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Properties

Label 50.2.d.b.31.1
Level $50$
Weight $2$
Character 50.31
Analytic conductor $0.399$
Analytic rank $0$
Dimension $8$
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] = [50,2,Mod(11,50)] 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("50.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 50.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.399252010106\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.58140625.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 31.1
Root \(-0.983224 + 0.644389i\) of defining polynomial
Character \(\chi\) \(=\) 50.31
Dual form 50.2.d.b.21.1

$q$-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+(0.309017 + 0.951057i) q^{2} +(-2.39991 + 1.74363i) q^{3} +(-0.809017 + 0.587785i) q^{4} +(2.15743 - 0.587785i) q^{5} +(-2.39991 - 1.74363i) q^{6} +1.83337 q^{7} +(-0.809017 - 0.587785i) q^{8} +(1.79224 - 5.51595i) q^{9} +(1.22570 + 1.87020i) q^{10} +(-0.566541 - 1.74363i) q^{11} +(0.916683 - 2.82126i) q^{12} +(-0.747156 + 2.29951i) q^{13} +(0.566541 + 1.74363i) q^{14} +(-4.15275 + 5.17240i) q^{15} +(0.309017 - 0.951057i) q^{16} +(-2.25284 - 1.63679i) q^{17} +5.79981 q^{18} +(1.35294 + 0.982966i) q^{19} +(-1.39991 + 1.74363i) q^{20} +(-4.39991 + 3.19672i) q^{21} +(1.48322 - 1.07763i) q^{22} +(-2.39991 - 7.38615i) q^{23} +2.96645 q^{24} +(4.30902 - 2.53621i) q^{25} -2.41785 q^{26} +(2.56654 + 7.89900i) q^{27} +(-1.48322 + 1.07763i) q^{28} +(-6.13597 + 4.45805i) q^{29} +(-6.20252 - 2.35114i) q^{30} +(4.28304 + 3.11181i) q^{31} +1.00000 q^{32} +(4.39991 + 3.19672i) q^{33} +(0.860510 - 2.64838i) q^{34} +(3.95536 - 1.07763i) q^{35} +(1.79224 + 5.51595i) q^{36} +(-0.406315 + 1.25051i) q^{37} +(-0.516776 + 1.59047i) q^{38} +(-2.21640 - 6.82138i) q^{39} +(-2.09089 - 0.792578i) q^{40} +(1.08621 - 3.34301i) q^{41} +(-4.39991 - 3.19672i) q^{42} -4.30550 q^{43} +(1.48322 + 1.07763i) q^{44} +(0.624442 - 12.9537i) q^{45} +(6.28304 - 4.56489i) q^{46} +(-1.48322 + 1.07763i) q^{47} +(0.916683 + 2.82126i) q^{48} -3.63877 q^{49} +(3.74364 + 3.31439i) q^{50} +8.26057 q^{51} +(-0.747156 - 2.29951i) q^{52} +(5.27267 - 3.83082i) q^{53} +(-6.71929 + 4.88185i) q^{54} +(-2.24716 - 3.42877i) q^{55} +(-1.48322 - 1.07763i) q^{56} -4.96086 q^{57} +(-6.13597 - 4.45805i) q^{58} +(-2.79981 + 8.61694i) q^{59} +(0.319385 - 6.62549i) q^{60} +(0.799717 + 2.46127i) q^{61} +(-1.63597 + 5.03501i) q^{62} +(3.28583 - 10.1128i) q^{63} +(0.309017 + 0.951057i) q^{64} +(-0.260320 + 5.40020i) q^{65} +(-1.68061 + 5.17240i) q^{66} +(-7.68574 - 5.58402i) q^{67} +2.78467 q^{68} +(18.6383 + 13.5415i) q^{69} +(2.24716 + 3.42877i) q^{70} +(-0.247156 + 0.179569i) q^{71} +(-4.69215 + 3.40904i) q^{72} +(4.61920 + 14.2164i) q^{73} -1.31486 q^{74} +(-5.91901 + 13.6000i) q^{75} -1.67232 