Properties

Label 121.4.a.g
Level $121$
Weight $4$
Character orbit 121.a
Self dual yes
Analytic conductor $7.139$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{5}, \sqrt{37})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 21x^{2} + 64 \) 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)
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 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.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1 + 1) q^{2} + (\beta_{2} + 2) q^{3} + (2 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{4} + ( - 3 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 3) q^{5} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 11) q^{6} + ( - 3 \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 11) q^{7} + (6 \beta_{3} - 3 \beta_{2} + 13 \beta_1 + 7) q^{8} + (3 \beta_{2} - 14) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1 + 1) q^{2} + (\beta_{2} + 2) q^{3} + (2 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{4} + ( - 3 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 3) q^{5} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 11) q^{6} + ( - 3 \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 11) q^{7} + (6 \beta_{3} - 3 \beta_{2} + 13 \beta_1 + 7) q^{8} + (3 \beta_{2} - 14) q^{9} + ( - 6 \beta_{2} - 28 \beta_1 + 16) q^{10} + (5 \beta_{3} + 4 \beta_{2} + 19 \beta_1 + 15) q^{12} + ( - 7 \beta_{3} - 4 \beta_{2} + 14 \beta_1 - 1) q^{13} + ( - 4 \beta_{3} + 8 \beta_{2} - 24 \beta_1 + 50) q^{14} + ( - 2 \beta_{3} - \beta_{2} - 23 \beta_1 + 12) q^{15} + (6 \beta_{3} + 5 \beta_{2} + 69 \beta_1 - 25) q^{16} + ( - 2 \beta_{3} - 7 \beta_{2} - 16 \beta_1 + 63) q^{17} + (3 \beta_{3} - 14 \beta_{2} - 17 \beta_1 + 13) q^{18} + ( - 12 \beta_{3} + 6 \beta_{2} + 51 \beta_1 + 19) q^{19} + ( - 10 \beta_{3} - 38 \beta_1 - 42) q^{20} + ( - 9 \beta_{3} + 16 \beta_{2} - 30 \beta_1 + 67) q^{21} + ( - 10 \beta_{3} - 26 \beta_{2} - 16 \beta_1 - 4) q^{23} + (25 \beta_{3} + 4 \beta_{2} + 67 \beta_1 - 13) q^{24} + (19 \beta_{3} - 31 \beta_{2} - 51 \beta_1 + 2) q^{25} + ( - 4 \beta_{3} - 8 \beta_{2} - 46 \beta_1 - 30) q^{26} + ( - 38 \beta_{2} - 55) q^{27} + (6 \beta_{2} + 6 \beta_1 + 6) q^{28} + (31 \beta_{3} + \beta_{2} - 19 \beta_1 + 93) q^{29} + ( - 28 \beta_{3} + 10 \beta_{2} - 28 \beta_1 - 22) q^{30} + (41 \beta_{3} - 40 \beta_{2} + 48 \beta_1 - 45) q^{31} + (38 \beta_{3} + 5 \beta_{2} - 11 \beta_1 + 39) q^{32} + ( - 27 \beta_{3} + 61 \beta_{2} + 36 \beta_1 - 18) q^{34} + ( - 4 \beta_{3} + 3 \beta_{2} - 81 \beta_1 + 132) q^{35} + ( - 25 \beta_{3} - 8 \beta_{2} + 37 \beta_1 - 15) q^{36} + ( - 35 \beta_{3} + 17 \beta_{2} + 43 \beta_1 + 19) q^{37} + (33 \beta_{3} + 7 \beta_{2} - 44 \beta_1 + 112) q^{38} + ( - 5 \beta_{2} - 49 \beta_1 - 38) q^{39} + ( - 