Properties

Label 315.6.a.c
Level $315$
Weight $6$
Character orbit 315.a
Self dual yes
Analytic conductor $50.521$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 315.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(50.5209032411\)
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.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta - 16) q^{4} + 25 q^{5} - 49 q^{7} + (47 \beta - 16) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta - 16) q^{4} + 25 q^{5} - 49 q^{7} + (47 \beta - 16) q^{8} - 25 \beta q^{10} + ( - 97 \beta + 349) q^{11} + (53 \beta - 315) q^{13} + 49 \beta q^{14} + ( - 63 \beta - 240) q^{16} + ( - 251 \beta + 105) q^{17} + ( - 86 \beta + 358) q^{19} + (25 \beta - 400) q^{20} + ( - 252 \beta + 1552) q^{22} + (902 \beta - 230) q^{23} + 625 q^{25} + (262 \beta - 848) q^{26} + ( - 49 \beta + 784) q^{28} + (945 \beta - 3415) q^{29} + (924 \beta - 660) q^{31} + ( - 1201 \beta + 1520) q^{32} + (146 \beta + 4016) q^{34} - 1225 q^{35} + ( - 1260 \beta - 3822) q^{37} + ( - 272 \beta + 1376) q^{38} + (1175 \beta - 400) q^{40} + (3818 \beta - 2796) q^{41} + (922 \beta - 14022) q^{43} + (1804 \beta - 7136) q^{44} + ( - 672 \beta - 14432) q^{46} + (1575 \beta + 9857) q^{47} + 2401 q^{49} - 625 \beta q^{50} + ( - 1110 \beta + 5888) q^{52} + (454 \beta + 27564) q^{53} + ( - 2425 \beta + 8725) q^{55} + ( - 2303 \beta + 784) q^{56} + (2470 \beta - 15120) q^{58} + ( - 5184 \beta - 27208) q^{59} + (5706 \beta - 28776) q^{61} + ( - 264 \beta - 14784) q^{62} + (1697 \beta + 26896) q^{64} + (1325 \beta - 7875) q^{65} + ( - 4568 \beta - 20388) q^{67} + (3870 \beta - 5696) q^{68} + 1225 \beta q^{70} + ( - 5304 \beta - 37720) q^{71} + ( - 4192 \beta - 4670) q^{73} + (5082 \beta + 20160) q^{74} + (1648 \beta - 7104) q^{76} + (4753 \beta - 17101) q^{77} + (17635 \beta - 34715) q^{79} + ( - 1575 \beta - 6000) q^{80} + ( - 1022 \beta - 61088) q^{82} + (3924 \beta - 56876) q^{83} + ( - 6275 \beta + 2625) q^{85} + (13100 \beta - 14752) q^{86} + (13396 \beta - 78528) q^{88} + (5722 \beta + 15964) q^{89} + ( - 2597 \beta + 15435) q^{91} + ( - 13760 \beta + 18112) q^{92} + ( - 11432 \beta - 25200) q^{94} + ( - 2150 \beta + 8950) q^{95} + (13943 \beta - 55141) q^{97} - 2401 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 31 q^{4} + 50 q^{5} - 98 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 31 q^{4} + 50 q^{5} - 98 q^{7} + 15 q^{8} - 25 q^{10} + 601 q^{11} - 577 q^{13} + 49 q^{14} - 543 q^{16} - 41 q^{17} + 630 q^{19} - 775 q^{20} + 2852 q^{22} + 442 q^{23} + 1250 q^{25} - 1434 q^{26} + 1519 q^{28} - 5885 q^{29} - 396 q^{31} + 1839 q^{32} + 8178 q^{34} - 2450 q^{35} - 8904 q^{37} + 2480 q^{38} + 375 q^{40} - 1774 q^{41} - 27122 q^{43} - 12468 q^{44} - 29536 q^{46} + 21289 q^{47} + 4802 q^{49} - 625 q^{50} + 10666 q^{52} + 55582 q^{53} + 15025 q^{55} - 735 q^{56} - 27770 q^{58} - 59600 q^{59} - 51846 q^{61} - 29832 q^{62} + 55489 q^{64} - 14425 q^{65} - 45344 q^{67} - 7522 q^{68} + 1225 q^{70} - 80744 q^{71} - 13532 q^{73} + 45402 q^{74} - 12560 q^{76} - 29449 q^{77} - 51795 q^{79} - 13575 q^{80} - 123198 q^{82} - 109828 q^{83} - 1025 q^{85} - 16404 q^{86} - 143660 q^{88} + 37650 q^{89} + 28273 q^{91} + 22464 q^{92} - 61832 q^{94} + 15750 q^{95} - 96339 q^{97} - 2401 q^{98}+O(q^{100}) \) 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
4.53113
−3.53113
−4.53113 0 −11.4689 25.0000 0 −49.0000 196.963 0 −113.278
1.2 3.53113 0 −19.5311 25.0000 0 −49.0000 −181.963 0 88.2782
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)

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 315.6.a.c 2
3.b odd 2 1 35.6.a.b 2
12.b even 2 1 560.6.a.l 2
15.d odd 2 1 175.6.a.d 2
15.e even 4 2 175.6.b.d 4
21.c even 2 1 245.6.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.b 2 3.b odd 2 1
175.6.a.d 2 15.d odd 2 1
175.6.b.d 4 15.e even 4 2
245.6.a.c 2 21.c even 2 1
315.6.a.c 2 1.a even 1 1 trivial
560.6.a.l 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} - 16 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(315))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 16 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 25)^{2} \) Copy content Toggle raw display
$7$ \( (T + 49)^{2} \) 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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