Properties

Label 138.4.a.e
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: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 8\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + (\beta + 4) q^{5} - 6 q^{6} + ( - \beta + 6) q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + (\beta + 4) q^{5} - 6 q^{6} + ( - \beta + 6) q^{7} + 8 q^{8} + 9 q^{9} + (2 \beta + 8) q^{10} + ( - 4 \beta + 24) q^{11} - 12 q^{12} + (4 \beta + 42) q^{13} + ( - 2 \beta + 12) q^{14} + ( - 3 \beta - 12) q^{15} + 16 q^{16} + ( - 3 \beta + 52) q^{17} + 18 q^{18} + (3 \beta + 14) q^{19} + (4 \beta + 16) q^{20} + (3 \beta - 18) q^{21} + ( - 8 \beta + 48) q^{22} + 23 q^{23} - 24 q^{24} + (8 \beta + 19) q^{25} + (8 \beta + 84) q^{26} - 27 q^{27} + ( - 4 \beta + 24) q^{28} + (10 \beta + 66) q^{29} + ( - 6 \beta - 24) q^{30} + ( - 10 \beta - 4) q^{31} + 32 q^{32} + (12 \beta - 72) q^{33} + ( - 6 \beta + 104) q^{34} + (2 \beta - 104) q^{35} + 36 q^{36} + ( - 22 \beta - 162) q^{37} + (6 \beta + 28) q^{38} + ( - 12 \beta - 126) q^{39} + (8 \beta + 32) q^{40} + ( - 12 \beta - 70) q^{41} + (6 \beta - 36) q^{42} + (11 \beta - 274) q^{43} + ( - 16 \beta + 96) q^{44} + (9 \beta + 36) q^{45} + 46 q^{46} + ( - 6 \beta - 232) q^{47} - 48 q^{48} + ( - 12 \beta - 179) q^{49} + (16 \beta + 38) q^{50} + (9 \beta - 156) q^{51} + (16 \beta + 168) q^{52} + ( - 41 \beta + 168) q^{53} - 54 q^{54} + (8 \beta - 416) q^{55} + ( - 8 \beta + 48) q^{56} + ( - 9 \beta - 42) q^{57} + (20 \beta + 132) q^{58} + (62 \beta + 36) q^{59} + ( - 12 \beta - 48) q^{60} + (46 \beta + 30) q^{61} + ( - 20 \beta - 8) q^{62} + ( - 9 \beta + 54) q^{63} + 64 q^{64} + (58 \beta + 680) q^{65} + (24 \beta - 144) q^{66} + ( - 27 \beta - 222) q^{67} + ( - 12 \beta + 208) q^{68} - 69 q^{69} + (4 \beta - 208) q^{70} + ( - 56 \beta + 336) q^{71} + 72 q^{72} + ( - 52 \beta - 126) q^{73} + ( - 44 \beta - 324) q^{74} + ( - 24 \beta - 57) q^{75} + (12 \beta + 56) q^{76} + ( - 48 \beta + 656) q^{77} + ( - 24 \beta - 252) q^{78} + (\beta - 846) q^{79} + (16 \beta + 64) q^{80} + 81 q^{81} + ( - 24 \beta - 140) q^{82} + (88 \beta - 200) q^{83} + (12 \beta - 72) q^{84} + (40 \beta - 176) q^{85} + (22 \beta - 548) q^{86} + ( - 30 \beta - 198) q^{87} + ( - 32 \beta + 192) q^{88} + (19 \beta - 168) q^{89} + (18 \beta + 72) q^{90} + ( - 18 \beta - 260) q^{91} + 92 q^{92} + (30 \beta + 12) q^{93} + ( - 12 \beta - 464) q^{94} + (26 \beta + 440) q^{95} - 96 q^{96} + (82 \beta + 110) q^{97} + ( - 24 \beta - 358) q^{98} + ( - 36 \beta + 216) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 6 q^{3} + 8 q^{4} + 8 q^{5} - 12 q^{6} + 12 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} - 6 q^{3} + 8 q^{4} + 8 q^{5} - 12 q^{6} + 12 q^{7} + 16 q^{8} + 18 q^{9} + 16 q^{10} + 48 q^{11} - 24 q^{12} + 84 q^{13} + 24 q^{14} - 24 q^{15} + 32 q^{16} + 104 q^{17} + 36 q^{18} + 28 q^{19} + 32 q^{20} - 36 q^{21} + 96 q^{22} + 46 q^{23} - 48 q^{24} + 38 q^{25} + 168 q^{26} - 54 q^{27} + 48 q^{28} + 