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,-128,0,688] 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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-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)
 
\(f(q)\) \(=\) \(q-8.00000i q^{2} +43.0000i q^{3} -64.0000 q^{4} +344.000 q^{6} -974.000i q^{7} +512.000i q^{8} +338.000 q^{9} +87.0000 q^{11} -2752.00i q^{12} +14828.0i q^{13} -7792.00 q^{14} +4096.00 q^{16} +35571.0i q^{17} -2704.00i q^{18} -20615.0 q^{19} +41882.0 q^{21} -696.000i q^{22} +22218.0i q^{23} -22016.0 q^{24} +118624. q^{26} +108575. i q^{27} +62336.0i q^{28} +5760.00 q^{29} +302942. q^{31} -32768.0i q^{32} +3741.00i q^{33} +284568. q^{34} -21632.0 q^{36} +199366. i q^{37} +164920. i q^{38} -637604. q^{39} -668523. q^{41} -335056. i q^{42} -143212. i q^{43} -5568.00 q^{44} +177744. q^{46} +338316. i q^{47} +176128. i q^{48} -125133. q^{49} -1.52955e6 q^{51} -948992. i q^{52} -1.09432e6i q^{53} +868600. q^{54} +498688. q^{56} -886445. i q^{57} -46080.0i q^{58} +2.13552e6 q^{59} -1.93932e6 q^{61} -2.42354e6i q^{62} -329212. i q^{63} -262144. q^{64} +29928.0 q^{66} +3.34875e6i q^{67} -2.27654e6i q^{68} -955374. q^{69} +3.00565e6 q^{71} +173056. i q^{72} -3.04840e6i q^{73} +1.59493e6 q^{74} +1.31936e6 q^{76} -84738.0i q^{77} +5.10083e6i q^{78} -5.48513e6 q^{79} -3.92952e6 q^{81} +5.34818e6i q^{82} +5.20593e6i q^{83} -2.68045e6 q^{84} -1.14570e6 q^{86} +247680. i q^{87} +44544.0i q^{88} +832665. q^{89} +1.44425e7 q^{91} -1.42195e6i q^{92} +1.30265e7i q^{93} +2.70653e6 q^{94} +1.40902e6 q^{96} -5.31475e6i q^{97} +1.00106e6i q^{98} +29406.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 128 q^{4} + 688 q^{6} + 676 q^{9} + 174 q^{11} - 15584 q^{14} + 8192 q^{16} - 41230 q^{19} + 83764 q^{21} - 44032 q^{24} + 237248 q^{26} + 11520 q^{29} + 605884 q^{31} + 569136 q^{34} - 43264 q^{36}+ \cdots + 58812 q^{99}+O(q^{100}) \) 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
\(3\) 43.0000i 0.919484i 0.888053 + 0.459742i \(0.152058\pi\)
−0.888053 + 0.459742i \(0.847942\pi\)
\(4\) −64.0000 −0.500000
\(5\) 0 0
\(6\) 344.000 0.650173
\(7\) − 974.000i − 1.07329i −0.843809 − 0.536643i \(-0.819692\pi\)
0.843809 − 0.536643i \(-0.180308\pi\)
\(8\) 512.000i 0.353553i
\(9\) 338.000 0.154550
\(10\) 0 0
\(11\) 87.0000 0.0197081 0.00985405 − 0.999951i \(-0.496863\pi\)
0.00985405 + 0.999951i \(0.496863\pi\)
\(12\) − 2752.00i − 0.459742i
\(13\) 14828.0i 1.87189i 0.352143 + 0.935946i \(0.385453\pi\)
−0.352143 + 0.935946i \(0.614547\pi\)
\(14\) −7792.00 −0.758928
\(15\) 0 0
\(16\) 4096.00 0.250000
\(17\) 35571.0i 1.75600i 0.478659 + 0.878001i \(0.341123\pi\)
−0.478659 + 0.878001i \(0.658877\pi\)
\(18\) − 2704.00i − 0.109283i
\(19\) −20615.0 −0.689518 −0.344759 − 0.938691i \(-0.612039\pi\)
−0.344759 + 0.938691i \(0.612039\pi\)
\(20\) 0 0
\(21\) 41882.0 0.986870
\(22\) − 696.000i − 0.0139357i
\(23\) 22218.0i 0.380765i 0.981710 + 0.190383i \(0.0609729\pi\)
−0.981710 + 0.190383i \(0.939027\pi\)
\(24\) −22016.0 −0.325087
\(25\) 0 0
\(26\) 118624. 1.32363
\(27\) 108575.i 1.06159i
\(28\) 62336.0i 0.536643i
\(29\) 5760.00 0.0438560 0.0219280 − 0.999760i \(-0.493020\pi\)
0.0219280 + 0.999760i \(0.493020\pi\)
\(30\) 0 0
\(31\) 302942. 1.82639 0.913195 − 0.407523i \(-0.133607\pi\)
0.913195 + 0.407523i \(0.133607\pi\)
\(32\) − 32768.0i − 0.176777i
\(33\) 3741.00i 0.0181213i
\(34\) 284568. 1.24168
\(35\) 0 0
\(36\) −21632.0 −0.0772748
