Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1150,2,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, 2); 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, 2, names="a")
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,-6,0,0,-10,0,-14] 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: \(9.18279623245\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.2
Root \(1.30278i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.2.b.f.599.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +0.302776i q^{3} -1.00000 q^{4} +0.302776 q^{6} +3.30278i q^{7} +1.00000i q^{8} +2.90833 q^{9} -1.69722 q^{11} -0.302776i q^{12} -3.30278i q^{13} +3.30278 q^{14} +1.00000 q^{16} +6.90833i q^{17} -2.90833i q^{18} -5.90833 q^{19} -1.00000 q^{21} +1.69722i q^{22} +1.00000i q^{23} -0.302776 q^{24} -3.30278 q^{26} +1.78890i q^{27} -3.30278i q^{28} +2.60555 q^{29} -7.90833 q^{31} -1.00000i q^{32} -0.513878i q^{33} +6.90833 q^{34} -2.90833 q^{36} +8.00000i q^{37} +5.90833i q^{38} +1.00000 q^{39} +0.908327 q^{41} +1.00000i q^{42} +9.21110i q^{43} +1.69722 q^{44} +1.00000 q^{46} -2.60555i q^{47} +0.302776i q^{48} -3.90833 q^{49} -2.09167 q^{51} +3.30278i q^{52} +11.2111i q^{53} +1.78890 q^{54} -3.30278 q^{56} -1.78890i q^{57} -2.60555i q^{58} +3.39445 q^{59} +11.5139 q^{61} +7.90833i q^{62} +9.60555i q^{63} -1.00000 q^{64} -0.513878 q^{66} -4.00000i q^{67} -6.90833i q^{68} -0.302776 q^{69} -16.3028 q^{71} +2.90833i q^{72} +5.81665i q^{73} +8.00000 q^{74} +5.90833 q^{76} -5.60555i q^{77} -1.00000i q^{78} +14.4222 q^{79} +8.18335 q^{81} -0.908327i q^{82} -11.2111i q^{83} +1.00000 q^{84} +9.21110 q^{86} +0.788897i q^{87} -1.69722i q^{88} +10.9083 q^{91} -1.00000i q^{92} -2.39445i q^{93} -2.60555 q^{94} +0.302776 q^{96} +6.30278i q^{97} +3.90833i q^{98} -4.93608 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 6 q^{6} - 10 q^{9} - 14 q^{11} + 6 q^{14} + 4 q^{16} - 2 q^{19} - 4 q^{21} + 6 q^{24} - 6 q^{26} - 4 q^{29} - 10 q^{31} + 6 q^{34} + 10 q^{36} + 4 q^{39} - 18 q^{41} + 14 q^{44} + 4 q^{46}+ \cdots + 74 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0.302776i 0.174808i 0.996173 + 0.0874038i \(0.0278570\pi\)
−0.996173 + 0.0874038i \(0.972143\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0.302776 0.123608
\(7\) 3.30278i 1.24833i 0.781292 + 0.624166i \(0.214561\pi\)
−0.781292 + 0.624166i \(0.785439\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 2.90833 0.969442
\(10\) 0 0
\(11\) −1.69722 −0.511732 −0.255866 − 0.966712i \(-0.582361\pi\)
−0.255866 + 0.966712i \(0.582361\pi\)
\(12\) − 0.302776i − 0.0874038i
\(13\) − 3.30278i − 0.916025i −0.888946 − 0.458013i \(-0.848561\pi\)
0.888946 − 0.458013i \(-0.151439\pi\)
\(14\) 3.30278 0.882704
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.90833i 1.67552i 0.546042 + 0.837758i \(0.316134\pi\)
−0.546042 + 0.837758i \(0.683866\pi\)
\(18\) − 2.90833i − 0.685499i
\(19\) −5.90833 −1.35546 −0.677732 − 0.735309i \(-0.737037\pi\)
−0.677732 + 0.735309i \(0.737037\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 1.69722i 0.361849i
\(23\) 1.00000i 0.208514i
\(24\) −0.302776 −0.0618038
\(25\) 0 0
\(26\) −3.30278 −0.647728
\(27\) 1.78890i 0.344273i
\(28\) − 3.30278i − 0.624166i
\(29\) 2.60555 0.483839 0.241919 − 0.970296i \(-0.422223\pi\)
0.241919 + 0.970296i \(0.422223\pi\)
\(30\) 0 0
\(31\) −7.90833 −1.42038 −0.710189 − 0.704011i \(-0.751390\pi\)
