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Properties

Label 6171.2.a.bs
Level $6171$
Weight $2$
Character orbit 6171.a
Self dual yes
Analytic conductor $49.276$
Analytic rank $0$
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] = [6171,2,Mod(1,6171)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6171, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6171.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6171 = 3 \cdot 11^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6171.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: \(49.2756830873\)
Analytic rank: \(0\)
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: \( x^{20} - 3 x^{19} - 28 x^{18} + 85 x^{17} + 320 x^{16} - 989 x^{15} - 1923 x^{14} + 6124 x^{13} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 561)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} - \beta_{12} q^{5} - \beta_1 q^{6} - \beta_{8} q^{7} + ( - \beta_{4} + \beta_{3} + 2 \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} - \beta_{12} q^{5} - \beta_1 q^{6} - \beta_{8} q^{7} + ( - \beta_{4} + \beta_{3} + 2 \beta_1) q^{8} + q^{9} + (\beta_{19} + \beta_{17} - \beta_{13} + \cdots - 3) q^{10}+ \cdots + (\beta_{19} + \beta_{18} + \cdots + 6 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 3 q^{2} - 20 q^{3} + 25 q^{4} + 7 q^{5} - 3 q^{6} - 5 q^{7} + 12 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 3 q^{2} - 20 q^{3} + 25 q^{4} + 7 q^{5} - 3 q^{6} - 5 q^{7} + 12 q^{8} + 20 q^{9} + 4 q^{10} - 25 q^{12} + 8 q^{13} + 3 q^{14} - 7 q^{15} + 47 q^{16} + 20 q^{17} + 3 q^{18} + 6 q^{19} + 26 q^{20} + 5 q^{21} - 16 q^{23} - 12 q^{24} + 31 q^{25} + 15 q^{26} - 20 q^{27} - 15 q^{28} + 4 q^{29} - 4 q^{30} + 27 q^{31} + 47 q^{32} + 3 q^{34} - 2 q^{35} + 25 q^{36} + 36 q^{37} - 8 q^{39} - 31 q^{40} + 6 q^{41} - 3 q^{42} + 46 q^{43} + 7 q^{45} - 25 q^{46} - 24 q^{47} - 47 q^{48} + 73 q^{49} - 20 q^{51} + 8 q^{52} + 6 q^{53} - 3 q^{54} + 39 q^{56} - 6 q^{57} - 25 q^{58} + 4 q^{59} - 26 q^{60} - 38 q^{61} - 9 q^{62} - 5 q^{63} + 100 q^{64} + 26 q^{65} + 21 q^{67} + 25 q^{68} + 16 q^{69} + 9 q^{70} - 24 q^{71} + 12 q^{72} + 5 q^{73} - 94 q^{74} - 31 q^{75} + 81 q^{76} - 15 q^{78} - 47 q^{79} + 17 q^{80} + 20 q^{81} + 75 q^{82} + 6 q^{83} + 15 q^{84} + 7 q^{85} + 54 q^{86} - 4 q^{87} + 44 q^{89} + 4 q^{90} + 37 q^{91} - 28 q^{92} - 27 q^{93} - 36 q^{94} + 18 q^{95} - 47 q^{96} + 19 q^{97} + 81 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 3 x^{19} - 28 x^{18} + 85 x^{17} + 320 x^{16} - 989 x^{15} - 1923 x^{14} + 6124 x^{13} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1063572 \nu^{19} + 3267296 \nu^{18} + 29516553 \nu^{17} - 92482639 \nu^{16} - 332768308 \nu^{15} + \cdots - 168384556 ) / 4750974 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1063572 \nu^{19} + 3267296 \nu^{18} + 29516553 \nu^{17} - 92482639 \nu^{16} - 332768308 \nu^{15} + \cdots - 168384556 ) / 4750974 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1980029 \nu^{19} - 5659909 \nu^{18} - 56276434 \nu^{17} + 161392690 \nu^{16} + 654134596 \nu^{15} + \cdots + 299891768 ) / 4750974 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 22566295 \nu^{19} - 68863087 \nu^{18} - 628232294 \nu^{17} + 1952047839 \nu^{16} + \cdots + 3562844072 ) / 19003896 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8280122 \nu^{19} + 26023794 \nu^{18} + 227724259 \nu^{17} - 734295830 \nu^{16} + \cdots - 1305190500 ) / 4750974 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9420659 \nu^{19} - 29116951 \nu^{18} - 260963923 \nu^{17} + 823990813 \nu^{16} + 2935003216 \nu^{15} + \cdots + 1515449264 ) / 4750974 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 42096139 \nu^{19} - 130542705 \nu^{18} - 1165622708 \nu^{17} + 3696238027 \nu^{16} + \cdots + 6805713192 ) / 19003896 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 25871103 \nu^{19} + 80174474 \nu^{18} + 715943643 \nu^{17} - 2268103105 \nu^{16} + \cdots - 4115315692 ) / 9501948 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 26584379 \nu^{19} - 82624487 \nu^{18} - 734801074 \nu^{17} + 2336271777 \nu^{16} + \cdots + 4246914328 ) / 9501948 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18377413 \nu^{19} + 57111883 \nu^{18} + 508470053 \nu^{17} - 1616712447 \nu^{16} + \cdots - 2981902382 ) / 4750974 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 76060985 \nu^{19} + 235147791 \nu^{18} + 2108932420 \nu^{17} - 6662002949 \nu^{16} + \cdots - 12362661360 ) / 19003896 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 99648305 \nu^{19} + 309094685 \nu^{18} + 2757848326 \nu^{17} - 8747687085 \nu^{16} + \cdots - 16087309888 ) / 19003896 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 56327789 \nu^{19} + 174653373 \nu^{18} + 1560229216 \nu^{17} - 4946884529 \nu^{16} + \cdots - 9174943116 ) / 9501948 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 59409774 \nu^{19} + 184352467 \nu^{18} + 1644415671 \nu^{17} - 5218280798 \nu^{16} + \cdots - 9584236364 ) / 9501948 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 125982097 \nu^{19} - 390574263 \nu^{18} - 3488552048 \nu^{17} + 11058415153 \nu^{16} + \cdots + 20406082656 ) / 19003896 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 64617423 \nu^{19} - 200433331 \nu^{18} - 1788911160 \nu^{17} + 5674622729 \nu^{16} + \cdots + 10496607776 ) / 9501948 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 200416523 \nu^{19} + 621142095 \nu^{18} + 5548780786 \nu^{17} - 17581919219 \nu^{16} + \cdots - 32338923336 ) / 19003896 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{19} + \beta_{13} + \beta_{12} + \beta_{10} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} + 8\beta_{2} + \beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{19} - \beta_{18} - \beta_{17} - \beta_{15} - \beta_{14} + 2 \beta_{12} - \beta_{11} + \beta_{9} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 13 \beta_{19} - 3 \beta_{18} + \beta_{16} - \beta_{15} + \beta_{14} + 11 \beta_{13} + 10 \beta_{12} + \cdots + 114 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 13 \beta_{19} - 11 \beta_{18} - 15 \beta_{17} - 12 \beta_{15} - 12 \beta_{14} - \beta_{13} + \cdots + 49 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 125 \beta_{19} - 43 \beta_{18} - \beta_{17} + 14 \beta_{16} - 15 \beta_{15} + 15 \beta_{14} + \cdots + 814 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 123 \beta_{19} - 90 \beta_{18} - 162 \beta_{17} + \beta_{16} - 107 \beta_{15} - 112 \beta_{14} + \cdots + 453 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1078 \beta_{19} - 442 \beta_{18} - 23 \beta_{17} + 141 \beta_{16} - 160 \beta_{15} + 160 \beta_{14} + \cdots + 5975 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1045 \beta_{19} - 673 \beta_{18} - 1537 \beta_{17} + 19 \beta_{16} - 863 \beta_{15} - 954 \beta_{14} + \cdots + 3822 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 8849 \beta_{19} - 3998 \beta_{18} - 357 \beta_{17} + 1247 \beta_{16} - 1492 \beta_{15} + 1498 \beta_{14} + \cdots + 44492 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 8509 \beta_{19} - 4903 \beta_{18} - 13660 \beta_{17} + 229 \beta_{16} - 6692 \beta_{15} + \cdots + 31042 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 70840 \beta_{19} - 33943 \beta_{18} - 4560 \beta_{17} + 10323 \beta_{16} - 12998 \beta_{15} + \cdots + 334126 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 68137 \beta_{19} - 35701 \beta_{18} - 116945 \beta_{17} + 2245 \beta_{16} - 51137 \beta_{15} + \cdots + 247577 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 559386 \beta_{19} - 278103 \beta_{18} - 51579 \beta_{17} + 82293 \beta_{16} - 108924 \beta_{15} + \cdots + 2522974 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 542470 \beta_{19} - 262577 \beta_{18} - 978119 \beta_{17} + 19602 \beta_{16} - 389289 \beta_{15} + \cdots + 1955995 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 4382490 \beta_{19} - 2229969 \beta_{18} - 537217 \beta_{17} + 641191 \beta_{16} - 891326 \beta_{15} + \cdots + 19121832 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 4313417 \beta_{19} - 1957505 \beta_{18} - 8055249 \beta_{17} + 159254 \beta_{16} - 2966417 \beta_{15} + \cdots + 15374733 \) 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
−2.75020
−2.39816
−2.30762
−1.81842
−1.60278
−1.58895
−0.933672
−0.641589
−0.0990968
−0.0986252
0.270559
0.829279
1.07384
1.26543
1.44286
1.97974
