Properties

Label 363.6.a.f
Level $363$
Weight $6$
Character orbit 363.a
Self dual yes
Analytic conductor $58.219$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -6 - \beta ) q^{2} + 9 q^{3} + ( 12 + 13 \beta ) q^{4} + ( 34 - 10 \beta ) q^{5} + ( -54 - 9 \beta ) q^{6} + ( -104 + 62 \beta ) q^{7} + ( 16 - 71 \beta ) q^{8} + 81 q^{9} +O(q^{10})\) \( q + ( -6 - \beta ) q^{2} + 9 q^{3} + ( 12 + 13 \beta ) q^{4} + ( 34 - 10 \beta ) q^{5} + ( -54 - 9 \beta ) q^{6} + ( -104 + 62 \beta ) q^{7} + ( 16 - 71 \beta ) q^{8} + 81 q^{9} + ( -124 + 36 \beta ) q^{10} + ( 108 + 117 \beta ) q^{12} + ( 102 - 74 \beta ) q^{13} + ( 128 - 330 \beta ) q^{14} + ( 306 - 90 \beta ) q^{15} + ( 88 + 65 \beta ) q^{16} + ( 178 + 372 \beta ) q^{17} + ( -486 - 81 \beta ) q^{18} + ( 840 - 852 \beta ) q^{19} + ( -632 + 192 \beta ) q^{20} + ( -936 + 558 \beta ) q^{21} + ( -284 + 330 \beta ) q^{23} + ( 144 - 639 \beta ) q^{24} + ( -1169 - 580 \beta ) q^{25} + ( -20 + 416 \beta ) q^{26} + 729 q^{27} + ( 5200 + 198 \beta ) q^{28} + ( 398 - 1492 \beta ) q^{29} + ( -1116 + 324 \beta ) q^{30} + ( -4440 - 1600 \beta ) q^{31} + ( -1560 + 1729 \beta ) q^{32} + ( -4044 - 2782 \beta ) q^{34} + ( -8496 + 2528 \beta ) q^{35} + ( 972 + 1053 \beta ) q^{36} + ( -2362 + 2816 \beta ) q^{37} + ( 1776 + 5124 \beta ) q^{38} + ( 918 - 666 \beta ) q^{39} + ( 6224 - 1864 \beta ) q^{40} + ( -18238 - 8 \beta ) q^{41} + ( 1152 - 2970 \beta ) q^{42} + ( -3328 - 3112 \beta ) q^{43} + ( 2754 - 810 \beta ) q^{45} + ( -936 - 2026 \beta ) q^{46} + ( 21676 + 390 \beta ) q^{47} + ( 792 + 585 \beta ) q^{48} + ( 24761 - 9052 \beta ) q^{49} + ( 11654 + 5229 \beta ) q^{50} + ( 1602 + 3348 \beta ) q^{51} + ( -6472 - 524 \beta ) q^{52} + ( -9638 + 7102 \beta ) q^{53} + ( -4374 - 729 \beta ) q^{54} + ( -36880 + 3974 \beta ) q^{56} + ( 7560 - 7668 \beta ) q^{57} + ( 9548 + 10046 \beta ) q^{58} + ( -404 - 1980 \beta ) q^{59} + ( -5688 + 1728 \beta ) q^{60} + ( 11638 + 2026 \beta ) q^{61} + ( 39440 + 15640 \beta ) q^{62} + ( -8424 + 5022 \beta ) q^{63} + ( -7288 - 12623 \beta ) q^{64} + ( 9388 - 2796 \beta ) q^{65} + ( -26612 + 12704 \beta ) q^{67} + ( 40824 + 11614 \beta ) q^{68} + ( -2556 + 2970 \beta ) q^{69} + ( 30752 - 9200 \beta ) q^{70} + ( 13516 + 4354 \beta ) q^{71} + ( 1296 - 5751 \beta ) q^{72} + ( 20606 + 5568 \beta ) q^{73} + ( -8356 - 17350 \beta ) q^{74} + ( -10521 - 5220 \beta ) q^{75} + ( -78528 - 10380 \beta ) q^{76} + ( -180 + 3744 \beta ) q^{78} + ( 2712 + 11426 \beta ) q^{79} + ( -2208 + 680 \beta ) q^{80} + 6561 q^{81} + ( 109492 + 18294 \beta ) q^{82} + ( -50700 + 21960 \beta ) q^{83} + ( 46800 + 1782 \beta ) q^{84} + ( -23708 + 7148 \beta ) q^{85} + ( 44864 + 25112 \beta ) q^{86} + ( 3582 - 13428 \beta ) q^{87} + ( -13750 - 26704 \beta ) q^{89} + ( -10044 + 2916 \beta ) q^{90} + ( -47312 + 9432 \beta ) q^{91} + ( 30912 + 4558 \beta ) q^{92} + ( -39960 - 14400 \beta ) q^{93} + ( -133176 - 24406 \beta ) q^{94} + ( 96720 - 28848 \beta ) q^{95} + ( -14040 + 15561 \beta ) q^{96} + ( -115822 - 9924 \beta ) q^{97} + ( -76150 + 38603 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 