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Properties

Label 25.2.d.a.11.1
Level $25$
Weight $2$
Character 25.11
Analytic conductor $0.200$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [25,2,Mod(6,25)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("25.6"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(25, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 25.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.199626005053\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 11.1
Root \(0.809017 + 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 25.11
Dual form 25.2.d.a.16.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.363271i) q^{2} +(0.309017 - 0.951057i) q^{3} +(-0.500000 + 1.53884i) q^{4} +(-1.80902 - 1.31433i) q^{5} +(0.190983 + 0.587785i) q^{6} -1.61803 q^{7} +(-0.690983 - 2.12663i) q^{8} +(1.61803 + 1.17557i) q^{9} +1.38197 q^{10} +(0.618034 - 0.449028i) q^{11} +(1.30902 + 0.951057i) q^{12} +(3.92705 + 2.85317i) q^{13} +(0.809017 - 0.587785i) q^{14} +(-1.80902 + 1.31433i) q^{15} +(-1.50000 - 1.08981i) q^{16} +(-0.236068 - 0.726543i) q^{17} -1.23607 q^{18} +(-1.80902 - 5.56758i) q^{19} +(2.92705 - 2.12663i) q^{20} +(-0.500000 + 1.53884i) q^{21} +(-0.145898 + 0.449028i) q^{22} +(-6.66312 + 4.84104i) q^{23} -2.23607 q^{24} +(1.54508 + 4.75528i) q^{25} -3.00000 q^{26} +(4.04508 - 2.93893i) q^{27} +(0.809017 - 2.48990i) q^{28} +(-0.427051 + 1.31433i) q^{29} +(0.427051 - 1.31433i) q^{30} +(-0.927051 - 2.85317i) q^{31} +5.61803 q^{32} +(-0.236068 - 0.726543i) q^{33} +(0.381966 + 0.277515i) q^{34} +(2.92705 + 2.12663i) q^{35} +(-2.61803 + 1.90211i) q^{36} +(-3.42705 - 2.48990i) q^{37} +(2.92705 + 2.12663i) q^{38} +(3.92705 - 2.85317i) q^{39} +(-1.54508 + 4.75528i) q^{40} +(4.23607 + 3.07768i) q^{41} +(-0.309017 - 0.951057i) q^{42} +1.85410 q^{43} +(0.381966 + 1.17557i) q^{44} +(-1.38197 - 4.25325i) q^{45} +(1.57295 - 4.84104i) q^{46} +(-0.500000 + 1.53884i) q^{47} +(-1.50000 + 1.08981i) q^{48} -4.38197 q^{49} +(-2.50000 - 1.81636i) q^{50} -0.763932 q^{51} +(-6.35410 + 4.61653i) q^{52} +(1.69098 - 5.20431i) q^{53} +(-0.954915 + 2.93893i) q^{54} -1.70820 q^{55} +(1.11803 + 3.44095i) q^{56} -5.85410 q^{57} +(-0.263932 - 0.812299i) q^{58} +(3.35410 + 2.43690i) q^{59} +(-1.11803 - 3.44095i) q^{60} +(3.80902 - 2.76741i) q^{61} +(1.50000 + 1.08981i) q^{62} +(-2.61803 - 1.90211i) q^{63} +(0.190983 - 0.138757i) q^{64} +(-3.35410 - 10.3229i) q^{65} +(0.381966 + 0.277515i) q^{66} +(2.85410 + 8.78402i) q^{67} +1.23607 q^{68} +(2.54508 + 7.83297i) q^{69} -2.23607 q^{70} +(-1.35410 + 4.16750i) q^{71} +(1.38197 - 4.25325i) q^{72} +(7.28115 - 5.29007i) q^{73} +2.61803 q^{74} +5.00000 q^{75} +9.47214 q^{76} +(-1.00000 + 0.726543i) q^{77} +(-0.927051 + 2.85317i) q^{78} +(0.954915 - 2.93893i) q^{79} +(1.28115 + 3.94298i) q^{80} +(0.309017 + 0.951057i) q^{81} -3.23607 q^{82} +(-0.545085 - 1.67760i) q^{83} +(-2.11803 - 1.53884i) q^{84} +(-0.527864 + 1.62460i) q^{85} +(-0.927051 + 0.673542i) q^{86} +(1.11803 + 0.812299i) q^{87} +(-1.38197 - 1.00406i) q^{88} +(-7.23607 + 5.25731i) q^{89} +(2.23607 + 1.62460i) q^{90} +(-6.35410 - 4.61653i) q^{91} +(-4.11803 - 12.6740i) q^{92} -3.00000 q^{93} +(-0.309017 - 0.951057i) q^{94} +(-4.04508 + 12.4495i) q^{95} +(1.73607 - 5.34307i) q^{96} +(0.881966 - 2.71441i) q^{97} +(2.19098 - 1.59184i) q^{98} +1.52786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - q^{3} - 2 q^{4} - 5 q^{5} + 3 q^{6} - 2 q^{7} - 5 q^{8} + 2 q^{9} + 10 q^{10} - 2 q^{11} + 3 q^{12} + 9 q^{13} + q^{14} - 5 q^{15} - 6 q^{16} + 8 q^{17} + 4 q^{18} - 5 q^{19} + 5 q^{20}+ \cdots + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{4}{5}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 + 0.363271i −0.353553 + 0.256872i −0.750358 − 0.661031i \(-0.770119\pi\)
0.396805 + 0.917903i \(0.370119\pi\)
\(3\) 0.309017 − 0.951057i 0.178411 − 0.549093i −0.821362 − 0.570408i \(-0.806785\pi\)
0.999773 + 0.0213149i \(0.00678525\pi\)
\(4\) −0.500000 + 1.53884i −0.250000 + 0.769421i
\(5\) −1.80902 − 1.31433i −0.809017 − 0.587785i
\(6\) 0.190983 + 0.587785i 0.0779685 + 0.239962i
\(7\) −1.61803 −0.611559 −0.305780 − 0.952102i \(-0.598917\pi\)
−0.305780 + 0.952102i \(0.598917\pi\)
\(8\) −0.690983 − 2.12663i −0.244299 − 0.751876i
\(9\) 1.61803 + 1.17557i 0.539345 + 0.391857i
\(10\) 1.38197 0.437016
\(11\) 0.618034 − 0.449028i 0.186344 − 0.135387i −0.490702 − 0.871327i \(-0.663260\pi\)
0.677046 + 0.735940i \(0.263260\pi\)
\(12\) 1.30902 + 0.951057i 0.377881 + 0.274546i
\(13\) 3.92705 + 2.85317i 1.08917 + 0.791327i 0.979259 − 0.202615i \(-0.0649439\pi\)
0.109909 + 0.993942i \(0.464944\pi\)
\(14\) 0.809017 − 0.587785i 0.216219 − 0.157092i
\(15\) −1.80902 + 1.31433i −0.467086 + 0.339358i
\(16\) −1.50000 − 1.08981i −0.375000 − 0.272453i
\(17\) −0.236068 − 0.726543i −0.0572549 − 0.176212i 0.918339 − 0.395794i \(-0.129531\pi\)
−0.975594 + 0.219582i \(0.929531\pi\)
\(18\) −1.23607 −0.291344
\(19\) −1.80902 − 5.56758i −0.415017 − 1.27729i −0.912236 − 0.409666i \(-0.865645\pi\)
0.497219 − 0.867625i \(-0.334355\pi\)
\(20\) 2.92705 − 2.12663i 0.654508 − 0.475528i
\(21\) −0.500000 + 1.53884i −0.109109 + 0.335803i
\(22\) −0.145898 + 0.449028i −0.0311056 + 0.0957331i
\(23\) −6.66312 + 4.84104i −1.38936 + 1.00943i −0.393421 + 0.919359i \(0.628708\pi\)
−0.995936 + 0.0900679i \(0.971292\pi\)
\(24\) −2.23607 −0.456435
\(25\) 1.54508 + 4.75528i 0.309017 + 0.951057i
\(26\) −3.00000 −0.588348
\(27\) 4.04508 − 2.93893i 0.778477 − 0.565597i
\(28\) 0.809017 − 2.48990i 0.152890 − 0.470547i
\(29\) −0.427051 + 1.31433i −0.0793014 + 0.244065i −0.982846 − 0.184430i \(-0.940956\pi\)
0.903544 + 0.428495i \(0.140956\pi\)
\(30\) 0.427051 − 1.31433i 0.0779685 − 0.239962i
\(31\) −0.927051 − 2.85317i −0.166503 − 0.512444i 0.832641 − 0.553814i \(-0.186828\pi\)
−0.999144 + 0.0413693i \(0.986828\pi\)
\(32\) 5.61803 0.993137
