Properties

Label 121.4.a.f
Level $121$
Weight $4$
Character orbit 121.a
Self dual yes
Analytic conductor $7.139$
Analytic rank $1$
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: \(1\)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1 - 1) q^{2} + ( - \beta_{2} + 1) q^{3} + (2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 3) q^{4} + ( - 3 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{5} + ( - \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 10) q^{6} + (3 \beta_{3} + 2 \beta_{2} - 6 \beta_1 - 3) q^{7} + ( - 6 \beta_{3} + 3 \beta_{2} + 19 \beta_1 - 23) q^{8} + ( - 3 \beta_{2} - 17) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1 - 1) q^{2} + ( - \beta_{2} + 1) q^{3} + (2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 3) q^{4} + ( - 3 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{5} + ( - \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 10) q^{6} + (3 \beta_{3} + 2 \beta_{2} - 6 \beta_1 - 3) q^{7} + ( - 6 \beta_{3} + 3 \beta_{2} + 19 \beta_1 - 23) q^{8} + ( - 3 \beta_{2} - 17) q^{9} + ( - 6 \beta_{2} - 28 \beta_1 + 6) q^{10} + (5 \beta_{3} - 9 \beta_{2} - 24 \beta_1 + 30) q^{12} + (7 \beta_{3} - 11 \beta_{2} + 7 \beta_1 - 17) q^{13} + ( - 4 \beta_{3} - 4 \beta_{2} + 28 \beta_1 + 18) q^{14} + ( - 2 \beta_{3} + 3 \beta_{2} + 25 \beta_1 - 10) q^{15} + (6 \beta_{3} - 11 \beta_{2} - 75 \beta_1 + 39) q^{16} + (2 \beta_{3} - 9 \beta_{2} - 18 \beta_1 - 54) q^{17} + ( - 3 \beta_{3} - 11 \beta_{2} - 14 \beta_1 - 10) q^{18} + (12 \beta_{3} - 6 \beta_{2} + 39 \beta_1 - 64) q^{19} + ( - 10 \beta_{3} + 10 \beta_{2} + 48 \beta_1 - 80) q^{20} + (9 \beta_{3} + 7 \beta_{2} - 39 \beta_1 - 21) q^{21} + ( - 10 \beta_{3} + 36 \beta_{2} + 26 \beta_1 + 6) q^{23} + ( - 25 \beta_{3} + 29 \beta_{2} + 92 \beta_1 - 50) q^{24} + (19 \beta_{3} + 12 \beta_{2} + 32 \beta_1 - 18) q^{25} + ( - 4 \beta_{3} + 12 \beta_{2} + 50 \beta_1 - 68) q^{26} + (38 \beta_{2} - 17) q^{27} + (6 \beta_{2} + 6 \beta_1 - 6) q^{28} + ( - 31 \beta_{3} + 32 \beta_{2} + 12 \beta_1 - 73) q^{29} + (28 \beta_{3} - 18 \beta_{2} - 56 \beta_1 + 60) q^{30} + (41 \beta_{3} - \beta_{2} - 89 \beta_1 + 43) q^{31} + ( - 38 \beta_{3} + 43 \beta_{2} + 27 \beta_1 - 23) q^{32} + ( - 27 \beta_{3} - 34 \beta_{2} - 9 \beta_1 - 43) q^{34} + (4 \beta_{3} - \beta_{2} - 85 \beta_1 - 48) q^{35} + ( - 25 \beta_{3} + 33 \beta_{2} - 12 \beta_1 + 30) q^{36} + ( - 35 \beta_{3} + 18 \beta_{2} - 8 \beta_1 + 45) q^{37} + (33 \beta_{3} - 40 \beta_{2} + 11 \beta_1 + 61) q^{38} + ( - 5 \beta_{2} - 49 \beta_1 + 82) q^{39} + (58 \beta_{3} - 62 \beta_{2} - 4 \beta_1 + 160) q^{40} + ( - 62 \beta_{3} + 47 \beta_{2} + 40 \beta_1 - 19) q^{41} + ( - 32 \beta_{3} - 26 \beta_{2} + 92 \beta_1 + 54) q^{42} + (31 \beta_{3} - 39 \beta_{2} + 168 \beta_1 - 30) q^{43} + (54 \beta_{3} - 11 \beta_{2} + 95 \beta_1 - 10) q^{45} + (62 \beta_{3} - 76 \beta_{2} - 146 \beta_1 + 334) q^{46} + ( - 9 \beta_{3} - 28 \beta_{2} - 80 \beta_1 - 189) q^{47} + (81 \beta_{3} - 61 \beta_{2} - 204 \beta_1 + 138) q^{48} + ( - 69 \beta_{3} - 43 \beta_{2} + 285 \beta_1 - 208) q^{49} + (44 \beta_{3} - 23 \beta_{2} + 109 \beta_1 + 177) q^{50} + (20 \beta_{3} + 36 \beta_{2} - 54 \beta_1 + 27) q^{51} + (6 \beta_{3} - 8 \beta_{2} - 222 \beta_1 + 358) q^{52} + ( - 41 \beta_{3} + 39 \beta_{2} - 197 \beta_1 + 331) q^{53} + (38 \beta_{3} - 93 \beta_{2} - 55 \beta_1 + 359) q^{54} + (44 \beta_{3} + 14 \beta_{2} - 242 \beta_1 - 78) q^{56} + ( - 27 \beta_{3} + 52 \beta_{2} - 30 \beta_1 - 10) q^{57} + (44 \beta_{3} - 168 \beta_{2} - 396 \beta_1 + 342) q^{58} + (32 \beta_{3} - 34 \beta_{2} - 169 \beta_1 + 104) q^{59} + ( - 58 \beta_{3} + 100 \beta_{2} + 186 \beta_1 - 170) q^{60} + ( - 147 \beta_{3} + 105 \beta_{2} + 189 \beta_1 + 21) q^{61} + ( - 90 \beta_{3} + 86 \beta_{2} + 502 \beta_1 - 100) q^{62} + ( - 33 \beta_{3} - 19 \beta_{2} + 3 \beta_1 - 3) q^{63} + (22 \beta_{3} - 59 \beta_{2} + 165 \beta_1 + 87) q^{64} + (13 \beta_{3} - 44 \beta_{2} + 156 \beta_1 - 327) q^{65} + ( - 63 \beta_{3} - 46 \beta_{2} + 309 \beta_1 - 151) q^{67} + ( - 59 \beta_{3} + 70 \beta_{2} - 99 \beta_1 + 133) q^{68} + ( - 36 \beta_{3} + 66 \beta_{2} + 142 \beta_1 - 318) q^{69} + ( - 86 \beta_{3} - 42 \beta_{2} + 74 \beta_1 - 42) q^{70} + (77 \beta_{3} + 43 \beta_{2} + 253 \beta_1 + 33) q^{71} + (45 \beta_{3} + 27 \beta_{2} - 104 \beta_1 + 310) q^{72} + (44 \beta_{3} - 95 \beta_{2} - 480 \beta_1 + 249) q^{73} + (10 \beta_{3} - 26 \beta_{2} - 280 \beta_1 + 74) q^{74} + ( - 13 \beta_{3} + 42 \beta_{2} - 107 \beta_1 - 126) q^{75} + ( - 125 \beta_{3} + 222 \beta_{2} + 75 \beta_1 + 135) q^{76} + ( - 54 \beta_{3} + 92 \beta_{2} + 136 \beta_1 - 176) q^{78} + ( - 13 \beta_{3} - 115 \beta_{2} + 81 \beta_1 - 339) q^{79} + (14 \beta_{3} + 262 \beta_{2} + 364 \beta_1 - 24) q^{80} + (174 \beta_{2} + 100) q^{81} + (87 \beta_{3} - 175 \beta_{2} - 664 \beta_1 + 420) q^{82} + (208 \beta_{3} - 194 \beta_{2} - 283 \beta_1 - 429) q^{83} + ( - 6 \beta_{3} + 18 \beta_{2} + 12 \beta_1 - 60) q^{84} + (197 \beta_{3} + 10 \beta_{2} + 306 \beta_1 - 17) q^{85} + (129 \beta_{3} + 79 \beta_{2} + 120 \beta_1 - 122) q^{86} + ( - 43 \beta_{3} + 137 \beta_{2} + 303 \beta_1 - 361) q^{87} + (25 \beta_{3} - 57 \beta_{2} - 428 \beta_1 + 652) q^{89} + (84 \beta_{3} + 66 \beta_{2} + 392 \beta_1 + 60) q^{90} + (8 \beta_{3} + 21 \beta_{2} - 23 \beta_1) q^{91} + ( - 142 \beta_{3} + 260 \beta_{2} + 906 \beta_1 - 1150) q^{92} + (130 \beta_{3} - 45 \beta_{2} - 547 \beta_1 + 52) q^{93} + ( - 108 \beta_{3} - 142 \beta_{2} - 162 \beta_1 - 152) q^{94} + (60 \beta_{3} - 217 \beta_{2} - 113 \beta_1 - 518) q^{95} + ( - 65 \beta_{3} + 109 \beta_{2} + 396 \beta_1 - 410) q^{96} + (50 \beta_{3} - 194 \beta_{2} - 471 \beta_1 - 317) q^{97} + (242 \beta_{3} - 191 \beta_{2} - 1071 \beta_1 + 37) 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
−5.15942 4.54138 18.6196 −8.63533 −23.4309 −1.66258 −54.7907 −6.37586 44.5532
1.2 −2.92335 4.54138 0.545959 6.17671 −13.2760 −32.1271 21.7907 −6.37586 −18.0567
1.3 0.923347 −1.54138 −7.14743 8.72550 −1.42323 −0.775116 −13.9863 −24.6241 8.05666
1.4 3.15942 −1.54138 1.98190 −17.2669 −4.86986 9.56478 −19.0137 −24.6241 −54.5532
\(n\): 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.f 4
3.b odd 2 1 1089.4.a.bh 4
4.b odd 2 1 1936.4.a.bl 4
11.b odd 2 1 121.4.a.g 4
11.c even 5 2 121.4.c.b 8
11.c even 5 2 121.4.c.i 8
11.d odd 10 2 11.4.c.a 8
11.d odd 10 2 121.4.c.h 8
33.d even 2 1 1089.4.a.y 4
33.f even 10 2 99.4.f.c 8
44.c even 2 1 1936.4.a.bk 4
44.g even 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.d odd 10 2
99.4.f.c 8 33.f even 10 2
121.4.a.f 4 1.a even 1 1 trivial
121.4.a.g 4 11.b odd 2 1
121.4.c.b 8 11.c even 5 2
121.4.c.h 8 11.d odd 10 2
121.4.c.i 8 11.c even 5 2
176.4.m.c 8 44.g even 10 2
1089.4.a.y 4 33.d even 2 1
1089.4.a.bh 4 3.b odd 2 1
1936.4.a.bk 4 44.c even 2 1
1936.4.a.bl 4 4.b odd 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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