58 \beta_{3} - 4 \beta_{2} + 54 \beta_1 - 218) q^{40} + (62 \beta_{3} - 15 \beta_{2} - 22 \beta_1 + 26) q^{41} + ( - 32 \beta_{3} + 58 \beta_{2} - 60 \beta_1 + 172) q^{42} + ( - 31 \beta_{3} - 8 \beta_{2} + 199 \beta_1 - 177) q^{43} + (54 \beta_{3} - 43 \beta_{2} - 149 \beta_1 + 96) q^{45} + ( - 62 \beta_{3} - 14 \beta_{2} - 84 \beta_1 - 264) q^{46} + ( - 9 \beta_{3} + 37 \beta_{2} + 89 \beta_1 - 241) q^{47} + (81 \beta_{3} - 20 \beta_{2} + 123 \beta_1 - 5) q^{48} + ( - 69 \beta_{3} + 112 \beta_{2} - 216 \beta_1 + 120) q^{49} + ( - 44 \beta_{3} + 21 \beta_{2} + 153 \beta_1 - 309) q^{50} + ( - 20 \beta_{3} + 56 \beta_{2} - 34 \beta_1 + 63) q^{51} + ( - 6 \beta_{3} - 2 \beta_{2} - 216 \beta_1 - 144) q^{52} + ( - 41 \beta_{3} + 2 \beta_{2} + 238 \beta_1 + 95) q^{53} + ( - 38 \beta_{3} - 55 \beta_{2} - 17 \beta_1 - 397) q^{54} + (44 \beta_{3} - 58 \beta_{2} + 198 \beta_1 - 334) q^{56} + (27 \beta_{3} + 25 \beta_{2} - 57 \beta_1 + 92) q^{57} + (44 \beta_{3} + 124 \beta_{2} + 352 \beta_1 + 114) q^{58} + (32 \beta_{3} + 2 \beta_{2} + 137 \beta_1 - 31) q^{59} + ( - 58 \beta_{3} - 42 \beta_{2} - 128 \beta_1 - 84) q^{60} + (147 \beta_{3} - 42 \beta_{2} + 42 \beta_1 - 105) q^{61} + (90 \beta_{3} - 4 \beta_{2} + 412 \beta_1 - 316) q^{62} + (33 \beta_{3} - 52 \beta_{2} - 30 \beta_1 - 19) q^{63} + (22 \beta_{3} + 37 \beta_{2} - 187 \beta_1 + 311) q^{64} + ( - 13 \beta_{3} - 31 \beta_{2} + 169 \beta_1 + 127) q^{65} + ( - 63 \beta_{3} + 109 \beta_{2} - 246 \beta_1 + 204) q^{67} + (59 \beta_{3} + 11 \beta_{2} - 158 \beta_1 + 36) q^{68} + ( - 36 \beta_{3} - 30 \beta_{2} - 106 \beta_1 - 242) q^{69} + ( - 86 \beta_{3} + 128 \beta_{2} + 12 \beta_1 + 74) q^{70} + (77 \beta_{3} - 120 \beta_{2} - 330 \beta_1 + 243) q^{71} + ( - 45 \beta_{3} + 72 \beta_{2} - 59 \beta_1 - 179) q^{72} + ( - 44 \beta_{3} - 51 \beta_{2} - 436 \beta_1 + 136) q^{73} + ( - 10 \beta_{3} - 16 \beta_{2} - 270 \beta_1 + 180) q^{74} + ( - 13 \beta_{3} - 29 \beta_{2} + 120 \beta_1 - 275) q^{75} + (125 \beta_{3} + 97 \beta_{2} - 50 \beta_1 + 12) q^{76} + ( - 54 \beta_{3} - 38 \beta_{2} - 82 \beta_1 - 132) q^{78} + (13 \beta_{3} - 128 \beta_{2} + 68 \beta_1 + 143) q^{79} + (14 \beta_{3} - 276 \beta_{2} - 378 \beta_1 + 78) q^{80} + ( - 174 \beta_{2} - 74) q^{81} + (87 \beta_{3} + 88 \beta_{2} + 577 \beta_1 - 69) q^{82} + ( - 208 \beta_{3} + 14 \beta_{2} - 75 \beta_1 + 518) q^{83} + (6 \beta_{3} + 12 \beta_{2} + 6 \beta_1 + 66) q^{84} + ( - 197 \beta_{3} + 207 \beta_{2} + 503 \beta_1 - 279) q^{85} + (129 \beta_{3} - 208 \beta_{2} - 249 \beta_1 - 81) q^{86} + (43 \beta_{3} + 94 \beta_{2} + 260 \beta_1 + 195) q^{87} + (25 \beta_{3} + 32 \beta_{2} + 403 \beta_1 + 281) q^{89} + ( - 84 \beta_{3} + 150 \beta_{2} + 476 \beta_1 - 386) q^{90} + (8 \beta_{3} - 29 \beta_{2} + 