Properties

Label 72.6.d.b
Level $72$
Weight $6$
Character orbit 72.d
Analytic conductor $11.548$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.37");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(11.5476350265\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.218489.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 8x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + \beta_{2} + \beta_1 + 5) q^{4} + (2 \beta_{2} - 4 \beta_1 + 2) q^{5} + ( - 4 \beta_{3} + 4 \beta_{2} + 20 \beta_1 + 12) q^{7} + (10 \beta_{3} - 6 \beta_{2} + 2 \beta_1 + 66) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + \beta_{2} + \beta_1 + 5) q^{4} + (2 \beta_{2} - 4 \beta_1 + 2) q^{5} + ( - 4 \beta_{3} + 4 \beta_{2} + 20 \beta_1 + 12) q^{7} + (10 \beta_{3} - 6 \beta_{2} + 2 \beta_1 + 66) q^{8} + (4 \beta_{3} - 12 \beta_{2} - 8 \beta_1 + 164) q^{10} + (5 \beta_{3} - 16 \beta_{2} + 37 \beta_1 - 16) q^{11} + (32 \beta_{3} - 6 \beta_{2} + 44 \beta_1 - 6) q^{13} + (16 \beta_{3} + 16 \beta_{2} + 24 \beta_1 + 592) q^{14} + (28 \beta_{3} - 4 \beta_{2} + 76 \beta_1 - 852) q^{16} + (8 \beta_{3} - 8 \beta_{2} - 40 \beta_1 - 26) q^{17} + (7 \beta_{3} - 48 \beta_{2} + 103 \beta_1 - 48) q^{19} + ( - 36 \beta_{3} + 28 \beta_{2} + 188 \beta_1 - 1268) q^{20} + ( - 2 \beta_{3} + 86 \beta_{2} + 64 \beta_1 - 1442) q^{22} + ( - 52 \beta_{3} + 52 \beta_{2} + 260 \beta_1 - 740) q^{23} + ( - 80 \beta_{3} + 80 \beta_{2} + 400 \beta_1 + 149) q^{25} + (180 \beta_{3} - 28 \beta_{2} + 24 \beta_1 - 1324) q^{26} + (168 \beta_{3} - 88 \beta_{2} + 552 \beta_1 + 1096) q^{28} + ( - 256 \beta_{3} - 82 \beta_{2} - 92 \beta_1 - 82) q^{29} + (48 \beta_{3} - 48 \beta_{2} - 240 \beta_1 - 3088) q^{31} + (200 \beta_{3} + 8 \beta_{2} - 792 \beta_1 - 856) q^{32} + ( - 32 \beta_{3} - 32 \beta_{2} - 50 \beta_1 - 1184) q^{34} + ( - 16 \beta_{3} - 224 \beta_{2} + 432 \beta_1 - 224) q^{35} + ( - 160 \beta_{3} - 114 \beta_{2} + 68 \beta_1 - 114) q^{37} + ( - 54 \beta_{3} + 274 \beta_{2} + 192 \beta_1 - 4118) q^{38} + (120 \beta_{3} + 184 \beta_{2} - 1128 \beta_1 + 4632) q^{40} + ( - 272 \beta_{3} + 272 \beta_{2} + 1360 \beta_1 + 326) q^{41} + ( - 421 \beta_{3} - 96 \beta_{2} - 229 \beta_1 - 96) q^{43} + (398 \beta_{3} - 274 \beta_{2} - 1634 \beta_1 + 8294) q^{44} + (208 \beta_{3} + 208 \beta_{2} - 584 \beta_1 + 7696) q^{46} + (216 \beta_{3} - 216 \beta_{2} - 1080 \beta_1 + 14328) q^{47} + ( - 192 \beta_{3} + 192 \beta_{2} + 960 \beta_1 + 1881) q^{49} + (320 \beta_{3} + 320 \beta_{2} + 389 \beta_1 + 11840) q^{50} + (812 \beta_{3} - 404 \beta_{2} - 1396 \beta_1 - 8036) q^{52} + ( - 928 \beta_{3} - 262 \beta_{2} - 404 \beta_1 - 262) q^{53} + (620 \beta_{3} - 620 \beta_{2} - 3100 \beta_1 + 23228) q^{55} + (1040 \beta_{3} + 400 \beta_{2} + 1744 \beta_1 - 10544) q^{56} + ( - 1700 \beta_{3} + 1004 \beta_{2} + 328 \beta_1 - 68) q^{58} + ( - 1427 \beta_{3} - 256 \beta_{2} - 915 \beta_1 - 256) q^{59} + ( - 2016 \beta_{3} + 1314 \beta_{2} - 4644 \beta_1 + 1314) q^{61} + ( - 192 \beta_{3} - 192 \beta_{2} - 3232 \beta_1 - 7104) q^{62} + (240 \beta_{3} - 1424 \beta_{2} - 1872 \beta_1 - 9424) q^{64} + (112 \beta_{3} - 112 \beta_{2} - 