Properties

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

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{154}) \)
Defining polynomial: \( x^{2} - 154 \) Copy content Toggle raw display
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 \(\beta = \sqrt{154}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 q^{2} + 9 q^{3} + 16 q^{4} + (3 \beta - 2) q^{5} - 36 q^{6} + ( - 5 \beta - 20) q^{7} - 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + 9 q^{3} + 16 q^{4} + (3 \beta - 2) q^{5} - 36 q^{6} + ( - 5 \beta - 20) q^{7} - 64 q^{8} + 81 q^{9} + ( - 12 \beta + 8) q^{10} + ( - 20 \beta - 80) q^{11} + 144 q^{12} + (12 \beta - 728) q^{13} + (20 \beta + 80) q^{14} + (27 \beta - 18) q^{15} + 256 q^{16} + ( - 135 \beta - 218) q^{17} - 324 q^{18} + (177 \beta - 224) q^{19} + (48 \beta - 32) q^{20} + ( - 45 \beta - 180) q^{21} + (80 \beta + 320) q^{22} + 529 q^{23} - 576 q^{24} + ( - 12 \beta - 1735) q^{25} + ( - 48 \beta + 2912) q^{26} + 729 q^{27} + ( - 80 \beta - 320) q^{28} + (282 \beta + 3182) q^{29} + ( - 108 \beta + 72) q^{30} + ( - 566 \beta - 1180) q^{31} - 1024 q^{32} + ( - 180 \beta - 720) q^{33} + (540 \beta + 872) q^{34} + ( - 50 \beta - 2270) q^{35} + 1296 q^{36} + (402 \beta - 9694) q^{37} + ( - 708 \beta + 896) q^{38} + (108 \beta - 6552) q^{39} + ( - 192 \beta + 128) q^{40} + (896 \beta - 8738) q^{41} + (180 \beta + 720) q^{42} + ( - 509 \beta - 11880) q^{43} + ( - 320 \beta - 1280) q^{44} + (243 \beta - 162) q^{45} - 2116 q^{46} + ( - 950 \beta - 17842) q^{47} + 2304 q^{48} + (200 \beta - 12557) q^{49} + (48 \beta + 6940) q^{50} + ( - 1215 \beta - 1962) q^{51} + (192 \beta - 11648) q^{52} + (923 \beta + 738) q^{53} - 2916 q^{54} + ( - 200 \beta - 9080) q^{55} + (320 \beta + 1280) q^{56} + (1593 \beta - 2016) q^{57} + ( - 1128 \beta - 12728) q^{58} + (2830 \beta + 11546) q^{59} + (432 \beta - 288) q^{60} + (238 \beta - 35442) q^{61} + (2264 \beta + 4720) q^{62} + ( - 405 \beta - 1620) q^{63} + 4096 q^{64} + ( - 2208 \beta + 7000) q^{65} + (720 \beta + 2880) q^{66} + (495 \beta - 40912) q^{67} + ( - 2160 \beta - 3488) q^{68} + 4761 q^{69} + (200 \beta + 9080) q^{70} + ( - 4392 \beta + 7776) q^{71} - 5184 q^{72} + ( - 4116 \beta + 24822) q^{73} + ( - 1608 \beta + 38776) q^{74} + ( - 108 \beta - 15615) q^{75} + (2832 \beta - 3584) q^{76} + (800 \beta + 17000) q^{77} + ( - 432 \beta + 26208) q^{78} + (911 \beta + 51340) q^{79} + (768 \beta - 512) q^{80} + 6561 q^{81} + ( - 3584 \beta + 34952) q^{82} + (5492 \beta + 19548) q^{83} + ( - 720 \beta - 2880) q^{84} + ( - 384 \beta - 61934) q^{85} + (2036 \beta + 47520) q^{86} + (2538 \beta + 28638) q^{87} + (1280 \beta + 5120) q^{88} + ( - 4791 \beta - 10614) q^{89} + ( - 972 \beta + 648) q^{90} + (3400 \beta + 5320) q^{91} + 8464 q^{92} + ( - 5094 \beta - 10620) q^{93} + (3800 \beta + 71368) q^{94} + ( - 1026 \beta + 82222) q^{95} - 9216 q^{96} + (7042 \beta - 12886) q^{97} + ( - 800 \beta + 50228) q^{98} + ( - 1620 \beta - 6480) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} + 18 q^{3} + 32 q^{4} - 4 q^{5} - 72 q^{6} - 40 q^{7} - 128 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} + 18 q^{3} + 32 q^{4} - 4 q^{5} - 72 q^{6} - 40 q^{7} - 128 q^{8} + 162 q^{9} + 16 q^{10} - 160 q^{11} + 288 q^{12} - 1456 q^{13} + 160 q^{14} - 36 q^{15} + 