Properties

Label 38.4.a.c
Level $38$
Weight $4$
Character orbit 38.a
Self dual yes
Analytic conductor $2.242$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.24207258022\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{73}) \)
Defining polynomial: \( x^{2} - x - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{73})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( - \beta + 5) q^{3} + 4 q^{4} + (3 \beta - 6) q^{5} + ( - 2 \beta + 10) q^{6} + (4 \beta - 11) q^{7} + 8 q^{8} + ( - 9 \beta + 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + ( - \beta + 5) q^{3} + 4 q^{4} + (3 \beta - 6) q^{5} + ( - 2 \beta + 10) q^{6} + (4 \beta - 11) q^{7} + 8 q^{8} + ( - 9 \beta + 16) q^{9} + (6 \beta - 12) q^{10} + ( - \beta - 8) q^{11} + ( - 4 \beta + 20) q^{12} + ( - 13 \beta + 15) q^{13} + (8 \beta - 22) q^{14} + (18 \beta - 84) q^{15} + 16 q^{16} + (2 \beta - 41) q^{17} + ( - 18 \beta + 32) q^{18} + 19 q^{19} + (12 \beta - 24) q^{20} + (27 \beta - 127) q^{21} + ( - 2 \beta - 16) q^{22} + ( - 13 \beta + 43) q^{23} + ( - 8 \beta + 40) q^{24} + ( - 27 \beta + 73) q^{25} + ( - 26 \beta + 30) q^{26} + ( - 25 \beta + 107) q^{27} + (16 \beta - 44) q^{28} + (21 \beta - 9) q^{29} + (36 \beta - 168) q^{30} + (44 \beta + 84) q^{31} + 32 q^{32} + (4 \beta - 22) q^{33} + (4 \beta - 82) q^{34} + ( - 45 \beta + 282) q^{35} + ( - 36 \beta + 64) q^{36} + (28 \beta + 82) q^{37} + 38 q^{38} + ( - 67 \beta + 309) q^{39} + (24 \beta - 48) q^{40} + ( - 10 \beta - 20) q^{41} + (54 \beta - 254) q^{42} + ( - 7 \beta + 342) q^{43} + ( - 4 \beta - 32) q^{44} + (75 \beta - 582) q^{45} + ( - 26 \beta + 86) q^{46} + (71 \beta - 230) q^{47} + ( - 16 \beta + 80) q^{48} + ( - 72 \beta + 66) q^{49} + ( - 54 \beta + 146) q^{50} + (49 \beta - 241) q^{51} + ( - 52 \beta + 60) q^{52} + ( - 17 \beta - 601) q^{53} + ( - 50 \beta + 214) q^{54} + ( - 21 \beta - 6) q^{55} + (32 \beta - 88) q^{56} + ( - 19 \beta + 95) q^{57} + (42 \beta - 18) q^{58} + ( - 25 \beta - 131) q^{59} + (72 \beta - 336) q^{60} + ( - 111 \beta + 212) q^{61} + (88 \beta + 168) q^{62} + (127 \beta - 824) q^{63} + 64 q^{64} + (84 \beta - 792) q^{65} + (8 \beta - 44) q^{66} + (77 \beta + 573) q^{67} + (8 \beta - 164) q^{68} + ( - 95 \beta + 449) q^{69} + ( - 90 \beta + 564) q^{70} + ( - 116 \beta + 158) q^{71} + ( - 72 \beta + 128) q^{72} + (184 \beta + 97) q^{73} + (56 \beta + 164) q^{74} + ( - 181 \beta + 851) q^{75} + 76 q^{76} + ( - 25 \beta + 16) q^{77} + ( - 134 \beta + 618) q^{78} + (58 \beta + 646) q^{79} + (48 \beta - 96) q^{80} + (36 \beta + 553) q^{81} + ( - 20 \beta - 40) q^{82} + ( - 194 \beta - 238) q^{83} + (108 \beta - 508) q^{84} + ( - 129 \beta + 354) q^{85} + ( - 14 \beta + 684) q^{86} + (93 \beta - 423) q^{87} + ( - 8 \beta - 64) q^{88} + (188 \beta - 212) q^{89} + (150 \beta - 1164) q^{90} + (151 \beta - 1101) q^{91} + ( - 52 \beta + 172) q^{92} + (92 \beta - 372) q^{93} + (142 \beta - 460) q^{94} + (57 \beta - 114) q^{95} + ( - 32 \beta + 160) q^{96} + ( - 102 \beta + 698) q^{97} + ( - 144 \beta + 132) q^{98} + (65 \beta + 34) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 9 q^{3} + 8 q^{4} - 9 q^{5} + 18 q^{6} - 18 q^{7} + 16 q^{8} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 9 q^{3} + 8 q^{4} - 9 q^{5} + 18 q^{6} - 18 q^{7} + 16 q^{8} + 23 q^{9} - 18 q^{10} - 17 q^{11} + 36 q^{12} + 17 q^{13} - 36 q^{14} - 150 q^{15} + 32 q^{16} - 80 q^{17} + 46 q^{18} + 38 q^{19} - 36 q^{20} - 227 q^{21} - 34 q^{22} + 73 q^{23} + 72 q^{24} + 119 q^{25} + 34 q^{26} + 