Properties

Label 80.12.a.j
Level 80
Weight 12
Character orbit 80.a
Self dual yes
Analytic conductor 61.467
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 80.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.4674544448\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{151}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 32\sqrt{151}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 110 + \beta ) q^{3} -3125 q^{5} + ( -28950 + 33 \beta ) q^{7} + ( -10423 + 220 \beta ) q^{9} +O(q^{10})\) \( q + ( 110 + \beta ) q^{3} -3125 q^{5} + ( -28950 + 33 \beta ) q^{7} + ( -10423 + 220 \beta ) q^{9} + ( 309088 - 1650 \beta ) q^{11} + ( 1707130 + 804 \beta ) q^{13} + ( -343750 - 3125 \beta ) q^{15} + ( 658970 - 7908 \beta ) q^{17} + ( -2662660 + 17160 \beta ) q^{19} + ( 1918092 - 25320 \beta ) q^{21} + ( -29471970 + 2091 \beta ) q^{23} + 9765625 q^{25} + ( 13384580 - 163370 \beta ) q^{27} + ( 47070190 - 143640 \beta ) q^{29} + ( -122271732 - 450450 \beta ) q^{31} + ( -221129920 + 127588 \beta ) q^{33} + ( 90468750 - 103125 \beta ) q^{35} + ( 10501610 - 1189608 \beta ) q^{37} + ( 312101996 + 1795570 \beta ) q^{39} + ( -372871658 + 1415700 \beta ) q^{41} + ( -314975050 - 869319 \beta ) q^{43} + ( 32571875 - 687500 \beta ) q^{45} + ( 701030770 - 1281567 \beta ) q^{47} + ( -970838707 - 1910700 \beta ) q^{49} + ( -1150279892 - 210910 \beta ) q^{51} + ( 569160290 + 11672124 \beta ) q^{53} + ( -965900000 + 5156250 \beta ) q^{55} + ( 2360455240 - 775060 \beta ) q^{57} + ( -3658757780 + 10988580 \beta ) q^{59} + ( -758212838 + 3348000 \beta ) q^{61} + ( 1424316090 - 6712959 \beta ) q^{63} + ( -5334781250 - 2512500 \beta ) q^{65} + ( -7867145070 - 5730717 \beta ) q^{67} + ( -2918597916 - 29241960 \beta ) q^{69} + ( -16469235772 - 3425250 \beta ) q^{71} + ( -14991424430 - 21226116 \beta ) q^{73} + ( 1074218750 + 9765625 \beta ) q^{75} + ( -17367374400 + 57967404 \beta ) q^{77} + ( 1651411560 - 35977260 \beta ) q^{79} + ( -21942215899 - 43558460 \beta ) q^{81} + ( -6649551210 - 68801139 \beta ) q^{83} + ( -2059281250 + 24712500 \beta ) q^{85} + ( -17032470460 + 31269790 \beta ) q^{87} + ( -6337385430 + 90955080 \beta ) q^{89} + ( -45318929532 + 33059490 \beta ) q^{91} + ( -83100271320 - 171821232 \beta ) q^{93} + ( 8320812500 - 53625000 \beta ) q^{95} + ( -1540351870 - 284116908 \beta ) q^{97} + ( -59350136224 + 85197310 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 220q^{3} - 6250q^{5} - 57900q^{7} - 20846q^{9} + O(q^{10}) \) \( 2q + 220q^{3} - 6250q^{5} - 57900q^{7} - 20846q^{9} + 618176q^{11} + 3414260q^{13} - 687500q^{15} + 1317940q^{17} - 5325320q^{19} + 3836184q^{21} - 58943940q^{23} + 19531250q^{25} + 26769160q^{27} + 94140380q^{29} - 244543464q^{31} - 442259840q^{33} + 180937500q^{35} + 21003220q^{37} + 624203992q^{39} - 745743316q^{41} - 629950100q^{43} + 65143750q^{45} + 1402061540q^{47} - 1941677414q^{49} - 2300559784q^{51} + 1138320580q^{53} - 1931800000q^{55} + 4720910480q^{57} - 7317515560q^{59} - 1516425676q^{61} + 2848632180q^{63} - 10669562500q^{65} - 15734290140q^{67} - 5837195832q^{69} - 32938471544q^{71} - 29982848860q^{73} + 2148437500q^{75} - 34734748800q^{77} + 3302823120q^{79} - 43884431798q^{81} - 13299102420q^{83} - 4118562500q^{85} - 34064940920q^{87} - 12674770860q^{89} - 90637859064q^{91} - 166200542640q^{93} + 16641625000q^{95} - 3080703740q^{97} - 118700272448q^{99} + O(q^{100}) \)

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
−12.2882
12.2882
0 −283.223 0 −3125.00 0 −41926.3 0 −96932.0 0
