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: \( x^{2} - x - 8 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta - 6) q^{2} + 9 q^{3} + (13 \beta + 12) q^{4} + ( - 10 \beta + 34) q^{5} + ( - 9 \beta - 54) q^{6} + (62 \beta - 104) q^{7} + ( - 71 \beta + 16) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 6) q^{2} + 9 q^{3} + (13 \beta + 12) q^{4} + ( - 10 \beta + 34) q^{5} + ( - 9 \beta - 54) q^{6} + (62 \beta - 104) q^{7} + ( - 71 \beta + 16) q^{8} + 81 q^{9} + (36 \beta - 124) q^{10} + (117 \beta + 108) q^{12} + ( - 74 \beta + 102) q^{13} + ( - 330 \beta + 128) q^{14} + ( - 90 \beta + 306) q^{15} + (65 \beta + 88) q^{16} + (372 \beta + 178) q^{17} + ( - 81 \beta - 486) q^{18} + ( - 852 \beta + 840) q^{19} + (192 \beta - 632) q^{20} + (558 \beta - 936) q^{21} + (330 \beta - 284) q^{23} + ( - 639 \beta + 144) q^{24} + ( - 580 \beta - 1169) q^{25} + (416 \beta - 20) q^{26} + 729 q^{27} + (198 \beta + 5200) q^{28} + ( - 1492 \beta + 398) q^{29} + (324 \beta - 1116) q^{30} + ( - 1600 \beta - 4440) q^{31} + (1729 \beta - 1560) q^{32} + ( - 2782 \beta - 4044) q^{34} + (2528 \beta - 8496) q^{35} + (1053 \beta + 972) q^{36} + (2816 \beta - 2362) q^{37} + (5124 \beta + 1776) q^{38} + ( - 666 \beta + 918) q^{39} + ( - 1864 \beta + 6224) q^{40} + ( - 8 \beta - 18238) q^{41} + ( - 2970 \beta + 1152) q^{42} + ( - 3112 \beta - 3328) q^{43} + ( - 810 \beta + 2754) q^{45} + ( - 2026 \beta - 936) q^{46} + (390 \beta + 21676) q^{47} + (585 \beta + 792) q^{48} + ( - 9052 \beta + 24761) q^{49} + (5229 \beta + 11654) q^{50} + (3348 \beta + 1602) q^{51} + ( - 524 \beta - 6472) q^{52} + (7102 \beta - 9638) q^{53} + ( - 729 \beta - 4374) q^{54} + (3974 \beta - 36880) q^{56} + ( - 7668 \beta + 7560) q^{57} + (10046 \beta + 9548) q^{58} + ( - 1980 \beta - 404) q^{59} + (1728 \beta - 5688) q^{60} + (2026 \beta + 11638) q^{61} + (15640 \beta + 39440) q^{62} + (5022 \beta - 8424) q^{63} + ( - 12623 \beta - 7288) q^{64} + ( - 2796 \beta + 9388) q^{65} + (12704 \beta - 26612) q^{67} + (11614 \beta + 40824) q^{68} + (2970 \beta - 2556) q^{69} + ( - 9200 \beta + 30752) q^{70} + (4354 \beta + 13516) q^{71} + ( - 5751 \beta + 1296) q^{72} + (5568 \beta + 20606) q^{73} + ( - 17350 \beta - 8356) q^{74} + ( - 5220 \beta - 10521) q^{75} + ( - 10380 \beta - 78528) q^{76} + (3744 \beta - 180) q^{78} + (11426 \beta + 2712) q^{79} + (680 \beta - 2208) q^{80} + 6561 q^{81} + (18294 \beta + 109492) q^{82} + (21960 \beta - 50700) q^{83} + (1782 \beta + 46800) q^{84} + (7148 \beta - 23708) q^{85} + (25112 \beta + 44864) q^{86} + ( - 13428 \beta + 3582) q^{87} + ( - 26704 \beta - 13750) q^{89} + (2916 \beta - 10044) q^{90} + (9432 \beta - 47312) q^{91} + (4558 \beta + 30912) q^{92} + ( - 14400 \beta - 39960) q^{93} + ( - 24406 \beta - 133176) q^{94} + ( - 28848 \beta + 96720) q^{95} + (15561 \beta - 14040) q^{96} + ( - 9924 \beta - 115822) q^{97} + (38603 \beta - 76150) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 13 q^{2} + 18 q^{3} + 37 q^{4} + 58 q^{5} - 117 q^{6} - 146 q^{7} - 39 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 13 q^{2} + 18 q^{3} + 37 q^{4} + 58 q^{5} - 117 q^{6} - 146 q^{7} - 39 q^{8} + 162 q^{9} - 212 q^{10} + 333 q^{12} + 130 q^{13} - 74 q^{14} + 522 q^{15} + 241 q^{16} + 728 q^{17} - 1053 q^{18} + 828 q^{19} - 1072 q^{20} - 1314 q^{21} - 238 q^{23} - 351 q^{24} - 2918 q^{25} + 376 q^{26} + 1458 q^{27} + 10598 q^{28} - 696 q^{29} - 1908 q^{30} - 10480 q^{31} - 1391 q^{32} - 10870 q^{34} - 14464 q^{35} + 2997 q^{36} - 1908 q^{37} + 8676 q^{38} + 1170 q^{39} + 10584 q^{40} - 36484 q^{41} - 666 q^{42} - 9768 q^{43} + 4698 q^{45} - 3898 q^{46} + 43742 q^{47} + 2169 q^{48} + 40470 q^{49} + 28537 q^{50} + 6552 q^{51} - 13468 q^{52} - 12174 q^{53} - 9477 q^{54} - 69786 q^{56} + 7452 q^{57} + 29142 q^{58} - 2788 q^{59} - 9648 q^{60} + 25302 q^{61} + 94520 q^{62} - 11826 q^{63} - 27199 q^{64} + 15980 q^{65} - 40520 q^{67} + 93262 q^{68} - 2142 q^{69} + 52304 q^{70} + 31386 q^{71} - 3159 q^{72} + 46780 q^{73} - 34062 q^{74} - 26262 q^{75} - 167436 q^{76} + 3384 q^{78} + 16850 q^{79} - 3736 q^{80} + 13122 q^{81} + 237278 q^{82} - 79440 q^{83} + 95382 q^{84} - 40268 q^{85} + 114840 q^{86} - 6264 q^{87} - 54204 q^{89} - 17172 q^{90} - 85192 q^{91} + 66382 q^{92} - 94320 q^{93} - 290758 q^{94} + 164592 q^{95} - 12519 q^{96} - 241568 q^{97} - 113697 q^{98}+O(q^{100}) \) 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
3.37228
−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 \( T_{2}^{2} + 13T_{2} + 34 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(363))\). Copy content Toggle raw display

Hecke characteristic polynomials

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