Properties

Label 245.6.a.f
Level $245$
Weight $6$
Character orbit 245.a
Self dual yes
Analytic conductor $39.294$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.2940358542\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 2 x^{4} - 128 x^{3} + 288 x^{2} + 3551 x - 6510\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + ( -1 - \beta_{1} - \beta_{2} ) q^{3} + ( 21 - 3 \beta_{1} + \beta_{3} ) q^{4} + 25 q^{5} + ( -61 + 4 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{6} + ( -138 + 11 \beta_{1} + 8 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{8} + ( 53 + \beta_{1} + 7 \beta_{2} + 8 \beta_{3} - \beta_{4} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + ( -1 - \beta_{1} - \beta_{2} ) q^{3} + ( 21 - 3 \beta_{1} + \beta_{3} ) q^{4} + 25 q^{5} + ( -61 + 4 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{6} + ( -138 + 11 \beta_{1} + 8 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{8} + ( 53 + \beta_{1} + 7 \beta_{2} + 8 \beta_{3} - \beta_{4} ) q^{9} + ( -25 + 25 \beta_{1} ) q^{10} + ( 207 - 7 \beta_{1} + 13 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} ) q^{11} + ( 275 - 80 \beta_{1} - 32 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} ) q^{12} + ( -214 - 9 \beta_{1} + 9 \beta_{2} - 16 \beta_{3} - 3 \beta_{4} ) q^{13} + ( -25 - 25 \beta_{1} - 25 \beta_{2} ) q^{15} + ( 112 - 137 \beta_{1} - 44 \beta_{2} - \beta_{3} - 13 \beta_{4} ) q^{16} + ( -694 + 71 \beta_{1} - 19 \beta_{2} - 40 \beta_{3} - 5 \beta_{4} ) q^{17} + ( 110 + 159 \beta_{1} + 114 \beta_{2} + 11 \beta_{3} + 3 \beta_{4} ) q^{18} + ( -584 + 16 \beta_{1} + 112 \beta_{2} + 28 \beta_{3} + 14 \beta_{4} ) q^{19} + ( 525 - 75 \beta_{1} + 25 \beta_{3} ) q^{20} + ( -473 + 250 \beta_{1} + 202 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{22} + ( -616 + 160 \beta_{1} - 16 \beta_{2} + 16 \beta_{3} + 28 \beta_{4} ) q^{23} + ( -2820 + 470 \beta_{1} + 140 \beta_{2} - 60 \beta_{3} + 10 \beta_{4} ) q^{24} + 625 q^{25} + ( -321 - 476 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 19 \beta_{4} ) q^{26} + ( -541 - 401 \beta_{1} + 7 \beta_{2} - 76 \beta_{3} + 2 \beta_{4} ) q^{27} + ( 60 + 113 \beta_{1} + 147 \beta_{2} - 16 \beta_{3} + 9 \beta_{4} ) q^{29} + ( -1525 + 100 \beta_{1} - 50 \beta_{2} - 75 \beta_{3} + 25 \beta_{4} ) q^{30} + ( -2254 + 286 \beta_{1} - 218 \beta_{2} + 8 \beta_{3} + 44 \beta_{4} ) q^{31} + ( -3462 + 163 \beta_{1} + 116 \beta_{2} - 181 \beta_{3} + 37 \beta_{4} ) q^{32} + ( -2034 - 1159 \beta_{1} - 301 \beta_{2} - 56 \beta_{3} - 3 \beta_{4} ) q^{33} + ( 