Properties

Label 121.3.b.c
Level $121$
Weight $3$
Character orbit 121.b
Analytic conductor $3.297$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [121,3,Mod(120,121)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(121, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("121.120");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 121.b (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: \(3.29701119876\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.523388583936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 28x^{6} + 262x^{4} + 948x^{2} + 1089 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 11 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{5} + \beta_{3} - 1) q^{3} + (\beta_{5} + \beta_{4} - 3) q^{4} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{5} + ( - \beta_{7} + \beta_{2} - \beta_1) q^{6} + ( - \beta_{6} + \beta_{2}) q^{7} + (\beta_{7} + \beta_{6} + 2 \beta_{2}) q^{8} + (4 \beta_{5} + \beta_{4} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{5} + \beta_{3} - 1) q^{3} + (\beta_{5} + \beta_{4} - 3) q^{4} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{5} + ( - \beta_{7} + \beta_{2} - \beta_1) q^{6} + ( - \beta_{6} + \beta_{2}) q^{7} + (\beta_{7} + \beta_{6} + 2 \beta_{2}) q^{8} + (4 \beta_{5} + \beta_{4} + 2) q^{9} + (2 \beta_{7} - \beta_{6} + \beta_{2} - \beta_1) q^{10} + (3 \beta_{5} - \beta_{3} + 7) q^{12} + (\beta_{7} - \beta_{6} - 2 \beta_{2} - 4 \beta_1) q^{13} + (6 \beta_{5} - \beta_{4} - 3 \beta_{3} + 1) q^{14} + (\beta_{5} - 3 \beta_{4} - \beta_{3} - 8) q^{15} + ( - 2 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} - 5) q^{16} + ( - \beta_{7} - \beta_{6} + \beta_{2} + 3 \beta_1) q^{17} + (\beta_{7} + 4 \beta_{6} + 2 \beta_{2} + \beta_1) q^{18} + ( - 4 \beta_{6} + \beta_{2} + \beta_1) q^{19} + ( - 3 \beta_{5} - 4 \beta_{4} + \beta_{3} + 6) q^{20} + ( - \beta_{7} + 4 \beta_{6} - 2 \beta_{2} + 4 \beta_1) q^{21} + ( - 7 \beta_{5} - \beta_{4} + 5 \beta_{3} - 4) q^{23} + ( - 3 \beta_{7} + 4 \beta_{6} + 3 \beta_{2} + 3 \beta_1) q^{24} + ( - 8 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} - 1) q^{25} + (2 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 19) q^{26} + ( - \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 14) q^{27} + (2 \beta_{7} + 5 \beta_{6} - \beta_{2} + 2 \beta_1) q^{28} + ( - 3 \beta_{7} - 2 \beta_{6} - 6 \beta_{2} - 5 \beta_1) q^{29} + ( - 2 \beta_{7} + 2 \beta_{6} - 7 \beta_{2} - 5 \beta_1) q^{30} + (2 \beta_{4} - 5 \beta_{3} - 12) q^{31} + ( - 4 \beta_{7} - 2 \beta_{6} + 4 \beta_{2} - \beta_1) q^{32} + (9 \beta_{5} + 5 \beta_{4} - 7 \beta_{3} - 19) q^{34} + (5 \beta_{7} - \beta_{6} + \beta_{2} + 4 \beta_1) q^{35} + ( - 7 \beta_{5} - 6 \beta_{4} + 10 \beta_{3} + 14) q^{36} + ( - 13 \beta_{5} + 10 \beta_{4} - \beta_{3}) q^{37} + (25 \beta_{5} + 3 \beta_{4} - 9 \beta_{3} - 12) q^{38} + (5 \beta_{7} - 6 \beta_{6} - 11 \beta_{2} - 9 \beta_1) q^{39} + (3 \beta_{7} - 8 \beta_{6} - 3 \beta_{2} + 6 \beta_1) q^{40} + ( - 2 \beta_{7} + 8 \beta_{6} - \beta_{2} + 6 \beta_1) q^{41} + ( - 20 \beta_{5} + 7 \beta_{4} + 6 \beta_{3} - 25) q^{42} + ( - 3 \beta_{7} - 7 \beta_{6} - 10 \beta_{2} - 13 \beta_1) q^{43} + ( - 13 \beta_{5} - 5 \beta_{4} - \beta_{3} - 9) q^{45} + ( - 6 \beta_{7} - 12 \beta_{6} + 3 \beta_{2} - 3 \beta_1) q^{46} + ( - 15 \beta_{5} - 7 \beta_{4} + 3 \beta_{3} - 32) q^{47} + ( - 9 \beta_{5} + 2 \beta_{4} - 11 \beta_{3} + 27) q^{48} + ( - 2 \beta_{5} + \beta_{4} + 10 \beta_{3} + 6) q^{49} + (7 \beta_{7} - 6 \beta_{6} + 8 \beta_{2} - 6 \beta_1) q^{50} + ( - 3 \beta_{7} + 5 \beta_{6} + 3 \beta_{2} + 6 \beta_1) q^{51} + ( - 2 \beta_{7} - 6 \beta_{6} - 8 \beta_{2} + 5 \beta_1) q^{52} + (8 \beta_{5} + 5 \beta_{3} + 31) q^{53} + (5 \beta_{7} + \beta_{6} + 4 \beta_{2} + 11 \beta_1) q^{54} + ( - 4 \beta_{5} - 11 \beta_{4} + 7 \beta_{3} - 5) q^{56} + ( - 2 \beta_{7} + 7 \beta_{6} - 10 \beta_{2} + 3 \beta_1) q^{57} + (7 \beta_{5} + 18 \beta_{4} - 10 \beta_{3} + 16) q^{58} + (18 \beta_{5} - 6 \beta_{4} + 2 \beta_{3} - 8) q^{59} + ( - 13 \beta_{5} + \beta_{4} - \beta_{3} - 12) q^{60} + ( - 10 \beta_{7} + 13 \beta_{6} - 6 \beta_{2} + 7 \beta_1) q^{61} + (7 \beta_{7} + 5 \beta_{6} - \beta_{2} - 14 \beta_1) q^{62} + ( - \beta_{7} + 7 \beta_{2} - 7 \beta_1) q^{63} + (3 \beta_{5} - 11 \beta_{4} - 8 \beta_{3} - 1) q^{64} + ( - 7 \beta_{7} + 6 \beta_{6} + 7 \beta_{2} + 12 \beta_1) q^{65} + (12 \beta_{5} + 19 \beta_{4} - 7 \beta_{3} + 39) q^{67} + (8 \beta_{7} + 12 \beta_{6} + 7 \beta_{2} - 12 \beta_1) q^{68} + (15 \beta_{5} - 6 \beta_{4} - 12 \beta_{3} + 39) q^{69} + (10 \beta_{5} - 12 \beta_{4} + 17 \beta_{3} - 32) q^{70} + (\beta_{5} - 19 \beta_{4} + 34) q^{71} + ( - 12 \beta_{7} - \beta_{6} + 6 \beta_{2} + 24 \beta_1) q^{72} + ( - 9 \beta_{7} + 7 \beta_{6} + 2 \beta_{2} + 7 \beta_1) q^{73} + (11 \beta_{7} - 12 \beta_{6} + 19 \beta_{2} - 10 \beta_1) q^{74} + (19 \beta_{5} - 13 \beta_{4} - 4 \beta_{3} - 10) q^{75} + (12 \beta_{7} + 18 \beta_{6} + \beta_{2} - 11 \beta_1) q^{76} + (27 \beta_{5} + 4 \beta_{4} + 19 \beta_{3} + 13) q^{78} + (2 \beta_{7} + 5 \beta_{6} + 5 \beta_{2} + 4 \beta_1) q^{79} + (42 \beta_{5} - 5 \beta_{4} + 3 \beta_{3} - 46) q^{80} + ( - 14 \beta_{5} - 13 \beta_{4} + 16 \beta_{3} - 42) q^{81} + ( - 42 \beta_{5} + 6 \beta_{4} + 9 \beta_{3} - 27) q^{82} + (6 \beta_{7} + 6 \beta_{6} + 27 \beta_{2} + 5 \beta_1) q^{83} + ( - 3 \beta_{7} - 10 \beta_{6} + 12 \beta_{2} - 16 \beta_1) q^{84} + (9 \beta_{7} - \beta_{6} - 6 \beta_1) q^{85} + (29 \beta_{5} + 23 \beta_{4} - 16 \beta_{3} + 50) q^{86} + (14 \beta_{7} - 13 \beta_{6} - 11 \beta_{2} - 4 \beta_1) q^{87} + ( - 32 \beta_{5} - 3 \beta_{4} - 18 \beta_{3} + 6) q^{89} + ( - 4 \beta_{7} - 12 \beta_{6} - 11 \beta_{2} - 4 \beta_1) q^{90} + ( - 2 \beta_{5} + \beta_{4} + 9 \beta_{3} + 13) q^{91} + (41 \beta_{5} + 17 \beta_{4} - 31 \beta_{3} - 