\(33\) −0.236068 − 0.726543i −0.0410942 − 0.126475i
\(34\) 0.381966 + 0.277515i 0.0655066 + 0.0475934i
\(35\) 2.92705 + 2.12663i 0.494762 + 0.359466i
\(36\) −2.61803 + 1.90211i −0.436339 + 0.317019i
\(37\) −3.42705 − 2.48990i −0.563404 − 0.409337i 0.269299 − 0.963057i \(-0.413208\pi\)
−0.832703 + 0.553720i \(0.813208\pi\)
\(38\) 2.92705 + 2.12663i 0.474830 + 0.344984i
\(39\) 3.92705 − 2.85317i 0.628831 − 0.456873i
\(40\) −1.54508 + 4.75528i −0.244299 + 0.751876i
\(41\) 4.23607 + 3.07768i 0.661563 + 0.480653i 0.867190 − 0.497977i \(-0.165924\pi\)
−0.205628 + 0.978630i \(0.565924\pi\)
\(42\) −0.309017 − 0.951057i −0.0476824 − 0.146751i
\(43\) 1.85410 0.282748 0.141374 − 0.989956i \(-0.454848\pi\)
0.141374 + 0.989956i \(0.454848\pi\)
\(44\) 0.381966 + 1.17557i 0.0575835 + 0.177224i
\(45\) −1.38197 − 4.25325i −0.206011 − 0.634038i
\(46\) 1.57295 − 4.84104i 0.231919 − 0.713772i
\(47\) −0.500000 + 1.53884i −0.0729325 + 0.224463i −0.980877 − 0.194626i \(-0.937651\pi\)
0.907945 + 0.419089i \(0.137651\pi\)
\(48\) −1.50000 + 1.08981i −0.216506 + 0.157301i
\(49\) −4.38197 −0.625995
\(50\) −2.50000 − 1.81636i −0.353553 − 0.256872i
\(51\) −0.763932 −0.106972
\(52\) −6.35410 + 4.61653i −0.881155 + 0.640197i
\(53\) 1.69098 − 5.20431i 0.232274 − 0.714867i −0.765197 − 0.643796i \(-0.777358\pi\)
0.997471 − 0.0710707i \(-0.0226416\pi\)
\(54\) −0.954915 + 2.93893i −0.129947 + 0.399937i
\(55\) −1.70820 −0.230334
\(56\) 1.11803 + 3.44095i 0.149404 + 0.459817i
\(57\) −5.85410 −0.775395
\(58\) −0.263932 − 0.812299i −0.0346560 − 0.106660i
\(59\) 3.35410 + 2.43690i 0.436667 + 0.317257i 0.784309 − 0.620370i \(-0.213018\pi\)
−0.347642 + 0.937627i \(0.613018\pi\)
\(60\) −1.11803 − 3.44095i −0.144338 − 0.444225i
\(61\) 3.80902 − 2.76741i 0.487695 − 0.354331i −0.316602 − 0.948558i \(-0.602542\pi\)
0.804297 + 0.594227i \(0.202542\pi\)
\(62\) 1.50000 + 1.08981i 0.190500 + 0.138406i
\(63\) −2.61803 − 1.90211i −0.329841 − 0.239644i
\(64\) 0.190983 − 0.138757i 0.0238729 − 0.0173447i
\(65\) −3.35410 − 10.3229i −0.416025 − 1.28039i
\(66\) 0.381966 + 0.277515i 0.0470168 + 0.0341597i
\(67\) 2.85410 + 8.78402i 0.348684 + 1.07314i 0.959582 + 0.281430i \(0.0908086\pi\)
−0.610898 + 0.791709i \(0.709191\pi\)
\(68\) 1.23607 0.149895
\(69\) 2.54508 + 7.83297i 0.306392 + 0.942978i
\(70\) −2.23607 −0.267261
\(71\) −1.35410 + 4.16750i −0.160702 + 0.494591i −0.998694 − 0.0510922i \(-0.983730\pi\)
0.837992 + 0.545683i \(0.183730\pi\)
\(72\) 1.38197 − 4.25325i 0.162866 − 0.501251i
\(73\) 7.28115 − 5.29007i 0.852194 − 0.619156i −0.0735557 − 0.997291i \(-0.523435\pi\)
0.925750 + 0.378136i \(0.123435\pi\)
\(74\) 2.61803 0.304340
\(75\) 5.00000 0.577350
\(76\) 9.47214 1.08653
\(77\) −1.00000 + 0.726543i −0.113961 + 0.0827972i
\(78\) −0.927051 + 2.85317i −0.104968 + 0.323058i
\(79\) 0.954915 − 2.93893i 0.107436 − 0.330655i −0.882858 − 0.469640i \(-0.844384\pi\)
0.990295 + 0.138985i \(0.0443839\pi\)