3841 - 1414 \beta_{1} - 178 \beta_{2} + 13 \beta_{3} - 11 \beta_{4} ) q^{34} + ( 7724 - 409 \beta_{1} - 16 \beta_{2} + 143 \beta_{3} - 77 \beta_{4} ) q^{36} + ( -386 - 78 \beta_{1} - 354 \beta_{2} - 296 \beta_{3} - 18 \beta_{4} ) q^{37} + ( 2942 - 518 \beta_{1} - 56 \beta_{2} + 296 \beta_{3} - 112 \beta_{4} ) q^{38} + ( -2839 + 441 \beta_{1} + 537 \beta_{2} + 68 \beta_{3} - 6 \beta_{4} ) q^{39} + ( -3450 + 275 \beta_{1} + 200 \beta_{2} - 75 \beta_{3} + 25 \beta_{4} ) q^{40} + ( 2016 - 754 \beta_{1} + 122 \beta_{2} - 72 \beta_{3} + 94 \beta_{4} ) q^{41} + ( -3278 - 322 \beta_{1} - 370 \beta_{2} - 80 \beta_{3} + 100 \beta_{4} ) q^{43} + ( 8875 - 1306 \beta_{1} + 48 \beta_{2} + 522 \beta_{3} - 69 \beta_{4} ) q^{44} + ( 1325 + 25 \beta_{1} + 175 \beta_{2} + 200 \beta_{3} - 25 \beta_{4} ) q^{45} + ( 9308 - 660 \beta_{1} - 912 \beta_{2} + 240 \beta_{3} - 24 \beta_{4} ) q^{46} + ( -11421 - 1471 \beta_{1} + 53 \beta_{2} + 132 \beta_{3} + 16 \beta_{4} ) q^{47} + ( 19590 - 2590 \beta_{1} + 464 \beta_{2} + 662 \beta_{3} - 124 \beta_{4} ) q^{48} + ( -625 + 625 \beta_{1} ) q^{50} + ( -3271 + 2099 \beta_{1} + 1343 \beta_{2} + 92 \beta_{3} + 76 \beta_{4} ) q^{51} + ( -17909 + 896 \beta_{1} + 368 \beta_{2} - 44 \beta_{3} + 133 \beta_{4} ) q^{52} + ( -1770 - 340 \beta_{1} - 420 \beta_{2} + 80 \beta_{3} - 120 \beta_{4} ) q^{53} + ( -20743 - 978 \beta_{1} - 666 \beta_{2} - 379 \beta_{3} - 87 \beta_{4} ) q^{54} + ( 5175 - 175 \beta_{1} + 325 \beta_{2} + 100 \beta_{3} - 100 \beta_{4} ) q^{55} + ( -24572 + 1318 \beta_{1} - 374 \beta_{2} - 776 \beta_{3} + 2 \beta_{4} ) q^{57} + ( 7309 - 872 \beta_{1} - 158 \beta_{2} + 443 \beta_{3} - 181 \beta_{4} ) q^{58} + ( -10556 - 396 \beta_{1} - 492 \beta_{2} + 292 \beta_{3} + 6 \beta_{4} ) q^{59} + ( 6875 - 2000 \beta_{1} - 800 \beta_{2} + 100 \beta_{3} - 75 \beta_{4} ) q^{60} + ( -16650 + 210 \beta_{1} + 1470 \beta_{2} + 600 \beta_{3} - 90 \beta_{4} ) q^{61} + ( 15662 - 2088 \beta_{1} - 1956 \beta_{2} + 26 \beta_{3} + 138 \beta_{4} ) q^{62} + ( 8802 - 2685 \beta_{1} - 1140 \beta_{2} + 575 \beta_{3} + 45 \beta_{4} ) q^{64} + ( -5350 - 225 \beta_{1} + 225 \beta_{2} - 400 \beta_{3} - 75 \beta_{4} ) q^{65} + ( -61681 + 294 \beta_{1} - 942 \beta_{2} - 1773 \beta_{3} + 251 \beta_{4} ) q^{66} + ( -70 - 322 \beta_{1} - 82 \beta_{2} + 232 \beta_{3} + 196 \beta_{4} ) q^{67} + ( -57015 + 5150 \beta_{1} + 752 \beta_{2} - 534 \beta_{3} + 373 \beta_{4} ) q^{68} + ( -6112 + 5188 \beta_{1} - 164 \beta_{2} - 536 \beta_{3} + 132 \beta_{4} ) q^{69} + ( 1560 - 2196 \beta_{1} - 1804 \beta_{2} + 