Properties

Label 6171.2.a.bk
Level $6171$
Weight $2$
Character orbit 6171.a
Self dual yes
Analytic conductor $49.276$
Analytic rank $1$
Dimension $12$
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: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 16 x^{10} + 15 x^{9} + 89 x^{8} - 78 x^{7} - 201 x^{6} + 157 x^{5} + 159 x^{4} - 92 x^{3} - 29 x^{2} + 17 x - 1 \) 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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{10} + \beta_{9} + \beta_{7} + \beta_{3} + \beta_1) q^{4} + (\beta_{11} - \beta_{9} - 1) q^{5} - \beta_1 q^{6} - \beta_{2} q^{7} + ( - \beta_{11} - \beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + (\beta_{10} + \beta_{9} + \beta_{7} + \beta_{3} + \beta_1) q^{4} + (\beta_{11} - \beta_{9} - 1) q^{5} - \beta_1 q^{6} - \beta_{2} q^{7} + ( - \beta_{11} - \beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_1) q^{8} + q^{9} + (\beta_{8} - \beta_{5} - \beta_{4} + 2 \beta_1 + 1) q^{10} + (\beta_{10} + \beta_{9} + \beta_{7} + \beta_{3} + \beta_1) q^{12} + (\beta_{11} - \beta_{10} + \beta_{7} + 2 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 - 2) q^{13} + ( - \beta_{11} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} - 1) q^{14} + (\beta_{11} - \beta_{9} - 1) q^{15} + (\beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{2} + 2) q^{16} + q^{17} - \beta_1 q^{18} + (\beta_{11} + \beta_{10} + \beta_{8} - \beta_{6} + \beta_{5} + 2 \beta_{3} + \beta_{2} + 3 \beta_1) q^{19} + ( - 4 \beta_{10} - 2 \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} + \cdots - 2) q^{20}+ \cdots + (3 \beta_{11} - \beta_{10} - \beta_{9} + 3 \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{4} + 2 \beta_{3} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} + 12 q^{3} + 9 q^{4} - 14 q^{5} - q^{6} - 5 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{2} + 12 q^{3} + 9 q^{4} - 14 q^{5} - q^{6} - 5 q^{7} + 12 q^{9} + 8 q^{10} + 9 q^{12} - 7 q^{13} - 13 q^{14} - 14 q^{15} + 11 q^{16} + 12 q^{17} - q^{18} + 3 q^{19} - 36 q^{20} - 5 q^{21} - 39 q^{23} + 8 q^{25} - 25 q^{26} + 12 q^{27} - 9 q^{28} + 10 q^{29} + 8 q^{30} - 25 q^{31} - 11 q^{32} - q^{34} + 12 q^{35} + 9 q^{36} - 10 q^{37} - 34 q^{38} - 7 q^{39} - 13 q^{40} + 17 q^{41} - 13 q^{42} - 18 q^{43} - 14 q^{45} + 19 q^{46} - 38 q^{47} + 11 q^{48} - 15 q^{49} - 18 q^{50} + 12 q^{51} + 32 q^{52} - 40 q^{53} - q^{54} - 25 q^{56} + 3 q^{57} + 3 q^{58} - 18 q^{59} - 36 q^{60} - 22 q^{61} + 3 q^{62} - 5 q^{63} - 20 q^{64} + 3 q^{65} - 9 q^{67} + 9 q^{68} - 39 q^{69} + 13 q^{70} - 24 q^{71} + q^{73} - 4 q^{74} + 8 q^{75} + 29 q^{76} - 25 q^{78} - 3 q^{79} - 43 q^{80} + 12 q^{81} + q^{82} + 4 q^{83} - 9 q^{84} - 14 q^{85} - 4 q^{86} + 10 q^{87} - 62 q^{89} + 8 q^{90} - 5 q^{91} - 52 q^{92} - 25 q^{93} - 16 q^{94} - 4 q^{95} - 11 q^{96} - 5 q^{97} - 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 16 x^{10} + 15 x^{9} + 89 x^{8} - 78 x^{7} - 201 x^{6} + 157 x^{5} + 159 x^{4} - 92 x^{3} - 29 x^{2} + 17 x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{11} + 2 \nu^{10} + 82 \nu^{9} - 25 \nu^{8} - 472 \nu^{7} + 94 \nu^{6} + 1123 \nu^{5} - 48 \nu^{4} - 955 \nu^{3} - 217 \nu^{2} + 122 \nu + 29 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11 \nu^{11} + 10 \nu^{10} + 174 \nu^{9} - 147 \nu^{8} - 948 \nu^{7} + 742 \nu^{6} + 2053 \nu^{5} - 1400 \nu^{4} - 1461 \nu^{3} + 629 \nu^{2} + 206 \nu - 65 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 39 \nu^{11} - 26 \nu^{10} - 638 \nu^{9} + 367 \nu^{8} + 3668 \nu^{7} - 1734 \nu^{6} - 8753 \nu^{5} + 2784 \nu^{4} + 7657 \nu^{3} - 353 \nu^{2} - 1414 \nu + 53 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 73 \nu^{11} + 38 \nu^{10} + 1186 \nu^{9} - 529 \nu^{8} - 6748 \nu^{7} + 2490 \nu^{6} + 15871 \nu^{5} - 3952 \nu^{4} - 13623 \nu^{3} + 255 \nu^{2} + 2554 \nu - 75 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 103 \nu^{11} - 58 \nu^{10} - 1678 \nu^{9} + 815 \nu^{8} + 9588 \nu^{7} - 3878 \nu^{6} - 22689 \nu^{5} + 6368 \nu^{4} + 19609 \nu^{3} - 1041 \nu^{2} - 3558 \nu + 245 ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 107 \nu^{11} - 66 \nu^{10} - 1734 \nu^{9} + 931 \nu^{8} + 9828 \nu^{7} - 4446 \nu^{6} - 22925 \nu^{5} + 7392 \nu^{4} + 19221 \nu^{3} - 1421 \nu^{2} - 3262 \nu + 193 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 245 \nu^{11} - 142 \nu^{10} - 3978 \nu^{9} + 1997 \nu^{8} + 22620 \nu^{7} - 9522 \nu^{6} - 53123 \nu^{5} + 15776 \nu^{4} + 45323 \nu^{3} - 2931 \nu^{2} - 8146 \nu + 575 ) / 32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 247 \nu^{11} - 138 \nu^{10} - 4014 \nu^{9} + 1935 \nu^{8} + 22852 \nu^{7} - 9206 \nu^{6} - 53777 \nu^{5} + 15200 \nu^{4} + 46089 \nu^{3} - 2705 \nu^{2} - 8390 \nu + 597 ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 83 \nu^{11} + 46 \nu^{10} + 1350 \nu^{9} - 643 \nu^{8} - 7696 \nu^{7} + 3042 \nu^{6} + 18149 \nu^{5} - 4948 \nu^{4} - 15597 \nu^{3} + 725 \nu^{2} + 2802 \nu - 181 ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 191 \nu^{11} + 106 \nu^{10} + 3102 \nu^{9} - 1487 \nu^{8} - 17644 \nu^{7} + 7078 \nu^{6} + 41465 \nu^{5} - 11656 \nu^{4} - 35433 \nu^{3} + 1937 \nu^{2} + 6342 \nu - 421 ) / 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{7} + \beta_{3} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{8} + \beta_{6} - \beta_{5} - \beta_{4} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{10} + 7\beta_{9} + \beta_{8} + 5\beta_{7} + \beta_{5} + 6\beta_{3} - \beta_{2} + 6\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{11} - \beta_{9} + 9\beta_{8} + 2\beta_{7} + 11\beta_{6} - 8\beta_{5} - 11\beta_{4} + \beta_{3} + 28\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 48 \beta_{10} + 46 \beta_{9} + 12 \beta_{8} + 26 \beta_{7} + \beta_{6} + 9 \beta_{5} + 37 \beta_{3} - 11 \beta_{2} + 38 \beta _1 + 59 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 67 \beta_{11} + 6 \beta_{10} - 8 \beta_{9} + 73 \beta_{8} + 24 \beta_{7} + 87 \beta_{6} - 55 \beta_{5} - 89 \beta_{4} + 18 \beta_{3} + 172 \beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{11} + 326 \beta_{10} + 298 \beta_{9} + 105 \beta_{8} + 146 \beta_{7} + 22 \beta_{6} + 63 \beta_{5} - 6 \beta_{4} + 235 \beta_{3} - 89 \beta_{2} + 254 \beta _1 + 368 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 468 \beta_{11} + 99 \beta_{10} - 38 \beta_{9} + 562 \beta_{8} + 213 \beta_{7} + 620 \beta_{6} - 361 \beta_{5} - 652 \beta_{4} + 198 \beta_{3} - 6 \beta_{2} + 1116 \beta _1 + 193 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 43 \beta_{11} + 2205 \beta_{10} + 1931 \beta_{9} + 830 \beta_{8} + 877 \beta_{7} + 276 \beta_{6} + 399 \beta_{5} - 113 \beta_{4} + 1527 \beta_{3} - 652 \beta_{2} + 1757 \beta _1 + 2367 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3188 \beta_{11} + 1101 \beta_{10} - 56 \beta_{9} + 4179 \beta_{8} + 1695 \beta_{7} + 4254 \beta_{6} - 2330 \beta_{5} - 4586 \beta_{4} + 1803 \beta_{3} - 113 \beta_{2} + 7470 \beta _1 + 1824 \) Copy content Toggle raw display

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
2.66943
2.13144
1.72357
1.52953
0.441507
0.435069
0.0682851
−0.578221
−0.859524
−1.84345
−2.23120
−2.48644
−2.66943 1.00000 5.12588 −3.14556 −2.66943 2.02309 −8.34433 1.00000 8.39687
1.2 −2.13144 1.00000 2.54304 −0.603451 −2.13144 0.204177 −1.15745 1.00000 1.28622
1.3 −1.72357 1.00000 0.970688 −3.92121 −1.72357 −3.53774 1.77409 1.00000 6.75848
1.4 −1.52953 1.00000 0.339465 0.0123642 −1.52953 −0.129260 2.53984 1.00000 −0.0189114
