Properties

Label 29.4.a.a
Level $29$
Weight $4$
Character orbit 29.a
Self dual yes
Analytic conductor $1.711$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 29.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.71105539017\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + ( -5 - 3 \beta ) q^{3} + ( -5 - 2 \beta ) q^{4} + ( -5 + 4 \beta ) q^{5} + ( -1 - 2 \beta ) q^{6} + ( -8 + 10 \beta ) q^{7} + ( 9 - 11 \beta ) q^{8} + ( 16 + 30 \beta ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + ( -5 - 3 \beta ) q^{3} + ( -5 - 2 \beta ) q^{4} + ( -5 + 4 \beta ) q^{5} + ( -1 - 2 \beta ) q^{6} + ( -8 + 10 \beta ) q^{7} + ( 9 - 11 \beta ) q^{8} + ( 16 + 30 \beta ) q^{9} + ( 13 - 9 \beta ) q^{10} + ( -13 - 37 \beta ) q^{11} + ( 37 + 25 \beta ) q^{12} + ( -13 - 26 \beta ) q^{13} + ( 28 - 18 \beta ) q^{14} + ( 1 - 5 \beta ) q^{15} + ( 9 + 36 \beta ) q^{16} + ( 30 + 18 \beta ) q^{17} + ( 44 - 14 \beta ) q^{18} + ( -110 + 32 \beta ) q^{19} + ( 9 - 10 \beta ) q^{20} + ( -20 - 26 \beta ) q^{21} + ( -61 + 24 \beta ) q^{22} + ( 26 + 48 \beta ) q^{23} + ( 21 + 28 \beta ) q^{24} + ( -68 - 40 \beta ) q^{25} + ( -39 + 13 \beta ) q^{26} + ( -125 - 117 \beta ) q^{27} -34 \beta q^{28} + 29 q^{29} + ( -11 + 6 \beta ) q^{30} + ( -147 - 63 \beta ) q^{31} + ( -9 + 61 \beta ) q^{32} + ( 287 + 224 \beta ) q^{33} + ( 6 + 12 \beta ) q^{34} + ( 120 - 82 \beta ) q^{35} + ( -200 - 182 \beta ) q^{36} + ( 156 - 56 \beta ) q^{37} + ( 174 - 142 \beta ) q^{38} + ( 221 + 169 \beta ) q^{39} + ( -133 + 91 \beta ) q^{40} + ( 20 + 138 \beta ) q^{41} + ( -32 + 6 \beta ) q^{42} + ( -161 + 171 \beta ) q^{43} + ( 213 + 211 \beta ) q^{44} + ( 160 - 86 \beta ) q^{45} + ( 70 - 22 \beta ) q^{46} + ( -65 - 207 \beta ) q^{47} + ( -261 - 207 \beta ) q^{48} + ( -79 - 160 \beta ) q^{49} + ( -12 - 28 \beta ) q^{50} + ( -258 - 180 \beta ) q^{51} + ( 169 + 156 \beta ) q^{52} + ( 501 - 122 \beta ) q^{53} + ( -109 - 8 \beta ) q^{54} + ( -231 + 133 \beta ) q^{55} + ( -292 + 178 \beta ) q^{56} + ( 358 + 170 \beta ) q^{57} + ( -29 + 29 \beta ) q^{58} + ( -450 + 248 \beta ) q^{59} + ( 15 + 23 \beta ) q^{60} + ( -474 - 178 \beta ) q^{61} + ( 21 - 84 \beta ) q^{62} + ( 472 - 80 \beta ) q^{63} + ( 59 - 358 \beta ) q^{64} + ( -143 + 78 \beta ) q^{65} + ( 161 + 63 \beta ) q^{66} + ( 160 + 484 \beta ) q^{67} + ( -222 - 150 \beta ) q^{68} + ( -418 - 318 \beta ) q^{69} + ( -284 + 202 \beta ) q^{70} + ( -330 - 34 \beta ) q^{71} + ( -516 + 94 \beta ) q^{72} + ( 324 - 640 \beta ) q^{73} + ( -268 + 212 \beta ) q^{74} + ( 580 + 404 \beta ) q^{75} + ( 422 + 60 \beta ) q^{76} + ( -636 + 166 \beta ) q^{77} + ( 117 + 52 \beta ) q^{78} + ( 129 - 341 \beta ) q^{79} + ( 243 - 144 \beta ) q^{80} + ( 895 + 150 \beta ) q^{81} + ( 256 - 118 \beta ) q^{82} + ( 606 + 64 \beta ) q^{83} + ( 204 + 170 \beta ) q^{84} + ( -6 + 30 \beta ) q^{85} + ( 503 - 332 \beta ) q^{86} + ( -145 - 87 \beta ) q^{87} + ( 697 - 190 \beta ) q^{88} + ( 380 + 522 \beta ) q^{89} + ( -332 + 246 \beta ) q^{90} + ( -416 + 78 \beta ) q^{91} + ( -322 - 292 \beta ) q^{92} + ( 1113 + 756 \beta ) q^{93} + ( -349 + 142 \beta ) q^{94} + ( 806 - 600 \beta ) q^{95} + ( -321 - 278 \beta ) q^{96} + ( 12 - 578 \beta ) q^{97} + ( -241 + 81 \beta ) q^{98} + ( -2428 - 982 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 10q^{3} - 10q^{4} - 10q^{5} - 2q^{6} - 16q^{7} + 18q^{8} + 32q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 10q^{3} - 10q^{4} - 10q^{5} - 2q^{6} - 16q^{7} + 18q^{8} + 32q^{9} + 26q^{10} - 26q^{11} + 74q^{12} - 26q^{13} + 56q^{14} + 2q^{15} + 18q^{16} + 60q^{17} + 88q^{18} - 220q^{19} + 18q^{20} - 40q^{21} - 122q^{22} + 52q^{23} + 42q^{24} - 136q^{25} - 78q^{26} - 250q^{27} + 58q^{29} - 22q^{30} - 294q^{31} - 18q^{32} + 574q^{33} + 12q^{34} + 240q^{35} - 400q^{36} + 312q^{37} + 348q^{38} + 442q^{39} - 266q^{40} + 40q^{41} - 64q^{42} - 322q^{43} + 426q^{44} + 320q^{45} + 140q^{46} - 130q^{47} - 522q^{48} - 158q^{49} - 24q^{50} - 516q^{51} + 338q^{52} + 1002q^{53} - 218q^{54} - 462q^{55} - 584q^{56} + 716q^{57} - 58q^{58} - 900q^{59} + 30q^{60} - 948q^{61} + 42q^{62} + 944q^{63} + 118q^{64} - 286q^{65} + 322q^{66} + 320q^{67} - 444q^{68} - 836q^{69} - 568q^{70} - 660q^{71} - 1032q^{72} + 648q^{73} - 536q^{74} + 1160q^{75} + 844q^{76} - 1272q^{77} + 234q^{78} + 258q^{79} + 486q^{80} + 1790q^{81} + 512q^{82} + 1212q^{83} + 408q^{84} - 12q^{85} + 1006q^{86} - 290q^{87} + 1394q^{88} + 760q^{89} - 664q^{90} - 832q^{91} - 644q^{92} + 2226q^{93} - 698q^{94} + 1612q^{95} - 642q^{96} + 24q^{97} - 482q^{98} - 4856q^{99} + O(q^{100}) \)

