Properties

Label 207.6.a.f
Level $207$
Weight $6$
Character orbit 207.a
Self dual yes
Analytic conductor $33.199$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(33.1994507013\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - 113x^{3} - 257x^{2} + 1404x + 2197 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - 2) q^{2} + (\beta_{3} - \beta_{2} + 24) q^{4} + ( - \beta_{4} - \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 18) q^{5} + ( - \beta_{4} + 5 \beta_{2} - 5 \beta_1 + 52) q^{7} + (\beta_{4} + 22 \beta_{2} - 10 \beta_1 - 60) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 2) q^{2} + (\beta_{3} - \beta_{2} + 24) q^{4} + ( - \beta_{4} - \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 18) q^{5} + ( - \beta_{4} + 5 \beta_{2} - 5 \beta_1 + 52) q^{7} + (\beta_{4} + 22 \beta_{2} - 10 \beta_1 - 60) q^{8} + (4 \beta_{4} - 6 \beta_{3} - 52 \beta_{2} + 22 \beta_1 - 12) q^{10} + (6 \beta_{4} + 5 \beta_{3} - 14 \beta_{2} - 212) q^{11} + (\beta_{4} - 10 \beta_{3} - 33 \beta_{2} - 22 \beta_1 - 182) q^{13} + (13 \beta_{4} + 11 \beta_{3} + 65 \beta_{2} + 12 \beta_1 + 48) q^{14} + (2 \beta_{4} - \beta_{3} + \beta_{2} - 12 \beta_1 + 244) q^{16} + ( - 5 \beta_{4} - 8 \beta_{3} - 71 \beta_{2} - 13 \beta_1 - 478) q^{17} + (23 \beta_{4} - 9 \beta_{3} + 53 \beta_{2} + 35 \beta_1 + 400) q^{19} + ( - 28 \beta_{4} - 52 \beta_{3} - 188 \beta_{2} - 84 \beta_1 - 1480) q^{20} + ( - 43 \beta_{4} - 15 \beta_{3} - 89 \beta_{2} - 122 \beta_1 - 496) q^{22} + 529 q^{23} + (11 \beta_{4} + 24 \beta_{3} + 341 \beta_{2} + 106 \beta_1 + 2275) q^{25} + (4 \beta_{4} - 22 \beta_{3} - 506 \beta_{2} + 88 \beta_1 - 1652) q^{26} + ( - 73 \beta_{4} + 51 \beta_{3} + 243 \beta_{2} - 106 \beta_1 + 1488) q^{28} + (26 \beta_{4} + 96 \beta_{3} - 242 \beta_{2} - 90 \beta_1 - 198) q^{29} + (20 \beta_{4} - 36 \beta_{3} - 316 \beta_{2} + 50 \beta_1 - 1344) q^{31} + ( - 37 \beta_{4} + 10 \beta_{3} - 484 \beta_{2} + 306 \beta_1 + 1196) q^{32} + (45 \beta_{4} - 61 \beta_{3} - 769 \beta_{2} + 140 \beta_1 - 2796) q^{34} + ( - 21 \beta_{4} - 2 \beta_{3} - 979 \beta_{2} + 220 \beta_1 - 828) q^{35} + ( - 30 \beta_{4} - 9 \beta_{3} - 530 \beta_{2} + 114 \beta_1 - 1442) q^{37} + ( - 228 \beta_{4} - 14 \beta_{3} + 70 \beta_{2} - 186 \beta_1 + 2736) q^{38} + (128 \beta_{4} + 64 \beta_{3} - 1448 \beta_{2} + 152 \beta_1 - 6864) q^{40} + (204 \beta_{4} - 34 \beta_{3} - 1036 \beta_{2} + 132 \beta_1 - 3754) q^{41} + (19 \beta_{4} + 361 \beta_{3} - 295 \beta_{2} + 135 \beta_1 - 784) q^{43} + (259 \beta_{4} - 99 \beta_{3} - 351 \beta_{2} + 666 \beta_1 + 1096) q^{44} + (529 \beta_{2} - 1058) q^{46} + ( - 161 \beta_{4} + 102 \beta_{3} + 921 \beta_{2} - 1116 \beta_1 - 540) q^{47} + ( - 65 \beta_{4} + 116 \beta_{3} - 127 \beta_{2} - 338 \beta_1 - 5571) q^{49} + ( - 170 \beta_{4} + 248 \beta_{3} + 3221 \beta_{2} - 372 \beta_1 + 15018) q^{50} + ( - 174 \beta_{4} - 300 \beta_{3} - 1884 \beta_{2} + 876 \beta_1 - 14592) q^{52} + (335 \beta_{4} + 55 \beta_{3} + 1461 \beta_{2} - 235 \beta_1 + 1846) q^{53} + (138 \beta_{4} + 144 \beta_{3} + 2006 \beta_{2} - 1932 \beta_1 - 16160) q^{55} + (325 \beta_{4} + 121 \beta_{3} + 1557 \beta_{2} - 18 \beta_1 + 5232) q^{56} + ( - 22 \beta_{4} - 82 \beta_{3} + 2548 \beta_{2} - 1272 \beta_1 - 16964) q^{58} + ( - 261 \beta_{4} - 324 \beta_{3} - 419 \beta_{2} - 1416 \beta_1 - 14872) q^{59} + (44 \beta_{4} + 133 \beta_{3} + 1860 \beta_{2} + 242 \beta_1 - 9650) q^{61} + ( - 246 \beta_{4} - 422 \beta_{3} - 2886 \beta_{2} + 120 \beta_1 - 11920) q^{62} + ( - 64 \beta_{4} - 711 \beta_{3} + 795 \beta_{2} + 728 \beta_1 - 27820) q^{64} + (196 \beta_{4} + 1202 \beta_{3} + 3140 \beta_{2} - 762 \beta_1 + 3700) q^{65} + ( - 31 \beta_{4} + 47 \beta_{3} + 4643 \beta_{2} - 563 \beta_1 - 2152) q^{67} + ( - 401 \beta_{4} - 759 \beta_{3} - 3459 \beta_{2} + 486 \beta_1 - 14816) q^{68} + ( - 54 \beta_{4} - 1180 \beta_{3} - 2026 \beta_{2} + 272 \beta_1 - 43672) q^{70} + (64 \beta_{4} + 58 \beta_{3} - 2272 \beta_{2} + 4352 \beta_1 - 6648) q^{71} + (64 \beta_{4} - 88 \beta_{3} + 3552 \beta_{2} + 1156 \beta_1 + 7914) q^{73} + (117 \beta_{4} - 623 \beta_{3} - 2275 \beta_{2} + 450 \beta_1 - 21364) q^{74} + (1260 \beta_{4} + 758 \beta_{3} + 1546 \beta_{2} + 1756 \beta_1 - 16024) q^{76} + (186 \beta_{4} - 330 \beta_{3} + 2342 \beta_{2} + 740 \beta_1 - 32264) q^{77} + ( - 251 \beta_{4} + 624 \beta_{3} - 4569 \beta_{2} + 4233 \beta_1 + 11836) q^{79} + ( - 216 \beta_{4} - 848 \beta_{2} + 512 \beta_1 - 13632) q^{80} + ( - 1798 \beta_{4} - 1406 \beta_{3} - 6588 \beta_{2} + \cdots - 44828) q^{82}+ \cdots + (974 \beta_{4} + 392 \beta_{3} - 1569 \beta_{2} - 380 \beta_1 - 5578) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 8 q^{2} + 118 q^{4} - 94 q^{5} + 272 q^{7} - 258 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 8 q^{2} + 118 q^{4} - 94 q^{5} + 272 q^{7} - 258 q^{8} - 172 q^{10} - 1100 q^{11} - 978 q^{13} + 344 q^{14} + 1218 q^{16} - 2522 q^{17} + 2060 q^{19} - 7720 q^{20} - 2572 q^{22} + 2645 q^{23} + 12035 q^{25} - 9280 q^{26} + 8072 q^{28} - 1526 q^{29} - 7392 q^{31} + 5086 q^{32} - 15608 q^{34} - 6056 q^{35} - 8210 q^{37} + 14276 q^{38} - 37472 q^{40} - 21250 q^{41} - 4548 q^{43} + 4260 q^{44} - 4232 q^{46} - 536 q^{47} - 27979 q^{49} + 81872 q^{50} - 76380 q^{52} + 11482 q^{53} - 77064 q^{55} + 28624 q^{56} - 79680 q^{58} - 74676 q^{59} - 44618 q^{61} - 64880 q^{62} - 137382 q^{64} + 24388 q^{65} - 1412 q^{67} - 80196 q^{68} - 222304 q^{70} - 37912 q^{71} + 46546 q^{73} - 111604 q^{74} - 79548 q^{76} - 157008 q^{77} + 50544 q^{79} - 69424 q^{80} - 233720 q^{82} - 89588 q^{83} + 147892 q^{85} - 77428 q^{86} + 54484 q^{88} - 280410 q^{89} - 27416 q^{91} + 62422 q^{92} + 113632 q^{94} - 203120 q^{95} + 90074 q^{97} - 32976 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 113x^{3} - 257x^{2} + 1404x + 2197 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{4} - 13\nu^{3} - 700\nu^{2} - 499\nu + 5941 ) / 468 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} + 13\nu^{3} - 56\nu^{2} - 419\nu + 4511 ) / 117 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23\nu^{4} - 221\nu^{3} - 1676\nu^{2} + 11821\nu + 18005 ) / 468 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{4} - 4\beta_{3} + \beta_{2} + 6\beta _1 + 180 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{4} - 7\beta_{3} + 19\beta_{2} + 86\beta _1 + 298 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -63\beta_{4} - 213\beta_{3} + 219\beta_{2} + 531\beta _1 + 7856 ) / 2 \) 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
