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: \(x^{5} - 113 x^{3} - 257 x^{2} + 1404 x + 2197\)
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.

\(f(q)\) \(=\) \( q + ( -2 + \beta_{2} ) q^{2} + ( 24 - \beta_{2} + \beta_{3} ) q^{4} + ( -18 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{5} + ( 52 - 5 \beta_{1} + 5 \beta_{2} - \beta_{4} ) q^{7} + ( -60 - 10 \beta_{1} + 22 \beta_{2} + \beta_{4} ) q^{8} +O(q^{10})\) \( q + ( -2 + \beta_{2} ) q^{2} + ( 24 - \beta_{2} + \beta_{3} ) q^{4} + ( -18 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{5} + ( 52 - 5 \beta_{1} + 5 \beta_{2} - \beta_{4} ) q^{7} + ( -60 - 10 \beta_{1} + 22 \beta_{2} + \beta_{4} ) q^{8} + ( -12 + 22 \beta_{1} - 52 \beta_{2} - 6 \beta_{3} + 4 \beta_{4} ) q^{10} + ( -212 - 14 \beta_{2} + 5 \beta_{3} + 6 \beta_{4} ) q^{11} + ( -182 - 22 \beta_{1} - 33 \beta_{2} - 10 \beta_{3} + \beta_{4} ) q^{13} + ( 48 + 12 \beta_{1} + 65 \beta_{2} + 11 \beta_{3} + 13 \beta_{4} ) q^{14} + ( 244 - 12 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{16} + ( -478 - 13 \beta_{1} - 71 \beta_{2} - 8 \beta_{3} - 5 \beta_{4} ) q^{17} + ( 400 + 35 \beta_{1} + 53 \beta_{2} - 9 \beta_{3} + 23 \beta_{4} ) q^{19} + ( -1480 - 84 \beta_{1} - 188 \beta_{2} - 52 \beta_{3} - 28 \beta_{4} ) q^{20} + ( -496 - 122 \beta_{1} - 89 \beta_{2} - 15 \beta_{3} - 43 \beta_{4} ) q^{22} + 529 q^{23} + ( 2275 + 106 \beta_{1} + 341 \beta_{2} + 24 \beta_{3} + 11 \beta_{4} ) q^{25} + ( -1652 + 88 \beta_{1} - 506 \beta_{2} - 22 \beta_{3} + 4 \beta_{4} ) q^{26} + ( 1488 - 106 \beta_{1} + 243 \beta_{2} + 51 \beta_{3} - 73 \beta_{4} ) q^{28} + ( -198 - 90 \beta_{1} - 242 \beta_{2} + 96 \beta_{3} + 26 \beta_{4} ) q^{29} + ( -1344 + 50 \beta_{1} - 316 \beta_{2} - 36 \beta_{3} + 20 \beta_{4} ) q^{31} + ( 1196 + 306 \beta_{1} - 484 \beta_{2} + 10 \beta_{3} - 37 \beta_{4} ) q^{32} + ( -2796 + 140 \beta_{1} - 769 \beta_{2} - 61 \beta_{3} + 45 \beta_{4} ) q^{34} + ( -828 + 220 \beta_{1} - 979 \beta_{2} - 2 \beta_{3} - 21 \beta_{4} ) q^{35} + ( -1442 + 114 \beta_{1} - 530 \beta_{2} - 9 \beta_{3} - 30 \beta_{4} ) q^{37} + ( 2736 - 186 \beta_{1} + 70 \beta_{2} - 14 \beta_{3} - 228 \beta_{4} ) q^{38} + ( -6864 + 152 \beta_{1} - 1448 \beta_{2} + 64 \beta_{3} + 128 \beta_{4} ) q^{40} + ( -3754 + 132 \beta_{1} - 1036 \beta_{2} - 34 \beta_{3} + 204 \beta_{4} ) q^{41} + ( -784 + 135 \beta_{1} - 295 \beta_{2} + 361 \beta_{3} + 19 \beta_{4} ) q^{43} + ( 1096 + 666 \beta_{1} - 351 \beta_{2} - 99 \beta_{3} + 259 \beta_{4} ) q^{44} + ( -1058 + 