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Properties

Label 72.8.d.d
Level $72$
Weight $8$
Character orbit 72.d
Analytic conductor $22.492$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 72.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.4917218349\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \( x^{14} - 6 x^{13} - 52 x^{12} + 300 x^{11} - 1005 x^{10} - 23250 x^{9} + 349930 x^{8} + 2867784 x^{7} - 20463993 x^{6} - 78987210 x^{5} + 94608296 x^{4} + \cdots + 3813237677250 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{42}\cdot 3^{18} \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} + ( - \beta_{2} + \beta_1 - 15) q^{4} + (\beta_{3} - \beta_{2} - 3 \beta_1) q^{5} + (\beta_{8} - 3 \beta_{2} - 4 \beta_1 + 98) q^{7} + ( - \beta_{13} - \beta_{9} - \beta_{8} - \beta_{4} - \beta_{2} - 15 \beta_1 + 30) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{2} + ( - \beta_{2} + \beta_1 - 15) q^{4} + (\beta_{3} - \beta_{2} - 3 \beta_1) q^{5} + (\beta_{8} - 3 \beta_{2} - 4 \beta_1 + 98) q^{7} + ( - \beta_{13} - \beta_{9} - \beta_{8} - \beta_{4} - \beta_{2} - 15 \beta_1 + 30) q^{8} + ( - \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \cdots + 358) q^{10}+ \cdots + (2232 \beta_{13} - 1560 \beta_{12} + 4056 \beta_{11} + \cdots - 3783131) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 14 q^{2} - 208 q^{4} + 1372 q^{7} + 428 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 14 q^{2} - 208 q^{4} + 1372 q^{7} + 428 q^{8} + 5020 q^{10} - 4636 q^{14} - 43336 q^{16} + 2908 q^{17} - 175096 q^{20} - 128480 q^{22} + 143416 q^{23} - 202626 q^{25} - 424984 q^{26} + 567520 q^{28} - 89468 q^{31} + 893944 q^{32} + 1109820 q^{34} + 823816 q^{38} - 860888 q^{40} + 441284 q^{41} - 1275264 q^{44} - 2167992 q^{46} + 1056408 q^{47} + 2158134 q^{49} - 324610 q^{50} - 2059248 q^{52} + 4757504 q^{55} - 1643704 q^{56} - 5494676 q^{58} - 5767172 q^{62} + 3852224 q^{64} + 2520464 q^{65} + 3735840 q^{68} + 12890312 q^{70} - 5172696 q^{71} - 5446196 q^{73} + 6468800 q^{74} - 9084624 q^{76} - 14373548 q^{79} - 14369088 q^{80} - 7935708 q^{82} - 4738312 q^{86} + 12598720 q^{88} + 11952620 q^{89} - 11004480 q^{92} - 15440088 q^{94} + 69327376 q^{95} + 133732 q^{97} - 53030538 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 6 x^{13} - 52 x^{12} + 300 x^{11} - 1005 x^{10} - 23250 x^{9} + 349930 x^{8} + 2867784 x^{7} - 20463993 x^{6} - 78987210 x^{5} + 94608296 x^{4} + \cdots + 3813237677250 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 21691079 \nu^{13} - 270184387 \nu^{12} + 782417201 \nu^{11} + 1515938117 \nu^{10} - 53036710294 \nu^{9} - 206804744724 \nu^{8} + \cdots + 95\!\cdots\!30 ) / 41\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 166890813 \nu^{13} + 2670091729 \nu^{12} - 17159192555 \nu^{11} + 71557375481 \nu^{10} + 211250383698 \nu^{9} + \cdots - 94\!\cdots\!82 ) / 41\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 492629525 \nu^{13} - 5330686601 \nu^{12} + 12047622611 \nu^{11} - 39149525617 \nu^{10} + 861347903166 \nu^{9} + \cdots + 29\!\cdots\!38 ) / 41\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 26782385 \nu^{13} + 628599829 \nu^{12} - 4700265751 \nu^{11} + 28473483741 \nu^{10} - 108938811238 \nu^{9} + \cdots - 53\!