Properties

Label 245.6.a.c
Level $245$
Weight $6$
Character orbit 245.a
Self dual yes
Analytic conductor $39.294$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{65}) \)
Defining polynomial: \( x^{2} - x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{65})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + (3 \beta - 3) q^{3} + (\beta - 16) q^{4} + 25 q^{5} + 48 q^{6} + ( - 47 \beta + 16) q^{8} + ( - 9 \beta - 90) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (3 \beta - 3) q^{3} + (\beta - 16) q^{4} + 25 q^{5} + 48 q^{6} + ( - 47 \beta + 16) q^{8} + ( - 9 \beta - 90) q^{9} + 25 \beta q^{10} + (97 \beta - 349) q^{11} + ( - 48 \beta + 96) q^{12} + ( - 53 \beta + 315) q^{13} + (75 \beta - 75) q^{15} + ( - 63 \beta - 240) q^{16} + ( - 251 \beta + 105) q^{17} + ( - 99 \beta - 144) q^{18} + (86 \beta - 358) q^{19} + (25 \beta - 400) q^{20} + ( - 252 \beta + 1552) q^{22} + ( - 902 \beta + 230) q^{23} + (48 \beta - 2304) q^{24} + 625 q^{25} + (262 \beta - 848) q^{26} + ( - 999 \beta + 567) q^{27} + ( - 945 \beta + 3415) q^{29} + 1200 q^{30} + ( - 924 \beta + 660) q^{31} + (1201 \beta - 1520) q^{32} + ( - 1047 \beta + 5703) q^{33} + ( - 146 \beta - 4016) q^{34} + (45 \beta + 1296) q^{36} + ( - 1260 \beta - 3822) q^{37} + ( - 272 \beta + 1376) q^{38} + (945 \beta - 3489) q^{39} + ( - 1175 \beta + 400) q^{40} + (3818 \beta - 2796) q^{41} + (922 \beta - 14022) q^{43} + ( - 1804 \beta + 7136) q^{44} + ( - 225 \beta - 2250) q^{45} + ( - 672 \beta - 14432) q^{46} + (1575 \beta + 9857) q^{47} + ( - 720 \beta - 2304) q^{48} + 625 \beta q^{50} + (315 \beta - 12363) q^{51} + (1110 \beta - 5888) q^{52} + ( - 454 \beta - 27564) q^{53} + ( - 432 \beta - 15984) q^{54} + (2425 \beta - 8725) q^{55} + ( - 1074 \beta + 5202) q^{57} + (2470 \beta - 15120) q^{58} + ( - 5184 \beta - 27208) q^{59} + ( - 1200 \beta + 2400) q^{60} + ( - 5706 \beta + 28776) q^{61} + ( - 264 \beta - 14784) q^{62} + (1697 \beta + 26896) q^{64} + ( - 1325 \beta + 7875) q^{65} + (4656 \beta - 16752) q^{66} + ( - 4568 \beta - 20388) q^{67} + (3870 \beta - 5696) q^{68} + (690 \beta - 43986) q^{69} + (5304 \beta + 37720) q^{71} + (4509 \beta + 5328) q^{72} + (4192 \beta + 4670) q^{73} + ( - 5082 \beta - 20160) q^{74} + (1875 \beta - 1875) q^{75} + ( - 1648 \beta + 7104) q^{76} + ( - 2544 \beta + 15120) q^{78} + (17635 \beta - 34715) q^{79} + ( - 1575 \beta - 6000) q^{80} + (3888 \beta - 27783) q^{81} + (1022 \beta + 61088) q^{82} + (3924 \beta - 56876) q^{83} + ( - 6275 \beta + 2625) q^{85} + ( - 13100 \beta + 14752) q^{86} + (10245 \beta - 55605) q^{87} + (13396 \beta - 78528) q^{88} + (5722 \beta + 15964) q^{89} + ( - 2475 \beta - 3600) q^{90} + (13760 \beta - 18112) q^{92} + (1980 \beta - 46332) q^{93} + (11432 \beta + 25200) q^{94} + (2150 \beta - 8950) q^{95} + ( - 4560 \beta + 62208) q^{96} + ( - 13943 \beta + 55141) q^{97} + ( - 6462 \beta + 17442) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 3 q^{3} - 31 q^{4} + 50 q^{5} + 96 q^{6} - 15 q^{8} - 189 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 3 q^{3} - 31 q^{4} + 50 q^{5} + 96 q^{6} - 15 q^{8} - 189 q^{9} + 25 q^{10} - 601 q^{11} + 144 q^{12} + 577 q^{13} - 75 q^{15} - 543 q^{16} - 41 q^{17} - 387 q^{18} - 630 q^{19} - 775 q^{20} + 2852 q^{22} - 442 q^{23} - 4560 q^{24} + 1250 q^{25} - 1434 q^{26} + 135 q^{27} + 5885 q^{29} + 2400 q^{30} + 396 q^{31} - 1839 q^{32} + 10359 q^{33} - 8178 q^{34} + 2637 q^{36} - 8904 q^{37} + 2480 q^{38} - 6033 q^{39} - 375 q^{40} - 1774 q^{41} - 27122 q^{43} + 12468 q^{44} - 4725 q^{45} - 29536 q^{46} + 21289 q^{47} - 5328 q^{48} + 625 q^{50} - 24411 q^{51} - 10666 q^{52} - 55582 q^{53} - 32400 q^{54} - 15025 q^{55} + 9330 q^{57} - 27770 q^{58} - 59600 q^{59} + 3600 q^{60} + 51846 q^{61} - 29832 q^{62} + 55489 q^{64} + 14425 q^{65} - 28848 q^{66} - 45344 q^{67} - 7522 q^{68} - 87282 q^{69} + 80744 q^{71} + 15165 q^{72} + 13532 q^{73} - 45402 q^{74} - 1875 q^{75} + 12560 q^{76} + 27696 q^{78} - 51795 q^{79} - 13575 q^{80} - 51678 q^{81} + 123198 q^{82} - 109828 q^{83} - 1025 q^{85} + 16404 q^{86} - 100965 q^{87} - 143660 q^{88} + 37650 q^{89} - 9675 q^{90} - 22464 q^{92} - 90684 q^{93} + 61832 q^{94} - 15750 q^{95} + 119856 q^{96} + 96339 q^{97} + 28422 q^{99}+O(q^{100}) \) 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
−3.53113
4.53113
−3.53113 −13.5934 −19.5311 25.0000 48.0000 0 181.963 −58.2198 −88.2782
1.2 4.53113 10.5934 −11.4689 25.0000 48.0000 0 −196.963 −130.780 113.278
\(n\): e.g. 2-40 or 990-1000
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.c 2
7.b odd 2 1 35.6.a.b 2
21.c even 2 1 315.6.a.c 2
28.d even 2 1 560.6.a.l 2
35.c odd 2 1 175.6.a.d 2
35.f even 4 2 175.6.b.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.b 2 7.b odd 2 1
175.6.a.d 2 35.c odd 2 1
175.6.b.d 4 35.f even 4 2
245.6.a.c 2 1.a even 1 1 trivial
315.6.a.c 2 21.c even 2 1
560.6.a.l 2 28.d even 2 1