132 q^{29} - 48 q^{30} - 8 q^{31} + 64 q^{32} - 144 q^{33} + 208 q^{34} - 208 q^{35} + 72 q^{36} - 324 q^{37} + 56 q^{38} - 252 q^{39} + 64 q^{40} - 140 q^{41} - 72 q^{42} - 548 q^{43} + 192 q^{44} + 72 q^{45} + 92 q^{46} - 464 q^{47} - 96 q^{48} - 358 q^{49} + 76 q^{50} - 312 q^{51} + 336 q^{52} + 336 q^{53} - 108 q^{54} - 832 q^{55} + 96 q^{56} - 84 q^{57} + 264 q^{58} + 72 q^{59} - 96 q^{60} + 60 q^{61} - 16 q^{62} + 108 q^{63} + 128 q^{64} + 1360 q^{65} - 288 q^{66} - 444 q^{67} + 416 q^{68} - 138 q^{69} - 416 q^{70} + 672 q^{71} + 144 q^{72} - 252 q^{73} - 648 q^{74} - 114 q^{75} + 112 q^{76} + 1312 q^{77} - 504 q^{78} - 1692 q^{79} + 128 q^{80} + 162 q^{81} - 280 q^{82} - 400 q^{83} - 144 q^{84} - 352 q^{85} - 1096 q^{86} - 396 q^{87} + 384 q^{88} - 336 q^{89} + 144 q^{90} - 520 q^{91} + 184 q^{92} + 24 q^{93} - 928 q^{94} + 880 q^{95} - 192 q^{96} + 220 q^{97} - 716 q^{98} + 432 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
2.00000 −3.00000 4.00000 −7.31371 −6.00000 17.3137 8.00000 9.00000 −14.6274
1.2 2.00000 −3.00000 4.00000 15.3137 −6.00000 −5.31371 8.00000 9.00000 30.6274
\(n\): 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.e 2
3.b odd 2 1 414.4.a.g 2
4.b odd 2 1 1104.4.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.4.a.e 2 1.a even 1 1 trivial
414.4.a.g 2 3.b odd 2 1
1104.4.a.p 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 8T_{5} - 112 \) 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
$5$ \( T^{2} - 8T - 112 \) Copy content Toggle raw display
$7$ \( T^{2} - 12T - 92 \) Copy content Toggle raw display
$11$ \( T^{2} - 48T - 1472 \) Copy content Toggle raw display
$13$ \( T^{2} - 84T - 284 \) Copy content Toggle raw display
$17$ \( T^{2} - 104T + 1552 \) Copy content Toggle raw display
$19$ \( T^{2} - 28T - 956 \) Copy content Toggle raw display
$23$ \( (T - 23)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 132T - 8444 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T - 12784 \) Copy content Toggle raw display
$37$ \( T^{2} + 324T - 35708 \) Copy content Toggle raw display
$41$ \( T^{2} + 140T - 13532 \) Copy content Toggle raw display
$43$ \( T^{2} + 548T + 59588 \) Copy content Toggle raw display
$47$ \( T^{2} + 464T + 49216 \) Copy content Toggle raw display
$53$ \( T^{2} - 336T - 186944 \) Copy content Toggle raw display
$59$ \( T^{2} - 72T - 490736 \) Copy content Toggle raw display
$61$ \( T^{2} - 60T - 269948 \) Copy content Toggle raw display
$67$ \( T^{2} + 444T - 44028 \) Copy content Toggle raw display
$71$ \( T^{2} - 672T - 288512 \) Copy content Toggle raw display
$73$ \( T^{2} + 252T - 330236 \) Copy content Toggle raw display
$79$ \( T^{2} + 1692 T + 715588 \) Copy content Toggle raw display
$83$ \( T^{2} + 400T - 951232 \) Copy content Toggle raw display
$89$ \( T^{2} + 336T - 17984 \) Copy content Toggle raw display
$97$ \( T^{2} - 220T - 848572 \) Copy content Toggle raw display
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