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Properties

Label 273.12.a.b
Level $273$
Weight $12$
Character orbit 273.a
Self dual yes
Analytic conductor $209.758$
Analytic rank $0$
Dimension $15$
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: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 5 x^{14} - 23794 x^{13} + 53918 x^{12} + 223391819 x^{11} - 109456151 x^{10} + \cdots + 30\!\cdots\!88 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: multiple of \( 2^{16}\cdot 3^{6}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 7) q^{2} - 243 q^{3} + (\beta_{2} - 10 \beta_1 + 1174) q^{4} + ( - \beta_{3} - 21 \beta_1 + 413) q^{5} + (243 \beta_1 - 1701) q^{6} - 16807 q^{7} + ( - \beta_{4} - 3 \beta_{3} + \cdots + 26964) q^{8}+ \cdots + 59049 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 7) q^{2} - 243 q^{3} + (\beta_{2} - 10 \beta_1 + 1174) q^{4} + ( - \beta_{3} - 21 \beta_1 + 413) q^{5} + (243 \beta_1 - 1701) q^{6} - 16807 q^{7} + ( - \beta_{4} - 3 \beta_{3} + \cdots + 26964) q^{8}+ \cdots + (59049 \beta_{11} + 59049 \beta_{8} + \cdots - 1517264055) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 100 q^{2} - 3645 q^{3} + 17558 q^{4} + 6095 q^{5} - 24300 q^{6} - 252105 q^{7} + 399402 q^{8} + 885735 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 100 q^{2} - 3645 q^{3} + 17558 q^{4} + 6095 q^{5} - 24300 q^{6} - 252105 q^{7} + 399402 q^{8} + 885735 q^{9} + 1022855 q^{10} - 388738 q^{11} - 4266594 q^{12} - 5569395 q^{13} - 1680700 q^{14} - 1481085 q^{15} + 14897298 q^{16} + 8975160 q^{17} + 5904900 q^{18} + 10460989 q^{19} + 10830745 q^{20} + 61261515 q^{21} + 28740323 q^{22} + 31292367 q^{23} - 97054686 q^{24} - 31523900 q^{25} - 37129300 q^{26} - 215233605 q^{27} - 295097306 q^{28} - 266712559 q^{29} - 248553765 q^{30} + 38726725 q^{31} + 390421586 q^{32} + 94463334 q^{33} + 314361153 q^{34} - 102438665 q^{35} + 1036782342 q^{36} + 480805308 q^{37} + 570943153 q^{38} + 1353362985 q^{39} + 3438272645 q^{40} + 2051020930 q^{41} + 408410100 q^{42} - 775705061 q^{43} + 1568731765 q^{44} + 359903655 q^{45} + 1126037693 q^{46} + 501527679 q^{47} - 3620043414 q^{48} + 4237128735 q^{49} - 305604385 q^{50} - 2180963880 q^{51} - 6519162494 q^{52} - 1656880767 q^{53} - 1434890700 q^{54} - 1743230910 q^{55} - 6712749414 q^{56} - 2542020327 q^{57} - 13398884899 q^{58} - 865195488 q^{59} - 2631871035 q^{60} + 210684974 q^{61} - 9262636236 q^{62} - 14886548145 q^{63} + 1184466146 q^{64} - 2263030835 q^{65} - 6983898489 q^{66} + 1630564138 q^{67} + 22586366835 q^{68} - 7604045181 q^{69} - 17191123985 q^{70} + 7284674496 q^{71} + 23584288698 q^{72} - 2923610359 q^{73} - 67219804647 q^{74} + 7660307700 q^{75} + 67374882771 q^{76} + 6533519566 q^{77} + 9022419900 q^{78} + 76702645855 q^{79} + 95956085605 q^{80} + 52301766015 q^{81} + 94190708466 q^{82} + 8683989747 q^{83} + 71708645358 q^{84} - 89514225460 q^{85} + 51690787061 q^{86} + 64811151837 q^{87} + 179292998025 q^{88} - 5477740471 q^{89} + 60398564895 q^{90} + 93604821765 q^{91} + 122306479383 q^{92} - 9410594175 q^{93} + 219952417436 q^{94} - 6973459855 q^{95} - 94872445398 q^{96} + 200031380073 q^{97} + 28247524900 q^{98} - 22954590162 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 5 x^{14} - 23794 x^{13} + 53918 x^{12} + 223391819 x^{11} - 109456151 x^{10} + \cdots + 30\!\cdots\!88 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4\nu - 3173 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\!\cdots\!13 \nu^{14} + \cdots - 26\!\cdots\!32 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!13 \nu^{14} + \cdots + 26\!\cdots\!32 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 62\!\cdots\!07 \nu^{14} + \cdots - 14\!\cdots\!56 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 67\!\cdots\!87 \nu^{14} + \cdots - 13\!\cdots\!92 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17\!\cdots\!17 \nu^{14} + \cdots + 38\!\cdots\!56 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 25\!\cdots\!85 \nu^{14} + \cdots - 66\!\cdots\!96 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10\!\cdots\!23 \nu^{14} + \cdots + 27\!\cdots\!80 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13\!\cdots\!35 \nu^{14} + \cdots - 33\!\cdots\!76 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 89\!\cdots\!73 \nu^{14} + \cdots - 22\!\cdots\!32 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 90\!\cdots\!79 \nu^{14} + \cdots - 22\!\cdots\!04 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 26\!\cdots\!69 \nu^{14} + \cdots + 62\!\cdots\!16 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 34\!\cdots\!01 \nu^{14} + \cdots - 85\!