Properties

Label 121.3.d.c
Level $121$
Weight $3$
Character orbit 121.d
Analytic conductor $3.297$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [121,3,Mod(40,121)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(121, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([7]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("121.40");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 121.d (of order \(10\), degree \(4\), not 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: \(3.29701119876\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2) q^{2} + (3 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 3) q^{3} + (4 \zeta_{10}^{2} - \zeta_{10} + 4) q^{4} + 4 \zeta_{10}^{2} q^{5} + (7 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + \zeta_{10} - 8) q^{6} + ( - 4 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 2 \zeta_{10} - 2) q^{7} + ( - \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 3 \zeta_{10} - 1) q^{8} + ( - \zeta_{10}^{3} - 5 \zeta_{10} + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2) q^{2} + (3 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 3) q^{3} + (4 \zeta_{10}^{2} - \zeta_{10} + 4) q^{4} + 4 \zeta_{10}^{2} q^{5} + (7 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + \zeta_{10} - 8) q^{6} + ( - 4 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 2 \zeta_{10} - 2) q^{7} + ( - \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 3 \zeta_{10} - 1) q^{8} + ( - \zeta_{10}^{3} - 5 \zeta_{10} + 5) q^{9} + (8 \zeta_{10}^{3} + 8 \zeta_{10} - 4) q^{10} + (11 \zeta_{10}^{3} - 11 \zeta_{10}^{2} - 13) q^{12} + ( - 4 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 2 \zeta_{10} + 6) q^{13} + (4 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 4) q^{14} + ( - 4 \zeta_{10}^{2} - 8 \zeta_{10} - 4) q^{15} + ( - 8 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 8 \zeta_{10}) q^{16} + ( - 13 \zeta_{10}^{3} + 13 \zeta_{10}^{2} - 13 \zeta_{10} + 26) q^{17} + ( - 12 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 6 \zeta_{10} + 7) q^{18} + (2 \zeta_{10}^{3} + 5 \zeta_{10}^{2} + 5 \zeta_{10} + 2) q^{19} + (12 \zeta_{10}^{3} + 16 \zeta_{10} - 16) q^{20} + ( - 4 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 12 \zeta_{10} + 6) q^{21} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 6) q^{23} + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} - 7 \zeta_{10} - 9) q^{24} + ( - 9 \zeta_{10}^{3} + 9 \zeta_{10}^{2} - 9 \zeta_{10} + 9) q^{25} + (10 \zeta_{10}^{2} + 10 \zeta_{10} + 10) q^{26} + (2 \zeta_{10}^{3} - 21 \zeta_{10}^{2} + 2 \zeta_{10}) q^{27} + ( - 2 \zeta_{10}^{3} - 10 \zeta_{10}^{2} + 22 \zeta_{10} - 20) q^{28} + (16 \zeta_{10}^{3} - 18 \zeta_{10}^{2} + 8 \zeta_{10} + 2) q^{29} + ( - 12 \zeta_{10}^{3} - 16 \zeta_{10}^{2} - 16 \zeta_{10} - 12) q^{30} + ( - 22 \zeta_{10}^{3} - 18 \zeta_{10} + 18) q^{31} + ( - 22 \zeta_{10}^{3} - 24 \zeta_{10}^{2} + 2 \zeta_{10} - 1) q^{32} + ( - 39 \zeta_{10}^{3} + 39 \zeta_{10}^{2} + 52) q^{34} + (16 \zeta_{10}^{3} - 32 \zeta_{10}^{2} + 24 \zeta_{10} - 8) q^{35} + ( - 23 \zeta_{10}^{3} + 24 \zeta_{10}^{2} - 24 \zeta_{10} + 23) q^{36} + ( - 36 \zeta_{10}^{2} + 18 \zeta_{10} - 36) q^{37} + (17 \zeta_{10}^{3} + 9 \zeta_{10}^{2} + 17 \zeta_{10}) q^{38} + (20 \zeta_{10}^{3} - 10 \zeta_{10}^{2} - 20) q^{39} + (24 \zeta_{10}^{3} - 16 \zeta_{10}^{2} + 12 \zeta_{10} - 8) q^{40} + (27 \zeta_{10}^{3} - 16 \zeta_{10}^{2} - 16 \zeta_{10} + 27) q^{41} - 10 \zeta_{10}^{3} q^{42} + (17 \zeta_{10}^{3} + 19 \zeta_{10}^{2} - 2 \zeta_{10} + 1) q^{43} + ( - 20 \zeta_{10}^{3} + 20 \zeta_{10}^{2} + 4) q^{45} + (6 \zeta_{10}^{3} - 12 \zeta_{10}^{2} - 2 \zeta_{10} - 14) q^{46} + (24 \zeta_{10}^{3} - 26 \zeta_{10}^{2} + 26 \zeta_{10} - 24) q^{47} + (19 \zeta_{10}^{2} - 2 \zeta_{10} + 19) q^{48} + (44 \zeta_{10}^{3} - 23 \zeta_{10}^{2} + 44 \zeta_{10}) q^{49} + ( - 18 \zeta_{10}^{3} + 9 \zeta_{10}^{2} + 18) q^{50} + (78 \zeta_{10}^{3} - 26 \zeta_{10}^{2} + 39 \zeta_{10} - 52) q^{51} + (4 \zeta_{10}^{3} + 22 \zeta_{10}^{2} + 22 \zeta_{10} + 4) q^{52} + (60 \zeta_{10}^{3} + 30 \zeta_{10} - 30) q^{53} + ( - 36 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 38 \zeta_{10} + 19) q^{54} + (26 \zeta_{10}^{3} - 26 \zeta_{10}^{2} + 12) q^{56} + (9 \zeta_{10}^{3} - 18 \zeta_{10}^{2} - 8 \zeta_{10} - 26) q^{57} + (2 \zeta_{10}^{3} + 12 \zeta_{10}^{2} - 12 \zeta_{10} - 2) q^{58} + ( - 15 \zeta_{10}^{2} - 7 \zeta_{10} - 15) q^{59} + ( - 44 \zeta_{10}^{3} - 8 \zeta_{10}^{2} - 44 \zeta_{10}) q^{60} + ( - 14 \zeta_{10}^{3} + 16 \zeta_{10}^{2} - 18 \zeta_{10} + 32) q^{61} + ( - 80 \zeta_{10}^{3} + 18 \zeta_{10}^{2} - 40 \zeta_{10} + 62) q^{62} + ( - 26 \zeta_{10}^{3} + 18 \zeta_{10}^{2} + 18 \zeta_{10} - 26) q^{63} + ( - 77 \zeta_{10}^{3} - 36 \zeta_{10} + 36) q^{64} + (24 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 32 \zeta_{10} - 16) q^{65} + (17 \zeta_{10}^{3} - 17 \zeta_{10}^{2} - 66) q^{67} + ( - 52 \zeta_{10}^{3} + 104 \zeta_{10}^{2} - 13 \zeta_{10} + 91) q^{68} + ( - 20 \zeta_{10}^{3} + 10 \zeta_{10}^{2} - 10 \zeta_{10} + 20) q^{69} + (8 \zeta_{10}^{2} - 24 \zeta_{10} + 8) q^{70} + ( - 62 \zeta_{10}^{3} + 38 \zeta_{10}^{2} - 62 \zeta_{10}) q^{71} + ( - 17 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 21 \zeta_{10} - 4) q^{72} + ( - 120 \zeta_{10}^{3} + 77 \zeta_{10}^{2} - 60 \zeta_{10} + 43) q^{73} + ( - 18 \zeta_{10}^{3} - 54 \zeta_{10}^{2} - 54 \zeta_{10} - 18) q^{74} + (27 \zeta_{10}^{3} + 9 \zeta_{10} - 9) q^{75} + (41 \zeta_{10}^{3} + 5 \zeta_{10}^{2} + 36 \zeta_{10} - 18) q^{76} + (40 \zeta_{10}^{3} - 40 \zeta_{10}^{2} - 70) q^{78} + (14 \zeta_{10}^{3} - 28 \zeta_{10}^{2} - 12 \zeta_{10} - 40) q^{79} + ( - 12 \zeta_{10}^{3} - 20 \zeta_{10}^{2} + 20 \zeta_{10} + 12) q^{80} + (60 \zeta_{10}^{2} + 4 \zeta_{10} + 60) q^{81} + ( - 21 \zeta_{10}^{3} + 38 \zeta_{10}^{2} - 21 \zeta_{10}) q^{82} + (47 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 43 \zeta_{10} - 4) q^{83} + ( - 36 \zeta_{10}^{3} + 32 \zeta_{10}^{2} - 18 \zeta_{10} + 