Properties

 Label 4025.2.a.bd Level $4025$ Weight $2$ Character orbit 4025.a Self dual yes Analytic conductor $32.140$ Analytic rank $0$ Dimension $21$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$4025 = 5^{2} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4025.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$32.1397868136$$ Analytic rank: $$0$$ Dimension: $$21$$ Twist minimal: no (minimal twist has level 805) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$21 q - 2 q^{2} + q^{3} + 30 q^{4} + 6 q^{6} - 21 q^{7} - 6 q^{8} + 30 q^{9}+O(q^{10})$$ 21 * q - 2 * q^2 + q^3 + 30 * q^4 + 6 * q^6 - 21 * q^7 - 6 * q^8 + 30 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$21 q - 2 q^{2} + q^{3} + 30 q^{4} + 6 q^{6} - 21 q^{7} - 6 q^{8} + 30 q^{9} + 7 q^{11} + 22 q^{12} + 3 q^{13} + 2 q^{14} + 56 q^{16} - 7 q^{17} + 24 q^{19} - q^{21} - 4 q^{22} - 21 q^{23} + 24 q^{24} - 2 q^{26} + 19 q^{27} - 30 q^{28} + 11 q^{29} + 46 q^{31} + 6 q^{32} + 3 q^{33} + 28 q^{34} + 58 q^{36} - 24 q^{37} + 4 q^{38} + 31 q^{39} + 14 q^{41} - 6 q^{42} - 18 q^{43} + 12 q^{44} + 2 q^{46} + 25 q^{47} + 36 q^{48} + 21 q^{49} + 17 q^{51} + 8 q^{52} - 22 q^{53} - 6 q^{54} + 6 q^{56} - 40 q^{57} - 6 q^{58} + 10 q^{59} + 38 q^{61} + 54 q^{62} - 30 q^{63} + 100 q^{64} + 38 q^{66} - 12 q^{67} - 18 q^{68} - q^{69} + 56 q^{71} - 42 q^{72} + 40 q^{73} - 20 q^{74} + 60 q^{76} - 7 q^{77} - 38 q^{78} + 49 q^{79} + 57 q^{81} - 16 q^{82} + 2 q^{83} - 22 q^{84} + 16 q^{86} + 23 q^{87} - 12 q^{88} + 28 q^{89} - 3 q^{91} - 30 q^{92} + 30 q^{93} + 66 q^{94} + 46 q^{96} + q^{97} - 2 q^{98} - 2 q^{99}+O(q^{100})$$ 21 * q - 2 * q^2 + q^3 + 30 * q^4 + 6 * q^6 - 21 * q^7 - 6 * q^8 + 30 * q^9 + 7 * q^11 + 22 * q^12 + 3 * q^13 + 2 * q^14 + 56 * q^16 - 7 * q^17 + 24 * q^19 - q^21 - 4 * q^22 - 21 * q^23 + 24 * q^24 - 2 * q^26 + 19 * q^27 - 30 * q^28 + 11 * q^29 + 46 * q^31 + 6 * q^32 + 3 * q^33 + 28 * q^34 + 58 * q^36 - 24 * q^37 + 4 * q^38 + 31 * q^39 + 14 * q^41 - 6 * q^42 - 18 * q^43 + 12 * q^44 + 2 * q^46 + 25 * q^47 + 36 * q^48 + 21 * q^49 + 17 * q^51 + 8 * q^52 - 22 * q^53 - 6 * q^54 + 6 * q^56 - 40 * q^57 - 6 * q^58 + 10 * q^59 + 38 * q^61 + 54 * q^62 - 30 * q^63 + 100 * q^64 + 38 * q^66 - 12 * q^67 - 18 * q^68 - q^69 + 56 * q^71 - 42 * q^72 + 40 * q^73 - 20 * q^74 + 60 * q^76 - 7 * q^77 - 38 * q^78 + 49 * q^79 + 57 * q^81 - 16 * q^82 + 2 * q^83 - 22 * q^84 + 16 * q^86 + 23 * q^87 - 12 * q^88 + 28 * q^89 - 3 * q^91 - 30 * q^92 + 30 * q^93 + 66 * q^94 + 46 * q^96 + q^97 - 2 * q^98 - 2 * q^99

