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Properties

Label 625.2.d.o.501.1
Level $625$
Weight $2$
Character 625.501
Analytic conductor $4.991$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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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] = [625,2,Mod(126,625)] 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("625.126"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(625, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.d (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,-8,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.99065012633\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 501.1
Root \(-0.917186 - 1.66637i\) of defining polynomial
Character \(\chi\) \(=\) 625.501
Dual form 625.2.d.o.126.1

$q$-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+(-0.713605 + 2.19625i) q^{2} +(0.384204 + 0.279141i) q^{3} +(-2.69625 - 1.95894i) q^{4} +(-0.887234 + 0.644613i) q^{6} -3.03582 q^{7} +(2.48990 - 1.80902i) q^{8} +(-0.857358 - 2.63868i) q^{9} +(0.618034 - 1.90211i) q^{11} +(-0.489091 - 1.50527i) q^{12} +(0.441032 + 1.35736i) q^{13} +(2.16637 - 6.66742i) q^{14} +(0.136498 + 0.420099i) q^{16} +(1.50497 - 1.09343i) q^{17} +6.40701 q^{18} +(-0.730800 + 0.530958i) q^{19} +(-1.16637 - 0.847421i) q^{21} +(3.73648 + 2.71472i) q^{22} +(1.02882 - 3.16637i) q^{23} +1.46160 q^{24} -3.29582 q^{26} +(0.847421 - 2.60809i) q^{27} +(8.18532 + 5.94699i) q^{28} +(-3.20619 - 2.32943i) q^{29} +(5.21004 - 3.78532i) q^{31} +5.13532 q^{32} +(0.768409 - 0.558282i) q^{33} +(1.32748 + 4.08557i) q^{34} +(-2.85736 + 8.79404i) q^{36} +(-1.18051 - 3.63324i) q^{37} +(-0.644613 - 1.98391i) q^{38} +(-0.209447 + 0.644613i) q^{39} +(-0.566805 - 1.74445i) q^{41} +(2.69348 - 1.95693i) q^{42} +3.59445 q^{43} +(-5.39250 + 3.91788i) q^{44} +(6.21998 + 4.51908i) q^{46} +(-3.88324 - 2.82134i) q^{47} +(-0.0648235 + 0.199506i) q^{48} +2.21619 q^{49} +0.883436 q^{51} +(1.46985 - 4.52373i) q^{52} +(-7.68949 - 5.58674i) q^{53} +(5.12330 + 3.72230i) q^{54} +(-7.55888 + 5.49184i) q^{56} -0.428989 q^{57} +(7.40398 - 5.37930i) q^{58} +(3.28968 + 10.1246i) q^{59} +(4.41097 - 13.5756i) q^{61} +(4.59559 + 14.1438i) q^{62} +(2.60278 + 8.01054i) q^{63} +(-3.93759 + 12.1186i) q^{64} +(0.677786 + 2.08601i) q^{66} +(-8.64854 + 6.28353i) q^{67} -6.19974 q^{68} +(1.27914 - 0.929350i) q^{69} +(-10.0802 - 7.32371i) q^{71} +(-6.90814 - 5.01906i) q^{72} +(-0.0827026 + 0.254532i) q^{73} +8.82193 q^{74} +3.01054 q^{76} +(-1.87624 + 5.77447i) q^{77} +(-1.26627 - 0.919998i) q^{78} +(6.93470 + 5.03835i) q^{79} +(-5.68017 + 4.12688i) q^{81} +4.23572 q^{82} +(-10.2083 + 7.41677i) q^{83} +(1.48479 + 