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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [50,22,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, 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([1])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 50.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-8388608,0,198172672] 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: \(139.738672144\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 631153566 x^{6} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{4}\cdot 5^{14}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.3
Root \(-3842.38i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.22.b.h.49.6

$q$-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)
 
\(f(q)\) \(=\) \(q-1024.00i q^{2} +62614.8i q^{3} -1.04858e6 q^{4} +6.41176e7 q^{6} +1.35870e9i q^{7} +1.07374e9i q^{8} +6.53974e9 q^{9} +6.02123e10 q^{11} -6.56564e10i q^{12} -6.48747e11i q^{13} +1.39130e12 q^{14} +1.09951e12 q^{16} -1.36827e12i q^{17} -6.69669e12i q^{18} -3.73922e13 q^{19} -8.50745e13 q^{21} -6.16574e13i q^{22} -3.31105e14i q^{23} -6.72321e13 q^{24} -6.64316e14 q^{26} +1.06446e15i q^{27} -1.42470e15i q^{28} -2.94996e15 q^{29} +6.11374e15 q^{31} -1.12590e15i q^{32} +3.77018e15i q^{33} -1.40111e15 q^{34} -6.85741e15 q^{36} -2.13670e16i q^{37} +3.82896e16i q^{38} +4.06211e16 q^{39} +7.53942e16 q^{41} +8.71163e16i q^{42} +1.11574e17i q^{43} -6.31372e16 q^{44} -3.39051e17 q^{46} -3.03021e17i q^{47} +6.88457e16i q^{48} -1.28751e18 q^{49} +8.56742e16 q^{51} +6.80260e17i q^{52} -1.33812e18i q^{53} +1.09000e18 q^{54} -1.45889e18 q^{56} -2.34131e18i q^{57} +3.02076e18i q^{58} +8.63657e17 q^{59} +4.07764e18 q^{61} -6.26047e18i q^{62} +8.88552e18i q^{63} -1.15292e18 q^{64} +3.86067e18 q^{66} -2.05942e19i q^{67} +1.43474e18i q^{68} +2.07321e19 q^{69} -2.84243e19 q^{71} +7.02199e18i q^{72} +8.35063e18i q^{73} -2.18798e19 q^{74} +3.92086e19 q^{76} +8.18103e19i q^{77} -4.15961e19i q^{78} +2.13799e19 q^{79} +1.75716e18 q^{81} -7.72037e19i q^{82} +1.83840e20i q^{83} +8.92071e19 q^{84} +1.14252e20 q^{86} -1.84711e20i q^{87} +6.46525e19i q^{88} +3.15500e20 q^{89} +8.81449e20 q^{91} +3.47188e20i q^{92} +3.82811e20i q^{93} -3.10294e20 q^{94} +7.04980e19 q^{96} +1.08447e20i q^{97} +1.31841e21i q^{98} +3.93773e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8388608 q^{4} + 198172672 q^{6} - 47229523424 q^{9} + 175118508216 q^{11} - 333333561344 q^{14} + 8796093022208 q^{16} - 167026213839880 q^{19} - 8313944227904 q^{21} - 207799107715072 q^{24} - 200905406660608 q^{26}+ \cdots + 56\!\cdots\!52 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1024.00i − 0.707107i
\(3\) 62614.8i 0.612215i 0.951997 + 0.306107i \(0.0990267\pi\)
−0.951997 + 0.306107i \(0.900973\pi\)
\(4\) −1.04858e6 −0.500000
\(5\) 0 0
\(6\) 6.41176e7 0.432901
\(7\) 1.35870e9i 1.81800i 0.416801 + 0.908998i \(0.363151\pi\)
−0.416801 + 0.908998i \(0.636849\pi\)
\(8\) 1.07374e9i 0.353553i
\(9\) 6.53974e9 0.625193
\(10\) 0 0
\(11\) 6.02123e10 0.699942 0.349971 − 0.936760i \(-0.386191\pi\)
0.349971 + 0.936760i \(0.386191\pi\)
\(12\) − 6.56564e10i − 0.306107i
\(13\) − 6.48747e11i − 1.30518i −0.757712 − 0.652589i \(-0.773683\pi\)
0.757712 − 0.652589i \(-0.226317\pi\)
\(14\) 1.39130e12 1.28552
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) − 1.36827e12i − 0.164611i −0.996607 − 0.0823056i \(-0.973772\pi\)
0.996607 − 0.0823056i \(-0.0262284\pi\)
\(18\) − 6.69669e12i − 0.442078i
\(19\) −3.73922e13 −1.39916 −0.699581 − 0.714553i \(-0.746630\pi\)
−0.699581 + 0.714553i \(0.746630\pi\)
\(20\) 0 0
\(21\) −8.50745e13 −1.11300