232 \beta_{3} + 112 \beta_{4} ) q^{71} + ( -32826 + 5867 \beta_{1} + 236 \beta_{2} - 1101 \beta_{3} + 217 \beta_{4} ) q^{72} + ( -19228 + 1502 \beta_{1} + 2498 \beta_{2} + 1088 \beta_{3} - 326 \beta_{4} ) q^{73} + ( -9552 - 3886 \beta_{1} - 2428 \beta_{2} - 858 \beta_{3} + 94 \beta_{4} ) q^{74} + ( -625 - 625 \beta_{1} - 625 \beta_{2} ) q^{75} + ( -11358 + 8482 \beta_{1} + 2704 \beta_{2} - 1974 \beta_{3} + 128 \beta_{4} ) q^{76} + ( 31527 - 4238 \beta_{1} + 1834 \beta_{2} + 1491 \beta_{3} - 457 \beta_{4} ) q^{78} + ( 16467 - 7321 \beta_{1} + 767 \beta_{2} + 412 \beta_{3} + 274 \beta_{4} ) q^{79} + ( 2800 - 3425 \beta_{1} - 1100 \beta_{2} - 25 \beta_{3} - 325 \beta_{4} ) q^{80} + ( 2369 + 3220 \beta_{1} + 1540 \beta_{2} - 400 \beta_{3} - 160 \beta_{4} ) q^{81} + ( -39098 + 1912 \beta_{1} - 3716 \beta_{2} - 134 \beta_{3} - 382 \beta_{4} ) q^{82} + ( 23786 + 6686 \beta_{1} - 1594 \beta_{2} + 260 \beta_{3} - 470 \beta_{4} ) q^{83} + ( -17350 + 1775 \beta_{1} - 475 \beta_{2} - 1000 \beta_{3} - 125 \beta_{4} ) q^{85} + ( -16226 - 2904 \beta_{1} - 4980 \beta_{2} - 662 \beta_{3} + 90 \beta_{4} ) q^{86} + ( -43815 + 2195 \beta_{1} - 529 \beta_{2} - 1012 \beta_{3} + 104 \beta_{4} ) q^{87} + ( -58552 + 11764 \beta_{1} + 292 \beta_{2} - 1582 \beta_{3} + 644 \beta_{4} ) q^{88} + ( -20208 - 3268 \beta_{1} - 1996 \beta_{2} + 2376 \beta_{3} + 28 \beta_{4} ) q^{89} + ( 2750 + 3975 \beta_{1} + 2850 \beta_{2} + 275 \beta_{3} + 75 \beta_{4} ) q^{90} + ( -31716 + 12108 \beta_{1} + 1472 \beta_{2} - 3092 \beta_{3} + 304 \beta_{4} ) q^{92} + ( 32588 + 10978 \beta_{1} + 1846 \beta_{2} + 104 \beta_{3} + 242 \beta_{4} ) q^{93} + ( -63377 - 6542 \beta_{1} + 586 \beta_{2} - 1301 \beta_{3} + 47 \beta_{4} ) q^{94} + ( -14600 + 400 \beta_{1} + 2800 \beta_{2} + 700 \beta_{3} + 350 \beta_{4} ) q^{95} + ( -56616 + 19054 \beta_{1} + 6208 \beta_{2} - 238 \beta_{3} + 126 \beta_{4} ) q^{96} + ( -41914 + 12571 \beta_{1} - 2975 \beta_{2} - 808 \beta_{3} + 131 \beta_{4} ) q^{97} + ( 87810 + 3442 \beta_{1} + 4378 \beta_{2} + 4296 \beta_{3} - 184 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{2} - 7 q^{3} + 101 q^{4} + 125 q^{5} - 304 q^{6} - 675 q^{8} + 284 q^{9} + O(q^{10}) \) \( 5 q - 3 q^{2} - 7 q^{3} + 101 q^{4} + 125 q^{5} - 304 q^{6} - 675 q^{8} + 284 q^{9} - 75 q^{10} + 1033 q^{11} + 1226 q^{12} - 1117 q^{13} - 175 q^{15} + 297 q^{16} - 3403 q^{17} + 887 q^{18} - 2846 q^{19} + 2525 q^{20} - 1858 q^{22} - 2756 q^{23} - 13290 q^{24} + 3125 q^{25} - 2544 q^{26} - 3661 q^{27} + 485 q^{29} - 7600 q^{30} - 10726 