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Properties

Label 50.12.d.b.21.3
Level $50$
Weight $12$
Character 50.21
Analytic conductor $38.417$
Analytic rank $0$
Dimension $56$
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(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, 12); 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, 12, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 50.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [56] 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: \(38.4171590280\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 21.3
Character \(\chi\) \(=\) 50.21
Dual form 50.12.d.b.31.3

$q$-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+(9.88854 - 30.4338i) q^{2} +(-486.674 - 353.590i) q^{3} +(-828.433 - 601.892i) q^{4} +(-1629.59 - 6795.04i) q^{5} +(-15573.6 + 11314.9i) q^{6} +3121.65 q^{7} +(-26509.9 + 19260.5i) q^{8} +(57084.8 + 175689. i) q^{9} +(-222913. - 17598.3i) q^{10} +(-109826. + 338011. i) q^{11} +(190354. + 585851. i) q^{12} +(284110. + 874402. i) q^{13} +(30868.6 - 95003.8i) q^{14} +(-1.60957e6 + 3.88318e6i) q^{15} +(324028. + 997255. i) q^{16} +(-1.94915e6 + 1.41614e6i) q^{17} +5.91137e6 q^{18} +(-2.33727e6 + 1.69812e6i) q^{19} +(-2.73987e6 + 6.61008e6i) q^{20} +(-1.51923e6 - 1.10378e6i) q^{21} +(9.20093e6 + 6.68487e6i) q^{22} +(-1.07266e6 + 3.30130e6i) q^{23} +1.97120e7 q^{24} +(-4.35170e7 + 2.21463e7i) q^{25} +2.94208e7 q^{26} +(1.40971e6 - 4.33864e6i) q^{27} +(-2.58608e6 - 1.87890e6i) q^{28} +(-1.54533e6 - 1.12275e6i) q^{29} +(1.02264e8 + 8.73844e7i) q^{30} +(2.24017e8 - 1.62758e8i) q^{31} +3.35544e7 q^{32} +(1.72967e8 - 1.25668e8i) q^{33} +(2.38243e7 + 7.33237e7i) q^{34} +(-5.08703e6 - 2.12118e7i) q^{35} +(5.84549e7 - 1.79906e8i) q^{36} +(3.90695e7 + 1.20244e8i) q^{37} +(2.85682e7 + 8.79239e7i) q^{38} +(1.70910e8 - 5.26007e8i) q^{39} +(1.74076e8 + 1.48749e8i) q^{40} +(-1.34607e8 - 4.14278e8i) q^{41} +(-4.86153e7 + 3.53211e7i) q^{42} -4.46722e8 q^{43} +(2.94430e8 - 2.13916e8i) q^{44} +(1.10079e9 - 6.74196e8i) q^{45} +(8.98640e7 + 6.52900e7i) q^{46} +(-9.81911e8 - 7.13400e8i) q^{47} +(1.94923e8 - 5.99911e8i) q^{48} -1.96758e9 q^{49} +(2.43677e8 + 1.54338e9i) q^{50} +1.44934e9 q^{51} +(2.90929e8 - 8.95387e8i) q^{52} +(3.51028e9 + 2.55037e9i) q^{53} +(-1.18101e8 - 8.58057e7i) q^{54} +(2.47577e9 + 1.95454e8i) q^{55} +(-8.27546e7 + 6.01247e7i) q^{56} +1.73793e9 q^{57} +(-4.94506e7 + 3.59280e7i) q^{58} +(-3.90044e8 - 1.20043e9i) q^{59} +(3.67068e9 - 2.24816e9i) q^{60} +(1.79302e9 - 5.51836e9i) q^{61} +(-2.73813e9 - 8.42711e9i) q^{62} +(1.78199e8 + 5.48440e8i) q^{63} +(3.31804e8 - 1.02119e9i) q^{64} +(5.47861e9 - 3.35546e9i) q^{65} +(-2.11416e9 - 6.50671e9i) q^{66} +(6.20425e9 - 4.50765e9i) q^{67} +2.46711e9 q^{68} +(1.68934e9 - 1.22738e9i) q^{69} +(-6.95858e8 - 5.49358e7i) q^{70} +(4.29969e9 + 3.12391e9i) q^{71} +(-4.89718e9 - 3.55801e9i) q^{72} +(-5.43059e9 + 1.67136e10i) q^{73} +4.04581e9 q^{74} +(2.90093e10 + 4.60911e9i) q^{75} +2.95836e9 q^{76} +(-3.42840e8 + 1.05515e9i) q^{77} +(-1.43184e10 - 1.04029e10i) q^{78} +(3.23402e10 + 2.34965e10i) q^{79} +(6.24835e9 - 3.82690e9i) q^{80} +(2.42545e10 - 1.76219e10i) q^{81} -1.39391e10 q^{82} +(-4.74630e10 + 3.44839e10i) q^{83} +(5.94221e8 + 1.82882e9i) q^{84} +(1.27991e10 + 1.09368e10i) q^{85} +(-4.41743e9 + 1.35954e10i) q^{86} +(3.55081e8 + 1.09283e9i) q^{87} +(-3.59879e9 - 1.10759e10i) q^{88} +(-2.79640e10 + 8.60644e10i) q^{89} +(-9.63314e9 - 4.01680e10i) q^{90} +(8.86894e8 + 2.72958e9i) q^{91} +(2.87565e9 - 2.08928e9i) q^{92} -1.66572e11 q^{93} +(-3.14212e10 + 2.28288e10i) q^{94} +(1.53476e10 + 1.31146e10i) q^{95} +(-1.63301e10 - 1.18645e10i) q^{96} +(-4.95689e10 - 3.60139e10i) q^{97} +(-1.94565e10 + 5.98810e10i) q^{98} -6.56542e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 448 q^{2} - 263 q^{3} - 14336 q^{4} + 1770 q^{5} - 8416 q^{6} - 111844 q^{7} - 458752 q^{8} - 1174523 q^{9} + 304960 q^{10} + 207277 q^{11} + 1026048 q^{12} + 893677 q^{13} - 1270048 q^{14} + 4696640 q^{15}+ \cdots - 505737997606 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(27\)
\(\chi(n)\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.88854 − 30.4338i 0.218508 − 0.672499i
\(3\) −486.674 − 353.590i −1.15630 − 0.840103i −0.166997 − 0.985957i \(-0.553407\pi\)
−0.989306 + 0.145854i \(0.953407\pi\)
\(4\) −828.433 − 601.892i −0.404508 − 0.293893i
\(5\) −1629.59 − 6795.04i −0.233208 − 0.972427i
\(6\) −15573.6 + 11314.9i −0.817629 + 0.594043i
\(7\) 3121.65 0.0702013 0.0351007 − 0.999384i \(-0.488825\pi\)
0.0351007 + 0.999384i \(0.488825\pi\)
\(8\) −26509.9 + 19260.5i −0.286031 + 0.207813i
\(9\) 57084.8 + 175689.i 0.322246 + 0.991770i
\(10\) −222913. − 17598.3i −0.704913 − 0.0556507i
\(11\) −109826. + 338011.i −0.205611 + 0.632806i 0.794077 + 0.607818i \(0.207955\pi\)
−0.999688 + 0.0249884i \(0.992045\pi\)
\(12\) 190354. + 585851.i 0.220834 + 0.679658i
\(13\) 284110. + 874402.i 0.212226 + 0.653165i 0.999339 + 0.0363556i \(0.0115749\pi\)
−0.787113 + 0.616809i \(0.788425\pi\)
\(14\) 30868.6 − 95003.8i 0.0153396 − 0.0472103i
\(15\) −1.60957e6 + 3.88318e6i −0.547279 + 1.32034i
\(16\) 324028. + 997255.i 0.0772542 + 0.237764i
\(17\) −1.94915e6 + 1.41614e6i −0.332948 + 0.241901i −0.741681 − 0.670753i \(-0.765971\pi\)
0.408732 + 0.912654i \(0.365971\pi\)
\(18\) 5.91137e6 0.737377
\(19\) −2.33727e6 + 1.69812e6i −0.216553 + 0.157335i −0.690773 − 0.723071i \(-0.742730\pi\)
0.474221 + 0.880406i \(0.342730\pi\)
