Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [114,2,Mod(25,114)] 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("114.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(114, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 14])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 114 = 2 \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 114.i (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.910294583043\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 73.1
Root \(-0.766044 - 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 114.73
Dual form 114.2.i.a.25.1

$q$-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+(-0.766044 - 0.642788i) q^{2} +(-0.939693 + 0.342020i) q^{3} +(0.173648 + 0.984808i) q^{4} +(-0.386659 + 2.19285i) q^{5} +(0.939693 + 0.342020i) q^{6} +(1.32635 + 2.29731i) q^{7} +(0.500000 - 0.866025i) q^{8} +(0.766044 - 0.642788i) q^{9} +(1.70574 - 1.43128i) q^{10} +(-1.11334 + 1.92836i) q^{11} +(-0.500000 - 0.866025i) q^{12} +(4.97178 + 1.80958i) q^{13} +(0.460637 - 2.61240i) q^{14} +(-0.386659 - 2.19285i) q^{15} +(-0.939693 + 0.342020i) q^{16} +(-2.61334 - 2.19285i) q^{17} -1.00000 q^{18} +(-4.29813 - 0.725293i) q^{19} -2.22668 q^{20} +(-2.03209 - 1.70513i) q^{21} +(2.09240 - 0.761570i) q^{22} +(0.386659 + 2.19285i) q^{23} +(-0.173648 + 0.984808i) q^{24} +(0.0393628 + 0.0143269i) q^{25} +(-2.64543 - 4.58202i) q^{26} +(-0.500000 + 0.866025i) q^{27} +(-2.03209 + 1.70513i) q^{28} +(3.68866 - 3.09516i) q^{29} +(-1.11334 + 1.92836i) q^{30} +(-5.15657 - 8.93145i) q^{31} +(0.939693 + 0.342020i) q^{32} +(0.386659 - 2.19285i) q^{33} +(0.592396 + 3.35965i) q^{34} +(-5.55051 + 2.02022i) q^{35} +(0.766044 + 0.642788i) q^{36} +2.30541 q^{37} +(2.82635 + 3.31839i) q^{38} -5.29086 q^{39} +(1.70574 + 1.43128i) q^{40} +(6.79813 - 2.47432i) q^{41} +(0.460637 + 2.61240i) q^{42} +(-1.02822 + 5.83132i) q^{43} +(-2.09240 - 0.761570i) q^{44} +(1.11334 + 1.92836i) q^{45} +(1.11334 - 1.92836i) q^{46} +(8.43242 - 7.07564i) q^{47} +(0.766044 - 0.642788i) q^{48} +(-0.0184183 + 0.0319015i) q^{49} +(-0.0209445 - 0.0362770i) q^{50} +(3.20574 + 1.16679i) q^{51} +(-0.918748 + 5.21048i) q^{52} +(1.70574 + 9.67372i) q^{53} +(0.939693 - 0.342020i) q^{54} +(-3.79813 - 3.18701i) q^{55} +2.65270 q^{56} +(4.28699 - 0.788496i) q^{57} -4.81521 q^{58} +(3.79813 + 3.18701i) q^{59} +(2.09240 - 0.761570i) q^{60} +(-0.990200 - 5.61570i) q^{61} +(-1.79086 + 10.1565i) q^{62} +(2.49273 + 0.907278i) q^{63} +(-0.500000 - 0.866025i) q^{64} +(-5.89053 + 10.2027i) q^{65} +(-1.70574 + 1.43128i) q^{66} +(6.56805 - 5.51125i) q^{67} +(1.70574 - 2.95442i) q^{68} +(-1.11334 - 1.92836i) q^{69} +(5.55051 + 2.02022i) q^{70} +(0.764700 - 