Properties

Label 50.22.a.i.1.2
Level $50$
Weight $22$
Character 50.1
Self dual yes
Analytic conductor $139.739$
Analytic rank $0$
Dimension $4$
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] = [50,22,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, 22); 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, 22, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-4096,-96764] 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: \(139.738672144\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 315576783x^{2} - 792388522562x + 1396510961585520 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 5^{7}\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3842.38\) 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)
 
\(f(q)\) \(=\) \(q-1024.00 q^{2} -62614.8 q^{3} +1.04858e6 q^{4} +6.41176e7 q^{6} +1.35870e9 q^{7} -1.07374e9 q^{8} -6.53974e9 q^{9} +6.02123e10 q^{11} -6.56564e10 q^{12} +6.48747e11 q^{13} -1.39130e12 q^{14} +1.09951e12 q^{16} -1.36827e12 q^{17} +6.69669e12 q^{18} +3.73922e13 q^{19} -8.50745e13 q^{21} -6.16574e13 q^{22} +3.31105e14 q^{23} +6.72321e13 q^{24} -6.64316e14 q^{26} +1.06446e15 q^{27} +1.42470e15 q^{28} +2.94996e15 q^{29} +6.11374e15 q^{31} -1.12590e15 q^{32} -3.77018e15 q^{33} +1.40111e15 q^{34} -6.85741e15 q^{36} -2.13670e16 q^{37} -3.82896e16 q^{38} -4.06211e16 q^{39} +7.53942e16 q^{41} +8.71163e16 q^{42} -1.11574e17 q^{43} +6.31372e16 q^{44} -3.39051e17 q^{46} -3.03021e17 q^{47} -6.88457e16 q^{48} +1.28751e18 q^{49} +8.56742e16 q^{51} +6.80260e17 q^{52} +1.33812e18 q^{53} -1.09000e18 q^{54} -1.45889e18 q^{56} -2.34131e18 q^{57} -3.02076e18 q^{58} -8.63657e17 q^{59} +4.07764e18 q^{61} -6.26047e18 q^{62} -8.88552e18 q^{63} +1.15292e18 q^{64} +3.86067e18 q^{66} -2.05942e19 q^{67} -1.43474e18 q^{68} -2.07321e19 q^{69} -2.84243e19 q^{71} +7.02199e18 q^{72} -8.35063e18 q^{73} +2.18798e19 q^{74} +3.92086e19 q^{76} +8.18103e19 q^{77} +4.15961e19 q^{78} -2.13799e19 q^{79} +1.75716e18 q^{81} -7.72037e19 q^{82} -1.83840e20 q^{83} -8.92071e19 q^{84} +1.14252e20 q^{86} -1.84711e20 q^{87} -6.46525e19 q^{88} -3.15500e20 q^{89} +8.81449e20 q^{91} +3.47188e20 q^{92} -3.82811e20 q^{93} +3.10294e20 q^{94} +7.04980e19 q^{96} +1.08447e20 q^{97} -1.31841e21 q^{98} -3.93773e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4096 q^{2} - 96764 q^{3} + 4194304 q^{4} + 99086336 q^{6} - 162760528 q^{7} - 4294967296 q^{8} + 23614761712 q^{9} + 87559254108 q^{11} - 101464408064 q^{12} + 98098343096 q^{13} + 166666780672 q^{14}+ \cdots - 28\!\cdots\!76 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1024.00 −0.707107
\(3\) −62614.8 −0.612215 −0.306107 − 0.951997i \(-0.599027\pi\)
−0.306107 + 0.951997i \(0.599027\pi\)
\(4\) 1.04858e6 0.500000
\(5\) 0 0
\(6\) 6.41176e7 0.432901
\(7\) 1.35870e9 1.81800 0.908998 − 0.416801i \(-0.136849\pi\)
0.908998 + 0.416801i \(0.136849\pi\)
\(8\) −1.07374e9 −0.353553
\(9\) −6.53974e9 −0.625193
\(10\) 0 0
\(11\) 6.02123e10 0.699942 0.349971 − 0.936760i \(-0.386191\pi\)
0.349971 + 0.936760i \(0.386191\pi\)
\(12\) −6.56564e10 −0.306107
\(13\) 6.48747e11 1.30518 0.652589 − 0.757712i \(-0.273683\pi\)
0.652589 + 0.757712i \(0.273683\pi\)
\(14\) −1.39130e12 −1.28552
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) −1.36827e12 −0.164611 −0.0823056 − 0.996607i \(-0.526228\pi\)
−0.0823056 + 0.996607i \(0.526228\pi\)
\(18\) 6.69669e12 0.442078
\(19\) 3.73922e13 1.39916 0.699581 − 0.714553i \(-0.253370\pi\)
0.699581 + 0.714553i \(0.253370\pi\)
\(20\) 0 0
\(21\) −8.50745e13 −1.11300
\(22\) −6.16574e13 −0.494934