\cdots\!36 ) / 40\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4\beta _1 + 3173 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 3\beta_{3} + 13\beta_{2} + 5044\beta _1 + 11340 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{13} + 4 \beta_{11} + 2 \beta_{9} + 4 \beta_{7} + 4 \beta_{6} + 8 \beta_{5} + 19 \beta_{4} + \cdots + 15984867 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 88 \beta_{14} + 52 \beta_{13} - 56 \beta_{12} - 64 \beta_{11} + 56 \beta_{10} - 36 \beta_{9} + \cdots + 170893812 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 584 \beta_{14} + 15684 \beta_{13} - 9240 \beta_{12} + 48000 \beta_{11} - 3304 \beta_{10} + \cdots + 92998697529 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1304792 \beta_{14} + 536608 \beta_{13} - 1147064 \beta_{12} - 450536 \beta_{11} + 761080 \beta_{10} + \cdots + 1769841047100 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5517512 \beta_{14} + 97413918 \beta_{13} - 150285288 \beta_{12} + 426586052 \beta_{11} + \cdots + 584565781498111 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 13727902976 \beta_{14} + 4223615604 \beta_{13} - 14425879680 \beta_{12} - 1130781208 \beta_{11} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 189996515184 \beta_{14} + 560953092816 \beta_{13} - 1698994598448 \beta_{12} + 3445801855376 \beta_{11} + \cdots + 38\!\cdots\!37 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 125924467343888 \beta_{14} + 29752580903568 \beta_{13} - 150425229640016 \beta_{12} + \cdots + 13\!\cdots\!72 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 28\!\cdots\!68 \beta_{14} + \cdots + 26\!\cdots\!87 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 10\!\cdots\!48 \beta_{14} + \cdots + 11\!\cdots\!80 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 32\!\cdots\!68 \beta_{14} + \cdots + 18\!\cdots\!49 \) 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
88.3028
79.7906
61.6374
59.3426
43.9584
25.0592
12.3226
12.1963
−23.6753
−27.8412
−46.1319
−61.8171
−64.7086
−73.2657
−80.1701
−81.3028 −243.000 4562.14 −1608.20 19756.6 −16807.0 −204407. 59049.0 130751.
1.2 −72.7906 −243.000 3250.47 −9325.43 17688.1 −16807.0 −87528.5 59049.0 678804.
1.3 −54.6374 −243.000 937.249 7756.90 13276.9 −16807.0 60688.6 59049.0 −423817.
1.4 −52.3426 −243.000 691.747 6665.38 12719.2 −16807.0 70989.8 59049.0 −348883.
1.5 −36.9584 −243.000 −682.074 −10594.0 8980.90 −16807.0 100899. 59049.0 391538.
1.6 −18.0592 −243.000 −1721.86 5325.80 4388.39 −16807.0 68080.8 59049.0 −96179.8
1.7 −5.32255 −243.000 −2019.67 −4298.90 1293.38 −16807.0 21650.4 59049.0 22881.1
1.8 −5.19627 −243.000 −2021.00 1560.79 1262.69 −16807.0 21143.6 59049.0 −8110.26
1.9 30.6753 −243.000 −1107.03 9054.72 −7454.10 −16807.0 −96781.4 59049.0 277756.
1.10 34.8412 −243.000 −834.091 −4809.24 −8466.41 −16807.0 −100416. 59049.0 −167560.
1.11 53.1319 −243.000 774.995 4221.10 −12911.0 −16807.0 −67637.1 59049.0 224275.
1.12 68.8171 −243.000 2687.79 −3402.19 −16722.6 −16807.0 44028.5 59049.0 −234129.
1.13 71.7086 −243.000 3094.12 −9678.37 −17425.2 −16807.0 75015.6 59049.0 −694022.
1.14 80.2657 −243.000 4394.59 8365.52 −19504.6 −16807.0 188351. 59049.0 671465.
1.15 87.1701 −243.000 5550.62 6861.14 −21182.3 −16807.0 305324. 59049.0 598086.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.b ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.12.a.b ✓ 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} - 100 T_{2}^{14} - 19139 T_{2}^{13} + 1977566 T_{2}^{12} + 139633368 T_{2}^{11} + \cdots - 61\!\cdots\!44 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(273))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + \cdots - 61\!\cdots\!44 \) Copy content Toggle raw display
$3$ \( (T + 243)^{15} \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T + 16807)^{15} \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 49\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( (T + 371293)^{15} \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots - 16\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 14\!\cdots\!08 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 16\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 93\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 12\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 11\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 15\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 12\!\cdots\!40 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 75\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 12\!\cdots\!40 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 59\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 43\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 33\!\cdots\!20 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 53\!\cdots\!60 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 20\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 13\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 59\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 54\!\cdots\!64 \) Copy content Toggle raw display
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