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 1.3
Root \(-2.16288\) 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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -3.16288 q^{3} +4.00000 q^{4} -6.32576 q^{6} +3.52475 q^{7} +8.00000 q^{8} -16.9962 q^{9} -16.9719 q^{11} -12.6515 q^{12} +46.0060 q^{13} +7.04951 q^{14} +16.0000 q^{16} +16.2568 q^{17} -33.9924 q^{18} -31.1330 q^{19} -11.1484 q^{21} -33.9437 q^{22} -23.0000 q^{23} -25.3030 q^{24} +92.0120 q^{26} +139.155 q^{27} +14.0990 q^{28} -11.4072 q^{29} -82.7344 q^{31} +32.0000 q^{32} +53.6800 q^{33} +32.5135 q^{34} -67.9848 q^{36} -296.932 q^{37} -62.2659 q^{38} -145.511 q^{39} +407.814 q^{41} -22.2967 q^{42} -549.918 q^{43} -67.8875 q^{44} -46.0000 q^{46} +503.804 q^{47} -50.6061 q^{48} -330.576 q^{49} -51.4182 q^{51} +184.024 q^{52} -605.711 q^{53} +278.309 q^{54} +28.1980 q^{56} +98.4698 q^{57} -22.8143 q^{58} +522.579 q^{59} +242.576 q^{61} -165.469 q^{62} -59.9074 q^{63} +64.0000 q^{64} +107.360 q^{66} -1014.23 q^{67} +65.0271 q^{68} +72.7462 q^{69} -334.393 q^{71} -135.970 q^{72} -181.460 q^{73} -593.864 q^{74} -124.532 q^{76} -59.8216 q^{77} -291.023 q^{78} -461.511 q^{79} +18.7681 q^{81} +815.628 q^{82} -817.840 q^{83} -44.5935 q^{84} -1099.84 q^{86} +36.0795 q^{87} -135.775 q^{88} -774.536 q^{89} +162.160 q^{91} -92.0000 q^{92} +261.679 q^{93} +1007.61 q^{94} -101.212 q^{96} +1402.97 q^{97} -661.152 q^{98} +288.457 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 12 q^{2} - 5 q^{3} + 24 q^{4} - 10 q^{6} - 42 q^{7} + 48 q^{8} + 61 q^{9} - 49 q^{11} - 20 q^{12} - 16 q^{13} - 84 q^{14} + 96 q^{16} - 175 q^{17} + 122 q^{18} - 229 q^{19} + 92 q^{21} - 98 q^{22}+ \cdots - 142 q^{99}+O(q^{100}) \) 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
\(3\) −3.16288 −0.608696 −0.304348 − 0.952561i \(-0.598439\pi\)
−0.304348 + 0.952561i \(0.598439\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) −6.32576 −0.430413
\(7\) 3.52475 0.190319 0.0951594 − 0.995462i \(-0.469664\pi\)
0.0951594 + 0.995462i \(0.469664\pi\)
\(8\) 8.00000 0.353553
\(9\) −16.9962 −0.629489
\(10\) 0 0
\(11\) −16.9719 −0.465201 −0.232601 − 0.972572i \(-0.574723\pi\)
−0.232601 + 0.972572i \(0.574723\pi\)
\(12\) −12.6515 −0.304348
\(13\) 46.0060 0.981520 0.490760 − 0.871295i \(-0.336719\pi\)
0.490760 + 0.871295i \(0.336719\pi\)
\(14\) 7.04951 0.134576
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 16.2568 0.231932 0.115966 − 0.993253i \(-0.463004\pi\)
0.115966 + 0.993253i \(0.463004\pi\)
\(18\) −33.9924 −0.445116
\(19\) −31.1330 −0.375915 −0.187958 − 0.982177i \(-0.560187\pi\)
−0.187958 + 0.982177i \(0.560187\pi\)
\(20\) 0 0
\(21\) −11.1484 −0.115846
\(22\) −33.9437 −0.328947
\(23\) −23.0000 −0.208514
\(24\) −25.3030 −0.215207
\(25\) 0 0
\(26\) 92.0120 0.694040
\(27\) 139.155 0.991864
\(28\) 14.0990 0.0951594
\(29\) −11.4072 −0.0730433 −0.0365217 − 0.999333i \(-0.511628\pi\)
−0.0365217 + 0.999333i \(0.511628\pi\)
\(30\) 0 0
\(31\) −82.7344 −0.479340 −0.239670 − 0.970854i \(-0.577039\pi\)