\(20\) −2.73987e6 + 6.61008e6i −0.191454 + 0.461893i
\(21\) −1.51923e6 − 1.10378e6i −0.0811740 − 0.0589764i
\(22\) 9.20093e6 + 6.68487e6i 0.380633 + 0.276546i
\(23\) −1.07266e6 + 3.30130e6i −0.0347502 + 0.106950i −0.966927 − 0.255053i \(-0.917907\pi\)
0.932177 + 0.362003i \(0.117907\pi\)
\(24\) 1.97120e7 0.505323
\(25\) −4.35170e7 + 2.21463e7i −0.891228 + 0.453556i
\(26\) 2.94208e7 0.485625
\(27\) 1.40971e6 − 4.33864e6i 0.0189073 − 0.0581907i
\(28\) −2.58608e6 − 1.87890e6i −0.0283970 − 0.0206317i
\(29\) −1.54533e6 − 1.12275e6i −0.0139905 − 0.0101647i 0.580768 − 0.814069i \(-0.302752\pi\)
−0.594759 + 0.803904i \(0.702752\pi\)
\(30\) 1.02264e8 + 8.73844e7i 0.768341 + 0.656549i
\(31\) 2.24017e8 − 1.62758e8i 1.40537 − 1.02106i 0.411395 − 0.911457i \(-0.365042\pi\)
0.993975 − 0.109604i \(-0.0349583\pi\)
\(32\) 3.35544e7 0.176777
\(33\) 1.72967e8 − 1.25668e8i 0.769371 − 0.558981i
\(34\) 2.38243e7 + 7.33237e7i 0.0899263 + 0.276765i
\(35\) −5.08703e6 − 2.12118e7i −0.0163715 − 0.0682657i
\(36\) 5.84549e7 − 1.79906e8i 0.161123 − 0.495885i
\(37\) 3.90695e7 + 1.20244e8i 0.0926251 + 0.285071i 0.986627 − 0.162991i \(-0.0521143\pi\)
−0.894002 + 0.448062i \(0.852114\pi\)
\(38\) 2.85682e7 + 8.79239e7i 0.0584888 + 0.180010i
\(39\) 1.70910e8 − 5.26007e8i 0.303328 − 0.933548i
\(40\) 1.74076e8 + 1.48749e8i 0.268788 + 0.229680i
\(41\) −1.34607e8 − 4.14278e8i −0.181450 − 0.558445i 0.818419 − 0.574621i \(-0.194851\pi\)
−0.999869 + 0.0161764i \(0.994851\pi\)
\(42\) −4.86153e7 + 3.53211e7i −0.0573987 + 0.0417026i
\(43\) −4.46722e8 −0.463405 −0.231702 − 0.972787i \(-0.574430\pi\)
−0.231702 + 0.972787i \(0.574430\pi\)
\(44\) 2.94430e8 − 2.13916e8i 0.269148 − 0.195548i
\(45\) 1.10079e9 − 6.74196e8i 0.889273 − 0.544649i
\(46\) 8.98640e7 + 6.52900e7i 0.0643306 + 0.0467389i
\(47\) −9.81911e8 − 7.13400e8i −0.624502 − 0.453727i 0.229989 − 0.973193i \(-0.426131\pi\)
−0.854491 + 0.519466i \(0.826131\pi\)
\(48\) 1.94923e8 − 5.99911e8i 0.110417 − 0.339829i
\(49\) −1.96758e9 −0.995072
\(50\) 2.43677e8 + 1.54338e9i 0.110276 + 0.698455i
\(51\) 1.44934e9 0.588211
\(52\) 2.90929e8 − 8.95387e8i 0.106113 − 0.326582i
\(53\) 3.51028e9 + 2.55037e9i 1.15299 + 0.837695i 0.988875 − 0.148746i \(-0.0475238\pi\)
0.164113 + 0.986442i \(0.447524\pi\)
\(54\) −1.18101e8 − 8.58057e7i −0.0350017 − 0.0254303i
\(55\) 2.47577e9 + 1.95454e8i 0.663308 + 0.0523660i
\(56\) −8.27546e7 + 6.01247e7i −0.0200797 + 0.0145888i
\(57\) 1.73793e9 0.382578
\(58\) −4.94506e7 + 3.59280e7i −0.00989277 + 0.00718752i
\(59\) −3.90044e8 − 1.20043e9i −0.0710277 − 0.218601i 0.909241 − 0.416270i \(-0.136663\pi\)
−0.980269 + 0.197669i \(0.936663\pi\)
\(60\) 3.67068e9 − 2.24816e9i 0.609417 − 0.373247i
\(61\) 1.79302e9 − 5.51836e9i 0.271814 − 0.836558i −0.718231 − 0.695805i \(-0.755048\pi\)
