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Properties

Label 273.12.a.d
Level $273$
Weight $12$
Character orbit 273.a
Self dual yes
Analytic conductor $209.758$
Analytic rank $1$
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: \(1\)
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: \( x^{17} - 7 x^{16} - 27968 x^{15} + 171016 x^{14} + 319028471 x^{13} - 1580922081 x^{12} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: multiple of \( 2^{17}\cdot 3^{9}\cdot 7^{5} \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + 243 q^{3} + (\beta_{2} - \beta_1 + 1246) q^{4} + (\beta_{4} - 9 \beta_1 - 373) q^{5} + (243 \beta_1 - 243) q^{6} - 16807 q^{7} + (\beta_{4} + \beta_{3} - 6 \beta_{2} + \cdots - 3520) q^{8}+ \cdots + 59049 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + 243 q^{3} + (\beta_{2} - \beta_1 + 1246) q^{4} + (\beta_{4} - 9 \beta_1 - 373) q^{5} + (243 \beta_1 - 243) q^{6} - 16807 q^{7} + (\beta_{4} + \beta_{3} - 6 \beta_{2} + \cdots - 3520) q^{8}+ \cdots + ( - 59049 \beta_{8} - 59049 \beta_{7} + \cdots - 998872884) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q - 10 q^{2} + 4131 q^{3} + 21172 q^{4} - 6405 q^{5} - 2430 q^{6} - 285719 q^{7} - 52368 q^{8} + 1003833 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q - 10 q^{2} + 4131 q^{3} + 21172 q^{4} - 6405 q^{5} - 2430 q^{6} - 285719 q^{7} - 52368 q^{8} + 1003833 q^{9} - 527345 q^{10} - 291538 q^{11} + 5144796 q^{12} - 6311981 q^{13} + 168070 q^{14} - 1556415 q^{15} + 16350148 q^{16} - 9278416 q^{17} - 590490 q^{18} - 1961467 q^{19} - 12100505 q^{20} - 69429717 q^{21} - 31039541 q^{22} + 33267339 q^{23} - 12725424 q^{24} + 210926674 q^{25} + 3712930 q^{26} + 243931419 q^{27} - 355837804 q^{28} - 80052755 q^{29} - 128144835 q^{30} - 277207587 q^{31} - 598963644 q^{32} - 70843734 q^{33} - 167909439 q^{34} + 107648835 q^{35} + 1250185428 q^{36} - 71963520 q^{37} + 1131189079 q^{38} - 1533811383 q^{39} + 618371277 q^{40} - 86744162 q^{41} + 40841010 q^{42} - 866138645 q^{43} + 1809436635 q^{44} - 378208845 q^{45} + 1664942397 q^{46} + 460740083 q^{47} + 3973085964 q^{48} + 4802079233 q^{49} - 4662633641 q^{50} - 2254655088 q^{51} - 7861015396 q^{52} - 5981625787 q^{53} - 143489070 q^{54} - 10019177398 q^{55} + 880148976 q^{56} - 476636481 q^{57} - 22147081307 q^{58} - 8556315252 q^{59} - 2940422715 q^{60} + 1279540394 q^{61} - 11043995896 q^{62} - 16871421231 q^{63} - 9178730272 q^{64} + 2378131665 q^{65} - 7542608463 q^{66} - 28221361318 q^{67} - 75930275803 q^{68} + 8083963377 q^{69} + 8863087415 q^{70} - 54038549984 q^{71} - 3092278032 q^{72} - 20877618147 q^{73} - 94723156781 q^{74} + 51255181782 q^{75} - 74563238549 q^{76} + 4899879166 q^{77} + 902241990 q^{78} - 68638438265 q^{79} - 292371455125 q^{80} + 59275334817 q^{81} - 305606451022 q^{82} - 204094126985 q^{83} - 86468586372 q^{84} - 187886195916 q^{85} - 313944604505 q^{86} - 19452819465 q^{87} - 277518424767 q^{88} - 172724189963 q^{89} - 31139194905 q^{90} + 106085464667 q^{91} - 295734666351 q^{92} - 67361443641 q^{93} - 296263483140 q^{94} - 53779729719 q^{95} - 145548165492 q^{96} - 24546366723 q^{97} - 2824752490 q^{98} - 17215027362 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 7 x^{16} - 27968 x^{15} + 171016 x^{14} + 319028471 x^{13} - 1580922081 x^{12} + \cdots + 63\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3293 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\!\cdots\!71 \nu^{16} + \cdots - 48\!\cdots\!00 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15\!\cdots\!71 \nu^{16} + \cdots + 45\!\cdots\!40 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 44\!\cdots\!91 \nu^{16} + \cdots + 11\!\cdots\!00 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 31\!\cdots\!64 \nu^{16} + \cdots + 30\!\cdots\!68 ) / 15\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 26\!\cdots\!09 \nu^{16} + \cdots - 10\!\cdots\!60 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 22\!\cdots\!09 \nu^{16} + \cdots + 36\!\cdots\!60 ) / 88\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 83\!\cdots\!37 \nu^{16} + \cdots - 10\!\cdots\!00 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 98\!\cdots\!37 \nu^{16} + \cdots + 45\!\cdots\!28 ) / 24\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 66\!\cdots\!11 \nu^{16} + \cdots - 23\!\cdots\!40 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!67 \nu^{16} + \cdots + 11\!\cdots\!60 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 19\!\cdots\!13 \nu^{16} + \cdots - 11\!\cdots\!00 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 56\!\cdots\!87 \nu^{16} + \cdots - 23\!\cdots\!40 ) / 61\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 21\!\cdots\!59 \nu^{16} + \cdots + 45\!\cdots\!60 ) / 11\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 82\!\cdots\!17 \nu^{16} + \cdots + 46\!