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Properties

Label 50.26.a.c.1.2
Level $50$
Weight $26$
Character 50.1
Self dual yes
Analytic conductor $197.998$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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] = [50,26,Mod(1,50)] 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("50.1"); S:= CuspForms(chi, 26); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 26, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-8192,-379848] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(197.998389976\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{106705}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 26676 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-162.829\) of defining polynomial
Character \(\chi\) \(=\) 50.1

$q$-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)
 
\(f(q)\) \(=\) \(q-4096.00 q^{2} +1.37803e6 q^{3} +1.67772e7 q^{4} -5.64442e9 q^{6} -3.00388e10 q^{7} -6.87195e10 q^{8} +1.05168e12 q^{9} +5.73599e12 q^{11} +2.31195e13 q^{12} +1.07343e13 q^{13} +1.23039e14 q^{14} +2.81475e14 q^{16} -2.97473e15 q^{17} -4.30769e15 q^{18} -5.42738e15 q^{19} -4.13944e16 q^{21} -2.34946e16 q^{22} +1.04540e17 q^{23} -9.46976e16 q^{24} -4.39677e16 q^{26} +2.81659e17 q^{27} -5.03967e17 q^{28} +3.09182e18 q^{29} -4.26809e18 q^{31} -1.15292e18 q^{32} +7.90437e18 q^{33} +1.21845e19 q^{34} +1.76443e19 q^{36} +4.51028e19 q^{37} +2.22305e19 q^{38} +1.47922e19 q^{39} -7.56724e19 q^{41} +1.69551e20 q^{42} +1.39036e20 q^{43} +9.62339e19 q^{44} -4.28196e20 q^{46} -4.67328e20 q^{47} +3.87881e20 q^{48} -4.38741e20 q^{49} -4.09927e21 q^{51} +1.80092e20 q^{52} +1.24315e21 q^{53} -1.15368e21 q^{54} +2.06425e21 q^{56} -7.47909e21 q^{57} -1.26641e22 q^{58} -1.32468e22 q^{59} -2.36984e20 q^{61} +1.74821e22 q^{62} -3.15912e22 q^{63} +4.72237e21 q^{64} -3.23763e22 q^{66} +5.36922e22 q^{67} -4.99076e22 q^{68} +1.44059e23 q^{69} -2.27032e23 q^{71} -7.22710e22 q^{72} -2.78528e23 q^{73} -1.84741e23 q^{74} -9.10563e22 q^{76} -1.72302e23 q^{77} -6.05889e22 q^{78} +6.36958e23 q^{79} -5.02942e23 q^{81} +3.09954e23 q^{82} -1.19662e24 q^{83} -6.94482e23 q^{84} -5.69491e23 q^{86} +4.26063e24 q^{87} -3.94174e23 q^{88} -3.22134e24 q^{89} -3.22445e23 q^{91} +1.75389e24 q^{92} -5.88156e24 q^{93} +1.91418e24 q^{94} -1.58876e24 q^{96} -3.58465e24 q^{97} +1.79708e24 q^{98} +6.03243e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8192 q^{2} - 379848 q^{3} + 33554432 q^{4} + 1555857408 q^{6} + 376536944 q^{7} - 137438953472 q^{8} + 3294531432666 q^{9} + 8323034610264 q^{11} - 6372791943168 q^{12} + 106467053152292 q^{13} - 1542295322624 q^{14}+ \cdots + 11\!\cdots\!12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4096.00 −0.707107
\(3\) 1.37803e6 1.49707 0.748537 − 0.663093i \(-0.230757\pi\)
0.748537 + 0.663093i \(0.230757\pi\)
\(4\) 1.67772e7 0.500000
\(5\) 0 0
\(6\) −5.64442e9 −1.05859
\(7\) −3.00388e10 −0.820270 −0.410135 − 0.912025i \(-0.634518\pi\)
−0.410135 + 0.912025i \(0.634518\pi\)
\(8\) −6.87195e10 −0.353553
\(9\) 1.05168e12 1.24123
\(10\) 0 0
\(11\) 5.73599e12 0.551061 0.275531 − 0.961292i \(-0.411146\pi\)
0.275531 + 0.961292i \(0.411146\pi\)
\(12\) 2.31195e13 0.748537
\(13\) 1.07343e13 0.127786 0.0638928 − 0.997957i \(-0.479648\pi\)
0.0638928 + 0.997957i \(0.479648\pi\)
\(14\) 1.23039e14 0.580018
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) −2.97473e15 −1.23833 −0.619164 − 0.785262i \(-0.712528\pi\)
−0.619164 + 0.785262i \(0.712528\pi\)
\(18\) −4.30769e15 −0.877683
\(19\) −5.42738e15 −0.562561 −0.281281 − 0.959626i \(-0.590759\pi\)
−0.281281 + 0.959626i \(0.590759\pi\)
\(20\) 0 0
\(21\) −4.13944e16 −1.22800
