Properties

Label 841.2.d.g
Level $841$
Weight $2$
Character orbit 841.d
Analytic conductor $6.715$
Analytic rank $0$
Dimension $12$
Inner twists $6$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [841,2,Mod(190,841)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(841, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("841.190");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 841 = 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 841.d (of order \(7\), degree \(6\), not minimal)

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: no
Analytic conductor: \(6.71541880999\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
Coefficient field: 12.0.4413675765625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 2x^{10} - 3x^{9} + 5x^{8} - 8x^{7} + 13x^{6} + 8x^{5} + 5x^{4} + 3x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{10} q^{3} + ( - \beta_{5} + \beta_{4}) q^{4} + (2 \beta_{3} + 3 \beta_1) q^{5} + (\beta_{11} + \beta_{10} + \cdots - \beta_1) q^{6}+ \cdots + (2 \beta_{7} - \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{10} q^{3} + ( - \beta_{5} + \beta_{4}) q^{4} + (2 \beta_{3} + 3 \beta_1) q^{5} + (\beta_{11} + \beta_{10} + \cdots - \beta_1) q^{6}+ \cdots + ( - \beta_{2} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} + q^{3} + q^{4} - q^{5} + 3 q^{6} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{2} + q^{3} + q^{4} - q^{5} + 3 q^{6} + 3 q^{9} + 7 q^{10} - 5 q^{11} - 12 q^{12} - 4 q^{13} - 5 q^{14} - 7 q^{15} + 3 q^{16} + 66 q^{17} - q^{18} - 3 q^{19} + 8 q^{20} + 5 q^{21} - 5 q^{22} - 2 q^{23} - 5 q^{24} - 13 q^{25} - 7 q^{26} - 2 q^{27} + 30 q^{28} + 36 q^{30} - 9 q^{31} + 9 q^{32} + 5 q^{33} - 8 q^{34} + 15 q^{35} - 4 q^{36} - 4 q^{37} + 6 q^{38} + 7 q^{39} - 15 q^{40} - 6 q^{41} + 5 q^{42} - 10 q^{43} + 9 q^{45} + 36 q^{46} - 14 q^{47} - 9 q^{48} + 4 q^{49} + q^{50} + 8 q^{51} - 3 q^{52} + 4 q^{53} - 11 q^{54} + 5 q^{55} + 10 q^{56} + 36 q^{57} + 6 q^{59} + q^{60} + q^{61} + 8 q^{62} - 5 q^{63} - 4 q^{64} + 13 q^{65} + 10 q^{66} + 12 q^{67} + 3 q^{68} + 6 q^{69} - 30 q^{70} - 12 q^{71} + 5 q^{72} - 14 q^{73} - 17 q^{74} + 6 q^{75} + 9 q^{76} - 5 q^{77} + 11 q^{78} - q^{79} - 21 q^{80} + 2 q^{81} - 7 q^{82} + 2 q^{83} + 2 q^{85} - 30 q^{88} - 4 q^{89} - 8 q^{90} - 10 q^{91} - 4 q^{92} - 8 q^{93} - 7 q^{94} - 24 q^{95} - 2 q^{96} - 13 q^{97} + 2 q^{98} - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 2x^{10} - 3x^{9} + 5x^{8} - 8x^{7} + 13x^{6} + 8x^{5} + 5x^{4} + 3x^{3} + 2x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 8 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} + 21\nu ) / 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{9} - 21\nu^{2} ) / 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} + 34\nu^{2} ) / 13 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{10} + 34\nu^{3} ) / 13 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{10} - 55\nu^{3} ) / 13 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -2\nu^{11} - 55\nu^{4} ) / 13 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{11} - 89\nu^{4} ) / 13 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3 \nu^{11} + 6 \nu^{10} - 9 \nu^{9} + 15 \nu^{8} - 24 \nu^{7} + 39 \nu^{6} - 65 \nu^{5} + 15 \nu^{4} + \cdots + 3 ) / 13 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 8 \nu^{11} - 8 \nu^{10} + 16 \nu^{9} - 24 \nu^{8} + 40 \nu^{7} - 65 \nu^{6} + 104 \nu^{5} + 64 \nu^{4} + \cdots + 8 ) / 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{9} + 3\beta_{8} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{11} - 5\beta_{10} - 3\beta_{9} - 3\beta_{7} + 3\beta_{5} + 3\beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{11} - 8\beta_{10} - 8\beta_{8} + 8\beta_{6} - 8\beta_{4} + 8\beta_{2} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13\beta_{2} - 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13\beta_{3} - 21\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -21\beta_{5} - 34\beta_{4} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -34\beta_{7} - 55\beta_{6} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 55\beta_{9} - 89\beta_{8} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/841\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
190.1
1.45780 + 0.702039i
−0.556829 0.268155i
1.45780 0.702039i
−0.556829 + 0.268155i
0.385338 0.483198i
−1.00883 + 1.26503i
0.360046 1.57747i
−0.137526 + 0.602539i
0.360046 + 1.57747i
−0.137526 0.602539i
0.385338 + 0.483198i
−1.00883 1.26503i
−1.45780 0.702039i −1.00883 + 1.26503i 0.385338 + 0.483198i 2.57146 + 1.23835i 2.35877 1.13592i 1.39417 1.74823i 0.497572 + 2.18001i 0.0849954 + 0.372389i −2.87930 3.61052i