3.40352
−5.17654
−7.90234
11.0973
−1.42196
−9.34908 0 55.4053 −70.5203 0 −89.7132 −218.818 0 659.300
1.2 −9.27315 0 53.9914 37.4928 0 154.850 −203.929 0 −347.676
1.3 −2.24792 0 −26.9469 −53.3906 0 89.8688 132.508 0 120.018
1.4 3.54286 0 −19.4482 92.1306 0 −8.98894 −182.273 0 326.406
1.5 9.32729 0 54.9984 −99.7124 0 125.983 214.513 0 −930.047
\(n\): 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 207.6.a.f 5
3.b odd 2 1 69.6.a.e 5
12.b even 2 1 1104.6.a.r 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.6.a.e 5 3.b odd 2 1
207.6.a.f 5 1.a even 1 1 trivial
1104.6.a.r 5 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + 8T_{2}^{4} - 107T_{2}^{3} - 770T_{2}^{2} + 1740T_{2} + 6440 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(207))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 8 T^{4} - 107 T^{3} + \cdots + 6440 \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( T^{5} + 94 T^{4} + \cdots + 1296820224 \) Copy content Toggle raw display
$7$ \( T^{5} - 272 T^{4} + \cdots - 1413831904 \) Copy content Toggle raw display
$11$ \( T^{5} + 1100 T^{4} + \cdots - 5578745996288 \) Copy content Toggle raw display
$13$ \( T^{5} + 978 T^{4} + \cdots - 2474410088160 \) Copy content Toggle raw display
$17$ \( T^{5} + 2522 T^{4} + \cdots - 13327498013440 \) Copy content Toggle raw display
$19$ \( T^{5} - 2060 T^{4} + \cdots + 30\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( (T - 529)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} + 1526 T^{4} + \cdots - 32\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{5} + 7392 T^{4} + \cdots + 16\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( T^{5} + 8210 T^{4} + \cdots + 84\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{5} + 21250 T^{4} + \cdots + 22\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{5} + 4548 T^{4} + \cdots - 23\!\cdots\!72 \) Copy content Toggle raw display
$47$ \( T^{5} + 536 T^{4} + \cdots - 94\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{5} - 11482 T^{4} + \cdots - 77\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{5} + 74676 T^{4} + \cdots + 10\!\cdots\!88 \) Copy content Toggle raw display
$61$ \( T^{5} + 44618 T^{4} + \cdots + 11\!\cdots\!60 \) Copy content Toggle raw display
$67$ \( T^{5} + 1412 T^{4} + \cdots - 11\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{5} + 37912 T^{4} + \cdots + 42\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{5} - 46546 T^{4} + \cdots + 27\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{5} - 50544 T^{4} + \cdots - 39\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{5} + 89588 T^{4} + \cdots + 29\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{5} + 280410 T^{4} + \cdots - 28\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{5} - 90074 T^{4} + \cdots + 55\!\cdots\!56 \) Copy content Toggle raw display
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