\cdots\!22 ) / 16\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 136458837 \nu^{13} + 1839828681 \nu^{12} - 4871646099 \nu^{11} - 27635439951 \nu^{10} + 295776184770 \nu^{9} + \cdots - 55\!\cdots\!34 ) / 69\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 907013959 \nu^{13} + 7107775043 \nu^{12} + 14703813455 \nu^{11} - 626741066437 \nu^{10} + 4316482827926 \nu^{9} + \cdots - 59\!\cdots\!14 ) / 41\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1068982365 \nu^{13} + 23519404401 \nu^{12} - 172128800203 \nu^{11} + 271435164313 \nu^{10} + 4040669859986 \nu^{9} + \cdots - 41\!\cdots\!02 ) / 41\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1197811243 \nu^{13} + 11406180407 \nu^{12} - 29346275821 \nu^{11} + 4316955983 \nu^{10} + 2168133588542 \nu^{9} + \cdots - 51\!\cdots\!50 ) / 41\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 149795513 \nu^{13} + 1775346621 \nu^{12} - 1910006095 \nu^{11} - 26079690555 \nu^{10} + 399012730218 \nu^{9} + \cdots - 75\!\cdots\!38 ) / 52\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 172051509 \nu^{13} - 2404852809 \nu^{12} + 7188831539 \nu^{11} + 65785727119 \nu^{10} - 1183451299138 \nu^{9} + \cdots + 88\!\cdots\!18 ) / 52\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1513426675 \nu^{13} + 17693523359 \nu^{12} + 18463120539 \nu^{11} - 613756439561 \nu^{10} + 1692810357390 \nu^{9} + \cdots - 10\!\cdots\!22 ) / 41\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 59960501 \nu^{13} + 905146077 \nu^{12} - 3961879171 \nu^{11} + 12082154925 \nu^{10} - 16745842566 \nu^{9} + \cdots - 44\!\cdots\!10 ) / 13\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2715351137 \nu^{13} - 33639911173 \nu^{12} + 83405896423 \nu^{11} + 519337332915 \nu^{10} - 9612896802170 \nu^{9} + \cdots + 11\!\cdots\!06 ) / 41\!\cdots\!72 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( - 9 \beta_{13} + 9 \beta_{12} - 9 \beta_{10} - 36 \beta_{9} - 16 \beta_{7} - 7 \beta_{6} + 18 \beta_{5} + 2 \beta_{4} + 13 \beta_{3} + 7 \beta_{2} - 74 \beta _1 + 2226 ) / 5184 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 27 \beta_{13} + 135 \beta_{12} - 72 \beta_{11} + 81 \beta_{10} + 9 \beta_{9} - 32 \beta_{7} - 5 \beta_{6} + 214 \beta_{5} - 86 \beta_{4} + 323 \beta_{3} - 103 \beta_{2} + 1058 \beta _1 + 51810 ) / 5184 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 135 \beta_{13} + 891 \beta_{12} - 792 \beta_{11} - 243 \beta_{10} - 387 \beta_{9} + 112 \beta_{7} - 1481 \beta_{6} + 1614 \beta_{5} - 1346 \beta_{4} - 1297 \beta_{3} - 5971 \beta_{2} - 20686 \beta _1 + 91374 ) / 5184 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4533 \beta_{13} + 1551 \beta_{12} + 168 \beta_{11} - 1767 \beta_{10} - 393 \beta_{9} + 2592 \beta_{8} - 672 \beta_{7} + 267 \beta_{6} + 3638 \beta_{5} - 54 \beta_{4} + 6363 \beta_{3} - 5343 \beta_{2} + \cdots + 1360866 ) / 1728 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 28233 \beta_{13} + 35811 \beta_{12} + 79128 \beta_{11} - 13131 \beta_{10} - 67005 \beta_{9} + 137376 \beta_{8} + 18672 \beta_{7} - 57 \beta_{6} - 99794 \beta_{5} + 20958 \beta_{4} + \cdots + 59148270 ) / 5184 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 375219 \beta_{13} + 427977 \beta_{12} + 10008 \beta_{11} - 