\(37\) 199366.i 0.647061i 0.946218 + 0.323530i \(0.104870\pi\)
−0.946218 + 0.323530i \(0.895130\pi\)
\(38\) 164920.i 0.487563i
\(39\) −637604. −1.72117
\(40\) 0 0
\(41\) −668523. −1.51486 −0.757431 − 0.652916i \(-0.773546\pi\)
−0.757431 + 0.652916i \(0.773546\pi\)
\(42\) − 335056.i − 0.697822i
\(43\) − 143212.i − 0.274688i −0.990523 − 0.137344i \(-0.956143\pi\)
0.990523 − 0.137344i \(-0.0438566\pi\)
\(44\) −5568.00 −0.00985405
\(45\) 0 0
\(46\) 177744. 0.269242
\(47\) 338316.i 0.475313i 0.971349 + 0.237657i \(0.0763793\pi\)
−0.971349 + 0.237657i \(0.923621\pi\)
\(48\) 176128.i 0.229871i
\(49\) −125133. −0.151945
\(50\) 0 0
\(51\) −1.52955e6 −1.61461
\(52\) − 948992.i − 0.935946i
\(53\) − 1.09432e6i − 1.00967i −0.863216 − 0.504835i \(-0.831553\pi\)
0.863216 − 0.504835i \(-0.168447\pi\)
\(54\) 868600. 0.750657
\(55\) 0 0
\(56\) 498688. 0.379464
\(57\) − 886445.i − 0.634001i
\(58\) − 46080.0i − 0.0310109i
\(59\) 2.13552e6 1.35370 0.676849 − 0.736122i \(-0.263345\pi\)
0.676849 + 0.736122i \(0.263345\pi\)
\(60\) 0 0
\(61\) −1.93932e6 −1.09394 −0.546971 − 0.837151i \(-0.684219\pi\)
−0.546971 + 0.837151i \(0.684219\pi\)
\(62\) − 2.42354e6i − 1.29145i
\(63\) − 329212.i − 0.165876i
\(64\) −262144. −0.125000
\(65\) 0 0
\(66\) 29928.0 0.0128137
\(67\) 3.34875e6i 1.36026i 0.733093 + 0.680129i \(0.238076\pi\)
−0.733093 + 0.680129i \(0.761924\pi\)
\(68\) − 2.27654e6i − 0.878001i
\(69\) −955374. −0.350108
\(70\) 0 0
\(71\) 3.00565e6 0.996631 0.498316 − 0.866996i \(-0.333952\pi\)
0.498316 + 0.866996i \(0.333952\pi\)
\(72\) 173056.i 0.0546415i
\(73\) − 3.04840e6i − 0.917152i −0.888655 − 0.458576i \(-0.848360\pi\)
0.888655 − 0.458576i \(-0.151640\pi\)
\(74\) 1.59493e6 0.457541
\(75\) 0 0
\(76\) 1.31936e6 0.344759
\(77\) − 84738.0i − 0.0211524i
\(78\) 5.10083e6i 1.21705i
\(79\) −5.48513e6 −1.25168 −0.625838 − 0.779953i \(-0.715243\pi\)
−0.625838 + 0.779953i \(0.715243\pi\)
\(80\) 0 0
\(81\) −3.92952e6 −0.821565
\(82\) 5.34818e6i 1.07117i
\(83\) 5.20593e6i 0.999368i 0.866208 + 0.499684i \(0.166550\pi\)
−0.866208 + 0.499684i \(0.833450\pi\)
\(84\) −2.68045e6 −0.493435
\(85\) 0 0
\(86\) −1.14570e6 −0.194234
\(87\) 247680.i 0.0403249i
\(88\) 44544.0i 0.00696787i
\(89\) 832665. 0.125200 0.0626001 − 0.998039i \(-0.480061\pi\)
0.0626001 + 0.998039i \(0.480061\pi\)
\(90\) 0 0
\(91\) 1.44425e7 2.00908
\(92\) − 1.42195e6i − 0.190383i
\(93\) 1.30265e7i 1.67934i
\(94\) 2.70653e6 0.336097
\(95\) 0 0
\(96\) 1.40902e6 0.162543
\(97\) − 5.31475e6i − 0.591265i −0.955302 − 0.295632i \(-0.904470\pi\)
0.955302 − 0.295632i \(-0.0955304\pi\)
\(98\) 1.00106e6i 0.107441i
\(99\) 29406.0 0.00304588
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.e.49.1 2
3.2 odd 2 450.8.c.i.199.2 2
4.3 odd 2 400.8.c.g.49.1 2
5.2 odd 4 50.8.a.h.1.1 yes 1
5.3 odd 4 50.8.a.a.1.1 ✓ 1
5.4 even 2 inner 50.8.b.e.49.2 2
15.2 even 4 450.8.a.j.1.1 1
15.8 even 4 450.8.a.q.1.1 1
15.14 odd 2 450.8.c.i.199.1 2
20.3 even 4 400.8.a.o.1.1 1
20.7 even 4 400.8.a.f.1.1 1
20.19 odd 2 400.8.c.g.49.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.8.a.a.1.1 ✓ 1 5.3 odd 4
50.8.a.h.1.1 yes 1 5.2 odd 4
50.8.b.e.49.1 2 1.1 even 1 trivial
50.8.b.e.49.2 2 5.4 even 2 inner
400.8.a.f.1.1 1 20.7 even 4
400.8.a.o.1.1 1 20.3 even 4
400.8.c.g.49.1 2 4.3 odd 2
400.8.c.g.49.2 2 20.19 odd 2
450.8.a.j.1.1 1 15.2 even 4
450.8.a.q.1.1 1 15.8 even 4
450.8.c.i.199.1 2 15.14 odd 2
450.8.c.i.199.2 2 3.2 odd 2