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Properties

Label 841.2.b.e
Level $841$
Weight $2$
Character orbit 841.b
Analytic conductor $6.715$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [841,2,Mod(840,841)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(841, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("841.840");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 841 = 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 841.b (of order \(2\), degree \(1\), 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: \(12\)
Coefficient field: 12.0.7877952219361.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3x^{11} + 13x^{9} - 18x^{8} - 14x^{7} + 57x^{6} - 28x^{5} - 72x^{4} + 104x^{3} - 96x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 29)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{11} - \beta_1) q^{2} + (\beta_{11} - \beta_{5}) q^{3} + ( - \beta_{8} + \beta_{7} + \beta_{3} - 1) q^{4} + ( - \beta_{6} - 1) q^{5} + (\beta_{10} - \beta_{8} + \beta_{7} + \cdots - 3) q^{6}+ \cdots + (2 \beta_{10} - \beta_{8} + \beta_{6} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{11} - \beta_1) q^{2} + (\beta_{11} - \beta_{5}) q^{3} + ( - \beta_{8} + \beta_{7} + \beta_{3} - 1) q^{4} + ( - \beta_{6} - 1) q^{5} + (\beta_{10} - \beta_{8} + \beta_{7} + \cdots - 3) q^{6}+ \cdots + ( - \beta_{9} - 4 \beta_{5} + \cdots - 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{4} - 8 q^{5} - 24 q^{6} + 10 q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{4} - 8 q^{5} - 24 q^{6} + 10 q^{7} - 10 q^{9} - 12 q^{13} + 16 q^{16} + 24 q^{20} + 38 q^{22} + 30 q^{23} + 10 q^{24} - 8 q^{25} + 12 q^{28} + 2 q^{30} + 4 q^{33} - 6 q^{34} - 44 q^{35} + 16 q^{36} - 6 q^{42} - 12 q^{45} + 6 q^{49} - 8 q^{51} - 6 q^{52} + 32 q^{53} + 32 q^{54} - 14 q^{57} + 44 q^{59} + 16 q^{62} - 34 q^{63} + 2 q^{64} + 8 q^{65} - 30 q^{67} - 70 q^{71} + 56 q^{74} - 4 q^{78} - 34 q^{80} + 8 q^{81} - 62 q^{82} + 82 q^{83} - 44 q^{86} - 66 q^{88} + 32 q^{91} + 22 q^{92} + 12 q^{93} + 10 q^{94} + 58 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3x^{11} + 13x^{9} - 18x^{8} - 14x^{7} + 57x^{6} - 28x^{5} - 72x^{4} + 104x^{3} - 96x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{10} + 5\nu^{9} - 19\nu^{7} + 18\nu^{6} + 22\nu^{5} - 59\nu^{4} + 8\nu^{3} + 76\nu^{2} - 56\nu - 16 ) / 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} + 9 \nu^{10} - 22 \nu^{9} - 9 \nu^{8} + 72 \nu^{7} - 66 \nu^{6} - 77 \nu^{5} + 202 \nu^{4} + \cdots - 224 ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + \nu^{10} - 7\nu^{8} + 2\nu^{7} + 22\nu^{6} - 39\nu^{5} - 16\nu^{4} + 84\nu^{3} - 24\nu^{2} - 80\nu + 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 5 \nu^{10} - 6 \nu^{9} + 25 \nu^{8} - 20 \nu^{7} - 54 \nu^{6} + 85 \nu^{5} + 26 \nu^{4} + \cdots - 96 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{11} + 5\nu^{9} - 13\nu^{8} - \nu^{7} + 32\nu^{6} - 35\nu^{5} - 31\nu^{4} + 80\nu^{3} - 28\nu^{2} - 88\nu + 80 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5 \nu^{11} - 21 \nu^{10} + 6 \nu^{9} + 77 \nu^{8} - 112 \nu^{7} - 70 \nu^{6} + 305 \nu^{5} + \cdots - 416 ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9 \nu^{11} - 5 \nu^{10} - 30 \nu^{9} + 65 \nu^{8} + 4 \nu^{7} - 198 \nu^{6} + 189 \nu^{5} + 178 \nu^{4} + \cdots - 352 ) / 128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11 \nu^{11} - 11 \nu^{10} - 38 \nu^{9} + 83 \nu^{8} + 32 \nu^{7} - 234 \nu^{6} + 95 \nu^{5} + \cdots - 96 ) / 128 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11 \nu^{11} + 35 \nu^{10} + 14 \nu^{9} - 163 \nu^{8} + 152 \nu^{7} + 298 \nu^{6} - 591 \nu^{5} + \cdots + 736 ) / 128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15 \nu^{11} - 23 \nu^{10} - 38 \nu^{9} + 135 \nu^{8} - 40 \nu^{7} - 290 \nu^{6} + 323 \nu^{5} + \cdots - 352 ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4 \nu^{11} - 7 \nu^{10} - 11 \nu^{9} + 36 \nu^{8} - 11 \nu^{7} - 78 \nu^{6} + 98 \nu^{5} + 61 \nu^{4} + \cdots - 144 ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{10} + \beta_{8} - \beta_{5} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - 2\beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - 2\beta_{6} - 2\beta_{5} + \beta_{3} - \beta_{2} + \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{11} - 2 \beta_{10} - 3 \beta_{9} - \beta_{8} - 2 \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{11} - 5\beta_{10} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{3} - 4\beta_{2} - 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5 \beta_{11} - 3 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + 5 \beta_{6} - \beta_{4} + \cdots + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5\beta_{11} + 5\beta_{9} - 2\beta_{5} - 5\beta_{4} - 14\beta_{3} - 5\beta_{2} + \beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4 \beta_{11} + 9 \beta_{10} + 4 \beta_{9} + \beta_{8} - 10 \beta_{7} + 4 \beta_{6} - 11 \beta_{5} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 8 \beta_{11} + 6 \beta_{10} - 11 \beta_{9} + 4 \beta_{8} - \beta_{7} - 7 \beta_{6} + 3 \beta_{5} + \cdots - 20 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 15 \beta_{11} + 8 \beta_{10} - 7 \beta_{9} - 9 \beta_{8} + 5 \beta_{7} - 14 \beta_{5} - 2 \beta_{4} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 10 \beta_{11} - 18 \beta_{10} - 7 \beta_{9} + \beta_{8} + 36 \beta_{7} + 7 \beta_{6} + 21 \beta_{5} + \cdots - 18 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
840.1
0.911180 + 1.08155i
1.38491 − 0.286410i
1.23295 − 0.692694i
−1.41140 + 0.0891373i
0.639551 + 1.26134i
−1.25719 + 0.647667i
−1.25719 − 0.647667i
0.639551 − 1.26134i
−1.41140 − 0.0891373i
1.23295 + 0.692694i
1.38491 + 0.286410i
0.911180 − 1.08155i
− 2.60244i − 0.439339i −4.77269 −2.58042 −1.14335 0.0751311 7.21577i 2.80698 6.71540i
840.2 − 2.26775i − 2.84057i −3.14268 −0.0578828 −6.44169 −1.56196 2.59131i −5.06883 0.131264i
840.3 − 1.54928i − 2.93466i −0.400257 0.454328 −4.54661 3.41326 − 2.47844i −5.61226 − 0.703879i
840.4 − 1.16308i − 0.984809i 0.647237 −1.89937 −1.14541 1.52574 − 3.07896i 2.03015 2.20913i
840.5 − 0.549876i 1.97280i 1.69764 −2.74405 1.08480 4.47381 − 2.03324i −0.891943 1.50889i
840.6 − 0.171009i 1.12432i 1.97076 2.82740 0.192270 −2.92599 − 0.679037i 1.73590 − 0.483513i
840.7 0.171009i − 1.12432i 1.97076 2.82740 0.192270 −2.92599 0.679037i 1.73590 0.483513i
840.8 0.549876i − 1.97280i 1.69764 −2.74405 1.08480 4.47381 2.03324i −0.891943 − 1.50889i
840.9 1.16308i 0.984809i 0.647237 −1.89937 −1.14541 1.52574 3.07896i 2.03015 − 2.20913i
840.10 1.54928i 2.93466i −0.400257 0.454328 −4.54661 3.41326 2.47844i −5.61226 0.703879i
840.11 2.26775i 2.84057i −3.14268 −0.0578828 −6.44169 −1.56196 − 2.59131i −5.06883 − 0.131264i
840.12 2.60244i 0.439339i −4.77269 −2.58042 −1.14335 0.0751311 − 7.21577i 2.80698 − 6.71540i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 840.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 841.2.b.e 12
29.b even 2 1 inner 841.2.b.e 12
29.c odd 4 2 841.2.a.k 12
29.d even 7 1 29.2.e.a ✓ 12