q^{31} - 17383 q^{32} - 12597 q^{33} + 16414 q^{34} + 38165 q^{36} - 2660 q^{37} + 14378 q^{38} - 13171 q^{39} - 16875 q^{40} + 8334 q^{41} - 17294 q^{43} + 42876 q^{44} + 7100 q^{45} + 45724 q^{46} - 59799 q^{47} + 94218 q^{48} - 1875 q^{50} - 12049 q^{51} - 87974 q^{52} - 9250 q^{53} - 106342 q^{54} + 25825 q^{55} - 121778 q^{57} + 35868 q^{58} - 52994 q^{59} + 30650 q^{60} - 81540 q^{61} + 74048 q^{62} + 39745 q^{64} - 27925 q^{65} - 311614 q^{66} - 726 q^{67} - 276216 q^{68} - 21388 q^{69} + 3760 q^{71} - 154815 q^{72} - 90634 q^{73} - 57342 q^{74} - 4375 q^{75} - 43902 q^{76} + 152598 q^{78} + 68243 q^{79} + 7425 q^{80} + 17645 q^{81} - 191552 q^{82} + 133292 q^{83} - 85075 q^{85} - 88352 q^{86} - 216813 q^{87} - 273040 q^{88} - 102852 q^{89} + 22175 q^{90} - 140852 q^{92} + 184862 q^{93} - 332618 q^{94} - 71150 q^{95} - 245574 q^{96} - 186175 q^{97} + 454710 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 128 x^{3} + 288 x^{2} + 3551 x - 6510\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} + 9 \nu^{3} - 89 \nu^{2} - 571 \nu + 1350 \)\()/60\)
\(\beta_{3}\)\(=\)\( \nu^{2} + \nu - 52 \)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{4} - 3 \nu^{3} + 178 \nu^{2} + 107 \nu - 2025 \)\()/15\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - \beta_{1} + 52\)
\(\nu^{3}\)\(=\)\(\beta_{4} + 8 \beta_{2} + 69 \beta_{1} - 45\)
\(\nu^{4}\)\(=\)\(-9 \beta_{4} + 89 \beta_{3} - 12 \beta_{2} - 139 \beta_{1} + 3683\)

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
−9.65317
−6.03043
1.78024
7.31036
8.59300
−10.6532 22.7173 81.4900 25.0000 −242.012 0 −527.226 273.078 −266.329
1.2 −7.03043 −10.0622 17.4270 25.0000 70.7416 0 102.455 −141.752 −175.761
1.3 0.780243 −4.65090 −31.3912 25.0000 −3.62883 0 −49.4606 −221.369 19.5061
1.4 6.31036 11.8302 7.82066 25.0000 74.6531 0 −152.580 −103.046 157.759
1.5 7.59300 −26.8345 25.6536 25.0000 −203.754 0 −48.1883 477.089 189.825
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(-1\)

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 245.6.a.f 5
7.b odd 2 1 245.6.a.g yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.6.a.f 5 1.a even 1 1 trivial
245.6.a.g yes 5 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(245))\):

\( T_{2}^{5} + 3 T_{2}^{4} - 126 T_{2}^{3} - 98 T_{2}^{2} + 3740 T_{2} - 2800 \)
\( T_{3}^{5} + 7 T_{3}^{4} - 725 T_{3}^{3} - 2835 T_{3}^{2} + 75300 T_{3} + 337500 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2800 + 3740 T - 98 T^{2} - 126 T^{3} + 3 T^{4} + T^{5} \)