q^{76} +(-1.03868 - 3.19672i) q^{77} +(5.80261 - 4.21584i) q^{78} +(2.79981 - 2.03418i) q^{79} +(0.107666 - 2.23347i) q^{80} +(-5.85599 - 4.25462i) q^{81} +3.51505 q^{82} +(5.15555 + 3.74572i) q^{83} +(1.68061 - 5.17240i) q^{84} +(-5.82243 - 2.20707i) q^{85} +(-1.33047 - 4.09478i) q^{86} +(6.95256 - 21.3978i) q^{87} +(-0.566541 + 1.74363i) q^{88} +(-1.02608 - 3.15794i) q^{89} +(12.5127 - 3.40904i) q^{90} +(-1.36981 + 4.21584i) q^{91} +(6.28304 + 4.56489i) q^{92} -15.7047 q^{93} +(-1.48322 - 1.07763i) q^{94} +(3.49664 + 1.32545i) q^{95} +(-2.39991 + 1.74363i) q^{96} +(-8.97214 + 6.51864i) q^{97} +(-1.12444 - 3.46068i) q^{98} -10.6332 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 3 q^{3} - 2 q^{4} - 3 q^{6} + 4 q^{7} - 2 q^{8} - q^{9} + q^{11} + 2 q^{12} - 13 q^{13} - q^{14} - 10 q^{15} - 2 q^{16} - 11 q^{17} + 14 q^{18} + 20 q^{19} + 5 q^{20} - 19 q^{21} + q^{22}+ \cdots - 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(e\left(\frac{2}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.309017 + 0.951057i 0.218508 + 0.672499i
\(3\) −2.39991 + 1.74363i −1.38559 + 1.00669i −0.389254 + 0.921131i \(0.627267\pi\)
−0.996333 + 0.0855571i \(0.972733\pi\)
\(4\) −0.809017 + 0.587785i −0.404508 + 0.293893i
\(5\) 2.15743 − 0.587785i 0.964832 − 0.262866i
\(6\) −2.39991 − 1.74363i −0.979758 − 0.711836i
\(7\) 1.83337 0.692947 0.346474 − 0.938060i \(-0.387379\pi\)
0.346474 + 0.938060i \(0.387379\pi\)
\(8\) −0.809017 − 0.587785i −0.286031 − 0.207813i
\(9\) 1.79224 − 5.51595i 0.597414 − 1.83865i
\(10\) 1.22570 + 1.87020i 0.387600 + 0.591410i
\(11\) −0.566541 − 1.74363i −0.170819 − 0.525726i 0.828599 − 0.559842i \(-0.189138\pi\)
−0.999418 + 0.0341166i \(0.989138\pi\)
\(12\) 0.916683 − 2.82126i 0.264624 − 0.814428i
\(13\) −0.747156 + 2.29951i −0.207224 + 0.637769i 0.792391 + 0.610014i \(0.208836\pi\)
−0.999615 + 0.0277557i \(0.991164\pi\)
\(14\) 0.566541 + 1.74363i 0.151414 + 0.466006i
\(15\) −4.15275 + 5.17240i −1.07224 + 1.33551i
\(16\) 0.309017 − 0.951057i 0.0772542 − 0.237764i
\(17\) −2.25284 − 1.63679i −0.546395 − 0.396979i 0.280060 − 0.959983i \(-0.409646\pi\)
−0.826455 + 0.563003i \(0.809646\pi\)
\(18\) 5.79981 1.36703
\(19\) 1.35294 + 0.982966i 0.310385 + 0.225508i 0.732062 − 0.681238i \(-0.238558\pi\)
−0.421677 + 0.906746i \(0.638558\pi\)
\(20\) −1.39991 + 1.74363i −0.313029 + 0.389889i
\(21\) −4.39991 + 3.19672i −0.960138 + 0.697581i
\(22\) 1.48322 − 1.07763i 0.316224 − 0.229750i
\(23\) −2.39991 − 7.38615i −0.500415 − 1.54012i −0.808344 − 0.588710i \(-0.799636\pi\)
0.307929 − 0.951409i \(-0.400364\pi\)
\(24\) 2.96645 0.605524
\(25\) 4.30902 − 2.53621i 0.861803 − 0.507242i
\(26\) −2.41785 −0.474179
\(27\) 2.56654 + 7.89900i 0.493931 + 1.52016i
\(28\) −1.48322 + 1.07763i −0.280303 + 0.203652i
