Properties

Label 363.6.a.g
Level $363$
Weight $6$
Character orbit 363.a
Self dual yes
Analytic conductor $58.219$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 363.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(58.2193265921\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{313}) \)
Defining polynomial: \(x^{2} - x - 78\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{313})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} -9 q^{3} + ( 46 + \beta ) q^{4} + ( -24 + 10 \beta ) q^{5} + 9 \beta q^{6} + ( 10 - 2 \beta ) q^{7} + ( -78 - 15 \beta ) q^{8} + 81 q^{9} +O(q^{10})\) \( q -\beta q^{2} -9 q^{3} + ( 46 + \beta ) q^{4} + ( -24 + 10 \beta ) q^{5} + 9 \beta q^{6} + ( 10 - 2 \beta ) q^{7} + ( -78 - 15 \beta ) q^{8} + 81 q^{9} + ( -780 + 14 \beta ) q^{10} + ( -414 - 9 \beta ) q^{12} + ( -20 + 106 \beta ) q^{13} + ( 156 - 8 \beta ) q^{14} + ( 216 - 90 \beta ) q^{15} + ( -302 + 61 \beta ) q^{16} + ( 462 - 4 \beta ) q^{17} -81 \beta q^{18} + ( 1468 - 4 \beta ) q^{19} + ( -324 + 446 \beta ) q^{20} + ( -90 + 18 \beta ) q^{21} + ( 2610 + 26 \beta ) q^{23} + ( 702 + 135 \beta ) q^{24} + ( 5251 - 380 \beta ) q^{25} + ( -8268 - 86 \beta ) q^{26} -729 q^{27} + ( 304 - 84 \beta ) q^{28} + ( 6234 + 132 \beta ) q^{29} + ( 7020 - 126 \beta ) q^{30} + ( 4664 + 608 \beta ) q^{31} + ( -2262 + 721 \beta ) q^{32} + ( 312 - 458 \beta ) q^{34} + ( -1800 + 128 \beta ) q^{35} + ( 3726 + 81 \beta ) q^{36} + ( 3158 - 320 \beta ) q^{37} + ( 312 - 1464 \beta ) q^{38} + ( 180 - 954 \beta ) q^{39} + ( -9828 - 570 \beta ) q^{40} + ( -12486 + 728 \beta ) q^{41} + ( -1404 + 72 \beta ) q^{42} + ( -9560 - 1240 \beta ) q^{43} + ( -1944 + 810 \beta ) q^{45} + ( -2028 - 2636 \beta ) q^{46} + ( -2514 - 778 \beta ) q^{47} + ( 2718 - 549 \beta ) q^{48} + ( -16395 - 36 \beta ) q^{49} + ( 29640 - 4871 \beta ) q^{50} + ( -4158 + 36 \beta ) q^{51} + ( 7348 + 4962 \beta ) q^{52} + ( 20088 + 594 \beta ) q^{53} + 729 \beta q^{54} + ( 1560 + 36 \beta ) q^{56} + ( -13212 + 36 \beta ) q^{57} + ( -10296 - 6366 \beta ) q^{58} + ( 10944 - 3676 \beta ) q^{59} + ( 2916 - 4014 \beta ) q^{60} + ( 7072 - 2746 \beta ) q^{61} + ( -47424 - 5272 \beta ) q^{62} + ( 810 - 162 \beta ) q^{63} + ( -46574 - 411 \beta ) q^{64} + ( 83160 - 1684 \beta ) q^{65} + ( 32300 + 768 \beta ) q^{67} + ( 20940 + 274 \beta ) q^{68} + ( -23490 - 234 \beta ) q^{69} + ( -9984 + 1672 \beta ) q^{70} + ( 32274 - 3102 \beta ) q^{71} + ( -6318 - 1215 \beta ) q^{72} + ( -26546 - 320 \beta ) q^{73} + ( 24960 - 2838 \beta ) q^{74} + ( -47259 + 3420 \beta ) q^{75} + ( 67216 + 1280 \beta ) q^{76} + ( 74412 + 774 \beta ) q^{78} + ( -9626 + 2130 \beta ) q^{79} + ( 54828 - 3874 \beta ) q^{80} + 6561 q^{81} + ( -56784 + 11758 \beta ) q^{82} + ( 5388 + 3528 \beta ) q^{83} + ( -2736 + 756 \beta ) q^{84} + ( -14208 + 4676 \beta ) q^{85} + ( 96720 + 10800 \beta ) q^{86} + ( -56106 - 1188 \beta ) q^{87} + ( -30582 + 3024 \beta ) q^{89} + ( -63180 + 1134 \beta ) q^{90} + ( -16736 + 888 \beta ) q^{91} + ( 122088 + 3832 \beta ) q^{92} + ( -41976 - 5472 \beta ) q^{93} + ( 60684 + 3292 \beta ) q^{94} + ( -38352 + 14736 \beta ) q^{95} + ( 20358 - 6489 \beta ) q^{96} + ( -92074 + 1092 \beta ) q^{97} + ( 2808 + 16431 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - 18q^{3} + 93q^{4} - 38q^{5} + 9q^{6} + 18q^{7} - 171q^{8} + 162q^{9} + O(q^{10}) \) \( 2q - q^{2} - 18q^{3} + 93q^{4} - 38q^{5} + 9q^{6} + 18q^{7} - 171q^{8} + 162q^{9} - 1546q^{10} - 837q^{12} + 66q^{13} + 304q^{14} + 342q^{15} - 543q^{16} + 920q^{17} - 81q^{18} + 2932q^{19} - 202q^{20} - 162q^{21} + 5246q^{23} + 1539q^{24} + 10122q^{25} - 16622q^{26} - 1458q^{27} + 524q^{28} + 12600q^{29} + 13914q^{30} + 9936q^{31} - 3803q^{32} + 166q^{34} - 3472q^{35} + 7533q^{36} + 5996q^{37} - 840q^{38} - 594q^{39} - 20226q^{40} - 24244q^{41} - 2736q^{42} - 20360q^{43} - 3078q^{45} - 6692q^{46} - 5806q^{47} + 4887q^{48} - 32826q^{49} + 54409q^{50} - 8280q^{51} + 19658q^{52} + 40770q^{53} + 729q^{54} + 3156q^{56} - 26388q^{57} - 26958q^{58} + 18212q^{59} + 1818q^{60} + 11398q^{61} - 100120q^{62} + 1458q^{63} - 93559q^{64} + 164636q^{65} + 65368q^{67} + 42154q^{68} - 47214q^{69} - 18296q^{70} + 61446q^{71} - 13851q^{72} - 53412q^{73} + 47082q^{74} - 91098q^{75} + 135712q^{76} + 149598q^{78} - 17122q^{79} + 105782q^{80} + 13122q^{81} - 101810q^{82} + 14304q^{83} - 4716q^{84} - 23740q^{85} + 204240q^{86} - 113400q^{87} - 58140q^{89} - 125226q^{90} - 32584q^{91} + 248008q^{92} - 89424q^{93} + 124660q^{94} - 61968q^{95} + 34227q^{96} - 183056q^{97} + 22047q^{98} + O(q^{100}) \)

