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Properties

Label 50.12.b.f.49.2
Level $50$
Weight $12$
Character 50.49
Analytic conductor $38.417$
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] = [50,12,Mod(49,50)] 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("50.49"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 50.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4096,0,38656] 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: \(38.4171590280\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{1969})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 985x^{2} + 242064 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(-22.6867i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.12.b.f.49.3

$q$-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)
 
\(f(q)\) \(=\) \(q-32.0000i q^{2} +745.734i q^{3} -1024.00 q^{4} +23863.5 q^{6} +71494.9i q^{7} +32768.0i q^{8} -378972. q^{9} -345651. q^{11} -763632. i q^{12} +1.50956e6i q^{13} +2.28784e6 q^{14} +1.04858e6 q^{16} +5.39291e6i q^{17} +1.21271e7i q^{18} +1.11633e7 q^{19} -5.33162e7 q^{21} +1.10608e7i q^{22} -5.27646e6i q^{23} -2.44362e7 q^{24} +4.83060e7 q^{26} -1.50508e8i q^{27} -7.32108e7i q^{28} +1.86291e7 q^{29} +7.10448e7 q^{31} -3.35544e7i q^{32} -2.57763e8i q^{33} +1.72573e8 q^{34} +3.88068e8 q^{36} -3.23164e8i q^{37} -3.57225e8i q^{38} -1.12573e9 q^{39} -9.11277e8 q^{41} +1.70612e9i q^{42} -1.16431e9i q^{43} +3.53946e8 q^{44} -1.68847e8 q^{46} -2.81949e8i q^{47} +7.81959e8i q^{48} -3.13420e9 q^{49} -4.02168e9 q^{51} -1.54579e9i q^{52} +4.05957e9i q^{53} -4.81626e9 q^{54} -2.34275e9 q^{56} +8.32485e9i q^{57} -5.96132e8i q^{58} -4.89828e9 q^{59} +1.07565e10 q^{61} -2.27343e9i q^{62} -2.70946e10i q^{63} -1.07374e9 q^{64} -8.24843e9 q^{66} -3.70812e9i q^{67} -5.52234e9i q^{68} +3.93484e9 q^{69} +3.45274e9 q^{71} -1.24182e10i q^{72} -2.21136e10i q^{73} -1.03413e10 q^{74} -1.14312e10 q^{76} -2.47123e10i q^{77} +3.60235e10i q^{78} -7.02672e9 q^{79} +4.51052e10 q^{81} +2.91609e10i q^{82} +5.55656e10i q^{83} +5.45958e10 q^{84} -3.72580e10 q^{86} +1.38924e10i q^{87} -1.13263e10i q^{88} +9.29706e9 q^{89} -1.07926e11 q^{91} +5.40310e9i q^{92} +5.29805e10i q^{93} -9.02235e9 q^{94} +2.50227e10 q^{96} -4.71888e10i q^{97} +1.00294e11i q^{98} +1.30992e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4096 q^{4} + 38656 q^{6} - 443828 q^{9} + 843168 q^{11} - 901888 q^{14} + 4194304 q^{16} + 57794800 q^{19} - 130893632 q^{21} - 39583744 q^{24} + 110753536 q^{26} - 116452440 q^{29} + 82826768 q^{31}+ \cdots + 502985449824 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(27\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 32.0000i − 0.707107i
\(3\) 745.734i 1.77181i 0.463867 + 0.885905i \(0.346462\pi\)
−0.463867 + 0.885905i \(0.653538\pi\)
\(4\) −1024.00 −0.500000
\(5\) 0 0
\(6\) 23863.5 1.25286
\(7\) 71494.9i 1.60782i 0.594754 + 0.803908i \(0.297249\pi\)
−0.594754 + 0.803908i \(0.702751\pi\)
\(8\) 32768.0i 0.353553i
\(9\) −378972. −2.13931
\(10\) 0 0
\(11\) −345651. −0.647109 −0.323555 − 0.946209i \(-0.604878\pi\)
−0.323555 + 0.946209i \(0.604878\pi\)
\(12\) − 763632.i − 0.885905i
\(13\) 1.50956e6i 1.12762i 0.825904 + 0.563810i \(0.190665\pi\)
−0.825904 + 0.563810i \(0.809335\pi\)
\(14\) 2.28784e6 1.13690
\(15\) 0 0
\(16\) 1.04858e6 0.250000
\(17\) 5.39291e6i 0.921200i 0.887608 + 0.460600i \(0.152366\pi\)
−0.887608 + 0.460600i \(0.847634\pi\)
\(18\) 1.21271e7i 1.51272i
\(19\) 1.11633e7 1.03430 0.517151 − 0.855894i \(-0.326993\pi\)
0.517151 + 0.855894i \(0.326993\pi\)
\(20\) 0 0
\(21\) −5.33162e7 −2.84874
\(22\) 1.10608e7i 0.457575i
\(23\) − 5.27646e6i − 0.170938i −0.996341 − 0.0854692i \(-0.972761\pi\)
