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Properties

Label 273.12.a.c
Level $273$
Weight $12$
Character orbit 273.a
Self dual yes
Analytic conductor $209.758$
Analytic rank $1$
Dimension $16$
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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} - 26640 x^{14} - 38138 x^{13} + 279954048 x^{12} + 940095168 x^{11} + \cdots + 45\!\cdots\!68 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{12}\cdot 5\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 4) q^{2} - 243 q^{3} + (\beta_{2} - 4 \beta_1 + 1298) q^{4} + (\beta_{3} + 20 \beta_1 - 199) q^{5} + ( - 243 \beta_1 + 972) q^{6} + 16807 q^{7} + (\beta_{4} - 2 \beta_{3} + \cdots - 11845) q^{8}+ \cdots + 59049 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 4) q^{2} - 243 q^{3} + (\beta_{2} - 4 \beta_1 + 1298) q^{4} + (\beta_{3} + 20 \beta_1 - 199) q^{5} + ( - 243 \beta_1 + 972) q^{6} + 16807 q^{7} + (\beta_{4} - 2 \beta_{3} + \cdots - 11845) q^{8}+ \cdots + (59049 \beta_{9} - 59049 \beta_{4} + \cdots - 1720805958) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 63 q^{2} - 3888 q^{3} + 20761 q^{4} - 3168 q^{5} + 15309 q^{6} + 268912 q^{7} - 187965 q^{8} + 944784 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 63 q^{2} - 3888 q^{3} + 20761 q^{4} - 3168 q^{5} + 15309 q^{6} + 268912 q^{7} - 187965 q^{8} + 944784 q^{9} + 1056711 q^{10} - 466884 q^{11} - 5044923 q^{12} - 5940688 q^{13} - 1058841 q^{14} + 769824 q^{15} + 40058265 q^{16} + 1753452 q^{17} - 3720087 q^{18} + 4237800 q^{19} - 20710503 q^{20} - 65345616 q^{21} - 37479661 q^{22} - 150481440 q^{23} + 45675495 q^{24} + 150983176 q^{25} + 23391459 q^{26} - 229582512 q^{27} + 348930127 q^{28} - 111411432 q^{29} - 256780773 q^{30} - 90536236 q^{31} - 609941313 q^{32} + 113452812 q^{33} + 443745577 q^{34} - 53244576 q^{35} + 1225916289 q^{36} - 1756337900 q^{37} - 4378868973 q^{38} + 1443587184 q^{39} + 57535383 q^{40} + 649575720 q^{41} + 257298363 q^{42} - 1889554520 q^{43} - 4979587689 q^{44} - 187067232 q^{45} - 3204000867 q^{46} - 4036082940 q^{47} - 9734158395 q^{48} + 4519603984 q^{49} - 15928886862 q^{50} - 426088836 q^{51} - 7708413973 q^{52} + 511144020 q^{53} + 903981141 q^{54} - 8519365572 q^{55} - 3159127755 q^{56} - 1029785400 q^{57} - 7851149577 q^{58} + 3728287296 q^{59} + 5032652229 q^{60} + 26227205052 q^{61} + 2996618490 q^{62} + 15878984688 q^{63} + 51104408465 q^{64} + 1176256224 q^{65} + 9107557623 q^{66} - 5295680024 q^{67} + 7919540325 q^{68} + 36566989920 q^{69} + 17760141777 q^{70} + 17082800928 q^{71} - 11099145285 q^{72} + 21076301488 q^{73} + 52794333675 q^{74} - 36688911768 q^{75} + 81459179177 q^{76} - 7846919388 q^{77} - 5684124537 q^{78} - 83677977852 q^{79} - 163988465427 q^{80} + 55788550416 q^{81} - 61543728760 q^{82} - 72857072340 q^{83} - 84790020861 q^{84} - 12608565060 q^{85} - 20177818011 q^{86} + 27072977976 q^{87} - 202213679643 q^{88} + 32238476676 q^{89} + 62397727839 q^{90} - 99845143216 q^{91} - 487057197033 q^{92} + 22000305348 q^{93} + 183904776602 q^{94} - 143022915504 q^{95} + 148215739059 q^{96} + 6973535140 q^{97} - 17795940687 q^{98} - 27569033316 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} - 26640 x^{14} - 38138 x^{13} + 279954048 x^{12} + 940095168 x^{11} + \cdots + 45\!\cdots\!68 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4\nu - 3330 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14\!\cdots\!61 \nu^{15} + \cdots + 17\!\cdots\!16 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\!\cdots\!61 \nu^{15} + \cdots + 17\!\cdots\!16 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 28\!\cdots\!61 \nu^{15} + \cdots - 34\!\cdots\!16 ) / 24\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 51\!\cdots\!41 \nu^{15} + \cdots - 62\!\cdots\!96 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 88\!\cdots\!71 \nu^{15} + \cdots - 10\!\cdots\!96 ) / 77\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 14\!\cdots\!17 \nu^{15} + \cdots + 17\!\cdots\!72 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 81\!\cdots\!79 \nu^{15} + \cdots - 99\!\cdots\!24 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 61\!\cdots\!71 \nu^{15} + \cdots + 75\!\cdots\!36 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 45\!\cdots\!39 \nu^{15} + \cdots - 55\!\cdots\!64 ) / 24\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 78\!\cdots\!29 \nu^{15} + \cdots + 96\!\cdots\!04 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17\!\cdots\!93 \nu^{15} + \cdots + 21\!\cdots\!68 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 49\!\cdots\!03 \nu^{15} + \cdots - 60\!\cdots\!08 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 12\!\cdots\!89 \nu^{15} + \cdots + 14\!