4.33683i) q^{71} +(-0.173648 - 0.984808i) q^{72} +(2.62701 - 0.956154i) q^{73} +(-1.76604 - 1.48189i) q^{74} -0.0418891 q^{75} +(-0.0320889 - 4.35878i) q^{76} -5.90673 q^{77} +(4.05303 + 3.40090i) q^{78} +(-12.9684 + 4.72010i) q^{79} +(-0.386659 - 2.19285i) q^{80} +(0.173648 - 0.984808i) q^{81} +(-6.79813 - 2.47432i) q^{82} +(5.25150 + 9.09586i) q^{83} +(1.32635 - 2.29731i) q^{84} +(5.81908 - 4.88279i) q^{85} +(4.53596 - 3.80612i) q^{86} +(-2.40760 + 4.17009i) q^{87} +(1.11334 + 1.92836i) q^{88} +(-7.34389 - 2.67296i) q^{89} +(0.386659 - 2.19285i) q^{90} +(2.43717 + 13.8219i) q^{91} +(-2.09240 + 0.761570i) q^{92} +(7.90033 + 6.62916i) q^{93} -11.0077 q^{94} +(3.25237 - 9.14473i) q^{95} -1.00000 q^{96} +(-13.6270 - 11.4344i) q^{97} +(0.0346151 - 0.0125989i) q^{98} +(0.386659 + 2.19285i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{5} + 9 q^{7} + 3 q^{8} - 3 q^{12} + 15 q^{13} - 6 q^{14} - 9 q^{15} - 9 q^{17} - 6 q^{18} - 12 q^{19} - 3 q^{21} + 9 q^{22} + 9 q^{23} + 9 q^{25} - 3 q^{27} - 3 q^{28} - 9 q^{29} - 9 q^{31} + 9 q^{33}+ \cdots + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.766044 − 0.642788i −0.541675 − 0.454519i
\(3\) −0.939693 + 0.342020i −0.542532 + 0.197465i
\(4\) 0.173648 + 0.984808i 0.0868241 + 0.492404i
\(5\) −0.386659 + 2.19285i −0.172919 + 0.980674i 0.767599 + 0.640930i \(0.221451\pi\)
−0.940518 + 0.339743i \(0.889660\pi\)
\(6\) 0.939693 + 0.342020i 0.383628 + 0.139629i
\(7\) 1.32635 + 2.29731i 0.501314 + 0.868301i 0.999999 + 0.00151779i \(0.000483127\pi\)
−0.498685 + 0.866783i \(0.666184\pi\)
\(8\) 0.500000 − 0.866025i 0.176777 − 0.306186i
\(9\) 0.766044 − 0.642788i 0.255348 − 0.214263i
\(10\) 1.70574 − 1.43128i 0.539401 − 0.452612i
\(11\) −1.11334 + 1.92836i −0.335685 + 0.581423i −0.983616 − 0.180276i \(-0.942301\pi\)
0.647931 + 0.761699i \(0.275634\pi\)
\(12\) −0.500000 − 0.866025i −0.144338 − 0.250000i
\(13\) 4.97178 + 1.80958i 1.37892 + 0.501887i 0.921852 − 0.387542i \(-0.126676\pi\)
0.457072 + 0.889430i \(0.348898\pi\)
\(14\) 0.460637 − 2.61240i 0.123110 − 0.698194i
\(15\) −0.386659 − 2.19285i −0.0998350 − 0.566192i
\(16\) −0.939693 + 0.342020i −0.234923 + 0.0855050i
\(17\) −2.61334 − 2.19285i −0.633828 − 0.531845i 0.268288 − 0.963339i \(-0.413542\pi\)
−0.902116 + 0.431494i \(0.857987\pi\)
\(18\) −1.00000 −0.235702
\(19\) −4.29813 − 0.725293i −0.986059 − 0.166394i
\(20\) −2.22668 −0.497901
\(21\) −2.03209 − 1.70513i −0.443438 − 0.372089i
\(22\) 2.09240 − 0.761570i 0.446100 − 0.162367i
\(23\) 0.386659 + 2.19285i 0.0806240 + 0.457242i 0.998215 + 0.0597166i \(0.0190197\pi\)
−0.917591 + 0.397525i \(0.869869\pi\)
\(24\) −0.173648 + 0.984808i −0.0354458 + 0.201023i
\(25\) 0.0393628 + 0.0143269i 0.00787257 + 0.00286538i