0.996341 − 0.0854692i \(-0.0272389\pi\)
\(24\) −2.44362e7 −0.626429
\(25\) 0 0
\(26\) 4.83060e7 0.797348
\(27\) − 1.50508e8i − 2.01864i
\(28\) − 7.32108e7i − 0.803908i
\(29\) 1.86291e7 0.168657 0.0843284 − 0.996438i \(-0.473126\pi\)
0.0843284 + 0.996438i \(0.473126\pi\)
\(30\) 0 0
\(31\) 7.10448e7 0.445700 0.222850 − 0.974853i \(-0.428464\pi\)
0.222850 + 0.974853i \(0.428464\pi\)
\(32\) − 3.35544e7i − 0.176777i
\(33\) − 2.57763e8i − 1.14655i
\(34\) 1.72573e8 0.651387
\(35\) 0 0
\(36\) 3.88068e8 1.06966
\(37\) − 3.23164e8i − 0.766150i −0.923717 − 0.383075i \(-0.874865\pi\)
0.923717 − 0.383075i \(-0.125135\pi\)
\(38\) − 3.57225e8i − 0.731362i
\(39\) −1.12573e9 −1.99793
\(40\) 0 0
\(41\) −9.11277e8 −1.22840 −0.614199 − 0.789151i \(-0.710521\pi\)
−0.614199 + 0.789151i \(0.710521\pi\)
\(42\) 1.70612e9i 2.01437i
\(43\) − 1.16431e9i − 1.20779i −0.797063 − 0.603897i \(-0.793614\pi\)
0.797063 − 0.603897i \(-0.206386\pi\)
\(44\) 3.53946e8 0.323555
\(45\) 0 0
\(46\) −1.68847e8 −0.120872
\(47\) − 2.81949e8i − 0.179321i −0.995972 − 0.0896606i \(-0.971422\pi\)
0.995972 − 0.0896606i \(-0.0285782\pi\)
\(48\) 7.81959e8i 0.442952i
\(49\) −3.13420e9 −1.58507
\(50\) 0 0
\(51\) −4.02168e9 −1.63219
\(52\) − 1.54579e9i − 0.563810i
\(53\) 4.05957e9i 1.33341i 0.745323 + 0.666704i \(0.232295\pi\)
−0.745323 + 0.666704i \(0.767705\pi\)
\(54\) −4.81626e9 −1.42740
\(55\) 0 0
\(56\) −2.34275e9 −0.568449
\(57\) 8.32485e9i 1.83259i
\(58\) − 5.96132e8i − 0.119258i
\(59\) −4.89828e9 −0.891986 −0.445993 − 0.895036i \(-0.647149\pi\)
−0.445993 + 0.895036i \(0.647149\pi\)
\(60\) 0 0
\(61\) 1.07565e10 1.63064 0.815320 − 0.579011i \(-0.196561\pi\)
0.815320 + 0.579011i \(0.196561\pi\)
\(62\) − 2.27343e9i − 0.315158i
\(63\) − 2.70946e10i − 3.43962i
\(64\) −1.07374e9 −0.125000
\(65\) 0 0
\(66\) −8.24843e9 −0.810736
\(67\) − 3.70812e9i − 0.335539i −0.985826 − 0.167769i \(-0.946344\pi\)
0.985826 − 0.167769i \(-0.0536564\pi\)
\(68\) − 5.52234e9i − 0.460600i
\(69\) 3.93484e9 0.302870
\(70\) 0 0
\(71\) 3.45274e9 0.227113 0.113557 − 0.993532i \(-0.463776\pi\)
0.113557 + 0.993532i \(0.463776\pi\)
\(72\) − 1.24182e10i − 0.756360i
\(73\) − 2.21136e10i − 1.24848i −0.781231 − 0.624242i \(-0.785408\pi\)
0.781231 − 0.624242i \(-0.214592\pi\)
\(74\) −1.03413e10 −0.541750
\(75\) 0 0
\(76\) −1.14312e10 −0.517151
\(77\) − 2.47123e10i − 1.04043i
\(78\) 3.60235e10i 1.41275i
\(79\) −7.02672e9 −0.256923 −0.128462 − 0.991714i \(-0.541004\pi\)
−0.128462 + 0.991714i \(0.541004\pi\)
\(80\) 0 0
\(81\) 4.51052e10 1.43734
\(82\) 2.91609e10i 0.868609i
\(83\) 5.55656e10i 1.54838i 0.632956 + 0.774188i \(0.281841\pi\)
−0.632956 + 0.774188i \(0.718159\pi\)
\(84\) 5.45958e10 1.42437
\(85\) 0 0
\(86\) −3.72580e10 −0.854039
\(87\) 1.38924e10i 0.298828i
\(88\) − 1.13263e10i − 0.228788i
\(89\) 9.29706e9 0.176482 0.0882410 − 0.996099i \(-0.471875\pi\)
0.0882410 + 0.996099i \(0.471875\pi\)
\(90\) 0 0
\(91\) −1.07926e11 −1.81301
\(92\) 5.40310e9i 0.0854692i
\(93\) 5.29805e10i 0.789696i
\(94\) −9.02235e9 −0.126799
\(95\) 0 0
\(96\) 2.50227e10 0.313215
\(97\) − 4.71888e10i − 0.557949i −0.960298 − 0.278975i \(-0.910005\pi\)
0.960298 − 0.278975i \(-0.0899945\pi\)
\(98\) 1.00294e11i 1.12081i
\(99\) 1.30992e11 1.38437
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 50.12.b.f.49.2 4
5.2 odd 4 10.12.a.d.1.2 ✓ 2
5.3 odd 4 50.12.a.f.1.1 2
5.4 even 2 inner 50.12.b.f.49.3 4
15.2 even 4 90.12.a.l.1.1 2
20.7 even 4 80.12.a.g.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.12.a.d.1.2 ✓ 2 5.2 odd 4
50.12.a.f.1.1 2 5.3 odd 4
50.12.b.f.49.2 4 1.1 even 1 trivial
50.12.b.f.49.3 4 5.4 even 2 inner
80.12.a.g.1.1 2 20.7 even 4
90.12.a.l.1.1 2 15.2 even 4