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Properties

Label 841.2.d.j
Level $841$
Weight $2$
Character orbit 841.d
Analytic conductor $6.715$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $6$

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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: 12.0.74049191673856.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 2x^{10} + 4x^{8} + 8x^{6} + 16x^{4} + 32x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 29)
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.

\(f(q)\) \(=\) \( q + (\beta_{9} - \beta_{2}) q^{2} + (\beta_{10} - \beta_{3}) q^{3} + ( - 2 \beta_{11} + \beta_{4}) q^{4} - \beta_{2} q^{5} + (3 \beta_{10} + 3 \beta_{8} + 3 \beta_{6} + \cdots + 3) q^{6}+ \cdots + (2 \beta_{11} + 2 \beta_{9} + \cdots + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} - \beta_{2}) q^{2} + (\beta_{10} - \beta_{3}) q^{3} + ( - 2 \beta_{11} + \beta_{4}) q^{4} - \beta_{2} q^{5} + (3 \beta_{10} + 3 \beta_{8} + 3 \beta_{6} + \cdots + 3) q^{6}+ \cdots + ( - 2 \beta_{7} - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{5} + 6 q^{6} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{5} + 6 q^{6} + 6 q^{8} - 2 q^{10} - 2 q^{11} + 60 q^{12} + 2 q^{13} - 8 q^{14} + 2 q^{15} - 6 q^{16} - 24 q^{17} + 8 q^{18} - 12 q^{19} + 2 q^{20} + 8 q^{21} - 2 q^{22} + 4 q^{23} + 10 q^{24} + 8 q^{25} - 10 q^{26} - 2 q^{27} - 96 q^{28} + 36 q^{30} - 6 q^{31} - 6 q^{32} + 2 q^{33} + 4 q^{34} - 16 q^{36} + 8 q^{37} + 12 q^{38} + 10 q^{39} - 6 q^{40} + 48 q^{41} - 16 q^{42} - 10 q^{43} + 6 q^{44} - 72 q^{46} - 2 q^{47} - 6 q^{48} - 2 q^{49} - 8 q^{50} - 4 q^{51} + 18 q^{52} - 2 q^{53} - 2 q^{54} + 2 q^{55} - 8 q^{56} + 72 q^{57} + 24 q^{59} + 10 q^{60} + 4 q^{61} + 26 q^{62} + 16 q^{63} + 14 q^{64} - 2 q^{65} - 2 q^{66} - 12 q^{68} - 12 q^{69} - 48 q^{70} + 12 q^{71} + 8 q^{72} - 8 q^{73} - 8 q^{74} - 48 q^{75} - 12 q^{76} - 8 q^{77} - 22 q^{78} + 2 q^{79} + 6 q^{80} + 2 q^{81} - 16 q^{82} - 4 q^{83} + 24 q^{84} - 4 q^{85} - 36 q^{86} - 12 q^{88} + 8 q^{89} - 8 q^{90} - 16 q^{91} - 28 q^{92} - 26 q^{93} - 10 q^{94} + 12 q^{95} - 2 q^{96} + 8 q^{97} + 2 q^{98} - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 2x^{10} + 4x^{8} + 8x^{6} + 16x^{4} + 32x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{8} ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16\beta_{8} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 16\beta_{9} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 32\beta_{10} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 32\beta_{11} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(\beta_{4}\)

