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 = [6,12,-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: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 109x^{4} + 94x^{3} + 2808x^{2} + 81x - 9774 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(6.96868\) 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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +5.96868 q^{3} +4.00000 q^{4} +11.9374 q^{6} -15.2484 q^{7} +8.00000 q^{8} +8.62516 q^{9} +12.0268 q^{11} +23.8747 q^{12} -90.1983 q^{13} -30.4968 q^{14} +16.0000 q^{16} -83.9831 q^{17} +17.2503 q^{18} +74.8503 q^{19} -91.0130 q^{21} +24.0537 q^{22} -23.0000 q^{23} +47.7495 q^{24} -180.397 q^{26} -109.674 q^{27} -60.9937 q^{28} +94.5321 q^{29} -178.393 q^{31} +32.0000 q^{32} +71.7844 q^{33} -167.966 q^{34} +34.5007 q^{36} -296.487 q^{37} +149.701 q^{38} -538.365 q^{39} +193.074 q^{41} -182.026 q^{42} -15.4576 q^{43} +48.1074 q^{44} -46.0000 q^{46} -128.899 q^{47} +95.4989 q^{48} -110.486 q^{49} -501.269 q^{51} -360.793 q^{52} -342.961 q^{53} -219.347 q^{54} -121.987 q^{56} +446.758 q^{57} +189.064 q^{58} +725.012 q^{59} -784.288 q^{61} -356.786 q^{62} -131.520 q^{63} +64.0000 q^{64} +143.569 q^{66} +782.822 q^{67} -335.932 q^{68} -137.280 q^{69} -141.768 q^{71} +69.0013 q^{72} -404.340 q^{73} -592.973 q^{74} +299.401 q^{76} -183.390 q^{77} -1076.73 q^{78} -721.038 q^{79} -887.486 q^{81} +386.147 q^{82} +985.391 q^{83} -364.052 q^{84} -30.9153 q^{86} +564.232 q^{87} +96.2148 q^{88} +463.054 q^{89} +1375.38 q^{91} -92.0000 q^{92} -1064.77 q^{93} -257.798 q^{94} +190.998 q^{96} +1680.05 q^{97} -220.971 q^{98} +103.734 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 12 q^{2} - 5 q^{3} + 24 q^{4} - 10 q^{6} - 42 q^{7} + 48 q^{8} + 61 q^{9} - 49 q^{11} - 20 q^{12} - 16 q^{13} - 84 q^{14} + 96 q^{16} - 175 q^{17} + 122 q^{18} - 229 q^{19} + 92 q^{21} - 98 q^{22}+ \cdots - 142 q^{99}+O(q^{100}) \) 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
\(3\) 5.96868 1.14867 0.574337 − 0.818619i \(-0.305260\pi\)
0.574337 + 0.818619i \(0.305260\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) 11.9374 0.812235
\(7\) −15.2484 −0.823337 −0.411669 − 0.911334i \(-0.635054\pi\)
−0.411669 + 0.911334i \(0.635054\pi\)
\(8\) 8.00000 0.353553
\(9\) 8.62516 0.319451
\(10\) 0 0
\(11\) 12.0268 0.329657 0.164829 − 0.986322i \(-0.447293\pi\)
0.164829 + 0.986322i \(0.447293\pi\)
\(12\) 23.8747 0.574337
\(13\) −90.1983 −1.92435 −0.962173 − 0.272438i \(-0.912170\pi\)
−0.962173 + 0.272438i \(0.912170\pi\)
\(14\) −30.4968 −0.582187
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −83.9831 −1.19817 −0.599085 − 0.800685i \(-0.704469\pi\)
−0.599085 + 0.800685i \(0.704469\pi\)
\(18\) 17.2503 0.225886
\(19\) 74.8503 0.903781 0.451890 − 0.892073i \(-0.350750\pi\)
0.451890 + 0.892073i \(0.350750\pi\)
\(20\) 0 0
\(21\) −91.0130 −0.945746
\(22\) 24.0537 0.233103
\(23\) −23.0000 −0.208514
\(24\) 47.7495 0.406117
\(25\) 0 0
\(26\) −180.397 −1.36072
\(27\) −109.674 −0.781729
\(28\) −60.9937 −0.411669
\(29\) 94.5321 0.605316 0.302658 − 0.953099i \(-0.402126\pi\)