\cdots\!40 ) / 24\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3293 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} - 3\beta_{2} + 5162\beta _1 + 2264 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + \cdots + 17000390 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 18 \beta_{16} - 70 \beta_{15} - 18 \beta_{14} - 15 \beta_{13} + 51 \beta_{12} + 8 \beta_{11} + \cdots - 206643 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 134 \beta_{16} + 10785 \beta_{15} - 484 \beta_{14} - 3429 \beta_{13} - 849 \beta_{12} + \cdots + 99105895723 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 211472 \beta_{16} - 892576 \beta_{15} - 183480 \beta_{14} - 138736 \beta_{13} + 649200 \beta_{12} + \cdots - 112575106160 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 877784 \beta_{16} + 97111251 \beta_{15} - 6128392 \beta_{14} - 47177824 \beta_{13} + \cdots + 611164274342048 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1872883998 \beta_{16} - 8527644506 \beta_{15} - 1537237678 \beta_{14} - 973873021 \beta_{13} + \cdots - 14\!\cdots\!69 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 28071107702 \beta_{16} + 813839902033 \beta_{15} - 60688998468 \beta_{14} - 462647030759 \beta_{13} + \cdots + 39\!\cdots\!21 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 15057372010244 \beta_{16} - 73116260639124 \beta_{15} - 12237792576892 \beta_{14} + \cdots - 13\!\cdots\!98 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 365307019680444 \beta_{16} + \cdots + 25\!\cdots\!46 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 11\!\cdots\!02 \beta_{16} + \cdots - 12\!\cdots\!43 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 37\!\cdots\!74 \beta_{16} + \cdots + 17\!\cdots\!75 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 87\!\cdots\!20 \beta_{16} + \cdots - 10\!\cdots\!52 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 33\!\cdots\!96 \beta_{16} + \cdots + 12\!\cdots\!08 \) 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
−86.2391
−72.6846
−72.6710
−70.7113
−40.7493
−36.5654
−35.5886
−10.3093
−9.65646
24.1271
24.1945
35.2995
58.7043
66.1921
74.0782
74.3829
85.1965
−87.2391 243.000 5562.67 −8148.17 −21199.1 −16807.0 −306616. 59049.0 710839.
1.2 −73.6846 243.000 3381.42 13105.2 −17905.4 −16807.0 −98252.2 59049.0 −965649.
1.3 −73.6710 243.000 3379.42 −3108.08 −17902.1 −16807.0 −98086.9 59049.0 228976.
1.4 −71.7113 243.000 3094.52 −489.938 −17425.9 −16807.0 −75047.2 59049.0 35134.1
1.5 −41.7493 243.000 −304.992 10129.7 −10145.1 −16807.0 98235.9 59049.0 −422908.
1.6 −37.5654 243.000 −636.843 5731.85 −9128.38 −16807.0 100857. 59049.0 −215319.
1.7 −36.5886 243.000 −709.271 −10702.0 −8891.04 −16807.0 100885. 59049.0 391570.
1.8 −11.3093 243.000 −1920.10 −7263.56 −2748.15 −16807.0 44876.3 59049.0 82145.5
1.9 −10.6565 243.000 −1934.44 4166.03 −2589.52 −16807.0 42438.7 59049.0 −44395.1
1.10 23.1271 243.000 −1513.14 −3771.74 5619.89 −16807.0 −82358.9 59049.0 −87229.6
1.11 23.1945 243.000 −1510.02 −12055.9 5636.26 −16807.0 −82526.4 59049.0 −279630.
1.12 34.2995 243.000 −871.546 4933.24 8334.77 −16807.0 −100139. 59049.0 169208.
1.13 57.7043 243.000 1281.78 5600.30 14022.1 −16807.0 −44214.0 59049.0 323162.
1.14 65.1921 243.000 2202.01 10158.7 15841.7 −16807.0 10040.5 59049.0 662269.
1.15 73.0782 243.000 3292.43 −11796.1 17758.0 −16807.0 90940.6 59049.0 −862040.
1.16 73.3829 243.000 3337.05 902.926 17832.0 −16807.0 94593.8 59049.0 66259.3
1.17 84.1965 243.000 5041.06 −3797.49 20459.8 −16807.0 252005. 59049.0 −319736.
\(n\): 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.d ✓ 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.12.a.d ✓ 17 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{17} + 10 T_{2}^{16} - 27944 T_{2}^{15} - 248664 T_{2}^{14} + 318484515 T_{2}^{13} + \cdots + 73\!\cdots\!88 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(273))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} + \cdots + 73\!\cdots\!88 \) Copy content Toggle raw display
$3$ \( (T - 243)^{17} \) Copy content Toggle raw display
$5$ \( T^{17} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T + 16807)^{17} \) Copy content Toggle raw display
$11$ \( T^{17} + \cdots - 13\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( (T + 371293)^{17} \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots - 68\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots - 85\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{17} + \cdots - 15\!\cdots\!12 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots - 99\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots + 19\!\cdots\!80 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots - 45\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots + 10\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots + 17\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 42\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots + 79\!\cdots\!24 \) Copy content Toggle raw display
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