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Properties

Label 334.8.a.d
Level $334$
Weight $8$
Character orbit 334.a
Self dual yes
Analytic conductor $104.337$
Analytic rank $0$
Dimension $26$
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] = [334,8,Mod(1,334)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(334, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("334.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 334 = 2 \cdot 167 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 334.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: \(104.336598512\)
Analytic rank: \(0\)
Dimension: \(26\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 26 q + 208 q^{2} + 137 q^{3} + 1664 q^{4} + 500 q^{5} + 1096 q^{6} + 2922 q^{7} + 13312 q^{8} + 22519 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q + 208 q^{2} + 137 q^{3} + 1664 q^{4} + 500 q^{5} + 1096 q^{6} + 2922 q^{7} + 13312 q^{8} + 22519 q^{9} + 4000 q^{10} + 21435 q^{11} + 8768 q^{12} + 6800 q^{13} + 23376 q^{14} + 43597 q^{15} + 106496 q^{16} + 58905 q^{17} + 180152 q^{18} + 71838 q^{19} + 32000 q^{20} + 51483 q^{21} + 171480 q^{22} + 168380 q^{23} + 70144 q^{24} + 589644 q^{25} + 54400 q^{26} + 411944 q^{27} + 187008 q^{28} + 518486 q^{29} + 348776 q^{30} + 267870 q^{31} + 851968 q^{32} - 233432 q^{33} + 471240 q^{34} + 324868 q^{35} + 1441216 q^{36} + 782323 q^{37} + 574704 q^{38} + 1731227 q^{39} + 256000 q^{40} + 1104794 q^{41} + 411864 q^{42} + 390697 q^{43} + 1371840 q^{44} - 960905 q^{45} + 1347040 q^{46} + 891666 q^{47} + 561152 q^{48} + 3982788 q^{49} + 4717152 q^{50} + 2228648 q^{51} + 435200 q^{52} + 3879373 q^{53} + 3295552 q^{54} + 3562159 q^{55} + 1496064 q^{56} + 3405755 q^{57} + 4147888 q^{58} + 5439346 q^{59} + 2790208 q^{60} + 7812129 q^{61} + 2142960 q^{62} + 19067325 q^{63} + 6815744 q^{64} + 15958866 q^{65} - 1867456 q^{66} + 13633079 q^{67} + 3769920 q^{68} + 21874217 q^{69} + 2598944 q^{70} + 25825226 q^{71} + 11529728 q^{72} + 8190073 q^{73} + 6258584 q^{74} + 24925472 q^{75} + 4597632 q^{76} + 24054065 q^{77} + 13849816 q^{78} + 19917590 q^{79} + 2048000 q^{80} + 38367026 q^{81} + 8838352 q^{82} + 16963813 q^{83} + 3294912 q^{84} + 15582667 q^{85} + 3125576 q^{86} - 12587015 q^{87} + 10974720 q^{88} + 7440924 q^{89} - 7687240 q^{90} + 13392476 q^{91} + 10776320 q^{92} + 11406225 q^{93} + 7133328 q^{94} + 8569432 q^{95} + 4489216 q^{96} - 4892614 q^{97} + 31862304 q^{98} + 73733120 q^{99}+O(q^{100}) \) 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 8.00000 −88.1395 64.0000 −408.697 −705.116 1430.94 512.000 5581.58 −3269.57
1.2 8.00000 −83.8350 64.0000 −191.120 −670.680 −280.394 512.000 4841.30 −1528.96
1.3 8.00000 −77.4267 64.0000 159.058 −619.413 796.278 512.000 3807.89 1272.47
1.4 8.00000 −58.8853 64.0000 −21.1768 −471.083 689.964 512.000 1280.48 −169.415
1.5 8.00000 −53.9296 64.0000 430.227 −431.437 −148.049 512.000 721.405 3441.82
1.6 8.00000 −48.2829 64.0000 −176.663 −386.263 −183.959 512.000 144.239 −1413.31
1.7 8.00000 −43.2447 64.0000 463.085 −345.957 −662.865 512.000 −316.897 3704.68
1.8 8.00000 −36.8130 64.0000 64.4320 −294.504 −1576.32 512.000 −831.803 515.456
1.9 8.00000 −18.0679 64.0000 −327.147 −144.543 337.089 512.000 −1860.55 −2617.18
1.10 8.00000 −15.4105 64.0000 −136.769 −123.284 −1574.81 512.000 −1949.52 −1094.15
1.11 8.00000 −9.78855 64.0000 −22.0889 −78.3084 1242.65 512.000 −2091.18 −176.711
1.12 8.00000 −9.53553 64.0000 467.207 −76.2842 1632.44 512.000 −2096.07 3737.66
1.13 8.00000 −0.0775814 64.0000 −368.412 −0.620651 −1111.96 512.000 −2186.99 −2947.29
1.14 8.00000 1.80492 64.0000 109.686 14.4393 373.116 512.000 −2183.74 877.487
1.15 8.00000 10.0614 64.0000 −424.861 80.4911 543.435 512.000 −2085.77 −3398.89
1.16 8.00000 22.9471 64.0000 −12.0358 183.576 −1308.34 512.000 −1660.43 −96.2864
1.17 8.00000 26.5272 64.0000 482.934 212.217 −227.531 512.000 −1483.31 3863.47
1.18 8.00000 46.7449 64.0000 −129.110 373.959 1181.93 512.000 −1.91519 −1032.88
1.19 8.00000 49.0230 64.0000 205.797 392.184 1096.16 512.000 216.256 1646.38
1.20 8.00000 57.8349 64.0000 491.687 462.679 469.077 512.000 1157.88 3933.49
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.26
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(167\) \( -1 \)

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 334.8.a.d ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
334.8.a.d ✓ 26 1.a even 1 1 trivial