Properties

Label 207.6.a.g
Level $207$
Weight $6$
Character orbit 207.a
Self dual yes
Analytic conductor $33.199$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 2x^{5} - 149x^{4} + 215x^{3} + 6182x^{2} - 4625x - 79150 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2\cdot 3^{2}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 20) q^{4} + (\beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 - 6) q^{5} + ( - 2 \beta_{5} + \beta_{4} + 3 \beta_{2} + 11 \beta_1 + 48) q^{7} + ( - 3 \beta_{5} - 8 \beta_{2} + 7 \beta_1 - 69) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 20) q^{4} + (\beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 - 6) q^{5} + ( - 2 \beta_{5} + \beta_{4} + 3 \beta_{2} + 11 \beta_1 + 48) q^{7} + ( - 3 \beta_{5} - 8 \beta_{2} + 7 \beta_1 - 69) q^{8} + ( - 10 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 22 \beta_1 - 4) q^{10} + ( - \beta_{4} + 8 \beta_{3} + 11 \beta_{2} - 11 \beta_1 + 20) q^{11} + (14 \beta_{5} - 4 \beta_{4} + 5 \beta_{3} + 8 \beta_{2} - 22 \beta_1 + 138) q^{13} + (12 \beta_{4} + 14 \beta_{3} - 6 \beta_{2} + 66 \beta_1 + 474) q^{14} + (6 \beta_{5} + \beta_{4} + 20 \beta_{3} + 45 \beta_{2} - 84 \beta_1 - 271) q^{16} + (18 \beta_{5} + 13 \beta_{4} - 26 \beta_{3} + 23 \beta_{2} - 53 \beta_1 + 70) q^{17} + (30 \beta_{5} + 2 \beta_{4} + 24 \beta_{3} + 40 \beta_{2} + 6 \beta_1 + 460) q^{19} + ( - 2 \beta_{4} + 50 \beta_{3} + 40 \beta_{2} + 24 \beta_1 + 1210) q^{20} + ( - 20 \beta_{5} - 2 \beta_{3} - 86 \beta_{2} + 192 \beta_1 - 412) q^{22} - 529 q^{23} + ( - 90 \beta_{4} - 50 \beta_{2} + 250 \beta_1 + 1591) q^{25} + ( - 27 \beta_{5} - 51 \beta_{4} - 111 \beta_{3} - 116 \beta_{2} + \cdots - 885) q^{26}+ \cdots + ( - 1764 \beta_{5} + 852 \beta_{4} + 2200 \beta_{3} + \cdots + 55455) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{2} + 112 q^{4} - 42 q^{5} + 300 q^{7} - 393 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{2} + 112 q^{4} - 42 q^{5} + 300 q^{7} - 393 q^{8} - 10 q^{10} + 58 q^{11} + 792 q^{13} + 2984 q^{14} - 1904 q^{16} + 400 q^{17} + 2738 q^{19} + 7124 q^{20} - 1972 q^{22} - 3174 q^{23} + 9966 q^{25} - 4511 q^{26} + 9570 q^{28} - 11244 q^{29} + 13748 q^{31} - 6600 q^{32} - 16226 q^{34} + 4296 q^{35} + 25426 q^{37} + 8028 q^{38} + 10230 q^{40} + 14268 q^{41} - 18082 q^{43} + 51146 q^{44} + 2116 q^{46} + 23084 q^{47} + 37422 q^{49} + 67436 q^{50} + 36807 q^{52} - 17522 q^{53} + 47576 q^{55} - 44946 q^{56} + 141001 q^{58} + 36392 q^{59} + 27062 q^{61} - 48971 q^{62} + 89451 q^{64} - 7108 q^{65} + 37138 q^{67} - 17260 q^{68} + 248380 q^{70} + 158556 q^{71} + 112228 q^{73} + 66878 q^{74} + 157816 q^{76} + 89760 q^{77} + 36844 q^{79} + 158530 q^{80} + 150039 q^{82} + 76350 q^{83} - 102132 q^{85} + 100578 q^{86} - 219028 q^{88} - 16100 q^{89} - 250592 q^{91} - 59248 q^{92} + 12887 q^{94} + 190096 q^{95} + 259432 q^{97} + 325816 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 149x^{4} + 215x^{3} + 6182x^{2} - 4625x - 79150 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{5} + 45\nu^{4} + 274\nu^{3} - 5031\nu^{2} - 2171\nu + 101838 ) / 412 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{5} - 45\nu^{4} - 274\nu^{3} + 5443\nu^{2} + 2171\nu - 122850 ) / 412 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 27\nu^{5} + 7\nu^{4} - 3290\nu^{3} - 2513\nu^{2} + 75571\nu + 106866 ) / 412 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} - 30\nu^{4} - 217\nu^{3} + 3457\nu^{2} + 3782\nu - 72424 ) / 103 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 51 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{5} + 3\beta_{3} - 5\beta_{2} + 68\beta _1 + 21 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{5} + \beta_{4} + 122\beta_{3} + 115\beta_{2} + 3474 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -364\beta_{5} + 15\beta_{4} + 427\beta_{3} - 546\beta_{2} + 5487\beta _1 + 2447 \) 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
