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Properties

Label 273.12.a.f
Level $273$
Weight $12$
Character orbit 273.a
Self dual yes
Analytic conductor $209.758$
Analytic rank $0$
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: \(0\)
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: \( x^{18} - x^{17} - 30609 x^{16} + 82531 x^{15} + 387432317 x^{14} - 1631290103 x^{13} + \cdots + 29\!\cdots\!20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: multiple of \( 2^{18}\cdot 3^{14}\cdot 7^{5} \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} + 243 q^{3} + (\beta_{2} - \beta_1 + 1354) q^{4} + (\beta_{4} - \beta_1 - 176) q^{5} + (243 \beta_1 + 243) q^{6} + 16807 q^{7} + (\beta_{4} + \beta_{3} + 1190 \beta_1 - 2839) q^{8} + 59049 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{2} + 243 q^{3} + (\beta_{2} - \beta_1 + 1354) q^{4} + (\beta_{4} - \beta_1 - 176) q^{5} + (243 \beta_1 + 243) q^{6} + 16807 q^{7} + (\beta_{4} + \beta_{3} + 1190 \beta_1 - 2839) q^{8} + 59049 q^{9} + ( - \beta_{5} + 4 \beta_{4} + \cdots - 5185) q^{10}+ \cdots + ( - 59049 \beta_{6} + \cdots + 4502840544) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 19 q^{2} + 4374 q^{3} + 24375 q^{4} - 3168 q^{5} + 4617 q^{6} + 302526 q^{7} - 49911 q^{8} + 1062882 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 19 q^{2} + 4374 q^{3} + 24375 q^{4} - 3168 q^{5} + 4617 q^{6} + 302526 q^{7} - 49911 q^{8} + 1062882 q^{9} - 93489 q^{10} + 1373240 q^{11} + 5923125 q^{12} - 6683274 q^{13} + 319333 q^{14} - 769824 q^{15} + 22874139 q^{16} + 538160 q^{17} + 1121931 q^{18} + 8588040 q^{19} - 5241753 q^{20} + 73513818 q^{21} + 32463867 q^{22} - 19779608 q^{23} - 12128373 q^{24} + 100586646 q^{25} - 7054567 q^{26} + 258280326 q^{27} + 409670625 q^{28} + 217263476 q^{29} - 22717827 q^{30} + 434904516 q^{31} - 98482539 q^{32} + 333697320 q^{33} + 761623977 q^{34} - 53244576 q^{35} + 1439319375 q^{36} + 2296063512 q^{37} - 551120687 q^{38} - 1624035582 q^{39} + 1071383655 q^{40} + 680230800 q^{41} + 77597919 q^{42} + 1376958000 q^{43} + 5621494757 q^{44} - 187067232 q^{45} + 2316406173 q^{46} + 6746473896 q^{47} + 5558415777 q^{48} + 5084554482 q^{49} + 3613153318 q^{50} + 130772880 q^{51} - 9050266875 q^{52} + 8402843056 q^{53} + 272629233 q^{54} + 7363981236 q^{55} - 838854177 q^{56} + 2086893720 q^{57} + 4783138431 q^{58} + 2753373124 q^{59} - 1273745979 q^{60} + 2983465632 q^{61} + 15873384142 q^{62} + 17863857774 q^{63} + 56288390271 q^{64} + 1176256224 q^{65} + 7888719681 q^{66} + 19901432904 q^{67} + 58515314007 q^{68} - 4806444744 q^{69} - 1571269623 q^{70} + 24957812204 q^{71} - 2947194639 q^{72} + 30502187748 q^{73} + 38409721181 q^{74} + 24442554978 q^{75} + 109584856041 q^{76} + 23080044680 q^{77} - 1714259781 q^{78} + 68920908564 q^{79} + 178910341659 q^{80} + 62762119218 q^{81} + 151003817328 q^{82} + 35169061736 q^{83} + 99549961875 q^{84} + 73852445220 q^{85} + 276199974147 q^{86} + 52795024668 q^{87} + 142709055213 q^{88} + 121317306868 q^{89} - 5520431961 q^{90} - 112325786118 q^{91} + 239921140145 q^{92} + 105681797388 q^{93} + 234509546754 q^{94} + 176420904408 q^{95} - 23931256977 q^{96} + 167287034880 q^{97} + 5367029731 q^{98} + 81088448760 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{17} - 30609 x^{16} + 82531 x^{15} + 387432317 x^{14} - 1631290103 x^{13} + \cdots + 29\!\cdots\!20 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3\nu - 3401 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 53\!\cdots\!67 \nu^{17} + \cdots + 41\!\cdots\!80 ) / 49\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 53\!\cdots\!67 \nu^{17} + \cdots - 47\!\cdots\!52 ) / 49\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 54\!\cdots\!27 \nu^{17} + \cdots - 62\!\cdots\!76 ) / 70\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 94\!