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Properties

Label 1341.2.a.e
Level $1341$
Weight $2$
Character orbit 1341.a
Self dual yes
Analytic conductor $10.708$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1341,2,Mod(1,1341)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1341, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1341.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1341 = 3^{2} \cdot 149 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1341.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: \(10.7079389111\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 15x^{7} + 12x^{6} + 75x^{5} - 48x^{4} - 137x^{3} + 76x^{2} + 68x - 39 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 149)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \cdots + 2) q^{4}+ \cdots + (\beta_{8} - \beta_{7} + 2 \beta_{6} + \cdots + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \cdots + 2) q^{4}+ \cdots + ( - \beta_{8} + 9 \beta_{7} + \cdots - 11) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + q^{2} + 13 q^{4} + q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + q^{2} + 13 q^{4} + q^{5} + 3 q^{7} + 6 q^{8} - 6 q^{10} - 5 q^{11} + 7 q^{13} + 8 q^{14} + 13 q^{16} + 5 q^{17} + 30 q^{19} + 10 q^{20} - 7 q^{22} + 4 q^{23} + 6 q^{25} + 15 q^{26} - 12 q^{28} + 16 q^{29} + 22 q^{31} + 38 q^{32} + 9 q^{34} + 11 q^{35} - 7 q^{37} + 18 q^{38} - 7 q^{40} - 6 q^{41} + 4 q^{43} - 6 q^{44} + q^{46} + 6 q^{47} + 14 q^{49} + 16 q^{50} + 50 q^{52} + 2 q^{53} - 2 q^{55} - 7 q^{56} + 2 q^{58} - 43 q^{59} + q^{61} - 33 q^{62} + 18 q^{64} + 20 q^{65} + 33 q^{67} + 16 q^{68} - 3 q^{70} - 15 q^{71} - 11 q^{73} - 33 q^{74} + 59 q^{76} + 30 q^{77} + q^{79} - 65 q^{80} + 5 q^{82} + 4 q^{83} - 34 q^{85} + 7 q^{86} - 37 q^{88} + 19 q^{89} + 62 q^{91} - 17 q^{92} + 17 q^{94} + 21 q^{95} - q^{97} - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 15x^{7} + 12x^{6} + 75x^{5} - 48x^{4} - 137x^{3} + 76x^{2} + 68x - 39 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{8} + \nu^{7} + 14\nu^{6} - 9\nu^{5} - 63\nu^{4} + 19\nu^{3} + 92\nu^{2} - 3\nu - 26 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} - 15\nu^{6} - 3\nu^{5} + 72\nu^{4} + 24\nu^{3} - 117\nu^{2} - 37\nu + 51 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{8} + 17\nu^{6} + \nu^{5} - 94\nu^{4} - 12\nu^{3} + 185\nu^{2} + 31\nu - 91 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{8} + \nu^{7} + 29\nu^{6} - 8\nu^{5} - 137\nu^{4} + 13\nu^{3} + 225\nu^{2} + 2\nu - 87 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{8} - \nu^{7} - 29\nu^{6} + 6\nu^{5} + 139\nu^{4} + 5\nu^{3} - 237\nu^{2} - 38\nu + 101 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{8} + 16\nu^{6} + \nu^{5} - 82\nu^{4} - 9\nu^{3} + 143\nu^{2} + 16\nu - 56 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -3\nu^{8} + \nu^{7} + 46\nu^{6} - 5\nu^{5} - 233\nu^{4} - 13\nu^{3} + 418\nu^{2} + 49\nu - 180 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - \beta_{7} + 2\beta_{6} + \beta_{5} - \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{7} + 8\beta_{6} + 7\beta_{5} + 7\beta_{4} - 8\beta_{3} + \beta_{2} + \beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{8} - 10\beta_{7} + 18\beta_{6} + 8\beta_{5} + \beta_{4} - 11\beta_{3} + \beta_{2} + 28\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{8} - 47\beta_{7} + 60\beta_{6} + 45\beta_{5} + 46\beta_{4} - 57\beta_{3} + 12\beta_{2} + 12\beta _1 + 134 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 68\beta_{8} - 86\beta_{7} + 143\beta_{6} + 60\beta_{5} + 20\beta_{4} - 95\beta_{3} + 19\beta_{2} + 164\beta _1 + 136 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 48 \beta_{8} - 324 \beta_{7} + 447 \beta_{6} + 288 \beta_{5} + 306 \beta_{4} - 401 \beta_{3} + \cdots + 861 \) 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
−2.37103
−2.25776
−1.59227
−0.917233
0.652876
0.717583
1.60510
2.40862
2.75412
−2.37103 0 3.62178 1.07225 0 2.43457 −3.84529 0 −2.54234
1.2 −2.25776 0 3.09749 −0.546634 0 −5.17651 −2.47787 0 1.23417
1.3 −1.59227 0 0.535312 4.15709 0 0.321851 2.33217 0 −6.61920
1.4 −0.917233 0 −1.15868 −2.00784 0 1.87253 2.89725 0 1.84166
1.5 0.652876 0 −1.57375 −1.28287 0 3.13476 −2.33322 0 −0.837555
1.6 0.717583 0 −1.48508 −2.44082 0 −2.97033 −2.50083 0 −1.75149
1.7 1.60510 0 0.576343 1.61810 0 4.58029 −2.28511 0 2.59720
1.8 2.40862 0 3.80143 3.20904 0 −0.999497 4.33897 0 7.72936
1.9 2.75412 0 5.58515 −2.77832 0 −0.197676 9.87393 0 −7.65181
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(149\) \(1\)

