Properties

Label 11.4.a.a
Level 11
Weight 4
Character orbit 11.a
Self dual yes
Analytic conductor 0.649
Analytic rank 0
Dimension 2
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 11.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.649021010063\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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} -11 q^{11} + ( -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} + ( -11 - 11 \beta ) q^{22} + ( -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} + ( 11 + 44 \beta ) q^{33} + ( -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} + ( 44 - 22 \beta ) q^{44} + ( 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} + ( -11 - 88 \beta ) q^{55} + ( 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} + ( 143 + 55 \beta ) q^{66} + ( 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} + ( -110 + 44 \beta ) q^{77} + ( -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} + ( 66 + 110 \beta ) q^{88} + ( -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} + ( -242 - 88 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} - 8q^{4} + 2q^{5} - 26q^{6} + 20q^{7} - 12q^{8} + 44q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} - 8q^{4} + 2q^{5} - 26q^{6} + 20q^{7} - 12q^{8} + 44q^{9} + 50q^{10} - 22q^{11} - 40q^{12} + 80q^{13} - 4q^{14} - 194q^{15} - 8q^{16} - 124q^{17} + 92q^{18} + 72q^{19} + 88q^{20} + 76q^{21} - 22q^{22} - 98q^{23} + 252q^{24} + 136q^{25} - 40q^{26} - 182q^{27} - 128q^{28} + 144q^{29} - 266q^{30} - 34q^{31} - 104q^{32} + 22q^{33} - 52q^{34} - 172q^{35} - 80q^{36} + 54q^{37} + 432q^{38} + 400q^{39} - 492q^{40} + 536q^{41} - 140q^{42} - 60q^{43} + 88q^{44} + 428q^{45} - 314q^{46} - 272q^{47} + 776q^{48} - 390q^{49} + 232q^{50} - 164q^{51} - 560q^{52} - 492q^{53} - 110q^{54} - 22q^{55} + 120q^{56} - 1512q^{57} - 192q^{58} + 634q^{59} + 632q^{60} + 840q^{61} + 134q^{62} + 248q^{63} + 224q^{64} - 880q^{65} + 286q^{66} + 754q^{67} + 640q^{68} + 962q^{69} + 284q^{70} - 678q^{71} - 744q^{72} - 400q^{73} + 6q^{74} - 520q^{75} + 432q^{76} - 220q^{77} - 440q^{78} + 316q^{79} - 1544q^{80} - 1294q^{81} + 512q^{82} + 468q^{83} - 736q^{84} + 452q^{85} - 156q^{86} + 1200q^{87} + 132q^{88} - 1842q^{89} + 1532q^{90} + 1280q^{91} - 40q^{92} - 638q^{93} - 992q^{94} + 2952q^{95} - 952q^{96} + 2194q^{97} - 870q^{98} - 484q^{99} + O(q^{100}) \)

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
−0.732051 5.92820 −7.46410 −12.8564 −4.33975 16.9282 11.3205 8.14359 9.41154
1.2 2.73205 −7.92820 −0.535898 14.8564 −21.6603 3.07180 −23.3205 35.8564 40.5885
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 11.4.a.a 2
3.b odd 2 1 99.4.a.c 2
4.b odd 2 1 176.4.a.i 2
5.b even 2 1 275.4.a.b 2
5.c odd 4 2 275.4.b.c 4
7.b odd 2 1 539.4.a.e 2
8.b even 2 1 704.4.a.p 2
8.d odd 2 1 704.4.a.n 2
11.b odd 2 1 121.4.a.c 2
11.c even 5 4 121.4.c.c 8
11.d odd 10 4 121.4.c.f 8
12.b even 2 1 1584.4.a.bc 2
13.b even 2 1 1859.4.a.a 2
15.d odd 2 1 2475.4.a.q 2
33.d even 2 1 1089.4.a.v 2
44.c even 2 1 1936.4.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 1.a even 1 1 trivial
99.4.a.c 2 3.b odd 2 1
121.4.a.c 2 11.b odd 2 1
121.4.c.c 8 11.c even 5 4
121.4.c.f 8 11.d odd 10 4
176.4.a.i 2 4.b odd 2 1
275.4.a.b 2 5.b even 2 1
275.4.b.c 4 5.c odd 4 2
539.4.a.e 2 7.b odd 2 1
704.4.a.n 2 8.d odd 2 1
704.4.a.p 2 8.b even 2 1
1089.4.a.v 2 33.d even 2 1
1584.4.a.bc 2 12.b even 2 1
1859.4.a.a 2 13.b even 2 1
1936.4.a.w 2 44.c even 2 1
2475.4.a.q 2 15.d odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(\Gamma_0(11))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 14 T^{2} - 16 T^{3} + 64 T^{4} \)
$3$ \( 1 + 2 T + 7 T^{2} + 54 T^{3} + 729 T^{4} \)
$5$ \( 1 - 2 T + 59 T^{2} - 250 T^{3} + 15625 T^{4} \)
$7$ \( 1 - 20 T + 738 T^{2} - 6860 T^{3} + 117649 T^{4} \)
$11$ \( ( 1 + 11 T )^{2} \)
$13$ \( 1 - 80 T + 4794 T^{2} - 175760 T^{3} + 4826809 T^{4} \)
$17$ \( 1 + 124 T + 13238 T^{2} + 609212 T^{3} + 24137569 T^{4} \)
$19$ \( 1 - 72 T + 4214 T^{2} - 493848 T^{3} + 47045881 T^{4} \)
$23$ \( 1 + 98 T + 22847 T^{2} + 1192366 T^{3} + 148035889 T^{4} \)
$29$ \( 1 - 144 T + 44554 T^{2} - 3512016 T^{3} + 594823321 T^{4} \)
$31$ \( 1 + 34 T + 57519 T^{2} + 1012894 T^{3} + 887503681 T^{4} \)
$37$ \( 1 - 54 T + 101843 T^{2} - 2735262 T^{3} + 2565726409 T^{4} \)
$41$ \( 1 - 536 T + 209618 T^{2} - 36941656 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 + 60 T + 159146 T^{2} + 4770420 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 + 272 T + 182942 T^{2} + 28239856 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 + 492 T + 348862 T^{2} + 73247484 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 - 634 T + 458975 T^{2} - 130210286 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 - 840 T + 528794 T^{2} - 190664040 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 - 754 T + 742455 T^{2} - 226775302 T^{3} + 90458382169 T^{4} \)
$71$ \( 1 + 678 T + 813415 T^{2} + 242663658 T^{3} + 128100283921 T^{4} \)
$73$ \( 1 + 400 T + 160962 T^{2} + 155606800 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 - 316 T - 279966 T^{2} - 155800324 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 - 468 T + 1155130 T^{2} - 267596316 T^{3} + 326940373369 T^{4} \)
$89$ \( 1 + 1842 T + 1935427 T^{2} + 1298552898 T^{3} + 496981290961 T^{4} \)
$97$ \( 1 - 2194 T + 2966547 T^{2} - 2002404562 T^{3} + 832972004929 T^{4} \)
show more
show less