Properties

Label 69.6.a.a
Level $69$
Weight $6$
Character orbit 69.a
Self dual yes
Analytic conductor $11.066$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0664835671\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{29}) \)
Defining polynomial: \( x^{2} - x - 7 \) 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 \(\beta = \sqrt{29}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta q^{2} - 9 q^{3} + 84 q^{4} + ( - \beta + 47) q^{5} + 18 \beta q^{6} + (11 \beta - 59) q^{7} - 104 \beta q^{8} + 81 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta q^{2} - 9 q^{3} + 84 q^{4} + ( - \beta + 47) q^{5} + 18 \beta q^{6} + (11 \beta - 59) q^{7} - 104 \beta q^{8} + 81 q^{9} + ( - 94 \beta + 58) q^{10} + ( - 108 \beta + 160) q^{11} - 756 q^{12} + (74 \beta - 144) q^{13} + (118 \beta - 638) q^{14} + (9 \beta - 423) q^{15} + 3344 q^{16} + ( - 13 \beta - 905) q^{17} - 162 \beta q^{18} + (311 \beta + 365) q^{19} + ( - 84 \beta + 3948) q^{20} + ( - 99 \beta + 531) q^{21} + ( - 320 \beta + 6264) q^{22} + 529 q^{23} + 936 \beta q^{24} + ( - 94 \beta - 887) q^{25} + (288 \beta - 4292) q^{26} - 729 q^{27} + (924 \beta - 4956) q^{28} + ( - 266 \beta + 4104) q^{29} + (846 \beta - 522) q^{30} + (1514 \beta + 886) q^{31} - 3360 \beta q^{32} + (972 \beta - 1440) q^{33} + (1810 \beta + 754) q^{34} + (576 \beta - 3092) q^{35} + 6804 q^{36} + ( - 206 \beta - 11556) q^{37} + ( - 730 \beta - 18038) q^{38} + ( - 666 \beta + 1296) q^{39} + ( - 4888 \beta + 3016) q^{40} + (1372 \beta + 2758) q^{41} + ( - 1062 \beta + 5742) q^{42} + (2667 \beta + 5161) q^{43} + ( - 9072 \beta + 13440) q^{44} + ( - 81 \beta + 3807) q^{45} - 1058 \beta q^{46} + (1152 \beta + 21476) q^{47} - 30096 q^{48} + ( - 1298 \beta - 9817) q^{49} + (1774 \beta + 5452) q^{50} + (117 \beta + 8145) q^{51} + (6216 \beta - 12096) q^{52} + (4369 \beta - 12675) q^{53} + 1458 \beta q^{54} + ( - 5236 \beta + 10652) q^{55} + (6136 \beta - 33176) q^{56} + ( - 2799 \beta - 3285) q^{57} + ( - 8208 \beta + 15428) q^{58} + (508 \beta + 9172) q^{59} + (756 \beta - 35532) q^{60} + (3946 \beta + 18612) q^{61} + ( - 1772 \beta - 87812) q^{62} + (891 \beta - 4779) q^{63} + 87872 q^{64} + (3622 \beta - 8914) q^{65} + (2880 \beta - 56376) q^{66} + ( - 4607 \beta - 3741) q^{67} + ( - 1092 \beta - 76020) q^{68} - 4761 q^{69} + (6184 \beta - 33408) q^{70} + ( - 2088 \beta + 63424) q^{71} - 8424 \beta q^{72} + ( - 2212 \beta + 68830) q^{73} + (23112 \beta + 11948) q^{74} + (846 \beta + 7983) q^{75} + (26124 \beta + 30660) q^{76} + (8132 \beta - 43892) q^{77} + ( - 2592 \beta + 38628) q^{78} + ( - 1479 \beta + 31143) q^{79} + ( - 3344 \beta + 157168) q^{80} + 6561 q^{81} + ( - 5516 \beta - 79576) q^{82} + ( - 11124 \beta + 41560) q^{83} + ( - 8316 \beta + 44604) q^{84} + (294 \beta - 42158) q^{85} + ( - 10322 \beta - 154686) q^{86} + (2394 \beta - 36936) q^{87} + ( - 16640 \beta + 325728) q^{88} + (2317 \beta + 34885) q^{89} + ( - 7614 \beta + 4698) q^{90} + ( - 5950 \beta + 32102) q^{91} + 44436 q^{92} + ( - 13626 \beta - 7974) q^{93} + ( - 42952 \beta - 66816) q^{94} + (14252 \beta + 8136) q^{95} + 30240 \beta q^{96} + ( - 10274 \beta - 85052) q^{97} + (19634 \beta + 75284) q^{98} + ( - 8748 \beta + 12960) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{3} + 168 q^{4} + 94 q^{5} - 118 q^{7} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{3} + 168 q^{4} + 94 q^{5} - 118 q^{7} + 162 q^{9} + 116 q^{10} + 320 q^{11} - 1512 q^{12} - 288 q^{13} - 1276 q^{14} - 846 q^{15} + 6688 q^{16} - 