189 q^{27} - 72 q^{28} + 3 q^{29} - 300 q^{30} + 212 q^{31} + 64 q^{32} - 40 q^{33} - 160 q^{34} + 519 q^{35} + 92 q^{36} + 192 q^{37} + 76 q^{38} + 551 q^{39} - 72 q^{40} - 50 q^{41} - 454 q^{42} + 677 q^{43} - 68 q^{44} - 1089 q^{45} + 146 q^{46} - 389 q^{47} + 144 q^{48} + 60 q^{49} + 238 q^{50} - 433 q^{51} + 68 q^{52} - 1219 q^{53} + 378 q^{54} - 33 q^{55} - 144 q^{56} + 171 q^{57} + 6 q^{58} - 287 q^{59} - 600 q^{60} + 313 q^{61} + 424 q^{62} - 1521 q^{63} + 128 q^{64} - 1500 q^{65} - 80 q^{66} + 1223 q^{67} - 320 q^{68} + 803 q^{69} + 1038 q^{70} + 200 q^{71} + 184 q^{72} + 378 q^{73} + 384 q^{74} + 1521 q^{75} + 152 q^{76} + 7 q^{77} + 1102 q^{78} + 1350 q^{79} - 144 q^{80} + 1142 q^{81} - 100 q^{82} - 670 q^{83} - 908 q^{84} + 579 q^{85} + 1354 q^{86} - 753 q^{87} - 136 q^{88} - 236 q^{89} - 2178 q^{90} - 2051 q^{91} + 292 q^{92} - 652 q^{93} - 778 q^{94} - 171 q^{95} + 288 q^{96} + 1294 q^{97} + 120 q^{98} + 133 q^{99}+O(q^{100}) \) 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
4.77200
−3.77200
2.00000 0.227998 4.00000 8.31601 0.455996 8.08801 8.00000 −26.9480 16.6320
1.2 2.00000 8.77200 4.00000 −17.3160 17.5440 −26.0880 8.00000 49.9480 −34.6320
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)

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 38.4.a.c 2
3.b odd 2 1 342.4.a.h 2
4.b odd 2 1 304.4.a.c 2
5.b even 2 1 950.4.a.e 2
5.c odd 4 2 950.4.b.i 4
7.b odd 2 1 1862.4.a.e 2
8.b even 2 1 1216.4.a.g 2
8.d odd 2 1 1216.4.a.p 2
19.b odd 2 1 722.4.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.a.c 2 1.a even 1 1 trivial
304.4.a.c 2 4.b odd 2 1
342.4.a.h 2 3.b odd 2 1
722.4.a.f 2 19.b odd 2 1
950.4.a.e 2 5.b even 2 1
950.4.b.i 4 5.c odd 4 2
1216.4.a.g 2 8.b even 2 1
1216.4.a.p 2 8.d odd 2 1
1862.4.a.e 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 9T_{3} + 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(38))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 9T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} + 9T - 144 \) Copy content Toggle raw display
$7$ \( T^{2} + 18T - 211 \) Copy content Toggle raw display
$11$ \( T^{2} + 17T + 54 \) Copy content Toggle raw display
$13$ \( T^{2} - 17T - 3012 \) Copy content Toggle raw display
$17$ \( T^{2} + 80T + 1527 \) Copy content Toggle raw display
$19$ \( (T - 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 73T - 1752 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T - 8046 \) Copy content Toggle raw display
$31$ \( T^{2} - 212T - 24096 \) Copy content Toggle raw display
$37$ \( T^{2} - 192T - 5092 \) Copy content Toggle raw display
$41$ \( T^{2} + 50T - 1200 \) Copy content Toggle raw display
$43$ \( T^{2} - 677T + 113688 \) Copy content Toggle raw display
$47$ \( T^{2} + 389T - 54168 \) Copy content Toggle raw display
$53$ \( T^{2} + 1219 T + 366216 \) Copy content Toggle raw display
$59$ \( T^{2} + 287T + 9186 \) Copy content Toggle raw display
$61$ \( T^{2} - 313T - 200366 \) Copy content Toggle raw display
$67$ \( T^{2} - 1223 T + 265728 \) Copy content Toggle raw display
$71$ \( T^{2} - 200T - 235572 \) Copy content Toggle raw display
$73$ \( T^{2} - 378T - 582151 \) Copy content Toggle raw display
$79$ \( T^{2} - 1350 T + 394232 \) Copy content Toggle raw display
$83$ \( T^{2} + 670T - 574632 \) Copy content Toggle raw display
$89$ \( T^{2} + 236T - 631104 \) Copy content Toggle raw display
$97$ \( T^{2} - 1294 T + 228736 \) Copy content Toggle raw display
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