q^65 - 5760 * q^66 - 46124 * q^67 + 19456 * q^68 + 9522 * q^69 + 53680 * q^70 - 16320 * q^71 + 10368 * q^72 + 9756 * q^73 + 87472 * q^74 - 34218 * q^75 + 46784 * q^76 - 49120 * q^77 - 22752 * q^78 - 3028 * q^79 + 10240 * q^80 + 13122 * q^81 + 38128 * q^82 + 61560 * q^83 + 14400 * q^84 + 15068 * q^85 + 26448 * q^86 - 47700 * q^87 + 10240 * q^88 + 65592 * q^89 + 12960 * q^90 - 154960 * q^91 - 16928 * q^92 - 22608 * q^93 + 15440 * q^94 - 6284 * q^95 - 18432 * q^96 - 106724 * q^97 - 11656 * q^98 + 12960 * q^99

## 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
 −22.6716 22.6716
4.00000 −9.00000 16.0000 −48.0147 −36.0000 −163.358 64.0000 81.0000 −192.059
1.2 4.00000 −9.00000 16.0000 88.0147 −36.0000 63.3578 64.0000 81.0000 352.059
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$23$$ $$1$$

## 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 138.6.a.f 2
3.b odd 2 1 414.6.a.e 2
4.b odd 2 1 1104.6.a.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.6.a.f 2 1.a even 1 1 trivial
414.6.a.e 2 3.b odd 2 1
1104.6.a.g 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 40T_{5} - 4226$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(138))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 4)^{2}$$
$3$ $$(T + 9)^{2}$$
$5$ $$T^{2} - 40T - 4226$$
$7$ $$T^{2} + 100T - 10350$$
$11$ $$T^{2} - 160T - 26496$$
$13$ $$T^{2} - 632T - 196208$$
$17$ $$T^{2} - 1216 T + 365038$$
$19$ $$T^{2} - 2924 T + 1910770$$
$23$ $$(T + 529)^{2}$$
$29$ $$T^{2} - 5300 T + 1674844$$
$31$ $$T^{2} - 2512 T - 15448200$$
$37$ $$T^{2} - 21868 T + 103990492$$
$41$ $$T^{2} - 9532 T + 14293380$$
$43$ $$T^{2} - 6612 T - 23549998$$
$47$ $$T^{2} - 3860 T - 18087204$$
$53$ $$T^{2} - 27744 T + 21306878$$
$59$ $$T^{2} + 59140 T + 873099900$$
$61$ $$T^{2} + 5724 T + 7103420$$
$67$ $$T^{2} + 46124 T - 1139059982$$
$71$ $$T^{2} + 16320 T - 275664384$$
$73$ $$T^{2} - 9756 T - 2090176092$$
$79$ $$T^{2} + 3028 T - 1683484398$$
$83$ $$T^{2} - 61560 T - 106612336$$
$89$ $$T^{2} - 65592 T - 10138998258$$
$97$ $$T^{2} + 106724 T - 11623465860$$