Properties

Label 72.5.b.d
Level $72$
Weight $5$
Character orbit 72.b
Analytic conductor $7.443$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,5,Mod(19,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.19");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 72.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.44263734204\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 20x^{6} - 6x^{5} + 121x^{4} + 18x^{3} - 114x^{2} + 72x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} + (\beta_{5} + \beta_1 + 1) q^{4} + ( - \beta_{3} + \beta_{2} + 3 \beta_1) q^{5} + (\beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{7} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 23) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{2} + (\beta_{5} + \beta_1 + 1) q^{4} + ( - \beta_{3} + \beta_{2} + 3 \beta_1) q^{5} + (\beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{7} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 23) q^{8} + (\beta_{7} - 5 \beta_{6} + 5 \beta_{5} + 2 \beta_{2} + 4 \beta_1 - 42) q^{10} + (4 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} + 8 \beta_1 - 24) q^{11} + ( - 2 \beta_{7} - 12 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + \cdots + 4) q^{13}+ \cdots + ( - 112 \beta_{7} + 112 \beta_{6} - 368 \beta_{5} - 224 \beta_{4} + \cdots - 5915) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{2} + 8 q^{4} + 180 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{2} + 8 q^{4} + 180 q^{8} - 324 q^{10} - 192 q^{11} - 420 q^{14} - 712 q^{16} - 240 q^{17} - 704 q^{19} - 168 q^{20} + 592 q^{22} - 664 q^{25} - 1008 q^{26} - 528 q^{28} - 3624 q^{32} + 2716 q^{34} + 5184 q^{35} + 6360 q^{38} + 408 q^{40} - 720 q^{41} + 10048 q^{43} + 6720 q^{44} + 2616 q^{46} - 1240 q^{49} - 5394 q^{50} + 2448 q^{52} - 7512 q^{56} - 10740 q^{58} - 13056 q^{59} + 8724 q^{62} - 17632 q^{64} + 1344 q^{65} - 6656 q^{67} + 5616 q^{68} + 19800 q^{70} - 16880 q^{73} - 17400 q^{74} + 14320 q^{76} - 28512 q^{80} - 9740 q^{82} + 24000 q^{83} + 34344 q^{86} - 19616 q^{88} - 15600 q^{89} + 1344 q^{91} + 48096 q^{92} + 12120 q^{94} - 12176 q^{97} - 47778 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 20x^{6} - 6x^{5} + 121x^{4} + 18x^{3} - 114x^{2} + 72x + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -280\nu^{7} + 1019\nu^{6} - 5633\nu^{5} + 20827\nu^{4} - 39823\nu^{3} + 101396\nu^{2} + 61086\nu - 151668 ) / 32244 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 279\nu^{7} + 165\nu^{6} + 6371\nu^{5} + 2845\nu^{4} + 48212\nu^{3} + 31512\nu^{2} + 12660\nu + 24108 ) / 5374 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3503 \nu^{7} + 1511 \nu^{6} - 72229 \nu^{5} + 63997 \nu^{4} - 468590 \nu^{3} + 172202 \nu^{2} + 106164 \nu - 714696 ) / 64488 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2777 \nu^{7} - 920 \nu^{6} - 51808 \nu^{5} - 2998 \nu^{4} - 279485 \nu^{3} - 188894 \nu^{2} + 623634 \nu + 30708 ) / 32244 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1138 \nu^{7} - 1205 \nu^{6} + 22683 \nu^{5} - 33805 \nu^{4} + 145327 \nu^{3} - 164668 \nu^{2} - 126166 \nu + 86776 ) / 10748 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7859 \nu^{7} - 5349 \nu^{6} + 162511 \nu^{5} - 154275 \nu^{4} + 1119560 \nu^{3} - 619098 \nu^{2} + 96696 \nu + 543984 ) / 64488 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11979 \nu^{7} + 1537 \nu^{6} + 244245 \nu^{5} - 42709 \nu^{4} + 1439514 \nu^{3} + 205894 \nu^{2} - 1578300 \nu + 127056 ) / 64488 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 7\beta_{6} - 7\beta_{5} + \beta_{4} - 2\beta_{3} - 6\beta_{2} + \beta _1 + 4 ) / 72 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - \beta_{6} - 5\beta_{5} - 2\beta_{4} - 10\beta_{3} - 14\beta _1 - 124 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -37\beta_{7} - 25\beta_{6} + 61\beta_{5} - 25\beta_{4} + 20\beta_{3} + 60\beta_{2} + 83\beta _1 + 158 ) / 72 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 19\beta_{7} + 43\beta_{6} - 13\beta_{5} + 6\beta_{4} + 118\beta_{3} - 12\beta_{2} - 18\beta _1 + 976 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 541\beta_{7} - 119\beta_{6} - 529\beta_{5} + 589\beta_{4} - 452\beta_{3} - 312\beta_{2} - 1427\beta _1 - 6674 ) / 72 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -105\beta_{7} - 269\beta_{6} + 311\beta_{5} + 6\beta_{4} - 382\beta_{3} + 132\beta_{2} + 890\beta _1 - 680 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 5689 \beta_{7} + 7175 \beta_{6} + 2293 \beta_{5} - 8713 \beta_{4} + 8768 \beta_{3} - 1920 \beta_{2} + 18755 \beta _1 + 134474 ) / 72 \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.866025 + 0.339683i
