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 $1$

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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: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + ( - 4 \beta - 1) q^{3} + (2 \beta - 4) q^{4} + (8 \beta + 1) q^{5} + ( - 5 \beta - 13) q^{6} + ( - 4 \beta + 10) q^{7} + ( - 10 \beta - 6) q^{8} + (8 \beta + 22) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + ( - 4 \beta - 1) q^{3} + (2 \beta - 4) q^{4} + (8 \beta + 1) q^{5} + ( - 5 \beta - 13) q^{6} + ( - 4 \beta + 10) q^{7} + ( - 10 \beta - 6) q^{8} + (8 \beta + 22) q^{9} + (9 \beta + 25) q^{10} - 11 q^{11} + (14 \beta - 20) q^{12} + ( - 20 \beta + 40) q^{13} + (6 \beta - 2) q^{14} + ( - 12 \beta - 97) q^{15} + ( - 32 \beta - 4) q^{16} + (12 \beta - 62) q^{17} + (30 \beta + 46) q^{18} + (60 \beta + 36) q^{19} + ( - 30 \beta + 44) q^{20} + ( - 36 \beta + 38) q^{21} + ( - 11 \beta - 11) q^{22} + ( - 36 \beta - 49) q^{23} + (34 \beta + 126) q^{24} + (16 \beta + 68) q^{25} + (20 \beta - 20) q^{26} + (12 \beta - 91) q^{27} + (36 \beta - 64) q^{28} + ( - 56 \beta + 72) q^{29} + ( - 109 \beta - 133) q^{30} + (28 \beta - 17) q^{31} + (44 \beta - 52) q^{32} + (44 \beta + 11) q^{33} + ( - 50 \beta - 26) q^{34} + (76 \beta - 86) q^{35} + (12 \beta - 40) q^{36} + ( - 8 \beta + 27) q^{37} + (96 \beta + 216) q^{38} + ( - 140 \beta + 200) q^{39} + ( - 58 \beta - 246) q^{40} + ( - 4 \beta + 268) q^{41} + (2 \beta - 70) q^{42} + ( - 16 \beta - 30) q^{43} + ( - 22 \beta + 44) q^{44} + (184 \beta + 214) q^{45} + ( - 85 \beta - 157) q^{46} + ( - 120 \beta - 136) q^{47} + (48 \beta + 388) q^{48} + ( - 80 \beta - 195) q^{49} + (84 \beta + 116) q^{50} + (236 \beta - 82) q^{51} + (160 \beta - 280) q^{52} + ( - 56 \beta - 246) q^{53} + ( - 79 \beta - 55) q^{54} + ( - 88 \beta - 11) q^{55} + ( - 76 \beta + 60) q^{56} + ( - 204 \beta - 756) q^{57} + (16 \beta - 96) q^{58} + ( - 132 \beta + 317) q^{59} + ( - 146 \beta + 316) q^{60} + (184 \beta + 420) q^{61} + (11 \beta + 67) q^{62} + ( - 8 \beta + 124) q^{63} + (248 \beta + 112) q^{64} + (300 \beta - 440) q^{65} + (55 \beta + 143) q^{66} + ( - 20 \beta + 377) q^{67} + ( - 172 \beta + 320) q^{68} + (232 \beta + 481) q^{69} + ( - 10 \beta + 142) q^{70} + (76 \beta - 339) q^{71} + ( - 268 \beta - 372) q^{72} + ( - 468 \beta - 200) q^{73} + (19 \beta + 3) q^{74} + ( - 288 \beta - 260) q^{75} + ( - 168 \beta + 216) q^{76} + (44 \beta - 110) q^{77} + (60 \beta - 220) q^{78} + (656 \beta + 158) q^{79} + ( - 64 \beta - 772) q^{80} + (136 \beta - 647) q^{81} + (264 \beta + 256) q^{82} + (120 \beta + 234) q^{83} + (220 \beta - 368) q^{84} + ( - 484 \beta + 226) q^{85} + ( - 46 \beta - 78) q^{86} + ( - 232 \beta + 600) q^{87} + (110 \beta + 66) q^{88} + ( - 328 \beta - 921) q^{89} + (398 \beta + 766) q^{90} + ( - 360 \beta + 640) q^{91} + (46 \beta - 20) q^{92} + (40 \beta - 319) q^{93} + ( - 256 \beta - 496) q^{94} + (348 \beta + 1476) q^{95} + (164 \beta - 476) q^{96} + (144 \beta + 1097) q^{97} + ( - 275 \beta - 435) q^{98} + ( - 88 \beta - 242) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} - 8 q^{4} + 2 q^{5} - 26 q^{6} + 20 q^{7} - 12 q^{8} + 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} - 8 q^{4} + 2 q^{5} - 26 q^{6} + 20 q^{7} - 12 q^{8} + 44 q^{9} + 50 q^{10} - 22 q^{11} - 40 q^{12} + 80 q^{13} - 4 q^{14} - 194 q^{15} - 8 q^{16} - 124 q^{17} + 92 q^{18} + 72 q^{19} + 88 q^{20} + 76 q^{21} - 22 q^{22} - 98 q^{23} + 252 q^{24} + 136 q^{25} - 40 q^{26} - 182 q^{27} - 128 q^{28} + 144 q^{29} - 266 q^{30} - 34 q^{31} - 104 q^{32} + 22 q^{33} - 52 q^{34} - 172 q^{35} - 80 q^{36} + 54 q^{37} + 432 q^{38} + 400 q^{39} - 492 q^{40} + 536 q^{41} - 140 q^{42} - 60 q^{43} + 88 q^{44} + 428 q^{45} - 314 q^{46} - 272 q^{47} + 776 q^{48} - 390 q^{49} + 232 q^{50} - 164 q^{51} - 560 q^{52} - 492 q^{53} - 110 q^{54} - 22 q^{55} + 120 q^{56} - 1512 q^{57} - 192 q^{58} + 634 q^{59} + 632 q^{60} + 840 q^{61} + 134 q^{62} + 248 q^{63} + 224 q^{64} - 880 q^{65} + 286 q^{66} + 754 q^{67} + 640 q^{68} + 962 q^{69} + 284 q^{70} - 678 q^{71} - 744 q^{72} - 400 q^{73} + 6 q^{74} - 520 q^{75} + 432 q^{76} - 220 q^{77} - 440 q^{78} + 316 q^{79} - 1544 q^{80} - 1294 q^{81} + 512 q^{82} + 468 q^{83} - 736 q^{84} + 452 q^{85} - 156 q^{86} + 1200 q^{87} + 132 q^{88} - 1842 q^{89} + 1532 q^{90} + 1280 q^{91} - 40 q^{92} - 638 q^{93} - 992 q^{94} + 2952 q^{95} - 952 q^{96} + 2194 q^{97} - 870 q^{98} - 484 q^{99}+O(q^{100}) \) 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.

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:

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 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

Hecke kernels

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

Hecke characteristic polynomials

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