↑

Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [841,2,Mod(63,841)]
 
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([11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("841.63");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 841 = 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 841.e (of order \(14\), 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 \(\Q(\zeta_{14})\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 29)
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{11} - \beta_{8} - \beta_{4}) q^{2} + (\beta_{11} + \beta_{10} + \cdots - \beta_1) q^{3}+ \cdots + (3 \beta_{11} - \beta_{10} - \beta_{9} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{11} - \beta_{8} - \beta_{4}) q^{2} + (\beta_{11} + \beta_{10} + \cdots - \beta_1) q^{3}+ \cdots + ( - \beta_{11} - 3 \beta_{10} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{4} - q^{5} + 4 q^{6} + 10 q^{7} + 14 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{4} - q^{5} + 4 q^{6} + 10 q^{7} + 14 q^{8} + 18 q^{9} + 14 q^{10} - 5 q^{13} + 7 q^{15} + 16 q^{16} - 21 q^{18} + 31 q^{20} - 25 q^{22} - 12 q^{23} + 31 q^{24} - 15 q^{25} - 21 q^{26} + 42 q^{27} + 12 q^{28} + 2 q^{30} - 14 q^{31} + 14 q^{32} - 3 q^{33} - 20 q^{34} - 9 q^{35} + 16 q^{36} + 49 q^{37} - 7 q^{38} + 35 q^{39} + 35 q^{40} - 20 q^{42} + 14 q^{43} - 35 q^{44} - 5 q^{45} - 28 q^{47} - 14 q^{48} - 22 q^{49} + 14 q^{50} + 6 q^{51} + 15 q^{52} + 4 q^{53} + 11 q^{54} + 28 q^{55} - 21 q^{56} - 14 q^{57} + 44 q^{59} + 7 q^{60} - 35 q^{61} - 12 q^{62} + 36 q^{63} + 44 q^{64} + 29 q^{65} - 14 q^{66} + 26 q^{67} + 7 q^{68} + 28 q^{71} + 42 q^{72} + 14 q^{73} - 21 q^{74} + 14 q^{76} + 21 q^{77} - 4 q^{78} + 14 q^{79} + 29 q^{80} - 27 q^{81} + 22 q^{82} - 30 q^{83} + 21 q^{85} - 44 q^{86} - 66 q^{88} - 21 q^{89} - 14 q^{90} - 17 q^{91} - 6 q^{92} - 23 q^{93} - 53 q^{94} - 42 q^{95} - 54 q^{96} - 28 q^{97} - 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( \nu^{11} + 7 \nu^{10} - 10 \nu^{9} - 7 \nu^{8} + 40 \nu^{7} - 14 \nu^{6} - 83 \nu^{5} + 102 \nu^{4} + \cdots + 96 ) / 128 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} + 5 \nu^{10} - 10 \nu^{9} + 7 \nu^{8} + 4 \nu^{7} - 58 \nu^{6} + 59 \nu^{5} + 38 \nu^{4} + \cdots - 288 ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{11} + 5 \nu^{10} - 10 \nu^{9} + 7 \nu^{8} + 36 \nu^{7} - 58 \nu^{6} - 5 \nu^{5} + 134 \nu^{4} + \cdots - 32 ) / 128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 3 \nu^{10} + 4 \nu^{9} + \nu^{8} - 18 \nu^{7} + 22 \nu^{6} + 17 \nu^{5} - 52 \nu^{4} + \cdots + 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{11} + 5 \nu^{10} - 10 \nu^{9} - 25 \nu^{8} + 68 \nu^{7} + 6 \nu^{6} - 165 \nu^{5} + 134 \nu^{4} + \cdots + 224 ) / 128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3 \nu^{11} + \nu^{10} - 18 \nu^{9} + 11 \nu^{8} + 36 \nu^{7} - 50 \nu^{6} - 49 \nu^{5} + 94 \nu^{4} + \cdots + 96 ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{10} + 3\nu^{9} - 9\nu^{7} + 10\nu^{6} + 6\nu^{5} - 29\nu^{4} + 16\nu^{3} + 20\nu^{2} - 40\nu + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{11} + 5 \nu^{10} - 10 \nu^{9} - 9 \nu^{8} + 36 \nu^{7} - 26 \nu^{6} - 53 \nu^{5} + 86 \nu^{4} + \cdots - 32 ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} - 5\nu^{9} + 5\nu^{8} + 9\nu^{7} - 16\nu^{6} - 5\nu^{5} + 31\nu^{4} - 12\nu^{2} + 8\nu + 16 ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3 \nu^{11} - 11 \nu^{10} - 14 \nu^{9} + 59 \nu^{8} - 40 \nu^{7} - 122 \nu^{6} + 199 \nu^{5} + \cdots - 352 ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 15 \nu^{11} + 31 \nu^{10} + 30 \nu^{9} - 151 \nu^{8} + 80 \nu^{7} + 290 \nu^{6} - 435 \nu^{5} + \cdots + 480 ) / 128 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{8} + \beta_{5} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{6} - \beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + \beta_{9} + 2\beta_{8} - 3\beta_{6} - \beta_{5} - \beta_{3} - 2\beta_{2} + 2\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{10} + \beta_{8} - \beta_{7} - \beta_{5} + 5\beta_{4} + 4\beta_{3} - 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -5\beta_{10} + \beta_{9} - 6\beta_{8} - 5\beta_{7} + 5\beta_{6} + 5\beta_{3} + \beta_{2} - 10\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4 \beta_{11} + 4 \beta_{10} + 10 \beta_{9} - 11 \beta_{8} + 4 \beta_{7} + 4 \beta_{6} + \beta_{5} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 6 \beta_{11} - 10 \beta_{10} - 3 \beta_{9} - 4 \beta_{8} - 11 \beta_{6} + 4 \beta_{5} - 15 \beta_{4} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2\beta_{11} + 9\beta_{9} + 9\beta_{7} - 7\beta_{6} - 9\beta_{5} + \beta_{4} + 9\beta_{3} + 5\beta_{2} + 18\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 18 \beta_{11} - 29 \beta_{10} - 29 \beta_{9} - 8 \beta_{8} - 28 \beta_{7} - 7 \beta_{6} + \beta_{5} + \cdots + 11 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
1.38491 + 0.286410i
0.639551 − 1.26134i
−1.25719 + 0.647667i
0.911180 − 1.08155i
−1.25719 − 0.647667i
0.911180 + 1.08155i
1.38491 − 0.286410i
0.639551 + 1.26134i
1.23295 + 0.692694i
−1.41140 + 0.0891373i
1.23295 − 0.692694i
−1.41140 − 0.0891373i
−1.77300 + 1.41392i 2.76935 − 0.632086i 0.699312 − 3.06388i −0.0360893 − 0.0452546i −4.01633 + 5.03632i 0.347569 + 1.52280i 1.12433 + 2.33469i 4.56686 − 2.19928i 0.127973 + 0.0292089i
63.2 −0.429910 + 0.342842i −1.92334 + 0.438989i −0.377760 + 1.65507i −1.71089 − 2.14539i 0.676359 − 0.848127i −0.995517 − 4.36165i −0.882190 − 1.83189i 0.803613 − 0.387000i 1.47106 + 0.335760i
196.1 −0.166722 + 0.0380532i 0.487826 − 1.01298i −1.77559 + 0.855079i −0.629156 − 2.75651i −0.0427841 + 0.187449i 2.63622 + 1.26954i 0.530892 − 0.423373i 1.08231 + 1.35718i 0.209788 + 0.435630i
196.2 2.53719 − 0.579097i 0.190622 − 0.395831i 4.30005 − 2.07079i 0.574198 + 2.51573i 0.254420 − 1.11469i −0.0676908 − 0.0325982i 5.64151 − 4.49896i 1.75012 + 2.19459i 2.91370 + 6.05036i
236.1 −0.166722 − 0.0380532i 0.487826 + 1.01298i −1.77559 − 0.855079i −0.629156 + 2.75651i −0.0427841 − 0.187449i 2.63622 − 1.26954i 0.530892 + 0.423373i 1.08231 − 1.35718i 0.209788 − 0.435630i
