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Properties

Label 72.13.b.b
Level $72$
Weight $13$
Character orbit 72.b
Analytic conductor $65.808$
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] = [72,13,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, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.19");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 13 \)
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: \(65.8075548439\)
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: \( x^{10} - 5 x^{9} + 468 x^{8} + 1496 x^{7} + 710096 x^{6} + 29155008 x^{5} + 143571008 x^{4} + \cdots + 824967906703360 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{45}\cdot 3^{3}\cdot 23 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 11) q^{2} + (\beta_{3} - \beta_{2} - 12 \beta_1 - 244) q^{4} + (\beta_{5} + \beta_{2} + 48 \beta_1) q^{5} + ( - \beta_{8} + \beta_{7} + \cdots - 323 \beta_1) q^{7}+ \cdots + ( - \beta_{9} - \beta_{8} + \cdots - 2710) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 11) q^{2} + (\beta_{3} - \beta_{2} - 12 \beta_1 - 244) q^{4} + (\beta_{5} + \beta_{2} + 48 \beta_1) q^{5} + ( - \beta_{8} + \beta_{7} + \cdots - 323 \beta_1) q^{7}+ \cdots + (2172032 \beta_{9} + \cdots - 237815190229) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 110 q^{2} - 2444 q^{4} - 27160 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 110 q^{2} - 2444 q^{4} - 27160 q^{8} + 1873200 q^{10} + 4591444 q^{11} - 12728736 q^{14} - 38294000 q^{16} - 42876500 q^{17} + 36062828 q^{19} - 17615520 q^{20} - 227004580 q^{22} - 706946390 q^{25} - 151295184 q^{26} - 132075840 q^{28} + 2651204000 q^{32} + 4454663012 q^{34} + 2838470400 q^{35} - 1644178460 q^{38} + 8180322240 q^{40} + 12339248044 q^{41} + 25081495340 q^{43} + 27950589832 q^{44} - 16813594656 q^{46} - 39532486838 q^{49} - 24888425650 q^{50} - 36172521120 q^{52} - 154450364544 q^{56} + 193270394640 q^{58} + 109448026708 q^{59} + 299237961600 q^{62} + 276213192256 q^{64} - 96485235840 q^{65} - 49860896020 q^{67} + 296812951960 q^{68} - 799057954560 q^{70} - 58835592940 q^{73} - 742739480496 q^{74} - 699737494024 q^{76} - 1981932232320 q^{80} + 1654109754980 q^{82} + 146236977940 q^{83} + 2261898070564 q^{86} + 2170357811600 q^{88} - 913341514388 q^{89} + 361546645248 q^{91} + 2649411172800 q^{92} - 2517413216064 q^{94} - 1474226441260 q^{97} - 2377890492370 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 5 x^{9} + 468 x^{8} + 1496 x^{7} + 710096 x^{6} + 29155008 x^{5} + 143571008 x^{4} + \cdots + 824967906703360 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2311 \nu^{9} + 15111 \nu^{8} - 1567498 \nu^{7} + 64737964 \nu^{6} + 1755234488 \nu^{5} + \cdots + 75\!\cdots\!96 ) / 11819749998592 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2311 \nu^{9} + 15111 \nu^{8} - 1567498 \nu^{7} + 64737964 \nu^{6} + 1755234488 \nu^{5} + \cdots + 80\!\cdots\!88 ) / 11819749998592 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4181 \nu^{9} - 42581 \nu^{8} + 6006942 \nu^{7} - 182270212 \nu^{6} + \cdots - 10\!\cdots\!40 ) / 5909874999296 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10135 \nu^{9} - 622185 \nu^{8} + 7062422 \nu^{7} + 358257644 \nu^{6} + \cdots - 42\!\cdots\!60 ) / 11819749998592 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 45867 \nu^{9} - 139989 \nu^{8} - 15342690 \nu^{7} + 728505596 \nu^{6} + \cdots + 12\!\cdots\!84 ) / 5909874999296 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 73249 \nu^{9} + 5137375 \nu^{8} - 102710042 \nu^{7} - 916635508 \nu^{6} + \cdots + 30\!\cdots\!92 ) / 11819749998592 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 37967 \nu^{9} - 1506737 \nu^{8} + 26451014 \nu^{7} + 205390668 \nu^{6} + \cdots - 17\!\cdots\!64 ) / 2954937499648 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 88705 \nu^{9} - 1656703 \nu^{8} - 7265062 \nu^{7} + 2139474676 \nu^{6} + \cdots + 37\!