Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1850,2,Mod(149,1850)] 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("1850.149"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1850, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,4,0,0,-16,0,4] 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: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.1
Root \(-1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1850.149
Dual form 1850.2.b.k.149.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.44949i q^{3} -1.00000 q^{4} -1.44949 q^{6} -2.44949i q^{7} +1.00000i q^{8} +0.898979 q^{9} +3.44949 q^{11} +1.44949i q^{12} -0.449490i q^{13} -2.44949 q^{14} +1.00000 q^{16} -3.44949i q^{17} -0.898979i q^{18} +5.00000 q^{19} -3.55051 q^{21} -3.44949i q^{22} +2.00000i q^{23} +1.44949 q^{24} -0.449490 q^{26} -5.65153i q^{27} +2.44949i q^{28} +0.898979 q^{29} +4.44949 q^{31} -1.00000i q^{32} -5.00000i q^{33} -3.44949 q^{34} -0.898979 q^{36} +1.00000i q^{37} -5.00000i q^{38} -0.651531 q^{39} +1.00000 q^{41} +3.55051i q^{42} -1.10102i q^{43} -3.44949 q^{44} +2.00000 q^{46} -9.79796i q^{47} -1.44949i q^{48} +1.00000 q^{49} -5.00000 q^{51} +0.449490i q^{52} +6.00000i q^{53} -5.65153 q^{54} +2.44949 q^{56} -7.24745i q^{57} -0.898979i q^{58} +2.00000 q^{59} -6.44949 q^{61} -4.44949i q^{62} -2.20204i q^{63} -1.00000 q^{64} -5.00000 q^{66} -4.55051i q^{67} +3.44949i q^{68} +2.89898 q^{69} -7.55051 q^{71} +0.898979i q^{72} +12.7980i q^{73} +1.00000 q^{74} -5.00000 q^{76} -8.44949i q^{77} +0.651531i q^{78} -7.79796 q^{79} -5.49490 q^{81} -1.00000i q^{82} +3.44949i q^{83} +3.55051 q^{84} -1.10102 q^{86} -1.30306i q^{87} +3.44949i q^{88} -14.3485 q^{89} -1.10102 q^{91} -2.00000i q^{92} -6.44949i q^{93} -9.79796 q^{94} -1.44949 q^{96} -14.0000i q^{97} -1.00000i q^{98} +3.10102 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{6} - 16 q^{9} + 4 q^{11} + 4 q^{16} + 20 q^{19} - 24 q^{21} - 4 q^{24} + 8 q^{26} - 16 q^{29} + 8 q^{31} - 4 q^{34} + 16 q^{36} - 32 q^{39} + 4 q^{41} - 4 q^{44} + 8 q^{46} + 4 q^{49}+ \cdots + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) − 1.44949i − 0.836863i −0.908248 − 0.418432i \(-0.862580\pi\)
0.908248 − 0.418432i \(-0.137420\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.44949 −0.591752
\(7\) − 2.44949i − 0.925820i −0.886405 − 0.462910i \(-0.846805\pi\)
0.886405 − 0.462910i \(-0.153195\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0.898979 0.299660
\(10\) 0 0
\(11\) 3.44949 1.04006 0.520030 − 0.854148i \(-0.325921\pi\)
0.520030 + 0.854148i \(0.325921\pi\)
\(12\) 1.44949i 0.418432i
\(13\) − 0.449490i − 0.124666i −0.998055 − 0.0623330i \(-0.980146\pi\)
0.998055 − 0.0623330i \(-0.0198541\pi\)
\(14\) −2.44949 −0.654654
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 3.44949i − 0.836624i −0.908303 − 0.418312i \(-0.862622\pi\)
0.908303 − 0.418312i \(-0.137378\pi\)
\(18\) − 0.898979i − 0.211891i
\(19\) 5.00000 1.14708 0.573539 − 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) −3.55051 −0.774785
\(22\) − 3.44949i − 0.735434i
\(23\) 2.00000i 0.417029i 0.978019 + 0.208514i \(0.0668628\pi\)
−0.978019 + 0.208514i \(0.933137\pi\)
\(24\) 1.44949 0.295876
\(25\) 0 0
\(26\) −0.449490 −0.0881522
