Properties

Label 69.4.a.b
Level $69$
Weight $4$
Character orbit 69.a
Self dual yes
Analytic conductor $4.071$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.07113179040\)
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: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} -3 q^{3} + ( 1 - 2 \beta ) q^{4} + ( 4 - 4 \beta ) q^{5} + ( 3 - 3 \beta ) q^{6} + ( -14 + \beta ) q^{7} + ( -9 - 5 \beta ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} -3 q^{3} + ( 1 - 2 \beta ) q^{4} + ( 4 - 4 \beta ) q^{5} + ( 3 - 3 \beta ) q^{6} + ( -14 + \beta ) q^{7} + ( -9 - 5 \beta ) q^{8} + 9 q^{9} + ( -36 + 8 \beta ) q^{10} + ( -36 + 7 \beta ) q^{11} + ( -3 + 6 \beta ) q^{12} + ( -30 + 6 \beta ) q^{13} + ( 22 - 15 \beta ) q^{14} + ( -12 + 12 \beta ) q^{15} + ( -39 + 12 \beta ) q^{16} + ( 48 - 5 \beta ) q^{17} + ( -9 + 9 \beta ) q^{18} + ( -74 + 24 \beta ) q^{19} + ( 68 - 12 \beta ) q^{20} + ( 42 - 3 \beta ) q^{21} + ( 92 - 43 \beta ) q^{22} + 23 q^{23} + ( 27 + 15 \beta ) q^{24} + ( 19 - 32 \beta ) q^{25} + ( 78 - 36 \beta ) q^{26} -27 q^{27} + ( -30 + 29 \beta ) q^{28} + ( -102 + 66 \beta ) q^{29} + ( 108 - 24 \beta ) q^{30} + ( -84 - 54 \beta ) q^{31} + ( 207 - 11 \beta ) q^{32} + ( 108 - 21 \beta ) q^{33} + ( -88 + 53 \beta ) q^{34} + ( -88 + 60 \beta ) q^{35} + ( 9 - 18 \beta ) q^{36} + ( 58 - 43 \beta ) q^{37} + ( 266 - 98 \beta ) q^{38} + ( 90 - 18 \beta ) q^{39} + ( 124 + 16 \beta ) q^{40} + ( 2 + 26 \beta ) q^{41} + ( -66 + 45 \beta ) q^{42} + ( -210 - 18 \beta ) q^{43} + ( -148 + 79 \beta ) q^{44} + ( 36 - 36 \beta ) q^{45} + ( -23 + 23 \beta ) q^{46} + ( -24 + 88 \beta ) q^{47} + ( 117 - 36 \beta ) q^{48} + ( -139 - 28 \beta ) q^{49} + ( -275 + 51 \beta ) q^{50} + ( -144 + 15 \beta ) q^{51} + ( -126 + 66 \beta ) q^{52} + ( 16 + 10 \beta ) q^{53} + ( 27 - 27 \beta ) q^{54} + ( -368 + 172 \beta ) q^{55} + ( 86 + 61 \beta ) q^{56} + ( 222 - 72 \beta ) q^{57} + ( 630 - 168 \beta ) q^{58} + ( -20 - 158 \beta ) q^{59} + ( -204 + 36 \beta ) q^{60} + ( -382 - 49 \beta ) q^{61} + ( -348 - 30 \beta ) q^{62} + ( -126 + 9 \beta ) q^{63} + ( 17 + 122 \beta ) q^{64} + ( -312 + 144 \beta ) q^{65} + ( -276 + 129 \beta ) q^{66} + ( -494 - 146 \beta ) q^{67} + ( 128 - 101 \beta ) q^{68} -69 q^{69} + ( 568 - 148 \beta ) q^{70} + ( -112 - 286 \beta ) q^{71} + ( -81 - 45 \beta ) q^{72} + ( 410 - 56 \beta ) q^{73} + ( -402 + 101 \beta ) q^{74} + ( -57 + 96 \beta ) q^{75} + ( -458 + 172 \beta ) q^{76} + ( 560 - 134 \beta ) q^{77} + ( -234 + 108 \beta ) q^{78} + ( 886 + 59 \beta ) q^{79} + ( -540 + 204 \beta ) q^{80} + 81 q^{81} + ( 206 - 24 \beta ) q^{82} + ( 740 - 211 \beta ) q^{83} + ( 90 - 87 \beta ) q^{84} + ( 352 - 212 \beta ) q^{85} + ( 66 - 192 \beta ) q^{86} + ( 306 - 198 \beta ) q^{87} + ( 44 + 117 \beta ) q^{88} + ( 372 + 251 \beta ) q^{89} + ( -324 + 72 \beta ) q^{90} + ( 468 - 114 \beta ) q^{91} + ( 23 - 46 \beta ) q^{92} + ( 252 + 162 \beta ) q^{93} + ( 728 - 112 \beta ) q^{94} + ( -1064 + 392 \beta ) q^{95} + ( -621 + 33 \beta ) q^{96} + ( -130 + 456 \beta ) q^{97} + ( -85 - 111 \beta ) q^{98} + ( -324 + 63 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 6q^{3} + 2q^{4} + 8q^{5} + 6q^{6} - 28q^{7} - 18q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 6q^{3} + 2q^{4} + 8q^{5} + 6q^{6} - 28q^{7} - 18q^{8} + 18q^{9} - 72q^{10} - 72q^{11} - 6q^{12} - 60q^{13} + 44q^{14} - 24q^{15} - 78q^{16} + 96q^{17} - 18q^{18} - 148q^{19} + 136q^{20} + 84q^{21} + 184q^{22} + 46q^{23} + 54q^{24} + 38q^{25} + 156q^{26} - 54q^{27} - 60q^{28} - 204q^{29} + 216q^{30} - 168q^{31} + 414q^{32} + 216q^{33} - 176q^{34} - 176q^{35} + 18q^{36} + 116q^{37} + 532q^{38} + 180q^{39} + 248q^{40} + 4q^{41} - 132q^{42} - 420q^{43} - 296q^{44} + 72q^{45} - 46q^{46} - 48q^{47} + 234q^{48} - 278q^{49} - 550q^{50} - 288q^{51} - 252q^{52} + 32q^{53} + 54q^{54} - 736q^{55} + 172q^{56} + 444q^{57} + 1260q^{58} - 40q^{59} - 408q^{60} - 764q^{61} - 696q^{62} - 252q^{63} + 34q^{64} - 624q^{65} - 552q^{66} - 988q^{67} + 256q^{68} - 138q^{69} + 1136q^{70} - 224q^{71} - 162q^{72} + 820q^{73} - 804q^{74} - 114q^{75} - 916q^{76} + 1120q^{77} - 468q^{78} + 1772q^{79} - 1080q^{80} + 162q^{81} + 412q^{82} + 1480q^{83} + 180q^{84} + 704q^{85} + 132q^{86} + 612q^{87} + 88q^{88} + 744q^{89} - 648q^{90} + 936q^{91} + 46q^{92} + 504q^{93} + 1456q^{94} - 2128q^{95} - 1242q^{96} - 260q^{97} - 170q^{98} - 648q^{99} + O(q^{100}) \)

