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Properties

Label 625.2.d.n.501.1
Level $625$
Weight $2$
Character 625.501
Analytic conductor $4.991$
Analytic rank $0$
Dimension $16$
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] = [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,-5,-8,0,-3] 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: \( x^{16} + 25x^{14} + 239x^{12} + 1165x^{10} + 3166x^{8} + 4820x^{6} + 3809x^{4} + 1205x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 501.1
Root \(-0.991969i\) of defining polynomial
Character \(\chi\) \(=\) 625.501
Dual form 625.2.d.n.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+(-0.520202 + 1.60102i) q^{2} +(-0.574677 - 0.417528i) q^{3} +(-0.674615 - 0.490137i) q^{4} +(0.967418 - 0.702870i) q^{6} +4.59110 q^{7} +(-1.58816 + 1.15387i) q^{8} +(-0.771126 - 2.37328i) q^{9} +(1.20974 - 3.72319i) q^{11} +(0.183041 + 0.563341i) q^{12} +(0.177048 + 0.544898i) q^{13} +(-2.38830 + 7.35044i) q^{14} +(-1.53656 - 4.72903i) q^{16} +(-0.188186 + 0.136725i) q^{17} +4.20081 q^{18} +(4.49012 - 3.26227i) q^{19} +(-2.63840 - 1.91691i) q^{21} +(5.33158 + 3.87362i) q^{22} +(1.52428 - 4.69124i) q^{23} +1.39445 q^{24} -0.964492 q^{26} +(-1.20628 + 3.71256i) q^{27} +(-3.09723 - 2.25027i) q^{28} +(-3.34174 - 2.42792i) q^{29} +(2.82777 - 2.05449i) q^{31} +4.44444 q^{32} +(-2.24974 + 1.63453i) q^{33} +(-0.121005 - 0.372415i) q^{34} +(-0.643019 + 1.97901i) q^{36} +(1.67378 + 5.15138i) q^{37} +(2.88717 + 8.88581i) q^{38} +(0.125764 - 0.387063i) q^{39} +(3.22079 + 9.91257i) q^{41} +(4.44152 - 3.22695i) q^{42} -1.38833 q^{43} +(-2.64098 + 1.91878i) q^{44} +(6.71783 + 4.88079i) q^{46} +(0.744634 + 0.541008i) q^{47} +(-1.09148 + 3.35922i) q^{48} +14.0782 q^{49} +0.165233 q^{51} +(0.147635 - 0.454374i) q^{52} +(0.996044 + 0.723668i) q^{53} +(-5.31636 - 3.86256i) q^{54} +(-7.29141 + 5.29752i) q^{56} -3.94246 q^{57} +(5.62552 - 4.08718i) q^{58} +(1.39299 + 4.28718i) q^{59} +(-3.60276 + 11.0881i) q^{61} +(1.81827 + 5.59606i) q^{62} +(-3.54032 - 10.8960i) q^{63} +(0.761103 - 2.34244i) q^{64} +(-1.44660 - 4.45216i) q^{66} +(-2.39022 + 1.73660i) q^{67} +0.193968 q^{68} +(-2.83469 + 2.05952i) q^{69} +(-2.59331 - 1.88415i) q^{71} +(3.96312 + 2.87938i) q^{72} +(3.18047 - 9.78847i) q^{73} -9.11816 q^{74} -4.62806 q^{76} +(5.55402 - 17.0935i) q^{77} +(0.554272 + 0.402702i) q^{78} +(7.77877 + 5.65161i) q^{79} +(-3.81318 + 2.77044i) q^{81} -17.5457 q^{82} +(-8.48125 + 6.16199i) q^{83} +(0.840358 + 2.58636i) q^{84} +(0.722211 - 2.22274i) q^{86} +(0.906701 + 2.79054i) q^{87} +(2.37480 + 