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Properties

Label 25.8.d.a
Level $25$
Weight $8$
Character orbit 25.d
Analytic conductor $7.810$
Analytic rank $0$
Dimension $68$
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] = [25,8,Mod(6,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.6");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 25.d (of order \(5\), degree \(4\), minimal)

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: no
Analytic conductor: \(7.80962563710\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(17\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 11 q^{2} + 23 q^{3} - 1091 q^{4} + 475 q^{5} - 179 q^{6} + 1734 q^{7} + 220 q^{8} - 14554 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 11 q^{2} + 23 q^{3} - 1091 q^{4} + 475 q^{5} - 179 q^{6} + 1734 q^{7} + 220 q^{8} - 14554 q^{9} + 4245 q^{10} + 466 q^{11} + 9979 q^{12} - 7407 q^{13} - 20177 q^{14} - 9285 q^{15} - 68527 q^{16} + 71504 q^{17} + 116208 q^{18} - 48685 q^{19} - 138265 q^{20} - 2274 q^{21} - 271952 q^{22} - 63227 q^{23} + 553980 q^{24} + 276615 q^{25} + 675246 q^{26} - 270235 q^{27} - 782613 q^{28} - 99655 q^{29} - 1011285 q^{30} + 39051 q^{31} + 1045974 q^{32} + 1766456 q^{33} + 605893 q^{34} - 476935 q^{35} - 2928227 q^{36} + 83909 q^{37} + 1937525 q^{38} + 1316257 q^{39} + 2903410 q^{40} - 1964944 q^{41} - 3462547 q^{42} - 3439822 q^{43} - 3557652 q^{44} - 2121790 q^{45} + 2982521 q^{46} + 1663474 q^{47} + 1678378 q^{48} + 6584354 q^{49} + 408485 q^{50} + 3922076 q^{51} - 5301176 q^{52} + 788553 q^{53} - 6946145 q^{54} - 2630560 q^{55} - 415350 q^{56} - 14598410 q^{57} + 4727030 q^{58} - 3625740 q^{59} + 29158890 q^{60} - 4161559 q^{61} + 19658928 q^{62} + 2426738 q^{63} - 4408596 q^{64} - 2030220 q^{65} - 6985528 q^{66} + 8470434 q^{67} - 29099218 q^{68} - 360673 q^{69} - 22417320 q^{70} - 4808044 q^{71} + 25676065 q^{72} + 5517633 q^{73} + 14050068 q^{74} + 257240 q^{75} + 22502000 q^{76} - 9936772 q^{77} + 14948561 q^{78} - 5343505 q^{79} - 3987180 q^{80} - 5865817 q^{81} - 76614542 q^{82} - 645927 q^{83} + 5992628 q^{84} + 48268560 q^{85} - 27540609 q^{86} + 65370380 q^{87} - 3612320 q^{88} + 11153500 q^{89} + 7267975 q^{90} - 9123484 q^{91} - 33863736 q^{92} - 118704764 q^{93} - 30761227 q^{94} - 26382485 q^{95} + 1256206 q^{96} + 55538244 q^{97} + 102201752 q^{98} + 71266232 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −6.77922 − 20.8643i −20.4035 + 14.8240i −285.807 + 207.651i 278.853 + 19.1271i 447.612 + 325.209i −253.809 3998.26 + 2904.90i −479.269 + 1475.04i −1491.33 − 5947.74i
6.2 −5.60918 − 17.2633i 68.4023 − 49.6972i −163.004 + 118.429i −18.3423 − 278.906i −1241.62 − 902.088i 935.944 1079.11 + 784.017i 1533.25 − 4718.84i −4711.95 + 1881.08i
6.3 −5.04668 − 15.5321i −52.9300 + 38.4559i −112.223 + 81.5346i −278.209 − 26.9206i 864.422 + 628.039i 1099.17 141.570 + 102.857i 646.911 − 1990.99i 985.899 + 4457.03i
6.4 −4.92111 − 15.1456i 26.8202 − 19.4860i −101.618 + 73.8300i −162.557 + 227.377i −427.112 − 310.315i −1066.72 −30.8290 − 22.3986i −336.203 + 1034.73i 4243.72 + 1343.08i