560 \beta_1 + 5216) q^{65} + (1359 \beta_{3} - 1584 \beta_{2} + 4527 \beta_1 - 1584) q^{67} + ( - 338 \beta_{3} + 174 \beta_{2} - 1106 \beta_1 - 2202) q^{68} + ( - 544 \beta_{3} + 1376 \beta_{2} + 896 \beta_1 - 17952) q^{70} + (228 \beta_{3} - 228 \beta_{2} - 1140 \beta_1 - 50988) q^{71} + ( - 1208 \beta_{3} + 1208 \beta_{2} + 6040 \beta_1 + 6370) q^{73} + ( - 1188 \beta_{3} + 1004 \beta_{2} + 456 \beta_1 - 5188) q^{74} + (1018 \beta_{3} - 742 \beta_{2} - 4694 \beta_1 + 27842) q^{76} + (928 \beta_{3} + 1672 \beta_{2} - 2416 \beta_1 + 1672) q^{77} + (344 \beta_{3} - 344 \beta_{2} - 1720 \beta_1 - 60936) q^{79} + (208 \beta_{3} - 2224 \beta_{2} + 2832 \beta_1 + 7568) q^{80} + (1088 \beta_{3} + 1088 \beta_{2} + 1142 \beta_1 + 40256) q^{82} + (2569 \beta_{3} - 1408 \beta_{2} + 5385 \beta_1 - 1408) q^{83} + (32 \beta_{3} + 444 \beta_{2} - 856 \beta_1 + 444) q^{85} + ( - 2718 \beta_{3} + 1418 \beta_{2} + 384 \beta_1 + 3074) q^{86} + ( - 740 \beta_{3} - 1732 \beta_{2} + 7084 \beta_1 - 45716) q^{88} + ( - 2504 \beta_{3} + 2504 \beta_{2} + 12520 \beta_1 + 13646) q^{89} + (5168 \beta_{3} - 96 \beta_{2} + 5360 \beta_1 - 96) q^{91} + (1288 \beta_{3} - 2040 \beta_{2} + 6280 \beta_1 + 9768) q^{92} + ( - 864 \beta_{3} - 864 \beta_{2} + 13680 \beta_1 - 31968) q^{94} + (1892 \beta_{3} - 1892 \beta_{2} - 9460 \beta_1 + 70612) q^{95} + ( - 1096 \beta_{3} + 1096 \beta_{2} + 5480 \beta_1 - 28182) q^{97} + (768 \beta_{3} + 768 \beta_{2} + 2457 \beta_1 + 28416) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 20 q^{4} + 96 q^{7} + 248 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 20 q^{4} + 96 q^{7} + 248 q^{8} + 632 q^{10} + 2384 q^{14} - 3312 q^{16} - 200 q^{17} - 4624 q^{20} - 5636 q^{22} - 2336 q^{23} + 1556 q^{25} - 5608 q^{26} + 5152 q^{28} - 12928 q^{31} - 5408 q^{32} - 4772 q^{34} - 15980 q^{38} + 16032 q^{40} + 4568 q^{41} + 29112 q^{44} + 29200 q^{46} + 54720 q^{47} + 9828 q^{49} + 47498 q^{50} - 36560 q^{52} + 85472 q^{55} - 40768 q^{56} + 3784 q^{58} - 34496 q^{62} - 41920 q^{64} + 19520 q^{65} - 10344 q^{68} - 68928 q^{70} - 206688 q^{71} + 39976 q^{73} - 17464 q^{74} + 99944 q^{76} - 247872 q^{79} + 35520 q^{80} + 161132 q^{82} + 18500 q^{86} - 167216 q^{88} + 84632 q^{89} + 49056 q^{92} - 98784 q^{94} + 259744 q^{95} - 99576 q^{97} + 117042 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 8x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 5\nu^{2} - 2\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 3\nu^{2} - 2\nu - 12 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta _1 + 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{3} - 3\beta_{2} + \beta _1 + 33 ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−1.88600 2.10784i
−1.88600 + 2.10784i
2.38600 1.51888i
2.38600 + 1.51888i
−3.77200 4.21569i 0 −3.54400 + 31.8031i 73.9600i 0 −112.704 147.440 105.021i 0 311.792 278.977i
37.2 −3.77200 + 4.21569i 0 −3.54400 31.8031i 73.9600i 0 −112.704 147.440 + 105.021i 0 311.792 + 278.977i
37.3 4.77200 3.03776i 0 13.5440 28.9924i 1.38521i 0 160.704 −23.4400 179.495i 0 4.20793 + 6.61022i
37.4 4.77200 + 3.03776i 0 13.5440 + 28.9924i 1.38521i 0 160.704 −23.4400 + 179.495i 0 4.20793 6.61022i