\(27\) − 5.65153i − 1.08764i
\(28\) 2.44949i 0.462910i
\(29\) 0.898979 0.166936 0.0834681 − 0.996510i \(-0.473400\pi\)
0.0834681 + 0.996510i \(0.473400\pi\)
\(30\) 0 0
\(31\) 4.44949 0.799152 0.399576 − 0.916700i \(-0.369157\pi\)
0.399576 + 0.916700i \(0.369157\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 5.00000i − 0.870388i
\(34\) −3.44949 −0.591583
\(35\) 0 0
\(36\) −0.898979 −0.149830
\(37\) 1.00000i 0.164399i
\(38\) − 5.00000i − 0.811107i
\(39\) −0.651531 −0.104328
\(40\) 0 0
\(41\) 1.00000 0.156174 0.0780869 − 0.996947i \(-0.475119\pi\)
0.0780869 + 0.996947i \(0.475119\pi\)
\(42\) 3.55051i 0.547856i
\(43\) − 1.10102i − 0.167904i −0.996470 − 0.0839520i \(-0.973246\pi\)
0.996470 − 0.0839520i \(-0.0267543\pi\)
\(44\) −3.44949 −0.520030
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) − 9.79796i − 1.42918i −0.699544 − 0.714590i \(-0.746613\pi\)
0.699544 − 0.714590i \(-0.253387\pi\)
\(48\) − 1.44949i − 0.209216i
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −5.00000 −0.700140
\(52\) 0.449490i 0.0623330i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) −5.65153 −0.769076
\(55\) 0 0
\(56\) 2.44949 0.327327
\(57\) − 7.24745i − 0.959948i
\(58\) − 0.898979i − 0.118042i
\(59\) 2.00000 0.260378 0.130189 − 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 0 0
\(61\) −6.44949 −0.825773 −0.412886 − 0.910783i \(-0.635479\pi\)
−0.412886 + 0.910783i \(0.635479\pi\)
\(62\) − 4.44949i − 0.565086i
\(63\) − 2.20204i − 0.277431i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −5.00000 −0.615457
\(67\) − 4.55051i − 0.555933i −0.960591 − 0.277967i \(-0.910340\pi\)
0.960591 − 0.277967i \(-0.0896605\pi\)
\(68\) 3.44949i 0.418312i
\(69\) 2.89898 0.348996
\(70\) 0 0
\(71\) −7.55051 −0.896081 −0.448040 − 0.894013i \(-0.647878\pi\)
−0.448040 + 0.894013i \(0.647878\pi\)
\(72\) 0.898979i 0.105946i
\(73\) 12.7980i 1.49789i 0.662633 + 0.748944i \(0.269439\pi\)
−0.662633 + 0.748944i \(0.730561\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −5.00000 −0.573539
\(77\) − 8.44949i − 0.962909i
\(78\) 0.651531i 0.0737713i
\(79\) −7.79796 −0.877339 −0.438669 − 0.898648i \(-0.644550\pi\)
−0.438669 + 0.898648i \(0.644550\pi\)
\(80\) 0 0
\(81\) −5.49490 −0.610544
\(82\) − 1.00000i − 0.110432i
\(83\) 3.44949i 0.378631i 0.981916 + 0.189315i \(0.0606268\pi\)
−0.981916 + 0.189315i \(0.939373\pi\)
\(84\) 3.55051 0.387392
\(85\) 0 0
\(86\) −1.10102 −0.118726
\(87\) − 1.30306i − 0.139703i
\(88\) 3.44949i 0.367717i
\(89\) −14.3485 −1.52093 −0.760467 − 0.649376i \(-0.775030\pi\)
−0.760467 + 0.649376i \(0.775030\pi\)
\(90\) 0 0
\(91\) −1.10102 −0.115418
\(92\) − 2.00000i − 0.208514i
\(93\) − 6.44949i − 0.668781i
\(94\) −9.79796 −1.01058
\(95\) 0 0
\(96\) −1.44949 −0.147938
\(97\) − 14.0000i − 1.42148i −0.703452 − 0.710742i \(-0.748359\pi\)
0.703452 − 0.710742i \(-0.251641\pi\)
\(98\) − 1.00000i − 0.101015i
\(99\) 3.10102 0.311664
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 1850.2.b.k.149.1 4
5.2 odd 4 1850.2.a.w.1.1 yes 2
5.3 odd 4 1850.2.a.r.1.2 ✓ 2
5.4 even 2 inner 1850.2.b.k.149.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.r.1.2 ✓ 2 5.3 odd 4
1850.2.a.w.1.1 yes 2 5.2 odd 4
1850.2.b.k.149.1 4 1.1 even 1 trivial
1850.2.b.k.149.4 4 5.4 even 2 inner