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: \(x^{2} - x - 7\)
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.

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

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
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 \(S_{6}^{\mathrm{new}}(\Gamma_0(69))\).

Hecke characteristic polynomials

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