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Properties

Label 2005.4.a.d
Level $2005$
Weight $4$
Character orbit 2005.a
Self dual yes
Analytic conductor $118.299$
Analytic rank $0$
Dimension $104$
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] = [2005,4,Mod(1,2005)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2005, 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("2005.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2005 = 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2005.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: \(118.298829562\)
Analytic rank: \(0\)
Dimension: \(104\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 104 q + 21 q^{2} + 48 q^{3} + 451 q^{4} + 520 q^{5} + 64 q^{6} + 258 q^{7} + 273 q^{8} + 1088 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 104 q + 21 q^{2} + 48 q^{3} + 451 q^{4} + 520 q^{5} + 64 q^{6} + 258 q^{7} + 273 q^{8} + 1088 q^{9} + 105 q^{10} + 422 q^{11} + 336 q^{12} + 366 q^{13} + 244 q^{14} + 240 q^{15} + 1967 q^{16} + 590 q^{17} + 720 q^{18} + 354 q^{19} + 2255 q^{20} + 438 q^{21} + 718 q^{22} + 1300 q^{23} + 1019 q^{24} + 2600 q^{25} + 275 q^{26} + 1872 q^{27} + 2266 q^{28} + 948 q^{29} + 320 q^{30} + 628 q^{31} + 2900 q^{32} + 1196 q^{33} + 776 q^{34} + 1290 q^{35} + 6055 q^{36} + 1122 q^{37} + 2247 q^{38} + 890 q^{39} + 1365 q^{40} + 966 q^{41} + 2884 q^{42} + 3594 q^{43} + 3978 q^{44} + 5440 q^{45} + 793 q^{46} + 2954 q^{47} + 2600 q^{48} + 6152 q^{49} + 525 q^{50} + 3402 q^{51} + 4189 q^{52} + 3742 q^{53} + 2176 q^{54} + 2110 q^{55} + 3392 q^{56} + 3222 q^{57} + 2467 q^{58} + 4348 q^{59} + 1680 q^{60} + 1412 q^{61} + 2903 q^{62} + 8500 q^{63} + 9199 q^{64} + 1830 q^{65} + 1126 q^{66} + 5968 q^{67} + 6948 q^{68} + 1788 q^{69} + 1220 q^{70} + 4920 q^{71} + 9051 q^{72} + 5172 q^{73} + 4391 q^{74} + 1200 q^{75} + 1893 q^{76} + 4972 q^{77} - 448 q^{78} + 2378 q^{79} + 9835 q^{80} + 13168 q^{81} + 5594 q^{82} + 10396 q^{83} + 3972 q^{84} + 2950 q^{85} + 5421 q^{86} + 9462 q^{87} + 7830 q^{88} + 3920 q^{89} + 3600 q^{90} + 3900 q^{91} + 11173 q^{92} + 4272 q^{93} + 1540 q^{94} + 1770 q^{95} + 6063 q^{96} + 3662 q^{97} + 5844 q^{98} + 9998 q^{99}+O(q^{100}) \) 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 −5.48242 −7.07161 22.0570 5.00000 38.7696 33.3636 −77.0662 23.0077 −27.4121
1.2 −5.42567 0.00989723 21.4379 5.00000 −0.0536991 0.0218356 −72.9094 −26.9999 −27.1283
1.3 −5.32751 −2.91767 20.3824 5.00000 15.5439 −3.41579 −65.9671 −18.4872 −26.6375
1.4 −5.10379 5.59212 18.0487 5.00000 −28.5411 −14.8359 −51.2866 4.27185 −25.5190
1.5 −5.08796 2.43316 17.8874 5.00000 −12.3798 13.4878 −50.3066 −21.0797 −25.4398
1.6 −5.05494 −5.02563 17.5524 5.00000 25.4043 −26.7280 −48.2868 −1.74300 −25.2747
1.7 −5.04018 9.56525 17.4034 5.00000 −48.2106 3.67647 −47.3951 64.4941 −25.2009
1.8 −5.02026 −5.26479 17.2030 5.00000 26.4306 −13.3527 −46.2012 0.718061 −25.1013
1.9 −4.85977 9.07507 15.6173 5.00000 −44.1028 1.26485 −37.0185 55.3569 −24.2988
1.10 −4.81034 8.43952 15.1394 5.00000 −40.5970 19.5565 −34.3429 44.2255 −24.0517
1.11 −4.69195 2.72727 14.0144 5.00000 −12.7962 32.1155 −28.2192 −19.5620 −23.4597
1.12 −4.63720 2.16669 13.5037 5.00000 −10.0474 −10.5548 −25.5216 −22.3055 −23.1860
1.13 −4.50268 −10.3127 12.2741 5.00000 46.4346 13.9886 −19.2451 79.3510 −22.5134
1.14 −4.22847 8.51704 9.87997 5.00000 −36.0140 −19.1069 −7.94939 45.5399 −21.1424
1.15 −4.19383 −1.43068 9.58817 5.00000 6.00003 −25.4667 −6.66051 −24.9531 −20.9691
1.16 −4.11941 −8.10164 8.96952 5.00000 33.3739 26.0678 −3.99387 38.6365 −20.5970
1.17 −4.10002 −1.67242 8.81017 5.00000 6.85696 18.1133 −3.32171 −24.2030 −20.5001
1.18 −3.97439 0.963150 7.79574 5.00000 −3.82793 −22.7929 0.811822 −26.0723 −19.8719
1.19 −3.85371 −6.79615 6.85110 5.00000 26.1904 −6.32498 4.42754 19.1876 −19.2686
1.20 −3.57494 −7.29123 4.78017 5.00000 26.0657 25.2608 11.5107 26.1620 −17.8747
See next 80 embeddings (of 104 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.104
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(401\) \(-1\)

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 2005.4.a.d ✓ 104
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2005.4.a.d ✓ 104 1.a even 1 1 trivial