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Properties

Label 72.14.d.c
Level $72$
Weight $14$
Character orbit 72.d
Analytic conductor $77.206$
Analytic rank $0$
Dimension $10$
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] = [72,14,Mod(37,72)]
 
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, 0]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.37");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 72.d (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: \(77.2062688454\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 5 x^{9} + 752 x^{8} + 708 x^{7} - 743866 x^{6} + 96647426 x^{5} + 2540283092 x^{4} + \cdots + 31\!\cdots\!68 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 8)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 11) q^{2} + (\beta_{5} + \beta_{3} - 12 \beta_1 - 472) q^{4} + (\beta_{4} + 3 \beta_{3} + 56 \beta_1) q^{5} + ( - \beta_{9} - 2 \beta_{5} + \cdots + 58697) q^{7}+ \cdots + ( - \beta_{8} + \beta_{7} + \cdots + 27077) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 11) q^{2} + (\beta_{5} + \beta_{3} - 12 \beta_1 - 472) q^{4} + (\beta_{4} + 3 \beta_{3} + 56 \beta_1) q^{5} + ( - \beta_{9} - 2 \beta_{5} + \cdots + 58697) q^{7}+ \cdots + (16022784 \beta_{9} + \cdots - 5432359791379) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 110 q^{2} - 4716 q^{4} + 586960 q^{7} + 270712 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 110 q^{2} - 4716 q^{4} + 586960 q^{7} + 270712 q^{8} - 4542088 q^{10} - 1408688 q^{14} + 56624912 q^{16} - 217326004 q^{17} - 21655184 q^{20} - 177987876 q^{22} + 78679952 q^{23} - 3076402574 q^{25} - 3734872040 q^{26} - 1653812448 q^{28} + 648233792 q^{31} + 11298380000 q^{32} - 6096822724 q^{34} + 18764968628 q^{38} + 7466802592 q^{40} - 59324640356 q^{41} - 13325704392 q^{44} - 55046867440 q^{46} + 10176534816 q^{47} + 182708552058 q^{49} - 326454435302 q^{50} - 53296499536 q^{52} - 123010753008 q^{55} + 462152447680 q^{56} + 766482705096 q^{58} + 1665308528960 q^{62} - 2180548996032 q^{64} - 1577231990240 q^{65} - 2338280915304 q^{68} - 6070110714688 q^{70} - 726361179984 q^{71} - 633240365532 q^{73} - 7528513982264 q^{74} + 10338420845032 q^{76} + 5445103565344 q^{79} + 15406871881920 q^{80} + 12273334206796 q^{82} + 26794541719396 q^{86} - 27677491769136 q^{88} - 5506344808004 q^{89} - 33971694298464 q^{92} - 45356008560096 q^{94} + 14214732035504 q^{95} + 1361133320788 q^{97} - 54325451514942 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 5 x^{9} + 752 x^{8} + 708 x^{7} - 743866 x^{6} + 96647426 x^{5} + 2540283092 x^{4} + \cdots + 31\!\cdots\!68 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{9} - 10 \nu^{8} + 802 \nu^{7} - 3302 \nu^{6} - 727356 \nu^{5} + 100284206 \nu^{4} + \cdots - 785602817817372 ) / 8796093022208 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 9909 \nu^{9} - 1478382 \nu^{8} - 145765642 \nu^{7} - 6912370530 \nu^{6} + \cdots - 72\!\cdots\!68 ) / 58\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 40121 \nu^{9} - 167622 \nu^{8} + 57095790 \nu^{7} + 2414437430 \nu^{6} + \cdots + 19\!\cdots\!08 ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 60463 \nu^{9} + 7721686 \nu^{8} + 513744322 \nu^{7} - 34067281094 \nu^{6} + \cdots - 19\!\cdots\!52 ) / 17\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 179605 \nu^{9} - 7926830 \nu^{8} - 159680714 \nu^{7} - 13495557154 \nu^{6} + \cdots + 19\!\cdots\!16 ) / 23\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 405715 \nu^{9} + 35804610 \nu^{8} + 1199133446 \nu^{7} - 53192514450 \nu^{6} + \cdots - 13\!\cdots\!80 ) / 58\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5289667 \nu^{9} + 1169301602 \nu^{8} + 45864817382 \nu^{7} + 1390809750734 \nu^{6} + \cdots + 89\!\cdots\!56 ) / 69\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18889069 \nu^{9} - 1797759422 \nu^{8} - 49111091066 \nu^{7} + 597922173934 \nu^{6} + \cdots + 17\!\cdots\!80 ) / 13\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 28179021 \nu^{9} - 1506164990 \nu^{8} - 128265727290 \nu^{7} - 4466585299666 \nu^{6} + \cdots + 11\!