Properties

Label 141.4.a.a
Level $141$
Weight $4$
Character orbit 141.a
Self dual yes
Analytic conductor $8.319$
Analytic rank $0$
Dimension $2$
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] = [141,4,Mod(1,141)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(141, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("141.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 141 = 3 \cdot 47 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 141.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: \(8.31926931081\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{111}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 111 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\beta = \sqrt{111}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} + 3 q^{3} + 8 q^{4} + (\beta + 3) q^{5} + 12 q^{6} + ( - 2 \beta + 10) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 3 q^{3} + 8 q^{4} + (\beta + 3) q^{5} + 12 q^{6} + ( - 2 \beta + 10) q^{7} + 9 q^{9} + (4 \beta + 12) q^{10} + ( - 2 \beta + 42) q^{11} + 24 q^{12} + (4 \beta + 6) q^{13} + ( - 8 \beta + 40) q^{14} + (3 \beta + 9) q^{15} - 64 q^{16} + ( - 8 \beta - 30) q^{17} + 36 q^{18} + (\beta - 19) q^{19} + (8 \beta + 24) q^{20} + ( - 6 \beta + 30) q^{21} + ( - 8 \beta + 168) q^{22} + (6 \beta - 118) q^{23} + (6 \beta - 5) q^{25} + (16 \beta + 24) q^{26} + 27 q^{27} + ( - 16 \beta + 80) q^{28} + (7 \beta + 49) q^{29} + (12 \beta + 36) q^{30} + (21 \beta - 99) q^{31} - 256 q^{32} + ( - 6 \beta + 126) q^{33} + ( - 32 \beta - 120) q^{34} + (4 \beta - 192) q^{35} + 72 q^{36} + ( - 14 \beta - 42) q^{37} + (4 \beta - 76) q^{38} + (12 \beta + 18) q^{39} + ( - 35 \beta + 83) q^{41} + ( - 24 \beta + 120) q^{42} + (19 \beta + 51) q^{43} + ( - 16 \beta + 336) q^{44} + (9 \beta + 27) q^{45} + (24 \beta - 472) q^{46} + 47 q^{47} - 192 q^{48} + ( - 40 \beta + 201) q^{49} + (24 \beta - 20) q^{50} + ( - 24 \beta - 90) q^{51} + (32 \beta + 48) q^{52} + (14 \beta - 428) q^{53} + 108 q^{54} + (36 \beta - 96) q^{55} + (3 \beta - 57) q^{57} + (28 \beta + 196) q^{58} + ( - 48 \beta + 276) q^{59} + (24 \beta + 72) q^{60} + ( - 22 \beta - 34) q^{61} + (84 \beta - 396) q^{62} + ( - 18 \beta + 90) q^{63} - 512 q^{64} + (18 \beta + 462) q^{65} + ( - 24 \beta + 504) q^{66} + (\beta - 215) q^{67} + ( - 64 \beta - 240) q^{68} + (18 \beta - 354) q^{69} + (16 \beta - 768) q^{70} + (24 \beta + 702) q^{71} + ( - 38 \beta - 376) q^{73} + ( - 56 \beta - 168) q^{74} + (18 \beta - 15) q^{75} + (8 \beta - 152) q^{76} + ( - 104 \beta + 864) q^{77} + (48 \beta + 72) q^{78} + (62 \beta + 194) q^{79} + ( - 64 \beta - 192) q^{80} + 81 q^{81} + ( - 140 \beta + 332) q^{82} + (76 \beta + 50) q^{83} + ( - 48 \beta + 240) q^{84} + ( - 54 \beta - 978) q^{85} + (76 \beta + 204) q^{86} + (21 \beta + 147) q^{87} + (98 \beta + 472) q^{89} + (36 \beta + 108) q^{90} + (28 \beta - 828) q^{91} + (48 \beta - 944) q^{92} + (63 \beta - 297) q^{93} + 188 q^{94} + ( - 16 \beta + 54) q^{95} - 768 q^{96} + (38 \beta - 774) q^{97} + ( - 160 \beta + 804) q^{98} + ( - 18 \beta + 378) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} + 6 q^{3} + 16 q^{4} + 6 q^{5} + 24 q^{6} + 20 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{2} + 6 q^{3} + 16 q^{4} + 6 q^{5} + 24 q^{6} + 20 q^{7} + 18 q^{9} + 24 q^{10} + 84 