0.302658 + 0.953099i \(0.402126\pi\)
\(30\) 0 0
\(31\) −178.393 −1.03356 −0.516779 − 0.856119i \(-0.672869\pi\)
−0.516779 + 0.856119i \(0.672869\pi\)
\(32\) 32.0000 0.176777
\(33\) 71.7844 0.378669
\(34\) −167.966 −0.847234
\(35\) 0 0
\(36\) 34.5007 0.159725
\(37\) −296.487 −1.31735 −0.658677 − 0.752426i \(-0.728884\pi\)
−0.658677 + 0.752426i \(0.728884\pi\)
\(38\) 149.701 0.639069
\(39\) −538.365 −2.21045
\(40\) 0 0
\(41\) 193.074 0.735440 0.367720 − 0.929937i \(-0.380139\pi\)
0.367720 + 0.929937i \(0.380139\pi\)
\(42\) −182.026 −0.668743
\(43\) −15.4576 −0.0548202 −0.0274101 − 0.999624i \(-0.508726\pi\)
−0.0274101 + 0.999624i \(0.508726\pi\)
\(44\) 48.1074 0.164829
\(45\) 0 0
\(46\) −46.0000 −0.147442
\(47\) −128.899 −0.400040 −0.200020 − 0.979792i \(-0.564101\pi\)
−0.200020 + 0.979792i \(0.564101\pi\)
\(48\) 95.4989 0.287168
\(49\) −110.486 −0.322116
\(50\) 0 0
\(51\) −501.269 −1.37631
\(52\) −360.793 −0.962173
\(53\) −342.961 −0.888855 −0.444428 − 0.895815i \(-0.646593\pi\)
−0.444428 + 0.895815i \(0.646593\pi\)
\(54\) −219.347 −0.552766
\(55\) 0 0
\(56\) −121.987 −0.291094
\(57\) 446.758 1.03815
\(58\) 189.064 0.428023
\(59\) 725.012 1.59980 0.799902 − 0.600130i \(-0.204885\pi\)
0.799902 + 0.600130i \(0.204885\pi\)
\(60\) 0 0
\(61\) −784.288 −1.64619 −0.823097 − 0.567901i \(-0.807756\pi\)
−0.823097 + 0.567901i \(0.807756\pi\)
\(62\) −356.786 −0.730837
\(63\) −131.520 −0.263016
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 143.569 0.267759
\(67\) 782.822 1.42742 0.713708 − 0.700443i \(-0.247014\pi\)
0.713708 + 0.700443i \(0.247014\pi\)
\(68\) −335.932 −0.599085
\(69\) −137.280 −0.239515
\(70\) 0 0
\(71\) −141.768 −0.236968 −0.118484 − 0.992956i \(-0.537803\pi\)
−0.118484 + 0.992956i \(0.537803\pi\)
\(72\) 69.0013 0.112943
\(73\) −404.340 −0.648280 −0.324140 − 0.946009i \(-0.605075\pi\)
−0.324140 + 0.946009i \(0.605075\pi\)
\(74\) −592.973 −0.931510
\(75\) 0 0
\(76\) 299.401 0.451890
\(77\) −183.390 −0.271419
\(78\) −1076.73 −1.56302
\(79\) −721.038 −1.02687 −0.513437 − 0.858127i \(-0.671628\pi\)
−0.513437 + 0.858127i \(0.671628\pi\)
\(80\) 0 0
\(81\) −887.486 −1.21740
\(82\) 386.147 0.520035
\(83\) 985.391 1.30314 0.651570 − 0.758588i \(-0.274110\pi\)
0.651570 + 0.758588i \(0.274110\pi\)
\(84\) −364.052 −0.472873
\(85\) 0 0
\(86\) −30.9153 −0.0387637
\(87\) 564.232 0.695311
\(88\) 96.2148 0.116551
\(89\) 463.054 0.551502 0.275751 − 0.961229i \(-0.411074\pi\)
0.275751 + 0.961229i \(0.411074\pi\)
\(90\) 0 0
\(91\) 1375.38 1.58439
\(92\) −92.0000 −0.104257
\(93\) −1064.77 −1.18722
\(94\) −257.798 −0.282871
\(95\) 0 0
\(96\) 190.998 0.203059
\(97\) 1680.05 1.75859 0.879295 − 0.476277i \(-0.158014\pi\)
0.879295 + 0.476277i \(0.158014\pi\)
\(98\) −220.971 −0.227770
\(99\) 103.734 0.105309
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.x.1.5 yes 6
5.2 odd 4 1150.4.b.s.599.8 12
5.3 odd 4 1150.4.b.s.599.5 12
5.4 even 2 1150.4.a.w.1.2 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.4.a.w.1.2 ✓ 6 5.4 even 2
1150.4.a.x.1.5 yes 6 1.1 even 1 trivial
1150.4.b.s.599.5 12 5.3 odd 4
1150.4.b.s.599.8 12 5.2 odd 4