\(29\) −6.13597 + 4.45805i −1.13942 + 0.827838i −0.987039 − 0.160483i \(-0.948695\pi\)
−0.152383 + 0.988321i \(0.548695\pi\)
\(30\) −6.20252 − 2.35114i −1.13242 − 0.429258i
\(31\) 4.28304 + 3.11181i 0.769256 + 0.558897i 0.901735 − 0.432288i \(-0.142294\pi\)
−0.132479 + 0.991186i \(0.542294\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.39991 + 3.19672i 0.765925 + 0.556477i
\(34\) 0.860510 − 2.64838i 0.147576 − 0.454193i
\(35\) 3.95536 − 1.07763i 0.668578 − 0.182152i
\(36\) 1.79224 + 5.51595i 0.298707 + 0.919325i
\(37\) −0.406315 + 1.25051i −0.0667977 + 0.205582i −0.978884 − 0.204416i \(-0.934471\pi\)
0.912086 + 0.409998i \(0.134471\pi\)
\(38\) −0.516776 + 1.59047i −0.0838321 + 0.258009i
\(39\) −2.21640 − 6.82138i −0.354908 − 1.09229i
\(40\) −2.09089 − 0.792578i −0.330599 − 0.125318i
\(41\) 1.08621 − 3.34301i 0.169637 − 0.522090i −0.829711 − 0.558194i \(-0.811495\pi\)
0.999348 + 0.0361034i \(0.0114946\pi\)
\(42\) −4.39991 − 3.19672i −0.678920 − 0.493265i
\(43\) −4.30550 −0.656583 −0.328291 − 0.944576i \(-0.606473\pi\)
−0.328291 + 0.944576i \(0.606473\pi\)
\(44\) 1.48322 + 1.07763i 0.223604 + 0.162458i
\(45\) 0.624442 − 12.9537i 0.0930863 − 1.93103i
\(46\) 6.28304 − 4.56489i 0.926383 − 0.673057i
\(47\) −1.48322 + 1.07763i −0.216350 + 0.157188i −0.690682 − 0.723158i \(-0.742690\pi\)
0.474332 + 0.880346i \(0.342690\pi\)
\(48\) 0.916683 + 2.82126i 0.132312 + 0.407214i
\(49\) −3.63877 −0.519824
\(50\) 3.74364 + 3.31439i 0.529431 + 0.468725i
\(51\) 8.26057 1.15671
\(52\) −0.747156 − 2.29951i −0.103612 − 0.318885i
\(53\) 5.27267 − 3.83082i 0.724257 − 0.526203i −0.163485 − 0.986546i \(-0.552273\pi\)
0.887741 + 0.460342i \(0.152273\pi\)
\(54\) −6.71929 + 4.88185i −0.914380 + 0.664336i
\(55\) −2.24716 − 3.42877i −0.303006 − 0.462335i
\(56\) −1.48322 − 1.07763i −0.198204 − 0.144004i
\(57\) −4.96086 −0.657082
\(58\) −6.13597 − 4.45805i −0.805693 − 0.585370i
\(59\) −2.79981 + 8.61694i −0.364505 + 1.12183i 0.585786 + 0.810466i \(0.300786\pi\)
−0.950291 + 0.311364i \(0.899214\pi\)
\(60\) 0.319385 − 6.62549i 0.0412324 − 0.855347i
\(61\) 0.799717 + 2.46127i 0.102393 + 0.315134i 0.989110 − 0.147180i \(-0.0470195\pi\)
−0.886717 + 0.462313i \(0.847019\pi\)
\(62\) −1.63597 + 5.03501i −0.207769 + 0.639447i
\(63\) 3.28583 − 10.1128i 0.413976 − 1.27409i
\(64\) 0.309017 + 0.951057i 0.0386271 + 0.118882i
\(65\) −0.260320 + 5.40020i −0.0322887 + 0.669812i
\(66\) −1.68061 + 5.17240i −0.206869 + 0.636679i
\(67\) −7.68574 − 5.58402i −0.938963 − 0.682196i 0.00920814 − 0.999958i \(-0.497069\pi\)
−0.948171 + 0.317761i \(0.897069\pi\)
\(68\) 2.78467 0.337691
\(69\) 18.6383 + 13.5415i 2.24379 + 1.63021i
\(70\) 2.24716 + 3.42877i 0.268587 + 0.409816i
\(71\) −0.247156 + 0.179569i −0.0293320 + 0.0213110i −0.602355 − 0.798229i \(-0.705771\pi\)
0.573023 + 0.819540i \(0.305771\pi\)
\(72\) −4.69215 + 3.40904i −0.552975 + 0.401760i
