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Properties

Label 72.5.h.a
Level $72$
Weight $5$
Character orbit 72.h
Analytic conductor $7.443$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,5,Mod(53,72)]
 
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([0, 1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.53");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 72.h (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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} + 27 x^{14} - 78 x^{13} + 215 x^{12} - 396 x^{11} + 900 x^{10} - 6864 x^{9} + \cdots + 20699712 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{14} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} - 2) q^{4} + \beta_{5} q^{5} + ( - \beta_{11} + \beta_{2}) q^{7} + ( - \beta_{7} - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} - 2) q^{4} + \beta_{5} q^{5} + ( - \beta_{11} + \beta_{2}) q^{7} + ( - \beta_{7} - 2 \beta_1) q^{8} + (\beta_{13} - \beta_{11} + \beta_{2} - 3) q^{10} + (\beta_{12} - \beta_{7} + \cdots + 2 \beta_1) q^{11}+ \cdots + (32 \beta_{15} + 192 \beta_{9} + \cdots + 467 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 28 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 28 q^{4} - 52 q^{10} - 72 q^{16} + 616 q^{22} + 2000 q^{25} - 2936 q^{28} - 256 q^{31} - 2972 q^{34} + 2544 q^{40} + 12088 q^{46} + 8496 q^{49} - 18536 q^{52} - 11776 q^{55} - 12140 q^{58} + 22112 q^{64} + 40968 q^{70} + 7360 q^{73} - 43600 q^{76} - 13056 q^{79} - 45772 q^{82} + 58048 q^{88} + 85800 q^{94} - 6528 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 27 x^{14} - 78 x^{13} + 215 x^{12} - 396 x^{11} + 900 x^{10} - 6864 x^{9} + \cdots + 20699712 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 72\!\cdots\!09 \nu^{15} + \cdots + 14\!\cdots\!80 ) / 18\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 72\!\cdots\!09 \nu^{15} + \cdots - 13\!\cdots\!68 ) / 46\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 28\!\cdots\!51 \nu^{15} + \cdots + 13\!\cdots\!08 ) / 75\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 41\!\cdots\!43 \nu^{15} + \cdots + 29\!\cdots\!60 ) / 10\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 39\!\cdots\!60 \nu^{15} + \cdots + 21\!\cdots\!64 ) / 86\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\!\cdots\!62 \nu^{15} + \cdots - 70\!\cdots\!16 ) / 37\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 59\!\cdots\!77 \nu^{15} + \cdots - 76\!\cdots\!88 ) / 93\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 13\!\cdots\!31 \nu^{15} + \cdots + 24\!\cdots\!84 ) / 20\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14\!\cdots\!93 \nu^{15} + \cdots + 17\!\cdots\!52 ) / 20\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 32\!\cdots\!23 \nu^{15} + \cdots - 15\!\cdots\!76 ) / 34\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 19\!\cdots\!07 \nu^{15} + \cdots - 11\!\cdots\!20 ) / 18\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 24\!\cdots\!92 \nu^{15} + \cdots - 15\!\cdots\!48 ) / 18\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 21\!\cdots\!25 \nu^{15} + \cdots - 15\!\cdots\!64 ) / 14\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 38\!\cdots\!73 \nu^{15} + \cdots - 32\!\cdots\!68 ) / 16\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 36\!\cdots\!67 \nu^{15} + \cdots + 38\!\cdots\!88 ) / 93\!\cdots\!12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + 2\beta_{9} - \beta_{8} + \beta_{3} + 2\beta_{2} + 2\beta _1 + 21 ) / 54 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 4\beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9 \beta_{14} - 9 \beta_{13} - 18 \beta_{11} - 10 \beta_{10} + 4 \beta_{9} + \beta_{8} - 54 \beta_{4} + \cdots - 255 ) / 108 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{14} - \beta_{13} - 2\beta_{11} + \beta_{8} + 8\beta_{7} + \beta_{3} - 22\beta_{2} - 16\beta _1 - 37 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 45 \beta_{15} - 108 \beta_{14} + 18 \beta_{13} + 180 \beta_{12} + 162 \beta_{11} - 17 \beta_{10} + \cdots - 1563 ) / 54 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 12 \beta_{15} - 37 \beta_{14} + 17 \beta_{13} - 48 \beta_{12} + 62 \beta_{11} + 10 \beta_{10} + \cdots - 345 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2268 \beta_{15} + 3267 \beta_{14} + 4005 \beta_{13} - 6048 \beta_{12} + 594 \beta_{11} + 1040 \beta_{10} + \cdots + 252249 ) / 108 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 200 \beta_{15} + 365 \beta_{14} + 209 \beta_{13} + 352 \beta_{12} - 232 \beta_{11} - 257 \beta_{10} + \cdots + 17008 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 16254 \beta_{15} - 56511 \beta_{14} - 8397 \beta_{13} + 58536 \beta_{12} - 102870 \beta_{11} + \cdots - 8448267 ) / 108 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 5496 \beta_{15} - 12095 \beta_{14} - 6385 \beta_{13} - 3584 \beta_{12} - 13010 \beta_{11} + \cdots - 353137 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 81081 \beta_{15} + 252819 \beta_{14} - 342081 \beta_{13} - 198396 \beta_{12} + 80856 \beta_{11} + \cdots + 76182024 ) / 54 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 1128 \beta_{15} + 83065 \beta_{14} + 9555 \beta_{13} + 88864 \beta_{12} + 291874 \beta_{11} + \cdots + 5691585 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1389024 \beta_{15} - 8085825 \beta_{14} + 7771905 \beta_{13} + 35568 \beta_{12} + 30318786 \beta_{11} + \cdots - 1625149047 ) / 108 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 440698 \beta_{15} + 574700 \beta_{14} + 271734 \beta_{13} - 2053160 \beta_{12} - 988522 \beta_{11} + \cdots - 42437233 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 164493342 \beta_{15} + 355143879 \beta_{14} + 11802573 \beta_{13} + 132288120 \beta_{12} + \cdots + 15088454223 ) / 108 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−0.511776 + 4.19595i