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 1341.2.a.e 9
3.b odd 2 1 149.2.a.b ✓ 9
12.b even 2 1 2384.2.a.j 9
15.d odd 2 1 3725.2.a.c 9
21.c even 2 1 7301.2.a.j 9
24.f even 2 1 9536.2.a.w 9
24.h odd 2 1 9536.2.a.v 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
149.2.a.b ✓ 9 3.b odd 2 1
1341.2.a.e 9 1.a even 1 1 trivial
2384.2.a.j 9 12.b even 2 1
3725.2.a.c 9 15.d odd 2 1
7301.2.a.j 9 21.c even 2 1
9536.2.a.v 9 24.h odd 2 1
9536.2.a.w 9 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1341))\):

\( T_{2}^{9} - T_{2}^{8} - 15T_{2}^{7} + 12T_{2}^{6} + 75T_{2}^{5} - 48T_{2}^{4} - 137T_{2}^{3} + 76T_{2}^{2} + 68T_{2} - 39 \) Copy content Toggle raw display
\( T_{5}^{9} - T_{5}^{8} - 25T_{5}^{7} + 4T_{5}^{6} + 202T_{5}^{5} + 83T_{5}^{4} - 529T_{5}^{3} - 305T_{5}^{2} + 392T_{5} + 221 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - T^{8} + \cdots - 39 \) Copy content Toggle raw display
$3$ \( T^{9} \) Copy content Toggle raw display
$5$ \( T^{9} - T^{8} + \cdots + 221 \) Copy content Toggle raw display
$7$ \( T^{9} - 3 T^{8} + \cdots - 64 \) Copy content Toggle raw display
$11$ \( T^{9} + 5 T^{8} + \cdots - 981 \) Copy content Toggle raw display
$13$ \( T^{9} - 7 T^{8} + \cdots - 64 \) Copy content Toggle raw display
$17$ \( T^{9} - 5 T^{8} + \cdots - 24053 \) Copy content Toggle raw display
$19$ \( T^{9} - 30 T^{8} + \cdots + 145856 \) Copy content Toggle raw display
$23$ \( T^{9} - 4 T^{8} + \cdots + 6341 \) Copy content Toggle raw display
$29$ \( T^{9} - 16 T^{8} + \cdots - 2861 \) Copy content Toggle raw display
$31$ \( T^{9} - 22 T^{8} + \cdots + 161984 \) Copy content Toggle raw display
$37$ \( T^{9} + 7 T^{8} + \cdots - 75969 \) Copy content Toggle raw display
$41$ \( T^{9} + 6 T^{8} + \cdots - 35328 \) Copy content Toggle raw display
$43$ \( T^{9} - 4 T^{8} + \cdots + 109051 \) Copy content Toggle raw display
$47$ \( T^{9} - 6 T^{8} + \cdots - 1225536 \) Copy content Toggle raw display
$53$ \( T^{9} - 2 T^{8} + \cdots + 43997 \) Copy content Toggle raw display
$59$ \( T^{9} + 43 T^{8} + \cdots + 13589 \) Copy content Toggle raw display
$61$ \( T^{9} - T^{8} + \cdots + 1028703 \) Copy content Toggle raw display
$67$ \( T^{9} - 33 T^{8} + \cdots - 8246976 \) Copy content Toggle raw display
$71$ \( T^{9} + 15 T^{8} + \cdots - 2931 \) Copy content Toggle raw display
$73$ \( T^{9} + 11 T^{8} + \cdots - 3257073 \) Copy content Toggle raw display
$79$ \( T^{9} - T^{8} + \cdots + 468778432 \) Copy content Toggle raw display
$83$ \( T^{9} - 4 T^{8} + \cdots - 2245797 \) Copy content Toggle raw display
$89$ \( T^{9} - 19 T^{8} + \cdots + 239936 \) Copy content Toggle raw display
$97$ \( T^{9} + T^{8} + \cdots + 3173696 \) Copy content Toggle raw display
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