Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [522,2,Mod(115,522)] 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("522.115"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 463.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 522.463
Dual form 522.2.h.a.115.1

$q$-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+(-0.866025 + 0.500000i) q^{2} +(0.866025 - 1.50000i) q^{3} +(0.500000 - 0.866025i) q^{4} +(1.00000 - 1.73205i) q^{5} +1.73205i q^{6} +(-0.500000 - 0.866025i) q^{7} +1.00000i q^{8} +(-1.50000 - 2.59808i) q^{9} +2.00000i q^{10} +(-0.866025 + 0.500000i) q^{11} +(-0.866025 - 1.50000i) q^{12} +(2.00000 - 3.46410i) q^{13} +(0.866025 + 0.500000i) q^{14} +(-1.73205 - 3.00000i) q^{15} +(-0.500000 - 0.866025i) q^{16} +2.00000i q^{17} +(2.59808 + 1.50000i) q^{18} -1.00000i q^{19} +(-1.00000 - 1.73205i) q^{20} -1.73205 q^{21} +(0.500000 - 0.866025i) q^{22} +(1.50000 + 0.866025i) q^{24} +(0.500000 + 0.866025i) q^{25} +4.00000i q^{26} -5.19615 q^{27} -1.00000 q^{28} +(-5.33013 + 0.767949i) q^{29} +(3.00000 + 1.73205i) q^{30} +(-3.46410 - 2.00000i) q^{31} +(0.866025 + 0.500000i) q^{32} +1.73205i q^{33} +(-1.00000 - 1.73205i) q^{34} -2.00000 q^{35} -3.00000 q^{36} -2.00000i q^{37} +(0.500000 + 0.866025i) q^{38} +(-3.46410 - 6.00000i) q^{39} +(1.73205 + 1.00000i) q^{40} +(1.73205 + 1.00000i) q^{41} +(1.50000 - 0.866025i) q^{42} +(7.79423 - 4.50000i) q^{43} +1.00000i q^{44} -6.00000 q^{45} +(-5.19615 + 3.00000i) q^{47} -1.73205 q^{48} +(3.00000 - 5.19615i) q^{49} +(-0.866025 - 0.500000i) q^{50} +(3.00000 + 1.73205i) q^{51} +(-2.00000 - 3.46410i) q^{52} +2.00000 q^{53} +(4.50000 - 2.59808i) q^{54} +2.00000i q^{55} +(0.866025 - 0.500000i) q^{56} +(-1.50000 - 0.866025i) q^{57} +(4.23205 - 3.33013i) q^{58} +(-2.00000 + 3.46410i) q^{59} -3.46410 q^{60} +(11.2583 - 6.50000i) q^{61} +4.00000 q^{62} +(-1.50000 + 2.59808i) q^{63} -1.00000 q^{64} +(-4.00000 - 6.92820i) q^{65} +(-0.866025 - 1.50000i) q^{66} +(-1.00000 + 1.73205i) q^{67} +(1.73205 + 1.00000i) q^{68} +(1.73205 - 1.00000i) q^{70} +7.00000 q^{71} +(2.59808 - 1.50000i) q^{72} -4.00000i q^{73} +(1.00000 + 1.73205i) q^{74} +1.73205 q^{75} +(-0.866025 - 0.500000i) q^{76} +(0.866025 + 0.500000i) q^{77} +(6.00000 + 3.46410i) q^{78} +(3.46410 - 2.00000i) q^{79} -2.00000 q^{80} +(-4.50000 + 7.79423i) q^{81} -2.00000 q^{82} +(2.00000 + 3.46410i) q^{83} +(-0.866025 + 1.50000i) q^{84} +(3.46410 + 2.00000i) q^{85} +(-4.50000 + 7.79423i) q^{86} +(-3.46410 + 8.66025i) q^{87} +(-0.500000 - 0.866025i) q^{88} -2.00000i q^{89} +(5.19615 - 3.00000i) q^{90} -4.00000 q^{91} +(-6.00000 + 3.46410i) q^{93} +(3.00000 - 5.19615i) q^{94} +(-1.73205 - 1.00000i) q^{95} +(1.50000 - 0.866025i) q^{96} +(6.92820 - 4.00000i) q^{97} +6.00000i q^{98} +(2.59808 + 1.50000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 4 q^{5} - 2 q^{7} - 6 q^{9} + 8 q^{13} - 2 q^{16} - 4 q^{20} + 2 q^{22} + 6 q^{24} + 2 q^{25} - 4 q^{28} - 4 q^{29} + 12 q^{30} - 4 q^{34} - 8 q^{35} - 12 q^{36} + 2 q^{38} + 6 q^{42} - 24 q^{45}+ \cdots + 6 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/522\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(407\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.612372 + 0.353553i