\(26\) −2.64543 − 4.58202i −0.518811 − 0.898608i
\(27\) −0.500000 + 0.866025i −0.0962250 + 0.166667i
\(28\) −2.03209 + 1.70513i −0.384029 + 0.322238i
\(29\) 3.68866 − 3.09516i 0.684968 − 0.574756i −0.232486 − 0.972600i \(-0.574686\pi\)
0.917453 + 0.397844i \(0.130241\pi\)
\(30\) −1.11334 + 1.92836i −0.203267 + 0.352069i
\(31\) −5.15657 − 8.93145i −0.926148 − 1.60414i −0.789704 − 0.613488i \(-0.789766\pi\)
−0.136444 − 0.990648i \(-0.543567\pi\)
\(32\) 0.939693 + 0.342020i 0.166116 + 0.0604612i
\(33\) 0.386659 − 2.19285i 0.0673087 − 0.381727i
\(34\) 0.592396 + 3.35965i 0.101595 + 0.576175i
\(35\) −5.55051 + 2.02022i −0.938207 + 0.341479i
\(36\) 0.766044 + 0.642788i 0.127674 + 0.107131i
\(37\) 2.30541 0.379007 0.189503 − 0.981880i \(-0.439312\pi\)
0.189503 + 0.981880i \(0.439312\pi\)
\(38\) 2.82635 + 3.31839i 0.458495 + 0.538315i
\(39\) −5.29086 −0.847216
\(40\) 1.70574 + 1.43128i 0.269701 + 0.226306i
\(41\) 6.79813 − 2.47432i 1.06169 − 0.386424i 0.248627 − 0.968599i \(-0.420021\pi\)
0.813063 + 0.582176i \(0.197798\pi\)
\(42\) 0.460637 + 2.61240i 0.0710779 + 0.403103i
\(43\) −1.02822 + 5.83132i −0.156802 + 0.889267i 0.800318 + 0.599576i \(0.204664\pi\)
−0.957120 + 0.289692i \(0.906447\pi\)
\(44\) −2.09240 − 0.761570i −0.315441 − 0.114811i
\(45\) 1.11334 + 1.92836i 0.165967 + 0.287463i
\(46\) 1.11334 − 1.92836i 0.164153 − 0.284322i
\(47\) 8.43242 − 7.07564i 1.22999 − 1.03209i 0.231755 − 0.972774i \(-0.425553\pi\)
0.998239 − 0.0593140i \(-0.0188913\pi\)
\(48\) 0.766044 − 0.642788i 0.110569 − 0.0927784i
\(49\) −0.0184183 + 0.0319015i −0.00263119 + 0.00455735i
\(50\) −0.0209445 − 0.0362770i −0.00296200 − 0.00513034i
\(51\) 3.20574 + 1.16679i 0.448893 + 0.163384i
\(52\) −0.918748 + 5.21048i −0.127407 + 0.722563i
\(53\) 1.70574 + 9.67372i 0.234301 + 1.32879i 0.844081 + 0.536216i \(0.180147\pi\)
−0.609780 + 0.792571i \(0.708742\pi\)
\(54\) 0.939693 − 0.342020i 0.127876 − 0.0465430i
\(55\) −3.79813 − 3.18701i −0.512140 − 0.429737i
\(56\) 2.65270 0.354482
\(57\) 4.28699 − 0.788496i 0.567826 − 0.104439i
\(58\) −4.81521 −0.632268
\(59\) 3.79813 + 3.18701i 0.494475 + 0.414914i 0.855627 − 0.517593i \(-0.173172\pi\)
−0.361152 + 0.932507i \(0.617616\pi\)
\(60\) 2.09240 − 0.761570i 0.270127 − 0.0983183i
\(61\) −0.990200 − 5.61570i −0.126782 − 0.719017i −0.980233 − 0.197844i \(-0.936606\pi\)
0.853451 − 0.521173i \(-0.174505\pi\)
\(62\) −1.79086 + 10.1565i −0.227439 + 1.28987i
\(63\) 2.49273 + 0.907278i 0.314054 + 0.114306i
\(64\) −0.500000 − 0.866025i −0.0625000 − 0.108253i
\(65\) −5.89053 + 10.2027i −0.730630 + 1.26549i
\(66\) −1.70574 + 1.43128i −0.209962 + 0.176179i
\(67\) 6.56805 − 5.51125i 0.802415 − 0.673306i −0.146370 − 0.989230i \(-0.546759\pi\)
