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 $1$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{514}) \)
Defining polynomial: \( x^{2} - 514 \) 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 \(\beta = \sqrt{514}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} - 9 q^{3} + 16 q^{4} + (3 \beta + 20) q^{5} - 36 q^{6} + (5 \beta - 50) q^{7} + 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 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}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} - 18 q^{3} + 32 q^{4} + 40 q^{5} - 72 q^{6} - 100 q^{7} + 128 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{2} - 18 q^{3} + 32 q^{4} + 40 q^{5} - 72 q^{6} - 100 q^{7} + 128 q^{8} + 162 q^{9} + 160 q^{10} + 160 q^{11} - 288 q^{12} + 632 q^{13} - 400 q^{14} - 360 q^{15} + 512 q^{16} + 1216 q^{17} + 648 q^{18} + 2924 q^{19} + 640 q^{20} + 900 q^{21} + 640 q^{22} - 1058 q^{23} - 1152 q^{24} + 3802 q^{25} + 2528 q^{26} - 1458 q^{27} - 1600 q^{28} + 5300 q^{29} - 1440 q^{30} + 2512 q^{31} + 2048 q^{32} - 1440 q^{33} + 4864 q^{34} + 13420 q^{35} + 2592 q^{36} + 21868 q^{37} + 11696 q^{38} - 5688 q^{39} + 2560 q^{40} + 9532 q^{41} + 3600 q^{42} + 6612 q^{43} + 2560 q^{44} + 3240 q^{45} - 4232 q^{46} + 3860 q^{47} - 4608 q^{48} - 2914 q^{49} + 15208 q^{50} - 10944 q^{51} + 10112 q^{52} + 27744 q^{53} - 5832 q^{54} - 21472 q^{55} - 6400 q^{56} - 26316 q^{57} + 21200 q^{58} - 59140 q^{59} - 5760 q^{60} - 5724 q^{61} + 10048 q^{62} - 8100 q^{63} + 8192 q^{64} - 61376 q^{65} - 5760 q^{66} - 46124 q^{67} + 19456 q^{68} + 9522 q^{69} + 53680 q^{70} - 16320 q^{71} + 10368 q^{72} + 9756 q^{73} + 87472 q^{74} - 34218 q^{75} + 46784 q^{76} - 49120 q^{77} - 22752 q^{78} - 3028 q^{79} + 10240 q^{80} + 13122 q^{81} + 38128 q^{82} + 61560 q^{83} + 14400 q^{84} + 15068 q^{85} + 26448 q^{86} - 47700 q^{87} + 10240 q^{88} + 65592 q^{89} + 12960 q^{90} - 154960 q^{91} - 16928 q^{92} - 22608 q^{93} + 15440 q^{94} - 6284 q^{95} - 18432 q^{96} - 106724 q^{97} - 11656 q^{98} + 12960 q^{99}+O(q^{100}) \) 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
−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:

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 \(S_{6}^{\mathrm{new}}(\Gamma_0(138))\). 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
$5$ \( T^{2} - 40T - 4226 \) Copy content Toggle raw display
$7$ \( T^{2} + 100T - 10350 \) Copy content Toggle raw display
$11$ \( T^{2} - 160T - 26496 \) Copy content Toggle raw display
$13$ \( T^{2} - 632T - 196208 \) Copy content Toggle raw display
$17$ \( T^{2} - 1216 T + 365038 \) Copy content Toggle raw display
$19$ \( T^{2} - 2924 T + 1910770 \) Copy content Toggle raw display
$23$ \( (T + 529)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 5300 T + 1674844 \) Copy content Toggle raw display
$31$ \( T^{2} - 2512 T - 15448200 \) Copy content Toggle raw display
$37$ \( T^{2} - 21868 T + 103990492 \) Copy content Toggle raw display
$41$ \( T^{2} - 9532 T + 14293380 \) Copy content Toggle raw display
$43$ \( T^{2} - 6612 T - 23549998 \) Copy content Toggle raw display
$47$ \( T^{2} - 3860 T - 18087204 \) Copy content Toggle raw display
$53$ \( T^{2} - 27744 T + 21306878 \) Copy content Toggle raw display
$59$ \( T^{2} + 59140 T + 873099900 \) Copy content Toggle raw display
$61$ \( T^{2} + 5724 T + 7103420 \) Copy content Toggle raw display
$67$ \( T^{2} + 46124 T - 1139059982 \) Copy content Toggle raw display
$71$ \( T^{2} + 16320 T - 275664384 \) Copy content Toggle raw display
$73$ \( T^{2} - 9756 T - 2090176092 \) Copy content Toggle raw display
$79$ \( T^{2} + 3028 T - 1683484398 \) Copy content Toggle raw display
$83$ \( T^{2} - 61560 T - 106612336 \) Copy content Toggle raw display
$89$ \( T^{2} - 65592 T - 10138998258 \) Copy content Toggle raw display
$97$ \( T^{2} + 106724 T - 11623465860 \) Copy content Toggle raw display
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