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
−3.82843 −3.00000 6.65685 15.3137 11.4853 −16.8284 5.14214 9.00000 −58.6274
1.2 1.82843 −3.00000 −4.65685 −7.31371 −5.48528 −11.1716 −23.1421 9.00000 −13.3726
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(23\) \(-1\)

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 69.4.a.b 2
3.b odd 2 1 207.4.a.b 2
4.b odd 2 1 1104.4.a.q 2
5.b even 2 1 1725.4.a.m 2
23.b odd 2 1 1587.4.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.4.a.b 2 1.a even 1 1 trivial
207.4.a.b 2 3.b odd 2 1
1104.4.a.q 2 4.b odd 2 1
1587.4.a.c 2 23.b odd 2 1
1725.4.a.m 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2 T_{2} - 7 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(69))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -7 + 2 T + T^{2} \)
$3$ \( ( 3 + T )^{2} \)
$5$ \( -112 - 8 T + T^{2} \)
$7$ \( 188 + 28 T + T^{2} \)
$11$ \( 904 + 72 T + T^{2} \)
$13$ \( 612 + 60 T + T^{2} \)
$17$ \( 2104 - 96 T + T^{2} \)
$19$ \( 868 + 148 T + T^{2} \)
$23$ \( ( -23 + T )^{2} \)
$29$ \( -24444 + 204 T + T^{2} \)
$31$ \( -16272 + 168 T + T^{2} \)
$37$ \( -11428 - 116 T + T^{2} \)
$41$ \( -5404 - 4 T + T^{2} \)
$43$ \( 41508 + 420 T + T^{2} \)
$47$ \( -61376 + 48 T + T^{2} \)
$53$ \( -544 - 32 T + T^{2} \)
$59$ \( -199312 + 40 T + T^{2} \)
$61$ \( 126716 + 764 T + T^{2} \)
$67$ \( 73508 + 988 T + T^{2} \)
$71$ \( -641824 + 224 T + T^{2} \)
$73$ \( 143012 - 820 T + T^{2} \)
$79$ \( 757148 - 1772 T + T^{2} \)
$83$ \( 191432 - 1480 T + T^{2} \)
$89$ \( -365624 - 744 T + T^{2} \)
$97$ \( -1646588 + 260 T + T^{2} \)
show more
show less