4.56972i) q^{84} +(-2.56502 + 7.89432i) q^{86} +(-0.581593 - 1.78996i) q^{87} +(-1.90211 - 5.85410i) q^{88} +(-1.47338 + 4.53460i) q^{89} +(-1.33889 - 4.12069i) q^{91} +(-8.97669 + 6.52195i) q^{92} +3.05836 q^{93} +(8.96746 - 6.51524i) q^{94} +(1.97301 + 1.43348i) q^{96} +(-8.05623 - 5.85319i) q^{97} +(-1.58148 + 4.86730i) q^{98} -5.54893 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{4} + 12 q^{6} - 2 q^{9} - 8 q^{11} + 14 q^{14} - 4 q^{16} - 20 q^{19} + 2 q^{21} + 40 q^{24} + 12 q^{26} - 30 q^{29} + 2 q^{31} + 24 q^{34} - 34 q^{36} - 24 q^{39} - 18 q^{41} - 16 q^{44} + 32 q^{46}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/625\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.713605 + 2.19625i −0.504595 + 1.55298i 0.296855 + 0.954922i \(0.404062\pi\)
−0.801450 + 0.598061i \(0.795938\pi\)
\(3\) 0.384204 + 0.279141i 0.221821 + 0.161162i 0.693146 − 0.720797i \(-0.256224\pi\)
−0.471325 + 0.881959i \(0.656224\pi\)
\(4\) −2.69625 − 1.95894i −1.34813 − 0.979470i
\(5\) 0 0
\(6\) −0.887234 + 0.644613i −0.362212 + 0.263162i
\(7\) −3.03582 −1.14743 −0.573716 − 0.819055i \(-0.694498\pi\)
−0.573716 + 0.819055i \(0.694498\pi\)
\(8\) 2.48990 − 1.80902i 0.880312 − 0.639584i
\(9\) −0.857358 − 2.63868i −0.285786 − 0.879558i
\(10\) 0 0
\(11\) 0.618034 − 1.90211i 0.186344 − 0.573509i −0.813625 − 0.581390i \(-0.802509\pi\)
0.999969 + 0.00788181i \(0.00250889\pi\)
\(12\) −0.489091 − 1.50527i −0.141188 − 0.434533i
\(13\) 0.441032 + 1.35736i 0.122320 + 0.376463i 0.993403 − 0.114673i \(-0.0365819\pi\)
−0.871083 + 0.491136i \(0.836582\pi\)
\(14\) 2.16637 − 6.66742i 0.578988 − 1.78194i
\(15\) 0 0
\(16\) 0.136498 + 0.420099i 0.0341246 + 0.105025i
\(17\) 1.50497 − 1.09343i 0.365009 − 0.265195i −0.390129 − 0.920760i \(-0.627570\pi\)
0.755138 + 0.655565i \(0.227570\pi\)
\(18\) 6.40701 1.51015
\(19\) −0.730800 + 0.530958i −0.167657 + 0.121810i −0.668450 − 0.743757i \(-0.733042\pi\)
0.500793 + 0.865567i \(0.333042\pi\)
\(20\) 0 0
\(21\) −1.16637 − 0.847421i −0.254524 − 0.184922i
\(22\) 3.73648 + 2.71472i 0.796621 + 0.578779i
\(23\) 1.02882 − 3.16637i 0.214523 − 0.660235i −0.784664 − 0.619922i \(-0.787164\pi\)
0.999187 − 0.0403132i \(-0.0128356\pi\)
\(24\) 1.46160 0.298348
\(25\) 0 0
\(26\) −3.29582 −0.646364
\(27\) 0.847421 − 2.60809i 0.163086 − 0.501928i
\(28\) 8.18532 + 5.94699i 1.54688 + 1.12387i
\(29\) −3.20619 − 2.32943i −0.595375 − 0.432565i 0.248859 − 0.968540i \(-0.419944\pi\)
−0.844234 + 0.535974i \(0.819944\pi\)
\(30\) 0 0
\(31\) 5.21004 − 3.78532i 0.935751 − 0.679863i −0.0116431 − 0.999932i \(-0.503706\pi\)
0.947394 + 0.320069i \(0.103706\pi\)
\(32\) 5.13532 0.907805
\(33\) 0.768409 − 0.558282i 0.133763 − 0.0971844i
\(34\) 1.32748 + 4.08557i 0.227661 + 0.700669i