\(73\) 4.61920 + 14.2164i 0.540636 + 1.66391i 0.731145 + 0.682222i \(0.238986\pi\)
−0.190509 + 0.981685i \(0.561014\pi\)
\(74\) −1.31486 −0.152850
\(75\) −5.91901 + 13.6000i −0.683469 + 1.57040i
\(76\) −1.67232 −0.191829
\(77\) −1.03868 − 3.19672i −0.118368 − 0.364300i
\(78\) 5.80261 − 4.21584i 0.657016 − 0.477350i
\(79\) 2.79981 − 2.03418i 0.315004 − 0.228864i −0.419037 − 0.907969i \(-0.637632\pi\)
0.734041 + 0.679106i \(0.237632\pi\)
\(80\) 0.107666 − 2.23347i 0.0120374 − 0.249710i
\(81\) −5.85599 − 4.25462i −0.650665 − 0.472736i
\(82\) 3.51505 0.388172
\(83\) 5.15555 + 3.74572i 0.565895 + 0.411147i 0.833612 − 0.552351i \(-0.186269\pi\)
−0.267717 + 0.963498i \(0.586269\pi\)
\(84\) 1.68061 − 5.17240i 0.183370 − 0.564355i
\(85\) −5.82243 − 2.20707i −0.631532 − 0.239390i
\(86\) −1.33047 − 4.09478i −0.143469 − 0.441551i
\(87\) 6.95256 − 21.3978i 0.745393 − 2.29408i
\(88\) −0.566541 + 1.74363i −0.0603935 + 0.185872i
\(89\) −1.02608 − 3.15794i −0.108764 − 0.334741i 0.881832 − 0.471565i \(-0.156311\pi\)
−0.990595 + 0.136824i \(0.956311\pi\)
\(90\) 12.5127 − 3.40904i 1.31895 − 0.359345i
\(91\) −1.36981 + 4.21584i −0.143595 + 0.441940i
\(92\) 6.28304 + 4.56489i 0.655052 + 0.475923i
\(93\) −15.7047 −1.62851
\(94\) −1.48322 − 1.07763i −0.152983 − 0.111149i
\(95\) 3.49664 + 1.32545i 0.358748 + 0.135988i
\(96\) −2.39991 + 1.74363i −0.244939 + 0.177959i
\(97\) −8.97214 + 6.51864i −0.910982 + 0.661867i −0.941263 − 0.337674i \(-0.890360\pi\)
0.0302807 + 0.999541i \(0.490360\pi\)
\(98\) −1.12444 − 3.46068i −0.113586 − 0.349581i
\(99\) −10.6332 −1.06867
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 50.2.d.b.31.1 yes 8
3.2 odd 2 450.2.h.e.181.1 8
4.3 odd 2 400.2.u.d.81.2 8
5.2 odd 4 250.2.e.c.99.2 16
5.3 odd 4 250.2.e.c.99.3 16
5.4 even 2 250.2.d.d.151.2 8
25.2 odd 20 1250.2.b.e.1249.5 8
25.3 odd 20 250.2.e.c.149.2 16
25.4 even 10 250.2.d.d.101.2 8
25.11 even 5 1250.2.a.l.1.4 4
25.14 even 10 1250.2.a.f.1.1 4
25.21 even 5 inner 50.2.d.b.21.1 ✓ 8
25.22 odd 20 250.2.e.c.149.3 16
25.23 odd 20 1250.2.b.e.1249.4 8
75.71 odd 10 450.2.h.e.271.1 8
100.11 odd 10 10000.2.a.t.1.1 4
100.39 odd 10 10000.2.a.x.1.4 4
100.71 odd 10 400.2.u.d.321.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.2.d.b.21.1 ✓ 8 25.21 even 5 inner
50.2.d.b.31.1 yes 8 1.1 even 1 trivial
250.2.d.d.101.2 8 25.4 even 10
250.2.d.d.151.2 8 5.4 even 2
250.2.e.c.99.2 16 5.2 odd 4
250.2.e.c.99.3 16 5.3 odd 4
250.2.e.c.149.2 16 25.3 odd 20
250.2.e.c.149.3 16 25.22 odd 20
400.2.u.d.81.2 8 4.3 odd 2
400.2.u.d.321.2 8 100.71 odd 10
450.2.h.e.181.1 8 3.2 odd 2
450.2.h.e.271.1 8 75.71 odd 10
1250.2.a.f.1.1 4 25.14 even 10
1250.2.a.l.1.4 4 25.11 even 5
1250.2.b.e.1249.4 8 25.23 odd 20
1250.2.b.e.1249.5 8 25.2 odd 20
10000.2.a.t.1.1 4 100.11 odd 10
10000.2.a.x.1.4 4 100.39 odd 10