Properties

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

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.7925.1
Defining polynomial: \( x^{3} - x^{2} - 13x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 23)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (4 \beta_{2} - 6 \beta_1 + 2) q^{4} + ( - 2 \beta_{2} + 3 \beta_1 + 21) q^{5} + ( - 4 \beta_{2} + 17 \beta_1 - 87) q^{7} + (16 \beta_{2} - 8 \beta_1 + 112) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (4 \beta_{2} - 6 \beta_1 + 2) q^{4} + ( - 2 \beta_{2} + 3 \beta_1 + 21) q^{5} + ( - 4 \beta_{2} + 17 \beta_1 - 87) q^{7} + (16 \beta_{2} - 8 \beta_1 + 112) q^{8} + ( - 8 \beta_{2} - 2 \beta_1 - 50) q^{10} + ( - 28 \beta_{2} - 3 \beta_1 - 37) q^{11} + ( - 99 \beta_{2} + 45 \beta_1 - 324) q^{13} + ( - 60 \beta_{2} + 180 \beta_1 - 592) q^{14} + ( - 128 \beta_{2} + 8 \beta_1 + 88) q^{16} + (146 \beta_{2} + 57 \beta_1 + 269) q^{17} + (200 \beta_{2} + 40 \beta_1 + 498) q^{19} + (88 \beta_{2} - 40 \beta_1 - 544) q^{20} + (68 \beta_{2} + 78 \beta_1 + 454) q^{22} + 529 q^{23} + ( - 88 \beta_{2} + 86 \beta_1 - 2391) q^{25} + (18 \beta_{2} + 747 \beta_1 - 423) q^{26} + ( - 472 \beta_{2} + 1068 \beta_1 - 2908) q^{28} + ( - 523 \beta_{2} + 745 \beta_1 + 704) q^{29} + (479 \beta_{2} - 914 \beta_1 - 1471) q^{31} + ( - 288 \beta_{2} + 464 \beta_1 - 1968) q^{32} + ( - 520 \beta_{2} - 276 \beta_1 - 3656) q^{34} + (148 \beta_{2} - 160 \beta_1 - 460) q^{35} + (1662 \beta_{2} - 1117 \beta_1 + 2053) q^{37} + ( - 560 \beta_{2} - 698 \beta_1 - 3622) q^{38} + (240 \beta_{2} + 232 \beta_1 + 1144) q^{40} + (1847 \beta_{2} - 1171 \beta_1 + 3258) q^{41} + ( - 1150 \beta_{2} + 2050 \beta_1 - 4510) q^{43} + (448 \beta_{2} - 104 \beta_1 - 1888) q^{44} + ( - 529 \beta_1 + 529) q^{46} + (1021 \beta_{2} + 682 \beta_1 - 7613) q^{47} + (1428 \beta_{2} - 4306 \beta_1 - 949) q^{49} + ( - 168 \beta_{2} + 2997 \beta_1 - 3997) q^{50} + (144 \beta_{2} + 2682 \beta_1 - 14958) q^{52} + ( - 534 \beta_{2} - 4444 \beta_1 - 7006) q^{53} + ( - 840 \beta_{2} - 14 \beta_1 + 130) q^{55} + ( - 1408 \beta_{2} + 3432 \beta_1 - 12600) q^{56} + ( - 1934 \beta_{2} + 4067 \beta_1 - 16559) q^{58} + ( - 1980 \beta_{2} - 934 \beta_1 - 17710) q^{59} + ( - 42 \beta_{2} + 1322 \beta_1 - 23320) q^{61} + (2698 \beta_{2} - 4057 \beta_1 + 21985) q^{62} + (2816 \beta_{2} + 4608 \beta_1 - 16064) q^{64} + ( - 2250 \beta_{2} - 351 \beta_1 + 351) q^{65} + ( - 1436 \beta_{2} - 4323 \beta_1 - 21949) q^{67} + ( - 2528 \beta_{2} + 1492 \beta_1 + 4124) q^{68} + (344 \beta_{2} - 636 \beta_1 + 2748) q^{70} + ( - 2227 \beta_{2} + 1628 \beta_1 - 31515) q^{71} + ( - 21 \beta_{2} + 3241 \beta_1 - 26170) q^{73} + (1144 \beta_{2} - 10962 \beta_1 + 15646) q^{74} + ( - 2488 \beta_{2} - 28 \beta_1 + 11316) q^{76} + (876 \beta_{2} - 532 \beta_1 - 368) q^{77} + ( - 6662 \beta_{2} - 4778 \beta_1 + 20036) q^{79} + ( - 4224 \beta_{2} + 816 \beta_1 + 7536) q^{80} + (990 \beta_{2} - 12807 \beta_1 + 16043) q^{82} + (13758 \beta_{2} - 5867 \beta_1 - 9839) q^{83} + (4476 \beta_{2} + 508 \beta_1 + 3856) q^{85} + ( - 5900 \beta_{2} + 17060 \beta_1 - 56060) q^{86} + ( - 2656 \beta_{2} - 2024 \beta_1 - 19256) q^{88} + ( - 8666 \beta_{2} + 5964 \beta_1 - 4304) q^{89} + (6984 \beta_{2} - 14229 \beta_1 + 43587) q^{91} + (2116 \beta_{2} - 3174 \beta_1 + 1058) q^{92} + ( - 4770 \beta_{2} + 8981 \beta_1 - 44413) q^{94} + (5644 \beta_{2} + 894 \beta_1 + 5298) q^{95} + (618 \beta_{2} + 3351 \beta_1 - 90313) q^{97} + (14368 \beta_{2} - 23437 \beta_1 + 121157) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{2} + 16 q^{4} + 58 q^{5} - 282 q^{7} + 360 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 4 q^{2} + 16 q^{4} + 58 q^{5} - 282 q^{7} + 360 q^{8} - 156 q^{10} - 