7.30889i) q^{88} +(2.24293 - 6.90303i) q^{89} +(0.812846 + 2.50168i) q^{91} +(-3.32765 + 2.41768i) q^{92} -2.48286 q^{93} +(-1.25352 + 0.910739i) q^{94} +(-2.55412 - 1.85568i) q^{96} +(-6.73079 - 4.89020i) q^{97} +(-7.32353 + 22.5395i) q^{98} -9.76903 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 5 q^{3} - 8 q^{4} - 3 q^{6} + 20 q^{7} - 10 q^{8} + 3 q^{9} + 2 q^{11} - 25 q^{12} - 5 q^{13} + 9 q^{14} - 14 q^{16} + 10 q^{17} - 10 q^{18} + 7 q^{21} + 40 q^{22} - 15 q^{23} + 10 q^{24} + 22 q^{26}+ \cdots - 44 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.520202 + 1.60102i −0.367839 + 1.13209i 0.580346 + 0.814370i \(0.302917\pi\)
−0.948184 + 0.317721i \(0.897083\pi\)
\(3\) −0.574677 − 0.417528i −0.331790 − 0.241060i 0.409400 − 0.912355i \(-0.365738\pi\)
−0.741190 + 0.671295i \(0.765738\pi\)
\(4\) −0.674615 − 0.490137i −0.337308 − 0.245068i
\(5\) 0 0
\(6\) 0.967418 − 0.702870i 0.394947 − 0.286946i
\(7\) 4.59110 1.73527 0.867637 − 0.497198i \(-0.165638\pi\)
0.867637 + 0.497198i \(0.165638\pi\)
\(8\) −1.58816 + 1.15387i −0.561500 + 0.407953i
\(9\) −0.771126 − 2.37328i −0.257042 − 0.791094i
\(10\) 0 0
\(11\) 1.20974 − 3.72319i 0.364749 − 1.12258i −0.585389 − 0.810753i \(-0.699058\pi\)
0.950138 − 0.311830i \(-0.100942\pi\)
\(12\) 0.183041 + 0.563341i 0.0528393 + 0.162623i
\(13\) 0.177048 + 0.544898i 0.0491043 + 0.151127i 0.972602 − 0.232476i \(-0.0746828\pi\)
−0.923498 + 0.383604i \(0.874683\pi\)
\(14\) −2.38830 + 7.35044i −0.638301 + 1.96449i
\(15\) 0 0
\(16\) −1.53656 − 4.72903i −0.384139 − 1.18226i
\(17\) −0.188186 + 0.136725i −0.0456419 + 0.0331608i −0.610372 − 0.792115i \(-0.708980\pi\)
0.564730 + 0.825275i \(0.308980\pi\)
\(18\) 4.20081 0.990140
\(19\) 4.49012 − 3.26227i 1.03010 − 0.748415i 0.0617752 − 0.998090i \(-0.480324\pi\)
0.968330 + 0.249675i \(0.0803238\pi\)
\(20\) 0 0
\(21\) −2.63840 − 1.91691i −0.575747 − 0.418305i
\(22\) 5.33158 + 3.87362i 1.13670 + 0.825858i
\(23\) 1.52428 − 4.69124i 0.317834 − 0.978192i −0.656738 − 0.754118i \(-0.728064\pi\)
0.974572 − 0.224073i \(-0.0719356\pi\)
\(24\) 1.39445 0.284641
\(25\) 0 0
\(26\) −0.964492 −0.189152
\(27\) −1.20628 + 3.71256i −0.232149 + 0.714482i
\(28\) −3.09723 − 2.25027i −0.585321 − 0.425261i
\(29\) −3.34174 − 2.42792i −0.620546 − 0.450853i 0.232566 − 0.972581i \(-0.425288\pi\)
−0.853112 + 0.521727i \(0.825288\pi\)
\(30\) 0 0
\(31\) 2.82777 − 2.05449i 0.507882 − 0.368998i −0.304138 − 0.952628i \(-0.598368\pi\)
0.812020 + 0.583630i \(0.198368\pi\)
\(32\) 4.44444 0.785674
\(33\) −2.24974 + 1.63453i −0.391630 + 0.284536i
\(34\) −0.121005 − 0.372415i −0.0207522 − 0.0638686i