\cdots\!81 \nu^{17} + \cdots - 22\!\cdots\!00 ) / 49\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 52\!\cdots\!15 \nu^{17} + \cdots - 45\!\cdots\!68 ) / 22\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20\!\cdots\!03 \nu^{17} + \cdots + 70\!\cdots\!04 ) / 70\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 44\!\cdots\!65 \nu^{17} + \cdots - 84\!\cdots\!36 ) / 10\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 16\!\cdots\!17 \nu^{17} + \cdots + 31\!\cdots\!08 ) / 35\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 89\!\cdots\!19 \nu^{17} + \cdots + 12\!\cdots\!92 ) / 70\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 32\!\cdots\!75 \nu^{17} + \cdots + 11\!\cdots\!12 ) / 12\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 22\!\cdots\!79 \nu^{17} + \cdots + 74\!\cdots\!44 ) / 49\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 22\!\cdots\!21 \nu^{17} + \cdots + 55\!\cdots\!68 ) / 49\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 25\!\cdots\!01 \nu^{17} + \cdots - 43\!\cdots\!00 ) / 49\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 28\!\cdots\!21 \nu^{17} + \cdots + 40\!\cdots\!28 ) / 49\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 48\!\cdots\!67 \nu^{17} + \cdots - 54\!\cdots\!24 ) / 49\!\cdots\!12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3\beta _1 + 3401 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} - 3\beta_{2} + 5292\beta _1 - 8947 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + \beta_{5} + 186\beta_{4} - 10\beta_{3} + 7663\beta_{2} - 31484\beta _1 + 17993636 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{17} - \beta_{16} + 11 \beta_{15} + 15 \beta_{14} + 9 \beta_{13} + 5 \beta_{12} + \cdots - 97759730 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 220 \beta_{17} + 126 \beta_{16} + 338 \beta_{15} - 232 \beta_{14} - 1152 \beta_{13} + \cdots + 110518074507 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 32918 \beta_{17} - 31278 \beta_{16} + 181012 \beta_{15} + 192484 \beta_{14} + 126368 \beta_{13} + \cdots - 970613286685 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3752874 \beta_{17} + 2225702 \beta_{16} + 4319476 \beta_{15} - 2433752 \beta_{14} + \cdots + 737970264131878 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 115751154 \beta_{17} - 444872353 \beta_{16} + 2082957141 \beta_{15} + 1815021901 \beta_{14} + \cdots - 92\!\cdots\!44 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 43398989898 \beta_{17} + 31085629944 \beta_{16} + 40061403910 \beta_{15} - 19622278332 \beta_{14} + \cdots + 51\!\cdots\!85 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 787738389388 \beta_{17} - 4841511573458 \beta_{16} + 20791369901354 \beta_{15} + 15263562556062 \beta_{14} + \cdots - 85\!\cdots\!71 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 434751096801704 \beta_{17} + 376400903222904 \beta_{16} + 325145162921264 \beta_{15} + \cdots + 37\!\cdots\!08 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 21\!\cdots\!36 \beta_{17} + \cdots - 75\!\cdots\!82 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 40\!\cdots\!44 \beta_{17} + \cdots + 27\!\cdots\!15 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 28\!\cdots\!10 \beta_{17} + \cdots - 66\!\cdots\!93 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 36\!\cdots\!74 \beta_{17} + \cdots + 20\!\cdots\!86 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 31\!\cdots\!94 \beta_{17} + \cdots - 56\!\cdots\!08 \) 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
−89.8558
−83.8568
−66.3040
−60.3196
−59.0414
−56.0281
−30.7939
−22.3496
1.09911
5.51341
25.4866
34.9599
46.3622
58.3132
61.2171
69.6552
80.4736
86.4689
−88.8558 243.000 5847.36 6346.98 −21592.0 16807.0 −337595. 59049.0 −563966.