\cdots\!24 ) / 24\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4\beta _1 + 3330 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 2\beta_{3} + 6\beta_{2} + 5629\beta _1 + 11795 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} - \beta_{14} - 2 \beta_{13} + 5 \beta_{12} + \beta_{11} - \beta_{9} + 3 \beta_{8} + \cdots + 18737114 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 223 \beta_{15} + 231 \beta_{14} - 76 \beta_{13} - \beta_{12} - 67 \beta_{11} - 10 \beta_{10} + \cdots + 155790112 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4519 \beta_{15} - 11911 \beta_{14} - 33224 \beta_{13} + 65461 \beta_{12} + 14315 \beta_{11} + \cdots + 120038018864 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3016319 \beta_{15} + 2826975 \beta_{14} - 1359700 \beta_{13} + 515327 \beta_{12} - 667675 \beta_{11} + \cdots + 1631988243260 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 19632589 \beta_{15} - 90806243 \beta_{14} - 371302268 \beta_{13} + 643084029 \beta_{12} + \cdots + 819405787445428 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 30318886279 \beta_{15} + 26065139479 \beta_{14} - 17434683404 \beta_{13} + 10583726927 \beta_{12} + \cdots + 16\!\cdots\!88 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 658708717629 \beta_{15} - 531994143347 \beta_{14} - 3594417665428 \beta_{13} + 5713230246213 \beta_{12} + \cdots + 58\!\cdots\!44 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 275044655337567 \beta_{15} + 218121123242959 \beta_{14} - 192481955056812 \beta_{13} + \cdots + 15\!\cdots\!48 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 92\!\cdots\!13 \beta_{15} + \cdots + 43\!\cdots\!36 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 23\!\cdots\!03 \beta_{15} + \cdots + 14\!\cdots\!88 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 10\!\cdots\!05 \beta_{15} + \cdots + 32\!\cdots\!20 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 20\!\cdots\!27 \beta_{15} + \cdots + 13\!\cdots\!80 \) 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
−84.0758
−81.4415
−68.0202
−65.6788
−37.3711
−37.3707
−23.9940
−6.60872
9.73282
18.3601
25.8857
41.2629
67.4693
72.4378
78.0478
92.3644
−88.0758 −243.000 5709.35 −13401.9 21402.4 16807.0 −322476. 59049.0 1.18039e6
1.2 −85.4415 −243.000 5252.25 8053.09 20762.3 16807.0 −273776. 59049.0 −688068.
1.3 −72.0202 −243.000 3138.91 10386.5 17500.9 16807.0 −78567.7 59049.0 −748037.
1.4 −69.6788 −243.000 2807.14 −8533.64 16932.0 16807.0 −52895.7 59049.0 594614.
1.5 −41.3711 −243.000 −336.429 −4784.52 10053.2 16807.0 98646.5 59049.0 197941.
1.6 −41.3707 −243.000 −336.467 −335.043 10053.1 16807.0 98647.0 59049.0 13861.0
1.7 −27.9940 −243.000 −1264.33 −7765.11 6802.55 16807.0 92725.6 59049.0 217377.
1.8 −10.6087 −243.000 −1935.46 7850.01 2577.92 16807.0 42259.4 59049.0 −83278.5
1.9 5.73282 −243.000 −2015.13 −168.612 −1393.07 16807.0 −23293.2 59049.0 −966.623
1.10 14.3601 −243.000 −1841.79 4424.64 −3489.51 16807.0 −55857.8 59049.0 63538.3
1.11 21.8857 −243.000 −1569.02 −9779.77 −5318.23 16807.0 −79161.0 59049.0 −214037.
1.12 37.2629 −243.000 −659.475 3035.84 −9054.89 16807.0 −100888. 59049.0 113124.
1.13 63.4693 −243.000 1980.36 12941.9 −15423.1 16807.0 −4293.20 59049.0 821414.
1.14 68.4378 −243.000 2635.73 3481.49 −16630.4 16807.0 40223.3 59049.0 238265.
1.15 74.0478 −243.000 3435.08 −7551.16 −17993.6 16807.0 102710. 59049.0 −559147.
1.16 88.3644 −243.000 5760.27 −1021.65 −21472.6 16807.0 328033. 59049.0 −90277.5
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
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.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.12.a.c ✓ 16 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 63 T_{2}^{15} - 24780 T_{2}^{14} - 1495818 T_{2}^{13} + 239619832 T_{2}^{12} + \cdots + 36\!\cdots\!08 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(273))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 36\!\cdots\!08 \) Copy content Toggle raw display
$3$ \( (T + 243)^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots - 71\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T - 16807)^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T + 371293)^{16} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 19\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots - 18\!\cdots\!12 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 53\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots - 99\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 73\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 51\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots - 71\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots - 53\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots - 62\!\cdots\!60 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots - 67\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots - 86\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots - 54\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 37\!\cdots\!40 \) Copy content Toggle raw display
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