\(n\): e.g. 2-40 or 990-1000
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 72.6.d.b 4
3.b odd 2 1 8.6.b.a 4
4.b odd 2 1 288.6.d.b 4
8.b even 2 1 inner 72.6.d.b 4
8.d odd 2 1 288.6.d.b 4
12.b even 2 1 32.6.b.a 4
15.d odd 2 1 200.6.d.a 4
15.e even 4 2 200.6.f.a 8
24.f even 2 1 32.6.b.a 4
24.h odd 2 1 8.6.b.a 4
48.i odd 4 2 256.6.a.k 4
48.k even 4 2 256.6.a.n 4
60.h even 2 1 800.6.d.a 4
60.l odd 4 2 800.6.f.a 8
120.i odd 2 1 200.6.d.a 4
120.m even 2 1 800.6.d.a 4
120.q odd 4 2 800.6.f.a 8
120.w even 4 2 200.6.f.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.6.b.a 4 3.b odd 2 1
8.6.b.a 4 24.h odd 2 1
32.6.b.a 4 12.b even 2 1
32.6.b.a 4 24.f even 2 1
72.6.d.b 4 1.a even 1 1 trivial
72.6.d.b 4 8.b even 2 1 inner
200.6.d.a 4 15.d odd 2 1
200.6.d.a 4 120.i odd 2 1
200.6.f.a 8 15.e even 4 2
200.6.f.a 8 120.w even 4 2
256.6.a.k 4 48.i odd 4 2
256.6.a.n 4 48.k even 4 2
288.6.d.b 4 4.b odd 2 1
288.6.d.b 4 8.d odd 2 1
800.6.d.a 4 60.h even 2 1
800.6.d.a 4 120.m even 2 1
800.6.f.a 8 60.l odd 4 2
800.6.f.a 8 120.q odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 5472T_{5}^{2} + 10496 \) acting on \(S_{6}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} - 8 T^{2} - 64 T + 1024 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 5472 T^{2} + 10496 \) Copy content Toggle raw display
$7$ \( (T^{2} - 48 T - 18112)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 347768 T^{2} + \cdots + 5520765456 \) Copy content Toggle raw display
$13$ \( T^{4} + 590944 T^{2} + \cdots + 7999305984 \) Copy content Toggle raw display
$17$ \( (T^{2} + 100 T - 72252)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 3109816 T^{2} + \cdots + 120994976016 \) Copy content Toggle raw display
$23$ \( (T^{2} + 1168 T - 2817216)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 535633608132864 \) Copy content Toggle raw display
$31$ \( (T^{2} + 6464 T + 7754752)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 306881230162176 \) Copy content Toggle raw display
$41$ \( (T^{2} - 2284 T - 85109148)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 121686904 T^{2} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( (T^{2} - 27360 T + 132648192)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 633629792 T^{2} + \cdots + 76\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{4} + 1322273016 T^{2} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{4} + 4119483744 T^{2} + \cdots + 41\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{4} + 4033664568 T^{2} + \cdots + 32\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{2} + 103344 T + 2609278272)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 19988 T - 1604540316)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 123936 T + 3701816576)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 5708307384 T^{2} + \cdots + 72\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{2} - 42316 T - 6875717724)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 49788 T - 783309052)^{2} \) Copy content Toggle raw display
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