Properties

Label 15.8.a.c
Level $15$
Weight $8$
Character orbit 15.a
Self dual yes
Analytic conductor $4.686$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 15.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.68577538226\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{601}) \)
Defining polynomial: \( x^{2} - x - 150 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{601})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 4) q^{2} + 27 q^{3} + ( - 7 \beta + 38) q^{4} + 125 q^{5} + ( - 27 \beta + 108) q^{6} + (56 \beta + 624) q^{7} + (69 \beta + 690) q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 4) q^{2} + 27 q^{3} + ( - 7 \beta + 38) q^{4} + 125 q^{5} + ( - 27 \beta + 108) q^{6} + (56 \beta + 624) q^{7} + (69 \beta + 690) q^{8} + 729 q^{9} + ( - 125 \beta + 500) q^{10} + (464 \beta + 1492) q^{11} + ( - 189 \beta + 1026) q^{12} + ( - 824 \beta - 4082) q^{13} + ( - 456 \beta - 5904) q^{14} + 3375 q^{15} + (413 \beta - 12454) q^{16} + (1400 \beta - 3446) q^{17} + ( - 729 \beta + 2916) q^{18} + (2056 \beta - 25820) q^{19} + ( - 875 \beta + 4750) q^{20} + (1512 \beta + 16848) q^{21} + ( - 100 \beta - 63632) q^{22} + ( - 5208 \beta + 48528) q^{23} + (1863 \beta + 18630) q^{24} + 15625 q^{25} + (1610 \beta + 107272) q^{26} + 19683 q^{27} + ( - 2632 \beta - 35088) q^{28} + ( - 8288 \beta + 95030) q^{29} + ( - 3375 \beta + 13500) q^{30} + (2168 \beta + 151032) q^{31} + (4861 \beta - 200086) q^{32} + (12528 \beta + 40284) q^{33} + (7646 \beta - 223784) q^{34} + (7000 \beta + 78000) q^{35} + ( - 5103 \beta + 27702) q^{36} + ( - 14424 \beta - 243946) q^{37} + (31988 \beta - 411680) q^{38} + ( - 22248 \beta - 110214) q^{39} + (8625 \beta + 86250) q^{40} + ( - 21392 \beta + 326282) q^{41} + ( - 12312 \beta - 159408) q^{42} + ( - 7136 \beta + 180388) q^{43} + (3940 \beta - 430504) q^{44} + 91125 q^{45} + ( - 64152 \beta + 975312) q^{46} + (5912 \beta - 236696) q^{47} + (11151 \beta - 336258) q^{48} + (73024 \beta + 36233) q^{49} + ( - 15625 \beta + 62500) q^{50} + (37800 \beta - 93042) q^{51} + (3030 \beta + 710084) q^{52} + (13232 \beta - 290642) q^{53} + ( - 19683 \beta + 78732) q^{54} + (58000 \beta + 186500) q^{55} + (85560 \beta + 1010160) q^{56} + (55512 \beta - 697140) q^{57} + ( - 119894 \beta + 1623320) q^{58} + ( - 149776 \beta + 218500) q^{59} + ( - 23625 \beta + 128250) q^{60} + (1456 \beta - 1257818) q^{61} + ( - 144528 \beta + 278928) q^{62} + (40824 \beta + 454896) q^{63} + (161805 \beta + 64618) q^{64} + ( - 103000 \beta - 510250) q^{65} + ( - 2700 \beta - 1718064) q^{66} + (129920 \beta - 2601876) q^{67} + (67522 \beta - 1600948) q^{68} + ( - 140616 \beta + 1310256) q^{69} + ( - 57000 \beta - 738000) q^{70} + (76480 \beta - 1912648) q^{71} + (50301 \beta + 503010) q^{72} + ( - 388208 \beta - 544502) q^{73} + (200674 \beta + 1187816) q^{74} + 421875 q^{75} + (244476 \beta - 3139960) q^{76} + (399072 \beta + 4828608) q^{77} + (43470 \beta + 2896344) q^{78} + ( - 80440 \beta - 2273640) q^{79} + (51625 \beta - 1556750) q^{80} + 531441 q^{81} + ( - 390458 \beta + 4513928) q^{82} + ( - 91872 \beta - 2990532) q^{83} + ( - 71064 \beta - 947376) q^{84} + (175000 \beta - 430750) q^{85} + ( - 201796 \beta + 1791952) q^{86} + ( - 223776 \beta + 2565810) q^{87} + (455124 \beta + 5831880) q^{88} + (20496 \beta + 8247930) q^{89} + ( - 91125 \beta + 364500) q^{90} + ( - 788912 \beta - 9468768) q^{91} + ( - 501144 \beta + 7312464) q^{92} + (58536 \beta + 4077864) q^{93} + (254432 \beta - 1833584) q^{94} + (257000 \beta - 3227500) q^{95} + (131247 \beta - 5402322) q^{96} + (428640 \beta + 1147394) q^{97} + (182839 \beta - 10808668) q^{98} + (338256 \beta + 1087668) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 7 q^{2} + 54 q^{3} + 69 q^{4} + 250 q^{5} + 189 q^{6} + 1304 q^{7} + 1449 q^{8} + 1458 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 7 q^{2} + 54 q^{3} + 69 q^{4} + 250 q^{5} + 189 q^{6} + 1304 q^{7} + 1449 q^{8} + 1458 q^{9} + 875 q^{10} + 3448 q^{11} + 1863 q^{12} - 8988 q^{13} - 12264 q^{14} + 6750 q^{15} - 24495 