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Properties

Label 6171.2.a.bq
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: \( x^{20} - x^{19} - 34 x^{18} + 31 x^{17} + 488 x^{16} - 395 x^{15} - 3853 x^{14} + 2660 x^{13} + \cdots + 400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{13} + 1) q^{5} - \beta_1 q^{6} - \beta_{15} q^{7} + ( - \beta_{3} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{13} + 1) q^{5} - \beta_1 q^{6} - \beta_{15} q^{7} + ( - \beta_{3} - \beta_1) q^{8} + q^{9} + ( - \beta_{19} - \beta_{16} + \cdots + \beta_{2}) q^{10}+ \cdots + ( - \beta_{18} + \beta_{16} + \cdots - 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{2} + 20 q^{3} + 29 q^{4} + 17 q^{5} - q^{6} - q^{7} - 6 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - q^{2} + 20 q^{3} + 29 q^{4} + 17 q^{5} - q^{6} - q^{7} - 6 q^{8} + 20 q^{9} - 6 q^{10} + 29 q^{12} - 8 q^{13} + 21 q^{14} + 17 q^{15} + 39 q^{16} - 20 q^{17} - q^{18} - 6 q^{19} + 32 q^{20} - q^{21} + 36 q^{23} - 6 q^{24} + 39 q^{25} + 17 q^{26} + 20 q^{27} + 7 q^{28} + 4 q^{29} - 6 q^{30} + 7 q^{31} + 7 q^{32} + q^{34} + 2 q^{35} + 29 q^{36} + 12 q^{37} + 24 q^{38} - 8 q^{39} + 15 q^{40} + 6 q^{41} + 21 q^{42} + 6 q^{43} + 17 q^{45} - 9 q^{46} + 52 q^{47} + 39 q^{48} + 45 q^{49} - 18 q^{50} - 20 q^{51} - 36 q^{52} + 60 q^{53} - q^{54} + 51 q^{56} - 6 q^{57} - 15 q^{58} + 42 q^{59} + 32 q^{60} + 18 q^{61} + 55 q^{62} - q^{63} + 40 q^{64} + 8 q^{65} - 15 q^{67} - 29 q^{68} + 36 q^{69} - 31 q^{70} + 50 q^{71} - 6 q^{72} + 17 q^{73} - 2 q^{74} + 39 q^{75} - 11 q^{76} + 17 q^{78} - 3 q^{79} + 37 q^{80} + 20 q^{81} - 5 q^{82} - 4 q^{83} + 7 q^{84} - 17 q^{85} + 32 q^{86} + 4 q^{87} + 16 q^{89} - 6 q^{90} + 13 q^{91} + 86 q^{92} + 7 q^{93} + 24 q^{94} + 64 q^{95} + 7 q^{96} + 15 q^{97} + 69 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} - 34 x^{18} + 31 x^{17} + 488 x^{16} - 395 x^{15} - 3853 x^{14} + 2660 x^{13} + \cdots + 400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 189183 \nu^{19} + 19027439 \nu^{18} - 26587060 \nu^{17} - 584983719 \nu^{16} + 698943150 \nu^{15} + \cdots - 3214885400 ) / 727694520 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 307057 \nu^{19} + 5036339 \nu^{18} + 7719557 \nu^{17} - 156002336 \nu^{16} + \cdots - 1997633820 ) / 181923630 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 895953 \nu^{19} + 13364672 \nu^{18} - 31407263 \nu^{17} - 415879041 \nu^{16} + 458713863 \nu^{15} + \cdots - 506093380 ) / 363847260 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 576242 \nu^{19} + 1181247 \nu^{18} + 19174203 \nu^{17} - 38833457 \nu^{16} - 264649881 \nu^{15} + \cdots - 309774100 ) / 181923630 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3097741 \nu^{19} - 5402709 \nu^{18} - 100598206 \nu^{17} + 172726783 \nu^{16} + 1356363780 \nu^{15} + \cdots + 722071440 ) / 727694520 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 6873203 \nu^{19} + 14536027 \nu^{18} + 217914798 \nu^{17} - 451955489 \nu^{16} + \cdots - 3427310680 ) / 727694520 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4567405 \nu^{19} - 12638857 \nu^{18} - 132827816 \nu^{17} + 383454771 \nu^{16} + \cdots - 233107420 ) / 363847260 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11574353 \nu^{19} + 11941175 \nu^{18} + 364597464 \nu^{17} - 375945055 \nu^{16} + \cdots + 1788689800 ) / 727694520 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 613638 \nu^{19} + 1261204 \nu^{18} + 19613991 \nu^{17} - 40039216 \nu^{16} - 260170622 \nu^{15} + \cdots - 261373700 ) / 36384726 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 6928637 \nu^{19} + 2005221 \nu^{18} + 221254676 \nu^{17} - 59470505 \nu^{16} + \cdots - 1547890440 ) / 363847260 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 7449737 \nu^{19} - 9860693 \nu^{18} - 237483564 \nu^{17} + 304733823 \nu^{16} + 3137533598 \nu^{15} + \cdots - 373844820 ) / 363847260 