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Properties

Label 625.8.a.d
Level $625$
Weight $8$
Character orbit 625.a
Self dual yes
Analytic conductor $195.241$
Analytic rank $0$
Dimension $34$
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] = [625,8,Mod(1,625)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(625, 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("625.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 625.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: \(195.240640928\)
Analytic rank: \(0\)
Dimension: \(34\)
Twist minimal: no (minimal twist has level 25)
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.

\(\operatorname{Tr}(f)(q) = \) \( 34 q + 17 q^{2} + 14 q^{3} + 2177 q^{4} + 993 q^{6} + 867 q^{7} + 2700 q^{8} + 23708 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q + 17 q^{2} + 14 q^{3} + 2177 q^{4} + 993 q^{6} + 867 q^{7} + 2700 q^{8} + 23708 q^{9} + 8718 q^{11} + 2377 q^{12} + 8769 q^{13} + 17674 q^{14} + 139109 q^{16} - 48258 q^{17} + 58104 q^{18} + 61955 q^{19} + 96943 q^{21} - 52556 q^{22} + 55584 q^{23} + 276990 q^{24} + 337623 q^{26} - 24370 q^{27} + 392941 q^{28} + 449815 q^{29} + 450438 q^{31} + 522987 q^{32} - 689282 q^{33} + 639889 q^{34} + 2024094 q^{36} - 373008 q^{37} + 1272835 q^{38} + 877321 q^{39} + 926418 q^{41} + 580554 q^{42} - 1719911 q^{43} + 1809244 q^{44} + 1186623 q^{46} + 640247 q^{47} + 3924149 q^{48} + 3292177 q^{49} + 1961038 q^{51} + 3900212 q^{52} + 1609214 q^{53} + 3802395 q^{54} + 5709425 q^{56} - 7299205 q^{57} - 4001010 q^{58} + 6056265 q^{59} + 2885628 q^{61} - 14351151 q^{62} + 8834194 q^{63} + 12281872 q^{64} + 8819186 q^{66} - 164318 q^{67} - 14549609 q^{68} + 5884976 q^{69} + 10286173 q^{71} + 41997930 q^{72} + 20595894 q^{73} + 7025034 q^{74} + 11251000 q^{76} + 14569234 q^{77} - 32702012 q^{78} + 11772835 q^{79} + 24408674 q^{81} - 38307271 q^{82} + 2455444 q^{83} + 17190739 q^{84} + 7636893 q^{86} - 39372965 q^{87} + 45397760 q^{88} + 17788560 q^{89} + 15865973 q^{91} + 37183957 q^{92} - 59352382 q^{93} + 29349634 q^{94} + 38027693 q^{96} + 3708847 q^{97} + 6000776 q^{98} + 35633116 q^{99}+O(q^{100}) \) 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 −21.9380 25.2201 353.277 0 −553.279 −253.809 −4942.12 −1550.95 0
1.2 −21.8450 −17.1614 349.203 0 374.891 −851.881 −4832.18 −1892.49 0
1.3 −18.1517 −84.5499 201.484 0 1534.72 935.944 −1333.85 4961.69 0
1.4 −17.7272 42.9355 186.254 0 −761.125 6.21587 −1032.67 −343.547 0
1.5 −16.3314 65.4251 138.715 0 −1068.48 1099.17 −174.991 2093.45 0
1.6 −16.1061 −50.9824 131.408 0 821.130 1547.95 −54.8863 412.208 0
1.7 −15.9250 −33.1515 125.607 0 527.940 −1066.72 38.1067 −1087.98 0
1.8 −13.8299 84.1734 63.2650 0 −1164.11 483.061 895.276 4898.16 0
1.9 −13.3221 −80.9466 49.4794 0 1078.38 −1424.61 1046.06 4365.35 0
1.10 −10.7721 25.2396 −11.9618 0 −271.884 −696.703 1507.68 −1549.96 0
1.11 −9.46927 −28.1823 −38.3329 0 266.865 1510.43 1575.05 −1392.76 0
1.12 −8.03340 −11.2001 −63.4645 0 89.9748 756.043 1538.11 −2061.56 0
1.13 −7.51166 8.67312 −71.5749 0 −65.1495 −1153.54 1499.14 −2111.78 0
1.14 −5.43929 84.4206 −98.4141 0 −459.188 −821.987 1231.53 4939.84 0
1.15 −0.788080 45.3350 −127.379 0 −35.7276 −652.973 201.259 −131.737 0
1.16 −0.281457 −42.8424 −127.921 0 12.0583 −32.0119 72.0307 −351.525 0
1.17 0.799437 21.8043 −127.361 0 17.4312 544.176 −204.145 −1711.57 0
1.18 1.10632 −79.9785 −126.776 0 −88.4817 −1498.89 −281.864 4209.56 0
1.19 3.78736 −65.3865 −113.656 0 −247.642 −294.102 −915.238 2088.40 0
1.20 3.82054 62.4623 −113.403 0 238.640 1509.17 −922.293 1714.54 0
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.34
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)

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 625.8.a.d 34
5.b even 2 1 625.8.a.c 34
25.d even 5 2 25.8.d.a ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.8.d.a ✓ 68 25.d even 5 2
625.8.a.c 34 5.b even 2 1
625.8.a.d 34 1.a even 1 1 trivial