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Properties

Label 50.8.b.c.49.2
Level $50$
Weight $8$
Character 50.49
Analytic conductor $15.619$
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] = [50,8,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, 8); 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, 8, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 50.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-128,0,-192] 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: \(15.6192512742\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.8.b.c.49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+8.00000i q^{2} +12.0000i q^{3} -64.0000 q^{4} -96.0000 q^{6} -1016.00i q^{7} -512.000i q^{8} +2043.00 q^{9} +1092.00 q^{11} -768.000i q^{12} +1382.00i q^{13} +8128.00 q^{14} +4096.00 q^{16} -14706.0i q^{17} +16344.0i q^{18} +39940.0 q^{19} +12192.0 q^{21} +8736.00i q^{22} +68712.0i q^{23} +6144.00 q^{24} -11056.0 q^{26} +50760.0i q^{27} +65024.0i q^{28} +102570. q^{29} +227552. q^{31} +32768.0i q^{32} +13104.0i q^{33} +117648. q^{34} -130752. q^{36} -160526. i q^{37} +319520. i q^{38} -16584.0 q^{39} +10842.0 q^{41} +97536.0i q^{42} -630748. i q^{43} -69888.0 q^{44} -549696. q^{46} -472656. i q^{47} +49152.0i q^{48} -208713. q^{49} +176472. q^{51} -88448.0i q^{52} -1.49402e6i q^{53} -406080. q^{54} -520192. q^{56} +479280. i q^{57} +820560. i q^{58} -2.64066e6 q^{59} +827702. q^{61} +1.82042e6i q^{62} -2.07569e6i q^{63} -262144. q^{64} -104832. q^{66} +126004. i q^{67} +941184. i q^{68} -824544. q^{69} -1.41473e6 q^{71} -1.04602e6i q^{72} +980282. i q^{73} +1.28421e6 q^{74} -2.55616e6 q^{76} -1.10947e6i q^{77} -132672. i q^{78} +3.56680e6 q^{79} +3.85892e6 q^{81} +86736.0i q^{82} +5.67289e6i q^{83} -780288. q^{84} +5.04598e6 q^{86} +1.23084e6i q^{87} -559104. i q^{88} +1.19512e7 q^{89} +1.40411e6 q^{91} -4.39757e6i q^{92} +2.73062e6i q^{93} +3.78125e6 q^{94} -393216. q^{96} -8.68215e6i q^{97} -1.66970e6i q^{98} +2.23096e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 128 q^{4} - 192 q^{6} + 4086 q^{9} + 2184 q^{11} + 16256 q^{14} + 8192 q^{16} + 79880 q^{19} + 24384 q^{21} + 12288 q^{24} - 22112 q^{26} + 205140 q^{29} + 455104 q^{31} + 235296 q^{34} - 261504 q^{36}+ \cdots + 4461912 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000i 0.707107i
\(3\) 12.0000i 0.256600i 0.991735 + 0.128300i \(0.0409521\pi\)
−0.991735 + 0.128300i \(0.959048\pi\)
\(4\) −64.0000 −0.500000
\(5\) 0 0
\(6\) −96.0000 −0.181444
\(7\) − 1016.00i − 1.11957i −0.828638 − 0.559784i \(-0.810884\pi\)
0.828638 − 0.559784i \(-0.189116\pi\)
\(8\) − 512.000i − 0.353553i
\(9\) 2043.00 0.934156
\(10\) 0 0
\(11\) 1092.00 0.247371 0.123685 − 0.992321i \(-0.460529\pi\)
0.123685 + 0.992321i \(0.460529\pi\)
\(12\) − 768.000i − 0.128300i
\(13\) 1382.00i 0.174464i 0.996188 + 0.0872321i \(0.0278022\pi\)