\(35\) 0 0
\(36\) −2.85736 + 8.79404i −0.476226 + 1.46567i
\(37\) −1.18051 − 3.63324i −0.194075 − 0.597301i −0.999986 − 0.00526493i \(-0.998324\pi\)
0.805911 − 0.592037i \(-0.201676\pi\)
\(38\) −0.644613 − 1.98391i −0.104570 − 0.321833i
\(39\) −0.209447 + 0.644613i −0.0335384 + 0.103221i
\(40\) 0 0
\(41\) −0.566805 − 1.74445i −0.0885201 − 0.272437i 0.896991 − 0.442049i \(-0.145748\pi\)
−0.985511 + 0.169613i \(0.945748\pi\)
\(42\) 2.69348 − 1.95693i 0.415613 − 0.301960i
\(43\) 3.59445 0.548149 0.274074 − 0.961708i \(-0.411629\pi\)
0.274074 + 0.961708i \(0.411629\pi\)
\(44\) −5.39250 + 3.91788i −0.812950 + 0.590643i
\(45\) 0 0
\(46\) 6.21998 + 4.51908i 0.917086 + 0.666302i
\(47\) −3.88324 − 2.82134i −0.566428 − 0.411534i 0.267378 − 0.963592i \(-0.413843\pi\)
−0.833806 + 0.552057i \(0.813843\pi\)
\(48\) −0.0648235 + 0.199506i −0.00935647 + 0.0287962i
\(49\) 2.21619 0.316598
\(50\) 0 0
\(51\) 0.883436 0.123706
\(52\) 1.46985 − 4.52373i 0.203831 − 0.627329i
\(53\) −7.68949 − 5.58674i −1.05623 − 0.767398i −0.0828447 − 0.996562i \(-0.526401\pi\)
−0.973388 + 0.229165i \(0.926401\pi\)
\(54\) 5.12330 + 3.72230i 0.697193 + 0.506540i
\(55\) 0 0
\(56\) −7.55888 + 5.49184i −1.01010 + 0.733879i
\(57\) −0.428989 −0.0568209
\(58\) 7.40398 − 5.37930i 0.972190 − 0.706337i
\(59\) 3.28968 + 10.1246i 0.428279 + 1.31811i 0.899819 + 0.436263i \(0.143698\pi\)
−0.471540 + 0.881845i \(0.656302\pi\)
\(60\) 0 0
\(61\) 4.41097 − 13.5756i 0.564766 − 1.73817i −0.103879 − 0.994590i \(-0.533125\pi\)
0.668645 − 0.743582i \(-0.266875\pi\)
\(62\) 4.59559 + 14.1438i 0.583641 + 1.79626i
\(63\) 2.60278 + 8.01054i 0.327920 + 1.00923i
\(64\) −3.93759 + 12.1186i −0.492198 + 1.51483i
\(65\) 0 0
\(66\) 0.677786 + 2.08601i 0.0834297 + 0.256770i
\(67\) −8.64854 + 6.28353i −1.05659 + 0.767656i −0.973454 − 0.228883i \(-0.926493\pi\)
−0.0831333 + 0.996538i \(0.526493\pi\)
\(68\) −6.19974 −0.751828
\(69\) 1.27914 − 0.929350i 0.153990 − 0.111881i
\(70\) 0 0
\(71\) −10.0802 − 7.32371i −1.19630 − 0.869165i −0.202387 − 0.979306i \(-0.564870\pi\)
−0.993916 + 0.110141i \(0.964870\pi\)
\(72\) −6.90814 − 5.01906i −0.814132 − 0.591502i
\(73\) −0.0827026 + 0.254532i −0.00967961 + 0.0297908i −0.955780 − 0.294084i \(-0.904986\pi\)
0.946100 + 0.323874i \(0.104986\pi\)
\(74\) 8.82193 1.02553
\(75\) 0 0
\(76\) 3.01054 0.345332
\(77\) −1.87624 + 5.77447i −0.213817 + 0.658062i
\(78\) −1.26627 − 0.919998i −0.143377 − 0.104169i
\(79\) 6.93470 + 5.03835i 0.780214 + 0.566859i 0.905043 − 0.425319i \(-0.139838\pi\)
−0.124829 + 0.992178i \(0.539838\pi\)
\(80\) 0 0
\(81\) −5.68017 + 4.12688i −0.631129 + 0.458542i
\(82\) 4.23572 0.467757
\(83\) −10.2083 + 7.41677i −1.12051 + 0.814097i −0.984286 − 0.176582i \(-0.943496\pi\)
