Properties

Label 121.4.a.c
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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Newspace parameters

Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 121.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.13923111069\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + ( -1 + 4 \beta ) q^{3} + ( -4 - 2 \beta ) q^{4} + ( 1 - 8 \beta ) q^{5} + ( 13 - 5 \beta ) q^{6} + ( -10 - 4 \beta ) q^{7} + ( 6 - 10 \beta ) q^{8} + ( 22 - 8 \beta ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + ( -1 + 4 \beta ) q^{3} + ( -4 - 2 \beta ) q^{4} + ( 1 - 8 \beta ) q^{5} + ( 13 - 5 \beta ) q^{6} + ( -10 - 4 \beta ) q^{7} + ( 6 - 10 \beta ) q^{8} + ( 22 - 8 \beta ) q^{9} + ( -25 + 9 \beta ) q^{10} + ( -20 - 14 \beta ) q^{12} + ( -40 - 20 \beta ) q^{13} + ( -2 - 6 \beta ) q^{14} + ( -97 + 12 \beta ) q^{15} + ( -4 + 32 \beta ) q^{16} + ( 62 + 12 \beta ) q^{17} + ( -46 + 30 \beta ) q^{18} + ( -36 + 60 \beta ) q^{19} + ( 44 + 30 \beta ) q^{20} + ( -38 - 36 \beta ) q^{21} + ( -49 + 36 \beta ) q^{23} + ( -126 + 34 \beta ) q^{24} + ( 68 - 16 \beta ) q^{25} + ( -20 - 20 \beta ) q^{26} + ( -91 - 12 \beta ) q^{27} + ( 64 + 36 \beta ) q^{28} + ( -72 - 56 \beta ) q^{29} + ( 133 - 109 \beta ) q^{30} + ( -17 - 28 \beta ) q^{31} + ( 52 + 44 \beta ) q^{32} + ( -26 + 50 \beta ) q^{34} + ( 86 + 76 \beta ) q^{35} + ( -40 - 12 \beta ) q^{36} + ( 27 + 8 \beta ) q^{37} + ( 216 - 96 \beta ) q^{38} + ( -200 - 140 \beta ) q^{39} + ( 246 - 58 \beta ) q^{40} + ( -268 - 4 \beta ) q^{41} + ( -70 - 2 \beta ) q^{42} + ( 30 - 16 \beta ) q^{43} + ( 214 - 184 \beta ) q^{45} + ( 157 - 85 \beta ) q^{46} + ( -136 + 120 \beta ) q^{47} + ( 388 - 48 \beta ) q^{48} + ( -195 + 80 \beta ) q^{49} + ( -116 + 84 \beta ) q^{50} + ( 82 + 236 \beta ) q^{51} + ( 280 + 160 \beta ) q^{52} + ( -246 + 56 \beta ) q^{53} + ( 55 - 79 \beta ) q^{54} + ( 60 + 76 \beta ) q^{56} + ( 756 - 204 \beta ) q^{57} + ( -96 - 16 \beta ) q^{58} + ( 317 + 132 \beta ) q^{59} + ( 316 + 146 \beta ) q^{60} + ( -420 + 184 \beta ) q^{61} + ( -67 + 11 \beta ) q^{62} + ( -124 - 8 \beta ) q^{63} + ( 112 - 248 \beta ) q^{64} + ( 440 + 300 \beta ) q^{65} + ( 377 + 20 \beta ) q^{67} + ( -320 - 172 \beta ) q^{68} + ( 481 - 232 \beta ) q^{69} + ( 142 + 10 \beta ) q^{70} + ( -339 - 76 \beta ) q^{71} + ( 372 - 268 \beta ) q^{72} + ( 200 - 468 \beta ) q^{73} + ( -3 + 19 \beta ) q^{74} + ( -260 + 288 \beta ) q^{75} + ( -216 - 168 \beta ) q^{76} + ( -220 - 60 \beta ) q^{78} + ( -158 + 656 \beta ) q^{79} + ( -772 + 64 \beta ) q^{80} + ( -647 - 136 \beta ) q^{81} + ( 256 - 264 \beta ) q^{82} + ( -234 + 120 \beta ) q^{83} + ( 368 + 220 \beta ) q^{84} + ( -226 - 484 \beta ) q^{85} + ( -78 + 46 \beta ) q^{86} + ( -600 - 232 \beta ) q^{87} + ( -921 + 328 \beta ) q^{89} + ( -766 + 398 \beta ) q^{90} + ( 640 + 360 \beta ) q^{91} + ( -20 - 46 \beta ) q^{92} + ( -319 - 40 \beta ) q^{93} + ( 496 - 256 \beta ) q^{94} + ( -1476 + 348 \beta ) q^{95} + ( 476 + 164 \beta ) q^{96} + ( 1097 - 144 \beta ) q^{97} + ( 435 - 275 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} - 8 q^{4} + 2 q^{5} + 26 q^{6} - 20 q^{7} + 12 q^{8} + 44 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} - 2 q^{3} - 8 q^{4} + 2 q^{5} + 26 q^{6} - 20 q^{7} + 12 q^{8} + 44 q^{9} - 50 q^{10} - 40 q^{12} - 80 q^{13} - 4 q^{14} - 194 q^{15} - 8 q^{16} + 124 q^{17} - 92 q^{18} - 72 q^{19} + 88 q^{20} - 76 q^{21} - 98 q^{23} - 252 q^{24} + 136 q^{25} - 40 q^{26} - 182 q^{27} + 128 q^{28} - 144 q^{29} + 266 q^{30} - 34 q^{31} + 104 q^{32} - 52 q^{34} + 172 q^{35} - 80 q^{36} + 54 q^{37} + 432 q^{38} - 400 q^{39} + 492 q^{40} - 536 q^{41} - 140 q^{42} + 60 q^{43} + 428 q^{45} + 314 q^{46} - 272 q^{47} + 776 q^{48} - 390 q^{49} - 232 q^{50} + 164 q^{51} + 560 q^{52} - 492 q^{53} + 110 q^{54} + 120 q^{56} + 1512 q^{57} - 192 q^{58} + 634 q^{59} + 632 q^{60} - 840 q^{61} - 134 q^{62} - 248 q^{63} + 224 q^{64} + 880 q^{65} + 754 q^{67} - 640 q^{68} + 962 q^{69} + 284 q^{70} - 678 q^{71} + 744 q^{72} + 400 q^{73} - 6 q^{74} - 520 q^{75} - 432 q^{76} - 440 q^{78} - 316 q^{79} - 1544 q^{80} - 1294 q^{81} + 512 q^{82} - 468 q^{83} + 736 q^{84} - 452 q^{85} - 156 q^{86} - 1200 q^{87} - 1842 q^{89} - 1532 q^{90} + 1280 q^{91} - 40 q^{92} - 638 q^{93} + 992 q^{94} - 2952 q^{95} + 952 q^{96} + 2194 q^{97} + 870 q^{98} + O(q^{100}) \)