\(3\) 0.866025 − 1.50000i 0.500000 − 0.866025i
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 1.00000 − 1.73205i 0.447214 − 0.774597i −0.550990 − 0.834512i \(-0.685750\pi\)
0.998203 + 0.0599153i \(0.0190830\pi\)
\(6\) 1.73205i 0.707107i
\(7\) −0.500000 − 0.866025i −0.188982 − 0.327327i 0.755929 − 0.654654i \(-0.227186\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.50000 − 2.59808i −0.500000 − 0.866025i
\(10\) 2.00000i 0.632456i
\(11\) −0.866025 + 0.500000i −0.261116 + 0.150756i −0.624844 − 0.780750i \(-0.714837\pi\)
0.363727 + 0.931505i \(0.381504\pi\)
\(12\) −0.866025 − 1.50000i −0.250000 − 0.433013i
\(13\) 2.00000 − 3.46410i 0.554700 − 0.960769i −0.443227 − 0.896410i \(-0.646166\pi\)
0.997927 − 0.0643593i \(-0.0205004\pi\)
\(14\) 0.866025 + 0.500000i 0.231455 + 0.133631i
\(15\) −1.73205 − 3.00000i −0.447214 − 0.774597i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 2.59808 + 1.50000i 0.612372 + 0.353553i
\(19\) − 1.00000i − 0.229416i −0.993399 − 0.114708i \(-0.963407\pi\)
0.993399 − 0.114708i \(-0.0365932\pi\)
\(20\) −1.00000 − 1.73205i −0.223607 − 0.387298i
\(21\) −1.73205 −0.377964
\(22\) 0.500000 − 0.866025i 0.106600 − 0.184637i
\(23\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 1.50000 + 0.866025i 0.306186 + 0.176777i
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) 4.00000i 0.784465i
\(27\) −5.19615 −1.00000
\(28\) −1.00000 −0.188982
\(29\) −5.33013 + 0.767949i −0.989780 + 0.142605i
\(30\) 3.00000 + 1.73205i 0.547723 + 0.316228i
\(31\) −3.46410 − 2.00000i −0.622171 − 0.359211i 0.155543 − 0.987829i \(-0.450287\pi\)
−0.777714 + 0.628619i \(0.783621\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) 1.73205i 0.301511i
\(34\) −1.00000 − 1.73205i −0.171499 − 0.297044i
\(35\) −2.00000 −0.338062
\(36\) −3.00000 −0.500000
\(37\) − 2.00000i − 0.328798i −0.986394 − 0.164399i \(-0.947432\pi\)
0.986394 − 0.164399i \(-0.0525685\pi\)
\(38\) 0.500000 + 0.866025i 0.0811107 + 0.140488i
\(39\) −3.46410 − 6.00000i −0.554700 − 0.960769i
\(40\) 1.73205 + 1.00000i 0.273861 + 0.158114i
\(41\) 1.73205 + 1.00000i 0.270501 + 0.156174i 0.629115 − 0.777312i \(-0.283417\pi\)
−0.358614 + 0.933486i \(0.616751\pi\)
\(42\) 1.50000 − 0.866025i 0.231455 − 0.133631i
\(43\) 7.79423 − 4.50000i 1.18861 − 0.686244i 0.230618 − 0.973044i \(-0.425925\pi\)
0.957990 + 0.286801i \(0.0925917\pi\)
\(44\) 1.00000i 0.150756i
\(45\) −6.00000 −0.894427
\(46\) 0 0
\(47\) −5.19615 + 3.00000i −0.757937 + 0.437595i −0.828554 − 0.559908i \(-0.810836\pi\)
0.0706177 + 0.997503i \(0.477503\pi\)
\(48\) −1.73205 −0.250000
\(49\) 3.00000 − 5.19615i 0.428571 − 0.742307i
\(50\) −0.866025 − 0.500000i −0.122474 − 0.0707107i
\(51\) 3.00000 + 1.73205i 0.420084 + 0.242536i
\(52\) −2.00000 − 3.46410i −0.277350 − 0.480384i
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 4.50000 − 2.59808i 0.612372 − 0.353553i
\(55\) 2.00000i 0.269680i
\(56\) 0.866025 − 0.500000i 0.115728 − 0.0668153i
\(57\) −1.50000 − 0.866025i −0.198680 − 0.114708i