−0.239670 + 0.970854i \(0.577039\pi\)
\(32\) 32.0000 0.176777
\(33\) 53.6800 0.283166
\(34\) 32.5135 0.164001
\(35\) 0 0
\(36\) −67.9848 −0.314744
\(37\) −296.932 −1.31933 −0.659666 − 0.751559i \(-0.729302\pi\)
−0.659666 + 0.751559i \(0.729302\pi\)
\(38\) −62.2659 −0.265812
\(39\) −145.511 −0.597448
\(40\) 0 0
\(41\) 407.814 1.55341 0.776705 − 0.629864i \(-0.216890\pi\)
0.776705 + 0.629864i \(0.216890\pi\)
\(42\) −22.2967 −0.0819157
\(43\) −549.918 −1.95027 −0.975136 − 0.221609i \(-0.928869\pi\)
−0.975136 + 0.221609i \(0.928869\pi\)
\(44\) −67.8875 −0.232601
\(45\) 0 0
\(46\) −46.0000 −0.147442
\(47\) 503.804 1.56356 0.781781 − 0.623553i \(-0.214311\pi\)
0.781781 + 0.623553i \(0.214311\pi\)
\(48\) −50.6061 −0.152174
\(49\) −330.576 −0.963779
\(50\) 0 0
\(51\) −51.4182 −0.141176
\(52\) 184.024 0.490760
\(53\) −605.711 −1.56983 −0.784913 − 0.619606i \(-0.787293\pi\)
−0.784913 + 0.619606i \(0.787293\pi\)
\(54\) 278.309 0.701354
\(55\) 0 0
\(56\) 28.1980 0.0672878
\(57\) 98.4698 0.228818
\(58\) −22.8143 −0.0516494
\(59\) 522.579 1.15312 0.576559 − 0.817056i \(-0.304395\pi\)
0.576559 + 0.817056i \(0.304395\pi\)
\(60\) 0 0
\(61\) 242.576 0.509158 0.254579 − 0.967052i \(-0.418063\pi\)
0.254579 + 0.967052i \(0.418063\pi\)
\(62\) −165.469 −0.338945
\(63\) −59.9074 −0.119804
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 107.360 0.200229
\(67\) −1014.23 −1.84937 −0.924687 − 0.380728i \(-0.875674\pi\)
−0.924687 + 0.380728i \(0.875674\pi\)
\(68\) 65.0271 0.115966
\(69\) 72.7462 0.126922
\(70\) 0 0
\(71\) −334.393 −0.558946 −0.279473 − 0.960154i \(-0.590160\pi\)
−0.279473 + 0.960154i \(0.590160\pi\)
\(72\) −135.970 −0.222558
\(73\) −181.460 −0.290935 −0.145468 − 0.989363i \(-0.546469\pi\)
−0.145468 + 0.989363i \(0.546469\pi\)
\(74\) −593.864 −0.932909
\(75\) 0 0
\(76\) −124.532 −0.187958
\(77\) −59.8216 −0.0885365
\(78\) −291.023 −0.422459
\(79\) −461.511 −0.657267 −0.328633 − 0.944458i \(-0.606588\pi\)
−0.328633 + 0.944458i \(0.606588\pi\)
\(80\) 0 0
\(81\) 18.7681 0.0257450
\(82\) 815.628 1.09843
\(83\) −817.840 −1.08156 −0.540781 − 0.841164i \(-0.681871\pi\)
−0.540781 + 0.841164i \(0.681871\pi\)
\(84\) −44.5935 −0.0579232
\(85\) 0 0
\(86\) −1099.84 −1.37905
\(87\) 36.0795 0.0444612
\(88\) −135.775 −0.164473
\(89\) −774.536 −0.922479 −0.461240 − 0.887276i \(-0.652595\pi\)
−0.461240 + 0.887276i \(0.652595\pi\)
\(90\) 0 0
\(91\) 162.160 0.186802
\(92\) −92.0000 −0.104257
\(93\) 261.679 0.291773
\(94\) 1007.61 1.10561
\(95\) 0 0
\(96\) −101.212 −0.107603
\(97\) 1402.97 1.46856 0.734278 − 0.678848i \(-0.237521\pi\)
0.734278 + 0.678848i \(0.237521\pi\)
\(98\) −661.152 −0.681495
\(99\) 288.457 0.292839
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.1.3 yes 6
5.2 odd 4 1150.4.b.s.599.10 12
5.3 odd 4 1150.4.b.s.599.3 12
5.4 even 2 1150.4.a.w.1.4 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.4.a.w.1.4 ✓ 6 5.4 even 2
1150.4.a.x.1.3 yes 6 1.1 even 1 trivial
1150.4.b.s.599.3 12 5.3 odd 4
1150.4.b.s.599.10 12 5.2 odd 4