−0.996188 + 0.0872321i \(0.972198\pi\)
\(14\) 8128.00 0.791654
\(15\) 0 0
\(16\) 4096.00 0.250000
\(17\) − 14706.0i − 0.725978i −0.931793 − 0.362989i \(-0.881756\pi\)
0.931793 − 0.362989i \(-0.118244\pi\)
\(18\) 16344.0i 0.660548i
\(19\) 39940.0 1.33589 0.667945 − 0.744211i \(-0.267174\pi\)
0.667945 + 0.744211i \(0.267174\pi\)
\(20\) 0 0
\(21\) 12192.0 0.287281
\(22\) 8736.00i 0.174917i
\(23\) 68712.0i 1.17757i 0.808291 + 0.588783i \(0.200393\pi\)
−0.808291 + 0.588783i \(0.799607\pi\)
\(24\) 6144.00 0.0907218
\(25\) 0 0
\(26\) −11056.0 −0.123365
\(27\) 50760.0i 0.496305i
\(28\) 65024.0i 0.559784i
\(29\) 102570. 0.780957 0.390479 − 0.920612i \(-0.372310\pi\)
0.390479 + 0.920612i \(0.372310\pi\)
\(30\) 0 0
\(31\) 227552. 1.37188 0.685938 − 0.727660i \(-0.259392\pi\)
0.685938 + 0.727660i \(0.259392\pi\)
\(32\) 32768.0i 0.176777i
\(33\) 13104.0i 0.0634753i
\(34\) 117648. 0.513344
\(35\) 0 0
\(36\) −130752. −0.467078
\(37\) − 160526.i − 0.521002i −0.965474 − 0.260501i \(-0.916112\pi\)
0.965474 − 0.260501i \(-0.0838877\pi\)
\(38\) 319520.i 0.944616i
\(39\) −16584.0 −0.0447675
\(40\) 0 0
\(41\) 10842.0 0.0245678 0.0122839 − 0.999925i \(-0.496090\pi\)
0.0122839 + 0.999925i \(0.496090\pi\)
\(42\) 97536.0i 0.203139i
\(43\) − 630748.i − 1.20981i −0.796299 − 0.604904i \(-0.793212\pi\)
0.796299 − 0.604904i \(-0.206788\pi\)
\(44\) −69888.0 −0.123685
\(45\) 0 0
\(46\) −549696. −0.832665
\(47\) − 472656.i − 0.664053i −0.943270 − 0.332026i \(-0.892268\pi\)
0.943270 − 0.332026i \(-0.107732\pi\)
\(48\) 49152.0i 0.0641500i
\(49\) −208713. −0.253433
\(50\) 0 0
\(51\) 176472. 0.186286
\(52\) − 88448.0i − 0.0872321i
\(53\) − 1.49402e6i − 1.37845i −0.724548 − 0.689224i \(-0.757952\pi\)
0.724548 − 0.689224i \(-0.242048\pi\)
\(54\) −406080. −0.350940
\(55\) 0 0
\(56\) −520192. −0.395827
\(57\) 479280.i 0.342789i
\(58\) 820560.i 0.552220i
\(59\) −2.64066e6 −1.67390 −0.836952 − 0.547277i \(-0.815665\pi\)
−0.836952 + 0.547277i \(0.815665\pi\)
\(60\) 0 0
\(61\) 827702. 0.466895 0.233448 − 0.972369i \(-0.424999\pi\)
0.233448 + 0.972369i \(0.424999\pi\)
\(62\) 1.82042e6i 0.970063i
\(63\) − 2.07569e6i − 1.04585i
\(64\) −262144. −0.125000
\(65\) 0 0
\(66\) −104832. −0.0448838
\(67\) 126004.i 0.0511826i 0.999672 + 0.0255913i \(0.00814686\pi\)
−0.999672 + 0.0255913i \(0.991853\pi\)
\(68\) 941184.i 0.362989i
\(69\) −824544. −0.302164
\(70\) 0 0
\(71\) −1.41473e6 −0.469104 −0.234552 − 0.972104i \(-0.575362\pi\)
−0.234552 + 0.972104i \(0.575362\pi\)
\(72\) − 1.04602e6i − 0.330274i
\(73\) 980282.i 0.294931i 0.989067 + 0.147466i \(0.0471116\pi\)
−0.989067 + 0.147466i \(0.952888\pi\)
\(74\) 1.28421e6 0.368404
\(75\) 0 0
\(76\) −2.55616e6 −0.667945
\(77\) − 1.10947e6i − 0.276948i
\(78\) − 132672.i − 0.0316554i
\(79\) 3.56680e6 0.813924 0.406962 − 0.913445i \(-0.366588\pi\)
0.406962 + 0.913445i \(0.366588\pi\)