1.2 −82.8568 243.000 4817.25 −5994.77 −20134.2 16807.0 −229451. 59049.0 496707.
1.3 −65.3040 243.000 2216.61 4477.12 −15868.9 16807.0 −11011.0 59049.0 −292374.
1.4 −59.3196 243.000 1470.81 −13283.0 −14414.7 16807.0 34238.4 59049.0 787941.
1.5 −58.0414 243.000 1320.80 −4542.56 −14104.1 16807.0 42207.7 59049.0 263656.
1.6 −55.0281 243.000 980.088 6264.12 −13371.8 16807.0 58765.1 59049.0 −344703.
1.7 −29.7939 243.000 −1160.32 956.944 −7239.92 16807.0 95588.5 59049.0 −28511.1
1.8 −21.3496 243.000 −1592.19 10851.9 −5187.95 16807.0 77716.7 59049.0 −231684.
1.9 2.09911 243.000 −2043.59 −6598.78 510.084 16807.0 −8588.72 59049.0 −13851.6
1.10 6.51341 243.000 −2005.58 4699.91 1582.76 16807.0 −26402.6 59049.0 30612.5
1.11 26.4866 243.000 −1346.46 −5189.57 6436.23 16807.0 −89907.6 59049.0 −137454.
1.12 35.9599 243.000 −754.885 −89.1090 8738.26 16807.0 −100791. 59049.0 −3204.35
1.13 47.3622 243.000 195.174 10193.4 11509.0 16807.0 −87753.8 59049.0 482780.
1.14 59.3132 243.000 1470.05 −10452.0 14413.1 16807.0 −34279.8 59049.0 −619944.
1.15 62.2171 243.000 1822.97 −9554.60 15118.8 16807.0 −14000.6 59049.0 −594459.
1.16 70.6552 243.000 2944.16 1509.32 17169.2 16807.0 63318.5 59049.0 106641.
1.17 81.4736 243.000 4589.95 10786.3 19798.1 16807.0 207101. 59049.0 878801.
1.18 87.4689 243.000 5602.81 −3549.59 21254.9 16807.0 310935. 59049.0 −310479.
\(n\): 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.f ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.12.a.f ✓ 18 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - 19 T_{2}^{17} - 30439 T_{2}^{16} + 571323 T_{2}^{15} + 382525012 T_{2}^{14} + \cdots + 66\!\cdots\!32 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(273))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + \cdots + 66\!\cdots\!32 \) Copy content Toggle raw display
$3$ \( (T - 243)^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots - 56\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T - 16807)^{18} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots - 33\!\cdots\!60 \) Copy content Toggle raw display
$13$ \( (T + 371293)^{18} \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots - 30\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 16\!\cdots\!60 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots - 39\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 26\!\cdots\!32 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 14\!\cdots\!40 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 71\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots - 55\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 16\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 33\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 37\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 78\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 36\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 29\!\cdots\!00 \) Copy content Toggle raw display
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