\(22\) − 6.16574e13i − 0.494934i
\(23\) − 3.31105e14i − 1.66657i −0.552845 − 0.833284i \(-0.686458\pi\)
0.552845 − 0.833284i \(-0.313542\pi\)
\(24\) −6.72321e13 −0.216451
\(25\) 0 0
\(26\) −6.64316e14 −0.922900
\(27\) 1.06446e15i 0.994967i
\(28\) − 1.42470e15i − 0.908998i
\(29\) −2.94996e15 −1.30208 −0.651039 − 0.759044i \(-0.725667\pi\)
−0.651039 + 0.759044i \(0.725667\pi\)
\(30\) 0 0
\(31\) 6.11374e15 1.33971 0.669853 − 0.742494i \(-0.266357\pi\)
0.669853 + 0.742494i \(0.266357\pi\)
\(32\) − 1.12590e15i − 0.176777i
\(33\) 3.77018e15i 0.428515i
\(34\) −1.40111e15 −0.116398
\(35\) 0 0
\(36\) −6.85741e15 −0.312596
\(37\) − 2.13670e16i − 0.730509i −0.930908 − 0.365254i \(-0.880982\pi\)
0.930908 − 0.365254i \(-0.119018\pi\)
\(38\) 3.82896e16i 0.989358i
\(39\) 4.06211e16 0.799050
\(40\) 0 0
\(41\) 7.53942e16 0.877218 0.438609 − 0.898678i \(-0.355471\pi\)
0.438609 + 0.898678i \(0.355471\pi\)
\(42\) 8.71163e16i 0.787013i
\(43\) 1.11574e17i 0.787310i 0.919258 + 0.393655i \(0.128790\pi\)
−0.919258 + 0.393655i \(0.871210\pi\)
\(44\) −6.31372e16 −0.349971
\(45\) 0 0
\(46\) −3.39051e17 −1.17844
\(47\) − 3.03021e17i − 0.840322i −0.907450 − 0.420161i \(-0.861974\pi\)
0.907450 − 0.420161i \(-0.138026\pi\)
\(48\) 6.88457e16i 0.153054i
\(49\) −1.28751e18 −2.30511
\(50\) 0 0
\(51\) 8.56742e16 0.100777
\(52\) 6.80260e17i 0.652589i
\(53\) − 1.33812e18i − 1.05099i −0.850798 − 0.525493i \(-0.823881\pi\)
0.850798 − 0.525493i \(-0.176119\pi\)
\(54\) 1.09000e18 0.703548
\(55\) 0 0
\(56\) −1.45889e18 −0.642759
\(57\) − 2.34131e18i − 0.856588i
\(58\) 3.02076e18i 0.920708i
\(59\) 8.63657e17 0.219986 0.109993 − 0.993932i \(-0.464917\pi\)
0.109993 + 0.993932i \(0.464917\pi\)
\(60\) 0 0
\(61\) 4.07764e18 0.731889 0.365944 − 0.930637i \(-0.380746\pi\)
0.365944 + 0.930637i \(0.380746\pi\)
\(62\) − 6.26047e18i − 0.947315i
\(63\) 8.88552e18i 1.13660i
\(64\) −1.15292e18 −0.125000
\(65\) 0 0
\(66\) 3.86067e18 0.303006
\(67\) − 2.05942e19i − 1.38026i −0.723687 − 0.690129i \(-0.757554\pi\)
0.723687 − 0.690129i \(-0.242446\pi\)
\(68\) 1.43474e18i 0.0823056i
\(69\) 2.07321e19 1.02030
\(70\) 0 0
\(71\) −2.84243e19 −1.03628 −0.518140 − 0.855296i \(-0.673375\pi\)
−0.518140 + 0.855296i \(0.673375\pi\)
\(72\) 7.02199e18i 0.221039i
\(73\) 8.35063e18i 0.227420i 0.993514 + 0.113710i \(0.0362735\pi\)
−0.993514 + 0.113710i \(0.963726\pi\)
\(74\) −2.18798e19 −0.516548
\(75\) 0 0
\(76\) 3.92086e19 0.699581
\(77\) 8.18103e19i 1.27249i
\(78\) − 4.15961e19i − 0.565013i
\(79\) 2.13799e19 0.254052 0.127026 − 0.991899i \(-0.459457\pi\)
0.127026 + 0.991899i \(0.459457\pi\)
\(80\) 0 0
\(81\) 1.75716e18 0.0160590
\(82\) − 7.72037e19i − 0.620287i
\(83\) 1.83840e20i 1.30053i 0.759707 + 0.650265i \(0.225342\pi\)
−0.759707 + 0.650265i \(0.774658\pi\)
\(84\) 8.92071e19 0.556502
\(85\) 0 0
\(86\) 1.14252e20 0.556712
\(87\) − 1.84711e20i − 0.797151i
\(88\) 6.46525e19i 0.247467i
\(89\) 3.15500e20 1.07252 0.536259 − 0.844054i \(-0.319837\pi\)
0.536259 + 0.844054i \(0.319837\pi\)
\(90\) 0 0
\(91\) 8.81449e20 2.37281
\(92\) 3.47188e20i 0.833284i
\(93\) 3.82811e20i 0.820188i
\(94\) −3.10294e20 −0.594197
\(95\) 0 0
\(96\) 7.04980e19 0.108225
\(97\) 1.08447e20i 0.149318i 0.997209 + 0.0746592i \(0.0237869\pi\)
−0.997209 + 0.0746592i \(0.976213\pi\)
\(98\) 1.31841e21i 1.62996i
\(99\) 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.b.h.49.3 8
5.2 odd 4 50.22.a.j.1.3 yes 4
5.3 odd 4 50.22.a.i.1.2 ✓ 4
5.4 even 2 inner 50.22.b.h.49.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.22.a.i.1.2 ✓ 4 5.3 odd 4
50.22.a.j.1.3 yes 4 5.2 odd 4
50.22.b.h.49.3 8 1.1 even 1 trivial
50.22.b.h.49.6 8 5.4 even 2 inner