529 \beta_{2} ) q^{46} + ( -540 - 1116 \beta_{1} + 921 \beta_{2} + 102 \beta_{3} - 161 \beta_{4} ) q^{47} + ( -5571 - 338 \beta_{1} - 127 \beta_{2} + 116 \beta_{3} - 65 \beta_{4} ) q^{49} + ( 15018 - 372 \beta_{1} + 3221 \beta_{2} + 248 \beta_{3} - 170 \beta_{4} ) q^{50} + ( -14592 + 876 \beta_{1} - 1884 \beta_{2} - 300 \beta_{3} - 174 \beta_{4} ) q^{52} + ( 1846 - 235 \beta_{1} + 1461 \beta_{2} + 55 \beta_{3} + 335 \beta_{4} ) q^{53} + ( -16160 - 1932 \beta_{1} + 2006 \beta_{2} + 144 \beta_{3} + 138 \beta_{4} ) q^{55} + ( 5232 - 18 \beta_{1} + 1557 \beta_{2} + 121 \beta_{3} + 325 \beta_{4} ) q^{56} + ( -16964 - 1272 \beta_{1} + 2548 \beta_{2} - 82 \beta_{3} - 22 \beta_{4} ) q^{58} + ( -14872 - 1416 \beta_{1} - 419 \beta_{2} - 324 \beta_{3} - 261 \beta_{4} ) q^{59} + ( -9650 + 242 \beta_{1} + 1860 \beta_{2} + 133 \beta_{3} + 44 \beta_{4} ) q^{61} + ( -11920 + 120 \beta_{1} - 2886 \beta_{2} - 422 \beta_{3} - 246 \beta_{4} ) q^{62} + ( -27820 + 728 \beta_{1} + 795 \beta_{2} - 711 \beta_{3} - 64 \beta_{4} ) q^{64} + ( 3700 - 762 \beta_{1} + 3140 \beta_{2} + 1202 \beta_{3} + 196 \beta_{4} ) q^{65} + ( -2152 - 563 \beta_{1} + 4643 \beta_{2} + 47 \beta_{3} - 31 \beta_{4} ) q^{67} + ( -14816 + 486 \beta_{1} - 3459 \beta_{2} - 759 \beta_{3} - 401 \beta_{4} ) q^{68} + ( -43672 + 272 \beta_{1} - 2026 \beta_{2} - 1180 \beta_{3} - 54 \beta_{4} ) q^{70} + ( -6648 + 4352 \beta_{1} - 2272 \beta_{2} + 58 \beta_{3} + 64 \beta_{4} ) q^{71} + ( 7914 + 1156 \beta_{1} + 3552 \beta_{2} - 88 \beta_{3} + 64 \beta_{4} ) q^{73} + ( -21364 + 450 \beta_{1} - 2275 \beta_{2} - 623 \beta_{3} + 117 \beta_{4} ) q^{74} + ( -16024 + 1756 \beta_{1} + 1546 \beta_{2} + 758 \beta_{3} + 1260 \beta_{4} ) q^{76} + ( -32264 + 740 \beta_{1} + 2342 \beta_{2} - 330 \beta_{3} + 186 \beta_{4} ) q^{77} + ( 11836 + 4233 \beta_{1} - 4569 \beta_{2} + 624 \beta_{3} - 251 \beta_{4} ) q^{79} + ( -13632 + 512 \beta_{1} - 848 \beta_{2} - 216 \beta_{4} ) q^{80} + ( -44828 - 2108 \beta_{1} - 6588 \beta_{2} - 1406 \beta_{3} - 1798 \beta_{4} ) q^{82} + ( -21124 + 1080 \beta_{1} + 7684 \beta_{2} - 781 \beta_{3} - 332 \beta_{4} ) q^{83} + ( 27752 - 2490 \beta_{1} + 4987 \beta_{2} + 1568 \beta_{3} + 421 \beta_{4} ) q^{85} + ( -19424 - 3838 \beta_{1} + 9920 \beta_{2} - 88 \beta_{3} + 74 \beta_{4} ) q^{86} + ( 10680 + 1786 \beta_{1} - 919 \beta_{2} - 895 \beta_{3} - 1461 \beta_{4} ) q^{88} + ( -59454 - 2883 \beta_{1} + 8269 \beta_{2} + 558 \beta_{3} - 161 \beta_{4} ) q^{89} + ( -3168 - 38 \beta_{1} - 5602 \beta_{2} - 1152 \beta_{3} + 