↑

Properties

Label 6171.2.a.bn
Level $6171$
Weight $2$
Character orbit 6171.a
Self dual yes
Analytic conductor $49.276$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} - 19 x^{12} + 36 x^{11} + 136 x^{10} - 244 x^{9} - 449 x^{8} + 778 x^{7} + 638 x^{6} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} - \beta_{5} q^{5} + \beta_1 q^{6} + (\beta_{13} - \beta_{12} + \beta_{9}) q^{7} + ( - \beta_{5} - \beta_{4} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} - \beta_{5} q^{5} + \beta_1 q^{6} + (\beta_{13} - \beta_{12} + \beta_{9}) q^{7} + ( - \beta_{5} - \beta_{4} - \beta_1) q^{8} + q^{9} + (\beta_{5} + \beta_{4} + \beta_{3}) q^{10} + ( - \beta_{2} - 1) q^{12} + (\beta_{12} + \beta_{8} + \beta_{5} + 2) q^{13} + ( - \beta_{13} + \beta_{12} + \cdots - 2 \beta_{2}) q^{14}+ \cdots + ( - \beta_{13} - 2 \beta_{12} + \cdots - 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{2} - 14 q^{3} + 14 q^{4} + 2 q^{6} + 6 q^{7} - 6 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{2} - 14 q^{3} + 14 q^{4} + 2 q^{6} + 6 q^{7} - 6 q^{8} + 14 q^{9} + 8 q^{10} - 14 q^{12} + 20 q^{13} + 4 q^{14} + 14 q^{16} + 14 q^{17} - 2 q^{18} + 16 q^{19} - 16 q^{20} - 6 q^{21} - 2 q^{23} + 6 q^{24} + 22 q^{25} - 4 q^{26} - 14 q^{27} - 30 q^{28} + 22 q^{29} - 8 q^{30} - 34 q^{32} - 2 q^{34} + 2 q^{35} + 14 q^{36} + 8 q^{37} - 16 q^{38} - 20 q^{39} + 74 q^{40} + 24 q^{41} - 4 q^{42} + 12 q^{43} + 26 q^{46} + 28 q^{47} - 14 q^{48} + 28 q^{49} - 50 q^{50} - 14 q^{51} + 48 q^{52} + 18 q^{53} + 2 q^{54} + 6 q^{56} - 16 q^{57} + 20 q^{59} + 16 q^{60} + 64 q^{61} + 62 q^{62} + 6 q^{63} - 4 q^{64} - 30 q^{65} + 10 q^{67} + 14 q^{68} + 2 q^{69} - 44 q^{70} - 8 q^{71} - 6 q^{72} + 44 q^{73} + 50 q^{74} - 22 q^{75} + 24 q^{76} + 4 q^{78} + 8 q^{79} - 102 q^{80} + 14 q^{81} - 46 q^{82} + 12 q^{83} + 30 q^{84} + 12 q^{86} - 22 q^{87} - 8 q^{89} + 8 q^{90} - 20 q^{91} - 20 q^{92} - 16 q^{94} - 30 q^{95} + 34 q^{96} - 14 q^{97} - 90 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} - 19 x^{12} + 36 x^{11} + 136 x^{10} - 244 x^{9} - 449 x^{8} + 778 x^{7} + 638 x^{6} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 51 \nu^{13} - 207 \nu^{12} - 592 \nu^{11} + 3104 \nu^{10} + 1406 \nu^{9} - 15978 \nu^{8} + 4526 \nu^{7} + \cdots - 92 ) / 209 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 23 \nu^{13} - 5 \nu^{12} + 644 \nu^{11} - 236 \nu^{10} - 6232 \nu^{9} + 4206 \nu^{8} + 26305 \nu^{7} + \cdots + 648 ) / 209 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23 \nu^{13} + 5 \nu^{12} - 644 \nu^{11} + 236 \nu^{10} + 6232 \nu^{9} - 4206 \nu^{8} - 26305 \nu^{7} + \cdots - 648 ) / 209 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 93 \nu^{13} - 107 \nu^{12} - 1768 \nu^{11} + 1554 \nu^{10} + 12768 \nu^{9} - 7511 \nu^{8} - 43382 \nu^{7} + \cdots + 742 ) / 209 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 221 \nu^{13} - 270 \nu^{12} - 4307 \nu^{11} + 4394 \nu^{10} + 31730 \nu^{9} - 25766 \nu^{8} + \cdots - 956 ) / 418 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 136 \nu^{13} - 343 \nu^{12} - 2136 \nu^{11} + 5212 \nu^{10} + 11970 \nu^{9} - 27560 \nu^{8} + \cdots - 733 ) / 209 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 