224721 \beta_{10} + 188289 \beta_{9} + 1218240 \beta_{8} - 84064 \beta_{7} + 441005 \beta_{6} + 193274 \beta_{5} + \cdots - 134979234 ) / 5184 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2045673 \beta_{13} + 1855827 \beta_{12} - 165384 \beta_{11} + 2623365 \beta_{10} + 3234879 \beta_{9} + 8999424 \beta_{8} + 2969328 \beta_{7} + 1948359 \beta_{6} + \cdots - 5879743938 ) / 5184 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 8231313 \beta_{13} + 2397795 \beta_{12} - 1354488 \beta_{11} + 990309 \beta_{10} + 17113947 \beta_{9} + 11042784 \beta_{8} + 13734368 \beta_{7} + 8425967 \beta_{6} + \cdots - 5898236454 ) / 1728 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 154797201 \beta_{13} - 42135381 \beta_{12} + 133399224 \beta_{11} - 169790643 \beta_{10} + 350444007 \beta_{9} + 255817440 \beta_{8} + 143369456 \beta_{7} + \cdots - 89174752386 ) / 5184 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 90051723 \beta_{13} - 30935439 \beta_{12} + 1785733272 \beta_{11} - 219360825 \beta_{10} + 5588196705 \beta_{9} + 2239042176 \beta_{8} - 311056736 \beta_{7} + \cdots + 741190189038 ) / 5184 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 16746813639 \beta_{13} + 5288240547 \beta_{12} + 8044070328 \beta_{11} + 452918709 \beta_{10} + 51171467895 \beta_{9} - 13418955072 \beta_{8} + \cdots - 4174643792514 ) / 5184 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 14070738661 \beta_{13} + 6573049441 \beta_{12} + 5758500056 \beta_{11} + 5460206711 \beta_{10} + 35427013849 \beta_{9} - 51469850016 \beta_{8} + \cdots - 4268598731138 ) / 576 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1398515170815 \beta_{13} + 972388573371 \beta_{12} + 1256975130168 \beta_{11} - 339276246147 \beta_{10} + 759639983895 \beta_{9} + \cdots + 230969302740414 ) / 5184 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
8.85262 + 1.52851i
8.85262 − 1.52851i
1.24645 − 7.99620i
1.24645 + 7.99620i
7.97707 − 3.91414i
7.97707 + 3.91414i
−5.80663 − 4.20354i
−5.80663 + 4.20354i
2.71713 − 7.81354i
2.71713 + 7.81354i
−6.99438 − 0.299706i
−6.99438 + 0.299706i
−4.99225 + 5.30027i
−4.99225 − 5.30027i
−9.38113 − 6.32411i 0 48.0111 + 118.655i 425.308i 0 1664.03 299.987 − 1416.74i 0 2689.70 − 3989.87i
37.2 −9.38113 + 6.32411i 0 48.0111 − 118.655i − 425.308i 0 1664.03 299.987 + 1416.74i 0 2689.70 + 3989.87i
37.3 −8.24265 − 7.74976i 0 7.88255 + 127.757i − 76.0929i 0 −222.735 925.113 − 1114.14i 0 −589.701 + 627.207i
37.4 −8.24265 + 7.74976i 0 7.88255 − 127.757i 76.0929i 0 −222.735 925.113 + 1114.14i 0 −589.701 − 627.207i
37.5 −3.06293 − 10.8912i 0 −109.237 + 66.7181i 23.0228i 0 −1547.56 1061.23 + 985.368i 0 250.747 − 70.5175i
37.6 −3.06293 + 10.8912i 0 −109.237 − 66.7181i − 23.0228i 0 −1547.56 1061.23 − 985.368i 0 250.747 + 70.5175i
37.7 2.60309 − 11.0102i 0 −114.448 − 57.3210i 468.400i 0 81.2421 −929.033 + 1110.88i 0 5157.17 + 1219.29i
37.8 2.60309 + 11.0102i 0 −114.448 + 57.3210i − 468.400i 0 81.2421 −929.033 − 1110.88i 0 5157.17 − 1219.29i
37.9 6.09641 − 9.53067i 0 −53.6675 − 116.206i − 137.155i 0 808.153 −1434.70 − 196.951i 0 −1307.18 − 836.155i
37.10 6.09641 + 9.53067i 0 −53.6675 + 116.206i 137.155i 0 808.153 −1434.70 + 196.951i 0 −1307.18 + 836.155i
