Properties

Label 36.3.f.b
Level $36$
Weight $3$
Character orbit 36.f
Analytic conductor $0.981$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [36,3,Mod(7,36)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(36, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("36.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 36.f (of order \(6\), degree \(2\), 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: \(0.980928951697\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} - 3 \zeta_{6} q^{3} + 4 q^{4} + (4 \zeta_{6} - 4) q^{5} - 6 \zeta_{6} q^{6} + (2 \zeta_{6} - 4) q^{7} + 8 q^{8} + (9 \zeta_{6} - 9) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 3 \zeta_{6} q^{3} + 4 q^{4} + (4 \zeta_{6} - 4) q^{5} - 6 \zeta_{6} q^{6} + (2 \zeta_{6} - 4) q^{7} + 8 q^{8} + (9 \zeta_{6} - 9) q^{9} + (8 \zeta_{6} - 8) q^{10} + (7 \zeta_{6} - 14) q^{11} - 12 \zeta_{6} q^{12} + ( - 22 \zeta_{6} + 22) q^{13} + (4 \zeta_{6} - 8) q^{14} + 12 q^{15} + 16 q^{16} - 11 q^{17} + (18 \zeta_{6} - 18) q^{18} + ( - 18 \zeta_{6} + 9) q^{19} + (16 \zeta_{6} - 16) q^{20} + (6 \zeta_{6} + 6) q^{21} + (14 \zeta_{6} - 28) q^{22} + (14 \zeta_{6} + 14) q^{23} - 24 \zeta_{6} q^{24} + 9 \zeta_{6} q^{25} + ( - 44 \zeta_{6} + 44) q^{26} + 27 q^{27} + (8 \zeta_{6} - 16) q^{28} - 34 \zeta_{6} q^{29} + 24 q^{30} + (4 \zeta_{6} + 4) q^{31} + 32 q^{32} + (21 \zeta_{6} + 21) q^{33} - 22 q^{34} + ( - 16 \zeta_{6} + 8) q^{35} + (36 \zeta_{6} - 36) q^{36} - 16 q^{37} + ( - 36 \zeta_{6} + 18) q^{38} - 66 q^{39} + (32 \zeta_{6} - 32) q^{40} + (13 \zeta_{6} - 13) q^{41} + (12 \zeta_{6} + 12) q^{42} + (29 \zeta_{6} - 58) q^{43} + (28 \zeta_{6} - 56) q^{44} - 36 \zeta_{6} q^{45} + (28 \zeta_{6} + 28) q^{46} + ( - 2 \zeta_{6} + 4) q^{47} - 48 \zeta_{6} q^{48} + (37 \zeta_{6} - 37) q^{49} + 18 \zeta_{6} q^{50} + 33 \zeta_{6} q^{51} + ( - 88 \zeta_{6} + 88) q^{52} + 52 q^{53} + 54 q^{54} + ( - 56 \zeta_{6} + 28) q^{55} + (16 \zeta_{6} - 32) q^{56} + (27 \zeta_{6} - 54) q^{57} - 68 \zeta_{6} q^{58} + ( - 31 \zeta_{6} - 31) q^{59} + 48 q^{60} + 16 \zeta_{6} q^{61} + (8 \zeta_{6} + 8) q^{62} + ( - 36 \zeta_{6} + 18) q^{63} + 64 q^{64} + 88 \zeta_{6} q^{65} + (42 \zeta_{6} + 42) q^{66} + (67 \zeta_{6} + 67) q^{67} - 44 q^{68} + ( - 84 \zeta_{6} + 42) q^{69} + ( - 32 \zeta_{6} + 16) q^{70} + (72 \zeta_{6} - 72) q^{72} - 25 q^{73} - 32 q^{74} + ( - 27 \zeta_{6} + 27) q^{75} + ( - 72 \zeta_{6} + 36) q^{76} + ( - 42 \zeta_{6} + 42) q^{77} - 132 q^{78} + ( - 16 \zeta_{6} + 32) q^{79} + (64 \zeta_{6} - 64) q^{80} - 81 \zeta_{6} q^{81} + (26 \zeta_{6} - 26) q^{82} + ( - 20 \zeta_{6} + 40) q^{83} + (24 \zeta_{6} + 24) q^{84} + ( - 44 \zeta_{6} + 44) q^{85} + (58 \zeta_{6} - 116) q^{86} + (102 \zeta_{6} - 102) q^{87} + (56 \zeta_{6} - 112) q^{88} - 2 q^{89} - 72 \zeta_{6} q^{90} + (88 \zeta_{6} - 44) q^{91} + (56 \zeta_{6} + 56) q^{92} + ( - 24 \zeta_{6} + 12) q^{93} + ( - 4 \zeta_{6} + 8) q^{94} + (36 \zeta_{6} + 36) q^{95} - 96 \zeta_{6} q^{96} + 43 \zeta_{6} q^{97} + (74 \zeta_{6} - 74) q^{98} + ( - 126 \zeta_{6} + 63) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 3 q^{3} + 8 q^{4} - 4 q^{5} - 6 q^{6} - 6 q^{7} + 16 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} - 3 q^{3} + 8 q^{4} - 4 q^{5} - 6 q^{6} - 6 q^{7} + 16 q^{8} - 9 q^{9} - 8 q^{10} - 21 q^{11} - 12 q^{12} + 22 q^{13} - 12 q^{14} + 24 q^{15} + 32 q^{16} - 22 q^{17} - 18 q^{18} - 16 q^{20} + 18 q^{21} - 42 q^{22} + 42 q^{23} - 24 q^{24} + 9 q^{25} + 44 q^{26} + 54 q^{27} - 24 q^{28} - 34 q^{29} + 48 q^{30} + 12 q^{31} + 64 q^{32} + 63 q^{33} - 44 q^{34} - 36 q^{36} - 32 q^{37} - 132 q^{39} - 32 q^{40} - 