124 \nu^{13} + 73 \nu^{12} + 2636 \nu^{11} - 1236 \nu^{10} - 21204 \nu^{9} + 7646 \nu^{8} + \cdots + 822 ) / 209 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13 \nu^{13} + 17 \nu^{12} + 250 \nu^{11} - 273 \nu^{10} - 1824 \nu^{9} + 1576 \nu^{8} + 6247 \nu^{7} + \cdots + 54 ) / 19 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 27 \nu^{13} + 28 \nu^{12} + 547 \nu^{11} - 472 \nu^{10} - 4218 \nu^{9} + 2940 \nu^{8} + 15275 \nu^{7} + \cdots + 232 ) / 38 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 153 \nu^{13} + 203 \nu^{12} + 3030 \nu^{11} - 3460 \nu^{10} - 22819 \nu^{9} + 21809 \nu^{8} + \cdots + 1739 ) / 209 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 387 \nu^{13} + 452 \nu^{12} + 7701 \nu^{11} - 7424 \nu^{10} - 58444 \nu^{9} + 44382 \nu^{8} + \cdots + 3034 ) / 418 \) 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
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{10} - \beta_{7} + \beta_{6} + \beta_{5} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} - \beta_{12} - \beta_{11} - \beta_{8} + 8\beta_{5} + 9\beta_{4} + \beta_{3} + 28\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{13} - \beta_{12} + 10 \beta_{11} - 11 \beta_{10} - \beta_{9} - 11 \beta_{7} + 9 \beta_{6} + \cdots + 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{13} - 10 \beta_{12} - 12 \beta_{11} - \beta_{9} - 11 \beta_{8} + 57 \beta_{5} + 67 \beta_{4} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 12 \beta_{13} - 10 \beta_{12} + 78 \beta_{11} - 92 \beta_{10} - 13 \beta_{9} - \beta_{8} - 90 \beta_{7} + \cdots + 504 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 104 \beta_{13} - 73 \beta_{12} - 105 \beta_{11} - 5 \beta_{10} - 15 \beta_{9} - 91 \beta_{8} + \cdots + 110 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 109 \beta_{13} - 69 \beta_{12} + 551 \beta_{11} - 699 \beta_{10} - 121 \beta_{9} - 18 \beta_{8} + \cdots + 3165 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 808 \beta_{13} - 470 \beta_{12} - 823 \beta_{11} - 99 \beta_{10} - 157 \beta_{9} - 682 \beta_{8} + \cdots + 921 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 907 \beta_{13} - 387 \beta_{12} + 3690 \beta_{11} - 5093 \beta_{10} - 995 \beta_{9} - 211 \beta_{8} + \cdots + 20468 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 6000 \beta_{13} - 2806 \beta_{12} - 6153 \beta_{11} - 1271 \beta_{10} - 1420 \beta_{9} - 4899 \beta_{8} + \cdots + 7393 \) 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
2.66608
2.58597
2.02626
1.87481
0.950865
0.926602
0.347590
0.158853
−0.160725
−1.11087
−1.82410
−1.92531
−1.95461
−2.56141
−2.66608 −1.00000 5.10797 −4.17179 2.66608 −3.51874 −8.28610 1.00000 11.1223
1.2 −2.58597 −1.00000 4.68725 −2.30773 2.58597 1.99789 −6.94915 1.00000 5.96773
1.3 −2.02626 −1.00000 2.10572 3.45577 2.02626 2.90628 −0.214207 1.00000 −7.00228
1.4 −1.87481 −1.00000 1.51492 0.415159 1.87481 −5.01369 0.909438 1.00000 −0.778345
1.5 −0.950865 −1.00000 −1.09586 2.37917 0.950865 −1.47437 2.94374 1.00000 −2.26227
1.6 −0.926602 −1.00000 −1.14141 2.19684 0.926602 5.13724 2.91083 1.00000 −2.03560
1.7 −0.347590 −1.00000 −1.87918 −1.45808 0.347590 2.71605 1.34837 1.00000 0.506815