15 \beta_1 - 44) q^{91} + ( - 142 \beta_{3} - 118 \beta_{2} - 764 \beta_1 - 504) q^{92} + (130 \beta_{3} - 85 \beta_{2} + 417 \beta_1 - 450) q^{93} + (108 \beta_{3} - 250 \beta_{2} - 270 \beta_1 + 172) q^{94} + ( - 60 \beta_{3} - 157 \beta_{2} - 53 \beta_1 + 414) q^{95} + (65 \beta_{3} + 44 \beta_{2} + 331 \beta_1 + 123) q^{96} + (50 \beta_{3} + 144 \beta_{2} + 421 \beta_1 - 594) q^{97} + ( - 242 \beta_{3} + 51 \beta_{2} - 829 \beta_1 + 843) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 6 q^{3} + 14 q^{4} - 11 q^{5} + 43 q^{6} + 25 q^{7} + 66 q^{8} - 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 6 q^{3} + 14 q^{4} - 11 q^{5} + 43 q^{6} + 25 q^{7} + 66 q^{8} - 62 q^{9} + 20 q^{10} + 95 q^{12} + 25 q^{13} + 132 q^{14} + 2 q^{15} + 34 q^{16} + 232 q^{17} + 49 q^{18} + 154 q^{19} - 254 q^{20} + 167 q^{21} - 6 q^{23} + 99 q^{24} - 13 q^{25} - 200 q^{26} - 144 q^{27} + 24 q^{28} + 363 q^{29} - 192 q^{30} + 37 q^{31} + 162 q^{32} - 149 q^{34} + 356 q^{35} + 5 q^{36} + 93 q^{37} + 379 q^{38} - 240 q^{39} - 814 q^{40} + 152 q^{41} + 420 q^{42} - 325 q^{43} + 226 q^{45} - 1258 q^{46} - 869 q^{47} + 347 q^{48} - 245 q^{49} - 1016 q^{50} + 52 q^{51} - 1010 q^{52} + 811 q^{53} - 1550 q^{54} - 780 q^{56} + 231 q^{57} + 956 q^{58} + 178 q^{59} - 566 q^{60} - 105 q^{61} - 342 q^{62} + q^{63} + 818 q^{64} + 895 q^{65} + 43 q^{67} - 135 q^{68} - 1156 q^{69} - 22 q^{70} + 629 q^{71} - 1023 q^{72} - 270 q^{73} + 202 q^{74} - 815 q^{75} - 121 q^{76} - 670 q^{78} + 977 q^{79} + 122 q^{80} + 52 q^{81} + 789 q^{82} + 1686 q^{83} + 258 q^{84} - 721 q^{85} - 277 q^{86} + 1155 q^{87} + 1891 q^{89} - 976 q^{90} - 80 q^{91} - 3450 q^{92} - 666 q^{93} + 756 q^{94} + 1804 q^{95} + 1131 q^{96} - 1772 q^{97} + 1370 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 21x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 13\nu + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 29\nu - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{2} + \nu - 10 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{3} - \beta_{2} - \beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 13\beta_{2} + 29\beta _1 - 8 \) 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
−4.15942
−1.92335
1.92335
4.15942
−3.15942 −1.54138 1.98190 −17.2669 4.86986 −9.56478 19.0137 −24.6241 54.5532
1.2 −0.923347 −1.54138 −7.14743 8.72550 1.42323 0.775116 13.9863 −24.6241 −8.05666
1.3 2.92335 4.54138 0.545959 6.17671 13.2760 32.1271 −21.7907 −6.37586 18.0567
1.4 5.15942 4.54138 18.6196 −8.63533 23.4309 1.66258 54.7907 −6.37586 −44.5532
\(n\): 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.g 4
3.b odd 2 1 1089.4.a.y 4
4.b odd 2 1 1936.4.a.bk 4
11.b odd 2 1 121.4.a.f 4
11.c even 5 2 11.4.c.a 8
11.c even 5 2 121.4.c.h 8
11.d odd 10 2 121.4.c.b 8
