Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1150,4,Mod(1,1150)] 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("1150.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,10,5] 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: \(67.8521965066\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 107x^{3} - 3x^{2} + 2151x - 2916 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-8.48510\) of defining polynomial
Character \(\chi\) \(=\) 1150.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +9.48510 q^{3} +4.00000 q^{4} +18.9702 q^{6} -8.59355 q^{7} +8.00000 q^{8} +62.9670 q^{9} +44.0665 q^{11} +37.9404 q^{12} -8.11744 q^{13} -17.1871 q^{14} +16.0000 q^{16} +87.4729 q^{17} +125.934 q^{18} -150.610 q^{19} -81.5106 q^{21} +88.1330 q^{22} +23.0000 q^{23} +75.8808 q^{24} -16.2349 q^{26} +341.151 q^{27} -34.3742 q^{28} +85.9907 q^{29} +209.918 q^{31} +32.0000 q^{32} +417.975 q^{33} +174.946 q^{34} +251.868 q^{36} +196.389 q^{37} -301.219 q^{38} -76.9947 q^{39} -38.3709 q^{41} -163.021 q^{42} -399.692 q^{43} +176.266 q^{44} +46.0000 q^{46} +127.838 q^{47} +151.762 q^{48} -269.151 q^{49} +829.689 q^{51} -32.4698 q^{52} -594.405 q^{53} +682.302 q^{54} -68.7484 q^{56} -1428.55 q^{57} +171.981 q^{58} -459.633 q^{59} +582.395 q^{61} +419.837 q^{62} -541.110 q^{63} +64.0000 q^{64} +835.950 q^{66} +344.321 q^{67} +349.892 q^{68} +218.157 q^{69} +478.640 q^{71} +503.736 q^{72} +726.308 q^{73} +392.777 q^{74} -602.439 q^{76} -378.688 q^{77} -153.989 q^{78} +475.430 q^{79} +1535.74 q^{81} -76.7418 q^{82} +374.770 q^{83} -326.042 q^{84} -799.384 q^{86} +815.630 q^{87} +352.532 q^{88} -152.315 q^{89} +69.7576 q^{91} +92.0000 q^{92} +1991.10 q^{93} +255.677 q^{94} +303.523 q^{96} -497.667 q^{97} -538.302 q^{98} +2774.74 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 10 q^{2} + 5 q^{3} + 20 q^{4} + 10 q^{6} - 3 q^{7} + 40 q^{8} + 84 q^{9} - 26 q^{11} + 20 q^{12} - 61 q^{13} - 6 q^{14} + 80 q^{16} + 231 q^{17} + 168 q^{18} + 74 q^{19} - 88 q^{21} - 52 q^{22} + 115 q^{23}+ \cdots + 2397 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 9.48510 1.82541 0.912704 − 0.408622i \(-0.133991\pi\)
0.912704 + 0.408622i \(0.133991\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) 18.9702 1.29076
\(7\) −8.59355 −0.464008 −0.232004 − 0.972715i \(-0.574528\pi\)
−0.232004 + 0.972715i \(0.574528\pi\)
\(8\) 8.00000 0.353553
\(9\) 62.9670 2.33211
\(10\) 0 0
\(11\) 44.0665 1.20787 0.603934 − 0.797034i \(-0.293599\pi\)
0.603934 + 0.797034i \(0.293599\pi\)
\(12\) 37.9404 0.912704
\(13\) −8.11744 −0.173183 −0.0865913 − 0.996244i \(-0.527597\pi\)
−0.0865913 + 0.996244i \(0.527597\pi\)
\(14\) −17.1871 −0.328103
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 87.4729 1.24796 0.623980 − 0.781441i \(-0.285515\pi\)
0.623980 + 0.781441i \(0.285515\pi\)
\(18\) 125.934 1.64905
\(19\) −150.610 −1.81854 −0.909269 − 0.416209i \(-0.863358\pi\)
−0.909269 + 0.416209i \(0.863358\pi\)
\(20\) 0 0
\(21\) −81.5106 −0.847004
\(22\) 88.1330 0.854092
\(23\) 23.0000 0.208514
\(24\) 75.8808 0.645379
\(25\) 0 0
\(26\) −16.2349 −0.122459
\(27\) 341.151 2.43165
\(28\) −34.3742 −0.232004