6.5 −3.32876 − 10.2449i −20.4193 + 14.8355i 9.67727 − 7.03095i 58.2563 − 273.370i 219.959 + 159.809i −696.703 −1219.74 − 886.194i −478.964 + 1474.10i −2994.57 + 313.156i
6.6 −2.92617 − 9.00581i 22.7999 − 16.5651i 31.0120 − 22.5315i 233.049 + 154.315i −215.899 − 156.860i 1510.43 −1274.24 − 925.792i −430.387 + 1324.59i 707.789 − 2550.35i
6.7 −1.68083 − 5.17307i −68.2977 + 49.6212i 79.6187 − 57.8464i 181.166 + 212.847i 371.491 + 269.904i −821.987 −996.330 − 723.876i 1526.49 − 4698.06i 796.560 − 1294.95i
6.8 −0.0869750 − 0.267681i 34.6603 − 25.1822i 103.490 − 75.1899i −214.708 − 178.957i −9.75537 − 7.08769i −32.0119 −58.2740 − 42.3386i −108.627 + 334.320i −29.2293 + 73.0380i
6.9 0.247040 + 0.760310i −17.6400 + 12.8162i 103.037 − 74.8609i −179.724 + 214.066i −14.1021 − 10.2458i 544.176 165.157 + 119.993i −528.905 + 1627.80i −207.155 − 83.7630i
6.10 0.341871 + 1.05217i 64.7040 − 47.0102i 102.564 − 74.5171i 265.518 + 87.3222i 71.5832 + 52.0083i −1498.89 228.032 + 165.675i 1300.83 − 4003.53i −1.10492 + 309.224i
6.11 1.97156 + 6.06785i −27.8056 + 20.2019i 70.6225 − 51.3102i 236.094 − 149.614i −177.403 − 128.891i 251.293 1111.27 + 807.382i −310.789 + 956.509i 1373.31 + 1137.61i
6.12 3.70716 + 11.4095i −56.4422 + 41.0076i −12.8786 + 9.35683i −230.703 − 157.801i −677.115 − 491.953i −24.3113 1087.80 + 790.333i 828.272 − 2549.16i 945.171 − 3217.19i
6.13 4.26313 + 13.1206i −1.52497 + 1.10796i −50.4211 + 36.6331i 18.8507 + 278.872i −21.0382 − 15.2852i −1238.47 733.013 + 532.565i −674.722 + 2076.58i −3578.60 + 1436.20i
6.14 4.50567 + 13.8670i 60.2452 − 43.7707i −68.4392 + 49.7240i −222.360 + 169.354i 878.414 + 638.205i 1220.94 512.000 + 371.990i 1037.79 − 3193.98i −3350.32 − 2320.42i
6.15 4.60206 + 14.1637i 31.1717 − 22.6475i −75.8764 + 55.1275i 229.190 − 159.991i 464.226 + 337.280i 799.171 412.192 + 299.475i −217.058 + 668.036i 3320.80 + 2509.89i
6.16 6.43113 + 19.7930i 13.2953 − 9.65964i −246.849 + 179.346i −144.484 − 239.268i 276.697 + 201.032i −1305.03 −2982.18 − 2166.68i −592.362 + 1823.10i 3806.64 − 4398.54i
6.17 6.59046 + 20.2833i −51.4446 + 37.3767i −264.426 + 192.117i 176.749 + 216.529i −1097.17 − 797.139i 1543.62 −3430.94 − 2492.72i 573.708 − 1765.69i −3227.08 + 5012.09i
11.1 −17.1371 + 12.4508i 26.1576 − 80.5049i 99.1029 − 305.007i −220.555 − 171.698i 554.087 + 1705.31i 80.6154 1261.40 + 3882.18i −4027.50 − 2926.15i 5917.47 + 196.310i
11.2 −16.6693 + 12.1109i −25.4123 + 78.2109i 91.6358 − 282.026i 120.034 − 252.422i −523.603 − 1611.49i 289.141 1073.11 + 3302.69i −3701.84 − 2689.55i 1056.18 + 5661.42i
11.3 −14.2783 + 10.3738i 2.44290 − 7.51846i 56.6997 − 174.504i 156.133 + 231.835i 43.1144 + 132.693i 929.832 302.599 + 931.304i 1718.76 + 1248.75i −4634.31 − 1690.51i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.8.d.a ✓ 68
25.d even 5 1 inner 25.8.d.a ✓ 68
25.d even 5 1 625.8.a.d 34
25.e even 10 1 625.8.a.c 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.8.d.a ✓ 68 1.a even 1 1 trivial
25.8.d.a ✓ 68 25.d even 5 1 inner
625.8.a.c 34 25.e even 10 1
625.8.a.d 34 25.d even 5 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(25, [\chi])\).