\(23\) 3.31105e14 1.66657 0.833284 − 0.552845i \(-0.186458\pi\)
0.833284 + 0.552845i \(0.186458\pi\)
\(24\) 6.72321e13 0.216451
\(25\) 0 0
\(26\) −6.64316e14 −0.922900
\(27\) 1.06446e15 0.994967
\(28\) 1.42470e15 0.908998
\(29\) 2.94996e15 1.30208 0.651039 − 0.759044i \(-0.274333\pi\)
0.651039 + 0.759044i \(0.274333\pi\)
\(30\) 0 0
\(31\) 6.11374e15 1.33971 0.669853 − 0.742494i \(-0.266357\pi\)
0.669853 + 0.742494i \(0.266357\pi\)
\(32\) −1.12590e15 −0.176777
\(33\) −3.77018e15 −0.428515
\(34\) 1.40111e15 0.116398
\(35\) 0 0
\(36\) −6.85741e15 −0.312596
\(37\) −2.13670e16 −0.730509 −0.365254 − 0.930908i \(-0.619018\pi\)
−0.365254 + 0.930908i \(0.619018\pi\)
\(38\) −3.82896e16 −0.989358
\(39\) −4.06211e16 −0.799050
\(40\) 0 0
\(41\) 7.53942e16 0.877218 0.438609 − 0.898678i \(-0.355471\pi\)
0.438609 + 0.898678i \(0.355471\pi\)
\(42\) 8.71163e16 0.787013
\(43\) −1.11574e17 −0.787310 −0.393655 − 0.919258i \(-0.628790\pi\)
−0.393655 + 0.919258i \(0.628790\pi\)
\(44\) 6.31372e16 0.349971
\(45\) 0 0
\(46\) −3.39051e17 −1.17844
\(47\) −3.03021e17 −0.840322 −0.420161 − 0.907450i \(-0.638026\pi\)
−0.420161 + 0.907450i \(0.638026\pi\)
\(48\) −6.88457e16 −0.153054
\(49\) 1.28751e18 2.30511
\(50\) 0 0
\(51\) 8.56742e16 0.100777
\(52\) 6.80260e17 0.652589
\(53\) 1.33812e18 1.05099 0.525493 − 0.850798i \(-0.323881\pi\)
0.525493 + 0.850798i \(0.323881\pi\)
\(54\) −1.09000e18 −0.703548
\(55\) 0 0
\(56\) −1.45889e18 −0.642759
\(57\) −2.34131e18 −0.856588
\(58\) −3.02076e18 −0.920708
\(59\) −8.63657e17 −0.219986 −0.109993 − 0.993932i \(-0.535083\pi\)
−0.109993 + 0.993932i \(0.535083\pi\)
\(60\) 0 0
\(61\) 4.07764e18 0.731889 0.365944 − 0.930637i \(-0.380746\pi\)
0.365944 + 0.930637i \(0.380746\pi\)
\(62\) −6.26047e18 −0.947315
\(63\) −8.88552e18 −1.13660
\(64\) 1.15292e18 0.125000
\(65\) 0 0
\(66\) 3.86067e18 0.303006
\(67\) −2.05942e19 −1.38026 −0.690129 − 0.723687i \(-0.742446\pi\)
−0.690129 + 0.723687i \(0.742446\pi\)
\(68\) −1.43474e18 −0.0823056
\(69\) −2.07321e19 −1.02030
\(70\) 0 0
\(71\) −2.84243e19 −1.03628 −0.518140 − 0.855296i \(-0.673375\pi\)
−0.518140 + 0.855296i \(0.673375\pi\)
\(72\) 7.02199e18 0.221039
\(73\) −8.35063e18 −0.227420 −0.113710 − 0.993514i \(-0.536274\pi\)
−0.113710 + 0.993514i \(0.536274\pi\)
\(74\) 2.18798e19 0.516548
\(75\) 0 0
\(76\) 3.92086e19 0.699581
\(77\) 8.18103e19 1.27249
\(78\) 4.15961e19 0.565013
\(79\) −2.13799e19 −0.254052 −0.127026 − 0.991899i \(-0.540543\pi\)
−0.127026 + 0.991899i \(0.540543\pi\)
\(80\) 0 0
\(81\) 1.75716e18 0.0160590
\(82\) −7.72037e19 −0.620287
\(83\) −1.83840e20 −1.30053 −0.650265 − 0.759707i \(-0.725342\pi\)
−0.650265 + 0.759707i \(0.725342\pi\)
\(84\) −8.92071e19 −0.556502
\(85\) 0 0
\(86\) 1.14252e20 0.556712
\(87\) −1.84711e20 −0.797151
\(88\) −6.46525e19 −0.247467
\(89\) −3.15500e20 −1.07252 −0.536259 − 0.844054i \(-0.680163\pi\)
−0.536259 + 0.844054i \(0.680163\pi\)
\(90\) 0 0
\(91\) 8.81449e20 2.37281
\(92\) 3.47188e20 0.833284
\(93\) −3.82811e20 −0.820188
\(94\) 3.10294e20 0.594197
\(95\) 0 0
\(96\) 7.04980e19 0.108225
\(97\) 1.08447e20 0.149318 0.0746592 − 0.997209i \(-0.476213\pi\)
0.0746592 + 0.997209i \(0.476213\pi\)
\(98\) −1.31841e21 −1.62996
\(99\) −3.93773e20 −0.437599
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 50.22.a.i.1.2 ✓ 4
5.2 odd 4 50.22.b.h.49.3 8
5.3 odd 4 50.22.b.h.49.6 8
5.4 even 2 50.22.a.j.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.22.a.i.1.2 ✓ 4 1.1 even 1 trivial
50.22.a.j.1.3 yes 4 5.4 even 2
50.22.b.h.49.3 8 5.2 odd 4
50.22.b.h.49.6 8 5.3 odd 4