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Properties

Label 273.12.a.g
Level $273$
Weight $12$
Character orbit 273.a
Self dual yes
Analytic conductor $209.758$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [273,12,Mod(1,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 273.a (trivial)

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: yes
Analytic conductor: \(209.757688293\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 27936 x^{16} - 16248 x^{15} + 322705437 x^{14} + 384273264 x^{13} - 1999754084562 x^{12} + \cdots - 34\!\cdots\!80 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: multiple of \( 2^{18}\cdot 3^{12}\cdot 7^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{2} - 243 q^{3} + (\beta_{2} - 3 \beta_1 + 1060) q^{4} + (\beta_{4} + 12 \beta_1 + 176) q^{5} + (243 \beta_1 - 486) q^{6} - 16807 q^{7} + ( - \beta_{3} + 6 \beta_{2} + \cdots + 7732) q^{8}+ \cdots + 59049 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 2) q^{2} - 243 q^{3} + (\beta_{2} - 3 \beta_1 + 1060) q^{4} + (\beta_{4} + 12 \beta_1 + 176) q^{5} + (243 \beta_1 - 486) q^{6} - 16807 q^{7} + ( - \beta_{3} + 6 \beta_{2} + \cdots + 7732) q^{8}+ \cdots + (59049 \beta_{5} - 236196 \beta_{4} + \cdots - 2533202100) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 36 q^{2} - 4374 q^{3} + 19080 q^{4} + 3168 q^{5} - 8748 q^{6} - 302526 q^{7} + 139176 q^{8} + 1062882 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 36 q^{2} - 4374 q^{3} + 19080 q^{4} + 3168 q^{5} - 8748 q^{6} - 302526 q^{7} + 139176 q^{8} + 1062882 q^{9} - 647757 q^{10} - 772200 q^{11} - 4636440 q^{12} + 6683274 q^{13} - 605052 q^{14} - 769824 q^{15} + 2747244 q^{16} - 12733776 q^{17} + 2125764 q^{18} - 18224136 q^{19} + 5241753 q^{20} + 73513818 q^{21} + 46885959 q^{22} - 48574872 q^{23} - 33819768 q^{24} + 222942102 q^{25} + 13366548 q^{26} - 258280326 q^{27} - 320677560 q^{28} + 101923596 q^{29} + 157404951 q^{30} + 143268972 q^{31} + 459446184 q^{32} + 187644600 q^{33} - 565492125 q^{34} - 53244576 q^{35} + 1126654920 q^{36} + 587619840 q^{37} - 513236385 q^{38} - 1624035582 q^{39} - 23504997 q^{40} - 1911061800 q^{41} + 147027636 q^{42} + 569800896 q^{43} - 2266629273 q^{44} + 187067232 q^{45} + 1214920779 q^{46} - 2537708904 q^{47} - 667580292 q^{48} + 5084554482 q^{49} + 7457084091 q^{50} + 3094307568 q^{51} + 7084270440 q^{52} - 3333522024 q^{53} - 516560652 q^{54} - 5501144244 q^{55} - 2339131032 q^{56} + 4428465048 q^{57} + 6326427375 q^{58} - 5740477668 q^{59} - 1273745979 q^{60} - 5842045680 q^{61} - 25541889030 q^{62} - 17863857774 q^{63} + 2910666876 q^{64} + 1176256224 q^{65} - 11393288037 q^{66} - 4425288432 q^{67} - 1050976881 q^{68} + 11803693896 q^{69} + 10886851899 q^{70} - 12474559188 q^{71} + 8218203624 q^{72} - 1956619452 q^{73} + 62253202059 q^{74} - 54174930786 q^{75} - 35495777955 q^{76} + 12978365400 q^{77} - 3248071164 q^{78} + 22032383364 q^{79} + 18996992541 q^{80} + 62762119218 q^{81} - 121996533942 q^{82} - 24683921640 q^{83} + 77924647080 q^{84} - 78868639476 q^{85} + 100239949341 q^{86} - 24767433828 q^{87} - 35881646991 q^{88} - 87843698028 q^{89} - 38249403093 q^{90} - 112325786118 q^{91} + 32257813761 q^{92} - 34814360196 q^{93} - 36434297946 q^{94} - 250179495480 q^{95} - 111645422712 q^{96} + 63534965208 q^{97} + 10169108964 q^{98} - 45597637800 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 27936 x^{16} - 16248 x^{15} + 322705437 x^{14} + 384273264 x^{13} - 1999754084562 x^{12} + \cdots - 34\!\cdots\!80 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3104 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4833\nu - 2708 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 76\!\cdots\!79 \nu^{17} + \cdots + 10\!\cdots\!84 ) / 74\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 45\!\cdots\!89 \nu^{17} + \cdots + 87\!\cdots\!24 ) / 14\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 33\!\cdots\!29 \nu^{17} + \cdots + 31\!\cdots\!40 ) / 93\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 85\!\cdots\!23 \nu^{17} + \cdots - 53\!\cdots\!76 ) / 14\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 51\!\cdots\!73 \nu^{17} + \cdots - 19\!\cdots\!84 ) / 74\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19\!\cdots\!69 \nu^{17} + \cdots - 42\!\cdots\!04 ) / 14\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21\!\cdots\!51 \nu^{17} + \cdots + 16\!\cdots\!04 ) / 14\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 18\!\cdots\!34 \nu^{17} + \cdots + 40\!\cdots\!80 ) / 11\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 30\!\cdots\!37 \nu^{17} + \cdots - 34\!\cdots\!96 ) / 14\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 16\!\cdots\!51 \nu^{17} + \cdots + 18\!\cdots\!68 ) / 74\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 17\!\cdots\!65 \nu^{17} + \cdots + 11\!\cdots\!80 ) / 74\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 20\!\cdots\!79 \nu^{17} + \cdots - 13\!\cdots\!40 ) / 74\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 22\!\cdots\!83 \nu^{17} + \cdots - 34\!\cdots\!44 ) / 74\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 14\!\cdots\!97 \nu^{17} + \cdots + 42\!\cdots\!60 ) / 46\!