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Properties

Label 141.12.a.a
Level $141$
Weight $12$
Character orbit 141.a
Self dual yes
Analytic conductor $108.336$
Analytic rank $1$
Dimension $20$
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] = [141,12,Mod(1,141)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(141, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("141.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 141 = 3 \cdot 47 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 141.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: \(108.336388459\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 3 x^{19} - 28908 x^{18} + 116178 x^{17} + 345460466 x^{16} - 1626785346 x^{15} + \cdots - 40\!\cdots\!08 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: multiple of \( 2^{27}\cdot 3^{9} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} - 243 q^{3} + (\beta_{2} + 845) q^{4} + (\beta_{3} + 3 \beta_1 + 395) q^{5} + (243 \beta_1 + 243) q^{6} + (\beta_{4} - 12 \beta_1 - 2931) q^{7} + ( - \beta_{5} + \beta_{4} + 2 \beta_{2} + \cdots + 588) q^{8}+ \cdots + 59049 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} - 243 q^{3} + (\beta_{2} + 845) q^{4} + (\beta_{3} + 3 \beta_1 + 395) q^{5} + (243 \beta_1 + 243) q^{6} + (\beta_{4} - 12 \beta_1 - 2931) q^{7} + ( - \beta_{5} + \beta_{4} + 2 \beta_{2} + \cdots + 588) q^{8}+ \cdots + ( - 59049 \beta_{15} + \cdots - 2543889969) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 23 q^{2} - 4860 q^{3} + 16891 q^{4} + 7901 q^{5} + 5589 q^{6} - 58651 q^{7} + 9039 q^{8} + 1180980 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 23 q^{2} - 4860 q^{3} + 16891 q^{4} + 7901 q^{5} + 5589 q^{6} - 58651 q^{7} + 9039 q^{8} + 1180980 q^{9} - 170300 q^{10} - 854929 q^{11} - 4104513 q^{12} - 520582 q^{13} + 763370 q^{14} - 1919943 q^{15} + 17592875 q^{16} - 2233550 q^{17} - 1358127 q^{18} - 26102926 q^{19} - 14903066 q^{20} + 14252193 q^{21} - 125830698 q^{22} - 37956247 q^{23} - 2196477 q^{24} + 111448343 q^{25} - 47825854 q^{26} - 286978140 q^{27} - 82681732 q^{28} + 42709597 q^{29} + 41382900 q^{30} - 107652956 q^{31} + 269440659 q^{32} + 207747747 q^{33} + 300567782 q^{34} + 95295957 q^{35} + 997396659 q^{36} + 152003289 q^{37} + 1571079296 q^{38} + 126501426 q^{39} - 499629638 q^{40} - 1059422400 q^{41} - 185498910 q^{42} - 856712596 q^{43} - 3495256016 q^{44} + 466546149 q^{45} - 1415819734 q^{46} + 4586900140 q^{47} - 4275068625 q^{48} - 926041453 q^{49} - 945869927 q^{50} + 542752650 q^{51} + 8734530718 q^{52} + 5760147696 q^{53} + 330024861 q^{54} + 15045778187 q^{55} + 22329044624 q^{56} + 6343011018 q^{57} + 28940078284 q^{58} + 4071312342 q^{59} + 3621445038 q^{60} + 5218557164 q^{61} + 29147389816 q^{62} - 3463282899 q^{63} + 52899812155 q^{64} + 20769281248 q^{65} + 30576859614 q^{66} + 3848279932 q^{67} + 18679129698 q^{68} + 9223368021 q^{69} + 400234666 q^{70} + 3253573118 q^{71} + 533743911 q^{72} - 31066581698 q^{73} - 39318576592 q^{74} - 27081947349 q^{75} + 4754484472 q^{76} - 6384609967 q^{77} + 11621682522 q^{78} - 148328817099 q^{79} - 15916527242 q^{80} + 69735688020 q^{81} - 213751985666 q^{82} - 172629153370 q^{83} + 20091660876 q^{84} - 109335451732 q^{85} - 102290100600 q^{86} - 10378432071 q^{87} - 499401992116 q^{88} - 98793989256 q^{89} - 10056044700 q^{90} - 374562988772 q^{91} - 390301536296 q^{92} + 26159668308 q^{93} - 5274935161 q^{94} - 180065577642 q^{95} - 65474080137 q^{96} - 137234750027 q^{97} - 258984423961 q^{98} - 50482702521 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 3 x^{19} - 28908 x^{18} + 116178 x^{17} + 345460466 x^{16} - 1626785346 x^{15} + \cdots - 40\!\cdots\!08 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2\nu - 2892 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\!\cdots\!25 \nu^{19} + \cdots + 70\!\cdots\!80 ) / 15\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 57\!\cdots\!73 \nu^{19} + \cdots + 56\!\cdots\!60 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 57\!\cdots\!73 \nu^{19} + \cdots + 56\!\cdots\!40 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 78\!\cdots\!09 \nu^{19} + \cdots - 51\!