\(80\) 0 0
\(81\) 3.85892e6 0.806805
\(82\) 86736.0i 0.0173720i
\(83\) 5.67289e6i 1.08901i 0.838758 + 0.544504i \(0.183282\pi\)
−0.838758 + 0.544504i \(0.816718\pi\)
\(84\) −780288. −0.143641
\(85\) 0 0
\(86\) 5.04598e6 0.855463
\(87\) 1.23084e6i 0.200394i
\(88\) − 559104.i − 0.0874587i
\(89\) 1.19512e7 1.79699 0.898496 − 0.438982i \(-0.144661\pi\)
0.898496 + 0.438982i \(0.144661\pi\)
\(90\) 0 0
\(91\) 1.40411e6 0.195325
\(92\) − 4.39757e6i − 0.588783i
\(93\) 2.73062e6i 0.352023i
\(94\) 3.78125e6 0.469556
\(95\) 0 0
\(96\) −393216. −0.0453609
\(97\) − 8.68215e6i − 0.965886i −0.875652 − 0.482943i \(-0.839568\pi\)
0.875652 − 0.482943i \(-0.160432\pi\)
\(98\) − 1.66970e6i − 0.179204i
\(99\) 2.23096e6 0.231083
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 50.8.b.c.49.2 2
3.2 odd 2 450.8.c.g.199.1 2
4.3 odd 2 400.8.c.j.49.1 2
5.2 odd 4 2.8.a.a.1.1 ✓ 1
5.3 odd 4 50.8.a.g.1.1 1
5.4 even 2 inner 50.8.b.c.49.1 2
15.2 even 4 18.8.a.b.1.1 1
15.8 even 4 450.8.a.c.1.1 1
15.14 odd 2 450.8.c.g.199.2 2
20.3 even 4 400.8.a.l.1.1 1
20.7 even 4 16.8.a.b.1.1 1
20.19 odd 2 400.8.c.j.49.2 2
35.2 odd 12 98.8.c.d.67.1 2
35.12 even 12 98.8.c.e.67.1 2
35.17 even 12 98.8.c.e.79.1 2
35.27 even 4 98.8.a.a.1.1 1
35.32 odd 12 98.8.c.d.79.1 2
40.27 even 4 64.8.a.e.1.1 1
40.37 odd 4 64.8.a.c.1.1 1
45.2 even 12 162.8.c.a.109.1 2
45.7 odd 12 162.8.c.l.109.1 2
45.22 odd 12 162.8.c.l.55.1 2
45.32 even 12 162.8.c.a.55.1 2
55.32 even 4 242.8.a.e.1.1 1
60.47 odd 4 144.8.a.i.1.1 1
65.12 odd 4 338.8.a.d.1.1 1
65.47 even 4 338.8.b.d.337.1 2
65.57 even 4 338.8.b.d.337.2 2
80.27 even 4 256.8.b.f.129.2 2
80.37 odd 4 256.8.b.b.129.1 2
80.67 even 4 256.8.b.f.129.1 2
80.77 odd 4 256.8.b.b.129.2 2
85.67 odd 4 578.8.a.b.1.1 1
120.77 even 4 576.8.a.g.1.1 1
120.107 odd 4 576.8.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.8.a.a.1.1 ✓ 1 5.2 odd 4
16.8.a.b.1.1 1 20.7 even 4
18.8.a.b.1.1 1 15.2 even 4
50.8.a.g.1.1 1 5.3 odd 4
50.8.b.c.49.1 2 5.4 even 2 inner
50.8.b.c.49.2 2 1.1 even 1 trivial
64.8.a.c.1.1 1 40.37 odd 4
64.8.a.e.1.1 1 40.27 even 4
98.8.a.a.1.1 1 35.27 even 4
98.8.c.d.67.1 2 35.2 odd 12
98.8.c.d.79.1 2 35.32 odd 12
98.8.c.e.67.1 2 35.12 even 12
98.8.c.e.79.1 2 35.17 even 12
144.8.a.i.1.1 1 60.47 odd 4
162.8.c.a.55.1 2 45.32 even 12
162.8.c.a.109.1 2 45.2 even 12
162.8.c.l.55.1 2 45.22 odd 12
162.8.c.l.109.1 2 45.7 odd 12
242.8.a.e.1.1 1 55.32 even 4
256.8.b.b.129.1 2 80.37 odd 4
256.8.b.b.129.2 2 80.77 odd 4
256.8.b.f.129.1 2 80.67 even 4
256.8.b.f.129.2 2 80.27 even 4
338.8.a.d.1.1 1 65.12 odd 4
338.8.b.d.337.1 2 65.47 even 4
338.8.b.d.337.2 2 65.57 even 4
400.8.a.l.1.1 1 20.3 even 4
400.8.c.j.49.1 2 4.3 odd 2
400.8.c.j.49.2 2 20.19 odd 2
450.8.a.c.1.1 1 15.8 even 4
450.8.c.g.199.1 2 3.2 odd 2
450.8.c.g.199.2 2 15.14 odd 2
576.8.a.f.1.1 1 120.107 odd 4
576.8.a.g.1.1 1 120.77 even 4
578.8.a.b.1.1 1 85.67 odd 4