29.d even 7 1 841.2.e.a 12
29.d even 7 1 841.2.e.e 12
29.d even 7 1 841.2.e.f 12
29.d even 7 1 841.2.e.h 12
29.d even 7 1 841.2.e.i 12
29.e even 14 1 29.2.e.a ✓ 12
29.e even 14 1 841.2.e.a 12
29.e even 14 1 841.2.e.e 12
29.e even 14 1 841.2.e.f 12
29.e even 14 1 841.2.e.h 12
29.e even 14 1 841.2.e.i 12
29.f odd 28 4 841.2.d.k 24
29.f odd 28 4 841.2.d.l 24
29.f odd 28 4 841.2.d.m 24
87.f even 4 2 7569.2.a.bp 12
87.h odd 14 1 261.2.o.a 12
87.j odd 14 1 261.2.o.a 12
116.h odd 14 1 464.2.y.d 12
116.j odd 14 1 464.2.y.d 12
145.l even 14 1 725.2.q.a 12
145.n even 14 1 725.2.q.a 12
145.p odd 28 2 725.2.p.a 24
145.q odd 28 2 725.2.p.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.e.a ✓ 12 29.d even 7 1
29.2.e.a ✓ 12 29.e even 14 1
261.2.o.a 12 87.h odd 14 1
261.2.o.a 12 87.j odd 14 1
464.2.y.d 12 116.h odd 14 1
464.2.y.d 12 116.j odd 14 1
725.2.p.a 24 145.p odd 28 2
725.2.p.a 24 145.q odd 28 2
725.2.q.a 12 145.l even 14 1
725.2.q.a 12 145.n even 14 1
841.2.a.k 12 29.c odd 4 2
841.2.b.e 12 1.a even 1 1 trivial
841.2.b.e 12 29.b even 2 1 inner
841.2.d.k 24 29.f odd 28 4
841.2.d.l 24 29.f odd 28 4
841.2.d.m 24 29.f odd 28 4
841.2.e.a 12 29.d even 7 1
841.2.e.a 12 29.e even 14 1
841.2.e.e 12 29.d even 7 1
841.2.e.e 12 29.e even 14 1
841.2.e.f 12 29.d even 7 1
841.2.e.f 12 29.e even 14 1
841.2.e.h 12 29.d even 7 1
841.2.e.h 12 29.e even 14 1
841.2.e.i 12 29.d even 7 1
841.2.e.i 12 29.e even 14 1
7569.2.a.bp 12 87.f even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 16T_{2}^{10} + 88T_{2}^{8} + 197T_{2}^{6} + 170T_{2}^{4} + 39T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(841, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 16 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} + 23 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( (T^{6} + 4 T^{5} - 5 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 5 T^{5} - 10 T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 51 T^{10} + \cdots + 10816 \) Copy content Toggle raw display
$13$ \( (T^{6} + 6 T^{5} - 13 T^{4} + \cdots + 29)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 71 T^{10} + \cdots + 53824 \) Copy content Toggle raw display
$19$ \( T^{12} + 63 T^{10} + \cdots + 3136 \) Copy content Toggle raw display
$23$ \( (T^{6} - 15 T^{5} + 78 T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} + 95 T^{10} + \cdots + 817216 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 110103049 \) Copy content Toggle raw display
$41$ \( T^{12} + 99 T^{10} + \cdots + 107584 \) Copy content Toggle raw display
$43$ \( T^{12} + 263 T^{10} + \cdots + 24364096 \) Copy content Toggle raw display
$47$ \( T^{12} + 387 T^{10} + \cdots + 11343424 \) Copy content Toggle raw display
$53$ \( (T^{6} - 16 T^{5} + \cdots - 97)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 22 T^{5} + \cdots + 1856)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 325694209 \) Copy content Toggle raw display
$67$ \( (T^{6} + 15 T^{5} + \cdots + 20392)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 35 T^{5} + \cdots - 51688)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 210 T^{10} + \cdots + 625681 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 30056463424 \) Copy content Toggle raw display
$83$ \( (T^{6} - 41 T^{5} + \cdots - 20488)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 203946961 \) Copy content Toggle raw display
$97$ \( T^{12} + 478 T^{10} + \cdots + 1697809 \) Copy content Toggle raw display
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