\cdots\!92 ) / 86\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 71\!\cdots\!17 \nu^{19} + \cdots + 60\!\cdots\!96 ) / 78\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 56\!\cdots\!37 \nu^{19} + \cdots - 20\!\cdots\!64 ) / 52\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 99\!\cdots\!31 \nu^{19} + \cdots + 10\!\cdots\!20 ) / 78\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 92\!\cdots\!69 \nu^{19} + \cdots - 15\!\cdots\!36 ) / 65\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24\!\cdots\!61 \nu^{19} + \cdots + 87\!\cdots\!48 ) / 15\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 71\!\cdots\!95 \nu^{19} + \cdots - 92\!\cdots\!96 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 66\!\cdots\!73 \nu^{19} + \cdots - 18\!\cdots\!76 ) / 26\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 24\!\cdots\!49 \nu^{19} + \cdots + 27\!\cdots\!04 ) / 97\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 78\!\cdots\!95 \nu^{19} + \cdots + 92\!\cdots\!32 ) / 26\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 28\!\cdots\!83 \nu^{19} + \cdots - 18\!\cdots\!28 ) / 86\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 26\!\cdots\!55 \nu^{19} + \cdots + 64\!\cdots\!64 ) / 78\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 94\!\cdots\!57 \nu^{19} + \cdots - 10\!\cdots\!08 ) / 26\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 14\!\cdots\!43 \nu^{19} + \cdots + 25\!\cdots\!00 ) / 32\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2\beta _1 + 2892 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} - 5\beta_{2} + 5002\beta _1 - 5169 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7 \beta_{19} - 9 \beta_{18} - 8 \beta_{17} + 7 \beta_{16} + 11 \beta_{15} + 3 \beta_{14} + \cdots + 14464672 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 10 \beta_{19} - 145 \beta_{18} - 19 \beta_{17} + 121 \beta_{16} + 115 \beta_{15} - 175 \beta_{14} + \cdots - 69843212 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 81590 \beta_{19} - 97685 \beta_{18} - 86494 \beta_{17} + 69417 \beta_{16} + 114317 \beta_{15} + \cdots + 86716164605 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 197750 \beta_{19} - 1938449 \beta_{18} - 72072 \beta_{17} + 1720981 \beta_{16} + 1408821 \beta_{15} + \cdots - 634014749110 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 730474581 \beta_{19} - 806428546 \beta_{18} - 727542064 \beta_{17} + 547917428 \beta_{16} + \cdots + 565789155561667 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3680678856 \beta_{19} - 18061265406 \beta_{18} + 1137359449 \beta_{17} + 16393497818 \beta_{16} + \cdots - 54\!\cdots\!57 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5998533606244 \beta_{19} - 6073505432930 \beta_{18} - 5617059945294 \beta_{17} + 4026214265034 \beta_{16} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 52285436832124 \beta_{19} - 144350107260242 \beta_{18} + 25972948414508 \beta_{17} + \cdots - 45\!\cdots\!11 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 47\!\cdots\!95 \beta_{19} + \cdots + 26\!\cdots\!66 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 61\!\cdots\!46 \beta_{19} + \cdots - 38\!\cdots\!90 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 37\!\cdots\!02 \beta_{19} + \cdots + 19\!\cdots\!55 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 64\!\cdots\!14 \beta_{19} + \cdots - 31\!\cdots\!28 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 28\!\cdots\!29 \beta_{19} + \cdots + 13\!\cdots\!05 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 62\!\cdots\!44 \beta_{19} + \cdots - 26\!\cdots\!35 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 22\!\cdots\!08 \beta_{19} + \cdots + 98\!\cdots\!02 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 56\!\cdots\!76 \beta_{19} + \cdots - 21\!\cdots\!53 \) 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
84.6403
81.4306
64.5232
60.5747
43.6594
43.2876
42.6752
36.4620
18.3816
−5.10712
−11.8439
−12.3707
−17.0703
−32.7898
−43.6953
−55.2953
−56.0395
−68.7313
−82.1740
−87.5172
−85.6403 −243.000 5286.26 7141.09 20810.6 −15614.4 −277326. 59049.0 −611565.
