Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [363,4,Mod(1,363)] 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("363.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 363.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,6,-10,-6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(21.4176933321\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 363.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205 q^{2} +3.00000 q^{3} -5.00000 q^{4} -3.00000 q^{5} +5.19615 q^{6} +3.46410 q^{7} -22.5167 q^{8} +9.00000 q^{9} -5.19615 q^{10} -15.0000 q^{12} +1.73205 q^{13} +6.00000 q^{14} -9.00000 q^{15} +1.00000 q^{16} +12.1244 q^{17} +15.5885 q^{18} -131.636 q^{19} +15.0000 q^{20} +10.3923 q^{21} -174.000 q^{23} -67.5500 q^{24} -116.000 q^{25} +3.00000 q^{26} +27.0000 q^{27} -17.3205 q^{28} -53.6936 q^{29} -15.5885 q^{30} -20.0000 q^{31} +181.865 q^{32} +21.0000 q^{34} -10.3923 q^{35} -45.0000 q^{36} -299.000 q^{37} -228.000 q^{38} +5.19615 q^{39} +67.5500 q^{40} -303.109 q^{41} +18.0000 q^{42} +495.367 q^{43} -27.0000 q^{45} -301.377 q^{46} -384.000 q^{47} +3.00000 q^{48} -331.000 q^{49} -200.918 q^{50} +36.3731 q^{51} -8.66025 q^{52} +255.000 q^{53} +46.7654 q^{54} -78.0000 q^{56} -394.908 q^{57} -93.0000 q^{58} +570.000 q^{59} +45.0000 q^{60} +374.123 q^{61} -34.6410 q^{62} +31.1769 q^{63} +307.000 q^{64} -5.19615 q^{65} +46.0000 q^{67} -60.6218 q^{68} -522.000 q^{69} -18.0000 q^{70} -630.000 q^{71} -202.650 q^{72} +575.041 q^{73} -517.883 q^{74} -348.000 q^{75} +658.179 q^{76} +9.00000 q^{78} +249.415 q^{79} -3.00000 q^{80} +81.0000 q^{81} -525.000 q^{82} +1288.65 q^{83} -51.9615 q^{84} -36.3731 q^{85} +858.000 q^{86} -161.081 q^{87} -207.000 q^{89} -46.7654 q^{90} +6.00000 q^{91} +870.000 q^{92} -60.0000 q^{93} -665.108 q^{94} +394.908 q^{95} +545.596 q^{96} -1615.00 q^{97} -573.309 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 10 q^{4} - 6 q^{5} + 18 q^{9} - 30 q^{12} + 12 q^{14} - 18 q^{15} + 2 q^{16} + 30 q^{20} - 348 q^{23} - 232 q^{25} + 6 q^{26} + 54 q^{27} - 40 q^{31} + 42 q^{34} - 90 q^{36} - 598 q^{37}+ \cdots - 3230 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 0.612372 0.306186 − 0.951972i \(-0.400947\pi\)
0.306186 + 0.951972i \(0.400947\pi\)
\(3\) 3.00000 0.577350
\(4\) −5.00000 −0.625000
\(5\) −3.00000 −0.268328 −0.134164 − 0.990959i \(-0.542835\pi\)
−0.134164 + 0.990959i \(0.542835\pi\)
\(6\) 5.19615 0.353553
\(7\) 3.46410 0.187044 0.0935220 − 0.995617i \(-0.470187\pi\)
0.0935220 + 0.995617i \(0.470187\pi\)
\(8\) −22.5167 −0.995105
\(9\) 9.00000 0.333333
\(10\) −5.19615 −0.164317
\(11\) 0 0
\(12\) −15.0000 −0.360844
\(13\) 1.73205 0.0369527 0.0184763 − 0.999829i \(-0.494118\pi\)
0.0184763 + 0.999829i \(0.494118\pi\)
\(14\) 6.00000 0.114541
\(15\) −9.00000 −0.154919
\(16\) 1.00000 0.0156250
\(17\) 12.1244 0.172976 0.0864879 − 0.996253i \(-0.472436\pi\)
0.0864879 + 0.996253i \(0.472436\pi\)
\(18\) 15.5885 0.204124
\(19\) −131.636 −1.58944 −0.794719 − 0.606977i \(-0.792382\pi\)
−0.794719 + 0.606977i \(0.792382\pi\)
\(20\) 15.0000 0.167705
\(21\) 10.3923 0.107990
\(22\) 0 0
\(23\) −174.000 −1.57746 −0.788728 − 0.614742i \(-0.789260\pi\)
−0.788728 + 0.614742i \(0.789260\pi\)
\(24\) −67.5500 −0.574524
\(25\) −116.000 −0.928000
\(26\) 3.00000 0.0226288
\(27\) 27.0000 0.192450