236.2 2.53719 + 0.579097i 0.190622 + 0.395831i 4.30005 + 2.07079i 0.574198 − 2.51573i 0.254420 + 1.11469i −0.0676908 + 0.0325982i 5.64151 + 4.49896i 1.75012 − 2.19459i 2.91370 − 6.05036i
267.1 −1.77300 − 1.41392i 2.76935 + 0.632086i 0.699312 + 3.06388i −0.0360893 + 0.0452546i −4.01633 − 5.03632i 0.347569 − 1.52280i 1.12433 − 2.33469i 4.56686 + 2.19928i 0.127973 − 0.0292089i
267.2 −0.429910 − 0.342842i −1.92334 − 0.438989i −0.377760 − 1.65507i −1.71089 + 2.14539i 0.676359 + 0.848127i −0.995517 + 4.36165i −0.882190 + 1.83189i 0.803613 + 0.387000i 1.47106 − 0.335760i
270.1 −0.672206 − 1.39585i −2.29441 + 1.82973i −0.249556 + 0.312934i −0.409335 + 0.197125i 4.09635 + 1.97270i 2.12813 + 2.66859i −2.41630 − 0.551506i 1.24884 − 5.47155i 0.550315 + 0.438862i
270.2 0.504643 + 1.04790i 0.769955 − 0.614018i 0.403546 − 0.506030i 1.71127 − 0.824106i 1.03198 + 0.496977i 0.951284 + 1.19287i 3.00176 + 0.685132i −0.451751 + 1.97925i 1.72716 + 1.37737i
651.1 −0.672206 + 1.39585i −2.29441 − 1.82973i −0.249556 − 0.312934i −0.409335 − 0.197125i 4.09635 − 1.97270i 2.12813 − 2.66859i −2.41630 + 0.551506i 1.24884 + 5.47155i 0.550315 − 0.438862i
651.2 0.504643 − 1.04790i 0.769955 + 0.614018i 0.403546 + 0.506030i 1.71127 + 0.824106i 1.03198 − 0.496977i 0.951284 − 1.19287i 3.00176 − 0.685132i −0.451751 − 1.97925i 1.72716 − 1.37737i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.e even 14 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 5 T_{2}^{10} - 14 T_{2}^{9} + 11 T_{2}^{8} + 28 T_{2}^{7} + 85 T_{2}^{6} + 56 T_{2}^{5} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(841, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 5 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} - 12 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{12} - 10 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{12} + 2 T^{10} + \cdots + 10816 \) Copy content Toggle raw display
$13$ \( T^{12} + 5 T^{11} + \cdots + 841 \) Copy content Toggle raw display
$17$ \( T^{12} + 71 T^{10} + \cdots + 53824 \) Copy content Toggle raw display
$19$ \( T^{12} - 14 T^{10} + \cdots + 3136 \) Copy content Toggle raw display
$23$ \( T^{12} + 12 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} + 14 T^{11} + \cdots + 817216 \) 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
$43$ \( T^{12} - 14 T^{11} + \cdots + 24364096 \) Copy content Toggle raw display
$47$ \( T^{12} + 28 T^{11} + \cdots + 11343424 \) Copy content Toggle raw display
$53$ \( T^{12} - 4 T^{11} + \cdots + 9409 \) 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
$67$ \( T^{12} + \cdots + 415833664 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 2671649344 \) Copy content Toggle raw display
$73$ \( T^{12} - 14 T^{11} + \cdots + 625681 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 30056463424 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 419758144 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 203946961 \) Copy content Toggle raw display
$97$ \( T^{12} + 28 T^{11} + \cdots + 1697809 \) Copy content Toggle raw display
show more
show less