\(35\) 0 0
\(36\) −0.643019 + 1.97901i −0.107170 + 0.329835i
\(37\) 1.67378 + 5.15138i 0.275169 + 0.846882i 0.989175 + 0.146742i \(0.0468788\pi\)
−0.714006 + 0.700139i \(0.753121\pi\)
\(38\) 2.88717 + 8.88581i 0.468361 + 1.44147i
\(39\) 0.125764 − 0.387063i 0.0201384 − 0.0619797i
\(40\) 0 0
\(41\) 3.22079 + 9.91257i 0.503003 + 1.54808i 0.804103 + 0.594491i \(0.202646\pi\)
−0.301100 + 0.953593i \(0.597354\pi\)
\(42\) 4.44152 − 3.22695i 0.685341 − 0.497929i
\(43\) −1.38833 −0.211718 −0.105859 − 0.994381i \(-0.533759\pi\)
−0.105859 + 0.994381i \(0.533759\pi\)
\(44\) −2.64098 + 1.91878i −0.398142 + 0.289267i
\(45\) 0 0
\(46\) 6.71783 + 4.88079i 0.990491 + 0.719634i
\(47\) 0.744634 + 0.541008i 0.108616 + 0.0789142i 0.640767 − 0.767735i \(-0.278616\pi\)
−0.532151 + 0.846649i \(0.678616\pi\)
\(48\) −1.09148 + 3.35922i −0.157541 + 0.484862i
\(49\) 14.0782 2.01118
\(50\) 0 0
\(51\) 0.165233 0.0231373
\(52\) 0.147635 − 0.454374i 0.0204733 − 0.0630103i
\(53\) 0.996044 + 0.723668i 0.136817 + 0.0994034i 0.654089 − 0.756418i \(-0.273052\pi\)
−0.517272 + 0.855821i \(0.673052\pi\)
\(54\) −5.31636 − 3.86256i −0.723466 − 0.525628i
\(55\) 0 0
\(56\) −7.29141 + 5.29752i −0.974356 + 0.707911i
\(57\) −3.94246 −0.522191
\(58\) 5.62552 − 4.08718i 0.738668 − 0.536673i
\(59\) 1.39299 + 4.28718i 0.181352 + 0.558143i 0.999866 − 0.0163426i \(-0.00520224\pi\)
−0.818515 + 0.574485i \(0.805202\pi\)
\(60\) 0 0
\(61\) −3.60276 + 11.0881i −0.461286 + 1.41969i 0.402309 + 0.915504i \(0.368208\pi\)
−0.863594 + 0.504187i \(0.831792\pi\)
\(62\) 1.81827 + 5.59606i 0.230921 + 0.710700i
\(63\) −3.54032 − 10.8960i −0.446038 − 1.37277i
\(64\) 0.761103 − 2.34244i 0.0951379 − 0.292804i
\(65\) 0 0
\(66\) −1.44660 − 4.45216i −0.178064 − 0.548023i
\(67\) −2.39022 + 1.73660i −0.292012 + 0.212159i −0.724140 − 0.689653i \(-0.757763\pi\)
0.432128 + 0.901812i \(0.357763\pi\)
\(68\) 0.193968 0.0235220
\(69\) −2.83469 + 2.05952i −0.341257 + 0.247938i
\(70\) 0 0
\(71\) −2.59331 − 1.88415i −0.307770 − 0.223608i 0.423169 − 0.906051i \(-0.360918\pi\)
−0.730939 + 0.682443i \(0.760918\pi\)
\(72\) 3.96312 + 2.87938i 0.467058 + 0.339338i
\(73\) 3.18047 − 9.78847i 0.372245 − 1.14565i −0.573073 − 0.819504i \(-0.694249\pi\)
0.945318 − 0.326149i \(-0.105751\pi\)
\(74\) −9.11816 −1.05996
\(75\) 0 0
\(76\) −4.62806 −0.530875
\(77\) 5.55402 − 17.0935i 0.632940 − 1.94799i
\(78\) 0.554272 + 0.402702i 0.0627589 + 0.0455970i
\(79\) 7.77877 + 5.65161i 0.875180 + 0.635856i 0.931972 − 0.362531i \(-0.118087\pi\)
−0.0567919 + 0.998386i \(0.518087\pi\)
\(80\) 0 0
