Properties

Label 121.4.a.b
Level $121$
Weight $4$
Character orbit 121.a
Self dual yes
Analytic conductor $7.139$
Analytic rank $1$
Dimension $2$
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] = [121,4,Mod(1,121)]
 
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([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("121.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 121.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: \(7.13923111069\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + (\beta - 4) q^{3} + ( - 2 \beta + 5) q^{4} + (\beta - 5) q^{5} + ( - 5 \beta + 16) q^{6} + ( - 7 \beta - 4) q^{7} + ( - \beta - 21) q^{8} + ( - 8 \beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} + (\beta - 4) q^{3} + ( - 2 \beta + 5) q^{4} + (\beta - 5) q^{5} + ( - 5 \beta + 16) q^{6} + ( - 7 \beta - 4) q^{7} + ( - \beta - 21) q^{8} + ( - 8 \beta + 1) q^{9} + ( - 6 \beta + 17) q^{10} + (13 \beta - 44) q^{12} + (\beta + 65) q^{13} + (3 \beta - 80) q^{14} + ( - 9 \beta + 32) q^{15} + ( - 4 \beta - 31) q^{16} + (18 \beta - 7) q^{17} + (9 \beta - 97) q^{18} + ( - 9 \beta - 24) q^{19} + (15 \beta - 49) q^{20} + (24 \beta - 68) q^{21} + ( - 33 \beta - 64) q^{23} + ( - 17 \beta + 72) q^{24} + ( - 10 \beta - 88) q^{25} + (64 \beta - 53) q^{26} + (6 \beta + 8) q^{27} + ( - 27 \beta + 148) q^{28} + (37 \beta - 15) q^{29} + (41 \beta - 140) q^{30} + (47 \beta - 92) q^{31} + ( - 19 \beta + 151) q^{32} + ( - 25 \beta + 223) q^{34} + (31 \beta - 64) q^{35} + ( - 42 \beta + 197) q^{36} + ( - 79 \beta + 63) q^{37} + ( - 15 \beta - 84) q^{38} + (61 \beta - 248) q^{39} + ( - 16 \beta + 93) q^{40} + (2 \beta + 185) q^{41} + ( - 92 \beta + 356) q^{42} + ( - 22 \beta - 132) q^{43} + (41 \beta - 101) q^{45} + ( - 31 \beta - 332) q^{46} + (111 \beta + 128) q^{47} + ( - 15 \beta + 76) q^{48} + (56 \beta + 261) q^{49} + ( - 78 \beta - 32) q^{50} + ( - 79 \beta + 244) q^{51} + ( - 125 \beta + 301) q^{52} + ( - 85 \beta - 81) q^{53} + (2 \beta + 64) q^{54} + (151 \beta + 168) q^{56} + (12 \beta - 12) q^{57} + ( - 52 \beta + 459) q^{58} + (42 \beta - 652) q^{59} + ( - 109 \beta + 376) q^{60} + ( - 92 \beta - 150) q^{61} + ( - 139 \beta + 656) q^{62} + (25 \beta + 668) q^{63} + (202 \beta - 131) q^{64} + (60 \beta - 313) q^{65} + (11 \beta - 328) q^{67} + (104 \beta - 467) q^{68} + (68 \beta - 140) q^{69} + ( - 95 \beta + 436) q^{70} + (56 \beta - 588) q^{71} + (167 \beta + 75) q^{72} + (180 \beta - 334) q^{73} + (142 \beta - 1011) q^{74} + ( - 48 \beta + 232) q^{75} + (3 \beta + 96) q^{76} + ( - 309 \beta + 980) q^{78} + ( - 109 \beta + 208) q^{79} + ( - 11 \beta + 107) q^{80} + (200 \beta + 13) q^{81} + (183 \beta - 161) q^{82} + (51 \beta - 480) q^{83} + (256 \beta - 916) q^{84} + ( - 97 \beta + 251) q^{85} + ( - 110 \beta - 132) q^{86} + ( - 163 \beta + 504) q^{87} + ( - 176 \beta - 537) q^{89} + ( - 142 \beta + 593) q^{90} + ( - 459 \beta - 344) q^{91} + ( - 37 \beta + 472) q^{92} + ( - 280 \beta + 932) q^{93} + (17 \beta + 1204) q^{94} + (21 \beta + 12) q^{95} + (227 \beta - 832) q^{96} + (234 \beta - 169) q^{97} + (205 \beta + 411) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 8 q^{3} + 10 q^{4} - 10 q^{5} + 32 q^{6} - 8 q^{7} - 42 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 8 q^{3} + 10 q^{4} - 10 q^{5} + 32 q^{6} - 8 q^{7} - 42 q^{8} + 2 q^{9} + 34 q^{10} - 88 q^{12} + 130 q^{13} - 160 q^{14} + 64 q^{15} - 62 q^{16} - 14 q^{17} - 194 q^{18} - 48 q^{19} - 98 q^{20} - 136 q^{21} - 128 q^{23} + 144 q^{24} - 176 q^{25} - 106 q^{26} + 16 q^{27} + 296 q^{28} - 30 q^{29} - 280 q^{30} - 184 q^{31} + 302 q^{32} + 446 q^{34} - 128 q^{35} + 394 q^{36} + 126 q^{37} - 168 q^{38} - 496 q^{39} + 186 q^{40} + 370 q^{41} + 712 q^{42} - 264 q^{43} - 202 q^{45} - 664 q^{46} + 256 q^{47} + 152 q^{48} + 522 q^{49} - 64 q^{50} + 488 q^{51} + 602 q^{52} - 162 q^{53} + 128 q^{54} + 336 q^{56} - 24 q^{57} + 918 q^{58} - 1304 q^{59} + 752 q^{60} - 300 q^{61} + 1312 q^{62} + 1336 q^{63} - 262 q^{64} - 626 q^{65} - 656 q^{67} - 934 q^{68} - 280 q^{69} + 872 q^{70} - 1176 q^{71} + 150 q^{72} - 668 q^{73} - 2022 q^{74} + 464 q^{75} + 192 q^{76} + 1960 q^{78} + 416 q^{79} + 214 q^{80} + 26 q^{81} - 322 q^{82} - 960 q^{83} - 1832 q^{84} + 502 q^{85} - 264 q^{86} + 1008 q^{87} - 1074 q^{89} + 1186 q^{90} - 688 q^{91} + 944 q^{92} + 1864 q^{93} + 2408 q^{94} + 24 q^{95} - 1664 q^{96} - 338 q^{97} + 822 q^{98}+O(q^{100}) \) 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
−1.73205
1.73205
−4.46410 −7.46410 11.9282 −8.46410 33.3205 20.2487 −17.5359 28.7128 37.7846
1.2 2.46410 −0.535898 −1.92820 −1.53590 −1.32051 −28.2487 −24.4641 −26.7128 −3.78461
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)

