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: \( x^{3} - x^{2} - 11x + 8 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 3) q^{2} + ( - 9 \beta_{2} - 4 \beta_1 + 9) q^{4} + ( - 6 \beta_{2} - 11 \beta_1 + 17) q^{5} + ( - 8 \beta_{2} - 26 \beta_1 - 44) q^{7} + ( - 35 \beta_{2} - 32 \beta_1 + 171) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 3) q^{2} + ( - 9 \beta_{2} - 4 \beta_1 + 9) q^{4} + ( - 6 \beta_{2} - 11 \beta_1 + 17) q^{5} + ( - 8 \beta_{2} - 26 \beta_1 - 44) q^{7} + ( - 35 \beta_{2} - 32 \beta_1 + 171) q^{8} + ( - 64 \beta_{2} - 13 \beta_1 + 111) q^{10} + (42 \beta_{2} - 49 \beta_1 + 95) q^{11} + ( - 132 \beta_{2} - 66 \beta_1 - 264) q^{13} + ( - 30 \beta_{2} - 6 \beta_1 - 188) q^{14} + ( - 125 \beta_{2} + 20 \beta_1 + 961) q^{16} + ( - 72 \beta_{2} + 68 \beta_1 + 896) q^{17} + ( - 10 \beta_{2} - 143 \beta_1 - 993) q^{19} + ( - 316 \beta_{2} + 109 \beta_1 + 1681) q^{20} + (108 \beta_{2} + 217 \beta_1 - 1647) q^{22} - 529 q^{23} + (40 \beta_{2} + 10 \beta_1 + 241) q^{25} + ( - 594 \beta_{2} - 462 \beta_1 + 2640) q^{26} + (258 \beta_{2} + 718 \beta_1 + 1732) q^{28} + ( - 200 \beta_{2} + 254 \beta_1 + 5608) q^{29} + (560 \beta_{2} - 158 \beta_1 - 5158) q^{31} + ( - 571 \beta_{2} + 504 \beta_1 + 1651) q^{32} + ( - 1260 \beta_{2} - 356 \beta_1 + 5808) q^{34} + (884 \beta_{2} + 826 \beta_1 + 6154) q^{35} + (370 \beta_{2} - 105 \beta_1 + 5133) q^{37} + (790 \beta_{2} + 103 \beta_1 - 4375) q^{38} + ( - 1420 \beta_{2} - 957 \beta_1 + 12911) q^{40} + ( - 1204 \beta_{2} - 1048 \beta_1 - 4150) q^{41} + (1538 \beta_{2} + 163 \beta_1 + 593) q^{43} + (1168 \beta_{2} + 1783 \beta_1 - 8833) q^{44} + (529 \beta_{2} - 1587) q^{46} + ( - 284 \beta_{2} - 2000 \beta_1 - 10548) q^{47} + (2800 \beta_{2} + 3696 \beta_1 + 1789) q^{49} + (9 \beta_{2} + 150 \beta_1 - 437) q^{50} + ( - 2442 \beta_{2} + 198 \beta_1 + 29832) q^{52} + ( - 3166 \beta_{2} - 5619 \beta_1 + 13877) q^{53} + (3224 \beta_{2} - 1542 \beta_1 + 11198) q^{55} + (1494 \beta_{2} + 506 \beta_1 + 11572) q^{56} + ( - 6554 \beta_{2} - 1054 \beta_1 + 26272) q^{58} + (752 \beta_{2} + 4164 \beta_1 + 10904) q^{59} + ( - 986 \beta_{2} + 1545 \beta_1 + 18847) q^{61} + (8360 \beta_{2} + 2398 \beta_1 - 35290) q^{62} + ( - 573 \beta_{2} - 3428 \beta_1 - 1479) q^{64} + ( - 1980 \beta_{2} + 6006 \beta_1 + 19734) q^{65} + ( - 6138 \beta_{2} - 9559 \beta_1 + 13323) q^{67} + ( - 11420 \beta_{2} - 6860 \beta_1 + 24800) q^{68} + ( - 24 \beta_{2} + 2710 \beta_1 + 86) q^{70} + (1364 \beta_{2} - 4708 \beta_1 - 9832) q^{71} + (7808 \beta_{2} + 8490 \beta_1 - 30240) q^{73} + ( - 3018 \beta_{2} + 1585 \beta_1 + 2299) q^{74} + (9538 \beta_{2} + 7633 \beta_1 - 5393) q^{76} + (4196 \beta_{2} - 1770 \beta_1 + 30414) q^{77} + (32 \beta_{2} - 9526 \beta_1 - 19652) q^{79} + ( - 12276 \beta_{2} - 8211 \beta_1 + 18897) q^{80} + ( - 4122 \beta_{2} - 3768 \beta_1 + 13502) q^{82} + (16006 \beta_{2} + 6321 \beta_1 + 31417) q^{83} + ( - 11272 \beta_{2} - 8892 \beta_1 + 2756) q^{85} + (8798 \beta_{2} + 5989 \beta_1 - 45481) q^{86} + (14168 \beta_{2} - 4055 \beta_1 + 10225) q^{88} + (8868 \beta_{2} - 1622 \beta_1 - 35534) q^{89} + (7656 \beta_{2} + 21384 \beta_1 + 47652) q^{91} + (4761 \beta_{2} + 2116 \beta_1 - 4761) q^{92} + (6844 \beta_{2} + 864 \beta_1 - 46556) q^{94} + (10932 \beta_{2} + 12124 \beta_1 + 19040) q^{95} + ( - 10068 \beta_{2} + 7014 \beta_1 - 39412) q^{97} + (18707 \beta_{2} + 7504 \beta_1 - 