190.2 0.556829 + 0.268155i 0.385338 0.483198i −1.00883 1.26503i −3.47243 1.67223i 0.344139 0.165729i −1.39417 + 1.74823i −0.497572 2.18001i 0.582567 + 2.55239i −1.48513 1.86230i
571.1 −1.45780 + 0.702039i −1.00883 1.26503i 0.385338 0.483198i 2.57146 1.23835i 2.35877 + 1.13592i 1.39417 + 1.74823i 0.497572 2.18001i 0.0849954 0.372389i −2.87930 + 3.61052i
571.2 0.556829 0.268155i 0.385338 + 0.483198i −1.00883 + 1.26503i −3.47243 + 1.67223i 0.344139 + 0.165729i −1.39417 1.74823i −0.497572 + 2.18001i 0.582567 2.55239i −1.48513 + 1.86230i
574.1 −0.385338 + 0.483198i −0.137526 + 0.602539i 0.360046 + 1.57747i 2.40299 3.01326i −0.238152 0.298633i 0.497572 2.18001i −2.01463 0.970194i 2.35877 + 1.13592i 0.530037 + 2.32225i
574.2 1.00883 1.26503i 0.360046 1.57747i −0.137526 0.602539i −1.77950 + 2.23143i −1.63232 2.04686i −0.497572 + 2.18001i 2.01463 + 0.970194i 0.344139 + 0.165729i 1.02761 + 4.50225i
605.1 −0.360046 + 1.57747i 1.45780 0.702039i −0.556829 0.268155i 0.635097 2.78254i 0.582567 + 2.55239i −2.01463 + 0.970194i −1.39417 + 1.74823i −0.238152 + 0.298633i 4.16070 + 2.00369i
605.2 0.137526 0.602539i −0.556829 + 0.268155i 1.45780 + 0.702039i −0.857618 + 3.75747i 0.0849954 + 0.372389i 2.01463 0.970194i 1.39417 1.74823i −1.63232 + 2.04686i 2.14608 + 1.03350i
645.1 −0.360046 1.57747i 1.45780 + 0.702039i −0.556829 + 0.268155i 0.635097 + 2.78254i 0.582567 2.55239i −2.01463 0.970194i −1.39417 1.74823i −0.238152 0.298633i 4.16070 2.00369i
645.2 0.137526 + 0.602539i −0.556829 0.268155i 1.45780 0.702039i −0.857618 3.75747i 0.0849954 0.372389i 2.01463 + 0.970194i 1.39417 + 1.74823i −1.63232 2.04686i 2.14608 1.03350i
778.1 −0.385338 0.483198i −0.137526 0.602539i 0.360046 1.57747i 2.40299 + 3.01326i −0.238152 + 0.298633i 0.497572 + 2.18001i −2.01463 + 0.970194i 2.35877 1.13592i 0.530037 2.32225i
778.2 1.00883 + 1.26503i 0.360046 + 1.57747i −0.137526 + 0.602539i −1.77950 2.23143i −1.63232 + 2.04686i −0.497572 2.18001i 2.01463 0.970194i 0.344139 0.165729i 1.02761 4.50225i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 190.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 5 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 841.2.d.g 12
29.b even 2 1 841.2.d.i 12
29.c odd 4 2 841.2.e.j 24
29.d even 7 1 841.2.a.c yes 2
29.d even 7 5 inner 841.2.d.g 12
29.e even 14 1 841.2.a.a 2
29.e even 14 5 841.2.d.i 12
29.f odd 28 2 841.2.b.b 4
29.f odd 28 10 841.2.e.j 24
87.h odd 14 1 7569.2.a.l 2
87.j odd 14 1 7569.2.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
841.2.a.a 2 29.e even 14 1
841.2.a.c yes 2 29.d even 7 1
841.2.b.b 4 29.f odd 28 2
841.2.d.g 12 1.a even 1 1 trivial
841.2.d.g 12 29.d even 7 5 inner
841.2.d.i 12 29.b even 2 1
841.2.d.i 12 29.e even 14 5
841.2.e.j 24 29.c odd 4 2
841.2.e.j 24 29.f odd 28 10
7569.2.a.d 2 87.j odd 14 1
7569.2.a.l 2 87.h odd 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + T_{2}^{11} + 2 T_{2}^{10} + 3 T_{2}^{9} + 5 T_{2}^{8} + 8 T_{2}^{7} + 13 T_{2}^{6} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(841, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} - T^{11} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} + T^{11} + \cdots + 1771561 \) Copy content Toggle raw display
$7$ \( T^{12} + 5 T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$11$ \( T^{12} + 5 T^{11} + \cdots + 15625 \) Copy content Toggle raw display
$13$ \( T^{12} + 4 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( (T^{2} - 11 T + 29)^{6} \) Copy content Toggle raw display
$19$ \( T^{12} + 3 T^{11} + \cdots + 531441 \) Copy content Toggle raw display
$23$ \( T^{12} + 2 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} + 9 T^{11} + \cdots + 1771561 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 4750104241 \) Copy content Toggle raw display
$41$ \( (T^{2} + T - 11)^{6} \) Copy content Toggle raw display
$43$ \( T^{12} + 10 T^{11} + \cdots + 64000000 \) Copy content Toggle raw display
$47$ \( (T^{6} + 7 T^{5} + \cdots + 117649)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 2 T^{5} + 4 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - T - 31)^{6} \) Copy content Toggle raw display
$61$ \( T^{12} - T^{11} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{12} - 12 T^{11} + \cdots + 16777216 \) Copy content Toggle raw display
$71$ \( T^{12} + 12 T^{11} + \cdots + 16777216 \) Copy content Toggle raw display
$73$ \( T^{12} + 14 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 887503681 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 243087455521 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 4750104241 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 42180533641 \) Copy content Toggle raw display
show more
show less