13 q^{41} + 36 q^{42} - 87 q^{43} - 84 q^{44} - 36 q^{45} + 84 q^{46} + 6 q^{47} - 48 q^{48} - 37 q^{49} + 18 q^{50} + 33 q^{51} + 88 q^{52} + 104 q^{53} + 108 q^{54} - 48 q^{56} - 81 q^{57} - 68 q^{58} - 93 q^{59} + 96 q^{60} + 16 q^{61} + 24 q^{62} + 128 q^{64} + 88 q^{65} + 126 q^{66} + 201 q^{67} - 88 q^{68} - 72 q^{72} - 50 q^{73} - 64 q^{74} + 27 q^{75} + 42 q^{77} - 264 q^{78} + 48 q^{79} - 64 q^{80} - 81 q^{81} - 26 q^{82} + 60 q^{83} + 72 q^{84} + 44 q^{85} - 174 q^{86} - 102 q^{87} - 168 q^{88} - 4 q^{89} - 72 q^{90} + 168 q^{92} + 12 q^{94} + 108 q^{95} - 96 q^{96} + 43 q^{97} - 74 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(-1\) \(-1 + \zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.500000 0.866025i
0.500000 + 0.866025i
2.00000 −1.50000 + 2.59808i 4.00000 −2.00000 3.46410i −3.00000 + 5.19615i −3.00000 1.73205i 8.00000 −4.50000 7.79423i −4.00000 6.92820i
31.1 2.00000 −1.50000 2.59808i 4.00000 −2.00000 + 3.46410i −3.00000 5.19615i −3.00000 + 1.73205i 8.00000 −4.50000 + 7.79423i −4.00000 + 6.92820i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.3.f.b yes 2
3.b odd 2 1 108.3.f.a 2
4.b odd 2 1 36.3.f.a 2
8.b even 2 1 576.3.o.b 2
8.d odd 2 1 576.3.o.a 2
9.c even 3 1 36.3.f.a 2
9.c even 3 1 324.3.d.b 2
9.d odd 6 1 108.3.f.b 2
9.d odd 6 1 324.3.d.c 2
12.b even 2 1 108.3.f.b 2
24.f even 2 1 1728.3.o.b 2
24.h odd 2 1 1728.3.o.a 2
36.f odd 6 1 inner 36.3.f.b yes 2
36.f odd 6 1 324.3.d.b 2
36.h even 6 1 108.3.f.a 2
36.h even 6 1 324.3.d.c 2
72.j odd 6 1 1728.3.o.b 2
72.l even 6 1 1728.3.o.a 2
72.n even 6 1 576.3.o.a 2
72.p odd 6 1 576.3.o.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.f.a 2 4.b odd 2 1
36.3.f.a 2 9.c even 3 1
36.3.f.b yes 2 1.a even 1 1 trivial
36.3.f.b yes 2 36.f odd 6 1 inner
108.3.f.a 2 3.b odd 2 1
108.3.f.a 2 36.h even 6 1
108.3.f.b 2 9.d odd 6 1
108.3.f.b 2 12.b even 2 1
324.3.d.b 2 9.c even 3 1
324.3.d.b 2 36.f odd 6 1
324.3.d.c 2 9.d odd 6 1
324.3.d.c 2 36.h even 6 1
576.3.o.a 2 8.d odd 2 1
576.3.o.a 2 72.n even 6 1
576.3.o.b 2 8.b even 2 1
576.3.o.b 2 72.p odd 6 1
1728.3.o.a 2 24.h odd 2 1
1728.3.o.a 2 72.l even 6 1
1728.3.o.b 2 24.f even 2 1
1728.3.o.b 2 72.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(36, [\chi])\):

\( T_{5}^{2} + 4T_{5} + 16 \) Copy content Toggle raw display
\( T_{7}^{2} + 6T_{7} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$7$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$11$ \( T^{2} + 21T + 147 \) Copy content Toggle raw display
$13$ \( T^{2} - 22T + 484 \) Copy content Toggle raw display
$17$ \( (T + 11)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 243 \) Copy content Toggle raw display
$23$ \( T^{2} - 42T + 588 \) Copy content Toggle raw display
$29$ \( T^{2} + 34T + 1156 \) Copy content Toggle raw display
$31$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$37$ \( (T + 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$43$ \( T^{2} + 87T + 2523 \) Copy content Toggle raw display
$47$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$53$ \( (T - 52)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 93T + 2883 \) Copy content Toggle raw display
$61$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$67$ \( T^{2} - 201T + 13467 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 25)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 48T + 768 \) Copy content Toggle raw display
$83$ \( T^{2} - 60T + 1200 \) Copy content Toggle raw display
$89$ \( (T + 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 43T + 1849 \) Copy content Toggle raw display
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