−0.136222 + 0.990678i \(0.543496\pi\)
\(84\) 1.48479 + 4.56972i 0.162004 + 0.498597i
\(85\) 0 0
\(86\) −2.56502 + 7.89432i −0.276593 + 0.851266i
\(87\) −0.581593 − 1.78996i −0.0623533 − 0.191904i
\(88\) −1.90211 − 5.85410i −0.202766 − 0.624049i
\(89\) −1.47338 + 4.53460i −0.156178 + 0.480666i −0.998278 − 0.0586546i \(-0.981319\pi\)
0.842100 + 0.539321i \(0.181319\pi\)
\(90\) 0 0
\(91\) −1.33889 − 4.12069i −0.140354 − 0.431966i
\(92\) −8.97669 + 6.52195i −0.935885 + 0.679960i
\(93\) 3.05836 0.317137
\(94\) 8.96746 − 6.51524i 0.924923 − 0.671996i
\(95\) 0 0
\(96\) 1.97301 + 1.43348i 0.201370 + 0.146304i
\(97\) −8.05623 − 5.85319i −0.817986 − 0.594302i 0.0981488 − 0.995172i \(-0.468708\pi\)
−0.916135 + 0.400870i \(0.868708\pi\)
\(98\) −1.58148 + 4.86730i −0.159754 + 0.491671i
\(99\) −5.54893 −0.557689
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 625.2.d.o.501.1 16
5.2 odd 4 625.2.e.i.124.1 8
5.3 odd 4 625.2.e.a.124.2 8
5.4 even 2 inner 625.2.d.o.501.4 16
25.2 odd 20 25.2.e.a.9.1 ✓ 8
25.3 odd 20 25.2.e.a.14.1 yes 8
25.4 even 10 125.2.d.b.51.1 16
25.6 even 5 inner 625.2.d.o.126.1 16
25.8 odd 20 625.2.e.i.499.1 8
25.9 even 10 625.2.a.f.1.8 8
25.11 even 5 125.2.d.b.76.4 16
25.12 odd 20 625.2.b.c.624.1 8
25.13 odd 20 625.2.b.c.624.8 8
25.14 even 10 125.2.d.b.76.1 16
25.16 even 5 625.2.a.f.1.1 8
25.17 odd 20 625.2.e.a.499.2 8
25.19 even 10 inner 625.2.d.o.126.4 16
25.21 even 5 125.2.d.b.51.4 16
25.22 odd 20 125.2.e.b.74.2 8
25.23 odd 20 125.2.e.b.49.2 8
75.2 even 20 225.2.m.a.109.2 8
75.41 odd 10 5625.2.a.x.1.8 8
75.53 even 20 225.2.m.a.64.2 8
75.59 odd 10 5625.2.a.x.1.1 8
100.3 even 20 400.2.y.c.289.1 8
100.27 even 20 400.2.y.c.209.1 8
100.59 odd 10 10000.2.a.bj.1.4 8
100.91 odd 10 10000.2.a.bj.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.2.e.a.9.1 ✓ 8 25.2 odd 20
25.2.e.a.14.1 yes 8 25.3 odd 20
125.2.d.b.51.1 16 25.4 even 10
125.2.d.b.51.4 16 25.21 even 5
125.2.d.b.76.1 16 25.14 even 10
125.2.d.b.76.4 16 25.11 even 5
125.2.e.b.49.2 8 25.23 odd 20
125.2.e.b.74.2 8 25.22 odd 20
225.2.m.a.64.2 8 75.53 even 20
225.2.m.a.109.2 8 75.2 even 20
400.2.y.c.209.1 8 100.27 even 20
400.2.y.c.289.1 8 100.3 even 20
625.2.a.f.1.1 8 25.16 even 5
625.2.a.f.1.8 8 25.9 even 10
625.2.b.c.624.1 8 25.12 odd 20
625.2.b.c.624.8 8 25.13 odd 20
625.2.d.o.126.1 16 25.6 even 5 inner
625.2.d.o.126.4 16 25.19 even 10 inner
625.2.d.o.501.1 16 1.1 even 1 trivial
625.2.d.o.501.4 16 5.4 even 2 inner
625.2.e.a.124.2 8 5.3 odd 4
625.2.e.a.499.2 8 25.17 odd 20
625.2.e.i.124.1 8 5.2 odd 4
625.2.e.i.499.1 8 25.8 odd 20
5625.2.a.x.1.1 8 75.59 odd 10
5625.2.a.x.1.8 8 75.41 odd 10
10000.2.a.bj.1.4 8 100.59 odd 10
10000.2.a.bj.1.5 8 100.91 odd 10