Properties

Label 138.4.a.d
Level $138$
Weight $4$
Character orbit 138.a
Self dual yes
Analytic conductor $8.142$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 138.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.14226358079\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{277}) \)
Defining polynomial: \( x^{2} - x - 69 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{277}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + 3 q^{3} + 4 q^{4} + ( - \beta + 1) q^{5} - 6 q^{6} + 14 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 3 q^{3} + 4 q^{4} + ( - \beta + 1) q^{5} - 6 q^{6} + 14 q^{7} - 8 q^{8} + 9 q^{9} + (2 \beta - 2) q^{10} + (\beta - 1) q^{11} + 12 q^{12} + (2 \beta + 24) q^{13} - 28 q^{14} + ( - 3 \beta + 3) q^{15} + 16 q^{16} + (2 \beta - 2) q^{17} - 18 q^{18} + ( - \beta + 39) q^{19} + ( - 4 \beta + 4) q^{20} + 42 q^{21} + ( - 2 \beta + 2) q^{22} + 23 q^{23} - 24 q^{24} + ( - 2 \beta + 153) q^{25} + ( - 4 \beta - 48) q^{26} + 27 q^{27} + 56 q^{28} + (10 \beta + 92) q^{29} + (6 \beta - 6) q^{30} + ( - 2 \beta + 154) q^{31} - 32 q^{32} + (3 \beta - 3) q^{33} + ( - 4 \beta + 4) q^{34} + ( - 14 \beta + 14) q^{35} + 36 q^{36} + ( - 3 \beta + 185) q^{37} + (2 \beta - 78) q^{38} + (6 \beta + 72) q^{39} + (8 \beta - 8) q^{40} + (24 \beta + 6) q^{41} - 84 q^{42} + ( - 7 \beta + 93) q^{43} + (4 \beta - 4) q^{44} + ( - 9 \beta + 9) q^{45} - 46 q^{46} + ( - 12 \beta - 264) q^{47} + 48 q^{48} - 147 q^{49} + (4 \beta - 306) q^{50} + (6 \beta - 6) q^{51} + (8 \beta + 96) q^{52} + ( - 5 \beta - 463) q^{53} - 54 q^{54} + (2 \beta - 278) q^{55} - 112 q^{56} + ( - 3 \beta + 117) q^{57} + ( - 20 \beta - 184) q^{58} - 396 q^{59} + ( - 12 \beta + 12) q^{60} + ( - 37 \beta - 21) q^{61} + (4 \beta - 308) q^{62} + 126 q^{63} + 64 q^{64} + ( - 22 \beta - 530) q^{65} + ( - 6 \beta + 6) q^{66} + (51 \beta - 217) q^{67} + (8 \beta - 8) q^{68} + 69 q^{69} + (28 \beta - 28) q^{70} + ( - 36 \beta + 312) q^{71} - 72 q^{72} + (46 \beta - 176) q^{73} + (6 \beta - 370) q^{74} + ( - 6 \beta + 459) q^{75} + ( - 4 \beta + 156) q^{76} + (14 \beta - 14) q^{77} + ( - 12 \beta - 144) q^{78} + (16 \beta - 254) q^{79} + ( - 16 \beta + 16) q^{80} + 81 q^{81} + ( - 48 \beta - 12) q^{82} + (55 \beta + 497) q^{83} + 168 q^{84} + (4 \beta - 556) q^{85} + (14 \beta - 186) q^{86} + (30 \beta + 276) q^{87} + ( - 8 \beta + 8) q^{88} - 36 \beta q^{89} + (18 \beta - 18) q^{90} + (28 \beta + 336) q^{91} + 92 q^{92} + ( - 6 \beta + 462) q^{93} + (24 \beta + 528) q^{94} + ( - 40 \beta + 316) q^{95} - 96 q^{96} + ( - 26 \beta - 392) q^{97} + 294 q^{98} + (9 \beta - 9) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} + 2 q^{5} - 12 q^{6} + 28 q^{7} - 16 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} + 2 q^{5} - 12 q^{6} + 28 q^{7} - 16 q^{8} + 18 q^{9} - 4 q^{10} - 2 q^{11} + 24 q^{12} + 48 q^{13} - 56 q^{14} + 6 q^{15} + 32 q^{16} - 4 q^{17} - 36 q^{18} + 78 q^{19} + 8 q^{20} + 84 q^{21} + 4 q^{22} + 46 q^{23} - 48 q^{24} + 306 q^{25} - 96 q^{26} + 54 q^{27} + 112 q^{28} + 184 q^{29} - 12 