4) q^{92} + ( - 25 \beta_{5} - 2 \beta_{4} - 10 \beta_{3} - 27) q^{93} + ( - 10 \beta_{7} - 18 \beta_{6} - 11 \beta_{2} - 25 \beta_1) q^{94} + (16 \beta_{7} + \beta_{6} + 11 \beta_{2} + 12 \beta_1) q^{95} + (\beta_{7} + 18 \beta_{6} + 5 \beta_{2} + 37 \beta_1) q^{96} + ( - 21 \beta_{5} - \beta_{4} + 32 \beta_{3} + 3) q^{97} + ( - 9 \beta_{7} - 12 \beta_{6} + 12 \beta_{2} + 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} - 24 q^{4} - 4 q^{5} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} - 24 q^{4} - 4 q^{5} + 16 q^{9} + 52 q^{12} - 4 q^{14} - 68 q^{15} - 24 q^{16} + 52 q^{20} - 12 q^{23} - 16 q^{25} + 168 q^{26} + 104 q^{27} - 116 q^{31} - 180 q^{34} + 152 q^{36} - 4 q^{37} - 132 q^{38} - 176 q^{42} - 76 q^{45} - 244 q^{47} + 172 q^{48} + 88 q^{49} + 268 q^{53} - 12 q^{56} + 88 q^{58} - 56 q^{59} - 100 q^{60} - 40 q^{64} + 284 q^{67} + 264 q^{69} - 188 q^{70} + 272 q^{71} - 96 q^{75} + 180 q^{78} - 356 q^{80} - 272 q^{81} - 180 q^{82} + 336 q^{86} - 24 q^{89} + 140 q^{91} - 156 q^{92} - 256 q^{93} + 152 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 28x^{6} + 262x^{4} + 948x^{2} + 1089 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 13\nu^{5} + 41\nu^{3} + 489\nu ) / 108 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 13\nu^{4} + 5\nu^{2} + 201 ) / 36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 22\nu^{4} - 121\nu^{2} - 114 ) / 18 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 22\nu^{4} + 139\nu^{2} + 240 ) / 18 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 22\nu^{5} + 139\nu^{3} + 240\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} - 53\nu^{5} - 404\nu^{3} - 777\nu ) / 54 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + 2\beta_{2} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -14\beta_{5} - 16\beta_{4} + 4\beta_{3} + 63 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -20\beta_{7} - 18\beta_{6} - 28\beta_{2} + 79\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 187\beta_{5} + 213\beta_{4} - 88\beta_{3} - 653 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 301\beta_{7} + 275\beta_{6} + 338\beta_{2} - 866\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
120.1
3.57626i
2.89067i
2.16205i
1.47646i
1.47646i
2.16205i
2.89067i
3.57626i
3.57626i 0.119159 −8.78965 −3.44471 0.426142i 3.88348i 17.1290i −8.98580 12.3192i
120.2 2.89067i −3.85230 −4.35597 −1.96778 11.1357i 9.76707i 1.02900i 5.84018 5.68820i
120.3 2.16205i −2.85121 −0.674453 7.64086 6.16445i 6.05557i 7.18999i −0.870605 16.5199i
120.4 1.47646i 4.58435 1.82008 −4.22837 6.76859i 2.20298i 8.59309i 12.0162 6.24301i
120.5 1.47646i 4.58435 1.82008 −4.22837 6.76859i 2.20298i 8.59309i 12.0162 6.24301i
120.6 2.16205i −2.85121 −0.674453 7.64086 6.16445i 6.05557i 7.18999i −0.870605 16.5199i
120.7 2.89067i −3.85230 −4.35597 −1.96778 11.1357i 9.76707i 1.02900i 5.84018 5.68820i