Hecke kernels

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

\( T_{2}^{2} - T_{2} - 16 \) Copy content Toggle raw display
\( T_{3}^{2} + 3T_{3} - 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 16 \) Copy content Toggle raw display
$3$ \( T^{2} + 3T - 144 \) Copy content Toggle raw display
$5$ \( (T - 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 601T - 62596 \) Copy content Toggle raw display
$13$ \( T^{2} - 577T + 37586 \) Copy content Toggle raw display
$17$ \( T^{2} + 41T - 1023346 \) Copy content Toggle raw display
$19$ \( T^{2} + 630T - 20960 \) Copy content Toggle raw display
$23$ \( T^{2} + 442 T - 13172224 \) Copy content Toggle raw display
$29$ \( T^{2} - 5885 T - 5853350 \) Copy content Toggle raw display
$31$ \( T^{2} - 396 T - 13834656 \) Copy content Toggle raw display
$37$ \( T^{2} + 8904 T - 5978196 \) Copy content Toggle raw display
$41$ \( T^{2} + 1774 T - 236091496 \) Copy content Toggle raw display
$43$ \( T^{2} + 27122 T + 170086856 \) Copy content Toggle raw display
$47$ \( T^{2} - 21289 T + 72995224 \) Copy content Toggle raw display
$53$ \( T^{2} + 55582 T + 768990296 \) Copy content Toggle raw display
$59$ \( T^{2} + 59600 T + 451339840 \) Copy content Toggle raw display
$61$ \( T^{2} - 51846 T + 142927344 \) Copy content Toggle raw display
$67$ \( T^{2} + 45344 T + 174936944 \) Copy content Toggle raw display
$71$ \( T^{2} - 80744 T + 1172746624 \) Copy content Toggle raw display
$73$ \( T^{2} - 13532 T - 239780284 \) Copy content Toggle raw display
$79$ \( T^{2} + 51795 T - 4382959400 \) Copy content Toggle raw display
$83$ \( T^{2} + 109828 T + 2765333536 \) Copy content Toggle raw display
$89$ \( T^{2} - 37650 T - 177665240 \) Copy content Toggle raw display
$97$ \( T^{2} - 96339 T - 838817066 \) Copy content Toggle raw display
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