q^{30} + 308 q^{31} - 64 q^{32} - 6 q^{33} + 8 q^{34} + 28 q^{35} + 72 q^{36} + 370 q^{37} - 156 q^{38} + 144 q^{39} - 16 q^{40} + 12 q^{41} - 168 q^{42} + 186 q^{43} - 8 q^{44} + 18 q^{45} - 92 q^{46} - 528 q^{47} + 96 q^{48} - 294 q^{49} - 612 q^{50} - 12 q^{51} + 192 q^{52} - 926 q^{53} - 108 q^{54} - 556 q^{55} - 224 q^{56} + 234 q^{57} - 368 q^{58} - 792 q^{59} + 24 q^{60} - 42 q^{61} - 616 q^{62} + 252 q^{63} + 128 q^{64} - 1060 q^{65} + 12 q^{66} - 434 q^{67} - 16 q^{68} + 138 q^{69} - 56 q^{70} + 624 q^{71} - 144 q^{72} - 352 q^{73} - 740 q^{74} + 918 q^{75} + 312 q^{76} - 28 q^{77} - 288 q^{78} - 508 q^{79} + 32 q^{80} + 162 q^{81} - 24 q^{82} + 994 q^{83} + 336 q^{84} - 1112 q^{85} - 372 q^{86} + 552 q^{87} + 16 q^{88} - 36 q^{90} + 672 q^{91} + 184 q^{92} + 924 q^{93} + 1056 q^{94} + 632 q^{95} - 192 q^{96} - 784 q^{97} + 588 q^{98} - 18 q^{99}+O(q^{100}) \) 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
8.82166
−7.82166
−2.00000 3.00000 4.00000 −15.6433 −6.00000 14.0000 −8.00000 9.00000 31.2866
1.2 −2.00000 3.00000 4.00000 17.6433 −6.00000 14.0000 −8.00000 9.00000 −35.2866
\(n\): 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 138.4.a.d 2
3.b odd 2 1 414.4.a.k 2
4.b odd 2 1 1104.4.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.4.a.d 2 1.a even 1 1 trivial
414.4.a.k 2 3.b odd 2 1
1104.4.a.k 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 2T_{5} - 276 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(138))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{2} \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 2T - 276 \) Copy content Toggle raw display
$7$ \( (T - 14)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 2T - 276 \) Copy content Toggle raw display
$13$ \( T^{2} - 48T - 532 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 1104 \) Copy content Toggle raw display
$19$ \( T^{2} - 78T + 1244 \) Copy content Toggle raw display
$23$ \( (T - 23)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 184T - 19236 \) Copy content Toggle raw display
$31$ \( T^{2} - 308T + 22608 \) Copy content Toggle raw display
$37$ \( T^{2} - 370T + 31732 \) Copy content Toggle raw display
$41$ \( T^{2} - 12T - 159516 \) Copy content Toggle raw display
$43$ \( T^{2} - 186T - 4924 \) Copy content Toggle raw display
$47$ \( T^{2} + 528T + 29808 \) Copy content Toggle raw display
$53$ \( T^{2} + 926T + 207444 \) Copy content Toggle raw display
$59$ \( (T + 396)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 42T - 378772 \) Copy content Toggle raw display
$67$ \( T^{2} + 434T - 673388 \) Copy content Toggle raw display
$71$ \( T^{2} - 624T - 261648 \) Copy content Toggle raw display
$73$ \( T^{2} + 352T - 555156 \) Copy content Toggle raw display
$79$ \( T^{2} + 508T - 6396 \) Copy content Toggle raw display
$83$ \( T^{2} - 994T - 590916 \) Copy content Toggle raw display
$89$ \( T^{2} - 358992 \) Copy content Toggle raw display
$97$ \( T^{2} + 784T - 33588 \) Copy content Toggle raw display
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