−9.33646
−5.29078
−4.81709
5.64307
6.03655
9.76471
−10.3365 0 74.8423 32.5264 0 43.6495 −442.838 0 −336.208
1.2 −6.29078 0 7.57397 45.2266 0 −206.967 153.659 0 −284.511
1.3 −5.81709 0 1.83858 −83.3054 0 84.5782 175.452 0 484.596
1.4 4.64307 0 −10.4419 −9.65155 0 199.091 −197.061 0 −44.8128
1.5 5.03655 0 −6.63314 −108.846 0 −33.8609 −194.578 0 −548.208
1.6 8.76471 0 44.8202 82.0499 0 213.509 112.365 0 719.144
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.g 6
3.b odd 2 1 23.6.a.b 6
12.b even 2 1 368.6.a.h 6
15.d odd 2 1 575.6.a.c 6
69.c even 2 1 529.6.a.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.6.a.b 6 3.b odd 2 1
207.6.a.g 6 1.a even 1 1 trivial
368.6.a.h 6 12.b even 2 1
529.6.a.c 6 69.c even 2 1
575.6.a.c 6 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 4T_{2}^{5} - 144T_{2}^{4} - 381T_{2}^{3} + 5928T_{2}^{2} + 7784T_{2} - 77528 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(207))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 4 T^{5} - 144 T^{4} + \cdots - 77528 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 42 T^{5} + \cdots - 10563094144 \) Copy content Toggle raw display
$7$ \( T^{6} - 300 T^{5} + \cdots + 1099779388928 \) Copy content Toggle raw display
$11$ \( T^{6} - 58 T^{5} + \cdots + 97362234604672 \) Copy content Toggle raw display
$13$ \( T^{6} - 792 T^{5} + \cdots - 83\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{6} - 400 T^{5} + \cdots - 22\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{6} - 2738 T^{5} + \cdots - 18\!\cdots\!48 \) Copy content Toggle raw display
$23$ \( (T + 529)^{6} \) Copy content Toggle raw display
$29$ \( T^{6} + 11244 T^{5} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{6} - 13748 T^{5} + \cdots + 76\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{6} - 25426 T^{5} + \cdots - 24\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{6} - 14268 T^{5} + \cdots - 63\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{6} + 18082 T^{5} + \cdots - 11\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{6} - 23084 T^{5} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{6} + 17522 T^{5} + \cdots - 60\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{6} - 36392 T^{5} + \cdots - 35\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} - 27062 T^{5} + \cdots - 14\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{6} - 37138 T^{5} + \cdots - 47\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{6} - 158556 T^{5} + \cdots + 63\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{6} - 112228 T^{5} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{6} - 36844 T^{5} + \cdots + 19\!\cdots\!92 \) Copy content Toggle raw display
$83$ \( T^{6} - 76350 T^{5} + \cdots + 75\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{6} + 16100 T^{5} + \cdots - 71\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} - 259432 T^{5} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
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