−0.511776 − 4.19595i
1.90485 − 3.50504i
1.90485 + 3.50504i
−2.57845 + 2.57680i
−2.57845 − 2.57680i
2.68538 − 2.30232i
2.68538 + 2.30232i
2.68538 − 0.526103i
2.68538 + 0.526103i
−2.57845 + 0.251626i
−2.57845 − 0.251626i
1.90485 + 0.676615i
1.90485 − 0.676615i
−0.511776 − 1.36753i
−0.511776 + 1.36753i
−3.93398 − 0.723761i 0 14.9523 + 5.69452i −10.6666 0 17.1283 −54.7007 − 33.2240i 0 41.9623 + 7.72010i
53.2 −3.93398 + 0.723761i 0 14.9523 − 5.69452i −10.6666 0 17.1283 −54.7007 + 33.2240i 0 41.9623 − 7.72010i
53.3 −2.95688 − 2.69386i 0 1.48626 + 15.9308i 24.5089 0 −84.8713 38.5207 − 51.1093i 0 −72.4697 − 66.0234i
53.4 −2.95688 + 2.69386i 0 1.48626 − 15.9308i 24.5089 0 −84.8713 38.5207 + 51.1093i 0 −72.4697 + 66.0234i
53.5 −1.64415 − 3.64648i 0 −10.5936 + 11.9907i 21.8389 0 2.75144 61.1411 + 18.9147i 0 −35.9064 − 79.6350i
53.6 −1.64415 + 3.64648i 0 −10.5936 − 11.9907i 21.8389 0 2.75144 61.1411 − 18.9147i 0 −35.9064 + 79.6350i
53.7 −1.25598 − 3.79770i 0 −12.8450 + 9.53966i −42.5276 0 64.9916 52.3618 + 36.8000i 0 53.4138 + 161.507i
53.8 −1.25598 + 3.79770i 0 −12.8450 − 9.53966i −42.5276 0 64.9916 52.3618 − 36.8000i 0 53.4138 − 161.507i
53.9 1.25598 − 3.79770i 0 −12.8450 − 9.53966i 42.5276 0 64.9916 −52.3618 + 36.8000i 0 53.4138 − 161.507i
53.10 1.25598 + 3.79770i 0 −12.8450 + 9.53966i 42.5276 0 64.9916 −52.3618 − 36.8000i 0 53.4138 + 161.507i
53.11 1.64415 − 3.64648i 0 −10.5936 − 11.9907i −21.8389 0 2.75144 −61.1411 + 18.9147i 0 −35.9064 + 79.6350i
53.12 1.64415 + 3.64648i 0 −10.5936 + 11.9907i −21.8389 0 2.75144 −61.1411 − 18.9147i 0 −35.9064 − 79.6350i
53.13 2.95688 − 2.69386i 0 1.48626 − 15.9308i −24.5089 0 −84.8713 −38.5207 − 51.1093i 0 −72.4697 + 66.0234i
53.14 2.95688 + 2.69386i 0 1.48626 + 15.9308i −24.5089 0 −84.8713 −38.5207 + 51.1093i 0 −72.4697 − 66.0234i
53.15 3.93398 − 0.723761i 0 14.9523 − 5.69452i 10.6666 0 17.1283 54.7007 − 33.2240i 0 41.9623 − 7.72010i
53.16 3.93398 + 0.723761i 0 14.9523 + 5.69452i 10.6666 0 17.1283 54.7007 + 33.2240i 0 41.9623 + 7.72010i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h 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.h.a ✓ 16
3.b odd 2 1 inner 72.5.h.a ✓ 16
4.b odd 2 1 288.5.h.a 16
8.b even 2 1 inner 72.5.h.a ✓ 16
8.d odd 2 1 288.5.h.a 16
12.b even 2 1 288.5.h.a 16
24.f even 2 1 288.5.h.a 16
24.h odd 2 1 inner 72.5.h.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.5.h.a ✓ 16 1.a even 1 1 trivial
72.5.h.a ✓ 16 3.b odd 2 1 inner
72.5.h.a ✓ 16 8.b even 2 1 inner
72.5.h.a ✓ 16 24.h odd 2 1 inner
288.5.h.a 16 4.b odd 2 1
288.5.h.a 16 8.d odd 2 1
288.5.h.a 16 12.b even 2 1
288.5.h.a 16 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(72, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 4294967296 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 3000 T^{6} + \cdots + 58953050128)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 5864 T^{2} + \cdots - 259952)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 14\!\cdots\!48)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 34\!\cdots\!48)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 58\!\cdots\!36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 56\!\cdots\!88)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 72\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 10\!\cdots\!48)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 64 T^{3} + \cdots - 277423820912)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 11\!\cdots\!28)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 64\!\cdots\!44)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 29\!\cdots\!08)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 17\!\cdots\!16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 39\!\cdots\!48)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 13\!\cdots\!92)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 75\!\cdots\!12)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 18\!\cdots\!12)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 83\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 808253423808768)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 415145221190032)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 13\!\cdots\!32)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 85\!\cdots\!44)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 28\!\cdots\!88)^{4} \) Copy content Toggle raw display
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