\cdots\!96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3104 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4833\beta _1 + 2708 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{16} - \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} + \cdots + 15001026 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 7 \beta_{17} + 10 \beta_{16} - 11 \beta_{15} + \beta_{14} - 6 \beta_{13} + 27 \beta_{12} + \cdots + 19341869 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1543 \beta_{17} - 10535 \beta_{16} - 913 \beta_{15} - 421 \beta_{14} + 1004 \beta_{13} + \cdots + 84019655693 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10502 \beta_{17} + 128042 \beta_{16} - 104360 \beta_{15} + 45194 \beta_{14} - 29956 \beta_{13} + \cdots + 7416825606 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 21229308 \beta_{17} - 84853467 \beta_{16} - 12627550 \beta_{15} - 3226088 \beta_{14} + \cdots + 504658174808746 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 685518187 \beta_{17} + 1137387734 \beta_{16} - 775660921 \beta_{15} + 561908803 \beta_{14} + \cdots - 11\!\cdots\!81 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 202058831557 \beta_{17} - 627065325575 \beta_{16} - 124920052235 \beta_{15} - 13494592047 \beta_{14} + \cdots + 31\!\cdots\!27 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8617848176036 \beta_{17} + 8834324612262 \beta_{16} - 5161286429982 \beta_{15} + 5077647534368 \beta_{14} + \cdots - 15\!\cdots\!68 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 16\!\cdots\!30 \beta_{17} + \cdots + 20\!\cdots\!64 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 80\!\cdots\!73 \beta_{17} + \cdots - 15\!\cdots\!99 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 12\!\cdots\!63 \beta_{17} + \cdots + 13\!\cdots\!37 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 66\!\cdots\!42 \beta_{17} + \cdots - 13\!\cdots\!94 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 92\!\cdots\!80 \beta_{17} + \cdots + 85\!\cdots\!62 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 51\!\cdots\!19 \beta_{17} + \cdots - 11\!\cdots\!21 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
82.2306
75.9857
74.3929
64.3235
50.2499
42.8497
27.9731
20.7287
16.8843
−10.0232
−28.2358
−29.9256
−50.1412
−53.9043
−58.6838
−60.3621
−81.3357
−83.0067
−80.2306 −243.000 4388.95 −4941.56 19496.0 −16807.0 −187816. 59049.0 396464.
1.2 −73.9857 −243.000 3425.89 11417.2 17978.5 −16807.0 −101944. 59049.0 −844709.
1.3 −72.3929 −243.000 3192.73 5196.81 17591.5 −16807.0 −82870.7 59049.0 −376212.
1.4 −62.3235 −243.000 1836.22 −8920.41 15144.6 −16807.0 13198.7 59049.0 555951.
1.5 −48.2499 −243.000 280.055 1480.96 11724.7 −16807.0 85303.2 59049.0 −71456.2
1.6 −40.8497 −243.000 −379.306 −5304.08 9926.47 −16807.0 99154.6 59049.0 216670.
1.7 −25.9731 −243.000 −1373.40 12247.4 6311.46 −16807.0 88864.3 59049.0 −318104.
1.8 −18.7287 −243.000 −1697.23 −4235.28 4551.09 −16807.0 70143.5 59049.0 79321.5
1.9 −14.8843 −243.000 −1826.46 482.493 3616.88 −16807.0 57668.5 59049.0 −7181.56
1.10 12.0232 −243.000 −1903.44 7714.56 −2921.64 −16807.0 −47509.1 59049.0 92753.9
1.11 30.2358 −243.000 −1133.80 −10037.4 −7347.30 −16807.0 −96204.2 59049.0 −303490.
1.12 31.9256 −243.000 −1028.76 −3373.95 −7757.92 −16807.0 −98227.3 59049.0 −107715.
1.13 52.1412 −243.000 670.706 −3983.33 −12670.3 −16807.0 −71813.8 59049.0 −207696.
1.14 55.9043 −243.000 1077.29 12152.1 −13584.7 −16807.0 −54266.9 59049.0 679355.
1.15 60.6838 −243.000 1634.53 −13650.9 −14746.2 −16807.0 −25091.0 59049.0 −828391.
1.16 62.3621 −243.000 1841.03 8135.37 −15154.0 −16807.0 −12907.0 59049.0 507339.
1.17 83.3357 −243.000 4896.84 4568.23 −20250.6 −16807.0 237410. 59049.0 380697.
1.18 85.0067 −243.000 5178.15 −5780.18 −20656.6 −16807.0 266084. 59049.0 −491354.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.12.a.g ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.12.a.g ✓ 18 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - 36 T_{2}^{17} - 27324 T_{2}^{16} + 903672 T_{2}^{15} + 308857677 T_{2}^{14} + \cdots - 34\!\cdots\!68 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(273))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + \cdots - 34\!\cdots\!68 \) Copy content Toggle raw display
$3$ \( (T + 243)^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T + 16807)^{18} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T - 371293)^{18} \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots - 14\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 17\!\cdots\!80 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 38\!\cdots\!20 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 12\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots - 55\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 43\!\cdots\!20 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots - 16\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 10\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 17\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 23\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 94\!\cdots\!88 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 65\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 14\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 63\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 36\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 96\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 51\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 99\!\cdots\!60 \) Copy content Toggle raw display
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