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
−2.41421 −0.757359 −2.17157 −10.6569 1.82843 −22.1421 24.5563 −26.4264 25.7279
1.2 0.414214 −9.24264 −7.82843 0.656854 −3.82843 6.14214 −6.55635 58.4264 0.272078
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(29\) \(-1\)

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 29.4.a.a 2
3.b odd 2 1 261.4.a.b 2
4.b odd 2 1 464.4.a.f 2
5.b even 2 1 725.4.a.b 2
7.b odd 2 1 1421.4.a.c 2
8.b even 2 1 1856.4.a.n 2
8.d odd 2 1 1856.4.a.h 2
29.b even 2 1 841.4.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.4.a.a 2 1.a even 1 1 trivial
261.4.a.b 2 3.b odd 2 1
464.4.a.f 2 4.b odd 2 1
725.4.a.b 2 5.b even 2 1
841.4.a.a 2 29.b even 2 1
1421.4.a.c 2 7.b odd 2 1
1856.4.a.h 2 8.d odd 2 1
1856.4.a.n 2 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2 T_{2} - 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(29))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + 2 T + T^{2} \)
$3$ \( 7 + 10 T + T^{2} \)
$5$ \( -7 + 10 T + T^{2} \)
$7$ \( -136 + 16 T + T^{2} \)
$11$ \( -2569 + 26 T + T^{2} \)
$13$ \( -1183 + 26 T + T^{2} \)
$17$ \( 252 - 60 T + T^{2} \)
$19$ \( 10052 + 220 T + T^{2} \)
$23$ \( -3932 - 52 T + T^{2} \)
$29$ \( ( -29 + T )^{2} \)
$31$ \( 13671 + 294 T + T^{2} \)
$37$ \( 18064 - 312 T + T^{2} \)
$41$ \( -37688 - 40 T + T^{2} \)
$43$ \( -32561 + 322 T + T^{2} \)
$47$ \( -81473 + 130 T + T^{2} \)
$53$ \( 221233 - 1002 T + T^{2} \)
$59$ \( 79492 + 900 T + T^{2} \)
$61$ \( 161308 + 948 T + T^{2} \)
$67$ \( -442912 - 320 T + T^{2} \)
$71$ \( 106588 + 660 T + T^{2} \)
$73$ \( -714224 - 648 T + T^{2} \)
$79$ \( -215921 - 258 T + T^{2} \)
$83$ \( 359044 - 1212 T + T^{2} \)
$89$ \( -400568 - 760 T + T^{2} \)
$97$ \( -668024 - 24 T + T^{2} \)
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