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Properties

Label 141.10.a.c
Level $141$
Weight $10$
Character orbit 141.a
Self dual yes
Analytic conductor $72.620$
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] = [141,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("141.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 141 = 3 \cdot 47 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(72.6200528991\)
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: \( x^{18} - x^{17} - 6997 x^{16} + 5091 x^{15} + 19581988 x^{14} - 6841278 x^{13} - 28177439860 x^{12} + \cdots + 12\!\cdots\!72 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: multiple of \( 2^{24}\cdot 3^{7} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 81 q^{3} + (\beta_{2} + 266) q^{4} + ( - \beta_{4} + 57) q^{5} - 81 \beta_1 q^{6} + (\beta_{7} - \beta_{2} - 4 \beta_1 + 888) q^{7} + (\beta_{4} + \beta_{3} - \beta_{2} + \cdots + 240) q^{8}+ \cdots + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 81 q^{3} + (\beta_{2} + 266) q^{4} + ( - \beta_{4} + 57) q^{5} - 81 \beta_1 q^{6} + (\beta_{7} - \beta_{2} - 4 \beta_1 + 888) q^{7} + (\beta_{4} + \beta_{3} - \beta_{2} + \cdots + 240) q^{8}+ \cdots + (6561 \beta_{16} - 6561 \beta_{15} + \cdots + 16842087) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + q^{2} - 1458 q^{3} + 4779 q^{4} + 1030 q^{5} - 81 q^{6} + 15990 q^{7} + 4695 q^{8} + 118098 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + q^{2} - 1458 q^{3} + 4779 q^{4} + 1030 q^{5} - 81 q^{6} + 15990 q^{7} + 4695 q^{8} + 118098 q^{9} + 6764 q^{10} + 46414 q^{11} - 387099 q^{12} + 170560 q^{13} - 62612 q^{14} - 83430 q^{15} + 2817963 q^{16} - 66968 q^{17} + 6561 q^{18} + 1520800 q^{19} + 581564 q^{20} - 1295190 q^{21} + 1163416 q^{22} + 385934 q^{23} - 380295 q^{24} + 4639296 q^{25} - 9222658 q^{26} - 9565938 q^{27} - 6792936 q^{28} + 4840174 q^{29} - 547884 q^{30} - 2878564 q^{31} - 7938689 q^{32} - 3759534 q^{33} - 6797862 q^{34} - 9837034 q^{35} + 31355019 q^{36} + 20491158 q^{37} + 26896510 q^{38} - 13815360 q^{39} - 40360688 q^{40} - 2789684 q^{41} + 5071572 q^{42} + 7449864 q^{43} - 86864740 q^{44} + 6757830 q^{45} - 117180500 q^{46} + 87834258 q^{47} - 228255003 q^{48} - 24639808 q^{49} - 316151541 q^{50} + 5424408 q^{51} + 56116738 q^{52} - 151952976 q^{53} - 531441 q^{54} + 271521874 q^{55} - 32724636 q^{56} - 123184800 q^{57} + 155119108 q^{58} + 353597432 q^{59} - 47106684 q^{60} + 514582564 q^{61} + 307304506 q^{62} + 104910390 q^{63} + 1537823883 q^{64} + 3648752 q^{65} - 94236696 q^{66} + 536810272 q^{67} + 855459942 q^{68} - 31260654 q^{69} + 2016047312 q^{70} + 680924104 q^{71} + 30803895 q^{72} + 788724356 q^{73} + 1166542642 q^{74} - 375782976 q^{75} + 2088341270 q^{76} + 712051374 q^{77} + 747035298 q^{78} + 1740465906 q^{79} + 2730484544 q^{80} + 774840978 q^{81} + 3855475716 q^{82} + 1996677328 q^{83} + 550227816 q^{84} + 1712406828 q^{85} + 1747596798 q^{86} - 392054094 q^{87} + 6327452000 q^{88} + 1432351408 q^{89} + 44378604 q^{90} + 573194340 q^{91} + 5418051720 q^{92} + 233163684 q^{93} + 4879681 q^{94} + 1102428080 q^{95} + 643033809 q^{96} + 422334166 q^{97} + 5352758569 q^{98} + 304522254 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{17} - 6997 x^{16} + 5091 x^{15} + 19581988 x^{14} - 6841278 x^{13} - 28177439860 x^{12} + \cdots + 12\!\cdots\!72 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 778 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 35\!\cdots\!69 \nu^{17} + \cdots - 12\!\cdots\!60 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 35\!\cdots\!69 \nu^{17} + \cdots + 75\!\cdots\!60 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\!\cdots\!35 \nu^{17} + \cdots - 12\!\cdots\!68 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\!\cdots\!83 \nu^{17} + \cdots - 36\!\cdots\!36 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 23\!\cdots\!47 \nu^{17} + \cdots - 16\!\cdots\!72 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 87\!\cdots\!89 \nu^{17} + \cdots - 13\!\cdots\!84 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13\!\cdots\!81 \nu^{17} + \cdots + 21\!\cdots\!48 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 55\!\cdots\!09 \nu^{17} + \cdots - 67\!\cdots\!84 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11\!\cdots\!15 \nu^{17} + \cdots + 37\!\cdots\!28 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 40\!\cdots\!75 \nu^{17} + \cdots - 11\!\cdots\!04 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 28\!\cdots\!15 \nu^{17} + \cdots - 68\!\cdots\!64 ) / 92\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11\!\cdots\!99 \nu^{17} + \cdots - 20\!\cdots\!20 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 26\!\cdots\!91 \nu^{17} + \cdots + 36\!\cdots\!28 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 20\!\cdots\!33 \nu^{17} + \cdots - 31\!\cdots\!32 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 40\!\cdots\!79 \nu^{17} + \cdots + 32\!