Properties

Label 1150.4.a.w.1.4
Level $1150$
Weight $4$
Character 1150.1
Self dual yes
Analytic conductor $67.852$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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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: \(0\)
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 1.4
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 \(-0.401561\pi\)
0.304348 + 0.952561i \(0.401561\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) −6.32576 −0.430413
\(7\) −3.52475 −0.190319 −0.0951594 − 0.995462i \(-0.530336\pi\)
−0.0951594 + 0.995462i \(0.530336\pi\)
\(8\) −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 \(-0.663281\pi\)
−0.490760 + 0.871295i \(0.663281\pi\)
\(14\) 7.04951 0.134576
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −16.2568 −0.231932 −0.115966 − 0.993253i \(-0.536996\pi\)
−0.115966 + 0.993253i \(0.536996\pi\)
\(18\) 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 \(-0.270698\pi\)
0.659666 + 0.751559i \(0.270698\pi\)
\(38\) 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 \(-0.0711309\pi\)
0.975136 + 0.221609i \(0.0711309\pi\)
\(44\) −67.8875 −0.232601
\(45\) 0 0
\(46\) −46.0000 −0.147442
\(47\) −503.804 −1.56356 −0.781781 − 0.623553i \(-0.785689\pi\)
−0.781781 + 0.623553i \(0.785689\pi\)
\(48\) 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 \(-0.212707\pi\)
0.784913 + 0.619606i \(0.212707\pi\)
\(54\) 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 \(-0.124326\pi\)
0.924687 + 0.380728i \(0.124326\pi\)
\(68\) −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 \(-0.453531\pi\)
0.145468 + 0.989363i \(0.453531\pi\)
\(74\) −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 \(-0.318129\pi\)
0.540781 + 0.841164i \(0.318129\pi\)
\(84\) −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 \(-0.762479\pi\)
−0.734278 + 0.678848i \(0.762479\pi\)
\(98\) 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.w.1.4 ✓ 6
5.2 odd 4 1150.4.b.s.599.3 12
5.3 odd 4 1150.4.b.s.599.10 12
5.4 even 2 1150.4.a.x.1.3 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.4.a.w.1.4 ✓ 6 1.1 even 1 trivial
1150.4.a.x.1.3 yes 6 5.4 even 2
1150.4.b.s.599.3 12 5.2 odd 4
1150.4.b.s.599.10 12 5.3 odd 4