\(28\) −17.3205 −0.116902
\(29\) −53.6936 −0.343815 −0.171908 − 0.985113i \(-0.554993\pi\)
−0.171908 + 0.985113i \(0.554993\pi\)
\(30\) −15.5885 −0.0948683
\(31\) −20.0000 −0.115874 −0.0579372 − 0.998320i \(-0.518452\pi\)
−0.0579372 + 0.998320i \(0.518452\pi\)
\(32\) 181.865 1.00467
\(33\) 0 0
\(34\) 21.0000 0.105926
\(35\) −10.3923 −0.0501891
\(36\) −45.0000 −0.208333
\(37\) −299.000 −1.32852 −0.664261 − 0.747501i \(-0.731254\pi\)
−0.664261 + 0.747501i \(0.731254\pi\)
\(38\) −228.000 −0.973329
\(39\) 5.19615 0.0213346
\(40\) 67.5500 0.267015
\(41\) −303.109 −1.15458 −0.577288 − 0.816540i \(-0.695889\pi\)
−0.577288 + 0.816540i \(0.695889\pi\)
\(42\) 18.0000 0.0661300
\(43\) 495.367 1.75681 0.878403 − 0.477920i \(-0.158609\pi\)
0.878403 + 0.477920i \(0.158609\pi\)
\(44\) 0 0
\(45\) −27.0000 −0.0894427
\(46\) −301.377 −0.965991
\(47\) −384.000 −1.19175 −0.595874 − 0.803078i \(-0.703194\pi\)
−0.595874 + 0.803078i \(0.703194\pi\)
\(48\) 3.00000 0.00902110
\(49\) −331.000 −0.965015
\(50\) −200.918 −0.568282
\(51\) 36.3731 0.0998676
\(52\) −8.66025 −0.0230954
\(53\) 255.000 0.660886 0.330443 − 0.943826i \(-0.392802\pi\)
0.330443 + 0.943826i \(0.392802\pi\)
\(54\) 46.7654 0.117851
\(55\) 0 0
\(56\) −78.0000 −0.186128
\(57\) −394.908 −0.917663
\(58\) −93.0000 −0.210543
\(59\) 570.000 1.25776 0.628879 − 0.777503i \(-0.283514\pi\)
0.628879 + 0.777503i \(0.283514\pi\)
\(60\) 45.0000 0.0968246
\(61\) 374.123 0.785271 0.392636 − 0.919694i \(-0.371563\pi\)
0.392636 + 0.919694i \(0.371563\pi\)
\(62\) −34.6410 −0.0709583
\(63\) 31.1769 0.0623480
\(64\) 307.000 0.599609
\(65\) −5.19615 −0.00991544
\(66\) 0 0
\(67\) 46.0000 0.0838775 0.0419388 − 0.999120i \(-0.486647\pi\)
0.0419388 + 0.999120i \(0.486647\pi\)
\(68\) −60.6218 −0.108110
\(69\) −522.000 −0.910745
\(70\) −18.0000 −0.0307344
\(71\) −630.000 −1.05306 −0.526530 − 0.850157i \(-0.676507\pi\)
−0.526530 + 0.850157i \(0.676507\pi\)
\(72\) −202.650 −0.331702
\(73\) 575.041 0.921965 0.460982 − 0.887409i \(-0.347497\pi\)
0.460982 + 0.887409i \(0.347497\pi\)
\(74\) −517.883 −0.813550
\(75\) −348.000 −0.535781
\(76\) 658.179 0.993399
\(77\) 0 0
\(78\) 9.00000 0.0130647
\(79\) 249.415 0.355208 0.177604 − 0.984102i \(-0.443165\pi\)
0.177604 + 0.984102i \(0.443165\pi\)
\(80\) −3.00000 −0.00419263
\(81\) 81.0000 0.111111
\(82\) −525.000 −0.707031
\(83\) 1288.65 1.70418 0.852092 − 0.523392i \(-0.175334\pi\)
0.852092 + 0.523392i \(0.175334\pi\)
\(84\) −51.9615 −0.0674937
\(85\) −36.3731 −0.0464143
\(86\) 858.000 1.07582
\(87\) −161.081 −0.198502
\(88\) 0 0
\(89\) −207.000 −0.246539 −0.123269 − 0.992373i \(-0.539338\pi\)
−0.123269 + 0.992373i \(0.539338\pi\)
\(90\) −46.7654 −0.0547723
\(91\) 6.00000 0.00691177
\(92\) 870.000 0.985911
\(93\) −60.0000 −0.0669001
\(94\) −665.108 −0.729794
\(95\) 394.908 0.426491
\(96\) 545.596 0.580049
\(97\) −1615.00 −1.69050 −0.845250 − 0.534372i \(-0.820548\pi\)
−0.845250 + 0.534372i \(0.820548\pi\)
\(98\) −573.309 −0.590948
\(99\) 0 0
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 363.4.a.m.1.2 yes 2
3.2 odd 2 1089.4.a.n.1.1 2
11.10 odd 2 inner 363.4.a.m.1.1 ✓ 2
33.32 even 2 1089.4.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
363.4.a.m.1.1 ✓ 2 11.10 odd 2 inner
363.4.a.m.1.2 yes 2 1.1 even 1 trivial
1089.4.a.n.1.1 2 3.2 odd 2
1089.4.a.n.1.2 2 33.32 even 2