39881) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 8 q^{2} + 22 q^{4} + 56 q^{5} - 114 q^{7} + 510 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 8 q^{2} + 22 q^{4} + 56 q^{5} - 114 q^{7} + 510 q^{8} + 282 q^{10} + 376 q^{11} - 858 q^{13} - 588 q^{14} + 2738 q^{16} + 2548 q^{17} - 2846 q^{19} + 4618 q^{20} - 5050 q^{22} - 1587 q^{23} + 753 q^{25} + 7788 q^{26} + 4736 q^{28} + 16370 q^{29} - 14756 q^{31} + 3878 q^{32} + 16520 q^{34} + 18520 q^{35} + 15874 q^{37} - 12438 q^{38} + 38270 q^{40} - 12606 q^{41} + 3154 q^{43} - 27114 q^{44} - 4232 q^{46} - 29928 q^{47} + 4471 q^{49} - 1452 q^{50} + 86856 q^{52} + 44084 q^{53} + 38360 q^{55} + 35704 q^{56} + 73316 q^{58} + 29300 q^{59} + 54010 q^{61} - 99908 q^{62} - 1582 q^{64} + 51216 q^{65} + 43390 q^{67} + 69840 q^{68} - 2476 q^{70} - 23424 q^{71} - 91402 q^{73} + 2294 q^{74} - 14274 q^{76} + 97208 q^{77} - 49398 q^{79} + 52626 q^{80} + 40152 q^{82} + 103936 q^{83} + 5888 q^{85} - 133634 q^{86} + 48898 q^{88} - 96112 q^{89} + 129228 q^{91} - 11638 q^{92} - 133688 q^{94} + 55928 q^{95} - 135318 q^{97} - 108440 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} + \beta _1 + 15 ) / 2 \) 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
−3.20733
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 \( T_{2}^{3} - 8T_{2}^{2} - 27T_{2} + 46 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(207))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 8 T^{2} - 27 T + 46 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 56 T^{2} - 3496 T + 195576 \) Copy content Toggle raw display
$7$ \( T^{3} + 114 T^{2} - 20948 T + 51736 \) Copy content Toggle raw display
$11$ \( T^{3} - 376 T^{2} + \cdots - 21141352 \) Copy content Toggle raw display
$13$ \( T^{3} + 858 T^{2} + \cdots - 368282376 \) Copy content Toggle raw display
$17$ \( T^{3} - 2548 T^{2} + \cdots + 129112640 \) Copy content Toggle raw display
$19$ \( T^{3} + 2846 T^{2} + \cdots - 4313168 \) Copy content Toggle raw display
$23$ \( (T + 529)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 16370 T^{2} + \cdots - 117835741080 \) Copy content Toggle raw display
$31$ \( T^{3} + 14756 T^{2} + \cdots + 16664141952 \) Copy content Toggle raw display
$37$ \( T^{3} - 15874 T^{2} + \cdots - 103469473312 \) Copy content Toggle raw display
$41$ \( T^{3} + 12606 T^{2} + \cdots - 213582513784 \) Copy content Toggle raw display
$43$ \( T^{3} - 3154 T^{2} + \cdots + 409945701888 \) Copy content Toggle raw display
$47$ \( T^{3} + 29928 T^{2} + \cdots - 524672802816 \) Copy content Toggle raw display
$53$ \( T^{3} - 44084 T^{2} + \cdots + 30187666172280 \) Copy content Toggle raw display
$59$ \( T^{3} - 29300 T^{2} + \cdots + 4070512924224 \) Copy content Toggle raw display
$61$ \( T^{3} - 54010 T^{2} + \cdots - 694379910768 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 128918418373088 \) Copy content Toggle raw display
$71$ \( T^{3} + 23424 T^{2} + \cdots - 26245560332032 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 119459239092680 \) Copy content Toggle raw display
$79$ \( T^{3} + 49398 T^{2} + \cdots - 93978622829240 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 694250483317800 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 102645237296960 \) Copy content Toggle raw display
$97$ \( T^{3} + 135318 T^{2} + \cdots - 82905816948920 \) Copy content Toggle raw display
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