↑

Properties

Label 451.8.a.a
Level $451$
Weight $8$
Character orbit 451.a
Self dual yes
Analytic conductor $140.886$
Analytic rank $1$
Dimension $54$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [451,8,Mod(1,451)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(451, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("451.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 451 = 11 \cdot 41 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 451.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: \(140.885646493\)
Analytic rank: \(1\)
Dimension: \(54\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 54 q - 24 q^{2} + 8 q^{3} + 2788 q^{4} - 1718 q^{5} - 270 q^{6} + 1901 q^{7} - 3390 q^{8} + 30228 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q - 24 q^{2} + 8 q^{3} + 2788 q^{4} - 1718 q^{5} - 270 q^{6} + 1901 q^{7} - 3390 q^{8} + 30228 q^{9} + 1250 q^{10} - 71874 q^{11} + 10868 q^{12} - 5445 q^{13} - 5387 q^{14} - 743 q^{15} + 105580 q^{16} - 25627 q^{17} + 35051 q^{18} - 2065 q^{19} - 143639 q^{20} - 160037 q^{21} + 31944 q^{22} - 105507 q^{23} - 37232 q^{24} + 630898 q^{25} - 76207 q^{26} + 47204 q^{27} + 319896 q^{28} - 751488 q^{29} - 623568 q^{30} - 527137 q^{31} - 456079 q^{32} - 10648 q^{33} - 908976 q^{34} - 1209951 q^{35} + 345230 q^{36} - 1064512 q^{37} + 34075 q^{38} + 107672 q^{39} - 2404307 q^{40} + 3721734 q^{41} - 1850103 q^{42} - 2041834 q^{43} - 3710828 q^{44} - 3665888 q^{45} - 1628252 q^{46} - 364486 q^{47} - 1600109 q^{48} + 4549289 q^{49} + 205918 q^{50} + 1080749 q^{51} - 1450823 q^{52} - 7794462 q^{53} - 8760615 q^{54} + 2286658 q^{55} - 8099195 q^{56} - 1579772 q^{57} + 1326060 q^{58} - 3712705 q^{59} + 215484 q^{60} - 197777 q^{61} - 1309205 q^{62} - 1280680 q^{63} - 8911884 q^{64} - 10420709 q^{65} + 359370 q^{66} - 6612598 q^{67} - 11035431 q^{68} - 3892232 q^{69} + 812744 q^{70} + 513621 q^{71} + 4436458 q^{72} + 3190287 q^{73} - 4386919 q^{74} - 3038310 q^{75} - 3703121 q^{76} - 2530231 q^{77} - 23319295 q^{78} + 5016878 q^{79} - 6152612 q^{80} - 6210206 q^{81} - 1654104 q^{82} + 858376 q^{83} + 26574 q^{84} - 4366665 q^{85} - 37167776 q^{86} - 2920705 q^{87} + 4512090 q^{88} - 28880202 q^{89} - 37776079 q^{90} - 6491258 q^{91} - 21390002 q^{92} - 27138221 q^{93} - 31653654 q^{94} - 17193356 q^{95} - 26636764 q^{96} - 26228973 q^{97} - 70251388 q^{98} - 40233468 q^{99}+O(q^{100}) \) 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 −21.5560 −14.3881 336.659 320.967 310.148 492.440 −4497.84 −1979.98 −6918.75
1.2 −21.5544 −30.7870 336.592 −245.092 663.596 1557.79 −4496.06 −1239.16 5282.80
1.3 −20.6133 75.2708 296.907 −113.652 −1551.58 872.880 −3481.72 3478.69 2342.74
1.4 −19.4768 16.6221 251.345 −389.117 −323.745 −602.113 −2402.36 −1910.71 7578.75
1.5 −19.2362 −53.0500 242.032 296.068 1020.48 −1298.28 −2193.54 627.299 −5695.22
1.6 −17.9687 −13.0357 194.873 344.730 234.234 −299.397 −1201.62 −2017.07 −6194.35
1.7 −17.6638 63.8270 184.009 −209.813 −1127.42 −1169.48 −989.323 1886.88 3706.08
1.8 −17.3902 −85.4642 174.419 −343.223 1486.24 −749.260 −807.240 5117.13 5968.72
1.9 −17.3543 58.5096 173.172 404.331 −1015.39 115.139 −783.933 1236.37 −7016.88
1.10 −16.1014 44.6533 131.256 265.542 −718.981 999.318 −52.4207 −193.083 −4275.60
1.11 −15.8079 −33.3706 121.891 −291.406 527.521 −817.289 96.5701 −1073.40 4606.53
1.12 −15.1354 17.2429 101.080 −539.246 −260.978 1334.24 407.443 −1889.68 8161.70
1.13 −13.2842 −62.9534 48.4709 −6.44373 836.288 1137.81 1056.48 1776.13 85.6000
1.14 −12.3307 90.7264 24.0472 −0.501745 −1118.72 −1194.51 1281.82 6044.28 6.18689
1.15 −11.6148 2.28314 6.90297 −20.1261 −26.5182 −526.201 1406.51 −2181.79 233.760
1.16 −11.6045 0.368038 6.66420 −113.297 −4.27090 514.279 1408.04 −2186.86 1314.75
1.17 −8.54979 −62.6897 −54.9011 −371.584 535.983 1285.12 1563.77 1742.99 3176.97
1.18 −8.45724 50.6945 −56.4750 −119.499 −428.736 1430.24 1560.15 382.929 1010.63
1.19 −7.91206 −47.2169 −65.3993 323.107 373.583 −1179.34 1530.19 42.4367 −2556.44
1.20 −7.66860 −68.9222 −69.1926 −2.76331 528.537 437.832 1512.19 2563.27 21.1907
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.54
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \( +1 \)
\(41\) \( -1 \)

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 451.8.a.a ✓ 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
451.8.a.a ✓ 54 1.a even 1 1 trivial