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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1441,4,Mod(1,1441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1441, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1441.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1441 = 11 \cdot 131 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1441.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: \(85.0217523183\)
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.

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

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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