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(599,1150)] 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("1150.599"); 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([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,-48,0,-20,0,0,-122,0,-98] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(67.8521965066\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 219x^{10} + 17685x^{8} + 640366x^{6} + 10000368x^{4} + 54897345x^{2} + 95531076 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.3
Root \(-2.16288i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.4.b.s.599.10

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} -3.16288i q^{3} -4.00000 q^{4} -6.32576 q^{6} -3.52475i q^{7} +8.00000i q^{8} +16.9962 q^{9} -16.9719 q^{11} +12.6515i q^{12} +46.0060i q^{13} -7.04951 q^{14} +16.0000 q^{16} -16.2568i q^{17} -33.9924i q^{18} +31.1330 q^{19} -11.1484 q^{21} +33.9437i q^{22} -23.0000i q^{23} +25.3030 q^{24} +92.0120 q^{26} -139.155i q^{27} +14.0990i q^{28} +11.4072 q^{29} -82.7344 q^{31} -32.0000i q^{32} +53.6800i q^{33} -32.5135 q^{34} -67.9848 q^{36} +296.932i q^{37} -62.2659i q^{38} +145.511 q^{39} +407.814 q^{41} +22.2967i q^{42} -549.918i q^{43} +67.8875 q^{44} -46.0000 q^{46} -503.804i q^{47} -50.6061i q^{48} +330.576 q^{49} -51.4182 q^{51} -184.024i q^{52} -605.711i q^{53} -278.309 q^{54} +28.1980 q^{56} -98.4698i q^{57} -22.8143i q^{58} -522.579 q^{59} +242.576 q^{61} +165.469i q^{62} -59.9074i q^{63} -64.0000 q^{64} +107.360 q^{66} +1014.23i q^{67} +65.0271i q^{68} -72.7462 q^{69} -334.393 q^{71} +135.970i q^{72} -181.460i q^{73} +593.864 q^{74} -124.532 q^{76} +59.8216i q^{77} -291.023i q^{78} +461.511 q^{79} +18.7681 q^{81} -815.628i q^{82} -817.840i q^{83} +44.5935 q^{84} -1099.84 q^{86} -36.0795i q^{87} -135.775i q^{88} +774.536 q^{89} +162.160 q^{91} +92.0000i q^{92} +261.679i q^{93} -1007.61 q^{94} -101.212 q^{96} -1402.97i q^{97} -661.152i q^{98} -288.457 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 48 q^{4} - 20 q^{6} - 122 q^{9} - 98 q^{11} + 168 q^{14} + 192 q^{16} + 458 q^{19} + 184 q^{21} + 80 q^{24} - 64 q^{26} + 364 q^{29} + 228 q^{31} + 700 q^{34} + 488 q^{36} + 286 q^{39} + 486 q^{41}+ \cdots + 284 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 0.707107i
\(3\) − 3.16288i − 0.608696i −0.952561 − 0.304348i \(-0.901561\pi\)
0.952561 − 0.304348i \(-0.0984386\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) −6.32576 −0.430413
\(7\) − 3.52475i − 0.190319i −0.995462 − 0.0951594i \(-0.969664\pi\)
0.995462 − 0.0951594i \(-0.0303361\pi\)
\(8\) 8.00000i 0.353553i
\(9\) 16.9962 0.629489
\(10\) 0 0
\(11\) −16.9719 −0.465201 −0.232601 − 0.972572i \(-0.574723\pi\)
−0.232601 + 0.972572i \(0.574723\pi\)
\(12\) 12.6515i 0.304348i
\(13\) 46.0060i 0.981520i 0.871295 + 0.490760i \(0.163281\pi\)
−0.871295 + 0.490760i \(0.836719\pi\)
\(14\) −7.04951 −0.134576
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 16.2568i − 0.231932i −0.993253 − 0.115966i \(-0.963004\pi\)
0.993253 − 0.115966i \(-0.0369964\pi\)
\(18\) − 33.9924i − 0.445116i
\(19\) 31.1330 0.375915 0.187958 − 0.982177i \(-0.439813\pi\)
0.187958 + 0.982177i \(0.439813\pi\)
\(20\) 0 0
\(21\) −11.1484 −0.115846
\(22\) 33.9437i 0.328947i
\(23\) − 23.0000i − 0.208514i
\(24\) 25.3030 0.215207
\(25\) 0 0
\(26\) 92.0120 0.694040