512 q^{16} - 436 q^{17} - 648 q^{18} - 448 q^{19} - 64 q^{20} - 360 q^{21} + 640 q^{22} + 1058 q^{23} - 1152 q^{24} - 3470 q^{25} + 5824 q^{26} + 1458 q^{27} - 640 q^{28} + 6364 q^{29} + 144 q^{30} - 2360 q^{31} - 2048 q^{32} - 1440 q^{33} + 1744 q^{34} - 4540 q^{35} + 2592 q^{36} - 19388 q^{37} + 1792 q^{38} - 13104 q^{39} + 256 q^{40} - 17476 q^{41} + 1440 q^{42} - 23760 q^{43} - 2560 q^{44} - 324 q^{45} - 4232 q^{46} - 35684 q^{47} + 4608 q^{48} - 25114 q^{49} + 13880 q^{50} - 3924 q^{51} - 23296 q^{52} + 1476 q^{53} - 5832 q^{54} - 18160 q^{55} + 2560 q^{56} - 4032 q^{57} - 25456 q^{58} + 23092 q^{59} - 576 q^{60} - 70884 q^{61} + 9440 q^{62} - 3240 q^{63} + 8192 q^{64} + 14000 q^{65} + 5760 q^{66} - 81824 q^{67} - 6976 q^{68} + 9522 q^{69} + 18160 q^{70} + 15552 q^{71} - 10368 q^{72} + 49644 q^{73} + 77552 q^{74} - 31230 q^{75} - 7168 q^{76} + 34000 q^{77} + 52416 q^{78} + 102680 q^{79} - 1024 q^{80} + 13122 q^{81} + 69904 q^{82} + 39096 q^{83} - 5760 q^{84} - 123868 q^{85} + 95040 q^{86} + 57276 q^{87} + 10240 q^{88} - 21228 q^{89} + 1296 q^{90} + 10640 q^{91} + 16928 q^{92} - 21240 q^{93} + 142736 q^{94} + 164444 q^{95} - 18432 q^{96} - 25772 q^{97} + 100456 q^{98} - 12960 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
−12.4097
12.4097
−4.00000 9.00000 16.0000 −39.2290 −36.0000 42.0484 −64.0000 81.0000 156.916
1.2 −4.00000 9.00000 16.0000 35.2290 −36.0000 −82.0484 −64.0000 81.0000 −140.916
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 4T_{5} - 1382 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(138))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T - 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4T - 1382 \) Copy content Toggle raw display
$7$ \( T^{2} + 40T - 3450 \) Copy content Toggle raw display
$11$ \( T^{2} + 160T - 55200 \) Copy content Toggle raw display
$13$ \( T^{2} + 1456 T + 507808 \) Copy content Toggle raw display
$17$ \( T^{2} + 436 T - 2759126 \) Copy content Toggle raw display
$19$ \( T^{2} + 448 T - 4774490 \) Copy content Toggle raw display
$23$ \( (T - 529)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 6364 T - 2121572 \) Copy content Toggle raw display
$31$ \( T^{2} + 2360 T - 47942424 \) Copy content Toggle raw display
$37$ \( T^{2} + 19388 T + 69086620 \) Copy content Toggle raw display
$41$ \( T^{2} + 17476 T - 47281020 \) Copy content Toggle raw display
$43$ \( T^{2} + 23760 T + 101235926 \) Copy content Toggle raw display
$47$ \( T^{2} + 35684 T + 179351964 \) Copy content Toggle raw display
$53$ \( T^{2} - 1476 T - 130652422 \) Copy content Toggle raw display
$59$ \( T^{2} - 23092 T - 1100060484 \) Copy content Toggle raw display
$61$ \( T^{2} + 70884 T + 1247412188 \) Copy content Toggle raw display
$67$ \( T^{2} + 81824 T + 1636057894 \) Copy content Toggle raw display
$71$ \( T^{2} - 15552 T - 2910142080 \) Copy content Toggle raw display
$73$ \( T^{2} - 49644 T - 1992852540 \) Copy content Toggle raw display
$79$ \( T^{2} - 102680 T + 2507987766 \) Copy content Toggle raw display
$83$ \( T^{2} - 39096 T - 4262833552 \) Copy content Toggle raw display
$89$ \( T^{2} + 21228 T - 3422209878 \) Copy content Toggle raw display
$97$ \( T^{2} + 25772 T - 7470774660 \) Copy content Toggle raw display
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