1810 q^{17} + 730 q^{19} + 7896 q^{20} + 1062 q^{21} + 12528 q^{22} + 1058 q^{23} - 1774 q^{25} - 8584 q^{26} - 1458 q^{27} - 9912 q^{28} + 8208 q^{29} - 1044 q^{30} + 1772 q^{31} - 2880 q^{33} + 1508 q^{34} - 6184 q^{35} + 13608 q^{36} - 23112 q^{37} - 36076 q^{38} + 2592 q^{39} + 6032 q^{40} + 5516 q^{41} + 11484 q^{42} + 10322 q^{43} + 26880 q^{44} + 7614 q^{45} + 42952 q^{47} - 60192 q^{48} - 19634 q^{49} + 10904 q^{50} + 16290 q^{51} - 24192 q^{52} - 25350 q^{53} + 21304 q^{55} - 66352 q^{56} - 6570 q^{57} + 30856 q^{58} + 18344 q^{59} - 71064 q^{60} + 37224 q^{61} - 175624 q^{62} - 9558 q^{63} + 175744 q^{64} - 17828 q^{65} - 112752 q^{66} - 7482 q^{67} - 152040 q^{68} - 9522 q^{69} - 66816 q^{70} + 126848 q^{71} + 137660 q^{73} + 23896 q^{74} + 15966 q^{75} + 61320 q^{76} - 87784 q^{77} + 77256 q^{78} + 62286 q^{79} + 314336 q^{80} + 13122 q^{81} - 159152 q^{82} + 83120 q^{83} + 89208 q^{84} - 84316 q^{85} - 309372 q^{86} - 73872 q^{87} + 651456 q^{88} + 69770 q^{89} + 9396 q^{90} + 64204 q^{91} + 88872 q^{92} - 15948 q^{93} - 133632 q^{94} + 16272 q^{95} - 170104 q^{97} + 150568 q^{98} + 25920 q^{99}+O(q^{100}) \) 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
3.19258
−2.19258
−10.7703 −9.00000 84.0000 41.6148 96.9330 0.236813 −560.057 81.0000 −448.205
1.2 10.7703 −9.00000 84.0000 52.3852 −96.9330 −118.237 560.057 81.0000 564.205
\(n\): 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 69.6.a.a 2
3.b odd 2 1 207.6.a.a 2
4.b odd 2 1 1104.6.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.6.a.a 2 1.a even 1 1 trivial
207.6.a.a 2 3.b odd 2 1
1104.6.a.h 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 116 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(69))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 116 \) Copy content Toggle raw display
$3$ \( (T + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 94T + 2180 \) Copy content Toggle raw display
$7$ \( T^{2} + 118T - 28 \) Copy content Toggle raw display
$11$ \( T^{2} - 320T - 312656 \) Copy content Toggle raw display
$13$ \( T^{2} + 288T - 138068 \) Copy content Toggle raw display
$17$ \( T^{2} + 1810 T + 814124 \) Copy content Toggle raw display
$19$ \( T^{2} - 730 T - 2671684 \) Copy content Toggle raw display
$23$ \( (T - 529)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 8208 T + 14790892 \) Copy content Toggle raw display
$31$ \( T^{2} - 1772 T - 65688688 \) Copy content Toggle raw display
$37$ \( T^{2} + 23112 T + 132310492 \) Copy content Toggle raw display
$41$ \( T^{2} - 5516 T - 46982572 \) Copy content Toggle raw display
$43$ \( T^{2} - 10322 T - 179637860 \) Copy content Toggle raw display
$47$ \( T^{2} - 42952 T + 422732560 \) Copy content Toggle raw display
$53$ \( T^{2} + 25350 T - 392901044 \) Copy content Toggle raw display
$59$ \( T^{2} - 18344 T + 76641728 \) Copy content Toggle raw display
$61$ \( T^{2} - 37224 T - 105150020 \) Copy content Toggle raw display
$67$ \( T^{2} + 7482 T - 601513940 \) Copy content Toggle raw display
$71$ \( T^{2} - 126848 T + 3896171200 \) Copy content Toggle raw display
$73$ \( T^{2} - 137660 T + 4595673524 \) Copy content Toggle raw display
$79$ \( T^{2} - 62286 T + 906450660 \) Copy content Toggle raw display
$83$ \( T^{2} - 83120 T - 1861324304 \) Copy content Toggle raw display
$89$ \( T^{2} - 69770 T + 1061277044 \) Copy content Toggle raw display
$97$ \( T^{2} + 170104 T + 4172745500 \) Copy content Toggle raw display
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