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Properties

Label 273.12.a.e
Level $273$
Weight $12$
Character orbit 273.a
Self dual yes
Analytic conductor $209.758$
Analytic rank $0$
Dimension $17$
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: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 3 x^{16} - 26373 x^{15} + 3751 x^{14} + 279889464 x^{13} + 728094320 x^{12} + \cdots + 15\!\cdots\!12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{10}\cdot 5^{2}\cdot 7^{5} \)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 4) q^{2} - 243 q^{3} + (\beta_{2} - 4 \beta_1 + 1071) q^{4} + ( - \beta_{3} + 12 \beta_1 + 375) q^{5} + (243 \beta_1 - 972) q^{6} + 16807 q^{7} + ( - \beta_{4} - \beta_{3} + \cdots + 8524) q^{8}+ \cdots + 59049 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 4) q^{2} - 243 q^{3} + (\beta_{2} - 4 \beta_1 + 1071) q^{4} + ( - \beta_{3} + 12 \beta_1 + 375) q^{5} + (243 \beta_1 - 972) q^{6} + 16807 q^{7} + ( - \beta_{4} - \beta_{3} + \cdots + 8524) q^{8}+ \cdots + (59049 \beta_{9} - 118098 \beta_{3} + \cdots + 1526534748) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 65 q^{2} - 4131 q^{3} + 18187 q^{4} + 6405 q^{5} - 15795 q^{6} + 285719 q^{7} + 141633 q^{8} + 1003833 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 65 q^{2} - 4131 q^{3} + 18187 q^{4} + 6405 q^{5} - 15795 q^{6} + 285719 q^{7} + 141633 q^{8} + 1003833 q^{9} - 613901 q^{10} + 438062 q^{11} - 4419441 q^{12} + 6311981 q^{13} + 1092455 q^{14} - 1556415 q^{15} + 19519603 q^{16} + 2762228 q^{17} + 3838185 q^{18} - 14542929 q^{19} + 12100505 q^{20} - 69429717 q^{21} + 21895031 q^{22} + 62513631 q^{23} - 34416819 q^{24} + 333282130 q^{25} + 24134045 q^{26} - 243931419 q^{27} + 305668909 q^{28} + 220284497 q^{29} + 149177943 q^{30} + 173032535 q^{31} + 819547889 q^{32} - 106449066 q^{33} - 436107701 q^{34} + 107648835 q^{35} + 1073924163 q^{36} - 860590864 q^{37} + 2003988453 q^{38} - 1533811383 q^{39} + 960293889 q^{40} + 591154594 q^{41} - 265466565 q^{42} + 2778986163 q^{43} + 3898280025 q^{44} + 378208845 q^{45} + 674960931 q^{46} + 5210262301 q^{47} - 4743263529 q^{48} + 4802079233 q^{49} + 9432313682 q^{50} - 671221404 q^{51} + 6752705791 q^{52} + 8935599793 q^{53} - 932678955 q^{54} + 2055022390 q^{55} + 2380425831 q^{56} + 3533931747 q^{57} + 2381725333 q^{58} + 9216410516 q^{59} - 2940422715 q^{60} - 6852607234 q^{61} - 17877233868 q^{62} + 16871421231 q^{63} - 11612626117 q^{64} + 2378131665 q^{65} - 5320492533 q^{66} - 1245305818 q^{67} - 52210558883 q^{68} - 15190812333 q^{69} - 10317834107 q^{70} - 9146008296 q^{71} + 8363287017 q^{72} - 28053584441 q^{73} + 42320360889 q^{74} - 80987557590 q^{75} - 39845809257 q^{76} + 7362508034 q^{77} - 5864572935 q^{78} + 103080223 q^{79} - 62858645595 q^{80} + 59275334817 q^{81} - 35366116032 q^{82} + 806358017 q^{83} - 74277544887 q^{84} - 21577148104 q^{85} + 123046587361 q^{86} - 53529132771 q^{87} + 112491232797 q^{88} + 100182395651 q^{89} - 36250240149 q^{90} + 106085464667 q^{91} - 17063348335 q^{92} - 42046906005 q^{93} - 134035491204 q^{94} + 393873949589 q^{95} - 199150137027 q^{96} + 15332439047 q^{97} + 18360891185 q^{98} + 25867123038 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 3 x^{16} - 26373 x^{15} + 3751 x^{14} + 279889464 x^{13} + 728094320 x^{12} + \cdots + 15\!\cdots\!12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4\nu - 3103 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\!\cdots\!89 \nu^{16} + \cdots + 46\!\cdots\!60 ) / 46\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15\!\cdots\!89 \nu^{16} + \cdots - 60\!\cdots\!52 ) / 46\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 68\!\cdots\!47 \nu^{16} + \cdots - 53\!\cdots\!56 ) / 46\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 24\!\cdots\!35 \nu^{16} + \cdots + 42\!\cdots\!24 ) / 15\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21\!\cdots\!01 \nu^{16} + \cdots - 70\!\cdots\!44 ) / 78\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 64\!\cdots\!13 \nu^{16} + \cdots + 10\!\cdots\!68 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 28\!\cdots\!35 \nu^{16} + \cdots - 38\!\cdots\!56 ) / 46\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 96\!\cdots\!81 \nu^{16} + \cdots - 20\!\cdots\!76 ) / 78\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 49\!\cdots\!91 \nu^{16} + \cdots - 68\!\cdots\!16 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 30\!\cdots\!11 \nu^{16} + \cdots - 14\!\cdots\!36 ) / 23\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 31\!\cdots\!11 \nu^{16} + \cdots + 70\!\cdots\!24 ) / 23\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 38\!\cdots\!59 \nu^{16} + \cdots - 97\!\cdots\!04 ) / 23\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 36\!\cdots\!67 \nu^{16} + \cdots + 28\!\cdots\!52 ) / 14\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 10\!\cdots\!21 \nu^{16} + \cdots - 87\!