Properties

Label 3.10.a.b
Level $3$
Weight $10$
Character orbit 3.a
Self dual yes
Analytic conductor $1.545$
Analytic rank $0$
Dimension $1$
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] = [3,10,Mod(1,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 3.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: \(1.54510750849\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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)
 
\(f(q)\) \(=\) \( q + 18 q^{2} + 81 q^{3} - 188 q^{4} - 1530 q^{5} + 1458 q^{6} + 9128 q^{7} - 12600 q^{8} + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 18 q^{2} + 81 q^{3} - 188 q^{4} - 1530 q^{5} + 1458 q^{6} + 9128 q^{7} - 12600 q^{8} + 6561 q^{9} - 27540 q^{10} + 21132 q^{11} - 15228 q^{12} + 31214 q^{13} + 164304 q^{14} - 123930 q^{15} - 130544 q^{16} - 279342 q^{17} + 118098 q^{18} + 144020 q^{19} + 287640 q^{20} + 739368 q^{21} + 380376 q^{22} - 1763496 q^{23} - 1020600 q^{24} + 387775 q^{25} + 561852 q^{26} + 531441 q^{27} - 1716064 q^{28} + 4692510 q^{29} - 2230740 q^{30} - 369088 q^{31} + 4101408 q^{32} + 1711692 q^{33} - 5028156 q^{34} - 13965840 q^{35} - 1233468 q^{36} + 9347078 q^{37} + 2592360 q^{38} + 2528334 q^{39} + 19278000 q^{40} - 7226838 q^{41} + 13308624 q^{42} - 23147476 q^{43} - 3972816 q^{44} - 10038330 q^{45} - 31742928 q^{46} + 22971888 q^{47} - 10574064 q^{48} + 42966777 q^{49} + 6979950 q^{50} - 22626702 q^{51} - 5868232 q^{52} + 78477174 q^{53} + 9565938 q^{54} - 32331960 q^{55} - 115012800 q^{56} + 11665620 q^{57} + 84465180 q^{58} - 20310660 q^{59} + 23298840 q^{60} - 179339938 q^{61} - 6643584 q^{62} + 59888808 q^{63} + 140663872 q^{64} - 47757420 q^{65} + 30810456 q^{66} + 274528388 q^{67} + 52516296 q^{68} - 142843176 q^{69} - 251385120 q^{70} - 36342648 q^{71} - 82668600 q^{72} - 247089526 q^{73} + 168247404 q^{74} + 31409775 q^{75} - 27075760 q^{76} + 192892896 q^{77} + 45510012 q^{78} + 191874800 q^{79} + 199732320 q^{80} + 43046721 q^{81} - 130083084 q^{82} - 276159276 q^{83} - 139001184 q^{84} + 427393260 q^{85} - 416654568 q^{86} + 380093310 q^{87} - 266263200 q^{88} - 678997350 q^{89} - 180689940 q^{90} + 284921392 q^{91} + 331537248 q^{92} - 29896128 q^{93} + 413493984 q^{94} - 220350600 q^{95} + 332214048 q^{96} - 567657502 q^{97} + 773401986 q^{98} + 138647052 q^{99}+O(q^{100}) \) 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
0
18.0000 81.0000 −188.000 −1530.00 1458.00 9128.00 −12600.0 6561.00 −27540.0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 3.10.a.b 1
3.b odd 2 1 9.10.a.a 1
4.b odd 2 1 48.10.a.a 1
5.b even 2 1 75.10.a.b 1
5.c odd 4 2 75.10.b.c 2
7.b odd 2 1 147.10.a.c 1
8.b even 2 1 192.10.a.g 1
8.d odd 2 1 192.10.a.n 1
9.c even 3 2 81.10.c.b 2
9.d odd 6 2 81.10.c.d 2
11.b odd 2 1 363.10.a.a 1
12.b even 2 1 144.10.a.m 1
15.d odd 2 1 225.10.a.e 1
15.e even 4 2 225.10.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.10.a.b 1 1.a even 1 1 trivial
9.10.a.a 1 3.b odd 2 1
48.10.a.a 1 4.b odd 2 1
75.10.a.b 1 5.b even 2 1
75.10.b.c 2 5.c odd 4 2
81.10.c.b 2 9.c even 3 2
81.10.c.d 2 9.d odd 6 2
144.10.a.m 1 12.b even 2 1
147.10.a.c 1 7.b odd 2 1
192.10.a.g 1 8.b even 2 1
192.10.a.n 1 8.d odd 2 1
225.10.a.e 1 15.d odd 2 1
225.10.b.c 2 15.e even 4 2
363.10.a.a 1 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 18 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 18 \) Copy content Toggle raw display
$3$ \( T - 81 \) Copy content Toggle raw display
$5$ \( T + 1530 \) Copy content Toggle raw display
$7$ \( T - 9128 \) Copy content Toggle raw display
$11$ \( T - 21132 \) Copy content Toggle raw display
$13$ \( T - 31214 \) Copy content Toggle raw display
$17$ \( T + 279342 \) Copy content Toggle raw display
$19$ \( T - 144020 \) Copy content Toggle raw display
$23$ \( T + 1763496 \) Copy content Toggle raw display
$29$ \( T - 4692510 \) Copy content Toggle raw display
$31$ \( T + 369088 \) Copy content Toggle raw display
$37$ \( T - 9347078 \) Copy content Toggle raw display
$41$ \( T + 7226838 \) Copy content Toggle raw display
$43$ \( T + 23147476 \) Copy content Toggle raw display
$47$ \( T - 22971888 \) Copy content Toggle raw display
$53$ \( T - 78477174 \) Copy content Toggle raw display
$59$ \( T + 20310660 \) Copy content Toggle raw display
$61$ \( T + 179339938 \) Copy content Toggle raw display
$67$ \( T - 274528388 \) Copy content Toggle raw display
$71$ \( T + 36342648 \) Copy content Toggle raw display
$73$ \( T + 247089526 \) Copy content Toggle raw display
$79$ \( T - 191874800 \) Copy content Toggle raw display
$83$ \( T + 276159276 \) Copy content Toggle raw display
$89$ \( T + 678997350 \) Copy content Toggle raw display
$97$ \( T + 567657502 \) Copy content Toggle raw display
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