Properties

Label 5.16.a.b
Level $5$
Weight $16$
Character orbit 5.a
Self dual yes
Analytic conductor $7.135$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 5.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.13467525500\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \( x^{3} - x^{2} - 1972x + 21070 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 5 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (8 \beta_{2} - 16 \beta_1 + 1170) q^{3} + ( - 9 \beta_{2} - 143 \beta_1 + 6744) q^{4} - 78125 q^{5} + ( - 1256 \beta_{2} - 3362 \beta_1 + 690186) q^{6} + (4968 \beta_{2} + 10960 \beta_1 - 296426) q^{7} + ( - 36 \beta_{2} + 5628 \beta_1 + 5560104) q^{8} + ( - 18848 \beta_{2} - 10496 \beta_1 + 15960729) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (8 \beta_{2} - 16 \beta_1 + 1170) q^{3} + ( - 9 \beta_{2} - 143 \beta_1 + 6744) q^{4} - 78125 q^{5} + ( - 1256 \beta_{2} - 3362 \beta_1 + 690186) q^{6} + (4968 \beta_{2} + 10960 \beta_1 - 296426) q^{7} + ( - 36 \beta_{2} + 5628 \beta_1 + 5560104) q^{8} + ( - 18848 \beta_{2} - 10496 \beta_1 + 15960729) q^{9} + (78125 \beta_1 - 78125) q^{10} + (55440 \beta_{2} - 161920 \beta_1 + 40922992) q^{11} + ( - 117818 \beta_{2} - 655862 \beta_1 + 86263728) q^{12} + (468000 \beta_{2} + 311168 \beta_1 - 30044118) q^{13} + ( - 591912 \beta_{2} + 1902426 \beta_1 - 398039346) q^{14} + ( - 625000 \beta_{2} + 1250000 \beta_1 - 91406250) q^{15} + (350568 \beta_{2} - 75464 \beta_1 - 438050976) q^{16} + (1907424 \beta_{2} - 1452928 \beta_1 - 1289506310) q^{17} + (2525408 \beta_{2} - 17639641 \beta_1 + 296753145) q^{18} + ( - 7803936 \beta_{2} - 3098272 \beta_1 + 1219994004) q^{19} + (703125 \beta_{2} + 11171875 \beta_1 - 526875000) q^{20} + (6523968 \beta_{2} + 79767936 \beta_1 + 3227700300) q^{21} + ( - 9163440 \beta_{2} - 63361232 \beta_1 + 6832445312) q^{22} + ( - 11330568 \beta_{2} + 18859632 \beta_1 + 8901862434) q^{23} + (51630552 \beta_{2} - 70408296 \beta_1 + 2546915472) q^{24} + 6103515625 q^{25} + ( - 62251488 \beta_{2} + 78909974 \beta_1 - 8999462966) q^{26} + (18437936 \beta_{2} - 206415328 \beta_1 - 33093483204) q^{27} + ( - 63393822 \beta_{2} + 303127438 \beta_1 - 70057040624) q^{28} + (15846336 \beta_{2} - 449184128 \beta_1 + 48370531646) q^{29} + (98125000 \beta_{2} + 262656250 \beta_1 - 53920781250) q^{30} + (108915120 \beta_{2} + 471103840 \beta_1 - 8431121228) q^{31} + ( - 48228480 \beta_{2} + 246422464 \beta_1 - 177159095104) q^{32} + (67387936 \beta_{2} - 768555392 \beta_1 + 283606577760) q^{33} + ( - 278208288 \beta_{2} + 1102264774 \beta_1 + 69669379418) q^{34} + ( - 388125000 \beta_{2} - 856250000 \beta_1 + 23158281250) q^{35} + (107822783 \beta_{2} - 2432395159 \beta_1 + 192198464664) q^{36} + (163094976 \beta_{2} + 906686976 \beta_1 + 140183343962) q^{37} + (1056862656 \beta_{2} - 1737987988 \beta_1 + 68188853716) q^{38} + ( - 314100592 \beta_{2} + 5134834016 \beta_1 + 798562731252) q^{39} + (2812500 \beta_{2} - 439687500 \beta_1 - 434383125000) q^{40} + ( - 191270880 \beta_{2} - 3845980160 \beta_1 + 89989506422) q^{41} + ( - 188920128 \beta_{2} + 8164586292 \beta_1 - 3102130426356) q^{42} + ( - 1476432504 \beta_{2} + 5073312560 \beta_1 + 784554249738) q^{43} + ( - 1113190848 \beta_{2} - 10615580096 \beta_1 + 1104227239808) q^{44} + (1472500000 \beta_{2} + 820000000 \beta_1 - 1246931953125) q^{45} + (1744685640 \beta_{2} - 6337100370 \beta_1 - 816764743158) q^{46} + (801928872 \beta_{2} - 13707131632 \beta_1 - 2967991804082) q^{47} + ( - 3949661168 \beta_{2} + 9462698032 \beta_1 + 324594331584) q^{48} + (11441544480 \beta_{2} - 1200221440 \beta_1 + 5871319044973) q^{49} + ( - 6103515625 \beta_1 + 6103515625) q^{50} + ( - 14056160624 \beta_{2} + \cdots + 3764158749588) q^{51}+ \cdots + (434642042384 \beta_{2} + \cdots + 425085777531888) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{2} + 3518 q^{3} + 20384 q^{4} - 234375 q^{5} + 2075176 q^{6} - 905206 q^{7} + 16674720 q^{8} + 47911531 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 4 q^{2} + 3518 q^{3} + 20384 q^{4} - 234375 q^{5} + 2075176 q^{6} - 905206 q^{7} + 16674720 q^{8} + 47911531 q^{9} - 312500 q^{10} + 122875456 q^{11} + 259564864 q^{12} - 90911522 q^{13} - 1195428552 q^{14} - 274843750 q^{15} - 1314428032 q^{16} - 3868973426 q^{17} + 905373668 q^{18} + 3670884220 q^{19} - 1592500000 q^{20} + 9596808996 q^{21} + 20569860608 q^{22} + 26698058238 q^{23} + 7659524160 q^{24} + 18310546875 q^{25} - 27015047384 q^{26} - 99092472220 q^{27} - 210410855488 q^{28} + 145544932730 q^{29} - 162123125000 q^{30} - 25873382644 q^{31} - 531675479296 q^{32} + 851520900736 q^{33} + 208184081768 q^{34} + 70719218750 q^{35} + 578919966368 q^{36} + 419480249934 q^{37} + 