Properties

Label 180.3.c.a
Level $180$
Weight $3$
Character orbit 180.c
Analytic conductor $4.905$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [180,3,Mod(91,180)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(180, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("180.91");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 180.c (of order \(2\), degree \(1\), 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: \(4.90464475849\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + ( - \beta_{3} - \beta_1) q^{5} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{7} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + ( - \beta_{3} - \beta_1) q^{5} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{7} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{8} + ( - \beta_{3} - \beta_{2} - 3) q^{10} + ( - 4 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 4) q^{11} + (2 \beta_{3} + 2 \beta_1 - 4) q^{13} + ( - \beta_{3} + 3 \beta_{2} + \beta_1 - 11) q^{14} + ( - 8 \beta_{3} - 8) q^{16} + (8 \beta_{3} + 8 \beta_1 + 6) q^{17} + ( - 4 \beta_{3} + 4 \beta_1 - 4) q^{19} + (3 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{20} + (10 \beta_{3} + 2 \beta_{2} + 6 \beta_1 - 18) q^{22} + (2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 2) q^{23} + 5 q^{25} + (2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 6) q^{26} + ( - 8 \beta_{3} + 4 \beta_{2} - 12 \beta_1 - 8) q^{28} + ( - 4 \beta_{3} - 4 \beta_1 + 2) q^{29} + (4 \beta_{3} - 10 \beta_{2} - 4 \beta_1 + 4) q^{31} - 32 q^{32} + (8 \beta_{3} + 8 \beta_{2} + 6 \beta_1 + 24) q^{34} - 5 \beta_{2} q^{35} + (10 \beta_{3} + 10 \beta_1 + 4) q^{37} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 20) q^{38} + (2 \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 14) q^{40} + (6 \beta_{3} + 6 \beta_1 + 28) q^{41} + (4 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 4) q^{43} + (8 \beta_{2} - 24 \beta_1 + 32) q^{44} + (7 \beta_{3} - 5 \beta_{2} + \beta_1 + 13) q^{46} + ( - 6 \beta_{3} - 13 \beta_{2} + 6 \beta_1 - 6) q^{47} + ( - 10 \beta_{3} - 10 \beta_1 - 1) q^{49} + 5 \beta_1 q^{50} + ( - 10 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 10) q^{52} + ( - 10 \beta_{3} - 10 \beta_1 + 44) q^{53} + ( - 8 \beta_{3} - 6 \beta_{2} + 8 \beta_1 - 8) q^{55} + ( - 24 \beta_{3} - 8 \beta_{2} - 16 \beta_1 - 24) q^{56} + ( - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 12) q^{58} + ( - 12 \beta_{3} + 12 \beta_{2} + 12 \beta_1 - 12) q^{59} + ( - 26 \beta_{3} - 26 \beta_1 + 32) q^{61} + (26 \beta_{3} - 14 \beta_{2} + 6 \beta_1 + 30) q^{62} - 32 \beta_1 q^{64} + (4 \beta_{3} + 4 \beta_1 - 10) q^{65} + (20 \beta_{3} - 11 \beta_{2} - 20 \beta_1 + 20) q^{67} + ( - 18 \beta_{3} + 14 \beta_{2} + 14 \beta_1 + 18) q^{68} + (15 \beta_{3} - 5 \beta_{2} + 5 \beta_1 + 5) q^{70} + (20 \beta_{3} + 2 \beta_{2} - 20 \beta_1 + 20) q^{71} + (32 \beta_{3} + 32 \beta_1 + 66) q^{73} + (10 \beta_{3} + 10 \beta_{2} + 4 \beta_1 + 30) q^{74} + ( - 8 \beta_{3} + 8 \beta_{2} - 24 \beta_1 + 8) q^{76} + (20 \beta_{3} + 20 \beta_1 - 60) q^{77} + ( - 16 \beta_{3} + 20 \beta_{2} + 16 \beta_1 - 16) q^{79} + ( - 8 \beta_{2} + 8 \beta_1 + 16) q^{80} + (6 \beta_{3} + 6 \beta_{2} + 28 \beta_1 + 18) q^{82} + (12 \beta_{3} + 13 \beta_{2} - 12 \beta_1 + 12) q^{83} + ( - 6 \beta_{3} - 6 \beta_1 - 40) q^{85} + ( - 13 \beta_{3} - \beta_{2} - 7 \beta_1 + 17) q^{86} + ( - 48 \beta_{3} - 16 \beta_{2} + 16) q^{88} + ( - 40 \beta_{3} - 40 \beta_1 + 22) q^{89} + (8 \beta_{3} + 6 \beta_{2} - 8 \beta_1 + 8) q^{91} + (16 \beta_{3} - 4 \beta_{2} + 12 \beta_1 + 32) q^{92} + (45 \beta_{3} - 7 \beta_{2} + 19 \beta_1 - 17) q^{94} + ( - 4 \beta_{3} - 8 \beta_{2} + 4 \beta_1 - 4) q^{95} + (12 \beta_{3} + 12 \beta_1 - 66) q^{97} + ( - 10 \beta_{3} - 10 \beta_{2} - \beta_1 - 30) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 4 q^{4} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 4 q^{4} + 8 q^{8} - 10 q^{10} - 16 q^{13} - 40 q^{14} - 16 q^{16} + 24 q^{17} - 20 q^{20} - 80 q^{22} + 20 q^{25} + 12 q^{26} - 40 q^{28} + 8 q^{29} - 128 q^{32} + 92 q^{34} + 16 q^{37} - 80 q^{38} + 40 q^{40} + 112 q^{41} + 80 q^{44} + 40 q^{46} - 4 q^{49} + 10 q^{50} + 56 q^{52} + 176 q^{53} - 80 q^{56} - 36 q^{58} + 128 q^{61} + 80 q^{62} - 64 q^{64} - 40 q^{65} + 136 q^{68} + 264 q^{73} + 108 q^{74} - 240 q^{77} + 80 q^{80} + 116 q^{82} - 160 q^{85} + 80 q^{86} + 160 q^{88} + 88 q^{89} + 120 q^{92} - 120 q^{94} - 264 q^{97} - 102 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{10} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{10}^{3} + 2\zeta_{10}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{10}^{3} + 2\zeta_{10}^{2} - 2\zeta_{10} + 1 \) Copy content Toggle raw display