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
9.34590
−8.34590
−9.34590 −9.00000 55.3459 69.4590 84.1131 −8.69181 −218.189 81.0000 −649.157
1.2 8.34590 −9.00000 37.6541 −107.459 −75.1131 26.6918 47.1885 81.0000 −896.843
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(-1\)

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 363.6.a.g 2
3.b odd 2 1 1089.6.a.o 2
11.b odd 2 1 33.6.a.d 2
33.d even 2 1 99.6.a.e 2
44.c even 2 1 528.6.a.q 2
55.d odd 2 1 825.6.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.d 2 11.b odd 2 1
99.6.a.e 2 33.d even 2 1
363.6.a.g 2 1.a even 1 1 trivial
528.6.a.q 2 44.c even 2 1
825.6.a.d 2 55.d odd 2 1
1089.6.a.o 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} - 78 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(363))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -78 + T + T^{2} \)
$3$ \( ( 9 + T )^{2} \)
$5$ \( -7464 + 38 T + T^{2} \)
$7$ \( -232 - 18 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( -878128 - 66 T + T^{2} \)
$17$ \( 210348 - 920 T + T^{2} \)
$19$ \( 2147904 - 2932 T + T^{2} \)
$23$ \( 6827232 - 5246 T + T^{2} \)
$29$ \( 38326572 - 12600 T + T^{2} \)
$31$ \( -4245184 - 9936 T + T^{2} \)
$37$ \( 975204 - 5996 T + T^{2} \)
$41$ \( 105471636 + 24244 T + T^{2} \)
$43$ \( -16684800 + 20360 T + T^{2} \)
$47$ \( -38936064 + 5806 T + T^{2} \)
$53$ \( 387938808 - 40770 T + T^{2} \)
$59$ \( -974471136 - 18212 T + T^{2} \)
$61$ \( -557566776 - 11398 T + T^{2} \)
$67$ \( 1022090128 - 65368 T + T^{2} \)
$71$ \( 190949616 - 61446 T + T^{2} \)
$73$ \( 705197636 + 53412 T + T^{2} \)
$79$ \( -281721704 + 17122 T + T^{2} \)
$83$ \( -922809744 - 14304 T + T^{2} \)
$89$ \( 129501828 + 58140 T + T^{2} \)
$97$ \( 8284064476 + 183056 T + T^{2} \)
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