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
−0.314692 + 1.37876i
0.314692 − 1.37876i
−0.314692 − 1.37876i
0.314692 + 1.37876i
1.27416 − 0.613604i
−1.27416 + 0.613604i
0.881748 + 1.10568i
−0.881748 − 1.10568i
0.881748 − 1.10568i
−0.881748 + 1.10568i
1.27416 + 0.613604i
−1.27416 − 0.613604i
−0.373194 − 0.179721i −0.258258 + 0.323845i −1.14001 − 1.42952i 0.900969 + 0.433884i 0.154582 − 0.0744427i 1.76350 − 2.21135i 0.352871 + 1.54603i 0.629384 + 2.75751i −0.258258 − 0.323845i
190.2 2.17513 + 1.04749i 1.50524 − 1.88751i 2.38699 + 2.99318i 0.900969 + 0.433884i 5.25123 − 2.52886i −1.76350 + 2.21135i 0.982255 + 4.30354i −0.629384 − 2.75751i 1.50524 + 1.88751i
571.1 −0.373194 + 0.179721i −0.258258 − 0.323845i −1.14001 + 1.42952i 0.900969 − 0.433884i 0.154582 + 0.0744427i 1.76350 + 2.21135i 0.352871 − 1.54603i 0.629384 − 2.75751i −0.258258 + 0.323845i
571.2 2.17513 − 1.04749i 1.50524 + 1.88751i 2.38699 − 2.99318i 0.900969 − 0.433884i 5.25123 + 2.52886i −1.76350 − 2.21135i 0.982255 − 4.30354i −0.629384 + 2.75751i 1.50524 − 1.88751i
574.1 −1.50524 + 1.88751i −0.537213 + 2.35368i −0.851905 − 3.73244i −0.623490 + 0.781831i −3.63396 − 4.55685i 0.629384 − 2.75751i 3.97707 + 1.91526i −2.54832 − 1.22721i −0.537213 − 2.35368i
574.2 0.258258 − 0.323845i 0.0921712 − 0.403828i 0.406863 + 1.78258i −0.623490 + 0.781831i −0.106974 − 0.134141i −0.629384 + 2.75751i 1.42874 + 0.688047i 2.54832 + 1.22721i 0.0921712 + 0.403828i
605.1 −0.0921712 + 0.403828i 0.373194 − 0.179721i 1.64736 + 0.793325i 0.222521 − 0.974928i 0.0381786 + 0.167271i −2.54832 + 1.22721i −0.988722 + 1.23982i −1.76350 + 2.21135i 0.373194 + 0.179721i
605.2 0.537213 − 2.35368i −2.17513 + 1.04749i −3.44929 − 1.66109i 0.222521 − 0.974928i 1.29695 + 5.68230i 2.54832 − 1.22721i −2.75222 + 3.45117i 1.76350 − 2.21135i −2.17513 − 1.04749i
645.1 −0.0921712 − 0.403828i 0.373194 + 0.179721i 1.64736 − 0.793325i 0.222521 + 0.974928i 0.0381786 − 0.167271i −2.54832 − 1.22721i −0.988722 − 1.23982i −1.76350 − 2.21135i 0.373194 − 0.179721i
645.2 0.537213 + 2.35368i −2.17513 − 1.04749i −3.44929 + 1.66109i 0.222521 + 0.974928i 1.29695 − 5.68230i 2.54832 + 1.22721i −2.75222 − 3.45117i 1.76350 + 2.21135i −2.17513 + 1.04749i
778.1 −1.50524 − 1.88751i −0.537213 − 2.35368i −0.851905 + 3.73244i −0.623490 − 0.781831i −3.63396 + 4.55685i 0.629384 + 2.75751i 3.97707 − 1.91526i −2.54832 + 1.22721i −0.537213 + 2.35368i
778.2 0.258258 + 0.323845i 0.0921712 + 0.403828i 0.406863 − 1.78258i −0.623490 − 0.781831i −0.106974 + 0.134141i −0.629384 − 2.75751i 1.42874 − 0.688047i 2.54832 − 1.22721i 0.0921712 − 0.403828i
\(n\): 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.j 12
29.b even 2 1 841.2.d.f 12
29.c odd 4 2 841.2.e.k 24
29.d even 7 1 29.2.a.a ✓ 2
29.d even 7 5 inner 841.2.d.j 12
29.e even 14 1 841.2.a.d 2
29.e even 14 5 841.2.d.f 12
29.f odd 28 2 841.2.b.a 4
29.f odd 28 10 841.2.e.k 24
87.h odd 14 1 7569.2.a.c 2
87.j odd 14 1 261.2.a.d 2
116.j odd 14 1 464.2.a.h 2
145.n even 14 1 725.2.a.b 2
145.p odd 28 2 725.2.b.b 4
203.n odd 14 1 1421.2.a.j 2
232.p odd 14 1 1856.2.a.w 2
232.s even 14 1 1856.2.a.r 2
319.m odd 14 1 3509.2.a.j 2
348.s even 14 1 4176.2.a.bq 2
377.w even 14 1 4901.2.a.g 2
435.w odd 14 1 6525.2.a.o 2
493.p even 14 1 8381.2.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.a.a ✓ 2 29.d even 7 1
261.2.a.d 2 87.j odd 14 1
464.2.a.h 2 116.j odd 14 1
725.2.a.b 2 145.n even 14 1
725.2.b.b 4 145.p odd 28 2
841.2.a.d 2 29.e even 14 1
841.2.b.a 4 29.f odd 28 2
841.2.d.f 12 29.b even 2 1
841.2.d.f 12 29.e even 14 5
841.2.d.j 12 1.a even 1 1 trivial
841.2.d.j 12 29.d even 7 5 inner
841.2.e.k 24 29.c odd 4 2
841.2.e.k 24 29.f odd 28 10
1421.2.a.j 2 203.n odd 14 1
1856.2.a.r 2 232.s even 14 1
1856.2.a.w 2 232.p odd 14 1
3509.2.a.j 2 319.m odd 14 1
4176.2.a.bq 2 348.s even 14 1
4901.2.a.g 2 377.w even 14 1
6525.2.a.o 2 435.w odd 14 1
7569.2.a.c 2 87.h odd 14 1
8381.2.a.e 2 493.p even 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 2 T_{2}^{11} + 5 T_{2}^{10} - 12 T_{2}^{9} + 29 T_{2}^{8} - 70 T_{2}^{7} + 169 T_{2}^{6} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(841, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + 8 T^{10} + \cdots + 262144 \) Copy content Toggle raw display
$11$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{12} - 2 T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T - 4)^{6} \) Copy content Toggle raw display
$19$ \( (T^{6} + 6 T^{5} + \cdots + 46656)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 481890304 \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 4750104241 \) Copy content Toggle raw display
$37$ \( (T^{6} - 4 T^{5} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 8 T - 56)^{6} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 148035889 \) Copy content Toggle raw display
$47$ \( T^{12} + 2 T^{11} + \cdots + 24137569 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 128100283921 \) Copy content Toggle raw display
$59$ \( (T^{2} - 4 T - 28)^{6} \) Copy content Toggle raw display
$61$ \( T^{12} - 4 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 1073741824 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 481890304 \) Copy content Toggle raw display
$73$ \( (T^{6} + 4 T^{5} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 481890304 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 30840979456 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 30840979456 \) Copy content Toggle raw display
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