1.2 −82.4306 −243.000 4746.80 −5399.65 20030.6 −57638.9 −222464. 59049.0 445096.
1.3 −65.5232 −243.000 2245.29 −2512.50 15922.1 59192.0 −12927.1 59049.0 164627.
1.4 −61.5747 −243.000 1743.44 −4119.30 14962.6 24061.0 18753.3 59049.0 253644.
1.5 −44.6594 −243.000 −53.5414 10077.8 10852.2 −73037.1 93853.5 59049.0 −450068.
1.6 −44.2876 −243.000 −86.6123 −12572.5 10761.9 −8114.29 94536.8 59049.0 556807.
1.7 −43.6752 −243.000 −140.476 6264.53 10613.1 36631.8 95582.1 59049.0 −273605.
1.8 −37.4620 −243.000 −644.602 4035.21 9103.25 −4719.52 100870. 59049.0 −151167.
1.9 −19.3816 −243.000 −1672.35 7232.66 4709.74 76767.7 72106.5 59049.0 −140181.
1.10 4.10712 −243.000 −2031.13 −5375.15 −998.030 −47545.9 −16753.5 59049.0 −22076.4
1.11 10.8439 −243.000 −1930.41 9437.85 −2635.08 −16851.6 −43141.6 59049.0 102344.
1.12 11.3707 −243.000 −1918.71 −402.515 −2763.07 40077.6 −45104.1 59049.0 −4576.86
1.13 16.0703 −243.000 −1789.74 −11934.2 −3905.09 −35836.2 −61673.8 59049.0 −191786.
1.14 31.7898 −243.000 −1037.41 7156.41 −7724.93 −83197.3 −98084.6 59049.0 227501.
1.15 42.6953 −243.000 −225.112 −10150.8 −10375.0 5031.74 −97051.2 59049.0 −433391.
1.16 54.2953 −243.000 899.984 12262.1 −13193.8 31691.8 −62331.9 59049.0 665777.
1.17 55.0395 −243.000 981.347 4219.43 −13374.6 −18334.7 −58708.1 59049.0 232235.
1.18 67.7313 −243.000 2539.53 −4611.49 −16458.7 59652.9 33292.3 59049.0 −312343.
1.19 81.1740 −243.000 4541.22 −3523.84 −19725.3 −32115.7 202384. 59049.0 −286044.
1.20 86.5172 −243.000 5437.22 675.831 −21023.7 1248.08 293226. 59049.0 58471.0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(47\) \(-1\)

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 141.12.a.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.12.a.a ✓ 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 23 T_{2}^{19} - 28661 T_{2}^{18} - 634869 T_{2}^{17} + 339070268 T_{2}^{16} + \cdots - 28\!\cdots\!64 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(141))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots - 28\!\cdots\!64 \) Copy content Toggle raw display
$3$ \( (T + 243)^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots - 67\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots - 30\!\cdots\!80 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 78\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots - 33\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 60\!\cdots\!32 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots - 38\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 35\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots - 38\!\cdots\!08 \) Copy content Toggle raw display
$47$ \( (T - 229345007)^{20} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 46\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 28\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots - 10\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 12\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 81\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 12\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots - 35\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
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