Properties

Label 160.6.a.f
Level 160
Weight 6
Character orbit 160.a
Self dual yes
Analytic conductor 25.661
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.6614111701\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.39180.1
Defining polynomial: \(x^{3} - x^{2} - 36 x - 24\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q + ( -3 - \beta_{1} ) q^{3} -25 q^{5} + ( 2 + \beta_{1} - \beta_{2} ) q^{7} + ( 151 + 12 \beta_{1} + 2 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -3 - \beta_{1} ) q^{3} -25 q^{5} + ( 2 + \beta_{1} - \beta_{2} ) q^{7} + ( 151 + 12 \beta_{1} + 2 \beta_{2} ) q^{9} + ( 131 + 3 \beta_{2} ) q^{11} + ( -124 + 20 \beta_{1} - 2 \beta_{2} ) q^{13} + ( 75 + 25 \beta_{1} ) q^{15} + ( 368 + 60 \beta_{1} - 6 \beta_{2} ) q^{17} + ( -1047 - 50 \beta_{1} - \beta_{2} ) q^{19} + ( -296 + 72 \beta_{1} - 4 \beta_{2} ) q^{21} + ( -2054 + 43 \beta_{1} - 7 \beta_{2} ) q^{23} + 625 q^{25} + ( -4534 - 182 \beta_{1} - 20 \beta_{2} ) q^{27} + ( 138 - 40 \beta_{1} + 52 \beta_{2} ) q^{29} + ( -1196 + 270 \beta_{1} + 42 \beta_{2} ) q^{31} + ( -678 - 380 \beta_{1} + 6 \beta_{2} ) q^{33} + ( -50 - 25 \beta_{1} + 25 \beta_{2} ) q^{35} + ( -3762 - 200 \beta_{1} - 76 \beta_{2} ) q^{37} + ( -7138 + 110 \beta_{1} - 44 \beta_{2} ) q^{39} + ( 4120 + 68 \beta_{1} + 22 \beta_{2} ) q^{41} + ( -8713 - 221 \beta_{1} + 14 \beta_{2} ) q^{43} + ( -3775 - 300 \beta_{1} - 50 \beta_{2} ) q^{45} + ( -12288 + 77 \beta_{1} + 25 \beta_{2} ) q^{47} + ( -1153 - 316 \beta_{1} - 74 \beta_{2} ) q^{49} + ( -23634 - 410 \beta_{1} - 132 \beta_{2} ) q^{51} + ( -6712 - 1140 \beta_{1} + 114 \beta_{2} ) q^{53} + ( -3275 - 75 \beta_{2} ) q^{55} + ( 22486 + 1580 \beta_{1} + 98 \beta_{2} ) q^{57} + ( -11765 + 70 \beta_{1} + 185 \beta_{2} ) q^{59} + ( -8718 + 1984 \beta_{1} + 32 \beta_{2} ) q^{61} + ( -26938 - 263 \beta_{1} + 91 \beta_{2} ) q^{63} + ( 3100 - 500 \beta_{1} + 50 \beta_{2} ) q^{65} + ( 2381 + 2345 \beta_{1} + 82 \beta_{2} ) q^{67} + ( -9728 + 2248 \beta_{1} - 100 \beta_{2} ) q^{69} + ( -29002 - 1030 \beta_{1} - 56 \beta_{2} ) q^{71} + ( 23080 - 2340 \beta_{1} - 150 \beta_{2} ) q^{73} + ( -1875 - 625 \beta_{1} ) q^{75} + ( -45818 + 860 \beta_{1} + 106 \beta_{2} ) q^{77} + ( 31272 - 700 \beta_{1} - 164 \beta_{2} ) q^{79} + ( 48879 + 4916 \beta_{1} - 162 \beta_{2} ) q^{81} + ( -10935 + 2923 \beta_{1} - 376 \beta_{2} ) q^{83} + ( -9200 - 1500 \beta_{1} + 150 \beta_{2} ) q^{85} + ( 10046 - 4094 \beta_{1} + 184 \beta_{2} ) q^{87} + ( 57774 - 600 \beta_{1} - 36 \beta_{2} ) q^{89} + ( 36272 - 2110 \beta_{1} + 106 \beta_{2} ) q^{91} + ( -104352 - 4720 \beta_{1} - 456 \beta_{2} ) q^{93} + ( 26175 + 1250 \beta_{1} + 25 \beta_{2} ) q^{95} + ( 57300 - 1420 \beta_{1} + 