4) q^{84} + (52 \zeta_{10}^{2} + 52 \zeta_{10}) q^{85} + (69 \zeta_{10}^{3} + 53 \zeta_{10} - 53) q^{86} + ( - 2 \zeta_{10}^{3} - 26 \zeta_{10}^{2} + 24 \zeta_{10} - 12) q^{87} + (83 \zeta_{10}^{3} - 83 \zeta_{10}^{2} - 11) q^{89} + ( - 4 \zeta_{10}^{3} + 8 \zeta_{10}^{2} + 20 \zeta_{10} + 28) q^{90} + (20 \zeta_{10}^{3} - 40 \zeta_{10}^{2} + 40 \zeta_{10} - 20) q^{91} + ( - 22 \zeta_{10}^{2} - 4 \zeta_{10} - 22) q^{92} + (58 \zeta_{10}^{3} + 26 \zeta_{10}^{2} + 58 \zeta_{10}) q^{93} + (46 \zeta_{10}^{3} - 22 \zeta_{10}^{2} - 2 \zeta_{10} - 44) q^{94} + (40 \zeta_{10}^{3} - 12 \zeta_{10}^{2} + 20 \zeta_{10} - 28) q^{95} + (21 \zeta_{10}^{3} + 68 \zeta_{10}^{2} + 68 \zeta_{10} + 21) q^{96} + ( - 31 \zeta_{10}^{3} - 67 \zeta_{10} + 67) q^{97} + (86 \zeta_{10}^{3} + 44 \zeta_{10}^{2} + 42 \zeta_{10} - 21) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{2} - 5 q^{3} + 11 q^{4} - 4 q^{5} - 20 q^{6} - 20 q^{7} - 5 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 5 q^{2} - 5 q^{3} + 11 q^{4} - 4 q^{5} - 20 q^{6} - 20 q^{7} - 5 q^{8} + 14 q^{9} - 30 q^{12} + 10 q^{13} - 20 q^{15} - 21 q^{16} + 65 q^{17} + 5 q^{18} + 10 q^{19} - 36 q^{20} - 20 q^{23} - 40 q^{24} + 9 q^{25} + 40 q^{26} + 25 q^{27} - 50 q^{28} + 50 q^{29} - 60 q^{30} + 32 q^{31} + 130 q^{34} + 40 q^{35} + 21 q^{36} - 90 q^{37} + 25 q^{38} - 50 q^{39} + 20 q^{40} + 135 q^{41} - 10 q^{42} - 24 q^{45} - 40 q^{46} - 20 q^{47} + 55 q^{48} + 111 q^{49} + 45 q^{50} - 65 q^{51} + 20 q^{52} - 30 q^{53} + 100 q^{56} - 85 q^{57} - 30 q^{58} - 52 q^{59} - 80 q^{60} + 80 q^{61} + 110 q^{62} - 130 q^{63} + 31 q^{64} - 230 q^{67} + 195 q^{68} + 40 q^{69} - 162 q^{71} - 10 q^{72} - 85 q^{73} - 90 q^{74} - 200 q^{78} - 130 q^{79} + 76 q^{80} + 184 q^{81} - 80 q^{82} - 10 q^{83} - 70 q^{84} - 90 q^{86} + 122 q^{89} + 120 q^{90} + 20 q^{91} - 70 q^{92} + 90 q^{93} - 110 q^{94} - 40 q^{95} + 105 q^{96} + 170 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1
−0.309017 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
−0.309017 + 0.951057i
−0.427051 + 0.587785i 0.427051 1.31433i 1.07295 + 3.30220i −3.23607 + 2.35114i 0.590170 + 0.812299i −9.47214 + 3.07768i −5.16312 1.67760i 5.73607 + 4.16750i 2.90617i
94.1 2.92705 0.951057i −2.92705 2.12663i 4.42705 3.21644i 1.23607 3.80423i −10.5902 3.44095i −0.527864 0.726543i 2.66312 3.66547i 1.26393 + 3.88998i 12.3107i
112.1 2.92705 + 0.951057i −2.92705 + 2.12663i 4.42705 + 3.21644i 1.23607 + 3.80423i −10.5902 + 3.44095i −0.527864 + 0.726543i 2.66312 + 3.66547i 1.26393 3.88998i 12.3107i
118.1 −0.427051 0.587785i 0.427051 + 1.31433i 1.07295 3.30220i −3.23607 2.35114i 0.590170 0.812299i −9.47214 3.07768i −5.16312 + 1.67760i 5.73607 4.16750i 2.90617i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.3.d.c 4
11.b odd 2 1 121.3.d.a 4
11.c even 5 1 11.3.d.a 4
11.c even 5 1 121.3.b.b 4
11.c even 5 1 121.3.d.a 4
11.c even 5 1 121.3.d.d 4
11.d odd 10 1 11.3.d.a 4
11.d odd 10 1 121.3.b.b 4
11.d odd 10 1 inner 121.3.d.c 4
11.d odd 10 1 121.3.d.d 4