186 \beta_{4} ) q^{91} + ( 12696 - 529 \beta_{2} + 529 \beta_{3} ) q^{92} + ( 21672 + 912 \beta_{1} + 5142 \beta_{2} + 2300 \beta_{3} + 2506 \beta_{4} ) q^{94} + ( -42764 + 3664 \beta_{1} + 4831 \beta_{2} - 1130 \beta_{3} - 519 \beta_{4} ) q^{95} + ( 20826 + 1122 \beta_{1} - 7270 \beta_{2} - 958 \beta_{3} - 242 \beta_{4} ) q^{97} + ( -5578 - 380 \beta_{1} - 1569 \beta_{2} + 392 \beta_{3} + 974 \beta_{4} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 8 q^{2} + 118 q^{4} - 94 q^{5} + 272 q^{7} - 258 q^{8} + O(q^{10}) \) \( 5 q - 8 q^{2} + 118 q^{4} - 94 q^{5} + 272 q^{7} - 258 q^{8} - 172 q^{10} - 1100 q^{11} - 978 q^{13} + 344 q^{14} + 1218 q^{16} - 2522 q^{17} + 2060 q^{19} - 7720 q^{20} - 2572 q^{22} + 2645 q^{23} + 12035 q^{25} - 9280 q^{26} + 8072 q^{28} - 1526 q^{29} - 7392 q^{31} + 5086 q^{32} - 15608 q^{34} - 6056 q^{35} - 8210 q^{37} + 14276 q^{38} - 37472 q^{40} - 21250 q^{41} - 4548 q^{43} + 4260 q^{44} - 4232 q^{46} - 536 q^{47} - 27979 q^{49} + 81872 q^{50} - 76380 q^{52} + 11482 q^{53} - 77064 q^{55} + 28624 q^{56} - 79680 q^{58} - 74676 q^{59} - 44618 q^{61} - 64880 q^{62} - 137382 q^{64} + 24388 q^{65} - 1412 q^{67} - 80196 q^{68} - 222304 q^{70} - 37912 q^{71} + 46546 q^{73} - 111604 q^{74} - 79548 q^{76} - 157008 q^{77} + 50544 q^{79} - 69424 q^{80} - 233720 q^{82} - 89588 q^{83} + 147892 q^{85} - 77428 q^{86} + 54484 q^{88} - 280410 q^{89} - 27416 q^{91} + 62422 q^{92} + 113632 q^{94} - 203120 q^{95} + 90074 q^{97} - 32976 q^{98} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( 7 \nu^{4} - 13 \nu^{3} - 700 \nu^{2} - 499 \nu + 5941 \)\()/468\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{4} + 13 \nu^{3} - 56 \nu^{2} - 419 \nu + 4511 \)\()/117\)
\(\beta_{4}\)\(=\)\((\)\( 23 \nu^{4} - 221 \nu^{3} - 1676 \nu^{2} + 11821 \nu + 18005 \)\()/468\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{4} - 4 \beta_{3} + \beta_{2} + 6 \beta_{1} + 180\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-7 \beta_{4} - 7 \beta_{3} + 19 \beta_{2} + 86 \beta_{1} + 298\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-63 \beta_{4} - 213 \beta_{3} + 219 \beta_{2} + 531 \beta_{1} + 7856\)\()/2\)

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 \( T_{2}^{5} + 8 T_{2}^{4} - 107 T_{2}^{3} - 770 T_{2}^{2} + 1740 T_{2} + 6440 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(207))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 6440 + 1740 T - 770 T^{2} - 107 T^{3} + 8 T^{4} + T^{5} \)
$3$ \( T^{5} \)
