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Properties

Label 1441.4.a.d
Level $1441$
Weight $4$
Character orbit 1441.a
Self dual yes
Analytic conductor $85.022$
Analytic rank $0$
Dimension $86$
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: \(0\)
Dimension: \(86\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 86 q + 18 q^{2} + 12 q^{3} + 372 q^{4} + 40 q^{5} + 57 q^{6} + 109 q^{7} + 198 q^{8} + 900 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 86 q + 18 q^{2} + 12 q^{3} + 372 q^{4} + 40 q^{5} + 57 q^{6} + 109 q^{7} + 198 q^{8} + 900 q^{9} + 122 q^{10} + 946 q^{11} + 96 q^{12} + 304 q^{13} + 246 q^{14} + 436 q^{15} + 1712 q^{16} + 352 q^{17} + 523 q^{18} + 483 q^{19} + 382 q^{20} + 886 q^{21} + 198 q^{22} + 794 q^{23} + 694 q^{24} + 2576 q^{25} + 352 q^{26} + 132 q^{27} + 892 q^{28} + 1675 q^{29} + 587 q^{30} + 550 q^{31} + 2187 q^{32} + 132 q^{33} + 1012 q^{34} + 1590 q^{35} + 3275 q^{36} - 63 q^{37} - 893 q^{38} + 2790 q^{39} - 884 q^{40} + 1407 q^{41} + 1171 q^{42} + 946 q^{43} + 4092 q^{44} + 1000 q^{45} + 2684 q^{46} + 1530 q^{47} - 2776 q^{48} + 4759 q^{49} + 4256 q^{50} + 1686 q^{51} + 915 q^{52} + 1350 q^{53} - 356 q^{54} + 440 q^{55} + 1861 q^{56} + 1240 q^{57} - 348 q^{58} + 3882 q^{59} - 1797 q^{60} + 4526 q^{61} - 666 q^{62} + 4781 q^{63} + 9288 q^{64} + 1002 q^{65} + 627 q^{66} + 1615 q^{67} + 3791 q^{68} + 12 q^{69} - 735 q^{70} + 7256 q^{71} + 6346 q^{72} + 1352 q^{73} + 2886 q^{74} + 3004 q^{75} + 2863 q^{76} + 1199 q^{77} + 4405 q^{78} + 6038 q^{79} - 897 q^{80} + 11410 q^{81} - 1179 q^{82} + 3827 q^{83} + 5109 q^{84} + 4210 q^{85} + 5696 q^{86} + 2524 q^{87} + 2178 q^{88} + 3763 q^{89} + 3504 q^{90} + 3578 q^{91} + 5578 q^{92} + 954 q^{93} + 4669 q^{94} + 11334 q^{95} + 3633 q^{96} - 834 q^{97} + 2231 q^{98} + 9900 q^{99}+O(q^{100}) \) 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 −5.44803 −3.88459 21.6811 −9.18308 21.1634 −29.6118 −74.5350 −11.9100 50.0297
1.2 −5.39326 −8.27033 21.0873 0.267878 44.6041 −0.0391150 −70.5833 41.3983 −1.44474
1.3 −5.35765 −0.992257 20.7044 12.4738 5.31616 −13.0068 −68.0656 −26.0154 −66.8302
1.4 −5.33564 6.10889 20.4691 −12.5048 −32.5949 0.447911 −66.5306 10.3186 66.7211
1.5 −5.26457 4.77883 19.7157 13.4428 −25.1585 4.58357 −61.6778 −4.16274 −70.7704
1.6 −4.82907 −5.87907 15.3199 13.1247 28.3904 34.8270 −35.3483 7.56342 −63.3802
1.7 −4.78186 −8.90821 14.8662 6.48276 42.5979 23.0625 −32.8334 52.3562 −30.9997
1.8 −4.70408 2.60101 14.1283 1.32450 −12.2353 12.9890 −28.8281 −20.2348 −6.23055
1.9 −4.59033 8.77021 13.0711 13.7875 −40.2581 23.7020 −23.2780 49.9166 −63.2893
1.10 −4.56967 1.59222 12.8819 −18.4972 −7.27590 −3.34022 −22.3085 −24.4648 84.5259
1.11 −4.42178 −3.92510 11.5521 −16.0301 17.3559 11.2575 −15.7066 −11.5936 70.8815
1.12 −4.38725 8.91769 11.2480 −8.28617 −39.1241 10.2059 −14.2496 52.5252 36.3535
1.13 −4.20746 −1.37725 9.70268 −7.90828 5.79473 16.5208 −7.16395 −25.1032 33.2737
1.14 −4.12933 2.52725 9.05141 10.7343 −10.4358 −34.6796 −4.34160 −20.6130 −44.3255
1.15 −3.92609 −4.21599 7.41420 4.12077 16.5524 −24.0177 2.29992 −9.22539 −16.1785
1.16 −3.82615 −7.90799 6.63941 −15.5303 30.2571 −7.02738 5.20581 35.5362 59.4213
1.17 −3.67637 8.93540 5.51572 −3.28699 −32.8499 −18.1345 9.13314 52.8413 12.0842
1.18 −3.65657 −8.65273 5.37050 11.2423 31.6393 −23.6367 9.61496 47.8697 −41.1082
1.19 −3.53367 4.67986 4.48684 −2.67374 −16.5371 34.9886 12.4144 −5.09892 9.44813
1.20 −3.45701 2.02230 3.95093 13.8534 −6.99111 −8.13275 13.9977 −22.9103 −47.8914
See all 86 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.86
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.d ✓ 86
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1441.4.a.d ✓ 86 1.a even 1 1 trivial