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Properties

Label 150.10.c.f.49.1
Level $150$
Weight $10$
Character 150.49
Analytic conductor $77.255$
Analytic rank $0$
Dimension $2$
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] = [150,10,Mod(49,150)] 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("150.49"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 150.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-512,0,2592,0,0,-13122,0,-167496] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(77.2553754246\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 150.49
Dual form 150.10.c.f.49.2

$q$-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)
 
\(f(q)\) \(=\) \(q-16.0000i q^{2} +81.0000i q^{3} -256.000 q^{4} +1296.00 q^{6} -7168.00i q^{7} +4096.00i q^{8} -6561.00 q^{9} -83748.0 q^{11} -20736.0i q^{12} -128126. i q^{13} -114688. q^{14} +65536.0 q^{16} +560802. i q^{17} +104976. i q^{18} +577660. q^{19} +580608. q^{21} +1.33997e6i q^{22} -2.43130e6i q^{23} -331776. q^{24} -2.05002e6 q^{26} -531441. i q^{27} +1.83501e6i q^{28} -5.79171e6 q^{29} +4.14531e6 q^{31} -1.04858e6i q^{32} -6.78359e6i q^{33} +8.97283e6 q^{34} +1.67962e6 q^{36} -7.01166e6i q^{37} -9.24256e6i q^{38} +1.03782e7 q^{39} -8.88140e6 q^{41} -9.28973e6i q^{42} +1.57307e7i q^{43} +2.14395e7 q^{44} -3.89007e7 q^{46} +6.05521e7i q^{47} +5.30842e6i q^{48} -1.10266e7 q^{49} -4.54250e7 q^{51} +3.28003e7i q^{52} -3.02734e7i q^{53} -8.50306e6 q^{54} +2.93601e7 q^{56} +4.67905e7i q^{57} +9.26674e7i q^{58} -4.59577e7 q^{59} +3.75951e7 q^{61} -6.63250e7i q^{62} +4.70292e7i q^{63} -1.67772e7 q^{64} -1.08537e8 q^{66} +1.96784e8i q^{67} -1.43565e8i q^{68} +1.96935e8 q^{69} +5.60480e7 q^{71} -2.68739e7i q^{72} +1.59688e8i q^{73} -1.12187e8 q^{74} -1.47881e8 q^{76} +6.00306e8i q^{77} -1.66051e8i q^{78} -2.01923e8 q^{79} +4.30467e7 q^{81} +1.42102e8i q^{82} +3.62955e8i q^{83} -1.48636e8 q^{84} +2.51691e8 q^{86} -4.69129e8i q^{87} -3.43032e8i q^{88} +2.72479e8 q^{89} -9.18407e8 q^{91} +6.22412e8i q^{92} +3.35770e8i q^{93} +9.68833e8 q^{94} +8.49347e7 q^{96} -6.00852e8i q^{97} +1.76426e8i q^{98} +5.49471e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 512 q^{4} + 2592 q^{6} - 13122 q^{9} - 167496 q^{11} - 229376 q^{14} + 131072 q^{16} + 1155320 q^{19} + 1161216 q^{21} - 663552 q^{24} - 4100032 q^{26} - 11583420 q^{29} + 8290624 q^{31} + 17945664 q^{34}+ \cdots + 1098941256 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 16.0000i − 0.707107i
\(3\) 81.0000i 0.577350i
\(4\) −256.000 −0.500000
\(5\) 0 0
\(6\) 1296.00 0.408248
\(7\) − 7168.00i − 1.12838i −0.825644 − 0.564192i \(-0.809188\pi\)
0.825644 − 0.564192i \(-0.190812\pi\)
\(8\) 4096.00i 0.353553i
\(9\) −6561.00 −0.333333
\(10\) 0 0
\(11\) −83748.0 −1.72468 −0.862338 − 0.506334i \(-0.831000\pi\)
−0.862338 + 0.506334i \(0.831000\pi\)
\(12\) − 20736.0i − 0.288675i
\(13\) − 128126.i − 1.24421i −0.782936 − 0.622103i \(-0.786279\pi\)
0.782936 − 0.622103i \(-0.213721\pi\)
\(14\) −114688. −0.797888
\(15\) 0 0
\(16\) 65536.0 0.250000
\(17\) 560802.i 1.62851i 0.580510 + 0.814253i \(0.302853\pi\)
−0.580510 + 0.814253i \(0.697147\pi\)
\(18\) 104976.i 0.235702i
\(19\) 577660. 1.01691 0.508453 − 0.861090i \(-0.330217\pi\)
0.508453 + 0.861090i \(0.330217\pi\)
\(20\) 0 0
\(21\) 580608. 0.651473
\(22\) 1.33997e6i 1.21953i
\(23\) − 2.43130e6i − 1.81160i −0.423704 − 0.905801i \(-0.639270\pi\)
0.423704 − 0.905801i \(-0.360730\pi\)
\(24\) −331776. −0.204124
\(25\) 0 0
\(26\) −2.05002e6 −0.879786
