Properties

Label 207.6.a.c
Level $207$
Weight $6$
Character orbit 207.a
Self dual yes
Analytic conductor $33.199$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 207.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.1994507013\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.5333.1
Defining polynomial: \(x^{3} - x^{2} - 11 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 - \beta_{2} ) q^{2} + ( 9 - 4 \beta_{1} - 9 \beta_{2} ) q^{4} + ( 17 - 11 \beta_{1} - 6 \beta_{2} ) q^{5} + ( -44 - 26 \beta_{1} - 8 \beta_{2} ) q^{7} + ( 171 - 32 \beta_{1} - 35 \beta_{2} ) q^{8} +O(q^{10})\) \( q + ( 3 - \beta_{2} ) q^{2} + ( 9 - 4 \beta_{1} - 9 \beta_{2} ) q^{4} + ( 17 - 11 \beta_{1} - 6 \beta_{2} ) q^{5} + ( -44 - 26 \beta_{1} - 8 \beta_{2} ) q^{7} + ( 171 - 32 \beta_{1} - 35 \beta_{2} ) q^{8} + ( 111 - 13 \beta_{1} - 64 \beta_{2} ) q^{10} + ( 95 - 49 \beta_{1} + 42 \beta_{2} ) q^{11} + ( -264 - 66 \beta_{1} - 132 \beta_{2} ) q^{13} + ( -188 - 6 \beta_{1} - 30 \beta_{2} ) q^{14} + ( 961 + 20 \beta_{1} - 125 \beta_{2} ) q^{16} + ( 896 + 68 \beta_{1} - 72 \beta_{2} ) q^{17} + ( -993 - 143 \beta_{1} - 10 \beta_{2} ) q^{19} + ( 1681 + 109 \beta_{1} - 316 \beta_{2} ) q^{20} + ( -1647 + 217 \beta_{1} + 108 \beta_{2} ) q^{22} -529 q^{23} + ( 241 + 10 \beta_{1} + 40 \beta_{2} ) q^{25} + ( 2640 - 462 \beta_{1} - 594 \beta_{2} ) q^{26} + ( 1732 + 718 \beta_{1} + 258 \beta_{2} ) q^{28} + ( 5608 + 254 \beta_{1} - 200 \beta_{2} ) q^{29} + ( -5158 - 158 \beta_{1} + 560 \beta_{2} ) q^{31} + ( 1651 + 504 \beta_{1} - 571 \beta_{2} ) q^{32} + ( 5808 - 356 \beta_{1} - 1260 \beta_{2} ) q^{34} + ( 6154 + 826 \beta_{1} + 884 \beta_{2} ) q^{35} + ( 5133 - 105 \beta_{1} + 370 \beta_{2} ) q^{37} + ( -4375 + 103 \beta_{1} + 790 \beta_{2} ) q^{38} + ( 12911 - 957 \beta_{1} - 1420 \beta_{2} ) q^{40} + ( -4150 - 1048 \beta_{1} - 1204 \beta_{2} ) q^{41} + ( 593 + 163 \beta_{1} + 1538 \beta_{2} ) q^{43} + ( -8833 + 1783 \beta_{1} + 1168 \beta_{2} ) q^{44} + ( -1587 + 529 \beta_{2} ) q^{46} + ( -10548 - 2000 \beta_{1} - 284 \beta_{2} ) q^{47} + ( 1789 + 3696 \beta_{1} + 2800 \beta_{2} ) q^{49} + ( -437 + 150 \beta_{1} + 9 \beta_{2} ) q^{50} + ( 29832 + 198 \beta_{1} - 2442 \beta_{2} ) q^{52} + ( 13877 - 5619 \beta_{1} - 3166 \beta_{2} ) q^{53} + ( 11198 - 1542 \beta_{1} + 3224 \beta_{2} ) q^{55} + ( 11572 + 506 \beta_{1} + 1494 \beta_{2} ) q^{56} + ( 26272 - 1054 \beta_{1} - 6554 \beta_{2} ) q^{58} + ( 10904 + 4164 \beta_{1} + 752 \beta_{2} ) q^{59} + ( 18847 + 1545 \beta_{1} - 986 \beta_{2} ) q^{61} + ( -35290 + 2398 \beta_{1} + 8360 \beta_{2} ) q^{62} + ( -1479 - 3428 \beta_{1} - 573 \beta_{2} ) q^{64} + ( 19734 + 6006 \beta_{1} - 1980 \beta_{2} ) q^{65} + ( 13323 - 9559 \beta_{1} - 6138 \beta_{2} ) q^{67} + ( 24800 - 6860 \beta_{1} - 11420 \beta_{2} ) q^{68} + ( 86 + 2710 \beta_{1} - 24 \beta_{2} ) q^{70} + ( -9832 - 4708 \beta_{1} + 1364 \beta_{2} ) q^{71} + ( -30240 + 8490 \beta_{1} + 7808 \beta_{2} ) q^{73} + ( 2299 + 1585 \beta_{1} - 3018 \beta_{2} ) q^{74} + ( -5393 + 7633 \beta_{1} + 9538 \beta_{2} ) q^{76} + ( 30414 - 1770 \beta_{1} + 4196 \beta_{2} ) q^{77} + ( -19652 - 9526 \beta_{1} + 32 \beta_{2} ) q^{79} + ( 18897 - 8211 \beta_{1} - 12276 \beta_{2} ) q^{80} + ( 13502 - 3768 \beta_{1} - 4122 \beta_{2} ) q^{82} + ( 31417 + 6321 \beta_{1} + 16006 \beta_{2} ) q^{83} + ( 2756 - 8892 \beta_{1} - 11272 \beta_{2} ) q^{85} + ( -45481 + 5989 \beta_{1} + 8798 \beta_{2} ) q^{86} + ( 10225 - 4055 \beta_{1} + 14168 \beta_{2} ) q^{88} + ( -35534 - 1622 \beta_{1} + 8868 \beta_{2} ) q^{89} + ( 47652 + 21384 \beta_{1} + 7656 \beta_{2} ) q^{91} + ( -4761 + 2116 \beta_{1} + 4761 \beta_{2} ) q^{92} + ( -46556 + 864 \beta_{1} + 6844 \beta_{2} ) q^{94} + ( 19040 + 12124 \beta_{1} + 10932 \beta_{2} ) q^{95} + ( -39412 + 7014 \beta_{1} - 10068 \beta_{2} ) q^{97} + ( -39881 + 7504 \beta_{1} + 18707 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 8q^{2} + 22q^{4} + 56q^{5} - 114q^{7} + 510q^{8} + O(q^{10}) \) \( 3q + 8q^{2} + 22q^{4} + 56q^{5} - 114q^{7} + 510q^{8} + 282q^{10} + 376q^{11} - 858q^{13} - 588q^{14} + 2738q^{16} + 2548q^{17} - 2846q^{19} + 4618q^{20} - 5050q^{22} - 1587q^{23} + 753q^{25} + 7788q^{26} + 4736q^{28} + 16370q^{29} - 14756q^{31} + 3878q^{32} + 16520q^{34} + 18520q^{35} + 15874q^{37} - 12438q^{38} + 38270q^{40} - 12606q^{41} + 3154q^{43} - 27114q^{44} - 4232q^{46} - 29928q^{47} + 4471q^{49} - 1452q^{50} + 86856q^{52} + 44084q^{53} + 38360q^{55} + 35704q^{56} + 73316q^{58} + 29300q^{59} + 54010q^{61} - 99908q^{62} - 1582q^{64} + 51216q^{65} + 43390q^{67} + 69840q^{68} - 2476q^{70} - 23424q^{71} - 91402q^{73} + 2294q^{74} - 14274q^{76} + 97208q^{77} - 49398q^{79} + 52626q^{80} + 40152q^{82} + 103936q^{83} + 5888q^{85} - 133634q^{86} + 48898q^{88} - 96112q^{89} + 129228q^{91} - 11638q^{92} - 133688q^{94} + 55928q^{95} - 135318q^{97} - 108440q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 11 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 7 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{2} + \beta_{1} + 15\)\()/2\)

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
−3.20733
3.49331
0.714018
−3.49429 0 −19.7900 59.5955 0 96.8268 180.969 0 −208.244
1.2 1.29009 0 −30.3357 −59.1123 0 −213.331 −80.4187 0 −76.2602
1.3 10.2042 0 72.1256 55.5168 0 2.50462 409.450 0 566.504
\(n\): 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 207.6.a.c 3
3.b odd 2 1 69.6.a.b 3
12.b even 2 1 1104.6.a.i 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.6.a.b 3 3.b odd 2 1
207.6.a.c 3 1.a even 1 1 trivial
1104.6.a.i 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 8 T_{2}^{2} - 27 T_{2} + 46 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(207))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 46 - 27 T - 8 T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( 195576 - 3496 T - 56 T^{2} + T^{3} \)
$7$ \( 51736 - 20948 T + 114 T^{2} + T^{3} \)
$11$ \( -21141352 - 230672 T - 376 T^{2} + T^{3} \)
$13$ \( -368282376 - 439956 T + 858 T^{2} + T^{3} \)
$17$ \( 129112640 + 1504816 T - 2548 T^{2} + T^{3} \)
$19$ \( -4313168 + 1826204 T + 2846 T^{2} + T^{3} \)
$23$ \( ( 529 + T )^{3} \)
$29$ \( -117835741080 + 82401708 T - 16370 T^{2} + T^{3} \)
$31$ \( 16664141952 + 52692624 T + 14756 T^{2} + T^{3} \)
$37$ \( -103469473312 + 75298092 T - 15874 T^{2} + T^{3} \)
$41$ \( -213582513784 - 15571268 T + 12606 T^{2} + T^{3} \)
$43$ \( 409945701888 - 102023460 T - 3154 T^{2} + T^{3} \)
$47$ \( -524672802816 + 136428688 T + 29928 T^{2} + T^{3} \)
$53$ \( 30187666172280 - 544538824 T - 44084 T^{2} + T^{3} \)
$59$ \( 4070512924224 - 399858640 T - 29300 T^{2} + T^{3} \)
$61$ \( -694379910768 + 755208380 T - 54010 T^{2} + T^{3} \)
$67$ \( 128918418373088 - 2949665124 T - 43390 T^{2} + T^{3} \)
$71$ \( -26245560332032 - 1173004304 T + 23424 T^{2} + T^{3} \)
$73$ \( -119459239092680 - 733701012 T + 91402 T^{2} + T^{3} \)
$79$ \( -93978622829240 - 3312817172 T + 49398 T^{2} + T^{3} \)
$83$ \( 694250483317800 - 6478606576 T - 103936 T^{2} + T^{3} \)
$89$ \( -102645237296960 - 1426041824 T + 96112 T^{2} + T^{3} \)
$97$ \( -82905816948920 - 3897620052 T + 135318 T^{2} + T^{3} \)
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