37.11 7.69468 − 8.29409i 0 −9.58387 − 127.641i − 455.347i 0 −743.502 −1132.41 − 902.665i 0 −3776.69 − 3503.75i
37.12 7.69468 + 8.29409i 0 −9.58387 + 127.641i 455.347i 0 −743.502 −1132.41 + 902.665i 0 −3776.69 + 3503.75i
37.13 11.2925 − 0.691979i 0 127.042 − 15.6284i 124.215i 0 646.373 1423.81 − 264.394i 0 85.9543 + 1402.70i
37.14 11.2925 + 0.691979i 0 127.042 + 15.6284i − 124.215i 0 646.373 1423.81 + 264.394i 0 85.9543 − 1402.70i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.8.d.d 14
3.b odd 2 1 24.8.d.a ✓ 14
4.b odd 2 1 288.8.d.d 14
8.b even 2 1 inner 72.8.d.d 14
8.d odd 2 1 288.8.d.d 14
12.b even 2 1 96.8.d.a 14
24.f even 2 1 96.8.d.a 14
24.h odd 2 1 24.8.d.a ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.d.a ✓ 14 3.b odd 2 1
24.8.d.a ✓ 14 24.h odd 2 1
72.8.d.d 14 1.a even 1 1 trivial
72.8.d.d 14 8.b even 2 1 inner
96.8.d.a 14 12.b even 2 1
96.8.d.a 14 24.f even 2 1
288.8.d.d 14 4.b odd 2 1
288.8.d.d 14 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} + 648188 T_{5}^{12} + 147837846096 T_{5}^{10} + \cdots + 73\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 562949953421312 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 648188 T^{12} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{7} - 686 T^{6} + \cdots + 18\!\cdots\!56)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + 146207744 T^{12} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + 491467248 T^{12} + \cdots + 82\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( (T^{7} - 1454 T^{6} + \cdots - 20\!\cdots\!36)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + 5540192496 T^{12} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( (T^{7} - 71708 T^{6} + \cdots + 23\!\cdots\!68)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + 109525673372 T^{12} + \cdots + 30\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( (T^{7} + 44734 T^{6} + \cdots - 33\!\cdots\!12)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + 473007845568 T^{12} + \cdots + 88\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{7} - 220642 T^{6} + \cdots - 38\!\cdots\!28)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + 2638324171248 T^{12} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( (T^{7} - 528204 T^{6} + \cdots - 43\!\cdots\!56)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + 9574301782364 T^{12} + \cdots + 98\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{14} + 25057896090608 T^{12} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{14} + 17284536099072 T^{12} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{14} + 41923367956080 T^{12} + \cdots + 45\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{7} + 2586348 T^{6} + \cdots + 12\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} + 2723098 T^{6} + \cdots - 49\!\cdots\!48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} + 7186774 T^{6} + \cdots + 64\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + 231376940860160 T^{12} + \cdots + 31\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{7} - 5976310 T^{6} + \cdots + 78\!\cdots\!80)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} - 66866 T^{6} + \cdots - 18\!\cdots\!04)^{2} \) Copy content Toggle raw display
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