1.5 −0.441507 1.00000 −1.80507 −4.28875 −0.441507 3.18006 1.67996 1.00000 1.89351
1.6 −0.435069 1.00000 −1.81072 1.52949 −0.435069 2.82480 1.65792 1.00000 −0.665433
1.7 −0.0682851 1.00000 −1.99534 −1.22357 −0.0682851 −4.50194 0.272822 1.00000 0.0835515
1.8 0.578221 1.00000 −1.66566 0.295515 0.578221 −0.675716 −2.11956 1.00000 0.170873
1.9 0.859524 1.00000 −1.26122 2.41916 0.859524 1.04339 −2.80310 1.00000 2.07932
1.10 1.84345 1.00000 1.39833 −0.675257 1.84345 −1.70831 −1.10916 1.00000 −1.24481
1.11 2.23120 1.00000 2.97825 −0.773638 2.23120 −2.99730 2.18267 1.00000 −1.72614
1.12 2.48644 1.00000 4.18236 −3.62508 2.48644 −0.725250 5.42630 1.00000 −9.01354
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.bk 12
11.b odd 2 1 6171.2.a.bl 12
11.c even 5 2 561.2.m.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
561.2.m.d 24 11.c even 5 2
6171.2.a.bk 12 1.a even 1 1 trivial
6171.2.a.bl 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6171))\):

\( T_{2}^{12} + T_{2}^{11} - 16 T_{2}^{10} - 15 T_{2}^{9} + 89 T_{2}^{8} + 78 T_{2}^{7} - 201 T_{2}^{6} - 157 T_{2}^{5} + 159 T_{2}^{4} + 92 T_{2}^{3} - 29 T_{2}^{2} - 17 T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{12} + 14 T_{5}^{11} + 64 T_{5}^{10} + 49 T_{5}^{9} - 428 T_{5}^{8} - 1108 T_{5}^{7} - 245 T_{5}^{6} + 1884 T_{5}^{5} + 2147 T_{5}^{4} + 624 T_{5}^{3} - 160 T_{5}^{2} - 79 T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{12} + 5 T_{7}^{11} - 22 T_{7}^{10} - 117 T_{7}^{9} + 153 T_{7}^{8} + 893 T_{7}^{7} - 301 T_{7}^{6} - 2418 T_{7}^{5} - 340 T_{7}^{4} + 1741 T_{7}^{3} + 649 T_{7}^{2} - 100 T_{7} - 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + T^{11} - 16 T^{10} - 15 T^{9} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 14 T^{11} + 64 T^{10} + 49 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{12} + 5 T^{11} - 22 T^{10} - 117 T^{9} + \cdots - 20 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + 7 T^{11} - 43 T^{10} + \cdots + 1471 \) Copy content Toggle raw display
$17$ \( (T - 1)^{12} \) Copy content Toggle raw display
$19$ \( T^{12} - 3 T^{11} - 104 T^{10} + \cdots + 174875 \) Copy content Toggle raw display
$23$ \( T^{12} + 39 T^{11} + 595 T^{10} + \cdots - 7188731 \) Copy content Toggle raw display
$29$ \( T^{12} - 10 T^{11} - 130 T^{10} + \cdots + 14530979 \) Copy content Toggle raw display
$31$ \( T^{12} + 25 T^{11} + 93 T^{10} + \cdots - 38588656 \) Copy content Toggle raw display
$37$ \( T^{12} + 10 T^{11} - 153 T^{10} + \cdots - 6456004 \) Copy content Toggle raw display
$41$ \( T^{12} - 17 T^{11} + \cdots - 417799076 \) Copy content Toggle raw display
$43$ \( T^{12} + 18 T^{11} - 69 T^{10} + \cdots - 10894336 \) Copy content Toggle raw display
$47$ \( T^{12} + 38 T^{11} + \cdots + 469534724 \) Copy content Toggle raw display
$53$ \( T^{12} + 40 T^{11} + \cdots - 182965504 \) Copy content Toggle raw display
$59$ \( T^{12} + 18 T^{11} - 223 T^{10} + \cdots - 15792284 \) Copy content Toggle raw display
$61$ \( T^{12} + 22 T^{11} - 25 T^{10} + \cdots - 3217580 \) Copy content Toggle raw display
$67$ \( T^{12} + 9 T^{11} + \cdots - 1284103004 \) Copy content Toggle raw display
$71$ \( T^{12} + 24 T^{11} + \cdots - 134992955 \) Copy content Toggle raw display
$73$ \( T^{12} - T^{11} - 404 T^{10} + \cdots - 4520659120 \) Copy content Toggle raw display
$79$ \( T^{12} + 3 T^{11} + \cdots + 8586213680 \) Copy content Toggle raw display
$83$ \( T^{12} - 4 T^{11} - 385 T^{10} + \cdots + 200050756 \) Copy content Toggle raw display
$89$ \( T^{12} + 62 T^{11} + \cdots - 833986100 \) Copy content Toggle raw display
$97$ \( T^{12} + 5 T^{11} - 491 T^{10} + \cdots + 54526364 \) Copy content Toggle raw display
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