\(\zeta_{10}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{10}^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta _1 - 1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{10}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} - \beta _1 + 1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
−0.618034 1.90211i 0 −3.23607 + 2.35114i 2.23607 0 5.25731i 6.47214 + 4.70228i 0 −1.38197 4.25325i
91.2 −0.618034 + 1.90211i 0 −3.23607 2.35114i 2.23607 0 5.25731i 6.47214 4.70228i 0 −1.38197 + 4.25325i
91.3 1.61803 1.17557i 0 1.23607 3.80423i −2.23607 0 8.50651i −2.47214 7.60845i 0 −3.61803 + 2.62866i
91.4 1.61803 + 1.17557i 0 1.23607 + 3.80423i −2.23607 0 8.50651i −2.47214 + 7.60845i 0 −3.61803 2.62866i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.c.a 4
3.b odd 2 1 20.3.b.a 4
4.b odd 2 1 inner 180.3.c.a 4
5.b even 2 1 900.3.c.k 4
5.c odd 4 2 900.3.f.e 8
8.b even 2 1 2880.3.e.e 4
8.d odd 2 1 2880.3.e.e 4
12.b even 2 1 20.3.b.a 4
15.d odd 2 1 100.3.b.f 4
15.e even 4 2 100.3.d.b 8
20.d odd 2 1 900.3.c.k 4
20.e even 4 2 900.3.f.e 8
24.f even 2 1 320.3.b.c 4
24.h odd 2 1 320.3.b.c 4
48.i odd 4 2 1280.3.g.e 8
48.k even 4 2 1280.3.g.e 8
60.h even 2 1 100.3.b.f 4
60.l odd 4 2 100.3.d.b 8
120.i odd 2 1 1600.3.b.s 4
120.m even 2 1 1600.3.b.s 4
120.q odd 4 2 1600.3.h.n 8
120.w even 4 2 1600.3.h.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.b.a 4 3.b odd 2 1
20.3.b.a 4 12.b even 2 1
100.3.b.f 4 15.d odd 2 1
100.3.b.f 4 60.h even 2 1
100.3.d.b 8 15.e even 4 2
100.3.d.b 8 60.l odd 4 2
180.3.c.a 4 1.a even 1 1 trivial
180.3.c.a 4 4.b odd 2 1 inner
320.3.b.c 4 24.f even 2 1
320.3.b.c 4 24.h odd 2 1
900.3.c.k 4 5.b even 2 1
900.3.c.k 4 20.d odd 2 1
900.3.f.e 8 5.c odd 4 2
900.3.f.e 8 20.e even 4 2
1280.3.g.e 8 48.i odd 4 2
1280.3.g.e 8 48.k even 4 2
1600.3.b.s 4 120.i odd 2 1
1600.3.b.s 4 120.m even 2 1
1600.3.h.n 8 120.q odd 4 2
1600.3.h.n 8 120.w even 4 2
2880.3.e.e 4 8.b even 2 1
2880.3.e.e 4 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 100T_{7}^{2} + 2000 \) acting on \(S_{3}^{\mathrm{new}}(180, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 100T^{2} + 2000 \) Copy content Toggle raw display
$11$ \( T^{4} + 400T^{2} + 1280 \) Copy content Toggle raw display
$13$ \( (T^{2} + 8 T - 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 12 T - 284)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 320 T^{2} + 20480 \) Copy content Toggle raw display
$23$ \( T^{4} + 260T^{2} + 80 \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 76)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2320 T^{2} + 154880 \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T - 484)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 56 T + 604)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 500T^{2} + 2000 \) Copy content Toggle raw display
$47$ \( T^{4} + 4100 T^{2} + \cdots + 3561680 \) Copy content Toggle raw display
$53$ \( (T^{2} - 88 T + 1436)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 5760 T^{2} + \cdots + 1658880 \) Copy content Toggle raw display
$61$ \( (T^{2} - 64 T - 2356)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 10420 T^{2} + \cdots + 19920080 \) Copy content Toggle raw display
$71$ \( T^{4} + 8080 T^{2} + \cdots + 10138880 \) Copy content Toggle raw display
$73$ \( (T^{2} - 132 T - 764)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 13120 T^{2} + \cdots + 2478080 \) Copy content Toggle raw display
$83$ \( T^{4} + 6260 T^{2} + \cdots + 2620880 \) Copy content Toggle raw display
$89$ \( (T^{2} - 44 T - 7516)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 132 T + 3636)^{2} \) Copy content Toggle raw display
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