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 3923640 \nu^{19} + 5949855 \nu^{18} + 122616991 \nu^{17} - 181921046 \nu^{16} + \cdots - 481380560 ) / 181923630 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 8258897 \nu^{19} + 9380860 \nu^{18} + 256509213 \nu^{17} - 288250827 \nu^{16} + \cdots - 3392691780 ) / 363847260 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 8392783 \nu^{19} + 18847999 \nu^{18} + 261852990 \nu^{17} - 594082901 \nu^{16} + \cdots - 3373188640 ) / 363847260 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 22188275 \nu^{19} - 47823695 \nu^{18} - 701000998 \nu^{17} + 1516226413 \nu^{16} + \cdots + 13073712240 ) / 727694520 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 14713139 \nu^{19} - 42956609 \nu^{18} - 465187352 \nu^{17} + 1337645255 \nu^{16} + \cdots + 8081022720 ) / 363847260 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{18} + \beta_{17} - \beta_{8} + \beta_{7} + 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{18} + 2 \beta_{17} + \beta_{15} + \beta_{14} - \beta_{11} + \beta_{8} + \beta_{7} + \cdots + 29 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12 \beta_{18} + 12 \beta_{17} + \beta_{15} + \beta_{14} + \beta_{12} - 2 \beta_{11} - \beta_{10} + \cdots + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{19} + 14 \beta_{18} + 26 \beta_{17} - \beta_{16} + 14 \beta_{15} + 13 \beta_{14} + 2 \beta_{12} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 109 \beta_{18} + 109 \beta_{17} + 16 \beta_{15} + 17 \beta_{14} + \beta_{13} + 15 \beta_{12} + \cdots + 510 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 16 \beta_{19} + 141 \beta_{18} + 249 \beta_{17} - 16 \beta_{16} + 141 \beta_{15} + 123 \beta_{14} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 899 \beta_{18} + 899 \beta_{17} + 2 \beta_{16} + 175 \beta_{15} + 194 \beta_{14} + 17 \beta_{13} + \cdots + 3158 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 171 \beta_{19} + 1254 \beta_{18} + 2132 \beta_{17} - 177 \beta_{16} + 1253 \beta_{15} + 1038 \beta_{14} + \cdots - 36 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2 \beta_{19} + 7094 \beta_{18} + 7095 \beta_{17} + 48 \beta_{16} + 1650 \beta_{15} + 1882 \beta_{14} + \cdots + 19959 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1537 \beta_{19} + 10505 \beta_{18} + 17306 \beta_{17} - 1681 \beta_{16} + 10472 \beta_{15} + 8311 \beta_{14} + \cdots + 161 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 61 \beta_{19} + 54696 \beta_{18} + 54728 \beta_{17} + 709 \beta_{16} + 14471 \beta_{15} + \cdots + 128105 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 12569 \beta_{19} + 85202 \beta_{18} + 136479 \beta_{17} - 14693 \beta_{16} + 84592 \beta_{15} + \cdots + 5177 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 1092 \beta_{19} + 416286 \beta_{18} + 416865 \beta_{17} + 8360 \beta_{16} + 122002 \beta_{15} + \cdots + 832934 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 96902 \beta_{19} + 678178 \beta_{18} + 1058412 \beta_{17} - 121894 \beta_{16} + 669598 \beta_{15} + \cdots + 70819 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 15104 \beta_{19} + 3145000 \beta_{18} + 3153005 \beta_{17} + 86550 \beta_{16} + 1004870 \beta_{15} + \cdots + 5478390 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 717424 \beta_{19} + 5336275 \beta_{18} + 8125441 \beta_{17} - 975606 \beta_{16} + 5233212 \beta_{15} + \cdots + 773715 \) 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
2.75744
2.59952
2.43131
2.20274
2.18586
1.65364
1.54979
0.943270
0.595184
0.217824
−0.200382
−0.702657
−1.07388
−1.19251
−1.40167
−1.52034
−2.31327
−2.52176
−2.57844
−2.63168
−2.75744 1.00000 5.60350 −1.22544 −2.75744 −3.45190 −9.93645 1.00000 3.37909
1.2 −2.59952 1.00000 4.75752 3.83863 −2.59952 2.15412 −7.16824 1.00000 −9.97861
1.3 −2.43131 1.00000 3.91125 −0.530533 −2.43131 −1.61213 −4.64683 1.00000 1.28989
