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Properties

Label 625.8.a.a
Level $625$
Weight $8$
Character orbit 625.a
Self dual yes
Analytic conductor $195.241$
Analytic rank $1$
Dimension $22$
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: \(1\)
Dimension: \(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 23 q^{2} - 121 q^{3} + 1061 q^{4} - 431 q^{6} - 843 q^{7} - 4980 q^{8} + 14799 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 23 q^{2} - 121 q^{3} + 1061 q^{4} - 431 q^{6} - 843 q^{7} - 4980 q^{8} + 14799 q^{9} - 781 q^{11} - 12608 q^{12} - 14686 q^{13} + 20762 q^{14} + 16117 q^{16} - 45648 q^{17} - 47171 q^{18} + 6185 q^{19} + 14149 q^{21} + 71124 q^{22} - 126921 q^{23} + 271570 q^{24} + 304129 q^{26} - 546520 q^{27} - 2019 q^{28} + 59330 q^{29} + 394804 q^{31} + 74397 q^{32} - 49067 q^{33} - 286938 q^{34} - 287278 q^{36} + 792122 q^{37} - 1338860 q^{38} + 635223 q^{39} - 160466 q^{41} - 3420191 q^{42} - 1527256 q^{43} - 1154853 q^{44} + 2653604 q^{46} - 1300863 q^{47} - 1885241 q^{48} + 1652981 q^{49} + 3408539 q^{51} - 1423303 q^{52} - 755656 q^{53} + 3117755 q^{54} - 2132625 q^{56} - 3026890 q^{57} + 5941470 q^{58} - 1548370 q^{59} - 6029951 q^{61} - 79936 q^{62} + 6962459 q^{63} - 5858224 q^{64} - 5380407 q^{66} - 7608838 q^{67} - 10737124 q^{68} + 13519553 q^{69} + 9483549 q^{71} - 6806340 q^{72} - 13548801 q^{73} - 15016023 q^{74} + 19635315 q^{76} + 2145019 q^{77} - 17222402 q^{78} + 10769160 q^{79} + 2757382 q^{81} - 12087571 q^{82} + 9632744 q^{83} + 19168542 q^{84} - 8511651 q^{86} - 298330 q^{87} + 21641425 q^{88} - 10850545 q^{89} - 6648131 q^{91} - 52978503 q^{92} - 51294822 q^{93} + 5863777 q^{94} + 2611654 q^{96} - 11579993 q^{97} - 7468074 q^{98} - 13552997 q^{99}+O(q^{100}) \) 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 −21.3970 −45.8462 329.831 0 980.970 104.481 −4318.57 −85.1275 0
1.2 −19.9369 7.59174 269.481 0 −151.356 250.277 −2820.70 −2129.37 0
1.3 −17.3699 −85.0611 173.712 0 1477.50 −1277.84 −794.018 5048.39 0
1.4 −16.8316 60.3488 155.304 0 −1015.77 1500.12 −459.578 1454.98 0
1.5 −16.3997 21.4355 140.949 0 −351.535 −1450.75 −212.357 −1727.52 0
1.6 −14.6488 70.4760 86.5879 0 −1032.39 −98.3469 606.638 2779.87 0
1.7 −9.65795 −78.6437 −34.7239 0 759.537 34.6342 1571.58 3997.83 0
1.8 −9.55882 −19.6493 −36.6289 0 187.824 275.298 1573.66 −1800.91 0
1.9 −8.75606 −29.2735 −51.3314 0 256.321 528.656 1570.24 −1330.06 0
1.10 −3.59862 41.3836 −115.050 0 −148.924 −1266.89 874.645 −474.397 0
1.11 −3.07018 68.4413 −118.574 0 −210.127 −698.276 757.028 2497.20 0
1.12 2.72966 −0.659241 −120.549 0 −1.79951 767.270 −678.454 −2186.57 0
1.13 3.20926 62.8158 −117.701 0 201.593 1045.66 −788.518 1758.83 0
1.14 5.27101 −89.1473 −100.216 0 −469.896 1076.06 −1202.93 5760.24 0
1.15 5.97357 −39.4745 −92.3164 0 −235.804 −1669.35 −1316.08 −628.764 0
1.16 7.52269 −47.0815 −71.4092 0 −354.180 −1676.07 −1500.09 29.6718 0
1.17 9.79880 0.219481 −31.9834 0 2.15065 700.879 −1567.65 −2186.95 0
1.18 13.4550 −79.3155 53.0377 0 −1067.19 1416.14 −1008.62 4103.95 0
1.19 14.0235 58.7448 68.6591 0 823.809 136.950 −832.168 1263.95 0
1.20 17.2747 56.9787 170.417 0 984.292 −464.211 732.735 1059.57 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
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.a ✓ 22
5.b even 2 1 625.8.a.b yes 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
625.8.a.a ✓ 22 1.a even 1 1 trivial
625.8.a.b yes 22 5.b even 2 1