\(80\) 1.28115 + 3.94298i 0.143237 + 0.440839i
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) −3.23607 −0.357364
\(83\) −0.545085 − 1.67760i −0.0598308 − 0.184140i 0.916674 − 0.399636i \(-0.130863\pi\)
−0.976505 + 0.215495i \(0.930863\pi\)
\(84\) −2.11803 − 1.53884i −0.231096 − 0.167901i
\(85\) −0.527864 + 1.62460i −0.0572549 + 0.176212i
\(86\) −0.927051 + 0.673542i −0.0999665 + 0.0726299i
\(87\) 1.11803 + 0.812299i 0.119866 + 0.0870876i
\(88\) −1.38197 − 1.00406i −0.147318 − 0.107033i
\(89\) −7.23607 + 5.25731i −0.767022 + 0.557274i −0.901056 − 0.433703i \(-0.857207\pi\)
0.134034 + 0.990977i \(0.457207\pi\)
\(90\) 2.23607 + 1.62460i 0.235702 + 0.171248i
\(91\) −6.35410 − 4.61653i −0.666091 − 0.483943i
\(92\) −4.11803 − 12.6740i −0.429335 − 1.32136i
\(93\) −3.00000 −0.311086
\(94\) −0.309017 − 0.951057i −0.0318727 − 0.0980940i
\(95\) −4.04508 + 12.4495i −0.415017 + 1.27729i
\(96\) 1.73607 − 5.34307i 0.177187 − 0.545325i
\(97\) 0.881966 − 2.71441i 0.0895501 − 0.275607i −0.896245 − 0.443559i \(-0.853716\pi\)
0.985795 + 0.167953i \(0.0537155\pi\)
\(98\) 2.19098 − 1.59184i 0.221323 − 0.160800i
\(99\) 1.52786 0.153556
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 25.2.d.a.11.1 ✓ 4
3.2 odd 2 225.2.h.b.136.1 4
4.3 odd 2 400.2.u.b.161.1 4
5.2 odd 4 125.2.e.a.74.1 8
5.3 odd 4 125.2.e.a.74.2 8
5.4 even 2 125.2.d.a.51.1 4
25.2 odd 20 625.2.e.c.499.1 8
25.3 odd 20 625.2.b.a.624.2 4
25.4 even 10 625.2.a.c.1.1 2
25.6 even 5 625.2.d.h.501.1 4
25.8 odd 20 625.2.e.c.124.1 8
25.9 even 10 125.2.d.a.76.1 4
25.11 even 5 625.2.d.h.126.1 4
25.12 odd 20 125.2.e.a.49.2 8
25.13 odd 20 125.2.e.a.49.1 8
25.14 even 10 625.2.d.b.126.1 4
25.16 even 5 inner 25.2.d.a.16.1 yes 4
25.17 odd 20 625.2.e.c.124.2 8
25.19 even 10 625.2.d.b.501.1 4
25.21 even 5 625.2.a.b.1.2 2
25.22 odd 20 625.2.b.a.624.3 4
25.23 odd 20 625.2.e.c.499.2 8
75.29 odd 10 5625.2.a.d.1.2 2
75.41 odd 10 225.2.h.b.91.1 4
75.71 odd 10 5625.2.a.f.1.1 2
100.71 odd 10 10000.2.a.c.1.2 2
100.79 odd 10 10000.2.a.l.1.1 2
100.91 odd 10 400.2.u.b.241.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.2.d.a.11.1 ✓ 4 1.1 even 1 trivial
25.2.d.a.16.1 yes 4 25.16 even 5 inner
125.2.d.a.51.1 4 5.4 even 2
125.2.d.a.76.1 4 25.9 even 10
125.2.e.a.49.1 8 25.13 odd 20
125.2.e.a.49.2 8 25.12 odd 20
125.2.e.a.74.1 8 5.2 odd 4
125.2.e.a.74.2 8 5.3 odd 4
225.2.h.b.91.1 4 75.41 odd 10
225.2.h.b.136.1 4 3.2 odd 2
400.2.u.b.161.1 4 4.3 odd 2
400.2.u.b.241.1 4 100.91 odd 10
625.2.a.b.1.2 2 25.21 even 5
625.2.a.c.1.1 2 25.4 even 10
625.2.b.a.624.2 4 25.3 odd 20
625.2.b.a.624.3 4 25.22 odd 20
625.2.d.b.126.1 4 25.14 even 10
625.2.d.b.501.1 4 25.19 even 10
625.2.d.h.126.1 4 25.11 even 5
625.2.d.h.501.1 4 25.6 even 5
625.2.e.c.124.1 8 25.8 odd 20
625.2.e.c.124.2 8 25.17 odd 20
625.2.e.c.499.1 8 25.2 odd 20
625.2.e.c.499.2 8 25.23 odd 20
5625.2.a.d.1.2 2 75.29 odd 10
5625.2.a.f.1.1 2 75.71 odd 10
10000.2.a.c.1.2 2 100.71 odd 10
10000.2.a.l.1.1 2 100.79 odd 10