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 121.4.a.b 2
3.b odd 2 1 1089.4.a.x 2
4.b odd 2 1 1936.4.a.z 2
11.b odd 2 1 121.4.a.e yes 2
11.c even 5 4 121.4.c.g 8
11.d odd 10 4 121.4.c.d 8
33.d even 2 1 1089.4.a.k 2
44.c even 2 1 1936.4.a.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.4.a.b 2 1.a even 1 1 trivial
121.4.a.e yes 2 11.b odd 2 1
121.4.c.d 8 11.d odd 10 4
121.4.c.g 8 11.c even 5 4
1089.4.a.k 2 33.d even 2 1
1089.4.a.x 2 3.b odd 2 1
1936.4.a.y 2 44.c even 2 1
1936.4.a.z 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2T_{2} - 11 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(121))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 11 \) Copy content Toggle raw display
$3$ \( T^{2} + 8T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 10T + 13 \) Copy content Toggle raw display
$7$ \( T^{2} + 8T - 572 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 130T + 4213 \) Copy content Toggle raw display
$17$ \( T^{2} + 14T - 3839 \) Copy content Toggle raw display
$19$ \( T^{2} + 48T - 396 \) Copy content Toggle raw display
$23$ \( T^{2} + 128T - 8972 \) Copy content Toggle raw display
$29$ \( T^{2} + 30T - 16203 \) Copy content Toggle raw display
$31$ \( T^{2} + 184T - 18044 \) Copy content Toggle raw display
$37$ \( T^{2} - 126T - 70923 \) Copy content Toggle raw display
$41$ \( T^{2} - 370T + 34177 \) Copy content Toggle raw display
$43$ \( T^{2} + 264T + 11616 \) Copy content Toggle raw display
$47$ \( T^{2} - 256T - 131468 \) Copy content Toggle raw display
$53$ \( T^{2} + 162T - 80139 \) Copy content Toggle raw display
$59$ \( T^{2} + 1304 T + 403936 \) Copy content Toggle raw display
$61$ \( T^{2} + 300T - 79068 \) Copy content Toggle raw display
$67$ \( T^{2} + 656T + 106132 \) Copy content Toggle raw display
$71$ \( T^{2} + 1176 T + 308112 \) Copy content Toggle raw display
$73$ \( T^{2} + 668T - 277244 \) Copy content Toggle raw display
$79$ \( T^{2} - 416T - 99308 \) Copy content Toggle raw display
$83$ \( T^{2} + 960T + 199188 \) Copy content Toggle raw display
$89$ \( T^{2} + 1074T - 83343 \) Copy content Toggle raw display
$97$ \( T^{2} + 338T - 628511 \) Copy content Toggle raw display
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