\(81\) −3.81318 + 2.77044i −0.423687 + 0.307827i
\(82\) −17.5457 −1.93759
\(83\) −8.48125 + 6.16199i −0.930938 + 0.676366i −0.946222 − 0.323517i \(-0.895135\pi\)
0.0152844 + 0.999883i \(0.495135\pi\)
\(84\) 0.840358 + 2.58636i 0.0916906 + 0.282195i
\(85\) 0 0
\(86\) 0.722211 − 2.22274i 0.0778780 − 0.239684i
\(87\) 0.906701 + 2.79054i 0.0972086 + 0.299177i
\(88\) 2.37480 + 7.30889i 0.253155 + 0.779130i
\(89\) 2.24293 − 6.90303i 0.237750 − 0.731720i −0.758994 − 0.651097i \(-0.774309\pi\)
0.996745 − 0.0806230i \(-0.0256910\pi\)
\(90\) 0 0
\(91\) 0.812846 + 2.50168i 0.0852094 + 0.262248i
\(92\) −3.32765 + 2.41768i −0.346932 + 0.252061i
\(93\) −2.48286 −0.257461
\(94\) −1.25352 + 0.910739i −0.129291 + 0.0939356i
\(95\) 0 0
\(96\) −2.55412 − 1.85568i −0.260679 − 0.189394i
\(97\) −6.73079 − 4.89020i −0.683408 − 0.496525i 0.191079 − 0.981575i \(-0.438801\pi\)
−0.874487 + 0.485050i \(0.838801\pi\)
\(98\) −7.32353 + 22.5395i −0.739788 + 2.27683i
\(99\) −9.76903 −0.981824
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.n.501.1 16
5.2 odd 4 625.2.e.k.124.3 32
5.3 odd 4 625.2.e.k.124.6 32
5.4 even 2 625.2.d.p.501.4 16
25.2 odd 20 625.2.e.j.249.3 32
25.3 odd 20 625.2.e.j.374.3 32
25.4 even 10 625.2.d.q.251.1 16
25.6 even 5 inner 625.2.d.n.126.1 16
25.8 odd 20 625.2.e.k.499.3 32
25.9 even 10 625.2.a.e.1.7 ✓ 8
25.11 even 5 625.2.d.m.376.4 16
25.12 odd 20 625.2.b.d.624.5 16
25.13 odd 20 625.2.b.d.624.12 16
25.14 even 10 625.2.d.q.376.1 16
25.16 even 5 625.2.a.g.1.2 yes 8
25.17 odd 20 625.2.e.k.499.6 32
25.19 even 10 625.2.d.p.126.4 16
25.21 even 5 625.2.d.m.251.4 16
25.22 odd 20 625.2.e.j.374.6 32
25.23 odd 20 625.2.e.j.249.6 32
75.41 odd 10 5625.2.a.s.1.7 8
75.59 odd 10 5625.2.a.be.1.2 8
100.59 odd 10 10000.2.a.bn.1.4 8
100.91 odd 10 10000.2.a.be.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
625.2.a.e.1.7 ✓ 8 25.9 even 10
625.2.a.g.1.2 yes 8 25.16 even 5
625.2.b.d.624.5 16 25.12 odd 20
625.2.b.d.624.12 16 25.13 odd 20
625.2.d.m.251.4 16 25.21 even 5
625.2.d.m.376.4 16 25.11 even 5
625.2.d.n.126.1 16 25.6 even 5 inner
625.2.d.n.501.1 16 1.1 even 1 trivial
625.2.d.p.126.4 16 25.19 even 10
625.2.d.p.501.4 16 5.4 even 2
625.2.d.q.251.1 16 25.4 even 10
625.2.d.q.376.1 16 25.14 even 10
625.2.e.j.249.3 32 25.2 odd 20
625.2.e.j.249.6 32 25.23 odd 20
625.2.e.j.374.3 32 25.3 odd 20
625.2.e.j.374.6 32 25.22 odd 20
625.2.e.k.124.3 32 5.2 odd 4
625.2.e.k.124.6 32 5.3 odd 4
625.2.e.k.499.3 32 25.8 odd 20
625.2.e.k.499.6 32 25.17 odd 20
5625.2.a.s.1.7 8 75.41 odd 10
5625.2.a.be.1.2 8 75.59 odd 10
10000.2.a.be.1.5 8 100.91 odd 10
10000.2.a.bn.1.4 8 100.59 odd 10