120.8 3.57626i 0.119159 −8.78965 −3.44471 0.426142i 3.88348i 17.1290i −8.98580 12.3192i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 120.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.3.b.c 8
3.b odd 2 1 1089.3.c.k 8
11.b odd 2 1 inner 121.3.b.c 8
11.c even 5 4 121.3.d.f 32
11.d odd 10 4 121.3.d.f 32
33.d even 2 1 1089.3.c.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.3.b.c 8 1.a even 1 1 trivial
121.3.b.c 8 11.b odd 2 1 inner
121.3.d.f 32 11.c even 5 4
121.3.d.f 32 11.d odd 10 4
1089.3.c.k 8 3.b odd 2 1
1089.3.c.k 8 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 28T_{2}^{6} + 262T_{2}^{4} + 948T_{2}^{2} + 1089 \) acting on \(S_{3}^{\mathrm{new}}(121, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 28 T^{6} + 262 T^{4} + \cdots + 1089 \) Copy content Toggle raw display
$3$ \( (T^{4} + 2 T^{3} - 20 T^{2} - 48 T + 6)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 2 T^{3} - 44 T^{2} - 198 T - 219)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 152 T^{6} + 6204 T^{4} + \cdots + 256036 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 580 T^{6} + 57238 T^{4} + \cdots + 5803281 \) Copy content Toggle raw display
$17$ \( T^{8} + 600 T^{6} + \cdots + 36469521 \) Copy content Toggle raw display
$19$ \( T^{8} + 1552 T^{6} + \cdots + 1961249796 \) Copy content Toggle raw display
$23$ \( (T^{4} + 6 T^{3} - 984 T^{2} + 3348 T - 162)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 3028 T^{6} + \cdots + 36874368729 \) Copy content Toggle raw display
$31$ \( (T^{4} + 58 T^{3} + 760 T^{2} + \cdots - 13458)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{3} - 3416 T^{2} + \cdots + 291069)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 5232 T^{6} + \cdots + 534239660889 \) Copy content Toggle raw display
$43$ \( T^{8} + 10216 T^{6} + \cdots + 20286952842816 \) Copy content Toggle raw display
$47$ \( (T^{4} + 122 T^{3} + 2788 T^{2} + \cdots - 3669186)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 134 T^{3} + 6052 T^{2} + \cdots + 730761)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 28 T^{3} - 2288 T^{2} + \cdots + 233952)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 21416 T^{6} + \cdots + 87785731315216 \) Copy content Toggle raw display
$67$ \( (T^{4} - 142 T^{3} - 2384 T^{2} + \cdots - 23025234)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 136 T^{3} - 2456 T^{2} + \cdots + 10616856)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 10136 T^{6} + \cdots + 4549825513024 \) Copy content Toggle raw display
$79$ \( T^{8} + 3520 T^{6} + \cdots + 163631576196 \) Copy content Toggle raw display
$83$ \( T^{8} + 43120 T^{6} + \cdots + 75\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{4} + 12 T^{3} - 10590 T^{2} + \cdots - 6209127)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 76 T^{3} - 26106 T^{2} + \cdots - 65182079)^{2} \) Copy content Toggle raw display
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