−0.866025 0.339683i
0.866025 0.663939i
0.866025 + 0.663939i
0.866025 3.05404i
0.866025 + 3.05404i
−0.866025 + 3.62169i
−0.866025 3.62169i
−2.93333 2.71948i 0 1.20889 + 15.9543i 3.88698i 0 23.9200i 39.8412 50.0867i 0 10.5706 11.4018i
19.2 −2.93333 + 2.71948i 0 1.20889 15.9543i 3.88698i 0 23.9200i 39.8412 + 50.0867i 0 10.5706 + 11.4018i
19.3 −1.03857 3.86282i 0 −13.8428 + 8.02360i 34.2464i 0 9.22331i 45.3703 + 45.1390i 0 −132.288 + 35.5672i
19.4 −1.03857 + 3.86282i 0 −13.8428 8.02360i 34.2464i 0 9.22331i 45.3703 45.1390i 0 −132.288 35.5672i
19.5 3.40459 2.09970i 0 7.18250 14.2973i 13.7025i 0 88.3075i −5.56649 63.7575i 0 28.7711 + 46.6513i
19.6 3.40459 + 2.09970i 0 7.18250 + 14.2973i 13.7025i 0 88.3075i −5.56649 + 63.7575i 0 28.7711 46.6513i
19.7 3.56731 1.80951i 0 9.45137 12.9101i 38.1617i 0 42.0542i 10.3550 63.1567i 0 −69.0539 136.135i
19.8 3.56731 + 1.80951i 0 9.45137 + 12.9101i 38.1617i 0 42.0542i 10.3550 + 63.1567i 0 −69.0539 + 136.135i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.5.b.d 8
3.b odd 2 1 24.5.b.a 8
4.b odd 2 1 288.5.b.d 8
8.b even 2 1 288.5.b.d 8
8.d odd 2 1 inner 72.5.b.d 8
12.b even 2 1 96.5.b.a 8
24.f even 2 1 24.5.b.a 8
24.h odd 2 1 96.5.b.a 8
48.i odd 4 2 768.5.g.k 16
48.k even 4 2 768.5.g.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.5.b.a 8 3.b odd 2 1
24.5.b.a 8 24.f even 2 1
72.5.b.d 8 1.a even 1 1 trivial
72.5.b.d 8 8.d odd 2 1 inner
96.5.b.a 8 12.b even 2 1
96.5.b.a 8 24.h odd 2 1
288.5.b.d 8 4.b odd 2 1
288.5.b.d 8 8.b even 2 1
768.5.g.k 16 48.i odd 4 2
768.5.g.k 16 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 2832T_{5}^{6} + 2244192T_{5}^{4} + 353952000T_{5}^{2} + 4845166848 \) acting on \(S_{5}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 6 T^{7} + 14 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 2832 T^{6} + \cdots + 4845166848 \) Copy content Toggle raw display
$7$ \( T^{8} + 10224 T^{6} + \cdots + 671288262912 \) Copy content Toggle raw display
$11$ \( (T^{4} + 96 T^{3} - 40064 T^{2} + \cdots + 193957888)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 155136 T^{6} + \cdots + 56\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( (T^{4} + 120 T^{3} - 96872 T^{2} + \cdots - 1023349232)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 352 T^{3} - 220128 T^{2} + \cdots - 5712629504)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 1644480 T^{6} + \cdots + 64\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{8} + 4596240 T^{6} + \cdots + 22\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( T^{8} + 2201712 T^{6} + \cdots + 15\!\cdots\!12 \) Copy content Toggle raw display
$37$ \( T^{8} + 6796224 T^{6} + \cdots + 78\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( (T^{4} + 360 T^{3} + \cdots + 226033840912)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 5024 T^{3} + \cdots + 2240667904)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 14015424 T^{6} + \cdots + 13\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{8} + 27560592 T^{6} + \cdots + 86\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( (T^{4} + 6528 T^{3} + \cdots - 21853379868416)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 79948992 T^{6} + \cdots + 34\!\cdots\!12 \) Copy content Toggle raw display
$67$ \( (T^{4} + 3328 T^{3} + \cdots - 30681172039424)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 82031040 T^{6} + \cdots + 64\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( (T^{4} + 8440 T^{3} + \cdots - 35664142829552)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 162047472 T^{6} + \cdots + 43\!\cdots\!52 \) Copy content Toggle raw display
$83$ \( (T^{4} - 12000 T^{3} + \cdots + 15292937408512)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 7800 T^{3} + \cdots + 219207569912848)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 6088 T^{3} + \cdots - 29\!\cdots\!68)^{2} \) Copy content Toggle raw display
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