\(22\) −2.34946e16 −0.389659
\(23\) 1.04540e17 0.994683 0.497341 − 0.867555i \(-0.334310\pi\)
0.497341 + 0.867555i \(0.334310\pi\)
\(24\) −9.46976e16 −0.529296
\(25\) 0 0
\(26\) −4.39677e16 −0.0903581
\(27\) 2.81659e17 0.361141
\(28\) −5.03967e17 −0.410135
\(29\) 3.09182e18 1.62270 0.811352 − 0.584558i \(-0.198732\pi\)
0.811352 + 0.584558i \(0.198732\pi\)
\(30\) 0 0
\(31\) −4.26809e18 −0.973222 −0.486611 − 0.873619i \(-0.661767\pi\)
−0.486611 + 0.873619i \(0.661767\pi\)
\(32\) −1.15292e18 −0.176777
\(33\) 7.90437e18 0.824980
\(34\) 1.21845e19 0.875630
\(35\) 0 0
\(36\) 1.76443e19 0.620616
\(37\) 4.51028e19 1.12637 0.563186 − 0.826330i \(-0.309576\pi\)
0.563186 + 0.826330i \(0.309576\pi\)
\(38\) 2.22305e19 0.397791
\(39\) 1.47922e19 0.191305
\(40\) 0 0
\(41\) −7.56724e19 −0.523769 −0.261884 − 0.965099i \(-0.584344\pi\)
−0.261884 + 0.965099i \(0.584344\pi\)
\(42\) 1.69551e20 0.868330
\(43\) 1.39036e20 0.530605 0.265302 − 0.964165i \(-0.414528\pi\)
0.265302 + 0.964165i \(0.414528\pi\)
\(44\) 9.62339e19 0.275531
\(45\) 0 0
\(46\) −4.28196e20 −0.703347
\(47\) −4.67328e20 −0.586676 −0.293338 − 0.956009i \(-0.594766\pi\)
−0.293338 + 0.956009i \(0.594766\pi\)
\(48\) 3.87881e20 0.374269
\(49\) −4.38741e20 −0.327158
\(50\) 0 0
\(51\) −4.09927e21 −1.85387
\(52\) 1.80092e20 0.0638928
\(53\) 1.24315e21 0.347595 0.173797 − 0.984781i \(-0.444396\pi\)
0.173797 + 0.984781i \(0.444396\pi\)
\(54\) −1.15368e21 −0.255365
\(55\) 0 0
\(56\) 2.06425e21 0.290009
\(57\) −7.47909e21 −0.842196
\(58\) −1.26641e22 −1.14743
\(59\) −1.32468e22 −0.969303 −0.484652 − 0.874707i \(-0.661054\pi\)
−0.484652 + 0.874707i \(0.661054\pi\)
\(60\) 0 0
\(61\) −2.36984e20 −0.0114313 −0.00571565 − 0.999984i \(-0.501819\pi\)
−0.00571565 + 0.999984i \(0.501819\pi\)
\(62\) 1.74821e22 0.688172
\(63\) −3.15912e22 −1.01814
\(64\) 4.72237e21 0.125000
\(65\) 0 0
\(66\) −3.23763e22 −0.583349
\(67\) 5.36922e22 0.801634 0.400817 − 0.916158i \(-0.368726\pi\)
0.400817 + 0.916158i \(0.368726\pi\)
\(68\) −4.99076e22 −0.619164
\(69\) 1.44059e23 1.48911
\(70\) 0 0
\(71\) −2.27032e23 −1.64194 −0.820971 − 0.570970i \(-0.806567\pi\)
−0.820971 + 0.570970i \(0.806567\pi\)
\(72\) −7.22710e22 −0.438842
\(73\) −2.78528e23 −1.42342 −0.711710 − 0.702473i \(-0.752079\pi\)
−0.711710 + 0.702473i \(0.752079\pi\)
\(74\) −1.84741e23 −0.796465
\(75\) 0 0
\(76\) −9.10563e22 −0.281281
\(77\) −1.72302e23 −0.452019
\(78\) −6.05889e22 −0.135273
\(79\) 6.36958e23 1.21275 0.606376 − 0.795178i \(-0.292622\pi\)
0.606376 + 0.795178i \(0.292622\pi\)
\(80\) 0 0
\(81\) −5.02942e23 −0.700576
\(82\) 3.09954e23 0.370360
\(83\) −1.19662e24 −1.22880 −0.614398 − 0.788996i \(-0.710601\pi\)
−0.614398 + 0.788996i \(0.710601\pi\)
\(84\) −6.94482e23 −0.614002
\(85\) 0 0
\(86\) −5.69491e23 −0.375194
\(87\) 4.26063e24 2.42931
\(88\) −3.94174e23 −0.194830
\(89\) −3.22134e24 −1.38249 −0.691244 − 0.722621i \(-0.742937\pi\)
−0.691244 + 0.722621i \(0.742937\pi\)
\(90\) 0 0
\(91\) −3.22445e23 −0.104819
\(92\) 1.75389e24 0.497341
\(93\) −5.88156e24 −1.45699
\(94\) 1.91418e24 0.414843
\(95\) 0 0
\(96\) −1.58876e24 −0.264648
\(97\) −3.58465e24 −0.524566 −0.262283 − 0.964991i \(-0.584475\pi\)
−0.262283 + 0.964991i \(0.584475\pi\)
\(98\) 1.79708e24 0.231335
\(99\) 6.03243e24 0.683994
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 50.26.a.c.1.2 2
5.2 odd 4 50.26.b.e.49.1 4
5.3 odd 4 50.26.b.e.49.4 4
5.4 even 2 2.26.a.b.1.1 ✓ 2
15.14 odd 2 18.26.a.e.1.1 2
20.19 odd 2 16.26.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.26.a.b.1.1 ✓ 2 5.4 even 2
16.26.a.c.1.2 2 20.19 odd 2
18.26.a.e.1.1 2 15.14 odd 2
50.26.a.c.1.2 2 1.1 even 1 trivial
50.26.b.e.49.1 4 5.2 odd 4
50.26.b.e.49.4 4 5.3 odd 4