205247686480 q^{38} + 2390867460332 q^{39} - 1302712500000 q^{40} + 274005770306 q^{41} - 9314366945232 q^{42} + 2350065869158 q^{43} + 3324410490368 q^{44} - 3743088359375 q^{45} - 2445701814744 q^{46} - 8891070209486 q^{47} + 968269957888 q^{48} + 17603715811879 q^{49} + 24414062500 q^{50} + 11276036492236 q^{51} - 7757281361856 q^{52} + 8749242811318 q^{53} + 24761205955120 q^{54} - 9599645000000 q^{55} + 1555780658880 q^{56} - 41669191785640 q^{57} + 53844260003320 q^{58} - 14173516437140 q^{59} - 20278505000000 q^{60} - 38066837721794 q^{61} - 53610636798192 q^{62} - 97119517183302 q^{63} + 12081926129664 q^{64} + 7102462656250 q^{65} + 93794721354752 q^{66} + 144391638065474 q^{67} - 9729892224448 q^{68} - 83804251999188 q^{69} + 93392855625000 q^{70} - 126512337318844 q^{71} + 262100048315040 q^{72} + 199804772078038 q^{73} - 103551095018392 q^{74} + 21472167968750 q^{75} + 108919950456960 q^{76} - 24166822365312 q^{77} - 612963962524784 q^{78} - 73797562093720 q^{79} + 102689690000000 q^{80} - 252524358845777 q^{81} + 452714654584408 q^{82} - 219109046205402 q^{83} - 12\!\cdots\!12 q^{84}+ \cdots + 12\!\cdots\!12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 1972x + 21070 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{2} + 78\nu + 2597 ) / 21 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{2} + 42\nu - 2645 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 7\beta _1 + 16 ) / 40 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 39\beta_{2} - 147\beta _1 + 52564 ) / 40 \) 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
11.3631
38.1900
−48.5531
−152.575 −6379.22 −9488.76 −78125.0 973313. −1.77538e6 6.44734e6 2.63456e7 1.19200e7
1.2 −125.613 4146.67 −16989.4 −78125.0 −520875. 4.19779e6 6.25017e6 2.84597e6 9.81350e6
1.3 282.188 5750.55 46862.2 −78125.0 1.62274e6 −3.32761e6 3.97721e6 1.87200e7 −2.20460e7
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)

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 5.16.a.b 3
3.b odd 2 1 45.16.a.f 3
4.b odd 2 1 80.16.a.g 3
5.b even 2 1 25.16.a.c 3
5.c odd 4 2 25.16.b.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.16.a.b 3 1.a even 1 1 trivial
25.16.a.c 3 5.b even 2 1
25.16.b.c 6 5.c odd 4 2
45.16.a.f 3 3.b odd 2 1
80.16.a.g 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 4T_{2}^{2} - 59336T_{2} - 5408256 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(5))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 4 T^{2} - 59336 T - 5408256 \) Copy content Toggle raw display
$3$ \( T^{3} - 3518 T^{2} + \cdots + 152116768152 \) Copy content Toggle raw display
$5$ \( (T + 78125)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 905206 T^{2} + \cdots - 24\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{3} - 122875456 T^{2} + \cdots + 92\!\cdots\!92 \) Copy content Toggle raw display
$13$ \( T^{3} + 90911522 T^{2} + \cdots - 95\!\cdots\!08 \) Copy content Toggle raw display
$17$ \( T^{3} + 3868973426 T^{2} + \cdots + 65\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{3} - 3670884220 T^{2} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{3} - 26698058238 T^{2} + \cdots - 27\!\cdots\!68 \) Copy content Toggle raw display
$29$ \( T^{3} - 145544932730 T^{2} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{3} + 25873382644 T^{2} + \cdots - 90\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( T^{3} - 419480249934 T^{2} + \cdots + 70\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{3} - 274005770306 T^{2} + \cdots - 22\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( T^{3} - 2350065869158 T^{2} + \cdots + 82\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{3} + 8891070209486 T^{2} + \cdots - 20\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{3} - 8749242811318 T^{2} + \cdots + 14\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{3} + 14173516437140 T^{2} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} + 38066837721794 T^{2} + \cdots - 18\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{3} - 144391638065474 T^{2} + \cdots - 97\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{3} + 126512337318844 T^{2} + \cdots + 13\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( T^{3} - 199804772078038 T^{2} + \cdots + 74\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{3} + 73797562093720 T^{2} + \cdots - 65\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} + 219109046205402 T^{2} + \cdots + 10\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{3} + 360765737744610 T^{2} + \cdots - 25\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + 509209931654586 T^{2} + \cdots - 21\!\cdots\!96 \) Copy content Toggle raw display
show more
show less