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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,-1392] 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: \(1\)
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: yes
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.a.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)
 
\(f(q)\) \(=\) \(q+8.00000i q^{2} +87.0000i q^{3} -64.0000 q^{4} -696.000 q^{6} -1366.00i q^{7} -512.000i q^{8} -5382.00 q^{9} -1083.00 q^{11} -5568.00i q^{12} -5468.00i q^{13} +10928.0 q^{14} +4096.00 q^{16} +25269.0i q^{17} -43056.0i q^{18} -33485.0 q^{19} +118842. q^{21} -8664.00i q^{22} -5838.00i q^{23} +44544.0 q^{24} +43744.0 q^{26} -277965. i q^{27} +87424.0i q^{28} -125280. q^{29} -73798.0 q^{31} +32768.0i q^{32} -94221.0i q^{33} -202152. q^{34} +344448. q^{36} -395926. i q^{37} -267880. i q^{38} +475716. q^{39} -22683.0 q^{41} +950736. i q^{42} -100148. i q^{43} +69312.0 q^{44} +46704.0 q^{46} +1.14524e6i q^{47} +356352. i q^{48} -1.04241e6 q^{49} -2.19840e6 q^{51} +349952. i q^{52} +354882. i q^{53} +2.22372e6 q^{54} -699392. q^{56} -2.91320e6i q^{57} -1.00224e6i q^{58} -1.09836e6 q^{59} -422998. q^{61} -590384. i q^{62} +7.35181e6i q^{63} -262144. q^{64} +753768. q^{66} +2.55858e6i q^{67} -1.61722e6i q^{68} +507906. q^{69} -2.28743e6 q^{71} +2.75558e6i q^{72} -6.37244e6i q^{73} +3.16741e6 q^{74} +2.14304e6 q^{76} +1.47938e6i q^{77} +3.80573e6i q^{78} +2.01925e6 q^{79} +1.24125e7 q^{81} -181464. i q^{82} -7.97298e6i q^{83} -7.60589e6 q^{84} +801184. q^{86} -1.08994e7i q^{87} +554496. i q^{88} -2.18594e6 q^{89} -7.46929e6 q^{91} +373632. i q^{92} -6.42043e6i q^{93} -9.16195e6 q^{94} -2.85082e6 q^{96} -5.82365e6i q^{97} -8.33930e6i q^{98} +5.82871e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 128 q^{4} - 1392 q^{6} - 10764 q^{9} - 2166 q^{11} + 21856 q^{14} + 8192 q^{16} - 66970 q^{19} + 237684 q^{21} + 89088 q^{24} + 87488 q^{26} - 250560 q^{29} - 147596 q^{31} - 404304 q^{34} + 688896 q^{36}+ \cdots + 11657412 q^{99}+O(q^{100}) \) 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
\(3\) 87.0000i 1.86035i 0.367115 + 0.930175i \(0.380345\pi\)
−0.367115 + 0.930175i \(0.619655\pi\)
\(4\) −64.0000 −0.500000
\(5\) 0 0
\(6\) −696.000 −1.31547
\(7\) − 1366.00i − 1.50525i −0.658452 − 0.752623i \(-0.728788\pi\)
0.658452 − 0.752623i \(-0.271212\pi\)
\(8\) − 512.000i − 0.353553i
\(9\) −5382.00 −2.46091
\(10\) 0 0
\(11\) −1083.00 −0.245332 −0.122666 − 0.992448i \(-0.539144\pi\)
−0.122666 + 0.992448i \(0.539144\pi\)
\(12\) − 5568.00i − 0.930175i
\(13\) − 5468.00i − 0.690282i −0.938551 − 0.345141i \(-0.887831\pi\)
0.938551 − 0.345141i \(-0.112169\pi\)
\(14\) 10928.0 1.06437
\(15\) 0 0
\(16\) 4096.00 0.250000
\(17\) 25269.0i 1.24743i 0.781651 + 0.623716i \(0.214378\pi\)
−0.781651 + 0.623716i \(0.785622\pi\)
\(18\) − 43056.0i − 1.74012i
\(19\) −33485.0 −1.11999 −0.559993 − 0.828497i \(-0.689196\pi\)
−0.559993 + 0.828497i \(0.689196\pi\)
\(20\) 0 0
\(21\) 118842. 2.80029
\(22\) − 8664.00i − 0.173476i
\(23\) − 5838.00i − 0.100050i −0.998748 − 0.0500250i \(-0.984070\pi\)
0.998748 − 0.0500250i \(-0.0159301\pi\)
\(24\) 44544.0 0.657733
\(25\) 0 0
\(26\) 43744.0 0.488103
\(27\) − 277965.i − 2.71780i
\(28\) 87424.0i 0.752623i
\(29\) −125280. −0.953869 −0.476935 − 0.878939i \(-0.658252\pi\)
−0.476935 + 0.878939i \(0.658252\pi\)
\(30\) 0 0
\(31\) −73798.0 −0.444917 −0.222458 − 0.974942i \(-0.571408\pi\)
−0.222458 + 0.974942i \(0.571408\pi\)
\(32\) 32768.0i 0.176777i