430 \beta_{2} ) q^{97} + ( 115931 + 3600 \beta_{1} + 43 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 10q^{3} - 75q^{5} + 6q^{7} + 467q^{9} + O(q^{10}) \) \( 3q - 10q^{3} - 75q^{5} + 6q^{7} + 467q^{9} + 396q^{11} - 354q^{13} + 250q^{15} + 1158q^{17} - 3192q^{19} - 820q^{21} - 6126q^{23} + 1875q^{25} - 13804q^{27} + 426q^{29} - 3276q^{31} - 2408q^{33} - 150q^{35} - 11562q^{37} - 21348q^{39} + 12450q^{41} - 26346q^{43} - 11675q^{45} - 36762q^{47} - 3849q^{49} - 71444q^{51} - 21162q^{53} - 9900q^{55} + 69136q^{57} - 35040q^{59} - 24138q^{61} - 80986q^{63} + 8850q^{65} + 9570q^{67} - 27036q^{69} - 88092q^{71} + 66750q^{73} - 6250q^{75} - 136488q^{77} + 92952q^{79} + 151391q^{81} - 30258q^{83} - 28950q^{85} + 26228q^{87} + 172686q^{89} + 106812q^{91} - 318232q^{93} + 79800q^{95} + 170910q^{97} + 351436q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 36 x - 24\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 4 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 8 \nu^{2} - 16 \nu - 189 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 4 \beta_{1} + 193\)\()/8\)

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
6.80681
−0.688934
−5.11788
0 −29.2272 0 −25.0000 0 −44.5253 0 611.232 0
1.2 0 0.755735 0 −25.0000 0 172.424 0 −242.429 0
1.3 0 18.4715 0 −25.0000 0 −121.899 0 98.1968 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 160.6.a.f 3
4.b odd 2 1 160.6.a.g yes 3
5.b even 2 1 800.6.a.o 3
5.c odd 4 2 800.6.c.k 6
8.b even 2 1 320.6.a.y 3
8.d odd 2 1 320.6.a.x 3
20.d odd 2 1 800.6.a.n 3
20.e even 4 2 800.6.c.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.f 3 1.a even 1 1 trivial
160.6.a.g yes 3 4.b odd 2 1
320.6.a.x 3 8.d odd 2 1
320.6.a.y 3 8.b even 2 1
800.6.a.n 3 20.d odd 2 1
800.6.a.o 3 5.b even 2 1
800.6.c.j 6 20.e even 4 2
800.6.c.k 6 5.c odd 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + 10 T_{3}^{2} - 548 T_{3} + 408 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(160))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 10 T + 181 T^{2} + 5268 T^{3} + 43983 T^{4} + 590490 T^{5} + 14348907 T^{6} \)
$5$ \( ( 1 + 25 T )^{3} \)
$7$ \( 1 - 6 T + 27153 T^{2} - 1137532 T^{3} + 456360471 T^{4} - 1694851494 T^{5} + 4747561509943 T^{6} \)
$11$ \( 1 - 396 T + 327873 T^{2} - 67617992 T^{3} + 52804274523 T^{4} - 10271220141996 T^{5} + 4177248169415651 T^{6} \)
$13$ \( 1 + 354 T + 845379 T^{2} + 291738444 T^{3} + 313883305047 T^{4} + 48801906114546 T^{5} + 51185893014090757 T^{6} \)
$17$ \( 1 - 1158 T + 1914111 T^{2} - 544550612 T^{3} + 2717763902127 T^{4} - 2334520936719942 T^{5} + 2862423051509815793 T^{6} \)
$19$ \( 1 + 3192 T + 9330057 T^{2} + 15933739216 T^{3} + 23102144807643 T^{4} + 19570363494900792 T^{5} + 15181127029874798299 T^{6} \)
