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Properties

Label 625.8.a.g
Level $625$
Weight $8$
Character orbit 625.a
Self dual yes
Analytic conductor $195.241$
Analytic rank $1$
Dimension $64$
CM no
Inner twists $2$

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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: \(64\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 3582 q^{4} - 1982 q^{6} + 37148 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 3582 q^{4} - 1982 q^{6} + 37148 q^{9} - 17432 q^{11} - 34836 q^{14} + 164154 q^{16} - 123910 q^{19} - 193882 q^{21} - 544720 q^{24} - 708942 q^{26} - 666400 q^{29} - 900872 q^{31} - 911986 q^{34} + 309704 q^{36} - 1745894 q^{39} - 1862362 q^{41} - 3617976 q^{44} - 2373242 q^{46} + 3298162 q^{49} - 3922092 q^{51} - 7598090 q^{54} - 11353310 q^{56} - 12112530 q^{59} - 9276582 q^{61} - 4640288 q^{64} - 16518624 q^{66} - 17766944 q^{69} - 22846802 q^{71} - 20547876 q^{74} - 22436480 q^{76} - 23545670 q^{79} - 1824616 q^{81} - 31121786 q^{84} - 15273782 q^{86} - 29694630 q^{89} - 31731942 q^{91} + 3544204 q^{94} - 40852262 q^{96} - 57867624 q^{99}+O(q^{100}) \) 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 −21.2730 −11.4028 324.540 0 242.572 1219.10 −4181.00 −2056.98 0
1.2 −20.9781 −53.2496 312.080 0 1117.07 928.752 −3861.64 648.520 0
1.3 −20.8347 51.2562 306.085 0 −1067.91 983.801 −3710.35 440.196 0
1.4 −20.7994 80.3165 304.614 0 −1670.53 −664.316 −3673.47 4263.75 0
1.5 −19.3170 47.9467 245.146 0 −926.186 −1166.35 −2262.92 111.886 0
1.6 −18.3862 24.6524 210.053 0 −453.264 −1022.55 −1508.63 −1579.26 0
1.7 −18.2743 −0.950482 205.952 0 17.3694 −190.361 −1424.51 −2186.10 0
1.8 −18.2018 −49.5657 203.304 0 902.182 217.774 −1370.66 269.756 0
1.9 −18.0742 −44.2337 198.676 0 799.487 372.101 −1277.41 −230.384 0
1.10 −17.7550 70.5180 187.240 0 −1252.05 1074.80 −1051.80 2785.79 0
1.11 −17.5267 −27.8670 179.184 0 488.415 −1206.49 −897.076 −1410.43 0
1.12 −16.4875 −65.9475 143.839 0 1087.31 287.120 −261.139 2162.07 0
1.13 −14.8031 −37.3096 91.1325 0 552.298 198.096 545.755 −794.995 0
1.14 −13.2964 39.6856 48.7931 0 −527.673 876.288 1053.16 −612.056 0
1.15 −12.5105 29.2192 28.5132 0 −365.548 −461.143 1244.63 −1333.24 0
1.16 −11.8180 −85.4573 11.6654 0 1009.94 −865.676 1374.84 5115.96 0
1.17 −11.6716 72.4697 8.22735 0 −845.840 −1765.57 1397.94 3064.85 0
1.18 −10.4655 −52.3547 −18.4724 0 547.921 −497.010 1532.91 554.019 0
1.19 −9.85891 80.6313 −30.8019 0 −794.937 1036.46 1565.61 4314.41 0
1.20 −9.62339 18.0650 −35.3904 0 −173.847 933.124 1572.37 −1860.65 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.64
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 625.8.a.g 64
5.b even 2 1 inner 625.8.a.g 64
25.f odd 20 2 25.8.e.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.8.e.a ✓ 64 25.f odd 20 2
625.8.a.g 64 1.a even 1 1 trivial
625.8.a.g 64 5.b even 2 1 inner