$3$ \( 337500 + 75300 T - 2835 T^{2} - 725 T^{3} + 7 T^{4} + T^{5} \)
$5$ \( ( -25 + T )^{5} \)
$7$ \( T^{5} \)
$11$ \( -7633598722544 - 32476295560 T + 241453017 T^{2} - 23001 T^{3} - 1033 T^{4} + T^{5} \)
$13$ \( -7076320312500 - 202680540000 T - 699238965 T^{2} - 334165 T^{3} + 1117 T^{4} + T^{5} \)
$17$ \( -3252822240600000 - 10322528835600 T - 8863721415 T^{2} + 550355 T^{3} + 3403 T^{4} + T^{5} \)
$19$ \( 28909456807500000 + 14378709690000 T - 19598478000 T^{2} - 7428520 T^{3} + 2846 T^{4} + T^{5} \)
$23$ \( 26005593567462400 + 41701945155840 T - 37784820864 T^{2} - 16189984 T^{3} + 2756 T^{4} + T^{5} \)
$29$ \( -23738408137709676 + 41248918269192 T + 20665487981 T^{2} - 18175213 T^{3} - 485 T^{4} + T^{5} \)
$31$ \( -3160007607330000000 - 2806717664520000 T - 718731219000 T^{2} - 28136660 T^{3} + 10726 T^{4} + T^{5} \)
$37$ \( 1124678430489600000 + 936115379136000 T - 385480920960 T^{2} - 220817596 T^{3} + 2660 T^{4} + T^{5} \)
$41$ \( 3002899668750000000 + 1160901180000000 T - 683250525000 T^{2} - 248506100 T^{3} - 8334 T^{4} + T^{5} \)
$43$ \( 69138670934525558400 + 7675851960789120 T - 2426789603736 T^{2} - 185901764 T^{3} + 17294 T^{4} + T^{5} \)
$47$ \( -\)\(11\!\cdots\!00\)\( - 37550187566010000 T + 5067249935625 T^{2} + 1128477055 T^{3} + 59799 T^{4} + T^{5} \)
$53$ \( 31007066597284500000 + 5295658522410000 T - 2660682250000 T^{2} - 418721800 T^{3} + 9250 T^{4} + T^{5} \)
$59$ \( -29990549230267500000 - 70794491040630000 T - 2851939386000 T^{2} + 693859640 T^{3} + 52994 T^{4} + T^{5} \)
$61$ \( -\)\(14\!\cdots\!00\)\( - 2092407559128000000 T - 77961732120000 T^{2} + 765526500 T^{3} + 81540 T^{4} + T^{5} \)
$67$ \( \)\(26\!\cdots\!00\)\( + 96496374390791040 T + 2672858405384 T^{2} - 820777956 T^{3} + 726 T^{4} + T^{5} \)
$71$ \( \)\(77\!\cdots\!52\)\( + 2064197279221210112 T + 5006689354752 T^{2} - 3145558032 T^{3} - 3760 T^{4} + T^{5} \)
$73$ \( -\)\(12\!\cdots\!00\)\( - 18803093923304409600 T - 674330440335720 T^{2} - 4861128340 T^{3} + 90634 T^{4} + T^{5} \)
$79$ \( \)\(20\!\cdots\!44\)\( + 15027762877714824016 T + 166087299833755 T^{2} - 6288797561 T^{3} - 68243 T^{4} + T^{5} \)
$83$ \( \)\(88\!\cdots\!00\)\( - 64586338979300395200 T + 1355737753816560 T^{2} - 4351993300 T^{3} - 133292 T^{4} + T^{5} \)
$89$ \( \)\(29\!\cdots\!00\)\( + 12875525299082880000 T - 1192680829800000 T^{2} - 14729599760 T^{3} + 102852 T^{4} + T^{5} \)
$97$ \( \)\(57\!\cdots\!00\)\( - 34644465012198708000 T - 2523575434164675 T^{2} - 10066220645 T^{3} + 186175 T^{4} + T^{5} \)
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