\cdots\!36 ) / 21\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4\beta _1 + 3103 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + 3\beta_{2} + 5163\beta _1 + 12392 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{14} + \beta_{13} + \beta_{10} + 2 \beta_{9} - \beta_{8} - 3 \beta_{5} + 12 \beta_{4} + \cdots + 16020976 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 16 \beta_{16} - 8 \beta_{15} + 10 \beta_{14} + 18 \beta_{13} - 6 \beta_{12} - 6 \beta_{11} + \cdots + 118410786 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 648 \beta_{16} - 1676 \beta_{15} - 10357 \beta_{14} + 10045 \beta_{13} - 432 \beta_{12} + \cdots + 93498479848 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 227224 \beta_{16} - 154900 \beta_{15} + 82526 \beta_{14} + 240462 \beta_{13} - 92510 \beta_{12} + \cdots + 1047816320622 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 9130848 \beta_{16} - 24265296 \beta_{15} - 86905157 \beta_{14} + 79449789 \beta_{13} + \cdots + 582121097597308 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2377679392 \beta_{16} - 1968527072 \beta_{15} + 283597690 \beta_{14} + 2263680978 \beta_{13} + \cdots + 90\!\cdots\!14 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 96724505176 \beta_{16} - 251628334468 \beta_{15} - 681393884389 \beta_{14} + 585408271469 \beta_{13} + \cdots + 37\!\cdots\!72 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 21712333942088 \beta_{16} - 20564507801372 \beta_{15} - 1902853481242 \beta_{14} + \cdots + 76\!\cdots\!94 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 924181673943344 \beta_{16} + \cdots + 25\!\cdots\!56 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 18\!\cdots\!96 \beta_{16} + \cdots + 64\!\cdots\!78 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 83\!\cdots\!08 \beta_{16} + \cdots + 17\!\cdots\!64 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 14\!\cdots\!40 \beta_{16} + \cdots + 52\!\cdots\!02 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 72\!\cdots\!68 \beta_{16} + \cdots + 11\!\cdots\!64 \) 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
87.4506
77.7958
73.0591
58.9436
54.3528
41.1140
14.6493
0.230532
−0.0757053
−9.25448
−20.5065
−38.3168
−49.0270
−56.8260
−71.2614
−77.9476
−81.3804
−83.4506 −243.000 4915.99 2226.80 20278.5 16807.0 −239336. 59049.0 −185827.
1.2 −73.7958 −243.000 3397.82 2224.88 17932.4 16807.0 −99611.4 59049.0 −164187.
1.3 −69.0591 −243.000 2721.16 −10810.1 16781.4 16807.0 −46488.0 59049.0 746536.
1.4 −54.9436 −243.000 970.804 5119.82 13351.3 16807.0 59185.1 59049.0 −281302.
1.5 −50.3528 −243.000 487.409 13037.1 12235.7 16807.0 78580.2 59049.0 −656455.
1.6 −37.1140 −243.000 −670.551 4103.66 9018.70 16807.0 100896. 59049.0 −152303.
1.7 −10.6493 −243.000 −1934.59 −5373.56 2587.79 16807.0 42411.9 59049.0 57224.8
1.8 3.76947 −243.000 −2033.79 −9291.83 −915.981 16807.0 −15386.2 59049.0 −35025.2
1.9 4.07571 −243.000 −2031.39 −9600.51 −990.396 16807.0 −16626.4 59049.0 −39128.8
1.10 13.2545 −243.000 −1872.32 10613.9 −3220.84 16807.0 −51961.8 59049.0 140682.
1.11 24.5065 −243.000 −1447.43 11849.1 −5955.07 16807.0 −85660.7 59049.0 290379.
1.12 42.3168 −243.000 −257.292 −3376.04 −10283.0 16807.0 −97552.5 59049.0 −142863.
1.13 53.0270 −243.000 763.863 296.944 −12885.6 16807.0 −68093.9 59049.0 15746.0
1.14 60.8260 −243.000 1651.80 −11231.5 −14780.7 16807.0 −24099.4 59049.0 −683164.
1.15 75.2614 −243.000 3616.28 5194.49 −18288.5 16807.0 118031. 59049.0 390945.
1.16 81.9476 −243.000 4667.42 10648.5 −19913.3 16807.0 214655. 59049.0 872619.
1.17 85.3804 −243.000 5241.81 −9226.68 −20747.4 16807.0 272689. 59049.0 −787778.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
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.e ✓ 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.12.a.e ✓ 17 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{17} - 65 T_{2}^{16} - 24389 T_{2}^{15} + 1540869 T_{2}^{14} + 236294640 T_{2}^{13} + \cdots + 16\!\cdots\!72 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(273))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} + \cdots + 16\!\cdots\!72 \) Copy content Toggle raw display
$3$ \( (T + 243)^{17} \) Copy content Toggle raw display
$5$ \( T^{17} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T - 16807)^{17} \) Copy content Toggle raw display
$11$ \( T^{17} + \cdots + 85\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( (T - 371293)^{17} \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots + 50\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots - 32\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{17} + \cdots - 49\!\cdots\!20 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots - 60\!\cdots\!12 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots - 18\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots - 14\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots + 20\!\cdots\!88 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots - 43\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots + 55\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots - 39\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots + 41\!\cdots\!60 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots - 97\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 13\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots - 41\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots + 12\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots - 45\!\cdots\!04 \) Copy content Toggle raw display
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