Properties

Label 207.8.a.a
Level $207$
Weight $8$
Character orbit 207.a
Self dual yes
Analytic conductor $64.664$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 207.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.6637002752\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - 455x^{3} - 474x^{2} + 42284x + 127016 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{4} + \beta_{3} + 2 \beta_1 + 54) q^{4} + (\beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 54) q^{5} + ( - 4 \beta_{4} - 12 \beta_{3} - \beta_{2} + 3 \beta_1 - 96) q^{7} + (\beta_{4} - 11 \beta_{2} - 9 \beta_1 - 284) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{4} + \beta_{3} + 2 \beta_1 + 54) q^{4} + (\beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 54) q^{5} + ( - 4 \beta_{4} - 12 \beta_{3} - \beta_{2} + 3 \beta_1 - 96) q^{7} + (\beta_{4} - 11 \beta_{2} - 9 \beta_1 - 284) q^{8} + (4 \beta_{4} - 10 \beta_{3} + 8 \beta_{2} - 72 \beta_1 + 296) q^{10} + (5 \beta_{4} + 29 \beta_{3} - 4 \beta_{2} + 117 \beta_1 + 220) q^{11} + ( - 27 \beta_{4} + 82 \beta_{3} - 13 \beta_{2} - 372 \beta_1 - 172) q^{13} + (6 \beta_{4} + 23 \beta_{3} + 125 \beta_{2} + 589 \beta_1 - 1158) q^{14} + ( - 59 \beta_{4} + 83 \beta_{3} + 10 \beta_{2} - 108 \beta_1 - 4578) q^{16} + ( - 160 \beta_{4} - 76 \beta_{3} - 65 \beta_{2} + 1103 \beta_1 + 1126) q^{17} + (125 \beta_{4} + 127 \beta_{3} + 121 \beta_{2} - 598 \beta_1 - 1208) q^{19} + ( - 56 \beta_{4} + 92 \beta_{3} - 40 \beta_{2} + 196 \beta_1 + 5416) q^{20} + ( - 130 \beta_{4} - 83 \beta_{3} - 291 \beta_{2} - 1461 \beta_1 - 19598) q^{22} - 12167 q^{23} + ( - 19 \beta_{4} - 280 \beta_{3} + 257 \beta_{2} - 1198 \beta_1 - 52391) q^{25} + (138 \beta_{4} + 334 \beta_{3} - 780 \beta_{2} + 316 \beta_1 + 71120) q^{26} + ( - 718 \beta_{4} - 1325 \beta_{3} - 233 \beta_{2} + \cdots - 101182) q^{28}+ \cdots + (8554 \beta_{4} + 17220 \beta_{3} - 34650 \beta_{2} + \cdots - 2349564) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 270 q^{4} + 266 q^{5} - 496 q^{7} - 1422 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 270 q^{4} + 266 q^{5} - 496 q^{7} - 1422 q^{8} + 1452 q^{10} + 1148 q^{11} - 642 q^{13} - 5756 q^{14} - 22606 q^{16} + 5798 q^{17} - 6036 q^{19} + 27376 q^{20} - 97896 q^{22} - 60835 q^{23} - 262477 q^{25} + 355992 q^{26} - 507124 q^{28} + 169162 q^{29} - 199640 q^{31} + 284794 q^{32} - 1027740 q^{34} + 137680 q^{35} - 202002 q^{37} + 554924 q^{38} - 340904 q^{40} - 541282 q^{41} - 909596 q^{43} + 1236032 q^{44} - 80208 q^{47} + 850589 q^{49} + 941416 q^{50} + 146940 q^{52} + 278138 q^{53} - 933560 q^{55} + 539932 q^{56} - 3522712 q^{58} + 3177380 q^{59} + 147782 q^{61} - 4606456 q^{62} - 4142622 q^{64} - 3877332 q^{65} - 464916 q^{67} - 7513072 q^{68} + 2093200 q^{70} - 1576792 q^{71} - 38190 q^{73} - 12164864 q^{74} + 6889436 q^{76} - 10332384 q^{77} - 3913336 q^{79} - 6334776 q^{80} + 6799360 q^{82} - 15774716 q^{83} - 8520740 q^{85} - 24874084 q^{86} + 53216 q^{88} - 1116482 q^{89} - 27369552 q^{91} - 3285090 q^{92} - 7153744 q^{94} + 6067832 q^{95} - 15738566 q^{97} - 11730488 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 455x^{3} - 474x^{2} + 42284x + 127016 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{4} + 142\nu^{3} + 467\nu^{2} - 37904\nu - 76396 ) / 1552 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\nu^{4} - 10\nu^{3} - 3585\nu^{2} + 2560\nu + 117124 ) / 1552 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{4} + 10\nu^{3} + 5137\nu^{2} - 5664\nu - 399588 ) / 1552 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 2\beta _1 + 182 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + 11\beta_{2} + 265\beta _1 + 284 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 325\beta_{4} + 467\beta_{3} + 10\beta_{2} + 660\beta _1 + 48926 \) 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
18.2528
12.4672
−3.24502
−9.64907
−17.8260
−18.2528 0 205.166 55.2224 0 −1258.24 −1408.51 0 −1007.97
1.2 −12.4672 0 27.4323 40.8788 0 1126.70 1253.80 0 −509.647
1.3 3.24502 0 −117.470 −168.083 0 149.817 −796.555 0 −545.434
1.4 9.64907 0 −34.8955 306.939 0 733.069 −1571.79 0 2961.68
1.5 17.8260 0 189.767 31.0428 0 −1247.35 1101.05 0 553.369
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 207.8.a.a 5
3.b odd 2 1 69.8.a.a 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.8.a.a 5 3.b odd 2 1
207.8.a.a 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} - 455T_{2}^{3} + 474T_{2}^{2} + 42284T_{2} - 127016 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(207))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 455 T^{3} + 474 T^{2} + \cdots - 127016 \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 266 T^{4} + \cdots + 3615360000 \) Copy content Toggle raw display
$7$ \( T^{5} + \cdots - 194204974150720 \) Copy content Toggle raw display
$11$ \( T^{5} + \cdots - 154691619430400 \) Copy content Toggle raw display
$13$ \( T^{5} + 642 T^{4} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{5} - 5798 T^{4} + \cdots - 11\!\cdots\!80 \) Copy content Toggle raw display
$19$ \( T^{5} + 6036 T^{4} + \cdots - 14\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T + 12167)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} - 169162 T^{4} + \cdots - 24\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( T^{5} + 199640 T^{4} + \cdots - 23\!\cdots\!28 \) Copy content Toggle raw display
$37$ \( T^{5} + 202002 T^{4} + \cdots + 82\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{5} + 541282 T^{4} + \cdots - 16\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{5} + 909596 T^{4} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{5} + 80208 T^{4} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{5} - 278138 T^{4} + \cdots - 10\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{5} - 3177380 T^{4} + \cdots - 75\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{5} - 147782 T^{4} + \cdots - 14\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{5} + 464916 T^{4} + \cdots - 79\!\cdots\!40 \) Copy content Toggle raw display
$71$ \( T^{5} + 1576792 T^{4} + \cdots - 49\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( T^{5} + 38190 T^{4} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{5} + 3913336 T^{4} + \cdots - 36\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{5} + 15774716 T^{4} + \cdots - 18\!\cdots\!20 \) Copy content Toggle raw display
$89$ \( T^{5} + 1116482 T^{4} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{5} + 15738566 T^{4} + \cdots + 11\!\cdots\!20 \) Copy content Toggle raw display
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