\(33\) − 94221.0i − 0.456403i
\(34\) −202152. −0.882068
\(35\) 0 0
\(36\) 344448. 1.23045
\(37\) − 395926.i − 1.28501i −0.766280 − 0.642507i \(-0.777894\pi\)
0.766280 − 0.642507i \(-0.222106\pi\)
\(38\) − 267880.i − 0.791950i
\(39\) 475716. 1.28417
\(40\) 0 0
\(41\) −22683.0 −0.0513993 −0.0256996 − 0.999670i \(-0.508181\pi\)
−0.0256996 + 0.999670i \(0.508181\pi\)
\(42\) 950736.i 1.98010i
\(43\) − 100148.i − 0.192089i −0.995377 − 0.0960445i \(-0.969381\pi\)
0.995377 − 0.0960445i \(-0.0306191\pi\)
\(44\) 69312.0 0.122666
\(45\) 0 0
\(46\) 46704.0 0.0707460
\(47\) 1.14524e6i 1.60900i 0.593954 + 0.804499i \(0.297566\pi\)
−0.593954 + 0.804499i \(0.702434\pi\)
\(48\) 356352.i 0.465088i
\(49\) −1.04241e6 −1.26577
\(50\) 0 0
\(51\) −2.19840e6 −2.32066
\(52\) 349952.i 0.345141i
\(53\) 354882.i 0.327430i 0.986508 + 0.163715i \(0.0523477\pi\)
−0.986508 + 0.163715i \(0.947652\pi\)
\(54\) 2.22372e6 1.92177
\(55\) 0 0
\(56\) −699392. −0.532185
\(57\) − 2.91320e6i − 2.08357i
\(58\) − 1.00224e6i − 0.674487i
\(59\) −1.09836e6 −0.696246 −0.348123 − 0.937449i \(-0.613181\pi\)
−0.348123 + 0.937449i \(0.613181\pi\)
\(60\) 0 0
\(61\) −422998. −0.238607 −0.119304 − 0.992858i \(-0.538066\pi\)
−0.119304 + 0.992858i \(0.538066\pi\)
\(62\) − 590384.i − 0.314604i
\(63\) 7.35181e6i 3.70427i
\(64\) −262144. −0.125000
\(65\) 0 0
\(66\) 753768. 0.322726
\(67\) 2.55858e6i 1.03929i 0.854382 + 0.519645i \(0.173936\pi\)
−0.854382 + 0.519645i \(0.826064\pi\)
\(68\) − 1.61722e6i − 0.623716i
\(69\) 507906. 0.186128
\(70\) 0 0
\(71\) −2.28743e6 −0.758478 −0.379239 − 0.925299i \(-0.623814\pi\)
−0.379239 + 0.925299i \(0.623814\pi\)
\(72\) 2.75558e6i 0.870061i
\(73\) − 6.37244e6i − 1.91724i −0.284693 − 0.958619i \(-0.591892\pi\)
0.284693 − 0.958619i \(-0.408108\pi\)
\(74\) 3.16741e6 0.908642
\(75\) 0 0
\(76\) 2.14304e6 0.559993
\(77\) 1.47938e6i 0.369285i
\(78\) 3.80573e6i 0.908043i
\(79\) 2.01925e6 0.460782 0.230391 − 0.973098i \(-0.426000\pi\)
0.230391 + 0.973098i \(0.426000\pi\)
\(80\) 0 0
\(81\) 1.24125e7 2.59515
\(82\) − 181464.i − 0.0363448i
\(83\) − 7.97298e6i − 1.53055i −0.643703 − 0.765275i \(-0.722603\pi\)
0.643703 − 0.765275i \(-0.277397\pi\)
\(84\) −7.60589e6 −1.40014
\(85\) 0 0
\(86\) 801184. 0.135827
\(87\) − 1.08994e7i − 1.77453i
\(88\) 554496.i 0.0867379i
\(89\) −2.18594e6 −0.328679 −0.164340 − 0.986404i \(-0.552549\pi\)
−0.164340 + 0.986404i \(0.552549\pi\)
\(90\) 0 0
\(91\) −7.46929e6 −1.03904
\(92\) 373632.i 0.0500250i
\(93\) − 6.42043e6i − 0.827701i
\(94\) −9.16195e6 −1.13773
\(95\) 0 0
\(96\) −2.85082e6 −0.328867
\(97\) − 5.82365e6i − 0.647879i −0.946078 − 0.323939i \(-0.894993\pi\)
0.946078 − 0.323939i \(-0.105007\pi\)
\(98\) − 8.33930e6i − 0.895032i
\(99\) 5.82871e6 0.603739
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.a.49.2 2
3.2 odd 2 450.8.c.l.199.1 2
4.3 odd 2 400.8.c.a.49.1 2
5.2 odd 4 50.8.a.d.1.1 ✓ 1
5.3 odd 4 50.8.a.e.1.1 yes 1
5.4 even 2 inner 50.8.b.a.49.1 2
15.2 even 4 450.8.a.z.1.1 1
15.8 even 4 450.8.a.a.1.1 1
15.14 odd 2 450.8.c.l.199.2 2
20.3 even 4 400.8.a.s.1.1 1
20.7 even 4 400.8.a.a.1.1 1
20.19 odd 2 400.8.c.a.49.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.8.a.d.1.1 ✓ 1 5.2 odd 4
50.8.a.e.1.1 yes 1 5.3 odd 4
50.8.b.a.49.1 2 5.4 even 2 inner
50.8.b.a.49.2 2 1.1 even 1 trivial
400.8.a.a.1.1 1 20.7 even 4
400.8.a.s.1.1 1 20.3 even 4
400.8.c.a.49.1 2 4.3 odd 2
400.8.c.a.49.2 2 20.19 odd 2
450.8.a.a.1.1 1 15.8 even 4
450.8.a.z.1.1 1 15.2 even 4
450.8.c.l.199.1 2 3.2 odd 2
450.8.c.l.199.2 2 15.14 odd 2