\cdots\!92 ) / 13\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + 3\beta_{3} + \beta_{2} + 38\beta _1 + 512 ) / 1024 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{8} + 4\beta_{7} + 63\beta_{5} + 8\beta_{4} - 209\beta_{3} + \beta_{2} + 290\beta _1 - 151476 ) / 1024 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 224 \beta_{8} - 32 \beta_{7} + 352 \beta_{6} - 3373 \beta_{5} - 96 \beta_{4} + 2055 \beta_{3} + \cdots - 1358368 ) / 1024 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6144 \beta_{9} - 2476 \beta_{8} + 852 \beta_{7} - 9696 \beta_{6} - 202953 \beta_{5} + \cdots + 411504124 ) / 1024 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 184320 \beta_{9} - 314704 \beta_{8} + 105648 \beta_{7} + 169536 \beta_{6} + 6241787 \beta_{5} + \cdots - 45918421872 ) / 1024 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 14745600 \beta_{9} + 22025444 \beta_{8} - 928028 \beta_{7} - 14363328 \beta_{6} + \cdots - 2260666407316 ) / 1024 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 397750272 \beta_{9} - 107223296 \beta_{8} - 21666816 \beta_{7} + 719313952 \beta_{6} + \cdots + 164322052399296 ) / 1024 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 12615235584 \beta_{9} - 54365687564 \beta_{8} - 873594892 \beta_{7} - 18502748320 \beta_{6} + \cdots - 89\!\cdots\!84 ) / 1024 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 85621800960 \beta_{9} - 61190446384 \beta_{8} + 823270218448 \beta_{7} - 431355739392 \beta_{6} + \cdots + 46\!\cdots\!48 ) / 1024 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
37.4606 − 15.6556i
37.4606 + 15.6556i
27.9424 − 31.0290i
27.9424 + 31.0290i
1.33949 − 44.8086i
1.33949 + 44.8086i
−17.5296 − 43.4857i
−17.5296 + 43.4857i
−46.7129 − 17.5509i
−46.7129 + 17.5509i
−84.9212 − 31.3112i 0 6231.21 + 5317.98i − 2384.10i 0 −317448. −362649. − 646716.i 0 −74649.0 + 202460.i
37.2 −84.9212 + 31.3112i 0 6231.21 − 5317.98i 2384.10i 0 −317448. −362649. + 646716.i 0 −74649.0 − 202460.i
37.3 −65.8848 − 62.0580i 0 489.620 + 8177.36i − 25270.7i 0 608245. 475211. − 569148.i 0 −1.56825e6 + 1.66495e6i
37.4 −65.8848 + 62.0580i 0 489.620 − 8177.36i 25270.7i 0 608245. 475211. + 569148.i 0 −1.56825e6 − 1.66495e6i
37.5 −12.6790 − 89.6172i 0 −7870.49 + 2272.51i 45531.7i 0 −249036. 303446. + 676518.i 0 4.08042e6 − 577295.i
37.6 −12.6790 + 89.6172i 0 −7870.49 − 2272.51i − 45531.7i 0 −249036. 303446. − 676518.i 0 4.08042e6 + 577295.i
37.7 25.0592 − 86.9715i 0 −6936.08 − 4358.86i − 64905.5i 0 201238. −552909. + 494011.i 0 −5.64492e6 − 1.62648e6i
37.8 25.0592 + 86.9715i 0 −6936.08 + 4358.86i 64905.5i 0 201238. −552909. − 494011.i 0 −5.64492e6 + 1.62648e6i
37.9 83.4258 − 35.1018i 0 5727.73 − 5856.79i 26675.4i 0 50480.5 272257. − 689661.i 0 936352. + 2.22541e6i
37.10 83.4258 + 35.1018i 0 5727.73 + 5856.79i − 26675.4i 0 50480.5 272257. + 689661.i 0 936352. − 2.22541e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.10
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 72.14.d.c 10
3.b odd 2 1 8.14.b.b ✓ 10
8.b even 2 1 inner 72.14.d.c 10
12.b even 2 1 32.14.b.b 10
24.f even 2 1 32.14.b.b 10
24.h odd 2 1 8.14.b.b ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.14.b.b ✓ 10 3.b odd 2 1
8.14.b.b ✓ 10 24.h odd 2 1
32.14.b.b 10 12.b even 2 1
32.14.b.b 10 24.f even 2 1
72.14.d.c 10 1.a even 1 1 trivial
72.14.d.c 10 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 7641716912 T_{5}^{8} + \cdots + 22\!\cdots\!00 \) acting on \(S_{14}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 36\!\cdots\!32 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{5} + \cdots - 48\!\cdots\!64)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{5} + \cdots + 45\!\cdots\!36)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T^{5} + \cdots + 19\!\cdots\!36)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 12\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 92\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 24\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( (T^{5} + \cdots + 55\!\cdots\!44)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 98\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 38\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 48\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 21\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 23\!\cdots\!44)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 69\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 59\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 17\!\cdots\!56)^{2} \) Copy content Toggle raw display
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