\(27\) − 139.155i − 0.991864i
\(28\) 14.0990i 0.0951594i
\(29\) 11.4072 0.0730433 0.0365217 − 0.999333i \(-0.488372\pi\)
0.0365217 + 0.999333i \(0.488372\pi\)
\(30\) 0 0
\(31\) −82.7344 −0.479340 −0.239670 − 0.970854i \(-0.577039\pi\)
−0.239670 + 0.970854i \(0.577039\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) 53.6800i 0.283166i
\(34\) −32.5135 −0.164001
\(35\) 0 0
\(36\) −67.9848 −0.314744
\(37\) 296.932i 1.31933i 0.751559 + 0.659666i \(0.229302\pi\)
−0.751559 + 0.659666i \(0.770698\pi\)
\(38\) − 62.2659i − 0.265812i
\(39\) 145.511 0.597448
\(40\) 0 0
\(41\) 407.814 1.55341 0.776705 − 0.629864i \(-0.216890\pi\)
0.776705 + 0.629864i \(0.216890\pi\)
\(42\) 22.2967i 0.0819157i
\(43\) − 549.918i − 1.95027i −0.221609 − 0.975136i \(-0.571131\pi\)
0.221609 − 0.975136i \(-0.428869\pi\)
\(44\) 67.8875 0.232601
\(45\) 0 0
\(46\) −46.0000 −0.147442
\(47\) − 503.804i − 1.56356i −0.623553 − 0.781781i \(-0.714311\pi\)
0.623553 − 0.781781i \(-0.285689\pi\)
\(48\) − 50.6061i − 0.152174i
\(49\) 330.576 0.963779
\(50\) 0 0
\(51\) −51.4182 −0.141176
\(52\) − 184.024i − 0.490760i
\(53\) − 605.711i − 1.56983i −0.619606 − 0.784913i \(-0.712707\pi\)
0.619606 − 0.784913i \(-0.287293\pi\)
\(54\) −278.309 −0.701354
\(55\) 0 0
\(56\) 28.1980 0.0672878
\(57\) − 98.4698i − 0.228818i
\(58\) − 22.8143i − 0.0516494i
\(59\) −522.579 −1.15312 −0.576559 − 0.817056i \(-0.695605\pi\)
−0.576559 + 0.817056i \(0.695605\pi\)
\(60\) 0 0
\(61\) 242.576 0.509158 0.254579 − 0.967052i \(-0.418063\pi\)
0.254579 + 0.967052i \(0.418063\pi\)
\(62\) 165.469i 0.338945i
\(63\) − 59.9074i − 0.119804i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 107.360 0.200229
\(67\) 1014.23i 1.84937i 0.380728 + 0.924687i \(0.375674\pi\)
−0.380728 + 0.924687i \(0.624326\pi\)
\(68\) 65.0271i 0.115966i
\(69\) −72.7462 −0.126922
\(70\) 0 0
\(71\) −334.393 −0.558946 −0.279473 − 0.960154i \(-0.590160\pi\)
−0.279473 + 0.960154i \(0.590160\pi\)
\(72\) 135.970i 0.222558i
\(73\) − 181.460i − 0.290935i −0.989363 − 0.145468i \(-0.953531\pi\)
0.989363 − 0.145468i \(-0.0464686\pi\)
\(74\) 593.864 0.932909
\(75\) 0 0
\(76\) −124.532 −0.187958
\(77\) 59.8216i 0.0885365i
\(78\) − 291.023i − 0.422459i
\(79\) 461.511 0.657267 0.328633 − 0.944458i \(-0.393412\pi\)
0.328633 + 0.944458i \(0.393412\pi\)
\(80\) 0 0
\(81\) 18.7681 0.0257450
\(82\) − 815.628i − 1.09843i
\(83\) − 817.840i − 1.08156i −0.841164 − 0.540781i \(-0.818129\pi\)
0.841164 − 0.540781i \(-0.181871\pi\)
\(84\) 44.5935 0.0579232
\(85\) 0 0
\(86\) −1099.84 −1.37905
\(87\) − 36.0795i − 0.0444612i
\(88\) − 135.775i − 0.164473i
\(89\) 774.536 0.922479 0.461240 − 0.887276i \(-0.347405\pi\)
0.461240 + 0.887276i \(0.347405\pi\)
\(90\) 0 0
\(91\) 162.160 0.186802
\(92\) 92.0000i 0.104257i
\(93\) 261.679i 0.291773i
\(94\) −1007.61 −1.10561
\(95\) 0 0
\(96\) −101.212 −0.107603
\(97\) − 1402.97i − 1.46856i −0.678848 − 0.734278i \(-0.737521\pi\)
0.678848 − 0.734278i \(-0.262479\pi\)
\(98\) − 661.152i − 0.681495i
\(99\) −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.b.s.599.3 12
5.2 odd 4 1150.4.a.x.1.3 yes 6
5.3 odd 4 1150.4.a.w.1.4 ✓ 6
5.4 even 2 inner 1150.4.b.s.599.10 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.4.a.w.1.4 ✓ 6 5.3 odd 4
1150.4.a.x.1.3 yes 6 5.2 odd 4
1150.4.b.s.599.3 12 1.1 even 1 trivial
1150.4.b.s.599.10 12 5.4 even 2 inner