\(27\) − 531441.i − 0.192450i
\(28\) 1.83501e6i 0.564192i
\(29\) −5.79171e6 −1.52060 −0.760301 − 0.649570i \(-0.774949\pi\)
−0.760301 + 0.649570i \(0.774949\pi\)
\(30\) 0 0
\(31\) 4.14531e6 0.806176 0.403088 − 0.915161i \(-0.367937\pi\)
0.403088 + 0.915161i \(0.367937\pi\)
\(32\) − 1.04858e6i − 0.176777i
\(33\) − 6.78359e6i − 0.995742i
\(34\) 8.97283e6 1.15153
\(35\) 0 0
\(36\) 1.67962e6 0.166667
\(37\) − 7.01166e6i − 0.615054i −0.951539 − 0.307527i \(-0.900499\pi\)
0.951539 − 0.307527i \(-0.0995014\pi\)
\(38\) − 9.24256e6i − 0.719062i
\(39\) 1.03782e7 0.718342
\(40\) 0 0
\(41\) −8.88140e6 −0.490856 −0.245428 − 0.969415i \(-0.578928\pi\)
−0.245428 + 0.969415i \(0.578928\pi\)
\(42\) − 9.28973e6i − 0.460661i
\(43\) 1.57307e7i 0.701681i 0.936435 + 0.350840i \(0.114104\pi\)
−0.936435 + 0.350840i \(0.885896\pi\)
\(44\) 2.14395e7 0.862338
\(45\) 0 0
\(46\) −3.89007e7 −1.28100
\(47\) 6.05521e7i 1.81004i 0.425367 + 0.905021i \(0.360145\pi\)
−0.425367 + 0.905021i \(0.639855\pi\)
\(48\) 5.30842e6i 0.144338i
\(49\) −1.10266e7 −0.273250
\(50\) 0 0
\(51\) −4.54250e7 −0.940218
\(52\) 3.28003e7i 0.622103i
\(53\) − 3.02734e7i − 0.527011i −0.964658 − 0.263505i \(-0.915121\pi\)
0.964658 − 0.263505i \(-0.0848786\pi\)
\(54\) −8.50306e6 −0.136083
\(55\) 0 0
\(56\) 2.93601e7 0.398944
\(57\) 4.67905e7i 0.587111i
\(58\) 9.26674e7i 1.07523i
\(59\) −4.59577e7 −0.493769 −0.246885 − 0.969045i \(-0.579407\pi\)
−0.246885 + 0.969045i \(0.579407\pi\)
\(60\) 0 0
\(61\) 3.75951e7 0.347654 0.173827 − 0.984776i \(-0.444387\pi\)
0.173827 + 0.984776i \(0.444387\pi\)
\(62\) − 6.63250e7i − 0.570052i
\(63\) 4.70292e7i 0.376128i
\(64\) −1.67772e7 −0.125000
\(65\) 0 0
\(66\) −1.08537e8 −0.704096
\(67\) 1.96784e8i 1.19304i 0.802600 + 0.596518i \(0.203449\pi\)
−0.802600 + 0.596518i \(0.796551\pi\)
\(68\) − 1.43565e8i − 0.814253i
\(69\) 1.96935e8 1.04593
\(70\) 0 0
\(71\) 5.60480e7 0.261757 0.130878 − 0.991398i \(-0.458220\pi\)
0.130878 + 0.991398i \(0.458220\pi\)
\(72\) − 2.68739e7i − 0.117851i
\(73\) 1.59688e8i 0.658142i 0.944305 + 0.329071i \(0.106736\pi\)
−0.944305 + 0.329071i \(0.893264\pi\)
\(74\) −1.12187e8 −0.434909
\(75\) 0 0
\(76\) −1.47881e8 −0.508453
\(77\) 6.00306e8i 1.94610i
\(78\) − 1.66051e8i − 0.507945i
\(79\) −2.01923e8 −0.583263 −0.291632 − 0.956531i \(-0.594198\pi\)
−0.291632 + 0.956531i \(0.594198\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 1.42102e8i 0.347088i
\(83\) 3.62955e8i 0.839464i 0.907648 + 0.419732i \(0.137876\pi\)
−0.907648 + 0.419732i \(0.862124\pi\)
\(84\) −1.48636e8 −0.325736
\(85\) 0 0
\(86\) 2.51691e8 0.496163
\(87\) − 4.69129e8i − 0.877920i
\(88\) − 3.43032e8i − 0.609765i
\(89\) 2.72479e8 0.460339 0.230170 − 0.973151i \(-0.426072\pi\)
0.230170 + 0.973151i \(0.426072\pi\)
\(90\) 0 0
\(91\) −9.18407e8 −1.40394
\(92\) 6.22412e8i 0.905801i
\(93\) 3.35770e8i 0.465446i
\(94\) 9.68833e8 1.27989
\(95\) 0 0
\(96\) 8.49347e7 0.102062
\(97\) − 6.00852e8i − 0.689120i −0.938764 − 0.344560i \(-0.888028\pi\)
0.938764 − 0.344560i \(-0.111972\pi\)
\(98\) 1.76426e8i 0.193217i
\(99\) 5.49471e8 0.574892
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 150.10.c.f.49.1 2
5.2 odd 4 150.10.a.k.1.1 1
5.3 odd 4 30.10.a.b.1.1 ✓ 1
5.4 even 2 inner 150.10.c.f.49.2 2
15.8 even 4 90.10.a.f.1.1 1
20.3 even 4 240.10.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.10.a.b.1.1 ✓ 1 5.3 odd 4
90.10.a.f.1.1 1 15.8 even 4
150.10.a.k.1.1 1 5.2 odd 4
150.10.c.f.49.1 2 1.1 even 1 trivial
150.10.c.f.49.2 2 5.4 even 2 inner
240.10.a.i.1.1 1 20.3 even 4