−0.710189 + 0.704011i \(0.751390\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 0.513878i − 0.0894547i
\(34\) 6.90833 1.18477
\(35\) 0 0
\(36\) −2.90833 −0.484721
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 5.90833i 0.958457i
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 0.908327 0.141857 0.0709284 − 0.997481i \(-0.477404\pi\)
0.0709284 + 0.997481i \(0.477404\pi\)
\(42\) 1.00000i 0.154303i
\(43\) 9.21110i 1.40468i 0.711842 + 0.702340i \(0.247861\pi\)
−0.711842 + 0.702340i \(0.752139\pi\)
\(44\) 1.69722 0.255866
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) − 2.60555i − 0.380059i −0.981778 − 0.190029i \(-0.939142\pi\)
0.981778 − 0.190029i \(-0.0608583\pi\)
\(48\) 0.302776i 0.0437019i
\(49\) −3.90833 −0.558332
\(50\) 0 0
\(51\) −2.09167 −0.292893
\(52\) 3.30278i 0.458013i
\(53\) 11.2111i 1.53996i 0.638066 + 0.769982i \(0.279735\pi\)
−0.638066 + 0.769982i \(0.720265\pi\)
\(54\) 1.78890 0.243438
\(55\) 0 0
\(56\) −3.30278 −0.441352
\(57\) − 1.78890i − 0.236945i
\(58\) − 2.60555i − 0.342126i
\(59\) 3.39445 0.441920 0.220960 − 0.975283i \(-0.429081\pi\)
0.220960 + 0.975283i \(0.429081\pi\)
\(60\) 0 0
\(61\) 11.5139 1.47420 0.737101 − 0.675783i \(-0.236194\pi\)
0.737101 + 0.675783i \(0.236194\pi\)
\(62\) 7.90833i 1.00436i
\(63\) 9.60555i 1.21019i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −0.513878 −0.0632540
\(67\) − 4.00000i − 0.488678i −0.969690 − 0.244339i \(-0.921429\pi\)
0.969690 − 0.244339i \(-0.0785709\pi\)
\(68\) − 6.90833i − 0.837758i
\(69\) −0.302776 −0.0364499
\(70\) 0 0
\(71\) −16.3028 −1.93478 −0.967392 − 0.253285i \(-0.918489\pi\)
−0.967392 + 0.253285i \(0.918489\pi\)
\(72\) 2.90833i 0.342750i
\(73\) 5.81665i 0.680788i 0.940283 + 0.340394i \(0.110560\pi\)
−0.940283 + 0.340394i \(0.889440\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 5.90833 0.677732
\(77\) − 5.60555i − 0.638812i
\(78\) − 1.00000i − 0.113228i
\(79\) 14.4222 1.62262 0.811312 − 0.584613i \(-0.198754\pi\)
0.811312 + 0.584613i \(0.198754\pi\)
\(80\) 0 0
\(81\) 8.18335 0.909261
\(82\) − 0.908327i − 0.100308i
\(83\) − 11.2111i − 1.23058i −0.788301 − 0.615289i \(-0.789039\pi\)
0.788301 − 0.615289i \(-0.210961\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) 9.21110 0.993259
\(87\) 0.788897i 0.0845787i
\(88\) − 1.69722i − 0.180925i
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 10.9083 1.14350
\(92\) − 1.00000i − 0.104257i
\(93\) − 2.39445i − 0.248293i
\(94\) −2.60555 −0.268742
\(95\) 0 0
\(96\) 0.302776 0.0309019
\(97\) 6.30278i 0.639950i 0.947426 + 0.319975i \(0.103675\pi\)
−0.947426 + 0.319975i \(0.896325\pi\)
\(98\) 3.90833i 0.394801i
\(99\) −4.93608 −0.496095
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 1150.2.b.f.599.2 4
5.2 odd 4 1150.2.a.m.1.2 2
5.3 odd 4 230.2.a.b.1.1 ✓ 2
5.4 even 2 inner 1150.2.b.f.599.3 4
15.8 even 4 2070.2.a.w.1.2 2
20.3 even 4 1840.2.a.j.1.2 2
20.7 even 4 9200.2.a.ca.1.1 2
40.3 even 4 7360.2.a.bu.1.1 2
40.13 odd 4 7360.2.a.bc.1.2 2
115.68 even 4 5290.2.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.b.1.1 ✓ 2 5.3 odd 4
1150.2.a.m.1.2 2 5.2 odd 4
1150.2.b.f.599.2 4 1.1 even 1 trivial
1150.2.b.f.599.3 4 5.4 even 2 inner
1840.2.a.j.1.2 2 20.3 even 4
2070.2.a.w.1.2 2 15.8 even 4
5290.2.a.j.1.1 2 115.68 even 4
7360.2.a.bc.1.2 2 40.13 odd 4
7360.2.a.bu.1.1 2 40.3 even 4
9200.2.a.ca.1.1 2 20.7 even 4