↑

Properties

Label 625.8.a.e
Level $625$
Weight $8$
Character orbit 625.a
Self dual yes
Analytic conductor $195.241$
Analytic rank $1$
Dimension $48$
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] = [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: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 25 q^{2} - 95 q^{3} + 3419 q^{4} + 431 q^{6} - 4030 q^{7} - 3375 q^{8} + 36231 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 25 q^{2} - 95 q^{3} + 3419 q^{4} + 431 q^{6} - 4030 q^{7} - 3375 q^{8} + 36231 q^{9} + 781 q^{11} - 3925 q^{12} - 4290 q^{13} - 20762 q^{14} + 270603 q^{16} - 75075 q^{17} - 89950 q^{18} + 17750 q^{19} - 48034 q^{21} - 331305 q^{22} - 343890 q^{23} - 271570 q^{24} - 304129 q^{26} - 474740 q^{27} - 1146535 q^{28} - 59330 q^{29} - 385989 q^{31} - 1887300 q^{32} - 879805 q^{33} + 286938 q^{34} + 3553198 q^{36} - 935610 q^{37} - 984745 q^{38} - 294888 q^{39} + 160466 q^{41} + 783725 q^{42} + 146400 q^{43} + 2261658 q^{44} - 2639009 q^{46} - 4446810 q^{47} - 3994240 q^{48} + 7532484 q^{49} - 2294894 q^{51} - 4582065 q^{52} - 3977030 q^{53} - 3979475 q^{54} - 743430 q^{56} - 2455430 q^{57} - 14413560 q^{58} - 1614425 q^{59} + 7720866 q^{61} - 20362850 q^{62} - 26297840 q^{63} + 21801809 q^{64} + 945327 q^{66} - 3017910 q^{67} - 17494265 q^{68} - 13519553 q^{69} - 9483549 q^{71} - 21929370 q^{72} + 388070 q^{73} + 16144878 q^{74} - 13507955 q^{76} - 25473115 q^{77} + 3108110 q^{78} - 10950620 q^{79} + 34443488 q^{81} + 354040 q^{82} - 47217920 q^{83} - 27843102 q^{84} + 20021766 q^{86} - 56120960 q^{87} - 54397660 q^{88} + 10850545 q^{89} - 8553794 q^{91} + 20734425 q^{92} + 11206260 q^{93} - 52545997 q^{94} - 28125034 q^{96} - 32784020 q^{97} - 14131170 q^{98} + 27513602 q^{99}+O(q^{100}) \) 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 −22.4221 −74.0434 374.750 0 1660.21 992.609 −5532.66 3295.43 0
1.2 −22.3593 67.2375 371.938 0 −1503.38 −682.414 −5454.29 2333.88 0
1.3 −20.9466 63.5784 310.760 0 −1331.75 −1687.22 −3828.20 1855.21 0
1.4 −20.4387 51.2591 289.740 0 −1047.67 765.121 −3305.75 440.495 0
1.5 −19.8925 −60.7396 267.712 0 1208.26 −431.550 −2779.23 1502.29 0
1.6 −19.6229 −41.7393 257.059 0 819.047 −1210.87 −2532.51 −444.830 0
1.7 −18.8926 −78.3784 228.931 0 1480.77 −1117.51 −1906.86 3956.18 0
1.8 −18.0515 5.54184 197.856 0 −100.038 1727.85 −1261.01 −2156.29 0
1.9 −15.8809 70.5834 124.203 0 −1120.93 −519.540 60.3057 2795.02 0
1.10 −15.8794 −7.96952 124.157 0 126.552 832.039 61.0306 −2123.49 0
1.11 −14.6582 20.0017 86.8618 0 −293.189 1031.56 603.011 −1786.93 0
1.12 −14.0260 −18.0558 68.7286 0 253.250 −491.418 831.340 −1860.99 0
1.13 −13.2572 17.7308 47.7546 0 −235.062 −1160.50 1063.83 −1872.62 0
1.14 −11.1801 −35.6020 −3.00447 0 398.036 1081.39 1464.65 −919.496 0
1.15 −9.78625 −79.0302 −32.2293 0 773.410 741.503 1568.04 4058.78 0
1.16 −9.46721 57.5753 −38.3720 0 −545.077 −360.867 1575.08 1127.91 0
1.17 −8.04622 −69.2100 −63.2583 0 556.879 −1059.53 1538.91 2603.02 0
1.18 −7.31488 −1.48532 −74.4925 0 10.8650 −996.653 1481.21 −2184.79 0
1.19 −7.04810 86.5216 −78.3242 0 −609.813 609.260 1454.19 5298.98 0
1.20 −6.43383 79.6009 −86.6058 0 −512.139 599.534 1380.74 4149.31 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.48
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.e ✓ 48
5.b even 2 1 625.8.a.f yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
625.8.a.e ✓ 48 1.a even 1 1 trivial
625.8.a.f yes 48 5.b even 2 1