\cdots\!96 ) / 5909874999296 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - \beta_{2} + 12\beta _1 - 364 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} + \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} + 22 \beta_{3} + \cdots - 9098 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 11 \beta_{9} + 29 \beta_{8} + 74 \beta_{7} - 6 \beta_{6} + 378 \beta_{5} + 206 \beta_{4} + \cdots - 1982794 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1607 \beta_{9} + 1297 \beta_{8} + 722 \beta_{7} - 1686 \beta_{6} + 5346 \beta_{5} - 1910 \beta_{4} + \cdots - 124024130 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 42463 \beta_{9} + 1017 \beta_{8} - 43358 \beta_{7} - 111398 \beta_{6} + 50866 \beta_{5} + \cdots + 32940526 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 792039 \beta_{9} - 900815 \beta_{8} - 2897006 \beta_{7} - 1820150 \beta_{6} - 7591134 \beta_{5} + \cdots - 69696146690 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 9736769 \beta_{9} - 64761703 \beta_{8} - 97005342 \beta_{7} + 29738970 \beta_{6} - 384498702 \beta_{5} + \cdots - 7060971877458 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1757237801 \beta_{9} - 1786019007 \beta_{8} - 2047449166 \beta_{7} + 3449039594 \beta_{6} + \cdots - 135929362936226 ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
32.5710 + 17.8321i
32.5710 − 17.8321i
17.4695 + 29.8739i
17.4695 − 29.8739i
−10.5406 + 27.3936i
−10.5406 − 27.3936i
−11.5862 + 26.7344i
−11.5862 − 26.7344i
−25.4137 + 6.09761i
−25.4137 − 6.09761i
−53.1419 − 35.6642i 0 1552.12 + 3790.53i 21249.1i 0 − 149114.i 52703.6 − 256791.i 0 757833. − 1.12922e6i
19.2 −53.1419 + 35.6642i 0 1552.12 − 3790.53i − 21249.1i 0 149114.i 52703.6 + 256791.i 0 757833. + 1.12922e6i
19.3 −22.9390 − 59.7478i 0 −3043.60 + 2741.11i − 11036.2i 0 58770.2i 233593. + 118970.i 0 −659388. + 253159.i
19.4 −22.9390 + 59.7478i 0 −3043.60 − 2741.11i 11036.2i 0 − 58770.2i 233593. − 118970.i 0 −659388. − 253159.i
19.5 33.0812 − 54.7872i 0 −1907.27 − 3624.85i 2272.99i 0 92079.6i −261690. − 15420.3i 0 124531. + 75193.1i
19.6 33.0812 + 54.7872i 0 −1907.27 + 3624.85i − 2272.99i 0 − 92079.6i −261690. + 15420.3i 0 124531. − 75193.1i
19.7 35.1724 − 53.4687i 0 −1621.80 − 3761.25i 19079.0i 0 − 136156.i −258152. − 45576.2i 0 1.02013e6 + 671054.i
19.8 35.1724 + 53.4687i 0 −1621.80 + 3761.25i − 19079.0i 0 136156.i −258152. + 45576.2i 0 1.02013e6 − 671054.i
19.9 62.8274 − 12.1952i 0 3798.55 − 1532.39i − 25133.2i 0 − 190438.i 219965. − 142600.i 0 −306505. − 1.57906e6i
19.10 62.8274 + 12.1952i 0 3798.55 + 1532.39i 25133.2i 0 190438.i 219965. + 142600.i 0 −306505. + 1.57906e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.10
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 72.13.b.b 10
3.b odd 2 1 8.13.d.b ✓ 10
8.d odd 2 1 inner 72.13.b.b 10
12.b even 2 1 32.13.d.b 10
24.f even 2 1 8.13.d.b ✓ 10
24.h odd 2 1 32.13.d.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.13.d.b ✓ 10 3.b odd 2 1
8.13.d.b ✓ 10 24.f even 2 1
32.13.d.b 10 12.b even 2 1
32.13.d.b 10 24.h odd 2 1
72.13.b.b 10 1.a even 1 1 trivial
72.13.b.b 10 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 1574176320 T_{5}^{8} + \cdots + 65\!\cdots\!00 \) acting on \(S_{13}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots + 78\!\cdots\!52)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{5} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 43\!\cdots\!88)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 24\!\cdots\!68)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots + 45\!\cdots\!08)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots - 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 11\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
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