1.2 0 503.223 0 −3125.00 0 −15973.7 0 76086.0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 80.12.a.j 2
4.b odd 2 1 5.12.a.b 2
12.b even 2 1 45.12.a.d 2
20.d odd 2 1 25.12.a.c 2
20.e even 4 2 25.12.b.c 4
28.d even 2 1 245.12.a.b 2
60.h even 2 1 225.12.a.h 2
60.l odd 4 2 225.12.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.12.a.b 2 4.b odd 2 1
25.12.a.c 2 20.d odd 2 1
25.12.b.c 4 20.e even 4 2
45.12.a.d 2 12.b even 2 1
80.12.a.j 2 1.a even 1 1 trivial
225.12.a.h 2 60.h even 2 1
225.12.b.f 4 60.l odd 4 2
245.12.a.b 2 28.d even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 220 T_{3} - 142524 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(80))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 220 T + 211770 T^{2} - 38972340 T^{3} + 31381059609 T^{4} \)
$5$ \( ( 1 + 3125 T )^{2} \)
$7$ \( 1 + 57900 T + 4624370450 T^{2} + 114487218419700 T^{3} + 3909821048582988049 T^{4} \)
$11$ \( 1 - 618176 T + 245194892966 T^{2} - 176372827291625536 T^{3} + \)\(81\!\cdots\!21\)\( T^{4} \)
$13$ \( 1 - 3414260 T + 6398662197390 T^{2} - 6118901546944767620 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \)
$17$ \( 1 - 1317940 T + 59308395866630 T^{2} - 45168303019681836020 T^{3} + \)\(11\!\cdots\!89\)\( T^{4} \)
$19$ \( 1 + 5325320 T + 194538827137638 T^{2} + \)\(62\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!61\)\( T^{4} \)
$23$ \( 1 + 58943940 T + 2773540471931410 T^{2} + \)\(56\!\cdots\!80\)\( T^{3} + \)\(90\!\cdots\!29\)\( T^{4} \)
$29$ \( 1 - 94140380 T + 23426350431097358 T^{2} - \)\(11\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!41\)\( T^{4} \)
$31$ \( 1 + 244543464 T + 34393316207729486 T^{2} + \)\(62\!\cdots\!84\)\( T^{3} + \)\(64\!\cdots\!61\)\( T^{4} \)
$37$ \( 1 - 21003220 T + 137126715218410590 T^{2} - \)\(37\!\cdots\!60\)\( T^{3} + \)\(31\!\cdots\!69\)\( T^{4} \)
$41$ \( 1 + 745743316 T + 929792912462405846 T^{2} + \)\(41\!\cdots\!56\)\( T^{3} + \)\(30\!\cdots\!81\)\( T^{4} \)
$43$ \( 1 + 629950100 T + 1840945003918927050 T^{2} + \)\(58\!\cdots\!00\)\( T^{3} + \)\(86\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - 1402061540 T + 5181805952108806370 T^{2} - \)\(34\!\cdots\!20\)\( T^{3} + \)\(61\!\cdots\!09\)\( T^{4} \)
$53$ \( 1 - 1138320580 T - 2203723231625575330 T^{2} - \)\(10\!\cdots\!60\)\( T^{3} + \)\(85\!\cdots\!09\)\( T^{4} \)
$59$ \( 1 + 7317515560 T + 55027608950440780118 T^{2} + \)\(22\!\cdots\!40\)\( T^{3} + \)\(90\!\cdots\!81\)\( T^{4} \)
$61$ \( 1 + 1516425676 T + 85869525433683691566 T^{2} + \)\(65\!\cdots\!36\)\( T^{3} + \)\(18\!\cdots\!21\)\( T^{4} \)
$67$ \( 1 + 15734290140 T + \)\(30\!\cdots\!30\)\( T^{2} + \)\(19\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!89\)\( T^{4} \)
$71$ \( 1 + 32938471544 T + \)\(73\!\cdots\!26\)\( T^{2} + \)\(76\!\cdots\!24\)\( T^{3} + \)\(53\!\cdots\!41\)\( T^{4} \)
$73$ \( 1 + 29982848860 T + \)\(78\!\cdots\!10\)\( T^{2} + \)\(94\!\cdots\!20\)\( T^{3} + \)\(98\!\cdots\!29\)\( T^{4} \)
$79$ \( 1 - 3302823120 T + \)\(12\!\cdots\!58\)\( T^{2} - \)\(24\!\cdots\!80\)\( T^{3} + \)\(55\!\cdots\!41\)\( T^{4} \)
$83$ \( 1 + 13299102420 T + \)\(18\!\cdots\!30\)\( T^{2} + \)\(17\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!89\)\( T^{4} \)
$89$ \( 1 + 12674770860 T + \)\(43\!\cdots\!78\)\( T^{2} + \)\(35\!\cdots\!40\)\( T^{3} + \)\(77\!\cdots\!21\)\( T^{4} \)
$97$ \( 1 + 3080703740 T + \)\(18\!\cdots\!70\)\( T^{2} + \)\(22\!\cdots\!20\)\( T^{3} + \)\(51\!\cdots\!09\)\( T^{4} \)
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