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −2.72900 3.06108 5.44746 0 −8.35371 −1.00000 −9.40815 6.37023 0
1.2 −2.70441 −1.97894 5.31383 0 5.35185 −1.00000 −8.96197 0.916187 0
1.3 −2.57149 −0.285863 4.61257 0 0.735095 −1.00000 −6.71821 −2.91828 0
1.4 −2.35921 −2.50751 3.56589 0 5.91575 −1.00000 −3.69426 3.28760 0
1.5 −2.11197 1.59044 2.46043 0 −3.35897 −1.00000 −0.972410 −0.470497 0
1.6 −1.61751 2.88096 0.616354 0 −4.65999 −1.00000 2.23807 5.29992 0
1.7 −1.50084 0.816695 0.252529 0 −1.22573 −1.00000 2.62268 −2.33301 0
1.8 −1.30923 −2.43985 −0.285909 0 3.19433 −1.00000 2.99279 2.95286 0
1.9 −0.817573 −1.89500 −1.33157 0 1.54930 −1.00000 2.72381 0.591014 0
1.10 −0.475047 0.568069 −1.77433 0 −0.269859 −1.00000 1.79298 −2.67730 0
1.11 0.157646 −0.829461 −1.97515 0 −0.130761 −1.00000 −0.626665 −2.31200 0
1.12 0.294301 −3.07644 −1.91339 0 −0.905400 −1.00000 −1.15171 6.46449 0
1.13 0.556064 2.94180 −1.69079 0 1.63583 −1.00000 −2.05232 5.65418 0
1.14 0.685218 1.12744 −1.53048 0 0.772540 −1.00000 −2.41915 −1.72889 0
1.15 1.22234 −1.28664 −0.505878 0 −1.57271 −1.00000 −3.06304 −1.34457 0
1.16 1.53505 0.943091 0.356385 0 1.44769 −1.00000 −2.52303 −2.11058 0
1.17 1.67541 −3.24787 0.806990 0 −5.44150 −1.00000 −1.99878 7.54864 0
1.18 2.20248 2.83822 2.85093 0 6.25112 −1.00000 1.87416 5.05547 0
1.19 2.38755 −1.09554 3.70041 0 −2.61567 −1.00000 4.05982 −1.79978 0
1.20 2.68001 3.08395 5.18247 0 8.26504 −1.00000 8.52906 6.51078 0
See all 21 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.21 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$7$$ $$1$$
$$23$$ $$1$$

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 4025.2.a.bd 21
5.b even 2 1 4025.2.a.be 21
5.c odd 4 2 805.2.c.c 42

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.c.c 42 5.c odd 4 2
4025.2.a.bd 21 1.a even 1 1 trivial
4025.2.a.be 21 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4025))$$:

 $$T_{2}^{21} + 2 T_{2}^{20} - 34 T_{2}^{19} - 66 T_{2}^{18} + 485 T_{2}^{17} + 904 T_{2}^{16} - 3786 T_{2}^{15} - 6668 T_{2}^{14} + 17725 T_{2}^{13} + 28768 T_{2}^{12} - 51455 T_{2}^{11} - 73860 T_{2}^{10} + 92892 T_{2}^{9} + \cdots - 256$$ T2^21 + 2*T2^20 - 34*T2^19 - 66*T2^18 + 485*T2^17 + 904*T2^16 - 3786*T2^15 - 6668*T2^14 + 17725*T2^13 + 28768*T2^12 - 51455*T2^11 - 73860*T2^10 + 92892*T2^9 + 109792*T2^8 - 102128*T2^7 - 86892*T2^6 + 64924*T2^5 + 30006*T2^4 - 21061*T2^3 - 2370*T2^2 + 2373*T2 - 256 $$T_{3}^{21} - T_{3}^{20} - 46 T_{3}^{19} + 38 T_{3}^{18} + 890 T_{3}^{17} - 570 T_{3}^{16} - 9413 T_{3}^{15} + 4269 T_{3}^{14} + 59173 T_{3}^{13} - 16601 T_{3}^{12} - 225401 T_{3}^{11} + 31625 T_{3}^{10} + 512056 T_{3}^{9} + \cdots + 2848$$ T3^21 - T3^20 - 46*T3^19 + 38*T3^18 + 890*T3^17 - 570*T3^16 - 9413*T3^15 + 4269*T3^14 + 59173*T3^13 - 16601*T3^12 - 225401*T3^11 + 31625*T3^10 + 512056*T3^9 - 25260*T3^8 - 670132*T3^7 + 2564*T3^6 + 476224*T3^5 + 8520*T3^4 - 161380*T3^3 - 7900*T3^2 + 18256*T3 + 2848 $$T_{11}^{21} - 7 T_{11}^{20} - 121 T_{11}^{19} + 953 T_{11}^{18} + 5502 T_{11}^{17} - 53080 T_{11}^{16} - 104669 T_{11}^{15} + 1554057 T_{11}^{14} + 129492 T_{11}^{13} - 25459994 T_{11}^{12} + 27371520 T_{11}^{11} + \cdots + 73032192$$ T11^21 - 7*T11^20 - 121*T11^19 + 953*T11^18 + 5502*T11^17 - 53080*T11^16 - 104669*T11^15 + 1554057*T11^14 + 129492*T11^13 - 25459994*T11^12 + 27371520*T11^11 + 226587608*T11^10 - 443237200*T11^9 - 930136640*T11^8 + 2809458544*T11^7 + 665953328*T11^6 - 6703945920*T11^5 + 3344210144*T11^4 + 4044646016*T11^3 - 3498584832*T11^2 + 505446912*T11 + 73032192