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.

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
−2.73205 −7.92820 −0.535898 14.8564 21.6603 −3.07180 23.3205 35.8564 −40.5885
1.2 0.732051 5.92820 −7.46410 −12.8564 4.33975 −16.9282 −11.3205 8.14359 −9.41154
\(n\): 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.c 2
3.b odd 2 1 1089.4.a.v 2
4.b odd 2 1 1936.4.a.w 2
11.b odd 2 1 11.4.a.a 2
11.c even 5 4 121.4.c.f 8
11.d odd 10 4 121.4.c.c 8
33.d even 2 1 99.4.a.c 2
44.c even 2 1 176.4.a.i 2
55.d odd 2 1 275.4.a.b 2
55.e even 4 2 275.4.b.c 4
77.b even 2 1 539.4.a.e 2
88.b odd 2 1 704.4.a.p 2
88.g even 2 1 704.4.a.n 2
132.d odd 2 1 1584.4.a.bc 2
143.d odd 2 1 1859.4.a.a 2
165.d even 2 1 2475.4.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 11.b odd 2 1
99.4.a.c 2 33.d even 2 1
121.4.a.c 2 1.a even 1 1 trivial
121.4.c.c 8 11.d odd 10 4
121.4.c.f 8 11.c even 5 4
176.4.a.i 2 44.c even 2 1
275.4.a.b 2 55.d odd 2 1
275.4.b.c 4 55.e even 4 2
539.4.a.e 2 77.b even 2 1
704.4.a.n 2 88.g even 2 1
704.4.a.p 2 88.b odd 2 1
1089.4.a.v 2 3.b odd 2 1
1584.4.a.bc 2 132.d odd 2 1
1859.4.a.a 2 143.d odd 2 1
1936.4.a.w 2 4.b odd 2 1
2475.4.a.q 2 165.d even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + 2 T + T^{2} \)
$3$ \( -47 + 2 T + T^{2} \)
$5$ \( -191 - 2 T + T^{2} \)
$7$ \( 52 + 20 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 400 + 80 T + T^{2} \)
$17$ \( 3412 - 124 T + T^{2} \)
$19$ \( -9504 + 72 T + T^{2} \)
$23$ \( -1487 + 98 T + T^{2} \)
$29$ \( -4224 + 144 T + T^{2} \)
$31$ \( -2063 + 34 T + T^{2} \)
$37$ \( 537 - 54 T + T^{2} \)
$41$ \( 71776 + 536 T + T^{2} \)
$43$ \( 132 - 60 T + T^{2} \)
$47$ \( -24704 + 272 T + T^{2} \)
$53$ \( 51108 + 492 T + T^{2} \)
$59$ \( 48217 - 634 T + T^{2} \)
$61$ \( 74832 + 840 T + T^{2} \)
$67$ \( 140929 - 754 T + T^{2} \)
$71$ \( 97593 + 678 T + T^{2} \)
$73$ \( -617072 - 400 T + T^{2} \)
$79$ \( -1266044 + 316 T + T^{2} \)
$83$ \( 11556 + 468 T + T^{2} \)
$89$ \( 525489 + 1842 T + T^{2} \)
$97$ \( 1141201 - 2194 T + T^{2} \)
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