13q^{2} + 18q^{3} + 37q^{4} + 58q^{5} - 117q^{6} - 146q^{7} - 39q^{8} + 162q^{9} + O(q^{10}) \) \( 2q - 13q^{2} + 18q^{3} + 37q^{4} + 58q^{5} - 117q^{6} - 146q^{7} - 39q^{8} + 162q^{9} - 212q^{10} + 333q^{12} + 130q^{13} - 74q^{14} + 522q^{15} + 241q^{16} + 728q^{17} - 1053q^{18} + 828q^{19} - 1072q^{20} - 1314q^{21} - 238q^{23} - 351q^{24} - 2918q^{25} + 376q^{26} + 1458q^{27} + 10598q^{28} - 696q^{29} - 1908q^{30} - 10480q^{31} - 1391q^{32} - 10870q^{34} - 14464q^{35} + 2997q^{36} - 1908q^{37} + 8676q^{38} + 1170q^{39} + 10584q^{40} - 36484q^{41} - 666q^{42} - 9768q^{43} + 4698q^{45} - 3898q^{46} + 43742q^{47} + 2169q^{48} + 40470q^{49} + 28537q^{50} + 6552q^{51} - 13468q^{52} - 12174q^{53} - 9477q^{54} - 69786q^{56} + 7452q^{57} + 29142q^{58} - 2788q^{59} - 9648q^{60} + 25302q^{61} + 94520q^{62} - 11826q^{63} - 27199q^{64} + 15980q^{65} - 40520q^{67} + 93262q^{68} - 2142q^{69} + 52304q^{70} + 31386q^{71} - 3159q^{72} + 46780q^{73} - 34062q^{74} - 26262q^{75} - 167436q^{76} + 3384q^{78} + 16850q^{79} - 3736q^{80} + 13122q^{81} + 237278q^{82} - 79440q^{83} + 95382q^{84} - 40268q^{85} + 114840q^{86} - 6264q^{87} - 54204q^{89} - 17172q^{90} - 85192q^{91} + 66382q^{92} - 94320q^{93} - 290758q^{94} + 164592q^{95} - 12519q^{96} - 241568q^{97} - 113697q^{98} + O(q^{100}) \)

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
3.37228
−2.37228
−9.37228 9.00000 55.8397 0.277187 −84.3505 105.081 −223.432 81.0000 −2.59787
1.2 −3.62772 9.00000 −18.8397 57.7228 −32.6495 −251.081 184.432 81.0000 −209.402
\(n\): 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.f 2
3.b odd 2 1 1089.6.a.p 2
11.b odd 2 1 33.6.a.e 2
33.d even 2 1 99.6.a.d 2
44.c even 2 1 528.6.a.o 2
55.d odd 2 1 825.6.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.e 2 11.b odd 2 1
99.6.a.d 2 33.d even 2 1
363.6.a.f 2 1.a even 1 1 trivial
528.6.a.o 2 44.c even 2 1
825.6.a.c 2 55.d odd 2 1
1089.6.a.p 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 13 T_{2} + 34 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(363))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 34 + 13 T + T^{2} \)
$3$ \( ( -9 + T )^{2} \)
$5$ \( 16 - 58 T + T^{2} \)
$7$ \( -26384 + 146 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( -40952 - 130 T + T^{2} \)
$17$ \( -1009172 - 728 T + T^{2} \)
$19$ \( -5817312 - 828 T + T^{2} \)
$23$ \( -884264 + 238 T + T^{2} \)
$29$ \( -18243924 + 696 T + T^{2} \)
$31$ \( 6337600 + 10480 T + T^{2} \)
$37$ \( -64511196 + 1908 T + T^{2} \)
$41$ \( 332770036 + 36484 T + T^{2} \)
$43$ \( -56044032 + 9768 T + T^{2} \)
$47$ \( 477085816 - 43742 T + T^{2} \)
$53$ \( -379065264 + 12174 T + T^{2} \)
$59$ \( -30400064 + 2788 T + T^{2} \)
$61$ \( 126184224 - 25302 T + T^{2} \)
$67$ \( -921013232 + 40520 T + T^{2} \)
$71$ \( 89872392 - 31386 T + T^{2} \)
$73$ \( 291320452 - 46780 T + T^{2} \)
$79$ \( -1006085552 - 16850 T + T^{2} \)
$83$ \( -2400814800 + 79440 T + T^{2} \)
$89$ \( -5148586428 + 54204 T + T^{2} \)
$97$ \( 13776267004 + 241568 T + T^{2} \)
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