33.f even 10 1 99.3.k.a 4
33.f even 10 1 1089.3.c.e 4
33.h odd 10 1 99.3.k.a 4
33.h odd 10 1 1089.3.c.e 4
44.g even 10 1 176.3.n.a 4
44.h odd 10 1 176.3.n.a 4
55.h odd 10 1 275.3.x.e 4
55.j even 10 1 275.3.x.e 4
55.k odd 20 2 275.3.q.d 8
55.l even 20 2 275.3.q.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.3.d.a 4 11.c even 5 1
11.3.d.a 4 11.d odd 10 1
99.3.k.a 4 33.f even 10 1
99.3.k.a 4 33.h odd 10 1
121.3.b.b 4 11.c even 5 1
121.3.b.b 4 11.d odd 10 1
121.3.d.a 4 11.b odd 2 1
121.3.d.a 4 11.c even 5 1
121.3.d.c 4 1.a even 1 1 trivial
121.3.d.c 4 11.d odd 10 1 inner
121.3.d.d 4 11.c even 5 1
121.3.d.d 4 11.d odd 10 1
176.3.n.a 4 44.g even 10 1
176.3.n.a 4 44.h odd 10 1
275.3.q.d 8 55.k odd 20 2
275.3.q.d 8 55.l even 20 2
275.3.x.e 4 55.h odd 10 1
275.3.x.e 4 55.j even 10 1
1089.3.c.e 4 33.f even 10 1
1089.3.c.e 4 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 5T_{2}^{3} + 5T_{2}^{2} + 5T_{2} + 5 \) acting on \(S_{3}^{\mathrm{new}}(121, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 5 T^{3} + 5 T^{2} + 5 T + 5 \) Copy content Toggle raw display
$3$ \( T^{4} + 5 T^{3} + 10 T^{2} + 25 \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256 \) Copy content Toggle raw display
$7$ \( T^{4} + 20 T^{3} + 120 T^{2} + \cdots + 80 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 10T^{3} + 2000 \) Copy content Toggle raw display
$17$ \( T^{4} - 65 T^{3} + 1690 T^{2} + \cdots + 142805 \) Copy content Toggle raw display
$19$ \( T^{4} - 10 T^{3} + 90 T^{2} + \cdots + 605 \) Copy content Toggle raw display
$23$ \( (T^{2} + 10 T + 20)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 50 T^{3} + 600 T^{2} + \cdots + 9680 \) Copy content Toggle raw display
$31$ \( T^{4} - 32 T^{3} + 1384 T^{2} + \cdots + 55696 \) Copy content Toggle raw display
$37$ \( T^{4} + 90 T^{3} + 4860 T^{2} + \cdots + 2624400 \) Copy content Toggle raw display
$41$ \( T^{4} - 135 T^{3} + 5130 T^{2} + \cdots + 8405 \) Copy content Toggle raw display
$43$ \( T^{4} + 1625 T^{2} + 581405 \) Copy content Toggle raw display
$47$ \( T^{4} + 20 T^{3} + 640 T^{2} + \cdots + 384400 \) Copy content Toggle raw display
$53$ \( T^{4} + 30 T^{3} + 5400 T^{2} + \cdots + 810000 \) Copy content Toggle raw display
$59$ \( T^{4} + 52 T^{3} + 1054 T^{2} + \cdots + 5041 \) Copy content Toggle raw display
$61$ \( T^{4} - 80 T^{3} + 2720 T^{2} + \cdots + 403280 \) Copy content Toggle raw display
$67$ \( (T^{2} + 115 T + 2945)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 162 T^{3} + \cdots + 22619536 \) Copy content Toggle raw display
$73$ \( T^{4} + 85 T^{3} - 2210 T^{2} + \cdots + 93787805 \) Copy content Toggle raw display
$79$ \( T^{4} + 130 T^{3} + 4940 T^{2} + \cdots + 67280 \) Copy content Toggle raw display
$83$ \( T^{4} + 10 T^{3} - 410 T^{2} + \cdots + 22281605 \) Copy content Toggle raw display
$89$ \( (T^{2} - 61 T - 7681)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 170 T^{3} + \cdots + 31416025 \) Copy content Toggle raw display
show more
show less