# 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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 25.d (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 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.

 $$\operatorname{Tr}(f)(q) =$$ $$68q - 11q^{2} + 23q^{3} - 1091q^{4} + 475q^{5} - 179q^{6} + 1734q^{7} + 220q^{8} - 14554q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$68q - 11q^{2} + 23q^{3} - 1091q^{4} + 475q^{5} - 179q^{6} + 1734q^{7} + 220q^{8} - 14554q^{9} + 4245q^{10} + 466q^{11} + 9979q^{12} - 7407q^{13} - 20177q^{14} - 9285q^{15} - 68527q^{16} + 71504q^{17} + 116208q^{18} - 48685q^{19} - 138265q^{20} - 2274q^{21} - 271952q^{22} - 63227q^{23} + 553980q^{24} + 276615q^{25} + 675246q^{26} - 270235q^{27} - 782613q^{28} - 99655q^{29} - 1011285q^{30} + 39051q^{31} + 1045974q^{32} + 1766456q^{33} + 605893q^{34} - 476935q^{35} - 2928227q^{36} + 83909q^{37} + 1937525q^{38} + 1316257q^{39} + 2903410q^{40} - 1964944q^{41} - 3462547q^{42} - 3439822q^{43} - 3557652q^{44} - 2121790q^{45} + 2982521q^{46} + 1663474q^{47} + 1678378q^{48} + 6584354q^{49} + 408485q^{50} + 3922076q^{51} - 5301176q^{52} + 788553q^{53} - 6946145q^{54} - 2630560q^{55} - 415350q^{56} - 14598410q^{57} + 4727030q^{58} - 3625740q^{59} + 29158890q^{60} - 4161559q^{61} + 19658928q^{62} + 2426738q^{63} - 4408596q^{64} - 2030220q^{65} - 6985528q^{66} + 8470434q^{67} - 29099218q^{68} - 360673q^{69} - 22417320q^{70} - 4808044q^{71} + 25676065q^{72} + 5517633q^{73} + 14050068q^{74} + 257240q^{75} + 22502000q^{76} - 9936772q^{77} + 14948561q^{78} - 5343505q^{79} - 3987180q^{80} - 5865817q^{81} - 76614542q^{82} - 645927q^{83} + 5992628q^{84} + 48268560q^{85} - 27540609q^{86} + 65370380q^{87} - 3612320q^{88} + 11153500q^{89} + 7267975q^{90} - 9123484q^{91} - 33863736q^{92} - 118704764q^{93} - 30761227q^{94} - 26382485q^{95} + 1256206q^{96} + 55538244q^{97} + 102201752q^{98} + 71266232q^{99} + O(q^{100})$$

## 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.

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 21.17 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{8}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database