q^{11} + 48 q^{12} + 12 q^{13} + 80 q^{14} + 18 q^{15} - 128 q^{16} - 60 q^{17} + 72 q^{18} - 38 q^{19} + 48 q^{20} + 60 q^{21} + 336 q^{22} - 236 q^{23} - 10 q^{25} + 48 q^{26} + 54 q^{27} + 160 q^{28} + 98 q^{29} + 72 q^{30} - 198 q^{31} - 512 q^{32} + 252 q^{33} - 240 q^{34} - 384 q^{35} + 144 q^{36} - 84 q^{37} - 152 q^{38} + 36 q^{39} + 166 q^{41} + 240 q^{42} + 102 q^{43} + 672 q^{44} + 54 q^{45} - 944 q^{46} + 94 q^{47} - 384 q^{48} + 402 q^{49} - 40 q^{50} - 180 q^{51} + 96 q^{52} - 856 q^{53} + 216 q^{54} - 192 q^{55} - 114 q^{57} + 392 q^{58} + 552 q^{59} + 144 q^{60} - 68 q^{61} - 792 q^{62} + 180 q^{63} - 1024 q^{64} + 924 q^{65} + 1008 q^{66} - 430 q^{67} - 480 q^{68} - 708 q^{69} - 1536 q^{70} + 1404 q^{71} - 752 q^{73} - 336 q^{74} - 30 q^{75} - 304 q^{76} + 1728 q^{77} + 144 q^{78} + 388 q^{79} - 384 q^{80} + 162 q^{81} + 664 q^{82} + 100 q^{83} + 480 q^{84} - 1956 q^{85} + 408 q^{86} + 294 q^{87} + 944 q^{89} + 216 q^{90} - 1656 q^{91} - 1888 q^{92} - 594 q^{93} + 376 q^{94} + 108 q^{95} - 1536 q^{96} - 1548 q^{97} + 1608 q^{98} + 756 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
−10.5357
10.5357
4.00000 3.00000 8.00000 −7.53565 12.0000 31.0713 0 9.00000 −30.1426
1.2 4.00000 3.00000 8.00000 13.5357 12.0000 −11.0713 0 9.00000 54.1426
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(47\) \(-1\)

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 141.4.a.a 2
3.b odd 2 1 423.4.a.a 2
4.b odd 2 1 2256.4.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.4.a.a 2 1.a even 1 1 trivial
423.4.a.a 2 3.b odd 2 1
2256.4.a.h 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 4 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(141))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 4)^{2} \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 6T - 102 \) Copy content Toggle raw display
$7$ \( T^{2} - 20T - 344 \) Copy content Toggle raw display
$11$ \( T^{2} - 84T + 1320 \) Copy content Toggle raw display
$13$ \( T^{2} - 12T - 1740 \) Copy content Toggle raw display
$17$ \( T^{2} + 60T - 6204 \) Copy content Toggle raw display
$19$ \( T^{2} + 38T + 250 \) Copy content Toggle raw display
$23$ \( T^{2} + 236T + 9928 \) Copy content Toggle raw display
$29$ \( T^{2} - 98T - 3038 \) Copy content Toggle raw display
$31$ \( T^{2} + 198T - 39150 \) Copy content Toggle raw display
$37$ \( T^{2} + 84T - 19992 \) Copy content Toggle raw display
$41$ \( T^{2} - 166T - 129086 \) Copy content Toggle raw display
$43$ \( T^{2} - 102T - 37470 \) Copy content Toggle raw display
$47$ \( (T - 47)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 856T + 161428 \) Copy content Toggle raw display
$59$ \( T^{2} - 552T - 179568 \) Copy content Toggle raw display
$61$ \( T^{2} + 68T - 52568 \) Copy content Toggle raw display
$67$ \( T^{2} + 430T + 46114 \) Copy content Toggle raw display
$71$ \( T^{2} - 1404 T + 428868 \) Copy content Toggle raw display
$73$ \( T^{2} + 752T - 18908 \) Copy content Toggle raw display
$79$ \( T^{2} - 388T - 389048 \) Copy content Toggle raw display
$83$ \( T^{2} - 100T - 638636 \) Copy content Toggle raw display
$89$ \( T^{2} - 944T - 843260 \) Copy content Toggle raw display
$97$ \( T^{2} + 1548 T + 438792 \) Copy content Toggle raw display
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