1.8 −0.158853 −1.00000 −1.97477 2.83426 0.158853 −1.83463 0.631405 1.00000 −0.450232
1.9 0.160725 −1.00000 −1.97417 −4.09047 −0.160725 4.66194 −0.638748 1.00000 −0.657440
1.10 1.11087 −1.00000 −0.765968 −2.36838 −1.11087 −0.181091 −3.07263 1.00000 −2.63096
1.11 1.82410 −1.00000 1.32734 0.657004 −1.82410 3.24859 −1.22700 1.00000 1.19844
1.12 1.92531 −1.00000 1.70682 −1.29643 −1.92531 −1.16163 −0.564455 1.00000 −2.49603
1.13 1.95461 −1.00000 1.82049 3.45973 −1.95461 0.249388 −0.350867 1.00000 6.76241
1.14 2.56141 −1.00000 4.56084 0.294933 −2.56141 −1.73324 6.55937 1.00000 0.755446
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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.bn ✓ 14
11.b odd 2 1 6171.2.a.bo yes 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6171.2.a.bn ✓ 14 1.a even 1 1 trivial
6171.2.a.bo yes 14 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6171))\):

\( T_{2}^{14} + 2 T_{2}^{13} - 19 T_{2}^{12} - 36 T_{2}^{11} + 136 T_{2}^{10} + 244 T_{2}^{9} - 449 T_{2}^{8} + \cdots + 4 \) Copy content Toggle raw display
\( T_{5}^{14} - 46 T_{5}^{12} + 14 T_{5}^{11} + 794 T_{5}^{10} - 392 T_{5}^{9} - 6402 T_{5}^{8} + \cdots + 2512 \) Copy content Toggle raw display
\( T_{7}^{14} - 6 T_{7}^{13} - 45 T_{7}^{12} + 296 T_{7}^{11} + 622 T_{7}^{10} - 4838 T_{7}^{9} + \cdots - 5324 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 2 T^{13} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T + 1)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} - 46 T^{12} + \cdots + 2512 \) Copy content Toggle raw display
$7$ \( T^{14} - 6 T^{13} + \cdots - 5324 \) Copy content Toggle raw display
$11$ \( T^{14} \) Copy content Toggle raw display
$13$ \( T^{14} - 20 T^{13} + \cdots - 8279456 \) Copy content Toggle raw display
$17$ \( (T - 1)^{14} \) Copy content Toggle raw display
$19$ \( T^{14} - 16 T^{13} + \cdots - 73799 \) Copy content Toggle raw display
$23$ \( T^{14} + 2 T^{13} + \cdots - 83866112 \) Copy content Toggle raw display
$29$ \( T^{14} - 22 T^{13} + \cdots - 6438176 \) Copy content Toggle raw display
$31$ \( T^{14} - 225 T^{12} + \cdots + 6946756 \) Copy content Toggle raw display
$37$ \( T^{14} - 8 T^{13} + \cdots + 27280192 \) Copy content Toggle raw display
$41$ \( T^{14} - 24 T^{13} + \cdots + 89235712 \) Copy content Toggle raw display
$43$ \( T^{14} - 12 T^{13} + \cdots - 21663356 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 539562052 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 7857553792 \) Copy content Toggle raw display
$59$ \( T^{14} - 20 T^{13} + \cdots + 94369792 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 230353775996 \) Copy content Toggle raw display
$67$ \( T^{14} - 10 T^{13} + \cdots + 1497133 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 667842752 \) Copy content Toggle raw display
$73$ \( T^{14} - 44 T^{13} + \cdots + 322624 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 47141816896 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 29322388712 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 167662668256 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 6063172708 \) Copy content Toggle raw display
show more
show less