$5$ \( 1296820224 + 7019776 T - 941728 T^{2} - 9412 T^{3} + 94 T^{4} + T^{5} \)
$7$ \( -1413831904 - 136960224 T + 2364440 T^{2} + 8964 T^{3} - 272 T^{4} + T^{5} \)
$11$ \( -5578745996288 - 63217955584 T - 182462016 T^{2} + 144912 T^{3} + 1100 T^{4} + T^{5} \)
$13$ \( -2474410088160 - 99589342320 T - 490490640 T^{2} - 365096 T^{3} + 978 T^{4} + T^{5} \)
$17$ \( -13327498013440 - 314262009600 T - 337647680 T^{2} + 1404748 T^{3} + 2522 T^{4} + T^{5} \)
$19$ \( 3010484594372960 - 12155163321120 T + 14349684600 T^{2} - 4228108 T^{3} - 2060 T^{4} + T^{5} \)
$23$ \( ( -529 + T )^{5} \)
$29$ \( -321077683424449440 + 278387272580880 T + 35016763280 T^{2} - 49105544 T^{3} + 1526 T^{4} + T^{5} \)
$31$ \( 16557143540917248 - 11345767296000 T - 31920628544 T^{2} + 437104 T^{3} + 7392 T^{4} + T^{5} \)
$37$ \( 84165275968030880 + 14275410551760 T - 118822183280 T^{2} - 13572728 T^{3} + 8210 T^{4} + T^{5} \)
$41$ \( \)\(22\!\cdots\!12\)\( - 15270087820165296 T - 7880216624752 T^{2} - 293297688 T^{3} + 21250 T^{4} + T^{5} \)
$43$ \( -\)\(23\!\cdots\!72\)\( + 49174513307987040 T + 533272451576 T^{2} - 447278508 T^{3} + 4548 T^{4} + T^{5} \)
$47$ \( -\)\(94\!\cdots\!64\)\( + 193920884674686976 T + 2009810793728 T^{2} - 882704752 T^{3} + 536 T^{4} + T^{5} \)
$53$ \( -\)\(77\!\cdots\!36\)\( - 43031658091872448 T + 20148540401360 T^{2} - 974674548 T^{3} - 11482 T^{4} + T^{5} \)
$59$ \( \)\(10\!\cdots\!88\)\( - 402125524395219968 T - 49842241849920 T^{2} + 514568688 T^{3} + 74676 T^{4} + T^{5} \)
$61$ \( 1194743717850416160 - 125247918659625520 T - 11771098826800 T^{2} + 214301064 T^{3} + 44618 T^{4} + T^{5} \)
$67$ \( -\)\(11\!\cdots\!24\)\( + 650574262058706912 T - 27683158835768 T^{2} - 2709666380 T^{3} + 1412 T^{4} + T^{5} \)
$71$ \( \)\(42\!\cdots\!28\)\( + 11320178672395489280 T - 281487303913984 T^{2} - 7554678976 T^{3} + 37912 T^{4} + T^{5} \)
$73$ \( \)\(27\!\cdots\!68\)\( + 977910679949935696 T + 66492374421360 T^{2} - 2018921304 T^{3} - 46546 T^{4} + T^{5} \)
$79$ \( -\)\(39\!\cdots\!64\)\( + 2361477741365623584 T + 568163217678280 T^{2} - 8823710876 T^{3} - 50544 T^{4} + T^{5} \)
$83$ \( \)\(29\!\cdots\!24\)\( + 15328688118239528192 T - 636243818529216 T^{2} - 9243278864 T^{3} + 89588 T^{4} + T^{5} \)
$89$ \( -\)\(28\!\cdots\!76\)\( - \)\(10\!\cdots\!56\)\( T - 610285590892496 T^{2} + 19791083228 T^{3} + 280410 T^{4} + T^{5} \)
$97$ \( \)\(55\!\cdots\!56\)\( - 41635589139920255984 T + 1021916488616848 T^{2} - 6254051272 T^{3} - 90074 T^{4} + T^{5} \)
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