\(29\) 85.9907 0.550623 0.275312 − 0.961355i \(-0.411219\pi\)
0.275312 + 0.961355i \(0.411219\pi\)
\(30\) 0 0
\(31\) 209.918 1.21621 0.608104 − 0.793857i \(-0.291930\pi\)
0.608104 + 0.793857i \(0.291930\pi\)
\(32\) 32.0000 0.176777
\(33\) 417.975 2.20485
\(34\) 174.946 0.882440
\(35\) 0 0
\(36\) 251.868 1.16606
\(37\) 196.389 0.872597 0.436298 − 0.899802i \(-0.356289\pi\)
0.436298 + 0.899802i \(0.356289\pi\)
\(38\) −301.219 −1.28590
\(39\) −76.9947 −0.316129
\(40\) 0 0
\(41\) −38.3709 −0.146159 −0.0730796 − 0.997326i \(-0.523283\pi\)
−0.0730796 + 0.997326i \(0.523283\pi\)
\(42\) −163.021 −0.598922
\(43\) −399.692 −1.41750 −0.708750 − 0.705460i \(-0.750741\pi\)
−0.708750 + 0.705460i \(0.750741\pi\)
\(44\) 176.266 0.603934
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) 127.838 0.396748 0.198374 − 0.980126i \(-0.436434\pi\)
0.198374 + 0.980126i \(0.436434\pi\)
\(48\) 151.762 0.456352
\(49\) −269.151 −0.784697
\(50\) 0 0
\(51\) 829.689 2.27803
\(52\) −32.4698 −0.0865913
\(53\) −594.405 −1.54053 −0.770263 − 0.637727i \(-0.779875\pi\)
−0.770263 + 0.637727i \(0.779875\pi\)
\(54\) 682.302 1.71943
\(55\) 0 0
\(56\) −68.7484 −0.164052
\(57\) −1428.55 −3.31957
\(58\) 171.981 0.389350
\(59\) −459.633 −1.01422 −0.507112 − 0.861880i \(-0.669287\pi\)
−0.507112 + 0.861880i \(0.669287\pi\)
\(60\) 0 0
\(61\) 582.395 1.22243 0.611213 − 0.791466i \(-0.290682\pi\)
0.611213 + 0.791466i \(0.290682\pi\)
\(62\) 419.837 0.859989
\(63\) −541.110 −1.08212
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 835.950 1.55907
\(67\) 344.321 0.627843 0.313921 − 0.949449i \(-0.398357\pi\)
0.313921 + 0.949449i \(0.398357\pi\)
\(68\) 349.892 0.623980
\(69\) 218.157 0.380624
\(70\) 0 0
\(71\) 478.640 0.800057 0.400029 − 0.916503i \(-0.369000\pi\)
0.400029 + 0.916503i \(0.369000\pi\)
\(72\) 503.736 0.824526
\(73\) 726.308 1.16449 0.582246 − 0.813013i \(-0.302174\pi\)
0.582246 + 0.813013i \(0.302174\pi\)
\(74\) 392.777 0.617019
\(75\) 0 0
\(76\) −602.439 −0.909269
\(77\) −378.688 −0.560460
\(78\) −153.989 −0.223537
\(79\) 475.430 0.677089 0.338544 − 0.940950i \(-0.390065\pi\)
0.338544 + 0.940950i \(0.390065\pi\)
\(80\) 0 0
\(81\) 1535.74 2.10664
\(82\) −76.7418 −0.103350
\(83\) 374.770 0.495618 0.247809 − 0.968809i \(-0.420289\pi\)
0.247809 + 0.968809i \(0.420289\pi\)
\(84\) −326.042 −0.423502
\(85\) 0 0
\(86\) −799.384 −1.00232
\(87\) 815.630 1.00511
\(88\) 352.532 0.427046
\(89\) −152.315 −0.181409 −0.0907043 − 0.995878i \(-0.528912\pi\)
−0.0907043 + 0.995878i \(0.528912\pi\)
\(90\) 0 0
\(91\) 69.7576 0.0803581
\(92\) 92.0000 0.104257
\(93\) 1991.10 2.22007
\(94\) 255.677 0.280543
\(95\) 0 0
\(96\) 303.523 0.322690
\(97\) −497.667 −0.520932 −0.260466 − 0.965483i \(-0.583876\pi\)
−0.260466 + 0.965483i \(0.583876\pi\)
\(98\) −538.302 −0.554864
\(99\) 2774.74 2.81688
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 1150.4.a.v.1.5 yes 5
5.2 odd 4 1150.4.b.r.599.6 10
5.3 odd 4 1150.4.b.r.599.5 10
5.4 even 2 1150.4.a.q.1.1 ✓ 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.4.a.q.1.1 ✓ 5 5.4 even 2
1150.4.a.v.1.5 yes 5 1.1 even 1 trivial
1150.4.b.r.599.5 10 5.3 odd 4
1150.4.b.r.599.6 10 5.2 odd 4