11.d odd 10 2 121.4.c.i 8
33.d even 2 1 1089.4.a.bh 4
33.h odd 10 2 99.4.f.c 8
44.c even 2 1 1936.4.a.bl 4
44.h odd 10 2 176.4.m.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.c.a 8 11.c even 5 2
99.4.f.c 8 33.h odd 10 2
121.4.a.f 4 11.b odd 2 1
121.4.a.g 4 1.a even 1 1 trivial
121.4.c.b 8 11.d odd 10 2
121.4.c.h 8 11.c even 5 2
121.4.c.i 8 11.d odd 10 2
176.4.m.c 8 44.h odd 10 2
1089.4.a.y 4 3.b odd 2 1
1089.4.a.bh 4 33.d even 2 1
1936.4.a.bk 4 4.b odd 2 1
1936.4.a.bl 4 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 4T_{2}^{3} - 15T_{2}^{2} + 38T_{2} + 44 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(121))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4 T^{3} - 15 T^{2} + 38 T + 44 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T - 7)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 11 T^{3} - 183 T^{2} + \cdots + 8036 \) Copy content Toggle raw display
$7$ \( T^{4} - 25 T^{3} - 251 T^{2} + \cdots - 396 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 25 T^{3} - 2215 T^{2} + \cdots - 53900 \) Copy content Toggle raw display
$17$ \( T^{4} - 232 T^{3} + 18185 T^{2} + \cdots + 3099789 \) Copy content Toggle raw display
$19$ \( T^{4} - 154 T^{3} + 501 T^{2} + \cdots - 9492329 \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} - 21180 T^{2} + \cdots + 49883584 \) Copy content Toggle raw display
$29$ \( T^{4} - 363 T^{3} + \cdots - 98528364 \) Copy content Toggle raw display
$31$ \( T^{4} - 37 T^{3} - 57125 T^{2} + \cdots - 33222196 \) Copy content Toggle raw display
$37$ \( T^{4} - 93 T^{3} + \cdots + 163244164 \) Copy content Toggle raw display
$41$ \( T^{4} - 152 T^{3} + \cdots + 1659084581 \) Copy content Toggle raw display
$43$ \( T^{4} + 325 T^{3} + \cdots - 1288748736 \) Copy content Toggle raw display
$47$ \( T^{4} + 869 T^{3} + \cdots + 837687536 \) Copy content Toggle raw display
$53$ \( T^{4} - 811 T^{3} + \cdots - 847714576 \) Copy content Toggle raw display
$59$ \( T^{4} - 178 T^{3} + \cdots + 258148219 \) Copy content Toggle raw display
$61$ \( T^{4} + 105 T^{3} + \cdots + 47885889744 \) Copy content Toggle raw display
$67$ \( T^{4} - 43 T^{3} + \cdots - 8869996224 \) Copy content Toggle raw display
$71$ \( T^{4} - 629 T^{3} + \cdots - 744521796 \) Copy content Toggle raw display
$73$ \( T^{4} + 270 T^{3} + \cdots + 38050128809 \) Copy content Toggle raw display
$79$ \( T^{4} - 977 T^{3} + \cdots - 210591964 \) Copy content Toggle raw display
$83$ \( T^{4} - 1686 T^{3} + \cdots - 181259356509 \) Copy content Toggle raw display
$89$ \( T^{4} - 1891 T^{3} + \cdots - 2046678844 \) Copy content Toggle raw display
$97$ \( T^{4} + 1772 T^{3} + \cdots - 357121332099 \) Copy content Toggle raw display
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