q^{16} - 5492 q^{17} + 5103 q^{18} - 49584 q^{19} + 8625 q^{20} + 35208 q^{21} - 127364 q^{22} + 91848 q^{23} + 39123 q^{24} + 31250 q^{25} + 216154 q^{26} + 39366 q^{27} - 72808 q^{28} + 181772 q^{29} + 23625 q^{30} + 304232 q^{31} - 395311 q^{32} + 93096 q^{33} - 439922 q^{34} + 163000 q^{35} + 50301 q^{36} - 502316 q^{37} - 791372 q^{38} - 242676 q^{39} + 181125 q^{40} + 631172 q^{41} - 331128 q^{42} + 353640 q^{43} - 857068 q^{44} + 182250 q^{45} + 1886472 q^{46} - 467480 q^{47} - 661365 q^{48} + 145490 q^{49} + 109375 q^{50} - 148284 q^{51} + 1423198 q^{52} - 568052 q^{53} + 137781 q^{54} + 431000 q^{55} + 2105880 q^{56} - 1338768 q^{57} + 3126746 q^{58} + 287224 q^{59} + 232875 q^{60} - 2514180 q^{61} + 413328 q^{62} + 950616 q^{63} + 291041 q^{64} - 1123500 q^{65} - 3438828 q^{66} - 5073832 q^{67} - 3134374 q^{68} + 2479896 q^{69} - 1533000 q^{70} - 3748816 q^{71} + 1056321 q^{72} - 1477212 q^{73} + 2576306 q^{74} + 843750 q^{75} - 6035444 q^{76} + 10056288 q^{77} + 5836158 q^{78} - 4627720 q^{79} - 3061875 q^{80} + 1062882 q^{81} + 8637398 q^{82} - 6072936 q^{83} - 1965816 q^{84} - 686500 q^{85} + 3382108 q^{86} + 4907844 q^{87} + 12118884 q^{88} + 16516356 q^{89} + 637875 q^{90} - 19726448 q^{91} + 14123784 q^{92} + 8214264 q^{93} - 3412736 q^{94} - 6198000 q^{95} - 10673397 q^{96} + 2723428 q^{97} - 21434497 q^{98} + 2513592 q^{99}+O(q^{100}) \) 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
12.7577
−11.7577
−8.75765 27.0000 −51.3036 125.000 −236.457 1338.43 1570.28 729.000 −1094.71
1.2 15.7577 27.0000 120.304 125.000 425.457 −34.4284 −121.278 729.000 1969.71
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 15.8.a.c 2
3.b odd 2 1 45.8.a.i 2
4.b odd 2 1 240.8.a.p 2
5.b even 2 1 75.8.a.e 2
5.c odd 4 2 75.8.b.d 4
15.d odd 2 1 225.8.a.t 2
15.e even 4 2 225.8.b.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.8.a.c 2 1.a even 1 1 trivial
45.8.a.i 2 3.b odd 2 1
75.8.a.e 2 5.b even 2 1
75.8.b.d 4 5.c odd 4 2
225.8.a.t 2 15.d odd 2 1
225.8.b.n 4 15.e even 4 2
240.8.a.p 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 7T_{2} - 138 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(15))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 7T - 138 \) Copy content Toggle raw display
$3$ \( (T - 27)^{2} \) Copy content Toggle raw display
$5$ \( (T - 125)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 1304T - 46080 \) Copy content Toggle raw display
$11$ \( T^{2} - 3448 T - 29376048 \) Copy content Toggle raw display
$13$ \( T^{2} + 8988 T - 81820108 \) Copy content Toggle raw display
$17$ \( T^{2} + 5492 T - 286949484 \) Copy content Toggle raw display
$19$ \( T^{2} + 49584 T - 20483920 \) Copy content Toggle raw display
$23$ \( T^{2} - 91848 T - 1966256640 \) Copy content Toggle raw display
$29$ \( T^{2} - 181772 T - 2060549340 \) Copy content Toggle raw display
$31$ \( T^{2} - 304232 T + 22433068800 \) Copy content Toggle raw display
$37$ \( T^{2} + 502316 T + 31820561620 \) Copy content Toggle raw display
$41$ \( T^{2} - 631172 T + 30837469380 \) Copy content Toggle raw display
$43$ \( T^{2} - 353640 T + 23614207376 \) Copy content Toggle raw display
$47$ \( T^{2} + 467480 T + 49382888064 \) Copy content Toggle raw display
$53$ \( T^{2} + 568052 T + 54364123620 \) Copy content Toggle raw display
$59$ \( T^{2} - 287224 T - 3349911332400 \) Copy content Toggle raw display
$61$ \( T^{2} + 2514180 T + 1579956747716 \) Copy content Toggle raw display
$67$ \( T^{2} + 5073832 T + 3899842029456 \) Copy content Toggle raw display
$71$ \( T^{2} + 3748816 T + 2634564492864 \) Copy content Toggle raw display
$73$ \( T^{2} + 1477212 T - 22097955229180 \) Copy content Toggle raw display
$79$ \( T^{2} + 4627720 T + 4381741411200 \) Copy content Toggle raw display
$83$ \( T^{2} + 6072936 T + 7951958141328 \) Copy content Toggle raw display
$89$ \( T^{2} - 16516356 T + 68134385955780 \) Copy content Toggle raw display
$97$ \( T^{2} - 2723428 T - 25751505484604 \) Copy content Toggle raw display
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