0.990045 − 0.140753i \(-0.0449523\pi\)
\(62\) −2.73813e9 − 8.42711e9i −0.379578 − 1.16822i
\(63\) 1.78199e8 + 5.48440e8i 0.0226221 + 0.0696236i
\(64\) 3.31804e8 − 1.02119e9i 0.0386271 − 0.118882i
\(65\) 5.47861e9 − 3.35546e9i 0.585662 − 0.358698i
\(66\) −2.11416e9 − 6.50671e9i −0.207800 − 0.639543i
\(67\) 6.20425e9 − 4.50765e9i 0.561407 − 0.407886i −0.270567 − 0.962701i \(-0.587211\pi\)
0.831974 + 0.554815i \(0.187211\pi\)
\(68\) 2.46711e9 0.205773
\(69\) 1.68934e9 − 1.22738e9i 0.130031 − 0.0944729i
\(70\) −6.95858e8 − 5.49358e7i −0.0494859 − 0.00390675i
\(71\) 4.29969e9 + 3.12391e9i 0.282824 + 0.205484i 0.720148 − 0.693820i \(-0.244074\pi\)
−0.437324 + 0.899304i \(0.644074\pi\)
\(72\) −4.89718e9 − 3.55801e9i −0.298275 − 0.216710i
\(73\) −5.43059e9 + 1.67136e10i −0.306600 + 0.943616i 0.672476 + 0.740119i \(0.265231\pi\)
−0.979075 + 0.203497i \(0.934769\pi\)
\(74\) 4.04581e9 0.211949
\(75\) 2.90093e10 + 4.60911e9i 1.41156 + 0.224275i
\(76\) 2.95836e9 0.133837
\(77\) −3.42840e8 + 1.05515e9i −0.0144342 + 0.0444238i
\(78\) −1.43184e10 − 1.04029e10i −0.561530 − 0.407975i
\(79\) 3.23402e10 + 2.34965e10i 1.18248 + 0.859122i 0.992449 − 0.122656i \(-0.0391411\pi\)
0.190031 + 0.981778i \(0.439141\pi\)
\(80\) 6.24835e9 − 3.82690e9i 0.213192 − 0.130573i
\(81\) 2.42545e10 − 1.76219e10i 0.772901 − 0.561546i
\(82\) −1.39391e10 −0.415202
\(83\) −4.74630e10 + 3.44839e10i −1.32259 + 0.960918i −0.322694 + 0.946503i \(0.604589\pi\)
−0.999896 + 0.0144150i \(0.995411\pi\)
\(84\) 5.94221e8 + 1.82882e9i 0.0155029 + 0.0477129i
\(85\) 1.27991e10 + 1.09368e10i 0.312878 + 0.267354i
\(86\) −4.41743e9 + 1.35954e10i −0.101258 + 0.311639i
\(87\) 3.55081e8 + 1.09283e9i 0.00763786 + 0.0235069i
\(88\) −3.59879e9 − 1.10759e10i −0.0726945 − 0.223731i
\(89\) −2.79640e10 + 8.60644e10i −0.530829 + 1.63372i 0.221664 + 0.975123i \(0.428851\pi\)
−0.752493 + 0.658601i \(0.771149\pi\)
\(90\) −9.63314e9 − 4.01680e10i −0.171963 − 0.717045i
\(91\) 8.86894e8 + 2.72958e9i 0.0148986 + 0.0458530i
\(92\) 2.87565e9 − 2.08928e9i 0.0454886 − 0.0330494i
\(93\) −1.66572e11 −2.48283
\(94\) −3.14212e10 + 2.28288e10i −0.441590 + 0.320834i
\(95\) 1.53476e10 + 1.31146e10i 0.203498 + 0.173890i
\(96\) −1.63301e10 − 1.18645e10i −0.204407 − 0.148511i
\(97\) −4.95689e10 − 3.60139e10i −0.586090 − 0.425819i 0.254824 − 0.966987i \(-0.417982\pi\)
−0.840915 + 0.541168i \(0.817982\pi\)
\(98\) −1.94565e10 + 5.98810e10i −0.217431 + 0.669184i
\(99\) −6.56542e10 −0.693855
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.d.b.21.3 ✓ 56
25.6 even 5 inner 50.12.d.b.31.3 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.d.b.21.3 ✓ 56 1.1 even 1 trivial
50.12.d.b.31.3 yes 56 25.6 even 5 inner