1.4 −2.20274 1.00000 2.85206 4.08438 −2.20274 0.357998 −1.87688 1.00000 −8.99682
1.5 −2.18586 1.00000 2.77797 −0.765047 −2.18586 2.36983 −1.70052 1.00000 1.67228
1.6 −1.65364 1.00000 0.734539 4.09971 −1.65364 −5.05098 2.09262 1.00000 −6.77946
1.7 −1.54979 1.00000 0.401859 −2.69810 −1.54979 −0.967369 2.47679 1.00000 4.18150
1.8 −0.943270 1.00000 −1.11024 1.93902 −0.943270 −2.46442 2.93380 1.00000 −1.82902
1.9 −0.595184 1.00000 −1.64576 2.57263 −0.595184 4.73983 2.16990 1.00000 −1.53119
1.10 −0.217824 1.00000 −1.95255 0.529698 −0.217824 0.960997 0.860960 1.00000 −0.115381
1.11 0.200382 1.00000 −1.95985 −2.72678 0.200382 −0.220024 −0.793482 1.00000 −0.546398
1.12 0.702657 1.00000 −1.50627 4.43662 0.702657 0.917306 −2.46371 1.00000 3.11742
1.13 1.07388 1.00000 −0.846789 −2.62514 1.07388 −3.14120 −3.05710 1.00000 −2.81907
1.14 1.19251 1.00000 −0.577908 −0.343180 1.19251 −4.47826 −3.07419 1.00000 −0.409248
1.15 1.40167 1.00000 −0.0353175 1.15318 1.40167 4.89395 −2.85285 1.00000 1.61637
1.16 1.52034 1.00000 0.311435 2.45562 1.52034 −1.80873 −2.56719 1.00000 3.73338
1.17 2.31327 1.00000 3.35120 −3.50963 2.31327 4.19719 3.12568 1.00000 −8.11871
1.18 2.52176 1.00000 4.35926 3.24907 2.52176 1.14631 5.94949 1.00000 8.19338
1.19 2.57844 1.00000 4.64836 2.37174 2.57844 −3.45370 6.82864 1.00000 6.11540
1.20 2.63168 1.00000 4.92573 0.693545 2.63168 3.91119 7.69958 1.00000 1.82519
\(n\): 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.bq 20
11.b odd 2 1 6171.2.a.br 20
11.c even 5 2 561.2.m.f ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
561.2.m.f ✓ 40 11.c even 5 2
6171.2.a.bq 20 1.a even 1 1 trivial
6171.2.a.br 20 11.b odd 2 1

Hecke kernels

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

\( T_{2}^{20} + T_{2}^{19} - 34 T_{2}^{18} - 31 T_{2}^{17} + 488 T_{2}^{16} + 395 T_{2}^{15} - 3853 T_{2}^{14} + \cdots + 400 \) Copy content Toggle raw display
\( T_{5}^{20} - 17 T_{5}^{19} + 75 T_{5}^{18} + 274 T_{5}^{17} - 3069 T_{5}^{16} + 3746 T_{5}^{15} + \cdots + 131945 \) Copy content Toggle raw display
\( T_{7}^{20} + T_{7}^{19} - 92 T_{7}^{18} - 101 T_{7}^{17} + 3431 T_{7}^{16} + 4059 T_{7}^{15} + \cdots + 911104 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + T^{19} + \cdots + 400 \) Copy content Toggle raw display
$3$ \( (T - 1)^{20} \) Copy content Toggle raw display
$5$ \( T^{20} - 17 T^{19} + \cdots + 131945 \) Copy content Toggle raw display
$7$ \( T^{20} + T^{19} + \cdots + 911104 \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( T^{20} + 8 T^{19} + \cdots + 1280059 \) Copy content Toggle raw display
$17$ \( (T + 1)^{20} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots - 2045778851 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 250997484375 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 471359251 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 644525542144 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 14320451563264 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots - 160323550844 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots - 31190872064 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 202886220829696 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 25\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 15\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots - 82627684623616 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots - 45\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 16790174809621 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 379987655531264 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots - 335746181094400 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 88423657051136 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots - 515296036350720 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots - 24\!\cdots\!76 \) Copy content Toggle raw display
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