0.948784 + 0.315924i \(0.102314\pi\)
\(68\) 1.70574 − 2.95442i 0.206851 − 0.358276i
\(69\) −1.11334 − 1.92836i −0.134030 − 0.232148i
\(70\) 5.55051 + 2.02022i 0.663413 + 0.241462i
\(71\) 0.764700 − 4.33683i 0.0907532 − 0.514687i −0.905213 − 0.424958i \(-0.860289\pi\)
0.995966 − 0.0897290i \(-0.0286001\pi\)
\(72\) −0.173648 − 0.984808i −0.0204646 − 0.116061i
\(73\) 2.62701 − 0.956154i 0.307468 − 0.111909i −0.183678 − 0.982987i \(-0.558800\pi\)
0.491146 + 0.871077i \(0.336578\pi\)
\(74\) −1.76604 − 1.48189i −0.205298 − 0.172266i
\(75\) −0.0418891 −0.00483693
\(76\) −0.0320889 − 4.35878i −0.00368085 − 0.499986i
\(77\) −5.90673 −0.673134
\(78\) 4.05303 + 3.40090i 0.458916 + 0.385076i
\(79\) −12.9684 + 4.72010i −1.45906 + 0.531053i −0.945105 − 0.326766i \(-0.894041\pi\)
−0.513951 + 0.857819i \(0.671819\pi\)
\(80\) −0.386659 − 2.19285i −0.0432298 − 0.245168i
\(81\) 0.173648 − 0.984808i 0.0192942 − 0.109423i
\(82\) −6.79813 − 2.47432i −0.750728 − 0.273243i
\(83\) 5.25150 + 9.09586i 0.576427 + 0.998400i 0.995885 + 0.0906256i \(0.0288867\pi\)
−0.419458 + 0.907775i \(0.637780\pi\)
\(84\) 1.32635 − 2.29731i 0.144717 − 0.250657i
\(85\) 5.81908 − 4.88279i 0.631168 − 0.529613i
\(86\) 4.53596 − 3.80612i 0.489125 − 0.410425i
\(87\) −2.40760 + 4.17009i −0.258122 + 0.447081i
\(88\) 1.11334 + 1.92836i 0.118683 + 0.205564i
\(89\) −7.34389 − 2.67296i −0.778451 − 0.283333i −0.0779244 − 0.996959i \(-0.524829\pi\)
−0.700527 + 0.713626i \(0.747052\pi\)
\(90\) 0.386659 − 2.19285i 0.0407575 − 0.231147i
\(91\) 2.43717 + 13.8219i 0.255484 + 1.44892i
\(92\) −2.09240 + 0.761570i −0.218147 + 0.0793992i
\(93\) 7.90033 + 6.62916i 0.819226 + 0.687412i
\(94\) −11.0077 −1.13536
\(95\) 3.25237 − 9.14473i 0.333687 − 0.938230i
\(96\) −1.00000 −0.102062
\(97\) −13.6270 − 11.4344i −1.38361 − 1.16099i −0.967853 − 0.251515i \(-0.919071\pi\)
−0.415760 − 0.909474i \(-0.636484\pi\)
\(98\) 0.0346151 − 0.0125989i 0.00349665 − 0.00127268i
\(99\) 0.386659 + 2.19285i 0.0388607 + 0.220390i
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 114.2.i.a.73.1 yes 6
3.2 odd 2 342.2.u.e.73.1 6
4.3 odd 2 912.2.bo.a.529.1 6
19.5 even 9 2166.2.a.q.1.1 3
19.6 even 9 inner 114.2.i.a.25.1 ✓ 6
19.14 odd 18 2166.2.a.s.1.1 3
57.5 odd 18 6498.2.a.br.1.3 3
57.14 even 18 6498.2.a.bm.1.3 3
57.44 odd 18 342.2.u.e.253.1 6
76.63 odd 18 912.2.bo.a.481.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.a.25.1 ✓ 6 19.6 even 9 inner
114.2.i.a.73.1 yes 6 1.1 even 1 trivial
342.2.u.e.73.1 6 3.2 odd 2
342.2.u.e.253.1 6 57.44 odd 18
912.2.bo.a.481.1 6 76.63 odd 18
912.2.bo.a.529.1 6 4.3 odd 2
2166.2.a.q.1.1 3 19.5 even 9
2166.2.a.s.1.1 3 19.14 odd 18
6498.2.a.bm.1.3 3 57.14 even 18
6498.2.a.br.1.3 3 57.5 odd 18