$23$ \( 1 + 6126 T + 29722593 T^{2} + 84141222540 T^{3} + 191304803397399 T^{4} + 253778807694813774 T^{5} + \)\(26\!\cdots\!07\)\( T^{6} \)
$29$ \( 1 - 426 T - 939693 T^{2} + 141773503364 T^{3} - 19274183137257 T^{4} - 179221281385885626 T^{5} + \)\(86\!\cdots\!49\)\( T^{6} \)
$31$ \( 1 + 3276 T + 2292813 T^{2} - 41639420248 T^{3} + 65641289591763 T^{4} + 2685102268149104076 T^{5} + \)\(23\!\cdots\!51\)\( T^{6} \)
$37$ \( 1 + 11562 T + 90623691 T^{2} + 525372493468 T^{3} + 6284205331885287 T^{4} + 55596852513895170138 T^{5} + \)\(33\!\cdots\!93\)\( T^{6} \)
$41$ \( 1 - 12450 T + 384843783 T^{2} - 2886023186300 T^{3} + 44586538676848383 T^{4} - \)\(16\!\cdots\!50\)\( T^{5} + \)\(15\!\cdots\!01\)\( T^{6} \)
$43$ \( 1 + 26346 T + 640605069 T^{2} + 8098987233524 T^{3} + 94174353771597567 T^{4} + \)\(56\!\cdots\!54\)\( T^{5} + \)\(31\!\cdots\!07\)\( T^{6} \)
$47$ \( 1 + 36762 T + 1119958377 T^{2} + 18490559326820 T^{3} + 256856861812773639 T^{4} + \)\(19\!\cdots\!38\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} \)
$53$ \( 1 + 21162 T + 395789499 T^{2} - 881498066468 T^{3} + 165517384658528007 T^{4} + \)\(37\!\cdots\!38\)\( T^{5} + \)\(73\!\cdots\!57\)\( T^{6} \)
$59$ \( 1 + 35040 T + 1757220897 T^{2} + 50989349593920 T^{3} + 1256279917975876203 T^{4} + \)\(17\!\cdots\!40\)\( T^{5} + \)\(36\!\cdots\!99\)\( T^{6} \)
$61$ \( 1 + 24138 T + 393086643 T^{2} - 6904061162564 T^{3} + 331999524650307543 T^{4} + \)\(17\!\cdots\!38\)\( T^{5} + \)\(60\!\cdots\!01\)\( T^{6} \)
$67$ \( 1 - 9570 T + 659335509 T^{2} - 83080838420484 T^{3} + 890185424637524463 T^{4} - \)\(17\!\cdots\!30\)\( T^{5} + \)\(24\!\cdots\!43\)\( T^{6} \)
$71$ \( 1 + 88092 T + 7289446053 T^{2} + 329541325840584 T^{3} + 13151832521353701603 T^{4} + \)\(28\!\cdots\!92\)\( T^{5} + \)\(58\!\cdots\!51\)\( T^{6} \)
$73$ \( 1 - 66750 T + 3875077479 T^{2} - 111360074258500 T^{3} + 8033313042388954047 T^{4} - \)\(28\!\cdots\!50\)\( T^{5} + \)\(89\!\cdots\!57\)\( T^{6} \)
$79$ \( 1 - 92952 T + 11164448877 T^{2} - 573846024396496 T^{3} + 34353638858281213923 T^{4} - \)\(88\!\cdots\!52\)\( T^{5} + \)\(29\!\cdots\!99\)\( T^{6} \)
$83$ \( 1 + 30258 T + 4293702405 T^{2} + 426012342532708 T^{3} + 16913068282241846415 T^{4} + \)\(46\!\cdots\!42\)\( T^{5} + \)\(61\!\cdots\!07\)\( T^{6} \)
$89$ \( 1 - 172686 T + 26445328791 T^{2} - 2103593815517412 T^{3} + \)\(14\!\cdots\!59\)\( T^{4} - \)\(53\!\cdots\!86\)\( T^{5} + \)\(17\!\cdots\!49\)\( T^{6} \)
$97$ \( 1 - 170910 T + 30283966671 T^{2} - 2852314667192740 T^{3} + \)\(26\!\cdots\!47\)\( T^{4} - \)\(12\!\cdots\!90\)\( T^{5} + \)\(63\!\cdots\!93\)\( T^{6} \)
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