\(58\) 4.23205 − 3.33013i 0.555695 − 0.437267i
\(59\) −2.00000 + 3.46410i −0.260378 + 0.450988i −0.966342 − 0.257260i \(-0.917180\pi\)
0.705965 + 0.708247i \(0.250514\pi\)
\(60\) −3.46410 −0.447214
\(61\) 11.2583 − 6.50000i 1.44148 − 0.832240i 0.443533 − 0.896258i \(-0.353725\pi\)
0.997949 + 0.0640184i \(0.0203916\pi\)
\(62\) 4.00000 0.508001
\(63\) −1.50000 + 2.59808i −0.188982 + 0.327327i
\(64\) −1.00000 −0.125000
\(65\) −4.00000 − 6.92820i −0.496139 − 0.859338i
\(66\) −0.866025 − 1.50000i −0.106600 − 0.184637i
\(67\) −1.00000 + 1.73205i −0.122169 + 0.211604i −0.920623 − 0.390453i \(-0.872318\pi\)
0.798454 + 0.602056i \(0.205652\pi\)
\(68\) 1.73205 + 1.00000i 0.210042 + 0.121268i
\(69\) 0 0
\(70\) 1.73205 − 1.00000i 0.207020 − 0.119523i
\(71\) 7.00000 0.830747 0.415374 − 0.909651i \(-0.363651\pi\)
0.415374 + 0.909651i \(0.363651\pi\)
\(72\) 2.59808 − 1.50000i 0.306186 − 0.176777i
\(73\) − 4.00000i − 0.468165i −0.972217 − 0.234082i \(-0.924791\pi\)
0.972217 − 0.234082i \(-0.0752085\pi\)
\(74\) 1.00000 + 1.73205i 0.116248 + 0.201347i
\(75\) 1.73205 0.200000
\(76\) −0.866025 − 0.500000i −0.0993399 − 0.0573539i
\(77\) 0.866025 + 0.500000i 0.0986928 + 0.0569803i
\(78\) 6.00000 + 3.46410i 0.679366 + 0.392232i
\(79\) 3.46410 − 2.00000i 0.389742 − 0.225018i −0.292306 − 0.956325i \(-0.594423\pi\)
0.682048 + 0.731307i \(0.261089\pi\)
\(80\) −2.00000 −0.223607
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) −2.00000 −0.220863
\(83\) 2.00000 + 3.46410i 0.219529 + 0.380235i 0.954664 − 0.297686i \(-0.0962148\pi\)
−0.735135 + 0.677920i \(0.762881\pi\)
\(84\) −0.866025 + 1.50000i −0.0944911 + 0.163663i
\(85\) 3.46410 + 2.00000i 0.375735 + 0.216930i
\(86\) −4.50000 + 7.79423i −0.485247 + 0.840473i
\(87\) −3.46410 + 8.66025i −0.371391 + 0.928477i
\(88\) −0.500000 − 0.866025i −0.0533002 − 0.0923186i
\(89\) − 2.00000i − 0.212000i −0.994366 − 0.106000i \(-0.966196\pi\)
0.994366 − 0.106000i \(-0.0338043\pi\)
\(90\) 5.19615 − 3.00000i 0.547723 − 0.316228i
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) −6.00000 + 3.46410i −0.622171 + 0.359211i
\(94\) 3.00000 − 5.19615i 0.309426 − 0.535942i
\(95\) −1.73205 − 1.00000i −0.177705 − 0.102598i
\(96\) 1.50000 − 0.866025i 0.153093 − 0.0883883i
\(97\) 6.92820 − 4.00000i 0.703452 − 0.406138i −0.105180 − 0.994453i \(-0.533542\pi\)
0.808632 + 0.588315i \(0.200208\pi\)
\(98\) 6.00000i 0.606092i
\(99\) 2.59808 + 1.50000i 0.261116 + 0.150756i
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 522.2.h.a.463.1 yes 4
3.2 odd 2 1566.2.h.a.1333.2 4
9.2 odd 6 1566.2.h.a.289.1 4
9.7 even 3 inner 522.2.h.a.115.2 yes 4
29.28 even 2 inner 522.2.h.a.463.2 yes 4
87.86 odd 2 1566.2.h.a.1333.1 4
261.115 even 6 inner 522.2.h.a.115.1 ✓ 4
261.173 odd 6 1566.2.h.a.289.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
522.2.h.a.115.1 ✓ 4 261.115 even 6 inner
522.2.h.a.115.2 yes 4 9.7 even 3 inner
522.2.h.a.463.1 yes 4 1.1 even 1 trivial
522.2.h.a.463.2 yes 4 29.28 even 2 inner
1566.2.h.a.289.1 4 9.2 odd 6
1566.2.h.a.289.2 4 261.173 odd 6
1566.2.h.a.1333.1 4 87.86 odd 2
1566.2.h.a.1333.2 4 3.2 odd 2