Properties

Label 1441.4.a.b
Level $1441$
Weight $4$
Character orbit 1441.a
Self dual yes
Analytic conductor $85.022$
Analytic rank $1$
Dimension $79$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1441 = 11 \cdot 131 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(85.0217523183\)
Analytic rank: \(1\)
Dimension: \(79\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 79q - 20q^{2} - 12q^{3} + 288q^{4} - 40q^{5} - 111q^{6} - 101q^{7} - 258q^{8} + 585q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 79q - 20q^{2} - 12q^{3} + 288q^{4} - 40q^{5} - 111q^{6} - 101q^{7} - 258q^{8} + 585q^{9} - 178q^{10} + 869q^{11} - 144q^{12} - 242q^{13} - 342q^{14} - 524q^{15} + 928q^{16} - 260q^{17} - 611q^{18} - 543q^{19} - 578q^{20} - 710q^{21} - 220q^{22} - 908q^{23} - 1322q^{24} + 1701q^{25} - 844q^{26} - 732q^{27} - 1068q^{28} - 1747q^{29} - 973q^{30} - 1248q^{31} - 2069q^{32} - 132q^{33} - 76q^{34} - 1630q^{35} + 2155q^{36} - 535q^{37} + 1155q^{38} - 2514q^{39} - 298q^{40} - 2087q^{41} - 5q^{42} - 1008q^{43} + 3168q^{44} - 1160q^{45} - 1640q^{46} - 1960q^{47} + 3412q^{48} + 3670q^{49} - 2394q^{50} - 2994q^{51} - 2601q^{52} - 2466q^{53} + 1296q^{54} - 440q^{55} - 5195q^{56} - 3776q^{57} + 1068q^{58} - 2310q^{59} + 1599q^{60} - 3404q^{61} + 1534q^{62} - 3409q^{63} + 2568q^{64} - 3906q^{65} - 1221q^{66} - 2405q^{67} - 3145q^{68} - 2420q^{69} + 455q^{70} - 8978q^{71} - 7262q^{72} - 1868q^{73} - 2790q^{74} - 1196q^{75} - 5483q^{76} - 1111q^{77} + 349q^{78} - 9130q^{79} - 1697q^{80} + 4171q^{81} - 241q^{82} - 4639q^{83} - 1659q^{84} - 7634q^{85} - 5656q^{86} - 4412q^{87} - 2838q^{88} - 6561q^{89} - 6756q^{90} - 2742q^{91} - 5386q^{92} - 3234q^{93} - 5295q^{94} - 7930q^{95} - 12593q^{96} - 4520q^{97} - 3213q^{98} + 6435q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.60071 5.77825 23.3679 −13.4058 −32.3623 25.8324 −86.0714 6.38823 75.0821
1.2 −5.43622 10.0127 21.5525 2.58286 −54.4312 −19.3873 −73.6746 73.2537 −14.0410
1.3 −5.40814 5.08406 21.2480 19.9946 −27.4953 26.0035 −71.6468 −1.15237 −108.134
1.4 −5.36465 −2.69161 20.7795 −8.89356 14.4395 −3.71000 −68.5575 −19.7553 47.7108
1.5 −5.15789 −7.60590 18.6038 −18.9226 39.2304 23.8426 −54.6935 30.8496 97.6006
1.6 −5.03898 −3.26806 17.3913 11.8693 16.4677 −1.74461 −47.3228 −16.3198 −59.8090
1.7 −5.00956 6.33575 17.0957 3.89312 −31.7393 −28.7253 −45.5654 13.1417 −19.5028
1.8 −4.92029 −10.0554 16.2093 5.77408 49.4755 −6.22578 −40.3920 74.1112 −28.4102
1.9 −4.82837 −4.23811 15.3131 15.4265 20.4632 −6.47406 −35.3105 −9.03840 −74.4849
1.10 −4.81294 2.88277 15.1644 −12.4285 −13.8746 −12.7070 −34.4818 −18.6896 59.8178
1.11 −4.74082 −0.0634272 14.4754 −1.00948 0.300697 23.4744 −30.6988 −26.9960 4.78575
1.12 −4.36838 −5.97359 11.0827 −7.42634 26.0949 −29.1330 −13.4664 8.68373 32.4411
1.13 −4.33328 7.87388 10.7773 11.6477 −34.1197 −12.9990 −12.0350 34.9979 −50.4726
1.14 −4.09911 7.59971 8.80270 −11.2108 −31.1520 21.6950 −3.29036 30.7555 45.9544
1.15 −4.07584 3.89949 8.61251 −19.2550 −15.8937 −31.3149 −2.49649 −11.7940 78.4803
1.16 −4.04460 −8.35230 8.35876 −11.0565 33.7817 −15.6748 −1.45103 42.7609 44.7191
1.17 −3.95472 2.24647 7.63980 7.89434 −8.88416 22.8548 1.42449 −21.9534 −31.2199
1.18 −3.76895 2.47525 6.20498 20.7600 −9.32908 −15.9538 6.76534 −20.8732 −78.2432
1.19 −3.66084 −2.93605 5.40173 3.06305 10.7484 8.35536 9.51185 −18.3796 −11.2133
1.20 −3.36300 −7.69376 3.30976 21.7363 25.8741 −4.11624 15.7733 32.1939 −73.0991
See all 79 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.79
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(131\) \(1\)

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 1441.4.a.b 79
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1441.4.a.b 79 1.a even 1 1 trivial