2.19450
2.68833
2.71434
2.78024
−2.75020 −1.00000 5.56361 2.42464 2.75020 −2.28008 −9.80065 1.00000 −6.66826
1.2 −2.39816 −1.00000 3.75115 −1.69623 2.39816 2.85206 −4.19954 1.00000 4.06783
1.3 −2.30762 −1.00000 3.32509 3.77481 2.30762 −4.19225 −3.05780 1.00000 −8.71082
1.4 −1.81842 −1.00000 1.30666 −3.39364 1.81842 −3.58671 1.26079 1.00000 6.17107
1.5 −1.60278 −1.00000 0.568918 −1.65366 1.60278 2.84967 2.29372 1.00000 2.65046
1.6 −1.58895 −1.00000 0.524763 1.25005 1.58895 2.01165 2.34408 1.00000 −1.98627
1.7 −0.933672 −1.00000 −1.12826 4.10735 0.933672 4.10885 2.92077 1.00000 −3.83492
1.8 −0.641589 −1.00000 −1.58836 −1.31562 0.641589 −1.95224 2.30225 1.00000 0.844090
1.9 −0.0990968 −1.00000 −1.99018 −3.36358 0.0990968 −1.64314 0.395414 1.00000 0.333320
1.10 −0.0986252 −1.00000 −1.99027 0.554827 0.0986252 −3.61868 0.393541 1.00000 −0.0547199
1.11 0.270559 −1.00000 −1.92680 0.390082 −0.270559 2.10658 −1.06243 1.00000 0.105540
1.12 0.829279 −1.00000 −1.31230 −3.16223 −0.829279 2.63324 −2.74682 1.00000 −2.62237
1.13 1.07384 −1.00000 −0.846865 3.69259 −1.07384 0.561959 −3.05708 1.00000 3.96525
1.14 1.26543 −1.00000 −0.398690 0.930596 −1.26543 −4.89049 −3.03537 1.00000 1.17760
1.15 1.44286 −1.00000 0.0818317 1.85031 −1.44286 1.13770 −2.76764 1.00000 2.66973
1.16 1.97974 −1.00000 1.91937 −0.692096 −1.97974 −5.00586 −0.159633 1.00000 −1.37017
1.17 2.19450 −1.00000 2.81584 2.92001 −2.19450 4.06647 1.79035 1.00000 6.40796
1.18 2.68833 −1.00000 5.22712 −0.583128 −2.68833 4.92341 8.67557 1.00000 −1.56764
1.19 2.71434 −1.00000 5.36765 3.95163 −2.71434 −4.11016 9.14096 1.00000 10.7261
1.20 2.78024 −1.00000 5.72972 −2.98671 −2.78024 −0.971995 10.3695 1.00000 −8.30376
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(1\)
\(17\) \(-1\)

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 6171.2.a.bs 20
11.b odd 2 1 6171.2.a.bp 20
11.c even 5 2 561.2.m.e ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
561.2.m.e ✓ 40 11.c even 5 2
6171.2.a.bp 20 11.b odd 2 1
6171.2.a.bs 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6171))\):

\( T_{2}^{20} - 3 T_{2}^{19} - 28 T_{2}^{18} + 85 T_{2}^{17} + 320 T_{2}^{16} - 989 T_{2}^{15} - 1923 T_{2}^{14} + \cdots + 16 \) Copy content Toggle raw display
\( T_{5}^{20} - 7 T_{5}^{19} - 41 T_{5}^{18} + 356 T_{5}^{17} + 583 T_{5}^{16} - 7418 T_{5}^{15} + \cdots - 119809 \) Copy content Toggle raw display
\( T_{7}^{20} + 5 T_{7}^{19} - 94 T_{7}^{18} - 457 T_{7}^{17} + 3753 T_{7}^{16} + 17387 T_{7}^{15} + \cdots + 185657600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 3 T^{19} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T + 1)^{20} \) Copy content Toggle raw display
$5$ \( T^{20} - 7 T^{19} + \cdots - 119809 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 185657600 \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 8565341639 \) Copy content Toggle raw display
$17$ \( (T - 1)^{20} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots - 328714355 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots - 5293266399 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots - 527759797059 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots - 896028031744 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots - 410670520333056 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots - 1696923368100 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots - 133114770616320 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 4310977536 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 65449988096 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 14546273279744 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots - 77476506764544 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots - 26619824124 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 13190174354555 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 990503103744 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots - 4277692908544 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots - 3262543617024 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots - 15540482467584 \) Copy content Toggle raw display
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