136 q^{11} - 1116 q^{13} - 2016 q^{14} + 128 q^{16} + 896 q^{17} + 1654 q^{19} - 1504 q^{20} + 1352 q^{22} + 1587 q^{23} - 7347 q^{25} - 1998 q^{26} - 10264 q^{28} + 844 q^{29} - 3020 q^{31} - 6656 q^{32} - 11212 q^{34} - 1072 q^{35} + 8938 q^{37} - 10728 q^{38} + 3440 q^{40} + 12792 q^{41} - 16730 q^{43} - 5112 q^{44} + 2116 q^{46} - 22500 q^{47} + 2887 q^{49} - 15156 q^{50} - 47412 q^{52} - 17108 q^{53} - 436 q^{55} - 42640 q^{56} - 55678 q^{58} - 54176 q^{59} - 71324 q^{61} + 72710 q^{62} - 49984 q^{64} - 846 q^{65} - 62960 q^{67} + 8352 q^{68} + 9224 q^{70} - 98400 q^{71} - 81772 q^{73} + 59044 q^{74} + 31488 q^{76} + 304 q^{77} + 58224 q^{79} + 17568 q^{80} + 61926 q^{82} - 9892 q^{83} + 15536 q^{85} - 191140 q^{86} - 58400 q^{88} - 27542 q^{89} + 151974 q^{91} + 8464 q^{92} - 146990 q^{94} + 20644 q^{95} - 273672 q^{97} + 401276 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 13x + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} - \beta _1 + 17 ) / 2 \) 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
3.65736
0.917748
−3.57511
−5.31473 0 −3.75366 23.8768 0 −11.7843 190.021 0 −126.899
1.2 0.164504 0 −31.9729 37.9865 0 −43.8366 −10.5238 0 6.24894
1.3 9.15022 0 51.7266 −3.86330 0 −226.379 180.503 0 −35.3501
\(n\): 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.b 3
3.b odd 2 1 23.6.a.a 3
12.b even 2 1 368.6.a.e 3
15.d odd 2 1 575.6.a.b 3
69.c even 2 1 529.6.a.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.6.a.a 3 3.b odd 2 1
207.6.a.b 3 1.a even 1 1 trivial
368.6.a.e 3 12.b even 2 1
529.6.a.a 3 69.c even 2 1
575.6.a.b 3 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 4T_{2}^{2} - 48T_{2} + 8 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(207))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 4 T^{2} - 48 T + 8 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 58 T^{2} + 668 T + 3504 \) Copy content Toggle raw display
$7$ \( T^{3} + 282 T^{2} + 13108 T + 116944 \) Copy content Toggle raw display
$11$ \( T^{3} + 136 T^{2} - 43688 T - 840152 \) Copy content Toggle raw display
$13$ \( T^{3} + 1116 T^{2} + \cdots - 255615102 \) Copy content Toggle raw display
$17$ \( T^{3} - 896 T^{2} + \cdots - 220718408 \) Copy content Toggle raw display
$19$ \( T^{3} - 1654 T^{2} + \cdots + 460771768 \) Copy content Toggle raw display
$23$ \( (T - 529)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 844 T^{2} + \cdots + 33789223458 \) Copy content Toggle raw display
$31$ \( T^{3} + 3020 T^{2} + \cdots - 117638912880 \) Copy content Toggle raw display
$37$ \( T^{3} - 8938 T^{2} + \cdots + 1048082031344 \) Copy content Toggle raw display
$41$ \( T^{3} - 12792 T^{2} + \cdots + 1564944049486 \) Copy content Toggle raw display
$43$ \( T^{3} + 16730 T^{2} + \cdots - 95315904000 \) Copy content Toggle raw display
$47$ \( T^{3} + 22500 T^{2} + \cdots - 916008439440 \) Copy content Toggle raw display
$53$ \( T^{3} + 17108 T^{2} + \cdots - 7849670295504 \) Copy content Toggle raw display
$59$ \( T^{3} + 54176 T^{2} + \cdots + 1725012447168 \) Copy content Toggle raw display
$61$ \( T^{3} + 71324 T^{2} + \cdots + 11439907465152 \) Copy content Toggle raw display
$67$ \( T^{3} + 62960 T^{2} + \cdots - 11971711378840 \) Copy content Toggle raw display
$71$ \( T^{3} + 98400 T^{2} + \cdots + 24837760695040 \) Copy content Toggle raw display
$73$ \( T^{3} + 81772 T^{2} + \cdots + 7199078503954 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 235690469012368 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 297029282761704 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 125799322340896 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 693755159518744 \) Copy content Toggle raw display
show more
show less