\cdots\!68 ) / 44\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 778 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} - \beta_{2} + 1401\beta _1 + 240 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6 \beta_{17} - 4 \beta_{16} + 4 \beta_{15} - 6 \beta_{14} + 3 \beta_{13} - 3 \beta_{11} + \cdots + 1089641 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 122 \beta_{17} + 82 \beta_{16} + 53 \beta_{15} - 59 \beta_{14} - 46 \beta_{13} + 27 \beta_{12} + \cdots + 38920 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 15446 \beta_{17} - 11726 \beta_{16} + 13592 \beta_{15} - 16380 \beta_{14} + 7873 \beta_{13} + \cdots + 1785550799 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 485334 \beta_{17} + 369998 \beta_{16} + 311733 \beta_{15} - 198827 \beta_{14} - 200572 \beta_{13} + \cdots - 197828202 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 32378518 \beta_{17} - 28621786 \beta_{16} + 36310834 \beta_{15} - 33928334 \beta_{14} + \cdots + 3129073016441 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1359350190 \beta_{17} + 1080486750 \beta_{16} + 1023970655 \beta_{15} - 472289593 \beta_{14} + \cdots - 213720102288 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 64661341110 \beta_{17} - 65648759130 \beta_{16} + 87954470688 \beta_{15} - 63846280864 \beta_{14} + \cdots + 56\!\cdots\!47 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3328467201262 \beta_{17} + 2657684199158 \beta_{16} + 2719372601365 \beta_{15} - 965600889979 \beta_{14} + \cdots + 353003979867778 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 127330366441238 \beta_{17} - 145410127651154 \beta_{16} + 201693013086278 \beta_{15} + \cdots + 10\!\cdots\!45 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 76\!\cdots\!62 \beta_{17} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 24\!\cdots\!46 \beta_{17} + \cdots + 19\!\cdots\!91 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 16\!\cdots\!26 \beta_{17} + \cdots + 48\!\cdots\!54 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 48\!\cdots\!66 \beta_{17} + \cdots + 37\!\cdots\!89 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 36\!\cdots\!22 \beta_{17} + \cdots + 80\!\cdots\!44 \) 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
−44.8887
−40.0710
−38.3150
−26.5881
−22.7121
−19.3412
−11.8781
−8.66805
−2.63498
−0.591279
9.09809
11.1806
21.5003
23.1968
30.9353
34.2201
41.9229
44.6344
−44.8887 −81.0000 1503.00 1009.51 3635.99 2363.17 −44484.7 6561.00 −45315.7
1.2 −40.0710 −81.0000 1093.69 −2646.18 3245.75 3073.72 −23308.9 6561.00 106035.
1.3 −38.3150 −81.0000 956.037 2677.56 3103.51 −5881.40 −17013.3 6561.00 −102591.
1.4 −26.5881 −81.0000 194.929 493.880 2153.64 −1163.41 8430.32 6561.00 −13131.4
1.5 −22.7121 −81.0000 3.83988 −1770.65 1839.68 4743.77 11541.4 6561.00 40215.3
1.6 −19.3412 −81.0000 −137.920 −552.282 1566.63 −6810.01 12570.2 6561.00 10681.8
1.7 −11.8781 −81.0000 −370.910 −1157.09 962.129 7181.26 10487.3 6561.00 13744.1
1.8 −8.66805 −81.0000 −436.865 1232.85 702.112 −826.977 8224.81 6561.00 −10686.4
1.9 −2.63498 −81.0000 −505.057 956.680 213.433 12300.5 2679.92 6561.00 −2520.83
1.10 −0.591279 −81.0000 −511.650 −1124.95 47.8936 4257.14 605.263 6561.00 665.158
1.11 9.09809 −81.0000 −429.225 2238.51 −736.945 600.891 −8563.35 6561.00 20366.2
1.12 11.1806 −81.0000 −386.993 932.874 −905.632 −7973.75 −10051.3 6561.00 10430.1
1.13 21.5003 −81.0000 −49.7369 −259.395 −1741.52 −5696.97 −12077.5 6561.00 −5577.07
1.14 23.1968 −81.0000 26.0933 −831.457 −1878.94 10858.6 −11271.5 6561.00 −19287.2
1.15 30.9353 −81.0000 444.990 −1886.90 −2505.76 −3491.42 −2072.96 6561.00 −58371.8
1.16 34.2201 −81.0000 659.014 1700.72 −2771.83 6785.72 5030.84 6561.00 58198.7
1.17 41.9229 −81.0000 1245.53 −1172.59 −3395.76 −8421.17 30751.9 6561.00 −49158.6
1.18 44.6344 −81.0000 1480.23 1188.92 −3615.39 4090.36 43216.5 6561.00 53066.9
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

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

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 141.10.a.c ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.10.a.c ✓ 18 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - T_{2}^{17} - 6997 T_{2}^{16} + 5091 T_{2}^{15} + 19581988 T_{2}^{14} + \cdots + 12\!\cdots\!72 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(141))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + \cdots + 12\!\cdots\!72 \) Copy content Toggle raw display
$3$ \( (T + 81)^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots - 32\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 44\!\cdots\!08 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 21\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 93\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots - 49\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( (T - 4879681)^{18} \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 49\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 34\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots - 40\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 86\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 36\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 74\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 13\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 21\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
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